LMFDB - Embedded newform 3969.2.a.l.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3969,2,Mod(1,3969)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3969, 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("3969.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3969 = 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3969.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:	$31.6926245622$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.87939$ of defining polynomial
Character	$\chi$	$=$	3969.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.879385 q^{2} -1.22668 q^{4} +1.34730 q^{5} -2.83750 q^{8} +O(q^{10})$	$q+0.879385 q^{2} -1.22668 q^{4} +1.34730 q^{5} -2.83750 q^{8} +1.18479 q^{10} -1.65270 q^{11} +3.36959 q^{13} -0.0418891 q^{16} +0.467911 q^{17} +3.22668 q^{19} -1.65270 q^{20} -1.45336 q^{22} -8.94356 q^{23} -3.18479 q^{25} +2.96316 q^{26} -6.26857 q^{29} -9.23442 q^{31} +5.63816 q^{32} +0.411474 q^{34} +9.23442 q^{37} +2.83750 q^{38} -3.82295 q^{40} +3.41147 q^{41} -4.41147 q^{43} +2.02734 q^{44} -7.86484 q^{46} +9.35504 q^{47} -2.80066 q^{50} -4.13341 q^{52} +0.573978 q^{53} -2.22668 q^{55} -5.51249 q^{58} -10.3969 q^{59} -7.63816 q^{61} -8.12061 q^{62} +5.04189 q^{64} +4.53983 q^{65} +0.596267 q^{67} -0.573978 q^{68} +0.554378 q^{71} -2.04963 q^{73} +8.12061 q^{74} -3.95811 q^{76} -2.40373 q^{79} -0.0564370 q^{80} +3.00000 q^{82} -15.0496 q^{83} +0.630415 q^{85} -3.87939 q^{86} +4.68954 q^{88} +9.08647 q^{89} +10.9709 q^{92} +8.22668 q^{94} +4.34730 q^{95} +1.89899 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{8}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - 6 * q^8	$ 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{8} - 6 q^{11} + 3 q^{13} + 3 q^{16} + 6 q^{17} + 3 q^{19} - 6 q^{20} + 9 q^{22} - 12 q^{23} - 6 q^{25} - 3 q^{26} - 9 q^{29} + 3 q^{31} - 9 q^{34} - 3 q^{37} + 6 q^{38} + 9 q^{40} - 3 q^{43} - 15 q^{44} + 3 q^{47} + 6 q^{50} + 21 q^{52} - 6 q^{53} - 9 q^{58} - 3 q^{59} - 6 q^{61} - 30 q^{62} + 12 q^{64} - 15 q^{65} - 12 q^{67} + 6 q^{68} - 9 q^{71} + 21 q^{73} + 30 q^{74} - 15 q^{76} - 21 q^{79} - 15 q^{80} + 9 q^{82} - 18 q^{83} + 9 q^{85} - 6 q^{86} + 27 q^{88} + 12 q^{89} - 3 q^{92} + 18 q^{94} + 12 q^{95} + 3 q^{97}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - 6 * q^8 - 6 * q^11 + 3 * q^13 + 3 * q^16 + 6 * q^17 + 3 * q^19 - 6 * q^20 + 9 * q^22 - 12 * q^23 - 6 * q^25 - 3 * q^26 - 9 * q^29 + 3 * q^31 - 9 * q^34 - 3 * q^37 + 6 * q^38 + 9 * q^40 - 3 * q^43 - 15 * q^44 + 3 * q^47 + 6 * q^50 + 21 * q^52 - 6 * q^53 - 9 * q^58 - 3 * q^59 - 6 * q^61 - 30 * q^62 + 12 * q^64 - 15 * q^65 - 12 * q^67 + 6 * q^68 - 9 * q^71 + 21 * q^73 + 30 * q^74 - 15 * q^76 - 21 * q^79 - 15 * q^80 + 9 * q^82 - 18 * q^83 + 9 * q^85 - 6 * q^86 + 27 * q^88 + 12 * q^89 - 3 * q^92 + 18 * q^94 + 12 * q^95 + 3 * q^97

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 $

$2$	0.879385	0.621819	0.310910	−	0.950439i	$-0.399366\pi$
$2$	0.879385	0.621819	0.310910	+	0.950439i	$0.399366\pi$
$3$	0	0
$3$	0	0
$4$	−1.22668	−0.613341
$5$	1.34730	0.602529	0.301265	−	0.953541i	$-0.402591\pi$
$5$	1.34730	0.602529	0.301265	+	0.953541i	$0.402591\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−2.83750	−1.00321
$9$	0	0
$10$	1.18479	0.374664
$11$	−1.65270	−0.498309	−0.249154	−	0.968464i	$-0.580153\pi$
$11$	−1.65270	−0.498309	−0.249154	+	0.968464i	$0.580153\pi$
$12$	0	0
$13$	3.36959	0.934555	0.467277	−	0.884111i	$-0.345235\pi$
$13$	3.36959	0.934555	0.467277	+	0.884111i	$0.345235\pi$
$14$	0	0
$15$	0	0
$16$	−0.0418891	−0.0104723
$17$	0.467911	0.113485	0.0567426	−	0.998389i	$-0.481929\pi$
$17$	0.467911	0.113485	0.0567426	+	0.998389i	$0.481929\pi$
$18$	0	0
$19$	3.22668	0.740252	0.370126	−	0.928982i	$-0.379315\pi$
$19$	3.22668	0.740252	0.370126	+	0.928982i	$0.379315\pi$
$20$	−1.65270	−0.369556
$21$	0	0
$22$	−1.45336	−0.309858
$23$	−8.94356	−1.86486	−0.932431	−	0.361348i	$-0.882317\pi$
$23$	−8.94356	−1.86486	−0.932431	+	0.361348i	$0.882317\pi$
$24$	0	0
$25$	−3.18479	−0.636959
$26$	2.96316	0.581124
$27$	0	0
$28$	0	0
$29$	−6.26857	−1.16404	−0.582022	−	0.813173i	$-0.697738\pi$
$29$	−6.26857	−1.16404	−0.582022	+	0.813173i	$0.697738\pi$
$30$	0	0
$31$	−9.23442	−1.65855	−0.829276	−	0.558840i	$-0.811247\pi$
$31$	−9.23442	−1.65855	−0.829276	+	0.558840i	$0.811247\pi$
$32$	5.63816	0.996695
$33$	0	0
$34$	0.411474	0.0705672
$35$	0	0
$36$	0	0
$37$	9.23442	1.51813	0.759065	−	0.651015i	$-0.225657\pi$
$37$	9.23442	1.51813	0.759065	+	0.651015i	$0.225657\pi$
$38$	2.83750	0.460303
$39$	0	0
$40$	−3.82295	−0.604461
$41$	3.41147	0.532783	0.266391	−	0.963865i	$-0.414169\pi$
$41$	3.41147	0.532783	0.266391	+	0.963865i	$0.414169\pi$
$42$	0	0
$43$	−4.41147	−0.672743	−0.336372	−	0.941729i	$-0.609200\pi$
$43$	−4.41147	−0.672743	−0.336372	+	0.941729i	$0.609200\pi$
$44$	2.02734	0.305633
$45$	0	0
$46$	−7.86484	−1.15961
$47$	9.35504	1.36457	0.682286	−	0.731085i	$-0.260986\pi$
$47$	9.35504	1.36457	0.682286	+	0.731085i	$0.260986\pi$
$48$	0	0
$49$	0	0
$50$	−2.80066	−0.396073
$51$	0	0
$52$	−4.13341	−0.573201
$53$	0.573978	0.0788419	0.0394210	−	0.999223i	$-0.487449\pi$
$53$	0.573978	0.0788419	0.0394210	+	0.999223i	$0.487449\pi$
$54$	0	0
$55$	−2.22668	−0.300246
$56$	0	0
$57$	0	0
$58$	−5.51249	−0.723825
$59$	−10.3969	−1.35356	−0.676782	−	0.736183i	$-0.736626\pi$
$59$	−10.3969	−1.35356	−0.676782	+	0.736183i	$0.736626\pi$
$60$	0	0
$61$	−7.63816	−0.977966	−0.488983	−	0.872293i	$-0.662632\pi$
$61$	−7.63816	−0.977966	−0.488983	+	0.872293i	$0.662632\pi$
$62$	−8.12061	−1.03132
$63$	0	0
$64$	5.04189	0.630236
$65$	4.53983	0.563097
$66$	0	0
$67$	0.596267	0.0728456	0.0364228	−	0.999336i	$-0.488404\pi$
$67$	0.596267	0.0728456	0.0364228	+	0.999336i	$0.488404\pi$
$68$	−0.573978	−0.0696051
$69$	0	0
$70$	0	0
$71$	0.554378	0.0657925	0.0328963	−	0.999459i	$-0.489527\pi$
$71$	0.554378	0.0657925	0.0328963	+	0.999459i	$0.489527\pi$
$72$	0	0
$73$	−2.04963	−0.239891	−0.119946	−	0.992780i	$-0.538272\pi$
$73$	−2.04963	−0.239891	−0.119946	+	0.992780i	$0.538272\pi$
$74$	8.12061	0.944002
$75$	0	0
$76$	−3.95811	−0.454026
$77$	0	0
$78$	0	0
$79$	−2.40373	−0.270441	−0.135221	−	0.990816i	$-0.543174\pi$
$79$	−2.40373	−0.270441	−0.135221	+	0.990816i	$0.543174\pi$
$80$	−0.0564370	−0.00630985
$81$	0	0
$82$	3.00000	0.331295
$83$	−15.0496	−1.65191	−0.825956	−	0.563735i	$-0.809364\pi$
$83$	−15.0496	−1.65191	−0.825956	+	0.563735i	$0.809364\pi$
$84$	0	0
$85$	0.630415	0.0683781
$86$	−3.87939	−0.418325
$87$	0	0
$88$	4.68954	0.499907
$89$	9.08647	0.963164	0.481582	−	0.876401i	$-0.340062\pi$
$89$	9.08647	0.963164	0.481582	+	0.876401i	$0.340062\pi$
$90$	0	0
$91$	0	0
$92$	10.9709	1.14380
$93$	0	0
$94$	8.22668	0.848517
$95$	4.34730	0.446023
$96$	0	0
$97$	1.89899	0.192813	0.0964064	−	0.995342i	$-0.469265\pi$
$97$	1.89899	0.192813	0.0964064	+	0.995342i	$0.469265\pi$
$98$	0	0
$99$	0	0
$100$	3.90673	0.390673
$101$	−1.70914	−0.170066	−0.0850329	−	0.996378i	$-0.527100\pi$
$101$	−1.70914	−0.170066	−0.0850329	+	0.996378i	$0.527100\pi$
$102$	0	0
$103$	3.63816	0.358478	0.179239	−	0.983806i	$-0.442636\pi$
$103$	3.63816	0.358478	0.179239	+	0.983806i	$0.442636\pi$
$104$	−9.56118	−0.937551
$105$	0	0
$106$	0.504748	0.0490254
$107$	−7.12836	−0.689124	−0.344562	−	0.938764i	$-0.611973\pi$
$107$	−7.12836	−0.689124	−0.344562	+	0.938764i	$0.611973\pi$
$108$	0	0
$109$	0.403733	0.0386706	0.0193353	−	0.999813i	$-0.493845\pi$
$109$	0.403733	0.0386706	0.0193353	+	0.999813i	$0.493845\pi$
$110$	−1.95811	−0.186699
$111$	0	0
$112$	0	0
$113$	−14.3696	−1.35178	−0.675888	−	0.737004i	$-0.736240\pi$
$113$	−14.3696	−1.35178	−0.675888	+	0.737004i	$0.736240\pi$
$114$	0	0
$115$	−12.0496	−1.12363
$116$	7.68954	0.713956
$117$	0	0
$118$	−9.14290	−0.841672
$119$	0	0
$120$	0	0
$121$	−8.26857	−0.751688
$122$	−6.71688	−0.608118
$123$	0	0
$124$	11.3277	1.01726
$125$	−11.0273	−0.986315
$126$	0	0
$127$	−20.7716	−1.84318	−0.921589	−	0.388167i	$-0.873108\pi$
$127$	−20.7716	−1.84318	−0.921589	+	0.388167i	$0.873108\pi$
$128$	−6.84255	−0.604802
$129$	0	0
$130$	3.99226	0.350144
$131$	−7.16519	−0.626026	−0.313013	−	0.949749i	$-0.601338\pi$
$131$	−7.16519	−0.626026	−0.313013	+	0.949749i	$0.601338\pi$
$132$	0	0
$133$	0	0
$134$	0.524348	0.0452968
$135$	0	0
$136$	−1.32770	−0.113849
$137$	−2.56893	−0.219478	−0.109739	−	0.993960i	$-0.535002\pi$
$137$	−2.56893	−0.219478	−0.109739	+	0.993960i	$0.535002\pi$
$138$	0	0
$139$	6.13341	0.520229	0.260114	−	0.965578i	$-0.416240\pi$
$139$	6.13341	0.520229	0.260114	+	0.965578i	$0.416240\pi$
$140$	0	0
$141$	0	0
$142$	0.487511	0.0409111
$143$	−5.56893	−0.465697
$144$	0	0
$145$	−8.44562	−0.701371
$146$	−1.80241	−0.149169
$147$	0	0
$148$	−11.3277	−0.931131
$149$	−0.431074	−0.0353150	−0.0176575	−	0.999844i	$-0.505621\pi$
$149$	−0.431074	−0.0353150	−0.0176575	+	0.999844i	$0.505621\pi$
$150$	0	0
$151$	−2.47060	−0.201055	−0.100527	−	0.994934i	$-0.532053\pi$
$151$	−2.47060	−0.201055	−0.100527	+	0.994934i	$0.532053\pi$
$152$	−9.15570	−0.742625
$153$	0	0
$154$	0	0
$155$	−12.4415	−0.999326
$156$	0	0
$157$	−10.1334	−0.808734	−0.404367	−	0.914597i	$-0.632508\pi$
$157$	−10.1334	−0.808734	−0.404367	+	0.914597i	$0.632508\pi$
$158$	−2.11381	−0.168166
$159$	0	0
$160$	7.59627	0.600538
$161$	0	0
$162$	0	0
$163$	−2.59627	−0.203355	−0.101678	−	0.994817i	$-0.532421\pi$
$163$	−2.59627	−0.203355	−0.101678	+	0.994817i	$0.532421\pi$
$164$	−4.18479	−0.326777
$165$	0	0
$166$	−13.2344	−1.02719
$167$	−23.1830	−1.79396	−0.896979	−	0.442074i	$-0.854243\pi$
$167$	−23.1830	−1.79396	−0.896979	+	0.442074i	$0.854243\pi$
$168$	0	0
$169$	−1.64590	−0.126607
$170$	0.554378	0.0425188
$171$	0	0
$172$	5.41147	0.412621
$173$	−4.75196	−0.361285	−0.180643	−	0.983549i	$-0.557818\pi$
$173$	−4.75196	−0.361285	−0.180643	+	0.983549i	$0.557818\pi$
$174$	0	0
$175$	0	0
$176$	0.0692302	0.00521842
$177$	0	0
$178$	7.99050	0.598914
$179$	8.53209	0.637718	0.318859	−	0.947802i	$-0.396700\pi$
$179$	8.53209	0.637718	0.318859	+	0.947802i	$0.396700\pi$
$180$	0	0
$181$	17.2344	1.28102	0.640512	−	0.767948i	$-0.278722\pi$
$181$	17.2344	1.28102	0.640512	+	0.767948i	$0.278722\pi$
$182$	0	0
$183$	0	0
$184$	25.3773	1.87084
$185$	12.4415	0.914718
$186$	0	0
$187$	−0.773318	−0.0565506
$188$	−11.4757	−0.836948
$189$	0	0
$190$	3.82295	0.277346
$191$	−12.9094	−0.934092	−0.467046	−	0.884233i	$-0.654682\pi$
$191$	−12.9094	−0.934092	−0.467046	+	0.884233i	$0.654682\pi$
$192$	0	0
$193$	−0.638156	−0.0459355	−0.0229677	−	0.999736i	$-0.507311\pi$
$193$	−0.638156	−0.0459355	−0.0229677	+	0.999736i	$0.507311\pi$
$194$	1.66994	0.119895
$195$	0	0
$196$	0	0
$197$	−11.4456	−0.815467	−0.407733	−	0.913101i	$-0.633681\pi$
$197$	−11.4456	−0.815467	−0.407733	+	0.913101i	$0.633681\pi$
$198$	0	0
$199$	3.63816	0.257902	0.128951	−	0.991651i	$-0.458839\pi$
$199$	3.63816	0.257902	0.128951	+	0.991651i	$0.458839\pi$
$200$	9.03684	0.639001
$201$	0	0
$202$	−1.50299	−0.105750
$203$	0	0
$204$	0	0
$205$	4.59627	0.321017
$206$	3.19934	0.222909
$207$	0	0
$208$	−0.141149	−0.00978691
$209$	−5.33275	−0.368874
$210$	0	0
$211$	5.82295	0.400868	0.200434	−	0.979707i	$-0.435765\pi$
$211$	5.82295	0.400868	0.200434	+	0.979707i	$0.435765\pi$
$212$	−0.704088	−0.0483570
$213$	0	0
$214$	−6.26857	−0.428511
$215$	−5.94356	−0.405348
$216$	0	0
$217$	0	0
$218$	0.355037	0.0240461
$219$	0	0
$220$	2.73143	0.184153
$221$	1.57667	0.106058
$222$	0	0
$223$	−7.08378	−0.474365	−0.237182	−	0.971465i	$-0.576224\pi$
$223$	−7.08378	−0.474365	−0.237182	+	0.971465i	$0.576224\pi$
$224$	0	0
$225$	0	0
$226$	−12.6364	−0.840561
$227$	−11.9436	−0.792722	−0.396361	−	0.918095i	$-0.629727\pi$
$227$	−11.9436	−0.792722	−0.396361	+	0.918095i	$0.629727\pi$
$228$	0	0
$229$	17.5526	1.15991	0.579955	−	0.814649i	$-0.303070\pi$
$229$	17.5526	1.15991	0.579955	+	0.814649i	$0.303070\pi$
$230$	−10.5963	−0.698697
$231$	0	0
$232$	17.7870	1.16778
$233$	−16.2540	−1.06484	−0.532418	−	0.846481i	$-0.678717\pi$
$233$	−16.2540	−1.06484	−0.532418	+	0.846481i	$0.678717\pi$
$234$	0	0
$235$	12.6040	0.822195
$236$	12.7537	0.830196
$237$	0	0
$238$	0	0
$239$	15.0993	0.976690	0.488345	−	0.872651i	$-0.337601\pi$
$239$	15.0993	0.976690	0.488345	+	0.872651i	$0.337601\pi$
$240$	0	0
$241$	15.6382	1.00734	0.503671	−	0.863896i	$-0.331982\pi$
$241$	15.6382	1.00734	0.503671	+	0.863896i	$0.331982\pi$
$242$	−7.27126	−0.467414
$243$	0	0
$244$	9.36959	0.599826
$245$	0	0
$246$	0	0
$247$	10.8726	0.691806
$248$	26.2026	1.66387
$249$	0	0
$250$	−9.69728	−0.613310
$251$	19.0651	1.20338	0.601690	−	0.798730i	$-0.294494\pi$
$251$	19.0651	1.20338	0.601690	+	0.798730i	$0.294494\pi$
$252$	0	0
$253$	14.7811	0.929277
$254$	−18.2662	−1.14612
$255$	0	0
$256$	−16.1010	−1.00631
$257$	26.5817	1.65812	0.829061	−	0.559158i	$-0.188876\pi$
$257$	26.5817	1.65812	0.829061	+	0.559158i	$0.188876\pi$
$258$	0	0
$259$	0	0
$260$	−5.56893	−0.345370
$261$	0	0
$262$	−6.30096	−0.389275
$263$	0.734118	0.0452676	0.0226338	−	0.999744i	$-0.492795\pi$
$263$	0.734118	0.0452676	0.0226338	+	0.999744i	$0.492795\pi$
$264$	0	0
$265$	0.773318	0.0475046
$266$	0	0
$267$	0	0
$268$	−0.731429	−0.0446792
$269$	−20.8503	−1.27126	−0.635632	−	0.771992i	$-0.719261\pi$
$269$	−20.8503	−1.27126	−0.635632	+	0.771992i	$0.719261\pi$
$270$	0	0
$271$	−6.95811	−0.422675	−0.211338	−	0.977413i	$-0.567782\pi$
$271$	−6.95811	−0.422675	−0.211338	+	0.977413i	$0.567782\pi$
$272$	−0.0196004	−0.00118845
$273$	0	0
$274$	−2.25908	−0.136476
$275$	5.26352	0.317402
$276$	0	0
$277$	17.8726	1.07386	0.536930	−	0.843627i	$-0.319584\pi$
$277$	17.8726	1.07386	0.536930	+	0.843627i	$0.319584\pi$
$278$	5.39363	0.323488
$279$	0	0
$280$	0	0
$281$	−22.3105	−1.33093	−0.665465	−	0.746429i	$-0.731767\pi$
$281$	−22.3105	−1.33093	−0.665465	+	0.746429i	$0.731767\pi$
$282$	0	0
$283$	18.5945	1.10533	0.552665	−	0.833404i	$-0.313611\pi$
$283$	18.5945	1.10533	0.552665	+	0.833404i	$0.313611\pi$
$284$	−0.680045	−0.0403532
$285$	0	0
$286$	−4.89723	−0.289579
$287$	0	0
$288$	0	0
$289$	−16.7811	−0.987121
$290$	−7.42696	−0.436126
$291$	0	0
$292$	2.51424	0.147135
$293$	13.0915	0.764815	0.382407	−	0.923994i	$-0.375095\pi$
$293$	13.0915	0.764815	0.382407	+	0.923994i	$0.375095\pi$
$294$	0	0
$295$	−14.0077	−0.815562
$296$	−26.2026	−1.52300
$297$	0	0
$298$	−0.379081	−0.0219595
$299$	−30.1361	−1.74282
$300$	0	0
$301$	0	0
$302$	−2.17261	−0.125020
$303$	0	0
$304$	−0.135163	−0.00775211
$305$	−10.2909	−0.589253
$306$	0	0
$307$	−6.31046	−0.360157	−0.180078	−	0.983652i	$-0.557635\pi$
$307$	−6.31046	−0.360157	−0.180078	+	0.983652i	$0.557635\pi$
$308$	0	0
$309$	0	0
$310$	−10.9409	−0.621400
$311$	−9.52435	−0.540076	−0.270038	−	0.962850i	$-0.587036\pi$
$311$	−9.52435	−0.540076	−0.270038	+	0.962850i	$0.587036\pi$
$312$	0	0
$313$	17.6287	0.996431	0.498215	−	0.867053i	$-0.333989\pi$
$313$	17.6287	0.996431	0.498215	+	0.867053i	$0.333989\pi$
$314$	−8.91117	−0.502886
$315$	0	0
$316$	2.94862	0.165873
$317$	−8.07697	−0.453648	−0.226824	−	0.973936i	$-0.572834\pi$
$317$	−8.07697	−0.453648	−0.226824	+	0.973936i	$0.572834\pi$
$318$	0	0
$319$	10.3601	0.580054
$320$	6.79292	0.379736
$321$	0	0
$322$	0	0
$323$	1.50980	0.0840075
$324$	0	0
$325$	−10.7314	−0.595273
$326$	−2.28312	−0.126450
$327$	0	0
$328$	−9.68004	−0.534491
$329$	0	0
$330$	0	0
$331$	23.0496	1.26692	0.633461	−	0.773775i	$-0.281634\pi$
$331$	23.0496	1.26692	0.633461	+	0.773775i	$0.281634\pi$
$332$	18.4611	1.01318
$333$	0	0
$334$	−20.3868	−1.11552
$335$	0.803348	0.0438916
$336$	0	0
$337$	29.0232	1.58100	0.790498	−	0.612465i	$-0.209822\pi$
$337$	29.0232	1.58100	0.790498	+	0.612465i	$0.209822\pi$
$338$	−1.44738	−0.0787269
$339$	0	0
$340$	−0.773318	−0.0419391
$341$	15.2618	0.826471
$342$	0	0
$343$	0	0
$344$	12.5175	0.674901
$345$	0	0
$346$	−4.17881	−0.224654
$347$	−12.9463	−0.694991	−0.347496	−	0.937682i	$-0.612968\pi$
$347$	−12.9463	−0.694991	−0.347496	+	0.937682i	$0.612968\pi$
$348$	0	0
$349$	−1.46286	−0.0783050	−0.0391525	−	0.999233i	$-0.512466\pi$
$349$	−1.46286	−0.0783050	−0.0391525	+	0.999233i	$0.512466\pi$
$350$	0	0
$351$	0	0
$352$	−9.31820	−0.496662
$353$	14.3327	0.762855	0.381428	−	0.924399i	$-0.375433\pi$
$353$	14.3327	0.762855	0.381428	+	0.924399i	$0.375433\pi$
$354$	0	0
$355$	0.746911	0.0396419
$356$	−11.1462	−0.590747
$357$	0	0
$358$	7.50299	0.396546
$359$	20.9368	1.10500	0.552500	−	0.833513i	$-0.313674\pi$
$359$	20.9368	1.10500	0.552500	+	0.833513i	$0.313674\pi$
$360$	0	0
$361$	−8.58853	−0.452028
$362$	15.1557	0.796566
$363$	0	0
$364$	0	0
$365$	−2.76146	−0.144541
$366$	0	0
$367$	12.0574	0.629390	0.314695	−	0.949193i	$-0.398098\pi$
$367$	12.0574	0.629390	0.314695	+	0.949193i	$0.398098\pi$
$368$	0.374638	0.0195293
$369$	0	0
$370$	10.9409	0.568789
$371$	0	0
$372$	0	0
$373$	−0.781059	−0.0404417	−0.0202209	−	0.999796i	$-0.506437\pi$
$373$	−0.781059	−0.0404417	−0.0202209	+	0.999796i	$0.506437\pi$
$374$	−0.680045	−0.0351643
$375$	0	0
$376$	−26.5449	−1.36895
$377$	−21.1225	−1.08786
$378$	0	0
$379$	−6.92396	−0.355660	−0.177830	−	0.984061i	$-0.556908\pi$
$379$	−6.92396	−0.355660	−0.177830	+	0.984061i	$0.556908\pi$
$380$	−5.33275	−0.273564
$381$	0	0
$382$	−11.3523	−0.580837
$383$	7.73236	0.395105	0.197553	−	0.980292i	$-0.436701\pi$
$383$	7.73236	0.395105	0.197553	+	0.980292i	$0.436701\pi$
$384$	0	0
$385$	0	0
$386$	−0.561185	−0.0285636
$387$	0	0
$388$	−2.32945	−0.118260
$389$	−5.39961	−0.273771	−0.136886	−	0.990587i	$-0.543709\pi$
$389$	−5.39961	−0.273771	−0.136886	+	0.990587i	$0.543709\pi$
$390$	0	0
$391$	−4.18479	−0.211634
$392$	0	0
$393$	0	0
$394$	−10.0651	−0.507073
$395$	−3.23854	−0.162949
$396$	0	0
$397$	29.2344	1.46723	0.733617	−	0.679563i	$-0.237831\pi$
$397$	29.2344	1.46723	0.733617	+	0.679563i	$0.237831\pi$
$398$	3.19934	0.160368
$399$	0	0
$400$	0.133408	0.00667040
$401$	27.3979	1.36818	0.684092	−	0.729396i	$-0.260199\pi$
$401$	27.3979	1.36818	0.684092	+	0.729396i	$0.260199\pi$
$402$	0	0
$403$	−31.1162	−1.55001
$404$	2.09657	0.104308
$405$	0	0
$406$	0	0
$407$	−15.2618	−0.756498
$408$	0	0
$409$	9.02498	0.446256	0.223128	−	0.974789i	$-0.428373\pi$
$409$	9.02498	0.446256	0.223128	+	0.974789i	$0.428373\pi$
$410$	4.04189	0.199615
$411$	0	0
$412$	−4.46286	−0.219869
$413$	0	0
$414$	0	0
$415$	−20.2763	−0.995325
$416$	18.9982	0.931466
$417$	0	0
$418$	−4.68954	−0.229373
$419$	0.175297	0.00856382	0.00428191	−	0.999991i	$-0.498637\pi$
$419$	0.175297	0.00856382	0.00428191	+	0.999991i	$0.498637\pi$
$420$	0	0
$421$	−24.7050	−1.20405	−0.602025	−	0.798478i	$-0.705639\pi$
$421$	−24.7050	−1.20405	−0.602025	+	0.798478i	$0.705639\pi$
$422$	5.12061	0.249268
$423$	0	0
$424$	−1.62866	−0.0790947
$425$	−1.49020	−0.0722853
$426$	0	0
$427$	0	0
$428$	8.74422	0.422668
$429$	0	0
$430$	−5.22668	−0.252053
$431$	29.3191	1.41225	0.706126	−	0.708086i	$-0.250441\pi$
$431$	29.3191	1.41225	0.706126	+	0.708086i	$0.250441\pi$
$432$	0	0
$433$	−19.6554	−0.944578	−0.472289	−	0.881444i	$-0.656572\pi$
$433$	−19.6554	−0.944578	−0.472289	+	0.881444i	$0.656572\pi$
$434$	0	0
$435$	0	0
$436$	−0.495252	−0.0237183
$437$	−28.8580	−1.38047
$438$	0	0
$439$	21.9299	1.04666	0.523330	−	0.852130i	$-0.324690\pi$
$439$	21.9299	1.04666	0.523330	+	0.852130i	$0.324690\pi$
$440$	6.31820	0.301208
$441$	0	0
$442$	1.38650	0.0659489
$443$	18.7101	0.888942	0.444471	−	0.895793i	$-0.353392\pi$
$443$	18.7101	0.888942	0.444471	+	0.895793i	$0.353392\pi$
$444$	0	0
$445$	12.2422	0.580334
$446$	−6.22937	−0.294969
$447$	0	0
$448$	0	0
$449$	−6.68004	−0.315251	−0.157625	−	0.987499i	$-0.550384\pi$
$449$	−6.68004	−0.315251	−0.157625	+	0.987499i	$0.550384\pi$
$450$	0	0
$451$	−5.63816	−0.265490
$452$	17.6269	0.829100
$453$	0	0
$454$	−10.5030	−0.492930
$455$	0	0
$456$	0	0
$457$	−19.4287	−0.908837	−0.454418	−	0.890788i	$-0.650153\pi$
$457$	−19.4287	−0.908837	−0.454418	+	0.890788i	$0.650153\pi$
$458$	15.4355	0.721254
$459$	0	0
$460$	14.7811	0.689170
$461$	−0.965852	−0.0449842	−0.0224921	−	0.999747i	$-0.507160\pi$
$461$	−0.965852	−0.0449842	−0.0224921	+	0.999747i	$0.507160\pi$
$462$	0	0
$463$	−0.445622	−0.0207098	−0.0103549	−	0.999946i	$-0.503296\pi$
$463$	−0.445622	−0.0207098	−0.0103549	+	0.999946i	$0.503296\pi$
$464$	0.262585	0.0121902
$465$	0	0
$466$	−14.2935	−0.662136
$467$	−34.2148	−1.58327	−0.791637	−	0.610992i	$-0.790771\pi$
$467$	−34.2148	−1.58327	−0.791637	+	0.610992i	$0.790771\pi$
$468$	0	0
$469$	0	0
$470$	11.0838	0.511257
$471$	0	0
$472$	29.5012	1.35790
$473$	7.29086	0.335234
$474$	0	0
$475$	−10.2763	−0.471510
$476$	0	0
$477$	0	0
$478$	13.2781	0.607325
$479$	−21.7929	−0.995744	−0.497872	−	0.867251i	$-0.665885\pi$
$479$	−21.7929	−0.995744	−0.497872	+	0.867251i	$0.665885\pi$
$480$	0	0
$481$	31.1162	1.41878
$482$	13.7520	0.626385
$483$	0	0
$484$	10.1429	0.461041
$485$	2.55850	0.116175
$486$	0	0
$487$	19.3928	0.878772	0.439386	−	0.898298i	$-0.355196\pi$
$487$	19.3928	0.878772	0.439386	+	0.898298i	$0.355196\pi$
$488$	21.6732	0.981101
$489$	0	0
$490$	0	0
$491$	−26.1566	−1.18043	−0.590216	−	0.807245i	$-0.700957\pi$
$491$	−26.1566	−1.18043	−0.590216	+	0.807245i	$0.700957\pi$
$492$	0	0
$493$	−2.93313	−0.132102
$494$	9.56118	0.430178
$495$	0	0
$496$	0.386821	0.0173688
$497$	0	0
$498$	0	0
$499$	−14.3013	−0.640214	−0.320107	−	0.947381i	$-0.603719\pi$
$499$	−14.3013	−0.640214	−0.320107	+	0.947381i	$0.603719\pi$
$500$	13.5270	0.604947
$501$	0	0
$502$	16.7656	0.748284
$503$	18.7033	0.833937	0.416969	−	0.908921i	$-0.363092\pi$
$503$	18.7033	0.833937	0.416969	+	0.908921i	$0.363092\pi$
$504$	0	0
$505$	−2.30272	−0.102470
$506$	12.9982	0.577842
$507$	0	0
$508$	25.4801	1.13050
$509$	−25.6091	−1.13510	−0.567551	−	0.823338i	$-0.692109\pi$
$509$	−25.6091	−1.13510	−0.567551	+	0.823338i	$0.692109\pi$
$510$	0	0
$511$	0	0
$512$	−0.473897	−0.0209435
$513$	0	0
$514$	23.3756	1.03105
$515$	4.90167	0.215994
$516$	0	0
$517$	−15.4611	−0.679979
$518$	0	0
$519$	0	0
$520$	−12.8817	−0.564902
$521$	−21.2121	−0.929320	−0.464660	−	0.885489i	$-0.653824\pi$
$521$	−21.2121	−0.929320	−0.464660	+	0.885489i	$0.653824\pi$
$522$	0	0
$523$	−20.8057	−0.909770	−0.454885	−	0.890550i	$-0.650320\pi$
$523$	−20.8057	−0.909770	−0.454885	+	0.890550i	$0.650320\pi$
$524$	8.78941	0.383967
$525$	0	0
$526$	0.645572	0.0281483
$527$	−4.32089	−0.188221
$528$	0	0
$529$	56.9873	2.47771
$530$	0.680045	0.0295393
$531$	0	0
$532$	0	0
$533$	11.4953	0.497915
$534$	0	0
$535$	−9.60401	−0.415217
$536$	−1.69190	−0.0730791
$537$	0	0
$538$	−18.3354	−0.790497
$539$	0	0
$540$	0	0
$541$	26.7297	1.14920	0.574599	−	0.818435i	$-0.305158\pi$
$541$	26.7297	1.14920	0.574599	+	0.818435i	$0.305158\pi$
$542$	−6.11886	−0.262828
$543$	0	0
$544$	2.63816	0.113110
$545$	0.543948	0.0233002
$546$	0	0
$547$	36.7624	1.57185	0.785923	−	0.618324i	$-0.212188\pi$
$547$	36.7624	1.57185	0.785923	+	0.618324i	$0.212188\pi$
$548$	3.15125	0.134615
$549$	0	0
$550$	4.62866	0.197367
$551$	−20.2267	−0.861686
$552$	0	0
$553$	0	0
$554$	15.7169	0.667746
$555$	0	0
$556$	−7.52374	−0.319078
$557$	−32.3387	−1.37024	−0.685118	−	0.728432i	$-0.740249\pi$
$557$	−32.3387	−1.37024	−0.685118	+	0.728432i	$0.740249\pi$
$558$	0	0
$559$	−14.8648	−0.628716
$560$	0	0
$561$	0	0
$562$	−19.6195	−0.827598
$563$	−17.7419	−0.747730	−0.373865	−	0.927483i	$-0.621968\pi$
$563$	−17.7419	−0.747730	−0.373865	+	0.927483i	$0.621968\pi$
$564$	0	0
$565$	−19.3601	−0.814485
$566$	16.3517	0.687315
$567$	0	0
$568$	−1.57304	−0.0660035
$569$	26.6013	1.11519	0.557593	−	0.830115i	$-0.311725\pi$
$569$	26.6013	1.11519	0.557593	+	0.830115i	$0.311725\pi$
$570$	0	0
$571$	−10.0172	−0.419208	−0.209604	−	0.977786i	$-0.567218\pi$
$571$	−10.0172	−0.419208	−0.209604	+	0.977786i	$0.567218\pi$
$572$	6.83130	0.285631
$573$	0	0
$574$	0	0
$575$	28.4834	1.18784
$576$	0	0
$577$	32.9145	1.37025	0.685124	−	0.728427i	$-0.259748\pi$
$577$	32.9145	1.37025	0.685124	+	0.728427i	$0.259748\pi$
$578$	−14.7570	−0.613811
$579$	0	0
$580$	10.3601	0.430179
$581$	0	0
$582$	0	0
$583$	−0.948615	−0.0392876
$584$	5.81582	0.240660
$585$	0	0
$586$	11.5125	0.475577
$587$	−15.0729	−0.622123	−0.311062	−	0.950390i	$-0.600685\pi$
$587$	−15.0729	−0.622123	−0.311062	+	0.950390i	$0.600685\pi$
$588$	0	0
$589$	−29.7965	−1.22775
$590$	−12.3182	−0.507132
$591$	0	0
$592$	−0.386821	−0.0158983
$593$	41.0009	1.68371	0.841853	−	0.539706i	$-0.181465\pi$
$593$	41.0009	1.68371	0.841853	+	0.539706i	$0.181465\pi$
$594$	0	0
$595$	0	0
$596$	0.528791	0.0216601
$597$	0	0
$598$	−26.5012	−1.08372
$599$	−6.07367	−0.248164	−0.124082	−	0.992272i	$-0.539599\pi$
$599$	−6.07367	−0.248164	−0.124082	+	0.992272i	$0.539599\pi$
$600$	0	0
$601$	14.1352	0.576585	0.288293	−	0.957542i	$-0.406912\pi$
$601$	14.1352	0.576585	0.288293	+	0.957542i	$0.406912\pi$
$602$	0	0
$603$	0	0
$604$	3.03064	0.123315
$605$	−11.1402	−0.452914
$606$	0	0
$607$	−46.0898	−1.87073	−0.935363	−	0.353689i	$-0.884927\pi$
$607$	−46.0898	−1.87073	−0.935363	+	0.353689i	$0.884927\pi$
$608$	18.1925	0.737805
$609$	0	0
$610$	−9.04963	−0.366409
$611$	31.5226	1.27527
$612$	0	0
$613$	−26.4938	−1.07008	−0.535038	−	0.844828i	$-0.679703\pi$
$613$	−26.4938	−1.07008	−0.535038	+	0.844828i	$0.679703\pi$
$614$	−5.54933	−0.223953
$615$	0	0
$616$	0	0
$617$	2.24990	0.0905777	0.0452889	−	0.998974i	$-0.485579\pi$
$617$	2.24990	0.0905777	0.0452889	+	0.998974i	$0.485579\pi$
$618$	0	0
$619$	−6.19078	−0.248828	−0.124414	−	0.992230i	$-0.539705\pi$
$619$	−6.19078	−0.248828	−0.124414	+	0.992230i	$0.539705\pi$
$620$	15.2618	0.612927
$621$	0	0
$622$	−8.37557	−0.335830
$623$	0	0
$624$	0	0
$625$	1.06687	0.0426746
$626$	15.5024	0.619600
$627$	0	0
$628$	12.4305	0.496030
$629$	4.32089	0.172285
$630$	0	0
$631$	26.1661	1.04166	0.520829	−	0.853661i	$-0.325623\pi$
$631$	26.1661	1.04166	0.520829	+	0.853661i	$0.325623\pi$
$632$	6.82058	0.271308
$633$	0	0
$634$	−7.10277	−0.282087
$635$	−27.9855	−1.11057
$636$	0	0
$637$	0	0
$638$	9.11051	0.360689
$639$	0	0
$640$	−9.21894	−0.364411
$641$	−4.88888	−0.193099	−0.0965496	−	0.995328i	$-0.530781\pi$
$641$	−4.88888	−0.193099	−0.0965496	+	0.995328i	$0.530781\pi$
$642$	0	0
$643$	40.3678	1.59195	0.795976	−	0.605328i	$-0.206958\pi$
$643$	40.3678	1.59195	0.795976	+	0.605328i	$0.206958\pi$
$644$	0	0
$645$	0	0
$646$	1.32770	0.0522375
$647$	−2.28075	−0.0896657	−0.0448329	−	0.998995i	$-0.514276\pi$
$647$	−2.28075	−0.0896657	−0.0448329	+	0.998995i	$0.514276\pi$
$648$	0	0
$649$	17.1830	0.674493
$650$	−9.43706	−0.370152
$651$	0	0
$652$	3.18479	0.124726
$653$	−23.4793	−0.918815	−0.459407	−	0.888226i	$-0.651938\pi$
$653$	−23.4793	−0.918815	−0.459407	+	0.888226i	$0.651938\pi$
$654$	0	0
$655$	−9.65364	−0.377199
$656$	−0.142903	−0.00557944
$657$	0	0
$658$	0	0
$659$	47.9623	1.86835	0.934174	−	0.356818i	$-0.116138\pi$
$659$	47.9623	1.86835	0.934174	+	0.356818i	$0.116138\pi$
$660$	0	0
$661$	−29.3090	−1.13999	−0.569995	−	0.821648i	$-0.693055\pi$
$661$	−29.3090	−1.13999	−0.569995	+	0.821648i	$0.693055\pi$
$662$	20.2695	0.787797
$663$	0	0
$664$	42.7033	1.65721
$665$	0	0
$666$	0	0
$667$	56.0634	2.17078
$668$	28.4382	1.10031
$669$	0	0
$670$	0.706452	0.0272926
$671$	12.6236	0.487329
$672$	0	0
$673$	26.3182	1.01449	0.507246	−	0.861801i	$-0.330664\pi$
$673$	26.3182	1.01449	0.507246	+	0.861801i	$0.330664\pi$
$674$	25.5226	0.983094
$675$	0	0
$676$	2.01899	0.0776535
$677$	35.8907	1.37939	0.689697	−	0.724098i	$-0.257744\pi$
$677$	35.8907	1.37939	0.689697	+	0.724098i	$0.257744\pi$
$678$	0	0
$679$	0	0
$680$	−1.78880	−0.0685973
$681$	0	0
$682$	13.4210	0.513915
$683$	−35.0642	−1.34169	−0.670847	−	0.741596i	$-0.734069\pi$
$683$	−35.0642	−1.34169	−0.670847	+	0.741596i	$0.734069\pi$
$684$	0	0
$685$	−3.46110	−0.132242
$686$	0	0
$687$	0	0
$688$	0.184793	0.00704515
$689$	1.93407	0.0736821
$690$	0	0
$691$	−2.06687	−0.0786273	−0.0393136	−	0.999227i	$-0.512517\pi$
$691$	−2.06687	−0.0786273	−0.0393136	+	0.999227i	$0.512517\pi$
$692$	5.82915	0.221591
$693$	0	0
$694$	−11.3847	−0.432159
$695$	8.26352	0.313453
$696$	0	0
$697$	1.59627	0.0604629
$698$	−1.28642	−0.0486916
$699$	0	0
$700$	0	0
$701$	7.36009	0.277987	0.138993	−	0.990293i	$-0.455613\pi$
$701$	7.36009	0.277987	0.138993	+	0.990293i	$0.455613\pi$
$702$	0	0
$703$	29.7965	1.12380
$704$	−8.33275	−0.314052
$705$	0	0
$706$	12.6040	0.474358
$707$	0	0
$708$	0	0
$709$	9.10876	0.342086	0.171043	−	0.985264i	$-0.445286\pi$
$709$	9.10876	0.342086	0.171043	+	0.985264i	$0.445286\pi$
$710$	0.656822	0.0246501
$711$	0	0
$712$	−25.7828	−0.966252
$713$	82.5886	3.09297
$714$	0	0
$715$	−7.50299	−0.280596
$716$	−10.4662	−0.391139
$717$	0	0
$718$	18.4115	0.687110
$719$	−25.9537	−0.967908	−0.483954	−	0.875093i	$-0.660800\pi$
$719$	−25.9537	−0.967908	−0.483954	+	0.875093i	$0.660800\pi$
$720$	0	0
$721$	0	0
$722$	−7.55262	−0.281080
$723$	0	0
$724$	−21.1411	−0.785705
$725$	19.9641	0.741448
$726$	0	0
$727$	10.1601	0.376819	0.188409	−	0.982091i	$-0.439667\pi$
$727$	10.1601	0.376819	0.188409	+	0.982091i	$0.439667\pi$
$728$	0	0
$729$	0	0
$730$	−2.42839	−0.0898786
$731$	−2.06418	−0.0763464
$732$	0	0
$733$	−40.6614	−1.50186	−0.750931	−	0.660381i	$-0.770395\pi$
$733$	−40.6614	−1.50186	−0.750931	+	0.660381i	$0.770395\pi$
$734$	10.6031	0.391367
$735$	0	0
$736$	−50.4252	−1.85870
$737$	−0.985452	−0.0362996
$738$	0	0
$739$	−25.3618	−0.932951	−0.466475	−	0.884534i	$-0.654476\pi$
$739$	−25.3618	−0.932951	−0.466475	+	0.884534i	$0.654476\pi$
$740$	−15.2618	−0.561034
$741$	0	0
$742$	0	0
$743$	22.4442	0.823398	0.411699	−	0.911320i	$-0.364936\pi$
$743$	22.4442	0.823398	0.411699	+	0.911320i	$0.364936\pi$
$744$	0	0
$745$	−0.580785	−0.0212783
$746$	−0.686852	−0.0251474
$747$	0	0
$748$	0.948615	0.0346848
$749$	0	0
$750$	0	0
$751$	24.2172	0.883698	0.441849	−	0.897090i	$-0.354323\pi$
$751$	24.2172	0.883698	0.441849	+	0.897090i	$0.354323\pi$
$752$	−0.391874	−0.0142902
$753$	0	0
$754$	−18.5748	−0.676454
$755$	−3.32863	−0.121141
$756$	0	0
$757$	9.11793	0.331397	0.165698	−	0.986176i	$-0.447012\pi$
$757$	9.11793	0.331397	0.165698	+	0.986176i	$0.447012\pi$
$758$	−6.08883	−0.221156
$759$	0	0
$760$	−12.3354	−0.447453
$761$	−18.2722	−0.662366	−0.331183	−	0.943566i	$-0.607448\pi$
$761$	−18.2722	−0.662366	−0.331183	+	0.943566i	$0.607448\pi$
$762$	0	0
$763$	0	0
$764$	15.8357	0.572917
$765$	0	0
$766$	6.79973	0.245684
$767$	−35.0333	−1.26498
$768$	0	0
$769$	−18.5294	−0.668187	−0.334094	−	0.942540i	$-0.608430\pi$
$769$	−18.5294	−0.668187	−0.334094	+	0.942540i	$0.608430\pi$
$770$	0	0
$771$	0	0
$772$	0.782814	0.0281741
$773$	−2.96080	−0.106493	−0.0532463	−	0.998581i	$-0.516957\pi$
$773$	−2.96080	−0.106493	−0.0532463	+	0.998581i	$0.516957\pi$
$774$	0	0
$775$	29.4097	1.05643
$776$	−5.38836	−0.193431
$777$	0	0
$778$	−4.74834	−0.170236
$779$	11.0077	0.394393
$780$	0	0
$781$	−0.916222	−0.0327850
$782$	−3.68004	−0.131598
$783$	0	0
$784$	0	0
$785$	−13.6527	−0.487286
$786$	0	0
$787$	33.4020	1.19065	0.595326	−	0.803484i	$-0.297023\pi$
$787$	33.4020	1.19065	0.595326	+	0.803484i	$0.297023\pi$
$788$	14.0401	0.500159
$789$	0	0
$790$	−2.84793	−0.101325
$791$	0	0
$792$	0	0
$793$	−25.7374	−0.913962
$794$	25.7083	0.912354
$795$	0	0
$796$	−4.46286	−0.158182
$797$	49.3509	1.74810	0.874050	−	0.485837i	$-0.161485\pi$
$797$	49.3509	1.74810	0.874050	+	0.485837i	$0.161485\pi$
$798$	0	0
$799$	4.37733	0.154859
$800$	−17.9564	−0.634853
$801$	0	0
$802$	24.0933	0.850763
$803$	3.38743	0.119540
$804$	0	0
$805$	0	0
$806$	−27.3631	−0.963824
$807$	0	0
$808$	4.84968	0.170611
$809$	−19.8280	−0.697115	−0.348558	−	0.937287i	$-0.613328\pi$
$809$	−19.8280	−0.697115	−0.348558	+	0.937287i	$0.613328\pi$
$810$	0	0
$811$	23.8557	0.837686	0.418843	−	0.908059i	$-0.362436\pi$
$811$	23.8557	0.837686	0.418843	+	0.908059i	$0.362436\pi$
$812$	0	0
$813$	0	0
$814$	−13.4210	−0.470405
$815$	−3.49794	−0.122528
$816$	0	0
$817$	−14.2344	−0.497999
$818$	7.93643	0.277491
$819$	0	0
$820$	−5.63816	−0.196893
$821$	50.9427	1.77791	0.888957	−	0.457991i	$-0.151431\pi$
$821$	50.9427	1.77791	0.888957	+	0.457991i	$0.151431\pi$
$822$	0	0
$823$	13.6149	0.474587	0.237293	−	0.971438i	$-0.423740\pi$
$823$	13.6149	0.474587	0.237293	+	0.971438i	$0.423740\pi$
$824$	−10.3233	−0.359628
$825$	0	0
$826$	0	0
$827$	−36.2158	−1.25935	−0.629673	−	0.776861i	$-0.716811\pi$
$827$	−36.2158	−1.25935	−0.629673	+	0.776861i	$0.716811\pi$
$828$	0	0
$829$	−25.3259	−0.879606	−0.439803	−	0.898094i	$-0.644952\pi$
$829$	−25.3259	−0.879606	−0.439803	+	0.898094i	$0.644952\pi$
$830$	−17.8307	−0.618912
$831$	0	0
$832$	16.9891	0.588990
$833$	0	0
$834$	0	0
$835$	−31.2344	−1.08091
$836$	6.54158	0.226245
$837$	0	0
$838$	0.154154	0.00532515
$839$	8.71419	0.300847	0.150424	−	0.988622i	$-0.451936\pi$
$839$	8.71419	0.300847	0.150424	+	0.988622i	$0.451936\pi$
$840$	0	0
$841$	10.2950	0.354999
$842$	−21.7252	−0.748701
$843$	0	0
$844$	−7.14290	−0.245869
$845$	−2.21751	−0.0762847
$846$	0	0
$847$	0	0
$848$	−0.0240434	−0.000825654 0
$849$	0	0
$850$	−1.31046	−0.0449484
$851$	−82.5886	−2.83110
$852$	0	0
$853$	11.9813	0.410233	0.205117	−	0.978738i	$-0.434243\pi$
$853$	11.9813	0.410233	0.205117	+	0.978738i	$0.434243\pi$
$854$	0	0
$855$	0	0
$856$	20.2267	0.691334
$857$	6.50030	0.222046	0.111023	−	0.993818i	$-0.464587\pi$
$857$	6.50030	0.222046	0.111023	+	0.993818i	$0.464587\pi$
$858$	0	0
$859$	53.5526	1.82719	0.913596	−	0.406623i	$-0.133294\pi$
$859$	53.5526	1.82719	0.913596	+	0.406623i	$0.133294\pi$
$860$	7.29086	0.248616
$861$	0	0
$862$	25.7828	0.878166
$863$	−3.69965	−0.125937	−0.0629687	−	0.998016i	$-0.520057\pi$
$863$	−3.69965	−0.125937	−0.0629687	+	0.998016i	$0.520057\pi$
$864$	0	0
$865$	−6.40230	−0.217685
$866$	−17.2847	−0.587357
$867$	0	0
$868$	0	0
$869$	3.97266	0.134763
$870$	0	0
$871$	2.00917	0.0680782
$872$	−1.14559	−0.0387946
$873$	0	0
$874$	−25.3773	−0.858401
$875$	0	0
$876$	0	0
$877$	−11.7888	−0.398079	−0.199040	−	0.979991i	$-0.563782\pi$
$877$	−11.7888	−0.398079	−0.199040	+	0.979991i	$0.563782\pi$
$878$	19.2849	0.650833
$879$	0	0
$880$	0.0932736	0.00314425
$881$	49.4858	1.66722	0.833609	−	0.552355i	$-0.186271\pi$
$881$	49.4858	1.66722	0.833609	+	0.552355i	$0.186271\pi$
$882$	0	0
$883$	−21.5357	−0.724734	−0.362367	−	0.932035i	$-0.618031\pi$
$883$	−21.5357	−0.724734	−0.362367	+	0.932035i	$0.618031\pi$
$884$	−1.93407	−0.0650497
$885$	0	0
$886$	16.4534	0.552762
$887$	11.8848	0.399051	0.199526	−	0.979893i	$-0.436060\pi$
$887$	11.8848	0.399051	0.199526	+	0.979893i	$0.436060\pi$
$888$	0	0
$889$	0	0
$890$	10.7656	0.360863
$891$	0	0
$892$	8.68954	0.290947
$893$	30.1857	1.01013
$894$	0	0
$895$	11.4953	0.384244
$896$	0	0
$897$	0	0
$898$	−5.87433	−0.196029
$899$	57.8866	1.93063
$900$	0	0
$901$	0.268571	0.00894739
$902$	−4.95811	−0.165087
$903$	0	0
$904$	40.7736	1.35611
$905$	23.2199	0.771855
$906$	0	0
$907$	−26.0215	−0.864029	−0.432014	−	0.901867i	$-0.642197\pi$
$907$	−26.0215	−0.864029	−0.432014	+	0.901867i	$0.642197\pi$
$908$	14.6509	0.486209
$909$	0	0
$910$	0	0
$911$	4.03272	0.133610	0.0668050	−	0.997766i	$-0.478719\pi$
$911$	4.03272	0.133610	0.0668050	+	0.997766i	$0.478719\pi$
$912$	0	0
$913$	24.8726	0.823162
$914$	−17.0853	−0.565132
$915$	0	0
$916$	−21.5315	−0.711420
$917$	0	0
$918$	0	0
$919$	27.4270	0.904732	0.452366	−	0.891832i	$-0.350580\pi$
$919$	27.4270	0.904732	0.452366	+	0.891832i	$0.350580\pi$
$920$	34.1908	1.12724
$921$	0	0
$922$	−0.849356	−0.0279720
$923$	1.86802	0.0614867
$924$	0	0
$925$	−29.4097	−0.966986
$926$	−0.391874	−0.0128778
$927$	0	0
$928$	−35.3432	−1.16020
$929$	7.67675	0.251866	0.125933	−	0.992039i	$-0.459808\pi$
$929$	7.67675	0.251866	0.125933	+	0.992039i	$0.459808\pi$
$930$	0	0
$931$	0	0
$932$	19.9385	0.653108
$933$	0	0
$934$	−30.0880	−0.984510
$935$	−1.04189	−0.0340734
$936$	0	0
$937$	2.02465	0.0661425	0.0330713	−	0.999453i	$-0.489471\pi$
$937$	2.02465	0.0661425	0.0330713	+	0.999453i	$0.489471\pi$
$938$	0	0
$939$	0	0
$940$	−15.4611	−0.504286
$941$	−6.13928	−0.200135	−0.100067	−	0.994981i	$-0.531906\pi$
$941$	−6.13928	−0.200135	−0.100067	+	0.994981i	$0.531906\pi$
$942$	0	0
$943$	−30.5107	−0.993566
$944$	0.435518	0.0141749
$945$	0	0
$946$	6.41147	0.208455
$947$	−5.56448	−0.180821	−0.0904107	−	0.995905i	$-0.528818\pi$
$947$	−5.56448	−0.180821	−0.0904107	+	0.995905i	$0.528818\pi$
$948$	0	0
$949$	−6.90640	−0.224191
$950$	−9.03684	−0.293194
$951$	0	0
$952$	0	0
$953$	8.72018	0.282474	0.141237	−	0.989976i	$-0.454892\pi$
$953$	8.72018	0.282474	0.141237	+	0.989976i	$0.454892\pi$
$954$	0	0
$955$	−17.3928	−0.562818
$956$	−18.5220	−0.599044
$957$	0	0
$958$	−19.1644	−0.619173
$959$	0	0
$960$	0	0
$961$	54.2746	1.75079
$962$	27.3631	0.882222
$963$	0	0
$964$	−19.1830	−0.617844
$965$	−0.859785	−0.0276775
$966$	0	0
$967$	−57.7698	−1.85775	−0.928876	−	0.370391i	$-0.879224\pi$
$967$	−57.7698	−1.85775	−0.928876	+	0.370391i	$0.879224\pi$
$968$	23.4620	0.754098
$969$	0	0
$970$	2.24990	0.0722401
$971$	−30.7192	−0.985828	−0.492914	−	0.870078i	$-0.664068\pi$
$971$	−30.7192	−0.985828	−0.492914	+	0.870078i	$0.664068\pi$
$972$	0	0
$973$	0	0
$974$	17.0537	0.546437
$975$	0	0
$976$	0.319955	0.0102415
$977$	−10.3000	−0.329527	−0.164764	−	0.986333i	$-0.552686\pi$
$977$	−10.3000	−0.329527	−0.164764	+	0.986333i	$0.552686\pi$
$978$	0	0
$979$	−15.0172	−0.479953
$980$	0	0
$981$	0	0
$982$	−23.0018	−0.734015
$983$	13.6963	0.436846	0.218423	−	0.975854i	$-0.429909\pi$
$983$	13.6963	0.436846	0.218423	+	0.975854i	$0.429909\pi$
$984$	0	0
$985$	−15.4206	−0.491343
$986$	−2.57935	−0.0821434
$987$	0	0
$988$	−13.3372	−0.424313
$989$	39.4543	1.25457
$990$	0	0
$991$	57.9813	1.84184	0.920919	−	0.389754i	$-0.127440\pi$
$991$	57.9813	1.84184	0.920919	+	0.389754i	$0.127440\pi$
$992$	−52.0651	−1.65307
$993$	0	0
$994$	0	0
$995$	4.90167	0.155394
$996$	0	0
$997$	−16.2175	−0.513614	−0.256807	−	0.966463i	$-0.582670\pi$
$997$	−16.2175	−0.513614	−0.256807	+	0.966463i	$0.582670\pi$
$998$	−12.5763	−0.398097
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.l.1.3		3
3.2	odd	2		3969.2.a.q.1.1		3
7.6	odd	2		567.2.a.c.1.3		3
9.2	odd	6		441.2.f.c.148.3		6
9.4	even	3		1323.2.f.d.883.1		6
9.5	odd	6		441.2.f.c.295.3		6
9.7	even	3		1323.2.f.d.442.1		6
21.20	even	2		567.2.a.h.1.1		3
28.27	even	2		9072.2.a.bs.1.2		3
63.2	odd	6		441.2.g.b.67.3		6
63.4	even	3		1323.2.g.e.667.1		6
63.5	even	6		441.2.h.d.214.1		6
63.11	odd	6		441.2.h.e.373.1		6
63.13	odd	6		189.2.f.b.127.1		6
63.16	even	3		1323.2.g.e.361.1		6
63.20	even	6		63.2.f.a.22.3	✓	6
63.23	odd	6		441.2.h.e.214.1		6
63.25	even	3		1323.2.h.b.226.3		6
63.31	odd	6		1323.2.g.d.667.1		6
63.32	odd	6		441.2.g.b.79.3		6
63.34	odd	6		189.2.f.b.64.1		6
63.38	even	6		441.2.h.d.373.1		6
63.40	odd	6		1323.2.h.c.802.3		6
63.41	even	6		63.2.f.a.43.3	yes	6
63.47	even	6		441.2.g.c.67.3		6
63.52	odd	6		1323.2.h.c.226.3		6
63.58	even	3		1323.2.h.b.802.3		6
63.59	even	6		441.2.g.c.79.3		6
63.61	odd	6		1323.2.g.d.361.1		6
84.83	odd	2		9072.2.a.ca.1.2		3
252.83	odd	6		1008.2.r.h.337.2		6
252.139	even	6		3024.2.r.k.2017.2		6
252.167	odd	6		1008.2.r.h.673.2		6
252.223	even	6		3024.2.r.k.1009.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.f.a.22.3	✓	6	63.20	even	6
63.2.f.a.43.3	yes	6	63.41	even	6
189.2.f.b.64.1		6	63.34	odd	6
189.2.f.b.127.1		6	63.13	odd	6
441.2.f.c.148.3		6	9.2	odd	6
441.2.f.c.295.3		6	9.5	odd	6
441.2.g.b.67.3		6	63.2	odd	6
441.2.g.b.79.3		6	63.32	odd	6
441.2.g.c.67.3		6	63.47	even	6
441.2.g.c.79.3		6	63.59	even	6
441.2.h.d.214.1		6	63.5	even	6
441.2.h.d.373.1		6	63.38	even	6
441.2.h.e.214.1		6	63.23	odd	6
441.2.h.e.373.1		6	63.11	odd	6
567.2.a.c.1.3		3	7.6	odd	2
567.2.a.h.1.1		3	21.20	even	2
1008.2.r.h.337.2		6	252.83	odd	6
1008.2.r.h.673.2		6	252.167	odd	6
1323.2.f.d.442.1		6	9.7	even	3
1323.2.f.d.883.1		6	9.4	even	3
1323.2.g.d.361.1		6	63.61	odd	6
1323.2.g.d.667.1		6	63.31	odd	6
1323.2.g.e.361.1		6	63.16	even	3
1323.2.g.e.667.1		6	63.4	even	3
1323.2.h.b.226.3		6	63.25	even	3
1323.2.h.b.802.3		6	63.58	even	3
1323.2.h.c.226.3		6	63.52	odd	6
1323.2.h.c.802.3		6	63.40	odd	6
3024.2.r.k.1009.2		6	252.223	even	6
3024.2.r.k.2017.2		6	252.139	even	6
3969.2.a.l.1.3		3	1.1	even	1	trivial
3969.2.a.q.1.1		3	3.2	odd	2
9072.2.a.bs.1.2		3	28.27	even	2
9072.2.a.ca.1.2		3	84.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

\(f(q)\)	\(=\)	\(q+0.879385 q^{2} -1.22668 q^{4} +1.34730 q^{5} -2.83750 q^{8} +O(q^{10})\)	\(q+0.879385 q^{2} -1.22668 q^{4} +1.34730 q^{5} -2.83750 q^{8} +1.18479 q^{10} -1.65270 q^{11} +3.36959 q^{13} -0.0418891 q^{16} +0.467911 q^{17} +3.22668 q^{19} -1.65270 q^{20} -1.45336 q^{22} -8.94356 q^{23} -3.18479 q^{25} +2.96316 q^{26} -6.26857 q^{29} -9.23442 q^{31} +5.63816 q^{32} +0.411474 q^{34} +9.23442 q^{37} +2.83750 q^{38} -3.82295 q^{40} +3.41147 q^{41} -4.41147 q^{43} +2.02734 q^{44} -7.86484 q^{46} +9.35504 q^{47} -2.80066 q^{50} -4.13341 q^{52} +0.573978 q^{53} -2.22668 q^{55} -5.51249 q^{58} -10.3969 q^{59} -7.63816 q^{61} -8.12061 q^{62} +5.04189 q^{64} +4.53983 q^{65} +0.596267 q^{67} -0.573978 q^{68} +0.554378 q^{71} -2.04963 q^{73} +8.12061 q^{74} -3.95811 q^{76} -2.40373 q^{79} -0.0564370 q^{80} +3.00000 q^{82} -15.0496 q^{83} +0.630415 q^{85} -3.87939 q^{86} +4.68954 q^{88} +9.08647 q^{89} +10.9709 q^{92} +8.22668 q^{94} +4.34730 q^{95} +1.89899 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{8}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - 6 * q^8	\( 3 q - 3 q^{2} + 3 q^{4} + 3 q^{5} - 6 q^{8} - 6 q^{11} + 3 q^{13} + 3 q^{16} + 6 q^{17} + 3 q^{19} - 6 q^{20} + 9 q^{22} - 12 q^{23} - 6 q^{25} - 3 q^{26} - 9 q^{29} + 3 q^{31} - 9 q^{34} - 3 q^{37} + 6 q^{38} + 9 q^{40} - 3 q^{43} - 15 q^{44} + 3 q^{47} + 6 q^{50} + 21 q^{52} - 6 q^{53} - 9 q^{58} - 3 q^{59} - 6 q^{61} - 30 q^{62} + 12 q^{64} - 15 q^{65} - 12 q^{67} + 6 q^{68} - 9 q^{71} + 21 q^{73} + 30 q^{74} - 15 q^{76} - 21 q^{79} - 15 q^{80} + 9 q^{82} - 18 q^{83} + 9 q^{85} - 6 q^{86} + 27 q^{88} + 12 q^{89} - 3 q^{92} + 18 q^{94} + 12 q^{95} + 3 q^{97}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^4 + 3 * q^5 - 6 * q^8 - 6 * q^11 + 3 * q^13 + 3 * q^16 + 6 * q^17 + 3 * q^19 - 6 * q^20 + 9 * q^22 - 12 * q^23 - 6 * q^25 - 3 * q^26 - 9 * q^29 + 3 * q^31 - 9 * q^34 - 3 * q^37 + 6 * q^38 + 9 * q^40 - 3 * q^43 - 15 * q^44 + 3 * q^47 + 6 * q^50 + 21 * q^52 - 6 * q^53 - 9 * q^58 - 3 * q^59 - 6 * q^61 - 30 * q^62 + 12 * q^64 - 15 * q^65 - 12 * q^67 + 6 * q^68 - 9 * q^71 + 21 * q^73 + 30 * q^74 - 15 * q^76 - 21 * q^79 - 15 * q^80 + 9 * q^82 - 18 * q^83 + 9 * q^85 - 6 * q^86 + 27 * q^88 + 12 * q^89 - 3 * q^92 + 18 * q^94 + 12 * q^95 + 3 * q^97

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