LMFDB - Embedded newform 728.2.a.i.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [728,2,Mod(1,728)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(728, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("728.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 728 = 2^{3} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	728.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,1,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$5.81310926715$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.64268.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 12x^{2} + 6x + 32 $ x^4 - x^3 - 12x^2 + 6x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.19455$ of defining polynomial
Character	$\chi$	$=$	728.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.19455 q^{3} -1.70715 q^{5} -1.00000 q^{7} +1.81604 q^{9} +4.56246 q^{11} -1.00000 q^{13} -3.74642 q^{15} +7.37850 q^{17} +6.46416 q^{19} -2.19455 q^{21} +5.90170 q^{23} -2.08565 q^{25} -2.59825 q^{27} +2.69655 q^{29} +0.523191 q^{31} +10.0125 q^{33} +1.70715 q^{35} +2.17336 q^{37} -2.19455 q^{39} -4.98735 q^{41} +2.32864 q^{43} -3.10025 q^{45} -12.3266 q^{47} +1.00000 q^{49} +16.1925 q^{51} -4.71774 q^{53} -7.78879 q^{55} +14.1859 q^{57} -13.7822 q^{59} -0.137574 q^{61} -1.81604 q^{63} +1.70715 q^{65} -7.96215 q^{67} +12.9516 q^{69} -13.8392 q^{71} +2.52319 q^{73} -4.57706 q^{75} -4.56246 q^{77} +4.85531 q^{79} -11.1501 q^{81} -7.51054 q^{83} -12.5962 q^{85} +5.91772 q^{87} -10.8321 q^{89} +1.00000 q^{91} +1.14817 q^{93} -11.0353 q^{95} +17.6693 q^{97} +8.28563 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} + 2 q^{5} - 4 q^{7} + 13 q^{9} - q^{11} - 4 q^{13} + 10 q^{15} + 16 q^{17} - 10 q^{19} - q^{21} + 7 q^{23} + 14 q^{25} + 13 q^{27} + 4 q^{29} - q^{31} - q^{33} - 2 q^{35} + 5 q^{37} - q^{39}+ \cdots - 48 q^{99}+O(q^{100}) $ 4 * q + q^3 + 2 * q^5 - 4 * q^7 + 13 * q^9 - q^11 - 4 * q^13 + 10 * q^15 + 16 * q^17 - 10 * q^19 - q^21 + 7 * q^23 + 14 * q^25 + 13 * q^27 + 4 * q^29 - q^31 - q^33 - 2 * q^35 + 5 * q^37 - q^39 + 19 * q^41 + 14 * q^43 + 10 * q^45 - 13 * q^47 + 4 * q^49 + 16 * q^51 - 8 * q^53 + 2 * q^55 + 12 * q^57 - 26 * q^59 - q^61 - 13 * q^63 - 2 * q^65 + 5 * q^67 + 17 * q^69 - 18 * q^71 + 7 * q^73 + q^75 + q^77 + 9 * q^79 - 4 * q^81 + 12 * q^83 + 14 * q^85 - 46 * q^87 + 4 * q^89 + 4 * q^91 + 3 * q^93 - 26 * q^95 + 25 * q^97 - 48 * 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
$3$	2.19455	1.26702	0.633512	−	0.773733i	$-0.281613\pi$
$3$	2.19455	1.26702	0.633512	+	0.773733i	$0.281613\pi$
$4$	0	0
$5$	−1.70715	−0.763459	−0.381730	−	0.924274i	$-0.624671\pi$
$5$	−1.70715	−0.763459	−0.381730	+	0.924274i	$0.624671\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.81604	0.605348
$10$	0	0
$11$	4.56246	1.37563	0.687817	−	0.725884i	$-0.258569\pi$
$11$	4.56246	1.37563	0.687817	+	0.725884i	$0.258569\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−3.74642	−0.967320
$16$	0	0
$17$	7.37850	1.78955	0.894775	−	0.446517i	$-0.147336\pi$
$17$	7.37850	1.78955	0.894775	+	0.446517i	$0.147336\pi$
$18$	0	0
$19$	6.46416	1.48298	0.741489	−	0.670964i	$-0.234120\pi$
$19$	6.46416	1.48298	0.741489	+	0.670964i	$0.234120\pi$
$20$	0	0
$21$	−2.19455	−0.478890
$22$	0	0
$23$	5.90170	1.23059	0.615294	−	0.788298i	$-0.289037\pi$
$23$	5.90170	1.23059	0.615294	+	0.788298i	$0.289037\pi$
$24$	0	0
$25$	−2.08565	−0.417130
$26$	0	0
$27$	−2.59825	−0.500033
$28$	0	0
$29$	2.69655	0.500737	0.250369	−	0.968151i	$-0.419448\pi$
$29$	2.69655	0.500737	0.250369	+	0.968151i	$0.419448\pi$
$30$	0	0
$31$	0.523191	0.0939678	0.0469839	−	0.998896i	$-0.485039\pi$
$31$	0.523191	0.0939678	0.0469839	+	0.998896i	$0.485039\pi$
$32$	0	0
$33$	10.0125	1.74296
$34$	0	0
$35$	1.70715	0.288560
$36$	0	0
$37$	2.17336	0.357299	0.178649	−	0.983913i	$-0.442827\pi$
$37$	2.17336	0.357299	0.178649	+	0.983913i	$0.442827\pi$
$38$	0	0
$39$	−2.19455	−0.351409
$40$	0	0
$41$	−4.98735	−0.778893	−0.389446	−	0.921049i	$-0.627334\pi$
$41$	−4.98735	−0.778893	−0.389446	+	0.921049i	$0.627334\pi$
$42$	0	0
$43$	2.32864	0.355115	0.177557	−	0.984110i	$-0.443180\pi$
$43$	2.32864	0.355115	0.177557	+	0.984110i	$0.443180\pi$
$44$	0	0
$45$	−3.10025	−0.462158
$46$	0	0
$47$	−12.3266	−1.79802	−0.899008	−	0.437932i	$-0.855711\pi$
$47$	−12.3266	−1.79802	−0.899008	+	0.437932i	$0.855711\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	16.1925	2.26740
$52$	0	0
$53$	−4.71774	−0.648031	−0.324016	−	0.946052i	$-0.605033\pi$
$53$	−4.71774	−0.648031	−0.324016	+	0.946052i	$0.605033\pi$
$54$	0	0
$55$	−7.78879	−1.05024
$56$	0	0
$57$	14.1859	1.87897
$58$	0	0
$59$	−13.7822	−1.79429	−0.897145	−	0.441736i	$-0.854363\pi$
$59$	−13.7822	−1.79429	−0.897145	+	0.441736i	$0.854363\pi$
$60$	0	0
$61$	−0.137574	−0.0176146	−0.00880730	−	0.999961i	$-0.502803\pi$
$61$	−0.137574	−0.0176146	−0.00880730	+	0.999961i	$0.502803\pi$
$62$	0	0
$63$	−1.81604	−0.228800
$64$	0	0
$65$	1.70715	0.211745
$66$	0	0
$67$	−7.96215	−0.972732	−0.486366	−	0.873755i	$-0.661678\pi$
$67$	−7.96215	−0.972732	−0.486366	+	0.873755i	$0.661678\pi$
$68$	0	0
$69$	12.9516	1.55918
$70$	0	0
$71$	−13.8392	−1.64241	−0.821204	−	0.570634i	$-0.806697\pi$
$71$	−13.8392	−1.64241	−0.821204	+	0.570634i	$0.806697\pi$
$72$	0	0
$73$	2.52319	0.295317	0.147659	−	0.989038i	$-0.452826\pi$
$73$	2.52319	0.295317	0.147659	+	0.989038i	$0.452826\pi$
$74$	0	0
$75$	−4.57706	−0.528514
$76$	0	0
$77$	−4.56246	−0.519941
$78$	0	0
$79$	4.85531	0.546265	0.273133	−	0.961976i	$-0.411940\pi$
$79$	4.85531	0.546265	0.273133	+	0.961976i	$0.411940\pi$
$80$	0	0
$81$	−11.1501	−1.23890
$82$	0	0
$83$	−7.51054	−0.824389	−0.412194	−	0.911096i	$-0.635237\pi$
$83$	−7.51054	−0.824389	−0.412194	+	0.911096i	$0.635237\pi$
$84$	0	0
$85$	−12.5962	−1.36625
$86$	0	0
$87$	5.91772	0.634446
$88$	0	0
$89$	−10.8321	−1.14820	−0.574098	−	0.818786i	$-0.694647\pi$
$89$	−10.8321	−1.14820	−0.574098	+	0.818786i	$0.694647\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	1.14817	0.119059
$94$	0	0
$95$	−11.0353	−1.13219
$96$	0	0
$97$	17.6693	1.79405	0.897023	−	0.441985i	$-0.145725\pi$
$97$	17.6693	1.79405	0.897023	+	0.441985i	$0.145725\pi$
$98$	0	0
$99$	8.28563	0.832737
$100$	0	0
$101$	18.1692	1.80791	0.903954	−	0.427631i	$-0.140652\pi$
$101$	18.1692	1.80791	0.903954	+	0.427631i	$0.140652\pi$
$102$	0	0
$103$	15.1461	1.49239	0.746195	−	0.665727i	$-0.231879\pi$
$103$	15.1461	1.49239	0.746195	+	0.665727i	$0.231879\pi$
$104$	0	0
$105$	3.74642	0.365613
$106$	0	0
$107$	0.657284	0.0635420	0.0317710	−	0.999495i	$-0.489885\pi$
$107$	0.657284	0.0635420	0.0317710	+	0.999495i	$0.489885\pi$
$108$	0	0
$109$	7.76760	0.744001	0.372001	−	0.928232i	$-0.378672\pi$
$109$	7.76760	0.744001	0.372001	+	0.928232i	$0.378672\pi$
$110$	0	0
$111$	4.76955	0.452706
$112$	0	0
$113$	4.87650	0.458743	0.229371	−	0.973339i	$-0.426333\pi$
$113$	4.87650	0.458743	0.229371	+	0.973339i	$0.426333\pi$
$114$	0	0
$115$	−10.0751	−0.939504
$116$	0	0
$117$	−1.81604	−0.167893
$118$	0	0
$119$	−7.37850	−0.676386
$120$	0	0
$121$	9.81604	0.892368
$122$	0	0
$123$	−10.9450	−0.986875
$124$	0	0
$125$	12.0962	1.08192
$126$	0	0
$127$	11.6446	1.03329	0.516647	−	0.856199i	$-0.327180\pi$
$127$	11.6446	1.03329	0.516647	+	0.856199i	$0.327180\pi$
$128$	0	0
$129$	5.11032	0.449938
$130$	0	0
$131$	−15.4928	−1.35362	−0.676808	−	0.736160i	$-0.736637\pi$
$131$	−15.4928	−1.35362	−0.676808	+	0.736160i	$0.736637\pi$
$132$	0	0
$133$	−6.46416	−0.560513
$134$	0	0
$135$	4.43559	0.381755
$136$	0	0
$137$	5.16071	0.440909	0.220455	−	0.975397i	$-0.429246\pi$
$137$	5.16071	0.440909	0.220455	+	0.975397i	$0.429246\pi$
$138$	0	0
$139$	−9.76760	−0.828477	−0.414239	−	0.910168i	$-0.635952\pi$
$139$	−9.76760	−0.828477	−0.414239	+	0.910168i	$0.635952\pi$
$140$	0	0
$141$	−27.0513	−2.27813
$142$	0	0
$143$	−4.56246	−0.381532
$144$	0	0
$145$	−4.60341	−0.382293
$146$	0	0
$147$	2.19455	0.181003
$148$	0	0
$149$	8.38704	0.687093	0.343546	−	0.939136i	$-0.388372\pi$
$149$	8.38704	0.687093	0.343546	+	0.939136i	$0.388372\pi$
$150$	0	0
$151$	−12.5246	−1.01924	−0.509619	−	0.860400i	$-0.670214\pi$
$151$	−12.5246	−1.01924	−0.509619	+	0.860400i	$0.670214\pi$
$152$	0	0
$153$	13.3997	1.08330
$154$	0	0
$155$	−0.893163	−0.0717406
$156$	0	0
$157$	−5.19856	−0.414890	−0.207445	−	0.978247i	$-0.566515\pi$
$157$	−5.19856	−0.414890	−0.207445	+	0.978247i	$0.566515\pi$
$158$	0	0
$159$	−10.3533	−0.821071
$160$	0	0
$161$	−5.90170	−0.465119
$162$	0	0
$163$	16.8498	1.31978	0.659888	−	0.751364i	$-0.270604\pi$
$163$	16.8498	1.31978	0.659888	+	0.751364i	$0.270604\pi$
$164$	0	0
$165$	−17.0929	−1.33068
$166$	0	0
$167$	−22.1960	−1.71758	−0.858788	−	0.512331i	$-0.828782\pi$
$167$	−22.1960	−1.71758	−0.858788	+	0.512331i	$0.828782\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	11.7392	0.897718
$172$	0	0
$173$	−6.98941	−0.531395	−0.265697	−	0.964056i	$-0.585602\pi$
$173$	−6.98941	−0.531395	−0.265697	+	0.964056i	$0.585602\pi$
$174$	0	0
$175$	2.08565	0.157660
$176$	0	0
$177$	−30.2457	−2.27341
$178$	0	0
$179$	14.2106	1.06215	0.531074	−	0.847325i	$-0.321789\pi$
$179$	14.2106	1.06215	0.531074	+	0.847325i	$0.321789\pi$
$180$	0	0
$181$	−18.9516	−1.40866	−0.704329	−	0.709874i	$-0.748752\pi$
$181$	−18.9516	−1.40866	−0.704329	+	0.709874i	$0.748752\pi$
$182$	0	0
$183$	−0.301914	−0.0223181
$184$	0	0
$185$	−3.71025	−0.272783
$186$	0	0
$187$	33.6641	2.46177
$188$	0	0
$189$	2.59825	0.188995
$190$	0	0
$191$	3.34272	0.241870	0.120935	−	0.992660i	$-0.461411\pi$
$191$	3.34272	0.241870	0.120935	+	0.992660i	$0.461411\pi$
$192$	0	0
$193$	3.12492	0.224937	0.112468	−	0.993655i	$-0.464124\pi$
$193$	3.12492	0.224937	0.112468	+	0.993655i	$0.464124\pi$
$194$	0	0
$195$	3.74642	0.268286
$196$	0	0
$197$	−23.7867	−1.69473	−0.847367	−	0.531008i	$-0.821814\pi$
$197$	−23.7867	−1.69473	−0.847367	+	0.531008i	$0.821814\pi$
$198$	0	0
$199$	1.57511	0.111657	0.0558284	−	0.998440i	$-0.482220\pi$
$199$	1.57511	0.111657	0.0558284	+	0.998440i	$0.482220\pi$
$200$	0	0
$201$	−17.4733	−1.23247
$202$	0	0
$203$	−2.69655	−0.189261
$204$	0	0
$205$	8.51413	0.594653
$206$	0	0
$207$	10.7177	0.744934
$208$	0	0
$209$	29.4925	2.04004
$210$	0	0
$211$	−13.5463	−0.932567	−0.466284	−	0.884635i	$-0.654407\pi$
$211$	−13.5463	−0.932567	−0.466284	+	0.884635i	$0.654407\pi$
$212$	0	0
$213$	−30.3708	−2.08097
$214$	0	0
$215$	−3.97533	−0.271115
$216$	0	0
$217$	−0.523191	−0.0355165
$218$	0	0
$219$	5.53726	0.374174
$220$	0	0
$221$	−7.37850	−0.496332
$222$	0	0
$223$	9.56957	0.640826	0.320413	−	0.947278i	$-0.396178\pi$
$223$	9.56957	0.640826	0.320413	+	0.947278i	$0.396178\pi$
$224$	0	0
$225$	−3.78763	−0.252509
$226$	0	0
$227$	−12.7066	−0.843368	−0.421684	−	0.906743i	$-0.638561\pi$
$227$	−12.7066	−0.843368	−0.421684	+	0.906743i	$0.638561\pi$
$228$	0	0
$229$	−2.97480	−0.196581	−0.0982903	−	0.995158i	$-0.531337\pi$
$229$	−2.97480	−0.196581	−0.0982903	+	0.995158i	$0.531337\pi$
$230$	0	0
$231$	−10.0125	−0.658777
$232$	0	0
$233$	−9.06899	−0.594129	−0.297065	−	0.954857i	$-0.596008\pi$
$233$	−9.06899	−0.594129	−0.297065	+	0.954857i	$0.596008\pi$
$234$	0	0
$235$	21.0433	1.37271
$236$	0	0
$237$	10.6552	0.692131
$238$	0	0
$239$	12.3891	0.801384	0.400692	−	0.916213i	$-0.368770\pi$
$239$	12.3891	0.801384	0.400692	+	0.916213i	$0.368770\pi$
$240$	0	0
$241$	−22.9530	−1.47853	−0.739266	−	0.673414i	$-0.764827\pi$
$241$	−22.9530	−1.47853	−0.739266	+	0.673414i	$0.764827\pi$
$242$	0	0
$243$	−16.6747	−1.06968
$244$	0	0
$245$	−1.70715	−0.109066
$246$	0	0
$247$	−6.46416	−0.411304
$248$	0	0
$249$	−16.4822	−1.04452
$250$	0	0
$251$	−7.16935	−0.452526	−0.226263	−	0.974066i	$-0.572651\pi$
$251$	−7.16935	−0.452526	−0.226263	+	0.974066i	$0.572651\pi$
$252$	0	0
$253$	26.9263	1.69284
$254$	0	0
$255$	−27.6429	−1.73107
$256$	0	0
$257$	−0.813983	−0.0507749	−0.0253874	−	0.999678i	$-0.508082\pi$
$257$	−0.813983	−0.0507749	−0.0253874	+	0.999678i	$0.508082\pi$
$258$	0	0
$259$	−2.17336	−0.135046
$260$	0	0
$261$	4.89706	0.303120
$262$	0	0
$263$	22.0140	1.35744	0.678720	−	0.734398i	$-0.262535\pi$
$263$	22.0140	1.35744	0.678720	+	0.734398i	$0.262535\pi$
$264$	0	0
$265$	8.05387	0.494745
$266$	0	0
$267$	−23.7715	−1.45479
$268$	0	0
$269$	−6.77367	−0.412998	−0.206499	−	0.978447i	$-0.566207\pi$
$269$	−6.77367	−0.412998	−0.206499	+	0.978447i	$0.566207\pi$
$270$	0	0
$271$	−15.3341	−0.931479	−0.465739	−	0.884922i	$-0.654212\pi$
$271$	−15.3341	−0.931479	−0.465739	+	0.884922i	$0.654212\pi$
$272$	0	0
$273$	2.19455	0.132820
$274$	0	0
$275$	−9.51570	−0.573818
$276$	0	0
$277$	−20.6712	−1.24201	−0.621007	−	0.783805i	$-0.713276\pi$
$277$	−20.6712	−1.24201	−0.621007	+	0.783805i	$0.713276\pi$
$278$	0	0
$279$	0.950137	0.0568832
$280$	0	0
$281$	28.8925	1.72358	0.861792	−	0.507262i	$-0.169342\pi$
$281$	28.8925	1.72358	0.861792	+	0.507262i	$0.169342\pi$
$282$	0	0
$283$	−6.98735	−0.415355	−0.207677	−	0.978197i	$-0.566590\pi$
$283$	−6.98735	−0.415355	−0.207677	+	0.978197i	$0.566590\pi$
$284$	0	0
$285$	−24.2174	−1.43452
$286$	0	0
$287$	4.98735	0.294394
$288$	0	0
$289$	37.4423	2.20249
$290$	0	0
$291$	38.7761	2.27310
$292$	0	0
$293$	−1.94613	−0.113694	−0.0568470	−	0.998383i	$-0.518105\pi$
$293$	−1.94613	−0.113694	−0.0568470	+	0.998383i	$0.518105\pi$
$294$	0	0
$295$	23.5282	1.36987
$296$	0	0
$297$	−11.8544	−0.687862
$298$	0	0
$299$	−5.90170	−0.341304
$300$	0	0
$301$	−2.32864	−0.134221
$302$	0	0
$303$	39.8733	2.29066
$304$	0	0
$305$	0.234860	0.0134480
$306$	0	0
$307$	−4.24636	−0.242353	−0.121176	−	0.992631i	$-0.538667\pi$
$307$	−4.24636	−0.242353	−0.121176	+	0.992631i	$0.538667\pi$
$308$	0	0
$309$	33.2389	1.89089
$310$	0	0
$311$	4.20709	0.238562	0.119281	−	0.992861i	$-0.461941\pi$
$311$	4.20709	0.238562	0.119281	+	0.992861i	$0.461941\pi$
$312$	0	0
$313$	−5.08913	−0.287655	−0.143827	−	0.989603i	$-0.545941\pi$
$313$	−5.08913	−0.287655	−0.143827	+	0.989603i	$0.545941\pi$
$314$	0	0
$315$	3.10025	0.174679
$316$	0	0
$317$	9.29790	0.522222	0.261111	−	0.965309i	$-0.415911\pi$
$317$	9.29790	0.522222	0.261111	+	0.965309i	$0.415911\pi$
$318$	0	0
$319$	12.3029	0.688831
$320$	0	0
$321$	1.44244	0.0805092
$322$	0	0
$323$	47.6958	2.65387
$324$	0	0
$325$	2.08565	0.115691
$326$	0	0
$327$	17.0464	0.942667
$328$	0	0
$329$	12.3266	0.679586
$330$	0	0
$331$	−23.5981	−1.29707	−0.648535	−	0.761184i	$-0.724618\pi$
$331$	−23.5981	−1.29707	−0.648535	+	0.761184i	$0.724618\pi$
$332$	0	0
$333$	3.94692	0.216290
$334$	0	0
$335$	13.5926	0.742641
$336$	0	0
$337$	4.87650	0.265640	0.132820	−	0.991140i	$-0.457597\pi$
$337$	4.87650	0.265640	0.132820	+	0.991140i	$0.457597\pi$
$338$	0	0
$339$	10.7017	0.581237
$340$	0	0
$341$	2.38704	0.129265
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−22.1102	−1.19037
$346$	0	0
$347$	−33.1208	−1.77802	−0.889009	−	0.457890i	$-0.848605\pi$
$347$	−33.1208	−1.77802	−0.889009	+	0.457890i	$0.848605\pi$
$348$	0	0
$349$	0.589188	0.0315385	0.0157693	−	0.999876i	$-0.494980\pi$
$349$	0.589188	0.0315385	0.0157693	+	0.999876i	$0.494980\pi$
$350$	0	0
$351$	2.59825	0.138684
$352$	0	0
$353$	−3.08707	−0.164308	−0.0821541	−	0.996620i	$-0.526180\pi$
$353$	−3.08707	−0.164308	−0.0821541	+	0.996620i	$0.526180\pi$
$354$	0	0
$355$	23.6255	1.25391
$356$	0	0
$357$	−16.1925	−0.856997
$358$	0	0
$359$	18.5750	0.980351	0.490176	−	0.871624i	$-0.336933\pi$
$359$	18.5750	0.980351	0.490176	+	0.871624i	$0.336933\pi$
$360$	0	0
$361$	22.7853	1.19923
$362$	0	0
$363$	21.5418	1.13065
$364$	0	0
$365$	−4.30746	−0.225463
$366$	0	0
$367$	−25.3744	−1.32453	−0.662266	−	0.749269i	$-0.730405\pi$
$367$	−25.3744	−1.32453	−0.662266	+	0.749269i	$0.730405\pi$
$368$	0	0
$369$	−9.05724	−0.471501
$370$	0	0
$371$	4.71774	0.244933
$372$	0	0
$373$	−18.5392	−0.959924	−0.479962	−	0.877289i	$-0.659350\pi$
$373$	−18.5392	−0.959924	−0.479962	+	0.877289i	$0.659350\pi$
$374$	0	0
$375$	26.5458	1.37082
$376$	0	0
$377$	−2.69655	−0.138880
$378$	0	0
$379$	1.76064	0.0904380	0.0452190	−	0.998977i	$-0.485601\pi$
$379$	1.76064	0.0904380	0.0452190	+	0.998977i	$0.485601\pi$
$380$	0	0
$381$	25.5547	1.30921
$382$	0	0
$383$	4.50136	0.230009	0.115004	−	0.993365i	$-0.463312\pi$
$383$	4.50136	0.230009	0.115004	+	0.993365i	$0.463312\pi$
$384$	0	0
$385$	7.78879	0.396953
$386$	0	0
$387$	4.22892	0.214968
$388$	0	0
$389$	−27.9817	−1.41873	−0.709363	−	0.704843i	$-0.751017\pi$
$389$	−27.9817	−1.41873	−0.709363	+	0.704843i	$0.751017\pi$
$390$	0	0
$391$	43.5457	2.20220
$392$	0	0
$393$	−33.9998	−1.71506
$394$	0	0
$395$	−8.28873	−0.417051
$396$	0	0
$397$	3.64979	0.183178	0.0915889	−	0.995797i	$-0.470805\pi$
$397$	3.64979	0.183178	0.0915889	+	0.995797i	$0.470805\pi$
$398$	0	0
$399$	−14.1859	−0.710184
$400$	0	0
$401$	19.3174	0.964665	0.482333	−	0.875988i	$-0.339790\pi$
$401$	19.3174	0.964665	0.482333	+	0.875988i	$0.339790\pi$
$402$	0	0
$403$	−0.523191	−0.0260620
$404$	0	0
$405$	19.0349	0.945851
$406$	0	0
$407$	9.91588	0.491512
$408$	0	0
$409$	8.73234	0.431787	0.215893	−	0.976417i	$-0.430734\pi$
$409$	8.73234	0.431787	0.215893	+	0.976417i	$0.430734\pi$
$410$	0	0
$411$	11.3254	0.558642
$412$	0	0
$413$	13.7822	0.678178
$414$	0	0
$415$	12.8216	0.629387
$416$	0	0
$417$	−21.4355	−1.04970
$418$	0	0
$419$	−9.92636	−0.484935	−0.242467	−	0.970160i	$-0.577957\pi$
$419$	−9.92636	−0.484935	−0.242467	+	0.970160i	$0.577957\pi$
$420$	0	0
$421$	12.4697	0.607736	0.303868	−	0.952714i	$-0.401722\pi$
$421$	12.4697	0.607736	0.303868	+	0.952714i	$0.401722\pi$
$422$	0	0
$423$	−22.3856	−1.08843
$424$	0	0
$425$	−15.3890	−0.746475
$426$	0	0
$427$	0.137574	0.00665769
$428$	0	0
$429$	−10.0125	−0.483410
$430$	0	0
$431$	−0.154238	−0.00742937	−0.00371469	−	0.999993i	$-0.501182\pi$
$431$	−0.154238	−0.00742937	−0.00371469	+	0.999993i	$0.501182\pi$
$432$	0	0
$433$	33.6853	1.61881	0.809407	−	0.587249i	$-0.199789\pi$
$433$	33.6853	1.61881	0.809407	+	0.587249i	$0.199789\pi$
$434$	0	0
$435$	−10.1024	−0.484374
$436$	0	0
$437$	38.1495	1.82494
$438$	0	0
$439$	−14.5604	−0.694930	−0.347465	−	0.937693i	$-0.612957\pi$
$439$	−14.5604	−0.694930	−0.347465	+	0.937693i	$0.612957\pi$
$440$	0	0
$441$	1.81604	0.0864783
$442$	0	0
$443$	−9.82147	−0.466632	−0.233316	−	0.972401i	$-0.574958\pi$
$443$	−9.82147	−0.466632	−0.233316	+	0.972401i	$0.574958\pi$
$444$	0	0
$445$	18.4919	0.876601
$446$	0	0
$447$	18.4058	0.870563
$448$	0	0
$449$	37.6137	1.77510	0.887551	−	0.460709i	$-0.152405\pi$
$449$	37.6137	1.77510	0.887551	+	0.460709i	$0.152405\pi$
$450$	0	0
$451$	−22.7546	−1.07147
$452$	0	0
$453$	−27.4859	−1.29140
$454$	0	0
$455$	−1.70715	−0.0800323
$456$	0	0
$457$	22.0821	1.03296	0.516478	−	0.856301i	$-0.327243\pi$
$457$	22.0821	1.03296	0.516478	+	0.856301i	$0.327243\pi$
$458$	0	0
$459$	−19.1712	−0.894835
$460$	0	0
$461$	−40.2881	−1.87640	−0.938202	−	0.346089i	$-0.887510\pi$
$461$	−40.2881	−1.87640	−0.938202	+	0.346089i	$0.887510\pi$
$462$	0	0
$463$	−9.60432	−0.446351	−0.223175	−	0.974778i	$-0.571642\pi$
$463$	−9.60432	−0.446351	−0.223175	+	0.974778i	$0.571642\pi$
$464$	0	0
$465$	−1.96009	−0.0908970
$466$	0	0
$467$	19.0822	0.883018	0.441509	−	0.897257i	$-0.354443\pi$
$467$	19.0822	0.883018	0.441509	+	0.897257i	$0.354443\pi$
$468$	0	0
$469$	7.96215	0.367658
$470$	0	0
$471$	−11.4085	−0.525675
$472$	0	0
$473$	10.6243	0.488508
$474$	0	0
$475$	−13.4820	−0.618595
$476$	0	0
$477$	−8.56762	−0.392285
$478$	0	0
$479$	0.825105	0.0377000	0.0188500	−	0.999822i	$-0.494000\pi$
$479$	0.825105	0.0377000	0.0188500	+	0.999822i	$0.494000\pi$
$480$	0	0
$481$	−2.17336	−0.0990968
$482$	0	0
$483$	−12.9516	−0.589316
$484$	0	0
$485$	−30.1641	−1.36968
$486$	0	0
$487$	41.3466	1.87359	0.936797	−	0.349874i	$-0.113775\pi$
$487$	41.3466	1.87359	0.936797	+	0.349874i	$0.113775\pi$
$488$	0	0
$489$	36.9776	1.67219
$490$	0	0
$491$	−40.9283	−1.84707	−0.923534	−	0.383516i	$-0.874713\pi$
$491$	−40.9283	−1.84707	−0.923534	+	0.383516i	$0.874713\pi$
$492$	0	0
$493$	19.8965	0.896095
$494$	0	0
$495$	−14.1448	−0.635761
$496$	0	0
$497$	13.8392	0.620772
$498$	0	0
$499$	12.8306	0.574379	0.287189	−	0.957874i	$-0.407279\pi$
$499$	12.8306	0.574379	0.287189	+	0.957874i	$0.407279\pi$
$500$	0	0
$501$	−48.7101	−2.17621
$502$	0	0
$503$	−17.3931	−0.775520	−0.387760	−	0.921760i	$-0.626751\pi$
$503$	−17.3931	−0.775520	−0.387760	+	0.921760i	$0.626751\pi$
$504$	0	0
$505$	−31.0176	−1.38026
$506$	0	0
$507$	2.19455	0.0974633
$508$	0	0
$509$	7.19608	0.318961	0.159480	−	0.987201i	$-0.449018\pi$
$509$	7.19608	0.318961	0.159480	+	0.987201i	$0.449018\pi$
$510$	0	0
$511$	−2.52319	−0.111619
$512$	0	0
$513$	−16.7955	−0.741539
$514$	0	0
$515$	−25.8566	−1.13938
$516$	0	0
$517$	−56.2395	−2.47341
$518$	0	0
$519$	−15.3386	−0.673290
$520$	0	0
$521$	−24.9495	−1.09306	−0.546529	−	0.837440i	$-0.684051\pi$
$521$	−24.9495	−1.09306	−0.546529	+	0.837440i	$0.684051\pi$
$522$	0	0
$523$	25.5901	1.11898	0.559489	−	0.828838i	$-0.310997\pi$
$523$	25.5901	1.11898	0.559489	+	0.828838i	$0.310997\pi$
$524$	0	0
$525$	4.57706	0.199759
$526$	0	0
$527$	3.86036	0.168160
$528$	0	0
$529$	11.8300	0.514348
$530$	0	0
$531$	−25.0291	−1.08617
$532$	0	0
$533$	4.98735	0.216026
$534$	0	0
$535$	−1.12208	−0.0485117
$536$	0	0
$537$	31.1858	1.34577
$538$	0	0
$539$	4.56246	0.196519
$540$	0	0
$541$	25.5457	1.09829	0.549147	−	0.835726i	$-0.314953\pi$
$541$	25.5457	1.09829	0.549147	+	0.835726i	$0.314953\pi$
$542$	0	0
$543$	−41.5901	−1.78480
$544$	0	0
$545$	−13.2604	−0.568015
$546$	0	0
$547$	21.3567	0.913146	0.456573	−	0.889686i	$-0.349077\pi$
$547$	21.3567	0.913146	0.456573	+	0.889686i	$0.349077\pi$
$548$	0	0
$549$	−0.249841	−0.0106630
$550$	0	0
$551$	17.4309	0.742583
$552$	0	0
$553$	−4.85531	−0.206469
$554$	0	0
$555$	−8.14232	−0.345622
$556$	0	0
$557$	−0.651705	−0.0276136	−0.0138068	−	0.999905i	$-0.504395\pi$
$557$	−0.651705	−0.0276136	−0.0138068	+	0.999905i	$0.504395\pi$
$558$	0	0
$559$	−2.32864	−0.0984911
$560$	0	0
$561$	73.8776	3.11911
$562$	0	0
$563$	12.5479	0.528829	0.264415	−	0.964409i	$-0.414821\pi$
$563$	12.5479	0.528829	0.264415	+	0.964409i	$0.414821\pi$
$564$	0	0
$565$	−8.32490	−0.350231
$566$	0	0
$567$	11.1501	0.468261
$568$	0	0
$569$	20.0983	0.842565	0.421282	−	0.906930i	$-0.361580\pi$
$569$	20.0983	0.842565	0.421282	+	0.906930i	$0.361580\pi$
$570$	0	0
$571$	−14.2953	−0.598240	−0.299120	−	0.954215i	$-0.596693\pi$
$571$	−14.2953	−0.598240	−0.299120	+	0.954215i	$0.596693\pi$
$572$	0	0
$573$	7.33575	0.306456
$574$	0	0
$575$	−12.3089	−0.513316
$576$	0	0
$577$	−1.26418	−0.0526284	−0.0263142	−	0.999654i	$-0.508377\pi$
$577$	−1.26418	−0.0526284	−0.0263142	+	0.999654i	$0.508377\pi$
$578$	0	0
$579$	6.85779	0.285000
$580$	0	0
$581$	7.51054	0.311590
$582$	0	0
$583$	−21.5245	−0.891454
$584$	0	0
$585$	3.10025	0.128180
$586$	0	0
$587$	−28.8210	−1.18957	−0.594785	−	0.803885i	$-0.702763\pi$
$587$	−28.8210	−1.18957	−0.594785	+	0.803885i	$0.702763\pi$
$588$	0	0
$589$	3.38199	0.139352
$590$	0	0
$591$	−52.2011	−2.14727
$592$	0	0
$593$	−19.7534	−0.811176	−0.405588	−	0.914056i	$-0.632933\pi$
$593$	−19.7534	−0.811176	−0.405588	+	0.914056i	$0.632933\pi$
$594$	0	0
$595$	12.5962	0.516393
$596$	0	0
$597$	3.45666	0.141472
$598$	0	0
$599$	−14.1727	−0.579082	−0.289541	−	0.957166i	$-0.593503\pi$
$599$	−14.1727	−0.579082	−0.289541	+	0.957166i	$0.593503\pi$
$600$	0	0
$601$	−36.2922	−1.48039	−0.740195	−	0.672392i	$-0.765267\pi$
$601$	−36.2922	−1.48039	−0.740195	+	0.672392i	$0.765267\pi$
$602$	0	0
$603$	−14.4596	−0.588841
$604$	0	0
$605$	−16.7574	−0.681286
$606$	0	0
$607$	41.5780	1.68760	0.843799	−	0.536660i	$-0.180314\pi$
$607$	41.5780	1.68760	0.843799	+	0.536660i	$0.180314\pi$
$608$	0	0
$609$	−5.91772	−0.239798
$610$	0	0
$611$	12.3266	0.498680
$612$	0	0
$613$	−3.98334	−0.160885	−0.0804427	−	0.996759i	$-0.525633\pi$
$613$	−3.98334	−0.160885	−0.0804427	+	0.996759i	$0.525633\pi$
$614$	0	0
$615$	18.6847	0.753439
$616$	0	0
$617$	−33.5644	−1.35125	−0.675626	−	0.737244i	$-0.736127\pi$
$617$	−33.5644	−1.35125	−0.675626	+	0.737244i	$0.736127\pi$
$618$	0	0
$619$	6.75289	0.271421	0.135711	−	0.990749i	$-0.456668\pi$
$619$	6.75289	0.271421	0.135711	+	0.990749i	$0.456668\pi$
$620$	0	0
$621$	−15.3341	−0.615335
$622$	0	0
$623$	10.8321	0.433978
$624$	0	0
$625$	−10.2218	−0.408872
$626$	0	0
$627$	64.7226	2.58477
$628$	0	0
$629$	16.0362	0.639404
$630$	0	0
$631$	6.05323	0.240975	0.120488	−	0.992715i	$-0.461554\pi$
$631$	6.05323	0.240975	0.120488	+	0.992715i	$0.461554\pi$
$632$	0	0
$633$	−29.7281	−1.18158
$634$	0	0
$635$	−19.8791	−0.788878
$636$	0	0
$637$	−1.00000	−0.0396214
$638$	0	0
$639$	−25.1326	−0.994229
$640$	0	0
$641$	34.1807	1.35006	0.675029	−	0.737791i	$-0.264131\pi$
$641$	34.1807	1.35006	0.675029	+	0.737791i	$0.264131\pi$
$642$	0	0
$643$	−37.1500	−1.46505	−0.732527	−	0.680738i	$-0.761659\pi$
$643$	−37.1500	−1.46505	−0.732527	+	0.680738i	$0.761659\pi$
$644$	0	0
$645$	−8.72406	−0.343510
$646$	0	0
$647$	−11.7172	−0.460651	−0.230326	−	0.973114i	$-0.573979\pi$
$647$	−11.7172	−0.460651	−0.230326	+	0.973114i	$0.573979\pi$
$648$	0	0
$649$	−62.8808	−2.46829
$650$	0	0
$651$	−1.14817	−0.0450002
$652$	0	0
$653$	28.7529	1.12519	0.562594	−	0.826734i	$-0.309804\pi$
$653$	28.7529	1.12519	0.562594	+	0.826734i	$0.309804\pi$
$654$	0	0
$655$	26.4485	1.03343
$656$	0	0
$657$	4.58223	0.178770
$658$	0	0
$659$	−25.2740	−0.984536	−0.492268	−	0.870444i	$-0.663832\pi$
$659$	−25.2740	−0.984536	−0.492268	+	0.870444i	$0.663832\pi$
$660$	0	0
$661$	32.8491	1.27768	0.638842	−	0.769338i	$-0.279414\pi$
$661$	32.8491	1.27768	0.638842	+	0.769338i	$0.279414\pi$
$662$	0	0
$663$	−16.1925	−0.628864
$664$	0	0
$665$	11.0353	0.427929
$666$	0	0
$667$	15.9142	0.616202
$668$	0	0
$669$	21.0009	0.811941
$670$	0	0
$671$	−0.627678	−0.0242312
$672$	0	0
$673$	47.5870	1.83434	0.917172	−	0.398491i	$-0.130466\pi$
$673$	47.5870	1.83434	0.917172	+	0.398491i	$0.130466\pi$
$674$	0	0
$675$	5.41904	0.208579
$676$	0	0
$677$	33.5481	1.28936	0.644680	−	0.764453i	$-0.276991\pi$
$677$	33.5481	1.28936	0.644680	+	0.764453i	$0.276991\pi$
$678$	0	0
$679$	−17.6693	−0.678085
$680$	0	0
$681$	−27.8853	−1.06857
$682$	0	0
$683$	−29.7908	−1.13992	−0.569958	−	0.821674i	$-0.693040\pi$
$683$	−29.7908	−1.13992	−0.569958	+	0.821674i	$0.693040\pi$
$684$	0	0
$685$	−8.81009	−0.336616
$686$	0	0
$687$	−6.52835	−0.249072
$688$	0	0
$689$	4.71774	0.179732
$690$	0	0
$691$	−15.1959	−0.578078	−0.289039	−	0.957317i	$-0.593336\pi$
$691$	−15.1959	−0.578078	−0.289039	+	0.957317i	$0.593336\pi$
$692$	0	0
$693$	−8.28563	−0.314745
$694$	0	0
$695$	16.6747	0.632508
$696$	0	0
$697$	−36.7992	−1.39387
$698$	0	0
$699$	−19.9023	−0.752775
$700$	0	0
$701$	−28.2106	−1.06550	−0.532749	−	0.846273i	$-0.678841\pi$
$701$	−28.2106	−1.06550	−0.532749	+	0.846273i	$0.678841\pi$
$702$	0	0
$703$	14.0490	0.529866
$704$	0	0
$705$	46.1805	1.73926
$706$	0	0
$707$	−18.1692	−0.683325
$708$	0	0
$709$	21.8546	0.820765	0.410383	−	0.911913i	$-0.365395\pi$
$709$	21.8546	0.820765	0.410383	+	0.911913i	$0.365395\pi$
$710$	0	0
$711$	8.81746	0.330681
$712$	0	0
$713$	3.08771	0.115636
$714$	0	0
$715$	7.78879	0.291284
$716$	0	0
$717$	27.1885	1.01537
$718$	0	0
$719$	−45.2563	−1.68778	−0.843888	−	0.536520i	$-0.819739\pi$
$719$	−45.2563	−1.68778	−0.843888	+	0.536520i	$0.819739\pi$
$720$	0	0
$721$	−15.1461	−0.564070
$722$	0	0
$723$	−50.3714	−1.87333
$724$	0	0
$725$	−5.62407	−0.208873
$726$	0	0
$727$	44.8818	1.66457	0.832287	−	0.554345i	$-0.187031\pi$
$727$	44.8818	1.66457	0.832287	+	0.554345i	$0.187031\pi$
$728$	0	0
$729$	−3.14315	−0.116413
$730$	0	0
$731$	17.1819	0.635495
$732$	0	0
$733$	42.0205	1.55206	0.776032	−	0.630693i	$-0.217229\pi$
$733$	42.0205	1.55206	0.776032	+	0.630693i	$0.217229\pi$
$734$	0	0
$735$	−3.74642	−0.138189
$736$	0	0
$737$	−36.3270	−1.33812
$738$	0	0
$739$	29.2958	1.07767	0.538833	−	0.842413i	$-0.318866\pi$
$739$	29.2958	1.07767	0.538833	+	0.842413i	$0.318866\pi$
$740$	0	0
$741$	−14.1859	−0.521132
$742$	0	0
$743$	35.1128	1.28816	0.644081	−	0.764957i	$-0.277240\pi$
$743$	35.1128	1.28816	0.644081	+	0.764957i	$0.277240\pi$
$744$	0	0
$745$	−14.3179	−0.524567
$746$	0	0
$747$	−13.6395	−0.499042
$748$	0	0
$749$	−0.657284	−0.0240166
$750$	0	0
$751$	−17.8128	−0.649999	−0.325000	−	0.945714i	$-0.605364\pi$
$751$	−17.8128	−0.649999	−0.325000	+	0.945714i	$0.605364\pi$
$752$	0	0
$753$	−15.7335	−0.573360
$754$	0	0
$755$	21.3813	0.778147
$756$	0	0
$757$	−25.8850	−0.940807	−0.470404	−	0.882451i	$-0.655892\pi$
$757$	−25.8850	−0.940807	−0.470404	+	0.882451i	$0.655892\pi$
$758$	0	0
$759$	59.0910	2.14487
$760$	0	0
$761$	27.3306	0.990733	0.495367	−	0.868684i	$-0.335034\pi$
$761$	27.3306	0.990733	0.495367	+	0.868684i	$0.335034\pi$
$762$	0	0
$763$	−7.76760	−0.281206
$764$	0	0
$765$	−22.8752	−0.827056
$766$	0	0
$767$	13.7822	0.497647
$768$	0	0
$769$	−25.5192	−0.920245	−0.460123	−	0.887855i	$-0.652195\pi$
$769$	−25.5192	−0.920245	−0.460123	+	0.887855i	$0.652195\pi$
$770$	0	0
$771$	−1.78633	−0.0643330
$772$	0	0
$773$	−3.62797	−0.130489	−0.0652444	−	0.997869i	$-0.520783\pi$
$773$	−3.62797	−0.130489	−0.0652444	+	0.997869i	$0.520783\pi$
$774$	0	0
$775$	−1.09119	−0.0391968
$776$	0	0
$777$	−4.76955	−0.171107
$778$	0	0
$779$	−32.2390	−1.15508
$780$	0	0
$781$	−63.1407	−2.25935
$782$	0	0
$783$	−7.00632	−0.250385
$784$	0	0
$785$	8.87470	0.316752
$786$	0	0
$787$	13.3886	0.477251	0.238625	−	0.971112i	$-0.423303\pi$
$787$	13.3886	0.477251	0.238625	+	0.971112i	$0.423303\pi$
$788$	0	0
$789$	48.3107	1.71991
$790$	0	0
$791$	−4.87650	−0.173388
$792$	0	0
$793$	0.137574	0.00488541
$794$	0	0
$795$	17.6746	0.626854
$796$	0	0
$797$	−5.95519	−0.210944	−0.105472	−	0.994422i	$-0.533635\pi$
$797$	−5.95519	−0.210944	−0.105472	+	0.994422i	$0.533635\pi$
$798$	0	0
$799$	−90.9517	−3.21764
$800$	0	0
$801$	−19.6715	−0.695059
$802$	0	0
$803$	11.5120	0.406248
$804$	0	0
$805$	10.0751	0.355099
$806$	0	0
$807$	−14.8652	−0.523278
$808$	0	0
$809$	39.3205	1.38244	0.691218	−	0.722647i	$-0.257075\pi$
$809$	39.3205	1.38244	0.691218	+	0.722647i	$0.257075\pi$
$810$	0	0
$811$	−40.9353	−1.43743	−0.718716	−	0.695304i	$-0.755270\pi$
$811$	−40.9353	−1.43743	−0.718716	+	0.695304i	$0.755270\pi$
$812$	0	0
$813$	−33.6514	−1.18021
$814$	0	0
$815$	−28.7650	−1.00759
$816$	0	0
$817$	15.0527	0.526627
$818$	0	0
$819$	1.81604	0.0634577
$820$	0	0
$821$	12.7994	0.446701	0.223351	−	0.974738i	$-0.428300\pi$
$821$	12.7994	0.446701	0.223351	+	0.974738i	$0.428300\pi$
$822$	0	0
$823$	−2.76153	−0.0962609	−0.0481305	−	0.998841i	$-0.515326\pi$
$823$	−2.76153	−0.0962609	−0.0481305	+	0.998841i	$0.515326\pi$
$824$	0	0
$825$	−20.8827	−0.727041
$826$	0	0
$827$	7.37813	0.256563	0.128281	−	0.991738i	$-0.459054\pi$
$827$	7.37813	0.256563	0.128281	+	0.991738i	$0.459054\pi$
$828$	0	0
$829$	39.5245	1.37274	0.686371	−	0.727251i	$-0.259203\pi$
$829$	39.5245	1.37274	0.686371	+	0.727251i	$0.259203\pi$
$830$	0	0
$831$	−45.3641	−1.57366
$832$	0	0
$833$	7.37850	0.255650
$834$	0	0
$835$	37.8918	1.31130
$836$	0	0
$837$	−1.35938	−0.0469870
$838$	0	0
$839$	−13.5123	−0.466498	−0.233249	−	0.972417i	$-0.574936\pi$
$839$	−13.5123	−0.466498	−0.233249	+	0.972417i	$0.574936\pi$
$840$	0	0
$841$	−21.7286	−0.749262
$842$	0	0
$843$	63.4060	2.18382
$844$	0	0
$845$	−1.70715	−0.0587276
$846$	0	0
$847$	−9.81604	−0.337283
$848$	0	0
$849$	−15.3341	−0.526264
$850$	0	0
$851$	12.8265	0.439688
$852$	0	0
$853$	6.40680	0.219365	0.109682	−	0.993967i	$-0.465017\pi$
$853$	6.40680	0.219365	0.109682	+	0.993967i	$0.465017\pi$
$854$	0	0
$855$	−20.0405	−0.685371
$856$	0	0
$857$	27.5856	0.942306	0.471153	−	0.882052i	$-0.343838\pi$
$857$	27.5856	0.942306	0.471153	+	0.882052i	$0.343838\pi$
$858$	0	0
$859$	26.9943	0.921034	0.460517	−	0.887651i	$-0.347664\pi$
$859$	26.9943	0.921034	0.460517	+	0.887651i	$0.347664\pi$
$860$	0	0
$861$	10.9450	0.373004
$862$	0	0
$863$	29.1208	0.991284	0.495642	−	0.868527i	$-0.334933\pi$
$863$	29.1208	0.991284	0.495642	+	0.868527i	$0.334933\pi$
$864$	0	0
$865$	11.9319	0.405698
$866$	0	0
$867$	82.1690	2.79061
$868$	0	0
$869$	22.1522	0.751461
$870$	0	0
$871$	7.96215	0.269787
$872$	0	0
$873$	32.0882	1.08602
$874$	0	0
$875$	−12.0962	−0.408928
$876$	0	0
$877$	12.3126	0.415768	0.207884	−	0.978154i	$-0.433342\pi$
$877$	12.3126	0.415768	0.207884	+	0.978154i	$0.433342\pi$
$878$	0	0
$879$	−4.27087	−0.144053
$880$	0	0
$881$	7.93644	0.267386	0.133693	−	0.991023i	$-0.457316\pi$
$881$	7.93644	0.267386	0.133693	+	0.991023i	$0.457316\pi$
$882$	0	0
$883$	−18.1672	−0.611374	−0.305687	−	0.952132i	$-0.598886\pi$
$883$	−18.1672	−0.611374	−0.305687	+	0.952132i	$0.598886\pi$
$884$	0	0
$885$	51.6339	1.73565
$886$	0	0
$887$	−7.17531	−0.240923	−0.120462	−	0.992718i	$-0.538437\pi$
$887$	−7.17531	−0.240923	−0.120462	+	0.992718i	$0.538437\pi$
$888$	0	0
$889$	−11.6446	−0.390548
$890$	0	0
$891$	−50.8720	−1.70427
$892$	0	0
$893$	−79.6809	−2.66642
$894$	0	0
$895$	−24.2595	−0.810907
$896$	0	0
$897$	−12.9516	−0.432440
$898$	0	0
$899$	1.41081	0.0470532
$900$	0	0
$901$	−34.8099	−1.15968
$902$	0	0
$903$	−5.11032	−0.170061
$904$	0	0
$905$	32.3531	1.07545
$906$	0	0
$907$	48.8890	1.62333	0.811667	−	0.584120i	$-0.198560\pi$
$907$	48.8890	1.62333	0.811667	+	0.584120i	$0.198560\pi$
$908$	0	0
$909$	32.9961	1.09441
$910$	0	0
$911$	−42.2780	−1.40073	−0.700367	−	0.713783i	$-0.746980\pi$
$911$	−42.2780	−1.40073	−0.700367	+	0.713783i	$0.746980\pi$
$912$	0	0
$913$	−34.2665	−1.13406
$914$	0	0
$915$	0.515411	0.0170390
$916$	0	0
$917$	15.4928	0.511618
$918$	0	0
$919$	24.7334	0.815879	0.407940	−	0.913009i	$-0.366247\pi$
$919$	24.7334	0.815879	0.407940	+	0.913009i	$0.366247\pi$
$920$	0	0
$921$	−9.31884	−0.307066
$922$	0	0
$923$	13.8392	0.455522
$924$	0	0
$925$	−4.53288	−0.149040
$926$	0	0
$927$	27.5060	0.903416
$928$	0	0
$929$	−17.1260	−0.561885	−0.280942	−	0.959725i	$-0.590647\pi$
$929$	−17.1260	−0.561885	−0.280942	+	0.959725i	$0.590647\pi$
$930$	0	0
$931$	6.46416	0.211854
$932$	0	0
$933$	9.23266	0.302264
$934$	0	0
$935$	−57.4696	−1.87946
$936$	0	0
$937$	−17.8221	−0.582223	−0.291112	−	0.956689i	$-0.594025\pi$
$937$	−17.8221	−0.582223	−0.291112	+	0.956689i	$0.594025\pi$
$938$	0	0
$939$	−11.1683	−0.364465
$940$	0	0
$941$	−32.2675	−1.05189	−0.525946	−	0.850518i	$-0.676289\pi$
$941$	−32.2675	−1.05189	−0.525946	+	0.850518i	$0.676289\pi$
$942$	0	0
$943$	−29.4338	−0.958496
$944$	0	0
$945$	−4.43559	−0.144290
$946$	0	0
$947$	−19.4143	−0.630880	−0.315440	−	0.948946i	$-0.602152\pi$
$947$	−19.4143	−0.630880	−0.315440	+	0.948946i	$0.602152\pi$
$948$	0	0
$949$	−2.52319	−0.0819062
$950$	0	0
$951$	20.4047	0.661668
$952$	0	0
$953$	30.0644	0.973880	0.486940	−	0.873435i	$-0.338113\pi$
$953$	30.0644	0.973880	0.486940	+	0.873435i	$0.338113\pi$
$954$	0	0
$955$	−5.70651	−0.184658
$956$	0	0
$957$	26.9994	0.872765
$958$	0	0
$959$	−5.16071	−0.166648
$960$	0	0
$961$	−30.7263	−0.991170
$962$	0	0
$963$	1.19366	0.0384650
$964$	0	0
$965$	−5.33470	−0.171730
$966$	0	0
$967$	20.4288	0.656945	0.328473	−	0.944513i	$-0.393466\pi$
$967$	20.4288	0.656945	0.328473	+	0.944513i	$0.393466\pi$
$968$	0	0
$969$	104.671	3.36251
$970$	0	0
$971$	47.3661	1.52005	0.760026	−	0.649893i	$-0.225186\pi$
$971$	47.3661	1.52005	0.760026	+	0.649893i	$0.225186\pi$
$972$	0	0
$973$	9.76760	0.313135
$974$	0	0
$975$	4.57706	0.146583
$976$	0	0
$977$	−31.8816	−1.01998	−0.509991	−	0.860180i	$-0.670351\pi$
$977$	−31.8816	−1.01998	−0.509991	+	0.860180i	$0.670351\pi$
$978$	0	0
$979$	−49.4209	−1.57950
$980$	0	0
$981$	14.1063	0.450380
$982$	0	0
$983$	55.3529	1.76548	0.882742	−	0.469858i	$-0.155695\pi$
$983$	55.3529	1.76548	0.882742	+	0.469858i	$0.155695\pi$
$984$	0	0
$985$	40.6074	1.29386
$986$	0	0
$987$	27.0513	0.861052
$988$	0	0
$989$	13.7429	0.437000
$990$	0	0
$991$	19.9622	0.634119	0.317059	−	0.948406i	$-0.397305\pi$
$991$	19.9622	0.634119	0.317059	+	0.948406i	$0.397305\pi$
$992$	0	0
$993$	−51.7873	−1.64342
$994$	0	0
$995$	−2.68895	−0.0852455
$996$	0	0
$997$	54.7905	1.73523	0.867616	−	0.497234i	$-0.165651\pi$
$997$	54.7905	1.73523	0.867616	+	0.497234i	$0.165651\pi$
$998$	0	0
$999$	−5.64694	−0.178661

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	728.2.a.i.1.3	✓	4
3.2	odd	2		6552.2.a.br.1.3		4
4.3	odd	2		1456.2.a.v.1.2		4
7.6	odd	2		5096.2.a.s.1.2		4
8.3	odd	2		5824.2.a.ce.1.3		4
8.5	even	2		5824.2.a.cb.1.2		4
13.12	even	2		9464.2.a.z.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.a.i.1.3	✓	4	1.1	even	1	trivial
1456.2.a.v.1.2		4	4.3	odd	2
5096.2.a.s.1.2		4	7.6	odd	2
5824.2.a.cb.1.2		4	8.5	even	2
5824.2.a.ce.1.3		4	8.3	odd	2
6552.2.a.br.1.3		4	3.2	odd	2
9464.2.a.z.1.3		4	13.12	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 728 = 2^{3} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	728.a (trivial)

\(f(q)\)	\(=\)	\(q+2.19455 q^{3} -1.70715 q^{5} -1.00000 q^{7} +1.81604 q^{9} +4.56246 q^{11} -1.00000 q^{13} -3.74642 q^{15} +7.37850 q^{17} +6.46416 q^{19} -2.19455 q^{21} +5.90170 q^{23} -2.08565 q^{25} -2.59825 q^{27} +2.69655 q^{29} +0.523191 q^{31} +10.0125 q^{33} +1.70715 q^{35} +2.17336 q^{37} -2.19455 q^{39} -4.98735 q^{41} +2.32864 q^{43} -3.10025 q^{45} -12.3266 q^{47} +1.00000 q^{49} +16.1925 q^{51} -4.71774 q^{53} -7.78879 q^{55} +14.1859 q^{57} -13.7822 q^{59} -0.137574 q^{61} -1.81604 q^{63} +1.70715 q^{65} -7.96215 q^{67} +12.9516 q^{69} -13.8392 q^{71} +2.52319 q^{73} -4.57706 q^{75} -4.56246 q^{77} +4.85531 q^{79} -11.1501 q^{81} -7.51054 q^{83} -12.5962 q^{85} +5.91772 q^{87} -10.8321 q^{89} +1.00000 q^{91} +1.14817 q^{93} -11.0353 q^{95} +17.6693 q^{97} +8.28563 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} + 2 q^{5} - 4 q^{7} + 13 q^{9} - q^{11} - 4 q^{13} + 10 q^{15} + 16 q^{17} - 10 q^{19} - q^{21} + 7 q^{23} + 14 q^{25} + 13 q^{27} + 4 q^{29} - q^{31} - q^{33} - 2 q^{35} + 5 q^{37} - q^{39}+ \cdots - 48 q^{99}+O(q^{100}) \) 4 * q + q^3 + 2 * q^5 - 4 * q^7 + 13 * q^9 - q^11 - 4 * q^13 + 10 * q^15 + 16 * q^17 - 10 * q^19 - q^21 + 7 * q^23 + 14 * q^25 + 13 * q^27 + 4 * q^29 - q^31 - q^33 - 2 * q^35 + 5 * q^37 - q^39 + 19 * q^41 + 14 * q^43 + 10 * q^45 - 13 * q^47 + 4 * q^49 + 16 * q^51 - 8 * q^53 + 2 * q^55 + 12 * q^57 - 26 * q^59 - q^61 - 13 * q^63 - 2 * q^65 + 5 * q^67 + 17 * q^69 - 18 * q^71 + 7 * q^73 + q^75 + q^77 + 9 * q^79 - 4 * q^81 + 12 * q^83 + 14 * q^85 - 46 * q^87 + 4 * q^89 + 4 * q^91 + 3 * q^93 - 26 * q^95 + 25 * q^97 - 48 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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