LMFDB - Embedded newform 1449.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1449,2,Mod(1,1449)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1449, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1449.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1449 = 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1449.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:	$11.5703232529$
Analytic rank:	$0$
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:	no (minimal twist has level 483)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	1449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.30278 q^{2} +3.30278 q^{4} +0.697224 q^{5} -1.00000 q^{7} +3.00000 q^{8} +O(q^{10})$	\(q+2.30278 q^{2} +3.30278 q^{4} +0.697224 q^{5} -1.00000 q^{7} +3.00000 q^{8} +1.60555 q^{10} +5.00000 q^{11} +2.30278 q^{13} -2.30278 q^{14} +0.302776 q^{16} +5.60555 q^{17} -1.60555 q^{19} +2.30278 q^{20} +11.5139 q^{22} -1.00000 q^{23} -4.51388 q^{25} +5.30278 q^{26} -3.30278 q^{28} -6.21110 q^{29} +3.00000 q^{31} -5.30278 q^{32} +12.9083 q^{34} -0.697224 q^{35} -9.00000 q^{37} -3.69722 q^{38} +2.09167 q^{40} +12.2111 q^{41} +5.51388 q^{43} +16.5139 q^{44} -2.30278 q^{46} +8.60555 q^{47} +1.00000 q^{49} -10.3944 q^{50} +7.60555 q^{52} +12.5139 q^{53} +3.48612 q^{55} -3.00000 q^{56} -14.3028 q^{58} -3.90833 q^{59} -1.09167 q^{61} +6.90833 q^{62} -12.8167 q^{64} +1.60555 q^{65} -11.9083 q^{67} +18.5139 q^{68} -1.60555 q^{70} -0.908327 q^{71} -2.21110 q^{73} -20.7250 q^{74} -5.30278 q^{76} -5.00000 q^{77} -1.00000 q^{79} +0.211103 q^{80} +28.1194 q^{82} -5.60555 q^{83} +3.90833 q^{85} +12.6972 q^{86} +15.0000 q^{88} -10.9083 q^{89} -2.30278 q^{91} -3.30278 q^{92} +19.8167 q^{94} -1.11943 q^{95} -17.6056 q^{97} +2.30278 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 3 q^{4} + 5 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10}) $ 2 * q + q^2 + 3 * q^4 + 5 * q^5 - 2 * q^7 + 6 * q^8	$ 2 q + q^{2} + 3 q^{4} + 5 q^{5} - 2 q^{7} + 6 q^{8} - 4 q^{10} + 10 q^{11} + q^{13} - q^{14} - 3 q^{16} + 4 q^{17} + 4 q^{19} + q^{20} + 5 q^{22} - 2 q^{23} + 9 q^{25} + 7 q^{26} - 3 q^{28} + 2 q^{29} + 6 q^{31} - 7 q^{32} + 15 q^{34} - 5 q^{35} - 18 q^{37} - 11 q^{38} + 15 q^{40} + 10 q^{41} - 7 q^{43} + 15 q^{44} - q^{46} + 10 q^{47} + 2 q^{49} - 28 q^{50} + 8 q^{52} + 7 q^{53} + 25 q^{55} - 6 q^{56} - 25 q^{58} + 3 q^{59} - 13 q^{61} + 3 q^{62} - 4 q^{64} - 4 q^{65} - 13 q^{67} + 19 q^{68} + 4 q^{70} + 9 q^{71} + 10 q^{73} - 9 q^{74} - 7 q^{76} - 10 q^{77} - 2 q^{79} - 14 q^{80} + 31 q^{82} - 4 q^{83} - 3 q^{85} + 29 q^{86} + 30 q^{88} - 11 q^{89} - q^{91} - 3 q^{92} + 18 q^{94} + 23 q^{95} - 28 q^{97} + q^{98}+O(q^{100}) $ 2 * q + q^2 + 3 * q^4 + 5 * q^5 - 2 * q^7 + 6 * q^8 - 4 * q^10 + 10 * q^11 + q^13 - q^14 - 3 * q^16 + 4 * q^17 + 4 * q^19 + q^20 + 5 * q^22 - 2 * q^23 + 9 * q^25 + 7 * q^26 - 3 * q^28 + 2 * q^29 + 6 * q^31 - 7 * q^32 + 15 * q^34 - 5 * q^35 - 18 * q^37 - 11 * q^38 + 15 * q^40 + 10 * q^41 - 7 * q^43 + 15 * q^44 - q^46 + 10 * q^47 + 2 * q^49 - 28 * q^50 + 8 * q^52 + 7 * q^53 + 25 * q^55 - 6 * q^56 - 25 * q^58 + 3 * q^59 - 13 * q^61 + 3 * q^62 - 4 * q^64 - 4 * q^65 - 13 * q^67 + 19 * q^68 + 4 * q^70 + 9 * q^71 + 10 * q^73 - 9 * q^74 - 7 * q^76 - 10 * q^77 - 2 * q^79 - 14 * q^80 + 31 * q^82 - 4 * q^83 - 3 * q^85 + 29 * q^86 + 30 * q^88 - 11 * q^89 - q^91 - 3 * q^92 + 18 * q^94 + 23 * q^95 - 28 * q^97 + 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 $

$2$	2.30278	1.62831	0.814154	−	0.580649i	$-0.197201\pi$
$2$	2.30278	1.62831	0.814154	+	0.580649i	$0.197201\pi$
$3$	0	0
$3$	0	0
$4$	3.30278	1.65139
$5$	0.697224	0.311808	0.155904	−	0.987772i	$-0.450171\pi$
$5$	0.697224	0.311808	0.155904	+	0.987772i	$0.450171\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	3.00000	1.06066
$9$	0	0
$10$	1.60555	0.507720
$11$	5.00000	1.50756	0.753778	−	0.657129i	$-0.228229\pi$
$11$	5.00000	1.50756	0.753778	+	0.657129i	$0.228229\pi$
$12$	0	0
$13$	2.30278	0.638675	0.319338	−	0.947641i	$-0.396540\pi$
$13$	2.30278	0.638675	0.319338	+	0.947641i	$0.396540\pi$
$14$	−2.30278	−0.615443
$15$	0	0
$16$	0.302776	0.0756939
$17$	5.60555	1.35955	0.679773	−	0.733423i	$-0.262078\pi$
$17$	5.60555	1.35955	0.679773	+	0.733423i	$0.262078\pi$
$18$	0	0
$19$	−1.60555	−0.368339	−0.184169	−	0.982895i	$-0.558959\pi$
$19$	−1.60555	−0.368339	−0.184169	+	0.982895i	$0.558959\pi$
$20$	2.30278	0.514916
$21$	0	0
$22$	11.5139	2.45477
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.51388	−0.902776
$26$	5.30278	1.03996
$27$	0	0
$28$	−3.30278	−0.624166
$29$	−6.21110	−1.15337	−0.576686	−	0.816966i	$-0.695655\pi$
$29$	−6.21110	−1.15337	−0.576686	+	0.816966i	$0.695655\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	−5.30278	−0.937407
$33$	0	0
$34$	12.9083	2.21376
$35$	−0.697224	−0.117852
$36$	0	0
$37$	−9.00000	−1.47959	−0.739795	−	0.672832i	$-0.765078\pi$
$37$	−9.00000	−1.47959	−0.739795	+	0.672832i	$0.765078\pi$
$38$	−3.69722	−0.599769
$39$	0	0
$40$	2.09167	0.330723
$41$	12.2111	1.90705	0.953527	−	0.301308i	$-0.0974232\pi$
$41$	12.2111	1.90705	0.953527	+	0.301308i	$0.0974232\pi$
$42$	0	0
$43$	5.51388	0.840859	0.420429	−	0.907325i	$-0.361879\pi$
$43$	5.51388	0.840859	0.420429	+	0.907325i	$0.361879\pi$
$44$	16.5139	2.48956
$45$	0	0
$46$	−2.30278	−0.339526
$47$	8.60555	1.25525	0.627624	−	0.778516i	$-0.284027\pi$
$47$	8.60555	1.25525	0.627624	+	0.778516i	$0.284027\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−10.3944	−1.47000
$51$	0	0
$52$	7.60555	1.05470
$53$	12.5139	1.71891	0.859457	−	0.511209i	$-0.170802\pi$
$53$	12.5139	1.71891	0.859457	+	0.511209i	$0.170802\pi$
$54$	0	0
$55$	3.48612	0.470069
$56$	−3.00000	−0.400892
$57$	0	0
$58$	−14.3028	−1.87805
$59$	−3.90833	−0.508821	−0.254410	−	0.967096i	$-0.581881\pi$
$59$	−3.90833	−0.508821	−0.254410	+	0.967096i	$0.581881\pi$
$60$	0	0
$61$	−1.09167	−0.139774	−0.0698872	−	0.997555i	$-0.522264\pi$
$61$	−1.09167	−0.139774	−0.0698872	+	0.997555i	$0.522264\pi$
$62$	6.90833	0.877358
$63$	0	0
$64$	−12.8167	−1.60208
$65$	1.60555	0.199144
$66$	0	0
$67$	−11.9083	−1.45483	−0.727417	−	0.686196i	$-0.759279\pi$
$67$	−11.9083	−1.45483	−0.727417	+	0.686196i	$0.759279\pi$
$68$	18.5139	2.24514
$69$	0	0
$70$	−1.60555	−0.191900
$71$	−0.908327	−0.107799	−0.0538993	−	0.998546i	$-0.517165\pi$
$71$	−0.908327	−0.107799	−0.0538993	+	0.998546i	$0.517165\pi$
$72$	0	0
$73$	−2.21110	−0.258790	−0.129395	−	0.991593i	$-0.541304\pi$
$73$	−2.21110	−0.258790	−0.129395	+	0.991593i	$0.541304\pi$
$74$	−20.7250	−2.40923
$75$	0	0
$76$	−5.30278	−0.608270
$77$	−5.00000	−0.569803
$78$	0	0
$79$	−1.00000	−0.112509	−0.0562544	−	0.998416i	$-0.517916\pi$
$79$	−1.00000	−0.112509	−0.0562544	+	0.998416i	$0.517916\pi$
$80$	0.211103	0.0236020
$81$	0	0
$82$	28.1194	3.10527
$83$	−5.60555	−0.615289	−0.307645	−	0.951501i	$-0.599541\pi$
$83$	−5.60555	−0.615289	−0.307645	+	0.951501i	$0.599541\pi$
$84$	0	0
$85$	3.90833	0.423918
$86$	12.6972	1.36918
$87$	0	0
$88$	15.0000	1.59901
$89$	−10.9083	−1.15628	−0.578140	−	0.815937i	$-0.696221\pi$
$89$	−10.9083	−1.15628	−0.578140	+	0.815937i	$0.696221\pi$
$90$	0	0
$91$	−2.30278	−0.241396
$92$	−3.30278	−0.344338
$93$	0	0
$94$	19.8167	2.04393
$95$	−1.11943	−0.114851
$96$	0	0
$97$	−17.6056	−1.78757	−0.893786	−	0.448493i	$-0.851961\pi$
$97$	−17.6056	−1.78757	−0.893786	+	0.448493i	$0.851961\pi$
$98$	2.30278	0.232615
$99$	0	0
$100$	−14.9083	−1.49083
$101$	−9.30278	−0.925661	−0.462830	−	0.886447i	$-0.653166\pi$
$101$	−9.30278	−0.925661	−0.462830	+	0.886447i	$0.653166\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	6.90833	0.677417
$105$	0	0
$106$	28.8167	2.79892
$107$	−7.51388	−0.726394	−0.363197	−	0.931712i	$-0.618315\pi$
$107$	−7.51388	−0.726394	−0.363197	+	0.931712i	$0.618315\pi$
$108$	0	0
$109$	−3.90833	−0.374350	−0.187175	−	0.982327i	$-0.559933\pi$
$109$	−3.90833	−0.374350	−0.187175	+	0.982327i	$0.559933\pi$
$110$	8.02776	0.765417
$111$	0	0
$112$	−0.302776	−0.0286096
$113$	−5.30278	−0.498843	−0.249422	−	0.968395i	$-0.580240\pi$
$113$	−5.30278	−0.498843	−0.249422	+	0.968395i	$0.580240\pi$
$114$	0	0
$115$	−0.697224	−0.0650165
$116$	−20.5139	−1.90467
$117$	0	0
$118$	−9.00000	−0.828517
$119$	−5.60555	−0.513860
$120$	0	0
$121$	14.0000	1.27273
$122$	−2.51388	−0.227596
$123$	0	0
$124$	9.90833	0.889794
$125$	−6.63331	−0.593301
$126$	0	0
$127$	1.69722	0.150604	0.0753022	−	0.997161i	$-0.476008\pi$
$127$	1.69722	0.150604	0.0753022	+	0.997161i	$0.476008\pi$
$128$	−18.9083	−1.67128
$129$	0	0
$130$	3.69722	0.324268
$131$	−10.3944	−0.908167	−0.454084	−	0.890959i	$-0.650033\pi$
$131$	−10.3944	−0.908167	−0.454084	+	0.890959i	$0.650033\pi$
$132$	0	0
$133$	1.60555	0.139219
$134$	−27.4222	−2.36892
$135$	0	0
$136$	16.8167	1.44202
$137$	−16.8167	−1.43674	−0.718372	−	0.695659i	$-0.755112\pi$
$137$	−16.8167	−1.43674	−0.718372	+	0.695659i	$0.755112\pi$
$138$	0	0
$139$	15.9083	1.34933	0.674663	−	0.738126i	$-0.264289\pi$
$139$	15.9083	1.34933	0.674663	+	0.738126i	$0.264289\pi$
$140$	−2.30278	−0.194620
$141$	0	0
$142$	−2.09167	−0.175529
$143$	11.5139	0.962839
$144$	0	0
$145$	−4.33053	−0.359631
$146$	−5.09167	−0.421390
$147$	0	0
$148$	−29.7250	−2.44338
$149$	−8.60555	−0.704994	−0.352497	−	0.935813i	$-0.614667\pi$
$149$	−8.60555	−0.704994	−0.352497	+	0.935813i	$0.614667\pi$
$150$	0	0
$151$	16.6056	1.35134	0.675670	−	0.737204i	$-0.263854\pi$
$151$	16.6056	1.35134	0.675670	+	0.737204i	$0.263854\pi$
$152$	−4.81665	−0.390682
$153$	0	0
$154$	−11.5139	−0.927815
$155$	2.09167	0.168007
$156$	0	0
$157$	−3.81665	−0.304602	−0.152301	−	0.988334i	$-0.548668\pi$
$157$	−3.81665	−0.304602	−0.152301	+	0.988334i	$0.548668\pi$
$158$	−2.30278	−0.183199
$159$	0	0
$160$	−3.69722	−0.292291
$161$	1.00000	0.0788110
$162$	0	0
$163$	−13.7250	−1.07502	−0.537512	−	0.843256i	$-0.680636\pi$
$163$	−13.7250	−1.07502	−0.537512	+	0.843256i	$0.680636\pi$
$164$	40.3305	3.14929
$165$	0	0
$166$	−12.9083	−1.00188
$167$	−2.81665	−0.217959	−0.108980	−	0.994044i	$-0.534758\pi$
$167$	−2.81665	−0.217959	−0.108980	+	0.994044i	$0.534758\pi$
$168$	0	0
$169$	−7.69722	−0.592094
$170$	9.00000	0.690268
$171$	0	0
$172$	18.2111	1.38858
$173$	18.2111	1.38456	0.692282	−	0.721627i	$-0.256605\pi$
$173$	18.2111	1.38456	0.692282	+	0.721627i	$0.256605\pi$
$174$	0	0
$175$	4.51388	0.341217
$176$	1.51388	0.114113
$177$	0	0
$178$	−25.1194	−1.88278
$179$	2.69722	0.201600	0.100800	−	0.994907i	$-0.467860\pi$
$179$	2.69722	0.201600	0.100800	+	0.994907i	$0.467860\pi$
$180$	0	0
$181$	−6.81665	−0.506678	−0.253339	−	0.967378i	$-0.581529\pi$
$181$	−6.81665	−0.506678	−0.253339	+	0.967378i	$0.581529\pi$
$182$	−5.30278	−0.393068
$183$	0	0
$184$	−3.00000	−0.221163
$185$	−6.27502	−0.461349
$186$	0	0
$187$	28.0278	2.04959
$188$	28.4222	2.07290
$189$	0	0
$190$	−2.57779	−0.187013
$191$	4.60555	0.333246	0.166623	−	0.986021i	$-0.446714\pi$
$191$	4.60555	0.333246	0.166623	+	0.986021i	$0.446714\pi$
$192$	0	0
$193$	9.02776	0.649832	0.324916	−	0.945743i	$-0.394664\pi$
$193$	9.02776	0.649832	0.324916	+	0.945743i	$0.394664\pi$
$194$	−40.5416	−2.91072
$195$	0	0
$196$	3.30278	0.235913
$197$	7.09167	0.505261	0.252630	−	0.967563i	$-0.418704\pi$
$197$	7.09167	0.505261	0.252630	+	0.967563i	$0.418704\pi$
$198$	0	0
$199$	12.5139	0.887085	0.443543	−	0.896253i	$-0.353721\pi$
$199$	12.5139	0.887085	0.443543	+	0.896253i	$0.353721\pi$
$200$	−13.5416	−0.957538
$201$	0	0
$202$	−21.4222	−1.50726
$203$	6.21110	0.435934
$204$	0	0
$205$	8.51388	0.594635
$206$	−9.21110	−0.641768
$207$	0	0
$208$	0.697224	0.0483438
$209$	−8.02776	−0.555292
$210$	0	0
$211$	−27.4222	−1.88782	−0.943911	−	0.330199i	$-0.892884\pi$
$211$	−27.4222	−1.88782	−0.943911	+	0.330199i	$0.892884\pi$
$212$	41.3305	2.83859
$213$	0	0
$214$	−17.3028	−1.18279
$215$	3.84441	0.262187
$216$	0	0
$217$	−3.00000	−0.203653
$218$	−9.00000	−0.609557
$219$	0	0
$220$	11.5139	0.776266
$221$	12.9083	0.868308
$222$	0	0
$223$	19.9083	1.33316	0.666580	−	0.745433i	$-0.267757\pi$
$223$	19.9083	1.33316	0.666580	+	0.745433i	$0.267757\pi$
$224$	5.30278	0.354307
$225$	0	0
$226$	−12.2111	−0.812270
$227$	20.3305	1.34938	0.674692	−	0.738099i	$-0.264276\pi$
$227$	20.3305	1.34938	0.674692	+	0.738099i	$0.264276\pi$
$228$	0	0
$229$	2.51388	0.166122	0.0830609	−	0.996544i	$-0.473530\pi$
$229$	2.51388	0.166122	0.0830609	+	0.996544i	$0.473530\pi$
$230$	−1.60555	−0.105867
$231$	0	0
$232$	−18.6333	−1.22334
$233$	14.3305	0.938824	0.469412	−	0.882979i	$-0.344466\pi$
$233$	14.3305	0.938824	0.469412	+	0.882979i	$0.344466\pi$
$234$	0	0
$235$	6.00000	0.391397
$236$	−12.9083	−0.840261
$237$	0	0
$238$	−12.9083	−0.836723
$239$	14.0917	0.911515	0.455757	−	0.890104i	$-0.349369\pi$
$239$	14.0917	0.911515	0.455757	+	0.890104i	$0.349369\pi$
$240$	0	0
$241$	12.0278	0.774776	0.387388	−	0.921917i	$-0.373377\pi$
$241$	12.0278	0.774776	0.387388	+	0.921917i	$0.373377\pi$
$242$	32.2389	2.07239
$243$	0	0
$244$	−3.60555	−0.230822
$245$	0.697224	0.0445440
$246$	0	0
$247$	−3.69722	−0.235249
$248$	9.00000	0.571501
$249$	0	0
$250$	−15.2750	−0.966077
$251$	−23.8167	−1.50329	−0.751647	−	0.659566i	$-0.770740\pi$
$251$	−23.8167	−1.50329	−0.751647	+	0.659566i	$0.770740\pi$
$252$	0	0
$253$	−5.00000	−0.314347
$254$	3.90833	0.245230
$255$	0	0
$256$	−17.9083	−1.11927
$257$	−27.0278	−1.68595	−0.842973	−	0.537957i	$-0.819196\pi$
$257$	−27.0278	−1.68595	−0.842973	+	0.537957i	$0.819196\pi$
$258$	0	0
$259$	9.00000	0.559233
$260$	5.30278	0.328864
$261$	0	0
$262$	−23.9361	−1.47878
$263$	−10.3944	−0.640949	−0.320475	−	0.947257i	$-0.603842\pi$
$263$	−10.3944	−0.640949	−0.320475	+	0.947257i	$0.603842\pi$
$264$	0	0
$265$	8.72498	0.535971
$266$	3.69722	0.226691
$267$	0	0
$268$	−39.3305	−2.40249
$269$	−0.908327	−0.0553817	−0.0276908	−	0.999617i	$-0.508815\pi$
$269$	−0.908327	−0.0553817	−0.0276908	+	0.999617i	$0.508815\pi$
$270$	0	0
$271$	3.60555	0.219022	0.109511	−	0.993986i	$-0.465072\pi$
$271$	3.60555	0.219022	0.109511	+	0.993986i	$0.465072\pi$
$272$	1.69722	0.102909
$273$	0	0
$274$	−38.7250	−2.33946
$275$	−22.5694	−1.36099
$276$	0	0
$277$	−12.3028	−0.739202	−0.369601	−	0.929191i	$-0.620506\pi$
$277$	−12.3028	−0.739202	−0.369601	+	0.929191i	$0.620506\pi$
$278$	36.6333	2.19712
$279$	0	0
$280$	−2.09167	−0.125001
$281$	−27.8167	−1.65940	−0.829701	−	0.558208i	$-0.811489\pi$
$281$	−27.8167	−1.65940	−0.829701	+	0.558208i	$0.811489\pi$
$282$	0	0
$283$	17.3028	1.02854	0.514272	−	0.857627i	$-0.328062\pi$
$283$	17.3028	1.02854	0.514272	+	0.857627i	$0.328062\pi$
$284$	−3.00000	−0.178017
$285$	0	0
$286$	26.5139	1.56780
$287$	−12.2111	−0.720799
$288$	0	0
$289$	14.4222	0.848365
$290$	−9.97224	−0.585590
$291$	0	0
$292$	−7.30278	−0.427363
$293$	24.8444	1.45143	0.725713	−	0.687998i	$-0.241510\pi$
$293$	24.8444	1.45143	0.725713	+	0.687998i	$0.241510\pi$
$294$	0	0
$295$	−2.72498	−0.158655
$296$	−27.0000	−1.56934
$297$	0	0
$298$	−19.8167	−1.14795
$299$	−2.30278	−0.133173
$300$	0	0
$301$	−5.51388	−0.317815
$302$	38.2389	2.20040
$303$	0	0
$304$	−0.486122	−0.0278810
$305$	−0.761141	−0.0435828
$306$	0	0
$307$	18.2111	1.03936	0.519681	−	0.854360i	$-0.326051\pi$
$307$	18.2111	1.03936	0.519681	+	0.854360i	$0.326051\pi$
$308$	−16.5139	−0.940966
$309$	0	0
$310$	4.81665	0.273568
$311$	9.51388	0.539483	0.269741	−	0.962933i	$-0.413062\pi$
$311$	9.51388	0.539483	0.269741	+	0.962933i	$0.413062\pi$
$312$	0	0
$313$	1.21110	0.0684556	0.0342278	−	0.999414i	$-0.489103\pi$
$313$	1.21110	0.0684556	0.0342278	+	0.999414i	$0.489103\pi$
$314$	−8.78890	−0.495986
$315$	0	0
$316$	−3.30278	−0.185796
$317$	30.5139	1.71383	0.856915	−	0.515458i	$-0.172378\pi$
$317$	30.5139	1.71383	0.856915	+	0.515458i	$0.172378\pi$
$318$	0	0
$319$	−31.0555	−1.73877
$320$	−8.93608	−0.499542
$321$	0	0
$322$	2.30278	0.128329
$323$	−9.00000	−0.500773
$324$	0	0
$325$	−10.3944	−0.576580
$326$	−31.6056	−1.75047
$327$	0	0
$328$	36.6333	2.02274
$329$	−8.60555	−0.474439
$330$	0	0
$331$	17.6333	0.969214	0.484607	−	0.874732i	$-0.338963\pi$
$331$	17.6333	0.969214	0.484607	+	0.874732i	$0.338963\pi$
$332$	−18.5139	−1.01608
$333$	0	0
$334$	−6.48612	−0.354905
$335$	−8.30278	−0.453629
$336$	0	0
$337$	−15.6972	−0.855082	−0.427541	−	0.903996i	$-0.640620\pi$
$337$	−15.6972	−0.855082	−0.427541	+	0.903996i	$0.640620\pi$
$338$	−17.7250	−0.964112
$339$	0	0
$340$	12.9083	0.700052
$341$	15.0000	0.812296
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	16.5416	0.891865
$345$	0	0
$346$	41.9361	2.25450
$347$	−14.6333	−0.785557	−0.392779	−	0.919633i	$-0.628486\pi$
$347$	−14.6333	−0.785557	−0.392779	+	0.919633i	$0.628486\pi$
$348$	0	0
$349$	−10.0917	−0.540195	−0.270097	−	0.962833i	$-0.587056\pi$
$349$	−10.0917	−0.540195	−0.270097	+	0.962833i	$0.587056\pi$
$350$	10.3944	0.555607
$351$	0	0
$352$	−26.5139	−1.41319
$353$	11.0000	0.585471	0.292735	−	0.956193i	$-0.405434\pi$
$353$	11.0000	0.585471	0.292735	+	0.956193i	$0.405434\pi$
$354$	0	0
$355$	−0.633308	−0.0336125
$356$	−36.0278	−1.90947
$357$	0	0
$358$	6.21110	0.328267
$359$	35.5416	1.87582	0.937908	−	0.346884i	$-0.112760\pi$
$359$	35.5416	1.87582	0.937908	+	0.346884i	$0.112760\pi$
$360$	0	0
$361$	−16.4222	−0.864327
$362$	−15.6972	−0.825028
$363$	0	0
$364$	−7.60555	−0.398639
$365$	−1.54163	−0.0806928
$366$	0	0
$367$	21.5139	1.12302	0.561508	−	0.827472i	$-0.310222\pi$
$367$	21.5139	1.12302	0.561508	+	0.827472i	$0.310222\pi$
$368$	−0.302776	−0.0157833
$369$	0	0
$370$	−14.4500	−0.751218
$371$	−12.5139	−0.649688
$372$	0	0
$373$	16.2111	0.839379	0.419690	−	0.907668i	$-0.362139\pi$
$373$	16.2111	0.839379	0.419690	+	0.907668i	$0.362139\pi$
$374$	64.5416	3.33737
$375$	0	0
$376$	25.8167	1.33139
$377$	−14.3028	−0.736630
$378$	0	0
$379$	−22.4222	−1.15175	−0.575876	−	0.817537i	$-0.695339\pi$
$379$	−22.4222	−1.15175	−0.575876	+	0.817537i	$0.695339\pi$
$380$	−3.69722	−0.189664
$381$	0	0
$382$	10.6056	0.542627
$383$	−4.81665	−0.246120	−0.123060	−	0.992399i	$-0.539271\pi$
$383$	−4.81665	−0.246120	−0.123060	+	0.992399i	$0.539271\pi$
$384$	0	0
$385$	−3.48612	−0.177669
$386$	20.7889	1.05813
$387$	0	0
$388$	−58.1472	−2.95198
$389$	36.6333	1.85738	0.928691	−	0.370854i	$-0.120935\pi$
$389$	36.6333	1.85738	0.928691	+	0.370854i	$0.120935\pi$
$390$	0	0
$391$	−5.60555	−0.283485
$392$	3.00000	0.151523
$393$	0	0
$394$	16.3305	0.822720
$395$	−0.697224	−0.0350812
$396$	0	0
$397$	−27.3944	−1.37489	−0.687444	−	0.726237i	$-0.741267\pi$
$397$	−27.3944	−1.37489	−0.687444	+	0.726237i	$0.741267\pi$
$398$	28.8167	1.44445
$399$	0	0
$400$	−1.36669	−0.0683346
$401$	13.4222	0.670273	0.335136	−	0.942170i	$-0.391218\pi$
$401$	13.4222	0.670273	0.335136	+	0.942170i	$0.391218\pi$
$402$	0	0
$403$	6.90833	0.344128
$404$	−30.7250	−1.52862
$405$	0	0
$406$	14.3028	0.709835
$407$	−45.0000	−2.23057
$408$	0	0
$409$	24.0278	1.18810	0.594048	−	0.804430i	$-0.297529\pi$
$409$	24.0278	1.18810	0.594048	+	0.804430i	$0.297529\pi$
$410$	19.6056	0.968249
$411$	0	0
$412$	−13.2111	−0.650864
$413$	3.90833	0.192316
$414$	0	0
$415$	−3.90833	−0.191852
$416$	−12.2111	−0.598699
$417$	0	0
$418$	−18.4861	−0.904186
$419$	−15.1194	−0.738632	−0.369316	−	0.929304i	$-0.620408\pi$
$419$	−15.1194	−0.738632	−0.369316	+	0.929304i	$0.620408\pi$
$420$	0	0
$421$	9.69722	0.472614	0.236307	−	0.971678i	$-0.424063\pi$
$421$	9.69722	0.472614	0.236307	+	0.971678i	$0.424063\pi$
$422$	−63.1472	−3.07396
$423$	0	0
$424$	37.5416	1.82318
$425$	−25.3028	−1.22736
$426$	0	0
$427$	1.09167	0.0528298
$428$	−24.8167	−1.19956
$429$	0	0
$430$	8.85281	0.426921
$431$	8.72498	0.420268	0.210134	−	0.977673i	$-0.432610\pi$
$431$	8.72498	0.420268	0.210134	+	0.977673i	$0.432610\pi$
$432$	0	0
$433$	−21.0278	−1.01053	−0.505265	−	0.862964i	$-0.668605\pi$
$433$	−21.0278	−1.01053	−0.505265	+	0.862964i	$0.668605\pi$
$434$	−6.90833	−0.331610
$435$	0	0
$436$	−12.9083	−0.618197
$437$	1.60555	0.0768039
$438$	0	0
$439$	20.2111	0.964623	0.482312	−	0.876000i	$-0.339797\pi$
$439$	20.2111	0.964623	0.482312	+	0.876000i	$0.339797\pi$
$440$	10.4584	0.498583
$441$	0	0
$442$	29.7250	1.41387
$443$	32.8444	1.56049	0.780243	−	0.625477i	$-0.215096\pi$
$443$	32.8444	1.56049	0.780243	+	0.625477i	$0.215096\pi$
$444$	0	0
$445$	−7.60555	−0.360538
$446$	45.8444	2.17080
$447$	0	0
$448$	12.8167	0.605530
$449$	35.7250	1.68597	0.842983	−	0.537940i	$-0.180797\pi$
$449$	35.7250	1.68597	0.842983	+	0.537940i	$0.180797\pi$
$450$	0	0
$451$	61.0555	2.87499
$452$	−17.5139	−0.823784
$453$	0	0
$454$	46.8167	2.19721
$455$	−1.60555	−0.0752694
$456$	0	0
$457$	−10.8806	−0.508972	−0.254486	−	0.967077i	$-0.581906\pi$
$457$	−10.8806	−0.508972	−0.254486	+	0.967077i	$0.581906\pi$
$458$	5.78890	0.270497
$459$	0	0
$460$	−2.30278	−0.107367
$461$	−7.72498	−0.359788	−0.179894	−	0.983686i	$-0.557576\pi$
$461$	−7.72498	−0.359788	−0.179894	+	0.983686i	$0.557576\pi$
$462$	0	0
$463$	−18.8167	−0.874484	−0.437242	−	0.899344i	$-0.644045\pi$
$463$	−18.8167	−0.874484	−0.437242	+	0.899344i	$0.644045\pi$
$464$	−1.88057	−0.0873033
$465$	0	0
$466$	33.0000	1.52870
$467$	−23.6056	−1.09233	−0.546167	−	0.837676i	$-0.683914\pi$
$467$	−23.6056	−1.09233	−0.546167	+	0.837676i	$0.683914\pi$
$468$	0	0
$469$	11.9083	0.549875
$470$	13.8167	0.637315
$471$	0	0
$472$	−11.7250	−0.539686
$473$	27.5694	1.26764
$474$	0	0
$475$	7.24726	0.332527
$476$	−18.5139	−0.848582
$477$	0	0
$478$	32.4500	1.48423
$479$	18.2111	0.832087	0.416043	−	0.909345i	$-0.363416\pi$
$479$	18.2111	0.832087	0.416043	+	0.909345i	$0.363416\pi$
$480$	0	0
$481$	−20.7250	−0.944978
$482$	27.6972	1.26157
$483$	0	0
$484$	46.2389	2.10177
$485$	−12.2750	−0.557380
$486$	0	0
$487$	−0.816654	−0.0370061	−0.0185031	−	0.999829i	$-0.505890\pi$
$487$	−0.816654	−0.0370061	−0.0185031	+	0.999829i	$0.505890\pi$
$488$	−3.27502	−0.148253
$489$	0	0
$490$	1.60555	0.0725314
$491$	17.9361	0.809444	0.404722	−	0.914440i	$-0.367368\pi$
$491$	17.9361	0.809444	0.404722	+	0.914440i	$0.367368\pi$
$492$	0	0
$493$	−34.8167	−1.56806
$494$	−8.51388	−0.383057
$495$	0	0
$496$	0.908327	0.0407851
$497$	0.908327	0.0407440
$498$	0	0
$499$	31.9083	1.42841	0.714206	−	0.699935i	$-0.246788\pi$
$499$	31.9083	1.42841	0.714206	+	0.699935i	$0.246788\pi$
$500$	−21.9083	−0.979770
$501$	0	0
$502$	−54.8444	−2.44783
$503$	−17.7250	−0.790318	−0.395159	−	0.918613i	$-0.629310\pi$
$503$	−17.7250	−0.790318	−0.395159	+	0.918613i	$0.629310\pi$
$504$	0	0
$505$	−6.48612	−0.288629
$506$	−11.5139	−0.511854
$507$	0	0
$508$	5.60555	0.248706
$509$	33.4500	1.48264	0.741322	−	0.671150i	$-0.234199\pi$
$509$	33.4500	1.48264	0.741322	+	0.671150i	$0.234199\pi$
$510$	0	0
$511$	2.21110	0.0978134
$512$	−3.42221	−0.151242
$513$	0	0
$514$	−62.2389	−2.74524
$515$	−2.78890	−0.122894
$516$	0	0
$517$	43.0278	1.89236
$518$	20.7250	0.910603
$519$	0	0
$520$	4.81665	0.211224
$521$	−21.6333	−0.947772	−0.473886	−	0.880586i	$-0.657149\pi$
$521$	−21.6333	−0.947772	−0.473886	+	0.880586i	$0.657149\pi$
$522$	0	0
$523$	−0.422205	−0.0184617	−0.00923087	−	0.999957i	$-0.502938\pi$
$523$	−0.422205	−0.0184617	−0.00923087	+	0.999957i	$0.502938\pi$
$524$	−34.3305	−1.49974
$525$	0	0
$526$	−23.9361	−1.04366
$527$	16.8167	0.732545
$528$	0	0
$529$	1.00000	0.0434783
$530$	20.0917	0.872727
$531$	0	0
$532$	5.30278	0.229904
$533$	28.1194	1.21799
$534$	0	0
$535$	−5.23886	−0.226496
$536$	−35.7250	−1.54308
$537$	0	0
$538$	−2.09167	−0.0901784
$539$	5.00000	0.215365
$540$	0	0
$541$	−5.81665	−0.250077	−0.125039	−	0.992152i	$-0.539906\pi$
$541$	−5.81665	−0.250077	−0.125039	+	0.992152i	$0.539906\pi$
$542$	8.30278	0.356635
$543$	0	0
$544$	−29.7250	−1.27445
$545$	−2.72498	−0.116725
$546$	0	0
$547$	−7.72498	−0.330296	−0.165148	−	0.986269i	$-0.552810\pi$
$547$	−7.72498	−0.330296	−0.165148	+	0.986269i	$0.552810\pi$
$548$	−55.5416	−2.37262
$549$	0	0
$550$	−51.9722	−2.21610
$551$	9.97224	0.424832
$552$	0	0
$553$	1.00000	0.0425243
$554$	−28.3305	−1.20365
$555$	0	0
$556$	52.5416	2.22826
$557$	−5.02776	−0.213033	−0.106516	−	0.994311i	$-0.533970\pi$
$557$	−5.02776	−0.213033	−0.106516	+	0.994311i	$0.533970\pi$
$558$	0	0
$559$	12.6972	0.537035
$560$	−0.211103	−0.00892071
$561$	0	0
$562$	−64.0555	−2.70202
$563$	43.3305	1.82616	0.913082	−	0.407776i	$-0.133695\pi$
$563$	43.3305	1.82616	0.913082	+	0.407776i	$0.133695\pi$
$564$	0	0
$565$	−3.69722	−0.155543
$566$	39.8444	1.67479
$567$	0	0
$568$	−2.72498	−0.114338
$569$	−10.7889	−0.452294	−0.226147	−	0.974093i	$-0.572613\pi$
$569$	−10.7889	−0.452294	−0.226147	+	0.974093i	$0.572613\pi$
$570$	0	0
$571$	10.2111	0.427321	0.213661	−	0.976908i	$-0.431461\pi$
$571$	10.2111	0.427321	0.213661	+	0.976908i	$0.431461\pi$
$572$	38.0278	1.59002
$573$	0	0
$574$	−28.1194	−1.17368
$575$	4.51388	0.188242
$576$	0	0
$577$	−36.4222	−1.51628	−0.758138	−	0.652094i	$-0.773891\pi$
$577$	−36.4222	−1.51628	−0.758138	+	0.652094i	$0.773891\pi$
$578$	33.2111	1.38140
$579$	0	0
$580$	−14.3028	−0.593890
$581$	5.60555	0.232557
$582$	0	0
$583$	62.5694	2.59136
$584$	−6.63331	−0.274488
$585$	0	0
$586$	57.2111	2.36337
$587$	39.1472	1.61578	0.807889	−	0.589335i	$-0.200610\pi$
$587$	39.1472	1.61578	0.807889	+	0.589335i	$0.200610\pi$
$588$	0	0
$589$	−4.81665	−0.198467
$590$	−6.27502	−0.258338
$591$	0	0
$592$	−2.72498	−0.111996
$593$	−37.0278	−1.52055	−0.760274	−	0.649603i	$-0.774935\pi$
$593$	−37.0278	−1.52055	−0.760274	+	0.649603i	$0.774935\pi$
$594$	0	0
$595$	−3.90833	−0.160226
$596$	−28.4222	−1.16422
$597$	0	0
$598$	−5.30278	−0.216847
$599$	39.5139	1.61449	0.807247	−	0.590214i	$-0.200957\pi$
$599$	39.5139	1.61449	0.807247	+	0.590214i	$0.200957\pi$
$600$	0	0
$601$	40.9361	1.66982	0.834909	−	0.550388i	$-0.185520\pi$
$601$	40.9361	1.66982	0.834909	+	0.550388i	$0.185520\pi$
$602$	−12.6972	−0.517500
$603$	0	0
$604$	54.8444	2.23159
$605$	9.76114	0.396847
$606$	0	0
$607$	−24.4861	−0.993861	−0.496931	−	0.867790i	$-0.665540\pi$
$607$	−24.4861	−0.993861	−0.496931	+	0.867790i	$0.665540\pi$
$608$	8.51388	0.345283
$609$	0	0
$610$	−1.75274	−0.0709663
$611$	19.8167	0.801696
$612$	0	0
$613$	2.81665	0.113764	0.0568818	−	0.998381i	$-0.481884\pi$
$613$	2.81665	0.113764	0.0568818	+	0.998381i	$0.481884\pi$
$614$	41.9361	1.69240
$615$	0	0
$616$	−15.0000	−0.604367
$617$	−14.7250	−0.592805	−0.296403	−	0.955063i	$-0.595787\pi$
$617$	−14.7250	−0.592805	−0.296403	+	0.955063i	$0.595787\pi$
$618$	0	0
$619$	9.88057	0.397134	0.198567	−	0.980087i	$-0.436371\pi$
$619$	9.88057	0.397134	0.198567	+	0.980087i	$0.436371\pi$
$620$	6.90833	0.277445
$621$	0	0
$622$	21.9083	0.878444
$623$	10.9083	0.437033
$624$	0	0
$625$	17.9445	0.717779
$626$	2.78890	0.111467
$627$	0	0
$628$	−12.6056	−0.503016
$629$	−50.4500	−2.01157
$630$	0	0
$631$	−11.9722	−0.476607	−0.238304	−	0.971191i	$-0.576591\pi$
$631$	−11.9722	−0.476607	−0.238304	+	0.971191i	$0.576591\pi$
$632$	−3.00000	−0.119334
$633$	0	0
$634$	70.2666	2.79064
$635$	1.18335	0.0469597
$636$	0	0
$637$	2.30278	0.0912393
$638$	−71.5139	−2.83126
$639$	0	0
$640$	−13.1833	−0.521118
$641$	10.5416	0.416370	0.208185	−	0.978090i	$-0.433244\pi$
$641$	10.5416	0.416370	0.208185	+	0.978090i	$0.433244\pi$
$642$	0	0
$643$	43.5694	1.71821	0.859105	−	0.511800i	$-0.171021\pi$
$643$	43.5694	1.71821	0.859105	+	0.511800i	$0.171021\pi$
$644$	3.30278	0.130148
$645$	0	0
$646$	−20.7250	−0.815413
$647$	−16.3305	−0.642019	−0.321010	−	0.947076i	$-0.604022\pi$
$647$	−16.3305	−0.642019	−0.321010	+	0.947076i	$0.604022\pi$
$648$	0	0
$649$	−19.5416	−0.767076
$650$	−23.9361	−0.938850
$651$	0	0
$652$	−45.3305	−1.77528
$653$	−30.5139	−1.19410	−0.597050	−	0.802204i	$-0.703661\pi$
$653$	−30.5139	−1.19410	−0.597050	+	0.802204i	$0.703661\pi$
$654$	0	0
$655$	−7.24726	−0.283174
$656$	3.69722	0.144352
$657$	0	0
$658$	−19.8167	−0.772534
$659$	−42.6333	−1.66076	−0.830379	−	0.557199i	$-0.811876\pi$
$659$	−42.6333	−1.66076	−0.830379	+	0.557199i	$0.811876\pi$
$660$	0	0
$661$	20.8167	0.809674	0.404837	−	0.914389i	$-0.367328\pi$
$661$	20.8167	0.809674	0.404837	+	0.914389i	$0.367328\pi$
$662$	40.6056	1.57818
$663$	0	0
$664$	−16.8167	−0.652613
$665$	1.11943	0.0434096
$666$	0	0
$667$	6.21110	0.240495
$668$	−9.30278	−0.359935
$669$	0	0
$670$	−19.1194	−0.738648
$671$	−5.45837	−0.210718
$672$	0	0
$673$	−26.6333	−1.02664	−0.513319	−	0.858198i	$-0.671584\pi$
$673$	−26.6333	−1.02664	−0.513319	+	0.858198i	$0.671584\pi$
$674$	−36.1472	−1.39234
$675$	0	0
$676$	−25.4222	−0.977777
$677$	−37.1472	−1.42768	−0.713841	−	0.700308i	$-0.753046\pi$
$677$	−37.1472	−1.42768	−0.713841	+	0.700308i	$0.753046\pi$
$678$	0	0
$679$	17.6056	0.675639
$680$	11.7250	0.449632
$681$	0	0
$682$	34.5416	1.32267
$683$	1.42221	0.0544192	0.0272096	−	0.999630i	$-0.491338\pi$
$683$	1.42221	0.0544192	0.0272096	+	0.999630i	$0.491338\pi$
$684$	0	0
$685$	−11.7250	−0.447988
$686$	−2.30278	−0.0879204
$687$	0	0
$688$	1.66947	0.0636479
$689$	28.8167	1.09783
$690$	0	0
$691$	−31.5139	−1.19884	−0.599422	−	0.800433i	$-0.704603\pi$
$691$	−31.5139	−1.19884	−0.599422	+	0.800433i	$0.704603\pi$
$692$	60.1472	2.28645
$693$	0	0
$694$	−33.6972	−1.27913
$695$	11.0917	0.420731
$696$	0	0
$697$	68.4500	2.59273
$698$	−23.2389	−0.879604
$699$	0	0
$700$	14.9083	0.563482
$701$	25.5416	0.964694	0.482347	−	0.875980i	$-0.339784\pi$
$701$	25.5416	0.964694	0.482347	+	0.875980i	$0.339784\pi$
$702$	0	0
$703$	14.4500	0.544991
$704$	−64.0833	−2.41523
$705$	0	0
$706$	25.3305	0.953327
$707$	9.30278	0.349867
$708$	0	0
$709$	−47.3305	−1.77754	−0.888768	−	0.458358i	$-0.848438\pi$
$709$	−47.3305	−1.77754	−0.888768	+	0.458358i	$0.848438\pi$
$710$	−1.45837	−0.0547315
$711$	0	0
$712$	−32.7250	−1.22642
$713$	−3.00000	−0.112351
$714$	0	0
$715$	8.02776	0.300221
$716$	8.90833	0.332920
$717$	0	0
$718$	81.8444	3.05441
$719$	−14.2111	−0.529985	−0.264992	−	0.964251i	$-0.585369\pi$
$719$	−14.2111	−0.529985	−0.264992	+	0.964251i	$0.585369\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	−37.8167	−1.40739
$723$	0	0
$724$	−22.5139	−0.836722
$725$	28.0362	1.04124
$726$	0	0
$727$	−46.8722	−1.73839	−0.869196	−	0.494467i	$-0.835363\pi$
$727$	−46.8722	−1.73839	−0.869196	+	0.494467i	$0.835363\pi$
$728$	−6.90833	−0.256040
$729$	0	0
$730$	−3.55004	−0.131393
$731$	30.9083	1.14319
$732$	0	0
$733$	−26.6056	−0.982698	−0.491349	−	0.870963i	$-0.663496\pi$
$733$	−26.6056	−0.982698	−0.491349	+	0.870963i	$0.663496\pi$
$734$	49.5416	1.82862
$735$	0	0
$736$	5.30278	0.195463
$737$	−59.5416	−2.19324
$738$	0	0
$739$	−43.4222	−1.59731	−0.798656	−	0.601788i	$-0.794455\pi$
$739$	−43.4222	−1.59731	−0.798656	+	0.601788i	$0.794455\pi$
$740$	−20.7250	−0.761865
$741$	0	0
$742$	−28.8167	−1.05789
$743$	−30.5139	−1.11945	−0.559723	−	0.828680i	$-0.689092\pi$
$743$	−30.5139	−1.11945	−0.559723	+	0.828680i	$0.689092\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	37.3305	1.36677
$747$	0	0
$748$	92.5694	3.38467
$749$	7.51388	0.274551
$750$	0	0
$751$	8.69722	0.317366	0.158683	−	0.987330i	$-0.449275\pi$
$751$	8.69722	0.317366	0.158683	+	0.987330i	$0.449275\pi$
$752$	2.60555	0.0950147
$753$	0	0
$754$	−32.9361	−1.19946
$755$	11.5778	0.421359
$756$	0	0
$757$	−3.36669	−0.122365	−0.0611823	−	0.998127i	$-0.519487\pi$
$757$	−3.36669	−0.122365	−0.0611823	+	0.998127i	$0.519487\pi$
$758$	−51.6333	−1.87541
$759$	0	0
$760$	−3.35829	−0.121818
$761$	−12.0000	−0.435000	−0.217500	−	0.976060i	$-0.569790\pi$
$761$	−12.0000	−0.435000	−0.217500	+	0.976060i	$0.569790\pi$
$762$	0	0
$763$	3.90833	0.141491
$764$	15.2111	0.550318
$765$	0	0
$766$	−11.0917	−0.400758
$767$	−9.00000	−0.324971
$768$	0	0
$769$	−27.4500	−0.989871	−0.494935	−	0.868930i	$-0.664808\pi$
$769$	−27.4500	−0.989871	−0.494935	+	0.868930i	$0.664808\pi$
$770$	−8.02776	−0.289300
$771$	0	0
$772$	29.8167	1.07312
$773$	−31.8444	−1.14536	−0.572682	−	0.819778i	$-0.694097\pi$
$773$	−31.8444	−1.14536	−0.572682	+	0.819778i	$0.694097\pi$
$774$	0	0
$775$	−13.5416	−0.486430
$776$	−52.8167	−1.89601
$777$	0	0
$778$	84.3583	3.02439
$779$	−19.6056	−0.702442
$780$	0	0
$781$	−4.54163	−0.162512
$782$	−12.9083	−0.461601
$783$	0	0
$784$	0.302776	0.0108134
$785$	−2.66106	−0.0949774
$786$	0	0
$787$	32.1472	1.14592	0.572962	−	0.819582i	$-0.305794\pi$
$787$	32.1472	1.14592	0.572962	+	0.819582i	$0.305794\pi$
$788$	23.4222	0.834382
$789$	0	0
$790$	−1.60555	−0.0571230
$791$	5.30278	0.188545
$792$	0	0
$793$	−2.51388	−0.0892704
$794$	−63.0833	−2.23874
$795$	0	0
$796$	41.3305	1.46492
$797$	39.0278	1.38243	0.691217	−	0.722647i	$-0.257075\pi$
$797$	39.0278	1.38243	0.691217	+	0.722647i	$0.257075\pi$
$798$	0	0
$799$	48.2389	1.70657
$800$	23.9361	0.846268
$801$	0	0
$802$	30.9083	1.09141
$803$	−11.0555	−0.390141
$804$	0	0
$805$	0.697224	0.0245739
$806$	15.9083	0.560347
$807$	0	0
$808$	−27.9083	−0.981812
$809$	−25.9361	−0.911864	−0.455932	−	0.890015i	$-0.650694\pi$
$809$	−25.9361	−0.911864	−0.455932	+	0.890015i	$0.650694\pi$
$810$	0	0
$811$	−10.3944	−0.364998	−0.182499	−	0.983206i	$-0.558419\pi$
$811$	−10.3944	−0.364998	−0.182499	+	0.983206i	$0.558419\pi$
$812$	20.5139	0.719896
$813$	0	0
$814$	−103.625	−3.63205
$815$	−9.56939	−0.335201
$816$	0	0
$817$	−8.85281	−0.309721
$818$	55.3305	1.93459
$819$	0	0
$820$	28.1194	0.981973
$821$	19.3944	0.676871	0.338435	−	0.940990i	$-0.390102\pi$
$821$	19.3944	0.676871	0.338435	+	0.940990i	$0.390102\pi$
$822$	0	0
$823$	6.51388	0.227060	0.113530	−	0.993535i	$-0.463784\pi$
$823$	6.51388	0.227060	0.113530	+	0.993535i	$0.463784\pi$
$824$	−12.0000	−0.418040
$825$	0	0
$826$	9.00000	0.313150
$827$	9.90833	0.344546	0.172273	−	0.985049i	$-0.444889\pi$
$827$	9.90833	0.344546	0.172273	+	0.985049i	$0.444889\pi$
$828$	0	0
$829$	−30.4500	−1.05757	−0.528785	−	0.848756i	$-0.677352\pi$
$829$	−30.4500	−1.05757	−0.528785	+	0.848756i	$0.677352\pi$
$830$	−9.00000	−0.312395
$831$	0	0
$832$	−29.5139	−1.02321
$833$	5.60555	0.194221
$834$	0	0
$835$	−1.96384	−0.0679615
$836$	−26.5139	−0.917002
$837$	0	0
$838$	−34.8167	−1.20272
$839$	−38.3028	−1.32236	−0.661179	−	0.750228i	$-0.729944\pi$
$839$	−38.3028	−1.32236	−0.661179	+	0.750228i	$0.729944\pi$
$840$	0	0
$841$	9.57779	0.330269
$842$	22.3305	0.769561
$843$	0	0
$844$	−90.5694	−3.11753
$845$	−5.36669	−0.184620
$846$	0	0
$847$	−14.0000	−0.481046
$848$	3.78890	0.130111
$849$	0	0
$850$	−58.2666	−1.99853
$851$	9.00000	0.308516
$852$	0	0
$853$	−58.2389	−1.99406	−0.997030	−	0.0770106i	$-0.975462\pi$
$853$	−58.2389	−1.99406	−0.997030	+	0.0770106i	$0.975462\pi$
$854$	2.51388	0.0860231
$855$	0	0
$856$	−22.5416	−0.770457
$857$	50.2389	1.71613	0.858063	−	0.513544i	$-0.171668\pi$
$857$	50.2389	1.71613	0.858063	+	0.513544i	$0.171668\pi$
$858$	0	0
$859$	43.2111	1.47434	0.737172	−	0.675705i	$-0.236161\pi$
$859$	43.2111	1.47434	0.737172	+	0.675705i	$0.236161\pi$
$860$	12.6972	0.432972
$861$	0	0
$862$	20.0917	0.684325
$863$	5.76114	0.196112	0.0980558	−	0.995181i	$-0.468738\pi$
$863$	5.76114	0.196112	0.0980558	+	0.995181i	$0.468738\pi$
$864$	0	0
$865$	12.6972	0.431719
$866$	−48.4222	−1.64545
$867$	0	0
$868$	−9.90833	−0.336311
$869$	−5.00000	−0.169613
$870$	0	0
$871$	−27.4222	−0.929166
$872$	−11.7250	−0.397058
$873$	0	0
$874$	3.69722	0.125060
$875$	6.63331	0.224247
$876$	0	0
$877$	−14.7889	−0.499386	−0.249693	−	0.968325i	$-0.580330\pi$
$877$	−14.7889	−0.499386	−0.249693	+	0.968325i	$0.580330\pi$
$878$	46.5416	1.57070
$879$	0	0
$880$	1.05551	0.0355813
$881$	43.8167	1.47622	0.738110	−	0.674680i	$-0.235718\pi$
$881$	43.8167	1.47622	0.738110	+	0.674680i	$0.235718\pi$
$882$	0	0
$883$	−18.4861	−0.622108	−0.311054	−	0.950392i	$-0.600682\pi$
$883$	−18.4861	−0.622108	−0.311054	+	0.950392i	$0.600682\pi$
$884$	42.6333	1.43391
$885$	0	0
$886$	75.6333	2.54095
$887$	11.9361	0.400774	0.200387	−	0.979717i	$-0.435780\pi$
$887$	11.9361	0.400774	0.200387	+	0.979717i	$0.435780\pi$
$888$	0	0
$889$	−1.69722	−0.0569231
$890$	−17.5139	−0.587067
$891$	0	0
$892$	65.7527	2.20156
$893$	−13.8167	−0.462357
$894$	0	0
$895$	1.88057	0.0628605
$896$	18.9083	0.631683
$897$	0	0
$898$	82.2666	2.74527
$899$	−18.6333	−0.621456
$900$	0	0
$901$	70.1472	2.33694
$902$	140.597	4.68137
$903$	0	0
$904$	−15.9083	−0.529103
$905$	−4.75274	−0.157986
$906$	0	0
$907$	−8.14719	−0.270523	−0.135261	−	0.990810i	$-0.543187\pi$
$907$	−8.14719	−0.270523	−0.135261	+	0.990810i	$0.543187\pi$
$908$	67.1472	2.22836
$909$	0	0
$910$	−3.69722	−0.122562
$911$	24.2389	0.803069	0.401535	−	0.915844i	$-0.368477\pi$
$911$	24.2389	0.803069	0.401535	+	0.915844i	$0.368477\pi$
$912$	0	0
$913$	−28.0278	−0.927583
$914$	−25.0555	−0.828763
$915$	0	0
$916$	8.30278	0.274331
$917$	10.3944	0.343255
$918$	0	0
$919$	−10.3944	−0.342881	−0.171441	−	0.985194i	$-0.554842\pi$
$919$	−10.3944	−0.342881	−0.171441	+	0.985194i	$0.554842\pi$
$920$	−2.09167	−0.0689604
$921$	0	0
$922$	−17.7889	−0.585846
$923$	−2.09167	−0.0688483
$924$	0	0
$925$	40.6249	1.33574
$926$	−43.3305	−1.42393
$927$	0	0
$928$	32.9361	1.08118
$929$	−40.3305	−1.32320	−0.661601	−	0.749856i	$-0.730123\pi$
$929$	−40.3305	−1.32320	−0.661601	+	0.749856i	$0.730123\pi$
$930$	0	0
$931$	−1.60555	−0.0526198
$932$	47.3305	1.55036
$933$	0	0
$934$	−54.3583	−1.77866
$935$	19.5416	0.639080
$936$	0	0
$937$	−4.57779	−0.149550	−0.0747750	−	0.997200i	$-0.523824\pi$
$937$	−4.57779	−0.149550	−0.0747750	+	0.997200i	$0.523824\pi$
$938$	27.4222	0.895367
$939$	0	0
$940$	19.8167	0.646348
$941$	0.844410	0.0275270	0.0137635	−	0.999905i	$-0.495619\pi$
$941$	0.844410	0.0275270	0.0137635	+	0.999905i	$0.495619\pi$
$942$	0	0
$943$	−12.2111	−0.397648
$944$	−1.18335	−0.0385146
$945$	0	0
$946$	63.4861	2.06411
$947$	13.6333	0.443023	0.221511	−	0.975158i	$-0.428901\pi$
$947$	13.6333	0.443023	0.221511	+	0.975158i	$0.428901\pi$
$948$	0	0
$949$	−5.09167	−0.165283
$950$	16.6888	0.541457
$951$	0	0
$952$	−16.8167	−0.545031
$953$	51.3305	1.66276	0.831380	−	0.555705i	$-0.187552\pi$
$953$	51.3305	1.66276	0.831380	+	0.555705i	$0.187552\pi$
$954$	0	0
$955$	3.21110	0.103909
$956$	46.5416	1.50526
$957$	0	0
$958$	41.9361	1.35489
$959$	16.8167	0.543038
$960$	0	0
$961$	−22.0000	−0.709677
$962$	−47.7250	−1.53872
$963$	0	0
$964$	39.7250	1.27946
$965$	6.29437	0.202623
$966$	0	0
$967$	41.2666	1.32704	0.663522	−	0.748156i	$-0.269061\pi$
$967$	41.2666	1.32704	0.663522	+	0.748156i	$0.269061\pi$
$968$	42.0000	1.34993
$969$	0	0
$970$	−28.2666	−0.907586
$971$	35.0917	1.12615	0.563073	−	0.826407i	$-0.309619\pi$
$971$	35.0917	1.12615	0.563073	+	0.826407i	$0.309619\pi$
$972$	0	0
$973$	−15.9083	−0.509998
$974$	−1.88057	−0.0602574
$975$	0	0
$976$	−0.330532	−0.0105801
$977$	−31.1472	−0.996487	−0.498243	−	0.867037i	$-0.666021\pi$
$977$	−31.1472	−0.996487	−0.498243	+	0.867037i	$0.666021\pi$
$978$	0	0
$979$	−54.5416	−1.74316
$980$	2.30278	0.0735595
$981$	0	0
$982$	41.3028	1.31802
$983$	22.8167	0.727738	0.363869	−	0.931450i	$-0.381456\pi$
$983$	22.8167	0.727738	0.363869	+	0.931450i	$0.381456\pi$
$984$	0	0
$985$	4.94449	0.157544
$986$	−80.1749	−2.55329
$987$	0	0
$988$	−12.2111	−0.388487
$989$	−5.51388	−0.175331
$990$	0	0
$991$	35.5139	1.12814	0.564068	−	0.825729i	$-0.309236\pi$
$991$	35.5139	1.12814	0.564068	+	0.825729i	$0.309236\pi$
$992$	−15.9083	−0.505090
$993$	0	0
$994$	2.09167	0.0663438
$995$	8.72498	0.276600
$996$	0	0
$997$	−0.0277564	−0.000879053 0	−0.000439527	−	1.00000i	$-0.500140\pi$
$997$	−0.0277564	−0.000879053 0	−0.000439527		1.00000i	$0.500140\pi$
$998$	73.4777	2.32590
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1449.2.a.j.1.2		2
3.2	odd	2		483.2.a.f.1.1	✓	2
12.11	even	2		7728.2.a.x.1.2		2
21.20	even	2		3381.2.a.p.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.f.1.1	✓	2	3.2	odd	2
1449.2.a.j.1.2		2	1.1	even	1	trivial
3381.2.a.p.1.1		2	21.20	even	2
7728.2.a.x.1.2		2	12.11	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:	\( N \)	\(=\)	\( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1449.a (trivial)

\(f(q)\)	\(=\)	\(q+2.30278 q^{2} +3.30278 q^{4} +0.697224 q^{5} -1.00000 q^{7} +3.00000 q^{8} +O(q^{10})\)	\(q+2.30278 q^{2} +3.30278 q^{4} +0.697224 q^{5} -1.00000 q^{7} +3.00000 q^{8} +1.60555 q^{10} +5.00000 q^{11} +2.30278 q^{13} -2.30278 q^{14} +0.302776 q^{16} +5.60555 q^{17} -1.60555 q^{19} +2.30278 q^{20} +11.5139 q^{22} -1.00000 q^{23} -4.51388 q^{25} +5.30278 q^{26} -3.30278 q^{28} -6.21110 q^{29} +3.00000 q^{31} -5.30278 q^{32} +12.9083 q^{34} -0.697224 q^{35} -9.00000 q^{37} -3.69722 q^{38} +2.09167 q^{40} +12.2111 q^{41} +5.51388 q^{43} +16.5139 q^{44} -2.30278 q^{46} +8.60555 q^{47} +1.00000 q^{49} -10.3944 q^{50} +7.60555 q^{52} +12.5139 q^{53} +3.48612 q^{55} -3.00000 q^{56} -14.3028 q^{58} -3.90833 q^{59} -1.09167 q^{61} +6.90833 q^{62} -12.8167 q^{64} +1.60555 q^{65} -11.9083 q^{67} +18.5139 q^{68} -1.60555 q^{70} -0.908327 q^{71} -2.21110 q^{73} -20.7250 q^{74} -5.30278 q^{76} -5.00000 q^{77} -1.00000 q^{79} +0.211103 q^{80} +28.1194 q^{82} -5.60555 q^{83} +3.90833 q^{85} +12.6972 q^{86} +15.0000 q^{88} -10.9083 q^{89} -2.30278 q^{91} -3.30278 q^{92} +19.8167 q^{94} -1.11943 q^{95} -17.6056 q^{97} +2.30278 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 3 q^{4} + 5 q^{5} - 2 q^{7} + 6 q^{8}+O(q^{10}) \) 2 * q + q^2 + 3 * q^4 + 5 * q^5 - 2 * q^7 + 6 * q^8	\( 2 q + q^{2} + 3 q^{4} + 5 q^{5} - 2 q^{7} + 6 q^{8} - 4 q^{10} + 10 q^{11} + q^{13} - q^{14} - 3 q^{16} + 4 q^{17} + 4 q^{19} + q^{20} + 5 q^{22} - 2 q^{23} + 9 q^{25} + 7 q^{26} - 3 q^{28} + 2 q^{29} + 6 q^{31} - 7 q^{32} + 15 q^{34} - 5 q^{35} - 18 q^{37} - 11 q^{38} + 15 q^{40} + 10 q^{41} - 7 q^{43} + 15 q^{44} - q^{46} + 10 q^{47} + 2 q^{49} - 28 q^{50} + 8 q^{52} + 7 q^{53} + 25 q^{55} - 6 q^{56} - 25 q^{58} + 3 q^{59} - 13 q^{61} + 3 q^{62} - 4 q^{64} - 4 q^{65} - 13 q^{67} + 19 q^{68} + 4 q^{70} + 9 q^{71} + 10 q^{73} - 9 q^{74} - 7 q^{76} - 10 q^{77} - 2 q^{79} - 14 q^{80} + 31 q^{82} - 4 q^{83} - 3 q^{85} + 29 q^{86} + 30 q^{88} - 11 q^{89} - q^{91} - 3 q^{92} + 18 q^{94} + 23 q^{95} - 28 q^{97} + q^{98}+O(q^{100}) \) 2 * q + q^2 + 3 * q^4 + 5 * q^5 - 2 * q^7 + 6 * q^8 - 4 * q^10 + 10 * q^11 + q^13 - q^14 - 3 * q^16 + 4 * q^17 + 4 * q^19 + q^20 + 5 * q^22 - 2 * q^23 + 9 * q^25 + 7 * q^26 - 3 * q^28 + 2 * q^29 + 6 * q^31 - 7 * q^32 + 15 * q^34 - 5 * q^35 - 18 * q^37 - 11 * q^38 + 15 * q^40 + 10 * q^41 - 7 * q^43 + 15 * q^44 - q^46 + 10 * q^47 + 2 * q^49 - 28 * q^50 + 8 * q^52 + 7 * q^53 + 25 * q^55 - 6 * q^56 - 25 * q^58 + 3 * q^59 - 13 * q^61 + 3 * q^62 - 4 * q^64 - 4 * q^65 - 13 * q^67 + 19 * q^68 + 4 * q^70 + 9 * q^71 + 10 * q^73 - 9 * q^74 - 7 * q^76 - 10 * q^77 - 2 * q^79 - 14 * q^80 + 31 * q^82 - 4 * q^83 - 3 * q^85 + 29 * q^86 + 30 * q^88 - 11 * q^89 - q^91 - 3 * q^92 + 18 * q^94 + 23 * q^95 - 28 * q^97 + q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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