LMFDB - Embedded newform 1125.2.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1125,2,Mod(1,1125)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1125, 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("1125.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1125 = 3^{2} \cdot 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1125.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:	$8.98317022739$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 375)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	1125.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.381966 q^{2} -1.85410 q^{4} -3.61803 q^{7} -1.47214 q^{8} +O(q^{10})$	$q+0.381966 q^{2} -1.85410 q^{4} -3.61803 q^{7} -1.47214 q^{8} +1.76393 q^{11} -3.00000 q^{13} -1.38197 q^{14} +3.14590 q^{16} +5.61803 q^{17} -1.00000 q^{19} +0.673762 q^{22} +6.70820 q^{23} -1.14590 q^{26} +6.70820 q^{28} -0.236068 q^{29} +8.09017 q^{31} +4.14590 q^{32} +2.14590 q^{34} -5.00000 q^{37} -0.381966 q^{38} +10.8541 q^{41} +4.61803 q^{43} -3.27051 q^{44} +2.56231 q^{46} -13.1803 q^{47} +6.09017 q^{49} +5.56231 q^{52} -1.38197 q^{53} +5.32624 q^{56} -0.0901699 q^{58} +13.7984 q^{59} +6.09017 q^{61} +3.09017 q^{62} -4.70820 q^{64} +8.00000 q^{67} -10.4164 q^{68} -7.85410 q^{71} -15.8541 q^{73} -1.90983 q^{74} +1.85410 q^{76} -6.38197 q^{77} +7.23607 q^{79} +4.14590 q^{82} +7.32624 q^{83} +1.76393 q^{86} -2.59675 q^{88} -3.47214 q^{89} +10.8541 q^{91} -12.4377 q^{92} -5.03444 q^{94} +10.5623 q^{97} +2.32624 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} + 3 q^{4} - 5 q^{7} + 6 q^{8}+O(q^{10}) $ 2 * q + 3 * q^2 + 3 * q^4 - 5 * q^7 + 6 * q^8	$ 2 q + 3 q^{2} + 3 q^{4} - 5 q^{7} + 6 q^{8} + 8 q^{11} - 6 q^{13} - 5 q^{14} + 13 q^{16} + 9 q^{17} - 2 q^{19} + 17 q^{22} - 9 q^{26} + 4 q^{29} + 5 q^{31} + 15 q^{32} + 11 q^{34} - 10 q^{37} - 3 q^{38} + 15 q^{41} + 7 q^{43} + 27 q^{44} - 15 q^{46} - 4 q^{47} + q^{49} - 9 q^{52} - 5 q^{53} - 5 q^{56} + 11 q^{58} + 3 q^{59} + q^{61} - 5 q^{62} + 4 q^{64} + 16 q^{67} + 6 q^{68} - 9 q^{71} - 25 q^{73} - 15 q^{74} - 3 q^{76} - 15 q^{77} + 10 q^{79} + 15 q^{82} - q^{83} + 8 q^{86} + 44 q^{88} + 2 q^{89} + 15 q^{91} - 45 q^{92} + 19 q^{94} + q^{97} - 11 q^{98}+O(q^{100}) $ 2 * q + 3 * q^2 + 3 * q^4 - 5 * q^7 + 6 * q^8 + 8 * q^11 - 6 * q^13 - 5 * q^14 + 13 * q^16 + 9 * q^17 - 2 * q^19 + 17 * q^22 - 9 * q^26 + 4 * q^29 + 5 * q^31 + 15 * q^32 + 11 * q^34 - 10 * q^37 - 3 * q^38 + 15 * q^41 + 7 * q^43 + 27 * q^44 - 15 * q^46 - 4 * q^47 + q^49 - 9 * q^52 - 5 * q^53 - 5 * q^56 + 11 * q^58 + 3 * q^59 + q^61 - 5 * q^62 + 4 * q^64 + 16 * q^67 + 6 * q^68 - 9 * q^71 - 25 * q^73 - 15 * q^74 - 3 * q^76 - 15 * q^77 + 10 * q^79 + 15 * q^82 - q^83 + 8 * q^86 + 44 * q^88 + 2 * q^89 + 15 * q^91 - 45 * q^92 + 19 * q^94 + q^97 - 11 * 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 $

$2$	0.381966	0.270091	0.135045	−	0.990839i	$-0.456882\pi$
$2$	0.381966	0.270091	0.135045	+	0.990839i	$0.456882\pi$
$3$	0	0
$3$	0	0
$4$	−1.85410	−0.927051
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.61803	−1.36749	−0.683744	−	0.729722i	$-0.739650\pi$
$7$	−3.61803	−1.36749	−0.683744	+	0.729722i	$0.739650\pi$
$8$	−1.47214	−0.520479
$9$	0	0
$10$	0	0
$11$	1.76393	0.531846	0.265923	−	0.963994i	$-0.414323\pi$
$11$	1.76393	0.531846	0.265923	+	0.963994i	$0.414323\pi$
$12$	0	0
$13$	−3.00000	−0.832050	−0.416025	−	0.909353i	$-0.636577\pi$
$13$	−3.00000	−0.832050	−0.416025	+	0.909353i	$0.636577\pi$
$14$	−1.38197	−0.369346
$15$	0	0
$16$	3.14590	0.786475
$17$	5.61803	1.36257	0.681287	−	0.732017i	$-0.261421\pi$
$17$	5.61803	1.36257	0.681287	+	0.732017i	$0.261421\pi$
$18$	0	0
$19$	−1.00000	−0.229416	−0.114708	−	0.993399i	$-0.536593\pi$
$19$	−1.00000	−0.229416	−0.114708	+	0.993399i	$0.536593\pi$
$20$	0	0
$21$	0	0
$22$	0.673762	0.143647
$23$	6.70820	1.39876	0.699379	−	0.714751i	$-0.253460\pi$
$23$	6.70820	1.39876	0.699379	+	0.714751i	$0.253460\pi$
$24$	0	0
$25$	0	0
$26$	−1.14590	−0.224729
$27$	0	0
$28$	6.70820	1.26773
$29$	−0.236068	−0.0438367	−0.0219184	−	0.999760i	$-0.506977\pi$
$29$	−0.236068	−0.0438367	−0.0219184	+	0.999760i	$0.506977\pi$
$30$	0	0
$31$	8.09017	1.45304	0.726519	−	0.687147i	$-0.241137\pi$
$31$	8.09017	1.45304	0.726519	+	0.687147i	$0.241137\pi$
$32$	4.14590	0.732898
$33$	0	0
$34$	2.14590	0.368018
$35$	0	0
$36$	0	0
$37$	−5.00000	−0.821995	−0.410997	−	0.911636i	$-0.634819\pi$
$37$	−5.00000	−0.821995	−0.410997	+	0.911636i	$0.634819\pi$
$38$	−0.381966	−0.0619631
$39$	0	0
$40$	0	0
$41$	10.8541	1.69513	0.847563	−	0.530695i	$-0.178069\pi$
$41$	10.8541	1.69513	0.847563	+	0.530695i	$0.178069\pi$
$42$	0	0
$43$	4.61803	0.704244	0.352122	−	0.935954i	$-0.385460\pi$
$43$	4.61803	0.704244	0.352122	+	0.935954i	$0.385460\pi$
$44$	−3.27051	−0.493048
$45$	0	0
$46$	2.56231	0.377791
$47$	−13.1803	−1.92255	−0.961275	−	0.275591i	$-0.911127\pi$
$47$	−13.1803	−1.92255	−0.961275	+	0.275591i	$0.911127\pi$
$48$	0	0
$49$	6.09017	0.870024
$50$	0	0
$51$	0	0
$52$	5.56231	0.771353
$53$	−1.38197	−0.189828	−0.0949138	−	0.995485i	$-0.530258\pi$
$53$	−1.38197	−0.189828	−0.0949138	+	0.995485i	$0.530258\pi$
$54$	0	0
$55$	0	0
$56$	5.32624	0.711748
$57$	0	0
$58$	−0.0901699	−0.0118399
$59$	13.7984	1.79640	0.898198	−	0.439592i	$-0.144877\pi$
$59$	13.7984	1.79640	0.898198	+	0.439592i	$0.144877\pi$
$60$	0	0
$61$	6.09017	0.779766	0.389883	−	0.920864i	$-0.372515\pi$
$61$	6.09017	0.779766	0.389883	+	0.920864i	$0.372515\pi$
$62$	3.09017	0.392452
$63$	0	0
$64$	−4.70820	−0.588525
$65$	0	0
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	−10.4164	−1.26317
$69$	0	0
$70$	0	0
$71$	−7.85410	−0.932110	−0.466055	−	0.884756i	$-0.654325\pi$
$71$	−7.85410	−0.932110	−0.466055	+	0.884756i	$0.654325\pi$
$72$	0	0
$73$	−15.8541	−1.85558	−0.927791	−	0.373100i	$-0.878295\pi$
$73$	−15.8541	−1.85558	−0.927791	+	0.373100i	$0.878295\pi$
$74$	−1.90983	−0.222013
$75$	0	0
$76$	1.85410	0.212680
$77$	−6.38197	−0.727293
$78$	0	0
$79$	7.23607	0.814121	0.407061	−	0.913401i	$-0.366554\pi$
$79$	7.23607	0.814121	0.407061	+	0.913401i	$0.366554\pi$
$80$	0	0
$81$	0	0
$82$	4.14590	0.457838
$83$	7.32624	0.804159	0.402080	−	0.915605i	$-0.368288\pi$
$83$	7.32624	0.804159	0.402080	+	0.915605i	$0.368288\pi$
$84$	0	0
$85$	0	0
$86$	1.76393	0.190210
$87$	0	0
$88$	−2.59675	−0.276814
$89$	−3.47214	−0.368046	−0.184023	−	0.982922i	$-0.558912\pi$
$89$	−3.47214	−0.368046	−0.184023	+	0.982922i	$0.558912\pi$
$90$	0	0
$91$	10.8541	1.13782
$92$	−12.4377	−1.29672
$93$	0	0
$94$	−5.03444	−0.519263
$95$	0	0
$96$	0	0
$97$	10.5623	1.07244	0.536220	−	0.844078i	$-0.319852\pi$
$97$	10.5623	1.07244	0.536220	+	0.844078i	$0.319852\pi$
$98$	2.32624	0.234986
$99$	0	0
$100$	0	0
$101$	−1.52786	−0.152028	−0.0760141	−	0.997107i	$-0.524219\pi$
$101$	−1.52786	−0.152028	−0.0760141	+	0.997107i	$0.524219\pi$
$102$	0	0
$103$	−0.909830	−0.0896482	−0.0448241	−	0.998995i	$-0.514273\pi$
$103$	−0.909830	−0.0896482	−0.0448241	+	0.998995i	$0.514273\pi$
$104$	4.41641	0.433064
$105$	0	0
$106$	−0.527864	−0.0512707
$107$	−0.618034	−0.0597476	−0.0298738	−	0.999554i	$-0.509511\pi$
$107$	−0.618034	−0.0597476	−0.0298738	+	0.999554i	$0.509511\pi$
$108$	0	0
$109$	9.00000	0.862044	0.431022	−	0.902342i	$-0.358153\pi$
$109$	9.00000	0.862044	0.431022	+	0.902342i	$0.358153\pi$
$110$	0	0
$111$	0	0
$112$	−11.3820	−1.07549
$113$	−0.763932	−0.0718647	−0.0359323	−	0.999354i	$-0.511440\pi$
$113$	−0.763932	−0.0718647	−0.0359323	+	0.999354i	$0.511440\pi$
$114$	0	0
$115$	0	0
$116$	0.437694	0.0406389
$117$	0	0
$118$	5.27051	0.485190
$119$	−20.3262	−1.86330
$120$	0	0
$121$	−7.88854	−0.717140
$122$	2.32624	0.210608
$123$	0	0
$124$	−15.0000	−1.34704
$125$	0	0
$126$	0	0
$127$	6.41641	0.569364	0.284682	−	0.958622i	$-0.408112\pi$
$127$	6.41641	0.569364	0.284682	+	0.958622i	$0.408112\pi$
$128$	−10.0902	−0.891853
$129$	0	0
$130$	0	0
$131$	10.0902	0.881582	0.440791	−	0.897610i	$-0.354698\pi$
$131$	10.0902	0.881582	0.440791	+	0.897610i	$0.354698\pi$
$132$	0	0
$133$	3.61803	0.313723
$134$	3.05573	0.263975
$135$	0	0
$136$	−8.27051	−0.709190
$137$	16.2361	1.38714	0.693570	−	0.720389i	$-0.256037\pi$
$137$	16.2361	1.38714	0.693570	+	0.720389i	$0.256037\pi$
$138$	0	0
$139$	−1.76393	−0.149615	−0.0748074	−	0.997198i	$-0.523834\pi$
$139$	−1.76393	−0.149615	−0.0748074	+	0.997198i	$0.523834\pi$
$140$	0	0
$141$	0	0
$142$	−3.00000	−0.251754
$143$	−5.29180	−0.442522
$144$	0	0
$145$	0	0
$146$	−6.05573	−0.501176
$147$	0	0
$148$	9.27051	0.762031
$149$	11.9443	0.978513	0.489256	−	0.872140i	$-0.337268\pi$
$149$	11.9443	0.978513	0.489256	+	0.872140i	$0.337268\pi$
$150$	0	0
$151$	−4.18034	−0.340191	−0.170096	−	0.985428i	$-0.554408\pi$
$151$	−4.18034	−0.340191	−0.170096	+	0.985428i	$0.554408\pi$
$152$	1.47214	0.119406
$153$	0	0
$154$	−2.43769	−0.196435
$155$	0	0
$156$	0	0
$157$	−16.5623	−1.32182	−0.660908	−	0.750467i	$-0.729829\pi$
$157$	−16.5623	−1.32182	−0.660908	+	0.750467i	$0.729829\pi$
$158$	2.76393	0.219887
$159$	0	0
$160$	0	0
$161$	−24.2705	−1.91278
$162$	0	0
$163$	9.00000	0.704934	0.352467	−	0.935824i	$-0.385343\pi$
$163$	9.00000	0.704934	0.352467	+	0.935824i	$0.385343\pi$
$164$	−20.1246	−1.57147
$165$	0	0
$166$	2.79837	0.217196
$167$	4.47214	0.346064	0.173032	−	0.984916i	$-0.444644\pi$
$167$	4.47214	0.346064	0.173032	+	0.984916i	$0.444644\pi$
$168$	0	0
$169$	−4.00000	−0.307692
$170$	0	0
$171$	0	0
$172$	−8.56231	−0.652870
$173$	−3.32624	−0.252889	−0.126445	−	0.991974i	$-0.540357\pi$
$173$	−3.32624	−0.252889	−0.126445	+	0.991974i	$0.540357\pi$
$174$	0	0
$175$	0	0
$176$	5.54915	0.418283
$177$	0	0
$178$	−1.32624	−0.0994057
$179$	17.7639	1.32774	0.663869	−	0.747849i	$-0.268913\pi$
$179$	17.7639	1.32774	0.663869	+	0.747849i	$0.268913\pi$
$180$	0	0
$181$	−13.4164	−0.997234	−0.498617	−	0.866822i	$-0.666159\pi$
$181$	−13.4164	−0.997234	−0.498617	+	0.866822i	$0.666159\pi$
$182$	4.14590	0.307314
$183$	0	0
$184$	−9.87539	−0.728023
$185$	0	0
$186$	0	0
$187$	9.90983	0.724679
$188$	24.4377	1.78230
$189$	0	0
$190$	0	0
$191$	17.9443	1.29840	0.649201	−	0.760617i	$-0.275103\pi$
$191$	17.9443	1.29840	0.649201	+	0.760617i	$0.275103\pi$
$192$	0	0
$193$	−8.70820	−0.626830	−0.313415	−	0.949616i	$-0.601473\pi$
$193$	−8.70820	−0.626830	−0.313415	+	0.949616i	$0.601473\pi$
$194$	4.03444	0.289656
$195$	0	0
$196$	−11.2918	−0.806557
$197$	7.18034	0.511578	0.255789	−	0.966733i	$-0.417665\pi$
$197$	7.18034	0.511578	0.255789	+	0.966733i	$0.417665\pi$
$198$	0	0
$199$	5.29180	0.375125	0.187563	−	0.982253i	$-0.439941\pi$
$199$	5.29180	0.375125	0.187563	+	0.982253i	$0.439941\pi$
$200$	0	0
$201$	0	0
$202$	−0.583592	−0.0410614
$203$	0.854102	0.0599462
$204$	0	0
$205$	0	0
$206$	−0.347524	−0.0242132
$207$	0	0
$208$	−9.43769	−0.654386
$209$	−1.76393	−0.122014
$210$	0	0
$211$	−23.4164	−1.61205	−0.806026	−	0.591880i	$-0.798386\pi$
$211$	−23.4164	−1.61205	−0.806026	+	0.591880i	$0.798386\pi$
$212$	2.56231	0.175980
$213$	0	0
$214$	−0.236068	−0.0161373
$215$	0	0
$216$	0	0
$217$	−29.2705	−1.98701
$218$	3.43769	0.232830
$219$	0	0
$220$	0	0
$221$	−16.8541	−1.13373
$222$	0	0
$223$	−10.8885	−0.729151	−0.364575	−	0.931174i	$-0.618786\pi$
$223$	−10.8885	−0.729151	−0.364575	+	0.931174i	$0.618786\pi$
$224$	−15.0000	−1.00223
$225$	0	0
$226$	−0.291796	−0.0194100
$227$	6.18034	0.410204	0.205102	−	0.978741i	$-0.434247\pi$
$227$	6.18034	0.410204	0.205102	+	0.978741i	$0.434247\pi$
$228$	0	0
$229$	21.1246	1.39595	0.697977	−	0.716120i	$-0.254084\pi$
$229$	21.1246	1.39595	0.697977	+	0.716120i	$0.254084\pi$
$230$	0	0
$231$	0	0
$232$	0.347524	0.0228161
$233$	−12.5066	−0.819333	−0.409667	−	0.912235i	$-0.634355\pi$
$233$	−12.5066	−0.819333	−0.409667	+	0.912235i	$0.634355\pi$
$234$	0	0
$235$	0	0
$236$	−25.5836	−1.66535
$237$	0	0
$238$	−7.76393	−0.503261
$239$	24.0902	1.55826	0.779132	−	0.626860i	$-0.215660\pi$
$239$	24.0902	1.55826	0.779132	+	0.626860i	$0.215660\pi$
$240$	0	0
$241$	−15.7082	−1.01185	−0.505927	−	0.862576i	$-0.668850\pi$
$241$	−15.7082	−1.01185	−0.505927	+	0.862576i	$0.668850\pi$
$242$	−3.01316	−0.193693
$243$	0	0
$244$	−11.2918	−0.722883
$245$	0	0
$246$	0	0
$247$	3.00000	0.190885
$248$	−11.9098	−0.756275
$249$	0	0
$250$	0	0
$251$	−2.32624	−0.146831	−0.0734154	−	0.997301i	$-0.523390\pi$
$251$	−2.32624	−0.146831	−0.0734154	+	0.997301i	$0.523390\pi$
$252$	0	0
$253$	11.8328	0.743923
$254$	2.45085	0.153780
$255$	0	0
$256$	5.56231	0.347644
$257$	−3.18034	−0.198384	−0.0991921	−	0.995068i	$-0.531626\pi$
$257$	−3.18034	−0.198384	−0.0991921	+	0.995068i	$0.531626\pi$
$258$	0	0
$259$	18.0902	1.12407
$260$	0	0
$261$	0	0
$262$	3.85410	0.238107
$263$	−13.1459	−0.810611	−0.405305	−	0.914181i	$-0.632835\pi$
$263$	−13.1459	−0.810611	−0.405305	+	0.914181i	$0.632835\pi$
$264$	0	0
$265$	0	0
$266$	1.38197	0.0847338
$267$	0	0
$268$	−14.8328	−0.906058
$269$	−20.3820	−1.24271	−0.621355	−	0.783529i	$-0.713418\pi$
$269$	−20.3820	−1.24271	−0.621355	+	0.783529i	$0.713418\pi$
$270$	0	0
$271$	1.85410	0.112629	0.0563143	−	0.998413i	$-0.482065\pi$
$271$	1.85410	0.112629	0.0563143	+	0.998413i	$0.482065\pi$
$272$	17.6738	1.07163
$273$	0	0
$274$	6.20163	0.374654
$275$	0	0
$276$	0	0
$277$	−15.2705	−0.917516	−0.458758	−	0.888561i	$-0.651706\pi$
$277$	−15.2705	−0.917516	−0.458758	+	0.888561i	$0.651706\pi$
$278$	−0.673762	−0.0404096
$279$	0	0
$280$	0	0
$281$	−3.27051	−0.195102	−0.0975511	−	0.995231i	$-0.531101\pi$
$281$	−3.27051	−0.195102	−0.0975511	+	0.995231i	$0.531101\pi$
$282$	0	0
$283$	21.4721	1.27639	0.638193	−	0.769876i	$-0.279682\pi$
$283$	21.4721	1.27639	0.638193	+	0.769876i	$0.279682\pi$
$284$	14.5623	0.864114
$285$	0	0
$286$	−2.02129	−0.119521
$287$	−39.2705	−2.31806
$288$	0	0
$289$	14.5623	0.856606
$290$	0	0
$291$	0	0
$292$	29.3951	1.72022
$293$	19.4164	1.13432	0.567159	−	0.823608i	$-0.308042\pi$
$293$	19.4164	1.13432	0.567159	+	0.823608i	$0.308042\pi$
$294$	0	0
$295$	0	0
$296$	7.36068	0.427831
$297$	0	0
$298$	4.56231	0.264287
$299$	−20.1246	−1.16384
$300$	0	0
$301$	−16.7082	−0.963045
$302$	−1.59675	−0.0918825
$303$	0	0
$304$	−3.14590	−0.180430
$305$	0	0
$306$	0	0
$307$	22.1246	1.26272	0.631359	−	0.775491i	$-0.282497\pi$
$307$	22.1246	1.26272	0.631359	+	0.775491i	$0.282497\pi$
$308$	11.8328	0.674237
$309$	0	0
$310$	0	0
$311$	−19.0902	−1.08250	−0.541252	−	0.840860i	$-0.682050\pi$
$311$	−19.0902	−1.08250	−0.541252	+	0.840860i	$0.682050\pi$
$312$	0	0
$313$	19.9787	1.12926	0.564632	−	0.825343i	$-0.309018\pi$
$313$	19.9787	1.12926	0.564632	+	0.825343i	$0.309018\pi$
$314$	−6.32624	−0.357010
$315$	0	0
$316$	−13.4164	−0.754732
$317$	16.4164	0.922037	0.461019	−	0.887390i	$-0.347484\pi$
$317$	16.4164	0.922037	0.461019	+	0.887390i	$0.347484\pi$
$318$	0	0
$319$	−0.416408	−0.0233144
$320$	0	0
$321$	0	0
$322$	−9.27051	−0.516625
$323$	−5.61803	−0.312596
$324$	0	0
$325$	0	0
$326$	3.43769	0.190396
$327$	0	0
$328$	−15.9787	−0.882277
$329$	47.6869	2.62906
$330$	0	0
$331$	6.14590	0.337809	0.168905	−	0.985632i	$-0.445977\pi$
$331$	6.14590	0.337809	0.168905	+	0.985632i	$0.445977\pi$
$332$	−13.5836	−0.745496
$333$	0	0
$334$	1.70820	0.0934688
$335$	0	0
$336$	0	0
$337$	18.4721	1.00624	0.503121	−	0.864216i	$-0.332185\pi$
$337$	18.4721	1.00624	0.503121	+	0.864216i	$0.332185\pi$
$338$	−1.52786	−0.0831048
$339$	0	0
$340$	0	0
$341$	14.2705	0.772791
$342$	0	0
$343$	3.29180	0.177740
$344$	−6.79837	−0.366544
$345$	0	0
$346$	−1.27051	−0.0683030
$347$	−14.5066	−0.778754	−0.389377	−	0.921078i	$-0.627310\pi$
$347$	−14.5066	−0.778754	−0.389377	+	0.921078i	$0.627310\pi$
$348$	0	0
$349$	−18.9098	−1.01222	−0.506110	−	0.862469i	$-0.668917\pi$
$349$	−18.9098	−1.01222	−0.506110	+	0.862469i	$0.668917\pi$
$350$	0	0
$351$	0	0
$352$	7.31308	0.389789
$353$	33.3262	1.77378	0.886888	−	0.461984i	$-0.152862\pi$
$353$	33.3262	1.77378	0.886888	+	0.461984i	$0.152862\pi$
$354$	0	0
$355$	0	0
$356$	6.43769	0.341197
$357$	0	0
$358$	6.78522	0.358610
$359$	−11.5279	−0.608417	−0.304209	−	0.952605i	$-0.598392\pi$
$359$	−11.5279	−0.608417	−0.304209	+	0.952605i	$0.598392\pi$
$360$	0	0
$361$	−18.0000	−0.947368
$362$	−5.12461	−0.269344
$363$	0	0
$364$	−20.1246	−1.05482
$365$	0	0
$366$	0	0
$367$	20.9098	1.09148	0.545742	−	0.837953i	$-0.316248\pi$
$367$	20.9098	1.09148	0.545742	+	0.837953i	$0.316248\pi$
$368$	21.1033	1.10009
$369$	0	0
$370$	0	0
$371$	5.00000	0.259587
$372$	0	0
$373$	35.2705	1.82624	0.913119	−	0.407693i	$-0.133667\pi$
$373$	35.2705	1.82624	0.913119	+	0.407693i	$0.133667\pi$
$374$	3.78522	0.195729
$375$	0	0
$376$	19.4033	1.00065
$377$	0.708204	0.0364744
$378$	0	0
$379$	−2.56231	−0.131617	−0.0658084	−	0.997832i	$-0.520963\pi$
$379$	−2.56231	−0.131617	−0.0658084	+	0.997832i	$0.520963\pi$
$380$	0	0
$381$	0	0
$382$	6.85410	0.350686
$383$	−9.97871	−0.509888	−0.254944	−	0.966956i	$-0.582057\pi$
$383$	−9.97871	−0.509888	−0.254944	+	0.966956i	$0.582057\pi$
$384$	0	0
$385$	0	0
$386$	−3.32624	−0.169301
$387$	0	0
$388$	−19.5836	−0.994206
$389$	−18.0344	−0.914382	−0.457191	−	0.889368i	$-0.651145\pi$
$389$	−18.0344	−0.914382	−0.457191	+	0.889368i	$0.651145\pi$
$390$	0	0
$391$	37.6869	1.90591
$392$	−8.96556	−0.452829
$393$	0	0
$394$	2.74265	0.138172
$395$	0	0
$396$	0	0
$397$	−16.9443	−0.850409	−0.425204	−	0.905097i	$-0.639798\pi$
$397$	−16.9443	−0.850409	−0.425204	+	0.905097i	$0.639798\pi$
$398$	2.02129	0.101318
$399$	0	0
$400$	0	0
$401$	−13.6525	−0.681772	−0.340886	−	0.940105i	$-0.610727\pi$
$401$	−13.6525	−0.681772	−0.340886	+	0.940105i	$0.610727\pi$
$402$	0	0
$403$	−24.2705	−1.20900
$404$	2.83282	0.140938
$405$	0	0
$406$	0.326238	0.0161909
$407$	−8.81966	−0.437174
$408$	0	0
$409$	−16.7639	−0.828923	−0.414462	−	0.910067i	$-0.636030\pi$
$409$	−16.7639	−0.828923	−0.414462	+	0.910067i	$0.636030\pi$
$410$	0	0
$411$	0	0
$412$	1.68692	0.0831085
$413$	−49.9230	−2.45655
$414$	0	0
$415$	0	0
$416$	−12.4377	−0.609808
$417$	0	0
$418$	−0.673762	−0.0329548
$419$	−3.27051	−0.159775	−0.0798874	−	0.996804i	$-0.525456\pi$
$419$	−3.27051	−0.159775	−0.0798874	+	0.996804i	$0.525456\pi$
$420$	0	0
$421$	−26.0000	−1.26716	−0.633581	−	0.773676i	$-0.718416\pi$
$421$	−26.0000	−1.26716	−0.633581	+	0.773676i	$0.718416\pi$
$422$	−8.94427	−0.435400
$423$	0	0
$424$	2.03444	0.0988012
$425$	0	0
$426$	0	0
$427$	−22.0344	−1.06632
$428$	1.14590	0.0553891
$429$	0	0
$430$	0	0
$431$	7.76393	0.373975	0.186988	−	0.982362i	$-0.440128\pi$
$431$	7.76393	0.373975	0.186988	+	0.982362i	$0.440128\pi$
$432$	0	0
$433$	−27.3050	−1.31219	−0.656096	−	0.754677i	$-0.727793\pi$
$433$	−27.3050	−1.31219	−0.656096	+	0.754677i	$0.727793\pi$
$434$	−11.1803	−0.536673
$435$	0	0
$436$	−16.6869	−0.799158
$437$	−6.70820	−0.320897
$438$	0	0
$439$	1.38197	0.0659576	0.0329788	−	0.999456i	$-0.489501\pi$
$439$	1.38197	0.0659576	0.0329788	+	0.999456i	$0.489501\pi$
$440$	0	0
$441$	0	0
$442$	−6.43769	−0.306210
$443$	−22.4508	−1.06667	−0.533336	−	0.845903i	$-0.679062\pi$
$443$	−22.4508	−1.06667	−0.533336	+	0.845903i	$0.679062\pi$
$444$	0	0
$445$	0	0
$446$	−4.15905	−0.196937
$447$	0	0
$448$	17.0344	0.804802
$449$	4.50658	0.212679	0.106339	−	0.994330i	$-0.466087\pi$
$449$	4.50658	0.212679	0.106339	+	0.994330i	$0.466087\pi$
$450$	0	0
$451$	19.1459	0.901545
$452$	1.41641	0.0666222
$453$	0	0
$454$	2.36068	0.110792
$455$	0	0
$456$	0	0
$457$	17.6869	0.827359	0.413680	−	0.910423i	$-0.364243\pi$
$457$	17.6869	0.827359	0.413680	+	0.910423i	$0.364243\pi$
$458$	8.06888	0.377034
$459$	0	0
$460$	0	0
$461$	17.4721	0.813758	0.406879	−	0.913482i	$-0.366617\pi$
$461$	17.4721	0.813758	0.406879	+	0.913482i	$0.366617\pi$
$462$	0	0
$463$	0.673762	0.0313124	0.0156562	−	0.999877i	$-0.495016\pi$
$463$	0.673762	0.0313124	0.0156562	+	0.999877i	$0.495016\pi$
$464$	−0.742646	−0.0344765
$465$	0	0
$466$	−4.77709	−0.221294
$467$	4.90983	0.227200	0.113600	−	0.993527i	$-0.463762\pi$
$467$	4.90983	0.227200	0.113600	+	0.993527i	$0.463762\pi$
$468$	0	0
$469$	−28.9443	−1.33652
$470$	0	0
$471$	0	0
$472$	−20.3131	−0.934985
$473$	8.14590	0.374549
$474$	0	0
$475$	0	0
$476$	37.6869	1.72738
$477$	0	0
$478$	9.20163	0.420873
$479$	−19.5967	−0.895398	−0.447699	−	0.894184i	$-0.647756\pi$
$479$	−19.5967	−0.895398	−0.447699	+	0.894184i	$0.647756\pi$
$480$	0	0
$481$	15.0000	0.683941
$482$	−6.00000	−0.273293
$483$	0	0
$484$	14.6262	0.664826
$485$	0	0
$486$	0	0
$487$	6.94427	0.314675	0.157337	−	0.987545i	$-0.449709\pi$
$487$	6.94427	0.314675	0.157337	+	0.987545i	$0.449709\pi$
$488$	−8.96556	−0.405852
$489$	0	0
$490$	0	0
$491$	16.7984	0.758100	0.379050	−	0.925376i	$-0.376251\pi$
$491$	16.7984	0.758100	0.379050	+	0.925376i	$0.376251\pi$
$492$	0	0
$493$	−1.32624	−0.0597308
$494$	1.14590	0.0515564
$495$	0	0
$496$	25.4508	1.14278
$497$	28.4164	1.27465
$498$	0	0
$499$	17.2918	0.774087	0.387044	−	0.922061i	$-0.373496\pi$
$499$	17.2918	0.774087	0.387044	+	0.922061i	$0.373496\pi$
$500$	0	0
$501$	0	0
$502$	−0.888544	−0.0396577
$503$	13.8541	0.617724	0.308862	−	0.951107i	$-0.400052\pi$
$503$	13.8541	0.617724	0.308862	+	0.951107i	$0.400052\pi$
$504$	0	0
$505$	0	0
$506$	4.51973	0.200927
$507$	0	0
$508$	−11.8967	−0.527830
$509$	34.6525	1.53594	0.767972	−	0.640483i	$-0.221266\pi$
$509$	34.6525	1.53594	0.767972	+	0.640483i	$0.221266\pi$
$510$	0	0
$511$	57.3607	2.53749
$512$	22.3050	0.985749
$513$	0	0
$514$	−1.21478	−0.0535817
$515$	0	0
$516$	0	0
$517$	−23.2492	−1.02250
$518$	6.90983	0.303601
$519$	0	0
$520$	0	0
$521$	−0.326238	−0.0142927	−0.00714637	−	0.999974i	$-0.502275\pi$
$521$	−0.326238	−0.0142927	−0.00714637	+	0.999974i	$0.502275\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	−18.7082	−0.817272
$525$	0	0
$526$	−5.02129	−0.218938
$527$	45.4508	1.97987
$528$	0	0
$529$	22.0000	0.956522
$530$	0	0
$531$	0	0
$532$	−6.70820	−0.290838
$533$	−32.5623	−1.41043
$534$	0	0
$535$	0	0
$536$	−11.7771	−0.508693
$537$	0	0
$538$	−7.78522	−0.335645
$539$	10.7426	0.462719
$540$	0	0
$541$	17.0344	0.732368	0.366184	−	0.930542i	$-0.380664\pi$
$541$	17.0344	0.732368	0.366184	+	0.930542i	$0.380664\pi$
$542$	0.708204	0.0304200
$543$	0	0
$544$	23.2918	0.998628
$545$	0	0
$546$	0	0
$547$	1.50658	0.0644166	0.0322083	−	0.999481i	$-0.489746\pi$
$547$	1.50658	0.0644166	0.0322083	+	0.999481i	$0.489746\pi$
$548$	−30.1033	−1.28595
$549$	0	0
$550$	0	0
$551$	0.236068	0.0100568
$552$	0	0
$553$	−26.1803	−1.11330
$554$	−5.83282	−0.247813
$555$	0	0
$556$	3.27051	0.138701
$557$	20.2148	0.856528	0.428264	−	0.903654i	$-0.359125\pi$
$557$	20.2148	0.856528	0.428264	+	0.903654i	$0.359125\pi$
$558$	0	0
$559$	−13.8541	−0.585966
$560$	0	0
$561$	0	0
$562$	−1.24922	−0.0526953
$563$	30.2148	1.27340	0.636701	−	0.771111i	$-0.280299\pi$
$563$	30.2148	1.27340	0.636701	+	0.771111i	$0.280299\pi$
$564$	0	0
$565$	0	0
$566$	8.20163	0.344740
$567$	0	0
$568$	11.5623	0.485144
$569$	−29.6525	−1.24310	−0.621548	−	0.783376i	$-0.713496\pi$
$569$	−29.6525	−1.24310	−0.621548	+	0.783376i	$0.713496\pi$
$570$	0	0
$571$	31.4508	1.31618	0.658089	−	0.752941i	$-0.271365\pi$
$571$	31.4508	1.31618	0.658089	+	0.752941i	$0.271365\pi$
$572$	9.81153	0.410241
$573$	0	0
$574$	−15.0000	−0.626088
$575$	0	0
$576$	0	0
$577$	9.83282	0.409345	0.204673	−	0.978830i	$-0.434387\pi$
$577$	9.83282	0.409345	0.204673	+	0.978830i	$0.434387\pi$
$578$	5.56231	0.231361
$579$	0	0
$580$	0	0
$581$	−26.5066	−1.09968
$582$	0	0
$583$	−2.43769	−0.100959
$584$	23.3394	0.965791
$585$	0	0
$586$	7.41641	0.306369
$587$	−25.9443	−1.07083	−0.535417	−	0.844588i	$-0.679846\pi$
$587$	−25.9443	−1.07083	−0.535417	+	0.844588i	$0.679846\pi$
$588$	0	0
$589$	−8.09017	−0.333350
$590$	0	0
$591$	0	0
$592$	−15.7295	−0.646478
$593$	31.3607	1.28783	0.643914	−	0.765098i	$-0.277309\pi$
$593$	31.3607	1.28783	0.643914	+	0.765098i	$0.277309\pi$
$594$	0	0
$595$	0	0
$596$	−22.1459	−0.907131
$597$	0	0
$598$	−7.68692	−0.314341
$599$	38.4508	1.57106	0.785530	−	0.618824i	$-0.212391\pi$
$599$	38.4508	1.57106	0.785530	+	0.618824i	$0.212391\pi$
$600$	0	0
$601$	−9.00000	−0.367118	−0.183559	−	0.983009i	$-0.558762\pi$
$601$	−9.00000	−0.367118	−0.183559	+	0.983009i	$0.558762\pi$
$602$	−6.38197	−0.260110
$603$	0	0
$604$	7.75078	0.315375
$605$	0	0
$606$	0	0
$607$	−24.3607	−0.988769	−0.494385	−	0.869243i	$-0.664607\pi$
$607$	−24.3607	−0.988769	−0.494385	+	0.869243i	$0.664607\pi$
$608$	−4.14590	−0.168138
$609$	0	0
$610$	0	0
$611$	39.5410	1.59966
$612$	0	0
$613$	32.5967	1.31657	0.658285	−	0.752769i	$-0.271282\pi$
$613$	32.5967	1.31657	0.658285	+	0.752769i	$0.271282\pi$
$614$	8.45085	0.341049
$615$	0	0
$616$	9.39512	0.378540
$617$	6.23607	0.251055	0.125527	−	0.992090i	$-0.459938\pi$
$617$	6.23607	0.251055	0.125527	+	0.992090i	$0.459938\pi$
$618$	0	0
$619$	19.4164	0.780411	0.390206	−	0.920728i	$-0.372404\pi$
$619$	19.4164	0.780411	0.390206	+	0.920728i	$0.372404\pi$
$620$	0	0
$621$	0	0
$622$	−7.29180	−0.292374
$623$	12.5623	0.503298
$624$	0	0
$625$	0	0
$626$	7.63119	0.305004
$627$	0	0
$628$	30.7082	1.22539
$629$	−28.0902	−1.12003
$630$	0	0
$631$	−1.29180	−0.0514256	−0.0257128	−	0.999669i	$-0.508186\pi$
$631$	−1.29180	−0.0514256	−0.0257128	+	0.999669i	$0.508186\pi$
$632$	−10.6525	−0.423733
$633$	0	0
$634$	6.27051	0.249034
$635$	0	0
$636$	0	0
$637$	−18.2705	−0.723904
$638$	−0.159054	−0.00629699
$639$	0	0
$640$	0	0
$641$	9.59675	0.379049	0.189524	−	0.981876i	$-0.439305\pi$
$641$	9.59675	0.379049	0.189524	+	0.981876i	$0.439305\pi$
$642$	0	0
$643$	−22.9443	−0.904834	−0.452417	−	0.891807i	$-0.649438\pi$
$643$	−22.9443	−0.904834	−0.452417	+	0.891807i	$0.649438\pi$
$644$	45.0000	1.77325
$645$	0	0
$646$	−2.14590	−0.0844292
$647$	22.8541	0.898487	0.449244	−	0.893409i	$-0.351693\pi$
$647$	22.8541	0.898487	0.449244	+	0.893409i	$0.351693\pi$
$648$	0	0
$649$	24.3394	0.955405
$650$	0	0
$651$	0	0
$652$	−16.6869	−0.653510
$653$	−37.0132	−1.44844	−0.724218	−	0.689571i	$-0.757799\pi$
$653$	−37.0132	−1.44844	−0.724218	+	0.689571i	$0.757799\pi$
$654$	0	0
$655$	0	0
$656$	34.1459	1.33317
$657$	0	0
$658$	18.2148	0.710086
$659$	−29.0689	−1.13236	−0.566181	−	0.824281i	$-0.691580\pi$
$659$	−29.0689	−1.13236	−0.566181	+	0.824281i	$0.691580\pi$
$660$	0	0
$661$	−13.9787	−0.543709	−0.271854	−	0.962338i	$-0.587637\pi$
$661$	−13.9787	−0.543709	−0.271854	+	0.962338i	$0.587637\pi$
$662$	2.34752	0.0912391
$663$	0	0
$664$	−10.7852	−0.418548
$665$	0	0
$666$	0	0
$667$	−1.58359	−0.0613169
$668$	−8.29180	−0.320819
$669$	0	0
$670$	0	0
$671$	10.7426	0.414715
$672$	0	0
$673$	16.5836	0.639250	0.319625	−	0.947544i	$-0.396443\pi$
$673$	16.5836	0.639250	0.319625	+	0.947544i	$0.396443\pi$
$674$	7.05573	0.271776
$675$	0	0
$676$	7.41641	0.285246
$677$	−38.0689	−1.46311	−0.731553	−	0.681785i	$-0.761204\pi$
$677$	−38.0689	−1.46311	−0.731553	+	0.681785i	$0.761204\pi$
$678$	0	0
$679$	−38.2148	−1.46655
$680$	0	0
$681$	0	0
$682$	5.45085	0.208724
$683$	21.0000	0.803543	0.401771	−	0.915740i	$-0.368395\pi$
$683$	21.0000	0.803543	0.401771	+	0.915740i	$0.368395\pi$
$684$	0	0
$685$	0	0
$686$	1.25735	0.0480060
$687$	0	0
$688$	14.5279	0.553870
$689$	4.14590	0.157946
$690$	0	0
$691$	14.8197	0.563766	0.281883	−	0.959449i	$-0.409041\pi$
$691$	14.8197	0.563766	0.281883	+	0.959449i	$0.409041\pi$
$692$	6.16718	0.234441
$693$	0	0
$694$	−5.54102	−0.210334
$695$	0	0
$696$	0	0
$697$	60.9787	2.30973
$698$	−7.22291	−0.273391
$699$	0	0
$700$	0	0
$701$	−34.1803	−1.29097	−0.645487	−	0.763771i	$-0.723345\pi$
$701$	−34.1803	−1.29097	−0.645487	+	0.763771i	$0.723345\pi$
$702$	0	0
$703$	5.00000	0.188579
$704$	−8.30495	−0.313005
$705$	0	0
$706$	12.7295	0.479081
$707$	5.52786	0.207897
$708$	0	0
$709$	0.708204	0.0265972	0.0132986	−	0.999912i	$-0.495767\pi$
$709$	0.708204	0.0265972	0.0132986	+	0.999912i	$0.495767\pi$
$710$	0	0
$711$	0	0
$712$	5.11146	0.191560
$713$	54.2705	2.03245
$714$	0	0
$715$	0	0
$716$	−32.9361	−1.23088
$717$	0	0
$718$	−4.40325	−0.164328
$719$	−1.47214	−0.0549014	−0.0274507	−	0.999623i	$-0.508739\pi$
$719$	−1.47214	−0.0549014	−0.0274507	+	0.999623i	$0.508739\pi$
$720$	0	0
$721$	3.29180	0.122593
$722$	−6.87539	−0.255875
$723$	0	0
$724$	24.8754	0.924487
$725$	0	0
$726$	0	0
$727$	4.41641	0.163796	0.0818978	−	0.996641i	$-0.473902\pi$
$727$	4.41641	0.163796	0.0818978	+	0.996641i	$0.473902\pi$
$728$	−15.9787	−0.592211
$729$	0	0
$730$	0	0
$731$	25.9443	0.959584
$732$	0	0
$733$	7.05573	0.260609	0.130305	−	0.991474i	$-0.458404\pi$
$733$	7.05573	0.260609	0.130305	+	0.991474i	$0.458404\pi$
$734$	7.98684	0.294800
$735$	0	0
$736$	27.8115	1.02515
$737$	14.1115	0.519802
$738$	0	0
$739$	40.2705	1.48137	0.740687	−	0.671850i	$-0.234500\pi$
$739$	40.2705	1.48137	0.740687	+	0.671850i	$0.234500\pi$
$740$	0	0
$741$	0	0
$742$	1.90983	0.0701121
$743$	−40.0689	−1.46998	−0.734992	−	0.678075i	$-0.762814\pi$
$743$	−40.0689	−1.46998	−0.734992	+	0.678075i	$0.762814\pi$
$744$	0	0
$745$	0	0
$746$	13.4721	0.493250
$747$	0	0
$748$	−18.3738	−0.671814
$749$	2.23607	0.0817041
$750$	0	0
$751$	−0.785218	−0.0286530	−0.0143265	−	0.999897i	$-0.504560\pi$
$751$	−0.785218	−0.0286530	−0.0143265	+	0.999897i	$0.504560\pi$
$752$	−41.4640	−1.51204
$753$	0	0
$754$	0.270510	0.00985139
$755$	0	0
$756$	0	0
$757$	11.6525	0.423516	0.211758	−	0.977322i	$-0.432081\pi$
$757$	11.6525	0.423516	0.211758	+	0.977322i	$0.432081\pi$
$758$	−0.978714	−0.0355485
$759$	0	0
$760$	0	0
$761$	35.0689	1.27125	0.635623	−	0.772000i	$-0.280743\pi$
$761$	35.0689	1.27125	0.635623	+	0.772000i	$0.280743\pi$
$762$	0	0
$763$	−32.5623	−1.17883
$764$	−33.2705	−1.20368
$765$	0	0
$766$	−3.81153	−0.137716
$767$	−41.3951	−1.49469
$768$	0	0
$769$	−16.9443	−0.611026	−0.305513	−	0.952188i	$-0.598828\pi$
$769$	−16.9443	−0.611026	−0.305513	+	0.952188i	$0.598828\pi$
$770$	0	0
$771$	0	0
$772$	16.1459	0.581104
$773$	21.0000	0.755318	0.377659	−	0.925945i	$-0.376729\pi$
$773$	21.0000	0.755318	0.377659	+	0.925945i	$0.376729\pi$
$774$	0	0
$775$	0	0
$776$	−15.5492	−0.558182
$777$	0	0
$778$	−6.88854	−0.246966
$779$	−10.8541	−0.388889
$780$	0	0
$781$	−13.8541	−0.495739
$782$	14.3951	0.514768
$783$	0	0
$784$	19.1591	0.684252
$785$	0	0
$786$	0	0
$787$	14.4377	0.514648	0.257324	−	0.966325i	$-0.417159\pi$
$787$	14.4377	0.514648	0.257324	+	0.966325i	$0.417159\pi$
$788$	−13.3131	−0.474259
$789$	0	0
$790$	0	0
$791$	2.76393	0.0982741
$792$	0	0
$793$	−18.2705	−0.648805
$794$	−6.47214	−0.229688
$795$	0	0
$796$	−9.81153	−0.347760
$797$	−32.3262	−1.14505	−0.572527	−	0.819886i	$-0.694037\pi$
$797$	−32.3262	−1.14505	−0.572527	+	0.819886i	$0.694037\pi$
$798$	0	0
$799$	−74.0476	−2.61962
$800$	0	0
$801$	0	0
$802$	−5.21478	−0.184140
$803$	−27.9656	−0.986883
$804$	0	0
$805$	0	0
$806$	−9.27051	−0.326540
$807$	0	0
$808$	2.24922	0.0791274
$809$	−19.7984	−0.696074	−0.348037	−	0.937481i	$-0.613152\pi$
$809$	−19.7984	−0.696074	−0.348037	+	0.937481i	$0.613152\pi$
$810$	0	0
$811$	−0.0344419	−0.00120942	−0.000604709	−	1.00000i	$-0.500192\pi$
$811$	−0.0344419	−0.00120942	−0.000604709		1.00000i	$0.500192\pi$
$812$	−1.58359	−0.0555732
$813$	0	0
$814$	−3.36881	−0.118077
$815$	0	0
$816$	0	0
$817$	−4.61803	−0.161565
$818$	−6.40325	−0.223884
$819$	0	0
$820$	0	0
$821$	8.23607	0.287441	0.143720	−	0.989618i	$-0.454093\pi$
$821$	8.23607	0.287441	0.143720	+	0.989618i	$0.454093\pi$
$822$	0	0
$823$	−54.3607	−1.89489	−0.947447	−	0.319913i	$-0.896346\pi$
$823$	−54.3607	−1.89489	−0.947447	+	0.319913i	$0.896346\pi$
$824$	1.33939	0.0466600
$825$	0	0
$826$	−19.0689	−0.663491
$827$	17.3262	0.602492	0.301246	−	0.953546i	$-0.402597\pi$
$827$	17.3262	0.602492	0.301246	+	0.953546i	$0.402597\pi$
$828$	0	0
$829$	−45.1246	−1.56724	−0.783621	−	0.621239i	$-0.786630\pi$
$829$	−45.1246	−1.56724	−0.783621	+	0.621239i	$0.786630\pi$
$830$	0	0
$831$	0	0
$832$	14.1246	0.489683
$833$	34.2148	1.18547
$834$	0	0
$835$	0	0
$836$	3.27051	0.113113
$837$	0	0
$838$	−1.24922	−0.0431537
$839$	−0.527864	−0.0182239	−0.00911195	−	0.999958i	$-0.502900\pi$
$839$	−0.527864	−0.0182239	−0.00911195	+	0.999958i	$0.502900\pi$
$840$	0	0
$841$	−28.9443	−0.998078
$842$	−9.93112	−0.342249
$843$	0	0
$844$	43.4164	1.49445
$845$	0	0
$846$	0	0
$847$	28.5410	0.980681
$848$	−4.34752	−0.149295
$849$	0	0
$850$	0	0
$851$	−33.5410	−1.14977
$852$	0	0
$853$	−9.29180	−0.318145	−0.159073	−	0.987267i	$-0.550850\pi$
$853$	−9.29180	−0.318145	−0.159073	+	0.987267i	$0.550850\pi$
$854$	−8.41641	−0.288004
$855$	0	0
$856$	0.909830	0.0310974
$857$	−10.2148	−0.348930	−0.174465	−	0.984663i	$-0.555820\pi$
$857$	−10.2148	−0.348930	−0.174465	+	0.984663i	$0.555820\pi$
$858$	0	0
$859$	−8.81966	−0.300923	−0.150461	−	0.988616i	$-0.548076\pi$
$859$	−8.81966	−0.300923	−0.150461	+	0.988616i	$0.548076\pi$
$860$	0	0
$861$	0	0
$862$	2.96556	0.101007
$863$	32.2148	1.09660	0.548302	−	0.836280i	$-0.315274\pi$
$863$	32.2148	1.09660	0.548302	+	0.836280i	$0.315274\pi$
$864$	0	0
$865$	0	0
$866$	−10.4296	−0.354411
$867$	0	0
$868$	54.2705	1.84206
$869$	12.7639	0.432987
$870$	0	0
$871$	−24.0000	−0.813209
$872$	−13.2492	−0.448675
$873$	0	0
$874$	−2.56231	−0.0866713
$875$	0	0
$876$	0	0
$877$	33.9443	1.14622	0.573108	−	0.819480i	$-0.305737\pi$
$877$	33.9443	1.14622	0.573108	+	0.819480i	$0.305737\pi$
$878$	0.527864	0.0178145
$879$	0	0
$880$	0	0
$881$	−46.7639	−1.57552	−0.787758	−	0.615984i	$-0.788758\pi$
$881$	−46.7639	−1.57552	−0.787758	+	0.615984i	$0.788758\pi$
$882$	0	0
$883$	−7.94427	−0.267346	−0.133673	−	0.991025i	$-0.542677\pi$
$883$	−7.94427	−0.267346	−0.133673	+	0.991025i	$0.542677\pi$
$884$	31.2492	1.05103
$885$	0	0
$886$	−8.57546	−0.288098
$887$	54.3050	1.82338	0.911691	−	0.410877i	$-0.134777\pi$
$887$	54.3050	1.82338	0.911691	+	0.410877i	$0.134777\pi$
$888$	0	0
$889$	−23.2148	−0.778599
$890$	0	0
$891$	0	0
$892$	20.1885	0.675960
$893$	13.1803	0.441063
$894$	0	0
$895$	0	0
$896$	36.5066	1.21960
$897$	0	0
$898$	1.72136	0.0574425
$899$	−1.90983	−0.0636964
$900$	0	0
$901$	−7.76393	−0.258654
$902$	7.31308	0.243499
$903$	0	0
$904$	1.12461	0.0374040
$905$	0	0
$906$	0	0
$907$	−51.5410	−1.71139	−0.855696	−	0.517479i	$-0.826870\pi$
$907$	−51.5410	−1.71139	−0.855696	+	0.517479i	$0.826870\pi$
$908$	−11.4590	−0.380280
$909$	0	0
$910$	0	0
$911$	12.2705	0.406540	0.203270	−	0.979123i	$-0.434843\pi$
$911$	12.2705	0.406540	0.203270	+	0.979123i	$0.434843\pi$
$912$	0	0
$913$	12.9230	0.427688
$914$	6.75580	0.223462
$915$	0	0
$916$	−39.1672	−1.29412
$917$	−36.5066	−1.20555
$918$	0	0
$919$	−39.2705	−1.29541	−0.647707	−	0.761889i	$-0.724272\pi$
$919$	−39.2705	−1.29541	−0.647707	+	0.761889i	$0.724272\pi$
$920$	0	0
$921$	0	0
$922$	6.67376	0.219789
$923$	23.5623	0.775563
$924$	0	0
$925$	0	0
$926$	0.257354	0.00845718
$927$	0	0
$928$	−0.978714	−0.0321279
$929$	−28.9230	−0.948932	−0.474466	−	0.880274i	$-0.657359\pi$
$929$	−28.9230	−0.948932	−0.474466	+	0.880274i	$0.657359\pi$
$930$	0	0
$931$	−6.09017	−0.199597
$932$	23.1885	0.759564
$933$	0	0
$934$	1.87539	0.0613646
$935$	0	0
$936$	0	0
$937$	−29.8197	−0.974166	−0.487083	−	0.873356i	$-0.661939\pi$
$937$	−29.8197	−0.974166	−0.487083	+	0.873356i	$0.661939\pi$
$938$	−11.0557	−0.360982
$939$	0	0
$940$	0	0
$941$	−32.5623	−1.06150	−0.530750	−	0.847528i	$-0.678090\pi$
$941$	−32.5623	−1.06150	−0.530750	+	0.847528i	$0.678090\pi$
$942$	0	0
$943$	72.8115	2.37107
$944$	43.4083	1.41282
$945$	0	0
$946$	3.11146	0.101162
$947$	−38.8885	−1.26371	−0.631854	−	0.775087i	$-0.717706\pi$
$947$	−38.8885	−1.26371	−0.631854	+	0.775087i	$0.717706\pi$
$948$	0	0
$949$	47.5623	1.54394
$950$	0	0
$951$	0	0
$952$	29.9230	0.969810
$953$	7.92299	0.256651	0.128325	−	0.991732i	$-0.459040\pi$
$953$	7.92299	0.256651	0.128325	+	0.991732i	$0.459040\pi$
$954$	0	0
$955$	0	0
$956$	−44.6656	−1.44459
$957$	0	0
$958$	−7.48529	−0.241839
$959$	−58.7426	−1.89690
$960$	0	0
$961$	34.4508	1.11132
$962$	5.72949	0.184726
$963$	0	0
$964$	29.1246	0.938041
$965$	0	0
$966$	0	0
$967$	−20.4164	−0.656547	−0.328274	−	0.944583i	$-0.606467\pi$
$967$	−20.4164	−0.656547	−0.328274	+	0.944583i	$0.606467\pi$
$968$	11.6130	0.373256
$969$	0	0
$970$	0	0
$971$	−39.5967	−1.27072	−0.635360	−	0.772216i	$-0.719148\pi$
$971$	−39.5967	−1.27072	−0.635360	+	0.772216i	$0.719148\pi$
$972$	0	0
$973$	6.38197	0.204596
$974$	2.65248	0.0849908
$975$	0	0
$976$	19.1591	0.613266
$977$	−43.8541	−1.40302	−0.701509	−	0.712661i	$-0.747490\pi$
$977$	−43.8541	−1.40302	−0.701509	+	0.712661i	$0.747490\pi$
$978$	0	0
$979$	−6.12461	−0.195743
$980$	0	0
$981$	0	0
$982$	6.41641	0.204756
$983$	−2.03444	−0.0648886	−0.0324443	−	0.999474i	$-0.510329\pi$
$983$	−2.03444	−0.0648886	−0.0324443	+	0.999474i	$0.510329\pi$
$984$	0	0
$985$	0	0
$986$	−0.506578	−0.0161327
$987$	0	0
$988$	−5.56231	−0.176961
$989$	30.9787	0.985066
$990$	0	0
$991$	−32.1459	−1.02115	−0.510574	−	0.859834i	$-0.670567\pi$
$991$	−32.1459	−1.02115	−0.510574	+	0.859834i	$0.670567\pi$
$992$	33.5410	1.06493
$993$	0	0
$994$	10.8541	0.344271
$995$	0	0
$996$	0	0
$997$	−43.7214	−1.38467	−0.692335	−	0.721577i	$-0.743418\pi$
$997$	−43.7214	−1.38467	−0.692335	+	0.721577i	$0.743418\pi$
$998$	6.60488	0.209074
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1125.2.a.f.1.1		2
3.2	odd	2		375.2.a.a.1.2	✓	2
5.2	odd	4		1125.2.b.b.874.3		4
5.3	odd	4		1125.2.b.b.874.2		4
5.4	even	2		1125.2.a.a.1.2		2
12.11	even	2		6000.2.a.m.1.2		2
15.2	even	4		375.2.b.a.124.2		4
15.8	even	4		375.2.b.a.124.3		4
15.14	odd	2		375.2.a.d.1.1	yes	2
60.23	odd	4		6000.2.f.n.1249.1		4
60.47	odd	4		6000.2.f.n.1249.4		4
60.59	even	2		6000.2.a.q.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
375.2.a.a.1.2	✓	2	3.2	odd	2
375.2.a.d.1.1	yes	2	15.14	odd	2
375.2.b.a.124.2		4	15.2	even	4
375.2.b.a.124.3		4	15.8	even	4
1125.2.a.a.1.2		2	5.4	even	2
1125.2.a.f.1.1		2	1.1	even	1	trivial
1125.2.b.b.874.2		4	5.3	odd	4
1125.2.b.b.874.3		4	5.2	odd	4
6000.2.a.m.1.2		2	12.11	even	2
6000.2.a.q.1.1		2	60.59	even	2
6000.2.f.n.1249.1		4	60.23	odd	4
6000.2.f.n.1249.4		4	60.47	odd	4

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:	\( N \)	\(=\)	\( 1125 = 3^{2} \cdot 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1125.a (trivial)

\(f(q)\)	\(=\)	\(q+0.381966 q^{2} -1.85410 q^{4} -3.61803 q^{7} -1.47214 q^{8} +O(q^{10})\)	\(q+0.381966 q^{2} -1.85410 q^{4} -3.61803 q^{7} -1.47214 q^{8} +1.76393 q^{11} -3.00000 q^{13} -1.38197 q^{14} +3.14590 q^{16} +5.61803 q^{17} -1.00000 q^{19} +0.673762 q^{22} +6.70820 q^{23} -1.14590 q^{26} +6.70820 q^{28} -0.236068 q^{29} +8.09017 q^{31} +4.14590 q^{32} +2.14590 q^{34} -5.00000 q^{37} -0.381966 q^{38} +10.8541 q^{41} +4.61803 q^{43} -3.27051 q^{44} +2.56231 q^{46} -13.1803 q^{47} +6.09017 q^{49} +5.56231 q^{52} -1.38197 q^{53} +5.32624 q^{56} -0.0901699 q^{58} +13.7984 q^{59} +6.09017 q^{61} +3.09017 q^{62} -4.70820 q^{64} +8.00000 q^{67} -10.4164 q^{68} -7.85410 q^{71} -15.8541 q^{73} -1.90983 q^{74} +1.85410 q^{76} -6.38197 q^{77} +7.23607 q^{79} +4.14590 q^{82} +7.32624 q^{83} +1.76393 q^{86} -2.59675 q^{88} -3.47214 q^{89} +10.8541 q^{91} -12.4377 q^{92} -5.03444 q^{94} +10.5623 q^{97} +2.32624 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} + 3 q^{4} - 5 q^{7} + 6 q^{8}+O(q^{10}) \) 2 * q + 3 * q^2 + 3 * q^4 - 5 * q^7 + 6 * q^8	\( 2 q + 3 q^{2} + 3 q^{4} - 5 q^{7} + 6 q^{8} + 8 q^{11} - 6 q^{13} - 5 q^{14} + 13 q^{16} + 9 q^{17} - 2 q^{19} + 17 q^{22} - 9 q^{26} + 4 q^{29} + 5 q^{31} + 15 q^{32} + 11 q^{34} - 10 q^{37} - 3 q^{38} + 15 q^{41} + 7 q^{43} + 27 q^{44} - 15 q^{46} - 4 q^{47} + q^{49} - 9 q^{52} - 5 q^{53} - 5 q^{56} + 11 q^{58} + 3 q^{59} + q^{61} - 5 q^{62} + 4 q^{64} + 16 q^{67} + 6 q^{68} - 9 q^{71} - 25 q^{73} - 15 q^{74} - 3 q^{76} - 15 q^{77} + 10 q^{79} + 15 q^{82} - q^{83} + 8 q^{86} + 44 q^{88} + 2 q^{89} + 15 q^{91} - 45 q^{92} + 19 q^{94} + q^{97} - 11 q^{98}+O(q^{100}) \) 2 * q + 3 * q^2 + 3 * q^4 - 5 * q^7 + 6 * q^8 + 8 * q^11 - 6 * q^13 - 5 * q^14 + 13 * q^16 + 9 * q^17 - 2 * q^19 + 17 * q^22 - 9 * q^26 + 4 * q^29 + 5 * q^31 + 15 * q^32 + 11 * q^34 - 10 * q^37 - 3 * q^38 + 15 * q^41 + 7 * q^43 + 27 * q^44 - 15 * q^46 - 4 * q^47 + q^49 - 9 * q^52 - 5 * q^53 - 5 * q^56 + 11 * q^58 + 3 * q^59 + q^61 - 5 * q^62 + 4 * q^64 + 16 * q^67 + 6 * q^68 - 9 * q^71 - 25 * q^73 - 15 * q^74 - 3 * q^76 - 15 * q^77 + 10 * q^79 + 15 * q^82 - q^83 + 8 * q^86 + 44 * q^88 + 2 * q^89 + 15 * q^91 - 45 * q^92 + 19 * q^94 + q^97 - 11 * q^98

Introduction

L-functions

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