LMFDB - Embedded newform 891.2.a.p.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [891,2,Mod(1,891)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(891, 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("891.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 891 = 3^{4} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	891.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:	$7.11467082010$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.22545.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 5x + 4 $ x^4 - x^3 - 6x^2 + 5x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 99)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.45106$ of defining polynomial
Character	$\chi$	$=$	891.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.894434 q^{2} -1.19999 q^{4} -3.74893 q^{5} +1.45106 q^{7} -2.86218 q^{8} -3.35317 q^{10} -1.00000 q^{11} +5.75661 q^{13} +1.29787 q^{14} -0.160052 q^{16} +4.79655 q^{17} +0.702126 q^{19} +4.49867 q^{20} -0.894434 q^{22} +1.65105 q^{23} +9.05449 q^{25} +5.14891 q^{26} -1.74125 q^{28} +4.30555 q^{29} -3.30555 q^{31} +5.58120 q^{32} +4.29019 q^{34} -5.43991 q^{35} +9.73779 q^{37} +0.628005 q^{38} +10.7301 q^{40} -4.24760 q^{41} -4.10557 q^{43} +1.19999 q^{44} +1.47675 q^{46} -1.79655 q^{47} -4.89443 q^{49} +8.09864 q^{50} -6.90787 q^{52} +1.15318 q^{53} +3.74893 q^{55} -4.15318 q^{56} +3.85103 q^{58} +4.65105 q^{59} +2.54894 q^{61} -2.95660 q^{62} +5.31212 q^{64} -21.5811 q^{65} +8.94124 q^{67} -5.75580 q^{68} -4.86564 q^{70} -5.14204 q^{71} +10.5378 q^{73} +8.70981 q^{74} -0.842543 q^{76} -1.45106 q^{77} -1.08674 q^{79} +0.600024 q^{80} -3.79920 q^{82} +3.80342 q^{83} -17.9819 q^{85} -3.67216 q^{86} +2.86218 q^{88} +4.01536 q^{89} +8.35317 q^{91} -1.98123 q^{92} -1.60689 q^{94} -2.63222 q^{95} +3.29101 q^{97} -4.37775 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 11 q^{4} - 4 q^{5} + q^{7} + q^{10} - 4 q^{11} + 7 q^{13} - q^{14} + 17 q^{16} + 5 q^{17} + 9 q^{19} + 10 q^{20} + q^{22} - 14 q^{23} + 14 q^{25} - 22 q^{26} - q^{28} + 6 q^{29} - 2 q^{31}+ \cdots - 15 q^{98}+O(q^{100}) $ 4 * q - q^2 + 11 * q^4 - 4 * q^5 + q^7 + q^10 - 4 * q^11 + 7 * q^13 - q^14 + 17 * q^16 + 5 * q^17 + 9 * q^19 + 10 * q^20 + q^22 - 14 * q^23 + 14 * q^25 - 22 * q^26 - q^28 + 6 * q^29 - 2 * q^31 + 34 * q^32 + 16 * q^34 + 8 * q^35 + 3 * q^37 - 3 * q^38 + 12 * q^40 + 2 * q^41 - 21 * q^43 - 11 * q^44 + 2 * q^46 + 7 * q^47 - 15 * q^49 - 23 * q^50 - 10 * q^52 + 6 * q^53 + 4 * q^55 - 18 * q^56 - 21 * q^58 - 2 * q^59 + 15 * q^61 + 20 * q^62 + 16 * q^64 - 19 * q^65 + 14 * q^67 + 7 * q^68 - 38 * q^70 + 3 * q^71 + 22 * q^73 + 36 * q^74 + 42 * q^76 - q^77 + 11 * q^79 + 34 * q^80 - 17 * q^82 - 18 * q^83 + 13 * q^85 + 24 * q^86 + 6 * q^89 + 19 * q^91 - 67 * q^92 - 19 * q^94 + 30 * q^95 + 26 * q^97 - 15 * q^98

Coefficient data

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

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

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

$2$	0.894434	0.632460	0.316230	−	0.948683i	$-0.397583\pi$
$2$	0.894434	0.632460	0.316230	+	0.948683i	$0.397583\pi$
$3$	0	0
$3$	0	0
$4$	−1.19999	−0.599994
$5$	−3.74893	−1.67657	−0.838287	−	0.545230i	$-0.816442\pi$
$5$	−3.74893	−1.67657	−0.838287	+	0.545230i	$0.816442\pi$
$6$	0	0
$7$	1.45106	0.548448	0.274224	−	0.961666i	$-0.411579\pi$
$7$	1.45106	0.548448	0.274224	+	0.961666i	$0.411579\pi$
$8$	−2.86218	−1.01193
$9$	0	0
$10$	−3.35317	−1.06037
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.75661	1.59660	0.798298	−	0.602262i	$-0.205734\pi$
$13$	5.75661	1.59660	0.798298	+	0.602262i	$0.205734\pi$
$14$	1.29787	0.346872
$15$	0	0
$16$	−0.160052	−0.0400130
$17$	4.79655	1.16333	0.581667	−	0.813427i	$-0.302401\pi$
$17$	4.79655	1.16333	0.581667	+	0.813427i	$0.302401\pi$
$18$	0	0
$19$	0.702126	0.161079	0.0805393	−	0.996751i	$-0.474336\pi$
$19$	0.702126	0.161079	0.0805393	+	0.996751i	$0.474336\pi$
$20$	4.49867	1.00593
$21$	0	0
$22$	−0.894434	−0.190694
$23$	1.65105	0.344267	0.172133	−	0.985074i	$-0.444934\pi$
$23$	1.65105	0.344267	0.172133	+	0.985074i	$0.444934\pi$
$24$	0	0
$25$	9.05449	1.81090
$26$	5.14891	1.00978
$27$	0	0
$28$	−1.74125	−0.329066
$29$	4.30555	0.799521	0.399761	−	0.916620i	$-0.369093\pi$
$29$	4.30555	0.799521	0.399761	+	0.916620i	$0.369093\pi$
$30$	0	0
$31$	−3.30555	−0.593695	−0.296848	−	0.954925i	$-0.595935\pi$
$31$	−3.30555	−0.593695	−0.296848	+	0.954925i	$0.595935\pi$
$32$	5.58120	0.986626
$33$	0	0
$34$	4.29019	0.735762
$35$	−5.43991	−0.919513
$36$	0	0
$37$	9.73779	1.60088	0.800441	−	0.599411i	$-0.204599\pi$
$37$	9.73779	1.60088	0.800441	+	0.599411i	$0.204599\pi$
$38$	0.628005	0.101876
$39$	0	0
$40$	10.7301	1.69658
$41$	−4.24760	−0.663364	−0.331682	−	0.943391i	$-0.607616\pi$
$41$	−4.24760	−0.663364	−0.331682	+	0.943391i	$0.607616\pi$
$42$	0	0
$43$	−4.10557	−0.626093	−0.313046	−	0.949738i	$-0.601350\pi$
$43$	−4.10557	−0.626093	−0.313046	+	0.949738i	$0.601350\pi$
$44$	1.19999	0.180905
$45$	0	0
$46$	1.47675	0.217735
$47$	−1.79655	−0.262053	−0.131027	−	0.991379i	$-0.541827\pi$
$47$	−1.79655	−0.262053	−0.131027	+	0.991379i	$0.541827\pi$
$48$	0	0
$49$	−4.89443	−0.699205
$50$	8.09864	1.14532
$51$	0	0
$52$	−6.90787	−0.957949
$53$	1.15318	0.158402	0.0792009	−	0.996859i	$-0.474763\pi$
$53$	1.15318	0.158402	0.0792009	+	0.996859i	$0.474763\pi$
$54$	0	0
$55$	3.74893	0.505506
$56$	−4.15318	−0.554992
$57$	0	0
$58$	3.85103	0.505665
$59$	4.65105	0.605515	0.302757	−	0.953068i	$-0.402093\pi$
$59$	4.65105	0.605515	0.302757	+	0.953068i	$0.402093\pi$
$60$	0	0
$61$	2.54894	0.326359	0.163179	−	0.986596i	$-0.447825\pi$
$61$	2.54894	0.326359	0.163179	+	0.986596i	$0.447825\pi$
$62$	−2.95660	−0.375489
$63$	0	0
$64$	5.31212	0.664015
$65$	−21.5811	−2.67681
$66$	0	0
$67$	8.94124	1.09235	0.546173	−	0.837672i	$-0.316084\pi$
$67$	8.94124	1.09235	0.546173	+	0.837672i	$0.316084\pi$
$68$	−5.75580	−0.697993
$69$	0	0
$70$	−4.86564	−0.581555
$71$	−5.14204	−0.610248	−0.305124	−	0.952313i	$-0.598698\pi$
$71$	−5.14204	−0.610248	−0.305124	+	0.952313i	$0.598698\pi$
$72$	0	0
$73$	10.5378	1.23336	0.616678	−	0.787215i	$-0.288478\pi$
$73$	10.5378	1.23336	0.616678	+	0.787215i	$0.288478\pi$
$74$	8.70981	1.01249
$75$	0	0
$76$	−0.842543	−0.0966463
$77$	−1.45106	−0.165363
$78$	0	0
$79$	−1.08674	−0.122268	−0.0611340	−	0.998130i	$-0.519472\pi$
$79$	−1.08674	−0.122268	−0.0611340	+	0.998130i	$0.519472\pi$
$80$	0.600024	0.0670847
$81$	0	0
$82$	−3.79920	−0.419552
$83$	3.80342	0.417479	0.208740	−	0.977971i	$-0.433064\pi$
$83$	3.80342	0.417479	0.208740	+	0.977971i	$0.433064\pi$
$84$	0	0
$85$	−17.9819	−1.95041
$86$	−3.67216	−0.395979
$87$	0	0
$88$	2.86218	0.305109
$89$	4.01536	0.425627	0.212814	−	0.977093i	$-0.431737\pi$
$89$	4.01536	0.425627	0.212814	+	0.977093i	$0.431737\pi$
$90$	0	0
$91$	8.35317	0.875650
$92$	−1.98123	−0.206558
$93$	0	0
$94$	−1.60689	−0.165738
$95$	−2.63222	−0.270060
$96$	0	0
$97$	3.29101	0.334151	0.167075	−	0.985944i	$-0.446568\pi$
$97$	3.29101	0.334151	0.167075	+	0.985944i	$0.446568\pi$
$98$	−4.37775	−0.442219
$99$	0	0
$100$	−10.8653	−1.08653
$101$	−14.1931	−1.41226	−0.706131	−	0.708081i	$-0.749561\pi$
$101$	−14.1931	−1.41226	−0.706131	+	0.708081i	$0.749561\pi$
$102$	0	0
$103$	19.1777	1.88963	0.944817	−	0.327597i	$-0.106239\pi$
$103$	19.1777	1.88963	0.944817	+	0.327597i	$0.106239\pi$
$104$	−16.4764	−1.61565
$105$	0	0
$106$	1.03145	0.100183
$107$	10.0034	0.967066	0.483533	−	0.875326i	$-0.339353\pi$
$107$	10.0034	0.967066	0.483533	+	0.875326i	$0.339353\pi$
$108$	0	0
$109$	−7.40766	−0.709525	−0.354762	−	0.934956i	$-0.615438\pi$
$109$	−7.40766	−0.709525	−0.354762	+	0.934956i	$0.615438\pi$
$110$	3.35317	0.319712
$111$	0	0
$112$	−0.232244	−0.0219450
$113$	−1.64337	−0.154595	−0.0772974	−	0.997008i	$-0.524629\pi$
$113$	−1.64337	−0.154595	−0.0772974	+	0.997008i	$0.524629\pi$
$114$	0	0
$115$	−6.18965	−0.577188
$116$	−5.16661	−0.479708
$117$	0	0
$118$	4.16005	0.382964
$119$	6.96006	0.638028
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.27986	0.206409
$123$	0	0
$124$	3.96663	0.356214
$125$	−15.2000	−1.35953
$126$	0	0
$127$	−22.3309	−1.98155	−0.990773	−	0.135534i	$-0.956725\pi$
$127$	−22.3309	−1.98155	−0.990773	+	0.135534i	$0.956725\pi$
$128$	−6.41106	−0.566663
$129$	0	0
$130$	−19.3029	−1.69298
$131$	9.78887	0.855257	0.427629	−	0.903954i	$-0.359349\pi$
$131$	9.78887	0.855257	0.427629	+	0.903954i	$0.359349\pi$
$132$	0	0
$133$	1.01882	0.0883433
$134$	7.99735	0.690866
$135$	0	0
$136$	−13.7286	−1.17722
$137$	2.37621	0.203013	0.101507	−	0.994835i	$-0.467634\pi$
$137$	2.37621	0.203013	0.101507	+	0.994835i	$0.467634\pi$
$138$	0	0
$139$	−11.4867	−0.974291	−0.487145	−	0.873321i	$-0.661962\pi$
$139$	−11.4867	−0.974291	−0.487145	+	0.873321i	$0.661962\pi$
$140$	6.52783	0.551702
$141$	0	0
$142$	−4.59921	−0.385957
$143$	−5.75661	−0.481392
$144$	0	0
$145$	−16.1412	−1.34046
$146$	9.42536	0.780049
$147$	0	0
$148$	−11.6852	−0.960520
$149$	0.701315	0.0574540	0.0287270	−	0.999587i	$-0.490855\pi$
$149$	0.701315	0.0574540	0.0287270	+	0.999587i	$0.490855\pi$
$150$	0	0
$151$	−2.50554	−0.203898	−0.101949	−	0.994790i	$-0.532508\pi$
$151$	−2.50554	−0.203898	−0.101949	+	0.994790i	$0.532508\pi$
$152$	−2.00961	−0.163001
$153$	0	0
$154$	−1.29787	−0.104586
$155$	12.3923	0.995373
$156$	0	0
$157$	4.43570	0.354007	0.177004	−	0.984210i	$-0.443360\pi$
$157$	4.43570	0.354007	0.177004	+	0.984210i	$0.443360\pi$
$158$	−0.972019	−0.0773297
$159$	0	0
$160$	−20.9235	−1.65415
$161$	2.39576	0.188812
$162$	0	0
$163$	17.6986	1.38626	0.693131	−	0.720812i	$-0.256231\pi$
$163$	17.6986	1.38626	0.693131	+	0.720812i	$0.256231\pi$
$164$	5.09708	0.398015
$165$	0	0
$166$	3.40190	0.264039
$167$	8.90130	0.688804	0.344402	−	0.938822i	$-0.388082\pi$
$167$	8.90130	0.688804	0.344402	+	0.938822i	$0.388082\pi$
$168$	0	0
$169$	20.1386	1.54912
$170$	−16.0836	−1.23356
$171$	0	0
$172$	4.92663	0.375652
$173$	24.7309	1.88025	0.940126	−	0.340827i	$-0.110707\pi$
$173$	24.7309	1.88025	0.940126	+	0.340827i	$0.110707\pi$
$174$	0	0
$175$	13.1386	0.993183
$176$	0.160052	0.0120644
$177$	0	0
$178$	3.59147	0.269192
$179$	11.8587	0.886362	0.443181	−	0.896432i	$-0.353850\pi$
$179$	11.8587	0.886362	0.443181	+	0.896432i	$0.353850\pi$
$180$	0	0
$181$	−4.94546	−0.367593	−0.183796	−	0.982964i	$-0.558839\pi$
$181$	−4.94546	−0.367593	−0.183796	+	0.982964i	$0.558839\pi$
$182$	7.47136	0.553814
$183$	0	0
$184$	−4.72558	−0.348375
$185$	−36.5063	−2.68400
$186$	0	0
$187$	−4.79655	−0.350758
$188$	2.15584	0.157230
$189$	0	0
$190$	−2.35435	−0.170802
$191$	−0.323568	−0.0234126	−0.0117063	−	0.999931i	$-0.503726\pi$
$191$	−0.323568	−0.0234126	−0.0117063	+	0.999931i	$0.503726\pi$
$192$	0	0
$193$	14.4153	1.03764	0.518819	−	0.854884i	$-0.326372\pi$
$193$	14.4153	1.03764	0.518819	+	0.854884i	$0.326372\pi$
$194$	2.94359	0.211337
$195$	0	0
$196$	5.87326	0.419519
$197$	18.3267	1.30572	0.652860	−	0.757478i	$-0.273569\pi$
$197$	18.3267	1.30572	0.652860	+	0.757478i	$0.273569\pi$
$198$	0	0
$199$	−1.51211	−0.107190	−0.0535951	−	0.998563i	$-0.517068\pi$
$199$	−1.51211	−0.107190	−0.0535951	+	0.998563i	$0.517068\pi$
$200$	−25.9155	−1.83251
$201$	0	0
$202$	−12.6948	−0.893200
$203$	6.24760	0.438496
$204$	0	0
$205$	15.9240	1.11218
$206$	17.1532	1.19512
$207$	0	0
$208$	−0.921357	−0.0638846
$209$	−0.702126	−0.0485670
$210$	0	0
$211$	−11.0222	−0.758802	−0.379401	−	0.925232i	$-0.623870\pi$
$211$	−11.0222	−0.758802	−0.379401	+	0.925232i	$0.623870\pi$
$212$	−1.38381	−0.0950402
$213$	0	0
$214$	8.94738	0.611631
$215$	15.3915	1.04969
$216$	0	0
$217$	−4.79655	−0.325611
$218$	−6.62566	−0.448746
$219$	0	0
$220$	−4.49867	−0.303300
$221$	27.6119	1.85737
$222$	0	0
$223$	−23.3132	−1.56117	−0.780585	−	0.625050i	$-0.785079\pi$
$223$	−23.3132	−1.56117	−0.780585	+	0.625050i	$0.785079\pi$
$224$	8.09864	0.541113
$225$	0	0
$226$	−1.46988	−0.0977750
$227$	−5.97889	−0.396833	−0.198416	−	0.980118i	$-0.563580\pi$
$227$	−5.97889	−0.396833	−0.198416	+	0.980118i	$0.563580\pi$
$228$	0	0
$229$	9.42455	0.622792	0.311396	−	0.950280i	$-0.399203\pi$
$229$	9.42455	0.622792	0.311396	+	0.950280i	$0.399203\pi$
$230$	−5.53624	−0.365049
$231$	0	0
$232$	−12.3233	−0.809062
$233$	−27.1685	−1.77987	−0.889933	−	0.456091i	$-0.849249\pi$
$233$	−27.1685	−1.77987	−0.889933	+	0.456091i	$0.849249\pi$
$234$	0	0
$235$	6.73513	0.439352
$236$	−5.58120	−0.363305
$237$	0	0
$238$	6.22532	0.403527
$239$	−0.242578	−0.0156911	−0.00784553	−	0.999969i	$-0.502497\pi$
$239$	−0.242578	−0.0156911	−0.00784553	+	0.999969i	$0.502497\pi$
$240$	0	0
$241$	−12.7106	−0.818759	−0.409379	−	0.912364i	$-0.634255\pi$
$241$	−12.7106	−0.818759	−0.409379	+	0.912364i	$0.634255\pi$
$242$	0.894434	0.0574964
$243$	0	0
$244$	−3.05870	−0.195813
$245$	18.3489	1.17227
$246$	0	0
$247$	4.04186	0.257178
$248$	9.46108	0.600779
$249$	0	0
$250$	−13.5954	−0.859848
$251$	−23.6134	−1.49046	−0.745232	−	0.666805i	$-0.767661\pi$
$251$	−23.6134	−1.49046	−0.745232	+	0.666805i	$0.767661\pi$
$252$	0	0
$253$	−1.65105	−0.103800
$254$	−19.9735	−1.25325
$255$	0	0
$256$	−16.3585	−1.02241
$257$	−4.11671	−0.256793	−0.128397	−	0.991723i	$-0.540983\pi$
$257$	−4.11671	−0.256793	−0.128397	+	0.991723i	$0.540983\pi$
$258$	0	0
$259$	14.1301	0.878001
$260$	25.8971	1.60607
$261$	0	0
$262$	8.75549	0.540916
$263$	11.5923	0.714811	0.357405	−	0.933949i	$-0.383661\pi$
$263$	11.5923	0.714811	0.357405	+	0.933949i	$0.383661\pi$
$264$	0	0
$265$	−4.32320	−0.265572
$266$	0.911271	0.0558736
$267$	0	0
$268$	−10.7294	−0.655401
$269$	6.82534	0.416148	0.208074	−	0.978113i	$-0.433280\pi$
$269$	6.82534	0.416148	0.208074	+	0.978113i	$0.433280\pi$
$270$	0	0
$271$	−7.57539	−0.460172	−0.230086	−	0.973170i	$-0.573901\pi$
$271$	−7.57539	−0.460172	−0.230086	+	0.973170i	$0.573901\pi$
$272$	−0.767697	−0.0465484
$273$	0	0
$274$	2.12536	0.128398
$275$	−9.05449	−0.546006
$276$	0	0
$277$	−18.8034	−1.12978	−0.564892	−	0.825165i	$-0.691082\pi$
$277$	−18.8034	−1.12978	−0.564892	+	0.825165i	$0.691082\pi$
$278$	−10.2741	−0.616200
$279$	0	0
$280$	15.5700	0.930485
$281$	−24.8832	−1.48441	−0.742205	−	0.670173i	$-0.766220\pi$
$281$	−24.8832	−1.48441	−0.742205	+	0.670173i	$0.766220\pi$
$282$	0	0
$283$	−18.1412	−1.07838	−0.539192	−	0.842183i	$-0.681270\pi$
$283$	−18.1412	−1.07838	−0.539192	+	0.842183i	$0.681270\pi$
$284$	6.17039	0.366145
$285$	0	0
$286$	−5.14891	−0.304461
$287$	−6.16352	−0.363821
$288$	0	0
$289$	6.00687	0.353345
$290$	−14.4373	−0.847785
$291$	0	0
$292$	−12.6452	−0.740006
$293$	16.0775	0.939259	0.469630	−	0.882864i	$-0.344387\pi$
$293$	16.0775	0.939259	0.469630	+	0.882864i	$0.344387\pi$
$294$	0	0
$295$	−17.4364	−1.01519
$296$	−27.8713	−1.61998
$297$	0	0
$298$	0.627280	0.0363373
$299$	9.50443	0.549655
$300$	0	0
$301$	−5.95741	−0.343379
$302$	−2.24104	−0.128957
$303$	0	0
$304$	−0.112377	−0.00644524
$305$	−9.55581	−0.547164
$306$	0	0
$307$	−20.1343	−1.14913	−0.574563	−	0.818461i	$-0.694828\pi$
$307$	−20.1343	−1.14913	−0.574563	+	0.818461i	$0.694828\pi$
$308$	1.74125	0.0992170
$309$	0	0
$310$	11.0841	0.629534
$311$	13.0245	0.738553	0.369276	−	0.929320i	$-0.379606\pi$
$311$	13.0245	0.738553	0.369276	+	0.929320i	$0.379606\pi$
$312$	0	0
$313$	10.2518	0.579467	0.289734	−	0.957107i	$-0.406433\pi$
$313$	10.2518	0.579467	0.289734	+	0.957107i	$0.406433\pi$
$314$	3.96744	0.223895
$315$	0	0
$316$	1.30408	0.0733601
$317$	11.0176	0.618813	0.309406	−	0.950930i	$-0.399870\pi$
$317$	11.0176	0.618813	0.309406	+	0.950930i	$0.399870\pi$
$318$	0	0
$319$	−4.30555	−0.241065
$320$	−19.9148	−1.11327
$321$	0	0
$322$	2.14285	0.119416
$323$	3.36778	0.187388
$324$	0	0
$325$	52.1232	2.89127
$326$	15.8302	0.876755
$327$	0	0
$328$	12.1574	0.671280
$329$	−2.60689	−0.143723
$330$	0	0
$331$	−18.1443	−0.997302	−0.498651	−	0.866803i	$-0.666171\pi$
$331$	−18.1443	−0.997302	−0.498651	+	0.866803i	$0.666171\pi$
$332$	−4.56406	−0.250485
$333$	0	0
$334$	7.96163	0.435641
$335$	−33.5201	−1.83140
$336$	0	0
$337$	−1.67909	−0.0914656	−0.0457328	−	0.998954i	$-0.514562\pi$
$337$	−1.67909	−0.0914656	−0.0457328	+	0.998954i	$0.514562\pi$
$338$	18.0126	0.979757
$339$	0	0
$340$	21.5781	1.17024
$341$	3.30555	0.179006
$342$	0	0
$343$	−17.2595	−0.931925
$344$	11.7509	0.633564
$345$	0	0
$346$	22.1201	1.18918
$347$	18.7812	1.00823	0.504113	−	0.863637i	$-0.331819\pi$
$347$	18.7812	1.00823	0.504113	+	0.863637i	$0.331819\pi$
$348$	0	0
$349$	32.5089	1.74016	0.870082	−	0.492907i	$-0.164066\pi$
$349$	32.5089	1.74016	0.870082	+	0.492907i	$0.164066\pi$
$350$	11.7516	0.628149
$351$	0	0
$352$	−5.58120	−0.297479
$353$	−6.70975	−0.357124	−0.178562	−	0.983929i	$-0.557144\pi$
$353$	−6.70975	−0.357124	−0.178562	+	0.983929i	$0.557144\pi$
$354$	0	0
$355$	19.2771	1.02312
$356$	−4.81838	−0.255374
$357$	0	0
$358$	10.6068	0.560589
$359$	−14.7797	−0.780040	−0.390020	−	0.920806i	$-0.627532\pi$
$359$	−14.7797	−0.780040	−0.390020	+	0.920806i	$0.627532\pi$
$360$	0	0
$361$	−18.5070	−0.974054
$362$	−4.42338	−0.232488
$363$	0	0
$364$	−10.0237	−0.525385
$365$	−39.5055	−2.06781
$366$	0	0
$367$	26.5869	1.38782	0.693912	−	0.720060i	$-0.255886\pi$
$367$	26.5869	1.38782	0.693912	+	0.720060i	$0.255886\pi$
$368$	−0.264253	−0.0137751
$369$	0	0
$370$	−32.6525	−1.69752
$371$	1.67333	0.0868752
$372$	0	0
$373$	23.5563	1.21970	0.609848	−	0.792518i	$-0.291230\pi$
$373$	23.5563	1.21970	0.609848	+	0.792518i	$0.291230\pi$
$374$	−4.29019	−0.221841
$375$	0	0
$376$	5.14204	0.265180
$377$	24.7854	1.27651
$378$	0	0
$379$	31.6536	1.62594	0.812969	−	0.582307i	$-0.197850\pi$
$379$	31.6536	1.62594	0.812969	+	0.582307i	$0.197850\pi$
$380$	3.15863	0.162035
$381$	0	0
$382$	−0.289410	−0.0148075
$383$	35.8440	1.83155	0.915773	−	0.401697i	$-0.131580\pi$
$383$	35.8440	1.83155	0.915773	+	0.401697i	$0.131580\pi$
$384$	0	0
$385$	5.43991	0.277244
$386$	12.8936	0.656265
$387$	0	0
$388$	−3.94917	−0.200489
$389$	−3.92169	−0.198838	−0.0994188	−	0.995046i	$-0.531698\pi$
$389$	−3.92169	−0.198838	−0.0994188	+	0.995046i	$0.531698\pi$
$390$	0	0
$391$	7.91932	0.400497
$392$	14.0087	0.707548
$393$	0	0
$394$	16.3920	0.825817
$395$	4.07412	0.204991
$396$	0	0
$397$	−3.90292	−0.195882	−0.0979411	−	0.995192i	$-0.531226\pi$
$397$	−3.90292	−0.195882	−0.0979411	+	0.995192i	$0.531226\pi$
$398$	−1.35248	−0.0677936
$399$	0	0
$400$	−1.44919	−0.0724594
$401$	−31.6755	−1.58180	−0.790900	−	0.611946i	$-0.790387\pi$
$401$	−31.6755	−1.58180	−0.790900	+	0.611946i	$0.790387\pi$
$402$	0	0
$403$	−19.0288	−0.947892
$404$	17.0315	0.847349
$405$	0	0
$406$	5.58807	0.277331
$407$	−9.73779	−0.482684
$408$	0	0
$409$	−12.9163	−0.638670	−0.319335	−	0.947642i	$-0.603459\pi$
$409$	−12.9163	−0.638670	−0.319335	+	0.947642i	$0.603459\pi$
$410$	14.2429	0.703409
$411$	0	0
$412$	−23.0130	−1.13377
$413$	6.74893	0.332093
$414$	0	0
$415$	−14.2587	−0.699934
$416$	32.1288	1.57524
$417$	0	0
$418$	−0.628005	−0.0307167
$419$	−27.3608	−1.33666	−0.668331	−	0.743864i	$-0.732991\pi$
$419$	−27.3608	−1.33666	−0.668331	+	0.743864i	$0.732991\pi$
$420$	0	0
$421$	−19.5777	−0.954158	−0.477079	−	0.878861i	$-0.658304\pi$
$421$	−19.5777	−0.954158	−0.477079	+	0.878861i	$0.658304\pi$
$422$	−9.85865	−0.479912
$423$	0	0
$424$	−3.30061	−0.160292
$425$	43.4303	2.10668
$426$	0	0
$427$	3.69866	0.178991
$428$	−12.0040	−0.580234
$429$	0	0
$430$	13.7667	0.663888
$431$	−9.80495	−0.472288	−0.236144	−	0.971718i	$-0.575884\pi$
$431$	−9.80495	−0.472288	−0.236144	+	0.971718i	$0.575884\pi$
$432$	0	0
$433$	−1.98391	−0.0953409	−0.0476704	−	0.998863i	$-0.515180\pi$
$433$	−1.98391	−0.0953409	−0.0476704	+	0.998863i	$0.515180\pi$
$434$	−4.29019	−0.205936
$435$	0	0
$436$	8.88910	0.425711
$437$	1.15924	0.0554540
$438$	0	0
$439$	35.0617	1.67341	0.836703	−	0.547657i	$-0.184480\pi$
$439$	35.0617	1.67341	0.836703	+	0.547657i	$0.184480\pi$
$440$	−10.7301	−0.511538
$441$	0	0
$442$	24.6970	1.17472
$443$	−23.7121	−1.12660	−0.563298	−	0.826254i	$-0.690468\pi$
$443$	−23.7121	−1.12660	−0.563298	+	0.826254i	$0.690468\pi$
$444$	0	0
$445$	−15.0533	−0.713595
$446$	−20.8521	−0.987378
$447$	0	0
$448$	7.70818	0.364177
$449$	11.6575	0.550154	0.275077	−	0.961422i	$-0.411297\pi$
$449$	11.6575	0.550154	0.275077	+	0.961422i	$0.411297\pi$
$450$	0	0
$451$	4.24760	0.200012
$452$	1.97202	0.0927560
$453$	0	0
$454$	−5.34772	−0.250981
$455$	−31.3155	−1.46809
$456$	0	0
$457$	13.2003	0.617484	0.308742	−	0.951146i	$-0.400092\pi$
$457$	13.2003	0.617484	0.308742	+	0.951146i	$0.400092\pi$
$458$	8.42964	0.393891
$459$	0	0
$460$	7.42751	0.346310
$461$	30.2913	1.41080	0.705402	−	0.708807i	$-0.250766\pi$
$461$	30.2913	1.41080	0.705402	+	0.708807i	$0.250766\pi$
$462$	0	0
$463$	−27.2119	−1.26464	−0.632322	−	0.774706i	$-0.717898\pi$
$463$	−27.2119	−1.26464	−0.632322	+	0.774706i	$0.717898\pi$
$464$	−0.689112	−0.0319912
$465$	0	0
$466$	−24.3004	−1.12569
$467$	−31.1873	−1.44318	−0.721588	−	0.692322i	$-0.756588\pi$
$467$	−31.1873	−1.44318	−0.721588	+	0.692322i	$0.756588\pi$
$468$	0	0
$469$	12.9742	0.599095
$470$	6.02413	0.277872
$471$	0	0
$472$	−13.3121	−0.612740
$473$	4.10557	0.188774
$474$	0	0
$475$	6.35739	0.291697
$476$	−8.35199	−0.382813
$477$	0	0
$478$	−0.216970	−0.00992397
$479$	38.2771	1.74893	0.874464	−	0.485091i	$-0.161213\pi$
$479$	38.2771	1.74893	0.874464	+	0.485091i	$0.161213\pi$
$480$	0	0
$481$	56.0567	2.55596
$482$	−11.3688	−0.517832
$483$	0	0
$484$	−1.19999	−0.0545449
$485$	−12.3378	−0.560228
$486$	0	0
$487$	−15.8602	−0.718697	−0.359348	−	0.933204i	$-0.617001\pi$
$487$	−15.8602	−0.718697	−0.359348	+	0.933204i	$0.617001\pi$
$488$	−7.29553	−0.330253
$489$	0	0
$490$	16.4119	0.741413
$491$	7.00229	0.316009	0.158004	−	0.987438i	$-0.449494\pi$
$491$	7.00229	0.316009	0.158004	+	0.987438i	$0.449494\pi$
$492$	0	0
$493$	20.6518	0.930110
$494$	3.61518	0.162655
$495$	0	0
$496$	0.529060	0.0237555
$497$	−7.46139	−0.334689
$498$	0	0
$499$	24.6705	1.10440	0.552202	−	0.833710i	$-0.313788\pi$
$499$	24.6705	1.10440	0.552202	+	0.833710i	$0.313788\pi$
$500$	18.2398	0.815709
$501$	0	0
$502$	−21.1206	−0.942659
$503$	−6.05183	−0.269838	−0.134919	−	0.990857i	$-0.543077\pi$
$503$	−6.05183	−0.269838	−0.134919	+	0.990857i	$0.543077\pi$
$504$	0	0
$505$	53.2088	2.36776
$506$	−1.47675	−0.0656496
$507$	0	0
$508$	26.7968	1.18892
$509$	20.0560	0.888965	0.444482	−	0.895788i	$-0.353388\pi$
$509$	20.0560	0.888965	0.444482	+	0.895788i	$0.353388\pi$
$510$	0	0
$511$	15.2909	0.676432
$512$	−1.80948	−0.0799683
$513$	0	0
$514$	−3.68212	−0.162412
$515$	−71.8959	−3.16811
$516$	0	0
$517$	1.79655	0.0790121
$518$	12.6384	0.555300
$519$	0	0
$520$	61.7691	2.70875
$521$	−18.8721	−0.826804	−0.413402	−	0.910549i	$-0.635660\pi$
$521$	−18.8721	−0.826804	−0.413402	+	0.910549i	$0.635660\pi$
$522$	0	0
$523$	−16.0568	−0.702114	−0.351057	−	0.936354i	$-0.614178\pi$
$523$	−16.0568	−0.702114	−0.351057	+	0.936354i	$0.614178\pi$
$524$	−11.7465	−0.513149
$525$	0	0
$526$	10.3685	0.452089
$527$	−15.8552	−0.690666
$528$	0	0
$529$	−20.2740	−0.881480
$530$	−3.86682	−0.167964
$531$	0	0
$532$	−1.22258	−0.0530054
$533$	−24.4518	−1.05913
$534$	0	0
$535$	−37.5021	−1.62136
$536$	−25.5914	−1.10538
$537$	0	0
$538$	6.10481	0.263197
$539$	4.89443	0.210818
$540$	0	0
$541$	18.4357	0.792614	0.396307	−	0.918118i	$-0.370292\pi$
$541$	18.4357	0.792614	0.396307	+	0.918118i	$0.370292\pi$
$542$	−6.77568	−0.291041
$543$	0	0
$544$	26.7705	1.14778
$545$	27.7708	1.18957
$546$	0	0
$547$	−28.9669	−1.23853	−0.619267	−	0.785180i	$-0.712570\pi$
$547$	−28.9669	−1.23853	−0.619267	+	0.785180i	$0.712570\pi$
$548$	−2.85143	−0.121807
$549$	0	0
$550$	−8.09864	−0.345327
$551$	3.02304	0.128786
$552$	0	0
$553$	−1.57692	−0.0670576
$554$	−16.8184	−0.714544
$555$	0	0
$556$	13.7839	0.584569
$557$	11.7584	0.498219	0.249110	−	0.968475i	$-0.419862\pi$
$557$	11.7584	0.498219	0.249110	+	0.968475i	$0.419862\pi$
$558$	0	0
$559$	−23.6341	−0.999618
$560$	0.870668	0.0367925
$561$	0	0
$562$	−22.2564	−0.938830
$563$	0.565114	0.0238167	0.0119084	−	0.999929i	$-0.496209\pi$
$563$	0.565114	0.0238167	0.0119084	+	0.999929i	$0.496209\pi$
$564$	0	0
$565$	6.16086	0.259189
$566$	−16.2261	−0.682035
$567$	0	0
$568$	14.7174	0.617530
$569$	−6.23342	−0.261319	−0.130659	−	0.991427i	$-0.541709\pi$
$569$	−6.23342	−0.261319	−0.130659	+	0.991427i	$0.541709\pi$
$570$	0	0
$571$	−28.3934	−1.18823	−0.594114	−	0.804381i	$-0.702497\pi$
$571$	−28.3934	−1.18823	−0.594114	+	0.804381i	$0.702497\pi$
$572$	6.90787	0.288832
$573$	0	0
$574$	−5.51286	−0.230102
$575$	14.9494	0.623432
$576$	0	0
$577$	−25.5623	−1.06417	−0.532087	−	0.846690i	$-0.678592\pi$
$577$	−25.5623	−1.06417	−0.532087	+	0.846690i	$0.678592\pi$
$578$	5.37275	0.223477
$579$	0	0
$580$	19.3693	0.804266
$581$	5.51897	0.228966
$582$	0	0
$583$	−1.15318	−0.0477599
$584$	−30.1611	−1.24807
$585$	0	0
$586$	14.3803	0.594044
$587$	35.1865	1.45230	0.726151	−	0.687535i	$-0.241307\pi$
$587$	35.1865	1.45230	0.726151	+	0.687535i	$0.241307\pi$
$588$	0	0
$589$	−2.32091	−0.0956316
$590$	−15.5957	−0.642067
$591$	0	0
$592$	−1.55855	−0.0640561
$593$	40.8141	1.67604	0.838018	−	0.545643i	$-0.183714\pi$
$593$	40.8141	1.67604	0.838018	+	0.545643i	$0.183714\pi$
$594$	0	0
$595$	−26.0928	−1.06970
$596$	−0.841570	−0.0344720
$597$	0	0
$598$	8.50108	0.347635
$599$	−46.4126	−1.89637	−0.948184	−	0.317721i	$-0.897082\pi$
$599$	−46.4126	−1.89637	−0.948184	+	0.317721i	$0.897082\pi$
$600$	0	0
$601$	15.2945	0.623874	0.311937	−	0.950103i	$-0.399022\pi$
$601$	15.2945	0.623874	0.311937	+	0.950103i	$0.399022\pi$
$602$	−5.32851	−0.217174
$603$	0	0
$604$	3.00662	0.122338
$605$	−3.74893	−0.152416
$606$	0	0
$607$	−34.4948	−1.40010	−0.700051	−	0.714093i	$-0.746840\pi$
$607$	−34.4948	−1.40010	−0.700051	+	0.714093i	$0.746840\pi$
$608$	3.91870	0.158924
$609$	0	0
$610$	−8.54704	−0.346060
$611$	−10.3420	−0.418394
$612$	0	0
$613$	19.6568	0.793931	0.396965	−	0.917834i	$-0.370063\pi$
$613$	19.6568	0.793931	0.396965	+	0.917834i	$0.370063\pi$
$614$	−18.0088	−0.726776
$615$	0	0
$616$	4.15318	0.167336
$617$	36.0533	1.45145	0.725726	−	0.687984i	$-0.241504\pi$
$617$	36.0533	1.45145	0.725726	+	0.687984i	$0.241504\pi$
$618$	0	0
$619$	−8.81995	−0.354504	−0.177252	−	0.984166i	$-0.556721\pi$
$619$	−8.81995	−0.354504	−0.177252	+	0.984166i	$0.556721\pi$
$620$	−14.8706	−0.597218
$621$	0	0
$622$	11.6496	0.467105
$623$	5.82652	0.233434
$624$	0	0
$625$	11.7113	0.468451
$626$	9.16957	0.366490
$627$	0	0
$628$	−5.32278	−0.212402
$629$	46.7078	1.86236
$630$	0	0
$631$	−0.823111	−0.0327675	−0.0163838	−	0.999866i	$-0.505215\pi$
$631$	−0.823111	−0.0327675	−0.0163838	+	0.999866i	$0.505215\pi$
$632$	3.11045	0.123727
$633$	0	0
$634$	9.85456	0.391374
$635$	83.7169	3.32221
$636$	0	0
$637$	−28.1754	−1.11635
$638$	−3.85103	−0.152464
$639$	0	0
$640$	24.0346	0.950052
$641$	−31.0472	−1.22629	−0.613145	−	0.789970i	$-0.710096\pi$
$641$	−31.0472	−1.22629	−0.613145	+	0.789970i	$0.710096\pi$
$642$	0	0
$643$	−0.565925	−0.0223179	−0.0111589	−	0.999938i	$-0.503552\pi$
$643$	−0.565925	−0.0223179	−0.0111589	+	0.999938i	$0.503552\pi$
$644$	−2.87488	−0.113286
$645$	0	0
$646$	3.01226	0.118516
$647$	32.7343	1.28692	0.643458	−	0.765481i	$-0.277499\pi$
$647$	32.7343	1.28692	0.643458	+	0.765481i	$0.277499\pi$
$648$	0	0
$649$	−4.65105	−0.182569
$650$	46.6207	1.82861
$651$	0	0
$652$	−21.2381	−0.831749
$653$	7.23261	0.283034	0.141517	−	0.989936i	$-0.454802\pi$
$653$	7.23261	0.283034	0.141517	+	0.989936i	$0.454802\pi$
$654$	0	0
$655$	−36.6978	−1.43390
$656$	0.679837	0.0265432
$657$	0	0
$658$	−2.33169	−0.0908989
$659$	−36.5089	−1.42219	−0.711093	−	0.703098i	$-0.751800\pi$
$659$	−36.5089	−1.42219	−0.711093	+	0.703098i	$0.751800\pi$
$660$	0	0
$661$	14.1432	0.550105	0.275053	−	0.961429i	$-0.411305\pi$
$661$	14.1432	0.550105	0.275053	+	0.961429i	$0.411305\pi$
$662$	−16.2289	−0.630754
$663$	0	0
$664$	−10.8861	−0.422461
$665$	−3.81950	−0.148114
$666$	0	0
$667$	7.10866	0.275249
$668$	−10.6815	−0.413278
$669$	0	0
$670$	−29.9815	−1.15829
$671$	−2.54894	−0.0984009
$672$	0	0
$673$	15.5642	0.599958	0.299979	−	0.953946i	$-0.403020\pi$
$673$	15.5642	0.599958	0.299979	+	0.953946i	$0.403020\pi$
$674$	−1.50183	−0.0578484
$675$	0	0
$676$	−24.1661	−0.929464
$677$	34.1558	1.31271	0.656357	−	0.754451i	$-0.272097\pi$
$677$	34.1558	1.31271	0.656357	+	0.754451i	$0.272097\pi$
$678$	0	0
$679$	4.77544	0.183264
$680$	51.4675	1.97369
$681$	0	0
$682$	2.95660	0.113214
$683$	48.8978	1.87102	0.935512	−	0.353295i	$-0.114939\pi$
$683$	48.8978	1.87102	0.935512	+	0.353295i	$0.114939\pi$
$684$	0	0
$685$	−8.90825	−0.340367
$686$	−15.4375	−0.589406
$687$	0	0
$688$	0.657104	0.0250518
$689$	6.63842	0.252904
$690$	0	0
$691$	−34.1325	−1.29846	−0.649230	−	0.760592i	$-0.724909\pi$
$691$	−34.1325	−1.29846	−0.649230	+	0.760592i	$0.724909\pi$
$692$	−29.6767	−1.12814
$693$	0	0
$694$	16.7985	0.637663
$695$	43.0629	1.63347
$696$	0	0
$697$	−20.3738	−0.771714
$698$	29.0771	1.10058
$699$	0	0
$700$	−15.7661	−0.595904
$701$	22.2291	0.839581	0.419791	−	0.907621i	$-0.362104\pi$
$701$	22.2291	0.839581	0.419791	+	0.907621i	$0.362104\pi$
$702$	0	0
$703$	6.83715	0.257868
$704$	−5.31212	−0.200208
$705$	0	0
$706$	−6.00142	−0.225867
$707$	−20.5949	−0.774552
$708$	0	0
$709$	−20.4395	−0.767623	−0.383812	−	0.923411i	$-0.625389\pi$
$709$	−20.4395	−0.767623	−0.383812	+	0.923411i	$0.625389\pi$
$710$	17.2421	0.647086
$711$	0	0
$712$	−11.4927	−0.430706
$713$	−5.45762	−0.204389
$714$	0	0
$715$	21.5811	0.807089
$716$	−14.2303	−0.531812
$717$	0	0
$718$	−13.2194	−0.493344
$719$	−19.8844	−0.741563	−0.370782	−	0.928720i	$-0.620910\pi$
$719$	−19.8844	−0.741563	−0.370782	+	0.928720i	$0.620910\pi$
$720$	0	0
$721$	27.8279	1.03637
$722$	−16.5533	−0.616050
$723$	0	0
$724$	5.93449	0.220554
$725$	38.9846	1.44785
$726$	0	0
$727$	0.286730	0.0106342	0.00531712	−	0.999986i	$-0.498308\pi$
$727$	0.286730	0.0106342	0.00531712	+	0.999986i	$0.498308\pi$
$728$	−23.9083	−0.886099
$729$	0	0
$730$	−35.3350	−1.30781
$731$	−19.6925	−0.728355
$732$	0	0
$733$	10.2012	0.376789	0.188394	−	0.982093i	$-0.439672\pi$
$733$	10.2012	0.376789	0.188394	+	0.982093i	$0.439672\pi$
$734$	23.7802	0.877744
$735$	0	0
$736$	9.21481	0.339662
$737$	−8.94124	−0.329355
$738$	0	0
$739$	29.3420	1.07936	0.539681	−	0.841870i	$-0.318545\pi$
$739$	29.3420	1.07936	0.539681	+	0.841870i	$0.318545\pi$
$740$	43.8071	1.61038
$741$	0	0
$742$	1.49669	0.0549451
$743$	−47.8225	−1.75444	−0.877219	−	0.480090i	$-0.840604\pi$
$743$	−47.8225	−1.75444	−0.877219	+	0.480090i	$0.840604\pi$
$744$	0	0
$745$	−2.62918	−0.0963258
$746$	21.0695	0.771410
$747$	0	0
$748$	5.75580	0.210453
$749$	14.5155	0.530385
$750$	0	0
$751$	−20.0467	−0.731516	−0.365758	−	0.930710i	$-0.619190\pi$
$751$	−20.0467	−0.731516	−0.365758	+	0.930710i	$0.619190\pi$
$752$	0.287541	0.0104855
$753$	0	0
$754$	22.1689	0.807344
$755$	9.39311	0.341850
$756$	0	0
$757$	−34.7845	−1.26427	−0.632133	−	0.774860i	$-0.717820\pi$
$757$	−34.7845	−1.26427	−0.632133	+	0.774860i	$0.717820\pi$
$758$	28.3121	1.02834
$759$	0	0
$760$	7.53388	0.273283
$761$	6.55235	0.237522	0.118761	−	0.992923i	$-0.462108\pi$
$761$	6.55235	0.237522	0.118761	+	0.992923i	$0.462108\pi$
$762$	0	0
$763$	−10.7489	−0.389137
$764$	0.388278	0.0140474
$765$	0	0
$766$	32.0601	1.15838
$767$	26.7743	0.966762
$768$	0	0
$769$	−34.0222	−1.22687	−0.613435	−	0.789745i	$-0.710213\pi$
$769$	−34.0222	−1.22687	−0.613435	+	0.789745i	$0.710213\pi$
$770$	4.86564	0.175346
$771$	0	0
$772$	−17.2982	−0.622577
$773$	−5.09752	−0.183345	−0.0916725	−	0.995789i	$-0.529221\pi$
$773$	−5.09752	−0.183345	−0.0916725	+	0.995789i	$0.529221\pi$
$774$	0	0
$775$	−29.9301	−1.07512
$776$	−9.41944	−0.338138
$777$	0	0
$778$	−3.50769	−0.125757
$779$	−2.98235	−0.106854
$780$	0	0
$781$	5.14204	0.183997
$782$	7.08330	0.253298
$783$	0	0
$784$	0.783363	0.0279773
$785$	−16.6291	−0.593519
$786$	0	0
$787$	−25.1070	−0.894969	−0.447485	−	0.894292i	$-0.647680\pi$
$787$	−25.1070	−0.894969	−0.447485	+	0.894292i	$0.647680\pi$
$788$	−21.9918	−0.783425
$789$	0	0
$790$	3.64403	0.129649
$791$	−2.38462	−0.0847872
$792$	0	0
$793$	14.6733	0.521063
$794$	−3.49091	−0.123888
$795$	0	0
$796$	1.81451	0.0643135
$797$	6.79568	0.240715	0.120358	−	0.992731i	$-0.461596\pi$
$797$	6.79568	0.240715	0.120358	+	0.992731i	$0.461596\pi$
$798$	0	0
$799$	−8.61723	−0.304856
$800$	50.5349	1.78668
$801$	0	0
$802$	−28.3316	−1.00042
$803$	−10.5378	−0.371871
$804$	0	0
$805$	−8.98154	−0.316558
$806$	−17.0200	−0.599504
$807$	0	0
$808$	40.6231	1.42911
$809$	−17.0314	−0.598792	−0.299396	−	0.954129i	$-0.596785\pi$
$809$	−17.0314	−0.598792	−0.299396	+	0.954129i	$0.596785\pi$
$810$	0	0
$811$	2.28561	0.0802587	0.0401294	−	0.999194i	$-0.487223\pi$
$811$	2.28561	0.0802587	0.0401294	+	0.999194i	$0.487223\pi$
$812$	−7.49705	−0.263095
$813$	0	0
$814$	−8.70981	−0.305279
$815$	−66.3508	−2.32417
$816$	0	0
$817$	−2.88262	−0.100850
$818$	−11.5528	−0.403933
$819$	0	0
$820$	−19.1086	−0.667301
$821$	−22.0526	−0.769643	−0.384821	−	0.922991i	$-0.625737\pi$
$821$	−22.0526	−0.769643	−0.384821	+	0.922991i	$0.625737\pi$
$822$	0	0
$823$	−24.7113	−0.861381	−0.430691	−	0.902500i	$-0.641730\pi$
$823$	−24.7113	−0.861381	−0.430691	+	0.902500i	$0.641730\pi$
$824$	−54.8900	−1.91218
$825$	0	0
$826$	6.03647	0.210036
$827$	−19.8452	−0.690085	−0.345043	−	0.938587i	$-0.612136\pi$
$827$	−19.8452	−0.690085	−0.345043	+	0.938587i	$0.612136\pi$
$828$	0	0
$829$	32.9988	1.14610	0.573048	−	0.819522i	$-0.305761\pi$
$829$	32.9988	1.14610	0.573048	+	0.819522i	$0.305761\pi$
$830$	−12.7535	−0.442681
$831$	0	0
$832$	30.5798	1.06016
$833$	−23.4764	−0.813409
$834$	0	0
$835$	−33.3704	−1.15483
$836$	0.842543	0.0291399
$837$	0	0
$838$	−24.4724	−0.845386
$839$	20.2763	0.700017	0.350008	−	0.936747i	$-0.386179\pi$
$839$	20.2763	0.700017	0.350008	+	0.936747i	$0.386179\pi$
$840$	0	0
$841$	−10.4622	−0.360766
$842$	−17.5109	−0.603467
$843$	0	0
$844$	13.2265	0.455276
$845$	−75.4981	−2.59721
$846$	0	0
$847$	1.45106	0.0498589
$848$	−0.184569	−0.00633813
$849$	0	0
$850$	38.8455	1.33239
$851$	16.0775	0.551130
$852$	0	0
$853$	−7.52894	−0.257786	−0.128893	−	0.991659i	$-0.541142\pi$
$853$	−7.52894	−0.257786	−0.128893	+	0.991659i	$0.541142\pi$
$854$	3.30821	0.113205
$855$	0	0
$856$	−28.6315	−0.978605
$857$	11.5279	0.393785	0.196893	−	0.980425i	$-0.436915\pi$
$857$	11.5279	0.393785	0.196893	+	0.980425i	$0.436915\pi$
$858$	0	0
$859$	−2.62150	−0.0894445	−0.0447222	−	0.998999i	$-0.514240\pi$
$859$	−2.62150	−0.0894445	−0.0447222	+	0.998999i	$0.514240\pi$
$860$	−18.4696	−0.629808
$861$	0	0
$862$	−8.76988	−0.298703
$863$	−29.7385	−1.01231	−0.506156	−	0.862442i	$-0.668934\pi$
$863$	−29.7385	−1.01231	−0.506156	+	0.862442i	$0.668934\pi$
$864$	0	0
$865$	−92.7143	−3.15238
$866$	−1.77448	−0.0602993
$867$	0	0
$868$	5.75580	0.195365
$869$	1.08674	0.0368652
$870$	0	0
$871$	51.4712	1.74404
$872$	21.2020	0.717991
$873$	0	0
$874$	1.03686	0.0350725
$875$	−22.0560	−0.745631
$876$	0	0
$877$	2.12246	0.0716705	0.0358352	−	0.999358i	$-0.488591\pi$
$877$	2.12246	0.0716705	0.0358352	+	0.999358i	$0.488591\pi$
$878$	31.3604	1.05836
$879$	0	0
$880$	−0.600024	−0.0202268
$881$	−14.0627	−0.473783	−0.236892	−	0.971536i	$-0.576129\pi$
$881$	−14.0627	−0.473783	−0.236892	+	0.971536i	$0.576129\pi$
$882$	0	0
$883$	−20.5227	−0.690645	−0.345323	−	0.938484i	$-0.612231\pi$
$883$	−20.5227	−0.690645	−0.345323	+	0.938484i	$0.612231\pi$
$884$	−33.1339	−1.11441
$885$	0	0
$886$	−21.2089	−0.712527
$887$	−2.51373	−0.0844027	−0.0422013	−	0.999109i	$-0.513437\pi$
$887$	−2.51373	−0.0844027	−0.0422013	+	0.999109i	$0.513437\pi$
$888$	0	0
$889$	−32.4034	−1.08677
$890$	−13.4642	−0.451321
$891$	0	0
$892$	27.9756	0.936693
$893$	−1.26140	−0.0422112
$894$	0	0
$895$	−44.4575	−1.48605
$896$	−9.30282	−0.310785
$897$	0	0
$898$	10.4269	0.347950
$899$	−14.2322	−0.474672
$900$	0	0
$901$	5.53130	0.184274
$902$	3.79920	0.126500
$903$	0	0
$904$	4.70360	0.156439
$905$	18.5402	0.616296
$906$	0	0
$907$	33.2306	1.10341	0.551703	−	0.834041i	$-0.313978\pi$
$907$	33.2306	1.10341	0.551703	+	0.834041i	$0.313978\pi$
$908$	7.17460	0.238097
$909$	0	0
$910$	−28.0096	−0.928509
$911$	−5.54894	−0.183845	−0.0919223	−	0.995766i	$-0.529301\pi$
$911$	−5.54894	−0.183845	−0.0919223	+	0.995766i	$0.529301\pi$
$912$	0	0
$913$	−3.80342	−0.125875
$914$	11.8068	0.390534
$915$	0	0
$916$	−11.3094	−0.373672
$917$	14.2042	0.469064
$918$	0	0
$919$	−46.9009	−1.54712	−0.773560	−	0.633724i	$-0.781526\pi$
$919$	−46.9009	−1.54712	−0.773560	+	0.633724i	$0.781526\pi$
$920$	17.7159	0.584076
$921$	0	0
$922$	27.0935	0.892278
$923$	−29.6007	−0.974319
$924$	0	0
$925$	88.1707	2.89903
$926$	−24.3392	−0.799837
$927$	0	0
$928$	24.0302	0.788829
$929$	31.0100	1.01740	0.508702	−	0.860943i	$-0.330125\pi$
$929$	31.0100	1.01740	0.508702	+	0.860943i	$0.330125\pi$
$930$	0	0
$931$	−3.43651	−0.112627
$932$	32.6019	1.06791
$933$	0	0
$934$	−27.8950	−0.912752
$935$	17.9819	0.588072
$936$	0	0
$937$	−44.0640	−1.43951	−0.719755	−	0.694229i	$-0.755746\pi$
$937$	−44.0640	−1.43951	−0.719755	+	0.694229i	$0.755746\pi$
$938$	11.6046	0.378904
$939$	0	0
$940$	−8.08208	−0.263608
$941$	−2.12056	−0.0691283	−0.0345642	−	0.999402i	$-0.511004\pi$
$941$	−2.12056	−0.0691283	−0.0345642	+	0.999402i	$0.511004\pi$
$942$	0	0
$943$	−7.01299	−0.228374
$944$	−0.744409	−0.0242284
$945$	0	0
$946$	3.67216	0.119392
$947$	15.3247	0.497985	0.248993	−	0.968505i	$-0.419900\pi$
$947$	15.3247	0.497985	0.248993	+	0.968505i	$0.419900\pi$
$948$	0	0
$949$	60.6620	1.96917
$950$	5.68626	0.184487
$951$	0	0
$952$	−19.9209	−0.645641
$953$	−31.7696	−1.02912	−0.514559	−	0.857455i	$-0.672045\pi$
$953$	−31.7696	−1.02912	−0.514559	+	0.857455i	$0.672045\pi$
$954$	0	0
$955$	1.21303	0.0392529
$956$	0.291091	0.00941454
$957$	0	0
$958$	34.2364	1.10613
$959$	3.44802	0.111342
$960$	0	0
$961$	−20.0733	−0.647526
$962$	50.1390	1.61655
$963$	0	0
$964$	15.2525	0.491251
$965$	−54.0421	−1.73968
$966$	0	0
$967$	4.71821	0.151727	0.0758637	−	0.997118i	$-0.475829\pi$
$967$	4.71821	0.151727	0.0758637	+	0.997118i	$0.475829\pi$
$968$	−2.86218	−0.0919939
$969$	0	0
$970$	−11.0353	−0.354322
$971$	2.48946	0.0798905	0.0399452	−	0.999202i	$-0.487282\pi$
$971$	2.48946	0.0798905	0.0399452	+	0.999202i	$0.487282\pi$
$972$	0	0
$973$	−16.6679	−0.534348
$974$	−14.1859	−0.454547
$975$	0	0
$976$	−0.407963	−0.0130586
$977$	−47.9903	−1.53534	−0.767672	−	0.640842i	$-0.778585\pi$
$977$	−47.9903	−1.53534	−0.767672	+	0.640842i	$0.778585\pi$
$978$	0	0
$979$	−4.01536	−0.128331
$980$	−22.0185	−0.703354
$981$	0	0
$982$	6.26308	0.199863
$983$	0.497137	0.0158562	0.00792811	−	0.999969i	$-0.497476\pi$
$983$	0.497137	0.0158562	0.00792811	+	0.999969i	$0.497476\pi$
$984$	0	0
$985$	−68.7054	−2.18914
$986$	18.4717	0.588258
$987$	0	0
$988$	−4.85019	−0.154305
$989$	−6.77848	−0.215543
$990$	0	0
$991$	56.7182	1.80171	0.900856	−	0.434118i	$-0.142940\pi$
$991$	56.7182	1.80171	0.900856	+	0.434118i	$0.142940\pi$
$992$	−18.4490	−0.585755
$993$	0	0
$994$	−6.67372	−0.211678
$995$	5.66878	0.179712
$996$	0	0
$997$	51.1895	1.62119	0.810594	−	0.585609i	$-0.199144\pi$
$997$	51.1895	1.62119	0.810594	+	0.585609i	$0.199144\pi$
$998$	22.0662	0.698492
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	891.2.a.p.1.3		4
3.2	odd	2		891.2.a.q.1.2		4
9.2	odd	6		99.2.e.e.67.3	yes	8
9.4	even	3		297.2.e.e.100.2		8
9.5	odd	6		99.2.e.e.34.3	✓	8
9.7	even	3		297.2.e.e.199.2		8
11.10	odd	2		9801.2.a.bl.1.2		4
33.32	even	2		9801.2.a.bi.1.3		4
99.32	even	6		1089.2.e.i.727.2		8
99.65	even	6		1089.2.e.i.364.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.e.e.34.3	✓	8	9.5	odd	6
99.2.e.e.67.3	yes	8	9.2	odd	6
297.2.e.e.100.2		8	9.4	even	3
297.2.e.e.199.2		8	9.7	even	3
891.2.a.p.1.3		4	1.1	even	1	trivial
891.2.a.q.1.2		4	3.2	odd	2
1089.2.e.i.364.2		8	99.65	even	6
1089.2.e.i.727.2		8	99.32	even	6
9801.2.a.bi.1.3		4	33.32	even	2
9801.2.a.bl.1.2		4	11.10	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:	\( N \)	\(=\)	\( 891 = 3^{4} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	891.a (trivial)

\(f(q)\)	\(=\)	\(q+0.894434 q^{2} -1.19999 q^{4} -3.74893 q^{5} +1.45106 q^{7} -2.86218 q^{8} -3.35317 q^{10} -1.00000 q^{11} +5.75661 q^{13} +1.29787 q^{14} -0.160052 q^{16} +4.79655 q^{17} +0.702126 q^{19} +4.49867 q^{20} -0.894434 q^{22} +1.65105 q^{23} +9.05449 q^{25} +5.14891 q^{26} -1.74125 q^{28} +4.30555 q^{29} -3.30555 q^{31} +5.58120 q^{32} +4.29019 q^{34} -5.43991 q^{35} +9.73779 q^{37} +0.628005 q^{38} +10.7301 q^{40} -4.24760 q^{41} -4.10557 q^{43} +1.19999 q^{44} +1.47675 q^{46} -1.79655 q^{47} -4.89443 q^{49} +8.09864 q^{50} -6.90787 q^{52} +1.15318 q^{53} +3.74893 q^{55} -4.15318 q^{56} +3.85103 q^{58} +4.65105 q^{59} +2.54894 q^{61} -2.95660 q^{62} +5.31212 q^{64} -21.5811 q^{65} +8.94124 q^{67} -5.75580 q^{68} -4.86564 q^{70} -5.14204 q^{71} +10.5378 q^{73} +8.70981 q^{74} -0.842543 q^{76} -1.45106 q^{77} -1.08674 q^{79} +0.600024 q^{80} -3.79920 q^{82} +3.80342 q^{83} -17.9819 q^{85} -3.67216 q^{86} +2.86218 q^{88} +4.01536 q^{89} +8.35317 q^{91} -1.98123 q^{92} -1.60689 q^{94} -2.63222 q^{95} +3.29101 q^{97} -4.37775 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 11 q^{4} - 4 q^{5} + q^{7} + q^{10} - 4 q^{11} + 7 q^{13} - q^{14} + 17 q^{16} + 5 q^{17} + 9 q^{19} + 10 q^{20} + q^{22} - 14 q^{23} + 14 q^{25} - 22 q^{26} - q^{28} + 6 q^{29} - 2 q^{31}+ \cdots - 15 q^{98}+O(q^{100}) \) 4 * q - q^2 + 11 * q^4 - 4 * q^5 + q^7 + q^10 - 4 * q^11 + 7 * q^13 - q^14 + 17 * q^16 + 5 * q^17 + 9 * q^19 + 10 * q^20 + q^22 - 14 * q^23 + 14 * q^25 - 22 * q^26 - q^28 + 6 * q^29 - 2 * q^31 + 34 * q^32 + 16 * q^34 + 8 * q^35 + 3 * q^37 - 3 * q^38 + 12 * q^40 + 2 * q^41 - 21 * q^43 - 11 * q^44 + 2 * q^46 + 7 * q^47 - 15 * q^49 - 23 * q^50 - 10 * q^52 + 6 * q^53 + 4 * q^55 - 18 * q^56 - 21 * q^58 - 2 * q^59 + 15 * q^61 + 20 * q^62 + 16 * q^64 - 19 * q^65 + 14 * q^67 + 7 * q^68 - 38 * q^70 + 3 * q^71 + 22 * q^73 + 36 * q^74 + 42 * q^76 - q^77 + 11 * q^79 + 34 * q^80 - 17 * q^82 - 18 * q^83 + 13 * q^85 + 24 * q^86 + 6 * q^89 + 19 * q^91 - 67 * q^92 - 19 * q^94 + 30 * q^95 + 26 * q^97 - 15 * q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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