LMFDB - Embedded newform 8036.2.a.m.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8036,2,Mod(1,8036)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8036, 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("8036.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8036 = 2^{2} \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8036.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:	$64.1677830643$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 14x^{6} - 4x^{5} + 60x^{4} + 31x^{3} - 75x^{2} - 60x - 11 $ x^8 - 14x^6 - 4x^5 + 60x^4 + 31x^3 - 75x^2 - 60x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1148)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.71291$ of defining polynomial
Character	$\chi$	$=$	8036.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.71291 q^{3} +2.15120 q^{5} -0.0659453 q^{9} +O(q^{10})$	$q-1.71291 q^{3} +2.15120 q^{5} -0.0659453 q^{9} -3.02769 q^{11} +0.848387 q^{13} -3.68481 q^{15} -4.76096 q^{17} -3.90091 q^{19} +8.25272 q^{23} -0.372345 q^{25} +5.25168 q^{27} +0.0576440 q^{29} +4.30049 q^{31} +5.18615 q^{33} +0.475316 q^{37} -1.45321 q^{39} +1.00000 q^{41} +2.82667 q^{43} -0.141861 q^{45} +6.71976 q^{47} +8.15508 q^{51} -3.42941 q^{53} -6.51316 q^{55} +6.68191 q^{57} +5.72894 q^{59} +5.96715 q^{61} +1.82505 q^{65} -9.53549 q^{67} -14.1362 q^{69} -1.84489 q^{71} +3.89877 q^{73} +0.637793 q^{75} +5.11742 q^{79} -8.79782 q^{81} -7.41218 q^{83} -10.2418 q^{85} -0.0987388 q^{87} -0.305139 q^{89} -7.36635 q^{93} -8.39164 q^{95} -11.7545 q^{97} +0.199662 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{9}+O(q^{10}) $ 8 * q + 4 * q^9	$ 8 q + 4 q^{9} - 8 q^{11} - 7 q^{13} - q^{15} - q^{17} + 4 q^{19} - 3 q^{23} - 4 q^{25} - 12 q^{27} - 4 q^{29} + 4 q^{31} + 23 q^{33} - 31 q^{37} + 5 q^{39} + 8 q^{41} - 8 q^{43} + q^{45} + 24 q^{47} - 23 q^{51} - q^{53} + 2 q^{55} - 15 q^{57} + 4 q^{59} - 4 q^{61} - 24 q^{65} - 21 q^{69} + 8 q^{71} + 11 q^{73} - 15 q^{75} + 14 q^{79} - 28 q^{81} - 42 q^{83} - 20 q^{85} + 25 q^{87} - 11 q^{89} - 27 q^{93} - 15 q^{95} - 16 q^{97} - 8 q^{99}+O(q^{100}) $ 8 * q + 4 * q^9 - 8 * q^11 - 7 * q^13 - q^15 - q^17 + 4 * q^19 - 3 * q^23 - 4 * q^25 - 12 * q^27 - 4 * q^29 + 4 * q^31 + 23 * q^33 - 31 * q^37 + 5 * q^39 + 8 * q^41 - 8 * q^43 + q^45 + 24 * q^47 - 23 * q^51 - q^53 + 2 * q^55 - 15 * q^57 + 4 * q^59 - 4 * q^61 - 24 * q^65 - 21 * q^69 + 8 * q^71 + 11 * q^73 - 15 * q^75 + 14 * q^79 - 28 * q^81 - 42 * q^83 - 20 * q^85 + 25 * q^87 - 11 * q^89 - 27 * q^93 - 15 * q^95 - 16 * q^97 - 8 * 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
$3$	−1.71291	−0.988948	−0.494474	−	0.869192i	$-0.664639\pi$
$3$	−1.71291	−0.988948	−0.494474	+	0.869192i	$0.664639\pi$
$4$	0	0
$5$	2.15120	0.962045	0.481023	−	0.876708i	$-0.340265\pi$
$5$	2.15120	0.962045	0.481023	+	0.876708i	$0.340265\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−0.0659453	−0.0219818
$10$	0	0
$11$	−3.02769	−0.912882	−0.456441	−	0.889754i	$-0.650876\pi$
$11$	−3.02769	−0.912882	−0.456441	+	0.889754i	$0.650876\pi$
$12$	0	0
$13$	0.848387	0.235300	0.117650	−	0.993055i	$-0.462464\pi$
$13$	0.848387	0.235300	0.117650	+	0.993055i	$0.462464\pi$
$14$	0	0
$15$	−3.68481	−0.951413
$16$	0	0
$17$	−4.76096	−1.15470	−0.577351	−	0.816496i	$-0.695914\pi$
$17$	−4.76096	−1.15470	−0.577351	+	0.816496i	$0.695914\pi$
$18$	0	0
$19$	−3.90091	−0.894931	−0.447465	−	0.894301i	$-0.647673\pi$
$19$	−3.90091	−0.894931	−0.447465	+	0.894301i	$0.647673\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.25272	1.72081	0.860405	−	0.509610i	$-0.170210\pi$
$23$	8.25272	1.72081	0.860405	+	0.509610i	$0.170210\pi$
$24$	0	0
$25$	−0.372345	−0.0744690
$26$	0	0
$27$	5.25168	1.01069
$28$	0	0
$29$	0.0576440	0.0107042	0.00535211	−	0.999986i	$-0.498296\pi$
$29$	0.0576440	0.0107042	0.00535211	+	0.999986i	$0.498296\pi$
$30$	0	0
$31$	4.30049	0.772391	0.386196	−	0.922417i	$-0.373789\pi$
$31$	4.30049	0.772391	0.386196	+	0.922417i	$0.373789\pi$
$32$	0	0
$33$	5.18615	0.902793
$34$	0	0
$35$	0	0
$36$	0	0
$37$	0.475316	0.0781414	0.0390707	−	0.999236i	$-0.487560\pi$
$37$	0.475316	0.0781414	0.0390707	+	0.999236i	$0.487560\pi$
$38$	0	0
$39$	−1.45321	−0.232700
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	2.82667	0.431063	0.215531	−	0.976497i	$-0.430852\pi$
$43$	2.82667	0.431063	0.215531	+	0.976497i	$0.430852\pi$
$44$	0	0
$45$	−0.141861	−0.0211474
$46$	0	0
$47$	6.71976	0.980178	0.490089	−	0.871673i	$-0.336964\pi$
$47$	6.71976	0.980178	0.490089	+	0.871673i	$0.336964\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	8.15508	1.14194
$52$	0	0
$53$	−3.42941	−0.471065	−0.235533	−	0.971866i	$-0.575683\pi$
$53$	−3.42941	−0.471065	−0.235533	+	0.971866i	$0.575683\pi$
$54$	0	0
$55$	−6.51316	−0.878234
$56$	0	0
$57$	6.68191	0.885040
$58$	0	0
$59$	5.72894	0.745844	0.372922	−	0.927863i	$-0.378356\pi$
$59$	5.72894	0.745844	0.372922	+	0.927863i	$0.378356\pi$
$60$	0	0
$61$	5.96715	0.764015	0.382007	−	0.924159i	$-0.375233\pi$
$61$	5.96715	0.764015	0.382007	+	0.924159i	$0.375233\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.82505	0.226369
$66$	0	0
$67$	−9.53549	−1.16495	−0.582473	−	0.812850i	$-0.697915\pi$
$67$	−9.53549	−1.16495	−0.582473	+	0.812850i	$0.697915\pi$
$68$	0	0
$69$	−14.1362	−1.70179
$70$	0	0
$71$	−1.84489	−0.218948	−0.109474	−	0.993990i	$-0.534917\pi$
$71$	−1.84489	−0.218948	−0.109474	+	0.993990i	$0.534917\pi$
$72$	0	0
$73$	3.89877	0.456317	0.228158	−	0.973624i	$-0.426730\pi$
$73$	3.89877	0.456317	0.228158	+	0.973624i	$0.426730\pi$
$74$	0	0
$75$	0.637793	0.0736460
$76$	0	0
$77$	0	0
$78$	0	0
$79$	5.11742	0.575755	0.287877	−	0.957667i	$-0.407050\pi$
$79$	5.11742	0.575755	0.287877	+	0.957667i	$0.407050\pi$
$80$	0	0
$81$	−8.79782	−0.977535
$82$	0	0
$83$	−7.41218	−0.813592	−0.406796	−	0.913519i	$-0.633354\pi$
$83$	−7.41218	−0.813592	−0.406796	+	0.913519i	$0.633354\pi$
$84$	0	0
$85$	−10.2418	−1.11088
$86$	0	0
$87$	−0.0987388	−0.0105859
$88$	0	0
$89$	−0.305139	−0.0323447	−0.0161723	−	0.999869i	$-0.505148\pi$
$89$	−0.305139	−0.0323447	−0.0161723	+	0.999869i	$0.505148\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−7.36635	−0.763855
$94$	0	0
$95$	−8.39164	−0.860964
$96$	0	0
$97$	−11.7545	−1.19349	−0.596743	−	0.802433i	$-0.703539\pi$
$97$	−11.7545	−1.19349	−0.596743	+	0.802433i	$0.703539\pi$
$98$	0	0
$99$	0.199662	0.0200668
$100$	0	0
$101$	2.12301	0.211247	0.105623	−	0.994406i	$-0.466316\pi$
$101$	2.12301	0.211247	0.105623	+	0.994406i	$0.466316\pi$
$102$	0	0
$103$	−7.52336	−0.741299	−0.370650	−	0.928773i	$-0.620865\pi$
$103$	−7.52336	−0.741299	−0.370650	+	0.928773i	$0.620865\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−12.9029	−1.24737	−0.623684	−	0.781677i	$-0.714365\pi$
$107$	−12.9029	−1.24737	−0.623684	+	0.781677i	$0.714365\pi$
$108$	0	0
$109$	0.587473	0.0562697	0.0281349	−	0.999604i	$-0.491043\pi$
$109$	0.587473	0.0562697	0.0281349	+	0.999604i	$0.491043\pi$
$110$	0	0
$111$	−0.814172	−0.0772778
$112$	0	0
$113$	−1.87770	−0.176639	−0.0883197	−	0.996092i	$-0.528150\pi$
$113$	−1.87770	−0.176639	−0.0883197	+	0.996092i	$0.528150\pi$
$114$	0	0
$115$	17.7532	1.65550
$116$	0	0
$117$	−0.0559471	−0.00517231
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1.83310	−0.166646
$122$	0	0
$123$	−1.71291	−0.154448
$124$	0	0
$125$	−11.5570	−1.03369
$126$	0	0
$127$	11.9240	1.05808	0.529040	−	0.848597i	$-0.322552\pi$
$127$	11.9240	1.05808	0.529040	+	0.848597i	$0.322552\pi$
$128$	0	0
$129$	−4.84182	−0.426299
$130$	0	0
$131$	−19.1882	−1.67648	−0.838239	−	0.545303i	$-0.816415\pi$
$131$	−19.1882	−1.67648	−0.838239	+	0.545303i	$0.816415\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	11.2974	0.972326
$136$	0	0
$137$	−8.01114	−0.684438	−0.342219	−	0.939620i	$-0.611178\pi$
$137$	−8.01114	−0.684438	−0.342219	+	0.939620i	$0.611178\pi$
$138$	0	0
$139$	20.3407	1.72528	0.862639	−	0.505821i	$-0.168810\pi$
$139$	20.3407	1.72528	0.862639	+	0.505821i	$0.168810\pi$
$140$	0	0
$141$	−11.5103	−0.969345
$142$	0	0
$143$	−2.56865	−0.214801
$144$	0	0
$145$	0.124004	0.0102979
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−10.5821	−0.866923	−0.433462	−	0.901172i	$-0.642708\pi$
$149$	−10.5821	−0.866923	−0.433462	+	0.901172i	$0.642708\pi$
$150$	0	0
$151$	−11.5202	−0.937501	−0.468750	−	0.883331i	$-0.655296\pi$
$151$	−11.5202	−0.937501	−0.468750	+	0.883331i	$0.655296\pi$
$152$	0	0
$153$	0.313963	0.0253824
$154$	0	0
$155$	9.25121	0.743075
$156$	0	0
$157$	4.02606	0.321315	0.160657	−	0.987010i	$-0.448639\pi$
$157$	4.02606	0.321315	0.160657	+	0.987010i	$0.448639\pi$
$158$	0	0
$159$	5.87426	0.465859
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−19.0562	−1.49259	−0.746296	−	0.665614i	$-0.768170\pi$
$163$	−19.0562	−1.49259	−0.746296	+	0.665614i	$0.768170\pi$
$164$	0	0
$165$	11.1564	0.868528
$166$	0	0
$167$	11.7372	0.908252	0.454126	−	0.890937i	$-0.349952\pi$
$167$	11.7372	0.908252	0.454126	+	0.890937i	$0.349952\pi$
$168$	0	0
$169$	−12.2802	−0.944634
$170$	0	0
$171$	0.257247	0.0196722
$172$	0	0
$173$	−1.51029	−0.114825	−0.0574126	−	0.998351i	$-0.518285\pi$
$173$	−1.51029	−0.114825	−0.0574126	+	0.998351i	$0.518285\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−9.81315	−0.737601
$178$	0	0
$179$	−1.38268	−0.103346	−0.0516731	−	0.998664i	$-0.516455\pi$
$179$	−1.38268	−0.103346	−0.0516731	+	0.998664i	$0.516455\pi$
$180$	0	0
$181$	−3.21399	−0.238894	−0.119447	−	0.992841i	$-0.538112\pi$
$181$	−3.21399	−0.238894	−0.119447	+	0.992841i	$0.538112\pi$
$182$	0	0
$183$	−10.2212	−0.755571
$184$	0	0
$185$	1.02250	0.0751756
$186$	0	0
$187$	14.4147	1.05411
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−6.31469	−0.456915	−0.228457	−	0.973554i	$-0.573368\pi$
$191$	−6.31469	−0.456915	−0.228457	+	0.973554i	$0.573368\pi$
$192$	0	0
$193$	−15.1817	−1.09281	−0.546403	−	0.837523i	$-0.684003\pi$
$193$	−15.1817	−1.09281	−0.546403	+	0.837523i	$0.684003\pi$
$194$	0	0
$195$	−3.12614	−0.223868
$196$	0	0
$197$	−17.9461	−1.27861	−0.639304	−	0.768954i	$-0.720778\pi$
$197$	−17.9461	−1.27861	−0.639304	+	0.768954i	$0.720778\pi$
$198$	0	0
$199$	7.82193	0.554482	0.277241	−	0.960800i	$-0.410580\pi$
$199$	7.82193	0.554482	0.277241	+	0.960800i	$0.410580\pi$
$200$	0	0
$201$	16.3334	1.15207
$202$	0	0
$203$	0	0
$204$	0	0
$205$	2.15120	0.150246
$206$	0	0
$207$	−0.544228	−0.0378264
$208$	0	0
$209$	11.8108	0.816967
$210$	0	0
$211$	−2.30039	−0.158366	−0.0791828	−	0.996860i	$-0.525231\pi$
$211$	−2.30039	−0.158366	−0.0791828	+	0.996860i	$0.525231\pi$
$212$	0	0
$213$	3.16012	0.216528
$214$	0	0
$215$	6.08072	0.414702
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−6.67824	−0.451274
$220$	0	0
$221$	−4.03913	−0.271702
$222$	0	0
$223$	−6.01186	−0.402584	−0.201292	−	0.979531i	$-0.564514\pi$
$223$	−6.01186	−0.402584	−0.201292	+	0.979531i	$0.564514\pi$
$224$	0	0
$225$	0.0245544	0.00163696
$226$	0	0
$227$	24.8633	1.65023	0.825117	−	0.564961i	$-0.191109\pi$
$227$	24.8633	1.65023	0.825117	+	0.564961i	$0.191109\pi$
$228$	0	0
$229$	−24.0310	−1.58801	−0.794007	−	0.607908i	$-0.792009\pi$
$229$	−24.0310	−1.58801	−0.794007	+	0.607908i	$0.792009\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−7.77524	−0.509373	−0.254686	−	0.967024i	$-0.581972\pi$
$233$	−7.77524	−0.509373	−0.254686	+	0.967024i	$0.581972\pi$
$234$	0	0
$235$	14.4555	0.942975
$236$	0	0
$237$	−8.76567	−0.569391
$238$	0	0
$239$	19.2557	1.24555	0.622774	−	0.782402i	$-0.286006\pi$
$239$	19.2557	1.24555	0.622774	+	0.782402i	$0.286006\pi$
$240$	0	0
$241$	5.91887	0.381268	0.190634	−	0.981661i	$-0.438946\pi$
$241$	5.91887	0.381268	0.190634	+	0.981661i	$0.438946\pi$
$242$	0	0
$243$	−0.685198	−0.0439555
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3.30948	−0.210577
$248$	0	0
$249$	12.6964	0.804601
$250$	0	0
$251$	28.1909	1.77939	0.889697	−	0.456551i	$-0.150916\pi$
$251$	28.1909	1.77939	0.889697	+	0.456551i	$0.150916\pi$
$252$	0	0
$253$	−24.9867	−1.57090
$254$	0	0
$255$	17.5432	1.09860
$256$	0	0
$257$	4.27874	0.266900	0.133450	−	0.991056i	$-0.457394\pi$
$257$	4.27874	0.266900	0.133450	+	0.991056i	$0.457394\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−0.00380135	−0.000235297 0
$262$	0	0
$263$	−5.43795	−0.335318	−0.167659	−	0.985845i	$-0.553621\pi$
$263$	−5.43795	−0.335318	−0.167659	+	0.985845i	$0.553621\pi$
$264$	0	0
$265$	−7.37733	−0.453186
$266$	0	0
$267$	0.522675	0.0319872
$268$	0	0
$269$	13.8682	0.845561	0.422780	−	0.906232i	$-0.361054\pi$
$269$	13.8682	0.845561	0.422780	+	0.906232i	$0.361054\pi$
$270$	0	0
$271$	13.7110	0.832884	0.416442	−	0.909162i	$-0.363277\pi$
$271$	13.7110	0.832884	0.416442	+	0.909162i	$0.363277\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.12734	0.0679814
$276$	0	0
$277$	−7.62629	−0.458219	−0.229110	−	0.973401i	$-0.573581\pi$
$277$	−7.62629	−0.458219	−0.229110	+	0.973401i	$0.573581\pi$
$278$	0	0
$279$	−0.283597	−0.0169785
$280$	0	0
$281$	−26.0467	−1.55382	−0.776908	−	0.629614i	$-0.783213\pi$
$281$	−26.0467	−1.55382	−0.776908	+	0.629614i	$0.783213\pi$
$282$	0	0
$283$	3.46133	0.205755	0.102877	−	0.994694i	$-0.467195\pi$
$283$	3.46133	0.205755	0.102877	+	0.994694i	$0.467195\pi$
$284$	0	0
$285$	14.3741	0.851449
$286$	0	0
$287$	0	0
$288$	0	0
$289$	5.66671	0.333336
$290$	0	0
$291$	20.1343	1.18030
$292$	0	0
$293$	−11.2067	−0.654705	−0.327352	−	0.944902i	$-0.606156\pi$
$293$	−11.2067	−0.654705	−0.327352	+	0.944902i	$0.606156\pi$
$294$	0	0
$295$	12.3241	0.717536
$296$	0	0
$297$	−15.9005	−0.922638
$298$	0	0
$299$	7.00150	0.404907
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−3.63651	−0.208912
$304$	0	0
$305$	12.8365	0.735017
$306$	0	0
$307$	−0.582689	−0.0332558	−0.0166279	−	0.999862i	$-0.505293\pi$
$307$	−0.582689	−0.0332558	−0.0166279	+	0.999862i	$0.505293\pi$
$308$	0	0
$309$	12.8868	0.733106
$310$	0	0
$311$	6.24581	0.354167	0.177084	−	0.984196i	$-0.443334\pi$
$311$	6.24581	0.354167	0.177084	+	0.984196i	$0.443334\pi$
$312$	0	0
$313$	−17.7415	−1.00281	−0.501405	−	0.865213i	$-0.667183\pi$
$313$	−17.7415	−1.00281	−0.501405	+	0.865213i	$0.667183\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.9364	0.726583	0.363291	−	0.931676i	$-0.381653\pi$
$317$	12.9364	0.726583	0.363291	+	0.931676i	$0.381653\pi$
$318$	0	0
$319$	−0.174528	−0.00977169
$320$	0	0
$321$	22.1014	1.23358
$322$	0	0
$323$	18.5721	1.03338
$324$	0	0
$325$	−0.315893	−0.0175226
$326$	0	0
$327$	−1.00629	−0.0556478
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−18.9270	−1.04032	−0.520161	−	0.854068i	$-0.674128\pi$
$331$	−18.9270	−1.04032	−0.520161	+	0.854068i	$0.674128\pi$
$332$	0	0
$333$	−0.0313448	−0.00171768
$334$	0	0
$335$	−20.5127	−1.12073
$336$	0	0
$337$	16.2630	0.885903	0.442951	−	0.896546i	$-0.353931\pi$
$337$	16.2630	0.885903	0.442951	+	0.896546i	$0.353931\pi$
$338$	0	0
$339$	3.21633	0.174687
$340$	0	0
$341$	−13.0205	−0.705102
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−30.4097	−1.63720
$346$	0	0
$347$	2.71825	0.145923	0.0729615	−	0.997335i	$-0.476755\pi$
$347$	2.71825	0.145923	0.0729615	+	0.997335i	$0.476755\pi$
$348$	0	0
$349$	−20.9033	−1.11893	−0.559464	−	0.828855i	$-0.688993\pi$
$349$	−20.9033	−1.11893	−0.559464	+	0.828855i	$0.688993\pi$
$350$	0	0
$351$	4.45546	0.237815
$352$	0	0
$353$	10.0242	0.533533	0.266766	−	0.963761i	$-0.414045\pi$
$353$	10.0242	0.533533	0.266766	+	0.963761i	$0.414045\pi$
$354$	0	0
$355$	−3.96872	−0.210638
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−33.9417	−1.79137	−0.895686	−	0.444686i	$-0.853315\pi$
$359$	−33.9417	−1.79137	−0.895686	+	0.444686i	$0.853315\pi$
$360$	0	0
$361$	−3.78287	−0.199099
$362$	0	0
$363$	3.13994	0.164804
$364$	0	0
$365$	8.38704	0.438998
$366$	0	0
$367$	28.3396	1.47931	0.739656	−	0.672985i	$-0.234988\pi$
$367$	28.3396	1.47931	0.739656	+	0.672985i	$0.234988\pi$
$368$	0	0
$369$	−0.0659453	−0.00343297
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−31.3865	−1.62513	−0.812565	−	0.582871i	$-0.801929\pi$
$373$	−31.3865	−1.62513	−0.812565	+	0.582871i	$0.801929\pi$
$374$	0	0
$375$	19.7960	1.02226
$376$	0	0
$377$	0.0489044	0.00251870
$378$	0	0
$379$	−6.85269	−0.351999	−0.176000	−	0.984390i	$-0.556316\pi$
$379$	−6.85269	−0.351999	−0.176000	+	0.984390i	$0.556316\pi$
$380$	0	0
$381$	−20.4246	−1.04639
$382$	0	0
$383$	−37.9314	−1.93820	−0.969102	−	0.246659i	$-0.920667\pi$
$383$	−37.9314	−1.93820	−0.969102	+	0.246659i	$0.920667\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−0.186405	−0.00947551
$388$	0	0
$389$	18.7304	0.949668	0.474834	−	0.880075i	$-0.342508\pi$
$389$	18.7304	0.949668	0.474834	+	0.880075i	$0.342508\pi$
$390$	0	0
$391$	−39.2908	−1.98702
$392$	0	0
$393$	32.8676	1.65795
$394$	0	0
$395$	11.0086	0.553902
$396$	0	0
$397$	20.6693	1.03736	0.518681	−	0.854968i	$-0.326423\pi$
$397$	20.6693	1.03736	0.518681	+	0.854968i	$0.326423\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	36.6913	1.83227	0.916137	−	0.400866i	$-0.131291\pi$
$401$	36.6913	1.83227	0.916137	+	0.400866i	$0.131291\pi$
$402$	0	0
$403$	3.64848	0.181744
$404$	0	0
$405$	−18.9258	−0.940433
$406$	0	0
$407$	−1.43911	−0.0713339
$408$	0	0
$409$	−26.8678	−1.32853	−0.664264	−	0.747498i	$-0.731255\pi$
$409$	−26.8678	−1.32853	−0.664264	+	0.747498i	$0.731255\pi$
$410$	0	0
$411$	13.7223	0.676873
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−15.9451	−0.782713
$416$	0	0
$417$	−34.8418	−1.70621
$418$	0	0
$419$	−32.5057	−1.58801	−0.794004	−	0.607913i	$-0.792007\pi$
$419$	−32.5057	−1.58801	−0.794004	+	0.607913i	$0.792007\pi$
$420$	0	0
$421$	−39.9668	−1.94786	−0.973931	−	0.226844i	$-0.927159\pi$
$421$	−39.9668	−1.94786	−0.973931	+	0.226844i	$0.927159\pi$
$422$	0	0
$423$	−0.443136	−0.0215460
$424$	0	0
$425$	1.77272	0.0859895
$426$	0	0
$427$	0	0
$428$	0	0
$429$	4.39986	0.212427
$430$	0	0
$431$	34.2574	1.65012	0.825061	−	0.565043i	$-0.191141\pi$
$431$	34.2574	1.65012	0.825061	+	0.565043i	$0.191141\pi$
$432$	0	0
$433$	−10.9293	−0.525229	−0.262615	−	0.964901i	$-0.584585\pi$
$433$	−10.9293	−0.525229	−0.262615	+	0.964901i	$0.584585\pi$
$434$	0	0
$435$	−0.212407	−0.0101841
$436$	0	0
$437$	−32.1931	−1.54001
$438$	0	0
$439$	36.0170	1.71900	0.859499	−	0.511137i	$-0.170775\pi$
$439$	36.0170	1.71900	0.859499	+	0.511137i	$0.170775\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−15.2933	−0.726609	−0.363304	−	0.931670i	$-0.618352\pi$
$443$	−15.2933	−0.726609	−0.363304	+	0.931670i	$0.618352\pi$
$444$	0	0
$445$	−0.656415	−0.0311171
$446$	0	0
$447$	18.1263	0.857342
$448$	0	0
$449$	−22.7342	−1.07289	−0.536447	−	0.843934i	$-0.680234\pi$
$449$	−22.7342	−1.07289	−0.536447	+	0.843934i	$0.680234\pi$
$450$	0	0
$451$	−3.02769	−0.142568
$452$	0	0
$453$	19.7330	0.927139
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5.18370	−0.242483	−0.121242	−	0.992623i	$-0.538688\pi$
$457$	−5.18370	−0.242483	−0.121242	+	0.992623i	$0.538688\pi$
$458$	0	0
$459$	−25.0030	−1.16704
$460$	0	0
$461$	−15.9517	−0.742946	−0.371473	−	0.928444i	$-0.621147\pi$
$461$	−15.9517	−0.742946	−0.371473	+	0.928444i	$0.621147\pi$
$462$	0	0
$463$	−23.8765	−1.10964	−0.554818	−	0.831972i	$-0.687212\pi$
$463$	−23.8765	−1.10964	−0.554818	+	0.831972i	$0.687212\pi$
$464$	0	0
$465$	−15.8465	−0.734863
$466$	0	0
$467$	17.3183	0.801395	0.400697	−	0.916211i	$-0.368768\pi$
$467$	17.3183	0.801395	0.400697	+	0.916211i	$0.368768\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−6.89627	−0.317764
$472$	0	0
$473$	−8.55827	−0.393510
$474$	0	0
$475$	1.45249	0.0666446
$476$	0	0
$477$	0.226153	0.0103548
$478$	0	0
$479$	−35.3931	−1.61715	−0.808575	−	0.588393i	$-0.799761\pi$
$479$	−35.3931	−1.61715	−0.808575	+	0.588393i	$0.799761\pi$
$480$	0	0
$481$	0.403252	0.0183867
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−25.2862	−1.14819
$486$	0	0
$487$	−35.2364	−1.59671	−0.798357	−	0.602185i	$-0.794297\pi$
$487$	−35.2364	−1.59671	−0.798357	+	0.602185i	$0.794297\pi$
$488$	0	0
$489$	32.6414	1.47610
$490$	0	0
$491$	8.56552	0.386556	0.193278	−	0.981144i	$-0.438088\pi$
$491$	8.56552	0.386556	0.193278	+	0.981144i	$0.438088\pi$
$492$	0	0
$493$	−0.274440	−0.0123602
$494$	0	0
$495$	0.429512	0.0193051
$496$	0	0
$497$	0	0
$498$	0	0
$499$	13.6719	0.612040	0.306020	−	0.952025i	$-0.401003\pi$
$499$	13.6719	0.612040	0.306020	+	0.952025i	$0.401003\pi$
$500$	0	0
$501$	−20.1048	−0.898214
$502$	0	0
$503$	−21.0857	−0.940167	−0.470084	−	0.882622i	$-0.655776\pi$
$503$	−21.0857	−0.940167	−0.470084	+	0.882622i	$0.655776\pi$
$504$	0	0
$505$	4.56701	0.203229
$506$	0	0
$507$	21.0349	0.934194
$508$	0	0
$509$	−26.6892	−1.18298	−0.591489	−	0.806313i	$-0.701460\pi$
$509$	−26.6892	−1.18298	−0.591489	+	0.806313i	$0.701460\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−20.4864	−0.904495
$514$	0	0
$515$	−16.1843	−0.713163
$516$	0	0
$517$	−20.3453	−0.894787
$518$	0	0
$519$	2.58699	0.113556
$520$	0	0
$521$	−17.4396	−0.764041	−0.382020	−	0.924154i	$-0.624772\pi$
$521$	−17.4396	−0.764041	−0.382020	+	0.924154i	$0.624772\pi$
$522$	0	0
$523$	38.3651	1.67759	0.838795	−	0.544448i	$-0.183261\pi$
$523$	38.3651	1.67759	0.838795	+	0.544448i	$0.183261\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−20.4745	−0.891881
$528$	0	0
$529$	45.1074	1.96119
$530$	0	0
$531$	−0.377796	−0.0163950
$532$	0	0
$533$	0.848387	0.0367477
$534$	0	0
$535$	−27.7566	−1.20002
$536$	0	0
$537$	2.36840	0.102204
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−1.86191	−0.0800497	−0.0400249	−	0.999199i	$-0.512744\pi$
$541$	−1.86191	−0.0800497	−0.0400249	+	0.999199i	$0.512744\pi$
$542$	0	0
$543$	5.50527	0.236254
$544$	0	0
$545$	1.26377	0.0541340
$546$	0	0
$547$	32.7545	1.40048	0.700240	−	0.713908i	$-0.253076\pi$
$547$	32.7545	1.40048	0.700240	+	0.713908i	$0.253076\pi$
$548$	0	0
$549$	−0.393505	−0.0167944
$550$	0	0
$551$	−0.224864	−0.00957953
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−1.75145	−0.0743447
$556$	0	0
$557$	−33.9771	−1.43966	−0.719828	−	0.694152i	$-0.755779\pi$
$557$	−33.9771	−1.43966	−0.719828	+	0.694152i	$0.755779\pi$
$558$	0	0
$559$	2.39811	0.101429
$560$	0	0
$561$	−24.6910	−1.04246
$562$	0	0
$563$	−33.7187	−1.42107	−0.710536	−	0.703661i	$-0.751548\pi$
$563$	−33.7187	−1.42107	−0.710536	+	0.703661i	$0.751548\pi$
$564$	0	0
$565$	−4.03931	−0.169935
$566$	0	0
$567$	0	0
$568$	0	0
$569$	15.8428	0.664166	0.332083	−	0.943250i	$-0.392249\pi$
$569$	15.8428	0.664166	0.332083	+	0.943250i	$0.392249\pi$
$570$	0	0
$571$	−42.5373	−1.78013	−0.890065	−	0.455833i	$-0.849341\pi$
$571$	−42.5373	−1.78013	−0.890065	+	0.455833i	$0.849341\pi$
$572$	0	0
$573$	10.8165	0.451865
$574$	0	0
$575$	−3.07286	−0.128147
$576$	0	0
$577$	−4.79836	−0.199758	−0.0998791	−	0.995000i	$-0.531846\pi$
$577$	−4.79836	−0.199758	−0.0998791	+	0.995000i	$0.531846\pi$
$578$	0	0
$579$	26.0049	1.08073
$580$	0	0
$581$	0	0
$582$	0	0
$583$	10.3832	0.430027
$584$	0	0
$585$	−0.120353	−0.00497600
$586$	0	0
$587$	0.292839	0.0120868	0.00604339	−	0.999982i	$-0.498076\pi$
$587$	0.292839	0.0120868	0.00604339	+	0.999982i	$0.498076\pi$
$588$	0	0
$589$	−16.7758	−0.691237
$590$	0	0
$591$	30.7401	1.26448
$592$	0	0
$593$	2.67878	0.110004	0.0550022	−	0.998486i	$-0.482483\pi$
$593$	2.67878	0.110004	0.0550022	+	0.998486i	$0.482483\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−13.3982	−0.548354
$598$	0	0
$599$	−15.9623	−0.652204	−0.326102	−	0.945335i	$-0.605735\pi$
$599$	−15.9623	−0.652204	−0.326102	+	0.945335i	$0.605735\pi$
$600$	0	0
$601$	7.88205	0.321516	0.160758	−	0.986994i	$-0.448606\pi$
$601$	7.88205	0.321516	0.160758	+	0.986994i	$0.448606\pi$
$602$	0	0
$603$	0.628821	0.0256076
$604$	0	0
$605$	−3.94337	−0.160321
$606$	0	0
$607$	−33.5100	−1.36013	−0.680064	−	0.733153i	$-0.738048\pi$
$607$	−33.5100	−1.36013	−0.680064	+	0.733153i	$0.738048\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.70096	0.230636
$612$	0	0
$613$	33.5090	1.35342	0.676708	−	0.736251i	$-0.263406\pi$
$613$	33.5090	1.35342	0.676708	+	0.736251i	$0.263406\pi$
$614$	0	0
$615$	−3.68481	−0.148586
$616$	0	0
$617$	45.8922	1.84755	0.923775	−	0.382935i	$-0.125087\pi$
$617$	45.8922	1.84755	0.923775	+	0.382935i	$0.125087\pi$
$618$	0	0
$619$	29.8762	1.20082	0.600412	−	0.799691i	$-0.295003\pi$
$619$	29.8762	1.20082	0.600412	+	0.799691i	$0.295003\pi$
$620$	0	0
$621$	43.3407	1.73920
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−22.9996	−0.919985
$626$	0	0
$627$	−20.2307	−0.807938
$628$	0	0
$629$	−2.26296	−0.0902300
$630$	0	0
$631$	5.54997	0.220941	0.110470	−	0.993879i	$-0.464764\pi$
$631$	5.54997	0.220941	0.110470	+	0.993879i	$0.464764\pi$
$632$	0	0
$633$	3.94036	0.156615
$634$	0	0
$635$	25.6508	1.01792
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0.121662	0.00481286
$640$	0	0
$641$	26.0010	1.02698	0.513489	−	0.858096i	$-0.328353\pi$
$641$	26.0010	1.02698	0.513489	+	0.858096i	$0.328353\pi$
$642$	0	0
$643$	25.0891	0.989419	0.494709	−	0.869058i	$-0.335275\pi$
$643$	25.0891	0.989419	0.494709	+	0.869058i	$0.335275\pi$
$644$	0	0
$645$	−10.4157	−0.410119
$646$	0	0
$647$	−17.2904	−0.679757	−0.339878	−	0.940469i	$-0.610386\pi$
$647$	−17.2904	−0.679757	−0.339878	+	0.940469i	$0.610386\pi$
$648$	0	0
$649$	−17.3454	−0.680868
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9.29684	0.363814	0.181907	−	0.983316i	$-0.441773\pi$
$653$	9.29684	0.363814	0.181907	+	0.983316i	$0.441773\pi$
$654$	0	0
$655$	−41.2776	−1.61285
$656$	0	0
$657$	−0.257106	−0.0100306
$658$	0	0
$659$	5.40667	0.210614	0.105307	−	0.994440i	$-0.466417\pi$
$659$	5.40667	0.210614	0.105307	+	0.994440i	$0.466417\pi$
$660$	0	0
$661$	24.7223	0.961584	0.480792	−	0.876835i	$-0.340349\pi$
$661$	24.7223	0.961584	0.480792	+	0.876835i	$0.340349\pi$
$662$	0	0
$663$	6.91867	0.268699
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.475719	0.0184199
$668$	0	0
$669$	10.2978	0.398134
$670$	0	0
$671$	−18.0667	−0.697456
$672$	0	0
$673$	−19.7190	−0.760110	−0.380055	−	0.924964i	$-0.624095\pi$
$673$	−19.7190	−0.760110	−0.380055	+	0.924964i	$0.624095\pi$
$674$	0	0
$675$	−1.95544	−0.0752648
$676$	0	0
$677$	−17.6496	−0.678330	−0.339165	−	0.940727i	$-0.610144\pi$
$677$	−17.6496	−0.678330	−0.339165	+	0.940727i	$0.610144\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−42.5885	−1.63200
$682$	0	0
$683$	−45.9083	−1.75663	−0.878317	−	0.478080i	$-0.841333\pi$
$683$	−45.9083	−1.75663	−0.878317	+	0.478080i	$0.841333\pi$
$684$	0	0
$685$	−17.2336	−0.658460
$686$	0	0
$687$	41.1629	1.57046
$688$	0	0
$689$	−2.90946	−0.110842
$690$	0	0
$691$	−38.9277	−1.48088	−0.740439	−	0.672124i	$-0.765382\pi$
$691$	−38.9277	−1.48088	−0.740439	+	0.672124i	$0.765382\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	43.7569	1.65979
$696$	0	0
$697$	−4.76096	−0.180334
$698$	0	0
$699$	13.3183	0.503743
$700$	0	0
$701$	19.2696	0.727803	0.363902	−	0.931437i	$-0.381444\pi$
$701$	19.2696	0.727803	0.363902	+	0.931437i	$0.381444\pi$
$702$	0	0
$703$	−1.85417	−0.0699312
$704$	0	0
$705$	−24.7610	−0.932553
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−24.2495	−0.910709	−0.455355	−	0.890310i	$-0.650488\pi$
$709$	−24.2495	−0.910709	−0.455355	+	0.890310i	$0.650488\pi$
$710$	0	0
$711$	−0.337470	−0.0126561
$712$	0	0
$713$	35.4908	1.32914
$714$	0	0
$715$	−5.52568	−0.206649
$716$	0	0
$717$	−32.9832	−1.23178
$718$	0	0
$719$	−16.4335	−0.612867	−0.306434	−	0.951892i	$-0.599136\pi$
$719$	−16.4335	−0.612867	−0.306434	+	0.951892i	$0.599136\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−10.1385	−0.377054
$724$	0	0
$725$	−0.0214634	−0.000797132 0
$726$	0	0
$727$	−47.2145	−1.75109	−0.875545	−	0.483136i	$-0.839497\pi$
$727$	−47.2145	−1.75109	−0.875545	+	0.483136i	$0.839497\pi$
$728$	0	0
$729$	27.5671	1.02100
$730$	0	0
$731$	−13.4576	−0.497749
$732$	0	0
$733$	−0.860012	−0.0317653	−0.0158826	−	0.999874i	$-0.505056\pi$
$733$	−0.860012	−0.0317653	−0.0158826	+	0.999874i	$0.505056\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	28.8705	1.06346
$738$	0	0
$739$	17.8084	0.655094	0.327547	−	0.944835i	$-0.393778\pi$
$739$	17.8084	0.655094	0.327547	+	0.944835i	$0.393778\pi$
$740$	0	0
$741$	5.66884	0.208250
$742$	0	0
$743$	−2.23502	−0.0819950	−0.0409975	−	0.999159i	$-0.513054\pi$
$743$	−2.23502	−0.0819950	−0.0409975	+	0.999159i	$0.513054\pi$
$744$	0	0
$745$	−22.7643	−0.834020
$746$	0	0
$747$	0.488798	0.0178842
$748$	0	0
$749$	0	0
$750$	0	0
$751$	52.0736	1.90019	0.950097	−	0.311955i	$-0.100984\pi$
$751$	52.0736	1.90019	0.950097	+	0.311955i	$0.100984\pi$
$752$	0	0
$753$	−48.2884	−1.75973
$754$	0	0
$755$	−24.7822	−0.901918
$756$	0	0
$757$	3.36576	0.122331	0.0611653	−	0.998128i	$-0.480518\pi$
$757$	3.36576	0.122331	0.0611653	+	0.998128i	$0.480518\pi$
$758$	0	0
$759$	42.7999	1.55354
$760$	0	0
$761$	7.18908	0.260604	0.130302	−	0.991474i	$-0.458405\pi$
$761$	7.18908	0.260604	0.130302	+	0.991474i	$0.458405\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0.675396	0.0244190
$766$	0	0
$767$	4.86036	0.175497
$768$	0	0
$769$	3.07410	0.110855	0.0554275	−	0.998463i	$-0.482348\pi$
$769$	3.07410	0.110855	0.0554275	+	0.998463i	$0.482348\pi$
$770$	0	0
$771$	−7.32908	−0.263951
$772$	0	0
$773$	−18.7338	−0.673808	−0.336904	−	0.941539i	$-0.609380\pi$
$773$	−18.7338	−0.673808	−0.336904	+	0.941539i	$0.609380\pi$
$774$	0	0
$775$	−1.60127	−0.0575192
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−3.90091	−0.139765
$780$	0	0
$781$	5.58574	0.199874
$782$	0	0
$783$	0.302728	0.0108186
$784$	0	0
$785$	8.66086	0.309119
$786$	0	0
$787$	37.5567	1.33875	0.669377	−	0.742923i	$-0.266561\pi$
$787$	37.5567	1.33875	0.669377	+	0.742923i	$0.266561\pi$
$788$	0	0
$789$	9.31471	0.331612
$790$	0	0
$791$	0	0
$792$	0	0
$793$	5.06245	0.179773
$794$	0	0
$795$	12.6367	0.448177
$796$	0	0
$797$	−13.5950	−0.481559	−0.240779	−	0.970580i	$-0.577403\pi$
$797$	−13.5950	−0.481559	−0.240779	+	0.970580i	$0.577403\pi$
$798$	0	0
$799$	−31.9925	−1.13181
$800$	0	0
$801$	0.0201225	0.000710993 0
$802$	0	0
$803$	−11.8043	−0.416564
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−23.7550	−0.836216
$808$	0	0
$809$	22.6190	0.795242	0.397621	−	0.917550i	$-0.369836\pi$
$809$	22.6190	0.795242	0.397621	+	0.917550i	$0.369836\pi$
$810$	0	0
$811$	−36.5762	−1.28437	−0.642183	−	0.766552i	$-0.721971\pi$
$811$	−36.5762	−1.28437	−0.642183	+	0.766552i	$0.721971\pi$
$812$	0	0
$813$	−23.4857	−0.823679
$814$	0	0
$815$	−40.9936	−1.43594
$816$	0	0
$817$	−11.0266	−0.385771
$818$	0	0
$819$	0	0
$820$	0	0
$821$	43.8441	1.53017	0.765084	−	0.643930i	$-0.222697\pi$
$821$	43.8441	1.53017	0.765084	+	0.643930i	$0.222697\pi$
$822$	0	0
$823$	24.8684	0.866859	0.433429	−	0.901188i	$-0.357303\pi$
$823$	24.8684	0.866859	0.433429	+	0.901188i	$0.357303\pi$
$824$	0	0
$825$	−1.93104	−0.0672301
$826$	0	0
$827$	−19.4951	−0.677911	−0.338956	−	0.940802i	$-0.610074\pi$
$827$	−19.4951	−0.677911	−0.338956	+	0.940802i	$0.610074\pi$
$828$	0	0
$829$	−7.46834	−0.259386	−0.129693	−	0.991554i	$-0.541399\pi$
$829$	−7.46834	−0.259386	−0.129693	+	0.991554i	$0.541399\pi$
$830$	0	0
$831$	13.0631	0.453155
$832$	0	0
$833$	0	0
$834$	0	0
$835$	25.2491	0.873780
$836$	0	0
$837$	22.5848	0.780646
$838$	0	0
$839$	−0.549539	−0.0189722	−0.00948610	−	0.999955i	$-0.503020\pi$
$839$	−0.549539	−0.0189722	−0.00948610	+	0.999955i	$0.503020\pi$
$840$	0	0
$841$	−28.9967	−0.999885
$842$	0	0
$843$	44.6156	1.53664
$844$	0	0
$845$	−26.4172	−0.908780
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−5.92894	−0.203481
$850$	0	0
$851$	3.92265	0.134467
$852$	0	0
$853$	13.6543	0.467516	0.233758	−	0.972295i	$-0.424898\pi$
$853$	13.6543	0.467516	0.233758	+	0.972295i	$0.424898\pi$
$854$	0	0
$855$	0.553389	0.0189255
$856$	0	0
$857$	33.9066	1.15823	0.579115	−	0.815246i	$-0.303398\pi$
$857$	33.9066	1.15823	0.579115	+	0.815246i	$0.303398\pi$
$858$	0	0
$859$	−35.8208	−1.22219	−0.611095	−	0.791557i	$-0.709271\pi$
$859$	−35.8208	−1.22219	−0.611095	+	0.791557i	$0.709271\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	34.4639	1.17316	0.586582	−	0.809890i	$-0.300473\pi$
$863$	34.4639	1.17316	0.586582	+	0.809890i	$0.300473\pi$
$864$	0	0
$865$	−3.24893	−0.110467
$866$	0	0
$867$	−9.70656	−0.329652
$868$	0	0
$869$	−15.4939	−0.525596
$870$	0	0
$871$	−8.08979	−0.274112
$872$	0	0
$873$	0.775151	0.0262349
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−49.3159	−1.66528	−0.832640	−	0.553815i	$-0.813172\pi$
$877$	−49.3159	−1.66528	−0.832640	+	0.553815i	$0.813172\pi$
$878$	0	0
$879$	19.1961	0.647469
$880$	0	0
$881$	−52.6993	−1.77548	−0.887742	−	0.460342i	$-0.847727\pi$
$881$	−52.6993	−1.77548	−0.887742	+	0.460342i	$0.847727\pi$
$882$	0	0
$883$	47.2828	1.59119	0.795597	−	0.605827i	$-0.207157\pi$
$883$	47.2828	1.59119	0.795597	+	0.605827i	$0.207157\pi$
$884$	0	0
$885$	−21.1100	−0.709606
$886$	0	0
$887$	11.0043	0.369487	0.184744	−	0.982787i	$-0.440855\pi$
$887$	11.0043	0.369487	0.184744	+	0.982787i	$0.440855\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	26.6370	0.892374
$892$	0	0
$893$	−26.2132	−0.877191
$894$	0	0
$895$	−2.97441	−0.0994237
$896$	0	0
$897$	−11.9929	−0.400432
$898$	0	0
$899$	0.247897	0.00826784
$900$	0	0
$901$	16.3273	0.543940
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−6.91393	−0.229827
$906$	0	0
$907$	−50.3611	−1.67221	−0.836106	−	0.548568i	$-0.815173\pi$
$907$	−50.3611	−1.67221	−0.836106	+	0.548568i	$0.815173\pi$
$908$	0	0
$909$	−0.140002	−0.00464358
$910$	0	0
$911$	9.11340	0.301940	0.150970	−	0.988538i	$-0.451760\pi$
$911$	9.11340	0.301940	0.150970	+	0.988538i	$0.451760\pi$
$912$	0	0
$913$	22.4418	0.742714
$914$	0	0
$915$	−21.9878	−0.726893
$916$	0	0
$917$	0	0
$918$	0	0
$919$	23.7583	0.783714	0.391857	−	0.920026i	$-0.371833\pi$
$919$	23.7583	0.783714	0.391857	+	0.920026i	$0.371833\pi$
$920$	0	0
$921$	0.998093	0.0328883
$922$	0	0
$923$	−1.56518	−0.0515185
$924$	0	0
$925$	−0.176981	−0.00581911
$926$	0	0
$927$	0.496130	0.0162951
$928$	0	0
$929$	0.179643	0.00589390	0.00294695	−	0.999996i	$-0.499062\pi$
$929$	0.179643	0.00589390	0.00294695	+	0.999996i	$0.499062\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−10.6985	−0.350253
$934$	0	0
$935$	31.0089	1.01410
$936$	0	0
$937$	25.1739	0.822396	0.411198	−	0.911546i	$-0.365110\pi$
$937$	25.1739	0.822396	0.411198	+	0.911546i	$0.365110\pi$
$938$	0	0
$939$	30.3896	0.991727
$940$	0	0
$941$	−11.0109	−0.358947	−0.179473	−	0.983763i	$-0.557439\pi$
$941$	−11.0109	−0.358947	−0.179473	+	0.983763i	$0.557439\pi$
$942$	0	0
$943$	8.25272	0.268746
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−9.27919	−0.301533	−0.150767	−	0.988569i	$-0.548174\pi$
$947$	−9.27919	−0.301533	−0.150767	+	0.988569i	$0.548174\pi$
$948$	0	0
$949$	3.30767	0.107371
$950$	0	0
$951$	−22.1589	−0.718553
$952$	0	0
$953$	24.8414	0.804691	0.402345	−	0.915488i	$-0.368195\pi$
$953$	24.8414	0.804691	0.402345	+	0.915488i	$0.368195\pi$
$954$	0	0
$955$	−13.5842	−0.439573
$956$	0	0
$957$	0.298950	0.00966369
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−12.5058	−0.403412
$962$	0	0
$963$	0.850883	0.0274193
$964$	0	0
$965$	−32.6589	−1.05133
$966$	0	0
$967$	−24.4773	−0.787138	−0.393569	−	0.919295i	$-0.628760\pi$
$967$	−24.4773	−0.787138	−0.393569	+	0.919295i	$0.628760\pi$
$968$	0	0
$969$	−31.8123	−1.02196
$970$	0	0
$971$	−17.3746	−0.557576	−0.278788	−	0.960353i	$-0.589933\pi$
$971$	−17.3746	−0.557576	−0.278788	+	0.960353i	$0.589933\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0.541095	0.0173289
$976$	0	0
$977$	41.1045	1.31505	0.657525	−	0.753432i	$-0.271603\pi$
$977$	41.1045	1.31505	0.657525	+	0.753432i	$0.271603\pi$
$978$	0	0
$979$	0.923866	0.0295269
$980$	0	0
$981$	−0.0387411	−0.00123691
$982$	0	0
$983$	−30.1694	−0.962255	−0.481127	−	0.876651i	$-0.659773\pi$
$983$	−30.1694	−0.962255	−0.481127	+	0.876651i	$0.659773\pi$
$984$	0	0
$985$	−38.6057	−1.23008
$986$	0	0
$987$	0	0
$988$	0	0
$989$	23.3277	0.741777
$990$	0	0
$991$	8.95738	0.284540	0.142270	−	0.989828i	$-0.454560\pi$
$991$	8.95738	0.284540	0.142270	+	0.989828i	$0.454560\pi$
$992$	0	0
$993$	32.4202	1.02882
$994$	0	0
$995$	16.8265	0.533436
$996$	0	0
$997$	40.8257	1.29296	0.646482	−	0.762930i	$-0.276240\pi$
$997$	40.8257	1.29296	0.646482	+	0.762930i	$0.276240\pi$
$998$	0	0
$999$	2.49621	0.0789765

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8036.2.a.m.1.3		8
7.2	even	3		1148.2.i.d.165.6	✓	16
7.4	even	3		1148.2.i.d.821.6	yes	16
7.6	odd	2		8036.2.a.n.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.i.d.165.6	✓	16	7.2	even	3
1148.2.i.d.821.6	yes	16	7.4	even	3
8036.2.a.m.1.3		8	1.1	even	1	trivial
8036.2.a.n.1.6		8	7.6	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:	\( N \)	\(=\)	\( 8036 = 2^{2} \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8036.a (trivial)

\(f(q)\)	\(=\)	\(q-1.71291 q^{3} +2.15120 q^{5} -0.0659453 q^{9} +O(q^{10})\)	\(q-1.71291 q^{3} +2.15120 q^{5} -0.0659453 q^{9} -3.02769 q^{11} +0.848387 q^{13} -3.68481 q^{15} -4.76096 q^{17} -3.90091 q^{19} +8.25272 q^{23} -0.372345 q^{25} +5.25168 q^{27} +0.0576440 q^{29} +4.30049 q^{31} +5.18615 q^{33} +0.475316 q^{37} -1.45321 q^{39} +1.00000 q^{41} +2.82667 q^{43} -0.141861 q^{45} +6.71976 q^{47} +8.15508 q^{51} -3.42941 q^{53} -6.51316 q^{55} +6.68191 q^{57} +5.72894 q^{59} +5.96715 q^{61} +1.82505 q^{65} -9.53549 q^{67} -14.1362 q^{69} -1.84489 q^{71} +3.89877 q^{73} +0.637793 q^{75} +5.11742 q^{79} -8.79782 q^{81} -7.41218 q^{83} -10.2418 q^{85} -0.0987388 q^{87} -0.305139 q^{89} -7.36635 q^{93} -8.39164 q^{95} -11.7545 q^{97} +0.199662 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{9}+O(q^{10}) \) 8 * q + 4 * q^9	\( 8 q + 4 q^{9} - 8 q^{11} - 7 q^{13} - q^{15} - q^{17} + 4 q^{19} - 3 q^{23} - 4 q^{25} - 12 q^{27} - 4 q^{29} + 4 q^{31} + 23 q^{33} - 31 q^{37} + 5 q^{39} + 8 q^{41} - 8 q^{43} + q^{45} + 24 q^{47} - 23 q^{51} - q^{53} + 2 q^{55} - 15 q^{57} + 4 q^{59} - 4 q^{61} - 24 q^{65} - 21 q^{69} + 8 q^{71} + 11 q^{73} - 15 q^{75} + 14 q^{79} - 28 q^{81} - 42 q^{83} - 20 q^{85} + 25 q^{87} - 11 q^{89} - 27 q^{93} - 15 q^{95} - 16 q^{97} - 8 q^{99}+O(q^{100}) \) 8 * q + 4 * q^9 - 8 * q^11 - 7 * q^13 - q^15 - q^17 + 4 * q^19 - 3 * q^23 - 4 * q^25 - 12 * q^27 - 4 * q^29 + 4 * q^31 + 23 * q^33 - 31 * q^37 + 5 * q^39 + 8 * q^41 - 8 * q^43 + q^45 + 24 * q^47 - 23 * q^51 - q^53 + 2 * q^55 - 15 * q^57 + 4 * q^59 - 4 * q^61 - 24 * q^65 - 21 * q^69 + 8 * q^71 + 11 * q^73 - 15 * q^75 + 14 * q^79 - 28 * q^81 - 42 * q^83 - 20 * q^85 + 25 * q^87 - 11 * q^89 - 27 * q^93 - 15 * q^95 - 16 * q^97 - 8 * q^99

Introduction

L-functions

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Database

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