LMFDB - Embedded newform 5733.2.a.bf.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5733,2,Mod(1,5733)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5733, 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("5733.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5733 = 3^{2} \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5733.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:	$45.7782354788$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.17428.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 4x + 6 $ x^4 - x^3 - 6x^2 + 4x + 6
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 273)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.787711$ of defining polynomial
Character	$\chi$	$=$	5733.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.37951 q^{2} +3.66208 q^{4} -2.66208 q^{5} +3.95493 q^{8} +O(q^{10})$	$q+2.37951 q^{2} +3.66208 q^{4} -2.66208 q^{5} +3.95493 q^{8} -6.33445 q^{10} -1.57542 q^{11} -1.00000 q^{13} +2.08666 q^{16} +4.75902 q^{17} +2.23750 q^{19} -9.74873 q^{20} -3.74873 q^{22} -5.84568 q^{23} +2.08666 q^{25} -2.37951 q^{26} -4.23750 q^{29} -7.28055 q^{31} -2.94464 q^{32} +11.3242 q^{34} +10.4750 q^{37} +5.32415 q^{38} -10.5283 q^{40} -2.25127 q^{41} +0.913344 q^{43} -5.76931 q^{44} -13.9099 q^{46} -2.09695 q^{47} +4.96522 q^{50} -3.66208 q^{52} -1.08666 q^{53} +4.19389 q^{55} -10.0832 q^{58} -12.3344 q^{59} +7.51805 q^{61} -17.3242 q^{62} -11.1801 q^{64} +2.66208 q^{65} -15.6914 q^{67} +17.4279 q^{68} -10.0504 q^{71} -15.0797 q^{73} +24.9254 q^{74} +8.19389 q^{76} -11.7555 q^{79} -5.55484 q^{80} -5.35692 q^{82} +7.42110 q^{83} -12.6689 q^{85} +2.17331 q^{86} -6.23069 q^{88} -11.1371 q^{89} -21.4073 q^{92} -4.98971 q^{94} -5.95639 q^{95} +14.6047 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 7 q^{4} - 3 q^{5} - 3 q^{8}+O(q^{10}) $ 4 * q - q^2 + 7 * q^4 - 3 * q^5 - 3 * q^8	$ 4 q - q^{2} + 7 q^{4} - 3 q^{5} - 3 q^{8} + 4 q^{10} + 2 q^{11} - 4 q^{13} + 9 q^{16} - 2 q^{17} - 7 q^{19} - 32 q^{20} - 8 q^{22} - 3 q^{23} + 9 q^{25} + q^{26} - q^{29} - 3 q^{31} - 7 q^{32} + 30 q^{34} + 10 q^{37} + 6 q^{38} + 14 q^{40} - 16 q^{41} + 3 q^{43} + 12 q^{44} - 18 q^{46} + 5 q^{47} - 13 q^{50} - 7 q^{52} - 5 q^{53} - 10 q^{55} - 4 q^{58} - 20 q^{59} - 12 q^{61} - 54 q^{62} + 5 q^{64} + 3 q^{65} - 22 q^{67} - 10 q^{68} + 13 q^{73} + 6 q^{74} + 6 q^{76} + 11 q^{79} - 42 q^{80} - 10 q^{82} + q^{83} + 8 q^{85} + 10 q^{86} - 60 q^{88} - 5 q^{89} - 34 q^{92} - 34 q^{94} - 13 q^{95} + 17 q^{97}+O(q^{100}) $ 4 * q - q^2 + 7 * q^4 - 3 * q^5 - 3 * q^8 + 4 * q^10 + 2 * q^11 - 4 * q^13 + 9 * q^16 - 2 * q^17 - 7 * q^19 - 32 * q^20 - 8 * q^22 - 3 * q^23 + 9 * q^25 + q^26 - q^29 - 3 * q^31 - 7 * q^32 + 30 * q^34 + 10 * q^37 + 6 * q^38 + 14 * q^40 - 16 * q^41 + 3 * q^43 + 12 * q^44 - 18 * q^46 + 5 * q^47 - 13 * q^50 - 7 * q^52 - 5 * q^53 - 10 * q^55 - 4 * q^58 - 20 * q^59 - 12 * q^61 - 54 * q^62 + 5 * q^64 + 3 * q^65 - 22 * q^67 - 10 * q^68 + 13 * q^73 + 6 * q^74 + 6 * q^76 + 11 * q^79 - 42 * q^80 - 10 * q^82 + q^83 + 8 * q^85 + 10 * q^86 - 60 * q^88 - 5 * q^89 - 34 * q^92 - 34 * q^94 - 13 * q^95 + 17 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.37951	1.68257	0.841285	−	0.540593i	$-0.181800\pi$
$2$	2.37951	1.68257	0.841285	+	0.540593i	$0.181800\pi$
$3$	0	0
$3$	0	0
$4$	3.66208	1.83104
$5$	−2.66208	−1.19052	−0.595259	−	0.803534i	$-0.702950\pi$
$5$	−2.66208	−1.19052	−0.595259	+	0.803534i	$0.702950\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	3.95493	1.39828
$9$	0	0
$10$	−6.33445	−2.00313
$11$	−1.57542	−0.475007	−0.237504	−	0.971387i	$-0.576329\pi$
$11$	−1.57542	−0.475007	−0.237504	+	0.971387i	$0.576329\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	0	0
$16$	2.08666	0.521664
$17$	4.75902	1.15423	0.577116	−	0.816662i	$-0.304178\pi$
$17$	4.75902	1.15423	0.577116	+	0.816662i	$0.304178\pi$
$18$	0	0
$19$	2.23750	0.513317	0.256659	−	0.966502i	$-0.417378\pi$
$19$	2.23750	0.513317	0.256659	+	0.966502i	$0.417378\pi$
$20$	−9.74873	−2.17988
$21$	0	0
$22$	−3.74873	−0.799233
$23$	−5.84568	−1.21891	−0.609454	−	0.792821i	$-0.708611\pi$
$23$	−5.84568	−1.21891	−0.609454	+	0.792821i	$0.708611\pi$
$24$	0	0
$25$	2.08666	0.417331
$26$	−2.37951	−0.466661
$27$	0	0
$28$	0	0
$29$	−4.23750	−0.786884	−0.393442	−	0.919349i	$-0.628716\pi$
$29$	−4.23750	−0.786884	−0.393442	+	0.919349i	$0.628716\pi$
$30$	0	0
$31$	−7.28055	−1.30763	−0.653813	−	0.756656i	$-0.726832\pi$
$31$	−7.28055	−1.30763	−0.653813	+	0.756656i	$0.726832\pi$
$32$	−2.94464	−0.520544
$33$	0	0
$34$	11.3242	1.94208
$35$	0	0
$36$	0	0
$37$	10.4750	1.72208	0.861039	−	0.508538i	$-0.169814\pi$
$37$	10.4750	1.72208	0.861039	+	0.508538i	$0.169814\pi$
$38$	5.32415	0.863692
$39$	0	0
$40$	−10.5283	−1.66468
$41$	−2.25127	−0.351589	−0.175794	−	0.984427i	$-0.556249\pi$
$41$	−2.25127	−0.351589	−0.175794	+	0.984427i	$0.556249\pi$
$42$	0	0
$43$	0.913344	0.139284	0.0696418	−	0.997572i	$-0.477814\pi$
$43$	0.913344	0.139284	0.0696418	+	0.997572i	$0.477814\pi$
$44$	−5.76931	−0.869757
$45$	0	0
$46$	−13.9099	−2.05090
$47$	−2.09695	−0.305871	−0.152936	−	0.988236i	$-0.548873\pi$
$47$	−2.09695	−0.305871	−0.152936	+	0.988236i	$0.548873\pi$
$48$	0	0
$49$	0	0
$50$	4.96522	0.702189
$51$	0	0
$52$	−3.66208	−0.507839
$53$	−1.08666	−0.149264	−0.0746319	−	0.997211i	$-0.523778\pi$
$53$	−1.08666	−0.149264	−0.0746319	+	0.997211i	$0.523778\pi$
$54$	0	0
$55$	4.19389	0.565504
$56$	0	0
$57$	0	0
$58$	−10.0832	−1.32399
$59$	−12.3344	−1.60581	−0.802904	−	0.596108i	$-0.796713\pi$
$59$	−12.3344	−1.60581	−0.802904	+	0.596108i	$0.796713\pi$
$60$	0	0
$61$	7.51805	0.962587	0.481294	−	0.876559i	$-0.340167\pi$
$61$	7.51805	0.962587	0.481294	+	0.876559i	$0.340167\pi$
$62$	−17.3242	−2.20017
$63$	0	0
$64$	−11.1801	−1.39752
$65$	2.66208	0.330190
$66$	0	0
$67$	−15.6914	−1.91700	−0.958502	−	0.285084i	$-0.907978\pi$
$67$	−15.6914	−1.91700	−0.958502	+	0.285084i	$0.907978\pi$
$68$	17.4279	2.11345
$69$	0	0
$70$	0	0
$71$	−10.0504	−1.19277	−0.596383	−	0.802700i	$-0.703396\pi$
$71$	−10.0504	−1.19277	−0.596383	+	0.802700i	$0.703396\pi$
$72$	0	0
$73$	−15.0797	−1.76495	−0.882473	−	0.470363i	$-0.844123\pi$
$73$	−15.0797	−1.76495	−0.882473	+	0.470363i	$0.844123\pi$
$74$	24.9254	2.89752
$75$	0	0
$76$	8.19389	0.939904
$77$	0	0
$78$	0	0
$79$	−11.7555	−1.32260	−0.661301	−	0.750121i	$-0.729995\pi$
$79$	−11.7555	−1.32260	−0.661301	+	0.750121i	$0.729995\pi$
$80$	−5.55484	−0.621050
$81$	0	0
$82$	−5.35692	−0.591572
$83$	7.42110	0.814572	0.407286	−	0.913301i	$-0.366475\pi$
$83$	7.42110	0.814572	0.407286	+	0.913301i	$0.366475\pi$
$84$	0	0
$85$	−12.6689	−1.37413
$86$	2.17331	0.234354
$87$	0	0
$88$	−6.23069	−0.664193
$89$	−11.1371	−1.18053	−0.590264	−	0.807210i	$-0.700976\pi$
$89$	−11.1371	−1.18053	−0.590264	+	0.807210i	$0.700976\pi$
$90$	0	0
$91$	0	0
$92$	−21.4073	−2.23187
$93$	0	0
$94$	−4.98971	−0.514649
$95$	−5.95639	−0.611113
$96$	0	0
$97$	14.6047	1.48288	0.741442	−	0.671017i	$-0.234142\pi$
$97$	14.6047	1.48288	0.741442	+	0.671017i	$0.234142\pi$
$98$	0	0
$99$	0	0
$100$	7.64150	0.764150
$101$	−11.4349	−1.13781	−0.568906	−	0.822403i	$-0.692633\pi$
$101$	−11.4349	−1.13781	−0.568906	+	0.822403i	$0.692633\pi$
$102$	0	0
$103$	−11.1508	−1.09873	−0.549363	−	0.835584i	$-0.685129\pi$
$103$	−11.1508	−1.09873	−0.549363	+	0.835584i	$0.685129\pi$
$104$	−3.95493	−0.387813
$105$	0	0
$106$	−2.58571	−0.251147
$107$	−2.28403	−0.220805	−0.110403	−	0.993887i	$-0.535214\pi$
$107$	−2.28403	−0.220805	−0.110403	+	0.993887i	$0.535214\pi$
$108$	0	0
$109$	14.6689	1.40502	0.702512	−	0.711671i	$-0.252062\pi$
$109$	14.6689	1.40502	0.702512	+	0.711671i	$0.252062\pi$
$110$	9.97942	0.951500
$111$	0	0
$112$	0	0
$113$	4.23750	0.398630	0.199315	−	0.979935i	$-0.436128\pi$
$113$	4.23750	0.398630	0.199315	+	0.979935i	$0.436128\pi$
$114$	0	0
$115$	15.5617	1.45113
$116$	−15.5180	−1.44081
$117$	0	0
$118$	−29.3500	−2.70188
$119$	0	0
$120$	0	0
$121$	−8.51805	−0.774368
$122$	17.8893	1.61962
$123$	0	0
$124$	−26.6619	−2.39431
$125$	7.75555	0.693677
$126$	0	0
$127$	4.84916	0.430293	0.215147	−	0.976582i	$-0.430977\pi$
$127$	4.84916	0.430293	0.215147	+	0.976582i	$0.430977\pi$
$128$	−20.7140	−1.83087
$129$	0	0
$130$	6.33445	0.555568
$131$	−6.36721	−0.556305	−0.278153	−	0.960537i	$-0.589722\pi$
$131$	−6.36721	−0.556305	−0.278153	+	0.960537i	$0.589722\pi$
$132$	0	0
$133$	0	0
$134$	−37.3378	−3.22549
$135$	0	0
$136$	18.8216	1.61394
$137$	18.0258	1.54005	0.770024	−	0.638015i	$-0.220244\pi$
$137$	18.0258	1.54005	0.770024	+	0.638015i	$0.220244\pi$
$138$	0	0
$139$	−2.67585	−0.226962	−0.113481	−	0.993540i	$-0.536200\pi$
$139$	−2.67585	−0.226962	−0.113481	+	0.993540i	$0.536200\pi$
$140$	0	0
$141$	0	0
$142$	−23.9151	−2.00691
$143$	1.57542	0.131743
$144$	0	0
$145$	11.2805	0.936799
$146$	−35.8823	−2.96964
$147$	0	0
$148$	38.3603	3.15319
$149$	7.46471	0.611533	0.305766	−	0.952107i	$-0.401087\pi$
$149$	7.46471	0.611533	0.305766	+	0.952107i	$0.401087\pi$
$150$	0	0
$151$	13.0155	1.05919	0.529594	−	0.848251i	$-0.322344\pi$
$151$	13.0155	1.05919	0.529594	+	0.848251i	$0.322344\pi$
$152$	8.84916	0.717761
$153$	0	0
$154$	0	0
$155$	19.3814	1.55675
$156$	0	0
$157$	17.1233	1.36659	0.683294	−	0.730143i	$-0.260547\pi$
$157$	17.1233	1.36659	0.683294	+	0.730143i	$0.260547\pi$
$158$	−27.9725	−2.22537
$159$	0	0
$160$	7.83887	0.619717
$161$	0	0
$162$	0	0
$163$	−0.849158	−0.0665112	−0.0332556	−	0.999447i	$-0.510588\pi$
$163$	−0.849158	−0.0665112	−0.0332556	+	0.999447i	$0.510588\pi$
$164$	−8.24431	−0.643773
$165$	0	0
$166$	17.6586	1.37057
$167$	18.3781	1.42214	0.711068	−	0.703123i	$-0.248212\pi$
$167$	18.3781	1.42214	0.711068	+	0.703123i	$0.248212\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−30.1458	−2.31208
$171$	0	0
$172$	3.34474	0.255034
$173$	−12.2565	−0.931844	−0.465922	−	0.884826i	$-0.654277\pi$
$173$	−12.2565	−0.931844	−0.465922	+	0.884826i	$0.654277\pi$
$174$	0	0
$175$	0	0
$176$	−3.28736	−0.247794
$177$	0	0
$178$	−26.5008	−1.98632
$179$	−18.5146	−1.38384	−0.691922	−	0.721972i	$-0.743236\pi$
$179$	−18.5146	−1.38384	−0.691922	+	0.721972i	$0.743236\pi$
$180$	0	0
$181$	18.1664	1.35029	0.675147	−	0.737683i	$-0.264080\pi$
$181$	18.1664	1.35029	0.675147	+	0.737683i	$0.264080\pi$
$182$	0	0
$183$	0	0
$184$	−23.1193	−1.70438
$185$	−27.8853	−2.05016
$186$	0	0
$187$	−7.49747	−0.548269
$188$	−7.67918	−0.560062
$189$	0	0
$190$	−14.1733	−1.02824
$191$	−23.5151	−1.70149	−0.850747	−	0.525575i	$-0.823850\pi$
$191$	−23.5151	−1.70149	−0.850747	+	0.525575i	$0.823850\pi$
$192$	0	0
$193$	14.7344	1.06061	0.530303	−	0.847808i	$-0.322078\pi$
$193$	14.7344	1.06061	0.530303	+	0.847808i	$0.322078\pi$
$194$	34.7521	2.49505
$195$	0	0
$196$	0	0
$197$	−14.5008	−1.03314	−0.516570	−	0.856245i	$-0.672791\pi$
$197$	−14.5008	−1.03314	−0.516570	+	0.856245i	$0.672791\pi$
$198$	0	0
$199$	13.5180	0.958269	0.479135	−	0.877741i	$-0.340951\pi$
$199$	13.5180	0.958269	0.479135	+	0.877741i	$0.340951\pi$
$200$	8.25259	0.583546
$201$	0	0
$202$	−27.2094	−1.91445
$203$	0	0
$204$	0	0
$205$	5.99305	0.418572
$206$	−26.5336	−1.84868
$207$	0	0
$208$	−2.08666	−0.144684
$209$	−3.52500	−0.243830
$210$	0	0
$211$	4.06419	0.279790	0.139895	−	0.990166i	$-0.455323\pi$
$211$	4.06419	0.279790	0.139895	+	0.990166i	$0.455323\pi$
$212$	−3.97942	−0.273308
$213$	0	0
$214$	−5.43487	−0.371520
$215$	−2.43139	−0.165820
$216$	0	0
$217$	0	0
$218$	34.9048	2.36405
$219$	0	0
$220$	15.3584	1.03546
$221$	−4.75902	−0.320127
$222$	0	0
$223$	0.0641862	0.00429822	0.00214911	−	0.999998i	$-0.499316\pi$
$223$	0.0641862	0.00429822	0.00214911	+	0.999998i	$0.499316\pi$
$224$	0	0
$225$	0	0
$226$	10.0832	0.670723
$227$	−1.68614	−0.111913	−0.0559564	−	0.998433i	$-0.517821\pi$
$227$	−1.68614	−0.111913	−0.0559564	+	0.998433i	$0.517821\pi$
$228$	0	0
$229$	−0.648310	−0.0428415	−0.0214208	−	0.999771i	$-0.506819\pi$
$229$	−0.648310	−0.0428415	−0.0214208	+	0.999771i	$0.506819\pi$
$230$	37.0291	2.44163
$231$	0	0
$232$	−16.7590	−1.10028
$233$	−3.10724	−0.203562	−0.101781	−	0.994807i	$-0.532454\pi$
$233$	−3.10724	−0.203562	−0.101781	+	0.994807i	$0.532454\pi$
$234$	0	0
$235$	5.58223	0.364145
$236$	−45.1697	−2.94030
$237$	0	0
$238$	0	0
$239$	15.0659	0.974534	0.487267	−	0.873253i	$-0.337994\pi$
$239$	15.0659	0.974534	0.487267	+	0.873253i	$0.337994\pi$
$240$	0	0
$241$	−25.7280	−1.65729	−0.828643	−	0.559777i	$-0.810887\pi$
$241$	−25.7280	−1.65729	−0.828643	+	0.559777i	$0.810887\pi$
$242$	−20.2688	−1.30293
$243$	0	0
$244$	27.5317	1.76253
$245$	0	0
$246$	0	0
$247$	−2.23750	−0.142369
$248$	−28.7941	−1.82843
$249$	0	0
$250$	18.4544	1.16716
$251$	−11.6258	−0.733816	−0.366908	−	0.930257i	$-0.619584\pi$
$251$	−11.6258	−0.733816	−0.366908	+	0.930257i	$0.619584\pi$
$252$	0	0
$253$	9.20941	0.578991
$254$	11.5386	0.723998
$255$	0	0
$256$	−26.9289	−1.68305
$257$	−12.5651	−0.783791	−0.391896	−	0.920010i	$-0.628181\pi$
$257$	−12.5651	−0.783791	−0.391896	+	0.920010i	$0.628181\pi$
$258$	0	0
$259$	0	0
$260$	9.74873	0.604591
$261$	0	0
$262$	−15.1508	−0.936022
$263$	−9.37068	−0.577821	−0.288911	−	0.957356i	$-0.593293\pi$
$263$	−9.37068	−0.577821	−0.288911	+	0.957356i	$0.593293\pi$
$264$	0	0
$265$	2.89276	0.177701
$266$	0	0
$267$	0	0
$268$	−57.4630	−3.51011
$269$	−10.3918	−0.633600	−0.316800	−	0.948492i	$-0.602608\pi$
$269$	−10.3918	−0.633600	−0.316800	+	0.948492i	$0.602608\pi$
$270$	0	0
$271$	5.13026	0.311641	0.155821	−	0.987785i	$-0.450198\pi$
$271$	5.13026	0.311641	0.155821	+	0.987785i	$0.450198\pi$
$272$	9.93045	0.602122
$273$	0	0
$274$	42.8926	2.59124
$275$	−3.28736	−0.198235
$276$	0	0
$277$	8.04361	0.483293	0.241647	−	0.970364i	$-0.422312\pi$
$277$	8.04361	0.483293	0.241647	+	0.970364i	$0.422312\pi$
$278$	−6.36721	−0.381880
$279$	0	0
$280$	0	0
$281$	27.8525	1.66154	0.830770	−	0.556616i	$-0.187900\pi$
$281$	27.8525	1.66154	0.830770	+	0.556616i	$0.187900\pi$
$282$	0	0
$283$	−23.8197	−1.41594	−0.707968	−	0.706244i	$-0.750388\pi$
$283$	−23.8197	−1.41594	−0.707968	+	0.706244i	$0.750388\pi$
$284$	−36.8054	−2.18400
$285$	0	0
$286$	3.74873	0.221667
$287$	0	0
$288$	0	0
$289$	5.64831	0.332254
$290$	26.8422	1.57623
$291$	0	0
$292$	−55.2230	−3.23168
$293$	−16.4612	−0.961675	−0.480838	−	0.876810i	$-0.659667\pi$
$293$	−16.4612	−0.961675	−0.480838	+	0.876810i	$0.659667\pi$
$294$	0	0
$295$	32.8352	1.91174
$296$	41.4279	2.40795
$297$	0	0
$298$	17.7624	1.02895
$299$	5.84568	0.338064
$300$	0	0
$301$	0	0
$302$	30.9706	1.78216
$303$	0	0
$304$	4.66889	0.267779
$305$	−20.0136	−1.14598
$306$	0	0
$307$	−1.95639	−0.111657	−0.0558287	−	0.998440i	$-0.517780\pi$
$307$	−1.95639	−0.111657	−0.0558287	+	0.998440i	$0.517780\pi$
$308$	0	0
$309$	0	0
$310$	46.1182	2.61934
$311$	−6.67585	−0.378552	−0.189276	−	0.981924i	$-0.560614\pi$
$311$	−6.67585	−0.378552	−0.189276	+	0.981924i	$0.560614\pi$
$312$	0	0
$313$	6.78364	0.383434	0.191717	−	0.981450i	$-0.438594\pi$
$313$	6.78364	0.383434	0.191717	+	0.981450i	$0.438594\pi$
$314$	40.7451	2.29938
$315$	0	0
$316$	−43.0497	−2.42174
$317$	30.6947	1.72399	0.861993	−	0.506920i	$-0.169216\pi$
$317$	30.6947	1.72399	0.861993	+	0.506920i	$0.169216\pi$
$318$	0	0
$319$	6.67585	0.373776
$320$	29.7624	1.66377
$321$	0	0
$322$	0	0
$323$	10.6483	0.592488
$324$	0	0
$325$	−2.08666	−0.115747
$326$	−2.02058	−0.111910
$327$	0	0
$328$	−8.90361	−0.491620
$329$	0	0
$330$	0	0
$331$	−13.8267	−0.759983	−0.379992	−	0.924990i	$-0.624073\pi$
$331$	−13.8267	−0.759983	−0.379992	+	0.924990i	$0.624073\pi$
$332$	27.1766	1.49151
$333$	0	0
$334$	43.7308	2.39284
$335$	41.7716	2.28223
$336$	0	0
$337$	23.8633	1.29992	0.649959	−	0.759969i	$-0.274786\pi$
$337$	23.8633	1.29992	0.649959	+	0.759969i	$0.274786\pi$
$338$	2.37951	0.129428
$339$	0	0
$340$	−46.3945	−2.51609
$341$	11.4699	0.621132
$342$	0	0
$343$	0	0
$344$	3.61221	0.194758
$345$	0	0
$346$	−29.1645	−1.56789
$347$	13.6012	0.730152	0.365076	−	0.930978i	$-0.381043\pi$
$347$	13.6012	0.730152	0.365076	+	0.930978i	$0.381043\pi$
$348$	0	0
$349$	16.4039	0.878078	0.439039	−	0.898468i	$-0.355319\pi$
$349$	16.4039	0.878078	0.439039	+	0.898468i	$0.355319\pi$
$350$	0	0
$351$	0	0
$352$	4.63905	0.247262
$353$	−4.73322	−0.251924	−0.125962	−	0.992035i	$-0.540202\pi$
$353$	−4.73322	−0.251924	−0.125962	+	0.992035i	$0.540202\pi$
$354$	0	0
$355$	26.7550	1.42001
$356$	−40.7848	−2.16159
$357$	0	0
$358$	−44.0556	−2.32841
$359$	17.5479	0.926142	0.463071	−	0.886321i	$-0.346747\pi$
$359$	17.5479	0.926142	0.463071	+	0.886321i	$0.346747\pi$
$360$	0	0
$361$	−13.9936	−0.736505
$362$	43.2271	2.27196
$363$	0	0
$364$	0	0
$365$	40.1433	2.10120
$366$	0	0
$367$	11.1888	0.584052	0.292026	−	0.956410i	$-0.405671\pi$
$367$	11.1888	0.584052	0.292026	+	0.956410i	$0.405671\pi$
$368$	−12.1979	−0.635861
$369$	0	0
$370$	−66.3533	−3.44954
$371$	0	0
$372$	0	0
$373$	−5.43195	−0.281256	−0.140628	−	0.990063i	$-0.544912\pi$
$373$	−5.43195	−0.281256	−0.140628	+	0.990063i	$0.544912\pi$
$374$	−17.8403	−0.922501
$375$	0	0
$376$	−8.29328	−0.427693
$377$	4.23750	0.218242
$378$	0	0
$379$	10.8422	0.556927	0.278463	−	0.960447i	$-0.410175\pi$
$379$	10.8422	0.556927	0.278463	+	0.960447i	$0.410175\pi$
$380$	−21.8128	−1.11897
$381$	0	0
$382$	−55.9545	−2.86288
$383$	10.5353	0.538328	0.269164	−	0.963094i	$-0.413253\pi$
$383$	10.5353	0.538328	0.269164	+	0.963094i	$0.413253\pi$
$384$	0	0
$385$	0	0
$386$	35.0607	1.78454
$387$	0	0
$388$	53.4836	2.71522
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	−27.8197	−1.40690
$392$	0	0
$393$	0	0
$394$	−34.5048	−1.73833
$395$	31.2942	1.57458
$396$	0	0
$397$	35.4400	1.77868	0.889340	−	0.457246i	$-0.151164\pi$
$397$	35.4400	1.77868	0.889340	+	0.457246i	$0.151164\pi$
$398$	32.1664	1.61235
$399$	0	0
$400$	4.35413	0.217707
$401$	−23.7241	−1.18473	−0.592363	−	0.805671i	$-0.701805\pi$
$401$	−23.7241	−1.18473	−0.592363	+	0.805671i	$0.701805\pi$
$402$	0	0
$403$	7.28055	0.362670
$404$	−41.8754	−2.08338
$405$	0	0
$406$	0	0
$407$	−16.5025	−0.818000
$408$	0	0
$409$	3.95639	0.195631	0.0978156	−	0.995205i	$-0.468814\pi$
$409$	3.95639	0.195631	0.0978156	+	0.995205i	$0.468814\pi$
$410$	14.2605	0.704277
$411$	0	0
$412$	−40.8352	−2.01181
$413$	0	0
$414$	0	0
$415$	−19.7555	−0.969762
$416$	2.94464	0.144373
$417$	0	0
$418$	−8.38779	−0.410260
$419$	−18.0586	−0.882219	−0.441109	−	0.897453i	$-0.645415\pi$
$419$	−18.0586	−0.882219	−0.441109	+	0.897453i	$0.645415\pi$
$420$	0	0
$421$	24.4956	1.19384	0.596921	−	0.802300i	$-0.296391\pi$
$421$	24.4956	1.19384	0.596921	+	0.802300i	$0.296391\pi$
$422$	9.67078	0.470766
$423$	0	0
$424$	−4.29765	−0.208712
$425$	9.93045	0.481697
$426$	0	0
$427$	0	0
$428$	−8.36428	−0.404303
$429$	0	0
$430$	−5.78553	−0.279003
$431$	40.7779	1.96420	0.982101	−	0.188357i	$-0.0603163\pi$
$431$	40.7779	1.96420	0.982101	+	0.188357i	$0.0603163\pi$
$432$	0	0
$433$	18.0791	0.868828	0.434414	−	0.900713i	$-0.356955\pi$
$433$	18.0791	0.868828	0.434414	+	0.900713i	$0.356955\pi$
$434$	0	0
$435$	0	0
$436$	53.7186	2.57265
$437$	−13.0797	−0.625687
$438$	0	0
$439$	−20.7561	−0.990635	−0.495317	−	0.868712i	$-0.664948\pi$
$439$	−20.7561	−0.990635	−0.495317	+	0.868712i	$0.664948\pi$
$440$	16.5866	0.790734
$441$	0	0
$442$	−11.3242	−0.538635
$443$	8.24042	0.391514	0.195757	−	0.980652i	$-0.437284\pi$
$443$	8.24042	0.391514	0.195757	+	0.980652i	$0.437284\pi$
$444$	0	0
$445$	29.6478	1.40544
$446$	0.152732	0.00723206
$447$	0	0
$448$	0	0
$449$	−36.1061	−1.70395	−0.851975	−	0.523582i	$-0.824595\pi$
$449$	−36.1061	−1.70395	−0.851975	+	0.523582i	$0.824595\pi$
$450$	0	0
$451$	3.54669	0.167007
$452$	15.5180	0.729908
$453$	0	0
$454$	−4.01218	−0.188301
$455$	0	0
$456$	0	0
$457$	−6.54052	−0.305953	−0.152976	−	0.988230i	$-0.548886\pi$
$457$	−6.54052	−0.305953	−0.152976	+	0.988230i	$0.548886\pi$
$458$	−1.54266	−0.0720838
$459$	0	0
$460$	56.9880	2.65708
$461$	−14.6987	−0.684588	−0.342294	−	0.939593i	$-0.611204\pi$
$461$	−14.6987	−0.684588	−0.342294	+	0.939593i	$0.611204\pi$
$462$	0	0
$463$	21.7775	1.01208	0.506042	−	0.862509i	$-0.331108\pi$
$463$	21.7775	1.01208	0.506042	+	0.862509i	$0.331108\pi$
$464$	−8.84220	−0.410489
$465$	0	0
$466$	−7.39371	−0.342507
$467$	35.0086	1.62000	0.810001	−	0.586428i	$-0.199466\pi$
$467$	35.0086	1.62000	0.810001	+	0.586428i	$0.199466\pi$
$468$	0	0
$469$	0	0
$470$	13.2830	0.612699
$471$	0	0
$472$	−48.7819	−2.24537
$473$	−1.43890	−0.0661607
$474$	0	0
$475$	4.66889	0.214223
$476$	0	0
$477$	0	0
$478$	35.8496	1.63972
$479$	31.7814	1.45213	0.726064	−	0.687628i	$-0.241348\pi$
$479$	31.7814	1.45213	0.726064	+	0.687628i	$0.241348\pi$
$480$	0	0
$481$	−10.4750	−0.477619
$482$	−61.2201	−2.78850
$483$	0	0
$484$	−31.1938	−1.41790
$485$	−38.8789	−1.76540
$486$	0	0
$487$	−28.5817	−1.29516	−0.647580	−	0.761998i	$-0.724219\pi$
$487$	−28.5817	−1.29516	−0.647580	+	0.761998i	$0.724219\pi$
$488$	29.7334	1.34597
$489$	0	0
$490$	0	0
$491$	−21.7160	−0.980028	−0.490014	−	0.871715i	$-0.663008\pi$
$491$	−21.7160	−0.980028	−0.490014	+	0.871715i	$0.663008\pi$
$492$	0	0
$493$	−20.1664	−0.908247
$494$	−5.32415	−0.239545
$495$	0	0
$496$	−15.1920	−0.682141
$497$	0	0
$498$	0	0
$499$	−42.6895	−1.91104	−0.955522	−	0.294921i	$-0.904707\pi$
$499$	−42.6895	−1.91104	−0.955522	+	0.294921i	$0.904707\pi$
$500$	28.4014	1.27015
$501$	0	0
$502$	−27.6638	−1.23470
$503$	21.8922	0.976125	0.488063	−	0.872809i	$-0.337704\pi$
$503$	21.8922	0.976125	0.488063	+	0.872809i	$0.337704\pi$
$504$	0	0
$505$	30.4405	1.35458
$506$	21.9139	0.974192
$507$	0	0
$508$	17.7580	0.787883
$509$	−5.36546	−0.237820	−0.118910	−	0.992905i	$-0.537940\pi$
$509$	−5.36546	−0.237820	−0.118910	+	0.992905i	$0.537940\pi$
$510$	0	0
$511$	0	0
$512$	−22.6496	−1.00098
$513$	0	0
$514$	−29.8989	−1.31878
$515$	29.6844	1.30805
$516$	0	0
$517$	3.30357	0.145291
$518$	0	0
$519$	0	0
$520$	10.5283	0.461698
$521$	−11.5221	−0.504791	−0.252396	−	0.967624i	$-0.581218\pi$
$521$	−11.5221	−0.504791	−0.252396	+	0.967624i	$0.581218\pi$
$522$	0	0
$523$	3.62584	0.158547	0.0792734	−	0.996853i	$-0.474740\pi$
$523$	3.62584	0.158547	0.0792734	+	0.996853i	$0.474740\pi$
$524$	−23.3172	−1.01862
$525$	0	0
$526$	−22.2977	−0.972224
$527$	−34.6483	−1.50930
$528$	0	0
$529$	11.1720	0.485738
$530$	6.88336	0.298994
$531$	0	0
$532$	0	0
$533$	2.25127	0.0975132
$534$	0	0
$535$	6.08026	0.262872
$536$	−62.0583	−2.68051
$537$	0	0
$538$	−24.7275	−1.06608
$539$	0	0
$540$	0	0
$541$	37.2300	1.60064	0.800321	−	0.599572i	$-0.204662\pi$
$541$	37.2300	1.60064	0.800321	+	0.599572i	$0.204662\pi$
$542$	12.2075	0.524358
$543$	0	0
$544$	−14.0136	−0.600829
$545$	−39.0497	−1.67271
$546$	0	0
$547$	−4.34529	−0.185791	−0.0928956	−	0.995676i	$-0.529612\pi$
$547$	−4.34529	−0.185791	−0.0928956	+	0.995676i	$0.529612\pi$
$548$	66.0119	2.81989
$549$	0	0
$550$	−7.82232	−0.333545
$551$	−9.48140	−0.403921
$552$	0	0
$553$	0	0
$554$	19.1399	0.813175
$555$	0	0
$556$	−9.79915	−0.415577
$557$	−28.0052	−1.18662	−0.593310	−	0.804974i	$-0.702179\pi$
$557$	−28.0052	−1.18662	−0.593310	+	0.804974i	$0.702179\pi$
$558$	0	0
$559$	−0.913344	−0.0386303
$560$	0	0
$561$	0	0
$562$	66.2753	2.79566
$563$	−22.2525	−0.937829	−0.468915	−	0.883243i	$-0.655355\pi$
$563$	−22.2525	−0.937829	−0.468915	+	0.883243i	$0.655355\pi$
$564$	0	0
$565$	−11.2805	−0.474576
$566$	−56.6793	−2.38241
$567$	0	0
$568$	−39.7487	−1.66782
$569$	−3.10724	−0.130262	−0.0651311	−	0.997877i	$-0.520747\pi$
$569$	−3.10724	−0.130262	−0.0651311	+	0.997877i	$0.520747\pi$
$570$	0	0
$571$	16.5772	0.693733	0.346866	−	0.937915i	$-0.387246\pi$
$571$	16.5772	0.693733	0.346866	+	0.937915i	$0.387246\pi$
$572$	5.76931	0.241227
$573$	0	0
$574$	0	0
$575$	−12.1979	−0.508689
$576$	0	0
$577$	7.45253	0.310253	0.155126	−	0.987895i	$-0.450422\pi$
$577$	7.45253	0.310253	0.155126	+	0.987895i	$0.450422\pi$
$578$	13.4402	0.559039
$579$	0	0
$580$	41.3102	1.71531
$581$	0	0
$582$	0	0
$583$	1.71194	0.0709014
$584$	−59.6392	−2.46789
$585$	0	0
$586$	−39.1697	−1.61809
$587$	11.9892	0.494845	0.247423	−	0.968908i	$-0.420416\pi$
$587$	11.9892	0.494845	0.247423	+	0.968908i	$0.420416\pi$
$588$	0	0
$589$	−16.2902	−0.671227
$590$	78.1319	3.21664
$591$	0	0
$592$	21.8577	0.898347
$593$	−33.4767	−1.37473	−0.687363	−	0.726314i	$-0.741232\pi$
$593$	−33.4767	−1.37473	−0.687363	+	0.726314i	$0.741232\pi$
$594$	0	0
$595$	0	0
$596$	27.3363	1.11974
$597$	0	0
$598$	13.9099	0.568817
$599$	13.5863	0.555120	0.277560	−	0.960708i	$-0.410474\pi$
$599$	13.5863	0.555120	0.277560	+	0.960708i	$0.410474\pi$
$600$	0	0
$601$	6.78364	0.276710	0.138355	−	0.990383i	$-0.455818\pi$
$601$	6.78364	0.276710	0.138355	+	0.990383i	$0.455818\pi$
$602$	0	0
$603$	0	0
$604$	47.6638	1.93941
$605$	22.6757	0.921898
$606$	0	0
$607$	17.4905	0.709918	0.354959	−	0.934882i	$-0.384495\pi$
$607$	17.4905	0.709918	0.354959	+	0.934882i	$0.384495\pi$
$608$	−6.58863	−0.267204
$609$	0	0
$610$	−47.6227	−1.92819
$611$	2.09695	0.0848334
$612$	0	0
$613$	−38.2455	−1.54472	−0.772361	−	0.635184i	$-0.780924\pi$
$613$	−38.2455	−1.54472	−0.772361	+	0.635184i	$0.780924\pi$
$614$	−4.65526	−0.187871
$615$	0	0
$616$	0	0
$617$	34.1542	1.37500	0.687498	−	0.726187i	$-0.258709\pi$
$617$	34.1542	1.37500	0.687498	+	0.726187i	$0.258709\pi$
$618$	0	0
$619$	8.56805	0.344379	0.172190	−	0.985064i	$-0.444916\pi$
$619$	8.56805	0.344379	0.172190	+	0.985064i	$0.444916\pi$
$620$	70.9761	2.85047
$621$	0	0
$622$	−15.8853	−0.636941
$623$	0	0
$624$	0	0
$625$	−31.0791	−1.24317
$626$	16.1417	0.645154
$627$	0	0
$628$	62.7069	2.50228
$629$	49.8508	1.98768
$630$	0	0
$631$	−17.7119	−0.705101	−0.352551	−	0.935793i	$-0.614686\pi$
$631$	−17.7119	−0.705101	−0.352551	+	0.935793i	$0.614686\pi$
$632$	−46.4924	−1.84937
$633$	0	0
$634$	73.0384	2.90073
$635$	−12.9088	−0.512271
$636$	0	0
$637$	0	0
$638$	15.8853	0.628903
$639$	0	0
$640$	55.1422	2.17969
$641$	−16.6117	−0.656121	−0.328061	−	0.944657i	$-0.606395\pi$
$641$	−16.6117	−0.656121	−0.328061	+	0.944657i	$0.606395\pi$
$642$	0	0
$643$	−12.9500	−0.510698	−0.255349	−	0.966849i	$-0.582190\pi$
$643$	−12.9500	−0.510698	−0.255349	+	0.966849i	$0.582190\pi$
$644$	0	0
$645$	0	0
$646$	25.3378	0.996902
$647$	−23.6775	−0.930857	−0.465428	−	0.885086i	$-0.654100\pi$
$647$	−23.6775	−0.930857	−0.465428	+	0.885086i	$0.654100\pi$
$648$	0	0
$649$	19.4319	0.762771
$650$	−4.96522	−0.194752
$651$	0	0
$652$	−3.10968	−0.121785
$653$	−22.2811	−0.871927	−0.435963	−	0.899964i	$-0.643592\pi$
$653$	−22.2811	−0.871927	−0.435963	+	0.899964i	$0.643592\pi$
$654$	0	0
$655$	16.9500	0.662291
$656$	−4.69762	−0.183411
$657$	0	0
$658$	0	0
$659$	−15.5165	−0.604435	−0.302218	−	0.953239i	$-0.597727\pi$
$659$	−15.5165	−0.604435	−0.302218	+	0.953239i	$0.597727\pi$
$660$	0	0
$661$	−28.1444	−1.09469	−0.547346	−	0.836906i	$-0.684362\pi$
$661$	−28.1444	−1.09469	−0.547346	+	0.836906i	$0.684362\pi$
$662$	−32.9008	−1.27872
$663$	0	0
$664$	29.3500	1.13900
$665$	0	0
$666$	0	0
$667$	24.7711	0.959139
$668$	67.3018	2.60399
$669$	0	0
$670$	99.3961	3.84000
$671$	−11.8441	−0.457236
$672$	0	0
$673$	7.13477	0.275025	0.137513	−	0.990500i	$-0.456089\pi$
$673$	7.13477	0.275025	0.137513	+	0.990500i	$0.456089\pi$
$674$	56.7831	2.18720
$675$	0	0
$676$	3.66208	0.140849
$677$	−10.7660	−0.413770	−0.206885	−	0.978365i	$-0.566333\pi$
$677$	−10.7660	−0.413770	−0.206885	+	0.978365i	$0.566333\pi$
$678$	0	0
$679$	0	0
$680$	−50.1046	−1.92142
$681$	0	0
$682$	27.2928	1.04510
$683$	12.2020	0.466898	0.233449	−	0.972369i	$-0.424999\pi$
$683$	12.2020	0.466898	0.233449	+	0.972369i	$0.424999\pi$
$684$	0	0
$685$	−47.9861	−1.83345
$686$	0	0
$687$	0	0
$688$	1.90583	0.0726593
$689$	1.08666	0.0413983
$690$	0	0
$691$	−32.0367	−1.21873	−0.609366	−	0.792889i	$-0.708576\pi$
$691$	−32.0367	−1.21873	−0.609366	+	0.792889i	$0.708576\pi$
$692$	−44.8842	−1.70624
$693$	0	0
$694$	32.3643	1.22853
$695$	7.12331	0.270202
$696$	0	0
$697$	−10.7138	−0.405815
$698$	39.0332	1.47743
$699$	0	0
$700$	0	0
$701$	14.9513	0.564704	0.282352	−	0.959311i	$-0.408885\pi$
$701$	14.9513	0.564704	0.282352	+	0.959311i	$0.408885\pi$
$702$	0	0
$703$	23.4378	0.883973
$704$	17.6134	0.663830
$705$	0	0
$706$	−11.2627	−0.423879
$707$	0	0
$708$	0	0
$709$	−8.58279	−0.322333	−0.161167	−	0.986927i	$-0.551526\pi$
$709$	−8.58279	−0.322333	−0.161167	+	0.986927i	$0.551526\pi$
$710$	63.6638	2.38926
$711$	0	0
$712$	−44.0464	−1.65071
$713$	42.5598	1.59388
$714$	0	0
$715$	−4.19389	−0.156843
$716$	−67.8018	−2.53387
$717$	0	0
$718$	41.7554	1.55830
$719$	35.9644	1.34125	0.670623	−	0.741798i	$-0.266027\pi$
$719$	35.9644	1.34125	0.670623	+	0.741798i	$0.266027\pi$
$720$	0	0
$721$	0	0
$722$	−33.2979	−1.23922
$723$	0	0
$724$	66.5266	2.47244
$725$	−8.84220	−0.328391
$726$	0	0
$727$	31.5111	1.16868	0.584341	−	0.811508i	$-0.301353\pi$
$727$	31.5111	1.16868	0.584341	+	0.811508i	$0.301353\pi$
$728$	0	0
$729$	0	0
$730$	95.5215	3.53541
$731$	4.34662	0.160766
$732$	0	0
$733$	26.6047	0.982667	0.491334	−	0.870971i	$-0.336510\pi$
$733$	26.6047	0.982667	0.491334	+	0.870971i	$0.336510\pi$
$734$	26.6240	0.982708
$735$	0	0
$736$	17.2134	0.634496
$737$	24.7205	0.910591
$738$	0	0
$739$	−50.3397	−1.85177	−0.925887	−	0.377800i	$-0.876681\pi$
$739$	−50.3397	−1.85177	−0.925887	+	0.377800i	$0.876681\pi$
$740$	−102.118	−3.75393
$741$	0	0
$742$	0	0
$743$	−23.4021	−0.858540	−0.429270	−	0.903176i	$-0.641229\pi$
$743$	−23.4021	−0.858540	−0.429270	+	0.903176i	$0.641229\pi$
$744$	0	0
$745$	−19.8716	−0.728040
$746$	−12.9254	−0.473232
$747$	0	0
$748$	−27.4563	−1.00390
$749$	0	0
$750$	0	0
$751$	49.0305	1.78915	0.894574	−	0.446920i	$-0.147479\pi$
$751$	49.0305	1.78915	0.894574	+	0.446920i	$0.147479\pi$
$752$	−4.37561	−0.159562
$753$	0	0
$754$	10.0832	0.367208
$755$	−34.6483	−1.26098
$756$	0	0
$757$	−35.3883	−1.28621	−0.643106	−	0.765778i	$-0.722354\pi$
$757$	−35.3883	−1.28621	−0.643106	+	0.765778i	$0.722354\pi$
$758$	25.7992	0.937067
$759$	0	0
$760$	−23.5571	−0.854507
$761$	27.7051	1.00431	0.502155	−	0.864778i	$-0.332541\pi$
$761$	27.7051	1.00431	0.502155	+	0.864778i	$0.332541\pi$
$762$	0	0
$763$	0	0
$764$	−86.1142	−3.11550
$765$	0	0
$766$	25.0689	0.905775
$767$	12.3344	0.445371
$768$	0	0
$769$	−21.1383	−0.762265	−0.381133	−	0.924520i	$-0.624466\pi$
$769$	−21.1383	−0.762265	−0.381133	+	0.924520i	$0.624466\pi$
$770$	0	0
$771$	0	0
$772$	53.9586	1.94201
$773$	−41.4537	−1.49099	−0.745493	−	0.666513i	$-0.767786\pi$
$773$	−41.4537	−1.49099	−0.745493	+	0.666513i	$0.767786\pi$
$774$	0	0
$775$	−15.1920	−0.545713
$776$	57.7606	2.07349
$777$	0	0
$778$	−14.2771	−0.511858
$779$	−5.03721	−0.180477
$780$	0	0
$781$	15.8336	0.566572
$782$	−66.1974	−2.36721
$783$	0	0
$784$	0	0
$785$	−45.5836	−1.62695
$786$	0	0
$787$	28.7711	1.02558	0.512789	−	0.858515i	$-0.328612\pi$
$787$	28.7711	1.02558	0.512789	+	0.858515i	$0.328612\pi$
$788$	−53.1031	−1.89172
$789$	0	0
$790$	74.4649	2.64934
$791$	0	0
$792$	0	0
$793$	−7.51805	−0.266974
$794$	84.3298	2.99275
$795$	0	0
$796$	49.5041	1.75463
$797$	19.5357	0.691990	0.345995	−	0.938236i	$-0.387541\pi$
$797$	19.5357	0.691990	0.345995	+	0.938236i	$0.387541\pi$
$798$	0	0
$799$	−9.97942	−0.353046
$800$	−6.14446	−0.217239
$801$	0	0
$802$	−56.4518	−1.99338
$803$	23.7569	0.838362
$804$	0	0
$805$	0	0
$806$	17.3242	0.610217
$807$	0	0
$808$	−45.2241	−1.59098
$809$	36.1824	1.27211	0.636053	−	0.771645i	$-0.280566\pi$
$809$	36.1824	1.27211	0.636053	+	0.771645i	$0.280566\pi$
$810$	0	0
$811$	−0.794087	−0.0278842	−0.0139421	−	0.999903i	$-0.504438\pi$
$811$	−0.794087	−0.0278842	−0.0139421	+	0.999903i	$0.504438\pi$
$812$	0	0
$813$	0	0
$814$	−39.2680	−1.37634
$815$	2.26052	0.0791827
$816$	0	0
$817$	2.04361	0.0714967
$818$	9.41429	0.329163
$819$	0	0
$820$	21.9470	0.766422
$821$	15.8525	0.553256	0.276628	−	0.960977i	$-0.410783\pi$
$821$	15.8525	0.553256	0.276628	+	0.960977i	$0.410783\pi$
$822$	0	0
$823$	−20.0412	−0.698591	−0.349295	−	0.937013i	$-0.613579\pi$
$823$	−20.0412	−0.698591	−0.349295	+	0.937013i	$0.613579\pi$
$824$	−44.1008	−1.53633
$825$	0	0
$826$	0	0
$827$	−34.7848	−1.20959	−0.604794	−	0.796382i	$-0.706744\pi$
$827$	−34.7848	−1.20959	−0.604794	+	0.796382i	$0.706744\pi$
$828$	0	0
$829$	30.6793	1.06554	0.532769	−	0.846261i	$-0.321152\pi$
$829$	30.6793	1.06554	0.532769	+	0.846261i	$0.321152\pi$
$830$	−47.0086	−1.63169
$831$	0	0
$832$	11.1801	0.387601
$833$	0	0
$834$	0	0
$835$	−48.9238	−1.69308
$836$	−12.9088	−0.446461
$837$	0	0
$838$	−42.9706	−1.48439
$839$	39.8238	1.37487	0.687436	−	0.726245i	$-0.258736\pi$
$839$	39.8238	1.37487	0.687436	+	0.726245i	$0.258736\pi$
$840$	0	0
$841$	−11.0436	−0.380814
$842$	58.2875	2.00872
$843$	0	0
$844$	14.8834	0.512307
$845$	−2.66208	−0.0915782
$846$	0	0
$847$	0	0
$848$	−2.26748	−0.0778655
$849$	0	0
$850$	23.6296	0.810489
$851$	−61.2335	−2.09906
$852$	0	0
$853$	−24.7505	−0.847440	−0.423720	−	0.905793i	$-0.639276\pi$
$853$	−24.7505	−0.847440	−0.423720	+	0.905793i	$0.639276\pi$
$854$	0	0
$855$	0	0
$856$	−9.03317	−0.308748
$857$	6.47097	0.221044	0.110522	−	0.993874i	$-0.464748\pi$
$857$	6.47097	0.221044	0.110522	+	0.993874i	$0.464748\pi$
$858$	0	0
$859$	−31.0741	−1.06023	−0.530117	−	0.847925i	$-0.677852\pi$
$859$	−31.0741	−1.06023	−0.530117	+	0.847925i	$0.677852\pi$
$860$	−8.90395	−0.303622
$861$	0	0
$862$	97.0314	3.30490
$863$	−28.6391	−0.974885	−0.487442	−	0.873155i	$-0.662070\pi$
$863$	−28.6391	−0.974885	−0.487442	+	0.873155i	$0.662070\pi$
$864$	0	0
$865$	32.6277	1.10938
$866$	43.0195	1.46186
$867$	0	0
$868$	0	0
$869$	18.5199	0.628246
$870$	0	0
$871$	15.6914	0.531681
$872$	58.0145	1.96462
$873$	0	0
$874$	−31.1233	−1.05276
$875$	0	0
$876$	0	0
$877$	0.00110921	3.74555e−5 0	1.87278e−5	−	1.00000i	$-0.499994\pi$
$877$	0.00110921	3.74555e−5 0	1.87278e−5		1.00000i	$0.499994\pi$
$878$	−49.3894	−1.66681
$879$	0	0
$880$	8.75121	0.295003
$881$	23.9479	0.806824	0.403412	−	0.915019i	$-0.367824\pi$
$881$	23.9479	0.806824	0.403412	+	0.915019i	$0.367824\pi$
$882$	0	0
$883$	51.9965	1.74982	0.874911	−	0.484283i	$-0.160919\pi$
$883$	51.9965	1.74982	0.874911	+	0.484283i	$0.160919\pi$
$884$	−17.4279	−0.586164
$885$	0	0
$886$	19.6082	0.658750
$887$	10.8697	0.364970	0.182485	−	0.983209i	$-0.441586\pi$
$887$	10.8697	0.364970	0.182485	+	0.983209i	$0.441586\pi$
$888$	0	0
$889$	0	0
$890$	70.5472	2.36475
$891$	0	0
$892$	0.235055	0.00787021
$893$	−4.69192	−0.157009
$894$	0	0
$895$	49.2872	1.64749
$896$	0	0
$897$	0	0
$898$	−85.9148	−2.86701
$899$	30.8513	1.02895
$900$	0	0
$901$	−5.17142	−0.172285
$902$	8.43940	0.281001
$903$	0	0
$904$	16.7590	0.557397
$905$	−48.3603	−1.60755
$906$	0	0
$907$	−25.4892	−0.846354	−0.423177	−	0.906047i	$-0.639085\pi$
$907$	−25.4892	−0.846354	−0.423177	+	0.906047i	$0.639085\pi$
$908$	−6.17476	−0.204917
$909$	0	0
$910$	0	0
$911$	−42.2059	−1.39834	−0.699172	−	0.714953i	$-0.746448\pi$
$911$	−42.2059	−1.39834	−0.699172	+	0.714953i	$0.746448\pi$
$912$	0	0
$913$	−11.6914	−0.386928
$914$	−15.5632	−0.514786
$915$	0	0
$916$	−2.37416	−0.0784445
$917$	0	0
$918$	0	0
$919$	−29.4033	−0.969925	−0.484963	−	0.874535i	$-0.661167\pi$
$919$	−29.4033	−0.969925	−0.484963	+	0.874535i	$0.661167\pi$
$920$	61.5453	2.02909
$921$	0	0
$922$	−34.9758	−1.15187
$923$	10.0504	0.330814
$924$	0	0
$925$	21.8577	0.718677
$926$	51.8197	1.70290
$927$	0	0
$928$	12.4779	0.409608
$929$	−13.3759	−0.438849	−0.219425	−	0.975629i	$-0.570418\pi$
$929$	−13.3759	−0.438849	−0.219425	+	0.975629i	$0.570418\pi$
$930$	0	0
$931$	0	0
$932$	−11.3789	−0.372730
$933$	0	0
$934$	83.3033	2.72577
$935$	19.9588	0.652724
$936$	0	0
$937$	24.9981	0.816653	0.408326	−	0.912836i	$-0.366113\pi$
$937$	24.9981	0.816653	0.408326	+	0.912836i	$0.366113\pi$
$938$	0	0
$939$	0	0
$940$	20.4426	0.666763
$941$	10.5369	0.343493	0.171746	−	0.985141i	$-0.445059\pi$
$941$	10.5369	0.343493	0.171746	+	0.985141i	$0.445059\pi$
$942$	0	0
$943$	13.1602	0.428555
$944$	−25.7377	−0.837692
$945$	0	0
$946$	−3.42388	−0.111320
$947$	52.7779	1.71505	0.857525	−	0.514442i	$-0.172001\pi$
$947$	52.7779	1.71505	0.857525	+	0.514442i	$0.172001\pi$
$948$	0	0
$949$	15.0797	0.489508
$950$	11.1097	0.360446
$951$	0	0
$952$	0	0
$953$	−21.8486	−0.707746	−0.353873	−	0.935294i	$-0.615135\pi$
$953$	−21.8486	−0.707746	−0.353873	+	0.935294i	$0.615135\pi$
$954$	0	0
$955$	62.5991	2.02566
$956$	55.1726	1.78441
$957$	0	0
$958$	75.6241	2.44330
$959$	0	0
$960$	0	0
$961$	22.0064	0.709884
$962$	−24.9254	−0.803627
$963$	0	0
$964$	−94.2180	−3.03456
$965$	−39.2241	−1.26267
$966$	0	0
$967$	1.94143	0.0624323	0.0312162	−	0.999513i	$-0.490062\pi$
$967$	1.94143	0.0624323	0.0312162	+	0.999513i	$0.490062\pi$
$968$	−33.6883	−1.08278
$969$	0	0
$970$	−92.5127	−2.97040
$971$	−14.9077	−0.478412	−0.239206	−	0.970969i	$-0.576887\pi$
$971$	−14.9077	−0.478412	−0.239206	+	0.970969i	$0.576887\pi$
$972$	0	0
$973$	0	0
$974$	−68.0105	−2.17920
$975$	0	0
$976$	15.6876	0.502147
$977$	16.5353	0.529011	0.264505	−	0.964384i	$-0.414791\pi$
$977$	16.5353	0.529011	0.264505	+	0.964384i	$0.414791\pi$
$978$	0	0
$979$	17.5456	0.560759
$980$	0	0
$981$	0	0
$982$	−51.6734	−1.64897
$983$	−30.0694	−0.959065	−0.479533	−	0.877524i	$-0.659194\pi$
$983$	−30.0694	−0.959065	−0.479533	+	0.877524i	$0.659194\pi$
$984$	0	0
$985$	38.6023	1.22997
$986$	−47.9861	−1.52819
$987$	0	0
$988$	−8.19389	−0.260682
$989$	−5.33912	−0.169774
$990$	0	0
$991$	43.5902	1.38469	0.692345	−	0.721567i	$-0.256578\pi$
$991$	43.5902	1.38469	0.692345	+	0.721567i	$0.256578\pi$
$992$	21.4386	0.680677
$993$	0	0
$994$	0	0
$995$	−35.9861	−1.14084
$996$	0	0
$997$	−57.8577	−1.83237	−0.916186	−	0.400753i	$-0.868749\pi$
$997$	−57.8577	−1.83237	−0.916186	+	0.400753i	$0.868749\pi$
$998$	−101.580	−3.21546
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5733.2.a.bf.1.4		4
3.2	odd	2		1911.2.a.s.1.1		4
7.6	odd	2		819.2.a.k.1.4		4
21.20	even	2		273.2.a.e.1.1	✓	4
84.83	odd	2		4368.2.a.br.1.2		4
105.104	even	2		6825.2.a.bg.1.4		4
273.272	even	2		3549.2.a.w.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
273.2.a.e.1.1	✓	4	21.20	even	2
819.2.a.k.1.4		4	7.6	odd	2
1911.2.a.s.1.1		4	3.2	odd	2
3549.2.a.w.1.4		4	273.272	even	2
4368.2.a.br.1.2		4	84.83	odd	2
5733.2.a.bf.1.4		4	1.1	even	1	trivial
6825.2.a.bg.1.4		4	105.104	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:	\( N \)	\(=\)	\( 5733 = 3^{2} \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5733.a (trivial)

\(f(q)\)	\(=\)	\(q+2.37951 q^{2} +3.66208 q^{4} -2.66208 q^{5} +3.95493 q^{8} +O(q^{10})\)	\(q+2.37951 q^{2} +3.66208 q^{4} -2.66208 q^{5} +3.95493 q^{8} -6.33445 q^{10} -1.57542 q^{11} -1.00000 q^{13} +2.08666 q^{16} +4.75902 q^{17} +2.23750 q^{19} -9.74873 q^{20} -3.74873 q^{22} -5.84568 q^{23} +2.08666 q^{25} -2.37951 q^{26} -4.23750 q^{29} -7.28055 q^{31} -2.94464 q^{32} +11.3242 q^{34} +10.4750 q^{37} +5.32415 q^{38} -10.5283 q^{40} -2.25127 q^{41} +0.913344 q^{43} -5.76931 q^{44} -13.9099 q^{46} -2.09695 q^{47} +4.96522 q^{50} -3.66208 q^{52} -1.08666 q^{53} +4.19389 q^{55} -10.0832 q^{58} -12.3344 q^{59} +7.51805 q^{61} -17.3242 q^{62} -11.1801 q^{64} +2.66208 q^{65} -15.6914 q^{67} +17.4279 q^{68} -10.0504 q^{71} -15.0797 q^{73} +24.9254 q^{74} +8.19389 q^{76} -11.7555 q^{79} -5.55484 q^{80} -5.35692 q^{82} +7.42110 q^{83} -12.6689 q^{85} +2.17331 q^{86} -6.23069 q^{88} -11.1371 q^{89} -21.4073 q^{92} -4.98971 q^{94} -5.95639 q^{95} +14.6047 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 7 q^{4} - 3 q^{5} - 3 q^{8}+O(q^{10}) \) 4 * q - q^2 + 7 * q^4 - 3 * q^5 - 3 * q^8	\( 4 q - q^{2} + 7 q^{4} - 3 q^{5} - 3 q^{8} + 4 q^{10} + 2 q^{11} - 4 q^{13} + 9 q^{16} - 2 q^{17} - 7 q^{19} - 32 q^{20} - 8 q^{22} - 3 q^{23} + 9 q^{25} + q^{26} - q^{29} - 3 q^{31} - 7 q^{32} + 30 q^{34} + 10 q^{37} + 6 q^{38} + 14 q^{40} - 16 q^{41} + 3 q^{43} + 12 q^{44} - 18 q^{46} + 5 q^{47} - 13 q^{50} - 7 q^{52} - 5 q^{53} - 10 q^{55} - 4 q^{58} - 20 q^{59} - 12 q^{61} - 54 q^{62} + 5 q^{64} + 3 q^{65} - 22 q^{67} - 10 q^{68} + 13 q^{73} + 6 q^{74} + 6 q^{76} + 11 q^{79} - 42 q^{80} - 10 q^{82} + q^{83} + 8 q^{85} + 10 q^{86} - 60 q^{88} - 5 q^{89} - 34 q^{92} - 34 q^{94} - 13 q^{95} + 17 q^{97}+O(q^{100}) \) 4 * q - q^2 + 7 * q^4 - 3 * q^5 - 3 * q^8 + 4 * q^10 + 2 * q^11 - 4 * q^13 + 9 * q^16 - 2 * q^17 - 7 * q^19 - 32 * q^20 - 8 * q^22 - 3 * q^23 + 9 * q^25 + q^26 - q^29 - 3 * q^31 - 7 * q^32 + 30 * q^34 + 10 * q^37 + 6 * q^38 + 14 * q^40 - 16 * q^41 + 3 * q^43 + 12 * q^44 - 18 * q^46 + 5 * q^47 - 13 * q^50 - 7 * q^52 - 5 * q^53 - 10 * q^55 - 4 * q^58 - 20 * q^59 - 12 * q^61 - 54 * q^62 + 5 * q^64 + 3 * q^65 - 22 * q^67 - 10 * q^68 + 13 * q^73 + 6 * q^74 + 6 * q^76 + 11 * q^79 - 42 * q^80 - 10 * q^82 + q^83 + 8 * q^85 + 10 * q^86 - 60 * q^88 - 5 * q^89 - 34 * q^92 - 34 * q^94 - 13 * q^95 + 17 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more