LMFDB - Embedded newform 68.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [68,2,Mod(1,68)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(68, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("68.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 68 = 2^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	68.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:	$0.542982733745$
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			1.1
Root			$-1.73205$ of defining polynomial
Character	$\chi$	$=$	68.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.732051 q^{3} +3.46410 q^{5} +0.732051 q^{7} -2.46410 q^{9} +O(q^{10})$	$q-0.732051 q^{3} +3.46410 q^{5} +0.732051 q^{7} -2.46410 q^{9} -4.73205 q^{11} -1.46410 q^{13} -2.53590 q^{15} -1.00000 q^{17} +5.46410 q^{19} -0.535898 q^{21} -4.73205 q^{23} +7.00000 q^{25} +4.00000 q^{27} -3.46410 q^{29} -6.19615 q^{31} +3.46410 q^{33} +2.53590 q^{35} +11.4641 q^{37} +1.07180 q^{39} -6.00000 q^{41} +12.3923 q^{43} -8.53590 q^{45} +6.92820 q^{47} -6.46410 q^{49} +0.732051 q^{51} -0.928203 q^{53} -16.3923 q^{55} -4.00000 q^{57} +9.46410 q^{59} -7.46410 q^{61} -1.80385 q^{63} -5.07180 q^{65} +1.07180 q^{67} +3.46410 q^{69} +2.19615 q^{71} +2.00000 q^{73} -5.12436 q^{75} -3.46410 q^{77} -1.80385 q^{79} +4.46410 q^{81} -9.46410 q^{83} -3.46410 q^{85} +2.53590 q^{87} +9.46410 q^{89} -1.07180 q^{91} +4.53590 q^{93} +18.9282 q^{95} +8.92820 q^{97} +11.6603 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} - 6 q^{11} + 4 q^{13} - 12 q^{15} - 2 q^{17} + 4 q^{19} - 8 q^{21} - 6 q^{23} + 14 q^{25} + 8 q^{27} - 2 q^{31} + 12 q^{35} + 16 q^{37} + 16 q^{39} - 12 q^{41} + 4 q^{43} - 24 q^{45} - 6 q^{49} - 2 q^{51} + 12 q^{53} - 12 q^{55} - 8 q^{57} + 12 q^{59} - 8 q^{61} - 14 q^{63} - 24 q^{65} + 16 q^{67} - 6 q^{71} + 4 q^{73} + 14 q^{75} - 14 q^{79} + 2 q^{81} - 12 q^{83} + 12 q^{87} + 12 q^{89} - 16 q^{91} + 16 q^{93} + 24 q^{95} + 4 q^{97} + 6 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 - 6 * q^11 + 4 * q^13 - 12 * q^15 - 2 * q^17 + 4 * q^19 - 8 * q^21 - 6 * q^23 + 14 * q^25 + 8 * q^27 - 2 * q^31 + 12 * q^35 + 16 * q^37 + 16 * q^39 - 12 * q^41 + 4 * q^43 - 24 * q^45 - 6 * q^49 - 2 * q^51 + 12 * q^53 - 12 * q^55 - 8 * q^57 + 12 * q^59 - 8 * q^61 - 14 * q^63 - 24 * q^65 + 16 * q^67 - 6 * q^71 + 4 * q^73 + 14 * q^75 - 14 * q^79 + 2 * q^81 - 12 * q^83 + 12 * q^87 + 12 * q^89 - 16 * q^91 + 16 * q^93 + 24 * q^95 + 4 * q^97 + 6 * 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
$3$	−0.732051	−0.422650	−0.211325	−	0.977416i	$-0.567778\pi$
$3$	−0.732051	−0.422650	−0.211325	+	0.977416i	$0.567778\pi$
$4$	0	0
$5$	3.46410	1.54919	0.774597	−	0.632456i	$-0.217953\pi$
$5$	3.46410	1.54919	0.774597	+	0.632456i	$0.217953\pi$
$6$	0	0
$7$	0.732051	0.276689	0.138345	−	0.990384i	$-0.455822\pi$
$7$	0.732051	0.276689	0.138345	+	0.990384i	$0.455822\pi$
$8$	0	0
$9$	−2.46410	−0.821367
$10$	0	0
$11$	−4.73205	−1.42677	−0.713384	−	0.700774i	$-0.752838\pi$
$11$	−4.73205	−1.42677	−0.713384	+	0.700774i	$0.752838\pi$
$12$	0	0
$13$	−1.46410	−0.406069	−0.203034	−	0.979172i	$-0.565080\pi$
$13$	−1.46410	−0.406069	−0.203034	+	0.979172i	$0.565080\pi$
$14$	0	0
$15$	−2.53590	−0.654766
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	5.46410	1.25355	0.626775	−	0.779200i	$-0.284374\pi$
$19$	5.46410	1.25355	0.626775	+	0.779200i	$0.284374\pi$
$20$	0	0
$21$	−0.535898	−0.116943
$22$	0	0
$23$	−4.73205	−0.986701	−0.493350	−	0.869831i	$-0.664228\pi$
$23$	−4.73205	−0.986701	−0.493350	+	0.869831i	$0.664228\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	0	0
$27$	4.00000	0.769800
$28$	0	0
$29$	−3.46410	−0.643268	−0.321634	−	0.946864i	$-0.604232\pi$
$29$	−3.46410	−0.643268	−0.321634	+	0.946864i	$0.604232\pi$
$30$	0	0
$31$	−6.19615	−1.11286	−0.556431	−	0.830894i	$-0.687830\pi$
$31$	−6.19615	−1.11286	−0.556431	+	0.830894i	$0.687830\pi$
$32$	0	0
$33$	3.46410	0.603023
$34$	0	0
$35$	2.53590	0.428645
$36$	0	0
$37$	11.4641	1.88469	0.942343	−	0.334648i	$-0.108617\pi$
$37$	11.4641	1.88469	0.942343	+	0.334648i	$0.108617\pi$
$38$	0	0
$39$	1.07180	0.171625
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	12.3923	1.88981	0.944904	−	0.327346i	$-0.106154\pi$
$43$	12.3923	1.88981	0.944904	+	0.327346i	$0.106154\pi$
$44$	0	0
$45$	−8.53590	−1.27246
$46$	0	0
$47$	6.92820	1.01058	0.505291	−	0.862949i	$-0.331385\pi$
$47$	6.92820	1.01058	0.505291	+	0.862949i	$0.331385\pi$
$48$	0	0
$49$	−6.46410	−0.923443
$50$	0	0
$51$	0.732051	0.102508
$52$	0	0
$53$	−0.928203	−0.127499	−0.0637493	−	0.997966i	$-0.520306\pi$
$53$	−0.928203	−0.127499	−0.0637493	+	0.997966i	$0.520306\pi$
$54$	0	0
$55$	−16.3923	−2.21034
$56$	0	0
$57$	−4.00000	−0.529813
$58$	0	0
$59$	9.46410	1.23212	0.616061	−	0.787699i	$-0.288728\pi$
$59$	9.46410	1.23212	0.616061	+	0.787699i	$0.288728\pi$
$60$	0	0
$61$	−7.46410	−0.955680	−0.477840	−	0.878447i	$-0.658580\pi$
$61$	−7.46410	−0.955680	−0.477840	+	0.878447i	$0.658580\pi$
$62$	0	0
$63$	−1.80385	−0.227263
$64$	0	0
$65$	−5.07180	−0.629079
$66$	0	0
$67$	1.07180	0.130941	0.0654704	−	0.997855i	$-0.479145\pi$
$67$	1.07180	0.130941	0.0654704	+	0.997855i	$0.479145\pi$
$68$	0	0
$69$	3.46410	0.417029
$70$	0	0
$71$	2.19615	0.260635	0.130318	−	0.991472i	$-0.458400\pi$
$71$	2.19615	0.260635	0.130318	+	0.991472i	$0.458400\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	−5.12436	−0.591710
$76$	0	0
$77$	−3.46410	−0.394771
$78$	0	0
$79$	−1.80385	−0.202949	−0.101474	−	0.994838i	$-0.532356\pi$
$79$	−1.80385	−0.202949	−0.101474	+	0.994838i	$0.532356\pi$
$80$	0	0
$81$	4.46410	0.496011
$82$	0	0
$83$	−9.46410	−1.03882	−0.519410	−	0.854525i	$-0.673848\pi$
$83$	−9.46410	−1.03882	−0.519410	+	0.854525i	$0.673848\pi$
$84$	0	0
$85$	−3.46410	−0.375735
$86$	0	0
$87$	2.53590	0.271877
$88$	0	0
$89$	9.46410	1.00319	0.501596	−	0.865102i	$-0.332746\pi$
$89$	9.46410	1.00319	0.501596	+	0.865102i	$0.332746\pi$
$90$	0	0
$91$	−1.07180	−0.112355
$92$	0	0
$93$	4.53590	0.470351
$94$	0	0
$95$	18.9282	1.94199
$96$	0	0
$97$	8.92820	0.906522	0.453261	−	0.891378i	$-0.350261\pi$
$97$	8.92820	0.906522	0.453261	+	0.891378i	$0.350261\pi$
$98$	0	0
$99$	11.6603	1.17190
$100$	0	0
$101$	−9.46410	−0.941713	−0.470857	−	0.882210i	$-0.656055\pi$
$101$	−9.46410	−0.941713	−0.470857	+	0.882210i	$0.656055\pi$
$102$	0	0
$103$	2.92820	0.288524	0.144262	−	0.989539i	$-0.453919\pi$
$103$	2.92820	0.288524	0.144262	+	0.989539i	$0.453919\pi$
$104$	0	0
$105$	−1.85641	−0.181167
$106$	0	0
$107$	7.26795	0.702619	0.351310	−	0.936259i	$-0.385736\pi$
$107$	7.26795	0.702619	0.351310	+	0.936259i	$0.385736\pi$
$108$	0	0
$109$	6.39230	0.612272	0.306136	−	0.951988i	$-0.400964\pi$
$109$	6.39230	0.612272	0.306136	+	0.951988i	$0.400964\pi$
$110$	0	0
$111$	−8.39230	−0.796562
$112$	0	0
$113$	−7.85641	−0.739069	−0.369534	−	0.929217i	$-0.620483\pi$
$113$	−7.85641	−0.739069	−0.369534	+	0.929217i	$0.620483\pi$
$114$	0	0
$115$	−16.3923	−1.52859
$116$	0	0
$117$	3.60770	0.333532
$118$	0	0
$119$	−0.732051	−0.0671070
$120$	0	0
$121$	11.3923	1.03566
$122$	0	0
$123$	4.39230	0.396041
$124$	0	0
$125$	6.92820	0.619677
$126$	0	0
$127$	−20.3923	−1.80952	−0.904762	−	0.425917i	$-0.859952\pi$
$127$	−20.3923	−1.80952	−0.904762	+	0.425917i	$0.859952\pi$
$128$	0	0
$129$	−9.07180	−0.798727
$130$	0	0
$131$	−11.6603	−1.01876	−0.509381	−	0.860541i	$-0.670125\pi$
$131$	−11.6603	−1.01876	−0.509381	+	0.860541i	$0.670125\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	0	0
$135$	13.8564	1.19257
$136$	0	0
$137$	9.46410	0.808573	0.404286	−	0.914632i	$-0.367520\pi$
$137$	9.46410	0.808573	0.404286	+	0.914632i	$0.367520\pi$
$138$	0	0
$139$	−11.2679	−0.955735	−0.477867	−	0.878432i	$-0.658590\pi$
$139$	−11.2679	−0.955735	−0.477867	+	0.878432i	$0.658590\pi$
$140$	0	0
$141$	−5.07180	−0.427122
$142$	0	0
$143$	6.92820	0.579365
$144$	0	0
$145$	−12.0000	−0.996546
$146$	0	0
$147$	4.73205	0.390293
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−6.53590	−0.531884	−0.265942	−	0.963989i	$-0.585683\pi$
$151$	−6.53590	−0.531884	−0.265942	+	0.963989i	$0.585683\pi$
$152$	0	0
$153$	2.46410	0.199211
$154$	0	0
$155$	−21.4641	−1.72404
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	0.679492	0.0538872
$160$	0	0
$161$	−3.46410	−0.273009
$162$	0	0
$163$	−10.5885	−0.829352	−0.414676	−	0.909969i	$-0.636105\pi$
$163$	−10.5885	−0.829352	−0.414676	+	0.909969i	$0.636105\pi$
$164$	0	0
$165$	12.0000	0.934199
$166$	0	0
$167$	−11.6603	−0.902298	−0.451149	−	0.892449i	$-0.648986\pi$
$167$	−11.6603	−0.902298	−0.451149	+	0.892449i	$0.648986\pi$
$168$	0	0
$169$	−10.8564	−0.835108
$170$	0	0
$171$	−13.4641	−1.02963
$172$	0	0
$173$	−8.53590	−0.648972	−0.324486	−	0.945890i	$-0.605191\pi$
$173$	−8.53590	−0.648972	−0.324486	+	0.945890i	$0.605191\pi$
$174$	0	0
$175$	5.12436	0.387365
$176$	0	0
$177$	−6.92820	−0.520756
$178$	0	0
$179$	21.4641	1.60430	0.802151	−	0.597121i	$-0.203689\pi$
$179$	21.4641	1.60430	0.802151	+	0.597121i	$0.203689\pi$
$180$	0	0
$181$	11.4641	0.852120	0.426060	−	0.904695i	$-0.359901\pi$
$181$	11.4641	0.852120	0.426060	+	0.904695i	$0.359901\pi$
$182$	0	0
$183$	5.46410	0.403918
$184$	0	0
$185$	39.7128	2.91974
$186$	0	0
$187$	4.73205	0.346042
$188$	0	0
$189$	2.92820	0.212995
$190$	0	0
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	7.07180	0.509039	0.254520	−	0.967068i	$-0.418083\pi$
$193$	7.07180	0.509039	0.254520	+	0.967068i	$0.418083\pi$
$194$	0	0
$195$	3.71281	0.265880
$196$	0	0
$197$	−3.46410	−0.246807	−0.123404	−	0.992357i	$-0.539381\pi$
$197$	−3.46410	−0.246807	−0.123404	+	0.992357i	$0.539381\pi$
$198$	0	0
$199$	−6.19615	−0.439234	−0.219617	−	0.975586i	$-0.570481\pi$
$199$	−6.19615	−0.439234	−0.219617	+	0.975586i	$0.570481\pi$
$200$	0	0
$201$	−0.784610	−0.0553421
$202$	0	0
$203$	−2.53590	−0.177985
$204$	0	0
$205$	−20.7846	−1.45166
$206$	0	0
$207$	11.6603	0.810444
$208$	0	0
$209$	−25.8564	−1.78853
$210$	0	0
$211$	19.6603	1.35347	0.676734	−	0.736228i	$-0.263395\pi$
$211$	19.6603	1.35347	0.676734	+	0.736228i	$0.263395\pi$
$212$	0	0
$213$	−1.60770	−0.110157
$214$	0	0
$215$	42.9282	2.92768
$216$	0	0
$217$	−4.53590	−0.307917
$218$	0	0
$219$	−1.46410	−0.0989348
$220$	0	0
$221$	1.46410	0.0984861
$222$	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
$225$	−17.2487	−1.14991
$226$	0	0
$227$	−9.80385	−0.650704	−0.325352	−	0.945593i	$-0.605483\pi$
$227$	−9.80385	−0.650704	−0.325352	+	0.945593i	$0.605483\pi$
$228$	0	0
$229$	0.392305	0.0259242	0.0129621	−	0.999916i	$-0.495874\pi$
$229$	0.392305	0.0259242	0.0129621	+	0.999916i	$0.495874\pi$
$230$	0	0
$231$	2.53590	0.166850
$232$	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
$235$	24.0000	1.56559
$236$	0	0
$237$	1.32051	0.0857762
$238$	0	0
$239$	−20.7846	−1.34444	−0.672222	−	0.740349i	$-0.734660\pi$
$239$	−20.7846	−1.34444	−0.672222	+	0.740349i	$0.734660\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	−15.2679	−0.979439
$244$	0	0
$245$	−22.3923	−1.43059
$246$	0	0
$247$	−8.00000	−0.509028
$248$	0	0
$249$	6.92820	0.439057
$250$	0	0
$251$	−6.92820	−0.437304	−0.218652	−	0.975803i	$-0.570166\pi$
$251$	−6.92820	−0.437304	−0.218652	+	0.975803i	$0.570166\pi$
$252$	0	0
$253$	22.3923	1.40779
$254$	0	0
$255$	2.53590	0.158804
$256$	0	0
$257$	−7.60770	−0.474555	−0.237277	−	0.971442i	$-0.576255\pi$
$257$	−7.60770	−0.474555	−0.237277	+	0.971442i	$0.576255\pi$
$258$	0	0
$259$	8.39230	0.521472
$260$	0	0
$261$	8.53590	0.528359
$262$	0	0
$263$	−23.3205	−1.43800	−0.719002	−	0.695008i	$-0.755401\pi$
$263$	−23.3205	−1.43800	−0.719002	+	0.695008i	$0.755401\pi$
$264$	0	0
$265$	−3.21539	−0.197520
$266$	0	0
$267$	−6.92820	−0.423999
$268$	0	0
$269$	−20.5359	−1.25210	−0.626048	−	0.779785i	$-0.715329\pi$
$269$	−20.5359	−1.25210	−0.626048	+	0.779785i	$0.715329\pi$
$270$	0	0
$271$	14.9282	0.906824	0.453412	−	0.891301i	$-0.350207\pi$
$271$	14.9282	0.906824	0.453412	+	0.891301i	$0.350207\pi$
$272$	0	0
$273$	0.784610	0.0474867
$274$	0	0
$275$	−33.1244	−1.99747
$276$	0	0
$277$	−14.3923	−0.864750	−0.432375	−	0.901694i	$-0.642324\pi$
$277$	−14.3923	−0.864750	−0.432375	+	0.901694i	$0.642324\pi$
$278$	0	0
$279$	15.2679	0.914068
$280$	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
$283$	−10.5885	−0.629418	−0.314709	−	0.949188i	$-0.601907\pi$
$283$	−10.5885	−0.629418	−0.314709	+	0.949188i	$0.601907\pi$
$284$	0	0
$285$	−13.8564	−0.820783
$286$	0	0
$287$	−4.39230	−0.259270
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	−6.53590	−0.383141
$292$	0	0
$293$	30.0000	1.75262	0.876309	−	0.481749i	$-0.159998\pi$
$293$	30.0000	1.75262	0.876309	+	0.481749i	$0.159998\pi$
$294$	0	0
$295$	32.7846	1.90879
$296$	0	0
$297$	−18.9282	−1.09833
$298$	0	0
$299$	6.92820	0.400668
$300$	0	0
$301$	9.07180	0.522890
$302$	0	0
$303$	6.92820	0.398015
$304$	0	0
$305$	−25.8564	−1.48053
$306$	0	0
$307$	16.7846	0.957948	0.478974	−	0.877829i	$-0.341009\pi$
$307$	16.7846	0.957948	0.478974	+	0.877829i	$0.341009\pi$
$308$	0	0
$309$	−2.14359	−0.121945
$310$	0	0
$311$	21.1244	1.19785	0.598926	−	0.800804i	$-0.295594\pi$
$311$	21.1244	1.19785	0.598926	+	0.800804i	$0.295594\pi$
$312$	0	0
$313$	7.07180	0.399722	0.199861	−	0.979824i	$-0.435951\pi$
$313$	7.07180	0.399722	0.199861	+	0.979824i	$0.435951\pi$
$314$	0	0
$315$	−6.24871	−0.352075
$316$	0	0
$317$	15.4641	0.868550	0.434275	−	0.900780i	$-0.357005\pi$
$317$	15.4641	0.868550	0.434275	+	0.900780i	$0.357005\pi$
$318$	0	0
$319$	16.3923	0.917793
$320$	0	0
$321$	−5.32051	−0.296962
$322$	0	0
$323$	−5.46410	−0.304031
$324$	0	0
$325$	−10.2487	−0.568496
$326$	0	0
$327$	−4.67949	−0.258776
$328$	0	0
$329$	5.07180	0.279617
$330$	0	0
$331$	−15.3205	−0.842091	−0.421046	−	0.907039i	$-0.638337\pi$
$331$	−15.3205	−0.842091	−0.421046	+	0.907039i	$0.638337\pi$
$332$	0	0
$333$	−28.2487	−1.54802
$334$	0	0
$335$	3.71281	0.202853
$336$	0	0
$337$	34.7846	1.89484	0.947419	−	0.319995i	$-0.103681\pi$
$337$	34.7846	1.89484	0.947419	+	0.319995i	$0.103681\pi$
$338$	0	0
$339$	5.75129	0.312367
$340$	0	0
$341$	29.3205	1.58779
$342$	0	0
$343$	−9.85641	−0.532196
$344$	0	0
$345$	12.0000	0.646058
$346$	0	0
$347$	4.05256	0.217553	0.108776	−	0.994066i	$-0.465307\pi$
$347$	4.05256	0.217553	0.108776	+	0.994066i	$0.465307\pi$
$348$	0	0
$349$	−30.7846	−1.64786	−0.823931	−	0.566690i	$-0.808224\pi$
$349$	−30.7846	−1.64786	−0.823931	+	0.566690i	$0.808224\pi$
$350$	0	0
$351$	−5.85641	−0.312592
$352$	0	0
$353$	7.85641	0.418154	0.209077	−	0.977899i	$-0.432954\pi$
$353$	7.85641	0.418154	0.209077	+	0.977899i	$0.432954\pi$
$354$	0	0
$355$	7.60770	0.403775
$356$	0	0
$357$	0.535898	0.0283628
$358$	0	0
$359$	11.3205	0.597474	0.298737	−	0.954336i	$-0.403435\pi$
$359$	11.3205	0.597474	0.298737	+	0.954336i	$0.403435\pi$
$360$	0	0
$361$	10.8564	0.571390
$362$	0	0
$363$	−8.33975	−0.437723
$364$	0	0
$365$	6.92820	0.362639
$366$	0	0
$367$	19.6603	1.02626	0.513128	−	0.858312i	$-0.328486\pi$
$367$	19.6603	1.02626	0.513128	+	0.858312i	$0.328486\pi$
$368$	0	0
$369$	14.7846	0.769656
$370$	0	0
$371$	−0.679492	−0.0352775
$372$	0	0
$373$	12.3923	0.641649	0.320825	−	0.947139i	$-0.396040\pi$
$373$	12.3923	0.641649	0.320825	+	0.947139i	$0.396040\pi$
$374$	0	0
$375$	−5.07180	−0.261906
$376$	0	0
$377$	5.07180	0.261211
$378$	0	0
$379$	−8.73205	−0.448535	−0.224268	−	0.974528i	$-0.571999\pi$
$379$	−8.73205	−0.448535	−0.224268	+	0.974528i	$0.571999\pi$
$380$	0	0
$381$	14.9282	0.764795
$382$	0	0
$383$	−9.46410	−0.483593	−0.241797	−	0.970327i	$-0.577737\pi$
$383$	−9.46410	−0.483593	−0.241797	+	0.970327i	$0.577737\pi$
$384$	0	0
$385$	−12.0000	−0.611577
$386$	0	0
$387$	−30.5359	−1.55223
$388$	0	0
$389$	−25.1769	−1.27652	−0.638260	−	0.769821i	$-0.720346\pi$
$389$	−25.1769	−1.27652	−0.638260	+	0.769821i	$0.720346\pi$
$390$	0	0
$391$	4.73205	0.239310
$392$	0	0
$393$	8.53590	0.430579
$394$	0	0
$395$	−6.24871	−0.314407
$396$	0	0
$397$	−26.3923	−1.32459	−0.662296	−	0.749242i	$-0.730418\pi$
$397$	−26.3923	−1.32459	−0.662296	+	0.749242i	$0.730418\pi$
$398$	0	0
$399$	−2.92820	−0.146594
$400$	0	0
$401$	4.14359	0.206921	0.103461	−	0.994634i	$-0.467008\pi$
$401$	4.14359	0.206921	0.103461	+	0.994634i	$0.467008\pi$
$402$	0	0
$403$	9.07180	0.451898
$404$	0	0
$405$	15.4641	0.768417
$406$	0	0
$407$	−54.2487	−2.68901
$408$	0	0
$409$	−37.7128	−1.86478	−0.932389	−	0.361456i	$-0.882280\pi$
$409$	−37.7128	−1.86478	−0.932389	+	0.361456i	$0.882280\pi$
$410$	0	0
$411$	−6.92820	−0.341743
$412$	0	0
$413$	6.92820	0.340915
$414$	0	0
$415$	−32.7846	−1.60933
$416$	0	0
$417$	8.24871	0.403941
$418$	0	0
$419$	6.58846	0.321867	0.160934	−	0.986965i	$-0.448550\pi$
$419$	6.58846	0.321867	0.160934	+	0.986965i	$0.448550\pi$
$420$	0	0
$421$	26.2487	1.27928	0.639642	−	0.768673i	$-0.279083\pi$
$421$	26.2487	1.27928	0.639642	+	0.768673i	$0.279083\pi$
$422$	0	0
$423$	−17.0718	−0.830059
$424$	0	0
$425$	−7.00000	−0.339550
$426$	0	0
$427$	−5.46410	−0.264426
$428$	0	0
$429$	−5.07180	−0.244869
$430$	0	0
$431$	0.339746	0.0163650	0.00818249	−	0.999967i	$-0.497395\pi$
$431$	0.339746	0.0163650	0.00818249	+	0.999967i	$0.497395\pi$
$432$	0	0
$433$	24.3923	1.17222	0.586110	−	0.810232i	$-0.300659\pi$
$433$	24.3923	1.17222	0.586110	+	0.810232i	$0.300659\pi$
$434$	0	0
$435$	8.78461	0.421190
$436$	0	0
$437$	−25.8564	−1.23688
$438$	0	0
$439$	−13.8038	−0.658822	−0.329411	−	0.944187i	$-0.606850\pi$
$439$	−13.8038	−0.658822	−0.329411	+	0.944187i	$0.606850\pi$
$440$	0	0
$441$	15.9282	0.758486
$442$	0	0
$443$	20.7846	0.987507	0.493753	−	0.869602i	$-0.335625\pi$
$443$	20.7846	0.987507	0.493753	+	0.869602i	$0.335625\pi$
$444$	0	0
$445$	32.7846	1.55414
$446$	0	0
$447$	−4.39230	−0.207749
$448$	0	0
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	0	0
$451$	28.3923	1.33694
$452$	0	0
$453$	4.78461	0.224801
$454$	0	0
$455$	−3.71281	−0.174059
$456$	0	0
$457$	0.392305	0.0183512	0.00917562	−	0.999958i	$-0.497079\pi$
$457$	0.392305	0.0183512	0.00917562	+	0.999958i	$0.497079\pi$
$458$	0	0
$459$	−4.00000	−0.186704
$460$	0	0
$461$	−14.7846	−0.688588	−0.344294	−	0.938862i	$-0.611882\pi$
$461$	−14.7846	−0.688588	−0.344294	+	0.938862i	$0.611882\pi$
$462$	0	0
$463$	−29.8564	−1.38754	−0.693772	−	0.720194i	$-0.744053\pi$
$463$	−29.8564	−1.38754	−0.693772	+	0.720194i	$0.744053\pi$
$464$	0	0
$465$	15.7128	0.728664
$466$	0	0
$467$	21.4641	0.993240	0.496620	−	0.867968i	$-0.334574\pi$
$467$	21.4641	0.993240	0.496620	+	0.867968i	$0.334574\pi$
$468$	0	0
$469$	0.784610	0.0362299
$470$	0	0
$471$	−1.46410	−0.0674622
$472$	0	0
$473$	−58.6410	−2.69632
$474$	0	0
$475$	38.2487	1.75497
$476$	0	0
$477$	2.28719	0.104723
$478$	0	0
$479$	−0.339746	−0.0155234	−0.00776169	−	0.999970i	$-0.502471\pi$
$479$	−0.339746	−0.0155234	−0.00776169	+	0.999970i	$0.502471\pi$
$480$	0	0
$481$	−16.7846	−0.765312
$482$	0	0
$483$	2.53590	0.115387
$484$	0	0
$485$	30.9282	1.40438
$486$	0	0
$487$	−32.0526	−1.45244	−0.726220	−	0.687462i	$-0.758725\pi$
$487$	−32.0526	−1.45244	−0.726220	+	0.687462i	$0.758725\pi$
$488$	0	0
$489$	7.75129	0.350525
$490$	0	0
$491$	25.1769	1.13622	0.568109	−	0.822953i	$-0.307675\pi$
$491$	25.1769	1.13622	0.568109	+	0.822953i	$0.307675\pi$
$492$	0	0
$493$	3.46410	0.156015
$494$	0	0
$495$	40.3923	1.81550
$496$	0	0
$497$	1.60770	0.0721150
$498$	0	0
$499$	−26.9808	−1.20782	−0.603912	−	0.797051i	$-0.706392\pi$
$499$	−26.9808	−1.20782	−0.603912	+	0.797051i	$0.706392\pi$
$500$	0	0
$501$	8.53590	0.381356
$502$	0	0
$503$	2.87564	0.128219	0.0641093	−	0.997943i	$-0.479579\pi$
$503$	2.87564	0.128219	0.0641093	+	0.997943i	$0.479579\pi$
$504$	0	0
$505$	−32.7846	−1.45890
$506$	0	0
$507$	7.94744	0.352958
$508$	0	0
$509$	33.7128	1.49429	0.747147	−	0.664659i	$-0.231423\pi$
$509$	33.7128	1.49429	0.747147	+	0.664659i	$0.231423\pi$
$510$	0	0
$511$	1.46410	0.0647680
$512$	0	0
$513$	21.8564	0.964984
$514$	0	0
$515$	10.1436	0.446980
$516$	0	0
$517$	−32.7846	−1.44187
$518$	0	0
$519$	6.24871	0.274288
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	28.7846	1.25866	0.629332	−	0.777137i	$-0.283329\pi$
$523$	28.7846	1.25866	0.629332	+	0.777137i	$0.283329\pi$
$524$	0	0
$525$	−3.75129	−0.163720
$526$	0	0
$527$	6.19615	0.269909
$528$	0	0
$529$	−0.607695	−0.0264215
$530$	0	0
$531$	−23.3205	−1.01202
$532$	0	0
$533$	8.78461	0.380504
$534$	0	0
$535$	25.1769	1.08849
$536$	0	0
$537$	−15.7128	−0.678058
$538$	0	0
$539$	30.5885	1.31754
$540$	0	0
$541$	−2.39230	−0.102853	−0.0514266	−	0.998677i	$-0.516377\pi$
$541$	−2.39230	−0.102853	−0.0514266	+	0.998677i	$0.516377\pi$
$542$	0	0
$543$	−8.39230	−0.360148
$544$	0	0
$545$	22.1436	0.948527
$546$	0	0
$547$	39.2679	1.67898	0.839488	−	0.543378i	$-0.182855\pi$
$547$	39.2679	1.67898	0.839488	+	0.543378i	$0.182855\pi$
$548$	0	0
$549$	18.3923	0.784964
$550$	0	0
$551$	−18.9282	−0.806369
$552$	0	0
$553$	−1.32051	−0.0561537
$554$	0	0
$555$	−29.0718	−1.23403
$556$	0	0
$557$	−33.4641	−1.41792	−0.708960	−	0.705249i	$-0.750835\pi$
$557$	−33.4641	−1.41792	−0.708960	+	0.705249i	$0.750835\pi$
$558$	0	0
$559$	−18.1436	−0.767392
$560$	0	0
$561$	−3.46410	−0.146254
$562$	0	0
$563$	−0.679492	−0.0286372	−0.0143186	−	0.999897i	$-0.504558\pi$
$563$	−0.679492	−0.0286372	−0.0143186	+	0.999897i	$0.504558\pi$
$564$	0	0
$565$	−27.2154	−1.14496
$566$	0	0
$567$	3.26795	0.137241
$568$	0	0
$569$	19.8564	0.832424	0.416212	−	0.909268i	$-0.363357\pi$
$569$	19.8564	0.832424	0.416212	+	0.909268i	$0.363357\pi$
$570$	0	0
$571$	−13.1244	−0.549237	−0.274619	−	0.961553i	$-0.588552\pi$
$571$	−13.1244	−0.549237	−0.274619	+	0.961553i	$0.588552\pi$
$572$	0	0
$573$	−8.78461	−0.366982
$574$	0	0
$575$	−33.1244	−1.38138
$576$	0	0
$577$	−6.53590	−0.272093	−0.136047	−	0.990702i	$-0.543440\pi$
$577$	−6.53590	−0.272093	−0.136047	+	0.990702i	$0.543440\pi$
$578$	0	0
$579$	−5.17691	−0.215145
$580$	0	0
$581$	−6.92820	−0.287430
$582$	0	0
$583$	4.39230	0.181911
$584$	0	0
$585$	12.4974	0.516705
$586$	0	0
$587$	7.60770	0.314003	0.157002	−	0.987598i	$-0.449817\pi$
$587$	7.60770	0.314003	0.157002	+	0.987598i	$0.449817\pi$
$588$	0	0
$589$	−33.8564	−1.39503
$590$	0	0
$591$	2.53590	0.104313
$592$	0	0
$593$	−31.8564	−1.30819	−0.654093	−	0.756414i	$-0.726949\pi$
$593$	−31.8564	−1.30819	−0.654093	+	0.756414i	$0.726949\pi$
$594$	0	0
$595$	−2.53590	−0.103962
$596$	0	0
$597$	4.53590	0.185642
$598$	0	0
$599$	27.7128	1.13231	0.566157	−	0.824297i	$-0.308429\pi$
$599$	27.7128	1.13231	0.566157	+	0.824297i	$0.308429\pi$
$600$	0	0
$601$	−30.7846	−1.25573	−0.627865	−	0.778322i	$-0.716071\pi$
$601$	−30.7846	−1.25573	−0.627865	+	0.778322i	$0.716071\pi$
$602$	0	0
$603$	−2.64102	−0.107550
$604$	0	0
$605$	39.4641	1.60444
$606$	0	0
$607$	−35.2679	−1.43148	−0.715741	−	0.698366i	$-0.753911\pi$
$607$	−35.2679	−1.43148	−0.715741	+	0.698366i	$0.753911\pi$
$608$	0	0
$609$	1.85641	0.0752254
$610$	0	0
$611$	−10.1436	−0.410366
$612$	0	0
$613$	−20.1436	−0.813592	−0.406796	−	0.913519i	$-0.633354\pi$
$613$	−20.1436	−0.813592	−0.406796	+	0.913519i	$0.633354\pi$
$614$	0	0
$615$	15.2154	0.613544
$616$	0	0
$617$	47.5692	1.91506	0.957532	−	0.288326i	$-0.0930987\pi$
$617$	47.5692	1.91506	0.957532	+	0.288326i	$0.0930987\pi$
$618$	0	0
$619$	22.1962	0.892139	0.446069	−	0.894998i	$-0.352823\pi$
$619$	22.1962	0.892139	0.446069	+	0.894998i	$0.352823\pi$
$620$	0	0
$621$	−18.9282	−0.759563
$622$	0	0
$623$	6.92820	0.277573
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	18.9282	0.755920
$628$	0	0
$629$	−11.4641	−0.457104
$630$	0	0
$631$	34.5359	1.37485	0.687426	−	0.726254i	$-0.258740\pi$
$631$	34.5359	1.37485	0.687426	+	0.726254i	$0.258740\pi$
$632$	0	0
$633$	−14.3923	−0.572043
$634$	0	0
$635$	−70.6410	−2.80330
$636$	0	0
$637$	9.46410	0.374981
$638$	0	0
$639$	−5.41154	−0.214077
$640$	0	0
$641$	0.928203	0.0366618	0.0183309	−	0.999832i	$-0.494165\pi$
$641$	0.928203	0.0366618	0.0183309	+	0.999832i	$0.494165\pi$
$642$	0	0
$643$	1.41154	0.0556658	0.0278329	−	0.999613i	$-0.491139\pi$
$643$	1.41154	0.0556658	0.0278329	+	0.999613i	$0.491139\pi$
$644$	0	0
$645$	−31.4256	−1.23738
$646$	0	0
$647$	30.9282	1.21591	0.607957	−	0.793970i	$-0.291989\pi$
$647$	30.9282	1.21591	0.607957	+	0.793970i	$0.291989\pi$
$648$	0	0
$649$	−44.7846	−1.75795
$650$	0	0
$651$	3.32051	0.130141
$652$	0	0
$653$	10.3923	0.406682	0.203341	−	0.979108i	$-0.434820\pi$
$653$	10.3923	0.406682	0.203341	+	0.979108i	$0.434820\pi$
$654$	0	0
$655$	−40.3923	−1.57826
$656$	0	0
$657$	−4.92820	−0.192268
$658$	0	0
$659$	29.0718	1.13248	0.566238	−	0.824242i	$-0.308398\pi$
$659$	29.0718	1.13248	0.566238	+	0.824242i	$0.308398\pi$
$660$	0	0
$661$	−28.9282	−1.12518	−0.562588	−	0.826737i	$-0.690194\pi$
$661$	−28.9282	−1.12518	−0.562588	+	0.826737i	$0.690194\pi$
$662$	0	0
$663$	−1.07180	−0.0416251
$664$	0	0
$665$	13.8564	0.537328
$666$	0	0
$667$	16.3923	0.634713
$668$	0	0
$669$	6.14359	0.237525
$670$	0	0
$671$	35.3205	1.36353
$672$	0	0
$673$	27.8564	1.07379	0.536893	−	0.843650i	$-0.319598\pi$
$673$	27.8564	1.07379	0.536893	+	0.843650i	$0.319598\pi$
$674$	0	0
$675$	28.0000	1.07772
$676$	0	0
$677$	24.2487	0.931954	0.465977	−	0.884797i	$-0.345703\pi$
$677$	24.2487	0.931954	0.465977	+	0.884797i	$0.345703\pi$
$678$	0	0
$679$	6.53590	0.250825
$680$	0	0
$681$	7.17691	0.275020
$682$	0	0
$683$	−7.94744	−0.304100	−0.152050	−	0.988373i	$-0.548588\pi$
$683$	−7.94744	−0.304100	−0.152050	+	0.988373i	$0.548588\pi$
$684$	0	0
$685$	32.7846	1.25264
$686$	0	0
$687$	−0.287187	−0.0109569
$688$	0	0
$689$	1.35898	0.0517732
$690$	0	0
$691$	−25.8038	−0.981625	−0.490812	−	0.871265i	$-0.663300\pi$
$691$	−25.8038	−0.981625	−0.490812	+	0.871265i	$0.663300\pi$
$692$	0	0
$693$	8.53590	0.324252
$694$	0	0
$695$	−39.0333	−1.48062
$696$	0	0
$697$	6.00000	0.227266
$698$	0	0
$699$	−4.39230	−0.166132
$700$	0	0
$701$	4.39230	0.165895	0.0829475	−	0.996554i	$-0.473567\pi$
$701$	4.39230	0.165895	0.0829475	+	0.996554i	$0.473567\pi$
$702$	0	0
$703$	62.6410	2.36255
$704$	0	0
$705$	−17.5692	−0.661695
$706$	0	0
$707$	−6.92820	−0.260562
$708$	0	0
$709$	32.2487	1.21113	0.605563	−	0.795797i	$-0.292948\pi$
$709$	32.2487	1.21113	0.605563	+	0.795797i	$0.292948\pi$
$710$	0	0
$711$	4.44486	0.166695
$712$	0	0
$713$	29.3205	1.09806
$714$	0	0
$715$	24.0000	0.897549
$716$	0	0
$717$	15.2154	0.568229
$718$	0	0
$719$	−1.51666	−0.0565619	−0.0282809	−	0.999600i	$-0.509003\pi$
$719$	−1.51666	−0.0565619	−0.0282809	+	0.999600i	$0.509003\pi$
$720$	0	0
$721$	2.14359	0.0798316
$722$	0	0
$723$	−1.46410	−0.0544505
$724$	0	0
$725$	−24.2487	−0.900575
$726$	0	0
$727$	−36.7846	−1.36427	−0.682133	−	0.731228i	$-0.738947\pi$
$727$	−36.7846	−1.36427	−0.682133	+	0.731228i	$0.738947\pi$
$728$	0	0
$729$	−2.21539	−0.0820515
$730$	0	0
$731$	−12.3923	−0.458346
$732$	0	0
$733$	22.7846	0.841569	0.420784	−	0.907161i	$-0.361755\pi$
$733$	22.7846	0.841569	0.420784	+	0.907161i	$0.361755\pi$
$734$	0	0
$735$	16.3923	0.604639
$736$	0	0
$737$	−5.07180	−0.186822
$738$	0	0
$739$	−8.39230	−0.308716	−0.154358	−	0.988015i	$-0.549331\pi$
$739$	−8.39230	−0.308716	−0.154358	+	0.988015i	$0.549331\pi$
$740$	0	0
$741$	5.85641	0.215140
$742$	0	0
$743$	−41.9090	−1.53749	−0.768745	−	0.639555i	$-0.779119\pi$
$743$	−41.9090	−1.53749	−0.768745	+	0.639555i	$0.779119\pi$
$744$	0	0
$745$	20.7846	0.761489
$746$	0	0
$747$	23.3205	0.853253
$748$	0	0
$749$	5.32051	0.194407
$750$	0	0
$751$	15.2679	0.557135	0.278568	−	0.960417i	$-0.410140\pi$
$751$	15.2679	0.557135	0.278568	+	0.960417i	$0.410140\pi$
$752$	0	0
$753$	5.07180	0.184827
$754$	0	0
$755$	−22.6410	−0.823991
$756$	0	0
$757$	10.5359	0.382934	0.191467	−	0.981499i	$-0.438676\pi$
$757$	10.5359	0.382934	0.191467	+	0.981499i	$0.438676\pi$
$758$	0	0
$759$	−16.3923	−0.595003
$760$	0	0
$761$	−4.39230	−0.159221	−0.0796105	−	0.996826i	$-0.525368\pi$
$761$	−4.39230	−0.159221	−0.0796105	+	0.996826i	$0.525368\pi$
$762$	0	0
$763$	4.67949	0.169409
$764$	0	0
$765$	8.53590	0.308616
$766$	0	0
$767$	−13.8564	−0.500326
$768$	0	0
$769$	−20.3923	−0.735365	−0.367683	−	0.929951i	$-0.619849\pi$
$769$	−20.3923	−0.735365	−0.367683	+	0.929951i	$0.619849\pi$
$770$	0	0
$771$	5.56922	0.200571
$772$	0	0
$773$	−35.3205	−1.27039	−0.635195	−	0.772352i	$-0.719080\pi$
$773$	−35.3205	−1.27039	−0.635195	+	0.772352i	$0.719080\pi$
$774$	0	0
$775$	−43.3731	−1.55801
$776$	0	0
$777$	−6.14359	−0.220400
$778$	0	0
$779$	−32.7846	−1.17463
$780$	0	0
$781$	−10.3923	−0.371866
$782$	0	0
$783$	−13.8564	−0.495188
$784$	0	0
$785$	6.92820	0.247278
$786$	0	0
$787$	−1.12436	−0.0400790	−0.0200395	−	0.999799i	$-0.506379\pi$
$787$	−1.12436	−0.0400790	−0.0200395	+	0.999799i	$0.506379\pi$
$788$	0	0
$789$	17.0718	0.607772
$790$	0	0
$791$	−5.75129	−0.204492
$792$	0	0
$793$	10.9282	0.388072
$794$	0	0
$795$	2.35383	0.0834817
$796$	0	0
$797$	38.7846	1.37382	0.686911	−	0.726742i	$-0.258966\pi$
$797$	38.7846	1.37382	0.686911	+	0.726742i	$0.258966\pi$
$798$	0	0
$799$	−6.92820	−0.245102
$800$	0	0
$801$	−23.3205	−0.823990
$802$	0	0
$803$	−9.46410	−0.333981
$804$	0	0
$805$	−12.0000	−0.422944
$806$	0	0
$807$	15.0333	0.529198
$808$	0	0
$809$	19.8564	0.698114	0.349057	−	0.937101i	$-0.386502\pi$
$809$	19.8564	0.698114	0.349057	+	0.937101i	$0.386502\pi$
$810$	0	0
$811$	35.3731	1.24212	0.621058	−	0.783764i	$-0.286703\pi$
$811$	35.3731	1.24212	0.621058	+	0.783764i	$0.286703\pi$
$812$	0	0
$813$	−10.9282	−0.383269
$814$	0	0
$815$	−36.6795	−1.28483
$816$	0	0
$817$	67.7128	2.36897
$818$	0	0
$819$	2.64102	0.0922846
$820$	0	0
$821$	−45.0333	−1.57167	−0.785837	−	0.618434i	$-0.787767\pi$
$821$	−45.0333	−1.57167	−0.785837	+	0.618434i	$0.787767\pi$
$822$	0	0
$823$	−15.6603	−0.545882	−0.272941	−	0.962031i	$-0.587996\pi$
$823$	−15.6603	−0.545882	−0.272941	+	0.962031i	$0.587996\pi$
$824$	0	0
$825$	24.2487	0.844232
$826$	0	0
$827$	25.5167	0.887301	0.443651	−	0.896200i	$-0.353683\pi$
$827$	25.5167	0.887301	0.443651	+	0.896200i	$0.353683\pi$
$828$	0	0
$829$	12.1436	0.421764	0.210882	−	0.977511i	$-0.432366\pi$
$829$	12.1436	0.421764	0.210882	+	0.977511i	$0.432366\pi$
$830$	0	0
$831$	10.5359	0.365486
$832$	0	0
$833$	6.46410	0.223968
$834$	0	0
$835$	−40.3923	−1.39783
$836$	0	0
$837$	−24.7846	−0.856681
$838$	0	0
$839$	49.5167	1.70950	0.854752	−	0.519036i	$-0.173709\pi$
$839$	49.5167	1.70950	0.854752	+	0.519036i	$0.173709\pi$
$840$	0	0
$841$	−17.0000	−0.586207
$842$	0	0
$843$	−5.75129	−0.198085
$844$	0	0
$845$	−37.6077	−1.29374
$846$	0	0
$847$	8.33975	0.286557
$848$	0	0
$849$	7.75129	0.266024
$850$	0	0
$851$	−54.2487	−1.85962
$852$	0	0
$853$	−40.2487	−1.37809	−0.689045	−	0.724719i	$-0.741970\pi$
$853$	−40.2487	−1.37809	−0.689045	+	0.724719i	$0.741970\pi$
$854$	0	0
$855$	−46.6410	−1.59509
$856$	0	0
$857$	−52.6410	−1.79818	−0.899091	−	0.437761i	$-0.855772\pi$
$857$	−52.6410	−1.79818	−0.899091	+	0.437761i	$0.855772\pi$
$858$	0	0
$859$	7.32051	0.249773	0.124886	−	0.992171i	$-0.460143\pi$
$859$	7.32051	0.249773	0.124886	+	0.992171i	$0.460143\pi$
$860$	0	0
$861$	3.21539	0.109580
$862$	0	0
$863$	37.8564	1.28865	0.644324	−	0.764753i	$-0.277139\pi$
$863$	37.8564	1.28865	0.644324	+	0.764753i	$0.277139\pi$
$864$	0	0
$865$	−29.5692	−1.00538
$866$	0	0
$867$	−0.732051	−0.0248617
$868$	0	0
$869$	8.53590	0.289561
$870$	0	0
$871$	−1.56922	−0.0531710
$872$	0	0
$873$	−22.0000	−0.744587
$874$	0	0
$875$	5.07180	0.171458
$876$	0	0
$877$	30.3923	1.02628	0.513138	−	0.858306i	$-0.328483\pi$
$877$	30.3923	1.02628	0.513138	+	0.858306i	$0.328483\pi$
$878$	0	0
$879$	−21.9615	−0.740744
$880$	0	0
$881$	2.78461	0.0938159	0.0469079	−	0.998899i	$-0.485063\pi$
$881$	2.78461	0.0938159	0.0469079	+	0.998899i	$0.485063\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	0	0
$885$	−24.0000	−0.806751
$886$	0	0
$887$	42.5885	1.42998	0.714990	−	0.699134i	$-0.246431\pi$
$887$	42.5885	1.42998	0.714990	+	0.699134i	$0.246431\pi$
$888$	0	0
$889$	−14.9282	−0.500676
$890$	0	0
$891$	−21.1244	−0.707693
$892$	0	0
$893$	37.8564	1.26682
$894$	0	0
$895$	74.3538	2.48537
$896$	0	0
$897$	−5.07180	−0.169342
$898$	0	0
$899$	21.4641	0.715868
$900$	0	0
$901$	0.928203	0.0309229
$902$	0	0
$903$	−6.64102	−0.220999
$904$	0	0
$905$	39.7128	1.32010
$906$	0	0
$907$	−24.4449	−0.811678	−0.405839	−	0.913945i	$-0.633021\pi$
$907$	−24.4449	−0.811678	−0.405839	+	0.913945i	$0.633021\pi$
$908$	0	0
$909$	23.3205	0.773492
$910$	0	0
$911$	−26.1962	−0.867917	−0.433959	−	0.900933i	$-0.642884\pi$
$911$	−26.1962	−0.867917	−0.433959	+	0.900933i	$0.642884\pi$
$912$	0	0
$913$	44.7846	1.48215
$914$	0	0
$915$	18.9282	0.625747
$916$	0	0
$917$	−8.53590	−0.281880
$918$	0	0
$919$	−52.0000	−1.71532	−0.857661	−	0.514216i	$-0.828083\pi$
$919$	−52.0000	−1.71532	−0.857661	+	0.514216i	$0.828083\pi$
$920$	0	0
$921$	−12.2872	−0.404877
$922$	0	0
$923$	−3.21539	−0.105836
$924$	0	0
$925$	80.2487	2.63856
$926$	0	0
$927$	−7.21539	−0.236985
$928$	0	0
$929$	45.7128	1.49979	0.749894	−	0.661558i	$-0.230104\pi$
$929$	45.7128	1.49979	0.749894	+	0.661558i	$0.230104\pi$
$930$	0	0
$931$	−35.3205	−1.15758
$932$	0	0
$933$	−15.4641	−0.506272
$934$	0	0
$935$	16.3923	0.536086
$936$	0	0
$937$	27.8564	0.910029	0.455015	−	0.890484i	$-0.349634\pi$
$937$	27.8564	0.910029	0.455015	+	0.890484i	$0.349634\pi$
$938$	0	0
$939$	−5.17691	−0.168942
$940$	0	0
$941$	6.67949	0.217745	0.108873	−	0.994056i	$-0.465276\pi$
$941$	6.67949	0.217745	0.108873	+	0.994056i	$0.465276\pi$
$942$	0	0
$943$	28.3923	0.924581
$944$	0	0
$945$	10.1436	0.329971
$946$	0	0
$947$	52.7321	1.71356	0.856781	−	0.515681i	$-0.172461\pi$
$947$	52.7321	1.71356	0.856781	+	0.515681i	$0.172461\pi$
$948$	0	0
$949$	−2.92820	−0.0950535
$950$	0	0
$951$	−11.3205	−0.367093
$952$	0	0
$953$	−42.2487	−1.36857	−0.684285	−	0.729215i	$-0.739886\pi$
$953$	−42.2487	−1.36857	−0.684285	+	0.729215i	$0.739886\pi$
$954$	0	0
$955$	41.5692	1.34515
$956$	0	0
$957$	−12.0000	−0.387905
$958$	0	0
$959$	6.92820	0.223723
$960$	0	0
$961$	7.39230	0.238461
$962$	0	0
$963$	−17.9090	−0.577108
$964$	0	0
$965$	24.4974	0.788600
$966$	0	0
$967$	48.3923	1.55619	0.778096	−	0.628146i	$-0.216186\pi$
$967$	48.3923	1.55619	0.778096	+	0.628146i	$0.216186\pi$
$968$	0	0
$969$	4.00000	0.128499
$970$	0	0
$971$	−56.1051	−1.80050	−0.900249	−	0.435374i	$-0.856616\pi$
$971$	−56.1051	−1.80050	−0.900249	+	0.435374i	$0.856616\pi$
$972$	0	0
$973$	−8.24871	−0.264442
$974$	0	0
$975$	7.50258	0.240275
$976$	0	0
$977$	−42.0000	−1.34370	−0.671850	−	0.740688i	$-0.734500\pi$
$977$	−42.0000	−1.34370	−0.671850	+	0.740688i	$0.734500\pi$
$978$	0	0
$979$	−44.7846	−1.43132
$980$	0	0
$981$	−15.7513	−0.502900
$982$	0	0
$983$	29.9090	0.953948	0.476974	−	0.878917i	$-0.341734\pi$
$983$	29.9090	0.953948	0.476974	+	0.878917i	$0.341734\pi$
$984$	0	0
$985$	−12.0000	−0.382352
$986$	0	0
$987$	−3.71281	−0.118180
$988$	0	0
$989$	−58.6410	−1.86468
$990$	0	0
$991$	10.1962	0.323891	0.161946	−	0.986800i	$-0.448223\pi$
$991$	10.1962	0.323891	0.161946	+	0.986800i	$0.448223\pi$
$992$	0	0
$993$	11.2154	0.355910
$994$	0	0
$995$	−21.4641	−0.680458
$996$	0	0
$997$	42.3923	1.34258	0.671289	−	0.741196i	$-0.265741\pi$
$997$	42.3923	1.34258	0.671289	+	0.741196i	$0.265741\pi$
$998$	0	0
$999$	45.8564	1.45083

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	68.2.a.a.1.1	✓	2
3.2	odd	2		612.2.a.e.1.1		2
4.3	odd	2		272.2.a.e.1.2		2
5.2	odd	4		1700.2.e.c.749.3		4
5.3	odd	4		1700.2.e.c.749.2		4
5.4	even	2		1700.2.a.d.1.2		2
7.6	odd	2		3332.2.a.h.1.2		2
8.3	odd	2		1088.2.a.t.1.1		2
8.5	even	2		1088.2.a.p.1.2		2
11.10	odd	2		8228.2.a.k.1.1		2
12.11	even	2		2448.2.a.y.1.1		2
17.2	even	8		1156.2.e.d.905.2		8
17.3	odd	16		1156.2.h.f.757.2		16
17.4	even	4		1156.2.b.c.577.3		4
17.5	odd	16		1156.2.h.f.977.3		16
17.6	odd	16		1156.2.h.f.733.2		16
17.7	odd	16		1156.2.h.f.1001.3		16
17.8	even	8		1156.2.e.d.829.3		8
17.9	even	8		1156.2.e.d.829.2		8
17.10	odd	16		1156.2.h.f.1001.2		16
17.11	odd	16		1156.2.h.f.733.3		16
17.12	odd	16		1156.2.h.f.977.2		16
17.13	even	4		1156.2.b.c.577.2		4
17.14	odd	16		1156.2.h.f.757.3		16
17.15	even	8		1156.2.e.d.905.3		8
17.16	even	2		1156.2.a.a.1.2		2
20.19	odd	2		6800.2.a.bh.1.1		2
24.5	odd	2		9792.2.a.cr.1.2		2
24.11	even	2		9792.2.a.cs.1.2		2
68.67	odd	2		4624.2.a.x.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
68.2.a.a.1.1	✓	2	1.1	even	1	trivial
272.2.a.e.1.2		2	4.3	odd	2
612.2.a.e.1.1		2	3.2	odd	2
1088.2.a.p.1.2		2	8.5	even	2
1088.2.a.t.1.1		2	8.3	odd	2
1156.2.a.a.1.2		2	17.16	even	2
1156.2.b.c.577.2		4	17.13	even	4
1156.2.b.c.577.3		4	17.4	even	4
1156.2.e.d.829.2		8	17.9	even	8
1156.2.e.d.829.3		8	17.8	even	8
1156.2.e.d.905.2		8	17.2	even	8
1156.2.e.d.905.3		8	17.15	even	8
1156.2.h.f.733.2		16	17.6	odd	16
1156.2.h.f.733.3		16	17.11	odd	16
1156.2.h.f.757.2		16	17.3	odd	16
1156.2.h.f.757.3		16	17.14	odd	16
1156.2.h.f.977.2		16	17.12	odd	16
1156.2.h.f.977.3		16	17.5	odd	16
1156.2.h.f.1001.2		16	17.10	odd	16
1156.2.h.f.1001.3		16	17.7	odd	16
1700.2.a.d.1.2		2	5.4	even	2
1700.2.e.c.749.2		4	5.3	odd	4
1700.2.e.c.749.3		4	5.2	odd	4
2448.2.a.y.1.1		2	12.11	even	2
3332.2.a.h.1.2		2	7.6	odd	2
4624.2.a.x.1.1		2	68.67	odd	2
6800.2.a.bh.1.1		2	20.19	odd	2
8228.2.a.k.1.1		2	11.10	odd	2
9792.2.a.cr.1.2		2	24.5	odd	2
9792.2.a.cs.1.2		2	24.11	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:	\( N \)	\(=\)	\( 68 = 2^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	68.a (trivial)

\(f(q)\)	\(=\)	\(q-0.732051 q^{3} +3.46410 q^{5} +0.732051 q^{7} -2.46410 q^{9} +O(q^{10})\)	\(q-0.732051 q^{3} +3.46410 q^{5} +0.732051 q^{7} -2.46410 q^{9} -4.73205 q^{11} -1.46410 q^{13} -2.53590 q^{15} -1.00000 q^{17} +5.46410 q^{19} -0.535898 q^{21} -4.73205 q^{23} +7.00000 q^{25} +4.00000 q^{27} -3.46410 q^{29} -6.19615 q^{31} +3.46410 q^{33} +2.53590 q^{35} +11.4641 q^{37} +1.07180 q^{39} -6.00000 q^{41} +12.3923 q^{43} -8.53590 q^{45} +6.92820 q^{47} -6.46410 q^{49} +0.732051 q^{51} -0.928203 q^{53} -16.3923 q^{55} -4.00000 q^{57} +9.46410 q^{59} -7.46410 q^{61} -1.80385 q^{63} -5.07180 q^{65} +1.07180 q^{67} +3.46410 q^{69} +2.19615 q^{71} +2.00000 q^{73} -5.12436 q^{75} -3.46410 q^{77} -1.80385 q^{79} +4.46410 q^{81} -9.46410 q^{83} -3.46410 q^{85} +2.53590 q^{87} +9.46410 q^{89} -1.07180 q^{91} +4.53590 q^{93} +18.9282 q^{95} +8.92820 q^{97} +11.6603 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 2 q^{7} + 2 q^{9} - 6 q^{11} + 4 q^{13} - 12 q^{15} - 2 q^{17} + 4 q^{19} - 8 q^{21} - 6 q^{23} + 14 q^{25} + 8 q^{27} - 2 q^{31} + 12 q^{35} + 16 q^{37} + 16 q^{39} - 12 q^{41} + 4 q^{43} - 24 q^{45} - 6 q^{49} - 2 q^{51} + 12 q^{53} - 12 q^{55} - 8 q^{57} + 12 q^{59} - 8 q^{61} - 14 q^{63} - 24 q^{65} + 16 q^{67} - 6 q^{71} + 4 q^{73} + 14 q^{75} - 14 q^{79} + 2 q^{81} - 12 q^{83} + 12 q^{87} + 12 q^{89} - 16 q^{91} + 16 q^{93} + 24 q^{95} + 4 q^{97} + 6 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 2 * q^7 + 2 * q^9 - 6 * q^11 + 4 * q^13 - 12 * q^15 - 2 * q^17 + 4 * q^19 - 8 * q^21 - 6 * q^23 + 14 * q^25 + 8 * q^27 - 2 * q^31 + 12 * q^35 + 16 * q^37 + 16 * q^39 - 12 * q^41 + 4 * q^43 - 24 * q^45 - 6 * q^49 - 2 * q^51 + 12 * q^53 - 12 * q^55 - 8 * q^57 + 12 * q^59 - 8 * q^61 - 14 * q^63 - 24 * q^65 + 16 * q^67 - 6 * q^71 + 4 * q^73 + 14 * q^75 - 14 * q^79 + 2 * q^81 - 12 * q^83 + 12 * q^87 + 12 * q^89 - 16 * q^91 + 16 * q^93 + 24 * q^95 + 4 * q^97 + 6 * q^99

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