LMFDB - Embedded newform 2744.2.a.h.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2744,2,Mod(1,2744)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2744, 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("2744.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2744 = 2^{3} \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2744.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:	$21.9109503146$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 29x^{10} + 304x^{8} - 1393x^{6} + 2574x^{4} - 1164x^{2} + 8 $ x^12 - 29x^10 + 304x^8 - 1393x^6 + 2574x^4 - 1164*x^2 + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$3.35261$ of defining polynomial
Character	$\chi$	$=$	2744.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.35261 q^{3} +0.758778 q^{5} +8.24000 q^{9} +O(q^{10})$	$q+3.35261 q^{3} +0.758778 q^{5} +8.24000 q^{9} +5.02897 q^{11} +5.12977 q^{13} +2.54389 q^{15} -3.02627 q^{17} -5.21685 q^{19} +2.86521 q^{23} -4.42426 q^{25} +17.5677 q^{27} -0.478797 q^{29} -7.60413 q^{31} +16.8602 q^{33} -10.5609 q^{37} +17.1981 q^{39} +2.45606 q^{41} +1.38792 q^{43} +6.25233 q^{45} -5.40057 q^{47} -10.1459 q^{51} -2.61208 q^{53} +3.81587 q^{55} -17.4901 q^{57} +6.51749 q^{59} +1.01175 q^{61} +3.89235 q^{65} -1.10294 q^{67} +9.60592 q^{69} -8.66803 q^{71} -9.29821 q^{73} -14.8328 q^{75} -9.57594 q^{79} +34.1776 q^{81} -2.49082 q^{83} -2.29627 q^{85} -1.60522 q^{87} -14.1747 q^{89} -25.4937 q^{93} -3.95843 q^{95} +13.4016 q^{97} +41.4387 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 22 q^{9}+O(q^{10}) $ 12 * q + 22 * q^9	$ 12 q + 22 q^{9} + 18 q^{11} + 28 q^{15} + 20 q^{23} + 14 q^{25} - 18 q^{29} - 18 q^{37} + 36 q^{39} + 10 q^{43} + 48 q^{51} - 38 q^{53} + 12 q^{57} + 8 q^{65} + 42 q^{67} + 56 q^{71} + 56 q^{79} + 52 q^{81} + 8 q^{85} - 48 q^{93} + 84 q^{95} + 90 q^{99}+O(q^{100}) $ 12 * q + 22 * q^9 + 18 * q^11 + 28 * q^15 + 20 * q^23 + 14 * q^25 - 18 * q^29 - 18 * q^37 + 36 * q^39 + 10 * q^43 + 48 * q^51 - 38 * q^53 + 12 * q^57 + 8 * q^65 + 42 * q^67 + 56 * q^71 + 56 * q^79 + 52 * q^81 + 8 * q^85 - 48 * q^93 + 84 * q^95 + 90 * 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
$3$	3.35261	1.93563	0.967815	−	0.251661i	$-0.0809769\pi$
$3$	3.35261	1.93563	0.967815	+	0.251661i	$0.0809769\pi$
$4$	0	0
$5$	0.758778	0.339336	0.169668	−	0.985501i	$-0.445730\pi$
$5$	0.758778	0.339336	0.169668	+	0.985501i	$0.445730\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	8.24000	2.74667
$10$	0	0
$11$	5.02897	1.51629	0.758145	−	0.652086i	$-0.226106\pi$
$11$	5.02897	1.51629	0.758145	+	0.652086i	$0.226106\pi$
$12$	0	0
$13$	5.12977	1.42274	0.711371	−	0.702817i	$-0.248075\pi$
$13$	5.12977	1.42274	0.711371	+	0.702817i	$0.248075\pi$
$14$	0	0
$15$	2.54389	0.656829
$16$	0	0
$17$	−3.02627	−0.733979	−0.366990	−	0.930225i	$-0.619612\pi$
$17$	−3.02627	−0.733979	−0.366990	+	0.930225i	$0.619612\pi$
$18$	0	0
$19$	−5.21685	−1.19683	−0.598413	−	0.801187i	$-0.704202\pi$
$19$	−5.21685	−1.19683	−0.598413	+	0.801187i	$0.704202\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.86521	0.597437	0.298718	−	0.954341i	$-0.403441\pi$
$23$	2.86521	0.597437	0.298718	+	0.954341i	$0.403441\pi$
$24$	0	0
$25$	−4.42426	−0.884851
$26$	0	0
$27$	17.5677	3.38090
$28$	0	0
$29$	−0.478797	−0.0889104	−0.0444552	−	0.999011i	$-0.514155\pi$
$29$	−0.478797	−0.0889104	−0.0444552	+	0.999011i	$0.514155\pi$
$30$	0	0
$31$	−7.60413	−1.36574	−0.682871	−	0.730539i	$-0.739269\pi$
$31$	−7.60413	−1.36574	−0.682871	+	0.730539i	$0.739269\pi$
$32$	0	0
$33$	16.8602	2.93498
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−10.5609	−1.73621	−0.868105	−	0.496381i	$-0.834662\pi$
$37$	−10.5609	−1.73621	−0.868105	+	0.496381i	$0.834662\pi$
$38$	0	0
$39$	17.1981	2.75390
$40$	0	0
$41$	2.45606	0.383573	0.191786	−	0.981437i	$-0.438572\pi$
$41$	2.45606	0.383573	0.191786	+	0.981437i	$0.438572\pi$
$42$	0	0
$43$	1.38792	0.211655	0.105828	−	0.994384i	$-0.466251\pi$
$43$	1.38792	0.211655	0.105828	+	0.994384i	$0.466251\pi$
$44$	0	0
$45$	6.25233	0.932042
$46$	0	0
$47$	−5.40057	−0.787754	−0.393877	−	0.919163i	$-0.628866\pi$
$47$	−5.40057	−0.787754	−0.393877	+	0.919163i	$0.628866\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−10.1459	−1.42071
$52$	0	0
$53$	−2.61208	−0.358797	−0.179399	−	0.983776i	$-0.557415\pi$
$53$	−2.61208	−0.358797	−0.179399	+	0.983776i	$0.557415\pi$
$54$	0	0
$55$	3.81587	0.514532
$56$	0	0
$57$	−17.4901	−2.31661
$58$	0	0
$59$	6.51749	0.848505	0.424252	−	0.905544i	$-0.360537\pi$
$59$	6.51749	0.848505	0.424252	+	0.905544i	$0.360537\pi$
$60$	0	0
$61$	1.01175	0.129541	0.0647707	−	0.997900i	$-0.479368\pi$
$61$	1.01175	0.129541	0.0647707	+	0.997900i	$0.479368\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.89235	0.482787
$66$	0	0
$67$	−1.10294	−0.134745	−0.0673725	−	0.997728i	$-0.521462\pi$
$67$	−1.10294	−0.134745	−0.0673725	+	0.997728i	$0.521462\pi$
$68$	0	0
$69$	9.60592	1.15642
$70$	0	0
$71$	−8.66803	−1.02871	−0.514353	−	0.857579i	$-0.671968\pi$
$71$	−8.66803	−1.02871	−0.514353	+	0.857579i	$0.671968\pi$
$72$	0	0
$73$	−9.29821	−1.08827	−0.544136	−	0.838997i	$-0.683143\pi$
$73$	−9.29821	−1.08827	−0.544136	+	0.838997i	$0.683143\pi$
$74$	0	0
$75$	−14.8328	−1.71275
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−9.57594	−1.07738	−0.538688	−	0.842505i	$-0.681080\pi$
$79$	−9.57594	−1.07738	−0.538688	+	0.842505i	$0.681080\pi$
$80$	0	0
$81$	34.1776	3.79751
$82$	0	0
$83$	−2.49082	−0.273403	−0.136701	−	0.990612i	$-0.543650\pi$
$83$	−2.49082	−0.273403	−0.136701	+	0.990612i	$0.543650\pi$
$84$	0	0
$85$	−2.29627	−0.249065
$86$	0	0
$87$	−1.60522	−0.172098
$88$	0	0
$89$	−14.1747	−1.50251	−0.751256	−	0.660011i	$-0.770552\pi$
$89$	−14.1747	−1.50251	−0.751256	+	0.660011i	$0.770552\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−25.4937	−2.64357
$94$	0	0
$95$	−3.95843	−0.406126
$96$	0	0
$97$	13.4016	1.36073	0.680363	−	0.732875i	$-0.261822\pi$
$97$	13.4016	1.36073	0.680363	+	0.732875i	$0.261822\pi$
$98$	0	0
$99$	41.4387	4.16475
$100$	0	0
$101$	−0.773341	−0.0769503	−0.0384752	−	0.999260i	$-0.512250\pi$
$101$	−0.773341	−0.0769503	−0.0384752	+	0.999260i	$0.512250\pi$
$102$	0	0
$103$	−4.08136	−0.402148	−0.201074	−	0.979576i	$-0.564443\pi$
$103$	−4.08136	−0.402148	−0.201074	+	0.979576i	$0.564443\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	18.1130	1.75105	0.875525	−	0.483173i	$-0.160516\pi$
$107$	18.1130	1.75105	0.875525	+	0.483173i	$0.160516\pi$
$108$	0	0
$109$	15.6702	1.50094	0.750468	−	0.660907i	$-0.229828\pi$
$109$	15.6702	1.50094	0.750468	+	0.660907i	$0.229828\pi$
$110$	0	0
$111$	−35.4067	−3.36066
$112$	0	0
$113$	5.68514	0.534813	0.267406	−	0.963584i	$-0.413833\pi$
$113$	5.68514	0.534813	0.267406	+	0.963584i	$0.413833\pi$
$114$	0	0
$115$	2.17406	0.202732
$116$	0	0
$117$	42.2693	3.90780
$118$	0	0
$119$	0	0
$120$	0	0
$121$	14.2905	1.29914
$122$	0	0
$123$	8.23423	0.742455
$124$	0	0
$125$	−7.15092	−0.639597
$126$	0	0
$127$	−1.55518	−0.138000	−0.0689998	−	0.997617i	$-0.521981\pi$
$127$	−1.55518	−0.138000	−0.0689998	+	0.997617i	$0.521981\pi$
$128$	0	0
$129$	4.65314	0.409687
$130$	0	0
$131$	−3.12645	−0.273159	−0.136580	−	0.990629i	$-0.543611\pi$
$131$	−3.12645	−0.273159	−0.136580	+	0.990629i	$0.543611\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	13.3300	1.14726
$136$	0	0
$137$	7.34049	0.627140	0.313570	−	0.949565i	$-0.398475\pi$
$137$	7.34049	0.627140	0.313570	+	0.949565i	$0.398475\pi$
$138$	0	0
$139$	6.21902	0.527491	0.263745	−	0.964592i	$-0.415042\pi$
$139$	6.21902	0.527491	0.263745	+	0.964592i	$0.415042\pi$
$140$	0	0
$141$	−18.1060	−1.52480
$142$	0	0
$143$	25.7974	2.15729
$144$	0	0
$145$	−0.363301	−0.0301705
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.35097	0.684138	0.342069	−	0.939675i	$-0.388872\pi$
$149$	8.35097	0.684138	0.342069	+	0.939675i	$0.388872\pi$
$150$	0	0
$151$	−9.53097	−0.775619	−0.387810	−	0.921739i	$-0.626768\pi$
$151$	−9.53097	−0.775619	−0.387810	+	0.921739i	$0.626768\pi$
$152$	0	0
$153$	−24.9365	−2.01600
$154$	0	0
$155$	−5.76985	−0.463445
$156$	0	0
$157$	19.6257	1.56630	0.783150	−	0.621833i	$-0.213612\pi$
$157$	19.6257	1.56630	0.783150	+	0.621833i	$0.213612\pi$
$158$	0	0
$159$	−8.75730	−0.694499
$160$	0	0
$161$	0	0
$162$	0	0
$163$	16.3721	1.28236	0.641180	−	0.767390i	$-0.278445\pi$
$163$	16.3721	1.28236	0.641180	+	0.767390i	$0.278445\pi$
$164$	0	0
$165$	12.7931	0.995943
$166$	0	0
$167$	−4.39190	−0.339856	−0.169928	−	0.985457i	$-0.554353\pi$
$167$	−4.39190	−0.339856	−0.169928	+	0.985457i	$0.554353\pi$
$168$	0	0
$169$	13.3145	1.02419
$170$	0	0
$171$	−42.9868	−3.28728
$172$	0	0
$173$	−14.5647	−1.10733	−0.553667	−	0.832738i	$-0.686772\pi$
$173$	−14.5647	−1.10733	−0.553667	+	0.832738i	$0.686772\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	21.8506	1.64239
$178$	0	0
$179$	−3.33406	−0.249199	−0.124600	−	0.992207i	$-0.539765\pi$
$179$	−3.33406	−0.249199	−0.124600	+	0.992207i	$0.539765\pi$
$180$	0	0
$181$	−21.3608	−1.58773	−0.793867	−	0.608092i	$-0.791935\pi$
$181$	−21.3608	−1.58773	−0.793867	+	0.608092i	$0.791935\pi$
$182$	0	0
$183$	3.39201	0.250744
$184$	0	0
$185$	−8.01341	−0.589158
$186$	0	0
$187$	−15.2190	−1.11293
$188$	0	0
$189$	0	0
$190$	0	0
$191$	7.95909	0.575900	0.287950	−	0.957646i	$-0.407026\pi$
$191$	7.95909	0.575900	0.287950	+	0.957646i	$0.407026\pi$
$192$	0	0
$193$	15.2771	1.09967	0.549834	−	0.835274i	$-0.314691\pi$
$193$	15.2771	1.09967	0.549834	+	0.835274i	$0.314691\pi$
$194$	0	0
$195$	13.0495	0.934498
$196$	0	0
$197$	−21.1510	−1.50695	−0.753474	−	0.657478i	$-0.771623\pi$
$197$	−21.1510	−1.50695	−0.753474	+	0.657478i	$0.771623\pi$
$198$	0	0
$199$	8.58030	0.608241	0.304120	−	0.952634i	$-0.401637\pi$
$199$	8.58030	0.608241	0.304120	+	0.952634i	$0.401637\pi$
$200$	0	0
$201$	−3.69772	−0.260817
$202$	0	0
$203$	0	0
$204$	0	0
$205$	1.86361	0.130160
$206$	0	0
$207$	23.6093	1.64096
$208$	0	0
$209$	−26.2354	−1.81474
$210$	0	0
$211$	20.0909	1.38311	0.691556	−	0.722323i	$-0.256926\pi$
$211$	20.0909	1.38311	0.691556	+	0.722323i	$0.256926\pi$
$212$	0	0
$213$	−29.0605	−1.99120
$214$	0	0
$215$	1.05312	0.0718222
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−31.1733	−2.10649
$220$	0	0
$221$	−15.5241	−1.04426
$222$	0	0
$223$	−18.9190	−1.26691	−0.633456	−	0.773779i	$-0.718364\pi$
$223$	−18.9190	−1.26691	−0.633456	+	0.773779i	$0.718364\pi$
$224$	0	0
$225$	−36.4559	−2.43039
$226$	0	0
$227$	−8.00051	−0.531013	−0.265506	−	0.964109i	$-0.585539\pi$
$227$	−8.00051	−0.531013	−0.265506	+	0.964109i	$0.585539\pi$
$228$	0	0
$229$	−0.281230	−0.0185842	−0.00929209	−	0.999957i	$-0.502958\pi$
$229$	−0.281230	−0.0185842	−0.00929209	+	0.999957i	$0.502958\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−13.8423	−0.906839	−0.453419	−	0.891297i	$-0.649796\pi$
$233$	−13.8423	−0.906839	−0.453419	+	0.891297i	$0.649796\pi$
$234$	0	0
$235$	−4.09783	−0.267313
$236$	0	0
$237$	−32.1044	−2.08540
$238$	0	0
$239$	26.9785	1.74509	0.872547	−	0.488531i	$-0.162467\pi$
$239$	26.9785	1.74509	0.872547	+	0.488531i	$0.162467\pi$
$240$	0	0
$241$	−14.8793	−0.958463	−0.479232	−	0.877689i	$-0.659085\pi$
$241$	−14.8793	−0.958463	−0.479232	+	0.877689i	$0.659085\pi$
$242$	0	0
$243$	61.8811	3.96968
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−26.7612	−1.70278
$248$	0	0
$249$	−8.35074	−0.529206
$250$	0	0
$251$	13.8891	0.876669	0.438335	−	0.898812i	$-0.355568\pi$
$251$	13.8891	0.876669	0.438335	+	0.898812i	$0.355568\pi$
$252$	0	0
$253$	14.4090	0.905888
$254$	0	0
$255$	−7.69850	−0.482099
$256$	0	0
$257$	−5.15586	−0.321614	−0.160807	−	0.986986i	$-0.551410\pi$
$257$	−5.15586	−0.321614	−0.160807	+	0.986986i	$0.551410\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−3.94529	−0.244207
$262$	0	0
$263$	4.34121	0.267691	0.133845	−	0.991002i	$-0.457267\pi$
$263$	4.34121	0.267691	0.133845	+	0.991002i	$0.457267\pi$
$264$	0	0
$265$	−1.98199	−0.121753
$266$	0	0
$267$	−47.5222	−2.90831
$268$	0	0
$269$	12.3195	0.751136	0.375568	−	0.926795i	$-0.377448\pi$
$269$	12.3195	0.751136	0.375568	+	0.926795i	$0.377448\pi$
$270$	0	0
$271$	−17.7033	−1.07540	−0.537701	−	0.843136i	$-0.680707\pi$
$271$	−17.7033	−1.07540	−0.537701	+	0.843136i	$0.680707\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−22.2494	−1.34169
$276$	0	0
$277$	−2.84903	−0.171182	−0.0855909	−	0.996330i	$-0.527278\pi$
$277$	−2.84903	−0.171182	−0.0855909	+	0.996330i	$0.527278\pi$
$278$	0	0
$279$	−62.6581	−3.75124
$280$	0	0
$281$	−15.1411	−0.903240	−0.451620	−	0.892210i	$-0.649154\pi$
$281$	−15.1411	−0.903240	−0.451620	+	0.892210i	$0.649154\pi$
$282$	0	0
$283$	31.5635	1.87625	0.938127	−	0.346291i	$-0.112559\pi$
$283$	31.5635	1.87625	0.938127	+	0.346291i	$0.112559\pi$
$284$	0	0
$285$	−13.2711	−0.786110
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−7.84167	−0.461275
$290$	0	0
$291$	44.9304	2.63386
$292$	0	0
$293$	29.2932	1.71133	0.855665	−	0.517530i	$-0.173149\pi$
$293$	29.2932	1.71133	0.855665	+	0.517530i	$0.173149\pi$
$294$	0	0
$295$	4.94533	0.287928
$296$	0	0
$297$	88.3473	5.12643
$298$	0	0
$299$	14.6978	0.849998
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−2.59271	−0.148947
$304$	0	0
$305$	0.767694	0.0439581
$306$	0	0
$307$	12.1589	0.693942	0.346971	−	0.937876i	$-0.387210\pi$
$307$	12.1589	0.693942	0.346971	+	0.937876i	$0.387210\pi$
$308$	0	0
$309$	−13.6832	−0.778411
$310$	0	0
$311$	−16.4062	−0.930312	−0.465156	−	0.885229i	$-0.654002\pi$
$311$	−16.4062	−0.930312	−0.465156	+	0.885229i	$0.654002\pi$
$312$	0	0
$313$	−20.5871	−1.16365	−0.581825	−	0.813314i	$-0.697661\pi$
$313$	−20.5871	−1.16365	−0.581825	+	0.813314i	$0.697661\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−22.1389	−1.24344	−0.621722	−	0.783238i	$-0.713567\pi$
$317$	−22.1389	−1.24344	−0.621722	+	0.783238i	$0.713567\pi$
$318$	0	0
$319$	−2.40785	−0.134814
$320$	0	0
$321$	60.7258	3.38939
$322$	0	0
$323$	15.7876	0.878446
$324$	0	0
$325$	−22.6954	−1.25891
$326$	0	0
$327$	52.5362	2.90526
$328$	0	0
$329$	0	0
$330$	0	0
$331$	17.9499	0.986617	0.493308	−	0.869854i	$-0.335787\pi$
$331$	17.9499	0.986617	0.493308	+	0.869854i	$0.335787\pi$
$332$	0	0
$333$	−87.0222	−4.76879
$334$	0	0
$335$	−0.836883	−0.0457238
$336$	0	0
$337$	8.94662	0.487353	0.243677	−	0.969857i	$-0.421646\pi$
$337$	8.94662	0.487353	0.243677	+	0.969857i	$0.421646\pi$
$338$	0	0
$339$	19.0601	1.03520
$340$	0	0
$341$	−38.2409	−2.07086
$342$	0	0
$343$	0	0
$344$	0	0
$345$	7.28876	0.392414
$346$	0	0
$347$	23.6318	1.26862	0.634310	−	0.773079i	$-0.281284\pi$
$347$	23.6318	1.26862	0.634310	+	0.773079i	$0.281284\pi$
$348$	0	0
$349$	−2.61691	−0.140080	−0.0700399	−	0.997544i	$-0.522313\pi$
$349$	−2.61691	−0.140080	−0.0700399	+	0.997544i	$0.522313\pi$
$350$	0	0
$351$	90.1181	4.81015
$352$	0	0
$353$	−20.2502	−1.07781	−0.538905	−	0.842366i	$-0.681162\pi$
$353$	−20.2502	−1.07781	−0.538905	+	0.842366i	$0.681162\pi$
$354$	0	0
$355$	−6.57711	−0.349077
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−11.3168	−0.597277	−0.298639	−	0.954366i	$-0.596532\pi$
$359$	−11.3168	−0.597277	−0.298639	+	0.954366i	$0.596532\pi$
$360$	0	0
$361$	8.21550	0.432395
$362$	0	0
$363$	47.9105	2.51465
$364$	0	0
$365$	−7.05527	−0.369290
$366$	0	0
$367$	−13.0233	−0.679809	−0.339904	−	0.940460i	$-0.610395\pi$
$367$	−13.0233	−0.679809	−0.339904	+	0.940460i	$0.610395\pi$
$368$	0	0
$369$	20.2380	1.05355
$370$	0	0
$371$	0	0
$372$	0	0
$373$	12.2165	0.632546	0.316273	−	0.948668i	$-0.397568\pi$
$373$	12.2165	0.632546	0.316273	+	0.948668i	$0.397568\pi$
$374$	0	0
$375$	−23.9742	−1.23802
$376$	0	0
$377$	−2.45612	−0.126496
$378$	0	0
$379$	6.65140	0.341660	0.170830	−	0.985301i	$-0.445355\pi$
$379$	6.65140	0.341660	0.170830	+	0.985301i	$0.445355\pi$
$380$	0	0
$381$	−5.21390	−0.267116
$382$	0	0
$383$	−5.93177	−0.303099	−0.151550	−	0.988450i	$-0.548426\pi$
$383$	−5.93177	−0.303099	−0.151550	+	0.988450i	$0.548426\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	11.4364	0.581347
$388$	0	0
$389$	−3.46772	−0.175820	−0.0879101	−	0.996128i	$-0.528019\pi$
$389$	−3.46772	−0.175820	−0.0879101	+	0.996128i	$0.528019\pi$
$390$	0	0
$391$	−8.67090	−0.438506
$392$	0	0
$393$	−10.4818	−0.528735
$394$	0	0
$395$	−7.26601	−0.365593
$396$	0	0
$397$	−18.3134	−0.919124	−0.459562	−	0.888146i	$-0.651994\pi$
$397$	−18.3134	−0.919124	−0.459562	+	0.888146i	$0.651994\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−26.8018	−1.33842	−0.669210	−	0.743073i	$-0.733367\pi$
$401$	−26.8018	−1.33842	−0.669210	+	0.743073i	$0.733367\pi$
$402$	0	0
$403$	−39.0074	−1.94310
$404$	0	0
$405$	25.9332	1.28863
$406$	0	0
$407$	−53.1107	−2.63260
$408$	0	0
$409$	−14.6884	−0.726293	−0.363146	−	0.931732i	$-0.618298\pi$
$409$	−14.6884	−0.726293	−0.363146	+	0.931732i	$0.618298\pi$
$410$	0	0
$411$	24.6098	1.21391
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−1.88998	−0.0927753
$416$	0	0
$417$	20.8500	1.02103
$418$	0	0
$419$	28.1629	1.37585	0.687923	−	0.725783i	$-0.258523\pi$
$419$	28.1629	1.37585	0.687923	+	0.725783i	$0.258523\pi$
$420$	0	0
$421$	4.54473	0.221496	0.110748	−	0.993848i	$-0.464675\pi$
$421$	4.54473	0.221496	0.110748	+	0.993848i	$0.464675\pi$
$422$	0	0
$423$	−44.5007	−2.16370
$424$	0	0
$425$	13.3890	0.649462
$426$	0	0
$427$	0	0
$428$	0	0
$429$	86.4888	4.17572
$430$	0	0
$431$	−11.6877	−0.562975	−0.281488	−	0.959565i	$-0.590828\pi$
$431$	−11.6877	−0.562975	−0.281488	+	0.959565i	$0.590828\pi$
$432$	0	0
$433$	−28.3965	−1.36465	−0.682325	−	0.731049i	$-0.739031\pi$
$433$	−28.3965	−1.36465	−0.682325	+	0.731049i	$0.739031\pi$
$434$	0	0
$435$	−1.21801	−0.0583989
$436$	0	0
$437$	−14.9473	−0.715029
$438$	0	0
$439$	15.6464	0.746762	0.373381	−	0.927678i	$-0.378198\pi$
$439$	15.6464	0.746762	0.373381	+	0.927678i	$0.378198\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	30.3887	1.44381	0.721905	−	0.691992i	$-0.243267\pi$
$443$	30.3887	1.44381	0.721905	+	0.691992i	$0.243267\pi$
$444$	0	0
$445$	−10.7554	−0.509856
$446$	0	0
$447$	27.9975	1.32424
$448$	0	0
$449$	17.7195	0.836234	0.418117	−	0.908393i	$-0.362690\pi$
$449$	17.7195	0.836234	0.418117	+	0.908393i	$0.362690\pi$
$450$	0	0
$451$	12.3515	0.581608
$452$	0	0
$453$	−31.9536	−1.50131
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.47386	−0.302834	−0.151417	−	0.988470i	$-0.548384\pi$
$457$	−6.47386	−0.302834	−0.151417	+	0.988470i	$0.548384\pi$
$458$	0	0
$459$	−53.1646	−2.48151
$460$	0	0
$461$	24.3797	1.13548	0.567739	−	0.823209i	$-0.307818\pi$
$461$	24.3797	1.13548	0.567739	+	0.823209i	$0.307818\pi$
$462$	0	0
$463$	40.1970	1.86811	0.934057	−	0.357124i	$-0.116243\pi$
$463$	40.1970	1.86811	0.934057	+	0.357124i	$0.116243\pi$
$464$	0	0
$465$	−19.3441	−0.897059
$466$	0	0
$467$	−34.4906	−1.59604	−0.798018	−	0.602634i	$-0.794118\pi$
$467$	−34.4906	−1.59604	−0.798018	+	0.602634i	$0.794118\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	65.7973	3.03178
$472$	0	0
$473$	6.97979	0.320931
$474$	0	0
$475$	23.0807	1.05901
$476$	0	0
$477$	−21.5236	−0.985496
$478$	0	0
$479$	42.8908	1.95973	0.979864	−	0.199664i	$-0.0639850\pi$
$479$	42.8908	1.95973	0.979864	+	0.199664i	$0.0639850\pi$
$480$	0	0
$481$	−54.1752	−2.47018
$482$	0	0
$483$	0	0
$484$	0	0
$485$	10.1688	0.461743
$486$	0	0
$487$	18.1379	0.821906	0.410953	−	0.911657i	$-0.365196\pi$
$487$	18.1379	0.821906	0.410953	+	0.911657i	$0.365196\pi$
$488$	0	0
$489$	54.8892	2.48218
$490$	0	0
$491$	−15.7737	−0.711858	−0.355929	−	0.934513i	$-0.615836\pi$
$491$	−15.7737	−0.711858	−0.355929	+	0.934513i	$0.615836\pi$
$492$	0	0
$493$	1.44897	0.0652583
$494$	0	0
$495$	31.4428	1.41325
$496$	0	0
$497$	0	0
$498$	0	0
$499$	9.04791	0.405040	0.202520	−	0.979278i	$-0.435087\pi$
$499$	9.04791	0.405040	0.202520	+	0.979278i	$0.435087\pi$
$500$	0	0
$501$	−14.7243	−0.657835
$502$	0	0
$503$	−31.0366	−1.38385	−0.691927	−	0.721967i	$-0.743238\pi$
$503$	−31.0366	−1.38385	−0.691927	+	0.721967i	$0.743238\pi$
$504$	0	0
$505$	−0.586794	−0.0261120
$506$	0	0
$507$	44.6384	1.98246
$508$	0	0
$509$	32.3990	1.43606	0.718030	−	0.696012i	$-0.245044\pi$
$509$	32.3990	1.43606	0.718030	+	0.696012i	$0.245044\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−91.6479	−4.04635
$514$	0	0
$515$	−3.09685	−0.136463
$516$	0	0
$517$	−27.1593	−1.19446
$518$	0	0
$519$	−48.8298	−2.14339
$520$	0	0
$521$	2.37930	0.104239	0.0521195	−	0.998641i	$-0.483402\pi$
$521$	2.37930	0.104239	0.0521195	+	0.998641i	$0.483402\pi$
$522$	0	0
$523$	−28.4397	−1.24358	−0.621791	−	0.783184i	$-0.713595\pi$
$523$	−28.4397	−1.24358	−0.621791	+	0.783184i	$0.713595\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	23.0122	1.00243
$528$	0	0
$529$	−14.7906	−0.643069
$530$	0	0
$531$	53.7041	2.33056
$532$	0	0
$533$	12.5990	0.545725
$534$	0	0
$535$	13.7437	0.594194
$536$	0	0
$537$	−11.1778	−0.482357
$538$	0	0
$539$	0	0
$540$	0	0
$541$	12.5223	0.538377	0.269189	−	0.963087i	$-0.413244\pi$
$541$	12.5223	0.538377	0.269189	+	0.963087i	$0.413244\pi$
$542$	0	0
$543$	−71.6144	−3.07327
$544$	0	0
$545$	11.8902	0.509321
$546$	0	0
$547$	8.29269	0.354570	0.177285	−	0.984160i	$-0.443269\pi$
$547$	8.29269	0.354570	0.177285	+	0.984160i	$0.443269\pi$
$548$	0	0
$549$	8.33683	0.355807
$550$	0	0
$551$	2.49781	0.106410
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−26.8659	−1.14039
$556$	0	0
$557$	−15.3864	−0.651942	−0.325971	−	0.945380i	$-0.605691\pi$
$557$	−15.3864	−0.651942	−0.325971	+	0.945380i	$0.605691\pi$
$558$	0	0
$559$	7.11969	0.301131
$560$	0	0
$561$	−51.0235	−2.15421
$562$	0	0
$563$	1.71762	0.0723889	0.0361945	−	0.999345i	$-0.488476\pi$
$563$	1.71762	0.0723889	0.0361945	+	0.999345i	$0.488476\pi$
$564$	0	0
$565$	4.31376	0.181481
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−24.6312	−1.03259	−0.516297	−	0.856410i	$-0.672690\pi$
$569$	−24.6312	−1.03259	−0.516297	+	0.856410i	$0.672690\pi$
$570$	0	0
$571$	15.5744	0.651769	0.325884	−	0.945410i	$-0.394338\pi$
$571$	15.5744	0.651769	0.325884	+	0.945410i	$0.394338\pi$
$572$	0	0
$573$	26.6837	1.11473
$574$	0	0
$575$	−12.6764	−0.528643
$576$	0	0
$577$	−18.7712	−0.781455	−0.390728	−	0.920506i	$-0.627777\pi$
$577$	−18.7712	−0.781455	−0.390728	+	0.920506i	$0.627777\pi$
$578$	0	0
$579$	51.2181	2.12855
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−13.1361	−0.544041
$584$	0	0
$585$	32.0730	1.32606
$586$	0	0
$587$	−24.5850	−1.01473	−0.507365	−	0.861731i	$-0.669381\pi$
$587$	−24.5850	−1.01473	−0.507365	+	0.861731i	$0.669381\pi$
$588$	0	0
$589$	39.6696	1.63456
$590$	0	0
$591$	−70.9111	−2.91689
$592$	0	0
$593$	−34.8300	−1.43030	−0.715148	−	0.698973i	$-0.753641\pi$
$593$	−34.8300	−1.43030	−0.715148	+	0.698973i	$0.753641\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	28.7664	1.17733
$598$	0	0
$599$	7.52630	0.307516	0.153758	−	0.988109i	$-0.450862\pi$
$599$	7.52630	0.307516	0.153758	+	0.988109i	$0.450862\pi$
$600$	0	0
$601$	11.0767	0.451827	0.225914	−	0.974147i	$-0.427463\pi$
$601$	11.0767	0.451827	0.225914	+	0.974147i	$0.427463\pi$
$602$	0	0
$603$	−9.08819	−0.370100
$604$	0	0
$605$	10.8433	0.440844
$606$	0	0
$607$	24.3168	0.986990	0.493495	−	0.869749i	$-0.335719\pi$
$607$	24.3168	0.986990	0.493495	+	0.869749i	$0.335719\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−27.7037	−1.12077
$612$	0	0
$613$	27.8779	1.12598	0.562989	−	0.826464i	$-0.309651\pi$
$613$	27.8779	1.12598	0.562989	+	0.826464i	$0.309651\pi$
$614$	0	0
$615$	6.24795	0.251942
$616$	0	0
$617$	−25.5600	−1.02901	−0.514504	−	0.857488i	$-0.672024\pi$
$617$	−25.5600	−1.02901	−0.514504	+	0.857488i	$0.672024\pi$
$618$	0	0
$619$	−22.3330	−0.897637	−0.448819	−	0.893623i	$-0.648155\pi$
$619$	−22.3330	−0.897637	−0.448819	+	0.893623i	$0.648155\pi$
$620$	0	0
$621$	50.3350	2.01988
$622$	0	0
$623$	0	0
$624$	0	0
$625$	16.6953	0.667813
$626$	0	0
$627$	−87.9569	−3.51266
$628$	0	0
$629$	31.9603	1.27434
$630$	0	0
$631$	5.52027	0.219758	0.109879	−	0.993945i	$-0.464954\pi$
$631$	5.52027	0.219758	0.109879	+	0.993945i	$0.464954\pi$
$632$	0	0
$633$	67.3568	2.67719
$634$	0	0
$635$	−1.18003	−0.0468282
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−71.4246	−2.82551
$640$	0	0
$641$	2.00540	0.0792087	0.0396044	−	0.999215i	$-0.487390\pi$
$641$	2.00540	0.0792087	0.0396044	+	0.999215i	$0.487390\pi$
$642$	0	0
$643$	2.91626	0.115006	0.0575031	−	0.998345i	$-0.481686\pi$
$643$	2.91626	0.115006	0.0575031	+	0.998345i	$0.481686\pi$
$644$	0	0
$645$	3.53070	0.139021
$646$	0	0
$647$	40.1047	1.57668	0.788338	−	0.615242i	$-0.210942\pi$
$647$	40.1047	1.57668	0.788338	+	0.615242i	$0.210942\pi$
$648$	0	0
$649$	32.7762	1.28658
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−16.9230	−0.662246	−0.331123	−	0.943588i	$-0.607428\pi$
$653$	−16.9230	−0.662246	−0.331123	+	0.943588i	$0.607428\pi$
$654$	0	0
$655$	−2.37228	−0.0926927
$656$	0	0
$657$	−76.6172	−2.98912
$658$	0	0
$659$	−0.637912	−0.0248495	−0.0124248	−	0.999923i	$-0.503955\pi$
$659$	−0.637912	−0.0248495	−0.0124248	+	0.999923i	$0.503955\pi$
$660$	0	0
$661$	−33.7971	−1.31455	−0.657277	−	0.753649i	$-0.728292\pi$
$661$	−33.7971	−1.31455	−0.657277	+	0.753649i	$0.728292\pi$
$662$	0	0
$663$	−52.0462	−2.02131
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−1.37185	−0.0531183
$668$	0	0
$669$	−63.4281	−2.45227
$670$	0	0
$671$	5.08806	0.196423
$672$	0	0
$673$	24.8734	0.958800	0.479400	−	0.877597i	$-0.340854\pi$
$673$	24.8734	0.958800	0.479400	+	0.877597i	$0.340854\pi$
$674$	0	0
$675$	−77.7239	−2.99159
$676$	0	0
$677$	−25.6801	−0.986966	−0.493483	−	0.869755i	$-0.664277\pi$
$677$	−25.6801	−0.986966	−0.493483	+	0.869755i	$0.664277\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−26.8226	−1.02784
$682$	0	0
$683$	−15.5459	−0.594846	−0.297423	−	0.954746i	$-0.596127\pi$
$683$	−15.5459	−0.594846	−0.297423	+	0.954746i	$0.596127\pi$
$684$	0	0
$685$	5.56980	0.212811
$686$	0	0
$687$	−0.942854	−0.0359721
$688$	0	0
$689$	−13.3994	−0.510476
$690$	0	0
$691$	−21.1076	−0.802969	−0.401485	−	0.915866i	$-0.631506\pi$
$691$	−21.1076	−0.802969	−0.401485	+	0.915866i	$0.631506\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	4.71886	0.178996
$696$	0	0
$697$	−7.43272	−0.281534
$698$	0	0
$699$	−46.4078	−1.75530
$700$	0	0
$701$	0.491468	0.0185625	0.00928125	−	0.999957i	$-0.497046\pi$
$701$	0.491468	0.0185625	0.00928125	+	0.999957i	$0.497046\pi$
$702$	0	0
$703$	55.0949	2.07794
$704$	0	0
$705$	−13.7384	−0.517420
$706$	0	0
$707$	0	0
$708$	0	0
$709$	36.0023	1.35209	0.676047	−	0.736858i	$-0.263691\pi$
$709$	36.0023	1.35209	0.676047	+	0.736858i	$0.263691\pi$
$710$	0	0
$711$	−78.9057	−2.95919
$712$	0	0
$713$	−21.7874	−0.815945
$714$	0	0
$715$	19.5745	0.732046
$716$	0	0
$717$	90.4484	3.37786
$718$	0	0
$719$	−47.4758	−1.77055	−0.885274	−	0.465070i	$-0.846029\pi$
$719$	−47.4758	−1.77055	−0.885274	+	0.465070i	$0.846029\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−49.8847	−1.85523
$724$	0	0
$725$	2.11832	0.0786725
$726$	0	0
$727$	10.3685	0.384547	0.192273	−	0.981341i	$-0.438414\pi$
$727$	10.3685	0.384547	0.192273	+	0.981341i	$0.438414\pi$
$728$	0	0
$729$	104.931	3.88632
$730$	0	0
$731$	−4.20021	−0.155351
$732$	0	0
$733$	16.2362	0.599698	0.299849	−	0.953987i	$-0.403064\pi$
$733$	16.2362	0.599698	0.299849	+	0.953987i	$0.403064\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.54663	−0.204313
$738$	0	0
$739$	43.3712	1.59544	0.797718	−	0.603030i	$-0.206040\pi$
$739$	43.3712	1.59544	0.797718	+	0.603030i	$0.206040\pi$
$740$	0	0
$741$	−89.7199	−3.29594
$742$	0	0
$743$	15.2992	0.561273	0.280637	−	0.959814i	$-0.409454\pi$
$743$	15.2992	0.561273	0.280637	+	0.959814i	$0.409454\pi$
$744$	0	0
$745$	6.33653	0.232153
$746$	0	0
$747$	−20.5243	−0.750946
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−36.2989	−1.32456	−0.662282	−	0.749254i	$-0.730412\pi$
$751$	−36.2989	−1.32456	−0.662282	+	0.749254i	$0.730412\pi$
$752$	0	0
$753$	46.5646	1.69691
$754$	0	0
$755$	−7.23189	−0.263195
$756$	0	0
$757$	7.00732	0.254686	0.127343	−	0.991859i	$-0.459355\pi$
$757$	7.00732	0.254686	0.127343	+	0.991859i	$0.459355\pi$
$758$	0	0
$759$	48.3079	1.75347
$760$	0	0
$761$	39.9621	1.44862	0.724312	−	0.689473i	$-0.242158\pi$
$761$	39.9621	1.44862	0.724312	+	0.689473i	$0.242158\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−18.9213	−0.684099
$766$	0	0
$767$	33.4332	1.20720
$768$	0	0
$769$	27.7272	0.999869	0.499934	−	0.866063i	$-0.333357\pi$
$769$	27.7272	0.999869	0.499934	+	0.866063i	$0.333357\pi$
$770$	0	0
$771$	−17.2856	−0.622525
$772$	0	0
$773$	−37.2499	−1.33978	−0.669892	−	0.742458i	$-0.733660\pi$
$773$	−37.2499	−1.33978	−0.669892	+	0.742458i	$0.733660\pi$
$774$	0	0
$775$	33.6426	1.20848
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−12.8129	−0.459070
$780$	0	0
$781$	−43.5913	−1.55982
$782$	0	0
$783$	−8.41135	−0.300597
$784$	0	0
$785$	14.8915	0.531501
$786$	0	0
$787$	21.5993	0.769933	0.384967	−	0.922930i	$-0.374213\pi$
$787$	21.5993	0.769933	0.384967	+	0.922930i	$0.374213\pi$
$788$	0	0
$789$	14.5544	0.518150
$790$	0	0
$791$	0	0
$792$	0	0
$793$	5.19005	0.184304
$794$	0	0
$795$	−6.64484	−0.235668
$796$	0	0
$797$	−4.10701	−0.145478	−0.0727388	−	0.997351i	$-0.523174\pi$
$797$	−4.10701	−0.145478	−0.0727388	+	0.997351i	$0.523174\pi$
$798$	0	0
$799$	16.3436	0.578195
$800$	0	0
$801$	−116.799	−4.12690
$802$	0	0
$803$	−46.7604	−1.65014
$804$	0	0
$805$	0	0
$806$	0	0
$807$	41.3026	1.45392
$808$	0	0
$809$	3.56647	0.125390	0.0626951	−	0.998033i	$-0.480030\pi$
$809$	3.56647	0.125390	0.0626951	+	0.998033i	$0.480030\pi$
$810$	0	0
$811$	−24.7660	−0.869653	−0.434826	−	0.900514i	$-0.643190\pi$
$811$	−24.7660	−0.869653	−0.434826	+	0.900514i	$0.643190\pi$
$812$	0	0
$813$	−59.3524	−2.08158
$814$	0	0
$815$	12.4228	0.435151
$816$	0	0
$817$	−7.24055	−0.253315
$818$	0	0
$819$	0	0
$820$	0	0
$821$	11.4959	0.401208	0.200604	−	0.979672i	$-0.435710\pi$
$821$	11.4959	0.401208	0.200604	+	0.979672i	$0.435710\pi$
$822$	0	0
$823$	46.1776	1.60965	0.804826	−	0.593511i	$-0.202259\pi$
$823$	46.1776	1.60965	0.804826	+	0.593511i	$0.202259\pi$
$824$	0	0
$825$	−74.5937	−2.59702
$826$	0	0
$827$	−40.1060	−1.39462	−0.697311	−	0.716769i	$-0.745620\pi$
$827$	−40.1060	−1.39462	−0.697311	+	0.716769i	$0.745620\pi$
$828$	0	0
$829$	−16.3006	−0.566144	−0.283072	−	0.959099i	$-0.591354\pi$
$829$	−16.3006	−0.566144	−0.283072	+	0.959099i	$0.591354\pi$
$830$	0	0
$831$	−9.55170	−0.331345
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−3.33248	−0.115325
$836$	0	0
$837$	−133.587	−4.61744
$838$	0	0
$839$	−16.9921	−0.586634	−0.293317	−	0.956015i	$-0.594759\pi$
$839$	−16.9921	−0.586634	−0.293317	+	0.956015i	$0.594759\pi$
$840$	0	0
$841$	−28.7708	−0.992095
$842$	0	0
$843$	−50.7621	−1.74834
$844$	0	0
$845$	10.1028	0.347545
$846$	0	0
$847$	0	0
$848$	0	0
$849$	105.820	3.63174
$850$	0	0
$851$	−30.2593	−1.03728
$852$	0	0
$853$	8.80329	0.301419	0.150709	−	0.988578i	$-0.451844\pi$
$853$	8.80329	0.301419	0.150709	+	0.988578i	$0.451844\pi$
$854$	0	0
$855$	−32.6174	−1.11549
$856$	0	0
$857$	4.38370	0.149744	0.0748722	−	0.997193i	$-0.476145\pi$
$857$	4.38370	0.149744	0.0748722	+	0.997193i	$0.476145\pi$
$858$	0	0
$859$	21.4533	0.731978	0.365989	−	0.930619i	$-0.380731\pi$
$859$	21.4533	0.731978	0.365989	+	0.930619i	$0.380731\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	48.9506	1.66630	0.833149	−	0.553048i	$-0.186535\pi$
$863$	48.9506	1.66630	0.833149	+	0.553048i	$0.186535\pi$
$864$	0	0
$865$	−11.0514	−0.375758
$866$	0	0
$867$	−26.2901	−0.892858
$868$	0	0
$869$	−48.1571	−1.63362
$870$	0	0
$871$	−5.65781	−0.191707
$872$	0	0
$873$	110.429	3.73746
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−10.0259	−0.338551	−0.169276	−	0.985569i	$-0.554143\pi$
$877$	−10.0259	−0.338551	−0.169276	+	0.985569i	$0.554143\pi$
$878$	0	0
$879$	98.2088	3.31250
$880$	0	0
$881$	−41.5096	−1.39849	−0.699247	−	0.714880i	$-0.746481\pi$
$881$	−41.5096	−1.39849	−0.699247	+	0.714880i	$0.746481\pi$
$882$	0	0
$883$	35.9357	1.20933	0.604666	−	0.796479i	$-0.293307\pi$
$883$	35.9357	1.20933	0.604666	+	0.796479i	$0.293307\pi$
$884$	0	0
$885$	16.5798	0.557322
$886$	0	0
$887$	35.4867	1.19153	0.595764	−	0.803160i	$-0.296849\pi$
$887$	35.4867	1.19153	0.595764	+	0.803160i	$0.296849\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	171.878	5.75813
$892$	0	0
$893$	28.1740	0.942806
$894$	0	0
$895$	−2.52981	−0.0845622
$896$	0	0
$897$	49.2762	1.64528
$898$	0	0
$899$	3.64084	0.121429
$900$	0	0
$901$	7.90488	0.263350
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−16.2081	−0.538775
$906$	0	0
$907$	−9.36439	−0.310940	−0.155470	−	0.987841i	$-0.549689\pi$
$907$	−9.36439	−0.310940	−0.155470	+	0.987841i	$0.549689\pi$
$908$	0	0
$909$	−6.37233	−0.211357
$910$	0	0
$911$	−22.0274	−0.729800	−0.364900	−	0.931047i	$-0.618897\pi$
$911$	−22.0274	−0.729800	−0.364900	+	0.931047i	$0.618897\pi$
$912$	0	0
$913$	−12.5262	−0.414558
$914$	0	0
$915$	2.57378	0.0850866
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1.58868	−0.0524058	−0.0262029	−	0.999657i	$-0.508342\pi$
$919$	−1.58868	−0.0524058	−0.0262029	+	0.999657i	$0.508342\pi$
$920$	0	0
$921$	40.7639	1.34322
$922$	0	0
$923$	−44.4650	−1.46358
$924$	0	0
$925$	46.7243	1.53629
$926$	0	0
$927$	−33.6304	−1.10457
$928$	0	0
$929$	17.6434	0.578861	0.289430	−	0.957199i	$-0.406534\pi$
$929$	17.6434	0.578861	0.289430	+	0.957199i	$0.406534\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−55.0037	−1.80074
$934$	0	0
$935$	−11.5479	−0.377656
$936$	0	0
$937$	25.3335	0.827609	0.413804	−	0.910366i	$-0.364200\pi$
$937$	25.3335	0.827609	0.413804	+	0.910366i	$0.364200\pi$
$938$	0	0
$939$	−69.0204	−2.25240
$940$	0	0
$941$	−37.0670	−1.20835	−0.604175	−	0.796852i	$-0.706497\pi$
$941$	−37.0670	−1.20835	−0.604175	+	0.796852i	$0.706497\pi$
$942$	0	0
$943$	7.03713	0.229161
$944$	0	0
$945$	0	0
$946$	0	0
$947$	19.7855	0.642941	0.321471	−	0.946920i	$-0.395823\pi$
$947$	19.7855	0.642941	0.321471	+	0.946920i	$0.395823\pi$
$948$	0	0
$949$	−47.6976	−1.54833
$950$	0	0
$951$	−74.2231	−2.40685
$952$	0	0
$953$	−22.2707	−0.721420	−0.360710	−	0.932678i	$-0.617466\pi$
$953$	−22.2707	−0.721420	−0.360710	+	0.932678i	$0.617466\pi$
$954$	0	0
$955$	6.03918	0.195423
$956$	0	0
$957$	−8.07260	−0.260950
$958$	0	0
$959$	0	0
$960$	0	0
$961$	26.8228	0.865253
$962$	0	0
$963$	149.251	4.80955
$964$	0	0
$965$	11.5919	0.373156
$966$	0	0
$967$	5.63935	0.181349	0.0906746	−	0.995881i	$-0.471098\pi$
$967$	5.63935	0.181349	0.0906746	+	0.995881i	$0.471098\pi$
$968$	0	0
$969$	52.9297	1.70035
$970$	0	0
$971$	−32.8632	−1.05463	−0.527316	−	0.849669i	$-0.676802\pi$
$971$	−32.8632	−1.05463	−0.527316	+	0.849669i	$0.676802\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−76.0889	−2.43679
$976$	0	0
$977$	−20.0892	−0.642710	−0.321355	−	0.946959i	$-0.604138\pi$
$977$	−20.0892	−0.642710	−0.321355	+	0.946959i	$0.604138\pi$
$978$	0	0
$979$	−71.2840	−2.27825
$980$	0	0
$981$	129.123	4.12257
$982$	0	0
$983$	−22.5264	−0.718479	−0.359240	−	0.933245i	$-0.616964\pi$
$983$	−22.5264	−0.718479	−0.359240	+	0.933245i	$0.616964\pi$
$984$	0	0
$985$	−16.0489	−0.511361
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.97667	0.126451
$990$	0	0
$991$	−12.4984	−0.397024	−0.198512	−	0.980098i	$-0.563611\pi$
$991$	−12.4984	−0.397024	−0.198512	+	0.980098i	$0.563611\pi$
$992$	0	0
$993$	60.1791	1.90973
$994$	0	0
$995$	6.51054	0.206398
$996$	0	0
$997$	42.1882	1.33611	0.668056	−	0.744111i	$-0.267126\pi$
$997$	42.1882	1.33611	0.668056	+	0.744111i	$0.267126\pi$
$998$	0	0
$999$	−185.531	−5.86995

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2744.2.a.h.1.12	yes	12
4.3	odd	2		5488.2.a.w.1.1		12
7.6	odd	2	inner	2744.2.a.h.1.1	✓	12
28.27	even	2		5488.2.a.w.1.12		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2744.2.a.h.1.1	✓	12	7.6	odd	2	inner
2744.2.a.h.1.12	yes	12	1.1	even	1	trivial
5488.2.a.w.1.1		12	4.3	odd	2
5488.2.a.w.1.12		12	28.27	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:	\( N \)	\(=\)	\( 2744 = 2^{3} \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2744.a (trivial)

\(f(q)\)	\(=\)	\(q+3.35261 q^{3} +0.758778 q^{5} +8.24000 q^{9} +O(q^{10})\)	\(q+3.35261 q^{3} +0.758778 q^{5} +8.24000 q^{9} +5.02897 q^{11} +5.12977 q^{13} +2.54389 q^{15} -3.02627 q^{17} -5.21685 q^{19} +2.86521 q^{23} -4.42426 q^{25} +17.5677 q^{27} -0.478797 q^{29} -7.60413 q^{31} +16.8602 q^{33} -10.5609 q^{37} +17.1981 q^{39} +2.45606 q^{41} +1.38792 q^{43} +6.25233 q^{45} -5.40057 q^{47} -10.1459 q^{51} -2.61208 q^{53} +3.81587 q^{55} -17.4901 q^{57} +6.51749 q^{59} +1.01175 q^{61} +3.89235 q^{65} -1.10294 q^{67} +9.60592 q^{69} -8.66803 q^{71} -9.29821 q^{73} -14.8328 q^{75} -9.57594 q^{79} +34.1776 q^{81} -2.49082 q^{83} -2.29627 q^{85} -1.60522 q^{87} -14.1747 q^{89} -25.4937 q^{93} -3.95843 q^{95} +13.4016 q^{97} +41.4387 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 22 q^{9}+O(q^{10}) \) 12 * q + 22 * q^9	\( 12 q + 22 q^{9} + 18 q^{11} + 28 q^{15} + 20 q^{23} + 14 q^{25} - 18 q^{29} - 18 q^{37} + 36 q^{39} + 10 q^{43} + 48 q^{51} - 38 q^{53} + 12 q^{57} + 8 q^{65} + 42 q^{67} + 56 q^{71} + 56 q^{79} + 52 q^{81} + 8 q^{85} - 48 q^{93} + 84 q^{95} + 90 q^{99}+O(q^{100}) \) 12 * q + 22 * q^9 + 18 * q^11 + 28 * q^15 + 20 * q^23 + 14 * q^25 - 18 * q^29 - 18 * q^37 + 36 * q^39 + 10 * q^43 + 48 * q^51 - 38 * q^53 + 12 * q^57 + 8 * q^65 + 42 * q^67 + 56 * q^71 + 56 * q^79 + 52 * q^81 + 8 * q^85 - 48 * q^93 + 84 * q^95 + 90 * q^99

Introduction

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