LMFDB - Embedded newform 1521.2.a.q.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1521.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1521.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:	$12.1452461474$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 507)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.80194$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.80194 q^{2} +1.24698 q^{4} +1.44504 q^{5} -3.44504 q^{7} -1.35690 q^{8} +O(q^{10})$	$q+1.80194 q^{2} +1.24698 q^{4} +1.44504 q^{5} -3.44504 q^{7} -1.35690 q^{8} +2.60388 q^{10} -5.18598 q^{11} -6.20775 q^{14} -4.93900 q^{16} +0.753020 q^{17} -7.96077 q^{19} +1.80194 q^{20} -9.34481 q^{22} +2.82908 q^{23} -2.91185 q^{25} -4.29590 q^{28} +3.91185 q^{29} +4.89977 q^{31} -6.18598 q^{32} +1.35690 q^{34} -4.97823 q^{35} -6.24698 q^{37} -14.3448 q^{38} -1.96077 q^{40} +1.80194 q^{41} -7.09783 q^{43} -6.46681 q^{44} +5.09783 q^{46} +10.5526 q^{47} +4.86831 q^{49} -5.24698 q^{50} +3.08815 q^{53} -7.49396 q^{55} +4.67456 q^{56} +7.04892 q^{58} +1.87800 q^{59} +3.34481 q^{61} +8.82908 q^{62} -1.26875 q^{64} +4.54288 q^{67} +0.939001 q^{68} -8.97046 q^{70} -9.11960 q^{71} -2.95108 q^{73} -11.2567 q^{74} -9.92692 q^{76} +17.8659 q^{77} -9.43296 q^{79} -7.13706 q^{80} +3.24698 q^{82} -6.46681 q^{83} +1.08815 q^{85} -12.7899 q^{86} +7.03684 q^{88} +1.15883 q^{89} +3.52781 q^{92} +19.0151 q^{94} -11.5036 q^{95} -8.65817 q^{97} +8.77240 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} - q^{4} + 4 q^{5} - 10 q^{7}+O(q^{10}) $ 3 * q + q^2 - q^4 + 4 * q^5 - 10 * q^7	$ 3 q + q^{2} - q^{4} + 4 q^{5} - 10 q^{7} - q^{10} - q^{11} - q^{14} - 5 q^{16} + 7 q^{17} - 11 q^{19} + q^{20} - 5 q^{22} - 2 q^{23} - 5 q^{25} + q^{28} + 8 q^{29} - 8 q^{31} - 4 q^{32} - 18 q^{35} - 14 q^{37} - 20 q^{38} + 7 q^{40} + q^{41} - 3 q^{43} - 16 q^{44} - 3 q^{46} - 9 q^{47} + 17 q^{49} - 11 q^{50} + 13 q^{53} - 13 q^{55} - 7 q^{56} + 12 q^{58} - 14 q^{59} - 13 q^{61} + 16 q^{62} + 4 q^{64} - 5 q^{67} - 7 q^{68} + 8 q^{70} - 6 q^{71} - 18 q^{73} - 7 q^{74} - q^{76} + 15 q^{77} - 9 q^{79} - 16 q^{80} + 5 q^{82} - 16 q^{83} + 7 q^{85} - 15 q^{86} - 7 q^{88} - 5 q^{89} + 17 q^{92} + 32 q^{94} - 3 q^{95} - 5 q^{97} - 13 q^{98}+O(q^{100}) $ 3 * q + q^2 - q^4 + 4 * q^5 - 10 * q^7 - q^10 - q^11 - q^14 - 5 * q^16 + 7 * q^17 - 11 * q^19 + q^20 - 5 * q^22 - 2 * q^23 - 5 * q^25 + q^28 + 8 * q^29 - 8 * q^31 - 4 * q^32 - 18 * q^35 - 14 * q^37 - 20 * q^38 + 7 * q^40 + q^41 - 3 * q^43 - 16 * q^44 - 3 * q^46 - 9 * q^47 + 17 * q^49 - 11 * q^50 + 13 * q^53 - 13 * q^55 - 7 * q^56 + 12 * q^58 - 14 * q^59 - 13 * q^61 + 16 * q^62 + 4 * q^64 - 5 * q^67 - 7 * q^68 + 8 * q^70 - 6 * q^71 - 18 * q^73 - 7 * q^74 - q^76 + 15 * q^77 - 9 * q^79 - 16 * q^80 + 5 * q^82 - 16 * q^83 + 7 * q^85 - 15 * q^86 - 7 * q^88 - 5 * q^89 + 17 * q^92 + 32 * q^94 - 3 * q^95 - 5 * q^97 - 13 * q^98

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 $

$2$	1.80194	1.27416	0.637081	−	0.770797i	$-0.280142\pi$
$2$	1.80194	1.27416	0.637081	+	0.770797i	$0.280142\pi$
$3$	0	0
$3$	0	0
$4$	1.24698	0.623490
$5$	1.44504	0.646242	0.323121	−	0.946358i	$-0.395268\pi$
$5$	1.44504	0.646242	0.323121	+	0.946358i	$0.395268\pi$
$6$	0	0
$7$	−3.44504	−1.30210	−0.651052	−	0.759033i	$-0.725672\pi$
$7$	−3.44504	−1.30210	−0.651052	+	0.759033i	$0.725672\pi$
$8$	−1.35690	−0.479735
$9$	0	0
$10$	2.60388	0.823418
$11$	−5.18598	−1.56363	−0.781816	−	0.623509i	$-0.785706\pi$
$11$	−5.18598	−1.56363	−0.781816	+	0.623509i	$0.785706\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−6.20775	−1.65909
$15$	0	0
$16$	−4.93900	−1.23475
$17$	0.753020	0.182634	0.0913171	−	0.995822i	$-0.470892\pi$
$17$	0.753020	0.182634	0.0913171	+	0.995822i	$0.470892\pi$
$18$	0	0
$19$	−7.96077	−1.82633	−0.913163	−	0.407594i	$-0.866368\pi$
$19$	−7.96077	−1.82633	−0.913163	+	0.407594i	$0.866368\pi$
$20$	1.80194	0.402926
$21$	0	0
$22$	−9.34481	−1.99232
$23$	2.82908	0.589905	0.294952	−	0.955512i	$-0.404696\pi$
$23$	2.82908	0.589905	0.294952	+	0.955512i	$0.404696\pi$
$24$	0	0
$25$	−2.91185	−0.582371
$26$	0	0
$27$	0	0
$28$	−4.29590	−0.811848
$29$	3.91185	0.726413	0.363207	−	0.931709i	$-0.381682\pi$
$29$	3.91185	0.726413	0.363207	+	0.931709i	$0.381682\pi$
$30$	0	0
$31$	4.89977	0.880025	0.440013	−	0.897992i	$-0.354974\pi$
$31$	4.89977	0.880025	0.440013	+	0.897992i	$0.354974\pi$
$32$	−6.18598	−1.09354
$33$	0	0
$34$	1.35690	0.232706
$35$	−4.97823	−0.841474
$36$	0	0
$37$	−6.24698	−1.02700	−0.513499	−	0.858090i	$-0.671651\pi$
$37$	−6.24698	−1.02700	−0.513499	+	0.858090i	$0.671651\pi$
$38$	−14.3448	−2.32704
$39$	0	0
$40$	−1.96077	−0.310025
$41$	1.80194	0.281415	0.140708	−	0.990051i	$-0.455062\pi$
$41$	1.80194	0.281415	0.140708	+	0.990051i	$0.455062\pi$
$42$	0	0
$43$	−7.09783	−1.08241	−0.541205	−	0.840891i	$-0.682032\pi$
$43$	−7.09783	−1.08241	−0.541205	+	0.840891i	$0.682032\pi$
$44$	−6.46681	−0.974909
$45$	0	0
$46$	5.09783	0.751635
$47$	10.5526	1.53925	0.769625	−	0.638496i	$-0.220443\pi$
$47$	10.5526	1.53925	0.769625	+	0.638496i	$0.220443\pi$
$48$	0	0
$49$	4.86831	0.695473
$50$	−5.24698	−0.742035
$51$	0	0
$52$	0	0
$53$	3.08815	0.424189	0.212095	−	0.977249i	$-0.431971\pi$
$53$	3.08815	0.424189	0.212095	+	0.977249i	$0.431971\pi$
$54$	0	0
$55$	−7.49396	−1.01049
$56$	4.67456	0.624665
$57$	0	0
$58$	7.04892	0.925568
$59$	1.87800	0.244495	0.122248	−	0.992500i	$-0.460990\pi$
$59$	1.87800	0.244495	0.122248	+	0.992500i	$0.460990\pi$
$60$	0	0
$61$	3.34481	0.428260	0.214130	−	0.976805i	$-0.431308\pi$
$61$	3.34481	0.428260	0.214130	+	0.976805i	$0.431308\pi$
$62$	8.82908	1.12129
$63$	0	0
$64$	−1.26875	−0.158594
$65$	0	0
$66$	0	0
$67$	4.54288	0.555001	0.277500	−	0.960726i	$-0.410494\pi$
$67$	4.54288	0.555001	0.277500	+	0.960726i	$0.410494\pi$
$68$	0.939001	0.113871
$69$	0	0
$70$	−8.97046	−1.07218
$71$	−9.11960	−1.08230	−0.541149	−	0.840927i	$-0.682011\pi$
$71$	−9.11960	−1.08230	−0.541149	+	0.840927i	$0.682011\pi$
$72$	0	0
$73$	−2.95108	−0.345398	−0.172699	−	0.984975i	$-0.555249\pi$
$73$	−2.95108	−0.345398	−0.172699	+	0.984975i	$0.555249\pi$
$74$	−11.2567	−1.30856
$75$	0	0
$76$	−9.92692	−1.13870
$77$	17.8659	2.03601
$78$	0	0
$79$	−9.43296	−1.06129	−0.530645	−	0.847594i	$-0.678050\pi$
$79$	−9.43296	−1.06129	−0.530645	+	0.847594i	$0.678050\pi$
$80$	−7.13706	−0.797948
$81$	0	0
$82$	3.24698	0.358569
$83$	−6.46681	−0.709825	−0.354912	−	0.934900i	$-0.615489\pi$
$83$	−6.46681	−0.709825	−0.354912	+	0.934900i	$0.615489\pi$
$84$	0	0
$85$	1.08815	0.118026
$86$	−12.7899	−1.37917
$87$	0	0
$88$	7.03684	0.750129
$89$	1.15883	0.122836	0.0614181	−	0.998112i	$-0.480438\pi$
$89$	1.15883	0.122836	0.0614181	+	0.998112i	$0.480438\pi$
$90$	0	0
$91$	0	0
$92$	3.52781	0.367800
$93$	0	0
$94$	19.0151	1.96125
$95$	−11.5036	−1.18025
$96$	0	0
$97$	−8.65817	−0.879104	−0.439552	−	0.898217i	$-0.644863\pi$
$97$	−8.65817	−0.879104	−0.439552	+	0.898217i	$0.644863\pi$
$98$	8.77240	0.886146
$99$	0	0
$100$	−3.63102	−0.363102
$101$	8.47650	0.843443	0.421722	−	0.906725i	$-0.361426\pi$
$101$	8.47650	0.843443	0.421722	+	0.906725i	$0.361426\pi$
$102$	0	0
$103$	5.64742	0.556456	0.278228	−	0.960515i	$-0.410253\pi$
$103$	5.64742	0.556456	0.278228	+	0.960515i	$0.410253\pi$
$104$	0	0
$105$	0	0
$106$	5.56465	0.540486
$107$	6.73556	0.651151	0.325576	−	0.945516i	$-0.394442\pi$
$107$	6.73556	0.651151	0.325576	+	0.945516i	$0.394442\pi$
$108$	0	0
$109$	2.07606	0.198851	0.0994255	−	0.995045i	$-0.468300\pi$
$109$	2.07606	0.198851	0.0994255	+	0.995045i	$0.468300\pi$
$110$	−13.5036	−1.28752
$111$	0	0
$112$	17.0151	1.60777
$113$	−6.16852	−0.580286	−0.290143	−	0.956983i	$-0.593703\pi$
$113$	−6.16852	−0.580286	−0.290143	+	0.956983i	$0.593703\pi$
$114$	0	0
$115$	4.08815	0.381222
$116$	4.87800	0.452911
$117$	0	0
$118$	3.38404	0.311526
$119$	−2.59419	−0.237809
$120$	0	0
$121$	15.8944	1.44495
$122$	6.02715	0.545672
$123$	0	0
$124$	6.10992	0.548687
$125$	−11.4330	−1.02260
$126$	0	0
$127$	−14.2620	−1.26555	−0.632776	−	0.774335i	$-0.718085\pi$
$127$	−14.2620	−1.26555	−0.632776	+	0.774335i	$0.718085\pi$
$128$	10.0858	0.891463
$129$	0	0
$130$	0	0
$131$	−22.6015	−1.97470	−0.987350	−	0.158554i	$-0.949317\pi$
$131$	−22.6015	−1.97470	−0.987350	+	0.158554i	$0.949317\pi$
$132$	0	0
$133$	27.4252	2.37807
$134$	8.18598	0.707161
$135$	0	0
$136$	−1.02177	−0.0876161
$137$	−13.6353	−1.16495	−0.582473	−	0.812850i	$-0.697915\pi$
$137$	−13.6353	−1.16495	−0.582473	+	0.812850i	$0.697915\pi$
$138$	0	0
$139$	17.6015	1.49294	0.746469	−	0.665420i	$-0.231748\pi$
$139$	17.6015	1.49294	0.746469	+	0.665420i	$0.231748\pi$
$140$	−6.20775	−0.524651
$141$	0	0
$142$	−16.4330	−1.37902
$143$	0	0
$144$	0	0
$145$	5.65279	0.469439
$146$	−5.31767	−0.440093
$147$	0	0
$148$	−7.78986	−0.640322
$149$	12.7385	1.04358	0.521791	−	0.853073i	$-0.325264\pi$
$149$	12.7385	1.04358	0.521791	+	0.853073i	$0.325264\pi$
$150$	0	0
$151$	−15.6407	−1.27282	−0.636412	−	0.771350i	$-0.719582\pi$
$151$	−15.6407	−1.27282	−0.636412	+	0.771350i	$0.719582\pi$
$152$	10.8019	0.876153
$153$	0	0
$154$	32.1933	2.59421
$155$	7.08038	0.568710
$156$	0	0
$157$	−0.823708	−0.0657391	−0.0328695	−	0.999460i	$-0.510465\pi$
$157$	−0.823708	−0.0657391	−0.0328695	+	0.999460i	$0.510465\pi$
$158$	−16.9976	−1.35226
$159$	0	0
$160$	−8.93900	−0.706690
$161$	−9.74632	−0.768117
$162$	0	0
$163$	−6.26875	−0.491006	−0.245503	−	0.969396i	$-0.578953\pi$
$163$	−6.26875	−0.491006	−0.245503	+	0.969396i	$0.578953\pi$
$164$	2.24698	0.175460
$165$	0	0
$166$	−11.6528	−0.904432
$167$	−7.45042	−0.576531	−0.288265	−	0.957551i	$-0.593079\pi$
$167$	−7.45042	−0.576531	−0.288265	+	0.957551i	$0.593079\pi$
$168$	0	0
$169$	0	0
$170$	1.96077	0.150384
$171$	0	0
$172$	−8.85086	−0.674871
$173$	2.00969	0.152794	0.0763969	−	0.997077i	$-0.475658\pi$
$173$	2.00969	0.152794	0.0763969	+	0.997077i	$0.475658\pi$
$174$	0	0
$175$	10.0315	0.758307
$176$	25.6136	1.93070
$177$	0	0
$178$	2.08815	0.156513
$179$	−20.0368	−1.49762	−0.748812	−	0.662783i	$-0.769375\pi$
$179$	−20.0368	−1.49762	−0.748812	+	0.662783i	$0.769375\pi$
$180$	0	0
$181$	24.1226	1.79302	0.896509	−	0.443026i	$-0.146095\pi$
$181$	24.1226	1.79302	0.896509	+	0.443026i	$0.146095\pi$
$182$	0	0
$183$	0	0
$184$	−3.83877	−0.282998
$185$	−9.02715	−0.663689
$186$	0	0
$187$	−3.90515	−0.285573
$188$	13.1588	0.959707
$189$	0	0
$190$	−20.7289	−1.50383
$191$	7.08038	0.512318	0.256159	−	0.966635i	$-0.417543\pi$
$191$	7.08038	0.512318	0.256159	+	0.966635i	$0.417543\pi$
$192$	0	0
$193$	−9.76809	−0.703122	−0.351561	−	0.936165i	$-0.614349\pi$
$193$	−9.76809	−0.703122	−0.351561	+	0.936165i	$0.614349\pi$
$194$	−15.6015	−1.12012
$195$	0	0
$196$	6.07069	0.433621
$197$	23.4112	1.66798	0.833989	−	0.551781i	$-0.186052\pi$
$197$	23.4112	1.66798	0.833989	+	0.551781i	$0.186052\pi$
$198$	0	0
$199$	4.02475	0.285307	0.142654	−	0.989773i	$-0.454437\pi$
$199$	4.02475	0.285307	0.142654	+	0.989773i	$0.454437\pi$
$200$	3.95108	0.279384
$201$	0	0
$202$	15.2741	1.07468
$203$	−13.4765	−0.945865
$204$	0	0
$205$	2.60388	0.181863
$206$	10.1763	0.709016
$207$	0	0
$208$	0	0
$209$	41.2844	2.85570
$210$	0	0
$211$	−3.91185	−0.269303	−0.134652	−	0.990893i	$-0.542992\pi$
$211$	−3.91185	−0.269303	−0.134652	+	0.990893i	$0.542992\pi$
$212$	3.85086	0.264478
$213$	0	0
$214$	12.1371	0.829673
$215$	−10.2567	−0.699499
$216$	0	0
$217$	−16.8799	−1.14588
$218$	3.74094	0.253368
$219$	0	0
$220$	−9.34481	−0.630027
$221$	0	0
$222$	0	0
$223$	−7.44935	−0.498846	−0.249423	−	0.968395i	$-0.580241\pi$
$223$	−7.44935	−0.498846	−0.249423	+	0.968395i	$0.580241\pi$
$224$	21.3110	1.42390
$225$	0	0
$226$	−11.1153	−0.739378
$227$	−21.2500	−1.41041	−0.705205	−	0.709004i	$-0.749145\pi$
$227$	−21.2500	−1.41041	−0.705205	+	0.709004i	$0.749145\pi$
$228$	0	0
$229$	−9.29590	−0.614290	−0.307145	−	0.951663i	$-0.599374\pi$
$229$	−9.29590	−0.614290	−0.307145	+	0.951663i	$0.599374\pi$
$230$	7.36658	0.485738
$231$	0	0
$232$	−5.30798	−0.348486
$233$	−16.2107	−1.06200	−0.531000	−	0.847372i	$-0.678184\pi$
$233$	−16.2107	−1.06200	−0.531000	+	0.847372i	$0.678184\pi$
$234$	0	0
$235$	15.2489	0.994728
$236$	2.34183	0.152440
$237$	0	0
$238$	−4.67456	−0.303007
$239$	−13.5090	−0.873826	−0.436913	−	0.899504i	$-0.643928\pi$
$239$	−13.5090	−0.873826	−0.436913	+	0.899504i	$0.643928\pi$
$240$	0	0
$241$	−6.26875	−0.403806	−0.201903	−	0.979406i	$-0.564713\pi$
$241$	−6.26875	−0.403806	−0.201903	+	0.979406i	$0.564713\pi$
$242$	28.6407	1.84109
$243$	0	0
$244$	4.17092	0.267015
$245$	7.03492	0.449444
$246$	0	0
$247$	0	0
$248$	−6.64848	−0.422179
$249$	0	0
$250$	−20.6015	−1.30295
$251$	0.753020	0.0475302	0.0237651	−	0.999718i	$-0.492435\pi$
$251$	0.753020	0.0475302	0.0237651	+	0.999718i	$0.492435\pi$
$252$	0	0
$253$	−14.6716	−0.922394
$254$	−25.6993	−1.61252
$255$	0	0
$256$	20.7114	1.29446
$257$	−19.7265	−1.23050	−0.615252	−	0.788331i	$-0.710946\pi$
$257$	−19.7265	−1.23050	−0.615252	+	0.788331i	$0.710946\pi$
$258$	0	0
$259$	21.5211	1.33726
$260$	0	0
$261$	0	0
$262$	−40.7265	−2.51609
$263$	17.6093	1.08583	0.542917	−	0.839787i	$-0.317320\pi$
$263$	17.6093	1.08583	0.542917	+	0.839787i	$0.317320\pi$
$264$	0	0
$265$	4.46250	0.274129
$266$	49.4185	3.03004
$267$	0	0
$268$	5.66487	0.346037
$269$	16.3870	0.999135	0.499567	−	0.866275i	$-0.333492\pi$
$269$	16.3870	0.999135	0.499567	+	0.866275i	$0.333492\pi$
$270$	0	0
$271$	0.795233	0.0483070	0.0241535	−	0.999708i	$-0.492311\pi$
$271$	0.795233	0.0483070	0.0241535	+	0.999708i	$0.492311\pi$
$272$	−3.71917	−0.225508
$273$	0	0
$274$	−24.5700	−1.48433
$275$	15.1008	0.910614
$276$	0	0
$277$	−4.83340	−0.290411	−0.145205	−	0.989402i	$-0.546384\pi$
$277$	−4.83340	−0.290411	−0.145205	+	0.989402i	$0.546384\pi$
$278$	31.7168	1.90225
$279$	0	0
$280$	6.75494	0.403685
$281$	18.7748	1.12001	0.560005	−	0.828489i	$-0.310799\pi$
$281$	18.7748	1.12001	0.560005	+	0.828489i	$0.310799\pi$
$282$	0	0
$283$	−7.91723	−0.470631	−0.235315	−	0.971919i	$-0.575612\pi$
$283$	−7.91723	−0.470631	−0.235315	+	0.971919i	$0.575612\pi$
$284$	−11.3720	−0.674802
$285$	0	0
$286$	0	0
$287$	−6.20775	−0.366432
$288$	0	0
$289$	−16.4330	−0.966645
$290$	10.1860	0.598141
$291$	0	0
$292$	−3.67994	−0.215352
$293$	6.57912	0.384356	0.192178	−	0.981360i	$-0.438445\pi$
$293$	6.57912	0.384356	0.192178	+	0.981360i	$0.438445\pi$
$294$	0	0
$295$	2.71379	0.158003
$296$	8.47650	0.492687
$297$	0	0
$298$	22.9541	1.32969
$299$	0	0
$300$	0	0
$301$	24.4523	1.40941
$302$	−28.1836	−1.62178
$303$	0	0
$304$	39.3183	2.25506
$305$	4.83340	0.276759
$306$	0	0
$307$	24.8649	1.41911	0.709556	−	0.704649i	$-0.248895\pi$
$307$	24.8649	1.41911	0.709556	+	0.704649i	$0.248895\pi$
$308$	22.2784	1.26943
$309$	0	0
$310$	12.7584	0.724628
$311$	17.0804	0.968539	0.484270	−	0.874919i	$-0.339085\pi$
$311$	17.0804	0.968539	0.484270	+	0.874919i	$0.339085\pi$
$312$	0	0
$313$	15.6974	0.887269	0.443635	−	0.896208i	$-0.353689\pi$
$313$	15.6974	0.887269	0.443635	+	0.896208i	$0.353689\pi$
$314$	−1.48427	−0.0837622
$315$	0	0
$316$	−11.7627	−0.661704
$317$	32.7821	1.84123	0.920613	−	0.390477i	$-0.127690\pi$
$317$	32.7821	1.84123	0.920613	+	0.390477i	$0.127690\pi$
$318$	0	0
$319$	−20.2868	−1.13584
$320$	−1.83340	−0.102490
$321$	0	0
$322$	−17.5623	−0.978706
$323$	−5.99462	−0.333550
$324$	0	0
$325$	0	0
$326$	−11.2959	−0.625622
$327$	0	0
$328$	−2.44504	−0.135005
$329$	−36.3540	−2.00426
$330$	0	0
$331$	29.1618	1.60288	0.801439	−	0.598076i	$-0.204068\pi$
$331$	29.1618	1.60288	0.801439	+	0.598076i	$0.204068\pi$
$332$	−8.06398	−0.442569
$333$	0	0
$334$	−13.4252	−0.734594
$335$	6.56465	0.358665
$336$	0	0
$337$	−33.2911	−1.81348	−0.906741	−	0.421688i	$-0.861438\pi$
$337$	−33.2911	−1.81348	−0.906741	+	0.421688i	$0.861438\pi$
$338$	0	0
$339$	0	0
$340$	1.35690	0.0735880
$341$	−25.4101	−1.37604
$342$	0	0
$343$	7.34375	0.396525
$344$	9.63102	0.519270
$345$	0	0
$346$	3.62133	0.194684
$347$	0.873690	0.0469022	0.0234511	−	0.999725i	$-0.492535\pi$
$347$	0.873690	0.0469022	0.0234511	+	0.999725i	$0.492535\pi$
$348$	0	0
$349$	3.23191	0.173000	0.0865002	−	0.996252i	$-0.472432\pi$
$349$	3.23191	0.173000	0.0865002	+	0.996252i	$0.472432\pi$
$350$	18.0761	0.966206
$351$	0	0
$352$	32.0804	1.70989
$353$	8.14675	0.433608	0.216804	−	0.976215i	$-0.430437\pi$
$353$	8.14675	0.433608	0.216804	+	0.976215i	$0.430437\pi$
$354$	0	0
$355$	−13.1782	−0.699427
$356$	1.44504	0.0765871
$357$	0	0
$358$	−36.1051	−1.90822
$359$	2.64071	0.139371	0.0696857	−	0.997569i	$-0.477800\pi$
$359$	2.64071	0.139371	0.0696857	+	0.997569i	$0.477800\pi$
$360$	0	0
$361$	44.3739	2.33547
$362$	43.4674	2.28460
$363$	0	0
$364$	0	0
$365$	−4.26444	−0.223211
$366$	0	0
$367$	−2.90408	−0.151592	−0.0757960	−	0.997123i	$-0.524150\pi$
$367$	−2.90408	−0.151592	−0.0757960	+	0.997123i	$0.524150\pi$
$368$	−13.9729	−0.728385
$369$	0	0
$370$	−16.2664	−0.845648
$371$	−10.6388	−0.552339
$372$	0	0
$373$	−8.39852	−0.434859	−0.217429	−	0.976076i	$-0.569767\pi$
$373$	−8.39852	−0.434859	−0.217429	+	0.976076i	$0.569767\pi$
$374$	−7.03684	−0.363866
$375$	0	0
$376$	−14.3187	−0.738432
$377$	0	0
$378$	0	0
$379$	−15.7482	−0.808932	−0.404466	−	0.914553i	$-0.632543\pi$
$379$	−15.7482	−0.808932	−0.404466	+	0.914553i	$0.632543\pi$
$380$	−14.3448	−0.735873
$381$	0	0
$382$	12.7584	0.652776
$383$	12.7385	0.650909	0.325455	−	0.945558i	$-0.394483\pi$
$383$	12.7385	0.650909	0.325455	+	0.945558i	$0.394483\pi$
$384$	0	0
$385$	25.8170	1.31576
$386$	−17.6015	−0.895892
$387$	0	0
$388$	−10.7966	−0.548112
$389$	0.310371	0.0157365	0.00786823	−	0.999969i	$-0.497495\pi$
$389$	0.310371	0.0157365	0.00786823	+	0.999969i	$0.497495\pi$
$390$	0	0
$391$	2.13036	0.107737
$392$	−6.60579	−0.333643
$393$	0	0
$394$	42.1855	2.12528
$395$	−13.6310	−0.685851
$396$	0	0
$397$	−1.49098	−0.0748299	−0.0374150	−	0.999300i	$-0.511912\pi$
$397$	−1.49098	−0.0748299	−0.0374150	+	0.999300i	$0.511912\pi$
$398$	7.25236	0.363528
$399$	0	0
$400$	14.3817	0.719083
$401$	−23.8334	−1.19018	−0.595092	−	0.803658i	$-0.702884\pi$
$401$	−23.8334	−1.19018	−0.595092	+	0.803658i	$0.702884\pi$
$402$	0	0
$403$	0	0
$404$	10.5700	0.525878
$405$	0	0
$406$	−24.2838	−1.20519
$407$	32.3967	1.60585
$408$	0	0
$409$	−4.26742	−0.211010	−0.105505	−	0.994419i	$-0.533646\pi$
$409$	−4.26742	−0.211010	−0.105505	+	0.994419i	$0.533646\pi$
$410$	4.69202	0.231722
$411$	0	0
$412$	7.04221	0.346945
$413$	−6.46980	−0.318358
$414$	0	0
$415$	−9.34481	−0.458719
$416$	0	0
$417$	0	0
$418$	74.3919	3.63863
$419$	−29.6896	−1.45043	−0.725217	−	0.688521i	$-0.758260\pi$
$419$	−29.6896	−1.45043	−0.725217	+	0.688521i	$0.758260\pi$
$420$	0	0
$421$	−29.3991	−1.43282	−0.716412	−	0.697677i	$-0.754217\pi$
$421$	−29.3991	−1.43282	−0.716412	+	0.697677i	$0.754217\pi$
$422$	−7.04892	−0.343136
$423$	0	0
$424$	−4.19029	−0.203499
$425$	−2.19269	−0.106361
$426$	0	0
$427$	−11.5230	−0.557638
$428$	8.39911	0.405986
$429$	0	0
$430$	−18.4819	−0.891275
$431$	−33.0562	−1.59226	−0.796131	−	0.605124i	$-0.793123\pi$
$431$	−33.0562	−1.59226	−0.796131	+	0.605124i	$0.793123\pi$
$432$	0	0
$433$	−29.2664	−1.40645	−0.703226	−	0.710967i	$-0.748258\pi$
$433$	−29.2664	−1.40645	−0.703226	+	0.710967i	$0.748258\pi$
$434$	−30.4166	−1.46004
$435$	0	0
$436$	2.58881	0.123982
$437$	−22.5217	−1.07736
$438$	0	0
$439$	−2.13169	−0.101740	−0.0508699	−	0.998705i	$-0.516199\pi$
$439$	−2.13169	−0.101740	−0.0508699	+	0.998705i	$0.516199\pi$
$440$	10.1685	0.484765
$441$	0	0
$442$	0	0
$443$	22.9922	1.09239	0.546197	−	0.837657i	$-0.316075\pi$
$443$	22.9922	1.09239	0.546197	+	0.837657i	$0.316075\pi$
$444$	0	0
$445$	1.67456	0.0793819
$446$	−13.4233	−0.635610
$447$	0	0
$448$	4.37090	0.206505
$449$	−12.9379	−0.610579	−0.305289	−	0.952260i	$-0.598753\pi$
$449$	−12.9379	−0.610579	−0.305289	+	0.952260i	$0.598753\pi$
$450$	0	0
$451$	−9.34481	−0.440030
$452$	−7.69202	−0.361802
$453$	0	0
$454$	−38.2911	−1.79709
$455$	0	0
$456$	0	0
$457$	4.85325	0.227025	0.113513	−	0.993537i	$-0.463790\pi$
$457$	4.85325	0.227025	0.113513	+	0.993537i	$0.463790\pi$
$458$	−16.7506	−0.782705
$459$	0	0
$460$	5.09783	0.237688
$461$	18.8345	0.877208	0.438604	−	0.898680i	$-0.355473\pi$
$461$	18.8345	0.877208	0.438604	+	0.898680i	$0.355473\pi$
$462$	0	0
$463$	22.8767	1.06317	0.531585	−	0.847005i	$-0.321597\pi$
$463$	22.8767	1.06317	0.531585	+	0.847005i	$0.321597\pi$
$464$	−19.3207	−0.896939
$465$	0	0
$466$	−29.2107	−1.35316
$467$	−13.0000	−0.601568	−0.300784	−	0.953692i	$-0.597248\pi$
$467$	−13.0000	−0.601568	−0.300784	+	0.953692i	$0.597248\pi$
$468$	0	0
$469$	−15.6504	−0.722668
$470$	27.4776	1.26745
$471$	0	0
$472$	−2.54825	−0.117293
$473$	36.8092	1.69249
$474$	0	0
$475$	23.1806	1.06360
$476$	−3.23490	−0.148271
$477$	0	0
$478$	−24.3424	−1.11340
$479$	−38.0901	−1.74038	−0.870190	−	0.492717i	$-0.836004\pi$
$479$	−38.0901	−1.74038	−0.870190	+	0.492717i	$0.836004\pi$
$480$	0	0
$481$	0	0
$482$	−11.2959	−0.514514
$483$	0	0
$484$	19.8200	0.900909
$485$	−12.5114	−0.568114
$486$	0	0
$487$	21.2500	0.962928	0.481464	−	0.876466i	$-0.340105\pi$
$487$	21.2500	0.962928	0.481464	+	0.876466i	$0.340105\pi$
$488$	−4.53856	−0.205451
$489$	0	0
$490$	12.6765	0.572665
$491$	6.35019	0.286580	0.143290	−	0.989681i	$-0.454232\pi$
$491$	6.35019	0.286580	0.143290	+	0.989681i	$0.454232\pi$
$492$	0	0
$493$	2.94571	0.132668
$494$	0	0
$495$	0	0
$496$	−24.2000	−1.08661
$497$	31.4174	1.40926
$498$	0	0
$499$	4.65087	0.208202	0.104101	−	0.994567i	$-0.466804\pi$
$499$	4.65087	0.208202	0.104101	+	0.994567i	$0.466804\pi$
$500$	−14.2567	−0.637578
$501$	0	0
$502$	1.35690	0.0605612
$503$	−15.4752	−0.690004	−0.345002	−	0.938602i	$-0.612122\pi$
$503$	−15.4752	−0.690004	−0.345002	+	0.938602i	$0.612122\pi$
$504$	0	0
$505$	12.2489	0.545069
$506$	−26.4373	−1.17528
$507$	0	0
$508$	−17.7845	−0.789059
$509$	−20.5047	−0.908855	−0.454428	−	0.890784i	$-0.650156\pi$
$509$	−20.5047	−0.908855	−0.454428	+	0.890784i	$0.650156\pi$
$510$	0	0
$511$	10.1666	0.449744
$512$	17.1491	0.757892
$513$	0	0
$514$	−35.5459	−1.56786
$515$	8.16075	0.359606
$516$	0	0
$517$	−54.7254	−2.40682
$518$	38.7797	1.70388
$519$	0	0
$520$	0	0
$521$	42.0267	1.84122	0.920611	−	0.390481i	$-0.127691\pi$
$521$	42.0267	1.84122	0.920611	+	0.390481i	$0.127691\pi$
$522$	0	0
$523$	−29.9885	−1.31131	−0.655653	−	0.755062i	$-0.727607\pi$
$523$	−29.9885	−1.31131	−0.655653	+	0.755062i	$0.727607\pi$
$524$	−28.1836	−1.23121
$525$	0	0
$526$	31.7308	1.38353
$527$	3.68963	0.160723
$528$	0	0
$529$	−14.9963	−0.652012
$530$	8.04115	0.349285
$531$	0	0
$532$	34.1987	1.48270
$533$	0	0
$534$	0	0
$535$	9.73317	0.420802
$536$	−6.16421	−0.266253
$537$	0	0
$538$	29.5284	1.27306
$539$	−25.2470	−1.08746
$540$	0	0
$541$	−36.3803	−1.56411	−0.782056	−	0.623208i	$-0.785829\pi$
$541$	−36.3803	−1.56411	−0.782056	+	0.623208i	$0.785829\pi$
$542$	1.43296	0.0615509
$543$	0	0
$544$	−4.65817	−0.199717
$545$	3.00000	0.128506
$546$	0	0
$547$	−25.8159	−1.10381	−0.551905	−	0.833907i	$-0.686099\pi$
$547$	−25.8159	−1.10381	−0.551905	+	0.833907i	$0.686099\pi$
$548$	−17.0030	−0.726331
$549$	0	0
$550$	27.2107	1.16027
$551$	−31.1414	−1.32667
$552$	0	0
$553$	32.4969	1.38191
$554$	−8.70948	−0.370030
$555$	0	0
$556$	21.9487	0.930832
$557$	17.9903	0.762274	0.381137	−	0.924519i	$-0.375533\pi$
$557$	17.9903	0.762274	0.381137	+	0.924519i	$0.375533\pi$
$558$	0	0
$559$	0	0
$560$	24.5875	1.03901
$561$	0	0
$562$	33.8310	1.42707
$563$	39.1323	1.64923	0.824614	−	0.565695i	$-0.191392\pi$
$563$	39.1323	1.64923	0.824614	+	0.565695i	$0.191392\pi$
$564$	0	0
$565$	−8.91377	−0.375005
$566$	−14.2664	−0.599660
$567$	0	0
$568$	12.3744	0.519216
$569$	−30.6002	−1.28283	−0.641413	−	0.767196i	$-0.721651\pi$
$569$	−30.6002	−1.28283	−0.641413	+	0.767196i	$0.721651\pi$
$570$	0	0
$571$	−2.96184	−0.123949	−0.0619745	−	0.998078i	$-0.519740\pi$
$571$	−2.96184	−0.123949	−0.0619745	+	0.998078i	$0.519740\pi$
$572$	0	0
$573$	0	0
$574$	−11.1860	−0.466894
$575$	−8.23788	−0.343543
$576$	0	0
$577$	0.819396	0.0341119	0.0170560	−	0.999855i	$-0.494571\pi$
$577$	0.819396	0.0341119	0.0170560	+	0.999855i	$0.494571\pi$
$578$	−29.6112	−1.23166
$579$	0	0
$580$	7.04892	0.292690
$581$	22.2784	0.924265
$582$	0	0
$583$	−16.0151	−0.663276
$584$	4.00431	0.165700
$585$	0	0
$586$	11.8552	0.489732
$587$	−31.7995	−1.31251	−0.656254	−	0.754540i	$-0.727860\pi$
$587$	−31.7995	−1.31251	−0.656254	+	0.754540i	$0.727860\pi$
$588$	0	0
$589$	−39.0060	−1.60721
$590$	4.89008	0.201322
$591$	0	0
$592$	30.8538	1.26808
$593$	−4.26337	−0.175076	−0.0875379	−	0.996161i	$-0.527900\pi$
$593$	−4.26337	−0.175076	−0.0875379	+	0.996161i	$0.527900\pi$
$594$	0	0
$595$	−3.74871	−0.153682
$596$	15.8847	0.650663
$597$	0	0
$598$	0	0
$599$	−24.7278	−1.01035	−0.505175	−	0.863017i	$-0.668572\pi$
$599$	−24.7278	−1.01035	−0.505175	+	0.863017i	$0.668572\pi$
$600$	0	0
$601$	6.82371	0.278345	0.139172	−	0.990268i	$-0.455556\pi$
$601$	6.82371	0.278345	0.139172	+	0.990268i	$0.455556\pi$
$602$	44.0616	1.79582
$603$	0	0
$604$	−19.5036	−0.793592
$605$	22.9681	0.933785
$606$	0	0
$607$	31.9963	1.29869	0.649344	−	0.760494i	$-0.275043\pi$
$607$	31.9963	1.29869	0.649344	+	0.760494i	$0.275043\pi$
$608$	49.2452	1.99716
$609$	0	0
$610$	8.70948	0.352637
$611$	0	0
$612$	0	0
$613$	−33.5875	−1.35659	−0.678293	−	0.734792i	$-0.737280\pi$
$613$	−33.5875	−1.35659	−0.678293	+	0.734792i	$0.737280\pi$
$614$	44.8049	1.80818
$615$	0	0
$616$	−24.2422	−0.976746
$617$	−26.5870	−1.07035	−0.535176	−	0.844740i	$-0.679755\pi$
$617$	−26.5870	−1.07035	−0.535176	+	0.844740i	$0.679755\pi$
$618$	0	0
$619$	9.17928	0.368946	0.184473	−	0.982838i	$-0.440942\pi$
$619$	9.17928	0.368946	0.184473	+	0.982838i	$0.440942\pi$
$620$	8.82908	0.354585
$621$	0	0
$622$	30.7778	1.23408
$623$	−3.99223	−0.159945
$624$	0	0
$625$	−1.96184	−0.0784735
$626$	28.2857	1.13053
$627$	0	0
$628$	−1.02715	−0.0409876
$629$	−4.70410	−0.187565
$630$	0	0
$631$	−17.0043	−0.676931	−0.338465	−	0.940979i	$-0.609908\pi$
$631$	−17.0043	−0.676931	−0.338465	+	0.940979i	$0.609908\pi$
$632$	12.7995	0.509139
$633$	0	0
$634$	59.0713	2.34602
$635$	−20.6093	−0.817853
$636$	0	0
$637$	0	0
$638$	−36.5555	−1.44725
$639$	0	0
$640$	14.5743	0.576101
$641$	21.6649	0.855711	0.427856	−	0.903847i	$-0.359269\pi$
$641$	21.6649	0.855711	0.427856	+	0.903847i	$0.359269\pi$
$642$	0	0
$643$	−9.35557	−0.368948	−0.184474	−	0.982837i	$-0.559058\pi$
$643$	−9.35557	−0.368948	−0.184474	+	0.982837i	$0.559058\pi$
$644$	−12.1535	−0.478913
$645$	0	0
$646$	−10.8019	−0.424997
$647$	0.702775	0.0276289	0.0138145	−	0.999905i	$-0.495603\pi$
$647$	0.702775	0.0276289	0.0138145	+	0.999905i	$0.495603\pi$
$648$	0	0
$649$	−9.73928	−0.382300
$650$	0	0
$651$	0	0
$652$	−7.81700	−0.306137
$653$	37.3411	1.46127	0.730635	−	0.682768i	$-0.239224\pi$
$653$	37.3411	1.46127	0.730635	+	0.682768i	$0.239224\pi$
$654$	0	0
$655$	−32.6601	−1.27614
$656$	−8.89977	−0.347478
$657$	0	0
$658$	−65.5077	−2.55376
$659$	0.735562	0.0286534	0.0143267	−	0.999897i	$-0.495440\pi$
$659$	0.735562	0.0286534	0.0143267	+	0.999897i	$0.495440\pi$
$660$	0	0
$661$	−13.8485	−0.538643	−0.269321	−	0.963050i	$-0.586799\pi$
$661$	−13.8485	−0.538643	−0.269321	+	0.963050i	$0.586799\pi$
$662$	52.5478	2.04233
$663$	0	0
$664$	8.77479	0.340528
$665$	39.6305	1.53681
$666$	0	0
$667$	11.0670	0.428515
$668$	−9.29052	−0.359461
$669$	0	0
$670$	11.8291	0.456997
$671$	−17.3461	−0.669640
$672$	0	0
$673$	−6.35019	−0.244782	−0.122391	−	0.992482i	$-0.539056\pi$
$673$	−6.35019	−0.244782	−0.122391	+	0.992482i	$0.539056\pi$
$674$	−59.9885	−2.31067
$675$	0	0
$676$	0	0
$677$	−33.7241	−1.29612	−0.648061	−	0.761589i	$-0.724420\pi$
$677$	−33.7241	−1.29612	−0.648061	+	0.761589i	$0.724420\pi$
$678$	0	0
$679$	29.8278	1.14468
$680$	−1.47650	−0.0566212
$681$	0	0
$682$	−45.7875	−1.75329
$683$	19.2687	0.737298	0.368649	−	0.929569i	$-0.379820\pi$
$683$	19.2687	0.737298	0.368649	+	0.929569i	$0.379820\pi$
$684$	0	0
$685$	−19.7036	−0.752837
$686$	13.2330	0.505237
$687$	0	0
$688$	35.0562	1.33651
$689$	0	0
$690$	0	0
$691$	39.4010	1.49889	0.749443	−	0.662069i	$-0.230321\pi$
$691$	39.4010	1.49889	0.749443	+	0.662069i	$0.230321\pi$
$692$	2.50604	0.0952654
$693$	0	0
$694$	1.57434	0.0597610
$695$	25.4349	0.964800
$696$	0	0
$697$	1.35690	0.0513961
$698$	5.82371	0.220431
$699$	0	0
$700$	12.5090	0.472797
$701$	−18.3985	−0.694902	−0.347451	−	0.937698i	$-0.612953\pi$
$701$	−18.3985	−0.694902	−0.347451	+	0.937698i	$0.612953\pi$
$702$	0	0
$703$	49.7308	1.87563
$704$	6.57971	0.247982
$705$	0	0
$706$	14.6799	0.552487
$707$	−29.2019	−1.09825
$708$	0	0
$709$	38.4553	1.44422	0.722110	−	0.691778i	$-0.243172\pi$
$709$	38.4553	1.44422	0.722110	+	0.691778i	$0.243172\pi$
$710$	−23.7463	−0.891183
$711$	0	0
$712$	−1.57242	−0.0589288
$713$	13.8619	0.519131
$714$	0	0
$715$	0	0
$716$	−24.9855	−0.933753
$717$	0	0
$718$	4.75840	0.177582
$719$	−48.4999	−1.80874	−0.904371	−	0.426747i	$-0.859659\pi$
$719$	−48.4999	−1.80874	−0.904371	+	0.426747i	$0.859659\pi$
$720$	0	0
$721$	−19.4556	−0.724564
$722$	79.9590	2.97576
$723$	0	0
$724$	30.0804	1.11793
$725$	−11.3907	−0.423042
$726$	0	0
$727$	19.0344	0.705948	0.352974	−	0.935633i	$-0.385170\pi$
$727$	19.0344	0.705948	0.352974	+	0.935633i	$0.385170\pi$
$728$	0	0
$729$	0	0
$730$	−7.68425	−0.284407
$731$	−5.34481	−0.197685
$732$	0	0
$733$	−24.6213	−0.909410	−0.454705	−	0.890642i	$-0.650255\pi$
$733$	−24.6213	−0.909410	−0.454705	+	0.890642i	$0.650255\pi$
$734$	−5.23298	−0.193153
$735$	0	0
$736$	−17.5007	−0.645083
$737$	−23.5593	−0.867817
$738$	0	0
$739$	44.5115	1.63738	0.818692	−	0.574233i	$-0.194700\pi$
$739$	44.5115	1.63738	0.818692	+	0.574233i	$0.194700\pi$
$740$	−11.2567	−0.413803
$741$	0	0
$742$	−19.1704	−0.703769
$743$	10.4112	0.381950	0.190975	−	0.981595i	$-0.438835\pi$
$743$	10.4112	0.381950	0.190975	+	0.981595i	$0.438835\pi$
$744$	0	0
$745$	18.4077	0.674407
$746$	−15.1336	−0.554081
$747$	0	0
$748$	−4.86964	−0.178052
$749$	−23.2043	−0.847866
$750$	0	0
$751$	1.69979	0.0620263	0.0310131	−	0.999519i	$-0.490127\pi$
$751$	1.69979	0.0620263	0.0310131	+	0.999519i	$0.490127\pi$
$752$	−52.1191	−1.90059
$753$	0	0
$754$	0	0
$755$	−22.6015	−0.822552
$756$	0	0
$757$	27.4252	0.996786	0.498393	−	0.866951i	$-0.333924\pi$
$757$	27.4252	0.996786	0.498393	+	0.866951i	$0.333924\pi$
$758$	−28.3773	−1.03071
$759$	0	0
$760$	15.6093	0.566207
$761$	−5.02608	−0.182195	−0.0910977	−	0.995842i	$-0.529038\pi$
$761$	−5.02608	−0.182195	−0.0910977	+	0.995842i	$0.529038\pi$
$762$	0	0
$763$	−7.15213	−0.258924
$764$	8.82908	0.319425
$765$	0	0
$766$	22.9541	0.829364
$767$	0	0
$768$	0	0
$769$	−42.4456	−1.53063	−0.765314	−	0.643657i	$-0.777416\pi$
$769$	−42.4456	−1.53063	−0.765314	+	0.643657i	$0.777416\pi$
$770$	46.5206	1.67649
$771$	0	0
$772$	−12.1806	−0.438390
$773$	26.3593	0.948078	0.474039	−	0.880504i	$-0.342796\pi$
$773$	26.3593	0.948078	0.474039	+	0.880504i	$0.342796\pi$
$774$	0	0
$775$	−14.2674	−0.512501
$776$	11.7482	0.421737
$777$	0	0
$778$	0.559270	0.0200508
$779$	−14.3448	−0.513956
$780$	0	0
$781$	47.2941	1.69232
$782$	3.83877	0.137274
$783$	0	0
$784$	−24.0446	−0.858736
$785$	−1.19029	−0.0424834
$786$	0	0
$787$	−17.1424	−0.611062	−0.305531	−	0.952182i	$-0.598834\pi$
$787$	−17.1424	−0.611062	−0.305531	+	0.952182i	$0.598834\pi$
$788$	29.1933	1.03997
$789$	0	0
$790$	−24.5623	−0.873886
$791$	21.2508	0.755592
$792$	0	0
$793$	0	0
$794$	−2.68664	−0.0953455
$795$	0	0
$796$	5.01879	0.177886
$797$	−30.1629	−1.06842	−0.534212	−	0.845351i	$-0.679392\pi$
$797$	−30.1629	−1.06842	−0.534212	+	0.845351i	$0.679392\pi$
$798$	0	0
$799$	7.94630	0.281120
$800$	18.0127	0.636844
$801$	0	0
$802$	−42.9463	−1.51649
$803$	15.3043	0.540076
$804$	0	0
$805$	−14.0838	−0.496390
$806$	0	0
$807$	0	0
$808$	−11.5017	−0.404629
$809$	29.8504	1.04948	0.524742	−	0.851261i	$-0.324162\pi$
$809$	29.8504	1.04948	0.524742	+	0.851261i	$0.324162\pi$
$810$	0	0
$811$	−47.7362	−1.67624	−0.838122	−	0.545484i	$-0.816346\pi$
$811$	−47.7362	−1.67624	−0.838122	+	0.545484i	$0.816346\pi$
$812$	−16.8049	−0.589737
$813$	0	0
$814$	58.3769	2.04611
$815$	−9.05861	−0.317309
$816$	0	0
$817$	56.5042	1.97683
$818$	−7.68963	−0.268862
$819$	0	0
$820$	3.24698	0.113389
$821$	−17.9299	−0.625758	−0.312879	−	0.949793i	$-0.601293\pi$
$821$	−17.9299	−0.625758	−0.312879	+	0.949793i	$0.601293\pi$
$822$	0	0
$823$	54.3196	1.89346	0.946731	−	0.322026i	$-0.104364\pi$
$823$	54.3196	1.89346	0.946731	+	0.322026i	$0.104364\pi$
$824$	−7.66296	−0.266952
$825$	0	0
$826$	−11.6582	−0.405640
$827$	49.1041	1.70752	0.853758	−	0.520670i	$-0.174318\pi$
$827$	49.1041	1.70752	0.853758	+	0.520670i	$0.174318\pi$
$828$	0	0
$829$	−7.35796	−0.255553	−0.127776	−	0.991803i	$-0.540784\pi$
$829$	−7.35796	−0.255553	−0.127776	+	0.991803i	$0.540784\pi$
$830$	−16.8388	−0.584482
$831$	0	0
$832$	0	0
$833$	3.66594	0.127017
$834$	0	0
$835$	−10.7662	−0.372579
$836$	51.4808	1.78050
$837$	0	0
$838$	−53.4989	−1.84809
$839$	−36.5013	−1.26016	−0.630082	−	0.776529i	$-0.716979\pi$
$839$	−36.5013	−1.26016	−0.630082	+	0.776529i	$0.716979\pi$
$840$	0	0
$841$	−13.6974	−0.472324
$842$	−52.9754	−1.82565
$843$	0	0
$844$	−4.87800	−0.167908
$845$	0	0
$846$	0	0
$847$	−54.7569	−1.88147
$848$	−15.2524	−0.523768
$849$	0	0
$850$	−3.95108	−0.135521
$851$	−17.6732	−0.605831
$852$	0	0
$853$	9.73855	0.333441	0.166721	−	0.986004i	$-0.446682\pi$
$853$	9.73855	0.333441	0.166721	+	0.986004i	$0.446682\pi$
$854$	−20.7638	−0.710522
$855$	0	0
$856$	−9.13946	−0.312380
$857$	15.2030	0.519323	0.259662	−	0.965700i	$-0.416389\pi$
$857$	15.2030	0.519323	0.259662	+	0.965700i	$0.416389\pi$
$858$	0	0
$859$	−31.9885	−1.09143	−0.545717	−	0.837970i	$-0.683743\pi$
$859$	−31.9885	−1.09143	−0.545717	+	0.837970i	$0.683743\pi$
$860$	−12.7899	−0.436130
$861$	0	0
$862$	−59.5652	−2.02880
$863$	−35.2905	−1.20130	−0.600652	−	0.799511i	$-0.705092\pi$
$863$	−35.2905	−1.20130	−0.600652	+	0.799511i	$0.705092\pi$
$864$	0	0
$865$	2.90408	0.0987418
$866$	−52.7362	−1.79205
$867$	0	0
$868$	−21.0489	−0.714447
$869$	48.9191	1.65947
$870$	0	0
$871$	0	0
$872$	−2.81700	−0.0953958
$873$	0	0
$874$	−40.5827	−1.37273
$875$	39.3870	1.33152
$876$	0	0
$877$	15.3263	0.517532	0.258766	−	0.965940i	$-0.416684\pi$
$877$	15.3263	0.517532	0.258766	+	0.965940i	$0.416684\pi$
$878$	−3.84117	−0.129633
$879$	0	0
$880$	37.0127	1.24770
$881$	36.4306	1.22738	0.613689	−	0.789548i	$-0.289685\pi$
$881$	36.4306	1.22738	0.613689	+	0.789548i	$0.289685\pi$
$882$	0	0
$883$	−37.6819	−1.26810	−0.634048	−	0.773294i	$-0.718608\pi$
$883$	−37.6819	−1.26810	−0.634048	+	0.773294i	$0.718608\pi$
$884$	0	0
$885$	0	0
$886$	41.4306	1.39189
$887$	−4.24890	−0.142664	−0.0713320	−	0.997453i	$-0.522725\pi$
$887$	−4.24890	−0.142664	−0.0713320	+	0.997453i	$0.522725\pi$
$888$	0	0
$889$	49.1333	1.64788
$890$	3.01746	0.101145
$891$	0	0
$892$	−9.28919	−0.311025
$893$	−84.0066	−2.81117
$894$	0	0
$895$	−28.9541	−0.967828
$896$	−34.7458	−1.16078
$897$	0	0
$898$	−23.3134	−0.777977
$899$	19.1672	0.639262
$900$	0	0
$901$	2.32544	0.0774715
$902$	−16.8388	−0.560670
$903$	0	0
$904$	8.37004	0.278383
$905$	34.8582	1.15872
$906$	0	0
$907$	−12.6183	−0.418985	−0.209493	−	0.977810i	$-0.567181\pi$
$907$	−12.6183	−0.418985	−0.209493	+	0.977810i	$0.567181\pi$
$908$	−26.4983	−0.879376
$909$	0	0
$910$	0	0
$911$	−6.77777	−0.224558	−0.112279	−	0.993677i	$-0.535815\pi$
$911$	−6.77777	−0.224558	−0.112279	+	0.993677i	$0.535815\pi$
$912$	0	0
$913$	33.5368	1.10990
$914$	8.74525	0.289267
$915$	0	0
$916$	−11.5918	−0.383004
$917$	77.8631	2.57126
$918$	0	0
$919$	−20.4674	−0.675157	−0.337579	−	0.941297i	$-0.609608\pi$
$919$	−20.4674	−0.675157	−0.337579	+	0.941297i	$0.609608\pi$
$920$	−5.54719	−0.182885
$921$	0	0
$922$	33.9385	1.11771
$923$	0	0
$924$	0	0
$925$	18.1903	0.598093
$926$	41.2223	1.35465
$927$	0	0
$928$	−24.1987	−0.794360
$929$	−4.65220	−0.152634	−0.0763169	−	0.997084i	$-0.524316\pi$
$929$	−4.65220	−0.152634	−0.0763169	+	0.997084i	$0.524316\pi$
$930$	0	0
$931$	−38.7555	−1.27016
$932$	−20.2145	−0.662147
$933$	0	0
$934$	−23.4252	−0.766496
$935$	−5.64310	−0.184549
$936$	0	0
$937$	−41.8544	−1.36732	−0.683662	−	0.729798i	$-0.739614\pi$
$937$	−41.8544	−1.36732	−0.683662	+	0.729798i	$0.739614\pi$
$938$	−28.2010	−0.920797
$939$	0	0
$940$	19.0151	0.620203
$941$	30.3454	0.989232	0.494616	−	0.869112i	$-0.335309\pi$
$941$	30.3454	0.989232	0.494616	+	0.869112i	$0.335309\pi$
$942$	0	0
$943$	5.09783	0.166008
$944$	−9.27545	−0.301890
$945$	0	0
$946$	66.3279	2.15651
$947$	12.0325	0.391004	0.195502	−	0.980703i	$-0.437366\pi$
$947$	12.0325	0.391004	0.195502	+	0.980703i	$0.437366\pi$
$948$	0	0
$949$	0	0
$950$	41.7700	1.35520
$951$	0	0
$952$	3.52004	0.114085
$953$	−22.9825	−0.744478	−0.372239	−	0.928137i	$-0.621410\pi$
$953$	−22.9825	−0.744478	−0.372239	+	0.928137i	$0.621410\pi$
$954$	0	0
$955$	10.2314	0.331082
$956$	−16.8455	−0.544822
$957$	0	0
$958$	−68.6359	−2.21753
$959$	46.9743	1.51688
$960$	0	0
$961$	−6.99223	−0.225556
$962$	0	0
$963$	0	0
$964$	−7.81700	−0.251769
$965$	−14.1153	−0.454387
$966$	0	0
$967$	38.8883	1.25056	0.625281	−	0.780399i	$-0.284984\pi$
$967$	38.8883	1.25056	0.625281	+	0.780399i	$0.284984\pi$
$968$	−21.5670	−0.693191
$969$	0	0
$970$	−22.5448	−0.723870
$971$	57.5133	1.84569	0.922845	−	0.385171i	$-0.125857\pi$
$971$	57.5133	1.84569	0.922845	+	0.385171i	$0.125857\pi$
$972$	0	0
$973$	−60.6378	−1.94396
$974$	38.2911	1.22693
$975$	0	0
$976$	−16.5200	−0.528794
$977$	16.3690	0.523690	0.261845	−	0.965110i	$-0.415669\pi$
$977$	16.3690	0.523690	0.261845	+	0.965110i	$0.415669\pi$
$978$	0	0
$979$	−6.00969	−0.192070
$980$	8.77240	0.280224
$981$	0	0
$982$	11.4426	0.365150
$983$	−15.6963	−0.500635	−0.250318	−	0.968164i	$-0.580535\pi$
$983$	−15.6963	−0.500635	−0.250318	+	0.968164i	$0.580535\pi$
$984$	0	0
$985$	33.8301	1.07792
$986$	5.30798	0.169040
$987$	0	0
$988$	0	0
$989$	−20.0804	−0.638519
$990$	0	0
$991$	−11.2644	−0.357827	−0.178913	−	0.983865i	$-0.557258\pi$
$991$	−11.2644	−0.357827	−0.178913	+	0.983865i	$0.557258\pi$
$992$	−30.3099	−0.962340
$993$	0	0
$994$	56.6122	1.79563
$995$	5.81594	0.184378
$996$	0	0
$997$	−7.70112	−0.243897	−0.121948	−	0.992536i	$-0.538914\pi$
$997$	−7.70112	−0.243897	−0.121948	+	0.992536i	$0.538914\pi$
$998$	8.38059	0.265283
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.a.q.1.3		3
3.2	odd	2		507.2.a.j.1.1	✓	3
12.11	even	2		8112.2.a.by.1.2		3
13.5	odd	4		1521.2.b.m.1351.1		6
13.8	odd	4		1521.2.b.m.1351.6		6
13.12	even	2		1521.2.a.p.1.1		3
39.2	even	12		507.2.j.h.316.1		12
39.5	even	4		507.2.b.g.337.6		6
39.8	even	4		507.2.b.g.337.1		6
39.11	even	12		507.2.j.h.316.6		12
39.17	odd	6		507.2.e.j.484.1		6
39.20	even	12		507.2.j.h.361.1		12
39.23	odd	6		507.2.e.j.22.1		6
39.29	odd	6		507.2.e.k.22.3		6
39.32	even	12		507.2.j.h.361.6		12
39.35	odd	6		507.2.e.k.484.3		6
39.38	odd	2		507.2.a.k.1.3	yes	3
156.155	even	2		8112.2.a.cf.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
507.2.a.j.1.1	✓	3	3.2	odd	2
507.2.a.k.1.3	yes	3	39.38	odd	2
507.2.b.g.337.1		6	39.8	even	4
507.2.b.g.337.6		6	39.5	even	4
507.2.e.j.22.1		6	39.23	odd	6
507.2.e.j.484.1		6	39.17	odd	6
507.2.e.k.22.3		6	39.29	odd	6
507.2.e.k.484.3		6	39.35	odd	6
507.2.j.h.316.1		12	39.2	even	12
507.2.j.h.316.6		12	39.11	even	12
507.2.j.h.361.1		12	39.20	even	12
507.2.j.h.361.6		12	39.32	even	12
1521.2.a.p.1.1		3	13.12	even	2
1521.2.a.q.1.3		3	1.1	even	1	trivial
1521.2.b.m.1351.1		6	13.5	odd	4
1521.2.b.m.1351.6		6	13.8	odd	4
8112.2.a.by.1.2		3	12.11	even	2
8112.2.a.cf.1.2		3	156.155	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:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q+1.80194 q^{2} +1.24698 q^{4} +1.44504 q^{5} -3.44504 q^{7} -1.35690 q^{8} +O(q^{10})\)	\(q+1.80194 q^{2} +1.24698 q^{4} +1.44504 q^{5} -3.44504 q^{7} -1.35690 q^{8} +2.60388 q^{10} -5.18598 q^{11} -6.20775 q^{14} -4.93900 q^{16} +0.753020 q^{17} -7.96077 q^{19} +1.80194 q^{20} -9.34481 q^{22} +2.82908 q^{23} -2.91185 q^{25} -4.29590 q^{28} +3.91185 q^{29} +4.89977 q^{31} -6.18598 q^{32} +1.35690 q^{34} -4.97823 q^{35} -6.24698 q^{37} -14.3448 q^{38} -1.96077 q^{40} +1.80194 q^{41} -7.09783 q^{43} -6.46681 q^{44} +5.09783 q^{46} +10.5526 q^{47} +4.86831 q^{49} -5.24698 q^{50} +3.08815 q^{53} -7.49396 q^{55} +4.67456 q^{56} +7.04892 q^{58} +1.87800 q^{59} +3.34481 q^{61} +8.82908 q^{62} -1.26875 q^{64} +4.54288 q^{67} +0.939001 q^{68} -8.97046 q^{70} -9.11960 q^{71} -2.95108 q^{73} -11.2567 q^{74} -9.92692 q^{76} +17.8659 q^{77} -9.43296 q^{79} -7.13706 q^{80} +3.24698 q^{82} -6.46681 q^{83} +1.08815 q^{85} -12.7899 q^{86} +7.03684 q^{88} +1.15883 q^{89} +3.52781 q^{92} +19.0151 q^{94} -11.5036 q^{95} -8.65817 q^{97} +8.77240 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} - q^{4} + 4 q^{5} - 10 q^{7}+O(q^{10}) \) 3 * q + q^2 - q^4 + 4 * q^5 - 10 * q^7	\( 3 q + q^{2} - q^{4} + 4 q^{5} - 10 q^{7} - q^{10} - q^{11} - q^{14} - 5 q^{16} + 7 q^{17} - 11 q^{19} + q^{20} - 5 q^{22} - 2 q^{23} - 5 q^{25} + q^{28} + 8 q^{29} - 8 q^{31} - 4 q^{32} - 18 q^{35} - 14 q^{37} - 20 q^{38} + 7 q^{40} + q^{41} - 3 q^{43} - 16 q^{44} - 3 q^{46} - 9 q^{47} + 17 q^{49} - 11 q^{50} + 13 q^{53} - 13 q^{55} - 7 q^{56} + 12 q^{58} - 14 q^{59} - 13 q^{61} + 16 q^{62} + 4 q^{64} - 5 q^{67} - 7 q^{68} + 8 q^{70} - 6 q^{71} - 18 q^{73} - 7 q^{74} - q^{76} + 15 q^{77} - 9 q^{79} - 16 q^{80} + 5 q^{82} - 16 q^{83} + 7 q^{85} - 15 q^{86} - 7 q^{88} - 5 q^{89} + 17 q^{92} + 32 q^{94} - 3 q^{95} - 5 q^{97} - 13 q^{98}+O(q^{100}) \) 3 * q + q^2 - q^4 + 4 * q^5 - 10 * q^7 - q^10 - q^11 - q^14 - 5 * q^16 + 7 * q^17 - 11 * q^19 + q^20 - 5 * q^22 - 2 * q^23 - 5 * q^25 + q^28 + 8 * q^29 - 8 * q^31 - 4 * q^32 - 18 * q^35 - 14 * q^37 - 20 * q^38 + 7 * q^40 + q^41 - 3 * q^43 - 16 * q^44 - 3 * q^46 - 9 * q^47 + 17 * q^49 - 11 * q^50 + 13 * q^53 - 13 * q^55 - 7 * q^56 + 12 * q^58 - 14 * q^59 - 13 * q^61 + 16 * q^62 + 4 * q^64 - 5 * q^67 - 7 * q^68 + 8 * q^70 - 6 * q^71 - 18 * q^73 - 7 * q^74 - q^76 + 15 * q^77 - 9 * q^79 - 16 * q^80 + 5 * q^82 - 16 * q^83 + 7 * q^85 - 15 * q^86 - 7 * q^88 - 5 * q^89 + 17 * q^92 + 32 * q^94 - 3 * q^95 - 5 * q^97 - 13 * q^98

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