LMFDB - Embedded newform 6642.2.a.bu.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6642,2,Mod(1,6642)] 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("6642.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6642, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6642 = 2 \cdot 3^{4} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6642.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,-7,0,7,5,0,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$53.0366370225$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 15x^{5} + 11x^{4} + 58x^{3} - 12x^{2} - 60x - 6 $ x^7 - x^6 - 15x^5 + 11x^4 + 58x^3 - 12x^2 - 60*x - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 738)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$3.08538$ of defining polynomial
Character	$\chi$	$=$	6642.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{4} +0.571825 q^{5} -1.74317 q^{7} -1.00000 q^{8} -0.571825 q^{10} -3.95017 q^{11} -1.77039 q^{13} +1.74317 q^{14} +1.00000 q^{16} -5.91394 q^{17} -4.80043 q^{19} +0.571825 q^{20} +3.95017 q^{22} +6.31937 q^{23} -4.67302 q^{25} +1.77039 q^{26} -1.74317 q^{28} +9.23101 q^{29} -1.86357 q^{31} -1.00000 q^{32} +5.91394 q^{34} -0.996788 q^{35} +4.03784 q^{37} +4.80043 q^{38} -0.571825 q^{40} -1.00000 q^{41} -4.20700 q^{43} -3.95017 q^{44} -6.31937 q^{46} -10.3004 q^{47} -3.96136 q^{49} +4.67302 q^{50} -1.77039 q^{52} -4.65229 q^{53} -2.25881 q^{55} +1.74317 q^{56} -9.23101 q^{58} +15.0256 q^{59} -12.9575 q^{61} +1.86357 q^{62} +1.00000 q^{64} -1.01235 q^{65} -1.67811 q^{67} -5.91394 q^{68} +0.996788 q^{70} -13.2197 q^{71} +1.36938 q^{73} -4.03784 q^{74} -4.80043 q^{76} +6.88581 q^{77} +14.2153 q^{79} +0.571825 q^{80} +1.00000 q^{82} +14.6290 q^{83} -3.38174 q^{85} +4.20700 q^{86} +3.95017 q^{88} -0.353146 q^{89} +3.08609 q^{91} +6.31937 q^{92} +10.3004 q^{94} -2.74501 q^{95} +14.0488 q^{97} +3.96136 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 7 q^{2} + 7 q^{4} + 5 q^{5} + q^{7} - 7 q^{8} - 5 q^{10} + 10 q^{11} - 4 q^{13} - q^{14} + 7 q^{16} + 13 q^{17} - q^{19} + 5 q^{20} - 10 q^{22} + 15 q^{23} + 14 q^{25} + 4 q^{26} + q^{28} + 30 q^{29}+ \cdots + 12 q^{98}+O(q^{100}) $ 7 * q - 7 * q^2 + 7 * q^4 + 5 * q^5 + q^7 - 7 * q^8 - 5 * q^10 + 10 * q^11 - 4 * q^13 - q^14 + 7 * q^16 + 13 * q^17 - q^19 + 5 * q^20 - 10 * q^22 + 15 * q^23 + 14 * q^25 + 4 * q^26 + q^28 + 30 * q^29 + 13 * q^31 - 7 * q^32 - 13 * q^34 + 8 * q^35 + 7 * q^37 + q^38 - 5 * q^40 - 7 * q^41 - 5 * q^43 + 10 * q^44 - 15 * q^46 + 3 * q^47 - 12 * q^49 - 14 * q^50 - 4 * q^52 + 12 * q^53 - 13 * q^55 - q^56 - 30 * q^58 + 13 * q^59 - 55 * q^61 - 13 * q^62 + 7 * q^64 + 25 * q^65 - 15 * q^67 + 13 * q^68 - 8 * q^70 + 12 * q^71 - 7 * q^74 - q^76 - 5 * q^77 + 12 * q^79 + 5 * q^80 + 7 * q^82 + 36 * q^83 - 4 * q^85 + 5 * q^86 - 10 * q^88 + 16 * q^89 - 31 * q^91 + 15 * q^92 - 3 * q^94 + 43 * q^95 + q^97 + 12 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	0.571825	0.255728	0.127864	−	0.991792i	$-0.459188\pi$
$5$	0.571825	0.255728	0.127864	+	0.991792i	$0.459188\pi$
$6$	0	0
$7$	−1.74317	−0.658856	−0.329428	−	0.944181i	$-0.606856\pi$
$7$	−1.74317	−0.658856	−0.329428	+	0.944181i	$0.606856\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−0.571825	−0.180827
$11$	−3.95017	−1.19102	−0.595510	−	0.803348i	$-0.703050\pi$
$11$	−3.95017	−1.19102	−0.595510	+	0.803348i	$0.703050\pi$
$12$	0	0
$13$	−1.77039	−0.491018	−0.245509	−	0.969394i	$-0.578955\pi$
$13$	−1.77039	−0.491018	−0.245509	+	0.969394i	$0.578955\pi$
$14$	1.74317	0.465882
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.91394	−1.43434	−0.717170	−	0.696898i	$-0.754563\pi$
$17$	−5.91394	−1.43434	−0.717170	+	0.696898i	$0.754563\pi$
$18$	0	0
$19$	−4.80043	−1.10129	−0.550647	−	0.834738i	$-0.685619\pi$
$19$	−4.80043	−1.10129	−0.550647	+	0.834738i	$0.685619\pi$
$20$	0.571825	0.127864
$21$	0	0
$22$	3.95017	0.842179
$23$	6.31937	1.31768	0.658840	−	0.752283i	$-0.271047\pi$
$23$	6.31937	1.31768	0.658840	+	0.752283i	$0.271047\pi$
$24$	0	0
$25$	−4.67302	−0.934603
$26$	1.77039	0.347202
$27$	0	0
$28$	−1.74317	−0.329428
$29$	9.23101	1.71416	0.857078	−	0.515187i	$-0.172278\pi$
$29$	9.23101	1.71416	0.857078	+	0.515187i	$0.172278\pi$
$30$	0	0
$31$	−1.86357	−0.334707	−0.167354	−	0.985897i	$-0.553522\pi$
$31$	−1.86357	−0.334707	−0.167354	+	0.985897i	$0.553522\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	5.91394	1.01423
$35$	−0.996788	−0.168488
$36$	0	0
$37$	4.03784	0.663817	0.331908	−	0.943312i	$-0.392307\pi$
$37$	4.03784	0.663817	0.331908	+	0.943312i	$0.392307\pi$
$38$	4.80043	0.778733
$39$	0	0
$40$	−0.571825	−0.0904135
$41$	−1.00000	−0.156174
$41$	−1.00000	−0.156174
$42$	0	0
$43$	−4.20700	−0.641562	−0.320781	−	0.947153i	$-0.603945\pi$
$43$	−4.20700	−0.641562	−0.320781	+	0.947153i	$0.603945\pi$
$44$	−3.95017	−0.595510
$45$	0	0
$46$	−6.31937	−0.931741
$47$	−10.3004	−1.50247	−0.751233	−	0.660037i	$-0.770541\pi$
$47$	−10.3004	−1.50247	−0.751233	+	0.660037i	$0.770541\pi$
$48$	0	0
$49$	−3.96136	−0.565909
$50$	4.67302	0.660864
$51$	0	0
$52$	−1.77039	−0.245509
$53$	−4.65229	−0.639041	−0.319520	−	0.947579i	$-0.603522\pi$
$53$	−4.65229	−0.639041	−0.319520	+	0.947579i	$0.603522\pi$
$54$	0	0
$55$	−2.25881	−0.304577
$56$	1.74317	0.232941
$57$	0	0
$58$	−9.23101	−1.21209
$59$	15.0256	1.95616	0.978082	−	0.208219i	$-0.0667666\pi$
$59$	15.0256	1.95616	0.978082	+	0.208219i	$0.0667666\pi$
$60$	0	0
$61$	−12.9575	−1.65903	−0.829517	−	0.558481i	$-0.811384\pi$
$61$	−12.9575	−1.65903	−0.829517	+	0.558481i	$0.811384\pi$
$62$	1.86357	0.236674
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.01235	−0.125567
$66$	0	0
$67$	−1.67811	−0.205014	−0.102507	−	0.994732i	$-0.532686\pi$
$67$	−1.67811	−0.205014	−0.102507	+	0.994732i	$0.532686\pi$
$68$	−5.91394	−0.717170
$69$	0	0
$70$	0.996788	0.119139
$71$	−13.2197	−1.56889	−0.784446	−	0.620198i	$-0.787052\pi$
$71$	−13.2197	−1.56889	−0.784446	+	0.620198i	$0.787052\pi$
$72$	0	0
$73$	1.36938	0.160274	0.0801368	−	0.996784i	$-0.474464\pi$
$73$	1.36938	0.160274	0.0801368	+	0.996784i	$0.474464\pi$
$74$	−4.03784	−0.469389
$75$	0	0
$76$	−4.80043	−0.550647
$77$	6.88581	0.784711
$78$	0	0
$79$	14.2153	1.59935	0.799675	−	0.600433i	$-0.205005\pi$
$79$	14.2153	1.59935	0.799675	+	0.600433i	$0.205005\pi$
$80$	0.571825	0.0639320
$81$	0	0
$82$	1.00000	0.110432
$83$	14.6290	1.60574	0.802870	−	0.596155i	$-0.203305\pi$
$83$	14.6290	1.60574	0.802870	+	0.596155i	$0.203305\pi$
$84$	0	0
$85$	−3.38174	−0.366801
$86$	4.20700	0.453653
$87$	0	0
$88$	3.95017	0.421090
$89$	−0.353146	−0.0374334	−0.0187167	−	0.999825i	$-0.505958\pi$
$89$	−0.353146	−0.0374334	−0.0187167	+	0.999825i	$0.505958\pi$
$90$	0	0
$91$	3.08609	0.323510
$92$	6.31937	0.658840
$93$	0	0
$94$	10.3004	1.06240
$95$	−2.74501	−0.281632
$96$	0	0
$97$	14.0488	1.42644	0.713219	−	0.700942i	$-0.247237\pi$
$97$	14.0488	1.42644	0.713219	+	0.700942i	$0.247237\pi$
$98$	3.96136	0.400158
$99$	0	0
$100$	−4.67302	−0.467302
$101$	11.6537	1.15959	0.579795	−	0.814762i	$-0.303133\pi$
$101$	11.6537	1.15959	0.579795	+	0.814762i	$0.303133\pi$
$102$	0	0
$103$	10.7908	1.06325	0.531626	−	0.846979i	$-0.321581\pi$
$103$	10.7908	1.06325	0.531626	+	0.846979i	$0.321581\pi$
$104$	1.77039	0.173601
$105$	0	0
$106$	4.65229	0.451870
$107$	9.98635	0.965417	0.482708	−	0.875781i	$-0.339653\pi$
$107$	9.98635	0.965417	0.482708	+	0.875781i	$0.339653\pi$
$108$	0	0
$109$	−16.6089	−1.59085	−0.795423	−	0.606054i	$-0.792751\pi$
$109$	−16.6089	−1.59085	−0.795423	+	0.606054i	$0.792751\pi$
$110$	2.25881	0.215369
$111$	0	0
$112$	−1.74317	−0.164714
$113$	5.65037	0.531542	0.265771	−	0.964036i	$-0.414374\pi$
$113$	5.65037	0.531542	0.265771	+	0.964036i	$0.414374\pi$
$114$	0	0
$115$	3.61358	0.336968
$116$	9.23101	0.857078
$117$	0	0
$118$	−15.0256	−1.38322
$119$	10.3090	0.945024
$120$	0	0
$121$	4.60384	0.418531
$122$	12.9575	1.17311
$123$	0	0
$124$	−1.86357	−0.167354
$125$	−5.53127	−0.494732
$126$	0	0
$127$	4.59502	0.407742	0.203871	−	0.978998i	$-0.434648\pi$
$127$	4.59502	0.407742	0.203871	+	0.978998i	$0.434648\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	1.01235	0.0887892
$131$	7.67818	0.670846	0.335423	−	0.942068i	$-0.391121\pi$
$131$	7.67818	0.670846	0.335423	+	0.942068i	$0.391121\pi$
$132$	0	0
$133$	8.36796	0.725594
$134$	1.67811	0.144966
$135$	0	0
$136$	5.91394	0.507116
$137$	−17.1341	−1.46386	−0.731932	−	0.681378i	$-0.761381\pi$
$137$	−17.1341	−1.46386	−0.731932	+	0.681378i	$0.761381\pi$
$138$	0	0
$139$	5.08870	0.431618	0.215809	−	0.976436i	$-0.430761\pi$
$139$	5.08870	0.431618	0.215809	+	0.976436i	$0.430761\pi$
$140$	−0.996788	−0.0842439
$141$	0	0
$142$	13.2197	1.10937
$143$	6.99334	0.584812
$144$	0	0
$145$	5.27852	0.438357
$146$	−1.36938	−0.113331
$147$	0	0
$148$	4.03784	0.331908
$149$	9.57815	0.784673	0.392336	−	0.919822i	$-0.371667\pi$
$149$	9.57815	0.784673	0.392336	+	0.919822i	$0.371667\pi$
$150$	0	0
$151$	−11.2568	−0.916065	−0.458033	−	0.888935i	$-0.651446\pi$
$151$	−11.2568	−0.916065	−0.458033	+	0.888935i	$0.651446\pi$
$152$	4.80043	0.389366
$153$	0	0
$154$	−6.88581	−0.554875
$155$	−1.06564	−0.0855941
$156$	0	0
$157$	9.37615	0.748298	0.374149	−	0.927369i	$-0.377935\pi$
$157$	9.37615	0.748298	0.374149	+	0.927369i	$0.377935\pi$
$158$	−14.2153	−1.13091
$159$	0	0
$160$	−0.571825	−0.0452067
$161$	−11.0157	−0.868162
$162$	0	0
$163$	−9.75793	−0.764300	−0.382150	−	0.924100i	$-0.624816\pi$
$163$	−9.75793	−0.764300	−0.382150	+	0.924100i	$0.624816\pi$
$164$	−1.00000	−0.0780869
$165$	0	0
$166$	−14.6290	−1.13543
$167$	−2.55589	−0.197781	−0.0988904	−	0.995098i	$-0.531529\pi$
$167$	−2.55589	−0.197781	−0.0988904	+	0.995098i	$0.531529\pi$
$168$	0	0
$169$	−9.86572	−0.758902
$170$	3.38174	0.259367
$171$	0	0
$172$	−4.20700	−0.320781
$173$	7.44254	0.565845	0.282923	−	0.959143i	$-0.408696\pi$
$173$	7.44254	0.565845	0.282923	+	0.959143i	$0.408696\pi$
$174$	0	0
$175$	8.14586	0.615769
$176$	−3.95017	−0.297755
$177$	0	0
$178$	0.353146	0.0264694
$179$	20.6356	1.54238	0.771188	−	0.636607i	$-0.219663\pi$
$179$	20.6356	1.54238	0.771188	+	0.636607i	$0.219663\pi$
$180$	0	0
$181$	2.50863	0.186465	0.0932324	−	0.995644i	$-0.470280\pi$
$181$	2.50863	0.186465	0.0932324	+	0.995644i	$0.470280\pi$
$182$	−3.08609	−0.228756
$183$	0	0
$184$	−6.31937	−0.465870
$185$	2.30894	0.169757
$186$	0	0
$187$	23.3611	1.70833
$188$	−10.3004	−0.751233
$189$	0	0
$190$	2.74501	0.199144
$191$	−16.4651	−1.19138	−0.595688	−	0.803216i	$-0.703120\pi$
$191$	−16.4651	−1.19138	−0.595688	+	0.803216i	$0.703120\pi$
$192$	0	0
$193$	−17.3139	−1.24628	−0.623139	−	0.782111i	$-0.714143\pi$
$193$	−17.3139	−1.24628	−0.623139	+	0.782111i	$0.714143\pi$
$194$	−14.0488	−1.00864
$195$	0	0
$196$	−3.96136	−0.282954
$197$	22.5592	1.60728	0.803638	−	0.595119i	$-0.202895\pi$
$197$	22.5592	1.60728	0.803638	+	0.595119i	$0.202895\pi$
$198$	0	0
$199$	9.81568	0.695815	0.347908	−	0.937529i	$-0.386892\pi$
$199$	9.81568	0.695815	0.347908	+	0.937529i	$0.386892\pi$
$200$	4.67302	0.330432
$201$	0	0
$202$	−11.6537	−0.819954
$203$	−16.0912	−1.12938
$204$	0	0
$205$	−0.571825	−0.0399380
$206$	−10.7908	−0.751833
$207$	0	0
$208$	−1.77039	−0.122754
$209$	18.9625	1.31166
$210$	0	0
$211$	8.25850	0.568539	0.284269	−	0.958744i	$-0.408249\pi$
$211$	8.25850	0.568539	0.284269	+	0.958744i	$0.408249\pi$
$212$	−4.65229	−0.319520
$213$	0	0
$214$	−9.98635	−0.682653
$215$	−2.40567	−0.164065
$216$	0	0
$217$	3.24852	0.220524
$218$	16.6089	1.12490
$219$	0	0
$220$	−2.25881	−0.152289
$221$	10.4700	0.704286
$222$	0	0
$223$	−17.6663	−1.18302	−0.591511	−	0.806297i	$-0.701468\pi$
$223$	−17.6663	−1.18302	−0.591511	+	0.806297i	$0.701468\pi$
$224$	1.74317	0.116470
$225$	0	0
$226$	−5.65037	−0.375857
$227$	6.16256	0.409024	0.204512	−	0.978864i	$-0.434439\pi$
$227$	6.16256	0.409024	0.204512	+	0.978864i	$0.434439\pi$
$228$	0	0
$229$	−18.8703	−1.24698	−0.623491	−	0.781830i	$-0.714286\pi$
$229$	−18.8703	−1.24698	−0.623491	+	0.781830i	$0.714286\pi$
$230$	−3.61358	−0.238272
$231$	0	0
$232$	−9.23101	−0.606045
$233$	27.5072	1.80205	0.901027	−	0.433763i	$-0.142814\pi$
$233$	27.5072	1.80205	0.901027	+	0.433763i	$0.142814\pi$
$234$	0	0
$235$	−5.89002	−0.384223
$236$	15.0256	0.978082
$237$	0	0
$238$	−10.3090	−0.668233
$239$	18.0962	1.17055	0.585273	−	0.810837i	$-0.300987\pi$
$239$	18.0962	1.17055	0.585273	+	0.810837i	$0.300987\pi$
$240$	0	0
$241$	−12.9959	−0.837140	−0.418570	−	0.908185i	$-0.637469\pi$
$241$	−12.9959	−0.837140	−0.418570	+	0.908185i	$0.637469\pi$
$242$	−4.60384	−0.295946
$243$	0	0
$244$	−12.9575	−0.829517
$245$	−2.26521	−0.144719
$246$	0	0
$247$	8.49863	0.540755
$248$	1.86357	0.118337
$249$	0	0
$250$	5.53127	0.349828
$251$	15.6586	0.988359	0.494180	−	0.869360i	$-0.335469\pi$
$251$	15.6586	0.988359	0.494180	+	0.869360i	$0.335469\pi$
$252$	0	0
$253$	−24.9626	−1.56939
$254$	−4.59502	−0.288317
$255$	0	0
$256$	1.00000	0.0625000
$257$	27.0854	1.68954	0.844771	−	0.535128i	$-0.179737\pi$
$257$	27.0854	1.68954	0.844771	+	0.535128i	$0.179737\pi$
$258$	0	0
$259$	−7.03864	−0.437360
$260$	−1.01235	−0.0627835
$261$	0	0
$262$	−7.67818	−0.474359
$263$	13.2403	0.816430	0.408215	−	0.912886i	$-0.366151\pi$
$263$	13.2403	0.816430	0.408215	+	0.912886i	$0.366151\pi$
$264$	0	0
$265$	−2.66029	−0.163421
$266$	−8.36796	−0.513073
$267$	0	0
$268$	−1.67811	−0.102507
$269$	−9.00627	−0.549122	−0.274561	−	0.961570i	$-0.588533\pi$
$269$	−9.00627	−0.549122	−0.274561	+	0.961570i	$0.588533\pi$
$270$	0	0
$271$	8.43688	0.512504	0.256252	−	0.966610i	$-0.417512\pi$
$271$	8.43688	0.512504	0.256252	+	0.966610i	$0.417512\pi$
$272$	−5.91394	−0.358585
$273$	0	0
$274$	17.1341	1.03511
$275$	18.4592	1.11313
$276$	0	0
$277$	−0.111475	−0.00669791	−0.00334896	−	0.999994i	$-0.501066\pi$
$277$	−0.111475	−0.00669791	−0.00334896	+	0.999994i	$0.501066\pi$
$278$	−5.08870	−0.305200
$279$	0	0
$280$	0.996788	0.0595695
$281$	−7.73748	−0.461580	−0.230790	−	0.973004i	$-0.574131\pi$
$281$	−7.73748	−0.461580	−0.230790	+	0.973004i	$0.574131\pi$
$282$	0	0
$283$	−7.71175	−0.458416	−0.229208	−	0.973377i	$-0.573614\pi$
$283$	−7.71175	−0.458416	−0.229208	+	0.973377i	$0.573614\pi$
$284$	−13.2197	−0.784446
$285$	0	0
$286$	−6.99334	−0.413525
$287$	1.74317	0.102896
$288$	0	0
$289$	17.9746	1.05733
$290$	−5.27852	−0.309966
$291$	0	0
$292$	1.36938	0.0801368
$293$	−8.10536	−0.473520	−0.236760	−	0.971568i	$-0.576086\pi$
$293$	−8.10536	−0.473520	−0.236760	+	0.971568i	$0.576086\pi$
$294$	0	0
$295$	8.59201	0.500246
$296$	−4.03784	−0.234695
$297$	0	0
$298$	−9.57815	−0.554847
$299$	−11.1878	−0.647004
$300$	0	0
$301$	7.33351	0.422697
$302$	11.2568	0.647756
$303$	0	0
$304$	−4.80043	−0.275324
$305$	−7.40941	−0.424262
$306$	0	0
$307$	10.1284	0.578061	0.289031	−	0.957320i	$-0.406667\pi$
$307$	10.1284	0.578061	0.289031	+	0.957320i	$0.406667\pi$
$308$	6.88581	0.392356
$309$	0	0
$310$	1.06564	0.0605241
$311$	−17.8551	−1.01247	−0.506234	−	0.862396i	$-0.668963\pi$
$311$	−17.8551	−1.01247	−0.506234	+	0.862396i	$0.668963\pi$
$312$	0	0
$313$	−0.424994	−0.0240221	−0.0120110	−	0.999928i	$-0.503823\pi$
$313$	−0.424994	−0.0240221	−0.0120110	+	0.999928i	$0.503823\pi$
$314$	−9.37615	−0.529127
$315$	0	0
$316$	14.2153	0.799675
$317$	−14.8450	−0.833779	−0.416889	−	0.908957i	$-0.636880\pi$
$317$	−14.8450	−0.833779	−0.416889	+	0.908957i	$0.636880\pi$
$318$	0	0
$319$	−36.4641	−2.04159
$320$	0.571825	0.0319660
$321$	0	0
$322$	11.0157	0.613883
$323$	28.3894	1.57963
$324$	0	0
$325$	8.27306	0.458907
$326$	9.75793	0.540442
$327$	0	0
$328$	1.00000	0.0552158
$329$	17.9553	0.989909
$330$	0	0
$331$	−27.2787	−1.49937	−0.749686	−	0.661794i	$-0.769795\pi$
$331$	−27.2787	−1.49937	−0.749686	+	0.661794i	$0.769795\pi$
$332$	14.6290	0.802870
$333$	0	0
$334$	2.55589	0.139852
$335$	−0.959584	−0.0524277
$336$	0	0
$337$	19.9306	1.08569	0.542844	−	0.839833i	$-0.317347\pi$
$337$	19.9306	1.08569	0.542844	+	0.839833i	$0.317347\pi$
$338$	9.86572	0.536625
$339$	0	0
$340$	−3.38174	−0.183400
$341$	7.36143	0.398644
$342$	0	0
$343$	19.1075	1.03171
$344$	4.20700	0.226826
$345$	0	0
$346$	−7.44254	−0.400113
$347$	−18.1282	−0.973174	−0.486587	−	0.873632i	$-0.661758\pi$
$347$	−18.1282	−0.973174	−0.486587	+	0.873632i	$0.661758\pi$
$348$	0	0
$349$	31.4839	1.68529	0.842646	−	0.538467i	$-0.180996\pi$
$349$	31.4839	1.68529	0.842646	+	0.538467i	$0.180996\pi$
$350$	−8.14586	−0.435414
$351$	0	0
$352$	3.95017	0.210545
$353$	6.61288	0.351968	0.175984	−	0.984393i	$-0.443689\pi$
$353$	6.61288	0.351968	0.175984	+	0.984393i	$0.443689\pi$
$354$	0	0
$355$	−7.55936	−0.401209
$356$	−0.353146	−0.0187167
$357$	0	0
$358$	−20.6356	−1.09063
$359$	20.8524	1.10055	0.550275	−	0.834984i	$-0.314523\pi$
$359$	20.8524	1.10055	0.550275	+	0.834984i	$0.314523\pi$
$360$	0	0
$361$	4.04413	0.212849
$362$	−2.50863	−0.131851
$363$	0	0
$364$	3.08609	0.161755
$365$	0.783045	0.0409864
$366$	0	0
$367$	24.1222	1.25917	0.629584	−	0.776933i	$-0.283226\pi$
$367$	24.1222	1.25917	0.629584	+	0.776933i	$0.283226\pi$
$368$	6.31937	0.329420
$369$	0	0
$370$	−2.30894	−0.120036
$371$	8.10972	0.421036
$372$	0	0
$373$	8.10825	0.419829	0.209915	−	0.977720i	$-0.432681\pi$
$373$	8.10825	0.419829	0.209915	+	0.977720i	$0.432681\pi$
$374$	−23.3611	−1.20797
$375$	0	0
$376$	10.3004	0.531202
$377$	−16.3425	−0.841681
$378$	0	0
$379$	22.5064	1.15608	0.578039	−	0.816009i	$-0.303818\pi$
$379$	22.5064	1.15608	0.578039	+	0.816009i	$0.303818\pi$
$380$	−2.74501	−0.140816
$381$	0	0
$382$	16.4651	0.842429
$383$	3.44414	0.175987	0.0879936	−	0.996121i	$-0.471955\pi$
$383$	3.44414	0.175987	0.0879936	+	0.996121i	$0.471955\pi$
$384$	0	0
$385$	3.93748	0.200673
$386$	17.3139	0.881252
$387$	0	0
$388$	14.0488	0.713219
$389$	−29.7239	−1.50706	−0.753531	−	0.657412i	$-0.771651\pi$
$389$	−29.7239	−1.50706	−0.753531	+	0.657412i	$0.771651\pi$
$390$	0	0
$391$	−37.3724	−1.89000
$392$	3.96136	0.200079
$393$	0	0
$394$	−22.5592	−1.13652
$395$	8.12868	0.408999
$396$	0	0
$397$	−18.6701	−0.937027	−0.468513	−	0.883456i	$-0.655210\pi$
$397$	−18.6701	−0.937027	−0.468513	+	0.883456i	$0.655210\pi$
$398$	−9.81568	−0.492016
$399$	0	0
$400$	−4.67302	−0.233651
$401$	−11.3944	−0.569007	−0.284503	−	0.958675i	$-0.591829\pi$
$401$	−11.3944	−0.569007	−0.284503	+	0.958675i	$0.591829\pi$
$402$	0	0
$403$	3.29925	0.164347
$404$	11.6537	0.579795
$405$	0	0
$406$	16.0912	0.798593
$407$	−15.9502	−0.790620
$408$	0	0
$409$	11.8888	0.587864	0.293932	−	0.955826i	$-0.405036\pi$
$409$	11.8888	0.587864	0.293932	+	0.955826i	$0.405036\pi$
$410$	0.571825	0.0282404
$411$	0	0
$412$	10.7908	0.531626
$413$	−26.1921	−1.28883
$414$	0	0
$415$	8.36522	0.410632
$416$	1.77039	0.0868005
$417$	0	0
$418$	−18.9625	−0.927487
$419$	−5.48672	−0.268044	−0.134022	−	0.990978i	$-0.542789\pi$
$419$	−5.48672	−0.268044	−0.134022	+	0.990978i	$0.542789\pi$
$420$	0	0
$421$	−5.04657	−0.245955	−0.122977	−	0.992409i	$-0.539244\pi$
$421$	−5.04657	−0.245955	−0.122977	+	0.992409i	$0.539244\pi$
$422$	−8.25850	−0.402017
$423$	0	0
$424$	4.65229	0.225935
$425$	27.6359	1.34054
$426$	0	0
$427$	22.5871	1.09306
$428$	9.98635	0.482708
$429$	0	0
$430$	2.40567	0.116012
$431$	3.54756	0.170880	0.0854399	−	0.996343i	$-0.472770\pi$
$431$	3.54756	0.170880	0.0854399	+	0.996343i	$0.472770\pi$
$432$	0	0
$433$	−3.74015	−0.179740	−0.0898702	−	0.995953i	$-0.528645\pi$
$433$	−3.74015	−0.179740	−0.0898702	+	0.995953i	$0.528645\pi$
$434$	−3.24852	−0.155934
$435$	0	0
$436$	−16.6089	−0.795423
$437$	−30.3357	−1.45115
$438$	0	0
$439$	−10.8710	−0.518845	−0.259423	−	0.965764i	$-0.583532\pi$
$439$	−10.8710	−0.518845	−0.259423	+	0.965764i	$0.583532\pi$
$440$	2.25881	0.107684
$441$	0	0
$442$	−10.4700	−0.498006
$443$	−9.75286	−0.463372	−0.231686	−	0.972791i	$-0.574424\pi$
$443$	−9.75286	−0.463372	−0.231686	+	0.972791i	$0.574424\pi$
$444$	0	0
$445$	−0.201938	−0.00957277
$446$	17.6663	0.836523
$447$	0	0
$448$	−1.74317	−0.0823570
$449$	−37.7704	−1.78249	−0.891247	−	0.453518i	$-0.850169\pi$
$449$	−37.7704	−1.78249	−0.891247	+	0.453518i	$0.850169\pi$
$450$	0	0
$451$	3.95017	0.186006
$452$	5.65037	0.265771
$453$	0	0
$454$	−6.16256	−0.289223
$455$	1.76470	0.0827305
$456$	0	0
$457$	−16.2991	−0.762440	−0.381220	−	0.924484i	$-0.624496\pi$
$457$	−16.2991	−0.762440	−0.381220	+	0.924484i	$0.624496\pi$
$458$	18.8703	0.881750
$459$	0	0
$460$	3.61358	0.168484
$461$	−8.27402	−0.385360	−0.192680	−	0.981262i	$-0.561718\pi$
$461$	−8.27402	−0.385360	−0.192680	+	0.981262i	$0.561718\pi$
$462$	0	0
$463$	19.3692	0.900164	0.450082	−	0.892987i	$-0.351395\pi$
$463$	19.3692	0.900164	0.450082	+	0.892987i	$0.351395\pi$
$464$	9.23101	0.428539
$465$	0	0
$466$	−27.5072	−1.27424
$467$	−20.5070	−0.948950	−0.474475	−	0.880269i	$-0.657362\pi$
$467$	−20.5070	−0.948950	−0.474475	+	0.880269i	$0.657362\pi$
$468$	0	0
$469$	2.92523	0.135074
$470$	5.89002	0.271686
$471$	0	0
$472$	−15.0256	−0.691609
$473$	16.6184	0.764113
$474$	0	0
$475$	22.4325	1.02927
$476$	10.3090	0.472512
$477$	0	0
$478$	−18.0962	−0.827701
$479$	−5.43080	−0.248140	−0.124070	−	0.992273i	$-0.539595\pi$
$479$	−5.43080	−0.248140	−0.124070	+	0.992273i	$0.539595\pi$
$480$	0	0
$481$	−7.14855	−0.325946
$482$	12.9959	0.591947
$483$	0	0
$484$	4.60384	0.209266
$485$	8.03345	0.364780
$486$	0	0
$487$	2.16789	0.0982363	0.0491181	−	0.998793i	$-0.484359\pi$
$487$	2.16789	0.0982363	0.0491181	+	0.998793i	$0.484359\pi$
$488$	12.9575	0.586557
$489$	0	0
$490$	2.26521	0.102332
$491$	14.0211	0.632765	0.316383	−	0.948632i	$-0.397532\pi$
$491$	14.0211	0.632765	0.316383	+	0.948632i	$0.397532\pi$
$492$	0	0
$493$	−54.5916	−2.45868
$494$	−8.49863	−0.382371
$495$	0	0
$496$	−1.86357	−0.0836769
$497$	23.0442	1.03367
$498$	0	0
$499$	34.8634	1.56070	0.780350	−	0.625343i	$-0.215041\pi$
$499$	34.8634	1.56070	0.780350	+	0.625343i	$0.215041\pi$
$500$	−5.53127	−0.247366
$501$	0	0
$502$	−15.6586	−0.698875
$503$	16.7032	0.744760	0.372380	−	0.928080i	$-0.378542\pi$
$503$	16.7032	0.744760	0.372380	+	0.928080i	$0.378542\pi$
$504$	0	0
$505$	6.66390	0.296540
$506$	24.9626	1.10972
$507$	0	0
$508$	4.59502	0.203871
$509$	−23.3246	−1.03385	−0.516923	−	0.856032i	$-0.672923\pi$
$509$	−23.3246	−1.03385	−0.516923	+	0.856032i	$0.672923\pi$
$510$	0	0
$511$	−2.38706	−0.105597
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−27.0854	−1.19469
$515$	6.17047	0.271904
$516$	0	0
$517$	40.6883	1.78947
$518$	7.03864	0.309260
$519$	0	0
$520$	1.01235	0.0443946
$521$	2.52461	0.110605	0.0553027	−	0.998470i	$-0.482388\pi$
$521$	2.52461	0.110605	0.0553027	+	0.998470i	$0.482388\pi$
$522$	0	0
$523$	30.7486	1.34454	0.672270	−	0.740306i	$-0.265319\pi$
$523$	30.7486	1.34454	0.672270	+	0.740306i	$0.265319\pi$
$524$	7.67818	0.335423
$525$	0	0
$526$	−13.2403	−0.577303
$527$	11.0210	0.480084
$528$	0	0
$529$	16.9345	0.736282
$530$	2.66029	0.115556
$531$	0	0
$532$	8.36796	0.362797
$533$	1.77039	0.0766841
$534$	0	0
$535$	5.71044	0.246884
$536$	1.67811	0.0724832
$537$	0	0
$538$	9.00627	0.388288
$539$	15.6481	0.674009
$540$	0	0
$541$	23.3972	1.00593	0.502963	−	0.864308i	$-0.332243\pi$
$541$	23.3972	1.00593	0.502963	+	0.864308i	$0.332243\pi$
$542$	−8.43688	−0.362395
$543$	0	0
$544$	5.91394	0.253558
$545$	−9.49740	−0.406824
$546$	0	0
$547$	0.850280	0.0363553	0.0181777	−	0.999835i	$-0.494214\pi$
$547$	0.850280	0.0363553	0.0181777	+	0.999835i	$0.494214\pi$
$548$	−17.1341	−0.731932
$549$	0	0
$550$	−18.4592	−0.787103
$551$	−44.3128	−1.88779
$552$	0	0
$553$	−24.7797	−1.05374
$554$	0.111475	0.00473614
$555$	0	0
$556$	5.08870	0.215809
$557$	28.0828	1.18991	0.594954	−	0.803760i	$-0.297170\pi$
$557$	28.0828	1.18991	0.594954	+	0.803760i	$0.297170\pi$
$558$	0	0
$559$	7.44803	0.315018
$560$	−0.996788	−0.0421220
$561$	0	0
$562$	7.73748	0.326386
$563$	−21.6682	−0.913205	−0.456603	−	0.889671i	$-0.650934\pi$
$563$	−21.6682	−0.913205	−0.456603	+	0.889671i	$0.650934\pi$
$564$	0	0
$565$	3.23102	0.135930
$566$	7.71175	0.324149
$567$	0	0
$568$	13.2197	0.554687
$569$	46.0885	1.93213	0.966065	−	0.258300i	$-0.0831624\pi$
$569$	46.0885	1.93213	0.966065	+	0.258300i	$0.0831624\pi$
$570$	0	0
$571$	26.6870	1.11682	0.558409	−	0.829566i	$-0.311412\pi$
$571$	26.6870	1.11682	0.558409	+	0.829566i	$0.311412\pi$
$572$	6.99334	0.292406
$573$	0	0
$574$	−1.74317	−0.0727585
$575$	−29.5305	−1.23151
$576$	0	0
$577$	−5.32704	−0.221768	−0.110884	−	0.993833i	$-0.535368\pi$
$577$	−5.32704	−0.221768	−0.110884	+	0.993833i	$0.535368\pi$
$578$	−17.9746	−0.747646
$579$	0	0
$580$	5.27852	0.219179
$581$	−25.5008	−1.05795
$582$	0	0
$583$	18.3773	0.761111
$584$	−1.36938	−0.0566653
$585$	0	0
$586$	8.10536	0.334830
$587$	29.2750	1.20831	0.604153	−	0.796868i	$-0.293511\pi$
$587$	29.2750	1.20831	0.604153	+	0.796868i	$0.293511\pi$
$588$	0	0
$589$	8.94595	0.368611
$590$	−8.59201	−0.353727
$591$	0	0
$592$	4.03784	0.165954
$593$	11.9055	0.488899	0.244450	−	0.969662i	$-0.421393\pi$
$593$	11.9055	0.488899	0.244450	+	0.969662i	$0.421393\pi$
$594$	0	0
$595$	5.89494	0.241669
$596$	9.57815	0.392336
$597$	0	0
$598$	11.1878	0.457501
$599$	−36.1168	−1.47569	−0.737846	−	0.674969i	$-0.764157\pi$
$599$	−36.1168	−1.47569	−0.737846	+	0.674969i	$0.764157\pi$
$600$	0	0
$601$	−10.3361	−0.421618	−0.210809	−	0.977527i	$-0.567610\pi$
$601$	−10.3361	−0.421618	−0.210809	+	0.977527i	$0.567610\pi$
$602$	−7.33351	−0.298892
$603$	0	0
$604$	−11.2568	−0.458033
$605$	2.63259	0.107030
$606$	0	0
$607$	−11.1709	−0.453411	−0.226705	−	0.973963i	$-0.572795\pi$
$607$	−11.1709	−0.453411	−0.226705	+	0.973963i	$0.572795\pi$
$608$	4.80043	0.194683
$609$	0	0
$610$	7.40941	0.299998
$611$	18.2357	0.737738
$612$	0	0
$613$	15.1214	0.610746	0.305373	−	0.952233i	$-0.401219\pi$
$613$	15.1214	0.610746	0.305373	+	0.952233i	$0.401219\pi$
$614$	−10.1284	−0.408751
$615$	0	0
$616$	−6.88581	−0.277437
$617$	22.3959	0.901623	0.450812	−	0.892619i	$-0.351135\pi$
$617$	22.3959	0.901623	0.450812	+	0.892619i	$0.351135\pi$
$618$	0	0
$619$	−31.4893	−1.26566	−0.632830	−	0.774290i	$-0.718107\pi$
$619$	−31.4893	−1.26566	−0.632830	+	0.774290i	$0.718107\pi$
$620$	−1.06564	−0.0427970
$621$	0	0
$622$	17.8551	0.715923
$623$	0.615594	0.0246632
$624$	0	0
$625$	20.2022	0.808086
$626$	0.424994	0.0169862
$627$	0	0
$628$	9.37615	0.374149
$629$	−23.8795	−0.952139
$630$	0	0
$631$	21.9137	0.872369	0.436185	−	0.899857i	$-0.356329\pi$
$631$	21.9137	0.872369	0.436185	+	0.899857i	$0.356329\pi$
$632$	−14.2153	−0.565456
$633$	0	0
$634$	14.8450	0.589570
$635$	2.62755	0.104271
$636$	0	0
$637$	7.01315	0.277871
$638$	36.4641	1.44363
$639$	0	0
$640$	−0.571825	−0.0226034
$641$	20.8650	0.824117	0.412059	−	0.911157i	$-0.364810\pi$
$641$	20.8650	0.824117	0.412059	+	0.911157i	$0.364810\pi$
$642$	0	0
$643$	25.4941	1.00539	0.502694	−	0.864464i	$-0.332342\pi$
$643$	25.4941	1.00539	0.502694	+	0.864464i	$0.332342\pi$
$644$	−11.0157	−0.434081
$645$	0	0
$646$	−28.3894	−1.11697
$647$	−32.0475	−1.25992	−0.629959	−	0.776628i	$-0.716928\pi$
$647$	−32.0475	−1.25992	−0.629959	+	0.776628i	$0.716928\pi$
$648$	0	0
$649$	−59.3536	−2.32983
$650$	−8.27306	−0.324496
$651$	0	0
$652$	−9.75793	−0.382150
$653$	−6.78054	−0.265343	−0.132672	−	0.991160i	$-0.542356\pi$
$653$	−6.78054	−0.265343	−0.132672	+	0.991160i	$0.542356\pi$
$654$	0	0
$655$	4.39058	0.171554
$656$	−1.00000	−0.0390434
$657$	0	0
$658$	−17.9553	−0.699971
$659$	37.1841	1.44849	0.724243	−	0.689545i	$-0.242190\pi$
$659$	37.1841	1.44849	0.724243	+	0.689545i	$0.242190\pi$
$660$	0	0
$661$	49.4999	1.92532	0.962662	−	0.270708i	$-0.0872577\pi$
$661$	49.4999	1.92532	0.962662	+	0.270708i	$0.0872577\pi$
$662$	27.2787	1.06022
$663$	0	0
$664$	−14.6290	−0.567715
$665$	4.78501	0.185555
$666$	0	0
$667$	58.3342	2.25871
$668$	−2.55589	−0.0988904
$669$	0	0
$670$	0.959584	0.0370720
$671$	51.1842	1.97594
$672$	0	0
$673$	−9.72893	−0.375023	−0.187511	−	0.982262i	$-0.560042\pi$
$673$	−9.72893	−0.375023	−0.187511	+	0.982262i	$0.560042\pi$
$674$	−19.9306	−0.767698
$675$	0	0
$676$	−9.86572	−0.379451
$677$	−43.0573	−1.65483	−0.827414	−	0.561593i	$-0.810189\pi$
$677$	−43.0573	−1.65483	−0.827414	+	0.561593i	$0.810189\pi$
$678$	0	0
$679$	−24.4894	−0.939817
$680$	3.38174	0.129684
$681$	0	0
$682$	−7.36143	−0.281884
$683$	34.8567	1.33375	0.666877	−	0.745168i	$-0.267631\pi$
$683$	34.8567	1.33375	0.666877	+	0.745168i	$0.267631\pi$
$684$	0	0
$685$	−9.79770	−0.374351
$686$	−19.1075	−0.729528
$687$	0	0
$688$	−4.20700	−0.160390
$689$	8.23636	0.313780
$690$	0	0
$691$	−23.8259	−0.906380	−0.453190	−	0.891414i	$-0.649714\pi$
$691$	−23.8259	−0.906380	−0.453190	+	0.891414i	$0.649714\pi$
$692$	7.44254	0.282923
$693$	0	0
$694$	18.1282	0.688138
$695$	2.90985	0.110377
$696$	0	0
$697$	5.91394	0.224006
$698$	−31.4839	−1.19168
$699$	0	0
$700$	8.14586	0.307884
$701$	8.27804	0.312657	0.156329	−	0.987705i	$-0.450034\pi$
$701$	8.27804	0.312657	0.156329	+	0.987705i	$0.450034\pi$
$702$	0	0
$703$	−19.3834	−0.731058
$704$	−3.95017	−0.148878
$705$	0	0
$706$	−6.61288	−0.248879
$707$	−20.3144	−0.764003
$708$	0	0
$709$	6.53669	0.245491	0.122745	−	0.992438i	$-0.460830\pi$
$709$	6.53669	0.245491	0.122745	+	0.992438i	$0.460830\pi$
$710$	7.55936	0.283698
$711$	0	0
$712$	0.353146	0.0132347
$713$	−11.7766	−0.441038
$714$	0	0
$715$	3.99897	0.149553
$716$	20.6356	0.771188
$717$	0	0
$718$	−20.8524	−0.778206
$719$	40.0246	1.49267	0.746334	−	0.665572i	$-0.231812\pi$
$719$	40.0246	1.49267	0.746334	+	0.665572i	$0.231812\pi$
$720$	0	0
$721$	−18.8103	−0.700531
$722$	−4.04413	−0.150507
$723$	0	0
$724$	2.50863	0.0932324
$725$	−43.1367	−1.60206
$726$	0	0
$727$	45.1389	1.67411	0.837054	−	0.547120i	$-0.184276\pi$
$727$	45.1389	1.67411	0.837054	+	0.547120i	$0.184276\pi$
$728$	−3.08609	−0.114378
$729$	0	0
$730$	−0.783045	−0.0289818
$731$	24.8799	0.920218
$732$	0	0
$733$	−50.2128	−1.85465	−0.927326	−	0.374254i	$-0.877899\pi$
$733$	−50.2128	−1.85465	−0.927326	+	0.374254i	$0.877899\pi$
$734$	−24.1222	−0.890366
$735$	0	0
$736$	−6.31937	−0.232935
$737$	6.62881	0.244175
$738$	0	0
$739$	−16.1349	−0.593533	−0.296767	−	0.954950i	$-0.595908\pi$
$739$	−16.1349	−0.593533	−0.296767	+	0.954950i	$0.595908\pi$
$740$	2.30894	0.0848783
$741$	0	0
$742$	−8.10972	−0.297717
$743$	11.9862	0.439732	0.219866	−	0.975530i	$-0.429438\pi$
$743$	11.9862	0.439732	0.219866	+	0.975530i	$0.429438\pi$
$744$	0	0
$745$	5.47703	0.200663
$746$	−8.10825	−0.296864
$747$	0	0
$748$	23.3611	0.854165
$749$	−17.4079	−0.636070
$750$	0	0
$751$	4.09767	0.149526	0.0747630	−	0.997201i	$-0.476180\pi$
$751$	4.09767	0.149526	0.0747630	+	0.997201i	$0.476180\pi$
$752$	−10.3004	−0.375617
$753$	0	0
$754$	16.3425	0.595158
$755$	−6.43692	−0.234263
$756$	0	0
$757$	−10.8013	−0.392579	−0.196289	−	0.980546i	$-0.562889\pi$
$757$	−10.8013	−0.392579	−0.196289	+	0.980546i	$0.562889\pi$
$758$	−22.5064	−0.817471
$759$	0	0
$760$	2.74501	0.0995718
$761$	13.7238	0.497487	0.248743	−	0.968569i	$-0.419982\pi$
$761$	13.7238	0.497487	0.248743	+	0.968569i	$0.419982\pi$
$762$	0	0
$763$	28.9522	1.04814
$764$	−16.4651	−0.595688
$765$	0	0
$766$	−3.44414	−0.124442
$767$	−26.6011	−0.960511
$768$	0	0
$769$	21.7282	0.783539	0.391769	−	0.920063i	$-0.371863\pi$
$769$	21.7282	0.783539	0.391769	+	0.920063i	$0.371863\pi$
$770$	−3.93748	−0.141897
$771$	0	0
$772$	−17.3139	−0.623139
$773$	−24.6477	−0.886517	−0.443258	−	0.896394i	$-0.646178\pi$
$773$	−24.6477	−0.886517	−0.443258	+	0.896394i	$0.646178\pi$
$774$	0	0
$775$	8.70850	0.312819
$776$	−14.0488	−0.504322
$777$	0	0
$778$	29.7239	1.06565
$779$	4.80043	0.171993
$780$	0	0
$781$	52.2201	1.86858
$782$	37.3724	1.33643
$783$	0	0
$784$	−3.96136	−0.141477
$785$	5.36152	0.191361
$786$	0	0
$787$	−23.2123	−0.827430	−0.413715	−	0.910407i	$-0.635769\pi$
$787$	−23.2123	−0.827430	−0.413715	+	0.910407i	$0.635769\pi$
$788$	22.5592	0.803638
$789$	0	0
$790$	−8.12868	−0.289206
$791$	−9.84955	−0.350210
$792$	0	0
$793$	22.9398	0.814615
$794$	18.6701	0.662578
$795$	0	0
$796$	9.81568	0.347908
$797$	−21.2609	−0.753098	−0.376549	−	0.926397i	$-0.622889\pi$
$797$	−21.2609	−0.753098	−0.376549	+	0.926397i	$0.622889\pi$
$798$	0	0
$799$	60.9159	2.15505
$800$	4.67302	0.165216
$801$	0	0
$802$	11.3944	0.402349
$803$	−5.40927	−0.190889
$804$	0	0
$805$	−6.29907	−0.222013
$806$	−3.29925	−0.116211
$807$	0	0
$808$	−11.6537	−0.409977
$809$	−28.1507	−0.989725	−0.494862	−	0.868971i	$-0.664782\pi$
$809$	−28.1507	−0.989725	−0.494862	+	0.868971i	$0.664782\pi$
$810$	0	0
$811$	12.4579	0.437454	0.218727	−	0.975786i	$-0.429810\pi$
$811$	12.4579	0.437454	0.218727	+	0.975786i	$0.429810\pi$
$812$	−16.0912	−0.564691
$813$	0	0
$814$	15.9502	0.559053
$815$	−5.57983	−0.195453
$816$	0	0
$817$	20.1954	0.706548
$818$	−11.8888	−0.415683
$819$	0	0
$820$	−0.571825	−0.0199690
$821$	3.49158	0.121857	0.0609285	−	0.998142i	$-0.480594\pi$
$821$	3.49158	0.121857	0.0609285	+	0.998142i	$0.480594\pi$
$822$	0	0
$823$	28.4188	0.990618	0.495309	−	0.868717i	$-0.335055\pi$
$823$	28.4188	0.990618	0.495309	+	0.868717i	$0.335055\pi$
$824$	−10.7908	−0.375917
$825$	0	0
$826$	26.1921	0.911341
$827$	50.5457	1.75765	0.878823	−	0.477147i	$-0.158329\pi$
$827$	50.5457	1.75765	0.878823	+	0.477147i	$0.158329\pi$
$828$	0	0
$829$	6.29091	0.218492	0.109246	−	0.994015i	$-0.465156\pi$
$829$	6.29091	0.218492	0.109246	+	0.994015i	$0.465156\pi$
$830$	−8.36522	−0.290361
$831$	0	0
$832$	−1.77039	−0.0613772
$833$	23.4272	0.811706
$834$	0	0
$835$	−1.46152	−0.0505781
$836$	18.9625	0.655832
$837$	0	0
$838$	5.48672	0.189536
$839$	−6.76167	−0.233439	−0.116719	−	0.993165i	$-0.537238\pi$
$839$	−6.76167	−0.233439	−0.116719	+	0.993165i	$0.537238\pi$
$840$	0	0
$841$	56.2115	1.93833
$842$	5.04657	0.173916
$843$	0	0
$844$	8.25850	0.284269
$845$	−5.64147	−0.194072
$846$	0	0
$847$	−8.02527	−0.275752
$848$	−4.65229	−0.159760
$849$	0	0
$850$	−27.6359	−0.947904
$851$	25.5166	0.874699
$852$	0	0
$853$	19.9536	0.683199	0.341600	−	0.939846i	$-0.389031\pi$
$853$	19.9536	0.683199	0.341600	+	0.939846i	$0.389031\pi$
$854$	−22.5871	−0.772913
$855$	0	0
$856$	−9.98635	−0.341326
$857$	30.9581	1.05751	0.528754	−	0.848775i	$-0.322660\pi$
$857$	30.9581	1.05751	0.528754	+	0.848775i	$0.322660\pi$
$858$	0	0
$859$	21.6755	0.739558	0.369779	−	0.929120i	$-0.379433\pi$
$859$	21.6755	0.739558	0.369779	+	0.929120i	$0.379433\pi$
$860$	−2.40567	−0.0820326
$861$	0	0
$862$	−3.54756	−0.120830
$863$	12.7614	0.434402	0.217201	−	0.976127i	$-0.430307\pi$
$863$	12.7614	0.434402	0.217201	+	0.976127i	$0.430307\pi$
$864$	0	0
$865$	4.25583	0.144703
$866$	3.74015	0.127096
$867$	0	0
$868$	3.24852	0.110262
$869$	−56.1530	−1.90486
$870$	0	0
$871$	2.97090	0.100665
$872$	16.6089	0.562449
$873$	0	0
$874$	30.3357	1.02612
$875$	9.64194	0.325957
$876$	0	0
$877$	−52.9707	−1.78869	−0.894346	−	0.447376i	$-0.852359\pi$
$877$	−52.9707	−1.78869	−0.894346	+	0.447376i	$0.852359\pi$
$878$	10.8710	0.366879
$879$	0	0
$880$	−2.25881	−0.0761443
$881$	1.17039	0.0394313	0.0197156	−	0.999806i	$-0.493724\pi$
$881$	1.17039	0.0394313	0.0197156	+	0.999806i	$0.493724\pi$
$882$	0	0
$883$	−12.9169	−0.434688	−0.217344	−	0.976095i	$-0.569739\pi$
$883$	−12.9169	−0.434688	−0.217344	+	0.976095i	$0.569739\pi$
$884$	10.4700	0.352143
$885$	0	0
$886$	9.75286	0.327654
$887$	4.14949	0.139326	0.0696631	−	0.997571i	$-0.477808\pi$
$887$	4.14949	0.139326	0.0696631	+	0.997571i	$0.477808\pi$
$888$	0	0
$889$	−8.00990	−0.268643
$890$	0.201938	0.00676897
$891$	0	0
$892$	−17.6663	−0.591511
$893$	49.4463	1.65466
$894$	0	0
$895$	11.8000	0.394429
$896$	1.74317	0.0582352
$897$	0	0
$898$	37.7704	1.26041
$899$	−17.2027	−0.573741
$900$	0	0
$901$	27.5133	0.916602
$902$	−3.95017	−0.131526
$903$	0	0
$904$	−5.65037	−0.187928
$905$	1.43450	0.0476843
$906$	0	0
$907$	−18.3173	−0.608216	−0.304108	−	0.952638i	$-0.598358\pi$
$907$	−18.3173	−0.608216	−0.304108	+	0.952638i	$0.598358\pi$
$908$	6.16256	0.204512
$909$	0	0
$910$	−1.76470	−0.0584993
$911$	−33.3440	−1.10474	−0.552369	−	0.833600i	$-0.686276\pi$
$911$	−33.3440	−1.10474	−0.552369	+	0.833600i	$0.686276\pi$
$912$	0	0
$913$	−57.7870	−1.91247
$914$	16.2991	0.539127
$915$	0	0
$916$	−18.8703	−0.623491
$917$	−13.3844	−0.441991
$918$	0	0
$919$	−1.65034	−0.0544397	−0.0272198	−	0.999629i	$-0.508665\pi$
$919$	−1.65034	−0.0544397	−0.0272198	+	0.999629i	$0.508665\pi$
$920$	−3.61358	−0.119136
$921$	0	0
$922$	8.27402	0.272490
$923$	23.4040	0.770353
$924$	0	0
$925$	−18.8689	−0.620405
$926$	−19.3692	−0.636512
$927$	0	0
$928$	−9.23101	−0.303023
$929$	−7.07179	−0.232018	−0.116009	−	0.993248i	$-0.537010\pi$
$929$	−7.07179	−0.232018	−0.116009	+	0.993248i	$0.537010\pi$
$930$	0	0
$931$	19.0162	0.623232
$932$	27.5072	0.901027
$933$	0	0
$934$	20.5070	0.671009
$935$	13.3584	0.436868
$936$	0	0
$937$	−44.6691	−1.45928	−0.729638	−	0.683833i	$-0.760312\pi$
$937$	−44.6691	−1.45928	−0.729638	+	0.683833i	$0.760312\pi$
$938$	−2.92523	−0.0955120
$939$	0	0
$940$	−5.89002	−0.192111
$941$	4.34217	0.141551	0.0707754	−	0.997492i	$-0.477453\pi$
$941$	4.34217	0.141551	0.0707754	+	0.997492i	$0.477453\pi$
$942$	0	0
$943$	−6.31937	−0.205787
$944$	15.0256	0.489041
$945$	0	0
$946$	−16.6184	−0.540310
$947$	47.4683	1.54251	0.771256	−	0.636525i	$-0.219629\pi$
$947$	47.4683	1.54251	0.771256	+	0.636525i	$0.219629\pi$
$948$	0	0
$949$	−2.42433	−0.0786971
$950$	−22.4325	−0.727806
$951$	0	0
$952$	−10.3090	−0.334116
$953$	3.96085	0.128304	0.0641522	−	0.997940i	$-0.479566\pi$
$953$	3.96085	0.128304	0.0641522	+	0.997940i	$0.479566\pi$
$954$	0	0
$955$	−9.41518	−0.304668
$956$	18.0962	0.585273
$957$	0	0
$958$	5.43080	0.175461
$959$	29.8676	0.964475
$960$	0	0
$961$	−27.5271	−0.887971
$962$	7.14855	0.230478
$963$	0	0
$964$	−12.9959	−0.418570
$965$	−9.90050	−0.318708
$966$	0	0
$967$	14.5697	0.468531	0.234266	−	0.972173i	$-0.424731\pi$
$967$	14.5697	0.468531	0.234266	+	0.972173i	$0.424731\pi$
$968$	−4.60384	−0.147973
$969$	0	0
$970$	−8.03345	−0.257938
$971$	−50.9210	−1.63413	−0.817066	−	0.576544i	$-0.804401\pi$
$971$	−50.9210	−1.63413	−0.817066	+	0.576544i	$0.804401\pi$
$972$	0	0
$973$	−8.87046	−0.284374
$974$	−2.16789	−0.0694635
$975$	0	0
$976$	−12.9575	−0.414759
$977$	50.5222	1.61635	0.808175	−	0.588942i	$-0.200456\pi$
$977$	50.5222	1.61635	0.808175	+	0.588942i	$0.200456\pi$
$978$	0	0
$979$	1.39499	0.0445840
$980$	−2.26521	−0.0723594
$981$	0	0
$982$	−14.0211	−0.447433
$983$	−33.0892	−1.05538	−0.527691	−	0.849437i	$-0.676942\pi$
$983$	−33.0892	−1.05538	−0.527691	+	0.849437i	$0.676942\pi$
$984$	0	0
$985$	12.8999	0.411025
$986$	54.5916	1.73855
$987$	0	0
$988$	8.49863	0.270377
$989$	−26.5856	−0.845373
$990$	0	0
$991$	20.4427	0.649385	0.324692	−	0.945820i	$-0.394739\pi$
$991$	20.4427	0.649385	0.324692	+	0.945820i	$0.394739\pi$
$992$	1.86357	0.0591685
$993$	0	0
$994$	−23.0442	−0.730917
$995$	5.61285	0.177939
$996$	0	0
$997$	−3.31333	−0.104934	−0.0524671	−	0.998623i	$-0.516708\pi$
$997$	−3.31333	−0.104934	−0.0524671	+	0.998623i	$0.516708\pi$
$998$	−34.8634	−1.10358
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6642.2.a.bu.1.4		7
3.2	odd	2		6642.2.a.bv.1.4		7
9.2	odd	6		738.2.e.j.247.2	✓	14
9.4	even	3		2214.2.e.j.1477.4		14
9.5	odd	6		738.2.e.j.493.2	yes	14
9.7	even	3		2214.2.e.j.739.4		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
738.2.e.j.247.2	✓	14	9.2	odd	6
738.2.e.j.493.2	yes	14	9.5	odd	6
2214.2.e.j.739.4		14	9.7	even	3
2214.2.e.j.1477.4		14	9.4	even	3
6642.2.a.bu.1.4		7	1.1	even	1	trivial
6642.2.a.bv.1.4		7	3.2	odd	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:	\( N \)	\(=\)	\( 6642 = 2 \cdot 3^{4} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6642.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +0.571825 q^{5} -1.74317 q^{7} -1.00000 q^{8} -0.571825 q^{10} -3.95017 q^{11} -1.77039 q^{13} +1.74317 q^{14} +1.00000 q^{16} -5.91394 q^{17} -4.80043 q^{19} +0.571825 q^{20} +3.95017 q^{22} +6.31937 q^{23} -4.67302 q^{25} +1.77039 q^{26} -1.74317 q^{28} +9.23101 q^{29} -1.86357 q^{31} -1.00000 q^{32} +5.91394 q^{34} -0.996788 q^{35} +4.03784 q^{37} +4.80043 q^{38} -0.571825 q^{40} -1.00000 q^{41} -4.20700 q^{43} -3.95017 q^{44} -6.31937 q^{46} -10.3004 q^{47} -3.96136 q^{49} +4.67302 q^{50} -1.77039 q^{52} -4.65229 q^{53} -2.25881 q^{55} +1.74317 q^{56} -9.23101 q^{58} +15.0256 q^{59} -12.9575 q^{61} +1.86357 q^{62} +1.00000 q^{64} -1.01235 q^{65} -1.67811 q^{67} -5.91394 q^{68} +0.996788 q^{70} -13.2197 q^{71} +1.36938 q^{73} -4.03784 q^{74} -4.80043 q^{76} +6.88581 q^{77} +14.2153 q^{79} +0.571825 q^{80} +1.00000 q^{82} +14.6290 q^{83} -3.38174 q^{85} +4.20700 q^{86} +3.95017 q^{88} -0.353146 q^{89} +3.08609 q^{91} +6.31937 q^{92} +10.3004 q^{94} -2.74501 q^{95} +14.0488 q^{97} +3.96136 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 7 q^{2} + 7 q^{4} + 5 q^{5} + q^{7} - 7 q^{8} - 5 q^{10} + 10 q^{11} - 4 q^{13} - q^{14} + 7 q^{16} + 13 q^{17} - q^{19} + 5 q^{20} - 10 q^{22} + 15 q^{23} + 14 q^{25} + 4 q^{26} + q^{28} + 30 q^{29}+ \cdots + 12 q^{98}+O(q^{100}) \) 7 * q - 7 * q^2 + 7 * q^4 + 5 * q^5 + q^7 - 7 * q^8 - 5 * q^10 + 10 * q^11 - 4 * q^13 - q^14 + 7 * q^16 + 13 * q^17 - q^19 + 5 * q^20 - 10 * q^22 + 15 * q^23 + 14 * q^25 + 4 * q^26 + q^28 + 30 * q^29 + 13 * q^31 - 7 * q^32 - 13 * q^34 + 8 * q^35 + 7 * q^37 + q^38 - 5 * q^40 - 7 * q^41 - 5 * q^43 + 10 * q^44 - 15 * q^46 + 3 * q^47 - 12 * q^49 - 14 * q^50 - 4 * q^52 + 12 * q^53 - 13 * q^55 - q^56 - 30 * q^58 + 13 * q^59 - 55 * q^61 - 13 * q^62 + 7 * q^64 + 25 * q^65 - 15 * q^67 + 13 * q^68 - 8 * q^70 + 12 * q^71 - 7 * q^74 - q^76 - 5 * q^77 + 12 * q^79 + 5 * q^80 + 7 * q^82 + 36 * q^83 - 4 * q^85 + 5 * q^86 - 10 * q^88 + 16 * q^89 - 31 * q^91 + 15 * q^92 - 3 * q^94 + 43 * q^95 + q^97 + 12 * q^98

Introduction

L-functions

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Database

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