LMFDB - Embedded newform 6525.2.a.bn.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6525,2,Mod(1,6525)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$52.1023873189$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.331312.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 7x^{3} - 4x^{2} + 7x + 4 $ x^5 - 7x^3 - 4x^2 + 7*x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 435)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.05608$ 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)

$f(q)$	$=$	$q-2.45441 q^{2} +4.02413 q^{4} +2.54244 q^{7} -4.96805 q^{8} -1.37105 q^{11} -0.542444 q^{13} -6.24020 q^{14} +4.14537 q^{16} -0.0592276 q^{17} -0.567929 q^{19} +3.36512 q^{22} -8.45127 q^{23} +1.33138 q^{26} +10.2311 q^{28} +1.00000 q^{29} +7.33003 q^{31} -0.238340 q^{32} +0.145369 q^{34} -5.86600 q^{37} +1.39393 q^{38} +4.87876 q^{41} -7.46121 q^{43} -5.51729 q^{44} +20.7429 q^{46} -10.4950 q^{47} -0.535977 q^{49} -2.18287 q^{52} -7.49495 q^{53} -12.6310 q^{56} -2.45441 q^{58} +2.76839 q^{59} +0.0372209 q^{61} -17.9909 q^{62} -7.70575 q^{64} +12.5232 q^{67} -0.238340 q^{68} +2.45576 q^{71} +8.17238 q^{73} +14.3976 q^{74} -2.28542 q^{76} -3.48582 q^{77} +7.65282 q^{79} -11.9745 q^{82} +6.83127 q^{83} +18.3129 q^{86} +6.81145 q^{88} +10.5888 q^{89} -1.37913 q^{91} -34.0090 q^{92} +25.7589 q^{94} +3.11684 q^{97} +1.31551 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 8 q^{4} + 4 q^{7} - 18 q^{8} + 5 q^{11} + 6 q^{13} + 8 q^{14} + 14 q^{16} - 14 q^{17} + 2 q^{19} + 8 q^{22} - 13 q^{23} - 12 q^{26} - 4 q^{28} + 5 q^{29} + 2 q^{31} - 18 q^{32} - 6 q^{34}+ \cdots + 6 q^{98}+O(q^{100}) $ 5 * q - 2 * q^2 + 8 * q^4 + 4 * q^7 - 18 * q^8 + 5 * q^11 + 6 * q^13 + 8 * q^14 + 14 * q^16 - 14 * q^17 + 2 * q^19 + 8 * q^22 - 13 * q^23 - 12 * q^26 - 4 * q^28 + 5 * q^29 + 2 * q^31 - 18 * q^32 - 6 * q^34 + 17 * q^37 - 14 * q^38 + 19 * q^41 + 7 * q^43 + 22 * q^44 + 30 * q^46 - 18 * q^47 + 9 * q^49 + 20 * q^52 - 3 * q^53 + 16 * q^56 - 2 * q^58 + 22 * q^59 - 8 * q^61 + 4 * q^62 + 8 * q^64 + 10 * q^67 - 18 * q^68 + 18 * q^71 + 19 * q^73 - 2 * q^74 + 52 * q^76 - 8 * q^77 + 16 * q^79 + 22 * q^82 + 3 * q^83 - 26 * q^88 - 2 * q^89 - 36 * q^91 - 48 * q^92 + 2 * q^94 + 5 * q^97 + 6 * q^98

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 $

$2$	−2.45441	−1.73553	−0.867765	−	0.496974i	$-0.834444\pi$
$2$	−2.45441	−1.73553	−0.867765	+	0.496974i	$0.834444\pi$
$3$	0	0
$3$	0	0
$4$	4.02413	2.01207
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.54244	0.960954	0.480477	−	0.877007i	$-0.340464\pi$
$7$	2.54244	0.960954	0.480477	+	0.877007i	$0.340464\pi$
$8$	−4.96805	−1.75647
$9$	0	0
$10$	0	0
$11$	−1.37105	−0.413388	−0.206694	−	0.978406i	$-0.566270\pi$
$11$	−1.37105	−0.413388	−0.206694	+	0.978406i	$0.566270\pi$
$12$	0	0
$13$	−0.542444	−0.150447	−0.0752235	−	0.997167i	$-0.523967\pi$
$13$	−0.542444	−0.150447	−0.0752235	+	0.997167i	$0.523967\pi$
$14$	−6.24020	−1.66776
$15$	0	0
$16$	4.14537	1.03634
$17$	−0.0592276	−0.0143648	−0.00718240	−	0.999974i	$-0.502286\pi$
$17$	−0.0592276	−0.0143648	−0.00718240	+	0.999974i	$0.502286\pi$
$18$	0	0
$19$	−0.567929	−0.130292	−0.0651459	−	0.997876i	$-0.520751\pi$
$19$	−0.567929	−0.130292	−0.0651459	+	0.997876i	$0.520751\pi$
$20$	0	0
$21$	0	0
$22$	3.36512	0.717447
$23$	−8.45127	−1.76221	−0.881105	−	0.472920i	$-0.843200\pi$
$23$	−8.45127	−1.76221	−0.881105	+	0.472920i	$0.843200\pi$
$24$	0	0
$25$	0	0
$26$	1.33138	0.261105
$27$	0	0
$28$	10.2311	1.93350
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	7.33003	1.31651	0.658256	−	0.752794i	$-0.271294\pi$
$31$	7.33003	1.31651	0.658256	+	0.752794i	$0.271294\pi$
$32$	−0.238340	−0.0421329
$33$	0	0
$34$	0.145369	0.0249306
$35$	0	0
$36$	0	0
$37$	−5.86600	−0.964365	−0.482183	−	0.876071i	$-0.660156\pi$
$37$	−5.86600	−0.964365	−0.482183	+	0.876071i	$0.660156\pi$
$38$	1.39393	0.226125
$39$	0	0
$40$	0	0
$41$	4.87876	0.761935	0.380967	−	0.924589i	$-0.375591\pi$
$41$	4.87876	0.761935	0.380967	+	0.924589i	$0.375591\pi$
$42$	0	0
$43$	−7.46121	−1.13782	−0.568912	−	0.822398i	$-0.692635\pi$
$43$	−7.46121	−1.13782	−0.568912	+	0.822398i	$0.692635\pi$
$44$	−5.51729	−0.831763
$45$	0	0
$46$	20.7429	3.05837
$47$	−10.4950	−1.53085	−0.765423	−	0.643527i	$-0.777470\pi$
$47$	−10.4950	−1.53085	−0.765423	+	0.643527i	$0.777470\pi$
$48$	0	0
$49$	−0.535977	−0.0765682
$50$	0	0
$51$	0	0
$52$	−2.18287	−0.302709
$53$	−7.49495	−1.02951	−0.514755	−	0.857337i	$-0.672117\pi$
$53$	−7.49495	−1.02951	−0.514755	+	0.857337i	$0.672117\pi$
$54$	0	0
$55$	0	0
$56$	−12.6310	−1.68789
$57$	0	0
$58$	−2.45441	−0.322280
$59$	2.76839	0.360414	0.180207	−	0.983629i	$-0.442323\pi$
$59$	2.76839	0.360414	0.180207	+	0.983629i	$0.442323\pi$
$60$	0	0
$61$	0.0372209	0.00476564	0.00238282	−	0.999997i	$-0.499242\pi$
$61$	0.0372209	0.00476564	0.00238282	+	0.999997i	$0.499242\pi$
$62$	−17.9909	−2.28485
$63$	0	0
$64$	−7.70575	−0.963219
$65$	0	0
$66$	0	0
$67$	12.5232	1.52995	0.764977	−	0.644057i	$-0.222750\pi$
$67$	12.5232	1.52995	0.764977	+	0.644057i	$0.222750\pi$
$68$	−0.238340	−0.0289029
$69$	0	0
$70$	0	0
$71$	2.45576	0.291446	0.145723	−	0.989325i	$-0.453449\pi$
$71$	2.45576	0.291446	0.145723	+	0.989325i	$0.453449\pi$
$72$	0	0
$73$	8.17238	0.956505	0.478253	−	0.878222i	$-0.341270\pi$
$73$	8.17238	0.956505	0.478253	+	0.878222i	$0.341270\pi$
$74$	14.3976	1.67369
$75$	0	0
$76$	−2.28542	−0.262156
$77$	−3.48582	−0.397246
$78$	0	0
$79$	7.65282	0.861009	0.430505	−	0.902588i	$-0.358336\pi$
$79$	7.65282	0.861009	0.430505	+	0.902588i	$0.358336\pi$
$80$	0	0
$81$	0	0
$82$	−11.9745	−1.32236
$83$	6.83127	0.749829	0.374915	−	0.927059i	$-0.377672\pi$
$83$	6.83127	0.749829	0.374915	+	0.927059i	$0.377672\pi$
$84$	0	0
$85$	0	0
$86$	18.3129	1.97473
$87$	0	0
$88$	6.81145	0.726103
$89$	10.5888	1.12241	0.561206	−	0.827676i	$-0.310338\pi$
$89$	10.5888	1.12241	0.561206	+	0.827676i	$0.310338\pi$
$90$	0	0
$91$	−1.37913	−0.144573
$92$	−34.0090	−3.54568
$93$	0	0
$94$	25.7589	2.65683
$95$	0	0
$96$	0	0
$97$	3.11684	0.316467	0.158234	−	0.987402i	$-0.449420\pi$
$97$	3.11684	0.316467	0.158234	+	0.987402i	$0.449420\pi$
$98$	1.31551	0.132886
$99$	0	0
$100$	0	0
$101$	7.42866	0.739180	0.369590	−	0.929195i	$-0.379498\pi$
$101$	7.42866	0.739180	0.369590	+	0.929195i	$0.379498\pi$
$102$	0	0
$103$	−19.2261	−1.89440	−0.947202	−	0.320639i	$-0.896102\pi$
$103$	−19.2261	−1.89440	−0.947202	+	0.320639i	$0.896102\pi$
$104$	2.69489	0.264256
$105$	0	0
$106$	18.3957	1.78675
$107$	11.9636	1.15656	0.578280	−	0.815838i	$-0.303724\pi$
$107$	11.9636	1.15656	0.578280	+	0.815838i	$0.303724\pi$
$108$	0	0
$109$	9.27888	0.888756	0.444378	−	0.895839i	$-0.353425\pi$
$109$	9.27888	0.888756	0.444378	+	0.895839i	$0.353425\pi$
$110$	0	0
$111$	0	0
$112$	10.5394	0.995877
$113$	−7.26814	−0.683729	−0.341864	−	0.939749i	$-0.611058\pi$
$113$	−7.26814	−0.683729	−0.341864	+	0.939749i	$0.611058\pi$
$114$	0	0
$115$	0	0
$116$	4.02413	0.373631
$117$	0	0
$118$	−6.79476	−0.625509
$119$	−0.150583	−0.0138039
$120$	0	0
$121$	−9.12022	−0.829111
$122$	−0.0913553	−0.00827092
$123$	0	0
$124$	29.4970	2.64891
$125$	0	0
$126$	0	0
$127$	16.0629	1.42535	0.712675	−	0.701494i	$-0.247483\pi$
$127$	16.0629	1.42535	0.712675	+	0.701494i	$0.247483\pi$
$128$	19.3898	1.71383
$129$	0	0
$130$	0	0
$131$	11.7310	1.02494	0.512469	−	0.858705i	$-0.328731\pi$
$131$	11.7310	1.02494	0.512469	+	0.858705i	$0.328731\pi$
$132$	0	0
$133$	−1.44393	−0.125204
$134$	−30.7371	−2.65528
$135$	0	0
$136$	0.294246	0.0252314
$137$	12.5832	1.07506	0.537528	−	0.843246i	$-0.319358\pi$
$137$	12.5832	1.07506	0.537528	+	0.843246i	$0.319358\pi$
$138$	0	0
$139$	−21.5899	−1.83123	−0.915614	−	0.402058i	$-0.868295\pi$
$139$	−21.5899	−1.83123	−0.915614	+	0.402058i	$0.868295\pi$
$140$	0	0
$141$	0	0
$142$	−6.02745	−0.505813
$143$	0.743719	0.0621929
$144$	0	0
$145$	0	0
$146$	−20.0584	−1.66004
$147$	0	0
$148$	−23.6056	−1.94037
$149$	9.80926	0.803606	0.401803	−	0.915726i	$-0.368384\pi$
$149$	9.80926	0.803606	0.401803	+	0.915726i	$0.368384\pi$
$150$	0	0
$151$	23.7777	1.93500	0.967501	−	0.252866i	$-0.0813732\pi$
$151$	23.7777	1.93500	0.967501	+	0.252866i	$0.0813732\pi$
$152$	2.82150	0.228854
$153$	0	0
$154$	8.55564	0.689433
$155$	0	0
$156$	0	0
$157$	8.57059	0.684008	0.342004	−	0.939699i	$-0.388894\pi$
$157$	8.57059	0.684008	0.342004	+	0.939699i	$0.388894\pi$
$158$	−18.7832	−1.49431
$159$	0	0
$160$	0	0
$161$	−21.4869	−1.69340
$162$	0	0
$163$	22.1394	1.73409	0.867046	−	0.498228i	$-0.166016\pi$
$163$	22.1394	1.73409	0.867046	+	0.498228i	$0.166016\pi$
$164$	19.6328	1.53306
$165$	0	0
$166$	−16.7667	−1.30135
$167$	22.0420	1.70566	0.852829	−	0.522189i	$-0.174885\pi$
$167$	22.0420	1.70566	0.852829	+	0.522189i	$0.174885\pi$
$168$	0	0
$169$	−12.7058	−0.977366
$170$	0	0
$171$	0	0
$172$	−30.0249	−2.28938
$173$	−7.59608	−0.577519	−0.288760	−	0.957402i	$-0.593243\pi$
$173$	−7.59608	−0.577519	−0.288760	+	0.957402i	$0.593243\pi$
$174$	0	0
$175$	0	0
$176$	−5.68352	−0.428411
$177$	0	0
$178$	−25.9893	−1.94798
$179$	16.9100	1.26391	0.631956	−	0.775004i	$-0.282252\pi$
$179$	16.9100	1.26391	0.631956	+	0.775004i	$0.282252\pi$
$180$	0	0
$181$	−0.792853	−0.0589323	−0.0294661	−	0.999566i	$-0.509381\pi$
$181$	−0.792853	−0.0589323	−0.0294661	+	0.999566i	$0.509381\pi$
$182$	3.38496	0.250910
$183$	0	0
$184$	41.9863	3.09527
$185$	0	0
$186$	0	0
$187$	0.0812041	0.00593824
$188$	−42.2331	−3.08016
$189$	0	0
$190$	0	0
$191$	−17.0839	−1.23615	−0.618073	−	0.786121i	$-0.712086\pi$
$191$	−17.0839	−1.23615	−0.618073	+	0.786121i	$0.712086\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−7.65000	−0.549238
$195$	0	0
$196$	−2.15684	−0.154060
$197$	−3.76371	−0.268154	−0.134077	−	0.990971i	$-0.542807\pi$
$197$	−3.76371	−0.268154	−0.134077	+	0.990971i	$0.542807\pi$
$198$	0	0
$199$	−8.21249	−0.582168	−0.291084	−	0.956697i	$-0.594016\pi$
$199$	−8.21249	−0.582168	−0.291084	+	0.956697i	$0.594016\pi$
$200$	0	0
$201$	0	0
$202$	−18.2330	−1.28287
$203$	2.54244	0.178445
$204$	0	0
$205$	0	0
$206$	47.1887	3.28779
$207$	0	0
$208$	−2.24863	−0.155915
$209$	0.778660	0.0538610
$210$	0	0
$211$	−17.9980	−1.23903	−0.619515	−	0.784985i	$-0.712671\pi$
$211$	−17.9980	−1.23903	−0.619515	+	0.784985i	$0.712671\pi$
$212$	−30.1607	−2.07144
$213$	0	0
$214$	−29.3635	−2.00725
$215$	0	0
$216$	0	0
$217$	18.6362	1.26511
$218$	−22.7742	−1.54246
$219$	0	0
$220$	0	0
$221$	0.0321277	0.00216114
$222$	0	0
$223$	−5.81526	−0.389418	−0.194709	−	0.980861i	$-0.562376\pi$
$223$	−5.81526	−0.389418	−0.194709	+	0.980861i	$0.562376\pi$
$224$	−0.605965	−0.0404878
$225$	0	0
$226$	17.8390	1.18663
$227$	15.8357	1.05105	0.525527	−	0.850777i	$-0.323868\pi$
$227$	15.8357	1.05105	0.525527	+	0.850777i	$0.323868\pi$
$228$	0	0
$229$	7.89877	0.521965	0.260983	−	0.965343i	$-0.415953\pi$
$229$	7.89877	0.521965	0.260983	+	0.965343i	$0.415953\pi$
$230$	0	0
$231$	0	0
$232$	−4.96805	−0.326168
$233$	−8.85019	−0.579795	−0.289898	−	0.957058i	$-0.593621\pi$
$233$	−8.85019	−0.579795	−0.289898	+	0.957058i	$0.593621\pi$
$234$	0	0
$235$	0	0
$236$	11.1404	0.725176
$237$	0	0
$238$	0.369592	0.0239571
$239$	12.7957	0.827687	0.413844	−	0.910348i	$-0.364186\pi$
$239$	12.7957	0.827687	0.413844	+	0.910348i	$0.364186\pi$
$240$	0	0
$241$	−2.32107	−0.149513	−0.0747567	−	0.997202i	$-0.523818\pi$
$241$	−2.32107	−0.149513	−0.0747567	+	0.997202i	$0.523818\pi$
$242$	22.3848	1.43895
$243$	0	0
$244$	0.149782	0.00958879
$245$	0	0
$246$	0	0
$247$	0.308070	0.0196020
$248$	−36.4159	−2.31241
$249$	0	0
$250$	0	0
$251$	23.5133	1.48414	0.742072	−	0.670320i	$-0.233843\pi$
$251$	23.5133	1.48414	0.742072	+	0.670320i	$0.233843\pi$
$252$	0	0
$253$	11.5871	0.728476
$254$	−39.4249	−2.47374
$255$	0	0
$256$	−32.1789	−2.01118
$257$	25.8267	1.61103	0.805514	−	0.592577i	$-0.201889\pi$
$257$	25.8267	1.61103	0.805514	+	0.592577i	$0.201889\pi$
$258$	0	0
$259$	−14.9140	−0.926710
$260$	0	0
$261$	0	0
$262$	−28.7926	−1.77881
$263$	−0.151278	−0.00932820	−0.00466410	−	0.999989i	$-0.501485\pi$
$263$	−0.151278	−0.00932820	−0.00466410	+	0.999989i	$0.501485\pi$
$264$	0	0
$265$	0	0
$266$	3.54399	0.217296
$267$	0	0
$268$	50.3951	3.07837
$269$	16.8704	1.02861	0.514303	−	0.857608i	$-0.328051\pi$
$269$	16.8704	1.02861	0.514303	+	0.857608i	$0.328051\pi$
$270$	0	0
$271$	−12.2024	−0.741243	−0.370621	−	0.928784i	$-0.620855\pi$
$271$	−12.2024	−0.741243	−0.370621	+	0.928784i	$0.620855\pi$
$272$	−0.245520	−0.0148869
$273$	0	0
$274$	−30.8844	−1.86579
$275$	0	0
$276$	0	0
$277$	−19.0681	−1.14569	−0.572844	−	0.819664i	$-0.694160\pi$
$277$	−19.0681	−1.14569	−0.572844	+	0.819664i	$0.694160\pi$
$278$	52.9904	3.17815
$279$	0	0
$280$	0	0
$281$	22.3949	1.33597	0.667983	−	0.744176i	$-0.267158\pi$
$281$	22.3949	1.33597	0.667983	+	0.744176i	$0.267158\pi$
$282$	0	0
$283$	−6.66635	−0.396273	−0.198137	−	0.980174i	$-0.563489\pi$
$283$	−6.66635	−0.396273	−0.198137	+	0.980174i	$0.563489\pi$
$284$	9.88232	0.586408
$285$	0	0
$286$	−1.82539	−0.107938
$287$	12.4040	0.732184
$288$	0	0
$289$	−16.9965	−0.999794
$290$	0	0
$291$	0	0
$292$	32.8867	1.92455
$293$	−3.89339	−0.227454	−0.113727	−	0.993512i	$-0.536279\pi$
$293$	−3.89339	−0.227454	−0.113727	+	0.993512i	$0.536279\pi$
$294$	0	0
$295$	0	0
$296$	29.1426	1.69388
$297$	0	0
$298$	−24.0760	−1.39468
$299$	4.58434	0.265119
$300$	0	0
$301$	−18.9697	−1.09340
$302$	−58.3603	−3.35826
$303$	0	0
$304$	−2.35427	−0.135027
$305$	0	0
$306$	0	0
$307$	−11.1295	−0.635192	−0.317596	−	0.948226i	$-0.602876\pi$
$307$	−11.1295	−0.635192	−0.317596	+	0.948226i	$0.602876\pi$
$308$	−14.0274	−0.799286
$309$	0	0
$310$	0	0
$311$	4.62266	0.262127	0.131063	−	0.991374i	$-0.458161\pi$
$311$	4.62266	0.262127	0.131063	+	0.991374i	$0.458161\pi$
$312$	0	0
$313$	−4.85936	−0.274667	−0.137334	−	0.990525i	$-0.543853\pi$
$313$	−4.85936	−0.274667	−0.137334	+	0.990525i	$0.543853\pi$
$314$	−21.0358	−1.18712
$315$	0	0
$316$	30.7959	1.73241
$317$	−21.3076	−1.19675	−0.598377	−	0.801215i	$-0.704188\pi$
$317$	−21.3076	−1.19675	−0.598377	+	0.801215i	$0.704188\pi$
$318$	0	0
$319$	−1.37105	−0.0767642
$320$	0	0
$321$	0	0
$322$	52.7376	2.93895
$323$	0.0336371	0.00187162
$324$	0	0
$325$	0	0
$326$	−54.3392	−3.00957
$327$	0	0
$328$	−24.2379	−1.33832
$329$	−26.6828	−1.47107
$330$	0	0
$331$	5.36806	0.295055	0.147528	−	0.989058i	$-0.452868\pi$
$331$	5.36806	0.295055	0.147528	+	0.989058i	$0.452868\pi$
$332$	27.4899	1.50871
$333$	0	0
$334$	−54.1000	−2.96022
$335$	0	0
$336$	0	0
$337$	31.3599	1.70828	0.854142	−	0.520039i	$-0.174083\pi$
$337$	31.3599	1.70828	0.854142	+	0.520039i	$0.174083\pi$
$338$	31.1851	1.69625
$339$	0	0
$340$	0	0
$341$	−10.0498	−0.544230
$342$	0	0
$343$	−19.1598	−1.03453
$344$	37.0677	1.99855
$345$	0	0
$346$	18.6439	1.00230
$347$	−9.46310	−0.508006	−0.254003	−	0.967203i	$-0.581747\pi$
$347$	−9.46310	−0.508006	−0.254003	+	0.967203i	$0.581747\pi$
$348$	0	0
$349$	3.35186	0.179421	0.0897104	−	0.995968i	$-0.471406\pi$
$349$	3.35186	0.179421	0.0897104	+	0.995968i	$0.471406\pi$
$350$	0	0
$351$	0	0
$352$	0.326776	0.0174172
$353$	−22.6945	−1.20790	−0.603952	−	0.797021i	$-0.706408\pi$
$353$	−22.6945	−1.20790	−0.603952	+	0.797021i	$0.706408\pi$
$354$	0	0
$355$	0	0
$356$	42.6108	2.25837
$357$	0	0
$358$	−41.5041	−2.19356
$359$	−27.3201	−1.44190	−0.720949	−	0.692989i	$-0.756294\pi$
$359$	−27.3201	−1.44190	−0.720949	+	0.692989i	$0.756294\pi$
$360$	0	0
$361$	−18.6775	−0.983024
$362$	1.94599	0.102279
$363$	0	0
$364$	−5.54982	−0.290889
$365$	0	0
$366$	0	0
$367$	1.10003	0.0574212	0.0287106	−	0.999588i	$-0.490860\pi$
$367$	1.10003	0.0574212	0.0287106	+	0.999588i	$0.490860\pi$
$368$	−35.0336	−1.82625
$369$	0	0
$370$	0	0
$371$	−19.0555	−0.989312
$372$	0	0
$373$	11.2827	0.584195	0.292098	−	0.956388i	$-0.405647\pi$
$373$	11.2827	0.584195	0.292098	+	0.956388i	$0.405647\pi$
$374$	−0.199308	−0.0103060
$375$	0	0
$376$	52.1394	2.68889
$377$	−0.542444	−0.0279373
$378$	0	0
$379$	−2.56241	−0.131622	−0.0658111	−	0.997832i	$-0.520963\pi$
$379$	−2.56241	−0.131622	−0.0658111	+	0.997832i	$0.520963\pi$
$380$	0	0
$381$	0	0
$382$	41.9308	2.14537
$383$	−28.6218	−1.46251	−0.731253	−	0.682106i	$-0.761064\pi$
$383$	−28.6218	−1.46251	−0.731253	+	0.682106i	$0.761064\pi$
$384$	0	0
$385$	0	0
$386$	−34.3617	−1.74897
$387$	0	0
$388$	12.5426	0.636753
$389$	3.33465	0.169073	0.0845366	−	0.996420i	$-0.473059\pi$
$389$	3.33465	0.169073	0.0845366	+	0.996420i	$0.473059\pi$
$390$	0	0
$391$	0.500548	0.0253138
$392$	2.66276	0.134490
$393$	0	0
$394$	9.23770	0.465389
$395$	0	0
$396$	0	0
$397$	4.53377	0.227543	0.113772	−	0.993507i	$-0.463707\pi$
$397$	4.53377	0.227543	0.113772	+	0.993507i	$0.463707\pi$
$398$	20.1568	1.01037
$399$	0	0
$400$	0	0
$401$	23.6097	1.17901	0.589506	−	0.807764i	$-0.299323\pi$
$401$	23.6097	1.17901	0.589506	+	0.807764i	$0.299323\pi$
$402$	0	0
$403$	−3.97613	−0.198065
$404$	29.8939	1.48728
$405$	0	0
$406$	−6.24020	−0.309696
$407$	8.04260	0.398657
$408$	0	0
$409$	13.1652	0.650977	0.325489	−	0.945546i	$-0.394471\pi$
$409$	13.1652	0.650977	0.325489	+	0.945546i	$0.394471\pi$
$410$	0	0
$411$	0	0
$412$	−77.3683	−3.81166
$413$	7.03848	0.346341
$414$	0	0
$415$	0	0
$416$	0.129286	0.00633877
$417$	0	0
$418$	−1.91115	−0.0934774
$419$	24.6158	1.20256	0.601280	−	0.799039i	$-0.294658\pi$
$419$	24.6158	1.20256	0.601280	+	0.799039i	$0.294658\pi$
$420$	0	0
$421$	−8.07455	−0.393529	−0.196765	−	0.980451i	$-0.563044\pi$
$421$	−8.07455	−0.393529	−0.196765	+	0.980451i	$0.563044\pi$
$422$	44.1744	2.15038
$423$	0	0
$424$	37.2353	1.80831
$425$	0	0
$426$	0	0
$427$	0.0946320	0.00457956
$428$	48.1429	2.32707
$429$	0	0
$430$	0	0
$431$	−29.8230	−1.43652	−0.718261	−	0.695774i	$-0.755062\pi$
$431$	−29.8230	−1.43652	−0.718261	+	0.695774i	$0.755062\pi$
$432$	0	0
$433$	31.3189	1.50509	0.752545	−	0.658541i	$-0.228826\pi$
$433$	31.3189	1.50509	0.752545	+	0.658541i	$0.228826\pi$
$434$	−45.7409	−2.19563
$435$	0	0
$436$	37.3394	1.78823
$437$	4.79972	0.229602
$438$	0	0
$439$	6.80628	0.324846	0.162423	−	0.986721i	$-0.448069\pi$
$439$	6.80628	0.324846	0.162423	+	0.986721i	$0.448069\pi$
$440$	0	0
$441$	0	0
$442$	−0.0788545	−0.00375073
$443$	33.3117	1.58268	0.791342	−	0.611374i	$-0.209383\pi$
$443$	33.3117	1.58268	0.791342	+	0.611374i	$0.209383\pi$
$444$	0	0
$445$	0	0
$446$	14.2730	0.675847
$447$	0	0
$448$	−19.5915	−0.925609
$449$	−25.4825	−1.20259	−0.601297	−	0.799026i	$-0.705349\pi$
$449$	−25.4825	−1.20259	−0.601297	+	0.799026i	$0.705349\pi$
$450$	0	0
$451$	−6.68904	−0.314974
$452$	−29.2479	−1.37571
$453$	0	0
$454$	−38.8674	−1.82414
$455$	0	0
$456$	0	0
$457$	−25.1265	−1.17537	−0.587685	−	0.809090i	$-0.699961\pi$
$457$	−25.1265	−1.17537	−0.587685	+	0.809090i	$0.699961\pi$
$458$	−19.3868	−0.905887
$459$	0	0
$460$	0	0
$461$	−31.0592	−1.44657	−0.723286	−	0.690548i	$-0.757369\pi$
$461$	−31.0592	−1.44657	−0.723286	+	0.690548i	$0.757369\pi$
$462$	0	0
$463$	23.6230	1.09786	0.548928	−	0.835870i	$-0.315036\pi$
$463$	23.6230	1.09786	0.548928	+	0.835870i	$0.315036\pi$
$464$	4.14537	0.192444
$465$	0	0
$466$	21.7220	1.00625
$467$	−9.60493	−0.444463	−0.222232	−	0.974994i	$-0.571334\pi$
$467$	−9.60493	−0.444463	−0.222232	+	0.974994i	$0.571334\pi$
$468$	0	0
$469$	31.8396	1.47022
$470$	0	0
$471$	0	0
$472$	−13.7535	−0.633056
$473$	10.2297	0.470362
$474$	0	0
$475$	0	0
$476$	−0.605965	−0.0277744
$477$	0	0
$478$	−31.4060	−1.43648
$479$	26.4221	1.20726	0.603628	−	0.797266i	$-0.293721\pi$
$479$	26.4221	1.20726	0.603628	+	0.797266i	$0.293721\pi$
$480$	0	0
$481$	3.18198	0.145086
$482$	5.69687	0.259485
$483$	0	0
$484$	−36.7009	−1.66822
$485$	0	0
$486$	0	0
$487$	−21.1971	−0.960530	−0.480265	−	0.877123i	$-0.659460\pi$
$487$	−21.1971	−0.960530	−0.480265	+	0.877123i	$0.659460\pi$
$488$	−0.184915	−0.00837071
$489$	0	0
$490$	0	0
$491$	32.8715	1.48347	0.741734	−	0.670694i	$-0.234004\pi$
$491$	32.8715	1.48347	0.741734	+	0.670694i	$0.234004\pi$
$492$	0	0
$493$	−0.0592276	−0.00266748
$494$	−0.756129	−0.0340199
$495$	0	0
$496$	30.3857	1.36436
$497$	6.24364	0.280066
$498$	0	0
$499$	29.7674	1.33257	0.666286	−	0.745696i	$-0.267883\pi$
$499$	29.7674	1.33257	0.666286	+	0.745696i	$0.267883\pi$
$500$	0	0
$501$	0	0
$502$	−57.7112	−2.57578
$503$	1.39376	0.0621447	0.0310723	−	0.999517i	$-0.490108\pi$
$503$	1.39376	0.0621447	0.0310723	+	0.999517i	$0.490108\pi$
$504$	0	0
$505$	0	0
$506$	−28.4396	−1.26429
$507$	0	0
$508$	64.6391	2.86790
$509$	−3.84413	−0.170388	−0.0851940	−	0.996364i	$-0.527151\pi$
$509$	−3.84413	−0.170388	−0.0851940	+	0.996364i	$0.527151\pi$
$510$	0	0
$511$	20.7778	0.919157
$512$	40.2008	1.77664
$513$	0	0
$514$	−63.3894	−2.79599
$515$	0	0
$516$	0	0
$517$	14.3891	0.632833
$518$	36.6051	1.60833
$519$	0	0
$520$	0	0
$521$	−40.2368	−1.76281	−0.881403	−	0.472365i	$-0.843400\pi$
$521$	−40.2368	−1.76281	−0.881403	+	0.472365i	$0.843400\pi$
$522$	0	0
$523$	−7.45091	−0.325806	−0.162903	−	0.986642i	$-0.552086\pi$
$523$	−7.45091	−0.325806	−0.162903	+	0.986642i	$0.552086\pi$
$524$	47.2069	2.06224
$525$	0	0
$526$	0.371298	0.0161894
$527$	−0.434140	−0.0189114
$528$	0	0
$529$	48.4239	2.10539
$530$	0	0
$531$	0	0
$532$	−5.81055	−0.251919
$533$	−2.64646	−0.114631
$534$	0	0
$535$	0	0
$536$	−62.2160	−2.68732
$537$	0	0
$538$	−41.4069	−1.78518
$539$	0.734853	0.0316523
$540$	0	0
$541$	−0.126885	−0.00545520	−0.00272760	−	0.999996i	$-0.500868\pi$
$541$	−0.126885	−0.00545520	−0.00272760	+	0.999996i	$0.500868\pi$
$542$	29.9497	1.28645
$543$	0	0
$544$	0.0141163	0.000605231 0
$545$	0	0
$546$	0	0
$547$	16.5950	0.709549	0.354775	−	0.934952i	$-0.384558\pi$
$547$	16.5950	0.709549	0.354775	+	0.934952i	$0.384558\pi$
$548$	50.6365	2.16309
$549$	0	0
$550$	0	0
$551$	−0.567929	−0.0241946
$552$	0	0
$553$	19.4569	0.827390
$554$	46.8008	1.98838
$555$	0	0
$556$	−86.8804	−3.68455
$557$	4.14372	0.175575	0.0877875	−	0.996139i	$-0.472020\pi$
$557$	4.14372	0.175575	0.0877875	+	0.996139i	$0.472020\pi$
$558$	0	0
$559$	4.04729	0.171182
$560$	0	0
$561$	0	0
$562$	−54.9662	−2.31861
$563$	24.5690	1.03546	0.517730	−	0.855544i	$-0.326777\pi$
$563$	24.5690	1.03546	0.517730	+	0.855544i	$0.326777\pi$
$564$	0	0
$565$	0	0
$566$	16.3619	0.687744
$567$	0	0
$568$	−12.2004	−0.511916
$569$	−5.82478	−0.244187	−0.122094	−	0.992519i	$-0.538961\pi$
$569$	−5.82478	−0.244187	−0.122094	+	0.992519i	$0.538961\pi$
$570$	0	0
$571$	29.3945	1.23012	0.615061	−	0.788480i	$-0.289132\pi$
$571$	29.3945	1.23012	0.615061	+	0.788480i	$0.289132\pi$
$572$	2.99282	0.125136
$573$	0	0
$574$	−30.4445	−1.27073
$575$	0	0
$576$	0	0
$577$	17.4531	0.726583	0.363292	−	0.931675i	$-0.381653\pi$
$577$	17.4531	0.726583	0.363292	+	0.931675i	$0.381653\pi$
$578$	41.7164	1.73517
$579$	0	0
$580$	0	0
$581$	17.3681	0.720551
$582$	0	0
$583$	10.2760	0.425587
$584$	−40.6008	−1.68007
$585$	0	0
$586$	9.55597	0.394754
$587$	−28.4914	−1.17597	−0.587983	−	0.808873i	$-0.700078\pi$
$587$	−28.4914	−1.17597	−0.587983	+	0.808873i	$0.700078\pi$
$588$	0	0
$589$	−4.16293	−0.171531
$590$	0	0
$591$	0	0
$592$	−24.3168	−0.999412
$593$	29.6729	1.21852	0.609259	−	0.792971i	$-0.291467\pi$
$593$	29.6729	1.21852	0.609259	+	0.792971i	$0.291467\pi$
$594$	0	0
$595$	0	0
$596$	39.4738	1.61691
$597$	0	0
$598$	−11.2519	−0.460122
$599$	−23.6421	−0.965990	−0.482995	−	0.875623i	$-0.660451\pi$
$599$	−23.6421	−0.965990	−0.482995	+	0.875623i	$0.660451\pi$
$600$	0	0
$601$	−3.78887	−0.154551	−0.0772757	−	0.997010i	$-0.524622\pi$
$601$	−3.78887	−0.154551	−0.0772757	+	0.997010i	$0.524622\pi$
$602$	46.5595	1.89762
$603$	0	0
$604$	95.6846	3.89335
$605$	0	0
$606$	0	0
$607$	−17.1052	−0.694280	−0.347140	−	0.937813i	$-0.612847\pi$
$607$	−17.1052	−0.694280	−0.347140	+	0.937813i	$0.612847\pi$
$608$	0.135360	0.00548957
$609$	0	0
$610$	0	0
$611$	5.69293	0.230311
$612$	0	0
$613$	−29.7465	−1.20145	−0.600724	−	0.799457i	$-0.705121\pi$
$613$	−29.7465	−1.20145	−0.600724	+	0.799457i	$0.705121\pi$
$614$	27.3163	1.10240
$615$	0	0
$616$	17.3177	0.697752
$617$	41.7590	1.68115	0.840577	−	0.541692i	$-0.182216\pi$
$617$	41.7590	1.68115	0.840577	+	0.541692i	$0.182216\pi$
$618$	0	0
$619$	25.3412	1.01855	0.509274	−	0.860605i	$-0.329914\pi$
$619$	25.3412	1.01855	0.509274	+	0.860605i	$0.329914\pi$
$620$	0	0
$621$	0	0
$622$	−11.3459	−0.454929
$623$	26.9215	1.07859
$624$	0	0
$625$	0	0
$626$	11.9269	0.476694
$627$	0	0
$628$	34.4892	1.37627
$629$	0.347430	0.0138529
$630$	0	0
$631$	−4.21796	−0.167914	−0.0839572	−	0.996469i	$-0.526756\pi$
$631$	−4.21796	−0.167914	−0.0839572	+	0.996469i	$0.526756\pi$
$632$	−38.0196	−1.51234
$633$	0	0
$634$	52.2976	2.07700
$635$	0	0
$636$	0	0
$637$	0.290738	0.0115195
$638$	3.36512	0.133227
$639$	0	0
$640$	0	0
$641$	3.89870	0.153989	0.0769947	−	0.997032i	$-0.475468\pi$
$641$	3.89870	0.153989	0.0769947	+	0.997032i	$0.475468\pi$
$642$	0	0
$643$	37.3426	1.47265	0.736325	−	0.676628i	$-0.236560\pi$
$643$	37.3426	1.47265	0.736325	+	0.676628i	$0.236560\pi$
$644$	−86.4660	−3.40724
$645$	0	0
$646$	−0.0825592	−0.00324825
$647$	−18.2131	−0.716033	−0.358016	−	0.933715i	$-0.616547\pi$
$647$	−18.2131	−0.716033	−0.358016	+	0.933715i	$0.616547\pi$
$648$	0	0
$649$	−3.79561	−0.148991
$650$	0	0
$651$	0	0
$652$	89.0919	3.48911
$653$	11.0156	0.431073	0.215537	−	0.976496i	$-0.430850\pi$
$653$	11.0156	0.431073	0.215537	+	0.976496i	$0.430850\pi$
$654$	0	0
$655$	0	0
$656$	20.2243	0.789625
$657$	0	0
$658$	65.4906	2.55309
$659$	−7.67341	−0.298913	−0.149457	−	0.988768i	$-0.547752\pi$
$659$	−7.67341	−0.298913	−0.149457	+	0.988768i	$0.547752\pi$
$660$	0	0
$661$	27.1870	1.05745	0.528725	−	0.848793i	$-0.322670\pi$
$661$	27.1870	1.05745	0.528725	+	0.848793i	$0.322670\pi$
$662$	−13.1754	−0.512078
$663$	0	0
$664$	−33.9381	−1.31705
$665$	0	0
$666$	0	0
$667$	−8.45127	−0.327234
$668$	88.6998	3.43190
$669$	0	0
$670$	0	0
$671$	−0.0510318	−0.00197006
$672$	0	0
$673$	12.1400	0.467963	0.233981	−	0.972241i	$-0.424825\pi$
$673$	12.1400	0.467963	0.233981	+	0.972241i	$0.424825\pi$
$674$	−76.9702	−2.96478
$675$	0	0
$676$	−51.1296	−1.96652
$677$	28.1655	1.08249	0.541243	−	0.840866i	$-0.317954\pi$
$677$	28.1655	1.08249	0.541243	+	0.840866i	$0.317954\pi$
$678$	0	0
$679$	7.92439	0.304110
$680$	0	0
$681$	0	0
$682$	24.6665	0.944527
$683$	−12.4109	−0.474889	−0.237445	−	0.971401i	$-0.576310\pi$
$683$	−12.4109	−0.474889	−0.237445	+	0.971401i	$0.576310\pi$
$684$	0	0
$685$	0	0
$686$	47.0260	1.79546
$687$	0	0
$688$	−30.9295	−1.17917
$689$	4.06559	0.154887
$690$	0	0
$691$	39.1802	1.49048	0.745242	−	0.666794i	$-0.232334\pi$
$691$	39.1802	1.49048	0.745242	+	0.666794i	$0.232334\pi$
$692$	−30.5676	−1.16201
$693$	0	0
$694$	23.2263	0.881660
$695$	0	0
$696$	0	0
$697$	−0.288958	−0.0109450
$698$	−8.22684	−0.311390
$699$	0	0
$700$	0	0
$701$	−11.4674	−0.433118	−0.216559	−	0.976270i	$-0.569483\pi$
$701$	−11.4674	−0.433118	−0.216559	+	0.976270i	$0.569483\pi$
$702$	0	0
$703$	3.33147	0.125649
$704$	10.5650	0.398183
$705$	0	0
$706$	55.7015	2.09635
$707$	18.8870	0.710317
$708$	0	0
$709$	−0.356326	−0.0133821	−0.00669106	−	0.999978i	$-0.502130\pi$
$709$	−0.356326	−0.0133821	−0.00669106	+	0.999978i	$0.502130\pi$
$710$	0	0
$711$	0	0
$712$	−52.6057	−1.97148
$713$	−61.9480	−2.31997
$714$	0	0
$715$	0	0
$716$	68.0480	2.54307
$717$	0	0
$718$	67.0546	2.50246
$719$	15.7129	0.585992	0.292996	−	0.956114i	$-0.405348\pi$
$719$	15.7129	0.585992	0.292996	+	0.956114i	$0.405348\pi$
$720$	0	0
$721$	−48.8813	−1.82043
$722$	45.8421	1.70607
$723$	0	0
$724$	−3.19054	−0.118576
$725$	0	0
$726$	0	0
$727$	−19.6645	−0.729315	−0.364658	−	0.931142i	$-0.618814\pi$
$727$	−19.6645	−0.729315	−0.364658	+	0.931142i	$0.618814\pi$
$728$	6.85161	0.253937
$729$	0	0
$730$	0	0
$731$	0.441910	0.0163446
$732$	0	0
$733$	49.9707	1.84571	0.922855	−	0.385147i	$-0.125849\pi$
$733$	49.9707	1.84571	0.922855	+	0.385147i	$0.125849\pi$
$734$	−2.69993	−0.0996563
$735$	0	0
$736$	2.01427	0.0742471
$737$	−17.1700	−0.632464
$738$	0	0
$739$	28.3068	1.04128	0.520640	−	0.853776i	$-0.325693\pi$
$739$	28.3068	1.04128	0.520640	+	0.853776i	$0.325693\pi$
$740$	0	0
$741$	0	0
$742$	46.7700	1.71698
$743$	−0.243853	−0.00894609	−0.00447305	−	0.999990i	$-0.501424\pi$
$743$	−0.243853	−0.00894609	−0.00447305	+	0.999990i	$0.501424\pi$
$744$	0	0
$745$	0	0
$746$	−27.6923	−1.01389
$747$	0	0
$748$	0.326776	0.0119481
$749$	30.4167	1.11140
$750$	0	0
$751$	26.9186	0.982275	0.491137	−	0.871082i	$-0.336581\pi$
$751$	26.9186	0.982275	0.491137	+	0.871082i	$0.336581\pi$
$752$	−43.5055	−1.58648
$753$	0	0
$754$	1.33138	0.0484860
$755$	0	0
$756$	0	0
$757$	53.3150	1.93776	0.968882	−	0.247521i	$-0.0796160\pi$
$757$	53.3150	1.93776	0.968882	+	0.247521i	$0.0796160\pi$
$758$	6.28920	0.228434
$759$	0	0
$760$	0	0
$761$	−32.8369	−1.19034	−0.595169	−	0.803601i	$-0.702915\pi$
$761$	−32.8369	−1.19034	−0.595169	+	0.803601i	$0.702915\pi$
$762$	0	0
$763$	23.5910	0.854053
$764$	−68.7477	−2.48721
$765$	0	0
$766$	70.2497	2.53822
$767$	−1.50170	−0.0542231
$768$	0	0
$769$	33.5045	1.20820	0.604101	−	0.796908i	$-0.293532\pi$
$769$	33.5045	1.20820	0.604101	+	0.796908i	$0.293532\pi$
$770$	0	0
$771$	0	0
$772$	56.3378	2.02764
$773$	18.7198	0.673305	0.336653	−	0.941629i	$-0.390705\pi$
$773$	18.7198	0.673305	0.336653	+	0.941629i	$0.390705\pi$
$774$	0	0
$775$	0	0
$776$	−15.4846	−0.555865
$777$	0	0
$778$	−8.18459	−0.293432
$779$	−2.77079	−0.0992738
$780$	0	0
$781$	−3.36698	−0.120480
$782$	−1.22855	−0.0439329
$783$	0	0
$784$	−2.22182	−0.0793509
$785$	0	0
$786$	0	0
$787$	−29.4652	−1.05032	−0.525160	−	0.851003i	$-0.675995\pi$
$787$	−29.4652	−1.05032	−0.525160	+	0.851003i	$0.675995\pi$
$788$	−15.1457	−0.539543
$789$	0	0
$790$	0	0
$791$	−18.4788	−0.657032
$792$	0	0
$793$	−0.0201902	−0.000716977 0
$794$	−11.1277	−0.394908
$795$	0	0
$796$	−33.0481	−1.17136
$797$	−22.9951	−0.814529	−0.407264	−	0.913310i	$-0.633517\pi$
$797$	−22.9951	−0.814529	−0.407264	+	0.913310i	$0.633517\pi$
$798$	0	0
$799$	0.621591	0.0219903
$800$	0	0
$801$	0	0
$802$	−57.9478	−2.04621
$803$	−11.2048	−0.395407
$804$	0	0
$805$	0	0
$806$	9.75906	0.343748
$807$	0	0
$808$	−36.9060	−1.29835
$809$	−3.12325	−0.109808	−0.0549038	−	0.998492i	$-0.517485\pi$
$809$	−3.12325	−0.109808	−0.0549038	+	0.998492i	$0.517485\pi$
$810$	0	0
$811$	6.22609	0.218628	0.109314	−	0.994007i	$-0.465135\pi$
$811$	6.22609	0.218628	0.109314	+	0.994007i	$0.465135\pi$
$812$	10.2311	0.359042
$813$	0	0
$814$	−19.7398	−0.691881
$815$	0	0
$816$	0	0
$817$	4.23743	0.148249
$818$	−32.3128	−1.12979
$819$	0	0
$820$	0	0
$821$	−44.1951	−1.54242	−0.771211	−	0.636580i	$-0.780348\pi$
$821$	−44.1951	−1.54242	−0.771211	+	0.636580i	$0.780348\pi$
$822$	0	0
$823$	6.19976	0.216110	0.108055	−	0.994145i	$-0.465538\pi$
$823$	6.19976	0.216110	0.108055	+	0.994145i	$0.465538\pi$
$824$	95.5162	3.32746
$825$	0	0
$826$	−17.2753	−0.601085
$827$	−21.8002	−0.758066	−0.379033	−	0.925383i	$-0.623743\pi$
$827$	−21.8002	−0.758066	−0.379033	+	0.925383i	$0.623743\pi$
$828$	0	0
$829$	−3.45115	−0.119863	−0.0599316	−	0.998202i	$-0.519088\pi$
$829$	−3.45115	−0.119863	−0.0599316	+	0.998202i	$0.519088\pi$
$830$	0	0
$831$	0	0
$832$	4.17994	0.144913
$833$	0.0317447	0.00109989
$834$	0	0
$835$	0	0
$836$	3.13343	0.108372
$837$	0	0
$838$	−60.4172	−2.08708
$839$	−22.2149	−0.766945	−0.383473	−	0.923552i	$-0.625272\pi$
$839$	−22.2149	−0.766945	−0.383473	+	0.923552i	$0.625272\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	19.8183	0.682982
$843$	0	0
$844$	−72.4261	−2.49301
$845$	0	0
$846$	0	0
$847$	−23.1876	−0.796737
$848$	−31.0693	−1.06693
$849$	0	0
$850$	0	0
$851$	49.5752	1.69941
$852$	0	0
$853$	−19.3683	−0.663157	−0.331578	−	0.943428i	$-0.607581\pi$
$853$	−19.3683	−0.663157	−0.331578	+	0.943428i	$0.607581\pi$
$854$	−0.232266	−0.00794797
$855$	0	0
$856$	−59.4355	−2.03146
$857$	3.29114	0.112423	0.0562117	−	0.998419i	$-0.482098\pi$
$857$	3.29114	0.112423	0.0562117	+	0.998419i	$0.482098\pi$
$858$	0	0
$859$	−34.5765	−1.17974	−0.589868	−	0.807500i	$-0.700820\pi$
$859$	−34.5765	−1.17974	−0.589868	+	0.807500i	$0.700820\pi$
$860$	0	0
$861$	0	0
$862$	73.1978	2.49313
$863$	40.2651	1.37064	0.685320	−	0.728243i	$-0.259663\pi$
$863$	40.2651	1.37064	0.685320	+	0.728243i	$0.259663\pi$
$864$	0	0
$865$	0	0
$866$	−76.8694	−2.61213
$867$	0	0
$868$	74.9945	2.54548
$869$	−10.4924	−0.355931
$870$	0	0
$871$	−6.79315	−0.230177
$872$	−46.0979	−1.56107
$873$	0	0
$874$	−11.7805	−0.398480
$875$	0	0
$876$	0	0
$877$	47.7366	1.61195	0.805975	−	0.591949i	$-0.201641\pi$
$877$	47.7366	1.61195	0.805975	+	0.591949i	$0.201641\pi$
$878$	−16.7054	−0.563780
$879$	0	0
$880$	0	0
$881$	31.8103	1.07172	0.535858	−	0.844308i	$-0.319988\pi$
$881$	31.8103	1.07172	0.535858	+	0.844308i	$0.319988\pi$
$882$	0	0
$883$	6.64800	0.223723	0.111861	−	0.993724i	$-0.464319\pi$
$883$	6.64800	0.223723	0.111861	+	0.993724i	$0.464319\pi$
$884$	0.129286	0.00434836
$885$	0	0
$886$	−81.7605	−2.74680
$887$	40.4341	1.35764	0.678822	−	0.734303i	$-0.262491\pi$
$887$	40.4341	1.35764	0.678822	+	0.734303i	$0.262491\pi$
$888$	0	0
$889$	40.8390	1.36970
$890$	0	0
$891$	0	0
$892$	−23.4014	−0.783535
$893$	5.96038	0.199457
$894$	0	0
$895$	0	0
$896$	49.2974	1.64691
$897$	0	0
$898$	62.5445	2.08714
$899$	7.33003	0.244470
$900$	0	0
$901$	0.443908	0.0147887
$902$	16.4176	0.546648
$903$	0	0
$904$	36.1085	1.20095
$905$	0	0
$906$	0	0
$907$	−35.9572	−1.19394	−0.596969	−	0.802264i	$-0.703629\pi$
$907$	−35.9572	−1.19394	−0.596969	+	0.802264i	$0.703629\pi$
$908$	63.7251	2.11479
$909$	0	0
$910$	0	0
$911$	−11.4775	−0.380265	−0.190132	−	0.981758i	$-0.560892\pi$
$911$	−11.4775	−0.380265	−0.190132	+	0.981758i	$0.560892\pi$
$912$	0	0
$913$	−9.36603	−0.309970
$914$	61.6708	2.03989
$915$	0	0
$916$	31.7857	1.05023
$917$	29.8253	0.984919
$918$	0	0
$919$	18.2956	0.603517	0.301758	−	0.953384i	$-0.402426\pi$
$919$	18.2956	0.603517	0.301758	+	0.953384i	$0.402426\pi$
$920$	0	0
$921$	0	0
$922$	76.2321	2.51057
$923$	−1.33211	−0.0438471
$924$	0	0
$925$	0	0
$926$	−57.9806	−1.90536
$927$	0	0
$928$	−0.238340	−0.00782389
$929$	3.99065	0.130929	0.0654645	−	0.997855i	$-0.479147\pi$
$929$	3.99065	0.130929	0.0654645	+	0.997855i	$0.479147\pi$
$930$	0	0
$931$	0.304397	0.00997621
$932$	−35.6143	−1.16659
$933$	0	0
$934$	23.5744	0.771379
$935$	0	0
$936$	0	0
$937$	−28.6617	−0.936337	−0.468169	−	0.883639i	$-0.655086\pi$
$937$	−28.6617	−0.936337	−0.468169	+	0.883639i	$0.655086\pi$
$938$	−78.1474	−2.55160
$939$	0	0
$940$	0	0
$941$	−3.82741	−0.124770	−0.0623850	−	0.998052i	$-0.519871\pi$
$941$	−3.82741	−0.124770	−0.0623850	+	0.998052i	$0.519871\pi$
$942$	0	0
$943$	−41.2317	−1.34269
$944$	11.4760	0.373512
$945$	0	0
$946$	−25.1079	−0.816328
$947$	7.63690	0.248166	0.124083	−	0.992272i	$-0.460401\pi$
$947$	7.63690	0.248166	0.124083	+	0.992272i	$0.460401\pi$
$948$	0	0
$949$	−4.43306	−0.143903
$950$	0	0
$951$	0	0
$952$	0.748103	0.0242462
$953$	20.4244	0.661610	0.330805	−	0.943699i	$-0.392680\pi$
$953$	20.4244	0.661610	0.330805	+	0.943699i	$0.392680\pi$
$954$	0	0
$955$	0	0
$956$	51.4917	1.66536
$957$	0	0
$958$	−64.8506	−2.09523
$959$	31.9921	1.03308
$960$	0	0
$961$	22.7293	0.733203
$962$	−7.80989	−0.251801
$963$	0	0
$964$	−9.34030	−0.300831
$965$	0	0
$966$	0	0
$967$	−6.70817	−0.215720	−0.107860	−	0.994166i	$-0.534400\pi$
$967$	−6.70817	−0.215720	−0.107860	+	0.994166i	$0.534400\pi$
$968$	45.3097	1.45631
$969$	0	0
$970$	0	0
$971$	−28.0632	−0.900590	−0.450295	−	0.892880i	$-0.648681\pi$
$971$	−28.0632	−0.900590	−0.450295	+	0.892880i	$0.648681\pi$
$972$	0	0
$973$	−54.8910	−1.75973
$974$	52.0263	1.66703
$975$	0	0
$976$	0.154294	0.00493884
$977$	23.0313	0.736835	0.368418	−	0.929660i	$-0.379900\pi$
$977$	23.0313	0.736835	0.368418	+	0.929660i	$0.379900\pi$
$978$	0	0
$979$	−14.5178	−0.463991
$980$	0	0
$981$	0	0
$982$	−80.6800	−2.57460
$983$	−49.7645	−1.58724	−0.793620	−	0.608414i	$-0.791806\pi$
$983$	−49.7645	−1.58724	−0.793620	+	0.608414i	$0.791806\pi$
$984$	0	0
$985$	0	0
$986$	0.145369	0.00462949
$987$	0	0
$988$	1.23971	0.0394405
$989$	63.0567	2.00509
$990$	0	0
$991$	25.8862	0.822303	0.411152	−	0.911567i	$-0.365127\pi$
$991$	25.8862	0.822303	0.411152	+	0.911567i	$0.365127\pi$
$992$	−1.74704	−0.0554685
$993$	0	0
$994$	−15.3245	−0.486062
$995$	0	0
$996$	0	0
$997$	53.1804	1.68424	0.842120	−	0.539290i	$-0.181307\pi$
$997$	53.1804	1.68424	0.842120	+	0.539290i	$0.181307\pi$
$998$	−73.0615	−2.31272
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.bn.1.2		5
3.2	odd	2		2175.2.a.y.1.4		5
5.2	odd	4		1305.2.c.i.784.2		10
5.3	odd	4		1305.2.c.i.784.9		10
5.4	even	2		6525.2.a.br.1.4		5
15.2	even	4		435.2.c.d.349.9	yes	10
15.8	even	4		435.2.c.d.349.2	✓	10
15.14	odd	2		2175.2.a.x.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
435.2.c.d.349.2	✓	10	15.8	even	4
435.2.c.d.349.9	yes	10	15.2	even	4
1305.2.c.i.784.2		10	5.2	odd	4
1305.2.c.i.784.9		10	5.3	odd	4
2175.2.a.x.1.2		5	15.14	odd	2
2175.2.a.y.1.4		5	3.2	odd	2
6525.2.a.bn.1.2		5	1.1	even	1	trivial
6525.2.a.br.1.4		5	5.4	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-2.45441 q^{2} +4.02413 q^{4} +2.54244 q^{7} -4.96805 q^{8} -1.37105 q^{11} -0.542444 q^{13} -6.24020 q^{14} +4.14537 q^{16} -0.0592276 q^{17} -0.567929 q^{19} +3.36512 q^{22} -8.45127 q^{23} +1.33138 q^{26} +10.2311 q^{28} +1.00000 q^{29} +7.33003 q^{31} -0.238340 q^{32} +0.145369 q^{34} -5.86600 q^{37} +1.39393 q^{38} +4.87876 q^{41} -7.46121 q^{43} -5.51729 q^{44} +20.7429 q^{46} -10.4950 q^{47} -0.535977 q^{49} -2.18287 q^{52} -7.49495 q^{53} -12.6310 q^{56} -2.45441 q^{58} +2.76839 q^{59} +0.0372209 q^{61} -17.9909 q^{62} -7.70575 q^{64} +12.5232 q^{67} -0.238340 q^{68} +2.45576 q^{71} +8.17238 q^{73} +14.3976 q^{74} -2.28542 q^{76} -3.48582 q^{77} +7.65282 q^{79} -11.9745 q^{82} +6.83127 q^{83} +18.3129 q^{86} +6.81145 q^{88} +10.5888 q^{89} -1.37913 q^{91} -34.0090 q^{92} +25.7589 q^{94} +3.11684 q^{97} +1.31551 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 8 q^{4} + 4 q^{7} - 18 q^{8} + 5 q^{11} + 6 q^{13} + 8 q^{14} + 14 q^{16} - 14 q^{17} + 2 q^{19} + 8 q^{22} - 13 q^{23} - 12 q^{26} - 4 q^{28} + 5 q^{29} + 2 q^{31} - 18 q^{32} - 6 q^{34}+ \cdots + 6 q^{98}+O(q^{100}) \) 5 * q - 2 * q^2 + 8 * q^4 + 4 * q^7 - 18 * q^8 + 5 * q^11 + 6 * q^13 + 8 * q^14 + 14 * q^16 - 14 * q^17 + 2 * q^19 + 8 * q^22 - 13 * q^23 - 12 * q^26 - 4 * q^28 + 5 * q^29 + 2 * q^31 - 18 * q^32 - 6 * q^34 + 17 * q^37 - 14 * q^38 + 19 * q^41 + 7 * q^43 + 22 * q^44 + 30 * q^46 - 18 * q^47 + 9 * q^49 + 20 * q^52 - 3 * q^53 + 16 * q^56 - 2 * q^58 + 22 * q^59 - 8 * q^61 + 4 * q^62 + 8 * q^64 + 10 * q^67 - 18 * q^68 + 18 * q^71 + 19 * q^73 - 2 * q^74 + 52 * q^76 - 8 * q^77 + 16 * q^79 + 22 * q^82 + 3 * q^83 - 26 * q^88 - 2 * q^89 - 36 * q^91 - 48 * q^92 + 2 * q^94 + 5 * q^97 + 6 * q^98

Introduction

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