LMFDB - Embedded newform 3249.2.a.bi.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3249,2,Mod(1,3249)] 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("3249.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [6,0,0,12,0,0,-6,0,0,-12,0,0,-18,0,0,-12,0,0,0,0,0,18,0,0,12, 0,0,6,0,0,-24,0,0,-30,0,0,-24,0,0,-42,0,0,18,0,0,-6,0,0,0,0,0,-60,0,0, 0,0,0,-24,0,0,-18,0,0,-30,0,0,0,0,0,-42,0,0,0,0,0,0,0,0,-54,0,0,-66,0, 0,-48,0,0,0,0,0,-18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(91)] == traces)

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

Self dual:	yes
Analytic conductor:	$25.9433956167$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.21415104.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 12x^{4} + 45x^{2} - 51 $ x^6 - 12x^4 + 45x^2 - 51
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 171)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.08502$ 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)

$f(q)$	$=$	$q-2.08502 q^{2} +2.34730 q^{4} +0.724119 q^{5} +1.22668 q^{7} -0.724119 q^{8} -1.50980 q^{10} +1.83353 q^{11} -3.69459 q^{13} -2.55765 q^{14} -3.18479 q^{16} +6.64035 q^{17} +1.69972 q^{20} -3.82295 q^{22} -3.19443 q^{23} -4.47565 q^{25} +7.70329 q^{26} +2.87939 q^{28} -5.02796 q^{29} -2.81521 q^{31} +8.08858 q^{32} -13.8452 q^{34} +0.888263 q^{35} -7.41147 q^{37} -0.524348 q^{40} +10.0863 q^{41} -0.0641778 q^{43} +4.30385 q^{44} +6.66044 q^{46} -9.67063 q^{47} -5.49525 q^{49} +9.33181 q^{50} -8.67230 q^{52} -4.80681 q^{53} +1.32770 q^{55} -0.888263 q^{56} +10.4834 q^{58} -7.58562 q^{59} +4.17024 q^{61} +5.86976 q^{62} -10.4953 q^{64} -2.67532 q^{65} -7.51754 q^{67} +15.5869 q^{68} -1.85204 q^{70} -14.8163 q^{71} +3.90167 q^{73} +15.4531 q^{74} +2.24916 q^{77} -7.61081 q^{79} -2.30617 q^{80} -21.0300 q^{82} +12.3924 q^{83} +4.80840 q^{85} +0.133812 q^{86} -1.32770 q^{88} +13.6195 q^{89} -4.53209 q^{91} -7.49828 q^{92} +20.1634 q^{94} -4.10607 q^{97} +11.4577 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 12 q^{4} - 6 q^{7} - 12 q^{10} - 18 q^{13} - 12 q^{16} + 18 q^{22} + 12 q^{25} + 6 q^{28} - 24 q^{31} - 30 q^{34} - 24 q^{37} - 42 q^{40} + 18 q^{43} - 6 q^{46} - 60 q^{52} - 24 q^{58} - 18 q^{61}+ \cdots - 6 q^{94}+O(q^{100}) $ 6 * q + 12 * q^4 - 6 * q^7 - 12 * q^10 - 18 * q^13 - 12 * q^16 + 18 * q^22 + 12 * q^25 + 6 * q^28 - 24 * q^31 - 30 * q^34 - 24 * q^37 - 42 * q^40 + 18 * q^43 - 6 * q^46 - 60 * q^52 - 24 * q^58 - 18 * q^61 - 30 * q^64 - 42 * q^70 - 54 * q^79 - 66 * q^82 - 48 * q^85 - 18 * q^91 - 6 * q^94

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 $

$2$	−2.08502	−1.47433	−0.737165	−	0.675713i	$-0.763836\pi$
$2$	−2.08502	−1.47433	−0.737165	+	0.675713i	$0.763836\pi$
$3$	0	0
$3$	0	0
$4$	2.34730	1.17365
$5$	0.724119	0.323836	0.161918	−	0.986804i	$-0.448232\pi$
$5$	0.724119	0.323836	0.161918	+	0.986804i	$0.448232\pi$
$6$	0	0
$7$	1.22668	0.463642	0.231821	−	0.972758i	$-0.425532\pi$
$7$	1.22668	0.463642	0.231821	+	0.972758i	$0.425532\pi$
$8$	−0.724119	−0.256015
$9$	0	0
$10$	−1.50980	−0.477441
$11$	1.83353	0.552831	0.276416	−	0.961038i	$-0.410853\pi$
$11$	1.83353	0.552831	0.276416	+	0.961038i	$0.410853\pi$
$12$	0	0
$13$	−3.69459	−1.02470	−0.512348	−	0.858778i	$-0.671224\pi$
$13$	−3.69459	−1.02470	−0.512348	+	0.858778i	$0.671224\pi$
$14$	−2.55765	−0.683561
$15$	0	0
$16$	−3.18479	−0.796198
$17$	6.64035	1.61052	0.805260	−	0.592921i	$-0.202026\pi$
$17$	6.64035	1.61052	0.805260	+	0.592921i	$0.202026\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	1.69972	0.380069
$21$	0	0
$22$	−3.82295	−0.815055
$23$	−3.19443	−0.666085	−0.333043	−	0.942912i	$-0.608075\pi$
$23$	−3.19443	−0.666085	−0.333043	+	0.942912i	$0.608075\pi$
$24$	0	0
$25$	−4.47565	−0.895130
$26$	7.70329	1.51074
$27$	0	0
$28$	2.87939	0.544153
$29$	−5.02796	−0.933670	−0.466835	−	0.884345i	$-0.654606\pi$
$29$	−5.02796	−0.933670	−0.466835	+	0.884345i	$0.654606\pi$
$30$	0	0
$31$	−2.81521	−0.505626	−0.252813	−	0.967515i	$-0.581356\pi$
$31$	−2.81521	−0.505626	−0.252813	+	0.967515i	$0.581356\pi$
$32$	8.08858	1.42987
$33$	0	0
$34$	−13.8452	−2.37444
$35$	0.888263	0.150144
$36$	0	0
$37$	−7.41147	−1.21844	−0.609219	−	0.793002i	$-0.708517\pi$
$37$	−7.41147	−1.21844	−0.609219	+	0.793002i	$0.708517\pi$
$38$	0	0
$39$	0	0
$40$	−0.524348	−0.0829067
$41$	10.0863	1.57521	0.787605	−	0.616181i	$-0.211321\pi$
$41$	10.0863	1.57521	0.787605	+	0.616181i	$0.211321\pi$
$42$	0	0
$43$	−0.0641778	−0.00978702	−0.00489351	−	0.999988i	$-0.501558\pi$
$43$	−0.0641778	−0.00978702	−0.00489351	+	0.999988i	$0.501558\pi$
$44$	4.30385	0.648829
$45$	0	0
$46$	6.66044	0.982029
$47$	−9.67063	−1.41061	−0.705303	−	0.708905i	$-0.749189\pi$
$47$	−9.67063	−1.41061	−0.705303	+	0.708905i	$0.749189\pi$
$48$	0	0
$49$	−5.49525	−0.785036
$50$	9.33181	1.31972
$51$	0	0
$52$	−8.67230	−1.20263
$53$	−4.80681	−0.660267	−0.330133	−	0.943934i	$-0.607094\pi$
$53$	−4.80681	−0.660267	−0.330133	+	0.943934i	$0.607094\pi$
$54$	0	0
$55$	1.32770	0.179026
$56$	−0.888263	−0.118699
$57$	0	0
$58$	10.4834	1.37654
$59$	−7.58562	−0.987563	−0.493782	−	0.869586i	$-0.664386\pi$
$59$	−7.58562	−0.987563	−0.493782	+	0.869586i	$0.664386\pi$
$60$	0	0
$61$	4.17024	0.533945	0.266973	−	0.963704i	$-0.413977\pi$
$61$	4.17024	0.533945	0.266973	+	0.963704i	$0.413977\pi$
$62$	5.86976	0.745460
$63$	0	0
$64$	−10.4953	−1.31191
$65$	−2.67532	−0.331833
$66$	0	0
$67$	−7.51754	−0.918414	−0.459207	−	0.888329i	$-0.651866\pi$
$67$	−7.51754	−0.918414	−0.459207	+	0.888329i	$0.651866\pi$
$68$	15.5869	1.89018
$69$	0	0
$70$	−1.85204	−0.221362
$71$	−14.8163	−1.75837	−0.879184	−	0.476483i	$-0.841912\pi$
$71$	−14.8163	−1.75837	−0.879184	+	0.476483i	$0.841912\pi$
$72$	0	0
$73$	3.90167	0.456656	0.228328	−	0.973584i	$-0.426674\pi$
$73$	3.90167	0.456656	0.228328	+	0.973584i	$0.426674\pi$
$74$	15.4531	1.79638
$75$	0	0
$76$	0	0
$77$	2.24916	0.256316
$78$	0	0
$79$	−7.61081	−0.856284	−0.428142	−	0.903712i	$-0.640832\pi$
$79$	−7.61081	−0.856284	−0.428142	+	0.903712i	$0.640832\pi$
$80$	−2.30617	−0.257837
$81$	0	0
$82$	−21.0300	−2.32238
$83$	12.3924	1.36025	0.680123	−	0.733098i	$-0.261926\pi$
$83$	12.3924	1.36025	0.680123	+	0.733098i	$0.261926\pi$
$84$	0	0
$85$	4.80840	0.521544
$86$	0.133812	0.0144293
$87$	0	0
$88$	−1.32770	−0.141533
$89$	13.6195	1.44367	0.721833	−	0.692067i	$-0.243300\pi$
$89$	13.6195	1.44367	0.721833	+	0.692067i	$0.243300\pi$
$90$	0	0
$91$	−4.53209	−0.475092
$92$	−7.49828	−0.781749
$93$	0	0
$94$	20.1634	2.07970
$95$	0	0
$96$	0	0
$97$	−4.10607	−0.416908	−0.208454	−	0.978032i	$-0.566843\pi$
$97$	−4.10607	−0.416908	−0.208454	+	0.978032i	$0.566843\pi$
$98$	11.4577	1.15740
$99$	0	0
$100$	−10.5057	−1.05057
$101$	−10.6462	−1.05934	−0.529670	−	0.848204i	$-0.677684\pi$
$101$	−10.6462	−1.05934	−0.529670	+	0.848204i	$0.677684\pi$
$102$	0	0
$103$	0.864837	0.0852150	0.0426075	−	0.999092i	$-0.486434\pi$
$103$	0.864837	0.0852150	0.0426075	+	0.999092i	$0.486434\pi$
$104$	2.67532	0.262337
$105$	0	0
$106$	10.0223	0.973451
$107$	−5.61827	−0.543139	−0.271569	−	0.962419i	$-0.587543\pi$
$107$	−5.61827	−0.543139	−0.271569	+	0.962419i	$0.587543\pi$
$108$	0	0
$109$	−7.92902	−0.759462	−0.379731	−	0.925097i	$-0.623983\pi$
$109$	−7.92902	−0.759462	−0.379731	+	0.925097i	$0.623983\pi$
$110$	−2.76827	−0.263944
$111$	0	0
$112$	−3.90673	−0.369151
$113$	9.28534	0.873491	0.436746	−	0.899585i	$-0.356131\pi$
$113$	9.28534	0.873491	0.436746	+	0.899585i	$0.356131\pi$
$114$	0	0
$115$	−2.31315	−0.215702
$116$	−11.8021	−1.09580
$117$	0	0
$118$	15.8161	1.45599
$119$	8.14559	0.746705
$120$	0	0
$121$	−7.63816	−0.694378
$122$	−8.69503	−0.787211
$123$	0	0
$124$	−6.60813	−0.593427
$125$	−6.86150	−0.613711
$126$	0	0
$127$	19.0770	1.69281	0.846404	−	0.532542i	$-0.178763\pi$
$127$	19.0770	1.69281	0.846404	+	0.532542i	$0.178763\pi$
$128$	5.70561	0.504310
$129$	0	0
$130$	5.57810	0.489231
$131$	7.49828	0.655128	0.327564	−	0.944829i	$-0.393772\pi$
$131$	7.49828	0.655128	0.327564	+	0.944829i	$0.393772\pi$
$132$	0	0
$133$	0	0
$134$	15.6742	1.35404
$135$	0	0
$136$	−4.80840	−0.412317
$137$	−11.9198	−1.01838	−0.509188	−	0.860655i	$-0.670054\pi$
$137$	−11.9198	−1.01838	−0.509188	+	0.860655i	$0.670054\pi$
$138$	0	0
$139$	10.0942	0.856179	0.428090	−	0.903736i	$-0.359187\pi$
$139$	10.0942	0.856179	0.428090	+	0.903736i	$0.359187\pi$
$140$	2.08502	0.176216
$141$	0	0
$142$	30.8922	2.59241
$143$	−6.77416	−0.566484
$144$	0	0
$145$	−3.64084	−0.302356
$146$	−8.13506	−0.673262
$147$	0	0
$148$	−17.3969	−1.43002
$149$	5.24912	0.430024	0.215012	−	0.976611i	$-0.431021\pi$
$149$	5.24912	0.430024	0.215012	+	0.976611i	$0.431021\pi$
$150$	0	0
$151$	8.87258	0.722040	0.361020	−	0.932558i	$-0.382429\pi$
$151$	8.87258	0.722040	0.361020	+	0.932558i	$0.382429\pi$
$152$	0	0
$153$	0	0
$154$	−4.68954	−0.377894
$155$	−2.03854	−0.163740
$156$	0	0
$157$	−18.7442	−1.49595	−0.747976	−	0.663726i	$-0.768974\pi$
$157$	−18.7442	−1.49595	−0.747976	+	0.663726i	$0.768974\pi$
$158$	15.8687	1.26244
$159$	0	0
$160$	5.85710	0.463044
$161$	−3.91855	−0.308825
$162$	0	0
$163$	22.4192	1.75601	0.878004	−	0.478653i	$-0.158875\pi$
$163$	22.4192	1.75601	0.878004	+	0.478653i	$0.158875\pi$
$164$	23.6754	1.84874
$165$	0	0
$166$	−25.8384	−2.00545
$167$	−2.89648	−0.224136	−0.112068	−	0.993701i	$-0.535747\pi$
$167$	−2.89648	−0.224136	−0.112068	+	0.993701i	$0.535747\pi$
$168$	0	0
$169$	0.650015	0.0500012
$170$	−10.0256	−0.768928
$171$	0	0
$172$	−0.150644	−0.0114865
$173$	15.3657	1.16823	0.584117	−	0.811670i	$-0.301441\pi$
$173$	15.3657	1.16823	0.584117	+	0.811670i	$0.301441\pi$
$174$	0	0
$175$	−5.49020	−0.415020
$176$	−5.83942	−0.440163
$177$	0	0
$178$	−28.3969	−2.12844
$179$	22.8745	1.70972	0.854861	−	0.518857i	$-0.173642\pi$
$179$	22.8745	1.70972	0.854861	+	0.518857i	$0.173642\pi$
$180$	0	0
$181$	14.0865	1.04704	0.523519	−	0.852014i	$-0.324619\pi$
$181$	14.0865	1.04704	0.523519	+	0.852014i	$0.324619\pi$
$182$	9.44948	0.700442
$183$	0	0
$184$	2.31315	0.170528
$185$	−5.36679	−0.394574
$186$	0	0
$187$	12.1753	0.890346
$188$	−22.6998	−1.65556
$189$	0	0
$190$	0	0
$191$	−23.4078	−1.69373	−0.846865	−	0.531807i	$-0.821513\pi$
$191$	−23.4078	−1.69373	−0.846865	+	0.531807i	$0.821513\pi$
$192$	0	0
$193$	−22.2053	−1.59837	−0.799187	−	0.601082i	$-0.794736\pi$
$193$	−22.2053	−1.59837	−0.799187	+	0.601082i	$0.794736\pi$
$194$	8.56122	0.614660
$195$	0	0
$196$	−12.8990	−0.921356
$197$	−0.415628	−0.0296123	−0.0148061	−	0.999890i	$-0.504713\pi$
$197$	−0.415628	−0.0296123	−0.0148061	+	0.999890i	$0.504713\pi$
$198$	0	0
$199$	−15.6682	−1.11069	−0.555344	−	0.831621i	$-0.687413\pi$
$199$	−15.6682	−1.11069	−0.555344	+	0.831621i	$0.687413\pi$
$200$	3.24090	0.229167
$201$	0	0
$202$	22.1976	1.56182
$203$	−6.16771	−0.432889
$204$	0	0
$205$	7.30365	0.510109
$206$	−1.80320	−0.125635
$207$	0	0
$208$	11.7665	0.815861
$209$	0	0
$210$	0	0
$211$	−13.5175	−0.930586	−0.465293	−	0.885157i	$-0.654051\pi$
$211$	−13.5175	−0.930586	−0.465293	+	0.885157i	$0.654051\pi$
$212$	−11.2830	−0.774921
$213$	0	0
$214$	11.7142	0.800766
$215$	−0.0464723	−0.00316939
$216$	0	0
$217$	−3.45336	−0.234430
$218$	16.5321	1.11970
$219$	0	0
$220$	3.11650	0.210114
$221$	−24.5334	−1.65029
$222$	0	0
$223$	−7.82976	−0.524319	−0.262160	−	0.965025i	$-0.584435\pi$
$223$	−7.82976	−0.524319	−0.262160	+	0.965025i	$0.584435\pi$
$224$	9.92212	0.662949
$225$	0	0
$226$	−19.3601	−1.28781
$227$	19.1666	1.27213	0.636066	−	0.771635i	$-0.280561\pi$
$227$	19.1666	1.27213	0.636066	+	0.771635i	$0.280561\pi$
$228$	0	0
$229$	−5.45336	−0.360368	−0.180184	−	0.983633i	$-0.557669\pi$
$229$	−5.45336	−0.360368	−0.180184	+	0.983633i	$0.557669\pi$
$230$	4.82295	0.318016
$231$	0	0
$232$	3.64084	0.239033
$233$	−11.8895	−0.778905	−0.389452	−	0.921047i	$-0.627336\pi$
$233$	−11.8895	−0.778905	−0.389452	+	0.921047i	$0.627336\pi$
$234$	0	0
$235$	−7.00269	−0.456805
$236$	−17.8057	−1.15905
$237$	0	0
$238$	−16.9837	−1.10089
$239$	−29.5452	−1.91112	−0.955560	−	0.294796i	$-0.904748\pi$
$239$	−29.5452	−1.91112	−0.955560	+	0.294796i	$0.904748\pi$
$240$	0	0
$241$	1.38919	0.0894853	0.0447426	−	0.998999i	$-0.485753\pi$
$241$	1.38919	0.0894853	0.0447426	+	0.998999i	$0.485753\pi$
$242$	15.9257	1.02374
$243$	0	0
$244$	9.78880	0.626664
$245$	−3.97922	−0.254223
$246$	0	0
$247$	0	0
$248$	2.03854	0.129448
$249$	0	0
$250$	14.3063	0.904812
$251$	15.9722	1.00815	0.504077	−	0.863659i	$-0.331833\pi$
$251$	15.9722	1.00815	0.504077	+	0.863659i	$0.331833\pi$
$252$	0	0
$253$	−5.85710	−0.368233
$254$	−39.7758	−2.49576
$255$	0	0
$256$	9.09421	0.568388
$257$	0.174679	0.0108962	0.00544808	−	0.999985i	$-0.498266\pi$
$257$	0.174679	0.0108962	0.00544808	+	0.999985i	$0.498266\pi$
$258$	0	0
$259$	−9.09152	−0.564919
$260$	−6.27978	−0.389455
$261$	0	0
$262$	−15.6340	−0.965874
$263$	6.56354	0.404725	0.202363	−	0.979311i	$-0.435138\pi$
$263$	6.56354	0.404725	0.202363	+	0.979311i	$0.435138\pi$
$264$	0	0
$265$	−3.48070	−0.213818
$266$	0	0
$267$	0	0
$268$	−17.6459	−1.07789
$269$	0.502968	0.0306665	0.0153332	−	0.999882i	$-0.495119\pi$
$269$	0.502968	0.0306665	0.0153332	+	0.999882i	$0.495119\pi$
$270$	0	0
$271$	−6.33275	−0.384687	−0.192344	−	0.981328i	$-0.561609\pi$
$271$	−6.33275	−0.384687	−0.192344	+	0.981328i	$0.561609\pi$
$272$	−21.1481	−1.28229
$273$	0	0
$274$	24.8530	1.50142
$275$	−8.20626	−0.494856
$276$	0	0
$277$	−16.1489	−0.970293	−0.485146	−	0.874433i	$-0.661234\pi$
$277$	−16.1489	−0.970293	−0.485146	+	0.874433i	$0.661234\pi$
$278$	−21.0466	−1.26229
$279$	0	0
$280$	−0.643208	−0.0384390
$281$	12.0839	0.720867	0.360434	−	0.932785i	$-0.382629\pi$
$281$	12.0839	0.720867	0.360434	+	0.932785i	$0.382629\pi$
$282$	0	0
$283$	7.89393	0.469246	0.234623	−	0.972086i	$-0.424615\pi$
$283$	7.89393	0.469246	0.234623	+	0.972086i	$0.424615\pi$
$284$	−34.7782	−2.06371
$285$	0	0
$286$	14.1242	0.835184
$287$	12.3726	0.730333
$288$	0	0
$289$	27.0942	1.59378
$290$	7.59122	0.445772
$291$	0	0
$292$	9.15839	0.535954
$293$	8.08858	0.472540	0.236270	−	0.971687i	$-0.424075\pi$
$293$	8.08858	0.472540	0.236270	+	0.971687i	$0.424075\pi$
$294$	0	0
$295$	−5.49289	−0.319808
$296$	5.36679	0.311938
$297$	0	0
$298$	−10.9445	−0.633998
$299$	11.8021	0.682534
$300$	0	0
$301$	−0.0787257	−0.00453767
$302$	−18.4995	−1.06453
$303$	0	0
$304$	0	0
$305$	3.01975	0.172911
$306$	0	0
$307$	4.24628	0.242348	0.121174	−	0.992631i	$-0.461334\pi$
$307$	4.24628	0.242348	0.121174	+	0.992631i	$0.461334\pi$
$308$	5.27945	0.300825
$309$	0	0
$310$	4.25040	0.241407
$311$	−12.7313	−0.721923	−0.360962	−	0.932581i	$-0.617551\pi$
$311$	−12.7313	−0.721923	−0.360962	+	0.932581i	$0.617551\pi$
$312$	0	0
$313$	−14.5098	−0.820142	−0.410071	−	0.912054i	$-0.634496\pi$
$313$	−14.5098	−0.820142	−0.410071	+	0.912054i	$0.634496\pi$
$314$	39.0820	2.20553
$315$	0	0
$316$	−17.8648	−1.00498
$317$	−21.9454	−1.23258	−0.616288	−	0.787521i	$-0.711364\pi$
$317$	−21.9454	−1.23258	−0.616288	+	0.787521i	$0.711364\pi$
$318$	0	0
$319$	−9.21894	−0.516162
$320$	−7.59981	−0.424842
$321$	0	0
$322$	8.17024	0.455310
$323$	0	0
$324$	0	0
$325$	16.5357	0.917236
$326$	−46.7444	−2.58894
$327$	0	0
$328$	−7.30365	−0.403277
$329$	−11.8628	−0.654017
$330$	0	0
$331$	−15.8571	−0.871585	−0.435792	−	0.900047i	$-0.643532\pi$
$331$	−15.8571	−0.871585	−0.435792	+	0.900047i	$0.643532\pi$
$332$	29.0887	1.59645
$333$	0	0
$334$	6.03920	0.330450
$335$	−5.44359	−0.297415
$336$	0	0
$337$	15.6391	0.851915	0.425958	−	0.904743i	$-0.359937\pi$
$337$	15.6391	0.851915	0.425958	+	0.904743i	$0.359937\pi$
$338$	−1.35529	−0.0737182
$339$	0	0
$340$	11.2867	0.612109
$341$	−5.16178	−0.279526
$342$	0	0
$343$	−15.3277	−0.827618
$344$	0.0464723	0.00250562
$345$	0	0
$346$	−32.0378	−1.72236
$347$	−21.6313	−1.16123	−0.580614	−	0.814179i	$-0.697188\pi$
$347$	−21.6313	−1.16123	−0.580614	+	0.814179i	$0.697188\pi$
$348$	0	0
$349$	15.5466	0.832192	0.416096	−	0.909321i	$-0.363398\pi$
$349$	15.5466	0.832192	0.416096	+	0.909321i	$0.363398\pi$
$350$	11.4472	0.611876
$351$	0	0
$352$	14.8307	0.790478
$353$	−18.3855	−0.978559	−0.489280	−	0.872127i	$-0.662740\pi$
$353$	−18.3855	−0.978559	−0.489280	+	0.872127i	$0.662740\pi$
$354$	0	0
$355$	−10.7287	−0.569422
$356$	31.9690	1.69436
$357$	0	0
$358$	−47.6938	−2.52069
$359$	−23.7219	−1.25200	−0.625998	−	0.779825i	$-0.715308\pi$
$359$	−23.7219	−1.25200	−0.625998	+	0.779825i	$0.715308\pi$
$360$	0	0
$361$	0	0
$362$	−29.3705	−1.54368
$363$	0	0
$364$	−10.6382	−0.557591
$365$	2.82528	0.147882
$366$	0	0
$367$	−12.1634	−0.634926	−0.317463	−	0.948271i	$-0.602831\pi$
$367$	−12.1634	−0.634926	−0.317463	+	0.948271i	$0.602831\pi$
$368$	10.1736	0.530336
$369$	0	0
$370$	11.1898	0.581732
$371$	−5.89643	−0.306127
$372$	0	0
$373$	5.19253	0.268859	0.134430	−	0.990923i	$-0.457080\pi$
$373$	5.19253	0.268859	0.134430	+	0.990923i	$0.457080\pi$
$374$	−25.3857	−1.31266
$375$	0	0
$376$	7.00269	0.361136
$377$	18.5763	0.956727
$378$	0	0
$379$	−32.7128	−1.68034	−0.840171	−	0.542322i	$-0.817545\pi$
$379$	−32.7128	−1.68034	−0.840171	+	0.542322i	$0.817545\pi$
$380$	0	0
$381$	0	0
$382$	48.8057	2.49712
$383$	−14.4774	−0.739763	−0.369882	−	0.929079i	$-0.620602\pi$
$383$	−14.4774	−0.739763	−0.369882	+	0.929079i	$0.620602\pi$
$384$	0	0
$385$	1.62866	0.0830042
$386$	46.2985	2.35653
$387$	0	0
$388$	−9.63816	−0.489303
$389$	−29.1902	−1.48000	−0.740002	−	0.672605i	$-0.765175\pi$
$389$	−29.1902	−1.48000	−0.740002	+	0.672605i	$0.765175\pi$
$390$	0	0
$391$	−21.2121	−1.07274
$392$	3.97922	0.200981
$393$	0	0
$394$	0.866592	0.0436583
$395$	−5.51113	−0.277295
$396$	0	0
$397$	31.1002	1.56087	0.780437	−	0.625234i	$-0.214997\pi$
$397$	31.1002	1.56087	0.780437	+	0.625234i	$0.214997\pi$
$398$	32.6684	1.63752
$399$	0	0
$400$	14.2540	0.712701
$401$	23.8966	1.19334	0.596670	−	0.802487i	$-0.296490\pi$
$401$	23.8966	1.19334	0.596670	+	0.802487i	$0.296490\pi$
$402$	0	0
$403$	10.4010	0.518113
$404$	−24.9899	−1.24329
$405$	0	0
$406$	12.8598	0.638220
$407$	−13.5892	−0.673591
$408$	0	0
$409$	−23.0223	−1.13838	−0.569189	−	0.822206i	$-0.692743\pi$
$409$	−23.0223	−1.13838	−0.569189	+	0.822206i	$0.692743\pi$
$410$	−15.2282	−0.752069
$411$	0	0
$412$	2.03003	0.100012
$413$	−9.30514	−0.457876
$414$	0	0
$415$	8.97359	0.440496
$416$	−29.8840	−1.46518
$417$	0	0
$418$	0	0
$419$	9.38882	0.458674	0.229337	−	0.973347i	$-0.426344\pi$
$419$	9.38882	0.458674	0.229337	+	0.973347i	$0.426344\pi$
$420$	0	0
$421$	−20.2645	−0.987629	−0.493814	−	0.869567i	$-0.664398\pi$
$421$	−20.2645	−0.987629	−0.493814	+	0.869567i	$0.664398\pi$
$422$	28.1843	1.37199
$423$	0	0
$424$	3.48070	0.169038
$425$	−29.7199	−1.44163
$426$	0	0
$427$	5.11556	0.247559
$428$	−13.1877	−0.637454
$429$	0	0
$430$	0.0968956	0.00467272
$431$	10.4982	0.505682	0.252841	−	0.967508i	$-0.418635\pi$
$431$	10.4982	0.505682	0.252841	+	0.967508i	$0.418635\pi$
$432$	0	0
$433$	12.4902	0.600241	0.300120	−	0.953901i	$-0.402973\pi$
$433$	12.4902	0.600241	0.300120	+	0.953901i	$0.402973\pi$
$434$	7.20032	0.345626
$435$	0	0
$436$	−18.6117	−0.891341
$437$	0	0
$438$	0	0
$439$	−19.8794	−0.948791	−0.474396	−	0.880312i	$-0.657333\pi$
$439$	−19.8794	−0.948791	−0.474396	+	0.880312i	$0.657333\pi$
$440$	−0.961410	−0.0458334
$441$	0	0
$442$	51.1525	2.43308
$443$	−22.9619	−1.09095	−0.545476	−	0.838127i	$-0.683651\pi$
$443$	−22.9619	−1.09095	−0.545476	+	0.838127i	$0.683651\pi$
$444$	0	0
$445$	9.86215	0.467511
$446$	16.3252	0.773019
$447$	0	0
$448$	−12.8743	−0.608255
$449$	−8.38654	−0.395785	−0.197893	−	0.980224i	$-0.563410\pi$
$449$	−8.38654	−0.395785	−0.197893	+	0.980224i	$0.563410\pi$
$450$	0	0
$451$	18.4935	0.870825
$452$	21.7954	1.02517
$453$	0	0
$454$	−39.9627	−1.87554
$455$	−3.28177	−0.153852
$456$	0	0
$457$	25.5885	1.19698	0.598490	−	0.801130i	$-0.295767\pi$
$457$	25.5885	1.19698	0.598490	+	0.801130i	$0.295767\pi$
$458$	11.3704	0.531302
$459$	0	0
$460$	−5.42964	−0.253158
$461$	1.77653	0.0827411	0.0413705	−	0.999144i	$-0.486828\pi$
$461$	1.77653	0.0827411	0.0413705	+	0.999144i	$0.486828\pi$
$462$	0	0
$463$	36.3337	1.68857	0.844285	−	0.535895i	$-0.180026\pi$
$463$	36.3337	1.68857	0.844285	+	0.535895i	$0.180026\pi$
$464$	16.0130	0.743386
$465$	0	0
$466$	24.7897	1.14836
$467$	−21.9757	−1.01692	−0.508458	−	0.861087i	$-0.669784\pi$
$467$	−21.9757	−1.01692	−0.508458	+	0.861087i	$0.669784\pi$
$468$	0	0
$469$	−9.22163	−0.425815
$470$	14.6007	0.673481
$471$	0	0
$472$	5.49289	0.252831
$473$	−0.117672	−0.00541057
$474$	0	0
$475$	0	0
$476$	19.1201	0.876369
$477$	0	0
$478$	61.6023	2.81762
$479$	−41.3882	−1.89107	−0.945537	−	0.325513i	$-0.894463\pi$
$479$	−41.3882	−1.89107	−0.945537	+	0.325513i	$0.894463\pi$
$480$	0	0
$481$	27.3824	1.24853
$482$	−2.89648	−0.131931
$483$	0	0
$484$	−17.9290	−0.814955
$485$	−2.97328	−0.135010
$486$	0	0
$487$	−35.6195	−1.61407	−0.807037	−	0.590501i	$-0.798930\pi$
$487$	−35.6195	−1.61407	−0.807037	+	0.590501i	$0.798930\pi$
$488$	−3.01975	−0.136698
$489$	0	0
$490$	8.29673	0.374808
$491$	−11.5506	−0.521273	−0.260637	−	0.965437i	$-0.583932\pi$
$491$	−11.5506	−0.521273	−0.260637	+	0.965437i	$0.583932\pi$
$492$	0	0
$493$	−33.3874	−1.50369
$494$	0	0
$495$	0	0
$496$	8.96585	0.402579
$497$	−18.1748	−0.815253
$498$	0	0
$499$	−17.7023	−0.792465	−0.396233	−	0.918150i	$-0.629683\pi$
$499$	−17.7023	−0.792465	−0.396233	+	0.918150i	$0.629683\pi$
$500$	−16.1060	−0.720281
$501$	0	0
$502$	−33.3022	−1.48635
$503$	24.4764	1.09135	0.545674	−	0.837998i	$-0.316274\pi$
$503$	24.4764	1.09135	0.545674	+	0.837998i	$0.316274\pi$
$504$	0	0
$505$	−7.70914	−0.343052
$506$	12.2121	0.542896
$507$	0	0
$508$	44.7793	1.98676
$509$	−6.82063	−0.302319	−0.151160	−	0.988509i	$-0.548301\pi$
$509$	−6.82063	−0.302319	−0.151160	+	0.988509i	$0.548301\pi$
$510$	0	0
$511$	4.78611	0.211725
$512$	−30.3728	−1.34230
$513$	0	0
$514$	−0.364208	−0.0160645
$515$	0.626245	0.0275957
$516$	0	0
$517$	−17.7314	−0.779827
$518$	18.9560	0.832878
$519$	0	0
$520$	1.93725	0.0849541
$521$	20.4042	0.893925	0.446962	−	0.894553i	$-0.352506\pi$
$521$	20.4042	0.893925	0.446962	+	0.894553i	$0.352506\pi$
$522$	0	0
$523$	6.03508	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$523$	6.03508	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$524$	17.6007	0.768889
$525$	0	0
$526$	−13.6851	−0.596699
$527$	−18.6940	−0.814321
$528$	0	0
$529$	−12.7956	−0.556331
$530$	7.25733	0.315238
$531$	0	0
$532$	0	0
$533$	−37.2646	−1.61411
$534$	0	0
$535$	−4.06830	−0.175888
$536$	5.44359	0.235127
$537$	0	0
$538$	−1.04870	−0.0452125
$539$	−10.0757	−0.433992
$540$	0	0
$541$	−28.4739	−1.22419	−0.612094	−	0.790785i	$-0.709673\pi$
$541$	−28.4739	−1.22419	−0.612094	+	0.790785i	$0.709673\pi$
$542$	13.2039	0.567156
$543$	0	0
$544$	53.7110	2.30284
$545$	−5.74155	−0.245941
$546$	0	0
$547$	22.1702	0.947931	0.473966	−	0.880543i	$-0.342822\pi$
$547$	22.1702	0.947931	0.473966	+	0.880543i	$0.342822\pi$
$548$	−27.9793	−1.19522
$549$	0	0
$550$	17.1102	0.729581
$551$	0	0
$552$	0	0
$553$	−9.33605	−0.397009
$554$	33.6707	1.43053
$555$	0	0
$556$	23.6941	1.00485
$557$	13.5160	0.572693	0.286346	−	0.958126i	$-0.407559\pi$
$557$	13.5160	0.572693	0.286346	+	0.958126i	$0.407559\pi$
$558$	0	0
$559$	0.237111	0.0100287
$560$	−2.82893	−0.119544
$561$	0	0
$562$	−25.1952	−1.06280
$563$	−17.0760	−0.719666	−0.359833	−	0.933017i	$-0.617166\pi$
$563$	−17.0760	−0.719666	−0.359833	+	0.933017i	$0.617166\pi$
$564$	0	0
$565$	6.72369	0.282868
$566$	−16.4590	−0.691823
$567$	0	0
$568$	10.7287	0.450168
$569$	0.262018	0.0109844	0.00549219	−	0.999985i	$-0.498252\pi$
$569$	0.262018	0.0109844	0.00549219	+	0.999985i	$0.498252\pi$
$570$	0	0
$571$	−29.2253	−1.22304	−0.611519	−	0.791229i	$-0.709441\pi$
$571$	−29.2253	−1.22304	−0.611519	+	0.791229i	$0.709441\pi$
$572$	−15.9010	−0.664852
$573$	0	0
$574$	−25.7972	−1.07675
$575$	14.2972	0.596233
$576$	0	0
$577$	−29.2576	−1.21801	−0.609006	−	0.793166i	$-0.708431\pi$
$577$	−29.2576	−1.21801	−0.609006	+	0.793166i	$0.708431\pi$
$578$	−56.4919	−2.34975
$579$	0	0
$580$	−8.54614	−0.354859
$581$	15.2016	0.630667
$582$	0	0
$583$	−8.81345	−0.365016
$584$	−2.82528	−0.116911
$585$	0	0
$586$	−16.8648	−0.696680
$587$	34.6908	1.43184	0.715922	−	0.698180i	$-0.246007\pi$
$587$	34.6908	1.43184	0.715922	+	0.698180i	$0.246007\pi$
$588$	0	0
$589$	0	0
$590$	11.4528	0.471503
$591$	0	0
$592$	23.6040	0.970119
$593$	43.8282	1.79981	0.899904	−	0.436089i	$-0.143637\pi$
$593$	43.8282	1.79981	0.899904	+	0.436089i	$0.143637\pi$
$594$	0	0
$595$	5.89838	0.241810
$596$	12.3212	0.504697
$597$	0	0
$598$	−24.6076	−1.00628
$599$	10.2145	0.417352	0.208676	−	0.977985i	$-0.433085\pi$
$599$	10.2145	0.417352	0.208676	+	0.977985i	$0.433085\pi$
$600$	0	0
$601$	−9.98276	−0.407205	−0.203603	−	0.979054i	$-0.565265\pi$
$601$	−9.98276	−0.407205	−0.203603	+	0.979054i	$0.565265\pi$
$602$	0.164144	0.00669003
$603$	0	0
$604$	20.8266	0.847421
$605$	−5.53093	−0.224864
$606$	0	0
$607$	3.25166	0.131981	0.0659904	−	0.997820i	$-0.478979\pi$
$607$	3.25166	0.131981	0.0659904	+	0.997820i	$0.478979\pi$
$608$	0	0
$609$	0	0
$610$	−6.29624	−0.254927
$611$	35.7291	1.44544
$612$	0	0
$613$	−14.6759	−0.592755	−0.296378	−	0.955071i	$-0.595779\pi$
$613$	−14.6759	−0.592755	−0.296378	+	0.955071i	$0.595779\pi$
$614$	−8.85357	−0.357301
$615$	0	0
$616$	−1.62866	−0.0656206
$617$	−8.55069	−0.344238	−0.172119	−	0.985076i	$-0.555061\pi$
$617$	−8.55069	−0.344238	−0.172119	+	0.985076i	$0.555061\pi$
$618$	0	0
$619$	34.0137	1.36713	0.683564	−	0.729891i	$-0.260429\pi$
$619$	34.0137	1.36713	0.683564	+	0.729891i	$0.260429\pi$
$620$	−4.78507	−0.192173
$621$	0	0
$622$	26.5449	1.06435
$623$	16.7068	0.669344
$624$	0	0
$625$	17.4097	0.696389
$626$	30.2532	1.20916
$627$	0	0
$628$	−43.9982	−1.75572
$629$	−49.2148	−1.96232
$630$	0	0
$631$	−9.67736	−0.385249	−0.192625	−	0.981272i	$-0.561700\pi$
$631$	−9.67736	−0.385249	−0.192625	+	0.981272i	$0.561700\pi$
$632$	5.51113	0.219221
$633$	0	0
$634$	45.7565	1.81722
$635$	13.8140	0.548192
$636$	0	0
$637$	20.3027	0.804423
$638$	19.2216	0.760992
$639$	0	0
$640$	4.13154	0.163313
$641$	−2.90208	−0.114625	−0.0573126	−	0.998356i	$-0.518253\pi$
$641$	−2.90208	−0.114625	−0.0573126	+	0.998356i	$0.518253\pi$
$642$	0	0
$643$	28.2867	1.11552	0.557760	−	0.830002i	$-0.311661\pi$
$643$	28.2867	1.11552	0.557760	+	0.830002i	$0.311661\pi$
$644$	−9.19800	−0.362452
$645$	0	0
$646$	0	0
$647$	37.0738	1.45752	0.728761	−	0.684768i	$-0.240097\pi$
$647$	37.0738	1.45752	0.728761	+	0.684768i	$0.240097\pi$
$648$	0	0
$649$	−13.9085	−0.545956
$650$	−34.4772	−1.35231
$651$	0	0
$652$	52.6245	2.06094
$653$	35.5023	1.38931	0.694656	−	0.719342i	$-0.255557\pi$
$653$	35.5023	1.38931	0.694656	+	0.719342i	$0.255557\pi$
$654$	0	0
$655$	5.42964	0.212154
$656$	−32.1227	−1.25418
$657$	0	0
$658$	24.7341	0.964236
$659$	4.23070	0.164805	0.0824023	−	0.996599i	$-0.473741\pi$
$659$	4.23070	0.164805	0.0824023	+	0.996599i	$0.473741\pi$
$660$	0	0
$661$	25.5381	0.993316	0.496658	−	0.867946i	$-0.334560\pi$
$661$	25.5381	0.993316	0.496658	+	0.867946i	$0.334560\pi$
$662$	33.0623	1.28500
$663$	0	0
$664$	−8.97359	−0.348243
$665$	0	0
$666$	0	0
$667$	16.0615	0.621903
$668$	−6.79889	−0.263057
$669$	0	0
$670$	11.3500	0.438488
$671$	7.64628	0.295181
$672$	0	0
$673$	21.3773	0.824035	0.412018	−	0.911176i	$-0.364824\pi$
$673$	21.3773	0.824035	0.412018	+	0.911176i	$0.364824\pi$
$674$	−32.6078	−1.25600
$675$	0	0
$676$	1.52578	0.0586838
$677$	34.8513	1.33945	0.669723	−	0.742611i	$-0.266413\pi$
$677$	34.8513	1.33945	0.669723	+	0.742611i	$0.266413\pi$
$678$	0	0
$679$	−5.03684	−0.193296
$680$	−3.48185	−0.133523
$681$	0	0
$682$	10.7624	0.412113
$683$	10.8977	0.416990	0.208495	−	0.978023i	$-0.433144\pi$
$683$	10.8977	0.416990	0.208495	+	0.978023i	$0.433144\pi$
$684$	0	0
$685$	−8.63135	−0.329787
$686$	31.9585	1.22018
$687$	0	0
$688$	0.204393	0.00779241
$689$	17.7592	0.676572
$690$	0	0
$691$	8.19253	0.311659	0.155829	−	0.987784i	$-0.450195\pi$
$691$	8.19253	0.311659	0.155829	+	0.987784i	$0.450195\pi$
$692$	36.0679	1.37109
$693$	0	0
$694$	45.1016	1.71203
$695$	7.30941	0.277262
$696$	0	0
$697$	66.9763	2.53691
$698$	−32.4150	−1.22693
$699$	0	0
$700$	−12.8871	−0.487088
$701$	11.7754	0.444753	0.222376	−	0.974961i	$-0.428619\pi$
$701$	11.7754	0.444753	0.222376	+	0.974961i	$0.428619\pi$
$702$	0	0
$703$	0	0
$704$	−19.2434	−0.725263
$705$	0	0
$706$	38.3340	1.44272
$707$	−13.0595	−0.491155
$708$	0	0
$709$	25.9668	0.975203	0.487602	−	0.873066i	$-0.337872\pi$
$709$	25.9668	0.975203	0.487602	+	0.873066i	$0.337872\pi$
$710$	22.3696	0.839516
$711$	0	0
$712$	−9.86215	−0.369600
$713$	8.99299	0.336790
$714$	0	0
$715$	−4.90530	−0.183448
$716$	53.6933	2.00661
$717$	0	0
$718$	49.4606	1.84585
$719$	16.1468	0.602175	0.301088	−	0.953596i	$-0.402650\pi$
$719$	16.1468	0.602175	0.301088	+	0.953596i	$0.402650\pi$
$720$	0	0
$721$	1.06088	0.0395092
$722$	0	0
$723$	0	0
$724$	33.0651	1.22886
$725$	22.5034	0.835756
$726$	0	0
$727$	−1.59358	−0.0591025	−0.0295513	−	0.999563i	$-0.509408\pi$
$727$	−1.59358	−0.0591025	−0.0295513	+	0.999563i	$0.509408\pi$
$728$	3.28177	0.121631
$729$	0	0
$730$	−5.89075	−0.218026
$731$	−0.426163	−0.0157622
$732$	0	0
$733$	11.9831	0.442605	0.221303	−	0.975205i	$-0.428969\pi$
$733$	11.9831	0.442605	0.221303	+	0.975205i	$0.428969\pi$
$734$	25.3610	0.936091
$735$	0	0
$736$	−25.8384	−0.952417
$737$	−13.7837	−0.507728
$738$	0	0
$739$	−33.3500	−1.22680	−0.613400	−	0.789773i	$-0.710198\pi$
$739$	−33.3500	−1.22680	−0.613400	+	0.789773i	$0.710198\pi$
$740$	−12.5974	−0.463091
$741$	0	0
$742$	12.2942	0.451333
$743$	8.70922	0.319510	0.159755	−	0.987157i	$-0.448930\pi$
$743$	8.70922	0.319510	0.159755	+	0.987157i	$0.448930\pi$
$744$	0	0
$745$	3.80098	0.139257
$746$	−10.8265	−0.396387
$747$	0	0
$748$	28.5790	1.04495
$749$	−6.89183	−0.251822
$750$	0	0
$751$	−18.8607	−0.688237	−0.344119	−	0.938926i	$-0.611822\pi$
$751$	−18.8607	−0.688237	−0.344119	+	0.938926i	$0.611822\pi$
$752$	30.7990	1.12312
$753$	0	0
$754$	−38.7319	−1.41053
$755$	6.42480	0.233822
$756$	0	0
$757$	10.0273	0.364450	0.182225	−	0.983257i	$-0.441670\pi$
$757$	10.0273	0.364450	0.182225	+	0.983257i	$0.441670\pi$
$758$	68.2067	2.47738
$759$	0	0
$760$	0	0
$761$	10.0398	0.363942	0.181971	−	0.983304i	$-0.441752\pi$
$761$	10.0398	0.363942	0.181971	+	0.983304i	$0.441752\pi$
$762$	0	0
$763$	−9.72638	−0.352118
$764$	−54.9451	−1.98784
$765$	0	0
$766$	30.1857	1.09065
$767$	28.0258	1.01195
$768$	0	0
$769$	−40.1121	−1.44648	−0.723239	−	0.690598i	$-0.757347\pi$
$769$	−40.1121	−1.44648	−0.723239	+	0.690598i	$0.757347\pi$
$770$	−3.39578	−0.122376
$771$	0	0
$772$	−52.1225	−1.87593
$773$	40.3556	1.45149	0.725745	−	0.687964i	$-0.241495\pi$
$773$	40.3556	1.45149	0.725745	+	0.687964i	$0.241495\pi$
$774$	0	0
$775$	12.5999	0.452601
$776$	2.97328	0.106735
$777$	0	0
$778$	60.8621	2.18201
$779$	0	0
$780$	0	0
$781$	−27.1661	−0.972080
$782$	44.2277	1.58158
$783$	0	0
$784$	17.5012	0.625044
$785$	−13.5730	−0.484443
$786$	0	0
$787$	−48.8367	−1.74084	−0.870420	−	0.492310i	$-0.836153\pi$
$787$	−48.8367	−1.74084	−0.870420	+	0.492310i	$0.836153\pi$
$788$	−0.975603	−0.0347544
$789$	0	0
$790$	11.4908	0.408825
$791$	11.3902	0.404987
$792$	0	0
$793$	−15.4074	−0.547131
$794$	−64.8444	−2.30124
$795$	0	0
$796$	−36.7779	−1.30356
$797$	12.7171	0.450461	0.225231	−	0.974305i	$-0.427686\pi$
$797$	12.7171	0.450461	0.225231	+	0.974305i	$0.427686\pi$
$798$	0	0
$799$	−64.2164	−2.27181
$800$	−36.2017	−1.27992
$801$	0	0
$802$	−49.8248	−1.75938
$803$	7.15385	0.252454
$804$	0	0
$805$	−2.83750	−0.100009
$806$	−21.6864	−0.763869
$807$	0	0
$808$	7.70914	0.271207
$809$	11.0513	0.388544	0.194272	−	0.980948i	$-0.437766\pi$
$809$	11.0513	0.388544	0.194272	+	0.980948i	$0.437766\pi$
$810$	0	0
$811$	11.1857	0.392784	0.196392	−	0.980525i	$-0.437077\pi$
$811$	11.1857	0.392784	0.196392	+	0.980525i	$0.437077\pi$
$812$	−14.4774	−0.508059
$813$	0	0
$814$	28.3337	0.993095
$815$	16.2342	0.568658
$816$	0	0
$817$	0	0
$818$	48.0019	1.67835
$819$	0	0
$820$	17.1438	0.598689
$821$	−1.38757	−0.0484266	−0.0242133	−	0.999707i	$-0.507708\pi$
$821$	−1.38757	−0.0484266	−0.0242133	+	0.999707i	$0.507708\pi$
$822$	0	0
$823$	10.1875	0.355113	0.177557	−	0.984111i	$-0.443181\pi$
$823$	10.1875	0.355113	0.177557	+	0.984111i	$0.443181\pi$
$824$	−0.626245	−0.0218163
$825$	0	0
$826$	19.4014	0.675060
$827$	−24.0551	−0.836479	−0.418240	−	0.908337i	$-0.637353\pi$
$827$	−24.0551	−0.836479	−0.418240	+	0.908337i	$0.637353\pi$
$828$	0	0
$829$	46.4424	1.61301	0.806506	−	0.591226i	$-0.201356\pi$
$829$	46.4424	1.61301	0.806506	+	0.591226i	$0.201356\pi$
$830$	−18.7101	−0.649437
$831$	0	0
$832$	38.7757	1.34430
$833$	−36.4904	−1.26432
$834$	0	0
$835$	−2.09739	−0.0725833
$836$	0	0
$837$	0	0
$838$	−19.5758	−0.676236
$839$	−9.08033	−0.313488	−0.156744	−	0.987639i	$-0.550100\pi$
$839$	−9.08033	−0.313488	−0.156744	+	0.987639i	$0.550100\pi$
$840$	0	0
$841$	−3.71957	−0.128261
$842$	42.2517	1.45609
$843$	0	0
$844$	−31.7297	−1.09218
$845$	0.470688	0.0161922
$846$	0	0
$847$	−9.36959	−0.321943
$848$	15.3087	0.525703
$849$	0	0
$850$	61.9665	2.12543
$851$	23.6754	0.811584
$852$	0	0
$853$	37.2463	1.27529	0.637644	−	0.770331i	$-0.279909\pi$
$853$	37.2463	1.27529	0.637644	+	0.770331i	$0.279909\pi$
$854$	−10.6660	−0.364984
$855$	0	0
$856$	4.06830	0.139052
$857$	−21.2157	−0.724713	−0.362357	−	0.932039i	$-0.618028\pi$
$857$	−21.2157	−0.724713	−0.362357	+	0.932039i	$0.618028\pi$
$858$	0	0
$859$	7.65776	0.261279	0.130640	−	0.991430i	$-0.458297\pi$
$859$	7.65776	0.261279	0.130640	+	0.991430i	$0.458297\pi$
$860$	−0.109084	−0.00371975
$861$	0	0
$862$	−21.8890	−0.745542
$863$	−33.6954	−1.14701	−0.573503	−	0.819203i	$-0.694416\pi$
$863$	−33.6954	−1.14701	−0.573503	+	0.819203i	$0.694416\pi$
$864$	0	0
$865$	11.1266	0.378316
$866$	−26.0423	−0.884953
$867$	0	0
$868$	−8.10607	−0.275138
$869$	−13.9547	−0.473380
$870$	0	0
$871$	27.7743	0.941095
$872$	5.74155	0.194433
$873$	0	0
$874$	0	0
$875$	−8.41687	−0.284542
$876$	0	0
$877$	5.28218	0.178367	0.0891834	−	0.996015i	$-0.471574\pi$
$877$	5.28218	0.178367	0.0891834	+	0.996015i	$0.471574\pi$
$878$	41.4489	1.39883
$879$	0	0
$880$	−4.22844	−0.142541
$881$	50.7807	1.71084	0.855422	−	0.517931i	$-0.173298\pi$
$881$	50.7807	1.71084	0.855422	+	0.517931i	$0.173298\pi$
$882$	0	0
$883$	−9.77870	−0.329079	−0.164540	−	0.986370i	$-0.552614\pi$
$883$	−9.77870	−0.329079	−0.164540	+	0.986370i	$0.552614\pi$
$884$	−57.5871	−1.93686
$885$	0	0
$886$	47.8759	1.60842
$887$	20.8360	0.699604	0.349802	−	0.936824i	$-0.386249\pi$
$887$	20.8360	0.699604	0.349802	+	0.936824i	$0.386249\pi$
$888$	0	0
$889$	23.4014	0.784857
$890$	−20.5627	−0.689265
$891$	0	0
$892$	−18.3788	−0.615366
$893$	0	0
$894$	0	0
$895$	16.5639	0.553669
$896$	6.99897	0.233819
$897$	0	0
$898$	17.4861	0.583518
$899$	14.1548	0.472088
$900$	0	0
$901$	−31.9189	−1.06337
$902$	−38.5593	−1.28388
$903$	0	0
$904$	−6.72369	−0.223627
$905$	10.2003	0.339069
$906$	0	0
$907$	33.6067	1.11589	0.557946	−	0.829877i	$-0.311590\pi$
$907$	33.6067	1.11589	0.557946	+	0.829877i	$0.311590\pi$
$908$	44.9897	1.49303
$909$	0	0
$910$	6.84255	0.226828
$911$	0.821993	0.0272338	0.0136169	−	0.999907i	$-0.495665\pi$
$911$	0.821993	0.0272338	0.0136169	+	0.999907i	$0.495665\pi$
$912$	0	0
$913$	22.7219	0.751986
$914$	−53.3525	−1.76474
$915$	0	0
$916$	−12.8007	−0.422946
$917$	9.19800	0.303745
$918$	0	0
$919$	6.14115	0.202578	0.101289	−	0.994857i	$-0.467703\pi$
$919$	6.14115	0.202578	0.101289	+	0.994857i	$0.467703\pi$
$920$	1.67499	0.0552229
$921$	0	0
$922$	−3.70409	−0.121988
$923$	54.7401	1.80179
$924$	0	0
$925$	33.1712	1.09066
$926$	−75.7563	−2.48951
$927$	0	0
$928$	−40.6691	−1.33503
$929$	7.09684	0.232840	0.116420	−	0.993200i	$-0.462858\pi$
$929$	7.09684	0.232840	0.116420	+	0.993200i	$0.462858\pi$
$930$	0	0
$931$	0	0
$932$	−27.9081	−0.914160
$933$	0	0
$934$	45.8198	1.49927
$935$	8.81636	0.288326
$936$	0	0
$937$	−11.7784	−0.384783	−0.192391	−	0.981318i	$-0.561624\pi$
$937$	−11.7784	−0.384783	−0.192391	+	0.981318i	$0.561624\pi$
$938$	19.2273	0.627792
$939$	0	0
$940$	−16.4374	−0.536128
$941$	17.9234	0.584285	0.292142	−	0.956375i	$-0.405632\pi$
$941$	17.9234	0.584285	0.292142	+	0.956375i	$0.405632\pi$
$942$	0	0
$943$	−32.2199	−1.04922
$944$	24.1586	0.786296
$945$	0	0
$946$	0.245348	0.00797696
$947$	−39.0412	−1.26867	−0.634334	−	0.773059i	$-0.718725\pi$
$947$	−39.0412	−1.26867	−0.634334	+	0.773059i	$0.718725\pi$
$948$	0	0
$949$	−14.4151	−0.467934
$950$	0	0
$951$	0	0
$952$	−5.89838	−0.191167
$953$	7.71943	0.250057	0.125028	−	0.992153i	$-0.460098\pi$
$953$	7.71943	0.250057	0.125028	+	0.992153i	$0.460098\pi$
$954$	0	0
$955$	−16.9500	−0.548491
$956$	−69.3513	−2.24298
$957$	0	0
$958$	86.2951	2.78807
$959$	−14.6218	−0.472162
$960$	0	0
$961$	−23.0746	−0.744342
$962$	−57.0927	−1.84074
$963$	0	0
$964$	3.26083	0.105024
$965$	−16.0793	−0.517611
$966$	0	0
$967$	−35.1976	−1.13188	−0.565939	−	0.824447i	$-0.691486\pi$
$967$	−35.1976	−1.13188	−0.565939	+	0.824447i	$0.691486\pi$
$968$	5.53093	0.177771
$969$	0	0
$970$	6.19934	0.199049
$971$	1.40177	0.0449848	0.0224924	−	0.999747i	$-0.492840\pi$
$971$	1.40177	0.0449848	0.0224924	+	0.999747i	$0.492840\pi$
$972$	0	0
$973$	12.3824	0.396961
$974$	74.2672	2.37968
$975$	0	0
$976$	−13.2814	−0.425126
$977$	−16.0737	−0.514243	−0.257121	−	0.966379i	$-0.582774\pi$
$977$	−16.0737	−0.514243	−0.257121	+	0.966379i	$0.582774\pi$
$978$	0	0
$979$	24.9718	0.798103
$980$	−9.34040	−0.298368
$981$	0	0
$982$	24.0833	0.768528
$983$	2.23302	0.0712223	0.0356112	−	0.999366i	$-0.488662\pi$
$983$	2.23302	0.0712223	0.0356112	+	0.999366i	$0.488662\pi$
$984$	0	0
$985$	−0.300964	−0.00958952
$986$	69.6134	2.21694
$987$	0	0
$988$	0	0
$989$	0.205012	0.00651899
$990$	0	0
$991$	−19.6108	−0.622958	−0.311479	−	0.950253i	$-0.600824\pi$
$991$	−19.6108	−0.622958	−0.311479	+	0.950253i	$0.600824\pi$
$992$	−22.7710	−0.722981
$993$	0	0
$994$	37.8949	1.20195
$995$	−11.3456	−0.359681
$996$	0	0
$997$	7.92633	0.251029	0.125515	−	0.992092i	$-0.459942\pi$
$997$	7.92633	0.251029	0.125515	+	0.992092i	$0.459942\pi$
$998$	36.9097	1.16836
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3249.2.a.bi.1.2		6
3.2	odd	2	inner	3249.2.a.bi.1.5		6
19.9	even	9		171.2.u.d.100.2	yes	12
19.17	even	9		171.2.u.d.118.2	yes	12
19.18	odd	2		3249.2.a.bj.1.5		6
57.17	odd	18		171.2.u.d.118.1	yes	12
57.47	odd	18		171.2.u.d.100.1	✓	12
57.56	even	2		3249.2.a.bj.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
171.2.u.d.100.1	✓	12	57.47	odd	18
171.2.u.d.100.2	yes	12	19.9	even	9
171.2.u.d.118.1	yes	12	57.17	odd	18
171.2.u.d.118.2	yes	12	19.17	even	9
3249.2.a.bi.1.2		6	1.1	even	1	trivial
3249.2.a.bi.1.5		6	3.2	odd	2	inner
3249.2.a.bj.1.2		6	57.56	even	2
3249.2.a.bj.1.5		6	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:	\( N \)	\(=\)	\( 3249 = 3^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3249.a (trivial)

\(f(q)\)	\(=\)	\(q-2.08502 q^{2} +2.34730 q^{4} +0.724119 q^{5} +1.22668 q^{7} -0.724119 q^{8} -1.50980 q^{10} +1.83353 q^{11} -3.69459 q^{13} -2.55765 q^{14} -3.18479 q^{16} +6.64035 q^{17} +1.69972 q^{20} -3.82295 q^{22} -3.19443 q^{23} -4.47565 q^{25} +7.70329 q^{26} +2.87939 q^{28} -5.02796 q^{29} -2.81521 q^{31} +8.08858 q^{32} -13.8452 q^{34} +0.888263 q^{35} -7.41147 q^{37} -0.524348 q^{40} +10.0863 q^{41} -0.0641778 q^{43} +4.30385 q^{44} +6.66044 q^{46} -9.67063 q^{47} -5.49525 q^{49} +9.33181 q^{50} -8.67230 q^{52} -4.80681 q^{53} +1.32770 q^{55} -0.888263 q^{56} +10.4834 q^{58} -7.58562 q^{59} +4.17024 q^{61} +5.86976 q^{62} -10.4953 q^{64} -2.67532 q^{65} -7.51754 q^{67} +15.5869 q^{68} -1.85204 q^{70} -14.8163 q^{71} +3.90167 q^{73} +15.4531 q^{74} +2.24916 q^{77} -7.61081 q^{79} -2.30617 q^{80} -21.0300 q^{82} +12.3924 q^{83} +4.80840 q^{85} +0.133812 q^{86} -1.32770 q^{88} +13.6195 q^{89} -4.53209 q^{91} -7.49828 q^{92} +20.1634 q^{94} -4.10607 q^{97} +11.4577 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 12 q^{4} - 6 q^{7} - 12 q^{10} - 18 q^{13} - 12 q^{16} + 18 q^{22} + 12 q^{25} + 6 q^{28} - 24 q^{31} - 30 q^{34} - 24 q^{37} - 42 q^{40} + 18 q^{43} - 6 q^{46} - 60 q^{52} - 24 q^{58} - 18 q^{61}+ \cdots - 6 q^{94}+O(q^{100}) \) 6 * q + 12 * q^4 - 6 * q^7 - 12 * q^10 - 18 * q^13 - 12 * q^16 + 18 * q^22 + 12 * q^25 + 6 * q^28 - 24 * q^31 - 30 * q^34 - 24 * q^37 - 42 * q^40 + 18 * q^43 - 6 * q^46 - 60 * q^52 - 24 * q^58 - 18 * q^61 - 30 * q^64 - 42 * q^70 - 54 * q^79 - 66 * q^82 - 48 * q^85 - 18 * q^91 - 6 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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