LMFDB - Embedded newform 968.2.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [968,2,Mod(1,968)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(968, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("968.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 968 = 2^{3} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	968.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:	$7.72951891566$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	968.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.73205 q^{3} -3.73205 q^{5} +2.73205 q^{7} +4.46410 q^{9} +O(q^{10})$	$q+2.73205 q^{3} -3.73205 q^{5} +2.73205 q^{7} +4.46410 q^{9} +4.46410 q^{13} -10.1962 q^{15} +2.26795 q^{17} +3.26795 q^{19} +7.46410 q^{21} -2.19615 q^{23} +8.92820 q^{25} +4.00000 q^{27} -4.46410 q^{29} -4.73205 q^{31} -10.1962 q^{35} +5.73205 q^{37} +12.1962 q^{39} +7.73205 q^{41} +8.00000 q^{43} -16.6603 q^{45} -3.26795 q^{47} +0.464102 q^{49} +6.19615 q^{51} +1.19615 q^{53} +8.92820 q^{57} +13.4641 q^{59} +6.53590 q^{61} +12.1962 q^{63} -16.6603 q^{65} -0.196152 q^{67} -6.00000 q^{69} +2.53590 q^{71} -8.92820 q^{73} +24.3923 q^{75} -16.5885 q^{79} -2.46410 q^{81} +2.19615 q^{83} -8.46410 q^{85} -12.1962 q^{87} -16.4641 q^{89} +12.1962 q^{91} -12.9282 q^{93} -12.1962 q^{95} -11.9282 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 4 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 4 * q^5 + 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 4 q^{5} + 2 q^{7} + 2 q^{9} + 2 q^{13} - 10 q^{15} + 8 q^{17} + 10 q^{19} + 8 q^{21} + 6 q^{23} + 4 q^{25} + 8 q^{27} - 2 q^{29} - 6 q^{31} - 10 q^{35} + 8 q^{37} + 14 q^{39} + 12 q^{41} + 16 q^{43} - 16 q^{45} - 10 q^{47} - 6 q^{49} + 2 q^{51} - 8 q^{53} + 4 q^{57} + 20 q^{59} + 20 q^{61} + 14 q^{63} - 16 q^{65} + 10 q^{67} - 12 q^{69} + 12 q^{71} - 4 q^{73} + 28 q^{75} - 2 q^{79} + 2 q^{81} - 6 q^{83} - 10 q^{85} - 14 q^{87} - 26 q^{89} + 14 q^{91} - 12 q^{93} - 14 q^{95} - 10 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 - 4 * q^5 + 2 * q^7 + 2 * q^9 + 2 * q^13 - 10 * q^15 + 8 * q^17 + 10 * q^19 + 8 * q^21 + 6 * q^23 + 4 * q^25 + 8 * q^27 - 2 * q^29 - 6 * q^31 - 10 * q^35 + 8 * q^37 + 14 * q^39 + 12 * q^41 + 16 * q^43 - 16 * q^45 - 10 * q^47 - 6 * q^49 + 2 * q^51 - 8 * q^53 + 4 * q^57 + 20 * q^59 + 20 * q^61 + 14 * q^63 - 16 * q^65 + 10 * q^67 - 12 * q^69 + 12 * q^71 - 4 * q^73 + 28 * q^75 - 2 * q^79 + 2 * q^81 - 6 * q^83 - 10 * q^85 - 14 * q^87 - 26 * q^89 + 14 * q^91 - 12 * q^93 - 14 * q^95 - 10 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	2.73205	1.57735	0.788675	−	0.614810i	$-0.210767\pi$
$3$	2.73205	1.57735	0.788675	+	0.614810i	$0.210767\pi$
$4$	0	0
$5$	−3.73205	−1.66902	−0.834512	−	0.550990i	$-0.814250\pi$
$5$	−3.73205	−1.66902	−0.834512	+	0.550990i	$0.814250\pi$
$6$	0	0
$7$	2.73205	1.03262	0.516309	−	0.856402i	$-0.327306\pi$
$7$	2.73205	1.03262	0.516309	+	0.856402i	$0.327306\pi$
$8$	0	0
$9$	4.46410	1.48803
$10$	0	0
$11$	0	0
$11$	0	0
$12$	0	0
$13$	4.46410	1.23812	0.619060	−	0.785344i	$-0.287514\pi$
$13$	4.46410	1.23812	0.619060	+	0.785344i	$0.287514\pi$
$14$	0	0
$15$	−10.1962	−2.63264
$16$	0	0
$17$	2.26795	0.550058	0.275029	−	0.961436i	$-0.411312\pi$
$17$	2.26795	0.550058	0.275029	+	0.961436i	$0.411312\pi$
$18$	0	0
$19$	3.26795	0.749719	0.374859	−	0.927082i	$-0.377691\pi$
$19$	3.26795	0.749719	0.374859	+	0.927082i	$0.377691\pi$
$20$	0	0
$21$	7.46410	1.62880
$22$	0	0
$23$	−2.19615	−0.457929	−0.228965	−	0.973435i	$-0.573534\pi$
$23$	−2.19615	−0.457929	−0.228965	+	0.973435i	$0.573534\pi$
$24$	0	0
$25$	8.92820	1.78564
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	−4.46410	−0.828963	−0.414481	−	0.910058i	$-0.636037\pi$
$29$	−4.46410	−0.828963	−0.414481	+	0.910058i	$0.636037\pi$
$30$	0	0
$31$	−4.73205	−0.849901	−0.424951	−	0.905216i	$-0.639709\pi$
$31$	−4.73205	−0.849901	−0.424951	+	0.905216i	$0.639709\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−10.1962	−1.72346
$36$	0	0
$37$	5.73205	0.942343	0.471172	−	0.882042i	$-0.343831\pi$
$37$	5.73205	0.942343	0.471172	+	0.882042i	$0.343831\pi$
$38$	0	0
$39$	12.1962	1.95295
$40$	0	0
$41$	7.73205	1.20754	0.603772	−	0.797157i	$-0.293664\pi$
$41$	7.73205	1.20754	0.603772	+	0.797157i	$0.293664\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	−16.6603	−2.48356
$46$	0	0
$47$	−3.26795	−0.476679	−0.238340	−	0.971182i	$-0.576603\pi$
$47$	−3.26795	−0.476679	−0.238340	+	0.971182i	$0.576603\pi$
$48$	0	0
$49$	0.464102	0.0663002
$50$	0	0
$51$	6.19615	0.867635
$52$	0	0
$53$	1.19615	0.164304	0.0821521	−	0.996620i	$-0.473821\pi$
$53$	1.19615	0.164304	0.0821521	+	0.996620i	$0.473821\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	8.92820	1.18257
$58$	0	0
$59$	13.4641	1.75288	0.876438	−	0.481514i	$-0.159913\pi$
$59$	13.4641	1.75288	0.876438	+	0.481514i	$0.159913\pi$
$60$	0	0
$61$	6.53590	0.836836	0.418418	−	0.908255i	$-0.362585\pi$
$61$	6.53590	0.836836	0.418418	+	0.908255i	$0.362585\pi$
$62$	0	0
$63$	12.1962	1.53657
$64$	0	0
$65$	−16.6603	−2.06645
$66$	0	0
$67$	−0.196152	−0.0239638	−0.0119819	−	0.999928i	$-0.503814\pi$
$67$	−0.196152	−0.0239638	−0.0119819	+	0.999928i	$0.503814\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	0	0
$71$	2.53590	0.300956	0.150478	−	0.988613i	$-0.451919\pi$
$71$	2.53590	0.300956	0.150478	+	0.988613i	$0.451919\pi$
$72$	0	0
$73$	−8.92820	−1.04497	−0.522484	−	0.852649i	$-0.674994\pi$
$73$	−8.92820	−1.04497	−0.522484	+	0.852649i	$0.674994\pi$
$74$	0	0
$75$	24.3923	2.81658
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−16.5885	−1.86635	−0.933174	−	0.359426i	$-0.882973\pi$
$79$	−16.5885	−1.86635	−0.933174	+	0.359426i	$0.882973\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	0	0
$83$	2.19615	0.241059	0.120530	−	0.992710i	$-0.461541\pi$
$83$	2.19615	0.241059	0.120530	+	0.992710i	$0.461541\pi$
$84$	0	0
$85$	−8.46410	−0.918061
$86$	0	0
$87$	−12.1962	−1.30756
$88$	0	0
$89$	−16.4641	−1.74519	−0.872596	−	0.488443i	$-0.837565\pi$
$89$	−16.4641	−1.74519	−0.872596	+	0.488443i	$0.837565\pi$
$90$	0	0
$91$	12.1962	1.27850
$92$	0	0
$93$	−12.9282	−1.34059
$94$	0	0
$95$	−12.1962	−1.25130
$96$	0	0
$97$	−11.9282	−1.21113	−0.605563	−	0.795798i	$-0.707052\pi$
$97$	−11.9282	−1.21113	−0.605563	+	0.795798i	$0.707052\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−17.4641	−1.73774	−0.868872	−	0.495038i	$-0.835154\pi$
$101$	−17.4641	−1.73774	−0.868872	+	0.495038i	$0.835154\pi$
$102$	0	0
$103$	−6.53590	−0.644001	−0.322001	−	0.946739i	$-0.604355\pi$
$103$	−6.53590	−0.644001	−0.322001	+	0.946739i	$0.604355\pi$
$104$	0	0
$105$	−27.8564	−2.71851
$106$	0	0
$107$	−17.1244	−1.65547	−0.827737	−	0.561116i	$-0.810372\pi$
$107$	−17.1244	−1.65547	−0.827737	+	0.561116i	$0.810372\pi$
$108$	0	0
$109$	12.4641	1.19384	0.596922	−	0.802299i	$-0.296390\pi$
$109$	12.4641	1.19384	0.596922	+	0.802299i	$0.296390\pi$
$110$	0	0
$111$	15.6603	1.48641
$112$	0	0
$113$	5.00000	0.470360	0.235180	−	0.971952i	$-0.424432\pi$
$113$	5.00000	0.470360	0.235180	+	0.971952i	$0.424432\pi$
$114$	0	0
$115$	8.19615	0.764295
$116$	0	0
$117$	19.9282	1.84236
$118$	0	0
$119$	6.19615	0.568000
$120$	0	0
$121$	0	0
$122$	0	0
$123$	21.1244	1.90472
$124$	0	0
$125$	−14.6603	−1.31125
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	0	0
$129$	21.8564	1.92435
$130$	0	0
$131$	−8.00000	−0.698963	−0.349482	−	0.936943i	$-0.613642\pi$
$131$	−8.00000	−0.698963	−0.349482	+	0.936943i	$0.613642\pi$
$132$	0	0
$133$	8.92820	0.774173
$134$	0	0
$135$	−14.9282	−1.28482
$136$	0	0
$137$	−2.53590	−0.216656	−0.108328	−	0.994115i	$-0.534550\pi$
$137$	−2.53590	−0.216656	−0.108328	+	0.994115i	$0.534550\pi$
$138$	0	0
$139$	6.19615	0.525551	0.262775	−	0.964857i	$-0.415362\pi$
$139$	6.19615	0.525551	0.262775	+	0.964857i	$0.415362\pi$
$140$	0	0
$141$	−8.92820	−0.751890
$142$	0	0
$143$	0	0
$144$	0	0
$145$	16.6603	1.38356
$146$	0	0
$147$	1.26795	0.104579
$148$	0	0
$149$	−5.53590	−0.453518	−0.226759	−	0.973951i	$-0.572813\pi$
$149$	−5.53590	−0.453518	−0.226759	+	0.973951i	$0.572813\pi$
$150$	0	0
$151$	−10.5359	−0.857399	−0.428700	−	0.903447i	$-0.641028\pi$
$151$	−10.5359	−0.857399	−0.428700	+	0.903447i	$0.641028\pi$
$152$	0	0
$153$	10.1244	0.818506
$154$	0	0
$155$	17.6603	1.41851
$156$	0	0
$157$	−20.7846	−1.65879	−0.829396	−	0.558661i	$-0.811315\pi$
$157$	−20.7846	−1.65879	−0.829396	+	0.558661i	$0.811315\pi$
$158$	0	0
$159$	3.26795	0.259165
$160$	0	0
$161$	−6.00000	−0.472866
$162$	0	0
$163$	−23.5167	−1.84197	−0.920984	−	0.389602i	$-0.872613\pi$
$163$	−23.5167	−1.84197	−0.920984	+	0.389602i	$0.872613\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.53590	0.505763	0.252882	−	0.967497i	$-0.418622\pi$
$167$	6.53590	0.505763	0.252882	+	0.967497i	$0.418622\pi$
$168$	0	0
$169$	6.92820	0.532939
$170$	0	0
$171$	14.5885	1.11561
$172$	0	0
$173$	−2.53590	−0.192801	−0.0964004	−	0.995343i	$-0.530733\pi$
$173$	−2.53590	−0.192801	−0.0964004	+	0.995343i	$0.530733\pi$
$174$	0	0
$175$	24.3923	1.84388
$176$	0	0
$177$	36.7846	2.76490
$178$	0	0
$179$	16.0000	1.19590	0.597948	−	0.801535i	$-0.295983\pi$
$179$	16.0000	1.19590	0.597948	+	0.801535i	$0.295983\pi$
$180$	0	0
$181$	−2.80385	−0.208408	−0.104204	−	0.994556i	$-0.533230\pi$
$181$	−2.80385	−0.208408	−0.104204	+	0.994556i	$0.533230\pi$
$182$	0	0
$183$	17.8564	1.31998
$184$	0	0
$185$	−21.3923	−1.57279
$186$	0	0
$187$	0	0
$188$	0	0
$189$	10.9282	0.794910
$190$	0	0
$191$	4.39230	0.317816	0.158908	−	0.987293i	$-0.449203\pi$
$191$	4.39230	0.317816	0.158908	+	0.987293i	$0.449203\pi$
$192$	0	0
$193$	16.6603	1.19923	0.599616	−	0.800288i	$-0.295320\pi$
$193$	16.6603	1.19923	0.599616	+	0.800288i	$0.295320\pi$
$194$	0	0
$195$	−45.5167	−3.25952
$196$	0	0
$197$	5.92820	0.422367	0.211183	−	0.977446i	$-0.432268\pi$
$197$	5.92820	0.422367	0.211183	+	0.977446i	$0.432268\pi$
$198$	0	0
$199$	−6.53590	−0.463318	−0.231659	−	0.972797i	$-0.574415\pi$
$199$	−6.53590	−0.463318	−0.231659	+	0.972797i	$0.574415\pi$
$200$	0	0
$201$	−0.535898	−0.0377994
$202$	0	0
$203$	−12.1962	−0.856002
$204$	0	0
$205$	−28.8564	−2.01542
$206$	0	0
$207$	−9.80385	−0.681415
$208$	0	0
$209$	0	0
$210$	0	0
$211$	17.8564	1.22929	0.614643	−	0.788806i	$-0.289300\pi$
$211$	17.8564	1.22929	0.614643	+	0.788806i	$0.289300\pi$
$212$	0	0
$213$	6.92820	0.474713
$214$	0	0
$215$	−29.8564	−2.03619
$216$	0	0
$217$	−12.9282	−0.877624
$218$	0	0
$219$	−24.3923	−1.64828
$220$	0	0
$221$	10.1244	0.681038
$222$	0	0
$223$	−8.39230	−0.561990	−0.280995	−	0.959709i	$-0.590664\pi$
$223$	−8.39230	−0.561990	−0.280995	+	0.959709i	$0.590664\pi$
$224$	0	0
$225$	39.8564	2.65709
$226$	0	0
$227$	3.60770	0.239451	0.119726	−	0.992807i	$-0.461799\pi$
$227$	3.60770	0.239451	0.119726	+	0.992807i	$0.461799\pi$
$228$	0	0
$229$	12.8038	0.846102	0.423051	−	0.906106i	$-0.360959\pi$
$229$	12.8038	0.846102	0.423051	+	0.906106i	$0.360959\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.19615	−0.0783626	−0.0391813	−	0.999232i	$-0.512475\pi$
$233$	−1.19615	−0.0783626	−0.0391813	+	0.999232i	$0.512475\pi$
$234$	0	0
$235$	12.1962	0.795589
$236$	0	0
$237$	−45.3205	−2.94388
$238$	0	0
$239$	19.5167	1.26243	0.631214	−	0.775609i	$-0.282557\pi$
$239$	19.5167	1.26243	0.631214	+	0.775609i	$0.282557\pi$
$240$	0	0
$241$	−8.92820	−0.575116	−0.287558	−	0.957763i	$-0.592843\pi$
$241$	−8.92820	−0.575116	−0.287558	+	0.957763i	$0.592843\pi$
$242$	0	0
$243$	−18.7321	−1.20166
$244$	0	0
$245$	−1.73205	−0.110657
$246$	0	0
$247$	14.5885	0.928241
$248$	0	0
$249$	6.00000	0.380235
$250$	0	0
$251$	17.2679	1.08994	0.544972	−	0.838454i	$-0.316540\pi$
$251$	17.2679	1.08994	0.544972	+	0.838454i	$0.316540\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	−23.1244	−1.44810
$256$	0	0
$257$	−4.46410	−0.278463	−0.139232	−	0.990260i	$-0.544463\pi$
$257$	−4.46410	−0.278463	−0.139232	+	0.990260i	$0.544463\pi$
$258$	0	0
$259$	15.6603	0.973081
$260$	0	0
$261$	−19.9282	−1.23352
$262$	0	0
$263$	−2.33975	−0.144275	−0.0721375	−	0.997395i	$-0.522982\pi$
$263$	−2.33975	−0.144275	−0.0721375	+	0.997395i	$0.522982\pi$
$264$	0	0
$265$	−4.46410	−0.274228
$266$	0	0
$267$	−44.9808	−2.75278
$268$	0	0
$269$	−19.0526	−1.16166	−0.580828	−	0.814027i	$-0.697271\pi$
$269$	−19.0526	−1.16166	−0.580828	+	0.814027i	$0.697271\pi$
$270$	0	0
$271$	−6.53590	−0.397028	−0.198514	−	0.980098i	$-0.563611\pi$
$271$	−6.53590	−0.397028	−0.198514	+	0.980098i	$0.563611\pi$
$272$	0	0
$273$	33.3205	2.01665
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−13.7846	−0.828237	−0.414118	−	0.910223i	$-0.635910\pi$
$277$	−13.7846	−0.828237	−0.414118	+	0.910223i	$0.635910\pi$
$278$	0	0
$279$	−21.1244	−1.26468
$280$	0	0
$281$	7.85641	0.468674	0.234337	−	0.972155i	$-0.424708\pi$
$281$	7.85641	0.468674	0.234337	+	0.972155i	$0.424708\pi$
$282$	0	0
$283$	30.9282	1.83849	0.919245	−	0.393685i	$-0.128800\pi$
$283$	30.9282	1.83849	0.919245	+	0.393685i	$0.128800\pi$
$284$	0	0
$285$	−33.3205	−1.97374
$286$	0	0
$287$	21.1244	1.24693
$288$	0	0
$289$	−11.8564	−0.697436
$290$	0	0
$291$	−32.5885	−1.91037
$292$	0	0
$293$	22.3205	1.30398	0.651989	−	0.758228i	$-0.273935\pi$
$293$	22.3205	1.30398	0.651989	+	0.758228i	$0.273935\pi$
$294$	0	0
$295$	−50.2487	−2.92559
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−9.80385	−0.566971
$300$	0	0
$301$	21.8564	1.25978
$302$	0	0
$303$	−47.7128	−2.74103
$304$	0	0
$305$	−24.3923	−1.39670
$306$	0	0
$307$	−32.4449	−1.85173	−0.925863	−	0.377859i	$-0.876660\pi$
$307$	−32.4449	−1.85173	−0.925863	+	0.377859i	$0.876660\pi$
$308$	0	0
$309$	−17.8564	−1.01582
$310$	0	0
$311$	16.7846	0.951768	0.475884	−	0.879508i	$-0.342128\pi$
$311$	16.7846	0.951768	0.475884	+	0.879508i	$0.342128\pi$
$312$	0	0
$313$	−7.00000	−0.395663	−0.197832	−	0.980236i	$-0.563390\pi$
$313$	−7.00000	−0.395663	−0.197832	+	0.980236i	$0.563390\pi$
$314$	0	0
$315$	−45.5167	−2.56457
$316$	0	0
$317$	3.85641	0.216597	0.108299	−	0.994118i	$-0.465460\pi$
$317$	3.85641	0.216597	0.108299	+	0.994118i	$0.465460\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−46.7846	−2.61126
$322$	0	0
$323$	7.41154	0.412389
$324$	0	0
$325$	39.8564	2.21084
$326$	0	0
$327$	34.0526	1.88311
$328$	0	0
$329$	−8.92820	−0.492228
$330$	0	0
$331$	−1.07180	−0.0589113	−0.0294556	−	0.999566i	$-0.509377\pi$
$331$	−1.07180	−0.0589113	−0.0294556	+	0.999566i	$0.509377\pi$
$332$	0	0
$333$	25.5885	1.40224
$334$	0	0
$335$	0.732051	0.0399962
$336$	0	0
$337$	7.19615	0.391999	0.196000	−	0.980604i	$-0.437205\pi$
$337$	7.19615	0.391999	0.196000	+	0.980604i	$0.437205\pi$
$338$	0	0
$339$	13.6603	0.741923
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−17.8564	−0.964155
$344$	0	0
$345$	22.3923	1.20556
$346$	0	0
$347$	−24.3923	−1.30945	−0.654724	−	0.755868i	$-0.727215\pi$
$347$	−24.3923	−1.30945	−0.654724	+	0.755868i	$0.727215\pi$
$348$	0	0
$349$	17.3923	0.930989	0.465494	−	0.885051i	$-0.345877\pi$
$349$	17.3923	0.930989	0.465494	+	0.885051i	$0.345877\pi$
$350$	0	0
$351$	17.8564	0.953104
$352$	0	0
$353$	−7.92820	−0.421976	−0.210988	−	0.977489i	$-0.567668\pi$
$353$	−7.92820	−0.421976	−0.210988	+	0.977489i	$0.567668\pi$
$354$	0	0
$355$	−9.46410	−0.502302
$356$	0	0
$357$	16.9282	0.895936
$358$	0	0
$359$	25.2679	1.33359	0.666796	−	0.745241i	$-0.267665\pi$
$359$	25.2679	1.33359	0.666796	+	0.745241i	$0.267665\pi$
$360$	0	0
$361$	−8.32051	−0.437921
$362$	0	0
$363$	0	0
$364$	0	0
$365$	33.3205	1.74408
$366$	0	0
$367$	21.5167	1.12316	0.561580	−	0.827422i	$-0.310194\pi$
$367$	21.5167	1.12316	0.561580	+	0.827422i	$0.310194\pi$
$368$	0	0
$369$	34.5167	1.79687
$370$	0	0
$371$	3.26795	0.169663
$372$	0	0
$373$	−1.46410	−0.0758083	−0.0379042	−	0.999281i	$-0.512068\pi$
$373$	−1.46410	−0.0758083	−0.0379042	+	0.999281i	$0.512068\pi$
$374$	0	0
$375$	−40.0526	−2.06831
$376$	0	0
$377$	−19.9282	−1.02635
$378$	0	0
$379$	33.1769	1.70418	0.852092	−	0.523392i	$-0.175334\pi$
$379$	33.1769	1.70418	0.852092	+	0.523392i	$0.175334\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	19.3205	0.987232	0.493616	−	0.869680i	$-0.335675\pi$
$383$	19.3205	0.987232	0.493616	+	0.869680i	$0.335675\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	35.7128	1.81538
$388$	0	0
$389$	−30.1244	−1.52737	−0.763683	−	0.645592i	$-0.776611\pi$
$389$	−30.1244	−1.52737	−0.763683	+	0.645592i	$0.776611\pi$
$390$	0	0
$391$	−4.98076	−0.251888
$392$	0	0
$393$	−21.8564	−1.10251
$394$	0	0
$395$	61.9090	3.11498
$396$	0	0
$397$	11.8756	0.596022	0.298011	−	0.954563i	$-0.403677\pi$
$397$	11.8756	0.596022	0.298011	+	0.954563i	$0.403677\pi$
$398$	0	0
$399$	24.3923	1.22114
$400$	0	0
$401$	−9.24871	−0.461859	−0.230929	−	0.972971i	$-0.574177\pi$
$401$	−9.24871	−0.461859	−0.230929	+	0.972971i	$0.574177\pi$
$402$	0	0
$403$	−21.1244	−1.05228
$404$	0	0
$405$	9.19615	0.456960
$406$	0	0
$407$	0	0
$408$	0	0
$409$	27.7321	1.37126	0.685631	−	0.727949i	$-0.259526\pi$
$409$	27.7321	1.37126	0.685631	+	0.727949i	$0.259526\pi$
$410$	0	0
$411$	−6.92820	−0.341743
$412$	0	0
$413$	36.7846	1.81005
$414$	0	0
$415$	−8.19615	−0.402333
$416$	0	0
$417$	16.9282	0.828978
$418$	0	0
$419$	−5.66025	−0.276522	−0.138261	−	0.990396i	$-0.544151\pi$
$419$	−5.66025	−0.276522	−0.138261	+	0.990396i	$0.544151\pi$
$420$	0	0
$421$	12.5167	0.610025	0.305012	−	0.952348i	$-0.401339\pi$
$421$	12.5167	0.610025	0.305012	+	0.952348i	$0.401339\pi$
$422$	0	0
$423$	−14.5885	−0.709315
$424$	0	0
$425$	20.2487	0.982207
$426$	0	0
$427$	17.8564	0.864132
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4.39230	0.211570	0.105785	−	0.994389i	$-0.466264\pi$
$431$	4.39230	0.211570	0.105785	+	0.994389i	$0.466264\pi$
$432$	0	0
$433$	−9.00000	−0.432512	−0.216256	−	0.976337i	$-0.569385\pi$
$433$	−9.00000	−0.432512	−0.216256	+	0.976337i	$0.569385\pi$
$434$	0	0
$435$	45.5167	2.18236
$436$	0	0
$437$	−7.17691	−0.343318
$438$	0	0
$439$	36.5885	1.74627	0.873136	−	0.487477i	$-0.162083\pi$
$439$	36.5885	1.74627	0.873136	+	0.487477i	$0.162083\pi$
$440$	0	0
$441$	2.07180	0.0986570
$442$	0	0
$443$	14.9282	0.709260	0.354630	−	0.935007i	$-0.384607\pi$
$443$	14.9282	0.709260	0.354630	+	0.935007i	$0.384607\pi$
$444$	0	0
$445$	61.4449	2.91277
$446$	0	0
$447$	−15.1244	−0.715357
$448$	0	0
$449$	−1.14359	−0.0539695	−0.0269848	−	0.999636i	$-0.508591\pi$
$449$	−1.14359	−0.0539695	−0.0269848	+	0.999636i	$0.508591\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−28.7846	−1.35242
$454$	0	0
$455$	−45.5167	−2.13385
$456$	0	0
$457$	−3.58846	−0.167861	−0.0839305	−	0.996472i	$-0.526747\pi$
$457$	−3.58846	−0.167861	−0.0839305	+	0.996472i	$0.526747\pi$
$458$	0	0
$459$	9.07180	0.423435
$460$	0	0
$461$	−22.3205	−1.03957	−0.519785	−	0.854297i	$-0.673988\pi$
$461$	−22.3205	−1.03957	−0.519785	+	0.854297i	$0.673988\pi$
$462$	0	0
$463$	−36.7846	−1.70953	−0.854763	−	0.519019i	$-0.826298\pi$
$463$	−36.7846	−1.70953	−0.854763	+	0.519019i	$0.826298\pi$
$464$	0	0
$465$	48.2487	2.23748
$466$	0	0
$467$	−28.3923	−1.31384	−0.656920	−	0.753961i	$-0.728141\pi$
$467$	−28.3923	−1.31384	−0.656920	+	0.753961i	$0.728141\pi$
$468$	0	0
$469$	−0.535898	−0.0247455
$470$	0	0
$471$	−56.7846	−2.61650
$472$	0	0
$473$	0	0
$474$	0	0
$475$	29.1769	1.33873
$476$	0	0
$477$	5.33975	0.244490
$478$	0	0
$479$	13.0718	0.597266	0.298633	−	0.954368i	$-0.403469\pi$
$479$	13.0718	0.597266	0.298633	+	0.954368i	$0.403469\pi$
$480$	0	0
$481$	25.5885	1.16673
$482$	0	0
$483$	−16.3923	−0.745876
$484$	0	0
$485$	44.5167	2.02140
$486$	0	0
$487$	−18.1962	−0.824546	−0.412273	−	0.911060i	$-0.635265\pi$
$487$	−18.1962	−0.824546	−0.412273	+	0.911060i	$0.635265\pi$
$488$	0	0
$489$	−64.2487	−2.90543
$490$	0	0
$491$	−27.6603	−1.24829	−0.624145	−	0.781309i	$-0.714553\pi$
$491$	−27.6603	−1.24829	−0.624145	+	0.781309i	$0.714553\pi$
$492$	0	0
$493$	−10.1244	−0.455978
$494$	0	0
$495$	0	0
$496$	0	0
$497$	6.92820	0.310772
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	0	0
$501$	17.8564	0.797765
$502$	0	0
$503$	19.5167	0.870205	0.435102	−	0.900381i	$-0.356712\pi$
$503$	19.5167	0.870205	0.435102	+	0.900381i	$0.356712\pi$
$504$	0	0
$505$	65.1769	2.90033
$506$	0	0
$507$	18.9282	0.840631
$508$	0	0
$509$	42.7846	1.89639	0.948197	−	0.317682i	$-0.102905\pi$
$509$	42.7846	1.89639	0.948197	+	0.317682i	$0.102905\pi$
$510$	0	0
$511$	−24.3923	−1.07905
$512$	0	0
$513$	13.0718	0.577134
$514$	0	0
$515$	24.3923	1.07485
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−6.92820	−0.304114
$520$	0	0
$521$	−35.3205	−1.54742	−0.773710	−	0.633540i	$-0.781601\pi$
$521$	−35.3205	−1.54742	−0.773710	+	0.633540i	$0.781601\pi$
$522$	0	0
$523$	6.53590	0.285795	0.142897	−	0.989738i	$-0.454358\pi$
$523$	6.53590	0.285795	0.142897	+	0.989738i	$0.454358\pi$
$524$	0	0
$525$	66.6410	2.90845
$526$	0	0
$527$	−10.7321	−0.467495
$528$	0	0
$529$	−18.1769	−0.790301
$530$	0	0
$531$	60.1051	2.60834
$532$	0	0
$533$	34.5167	1.49508
$534$	0	0
$535$	63.9090	2.76303
$536$	0	0
$537$	43.7128	1.88635
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	0	0
$543$	−7.66025	−0.328733
$544$	0	0
$545$	−46.5167	−1.99255
$546$	0	0
$547$	−35.7128	−1.52697	−0.763485	−	0.645826i	$-0.776513\pi$
$547$	−35.7128	−1.52697	−0.763485	+	0.645826i	$0.776513\pi$
$548$	0	0
$549$	29.1769	1.24524
$550$	0	0
$551$	−14.5885	−0.621489
$552$	0	0
$553$	−45.3205	−1.92722
$554$	0	0
$555$	−58.4449	−2.48085
$556$	0	0
$557$	27.3205	1.15761	0.578804	−	0.815467i	$-0.303520\pi$
$557$	27.3205	1.15761	0.578804	+	0.815467i	$0.303520\pi$
$558$	0	0
$559$	35.7128	1.51049
$560$	0	0
$561$	0	0
$562$	0	0
$563$	21.1244	0.890285	0.445143	−	0.895460i	$-0.353153\pi$
$563$	21.1244	0.890285	0.445143	+	0.895460i	$0.353153\pi$
$564$	0	0
$565$	−18.6603	−0.785043
$566$	0	0
$567$	−6.73205	−0.282720
$568$	0	0
$569$	4.14359	0.173708	0.0868542	−	0.996221i	$-0.472319\pi$
$569$	4.14359	0.173708	0.0868542	+	0.996221i	$0.472319\pi$
$570$	0	0
$571$	−8.05256	−0.336989	−0.168495	−	0.985703i	$-0.553891\pi$
$571$	−8.05256	−0.336989	−0.168495	+	0.985703i	$0.553891\pi$
$572$	0	0
$573$	12.0000	0.501307
$574$	0	0
$575$	−19.6077	−0.817697
$576$	0	0
$577$	13.5359	0.563507	0.281753	−	0.959487i	$-0.409084\pi$
$577$	13.5359	0.563507	0.281753	+	0.959487i	$0.409084\pi$
$578$	0	0
$579$	45.5167	1.89161
$580$	0	0
$581$	6.00000	0.248922
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−74.3731	−3.07495
$586$	0	0
$587$	18.7321	0.773154	0.386577	−	0.922257i	$-0.373657\pi$
$587$	18.7321	0.773154	0.386577	+	0.922257i	$0.373657\pi$
$588$	0	0
$589$	−15.4641	−0.637187
$590$	0	0
$591$	16.1962	0.666221
$592$	0	0
$593$	−30.3731	−1.24727	−0.623636	−	0.781715i	$-0.714345\pi$
$593$	−30.3731	−1.24727	−0.623636	+	0.781715i	$0.714345\pi$
$594$	0	0
$595$	−23.1244	−0.948006
$596$	0	0
$597$	−17.8564	−0.730814
$598$	0	0
$599$	21.1244	0.863118	0.431559	−	0.902085i	$-0.357964\pi$
$599$	21.1244	0.863118	0.431559	+	0.902085i	$0.357964\pi$
$600$	0	0
$601$	−29.1962	−1.19094	−0.595468	−	0.803379i	$-0.703033\pi$
$601$	−29.1962	−1.19094	−0.595468	+	0.803379i	$0.703033\pi$
$602$	0	0
$603$	−0.875644	−0.0356590
$604$	0	0
$605$	0	0
$606$	0	0
$607$	36.1962	1.46916	0.734578	−	0.678524i	$-0.237380\pi$
$607$	36.1962	1.46916	0.734578	+	0.678524i	$0.237380\pi$
$608$	0	0
$609$	−33.3205	−1.35022
$610$	0	0
$611$	−14.5885	−0.590186
$612$	0	0
$613$	19.9282	0.804893	0.402446	−	0.915444i	$-0.368160\pi$
$613$	19.9282	0.804893	0.402446	+	0.915444i	$0.368160\pi$
$614$	0	0
$615$	−78.8372	−3.17902
$616$	0	0
$617$	−44.3205	−1.78428	−0.892138	−	0.451763i	$-0.850795\pi$
$617$	−44.3205	−1.78428	−0.892138	+	0.451763i	$0.850795\pi$
$618$	0	0
$619$	33.6603	1.35292	0.676460	−	0.736479i	$-0.263513\pi$
$619$	33.6603	1.35292	0.676460	+	0.736479i	$0.263513\pi$
$620$	0	0
$621$	−8.78461	−0.352514
$622$	0	0
$623$	−44.9808	−1.80212
$624$	0	0
$625$	10.0718	0.402872
$626$	0	0
$627$	0	0
$628$	0	0
$629$	13.0000	0.518344
$630$	0	0
$631$	32.0526	1.27599	0.637996	−	0.770040i	$-0.279764\pi$
$631$	32.0526	1.27599	0.637996	+	0.770040i	$0.279764\pi$
$632$	0	0
$633$	48.7846	1.93901
$634$	0	0
$635$	0	0
$636$	0	0
$637$	2.07180	0.0820876
$638$	0	0
$639$	11.3205	0.447832
$640$	0	0
$641$	−10.4641	−0.413307	−0.206654	−	0.978414i	$-0.566257\pi$
$641$	−10.4641	−0.413307	−0.206654	+	0.978414i	$0.566257\pi$
$642$	0	0
$643$	34.0526	1.34290	0.671451	−	0.741049i	$-0.265671\pi$
$643$	34.0526	1.34290	0.671451	+	0.741049i	$0.265671\pi$
$644$	0	0
$645$	−81.5692	−3.21178
$646$	0	0
$647$	16.3923	0.644448	0.322224	−	0.946663i	$-0.395570\pi$
$647$	16.3923	0.644448	0.322224	+	0.946663i	$0.395570\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	−35.3205	−1.38432
$652$	0	0
$653$	6.92820	0.271122	0.135561	−	0.990769i	$-0.456716\pi$
$653$	6.92820	0.271122	0.135561	+	0.990769i	$0.456716\pi$
$654$	0	0
$655$	29.8564	1.16659
$656$	0	0
$657$	−39.8564	−1.55495
$658$	0	0
$659$	2.98076	0.116114	0.0580570	−	0.998313i	$-0.481509\pi$
$659$	2.98076	0.116114	0.0580570	+	0.998313i	$0.481509\pi$
$660$	0	0
$661$	−17.3397	−0.674438	−0.337219	−	0.941426i	$-0.609486\pi$
$661$	−17.3397	−0.674438	−0.337219	+	0.941426i	$0.609486\pi$
$662$	0	0
$663$	27.6603	1.07424
$664$	0	0
$665$	−33.3205	−1.29211
$666$	0	0
$667$	9.80385	0.379606
$668$	0	0
$669$	−22.9282	−0.886456
$670$	0	0
$671$	0	0
$672$	0	0
$673$	−28.9282	−1.11510	−0.557550	−	0.830143i	$-0.688259\pi$
$673$	−28.9282	−1.11510	−0.557550	+	0.830143i	$0.688259\pi$
$674$	0	0
$675$	35.7128	1.37459
$676$	0	0
$677$	−16.8564	−0.647844	−0.323922	−	0.946084i	$-0.605002\pi$
$677$	−16.8564	−0.647844	−0.323922	+	0.946084i	$0.605002\pi$
$678$	0	0
$679$	−32.5885	−1.25063
$680$	0	0
$681$	9.85641	0.377698
$682$	0	0
$683$	−23.1244	−0.884829	−0.442414	−	0.896811i	$-0.645878\pi$
$683$	−23.1244	−0.884829	−0.442414	+	0.896811i	$0.645878\pi$
$684$	0	0
$685$	9.46410	0.361605
$686$	0	0
$687$	34.9808	1.33460
$688$	0	0
$689$	5.33975	0.203428
$690$	0	0
$691$	−21.1769	−0.805608	−0.402804	−	0.915286i	$-0.631964\pi$
$691$	−21.1769	−0.805608	−0.402804	+	0.915286i	$0.631964\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−23.1244	−0.877157
$696$	0	0
$697$	17.5359	0.664220
$698$	0	0
$699$	−3.26795	−0.123605
$700$	0	0
$701$	−12.6077	−0.476186	−0.238093	−	0.971242i	$-0.576522\pi$
$701$	−12.6077	−0.476186	−0.238093	+	0.971242i	$0.576522\pi$
$702$	0	0
$703$	18.7321	0.706493
$704$	0	0
$705$	33.3205	1.25492
$706$	0	0
$707$	−47.7128	−1.79443
$708$	0	0
$709$	−0.143594	−0.00539277	−0.00269638	−	0.999996i	$-0.500858\pi$
$709$	−0.143594	−0.00539277	−0.00269638	+	0.999996i	$0.500858\pi$
$710$	0	0
$711$	−74.0526	−2.77719
$712$	0	0
$713$	10.3923	0.389195
$714$	0	0
$715$	0	0
$716$	0	0
$717$	53.3205	1.99129
$718$	0	0
$719$	10.1962	0.380252	0.190126	−	0.981760i	$-0.439110\pi$
$719$	10.1962	0.380252	0.190126	+	0.981760i	$0.439110\pi$
$720$	0	0
$721$	−17.8564	−0.665007
$722$	0	0
$723$	−24.3923	−0.907160
$724$	0	0
$725$	−39.8564	−1.48023
$726$	0	0
$727$	−2.19615	−0.0814508	−0.0407254	−	0.999170i	$-0.512967\pi$
$727$	−2.19615	−0.0814508	−0.0407254	+	0.999170i	$0.512967\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	0	0
$731$	18.1436	0.671065
$732$	0	0
$733$	33.0000	1.21888	0.609441	−	0.792831i	$-0.291394\pi$
$733$	33.0000	1.21888	0.609441	+	0.792831i	$0.291394\pi$
$734$	0	0
$735$	−4.73205	−0.174544
$736$	0	0
$737$	0	0
$738$	0	0
$739$	16.7321	0.615498	0.307749	−	0.951468i	$-0.400424\pi$
$739$	16.7321	0.615498	0.307749	+	0.951468i	$0.400424\pi$
$740$	0	0
$741$	39.8564	1.46416
$742$	0	0
$743$	41.3731	1.51783	0.758915	−	0.651189i	$-0.225730\pi$
$743$	41.3731	1.51783	0.758915	+	0.651189i	$0.225730\pi$
$744$	0	0
$745$	20.6603	0.756933
$746$	0	0
$747$	9.80385	0.358704
$748$	0	0
$749$	−46.7846	−1.70947
$750$	0	0
$751$	−33.8564	−1.23544	−0.617719	−	0.786399i	$-0.711943\pi$
$751$	−33.8564	−1.23544	−0.617719	+	0.786399i	$0.711943\pi$
$752$	0	0
$753$	47.1769	1.71922
$754$	0	0
$755$	39.3205	1.43102
$756$	0	0
$757$	−38.1244	−1.38565	−0.692827	−	0.721104i	$-0.743635\pi$
$757$	−38.1244	−1.38565	−0.692827	+	0.721104i	$0.743635\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−23.1962	−0.840860	−0.420430	−	0.907325i	$-0.638121\pi$
$761$	−23.1962	−0.840860	−0.420430	+	0.907325i	$0.638121\pi$
$762$	0	0
$763$	34.0526	1.23279
$764$	0	0
$765$	−37.7846	−1.36611
$766$	0	0
$767$	60.1051	2.17027
$768$	0	0
$769$	14.2679	0.514515	0.257258	−	0.966343i	$-0.417181\pi$
$769$	14.2679	0.514515	0.257258	+	0.966343i	$0.417181\pi$
$770$	0	0
$771$	−12.1962	−0.439234
$772$	0	0
$773$	−39.8564	−1.43354	−0.716768	−	0.697312i	$-0.754379\pi$
$773$	−39.8564	−1.43354	−0.716768	+	0.697312i	$0.754379\pi$
$774$	0	0
$775$	−42.2487	−1.51762
$776$	0	0
$777$	42.7846	1.53489
$778$	0	0
$779$	25.2679	0.905318
$780$	0	0
$781$	0	0
$782$	0	0
$783$	−17.8564	−0.638136
$784$	0	0
$785$	77.5692	2.76856
$786$	0	0
$787$	4.78461	0.170553	0.0852765	−	0.996357i	$-0.472823\pi$
$787$	4.78461	0.170553	0.0852765	+	0.996357i	$0.472823\pi$
$788$	0	0
$789$	−6.39230	−0.227572
$790$	0	0
$791$	13.6603	0.485703
$792$	0	0
$793$	29.1769	1.03610
$794$	0	0
$795$	−12.1962	−0.432553
$796$	0	0
$797$	40.6410	1.43958	0.719789	−	0.694193i	$-0.244238\pi$
$797$	40.6410	1.43958	0.719789	+	0.694193i	$0.244238\pi$
$798$	0	0
$799$	−7.41154	−0.262202
$800$	0	0
$801$	−73.4974	−2.59690
$802$	0	0
$803$	0	0
$804$	0	0
$805$	22.3923	0.789225
$806$	0	0
$807$	−52.0526	−1.83234
$808$	0	0
$809$	13.7128	0.482117	0.241058	−	0.970511i	$-0.422505\pi$
$809$	13.7128	0.482117	0.241058	+	0.970511i	$0.422505\pi$
$810$	0	0
$811$	−17.0718	−0.599472	−0.299736	−	0.954022i	$-0.596899\pi$
$811$	−17.0718	−0.599472	−0.299736	+	0.954022i	$0.596899\pi$
$812$	0	0
$813$	−17.8564	−0.626252
$814$	0	0
$815$	87.7654	3.07429
$816$	0	0
$817$	26.1436	0.914649
$818$	0	0
$819$	54.4449	1.90246
$820$	0	0
$821$	13.1769	0.459877	0.229939	−	0.973205i	$-0.426147\pi$
$821$	13.1769	0.459877	0.229939	+	0.973205i	$0.426147\pi$
$822$	0	0
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−25.9090	−0.900943	−0.450471	−	0.892791i	$-0.648744\pi$
$827$	−25.9090	−0.900943	−0.450471	+	0.892791i	$0.648744\pi$
$828$	0	0
$829$	1.58846	0.0551694	0.0275847	−	0.999619i	$-0.491218\pi$
$829$	1.58846	0.0551694	0.0275847	+	0.999619i	$0.491218\pi$
$830$	0	0
$831$	−37.6603	−1.30642
$832$	0	0
$833$	1.05256	0.0364690
$834$	0	0
$835$	−24.3923	−0.844131
$836$	0	0
$837$	−18.9282	−0.654254
$838$	0	0
$839$	−53.2295	−1.83769	−0.918843	−	0.394624i	$-0.870875\pi$
$839$	−53.2295	−1.83769	−0.918843	+	0.394624i	$0.870875\pi$
$840$	0	0
$841$	−9.07180	−0.312821
$842$	0	0
$843$	21.4641	0.739263
$844$	0	0
$845$	−25.8564	−0.889487
$846$	0	0
$847$	0	0
$848$	0	0
$849$	84.4974	2.89994
$850$	0	0
$851$	−12.5885	−0.431527
$852$	0	0
$853$	−14.4641	−0.495241	−0.247621	−	0.968857i	$-0.579649\pi$
$853$	−14.4641	−0.495241	−0.247621	+	0.968857i	$0.579649\pi$
$854$	0	0
$855$	−54.4449	−1.86198
$856$	0	0
$857$	−26.1436	−0.893048	−0.446524	−	0.894772i	$-0.647338\pi$
$857$	−26.1436	−0.893048	−0.446524	+	0.894772i	$0.647338\pi$
$858$	0	0
$859$	54.6410	1.86433	0.932164	−	0.362037i	$-0.117919\pi$
$859$	54.6410	1.86433	0.932164	+	0.362037i	$0.117919\pi$
$860$	0	0
$861$	57.7128	1.96685
$862$	0	0
$863$	−14.5885	−0.496597	−0.248298	−	0.968684i	$-0.579871\pi$
$863$	−14.5885	−0.496597	−0.248298	+	0.968684i	$0.579871\pi$
$864$	0	0
$865$	9.46410	0.321789
$866$	0	0
$867$	−32.3923	−1.10010
$868$	0	0
$869$	0	0
$870$	0	0
$871$	−0.875644	−0.0296701
$872$	0	0
$873$	−53.2487	−1.80220
$874$	0	0
$875$	−40.0526	−1.35402
$876$	0	0
$877$	15.7846	0.533008	0.266504	−	0.963834i	$-0.414131\pi$
$877$	15.7846	0.533008	0.266504	+	0.963834i	$0.414131\pi$
$878$	0	0
$879$	60.9808	2.05683
$880$	0	0
$881$	−9.67949	−0.326110	−0.163055	−	0.986617i	$-0.552135\pi$
$881$	−9.67949	−0.326110	−0.163055	+	0.986617i	$0.552135\pi$
$882$	0	0
$883$	6.53590	0.219950	0.109975	−	0.993934i	$-0.464923\pi$
$883$	6.53590	0.219950	0.109975	+	0.993934i	$0.464923\pi$
$884$	0	0
$885$	−137.282	−4.61469
$886$	0	0
$887$	−43.2295	−1.45150	−0.725752	−	0.687957i	$-0.758508\pi$
$887$	−43.2295	−1.45150	−0.725752	+	0.687957i	$0.758508\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−10.6795	−0.357376
$894$	0	0
$895$	−59.7128	−1.99598
$896$	0	0
$897$	−26.7846	−0.894312
$898$	0	0
$899$	21.1244	0.704537
$900$	0	0
$901$	2.71281	0.0903769
$902$	0	0
$903$	59.7128	1.98712
$904$	0	0
$905$	10.4641	0.347839
$906$	0	0
$907$	17.8564	0.592912	0.296456	−	0.955046i	$-0.404195\pi$
$907$	17.8564	0.592912	0.296456	+	0.955046i	$0.404195\pi$
$908$	0	0
$909$	−77.9615	−2.58582
$910$	0	0
$911$	−13.1769	−0.436571	−0.218285	−	0.975885i	$-0.570046\pi$
$911$	−13.1769	−0.436571	−0.218285	+	0.975885i	$0.570046\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−66.6410	−2.20308
$916$	0	0
$917$	−21.8564	−0.721762
$918$	0	0
$919$	−12.0000	−0.395843	−0.197922	−	0.980218i	$-0.563419\pi$
$919$	−12.0000	−0.395843	−0.197922	+	0.980218i	$0.563419\pi$
$920$	0	0
$921$	−88.6410	−2.92082
$922$	0	0
$923$	11.3205	0.372619
$924$	0	0
$925$	51.1769	1.68269
$926$	0	0
$927$	−29.1769	−0.958296
$928$	0	0
$929$	−19.9282	−0.653823	−0.326912	−	0.945055i	$-0.606008\pi$
$929$	−19.9282	−0.653823	−0.326912	+	0.945055i	$0.606008\pi$
$930$	0	0
$931$	1.51666	0.0497065
$932$	0	0
$933$	45.8564	1.50127
$934$	0	0
$935$	0	0
$936$	0	0
$937$	36.2679	1.18482	0.592411	−	0.805636i	$-0.298176\pi$
$937$	36.2679	1.18482	0.592411	+	0.805636i	$0.298176\pi$
$938$	0	0
$939$	−19.1244	−0.624100
$940$	0	0
$941$	21.0000	0.684580	0.342290	−	0.939594i	$-0.388797\pi$
$941$	21.0000	0.684580	0.342290	+	0.939594i	$0.388797\pi$
$942$	0	0
$943$	−16.9808	−0.552970
$944$	0	0
$945$	−40.7846	−1.32672
$946$	0	0
$947$	−11.5167	−0.374241	−0.187121	−	0.982337i	$-0.559916\pi$
$947$	−11.5167	−0.374241	−0.187121	+	0.982337i	$0.559916\pi$
$948$	0	0
$949$	−39.8564	−1.29379
$950$	0	0
$951$	10.5359	0.341650
$952$	0	0
$953$	−17.1962	−0.557038	−0.278519	−	0.960431i	$-0.589844\pi$
$953$	−17.1962	−0.557038	−0.278519	+	0.960431i	$0.589844\pi$
$954$	0	0
$955$	−16.3923	−0.530443
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−6.92820	−0.223723
$960$	0	0
$961$	−8.60770	−0.277668
$962$	0	0
$963$	−76.4449	−2.46340
$964$	0	0
$965$	−62.1769	−2.00155
$966$	0	0
$967$	−18.7321	−0.602382	−0.301191	−	0.953564i	$-0.597384\pi$
$967$	−18.7321	−0.602382	−0.301191	+	0.953564i	$0.597384\pi$
$968$	0	0
$969$	20.2487	0.650482
$970$	0	0
$971$	39.5167	1.26815	0.634075	−	0.773272i	$-0.281381\pi$
$971$	39.5167	1.26815	0.634075	+	0.773272i	$0.281381\pi$
$972$	0	0
$973$	16.9282	0.542693
$974$	0	0
$975$	108.890	3.48726
$976$	0	0
$977$	26.1769	0.837474	0.418737	−	0.908108i	$-0.362473\pi$
$977$	26.1769	0.837474	0.418737	+	0.908108i	$0.362473\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	55.6410	1.77648
$982$	0	0
$983$	36.1051	1.15157	0.575787	−	0.817600i	$-0.304696\pi$
$983$	36.1051	1.15157	0.575787	+	0.817600i	$0.304696\pi$
$984$	0	0
$985$	−22.1244	−0.704941
$986$	0	0
$987$	−24.3923	−0.776416
$988$	0	0
$989$	−17.5692	−0.558669
$990$	0	0
$991$	17.8564	0.567227	0.283614	−	0.958939i	$-0.408467\pi$
$991$	17.8564	0.567227	0.283614	+	0.958939i	$0.408467\pi$
$992$	0	0
$993$	−2.92820	−0.0929237
$994$	0	0
$995$	24.3923	0.773288
$996$	0	0
$997$	57.9282	1.83460	0.917302	−	0.398192i	$-0.130362\pi$
$997$	57.9282	1.83460	0.917302	+	0.398192i	$0.130362\pi$
$998$	0	0
$999$	22.9282	0.725416

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	968.2.a.l.1.2	yes	2
3.2	odd	2		8712.2.a.bs.1.2		2
4.3	odd	2		1936.2.a.p.1.1		2
8.3	odd	2		7744.2.a.cw.1.2		2
8.5	even	2		7744.2.a.bv.1.1		2
11.2	odd	10		968.2.i.o.81.2		8
11.3	even	5		968.2.i.n.9.1		8
11.4	even	5		968.2.i.n.753.1		8
11.5	even	5		968.2.i.n.729.2		8
11.6	odd	10		968.2.i.o.729.2		8
11.7	odd	10		968.2.i.o.753.1		8
11.8	odd	10		968.2.i.o.9.1		8
11.9	even	5		968.2.i.n.81.2		8
11.10	odd	2		968.2.a.k.1.2	✓	2
33.32	even	2		8712.2.a.br.1.2		2
44.43	even	2		1936.2.a.q.1.1		2
88.21	odd	2		7744.2.a.bu.1.1		2
88.43	even	2		7744.2.a.cx.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
968.2.a.k.1.2	✓	2	11.10	odd	2
968.2.a.l.1.2	yes	2	1.1	even	1	trivial
968.2.i.n.9.1		8	11.3	even	5
968.2.i.n.81.2		8	11.9	even	5
968.2.i.n.729.2		8	11.5	even	5
968.2.i.n.753.1		8	11.4	even	5
968.2.i.o.9.1		8	11.8	odd	10
968.2.i.o.81.2		8	11.2	odd	10
968.2.i.o.729.2		8	11.6	odd	10
968.2.i.o.753.1		8	11.7	odd	10
1936.2.a.p.1.1		2	4.3	odd	2
1936.2.a.q.1.1		2	44.43	even	2
7744.2.a.bu.1.1		2	88.21	odd	2
7744.2.a.bv.1.1		2	8.5	even	2
7744.2.a.cw.1.2		2	8.3	odd	2
7744.2.a.cx.1.2		2	88.43	even	2
8712.2.a.br.1.2		2	33.32	even	2
8712.2.a.bs.1.2		2	3.2	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:	\( N \)	\(=\)	\( 968 = 2^{3} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	968.a (trivial)

\(f(q)\)	\(=\)	\(q+2.73205 q^{3} -3.73205 q^{5} +2.73205 q^{7} +4.46410 q^{9} +O(q^{10})\)	\(q+2.73205 q^{3} -3.73205 q^{5} +2.73205 q^{7} +4.46410 q^{9} +4.46410 q^{13} -10.1962 q^{15} +2.26795 q^{17} +3.26795 q^{19} +7.46410 q^{21} -2.19615 q^{23} +8.92820 q^{25} +4.00000 q^{27} -4.46410 q^{29} -4.73205 q^{31} -10.1962 q^{35} +5.73205 q^{37} +12.1962 q^{39} +7.73205 q^{41} +8.00000 q^{43} -16.6603 q^{45} -3.26795 q^{47} +0.464102 q^{49} +6.19615 q^{51} +1.19615 q^{53} +8.92820 q^{57} +13.4641 q^{59} +6.53590 q^{61} +12.1962 q^{63} -16.6603 q^{65} -0.196152 q^{67} -6.00000 q^{69} +2.53590 q^{71} -8.92820 q^{73} +24.3923 q^{75} -16.5885 q^{79} -2.46410 q^{81} +2.19615 q^{83} -8.46410 q^{85} -12.1962 q^{87} -16.4641 q^{89} +12.1962 q^{91} -12.9282 q^{93} -12.1962 q^{95} -11.9282 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 4 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 4 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 4 q^{5} + 2 q^{7} + 2 q^{9} + 2 q^{13} - 10 q^{15} + 8 q^{17} + 10 q^{19} + 8 q^{21} + 6 q^{23} + 4 q^{25} + 8 q^{27} - 2 q^{29} - 6 q^{31} - 10 q^{35} + 8 q^{37} + 14 q^{39} + 12 q^{41} + 16 q^{43} - 16 q^{45} - 10 q^{47} - 6 q^{49} + 2 q^{51} - 8 q^{53} + 4 q^{57} + 20 q^{59} + 20 q^{61} + 14 q^{63} - 16 q^{65} + 10 q^{67} - 12 q^{69} + 12 q^{71} - 4 q^{73} + 28 q^{75} - 2 q^{79} + 2 q^{81} - 6 q^{83} - 10 q^{85} - 14 q^{87} - 26 q^{89} + 14 q^{91} - 12 q^{93} - 14 q^{95} - 10 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 - 4 * q^5 + 2 * q^7 + 2 * q^9 + 2 * q^13 - 10 * q^15 + 8 * q^17 + 10 * q^19 + 8 * q^21 + 6 * q^23 + 4 * q^25 + 8 * q^27 - 2 * q^29 - 6 * q^31 - 10 * q^35 + 8 * q^37 + 14 * q^39 + 12 * q^41 + 16 * q^43 - 16 * q^45 - 10 * q^47 - 6 * q^49 + 2 * q^51 - 8 * q^53 + 4 * q^57 + 20 * q^59 + 20 * q^61 + 14 * q^63 - 16 * q^65 + 10 * q^67 - 12 * q^69 + 12 * q^71 - 4 * q^73 + 28 * q^75 - 2 * q^79 + 2 * q^81 - 6 * q^83 - 10 * q^85 - 14 * q^87 - 26 * q^89 + 14 * q^91 - 12 * q^93 - 14 * q^95 - 10 * q^97

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