LMFDB - Embedded newform 6633.2.a.w.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6633,2,Mod(1,6633)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6633, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("6633.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6633 = 3^{2} \cdot 11 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6633.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:	$52.9647716607$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - x^{16} - 26 x^{15} + 25 x^{14} + 272 x^{13} - 244 x^{12} - 1472 x^{11} + 1186 x^{10} + 4406 x^{9} + \cdots - 2 $ x^17 - x^16 - 26x^15 + 25x^14 + 272x^13 - 244x^12 - 1472x^11 + 1186x^10 + 4406x^9 - 3042x^8 - 7180x^7 + 4055x^6 + 5665x^5 - 2655x^4 - 1418x^3 + 810x^2 - 76*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 737)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.166502$ of defining polynomial
Character	$\chi$	$=$	6633.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.166502 q^{2} -1.97228 q^{4} +1.68184 q^{5} +1.74824 q^{7} +0.661391 q^{8} +O(q^{10})$	\(q-0.166502 q^{2} -1.97228 q^{4} +1.68184 q^{5} +1.74824 q^{7} +0.661391 q^{8} -0.280030 q^{10} -1.00000 q^{11} +4.74581 q^{13} -0.291085 q^{14} +3.83443 q^{16} +3.42532 q^{17} +8.33187 q^{19} -3.31706 q^{20} +0.166502 q^{22} +7.75315 q^{23} -2.17140 q^{25} -0.790186 q^{26} -3.44801 q^{28} +1.32256 q^{29} +4.05053 q^{31} -1.96122 q^{32} -0.570321 q^{34} +2.94026 q^{35} +11.3336 q^{37} -1.38727 q^{38} +1.11236 q^{40} +1.18117 q^{41} -4.11229 q^{43} +1.97228 q^{44} -1.29091 q^{46} -10.6028 q^{47} -3.94366 q^{49} +0.361542 q^{50} -9.36006 q^{52} -3.63284 q^{53} -1.68184 q^{55} +1.15627 q^{56} -0.220208 q^{58} +7.01431 q^{59} -2.79616 q^{61} -0.674420 q^{62} -7.34232 q^{64} +7.98172 q^{65} -1.00000 q^{67} -6.75568 q^{68} -0.489559 q^{70} -2.79463 q^{71} -9.48187 q^{73} -1.88707 q^{74} -16.4328 q^{76} -1.74824 q^{77} -11.6547 q^{79} +6.44892 q^{80} -0.196666 q^{82} +12.8719 q^{83} +5.76085 q^{85} +0.684703 q^{86} -0.661391 q^{88} +9.71865 q^{89} +8.29681 q^{91} -15.2914 q^{92} +1.76538 q^{94} +14.0129 q^{95} -14.0983 q^{97} +0.656627 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q - q^{2} + 19 q^{4} - 10 q^{5} + 20 q^{7}+O(q^{10}) $ 17 * q - q^2 + 19 * q^4 - 10 * q^5 + 20 * q^7	$ 17 q - q^{2} + 19 q^{4} - 10 q^{5} + 20 q^{7} + 10 q^{10} - 17 q^{11} + q^{13} + 11 q^{14} + 19 q^{16} - 2 q^{17} + 13 q^{19} - 3 q^{20} + q^{22} - 16 q^{23} + 33 q^{25} - 12 q^{26} + 44 q^{28} + 5 q^{29} + 16 q^{31} + 24 q^{32} + 4 q^{34} + 2 q^{35} + 29 q^{37} + 19 q^{38} + 31 q^{40} + 6 q^{41} + 19 q^{43} - 19 q^{44} - 33 q^{46} - 40 q^{47} + 23 q^{49} + 3 q^{50} - 28 q^{52} - 15 q^{53} + 10 q^{55} + 38 q^{56} - 12 q^{58} + 2 q^{59} - 6 q^{61} - 3 q^{62} - 4 q^{64} + 30 q^{65} - 17 q^{67} + 13 q^{68} + 71 q^{70} - 2 q^{71} + 41 q^{73} + 13 q^{74} + 21 q^{76} - 20 q^{77} + 41 q^{79} + 23 q^{80} - 8 q^{82} - 2 q^{83} - 36 q^{85} + 54 q^{86} - q^{89} + 16 q^{91} - 36 q^{92} + 12 q^{94} + 31 q^{95} + 3 q^{97} + 64 q^{98}+O(q^{100}) $ 17 * q - q^2 + 19 * q^4 - 10 * q^5 + 20 * q^7 + 10 * q^10 - 17 * q^11 + q^13 + 11 * q^14 + 19 * q^16 - 2 * q^17 + 13 * q^19 - 3 * q^20 + q^22 - 16 * q^23 + 33 * q^25 - 12 * q^26 + 44 * q^28 + 5 * q^29 + 16 * q^31 + 24 * q^32 + 4 * q^34 + 2 * q^35 + 29 * q^37 + 19 * q^38 + 31 * q^40 + 6 * q^41 + 19 * q^43 - 19 * q^44 - 33 * q^46 - 40 * q^47 + 23 * q^49 + 3 * q^50 - 28 * q^52 - 15 * q^53 + 10 * q^55 + 38 * q^56 - 12 * q^58 + 2 * q^59 - 6 * q^61 - 3 * q^62 - 4 * q^64 + 30 * q^65 - 17 * q^67 + 13 * q^68 + 71 * q^70 - 2 * q^71 + 41 * q^73 + 13 * q^74 + 21 * q^76 - 20 * q^77 + 41 * q^79 + 23 * q^80 - 8 * q^82 - 2 * q^83 - 36 * q^85 + 54 * q^86 - q^89 + 16 * q^91 - 36 * q^92 + 12 * q^94 + 31 * q^95 + 3 * q^97 + 64 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.166502	−0.117734	−0.0588672	−	0.998266i	$-0.518749\pi$
$2$	−0.166502	−0.117734	−0.0588672	+	0.998266i	$0.518749\pi$
$3$	0	0
$3$	0	0
$4$	−1.97228	−0.986139
$5$	1.68184	0.752143	0.376072	−	0.926591i	$-0.377275\pi$
$5$	1.68184	0.752143	0.376072	+	0.926591i	$0.377275\pi$
$6$	0	0
$7$	1.74824	0.660772	0.330386	−	0.943846i	$-0.392821\pi$
$7$	1.74824	0.660772	0.330386	+	0.943846i	$0.392821\pi$
$8$	0.661391	0.233837
$9$	0	0
$10$	−0.280030	−0.0885532
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.74581	1.31625	0.658126	−	0.752908i	$-0.271349\pi$
$13$	4.74581	1.31625	0.658126	+	0.752908i	$0.271349\pi$
$14$	−0.291085	−0.0777956
$15$	0	0
$16$	3.83443	0.958608
$17$	3.42532	0.830762	0.415381	−	0.909647i	$-0.363648\pi$
$17$	3.42532	0.830762	0.415381	+	0.909647i	$0.363648\pi$
$18$	0	0
$19$	8.33187	1.91146	0.955731	−	0.294242i	$-0.0950672\pi$
$19$	8.33187	1.91146	0.955731	+	0.294242i	$0.0950672\pi$
$20$	−3.31706	−0.741718
$21$	0	0
$22$	0.166502	0.0354983
$23$	7.75315	1.61664	0.808322	−	0.588741i	$-0.200376\pi$
$23$	7.75315	1.61664	0.808322	+	0.588741i	$0.200376\pi$
$24$	0	0
$25$	−2.17140	−0.434280
$26$	−0.790186	−0.154968
$27$	0	0
$28$	−3.44801	−0.651613
$29$	1.32256	0.245593	0.122796	−	0.992432i	$-0.460814\pi$
$29$	1.32256	0.245593	0.122796	+	0.992432i	$0.460814\pi$
$30$	0	0
$31$	4.05053	0.727497	0.363749	−	0.931497i	$-0.381497\pi$
$31$	4.05053	0.727497	0.363749	+	0.931497i	$0.381497\pi$
$32$	−1.96122	−0.346698
$33$	0	0
$34$	−0.570321	−0.0978093
$35$	2.94026	0.496995
$36$	0	0
$37$	11.3336	1.86324	0.931619	−	0.363436i	$-0.118396\pi$
$37$	11.3336	1.86324	0.931619	+	0.363436i	$0.118396\pi$
$38$	−1.38727	−0.225045
$39$	0	0
$40$	1.11236	0.175879
$41$	1.18117	0.184467	0.0922336	−	0.995737i	$-0.470599\pi$
$41$	1.18117	0.184467	0.0922336	+	0.995737i	$0.470599\pi$
$42$	0	0
$43$	−4.11229	−0.627118	−0.313559	−	0.949569i	$-0.601521\pi$
$43$	−4.11229	−0.627118	−0.313559	+	0.949569i	$0.601521\pi$
$44$	1.97228	0.297332
$45$	0	0
$46$	−1.29091	−0.190335
$47$	−10.6028	−1.54658	−0.773288	−	0.634055i	$-0.781389\pi$
$47$	−10.6028	−1.54658	−0.773288	+	0.634055i	$0.781389\pi$
$48$	0	0
$49$	−3.94366	−0.563381
$50$	0.361542	0.0511297
$51$	0	0
$52$	−9.36006	−1.29801
$53$	−3.63284	−0.499010	−0.249505	−	0.968374i	$-0.580268\pi$
$53$	−3.63284	−0.499010	−0.249505	+	0.968374i	$0.580268\pi$
$54$	0	0
$55$	−1.68184	−0.226780
$56$	1.15627	0.154513
$57$	0	0
$58$	−0.220208	−0.0289147
$59$	7.01431	0.913185	0.456592	−	0.889676i	$-0.349070\pi$
$59$	7.01431	0.913185	0.456592	+	0.889676i	$0.349070\pi$
$60$	0	0
$61$	−2.79616	−0.358011	−0.179006	−	0.983848i	$-0.557288\pi$
$61$	−2.79616	−0.358011	−0.179006	+	0.983848i	$0.557288\pi$
$62$	−0.674420	−0.0856515
$63$	0	0
$64$	−7.34232	−0.917790
$65$	7.98172	0.990010
$66$	0	0
$67$	−1.00000	−0.122169
$67$	−1.00000	−0.122169
$68$	−6.75568	−0.819247
$69$	0	0
$70$	−0.489559	−0.0585135
$71$	−2.79463	−0.331662	−0.165831	−	0.986154i	$-0.553031\pi$
$71$	−2.79463	−0.331662	−0.165831	+	0.986154i	$0.553031\pi$
$72$	0	0
$73$	−9.48187	−1.10977	−0.554884	−	0.831927i	$-0.687238\pi$
$73$	−9.48187	−1.10977	−0.554884	+	0.831927i	$0.687238\pi$
$74$	−1.88707	−0.219367
$75$	0	0
$76$	−16.4328	−1.88497
$77$	−1.74824	−0.199230
$78$	0	0
$79$	−11.6547	−1.31126	−0.655629	−	0.755083i	$-0.727597\pi$
$79$	−11.6547	−1.31126	−0.655629	+	0.755083i	$0.727597\pi$
$80$	6.44892	0.721011
$81$	0	0
$82$	−0.196666	−0.0217181
$83$	12.8719	1.41287	0.706435	−	0.707778i	$-0.250302\pi$
$83$	12.8719	1.41287	0.706435	+	0.707778i	$0.250302\pi$
$84$	0	0
$85$	5.76085	0.624852
$86$	0.684703	0.0738334
$87$	0	0
$88$	−0.661391	−0.0705045
$89$	9.71865	1.03018	0.515088	−	0.857137i	$-0.327759\pi$
$89$	9.71865	1.03018	0.515088	+	0.857137i	$0.327759\pi$
$90$	0	0
$91$	8.29681	0.869742
$92$	−15.2914	−1.59423
$93$	0	0
$94$	1.76538	0.182085
$95$	14.0129	1.43769
$96$	0	0
$97$	−14.0983	−1.43147	−0.715734	−	0.698373i	$-0.753908\pi$
$97$	−14.0983	−1.43147	−0.715734	+	0.698373i	$0.753908\pi$
$98$	0.656627	0.0663293
$99$	0	0
$100$	4.28260	0.428260
$101$	5.15458	0.512900	0.256450	−	0.966558i	$-0.417447\pi$
$101$	5.15458	0.512900	0.256450	+	0.966558i	$0.417447\pi$
$102$	0	0
$103$	1.87725	0.184971	0.0924854	−	0.995714i	$-0.470519\pi$
$103$	1.87725	0.184971	0.0924854	+	0.995714i	$0.470519\pi$
$104$	3.13884	0.307788
$105$	0	0
$106$	0.604875	0.0587506
$107$	8.61045	0.832404	0.416202	−	0.909272i	$-0.363361\pi$
$107$	8.61045	0.832404	0.416202	+	0.909272i	$0.363361\pi$
$108$	0	0
$109$	5.60087	0.536466	0.268233	−	0.963354i	$-0.413560\pi$
$109$	5.60087	0.536466	0.268233	+	0.963354i	$0.413560\pi$
$110$	0.280030	0.0266998
$111$	0	0
$112$	6.70350	0.633421
$113$	6.82662	0.642194	0.321097	−	0.947046i	$-0.395948\pi$
$113$	6.82662	0.642194	0.321097	+	0.947046i	$0.395948\pi$
$114$	0	0
$115$	13.0396	1.21595
$116$	−2.60845	−0.242188
$117$	0	0
$118$	−1.16789	−0.107513
$119$	5.98828	0.548944
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.465565	0.0421503
$123$	0	0
$124$	−7.98877	−0.717413
$125$	−12.0612	−1.07878
$126$	0	0
$127$	0.938271	0.0832581	0.0416291	−	0.999133i	$-0.486745\pi$
$127$	0.938271	0.0832581	0.0416291	+	0.999133i	$0.486745\pi$
$128$	5.14495	0.454754
$129$	0	0
$130$	−1.32897	−0.116558
$131$	−18.9446	−1.65520	−0.827599	−	0.561320i	$-0.810294\pi$
$131$	−18.9446	−1.65520	−0.827599	+	0.561320i	$0.810294\pi$
$132$	0	0
$133$	14.5661	1.26304
$134$	0.166502	0.0143836
$135$	0	0
$136$	2.26548	0.194263
$137$	−15.5849	−1.33151	−0.665754	−	0.746171i	$-0.731890\pi$
$137$	−15.5849	−1.33151	−0.665754	+	0.746171i	$0.731890\pi$
$138$	0	0
$139$	−16.4958	−1.39916	−0.699578	−	0.714556i	$-0.746629\pi$
$139$	−16.4958	−1.39916	−0.699578	+	0.714556i	$0.746629\pi$
$140$	−5.79901	−0.490106
$141$	0	0
$142$	0.465310	0.0390480
$143$	−4.74581	−0.396865
$144$	0	0
$145$	2.22433	0.184721
$146$	1.57875	0.130658
$147$	0	0
$148$	−22.3531	−1.83741
$149$	−7.69214	−0.630165	−0.315082	−	0.949064i	$-0.602032\pi$
$149$	−7.69214	−0.630165	−0.315082	+	0.949064i	$0.602032\pi$
$150$	0	0
$151$	−19.3189	−1.57215	−0.786077	−	0.618129i	$-0.787891\pi$
$151$	−19.3189	−1.57215	−0.786077	+	0.618129i	$0.787891\pi$
$152$	5.51062	0.446970
$153$	0	0
$154$	0.291085	0.0234563
$155$	6.81236	0.547182
$156$	0	0
$157$	13.9805	1.11577	0.557884	−	0.829919i	$-0.311614\pi$
$157$	13.9805	1.11577	0.557884	+	0.829919i	$0.311614\pi$
$158$	1.94053	0.154380
$159$	0	0
$160$	−3.29847	−0.260767
$161$	13.5544	1.06823
$162$	0	0
$163$	8.14837	0.638230	0.319115	−	0.947716i	$-0.396614\pi$
$163$	8.14837	0.638230	0.319115	+	0.947716i	$0.396614\pi$
$164$	−2.32959	−0.181910
$165$	0	0
$166$	−2.14319	−0.166343
$167$	18.4389	1.42685	0.713423	−	0.700733i	$-0.247144\pi$
$167$	18.4389	1.42685	0.713423	+	0.700733i	$0.247144\pi$
$168$	0	0
$169$	9.52276	0.732520
$170$	−0.959192	−0.0735667
$171$	0	0
$172$	8.11058	0.618426
$173$	17.6556	1.34233	0.671165	−	0.741308i	$-0.265794\pi$
$173$	17.6556	1.34233	0.671165	+	0.741308i	$0.265794\pi$
$174$	0	0
$175$	−3.79613	−0.286960
$176$	−3.83443	−0.289031
$177$	0	0
$178$	−1.61817	−0.121287
$179$	−8.28480	−0.619235	−0.309617	−	0.950861i	$-0.600201\pi$
$179$	−8.28480	−0.619235	−0.309617	+	0.950861i	$0.600201\pi$
$180$	0	0
$181$	16.8112	1.24956	0.624782	−	0.780799i	$-0.285188\pi$
$181$	16.8112	1.24956	0.624782	+	0.780799i	$0.285188\pi$
$182$	−1.38143	−0.102399
$183$	0	0
$184$	5.12786	0.378031
$185$	19.0614	1.40142
$186$	0	0
$187$	−3.42532	−0.250484
$188$	20.9116	1.52514
$189$	0	0
$190$	−2.33317	−0.169266
$191$	10.5242	0.761506	0.380753	−	0.924677i	$-0.375665\pi$
$191$	10.5242	0.761506	0.380753	+	0.924677i	$0.375665\pi$
$192$	0	0
$193$	6.43495	0.463198	0.231599	−	0.972811i	$-0.425604\pi$
$193$	6.43495	0.463198	0.231599	+	0.972811i	$0.425604\pi$
$194$	2.34739	0.168533
$195$	0	0
$196$	7.77800	0.555571
$197$	−26.3515	−1.87747	−0.938735	−	0.344640i	$-0.888001\pi$
$197$	−26.3515	−1.87747	−0.938735	+	0.344640i	$0.888001\pi$
$198$	0	0
$199$	−7.52414	−0.533372	−0.266686	−	0.963783i	$-0.585929\pi$
$199$	−7.52414	−0.533372	−0.266686	+	0.963783i	$0.585929\pi$
$200$	−1.43614	−0.101551
$201$	0	0
$202$	−0.858246	−0.0603860
$203$	2.31214	0.162281
$204$	0	0
$205$	1.98654	0.138746
$206$	−0.312565	−0.0217774
$207$	0	0
$208$	18.1975	1.26177
$209$	−8.33187	−0.576327
$210$	0	0
$211$	−7.49325	−0.515857	−0.257928	−	0.966164i	$-0.583040\pi$
$211$	−7.49325	−0.515857	−0.257928	+	0.966164i	$0.583040\pi$
$212$	7.16498	0.492093
$213$	0	0
$214$	−1.43365	−0.0980026
$215$	−6.91623	−0.471683
$216$	0	0
$217$	7.08130	0.480710
$218$	−0.932554	−0.0631605
$219$	0	0
$220$	3.31706	0.223636
$221$	16.2559	1.09349
$222$	0	0
$223$	−15.1318	−1.01330	−0.506649	−	0.862153i	$-0.669116\pi$
$223$	−15.1318	−1.01330	−0.506649	+	0.862153i	$0.669116\pi$
$224$	−3.42868	−0.229088
$225$	0	0
$226$	−1.13664	−0.0756084
$227$	9.44344	0.626783	0.313392	−	0.949624i	$-0.398535\pi$
$227$	9.44344	0.626783	0.313392	+	0.949624i	$0.398535\pi$
$228$	0	0
$229$	11.9753	0.791349	0.395674	−	0.918391i	$-0.370511\pi$
$229$	11.9753	0.791349	0.395674	+	0.918391i	$0.370511\pi$
$230$	−2.17111	−0.143159
$231$	0	0
$232$	0.874726	0.0574286
$233$	−14.1725	−0.928471	−0.464236	−	0.885712i	$-0.653671\pi$
$233$	−14.1725	−0.928471	−0.464236	+	0.885712i	$0.653671\pi$
$234$	0	0
$235$	−17.8322	−1.16325
$236$	−13.8342	−0.900527
$237$	0	0
$238$	−0.997058	−0.0646297
$239$	13.2998	0.860292	0.430146	−	0.902759i	$-0.358462\pi$
$239$	13.2998	0.860292	0.430146	+	0.902759i	$0.358462\pi$
$240$	0	0
$241$	−3.18335	−0.205058	−0.102529	−	0.994730i	$-0.532693\pi$
$241$	−3.18335	−0.205058	−0.102529	+	0.994730i	$0.532693\pi$
$242$	−0.166502	−0.0107031
$243$	0	0
$244$	5.51480	0.353049
$245$	−6.63263	−0.423743
$246$	0	0
$247$	39.5415	2.51597
$248$	2.67898	0.170116
$249$	0	0
$250$	2.00821	0.127010
$251$	−8.31279	−0.524698	−0.262349	−	0.964973i	$-0.584497\pi$
$251$	−8.31279	−0.524698	−0.262349	+	0.964973i	$0.584497\pi$
$252$	0	0
$253$	−7.75315	−0.487436
$254$	−0.156224	−0.00980235
$255$	0	0
$256$	13.8280	0.864249
$257$	0.449194	0.0280200	0.0140100	−	0.999902i	$-0.495540\pi$
$257$	0.449194	0.0280200	0.0140100	+	0.999902i	$0.495540\pi$
$258$	0	0
$259$	19.8139	1.23118
$260$	−15.7422	−0.976288
$261$	0	0
$262$	3.15431	0.194874
$263$	−19.5186	−1.20357	−0.601783	−	0.798659i	$-0.705543\pi$
$263$	−19.5186	−1.20357	−0.601783	+	0.798659i	$0.705543\pi$
$264$	0	0
$265$	−6.10988	−0.375327
$266$	−2.42528	−0.148703
$267$	0	0
$268$	1.97228	0.120476
$269$	−24.2855	−1.48071	−0.740357	−	0.672214i	$-0.765344\pi$
$269$	−24.2855	−1.48071	−0.740357	+	0.672214i	$0.765344\pi$
$270$	0	0
$271$	−0.702750	−0.0426890	−0.0213445	−	0.999772i	$-0.506795\pi$
$271$	−0.702750	−0.0426890	−0.0213445	+	0.999772i	$0.506795\pi$
$272$	13.1342	0.796375
$273$	0	0
$274$	2.59491	0.156764
$275$	2.17140	0.130940
$276$	0	0
$277$	−3.02879	−0.181982	−0.0909911	−	0.995852i	$-0.529003\pi$
$277$	−3.02879	−0.181982	−0.0909911	+	0.995852i	$0.529003\pi$
$278$	2.74658	0.164729
$279$	0	0
$280$	1.94466	0.116216
$281$	5.59512	0.333777	0.166888	−	0.985976i	$-0.446628\pi$
$281$	5.59512	0.333777	0.166888	+	0.985976i	$0.446628\pi$
$282$	0	0
$283$	31.3953	1.86626	0.933130	−	0.359540i	$-0.117066\pi$
$283$	31.3953	1.86626	0.933130	+	0.359540i	$0.117066\pi$
$284$	5.51178	0.327064
$285$	0	0
$286$	0.790186	0.0467247
$287$	2.06496	0.121891
$288$	0	0
$289$	−5.26718	−0.309834
$290$	−0.370355	−0.0217480
$291$	0	0
$292$	18.7009	1.09439
$293$	−14.7151	−0.859663	−0.429832	−	0.902909i	$-0.641427\pi$
$293$	−14.7151	−0.859663	−0.429832	+	0.902909i	$0.641427\pi$
$294$	0	0
$295$	11.7970	0.686846
$296$	7.49596	0.435694
$297$	0	0
$298$	1.28075	0.0741921
$299$	36.7950	2.12791
$300$	0	0
$301$	−7.18926	−0.414382
$302$	3.21664	0.185097
$303$	0	0
$304$	31.9480	1.83234
$305$	−4.70270	−0.269276
$306$	0	0
$307$	−0.138268	−0.00789138	−0.00394569	−	0.999992i	$-0.501256\pi$
$307$	−0.138268	−0.00789138	−0.00394569	+	0.999992i	$0.501256\pi$
$308$	3.44801	0.196469
$309$	0	0
$310$	−1.13427	−0.0644222
$311$	−2.61861	−0.148488	−0.0742439	−	0.997240i	$-0.523654\pi$
$311$	−2.61861	−0.148488	−0.0742439	+	0.997240i	$0.523654\pi$
$312$	0	0
$313$	−13.2199	−0.747232	−0.373616	−	0.927583i	$-0.621882\pi$
$313$	−13.2199	−0.747232	−0.373616	+	0.927583i	$0.621882\pi$
$314$	−2.32778	−0.131364
$315$	0	0
$316$	22.9863	1.29308
$317$	24.8110	1.39352	0.696761	−	0.717303i	$-0.254624\pi$
$317$	24.8110	1.39352	0.696761	+	0.717303i	$0.254624\pi$
$318$	0	0
$319$	−1.32256	−0.0740489
$320$	−12.3486	−0.690309
$321$	0	0
$322$	−2.25682	−0.125768
$323$	28.5393	1.58797
$324$	0	0
$325$	−10.3051	−0.571622
$326$	−1.35672	−0.0751416
$327$	0	0
$328$	0.781212	0.0431352
$329$	−18.5362	−1.02193
$330$	0	0
$331$	2.52664	0.138877	0.0694384	−	0.997586i	$-0.477879\pi$
$331$	2.52664	0.138877	0.0694384	+	0.997586i	$0.477879\pi$
$332$	−25.3869	−1.39329
$333$	0	0
$334$	−3.07011	−0.167989
$335$	−1.68184	−0.0918890
$336$	0	0
$337$	−30.4818	−1.66045	−0.830226	−	0.557427i	$-0.811789\pi$
$337$	−30.4818	−1.66045	−0.830226	+	0.557427i	$0.811789\pi$
$338$	−1.58555	−0.0862428
$339$	0	0
$340$	−11.3620	−0.616191
$341$	−4.05053	−0.219349
$342$	0	0
$343$	−19.1321	−1.03304
$344$	−2.71983	−0.146643
$345$	0	0
$346$	−2.93969	−0.158039
$347$	−23.5029	−1.26170	−0.630852	−	0.775903i	$-0.717294\pi$
$347$	−23.5029	−1.26170	−0.630852	+	0.775903i	$0.717294\pi$
$348$	0	0
$349$	9.89027	0.529414	0.264707	−	0.964329i	$-0.414725\pi$
$349$	9.89027	0.529414	0.264707	+	0.964329i	$0.414725\pi$
$350$	0.632061	0.0337851
$351$	0	0
$352$	1.96122	0.104533
$353$	−4.12703	−0.219660	−0.109830	−	0.993950i	$-0.535031\pi$
$353$	−4.12703	−0.219660	−0.109830	+	0.993950i	$0.535031\pi$
$354$	0	0
$355$	−4.70013	−0.249457
$356$	−19.1679	−1.01590
$357$	0	0
$358$	1.37943	0.0729053
$359$	8.44933	0.445939	0.222969	−	0.974825i	$-0.428425\pi$
$359$	8.44933	0.445939	0.222969	+	0.974825i	$0.428425\pi$
$360$	0	0
$361$	50.4200	2.65369
$362$	−2.79909	−0.147117
$363$	0	0
$364$	−16.3636	−0.857687
$365$	−15.9470	−0.834705
$366$	0	0
$367$	17.3072	0.903430	0.451715	−	0.892162i	$-0.350812\pi$
$367$	17.3072	0.903430	0.451715	+	0.892162i	$0.350812\pi$
$368$	29.7289	1.54973
$369$	0	0
$370$	−3.17376	−0.164996
$371$	−6.35108	−0.329731
$372$	0	0
$373$	−15.1856	−0.786281	−0.393140	−	0.919478i	$-0.628611\pi$
$373$	−15.1856	−0.786281	−0.393140	+	0.919478i	$0.628611\pi$
$374$	0.570321	0.0294906
$375$	0	0
$376$	−7.01259	−0.361647
$377$	6.27661	0.323262
$378$	0	0
$379$	24.5355	1.26031	0.630153	−	0.776471i	$-0.282992\pi$
$379$	24.5355	1.26031	0.630153	+	0.776471i	$0.282992\pi$
$380$	−27.6373	−1.41777
$381$	0	0
$382$	−1.75230	−0.0896555
$383$	−18.2429	−0.932170	−0.466085	−	0.884740i	$-0.654336\pi$
$383$	−18.2429	−0.932170	−0.466085	+	0.884740i	$0.654336\pi$
$384$	0	0
$385$	−2.94026	−0.149850
$386$	−1.07143	−0.0545343
$387$	0	0
$388$	27.8058	1.41163
$389$	−0.274138	−0.0138994	−0.00694968	−	0.999976i	$-0.502212\pi$
$389$	−0.274138	−0.0138994	−0.00694968	+	0.999976i	$0.502212\pi$
$390$	0	0
$391$	26.5570	1.34305
$392$	−2.60830	−0.131739
$393$	0	0
$394$	4.38758	0.221043
$395$	−19.6014	−0.986254
$396$	0	0
$397$	−5.24029	−0.263003	−0.131501	−	0.991316i	$-0.541980\pi$
$397$	−5.24029	−0.263003	−0.131501	+	0.991316i	$0.541980\pi$
$398$	1.25278	0.0627963
$399$	0	0
$400$	−8.32609	−0.416304
$401$	9.41635	0.470230	0.235115	−	0.971968i	$-0.424453\pi$
$401$	9.41635	0.470230	0.235115	+	0.971968i	$0.424453\pi$
$402$	0	0
$403$	19.2231	0.957570
$404$	−10.1663	−0.505790
$405$	0	0
$406$	−0.384976	−0.0191060
$407$	−11.3336	−0.561787
$408$	0	0
$409$	−16.3114	−0.806545	−0.403272	−	0.915080i	$-0.632127\pi$
$409$	−16.3114	−0.806545	−0.403272	+	0.915080i	$0.632127\pi$
$410$	−0.330762	−0.0163352
$411$	0	0
$412$	−3.70245	−0.182407
$413$	12.2627	0.603407
$414$	0	0
$415$	21.6485	1.06268
$416$	−9.30759	−0.456342
$417$	0	0
$418$	1.38727	0.0678536
$419$	21.7536	1.06273	0.531366	−	0.847143i	$-0.321679\pi$
$419$	21.7536	1.06273	0.531366	+	0.847143i	$0.321679\pi$
$420$	0	0
$421$	24.5002	1.19407	0.597034	−	0.802216i	$-0.296346\pi$
$421$	24.5002	1.19407	0.597034	+	0.802216i	$0.296346\pi$
$422$	1.24764	0.0607341
$423$	0	0
$424$	−2.40273	−0.116687
$425$	−7.43774	−0.360784
$426$	0	0
$427$	−4.88835	−0.236564
$428$	−16.9822	−0.820866
$429$	0	0
$430$	1.15156	0.0555333
$431$	−32.1375	−1.54801	−0.774004	−	0.633181i	$-0.781749\pi$
$431$	−32.1375	−1.54801	−0.774004	+	0.633181i	$0.781749\pi$
$432$	0	0
$433$	−11.7910	−0.566640	−0.283320	−	0.959025i	$-0.591436\pi$
$433$	−11.7910	−0.566640	−0.283320	+	0.959025i	$0.591436\pi$
$434$	−1.17905	−0.0565961
$435$	0	0
$436$	−11.0465	−0.529030
$437$	64.5982	3.09015
$438$	0	0
$439$	−23.8119	−1.13648	−0.568240	−	0.822863i	$-0.692375\pi$
$439$	−23.8119	−1.13648	−0.568240	+	0.822863i	$0.692375\pi$
$440$	−1.11236	−0.0530295
$441$	0	0
$442$	−2.70664	−0.128742
$443$	4.60849	0.218956	0.109478	−	0.993989i	$-0.465082\pi$
$443$	4.60849	0.218956	0.109478	+	0.993989i	$0.465082\pi$
$444$	0	0
$445$	16.3453	0.774840
$446$	2.51946	0.119300
$447$	0	0
$448$	−12.8361	−0.606450
$449$	−14.8029	−0.698594	−0.349297	−	0.937012i	$-0.613580\pi$
$449$	−14.8029	−0.698594	−0.349297	+	0.937012i	$0.613580\pi$
$450$	0	0
$451$	−1.18117	−0.0556189
$452$	−13.4640	−0.633293
$453$	0	0
$454$	−1.57235	−0.0737940
$455$	13.9539	0.654171
$456$	0	0
$457$	−10.0147	−0.468468	−0.234234	−	0.972180i	$-0.575258\pi$
$457$	−10.0147	−0.468468	−0.234234	+	0.972180i	$0.575258\pi$
$458$	−1.99390	−0.0931690
$459$	0	0
$460$	−25.7177	−1.19909
$461$	22.7232	1.05832	0.529162	−	0.848521i	$-0.322507\pi$
$461$	22.7232	1.05832	0.529162	+	0.848521i	$0.322507\pi$
$462$	0	0
$463$	−28.2804	−1.31430	−0.657151	−	0.753759i	$-0.728239\pi$
$463$	−28.2804	−1.31430	−0.657151	+	0.753759i	$0.728239\pi$
$464$	5.07125	0.235427
$465$	0	0
$466$	2.35974	0.109313
$467$	22.5742	1.04461	0.522304	−	0.852759i	$-0.325073\pi$
$467$	22.5742	1.04461	0.522304	+	0.852759i	$0.325073\pi$
$468$	0	0
$469$	−1.74824	−0.0807261
$470$	2.96910	0.136954
$471$	0	0
$472$	4.63920	0.213536
$473$	4.11229	0.189083
$474$	0	0
$475$	−18.0918	−0.830110
$476$	−11.8105	−0.541335
$477$	0	0
$478$	−2.21444	−0.101286
$479$	40.3623	1.84420	0.922100	−	0.386951i	$-0.126472\pi$
$479$	40.3623	1.84420	0.922100	+	0.386951i	$0.126472\pi$
$480$	0	0
$481$	53.7873	2.45249
$482$	0.530033	0.0241423
$483$	0	0
$484$	−1.97228	−0.0896490
$485$	−23.7112	−1.07667
$486$	0	0
$487$	24.0629	1.09039	0.545196	−	0.838308i	$-0.316455\pi$
$487$	24.0629	1.09039	0.545196	+	0.838308i	$0.316455\pi$
$488$	−1.84935	−0.0837163
$489$	0	0
$490$	1.10434	0.0498891
$491$	5.57187	0.251455	0.125728	−	0.992065i	$-0.459873\pi$
$491$	5.57187	0.251455	0.125728	+	0.992065i	$0.459873\pi$
$492$	0	0
$493$	4.53018	0.204029
$494$	−6.58373	−0.296216
$495$	0	0
$496$	15.5315	0.697385
$497$	−4.88568	−0.219153
$498$	0	0
$499$	40.2507	1.80187	0.900935	−	0.433955i	$-0.142882\pi$
$499$	40.2507	1.80187	0.900935	+	0.433955i	$0.142882\pi$
$500$	23.7880	1.06383
$501$	0	0
$502$	1.38409	0.0617751
$503$	−23.6309	−1.05365	−0.526824	−	0.849974i	$-0.676617\pi$
$503$	−23.6309	−1.05365	−0.526824	+	0.849974i	$0.676617\pi$
$504$	0	0
$505$	8.66920	0.385774
$506$	1.29091	0.0573881
$507$	0	0
$508$	−1.85053	−0.0821040
$509$	23.0173	1.02022	0.510111	−	0.860108i	$-0.329604\pi$
$509$	23.0173	1.02022	0.510111	+	0.860108i	$0.329604\pi$
$510$	0	0
$511$	−16.5766	−0.733304
$512$	−12.5923	−0.556505
$513$	0	0
$514$	−0.0747916	−0.00329892
$515$	3.15724	0.139125
$516$	0	0
$517$	10.6028	0.466310
$518$	−3.29905	−0.144952
$519$	0	0
$520$	5.27904	0.231501
$521$	22.1493	0.970378	0.485189	−	0.874409i	$-0.338751\pi$
$521$	22.1493	0.970378	0.485189	+	0.874409i	$0.338751\pi$
$522$	0	0
$523$	4.37693	0.191390	0.0956950	−	0.995411i	$-0.469493\pi$
$523$	4.37693	0.191390	0.0956950	+	0.995411i	$0.469493\pi$
$524$	37.3640	1.63225
$525$	0	0
$526$	3.24987	0.141701
$527$	13.8744	0.604377
$528$	0	0
$529$	37.1114	1.61354
$530$	1.01730	0.0441889
$531$	0	0
$532$	−28.7284	−1.24553
$533$	5.60559	0.242805
$534$	0	0
$535$	14.4814	0.626087
$536$	−0.661391	−0.0285677
$537$	0	0
$538$	4.04358	0.174331
$539$	3.94366	0.169866
$540$	0	0
$541$	−42.8333	−1.84155	−0.920774	−	0.390097i	$-0.872442\pi$
$541$	−42.8333	−1.84155	−0.920774	+	0.390097i	$0.872442\pi$
$542$	0.117009	0.00502597
$543$	0	0
$544$	−6.71781	−0.288024
$545$	9.41979	0.403499
$546$	0	0
$547$	19.0228	0.813354	0.406677	−	0.913572i	$-0.366687\pi$
$547$	19.0228	0.813354	0.406677	+	0.913572i	$0.366687\pi$
$548$	30.7377	1.31305
$549$	0	0
$550$	−0.361542	−0.0154162
$551$	11.0194	0.469441
$552$	0	0
$553$	−20.3752	−0.866443
$554$	0.504298	0.0214256
$555$	0	0
$556$	32.5343	1.37976
$557$	28.3961	1.20318	0.601590	−	0.798805i	$-0.294534\pi$
$557$	28.3961	1.20318	0.601590	+	0.798805i	$0.294534\pi$
$558$	0	0
$559$	−19.5162	−0.825446
$560$	11.2742	0.476424
$561$	0	0
$562$	−0.931596	−0.0392970
$563$	9.74594	0.410742	0.205371	−	0.978684i	$-0.434160\pi$
$563$	9.74594	0.410742	0.205371	+	0.978684i	$0.434160\pi$
$564$	0	0
$565$	11.4813	0.483022
$566$	−5.22738	−0.219723
$567$	0	0
$568$	−1.84834	−0.0775547
$569$	24.5954	1.03109	0.515546	−	0.856862i	$-0.327589\pi$
$569$	24.5954	1.03109	0.515546	+	0.856862i	$0.327589\pi$
$570$	0	0
$571$	13.6187	0.569924	0.284962	−	0.958539i	$-0.408019\pi$
$571$	13.6187	0.569924	0.284962	+	0.958539i	$0.408019\pi$
$572$	9.36006	0.391364
$573$	0	0
$574$	−0.343819	−0.0143507
$575$	−16.8352	−0.702076
$576$	0	0
$577$	−4.53826	−0.188930	−0.0944650	−	0.995528i	$-0.530114\pi$
$577$	−4.53826	−0.188930	−0.0944650	+	0.995528i	$0.530114\pi$
$578$	0.876994	0.0364782
$579$	0	0
$580$	−4.38700	−0.182160
$581$	22.5031	0.933585
$582$	0	0
$583$	3.63284	0.150457
$584$	−6.27122	−0.259505
$585$	0	0
$586$	2.45008	0.101212
$587$	−31.1986	−1.28770	−0.643852	−	0.765150i	$-0.722665\pi$
$587$	−31.1986	−1.28770	−0.643852	+	0.765150i	$0.722665\pi$
$588$	0	0
$589$	33.7485	1.39058
$590$	−1.96421	−0.0808654
$591$	0	0
$592$	43.4581	1.78612
$593$	25.7235	1.05634	0.528168	−	0.849140i	$-0.322879\pi$
$593$	25.7235	1.05634	0.528168	+	0.849140i	$0.322879\pi$
$594$	0	0
$595$	10.0713	0.412885
$596$	15.1710	0.621430
$597$	0	0
$598$	−6.12643	−0.250528
$599$	1.47502	0.0602679	0.0301339	−	0.999546i	$-0.490407\pi$
$599$	1.47502	0.0602679	0.0301339	+	0.999546i	$0.490407\pi$
$600$	0	0
$601$	7.62776	0.311143	0.155571	−	0.987825i	$-0.450278\pi$
$601$	7.62776	0.311143	0.155571	+	0.987825i	$0.450278\pi$
$602$	1.19702	0.0487871
$603$	0	0
$604$	38.1023	1.55036
$605$	1.68184	0.0683767
$606$	0	0
$607$	18.8683	0.765841	0.382920	−	0.923781i	$-0.374918\pi$
$607$	18.8683	0.765841	0.382920	+	0.923781i	$0.374918\pi$
$608$	−16.3406	−0.662700
$609$	0	0
$610$	0.783007	0.0317030
$611$	−50.3189	−2.03568
$612$	0	0
$613$	−6.61920	−0.267347	−0.133673	−	0.991025i	$-0.542677\pi$
$613$	−6.61920	−0.267347	−0.133673	+	0.991025i	$0.542677\pi$
$614$	0.0230219	0.000929087 0
$615$	0	0
$616$	−1.15627	−0.0465874
$617$	−38.4420	−1.54761	−0.773807	−	0.633421i	$-0.781650\pi$
$617$	−38.4420	−1.54761	−0.773807	+	0.633421i	$0.781650\pi$
$618$	0	0
$619$	30.4336	1.22323	0.611616	−	0.791155i	$-0.290520\pi$
$619$	30.4336	1.22323	0.611616	+	0.791155i	$0.290520\pi$
$620$	−13.4359	−0.539598
$621$	0	0
$622$	0.436003	0.0174821
$623$	16.9905	0.680711
$624$	0	0
$625$	−9.42801	−0.377121
$626$	2.20113	0.0879750
$627$	0	0
$628$	−27.5735	−1.10030
$629$	38.8213	1.54791
$630$	0	0
$631$	−18.3761	−0.731539	−0.365770	−	0.930705i	$-0.619194\pi$
$631$	−18.3761	−0.731539	−0.365770	+	0.930705i	$0.619194\pi$
$632$	−7.70832	−0.306621
$633$	0	0
$634$	−4.13106	−0.164066
$635$	1.57803	0.0626220
$636$	0	0
$637$	−18.7159	−0.741551
$638$	0.220208	0.00871811
$639$	0	0
$640$	8.65300	0.342040
$641$	−33.9334	−1.34029	−0.670145	−	0.742231i	$-0.733768\pi$
$641$	−33.9334	−1.34029	−0.670145	+	0.742231i	$0.733768\pi$
$642$	0	0
$643$	−45.4291	−1.79155	−0.895775	−	0.444507i	$-0.853379\pi$
$643$	−45.4291	−1.79155	−0.895775	+	0.444507i	$0.853379\pi$
$644$	−26.7329	−1.05343
$645$	0	0
$646$	−4.75184	−0.186959
$647$	−40.3376	−1.58583	−0.792916	−	0.609330i	$-0.791438\pi$
$647$	−40.3376	−1.58583	−0.792916	+	0.609330i	$0.791438\pi$
$648$	0	0
$649$	−7.01431	−0.275336
$650$	1.71581	0.0672996
$651$	0	0
$652$	−16.0709	−0.629383
$653$	36.0826	1.41202	0.706011	−	0.708201i	$-0.250493\pi$
$653$	36.0826	1.41202	0.706011	+	0.708201i	$0.250493\pi$
$654$	0	0
$655$	−31.8619	−1.24495
$656$	4.52910	0.176832
$657$	0	0
$658$	3.08631	0.120317
$659$	13.8394	0.539108	0.269554	−	0.962985i	$-0.413124\pi$
$659$	13.8394	0.539108	0.269554	+	0.962985i	$0.413124\pi$
$660$	0	0
$661$	17.1231	0.666012	0.333006	−	0.942925i	$-0.391937\pi$
$661$	17.1231	0.666012	0.333006	+	0.942925i	$0.391937\pi$
$662$	−0.420690	−0.0163506
$663$	0	0
$664$	8.51333	0.330381
$665$	24.4979	0.949988
$666$	0	0
$667$	10.2540	0.397036
$668$	−36.3667	−1.40707
$669$	0	0
$670$	0.280030	0.0108185
$671$	2.79616	0.107944
$672$	0	0
$673$	−0.932129	−0.0359309	−0.0179655	−	0.999839i	$-0.505719\pi$
$673$	−0.932129	−0.0359309	−0.0179655	+	0.999839i	$0.505719\pi$
$674$	5.07528	0.195492
$675$	0	0
$676$	−18.7815	−0.722366
$677$	−48.8124	−1.87601	−0.938007	−	0.346617i	$-0.887330\pi$
$677$	−48.8124	−1.87601	−0.938007	+	0.346617i	$0.887330\pi$
$678$	0	0
$679$	−24.6472	−0.945873
$680$	3.81018	0.146114
$681$	0	0
$682$	0.674420	0.0258249
$683$	−18.0063	−0.688990	−0.344495	−	0.938788i	$-0.611950\pi$
$683$	−18.0063	−0.688990	−0.344495	+	0.938788i	$0.611950\pi$
$684$	0	0
$685$	−26.2114	−1.00149
$686$	3.18553	0.121624
$687$	0	0
$688$	−15.7683	−0.601161
$689$	−17.2408	−0.656822
$690$	0	0
$691$	−10.7006	−0.407071	−0.203536	−	0.979068i	$-0.565243\pi$
$691$	−10.7006	−0.407071	−0.203536	+	0.979068i	$0.565243\pi$
$692$	−34.8217	−1.32372
$693$	0	0
$694$	3.91328	0.148546
$695$	−27.7434	−1.05237
$696$	0	0
$697$	4.04587	0.153248
$698$	−1.64675	−0.0623303
$699$	0	0
$700$	7.48701	0.282982
$701$	−46.2374	−1.74636	−0.873180	−	0.487397i	$-0.837946\pi$
$701$	−46.2374	−1.74636	−0.873180	+	0.487397i	$0.837946\pi$
$702$	0	0
$703$	94.4304	3.56151
$704$	7.34232	0.276724
$705$	0	0
$706$	0.687158	0.0258615
$707$	9.01143	0.338910
$708$	0	0
$709$	8.40222	0.315552	0.157776	−	0.987475i	$-0.449568\pi$
$709$	8.40222	0.315552	0.157776	+	0.987475i	$0.449568\pi$
$710$	0.782580	0.0293697
$711$	0	0
$712$	6.42783	0.240893
$713$	31.4044	1.17610
$714$	0	0
$715$	−7.98172	−0.298499
$716$	16.3399	0.610651
$717$	0	0
$718$	−1.40683	−0.0525024
$719$	29.0814	1.08455	0.542276	−	0.840201i	$-0.317563\pi$
$719$	29.0814	1.08455	0.542276	+	0.840201i	$0.317563\pi$
$720$	0	0
$721$	3.28188	0.122224
$722$	−8.39502	−0.312430
$723$	0	0
$724$	−33.1563	−1.23224
$725$	−2.87180	−0.106656
$726$	0	0
$727$	−22.4949	−0.834289	−0.417145	−	0.908840i	$-0.636969\pi$
$727$	−22.4949	−0.834289	−0.417145	+	0.908840i	$0.636969\pi$
$728$	5.48744	0.203378
$729$	0	0
$730$	2.65521	0.0982736
$731$	−14.0859	−0.520986
$732$	0	0
$733$	−19.9101	−0.735398	−0.367699	−	0.929945i	$-0.619854\pi$
$733$	−19.9101	−0.735398	−0.367699	+	0.929945i	$0.619854\pi$
$734$	−2.88168	−0.106365
$735$	0	0
$736$	−15.2056	−0.560487
$737$	1.00000	0.0368355
$738$	0	0
$739$	46.5061	1.71076	0.855378	−	0.518004i	$-0.173325\pi$
$739$	46.5061	1.71076	0.855378	+	0.518004i	$0.173325\pi$
$740$	−37.5944	−1.38200
$741$	0	0
$742$	1.05746	0.0388208
$743$	8.05407	0.295475	0.147738	−	0.989027i	$-0.452801\pi$
$743$	8.05407	0.295475	0.147738	+	0.989027i	$0.452801\pi$
$744$	0	0
$745$	−12.9370	−0.473974
$746$	2.52843	0.0925723
$747$	0	0
$748$	6.75568	0.247012
$749$	15.0531	0.550029
$750$	0	0
$751$	46.6397	1.70191	0.850954	−	0.525241i	$-0.176025\pi$
$751$	46.6397	1.70191	0.850954	+	0.525241i	$0.176025\pi$
$752$	−40.6557	−1.48256
$753$	0	0
$754$	−1.04507	−0.0380590
$755$	−32.4915	−1.18249
$756$	0	0
$757$	−22.9952	−0.835775	−0.417888	−	0.908499i	$-0.637229\pi$
$757$	−22.9952	−0.835775	−0.417888	+	0.908499i	$0.637229\pi$
$758$	−4.08521	−0.148381
$759$	0	0
$760$	9.26800	0.336186
$761$	−19.0457	−0.690405	−0.345202	−	0.938528i	$-0.612190\pi$
$761$	−19.0457	−0.690405	−0.345202	+	0.938528i	$0.612190\pi$
$762$	0	0
$763$	9.79165	0.354482
$764$	−20.7567	−0.750950
$765$	0	0
$766$	3.03748	0.109748
$767$	33.2886	1.20198
$768$	0	0
$769$	−38.1299	−1.37500	−0.687500	−	0.726185i	$-0.741292\pi$
$769$	−38.1299	−1.37500	−0.687500	+	0.726185i	$0.741292\pi$
$770$	0.489559	0.0176425
$771$	0	0
$772$	−12.6915	−0.456777
$773$	−13.5395	−0.486983	−0.243491	−	0.969903i	$-0.578293\pi$
$773$	−13.5395	−0.486983	−0.243491	+	0.969903i	$0.578293\pi$
$774$	0	0
$775$	−8.79533	−0.315938
$776$	−9.32450	−0.334730
$777$	0	0
$778$	0.0456445	0.00163643
$779$	9.84132	0.352602
$780$	0	0
$781$	2.79463	0.0999997
$782$	−4.42179	−0.158123
$783$	0	0
$784$	−15.1217	−0.540061
$785$	23.5131	0.839217
$786$	0	0
$787$	43.6451	1.55578	0.777890	−	0.628401i	$-0.216290\pi$
$787$	43.6451	1.55578	0.777890	+	0.628401i	$0.216290\pi$
$788$	51.9726	1.85145
$789$	0	0
$790$	3.26367	0.116116
$791$	11.9346	0.424344
$792$	0	0
$793$	−13.2700	−0.471233
$794$	0.872517	0.0309645
$795$	0	0
$796$	14.8397	0.525979
$797$	−4.76560	−0.168806	−0.0844032	−	0.996432i	$-0.526898\pi$
$797$	−4.76560	−0.168806	−0.0844032	+	0.996432i	$0.526898\pi$
$798$	0	0
$799$	−36.3180	−1.28484
$800$	4.25860	0.150564
$801$	0	0
$802$	−1.56784	−0.0553623
$803$	9.48187	0.334608
$804$	0	0
$805$	22.7963	0.803464
$806$	−3.20067	−0.112739
$807$	0	0
$808$	3.40919	0.119935
$809$	10.5195	0.369845	0.184923	−	0.982753i	$-0.440797\pi$
$809$	10.5195	0.369845	0.184923	+	0.982753i	$0.440797\pi$
$810$	0	0
$811$	−8.76094	−0.307638	−0.153819	−	0.988099i	$-0.549157\pi$
$811$	−8.76094	−0.307638	−0.153819	+	0.988099i	$0.549157\pi$
$812$	−4.56019	−0.160031
$813$	0	0
$814$	1.88707	0.0661417
$815$	13.7043	0.480040
$816$	0	0
$817$	−34.2631	−1.19871
$818$	2.71587	0.0949581
$819$	0	0
$820$	−3.91800	−0.136823
$821$	14.6659	0.511845	0.255922	−	0.966697i	$-0.417621\pi$
$821$	14.6659	0.511845	0.255922	+	0.966697i	$0.417621\pi$
$822$	0	0
$823$	21.2743	0.741575	0.370787	−	0.928718i	$-0.379088\pi$
$823$	21.2743	0.741575	0.370787	+	0.928718i	$0.379088\pi$
$824$	1.24159	0.0432530
$825$	0	0
$826$	−2.04176	−0.0710418
$827$	19.7147	0.685547	0.342774	−	0.939418i	$-0.388634\pi$
$827$	19.7147	0.685547	0.342774	+	0.939418i	$0.388634\pi$
$828$	0	0
$829$	−37.1830	−1.29142	−0.645709	−	0.763583i	$-0.723438\pi$
$829$	−37.1830	−1.29142	−0.645709	+	0.763583i	$0.723438\pi$
$830$	−3.60450	−0.125114
$831$	0	0
$832$	−34.8453	−1.20804
$833$	−13.5083	−0.468035
$834$	0	0
$835$	31.0114	1.07319
$836$	16.4328	0.568339
$837$	0	0
$838$	−3.62200	−0.125120
$839$	6.01653	0.207714	0.103857	−	0.994592i	$-0.466882\pi$
$839$	6.01653	0.207714	0.103857	+	0.994592i	$0.466882\pi$
$840$	0	0
$841$	−27.2508	−0.939684
$842$	−4.07933	−0.140583
$843$	0	0
$844$	14.7788	0.508706
$845$	16.0158	0.550960
$846$	0	0
$847$	1.74824	0.0600702
$848$	−13.9299	−0.478355
$849$	0	0
$850$	1.23840	0.0424767
$851$	87.8714	3.01219
$852$	0	0
$853$	−10.0514	−0.344153	−0.172077	−	0.985084i	$-0.555048\pi$
$853$	−10.0514	−0.344153	−0.172077	+	0.985084i	$0.555048\pi$
$854$	0.813918	0.0278517
$855$	0	0
$856$	5.69487	0.194647
$857$	−5.43658	−0.185710	−0.0928550	−	0.995680i	$-0.529599\pi$
$857$	−5.43658	−0.185710	−0.0928550	+	0.995680i	$0.529599\pi$
$858$	0	0
$859$	−34.8213	−1.18809	−0.594043	−	0.804433i	$-0.702469\pi$
$859$	−34.8213	−1.18809	−0.594043	+	0.804433i	$0.702469\pi$
$860$	13.6407	0.465145
$861$	0	0
$862$	5.35094	0.182254
$863$	6.65600	0.226573	0.113287	−	0.993562i	$-0.463862\pi$
$863$	6.65600	0.226573	0.113287	+	0.993562i	$0.463862\pi$
$864$	0	0
$865$	29.6940	1.00963
$866$	1.96322	0.0667131
$867$	0	0
$868$	−13.9663	−0.474046
$869$	11.6547	0.395359
$870$	0	0
$871$	−4.74581	−0.160806
$872$	3.70436	0.125446
$873$	0	0
$874$	−10.7557	−0.363817
$875$	−21.0858	−0.712830
$876$	0	0
$877$	−14.5582	−0.491597	−0.245798	−	0.969321i	$-0.579050\pi$
$877$	−14.5582	−0.491597	−0.245798	+	0.969321i	$0.579050\pi$
$878$	3.96472	0.133803
$879$	0	0
$880$	−6.44892	−0.217393
$881$	−14.9581	−0.503952	−0.251976	−	0.967733i	$-0.581080\pi$
$881$	−14.9581	−0.503952	−0.251976	+	0.967733i	$0.581080\pi$
$882$	0	0
$883$	−25.3245	−0.852238	−0.426119	−	0.904667i	$-0.640120\pi$
$883$	−25.3245	−0.852238	−0.426119	+	0.904667i	$0.640120\pi$
$884$	−32.0612	−1.07834
$885$	0	0
$886$	−0.767322	−0.0257787
$887$	32.0717	1.07686	0.538431	−	0.842669i	$-0.319017\pi$
$887$	32.0717	1.07686	0.538431	+	0.842669i	$0.319017\pi$
$888$	0	0
$889$	1.64032	0.0550146
$890$	−2.72151	−0.0912253
$891$	0	0
$892$	29.8440	0.999252
$893$	−88.3411	−2.95622
$894$	0	0
$895$	−13.9337	−0.465753
$896$	8.99460	0.300488
$897$	0	0
$898$	2.46471	0.0822485
$899$	5.35706	0.178668
$900$	0	0
$901$	−12.4437	−0.414558
$902$	0.196666	0.00654826
$903$	0	0
$904$	4.51506	0.150169
$905$	28.2738	0.939851
$906$	0	0
$907$	−8.78292	−0.291632	−0.145816	−	0.989312i	$-0.546581\pi$
$907$	−8.78292	−0.291632	−0.145816	+	0.989312i	$0.546581\pi$
$908$	−18.6251	−0.618095
$909$	0	0
$910$	−2.32336	−0.0770185
$911$	−8.73944	−0.289550	−0.144775	−	0.989465i	$-0.546246\pi$
$911$	−8.73944	−0.289550	−0.144775	+	0.989465i	$0.546246\pi$
$912$	0	0
$913$	−12.8719	−0.425996
$914$	1.66746	0.0551548
$915$	0	0
$916$	−23.6186	−0.780380
$917$	−33.1197	−1.09371
$918$	0	0
$919$	−13.0393	−0.430127	−0.215064	−	0.976600i	$-0.568996\pi$
$919$	−13.0393	−0.430127	−0.215064	+	0.976600i	$0.568996\pi$
$920$	8.62426	0.284334
$921$	0	0
$922$	−3.78344	−0.124601
$923$	−13.2628	−0.436550
$924$	0	0
$925$	−24.6099	−0.809167
$926$	4.70873	0.154739
$927$	0	0
$928$	−2.59382	−0.0851465
$929$	−24.9706	−0.819257	−0.409629	−	0.912252i	$-0.634342\pi$
$929$	−24.9706	−0.819257	−0.409629	+	0.912252i	$0.634342\pi$
$930$	0	0
$931$	−32.8581	−1.07688
$932$	27.9521	0.915602
$933$	0	0
$934$	−3.75864	−0.122986
$935$	−5.76085	−0.188400
$936$	0	0
$937$	−11.5428	−0.377088	−0.188544	−	0.982065i	$-0.560377\pi$
$937$	−11.5428	−0.377088	−0.188544	+	0.982065i	$0.560377\pi$
$938$	0.291085	0.00950425
$939$	0	0
$940$	35.1701	1.14712
$941$	−31.1762	−1.01632	−0.508158	−	0.861264i	$-0.669673\pi$
$941$	−31.1762	−1.01632	−0.508158	+	0.861264i	$0.669673\pi$
$942$	0	0
$943$	9.15776	0.298218
$944$	26.8959	0.875386
$945$	0	0
$946$	−0.684703	−0.0222616
$947$	18.0425	0.586301	0.293151	−	0.956066i	$-0.405296\pi$
$947$	18.0425	0.586301	0.293151	+	0.956066i	$0.405296\pi$
$948$	0	0
$949$	−44.9992	−1.46074
$950$	3.01232	0.0977325
$951$	0	0
$952$	3.96059	0.128363
$953$	32.8240	1.06327	0.531636	−	0.846973i	$-0.321577\pi$
$953$	32.8240	1.06327	0.531636	+	0.846973i	$0.321577\pi$
$954$	0	0
$955$	17.7001	0.572762
$956$	−26.2309	−0.848368
$957$	0	0
$958$	−6.72039	−0.217126
$959$	−27.2461	−0.879823
$960$	0	0
$961$	−14.5932	−0.470748
$962$	−8.95568	−0.288743
$963$	0	0
$964$	6.27845	0.202215
$965$	10.8226	0.348391
$966$	0	0
$967$	13.4197	0.431549	0.215774	−	0.976443i	$-0.430772\pi$
$967$	13.4197	0.431549	0.215774	+	0.976443i	$0.430772\pi$
$968$	0.661391	0.0212579
$969$	0	0
$970$	3.94795	0.126761
$971$	−12.6229	−0.405088	−0.202544	−	0.979273i	$-0.564921\pi$
$971$	−12.6229	−0.405088	−0.202544	+	0.979273i	$0.564921\pi$
$972$	0	0
$973$	−28.8386	−0.924523
$974$	−4.00651	−0.128377
$975$	0	0
$976$	−10.7217	−0.343192
$977$	25.7098	0.822529	0.411265	−	0.911516i	$-0.365087\pi$
$977$	25.7098	0.822529	0.411265	+	0.911516i	$0.365087\pi$
$978$	0	0
$979$	−9.71865	−0.310610
$980$	13.0814	0.417869
$981$	0	0
$982$	−0.927726	−0.0296049
$983$	−14.3262	−0.456933	−0.228467	−	0.973552i	$-0.573371\pi$
$983$	−14.3262	−0.456933	−0.228467	+	0.973552i	$0.573371\pi$
$984$	0	0
$985$	−44.3192	−1.41213
$986$	−0.754282	−0.0240212
$987$	0	0
$988$	−77.9868	−2.48109
$989$	−31.8832	−1.01383
$990$	0	0
$991$	9.33471	0.296527	0.148263	−	0.988948i	$-0.452632\pi$
$991$	9.33471	0.296527	0.148263	+	0.988948i	$0.452632\pi$
$992$	−7.94399	−0.252222
$993$	0	0
$994$	0.813474	0.0258018
$995$	−12.6544	−0.401172
$996$	0	0
$997$	8.70941	0.275830	0.137915	−	0.990444i	$-0.455960\pi$
$997$	8.70941	0.275830	0.137915	+	0.990444i	$0.455960\pi$
$998$	−6.70181	−0.212142
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6633.2.a.w.1.9		17
3.2	odd	2		737.2.a.f.1.9	✓	17
33.32	even	2		8107.2.a.o.1.9		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
737.2.a.f.1.9	✓	17	3.2	odd	2
6633.2.a.w.1.9		17	1.1	even	1	trivial
8107.2.a.o.1.9		17	33.32	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6633 = 3^{2} \cdot 11 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6633.a (trivial)

\(f(q)\)	\(=\)	\(q-0.166502 q^{2} -1.97228 q^{4} +1.68184 q^{5} +1.74824 q^{7} +0.661391 q^{8} +O(q^{10})\)	\(q-0.166502 q^{2} -1.97228 q^{4} +1.68184 q^{5} +1.74824 q^{7} +0.661391 q^{8} -0.280030 q^{10} -1.00000 q^{11} +4.74581 q^{13} -0.291085 q^{14} +3.83443 q^{16} +3.42532 q^{17} +8.33187 q^{19} -3.31706 q^{20} +0.166502 q^{22} +7.75315 q^{23} -2.17140 q^{25} -0.790186 q^{26} -3.44801 q^{28} +1.32256 q^{29} +4.05053 q^{31} -1.96122 q^{32} -0.570321 q^{34} +2.94026 q^{35} +11.3336 q^{37} -1.38727 q^{38} +1.11236 q^{40} +1.18117 q^{41} -4.11229 q^{43} +1.97228 q^{44} -1.29091 q^{46} -10.6028 q^{47} -3.94366 q^{49} +0.361542 q^{50} -9.36006 q^{52} -3.63284 q^{53} -1.68184 q^{55} +1.15627 q^{56} -0.220208 q^{58} +7.01431 q^{59} -2.79616 q^{61} -0.674420 q^{62} -7.34232 q^{64} +7.98172 q^{65} -1.00000 q^{67} -6.75568 q^{68} -0.489559 q^{70} -2.79463 q^{71} -9.48187 q^{73} -1.88707 q^{74} -16.4328 q^{76} -1.74824 q^{77} -11.6547 q^{79} +6.44892 q^{80} -0.196666 q^{82} +12.8719 q^{83} +5.76085 q^{85} +0.684703 q^{86} -0.661391 q^{88} +9.71865 q^{89} +8.29681 q^{91} -15.2914 q^{92} +1.76538 q^{94} +14.0129 q^{95} -14.0983 q^{97} +0.656627 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - q^{2} + 19 q^{4} - 10 q^{5} + 20 q^{7}+O(q^{10}) \) 17 * q - q^2 + 19 * q^4 - 10 * q^5 + 20 * q^7	\( 17 q - q^{2} + 19 q^{4} - 10 q^{5} + 20 q^{7} + 10 q^{10} - 17 q^{11} + q^{13} + 11 q^{14} + 19 q^{16} - 2 q^{17} + 13 q^{19} - 3 q^{20} + q^{22} - 16 q^{23} + 33 q^{25} - 12 q^{26} + 44 q^{28} + 5 q^{29} + 16 q^{31} + 24 q^{32} + 4 q^{34} + 2 q^{35} + 29 q^{37} + 19 q^{38} + 31 q^{40} + 6 q^{41} + 19 q^{43} - 19 q^{44} - 33 q^{46} - 40 q^{47} + 23 q^{49} + 3 q^{50} - 28 q^{52} - 15 q^{53} + 10 q^{55} + 38 q^{56} - 12 q^{58} + 2 q^{59} - 6 q^{61} - 3 q^{62} - 4 q^{64} + 30 q^{65} - 17 q^{67} + 13 q^{68} + 71 q^{70} - 2 q^{71} + 41 q^{73} + 13 q^{74} + 21 q^{76} - 20 q^{77} + 41 q^{79} + 23 q^{80} - 8 q^{82} - 2 q^{83} - 36 q^{85} + 54 q^{86} - q^{89} + 16 q^{91} - 36 q^{92} + 12 q^{94} + 31 q^{95} + 3 q^{97} + 64 q^{98}+O(q^{100}) \) 17 * q - q^2 + 19 * q^4 - 10 * q^5 + 20 * q^7 + 10 * q^10 - 17 * q^11 + q^13 + 11 * q^14 + 19 * q^16 - 2 * q^17 + 13 * q^19 - 3 * q^20 + q^22 - 16 * q^23 + 33 * q^25 - 12 * q^26 + 44 * q^28 + 5 * q^29 + 16 * q^31 + 24 * q^32 + 4 * q^34 + 2 * q^35 + 29 * q^37 + 19 * q^38 + 31 * q^40 + 6 * q^41 + 19 * q^43 - 19 * q^44 - 33 * q^46 - 40 * q^47 + 23 * q^49 + 3 * q^50 - 28 * q^52 - 15 * q^53 + 10 * q^55 + 38 * q^56 - 12 * q^58 + 2 * q^59 - 6 * q^61 - 3 * q^62 - 4 * q^64 + 30 * q^65 - 17 * q^67 + 13 * q^68 + 71 * q^70 - 2 * q^71 + 41 * q^73 + 13 * q^74 + 21 * q^76 - 20 * q^77 + 41 * q^79 + 23 * q^80 - 8 * q^82 - 2 * q^83 - 36 * q^85 + 54 * q^86 - q^89 + 16 * q^91 - 36 * q^92 + 12 * q^94 + 31 * q^95 + 3 * q^97 + 64 * q^98

Introduction

L-functions

Modular forms

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Database

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