LMFDB - Embedded newform 5625.2.a.bd.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5625,2,Mod(1,5625)] 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("5625.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5625, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,4,0,4,0,0,-8,12,0,0,-2,0,-16,-6,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(16)] == traces)

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

Self dual:	yes
Analytic conductor:	$44.9158511370$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.5444000000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 2x^{6} + 20x^{5} - 4x^{4} - 30x^{3} + 7x^{2} + 12x + 1 $ x^8 - 4x^7 - 2x^6 + 20x^5 - 4x^4 - 30x^3 + 7x^2 + 12*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 75)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$1.53655$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.53655 q^{2} +0.360976 q^{4} -1.49550 q^{7} -2.51844 q^{8} +2.35626 q^{11} -1.34951 q^{13} -2.29790 q^{14} -4.59165 q^{16} -2.19405 q^{17} +5.71069 q^{19} +3.62050 q^{22} +8.79501 q^{23} -2.07358 q^{26} -0.539839 q^{28} -7.90017 q^{29} -3.69717 q^{31} -2.01841 q^{32} -3.37126 q^{34} -9.75097 q^{37} +8.77474 q^{38} -1.85550 q^{41} +8.01874 q^{43} +0.850553 q^{44} +13.5139 q^{46} +6.66298 q^{47} -4.76349 q^{49} -0.487140 q^{52} +4.17153 q^{53} +3.76631 q^{56} -12.1390 q^{58} -11.0647 q^{59} -12.2372 q^{61} -5.68088 q^{62} +6.08192 q^{64} -4.31358 q^{67} -0.792000 q^{68} -5.77750 q^{71} -6.92684 q^{73} -14.9828 q^{74} +2.06142 q^{76} -3.52377 q^{77} -10.6687 q^{79} -2.85106 q^{82} +0.224003 q^{83} +12.3212 q^{86} -5.93408 q^{88} -0.429167 q^{89} +2.01818 q^{91} +3.17479 q^{92} +10.2380 q^{94} -12.4945 q^{97} -7.31933 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{2} + 4 q^{4} - 8 q^{7} + 12 q^{8} - 2 q^{11} - 16 q^{13} - 6 q^{14} + 16 q^{17} - 14 q^{19} - 12 q^{22} + 14 q^{23} - 6 q^{26} - 16 q^{28} - 2 q^{29} - 22 q^{31} - 2 q^{32} - 12 q^{34} - 28 q^{37}+ \cdots + 4 q^{98}+O(q^{100}) $ 8 * q + 4 * q^2 + 4 * q^4 - 8 * q^7 + 12 * q^8 - 2 * q^11 - 16 * q^13 - 6 * q^14 + 16 * q^17 - 14 * q^19 - 12 * q^22 + 14 * q^23 - 6 * q^26 - 16 * q^28 - 2 * q^29 - 22 * q^31 - 2 * q^32 - 12 * q^34 - 28 * q^37 - 16 * q^38 - 8 * q^41 - 20 * q^43 - 22 * q^44 - 2 * q^46 + 10 * q^47 - 16 * q^52 + 44 * q^53 - 30 * q^56 - 8 * q^58 - 14 * q^59 - 20 * q^61 + 16 * q^62 + 6 * q^64 - 16 * q^67 - 2 * q^68 - 16 * q^71 - 24 * q^73 - 26 * q^74 - 16 * q^76 + 46 * q^77 - 30 * q^79 - 16 * q^82 + 12 * q^83 - 32 * q^86 - 32 * q^88 - 16 * q^89 - 12 * q^91 - 2 * q^92 + 14 * q^94 - 16 * q^97 + 4 * 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 $

$2$	1.53655	1.08650	0.543251	−	0.839570i	$-0.317193\pi$
$2$	1.53655	1.08650	0.543251	+	0.839570i	$0.317193\pi$
$3$	0	0
$3$	0	0
$4$	0.360976	0.180488
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.49550	−0.565244	−0.282622	−	0.959231i	$-0.591204\pi$
$7$	−1.49550	−0.565244	−0.282622	+	0.959231i	$0.591204\pi$
$8$	−2.51844	−0.890402
$9$	0	0
$10$	0	0
$11$	2.35626	0.710438	0.355219	−	0.934783i	$-0.384406\pi$
$11$	2.35626	0.710438	0.355219	+	0.934783i	$0.384406\pi$
$12$	0	0
$13$	−1.34951	−0.374286	−0.187143	−	0.982333i	$-0.559923\pi$
$13$	−1.34951	−0.374286	−0.187143	+	0.982333i	$0.559923\pi$
$14$	−2.29790	−0.614140
$15$	0	0
$16$	−4.59165	−1.14791
$17$	−2.19405	−0.532135	−0.266068	−	0.963954i	$-0.585724\pi$
$17$	−2.19405	−0.532135	−0.266068	+	0.963954i	$0.585724\pi$
$18$	0	0
$19$	5.71069	1.31012	0.655061	−	0.755576i	$-0.272643\pi$
$19$	5.71069	1.31012	0.655061	+	0.755576i	$0.272643\pi$
$20$	0	0
$21$	0	0
$22$	3.62050	0.771892
$23$	8.79501	1.83389	0.916943	−	0.399018i	$-0.130649\pi$
$23$	8.79501	1.83389	0.916943	+	0.399018i	$0.130649\pi$
$24$	0	0
$25$	0	0
$26$	−2.07358	−0.406662
$27$	0	0
$28$	−0.539839	−0.102020
$29$	−7.90017	−1.46702	−0.733512	−	0.679676i	$-0.762120\pi$
$29$	−7.90017	−1.46702	−0.733512	+	0.679676i	$0.762120\pi$
$30$	0	0
$31$	−3.69717	−0.664031	−0.332016	−	0.943274i	$-0.607729\pi$
$31$	−3.69717	−0.664031	−0.332016	+	0.943274i	$0.607729\pi$
$32$	−2.01841	−0.356808
$33$	0	0
$34$	−3.37126	−0.578167
$35$	0	0
$36$	0	0
$37$	−9.75097	−1.60305	−0.801525	−	0.597962i	$-0.795977\pi$
$37$	−9.75097	−1.60305	−0.801525	+	0.597962i	$0.795977\pi$
$38$	8.77474	1.42345
$39$	0	0
$40$	0	0
$41$	−1.85550	−0.289780	−0.144890	−	0.989448i	$-0.546283\pi$
$41$	−1.85550	−0.289780	−0.144890	+	0.989448i	$0.546283\pi$
$42$	0	0
$43$	8.01874	1.22285	0.611423	−	0.791304i	$-0.290597\pi$
$43$	8.01874	1.22285	0.611423	+	0.791304i	$0.290597\pi$
$44$	0.850553	0.128226
$45$	0	0
$46$	13.5139	1.99252
$47$	6.66298	0.971895	0.485948	−	0.873988i	$-0.338475\pi$
$47$	6.66298	0.971895	0.485948	+	0.873988i	$0.338475\pi$
$48$	0	0
$49$	−4.76349	−0.680499
$50$	0	0
$51$	0	0
$52$	−0.487140	−0.0675542
$53$	4.17153	0.573003	0.286502	−	0.958080i	$-0.407508\pi$
$53$	4.17153	0.573003	0.286502	+	0.958080i	$0.407508\pi$
$54$	0	0
$55$	0	0
$56$	3.76631	0.503295
$57$	0	0
$58$	−12.1390	−1.59393
$59$	−11.0647	−1.44050	−0.720248	−	0.693716i	$-0.755972\pi$
$59$	−11.0647	−1.44050	−0.720248	+	0.693716i	$0.755972\pi$
$60$	0	0
$61$	−12.2372	−1.56682	−0.783408	−	0.621508i	$-0.786520\pi$
$61$	−12.2372	−1.56682	−0.783408	+	0.621508i	$0.786520\pi$
$62$	−5.68088	−0.721472
$63$	0	0
$64$	6.08192	0.760239
$65$	0	0
$66$	0	0
$67$	−4.31358	−0.526988	−0.263494	−	0.964661i	$-0.584875\pi$
$67$	−4.31358	−0.526988	−0.263494	+	0.964661i	$0.584875\pi$
$68$	−0.792000	−0.0960442
$69$	0	0
$70$	0	0
$71$	−5.77750	−0.685663	−0.342832	−	0.939397i	$-0.611386\pi$
$71$	−5.77750	−0.685663	−0.342832	+	0.939397i	$0.611386\pi$
$72$	0	0
$73$	−6.92684	−0.810725	−0.405362	−	0.914156i	$-0.632855\pi$
$73$	−6.92684	−0.810725	−0.405362	+	0.914156i	$0.632855\pi$
$74$	−14.9828	−1.74172
$75$	0	0
$76$	2.06142	0.236461
$77$	−3.52377	−0.401571
$78$	0	0
$79$	−10.6687	−1.20033	−0.600163	−	0.799878i	$-0.704898\pi$
$79$	−10.6687	−1.20033	−0.600163	+	0.799878i	$0.704898\pi$
$80$	0	0
$81$	0	0
$82$	−2.85106	−0.314846
$83$	0.224003	0.0245875	0.0122938	−	0.999924i	$-0.496087\pi$
$83$	0.224003	0.0245875	0.0122938	+	0.999924i	$0.496087\pi$
$84$	0	0
$85$	0	0
$86$	12.3212	1.32863
$87$	0	0
$88$	−5.93408	−0.632575
$89$	−0.429167	−0.0454916	−0.0227458	−	0.999741i	$-0.507241\pi$
$89$	−0.429167	−0.0454916	−0.0227458	+	0.999741i	$0.507241\pi$
$90$	0	0
$91$	2.01818	0.211563
$92$	3.17479	0.330995
$93$	0	0
$94$	10.2380	1.05597
$95$	0	0
$96$	0	0
$97$	−12.4945	−1.26863	−0.634315	−	0.773075i	$-0.718718\pi$
$97$	−12.4945	−1.26863	−0.634315	+	0.773075i	$0.718718\pi$
$98$	−7.31933	−0.739364
$99$	0	0
$100$	0	0
$101$	−8.19767	−0.815698	−0.407849	−	0.913049i	$-0.633721\pi$
$101$	−8.19767	−0.815698	−0.407849	+	0.913049i	$0.633721\pi$
$102$	0	0
$103$	−2.50005	−0.246337	−0.123169	−	0.992386i	$-0.539306\pi$
$103$	−2.50005	−0.246337	−0.123169	+	0.992386i	$0.539306\pi$
$104$	3.39865	0.333265
$105$	0	0
$106$	6.40975	0.622570
$107$	−1.81004	−0.174983	−0.0874914	−	0.996165i	$-0.527885\pi$
$107$	−1.81004	−0.174983	−0.0874914	+	0.996165i	$0.527885\pi$
$108$	0	0
$109$	−2.94778	−0.282346	−0.141173	−	0.989985i	$-0.545087\pi$
$109$	−2.94778	−0.282346	−0.141173	+	0.989985i	$0.545087\pi$
$110$	0	0
$111$	0	0
$112$	6.86679	0.648851
$113$	13.8365	1.30163	0.650813	−	0.759238i	$-0.274428\pi$
$113$	13.8365	1.30163	0.650813	+	0.759238i	$0.274428\pi$
$114$	0	0
$115$	0	0
$116$	−2.85178	−0.264781
$117$	0	0
$118$	−17.0014	−1.56510
$119$	3.28119	0.300787
$120$	0	0
$121$	−5.44806	−0.495278
$122$	−18.8031	−1.70235
$123$	0	0
$124$	−1.33459	−0.119850
$125$	0	0
$126$	0	0
$127$	−5.73995	−0.509338	−0.254669	−	0.967028i	$-0.581967\pi$
$127$	−5.73995	−0.509338	−0.254669	+	0.967028i	$0.581967\pi$
$128$	13.3820	1.18281
$129$	0	0
$130$	0	0
$131$	4.35756	0.380722	0.190361	−	0.981714i	$-0.439034\pi$
$131$	4.35756	0.380722	0.190361	+	0.981714i	$0.439034\pi$
$132$	0	0
$133$	−8.54031	−0.740539
$134$	−6.62802	−0.572574
$135$	0	0
$136$	5.52558	0.473814
$137$	1.97461	0.168702	0.0843510	−	0.996436i	$-0.473118\pi$
$137$	1.97461	0.168702	0.0843510	+	0.996436i	$0.473118\pi$
$138$	0	0
$139$	1.67910	0.142419	0.0712095	−	0.997461i	$-0.477314\pi$
$139$	1.67910	0.142419	0.0712095	+	0.997461i	$0.477314\pi$
$140$	0	0
$141$	0	0
$142$	−8.87740	−0.744975
$143$	−3.17978	−0.265907
$144$	0	0
$145$	0	0
$146$	−10.6434	−0.880855
$147$	0	0
$148$	−3.51987	−0.289331
$149$	7.38524	0.605023	0.302511	−	0.953146i	$-0.402175\pi$
$149$	7.38524	0.605023	0.302511	+	0.953146i	$0.402175\pi$
$150$	0	0
$151$	−4.26137	−0.346785	−0.173393	−	0.984853i	$-0.555473\pi$
$151$	−4.26137	−0.346785	−0.173393	+	0.984853i	$0.555473\pi$
$152$	−14.3820	−1.16653
$153$	0	0
$154$	−5.41444	−0.436308
$155$	0	0
$156$	0	0
$157$	−16.0573	−1.28152	−0.640758	−	0.767743i	$-0.721380\pi$
$157$	−16.0573	−1.28152	−0.640758	+	0.767743i	$0.721380\pi$
$158$	−16.3930	−1.30416
$159$	0	0
$160$	0	0
$161$	−13.1529	−1.03659
$162$	0	0
$163$	22.2938	1.74618	0.873092	−	0.487556i	$-0.162111\pi$
$163$	22.2938	1.74618	0.873092	+	0.487556i	$0.162111\pi$
$164$	−0.669790	−0.0523018
$165$	0	0
$166$	0.344191	0.0267144
$167$	−6.46601	−0.500355	−0.250177	−	0.968200i	$-0.580489\pi$
$167$	−6.46601	−0.500355	−0.250177	+	0.968200i	$0.580489\pi$
$168$	0	0
$169$	−11.1788	−0.859910
$170$	0	0
$171$	0	0
$172$	2.89458	0.220709
$173$	−11.8180	−0.898509	−0.449255	−	0.893404i	$-0.648310\pi$
$173$	−11.8180	−0.898509	−0.449255	+	0.893404i	$0.648310\pi$
$174$	0	0
$175$	0	0
$176$	−10.8191	−0.815520
$177$	0	0
$178$	−0.659435	−0.0494268
$179$	−15.5746	−1.16410	−0.582049	−	0.813154i	$-0.697749\pi$
$179$	−15.5746	−1.16410	−0.582049	+	0.813154i	$0.697749\pi$
$180$	0	0
$181$	−14.5797	−1.08370	−0.541851	−	0.840475i	$-0.682276\pi$
$181$	−14.5797	−1.08370	−0.541851	+	0.840475i	$0.682276\pi$
$182$	3.10103	0.229864
$183$	0	0
$184$	−22.1497	−1.63290
$185$	0	0
$186$	0	0
$187$	−5.16974	−0.378049
$188$	2.40518	0.175416
$189$	0	0
$190$	0	0
$191$	20.9884	1.51867	0.759333	−	0.650702i	$-0.225525\pi$
$191$	20.9884	1.51867	0.759333	+	0.650702i	$0.225525\pi$
$192$	0	0
$193$	−22.7094	−1.63466	−0.817328	−	0.576173i	$-0.804546\pi$
$193$	−22.7094	−1.63466	−0.817328	+	0.576173i	$0.804546\pi$
$194$	−19.1985	−1.37837
$195$	0	0
$196$	−1.71951	−0.122822
$197$	1.35341	0.0964268	0.0482134	−	0.998837i	$-0.484647\pi$
$197$	1.35341	0.0964268	0.0482134	+	0.998837i	$0.484647\pi$
$198$	0	0
$199$	−8.96061	−0.635201	−0.317600	−	0.948225i	$-0.602877\pi$
$199$	−8.96061	−0.635201	−0.317600	+	0.948225i	$0.602877\pi$
$200$	0	0
$201$	0	0
$202$	−12.5961	−0.886259
$203$	11.8147	0.829228
$204$	0	0
$205$	0	0
$206$	−3.84145	−0.267646
$207$	0	0
$208$	6.19646	0.429647
$209$	13.4558	0.930760
$210$	0	0
$211$	5.83983	0.402031	0.201015	−	0.979588i	$-0.435576\pi$
$211$	5.83983	0.402031	0.201015	+	0.979588i	$0.435576\pi$
$212$	1.50582	0.103420
$213$	0	0
$214$	−2.78121	−0.190119
$215$	0	0
$216$	0	0
$217$	5.52910	0.375340
$218$	−4.52939	−0.306769
$219$	0	0
$220$	0	0
$221$	2.96088	0.199171
$222$	0	0
$223$	7.82097	0.523731	0.261865	−	0.965104i	$-0.415662\pi$
$223$	7.82097	0.523731	0.261865	+	0.965104i	$0.415662\pi$
$224$	3.01853	0.201684
$225$	0	0
$226$	21.2604	1.41422
$227$	16.3090	1.08246	0.541232	−	0.840873i	$-0.317958\pi$
$227$	16.3090	1.08246	0.541232	+	0.840873i	$0.317958\pi$
$228$	0	0
$229$	−21.7088	−1.43456	−0.717278	−	0.696787i	$-0.754612\pi$
$229$	−21.7088	−1.43456	−0.717278	+	0.696787i	$0.754612\pi$
$230$	0	0
$231$	0	0
$232$	19.8961	1.30624
$233$	−13.5341	−0.886646	−0.443323	−	0.896362i	$-0.646201\pi$
$233$	−13.5341	−0.886646	−0.443323	+	0.896362i	$0.646201\pi$
$234$	0	0
$235$	0	0
$236$	−3.99408	−0.259993
$237$	0	0
$238$	5.04171	0.326805
$239$	10.5338	0.681377	0.340689	−	0.940176i	$-0.389340\pi$
$239$	10.5338	0.681377	0.340689	+	0.940176i	$0.389340\pi$
$240$	0	0
$241$	19.4838	1.25506	0.627530	−	0.778592i	$-0.284066\pi$
$241$	19.4838	1.25506	0.627530	+	0.778592i	$0.284066\pi$
$242$	−8.37120	−0.538121
$243$	0	0
$244$	−4.41735	−0.282792
$245$	0	0
$246$	0	0
$247$	−7.70661	−0.490360
$248$	9.31109	0.591255
$249$	0	0
$250$	0	0
$251$	20.9446	1.32201	0.661007	−	0.750380i	$-0.270129\pi$
$251$	20.9446	1.32201	0.661007	+	0.750380i	$0.270129\pi$
$252$	0	0
$253$	20.7233	1.30286
$254$	−8.81971	−0.553398
$255$	0	0
$256$	8.39819	0.524887
$257$	−1.67121	−0.104247	−0.0521237	−	0.998641i	$-0.516599\pi$
$257$	−1.67121	−0.104247	−0.0521237	+	0.998641i	$0.516599\pi$
$258$	0	0
$259$	14.5825	0.906115
$260$	0	0
$261$	0	0
$262$	6.69559	0.413655
$263$	7.55667	0.465964	0.232982	−	0.972481i	$-0.425152\pi$
$263$	7.55667	0.465964	0.232982	+	0.972481i	$0.425152\pi$
$264$	0	0
$265$	0	0
$266$	−13.1226	−0.804597
$267$	0	0
$268$	−1.55710	−0.0951151
$269$	11.2841	0.688005	0.344002	−	0.938969i	$-0.388217\pi$
$269$	11.2841	0.688005	0.344002	+	0.938969i	$0.388217\pi$
$270$	0	0
$271$	−10.9752	−0.666696	−0.333348	−	0.942804i	$-0.608178\pi$
$271$	−10.9752	−0.666696	−0.333348	+	0.942804i	$0.608178\pi$
$272$	10.0743	0.610845
$273$	0	0
$274$	3.03407	0.183295
$275$	0	0
$276$	0	0
$277$	5.75112	0.345551	0.172776	−	0.984961i	$-0.444726\pi$
$277$	5.75112	0.345551	0.172776	+	0.984961i	$0.444726\pi$
$278$	2.58001	0.154739
$279$	0	0
$280$	0	0
$281$	−8.49962	−0.507045	−0.253522	−	0.967330i	$-0.581589\pi$
$281$	−8.49962	−0.507045	−0.253522	+	0.967330i	$0.581589\pi$
$282$	0	0
$283$	17.2248	1.02391	0.511955	−	0.859013i	$-0.328922\pi$
$283$	17.2248	1.02391	0.511955	+	0.859013i	$0.328922\pi$
$284$	−2.08554	−0.123754
$285$	0	0
$286$	−4.88588	−0.288908
$287$	2.77489	0.163796
$288$	0	0
$289$	−12.1861	−0.716832
$290$	0	0
$291$	0	0
$292$	−2.50042	−0.146326
$293$	9.38764	0.548432	0.274216	−	0.961668i	$-0.411582\pi$
$293$	9.38764	0.548432	0.274216	+	0.961668i	$0.411582\pi$
$294$	0	0
$295$	0	0
$296$	24.5572	1.42736
$297$	0	0
$298$	11.3478	0.657359
$299$	−11.8689	−0.686397
$300$	0	0
$301$	−11.9920	−0.691207
$302$	−6.54779	−0.376783
$303$	0	0
$304$	−26.2215	−1.50390
$305$	0	0
$306$	0	0
$307$	−10.6465	−0.607627	−0.303814	−	0.952731i	$-0.598260\pi$
$307$	−10.6465	−0.607627	−0.303814	+	0.952731i	$0.598260\pi$
$308$	−1.27200	−0.0724788
$309$	0	0
$310$	0	0
$311$	27.3572	1.55128	0.775641	−	0.631174i	$-0.217427\pi$
$311$	27.3572	1.55128	0.775641	+	0.631174i	$0.217427\pi$
$312$	0	0
$313$	1.99509	0.112769	0.0563846	−	0.998409i	$-0.482043\pi$
$313$	1.99509	0.112769	0.0563846	+	0.998409i	$0.482043\pi$
$314$	−24.6729	−1.39237
$315$	0	0
$316$	−3.85116	−0.216645
$317$	8.64876	0.485763	0.242881	−	0.970056i	$-0.421907\pi$
$317$	8.64876	0.485763	0.242881	+	0.970056i	$0.421907\pi$
$318$	0	0
$319$	−18.6148	−1.04223
$320$	0	0
$321$	0	0
$322$	−20.2101	−1.12626
$323$	−12.5295	−0.697162
$324$	0	0
$325$	0	0
$326$	34.2554	1.89723
$327$	0	0
$328$	4.67295	0.258020
$329$	−9.96446	−0.549358
$330$	0	0
$331$	−3.94331	−0.216744	−0.108372	−	0.994110i	$-0.534564\pi$
$331$	−3.94331	−0.216744	−0.108372	+	0.994110i	$0.534564\pi$
$332$	0.0808598	0.00443776
$333$	0	0
$334$	−9.93532	−0.543637
$335$	0	0
$336$	0	0
$337$	−3.95471	−0.215427	−0.107713	−	0.994182i	$-0.534353\pi$
$337$	−3.95471	−0.215427	−0.107713	+	0.994182i	$0.534353\pi$
$338$	−17.1768	−0.934295
$339$	0	0
$340$	0	0
$341$	−8.71148	−0.471753
$342$	0	0
$343$	17.5923	0.949893
$344$	−20.1947	−1.08882
$345$	0	0
$346$	−18.1590	−0.976233
$347$	−9.99596	−0.536611	−0.268306	−	0.963334i	$-0.586464\pi$
$347$	−9.99596	−0.536611	−0.268306	+	0.963334i	$0.586464\pi$
$348$	0	0
$349$	18.4534	0.987789	0.493895	−	0.869522i	$-0.335573\pi$
$349$	18.4534	0.987789	0.493895	+	0.869522i	$0.335573\pi$
$350$	0	0
$351$	0	0
$352$	−4.75589	−0.253490
$353$	9.32398	0.496265	0.248133	−	0.968726i	$-0.420183\pi$
$353$	9.32398	0.496265	0.248133	+	0.968726i	$0.420183\pi$
$354$	0	0
$355$	0	0
$356$	−0.154919	−0.00821070
$357$	0	0
$358$	−23.9311	−1.26480
$359$	−28.9438	−1.52759	−0.763797	−	0.645457i	$-0.776667\pi$
$359$	−28.9438	−1.52759	−0.763797	+	0.645457i	$0.776667\pi$
$360$	0	0
$361$	13.6119	0.716418
$362$	−22.4024	−1.17744
$363$	0	0
$364$	0.728516	0.0381846
$365$	0	0
$366$	0	0
$367$	3.99869	0.208730	0.104365	−	0.994539i	$-0.466719\pi$
$367$	3.99869	0.208730	0.104365	+	0.994539i	$0.466719\pi$
$368$	−40.3836	−2.10514
$369$	0	0
$370$	0	0
$371$	−6.23850	−0.323887
$372$	0	0
$373$	3.17260	0.164271	0.0821354	−	0.996621i	$-0.473826\pi$
$373$	3.17260	0.164271	0.0821354	+	0.996621i	$0.473826\pi$
$374$	−7.94355	−0.410751
$375$	0	0
$376$	−16.7803	−0.865377
$377$	10.6613	0.549086
$378$	0	0
$379$	28.5206	1.46500	0.732501	−	0.680766i	$-0.238353\pi$
$379$	28.5206	1.46500	0.732501	+	0.680766i	$0.238353\pi$
$380$	0	0
$381$	0	0
$382$	32.2497	1.65004
$383$	−11.4611	−0.585636	−0.292818	−	0.956168i	$-0.594593\pi$
$383$	−11.4611	−0.585636	−0.292818	+	0.956168i	$0.594593\pi$
$384$	0	0
$385$	0	0
$386$	−34.8940	−1.77606
$387$	0	0
$388$	−4.51024	−0.228973
$389$	−34.2463	−1.73636	−0.868179	−	0.496252i	$-0.834709\pi$
$389$	−34.2463	−1.73636	−0.868179	+	0.496252i	$0.834709\pi$
$390$	0	0
$391$	−19.2967	−0.975876
$392$	11.9966	0.605917
$393$	0	0
$394$	2.07958	0.104768
$395$	0	0
$396$	0	0
$397$	20.0333	1.00545	0.502723	−	0.864448i	$-0.332332\pi$
$397$	20.0333	1.00545	0.502723	+	0.864448i	$0.332332\pi$
$398$	−13.7684	−0.690148
$399$	0	0
$400$	0	0
$401$	−4.98200	−0.248789	−0.124395	−	0.992233i	$-0.539699\pi$
$401$	−4.98200	−0.248789	−0.124395	+	0.992233i	$0.539699\pi$
$402$	0	0
$403$	4.98935	0.248537
$404$	−2.95916	−0.147224
$405$	0	0
$406$	18.1538	0.900958
$407$	−22.9758	−1.13887
$408$	0	0
$409$	23.8591	1.17976	0.589878	−	0.807493i	$-0.299176\pi$
$409$	23.8591	1.17976	0.589878	+	0.807493i	$0.299176\pi$
$410$	0	0
$411$	0	0
$412$	−0.902460	−0.0444610
$413$	16.5472	0.814233
$414$	0	0
$415$	0	0
$416$	2.72386	0.133548
$417$	0	0
$418$	20.6755	1.01127
$419$	−0.482550	−0.0235741	−0.0117871	−	0.999931i	$-0.503752\pi$
$419$	−0.482550	−0.0235741	−0.0117871	+	0.999931i	$0.503752\pi$
$420$	0	0
$421$	−17.7183	−0.863537	−0.431769	−	0.901984i	$-0.642110\pi$
$421$	−17.7183	−0.863537	−0.431769	+	0.901984i	$0.642110\pi$
$422$	8.97317	0.436807
$423$	0	0
$424$	−10.5057	−0.510203
$425$	0	0
$426$	0	0
$427$	18.3007	0.885634
$428$	−0.653380	−0.0315823
$429$	0	0
$430$	0	0
$431$	−26.8061	−1.29121	−0.645603	−	0.763673i	$-0.723394\pi$
$431$	−26.8061	−1.29121	−0.645603	+	0.763673i	$0.723394\pi$
$432$	0	0
$433$	9.23115	0.443621	0.221810	−	0.975090i	$-0.428803\pi$
$433$	9.23115	0.443621	0.221810	+	0.975090i	$0.428803\pi$
$434$	8.49573	0.407808
$435$	0	0
$436$	−1.06408	−0.0509601
$437$	50.2255	2.40261
$438$	0	0
$439$	1.00240	0.0478421	0.0239211	−	0.999714i	$-0.492385\pi$
$439$	1.00240	0.0478421	0.0239211	+	0.999714i	$0.492385\pi$
$440$	0	0
$441$	0	0
$442$	4.54954	0.216399
$443$	−26.2872	−1.24894	−0.624471	−	0.781048i	$-0.714685\pi$
$443$	−26.2872	−1.24894	−0.624471	+	0.781048i	$0.714685\pi$
$444$	0	0
$445$	0	0
$446$	12.0173	0.569035
$447$	0	0
$448$	−9.09548	−0.429721
$449$	−4.75449	−0.224378	−0.112189	−	0.993687i	$-0.535786\pi$
$449$	−4.75449	−0.224378	−0.112189	+	0.993687i	$0.535786\pi$
$450$	0	0
$451$	−4.37202	−0.205870
$452$	4.99464	0.234928
$453$	0	0
$454$	25.0595	1.17610
$455$	0	0
$456$	0	0
$457$	−15.9703	−0.747059	−0.373529	−	0.927618i	$-0.621852\pi$
$457$	−15.9703	−0.747059	−0.373529	+	0.927618i	$0.621852\pi$
$458$	−33.3566	−1.55865
$459$	0	0
$460$	0	0
$461$	−39.7558	−1.85161	−0.925806	−	0.377998i	$-0.876613\pi$
$461$	−39.7558	−1.85161	−0.925806	+	0.377998i	$0.876613\pi$
$462$	0	0
$463$	26.1209	1.21394	0.606971	−	0.794724i	$-0.292384\pi$
$463$	26.1209	1.21394	0.606971	+	0.794724i	$0.292384\pi$
$464$	36.2748	1.68402
$465$	0	0
$466$	−20.7957	−0.963343
$467$	3.85204	0.178251	0.0891256	−	0.996020i	$-0.471593\pi$
$467$	3.85204	0.178251	0.0891256	+	0.996020i	$0.471593\pi$
$468$	0	0
$469$	6.45095	0.297877
$470$	0	0
$471$	0	0
$472$	27.8657	1.28262
$473$	18.8942	0.868756
$474$	0	0
$475$	0	0
$476$	1.18443	0.0542884
$477$	0	0
$478$	16.1857	0.740318
$479$	14.2698	0.652004	0.326002	−	0.945369i	$-0.394298\pi$
$479$	14.2698	0.652004	0.326002	+	0.945369i	$0.394298\pi$
$480$	0	0
$481$	13.1590	0.599998
$482$	29.9377	1.36363
$483$	0	0
$484$	−1.96662	−0.0893919
$485$	0	0
$486$	0	0
$487$	24.8222	1.12480	0.562401	−	0.826865i	$-0.309878\pi$
$487$	24.8222	1.12480	0.562401	+	0.826865i	$0.309878\pi$
$488$	30.8187	1.39510
$489$	0	0
$490$	0	0
$491$	−11.4893	−0.518507	−0.259253	−	0.965809i	$-0.583476\pi$
$491$	−11.4893	−0.518507	−0.259253	+	0.965809i	$0.583476\pi$
$492$	0	0
$493$	17.3334	0.780656
$494$	−11.8416	−0.532777
$495$	0	0
$496$	16.9761	0.762250
$497$	8.64023	0.387567
$498$	0	0
$499$	−4.17487	−0.186893	−0.0934465	−	0.995624i	$-0.529788\pi$
$499$	−4.17487	−0.186893	−0.0934465	+	0.995624i	$0.529788\pi$
$500$	0	0
$501$	0	0
$502$	32.1824	1.43637
$503$	−38.7163	−1.72627	−0.863137	−	0.504970i	$-0.831503\pi$
$503$	−38.7163	−1.72627	−0.863137	+	0.504970i	$0.831503\pi$
$504$	0	0
$505$	0	0
$506$	31.8423	1.41556
$507$	0	0
$508$	−2.07199	−0.0919296
$509$	−30.1797	−1.33769	−0.668845	−	0.743402i	$-0.733211\pi$
$509$	−30.1797	−1.33769	−0.668845	+	0.743402i	$0.733211\pi$
$510$	0	0
$511$	10.3591	0.458258
$512$	−13.8597	−0.612519
$513$	0	0
$514$	−2.56790	−0.113265
$515$	0	0
$516$	0	0
$517$	15.6997	0.690471
$518$	22.4067	0.984496
$519$	0	0
$520$	0	0
$521$	25.4856	1.11654	0.558272	−	0.829658i	$-0.311464\pi$
$521$	25.4856	1.11654	0.558272	+	0.829658i	$0.311464\pi$
$522$	0	0
$523$	−3.89180	−0.170176	−0.0850882	−	0.996373i	$-0.527117\pi$
$523$	−3.89180	−0.170176	−0.0850882	+	0.996373i	$0.527117\pi$
$524$	1.57298	0.0687158
$525$	0	0
$526$	11.6112	0.506271
$527$	8.11178	0.353355
$528$	0	0
$529$	54.3522	2.36314
$530$	0	0
$531$	0	0
$532$	−3.08285	−0.133659
$533$	2.50400	0.108460
$534$	0	0
$535$	0	0
$536$	10.8635	0.469231
$537$	0	0
$538$	17.3386	0.747519
$539$	−11.2240	−0.483452
$540$	0	0
$541$	40.2148	1.72897	0.864484	−	0.502661i	$-0.167646\pi$
$541$	40.2148	1.72897	0.864484	+	0.502661i	$0.167646\pi$
$542$	−16.8639	−0.724367
$543$	0	0
$544$	4.42849	0.189870
$545$	0	0
$546$	0	0
$547$	−7.37923	−0.315513	−0.157757	−	0.987478i	$-0.550426\pi$
$547$	−7.37923	−0.315513	−0.157757	+	0.987478i	$0.550426\pi$
$548$	0.712786	0.0304487
$549$	0	0
$550$	0	0
$551$	−45.1154	−1.92198
$552$	0	0
$553$	15.9550	0.678478
$554$	8.83686	0.375442
$555$	0	0
$556$	0.606114	0.0257050
$557$	1.52499	0.0646160	0.0323080	−	0.999478i	$-0.489714\pi$
$557$	1.52499	0.0646160	0.0323080	+	0.999478i	$0.489714\pi$
$558$	0	0
$559$	−10.8213	−0.457694
$560$	0	0
$561$	0	0
$562$	−13.0601	−0.550905
$563$	13.7955	0.581411	0.290706	−	0.956813i	$-0.406110\pi$
$563$	13.7955	0.581411	0.290706	+	0.956813i	$0.406110\pi$
$564$	0	0
$565$	0	0
$566$	26.4667	1.11248
$567$	0	0
$568$	14.5503	0.610516
$569$	8.49904	0.356298	0.178149	−	0.984004i	$-0.442989\pi$
$569$	8.49904	0.356298	0.178149	+	0.984004i	$0.442989\pi$
$570$	0	0
$571$	39.3230	1.64562	0.822809	−	0.568319i	$-0.192406\pi$
$571$	39.3230	1.64562	0.822809	+	0.568319i	$0.192406\pi$
$572$	−1.14783	−0.0479930
$573$	0	0
$574$	4.26374	0.177965
$575$	0	0
$576$	0	0
$577$	−24.5832	−1.02341	−0.511707	−	0.859160i	$-0.670987\pi$
$577$	−24.5832	−1.02341	−0.511707	+	0.859160i	$0.670987\pi$
$578$	−18.7246	−0.778840
$579$	0	0
$580$	0	0
$581$	−0.334995	−0.0138980
$582$	0	0
$583$	9.82918	0.407083
$584$	17.4448	0.721871
$585$	0	0
$586$	14.4245	0.595872
$587$	−9.24270	−0.381487	−0.190744	−	0.981640i	$-0.561090\pi$
$587$	−9.24270	−0.381487	−0.190744	+	0.981640i	$0.561090\pi$
$588$	0	0
$589$	−21.1134	−0.869962
$590$	0	0
$591$	0	0
$592$	44.7730	1.84016
$593$	−6.07888	−0.249630	−0.124815	−	0.992180i	$-0.539834\pi$
$593$	−6.07888	−0.249630	−0.124815	+	0.992180i	$0.539834\pi$
$594$	0	0
$595$	0	0
$596$	2.66590	0.109199
$597$	0	0
$598$	−18.2372	−0.745773
$599$	−6.40129	−0.261550	−0.130775	−	0.991412i	$-0.541746\pi$
$599$	−6.40129	−0.261550	−0.130775	+	0.991412i	$0.541746\pi$
$600$	0	0
$601$	−38.4675	−1.56912	−0.784560	−	0.620052i	$-0.787111\pi$
$601$	−38.4675	−1.56912	−0.784560	+	0.620052i	$0.787111\pi$
$602$	−18.4263	−0.750999
$603$	0	0
$604$	−1.53825	−0.0625906
$605$	0	0
$606$	0	0
$607$	−5.22464	−0.212062	−0.106031	−	0.994363i	$-0.533814\pi$
$607$	−5.22464	−0.212062	−0.106031	+	0.994363i	$0.533814\pi$
$608$	−11.5265	−0.467462
$609$	0	0
$610$	0	0
$611$	−8.99173	−0.363766
$612$	0	0
$613$	−29.8153	−1.20423	−0.602114	−	0.798410i	$-0.705675\pi$
$613$	−29.8153	−1.20423	−0.602114	+	0.798410i	$0.705675\pi$
$614$	−16.3588	−0.660189
$615$	0	0
$616$	8.87439	0.357559
$617$	23.3025	0.938124	0.469062	−	0.883165i	$-0.344592\pi$
$617$	23.3025	0.938124	0.469062	+	0.883165i	$0.344592\pi$
$618$	0	0
$619$	−0.236492	−0.00950543	−0.00475272	−	0.999989i	$-0.501513\pi$
$619$	−0.236492	−0.00950543	−0.00475272	+	0.999989i	$0.501513\pi$
$620$	0	0
$621$	0	0
$622$	42.0356	1.68547
$623$	0.641818	0.0257139
$624$	0	0
$625$	0	0
$626$	3.06555	0.122524
$627$	0	0
$628$	−5.79632	−0.231298
$629$	21.3941	0.853039
$630$	0	0
$631$	−17.8789	−0.711748	−0.355874	−	0.934534i	$-0.615817\pi$
$631$	−17.8789	−0.711748	−0.355874	+	0.934534i	$0.615817\pi$
$632$	26.8685	1.06877
$633$	0	0
$634$	13.2892	0.527782
$635$	0	0
$636$	0	0
$637$	6.42836	0.254701
$638$	−28.6025	−1.13239
$639$	0	0
$640$	0	0
$641$	−10.8680	−0.429259	−0.214630	−	0.976695i	$-0.568854\pi$
$641$	−10.8680	−0.429259	−0.214630	+	0.976695i	$0.568854\pi$
$642$	0	0
$643$	−3.09039	−0.121873	−0.0609366	−	0.998142i	$-0.519409\pi$
$643$	−3.09039	−0.121873	−0.0609366	+	0.998142i	$0.519409\pi$
$644$	−4.74789	−0.187093
$645$	0	0
$646$	−19.2522	−0.757468
$647$	−5.53705	−0.217684	−0.108842	−	0.994059i	$-0.534714\pi$
$647$	−5.53705	−0.217684	−0.108842	+	0.994059i	$0.534714\pi$
$648$	0	0
$649$	−26.0712	−1.02338
$650$	0	0
$651$	0	0
$652$	8.04753	0.315166
$653$	38.8754	1.52131	0.760657	−	0.649154i	$-0.224877\pi$
$653$	38.8754	1.52131	0.760657	+	0.649154i	$0.224877\pi$
$654$	0	0
$655$	0	0
$656$	8.51978	0.332642
$657$	0	0
$658$	−15.3109	−0.596879
$659$	20.5389	0.800082	0.400041	−	0.916497i	$-0.368996\pi$
$659$	20.5389	0.800082	0.400041	+	0.916497i	$0.368996\pi$
$660$	0	0
$661$	38.4254	1.49457	0.747287	−	0.664502i	$-0.231356\pi$
$661$	38.4254	1.49457	0.747287	+	0.664502i	$0.231356\pi$
$662$	−6.05907	−0.235493
$663$	0	0
$664$	−0.564137	−0.0218928
$665$	0	0
$666$	0	0
$667$	−69.4821	−2.69036
$668$	−2.33408	−0.0903081
$669$	0	0
$670$	0	0
$671$	−28.8340	−1.11313
$672$	0	0
$673$	12.9819	0.500417	0.250208	−	0.968192i	$-0.419501\pi$
$673$	12.9819	0.500417	0.250208	+	0.968192i	$0.419501\pi$
$674$	−6.07660	−0.234062
$675$	0	0
$676$	−4.03529	−0.155204
$677$	7.23957	0.278239	0.139120	−	0.990276i	$-0.455573\pi$
$677$	7.23957	0.278239	0.139120	+	0.990276i	$0.455573\pi$
$678$	0	0
$679$	18.6855	0.717086
$680$	0	0
$681$	0	0
$682$	−13.3856	−0.512561
$683$	24.8800	0.952005	0.476003	−	0.879444i	$-0.342085\pi$
$683$	24.8800	0.952005	0.476003	+	0.879444i	$0.342085\pi$
$684$	0	0
$685$	0	0
$686$	27.0313	1.03206
$687$	0	0
$688$	−36.8192	−1.40372
$689$	−5.62950	−0.214467
$690$	0	0
$691$	−33.2706	−1.26567	−0.632836	−	0.774286i	$-0.718109\pi$
$691$	−33.2706	−1.26567	−0.632836	+	0.774286i	$0.718109\pi$
$692$	−4.26604	−0.162170
$693$	0	0
$694$	−15.3593	−0.583029
$695$	0	0
$696$	0	0
$697$	4.07105	0.154202
$698$	28.3546	1.07324
$699$	0	0
$700$	0	0
$701$	13.2163	0.499173	0.249586	−	0.968353i	$-0.419705\pi$
$701$	13.2163	0.499173	0.249586	+	0.968353i	$0.419705\pi$
$702$	0	0
$703$	−55.6847	−2.10019
$704$	14.3305	0.540103
$705$	0	0
$706$	14.3267	0.539194
$707$	12.2596	0.461069
$708$	0	0
$709$	6.25068	0.234749	0.117375	−	0.993088i	$-0.462552\pi$
$709$	6.25068	0.234749	0.117375	+	0.993088i	$0.462552\pi$
$710$	0	0
$711$	0	0
$712$	1.08083	0.0405058
$713$	−32.5167	−1.21776
$714$	0	0
$715$	0	0
$716$	−5.62205	−0.210106
$717$	0	0
$718$	−44.4735	−1.65973
$719$	−27.5016	−1.02564	−0.512818	−	0.858497i	$-0.671398\pi$
$719$	−27.5016	−1.02564	−0.512818	+	0.858497i	$0.671398\pi$
$720$	0	0
$721$	3.73882	0.139241
$722$	20.9154	0.778390
$723$	0	0
$724$	−5.26293	−0.195595
$725$	0	0
$726$	0	0
$727$	22.0397	0.817406	0.408703	−	0.912668i	$-0.365981\pi$
$727$	22.0397	0.817406	0.408703	+	0.912668i	$0.365981\pi$
$728$	−5.08266	−0.188376
$729$	0	0
$730$	0	0
$731$	−17.5935	−0.650720
$732$	0	0
$733$	−34.7134	−1.28217	−0.641085	−	0.767470i	$-0.721515\pi$
$733$	−34.7134	−1.28217	−0.641085	+	0.767470i	$0.721515\pi$
$734$	6.14417	0.226785
$735$	0	0
$736$	−17.7519	−0.654345
$737$	−10.1639	−0.374392
$738$	0	0
$739$	−7.58593	−0.279053	−0.139526	−	0.990218i	$-0.544558\pi$
$739$	−7.58593	−0.279053	−0.139526	+	0.990218i	$0.544558\pi$
$740$	0	0
$741$	0	0
$742$	−9.58575	−0.351904
$743$	27.5328	1.01008	0.505040	−	0.863096i	$-0.331478\pi$
$743$	27.5328	1.01008	0.505040	+	0.863096i	$0.331478\pi$
$744$	0	0
$745$	0	0
$746$	4.87484	0.178481
$747$	0	0
$748$	−1.86615	−0.0682334
$749$	2.70690	0.0989081
$750$	0	0
$751$	4.24930	0.155059	0.0775296	−	0.996990i	$-0.475297\pi$
$751$	4.24930	0.155059	0.0775296	+	0.996990i	$0.475297\pi$
$752$	−30.5940	−1.11565
$753$	0	0
$754$	16.3816	0.596584
$755$	0	0
$756$	0	0
$757$	45.6609	1.65957	0.829787	−	0.558081i	$-0.188462\pi$
$757$	45.6609	1.65957	0.829787	+	0.558081i	$0.188462\pi$
$758$	43.8232	1.59173
$759$	0	0
$760$	0	0
$761$	40.1268	1.45460	0.727298	−	0.686322i	$-0.240776\pi$
$761$	40.1268	1.45460	0.727298	+	0.686322i	$0.240776\pi$
$762$	0	0
$763$	4.40839	0.159594
$764$	7.57631	0.274101
$765$	0	0
$766$	−17.6105	−0.636295
$767$	14.9318	0.539157
$768$	0	0
$769$	−37.8350	−1.36437	−0.682183	−	0.731181i	$-0.738969\pi$
$769$	−37.8350	−1.36437	−0.682183	+	0.731181i	$0.738969\pi$
$770$	0	0
$771$	0	0
$772$	−8.19755	−0.295036
$773$	−26.2148	−0.942881	−0.471441	−	0.881898i	$-0.656266\pi$
$773$	−26.2148	−0.942881	−0.471441	+	0.881898i	$0.656266\pi$
$774$	0	0
$775$	0	0
$776$	31.4667	1.12959
$777$	0	0
$778$	−52.6211	−1.88656
$779$	−10.5962	−0.379646
$780$	0	0
$781$	−13.6133	−0.487121
$782$	−29.6503	−1.06029
$783$	0	0
$784$	21.8723	0.781153
$785$	0	0
$786$	0	0
$787$	2.41254	0.0859977	0.0429988	−	0.999075i	$-0.486309\pi$
$787$	2.41254	0.0859977	0.0429988	+	0.999075i	$0.486309\pi$
$788$	0.488551	0.0174039
$789$	0	0
$790$	0	0
$791$	−20.6924	−0.735737
$792$	0	0
$793$	16.5142	0.586437
$794$	30.7822	1.09242
$795$	0	0
$796$	−3.23457	−0.114646
$797$	8.11230	0.287353	0.143676	−	0.989625i	$-0.454108\pi$
$797$	8.11230	0.287353	0.143676	+	0.989625i	$0.454108\pi$
$798$	0	0
$799$	−14.6189	−0.517180
$800$	0	0
$801$	0	0
$802$	−7.65507	−0.270310
$803$	−16.3214	−0.575970
$804$	0	0
$805$	0	0
$806$	7.66638	0.270037
$807$	0	0
$808$	20.6453	0.726299
$809$	40.7091	1.43125	0.715627	−	0.698482i	$-0.246141\pi$
$809$	40.7091	1.43125	0.715627	+	0.698482i	$0.246141\pi$
$810$	0	0
$811$	−49.4990	−1.73814	−0.869072	−	0.494685i	$-0.835283\pi$
$811$	−49.4990	−1.73814	−0.869072	+	0.494685i	$0.835283\pi$
$812$	4.26482	0.149666
$813$	0	0
$814$	−35.3033	−1.23738
$815$	0	0
$816$	0	0
$817$	45.7925	1.60208
$818$	36.6606	1.28181
$819$	0	0
$820$	0	0
$821$	−42.2114	−1.47319	−0.736594	−	0.676335i	$-0.763567\pi$
$821$	−42.2114	−1.47319	−0.736594	+	0.676335i	$0.763567\pi$
$822$	0	0
$823$	49.2349	1.71622	0.858111	−	0.513464i	$-0.171638\pi$
$823$	49.2349	1.71622	0.858111	+	0.513464i	$0.171638\pi$
$824$	6.29622	0.219339
$825$	0	0
$826$	25.4255	0.884666
$827$	51.0011	1.77348	0.886742	−	0.462266i	$-0.152963\pi$
$827$	51.0011	1.77348	0.886742	+	0.462266i	$0.152963\pi$
$828$	0	0
$829$	−38.2342	−1.32793	−0.663964	−	0.747764i	$-0.731127\pi$
$829$	−38.2342	−1.32793	−0.663964	+	0.747764i	$0.731127\pi$
$830$	0	0
$831$	0	0
$832$	−8.20758	−0.284547
$833$	10.4513	0.362117
$834$	0	0
$835$	0	0
$836$	4.85724	0.167991
$837$	0	0
$838$	−0.741461	−0.0256133
$839$	16.7086	0.576845	0.288422	−	0.957503i	$-0.406869\pi$
$839$	16.7086	0.576845	0.288422	+	0.957503i	$0.406869\pi$
$840$	0	0
$841$	33.4127	1.15216
$842$	−27.2250	−0.938236
$843$	0	0
$844$	2.10804	0.0725618
$845$	0	0
$846$	0	0
$847$	8.14755	0.279953
$848$	−19.1542	−0.657757
$849$	0	0
$850$	0	0
$851$	−85.7599	−2.93981
$852$	0	0
$853$	18.2644	0.625361	0.312681	−	0.949858i	$-0.398773\pi$
$853$	18.2644	0.625361	0.312681	+	0.949858i	$0.398773\pi$
$854$	28.1199	0.962244
$855$	0	0
$856$	4.55846	0.155805
$857$	−53.4773	−1.82675	−0.913375	−	0.407119i	$-0.866534\pi$
$857$	−53.4773	−1.82675	−0.913375	+	0.407119i	$0.866534\pi$
$858$	0	0
$859$	−18.7575	−0.639998	−0.319999	−	0.947418i	$-0.603683\pi$
$859$	−18.7575	−0.639998	−0.319999	+	0.947418i	$0.603683\pi$
$860$	0	0
$861$	0	0
$862$	−41.1889	−1.40290
$863$	−51.2363	−1.74410	−0.872051	−	0.489414i	$-0.837211\pi$
$863$	−51.2363	−1.74410	−0.872051	+	0.489414i	$0.837211\pi$
$864$	0	0
$865$	0	0
$866$	14.1841	0.481995
$867$	0	0
$868$	1.99588	0.0677444
$869$	−25.1383	−0.852757
$870$	0	0
$871$	5.82121	0.197244
$872$	7.42378	0.251401
$873$	0	0
$874$	77.1739	2.61045
$875$	0	0
$876$	0	0
$877$	6.06306	0.204735	0.102368	−	0.994747i	$-0.467358\pi$
$877$	6.06306	0.204735	0.102368	+	0.994747i	$0.467358\pi$
$878$	1.54024	0.0519806
$879$	0	0
$880$	0	0
$881$	−22.6698	−0.763765	−0.381883	−	0.924211i	$-0.624724\pi$
$881$	−22.6698	−0.763765	−0.381883	+	0.924211i	$0.624724\pi$
$882$	0	0
$883$	5.53899	0.186402	0.0932009	−	0.995647i	$-0.470290\pi$
$883$	5.53899	0.186402	0.0932009	+	0.995647i	$0.470290\pi$
$884$	1.06881	0.0359480
$885$	0	0
$886$	−40.3915	−1.35698
$887$	−10.9716	−0.368390	−0.184195	−	0.982890i	$-0.558968\pi$
$887$	−10.9716	−0.368390	−0.184195	+	0.982890i	$0.558968\pi$
$888$	0	0
$889$	8.58408	0.287901
$890$	0	0
$891$	0	0
$892$	2.82319	0.0945272
$893$	38.0502	1.27330
$894$	0	0
$895$	0	0
$896$	−20.0127	−0.668577
$897$	0	0
$898$	−7.30550	−0.243788
$899$	29.2083	0.974151
$900$	0	0
$901$	−9.15254	−0.304915
$902$	−6.71781	−0.223679
$903$	0	0
$904$	−34.8463	−1.15897
$905$	0	0
$906$	0	0
$907$	19.3907	0.643856	0.321928	−	0.946764i	$-0.395669\pi$
$907$	19.3907	0.643856	0.321928	+	0.946764i	$0.395669\pi$
$908$	5.88715	0.195372
$909$	0	0
$910$	0	0
$911$	13.7190	0.454530	0.227265	−	0.973833i	$-0.427022\pi$
$911$	13.7190	0.454530	0.227265	+	0.973833i	$0.427022\pi$
$912$	0	0
$913$	0.527808	0.0174679
$914$	−24.5391	−0.811681
$915$	0	0
$916$	−7.83636	−0.258920
$917$	−6.51671	−0.215201
$918$	0	0
$919$	−32.4108	−1.06913	−0.534567	−	0.845126i	$-0.679525\pi$
$919$	−32.4108	−1.06913	−0.534567	+	0.845126i	$0.679525\pi$
$920$	0	0
$921$	0	0
$922$	−61.0867	−2.01178
$923$	7.79678	0.256634
$924$	0	0
$925$	0	0
$926$	40.1360	1.31895
$927$	0	0
$928$	15.9458	0.523446
$929$	4.33444	0.142208	0.0711042	−	0.997469i	$-0.477348\pi$
$929$	4.33444	0.142208	0.0711042	+	0.997469i	$0.477348\pi$
$930$	0	0
$931$	−27.2028	−0.891536
$932$	−4.88548	−0.160029
$933$	0	0
$934$	5.91884	0.193670
$935$	0	0
$936$	0	0
$937$	54.5925	1.78346	0.891730	−	0.452567i	$-0.149492\pi$
$937$	54.5925	1.78346	0.891730	+	0.452567i	$0.149492\pi$
$938$	9.91218	0.323644
$939$	0	0
$940$	0	0
$941$	−4.23853	−0.138172	−0.0690861	−	0.997611i	$-0.522008\pi$
$941$	−4.23853	−0.138172	−0.0690861	+	0.997611i	$0.522008\pi$
$942$	0	0
$943$	−16.3191	−0.531423
$944$	50.8051	1.65356
$945$	0	0
$946$	29.0318	0.943906
$947$	−12.6069	−0.409671	−0.204835	−	0.978796i	$-0.565666\pi$
$947$	−12.6069	−0.409671	−0.204835	+	0.978796i	$0.565666\pi$
$948$	0	0
$949$	9.34781	0.303443
$950$	0	0
$951$	0	0
$952$	−8.26348	−0.267821
$953$	31.1635	1.00948	0.504742	−	0.863270i	$-0.331588\pi$
$953$	31.1635	1.00948	0.504742	+	0.863270i	$0.331588\pi$
$954$	0	0
$955$	0	0
$956$	3.80247	0.122981
$957$	0	0
$958$	21.9262	0.708405
$959$	−2.95301	−0.0953578
$960$	0	0
$961$	−17.3309	−0.559062
$962$	20.2194	0.651900
$963$	0	0
$964$	7.03319	0.226524
$965$	0	0
$966$	0	0
$967$	1.55308	0.0499438	0.0249719	−	0.999688i	$-0.492050\pi$
$967$	1.55308	0.0499438	0.0249719	+	0.999688i	$0.492050\pi$
$968$	13.7206	0.440997
$969$	0	0
$970$	0	0
$971$	38.2623	1.22790	0.613948	−	0.789347i	$-0.289581\pi$
$971$	38.2623	1.22790	0.613948	+	0.789347i	$0.289581\pi$
$972$	0	0
$973$	−2.51108	−0.0805016
$974$	38.1405	1.22210
$975$	0	0
$976$	56.1890	1.79857
$977$	20.3661	0.651570	0.325785	−	0.945444i	$-0.394371\pi$
$977$	20.3661	0.651570	0.325785	+	0.945444i	$0.394371\pi$
$978$	0	0
$979$	−1.01123	−0.0323190
$980$	0	0
$981$	0	0
$982$	−17.6539	−0.563359
$983$	13.2557	0.422791	0.211395	−	0.977401i	$-0.432199\pi$
$983$	13.2557	0.422791	0.211395	+	0.977401i	$0.432199\pi$
$984$	0	0
$985$	0	0
$986$	26.6335	0.848185
$987$	0	0
$988$	−2.78190	−0.0885041
$989$	70.5249	2.24256
$990$	0	0
$991$	−2.24081	−0.0711816	−0.0355908	−	0.999366i	$-0.511331\pi$
$991$	−2.24081	−0.0711816	−0.0355908	+	0.999366i	$0.511331\pi$
$992$	7.46241	0.236932
$993$	0	0
$994$	13.2761	0.421093
$995$	0	0
$996$	0	0
$997$	−14.0404	−0.444663	−0.222331	−	0.974971i	$-0.571367\pi$
$997$	−14.0404	−0.444663	−0.222331	+	0.974971i	$0.571367\pi$
$998$	−6.41489	−0.203060
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.bd.1.6		8
3.2	odd	2		1875.2.a.m.1.3		8
5.4	even	2		5625.2.a.t.1.3		8
15.2	even	4		1875.2.b.h.1249.4		16
15.8	even	4		1875.2.b.h.1249.13		16
15.14	odd	2		1875.2.a.p.1.6		8
25.12	odd	20		225.2.m.b.19.1		16
25.23	odd	20		225.2.m.b.154.1		16
75.2	even	20		375.2.i.c.274.1		16
75.11	odd	10		375.2.g.e.226.2		16
75.14	odd	10		375.2.g.d.226.3		16
75.23	even	20		75.2.i.a.4.4	✓	16
75.38	even	20		375.2.i.c.349.1		16
75.41	odd	10		375.2.g.e.151.2		16
75.59	odd	10		375.2.g.d.151.3		16
75.62	even	20		75.2.i.a.19.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
75.2.i.a.4.4	✓	16	75.23	even	20
75.2.i.a.19.4	yes	16	75.62	even	20
225.2.m.b.19.1		16	25.12	odd	20
225.2.m.b.154.1		16	25.23	odd	20
375.2.g.d.151.3		16	75.59	odd	10
375.2.g.d.226.3		16	75.14	odd	10
375.2.g.e.151.2		16	75.41	odd	10
375.2.g.e.226.2		16	75.11	odd	10
375.2.i.c.274.1		16	75.2	even	20
375.2.i.c.349.1		16	75.38	even	20
1875.2.a.m.1.3		8	3.2	odd	2
1875.2.a.p.1.6		8	15.14	odd	2
1875.2.b.h.1249.4		16	15.2	even	4
1875.2.b.h.1249.13		16	15.8	even	4
5625.2.a.t.1.3		8	5.4	even	2
5625.2.a.bd.1.6		8	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q+1.53655 q^{2} +0.360976 q^{4} -1.49550 q^{7} -2.51844 q^{8} +2.35626 q^{11} -1.34951 q^{13} -2.29790 q^{14} -4.59165 q^{16} -2.19405 q^{17} +5.71069 q^{19} +3.62050 q^{22} +8.79501 q^{23} -2.07358 q^{26} -0.539839 q^{28} -7.90017 q^{29} -3.69717 q^{31} -2.01841 q^{32} -3.37126 q^{34} -9.75097 q^{37} +8.77474 q^{38} -1.85550 q^{41} +8.01874 q^{43} +0.850553 q^{44} +13.5139 q^{46} +6.66298 q^{47} -4.76349 q^{49} -0.487140 q^{52} +4.17153 q^{53} +3.76631 q^{56} -12.1390 q^{58} -11.0647 q^{59} -12.2372 q^{61} -5.68088 q^{62} +6.08192 q^{64} -4.31358 q^{67} -0.792000 q^{68} -5.77750 q^{71} -6.92684 q^{73} -14.9828 q^{74} +2.06142 q^{76} -3.52377 q^{77} -10.6687 q^{79} -2.85106 q^{82} +0.224003 q^{83} +12.3212 q^{86} -5.93408 q^{88} -0.429167 q^{89} +2.01818 q^{91} +3.17479 q^{92} +10.2380 q^{94} -12.4945 q^{97} -7.31933 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{2} + 4 q^{4} - 8 q^{7} + 12 q^{8} - 2 q^{11} - 16 q^{13} - 6 q^{14} + 16 q^{17} - 14 q^{19} - 12 q^{22} + 14 q^{23} - 6 q^{26} - 16 q^{28} - 2 q^{29} - 22 q^{31} - 2 q^{32} - 12 q^{34} - 28 q^{37}+ \cdots + 4 q^{98}+O(q^{100}) \) 8 * q + 4 * q^2 + 4 * q^4 - 8 * q^7 + 12 * q^8 - 2 * q^11 - 16 * q^13 - 6 * q^14 + 16 * q^17 - 14 * q^19 - 12 * q^22 + 14 * q^23 - 6 * q^26 - 16 * q^28 - 2 * q^29 - 22 * q^31 - 2 * q^32 - 12 * q^34 - 28 * q^37 - 16 * q^38 - 8 * q^41 - 20 * q^43 - 22 * q^44 - 2 * q^46 + 10 * q^47 - 16 * q^52 + 44 * q^53 - 30 * q^56 - 8 * q^58 - 14 * q^59 - 20 * q^61 + 16 * q^62 + 6 * q^64 - 16 * q^67 - 2 * q^68 - 16 * q^71 - 24 * q^73 - 26 * q^74 - 16 * q^76 + 46 * q^77 - 30 * q^79 - 16 * q^82 + 12 * q^83 - 32 * q^86 - 32 * q^88 - 16 * q^89 - 12 * q^91 - 2 * q^92 + 14 * q^94 - 16 * q^97 + 4 * q^98

Introduction

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