LMFDB - Embedded newform 1472.2.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1472,2,Mod(1,1472)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1472, 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("1472.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1472.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:	$11.7539791775$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13768.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 2x + 2 $ x^4 - x^3 - 5x^2 + 2x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 736)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.53744$ of defining polynomial
Character	$\chi$	$=$	1472.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.901180 q^{3} -2.78819 q^{5} -2.28669 q^{7} -2.18787 q^{9} -0.788195 q^{11} -5.18787 q^{13} +2.51267 q^{15} +4.28669 q^{17} -3.07489 q^{19} +2.06072 q^{21} +1.00000 q^{23} +2.77403 q^{25} +4.67521 q^{27} -3.61149 q^{29} +9.24860 q^{31} +0.710305 q^{33} +6.37575 q^{35} -2.78819 q^{37} +4.67521 q^{39} -2.76426 q^{41} +12.8772 q^{43} +6.10022 q^{45} -7.05096 q^{47} -1.77103 q^{49} -3.86308 q^{51} -0.727471 q^{53} +2.19764 q^{55} +2.77103 q^{57} +12.1781 q^{59} -3.27253 q^{61} +5.00300 q^{63} +14.4648 q^{65} -3.78519 q^{67} -0.901180 q^{69} +3.89818 q^{71} +8.56662 q^{73} -2.49990 q^{75} +1.80236 q^{77} -7.95214 q^{79} +2.35042 q^{81} -1.01417 q^{83} -11.9521 q^{85} +3.25460 q^{87} -4.37575 q^{89} +11.8631 q^{91} -8.33465 q^{93} +8.57339 q^{95} -17.4335 q^{97} +1.72447 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 6 q^{5} + 4 q^{7} + 10 q^{9} + 2 q^{11} - 2 q^{13} - 4 q^{15} + 4 q^{17} + 6 q^{19} - 4 q^{21} + 4 q^{23} + 12 q^{25} + 14 q^{27} - 6 q^{29} + 6 q^{31} - 12 q^{35} - 6 q^{37} + 14 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 - 6 * q^5 + 4 * q^7 + 10 * q^9 + 2 * q^11 - 2 * q^13 - 4 * q^15 + 4 * q^17 + 6 * q^19 - 4 * q^21 + 4 * q^23 + 12 * q^25 + 14 * q^27 - 6 * q^29 + 6 * q^31 - 12 * q^35 - 6 * q^37 + 14 * q^39 + 18 * q^41 + 22 * q^43 - 22 * q^45 + 14 * q^47 + 8 * q^49 + 8 * q^51 - 10 * q^53 + 20 * q^55 - 4 * q^57 - 6 * q^61 + 36 * q^63 - 4 * q^65 + 6 * q^67 + 2 * q^69 - 6 * q^71 - 6 * q^73 + 50 * q^75 - 4 * q^77 + 16 * q^79 + 2 * q^83 + 14 * q^87 + 20 * q^89 + 24 * q^91 + 38 * q^93 + 8 * q^95 - 4 * q^97 - 2 * q^99

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
$3$	−0.901180	−0.520297	−0.260148	−	0.965569i	$-0.583771\pi$
$3$	−0.901180	−0.520297	−0.260148	+	0.965569i	$0.583771\pi$
$4$	0	0
$5$	−2.78819	−1.24692	−0.623459	−	0.781856i	$-0.714273\pi$
$5$	−2.78819	−1.24692	−0.623459	+	0.781856i	$0.714273\pi$
$6$	0	0
$7$	−2.28669	−0.864289	−0.432145	−	0.901804i	$-0.642243\pi$
$7$	−2.28669	−0.864289	−0.432145	+	0.901804i	$0.642243\pi$
$8$	0	0
$9$	−2.18787	−0.729291
$10$	0	0
$11$	−0.788195	−0.237650	−0.118825	−	0.992915i	$-0.537913\pi$
$11$	−0.788195	−0.237650	−0.118825	+	0.992915i	$0.537913\pi$
$12$	0	0
$13$	−5.18787	−1.43886	−0.719429	−	0.694566i	$-0.755596\pi$
$13$	−5.18787	−1.43886	−0.719429	+	0.694566i	$0.755596\pi$
$14$	0	0
$15$	2.51267	0.648767
$16$	0	0
$17$	4.28669	1.03968	0.519838	−	0.854265i	$-0.325992\pi$
$17$	4.28669	1.03968	0.519838	+	0.854265i	$0.325992\pi$
$18$	0	0
$19$	−3.07489	−0.705428	−0.352714	−	0.935731i	$-0.614741\pi$
$19$	−3.07489	−0.705428	−0.352714	+	0.935731i	$0.614741\pi$
$20$	0	0
$21$	2.06072	0.449687
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	2.77403	0.554806
$26$	0	0
$27$	4.67521	0.899744
$28$	0	0
$29$	−3.61149	−0.670636	−0.335318	−	0.942105i	$-0.608844\pi$
$29$	−3.61149	−0.670636	−0.335318	+	0.942105i	$0.608844\pi$
$30$	0	0
$31$	9.24860	1.66110	0.830549	−	0.556946i	$-0.188027\pi$
$31$	9.24860	1.66110	0.830549	+	0.556946i	$0.188027\pi$
$32$	0	0
$33$	0.710305	0.123648
$34$	0	0
$35$	6.37575	1.07770
$36$	0	0
$37$	−2.78819	−0.458376	−0.229188	−	0.973382i	$-0.573607\pi$
$37$	−2.78819	−0.458376	−0.229188	+	0.973382i	$0.573607\pi$
$38$	0	0
$39$	4.67521	0.748633
$40$	0	0
$41$	−2.76426	−0.431705	−0.215853	−	0.976426i	$-0.569253\pi$
$41$	−2.76426	−0.431705	−0.215853	+	0.976426i	$0.569253\pi$
$42$	0	0
$43$	12.8772	1.96376	0.981881	−	0.189498i	$-0.0606862\pi$
$43$	12.8772	1.96376	0.981881	+	0.189498i	$0.0606862\pi$
$44$	0	0
$45$	6.10022	0.909367
$46$	0	0
$47$	−7.05096	−1.02849	−0.514244	−	0.857644i	$-0.671927\pi$
$47$	−7.05096	−1.02849	−0.514244	+	0.857644i	$0.671927\pi$
$48$	0	0
$49$	−1.77103	−0.253004
$50$	0	0
$51$	−3.86308	−0.540940
$52$	0	0
$53$	−0.727471	−0.0999258	−0.0499629	−	0.998751i	$-0.515910\pi$
$53$	−0.727471	−0.0999258	−0.0499629	+	0.998751i	$0.515910\pi$
$54$	0	0
$55$	2.19764	0.296330
$56$	0	0
$57$	2.77103	0.367032
$58$	0	0
$59$	12.1781	1.58545	0.792727	−	0.609576i	$-0.208660\pi$
$59$	12.1781	1.58545	0.792727	+	0.609576i	$0.208660\pi$
$60$	0	0
$61$	−3.27253	−0.419004	−0.209502	−	0.977808i	$-0.567184\pi$
$61$	−3.27253	−0.419004	−0.209502	+	0.977808i	$0.567184\pi$
$62$	0	0
$63$	5.00300	0.630319
$64$	0	0
$65$	14.4648	1.79414
$66$	0	0
$67$	−3.78519	−0.462435	−0.231218	−	0.972902i	$-0.574271\pi$
$67$	−3.78519	−0.462435	−0.231218	+	0.972902i	$0.574271\pi$
$68$	0	0
$69$	−0.901180	−0.108489
$70$	0	0
$71$	3.89818	0.462629	0.231314	−	0.972879i	$-0.425697\pi$
$71$	3.89818	0.462629	0.231314	+	0.972879i	$0.425697\pi$
$72$	0	0
$73$	8.56662	1.00265	0.501324	−	0.865260i	$-0.332847\pi$
$73$	8.56662	1.00265	0.501324	+	0.865260i	$0.332847\pi$
$74$	0	0
$75$	−2.49990	−0.288664
$76$	0	0
$77$	1.80236	0.205398
$78$	0	0
$79$	−7.95214	−0.894685	−0.447343	−	0.894363i	$-0.647630\pi$
$79$	−7.95214	−0.894685	−0.447343	+	0.894363i	$0.647630\pi$
$80$	0	0
$81$	2.35042	0.261158
$82$	0	0
$83$	−1.01417	−0.111319	−0.0556596	−	0.998450i	$-0.517726\pi$
$83$	−1.01417	−0.111319	−0.0556596	+	0.998450i	$0.517726\pi$
$84$	0	0
$85$	−11.9521	−1.29639
$86$	0	0
$87$	3.25460	0.348930
$88$	0	0
$89$	−4.37575	−0.463828	−0.231914	−	0.972736i	$-0.574499\pi$
$89$	−4.37575	−0.463828	−0.231914	+	0.972736i	$0.574499\pi$
$90$	0	0
$91$	11.8631	1.24359
$92$	0	0
$93$	−8.33465	−0.864263
$94$	0	0
$95$	8.57339	0.879611
$96$	0	0
$97$	−17.4335	−1.77010	−0.885050	−	0.465495i	$-0.845876\pi$
$97$	−17.4335	−1.77010	−0.885050	+	0.465495i	$0.845876\pi$
$98$	0	0
$99$	1.72447	0.173316
$100$	0	0
$101$	8.37575	0.833418	0.416709	−	0.909040i	$-0.363183\pi$
$101$	8.37575	0.833418	0.416709	+	0.909040i	$0.363183\pi$
$102$	0	0
$103$	20.0412	1.97472	0.987359	−	0.158502i	$-0.0506664\pi$
$103$	20.0412	1.97472	0.987359	+	0.158502i	$0.0506664\pi$
$104$	0	0
$105$	−5.74570	−0.560723
$106$	0	0
$107$	−6.22767	−0.602051	−0.301026	−	0.953616i	$-0.597329\pi$
$107$	−6.22767	−0.602051	−0.301026	+	0.953616i	$0.597329\pi$
$108$	0	0
$109$	14.2560	1.36548	0.682739	−	0.730663i	$-0.260789\pi$
$109$	14.2560	1.36548	0.682739	+	0.730663i	$0.260789\pi$
$110$	0	0
$111$	2.51267	0.238492
$112$	0	0
$113$	4.89442	0.460428	0.230214	−	0.973140i	$-0.426057\pi$
$113$	4.89442	0.460428	0.230214	+	0.973140i	$0.426057\pi$
$114$	0	0
$115$	−2.78819	−0.260000
$116$	0	0
$117$	11.3504	1.04935
$118$	0	0
$119$	−9.80236	−0.898581
$120$	0	0
$121$	−10.3787	−0.943523
$122$	0	0
$123$	2.49110	0.224615
$124$	0	0
$125$	6.20644	0.555121
$126$	0	0
$127$	7.32479	0.649970	0.324985	−	0.945719i	$-0.394641\pi$
$127$	7.32479	0.649970	0.324985	+	0.945719i	$0.394641\pi$
$128$	0	0
$129$	−11.6047	−1.02174
$130$	0	0
$131$	−9.12715	−0.797443	−0.398721	−	0.917072i	$-0.630546\pi$
$131$	−9.12715	−0.797443	−0.398721	+	0.917072i	$0.630546\pi$
$132$	0	0
$133$	7.03133	0.609694
$134$	0	0
$135$	−13.0354	−1.12191
$136$	0	0
$137$	−9.37875	−0.801281	−0.400640	−	0.916235i	$-0.631212\pi$
$137$	−9.37875	−0.801281	−0.400640	+	0.916235i	$0.631212\pi$
$138$	0	0
$139$	12.2799	1.04157	0.520785	−	0.853688i	$-0.325639\pi$
$139$	12.2799	1.04157	0.520785	+	0.853688i	$0.325639\pi$
$140$	0	0
$141$	6.35418	0.535119
$142$	0	0
$143$	4.08905	0.341944
$144$	0	0
$145$	10.0695	0.836228
$146$	0	0
$147$	1.59602	0.131637
$148$	0	0
$149$	−9.42231	−0.771905	−0.385953	−	0.922519i	$-0.626127\pi$
$149$	−9.42231	−0.771905	−0.385953	+	0.922519i	$0.626127\pi$
$150$	0	0
$151$	−5.64388	−0.459292	−0.229646	−	0.973274i	$-0.573757\pi$
$151$	−5.64388	−0.459292	−0.229646	+	0.973274i	$0.573757\pi$
$152$	0	0
$153$	−9.37875	−0.758227
$154$	0	0
$155$	−25.7869	−2.07125
$156$	0	0
$157$	21.6483	1.72772	0.863860	−	0.503731i	$-0.168040\pi$
$157$	21.6483	1.72772	0.863860	+	0.503731i	$0.168040\pi$
$158$	0	0
$159$	0.655583	0.0519911
$160$	0	0
$161$	−2.28669	−0.180217
$162$	0	0
$163$	10.0480	0.787017	0.393508	−	0.919321i	$-0.371261\pi$
$163$	10.0480	0.787017	0.393508	+	0.919321i	$0.371261\pi$
$164$	0	0
$165$	−1.98047	−0.154179
$166$	0	0
$167$	−23.3309	−1.80540	−0.902699	−	0.430272i	$-0.858418\pi$
$167$	−23.3309	−1.80540	−0.902699	+	0.430272i	$0.858418\pi$
$168$	0	0
$169$	13.9140	1.07031
$170$	0	0
$171$	6.72747	0.514463
$172$	0	0
$173$	13.1049	0.996348	0.498174	−	0.867077i	$-0.334004\pi$
$173$	13.1049	0.996348	0.498174	+	0.867077i	$0.334004\pi$
$174$	0	0
$175$	−6.34336	−0.479513
$176$	0	0
$177$	−10.9747	−0.824907
$178$	0	0
$179$	−18.0480	−1.34897	−0.674484	−	0.738290i	$-0.735634\pi$
$179$	−18.0480	−1.34897	−0.674484	+	0.738290i	$0.735634\pi$
$180$	0	0
$181$	6.78819	0.504563	0.252281	−	0.967654i	$-0.418819\pi$
$181$	6.78819	0.504563	0.252281	+	0.967654i	$0.418819\pi$
$182$	0	0
$183$	2.94914	0.218007
$184$	0	0
$185$	7.77403	0.571558
$186$	0	0
$187$	−3.37875	−0.247079
$188$	0	0
$189$	−10.6908	−0.777639
$190$	0	0
$191$	6.78989	0.491299	0.245650	−	0.969359i	$-0.420999\pi$
$191$	6.78989	0.491299	0.245650	+	0.969359i	$0.420999\pi$
$192$	0	0
$193$	−1.96490	−0.141437	−0.0707184	−	0.997496i	$-0.522529\pi$
$193$	−1.96490	−0.141437	−0.0707184	+	0.997496i	$0.522529\pi$
$194$	0	0
$195$	−13.0354	−0.933484
$196$	0	0
$197$	−16.9484	−1.20752	−0.603761	−	0.797166i	$-0.706332\pi$
$197$	−16.9484	−1.20752	−0.603761	+	0.797166i	$0.706332\pi$
$198$	0	0
$199$	−3.12039	−0.221198	−0.110599	−	0.993865i	$-0.535277\pi$
$199$	−3.12039	−0.221198	−0.110599	+	0.993865i	$0.535277\pi$
$200$	0	0
$201$	3.41114	0.240603
$202$	0	0
$203$	8.25836	0.579623
$204$	0	0
$205$	7.70730	0.538302
$206$	0	0
$207$	−2.18787	−0.152068
$208$	0	0
$209$	2.42361	0.167645
$210$	0	0
$211$	2.57939	0.177572	0.0887862	−	0.996051i	$-0.471701\pi$
$211$	2.57939	0.177572	0.0887862	+	0.996051i	$0.471701\pi$
$212$	0	0
$213$	−3.51296	−0.240704
$214$	0	0
$215$	−35.9043	−2.44865
$216$	0	0
$217$	−21.1487	−1.43567
$218$	0	0
$219$	−7.72007	−0.521674
$220$	0	0
$221$	−22.2388	−1.49595
$222$	0	0
$223$	22.3757	1.49839	0.749195	−	0.662349i	$-0.230440\pi$
$223$	22.3757	1.49839	0.749195	+	0.662349i	$0.230440\pi$
$224$	0	0
$225$	−6.06923	−0.404615
$226$	0	0
$227$	11.1983	0.743256	0.371628	−	0.928382i	$-0.378800\pi$
$227$	11.1983	0.743256	0.371628	+	0.928382i	$0.378800\pi$
$228$	0	0
$229$	6.25600	0.413408	0.206704	−	0.978404i	$-0.433726\pi$
$229$	6.25600	0.413408	0.206704	+	0.978404i	$0.433726\pi$
$230$	0	0
$231$	−1.62425	−0.106868
$232$	0	0
$233$	16.3973	1.07422	0.537112	−	0.843511i	$-0.319515\pi$
$233$	16.3973	1.07422	0.537112	+	0.843511i	$0.319515\pi$
$234$	0	0
$235$	19.6594	1.28244
$236$	0	0
$237$	7.16631	0.465502
$238$	0	0
$239$	8.86685	0.573549	0.286774	−	0.957998i	$-0.407417\pi$
$239$	8.86685	0.573549	0.286774	+	0.957998i	$0.407417\pi$
$240$	0	0
$241$	−20.9680	−1.35067	−0.675334	−	0.737512i	$-0.736000\pi$
$241$	−20.9680	−1.35067	−0.675334	+	0.737512i	$0.736000\pi$
$242$	0	0
$243$	−16.1438	−1.03562
$244$	0	0
$245$	4.93797	0.315475
$246$	0	0
$247$	15.9521	1.01501
$248$	0	0
$249$	0.913946	0.0579190
$250$	0	0
$251$	−22.5562	−1.42374	−0.711868	−	0.702313i	$-0.752151\pi$
$251$	−22.5562	−1.42374	−0.711868	+	0.702313i	$0.752151\pi$
$252$	0	0
$253$	−0.788195	−0.0495534
$254$	0	0
$255$	10.7710	0.674508
$256$	0	0
$257$	19.7896	1.23444	0.617220	−	0.786790i	$-0.288259\pi$
$257$	19.7896	1.23444	0.617220	+	0.786790i	$0.288259\pi$
$258$	0	0
$259$	6.37575	0.396170
$260$	0	0
$261$	7.90148	0.489089
$262$	0	0
$263$	−23.3444	−1.43948	−0.719739	−	0.694245i	$-0.755739\pi$
$263$	−23.3444	−1.43948	−0.719739	+	0.694245i	$0.755739\pi$
$264$	0	0
$265$	2.02833	0.124599
$266$	0	0
$267$	3.94334	0.241328
$268$	0	0
$269$	−28.1713	−1.71764	−0.858819	−	0.512280i	$-0.828801\pi$
$269$	−28.1713	−1.71764	−0.858819	+	0.512280i	$0.828801\pi$
$270$	0	0
$271$	7.60472	0.461954	0.230977	−	0.972959i	$-0.425808\pi$
$271$	7.60472	0.461954	0.230977	+	0.972959i	$0.425808\pi$
$272$	0	0
$273$	−10.6908	−0.647035
$274$	0	0
$275$	−2.18647	−0.131849
$276$	0	0
$277$	26.1087	1.56872	0.784359	−	0.620307i	$-0.212992\pi$
$277$	26.1087	1.56872	0.784359	+	0.620307i	$0.212992\pi$
$278$	0	0
$279$	−20.2348	−1.21142
$280$	0	0
$281$	14.8318	0.884788	0.442394	−	0.896821i	$-0.354129\pi$
$281$	14.8318	0.884788	0.442394	+	0.896821i	$0.354129\pi$
$282$	0	0
$283$	13.7912	0.819801	0.409901	−	0.912130i	$-0.365563\pi$
$283$	13.7912	0.819801	0.409901	+	0.912130i	$0.365563\pi$
$284$	0	0
$285$	−7.72617	−0.457659
$286$	0	0
$287$	6.32103	0.373118
$288$	0	0
$289$	1.37575	0.0809264
$290$	0	0
$291$	15.7107	0.920977
$292$	0	0
$293$	−4.28906	−0.250570	−0.125285	−	0.992121i	$-0.539984\pi$
$293$	−4.28906	−0.250570	−0.125285	+	0.992121i	$0.539984\pi$
$294$	0	0
$295$	−33.9549	−1.97693
$296$	0	0
$297$	−3.68497	−0.213824
$298$	0	0
$299$	−5.18787	−0.300023
$300$	0	0
$301$	−29.4463	−1.69726
$302$	0	0
$303$	−7.54806	−0.433625
$304$	0	0
$305$	9.12445	0.522464
$306$	0	0
$307$	−0.729167	−0.0416158	−0.0208079	−	0.999783i	$-0.506624\pi$
$307$	−0.729167	−0.0416158	−0.0208079	+	0.999783i	$0.506624\pi$
$308$	0	0
$309$	−18.0607	−1.02744
$310$	0	0
$311$	21.0314	1.19258	0.596291	−	0.802768i	$-0.296640\pi$
$311$	21.0314	1.19258	0.596291	+	0.802768i	$0.296640\pi$
$312$	0	0
$313$	−29.2264	−1.65197	−0.825986	−	0.563691i	$-0.809381\pi$
$313$	−29.2264	−1.65197	−0.825986	+	0.563691i	$0.809381\pi$
$314$	0	0
$315$	−13.9493	−0.785956
$316$	0	0
$317$	16.5315	0.928503	0.464252	−	0.885703i	$-0.346323\pi$
$317$	16.5315	0.928503	0.464252	+	0.885703i	$0.346323\pi$
$318$	0	0
$319$	2.84655	0.159376
$320$	0	0
$321$	5.61225	0.313245
$322$	0	0
$323$	−13.1811	−0.733417
$324$	0	0
$325$	−14.3913	−0.798286
$326$	0	0
$327$	−12.8472	−0.710453
$328$	0	0
$329$	16.1234	0.888911
$330$	0	0
$331$	−13.6303	−0.749192	−0.374596	−	0.927188i	$-0.622219\pi$
$331$	−13.6303	−0.749192	−0.374596	+	0.927188i	$0.622219\pi$
$332$	0	0
$333$	6.10022	0.334290
$334$	0	0
$335$	10.5539	0.576619
$336$	0	0
$337$	7.92381	0.431637	0.215819	−	0.976433i	$-0.430758\pi$
$337$	7.92381	0.431637	0.215819	+	0.976433i	$0.430758\pi$
$338$	0	0
$339$	−4.41075	−0.239559
$340$	0	0
$341$	−7.28969	−0.394759
$342$	0	0
$343$	20.0567	1.08296
$344$	0	0
$345$	2.51267	0.135277
$346$	0	0
$347$	−12.6779	−0.680586	−0.340293	−	0.940319i	$-0.610526\pi$
$347$	−12.6779	−0.680586	−0.340293	+	0.940319i	$0.610526\pi$
$348$	0	0
$349$	19.4591	1.04162	0.520811	−	0.853672i	$-0.325630\pi$
$349$	19.4591	1.04162	0.520811	+	0.853672i	$0.325630\pi$
$350$	0	0
$351$	−24.2544	−1.29460
$352$	0	0
$353$	−6.33465	−0.337160	−0.168580	−	0.985688i	$-0.553918\pi$
$353$	−6.33465	−0.337160	−0.168580	+	0.985688i	$0.553918\pi$
$354$	0	0
$355$	−10.8689	−0.576860
$356$	0	0
$357$	8.83369	0.467529
$358$	0	0
$359$	−17.9581	−0.947794	−0.473897	−	0.880580i	$-0.657153\pi$
$359$	−17.9581	−0.947794	−0.473897	+	0.880580i	$0.657153\pi$
$360$	0	0
$361$	−9.54506	−0.502371
$362$	0	0
$363$	9.35312	0.490912
$364$	0	0
$365$	−23.8854	−1.25022
$366$	0	0
$367$	−9.83475	−0.513370	−0.256685	−	0.966495i	$-0.582630\pi$
$367$	−9.83475	−0.513370	−0.256685	+	0.966495i	$0.582630\pi$
$368$	0	0
$369$	6.04786	0.314839
$370$	0	0
$371$	1.66350	0.0863648
$372$	0	0
$373$	17.6895	0.915926	0.457963	−	0.888971i	$-0.348579\pi$
$373$	17.6895	0.915926	0.457963	+	0.888971i	$0.348579\pi$
$374$	0	0
$375$	−5.59312	−0.288827
$376$	0	0
$377$	18.7359	0.964950
$378$	0	0
$379$	−6.57169	−0.337565	−0.168783	−	0.985653i	$-0.553984\pi$
$379$	−6.57169	−0.337565	−0.168783	+	0.985653i	$0.553984\pi$
$380$	0	0
$381$	−6.60096	−0.338177
$382$	0	0
$383$	2.00600	0.102502	0.0512509	−	0.998686i	$-0.483679\pi$
$383$	2.00600	0.102502	0.0512509	+	0.998686i	$0.483679\pi$
$384$	0	0
$385$	−5.02533	−0.256115
$386$	0	0
$387$	−28.1738	−1.43215
$388$	0	0
$389$	−29.2530	−1.48319	−0.741593	−	0.670850i	$-0.765929\pi$
$389$	−29.2530	−1.48319	−0.741593	+	0.670850i	$0.765929\pi$
$390$	0	0
$391$	4.28669	0.216787
$392$	0	0
$393$	8.22521	0.414907
$394$	0	0
$395$	22.1721	1.11560
$396$	0	0
$397$	14.1430	0.709817	0.354909	−	0.934901i	$-0.384512\pi$
$397$	14.1430	0.709817	0.354909	+	0.934901i	$0.384512\pi$
$398$	0	0
$399$	−6.33650	−0.317222
$400$	0	0
$401$	−14.8406	−0.741102	−0.370551	−	0.928812i	$-0.620831\pi$
$401$	−14.8406	−0.741102	−0.370551	+	0.928812i	$0.620831\pi$
$402$	0	0
$403$	−47.9806	−2.39008
$404$	0	0
$405$	−6.55342	−0.325642
$406$	0	0
$407$	2.19764	0.108933
$408$	0	0
$409$	29.5576	1.46153	0.730765	−	0.682629i	$-0.239163\pi$
$409$	29.5576	1.46153	0.730765	+	0.682629i	$0.239163\pi$
$410$	0	0
$411$	8.45194	0.416904
$412$	0	0
$413$	−27.8476	−1.37029
$414$	0	0
$415$	2.82769	0.138806
$416$	0	0
$417$	−11.0664	−0.541925
$418$	0	0
$419$	0.733472	0.0358324	0.0179162	−	0.999839i	$-0.494297\pi$
$419$	0.733472	0.0358324	0.0179162	+	0.999839i	$0.494297\pi$
$420$	0	0
$421$	−29.4635	−1.43596	−0.717982	−	0.696062i	$-0.754934\pi$
$421$	−29.4635	−1.43596	−0.717982	+	0.696062i	$0.754934\pi$
$422$	0	0
$423$	15.4266	0.750068
$424$	0	0
$425$	11.8914	0.576818
$426$	0	0
$427$	7.48327	0.362141
$428$	0	0
$429$	−3.68497	−0.177912
$430$	0	0
$431$	16.6077	0.799966	0.399983	−	0.916523i	$-0.369016\pi$
$431$	16.6077	0.799966	0.399983	+	0.916523i	$0.369016\pi$
$432$	0	0
$433$	33.4199	1.60606	0.803030	−	0.595939i	$-0.203220\pi$
$433$	33.4199	1.60606	0.803030	+	0.595939i	$0.203220\pi$
$434$	0	0
$435$	−9.07445	−0.435087
$436$	0	0
$437$	−3.07489	−0.147092
$438$	0	0
$439$	8.67521	0.414045	0.207023	−	0.978336i	$-0.433623\pi$
$439$	8.67521	0.414045	0.207023	+	0.978336i	$0.433623\pi$
$440$	0	0
$441$	3.87479	0.184514
$442$	0	0
$443$	−14.6900	−0.697943	−0.348972	−	0.937133i	$-0.613469\pi$
$443$	−14.6900	−0.697943	−0.348972	+	0.937133i	$0.613469\pi$
$444$	0	0
$445$	12.2004	0.578356
$446$	0	0
$447$	8.49120	0.401620
$448$	0	0
$449$	37.2966	1.76013	0.880067	−	0.474850i	$-0.157498\pi$
$449$	37.2966	1.76013	0.880067	+	0.474850i	$0.157498\pi$
$450$	0	0
$451$	2.17878	0.102595
$452$	0	0
$453$	5.08615	0.238968
$454$	0	0
$455$	−33.0766	−1.55065
$456$	0	0
$457$	16.7644	0.784204	0.392102	−	0.919922i	$-0.371748\pi$
$457$	16.7644	0.784204	0.392102	+	0.919922i	$0.371748\pi$
$458$	0	0
$459$	20.0412	0.935443
$460$	0	0
$461$	7.15954	0.333453	0.166727	−	0.986003i	$-0.446680\pi$
$461$	7.15954	0.333453	0.166727	+	0.986003i	$0.446680\pi$
$462$	0	0
$463$	−16.6813	−0.775246	−0.387623	−	0.921818i	$-0.626704\pi$
$463$	−16.6813	−0.775246	−0.387623	+	0.921818i	$0.626704\pi$
$464$	0	0
$465$	23.2386	1.07767
$466$	0	0
$467$	31.2187	1.44463	0.722314	−	0.691565i	$-0.243079\pi$
$467$	31.2187	1.44463	0.722314	+	0.691565i	$0.243079\pi$
$468$	0	0
$469$	8.65558	0.399678
$470$	0	0
$471$	−19.5090	−0.898927
$472$	0	0
$473$	−10.1498	−0.466687
$474$	0	0
$475$	−8.52983	−0.391376
$476$	0	0
$477$	1.59162	0.0728751
$478$	0	0
$479$	−19.0725	−0.871446	−0.435723	−	0.900081i	$-0.643507\pi$
$479$	−19.0725	−0.871446	−0.435723	+	0.900081i	$0.643507\pi$
$480$	0	0
$481$	14.4648	0.659538
$482$	0	0
$483$	2.06072	0.0937662
$484$	0	0
$485$	48.6079	2.20717
$486$	0	0
$487$	21.4267	0.970937	0.485468	−	0.874254i	$-0.338649\pi$
$487$	21.4267	0.970937	0.485468	+	0.874254i	$0.338649\pi$
$488$	0	0
$489$	−9.05502	−0.409482
$490$	0	0
$491$	38.8250	1.75215	0.876074	−	0.482177i	$-0.160154\pi$
$491$	38.8250	1.75215	0.876074	+	0.482177i	$0.160154\pi$
$492$	0	0
$493$	−15.4813	−0.697244
$494$	0	0
$495$	−4.80816	−0.216111
$496$	0	0
$497$	−8.91395	−0.399845
$498$	0	0
$499$	−24.6975	−1.10561	−0.552807	−	0.833309i	$-0.686443\pi$
$499$	−24.6975	−1.10561	−0.552807	+	0.833309i	$0.686443\pi$
$500$	0	0
$501$	21.0253	0.939343
$502$	0	0
$503$	−18.7934	−0.837954	−0.418977	−	0.907997i	$-0.637611\pi$
$503$	−18.7934	−0.837954	−0.418977	+	0.907997i	$0.637611\pi$
$504$	0	0
$505$	−23.3532	−1.03920
$506$	0	0
$507$	−12.5391	−0.556879
$508$	0	0
$509$	−27.3377	−1.21172	−0.605860	−	0.795571i	$-0.707171\pi$
$509$	−27.3377	−1.21172	−0.605860	+	0.795571i	$0.707171\pi$
$510$	0	0
$511$	−19.5893	−0.866577
$512$	0	0
$513$	−14.3757	−0.634705
$514$	0	0
$515$	−55.8787	−2.46231
$516$	0	0
$517$	5.55753	0.244420
$518$	0	0
$519$	−11.8099	−0.518397
$520$	0	0
$521$	4.92681	0.215847	0.107924	−	0.994159i	$-0.465580\pi$
$521$	4.92681	0.215847	0.107924	+	0.994159i	$0.465580\pi$
$522$	0	0
$523$	17.1699	0.750789	0.375395	−	0.926865i	$-0.377507\pi$
$523$	17.1699	0.750789	0.375395	+	0.926865i	$0.377507\pi$
$524$	0	0
$525$	5.71651	0.249489
$526$	0	0
$527$	39.6459	1.72700
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−26.6442	−1.15626
$532$	0	0
$533$	14.3407	0.621163
$534$	0	0
$535$	17.3639	0.750709
$536$	0	0
$537$	16.2645	0.701863
$538$	0	0
$539$	1.39592	0.0601263
$540$	0	0
$541$	16.7583	0.720494	0.360247	−	0.932857i	$-0.382692\pi$
$541$	16.7583	0.720494	0.360247	+	0.932857i	$0.382692\pi$
$542$	0	0
$543$	−6.11739	−0.262522
$544$	0	0
$545$	−39.7485	−1.70264
$546$	0	0
$547$	29.8786	1.27752	0.638759	−	0.769407i	$-0.279448\pi$
$547$	29.8786	1.27752	0.638759	+	0.769407i	$0.279448\pi$
$548$	0	0
$549$	7.15988	0.305576
$550$	0	0
$551$	11.1049	0.473085
$552$	0	0
$553$	18.1841	0.773267
$554$	0	0
$555$	−7.00580	−0.297380
$556$	0	0
$557$	−1.19634	−0.0506904	−0.0253452	−	0.999679i	$-0.508068\pi$
$557$	−1.19634	−0.0506904	−0.0253452	+	0.999679i	$0.508068\pi$
$558$	0	0
$559$	−66.8056	−2.82557
$560$	0	0
$561$	3.04486	0.128554
$562$	0	0
$563$	35.9910	1.51684	0.758419	−	0.651767i	$-0.225972\pi$
$563$	35.9910	1.51684	0.758419	+	0.651767i	$0.225972\pi$
$564$	0	0
$565$	−13.6466	−0.574116
$566$	0	0
$567$	−5.37469	−0.225716
$568$	0	0
$569$	−44.6942	−1.87368	−0.936838	−	0.349762i	$-0.886262\pi$
$569$	−44.6942	−1.87368	−0.936838	+	0.349762i	$0.886262\pi$
$570$	0	0
$571$	4.34978	0.182033	0.0910164	−	0.995849i	$-0.470988\pi$
$571$	4.34978	0.182033	0.0910164	+	0.995849i	$0.470988\pi$
$572$	0	0
$573$	−6.11891	−0.255621
$574$	0	0
$575$	2.77403	0.115685
$576$	0	0
$577$	−37.0186	−1.54110	−0.770552	−	0.637378i	$-0.780019\pi$
$577$	−37.0186	−1.54110	−0.770552	+	0.637378i	$0.780019\pi$
$578$	0	0
$579$	1.77073	0.0735891
$580$	0	0
$581$	2.31909	0.0962119
$582$	0	0
$583$	0.573389	0.0237473
$584$	0	0
$585$	−31.6472	−1.30845
$586$	0	0
$587$	16.5795	0.684309	0.342154	−	0.939644i	$-0.388843\pi$
$587$	16.5795	0.684309	0.342154	+	0.939644i	$0.388843\pi$
$588$	0	0
$589$	−28.4384	−1.17178
$590$	0	0
$591$	15.2735	0.628269
$592$	0	0
$593$	40.0028	1.64272	0.821359	−	0.570412i	$-0.193216\pi$
$593$	40.0028	1.64272	0.821359	+	0.570412i	$0.193216\pi$
$594$	0	0
$595$	27.3309	1.12046
$596$	0	0
$597$	2.81203	0.115089
$598$	0	0
$599$	41.8146	1.70850	0.854248	−	0.519865i	$-0.174018\pi$
$599$	41.8146	1.70850	0.854248	+	0.519865i	$0.174018\pi$
$600$	0	0
$601$	21.8348	0.890662	0.445331	−	0.895366i	$-0.353086\pi$
$601$	21.8348	0.890662	0.445331	+	0.895366i	$0.353086\pi$
$602$	0	0
$603$	8.28153	0.337250
$604$	0	0
$605$	28.9380	1.17650
$606$	0	0
$607$	45.2682	1.83738	0.918690	−	0.394979i	$-0.129248\pi$
$607$	45.2682	1.83738	0.918690	+	0.394979i	$0.129248\pi$
$608$	0	0
$609$	−7.44227	−0.301576
$610$	0	0
$611$	36.5795	1.47985
$612$	0	0
$613$	12.9858	0.524493	0.262246	−	0.965001i	$-0.415537\pi$
$613$	12.9858	0.524493	0.262246	+	0.965001i	$0.415537\pi$
$614$	0	0
$615$	−6.94567	−0.280076
$616$	0	0
$617$	−37.3566	−1.50392	−0.751960	−	0.659208i	$-0.770891\pi$
$617$	−37.3566	−1.50392	−0.751960	+	0.659208i	$0.770891\pi$
$618$	0	0
$619$	2.87125	0.115405	0.0577026	−	0.998334i	$-0.481622\pi$
$619$	2.87125	0.115405	0.0577026	+	0.998334i	$0.481622\pi$
$620$	0	0
$621$	4.67521	0.187610
$622$	0	0
$623$	10.0060	0.400882
$624$	0	0
$625$	−31.1749	−1.24700
$626$	0	0
$627$	−2.18411	−0.0872249
$628$	0	0
$629$	−11.9521	−0.476563
$630$	0	0
$631$	27.0571	1.07712	0.538562	−	0.842586i	$-0.318968\pi$
$631$	27.0571	1.07712	0.538562	+	0.842586i	$0.318968\pi$
$632$	0	0
$633$	−2.32449	−0.0923904
$634$	0	0
$635$	−20.4229	−0.810460
$636$	0	0
$637$	9.18787	0.364037
$638$	0	0
$639$	−8.52873	−0.337391
$640$	0	0
$641$	−13.7740	−0.544041	−0.272021	−	0.962291i	$-0.587692\pi$
$641$	−13.7740	−0.544041	−0.272021	+	0.962291i	$0.587692\pi$
$642$	0	0
$643$	39.3899	1.55339	0.776693	−	0.629879i	$-0.216896\pi$
$643$	39.3899	1.55339	0.776693	+	0.629879i	$0.216896\pi$
$644$	0	0
$645$	32.3562	1.27402
$646$	0	0
$647$	39.8672	1.56734	0.783671	−	0.621176i	$-0.213345\pi$
$647$	39.8672	1.56734	0.783671	+	0.621176i	$0.213345\pi$
$648$	0	0
$649$	−9.59872	−0.376783
$650$	0	0
$651$	19.0588	0.746973
$652$	0	0
$653$	23.1819	0.907177	0.453588	−	0.891211i	$-0.350144\pi$
$653$	23.1819	0.907177	0.453588	+	0.891211i	$0.350144\pi$
$654$	0	0
$655$	25.4483	0.994346
$656$	0	0
$657$	−18.7427	−0.731222
$658$	0	0
$659$	−41.4352	−1.61408	−0.807042	−	0.590493i	$-0.798933\pi$
$659$	−41.4352	−1.61408	−0.807042	+	0.590493i	$0.798933\pi$
$660$	0	0
$661$	−27.6287	−1.07463	−0.537317	−	0.843380i	$-0.680562\pi$
$661$	−27.6287	−1.07463	−0.537317	+	0.843380i	$0.680562\pi$
$662$	0	0
$663$	20.0412	0.778335
$664$	0	0
$665$	−19.6047	−0.760238
$666$	0	0
$667$	−3.61149	−0.139837
$668$	0	0
$669$	−20.1646	−0.779608
$670$	0	0
$671$	2.57939	0.0995762
$672$	0	0
$673$	50.6565	1.95267	0.976333	−	0.216273i	$-0.0693900\pi$
$673$	50.6565	1.95267	0.976333	+	0.216273i	$0.0693900\pi$
$674$	0	0
$675$	12.9692	0.499183
$676$	0	0
$677$	−19.7845	−0.760381	−0.380191	−	0.924908i	$-0.624142\pi$
$677$	−19.7845	−0.760381	−0.380191	+	0.924908i	$0.624142\pi$
$678$	0	0
$679$	39.8650	1.52988
$680$	0	0
$681$	−10.0917	−0.386713
$682$	0	0
$683$	12.7967	0.489650	0.244825	−	0.969567i	$-0.421269\pi$
$683$	12.7967	0.489650	0.244825	+	0.969567i	$0.421269\pi$
$684$	0	0
$685$	26.1498	0.999132
$686$	0	0
$687$	−5.63778	−0.215095
$688$	0	0
$689$	3.77403	0.143779
$690$	0	0
$691$	30.2936	1.15242	0.576211	−	0.817301i	$-0.304531\pi$
$691$	30.2936	1.15242	0.576211	+	0.817301i	$0.304531\pi$
$692$	0	0
$693$	−3.94334	−0.149795
$694$	0	0
$695$	−34.2388	−1.29875
$696$	0	0
$697$	−11.8496	−0.448834
$698$	0	0
$699$	−14.7769	−0.558915
$700$	0	0
$701$	28.3047	1.06905	0.534527	−	0.845151i	$-0.320490\pi$
$701$	28.3047	1.06905	0.534527	+	0.845151i	$0.320490\pi$
$702$	0	0
$703$	8.57339	0.323351
$704$	0	0
$705$	−17.7167	−0.667249
$706$	0	0
$707$	−19.1528	−0.720314
$708$	0	0
$709$	9.40344	0.353154	0.176577	−	0.984287i	$-0.443498\pi$
$709$	9.40344	0.353154	0.176577	+	0.984287i	$0.443498\pi$
$710$	0	0
$711$	17.3983	0.652486
$712$	0	0
$713$	9.24860	0.346363
$714$	0	0
$715$	−11.4011	−0.426376
$716$	0	0
$717$	−7.99063	−0.298415
$718$	0	0
$719$	47.8086	1.78296	0.891479	−	0.453062i	$-0.149668\pi$
$719$	47.8086	1.78296	0.891479	+	0.453062i	$0.149668\pi$
$720$	0	0
$721$	−45.8281	−1.70673
$722$	0	0
$723$	18.8959	0.702748
$724$	0	0
$725$	−10.0184	−0.372073
$726$	0	0
$727$	−2.21650	−0.0822055	−0.0411028	−	0.999155i	$-0.513087\pi$
$727$	−2.21650	−0.0822055	−0.0411028	+	0.999155i	$0.513087\pi$
$728$	0	0
$729$	7.49720	0.277674
$730$	0	0
$731$	55.2008	2.04168
$732$	0	0
$733$	−12.4124	−0.458464	−0.229232	−	0.973372i	$-0.573621\pi$
$733$	−12.4124	−0.458464	−0.229232	+	0.973372i	$0.573621\pi$
$734$	0	0
$735$	−4.45000	−0.164141
$736$	0	0
$737$	2.98347	0.109898
$738$	0	0
$739$	−6.72926	−0.247540	−0.123770	−	0.992311i	$-0.539498\pi$
$739$	−6.72926	−0.247540	−0.123770	+	0.992311i	$0.539498\pi$
$740$	0	0
$741$	−14.3757	−0.528106
$742$	0	0
$743$	0.436472	0.0160126	0.00800631	−	0.999968i	$-0.497451\pi$
$743$	0.436472	0.0160126	0.00800631	+	0.999968i	$0.497451\pi$
$744$	0	0
$745$	26.2712	0.962503
$746$	0	0
$747$	2.21887	0.0811841
$748$	0	0
$749$	14.2408	0.520346
$750$	0	0
$751$	11.6047	0.423462	0.211731	−	0.977328i	$-0.432090\pi$
$751$	11.6047	0.423462	0.211731	+	0.977328i	$0.432090\pi$
$752$	0	0
$753$	20.3272	0.740765
$754$	0	0
$755$	15.7362	0.572700
$756$	0	0
$757$	−30.9049	−1.12326	−0.561629	−	0.827389i	$-0.689825\pi$
$757$	−30.9049	−1.12326	−0.561629	+	0.827389i	$0.689825\pi$
$758$	0	0
$759$	0.710305	0.0257824
$760$	0	0
$761$	38.0042	1.37765	0.688825	−	0.724928i	$-0.258127\pi$
$761$	38.0042	1.37765	0.688825	+	0.724928i	$0.258127\pi$
$762$	0	0
$763$	−32.5991	−1.18017
$764$	0	0
$765$	26.1498	0.945447
$766$	0	0
$767$	−63.1785	−2.28124
$768$	0	0
$769$	28.3946	1.02394	0.511968	−	0.859005i	$-0.328917\pi$
$769$	28.3946	1.02394	0.511968	+	0.859005i	$0.328917\pi$
$770$	0	0
$771$	−17.8340	−0.642275
$772$	0	0
$773$	−4.55816	−0.163946	−0.0819729	−	0.996635i	$-0.526122\pi$
$773$	−4.55816	−0.163946	−0.0819729	+	0.996635i	$0.526122\pi$
$774$	0	0
$775$	25.6559	0.921586
$776$	0	0
$777$	−5.74570	−0.206126
$778$	0	0
$779$	8.49980	0.304537
$780$	0	0
$781$	−3.07252	−0.109944
$782$	0	0
$783$	−16.8844	−0.603401
$784$	0	0
$785$	−60.3596	−2.15433
$786$	0	0
$787$	51.2275	1.82606	0.913031	−	0.407890i	$-0.133735\pi$
$787$	51.2275	1.82606	0.913031	+	0.407890i	$0.133735\pi$
$788$	0	0
$789$	21.0375	0.748956
$790$	0	0
$791$	−11.1920	−0.397943
$792$	0	0
$793$	16.9775	0.602888
$794$	0	0
$795$	−1.82789	−0.0648286
$796$	0	0
$797$	−0.883250	−0.0312863	−0.0156432	−	0.999878i	$-0.504980\pi$
$797$	−0.883250	−0.0312863	−0.0156432	+	0.999878i	$0.504980\pi$
$798$	0	0
$799$	−30.2253	−1.06929
$800$	0	0
$801$	9.57359	0.338266
$802$	0	0
$803$	−6.75217	−0.238279
$804$	0	0
$805$	6.37575	0.224716
$806$	0	0
$807$	25.3875	0.893681
$808$	0	0
$809$	40.0333	1.40749	0.703747	−	0.710451i	$-0.251509\pi$
$809$	40.0333	1.40749	0.703747	+	0.710451i	$0.251509\pi$
$810$	0	0
$811$	−38.1647	−1.34014	−0.670071	−	0.742297i	$-0.733737\pi$
$811$	−38.1647	−1.34014	−0.670071	+	0.742297i	$0.733737\pi$
$812$	0	0
$813$	−6.85322	−0.240353
$814$	0	0
$815$	−28.0157	−0.981346
$816$	0	0
$817$	−39.5961	−1.38529
$818$	0	0
$819$	−25.9549	−0.906939
$820$	0	0
$821$	−34.0959	−1.18996	−0.594978	−	0.803742i	$-0.702839\pi$
$821$	−34.0959	−1.18996	−0.594978	+	0.803742i	$0.702839\pi$
$822$	0	0
$823$	3.14668	0.109686	0.0548432	−	0.998495i	$-0.482534\pi$
$823$	3.14668	0.109686	0.0548432	+	0.998495i	$0.482534\pi$
$824$	0	0
$825$	1.97041	0.0686008
$826$	0	0
$827$	−51.4704	−1.78980	−0.894900	−	0.446267i	$-0.852753\pi$
$827$	−51.4704	−1.78980	−0.894900	+	0.446267i	$0.852753\pi$
$828$	0	0
$829$	28.5195	0.990524	0.495262	−	0.868744i	$-0.335072\pi$
$829$	28.5195	0.990524	0.495262	+	0.868744i	$0.335072\pi$
$830$	0	0
$831$	−23.5286	−0.816199
$832$	0	0
$833$	−7.59186	−0.263042
$834$	0	0
$835$	65.0511	2.25118
$836$	0	0
$837$	43.2391	1.49456
$838$	0	0
$839$	−17.9905	−0.621102	−0.310551	−	0.950557i	$-0.600514\pi$
$839$	−17.9905	−0.621102	−0.310551	+	0.950557i	$0.600514\pi$
$840$	0	0
$841$	−15.9572	−0.550247
$842$	0	0
$843$	−13.3661	−0.460352
$844$	0	0
$845$	−38.7951	−1.33459
$846$	0	0
$847$	23.7330	0.815476
$848$	0	0
$849$	−12.4284	−0.426540
$850$	0	0
$851$	−2.78819	−0.0955781
$852$	0	0
$853$	16.2800	0.557417	0.278709	−	0.960376i	$-0.410094\pi$
$853$	16.2800	0.557417	0.278709	+	0.960376i	$0.410094\pi$
$854$	0	0
$855$	−18.7575	−0.641493
$856$	0	0
$857$	−55.5246	−1.89668	−0.948341	−	0.317251i	$-0.897240\pi$
$857$	−55.5246	−1.89668	−0.948341	+	0.317251i	$0.897240\pi$
$858$	0	0
$859$	23.9718	0.817906	0.408953	−	0.912555i	$-0.365894\pi$
$859$	23.9718	0.817906	0.408953	+	0.912555i	$0.365894\pi$
$860$	0	0
$861$	−5.69638	−0.194132
$862$	0	0
$863$	24.6632	0.839545	0.419773	−	0.907629i	$-0.362110\pi$
$863$	24.6632	0.839545	0.419773	+	0.907629i	$0.362110\pi$
$864$	0	0
$865$	−36.5391	−1.24237
$866$	0	0
$867$	−1.23980	−0.0421057
$868$	0	0
$869$	6.26783	0.212622
$870$	0	0
$871$	19.6371	0.665378
$872$	0	0
$873$	38.1422	1.29092
$874$	0	0
$875$	−14.1922	−0.479785
$876$	0	0
$877$	27.3457	0.923398	0.461699	−	0.887037i	$-0.347240\pi$
$877$	27.3457	0.923398	0.461699	+	0.887037i	$0.347240\pi$
$878$	0	0
$879$	3.86521	0.130370
$880$	0	0
$881$	14.8318	0.499694	0.249847	−	0.968285i	$-0.419620\pi$
$881$	14.8318	0.499694	0.249847	+	0.968285i	$0.419620\pi$
$882$	0	0
$883$	−19.2094	−0.646449	−0.323225	−	0.946322i	$-0.604767\pi$
$883$	−19.2094	−0.646449	−0.323225	+	0.946322i	$0.604767\pi$
$884$	0	0
$885$	30.5995	1.02859
$886$	0	0
$887$	−19.1022	−0.641390	−0.320695	−	0.947183i	$-0.603916\pi$
$887$	−19.1022	−0.641390	−0.320695	+	0.947183i	$0.603916\pi$
$888$	0	0
$889$	−16.7496	−0.561762
$890$	0	0
$891$	−1.85259	−0.0620640
$892$	0	0
$893$	21.6809	0.725524
$894$	0	0
$895$	50.3212	1.68205
$896$	0	0
$897$	4.67521	0.156101
$898$	0	0
$899$	−33.4012	−1.11399
$900$	0	0
$901$	−3.11845	−0.103891
$902$	0	0
$903$	26.5364	0.883078
$904$	0	0
$905$	−18.9268	−0.629148
$906$	0	0
$907$	51.3354	1.70456	0.852282	−	0.523083i	$-0.175218\pi$
$907$	51.3354	1.70456	0.852282	+	0.523083i	$0.175218\pi$
$908$	0	0
$909$	−18.3251	−0.607805
$910$	0	0
$911$	−9.61952	−0.318709	−0.159354	−	0.987221i	$-0.550941\pi$
$911$	−9.61952	−0.318709	−0.159354	+	0.987221i	$0.550941\pi$
$912$	0	0
$913$	0.799360	0.0264549
$914$	0	0
$915$	−8.22277	−0.271836
$916$	0	0
$917$	20.8710	0.689221
$918$	0	0
$919$	−37.9738	−1.25264	−0.626320	−	0.779566i	$-0.715440\pi$
$919$	−37.9738	−1.25264	−0.626320	+	0.779566i	$0.715440\pi$
$920$	0	0
$921$	0.657111	0.0216526
$922$	0	0
$923$	−20.2233	−0.665657
$924$	0	0
$925$	−7.73453	−0.254310
$926$	0	0
$927$	−43.8476	−1.44014
$928$	0	0
$929$	33.4705	1.09813	0.549066	−	0.835779i	$-0.314984\pi$
$929$	33.4705	1.09813	0.549066	+	0.835779i	$0.314984\pi$
$930$	0	0
$931$	5.44572	0.178476
$932$	0	0
$933$	−18.9531	−0.620497
$934$	0	0
$935$	9.42061	0.308087
$936$	0	0
$937$	−32.6908	−1.06796	−0.533981	−	0.845497i	$-0.679304\pi$
$937$	−32.6908	−1.06796	−0.533981	+	0.845497i	$0.679304\pi$
$938$	0	0
$939$	26.3382	0.859515
$940$	0	0
$941$	52.3648	1.70704	0.853521	−	0.521058i	$-0.174462\pi$
$941$	52.3648	1.70704	0.853521	+	0.521058i	$0.174462\pi$
$942$	0	0
$943$	−2.76426	−0.0900168
$944$	0	0
$945$	29.8080	0.969653
$946$	0	0
$947$	−8.69754	−0.282632	−0.141316	−	0.989965i	$-0.545133\pi$
$947$	−8.69754	−0.282632	−0.141316	+	0.989965i	$0.545133\pi$
$948$	0	0
$949$	−44.4426	−1.44267
$950$	0	0
$951$	−14.8979	−0.483097
$952$	0	0
$953$	−12.2455	−0.396671	−0.198335	−	0.980134i	$-0.563554\pi$
$953$	−12.2455	−0.396671	−0.198335	+	0.980134i	$0.563554\pi$
$954$	0	0
$955$	−18.9315	−0.612610
$956$	0	0
$957$	−2.56526	−0.0829230
$958$	0	0
$959$	21.4463	0.692538
$960$	0	0
$961$	54.5366	1.75924
$962$	0	0
$963$	13.6254	0.439071
$964$	0	0
$965$	5.47853	0.176360
$966$	0	0
$967$	3.32479	0.106918	0.0534590	−	0.998570i	$-0.482975\pi$
$967$	3.32479	0.106918	0.0534590	+	0.998570i	$0.482975\pi$
$968$	0	0
$969$	11.8786	0.381594
$970$	0	0
$971$	43.7697	1.40464	0.702319	−	0.711863i	$-0.252148\pi$
$971$	43.7697	1.40464	0.702319	+	0.711863i	$0.252148\pi$
$972$	0	0
$973$	−28.0804	−0.900218
$974$	0	0
$975$	12.9692	0.415346
$976$	0	0
$977$	31.9993	1.02375	0.511875	−	0.859060i	$-0.328951\pi$
$977$	31.9993	1.02375	0.511875	+	0.859060i	$0.328951\pi$
$978$	0	0
$979$	3.44894	0.110229
$980$	0	0
$981$	−31.1903	−0.995831
$982$	0	0
$983$	60.2517	1.92173	0.960865	−	0.277016i	$-0.0893455\pi$
$983$	60.2517	1.92173	0.960865	+	0.277016i	$0.0893455\pi$
$984$	0	0
$985$	47.2554	1.50568
$986$	0	0
$987$	−14.5301	−0.462497
$988$	0	0
$989$	12.8772	0.409473
$990$	0	0
$991$	3.94334	0.125264	0.0626321	−	0.998037i	$-0.480051\pi$
$991$	3.94334	0.125264	0.0626321	+	0.998037i	$0.480051\pi$
$992$	0	0
$993$	12.2834	0.389802
$994$	0	0
$995$	8.70024	0.275816
$996$	0	0
$997$	−33.8444	−1.07186	−0.535932	−	0.844261i	$-0.680040\pi$
$997$	−33.8444	−1.07186	−0.535932	+	0.844261i	$0.680040\pi$
$998$	0	0
$999$	−13.0354	−0.412422

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1472.2.a.z.1.2		4
4.3	odd	2		1472.2.a.y.1.3		4
8.3	odd	2		736.2.a.h.1.2	yes	4
8.5	even	2		736.2.a.g.1.3	✓	4
24.5	odd	2		6624.2.a.bf.1.2		4
24.11	even	2		6624.2.a.be.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
736.2.a.g.1.3	✓	4	8.5	even	2
736.2.a.h.1.2	yes	4	8.3	odd	2
1472.2.a.y.1.3		4	4.3	odd	2
1472.2.a.z.1.2		4	1.1	even	1	trivial
6624.2.a.be.1.2		4	24.11	even	2
6624.2.a.bf.1.2		4	24.5	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:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1472.a (trivial)

\(f(q)\)	\(=\)	\(q-0.901180 q^{3} -2.78819 q^{5} -2.28669 q^{7} -2.18787 q^{9} -0.788195 q^{11} -5.18787 q^{13} +2.51267 q^{15} +4.28669 q^{17} -3.07489 q^{19} +2.06072 q^{21} +1.00000 q^{23} +2.77403 q^{25} +4.67521 q^{27} -3.61149 q^{29} +9.24860 q^{31} +0.710305 q^{33} +6.37575 q^{35} -2.78819 q^{37} +4.67521 q^{39} -2.76426 q^{41} +12.8772 q^{43} +6.10022 q^{45} -7.05096 q^{47} -1.77103 q^{49} -3.86308 q^{51} -0.727471 q^{53} +2.19764 q^{55} +2.77103 q^{57} +12.1781 q^{59} -3.27253 q^{61} +5.00300 q^{63} +14.4648 q^{65} -3.78519 q^{67} -0.901180 q^{69} +3.89818 q^{71} +8.56662 q^{73} -2.49990 q^{75} +1.80236 q^{77} -7.95214 q^{79} +2.35042 q^{81} -1.01417 q^{83} -11.9521 q^{85} +3.25460 q^{87} -4.37575 q^{89} +11.8631 q^{91} -8.33465 q^{93} +8.57339 q^{95} -17.4335 q^{97} +1.72447 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 6 q^{5} + 4 q^{7} + 10 q^{9} + 2 q^{11} - 2 q^{13} - 4 q^{15} + 4 q^{17} + 6 q^{19} - 4 q^{21} + 4 q^{23} + 12 q^{25} + 14 q^{27} - 6 q^{29} + 6 q^{31} - 12 q^{35} - 6 q^{37} + 14 q^{39}+ \cdots - 2 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 - 6 * q^5 + 4 * q^7 + 10 * q^9 + 2 * q^11 - 2 * q^13 - 4 * q^15 + 4 * q^17 + 6 * q^19 - 4 * q^21 + 4 * q^23 + 12 * q^25 + 14 * q^27 - 6 * q^29 + 6 * q^31 - 12 * q^35 - 6 * q^37 + 14 * q^39 + 18 * q^41 + 22 * q^43 - 22 * q^45 + 14 * q^47 + 8 * q^49 + 8 * q^51 - 10 * q^53 + 20 * q^55 - 4 * q^57 - 6 * q^61 + 36 * q^63 - 4 * q^65 + 6 * q^67 + 2 * q^69 - 6 * q^71 - 6 * q^73 + 50 * q^75 - 4 * q^77 + 16 * q^79 + 2 * q^83 + 14 * q^87 + 20 * q^89 + 24 * q^91 + 38 * q^93 + 8 * q^95 - 4 * q^97 - 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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