LMFDB - Embedded newform 6525.2.a.bt.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6525,2,Mod(1,6525)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("6525.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 6525 = 3^{2} \cdot 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6525.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,14,0,0,0,0,0,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$52.1023873189$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.337383424.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 13x^{4} + 41x^{2} - 1 $ x^6 - 13x^4 + 41x^2 - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 145)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.156785$ of defining polynomial
Character	$\chi$	$=$	6525.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.156785 q^{2} -1.97542 q^{4} -1.09364 q^{7} +0.623285 q^{8} +4.40198 q^{11} -3.97108 q^{13} +0.171466 q^{14} +3.85312 q^{16} +6.22138 q^{17} +6.97542 q^{19} -0.690163 q^{22} -0.780070 q^{23} +0.622605 q^{26} +2.16040 q^{28} +1.00000 q^{29} +6.40198 q^{31} -1.85068 q^{32} -0.975419 q^{34} +1.09364 q^{37} -1.09364 q^{38} +0.376593 q^{43} -8.69575 q^{44} +0.122303 q^{46} -4.75115 q^{47} -5.80395 q^{49} +7.84455 q^{52} -11.2861 q^{53} -0.681649 q^{56} -0.156785 q^{58} -10.0000 q^{59} -1.14688 q^{61} -1.00373 q^{62} -7.41607 q^{64} +5.90782 q^{67} -12.2898 q^{68} -2.00000 q^{71} -8.72223 q^{73} -0.171466 q^{74} -13.7794 q^{76} -4.81418 q^{77} -5.54886 q^{79} +6.22138 q^{83} -0.0590441 q^{86} +2.74369 q^{88} +10.8040 q^{89} +4.34293 q^{91} +1.54096 q^{92} +0.744908 q^{94} +17.4176 q^{97} +0.909972 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 14 q^{4} + 10 q^{11} + 8 q^{14} + 42 q^{16} + 16 q^{19} + 46 q^{26} + 6 q^{29} + 22 q^{31} + 20 q^{34} + 2 q^{44} - 44 q^{46} - 2 q^{49} - 16 q^{56} - 60 q^{59} + 12 q^{61} + 38 q^{64} - 12 q^{71}+ \cdots + 2 q^{94}+O(q^{100}) $ 6 * q + 14 * q^4 + 10 * q^11 + 8 * q^14 + 42 * q^16 + 16 * q^19 + 46 * q^26 + 6 * q^29 + 22 * q^31 + 20 * q^34 + 2 * q^44 - 44 * q^46 - 2 * q^49 - 16 * q^56 - 60 * q^59 + 12 * q^61 + 38 * q^64 - 12 * q^71 - 8 * q^74 - 24 * q^76 + 2 * q^79 + 30 * q^86 + 32 * q^89 + 40 * q^91 + 2 * q^94

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 $

$2$	−0.156785	−0.110864	−0.0554318	−	0.998462i	$-0.517654\pi$
$2$	−0.156785	−0.110864	−0.0554318	+	0.998462i	$0.517654\pi$
$3$	0	0
$3$	0	0
$4$	−1.97542	−0.987709
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.09364	−0.413357	−0.206678	−	0.978409i	$-0.566265\pi$
$7$	−1.09364	−0.413357	−0.206678	+	0.978409i	$0.566265\pi$
$8$	0.623285	0.220365
$9$	0	0
$10$	0	0
$11$	4.40198	1.32725	0.663623	−	0.748067i	$-0.269018\pi$
$11$	4.40198	1.32725	0.663623	+	0.748067i	$0.269018\pi$
$12$	0	0
$13$	−3.97108	−1.10138	−0.550690	−	0.834710i	$-0.685635\pi$
$13$	−3.97108	−1.10138	−0.550690	+	0.834710i	$0.685635\pi$
$14$	0.171466	0.0458262
$15$	0	0
$16$	3.85312	0.963279
$17$	6.22138	1.50891	0.754454	−	0.656353i	$-0.227902\pi$
$17$	6.22138	1.50891	0.754454	+	0.656353i	$0.227902\pi$
$18$	0	0
$19$	6.97542	1.60027	0.800135	−	0.599819i	$-0.204761\pi$
$19$	6.97542	1.60027	0.800135	+	0.599819i	$0.204761\pi$
$20$	0	0
$21$	0	0
$22$	−0.690163	−0.147143
$23$	−0.780070	−0.162656	−0.0813279	−	0.996687i	$-0.525916\pi$
$23$	−0.780070	−0.162656	−0.0813279	+	0.996687i	$0.525916\pi$
$24$	0	0
$25$	0	0
$26$	0.622605	0.122103
$27$	0	0
$28$	2.16040	0.408276
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	6.40198	1.14983	0.574914	−	0.818214i	$-0.305035\pi$
$31$	6.40198	1.14983	0.574914	+	0.818214i	$0.305035\pi$
$32$	−1.85068	−0.327157
$33$	0	0
$34$	−0.975419	−0.167283
$35$	0	0
$36$	0	0
$37$	1.09364	0.179793	0.0898966	−	0.995951i	$-0.471346\pi$
$37$	1.09364	0.179793	0.0898966	+	0.995951i	$0.471346\pi$
$38$	−1.09364	−0.177412
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	0.376593	0.0574299	0.0287150	−	0.999588i	$-0.490858\pi$
$43$	0.376593	0.0574299	0.0287150	+	0.999588i	$0.490858\pi$
$44$	−8.69575	−1.31093
$45$	0	0
$46$	0.122303	0.0180326
$47$	−4.75115	−0.693027	−0.346513	−	0.938045i	$-0.612634\pi$
$47$	−4.75115	−0.693027	−0.346513	+	0.938045i	$0.612634\pi$
$48$	0	0
$49$	−5.80395	−0.829136
$50$	0	0
$51$	0	0
$52$	7.84455	1.08784
$53$	−11.2861	−1.55027	−0.775133	−	0.631798i	$-0.782317\pi$
$53$	−11.2861	−1.55027	−0.775133	+	0.631798i	$0.782317\pi$
$54$	0	0
$55$	0	0
$56$	−0.681649	−0.0910892
$57$	0	0
$58$	−0.156785	−0.0205869
$59$	−10.0000	−1.30189	−0.650945	−	0.759125i	$-0.725627\pi$
$59$	−10.0000	−1.30189	−0.650945	+	0.759125i	$0.725627\pi$
$60$	0	0
$61$	−1.14688	−0.146844	−0.0734218	−	0.997301i	$-0.523392\pi$
$61$	−1.14688	−0.146844	−0.0734218	+	0.997301i	$0.523392\pi$
$62$	−1.00373	−0.127474
$63$	0	0
$64$	−7.41607	−0.927009
$65$	0	0
$66$	0	0
$67$	5.90782	0.721754	0.360877	−	0.932613i	$-0.382477\pi$
$67$	5.90782	0.721754	0.360877	+	0.932613i	$0.382477\pi$
$68$	−12.2898	−1.49036
$69$	0	0
$70$	0	0
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	0	0
$73$	−8.72223	−1.02086	−0.510430	−	0.859919i	$-0.670514\pi$
$73$	−8.72223	−1.02086	−0.510430	+	0.859919i	$0.670514\pi$
$74$	−0.171466	−0.0199325
$75$	0	0
$76$	−13.7794	−1.58060
$77$	−4.81418	−0.548626
$78$	0	0
$79$	−5.54886	−0.624296	−0.312148	−	0.950034i	$-0.601048\pi$
$79$	−5.54886	−0.624296	−0.312148	+	0.950034i	$0.601048\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.22138	0.682886	0.341443	−	0.939903i	$-0.389084\pi$
$83$	6.22138	0.682886	0.341443	+	0.939903i	$0.389084\pi$
$84$	0	0
$85$	0	0
$86$	−0.0590441	−0.00636689
$87$	0	0
$88$	2.74369	0.292478
$89$	10.8040	1.14522	0.572608	−	0.819829i	$-0.305932\pi$
$89$	10.8040	1.14522	0.572608	+	0.819829i	$0.305932\pi$
$90$	0	0
$91$	4.34293	0.455263
$92$	1.54096	0.160657
$93$	0	0
$94$	0.744908	0.0768314
$95$	0	0
$96$	0	0
$97$	17.4176	1.76849	0.884244	−	0.467026i	$-0.154674\pi$
$97$	17.4176	1.76849	0.884244	+	0.467026i	$0.154674\pi$
$98$	0.909972	0.0919210
$99$	0	0
$100$	0	0
$101$	12.7548	1.26915	0.634574	−	0.772862i	$-0.281175\pi$
$101$	12.7548	1.26915	0.634574	+	0.772862i	$0.281175\pi$
$102$	0	0
$103$	5.15463	0.507901	0.253950	−	0.967217i	$-0.418270\pi$
$103$	5.15463	0.507901	0.253950	+	0.967217i	$0.418270\pi$
$104$	−2.47512	−0.242705
$105$	0	0
$106$	1.76949	0.171868
$107$	−1.09364	−0.105726	−0.0528631	−	0.998602i	$-0.516835\pi$
$107$	−1.09364	−0.105726	−0.0528631	+	0.998602i	$0.516835\pi$
$108$	0	0
$109$	15.3282	1.46818	0.734089	−	0.679053i	$-0.237609\pi$
$109$	15.3282	1.46818	0.734089	+	0.679053i	$0.237609\pi$
$110$	0	0
$111$	0	0
$112$	−4.21392	−0.398178
$113$	7.28814	0.685611	0.342805	−	0.939406i	$-0.388623\pi$
$113$	7.28814	0.685611	0.342805	+	0.939406i	$0.388623\pi$
$114$	0	0
$115$	0	0
$116$	−1.97542	−0.183413
$117$	0	0
$118$	1.56785	0.144332
$119$	−6.80395	−0.623717
$120$	0	0
$121$	8.37739	0.761581
$122$	0.179814	0.0162796
$123$	0	0
$124$	−12.6466	−1.13570
$125$	0	0
$126$	0	0
$127$	−17.7312	−1.57339	−0.786693	−	0.617345i	$-0.788208\pi$
$127$	−17.7312	−1.57339	−0.786693	+	0.617345i	$0.788208\pi$
$128$	4.86409	0.429929
$129$	0	0
$130$	0	0
$131$	−7.31835	−0.639407	−0.319704	−	0.947518i	$-0.603583\pi$
$131$	−7.31835	−0.639407	−0.319704	+	0.947518i	$0.603583\pi$
$132$	0	0
$133$	−7.62859	−0.661483
$134$	−0.926256	−0.0800163
$135$	0	0
$136$	3.87770	0.332510
$137$	9.78899	0.836330	0.418165	−	0.908371i	$-0.362673\pi$
$137$	9.78899	0.836330	0.418165	+	0.908371i	$0.362673\pi$
$138$	0	0
$139$	−2.75479	−0.233658	−0.116829	−	0.993152i	$-0.537273\pi$
$139$	−2.75479	−0.233658	−0.116829	+	0.993152i	$0.537273\pi$
$140$	0	0
$141$	0	0
$142$	0.313570	0.0263142
$143$	−17.4806	−1.46180
$144$	0	0
$145$	0	0
$146$	1.36751	0.113176
$147$	0	0
$148$	−2.16040	−0.177583
$149$	4.52428	0.370643	0.185322	−	0.982678i	$-0.440667\pi$
$149$	4.52428	0.370643	0.185322	+	0.982678i	$0.440667\pi$
$150$	0	0
$151$	2.80395	0.228182	0.114091	−	0.993470i	$-0.463604\pi$
$151$	2.80395	0.228182	0.114091	+	0.993470i	$0.463604\pi$
$152$	4.34768	0.352643
$153$	0	0
$154$	0.754790	0.0608227
$155$	0	0
$156$	0	0
$157$	4.97481	0.397033	0.198517	−	0.980098i	$-0.436388\pi$
$157$	4.97481	0.397033	0.198517	+	0.980098i	$0.436388\pi$
$158$	0.869977	0.0692117
$159$	0	0
$160$	0	0
$161$	0.853115	0.0672349
$162$	0	0
$163$	20.7615	1.62617	0.813084	−	0.582146i	$-0.197787\pi$
$163$	20.7615	1.62617	0.813084	+	0.582146i	$0.197787\pi$
$164$	0	0
$165$	0	0
$166$	−0.975419	−0.0757072
$167$	−21.9182	−1.69608	−0.848040	−	0.529932i	$-0.822217\pi$
$167$	−21.9182	−1.69608	−0.848040	+	0.529932i	$0.822217\pi$
$168$	0	0
$169$	2.76949	0.213038
$170$	0	0
$171$	0	0
$172$	−0.743929	−0.0567241
$173$	3.25404	0.247400	0.123700	−	0.992320i	$-0.460524\pi$
$173$	3.25404	0.247400	0.123700	+	0.992320i	$0.460524\pi$
$174$	0	0
$175$	0	0
$176$	16.9613	1.27851
$177$	0	0
$178$	−1.69390	−0.126963
$179$	7.19605	0.537858	0.268929	−	0.963160i	$-0.413330\pi$
$179$	7.19605	0.537858	0.268929	+	0.963160i	$0.413330\pi$
$180$	0	0
$181$	13.0345	0.968844	0.484422	−	0.874834i	$-0.339030\pi$
$181$	13.0345	0.968844	0.484422	+	0.874834i	$0.339030\pi$
$182$	−0.680906	−0.0504721
$183$	0	0
$184$	−0.486206	−0.0358436
$185$	0	0
$186$	0	0
$187$	27.3864	2.00269
$188$	9.38551	0.684509
$189$	0	0
$190$	0	0
$191$	10.1223	0.732424	0.366212	−	0.930531i	$-0.380654\pi$
$191$	10.1223	0.732424	0.366212	+	0.930531i	$0.380654\pi$
$192$	0	0
$193$	22.8589	1.64542	0.822710	−	0.568462i	$-0.192461\pi$
$193$	22.8589	1.64542	0.822710	+	0.568462i	$0.192461\pi$
$194$	−2.73081	−0.196061
$195$	0	0
$196$	11.4652	0.818945
$197$	−22.0711	−1.57250	−0.786251	−	0.617907i	$-0.787981\pi$
$197$	−22.0711	−1.57250	−0.786251	+	0.617907i	$0.787981\pi$
$198$	0	0
$199$	5.19605	0.368338	0.184169	−	0.982895i	$-0.441041\pi$
$199$	5.19605	0.368338	0.184169	+	0.982895i	$0.441041\pi$
$200$	0	0
$201$	0	0
$202$	−1.99976	−0.140702
$203$	−1.09364	−0.0767585
$204$	0	0
$205$	0	0
$206$	−0.808167	−0.0563077
$207$	0	0
$208$	−15.3010	−1.06094
$209$	30.7056	2.12395
$210$	0	0
$211$	1.64719	0.113397	0.0566985	−	0.998391i	$-0.481943\pi$
$211$	1.64719	0.113397	0.0566985	+	0.998391i	$0.481943\pi$
$212$	22.2948	1.53121
$213$	0	0
$214$	0.171466	0.0117212
$215$	0	0
$216$	0	0
$217$	−7.00145	−0.475290
$218$	−2.40323	−0.162768
$219$	0	0
$220$	0	0
$221$	−24.7056	−1.66188
$222$	0	0
$223$	9.47542	0.634521	0.317261	−	0.948338i	$-0.397237\pi$
$223$	9.47542	0.634521	0.317261	+	0.948338i	$0.397237\pi$
$224$	2.02398	0.135233
$225$	0	0
$226$	−1.14267	−0.0760093
$227$	−5.72800	−0.380181	−0.190090	−	0.981767i	$-0.560878\pi$
$227$	−5.72800	−0.380181	−0.190090	+	0.981767i	$0.560878\pi$
$228$	0	0
$229$	15.1469	1.00093	0.500467	−	0.865756i	$-0.333162\pi$
$229$	15.1469	1.00093	0.500467	+	0.865756i	$0.333162\pi$
$230$	0	0
$231$	0	0
$232$	0.623285	0.0409207
$233$	0.717046	0.0469753	0.0234876	−	0.999724i	$-0.492523\pi$
$233$	0.717046	0.0469753	0.0234876	+	0.999724i	$0.492523\pi$
$234$	0	0
$235$	0	0
$236$	19.7542	1.28589
$237$	0	0
$238$	1.06676	0.0691475
$239$	2.00000	0.129369	0.0646846	−	0.997906i	$-0.479396\pi$
$239$	2.00000	0.129369	0.0646846	+	0.997906i	$0.479396\pi$
$240$	0	0
$241$	17.3282	1.11621	0.558105	−	0.829770i	$-0.311529\pi$
$241$	17.3282	1.11621	0.558105	+	0.829770i	$0.311529\pi$
$242$	−1.31345	−0.0844316
$243$	0	0
$244$	2.26558	0.145039
$245$	0	0
$246$	0	0
$247$	−27.7000	−1.76251
$248$	3.99026	0.253382
$249$	0	0
$250$	0	0
$251$	−26.0099	−1.64173	−0.820865	−	0.571123i	$-0.806508\pi$
$251$	−26.0099	−1.64173	−0.820865	+	0.571123i	$0.806508\pi$
$252$	0	0
$253$	−3.43385	−0.215884
$254$	2.77998	0.174431
$255$	0	0
$256$	14.0695	0.879346
$257$	−15.0335	−0.937766	−0.468883	−	0.883260i	$-0.655343\pi$
$257$	−15.0335	−0.937766	−0.468883	+	0.883260i	$0.655343\pi$
$258$	0	0
$259$	−1.19605	−0.0743188
$260$	0	0
$261$	0	0
$262$	1.14741	0.0708870
$263$	12.0662	0.744032	0.372016	−	0.928226i	$-0.378667\pi$
$263$	12.0662	0.744032	0.372016	+	0.928226i	$0.378667\pi$
$264$	0	0
$265$	0	0
$266$	1.19605	0.0733344
$267$	0	0
$268$	−11.6704	−0.712884
$269$	10.8531	0.661726	0.330863	−	0.943679i	$-0.392660\pi$
$269$	10.8531	0.661726	0.330863	+	0.943679i	$0.392660\pi$
$270$	0	0
$271$	−12.7449	−0.774198	−0.387099	−	0.922038i	$-0.626523\pi$
$271$	−12.7449	−0.774198	−0.387099	+	0.922038i	$0.626523\pi$
$272$	23.9717	1.45350
$273$	0	0
$274$	−1.53476	−0.0927185
$275$	0	0
$276$	0	0
$277$	−20.0714	−1.20597	−0.602986	−	0.797752i	$-0.706022\pi$
$277$	−20.0714	−1.20597	−0.602986	+	0.797752i	$0.706022\pi$
$278$	0.431909	0.0259042
$279$	0	0
$280$	0	0
$281$	20.9361	1.24895	0.624473	−	0.781047i	$-0.285314\pi$
$281$	20.9361	1.24895	0.624473	+	0.781047i	$0.285314\pi$
$282$	0	0
$283$	−16.9703	−1.00878	−0.504389	−	0.863477i	$-0.668282\pi$
$283$	−16.9703	−1.00878	−0.504389	+	0.863477i	$0.668282\pi$
$284$	3.95084	0.234439
$285$	0	0
$286$	2.74069	0.162061
$287$	0	0
$288$	0	0
$289$	21.7056	1.27680
$290$	0	0
$291$	0	0
$292$	17.2301	1.00831
$293$	−9.47542	−0.553560	−0.276780	−	0.960933i	$-0.589267\pi$
$293$	−9.47542	−0.553560	−0.276780	+	0.960933i	$0.589267\pi$
$294$	0	0
$295$	0	0
$296$	0.681649	0.0396201
$297$	0	0
$298$	−0.709338	−0.0410909
$299$	3.09772	0.179146
$300$	0	0
$301$	−0.411857	−0.0237391
$302$	−0.439617	−0.0252971
$303$	0	0
$304$	26.8771	1.54151
$305$	0	0
$306$	0	0
$307$	−19.3812	−1.10614	−0.553072	−	0.833134i	$-0.686544\pi$
$307$	−19.3812	−1.10614	−0.553072	+	0.833134i	$0.686544\pi$
$308$	9.51001	0.541883
$309$	0	0
$310$	0	0
$311$	9.26919	0.525607	0.262804	−	0.964849i	$-0.415353\pi$
$311$	9.26919	0.525607	0.262804	+	0.964849i	$0.415353\pi$
$312$	0	0
$313$	14.5324	0.821422	0.410711	−	0.911766i	$-0.365281\pi$
$313$	14.5324	0.821422	0.410711	+	0.911766i	$0.365281\pi$
$314$	−0.779975	−0.0440165
$315$	0	0
$316$	10.9613	0.616623
$317$	−22.4116	−1.25876	−0.629380	−	0.777098i	$-0.716691\pi$
$317$	−22.4116	−1.25876	−0.629380	+	0.777098i	$0.716691\pi$
$318$	0	0
$319$	4.40198	0.246463
$320$	0	0
$321$	0	0
$322$	−0.133756	−0.00745390
$323$	43.3968	2.41466
$324$	0	0
$325$	0	0
$326$	−3.25509	−0.180283
$327$	0	0
$328$	0	0
$329$	5.19605	0.286467
$330$	0	0
$331$	7.89179	0.433772	0.216886	−	0.976197i	$-0.430410\pi$
$331$	7.89179	0.433772	0.216886	+	0.976197i	$0.430410\pi$
$332$	−12.2898	−0.674493
$333$	0	0
$334$	3.43644	0.188034
$335$	0	0
$336$	0	0
$337$	−33.4280	−1.82094	−0.910468	−	0.413579i	$-0.864279\pi$
$337$	−33.4280	−1.82094	−0.910468	+	0.413579i	$0.864279\pi$
$338$	−0.434214	−0.0236181
$339$	0	0
$340$	0	0
$341$	28.1813	1.52611
$342$	0	0
$343$	14.0029	0.756086
$344$	0.234725	0.0126555
$345$	0	0
$346$	−0.510183	−0.0274276
$347$	31.1069	1.66991	0.834954	−	0.550320i	$-0.185494\pi$
$347$	31.1069	1.66991	0.834954	+	0.550320i	$0.185494\pi$
$348$	0	0
$349$	−5.76949	−0.308834	−0.154417	−	0.988006i	$-0.549350\pi$
$349$	−5.76949	−0.308834	−0.154417	+	0.988006i	$0.549350\pi$
$350$	0	0
$351$	0	0
$352$	−8.14665	−0.434218
$353$	−13.5095	−0.719039	−0.359520	−	0.933137i	$-0.617059\pi$
$353$	−13.5095	−0.719039	−0.359520	+	0.933137i	$0.617059\pi$
$354$	0	0
$355$	0	0
$356$	−21.3423	−1.13114
$357$	0	0
$358$	−1.12823	−0.0596289
$359$	1.15677	0.0610518	0.0305259	−	0.999534i	$-0.490282\pi$
$359$	1.15677	0.0610518	0.0305259	+	0.999534i	$0.490282\pi$
$360$	0	0
$361$	29.6565	1.56087
$362$	−2.04361	−0.107410
$363$	0	0
$364$	−8.57911	−0.449667
$365$	0	0
$366$	0	0
$367$	10.5959	0.553104	0.276552	−	0.960999i	$-0.410808\pi$
$367$	10.5959	0.553104	0.276552	+	0.960999i	$0.410808\pi$
$368$	−3.00570	−0.156683
$369$	0	0
$370$	0	0
$371$	12.3429	0.640813
$372$	0	0
$373$	7.09907	0.367576	0.183788	−	0.982966i	$-0.441164\pi$
$373$	7.09907	0.367576	0.183788	+	0.982966i	$0.441164\pi$
$374$	−4.29377	−0.222026
$375$	0	0
$376$	−2.96132	−0.152719
$377$	−3.97108	−0.204521
$378$	0	0
$379$	18.1223	0.930880	0.465440	−	0.885079i	$-0.345896\pi$
$379$	18.1223	0.930880	0.465440	+	0.885079i	$0.345896\pi$
$380$	0	0
$381$	0	0
$382$	−1.58702	−0.0811992
$383$	−3.28092	−0.167647	−0.0838236	−	0.996481i	$-0.526713\pi$
$383$	−3.28092	−0.167647	−0.0838236	+	0.996481i	$0.526713\pi$
$384$	0	0
$385$	0	0
$386$	−3.58393	−0.182417
$387$	0	0
$388$	−34.4070	−1.74675
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	0	0
$391$	−4.85312	−0.245433
$392$	−3.61752	−0.182712
$393$	0	0
$394$	3.46042	0.174333
$395$	0	0
$396$	0	0
$397$	36.3054	1.82212	0.911058	−	0.412278i	$-0.135267\pi$
$397$	36.3054	1.82212	0.911058	+	0.412278i	$0.135267\pi$
$398$	−0.814661	−0.0408353
$399$	0	0
$400$	0	0
$401$	−26.0830	−1.30252	−0.651262	−	0.758853i	$-0.725760\pi$
$401$	−26.0830	−1.30252	−0.651262	+	0.758853i	$0.725760\pi$
$402$	0	0
$403$	−25.4228	−1.26640
$404$	−25.1960	−1.25355
$405$	0	0
$406$	0.171466	0.00850972
$407$	4.81418	0.238630
$408$	0	0
$409$	13.0486	0.645210	0.322605	−	0.946534i	$-0.395442\pi$
$409$	13.0486	0.645210	0.322605	+	0.946534i	$0.395442\pi$
$410$	0	0
$411$	0	0
$412$	−10.1825	−0.501658
$413$	10.9364	0.538145
$414$	0	0
$415$	0	0
$416$	7.34920	0.360324
$417$	0	0
$418$	−4.81418	−0.235469
$419$	−35.8525	−1.75151	−0.875755	−	0.482756i	$-0.839636\pi$
$419$	−35.8525	−1.75151	−0.875755	+	0.482756i	$0.839636\pi$
$420$	0	0
$421$	16.7056	0.814182	0.407091	−	0.913388i	$-0.366543\pi$
$421$	16.7056	0.814182	0.407091	+	0.913388i	$0.366543\pi$
$422$	−0.258254	−0.0125716
$423$	0	0
$424$	−7.03446	−0.341624
$425$	0	0
$426$	0	0
$427$	1.25428	0.0606988
$428$	2.16040	0.104427
$429$	0	0
$430$	0	0
$431$	26.3135	1.26748	0.633739	−	0.773547i	$-0.281519\pi$
$431$	26.3135	1.26748	0.633739	+	0.773547i	$0.281519\pi$
$432$	0	0
$433$	−10.7220	−0.515266	−0.257633	−	0.966243i	$-0.582943\pi$
$433$	−10.7220	−0.515266	−0.257633	+	0.966243i	$0.582943\pi$
$434$	1.09772	0.0526923
$435$	0	0
$436$	−30.2797	−1.45013
$437$	−5.44131	−0.260293
$438$	0	0
$439$	29.5587	1.41076	0.705381	−	0.708828i	$-0.250776\pi$
$439$	29.5587	1.41076	0.705381	+	0.708828i	$0.250776\pi$
$440$	0	0
$441$	0	0
$442$	3.87347	0.184242
$443$	−3.40697	−0.161870	−0.0809349	−	0.996719i	$-0.525791\pi$
$443$	−3.40697	−0.161870	−0.0809349	+	0.996719i	$0.525791\pi$
$444$	0	0
$445$	0	0
$446$	−1.48560	−0.0703453
$447$	0	0
$448$	8.11051	0.383186
$449$	−0.461020	−0.0217569	−0.0108784	−	0.999941i	$-0.503463\pi$
$449$	−0.461020	−0.0217569	−0.0108784	+	0.999941i	$0.503463\pi$
$450$	0	0
$451$	0	0
$452$	−14.3971	−0.677184
$453$	0	0
$454$	0.898063	0.0421482
$455$	0	0
$456$	0	0
$457$	−1.37262	−0.0642084	−0.0321042	−	0.999485i	$-0.510221\pi$
$457$	−1.37262	−0.0642084	−0.0321042	+	0.999485i	$0.510221\pi$
$458$	−2.37480	−0.110967
$459$	0	0
$460$	0	0
$461$	−14.5875	−0.679409	−0.339705	−	0.940532i	$-0.610327\pi$
$461$	−14.5875	−0.679409	−0.339705	+	0.940532i	$0.610327\pi$
$462$	0	0
$463$	18.8517	0.876112	0.438056	−	0.898948i	$-0.355667\pi$
$463$	18.8517	0.876112	0.438056	+	0.898948i	$0.355667\pi$
$464$	3.85312	0.178876
$465$	0	0
$466$	−0.112422	−0.00520785
$467$	29.9580	1.38629	0.693145	−	0.720798i	$-0.256225\pi$
$467$	29.9580	1.38629	0.693145	+	0.720798i	$0.256225\pi$
$468$	0	0
$469$	−6.46102	−0.298342
$470$	0	0
$471$	0	0
$472$	−6.23285	−0.286890
$473$	1.65776	0.0762237
$474$	0	0
$475$	0	0
$476$	13.4407	0.616051
$477$	0	0
$478$	−0.313570	−0.0143423
$479$	32.7647	1.49706	0.748528	−	0.663103i	$-0.230761\pi$
$479$	32.7647	1.49706	0.748528	+	0.663103i	$0.230761\pi$
$480$	0	0
$481$	−4.34293	−0.198021
$482$	−2.71680	−0.123747
$483$	0	0
$484$	−16.5489	−0.752221
$485$	0	0
$486$	0	0
$487$	14.1098	0.639375	0.319688	−	0.947523i	$-0.396422\pi$
$487$	14.1098	0.639375	0.319688	+	0.947523i	$0.396422\pi$
$488$	−0.714836	−0.0323591
$489$	0	0
$490$	0	0
$491$	−4.10821	−0.185401	−0.0927004	−	0.995694i	$-0.529550\pi$
$491$	−4.10821	−0.185401	−0.0927004	+	0.995694i	$0.529550\pi$
$492$	0	0
$493$	6.22138	0.280197
$494$	4.34293	0.195398
$495$	0	0
$496$	24.6676	1.10761
$497$	2.18728	0.0981129
$498$	0	0
$499$	30.0689	1.34607	0.673035	−	0.739611i	$-0.264990\pi$
$499$	30.0689	1.34607	0.673035	+	0.739611i	$0.264990\pi$
$500$	0	0
$501$	0	0
$502$	4.07795	0.182008
$503$	−14.8806	−0.663493	−0.331746	−	0.943369i	$-0.607638\pi$
$503$	−14.8806	−0.663493	−0.331746	+	0.943369i	$0.607638\pi$
$504$	0	0
$505$	0	0
$506$	0.538375	0.0239337
$507$	0	0
$508$	35.0264	1.55405
$509$	−1.93674	−0.0858445	−0.0429223	−	0.999078i	$-0.513667\pi$
$509$	−1.93674	−0.0858445	−0.0429223	+	0.999078i	$0.513667\pi$
$510$	0	0
$511$	9.53898	0.421980
$512$	−11.9341	−0.527416
$513$	0	0
$514$	2.35703	0.103964
$515$	0	0
$516$	0	0
$517$	−20.9145	−0.919817
$518$	0.187522	0.00823925
$519$	0	0
$520$	0	0
$521$	4.23051	0.185342	0.0926710	−	0.995697i	$-0.470460\pi$
$521$	4.23051	0.185342	0.0926710	+	0.995697i	$0.470460\pi$
$522$	0	0
$523$	4.40144	0.192462	0.0962308	−	0.995359i	$-0.469321\pi$
$523$	4.40144	0.192462	0.0962308	+	0.995359i	$0.469321\pi$
$524$	14.4568	0.631548
$525$	0	0
$526$	−1.89179	−0.0824861
$527$	39.8292	1.73499
$528$	0	0
$529$	−22.3915	−0.973543
$530$	0	0
$531$	0	0
$532$	15.0697	0.653353
$533$	0	0
$534$	0	0
$535$	0	0
$536$	3.68225	0.159049
$537$	0	0
$538$	−1.70160	−0.0733613
$539$	−25.5489	−1.10047
$540$	0	0
$541$	−32.4119	−1.39349	−0.696747	−	0.717317i	$-0.745370\pi$
$541$	−32.4119	−1.39349	−0.696747	+	0.717317i	$0.745370\pi$
$542$	1.99821	0.0858304
$543$	0	0
$544$	−11.5138	−0.493650
$545$	0	0
$546$	0	0
$547$	12.9093	0.551961	0.275980	−	0.961163i	$-0.410998\pi$
$547$	12.9093	0.551961	0.275980	+	0.961163i	$0.410998\pi$
$548$	−19.3374	−0.826051
$549$	0	0
$550$	0	0
$551$	6.97542	0.297163
$552$	0	0
$553$	6.06845	0.258057
$554$	3.14688	0.133698
$555$	0	0
$556$	5.44186	0.230786
$557$	23.0195	0.975369	0.487685	−	0.873020i	$-0.337842\pi$
$557$	23.0195	0.975369	0.487685	+	0.873020i	$0.337842\pi$
$558$	0	0
$559$	−1.49548	−0.0632522
$560$	0	0
$561$	0	0
$562$	−3.28247	−0.138463
$563$	−27.3234	−1.15154	−0.575771	−	0.817611i	$-0.695298\pi$
$563$	−27.3234	−1.15154	−0.575771	+	0.817611i	$0.695298\pi$
$564$	0	0
$565$	0	0
$566$	2.66068	0.111837
$567$	0	0
$568$	−1.24657	−0.0523049
$569$	0.803952	0.0337034	0.0168517	−	0.999858i	$-0.494636\pi$
$569$	0.803952	0.0337034	0.0168517	+	0.999858i	$0.494636\pi$
$570$	0	0
$571$	43.4604	1.81876	0.909381	−	0.415964i	$-0.136556\pi$
$571$	43.4604	1.81876	0.909381	+	0.415964i	$0.136556\pi$
$572$	34.5315	1.44384
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	1.27345	0.0530146	0.0265073	−	0.999649i	$-0.491561\pi$
$577$	1.27345	0.0530146	0.0265073	+	0.999649i	$0.491561\pi$
$578$	−3.40311	−0.141551
$579$	0	0
$580$	0	0
$581$	−6.80395	−0.282276
$582$	0	0
$583$	−49.6812	−2.05758
$584$	−5.43644	−0.224961
$585$	0	0
$586$	1.48560	0.0613696
$587$	31.8678	1.31533	0.657663	−	0.753312i	$-0.271545\pi$
$587$	31.8678	1.31533	0.657663	+	0.753312i	$0.271545\pi$
$588$	0	0
$589$	44.6565	1.84004
$590$	0	0
$591$	0	0
$592$	4.21392	0.173191
$593$	1.21043	0.0497064	0.0248532	−	0.999691i	$-0.492088\pi$
$593$	1.21043	0.0497064	0.0248532	+	0.999691i	$0.492088\pi$
$594$	0	0
$595$	0	0
$596$	−8.93735	−0.366088
$597$	0	0
$598$	−0.485676	−0.0198608
$599$	−24.2053	−0.989003	−0.494501	−	0.869177i	$-0.664649\pi$
$599$	−24.2053	−0.989003	−0.494501	+	0.869177i	$0.664649\pi$
$600$	0	0
$601$	4.68586	0.191140	0.0955702	−	0.995423i	$-0.469533\pi$
$601$	4.68586	0.191140	0.0955702	+	0.995423i	$0.469533\pi$
$602$	0.0645730	0.00263180
$603$	0	0
$604$	−5.53898	−0.225378
$605$	0	0
$606$	0	0
$607$	−0.502641	−0.0204016	−0.0102008	−	0.999948i	$-0.503247\pi$
$607$	−0.502641	−0.0204016	−0.0102008	+	0.999948i	$0.503247\pi$
$608$	−12.9093	−0.523540
$609$	0	0
$610$	0	0
$611$	18.8672	0.763286
$612$	0	0
$613$	22.9142	0.925496	0.462748	−	0.886490i	$-0.346863\pi$
$613$	22.9142	0.925496	0.462748	+	0.886490i	$0.346863\pi$
$614$	3.03868	0.122631
$615$	0	0
$616$	−3.00060	−0.120898
$617$	15.8574	0.638397	0.319198	−	0.947688i	$-0.396586\pi$
$617$	15.8574	0.638397	0.319198	+	0.947688i	$0.396586\pi$
$618$	0	0
$619$	−35.4014	−1.42290	−0.711451	−	0.702736i	$-0.751961\pi$
$619$	−35.4014	−1.42290	−0.711451	+	0.702736i	$0.751961\pi$
$620$	0	0
$621$	0	0
$622$	−1.45327	−0.0582707
$623$	−11.8156	−0.473383
$624$	0	0
$625$	0	0
$626$	−2.27846	−0.0910658
$627$	0	0
$628$	−9.82734	−0.392154
$629$	6.80395	0.271291
$630$	0	0
$631$	44.7056	1.77970	0.889851	−	0.456250i	$-0.150808\pi$
$631$	44.7056	1.77970	0.889851	+	0.456250i	$0.150808\pi$
$632$	−3.45852	−0.137573
$633$	0	0
$634$	3.51379	0.139551
$635$	0	0
$636$	0	0
$637$	23.0480	0.913194
$638$	−0.690163	−0.0273238
$639$	0	0
$640$	0	0
$641$	−10.0689	−0.397699	−0.198849	−	0.980030i	$-0.563720\pi$
$641$	−10.0689	−0.397699	−0.198849	+	0.980030i	$0.563720\pi$
$642$	0	0
$643$	13.9760	0.551161	0.275580	−	0.961278i	$-0.411130\pi$
$643$	13.9760	0.551161	0.275580	+	0.961278i	$0.411130\pi$
$644$	−1.68526	−0.0664085
$645$	0	0
$646$	−6.80395	−0.267698
$647$	−31.1607	−1.22505	−0.612527	−	0.790450i	$-0.709847\pi$
$647$	−31.1607	−1.22505	−0.612527	+	0.790450i	$0.709847\pi$
$648$	0	0
$649$	−44.0198	−1.72793
$650$	0	0
$651$	0	0
$652$	−41.0127	−1.60618
$653$	26.1129	1.02188	0.510939	−	0.859617i	$-0.329298\pi$
$653$	26.1129	1.02188	0.510939	+	0.859617i	$0.329298\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	−0.814661	−0.0317588
$659$	28.5974	1.11400	0.556999	−	0.830513i	$-0.311953\pi$
$659$	28.5974	1.11400	0.556999	+	0.830513i	$0.311953\pi$
$660$	0	0
$661$	−38.9994	−1.51690	−0.758450	−	0.651731i	$-0.774043\pi$
$661$	−38.9994	−1.51690	−0.758450	+	0.651731i	$0.774043\pi$
$662$	−1.23731	−0.0480895
$663$	0	0
$664$	3.87770	0.150484
$665$	0	0
$666$	0	0
$667$	−0.780070	−0.0302044
$668$	43.2976	1.67523
$669$	0	0
$670$	0	0
$671$	−5.04856	−0.194897
$672$	0	0
$673$	−38.6725	−1.49072	−0.745358	−	0.666665i	$-0.767721\pi$
$673$	−38.6725	−1.49072	−0.745358	+	0.666665i	$0.767721\pi$
$674$	5.24100	0.201876
$675$	0	0
$676$	−5.47090	−0.210419
$677$	−28.4800	−1.09458	−0.547288	−	0.836944i	$-0.684340\pi$
$677$	−28.4800	−1.09458	−0.547288	+	0.836944i	$0.684340\pi$
$678$	0	0
$679$	−19.0486	−0.731017
$680$	0	0
$681$	0	0
$682$	−4.41841	−0.169190
$683$	−12.4159	−0.475081	−0.237540	−	0.971378i	$-0.576341\pi$
$683$	−12.4159	−0.475081	−0.237540	+	0.971378i	$0.576341\pi$
$684$	0	0
$685$	0	0
$686$	−2.19544	−0.0838224
$687$	0	0
$688$	1.45106	0.0553211
$689$	44.8180	1.70743
$690$	0	0
$691$	10.8040	0.411002	0.205501	−	0.978657i	$-0.434118\pi$
$691$	10.8040	0.411002	0.205501	+	0.978657i	$0.434118\pi$
$692$	−6.42808	−0.244359
$693$	0	0
$694$	−4.87709	−0.185132
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0.904568	0.0342384
$699$	0	0
$700$	0	0
$701$	−8.47512	−0.320101	−0.160050	−	0.987109i	$-0.551166\pi$
$701$	−8.47512	−0.320101	−0.160050	+	0.987109i	$0.551166\pi$
$702$	0	0
$703$	7.62859	0.287718
$704$	−32.6454	−1.23037
$705$	0	0
$706$	2.11809	0.0797153
$707$	−13.9491	−0.524612
$708$	0	0
$709$	−32.8180	−1.23251	−0.616254	−	0.787548i	$-0.711350\pi$
$709$	−32.8180	−1.23251	−0.616254	+	0.787548i	$0.711350\pi$
$710$	0	0
$711$	0	0
$712$	6.73394	0.252365
$713$	−4.99399	−0.187026
$714$	0	0
$715$	0	0
$716$	−14.2152	−0.531247
$717$	0	0
$718$	−0.181363	−0.00676842
$719$	−39.7542	−1.48258	−0.741290	−	0.671184i	$-0.765786\pi$
$719$	−39.7542	−1.48258	−0.741290	+	0.671184i	$0.765786\pi$
$720$	0	0
$721$	−5.63731	−0.209944
$722$	−4.64968	−0.173043
$723$	0	0
$724$	−25.7485	−0.956936
$725$	0	0
$726$	0	0
$727$	−49.4844	−1.83527	−0.917637	−	0.397419i	$-0.869906\pi$
$727$	−49.4844	−1.83527	−0.917637	+	0.397419i	$0.869906\pi$
$728$	2.70689	0.100324
$729$	0	0
$730$	0	0
$731$	2.34293	0.0866565
$732$	0	0
$733$	46.2381	1.70784	0.853921	−	0.520403i	$-0.174218\pi$
$733$	46.2381	1.70784	0.853921	+	0.520403i	$0.174218\pi$
$734$	−1.66128	−0.0613191
$735$	0	0
$736$	1.44366	0.0532140
$737$	26.0061	0.957946
$738$	0	0
$739$	28.7155	1.05632	0.528159	−	0.849146i	$-0.322883\pi$
$739$	28.7155	1.05632	0.528159	+	0.849146i	$0.322883\pi$
$740$	0	0
$741$	0	0
$742$	−1.93518	−0.0710428
$743$	27.0536	0.992502	0.496251	−	0.868179i	$-0.334710\pi$
$743$	27.0536	0.992502	0.496251	+	0.868179i	$0.334710\pi$
$744$	0	0
$745$	0	0
$746$	−1.11303	−0.0407508
$747$	0	0
$748$	−54.0996	−1.97808
$749$	1.19605	0.0437026
$750$	0	0
$751$	−26.8771	−0.980759	−0.490380	−	0.871509i	$-0.663142\pi$
$751$	−26.8771	−0.980759	−0.490380	+	0.871509i	$0.663142\pi$
$752$	−18.3067	−0.667578
$753$	0	0
$754$	0.622605	0.0226739
$755$	0	0
$756$	0	0
$757$	7.28814	0.264892	0.132446	−	0.991190i	$-0.457717\pi$
$757$	7.28814	0.264892	0.132446	+	0.991190i	$0.457717\pi$
$758$	−2.84130	−0.103201
$759$	0	0
$760$	0	0
$761$	31.6079	1.14579	0.572893	−	0.819630i	$-0.305821\pi$
$761$	31.6079	1.14579	0.572893	+	0.819630i	$0.305821\pi$
$762$	0	0
$763$	−16.7636	−0.606882
$764$	−19.9958	−0.723422
$765$	0	0
$766$	0.514398	0.0185860
$767$	39.7108	1.43387
$768$	0	0
$769$	−0.293769	−0.0105936	−0.00529679	−	0.999986i	$-0.501686\pi$
$769$	−0.293769	−0.0105936	−0.00529679	+	0.999986i	$0.501686\pi$
$770$	0	0
$771$	0	0
$772$	−45.1559	−1.62520
$773$	28.4877	1.02463	0.512316	−	0.858797i	$-0.328788\pi$
$773$	28.4877	1.02463	0.512316	+	0.858797i	$0.328788\pi$
$774$	0	0
$775$	0	0
$776$	10.8561	0.389712
$777$	0	0
$778$	−1.88142	−0.0674521
$779$	0	0
$780$	0	0
$781$	−8.80395	−0.315030
$782$	0.760895	0.0272095
$783$	0	0
$784$	−22.3633	−0.798689
$785$	0	0
$786$	0	0
$787$	32.9883	1.17591	0.587954	−	0.808895i	$-0.299934\pi$
$787$	32.9883	1.17591	0.587954	+	0.808895i	$0.299934\pi$
$788$	43.5997	1.55317
$789$	0	0
$790$	0	0
$791$	−7.97060	−0.283402
$792$	0	0
$793$	4.55437	0.161731
$794$	−5.69213	−0.202006
$795$	0	0
$796$	−10.2644	−0.363811
$797$	−30.9194	−1.09522	−0.547611	−	0.836733i	$-0.684463\pi$
$797$	−30.9194	−1.09522	−0.547611	+	0.836733i	$0.684463\pi$
$798$	0	0
$799$	−29.5587	−1.04571
$800$	0	0
$801$	0	0
$802$	4.08942	0.144402
$803$	−38.3951	−1.35493
$804$	0	0
$805$	0	0
$806$	3.98590	0.140397
$807$	0	0
$808$	7.94987	0.279675
$809$	−21.0977	−0.741756	−0.370878	−	0.928682i	$-0.620943\pi$
$809$	−21.0977	−0.741756	−0.370878	+	0.928682i	$0.620943\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	2.16040	0.0758150
$813$	0	0
$814$	−0.754790	−0.0264554
$815$	0	0
$816$	0	0
$817$	2.62690	0.0919035
$818$	−2.04582	−0.0715303
$819$	0	0
$820$	0	0
$821$	15.2791	0.533243	0.266622	−	0.963801i	$-0.414093\pi$
$821$	15.2791	0.533243	0.266622	+	0.963801i	$0.414093\pi$
$822$	0	0
$823$	−7.78152	−0.271247	−0.135623	−	0.990760i	$-0.543304\pi$
$823$	−7.78152	−0.271247	−0.135623	+	0.990760i	$0.543304\pi$
$824$	3.21280	0.111923
$825$	0	0
$826$	−1.71466	−0.0596607
$827$	0.376593	0.0130954	0.00654772	−	0.999979i	$-0.497916\pi$
$827$	0.376593	0.0130954	0.00654772	+	0.999979i	$0.497916\pi$
$828$	0	0
$829$	−49.8723	−1.73214	−0.866068	−	0.499927i	$-0.833360\pi$
$829$	−49.8723	−1.73214	−0.866068	+	0.499927i	$0.833360\pi$
$830$	0	0
$831$	0	0
$832$	29.4498	1.02099
$833$	−36.1086	−1.25109
$834$	0	0
$835$	0	0
$836$	−60.6565	−2.09785
$837$	0	0
$838$	5.62113	0.194179
$839$	0.0590441	0.00203843	0.00101921	−	0.999999i	$-0.499676\pi$
$839$	0.0590441	0.00203843	0.00101921	+	0.999999i	$0.499676\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−2.61919	−0.0902632
$843$	0	0
$844$	−3.25388	−0.112003
$845$	0	0
$846$	0	0
$847$	−9.16185	−0.314805
$848$	−43.4867	−1.49334
$849$	0	0
$850$	0	0
$851$	−0.853115	−0.0292444
$852$	0	0
$853$	2.77983	0.0951795	0.0475897	−	0.998867i	$-0.484846\pi$
$853$	2.77983	0.0951795	0.0475897	+	0.998867i	$0.484846\pi$
$854$	−0.196652	−0.00672929
$855$	0	0
$856$	−0.681649	−0.0232983
$857$	−0.850802	−0.0290628	−0.0145314	−	0.999894i	$-0.504626\pi$
$857$	−0.850802	−0.0290628	−0.0145314	+	0.999894i	$0.504626\pi$
$858$	0	0
$859$	7.35342	0.250895	0.125448	−	0.992100i	$-0.459963\pi$
$859$	7.35342	0.250895	0.125448	+	0.992100i	$0.459963\pi$
$860$	0	0
$861$	0	0
$862$	−4.12556	−0.140517
$863$	−42.6167	−1.45069	−0.725345	−	0.688386i	$-0.758320\pi$
$863$	−42.6167	−1.45069	−0.725345	+	0.688386i	$0.758320\pi$
$864$	0	0
$865$	0	0
$866$	1.68105	0.0571242
$867$	0	0
$868$	13.8308	0.469448
$869$	−24.4260	−0.828594
$870$	0	0
$871$	−23.4604	−0.794926
$872$	9.55386	0.323535
$873$	0	0
$874$	0.853115	0.0288571
$875$	0	0
$876$	0	0
$877$	13.4196	0.453148	0.226574	−	0.973994i	$-0.427247\pi$
$877$	13.4196	0.453148	0.226574	+	0.973994i	$0.427247\pi$
$878$	−4.63436	−0.156402
$879$	0	0
$880$	0	0
$881$	38.3135	1.29082	0.645408	−	0.763838i	$-0.276687\pi$
$881$	38.3135	1.29082	0.645408	+	0.763838i	$0.276687\pi$
$882$	0	0
$883$	−31.5542	−1.06188	−0.530942	−	0.847408i	$-0.678162\pi$
$883$	−31.5542	−1.06188	−0.530942	+	0.847408i	$0.678162\pi$
$884$	48.8040	1.64145
$885$	0	0
$886$	0.534161	0.0179455
$887$	−23.3961	−0.785565	−0.392783	−	0.919631i	$-0.628488\pi$
$887$	−23.3961	−0.785565	−0.392783	+	0.919631i	$0.628488\pi$
$888$	0	0
$889$	19.3915	0.650370
$890$	0	0
$891$	0	0
$892$	−18.7179	−0.626722
$893$	−33.1413	−1.10903
$894$	0	0
$895$	0	0
$896$	−5.31956	−0.177714
$897$	0	0
$898$	0.0722810	0.00241205
$899$	6.40198	0.213518
$900$	0	0
$901$	−70.2152	−2.33921
$902$	0	0
$903$	0	0
$904$	4.54259	0.151084
$905$	0	0
$906$	0	0
$907$	32.4150	1.07632	0.538161	−	0.842842i	$-0.319119\pi$
$907$	32.4150	1.07632	0.538161	+	0.842842i	$0.319119\pi$
$908$	11.3152	0.375508
$909$	0	0
$910$	0	0
$911$	45.2749	1.50002	0.750011	−	0.661425i	$-0.230048\pi$
$911$	45.2749	1.50002	0.750011	+	0.661425i	$0.230048\pi$
$912$	0	0
$913$	27.3864	0.906357
$914$	0.215206	0.00711837
$915$	0	0
$916$	−29.9214	−0.988632
$917$	8.00364	0.264303
$918$	0	0
$919$	−5.90167	−0.194678	−0.0973391	−	0.995251i	$-0.531033\pi$
$919$	−5.90167	−0.194678	−0.0973391	+	0.995251i	$0.531033\pi$
$920$	0	0
$921$	0	0
$922$	2.28710	0.0753218
$923$	7.94216	0.261419
$924$	0	0
$925$	0	0
$926$	−2.95566	−0.0971289
$927$	0	0
$928$	−1.85068	−0.0607516
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	−40.4850	−1.32684
$932$	−1.41647	−0.0463979
$933$	0	0
$934$	−4.69695	−0.153689
$935$	0	0
$936$	0	0
$937$	43.0755	1.40721	0.703607	−	0.710589i	$-0.251571\pi$
$937$	43.0755	1.40721	0.703607	+	0.710589i	$0.251571\pi$
$938$	1.01299	0.0330753
$939$	0	0
$940$	0	0
$941$	−2.91637	−0.0950711	−0.0475355	−	0.998870i	$-0.515137\pi$
$941$	−2.91637	−0.0950711	−0.0475355	+	0.998870i	$0.515137\pi$
$942$	0	0
$943$	0	0
$944$	−38.5312	−1.25408
$945$	0	0
$946$	−0.259911	−0.00845043
$947$	−0.556407	−0.0180808	−0.00904041	−	0.999959i	$-0.502878\pi$
$947$	−0.556407	−0.0180808	−0.00904041	+	0.999959i	$0.502878\pi$
$948$	0	0
$949$	34.6367	1.12435
$950$	0	0
$951$	0	0
$952$	−4.24080	−0.137445
$953$	37.1661	1.20393	0.601964	−	0.798523i	$-0.294385\pi$
$953$	37.1661	1.20393	0.601964	+	0.798523i	$0.294385\pi$
$954$	0	0
$955$	0	0
$956$	−3.95084	−0.127779
$957$	0	0
$958$	−5.13700	−0.165969
$959$	−10.7056	−0.345703
$960$	0	0
$961$	9.98530	0.322106
$962$	0.680906	0.0219533
$963$	0	0
$964$	−34.2305	−1.10249
$965$	0	0
$966$	0	0
$967$	−24.2754	−0.780643	−0.390322	−	0.920679i	$-0.627636\pi$
$967$	−24.2754	−0.780643	−0.390322	+	0.920679i	$0.627636\pi$
$968$	5.22151	0.167826
$969$	0	0
$970$	0	0
$971$	17.1709	0.551039	0.275520	−	0.961295i	$-0.411150\pi$
$971$	17.1709	0.551039	0.275520	+	0.961295i	$0.411150\pi$
$972$	0	0
$973$	3.01275	0.0965842
$974$	−2.21220	−0.0708834
$975$	0	0
$976$	−4.41908	−0.141451
$977$	4.95785	0.158616	0.0793078	−	0.996850i	$-0.474729\pi$
$977$	4.95785	0.158616	0.0793078	+	0.996850i	$0.474729\pi$
$978$	0	0
$979$	47.5587	1.51998
$980$	0	0
$981$	0	0
$982$	0.644104	0.0205542
$983$	11.5651	0.368869	0.184434	−	0.982845i	$-0.440955\pi$
$983$	11.5651	0.368869	0.184434	+	0.982845i	$0.440955\pi$
$984$	0	0
$985$	0	0
$986$	−0.975419	−0.0310637
$987$	0	0
$988$	54.7190	1.74084
$989$	−0.293769	−0.00934132
$990$	0	0
$991$	−18.3429	−0.582682	−0.291341	−	0.956619i	$-0.594102\pi$
$991$	−18.3429	−0.582682	−0.291341	+	0.956619i	$0.594102\pi$
$992$	−11.8480	−0.376175
$993$	0	0
$994$	−0.342932	−0.0108771
$995$	0	0
$996$	0	0
$997$	32.2997	1.02294	0.511471	−	0.859300i	$-0.329101\pi$
$997$	32.2997	1.02294	0.511471	+	0.859300i	$0.329101\pi$
$998$	−4.71435	−0.149230
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.bt.1.3		6
3.2	odd	2		725.2.a.l.1.4		6
5.2	odd	4		1305.2.c.h.784.3		6
5.3	odd	4		1305.2.c.h.784.4		6
5.4	even	2	inner	6525.2.a.bt.1.4		6
15.2	even	4		145.2.b.c.59.4	yes	6
15.8	even	4		145.2.b.c.59.3	✓	6
15.14	odd	2		725.2.a.l.1.3		6
60.23	odd	4		2320.2.d.g.929.5		6
60.47	odd	4		2320.2.d.g.929.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.b.c.59.3	✓	6	15.8	even	4
145.2.b.c.59.4	yes	6	15.2	even	4
725.2.a.l.1.3		6	15.14	odd	2
725.2.a.l.1.4		6	3.2	odd	2
1305.2.c.h.784.3		6	5.2	odd	4
1305.2.c.h.784.4		6	5.3	odd	4
2320.2.d.g.929.2		6	60.47	odd	4
2320.2.d.g.929.5		6	60.23	odd	4
6525.2.a.bt.1.3		6	1.1	even	1	trivial
6525.2.a.bt.1.4		6	5.4	even	2	inner

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:	\( N \)	\(=\)	\( 6525 = 3^{2} \cdot 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6525.a (trivial)

\(f(q)\)	\(=\)	\(q-0.156785 q^{2} -1.97542 q^{4} -1.09364 q^{7} +0.623285 q^{8} +4.40198 q^{11} -3.97108 q^{13} +0.171466 q^{14} +3.85312 q^{16} +6.22138 q^{17} +6.97542 q^{19} -0.690163 q^{22} -0.780070 q^{23} +0.622605 q^{26} +2.16040 q^{28} +1.00000 q^{29} +6.40198 q^{31} -1.85068 q^{32} -0.975419 q^{34} +1.09364 q^{37} -1.09364 q^{38} +0.376593 q^{43} -8.69575 q^{44} +0.122303 q^{46} -4.75115 q^{47} -5.80395 q^{49} +7.84455 q^{52} -11.2861 q^{53} -0.681649 q^{56} -0.156785 q^{58} -10.0000 q^{59} -1.14688 q^{61} -1.00373 q^{62} -7.41607 q^{64} +5.90782 q^{67} -12.2898 q^{68} -2.00000 q^{71} -8.72223 q^{73} -0.171466 q^{74} -13.7794 q^{76} -4.81418 q^{77} -5.54886 q^{79} +6.22138 q^{83} -0.0590441 q^{86} +2.74369 q^{88} +10.8040 q^{89} +4.34293 q^{91} +1.54096 q^{92} +0.744908 q^{94} +17.4176 q^{97} +0.909972 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 14 q^{4} + 10 q^{11} + 8 q^{14} + 42 q^{16} + 16 q^{19} + 46 q^{26} + 6 q^{29} + 22 q^{31} + 20 q^{34} + 2 q^{44} - 44 q^{46} - 2 q^{49} - 16 q^{56} - 60 q^{59} + 12 q^{61} + 38 q^{64} - 12 q^{71}+ \cdots + 2 q^{94}+O(q^{100}) \) 6 * q + 14 * q^4 + 10 * q^11 + 8 * q^14 + 42 * q^16 + 16 * q^19 + 46 * q^26 + 6 * q^29 + 22 * q^31 + 20 * q^34 + 2 * q^44 - 44 * q^46 - 2 * q^49 - 16 * q^56 - 60 * q^59 + 12 * q^61 + 38 * q^64 - 12 * q^71 - 8 * q^74 - 24 * q^76 + 2 * q^79 + 30 * q^86 + 32 * q^89 + 40 * q^91 + 2 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more