LMFDB - Embedded newform 6498.2.a.bm.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6498,2,Mod(1,6498)] 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("6498.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6498, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 6498 = 2 \cdot 3^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6498.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-3,0,3,0,0,-9,-3,0,0,0] 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:	$51.8867912334$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 114)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	6498.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.41147 q^{5} -4.87939 q^{7} -1.00000 q^{8} +3.41147 q^{10} +3.41147 q^{11} -2.71688 q^{13} +4.87939 q^{14} +1.00000 q^{16} -1.18479 q^{17} -3.41147 q^{20} -3.41147 q^{22} +3.41147 q^{23} +6.63816 q^{25} +2.71688 q^{26} -4.87939 q^{28} +3.77332 q^{29} -6.61587 q^{31} -1.00000 q^{32} +1.18479 q^{34} +16.6459 q^{35} -6.75877 q^{37} +3.41147 q^{40} +6.55438 q^{41} +4.17024 q^{43} +3.41147 q^{44} -3.41147 q^{46} -2.85710 q^{47} +16.8084 q^{49} -6.63816 q^{50} -2.71688 q^{52} +0.630415 q^{53} -11.6382 q^{55} +4.87939 q^{56} -3.77332 q^{58} +11.6382 q^{59} +6.90167 q^{61} +6.61587 q^{62} +1.00000 q^{64} +9.26857 q^{65} -0.709141 q^{67} -1.18479 q^{68} -16.6459 q^{70} +7.95811 q^{71} -12.7442 q^{73} +6.75877 q^{74} -16.6459 q^{77} -0.297667 q^{79} -3.41147 q^{80} -6.55438 q^{82} -15.6040 q^{83} +4.04189 q^{85} -4.17024 q^{86} -3.41147 q^{88} +6.77332 q^{89} +13.2567 q^{91} +3.41147 q^{92} +2.85710 q^{94} +7.12567 q^{97} -16.8084 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} - 9 q^{7} - 3 q^{8} + 9 q^{14} + 3 q^{16} + 3 q^{25} - 9 q^{28} + 18 q^{29} - 9 q^{31} - 3 q^{32} + 9 q^{35} - 9 q^{37} + 9 q^{41} - 9 q^{43} - 9 q^{47} + 12 q^{49} - 3 q^{50} + 9 q^{53}+ \cdots - 12 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 - 9 * q^7 - 3 * q^8 + 9 * q^14 + 3 * q^16 + 3 * q^25 - 9 * q^28 + 18 * q^29 - 9 * q^31 - 3 * q^32 + 9 * q^35 - 9 * q^37 + 9 * q^41 - 9 * q^43 - 9 * q^47 + 12 * q^49 - 3 * q^50 + 9 * q^53 - 18 * q^55 + 9 * q^56 - 18 * q^58 + 18 * q^59 + 9 * q^61 + 9 * q^62 + 3 * q^64 + 18 * q^65 - 18 * q^67 - 9 * q^70 + 27 * q^71 - 9 * q^73 + 9 * q^74 - 9 * q^77 - 27 * q^79 - 9 * q^82 - 9 * q^83 + 9 * q^85 + 9 * q^86 + 27 * q^89 + 3 * q^91 + 9 * q^94 + 12 * 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
$5$	−3.41147	−1.52566	−0.762829	−	0.646601i	$-0.776190\pi$
$5$	−3.41147	−1.52566	−0.762829	+	0.646601i	$0.776190\pi$
$6$	0	0
$7$	−4.87939	−1.84423	−0.922117	−	0.386911i	$-0.873542\pi$
$7$	−4.87939	−1.84423	−0.922117	+	0.386911i	$0.873542\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.41147	1.07880
$11$	3.41147	1.02860	0.514299	−	0.857611i	$-0.328052\pi$
$11$	3.41147	1.02860	0.514299	+	0.857611i	$0.328052\pi$
$12$	0	0
$13$	−2.71688	−0.753527	−0.376764	−	0.926309i	$-0.622963\pi$
$13$	−2.71688	−0.753527	−0.376764	+	0.926309i	$0.622963\pi$
$14$	4.87939	1.30407
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.18479	−0.287354	−0.143677	−	0.989625i	$-0.545893\pi$
$17$	−1.18479	−0.287354	−0.143677	+	0.989625i	$0.545893\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	−3.41147	−0.762829
$21$	0	0
$22$	−3.41147	−0.727329
$23$	3.41147	0.711342	0.355671	−	0.934611i	$-0.384252\pi$
$23$	3.41147	0.711342	0.355671	+	0.934611i	$0.384252\pi$
$24$	0	0
$25$	6.63816	1.32763
$26$	2.71688	0.532824
$27$	0	0
$28$	−4.87939	−0.922117
$29$	3.77332	0.700688	0.350344	−	0.936621i	$-0.386065\pi$
$29$	3.77332	0.700688	0.350344	+	0.936621i	$0.386065\pi$
$30$	0	0
$31$	−6.61587	−1.18824	−0.594122	−	0.804375i	$-0.702501\pi$
$31$	−6.61587	−1.18824	−0.594122	+	0.804375i	$0.702501\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.18479	0.203190
$35$	16.6459	2.81367
$36$	0	0
$37$	−6.75877	−1.11114	−0.555568	−	0.831471i	$-0.687499\pi$
$37$	−6.75877	−1.11114	−0.555568	+	0.831471i	$0.687499\pi$
$38$	0	0
$39$	0	0
$40$	3.41147	0.539401
$41$	6.55438	1.02362	0.511811	−	0.859098i	$-0.328975\pi$
$41$	6.55438	1.02362	0.511811	+	0.859098i	$0.328975\pi$
$42$	0	0
$43$	4.17024	0.635956	0.317978	−	0.948098i	$-0.396996\pi$
$43$	4.17024	0.635956	0.317978	+	0.948098i	$0.396996\pi$
$44$	3.41147	0.514299
$45$	0	0
$46$	−3.41147	−0.502994
$47$	−2.85710	−0.416750	−0.208375	−	0.978049i	$-0.566818\pi$
$47$	−2.85710	−0.416750	−0.208375	+	0.978049i	$0.566818\pi$
$48$	0	0
$49$	16.8084	2.40120
$50$	−6.63816	−0.938777
$51$	0	0
$52$	−2.71688	−0.376764
$53$	0.630415	0.0865942	0.0432971	−	0.999062i	$-0.486214\pi$
$53$	0.630415	0.0865942	0.0432971	+	0.999062i	$0.486214\pi$
$54$	0	0
$55$	−11.6382	−1.56929
$56$	4.87939	0.652035
$57$	0	0
$58$	−3.77332	−0.495461
$59$	11.6382	1.51516	0.757579	−	0.652743i	$-0.226382\pi$
$59$	11.6382	1.51516	0.757579	+	0.652743i	$0.226382\pi$
$60$	0	0
$61$	6.90167	0.883669	0.441834	−	0.897097i	$-0.354328\pi$
$61$	6.90167	0.883669	0.441834	+	0.897097i	$0.354328\pi$
$62$	6.61587	0.840216
$63$	0	0
$64$	1.00000	0.125000
$65$	9.26857	1.14962
$66$	0	0
$67$	−0.709141	−0.0866353	−0.0433177	−	0.999061i	$-0.513793\pi$
$67$	−0.709141	−0.0866353	−0.0433177	+	0.999061i	$0.513793\pi$
$68$	−1.18479	−0.143677
$69$	0	0
$70$	−16.6459	−1.98957
$71$	7.95811	0.944454	0.472227	−	0.881477i	$-0.343450\pi$
$71$	7.95811	0.944454	0.472227	+	0.881477i	$0.343450\pi$
$72$	0	0
$73$	−12.7442	−1.49160	−0.745799	−	0.666171i	$-0.767932\pi$
$73$	−12.7442	−1.49160	−0.745799	+	0.666171i	$0.767932\pi$
$74$	6.75877	0.785691
$75$	0	0
$76$	0	0
$77$	−16.6459	−1.89698
$78$	0	0
$79$	−0.297667	−0.0334901	−0.0167450	−	0.999860i	$-0.505330\pi$
$79$	−0.297667	−0.0334901	−0.0167450	+	0.999860i	$0.505330\pi$
$80$	−3.41147	−0.381414
$81$	0	0
$82$	−6.55438	−0.723810
$83$	−15.6040	−1.71276	−0.856381	−	0.516344i	$-0.827293\pi$
$83$	−15.6040	−1.71276	−0.856381	+	0.516344i	$0.827293\pi$
$84$	0	0
$85$	4.04189	0.438404
$86$	−4.17024	−0.449689
$87$	0	0
$88$	−3.41147	−0.363664
$89$	6.77332	0.717970	0.358985	−	0.933343i	$-0.383123\pi$
$89$	6.77332	0.717970	0.358985	+	0.933343i	$0.383123\pi$
$90$	0	0
$91$	13.2567	1.38968
$92$	3.41147	0.355671
$93$	0	0
$94$	2.85710	0.294687
$95$	0	0
$96$	0	0
$97$	7.12567	0.723502	0.361751	−	0.932275i	$-0.382179\pi$
$97$	7.12567	0.723502	0.361751	+	0.932275i	$0.382179\pi$
$98$	−16.8084	−1.69790
$99$	0	0
$100$	6.63816	0.663816
$101$	4.64590	0.462284	0.231142	−	0.972920i	$-0.425754\pi$
$101$	4.64590	0.462284	0.231142	+	0.972920i	$0.425754\pi$
$102$	0	0
$103$	12.3824	1.22007	0.610036	−	0.792374i	$-0.291155\pi$
$103$	12.3824	1.22007	0.610036	+	0.792374i	$0.291155\pi$
$104$	2.71688	0.266412
$105$	0	0
$106$	−0.630415	−0.0612313
$107$	3.89899	0.376929	0.188465	−	0.982080i	$-0.439649\pi$
$107$	3.89899	0.376929	0.188465	+	0.982080i	$0.439649\pi$
$108$	0	0
$109$	11.5963	1.11072	0.555360	−	0.831610i	$-0.312580\pi$
$109$	11.5963	1.11072	0.555360	+	0.831610i	$0.312580\pi$
$110$	11.6382	1.10965
$111$	0	0
$112$	−4.87939	−0.461059
$113$	1.94087	0.182582	0.0912911	−	0.995824i	$-0.470901\pi$
$113$	1.94087	0.182582	0.0912911	+	0.995824i	$0.470901\pi$
$114$	0	0
$115$	−11.6382	−1.08526
$116$	3.77332	0.350344
$117$	0	0
$118$	−11.6382	−1.07138
$119$	5.78106	0.529949
$120$	0	0
$121$	0.638156	0.0580142
$122$	−6.90167	−0.624848
$123$	0	0
$124$	−6.61587	−0.594122
$125$	−5.58853	−0.499853
$126$	0	0
$127$	6.08378	0.539848	0.269924	−	0.962882i	$-0.413001\pi$
$127$	6.08378	0.539848	0.269924	+	0.962882i	$0.413001\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−9.26857	−0.812907
$131$	18.6040	1.62544	0.812720	−	0.582655i	$-0.197986\pi$
$131$	18.6040	1.62544	0.812720	+	0.582655i	$0.197986\pi$
$132$	0	0
$133$	0	0
$134$	0.709141	0.0612604
$135$	0	0
$136$	1.18479	0.101595
$137$	−4.54664	−0.388445	−0.194223	−	0.980957i	$-0.562218\pi$
$137$	−4.54664	−0.388445	−0.194223	+	0.980957i	$0.562218\pi$
$138$	0	0
$139$	−3.01455	−0.255691	−0.127845	−	0.991794i	$-0.540806\pi$
$139$	−3.01455	−0.255691	−0.127845	+	0.991794i	$0.540806\pi$
$140$	16.6459	1.40684
$141$	0	0
$142$	−7.95811	−0.667830
$143$	−9.26857	−0.775077
$144$	0	0
$145$	−12.8726	−1.06901
$146$	12.7442	1.05472
$147$	0	0
$148$	−6.75877	−0.555568
$149$	−1.18479	−0.0970620	−0.0485310	−	0.998822i	$-0.515454\pi$
$149$	−1.18479	−0.0970620	−0.0485310	+	0.998822i	$0.515454\pi$
$150$	0	0
$151$	3.63816	0.296069	0.148034	−	0.988982i	$-0.452705\pi$
$151$	3.63816	0.296069	0.148034	+	0.988982i	$0.452705\pi$
$152$	0	0
$153$	0	0
$154$	16.6459	1.34136
$155$	22.5699	1.81285
$156$	0	0
$157$	6.27126	0.500501	0.250250	−	0.968181i	$-0.419487\pi$
$157$	6.27126	0.500501	0.250250	+	0.968181i	$0.419487\pi$
$158$	0.297667	0.0236811
$159$	0	0
$160$	3.41147	0.269701
$161$	−16.6459	−1.31188
$162$	0	0
$163$	−19.2986	−1.51158	−0.755792	−	0.654812i	$-0.772748\pi$
$163$	−19.2986	−1.51158	−0.755792	+	0.654812i	$0.772748\pi$
$164$	6.55438	0.511811
$165$	0	0
$166$	15.6040	1.21111
$167$	−7.72193	−0.597541	−0.298771	−	0.954325i	$-0.596577\pi$
$167$	−7.72193	−0.597541	−0.298771	+	0.954325i	$0.596577\pi$
$168$	0	0
$169$	−5.61856	−0.432197
$170$	−4.04189	−0.309999
$171$	0	0
$172$	4.17024	0.317978
$173$	−14.4953	−1.10205	−0.551027	−	0.834488i	$-0.685764\pi$
$173$	−14.4953	−1.10205	−0.551027	+	0.834488i	$0.685764\pi$
$174$	0	0
$175$	−32.3901	−2.44846
$176$	3.41147	0.257150
$177$	0	0
$178$	−6.77332	−0.507682
$179$	−9.82295	−0.734202	−0.367101	−	0.930181i	$-0.619650\pi$
$179$	−9.82295	−0.734202	−0.367101	+	0.930181i	$0.619650\pi$
$180$	0	0
$181$	−18.9145	−1.40590	−0.702951	−	0.711239i	$-0.748135\pi$
$181$	−18.9145	−1.40590	−0.702951	+	0.711239i	$0.748135\pi$
$182$	−13.2567	−0.982653
$183$	0	0
$184$	−3.41147	−0.251497
$185$	23.0574	1.69521
$186$	0	0
$187$	−4.04189	−0.295572
$188$	−2.85710	−0.208375
$189$	0	0
$190$	0	0
$191$	10.0915	0.730197	0.365098	−	0.930969i	$-0.381035\pi$
$191$	10.0915	0.730197	0.365098	+	0.930969i	$0.381035\pi$
$192$	0	0
$193$	−2.52435	−0.181707	−0.0908533	−	0.995864i	$-0.528959\pi$
$193$	−2.52435	−0.181707	−0.0908533	+	0.995864i	$0.528959\pi$
$194$	−7.12567	−0.511593
$195$	0	0
$196$	16.8084	1.20060
$197$	−14.8571	−1.05852	−0.529262	−	0.848458i	$-0.677531\pi$
$197$	−14.8571	−1.05852	−0.529262	+	0.848458i	$0.677531\pi$
$198$	0	0
$199$	8.06923	0.572013	0.286006	−	0.958228i	$-0.407672\pi$
$199$	8.06923	0.572013	0.286006	+	0.958228i	$0.407672\pi$
$200$	−6.63816	−0.469388
$201$	0	0
$202$	−4.64590	−0.326884
$203$	−18.4115	−1.29223
$204$	0	0
$205$	−22.3601	−1.56170
$206$	−12.3824	−0.862721
$207$	0	0
$208$	−2.71688	−0.188382
$209$	0	0
$210$	0	0
$211$	−7.94356	−0.546857	−0.273429	−	0.961892i	$-0.588158\pi$
$211$	−7.94356	−0.546857	−0.273429	+	0.961892i	$0.588158\pi$
$212$	0.630415	0.0432971
$213$	0	0
$214$	−3.89899	−0.266529
$215$	−14.2267	−0.970252
$216$	0	0
$217$	32.2814	2.19140
$218$	−11.5963	−0.785398
$219$	0	0
$220$	−11.6382	−0.784644
$221$	3.21894	0.216529
$222$	0	0
$223$	−11.9682	−0.801451	−0.400726	−	0.916198i	$-0.631242\pi$
$223$	−11.9682	−0.801451	−0.400726	+	0.916198i	$0.631242\pi$
$224$	4.87939	0.326018
$225$	0	0
$226$	−1.94087	−0.129105
$227$	26.5945	1.76514	0.882570	−	0.470181i	$-0.155811\pi$
$227$	26.5945	1.76514	0.882570	+	0.470181i	$0.155811\pi$
$228$	0	0
$229$	−19.9026	−1.31520	−0.657601	−	0.753367i	$-0.728429\pi$
$229$	−19.9026	−1.31520	−0.657601	+	0.753367i	$0.728429\pi$
$230$	11.6382	0.767397
$231$	0	0
$232$	−3.77332	−0.247730
$233$	4.31046	0.282388	0.141194	−	0.989982i	$-0.454906\pi$
$233$	4.31046	0.282388	0.141194	+	0.989982i	$0.454906\pi$
$234$	0	0
$235$	9.74691	0.635818
$236$	11.6382	0.757579
$237$	0	0
$238$	−5.78106	−0.374730
$239$	−0.285807	−0.0184873	−0.00924366	−	0.999957i	$-0.502942\pi$
$239$	−0.285807	−0.0184873	−0.00924366	+	0.999957i	$0.502942\pi$
$240$	0	0
$241$	−9.56624	−0.616216	−0.308108	−	0.951351i	$-0.599696\pi$
$241$	−9.56624	−0.616216	−0.308108	+	0.951351i	$0.599696\pi$
$242$	−0.638156	−0.0410222
$243$	0	0
$244$	6.90167	0.441834
$245$	−57.3414	−3.66341
$246$	0	0
$247$	0	0
$248$	6.61587	0.420108
$249$	0	0
$250$	5.58853	0.353449
$251$	−29.6878	−1.87388	−0.936938	−	0.349495i	$-0.886353\pi$
$251$	−29.6878	−1.87388	−0.936938	+	0.349495i	$0.886353\pi$
$252$	0	0
$253$	11.6382	0.731685
$254$	−6.08378	−0.381730
$255$	0	0
$256$	1.00000	0.0625000
$257$	15.7965	0.985361	0.492681	−	0.870210i	$-0.336017\pi$
$257$	15.7965	0.985361	0.492681	+	0.870210i	$0.336017\pi$
$258$	0	0
$259$	32.9786	2.04919
$260$	9.26857	0.574812
$261$	0	0
$262$	−18.6040	−1.14936
$263$	−10.8648	−0.669955	−0.334977	−	0.942226i	$-0.608729\pi$
$263$	−10.8648	−0.669955	−0.334977	+	0.942226i	$0.608729\pi$
$264$	0	0
$265$	−2.15064	−0.132113
$266$	0	0
$267$	0	0
$268$	−0.709141	−0.0433177
$269$	−17.2935	−1.05441	−0.527203	−	0.849739i	$-0.676759\pi$
$269$	−17.2935	−1.05441	−0.527203	+	0.849739i	$0.676759\pi$
$270$	0	0
$271$	1.44562	0.0878153	0.0439077	−	0.999036i	$-0.486019\pi$
$271$	1.44562	0.0878153	0.0439077	+	0.999036i	$0.486019\pi$
$272$	−1.18479	−0.0718386
$273$	0	0
$274$	4.54664	0.274672
$275$	22.6459	1.36560
$276$	0	0
$277$	21.1908	1.27323	0.636615	−	0.771182i	$-0.280334\pi$
$277$	21.1908	1.27323	0.636615	+	0.771182i	$0.280334\pi$
$278$	3.01455	0.180801
$279$	0	0
$280$	−16.6459	−0.994783
$281$	−24.3678	−1.45366	−0.726831	−	0.686816i	$-0.759008\pi$
$281$	−24.3678	−1.45366	−0.726831	+	0.686816i	$0.759008\pi$
$282$	0	0
$283$	−4.12567	−0.245245	−0.122623	−	0.992453i	$-0.539131\pi$
$283$	−4.12567	−0.245245	−0.122623	+	0.992453i	$0.539131\pi$
$284$	7.95811	0.472227
$285$	0	0
$286$	9.26857	0.548062
$287$	−31.9813	−1.88780
$288$	0	0
$289$	−15.5963	−0.917427
$290$	12.8726	0.755904
$291$	0	0
$292$	−12.7442	−0.745799
$293$	2.78106	0.162471	0.0812356	−	0.996695i	$-0.474113\pi$
$293$	2.78106	0.162471	0.0812356	+	0.996695i	$0.474113\pi$
$294$	0	0
$295$	−39.7033	−2.31161
$296$	6.75877	0.392846
$297$	0	0
$298$	1.18479	0.0686332
$299$	−9.26857	−0.536015
$300$	0	0
$301$	−20.3482	−1.17285
$302$	−3.63816	−0.209352
$303$	0	0
$304$	0	0
$305$	−23.5449	−1.34818
$306$	0	0
$307$	25.0232	1.42815	0.714075	−	0.700069i	$-0.246847\pi$
$307$	25.0232	1.42815	0.714075	+	0.700069i	$0.246847\pi$
$308$	−16.6459	−0.948488
$309$	0	0
$310$	−22.5699	−1.28188
$311$	−23.9736	−1.35942	−0.679709	−	0.733482i	$-0.737894\pi$
$311$	−23.9736	−1.35942	−0.679709	+	0.733482i	$0.737894\pi$
$312$	0	0
$313$	−12.6382	−0.714351	−0.357175	−	0.934037i	$-0.616260\pi$
$313$	−12.6382	−0.714351	−0.357175	+	0.934037i	$0.616260\pi$
$314$	−6.27126	−0.353908
$315$	0	0
$316$	−0.297667	−0.0167450
$317$	−14.1334	−0.793811	−0.396906	−	0.917859i	$-0.629916\pi$
$317$	−14.1334	−0.793811	−0.396906	+	0.917859i	$0.629916\pi$
$318$	0	0
$319$	12.8726	0.720726
$320$	−3.41147	−0.190707
$321$	0	0
$322$	16.6459	0.927640
$323$	0	0
$324$	0	0
$325$	−18.0351	−1.00041
$326$	19.2986	1.06885
$327$	0	0
$328$	−6.55438	−0.361905
$329$	13.9409	0.768585
$330$	0	0
$331$	−29.4979	−1.62135	−0.810677	−	0.585494i	$-0.800901\pi$
$331$	−29.4979	−1.62135	−0.810677	+	0.585494i	$0.800901\pi$
$332$	−15.6040	−0.856381
$333$	0	0
$334$	7.72193	0.422525
$335$	2.41921	0.132176
$336$	0	0
$337$	22.2490	1.21198	0.605989	−	0.795473i	$-0.292777\pi$
$337$	22.2490	1.21198	0.605989	+	0.795473i	$0.292777\pi$
$338$	5.61856	0.305609
$339$	0	0
$340$	4.04189	0.219202
$341$	−22.5699	−1.22223
$342$	0	0
$343$	−47.8590	−2.58414
$344$	−4.17024	−0.224845
$345$	0	0
$346$	14.4953	0.779270
$347$	14.0250	0.752900	0.376450	−	0.926437i	$-0.377145\pi$
$347$	14.0250	0.752900	0.376450	+	0.926437i	$0.377145\pi$
$348$	0	0
$349$	−34.1121	−1.82598	−0.912988	−	0.407986i	$-0.866231\pi$
$349$	−34.1121	−1.82598	−0.912988	+	0.407986i	$0.866231\pi$
$350$	32.3901	1.73132
$351$	0	0
$352$	−3.41147	−0.181832
$353$	−30.4439	−1.62036	−0.810182	−	0.586179i	$-0.800632\pi$
$353$	−30.4439	−1.62036	−0.810182	+	0.586179i	$0.800632\pi$
$354$	0	0
$355$	−27.1489	−1.44091
$356$	6.77332	0.358985
$357$	0	0
$358$	9.82295	0.519159
$359$	−13.6459	−0.720203	−0.360101	−	0.932913i	$-0.617258\pi$
$359$	−13.6459	−0.720203	−0.360101	+	0.932913i	$0.617258\pi$
$360$	0	0
$361$	0	0
$362$	18.9145	0.994122
$363$	0	0
$364$	13.2567	0.694840
$365$	43.4766	2.27567
$366$	0	0
$367$	23.7665	1.24060	0.620301	−	0.784364i	$-0.287010\pi$
$367$	23.7665	1.24060	0.620301	+	0.784364i	$0.287010\pi$
$368$	3.41147	0.177835
$369$	0	0
$370$	−23.0574	−1.19870
$371$	−3.07604	−0.159700
$372$	0	0
$373$	23.2344	1.20303	0.601516	−	0.798860i	$-0.294563\pi$
$373$	23.2344	1.20303	0.601516	+	0.798860i	$0.294563\pi$
$374$	4.04189	0.209001
$375$	0	0
$376$	2.85710	0.147344
$377$	−10.2517	−0.527987
$378$	0	0
$379$	4.50030	0.231165	0.115583	−	0.993298i	$-0.463127\pi$
$379$	4.50030	0.231165	0.115583	+	0.993298i	$0.463127\pi$
$380$	0	0
$381$	0	0
$382$	−10.0915	−0.516327
$383$	4.36009	0.222790	0.111395	−	0.993776i	$-0.464468\pi$
$383$	4.36009	0.222790	0.111395	+	0.993776i	$0.464468\pi$
$384$	0	0
$385$	56.7870	2.89414
$386$	2.52435	0.128486
$387$	0	0
$388$	7.12567	0.361751
$389$	1.31046	0.0664429	0.0332215	−	0.999448i	$-0.489423\pi$
$389$	1.31046	0.0664429	0.0332215	+	0.999448i	$0.489423\pi$
$390$	0	0
$391$	−4.04189	−0.204407
$392$	−16.8084	−0.848952
$393$	0	0
$394$	14.8571	0.748490
$395$	1.01548	0.0510944
$396$	0	0
$397$	−19.0128	−0.954225	−0.477112	−	0.878842i	$-0.658317\pi$
$397$	−19.0128	−0.954225	−0.477112	+	0.878842i	$0.658317\pi$
$398$	−8.06923	−0.404474
$399$	0	0
$400$	6.63816	0.331908
$401$	35.8631	1.79092	0.895458	−	0.445145i	$-0.146848\pi$
$401$	35.8631	1.79092	0.895458	+	0.445145i	$0.146848\pi$
$402$	0	0
$403$	17.9745	0.895375
$404$	4.64590	0.231142
$405$	0	0
$406$	18.4115	0.913746
$407$	−23.0574	−1.14291
$408$	0	0
$409$	29.1088	1.43934	0.719668	−	0.694319i	$-0.244294\pi$
$409$	29.1088	1.43934	0.719668	+	0.694319i	$0.244294\pi$
$410$	22.3601	1.10429
$411$	0	0
$412$	12.3824	0.610036
$413$	−56.7870	−2.79431
$414$	0	0
$415$	53.2327	2.61309
$416$	2.71688	0.133206
$417$	0	0
$418$	0	0
$419$	−13.1584	−0.642829	−0.321415	−	0.946939i	$-0.604158\pi$
$419$	−13.1584	−0.642829	−0.321415	+	0.946939i	$0.604158\pi$
$420$	0	0
$421$	0.150644	0.00734195	0.00367098	−	0.999993i	$-0.498831\pi$
$421$	0.150644	0.00734195	0.00367098	+	0.999993i	$0.498831\pi$
$422$	7.94356	0.386687
$423$	0	0
$424$	−0.630415	−0.0306157
$425$	−7.86484	−0.381501
$426$	0	0
$427$	−33.6759	−1.62969
$428$	3.89899	0.188465
$429$	0	0
$430$	14.2267	0.686072
$431$	−30.7965	−1.48342	−0.741709	−	0.670722i	$-0.765984\pi$
$431$	−30.7965	−1.48342	−0.741709	+	0.670722i	$0.765984\pi$
$432$	0	0
$433$	29.4338	1.41450	0.707248	−	0.706965i	$-0.249936\pi$
$433$	29.4338	1.41450	0.707248	+	0.706965i	$0.249936\pi$
$434$	−32.2814	−1.54956
$435$	0	0
$436$	11.5963	0.555360
$437$	0	0
$438$	0	0
$439$	−19.8007	−0.945034	−0.472517	−	0.881322i	$-0.656654\pi$
$439$	−19.8007	−0.945034	−0.472517	+	0.881322i	$0.656654\pi$
$440$	11.6382	0.554827
$441$	0	0
$442$	−3.21894	−0.153109
$443$	18.5868	0.883084	0.441542	−	0.897241i	$-0.354432\pi$
$443$	18.5868	0.883084	0.441542	+	0.897241i	$0.354432\pi$
$444$	0	0
$445$	−23.1070	−1.09538
$446$	11.9682	0.566711
$447$	0	0
$448$	−4.87939	−0.230529
$449$	3.37908	0.159469	0.0797343	−	0.996816i	$-0.474593\pi$
$449$	3.37908	0.159469	0.0797343	+	0.996816i	$0.474593\pi$
$450$	0	0
$451$	22.3601	1.05290
$452$	1.94087	0.0912911
$453$	0	0
$454$	−26.5945	−1.24814
$455$	−45.2249	−2.12018
$456$	0	0
$457$	21.7912	1.01935	0.509674	−	0.860368i	$-0.329766\pi$
$457$	21.7912	1.01935	0.509674	+	0.860368i	$0.329766\pi$
$458$	19.9026	0.929988
$459$	0	0
$460$	−11.6382	−0.542632
$461$	−11.7638	−0.547896	−0.273948	−	0.961745i	$-0.588330\pi$
$461$	−11.7638	−0.547896	−0.273948	+	0.961745i	$0.588330\pi$
$462$	0	0
$463$	16.9932	0.789741	0.394870	−	0.918737i	$-0.370790\pi$
$463$	16.9932	0.789741	0.394870	+	0.918737i	$0.370790\pi$
$464$	3.77332	0.175172
$465$	0	0
$466$	−4.31046	−0.199678
$467$	11.7050	0.541644	0.270822	−	0.962629i	$-0.412705\pi$
$467$	11.7050	0.541644	0.270822	+	0.962629i	$0.412705\pi$
$468$	0	0
$469$	3.46017	0.159776
$470$	−9.74691	−0.449591
$471$	0	0
$472$	−11.6382	−0.535690
$473$	14.2267	0.654144
$474$	0	0
$475$	0	0
$476$	5.78106	0.264974
$477$	0	0
$478$	0.285807	0.0130725
$479$	−29.9796	−1.36980	−0.684901	−	0.728636i	$-0.740155\pi$
$479$	−29.9796	−1.36980	−0.684901	+	0.728636i	$0.740155\pi$
$480$	0	0
$481$	18.3628	0.837271
$482$	9.56624	0.435730
$483$	0	0
$484$	0.638156	0.0290071
$485$	−24.3090	−1.10382
$486$	0	0
$487$	−14.5107	−0.657544	−0.328772	−	0.944409i	$-0.606635\pi$
$487$	−14.5107	−0.657544	−0.328772	+	0.944409i	$0.606635\pi$
$488$	−6.90167	−0.312424
$489$	0	0
$490$	57.3414	2.59042
$491$	11.6554	0.526000	0.263000	−	0.964796i	$-0.415288\pi$
$491$	11.6554	0.526000	0.263000	+	0.964796i	$0.415288\pi$
$492$	0	0
$493$	−4.47060	−0.201346
$494$	0	0
$495$	0	0
$496$	−6.61587	−0.297061
$497$	−38.8307	−1.74179
$498$	0	0
$499$	−1.68685	−0.0755139	−0.0377569	−	0.999287i	$-0.512021\pi$
$499$	−1.68685	−0.0755139	−0.0377569	+	0.999287i	$0.512021\pi$
$500$	−5.58853	−0.249926
$501$	0	0
$502$	29.6878	1.32503
$503$	−17.2935	−0.771081	−0.385541	−	0.922691i	$-0.625985\pi$
$503$	−17.2935	−0.771081	−0.385541	+	0.922691i	$0.625985\pi$
$504$	0	0
$505$	−15.8494	−0.705287
$506$	−11.6382	−0.517379
$507$	0	0
$508$	6.08378	0.269924
$509$	32.4424	1.43799	0.718993	−	0.695017i	$-0.244603\pi$
$509$	32.4424	1.43799	0.718993	+	0.695017i	$0.244603\pi$
$510$	0	0
$511$	62.1840	2.75086
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−15.7965	−0.696756
$515$	−42.2422	−1.86141
$516$	0	0
$517$	−9.74691	−0.428669
$518$	−32.9786	−1.44900
$519$	0	0
$520$	−9.26857	−0.406454
$521$	31.1644	1.36534	0.682668	−	0.730729i	$-0.260820\pi$
$521$	31.1644	1.36534	0.682668	+	0.730729i	$0.260820\pi$
$522$	0	0
$523$	16.6604	0.728510	0.364255	−	0.931299i	$-0.381324\pi$
$523$	16.6604	0.728510	0.364255	+	0.931299i	$0.381324\pi$
$524$	18.6040	0.812720
$525$	0	0
$526$	10.8648	0.473729
$527$	7.83843	0.341447
$528$	0	0
$529$	−11.3618	−0.493993
$530$	2.15064	0.0934180
$531$	0	0
$532$	0	0
$533$	−17.8075	−0.771327
$534$	0	0
$535$	−13.3013	−0.575065
$536$	0.709141	0.0306302
$537$	0	0
$538$	17.2935	0.745578
$539$	57.3414	2.46987
$540$	0	0
$541$	36.2449	1.55829	0.779144	−	0.626845i	$-0.215654\pi$
$541$	36.2449	1.55829	0.779144	+	0.626845i	$0.215654\pi$
$542$	−1.44562	−0.0620948
$543$	0	0
$544$	1.18479	0.0507976
$545$	−39.5604	−1.69458
$546$	0	0
$547$	17.1976	0.735316	0.367658	−	0.929961i	$-0.380160\pi$
$547$	17.1976	0.735316	0.367658	+	0.929961i	$0.380160\pi$
$548$	−4.54664	−0.194223
$549$	0	0
$550$	−22.6459	−0.965624
$551$	0	0
$552$	0	0
$553$	1.45243	0.0617636
$554$	−21.1908	−0.900310
$555$	0	0
$556$	−3.01455	−0.127845
$557$	16.5699	0.702087	0.351044	−	0.936359i	$-0.385827\pi$
$557$	16.5699	0.702087	0.351044	+	0.936359i	$0.385827\pi$
$558$	0	0
$559$	−11.3301	−0.479210
$560$	16.6459	0.703418
$561$	0	0
$562$	24.3678	1.02789
$563$	−30.6973	−1.29374	−0.646868	−	0.762602i	$-0.723922\pi$
$563$	−30.6973	−1.29374	−0.646868	+	0.762602i	$0.723922\pi$
$564$	0	0
$565$	−6.62124	−0.278558
$566$	4.12567	0.173415
$567$	0	0
$568$	−7.95811	−0.333915
$569$	13.4534	0.563994	0.281997	−	0.959415i	$-0.409003\pi$
$569$	13.4534	0.563994	0.281997	+	0.959415i	$0.409003\pi$
$570$	0	0
$571$	−9.47565	−0.396544	−0.198272	−	0.980147i	$-0.563533\pi$
$571$	−9.47565	−0.396544	−0.198272	+	0.980147i	$0.563533\pi$
$572$	−9.26857	−0.387538
$573$	0	0
$574$	31.9813	1.33488
$575$	22.6459	0.944399
$576$	0	0
$577$	−18.7638	−0.781148	−0.390574	−	0.920571i	$-0.627723\pi$
$577$	−18.7638	−0.781148	−0.390574	+	0.920571i	$0.627723\pi$
$578$	15.5963	0.648719
$579$	0	0
$580$	−12.8726	−0.534505
$581$	76.1380	3.15873
$582$	0	0
$583$	2.15064	0.0890706
$584$	12.7442	0.527360
$585$	0	0
$586$	−2.78106	−0.114884
$587$	18.1753	0.750175	0.375087	−	0.926989i	$-0.377613\pi$
$587$	18.1753	0.750175	0.375087	+	0.926989i	$0.377613\pi$
$588$	0	0
$589$	0	0
$590$	39.7033	1.63456
$591$	0	0
$592$	−6.75877	−0.277784
$593$	32.6938	1.34257	0.671286	−	0.741198i	$-0.265742\pi$
$593$	32.6938	1.34257	0.671286	+	0.741198i	$0.265742\pi$
$594$	0	0
$595$	−19.7219	−0.808520
$596$	−1.18479	−0.0485310
$597$	0	0
$598$	9.26857	0.379020
$599$	8.83069	0.360812	0.180406	−	0.983592i	$-0.442259\pi$
$599$	8.83069	0.360812	0.180406	+	0.983592i	$0.442259\pi$
$600$	0	0
$601$	−33.0523	−1.34823	−0.674116	−	0.738625i	$-0.735475\pi$
$601$	−33.0523	−1.34823	−0.674116	+	0.738625i	$0.735475\pi$
$602$	20.3482	0.829332
$603$	0	0
$604$	3.63816	0.148034
$605$	−2.17705	−0.0885097
$606$	0	0
$607$	−5.16250	−0.209540	−0.104770	−	0.994497i	$-0.533411\pi$
$607$	−5.16250	−0.209540	−0.104770	+	0.994497i	$0.533411\pi$
$608$	0	0
$609$	0	0
$610$	23.5449	0.953304
$611$	7.76239	0.314033
$612$	0	0
$613$	−47.0883	−1.90188	−0.950940	−	0.309376i	$-0.899880\pi$
$613$	−47.0883	−1.90188	−0.950940	+	0.309376i	$0.899880\pi$
$614$	−25.0232	−1.00986
$615$	0	0
$616$	16.6459	0.670682
$617$	42.3851	1.70636	0.853179	−	0.521618i	$-0.174671\pi$
$617$	42.3851	1.70636	0.853179	+	0.521618i	$0.174671\pi$
$618$	0	0
$619$	−1.85616	−0.0746055	−0.0373027	−	0.999304i	$-0.511877\pi$
$619$	−1.85616	−0.0746055	−0.0373027	+	0.999304i	$0.511877\pi$
$620$	22.5699	0.906427
$621$	0	0
$622$	23.9736	0.961253
$623$	−33.0496	−1.32411
$624$	0	0
$625$	−14.1257	−0.565027
$626$	12.6382	0.505122
$627$	0	0
$628$	6.27126	0.250250
$629$	8.00774	0.319290
$630$	0	0
$631$	24.6483	0.981232	0.490616	−	0.871376i	$-0.336772\pi$
$631$	24.6483	0.981232	0.490616	+	0.871376i	$0.336772\pi$
$632$	0.297667	0.0118405
$633$	0	0
$634$	14.1334	0.561309
$635$	−20.7547	−0.823623
$636$	0	0
$637$	−45.6664	−1.80937
$638$	−12.8726	−0.509630
$639$	0	0
$640$	3.41147	0.134850
$641$	17.6382	0.696665	0.348333	−	0.937371i	$-0.386748\pi$
$641$	17.6382	0.696665	0.348333	+	0.937371i	$0.386748\pi$
$642$	0	0
$643$	29.8485	1.17711	0.588556	−	0.808457i	$-0.299697\pi$
$643$	29.8485	1.17711	0.588556	+	0.808457i	$0.299697\pi$
$644$	−16.6459	−0.655940
$645$	0	0
$646$	0	0
$647$	−23.9391	−0.941144	−0.470572	−	0.882362i	$-0.655952\pi$
$647$	−23.9391	−0.941144	−0.470572	+	0.882362i	$0.655952\pi$
$648$	0	0
$649$	39.7033	1.55849
$650$	18.0351	0.707394
$651$	0	0
$652$	−19.2986	−0.755792
$653$	26.4861	1.03648	0.518240	−	0.855235i	$-0.326587\pi$
$653$	26.4861	1.03648	0.518240	+	0.855235i	$0.326587\pi$
$654$	0	0
$655$	−63.4671	−2.47986
$656$	6.55438	0.255905
$657$	0	0
$658$	−13.9409	−0.543472
$659$	−5.29355	−0.206207	−0.103104	−	0.994671i	$-0.532877\pi$
$659$	−5.29355	−0.206207	−0.103104	+	0.994671i	$0.532877\pi$
$660$	0	0
$661$	−46.1070	−1.79335	−0.896677	−	0.442685i	$-0.854026\pi$
$661$	−46.1070	−1.79335	−0.896677	+	0.442685i	$0.854026\pi$
$662$	29.4979	1.14647
$663$	0	0
$664$	15.6040	0.605553
$665$	0	0
$666$	0	0
$667$	12.8726	0.498428
$668$	−7.72193	−0.298771
$669$	0	0
$670$	−2.41921	−0.0934624
$671$	23.5449	0.908940
$672$	0	0
$673$	−2.38144	−0.0917979	−0.0458990	−	0.998946i	$-0.514615\pi$
$673$	−2.38144	−0.0917979	−0.0458990	+	0.998946i	$0.514615\pi$
$674$	−22.2490	−0.856998
$675$	0	0
$676$	−5.61856	−0.216098
$677$	15.9486	0.612955	0.306478	−	0.951878i	$-0.400850\pi$
$677$	15.9486	0.612955	0.306478	+	0.951878i	$0.400850\pi$
$678$	0	0
$679$	−34.7689	−1.33431
$680$	−4.04189	−0.154999
$681$	0	0
$682$	22.5699	0.864245
$683$	−1.26083	−0.0482443	−0.0241222	−	0.999709i	$-0.507679\pi$
$683$	−1.26083	−0.0482443	−0.0241222	+	0.999709i	$0.507679\pi$
$684$	0	0
$685$	15.5107	0.592635
$686$	47.8590	1.82726
$687$	0	0
$688$	4.17024	0.158989
$689$	−1.71276	−0.0652511
$690$	0	0
$691$	−8.10876	−0.308472	−0.154236	−	0.988034i	$-0.549292\pi$
$691$	−8.10876	−0.308472	−0.154236	+	0.988034i	$0.549292\pi$
$692$	−14.4953	−0.551027
$693$	0	0
$694$	−14.0250	−0.532381
$695$	10.2841	0.390096
$696$	0	0
$697$	−7.76558	−0.294142
$698$	34.1121	1.29116
$699$	0	0
$700$	−32.3901	−1.22423
$701$	−10.2284	−0.386323	−0.193161	−	0.981167i	$-0.561874\pi$
$701$	−10.2284	−0.386323	−0.193161	+	0.981167i	$0.561874\pi$
$702$	0	0
$703$	0	0
$704$	3.41147	0.128575
$705$	0	0
$706$	30.4439	1.14577
$707$	−22.6691	−0.852560
$708$	0	0
$709$	0.204393	0.00767614	0.00383807	−	0.999993i	$-0.498778\pi$
$709$	0.204393	0.00767614	0.00383807	+	0.999993i	$0.498778\pi$
$710$	27.1489	1.01888
$711$	0	0
$712$	−6.77332	−0.253841
$713$	−22.5699	−0.845248
$714$	0	0
$715$	31.6195	1.18250
$716$	−9.82295	−0.367101
$717$	0	0
$718$	13.6459	0.509260
$719$	−15.9394	−0.594441	−0.297220	−	0.954809i	$-0.596060\pi$
$719$	−15.9394	−0.594441	−0.297220	+	0.954809i	$0.596060\pi$
$720$	0	0
$721$	−60.4184	−2.25010
$722$	0	0
$723$	0	0
$724$	−18.9145	−0.702951
$725$	25.0479	0.930255
$726$	0	0
$727$	−30.9632	−1.14836	−0.574180	−	0.818729i	$-0.694679\pi$
$727$	−30.9632	−1.14836	−0.574180	+	0.818729i	$0.694679\pi$
$728$	−13.2567	−0.491326
$729$	0	0
$730$	−43.4766	−1.60914
$731$	−4.94087	−0.182745
$732$	0	0
$733$	3.95811	0.146196	0.0730981	−	0.997325i	$-0.476711\pi$
$733$	3.95811	0.146196	0.0730981	+	0.997325i	$0.476711\pi$
$734$	−23.7665	−0.877238
$735$	0	0
$736$	−3.41147	−0.125749
$737$	−2.41921	−0.0891129
$738$	0	0
$739$	−25.8033	−0.949191	−0.474596	−	0.880204i	$-0.657406\pi$
$739$	−25.8033	−0.949191	−0.474596	+	0.880204i	$0.657406\pi$
$740$	23.0574	0.847606
$741$	0	0
$742$	3.07604	0.112925
$743$	31.9145	1.17083	0.585414	−	0.810734i	$-0.300932\pi$
$743$	31.9145	1.17083	0.585414	+	0.810734i	$0.300932\pi$
$744$	0	0
$745$	4.04189	0.148083
$746$	−23.2344	−0.850673
$747$	0	0
$748$	−4.04189	−0.147786
$749$	−19.0247	−0.695146
$750$	0	0
$751$	0.508045	0.0185388	0.00926942	−	0.999957i	$-0.497049\pi$
$751$	0.508045	0.0185388	0.00926942	+	0.999957i	$0.497049\pi$
$752$	−2.85710	−0.104188
$753$	0	0
$754$	10.2517	0.373343
$755$	−12.4115	−0.451700
$756$	0	0
$757$	−48.1694	−1.75075	−0.875374	−	0.483447i	$-0.839385\pi$
$757$	−48.1694	−1.75075	−0.875374	+	0.483447i	$0.839385\pi$
$758$	−4.50030	−0.163458
$759$	0	0
$760$	0	0
$761$	−5.15064	−0.186711	−0.0933554	−	0.995633i	$-0.529759\pi$
$761$	−5.15064	−0.186711	−0.0933554	+	0.995633i	$0.529759\pi$
$762$	0	0
$763$	−56.5827	−2.04843
$764$	10.0915	0.365098
$765$	0	0
$766$	−4.36009	−0.157536
$767$	−31.6195	−1.14171
$768$	0	0
$769$	19.5725	0.705804	0.352902	−	0.935660i	$-0.385195\pi$
$769$	19.5725	0.705804	0.352902	+	0.935660i	$0.385195\pi$
$770$	−56.7870	−2.04646
$771$	0	0
$772$	−2.52435	−0.0908533
$773$	22.3946	0.805476	0.402738	−	0.915315i	$-0.368059\pi$
$773$	22.3946	0.805476	0.402738	+	0.915315i	$0.368059\pi$
$774$	0	0
$775$	−43.9172	−1.57755
$776$	−7.12567	−0.255797
$777$	0	0
$778$	−1.31046	−0.0469823
$779$	0	0
$780$	0	0
$781$	27.1489	0.971464
$782$	4.04189	0.144538
$783$	0	0
$784$	16.8084	0.600300
$785$	−21.3942	−0.763593
$786$	0	0
$787$	−33.6887	−1.20087	−0.600437	−	0.799672i	$-0.705007\pi$
$787$	−33.6887	−1.20087	−0.600437	+	0.799672i	$0.705007\pi$
$788$	−14.8571	−0.529262
$789$	0	0
$790$	−1.01548	−0.0361292
$791$	−9.47028	−0.336724
$792$	0	0
$793$	−18.7510	−0.665869
$794$	19.0128	0.674739
$795$	0	0
$796$	8.06923	0.286006
$797$	−47.8631	−1.69540	−0.847699	−	0.530478i	$-0.822012\pi$
$797$	−47.8631	−1.69540	−0.847699	+	0.530478i	$0.822012\pi$
$798$	0	0
$799$	3.38507	0.119755
$800$	−6.63816	−0.234694
$801$	0	0
$802$	−35.8631	−1.26637
$803$	−43.4766	−1.53426
$804$	0	0
$805$	56.7870	2.00148
$806$	−17.9745	−0.633126
$807$	0	0
$808$	−4.64590	−0.163442
$809$	3.67406	0.129173	0.0645865	−	0.997912i	$-0.479427\pi$
$809$	3.67406	0.129173	0.0645865	+	0.997912i	$0.479427\pi$
$810$	0	0
$811$	18.7050	0.656822	0.328411	−	0.944535i	$-0.393487\pi$
$811$	18.7050	0.656822	0.328411	+	0.944535i	$0.393487\pi$
$812$	−18.4115	−0.646116
$813$	0	0
$814$	23.0574	0.808160
$815$	65.8367	2.30616
$816$	0	0
$817$	0	0
$818$	−29.1088	−1.01776
$819$	0	0
$820$	−22.3601	−0.780848
$821$	22.3164	0.778849	0.389425	−	0.921058i	$-0.372674\pi$
$821$	22.3164	0.778849	0.389425	+	0.921058i	$0.372674\pi$
$822$	0	0
$823$	−26.4037	−0.920376	−0.460188	−	0.887821i	$-0.652218\pi$
$823$	−26.4037	−0.920376	−0.460188	+	0.887821i	$0.652218\pi$
$824$	−12.3824	−0.431361
$825$	0	0
$826$	56.7870	1.97587
$827$	29.6705	1.03175	0.515873	−	0.856665i	$-0.327468\pi$
$827$	29.6705	1.03175	0.515873	+	0.856665i	$0.327468\pi$
$828$	0	0
$829$	−1.76827	−0.0614144	−0.0307072	−	0.999528i	$-0.509776\pi$
$829$	−1.76827	−0.0614144	−0.0307072	+	0.999528i	$0.509776\pi$
$830$	−53.2327	−1.84773
$831$	0	0
$832$	−2.71688	−0.0941909
$833$	−19.9145	−0.689995
$834$	0	0
$835$	26.3432	0.911643
$836$	0	0
$837$	0	0
$838$	13.1584	0.454549
$839$	26.7142	0.922276	0.461138	−	0.887328i	$-0.347441\pi$
$839$	26.7142	0.922276	0.461138	+	0.887328i	$0.347441\pi$
$840$	0	0
$841$	−14.7621	−0.509037
$842$	−0.150644	−0.00519154
$843$	0	0
$844$	−7.94356	−0.273429
$845$	19.1676	0.659384
$846$	0	0
$847$	−3.11381	−0.106992
$848$	0.630415	0.0216485
$849$	0	0
$850$	7.86484	0.269762
$851$	−23.0574	−0.790396
$852$	0	0
$853$	13.9263	0.476828	0.238414	−	0.971164i	$-0.423372\pi$
$853$	13.9263	0.476828	0.238414	+	0.971164i	$0.423372\pi$
$854$	33.6759	1.15237
$855$	0	0
$856$	−3.89899	−0.133265
$857$	−19.6691	−0.671884	−0.335942	−	0.941883i	$-0.609055\pi$
$857$	−19.6691	−0.671884	−0.335942	+	0.941883i	$0.609055\pi$
$858$	0	0
$859$	−43.3979	−1.48072	−0.740358	−	0.672213i	$-0.765344\pi$
$859$	−43.3979	−1.48072	−0.740358	+	0.672213i	$0.765344\pi$
$860$	−14.2267	−0.485126
$861$	0	0
$862$	30.7965	1.04893
$863$	−29.4192	−1.00144	−0.500721	−	0.865609i	$-0.666932\pi$
$863$	−29.4192	−1.00144	−0.500721	+	0.865609i	$0.666932\pi$
$864$	0	0
$865$	49.4502	1.68136
$866$	−29.4338	−1.00020
$867$	0	0
$868$	32.2814	1.09570
$869$	−1.01548	−0.0344479
$870$	0	0
$871$	1.92665	0.0652821
$872$	−11.5963	−0.392699
$873$	0	0
$874$	0	0
$875$	27.2686	0.921846
$876$	0	0
$877$	26.1652	0.883536	0.441768	−	0.897129i	$-0.354351\pi$
$877$	26.1652	0.883536	0.441768	+	0.897129i	$0.354351\pi$
$878$	19.8007	0.668240
$879$	0	0
$880$	−11.6382	−0.392322
$881$	−22.6287	−0.762379	−0.381189	−	0.924497i	$-0.624485\pi$
$881$	−22.6287	−0.762379	−0.381189	+	0.924497i	$0.624485\pi$
$882$	0	0
$883$	−8.28169	−0.278701	−0.139350	−	0.990243i	$-0.544501\pi$
$883$	−8.28169	−0.278701	−0.139350	+	0.990243i	$0.544501\pi$
$884$	3.21894	0.108265
$885$	0	0
$886$	−18.5868	−0.624435
$887$	40.3569	1.35505	0.677526	−	0.735499i	$-0.263052\pi$
$887$	40.3569	1.35505	0.677526	+	0.735499i	$0.263052\pi$
$888$	0	0
$889$	−29.6851	−0.995606
$890$	23.1070	0.774548
$891$	0	0
$892$	−11.9682	−0.400726
$893$	0	0
$894$	0	0
$895$	33.5107	1.12014
$896$	4.87939	0.163009
$897$	0	0
$898$	−3.37908	−0.112761
$899$	−24.9638	−0.832588
$900$	0	0
$901$	−0.746911	−0.0248832
$902$	−22.3601	−0.744510
$903$	0	0
$904$	−1.94087	−0.0645525
$905$	64.5262	2.14492
$906$	0	0
$907$	19.2513	0.639230	0.319615	−	0.947547i	$-0.396446\pi$
$907$	19.2513	0.639230	0.319615	+	0.947547i	$0.396446\pi$
$908$	26.5945	0.882570
$909$	0	0
$910$	45.2249	1.49919
$911$	−58.7684	−1.94708	−0.973542	−	0.228510i	$-0.926615\pi$
$911$	−58.7684	−1.94708	−0.973542	+	0.228510i	$0.926615\pi$
$912$	0	0
$913$	−53.2327	−1.76174
$914$	−21.7912	−0.720788
$915$	0	0
$916$	−19.9026	−0.657601
$917$	−90.7761	−2.99769
$918$	0	0
$919$	−53.7428	−1.77281	−0.886406	−	0.462909i	$-0.846806\pi$
$919$	−53.7428	−1.77281	−0.886406	+	0.462909i	$0.846806\pi$
$920$	11.6382	0.383699
$921$	0	0
$922$	11.7638	0.387421
$923$	−21.6212	−0.711672
$924$	0	0
$925$	−44.8658	−1.47518
$926$	−16.9932	−0.558431
$927$	0	0
$928$	−3.77332	−0.123865
$929$	31.0387	1.01835	0.509173	−	0.860664i	$-0.329951\pi$
$929$	31.0387	1.01835	0.509173	+	0.860664i	$0.329951\pi$
$930$	0	0
$931$	0	0
$932$	4.31046	0.141194
$933$	0	0
$934$	−11.7050	−0.383000
$935$	13.7888	0.450942
$936$	0	0
$937$	−20.7428	−0.677637	−0.338819	−	0.940852i	$-0.610027\pi$
$937$	−20.7428	−0.677637	−0.338819	+	0.940852i	$0.610027\pi$
$938$	−3.46017	−0.112979
$939$	0	0
$940$	9.74691	0.317909
$941$	−14.8631	−0.484523	−0.242261	−	0.970211i	$-0.577889\pi$
$941$	−14.8631	−0.484523	−0.242261	+	0.970211i	$0.577889\pi$
$942$	0	0
$943$	22.3601	0.728145
$944$	11.6382	0.378790
$945$	0	0
$946$	−14.2267	−0.462549
$947$	25.1999	0.818888	0.409444	−	0.912335i	$-0.365723\pi$
$947$	25.1999	0.818888	0.409444	+	0.912335i	$0.365723\pi$
$948$	0	0
$949$	34.6245	1.12396
$950$	0	0
$951$	0	0
$952$	−5.78106	−0.187365
$953$	−12.3851	−0.401192	−0.200596	−	0.979674i	$-0.564288\pi$
$953$	−12.3851	−0.401192	−0.200596	+	0.979674i	$0.564288\pi$
$954$	0	0
$955$	−34.4270	−1.11403
$956$	−0.285807	−0.00924366
$957$	0	0
$958$	29.9796	0.968596
$959$	22.1848	0.716384
$960$	0	0
$961$	12.7697	0.411926
$962$	−18.3628	−0.592040
$963$	0	0
$964$	−9.56624	−0.308108
$965$	8.61175	0.277222
$966$	0	0
$967$	−3.40879	−0.109619	−0.0548096	−	0.998497i	$-0.517455\pi$
$967$	−3.40879	−0.109619	−0.0548096	+	0.998497i	$0.517455\pi$
$968$	−0.638156	−0.0205111
$969$	0	0
$970$	24.3090	0.780516
$971$	−13.8972	−0.445983	−0.222992	−	0.974820i	$-0.571582\pi$
$971$	−13.8972	−0.445983	−0.222992	+	0.974820i	$0.571582\pi$
$972$	0	0
$973$	14.7091	0.471553
$974$	14.5107	0.464954
$975$	0	0
$976$	6.90167	0.220917
$977$	12.8402	0.410794	0.205397	−	0.978679i	$-0.434151\pi$
$977$	12.8402	0.410794	0.205397	+	0.978679i	$0.434151\pi$
$978$	0	0
$979$	23.1070	0.738503
$980$	−57.3414	−1.83170
$981$	0	0
$982$	−11.6554	−0.371939
$983$	−0.866592	−0.0276400	−0.0138200	−	0.999904i	$-0.504399\pi$
$983$	−0.866592	−0.0276400	−0.0138200	+	0.999904i	$0.504399\pi$
$984$	0	0
$985$	50.6846	1.61495
$986$	4.47060	0.142373
$987$	0	0
$988$	0	0
$989$	14.2267	0.452382
$990$	0	0
$991$	20.9709	0.666163	0.333081	−	0.942898i	$-0.391912\pi$
$991$	20.9709	0.666163	0.333081	+	0.942898i	$0.391912\pi$
$992$	6.61587	0.210054
$993$	0	0
$994$	38.8307	1.23163
$995$	−27.5280	−0.872695
$996$	0	0
$997$	39.7475	1.25882	0.629408	−	0.777075i	$-0.283297\pi$
$997$	39.7475	1.25882	0.629408	+	0.777075i	$0.283297\pi$
$998$	1.68685	0.0533964
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6498.2.a.bm.1.1		3
3.2	odd	2		2166.2.a.s.1.3		3
19.3	odd	18		342.2.u.e.199.1		6
19.13	odd	18		342.2.u.e.55.1		6
19.18	odd	2		6498.2.a.br.1.1		3
57.32	even	18		114.2.i.a.55.1	✓	6
57.41	even	18		114.2.i.a.85.1	yes	6
57.56	even	2		2166.2.a.q.1.3		3
228.155	odd	18		912.2.bo.a.769.1		6
228.203	odd	18		912.2.bo.a.625.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.i.a.55.1	✓	6	57.32	even	18
114.2.i.a.85.1	yes	6	57.41	even	18
342.2.u.e.55.1		6	19.13	odd	18
342.2.u.e.199.1		6	19.3	odd	18
912.2.bo.a.625.1		6	228.203	odd	18
912.2.bo.a.769.1		6	228.155	odd	18
2166.2.a.q.1.3		3	57.56	even	2
2166.2.a.s.1.3		3	3.2	odd	2
6498.2.a.bm.1.1		3	1.1	even	1	trivial
6498.2.a.br.1.1		3	19.18	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:	\( N \)	\(=\)	\( 6498 = 2 \cdot 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6498.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.41147 q^{5} -4.87939 q^{7} -1.00000 q^{8} +3.41147 q^{10} +3.41147 q^{11} -2.71688 q^{13} +4.87939 q^{14} +1.00000 q^{16} -1.18479 q^{17} -3.41147 q^{20} -3.41147 q^{22} +3.41147 q^{23} +6.63816 q^{25} +2.71688 q^{26} -4.87939 q^{28} +3.77332 q^{29} -6.61587 q^{31} -1.00000 q^{32} +1.18479 q^{34} +16.6459 q^{35} -6.75877 q^{37} +3.41147 q^{40} +6.55438 q^{41} +4.17024 q^{43} +3.41147 q^{44} -3.41147 q^{46} -2.85710 q^{47} +16.8084 q^{49} -6.63816 q^{50} -2.71688 q^{52} +0.630415 q^{53} -11.6382 q^{55} +4.87939 q^{56} -3.77332 q^{58} +11.6382 q^{59} +6.90167 q^{61} +6.61587 q^{62} +1.00000 q^{64} +9.26857 q^{65} -0.709141 q^{67} -1.18479 q^{68} -16.6459 q^{70} +7.95811 q^{71} -12.7442 q^{73} +6.75877 q^{74} -16.6459 q^{77} -0.297667 q^{79} -3.41147 q^{80} -6.55438 q^{82} -15.6040 q^{83} +4.04189 q^{85} -4.17024 q^{86} -3.41147 q^{88} +6.77332 q^{89} +13.2567 q^{91} +3.41147 q^{92} +2.85710 q^{94} +7.12567 q^{97} -16.8084 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} - 9 q^{7} - 3 q^{8} + 9 q^{14} + 3 q^{16} + 3 q^{25} - 9 q^{28} + 18 q^{29} - 9 q^{31} - 3 q^{32} + 9 q^{35} - 9 q^{37} + 9 q^{41} - 9 q^{43} - 9 q^{47} + 12 q^{49} - 3 q^{50} + 9 q^{53}+ \cdots - 12 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 - 9 * q^7 - 3 * q^8 + 9 * q^14 + 3 * q^16 + 3 * q^25 - 9 * q^28 + 18 * q^29 - 9 * q^31 - 3 * q^32 + 9 * q^35 - 9 * q^37 + 9 * q^41 - 9 * q^43 - 9 * q^47 + 12 * q^49 - 3 * q^50 + 9 * q^53 - 18 * q^55 + 9 * q^56 - 18 * q^58 + 18 * q^59 + 9 * q^61 + 9 * q^62 + 3 * q^64 + 18 * q^65 - 18 * q^67 - 9 * q^70 + 27 * q^71 - 9 * q^73 + 9 * q^74 - 9 * q^77 - 27 * q^79 - 9 * q^82 - 9 * q^83 + 9 * q^85 + 9 * q^86 + 27 * q^89 + 3 * q^91 + 9 * q^94 + 12 * q^97 - 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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