LMFDB - Embedded newform 3249.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3249,2,Mod(1,3249)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3249, 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("3249.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3249 = 3^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3249.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:	$25.9433956167$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 19)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.53209$ of defining polynomial
Character	$\chi$	$=$	3249.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.53209 q^{2} +4.41147 q^{4} +1.34730 q^{5} +1.53209 q^{7} -6.10607 q^{8} +O(q^{10})$	$q-2.53209 q^{2} +4.41147 q^{4} +1.34730 q^{5} +1.53209 q^{7} -6.10607 q^{8} -3.41147 q^{10} +1.18479 q^{11} -2.71688 q^{13} -3.87939 q^{14} +6.63816 q^{16} -3.87939 q^{17} +5.94356 q^{20} -3.00000 q^{22} +5.06418 q^{23} -3.18479 q^{25} +6.87939 q^{26} +6.75877 q^{28} -4.65270 q^{29} +3.83750 q^{31} -4.59627 q^{32} +9.82295 q^{34} +2.06418 q^{35} +4.10607 q^{37} -8.22668 q^{40} -9.98545 q^{41} -8.70233 q^{43} +5.22668 q^{44} -12.8229 q^{46} -0.573978 q^{47} -4.65270 q^{49} +8.06418 q^{50} -11.9855 q^{52} +2.94356 q^{53} +1.59627 q^{55} -9.35504 q^{56} +11.7811 q^{58} -3.93582 q^{59} -4.51754 q^{61} -9.71688 q^{62} -1.63816 q^{64} -3.66044 q^{65} +3.88713 q^{67} -17.1138 q^{68} -5.22668 q^{70} -6.93582 q^{71} +6.12836 q^{73} -10.3969 q^{74} +1.81521 q^{77} -9.80840 q^{79} +8.94356 q^{80} +25.2841 q^{82} -12.3182 q^{83} -5.22668 q^{85} +22.0351 q^{86} -7.23442 q^{88} -2.42602 q^{89} -4.16250 q^{91} +22.3405 q^{92} +1.45336 q^{94} -7.36959 q^{97} +11.7811 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - 6 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{8} - 6 q^{14} + 3 q^{16} - 6 q^{17} + 3 q^{20} - 9 q^{22} + 6 q^{23} - 6 q^{25} + 15 q^{26} + 9 q^{28} - 15 q^{29} + 9 q^{31} + 9 q^{34} - 3 q^{35} - 18 q^{40} - 12 q^{41} + 9 q^{44} - 18 q^{46} + 6 q^{47} - 15 q^{49} + 15 q^{50} - 18 q^{52} - 6 q^{53} - 9 q^{55} - 3 q^{56} + 18 q^{58} - 21 q^{59} + 9 q^{61} - 21 q^{62} + 12 q^{64} + 12 q^{65} - 18 q^{67} - 15 q^{68} - 9 q^{70} - 30 q^{71} - 3 q^{74} + 9 q^{77} + 9 q^{79} + 12 q^{80} + 18 q^{82} - 9 q^{85} + 21 q^{86} + 9 q^{88} - 15 q^{89} - 15 q^{91} + 24 q^{92} - 9 q^{94} - 15 q^{97} + 18 q^{98}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - 6 * q^8 - 6 * q^14 + 3 * q^16 - 6 * q^17 + 3 * q^20 - 9 * q^22 + 6 * q^23 - 6 * q^25 + 15 * q^26 + 9 * q^28 - 15 * q^29 + 9 * q^31 + 9 * q^34 - 3 * q^35 - 18 * q^40 - 12 * q^41 + 9 * q^44 - 18 * q^46 + 6 * q^47 - 15 * q^49 + 15 * q^50 - 18 * q^52 - 6 * q^53 - 9 * q^55 - 3 * q^56 + 18 * q^58 - 21 * q^59 + 9 * q^61 - 21 * q^62 + 12 * q^64 + 12 * q^65 - 18 * q^67 - 15 * q^68 - 9 * q^70 - 30 * q^71 - 3 * q^74 + 9 * q^77 + 9 * q^79 + 12 * q^80 + 18 * q^82 - 9 * q^85 + 21 * q^86 + 9 * q^88 - 15 * q^89 - 15 * q^91 + 24 * q^92 - 9 * q^94 - 15 * q^97 + 18 * 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 $

$2$	−2.53209	−1.79046	−0.895229	−	0.445607i	$-0.852988\pi$
$2$	−2.53209	−1.79046	−0.895229	+	0.445607i	$0.852988\pi$
$3$	0	0
$3$	0	0
$4$	4.41147	2.20574
$5$	1.34730	0.602529	0.301265	−	0.953541i	$-0.402591\pi$
$5$	1.34730	0.602529	0.301265	+	0.953541i	$0.402591\pi$
$6$	0	0
$7$	1.53209	0.579075	0.289538	−	0.957167i	$-0.406498\pi$
$7$	1.53209	0.579075	0.289538	+	0.957167i	$0.406498\pi$
$8$	−6.10607	−2.15882
$9$	0	0
$10$	−3.41147	−1.07880
$11$	1.18479	0.357228	0.178614	−	0.983919i	$-0.442839\pi$
$11$	1.18479	0.357228	0.178614	+	0.983919i	$0.442839\pi$
$12$	0	0
$13$	−2.71688	−0.753527	−0.376764	−	0.926309i	$-0.622963\pi$
$13$	−2.71688	−0.753527	−0.376764	+	0.926309i	$0.622963\pi$
$14$	−3.87939	−1.03681
$15$	0	0
$16$	6.63816	1.65954
$17$	−3.87939	−0.940889	−0.470445	−	0.882430i	$-0.655906\pi$
$17$	−3.87939	−0.940889	−0.470445	+	0.882430i	$0.655906\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	5.94356	1.32902
$21$	0	0
$22$	−3.00000	−0.639602
$23$	5.06418	1.05595	0.527977	−	0.849259i	$-0.322951\pi$
$23$	5.06418	1.05595	0.527977	+	0.849259i	$0.322951\pi$
$24$	0	0
$25$	−3.18479	−0.636959
$26$	6.87939	1.34916
$27$	0	0
$28$	6.75877	1.27729
$29$	−4.65270	−0.863985	−0.431993	−	0.901877i	$-0.642189\pi$
$29$	−4.65270	−0.863985	−0.431993	+	0.901877i	$0.642189\pi$
$30$	0	0
$31$	3.83750	0.689235	0.344617	−	0.938743i	$-0.388009\pi$
$31$	3.83750	0.689235	0.344617	+	0.938743i	$0.388009\pi$
$32$	−4.59627	−0.812513
$33$	0	0
$34$	9.82295	1.68462
$35$	2.06418	0.348910
$36$	0	0
$37$	4.10607	0.675033	0.337517	−	0.941320i	$-0.390413\pi$
$37$	4.10607	0.675033	0.337517	+	0.941320i	$0.390413\pi$
$38$	0	0
$39$	0	0
$40$	−8.22668	−1.30075
$41$	−9.98545	−1.55947	−0.779733	−	0.626112i	$-0.784645\pi$
$41$	−9.98545	−1.55947	−0.779733	+	0.626112i	$0.784645\pi$
$42$	0	0
$43$	−8.70233	−1.32709	−0.663547	−	0.748135i	$-0.730950\pi$
$43$	−8.70233	−1.32709	−0.663547	+	0.748135i	$0.730950\pi$
$44$	5.22668	0.787952
$45$	0	0
$46$	−12.8229	−1.89064
$47$	−0.573978	−0.0837233	−0.0418616	−	0.999123i	$-0.513329\pi$
$47$	−0.573978	−0.0837233	−0.0418616	+	0.999123i	$0.513329\pi$
$48$	0	0
$49$	−4.65270	−0.664672
$50$	8.06418	1.14045
$51$	0	0
$52$	−11.9855	−1.66208
$53$	2.94356	0.404329	0.202165	−	0.979352i	$-0.435202\pi$
$53$	2.94356	0.404329	0.202165	+	0.979352i	$0.435202\pi$
$54$	0	0
$55$	1.59627	0.215241
$56$	−9.35504	−1.25012
$57$	0	0
$58$	11.7811	1.54693
$59$	−3.93582	−0.512400	−0.256200	−	0.966624i	$-0.582471\pi$
$59$	−3.93582	−0.512400	−0.256200	+	0.966624i	$0.582471\pi$
$60$	0	0
$61$	−4.51754	−0.578412	−0.289206	−	0.957267i	$-0.593391\pi$
$61$	−4.51754	−0.578412	−0.289206	+	0.957267i	$0.593391\pi$
$62$	−9.71688	−1.23405
$63$	0	0
$64$	−1.63816	−0.204769
$65$	−3.66044	−0.454022
$66$	0	0
$67$	3.88713	0.474888	0.237444	−	0.971401i	$-0.423690\pi$
$67$	3.88713	0.474888	0.237444	+	0.971401i	$0.423690\pi$
$68$	−17.1138	−2.07535
$69$	0	0
$70$	−5.22668	−0.624708
$71$	−6.93582	−0.823131	−0.411565	−	0.911380i	$-0.635018\pi$
$71$	−6.93582	−0.823131	−0.411565	+	0.911380i	$0.635018\pi$
$72$	0	0
$73$	6.12836	0.717270	0.358635	−	0.933478i	$-0.383242\pi$
$73$	6.12836	0.717270	0.358635	+	0.933478i	$0.383242\pi$
$74$	−10.3969	−1.20862
$75$	0	0
$76$	0	0
$77$	1.81521	0.206862
$78$	0	0
$79$	−9.80840	−1.10353	−0.551766	−	0.833999i	$-0.686046\pi$
$79$	−9.80840	−1.10353	−0.551766	+	0.833999i	$0.686046\pi$
$80$	8.94356	0.999921
$81$	0	0
$82$	25.2841	2.79216
$83$	−12.3182	−1.35210	−0.676049	−	0.736857i	$-0.736309\pi$
$83$	−12.3182	−1.35210	−0.676049	+	0.736857i	$0.736309\pi$
$84$	0	0
$85$	−5.22668	−0.566913
$86$	22.0351	2.37610
$87$	0	0
$88$	−7.23442	−0.771192
$89$	−2.42602	−0.257158	−0.128579	−	0.991699i	$-0.541042\pi$
$89$	−2.42602	−0.257158	−0.128579	+	0.991699i	$0.541042\pi$
$90$	0	0
$91$	−4.16250	−0.436349
$92$	22.3405	2.32916
$93$	0	0
$94$	1.45336	0.149903
$95$	0	0
$96$	0	0
$97$	−7.36959	−0.748268	−0.374134	−	0.927375i	$-0.622060\pi$
$97$	−7.36959	−0.748268	−0.374134	+	0.927375i	$0.622060\pi$
$98$	11.7811	1.19007
$99$	0	0
$100$	−14.0496	−1.40496
$101$	−2.17024	−0.215947	−0.107974	−	0.994154i	$-0.534436\pi$
$101$	−2.17024	−0.215947	−0.107974	+	0.994154i	$0.534436\pi$
$102$	0	0
$103$	12.4757	1.22926	0.614631	−	0.788815i	$-0.289305\pi$
$103$	12.4757	1.22926	0.614631	+	0.788815i	$0.289305\pi$
$104$	16.5895	1.62673
$105$	0	0
$106$	−7.45336	−0.723935
$107$	−6.68004	−0.645784	−0.322892	−	0.946436i	$-0.604655\pi$
$107$	−6.68004	−0.645784	−0.322892	+	0.946436i	$0.604655\pi$
$108$	0	0
$109$	−9.45336	−0.905468	−0.452734	−	0.891646i	$-0.649551\pi$
$109$	−9.45336	−0.905468	−0.452734	+	0.891646i	$0.649551\pi$
$110$	−4.04189	−0.385379
$111$	0	0
$112$	10.1702	0.960998
$113$	1.31046	0.123278	0.0616388	−	0.998099i	$-0.480367\pi$
$113$	1.31046	0.123278	0.0616388	+	0.998099i	$0.480367\pi$
$114$	0	0
$115$	6.82295	0.636243
$116$	−20.5253	−1.90572
$117$	0	0
$118$	9.96585	0.917431
$119$	−5.94356	−0.544846
$120$	0	0
$121$	−9.59627	−0.872388
$122$	11.4388	1.03562
$123$	0	0
$124$	16.9290	1.52027
$125$	−11.0273	−0.986315
$126$	0	0
$127$	14.5030	1.28693	0.643466	−	0.765474i	$-0.277496\pi$
$127$	14.5030	1.28693	0.643466	+	0.765474i	$0.277496\pi$
$128$	13.3405	1.17914
$129$	0	0
$130$	9.26857	0.812907
$131$	19.8084	1.73067	0.865334	−	0.501196i	$-0.167106\pi$
$131$	19.8084	1.73067	0.865334	+	0.501196i	$0.167106\pi$
$132$	0	0
$133$	0	0
$134$	−9.84255	−0.850267
$135$	0	0
$136$	23.6878	2.03121
$137$	−10.2044	−0.871820	−0.435910	−	0.899990i	$-0.643573\pi$
$137$	−10.2044	−0.871820	−0.435910	+	0.899990i	$0.643573\pi$
$138$	0	0
$139$	−1.66044	−0.140837	−0.0704185	−	0.997518i	$-0.522433\pi$
$139$	−1.66044	−0.140837	−0.0704185	+	0.997518i	$0.522433\pi$
$140$	9.10607	0.769603
$141$	0	0
$142$	17.5621	1.47378
$143$	−3.21894	−0.269181
$144$	0	0
$145$	−6.26857	−0.520576
$146$	−15.5175	−1.28424
$147$	0	0
$148$	18.1138	1.48895
$149$	−11.2071	−0.918120	−0.459060	−	0.888405i	$-0.651814\pi$
$149$	−11.2071	−0.918120	−0.459060	+	0.888405i	$0.651814\pi$
$150$	0	0
$151$	11.0419	0.898576	0.449288	−	0.893387i	$-0.351678\pi$
$151$	11.0419	0.898576	0.449288	+	0.893387i	$0.351678\pi$
$152$	0	0
$153$	0	0
$154$	−4.59627	−0.370378
$155$	5.17024	0.415284
$156$	0	0
$157$	10.9932	0.877352	0.438676	−	0.898645i	$-0.355448\pi$
$157$	10.9932	0.877352	0.438676	+	0.898645i	$0.355448\pi$
$158$	24.8357	1.97583
$159$	0	0
$160$	−6.19253	−0.489563
$161$	7.75877	0.611477
$162$	0	0
$163$	−6.33275	−0.496019	−0.248010	−	0.968758i	$-0.579776\pi$
$163$	−6.33275	−0.496019	−0.248010	+	0.968758i	$0.579776\pi$
$164$	−44.0506	−3.43977
$165$	0	0
$166$	31.1908	2.42087
$167$	−13.7784	−1.06620	−0.533101	−	0.846051i	$-0.678973\pi$
$167$	−13.7784	−1.06620	−0.533101	+	0.846051i	$0.678973\pi$
$168$	0	0
$169$	−5.61856	−0.432197
$170$	13.2344	1.01503
$171$	0	0
$172$	−38.3901	−2.92722
$173$	25.2472	1.91951	0.959755	−	0.280838i	$-0.0906124\pi$
$173$	25.2472	1.91951	0.959755	+	0.280838i	$0.0906124\pi$
$174$	0	0
$175$	−4.87939	−0.368847
$176$	7.86484	0.592834
$177$	0	0
$178$	6.14290	0.460430
$179$	5.83069	0.435806	0.217903	−	0.975970i	$-0.430078\pi$
$179$	5.83069	0.435806	0.217903	+	0.975970i	$0.430078\pi$
$180$	0	0
$181$	−13.5621	−1.00806	−0.504032	−	0.863685i	$-0.668151\pi$
$181$	−13.5621	−1.00806	−0.504032	+	0.863685i	$0.668151\pi$
$182$	10.5398	0.781264
$183$	0	0
$184$	−30.9222	−2.27962
$185$	5.53209	0.406727
$186$	0	0
$187$	−4.59627	−0.336112
$188$	−2.53209	−0.184672
$189$	0	0
$190$	0	0
$191$	10.2841	0.744128	0.372064	−	0.928207i	$-0.378650\pi$
$191$	10.2841	0.744128	0.372064	+	0.928207i	$0.378650\pi$
$192$	0	0
$193$	−13.8007	−0.993393	−0.496697	−	0.867924i	$-0.665454\pi$
$193$	−13.8007	−0.993393	−0.496697	+	0.867924i	$0.665454\pi$
$194$	18.6604	1.33974
$195$	0	0
$196$	−20.5253	−1.46609
$197$	7.94087	0.565764	0.282882	−	0.959155i	$-0.408710\pi$
$197$	7.94087	0.565764	0.282882	+	0.959155i	$0.408710\pi$
$198$	0	0
$199$	27.0351	1.91647	0.958233	−	0.285988i	$-0.0923219\pi$
$199$	27.0351	1.91647	0.958233	+	0.285988i	$0.0923219\pi$
$200$	19.4466	1.37508
$201$	0	0
$202$	5.49525	0.386645
$203$	−7.12836	−0.500312
$204$	0	0
$205$	−13.4534	−0.939624
$206$	−31.5895	−2.20094
$207$	0	0
$208$	−18.0351	−1.25051
$209$	0	0
$210$	0	0
$211$	8.07192	0.555694	0.277847	−	0.960625i	$-0.410379\pi$
$211$	8.07192	0.555694	0.277847	+	0.960625i	$0.410379\pi$
$212$	12.9855	0.891845
$213$	0	0
$214$	16.9145	1.15625
$215$	−11.7246	−0.799613
$216$	0	0
$217$	5.87939	0.399119
$218$	23.9368	1.62120
$219$	0	0
$220$	7.04189	0.474764
$221$	10.5398	0.708986
$222$	0	0
$223$	15.4757	1.03633	0.518163	−	0.855282i	$-0.326616\pi$
$223$	15.4757	1.03633	0.518163	+	0.855282i	$0.326616\pi$
$224$	−7.04189	−0.470506
$225$	0	0
$226$	−3.31820	−0.220723
$227$	9.87258	0.655266	0.327633	−	0.944805i	$-0.393749\pi$
$227$	9.87258	0.655266	0.327633	+	0.944805i	$0.393749\pi$
$228$	0	0
$229$	20.1189	1.32949	0.664746	−	0.747070i	$-0.268540\pi$
$229$	20.1189	1.32949	0.664746	+	0.747070i	$0.268540\pi$
$230$	−17.2763	−1.13917
$231$	0	0
$232$	28.4097	1.86519
$233$	−3.53478	−0.231571	−0.115785	−	0.993274i	$-0.536939\pi$
$233$	−3.53478	−0.231571	−0.115785	+	0.993274i	$0.536939\pi$
$234$	0	0
$235$	−0.773318	−0.0504457
$236$	−17.3628	−1.13022
$237$	0	0
$238$	15.0496	0.975523
$239$	−11.9736	−0.774507	−0.387254	−	0.921973i	$-0.626576\pi$
$239$	−11.9736	−0.774507	−0.387254	+	0.921973i	$0.626576\pi$
$240$	0	0
$241$	−12.9017	−0.831070	−0.415535	−	0.909577i	$-0.636406\pi$
$241$	−12.9017	−0.831070	−0.415535	+	0.909577i	$0.636406\pi$
$242$	24.2986	1.56197
$243$	0	0
$244$	−19.9290	−1.27582
$245$	−6.26857	−0.400484
$246$	0	0
$247$	0	0
$248$	−23.4320	−1.48793
$249$	0	0
$250$	27.9222	1.76596
$251$	−14.3628	−0.906571	−0.453285	−	0.891366i	$-0.649748\pi$
$251$	−14.3628	−0.906571	−0.453285	+	0.891366i	$0.649748\pi$
$252$	0	0
$253$	6.00000	0.377217
$254$	−36.7229	−2.30420
$255$	0	0
$256$	−30.5030	−1.90644
$257$	4.97771	0.310501	0.155251	−	0.987875i	$-0.450382\pi$
$257$	4.97771	0.310501	0.155251	+	0.987875i	$0.450382\pi$
$258$	0	0
$259$	6.29086	0.390895
$260$	−16.1480	−1.00145
$261$	0	0
$262$	−50.1566	−3.09869
$263$	24.0428	1.48254	0.741272	−	0.671205i	$-0.234223\pi$
$263$	24.0428	1.48254	0.741272	+	0.671205i	$0.234223\pi$
$264$	0	0
$265$	3.96585	0.243620
$266$	0	0
$267$	0	0
$268$	17.1480	1.04748
$269$	−13.1111	−0.799399	−0.399700	−	0.916646i	$-0.630886\pi$
$269$	−13.1111	−0.799399	−0.399700	+	0.916646i	$0.630886\pi$
$270$	0	0
$271$	−26.5699	−1.61400	−0.807002	−	0.590549i	$-0.798911\pi$
$271$	−26.5699	−1.61400	−0.807002	+	0.590549i	$0.798911\pi$
$272$	−25.7520	−1.56144
$273$	0	0
$274$	25.8384	1.56096
$275$	−3.77332	−0.227540
$276$	0	0
$277$	−16.5107	−0.992034	−0.496017	−	0.868313i	$-0.665205\pi$
$277$	−16.5107	−0.992034	−0.496017	+	0.868313i	$0.665205\pi$
$278$	4.20439	0.252163
$279$	0	0
$280$	−12.6040	−0.753234
$281$	−19.3901	−1.15672	−0.578359	−	0.815783i	$-0.696307\pi$
$281$	−19.3901	−1.15672	−0.578359	+	0.815783i	$0.696307\pi$
$282$	0	0
$283$	−11.3105	−0.672337	−0.336169	−	0.941802i	$-0.609131\pi$
$283$	−11.3105	−0.672337	−0.336169	+	0.941802i	$0.609131\pi$
$284$	−30.5972	−1.81561
$285$	0	0
$286$	8.15064	0.481958
$287$	−15.2986	−0.903048
$288$	0	0
$289$	−1.95037	−0.114728
$290$	15.8726	0.932070
$291$	0	0
$292$	27.0351	1.58211
$293$	3.89899	0.227781	0.113891	−	0.993493i	$-0.463669\pi$
$293$	3.89899	0.227781	0.113891	+	0.993493i	$0.463669\pi$
$294$	0	0
$295$	−5.30272	−0.308736
$296$	−25.0719	−1.45728
$297$	0	0
$298$	28.3773	1.64385
$299$	−13.7588	−0.795690
$300$	0	0
$301$	−13.3327	−0.768487
$302$	−27.9590	−1.60886
$303$	0	0
$304$	0	0
$305$	−6.08647	−0.348510
$306$	0	0
$307$	−23.1753	−1.32268	−0.661342	−	0.750084i	$-0.730013\pi$
$307$	−23.1753	−1.32268	−0.661342	+	0.750084i	$0.730013\pi$
$308$	8.00774	0.456283
$309$	0	0
$310$	−13.0915	−0.743548
$311$	3.46110	0.196261	0.0981306	−	0.995174i	$-0.468714\pi$
$311$	3.46110	0.196261	0.0981306	+	0.995174i	$0.468714\pi$
$312$	0	0
$313$	−22.8898	−1.29381	−0.646904	−	0.762571i	$-0.723937\pi$
$313$	−22.8898	−1.29381	−0.646904	+	0.762571i	$0.723937\pi$
$314$	−27.8357	−1.57086
$315$	0	0
$316$	−43.2695	−2.43410
$317$	26.1206	1.46708	0.733540	−	0.679646i	$-0.237867\pi$
$317$	26.1206	1.46708	0.733540	+	0.679646i	$0.237867\pi$
$318$	0	0
$319$	−5.51249	−0.308640
$320$	−2.20708	−0.123380
$321$	0	0
$322$	−19.6459	−1.09482
$323$	0	0
$324$	0	0
$325$	8.65270	0.479966
$326$	16.0351	0.888101
$327$	0	0
$328$	60.9718	3.36661
$329$	−0.879385	−0.0484821
$330$	0	0
$331$	−19.0446	−1.04678	−0.523392	−	0.852092i	$-0.675334\pi$
$331$	−19.0446	−1.04678	−0.523392	+	0.852092i	$0.675334\pi$
$332$	−54.3414	−2.98237
$333$	0	0
$334$	34.8881	1.90899
$335$	5.23711	0.286134
$336$	0	0
$337$	−1.70140	−0.0926812	−0.0463406	−	0.998926i	$-0.514756\pi$
$337$	−1.70140	−0.0926812	−0.0463406	+	0.998926i	$0.514756\pi$
$338$	14.2267	0.773829
$339$	0	0
$340$	−23.0574	−1.25046
$341$	4.54664	0.246214
$342$	0	0
$343$	−17.8530	−0.963970
$344$	53.1370	2.86496
$345$	0	0
$346$	−63.9282	−3.43680
$347$	4.90167	0.263136	0.131568	−	0.991307i	$-0.457999\pi$
$347$	4.90167	0.263136	0.131568	+	0.991307i	$0.457999\pi$
$348$	0	0
$349$	−28.1293	−1.50573	−0.752863	−	0.658177i	$-0.771328\pi$
$349$	−28.1293	−1.50573	−0.752863	+	0.658177i	$0.771328\pi$
$350$	12.3550	0.660405
$351$	0	0
$352$	−5.44562	−0.290253
$353$	8.31996	0.442827	0.221413	−	0.975180i	$-0.428933\pi$
$353$	8.31996	0.442827	0.221413	+	0.975180i	$0.428933\pi$
$354$	0	0
$355$	−9.34461	−0.495960
$356$	−10.7023	−0.567223
$357$	0	0
$358$	−14.7638	−0.780292
$359$	−24.9290	−1.31570	−0.657852	−	0.753148i	$-0.728535\pi$
$359$	−24.9290	−1.31570	−0.657852	+	0.753148i	$0.728535\pi$
$360$	0	0
$361$	0	0
$362$	34.3405	1.80490
$363$	0	0
$364$	−18.3628	−0.962471
$365$	8.25671	0.432176
$366$	0	0
$367$	−2.58584	−0.134980	−0.0674898	−	0.997720i	$-0.521499\pi$
$367$	−2.58584	−0.134980	−0.0674898	+	0.997720i	$0.521499\pi$
$368$	33.6168	1.75240
$369$	0	0
$370$	−14.0077	−0.728228
$371$	4.50980	0.234137
$372$	0	0
$373$	23.3833	1.21074	0.605371	−	0.795943i	$-0.293025\pi$
$373$	23.3833	1.21074	0.605371	+	0.795943i	$0.293025\pi$
$374$	11.6382	0.601795
$375$	0	0
$376$	3.50475	0.180744
$377$	12.6408	0.651037
$378$	0	0
$379$	−25.4388	−1.30670	−0.653352	−	0.757054i	$-0.726638\pi$
$379$	−25.4388	−1.30670	−0.653352	+	0.757054i	$0.726638\pi$
$380$	0	0
$381$	0	0
$382$	−26.0401	−1.33233
$383$	−27.4807	−1.40420	−0.702099	−	0.712079i	$-0.747754\pi$
$383$	−27.4807	−1.40420	−0.702099	+	0.712079i	$0.747754\pi$
$384$	0	0
$385$	2.44562	0.124640
$386$	34.9445	1.77863
$387$	0	0
$388$	−32.5107	−1.65048
$389$	3.34224	0.169458	0.0847292	−	0.996404i	$-0.472997\pi$
$389$	3.34224	0.169458	0.0847292	+	0.996404i	$0.472997\pi$
$390$	0	0
$391$	−19.6459	−0.993536
$392$	28.4097	1.43491
$393$	0	0
$394$	−20.1070	−1.01298
$395$	−13.2148	−0.664910
$396$	0	0
$397$	−13.1233	−0.658640	−0.329320	−	0.944218i	$-0.606819\pi$
$397$	−13.1233	−0.658640	−0.329320	+	0.944218i	$0.606819\pi$
$398$	−68.4552	−3.43135
$399$	0	0
$400$	−21.1411	−1.05706
$401$	17.1138	0.854623	0.427311	−	0.904105i	$-0.359461\pi$
$401$	17.1138	0.854623	0.427311	+	0.904105i	$0.359461\pi$
$402$	0	0
$403$	−10.4260	−0.519357
$404$	−9.57398	−0.476323
$405$	0	0
$406$	18.0496	0.895788
$407$	4.86484	0.241141
$408$	0	0
$409$	8.79797	0.435032	0.217516	−	0.976057i	$-0.430205\pi$
$409$	8.79797	0.435032	0.217516	+	0.976057i	$0.430205\pi$
$410$	34.0651	1.68236
$411$	0	0
$412$	55.0360	2.71143
$413$	−6.03003	−0.296718
$414$	0	0
$415$	−16.5963	−0.814679
$416$	12.4875	0.612251
$417$	0	0
$418$	0	0
$419$	−6.84018	−0.334165	−0.167082	−	0.985943i	$-0.553435\pi$
$419$	−6.84018	−0.334165	−0.167082	+	0.985943i	$0.553435\pi$
$420$	0	0
$421$	4.82295	0.235056	0.117528	−	0.993070i	$-0.462503\pi$
$421$	4.82295	0.235056	0.117528	+	0.993070i	$0.462503\pi$
$422$	−20.4388	−0.994946
$423$	0	0
$424$	−17.9736	−0.872875
$425$	12.3550	0.599307
$426$	0	0
$427$	−6.92127	−0.334944
$428$	−29.4688	−1.42443
$429$	0	0
$430$	29.6878	1.43167
$431$	1.30365	0.0627947	0.0313974	−	0.999507i	$-0.490004\pi$
$431$	1.30365	0.0627947	0.0313974	+	0.999507i	$0.490004\pi$
$432$	0	0
$433$	−19.8239	−0.952675	−0.476337	−	0.879263i	$-0.658036\pi$
$433$	−19.8239	−0.952675	−0.476337	+	0.879263i	$0.658036\pi$
$434$	−14.8871	−0.714605
$435$	0	0
$436$	−41.7033	−1.99722
$437$	0	0
$438$	0	0
$439$	34.5672	1.64980	0.824901	−	0.565278i	$-0.191231\pi$
$439$	34.5672	1.64980	0.824901	+	0.565278i	$0.191231\pi$
$440$	−9.74691	−0.464666
$441$	0	0
$442$	−26.6878	−1.26941
$443$	−17.0101	−0.808174	−0.404087	−	0.914720i	$-0.632411\pi$
$443$	−17.0101	−0.808174	−0.404087	+	0.914720i	$0.632411\pi$
$444$	0	0
$445$	−3.26857	−0.154945
$446$	−39.1857	−1.85550
$447$	0	0
$448$	−2.50980	−0.118577
$449$	37.4097	1.76547	0.882737	−	0.469868i	$-0.155698\pi$
$449$	37.4097	1.76547	0.882737	+	0.469868i	$0.155698\pi$
$450$	0	0
$451$	−11.8307	−0.557085
$452$	5.78106	0.271918
$453$	0	0
$454$	−24.9982	−1.17323
$455$	−5.60813	−0.262913
$456$	0	0
$457$	9.11112	0.426200	0.213100	−	0.977030i	$-0.431644\pi$
$457$	9.11112	0.426200	0.213100	+	0.977030i	$0.431644\pi$
$458$	−50.9427	−2.38040
$459$	0	0
$460$	30.0993	1.40339
$461$	24.4483	1.13867	0.569336	−	0.822105i	$-0.307200\pi$
$461$	24.4483	1.13867	0.569336	+	0.822105i	$0.307200\pi$
$462$	0	0
$463$	−0.250725	−0.0116522	−0.00582609	−	0.999983i	$-0.501855\pi$
$463$	−0.250725	−0.0116522	−0.00582609	+	0.999983i	$0.501855\pi$
$464$	−30.8854	−1.43382
$465$	0	0
$466$	8.95037	0.414618
$467$	−15.3618	−0.710861	−0.355431	−	0.934703i	$-0.615666\pi$
$467$	−15.3618	−0.710861	−0.355431	+	0.934703i	$0.615666\pi$
$468$	0	0
$469$	5.95542	0.274996
$470$	1.95811	0.0903209
$471$	0	0
$472$	24.0324	1.10618
$473$	−10.3105	−0.474075
$474$	0	0
$475$	0	0
$476$	−26.2199	−1.20179
$477$	0	0
$478$	30.3182	1.38672
$479$	0.719246	0.0328632	0.0164316	−	0.999865i	$-0.494769\pi$
$479$	0.719246	0.0328632	0.0164316	+	0.999865i	$0.494769\pi$
$480$	0	0
$481$	−11.1557	−0.508656
$482$	32.6682	1.48800
$483$	0	0
$484$	−42.3337	−1.92426
$485$	−9.92902	−0.450853
$486$	0	0
$487$	−11.7469	−0.532303	−0.266152	−	0.963931i	$-0.585752\pi$
$487$	−11.7469	−0.532303	−0.266152	+	0.963931i	$0.585752\pi$
$488$	27.5844	1.24869
$489$	0	0
$490$	15.8726	0.717050
$491$	−0.0888306	−0.00400887	−0.00200443	−	0.999998i	$-0.500638\pi$
$491$	−0.0888306	−0.00400887	−0.00200443	+	0.999998i	$0.500638\pi$
$492$	0	0
$493$	18.0496	0.812914
$494$	0	0
$495$	0	0
$496$	25.4739	1.14381
$497$	−10.6263	−0.476655
$498$	0	0
$499$	14.6905	0.657636	0.328818	−	0.944393i	$-0.393350\pi$
$499$	14.6905	0.657636	0.328818	+	0.944393i	$0.393350\pi$
$500$	−48.6468	−2.17555
$501$	0	0
$502$	36.3678	1.62318
$503$	4.90404	0.218660	0.109330	−	0.994005i	$-0.465129\pi$
$503$	4.90404	0.218660	0.109330	+	0.994005i	$0.465129\pi$
$504$	0	0
$505$	−2.92396	−0.130115
$506$	−15.1925	−0.675390
$507$	0	0
$508$	63.9796	2.83863
$509$	6.41384	0.284288	0.142144	−	0.989846i	$-0.454600\pi$
$509$	6.41384	0.284288	0.142144	+	0.989846i	$0.454600\pi$
$510$	0	0
$511$	9.38919	0.415353
$512$	50.5553	2.23425
$513$	0	0
$514$	−12.6040	−0.555939
$515$	16.8084	0.740667
$516$	0	0
$517$	−0.680045	−0.0299083
$518$	−15.9290	−0.699881
$519$	0	0
$520$	22.3509	0.980153
$521$	−35.8135	−1.56902	−0.784508	−	0.620119i	$-0.787084\pi$
$521$	−35.8135	−1.56902	−0.784508	+	0.620119i	$0.787084\pi$
$522$	0	0
$523$	−38.7725	−1.69540	−0.847701	−	0.530474i	$-0.822014\pi$
$523$	−38.7725	−1.69540	−0.847701	+	0.530474i	$0.822014\pi$
$524$	87.3842	3.81740
$525$	0	0
$526$	−60.8786	−2.65443
$527$	−14.8871	−0.648493
$528$	0	0
$529$	2.64590	0.115039
$530$	−10.0419	−0.436192
$531$	0	0
$532$	0	0
$533$	27.1293	1.17510
$534$	0	0
$535$	−9.00000	−0.389104
$536$	−23.7351	−1.02520
$537$	0	0
$538$	33.1985	1.43129
$539$	−5.51249	−0.237440
$540$	0	0
$541$	9.49020	0.408016	0.204008	−	0.978969i	$-0.434603\pi$
$541$	9.49020	0.408016	0.204008	+	0.978969i	$0.434603\pi$
$542$	67.2772	2.88981
$543$	0	0
$544$	17.8307	0.764484
$545$	−12.7365	−0.545571
$546$	0	0
$547$	−14.2121	−0.607667	−0.303833	−	0.952725i	$-0.598267\pi$
$547$	−14.2121	−0.607667	−0.303833	+	0.952725i	$0.598267\pi$
$548$	−45.0164	−1.92301
$549$	0	0
$550$	9.55438	0.407400
$551$	0	0
$552$	0	0
$553$	−15.0273	−0.639028
$554$	41.8066	1.77619
$555$	0	0
$556$	−7.32501	−0.310650
$557$	−22.5398	−0.955043	−0.477522	−	0.878620i	$-0.658465\pi$
$557$	−22.5398	−0.955043	−0.477522	+	0.878620i	$0.658465\pi$
$558$	0	0
$559$	23.6432	1.00000
$560$	13.7023	0.579029
$561$	0	0
$562$	49.0975	2.07105
$563$	−42.9718	−1.81105	−0.905524	−	0.424296i	$-0.860522\pi$
$563$	−42.9718	−1.81105	−0.905524	+	0.424296i	$0.860522\pi$
$564$	0	0
$565$	1.76558	0.0742784
$566$	28.6391	1.20379
$567$	0	0
$568$	42.3506	1.77699
$569$	7.42696	0.311354	0.155677	−	0.987808i	$-0.450244\pi$
$569$	7.42696	0.311354	0.155677	+	0.987808i	$0.450244\pi$
$570$	0	0
$571$	4.04458	0.169260	0.0846301	−	0.996412i	$-0.473029\pi$
$571$	4.04458	0.169260	0.0846301	+	0.996412i	$0.473029\pi$
$572$	−14.2003	−0.593743
$573$	0	0
$574$	38.7374	1.61687
$575$	−16.1284	−0.672599
$576$	0	0
$577$	3.23442	0.134651	0.0673254	−	0.997731i	$-0.478553\pi$
$577$	3.23442	0.134651	0.0673254	+	0.997731i	$0.478553\pi$
$578$	4.93851	0.205415
$579$	0	0
$580$	−27.6536	−1.14825
$581$	−18.8726	−0.782966
$582$	0	0
$583$	3.48751	0.144438
$584$	−37.4201	−1.54846
$585$	0	0
$586$	−9.87258	−0.407832
$587$	40.8084	1.68434	0.842171	−	0.539210i	$-0.181277\pi$
$587$	40.8084	1.68434	0.842171	+	0.539210i	$0.181277\pi$
$588$	0	0
$589$	0	0
$590$	13.4270	0.552779
$591$	0	0
$592$	27.2567	1.12024
$593$	11.0642	0.454351	0.227176	−	0.973854i	$-0.427051\pi$
$593$	11.0642	0.454351	0.227176	+	0.973854i	$0.427051\pi$
$594$	0	0
$595$	−8.00774	−0.328285
$596$	−49.4397	−2.02513
$597$	0	0
$598$	34.8384	1.42465
$599$	−44.5577	−1.82058	−0.910289	−	0.413974i	$-0.864140\pi$
$599$	−44.5577	−1.82058	−0.910289	+	0.413974i	$0.864140\pi$
$600$	0	0
$601$	4.99907	0.203916	0.101958	−	0.994789i	$-0.467489\pi$
$601$	4.99907	0.203916	0.101958	+	0.994789i	$0.467489\pi$
$602$	33.7597	1.37594
$603$	0	0
$604$	48.7110	1.98202
$605$	−12.9290	−0.525639
$606$	0	0
$607$	31.1881	1.26589	0.632943	−	0.774199i	$-0.281847\pi$
$607$	31.1881	1.26589	0.632943	+	0.774199i	$0.281847\pi$
$608$	0	0
$609$	0	0
$610$	15.4115	0.623992
$611$	1.55943	0.0630878
$612$	0	0
$613$	16.3696	0.661161	0.330581	−	0.943778i	$-0.392755\pi$
$613$	16.3696	0.661161	0.330581	+	0.943778i	$0.392755\pi$
$614$	58.6819	2.36821
$615$	0	0
$616$	−11.0838	−0.446578
$617$	−16.0583	−0.646483	−0.323241	−	0.946317i	$-0.604773\pi$
$617$	−16.0583	−0.646483	−0.323241	+	0.946317i	$0.604773\pi$
$618$	0	0
$619$	23.8425	0.958313	0.479156	−	0.877730i	$-0.340943\pi$
$619$	23.8425	0.958313	0.479156	+	0.877730i	$0.340943\pi$
$620$	22.8084	0.916007
$621$	0	0
$622$	−8.76382	−0.351397
$623$	−3.71688	−0.148914
$624$	0	0
$625$	1.06687	0.0426746
$626$	57.9590	2.31651
$627$	0	0
$628$	48.4962	1.93521
$629$	−15.9290	−0.635131
$630$	0	0
$631$	21.4730	0.854825	0.427413	−	0.904057i	$-0.359425\pi$
$631$	21.4730	0.854825	0.427413	+	0.904057i	$0.359425\pi$
$632$	59.8907	2.38233
$633$	0	0
$634$	−66.1397	−2.62674
$635$	19.5398	0.775414
$636$	0	0
$637$	12.6408	0.500848
$638$	13.9581	0.552607
$639$	0	0
$640$	17.9736	0.710469
$641$	12.7537	0.503742	0.251871	−	0.967761i	$-0.418954\pi$
$641$	12.7537	0.503742	0.251871	+	0.967761i	$0.418954\pi$
$642$	0	0
$643$	28.6081	1.12819	0.564097	−	0.825708i	$-0.309224\pi$
$643$	28.6081	1.12819	0.564097	+	0.825708i	$0.309224\pi$
$644$	34.2276	1.34876
$645$	0	0
$646$	0	0
$647$	−16.7128	−0.657046	−0.328523	−	0.944496i	$-0.606551\pi$
$647$	−16.7128	−0.657046	−0.328523	+	0.944496i	$0.606551\pi$
$648$	0	0
$649$	−4.66313	−0.183044
$650$	−21.9094	−0.859358
$651$	0	0
$652$	−27.9368	−1.09409
$653$	−27.0000	−1.05659	−0.528296	−	0.849060i	$-0.677169\pi$
$653$	−27.0000	−1.05659	−0.528296	+	0.849060i	$0.677169\pi$
$654$	0	0
$655$	26.6878	1.04278
$656$	−66.2850	−2.58799
$657$	0	0
$658$	2.22668	0.0868051
$659$	−43.9009	−1.71013	−0.855067	−	0.518517i	$-0.826484\pi$
$659$	−43.9009	−1.71013	−0.855067	+	0.518517i	$0.826484\pi$
$660$	0	0
$661$	10.7561	0.418363	0.209182	−	0.977877i	$-0.432920\pi$
$661$	10.7561	0.418363	0.209182	+	0.977877i	$0.432920\pi$
$662$	48.2226	1.87422
$663$	0	0
$664$	75.2158	2.91894
$665$	0	0
$666$	0	0
$667$	−23.5621	−0.912329
$668$	−60.7829	−2.35176
$669$	0	0
$670$	−13.2608	−0.512311
$671$	−5.35235	−0.206625
$672$	0	0
$673$	−4.65776	−0.179543	−0.0897717	−	0.995962i	$-0.528614\pi$
$673$	−4.65776	−0.179543	−0.0897717	+	0.995962i	$0.528614\pi$
$674$	4.30810	0.165942
$675$	0	0
$676$	−24.7861	−0.953312
$677$	3.26857	0.125621	0.0628107	−	0.998025i	$-0.479994\pi$
$677$	3.26857	0.125621	0.0628107	+	0.998025i	$0.479994\pi$
$678$	0	0
$679$	−11.2909	−0.433303
$680$	31.9145	1.22386
$681$	0	0
$682$	−11.5125	−0.440836
$683$	6.21894	0.237961	0.118981	−	0.992897i	$-0.462037\pi$
$683$	6.21894	0.237961	0.118981	+	0.992897i	$0.462037\pi$
$684$	0	0
$685$	−13.7483	−0.525297
$686$	45.2053	1.72595
$687$	0	0
$688$	−57.7674	−2.20236
$689$	−7.99731	−0.304673
$690$	0	0
$691$	22.2175	0.845194	0.422597	−	0.906318i	$-0.361119\pi$
$691$	22.2175	0.845194	0.422597	+	0.906318i	$0.361119\pi$
$692$	111.377	4.23393
$693$	0	0
$694$	−12.4115	−0.471133
$695$	−2.23711	−0.0848584
$696$	0	0
$697$	38.7374	1.46728
$698$	71.2259	2.69594
$699$	0	0
$700$	−21.5253	−0.813579
$701$	27.7725	1.04895	0.524476	−	0.851425i	$-0.324261\pi$
$701$	27.7725	1.04895	0.524476	+	0.851425i	$0.324261\pi$
$702$	0	0
$703$	0	0
$704$	−1.94087	−0.0731495
$705$	0	0
$706$	−21.0669	−0.792862
$707$	−3.32501	−0.125050
$708$	0	0
$709$	−6.10782	−0.229384	−0.114692	−	0.993401i	$-0.536588\pi$
$709$	−6.10782	−0.229384	−0.114692	+	0.993401i	$0.536588\pi$
$710$	23.6614	0.887996
$711$	0	0
$712$	14.8135	0.555158
$713$	19.4338	0.727800
$714$	0	0
$715$	−4.33687	−0.162190
$716$	25.7219	0.961274
$717$	0	0
$718$	63.1225	2.35571
$719$	−38.7238	−1.44415	−0.722077	−	0.691813i	$-0.756812\pi$
$719$	−38.7238	−1.44415	−0.722077	+	0.691813i	$0.756812\pi$
$720$	0	0
$721$	19.1138	0.711835
$722$	0	0
$723$	0	0
$724$	−59.8289	−2.22352
$725$	14.8179	0.550323
$726$	0	0
$727$	−11.0779	−0.410857	−0.205428	−	0.978672i	$-0.565859\pi$
$727$	−11.0779	−0.410857	−0.205428	+	0.978672i	$0.565859\pi$
$728$	25.4165	0.941999
$729$	0	0
$730$	−20.9067	−0.773793
$731$	33.7597	1.24865
$732$	0	0
$733$	−15.8075	−0.583862	−0.291931	−	0.956439i	$-0.594298\pi$
$733$	−15.8075	−0.583862	−0.291931	+	0.956439i	$0.594298\pi$
$734$	6.54757	0.241675
$735$	0	0
$736$	−23.2763	−0.857976
$737$	4.60544	0.169643
$738$	0	0
$739$	1.54933	0.0569928	0.0284964	−	0.999594i	$-0.490928\pi$
$739$	1.54933	0.0569928	0.0284964	+	0.999594i	$0.490928\pi$
$740$	24.4047	0.897133
$741$	0	0
$742$	−11.4192	−0.419213
$743$	38.1634	1.40008	0.700040	−	0.714103i	$-0.253165\pi$
$743$	38.1634	1.40008	0.700040	+	0.714103i	$0.253165\pi$
$744$	0	0
$745$	−15.0993	−0.553194
$746$	−59.2086	−2.16778
$747$	0	0
$748$	−20.2763	−0.741375
$749$	−10.2344	−0.373958
$750$	0	0
$751$	25.3482	0.924970	0.462485	−	0.886627i	$-0.346958\pi$
$751$	25.3482	0.924970	0.462485	+	0.886627i	$0.346958\pi$
$752$	−3.81016	−0.138942
$753$	0	0
$754$	−32.0077	−1.16565
$755$	14.8767	0.541418
$756$	0	0
$757$	42.3705	1.53998	0.769991	−	0.638054i	$-0.220260\pi$
$757$	42.3705	1.53998	0.769991	+	0.638054i	$0.220260\pi$
$758$	64.4133	2.33960
$759$	0	0
$760$	0	0
$761$	2.85710	0.103570	0.0517848	−	0.998658i	$-0.483509\pi$
$761$	2.85710	0.103570	0.0517848	+	0.998658i	$0.483509\pi$
$762$	0	0
$763$	−14.4834	−0.524334
$764$	45.3678	1.64135
$765$	0	0
$766$	69.5836	2.51416
$767$	10.6932	0.386108
$768$	0	0
$769$	19.1206	0.689507	0.344754	−	0.938693i	$-0.387963\pi$
$769$	19.1206	0.689507	0.344754	+	0.938693i	$0.387963\pi$
$770$	−6.19253	−0.223163
$771$	0	0
$772$	−60.8813	−2.19116
$773$	2.51485	0.0904530	0.0452265	−	0.998977i	$-0.485599\pi$
$773$	2.51485	0.0904530	0.0452265	+	0.998977i	$0.485599\pi$
$774$	0	0
$775$	−12.2216	−0.439014
$776$	44.9992	1.61538
$777$	0	0
$778$	−8.46286	−0.303408
$779$	0	0
$780$	0	0
$781$	−8.21751	−0.294046
$782$	49.7452	1.77888
$783$	0	0
$784$	−30.8854	−1.10305
$785$	14.8111	0.528630
$786$	0	0
$787$	−2.72605	−0.0971733	−0.0485866	−	0.998819i	$-0.515472\pi$
$787$	−2.72605	−0.0971733	−0.0485866	+	0.998819i	$0.515472\pi$
$788$	35.0310	1.24793
$789$	0	0
$790$	33.4611	1.19049
$791$	2.00774	0.0713870
$792$	0	0
$793$	12.2736	0.435849
$794$	33.2294	1.17927
$795$	0	0
$796$	119.265	4.22722
$797$	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
$799$	2.22668	0.0787743
$800$	14.6382	0.517537
$801$	0	0
$802$	−43.3337	−1.53017
$803$	7.26083	0.256229
$804$	0	0
$805$	10.4534	0.368433
$806$	26.3996	0.929887
$807$	0	0
$808$	13.2517	0.466192
$809$	−54.7205	−1.92387	−0.961935	−	0.273278i	$-0.911892\pi$
$809$	−54.7205	−1.92387	−0.961935	+	0.273278i	$0.911892\pi$
$810$	0	0
$811$	2.31046	0.0811312	0.0405656	−	0.999177i	$-0.487084\pi$
$811$	2.31046	0.0811312	0.0405656	+	0.999177i	$0.487084\pi$
$812$	−31.4466	−1.10356
$813$	0	0
$814$	−12.3182	−0.431753
$815$	−8.53209	−0.298866
$816$	0	0
$817$	0	0
$818$	−22.2772	−0.778906
$819$	0	0
$820$	−59.3492	−2.07256
$821$	−1.11112	−0.0387783	−0.0193892	−	0.999812i	$-0.506172\pi$
$821$	−1.11112	−0.0387783	−0.0193892	+	0.999812i	$0.506172\pi$
$822$	0	0
$823$	20.6477	0.719732	0.359866	−	0.933004i	$-0.382822\pi$
$823$	20.6477	0.719732	0.359866	+	0.933004i	$0.382822\pi$
$824$	−76.1772	−2.65376
$825$	0	0
$826$	15.2686	0.531262
$827$	36.3054	1.26246	0.631231	−	0.775595i	$-0.282550\pi$
$827$	36.3054	1.26246	0.631231	+	0.775595i	$0.282550\pi$
$828$	0	0
$829$	7.14971	0.248320	0.124160	−	0.992262i	$-0.460376\pi$
$829$	7.14971	0.248320	0.124160	+	0.992262i	$0.460376\pi$
$830$	42.0232	1.45865
$831$	0	0
$832$	4.45067	0.154299
$833$	18.0496	0.625383
$834$	0	0
$835$	−18.5635	−0.642418
$836$	0	0
$837$	0	0
$838$	17.3200	0.598308
$839$	−34.6067	−1.19476	−0.597378	−	0.801960i	$-0.703791\pi$
$839$	−34.6067	−1.19476	−0.597378	+	0.801960i	$0.703791\pi$
$840$	0	0
$841$	−7.35235	−0.253529
$842$	−12.2121	−0.420858
$843$	0	0
$844$	35.6091	1.22571
$845$	−7.56986	−0.260411
$846$	0	0
$847$	−14.7023	−0.505178
$848$	19.5398	0.671001
$849$	0	0
$850$	−31.2841	−1.07303
$851$	20.7939	0.712804
$852$	0	0
$853$	33.2508	1.13849	0.569243	−	0.822169i	$-0.307236\pi$
$853$	33.2508	1.13849	0.569243	+	0.822169i	$0.307236\pi$
$854$	17.5253	0.599703
$855$	0	0
$856$	40.7888	1.39413
$857$	−3.88619	−0.132750	−0.0663749	−	0.997795i	$-0.521143\pi$
$857$	−3.88619	−0.132750	−0.0663749	+	0.997795i	$0.521143\pi$
$858$	0	0
$859$	1.65776	0.0565619	0.0282810	−	0.999600i	$-0.490997\pi$
$859$	1.65776	0.0565619	0.0282810	+	0.999600i	$0.490997\pi$
$860$	−51.7229	−1.76374
$861$	0	0
$862$	−3.30096	−0.112431
$863$	−52.7187	−1.79457	−0.897284	−	0.441455i	$-0.854463\pi$
$863$	−52.7187	−1.79457	−0.897284	+	0.441455i	$0.854463\pi$
$864$	0	0
$865$	34.0155	1.15656
$866$	50.1958	1.70572
$867$	0	0
$868$	25.9368	0.880351
$869$	−11.6209	−0.394213
$870$	0	0
$871$	−10.5609	−0.357841
$872$	57.7229	1.95474
$873$	0	0
$874$	0	0
$875$	−16.8949	−0.571151
$876$	0	0
$877$	21.1898	0.715530	0.357765	−	0.933812i	$-0.383539\pi$
$877$	21.1898	0.715530	0.357765	+	0.933812i	$0.383539\pi$
$878$	−87.5271	−2.95390
$879$	0	0
$880$	10.5963	0.357200
$881$	−32.1010	−1.08151	−0.540755	−	0.841180i	$-0.681862\pi$
$881$	−32.1010	−1.08151	−0.540755	+	0.841180i	$0.681862\pi$
$882$	0	0
$883$	−47.2968	−1.59167	−0.795833	−	0.605516i	$-0.792967\pi$
$883$	−47.2968	−1.59167	−0.795833	+	0.605516i	$0.792967\pi$
$884$	46.4962	1.56384
$885$	0	0
$886$	43.0711	1.44700
$887$	10.5631	0.354673	0.177336	−	0.984150i	$-0.443252\pi$
$887$	10.5631	0.354673	0.177336	+	0.984150i	$0.443252\pi$
$888$	0	0
$889$	22.2199	0.745231
$890$	8.27631	0.277423
$891$	0	0
$892$	68.2704	2.28586
$893$	0	0
$894$	0	0
$895$	7.85567	0.262586
$896$	20.4388	0.682813
$897$	0	0
$898$	−94.7247	−3.16101
$899$	−17.8547	−0.595489
$900$	0	0
$901$	−11.4192	−0.380429
$902$	29.9564	0.997438
$903$	0	0
$904$	−8.00175	−0.266134
$905$	−18.2722	−0.607388
$906$	0	0
$907$	−42.9205	−1.42515	−0.712575	−	0.701596i	$-0.752471\pi$
$907$	−42.9205	−1.42515	−0.712575	+	0.701596i	$0.752471\pi$
$908$	43.5526	1.44534
$909$	0	0
$910$	14.2003	0.470735
$911$	−55.1411	−1.82691	−0.913454	−	0.406942i	$-0.866595\pi$
$911$	−55.1411	−1.82691	−0.913454	+	0.406942i	$0.866595\pi$
$912$	0	0
$913$	−14.5945	−0.483008
$914$	−23.0702	−0.763093
$915$	0	0
$916$	88.7538	2.93251
$917$	30.3482	1.00219
$918$	0	0
$919$	−24.5577	−0.810083	−0.405041	−	0.914298i	$-0.632743\pi$
$919$	−24.5577	−0.810083	−0.405041	+	0.914298i	$0.632743\pi$
$920$	−41.6614	−1.37353
$921$	0	0
$922$	−61.9053	−2.03874
$923$	18.8438	0.620251
$924$	0	0
$925$	−13.0770	−0.429968
$926$	0.634858	0.0208627
$927$	0	0
$928$	21.3851	0.701999
$929$	22.2772	0.730893	0.365446	−	0.930832i	$-0.380916\pi$
$929$	22.2772	0.730893	0.365446	+	0.930832i	$0.380916\pi$
$930$	0	0
$931$	0	0
$932$	−15.5936	−0.510785
$933$	0	0
$934$	38.8976	1.27277
$935$	−6.19253	−0.202517
$936$	0	0
$937$	−9.55169	−0.312040	−0.156020	−	0.987754i	$-0.549866\pi$
$937$	−9.55169	−0.312040	−0.156020	+	0.987754i	$0.549866\pi$
$938$	−15.0797	−0.492368
$939$	0	0
$940$	−3.41147	−0.111270
$941$	−55.7256	−1.81660	−0.908301	−	0.418318i	$-0.862620\pi$
$941$	−55.7256	−1.81660	−0.908301	+	0.418318i	$0.862620\pi$
$942$	0	0
$943$	−50.5681	−1.64672
$944$	−26.1266	−0.850348
$945$	0	0
$946$	26.1070	0.848812
$947$	27.0428	0.878774	0.439387	−	0.898298i	$-0.355196\pi$
$947$	27.0428	0.878774	0.439387	+	0.898298i	$0.355196\pi$
$948$	0	0
$949$	−16.6500	−0.540482
$950$	0	0
$951$	0	0
$952$	36.2918	1.17622
$953$	23.1310	0.749288	0.374644	−	0.927169i	$-0.377765\pi$
$953$	23.1310	0.749288	0.374644	+	0.927169i	$0.377765\pi$
$954$	0	0
$955$	13.8557	0.448359
$956$	−52.8212	−1.70836
$957$	0	0
$958$	−1.82119	−0.0588401
$959$	−15.6340	−0.504849
$960$	0	0
$961$	−16.2736	−0.524956
$962$	28.2472	0.910727
$963$	0	0
$964$	−56.9154	−1.83312
$965$	−18.5936	−0.598548
$966$	0	0
$967$	39.0351	1.25528	0.627642	−	0.778502i	$-0.284020\pi$
$967$	39.0351	1.25528	0.627642	+	0.778502i	$0.284020\pi$
$968$	58.5954	1.88333
$969$	0	0
$970$	25.1411	0.807234
$971$	−41.2026	−1.32226	−0.661128	−	0.750273i	$-0.729922\pi$
$971$	−41.2026	−1.32226	−0.661128	+	0.750273i	$0.729922\pi$
$972$	0	0
$973$	−2.54395	−0.0815552
$974$	29.7442	0.953066
$975$	0	0
$976$	−29.9881	−0.959897
$977$	22.4938	0.719641	0.359821	−	0.933022i	$-0.382838\pi$
$977$	22.4938	0.719641	0.359821	+	0.933022i	$0.382838\pi$
$978$	0	0
$979$	−2.87433	−0.0918641
$980$	−27.6536	−0.883363
$981$	0	0
$982$	0.224927	0.00717771
$983$	−44.5461	−1.42080	−0.710401	−	0.703798i	$-0.751486\pi$
$983$	−44.5461	−1.42080	−0.710401	+	0.703798i	$0.751486\pi$
$984$	0	0
$985$	10.6987	0.340889
$986$	−45.7033	−1.45549
$987$	0	0
$988$	0	0
$989$	−44.0702	−1.40135
$990$	0	0
$991$	45.3296	1.43994	0.719971	−	0.694005i	$-0.244155\pi$
$991$	45.3296	1.43994	0.719971	+	0.694005i	$0.244155\pi$
$992$	−17.6382	−0.560012
$993$	0	0
$994$	26.9067	0.853430
$995$	36.4243	1.15473
$996$	0	0
$997$	−10.4911	−0.332258	−0.166129	−	0.986104i	$-0.553127\pi$
$997$	−10.4911	−0.332258	−0.166129	+	0.986104i	$0.553127\pi$
$998$	−37.1976	−1.17747
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3249.2.a.s.1.1		3
3.2	odd	2		361.2.a.h.1.3		3
12.11	even	2		5776.2.a.bi.1.2		3
15.14	odd	2		9025.2.a.x.1.1		3
19.14	odd	18		171.2.u.c.82.1		6
19.15	odd	18		171.2.u.c.73.1		6
19.18	odd	2		3249.2.a.z.1.3		3
57.2	even	18		361.2.e.g.99.1		6
57.5	odd	18		361.2.e.h.234.1		6
57.8	even	6		361.2.c.i.292.3		6
57.11	odd	6		361.2.c.h.292.1		6
57.14	even	18		19.2.e.a.6.1	✓	6
57.17	odd	18		361.2.e.a.99.1		6
57.23	odd	18		361.2.e.h.54.1		6
57.26	odd	6		361.2.c.h.68.1		6
57.29	even	18		361.2.e.g.62.1		6
57.32	even	18		361.2.e.f.245.1		6
57.35	odd	18		361.2.e.b.28.1		6
57.41	even	18		361.2.e.f.28.1		6
57.44	odd	18		361.2.e.b.245.1		6
57.47	odd	18		361.2.e.a.62.1		6
57.50	even	6		361.2.c.i.68.3		6
57.53	even	18		19.2.e.a.16.1	yes	6
57.56	even	2		361.2.a.g.1.1		3
228.71	odd	18		304.2.u.b.177.1		6
228.167	odd	18		304.2.u.b.225.1		6
228.227	odd	2		5776.2.a.br.1.2		3
285.14	even	18		475.2.l.a.101.1		6
285.53	odd	36		475.2.u.a.149.1		12
285.128	odd	36		475.2.u.a.424.2		12
285.167	odd	36		475.2.u.a.149.2		12
285.224	even	18		475.2.l.a.301.1		6
285.242	odd	36		475.2.u.a.424.1		12
285.284	even	2		9025.2.a.bd.1.3		3
399.53	even	18		931.2.x.a.814.1		6
399.110	odd	18		931.2.v.a.263.1		6
399.128	even	18		931.2.x.a.557.1		6
399.167	odd	18		931.2.w.a.491.1		6
399.185	odd	18		931.2.v.a.177.1		6
399.242	even	18		931.2.v.b.177.1		6
399.299	odd	18		931.2.x.b.557.1		6
399.338	even	18		931.2.v.b.263.1		6
399.356	odd	18		931.2.w.a.785.1		6
399.395	odd	18		931.2.x.b.814.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.2.e.a.6.1	✓	6	57.14	even	18
19.2.e.a.16.1	yes	6	57.53	even	18
171.2.u.c.73.1		6	19.15	odd	18
171.2.u.c.82.1		6	19.14	odd	18
304.2.u.b.177.1		6	228.71	odd	18
304.2.u.b.225.1		6	228.167	odd	18
361.2.a.g.1.1		3	57.56	even	2
361.2.a.h.1.3		3	3.2	odd	2
361.2.c.h.68.1		6	57.26	odd	6
361.2.c.h.292.1		6	57.11	odd	6
361.2.c.i.68.3		6	57.50	even	6
361.2.c.i.292.3		6	57.8	even	6
361.2.e.a.62.1		6	57.47	odd	18
361.2.e.a.99.1		6	57.17	odd	18
361.2.e.b.28.1		6	57.35	odd	18
361.2.e.b.245.1		6	57.44	odd	18
361.2.e.f.28.1		6	57.41	even	18
361.2.e.f.245.1		6	57.32	even	18
361.2.e.g.62.1		6	57.29	even	18
361.2.e.g.99.1		6	57.2	even	18
361.2.e.h.54.1		6	57.23	odd	18
361.2.e.h.234.1		6	57.5	odd	18
475.2.l.a.101.1		6	285.14	even	18
475.2.l.a.301.1		6	285.224	even	18
475.2.u.a.149.1		12	285.53	odd	36
475.2.u.a.149.2		12	285.167	odd	36
475.2.u.a.424.1		12	285.242	odd	36
475.2.u.a.424.2		12	285.128	odd	36
931.2.v.a.177.1		6	399.185	odd	18
931.2.v.a.263.1		6	399.110	odd	18
931.2.v.b.177.1		6	399.242	even	18
931.2.v.b.263.1		6	399.338	even	18
931.2.w.a.491.1		6	399.167	odd	18
931.2.w.a.785.1		6	399.356	odd	18
931.2.x.a.557.1		6	399.128	even	18
931.2.x.a.814.1		6	399.53	even	18
931.2.x.b.557.1		6	399.299	odd	18
931.2.x.b.814.1		6	399.395	odd	18
3249.2.a.s.1.1		3	1.1	even	1	trivial
3249.2.a.z.1.3		3	19.18	odd	2
5776.2.a.bi.1.2		3	12.11	even	2
5776.2.a.br.1.2		3	228.227	odd	2
9025.2.a.x.1.1		3	15.14	odd	2
9025.2.a.bd.1.3		3	285.284	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3249 = 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3249.a (trivial)

\(f(q)\)	\(=\)	\(q-2.53209 q^{2} +4.41147 q^{4} +1.34730 q^{5} +1.53209 q^{7} -6.10607 q^{8} +O(q^{10})\)	\(q-2.53209 q^{2} +4.41147 q^{4} +1.34730 q^{5} +1.53209 q^{7} -6.10607 q^{8} -3.41147 q^{10} +1.18479 q^{11} -2.71688 q^{13} -3.87939 q^{14} +6.63816 q^{16} -3.87939 q^{17} +5.94356 q^{20} -3.00000 q^{22} +5.06418 q^{23} -3.18479 q^{25} +6.87939 q^{26} +6.75877 q^{28} -4.65270 q^{29} +3.83750 q^{31} -4.59627 q^{32} +9.82295 q^{34} +2.06418 q^{35} +4.10607 q^{37} -8.22668 q^{40} -9.98545 q^{41} -8.70233 q^{43} +5.22668 q^{44} -12.8229 q^{46} -0.573978 q^{47} -4.65270 q^{49} +8.06418 q^{50} -11.9855 q^{52} +2.94356 q^{53} +1.59627 q^{55} -9.35504 q^{56} +11.7811 q^{58} -3.93582 q^{59} -4.51754 q^{61} -9.71688 q^{62} -1.63816 q^{64} -3.66044 q^{65} +3.88713 q^{67} -17.1138 q^{68} -5.22668 q^{70} -6.93582 q^{71} +6.12836 q^{73} -10.3969 q^{74} +1.81521 q^{77} -9.80840 q^{79} +8.94356 q^{80} +25.2841 q^{82} -12.3182 q^{83} -5.22668 q^{85} +22.0351 q^{86} -7.23442 q^{88} -2.42602 q^{89} -4.16250 q^{91} +22.3405 q^{92} +1.45336 q^{94} -7.36959 q^{97} +11.7811 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - 6 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{8} - 6 q^{14} + 3 q^{16} - 6 q^{17} + 3 q^{20} - 9 q^{22} + 6 q^{23} - 6 q^{25} + 15 q^{26} + 9 q^{28} - 15 q^{29} + 9 q^{31} + 9 q^{34} - 3 q^{35} - 18 q^{40} - 12 q^{41} + 9 q^{44} - 18 q^{46} + 6 q^{47} - 15 q^{49} + 15 q^{50} - 18 q^{52} - 6 q^{53} - 9 q^{55} - 3 q^{56} + 18 q^{58} - 21 q^{59} + 9 q^{61} - 21 q^{62} + 12 q^{64} + 12 q^{65} - 18 q^{67} - 15 q^{68} - 9 q^{70} - 30 q^{71} - 3 q^{74} + 9 q^{77} + 9 q^{79} + 12 q^{80} + 18 q^{82} - 9 q^{85} + 21 q^{86} + 9 q^{88} - 15 q^{89} - 15 q^{91} + 24 q^{92} - 9 q^{94} - 15 q^{97} + 18 q^{98}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - 6 * q^8 - 6 * q^14 + 3 * q^16 - 6 * q^17 + 3 * q^20 - 9 * q^22 + 6 * q^23 - 6 * q^25 + 15 * q^26 + 9 * q^28 - 15 * q^29 + 9 * q^31 + 9 * q^34 - 3 * q^35 - 18 * q^40 - 12 * q^41 + 9 * q^44 - 18 * q^46 + 6 * q^47 - 15 * q^49 + 15 * q^50 - 18 * q^52 - 6 * q^53 - 9 * q^55 - 3 * q^56 + 18 * q^58 - 21 * q^59 + 9 * q^61 - 21 * q^62 + 12 * q^64 + 12 * q^65 - 18 * q^67 - 15 * q^68 - 9 * q^70 - 30 * q^71 - 3 * q^74 + 9 * q^77 + 9 * q^79 + 12 * q^80 + 18 * q^82 - 9 * q^85 + 21 * q^86 + 9 * q^88 - 15 * q^89 - 15 * q^91 + 24 * q^92 - 9 * q^94 - 15 * q^97 + 18 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more