LMFDB - Embedded newform 9801.2.a.bi.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9801,2,Mod(1,9801)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9801, 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("9801.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9801 = 3^{4} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9801.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:	$78.2613790211$
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	$\chi$	$=$	9801.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.894434 q^{2} -1.19999 q^{4} +3.74893 q^{5} -1.45106 q^{7} -2.86218 q^{8} +3.35317 q^{10} -5.75661 q^{13} -1.29787 q^{14} -0.160052 q^{16} +4.79655 q^{17} -0.702126 q^{19} -4.49867 q^{20} -1.65105 q^{23} +9.05449 q^{25} -5.14891 q^{26} +1.74125 q^{28} +4.30555 q^{29} -3.30555 q^{31} +5.58120 q^{32} +4.29019 q^{34} -5.43991 q^{35} +9.73779 q^{37} -0.628005 q^{38} -10.7301 q^{40} -4.24760 q^{41} +4.10557 q^{43} -1.47675 q^{46} +1.79655 q^{47} -4.89443 q^{49} +8.09864 q^{50} +6.90787 q^{52} -1.15318 q^{53} +4.15318 q^{56} +3.85103 q^{58} -4.65105 q^{59} -2.54894 q^{61} -2.95660 q^{62} +5.31212 q^{64} -21.5811 q^{65} +8.94124 q^{67} -5.75580 q^{68} -4.86564 q^{70} +5.14204 q^{71} -10.5378 q^{73} +8.70981 q^{74} +0.842543 q^{76} +1.08674 q^{79} -0.600024 q^{80} -3.79920 q^{82} +3.80342 q^{83} +17.9819 q^{85} +3.67216 q^{86} -4.01536 q^{89} +8.35317 q^{91} +1.98123 q^{92} +1.60689 q^{94} -2.63222 q^{95} +3.29101 q^{97} -4.37775 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 11 q^{4} + 4 q^{5} - q^{7} - q^{10} - 7 q^{13} + q^{14} + 17 q^{16} + 5 q^{17} - 9 q^{19} - 10 q^{20} + 14 q^{23} + 14 q^{25} + 22 q^{26} + q^{28} + 6 q^{29} - 2 q^{31} + 34 q^{32} + 16 q^{34}+ \cdots - 15 q^{98}+O(q^{100}) $ 4 * q - q^2 + 11 * q^4 + 4 * q^5 - q^7 - q^10 - 7 * q^13 + q^14 + 17 * q^16 + 5 * q^17 - 9 * q^19 - 10 * q^20 + 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 - 2 * q^46 - 7 * q^47 - 15 * q^49 - 23 * q^50 + 10 * q^52 - 6 * q^53 + 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 - 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	$-0.183558\pi$
$5$	3.74893	1.67657	0.838287	+	0.545230i	$0.183558\pi$
$6$	0	0
$7$	−1.45106	−0.548448	−0.274224	−	0.961666i	$-0.588421\pi$
$7$	−1.45106	−0.548448	−0.274224	+	0.961666i	$0.588421\pi$
$8$	−2.86218	−1.01193
$9$	0	0
$10$	3.35317	1.06037
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−5.75661	−1.59660	−0.798298	−	0.602262i	$-0.794266\pi$
$13$	−5.75661	−1.59660	−0.798298	+	0.602262i	$0.794266\pi$
$14$	−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	$-0.525664\pi$
$19$	−0.702126	−0.161079	−0.0805393	+	0.996751i	$0.525664\pi$
$20$	−4.49867	−1.00593
$21$	0	0
$22$	0	0
$23$	−1.65105	−0.344267	−0.172133	−	0.985074i	$-0.555066\pi$
$23$	−1.65105	−0.344267	−0.172133	+	0.985074i	$0.555066\pi$
$24$	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	$-0.398650\pi$
$43$	4.10557	0.626093	0.313046	+	0.949738i	$0.398650\pi$
$44$	0	0
$45$	0	0
$46$	−1.47675	−0.217735
$47$	1.79655	0.262053	0.131027	−	0.991379i	$-0.458173\pi$
$47$	1.79655	0.262053	0.131027	+	0.991379i	$0.458173\pi$
$48$	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	$-0.525237\pi$
$53$	−1.15318	−0.158402	−0.0792009	+	0.996859i	$0.525237\pi$
$54$	0	0
$55$	0	0
$56$	4.15318	0.554992
$57$	0	0
$58$	3.85103	0.505665
$59$	−4.65105	−0.605515	−0.302757	−	0.953068i	$-0.597907\pi$
$59$	−4.65105	−0.605515	−0.302757	+	0.953068i	$0.597907\pi$
$60$	0	0
$61$	−2.54894	−0.326359	−0.163179	−	0.986596i	$-0.552175\pi$
$61$	−2.54894	−0.326359	−0.163179	+	0.986596i	$0.552175\pi$
$62$	−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	$-0.401302\pi$
$71$	5.14204	0.610248	0.305124	+	0.952313i	$0.401302\pi$
$72$	0	0
$73$	−10.5378	−1.23336	−0.616678	−	0.787215i	$-0.711522\pi$
$73$	−10.5378	−1.23336	−0.616678	+	0.787215i	$0.711522\pi$
$74$	8.70981	1.01249
$75$	0	0
$76$	0.842543	0.0966463
$77$	0	0
$78$	0	0
$79$	1.08674	0.122268	0.0611340	−	0.998130i	$-0.480528\pi$
$79$	1.08674	0.122268	0.0611340	+	0.998130i	$0.480528\pi$
$80$	−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
$88$	0	0
$89$	−4.01536	−0.425627	−0.212814	−	0.977093i	$-0.568263\pi$
$89$	−4.01536	−0.425627	−0.212814	+	0.977093i	$0.568263\pi$
$90$	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	$-0.384562\pi$
$109$	7.40766	0.709525	0.354762	+	0.934956i	$0.384562\pi$
$110$	0	0
$111$	0	0
$112$	0.232244	0.0219450
$113$	1.64337	0.154595	0.0772974	−	0.997008i	$-0.475371\pi$
$113$	1.64337	0.154595	0.0772974	+	0.997008i	$0.475371\pi$
$114$	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
$121$	0	0
$122$	−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	$-0.0432750\pi$
$127$	22.3309	1.98155	0.990773	+	0.135534i	$0.0432750\pi$
$128$	−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	$-0.532366\pi$
$137$	−2.37621	−0.203013	−0.101507	+	0.994835i	$0.532366\pi$
$138$	0	0
$139$	11.4867	0.974291	0.487145	−	0.873321i	$-0.338038\pi$
$139$	11.4867	0.974291	0.487145	+	0.873321i	$0.338038\pi$
$140$	6.52783	0.551702
$141$	0	0
$142$	4.59921	0.385957
$143$	0	0
$144$	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	$-0.467492\pi$
$151$	2.50554	0.203898	0.101949	+	0.994790i	$0.467492\pi$
$152$	2.00961	0.163001
$153$	0	0
$154$	0	0
$155$	−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	0
$177$	0	0
$178$	−3.59147	−0.269192
$179$	−11.8587	−0.886362	−0.443181	−	0.896432i	$-0.646150\pi$
$179$	−11.8587	−0.886362	−0.443181	+	0.896432i	$0.646150\pi$
$180$	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
$187$	0	0
$188$	−2.15584	−0.157230
$189$	0	0
$190$	−2.35435	−0.170802
$191$	0.323568	0.0234126	0.0117063	−	0.999931i	$-0.496274\pi$
$191$	0.323568	0.0234126	0.0117063	+	0.999931i	$0.496274\pi$
$192$	0	0
$193$	−14.4153	−1.03764	−0.518819	−	0.854884i	$-0.673628\pi$
$193$	−14.4153	−1.03764	−0.518819	+	0.854884i	$0.673628\pi$
$194$	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	0
$210$	0	0
$211$	11.0222	0.758802	0.379401	−	0.925232i	$-0.376130\pi$
$211$	11.0222	0.758802	0.379401	+	0.925232i	$0.376130\pi$
$212$	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
$220$	0	0
$221$	−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	$-0.365745\pi$
$241$	12.7106	0.818759	0.409379	+	0.912364i	$0.365745\pi$
$242$	0	0
$243$	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	$-0.232339\pi$
$251$	23.6134	1.49046	0.745232	+	0.666805i	$0.232339\pi$
$252$	0	0
$253$	0	0
$254$	19.9735	1.25325
$255$	0	0
$256$	−16.3585	−1.02241
$257$	4.11671	0.256793	0.128397	−	0.991723i	$-0.459017\pi$
$257$	4.11671	0.256793	0.128397	+	0.991723i	$0.459017\pi$
$258$	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	$-0.566720\pi$
$269$	−6.82534	−0.416148	−0.208074	+	0.978113i	$0.566720\pi$
$270$	0	0
$271$	7.57539	0.460172	0.230086	−	0.973170i	$-0.426099\pi$
$271$	7.57539	0.460172	0.230086	+	0.973170i	$0.426099\pi$
$272$	−0.767697	−0.0465484
$273$	0	0
$274$	−2.12536	−0.128398
$275$	0	0
$276$	0	0
$277$	18.8034	1.12978	0.564892	−	0.825165i	$-0.308918\pi$
$277$	18.8034	1.12978	0.564892	+	0.825165i	$0.308918\pi$
$278$	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	$-0.318730\pi$
$283$	18.1412	1.07838	0.539192	+	0.842183i	$0.318730\pi$
$284$	−6.17039	−0.366145
$285$	0	0
$286$	0	0
$287$	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	$-0.305172\pi$
$307$	20.1343	1.14913	0.574563	+	0.818461i	$0.305172\pi$
$308$	0	0
$309$	0	0
$310$	−11.0841	−0.629534
$311$	−13.0245	−0.738553	−0.369276	−	0.929320i	$-0.620394\pi$
$311$	−13.0245	−0.738553	−0.369276	+	0.929320i	$0.620394\pi$
$312$	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	$-0.600130\pi$
$317$	−11.0176	−0.618813	−0.309406	+	0.950930i	$0.600130\pi$
$318$	0	0
$319$	0	0
$320$	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	$-0.485438\pi$
$337$	1.67909	0.0914656	0.0457328	+	0.998954i	$0.485438\pi$
$338$	18.0126	0.979757
$339$	0	0
$340$	−21.5781	−1.17024
$341$	0	0
$342$	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	$-0.835934\pi$
$349$	−32.5089	−1.74016	−0.870082	+	0.492907i	$0.835934\pi$
$350$	−11.7516	−0.628149
$351$	0	0
$352$	0	0
$353$	6.70975	0.357124	0.178562	−	0.983929i	$-0.442856\pi$
$353$	6.70975	0.357124	0.178562	+	0.983929i	$0.442856\pi$
$354$	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	$-0.708770\pi$
$373$	−23.5563	−1.21970	−0.609848	+	0.792518i	$0.708770\pi$
$374$	0	0
$375$	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	$-0.868420\pi$
$383$	−35.8440	−1.83155	−0.915773	+	0.401697i	$0.868420\pi$
$384$	0	0
$385$	0	0
$386$	−12.8936	−0.656265
$387$	0	0
$388$	−3.94917	−0.200489
$389$	3.92169	0.198838	0.0994188	−	0.995046i	$-0.468302\pi$
$389$	3.92169	0.198838	0.0994188	+	0.995046i	$0.468302\pi$
$390$	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	$-0.209613\pi$
$401$	31.6755	1.58180	0.790900	+	0.611946i	$0.209613\pi$
$402$	0	0
$403$	19.0288	0.947892
$404$	17.0315	0.847349
$405$	0	0
$406$	−5.58807	−0.277331
$407$	0	0
$408$	0	0
$409$	12.9163	0.638670	0.319335	−	0.947642i	$-0.396541\pi$
$409$	12.9163	0.638670	0.319335	+	0.947642i	$0.396541\pi$
$410$	−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	0
$419$	27.3608	1.33666	0.668331	−	0.743864i	$-0.267009\pi$
$419$	27.3608	1.33666	0.668331	+	0.743864i	$0.267009\pi$
$420$	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	$-0.815520\pi$
$439$	−35.0617	−1.67341	−0.836703	+	0.547657i	$0.815520\pi$
$440$	0	0
$441$	0	0
$442$	−24.6970	−1.17472
$443$	23.7121	1.12660	0.563298	−	0.826254i	$-0.309532\pi$
$443$	23.7121	1.12660	0.563298	+	0.826254i	$0.309532\pi$
$444$	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	$-0.588703\pi$
$449$	−11.6575	−0.550154	−0.275077	+	0.961422i	$0.588703\pi$
$450$	0	0
$451$	0	0
$452$	−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	$-0.599908\pi$
$457$	−13.2003	−0.617484	−0.308742	+	0.951146i	$0.599908\pi$
$458$	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	$-0.243412\pi$
$467$	31.1873	1.44318	0.721588	+	0.692322i	$0.243412\pi$
$468$	0	0
$469$	−12.9742	−0.599095
$470$	6.02413	0.277872
$471$	0	0
$472$	13.3121	0.612740
$473$	0	0
$474$	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
$484$	0	0
$485$	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
$506$	0	0
$507$	0	0
$508$	−26.7968	−1.18892
$509$	−20.0560	−0.888965	−0.444482	−	0.895788i	$-0.646612\pi$
$509$	−20.0560	−0.888965	−0.444482	+	0.895788i	$0.646612\pi$
$510$	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
$517$	0	0
$518$	−12.6384	−0.555300
$519$	0	0
$520$	61.7691	2.70875
$521$	18.8721	0.826804	0.413402	−	0.910549i	$-0.364340\pi$
$521$	18.8721	0.826804	0.413402	+	0.910549i	$0.364340\pi$
$522$	0	0
$523$	16.0568	0.702114	0.351057	−	0.936354i	$-0.385822\pi$
$523$	16.0568	0.702114	0.351057	+	0.936354i	$0.385822\pi$
$524$	−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
$539$	0	0
$540$	0	0
$541$	−18.4357	−0.792614	−0.396307	−	0.918118i	$-0.629708\pi$
$541$	−18.4357	−0.792614	−0.396307	+	0.918118i	$0.629708\pi$
$542$	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	$-0.287430\pi$
$547$	28.9669	1.23853	0.619267	+	0.785180i	$0.287430\pi$
$548$	2.85143	0.121807
$549$	0	0
$550$	0	0
$551$	−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	$-0.297503\pi$
$571$	28.3934	1.18823	0.594114	+	0.804381i	$0.297503\pi$
$572$	0	0
$573$	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
$583$	0	0
$584$	30.1611	1.24807
$585$	0	0
$586$	14.3803	0.594044
$587$	−35.1865	−1.45230	−0.726151	−	0.687535i	$-0.758693\pi$
$587$	−35.1865	−1.45230	−0.726151	+	0.687535i	$0.758693\pi$
$588$	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	$-0.102918\pi$
$599$	46.4126	1.89637	0.948184	+	0.317721i	$0.102918\pi$
$600$	0	0
$601$	−15.2945	−0.623874	−0.311937	−	0.950103i	$-0.600978\pi$
$601$	−15.2945	−0.623874	−0.311937	+	0.950103i	$0.600978\pi$
$602$	−5.32851	−0.217174
$603$	0	0
$604$	−3.00662	−0.122338
$605$	0	0
$606$	0	0
$607$	34.4948	1.40010	0.700051	−	0.714093i	$-0.253160\pi$
$607$	34.4948	1.40010	0.700051	+	0.714093i	$0.253160\pi$
$608$	−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	$-0.629937\pi$
$613$	−19.6568	−0.793931	−0.396965	+	0.917834i	$0.629937\pi$
$614$	18.0088	0.726776
$615$	0	0
$616$	0	0
$617$	−36.0533	−1.45145	−0.725726	−	0.687984i	$-0.758496\pi$
$617$	−36.0533	−1.45145	−0.725726	+	0.687984i	$0.758496\pi$
$618$	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
$638$	0	0
$639$	0	0
$640$	−24.0346	−0.950052
$641$	31.0472	1.22629	0.613145	−	0.789970i	$-0.289904\pi$
$641$	31.0472	1.22629	0.613145	+	0.789970i	$0.289904\pi$
$642$	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	$-0.722501\pi$
$647$	−32.7343	−1.28692	−0.643458	+	0.765481i	$0.722501\pi$
$648$	0	0
$649$	0	0
$650$	−46.6207	−1.82861
$651$	0	0
$652$	−21.2381	−0.831749
$653$	−7.23261	−0.283034	−0.141517	−	0.989936i	$-0.545198\pi$
$653$	−7.23261	−0.283034	−0.141517	+	0.989936i	$0.545198\pi$
$654$	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
$671$	0	0
$672$	0	0
$673$	−15.5642	−0.599958	−0.299979	−	0.953946i	$-0.596980\pi$
$673$	−15.5642	−0.599958	−0.299979	+	0.953946i	$0.596980\pi$
$674$	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
$682$	0	0
$683$	−48.8978	−1.87102	−0.935512	−	0.353295i	$-0.885061\pi$
$683$	−48.8978	−1.87102	−0.935512	+	0.353295i	$0.885061\pi$
$684$	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
$704$	0	0
$705$	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
$715$	0	0
$716$	14.2303	0.531812
$717$	0	0
$718$	−13.2194	−0.493344
$719$	19.8844	0.741563	0.370782	−	0.928720i	$-0.379090\pi$
$719$	19.8844	0.741563	0.370782	+	0.928720i	$0.379090\pi$
$720$	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	$-0.560328\pi$
$733$	−10.2012	−0.376789	−0.188394	+	0.982093i	$0.560328\pi$
$734$	23.7802	0.877744
$735$	0	0
$736$	−9.21481	−0.339662
$737$	0	0
$738$	0	0
$739$	−29.3420	−1.07936	−0.539681	−	0.841870i	$-0.681455\pi$
$739$	−29.3420	−1.07936	−0.539681	+	0.841870i	$0.681455\pi$
$740$	−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
$748$	0	0
$749$	−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	$-0.289787\pi$
$769$	34.0222	1.22687	0.613435	+	0.789745i	$0.289787\pi$
$770$	0	0
$771$	0	0
$772$	17.2982	0.622577
$773$	5.09752	0.183345	0.0916725	−	0.995789i	$-0.470779\pi$
$773$	5.09752	0.183345	0.0916725	+	0.995789i	$0.470779\pi$
$774$	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
$781$	0	0
$782$	−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	$-0.352320\pi$
$787$	25.1070	0.894969	0.447485	+	0.894292i	$0.352320\pi$
$788$	−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	$-0.538404\pi$
$797$	−6.79568	−0.240715	−0.120358	+	0.992731i	$0.538404\pi$
$798$	0	0
$799$	8.61723	0.304856
$800$	50.5349	1.78668
$801$	0	0
$802$	28.3316	1.00042
$803$	0	0
$804$	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	$-0.512777\pi$
$811$	−2.28561	−0.0802587	−0.0401294	+	0.999194i	$0.512777\pi$
$812$	7.49705	0.263095
$813$	0	0
$814$	0	0
$815$	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	0
$837$	0	0
$838$	24.4724	0.845386
$839$	−20.2763	−0.700017	−0.350008	−	0.936747i	$-0.613821\pi$
$839$	−20.2763	−0.700017	−0.350008	+	0.936747i	$0.613821\pi$
$840$	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
$847$	0	0
$848$	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	$-0.458858\pi$
$853$	7.52894	0.257786	0.128893	+	0.991659i	$0.458858\pi$
$854$	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	$-0.331066\pi$
$863$	29.7385	1.01231	0.506156	+	0.862442i	$0.331066\pi$
$864$	0	0
$865$	92.7143	3.15238
$866$	−1.77448	−0.0602993
$867$	0	0
$868$	−5.75580	−0.195365
$869$	0	0
$870$	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	$-0.511409\pi$
$877$	−2.12246	−0.0716705	−0.0358352	+	0.999358i	$0.511409\pi$
$878$	−31.3604	−1.05836
$879$	0	0
$880$	0	0
$881$	14.0627	0.473783	0.236892	−	0.971536i	$-0.423871\pi$
$881$	14.0627	0.473783	0.236892	+	0.971536i	$0.423871\pi$
$882$	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
$902$	0	0
$903$	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	$-0.470699\pi$
$911$	5.54894	0.183845	0.0919223	+	0.995766i	$0.470699\pi$
$912$	0	0
$913$	0	0
$914$	−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	$-0.218474\pi$
$919$	46.9009	1.54712	0.773560	+	0.633724i	$0.218474\pi$
$920$	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	$-0.669875\pi$
$929$	−31.0100	−1.01740	−0.508702	+	0.860943i	$0.669875\pi$
$930$	0	0
$931$	3.43651	0.112627
$932$	32.6019	1.06791
$933$	0	0
$934$	27.8950	0.912752
$935$	0	0
$936$	0	0
$937$	44.0640	1.43951	0.719755	−	0.694229i	$-0.244254\pi$
$937$	44.0640	1.43951	0.719755	+	0.694229i	$0.244254\pi$
$938$	−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
$946$	0	0
$947$	−15.3247	−0.497985	−0.248993	−	0.968505i	$-0.580100\pi$
$947$	−15.3247	−0.497985	−0.248993	+	0.968505i	$0.580100\pi$
$948$	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	$-0.524171\pi$
$967$	−4.71821	−0.151727	−0.0758637	+	0.997118i	$0.524171\pi$
$968$	0	0
$969$	0	0
$970$	11.0353	0.354322
$971$	−2.48946	−0.0798905	−0.0399452	−	0.999202i	$-0.512718\pi$
$971$	−2.48946	−0.0798905	−0.0399452	+	0.999202i	$0.512718\pi$
$972$	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	$-0.221415\pi$
$977$	47.9903	1.53534	0.767672	+	0.640842i	$0.221415\pi$
$978$	0	0
$979$	0	0
$980$	22.0185	0.703354
$981$	0	0
$982$	6.26308	0.199863
$983$	−0.497137	−0.0158562	−0.00792811	−	0.999969i	$-0.502524\pi$
$983$	−0.497137	−0.0158562	−0.00792811	+	0.999969i	$0.502524\pi$
$984$	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	$-0.800856\pi$
$997$	−51.1895	−1.62119	−0.810594	+	0.585609i	$0.800856\pi$
$998$	22.0662	0.698492
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
99.2.e.e.34.3	✓	8	99.76	odd	6
99.2.e.e.67.3	yes	8	99.43	odd	6
297.2.e.e.100.2		8	99.32	even	6
297.2.e.e.199.2		8	99.65	even	6
891.2.a.p.1.3		4	33.32	even	2
891.2.a.q.1.2		4	11.10	odd	2
1089.2.e.i.364.2		8	9.7	even	3
1089.2.e.i.727.2		8	9.4	even	3
9801.2.a.bi.1.3		4	1.1	even	1	trivial
9801.2.a.bl.1.2		4	3.2	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:	\( N \)	\(=\)	\( 9801 = 3^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9801.a (trivial)

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

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