LMFDB - Embedded newform 6525.2.a.bo.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6525, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6525.1"); S:= CuspForms(chi, 2); N := Newforms(S);

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

Newform invariants

comment:select newform

sage:traces = [5,0,0,4,0,0,-10,-3,0,0,-2] 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:	$1$
Dimension:	$5$
Coefficient field:	5.5.294577.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 7x^{3} - x^{2} + 7x + 3 $ x^5 - 7x^3 - x^2 + 7x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 725)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.45914$ 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)

$f(q)$	$=$	$q-2.45914 q^{2} +4.04739 q^{4} -3.50903 q^{7} -5.03483 q^{8} +3.17006 q^{11} -4.61115 q^{13} +8.62920 q^{14} +4.28660 q^{16} -1.29916 q^{17} -4.04739 q^{19} -7.79563 q^{22} +2.57569 q^{23} +11.3395 q^{26} -14.2024 q^{28} +1.00000 q^{29} -0.423684 q^{31} -0.471703 q^{32} +3.19482 q^{34} +3.71028 q^{37} +9.95312 q^{38} +10.3443 q^{41} +4.57932 q^{43} +12.8305 q^{44} -6.33399 q^{46} -5.35453 q^{47} +5.31326 q^{49} -18.6631 q^{52} -0.0412739 q^{53} +17.6674 q^{56} -2.45914 q^{58} -2.86841 q^{59} -1.63042 q^{61} +1.04190 q^{62} -7.41322 q^{64} +0.464958 q^{67} -5.25821 q^{68} +4.33807 q^{71} +15.1207 q^{73} -9.12413 q^{74} -16.3814 q^{76} -11.1238 q^{77} +10.2945 q^{79} -25.4382 q^{82} +11.2606 q^{83} -11.2612 q^{86} -15.9607 q^{88} -16.6474 q^{89} +16.1806 q^{91} +10.4248 q^{92} +13.1676 q^{94} -3.96628 q^{97} -13.0661 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 4 q^{4} - 10 q^{7} - 3 q^{8} - 2 q^{11} - 8 q^{13} + 13 q^{14} + 6 q^{16} + 3 q^{17} - 4 q^{19} - 16 q^{22} + 3 q^{23} + 4 q^{26} - 10 q^{28} + 5 q^{29} + 5 q^{31} + 4 q^{32} - 9 q^{34} - 4 q^{37}+ \cdots - 51 q^{98}+O(q^{100}) $ 5 * q + 4 * q^4 - 10 * q^7 - 3 * q^8 - 2 * q^11 - 8 * q^13 + 13 * q^14 + 6 * q^16 + 3 * q^17 - 4 * q^19 - 16 * q^22 + 3 * q^23 + 4 * q^26 - 10 * q^28 + 5 * q^29 + 5 * q^31 + 4 * q^32 - 9 * q^34 - 4 * q^37 + 3 * q^38 + 3 * q^41 - 12 * q^43 + 19 * q^44 - q^47 + 13 * q^49 - 26 * q^52 + 12 * q^53 + 8 * q^56 + 15 * q^59 - 7 * q^61 + 10 * q^62 - 9 * q^64 - 17 * q^67 - 43 * q^68 + 17 * q^71 - 10 * q^73 - 12 * q^74 - 34 * q^76 - 8 * q^77 - 2 * q^79 - 31 * q^82 + 19 * q^83 - 31 * q^86 + 3 * q^88 - 14 * q^89 - 15 * q^91 - q^92 + 27 * q^94 + 9 * q^97 - 51 * 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 $

$2$	−2.45914	−1.73888	−0.869439	−	0.494040i	$-0.835519\pi$
$2$	−2.45914	−1.73888	−0.869439	+	0.494040i	$0.835519\pi$
$3$	0	0
$3$	0	0
$4$	4.04739	2.02370
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−3.50903	−1.32629	−0.663143	−	0.748492i	$-0.730778\pi$
$7$	−3.50903	−1.32629	−0.663143	+	0.748492i	$0.730778\pi$
$8$	−5.03483	−1.78008
$9$	0	0
$10$	0	0
$11$	3.17006	0.955808	0.477904	−	0.878412i	$-0.341397\pi$
$11$	3.17006	0.955808	0.477904	+	0.878412i	$0.341397\pi$
$12$	0	0
$13$	−4.61115	−1.27890	−0.639451	−	0.768831i	$-0.720838\pi$
$13$	−4.61115	−1.27890	−0.639451	+	0.768831i	$0.720838\pi$
$14$	8.62920	2.30625
$15$	0	0
$16$	4.28660	1.07165
$17$	−1.29916	−0.315092	−0.157546	−	0.987512i	$-0.550358\pi$
$17$	−1.29916	−0.315092	−0.157546	+	0.987512i	$0.550358\pi$
$18$	0	0
$19$	−4.04739	−0.928536	−0.464268	−	0.885695i	$-0.653682\pi$
$19$	−4.04739	−0.928536	−0.464268	+	0.885695i	$0.653682\pi$
$20$	0	0
$21$	0	0
$22$	−7.79563	−1.66203
$23$	2.57569	0.537068	0.268534	−	0.963270i	$-0.413461\pi$
$23$	2.57569	0.537068	0.268534	+	0.963270i	$0.413461\pi$
$24$	0	0
$25$	0	0
$26$	11.3395	2.22386
$27$	0	0
$28$	−14.2024	−2.68400
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−0.423684	−0.0760959	−0.0380480	−	0.999276i	$-0.512114\pi$
$31$	−0.423684	−0.0760959	−0.0380480	+	0.999276i	$0.512114\pi$
$32$	−0.471703	−0.0833861
$33$	0	0
$34$	3.19482	0.547907
$35$	0	0
$36$	0	0
$37$	3.71028	0.609967	0.304984	−	0.952358i	$-0.401349\pi$
$37$	3.71028	0.609967	0.304984	+	0.952358i	$0.401349\pi$
$38$	9.95312	1.61461
$39$	0	0
$40$	0	0
$41$	10.3443	1.61551	0.807757	−	0.589516i	$-0.200681\pi$
$41$	10.3443	1.61551	0.807757	+	0.589516i	$0.200681\pi$
$42$	0	0
$43$	4.57932	0.698340	0.349170	−	0.937059i	$-0.386464\pi$
$43$	4.57932	0.698340	0.349170	+	0.937059i	$0.386464\pi$
$44$	12.8305	1.93426
$45$	0	0
$46$	−6.33399	−0.933896
$47$	−5.35453	−0.781039	−0.390519	−	0.920595i	$-0.627705\pi$
$47$	−5.35453	−0.781039	−0.390519	+	0.920595i	$0.627705\pi$
$48$	0	0
$49$	5.31326	0.759037
$50$	0	0
$51$	0	0
$52$	−18.6631	−2.58811
$53$	−0.0412739	−0.00566941	−0.00283471	−	0.999996i	$-0.500902\pi$
$53$	−0.0412739	−0.00566941	−0.00283471	+	0.999996i	$0.500902\pi$
$54$	0	0
$55$	0	0
$56$	17.6674	2.36090
$57$	0	0
$58$	−2.45914	−0.322902
$59$	−2.86841	−0.373435	−0.186718	−	0.982414i	$-0.559785\pi$
$59$	−2.86841	−0.373435	−0.186718	+	0.982414i	$0.559785\pi$
$60$	0	0
$61$	−1.63042	−0.208754	−0.104377	−	0.994538i	$-0.533285\pi$
$61$	−1.63042	−0.208754	−0.104377	+	0.994538i	$0.533285\pi$
$62$	1.04190	0.132321
$63$	0	0
$64$	−7.41322	−0.926652
$65$	0	0
$66$	0	0
$67$	0.464958	0.0568037	0.0284018	−	0.999597i	$-0.490958\pi$
$67$	0.464958	0.0568037	0.0284018	+	0.999597i	$0.490958\pi$
$68$	−5.25821	−0.637651
$69$	0	0
$70$	0	0
$71$	4.33807	0.514834	0.257417	−	0.966300i	$-0.417129\pi$
$71$	4.33807	0.514834	0.257417	+	0.966300i	$0.417129\pi$
$72$	0	0
$73$	15.1207	1.76974	0.884872	−	0.465835i	$-0.154246\pi$
$73$	15.1207	1.76974	0.884872	+	0.465835i	$0.154246\pi$
$74$	−9.12413	−1.06066
$75$	0	0
$76$	−16.3814	−1.87907
$77$	−11.1238	−1.26768
$78$	0	0
$79$	10.2945	1.15822	0.579108	−	0.815251i	$-0.303401\pi$
$79$	10.2945	1.15822	0.579108	+	0.815251i	$0.303401\pi$
$80$	0	0
$81$	0	0
$82$	−25.4382	−2.80918
$83$	11.2606	1.23601	0.618007	−	0.786173i	$-0.287941\pi$
$83$	11.2606	1.23601	0.618007	+	0.786173i	$0.287941\pi$
$84$	0	0
$85$	0	0
$86$	−11.2612	−1.21433
$87$	0	0
$88$	−15.9607	−1.70142
$89$	−16.6474	−1.76463	−0.882313	−	0.470663i	$-0.844015\pi$
$89$	−16.6474	−1.76463	−0.882313	+	0.470663i	$0.844015\pi$
$90$	0	0
$91$	16.1806	1.69619
$92$	10.4248	1.08686
$93$	0	0
$94$	13.1676	1.35813
$95$	0	0
$96$	0	0
$97$	−3.96628	−0.402714	−0.201357	−	0.979518i	$-0.564535\pi$
$97$	−3.96628	−0.402714	−0.201357	+	0.979518i	$0.564535\pi$
$98$	−13.0661	−1.31987
$99$	0	0
$100$	0	0
$101$	2.82323	0.280922	0.140461	−	0.990086i	$-0.455142\pi$
$101$	2.82323	0.280922	0.140461	+	0.990086i	$0.455142\pi$
$102$	0	0
$103$	−7.04988	−0.694645	−0.347323	−	0.937746i	$-0.612909\pi$
$103$	−7.04988	−0.694645	−0.347323	+	0.937746i	$0.612909\pi$
$104$	23.2164	2.27655
$105$	0	0
$106$	0.101499	0.00985841
$107$	−3.25962	−0.315120	−0.157560	−	0.987509i	$-0.550363\pi$
$107$	−3.25962	−0.315120	−0.157560	+	0.987509i	$0.550363\pi$
$108$	0	0
$109$	−0.560643	−0.0536999	−0.0268499	−	0.999639i	$-0.508548\pi$
$109$	−0.560643	−0.0536999	−0.0268499	+	0.999639i	$0.508548\pi$
$110$	0	0
$111$	0	0
$112$	−15.0418	−1.42132
$113$	13.3505	1.25591	0.627953	−	0.778252i	$-0.283893\pi$
$113$	13.3505	1.25591	0.627953	+	0.778252i	$0.283893\pi$
$114$	0	0
$115$	0	0
$116$	4.04739	0.375791
$117$	0	0
$118$	7.05383	0.649358
$119$	4.55878	0.417903
$120$	0	0
$121$	−0.950746	−0.0864314
$122$	4.00944	0.362998
$123$	0	0
$124$	−1.71482	−0.153995
$125$	0	0
$126$	0	0
$127$	16.3437	1.45027	0.725135	−	0.688607i	$-0.241777\pi$
$127$	16.3437	1.45027	0.725135	+	0.688607i	$0.241777\pi$
$128$	19.1736	1.69472
$129$	0	0
$130$	0	0
$131$	−7.10277	−0.620571	−0.310286	−	0.950643i	$-0.600425\pi$
$131$	−7.10277	−0.620571	−0.310286	+	0.950643i	$0.600425\pi$
$132$	0	0
$133$	14.2024	1.23150
$134$	−1.14340	−0.0987746
$135$	0	0
$136$	6.54105	0.560890
$137$	0.338571	0.0289261	0.0144630	−	0.999895i	$-0.495396\pi$
$137$	0.338571	0.0289261	0.0144630	+	0.999895i	$0.495396\pi$
$138$	0	0
$139$	−13.1480	−1.11520	−0.557599	−	0.830110i	$-0.688277\pi$
$139$	−13.1480	−1.11520	−0.557599	+	0.830110i	$0.688277\pi$
$140$	0	0
$141$	0	0
$142$	−10.6679	−0.895234
$143$	−14.6176	−1.22239
$144$	0	0
$145$	0	0
$146$	−37.1840	−3.07737
$147$	0	0
$148$	15.0170	1.23439
$149$	14.2553	1.16784	0.583919	−	0.811812i	$-0.301518\pi$
$149$	14.2553	1.16784	0.583919	+	0.811812i	$0.301518\pi$
$150$	0	0
$151$	6.02626	0.490410	0.245205	−	0.969471i	$-0.421145\pi$
$151$	6.02626	0.490410	0.245205	+	0.969471i	$0.421145\pi$
$152$	20.3779	1.65287
$153$	0	0
$154$	27.3550	2.20433
$155$	0	0
$156$	0	0
$157$	−22.7417	−1.81499	−0.907494	−	0.420065i	$-0.862007\pi$
$157$	−22.7417	−1.81499	−0.907494	+	0.420065i	$0.862007\pi$
$158$	−25.3156	−2.01400
$159$	0	0
$160$	0	0
$161$	−9.03816	−0.712307
$162$	0	0
$163$	−24.5842	−1.92558	−0.962791	−	0.270247i	$-0.912895\pi$
$163$	−24.5842	−1.92558	−0.962791	+	0.270247i	$0.912895\pi$
$164$	41.8676	3.26931
$165$	0	0
$166$	−27.6915	−2.14928
$167$	−12.1715	−0.941860	−0.470930	−	0.882170i	$-0.656082\pi$
$167$	−12.1715	−0.941860	−0.470930	+	0.882170i	$0.656082\pi$
$168$	0	0
$169$	8.26271	0.635593
$170$	0	0
$171$	0	0
$172$	18.5343	1.41323
$173$	10.3682	0.788280	0.394140	−	0.919050i	$-0.371043\pi$
$173$	10.3682	0.788280	0.394140	+	0.919050i	$0.371043\pi$
$174$	0	0
$175$	0	0
$176$	13.5888	1.02429
$177$	0	0
$178$	40.9385	3.06847
$179$	−5.56095	−0.415645	−0.207822	−	0.978167i	$-0.566638\pi$
$179$	−5.56095	−0.415645	−0.207822	+	0.978167i	$0.566638\pi$
$180$	0	0
$181$	−18.2192	−1.35422	−0.677111	−	0.735881i	$-0.736768\pi$
$181$	−18.2192	−1.35422	−0.677111	+	0.735881i	$0.736768\pi$
$182$	−39.7905	−2.94947
$183$	0	0
$184$	−12.9682	−0.956026
$185$	0	0
$186$	0	0
$187$	−4.11841	−0.301168
$188$	−21.6719	−1.58059
$189$	0	0
$190$	0	0
$191$	9.51629	0.688574	0.344287	−	0.938864i	$-0.388121\pi$
$191$	9.51629	0.688574	0.344287	+	0.938864i	$0.388121\pi$
$192$	0	0
$193$	−7.95873	−0.572882	−0.286441	−	0.958098i	$-0.592472\pi$
$193$	−7.95873	−0.572882	−0.286441	+	0.958098i	$0.592472\pi$
$194$	9.75365	0.700271
$195$	0	0
$196$	21.5048	1.53606
$197$	−18.3498	−1.30737	−0.653686	−	0.756766i	$-0.726778\pi$
$197$	−18.3498	−1.30737	−0.653686	+	0.756766i	$0.726778\pi$
$198$	0	0
$199$	−18.8066	−1.33317	−0.666583	−	0.745431i	$-0.732244\pi$
$199$	−18.8066	−1.33317	−0.666583	+	0.745431i	$0.732244\pi$
$200$	0	0
$201$	0	0
$202$	−6.94273	−0.488489
$203$	−3.50903	−0.246285
$204$	0	0
$205$	0	0
$206$	17.3367	1.20790
$207$	0	0
$208$	−19.7662	−1.37054
$209$	−12.8305	−0.887501
$210$	0	0
$211$	4.73351	0.325868	0.162934	−	0.986637i	$-0.447904\pi$
$211$	4.73351	0.325868	0.162934	+	0.986637i	$0.447904\pi$
$212$	−0.167052	−0.0114732
$213$	0	0
$214$	8.01588	0.547955
$215$	0	0
$216$	0	0
$217$	1.48672	0.100925
$218$	1.37870	0.0933775
$219$	0	0
$220$	0	0
$221$	5.99062	0.402973
$222$	0	0
$223$	7.31989	0.490176	0.245088	−	0.969501i	$-0.421183\pi$
$223$	7.31989	0.490176	0.245088	+	0.969501i	$0.421183\pi$
$224$	1.65522	0.110594
$225$	0	0
$226$	−32.8307	−2.18387
$227$	−15.8457	−1.05172	−0.525858	−	0.850572i	$-0.676256\pi$
$227$	−15.8457	−1.05172	−0.525858	+	0.850572i	$0.676256\pi$
$228$	0	0
$229$	16.0440	1.06021	0.530107	−	0.847931i	$-0.322152\pi$
$229$	16.0440	1.06021	0.530107	+	0.847931i	$0.322152\pi$
$230$	0	0
$231$	0	0
$232$	−5.03483	−0.330553
$233$	18.9965	1.24450	0.622251	−	0.782818i	$-0.286218\pi$
$233$	18.9965	1.24450	0.622251	+	0.782818i	$0.286218\pi$
$234$	0	0
$235$	0	0
$236$	−11.6096	−0.755719
$237$	0	0
$238$	−11.2107	−0.726682
$239$	20.1139	1.30106	0.650531	−	0.759480i	$-0.274546\pi$
$239$	20.1139	1.30106	0.650531	+	0.759480i	$0.274546\pi$
$240$	0	0
$241$	−16.6785	−1.07435	−0.537177	−	0.843470i	$-0.680509\pi$
$241$	−16.6785	−1.07435	−0.537177	+	0.843470i	$0.680509\pi$
$242$	2.33802	0.150294
$243$	0	0
$244$	−6.59896	−0.422455
$245$	0	0
$246$	0	0
$247$	18.6631	1.18751
$248$	2.13318	0.135457
$249$	0	0
$250$	0	0
$251$	−5.87363	−0.370740	−0.185370	−	0.982669i	$-0.559348\pi$
$251$	−5.87363	−0.370740	−0.185370	+	0.982669i	$0.559348\pi$
$252$	0	0
$253$	8.16508	0.513334
$254$	−40.1915	−2.52184
$255$	0	0
$256$	−32.3242	−2.02026
$257$	−13.5371	−0.844421	−0.422210	−	0.906498i	$-0.638746\pi$
$257$	−13.5371	−0.844421	−0.422210	+	0.906498i	$0.638746\pi$
$258$	0	0
$259$	−13.0195	−0.808991
$260$	0	0
$261$	0	0
$262$	17.4667	1.07910
$263$	−5.19055	−0.320063	−0.160031	−	0.987112i	$-0.551160\pi$
$263$	−5.19055	−0.320063	−0.160031	+	0.987112i	$0.551160\pi$
$264$	0	0
$265$	0	0
$266$	−34.9258	−2.14144
$267$	0	0
$268$	1.88187	0.114953
$269$	14.6610	0.893894	0.446947	−	0.894560i	$-0.352511\pi$
$269$	14.6610	0.893894	0.446947	+	0.894560i	$0.352511\pi$
$270$	0	0
$271$	11.6971	0.710546	0.355273	−	0.934763i	$-0.384388\pi$
$271$	11.6971	0.710546	0.355273	+	0.934763i	$0.384388\pi$
$272$	−5.56898	−0.337669
$273$	0	0
$274$	−0.832596	−0.0502989
$275$	0	0
$276$	0	0
$277$	12.9375	0.777341	0.388670	−	0.921377i	$-0.372934\pi$
$277$	12.9375	0.777341	0.388670	+	0.921377i	$0.372934\pi$
$278$	32.3328	1.93919
$279$	0	0
$280$	0	0
$281$	7.94275	0.473825	0.236912	−	0.971531i	$-0.423865\pi$
$281$	7.94275	0.473825	0.236912	+	0.971531i	$0.423865\pi$
$282$	0	0
$283$	−28.3465	−1.68502	−0.842512	−	0.538677i	$-0.818924\pi$
$283$	−28.3465	−1.68502	−0.842512	+	0.538677i	$0.818924\pi$
$284$	17.5579	1.04187
$285$	0	0
$286$	35.9468	2.12558
$287$	−36.2985	−2.14264
$288$	0	0
$289$	−15.3122	−0.900717
$290$	0	0
$291$	0	0
$292$	61.1994	3.58142
$293$	−24.8491	−1.45170	−0.725849	−	0.687854i	$-0.758553\pi$
$293$	−24.8491	−1.45170	−0.725849	+	0.687854i	$0.758553\pi$
$294$	0	0
$295$	0	0
$296$	−18.6807	−1.08579
$297$	0	0
$298$	−35.0558	−2.03073
$299$	−11.8769	−0.686858
$300$	0	0
$301$	−16.0689	−0.926199
$302$	−14.8194	−0.852763
$303$	0	0
$304$	−17.3496	−0.995065
$305$	0	0
$306$	0	0
$307$	−9.99873	−0.570658	−0.285329	−	0.958430i	$-0.592103\pi$
$307$	−9.99873	−0.570658	−0.285329	+	0.958430i	$0.592103\pi$
$308$	−45.0224	−2.56539
$309$	0	0
$310$	0	0
$311$	−1.63825	−0.0928968	−0.0464484	−	0.998921i	$-0.514790\pi$
$311$	−1.63825	−0.0928968	−0.0464484	+	0.998921i	$0.514790\pi$
$312$	0	0
$313$	−1.93025	−0.109104	−0.0545522	−	0.998511i	$-0.517373\pi$
$313$	−1.93025	−0.109104	−0.0545522	+	0.998511i	$0.517373\pi$
$314$	55.9252	3.15604
$315$	0	0
$316$	41.6657	2.34388
$317$	29.1121	1.63510	0.817549	−	0.575859i	$-0.195332\pi$
$317$	29.1121	1.63510	0.817549	+	0.575859i	$0.195332\pi$
$318$	0	0
$319$	3.17006	0.177489
$320$	0	0
$321$	0	0
$322$	22.2261	1.23861
$323$	5.25821	0.292574
$324$	0	0
$325$	0	0
$326$	60.4561	3.34835
$327$	0	0
$328$	−52.0820	−2.87575
$329$	18.7892	1.03588
$330$	0	0
$331$	−3.88586	−0.213586	−0.106793	−	0.994281i	$-0.534058\pi$
$331$	−3.88586	−0.213586	−0.106793	+	0.994281i	$0.534058\pi$
$332$	45.5761	2.50132
$333$	0	0
$334$	29.9315	1.63778
$335$	0	0
$336$	0	0
$337$	−4.54398	−0.247527	−0.123763	−	0.992312i	$-0.539496\pi$
$337$	−4.54398	−0.247527	−0.123763	+	0.992312i	$0.539496\pi$
$338$	−20.3192	−1.10522
$339$	0	0
$340$	0	0
$341$	−1.34310	−0.0727331
$342$	0	0
$343$	5.91882	0.319586
$344$	−23.0561	−1.24310
$345$	0	0
$346$	−25.4969	−1.37072
$347$	13.3348	0.715848	0.357924	−	0.933751i	$-0.383485\pi$
$347$	13.3348	0.715848	0.357924	+	0.933751i	$0.383485\pi$
$348$	0	0
$349$	32.8243	1.75704	0.878522	−	0.477703i	$-0.158530\pi$
$349$	32.8243	1.75704	0.878522	+	0.477703i	$0.158530\pi$
$350$	0	0
$351$	0	0
$352$	−1.49532	−0.0797011
$353$	−18.4743	−0.983286	−0.491643	−	0.870797i	$-0.663603\pi$
$353$	−18.4743	−0.983286	−0.491643	+	0.870797i	$0.663603\pi$
$354$	0	0
$355$	0	0
$356$	−67.3788	−3.57107
$357$	0	0
$358$	13.6752	0.722756
$359$	25.8079	1.36209	0.681043	−	0.732243i	$-0.261526\pi$
$359$	25.8079	1.36209	0.681043	+	0.732243i	$0.261526\pi$
$360$	0	0
$361$	−2.61861	−0.137822
$362$	44.8036	2.35483
$363$	0	0
$364$	65.4894	3.43258
$365$	0	0
$366$	0	0
$367$	13.7885	0.719753	0.359876	−	0.933000i	$-0.382819\pi$
$367$	13.7885	0.719753	0.359876	+	0.933000i	$0.382819\pi$
$368$	11.0410	0.575549
$369$	0	0
$370$	0	0
$371$	0.144831	0.00751927
$372$	0	0
$373$	11.7048	0.606053	0.303026	−	0.952982i	$-0.402003\pi$
$373$	11.7048	0.606053	0.303026	+	0.952982i	$0.402003\pi$
$374$	10.1278	0.523694
$375$	0	0
$376$	26.9592	1.39031
$377$	−4.61115	−0.237486
$378$	0	0
$379$	−10.4270	−0.535598	−0.267799	−	0.963475i	$-0.586296\pi$
$379$	−10.4270	−0.535598	−0.267799	+	0.963475i	$0.586296\pi$
$380$	0	0
$381$	0	0
$382$	−23.4019	−1.19735
$383$	12.0391	0.615170	0.307585	−	0.951521i	$-0.400479\pi$
$383$	12.0391	0.615170	0.307585	+	0.951521i	$0.400479\pi$
$384$	0	0
$385$	0	0
$386$	19.5717	0.996171
$387$	0	0
$388$	−16.0531	−0.814972
$389$	−16.5646	−0.839858	−0.419929	−	0.907557i	$-0.637945\pi$
$389$	−16.5646	−0.839858	−0.419929	+	0.907557i	$0.637945\pi$
$390$	0	0
$391$	−3.34623	−0.169226
$392$	−26.7514	−1.35115
$393$	0	0
$394$	45.1249	2.27336
$395$	0	0
$396$	0	0
$397$	14.1044	0.707878	0.353939	−	0.935268i	$-0.384842\pi$
$397$	14.1044	0.707878	0.353939	+	0.935268i	$0.384842\pi$
$398$	46.2482	2.31821
$399$	0	0
$400$	0	0
$401$	−11.7748	−0.588007	−0.294003	−	0.955804i	$-0.594988\pi$
$401$	−11.7748	−0.588007	−0.294003	+	0.955804i	$0.594988\pi$
$402$	0	0
$403$	1.95367	0.0973193
$404$	11.4267	0.568501
$405$	0	0
$406$	8.62920	0.428260
$407$	11.7618	0.583011
$408$	0	0
$409$	−32.5326	−1.60864	−0.804318	−	0.594199i	$-0.797469\pi$
$409$	−32.5326	−1.60864	−0.804318	+	0.594199i	$0.797469\pi$
$410$	0	0
$411$	0	0
$412$	−28.5336	−1.40575
$413$	10.0653	0.495282
$414$	0	0
$415$	0	0
$416$	2.17509	0.106643
$417$	0	0
$418$	31.5520	1.54326
$419$	−6.80130	−0.332265	−0.166133	−	0.986103i	$-0.553128\pi$
$419$	−6.80130	−0.332265	−0.166133	+	0.986103i	$0.553128\pi$
$420$	0	0
$421$	−5.08494	−0.247825	−0.123912	−	0.992293i	$-0.539544\pi$
$421$	−5.08494	−0.247825	−0.123912	+	0.992293i	$0.539544\pi$
$422$	−11.6404	−0.566645
$423$	0	0
$424$	0.207807	0.0100920
$425$	0	0
$426$	0	0
$427$	5.72119	0.276868
$428$	−13.1930	−0.637706
$429$	0	0
$430$	0	0
$431$	−17.0921	−0.823295	−0.411648	−	0.911343i	$-0.635047\pi$
$431$	−17.0921	−0.823295	−0.411648	+	0.911343i	$0.635047\pi$
$432$	0	0
$433$	7.10031	0.341219	0.170610	−	0.985339i	$-0.445426\pi$
$433$	7.10031	0.341219	0.170610	+	0.985339i	$0.445426\pi$
$434$	−3.65605	−0.175496
$435$	0	0
$436$	−2.26914	−0.108672
$437$	−10.4248	−0.498687
$438$	0	0
$439$	12.1883	0.581718	0.290859	−	0.956766i	$-0.406059\pi$
$439$	12.1883	0.581718	0.290859	+	0.956766i	$0.406059\pi$
$440$	0	0
$441$	0	0
$442$	−14.7318	−0.700720
$443$	−0.994508	−0.0472505	−0.0236252	−	0.999721i	$-0.507521\pi$
$443$	−0.994508	−0.0472505	−0.0236252	+	0.999721i	$0.507521\pi$
$444$	0	0
$445$	0	0
$446$	−18.0007	−0.852357
$447$	0	0
$448$	26.0132	1.22901
$449$	17.6484	0.832881	0.416441	−	0.909163i	$-0.363277\pi$
$449$	17.6484	0.832881	0.416441	+	0.909163i	$0.363277\pi$
$450$	0	0
$451$	32.7921	1.54412
$452$	54.0345	2.54157
$453$	0	0
$454$	38.9669	1.82881
$455$	0	0
$456$	0	0
$457$	−4.27450	−0.199953	−0.0999763	−	0.994990i	$-0.531877\pi$
$457$	−4.27450	−0.199953	−0.0999763	+	0.994990i	$0.531877\pi$
$458$	−39.4544	−1.84358
$459$	0	0
$460$	0	0
$461$	−1.36149	−0.0634108	−0.0317054	−	0.999497i	$-0.510094\pi$
$461$	−1.36149	−0.0634108	−0.0317054	+	0.999497i	$0.510094\pi$
$462$	0	0
$463$	23.9028	1.11086	0.555429	−	0.831564i	$-0.312554\pi$
$463$	23.9028	1.11086	0.555429	+	0.831564i	$0.312554\pi$
$464$	4.28660	0.199000
$465$	0	0
$466$	−46.7151	−2.16404
$467$	13.0810	0.605317	0.302658	−	0.953099i	$-0.402126\pi$
$467$	13.0810	0.605317	0.302658	+	0.953099i	$0.402126\pi$
$468$	0	0
$469$	−1.63155	−0.0753379
$470$	0	0
$471$	0	0
$472$	14.4420	0.664745
$473$	14.5167	0.667479
$474$	0	0
$475$	0	0
$476$	18.4512	0.845708
$477$	0	0
$478$	−49.4631	−2.26239
$479$	−34.7395	−1.58729	−0.793643	−	0.608383i	$-0.791818\pi$
$479$	−34.7395	−1.58729	−0.793643	+	0.608383i	$0.791818\pi$
$480$	0	0
$481$	−17.1087	−0.780089
$482$	41.0147	1.86817
$483$	0	0
$484$	−3.84804	−0.174911
$485$	0	0
$486$	0	0
$487$	−18.3895	−0.833308	−0.416654	−	0.909065i	$-0.636797\pi$
$487$	−18.3895	−0.833308	−0.416654	+	0.909065i	$0.636797\pi$
$488$	8.20890	0.371600
$489$	0	0
$490$	0	0
$491$	−42.5512	−1.92031	−0.960155	−	0.279469i	$-0.909842\pi$
$491$	−42.5512	−1.92031	−0.960155	+	0.279469i	$0.909842\pi$
$492$	0	0
$493$	−1.29916	−0.0585112
$494$	−45.8953	−2.06493
$495$	0	0
$496$	−1.81616	−0.0815482
$497$	−15.2224	−0.682818
$498$	0	0
$499$	−17.4618	−0.781699	−0.390849	−	0.920455i	$-0.627819\pi$
$499$	−17.4618	−0.781699	−0.390849	+	0.920455i	$0.627819\pi$
$500$	0	0
$501$	0	0
$502$	14.4441	0.644672
$503$	14.4442	0.644036	0.322018	−	0.946734i	$-0.395639\pi$
$503$	14.4442	0.644036	0.322018	+	0.946734i	$0.395639\pi$
$504$	0	0
$505$	0	0
$506$	−20.0791	−0.892625
$507$	0	0
$508$	66.1494	2.93491
$509$	1.83780	0.0814591	0.0407295	−	0.999170i	$-0.487032\pi$
$509$	1.83780	0.0814591	0.0407295	+	0.999170i	$0.487032\pi$
$510$	0	0
$511$	−53.0589	−2.34719
$512$	41.1426	1.81827
$513$	0	0
$514$	33.2897	1.46834
$515$	0	0
$516$	0	0
$517$	−16.9742	−0.746523
$518$	32.0168	1.40674
$519$	0	0
$520$	0	0
$521$	−36.8440	−1.61417	−0.807083	−	0.590439i	$-0.798955\pi$
$521$	−36.8440	−1.61417	−0.807083	+	0.590439i	$0.798955\pi$
$522$	0	0
$523$	−30.1607	−1.31883	−0.659417	−	0.751777i	$-0.729197\pi$
$523$	−30.1607	−1.31883	−0.659417	+	0.751777i	$0.729197\pi$
$524$	−28.7477	−1.25585
$525$	0	0
$526$	12.7643	0.556550
$527$	0.550433	0.0239772
$528$	0	0
$529$	−16.3658	−0.711558
$530$	0	0
$531$	0	0
$532$	57.4827	2.49219
$533$	−47.6993	−2.06609
$534$	0	0
$535$	0	0
$536$	−2.34099	−0.101115
$537$	0	0
$538$	−36.0534	−1.55437
$539$	16.8433	0.725493
$540$	0	0
$541$	30.2558	1.30080	0.650400	−	0.759592i	$-0.274601\pi$
$541$	30.2558	1.30080	0.650400	+	0.759592i	$0.274601\pi$
$542$	−28.7648	−1.23555
$543$	0	0
$544$	0.612817	0.0262743
$545$	0	0
$546$	0	0
$547$	22.2206	0.950084	0.475042	−	0.879963i	$-0.342433\pi$
$547$	22.2206	0.950084	0.475042	+	0.879963i	$0.342433\pi$
$548$	1.37033	0.0585376
$549$	0	0
$550$	0	0
$551$	−4.04739	−0.172425
$552$	0	0
$553$	−36.1235	−1.53613
$554$	−31.8153	−1.35170
$555$	0	0
$556$	−53.2151	−2.25682
$557$	31.1531	1.32000	0.659999	−	0.751267i	$-0.270557\pi$
$557$	31.1531	1.32000	0.659999	+	0.751267i	$0.270557\pi$
$558$	0	0
$559$	−21.1159	−0.893109
$560$	0	0
$561$	0	0
$562$	−19.5324	−0.823923
$563$	−20.2319	−0.852675	−0.426337	−	0.904564i	$-0.640196\pi$
$563$	−20.2319	−0.852675	−0.426337	+	0.904564i	$0.640196\pi$
$564$	0	0
$565$	0	0
$566$	69.7082	2.93005
$567$	0	0
$568$	−21.8415	−0.916447
$569$	−23.1088	−0.968773	−0.484386	−	0.874854i	$-0.660957\pi$
$569$	−23.1088	−0.968773	−0.484386	+	0.874854i	$0.660957\pi$
$570$	0	0
$571$	−41.9425	−1.75524	−0.877620	−	0.479356i	$-0.840870\pi$
$571$	−41.9425	−1.75524	−0.877620	+	0.479356i	$0.840870\pi$
$572$	−59.1632	−2.47374
$573$	0	0
$574$	89.2634	3.72578
$575$	0	0
$576$	0	0
$577$	−19.8997	−0.828436	−0.414218	−	0.910178i	$-0.635945\pi$
$577$	−19.8997	−0.828436	−0.414218	+	0.910178i	$0.635945\pi$
$578$	37.6549	1.56624
$579$	0	0
$580$	0	0
$581$	−39.5138	−1.63931
$582$	0	0
$583$	−0.130841	−0.00541887
$584$	−76.1302	−3.15029
$585$	0	0
$586$	61.1075	2.52433
$587$	3.03862	0.125417	0.0627086	−	0.998032i	$-0.480026\pi$
$587$	3.03862	0.125417	0.0627086	+	0.998032i	$0.480026\pi$
$588$	0	0
$589$	1.71482	0.0706577
$590$	0	0
$591$	0	0
$592$	15.9045	0.653671
$593$	40.8122	1.67596	0.837979	−	0.545703i	$-0.183737\pi$
$593$	40.8122	1.67596	0.837979	+	0.545703i	$0.183737\pi$
$594$	0	0
$595$	0	0
$596$	57.6967	2.36335
$597$	0	0
$598$	29.2070	1.19436
$599$	−15.7500	−0.643530	−0.321765	−	0.946820i	$-0.604276\pi$
$599$	−15.7500	−0.643530	−0.321765	+	0.946820i	$0.604276\pi$
$600$	0	0
$601$	1.00387	0.0409489	0.0204745	−	0.999790i	$-0.493482\pi$
$601$	1.00387	0.0409489	0.0204745	+	0.999790i	$0.493482\pi$
$602$	39.5159	1.61055
$603$	0	0
$604$	24.3906	0.992441
$605$	0	0
$606$	0	0
$607$	30.6666	1.24472	0.622360	−	0.782731i	$-0.286174\pi$
$607$	30.6666	1.24472	0.622360	+	0.782731i	$0.286174\pi$
$608$	1.90917	0.0774269
$609$	0	0
$610$	0	0
$611$	24.6905	0.998873
$612$	0	0
$613$	3.13840	0.126759	0.0633793	−	0.997990i	$-0.479812\pi$
$613$	3.13840	0.126759	0.0633793	+	0.997990i	$0.479812\pi$
$614$	24.5883	0.992304
$615$	0	0
$616$	56.0065	2.25657
$617$	−48.0571	−1.93471	−0.967353	−	0.253434i	$-0.918440\pi$
$617$	−48.0571	−1.93471	−0.967353	+	0.253434i	$0.918440\pi$
$618$	0	0
$619$	42.0099	1.68852	0.844260	−	0.535934i	$-0.180040\pi$
$619$	42.0099	1.68852	0.844260	+	0.535934i	$0.180040\pi$
$620$	0	0
$621$	0	0
$622$	4.02870	0.161536
$623$	58.4163	2.34040
$624$	0	0
$625$	0	0
$626$	4.74677	0.189719
$627$	0	0
$628$	−92.0447	−3.67298
$629$	−4.82025	−0.192196
$630$	0	0
$631$	−20.9064	−0.832271	−0.416135	−	0.909303i	$-0.636616\pi$
$631$	−20.9064	−0.832271	−0.416135	+	0.909303i	$0.636616\pi$
$632$	−51.8309	−2.06172
$633$	0	0
$634$	−71.5909	−2.84324
$635$	0	0
$636$	0	0
$637$	−24.5002	−0.970734
$638$	−7.79563	−0.308632
$639$	0	0
$640$	0	0
$641$	−17.6324	−0.696439	−0.348220	−	0.937413i	$-0.613214\pi$
$641$	−17.6324	−0.696439	−0.348220	+	0.937413i	$0.613214\pi$
$642$	0	0
$643$	−9.33296	−0.368056	−0.184028	−	0.982921i	$-0.558914\pi$
$643$	−9.33296	−0.368056	−0.184028	+	0.982921i	$0.558914\pi$
$644$	−36.5810	−1.44149
$645$	0	0
$646$	−12.9307	−0.508751
$647$	6.08656	0.239287	0.119644	−	0.992817i	$-0.461825\pi$
$647$	6.08656	0.239287	0.119644	+	0.992817i	$0.461825\pi$
$648$	0	0
$649$	−9.09302	−0.356932
$650$	0	0
$651$	0	0
$652$	−99.5018	−3.89679
$653$	15.8120	0.618770	0.309385	−	0.950937i	$-0.399877\pi$
$653$	15.8120	0.618770	0.309385	+	0.950937i	$0.399877\pi$
$654$	0	0
$655$	0	0
$656$	44.3420	1.73127
$657$	0	0
$658$	−46.2053	−1.80127
$659$	−38.2114	−1.48850	−0.744252	−	0.667899i	$-0.767194\pi$
$659$	−38.2114	−1.48850	−0.744252	+	0.667899i	$0.767194\pi$
$660$	0	0
$661$	−32.2183	−1.25315	−0.626573	−	0.779362i	$-0.715543\pi$
$661$	−32.2183	−1.25315	−0.626573	+	0.779362i	$0.715543\pi$
$662$	9.55590	0.371401
$663$	0	0
$664$	−56.6953	−2.20021
$665$	0	0
$666$	0	0
$667$	2.57569	0.0997311
$668$	−49.2629	−1.90604
$669$	0	0
$670$	0	0
$671$	−5.16853	−0.199529
$672$	0	0
$673$	31.8086	1.22613	0.613065	−	0.790033i	$-0.289936\pi$
$673$	31.8086	1.22613	0.613065	+	0.790033i	$0.289936\pi$
$674$	11.1743	0.430418
$675$	0	0
$676$	33.4424	1.28625
$677$	10.6030	0.407507	0.203754	−	0.979022i	$-0.434686\pi$
$677$	10.6030	0.407507	0.203754	+	0.979022i	$0.434686\pi$
$678$	0	0
$679$	13.9178	0.534115
$680$	0	0
$681$	0	0
$682$	3.30288	0.126474
$683$	12.2980	0.470572	0.235286	−	0.971926i	$-0.424397\pi$
$683$	12.2980	0.470572	0.235286	+	0.971926i	$0.424397\pi$
$684$	0	0
$685$	0	0
$686$	−14.5552	−0.555722
$687$	0	0
$688$	19.6297	0.748376
$689$	0.190320	0.00725063
$690$	0	0
$691$	20.5877	0.783193	0.391596	−	0.920137i	$-0.371923\pi$
$691$	20.5877	0.783193	0.391596	+	0.920137i	$0.371923\pi$
$692$	41.9642	1.59524
$693$	0	0
$694$	−32.7921	−1.24477
$695$	0	0
$696$	0	0
$697$	−13.4389	−0.509036
$698$	−80.7196	−3.05528
$699$	0	0
$700$	0	0
$701$	31.8857	1.20431	0.602153	−	0.798380i	$-0.294310\pi$
$701$	31.8857	1.20431	0.602153	+	0.798380i	$0.294310\pi$
$702$	0	0
$703$	−15.0170	−0.566376
$704$	−23.5003	−0.885701
$705$	0	0
$706$	45.4309	1.70981
$707$	−9.90679	−0.372583
$708$	0	0
$709$	−37.0325	−1.39079	−0.695393	−	0.718630i	$-0.744769\pi$
$709$	−37.0325	−1.39079	−0.695393	+	0.718630i	$0.744769\pi$
$710$	0	0
$711$	0	0
$712$	83.8171	3.14118
$713$	−1.09128	−0.0408687
$714$	0	0
$715$	0	0
$716$	−22.5073	−0.841139
$717$	0	0
$718$	−63.4653	−2.36850
$719$	−2.30674	−0.0860268	−0.0430134	−	0.999074i	$-0.513696\pi$
$719$	−2.30674	−0.0860268	−0.0430134	+	0.999074i	$0.513696\pi$
$720$	0	0
$721$	24.7382	0.921299
$722$	6.43955	0.239655
$723$	0	0
$724$	−73.7402	−2.74053
$725$	0	0
$726$	0	0
$727$	−30.0583	−1.11480	−0.557401	−	0.830243i	$-0.688201\pi$
$727$	−30.0583	−1.11480	−0.557401	+	0.830243i	$0.688201\pi$
$728$	−81.4668	−3.01936
$729$	0	0
$730$	0	0
$731$	−5.94926	−0.220042
$732$	0	0
$733$	−49.7450	−1.83738	−0.918688	−	0.394985i	$-0.870750\pi$
$733$	−49.7450	−1.83738	−0.918688	+	0.394985i	$0.870750\pi$
$734$	−33.9079	−1.25156
$735$	0	0
$736$	−1.21496	−0.0447840
$737$	1.47394	0.0542934
$738$	0	0
$739$	−46.1269	−1.69681	−0.848404	−	0.529350i	$-0.822436\pi$
$739$	−46.1269	−1.69681	−0.848404	+	0.529350i	$0.822436\pi$
$740$	0	0
$741$	0	0
$742$	−0.356161	−0.0130751
$743$	−14.0745	−0.516345	−0.258172	−	0.966099i	$-0.583120\pi$
$743$	−14.0745	−0.516345	−0.258172	+	0.966099i	$0.583120\pi$
$744$	0	0
$745$	0	0
$746$	−28.7839	−1.05385
$747$	0	0
$748$	−16.6688	−0.609472
$749$	11.4381	0.417939
$750$	0	0
$751$	6.40760	0.233817	0.116908	−	0.993143i	$-0.462702\pi$
$751$	6.40760	0.233817	0.116908	+	0.993143i	$0.462702\pi$
$752$	−22.9527	−0.837000
$753$	0	0
$754$	11.3395	0.412960
$755$	0	0
$756$	0	0
$757$	−37.6194	−1.36730	−0.683651	−	0.729809i	$-0.739609\pi$
$757$	−37.6194	−1.36730	−0.683651	+	0.729809i	$0.739609\pi$
$758$	25.6415	0.931339
$759$	0	0
$760$	0	0
$761$	7.98770	0.289554	0.144777	−	0.989464i	$-0.453754\pi$
$761$	7.98770	0.289554	0.144777	+	0.989464i	$0.453754\pi$
$762$	0	0
$763$	1.96731	0.0712214
$764$	38.5161	1.39347
$765$	0	0
$766$	−29.6059	−1.06970
$767$	13.2267	0.477587
$768$	0	0
$769$	−12.3260	−0.444486	−0.222243	−	0.974991i	$-0.571338\pi$
$769$	−12.3260	−0.444486	−0.222243	+	0.974991i	$0.571338\pi$
$770$	0	0
$771$	0	0
$772$	−32.2121	−1.15934
$773$	−46.9639	−1.68917	−0.844587	−	0.535419i	$-0.820154\pi$
$773$	−46.9639	−1.68917	−0.844587	+	0.535419i	$0.820154\pi$
$774$	0	0
$775$	0	0
$776$	19.9695	0.716865
$777$	0	0
$778$	40.7347	1.46041
$779$	−41.8676	−1.50006
$780$	0	0
$781$	13.7519	0.492082
$782$	8.22886	0.294264
$783$	0	0
$784$	22.7758	0.813422
$785$	0	0
$786$	0	0
$787$	−35.1053	−1.25137	−0.625684	−	0.780076i	$-0.715180\pi$
$787$	−35.1053	−1.25137	−0.625684	+	0.780076i	$0.715180\pi$
$788$	−74.2690	−2.64572
$789$	0	0
$790$	0	0
$791$	−46.8471	−1.66569
$792$	0	0
$793$	7.51812	0.266976
$794$	−34.6847	−1.23091
$795$	0	0
$796$	−76.1177	−2.69792
$797$	−36.4490	−1.29109	−0.645546	−	0.763722i	$-0.723370\pi$
$797$	−36.4490	−1.29109	−0.645546	+	0.763722i	$0.723370\pi$
$798$	0	0
$799$	6.95639	0.246099
$800$	0	0
$801$	0	0
$802$	28.9560	1.02247
$803$	47.9334	1.69153
$804$	0	0
$805$	0	0
$806$	−4.80436	−0.169226
$807$	0	0
$808$	−14.2145	−0.500064
$809$	−50.3845	−1.77142	−0.885712	−	0.464235i	$-0.846329\pi$
$809$	−50.3845	−1.77142	−0.885712	+	0.464235i	$0.846329\pi$
$810$	0	0
$811$	44.6328	1.56727	0.783635	−	0.621221i	$-0.213363\pi$
$811$	44.6328	1.56727	0.783635	+	0.621221i	$0.213363\pi$
$812$	−14.2024	−0.498407
$813$	0	0
$814$	−28.9240	−1.01379
$815$	0	0
$816$	0	0
$817$	−18.5343	−0.648433
$818$	80.0025	2.79722
$819$	0	0
$820$	0	0
$821$	−25.6437	−0.894971	−0.447486	−	0.894291i	$-0.647680\pi$
$821$	−25.6437	−0.894971	−0.447486	+	0.894291i	$0.647680\pi$
$822$	0	0
$823$	4.74708	0.165473	0.0827364	−	0.996571i	$-0.473634\pi$
$823$	4.74708	0.165473	0.0827364	+	0.996571i	$0.473634\pi$
$824$	35.4950	1.23653
$825$	0	0
$826$	−24.7521	−0.861235
$827$	−16.8954	−0.587509	−0.293755	−	0.955881i	$-0.594905\pi$
$827$	−16.8954	−0.587509	−0.293755	+	0.955881i	$0.594905\pi$
$828$	0	0
$829$	−31.1447	−1.08170	−0.540850	−	0.841119i	$-0.681897\pi$
$829$	−31.1447	−1.08170	−0.540850	+	0.841119i	$0.681897\pi$
$830$	0	0
$831$	0	0
$832$	34.1835	1.18510
$833$	−6.90277	−0.239167
$834$	0	0
$835$	0	0
$836$	−51.9299	−1.79603
$837$	0	0
$838$	16.7254	0.577769
$839$	−20.3277	−0.701791	−0.350896	−	0.936415i	$-0.614123\pi$
$839$	−20.3277	−0.701791	−0.350896	+	0.936415i	$0.614123\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	12.5046	0.430937
$843$	0	0
$844$	19.1584	0.659458
$845$	0	0
$846$	0	0
$847$	3.33619	0.114633
$848$	−0.176925	−0.00607563
$849$	0	0
$850$	0	0
$851$	9.55654	0.327594
$852$	0	0
$853$	48.0623	1.64562	0.822810	−	0.568316i	$-0.192405\pi$
$853$	48.0623	1.64562	0.822810	+	0.568316i	$0.192405\pi$
$854$	−14.0692	−0.481439
$855$	0	0
$856$	16.4117	0.560939
$857$	−24.0426	−0.821280	−0.410640	−	0.911797i	$-0.634695\pi$
$857$	−24.0426	−0.821280	−0.410640	+	0.911797i	$0.634695\pi$
$858$	0	0
$859$	−38.8389	−1.32517	−0.662583	−	0.748988i	$-0.730540\pi$
$859$	−38.8389	−1.32517	−0.662583	+	0.748988i	$0.730540\pi$
$860$	0	0
$861$	0	0
$862$	42.0318	1.43161
$863$	43.5717	1.48320	0.741599	−	0.670843i	$-0.234068\pi$
$863$	43.5717	1.48320	0.741599	+	0.670843i	$0.234068\pi$
$864$	0	0
$865$	0	0
$866$	−17.4607	−0.593338
$867$	0	0
$868$	6.01733	0.204242
$869$	32.6340	1.10703
$870$	0	0
$871$	−2.14399	−0.0726464
$872$	2.82275	0.0955902
$873$	0	0
$874$	25.6362	0.867156
$875$	0	0
$876$	0	0
$877$	−33.4549	−1.12969	−0.564846	−	0.825197i	$-0.691064\pi$
$877$	−33.4549	−1.12969	−0.564846	+	0.825197i	$0.691064\pi$
$878$	−29.9729	−1.01154
$879$	0	0
$880$	0	0
$881$	17.1987	0.579440	0.289720	−	0.957111i	$-0.406438\pi$
$881$	17.1987	0.579440	0.289720	+	0.957111i	$0.406438\pi$
$882$	0	0
$883$	−31.2157	−1.05049	−0.525247	−	0.850950i	$-0.676027\pi$
$883$	−31.2157	−1.05049	−0.525247	+	0.850950i	$0.676027\pi$
$884$	24.2464	0.815494
$885$	0	0
$886$	2.44564	0.0821628
$887$	−50.3074	−1.68916	−0.844579	−	0.535430i	$-0.820149\pi$
$887$	−50.3074	−1.68916	−0.844579	+	0.535430i	$0.820149\pi$
$888$	0	0
$889$	−57.3505	−1.92347
$890$	0	0
$891$	0	0
$892$	29.6265	0.991968
$893$	21.6719	0.725222
$894$	0	0
$895$	0	0
$896$	−67.2806	−2.24769
$897$	0	0
$898$	−43.4001	−1.44828
$899$	−0.423684	−0.0141307
$900$	0	0
$901$	0.0536214	0.00178639
$902$	−80.6406	−2.68504
$903$	0	0
$904$	−67.2173	−2.23562
$905$	0	0
$906$	0	0
$907$	−40.2681	−1.33708	−0.668540	−	0.743676i	$-0.733080\pi$
$907$	−40.2681	−1.33708	−0.668540	+	0.743676i	$0.733080\pi$
$908$	−64.1338	−2.12835
$909$	0	0
$910$	0	0
$911$	46.0923	1.52710	0.763552	−	0.645746i	$-0.223453\pi$
$911$	46.0923	1.52710	0.763552	+	0.645746i	$0.223453\pi$
$912$	0	0
$913$	35.6968	1.18139
$914$	10.5116	0.347693
$915$	0	0
$916$	64.9362	2.14555
$917$	24.9238	0.823056
$918$	0	0
$919$	−10.9441	−0.361012	−0.180506	−	0.983574i	$-0.557773\pi$
$919$	−10.9441	−0.361012	−0.180506	+	0.983574i	$0.557773\pi$
$920$	0	0
$921$	0	0
$922$	3.34809	0.110264
$923$	−20.0035	−0.658423
$924$	0	0
$925$	0	0
$926$	−58.7804	−1.93165
$927$	0	0
$928$	−0.471703	−0.0154844
$929$	50.6115	1.66051	0.830255	−	0.557384i	$-0.188195\pi$
$929$	50.6115	1.66051	0.830255	+	0.557384i	$0.188195\pi$
$930$	0	0
$931$	−21.5048	−0.704793
$932$	76.8863	2.51849
$933$	0	0
$934$	−32.1681	−1.05257
$935$	0	0
$936$	0	0
$937$	−18.9658	−0.619585	−0.309793	−	0.950804i	$-0.600260\pi$
$937$	−18.9658	−0.619585	−0.309793	+	0.950804i	$0.600260\pi$
$938$	4.01222	0.131003
$939$	0	0
$940$	0	0
$941$	37.2758	1.21516	0.607579	−	0.794260i	$-0.292141\pi$
$941$	37.2758	1.21516	0.607579	+	0.794260i	$0.292141\pi$
$942$	0	0
$943$	26.6438	0.867642
$944$	−12.2957	−0.400192
$945$	0	0
$946$	−35.6987	−1.16066
$947$	25.3754	0.824591	0.412296	−	0.911050i	$-0.364727\pi$
$947$	25.3754	0.824591	0.412296	+	0.911050i	$0.364727\pi$
$948$	0	0
$949$	−69.7238	−2.26333
$950$	0	0
$951$	0	0
$952$	−22.9527	−0.743902
$953$	43.6572	1.41419	0.707097	−	0.707116i	$-0.250004\pi$
$953$	43.6572	1.41419	0.707097	+	0.707116i	$0.250004\pi$
$954$	0	0
$955$	0	0
$956$	81.4090	2.63295
$957$	0	0
$958$	85.4294	2.76010
$959$	−1.18806	−0.0383643
$960$	0	0
$961$	−30.8205	−0.994209
$962$	42.0727	1.35648
$963$	0	0
$964$	−67.5043	−2.17417
$965$	0	0
$966$	0	0
$967$	−28.4146	−0.913751	−0.456875	−	0.889531i	$-0.651032\pi$
$967$	−28.4146	−0.913751	−0.456875	+	0.889531i	$0.651032\pi$
$968$	4.78685	0.153855
$969$	0	0
$970$	0	0
$971$	−6.34773	−0.203708	−0.101854	−	0.994799i	$-0.532477\pi$
$971$	−6.34773	−0.203708	−0.101854	+	0.994799i	$0.532477\pi$
$972$	0	0
$973$	46.1366	1.47907
$974$	45.2225	1.44902
$975$	0	0
$976$	−6.98897	−0.223711
$977$	−6.78708	−0.217138	−0.108569	−	0.994089i	$-0.534627\pi$
$977$	−6.78708	−0.217138	−0.108569	+	0.994089i	$0.534627\pi$
$978$	0	0
$979$	−52.7733	−1.68664
$980$	0	0
$981$	0	0
$982$	104.640	3.33918
$983$	−33.2974	−1.06202	−0.531011	−	0.847365i	$-0.678188\pi$
$983$	−33.2974	−1.06202	−0.531011	+	0.847365i	$0.678188\pi$
$984$	0	0
$985$	0	0
$986$	3.19482	0.101744
$987$	0	0
$988$	75.5370	2.40315
$989$	11.7949	0.375056
$990$	0	0
$991$	23.2716	0.739247	0.369623	−	0.929182i	$-0.379487\pi$
$991$	23.2716	0.739247	0.369623	+	0.929182i	$0.379487\pi$
$992$	0.199853	0.00634534
$993$	0	0
$994$	37.4341	1.18734
$995$	0	0
$996$	0	0
$997$	18.5645	0.587945	0.293973	−	0.955814i	$-0.405023\pi$
$997$	18.5645	0.587945	0.293973	+	0.955814i	$0.405023\pi$
$998$	42.9412	1.35928
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6525.2.a.bo.1.1		5
3.2	odd	2		725.2.a.i.1.5	✓	5
5.4	even	2		6525.2.a.bp.1.5		5
15.2	even	4		725.2.b.g.349.10		10
15.8	even	4		725.2.b.g.349.1		10
15.14	odd	2		725.2.a.j.1.1	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
725.2.a.i.1.5	✓	5	3.2	odd	2
725.2.a.j.1.1	yes	5	15.14	odd	2
725.2.b.g.349.1		10	15.8	even	4
725.2.b.g.349.10		10	15.2	even	4
6525.2.a.bo.1.1		5	1.1	even	1	trivial
6525.2.a.bp.1.5		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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-2.45914 q^{2} +4.04739 q^{4} -3.50903 q^{7} -5.03483 q^{8} +3.17006 q^{11} -4.61115 q^{13} +8.62920 q^{14} +4.28660 q^{16} -1.29916 q^{17} -4.04739 q^{19} -7.79563 q^{22} +2.57569 q^{23} +11.3395 q^{26} -14.2024 q^{28} +1.00000 q^{29} -0.423684 q^{31} -0.471703 q^{32} +3.19482 q^{34} +3.71028 q^{37} +9.95312 q^{38} +10.3443 q^{41} +4.57932 q^{43} +12.8305 q^{44} -6.33399 q^{46} -5.35453 q^{47} +5.31326 q^{49} -18.6631 q^{52} -0.0412739 q^{53} +17.6674 q^{56} -2.45914 q^{58} -2.86841 q^{59} -1.63042 q^{61} +1.04190 q^{62} -7.41322 q^{64} +0.464958 q^{67} -5.25821 q^{68} +4.33807 q^{71} +15.1207 q^{73} -9.12413 q^{74} -16.3814 q^{76} -11.1238 q^{77} +10.2945 q^{79} -25.4382 q^{82} +11.2606 q^{83} -11.2612 q^{86} -15.9607 q^{88} -16.6474 q^{89} +16.1806 q^{91} +10.4248 q^{92} +13.1676 q^{94} -3.96628 q^{97} -13.0661 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 4 q^{4} - 10 q^{7} - 3 q^{8} - 2 q^{11} - 8 q^{13} + 13 q^{14} + 6 q^{16} + 3 q^{17} - 4 q^{19} - 16 q^{22} + 3 q^{23} + 4 q^{26} - 10 q^{28} + 5 q^{29} + 5 q^{31} + 4 q^{32} - 9 q^{34} - 4 q^{37}+ \cdots - 51 q^{98}+O(q^{100}) \) 5 * q + 4 * q^4 - 10 * q^7 - 3 * q^8 - 2 * q^11 - 8 * q^13 + 13 * q^14 + 6 * q^16 + 3 * q^17 - 4 * q^19 - 16 * q^22 + 3 * q^23 + 4 * q^26 - 10 * q^28 + 5 * q^29 + 5 * q^31 + 4 * q^32 - 9 * q^34 - 4 * q^37 + 3 * q^38 + 3 * q^41 - 12 * q^43 + 19 * q^44 - q^47 + 13 * q^49 - 26 * q^52 + 12 * q^53 + 8 * q^56 + 15 * q^59 - 7 * q^61 + 10 * q^62 - 9 * q^64 - 17 * q^67 - 43 * q^68 + 17 * q^71 - 10 * q^73 - 12 * q^74 - 34 * q^76 - 8 * q^77 - 2 * q^79 - 31 * q^82 + 19 * q^83 - 31 * q^86 + 3 * q^88 - 14 * q^89 - 15 * q^91 - q^92 + 27 * q^94 + 9 * q^97 - 51 * q^98

Introduction

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