LMFDB - Embedded newform 8325.2.a.cq.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8325,2,Mod(1,8325)] 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("8325.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8325 = 3^{2} \cdot 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8325.a (trivial)

Newform invariants

comment:select newform

sage:traces = [9,-5,0,11,0,0,8,-15,0,0,0,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$66.4754596827$
Analytic rank:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 4x^{8} - 6x^{7} + 30x^{6} + 15x^{5} - 70x^{4} - 22x^{3} + 44x^{2} + 4x - 4 $ x^9 - 4x^8 - 6x^7 + 30x^6 + 15x^5 - 70x^4 - 22x^3 + 44x^2 + 4x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 185)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$1.97415$ of defining polynomial
Character	$\chi$	$=$	8325.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.974151 q^{2} -1.05103 q^{4} +4.15169 q^{7} -2.97216 q^{8} +1.12356 q^{11} +0.328108 q^{13} +4.04438 q^{14} -0.793280 q^{16} +1.55144 q^{17} -4.32598 q^{19} +1.09452 q^{22} -1.73985 q^{23} +0.319627 q^{26} -4.36355 q^{28} -8.20254 q^{29} -10.1080 q^{31} +5.17155 q^{32} +1.51134 q^{34} -1.00000 q^{37} -4.21416 q^{38} +1.91727 q^{41} +4.22272 q^{43} -1.18089 q^{44} -1.69488 q^{46} -10.7452 q^{47} +10.2365 q^{49} -0.344851 q^{52} +2.11437 q^{53} -12.3395 q^{56} -7.99051 q^{58} -11.5917 q^{59} +9.72982 q^{61} -9.84669 q^{62} +6.62444 q^{64} -5.29915 q^{67} -1.63061 q^{68} -9.90260 q^{71} +8.12992 q^{73} -0.974151 q^{74} +4.54673 q^{76} +4.66467 q^{77} -0.0298662 q^{79} +1.86771 q^{82} -13.8190 q^{83} +4.11357 q^{86} -3.33940 q^{88} +9.37865 q^{89} +1.36220 q^{91} +1.82864 q^{92} -10.4674 q^{94} +10.3113 q^{97} +9.97194 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 5 q^{2} + 11 q^{4} + 8 q^{7} - 15 q^{8} + 6 q^{13} + 4 q^{14} + 11 q^{16} - 18 q^{17} - 4 q^{19} + 6 q^{22} - 16 q^{23} + 6 q^{26} - 20 q^{28} + 2 q^{29} - 6 q^{31} - 35 q^{32} + 6 q^{34} - 9 q^{37}+ \cdots - 21 q^{98}+O(q^{100}) $ 9 * q - 5 * q^2 + 11 * q^4 + 8 * q^7 - 15 * q^8 + 6 * q^13 + 4 * q^14 + 11 * q^16 - 18 * q^17 - 4 * q^19 + 6 * q^22 - 16 * q^23 + 6 * q^26 - 20 * q^28 + 2 * q^29 - 6 * q^31 - 35 * q^32 + 6 * q^34 - 9 * q^37 - 4 * q^38 - 2 * q^41 + 14 * q^43 - 4 * q^44 + 16 * q^46 - 34 * q^47 - 7 * q^49 + 16 * q^52 + 4 * q^53 + 24 * q^56 - 22 * q^58 - 10 * q^59 + 8 * q^61 - 26 * q^62 + 15 * q^64 + 16 * q^67 - 36 * q^68 - 8 * q^71 + 8 * q^73 + 5 * q^74 - 48 * q^76 - 14 * q^77 - 26 * q^79 - 4 * q^82 - 70 * q^83 + 16 * q^86 - 36 * q^88 + 8 * q^89 - 18 * q^91 - 16 * q^92 - 26 * q^94 - 2 * q^97 - 21 * 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 $

$2$	0.974151	0.688829	0.344415	−	0.938818i	$-0.388077\pi$
$2$	0.974151	0.688829	0.344415	+	0.938818i	$0.388077\pi$
$3$	0	0
$3$	0	0
$4$	−1.05103	−0.525515
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.15169	1.56919	0.784596	−	0.620007i	$-0.212870\pi$
$7$	4.15169	1.56919	0.784596	+	0.620007i	$0.212870\pi$
$8$	−2.97216	−1.05082
$9$	0	0
$10$	0	0
$11$	1.12356	0.338766	0.169383	−	0.985550i	$-0.445823\pi$
$11$	1.12356	0.338766	0.169383	+	0.985550i	$0.445823\pi$
$12$	0	0
$13$	0.328108	0.0910008	0.0455004	−	0.998964i	$-0.485512\pi$
$13$	0.328108	0.0910008	0.0455004	+	0.998964i	$0.485512\pi$
$14$	4.04438	1.08091
$15$	0	0
$16$	−0.793280	−0.198320
$17$	1.55144	0.376280	0.188140	−	0.982142i	$-0.439754\pi$
$17$	1.55144	0.376280	0.188140	+	0.982142i	$0.439754\pi$
$18$	0	0
$19$	−4.32598	−0.992449	−0.496224	−	0.868194i	$-0.665281\pi$
$19$	−4.32598	−0.992449	−0.496224	+	0.868194i	$0.665281\pi$
$20$	0	0
$21$	0	0
$22$	1.09452	0.233352
$23$	−1.73985	−0.362785	−0.181392	−	0.983411i	$-0.558060\pi$
$23$	−1.73985	−0.362785	−0.181392	+	0.983411i	$0.558060\pi$
$24$	0	0
$25$	0	0
$26$	0.319627	0.0626840
$27$	0	0
$28$	−4.36355	−0.824633
$29$	−8.20254	−1.52317	−0.761587	−	0.648063i	$-0.775579\pi$
$29$	−8.20254	−1.52317	−0.761587	+	0.648063i	$0.775579\pi$
$30$	0	0
$31$	−10.1080	−1.81545	−0.907723	−	0.419571i	$-0.862181\pi$
$31$	−10.1080	−1.81545	−0.907723	+	0.419571i	$0.862181\pi$
$32$	5.17155	0.914210
$33$	0	0
$34$	1.51134	0.259193
$35$	0	0
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	−4.21416	−0.683627
$39$	0	0
$40$	0	0
$41$	1.91727	0.299428	0.149714	−	0.988729i	$-0.452165\pi$
$41$	1.91727	0.299428	0.149714	+	0.988729i	$0.452165\pi$
$42$	0	0
$43$	4.22272	0.643959	0.321979	−	0.946747i	$-0.395652\pi$
$43$	4.22272	0.643959	0.321979	+	0.946747i	$0.395652\pi$
$44$	−1.18089	−0.178026
$45$	0	0
$46$	−1.69488	−0.249897
$47$	−10.7452	−1.56735	−0.783673	−	0.621174i	$-0.786656\pi$
$47$	−10.7452	−1.56735	−0.783673	+	0.621174i	$0.786656\pi$
$48$	0	0
$49$	10.2365	1.46236
$50$	0	0
$51$	0	0
$52$	−0.344851	−0.0478222
$53$	2.11437	0.290431	0.145215	−	0.989400i	$-0.453612\pi$
$53$	2.11437	0.290431	0.145215	+	0.989400i	$0.453612\pi$
$54$	0	0
$55$	0	0
$56$	−12.3395	−1.64894
$57$	0	0
$58$	−7.99051	−1.04921
$59$	−11.5917	−1.50911	−0.754557	−	0.656234i	$-0.772148\pi$
$59$	−11.5917	−1.50911	−0.754557	+	0.656234i	$0.772148\pi$
$60$	0	0
$61$	9.72982	1.24578	0.622888	−	0.782311i	$-0.285959\pi$
$61$	9.72982	1.24578	0.622888	+	0.782311i	$0.285959\pi$
$62$	−9.84669	−1.25053
$63$	0	0
$64$	6.62444	0.828054
$65$	0	0
$66$	0	0
$67$	−5.29915	−0.647395	−0.323697	−	0.946161i	$-0.604926\pi$
$67$	−5.29915	−0.647395	−0.323697	+	0.946161i	$0.604926\pi$
$68$	−1.63061	−0.197741
$69$	0	0
$70$	0	0
$71$	−9.90260	−1.17522	−0.587611	−	0.809144i	$-0.699931\pi$
$71$	−9.90260	−1.17522	−0.587611	+	0.809144i	$0.699931\pi$
$72$	0	0
$73$	8.12992	0.951535	0.475767	−	0.879571i	$-0.342170\pi$
$73$	8.12992	0.951535	0.475767	+	0.879571i	$0.342170\pi$
$74$	−0.974151	−0.113243
$75$	0	0
$76$	4.54673	0.521546
$77$	4.66467	0.531588
$78$	0	0
$79$	−0.0298662	−0.00336021	−0.00168011	−	0.999999i	$-0.500535\pi$
$79$	−0.0298662	−0.00336021	−0.00168011	+	0.999999i	$0.500535\pi$
$80$	0	0
$81$	0	0
$82$	1.86771	0.206255
$83$	−13.8190	−1.51683	−0.758416	−	0.651770i	$-0.774027\pi$
$83$	−13.8190	−1.51683	−0.758416	+	0.651770i	$0.774027\pi$
$84$	0	0
$85$	0	0
$86$	4.11357	0.443577
$87$	0	0
$88$	−3.33940	−0.355981
$89$	9.37865	0.994135	0.497067	−	0.867712i	$-0.334410\pi$
$89$	9.37865	0.994135	0.497067	+	0.867712i	$0.334410\pi$
$90$	0	0
$91$	1.36220	0.142798
$92$	1.82864	0.190649
$93$	0	0
$94$	−10.4674	−1.07963
$95$	0	0
$96$	0	0
$97$	10.3113	1.04695	0.523475	−	0.852041i	$-0.324635\pi$
$97$	10.3113	1.04695	0.523475	+	0.852041i	$0.324635\pi$
$98$	9.97194	1.00732
$99$	0	0
$100$	0	0
$101$	−15.0371	−1.49625	−0.748124	−	0.663559i	$-0.769045\pi$
$101$	−15.0371	−1.49625	−0.748124	+	0.663559i	$0.769045\pi$
$102$	0	0
$103$	−5.60494	−0.552271	−0.276135	−	0.961119i	$-0.589054\pi$
$103$	−5.60494	−0.552271	−0.276135	+	0.961119i	$0.589054\pi$
$104$	−0.975190	−0.0956253
$105$	0	0
$106$	2.05971	0.200057
$107$	−4.24636	−0.410511	−0.205255	−	0.978708i	$-0.565803\pi$
$107$	−4.24636	−0.410511	−0.205255	+	0.978708i	$0.565803\pi$
$108$	0	0
$109$	−3.14147	−0.300898	−0.150449	−	0.988618i	$-0.548072\pi$
$109$	−3.14147	−0.300898	−0.150449	+	0.988618i	$0.548072\pi$
$110$	0	0
$111$	0	0
$112$	−3.29345	−0.311202
$113$	8.26928	0.777908	0.388954	−	0.921257i	$-0.372836\pi$
$113$	8.26928	0.777908	0.388954	+	0.921257i	$0.372836\pi$
$114$	0	0
$115$	0	0
$116$	8.62111	0.800450
$117$	0	0
$118$	−11.2921	−1.03952
$119$	6.44111	0.590456
$120$	0	0
$121$	−9.73762	−0.885238
$122$	9.47831	0.858126
$123$	0	0
$124$	10.6238	0.954043
$125$	0	0
$126$	0	0
$127$	6.23393	0.553171	0.276586	−	0.960989i	$-0.410797\pi$
$127$	6.23393	0.553171	0.276586	+	0.960989i	$0.410797\pi$
$128$	−3.88990	−0.343822
$129$	0	0
$130$	0	0
$131$	−5.84789	−0.510933	−0.255466	−	0.966818i	$-0.582229\pi$
$131$	−5.84789	−0.510933	−0.255466	+	0.966818i	$0.582229\pi$
$132$	0	0
$133$	−17.9601	−1.55734
$134$	−5.16218	−0.445944
$135$	0	0
$136$	−4.61114	−0.395402
$137$	11.7743	1.00594	0.502971	−	0.864303i	$-0.332240\pi$
$137$	11.7743	1.00594	0.502971	+	0.864303i	$0.332240\pi$
$138$	0	0
$139$	−21.8186	−1.85063	−0.925316	−	0.379197i	$-0.876200\pi$
$139$	−21.8186	−1.85063	−0.925316	+	0.379197i	$0.876200\pi$
$140$	0	0
$141$	0	0
$142$	−9.64663	−0.809527
$143$	0.368648	0.0308279
$144$	0	0
$145$	0	0
$146$	7.91977	0.655445
$147$	0	0
$148$	1.05103	0.0863941
$149$	−8.13563	−0.666497	−0.333249	−	0.942839i	$-0.608145\pi$
$149$	−8.13563	−0.666497	−0.333249	+	0.942839i	$0.608145\pi$
$150$	0	0
$151$	6.24705	0.508377	0.254189	−	0.967155i	$-0.418192\pi$
$151$	6.24705	0.508377	0.254189	+	0.967155i	$0.418192\pi$
$152$	12.8575	1.04288
$153$	0	0
$154$	4.54409	0.366173
$155$	0	0
$156$	0	0
$157$	21.8540	1.74414	0.872068	−	0.489385i	$-0.162779\pi$
$157$	21.8540	1.74414	0.872068	+	0.489385i	$0.162779\pi$
$158$	−0.0290942	−0.00231461
$159$	0	0
$160$	0	0
$161$	−7.22334	−0.569279
$162$	0	0
$163$	13.1285	1.02830	0.514152	−	0.857699i	$-0.328107\pi$
$163$	13.1285	1.02830	0.514152	+	0.857699i	$0.328107\pi$
$164$	−2.01511	−0.157354
$165$	0	0
$166$	−13.4618	−1.04484
$167$	−14.0706	−1.08881	−0.544407	−	0.838821i	$-0.683245\pi$
$167$	−14.0706	−1.08881	−0.544407	+	0.838821i	$0.683245\pi$
$168$	0	0
$169$	−12.8923	−0.991719
$170$	0	0
$171$	0	0
$172$	−4.43820	−0.338410
$173$	−1.19192	−0.0906202	−0.0453101	−	0.998973i	$-0.514428\pi$
$173$	−1.19192	−0.0906202	−0.0453101	+	0.998973i	$0.514428\pi$
$174$	0	0
$175$	0	0
$176$	−0.891296	−0.0671840
$177$	0	0
$178$	9.13622	0.684789
$179$	2.81996	0.210774	0.105387	−	0.994431i	$-0.466392\pi$
$179$	2.81996	0.210774	0.105387	+	0.994431i	$0.466392\pi$
$180$	0	0
$181$	11.7419	0.872772	0.436386	−	0.899760i	$-0.356258\pi$
$181$	11.7419	0.872772	0.436386	+	0.899760i	$0.356258\pi$
$182$	1.32699	0.0983632
$183$	0	0
$184$	5.17113	0.381221
$185$	0	0
$186$	0	0
$187$	1.74314	0.127471
$188$	11.2935	0.823663
$189$	0	0
$190$	0	0
$191$	14.9337	1.08056	0.540281	−	0.841485i	$-0.318318\pi$
$191$	14.9337	1.08056	0.540281	+	0.841485i	$0.318318\pi$
$192$	0	0
$193$	11.8961	0.856301	0.428150	−	0.903707i	$-0.359165\pi$
$193$	11.8961	0.856301	0.428150	+	0.903707i	$0.359165\pi$
$194$	10.0447	0.721169
$195$	0	0
$196$	−10.7589	−0.768493
$197$	0.202721	0.0144433	0.00722163	−	0.999974i	$-0.497701\pi$
$197$	0.202721	0.0144433	0.00722163	+	0.999974i	$0.497701\pi$
$198$	0	0
$199$	22.8466	1.61955	0.809775	−	0.586740i	$-0.199589\pi$
$199$	22.8466	1.61955	0.809775	+	0.586740i	$0.199589\pi$
$200$	0	0
$201$	0	0
$202$	−14.6484	−1.03066
$203$	−34.0544	−2.39015
$204$	0	0
$205$	0	0
$206$	−5.46006	−0.380420
$207$	0	0
$208$	−0.260281	−0.0180473
$209$	−4.86050	−0.336207
$210$	0	0
$211$	2.18114	0.150156	0.0750781	−	0.997178i	$-0.476079\pi$
$211$	2.18114	0.150156	0.0750781	+	0.997178i	$0.476079\pi$
$212$	−2.22226	−0.152626
$213$	0	0
$214$	−4.13660	−0.282772
$215$	0	0
$216$	0	0
$217$	−41.9652	−2.84878
$218$	−3.06026	−0.207267
$219$	0	0
$220$	0	0
$221$	0.509041	0.0342418
$222$	0	0
$223$	−0.585984	−0.0392404	−0.0196202	−	0.999808i	$-0.506246\pi$
$223$	−0.585984	−0.0392404	−0.0196202	+	0.999808i	$0.506246\pi$
$224$	21.4707	1.43457
$225$	0	0
$226$	8.05553	0.535846
$227$	−9.34509	−0.620255	−0.310128	−	0.950695i	$-0.600372\pi$
$227$	−9.34509	−0.620255	−0.310128	+	0.950695i	$0.600372\pi$
$228$	0	0
$229$	−7.17597	−0.474201	−0.237101	−	0.971485i	$-0.576197\pi$
$229$	−7.17597	−0.474201	−0.237101	+	0.971485i	$0.576197\pi$
$230$	0	0
$231$	0	0
$232$	24.3793	1.60058
$233$	−3.90299	−0.255693	−0.127847	−	0.991794i	$-0.540807\pi$
$233$	−3.90299	−0.255693	−0.127847	+	0.991794i	$0.540807\pi$
$234$	0	0
$235$	0	0
$236$	12.1832	0.793061
$237$	0	0
$238$	6.27462	0.406723
$239$	−19.4873	−1.26053	−0.630264	−	0.776381i	$-0.717053\pi$
$239$	−19.4873	−1.26053	−0.630264	+	0.776381i	$0.717053\pi$
$240$	0	0
$241$	−17.6828	−1.13905	−0.569524	−	0.821975i	$-0.692872\pi$
$241$	−17.6828	−1.13905	−0.569524	+	0.821975i	$0.692872\pi$
$242$	−9.48591	−0.609778
$243$	0	0
$244$	−10.2263	−0.654673
$245$	0	0
$246$	0	0
$247$	−1.41939	−0.0903136
$248$	30.0426	1.90770
$249$	0	0
$250$	0	0
$251$	−20.5968	−1.30006	−0.650029	−	0.759910i	$-0.725243\pi$
$251$	−20.5968	−1.30006	−0.650029	+	0.759910i	$0.725243\pi$
$252$	0	0
$253$	−1.95483	−0.122899
$254$	6.07279	0.381041
$255$	0	0
$256$	−17.0382	−1.06489
$257$	−4.45425	−0.277848	−0.138924	−	0.990303i	$-0.544364\pi$
$257$	−4.45425	−0.277848	−0.138924	+	0.990303i	$0.544364\pi$
$258$	0	0
$259$	−4.15169	−0.257974
$260$	0	0
$261$	0	0
$262$	−5.69673	−0.351945
$263$	−25.8182	−1.59202	−0.796008	−	0.605286i	$-0.793059\pi$
$263$	−25.8182	−1.59202	−0.796008	+	0.605286i	$0.793059\pi$
$264$	0	0
$265$	0	0
$266$	−17.4959	−1.07274
$267$	0	0
$268$	5.56956	0.340215
$269$	2.98719	0.182132	0.0910660	−	0.995845i	$-0.470973\pi$
$269$	2.98719	0.182132	0.0910660	+	0.995845i	$0.470973\pi$
$270$	0	0
$271$	2.12040	0.128805	0.0644026	−	0.997924i	$-0.479486\pi$
$271$	2.12040	0.128805	0.0644026	+	0.997924i	$0.479486\pi$
$272$	−1.23073	−0.0746239
$273$	0	0
$274$	11.4699	0.692923
$275$	0	0
$276$	0	0
$277$	−17.7165	−1.06448	−0.532241	−	0.846593i	$-0.678650\pi$
$277$	−17.7165	−1.06448	−0.532241	+	0.846593i	$0.678650\pi$
$278$	−21.2546	−1.27477
$279$	0	0
$280$	0	0
$281$	27.8120	1.65912	0.829562	−	0.558415i	$-0.188590\pi$
$281$	27.8120	1.65912	0.829562	+	0.558415i	$0.188590\pi$
$282$	0	0
$283$	−33.2265	−1.97511	−0.987554	−	0.157279i	$-0.949728\pi$
$283$	−33.2265	−1.97511	−0.987554	+	0.157279i	$0.949728\pi$
$284$	10.4079	0.617596
$285$	0	0
$286$	0.359119	0.0212352
$287$	7.95993	0.469860
$288$	0	0
$289$	−14.5930	−0.858413
$290$	0	0
$291$	0	0
$292$	−8.54478	−0.500045
$293$	−10.7041	−0.625338	−0.312669	−	0.949862i	$-0.601223\pi$
$293$	−10.7041	−0.625338	−0.312669	+	0.949862i	$0.601223\pi$
$294$	0	0
$295$	0	0
$296$	2.97216	0.172754
$297$	0	0
$298$	−7.92534	−0.459103
$299$	−0.570860	−0.0330137
$300$	0	0
$301$	17.5314	1.01049
$302$	6.08557	0.350185
$303$	0	0
$304$	3.43172	0.196822
$305$	0	0
$306$	0	0
$307$	−28.2376	−1.61161	−0.805803	−	0.592184i	$-0.798266\pi$
$307$	−28.2376	−1.61161	−0.805803	+	0.592184i	$0.798266\pi$
$308$	−4.90270	−0.279357
$309$	0	0
$310$	0	0
$311$	−2.73696	−0.155199	−0.0775994	−	0.996985i	$-0.524726\pi$
$311$	−2.73696	−0.155199	−0.0775994	+	0.996985i	$0.524726\pi$
$312$	0	0
$313$	−9.88861	−0.558937	−0.279469	−	0.960155i	$-0.590158\pi$
$313$	−9.88861	−0.558937	−0.279469	+	0.960155i	$0.590158\pi$
$314$	21.2891	1.20141
$315$	0	0
$316$	0.0313903	0.00176584
$317$	−16.3820	−0.920104	−0.460052	−	0.887892i	$-0.652169\pi$
$317$	−16.3820	−0.920104	−0.460052	+	0.887892i	$0.652169\pi$
$318$	0	0
$319$	−9.21603	−0.515999
$320$	0	0
$321$	0	0
$322$	−7.03662	−0.392136
$323$	−6.71152	−0.373439
$324$	0	0
$325$	0	0
$326$	12.7891	0.708325
$327$	0	0
$328$	−5.69845	−0.314644
$329$	−44.6107	−2.45947
$330$	0	0
$331$	24.8361	1.36512	0.682558	−	0.730831i	$-0.260867\pi$
$331$	24.8361	1.36512	0.682558	+	0.730831i	$0.260867\pi$
$332$	14.5242	0.797118
$333$	0	0
$334$	−13.7069	−0.750007
$335$	0	0
$336$	0	0
$337$	−11.7557	−0.640374	−0.320187	−	0.947354i	$-0.603746\pi$
$337$	−11.7557	−0.640374	−0.320187	+	0.947354i	$0.603746\pi$
$338$	−12.5591	−0.683125
$339$	0	0
$340$	0	0
$341$	−11.3569	−0.615010
$342$	0	0
$343$	13.4371	0.725537
$344$	−12.5506	−0.676684
$345$	0	0
$346$	−1.16111	−0.0624218
$347$	−22.4552	−1.20546	−0.602730	−	0.797945i	$-0.705921\pi$
$347$	−22.4552	−1.20546	−0.602730	+	0.797945i	$0.705921\pi$
$348$	0	0
$349$	−23.4583	−1.25570	−0.627848	−	0.778336i	$-0.716064\pi$
$349$	−23.4583	−1.25570	−0.627848	+	0.778336i	$0.716064\pi$
$350$	0	0
$351$	0	0
$352$	5.81054	0.309703
$353$	9.68966	0.515729	0.257864	−	0.966181i	$-0.416981\pi$
$353$	9.68966	0.515729	0.257864	+	0.966181i	$0.416981\pi$
$354$	0	0
$355$	0	0
$356$	−9.85723	−0.522432
$357$	0	0
$358$	2.74707	0.145187
$359$	−1.53276	−0.0808959	−0.0404479	−	0.999182i	$-0.512878\pi$
$359$	−1.53276	−0.0808959	−0.0404479	+	0.999182i	$0.512878\pi$
$360$	0	0
$361$	−0.285869	−0.0150457
$362$	11.4384	0.601191
$363$	0	0
$364$	−1.43171	−0.0750422
$365$	0	0
$366$	0	0
$367$	21.4479	1.11957	0.559785	−	0.828638i	$-0.310884\pi$
$367$	21.4479	1.11957	0.559785	+	0.828638i	$0.310884\pi$
$368$	1.38019	0.0719474
$369$	0	0
$370$	0	0
$371$	8.77820	0.455742
$372$	0	0
$373$	−20.7877	−1.07635	−0.538174	−	0.842834i	$-0.680886\pi$
$373$	−20.7877	−1.07635	−0.538174	+	0.842834i	$0.680886\pi$
$374$	1.69808	0.0878056
$375$	0	0
$376$	31.9364	1.64700
$377$	−2.69132	−0.138610
$378$	0	0
$379$	35.2385	1.81008	0.905039	−	0.425328i	$-0.139841\pi$
$379$	35.2385	1.81008	0.905039	+	0.425328i	$0.139841\pi$
$380$	0	0
$381$	0	0
$382$	14.5477	0.744323
$383$	−10.4709	−0.535039	−0.267520	−	0.963552i	$-0.586204\pi$
$383$	−10.4709	−0.535039	−0.267520	+	0.963552i	$0.586204\pi$
$384$	0	0
$385$	0	0
$386$	11.5886	0.589845
$387$	0	0
$388$	−10.8374	−0.550187
$389$	−21.2760	−1.07874	−0.539369	−	0.842070i	$-0.681337\pi$
$389$	−21.2760	−1.07874	−0.539369	+	0.842070i	$0.681337\pi$
$390$	0	0
$391$	−2.69928	−0.136509
$392$	−30.4247	−1.53668
$393$	0	0
$394$	0.197481	0.00994893
$395$	0	0
$396$	0	0
$397$	7.65737	0.384313	0.192156	−	0.981364i	$-0.438452\pi$
$397$	7.65737	0.384313	0.192156	+	0.981364i	$0.438452\pi$
$398$	22.2560	1.11559
$399$	0	0
$400$	0	0
$401$	7.92690	0.395850	0.197925	−	0.980217i	$-0.436580\pi$
$401$	7.92690	0.395850	0.197925	+	0.980217i	$0.436580\pi$
$402$	0	0
$403$	−3.31651	−0.165207
$404$	15.8044	0.786300
$405$	0	0
$406$	−33.1742	−1.64641
$407$	−1.12356	−0.0556927
$408$	0	0
$409$	−25.1341	−1.24280	−0.621402	−	0.783492i	$-0.713437\pi$
$409$	−25.1341	−1.24280	−0.621402	+	0.783492i	$0.713437\pi$
$410$	0	0
$411$	0	0
$412$	5.89095	0.290226
$413$	−48.1253	−2.36809
$414$	0	0
$415$	0	0
$416$	1.69683	0.0831938
$417$	0	0
$418$	−4.73486	−0.231589
$419$	32.9831	1.61133	0.805664	−	0.592372i	$-0.201809\pi$
$419$	32.9831	1.61133	0.805664	+	0.592372i	$0.201809\pi$
$420$	0	0
$421$	21.2137	1.03389	0.516946	−	0.856018i	$-0.327069\pi$
$421$	21.2137	1.03389	0.516946	+	0.856018i	$0.327069\pi$
$422$	2.12477	0.103432
$423$	0	0
$424$	−6.28425	−0.305190
$425$	0	0
$426$	0	0
$427$	40.3952	1.95486
$428$	4.46305	0.215729
$429$	0	0
$430$	0	0
$431$	18.9998	0.915189	0.457595	−	0.889161i	$-0.348711\pi$
$431$	18.9998	0.915189	0.457595	+	0.889161i	$0.348711\pi$
$432$	0	0
$433$	14.4655	0.695168	0.347584	−	0.937649i	$-0.387002\pi$
$433$	14.4655	0.695168	0.347584	+	0.937649i	$0.387002\pi$
$434$	−40.8804	−1.96232
$435$	0	0
$436$	3.30177	0.158126
$437$	7.52658	0.360045
$438$	0	0
$439$	7.66347	0.365758	0.182879	−	0.983135i	$-0.441458\pi$
$439$	7.66347	0.365758	0.182879	+	0.983135i	$0.441458\pi$
$440$	0	0
$441$	0	0
$442$	0.495883	0.0235867
$443$	−9.40674	−0.446928	−0.223464	−	0.974712i	$-0.571736\pi$
$443$	−9.40674	−0.446928	−0.223464	+	0.974712i	$0.571736\pi$
$444$	0	0
$445$	0	0
$446$	−0.570837	−0.0270299
$447$	0	0
$448$	27.5026	1.29938
$449$	31.2681	1.47563	0.737817	−	0.675001i	$-0.235857\pi$
$449$	31.2681	1.47563	0.737817	+	0.675001i	$0.235857\pi$
$450$	0	0
$451$	2.15417	0.101436
$452$	−8.69125	−0.408802
$453$	0	0
$454$	−9.10353	−0.427250
$455$	0	0
$456$	0	0
$457$	−30.7079	−1.43646	−0.718228	−	0.695807i	$-0.755047\pi$
$457$	−30.7079	−1.43646	−0.718228	+	0.695807i	$0.755047\pi$
$458$	−6.99048	−0.326644
$459$	0	0
$460$	0	0
$461$	30.3777	1.41483	0.707414	−	0.706799i	$-0.249862\pi$
$461$	30.3777	1.41483	0.707414	+	0.706799i	$0.249862\pi$
$462$	0	0
$463$	23.6458	1.09892	0.549458	−	0.835521i	$-0.314834\pi$
$463$	23.6458	1.09892	0.549458	+	0.835521i	$0.314834\pi$
$464$	6.50691	0.302076
$465$	0	0
$466$	−3.80210	−0.176129
$467$	−4.40134	−0.203670	−0.101835	−	0.994801i	$-0.532471\pi$
$467$	−4.40134	−0.203670	−0.101835	+	0.994801i	$0.532471\pi$
$468$	0	0
$469$	−22.0004	−1.01589
$470$	0	0
$471$	0	0
$472$	34.4525	1.58581
$473$	4.74447	0.218151
$474$	0	0
$475$	0	0
$476$	−6.76980	−0.310293
$477$	0	0
$478$	−18.9836	−0.868289
$479$	6.87159	0.313971	0.156985	−	0.987601i	$-0.449822\pi$
$479$	6.87159	0.313971	0.156985	+	0.987601i	$0.449822\pi$
$480$	0	0
$481$	−0.328108	−0.0149604
$482$	−17.2257	−0.784609
$483$	0	0
$484$	10.2345	0.465205
$485$	0	0
$486$	0	0
$487$	−1.08477	−0.0491557	−0.0245778	−	0.999698i	$-0.507824\pi$
$487$	−1.08477	−0.0491557	−0.0245778	+	0.999698i	$0.507824\pi$
$488$	−28.9186	−1.30908
$489$	0	0
$490$	0	0
$491$	−25.5117	−1.15133	−0.575663	−	0.817687i	$-0.695256\pi$
$491$	−25.5117	−1.15133	−0.575663	+	0.817687i	$0.695256\pi$
$492$	0	0
$493$	−12.7258	−0.573140
$494$	−1.38270	−0.0622106
$495$	0	0
$496$	8.01845	0.360039
$497$	−41.1125	−1.84415
$498$	0	0
$499$	−13.8928	−0.621927	−0.310963	−	0.950422i	$-0.600652\pi$
$499$	−13.8928	−0.621927	−0.310963	+	0.950422i	$0.600652\pi$
$500$	0	0
$501$	0	0
$502$	−20.0644	−0.895517
$503$	15.0199	0.669706	0.334853	−	0.942270i	$-0.391313\pi$
$503$	15.0199	0.669706	0.334853	+	0.942270i	$0.391313\pi$
$504$	0	0
$505$	0	0
$506$	−1.90430	−0.0846564
$507$	0	0
$508$	−6.55204	−0.290700
$509$	−23.1670	−1.02686	−0.513429	−	0.858132i	$-0.671625\pi$
$509$	−23.1670	−1.02686	−0.513429	+	0.858132i	$0.671625\pi$
$510$	0	0
$511$	33.7529	1.49314
$512$	−8.81800	−0.389704
$513$	0	0
$514$	−4.33911	−0.191390
$515$	0	0
$516$	0	0
$517$	−12.0728	−0.530963
$518$	−4.04438	−0.177700
$519$	0	0
$520$	0	0
$521$	15.6402	0.685210	0.342605	−	0.939480i	$-0.388691\pi$
$521$	15.6402	0.685210	0.342605	+	0.939480i	$0.388691\pi$
$522$	0	0
$523$	−17.2031	−0.752239	−0.376120	−	0.926571i	$-0.622742\pi$
$523$	−17.2031	−0.752239	−0.376120	+	0.926571i	$0.622742\pi$
$524$	6.14630	0.268503
$525$	0	0
$526$	−25.1508	−1.09663
$527$	−15.6819	−0.683116
$528$	0	0
$529$	−19.9729	−0.868387
$530$	0	0
$531$	0	0
$532$	18.8766	0.818406
$533$	0.629072	0.0272482
$534$	0	0
$535$	0	0
$536$	15.7500	0.680294
$537$	0	0
$538$	2.90997	0.125458
$539$	11.5014	0.495399
$540$	0	0
$541$	37.5079	1.61259	0.806295	−	0.591514i	$-0.201469\pi$
$541$	37.5079	1.61259	0.806295	+	0.591514i	$0.201469\pi$
$542$	2.06559	0.0887248
$543$	0	0
$544$	8.02337	0.343999
$545$	0	0
$546$	0	0
$547$	11.2496	0.480999	0.240500	−	0.970649i	$-0.422689\pi$
$547$	11.2496	0.480999	0.240500	+	0.970649i	$0.422689\pi$
$548$	−12.3751	−0.528638
$549$	0	0
$550$	0	0
$551$	35.4840	1.51167
$552$	0	0
$553$	−0.123995	−0.00527282
$554$	−17.2586	−0.733246
$555$	0	0
$556$	22.9320	0.972534
$557$	14.2721	0.604730	0.302365	−	0.953192i	$-0.402224\pi$
$557$	14.2721	0.604730	0.302365	+	0.953192i	$0.402224\pi$
$558$	0	0
$559$	1.38551	0.0586007
$560$	0	0
$561$	0	0
$562$	27.0931	1.14285
$563$	−26.5150	−1.11748	−0.558738	−	0.829344i	$-0.688714\pi$
$563$	−26.5150	−1.11748	−0.558738	+	0.829344i	$0.688714\pi$
$564$	0	0
$565$	0	0
$566$	−32.3676	−1.36051
$567$	0	0
$568$	29.4321	1.23495
$569$	19.5570	0.819873	0.409937	−	0.912114i	$-0.365551\pi$
$569$	19.5570	0.819873	0.409937	+	0.912114i	$0.365551\pi$
$570$	0	0
$571$	−26.6827	−1.11664	−0.558318	−	0.829627i	$-0.688553\pi$
$571$	−26.6827	−1.11664	−0.558318	+	0.829627i	$0.688553\pi$
$572$	−0.387460	−0.0162005
$573$	0	0
$574$	7.75417	0.323653
$575$	0	0
$576$	0	0
$577$	−44.3351	−1.84569	−0.922847	−	0.385167i	$-0.874144\pi$
$577$	−44.3351	−1.84569	−0.922847	+	0.385167i	$0.874144\pi$
$578$	−14.2158	−0.591300
$579$	0	0
$580$	0	0
$581$	−57.3722	−2.38020
$582$	0	0
$583$	2.37562	0.0983880
$584$	−24.1635	−0.999891
$585$	0	0
$586$	−10.4274	−0.430751
$587$	−14.2646	−0.588764	−0.294382	−	0.955688i	$-0.595114\pi$
$587$	−14.2646	−0.588764	−0.294382	+	0.955688i	$0.595114\pi$
$588$	0	0
$589$	43.7269	1.80174
$590$	0	0
$591$	0	0
$592$	0.793280	0.0326036
$593$	−35.9243	−1.47523	−0.737617	−	0.675220i	$-0.764049\pi$
$593$	−35.9243	−1.47523	−0.737617	+	0.675220i	$0.764049\pi$
$594$	0	0
$595$	0	0
$596$	8.55079	0.350254
$597$	0	0
$598$	−0.556104	−0.0227408
$599$	−11.3383	−0.463270	−0.231635	−	0.972803i	$-0.574408\pi$
$599$	−11.3383	−0.463270	−0.231635	+	0.972803i	$0.574408\pi$
$600$	0	0
$601$	−14.1352	−0.576585	−0.288293	−	0.957542i	$-0.593088\pi$
$601$	−14.1352	−0.576585	−0.288293	+	0.957542i	$0.593088\pi$
$602$	17.0783	0.696058
$603$	0	0
$604$	−6.56583	−0.267160
$605$	0	0
$606$	0	0
$607$	29.9826	1.21696	0.608478	−	0.793570i	$-0.291780\pi$
$607$	29.9826	1.21696	0.608478	+	0.793570i	$0.291780\pi$
$608$	−22.3721	−0.907307
$609$	0	0
$610$	0	0
$611$	−3.52558	−0.142630
$612$	0	0
$613$	−11.8163	−0.477255	−0.238627	−	0.971111i	$-0.576697\pi$
$613$	−11.8163	−0.477255	−0.238627	+	0.971111i	$0.576697\pi$
$614$	−27.5077	−1.11012
$615$	0	0
$616$	−13.8642	−0.558603
$617$	14.3499	0.577704	0.288852	−	0.957374i	$-0.406726\pi$
$617$	14.3499	0.577704	0.288852	+	0.957374i	$0.406726\pi$
$618$	0	0
$619$	42.7627	1.71878	0.859388	−	0.511324i	$-0.170845\pi$
$619$	42.7627	1.71878	0.859388	+	0.511324i	$0.170845\pi$
$620$	0	0
$621$	0	0
$622$	−2.66621	−0.106905
$623$	38.9373	1.55999
$624$	0	0
$625$	0	0
$626$	−9.63300	−0.385012
$627$	0	0
$628$	−22.9691	−0.916569
$629$	−1.55144	−0.0618601
$630$	0	0
$631$	−25.5460	−1.01697	−0.508485	−	0.861071i	$-0.669794\pi$
$631$	−25.5460	−1.01697	−0.508485	+	0.861071i	$0.669794\pi$
$632$	0.0887674	0.00353098
$633$	0	0
$634$	−15.9585	−0.633794
$635$	0	0
$636$	0	0
$637$	3.35869	0.133076
$638$	−8.97781	−0.355435
$639$	0	0
$640$	0	0
$641$	15.7859	0.623507	0.311753	−	0.950163i	$-0.399084\pi$
$641$	15.7859	0.623507	0.311753	+	0.950163i	$0.399084\pi$
$642$	0	0
$643$	9.14608	0.360686	0.180343	−	0.983604i	$-0.442279\pi$
$643$	9.14608	0.360686	0.180343	+	0.983604i	$0.442279\pi$
$644$	7.59194	0.299164
$645$	0	0
$646$	−6.53803	−0.257236
$647$	−0.954028	−0.0375067	−0.0187534	−	0.999824i	$-0.505970\pi$
$647$	−0.954028	−0.0375067	−0.0187534	+	0.999824i	$0.505970\pi$
$648$	0	0
$649$	−13.0240	−0.511236
$650$	0	0
$651$	0	0
$652$	−13.7984	−0.540388
$653$	29.4039	1.15066	0.575332	−	0.817920i	$-0.304873\pi$
$653$	29.4039	1.15066	0.575332	+	0.817920i	$0.304873\pi$
$654$	0	0
$655$	0	0
$656$	−1.52093	−0.0593825
$657$	0	0
$658$	−43.4575	−1.69415
$659$	4.70710	0.183363	0.0916813	−	0.995788i	$-0.470776\pi$
$659$	4.70710	0.183363	0.0916813	+	0.995788i	$0.470776\pi$
$660$	0	0
$661$	7.77234	0.302309	0.151155	−	0.988510i	$-0.451701\pi$
$661$	7.77234	0.302309	0.151155	+	0.988510i	$0.451701\pi$
$662$	24.1941	0.940332
$663$	0	0
$664$	41.0723	1.59392
$665$	0	0
$666$	0	0
$667$	14.2712	0.552584
$668$	14.7886	0.572188
$669$	0	0
$670$	0	0
$671$	10.9320	0.422026
$672$	0	0
$673$	−6.61462	−0.254975	−0.127487	−	0.991840i	$-0.540691\pi$
$673$	−6.61462	−0.254975	−0.127487	+	0.991840i	$0.540691\pi$
$674$	−11.4518	−0.441109
$675$	0	0
$676$	13.5502	0.521163
$677$	43.0362	1.65402	0.827008	−	0.562190i	$-0.190041\pi$
$677$	43.0362	1.65402	0.827008	+	0.562190i	$0.190041\pi$
$678$	0	0
$679$	42.8092	1.64286
$680$	0	0
$681$	0	0
$682$	−11.0633	−0.423637
$683$	37.4802	1.43414	0.717070	−	0.697001i	$-0.245483\pi$
$683$	37.4802	1.43414	0.717070	+	0.697001i	$0.245483\pi$
$684$	0	0
$685$	0	0
$686$	13.0898	0.499771
$687$	0	0
$688$	−3.34980	−0.127710
$689$	0.693741	0.0264294
$690$	0	0
$691$	−23.8817	−0.908501	−0.454251	−	0.890874i	$-0.650093\pi$
$691$	−23.8817	−0.908501	−0.454251	+	0.890874i	$0.650093\pi$
$692$	1.25274	0.0476222
$693$	0	0
$694$	−21.8748	−0.830356
$695$	0	0
$696$	0	0
$697$	2.97454	0.112669
$698$	−22.8520	−0.864960
$699$	0	0
$700$	0	0
$701$	19.1253	0.722354	0.361177	−	0.932497i	$-0.382375\pi$
$701$	19.1253	0.722354	0.361177	+	0.932497i	$0.382375\pi$
$702$	0	0
$703$	4.32598	0.163158
$704$	7.44294	0.280516
$705$	0	0
$706$	9.43920	0.355249
$707$	−62.4295	−2.34790
$708$	0	0
$709$	2.67026	0.100284	0.0501419	−	0.998742i	$-0.484033\pi$
$709$	2.67026	0.100284	0.0501419	+	0.998742i	$0.484033\pi$
$710$	0	0
$711$	0	0
$712$	−27.8749	−1.04466
$713$	17.5864	0.658615
$714$	0	0
$715$	0	0
$716$	−2.96386	−0.110765
$717$	0	0
$718$	−1.49314	−0.0557234
$719$	9.55937	0.356504	0.178252	−	0.983985i	$-0.442956\pi$
$719$	9.55937	0.356504	0.178252	+	0.983985i	$0.442956\pi$
$720$	0	0
$721$	−23.2700	−0.866619
$722$	−0.278479	−0.0103639
$723$	0	0
$724$	−12.3411	−0.458654
$725$	0	0
$726$	0	0
$727$	24.7692	0.918639	0.459319	−	0.888271i	$-0.348093\pi$
$727$	24.7692	0.918639	0.459319	+	0.888271i	$0.348093\pi$
$728$	−4.04869	−0.150054
$729$	0	0
$730$	0	0
$731$	6.55131	0.242309
$732$	0	0
$733$	31.1904	1.15204	0.576022	−	0.817434i	$-0.304604\pi$
$733$	31.1904	1.15204	0.576022	+	0.817434i	$0.304604\pi$
$734$	20.8935	0.771192
$735$	0	0
$736$	−8.99775	−0.331661
$737$	−5.95391	−0.219315
$738$	0	0
$739$	8.25351	0.303610	0.151805	−	0.988410i	$-0.451491\pi$
$739$	8.25351	0.303610	0.151805	+	0.988410i	$0.451491\pi$
$740$	0	0
$741$	0	0
$742$	8.55130	0.313928
$743$	−13.6459	−0.500620	−0.250310	−	0.968166i	$-0.580532\pi$
$743$	−13.6459	−0.500620	−0.250310	+	0.968166i	$0.580532\pi$
$744$	0	0
$745$	0	0
$746$	−20.2504	−0.741420
$747$	0	0
$748$	−1.83209	−0.0669878
$749$	−17.6296	−0.644171
$750$	0	0
$751$	−7.48601	−0.273168	−0.136584	−	0.990628i	$-0.543612\pi$
$751$	−7.48601	−0.273168	−0.136584	+	0.990628i	$0.543612\pi$
$752$	8.52393	0.310836
$753$	0	0
$754$	−2.62175	−0.0954785
$755$	0	0
$756$	0	0
$757$	−2.13112	−0.0774568	−0.0387284	−	0.999250i	$-0.512331\pi$
$757$	−2.13112	−0.0774568	−0.0387284	+	0.999250i	$0.512331\pi$
$758$	34.3276	1.24683
$759$	0	0
$760$	0	0
$761$	−6.44482	−0.233625	−0.116812	−	0.993154i	$-0.537268\pi$
$761$	−6.44482	−0.233625	−0.116812	+	0.993154i	$0.537268\pi$
$762$	0	0
$763$	−13.0424	−0.472167
$764$	−15.6957	−0.567851
$765$	0	0
$766$	−10.2003	−0.368551
$767$	−3.80334	−0.137331
$768$	0	0
$769$	1.48402	0.0535153	0.0267577	−	0.999642i	$-0.491482\pi$
$769$	1.48402	0.0535153	0.0267577	+	0.999642i	$0.491482\pi$
$770$	0	0
$771$	0	0
$772$	−12.5032	−0.449999
$773$	35.2452	1.26768	0.633842	−	0.773463i	$-0.281477\pi$
$773$	35.2452	1.26768	0.633842	+	0.773463i	$0.281477\pi$
$774$	0	0
$775$	0	0
$776$	−30.6467	−1.10015
$777$	0	0
$778$	−20.7261	−0.743066
$779$	−8.29409	−0.297167
$780$	0	0
$781$	−11.1261	−0.398125
$782$	−2.62951	−0.0940311
$783$	0	0
$784$	−8.12044	−0.290016
$785$	0	0
$786$	0	0
$787$	−34.4009	−1.22626	−0.613130	−	0.789982i	$-0.710090\pi$
$787$	−34.4009	−1.22626	−0.613130	+	0.789982i	$0.710090\pi$
$788$	−0.213065	−0.00759014
$789$	0	0
$790$	0	0
$791$	34.3315	1.22069
$792$	0	0
$793$	3.19243	0.113366
$794$	7.45944	0.264726
$795$	0	0
$796$	−24.0124	−0.851097
$797$	35.2214	1.24761	0.623804	−	0.781581i	$-0.285586\pi$
$797$	35.2214	1.24761	0.623804	+	0.781581i	$0.285586\pi$
$798$	0	0
$799$	−16.6705	−0.589761
$800$	0	0
$801$	0	0
$802$	7.72200	0.272673
$803$	9.13444	0.322347
$804$	0	0
$805$	0	0
$806$	−3.23078	−0.113799
$807$	0	0
$808$	44.6928	1.57229
$809$	41.4435	1.45707	0.728537	−	0.685006i	$-0.240201\pi$
$809$	41.4435	1.45707	0.728537	+	0.685006i	$0.240201\pi$
$810$	0	0
$811$	−15.0206	−0.527443	−0.263722	−	0.964599i	$-0.584950\pi$
$811$	−15.0206	−0.527443	−0.263722	+	0.964599i	$0.584950\pi$
$812$	35.7922	1.25606
$813$	0	0
$814$	−1.09452	−0.0383628
$815$	0	0
$816$	0	0
$817$	−18.2674	−0.639096
$818$	−24.4845	−0.856079
$819$	0	0
$820$	0	0
$821$	−13.1002	−0.457202	−0.228601	−	0.973520i	$-0.573415\pi$
$821$	−13.1002	−0.457202	−0.228601	+	0.973520i	$0.573415\pi$
$822$	0	0
$823$	−52.6530	−1.83537	−0.917683	−	0.397313i	$-0.869943\pi$
$823$	−52.6530	−1.83537	−0.917683	+	0.397313i	$0.869943\pi$
$824$	16.6588	0.580337
$825$	0	0
$826$	−46.8813	−1.63121
$827$	−5.83378	−0.202860	−0.101430	−	0.994843i	$-0.532342\pi$
$827$	−5.83378	−0.202860	−0.101430	+	0.994843i	$0.532342\pi$
$828$	0	0
$829$	46.2971	1.60796	0.803982	−	0.594654i	$-0.202711\pi$
$829$	46.2971	1.60796	0.803982	+	0.594654i	$0.202711\pi$
$830$	0	0
$831$	0	0
$832$	2.17353	0.0753536
$833$	15.8814	0.550258
$834$	0	0
$835$	0	0
$836$	5.10852	0.176682
$837$	0	0
$838$	32.1305	1.10993
$839$	−35.6818	−1.23187	−0.615937	−	0.787795i	$-0.711222\pi$
$839$	−35.6818	−1.23187	−0.615937	+	0.787795i	$0.711222\pi$
$840$	0	0
$841$	38.2816	1.32006
$842$	20.6654	0.712175
$843$	0	0
$844$	−2.29245	−0.0789093
$845$	0	0
$846$	0	0
$847$	−40.4276	−1.38911
$848$	−1.67729	−0.0575982
$849$	0	0
$850$	0	0
$851$	1.73985	0.0596414
$852$	0	0
$853$	34.6928	1.18786	0.593929	−	0.804517i	$-0.297576\pi$
$853$	34.6928	1.18786	0.593929	+	0.804517i	$0.297576\pi$
$854$	39.3510	1.34656
$855$	0	0
$856$	12.6209	0.431373
$857$	−39.2348	−1.34024	−0.670118	−	0.742254i	$-0.733757\pi$
$857$	−39.2348	−1.34024	−0.670118	+	0.742254i	$0.733757\pi$
$858$	0	0
$859$	−2.59802	−0.0886434	−0.0443217	−	0.999017i	$-0.514113\pi$
$859$	−2.59802	−0.0886434	−0.0443217	+	0.999017i	$0.514113\pi$
$860$	0	0
$861$	0	0
$862$	18.5087	0.630409
$863$	28.2848	0.962826	0.481413	−	0.876494i	$-0.340124\pi$
$863$	28.2848	0.962826	0.481413	+	0.876494i	$0.340124\pi$
$864$	0	0
$865$	0	0
$866$	14.0916	0.478852
$867$	0	0
$868$	44.1066	1.49708
$869$	−0.0335565	−0.00113833
$870$	0	0
$871$	−1.73869	−0.0589134
$872$	9.33695	0.316189
$873$	0	0
$874$	7.33203	0.248009
$875$	0	0
$876$	0	0
$877$	−37.7220	−1.27378	−0.636890	−	0.770954i	$-0.719780\pi$
$877$	−37.7220	−1.27378	−0.636890	+	0.770954i	$0.719780\pi$
$878$	7.46538	0.251944
$879$	0	0
$880$	0	0
$881$	−28.5497	−0.961863	−0.480931	−	0.876758i	$-0.659701\pi$
$881$	−28.5497	−0.961863	−0.480931	+	0.876758i	$0.659701\pi$
$882$	0	0
$883$	−6.33313	−0.213127	−0.106563	−	0.994306i	$-0.533985\pi$
$883$	−6.33313	−0.213127	−0.106563	+	0.994306i	$0.533985\pi$
$884$	−0.535017	−0.0179946
$885$	0	0
$886$	−9.16359	−0.307857
$887$	29.7029	0.997326	0.498663	−	0.866796i	$-0.333825\pi$
$887$	29.7029	0.997326	0.498663	+	0.866796i	$0.333825\pi$
$888$	0	0
$889$	25.8813	0.868032
$890$	0	0
$891$	0	0
$892$	0.615886	0.0206214
$893$	46.4835	1.55551
$894$	0	0
$895$	0	0
$896$	−16.1497	−0.539523
$897$	0	0
$898$	30.4599	1.01646
$899$	82.9110	2.76524
$900$	0	0
$901$	3.28032	0.109283
$902$	2.09849	0.0698720
$903$	0	0
$904$	−24.5776	−0.817440
$905$	0	0
$906$	0	0
$907$	−51.0940	−1.69655	−0.848274	−	0.529558i	$-0.822358\pi$
$907$	−51.0940	−1.69655	−0.848274	+	0.529558i	$0.822358\pi$
$908$	9.82196	0.325953
$909$	0	0
$910$	0	0
$911$	−0.652657	−0.0216235	−0.0108117	−	0.999942i	$-0.503442\pi$
$911$	−0.652657	−0.0216235	−0.0108117	+	0.999942i	$0.503442\pi$
$912$	0	0
$913$	−15.5265	−0.513851
$914$	−29.9142	−0.989473
$915$	0	0
$916$	7.54215	0.249200
$917$	−24.2786	−0.801751
$918$	0	0
$919$	−14.7727	−0.487307	−0.243654	−	0.969862i	$-0.578346\pi$
$919$	−14.7727	−0.487307	−0.243654	+	0.969862i	$0.578346\pi$
$920$	0	0
$921$	0	0
$922$	29.5924	0.974575
$923$	−3.24912	−0.106946
$924$	0	0
$925$	0	0
$926$	23.0346	0.756965
$927$	0	0
$928$	−42.4199	−1.39250
$929$	−34.8843	−1.14452	−0.572259	−	0.820073i	$-0.693933\pi$
$929$	−34.8843	−1.14452	−0.572259	+	0.820073i	$0.693933\pi$
$930$	0	0
$931$	−44.2831	−1.45132
$932$	4.10216	0.134371
$933$	0	0
$934$	−4.28757	−0.140294
$935$	0	0
$936$	0	0
$937$	−4.40483	−0.143900	−0.0719498	−	0.997408i	$-0.522922\pi$
$937$	−4.40483	−0.143900	−0.0719498	+	0.997408i	$0.522922\pi$
$938$	−21.4318	−0.699772
$939$	0	0
$940$	0	0
$941$	−30.2905	−0.987443	−0.493721	−	0.869620i	$-0.664364\pi$
$941$	−30.2905	−0.987443	−0.493721	+	0.869620i	$0.664364\pi$
$942$	0	0
$943$	−3.33577	−0.108628
$944$	9.19548	0.299287
$945$	0	0
$946$	4.62183	0.150269
$947$	48.0223	1.56052	0.780258	−	0.625458i	$-0.215088\pi$
$947$	48.0223	1.56052	0.780258	+	0.625458i	$0.215088\pi$
$948$	0	0
$949$	2.66749	0.0865904
$950$	0	0
$951$	0	0
$952$	−19.1440	−0.620462
$953$	−20.3929	−0.660592	−0.330296	−	0.943877i	$-0.607149\pi$
$953$	−20.3929	−0.660592	−0.330296	+	0.943877i	$0.607149\pi$
$954$	0	0
$955$	0	0
$956$	20.4817	0.662426
$957$	0	0
$958$	6.69397	0.216272
$959$	48.8831	1.57852
$960$	0	0
$961$	71.1711	2.29584
$962$	−0.319627	−0.0103052
$963$	0	0
$964$	18.5851	0.598586
$965$	0	0
$966$	0	0
$967$	−9.28548	−0.298601	−0.149300	−	0.988792i	$-0.547702\pi$
$967$	−9.28548	−0.298601	−0.149300	+	0.988792i	$0.547702\pi$
$968$	28.9418	0.930225
$969$	0	0
$970$	0	0
$971$	−17.7722	−0.570338	−0.285169	−	0.958477i	$-0.592050\pi$
$971$	−17.7722	−0.570338	−0.285169	+	0.958477i	$0.592050\pi$
$972$	0	0
$973$	−90.5842	−2.90400
$974$	−1.05673	−0.0338599
$975$	0	0
$976$	−7.71847	−0.247062
$977$	−15.5526	−0.497572	−0.248786	−	0.968559i	$-0.580032\pi$
$977$	−15.5526	−0.497572	−0.248786	+	0.968559i	$0.580032\pi$
$978$	0	0
$979$	10.5375	0.336779
$980$	0	0
$981$	0	0
$982$	−24.8522	−0.793067
$983$	−52.6734	−1.68002	−0.840010	−	0.542570i	$-0.817451\pi$
$983$	−52.6734	−1.68002	−0.840010	+	0.542570i	$0.817451\pi$
$984$	0	0
$985$	0	0
$986$	−12.3968	−0.394795
$987$	0	0
$988$	1.49182	0.0474611
$989$	−7.34691	−0.233618
$990$	0	0
$991$	13.9043	0.441686	0.220843	−	0.975309i	$-0.429119\pi$
$991$	13.9043	0.441686	0.220843	+	0.975309i	$0.429119\pi$
$992$	−52.2739	−1.65970
$993$	0	0
$994$	−40.0498	−1.27030
$995$	0	0
$996$	0	0
$997$	16.3165	0.516749	0.258375	−	0.966045i	$-0.416813\pi$
$997$	16.3165	0.516749	0.258375	+	0.966045i	$0.416813\pi$
$998$	−13.5337	−0.428401
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8325.2.a.cq.1.7		9
3.2	odd	2		925.2.a.m.1.3		9
5.2	odd	4		1665.2.c.e.334.12		18
5.3	odd	4		1665.2.c.e.334.7		18
5.4	even	2		8325.2.a.cr.1.3		9
15.2	even	4		185.2.b.a.149.7	✓	18
15.8	even	4		185.2.b.a.149.12	yes	18
15.14	odd	2		925.2.a.l.1.7		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.b.a.149.7	✓	18	15.2	even	4
185.2.b.a.149.12	yes	18	15.8	even	4
925.2.a.l.1.7		9	15.14	odd	2
925.2.a.m.1.3		9	3.2	odd	2
1665.2.c.e.334.7		18	5.3	odd	4
1665.2.c.e.334.12		18	5.2	odd	4
8325.2.a.cq.1.7		9	1.1	even	1	trivial
8325.2.a.cr.1.3		9	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8325.a (trivial)

\(f(q)\)	\(=\)	\(q+0.974151 q^{2} -1.05103 q^{4} +4.15169 q^{7} -2.97216 q^{8} +1.12356 q^{11} +0.328108 q^{13} +4.04438 q^{14} -0.793280 q^{16} +1.55144 q^{17} -4.32598 q^{19} +1.09452 q^{22} -1.73985 q^{23} +0.319627 q^{26} -4.36355 q^{28} -8.20254 q^{29} -10.1080 q^{31} +5.17155 q^{32} +1.51134 q^{34} -1.00000 q^{37} -4.21416 q^{38} +1.91727 q^{41} +4.22272 q^{43} -1.18089 q^{44} -1.69488 q^{46} -10.7452 q^{47} +10.2365 q^{49} -0.344851 q^{52} +2.11437 q^{53} -12.3395 q^{56} -7.99051 q^{58} -11.5917 q^{59} +9.72982 q^{61} -9.84669 q^{62} +6.62444 q^{64} -5.29915 q^{67} -1.63061 q^{68} -9.90260 q^{71} +8.12992 q^{73} -0.974151 q^{74} +4.54673 q^{76} +4.66467 q^{77} -0.0298662 q^{79} +1.86771 q^{82} -13.8190 q^{83} +4.11357 q^{86} -3.33940 q^{88} +9.37865 q^{89} +1.36220 q^{91} +1.82864 q^{92} -10.4674 q^{94} +10.3113 q^{97} +9.97194 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 5 q^{2} + 11 q^{4} + 8 q^{7} - 15 q^{8} + 6 q^{13} + 4 q^{14} + 11 q^{16} - 18 q^{17} - 4 q^{19} + 6 q^{22} - 16 q^{23} + 6 q^{26} - 20 q^{28} + 2 q^{29} - 6 q^{31} - 35 q^{32} + 6 q^{34} - 9 q^{37}+ \cdots - 21 q^{98}+O(q^{100}) \) 9 * q - 5 * q^2 + 11 * q^4 + 8 * q^7 - 15 * q^8 + 6 * q^13 + 4 * q^14 + 11 * q^16 - 18 * q^17 - 4 * q^19 + 6 * q^22 - 16 * q^23 + 6 * q^26 - 20 * q^28 + 2 * q^29 - 6 * q^31 - 35 * q^32 + 6 * q^34 - 9 * q^37 - 4 * q^38 - 2 * q^41 + 14 * q^43 - 4 * q^44 + 16 * q^46 - 34 * q^47 - 7 * q^49 + 16 * q^52 + 4 * q^53 + 24 * q^56 - 22 * q^58 - 10 * q^59 + 8 * q^61 - 26 * q^62 + 15 * q^64 + 16 * q^67 - 36 * q^68 - 8 * q^71 + 8 * q^73 + 5 * q^74 - 48 * q^76 - 14 * q^77 - 26 * q^79 - 4 * q^82 - 70 * q^83 + 16 * q^86 - 36 * q^88 + 8 * q^89 - 18 * q^91 - 16 * q^92 - 26 * q^94 - 2 * q^97 - 21 * q^98

Introduction

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