LMFDB - Embedded newform 6498.2.a.bu.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6498,2,Mod(1,6498)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

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")

//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);

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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$51.8867912334$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{5}]$
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			1.3
Root			$-1.53209$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.53209 q^{5} +3.71688 q^{7} +1.00000 q^{8} +O(q^{10})$	$q+1.00000 q^{2} +1.00000 q^{4} +3.53209 q^{5} +3.71688 q^{7} +1.00000 q^{8} +3.53209 q^{10} +5.29086 q^{11} +0.226682 q^{13} +3.71688 q^{14} +1.00000 q^{16} +1.65270 q^{17} +3.53209 q^{20} +5.29086 q^{22} -8.68004 q^{23} +7.47565 q^{25} +0.226682 q^{26} +3.71688 q^{28} +0.120615 q^{29} -3.12061 q^{31} +1.00000 q^{32} +1.65270 q^{34} +13.1284 q^{35} -5.12836 q^{37} +3.53209 q^{40} +7.10607 q^{41} -5.35504 q^{43} +5.29086 q^{44} -8.68004 q^{46} +2.50980 q^{47} +6.81521 q^{49} +7.47565 q^{50} +0.226682 q^{52} +5.93582 q^{53} +18.6878 q^{55} +3.71688 q^{56} +0.120615 q^{58} +0.218941 q^{59} -1.57398 q^{61} -3.12061 q^{62} +1.00000 q^{64} +0.800660 q^{65} -15.4807 q^{67} +1.65270 q^{68} +13.1284 q^{70} -1.35504 q^{71} +2.42602 q^{73} -5.12836 q^{74} +19.6655 q^{77} +2.86484 q^{79} +3.53209 q^{80} +7.10607 q^{82} -1.92127 q^{83} +5.83750 q^{85} -5.35504 q^{86} +5.29086 q^{88} -12.1557 q^{89} +0.842549 q^{91} -8.68004 q^{92} +2.50980 q^{94} -17.5030 q^{97} +6.81521 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 3 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 + 3 * q^7 + 3 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 3 q^{7} + 3 q^{8} + 6 q^{10} - 6 q^{13} + 3 q^{14} + 3 q^{16} + 6 q^{17} + 6 q^{20} - 6 q^{23} + 3 q^{25} - 6 q^{26} + 3 q^{28} + 6 q^{29} - 15 q^{31} + 3 q^{32} + 6 q^{34} + 21 q^{35} + 3 q^{37} + 6 q^{40} + 9 q^{41} + 9 q^{43} - 6 q^{46} + 9 q^{47} + 24 q^{49} + 3 q^{50} - 6 q^{52} + 27 q^{53} + 12 q^{55} + 3 q^{56} + 6 q^{58} + 18 q^{59} + 3 q^{61} - 15 q^{62} + 3 q^{64} - 12 q^{65} - 12 q^{67} + 6 q^{68} + 21 q^{70} + 21 q^{71} + 15 q^{73} + 3 q^{74} + 21 q^{77} - 15 q^{79} + 6 q^{80} + 9 q^{82} + 3 q^{83} + 15 q^{85} + 9 q^{86} + 3 q^{89} - 15 q^{91} - 6 q^{92} + 9 q^{94} - 12 q^{97} + 24 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 + 3 * q^7 + 3 * q^8 + 6 * q^10 - 6 * q^13 + 3 * q^14 + 3 * q^16 + 6 * q^17 + 6 * q^20 - 6 * q^23 + 3 * q^25 - 6 * q^26 + 3 * q^28 + 6 * q^29 - 15 * q^31 + 3 * q^32 + 6 * q^34 + 21 * q^35 + 3 * q^37 + 6 * q^40 + 9 * q^41 + 9 * q^43 - 6 * q^46 + 9 * q^47 + 24 * q^49 + 3 * q^50 - 6 * q^52 + 27 * q^53 + 12 * q^55 + 3 * q^56 + 6 * q^58 + 18 * q^59 + 3 * q^61 - 15 * q^62 + 3 * q^64 - 12 * q^65 - 12 * q^67 + 6 * q^68 + 21 * q^70 + 21 * q^71 + 15 * q^73 + 3 * q^74 + 21 * q^77 - 15 * q^79 + 6 * q^80 + 9 * q^82 + 3 * q^83 + 15 * q^85 + 9 * q^86 + 3 * q^89 - 15 * q^91 - 6 * q^92 + 9 * q^94 - 12 * q^97 + 24 * q^98

Display 100 coefficients 10 coefficients

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
$5$	3.53209	1.57960	0.789799	−	0.613366i	$-0.210185\pi$
$5$	3.53209	1.57960	0.789799	+	0.613366i	$0.210185\pi$
$6$	0	0
$7$	3.71688	1.40485	0.702425	−	0.711758i	$-0.252101\pi$
$7$	3.71688	1.40485	0.702425	+	0.711758i	$0.252101\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	3.53209	1.11694
$11$	5.29086	1.59525	0.797627	−	0.603151i	$-0.206088\pi$
$11$	5.29086	1.59525	0.797627	+	0.603151i	$0.206088\pi$
$12$	0	0
$13$	0.226682	0.0628702	0.0314351	−	0.999506i	$-0.489992\pi$
$13$	0.226682	0.0628702	0.0314351	+	0.999506i	$0.489992\pi$
$14$	3.71688	0.993378
$15$	0	0
$16$	1.00000	0.250000
$17$	1.65270	0.400840	0.200420	−	0.979710i	$-0.435769\pi$
$17$	1.65270	0.400840	0.200420	+	0.979710i	$0.435769\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	3.53209	0.789799
$21$	0	0
$22$	5.29086	1.12802
$23$	−8.68004	−1.80991	−0.904957	−	0.425503i	$-0.860097\pi$
$23$	−8.68004	−1.80991	−0.904957	+	0.425503i	$0.860097\pi$
$24$	0	0
$25$	7.47565	1.49513
$26$	0.226682	0.0444559
$27$	0	0
$28$	3.71688	0.702425
$29$	0.120615	0.0223976	0.0111988	−	0.999937i	$-0.496435\pi$
$29$	0.120615	0.0223976	0.0111988	+	0.999937i	$0.496435\pi$
$30$	0	0
$31$	−3.12061	−0.560479	−0.280239	−	0.959930i	$-0.590414\pi$
$31$	−3.12061	−0.560479	−0.280239	+	0.959930i	$0.590414\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.65270	0.283436
$35$	13.1284	2.21910
$36$	0	0
$37$	−5.12836	−0.843096	−0.421548	−	0.906806i	$-0.638513\pi$
$37$	−5.12836	−0.843096	−0.421548	+	0.906806i	$0.638513\pi$
$38$	0	0
$39$	0	0
$40$	3.53209	0.558472
$41$	7.10607	1.10978	0.554891	−	0.831923i	$-0.312760\pi$
$41$	7.10607	1.10978	0.554891	+	0.831923i	$0.312760\pi$
$42$	0	0
$43$	−5.35504	−0.816636	−0.408318	−	0.912840i	$-0.633884\pi$
$43$	−5.35504	−0.816636	−0.408318	+	0.912840i	$0.633884\pi$
$44$	5.29086	0.797627
$45$	0	0
$46$	−8.68004	−1.27980
$47$	2.50980	0.366092	0.183046	−	0.983104i	$-0.441404\pi$
$47$	2.50980	0.366092	0.183046	+	0.983104i	$0.441404\pi$
$48$	0	0
$49$	6.81521	0.973601
$50$	7.47565	1.05722
$51$	0	0
$52$	0.226682	0.0314351
$53$	5.93582	0.815348	0.407674	−	0.913128i	$-0.366340\pi$
$53$	5.93582	0.815348	0.407674	+	0.913128i	$0.366340\pi$
$54$	0	0
$55$	18.6878	2.51986
$56$	3.71688	0.496689
$57$	0	0
$58$	0.120615	0.0158375
$59$	0.218941	0.0285037	0.0142518	−	0.999898i	$-0.495463\pi$
$59$	0.218941	0.0285037	0.0142518	+	0.999898i	$0.495463\pi$
$60$	0	0
$61$	−1.57398	−0.201527	−0.100764	−	0.994910i	$-0.532129\pi$
$61$	−1.57398	−0.201527	−0.100764	+	0.994910i	$0.532129\pi$
$62$	−3.12061	−0.396318
$63$	0	0
$64$	1.00000	0.125000
$65$	0.800660	0.0993096
$66$	0	0
$67$	−15.4807	−1.89127	−0.945635	−	0.325231i	$-0.894558\pi$
$67$	−15.4807	−1.89127	−0.945635	+	0.325231i	$0.894558\pi$
$68$	1.65270	0.200420
$69$	0	0
$70$	13.1284	1.56914
$71$	−1.35504	−0.160813	−0.0804067	−	0.996762i	$-0.525622\pi$
$71$	−1.35504	−0.160813	−0.0804067	+	0.996762i	$0.525622\pi$
$72$	0	0
$73$	2.42602	0.283944	0.141972	−	0.989871i	$-0.454656\pi$
$73$	2.42602	0.283944	0.141972	+	0.989871i	$0.454656\pi$
$74$	−5.12836	−0.596159
$75$	0	0
$76$	0	0
$77$	19.6655	2.24109
$78$	0	0
$79$	2.86484	0.322319	0.161160	−	0.986928i	$-0.448477\pi$
$79$	2.86484	0.322319	0.161160	+	0.986928i	$0.448477\pi$
$80$	3.53209	0.394900
$81$	0	0
$82$	7.10607	0.784734
$83$	−1.92127	−0.210887	−0.105444	−	0.994425i	$-0.533626\pi$
$83$	−1.92127	−0.210887	−0.105444	+	0.994425i	$0.533626\pi$
$84$	0	0
$85$	5.83750	0.633165
$86$	−5.35504	−0.577449
$87$	0	0
$88$	5.29086	0.564008
$89$	−12.1557	−1.28850	−0.644251	−	0.764814i	$-0.722831\pi$
$89$	−12.1557	−1.28850	−0.644251	+	0.764814i	$0.722831\pi$
$90$	0	0
$91$	0.842549	0.0883231
$92$	−8.68004	−0.904957
$93$	0	0
$94$	2.50980	0.258866
$95$	0	0
$96$	0	0
$97$	−17.5030	−1.77716	−0.888580	−	0.458722i	$-0.848307\pi$
$97$	−17.5030	−1.77716	−0.888580	+	0.458722i	$0.848307\pi$
$98$	6.81521	0.688440
$99$	0	0
$100$	7.47565	0.747565
$101$	−15.0155	−1.49410	−0.747048	−	0.664770i	$-0.768530\pi$
$101$	−15.0155	−1.49410	−0.747048	+	0.664770i	$0.768530\pi$
$102$	0	0
$103$	−6.66044	−0.656273	−0.328137	−	0.944630i	$-0.606421\pi$
$103$	−6.66044	−0.656273	−0.328137	+	0.944630i	$0.606421\pi$
$104$	0.226682	0.0222280
$105$	0	0
$106$	5.93582	0.576538
$107$	8.08647	0.781748	0.390874	−	0.920444i	$-0.372173\pi$
$107$	8.08647	0.781748	0.390874	+	0.920444i	$0.372173\pi$
$108$	0	0
$109$	−15.2490	−1.46059	−0.730293	−	0.683134i	$-0.760617\pi$
$109$	−15.2490	−1.46059	−0.730293	+	0.683134i	$0.760617\pi$
$110$	18.6878	1.78181
$111$	0	0
$112$	3.71688	0.351212
$113$	0.815207	0.0766883	0.0383441	−	0.999265i	$-0.487792\pi$
$113$	0.815207	0.0766883	0.0383441	+	0.999265i	$0.487792\pi$
$114$	0	0
$115$	−30.6587	−2.85894
$116$	0.120615	0.0111988
$117$	0	0
$118$	0.218941	0.0201551
$119$	6.14290	0.563119
$120$	0	0
$121$	16.9932	1.54484
$122$	−1.57398	−0.142501
$123$	0	0
$124$	−3.12061	−0.280239
$125$	8.74422	0.782107
$126$	0	0
$127$	−11.0642	−0.981787	−0.490894	−	0.871220i	$-0.663330\pi$
$127$	−11.0642	−0.981787	−0.490894	+	0.871220i	$0.663330\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0.800660	0.0702225
$131$	14.3996	1.25810	0.629050	−	0.777365i	$-0.283444\pi$
$131$	14.3996	1.25810	0.629050	+	0.777365i	$0.283444\pi$
$132$	0	0
$133$	0	0
$134$	−15.4807	−1.33733
$135$	0	0
$136$	1.65270	0.141718
$137$	0.472964	0.0404080	0.0202040	−	0.999796i	$-0.493568\pi$
$137$	0.472964	0.0404080	0.0202040	+	0.999796i	$0.493568\pi$
$138$	0	0
$139$	−10.1848	−0.863863	−0.431931	−	0.901906i	$-0.642168\pi$
$139$	−10.1848	−0.863863	−0.431931	+	0.901906i	$0.642168\pi$
$140$	13.1284	1.10955
$141$	0	0
$142$	−1.35504	−0.113712
$143$	1.19934	0.100294
$144$	0	0
$145$	0.426022	0.0353792
$146$	2.42602	0.200779
$147$	0	0
$148$	−5.12836	−0.421548
$149$	−4.88444	−0.400149	−0.200074	−	0.979781i	$-0.564118\pi$
$149$	−4.88444	−0.400149	−0.200074	+	0.979781i	$0.564118\pi$
$150$	0	0
$151$	−9.04189	−0.735818	−0.367909	−	0.929862i	$-0.619926\pi$
$151$	−9.04189	−0.735818	−0.367909	+	0.929862i	$0.619926\pi$
$152$	0	0
$153$	0	0
$154$	19.6655	1.58469
$155$	−11.0223	−0.885332
$156$	0	0
$157$	0.332748	0.0265562	0.0132781	−	0.999912i	$-0.495773\pi$
$157$	0.332748	0.0265562	0.0132781	+	0.999912i	$0.495773\pi$
$158$	2.86484	0.227914
$159$	0	0
$160$	3.53209	0.279236
$161$	−32.2627	−2.54266
$162$	0	0
$163$	3.94087	0.308673	0.154337	−	0.988018i	$-0.450676\pi$
$163$	3.94087	0.308673	0.154337	+	0.988018i	$0.450676\pi$
$164$	7.10607	0.554891
$165$	0	0
$166$	−1.92127	−0.149120
$167$	5.99319	0.463767	0.231884	−	0.972744i	$-0.425511\pi$
$167$	5.99319	0.463767	0.231884	+	0.972744i	$0.425511\pi$
$168$	0	0
$169$	−12.9486	−0.996047
$170$	5.83750	0.447716
$171$	0	0
$172$	−5.35504	−0.408318
$173$	12.2618	0.932245	0.466122	−	0.884720i	$-0.345651\pi$
$173$	12.2618	0.932245	0.466122	+	0.884720i	$0.345651\pi$
$174$	0	0
$175$	27.7861	2.10043
$176$	5.29086	0.398814
$177$	0	0
$178$	−12.1557	−0.911108
$179$	8.06418	0.602745	0.301372	−	0.953506i	$-0.402555\pi$
$179$	8.06418	0.602745	0.301372	+	0.953506i	$0.402555\pi$
$180$	0	0
$181$	−0.189845	−0.0141111	−0.00705553	−	0.999975i	$-0.502246\pi$
$181$	−0.189845	−0.0141111	−0.00705553	+	0.999975i	$0.502246\pi$
$182$	0.842549	0.0624539
$183$	0	0
$184$	−8.68004	−0.639901
$185$	−18.1138	−1.33175
$186$	0	0
$187$	8.74422	0.639441
$188$	2.50980	0.183046
$189$	0	0
$190$	0	0
$191$	−18.0378	−1.30517	−0.652584	−	0.757717i	$-0.726315\pi$
$191$	−18.0378	−1.30517	−0.652584	+	0.757717i	$0.726315\pi$
$192$	0	0
$193$	17.4388	1.25527	0.627637	−	0.778506i	$-0.284022\pi$
$193$	17.4388	1.25527	0.627637	+	0.778506i	$0.284022\pi$
$194$	−17.5030	−1.25664
$195$	0	0
$196$	6.81521	0.486801
$197$	10.4807	0.746719	0.373360	−	0.927687i	$-0.378206\pi$
$197$	10.4807	0.746719	0.373360	+	0.927687i	$0.378206\pi$
$198$	0	0
$199$	−13.6536	−0.967881	−0.483940	−	0.875101i	$-0.660795\pi$
$199$	−13.6536	−0.967881	−0.483940	+	0.875101i	$0.660795\pi$
$200$	7.47565	0.528608
$201$	0	0
$202$	−15.0155	−1.05649
$203$	0.448311	0.0314652
$204$	0	0
$205$	25.0993	1.75301
$206$	−6.66044	−0.464055
$207$	0	0
$208$	0.226682	0.0157175
$209$	0	0
$210$	0	0
$211$	15.9290	1.09660	0.548299	−	0.836282i	$-0.315275\pi$
$211$	15.9290	1.09660	0.548299	+	0.836282i	$0.315275\pi$
$212$	5.93582	0.407674
$213$	0	0
$214$	8.08647	0.552779
$215$	−18.9145	−1.28996
$216$	0	0
$217$	−11.5990	−0.787388
$218$	−15.2490	−1.03279
$219$	0	0
$220$	18.6878	1.25993
$221$	0.374638	0.0252008
$222$	0	0
$223$	−18.6159	−1.24661	−0.623305	−	0.781979i	$-0.714211\pi$
$223$	−18.6159	−1.24661	−0.623305	+	0.781979i	$0.714211\pi$
$224$	3.71688	0.248345
$225$	0	0
$226$	0.815207	0.0542268
$227$	−5.79292	−0.384490	−0.192245	−	0.981347i	$-0.561577\pi$
$227$	−5.79292	−0.384490	−0.192245	+	0.981347i	$0.561577\pi$
$228$	0	0
$229$	−8.12836	−0.537137	−0.268568	−	0.963261i	$-0.586551\pi$
$229$	−8.12836	−0.537137	−0.268568	+	0.963261i	$0.586551\pi$
$230$	−30.6587	−2.02157
$231$	0	0
$232$	0.120615	0.00791875
$233$	17.0273	1.11550	0.557749	−	0.830010i	$-0.311665\pi$
$233$	17.0273	1.11550	0.557749	+	0.830010i	$0.311665\pi$
$234$	0	0
$235$	8.86484	0.578278
$236$	0.218941	0.0142518
$237$	0	0
$238$	6.14290	0.398185
$239$	15.0196	0.971537	0.485769	−	0.874087i	$-0.338540\pi$
$239$	15.0196	0.971537	0.485769	+	0.874087i	$0.338540\pi$
$240$	0	0
$241$	21.4534	1.38193	0.690966	−	0.722887i	$-0.257185\pi$
$241$	21.4534	1.38193	0.690966	+	0.722887i	$0.257185\pi$
$242$	16.9932	1.09236
$243$	0	0
$244$	−1.57398	−0.100764
$245$	24.0719	1.53790
$246$	0	0
$247$	0	0
$248$	−3.12061	−0.198159
$249$	0	0
$250$	8.74422	0.553033
$251$	25.5672	1.61379	0.806893	−	0.590698i	$-0.201148\pi$
$251$	25.5672	1.61379	0.806893	+	0.590698i	$0.201148\pi$
$252$	0	0
$253$	−45.9249	−2.88727
$254$	−11.0642	−0.694228
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.79292	0.174217	0.0871087	−	0.996199i	$-0.472237\pi$
$257$	2.79292	0.174217	0.0871087	+	0.996199i	$0.472237\pi$
$258$	0	0
$259$	−19.0615	−1.18442
$260$	0.800660	0.0496548
$261$	0	0
$262$	14.3996	0.889611
$263$	−1.38413	−0.0853493	−0.0426746	−	0.999089i	$-0.513588\pi$
$263$	−1.38413	−0.0853493	−0.0426746	+	0.999089i	$0.513588\pi$
$264$	0	0
$265$	20.9659	1.28792
$266$	0	0
$267$	0	0
$268$	−15.4807	−0.945635
$269$	−17.0770	−1.04120	−0.520601	−	0.853800i	$-0.674292\pi$
$269$	−17.0770	−1.04120	−0.520601	+	0.853800i	$0.674292\pi$
$270$	0	0
$271$	−25.6878	−1.56042	−0.780211	−	0.625517i	$-0.784888\pi$
$271$	−25.6878	−1.56042	−0.780211	+	0.625517i	$0.784888\pi$
$272$	1.65270	0.100210
$273$	0	0
$274$	0.472964	0.0285728
$275$	39.5526	2.38511
$276$	0	0
$277$	22.2249	1.33537	0.667683	−	0.744446i	$-0.267286\pi$
$277$	22.2249	1.33537	0.667683	+	0.744446i	$0.267286\pi$
$278$	−10.1848	−0.610843
$279$	0	0
$280$	13.1284	0.784569
$281$	1.54395	0.0921042	0.0460521	−	0.998939i	$-0.485336\pi$
$281$	1.54395	0.0921042	0.0460521	+	0.998939i	$0.485336\pi$
$282$	0	0
$283$	10.2909	0.611728	0.305864	−	0.952075i	$-0.401055\pi$
$283$	10.2909	0.611728	0.305864	+	0.952075i	$0.401055\pi$
$284$	−1.35504	−0.0804067
$285$	0	0
$286$	1.19934	0.0709185
$287$	26.4124	1.55908
$288$	0	0
$289$	−14.2686	−0.839328
$290$	0.426022	0.0250169
$291$	0	0
$292$	2.42602	0.141972
$293$	−28.5030	−1.66516	−0.832581	−	0.553903i	$-0.813138\pi$
$293$	−28.5030	−1.66516	−0.832581	+	0.553903i	$0.813138\pi$
$294$	0	0
$295$	0.773318	0.0450243
$296$	−5.12836	−0.298080
$297$	0	0
$298$	−4.88444	−0.282948
$299$	−1.96761	−0.113790
$300$	0	0
$301$	−19.9040	−1.14725
$302$	−9.04189	−0.520302
$303$	0	0
$304$	0	0
$305$	−5.55943	−0.318332
$306$	0	0
$307$	25.5594	1.45875	0.729377	−	0.684112i	$-0.239810\pi$
$307$	25.5594	1.45875	0.729377	+	0.684112i	$0.239810\pi$
$308$	19.6655	1.12055
$309$	0	0
$310$	−11.0223	−0.626024
$311$	−10.3746	−0.588292	−0.294146	−	0.955761i	$-0.595035\pi$
$311$	−10.3746	−0.588292	−0.294146	+	0.955761i	$0.595035\pi$
$312$	0	0
$313$	−15.4365	−0.872520	−0.436260	−	0.899821i	$-0.643697\pi$
$313$	−15.4365	−0.872520	−0.436260	+	0.899821i	$0.643697\pi$
$314$	0.332748	0.0187781
$315$	0	0
$316$	2.86484	0.161160
$317$	28.3969	1.59493	0.797465	−	0.603365i	$-0.206174\pi$
$317$	28.3969	1.59493	0.797465	+	0.603365i	$0.206174\pi$
$318$	0	0
$319$	0.638156	0.0357299
$320$	3.53209	0.197450
$321$	0	0
$322$	−32.2627	−1.79793
$323$	0	0
$324$	0	0
$325$	1.69459	0.0939991
$326$	3.94087	0.218265
$327$	0	0
$328$	7.10607	0.392367
$329$	9.32863	0.514304
$330$	0	0
$331$	25.9222	1.42481	0.712407	−	0.701767i	$-0.247605\pi$
$331$	25.9222	1.42481	0.712407	+	0.701767i	$0.247605\pi$
$332$	−1.92127	−0.105444
$333$	0	0
$334$	5.99319	0.327933
$335$	−54.6792	−2.98745
$336$	0	0
$337$	23.5107	1.28071	0.640356	−	0.768079i	$-0.278787\pi$
$337$	23.5107	1.28071	0.640356	+	0.768079i	$0.278787\pi$
$338$	−12.9486	−0.704312
$339$	0	0
$340$	5.83750	0.316583
$341$	−16.5107	−0.894106
$342$	0	0
$343$	−0.686852	−0.0370865
$344$	−5.35504	−0.288724
$345$	0	0
$346$	12.2618	0.659196
$347$	2.40467	0.129089	0.0645446	−	0.997915i	$-0.479441\pi$
$347$	2.40467	0.129089	0.0645446	+	0.997915i	$0.479441\pi$
$348$	0	0
$349$	5.19759	0.278220	0.139110	−	0.990277i	$-0.455576\pi$
$349$	5.19759	0.278220	0.139110	+	0.990277i	$0.455576\pi$
$350$	27.7861	1.48523
$351$	0	0
$352$	5.29086	0.282004
$353$	−8.27631	−0.440504	−0.220252	−	0.975443i	$-0.570688\pi$
$353$	−8.27631	−0.440504	−0.220252	+	0.975443i	$0.570688\pi$
$354$	0	0
$355$	−4.78611	−0.254020
$356$	−12.1557	−0.644251
$357$	0	0
$358$	8.06418	0.426205
$359$	2.29591	0.121174	0.0605868	−	0.998163i	$-0.480703\pi$
$359$	2.29591	0.121174	0.0605868	+	0.998163i	$0.480703\pi$
$360$	0	0
$361$	0	0
$362$	−0.189845	−0.00997803
$363$	0	0
$364$	0.842549	0.0441615
$365$	8.56893	0.448518
$366$	0	0
$367$	29.8648	1.55893	0.779466	−	0.626445i	$-0.215491\pi$
$367$	29.8648	1.55893	0.779466	+	0.626445i	$0.215491\pi$
$368$	−8.68004	−0.452479
$369$	0	0
$370$	−18.1138	−0.941692
$371$	22.0627	1.14544
$372$	0	0
$373$	19.8530	1.02795	0.513974	−	0.857806i	$-0.328173\pi$
$373$	19.8530	1.02795	0.513974	+	0.857806i	$0.328173\pi$
$374$	8.74422	0.452153
$375$	0	0
$376$	2.50980	0.129433
$377$	0.0273411	0.00140814
$378$	0	0
$379$	−25.8256	−1.32657	−0.663287	−	0.748365i	$-0.730839\pi$
$379$	−25.8256	−1.32657	−0.663287	+	0.748365i	$0.730839\pi$
$380$	0	0
$381$	0	0
$382$	−18.0378	−0.922893
$383$	20.1343	1.02882	0.514408	−	0.857545i	$-0.328012\pi$
$383$	20.1343	1.02882	0.514408	+	0.857545i	$0.328012\pi$
$384$	0	0
$385$	69.4603	3.54002
$386$	17.4388	0.887612
$387$	0	0
$388$	−17.5030	−0.888580
$389$	−11.3327	−0.574593	−0.287297	−	0.957842i	$-0.592757\pi$
$389$	−11.3327	−0.574593	−0.287297	+	0.957842i	$0.592757\pi$
$390$	0	0
$391$	−14.3455	−0.725485
$392$	6.81521	0.344220
$393$	0	0
$394$	10.4807	0.528010
$395$	10.1189	0.509135
$396$	0	0
$397$	13.0942	0.657179	0.328590	−	0.944473i	$-0.393427\pi$
$397$	13.0942	0.657179	0.328590	+	0.944473i	$0.393427\pi$
$398$	−13.6536	−0.684395
$399$	0	0
$400$	7.47565	0.373783
$401$	21.2199	1.05967	0.529835	−	0.848101i	$-0.322254\pi$
$401$	21.2199	1.05967	0.529835	+	0.848101i	$0.322254\pi$
$402$	0	0
$403$	−0.707386	−0.0352374
$404$	−15.0155	−0.747048
$405$	0	0
$406$	0.448311	0.0222493
$407$	−27.1334	−1.34495
$408$	0	0
$409$	29.0797	1.43790	0.718948	−	0.695064i	$-0.244624\pi$
$409$	29.0797	1.43790	0.718948	+	0.695064i	$0.244624\pi$
$410$	25.0993	1.23956
$411$	0	0
$412$	−6.66044	−0.328137
$413$	0.813777	0.0400433
$414$	0	0
$415$	−6.78611	−0.333117
$416$	0.226682	0.0111140
$417$	0	0
$418$	0	0
$419$	−40.4962	−1.97837	−0.989184	−	0.146679i	$-0.953141\pi$
$419$	−40.4962	−1.97837	−0.989184	+	0.146679i	$0.953141\pi$
$420$	0	0
$421$	−31.2327	−1.52219	−0.761094	−	0.648642i	$-0.775337\pi$
$421$	−31.2327	−1.52219	−0.761094	+	0.648642i	$0.775337\pi$
$422$	15.9290	0.775412
$423$	0	0
$424$	5.93582	0.288269
$425$	12.3550	0.599307
$426$	0	0
$427$	−5.85029	−0.283115
$428$	8.08647	0.390874
$429$	0	0
$430$	−18.9145	−0.912137
$431$	26.0205	1.25337	0.626683	−	0.779275i	$-0.284412\pi$
$431$	26.0205	1.25337	0.626683	+	0.779275i	$0.284412\pi$
$432$	0	0
$433$	27.0642	1.30062	0.650311	−	0.759668i	$-0.274639\pi$
$433$	27.0642	1.30062	0.650311	+	0.759668i	$0.274639\pi$
$434$	−11.5990	−0.556768
$435$	0	0
$436$	−15.2490	−0.730293
$437$	0	0
$438$	0	0
$439$	2.94087	0.140360	0.0701801	−	0.997534i	$-0.477643\pi$
$439$	2.94087	0.140360	0.0701801	+	0.997534i	$0.477643\pi$
$440$	18.6878	0.890905
$441$	0	0
$442$	0.374638	0.0178197
$443$	−27.6049	−1.31155	−0.655775	−	0.754956i	$-0.727658\pi$
$443$	−27.6049	−1.31155	−0.655775	+	0.754956i	$0.727658\pi$
$444$	0	0
$445$	−42.9350	−2.03531
$446$	−18.6159	−0.881487
$447$	0	0
$448$	3.71688	0.175606
$449$	16.9222	0.798608	0.399304	−	0.916819i	$-0.369252\pi$
$449$	16.9222	0.798608	0.399304	+	0.916819i	$0.369252\pi$
$450$	0	0
$451$	37.5972	1.77038
$452$	0.815207	0.0383441
$453$	0	0
$454$	−5.79292	−0.271875
$455$	2.97596	0.139515
$456$	0	0
$457$	9.19160	0.429965	0.214982	−	0.976618i	$-0.431031\pi$
$457$	9.19160	0.429965	0.214982	+	0.976618i	$0.431031\pi$
$458$	−8.12836	−0.379813
$459$	0	0
$460$	−30.6587	−1.42947
$461$	15.4456	0.719374	0.359687	−	0.933073i	$-0.382883\pi$
$461$	15.4456	0.719374	0.359687	+	0.933073i	$0.382883\pi$
$462$	0	0
$463$	29.9813	1.39335	0.696675	−	0.717387i	$-0.254662\pi$
$463$	29.9813	1.39335	0.696675	+	0.717387i	$0.254662\pi$
$464$	0.120615	0.00559940
$465$	0	0
$466$	17.0273	0.788776
$467$	−28.8854	−1.33666	−0.668328	−	0.743867i	$-0.732990\pi$
$467$	−28.8854	−1.33666	−0.668328	+	0.743867i	$0.732990\pi$
$468$	0	0
$469$	−57.5399	−2.65695
$470$	8.86484	0.408904
$471$	0	0
$472$	0.218941	0.0100776
$473$	−28.3327	−1.30274
$474$	0	0
$475$	0	0
$476$	6.14290	0.281560
$477$	0	0
$478$	15.0196	0.686981
$479$	−13.6604	−0.624162	−0.312081	−	0.950056i	$-0.601026\pi$
$479$	−13.6604	−0.624162	−0.312081	+	0.950056i	$0.601026\pi$
$480$	0	0
$481$	−1.16250	−0.0530056
$482$	21.4534	0.977174
$483$	0	0
$484$	16.9932	0.772418
$485$	−61.8221	−2.80720
$486$	0	0
$487$	6.17799	0.279951	0.139976	−	0.990155i	$-0.455298\pi$
$487$	6.17799	0.279951	0.139976	+	0.990155i	$0.455298\pi$
$488$	−1.57398	−0.0712506
$489$	0	0
$490$	24.0719	1.08746
$491$	−4.10338	−0.185183	−0.0925914	−	0.995704i	$-0.529515\pi$
$491$	−4.10338	−0.185183	−0.0925914	+	0.995704i	$0.529515\pi$
$492$	0	0
$493$	0.199340	0.00897784
$494$	0	0
$495$	0	0
$496$	−3.12061	−0.140120
$497$	−5.03651	−0.225918
$498$	0	0
$499$	26.5790	1.18984	0.594920	−	0.803785i	$-0.297184\pi$
$499$	26.5790	1.18984	0.594920	+	0.803785i	$0.297184\pi$
$500$	8.74422	0.391054
$501$	0	0
$502$	25.5672	1.14112
$503$	25.9445	1.15681	0.578404	−	0.815750i	$-0.303676\pi$
$503$	25.9445	1.15681	0.578404	+	0.815750i	$0.303676\pi$
$504$	0	0
$505$	−53.0360	−2.36007
$506$	−45.9249	−2.04161
$507$	0	0
$508$	−11.0642	−0.490894
$509$	−25.1976	−1.11686	−0.558432	−	0.829551i	$-0.688597\pi$
$509$	−25.1976	−1.11686	−0.558432	+	0.829551i	$0.688597\pi$
$510$	0	0
$511$	9.01724	0.398899
$512$	1.00000	0.0441942
$513$	0	0
$514$	2.79292	0.123190
$515$	−23.5253	−1.03665
$516$	0	0
$517$	13.2790	0.584010
$518$	−19.0615	−0.837514
$519$	0	0
$520$	0.800660	0.0351112
$521$	−2.77601	−0.121619	−0.0608095	−	0.998149i	$-0.519368\pi$
$521$	−2.77601	−0.121619	−0.0608095	+	0.998149i	$0.519368\pi$
$522$	0	0
$523$	−11.2354	−0.491288	−0.245644	−	0.969360i	$-0.578999\pi$
$523$	−11.2354	−0.491288	−0.245644	+	0.969360i	$0.578999\pi$
$524$	14.3996	0.629050
$525$	0	0
$526$	−1.38413	−0.0603511
$527$	−5.15745	−0.224662
$528$	0	0
$529$	52.3432	2.27579
$530$	20.9659	0.910698
$531$	0	0
$532$	0	0
$533$	1.61081	0.0697721
$534$	0	0
$535$	28.5621	1.23485
$536$	−15.4807	−0.668665
$537$	0	0
$538$	−17.0770	−0.736240
$539$	36.0583	1.55314
$540$	0	0
$541$	−22.0128	−0.946404	−0.473202	−	0.880954i	$-0.656902\pi$
$541$	−22.0128	−0.946404	−0.473202	+	0.880954i	$0.656902\pi$
$542$	−25.6878	−1.10338
$543$	0	0
$544$	1.65270	0.0708591
$545$	−53.8607	−2.30714
$546$	0	0
$547$	−34.1557	−1.46039	−0.730196	−	0.683238i	$-0.760571\pi$
$547$	−34.1557	−1.46039	−0.730196	+	0.683238i	$0.760571\pi$
$548$	0.472964	0.0202040
$549$	0	0
$550$	39.5526	1.68653
$551$	0	0
$552$	0	0
$553$	10.6483	0.452810
$554$	22.2249	0.944247
$555$	0	0
$556$	−10.1848	−0.431931
$557$	37.8607	1.60421	0.802105	−	0.597183i	$-0.203713\pi$
$557$	37.8607	1.60421	0.802105	+	0.597183i	$0.203713\pi$
$558$	0	0
$559$	−1.21389	−0.0513420
$560$	13.1284	0.554774
$561$	0	0
$562$	1.54395	0.0651275
$563$	11.7246	0.494134	0.247067	−	0.968998i	$-0.420533\pi$
$563$	11.7246	0.494134	0.247067	+	0.968998i	$0.420533\pi$
$564$	0	0
$565$	2.87939	0.121137
$566$	10.2909	0.432557
$567$	0	0
$568$	−1.35504	−0.0568561
$569$	14.8972	0.624524	0.312262	−	0.949996i	$-0.398913\pi$
$569$	14.8972	0.624524	0.312262	+	0.949996i	$0.398913\pi$
$570$	0	0
$571$	12.9409	0.541559	0.270779	−	0.962641i	$-0.412719\pi$
$571$	12.9409	0.541559	0.270779	+	0.962641i	$0.412719\pi$
$572$	1.19934	0.0501469
$573$	0	0
$574$	26.4124	1.10243
$575$	−64.8890	−2.70606
$576$	0	0
$577$	22.7314	0.946322	0.473161	−	0.880976i	$-0.343113\pi$
$577$	22.7314	0.946322	0.473161	+	0.880976i	$0.343113\pi$
$578$	−14.2686	−0.593494
$579$	0	0
$580$	0.426022	0.0176896
$581$	−7.14115	−0.296265
$582$	0	0
$583$	31.4056	1.30069
$584$	2.42602	0.100390
$585$	0	0
$586$	−28.5030	−1.17745
$587$	10.5594	0.435834	0.217917	−	0.975967i	$-0.430074\pi$
$587$	10.5594	0.435834	0.217917	+	0.975967i	$0.430074\pi$
$588$	0	0
$589$	0	0
$590$	0.773318	0.0318370
$591$	0	0
$592$	−5.12836	−0.210774
$593$	10.7578	0.441771	0.220886	−	0.975300i	$-0.429105\pi$
$593$	10.7578	0.441771	0.220886	+	0.975300i	$0.429105\pi$
$594$	0	0
$595$	21.6973	0.889502
$596$	−4.88444	−0.200074
$597$	0	0
$598$	−1.96761	−0.0804614
$599$	−24.4311	−0.998227	−0.499113	−	0.866537i	$-0.666341\pi$
$599$	−24.4311	−0.998227	−0.499113	+	0.866537i	$0.666341\pi$
$600$	0	0
$601$	26.4115	1.07735	0.538673	−	0.842515i	$-0.318926\pi$
$601$	26.4115	1.07735	0.538673	+	0.842515i	$0.318926\pi$
$602$	−19.9040	−0.811228
$603$	0	0
$604$	−9.04189	−0.367909
$605$	60.0215	2.44022
$606$	0	0
$607$	16.3618	0.664107	0.332053	−	0.943261i	$-0.392259\pi$
$607$	16.3618	0.664107	0.332053	+	0.943261i	$0.392259\pi$
$608$	0	0
$609$	0	0
$610$	−5.55943	−0.225095
$611$	0.568926	0.0230163
$612$	0	0
$613$	5.94087	0.239950	0.119975	−	0.992777i	$-0.461719\pi$
$613$	5.94087	0.239950	0.119975	+	0.992777i	$0.461719\pi$
$614$	25.5594	1.03149
$615$	0	0
$616$	19.6655	0.792345
$617$	34.1284	1.37396	0.686978	−	0.726678i	$-0.258937\pi$
$617$	34.1284	1.37396	0.686978	+	0.726678i	$0.258937\pi$
$618$	0	0
$619$	32.3013	1.29830	0.649149	−	0.760661i	$-0.275125\pi$
$619$	32.3013	1.29830	0.649149	+	0.760661i	$0.275125\pi$
$620$	−11.0223	−0.442666
$621$	0	0
$622$	−10.3746	−0.415985
$623$	−45.1813	−1.81015
$624$	0	0
$625$	−6.49289	−0.259716
$626$	−15.4365	−0.616965
$627$	0	0
$628$	0.332748	0.0132781
$629$	−8.47565	−0.337946
$630$	0	0
$631$	18.0283	0.717694	0.358847	−	0.933396i	$-0.383170\pi$
$631$	18.0283	0.717694	0.358847	+	0.933396i	$0.383170\pi$
$632$	2.86484	0.113957
$633$	0	0
$634$	28.3969	1.12779
$635$	−39.0797	−1.55083
$636$	0	0
$637$	1.54488	0.0612105
$638$	0.638156	0.0252648
$639$	0	0
$640$	3.53209	0.139618
$641$	2.58853	0.102241	0.0511203	−	0.998693i	$-0.483721\pi$
$641$	2.58853	0.102241	0.0511203	+	0.998693i	$0.483721\pi$
$642$	0	0
$643$	16.1925	0.638571	0.319286	−	0.947659i	$-0.396557\pi$
$643$	16.1925	0.638571	0.319286	+	0.947659i	$0.396557\pi$
$644$	−32.2627	−1.27133
$645$	0	0
$646$	0	0
$647$	−0.477407	−0.0187688	−0.00938439	−	0.999956i	$-0.502987\pi$
$647$	−0.477407	−0.0187688	−0.00938439	+	0.999956i	$0.502987\pi$
$648$	0	0
$649$	1.15839	0.0454706
$650$	1.69459	0.0664674
$651$	0	0
$652$	3.94087	0.154337
$653$	−16.1019	−0.630118	−0.315059	−	0.949072i	$-0.602024\pi$
$653$	−16.1019	−0.630118	−0.315059	+	0.949072i	$0.602024\pi$
$654$	0	0
$655$	50.8607	1.98729
$656$	7.10607	0.277445
$657$	0	0
$658$	9.32863	0.363668
$659$	−14.5398	−0.566391	−0.283196	−	0.959062i	$-0.591395\pi$
$659$	−14.5398	−0.566391	−0.283196	+	0.959062i	$0.591395\pi$
$660$	0	0
$661$	14.5648	0.566505	0.283253	−	0.959045i	$-0.408586\pi$
$661$	14.5648	0.566505	0.283253	+	0.959045i	$0.408586\pi$
$662$	25.9222	1.00750
$663$	0	0
$664$	−1.92127	−0.0745599
$665$	0	0
$666$	0	0
$667$	−1.04694	−0.0405377
$668$	5.99319	0.231884
$669$	0	0
$670$	−54.6792	−2.11244
$671$	−8.32770	−0.321487
$672$	0	0
$673$	−24.8648	−0.958469	−0.479235	−	0.877687i	$-0.659086\pi$
$673$	−24.8648	−0.958469	−0.479235	+	0.877687i	$0.659086\pi$
$674$	23.5107	0.905600
$675$	0	0
$676$	−12.9486	−0.498024
$677$	46.4543	1.78538	0.892692	−	0.450668i	$-0.148814\pi$
$677$	46.4543	1.78538	0.892692	+	0.450668i	$0.148814\pi$
$678$	0	0
$679$	−65.0565	−2.49664
$680$	5.83750	0.223858
$681$	0	0
$682$	−16.5107	−0.632229
$683$	−26.0000	−0.994862	−0.497431	−	0.867503i	$-0.665723\pi$
$683$	−26.0000	−0.994862	−0.497431	+	0.867503i	$0.665723\pi$
$684$	0	0
$685$	1.67055	0.0638284
$686$	−0.686852	−0.0262241
$687$	0	0
$688$	−5.35504	−0.204159
$689$	1.34554	0.0512611
$690$	0	0
$691$	−16.0696	−0.611315	−0.305657	−	0.952142i	$-0.598876\pi$
$691$	−16.0696	−0.611315	−0.305657	+	0.952142i	$0.598876\pi$
$692$	12.2618	0.466122
$693$	0	0
$694$	2.40467	0.0912799
$695$	−35.9736	−1.36456
$696$	0	0
$697$	11.7442	0.444844
$698$	5.19759	0.196732
$699$	0	0
$700$	27.7861	1.05022
$701$	9.91117	0.374340	0.187170	−	0.982328i	$-0.440069\pi$
$701$	9.91117	0.374340	0.187170	+	0.982328i	$0.440069\pi$
$702$	0	0
$703$	0	0
$704$	5.29086	0.199407
$705$	0	0
$706$	−8.27631	−0.311483
$707$	−55.8108	−2.09898
$708$	0	0
$709$	−27.4715	−1.03172	−0.515858	−	0.856674i	$-0.672527\pi$
$709$	−27.4715	−1.03172	−0.515858	+	0.856674i	$0.672527\pi$
$710$	−4.78611	−0.179620
$711$	0	0
$712$	−12.1557	−0.455554
$713$	27.0871	1.01442
$714$	0	0
$715$	4.23618	0.158424
$716$	8.06418	0.301372
$717$	0	0
$718$	2.29591	0.0856827
$719$	−12.2635	−0.457352	−0.228676	−	0.973503i	$-0.573440\pi$
$719$	−12.2635	−0.457352	−0.228676	+	0.973503i	$0.573440\pi$
$720$	0	0
$721$	−24.7561	−0.921965
$722$	0	0
$723$	0	0
$724$	−0.189845	−0.00705553
$725$	0.901674	0.0334873
$726$	0	0
$727$	28.2594	1.04808	0.524042	−	0.851693i	$-0.324424\pi$
$727$	28.2594	1.04808	0.524042	+	0.851693i	$0.324424\pi$
$728$	0.842549	0.0312269
$729$	0	0
$730$	8.56893	0.317150
$731$	−8.85029	−0.327340
$732$	0	0
$733$	−22.5767	−0.833888	−0.416944	−	0.908932i	$-0.636899\pi$
$733$	−22.5767	−0.833888	−0.416944	+	0.908932i	$0.636899\pi$
$734$	29.8648	1.10233
$735$	0	0
$736$	−8.68004	−0.319951
$737$	−81.9062	−3.01705
$738$	0	0
$739$	26.4979	0.974743	0.487371	−	0.873195i	$-0.337956\pi$
$739$	26.4979	0.974743	0.487371	+	0.873195i	$0.337956\pi$
$740$	−18.1138	−0.665877
$741$	0	0
$742$	22.0627	0.809949
$743$	−27.6141	−1.01306	−0.506532	−	0.862221i	$-0.669073\pi$
$743$	−27.6141	−1.01306	−0.506532	+	0.862221i	$0.669073\pi$
$744$	0	0
$745$	−17.2523	−0.632074
$746$	19.8530	0.726869
$747$	0	0
$748$	8.74422	0.319720
$749$	30.0564	1.09824
$750$	0	0
$751$	−1.28993	−0.0470701	−0.0235350	−	0.999723i	$-0.507492\pi$
$751$	−1.28993	−0.0470701	−0.0235350	+	0.999723i	$0.507492\pi$
$752$	2.50980	0.0915230
$753$	0	0
$754$	0.0273411	0.000995706 0
$755$	−31.9368	−1.16230
$756$	0	0
$757$	−25.8871	−0.940884	−0.470442	−	0.882431i	$-0.655905\pi$
$757$	−25.8871	−0.940884	−0.470442	+	0.882431i	$0.655905\pi$
$758$	−25.8256	−0.938029
$759$	0	0
$760$	0	0
$761$	−11.3396	−0.411059	−0.205529	−	0.978651i	$-0.565892\pi$
$761$	−11.3396	−0.411059	−0.205529	+	0.978651i	$0.565892\pi$
$762$	0	0
$763$	−56.6786	−2.05190
$764$	−18.0378	−0.652584
$765$	0	0
$766$	20.1343	0.727483
$767$	0.0496299	0.00179203
$768$	0	0
$769$	21.4456	0.773349	0.386674	−	0.922216i	$-0.373624\pi$
$769$	21.4456	0.773349	0.386674	+	0.922216i	$0.373624\pi$
$770$	69.4603	2.50317
$771$	0	0
$772$	17.4388	0.627637
$773$	4.24722	0.152762	0.0763809	−	0.997079i	$-0.475664\pi$
$773$	4.24722	0.152762	0.0763809	+	0.997079i	$0.475664\pi$
$774$	0	0
$775$	−23.3286	−0.837989
$776$	−17.5030	−0.628321
$777$	0	0
$778$	−11.3327	−0.406299
$779$	0	0
$780$	0	0
$781$	−7.16931	−0.256538
$782$	−14.3455	−0.512996
$783$	0	0
$784$	6.81521	0.243400
$785$	1.17530	0.0419482
$786$	0	0
$787$	10.8203	0.385701	0.192850	−	0.981228i	$-0.438227\pi$
$787$	10.8203	0.385701	0.192850	+	0.981228i	$0.438227\pi$
$788$	10.4807	0.373360
$789$	0	0
$790$	10.1189	0.360013
$791$	3.03003	0.107735
$792$	0	0
$793$	−0.356792	−0.0126700
$794$	13.0942	0.464696
$795$	0	0
$796$	−13.6536	−0.483940
$797$	22.0327	0.780439	0.390219	−	0.920722i	$-0.372399\pi$
$797$	22.0327	0.780439	0.390219	+	0.920722i	$0.372399\pi$
$798$	0	0
$799$	4.14796	0.146744
$800$	7.47565	0.264304
$801$	0	0
$802$	21.2199	0.749300
$803$	12.8357	0.452963
$804$	0	0
$805$	−113.955	−4.01638
$806$	−0.707386	−0.0249166
$807$	0	0
$808$	−15.0155	−0.528243
$809$	−6.68685	−0.235097	−0.117549	−	0.993067i	$-0.537504\pi$
$809$	−6.68685	−0.235097	−0.117549	+	0.993067i	$0.537504\pi$
$810$	0	0
$811$	−25.4902	−0.895082	−0.447541	−	0.894263i	$-0.647700\pi$
$811$	−25.4902	−0.895082	−0.447541	+	0.894263i	$0.647700\pi$
$812$	0.448311	0.0157326
$813$	0	0
$814$	−27.1334	−0.951025
$815$	13.9195	0.487580
$816$	0	0
$817$	0	0
$818$	29.0797	1.01675
$819$	0	0
$820$	25.0993	0.876504
$821$	42.2894	1.47591	0.737956	−	0.674849i	$-0.235791\pi$
$821$	42.2894	1.47591	0.737956	+	0.674849i	$0.235791\pi$
$822$	0	0
$823$	−0.538896	−0.0187847	−0.00939237	−	0.999956i	$-0.502990\pi$
$823$	−0.538896	−0.0187847	−0.00939237	+	0.999956i	$0.502990\pi$
$824$	−6.66044	−0.232028
$825$	0	0
$826$	0.813777	0.0283149
$827$	−16.0232	−0.557182	−0.278591	−	0.960410i	$-0.589867\pi$
$827$	−16.0232	−0.557182	−0.278591	+	0.960410i	$0.589867\pi$
$828$	0	0
$829$	20.1034	0.698219	0.349110	−	0.937082i	$-0.386484\pi$
$829$	20.1034	0.698219	0.349110	+	0.937082i	$0.386484\pi$
$830$	−6.78611	−0.235549
$831$	0	0
$832$	0.226682	0.00785877
$833$	11.2635	0.390258
$834$	0	0
$835$	21.1685	0.732566
$836$	0	0
$837$	0	0
$838$	−40.4962	−1.39892
$839$	30.9709	1.06923	0.534617	−	0.845094i	$-0.320456\pi$
$839$	30.9709	1.06923	0.534617	+	0.845094i	$0.320456\pi$
$840$	0	0
$841$	−28.9855	−0.999498
$842$	−31.2327	−1.07635
$843$	0	0
$844$	15.9290	0.548299
$845$	−45.7357	−1.57335
$846$	0	0
$847$	63.1617	2.17026
$848$	5.93582	0.203837
$849$	0	0
$850$	12.3550	0.423774
$851$	44.5144	1.52593
$852$	0	0
$853$	−21.3054	−0.729483	−0.364742	−	0.931109i	$-0.618843\pi$
$853$	−21.3054	−0.729483	−0.364742	+	0.931109i	$0.618843\pi$
$854$	−5.85029	−0.200193
$855$	0	0
$856$	8.08647	0.276390
$857$	55.5518	1.89761	0.948807	−	0.315857i	$-0.102292\pi$
$857$	55.5518	1.89761	0.948807	+	0.315857i	$0.102292\pi$
$858$	0	0
$859$	−23.7205	−0.809333	−0.404667	−	0.914464i	$-0.632612\pi$
$859$	−23.7205	−0.809333	−0.404667	+	0.914464i	$0.632612\pi$
$860$	−18.9145	−0.644978
$861$	0	0
$862$	26.0205	0.886263
$863$	21.0820	0.717640	0.358820	−	0.933407i	$-0.383179\pi$
$863$	21.0820	0.717640	0.358820	+	0.933407i	$0.383179\pi$
$864$	0	0
$865$	43.3096	1.47257
$866$	27.0642	0.919678
$867$	0	0
$868$	−11.5990	−0.393694
$869$	15.1575	0.514181
$870$	0	0
$871$	−3.50919	−0.118904
$872$	−15.2490	−0.516395
$873$	0	0
$874$	0	0
$875$	32.5012	1.09874
$876$	0	0
$877$	51.2012	1.72894	0.864471	−	0.502683i	$-0.167654\pi$
$877$	51.2012	1.72894	0.864471	+	0.502683i	$0.167654\pi$
$878$	2.94087	0.0992497
$879$	0	0
$880$	18.6878	0.629965
$881$	34.1607	1.15090	0.575452	−	0.817835i	$-0.304826\pi$
$881$	34.1607	1.15090	0.575452	+	0.817835i	$0.304826\pi$
$882$	0	0
$883$	28.1189	0.946275	0.473137	−	0.880989i	$-0.343121\pi$
$883$	28.1189	0.946275	0.473137	+	0.880989i	$0.343121\pi$
$884$	0.374638	0.0126004
$885$	0	0
$886$	−27.6049	−0.927406
$887$	−35.5752	−1.19450	−0.597250	−	0.802055i	$-0.703740\pi$
$887$	−35.5752	−1.19450	−0.597250	+	0.802055i	$0.703740\pi$
$888$	0	0
$889$	−41.1242	−1.37926
$890$	−42.9350	−1.43918
$891$	0	0
$892$	−18.6159	−0.623305
$893$	0	0
$894$	0	0
$895$	28.4834	0.952095
$896$	3.71688	0.124172
$897$	0	0
$898$	16.9222	0.564701
$899$	−0.376392	−0.0125534
$900$	0	0
$901$	9.81016	0.326824
$902$	37.5972	1.25185
$903$	0	0
$904$	0.815207	0.0271134
$905$	−0.670549	−0.0222898
$906$	0	0
$907$	9.29591	0.308666	0.154333	−	0.988019i	$-0.450677\pi$
$907$	9.29591	0.308666	0.154333	+	0.988019i	$0.450677\pi$
$908$	−5.79292	−0.192245
$909$	0	0
$910$	2.97596	0.0986520
$911$	3.79055	0.125587	0.0627933	−	0.998027i	$-0.479999\pi$
$911$	3.79055	0.125587	0.0627933	+	0.998027i	$0.479999\pi$
$912$	0	0
$913$	−10.1652	−0.336419
$914$	9.19160	0.304031
$915$	0	0
$916$	−8.12836	−0.268568
$917$	53.5217	1.76744
$918$	0	0
$919$	−4.43107	−0.146168	−0.0730838	−	0.997326i	$-0.523284\pi$
$919$	−4.43107	−0.146168	−0.0730838	+	0.997326i	$0.523284\pi$
$920$	−30.6587	−1.01079
$921$	0	0
$922$	15.4456	0.508674
$923$	−0.307162	−0.0101104
$924$	0	0
$925$	−38.3378	−1.26054
$926$	29.9813	0.985248
$927$	0	0
$928$	0.120615	0.00395937
$929$	−30.4662	−0.999562	−0.499781	−	0.866152i	$-0.666586\pi$
$929$	−30.4662	−0.999562	−0.499781	+	0.866152i	$0.666586\pi$
$930$	0	0
$931$	0	0
$932$	17.0273	0.557749
$933$	0	0
$934$	−28.8854	−0.945158
$935$	30.8854	1.01006
$936$	0	0
$937$	−1.43107	−0.0467512	−0.0233756	−	0.999727i	$-0.507441\pi$
$937$	−1.43107	−0.0467512	−0.0233756	+	0.999727i	$0.507441\pi$
$938$	−57.5399	−1.87875
$939$	0	0
$940$	8.86484	0.289139
$941$	4.53478	0.147830	0.0739148	−	0.997265i	$-0.476451\pi$
$941$	4.53478	0.147830	0.0739148	+	0.997265i	$0.476451\pi$
$942$	0	0
$943$	−61.6810	−2.00861
$944$	0.218941	0.00712592
$945$	0	0
$946$	−28.3327	−0.921177
$947$	−0.413838	−0.0134479	−0.00672397	−	0.999977i	$-0.502140\pi$
$947$	−0.413838	−0.0134479	−0.00672397	+	0.999977i	$0.502140\pi$
$948$	0	0
$949$	0.549935	0.0178516
$950$	0	0
$951$	0	0
$952$	6.14290	0.199093
$953$	14.2257	0.460817	0.230409	−	0.973094i	$-0.425994\pi$
$953$	14.2257	0.460817	0.230409	+	0.973094i	$0.425994\pi$
$954$	0	0
$955$	−63.7110	−2.06164
$956$	15.0196	0.485769
$957$	0	0
$958$	−13.6604	−0.441349
$959$	1.75795	0.0567671
$960$	0	0
$961$	−21.2618	−0.685863
$962$	−1.16250	−0.0374806
$963$	0	0
$964$	21.4534	0.690966
$965$	61.5954	1.98283
$966$	0	0
$967$	−56.8343	−1.82767	−0.913834	−	0.406088i	$-0.866893\pi$
$967$	−56.8343	−1.82767	−0.913834	+	0.406088i	$0.866893\pi$
$968$	16.9932	0.546182
$969$	0	0
$970$	−61.8221	−1.98499
$971$	−9.06418	−0.290883	−0.145442	−	0.989367i	$-0.546460\pi$
$971$	−9.06418	−0.290883	−0.145442	+	0.989367i	$0.546460\pi$
$972$	0	0
$973$	−37.8557	−1.21360
$974$	6.17799	0.197955
$975$	0	0
$976$	−1.57398	−0.0503818
$977$	−2.65270	−0.0848675	−0.0424338	−	0.999099i	$-0.513511\pi$
$977$	−2.65270	−0.0848675	−0.0424338	+	0.999099i	$0.513511\pi$
$978$	0	0
$979$	−64.3141	−2.05549
$980$	24.0719	0.768949
$981$	0	0
$982$	−4.10338	−0.130944
$983$	−53.7975	−1.71587	−0.857936	−	0.513756i	$-0.828254\pi$
$983$	−53.7975	−1.71587	−0.857936	+	0.513756i	$0.828254\pi$
$984$	0	0
$985$	37.0188	1.17952
$986$	0.199340	0.00634829
$987$	0	0
$988$	0	0
$989$	46.4820	1.47804
$990$	0	0
$991$	−53.5918	−1.70240	−0.851200	−	0.524841i	$-0.824125\pi$
$991$	−53.5918	−1.70240	−0.851200	+	0.524841i	$0.824125\pi$
$992$	−3.12061	−0.0990796
$993$	0	0
$994$	−5.03651	−0.159748
$995$	−48.2259	−1.52886
$996$	0	0
$997$	18.1370	0.574406	0.287203	−	0.957870i	$-0.407275\pi$
$997$	18.1370	0.574406	0.287203	+	0.957870i	$0.407275\pi$
$998$	26.5790	0.841345
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6498.2.a.bu.1.3		3
3.2	odd	2		2166.2.a.p.1.1		3
19.14	odd	18		342.2.u.b.253.1		6
19.15	odd	18		342.2.u.b.73.1		6
19.18	odd	2		6498.2.a.bp.1.3		3
57.14	even	18		114.2.i.c.25.1	✓	6
57.53	even	18		114.2.i.c.73.1	yes	6
57.56	even	2		2166.2.a.r.1.1		3
228.71	odd	18		912.2.bo.d.481.1		6
228.167	odd	18		912.2.bo.d.529.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
114.2.i.c.25.1	✓	6	57.14	even	18
114.2.i.c.73.1	yes	6	57.53	even	18
342.2.u.b.73.1		6	19.15	odd	18
342.2.u.b.253.1		6	19.14	odd	18
912.2.bo.d.481.1		6	228.71	odd	18
912.2.bo.d.529.1		6	228.167	odd	18
2166.2.a.p.1.1		3	3.2	odd	2
2166.2.a.r.1.1		3	57.56	even	2
6498.2.a.bp.1.3		3	19.18	odd	2
6498.2.a.bu.1.3		3	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.53209 q^{5} +3.71688 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +3.53209 q^{5} +3.71688 q^{7} +1.00000 q^{8} +3.53209 q^{10} +5.29086 q^{11} +0.226682 q^{13} +3.71688 q^{14} +1.00000 q^{16} +1.65270 q^{17} +3.53209 q^{20} +5.29086 q^{22} -8.68004 q^{23} +7.47565 q^{25} +0.226682 q^{26} +3.71688 q^{28} +0.120615 q^{29} -3.12061 q^{31} +1.00000 q^{32} +1.65270 q^{34} +13.1284 q^{35} -5.12836 q^{37} +3.53209 q^{40} +7.10607 q^{41} -5.35504 q^{43} +5.29086 q^{44} -8.68004 q^{46} +2.50980 q^{47} +6.81521 q^{49} +7.47565 q^{50} +0.226682 q^{52} +5.93582 q^{53} +18.6878 q^{55} +3.71688 q^{56} +0.120615 q^{58} +0.218941 q^{59} -1.57398 q^{61} -3.12061 q^{62} +1.00000 q^{64} +0.800660 q^{65} -15.4807 q^{67} +1.65270 q^{68} +13.1284 q^{70} -1.35504 q^{71} +2.42602 q^{73} -5.12836 q^{74} +19.6655 q^{77} +2.86484 q^{79} +3.53209 q^{80} +7.10607 q^{82} -1.92127 q^{83} +5.83750 q^{85} -5.35504 q^{86} +5.29086 q^{88} -12.1557 q^{89} +0.842549 q^{91} -8.68004 q^{92} +2.50980 q^{94} -17.5030 q^{97} +6.81521 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 3 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 + 3 * q^7 + 3 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} + 3 q^{7} + 3 q^{8} + 6 q^{10} - 6 q^{13} + 3 q^{14} + 3 q^{16} + 6 q^{17} + 6 q^{20} - 6 q^{23} + 3 q^{25} - 6 q^{26} + 3 q^{28} + 6 q^{29} - 15 q^{31} + 3 q^{32} + 6 q^{34} + 21 q^{35} + 3 q^{37} + 6 q^{40} + 9 q^{41} + 9 q^{43} - 6 q^{46} + 9 q^{47} + 24 q^{49} + 3 q^{50} - 6 q^{52} + 27 q^{53} + 12 q^{55} + 3 q^{56} + 6 q^{58} + 18 q^{59} + 3 q^{61} - 15 q^{62} + 3 q^{64} - 12 q^{65} - 12 q^{67} + 6 q^{68} + 21 q^{70} + 21 q^{71} + 15 q^{73} + 3 q^{74} + 21 q^{77} - 15 q^{79} + 6 q^{80} + 9 q^{82} + 3 q^{83} + 15 q^{85} + 9 q^{86} + 3 q^{89} - 15 q^{91} - 6 q^{92} + 9 q^{94} - 12 q^{97} + 24 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 + 3 * q^7 + 3 * q^8 + 6 * q^10 - 6 * q^13 + 3 * q^14 + 3 * q^16 + 6 * q^17 + 6 * q^20 - 6 * q^23 + 3 * q^25 - 6 * q^26 + 3 * q^28 + 6 * q^29 - 15 * q^31 + 3 * q^32 + 6 * q^34 + 21 * q^35 + 3 * q^37 + 6 * q^40 + 9 * q^41 + 9 * q^43 - 6 * q^46 + 9 * q^47 + 24 * q^49 + 3 * q^50 - 6 * q^52 + 27 * q^53 + 12 * q^55 + 3 * q^56 + 6 * q^58 + 18 * q^59 + 3 * q^61 - 15 * q^62 + 3 * q^64 - 12 * q^65 - 12 * q^67 + 6 * q^68 + 21 * q^70 + 21 * q^71 + 15 * q^73 + 3 * q^74 + 21 * q^77 - 15 * q^79 + 6 * q^80 + 9 * q^82 + 3 * q^83 + 15 * q^85 + 9 * q^86 + 3 * q^89 - 15 * q^91 - 6 * q^92 + 9 * q^94 - 12 * q^97 + 24 * q^98

Introduction

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