LMFDB - Embedded newform 1215.2.a.k.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1215,2,Mod(1,1215)] 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("1215.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1215, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1215 = 3^{5} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1215.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,-1,0,3,2,0,-1,-6,0,-1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$9.70182384559$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	1215.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.30278 q^{2} +3.30278 q^{4} +1.00000 q^{5} +1.30278 q^{7} -3.00000 q^{8} -2.30278 q^{10} -2.30278 q^{11} +3.60555 q^{13} -3.00000 q^{14} +0.302776 q^{16} -5.30278 q^{17} -6.30278 q^{19} +3.30278 q^{20} +5.30278 q^{22} -3.00000 q^{23} +1.00000 q^{25} -8.30278 q^{26} +4.30278 q^{28} -7.60555 q^{29} +5.21110 q^{31} +5.30278 q^{32} +12.2111 q^{34} +1.30278 q^{35} -7.90833 q^{37} +14.5139 q^{38} -3.00000 q^{40} -6.69722 q^{41} -7.21110 q^{43} -7.60555 q^{44} +6.90833 q^{46} +6.69722 q^{47} -5.30278 q^{49} -2.30278 q^{50} +11.9083 q^{52} +8.51388 q^{53} -2.30278 q^{55} -3.90833 q^{56} +17.5139 q^{58} -1.39445 q^{59} +0.605551 q^{61} -12.0000 q^{62} -12.8167 q^{64} +3.60555 q^{65} +10.3028 q^{67} -17.5139 q^{68} -3.00000 q^{70} -1.60555 q^{71} +12.6056 q^{73} +18.2111 q^{74} -20.8167 q^{76} -3.00000 q^{77} -12.5139 q^{79} +0.302776 q^{80} +15.4222 q^{82} +14.7250 q^{83} -5.30278 q^{85} +16.6056 q^{86} +6.90833 q^{88} +4.81665 q^{89} +4.69722 q^{91} -9.90833 q^{92} -15.4222 q^{94} -6.30278 q^{95} -15.7250 q^{97} +12.2111 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{2} + 3 q^{4} + 2 q^{5} - q^{7} - 6 q^{8} - q^{10} - q^{11} - 6 q^{14} - 3 q^{16} - 7 q^{17} - 9 q^{19} + 3 q^{20} + 7 q^{22} - 6 q^{23} + 2 q^{25} - 13 q^{26} + 5 q^{28} - 8 q^{29} - 4 q^{31}+ \cdots + 10 q^{98}+O(q^{100}) $ 2 * q - q^2 + 3 * q^4 + 2 * q^5 - q^7 - 6 * q^8 - q^10 - q^11 - 6 * q^14 - 3 * q^16 - 7 * q^17 - 9 * q^19 + 3 * q^20 + 7 * q^22 - 6 * q^23 + 2 * q^25 - 13 * q^26 + 5 * q^28 - 8 * q^29 - 4 * q^31 + 7 * q^32 + 10 * q^34 - q^35 - 5 * q^37 + 11 * q^38 - 6 * q^40 - 17 * q^41 - 8 * q^44 + 3 * q^46 + 17 * q^47 - 7 * q^49 - q^50 + 13 * q^52 - q^53 - q^55 + 3 * q^56 + 17 * q^58 - 10 * q^59 - 6 * q^61 - 24 * q^62 - 4 * q^64 + 17 * q^67 - 17 * q^68 - 6 * q^70 + 4 * q^71 + 18 * q^73 + 22 * q^74 - 20 * q^76 - 6 * q^77 - 7 * q^79 - 3 * q^80 + 2 * q^82 - 3 * q^83 - 7 * q^85 + 26 * q^86 + 3 * q^88 - 12 * q^89 + 13 * q^91 - 9 * q^92 - 2 * q^94 - 9 * q^95 + q^97 + 10 * q^98

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.30278	−1.62831	−0.814154	−	0.580649i	$-0.802799\pi$
$2$	−2.30278	−1.62831	−0.814154	+	0.580649i	$0.802799\pi$
$3$	0	0
$3$	0	0
$4$	3.30278	1.65139
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.30278	0.492403	0.246201	−	0.969219i	$-0.420818\pi$
$7$	1.30278	0.492403	0.246201	+	0.969219i	$0.420818\pi$
$8$	−3.00000	−1.06066
$9$	0	0
$10$	−2.30278	−0.728202
$11$	−2.30278	−0.694313	−0.347156	−	0.937807i	$-0.612853\pi$
$11$	−2.30278	−0.694313	−0.347156	+	0.937807i	$0.612853\pi$
$12$	0	0
$13$	3.60555	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$13$	3.60555	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$14$	−3.00000	−0.801784
$15$	0	0
$16$	0.302776	0.0756939
$17$	−5.30278	−1.28611	−0.643056	−	0.765819i	$-0.722334\pi$
$17$	−5.30278	−1.28611	−0.643056	+	0.765819i	$0.722334\pi$
$18$	0	0
$19$	−6.30278	−1.44596	−0.722978	−	0.690871i	$-0.757227\pi$
$19$	−6.30278	−1.44596	−0.722978	+	0.690871i	$0.757227\pi$
$20$	3.30278	0.738523
$21$	0	0
$22$	5.30278	1.13056
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−8.30278	−1.62831
$27$	0	0
$28$	4.30278	0.813148
$29$	−7.60555	−1.41232	−0.706158	−	0.708055i	$-0.749573\pi$
$29$	−7.60555	−1.41232	−0.706158	+	0.708055i	$0.749573\pi$
$30$	0	0
$31$	5.21110	0.935942	0.467971	−	0.883744i	$-0.344985\pi$
$31$	5.21110	0.935942	0.467971	+	0.883744i	$0.344985\pi$
$32$	5.30278	0.937407
$33$	0	0
$34$	12.2111	2.09419
$35$	1.30278	0.220209
$36$	0	0
$37$	−7.90833	−1.30012	−0.650060	−	0.759882i	$-0.725256\pi$
$37$	−7.90833	−1.30012	−0.650060	+	0.759882i	$0.725256\pi$
$38$	14.5139	2.35446
$39$	0	0
$40$	−3.00000	−0.474342
$41$	−6.69722	−1.04593	−0.522965	−	0.852354i	$-0.675174\pi$
$41$	−6.69722	−1.04593	−0.522965	+	0.852354i	$0.675174\pi$
$42$	0	0
$43$	−7.21110	−1.09968	−0.549841	−	0.835269i	$-0.685312\pi$
$43$	−7.21110	−1.09968	−0.549841	+	0.835269i	$0.685312\pi$
$44$	−7.60555	−1.14658
$45$	0	0
$46$	6.90833	1.01858
$47$	6.69722	0.976891	0.488445	−	0.872595i	$-0.337564\pi$
$47$	6.69722	0.976891	0.488445	+	0.872595i	$0.337564\pi$
$48$	0	0
$49$	−5.30278	−0.757539
$50$	−2.30278	−0.325662
$51$	0	0
$52$	11.9083	1.65139
$53$	8.51388	1.16947	0.584736	−	0.811224i	$-0.301198\pi$
$53$	8.51388	1.16947	0.584736	+	0.811224i	$0.301198\pi$
$54$	0	0
$55$	−2.30278	−0.310506
$56$	−3.90833	−0.522272
$57$	0	0
$58$	17.5139	2.29968
$59$	−1.39445	−0.181542	−0.0907709	−	0.995872i	$-0.528933\pi$
$59$	−1.39445	−0.181542	−0.0907709	+	0.995872i	$0.528933\pi$
$60$	0	0
$61$	0.605551	0.0775329	0.0387664	−	0.999248i	$-0.487657\pi$
$61$	0.605551	0.0775329	0.0387664	+	0.999248i	$0.487657\pi$
$62$	−12.0000	−1.52400
$63$	0	0
$64$	−12.8167	−1.60208
$65$	3.60555	0.447214
$66$	0	0
$67$	10.3028	1.25868	0.629342	−	0.777128i	$-0.283324\pi$
$67$	10.3028	1.25868	0.629342	+	0.777128i	$0.283324\pi$
$68$	−17.5139	−2.12387
$69$	0	0
$70$	−3.00000	−0.358569
$71$	−1.60555	−0.190544	−0.0952719	−	0.995451i	$-0.530372\pi$
$71$	−1.60555	−0.190544	−0.0952719	+	0.995451i	$0.530372\pi$
$72$	0	0
$73$	12.6056	1.47537	0.737684	−	0.675146i	$-0.235919\pi$
$73$	12.6056	1.47537	0.737684	+	0.675146i	$0.235919\pi$
$74$	18.2111	2.11700
$75$	0	0
$76$	−20.8167	−2.38783
$77$	−3.00000	−0.341882
$78$	0	0
$79$	−12.5139	−1.40792	−0.703961	−	0.710239i	$-0.748587\pi$
$79$	−12.5139	−1.40792	−0.703961	+	0.710239i	$0.748587\pi$
$80$	0.302776	0.0338513
$81$	0	0
$82$	15.4222	1.70310
$83$	14.7250	1.61628	0.808138	−	0.588993i	$-0.200475\pi$
$83$	14.7250	1.61628	0.808138	+	0.588993i	$0.200475\pi$
$84$	0	0
$85$	−5.30278	−0.575167
$86$	16.6056	1.79062
$87$	0	0
$88$	6.90833	0.736430
$89$	4.81665	0.510564	0.255282	−	0.966867i	$-0.417832\pi$
$89$	4.81665	0.510564	0.255282	+	0.966867i	$0.417832\pi$
$90$	0	0
$91$	4.69722	0.492403
$92$	−9.90833	−1.03301
$93$	0	0
$94$	−15.4222	−1.59068
$95$	−6.30278	−0.646651
$96$	0	0
$97$	−15.7250	−1.59663	−0.798315	−	0.602240i	$-0.794275\pi$
$97$	−15.7250	−1.59663	−0.798315	+	0.602240i	$0.794275\pi$
$98$	12.2111	1.23351
$99$	0	0
$100$	3.30278	0.330278
$101$	−17.7250	−1.76370	−0.881851	−	0.471529i	$-0.843702\pi$
$101$	−17.7250	−1.76370	−0.881851	+	0.471529i	$0.843702\pi$
$102$	0	0
$103$	−8.81665	−0.868731	−0.434365	−	0.900737i	$-0.643027\pi$
$103$	−8.81665	−0.868731	−0.434365	+	0.900737i	$0.643027\pi$
$104$	−10.8167	−1.06066
$105$	0	0
$106$	−19.6056	−1.90426
$107$	2.09167	0.202210	0.101105	−	0.994876i	$-0.467762\pi$
$107$	2.09167	0.202210	0.101105	+	0.994876i	$0.467762\pi$
$108$	0	0
$109$	−16.9083	−1.61952	−0.809762	−	0.586758i	$-0.800404\pi$
$109$	−16.9083	−1.61952	−0.809762	+	0.586758i	$0.800404\pi$
$110$	5.30278	0.505600
$111$	0	0
$112$	0.394449	0.0372719
$113$	−6.21110	−0.584291	−0.292146	−	0.956374i	$-0.594369\pi$
$113$	−6.21110	−0.584291	−0.292146	+	0.956374i	$0.594369\pi$
$114$	0	0
$115$	−3.00000	−0.279751
$116$	−25.1194	−2.33228
$117$	0	0
$118$	3.21110	0.295606
$119$	−6.90833	−0.633285
$120$	0	0
$121$	−5.69722	−0.517929
$122$	−1.39445	−0.126247
$123$	0	0
$124$	17.2111	1.54560
$125$	1.00000	0.0894427
$126$	0	0
$127$	0.605551	0.0537340	0.0268670	−	0.999639i	$-0.491447\pi$
$127$	0.605551	0.0537340	0.0268670	+	0.999639i	$0.491447\pi$
$128$	18.9083	1.67128
$129$	0	0
$130$	−8.30278	−0.728202
$131$	10.8167	0.945055	0.472528	−	0.881316i	$-0.343342\pi$
$131$	10.8167	0.945055	0.472528	+	0.881316i	$0.343342\pi$
$132$	0	0
$133$	−8.21110	−0.711993
$134$	−23.7250	−2.04953
$135$	0	0
$136$	15.9083	1.36413
$137$	−15.0000	−1.28154	−0.640768	−	0.767734i	$-0.721384\pi$
$137$	−15.0000	−1.28154	−0.640768	+	0.767734i	$0.721384\pi$
$138$	0	0
$139$	11.0000	0.933008	0.466504	−	0.884519i	$-0.345513\pi$
$139$	11.0000	0.933008	0.466504	+	0.884519i	$0.345513\pi$
$140$	4.30278	0.363651
$141$	0	0
$142$	3.69722	0.310264
$143$	−8.30278	−0.694313
$144$	0	0
$145$	−7.60555	−0.631607
$146$	−29.0278	−2.40235
$147$	0	0
$148$	−26.1194	−2.14700
$149$	16.1194	1.32056	0.660278	−	0.751022i	$-0.270439\pi$
$149$	16.1194	1.32056	0.660278	+	0.751022i	$0.270439\pi$
$150$	0	0
$151$	−11.1194	−0.904886	−0.452443	−	0.891793i	$-0.649447\pi$
$151$	−11.1194	−0.904886	−0.452443	+	0.891793i	$0.649447\pi$
$152$	18.9083	1.53367
$153$	0	0
$154$	6.90833	0.556689
$155$	5.21110	0.418566
$156$	0	0
$157$	−2.81665	−0.224793	−0.112397	−	0.993663i	$-0.535853\pi$
$157$	−2.81665	−0.224793	−0.112397	+	0.993663i	$0.535853\pi$
$158$	28.8167	2.29253
$159$	0	0
$160$	5.30278	0.419221
$161$	−3.90833	−0.308019
$162$	0	0
$163$	−10.0000	−0.783260	−0.391630	−	0.920123i	$-0.628089\pi$
$163$	−10.0000	−0.783260	−0.391630	+	0.920123i	$0.628089\pi$
$164$	−22.1194	−1.72724
$165$	0	0
$166$	−33.9083	−2.63180
$167$	−18.9083	−1.46317	−0.731585	−	0.681750i	$-0.761219\pi$
$167$	−18.9083	−1.46317	−0.731585	+	0.681750i	$0.761219\pi$
$168$	0	0
$169$	0	0
$170$	12.2111	0.936549
$171$	0	0
$172$	−23.8167	−1.81600
$173$	−1.18335	−0.0899681	−0.0449841	−	0.998988i	$-0.514324\pi$
$173$	−1.18335	−0.0899681	−0.0449841	+	0.998988i	$0.514324\pi$
$174$	0	0
$175$	1.30278	0.0984806
$176$	−0.697224	−0.0525553
$177$	0	0
$178$	−11.0917	−0.831356
$179$	19.6056	1.46539	0.732694	−	0.680558i	$-0.238263\pi$
$179$	19.6056	1.46539	0.732694	+	0.680558i	$0.238263\pi$
$180$	0	0
$181$	−7.90833	−0.587821	−0.293911	−	0.955833i	$-0.594957\pi$
$181$	−7.90833	−0.587821	−0.293911	+	0.955833i	$0.594957\pi$
$182$	−10.8167	−0.801784
$183$	0	0
$184$	9.00000	0.663489
$185$	−7.90833	−0.581432
$186$	0	0
$187$	12.2111	0.892964
$188$	22.1194	1.61323
$189$	0	0
$190$	14.5139	1.05295
$191$	−7.60555	−0.550318	−0.275159	−	0.961399i	$-0.588731\pi$
$191$	−7.60555	−0.550318	−0.275159	+	0.961399i	$0.588731\pi$
$192$	0	0
$193$	5.00000	0.359908	0.179954	−	0.983675i	$-0.442405\pi$
$193$	5.00000	0.359908	0.179954	+	0.983675i	$0.442405\pi$
$194$	36.2111	2.59981
$195$	0	0
$196$	−17.5139	−1.25099
$197$	−20.7250	−1.47659	−0.738297	−	0.674476i	$-0.764370\pi$
$197$	−20.7250	−1.47659	−0.738297	+	0.674476i	$0.764370\pi$
$198$	0	0
$199$	23.4222	1.66036	0.830178	−	0.557498i	$-0.188239\pi$
$199$	23.4222	1.66036	0.830178	+	0.557498i	$0.188239\pi$
$200$	−3.00000	−0.212132
$201$	0	0
$202$	40.8167	2.87185
$203$	−9.90833	−0.695428
$204$	0	0
$205$	−6.69722	−0.467754
$206$	20.3028	1.41456
$207$	0	0
$208$	1.09167	0.0756939
$209$	14.5139	1.00395
$210$	0	0
$211$	−2.18335	−0.150308	−0.0751539	−	0.997172i	$-0.523945\pi$
$211$	−2.18335	−0.150308	−0.0751539	+	0.997172i	$0.523945\pi$
$212$	28.1194	1.93125
$213$	0	0
$214$	−4.81665	−0.329260
$215$	−7.21110	−0.491793
$216$	0	0
$217$	6.78890	0.460860
$218$	38.9361	2.63708
$219$	0	0
$220$	−7.60555	−0.512766
$221$	−19.1194	−1.28611
$222$	0	0
$223$	20.9083	1.40013	0.700063	−	0.714082i	$-0.253155\pi$
$223$	20.9083	1.40013	0.700063	+	0.714082i	$0.253155\pi$
$224$	6.90833	0.461582
$225$	0	0
$226$	14.3028	0.951406
$227$	−7.60555	−0.504798	−0.252399	−	0.967623i	$-0.581220\pi$
$227$	−7.60555	−0.504798	−0.252399	+	0.967623i	$0.581220\pi$
$228$	0	0
$229$	14.2111	0.939096	0.469548	−	0.882907i	$-0.344417\pi$
$229$	14.2111	0.939096	0.469548	+	0.882907i	$0.344417\pi$
$230$	6.90833	0.455522
$231$	0	0
$232$	22.8167	1.49799
$233$	22.8167	1.49477	0.747384	−	0.664392i	$-0.231309\pi$
$233$	22.8167	1.49477	0.747384	+	0.664392i	$0.231309\pi$
$234$	0	0
$235$	6.69722	0.436879
$236$	−4.60555	−0.299796
$237$	0	0
$238$	15.9083	1.03118
$239$	−20.0278	−1.29549	−0.647744	−	0.761858i	$-0.724287\pi$
$239$	−20.0278	−1.29549	−0.647744	+	0.761858i	$0.724287\pi$
$240$	0	0
$241$	−2.39445	−0.154240	−0.0771200	−	0.997022i	$-0.524572\pi$
$241$	−2.39445	−0.154240	−0.0771200	+	0.997022i	$0.524572\pi$
$242$	13.1194	0.843349
$243$	0	0
$244$	2.00000	0.128037
$245$	−5.30278	−0.338782
$246$	0	0
$247$	−22.7250	−1.44596
$248$	−15.6333	−0.992716
$249$	0	0
$250$	−2.30278	−0.145640
$251$	2.09167	0.132025	0.0660126	−	0.997819i	$-0.478972\pi$
$251$	2.09167	0.132025	0.0660126	+	0.997819i	$0.478972\pi$
$252$	0	0
$253$	6.90833	0.434323
$254$	−1.39445	−0.0874955
$255$	0	0
$256$	−17.9083	−1.11927
$257$	−9.90833	−0.618064	−0.309032	−	0.951052i	$-0.600005\pi$
$257$	−9.90833	−0.618064	−0.309032	+	0.951052i	$0.600005\pi$
$258$	0	0
$259$	−10.3028	−0.640183
$260$	11.9083	0.738523
$261$	0	0
$262$	−24.9083	−1.53884
$263$	−5.51388	−0.340000	−0.170000	−	0.985444i	$-0.554377\pi$
$263$	−5.51388	−0.340000	−0.170000	+	0.985444i	$0.554377\pi$
$264$	0	0
$265$	8.51388	0.523003
$266$	18.9083	1.15934
$267$	0	0
$268$	34.0278	2.07858
$269$	21.6972	1.32290	0.661452	−	0.749988i	$-0.269941\pi$
$269$	21.6972	1.32290	0.661452	+	0.749988i	$0.269941\pi$
$270$	0	0
$271$	−15.0278	−0.912871	−0.456436	−	0.889756i	$-0.650874\pi$
$271$	−15.0278	−0.912871	−0.456436	+	0.889756i	$0.650874\pi$
$272$	−1.60555	−0.0973508
$273$	0	0
$274$	34.5416	2.08674
$275$	−2.30278	−0.138863
$276$	0	0
$277$	−3.51388	−0.211128	−0.105564	−	0.994412i	$-0.533665\pi$
$277$	−3.51388	−0.211128	−0.105564	+	0.994412i	$0.533665\pi$
$278$	−25.3305	−1.51922
$279$	0	0
$280$	−3.90833	−0.233567
$281$	7.81665	0.466302	0.233151	−	0.972440i	$-0.425096\pi$
$281$	7.81665	0.466302	0.233151	+	0.972440i	$0.425096\pi$
$282$	0	0
$283$	3.11943	0.185431	0.0927154	−	0.995693i	$-0.470445\pi$
$283$	3.11943	0.185431	0.0927154	+	0.995693i	$0.470445\pi$
$284$	−5.30278	−0.314662
$285$	0	0
$286$	19.1194	1.13056
$287$	−8.72498	−0.515019
$288$	0	0
$289$	11.1194	0.654084
$290$	17.5139	1.02845
$291$	0	0
$292$	41.6333	2.43641
$293$	8.72498	0.509719	0.254859	−	0.966978i	$-0.417971\pi$
$293$	8.72498	0.509719	0.254859	+	0.966978i	$0.417971\pi$
$294$	0	0
$295$	−1.39445	−0.0811879
$296$	23.7250	1.37899
$297$	0	0
$298$	−37.1194	−2.15027
$299$	−10.8167	−0.625543
$300$	0	0
$301$	−9.39445	−0.541487
$302$	25.6056	1.47343
$303$	0	0
$304$	−1.90833	−0.109450
$305$	0.605551	0.0346738
$306$	0	0
$307$	−12.3028	−0.702156	−0.351078	−	0.936346i	$-0.614185\pi$
$307$	−12.3028	−0.702156	−0.351078	+	0.936346i	$0.614185\pi$
$308$	−9.90833	−0.564579
$309$	0	0
$310$	−12.0000	−0.681554
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	0	0
$313$	16.7250	0.945352	0.472676	−	0.881236i	$-0.343288\pi$
$313$	16.7250	0.945352	0.472676	+	0.881236i	$0.343288\pi$
$314$	6.48612	0.366033
$315$	0	0
$316$	−41.3305	−2.32502
$317$	6.21110	0.348850	0.174425	−	0.984670i	$-0.444193\pi$
$317$	6.21110	0.348850	0.174425	+	0.984670i	$0.444193\pi$
$318$	0	0
$319$	17.5139	0.980589
$320$	−12.8167	−0.716473
$321$	0	0
$322$	9.00000	0.501550
$323$	33.4222	1.85966
$324$	0	0
$325$	3.60555	0.200000
$326$	23.0278	1.27539
$327$	0	0
$328$	20.0917	1.10938
$329$	8.72498	0.481024
$330$	0	0
$331$	10.9361	0.601102	0.300551	−	0.953766i	$-0.402830\pi$
$331$	10.9361	0.601102	0.300551	+	0.953766i	$0.402830\pi$
$332$	48.6333	2.66910
$333$	0	0
$334$	43.5416	2.38249
$335$	10.3028	0.562901
$336$	0	0
$337$	19.5139	1.06299	0.531494	−	0.847062i	$-0.321631\pi$
$337$	19.5139	1.06299	0.531494	+	0.847062i	$0.321631\pi$
$338$	0	0
$339$	0	0
$340$	−17.5139	−0.949823
$341$	−12.0000	−0.649836
$342$	0	0
$343$	−16.0278	−0.865417
$344$	21.6333	1.16639
$345$	0	0
$346$	2.72498	0.146496
$347$	−13.1194	−0.704288	−0.352144	−	0.935946i	$-0.614547\pi$
$347$	−13.1194	−0.704288	−0.352144	+	0.935946i	$0.614547\pi$
$348$	0	0
$349$	−4.48612	−0.240137	−0.120068	−	0.992766i	$-0.538311\pi$
$349$	−4.48612	−0.240137	−0.120068	+	0.992766i	$0.538311\pi$
$350$	−3.00000	−0.160357
$351$	0	0
$352$	−12.2111	−0.650854
$353$	−24.0000	−1.27739	−0.638696	−	0.769460i	$-0.720526\pi$
$353$	−24.0000	−1.27739	−0.638696	+	0.769460i	$0.720526\pi$
$354$	0	0
$355$	−1.60555	−0.0852138
$356$	15.9083	0.843140
$357$	0	0
$358$	−45.1472	−2.38610
$359$	25.5416	1.34804	0.674018	−	0.738715i	$-0.264567\pi$
$359$	25.5416	1.34804	0.674018	+	0.738715i	$0.264567\pi$
$360$	0	0
$361$	20.7250	1.09079
$362$	18.2111	0.957154
$363$	0	0
$364$	15.5139	0.813148
$365$	12.6056	0.659805
$366$	0	0
$367$	34.0278	1.77623	0.888117	−	0.459617i	$-0.152013\pi$
$367$	34.0278	1.77623	0.888117	+	0.459617i	$0.152013\pi$
$368$	−0.908327	−0.0473498
$369$	0	0
$370$	18.2111	0.946750
$371$	11.0917	0.575851
$372$	0	0
$373$	−0.577795	−0.0299171	−0.0149585	−	0.999888i	$-0.504762\pi$
$373$	−0.577795	−0.0299171	−0.0149585	+	0.999888i	$0.504762\pi$
$374$	−28.1194	−1.45402
$375$	0	0
$376$	−20.0917	−1.03615
$377$	−27.4222	−1.41232
$378$	0	0
$379$	−11.1194	−0.571167	−0.285583	−	0.958354i	$-0.592187\pi$
$379$	−11.1194	−0.571167	−0.285583	+	0.958354i	$0.592187\pi$
$380$	−20.8167	−1.06787
$381$	0	0
$382$	17.5139	0.896088
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	0	0
$385$	−3.00000	−0.152894
$386$	−11.5139	−0.586041
$387$	0	0
$388$	−51.9361	−2.63666
$389$	−28.8167	−1.46106	−0.730531	−	0.682879i	$-0.760727\pi$
$389$	−28.8167	−1.46106	−0.730531	+	0.682879i	$0.760727\pi$
$390$	0	0
$391$	15.9083	0.804519
$392$	15.9083	0.803492
$393$	0	0
$394$	47.7250	2.40435
$395$	−12.5139	−0.629642
$396$	0	0
$397$	7.51388	0.377111	0.188555	−	0.982063i	$-0.439620\pi$
$397$	7.51388	0.377111	0.188555	+	0.982063i	$0.439620\pi$
$398$	−53.9361	−2.70357
$399$	0	0
$400$	0.302776	0.0151388
$401$	19.8167	0.989596	0.494798	−	0.869008i	$-0.335242\pi$
$401$	19.8167	0.989596	0.494798	+	0.869008i	$0.335242\pi$
$402$	0	0
$403$	18.7889	0.935942
$404$	−58.5416	−2.91256
$405$	0	0
$406$	22.8167	1.13237
$407$	18.2111	0.902691
$408$	0	0
$409$	−7.63331	−0.377443	−0.188721	−	0.982031i	$-0.560434\pi$
$409$	−7.63331	−0.377443	−0.188721	+	0.982031i	$0.560434\pi$
$410$	15.4222	0.761648
$411$	0	0
$412$	−29.1194	−1.43461
$413$	−1.81665	−0.0893917
$414$	0	0
$415$	14.7250	0.722821
$416$	19.1194	0.937407
$417$	0	0
$418$	−33.4222	−1.63473
$419$	5.02776	0.245622	0.122811	−	0.992430i	$-0.460809\pi$
$419$	5.02776	0.245622	0.122811	+	0.992430i	$0.460809\pi$
$420$	0	0
$421$	8.69722	0.423877	0.211938	−	0.977283i	$-0.432022\pi$
$421$	8.69722	0.423877	0.211938	+	0.977283i	$0.432022\pi$
$422$	5.02776	0.244747
$423$	0	0
$424$	−25.5416	−1.24041
$425$	−5.30278	−0.257222
$426$	0	0
$427$	0.788897	0.0381774
$428$	6.90833	0.333927
$429$	0	0
$430$	16.6056	0.800791
$431$	40.8167	1.96607	0.983035	−	0.183421i	$-0.0587170\pi$
$431$	40.8167	1.96607	0.983035	+	0.183421i	$0.0587170\pi$
$432$	0	0
$433$	−18.0278	−0.866359	−0.433179	−	0.901308i	$-0.642608\pi$
$433$	−18.0278	−0.866359	−0.433179	+	0.901308i	$0.642608\pi$
$434$	−15.6333	−0.750423
$435$	0	0
$436$	−55.8444	−2.67446
$437$	18.9083	0.904508
$438$	0	0
$439$	−20.1194	−0.960248	−0.480124	−	0.877201i	$-0.659408\pi$
$439$	−20.1194	−0.960248	−0.480124	+	0.877201i	$0.659408\pi$
$440$	6.90833	0.329342
$441$	0	0
$442$	44.0278	2.09419
$443$	9.27502	0.440669	0.220335	−	0.975424i	$-0.429285\pi$
$443$	9.27502	0.440669	0.220335	+	0.975424i	$0.429285\pi$
$444$	0	0
$445$	4.81665	0.228331
$446$	−48.1472	−2.27984
$447$	0	0
$448$	−16.6972	−0.788870
$449$	−39.2111	−1.85049	−0.925243	−	0.379375i	$-0.876139\pi$
$449$	−39.2111	−1.85049	−0.925243	+	0.379375i	$0.876139\pi$
$450$	0	0
$451$	15.4222	0.726203
$452$	−20.5139	−0.964892
$453$	0	0
$454$	17.5139	0.821967
$455$	4.69722	0.220209
$456$	0	0
$457$	13.5139	0.632153	0.316076	−	0.948734i	$-0.397634\pi$
$457$	13.5139	0.632153	0.316076	+	0.948734i	$0.397634\pi$
$458$	−32.7250	−1.52914
$459$	0	0
$460$	−9.90833	−0.461978
$461$	−33.0000	−1.53696	−0.768482	−	0.639872i	$-0.778987\pi$
$461$	−33.0000	−1.53696	−0.768482	+	0.639872i	$0.778987\pi$
$462$	0	0
$463$	−20.3305	−0.944840	−0.472420	−	0.881374i	$-0.656619\pi$
$463$	−20.3305	−0.944840	−0.472420	+	0.881374i	$0.656619\pi$
$464$	−2.30278	−0.106904
$465$	0	0
$466$	−52.5416	−2.43394
$467$	−31.3305	−1.44980	−0.724902	−	0.688852i	$-0.758115\pi$
$467$	−31.3305	−1.44980	−0.724902	+	0.688852i	$0.758115\pi$
$468$	0	0
$469$	13.4222	0.619780
$470$	−15.4222	−0.711373
$471$	0	0
$472$	4.18335	0.192554
$473$	16.6056	0.763524
$474$	0	0
$475$	−6.30278	−0.289191
$476$	−22.8167	−1.04580
$477$	0	0
$478$	46.1194	2.10945
$479$	14.7250	0.672802	0.336401	−	0.941719i	$-0.390790\pi$
$479$	14.7250	0.672802	0.336401	+	0.941719i	$0.390790\pi$
$480$	0	0
$481$	−28.5139	−1.30012
$482$	5.51388	0.251150
$483$	0	0
$484$	−18.8167	−0.855302
$485$	−15.7250	−0.714035
$486$	0	0
$487$	−5.18335	−0.234880	−0.117440	−	0.993080i	$-0.537469\pi$
$487$	−5.18335	−0.234880	−0.117440	+	0.993080i	$0.537469\pi$
$488$	−1.81665	−0.0822361
$489$	0	0
$490$	12.2111	0.551641
$491$	−24.6972	−1.11457	−0.557285	−	0.830321i	$-0.688157\pi$
$491$	−24.6972	−1.11457	−0.557285	+	0.830321i	$0.688157\pi$
$492$	0	0
$493$	40.3305	1.81640
$494$	52.3305	2.35446
$495$	0	0
$496$	1.57779	0.0708451
$497$	−2.09167	−0.0938244
$498$	0	0
$499$	1.78890	0.0800820	0.0400410	−	0.999198i	$-0.487251\pi$
$499$	1.78890	0.0800820	0.0400410	+	0.999198i	$0.487251\pi$
$500$	3.30278	0.147705
$501$	0	0
$502$	−4.81665	−0.214978
$503$	−12.6972	−0.566141	−0.283071	−	0.959099i	$-0.591353\pi$
$503$	−12.6972	−0.566141	−0.283071	+	0.959099i	$0.591353\pi$
$504$	0	0
$505$	−17.7250	−0.788751
$506$	−15.9083	−0.707211
$507$	0	0
$508$	2.00000	0.0887357
$509$	−33.8444	−1.50013	−0.750064	−	0.661366i	$-0.769977\pi$
$509$	−33.8444	−1.50013	−0.750064	+	0.661366i	$0.769977\pi$
$510$	0	0
$511$	16.4222	0.726476
$512$	3.42221	0.151242
$513$	0	0
$514$	22.8167	1.00640
$515$	−8.81665	−0.388508
$516$	0	0
$517$	−15.4222	−0.678268
$518$	23.7250	1.04242
$519$	0	0
$520$	−10.8167	−0.474342
$521$	5.93608	0.260065	0.130032	−	0.991510i	$-0.458492\pi$
$521$	5.93608	0.260065	0.130032	+	0.991510i	$0.458492\pi$
$522$	0	0
$523$	24.3305	1.06390	0.531950	−	0.846776i	$-0.321459\pi$
$523$	24.3305	1.06390	0.531950	+	0.846776i	$0.321459\pi$
$524$	35.7250	1.56065
$525$	0	0
$526$	12.6972	0.553625
$527$	−27.6333	−1.20373
$528$	0	0
$529$	−14.0000	−0.608696
$530$	−19.6056	−0.851611
$531$	0	0
$532$	−27.1194	−1.17578
$533$	−24.1472	−1.04593
$534$	0	0
$535$	2.09167	0.0904309
$536$	−30.9083	−1.33504
$537$	0	0
$538$	−49.9638	−2.15409
$539$	12.2111	0.525969
$540$	0	0
$541$	4.30278	0.184991	0.0924954	−	0.995713i	$-0.470516\pi$
$541$	4.30278	0.184991	0.0924954	+	0.995713i	$0.470516\pi$
$542$	34.6056	1.48644
$543$	0	0
$544$	−28.1194	−1.20561
$545$	−16.9083	−0.724273
$546$	0	0
$547$	−4.21110	−0.180054	−0.0900269	−	0.995939i	$-0.528695\pi$
$547$	−4.21110	−0.180054	−0.0900269	+	0.995939i	$0.528695\pi$
$548$	−49.5416	−2.11631
$549$	0	0
$550$	5.30278	0.226111
$551$	47.9361	2.04215
$552$	0	0
$553$	−16.3028	−0.693265
$554$	8.09167	0.343782
$555$	0	0
$556$	36.3305	1.54076
$557$	−9.00000	−0.381342	−0.190671	−	0.981654i	$-0.561066\pi$
$557$	−9.00000	−0.381342	−0.190671	+	0.981654i	$0.561066\pi$
$558$	0	0
$559$	−26.0000	−1.09968
$560$	0.394449	0.0166685
$561$	0	0
$562$	−18.0000	−0.759284
$563$	24.6333	1.03817	0.519085	−	0.854723i	$-0.326273\pi$
$563$	24.6333	1.03817	0.519085	+	0.854723i	$0.326273\pi$
$564$	0	0
$565$	−6.21110	−0.261303
$566$	−7.18335	−0.301939
$567$	0	0
$568$	4.81665	0.202102
$569$	−28.8167	−1.20806	−0.604028	−	0.796963i	$-0.706439\pi$
$569$	−28.8167	−1.20806	−0.604028	+	0.796963i	$0.706439\pi$
$570$	0	0
$571$	−38.8167	−1.62443	−0.812213	−	0.583361i	$-0.801737\pi$
$571$	−38.8167	−1.62443	−0.812213	+	0.583361i	$0.801737\pi$
$572$	−27.4222	−1.14658
$573$	0	0
$574$	20.0917	0.838610
$575$	−3.00000	−0.125109
$576$	0	0
$577$	20.6333	0.858976	0.429488	−	0.903073i	$-0.358694\pi$
$577$	20.6333	0.858976	0.429488	+	0.903073i	$0.358694\pi$
$578$	−25.6056	−1.06505
$579$	0	0
$580$	−25.1194	−1.04303
$581$	19.1833	0.795859
$582$	0	0
$583$	−19.6056	−0.811979
$584$	−37.8167	−1.56486
$585$	0	0
$586$	−20.0917	−0.829979
$587$	28.5416	1.17804	0.589020	−	0.808119i	$-0.299514\pi$
$587$	28.5416	1.17804	0.589020	+	0.808119i	$0.299514\pi$
$588$	0	0
$589$	−32.8444	−1.35333
$590$	3.21110	0.132199
$591$	0	0
$592$	−2.39445	−0.0984112
$593$	−21.6972	−0.890998	−0.445499	−	0.895282i	$-0.646974\pi$
$593$	−21.6972	−0.890998	−0.445499	+	0.895282i	$0.646974\pi$
$594$	0	0
$595$	−6.90833	−0.283214
$596$	53.2389	2.18075
$597$	0	0
$598$	24.9083	1.01858
$599$	34.7527	1.41996	0.709979	−	0.704223i	$-0.248704\pi$
$599$	34.7527	1.41996	0.709979	+	0.704223i	$0.248704\pi$
$600$	0	0
$601$	18.1833	0.741714	0.370857	−	0.928690i	$-0.379064\pi$
$601$	18.1833	0.741714	0.370857	+	0.928690i	$0.379064\pi$
$602$	21.6333	0.881708
$603$	0	0
$604$	−36.7250	−1.49432
$605$	−5.69722	−0.231625
$606$	0	0
$607$	−19.2111	−0.779755	−0.389877	−	0.920867i	$-0.627483\pi$
$607$	−19.2111	−0.779755	−0.389877	+	0.920867i	$0.627483\pi$
$608$	−33.4222	−1.35545
$609$	0	0
$610$	−1.39445	−0.0564596
$611$	24.1472	0.976891
$612$	0	0
$613$	28.5139	1.15166	0.575832	−	0.817568i	$-0.304678\pi$
$613$	28.5139	1.15166	0.575832	+	0.817568i	$0.304678\pi$
$614$	28.3305	1.14333
$615$	0	0
$616$	9.00000	0.362620
$617$	−34.8167	−1.40167	−0.700833	−	0.713326i	$-0.747188\pi$
$617$	−34.8167	−1.40167	−0.700833	+	0.713326i	$0.747188\pi$
$618$	0	0
$619$	24.5416	0.986412	0.493206	−	0.869913i	$-0.335825\pi$
$619$	24.5416	0.986412	0.493206	+	0.869913i	$0.335825\pi$
$620$	17.2111	0.691215
$621$	0	0
$622$	−13.8167	−0.553997
$623$	6.27502	0.251403
$624$	0	0
$625$	1.00000	0.0400000
$626$	−38.5139	−1.53932
$627$	0	0
$628$	−9.30278	−0.371221
$629$	41.9361	1.67210
$630$	0	0
$631$	13.7889	0.548927	0.274464	−	0.961597i	$-0.411500\pi$
$631$	13.7889	0.548927	0.274464	+	0.961597i	$0.411500\pi$
$632$	37.5416	1.49333
$633$	0	0
$634$	−14.3028	−0.568036
$635$	0.605551	0.0240306
$636$	0	0
$637$	−19.1194	−0.757539
$638$	−40.3305	−1.59670
$639$	0	0
$640$	18.9083	0.747417
$641$	21.6333	0.854464	0.427232	−	0.904142i	$-0.359489\pi$
$641$	21.6333	0.854464	0.427232	+	0.904142i	$0.359489\pi$
$642$	0	0
$643$	−1.42221	−0.0560863	−0.0280431	−	0.999607i	$-0.508928\pi$
$643$	−1.42221	−0.0560863	−0.0280431	+	0.999607i	$0.508928\pi$
$644$	−12.9083	−0.508659
$645$	0	0
$646$	−76.9638	−3.02810
$647$	38.2389	1.50332	0.751662	−	0.659548i	$-0.229252\pi$
$647$	38.2389	1.50332	0.751662	+	0.659548i	$0.229252\pi$
$648$	0	0
$649$	3.21110	0.126047
$650$	−8.30278	−0.325662
$651$	0	0
$652$	−33.0278	−1.29347
$653$	9.21110	0.360458	0.180229	−	0.983625i	$-0.442316\pi$
$653$	9.21110	0.360458	0.180229	+	0.983625i	$0.442316\pi$
$654$	0	0
$655$	10.8167	0.422642
$656$	−2.02776	−0.0791706
$657$	0	0
$658$	−20.0917	−0.783255
$659$	26.7889	1.04355	0.521774	−	0.853084i	$-0.325271\pi$
$659$	26.7889	1.04355	0.521774	+	0.853084i	$0.325271\pi$
$660$	0	0
$661$	−7.84441	−0.305112	−0.152556	−	0.988295i	$-0.548750\pi$
$661$	−7.84441	−0.305112	−0.152556	+	0.988295i	$0.548750\pi$
$662$	−25.1833	−0.978779
$663$	0	0
$664$	−44.1749	−1.71432
$665$	−8.21110	−0.318413
$666$	0	0
$667$	22.8167	0.883464
$668$	−62.4500	−2.41626
$669$	0	0
$670$	−23.7250	−0.916576
$671$	−1.39445	−0.0538321
$672$	0	0
$673$	51.1194	1.97051	0.985255	−	0.171095i	$-0.0547304\pi$
$673$	51.1194	1.97051	0.985255	+	0.171095i	$0.0547304\pi$
$674$	−44.9361	−1.73087
$675$	0	0
$676$	0	0
$677$	28.5416	1.09694	0.548472	−	0.836169i	$-0.315210\pi$
$677$	28.5416	1.09694	0.548472	+	0.836169i	$0.315210\pi$
$678$	0	0
$679$	−20.4861	−0.786185
$680$	15.9083	0.610056
$681$	0	0
$682$	27.6333	1.05813
$683$	−4.81665	−0.184304	−0.0921521	−	0.995745i	$-0.529375\pi$
$683$	−4.81665	−0.184304	−0.0921521	+	0.995745i	$0.529375\pi$
$684$	0	0
$685$	−15.0000	−0.573121
$686$	36.9083	1.40917
$687$	0	0
$688$	−2.18335	−0.0832393
$689$	30.6972	1.16947
$690$	0	0
$691$	15.6056	0.593663	0.296832	−	0.954930i	$-0.404070\pi$
$691$	15.6056	0.593663	0.296832	+	0.954930i	$0.404070\pi$
$692$	−3.90833	−0.148572
$693$	0	0
$694$	30.2111	1.14680
$695$	11.0000	0.417254
$696$	0	0
$697$	35.5139	1.34518
$698$	10.3305	0.391016
$699$	0	0
$700$	4.30278	0.162630
$701$	−19.6056	−0.740491	−0.370246	−	0.928934i	$-0.620727\pi$
$701$	−19.6056	−0.740491	−0.370246	+	0.928934i	$0.620727\pi$
$702$	0	0
$703$	49.8444	1.87992
$704$	29.5139	1.11235
$705$	0	0
$706$	55.2666	2.07999
$707$	−23.0917	−0.868452
$708$	0	0
$709$	−1.76114	−0.0661410	−0.0330705	−	0.999453i	$-0.510529\pi$
$709$	−1.76114	−0.0661410	−0.0330705	+	0.999453i	$0.510529\pi$
$710$	3.69722	0.138754
$711$	0	0
$712$	−14.4500	−0.541535
$713$	−15.6333	−0.585472
$714$	0	0
$715$	−8.30278	−0.310506
$716$	64.7527	2.41992
$717$	0	0
$718$	−58.8167	−2.19502
$719$	20.7250	0.772911	0.386456	−	0.922308i	$-0.373699\pi$
$719$	20.7250	0.772911	0.386456	+	0.922308i	$0.373699\pi$
$720$	0	0
$721$	−11.4861	−0.427766
$722$	−47.7250	−1.77614
$723$	0	0
$724$	−26.1194	−0.970721
$725$	−7.60555	−0.282463
$726$	0	0
$727$	−48.7250	−1.80711	−0.903555	−	0.428473i	$-0.859052\pi$
$727$	−48.7250	−1.80711	−0.903555	+	0.428473i	$0.859052\pi$
$728$	−14.0917	−0.522272
$729$	0	0
$730$	−29.0278	−1.07437
$731$	38.2389	1.41432
$732$	0	0
$733$	29.0000	1.07114	0.535570	−	0.844491i	$-0.320097\pi$
$733$	29.0000	1.07114	0.535570	+	0.844491i	$0.320097\pi$
$734$	−78.3583	−2.89226
$735$	0	0
$736$	−15.9083	−0.586389
$737$	−23.7250	−0.873921
$738$	0	0
$739$	−49.8444	−1.83356	−0.916778	−	0.399397i	$-0.869220\pi$
$739$	−49.8444	−1.83356	−0.916778	+	0.399397i	$0.869220\pi$
$740$	−26.1194	−0.960169
$741$	0	0
$742$	−25.5416	−0.937663
$743$	16.6056	0.609199	0.304599	−	0.952481i	$-0.401477\pi$
$743$	16.6056	0.609199	0.304599	+	0.952481i	$0.401477\pi$
$744$	0	0
$745$	16.1194	0.590570
$746$	1.33053	0.0487142
$747$	0	0
$748$	40.3305	1.47463
$749$	2.72498	0.0995686
$750$	0	0
$751$	21.6056	0.788398	0.394199	−	0.919025i	$-0.371022\pi$
$751$	21.6056	0.788398	0.394199	+	0.919025i	$0.371022\pi$
$752$	2.02776	0.0739447
$753$	0	0
$754$	63.1472	2.29968
$755$	−11.1194	−0.404677
$756$	0	0
$757$	0.880571	0.0320049	0.0160024	−	0.999872i	$-0.494906\pi$
$757$	0.880571	0.0320049	0.0160024	+	0.999872i	$0.494906\pi$
$758$	25.6056	0.930036
$759$	0	0
$760$	18.9083	0.685877
$761$	−42.3583	−1.53549	−0.767743	−	0.640757i	$-0.778620\pi$
$761$	−42.3583	−1.53549	−0.767743	+	0.640757i	$0.778620\pi$
$762$	0	0
$763$	−22.0278	−0.797458
$764$	−25.1194	−0.908789
$765$	0	0
$766$	27.6333	0.998432
$767$	−5.02776	−0.181542
$768$	0	0
$769$	31.7250	1.14403	0.572016	−	0.820242i	$-0.306162\pi$
$769$	31.7250	1.14403	0.572016	+	0.820242i	$0.306162\pi$
$770$	6.90833	0.248959
$771$	0	0
$772$	16.5139	0.594347
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	0	0
$775$	5.21110	0.187188
$776$	47.1749	1.69348
$777$	0	0
$778$	66.3583	2.37906
$779$	42.2111	1.51237
$780$	0	0
$781$	3.69722	0.132297
$782$	−36.6333	−1.31000
$783$	0	0
$784$	−1.60555	−0.0573411
$785$	−2.81665	−0.100531
$786$	0	0
$787$	43.8722	1.56387	0.781937	−	0.623358i	$-0.214232\pi$
$787$	43.8722	1.56387	0.781937	+	0.623358i	$0.214232\pi$
$788$	−68.4500	−2.43843
$789$	0	0
$790$	28.8167	1.02525
$791$	−8.09167	−0.287707
$792$	0	0
$793$	2.18335	0.0775329
$794$	−17.3028	−0.614053
$795$	0	0
$796$	77.3583	2.74189
$797$	17.5778	0.622637	0.311319	−	0.950306i	$-0.399229\pi$
$797$	17.5778	0.622637	0.311319	+	0.950306i	$0.399229\pi$
$798$	0	0
$799$	−35.5139	−1.25639
$800$	5.30278	0.187481
$801$	0	0
$802$	−45.6333	−1.61137
$803$	−29.0278	−1.02437
$804$	0	0
$805$	−3.90833	−0.137750
$806$	−43.2666	−1.52400
$807$	0	0
$808$	53.1749	1.87069
$809$	−3.06392	−0.107722	−0.0538608	−	0.998548i	$-0.517153\pi$
$809$	−3.06392	−0.107722	−0.0538608	+	0.998548i	$0.517153\pi$
$810$	0	0
$811$	47.2111	1.65781	0.828903	−	0.559392i	$-0.188965\pi$
$811$	47.2111	1.65781	0.828903	+	0.559392i	$0.188965\pi$
$812$	−32.7250	−1.14842
$813$	0	0
$814$	−41.9361	−1.46986
$815$	−10.0000	−0.350285
$816$	0	0
$817$	45.4500	1.59009
$818$	17.5778	0.614593
$819$	0	0
$820$	−22.1194	−0.772444
$821$	−17.0917	−0.596504	−0.298252	−	0.954487i	$-0.596403\pi$
$821$	−17.0917	−0.596504	−0.298252	+	0.954487i	$0.596403\pi$
$822$	0	0
$823$	−11.1833	−0.389827	−0.194913	−	0.980820i	$-0.562443\pi$
$823$	−11.1833	−0.389827	−0.194913	+	0.980820i	$0.562443\pi$
$824$	26.4500	0.921428
$825$	0	0
$826$	4.18335	0.145557
$827$	−16.3944	−0.570091	−0.285045	−	0.958514i	$-0.592009\pi$
$827$	−16.3944	−0.570091	−0.285045	+	0.958514i	$0.592009\pi$
$828$	0	0
$829$	−30.2389	−1.05024	−0.525119	−	0.851029i	$-0.675979\pi$
$829$	−30.2389	−1.05024	−0.525119	+	0.851029i	$0.675979\pi$
$830$	−33.9083	−1.17698
$831$	0	0
$832$	−46.2111	−1.60208
$833$	28.1194	0.974280
$834$	0	0
$835$	−18.9083	−0.654350
$836$	47.9361	1.65790
$837$	0	0
$838$	−11.5778	−0.399948
$839$	14.5778	0.503281	0.251641	−	0.967821i	$-0.419030\pi$
$839$	14.5778	0.503281	0.251641	+	0.967821i	$0.419030\pi$
$840$	0	0
$841$	28.8444	0.994635
$842$	−20.0278	−0.690202
$843$	0	0
$844$	−7.21110	−0.248216
$845$	0	0
$846$	0	0
$847$	−7.42221	−0.255030
$848$	2.57779	0.0885218
$849$	0	0
$850$	12.2111	0.418837
$851$	23.7250	0.813282
$852$	0	0
$853$	−13.6972	−0.468984	−0.234492	−	0.972118i	$-0.575343\pi$
$853$	−13.6972	−0.468984	−0.234492	+	0.972118i	$0.575343\pi$
$854$	−1.81665	−0.0621646
$855$	0	0
$856$	−6.27502	−0.214476
$857$	13.6056	0.464757	0.232378	−	0.972625i	$-0.425349\pi$
$857$	13.6056	0.464757	0.232378	+	0.972625i	$0.425349\pi$
$858$	0	0
$859$	−5.39445	−0.184056	−0.0920281	−	0.995756i	$-0.529335\pi$
$859$	−5.39445	−0.184056	−0.0920281	+	0.995756i	$0.529335\pi$
$860$	−23.8167	−0.812141
$861$	0	0
$862$	−93.9916	−3.20137
$863$	39.2111	1.33476	0.667381	−	0.744717i	$-0.267415\pi$
$863$	39.2111	1.33476	0.667381	+	0.744717i	$0.267415\pi$
$864$	0	0
$865$	−1.18335	−0.0402350
$866$	41.5139	1.41070
$867$	0	0
$868$	22.4222	0.761059
$869$	28.8167	0.977538
$870$	0	0
$871$	37.1472	1.25868
$872$	50.7250	1.71776
$873$	0	0
$874$	−43.5416	−1.47282
$875$	1.30278	0.0440419
$876$	0	0
$877$	−12.7889	−0.431850	−0.215925	−	0.976410i	$-0.569277\pi$
$877$	−12.7889	−0.431850	−0.215925	+	0.976410i	$0.569277\pi$
$878$	46.3305	1.56358
$879$	0	0
$880$	−0.697224	−0.0235034
$881$	10.6056	0.357310	0.178655	−	0.983912i	$-0.442825\pi$
$881$	10.6056	0.357310	0.178655	+	0.983912i	$0.442825\pi$
$882$	0	0
$883$	17.0000	0.572096	0.286048	−	0.958215i	$-0.407658\pi$
$883$	17.0000	0.572096	0.286048	+	0.958215i	$0.407658\pi$
$884$	−63.1472	−2.12387
$885$	0	0
$886$	−21.3583	−0.717546
$887$	19.1833	0.644114	0.322057	−	0.946720i	$-0.395626\pi$
$887$	19.1833	0.644114	0.322057	+	0.946720i	$0.395626\pi$
$888$	0	0
$889$	0.788897	0.0264588
$890$	−11.0917	−0.371794
$891$	0	0
$892$	69.0555	2.31215
$893$	−42.2111	−1.41254
$894$	0	0
$895$	19.6056	0.655341
$896$	24.6333	0.822941
$897$	0	0
$898$	90.2944	3.01316
$899$	−39.6333	−1.32184
$900$	0	0
$901$	−45.1472	−1.50407
$902$	−35.5139	−1.18248
$903$	0	0
$904$	18.6333	0.619735
$905$	−7.90833	−0.262882
$906$	0	0
$907$	48.3944	1.60691	0.803456	−	0.595365i	$-0.202992\pi$
$907$	48.3944	1.60691	0.803456	+	0.595365i	$0.202992\pi$
$908$	−25.1194	−0.833618
$909$	0	0
$910$	−10.8167	−0.358569
$911$	−38.9361	−1.29001	−0.645005	−	0.764178i	$-0.723145\pi$
$911$	−38.9361	−1.29001	−0.645005	+	0.764178i	$0.723145\pi$
$912$	0	0
$913$	−33.9083	−1.12220
$914$	−31.1194	−1.02934
$915$	0	0
$916$	46.9361	1.55081
$917$	14.0917	0.465348
$918$	0	0
$919$	−37.2111	−1.22748	−0.613741	−	0.789508i	$-0.710336\pi$
$919$	−37.2111	−1.22748	−0.613741	+	0.789508i	$0.710336\pi$
$920$	9.00000	0.296721
$921$	0	0
$922$	75.9916	2.50265
$923$	−5.78890	−0.190544
$924$	0	0
$925$	−7.90833	−0.260024
$926$	46.8167	1.53849
$927$	0	0
$928$	−40.3305	−1.32391
$929$	−28.3305	−0.929495	−0.464747	−	0.885443i	$-0.653855\pi$
$929$	−28.3305	−0.929495	−0.464747	+	0.885443i	$0.653855\pi$
$930$	0	0
$931$	33.4222	1.09537
$932$	75.3583	2.46844
$933$	0	0
$934$	72.1472	2.36073
$935$	12.2111	0.399346
$936$	0	0
$937$	11.2111	0.366251	0.183125	−	0.983090i	$-0.441379\pi$
$937$	11.2111	0.366251	0.183125	+	0.983090i	$0.441379\pi$
$938$	−30.9083	−1.00919
$939$	0	0
$940$	22.1194	0.721456
$941$	25.7527	0.839515	0.419758	−	0.907636i	$-0.362115\pi$
$941$	25.7527	0.839515	0.419758	+	0.907636i	$0.362115\pi$
$942$	0	0
$943$	20.0917	0.654275
$944$	−0.422205	−0.0137416
$945$	0	0
$946$	−38.2389	−1.24325
$947$	34.8167	1.13139	0.565695	−	0.824615i	$-0.308608\pi$
$947$	34.8167	1.13139	0.565695	+	0.824615i	$0.308608\pi$
$948$	0	0
$949$	45.4500	1.47537
$950$	14.5139	0.470892
$951$	0	0
$952$	20.7250	0.671700
$953$	−35.0278	−1.13466	−0.567330	−	0.823490i	$-0.692024\pi$
$953$	−35.0278	−1.13466	−0.567330	+	0.823490i	$0.692024\pi$
$954$	0	0
$955$	−7.60555	−0.246110
$956$	−66.1472	−2.13935
$957$	0	0
$958$	−33.9083	−1.09553
$959$	−19.5416	−0.631032
$960$	0	0
$961$	−3.84441	−0.124013
$962$	65.6611	2.11700
$963$	0	0
$964$	−7.90833	−0.254710
$965$	5.00000	0.160956
$966$	0	0
$967$	−26.1833	−0.842000	−0.421000	−	0.907061i	$-0.638321\pi$
$967$	−26.1833	−0.842000	−0.421000	+	0.907061i	$0.638321\pi$
$968$	17.0917	0.549347
$969$	0	0
$970$	36.2111	1.16267
$971$	−57.3583	−1.84072	−0.920358	−	0.391078i	$-0.872102\pi$
$971$	−57.3583	−1.84072	−0.920358	+	0.391078i	$0.872102\pi$
$972$	0	0
$973$	14.3305	0.459416
$974$	11.9361	0.382457
$975$	0	0
$976$	0.183346	0.00586877
$977$	7.60555	0.243323	0.121662	−	0.992572i	$-0.461178\pi$
$977$	7.60555	0.243323	0.121662	+	0.992572i	$0.461178\pi$
$978$	0	0
$979$	−11.0917	−0.354491
$980$	−17.5139	−0.559460
$981$	0	0
$982$	56.8722	1.81486
$983$	−28.8167	−0.919109	−0.459554	−	0.888150i	$-0.651991\pi$
$983$	−28.8167	−0.919109	−0.459554	+	0.888150i	$0.651991\pi$
$984$	0	0
$985$	−20.7250	−0.660353
$986$	−92.8722	−2.95765
$987$	0	0
$988$	−75.0555	−2.38783
$989$	21.6333	0.687899
$990$	0	0
$991$	20.0000	0.635321	0.317660	−	0.948205i	$-0.397103\pi$
$991$	20.0000	0.635321	0.317660	+	0.948205i	$0.397103\pi$
$992$	27.6333	0.877358
$993$	0	0
$994$	4.81665	0.152775
$995$	23.4222	0.742534
$996$	0	0
$997$	−42.7889	−1.35514	−0.677569	−	0.735459i	$-0.736966\pi$
$997$	−42.7889	−1.35514	−0.677569	+	0.735459i	$0.736966\pi$
$998$	−4.11943	−0.130398
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1215.2.a.k.1.1	✓	2
3.2	odd	2		1215.2.a.m.1.2	yes	2
5.4	even	2		6075.2.a.bo.1.2		2
9.2	odd	6		1215.2.e.l.811.1		4
9.4	even	3		1215.2.e.n.406.2		4
9.5	odd	6		1215.2.e.l.406.1		4
9.7	even	3		1215.2.e.n.811.2		4
15.14	odd	2		6075.2.a.bk.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1215.2.a.k.1.1	✓	2	1.1	even	1	trivial
1215.2.a.m.1.2	yes	2	3.2	odd	2
1215.2.e.l.406.1		4	9.5	odd	6
1215.2.e.l.811.1		4	9.2	odd	6
1215.2.e.n.406.2		4	9.4	even	3
1215.2.e.n.811.2		4	9.7	even	3
6075.2.a.bk.1.1		2	15.14	odd	2
6075.2.a.bo.1.2		2	5.4	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1215 = 3^{5} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1215.a (trivial)

\(f(q)\)	\(=\)	\(q-2.30278 q^{2} +3.30278 q^{4} +1.00000 q^{5} +1.30278 q^{7} -3.00000 q^{8} -2.30278 q^{10} -2.30278 q^{11} +3.60555 q^{13} -3.00000 q^{14} +0.302776 q^{16} -5.30278 q^{17} -6.30278 q^{19} +3.30278 q^{20} +5.30278 q^{22} -3.00000 q^{23} +1.00000 q^{25} -8.30278 q^{26} +4.30278 q^{28} -7.60555 q^{29} +5.21110 q^{31} +5.30278 q^{32} +12.2111 q^{34} +1.30278 q^{35} -7.90833 q^{37} +14.5139 q^{38} -3.00000 q^{40} -6.69722 q^{41} -7.21110 q^{43} -7.60555 q^{44} +6.90833 q^{46} +6.69722 q^{47} -5.30278 q^{49} -2.30278 q^{50} +11.9083 q^{52} +8.51388 q^{53} -2.30278 q^{55} -3.90833 q^{56} +17.5139 q^{58} -1.39445 q^{59} +0.605551 q^{61} -12.0000 q^{62} -12.8167 q^{64} +3.60555 q^{65} +10.3028 q^{67} -17.5139 q^{68} -3.00000 q^{70} -1.60555 q^{71} +12.6056 q^{73} +18.2111 q^{74} -20.8167 q^{76} -3.00000 q^{77} -12.5139 q^{79} +0.302776 q^{80} +15.4222 q^{82} +14.7250 q^{83} -5.30278 q^{85} +16.6056 q^{86} +6.90833 q^{88} +4.81665 q^{89} +4.69722 q^{91} -9.90833 q^{92} -15.4222 q^{94} -6.30278 q^{95} -15.7250 q^{97} +12.2111 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{2} + 3 q^{4} + 2 q^{5} - q^{7} - 6 q^{8} - q^{10} - q^{11} - 6 q^{14} - 3 q^{16} - 7 q^{17} - 9 q^{19} + 3 q^{20} + 7 q^{22} - 6 q^{23} + 2 q^{25} - 13 q^{26} + 5 q^{28} - 8 q^{29} - 4 q^{31}+ \cdots + 10 q^{98}+O(q^{100}) \) 2 * q - q^2 + 3 * q^4 + 2 * q^5 - q^7 - 6 * q^8 - q^10 - q^11 - 6 * q^14 - 3 * q^16 - 7 * q^17 - 9 * q^19 + 3 * q^20 + 7 * q^22 - 6 * q^23 + 2 * q^25 - 13 * q^26 + 5 * q^28 - 8 * q^29 - 4 * q^31 + 7 * q^32 + 10 * q^34 - q^35 - 5 * q^37 + 11 * q^38 - 6 * q^40 - 17 * q^41 - 8 * q^44 + 3 * q^46 + 17 * q^47 - 7 * q^49 - q^50 + 13 * q^52 - q^53 - q^55 + 3 * q^56 + 17 * q^58 - 10 * q^59 - 6 * q^61 - 24 * q^62 - 4 * q^64 + 17 * q^67 - 17 * q^68 - 6 * q^70 + 4 * q^71 + 18 * q^73 + 22 * q^74 - 20 * q^76 - 6 * q^77 - 7 * q^79 - 3 * q^80 + 2 * q^82 - 3 * q^83 - 7 * q^85 + 26 * q^86 + 3 * q^88 - 12 * q^89 + 13 * q^91 - 9 * q^92 - 2 * q^94 - 9 * q^95 + q^97 + 10 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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