LMFDB - Embedded newform 6525.2.a.bt.1.6

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)

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);

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")

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			1.6
Root			$2.77035$ 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)

$f(q)$	$=$	$q+2.77035 q^{2} +5.67486 q^{4} -1.86960 q^{7} +10.1807 q^{8} +3.25230 q^{11} +3.40121 q^{13} -5.17945 q^{14} +16.8543 q^{16} +2.40939 q^{17} -0.674860 q^{19} +9.01001 q^{22} -7.41031 q^{23} +9.42256 q^{26} -10.6097 q^{28} +1.00000 q^{29} +5.25230 q^{31} +26.3311 q^{32} +6.67486 q^{34} +1.86960 q^{37} -1.86960 q^{38} -3.46931 q^{43} +18.4563 q^{44} -20.5292 q^{46} -4.00910 q^{47} -3.50459 q^{49} +19.3014 q^{52} -0.877777 q^{53} -19.0338 q^{56} +2.77035 q^{58} -10.0000 q^{59} +11.8543 q^{61} +14.5507 q^{62} +39.2378 q^{64} +7.95010 q^{67} +13.6729 q^{68} -2.00000 q^{71} -0.607882 q^{73} +5.17945 q^{74} -3.82973 q^{76} -6.08050 q^{77} +8.60202 q^{79} +2.40939 q^{83} -9.61121 q^{86} +33.1105 q^{88} +8.50459 q^{89} -6.35891 q^{91} -42.0525 q^{92} -11.1066 q^{94} -13.1332 q^{97} -9.70897 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.77035	1.95894	0.979468	−	0.201600i	$-0.0646141\pi$
$2$	2.77035	1.95894	0.979468	+	0.201600i	$0.0646141\pi$
$3$	0	0
$3$	0	0
$4$	5.67486	2.83743
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.86960	−0.706643	−0.353321	−	0.935502i	$-0.614948\pi$
$7$	−1.86960	−0.706643	−0.353321	+	0.935502i	$0.614948\pi$
$8$	10.1807	3.59941
$9$	0	0
$10$	0	0
$11$	3.25230	0.980605	0.490302	−	0.871552i	$-0.336886\pi$
$11$	3.25230	0.980605	0.490302	+	0.871552i	$0.336886\pi$
$12$	0	0
$13$	3.40121	0.943327	0.471663	−	0.881779i	$-0.343654\pi$
$13$	3.40121	0.943327	0.471663	+	0.881779i	$0.343654\pi$
$14$	−5.17945	−1.38427
$15$	0	0
$16$	16.8543	4.21358
$17$	2.40939	0.584363	0.292181	−	0.956363i	$-0.405619\pi$
$17$	2.40939	0.584363	0.292181	+	0.956363i	$0.405619\pi$
$18$	0	0
$19$	−0.674860	−0.154823	−0.0774117	−	0.996999i	$-0.524666\pi$
$19$	−0.674860	−0.154823	−0.0774117	+	0.996999i	$0.524666\pi$
$20$	0	0
$21$	0	0
$22$	9.01001	1.92094
$23$	−7.41031	−1.54516	−0.772578	−	0.634920i	$-0.781033\pi$
$23$	−7.41031	−1.54516	−0.772578	+	0.634920i	$0.781033\pi$
$24$	0	0
$25$	0	0
$26$	9.42256	1.84792
$27$	0	0
$28$	−10.6097	−2.00505
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	5.25230	0.943340	0.471670	−	0.881775i	$-0.343651\pi$
$31$	5.25230	0.943340	0.471670	+	0.881775i	$0.343651\pi$
$32$	26.3311	4.65472
$33$	0	0
$34$	6.67486	1.14473
$35$	0	0
$36$	0	0
$37$	1.86960	0.307360	0.153680	−	0.988121i	$-0.450887\pi$
$37$	1.86960	0.307360	0.153680	+	0.988121i	$0.450887\pi$
$38$	−1.86960	−0.303289
$39$	0	0
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−3.46931	−0.529064	−0.264532	−	0.964377i	$-0.585218\pi$
$43$	−3.46931	−0.529064	−0.264532	+	0.964377i	$0.585218\pi$
$44$	18.4563	2.78240
$45$	0	0
$46$	−20.5292	−3.02686
$47$	−4.00910	−0.584787	−0.292393	−	0.956298i	$-0.594452\pi$
$47$	−4.00910	−0.584787	−0.292393	+	0.956298i	$0.594452\pi$
$48$	0	0
$49$	−3.50459	−0.500656
$50$	0	0
$51$	0	0
$52$	19.3014	2.67662
$53$	−0.877777	−0.120572	−0.0602859	−	0.998181i	$-0.519201\pi$
$53$	−0.877777	−0.120572	−0.0602859	+	0.998181i	$0.519201\pi$
$54$	0	0
$55$	0	0
$56$	−19.0338	−2.54349
$57$	0	0
$58$	2.77035	0.363765
$59$	−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
$61$	11.8543	1.51779	0.758895	−	0.651213i	$-0.225740\pi$
$61$	11.8543	1.51779	0.758895	+	0.651213i	$0.225740\pi$
$62$	14.5507	1.84794
$63$	0	0
$64$	39.2378	4.90473
$65$	0	0
$66$	0	0
$67$	7.95010	0.971259	0.485629	−	0.874165i	$-0.338590\pi$
$67$	7.95010	0.971259	0.485629	+	0.874165i	$0.338590\pi$
$68$	13.6729	1.65809
$69$	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
$73$	−0.607882	−0.0711472	−0.0355736	−	0.999367i	$-0.511326\pi$
$73$	−0.607882	−0.0711472	−0.0355736	+	0.999367i	$0.511326\pi$
$74$	5.17945	0.602099
$75$	0	0
$76$	−3.82973	−0.439301
$77$	−6.08050	−0.692937
$78$	0	0
$79$	8.60202	0.967803	0.483901	−	0.875123i	$-0.339219\pi$
$79$	8.60202	0.967803	0.483901	+	0.875123i	$0.339219\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	2.40939	0.264465	0.132232	−	0.991219i	$-0.457785\pi$
$83$	2.40939	0.264465	0.132232	+	0.991219i	$0.457785\pi$
$84$	0	0
$85$	0	0
$86$	−9.61121	−1.03640
$87$	0	0
$88$	33.1105	3.52960
$89$	8.50459	0.901485	0.450743	−	0.892654i	$-0.351159\pi$
$89$	8.50459	0.901485	0.450743	+	0.892654i	$0.351159\pi$
$90$	0	0
$91$	−6.35891	−0.666595
$92$	−42.0525	−4.38427
$93$	0	0
$94$	−11.1066	−1.14556
$95$	0	0
$96$	0	0
$97$	−13.1332	−1.33347	−0.666735	−	0.745295i	$-0.732309\pi$
$97$	−13.1332	−1.33347	−0.666735	+	0.745295i	$0.732309\pi$
$98$	−9.70897	−0.980754
$99$	0	0
$100$	0	0
$101$	−4.84513	−0.482108	−0.241054	−	0.970512i	$-0.577493\pi$
$101$	−4.84513	−0.482108	−0.241054	+	0.970512i	$0.577493\pi$
$102$	0	0
$103$	14.8887	1.46703	0.733514	−	0.679674i	$-0.237879\pi$
$103$	14.8887	1.46703	0.733514	+	0.679674i	$0.237879\pi$
$104$	34.6266	3.39542
$105$	0	0
$106$	−2.43175	−0.236193
$107$	−1.86960	−0.180741	−0.0903705	−	0.995908i	$-0.528805\pi$
$107$	−1.86960	−0.180741	−0.0903705	+	0.995908i	$0.528805\pi$
$108$	0	0
$109$	−8.77228	−0.840232	−0.420116	−	0.907470i	$-0.638011\pi$
$109$	−8.77228	−0.840232	−0.420116	+	0.907470i	$0.638011\pi$
$110$	0	0
$111$	0	0
$112$	−31.5108	−2.97749
$113$	−10.0699	−0.947299	−0.473650	−	0.880713i	$-0.657064\pi$
$113$	−10.0699	−0.947299	−0.473650	+	0.880713i	$0.657064\pi$
$114$	0	0
$115$	0	0
$116$	5.67486	0.526898
$117$	0	0
$118$	−27.7035	−2.55032
$119$	−4.50459	−0.412936
$120$	0	0
$121$	−0.422563	−0.0384148
$122$	32.8406	2.97325
$123$	0	0
$124$	29.8061	2.67666
$125$	0	0
$126$	0	0
$127$	18.6739	1.65704	0.828519	−	0.559961i	$-0.189184\pi$
$127$	18.6739	1.65704	0.828519	+	0.559961i	$0.189184\pi$
$128$	56.0404	4.95332
$129$	0	0
$130$	0	0
$131$	11.0338	0.964025	0.482012	−	0.876164i	$-0.339906\pi$
$131$	11.0338	0.964025	0.482012	+	0.876164i	$0.339906\pi$
$132$	0	0
$133$	1.26172	0.109405
$134$	22.0246	1.90263
$135$	0	0
$136$	24.5292	2.10336
$137$	−11.8714	−1.01425	−0.507123	−	0.861874i	$-0.669291\pi$
$137$	−11.8714	−1.01425	−0.507123	+	0.861874i	$0.669291\pi$
$138$	0	0
$139$	14.8451	1.25915	0.629574	−	0.776941i	$-0.283230\pi$
$139$	14.8451	1.25915	0.629574	+	0.776941i	$0.283230\pi$
$140$	0	0
$141$	0	0
$142$	−5.54071	−0.464966
$143$	11.0618	0.925030
$144$	0	0
$145$	0	0
$146$	−1.68405	−0.139373
$147$	0	0
$148$	10.6097	0.872114
$149$	−17.2769	−1.41538	−0.707688	−	0.706525i	$-0.750262\pi$
$149$	−17.2769	−1.41538	−0.707688	+	0.706525i	$0.750262\pi$
$150$	0	0
$151$	0.504595	0.0410633	0.0205317	−	0.999789i	$-0.493464\pi$
$151$	0.504595	0.0410633	0.0205317	+	0.999789i	$0.493464\pi$
$152$	−6.87052	−0.557273
$153$	0	0
$154$	−16.8451	−1.35742
$155$	0	0
$156$	0	0
$157$	−17.9519	−1.43272	−0.716360	−	0.697731i	$-0.754193\pi$
$157$	−17.9519	−1.43272	−0.716360	+	0.697731i	$0.754193\pi$
$158$	23.8306	1.89586
$159$	0	0
$160$	0	0
$161$	13.8543	1.09187
$162$	0	0
$163$	−5.45295	−0.427108	−0.213554	−	0.976931i	$-0.568504\pi$
$163$	−5.45295	−0.427108	−0.213554	+	0.976931i	$0.568504\pi$
$164$	0	0
$165$	0	0
$166$	6.67486	0.518070
$167$	1.51195	0.116998	0.0584992	−	0.998287i	$-0.481368\pi$
$167$	1.51195	0.116998	0.0584992	+	0.998287i	$0.481368\pi$
$168$	0	0
$169$	−1.43175	−0.110135
$170$	0	0
$171$	0	0
$172$	−19.6878	−1.50118
$173$	−8.74012	−0.664499	−0.332249	−	0.943192i	$-0.607808\pi$
$173$	−8.74012	−0.664499	−0.332249	+	0.943192i	$0.607808\pi$
$174$	0	0
$175$	0	0
$176$	54.8152	4.13185
$177$	0	0
$178$	23.5607	1.76595
$179$	9.49541	0.709720	0.354860	−	0.934919i	$-0.384529\pi$
$179$	9.49541	0.709720	0.354860	+	0.934919i	$0.384529\pi$
$180$	0	0
$181$	14.9363	1.11021	0.555105	−	0.831780i	$-0.312678\pi$
$181$	14.9363	1.11021	0.555105	+	0.831780i	$0.312678\pi$
$182$	−17.6164	−1.30582
$183$	0	0
$184$	−75.4418	−5.56165
$185$	0	0
$186$	0	0
$187$	7.83605	0.573029
$188$	−22.7511	−1.65929
$189$	0	0
$190$	0	0
$191$	−10.5292	−0.761864	−0.380932	−	0.924603i	$-0.624397\pi$
$191$	−10.5292	−0.761864	−0.380932	+	0.924603i	$0.624397\pi$
$192$	0	0
$193$	−18.1341	−1.30532	−0.652660	−	0.757651i	$-0.726347\pi$
$193$	−18.1341	−1.30532	−0.652660	+	0.757651i	$0.726347\pi$
$194$	−36.3835	−2.61218
$195$	0	0
$196$	−19.8881	−1.42058
$197$	−16.9798	−1.20976	−0.604879	−	0.796317i	$-0.706779\pi$
$197$	−16.9798	−1.20976	−0.604879	+	0.796317i	$0.706779\pi$
$198$	0	0
$199$	7.49541	0.531335	0.265668	−	0.964065i	$-0.414408\pi$
$199$	7.49541	0.531335	0.265668	+	0.964065i	$0.414408\pi$
$200$	0	0
$201$	0	0
$202$	−13.4227	−0.944419
$203$	−1.86960	−0.131220
$204$	0	0
$205$	0	0
$206$	41.2470	2.87381
$207$	0	0
$208$	57.3251	3.97478
$209$	−2.19484	−0.151821
$210$	0	0
$211$	18.0974	1.24588	0.622939	−	0.782270i	$-0.285938\pi$
$211$	18.0974	1.24588	0.622939	+	0.782270i	$0.285938\pi$
$212$	−4.98126	−0.342114
$213$	0	0
$214$	−5.17945	−0.354060
$215$	0	0
$216$	0	0
$217$	−9.81970	−0.666604
$218$	−24.3023	−1.64596
$219$	0	0
$220$	0	0
$221$	8.19484	0.551245
$222$	0	0
$223$	−6.33073	−0.423937	−0.211969	−	0.977276i	$-0.567987\pi$
$223$	−6.33073	−0.423937	−0.211969	+	0.977276i	$0.567987\pi$
$224$	−49.2286	−3.28923
$225$	0	0
$226$	−27.8973	−1.85570
$227$	24.8905	1.65204	0.826022	−	0.563638i	$-0.190599\pi$
$227$	24.8905	1.65204	0.826022	+	0.563638i	$0.190599\pi$
$228$	0	0
$229$	2.14569	0.141791	0.0708955	−	0.997484i	$-0.477414\pi$
$229$	2.14569	0.141791	0.0708955	+	0.997484i	$0.477414\pi$
$230$	0	0
$231$	0	0
$232$	10.1807	0.668393
$233$	5.33891	0.349763	0.174882	−	0.984589i	$-0.444046\pi$
$233$	5.33891	0.349763	0.174882	+	0.984589i	$0.444046\pi$
$234$	0	0
$235$	0	0
$236$	−56.7486	−3.69402
$237$	0	0
$238$	−12.4793	−0.808914
$239$	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
$241$	−6.77228	−0.436241	−0.218121	−	0.975922i	$-0.569993\pi$
$241$	−6.77228	−0.436241	−0.218121	+	0.975922i	$0.569993\pi$
$242$	−1.17065	−0.0752521
$243$	0	0
$244$	67.2716	4.30662
$245$	0	0
$246$	0	0
$247$	−2.29534	−0.146049
$248$	53.4719	3.39547
$249$	0	0
$250$	0	0
$251$	−20.2615	−1.27889	−0.639447	−	0.768835i	$-0.720837\pi$
$251$	−20.2615	−1.27889	−0.639447	+	0.768835i	$0.720837\pi$
$252$	0	0
$253$	−24.1005	−1.51519
$254$	51.7332	3.24603
$255$	0	0
$256$	76.7762	4.79851
$257$	−19.4376	−1.21248	−0.606242	−	0.795280i	$-0.707324\pi$
$257$	−19.4376	−1.21248	−0.606242	+	0.795280i	$0.707324\pi$
$258$	0	0
$259$	−3.49541	−0.217194
$260$	0	0
$261$	0	0
$262$	30.5674	1.88846
$263$	8.28808	0.511065	0.255533	−	0.966800i	$-0.417749\pi$
$263$	8.28808	0.511065	0.255533	+	0.966800i	$0.417749\pi$
$264$	0	0
$265$	0	0
$266$	3.49541	0.214317
$267$	0	0
$268$	45.1157	2.75588
$269$	23.8543	1.45442	0.727212	−	0.686413i	$-0.240816\pi$
$269$	23.8543	1.45442	0.727212	+	0.686413i	$0.240816\pi$
$270$	0	0
$271$	−0.893389	−0.0542695	−0.0271347	−	0.999632i	$-0.508638\pi$
$271$	−0.893389	−0.0542695	−0.0271347	+	0.999632i	$0.508638\pi$
$272$	40.6086	2.46226
$273$	0	0
$274$	−32.8881	−1.98684
$275$	0	0
$276$	0	0
$277$	−3.55706	−0.213723	−0.106862	−	0.994274i	$-0.534080\pi$
$277$	−3.55706	−0.213723	−0.106862	+	0.994274i	$0.534080\pi$
$278$	41.1262	2.46659
$279$	0	0
$280$	0	0
$281$	−7.76309	−0.463107	−0.231554	−	0.972822i	$-0.574381\pi$
$281$	−7.76309	−0.463107	−0.231554	+	0.972822i	$0.574381\pi$
$282$	0	0
$283$	−30.7889	−1.83021	−0.915105	−	0.403215i	$-0.867893\pi$
$283$	−30.7889	−1.83021	−0.915105	+	0.403215i	$0.867893\pi$
$284$	−11.3497	−0.673482
$285$	0	0
$286$	30.6450	1.81208
$287$	0	0
$288$	0	0
$289$	−11.1948	−0.658520
$290$	0	0
$291$	0	0
$292$	−3.44965	−0.201875
$293$	6.33073	0.369845	0.184923	−	0.982753i	$-0.440797\pi$
$293$	6.33073	0.369845	0.184923	+	0.982753i	$0.440797\pi$
$294$	0	0
$295$	0	0
$296$	19.0338	1.10632
$297$	0	0
$298$	−47.8631	−2.77263
$299$	−25.2040	−1.45759
$300$	0	0
$301$	6.48622	0.373859
$302$	1.39791	0.0804404
$303$	0	0
$304$	−11.3743	−0.652361
$305$	0	0
$306$	0	0
$307$	−12.5671	−0.717241	−0.358620	−	0.933483i	$-0.616753\pi$
$307$	−12.5671	−0.717241	−0.358620	+	0.933483i	$0.616753\pi$
$308$	−34.5060	−1.96616
$309$	0	0
$310$	0	0
$311$	−24.3835	−1.38266	−0.691330	−	0.722539i	$-0.742975\pi$
$311$	−24.3835	−1.38266	−0.691330	+	0.722539i	$0.742975\pi$
$312$	0	0
$313$	34.6618	1.95920	0.979601	−	0.200954i	$-0.0644042\pi$
$313$	34.6618	1.95920	0.979601	+	0.200954i	$0.0644042\pi$
$314$	−49.7332	−2.80661
$315$	0	0
$316$	48.8152	2.74607
$317$	−25.7880	−1.44840	−0.724199	−	0.689591i	$-0.757790\pi$
$317$	−25.7880	−1.44840	−0.724199	+	0.689591i	$0.757790\pi$
$318$	0	0
$319$	3.25230	0.182094
$320$	0	0
$321$	0	0
$322$	38.3814	2.13891
$323$	−1.62600	−0.0904730
$324$	0	0
$325$	0	0
$326$	−15.1066	−0.836678
$327$	0	0
$328$	0	0
$329$	7.49541	0.413235
$330$	0	0
$331$	−16.9609	−0.932257	−0.466128	−	0.884717i	$-0.654352\pi$
$331$	−16.9609	−0.932257	−0.466128	+	0.884717i	$0.654352\pi$
$332$	13.6729	0.750400
$333$	0	0
$334$	4.18864	0.229192
$335$	0	0
$336$	0	0
$337$	22.5952	1.23084	0.615420	−	0.788200i	$-0.288987\pi$
$337$	22.5952	1.23084	0.615420	+	0.788200i	$0.288987\pi$
$338$	−3.96646	−0.215747
$339$	0	0
$340$	0	0
$341$	17.0820	0.925044
$342$	0	0
$343$	19.6394	1.06043
$344$	−35.3198	−1.90432
$345$	0	0
$346$	−24.2132	−1.30171
$347$	12.0469	0.646714	0.323357	−	0.946277i	$-0.395189\pi$
$347$	12.0469	0.646714	0.323357	+	0.946277i	$0.395189\pi$
$348$	0	0
$349$	−1.56825	−0.0839464	−0.0419732	−	0.999119i	$-0.513364\pi$
$349$	−1.56825	−0.0839464	−0.0419732	+	0.999119i	$0.513364\pi$
$350$	0	0
$351$	0	0
$352$	85.6365	4.56444
$353$	7.66054	0.407730	0.203865	−	0.978999i	$-0.434650\pi$
$353$	7.66054	0.407730	0.203865	+	0.978999i	$0.434650\pi$
$354$	0	0
$355$	0	0
$356$	48.2624	2.55790
$357$	0	0
$358$	26.3056	1.39030
$359$	−17.5928	−0.928514	−0.464257	−	0.885701i	$-0.653679\pi$
$359$	−17.5928	−0.928514	−0.464257	+	0.885701i	$0.653679\pi$
$360$	0	0
$361$	−18.5446	−0.976030
$362$	41.3790	2.17483
$363$	0	0
$364$	−36.0859	−1.89142
$365$	0	0
$366$	0	0
$367$	9.88779	0.516138	0.258069	−	0.966126i	$-0.416914\pi$
$367$	9.88779	0.516138	0.258069	+	0.966126i	$0.416914\pi$
$368$	−124.896	−6.51064
$369$	0	0
$370$	0	0
$371$	1.64109	0.0852012
$372$	0	0
$373$	−16.2841	−0.843161	−0.421580	−	0.906791i	$-0.638524\pi$
$373$	−16.2841	−0.843161	−0.421580	+	0.906791i	$0.638524\pi$
$374$	21.7086	1.12253
$375$	0	0
$376$	−40.8152	−2.10489
$377$	3.40121	0.175171
$378$	0	0
$379$	−2.52917	−0.129915	−0.0649575	−	0.997888i	$-0.520691\pi$
$379$	−2.52917	−0.129915	−0.0649575	+	0.997888i	$0.520691\pi$
$380$	0	0
$381$	0	0
$382$	−29.1695	−1.49244
$383$	−5.60880	−0.286596	−0.143298	−	0.989680i	$-0.545771\pi$
$383$	−5.60880	−0.286596	−0.143298	+	0.989680i	$0.545771\pi$
$384$	0	0
$385$	0	0
$386$	−50.2378	−2.55704
$387$	0	0
$388$	−74.5288	−3.78363
$389$	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
$391$	−17.8543	−0.902931
$392$	−35.6791	−1.80207
$393$	0	0
$394$	−47.0400	−2.36984
$395$	0	0
$396$	0	0
$397$	−27.8660	−1.39856	−0.699278	−	0.714850i	$-0.746495\pi$
$397$	−27.8660	−1.39856	−0.699278	+	0.714850i	$0.746495\pi$
$398$	20.7649	1.04085
$399$	0	0
$400$	0	0
$401$	15.6174	0.779896	0.389948	−	0.920837i	$-0.372493\pi$
$401$	15.6174	0.779896	0.389948	+	0.920837i	$0.372493\pi$
$402$	0	0
$403$	17.8642	0.889878
$404$	−27.4954	−1.36795
$405$	0	0
$406$	−5.17945	−0.257052
$407$	6.08050	0.301399
$408$	0	0
$409$	−30.5538	−1.51079	−0.755393	−	0.655272i	$-0.772554\pi$
$409$	−30.5538	−1.51079	−0.755393	+	0.655272i	$0.772554\pi$
$410$	0	0
$411$	0	0
$412$	84.4913	4.16259
$413$	18.6960	0.919970
$414$	0	0
$415$	0	0
$416$	89.5576	4.39092
$417$	0	0
$418$	−6.08050	−0.297407
$419$	10.0492	0.490934	0.245467	−	0.969405i	$-0.421059\pi$
$419$	10.0492	0.490934	0.245467	+	0.969405i	$0.421059\pi$
$420$	0	0
$421$	−16.1948	−0.789288	−0.394644	−	0.918834i	$-0.629132\pi$
$421$	−16.1948	−0.789288	−0.394644	+	0.918834i	$0.629132\pi$
$422$	50.1363	2.44060
$423$	0	0
$424$	−8.93635	−0.433987
$425$	0	0
$426$	0	0
$427$	−22.1628	−1.07253
$428$	−10.6097	−0.512840
$429$	0	0
$430$	0	0
$431$	−11.1857	−0.538794	−0.269397	−	0.963029i	$-0.586824\pi$
$431$	−11.1857	−0.538794	−0.269397	+	0.963029i	$0.586824\pi$
$432$	0	0
$433$	−14.0306	−0.674267	−0.337134	−	0.941457i	$-0.609457\pi$
$433$	−14.0306	−0.674267	−0.337134	+	0.941457i	$0.609457\pi$
$434$	−27.2040	−1.30584
$435$	0	0
$436$	−49.7815	−2.38410
$437$	5.00092	0.239226
$438$	0	0
$439$	9.65947	0.461021	0.230511	−	0.973070i	$-0.425960\pi$
$439$	9.65947	0.461021	0.230511	+	0.973070i	$0.425960\pi$
$440$	0	0
$441$	0	0
$442$	22.7026	1.07985
$443$	−9.75160	−0.463313	−0.231656	−	0.972798i	$-0.574414\pi$
$443$	−9.75160	−0.463313	−0.231656	+	0.972798i	$0.574414\pi$
$444$	0	0
$445$	0	0
$446$	−17.5384	−0.830466
$447$	0	0
$448$	−73.3590	−3.46589
$449$	−8.86350	−0.418295	−0.209147	−	0.977884i	$-0.567069\pi$
$449$	−8.86350	−0.418295	−0.209147	+	0.977884i	$0.567069\pi$
$450$	0	0
$451$	0	0
$452$	−57.1454	−2.68790
$453$	0	0
$454$	68.9556	3.23625
$455$	0	0
$456$	0	0
$457$	−24.5041	−1.14625	−0.573127	−	0.819466i	$-0.694270\pi$
$457$	−24.5041	−1.14625	−0.573127	+	0.819466i	$0.694270\pi$
$458$	5.94431	0.277759
$459$	0	0
$460$	0	0
$461$	37.4173	1.74270	0.871348	−	0.490666i	$-0.163247\pi$
$461$	37.4173	1.74270	0.871348	+	0.490666i	$0.163247\pi$
$462$	0	0
$463$	−2.45534	−0.114109	−0.0570547	−	0.998371i	$-0.518171\pi$
$463$	−2.45534	−0.114109	−0.0570547	+	0.998371i	$0.518171\pi$
$464$	16.8543	0.782442
$465$	0	0
$466$	14.7907	0.685164
$467$	−34.4182	−1.59268	−0.796342	−	0.604846i	$-0.793235\pi$
$467$	−34.4182	−1.59268	−0.796342	+	0.604846i	$0.793235\pi$
$468$	0	0
$469$	−14.8635	−0.686333
$470$	0	0
$471$	0	0
$472$	−101.807	−4.68603
$473$	−11.2832	−0.518803
$474$	0	0
$475$	0	0
$476$	−25.5629	−1.17168
$477$	0	0
$478$	5.54071	0.253426
$479$	9.41636	0.430245	0.215122	−	0.976587i	$-0.430985\pi$
$479$	9.41636	0.430245	0.215122	+	0.976587i	$0.430985\pi$
$480$	0	0
$481$	6.35891	0.289941
$482$	−18.7616	−0.854568
$483$	0	0
$484$	−2.39798	−0.108999
$485$	0	0
$486$	0	0
$487$	−33.0909	−1.49949	−0.749745	−	0.661726i	$-0.769824\pi$
$487$	−33.0909	−1.49949	−0.749745	+	0.661726i	$0.769824\pi$
$488$	120.685	5.46314
$489$	0	0
$490$	0	0
$491$	−28.9609	−1.30699	−0.653494	−	0.756932i	$-0.726698\pi$
$491$	−28.9609	−1.30699	−0.653494	+	0.756932i	$0.726698\pi$
$492$	0	0
$493$	2.40939	0.108513
$494$	−6.35891	−0.286101
$495$	0	0
$496$	88.5239	3.97484
$497$	3.73920	0.167726
$498$	0	0
$499$	33.8727	1.51635	0.758175	−	0.652051i	$-0.226091\pi$
$499$	33.8727	1.51635	0.758175	+	0.652051i	$0.226091\pi$
$500$	0	0
$501$	0	0
$502$	−56.1315	−2.50527
$503$	−0.945870	−0.0421743	−0.0210871	−	0.999778i	$-0.506713\pi$
$503$	−0.945870	−0.0421743	−0.0210871	+	0.999778i	$0.506713\pi$
$504$	0	0
$505$	0	0
$506$	−66.7670	−2.96815
$507$	0	0
$508$	105.972	4.70173
$509$	−32.1404	−1.42460	−0.712299	−	0.701877i	$-0.752346\pi$
$509$	−32.1404	−1.42460	−0.712299	+	0.701877i	$0.752346\pi$
$510$	0	0
$511$	1.13650	0.0502757
$512$	100.616	4.44665
$513$	0	0
$514$	−53.8490	−2.37518
$515$	0	0
$516$	0	0
$517$	−13.0388	−0.573444
$518$	−9.68351	−0.425469
$519$	0	0
$520$	0	0
$521$	8.43175	0.369402	0.184701	−	0.982795i	$-0.440868\pi$
$521$	8.43175	0.369402	0.184701	+	0.982795i	$0.440868\pi$
$522$	0	0
$523$	21.8273	0.954442	0.477221	−	0.878783i	$-0.341644\pi$
$523$	21.8273	0.954442	0.477221	+	0.878783i	$0.341644\pi$
$524$	62.6151	2.73535
$525$	0	0
$526$	22.9609	1.00114
$527$	12.6548	0.551253
$528$	0	0
$529$	31.9127	1.38751
$530$	0	0
$531$	0	0
$532$	7.16007	0.310429
$533$	0	0
$534$	0	0
$535$	0	0
$536$	80.9372	3.49596
$537$	0	0
$538$	66.0849	2.84912
$539$	−11.3980	−0.490946
$540$	0	0
$541$	−25.5138	−1.09692	−0.548462	−	0.836176i	$-0.684786\pi$
$541$	−25.5138	−1.09692	−0.548462	+	0.836176i	$0.684786\pi$
$542$	−2.47500	−0.106310
$543$	0	0
$544$	63.4418	2.72005
$545$	0	0
$546$	0	0
$547$	17.7698	0.759781	0.379891	−	0.925031i	$-0.375962\pi$
$547$	17.7698	0.759781	0.379891	+	0.925031i	$0.375962\pi$
$548$	−67.3687	−2.87785
$549$	0	0
$550$	0	0
$551$	−0.674860	−0.0287500
$552$	0	0
$553$	−16.0823	−0.683890
$554$	−9.85431	−0.418670
$555$	0	0
$556$	84.2440	3.57274
$557$	−42.1665	−1.78665	−0.893326	−	0.449409i	$-0.851635\pi$
$557$	−42.1665	−1.78665	−0.893326	+	0.449409i	$0.851635\pi$
$558$	0	0
$559$	−11.7998	−0.499080
$560$	0	0
$561$	0	0
$562$	−21.5065	−0.907198
$563$	−5.76465	−0.242951	−0.121475	−	0.992594i	$-0.538763\pi$
$563$	−5.76465	−0.242951	−0.121475	+	0.992594i	$0.538763\pi$
$564$	0	0
$565$	0	0
$566$	−85.2961	−3.58526
$567$	0	0
$568$	−20.3613	−0.854342
$569$	−1.49541	−0.0626907	−0.0313453	−	0.999509i	$-0.509979\pi$
$569$	−1.49541	−0.0626907	−0.0313453	+	0.999509i	$0.509979\pi$
$570$	0	0
$571$	−7.03997	−0.294614	−0.147307	−	0.989091i	$-0.547060\pi$
$571$	−7.03997	−0.294614	−0.147307	+	0.989091i	$0.547060\pi$
$572$	62.7739	2.62471
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	34.7102	1.44501	0.722503	−	0.691368i	$-0.242991\pi$
$577$	34.7102	1.44501	0.722503	+	0.691368i	$0.242991\pi$
$578$	−31.0137	−1.29000
$579$	0	0
$580$	0	0
$581$	−4.50459	−0.186882
$582$	0	0
$583$	−2.85479	−0.118233
$584$	−6.18864	−0.256088
$585$	0	0
$586$	17.5384	0.724503
$587$	−37.4158	−1.54432	−0.772158	−	0.635430i	$-0.780823\pi$
$587$	−37.4158	−1.54432	−0.772158	+	0.635430i	$0.780823\pi$
$588$	0	0
$589$	−3.54456	−0.146051
$590$	0	0
$591$	0	0
$592$	31.5108	1.29509
$593$	32.6388	1.34032	0.670158	−	0.742218i	$-0.266226\pi$
$593$	32.6388	1.34032	0.670158	+	0.742218i	$0.266226\pi$
$594$	0	0
$595$	0	0
$596$	−98.0439	−4.01603
$597$	0	0
$598$	−69.8241	−2.85532
$599$	38.1466	1.55863	0.779314	−	0.626634i	$-0.215568\pi$
$599$	38.1466	1.55863	0.779314	+	0.626634i	$0.215568\pi$
$600$	0	0
$601$	−16.7178	−0.681934	−0.340967	−	0.940075i	$-0.610754\pi$
$601$	−16.7178	−0.681934	−0.340967	+	0.940075i	$0.610754\pi$
$602$	17.9691	0.732366
$603$	0	0
$604$	2.86350	0.116514
$605$	0	0
$606$	0	0
$607$	−0.673496	−0.0273364	−0.0136682	−	0.999907i	$-0.504351\pi$
$607$	−0.673496	−0.0273364	−0.0136682	+	0.999907i	$0.504351\pi$
$608$	−17.7698	−0.720660
$609$	0	0
$610$	0	0
$611$	−13.6358	−0.551645
$612$	0	0
$613$	26.4615	1.06877	0.534385	−	0.845241i	$-0.320543\pi$
$613$	26.4615	1.06877	0.534385	+	0.845241i	$0.320543\pi$
$614$	−34.8152	−1.40503
$615$	0	0
$616$	−61.9035	−2.49416
$617$	−27.9538	−1.12538	−0.562688	−	0.826669i	$-0.690233\pi$
$617$	−27.9538	−1.12538	−0.562688	+	0.826669i	$0.690233\pi$
$618$	0	0
$619$	24.6512	0.990814	0.495407	−	0.868661i	$-0.335019\pi$
$619$	24.6512	0.990814	0.495407	+	0.868661i	$0.335019\pi$
$620$	0	0
$621$	0	0
$622$	−67.5509	−2.70854
$623$	−15.9002	−0.637028
$624$	0	0
$625$	0	0
$626$	96.0255	3.83795
$627$	0	0
$628$	−101.875	−4.06524
$629$	4.50459	0.179610
$630$	0	0
$631$	11.8052	0.469956	0.234978	−	0.972001i	$-0.424498\pi$
$631$	11.8052	0.469956	0.234978	+	0.972001i	$0.424498\pi$
$632$	87.5742	3.48352
$633$	0	0
$634$	−71.4418	−2.83732
$635$	0	0
$636$	0	0
$637$	−11.9199	−0.472283
$638$	9.01001	0.356710
$639$	0	0
$640$	0	0
$641$	−13.8727	−0.547938	−0.273969	−	0.961738i	$-0.588337\pi$
$641$	−13.8727	−0.547938	−0.273969	+	0.961738i	$0.588337\pi$
$642$	0	0
$643$	5.29047	0.208636	0.104318	−	0.994544i	$-0.466734\pi$
$643$	5.29047	0.208636	0.104318	+	0.994544i	$0.466734\pi$
$644$	78.6213	3.09811
$645$	0	0
$646$	−4.50459	−0.177231
$647$	−40.7448	−1.60184	−0.800921	−	0.598769i	$-0.795657\pi$
$647$	−40.7448	−1.60184	−0.800921	+	0.598769i	$0.795657\pi$
$648$	0	0
$649$	−32.5230	−1.27664
$650$	0	0
$651$	0	0
$652$	−30.9447	−1.21189
$653$	−26.8742	−1.05167	−0.525834	−	0.850587i	$-0.676247\pi$
$653$	−26.8742	−1.05167	−0.525834	+	0.850587i	$0.676247\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	20.7649	0.809501
$659$	−29.1558	−1.13575	−0.567874	−	0.823116i	$-0.692234\pi$
$659$	−29.1558	−1.13575	−0.567874	+	0.823116i	$0.692234\pi$
$660$	0	0
$661$	19.9035	0.774155	0.387078	−	0.922047i	$-0.373485\pi$
$661$	19.9035	0.774155	0.387078	+	0.922047i	$0.373485\pi$
$662$	−46.9878	−1.82623
$663$	0	0
$664$	24.5292	0.951917
$665$	0	0
$666$	0	0
$667$	−7.41031	−0.286928
$668$	8.58012	0.331975
$669$	0	0
$670$	0	0
$671$	38.5538	1.48835
$672$	0	0
$673$	−8.71383	−0.335893	−0.167947	−	0.985796i	$-0.553714\pi$
$673$	−8.71383	−0.335893	−0.167947	+	0.985796i	$0.553714\pi$
$674$	62.5967	2.41114
$675$	0	0
$676$	−8.12499	−0.312500
$677$	−9.70565	−0.373018	−0.186509	−	0.982453i	$-0.559717\pi$
$677$	−9.70565	−0.373018	−0.186509	+	0.982453i	$0.559717\pi$
$678$	0	0
$679$	24.5538	0.942287
$680$	0	0
$681$	0	0
$682$	47.3233	1.81210
$683$	9.53014	0.364661	0.182330	−	0.983237i	$-0.441636\pi$
$683$	9.53014	0.364661	0.182330	+	0.983237i	$0.441636\pi$
$684$	0	0
$685$	0	0
$686$	54.4081	2.07731
$687$	0	0
$688$	−58.4728	−2.22925
$689$	−2.98550	−0.113739
$690$	0	0
$691$	8.50459	0.323530	0.161765	−	0.986829i	$-0.448281\pi$
$691$	8.50459	0.323530	0.161765	+	0.986829i	$0.448281\pi$
$692$	−49.5990	−1.88547
$693$	0	0
$694$	33.3743	1.26687
$695$	0	0
$696$	0	0
$697$	0	0
$698$	−4.34460	−0.164446
$699$	0	0
$700$	0	0
$701$	28.6266	1.08121	0.540606	−	0.841276i	$-0.318195\pi$
$701$	28.6266	1.08121	0.540606	+	0.841276i	$0.318195\pi$
$702$	0	0
$703$	−1.26172	−0.0475866
$704$	127.613	4.80960
$705$	0	0
$706$	21.2224	0.798716
$707$	9.05845	0.340678
$708$	0	0
$709$	14.9855	0.562792	0.281396	−	0.959592i	$-0.409202\pi$
$709$	14.9855	0.562792	0.281396	+	0.959592i	$0.409202\pi$
$710$	0	0
$711$	0	0
$712$	86.5824	3.24481
$713$	−38.9211	−1.45761
$714$	0	0
$715$	0	0
$716$	53.8851	2.01378
$717$	0	0
$718$	−48.7384	−1.81890
$719$	36.7486	1.37049	0.685246	−	0.728312i	$-0.259695\pi$
$719$	36.7486	1.37049	0.685246	+	0.728312i	$0.259695\pi$
$720$	0	0
$721$	−27.8359	−1.03666
$722$	−51.3750	−1.91198
$723$	0	0
$724$	84.7617	3.15014
$725$	0	0
$726$	0	0
$727$	−39.1647	−1.45254	−0.726270	−	0.687410i	$-0.758748\pi$
$727$	−39.1647	−1.45254	−0.726270	+	0.687410i	$0.758748\pi$
$728$	−64.7379	−2.39935
$729$	0	0
$730$	0	0
$731$	−8.35891	−0.309165
$732$	0	0
$733$	5.38071	0.198741	0.0993705	−	0.995051i	$-0.468317\pi$
$733$	5.38071	0.198741	0.0993705	+	0.995051i	$0.468317\pi$
$734$	27.3927	1.01108
$735$	0	0
$736$	−195.121	−7.19227
$737$	25.8561	0.952421
$738$	0	0
$739$	−9.93336	−0.365404	−0.182702	−	0.983168i	$-0.558484\pi$
$739$	−9.93336	−0.365404	−0.182702	+	0.983168i	$0.558484\pi$
$740$	0	0
$741$	0	0
$742$	4.54640	0.166904
$743$	−43.4963	−1.59573	−0.797863	−	0.602839i	$-0.794036\pi$
$743$	−43.4963	−1.59573	−0.797863	+	0.602839i	$0.794036\pi$
$744$	0	0
$745$	0	0
$746$	−45.1128	−1.65170
$747$	0	0
$748$	44.4685	1.62593
$749$	3.49541	0.127719
$750$	0	0
$751$	11.3743	0.415054	0.207527	−	0.978229i	$-0.433459\pi$
$751$	11.3743	0.415054	0.207527	+	0.978229i	$0.433459\pi$
$752$	−67.5705	−2.46404
$753$	0	0
$754$	9.42256	0.343149
$755$	0	0
$756$	0	0
$757$	−10.0699	−0.365998	−0.182999	−	0.983113i	$-0.558580\pi$
$757$	−10.0699	−0.365998	−0.182999	+	0.983113i	$0.558580\pi$
$758$	−7.00671	−0.254495
$759$	0	0
$760$	0	0
$761$	27.0092	0.979082	0.489541	−	0.871980i	$-0.337164\pi$
$761$	27.0092	0.979082	0.489541	+	0.871980i	$0.337164\pi$
$762$	0	0
$763$	16.4007	0.593744
$764$	−59.7516	−2.16174
$765$	0	0
$766$	−15.5384	−0.561424
$767$	−34.0121	−1.22811
$768$	0	0
$769$	25.7086	0.927077	0.463538	−	0.886077i	$-0.346580\pi$
$769$	25.7086	0.927077	0.463538	+	0.886077i	$0.346580\pi$
$770$	0	0
$771$	0	0
$772$	−102.908	−3.70375
$773$	−32.8185	−1.18040	−0.590200	−	0.807257i	$-0.700951\pi$
$773$	−32.8185	−1.18040	−0.590200	+	0.807257i	$0.700951\pi$
$774$	0	0
$775$	0	0
$776$	−133.704	−4.79970
$777$	0	0
$778$	33.2442	1.19186
$779$	0	0
$780$	0	0
$781$	−6.50459	−0.232753
$782$	−49.4628	−1.76878
$783$	0	0
$784$	−59.0675	−2.10955
$785$	0	0
$786$	0	0
$787$	−21.1973	−0.755602	−0.377801	−	0.925887i	$-0.623320\pi$
$787$	−21.1973	−0.755602	−0.377801	+	0.925887i	$0.623320\pi$
$788$	−96.3578	−3.43261
$789$	0	0
$790$	0	0
$791$	18.8267	0.669402
$792$	0	0
$793$	40.3190	1.43177
$794$	−77.1987	−2.73968
$795$	0	0
$796$	42.5354	1.50763
$797$	−21.7305	−0.769732	−0.384866	−	0.922972i	$-0.625752\pi$
$797$	−21.7305	−0.769732	−0.384866	+	0.922972i	$0.625752\pi$
$798$	0	0
$799$	−9.65947	−0.341727
$800$	0	0
$801$	0	0
$802$	43.2657	1.52777
$803$	−1.97701	−0.0697673
$804$	0	0
$805$	0	0
$806$	49.4901	1.74321
$807$	0	0
$808$	−49.3266	−1.73530
$809$	7.20403	0.253280	0.126640	−	0.991949i	$-0.459581\pi$
$809$	7.20403	0.253280	0.126640	+	0.991949i	$0.459581\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	−10.6097	−0.372328
$813$	0	0
$814$	16.8451	0.590421
$815$	0	0
$816$	0	0
$817$	2.34130	0.0819116
$818$	−84.6447	−2.95953
$819$	0	0
$820$	0	0
$821$	−24.1220	−0.841864	−0.420932	−	0.907092i	$-0.638297\pi$
$821$	−24.1220	−0.841864	−0.420932	+	0.907092i	$0.638297\pi$
$822$	0	0
$823$	−17.2300	−0.600600	−0.300300	−	0.953845i	$-0.597087\pi$
$823$	−17.2300	−0.600600	−0.300300	+	0.953845i	$0.597087\pi$
$824$	151.577	5.28043
$825$	0	0
$826$	51.7945	1.80216
$827$	−3.46931	−0.120640	−0.0603198	−	0.998179i	$-0.519212\pi$
$827$	−3.46931	−0.120640	−0.0603198	+	0.998179i	$0.519212\pi$
$828$	0	0
$829$	7.52619	0.261395	0.130698	−	0.991422i	$-0.458278\pi$
$829$	7.52619	0.261395	0.130698	+	0.991422i	$0.458278\pi$
$830$	0	0
$831$	0	0
$832$	133.456	4.62676
$833$	−8.44393	−0.292565
$834$	0	0
$835$	0	0
$836$	−12.4554	−0.430780
$837$	0	0
$838$	27.8397	0.961707
$839$	9.61121	0.331816	0.165908	−	0.986141i	$-0.446945\pi$
$839$	9.61121	0.331816	0.165908	+	0.986141i	$0.446945\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−44.8654	−1.54617
$843$	0	0
$844$	102.700	3.53509
$845$	0	0
$846$	0	0
$847$	0.790023	0.0271455
$848$	−14.7943	−0.508039
$849$	0	0
$850$	0	0
$851$	−13.8543	−0.474920
$852$	0	0
$853$	20.8330	0.713309	0.356654	−	0.934236i	$-0.383917\pi$
$853$	20.8330	0.713309	0.356654	+	0.934236i	$0.383917\pi$
$854$	−61.3989	−2.10103
$855$	0	0
$856$	−19.0338	−0.650561
$857$	33.0424	1.12871	0.564354	−	0.825533i	$-0.309125\pi$
$857$	33.0424	1.12871	0.564354	+	0.825533i	$0.309125\pi$
$858$	0	0
$859$	49.8061	1.69936	0.849680	−	0.527298i	$-0.176795\pi$
$859$	49.8061	1.69936	0.849680	+	0.527298i	$0.176795\pi$
$860$	0	0
$861$	0	0
$862$	−30.9882	−1.05546
$863$	9.03631	0.307599	0.153800	−	0.988102i	$-0.450849\pi$
$863$	9.03631	0.307599	0.153800	+	0.988102i	$0.450849\pi$
$864$	0	0
$865$	0	0
$866$	−38.8697	−1.32085
$867$	0	0
$868$	−55.7254	−1.89144
$869$	27.9763	0.949032
$870$	0	0
$871$	27.0400	0.916214
$872$	−89.3076	−3.02434
$873$	0	0
$874$	13.8543	0.468629
$875$	0	0
$876$	0	0
$877$	−24.0809	−0.813153	−0.406576	−	0.913617i	$-0.633278\pi$
$877$	−24.0809	−0.813153	−0.406576	+	0.913617i	$0.633278\pi$
$878$	26.7601	0.903111
$879$	0	0
$880$	0	0
$881$	0.814344	0.0274360	0.0137180	−	0.999906i	$-0.495633\pi$
$881$	0.814344	0.0274360	0.0137180	+	0.999906i	$0.495633\pi$
$882$	0	0
$883$	31.8751	1.07268	0.536341	−	0.844001i	$-0.319806\pi$
$883$	31.8751	1.07268	0.536341	+	0.844001i	$0.319806\pi$
$884$	46.5046	1.56412
$885$	0	0
$886$	−27.0154	−0.907600
$887$	45.6358	1.53230	0.766150	−	0.642661i	$-0.222170\pi$
$887$	45.6358	1.53230	0.766150	+	0.642661i	$0.222170\pi$
$888$	0	0
$889$	−34.9127	−1.17093
$890$	0	0
$891$	0	0
$892$	−35.9260	−1.20289
$893$	2.70558	0.0905387
$894$	0	0
$895$	0	0
$896$	−104.773	−3.50023
$897$	0	0
$898$	−24.5550	−0.819412
$899$	5.25230	0.175174
$900$	0	0
$901$	−2.11491	−0.0704577
$902$	0	0
$903$	0	0
$904$	−102.519	−3.40972
$905$	0	0
$906$	0	0
$907$	18.5820	0.617004	0.308502	−	0.951224i	$-0.400172\pi$
$907$	18.5820	0.617004	0.308502	+	0.951224i	$0.400172\pi$
$908$	141.250	4.68756
$909$	0	0
$910$	0	0
$911$	45.6296	1.51178	0.755888	−	0.654701i	$-0.227206\pi$
$911$	45.6296	1.51178	0.755888	+	0.654701i	$0.227206\pi$
$912$	0	0
$913$	7.83605	0.259335
$914$	−67.8851	−2.24544
$915$	0	0
$916$	12.1765	0.402322
$917$	−20.6287	−0.681221
$918$	0	0
$919$	24.6994	0.814759	0.407380	−	0.913259i	$-0.366443\pi$
$919$	24.6994	0.814759	0.407380	+	0.913259i	$0.366443\pi$
$920$	0	0
$921$	0	0
$922$	103.659	3.41383
$923$	−6.80243	−0.223905
$924$	0	0
$925$	0	0
$926$	−6.80217	−0.223533
$927$	0	0
$928$	26.3311	0.864360
$929$	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
$931$	2.36511	0.0775133
$932$	30.2975	0.992429
$933$	0	0
$934$	−95.3506	−3.11997
$935$	0	0
$936$	0	0
$937$	46.4389	1.51709	0.758546	−	0.651620i	$-0.225910\pi$
$937$	46.4389	1.51709	0.758546	+	0.651620i	$0.225910\pi$
$938$	−41.1772	−1.34448
$939$	0	0
$940$	0	0
$941$	14.2861	0.465712	0.232856	−	0.972511i	$-0.425193\pi$
$941$	14.2861	0.465712	0.232856	+	0.972511i	$0.425193\pi$
$942$	0	0
$943$	0	0
$944$	−168.543	−5.48561
$945$	0	0
$946$	−31.2585	−1.01630
$947$	−29.3713	−0.954440	−0.477220	−	0.878784i	$-0.658356\pi$
$947$	−29.3713	−0.954440	−0.477220	+	0.878784i	$0.658356\pi$
$948$	0	0
$949$	−2.06754	−0.0671151
$950$	0	0
$951$	0	0
$952$	−45.8598	−1.48632
$953$	22.5911	0.731796	0.365898	−	0.930655i	$-0.380762\pi$
$953$	22.5911	0.731796	0.365898	+	0.930655i	$0.380762\pi$
$954$	0	0
$955$	0	0
$956$	11.3497	0.367076
$957$	0	0
$958$	26.0867	0.842821
$959$	22.1948	0.716709
$960$	0	0
$961$	−3.41337	−0.110109
$962$	17.6164	0.567976
$963$	0	0
$964$	−38.4318	−1.23780
$965$	0	0
$966$	0	0
$967$	48.4316	1.55746	0.778728	−	0.627362i	$-0.215865\pi$
$967$	48.4316	1.55746	0.778728	+	0.627362i	$0.215865\pi$
$968$	−4.30197	−0.138270
$969$	0	0
$970$	0	0
$971$	−47.0829	−1.51096	−0.755482	−	0.655170i	$-0.772597\pi$
$971$	−47.0829	−1.51096	−0.755482	+	0.655170i	$0.772597\pi$
$972$	0	0
$973$	−27.7545	−0.889767
$974$	−91.6734	−2.93741
$975$	0	0
$976$	199.796	6.39532
$977$	51.1987	1.63799	0.818995	−	0.573800i	$-0.194532\pi$
$977$	51.1987	1.63799	0.818995	+	0.573800i	$0.194532\pi$
$978$	0	0
$979$	27.6595	0.884000
$980$	0	0
$981$	0	0
$982$	−80.2320	−2.56031
$983$	23.5123	0.749926	0.374963	−	0.927040i	$-0.377655\pi$
$983$	23.5123	0.749926	0.374963	+	0.927040i	$0.377655\pi$
$984$	0	0
$985$	0	0
$986$	6.67486	0.212571
$987$	0	0
$988$	−13.0257	−0.414404
$989$	25.7086	0.817487
$990$	0	0
$991$	−7.64109	−0.242727	−0.121364	−	0.992608i	$-0.538727\pi$
$991$	−7.64109	−0.242727	−0.121364	+	0.992608i	$0.538727\pi$
$992$	138.299	4.39099
$993$	0	0
$994$	10.3589	0.328565
$995$	0	0
$996$	0	0
$997$	3.71043	0.117510	0.0587552	−	0.998272i	$-0.481287\pi$
$997$	3.71043	0.117510	0.0587552	+	0.998272i	$0.481287\pi$
$998$	93.8393	2.97043
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.bt.1.6		6
3.2	odd	2		725.2.a.l.1.1		6
5.2	odd	4		1305.2.c.h.784.6		6
5.3	odd	4		1305.2.c.h.784.1		6
5.4	even	2	inner	6525.2.a.bt.1.1		6
15.2	even	4		145.2.b.c.59.1	✓	6
15.8	even	4		145.2.b.c.59.6	yes	6
15.14	odd	2		725.2.a.l.1.6		6
60.23	odd	4		2320.2.d.g.929.4		6
60.47	odd	4		2320.2.d.g.929.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.b.c.59.1	✓	6	15.2	even	4
145.2.b.c.59.6	yes	6	15.8	even	4
725.2.a.l.1.1		6	3.2	odd	2
725.2.a.l.1.6		6	15.14	odd	2
1305.2.c.h.784.1		6	5.3	odd	4
1305.2.c.h.784.6		6	5.2	odd	4
2320.2.d.g.929.3		6	60.47	odd	4
2320.2.d.g.929.4		6	60.23	odd	4
6525.2.a.bt.1.1		6	5.4	even	2	inner
6525.2.a.bt.1.6		6	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q+2.77035 q^{2} +5.67486 q^{4} -1.86960 q^{7} +10.1807 q^{8} +3.25230 q^{11} +3.40121 q^{13} -5.17945 q^{14} +16.8543 q^{16} +2.40939 q^{17} -0.674860 q^{19} +9.01001 q^{22} -7.41031 q^{23} +9.42256 q^{26} -10.6097 q^{28} +1.00000 q^{29} +5.25230 q^{31} +26.3311 q^{32} +6.67486 q^{34} +1.86960 q^{37} -1.86960 q^{38} -3.46931 q^{43} +18.4563 q^{44} -20.5292 q^{46} -4.00910 q^{47} -3.50459 q^{49} +19.3014 q^{52} -0.877777 q^{53} -19.0338 q^{56} +2.77035 q^{58} -10.0000 q^{59} +11.8543 q^{61} +14.5507 q^{62} +39.2378 q^{64} +7.95010 q^{67} +13.6729 q^{68} -2.00000 q^{71} -0.607882 q^{73} +5.17945 q^{74} -3.82973 q^{76} -6.08050 q^{77} +8.60202 q^{79} +2.40939 q^{83} -9.61121 q^{86} +33.1105 q^{88} +8.50459 q^{89} -6.35891 q^{91} -42.0525 q^{92} -11.1066 q^{94} -13.1332 q^{97} -9.70897 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 14 q^{4} + 10 q^{11} + 8 q^{14} + 42 q^{16} + 16 q^{19} + 46 q^{26} + 6 q^{29} + 22 q^{31} + 20 q^{34} + 2 q^{44} - 44 q^{46} - 2 q^{49} - 16 q^{56} - 60 q^{59} + 12 q^{61} + 38 q^{64} - 12 q^{71}+ \cdots + 2 q^{94}+O(q^{100}) \) 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

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