LMFDB - Embedded newform 539.2.a.e.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [539,2,Mod(1,539)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(539, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("539.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 539 = 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	539.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:	$4.30393666895$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	539.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.41421 q^{3} -1.00000 q^{4} -1.41421 q^{5} -1.41421 q^{6} +3.00000 q^{8} -1.00000 q^{9} +1.41421 q^{10} -1.00000 q^{11} -1.41421 q^{12} -2.00000 q^{15} -1.00000 q^{16} +1.00000 q^{18} -5.65685 q^{19} +1.41421 q^{20} +1.00000 q^{22} -2.00000 q^{23} +4.24264 q^{24} -3.00000 q^{25} -5.65685 q^{27} -6.00000 q^{29} +2.00000 q^{30} +4.24264 q^{31} -5.00000 q^{32} -1.41421 q^{33} +1.00000 q^{36} -10.0000 q^{37} +5.65685 q^{38} -4.24264 q^{40} +5.65685 q^{41} -8.00000 q^{43} +1.00000 q^{44} +1.41421 q^{45} +2.00000 q^{46} +9.89949 q^{47} -1.41421 q^{48} +3.00000 q^{50} +5.65685 q^{54} +1.41421 q^{55} -8.00000 q^{57} +6.00000 q^{58} -9.89949 q^{59} +2.00000 q^{60} +8.48528 q^{61} -4.24264 q^{62} +7.00000 q^{64} +1.41421 q^{66} +2.00000 q^{67} -2.82843 q^{69} +6.00000 q^{71} -3.00000 q^{72} -8.48528 q^{73} +10.0000 q^{74} -4.24264 q^{75} +5.65685 q^{76} -8.00000 q^{79} +1.41421 q^{80} -5.00000 q^{81} -5.65685 q^{82} -11.3137 q^{83} +8.00000 q^{86} -8.48528 q^{87} -3.00000 q^{88} +15.5563 q^{89} -1.41421 q^{90} +2.00000 q^{92} +6.00000 q^{93} -9.89949 q^{94} +8.00000 q^{95} -7.07107 q^{96} +12.7279 q^{97} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} - 2 q^{4} + 6 q^{8} - 2 q^{9} - 2 q^{11} - 4 q^{15} - 2 q^{16} + 2 q^{18} + 2 q^{22} - 4 q^{23} - 6 q^{25} - 12 q^{29} + 4 q^{30} - 10 q^{32} + 2 q^{36} - 20 q^{37} - 16 q^{43} + 2 q^{44}+ \cdots + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 - 2 * q^4 + 6 * q^8 - 2 * q^9 - 2 * q^11 - 4 * q^15 - 2 * q^16 + 2 * q^18 + 2 * q^22 - 4 * q^23 - 6 * q^25 - 12 * q^29 + 4 * q^30 - 10 * q^32 + 2 * q^36 - 20 * q^37 - 16 * q^43 + 2 * q^44 + 4 * q^46 + 6 * q^50 - 16 * q^57 + 12 * q^58 + 4 * q^60 + 14 * q^64 + 4 * q^67 + 12 * q^71 - 6 * q^72 + 20 * q^74 - 16 * q^79 - 10 * q^81 + 16 * q^86 - 6 * q^88 + 4 * q^92 + 12 * q^93 + 16 * q^95 + 2 * q^99

Coefficient data

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

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

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

$2$	−1.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	−1.00000	−0.500000
$5$	−1.41421	−0.632456	−0.316228	−	0.948683i	$-0.602416\pi$
$5$	−1.41421	−0.632456	−0.316228	+	0.948683i	$0.602416\pi$
$6$	−1.41421	−0.577350
$7$	0	0
$7$	0	0
$8$	3.00000	1.06066
$9$	−1.00000	−0.333333
$10$	1.41421	0.447214
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−1.41421	−0.408248
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	−1.00000	−0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	1.00000	0.235702
$19$	−5.65685	−1.29777	−0.648886	−	0.760886i	$-0.724765\pi$
$19$	−5.65685	−1.29777	−0.648886	+	0.760886i	$0.724765\pi$
$20$	1.41421	0.316228
$21$	0	0
$22$	1.00000	0.213201
$23$	−2.00000	−0.417029	−0.208514	−	0.978019i	$-0.566863\pi$
$23$	−2.00000	−0.417029	−0.208514	+	0.978019i	$0.566863\pi$
$24$	4.24264	0.866025
$25$	−3.00000	−0.600000
$26$	0	0
$27$	−5.65685	−1.08866
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	2.00000	0.365148
$31$	4.24264	0.762001	0.381000	−	0.924575i	$-0.375580\pi$
$31$	4.24264	0.762001	0.381000	+	0.924575i	$0.375580\pi$
$32$	−5.00000	−0.883883
$33$	−1.41421	−0.246183
$34$	0	0
$35$	0	0
$36$	1.00000	0.166667
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	5.65685	0.917663
$39$	0	0
$40$	−4.24264	−0.670820
$41$	5.65685	0.883452	0.441726	−	0.897150i	$-0.354366\pi$
$41$	5.65685	0.883452	0.441726	+	0.897150i	$0.354366\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	1.00000	0.150756
$45$	1.41421	0.210819
$46$	2.00000	0.294884
$47$	9.89949	1.44399	0.721995	−	0.691898i	$-0.243225\pi$
$47$	9.89949	1.44399	0.721995	+	0.691898i	$0.243225\pi$
$48$	−1.41421	−0.204124
$49$	0	0
$50$	3.00000	0.424264
$51$	0	0
$52$	0	0
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	5.65685	0.769800
$55$	1.41421	0.190693
$56$	0	0
$57$	−8.00000	−1.05963
$58$	6.00000	0.787839
$59$	−9.89949	−1.28880	−0.644402	−	0.764687i	$-0.722894\pi$
$59$	−9.89949	−1.28880	−0.644402	+	0.764687i	$0.722894\pi$
$60$	2.00000	0.258199
$61$	8.48528	1.08643	0.543214	−	0.839594i	$-0.317207\pi$
$61$	8.48528	1.08643	0.543214	+	0.839594i	$0.317207\pi$
$62$	−4.24264	−0.538816
$63$	0	0
$64$	7.00000	0.875000
$65$	0	0
$66$	1.41421	0.174078
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	0	0
$69$	−2.82843	−0.340503
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	−3.00000	−0.353553
$73$	−8.48528	−0.993127	−0.496564	−	0.868000i	$-0.665405\pi$
$73$	−8.48528	−0.993127	−0.496564	+	0.868000i	$0.665405\pi$
$74$	10.0000	1.16248
$75$	−4.24264	−0.489898
$76$	5.65685	0.648886
$77$	0	0
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	1.41421	0.158114
$81$	−5.00000	−0.555556
$82$	−5.65685	−0.624695
$83$	−11.3137	−1.24184	−0.620920	−	0.783874i	$-0.713241\pi$
$83$	−11.3137	−1.24184	−0.620920	+	0.783874i	$0.713241\pi$
$84$	0	0
$85$	0	0
$86$	8.00000	0.862662
$87$	−8.48528	−0.909718
$88$	−3.00000	−0.319801
$89$	15.5563	1.64897	0.824485	−	0.565884i	$-0.191465\pi$
$89$	15.5563	1.64897	0.824485	+	0.565884i	$0.191465\pi$
$90$	−1.41421	−0.149071
$91$	0	0
$92$	2.00000	0.208514
$93$	6.00000	0.622171
$94$	−9.89949	−1.02105
$95$	8.00000	0.820783
$96$	−7.07107	−0.721688
$97$	12.7279	1.29232	0.646162	−	0.763200i	$-0.276373\pi$
$97$	12.7279	1.29232	0.646162	+	0.763200i	$0.276373\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	3.00000	0.300000
$101$	−5.65685	−0.562878	−0.281439	−	0.959579i	$-0.590812\pi$
$101$	−5.65685	−0.562878	−0.281439	+	0.959579i	$0.590812\pi$
$102$	0	0
$103$	−9.89949	−0.975426	−0.487713	−	0.873004i	$-0.662169\pi$
$103$	−9.89949	−0.975426	−0.487713	+	0.873004i	$0.662169\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	5.65685	0.544331
$109$	−18.0000	−1.72409	−0.862044	−	0.506834i	$-0.830816\pi$
$109$	−18.0000	−1.72409	−0.862044	+	0.506834i	$0.830816\pi$
$110$	−1.41421	−0.134840
$111$	−14.1421	−1.34231
$112$	0	0
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	8.00000	0.749269
$115$	2.82843	0.263752
$116$	6.00000	0.557086
$117$	0	0
$118$	9.89949	0.911322
$119$	0	0
$120$	−6.00000	−0.547723
$121$	1.00000	0.0909091
$122$	−8.48528	−0.768221
$123$	8.00000	0.721336
$124$	−4.24264	−0.381000
$125$	11.3137	1.01193
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	3.00000	0.265165
$129$	−11.3137	−0.996116
$130$	0	0
$131$	19.7990	1.72985	0.864923	−	0.501905i	$-0.167367\pi$
$131$	19.7990	1.72985	0.864923	+	0.501905i	$0.167367\pi$
$132$	1.41421	0.123091
$133$	0	0
$134$	−2.00000	−0.172774
$135$	8.00000	0.688530
$136$	0	0
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	2.82843	0.240772
$139$	16.9706	1.43942	0.719712	−	0.694273i	$-0.244274\pi$
$139$	16.9706	1.43942	0.719712	+	0.694273i	$0.244274\pi$
$140$	0	0
$141$	14.0000	1.17901
$142$	−6.00000	−0.503509
$143$	0	0
$144$	1.00000	0.0833333
$145$	8.48528	0.704664
$146$	8.48528	0.702247
$147$	0	0
$148$	10.0000	0.821995
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	4.24264	0.346410
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	−16.9706	−1.37649
$153$	0	0
$154$	0	0
$155$	−6.00000	−0.481932
$156$	0	0
$157$	9.89949	0.790066	0.395033	−	0.918667i	$-0.370733\pi$
$157$	9.89949	0.790066	0.395033	+	0.918667i	$0.370733\pi$
$158$	8.00000	0.636446
$159$	0	0
$160$	7.07107	0.559017
$161$	0	0
$162$	5.00000	0.392837
$163$	22.0000	1.72317	0.861586	−	0.507611i	$-0.169471\pi$
$163$	22.0000	1.72317	0.861586	+	0.507611i	$0.169471\pi$
$164$	−5.65685	−0.441726
$165$	2.00000	0.155700
$166$	11.3137	0.878114
$167$	−16.9706	−1.31322	−0.656611	−	0.754230i	$-0.728011\pi$
$167$	−16.9706	−1.31322	−0.656611	+	0.754230i	$0.728011\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	0	0
$171$	5.65685	0.432590
$172$	8.00000	0.609994
$173$	−11.3137	−0.860165	−0.430083	−	0.902790i	$-0.641516\pi$
$173$	−11.3137	−0.860165	−0.430083	+	0.902790i	$0.641516\pi$
$174$	8.48528	0.643268
$175$	0	0
$176$	1.00000	0.0753778
$177$	−14.0000	−1.05230
$178$	−15.5563	−1.16600
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	−1.41421	−0.105409
$181$	−1.41421	−0.105118	−0.0525588	−	0.998618i	$-0.516738\pi$
$181$	−1.41421	−0.105118	−0.0525588	+	0.998618i	$0.516738\pi$
$182$	0	0
$183$	12.0000	0.887066
$184$	−6.00000	−0.442326
$185$	14.1421	1.03975
$186$	−6.00000	−0.439941
$187$	0	0
$188$	−9.89949	−0.721995
$189$	0	0
$190$	−8.00000	−0.580381
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	9.89949	0.714435
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	−12.7279	−0.913812
$195$	0	0
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	−1.00000	−0.0710669
$199$	4.24264	0.300753	0.150376	−	0.988629i	$-0.451951\pi$
$199$	4.24264	0.300753	0.150376	+	0.988629i	$0.451951\pi$
$200$	−9.00000	−0.636396
$201$	2.82843	0.199502
$202$	5.65685	0.398015
$203$	0	0
$204$	0	0
$205$	−8.00000	−0.558744
$206$	9.89949	0.689730
$207$	2.00000	0.139010
$208$	0	0
$209$	5.65685	0.391293
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	0	0
$213$	8.48528	0.581402
$214$	−8.00000	−0.546869
$215$	11.3137	0.771589
$216$	−16.9706	−1.15470
$217$	0	0
$218$	18.0000	1.21911
$219$	−12.0000	−0.810885
$220$	−1.41421	−0.0953463
$221$	0	0
$222$	14.1421	0.949158
$223$	−15.5563	−1.04173	−0.520865	−	0.853639i	$-0.674391\pi$
$223$	−15.5563	−1.04173	−0.520865	+	0.853639i	$0.674391\pi$
$224$	0	0
$225$	3.00000	0.200000
$226$	−14.0000	−0.931266
$227$	2.82843	0.187729	0.0938647	−	0.995585i	$-0.470078\pi$
$227$	2.82843	0.187729	0.0938647	+	0.995585i	$0.470078\pi$
$228$	8.00000	0.529813
$229$	−29.6985	−1.96253	−0.981266	−	0.192660i	$-0.938289\pi$
$229$	−29.6985	−1.96253	−0.981266	+	0.192660i	$0.938289\pi$
$230$	−2.82843	−0.186501
$231$	0	0
$232$	−18.0000	−1.18176
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	−14.0000	−0.913259
$236$	9.89949	0.644402
$237$	−11.3137	−0.734904
$238$	0	0
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	2.00000	0.129099
$241$	−16.9706	−1.09317	−0.546585	−	0.837404i	$-0.684072\pi$
$241$	−16.9706	−1.09317	−0.546585	+	0.837404i	$0.684072\pi$
$242$	−1.00000	−0.0642824
$243$	9.89949	0.635053
$244$	−8.48528	−0.543214
$245$	0	0
$246$	−8.00000	−0.510061
$247$	0	0
$248$	12.7279	0.808224
$249$	−16.0000	−1.01396
$250$	−11.3137	−0.715542
$251$	−15.5563	−0.981908	−0.490954	−	0.871185i	$-0.663352\pi$
$251$	−15.5563	−0.981908	−0.490954	+	0.871185i	$0.663352\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	8.00000	0.501965
$255$	0	0
$256$	−17.0000	−1.06250
$257$	1.41421	0.0882162	0.0441081	−	0.999027i	$-0.485955\pi$
$257$	1.41421	0.0882162	0.0441081	+	0.999027i	$0.485955\pi$
$258$	11.3137	0.704361
$259$	0	0
$260$	0	0
$261$	6.00000	0.371391
$262$	−19.7990	−1.22319
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	−4.24264	−0.261116
$265$	0	0
$266$	0	0
$267$	22.0000	1.34638
$268$	−2.00000	−0.122169
$269$	−9.89949	−0.603583	−0.301791	−	0.953374i	$-0.597585\pi$
$269$	−9.89949	−0.603583	−0.301791	+	0.953374i	$0.597585\pi$
$270$	−8.00000	−0.486864
$271$	14.1421	0.859074	0.429537	−	0.903049i	$-0.358677\pi$
$271$	14.1421	0.859074	0.429537	+	0.903049i	$0.358677\pi$
$272$	0	0
$273$	0	0
$274$	−14.0000	−0.845771
$275$	3.00000	0.180907
$276$	2.82843	0.170251
$277$	−18.0000	−1.08152	−0.540758	−	0.841178i	$-0.681862\pi$
$277$	−18.0000	−1.08152	−0.540758	+	0.841178i	$0.681862\pi$
$278$	−16.9706	−1.01783
$279$	−4.24264	−0.254000
$280$	0	0
$281$	−22.0000	−1.31241	−0.656205	−	0.754583i	$-0.727839\pi$
$281$	−22.0000	−1.31241	−0.656205	+	0.754583i	$0.727839\pi$
$282$	−14.0000	−0.833688
$283$	31.1127	1.84946	0.924729	−	0.380626i	$-0.124292\pi$
$283$	31.1127	1.84946	0.924729	+	0.380626i	$0.124292\pi$
$284$	−6.00000	−0.356034
$285$	11.3137	0.670166
$286$	0	0
$287$	0	0
$288$	5.00000	0.294628
$289$	−17.0000	−1.00000
$290$	−8.48528	−0.498273
$291$	18.0000	1.05518
$292$	8.48528	0.496564
$293$	14.1421	0.826192	0.413096	−	0.910687i	$-0.364447\pi$
$293$	14.1421	0.826192	0.413096	+	0.910687i	$0.364447\pi$
$294$	0	0
$295$	14.0000	0.815112
$296$	−30.0000	−1.74371
$297$	5.65685	0.328244
$298$	−6.00000	−0.347571
$299$	0	0
$300$	4.24264	0.244949
$301$	0	0
$302$	12.0000	0.690522
$303$	−8.00000	−0.459588
$304$	5.65685	0.324443
$305$	−12.0000	−0.687118
$306$	0	0
$307$	−19.7990	−1.12999	−0.564994	−	0.825095i	$-0.691122\pi$
$307$	−19.7990	−1.12999	−0.564994	+	0.825095i	$0.691122\pi$
$308$	0	0
$309$	−14.0000	−0.796432
$310$	6.00000	0.340777
$311$	1.41421	0.0801927	0.0400963	−	0.999196i	$-0.487234\pi$
$311$	1.41421	0.0801927	0.0400963	+	0.999196i	$0.487234\pi$
$312$	0	0
$313$	−24.0416	−1.35891	−0.679457	−	0.733716i	$-0.737784\pi$
$313$	−24.0416	−1.35891	−0.679457	+	0.733716i	$0.737784\pi$
$314$	−9.89949	−0.558661
$315$	0	0
$316$	8.00000	0.450035
$317$	−18.0000	−1.01098	−0.505490	−	0.862832i	$-0.668688\pi$
$317$	−18.0000	−1.01098	−0.505490	+	0.862832i	$0.668688\pi$
$318$	0	0
$319$	6.00000	0.335936
$320$	−9.89949	−0.553399
$321$	11.3137	0.631470
$322$	0	0
$323$	0	0
$324$	5.00000	0.277778
$325$	0	0
$326$	−22.0000	−1.21847
$327$	−25.4558	−1.40771
$328$	16.9706	0.937043
$329$	0	0
$330$	−2.00000	−0.110096
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	11.3137	0.620920
$333$	10.0000	0.547997
$334$	16.9706	0.928588
$335$	−2.82843	−0.154533
$336$	0	0
$337$	2.00000	0.108947	0.0544735	−	0.998515i	$-0.482652\pi$
$337$	2.00000	0.108947	0.0544735	+	0.998515i	$0.482652\pi$
$338$	13.0000	0.707107
$339$	19.7990	1.07533
$340$	0	0
$341$	−4.24264	−0.229752
$342$	−5.65685	−0.305888
$343$	0	0
$344$	−24.0000	−1.29399
$345$	4.00000	0.215353
$346$	11.3137	0.608229
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	8.48528	0.454859
$349$	25.4558	1.36262	0.681310	−	0.731995i	$-0.261411\pi$
$349$	25.4558	1.36262	0.681310	+	0.731995i	$0.261411\pi$
$350$	0	0
$351$	0	0
$352$	5.00000	0.266501
$353$	1.41421	0.0752710	0.0376355	−	0.999292i	$-0.488017\pi$
$353$	1.41421	0.0752710	0.0376355	+	0.999292i	$0.488017\pi$
$354$	14.0000	0.744092
$355$	−8.48528	−0.450352
$356$	−15.5563	−0.824485
$357$	0	0
$358$	−12.0000	−0.634220
$359$	4.00000	0.211112	0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000	0.211112	0.105556	+	0.994413i	$0.466338\pi$
$360$	4.24264	0.223607
$361$	13.0000	0.684211
$362$	1.41421	0.0743294
$363$	1.41421	0.0742270
$364$	0	0
$365$	12.0000	0.628109
$366$	−12.0000	−0.627250
$367$	7.07107	0.369107	0.184553	−	0.982822i	$-0.440916\pi$
$367$	7.07107	0.369107	0.184553	+	0.982822i	$0.440916\pi$
$368$	2.00000	0.104257
$369$	−5.65685	−0.294484
$370$	−14.1421	−0.735215
$371$	0	0
$372$	−6.00000	−0.311086
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	16.0000	0.826236
$376$	29.6985	1.53158
$377$	0	0
$378$	0	0
$379$	10.0000	0.513665	0.256833	−	0.966456i	$-0.417321\pi$
$379$	10.0000	0.513665	0.256833	+	0.966456i	$0.417321\pi$
$380$	−8.00000	−0.410391
$381$	−11.3137	−0.579619
$382$	0	0
$383$	−24.0416	−1.22847	−0.614235	−	0.789123i	$-0.710535\pi$
$383$	−24.0416	−1.22847	−0.614235	+	0.789123i	$0.710535\pi$
$384$	4.24264	0.216506
$385$	0	0
$386$	−2.00000	−0.101797
$387$	8.00000	0.406663
$388$	−12.7279	−0.646162
$389$	0	0		−	1.00000i	$-0.5\pi$
$389$	0	0			1.00000i	$0.5\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	28.0000	1.41241
$394$	18.0000	0.906827
$395$	11.3137	0.569254
$396$	−1.00000	−0.0502519
$397$	−15.5563	−0.780751	−0.390375	−	0.920656i	$-0.627655\pi$
$397$	−15.5563	−0.780751	−0.390375	+	0.920656i	$0.627655\pi$
$398$	−4.24264	−0.212664
$399$	0	0
$400$	3.00000	0.150000
$401$	28.0000	1.39825	0.699127	−	0.714998i	$-0.253572\pi$
$401$	28.0000	1.39825	0.699127	+	0.714998i	$0.253572\pi$
$402$	−2.82843	−0.141069
$403$	0	0
$404$	5.65685	0.281439
$405$	7.07107	0.351364
$406$	0	0
$407$	10.0000	0.495682
$408$	0	0
$409$	−19.7990	−0.978997	−0.489499	−	0.872004i	$-0.662820\pi$
$409$	−19.7990	−0.978997	−0.489499	+	0.872004i	$0.662820\pi$
$410$	8.00000	0.395092
$411$	19.7990	0.976612
$412$	9.89949	0.487713
$413$	0	0
$414$	−2.00000	−0.0982946
$415$	16.0000	0.785409
$416$	0	0
$417$	24.0000	1.17529
$418$	−5.65685	−0.276686
$419$	24.0416	1.17451	0.587255	−	0.809402i	$-0.300208\pi$
$419$	24.0416	1.17451	0.587255	+	0.809402i	$0.300208\pi$
$420$	0	0
$421$	−28.0000	−1.36464	−0.682318	−	0.731055i	$-0.739028\pi$
$421$	−28.0000	−1.36464	−0.682318	+	0.731055i	$0.739028\pi$
$422$	16.0000	0.778868
$423$	−9.89949	−0.481330
$424$	0	0
$425$	0	0
$426$	−8.48528	−0.411113
$427$	0	0
$428$	−8.00000	−0.386695
$429$	0	0
$430$	−11.3137	−0.545595
$431$	−16.0000	−0.770693	−0.385346	−	0.922772i	$-0.625918\pi$
$431$	−16.0000	−0.770693	−0.385346	+	0.922772i	$0.625918\pi$
$432$	5.65685	0.272166
$433$	9.89949	0.475739	0.237870	−	0.971297i	$-0.423551\pi$
$433$	9.89949	0.475739	0.237870	+	0.971297i	$0.423551\pi$
$434$	0	0
$435$	12.0000	0.575356
$436$	18.0000	0.862044
$437$	11.3137	0.541208
$438$	12.0000	0.573382
$439$	−2.82843	−0.134993	−0.0674967	−	0.997719i	$-0.521501\pi$
$439$	−2.82843	−0.134993	−0.0674967	+	0.997719i	$0.521501\pi$
$440$	4.24264	0.202260
$441$	0	0
$442$	0	0
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	14.1421	0.671156
$445$	−22.0000	−1.04290
$446$	15.5563	0.736614
$447$	8.48528	0.401340
$448$	0	0
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	−3.00000	−0.141421
$451$	−5.65685	−0.266371
$452$	−14.0000	−0.658505
$453$	−16.9706	−0.797347
$454$	−2.82843	−0.132745
$455$	0	0
$456$	−24.0000	−1.12390
$457$	22.0000	1.02912	0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000	1.02912	0.514558	+	0.857455i	$0.327956\pi$
$458$	29.6985	1.38772
$459$	0	0
$460$	−2.82843	−0.131876
$461$	−42.4264	−1.97599	−0.987997	−	0.154471i	$-0.950633\pi$
$461$	−42.4264	−1.97599	−0.987997	+	0.154471i	$0.950633\pi$
$462$	0	0
$463$	34.0000	1.58011	0.790057	−	0.613033i	$-0.210051\pi$
$463$	34.0000	1.58011	0.790057	+	0.613033i	$0.210051\pi$
$464$	6.00000	0.278543
$465$	−8.48528	−0.393496
$466$	10.0000	0.463241
$467$	−4.24264	−0.196326	−0.0981630	−	0.995170i	$-0.531297\pi$
$467$	−4.24264	−0.196326	−0.0981630	+	0.995170i	$0.531297\pi$
$468$	0	0
$469$	0	0
$470$	14.0000	0.645772
$471$	14.0000	0.645086
$472$	−29.6985	−1.36698
$473$	8.00000	0.367840
$474$	11.3137	0.519656
$475$	16.9706	0.778663
$476$	0	0
$477$	0	0
$478$	−12.0000	−0.548867
$479$	−19.7990	−0.904639	−0.452319	−	0.891856i	$-0.649403\pi$
$479$	−19.7990	−0.904639	−0.452319	+	0.891856i	$0.649403\pi$
$480$	10.0000	0.456435
$481$	0	0
$482$	16.9706	0.772988
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	−18.0000	−0.817338
$486$	−9.89949	−0.449050
$487$	26.0000	1.17817	0.589086	−	0.808070i	$-0.299488\pi$
$487$	26.0000	1.17817	0.589086	+	0.808070i	$0.299488\pi$
$488$	25.4558	1.15233
$489$	31.1127	1.40696
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	−8.00000	−0.360668
$493$	0	0
$494$	0	0
$495$	−1.41421	−0.0635642
$496$	−4.24264	−0.190500
$497$	0	0
$498$	16.0000	0.716977
$499$	−26.0000	−1.16392	−0.581960	−	0.813217i	$-0.697714\pi$
$499$	−26.0000	−1.16392	−0.581960	+	0.813217i	$0.697714\pi$
$500$	−11.3137	−0.505964
$501$	−24.0000	−1.07224
$502$	15.5563	0.694314
$503$	−2.82843	−0.126113	−0.0630567	−	0.998010i	$-0.520085\pi$
$503$	−2.82843	−0.126113	−0.0630567	+	0.998010i	$0.520085\pi$
$504$	0	0
$505$	8.00000	0.355995
$506$	−2.00000	−0.0889108
$507$	−18.3848	−0.816497
$508$	8.00000	0.354943
$509$	−15.5563	−0.689523	−0.344762	−	0.938690i	$-0.612040\pi$
$509$	−15.5563	−0.689523	−0.344762	+	0.938690i	$0.612040\pi$
$510$	0	0
$511$	0	0
$512$	11.0000	0.486136
$513$	32.0000	1.41283
$514$	−1.41421	−0.0623783
$515$	14.0000	0.616914
$516$	11.3137	0.498058
$517$	−9.89949	−0.435379
$518$	0	0
$519$	−16.0000	−0.702322
$520$	0	0
$521$	−21.2132	−0.929367	−0.464684	−	0.885477i	$-0.653832\pi$
$521$	−21.2132	−0.929367	−0.464684	+	0.885477i	$0.653832\pi$
$522$	−6.00000	−0.262613
$523$	−8.48528	−0.371035	−0.185518	−	0.982641i	$-0.559396\pi$
$523$	−8.48528	−0.371035	−0.185518	+	0.982641i	$0.559396\pi$
$524$	−19.7990	−0.864923
$525$	0	0
$526$	−8.00000	−0.348817
$527$	0	0
$528$	1.41421	0.0615457
$529$	−19.0000	−0.826087
$530$	0	0
$531$	9.89949	0.429601
$532$	0	0
$533$	0	0
$534$	−22.0000	−0.952033
$535$	−11.3137	−0.489134
$536$	6.00000	0.259161
$537$	16.9706	0.732334
$538$	9.89949	0.426798
$539$	0	0
$540$	−8.00000	−0.344265
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	−14.1421	−0.607457
$543$	−2.00000	−0.0858282
$544$	0	0
$545$	25.4558	1.09041
$546$	0	0
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	−14.0000	−0.598050
$549$	−8.48528	−0.362143
$550$	−3.00000	−0.127920
$551$	33.9411	1.44594
$552$	−8.48528	−0.361158
$553$	0	0
$554$	18.0000	0.764747
$555$	20.0000	0.848953
$556$	−16.9706	−0.719712
$557$	−22.0000	−0.932170	−0.466085	−	0.884740i	$-0.654336\pi$
$557$	−22.0000	−0.932170	−0.466085	+	0.884740i	$0.654336\pi$
$558$	4.24264	0.179605
$559$	0	0
$560$	0	0
$561$	0	0
$562$	22.0000	0.928014
$563$	28.2843	1.19204	0.596020	−	0.802970i	$-0.296748\pi$
$563$	28.2843	1.19204	0.596020	+	0.802970i	$0.296748\pi$
$564$	−14.0000	−0.589506
$565$	−19.7990	−0.832950
$566$	−31.1127	−1.30776
$567$	0	0
$568$	18.0000	0.755263
$569$	14.0000	0.586911	0.293455	−	0.955973i	$-0.405195\pi$
$569$	14.0000	0.586911	0.293455	+	0.955973i	$0.405195\pi$
$570$	−11.3137	−0.473879
$571$	40.0000	1.67395	0.836974	−	0.547243i	$-0.184323\pi$
$571$	40.0000	1.67395	0.836974	+	0.547243i	$0.184323\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.00000	0.250217
$576$	−7.00000	−0.291667
$577$	−18.3848	−0.765368	−0.382684	−	0.923879i	$-0.625000\pi$
$577$	−18.3848	−0.765368	−0.382684	+	0.923879i	$0.625000\pi$
$578$	17.0000	0.707107
$579$	2.82843	0.117545
$580$	−8.48528	−0.352332
$581$	0	0
$582$	−18.0000	−0.746124
$583$	0	0
$584$	−25.4558	−1.05337
$585$	0	0
$586$	−14.1421	−0.584206
$587$	−15.5563	−0.642079	−0.321040	−	0.947066i	$-0.604032\pi$
$587$	−15.5563	−0.642079	−0.321040	+	0.947066i	$0.604032\pi$
$588$	0	0
$589$	−24.0000	−0.988903
$590$	−14.0000	−0.576371
$591$	−25.4558	−1.04711
$592$	10.0000	0.410997
$593$	2.82843	0.116150	0.0580748	−	0.998312i	$-0.481504\pi$
$593$	2.82843	0.116150	0.0580748	+	0.998312i	$0.481504\pi$
$594$	−5.65685	−0.232104
$595$	0	0
$596$	−6.00000	−0.245770
$597$	6.00000	0.245564
$598$	0	0
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	−12.7279	−0.519615
$601$	22.6274	0.922992	0.461496	−	0.887142i	$-0.347313\pi$
$601$	22.6274	0.922992	0.461496	+	0.887142i	$0.347313\pi$
$602$	0	0
$603$	−2.00000	−0.0814463
$604$	12.0000	0.488273
$605$	−1.41421	−0.0574960
$606$	8.00000	0.324978
$607$	39.5980	1.60723	0.803616	−	0.595148i	$-0.202907\pi$
$607$	39.5980	1.60723	0.803616	+	0.595148i	$0.202907\pi$
$608$	28.2843	1.14708
$609$	0	0
$610$	12.0000	0.485866
$611$	0	0
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	19.7990	0.799022
$615$	−11.3137	−0.456213
$616$	0	0
$617$	−30.0000	−1.20775	−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000	−1.20775	−0.603877	+	0.797077i	$0.706378\pi$
$618$	14.0000	0.563163
$619$	9.89949	0.397894	0.198947	−	0.980010i	$-0.436248\pi$
$619$	9.89949	0.397894	0.198947	+	0.980010i	$0.436248\pi$
$620$	6.00000	0.240966
$621$	11.3137	0.454003
$622$	−1.41421	−0.0567048
$623$	0	0
$624$	0	0
$625$	−1.00000	−0.0400000
$626$	24.0416	0.960897
$627$	8.00000	0.319489
$628$	−9.89949	−0.395033
$629$	0	0
$630$	0	0
$631$	40.0000	1.59237	0.796187	−	0.605050i	$-0.206847\pi$
$631$	40.0000	1.59237	0.796187	+	0.605050i	$0.206847\pi$
$632$	−24.0000	−0.954669
$633$	−22.6274	−0.899359
$634$	18.0000	0.714871
$635$	11.3137	0.448971
$636$	0	0
$637$	0	0
$638$	−6.00000	−0.237542
$639$	−6.00000	−0.237356
$640$	−4.24264	−0.167705
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	−11.3137	−0.446516
$643$	18.3848	0.725025	0.362512	−	0.931979i	$-0.381919\pi$
$643$	18.3848	0.725025	0.362512	+	0.931979i	$0.381919\pi$
$644$	0	0
$645$	16.0000	0.629999
$646$	0	0
$647$	24.0416	0.945174	0.472587	−	0.881284i	$-0.343320\pi$
$647$	24.0416	0.945174	0.472587	+	0.881284i	$0.343320\pi$
$648$	−15.0000	−0.589256
$649$	9.89949	0.388589
$650$	0	0
$651$	0	0
$652$	−22.0000	−0.861586
$653$	36.0000	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$653$	36.0000	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$654$	25.4558	0.995402
$655$	−28.0000	−1.09405
$656$	−5.65685	−0.220863
$657$	8.48528	0.331042
$658$	0	0
$659$	16.0000	0.623272	0.311636	−	0.950202i	$-0.399123\pi$
$659$	16.0000	0.623272	0.311636	+	0.950202i	$0.399123\pi$
$660$	−2.00000	−0.0778499
$661$	41.0122	1.59519	0.797595	−	0.603194i	$-0.206105\pi$
$661$	41.0122	1.59519	0.797595	+	0.603194i	$0.206105\pi$
$662$	12.0000	0.466393
$663$	0	0
$664$	−33.9411	−1.31717
$665$	0	0
$666$	−10.0000	−0.387492
$667$	12.0000	0.464642
$668$	16.9706	0.656611
$669$	−22.0000	−0.850569
$670$	2.82843	0.109272
$671$	−8.48528	−0.327571
$672$	0	0
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	−2.00000	−0.0770371
$675$	16.9706	0.653197
$676$	13.0000	0.500000
$677$	−5.65685	−0.217411	−0.108705	−	0.994074i	$-0.534670\pi$
$677$	−5.65685	−0.217411	−0.108705	+	0.994074i	$0.534670\pi$
$678$	−19.7990	−0.760376
$679$	0	0
$680$	0	0
$681$	4.00000	0.153280
$682$	4.24264	0.162459
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	−5.65685	−0.216295
$685$	−19.7990	−0.756481
$686$	0	0
$687$	−42.0000	−1.60240
$688$	8.00000	0.304997
$689$	0	0
$690$	−4.00000	−0.152277
$691$	−18.3848	−0.699390	−0.349695	−	0.936864i	$-0.613715\pi$
$691$	−18.3848	−0.699390	−0.349695	+	0.936864i	$0.613715\pi$
$692$	11.3137	0.430083
$693$	0	0
$694$	−4.00000	−0.151838
$695$	−24.0000	−0.910372
$696$	−25.4558	−0.964901
$697$	0	0
$698$	−25.4558	−0.963518
$699$	−14.1421	−0.534905
$700$	0	0
$701$	−46.0000	−1.73740	−0.868698	−	0.495342i	$-0.835043\pi$
$701$	−46.0000	−1.73740	−0.868698	+	0.495342i	$0.835043\pi$
$702$	0	0
$703$	56.5685	2.13352
$704$	−7.00000	−0.263822
$705$	−19.7990	−0.745673
$706$	−1.41421	−0.0532246
$707$	0	0
$708$	14.0000	0.526152
$709$	−12.0000	−0.450669	−0.225335	−	0.974281i	$-0.572348\pi$
$709$	−12.0000	−0.450669	−0.225335	+	0.974281i	$0.572348\pi$
$710$	8.48528	0.318447
$711$	8.00000	0.300023
$712$	46.6690	1.74900
$713$	−8.48528	−0.317776
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	16.9706	0.633777
$718$	−4.00000	−0.149279
$719$	−9.89949	−0.369189	−0.184594	−	0.982815i	$-0.559097\pi$
$719$	−9.89949	−0.369189	−0.184594	+	0.982815i	$0.559097\pi$
$720$	−1.41421	−0.0527046
$721$	0	0
$722$	−13.0000	−0.483810
$723$	−24.0000	−0.892570
$724$	1.41421	0.0525588
$725$	18.0000	0.668503
$726$	−1.41421	−0.0524864
$727$	7.07107	0.262251	0.131126	−	0.991366i	$-0.458141\pi$
$727$	7.07107	0.262251	0.131126	+	0.991366i	$0.458141\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	−12.0000	−0.444140
$731$	0	0
$732$	−12.0000	−0.443533
$733$	−19.7990	−0.731292	−0.365646	−	0.930754i	$-0.619152\pi$
$733$	−19.7990	−0.731292	−0.365646	+	0.930754i	$0.619152\pi$
$734$	−7.07107	−0.260998
$735$	0	0
$736$	10.0000	0.368605
$737$	−2.00000	−0.0736709
$738$	5.65685	0.208232
$739$	0	0		−	1.00000i	$-0.5\pi$
$739$	0	0			1.00000i	$0.5\pi$
$740$	−14.1421	−0.519875
$741$	0	0
$742$	0	0
$743$	44.0000	1.61420	0.807102	−	0.590412i	$-0.201035\pi$
$743$	44.0000	1.61420	0.807102	+	0.590412i	$0.201035\pi$
$744$	18.0000	0.659912
$745$	−8.48528	−0.310877
$746$	−14.0000	−0.512576
$747$	11.3137	0.413947
$748$	0	0
$749$	0	0
$750$	−16.0000	−0.584237
$751$	10.0000	0.364905	0.182453	−	0.983215i	$-0.441596\pi$
$751$	10.0000	0.364905	0.182453	+	0.983215i	$0.441596\pi$
$752$	−9.89949	−0.360997
$753$	−22.0000	−0.801725
$754$	0	0
$755$	16.9706	0.617622
$756$	0	0
$757$	16.0000	0.581530	0.290765	−	0.956795i	$-0.406090\pi$
$757$	16.0000	0.581530	0.290765	+	0.956795i	$0.406090\pi$
$758$	−10.0000	−0.363216
$759$	2.82843	0.102665
$760$	24.0000	0.870572
$761$	−48.0833	−1.74302	−0.871508	−	0.490381i	$-0.836858\pi$
$761$	−48.0833	−1.74302	−0.871508	+	0.490381i	$0.836858\pi$
$762$	11.3137	0.409852
$763$	0	0
$764$	0	0
$765$	0	0
$766$	24.0416	0.868659
$767$	0	0
$768$	−24.0416	−0.867528
$769$	−19.7990	−0.713970	−0.356985	−	0.934110i	$-0.616195\pi$
$769$	−19.7990	−0.713970	−0.356985	+	0.934110i	$0.616195\pi$
$770$	0	0
$771$	2.00000	0.0720282
$772$	−2.00000	−0.0719816
$773$	29.6985	1.06818	0.534090	−	0.845428i	$-0.320654\pi$
$773$	29.6985	1.06818	0.534090	+	0.845428i	$0.320654\pi$
$774$	−8.00000	−0.287554
$775$	−12.7279	−0.457200
$776$	38.1838	1.37072
$777$	0	0
$778$	0	0
$779$	−32.0000	−1.14652
$780$	0	0
$781$	−6.00000	−0.214697
$782$	0	0
$783$	33.9411	1.21296
$784$	0	0
$785$	−14.0000	−0.499681
$786$	−28.0000	−0.998727
$787$	22.6274	0.806580	0.403290	−	0.915072i	$-0.367867\pi$
$787$	22.6274	0.806580	0.403290	+	0.915072i	$0.367867\pi$
$788$	18.0000	0.641223
$789$	11.3137	0.402779
$790$	−11.3137	−0.402524
$791$	0	0
$792$	3.00000	0.106600
$793$	0	0
$794$	15.5563	0.552074
$795$	0	0
$796$	−4.24264	−0.150376
$797$	24.0416	0.851598	0.425799	−	0.904818i	$-0.359993\pi$
$797$	24.0416	0.851598	0.425799	+	0.904818i	$0.359993\pi$
$798$	0	0
$799$	0	0
$800$	15.0000	0.530330
$801$	−15.5563	−0.549657
$802$	−28.0000	−0.988714
$803$	8.48528	0.299439
$804$	−2.82843	−0.0997509
$805$	0	0
$806$	0	0
$807$	−14.0000	−0.492823
$808$	−16.9706	−0.597022
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	−7.07107	−0.248452
$811$	−25.4558	−0.893876	−0.446938	−	0.894565i	$-0.647485\pi$
$811$	−25.4558	−0.893876	−0.446938	+	0.894565i	$0.647485\pi$
$812$	0	0
$813$	20.0000	0.701431
$814$	−10.0000	−0.350500
$815$	−31.1127	−1.08983
$816$	0	0
$817$	45.2548	1.58327
$818$	19.7990	0.692255
$819$	0	0
$820$	8.00000	0.279372
$821$	−54.0000	−1.88461	−0.942306	−	0.334751i	$-0.891348\pi$
$821$	−54.0000	−1.88461	−0.942306	+	0.334751i	$0.891348\pi$
$822$	−19.7990	−0.690569
$823$	−8.00000	−0.278862	−0.139431	−	0.990232i	$-0.544527\pi$
$823$	−8.00000	−0.278862	−0.139431	+	0.990232i	$0.544527\pi$
$824$	−29.6985	−1.03460
$825$	4.24264	0.147710
$826$	0	0
$827$	8.00000	0.278187	0.139094	−	0.990279i	$-0.455581\pi$
$827$	8.00000	0.278187	0.139094	+	0.990279i	$0.455581\pi$
$828$	−2.00000	−0.0695048
$829$	46.6690	1.62088	0.810442	−	0.585820i	$-0.199227\pi$
$829$	46.6690	1.62088	0.810442	+	0.585820i	$0.199227\pi$
$830$	−16.0000	−0.555368
$831$	−25.4558	−0.883053
$832$	0	0
$833$	0	0
$834$	−24.0000	−0.831052
$835$	24.0000	0.830554
$836$	−5.65685	−0.195646
$837$	−24.0000	−0.829561
$838$	−24.0416	−0.830504
$839$	−4.24264	−0.146472	−0.0732361	−	0.997315i	$-0.523333\pi$
$839$	−4.24264	−0.146472	−0.0732361	+	0.997315i	$0.523333\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	28.0000	0.964944
$843$	−31.1127	−1.07158
$844$	16.0000	0.550743
$845$	18.3848	0.632456
$846$	9.89949	0.340352
$847$	0	0
$848$	0	0
$849$	44.0000	1.51008
$850$	0	0
$851$	20.0000	0.685591
$852$	−8.48528	−0.290701
$853$	−5.65685	−0.193687	−0.0968435	−	0.995300i	$-0.530875\pi$
$853$	−5.65685	−0.193687	−0.0968435	+	0.995300i	$0.530875\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	24.0000	0.820303
$857$	2.82843	0.0966172	0.0483086	−	0.998832i	$-0.484617\pi$
$857$	2.82843	0.0966172	0.0483086	+	0.998832i	$0.484617\pi$
$858$	0	0
$859$	−21.2132	−0.723785	−0.361893	−	0.932220i	$-0.617869\pi$
$859$	−21.2132	−0.723785	−0.361893	+	0.932220i	$0.617869\pi$
$860$	−11.3137	−0.385794
$861$	0	0
$862$	16.0000	0.544962
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	28.2843	0.962250
$865$	16.0000	0.544016
$866$	−9.89949	−0.336399
$867$	−24.0416	−0.816497
$868$	0	0
$869$	8.00000	0.271381
$870$	−12.0000	−0.406838
$871$	0	0
$872$	−54.0000	−1.82867
$873$	−12.7279	−0.430775
$874$	−11.3137	−0.382692
$875$	0	0
$876$	12.0000	0.405442
$877$	−46.0000	−1.55331	−0.776655	−	0.629926i	$-0.783085\pi$
$877$	−46.0000	−1.55331	−0.776655	+	0.629926i	$0.783085\pi$
$878$	2.82843	0.0954548
$879$	20.0000	0.674583
$880$	−1.41421	−0.0476731
$881$	4.24264	0.142938	0.0714691	−	0.997443i	$-0.477231\pi$
$881$	4.24264	0.142938	0.0714691	+	0.997443i	$0.477231\pi$
$882$	0	0
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	0	0
$885$	19.7990	0.665536
$886$	−4.00000	−0.134383
$887$	−8.48528	−0.284908	−0.142454	−	0.989801i	$-0.545499\pi$
$887$	−8.48528	−0.284908	−0.142454	+	0.989801i	$0.545499\pi$
$888$	−42.4264	−1.42374
$889$	0	0
$890$	22.0000	0.737442
$891$	5.00000	0.167506
$892$	15.5563	0.520865
$893$	−56.0000	−1.87397
$894$	−8.48528	−0.283790
$895$	−16.9706	−0.567263
$896$	0	0
$897$	0	0
$898$	−12.0000	−0.400445
$899$	−25.4558	−0.849000
$900$	−3.00000	−0.100000
$901$	0	0
$902$	5.65685	0.188353
$903$	0	0
$904$	42.0000	1.39690
$905$	2.00000	0.0664822
$906$	16.9706	0.563809
$907$	2.00000	0.0664089	0.0332045	−	0.999449i	$-0.489429\pi$
$907$	2.00000	0.0664089	0.0332045	+	0.999449i	$0.489429\pi$
$908$	−2.82843	−0.0938647
$909$	5.65685	0.187626
$910$	0	0
$911$	−58.0000	−1.92163	−0.960813	−	0.277198i	$-0.910594\pi$
$911$	−58.0000	−1.92163	−0.960813	+	0.277198i	$0.910594\pi$
$912$	8.00000	0.264906
$913$	11.3137	0.374429
$914$	−22.0000	−0.727695
$915$	−16.9706	−0.561029
$916$	29.6985	0.981266
$917$	0	0
$918$	0	0
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	8.48528	0.279751
$921$	−28.0000	−0.922631
$922$	42.4264	1.39724
$923$	0	0
$924$	0	0
$925$	30.0000	0.986394
$926$	−34.0000	−1.11731
$927$	9.89949	0.325142
$928$	30.0000	0.984798
$929$	35.3553	1.15997	0.579986	−	0.814627i	$-0.303058\pi$
$929$	35.3553	1.15997	0.579986	+	0.814627i	$0.303058\pi$
$930$	8.48528	0.278243
$931$	0	0
$932$	10.0000	0.327561
$933$	2.00000	0.0654771
$934$	4.24264	0.138823
$935$	0	0
$936$	0	0
$937$	−59.3970	−1.94041	−0.970207	−	0.242277i	$-0.922106\pi$
$937$	−59.3970	−1.94041	−0.970207	+	0.242277i	$0.922106\pi$
$938$	0	0
$939$	−34.0000	−1.10955
$940$	14.0000	0.456630
$941$	−36.7696	−1.19865	−0.599327	−	0.800505i	$-0.704565\pi$
$941$	−36.7696	−1.19865	−0.599327	+	0.800505i	$0.704565\pi$
$942$	−14.0000	−0.456145
$943$	−11.3137	−0.368425
$944$	9.89949	0.322201
$945$	0	0
$946$	−8.00000	−0.260102
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	11.3137	0.367452
$949$	0	0
$950$	−16.9706	−0.550598
$951$	−25.4558	−0.825462
$952$	0	0
$953$	−18.0000	−0.583077	−0.291539	−	0.956559i	$-0.594167\pi$
$953$	−18.0000	−0.583077	−0.291539	+	0.956559i	$0.594167\pi$
$954$	0	0
$955$	0	0
$956$	−12.0000	−0.388108
$957$	8.48528	0.274290
$958$	19.7990	0.639676
$959$	0	0
$960$	−14.0000	−0.451848
$961$	−13.0000	−0.419355
$962$	0	0
$963$	−8.00000	−0.257796
$964$	16.9706	0.546585
$965$	−2.82843	−0.0910503
$966$	0	0
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	3.00000	0.0964237
$969$	0	0
$970$	18.0000	0.577945
$971$	29.6985	0.953070	0.476535	−	0.879156i	$-0.341893\pi$
$971$	29.6985	0.953070	0.476535	+	0.879156i	$0.341893\pi$
$972$	−9.89949	−0.317526
$973$	0	0
$974$	−26.0000	−0.833094
$975$	0	0
$976$	−8.48528	−0.271607
$977$	−12.0000	−0.383914	−0.191957	−	0.981403i	$-0.561483\pi$
$977$	−12.0000	−0.383914	−0.191957	+	0.981403i	$0.561483\pi$
$978$	−31.1127	−0.994874
$979$	−15.5563	−0.497183
$980$	0	0
$981$	18.0000	0.574696
$982$	20.0000	0.638226
$983$	−49.4975	−1.57872	−0.789362	−	0.613928i	$-0.789589\pi$
$983$	−49.4975	−1.57872	−0.789362	+	0.613928i	$0.789589\pi$
$984$	24.0000	0.765092
$985$	25.4558	0.811091
$986$	0	0
$987$	0	0
$988$	0	0
$989$	16.0000	0.508770
$990$	1.41421	0.0449467
$991$	26.0000	0.825917	0.412959	−	0.910750i	$-0.364495\pi$
$991$	26.0000	0.825917	0.412959	+	0.910750i	$0.364495\pi$
$992$	−21.2132	−0.673520
$993$	−16.9706	−0.538545
$994$	0	0
$995$	−6.00000	−0.190213
$996$	16.0000	0.506979
$997$	−16.9706	−0.537463	−0.268732	−	0.963215i	$-0.586604\pi$
$997$	−16.9706	−0.537463	−0.268732	+	0.963215i	$0.586604\pi$
$998$	26.0000	0.823016
$999$	56.5685	1.78975

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	539.2.a.e.1.2	yes	2
3.2	odd	2		4851.2.a.bc.1.2		2
4.3	odd	2		8624.2.a.bv.1.1		2
7.2	even	3		539.2.e.k.67.1		4
7.3	odd	6		539.2.e.k.177.2		4
7.4	even	3		539.2.e.k.177.1		4
7.5	odd	6		539.2.e.k.67.2		4
7.6	odd	2	inner	539.2.a.e.1.1	✓	2
11.10	odd	2		5929.2.a.r.1.2		2
21.20	even	2		4851.2.a.bc.1.1		2
28.27	even	2		8624.2.a.bv.1.2		2
77.76	even	2		5929.2.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
539.2.a.e.1.1	✓	2	7.6	odd	2	inner
539.2.a.e.1.2	yes	2	1.1	even	1	trivial
539.2.e.k.67.1		4	7.2	even	3
539.2.e.k.67.2		4	7.5	odd	6
539.2.e.k.177.1		4	7.4	even	3
539.2.e.k.177.2		4	7.3	odd	6
4851.2.a.bc.1.1		2	21.20	even	2
4851.2.a.bc.1.2		2	3.2	odd	2
5929.2.a.r.1.1		2	77.76	even	2
5929.2.a.r.1.2		2	11.10	odd	2
8624.2.a.bv.1.1		2	4.3	odd	2
8624.2.a.bv.1.2		2	28.27	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:	\( N \)	\(=\)	\( 539 = 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	539.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.41421 q^{3} -1.00000 q^{4} -1.41421 q^{5} -1.41421 q^{6} +3.00000 q^{8} -1.00000 q^{9} +1.41421 q^{10} -1.00000 q^{11} -1.41421 q^{12} -2.00000 q^{15} -1.00000 q^{16} +1.00000 q^{18} -5.65685 q^{19} +1.41421 q^{20} +1.00000 q^{22} -2.00000 q^{23} +4.24264 q^{24} -3.00000 q^{25} -5.65685 q^{27} -6.00000 q^{29} +2.00000 q^{30} +4.24264 q^{31} -5.00000 q^{32} -1.41421 q^{33} +1.00000 q^{36} -10.0000 q^{37} +5.65685 q^{38} -4.24264 q^{40} +5.65685 q^{41} -8.00000 q^{43} +1.00000 q^{44} +1.41421 q^{45} +2.00000 q^{46} +9.89949 q^{47} -1.41421 q^{48} +3.00000 q^{50} +5.65685 q^{54} +1.41421 q^{55} -8.00000 q^{57} +6.00000 q^{58} -9.89949 q^{59} +2.00000 q^{60} +8.48528 q^{61} -4.24264 q^{62} +7.00000 q^{64} +1.41421 q^{66} +2.00000 q^{67} -2.82843 q^{69} +6.00000 q^{71} -3.00000 q^{72} -8.48528 q^{73} +10.0000 q^{74} -4.24264 q^{75} +5.65685 q^{76} -8.00000 q^{79} +1.41421 q^{80} -5.00000 q^{81} -5.65685 q^{82} -11.3137 q^{83} +8.00000 q^{86} -8.48528 q^{87} -3.00000 q^{88} +15.5563 q^{89} -1.41421 q^{90} +2.00000 q^{92} +6.00000 q^{93} -9.89949 q^{94} +8.00000 q^{95} -7.07107 q^{96} +12.7279 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} - 2 q^{4} + 6 q^{8} - 2 q^{9} - 2 q^{11} - 4 q^{15} - 2 q^{16} + 2 q^{18} + 2 q^{22} - 4 q^{23} - 6 q^{25} - 12 q^{29} + 4 q^{30} - 10 q^{32} + 2 q^{36} - 20 q^{37} - 16 q^{43} + 2 q^{44}+ \cdots + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 - 2 * q^4 + 6 * q^8 - 2 * q^9 - 2 * q^11 - 4 * q^15 - 2 * q^16 + 2 * q^18 + 2 * q^22 - 4 * q^23 - 6 * q^25 - 12 * q^29 + 4 * q^30 - 10 * q^32 + 2 * q^36 - 20 * q^37 - 16 * q^43 + 2 * q^44 + 4 * q^46 + 6 * q^50 - 16 * q^57 + 12 * q^58 + 4 * q^60 + 14 * q^64 + 4 * q^67 + 12 * q^71 - 6 * q^72 + 20 * q^74 - 16 * q^79 - 10 * q^81 + 16 * q^86 - 6 * q^88 + 4 * q^92 + 12 * q^93 + 16 * q^95 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more