LMFDB - Embedded newform 2259.2.a.k.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2259,2,Mod(1,2259)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2259, 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("2259.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2259 = 3^{2} \cdot 251 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2259.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:	$18.0382058166$
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} - 2 x^{16} - 28 x^{15} + 54 x^{14} + 317 x^{13} - 582 x^{12} - 1867 x^{11} + 3178 x^{10} + \cdots + 256 $ x^17 - 2x^16 - 28x^15 + 54x^14 + 317x^13 - 582x^12 - 1867x^11 + 3178x^10 + 6186x^9 - 9216x^8 - 11921x^7 + 13680x^6 + 13752x^5 - 9400x^4 - 8800x^3 + 1920x^2 + 2240x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 251)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-0.622810$ of defining polynomial
Character	$\chi$	$=$	2259.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.622810 q^{2} -1.61211 q^{4} +1.69081 q^{5} +3.93205 q^{7} -2.24966 q^{8} +O(q^{10})$	\(q+0.622810 q^{2} -1.61211 q^{4} +1.69081 q^{5} +3.93205 q^{7} -2.24966 q^{8} +1.05305 q^{10} +3.01556 q^{11} +6.58110 q^{13} +2.44892 q^{14} +1.82311 q^{16} -2.00738 q^{17} +5.92036 q^{19} -2.72577 q^{20} +1.87812 q^{22} -5.10129 q^{23} -2.14116 q^{25} +4.09877 q^{26} -6.33889 q^{28} -3.88820 q^{29} -8.26372 q^{31} +5.63476 q^{32} -1.25022 q^{34} +6.64835 q^{35} -3.68109 q^{37} +3.68726 q^{38} -3.80374 q^{40} -1.89627 q^{41} +7.00055 q^{43} -4.86141 q^{44} -3.17714 q^{46} +9.72137 q^{47} +8.46101 q^{49} -1.33354 q^{50} -10.6094 q^{52} -6.65232 q^{53} +5.09874 q^{55} -8.84576 q^{56} -2.42161 q^{58} -9.91108 q^{59} +10.5438 q^{61} -5.14673 q^{62} -0.136826 q^{64} +11.1274 q^{65} -2.42143 q^{67} +3.23611 q^{68} +4.14066 q^{70} +15.6398 q^{71} +10.8434 q^{73} -2.29262 q^{74} -9.54426 q^{76} +11.8573 q^{77} +6.40419 q^{79} +3.08253 q^{80} -1.18101 q^{82} +6.22959 q^{83} -3.39410 q^{85} +4.36002 q^{86} -6.78398 q^{88} -3.23865 q^{89} +25.8772 q^{91} +8.22383 q^{92} +6.05457 q^{94} +10.0102 q^{95} +5.85568 q^{97} +5.26960 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q - 2 q^{2} + 26 q^{4} - 3 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) $ 17 * q - 2 * q^2 + 26 * q^4 - 3 * q^5 + 3 * q^7 - 6 * q^8	$ 17 q - 2 q^{2} + 26 q^{4} - 3 q^{5} + 3 q^{7} - 6 q^{8} + 7 q^{10} + q^{11} + 22 q^{13} + 7 q^{14} + 40 q^{16} + q^{17} + 13 q^{19} + 14 q^{20} + 4 q^{22} + 2 q^{23} + 32 q^{25} + 9 q^{26} - 10 q^{28} - 28 q^{29} + 12 q^{31} - 4 q^{32} - 21 q^{34} + 15 q^{35} + 27 q^{37} + 37 q^{38} - 7 q^{40} + q^{41} + 9 q^{43} + 43 q^{44} + 4 q^{46} + 20 q^{47} + 32 q^{49} + 28 q^{50} - q^{52} - q^{53} - 11 q^{55} + 61 q^{56} - 46 q^{58} + 20 q^{59} + 59 q^{61} + 73 q^{62} + 54 q^{64} + 14 q^{65} + 15 q^{67} + 20 q^{68} - 11 q^{70} + 26 q^{71} + 8 q^{73} - 2 q^{74} + 38 q^{76} + 33 q^{79} + 29 q^{80} + 10 q^{82} + 67 q^{85} - 11 q^{86} + 27 q^{88} - 11 q^{89} - 2 q^{91} - 28 q^{92} + 29 q^{94} + 8 q^{95} - 10 q^{97} - 22 q^{98}+O(q^{100}) $ 17 * q - 2 * q^2 + 26 * q^4 - 3 * q^5 + 3 * q^7 - 6 * q^8 + 7 * q^10 + q^11 + 22 * q^13 + 7 * q^14 + 40 * q^16 + q^17 + 13 * q^19 + 14 * q^20 + 4 * q^22 + 2 * q^23 + 32 * q^25 + 9 * q^26 - 10 * q^28 - 28 * q^29 + 12 * q^31 - 4 * q^32 - 21 * q^34 + 15 * q^35 + 27 * q^37 + 37 * q^38 - 7 * q^40 + q^41 + 9 * q^43 + 43 * q^44 + 4 * q^46 + 20 * q^47 + 32 * q^49 + 28 * q^50 - q^52 - q^53 - 11 * q^55 + 61 * q^56 - 46 * q^58 + 20 * q^59 + 59 * q^61 + 73 * q^62 + 54 * q^64 + 14 * q^65 + 15 * q^67 + 20 * q^68 - 11 * q^70 + 26 * q^71 + 8 * q^73 - 2 * q^74 + 38 * q^76 + 33 * q^79 + 29 * q^80 + 10 * q^82 + 67 * q^85 - 11 * q^86 + 27 * q^88 - 11 * q^89 - 2 * q^91 - 28 * q^92 + 29 * q^94 + 8 * q^95 - 10 * q^97 - 22 * 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 $

$2$	0.622810	0.440393	0.220197	−	0.975456i	$-0.429330\pi$
$2$	0.622810	0.440393	0.220197	+	0.975456i	$0.429330\pi$
$3$	0	0
$3$	0	0
$4$	−1.61211	−0.806054
$5$	1.69081	0.756153	0.378077	−	0.925774i	$-0.376586\pi$
$5$	1.69081	0.756153	0.378077	+	0.925774i	$0.376586\pi$
$6$	0	0
$7$	3.93205	1.48618	0.743088	−	0.669194i	$-0.233361\pi$
$7$	3.93205	1.48618	0.743088	+	0.669194i	$0.233361\pi$
$8$	−2.24966	−0.795374
$9$	0	0
$10$	1.05305	0.333005
$11$	3.01556	0.909226	0.454613	−	0.890689i	$-0.349778\pi$
$11$	3.01556	0.909226	0.454613	+	0.890689i	$0.349778\pi$
$12$	0	0
$13$	6.58110	1.82527	0.912634	−	0.408778i	$-0.134045\pi$
$13$	6.58110	1.82527	0.912634	+	0.408778i	$0.134045\pi$
$14$	2.44892	0.654501
$15$	0	0
$16$	1.82311	0.455777
$17$	−2.00738	−0.486861	−0.243431	−	0.969918i	$-0.578273\pi$
$17$	−2.00738	−0.486861	−0.243431	+	0.969918i	$0.578273\pi$
$18$	0	0
$19$	5.92036	1.35822	0.679112	−	0.734034i	$-0.262365\pi$
$19$	5.92036	1.35822	0.679112	+	0.734034i	$0.262365\pi$
$20$	−2.72577	−0.609500
$21$	0	0
$22$	1.87812	0.400417
$23$	−5.10129	−1.06369	−0.531847	−	0.846841i	$-0.678502\pi$
$23$	−5.10129	−1.06369	−0.531847	+	0.846841i	$0.678502\pi$
$24$	0	0
$25$	−2.14116	−0.428232
$26$	4.09877	0.803836
$27$	0	0
$28$	−6.33889	−1.19794
$29$	−3.88820	−0.722020	−0.361010	−	0.932562i	$-0.617568\pi$
$29$	−3.88820	−0.722020	−0.361010	+	0.932562i	$0.617568\pi$
$30$	0	0
$31$	−8.26372	−1.48421	−0.742104	−	0.670285i	$-0.766172\pi$
$31$	−8.26372	−1.48421	−0.742104	+	0.670285i	$0.766172\pi$
$32$	5.63476	0.996095
$33$	0	0
$34$	−1.25022	−0.214410
$35$	6.64835	1.12378
$36$	0	0
$37$	−3.68109	−0.605168	−0.302584	−	0.953123i	$-0.597849\pi$
$37$	−3.68109	−0.605168	−0.302584	+	0.953123i	$0.597849\pi$
$38$	3.68726	0.598153
$39$	0	0
$40$	−3.80374	−0.601425
$41$	−1.89627	−0.296147	−0.148073	−	0.988976i	$-0.547307\pi$
$41$	−1.89627	−0.296147	−0.148073	+	0.988976i	$0.547307\pi$
$42$	0	0
$43$	7.00055	1.06757	0.533787	−	0.845619i	$-0.320768\pi$
$43$	7.00055	1.06757	0.533787	+	0.845619i	$0.320768\pi$
$44$	−4.86141	−0.732885
$45$	0	0
$46$	−3.17714	−0.468443
$47$	9.72137	1.41801	0.709004	−	0.705205i	$-0.249145\pi$
$47$	9.72137	1.41801	0.709004	+	0.705205i	$0.249145\pi$
$48$	0	0
$49$	8.46101	1.20872
$50$	−1.33354	−0.188590
$51$	0	0
$52$	−10.6094	−1.47126
$53$	−6.65232	−0.913766	−0.456883	−	0.889527i	$-0.651034\pi$
$53$	−6.65232	−0.913766	−0.456883	+	0.889527i	$0.651034\pi$
$54$	0	0
$55$	5.09874	0.687515
$56$	−8.84576	−1.18206
$57$	0	0
$58$	−2.42161	−0.317973
$59$	−9.91108	−1.29031	−0.645156	−	0.764050i	$-0.723208\pi$
$59$	−9.91108	−1.29031	−0.645156	+	0.764050i	$0.723208\pi$
$60$	0	0
$61$	10.5438	1.34999	0.674995	−	0.737822i	$-0.264146\pi$
$61$	10.5438	1.34999	0.674995	+	0.737822i	$0.264146\pi$
$62$	−5.14673	−0.653635
$63$	0	0
$64$	−0.136826	−0.0171032
$65$	11.1274	1.38018
$66$	0	0
$67$	−2.42143	−0.295825	−0.147912	−	0.989000i	$-0.547255\pi$
$67$	−2.42143	−0.295825	−0.147912	+	0.989000i	$0.547255\pi$
$68$	3.23611	0.392436
$69$	0	0
$70$	4.14066	0.494904
$71$	15.6398	1.85610	0.928050	−	0.372456i	$-0.121484\pi$
$71$	15.6398	1.85610	0.928050	+	0.372456i	$0.121484\pi$
$72$	0	0
$73$	10.8434	1.26912	0.634562	−	0.772872i	$-0.281181\pi$
$73$	10.8434	1.26912	0.634562	+	0.772872i	$0.281181\pi$
$74$	−2.29262	−0.266512
$75$	0	0
$76$	−9.54426	−1.09480
$77$	11.8573	1.35127
$78$	0	0
$79$	6.40419	0.720528	0.360264	−	0.932850i	$-0.382687\pi$
$79$	6.40419	0.720528	0.360264	+	0.932850i	$0.382687\pi$
$80$	3.08253	0.344637
$81$	0	0
$82$	−1.18101	−0.130421
$83$	6.22959	0.683786	0.341893	−	0.939739i	$-0.388932\pi$
$83$	6.22959	0.683786	0.341893	+	0.939739i	$0.388932\pi$
$84$	0	0
$85$	−3.39410	−0.368142
$86$	4.36002	0.470153
$87$	0	0
$88$	−6.78398	−0.723175
$89$	−3.23865	−0.343296	−0.171648	−	0.985158i	$-0.554909\pi$
$89$	−3.23865	−0.343296	−0.171648	+	0.985158i	$0.554909\pi$
$90$	0	0
$91$	25.8772	2.71267
$92$	8.22383	0.857394
$93$	0	0
$94$	6.05457	0.624481
$95$	10.0102	1.02703
$96$	0	0
$97$	5.85568	0.594554	0.297277	−	0.954791i	$-0.403922\pi$
$97$	5.85568	0.594554	0.297277	+	0.954791i	$0.403922\pi$
$98$	5.26960	0.532310
$99$	0	0
$100$	3.45178	0.345178
$101$	−6.73240	−0.669899	−0.334949	−	0.942236i	$-0.608719\pi$
$101$	−6.73240	−0.669899	−0.334949	+	0.942236i	$0.608719\pi$
$102$	0	0
$103$	−1.87998	−0.185240	−0.0926199	−	0.995702i	$-0.529524\pi$
$103$	−1.87998	−0.185240	−0.0926199	+	0.995702i	$0.529524\pi$
$104$	−14.8052	−1.45177
$105$	0	0
$106$	−4.14313	−0.402417
$107$	−16.6182	−1.60654	−0.803271	−	0.595613i	$-0.796909\pi$
$107$	−16.6182	−1.60654	−0.803271	+	0.595613i	$0.796909\pi$
$108$	0	0
$109$	0.834566	0.0799369	0.0399685	−	0.999201i	$-0.487274\pi$
$109$	0.834566	0.0799369	0.0399685	+	0.999201i	$0.487274\pi$
$110$	3.17555	0.302777
$111$	0	0
$112$	7.16854	0.677364
$113$	5.91577	0.556509	0.278254	−	0.960507i	$-0.410244\pi$
$113$	5.91577	0.556509	0.278254	+	0.960507i	$0.410244\pi$
$114$	0	0
$115$	−8.62532	−0.804315
$116$	6.26820	0.581987
$117$	0	0
$118$	−6.17272	−0.568245
$119$	−7.89312	−0.723561
$120$	0	0
$121$	−1.90639	−0.173308
$122$	6.56676	0.594526
$123$	0	0
$124$	13.3220	1.19635
$125$	−12.0743	−1.07996
$126$	0	0
$127$	−10.5329	−0.934643	−0.467322	−	0.884087i	$-0.654781\pi$
$127$	−10.5329	−0.934643	−0.467322	+	0.884087i	$0.654781\pi$
$128$	−11.3547	−1.00363
$129$	0	0
$130$	6.93025	0.607823
$131$	17.3283	1.51398	0.756992	−	0.653424i	$-0.226668\pi$
$131$	17.3283	1.51398	0.756992	+	0.653424i	$0.226668\pi$
$132$	0	0
$133$	23.2792	2.01856
$134$	−1.50809	−0.130279
$135$	0	0
$136$	4.51592	0.387237
$137$	−13.8693	−1.18493	−0.592466	−	0.805596i	$-0.701845\pi$
$137$	−13.8693	−1.18493	−0.592466	+	0.805596i	$0.701845\pi$
$138$	0	0
$139$	−19.6739	−1.66872	−0.834361	−	0.551218i	$-0.814163\pi$
$139$	−19.6739	−1.66872	−0.834361	+	0.551218i	$0.814163\pi$
$140$	−10.7179	−0.905824
$141$	0	0
$142$	9.74061	0.817414
$143$	19.8457	1.65958
$144$	0	0
$145$	−6.57421	−0.545958
$146$	6.75338	0.558913
$147$	0	0
$148$	5.93432	0.487798
$149$	0.0299644	0.00245478	0.00122739	−	0.999999i	$-0.499609\pi$
$149$	0.0299644	0.00245478	0.00122739	+	0.999999i	$0.499609\pi$
$150$	0	0
$151$	−0.462488	−0.0376367	−0.0188184	−	0.999823i	$-0.505990\pi$
$151$	−0.462488	−0.0376367	−0.0188184	+	0.999823i	$0.505990\pi$
$152$	−13.3188	−1.08030
$153$	0	0
$154$	7.38487	0.595090
$155$	−13.9724	−1.12229
$156$	0	0
$157$	−5.93306	−0.473510	−0.236755	−	0.971569i	$-0.576084\pi$
$157$	−5.93306	−0.473510	−0.236755	+	0.971569i	$0.576084\pi$
$158$	3.98860	0.317316
$159$	0	0
$160$	9.52732	0.753201
$161$	−20.0585	−1.58083
$162$	0	0
$163$	−5.34504	−0.418656	−0.209328	−	0.977845i	$-0.567128\pi$
$163$	−5.34504	−0.418656	−0.209328	+	0.977845i	$0.567128\pi$
$164$	3.05698	0.238710
$165$	0	0
$166$	3.87985	0.301135
$167$	16.6417	1.28777	0.643887	−	0.765121i	$-0.277321\pi$
$167$	16.6417	1.28777	0.643887	+	0.765121i	$0.277321\pi$
$168$	0	0
$169$	30.3108	2.33160
$170$	−2.11388	−0.162127
$171$	0	0
$172$	−11.2856	−0.860522
$173$	18.7115	1.42261	0.711304	−	0.702884i	$-0.248105\pi$
$173$	18.7115	1.42261	0.711304	+	0.702884i	$0.248105\pi$
$174$	0	0
$175$	−8.41915	−0.636428
$176$	5.49769	0.414404
$177$	0	0
$178$	−2.01707	−0.151185
$179$	−5.50738	−0.411641	−0.205821	−	0.978590i	$-0.565986\pi$
$179$	−5.50738	−0.411641	−0.205821	+	0.978590i	$0.565986\pi$
$180$	0	0
$181$	22.3213	1.65913	0.829563	−	0.558413i	$-0.188589\pi$
$181$	22.3213	1.65913	0.829563	+	0.558413i	$0.188589\pi$
$182$	16.1166	1.19464
$183$	0	0
$184$	11.4762	0.846034
$185$	−6.22403	−0.457600
$186$	0	0
$187$	−6.05338	−0.442667
$188$	−15.6719	−1.14299
$189$	0	0
$190$	6.23446	0.452295
$191$	−9.45994	−0.684497	−0.342249	−	0.939609i	$-0.611189\pi$
$191$	−9.45994	−0.684497	−0.342249	+	0.939609i	$0.611189\pi$
$192$	0	0
$193$	−13.3834	−0.963356	−0.481678	−	0.876348i	$-0.659972\pi$
$193$	−13.3834	−0.963356	−0.481678	+	0.876348i	$0.659972\pi$
$194$	3.64697	0.261837
$195$	0	0
$196$	−13.6401	−0.974290
$197$	−5.83822	−0.415956	−0.207978	−	0.978134i	$-0.566688\pi$
$197$	−5.83822	−0.415956	−0.207978	+	0.978134i	$0.566688\pi$
$198$	0	0
$199$	−11.2099	−0.794652	−0.397326	−	0.917678i	$-0.630062\pi$
$199$	−11.2099	−0.794652	−0.397326	+	0.917678i	$0.630062\pi$
$200$	4.81687	0.340604
$201$	0	0
$202$	−4.19301	−0.295019
$203$	−15.2886	−1.07305
$204$	0	0
$205$	−3.20623	−0.223933
$206$	−1.17087	−0.0815784
$207$	0	0
$208$	11.9980	0.831914
$209$	17.8532	1.23493
$210$	0	0
$211$	−9.99147	−0.687841	−0.343921	−	0.938999i	$-0.611755\pi$
$211$	−9.99147	−0.687841	−0.343921	+	0.938999i	$0.611755\pi$
$212$	10.7243	0.736545
$213$	0	0
$214$	−10.3500	−0.707511
$215$	11.8366	0.807250
$216$	0	0
$217$	−32.4934	−2.20579
$218$	0.519776	0.0352037
$219$	0	0
$220$	−8.21972	−0.554174
$221$	−13.2108	−0.888652
$222$	0	0
$223$	2.52198	0.168884	0.0844421	−	0.996428i	$-0.473089\pi$
$223$	2.52198	0.168884	0.0844421	+	0.996428i	$0.473089\pi$
$224$	22.1562	1.48037
$225$	0	0
$226$	3.68440	0.245083
$227$	20.9064	1.38761	0.693805	−	0.720163i	$-0.255933\pi$
$227$	20.9064	1.38761	0.693805	+	0.720163i	$0.255933\pi$
$228$	0	0
$229$	−11.3615	−0.750786	−0.375393	−	0.926866i	$-0.622492\pi$
$229$	−11.3615	−0.750786	−0.375393	+	0.926866i	$0.622492\pi$
$230$	−5.37194	−0.354215
$231$	0	0
$232$	8.74711	0.574276
$233$	−6.49404	−0.425439	−0.212719	−	0.977113i	$-0.568232\pi$
$233$	−6.49404	−0.425439	−0.212719	+	0.977113i	$0.568232\pi$
$234$	0	0
$235$	16.4370	1.07223
$236$	15.9777	1.04006
$237$	0	0
$238$	−4.91591	−0.318651
$239$	9.49990	0.614497	0.307249	−	0.951629i	$-0.400592\pi$
$239$	9.49990	0.614497	0.307249	+	0.951629i	$0.400592\pi$
$240$	0	0
$241$	−5.85361	−0.377064	−0.188532	−	0.982067i	$-0.560373\pi$
$241$	−5.85361	−0.377064	−0.188532	+	0.982067i	$0.560373\pi$
$242$	−1.18732	−0.0763236
$243$	0	0
$244$	−16.9977	−1.08816
$245$	14.3060	0.913975
$246$	0	0
$247$	38.9625	2.47912
$248$	18.5905	1.18050
$249$	0	0
$250$	−7.52003	−0.475608
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	−15.3833	−0.967138
$254$	−6.55999	−0.411610
$255$	0	0
$256$	−6.79820	−0.424887
$257$	19.9382	1.24371	0.621856	−	0.783131i	$-0.286379\pi$
$257$	19.9382	1.24371	0.621856	+	0.783131i	$0.286379\pi$
$258$	0	0
$259$	−14.4742	−0.899386
$260$	−17.9385	−1.11250
$261$	0	0
$262$	10.7923	0.666748
$263$	−30.5342	−1.88282	−0.941409	−	0.337267i	$-0.890498\pi$
$263$	−30.5342	−1.88282	−0.941409	+	0.337267i	$0.890498\pi$
$264$	0	0
$265$	−11.2478	−0.690948
$266$	14.4985	0.888960
$267$	0	0
$268$	3.90361	0.238451
$269$	5.26126	0.320785	0.160392	−	0.987053i	$-0.448724\pi$
$269$	5.26126	0.320785	0.160392	+	0.987053i	$0.448724\pi$
$270$	0	0
$271$	18.5566	1.12724	0.563618	−	0.826036i	$-0.309409\pi$
$271$	18.5566	1.12724	0.563618	+	0.826036i	$0.309409\pi$
$272$	−3.65967	−0.221900
$273$	0	0
$274$	−8.63792	−0.521836
$275$	−6.45680	−0.389360
$276$	0	0
$277$	10.6839	0.641931	0.320966	−	0.947091i	$-0.395993\pi$
$277$	10.6839	0.641931	0.320966	+	0.947091i	$0.395993\pi$
$278$	−12.2531	−0.734894
$279$	0	0
$280$	−14.9565	−0.893822
$281$	22.1411	1.32083	0.660415	−	0.750901i	$-0.270380\pi$
$281$	22.1411	1.32083	0.660415	+	0.750901i	$0.270380\pi$
$282$	0	0
$283$	−2.25692	−0.134160	−0.0670799	−	0.997748i	$-0.521368\pi$
$283$	−2.25692	−0.134160	−0.0670799	+	0.997748i	$0.521368\pi$
$284$	−25.2130	−1.49612
$285$	0	0
$286$	12.3601	0.730868
$287$	−7.45621	−0.440126
$288$	0	0
$289$	−12.9704	−0.762966
$290$	−4.09448	−0.240436
$291$	0	0
$292$	−17.4807	−1.02298
$293$	−18.1554	−1.06065	−0.530326	−	0.847794i	$-0.677931\pi$
$293$	−18.1554	−1.06065	−0.530326	+	0.847794i	$0.677931\pi$
$294$	0	0
$295$	−16.7578	−0.975675
$296$	8.28120	0.481335
$297$	0	0
$298$	0.0186621	0.00108107
$299$	−33.5721	−1.94153
$300$	0	0
$301$	27.5265	1.58660
$302$	−0.288042	−0.0165749
$303$	0	0
$304$	10.7935	0.619047
$305$	17.8275	1.02080
$306$	0	0
$307$	3.85685	0.220122	0.110061	−	0.993925i	$-0.464895\pi$
$307$	3.85685	0.220122	0.110061	+	0.993925i	$0.464895\pi$
$308$	−19.1153	−1.08920
$309$	0	0
$310$	−8.70214	−0.494249
$311$	14.5608	0.825667	0.412833	−	0.910807i	$-0.364539\pi$
$311$	14.5608	0.825667	0.412833	+	0.910807i	$0.364539\pi$
$312$	0	0
$313$	1.26227	0.0713480	0.0356740	−	0.999363i	$-0.488642\pi$
$313$	1.26227	0.0713480	0.0356740	+	0.999363i	$0.488642\pi$
$314$	−3.69517	−0.208530
$315$	0	0
$316$	−10.3242	−0.580784
$317$	−4.21550	−0.236766	−0.118383	−	0.992968i	$-0.537771\pi$
$317$	−4.21550	−0.236766	−0.118383	+	0.992968i	$0.537771\pi$
$318$	0	0
$319$	−11.7251	−0.656480
$320$	−0.231346	−0.0129326
$321$	0	0
$322$	−12.4927	−0.696189
$323$	−11.8844	−0.661267
$324$	0	0
$325$	−14.0912	−0.781638
$326$	−3.32895	−0.184373
$327$	0	0
$328$	4.26595	0.235548
$329$	38.2249	2.10741
$330$	0	0
$331$	−20.2826	−1.11483	−0.557415	−	0.830234i	$-0.688207\pi$
$331$	−20.2826	−1.11483	−0.557415	+	0.830234i	$0.688207\pi$
$332$	−10.0428	−0.551168
$333$	0	0
$334$	10.3646	0.567127
$335$	−4.09418	−0.223689
$336$	0	0
$337$	34.1915	1.86253	0.931266	−	0.364340i	$-0.118706\pi$
$337$	34.1915	1.86253	0.931266	+	0.364340i	$0.118706\pi$
$338$	18.8779	1.02682
$339$	0	0
$340$	5.47166	0.296742
$341$	−24.9198	−1.34948
$342$	0	0
$343$	5.74478	0.310189
$344$	−15.7488	−0.849121
$345$	0	0
$346$	11.6537	0.626507
$347$	−13.9320	−0.747910	−0.373955	−	0.927447i	$-0.621999\pi$
$347$	−13.9320	−0.747910	−0.373955	+	0.927447i	$0.621999\pi$
$348$	0	0
$349$	−8.42095	−0.450763	−0.225382	−	0.974271i	$-0.572363\pi$
$349$	−8.42095	−0.450763	−0.225382	+	0.974271i	$0.572363\pi$
$350$	−5.24353	−0.280278
$351$	0	0
$352$	16.9920	0.905675
$353$	5.19599	0.276555	0.138277	−	0.990394i	$-0.455843\pi$
$353$	5.19599	0.276555	0.138277	+	0.990394i	$0.455843\pi$
$354$	0	0
$355$	26.4439	1.40350
$356$	5.22106	0.276715
$357$	0	0
$358$	−3.43005	−0.181284
$359$	−36.3620	−1.91911	−0.959556	−	0.281517i	$-0.909163\pi$
$359$	−36.3620	−1.91911	−0.959556	+	0.281517i	$0.909163\pi$
$360$	0	0
$361$	16.0507	0.844774
$362$	13.9019	0.730668
$363$	0	0
$364$	−41.7168	−2.18656
$365$	18.3341	0.959652
$366$	0	0
$367$	−4.87935	−0.254700	−0.127350	−	0.991858i	$-0.540647\pi$
$367$	−4.87935	−0.254700	−0.127350	+	0.991858i	$0.540647\pi$
$368$	−9.30020	−0.484806
$369$	0	0
$370$	−3.87639	−0.201524
$371$	−26.1572	−1.35802
$372$	0	0
$373$	−16.3744	−0.847833	−0.423917	−	0.905701i	$-0.639345\pi$
$373$	−16.3744	−0.847833	−0.423917	+	0.905701i	$0.639345\pi$
$374$	−3.77011	−0.194948
$375$	0	0
$376$	−21.8697	−1.12785
$377$	−25.5886	−1.31788
$378$	0	0
$379$	−12.4364	−0.638815	−0.319408	−	0.947617i	$-0.603484\pi$
$379$	−12.4364	−0.638815	−0.319408	+	0.947617i	$0.603484\pi$
$380$	−16.1375	−0.827839
$381$	0	0
$382$	−5.89175	−0.301448
$383$	16.0441	0.819817	0.409908	−	0.912127i	$-0.365561\pi$
$383$	16.0441	0.819817	0.409908	+	0.912127i	$0.365561\pi$
$384$	0	0
$385$	20.0485	1.02177
$386$	−8.33529	−0.424255
$387$	0	0
$388$	−9.43998	−0.479242
$389$	−8.27219	−0.419417	−0.209708	−	0.977764i	$-0.567251\pi$
$389$	−8.27219	−0.419417	−0.209708	+	0.977764i	$0.567251\pi$
$390$	0	0
$391$	10.2402	0.517871
$392$	−19.0344	−0.961381
$393$	0	0
$394$	−3.63610	−0.183184
$395$	10.8283	0.544830
$396$	0	0
$397$	−26.5169	−1.33084	−0.665422	−	0.746468i	$-0.731748\pi$
$397$	−26.5169	−1.33084	−0.665422	+	0.746468i	$0.731748\pi$
$398$	−6.98167	−0.349959
$399$	0	0
$400$	−3.90356	−0.195178
$401$	−23.4292	−1.17000	−0.585000	−	0.811033i	$-0.698906\pi$
$401$	−23.4292	−1.17000	−0.585000	+	0.811033i	$0.698906\pi$
$402$	0	0
$403$	−54.3844	−2.70908
$404$	10.8534	0.539975
$405$	0	0
$406$	−9.52189	−0.472563
$407$	−11.1006	−0.550235
$408$	0	0
$409$	−8.61597	−0.426032	−0.213016	−	0.977049i	$-0.568329\pi$
$409$	−8.61597	−0.426032	−0.213016	+	0.977049i	$0.568329\pi$
$410$	−1.99687	−0.0986184
$411$	0	0
$412$	3.03073	0.149313
$413$	−38.9709	−1.91763
$414$	0	0
$415$	10.5331	0.517047
$416$	37.0829	1.81814
$417$	0	0
$418$	11.1192	0.543856
$419$	12.2314	0.597545	0.298772	−	0.954324i	$-0.403423\pi$
$419$	12.2314	0.597545	0.298772	+	0.954324i	$0.403423\pi$
$420$	0	0
$421$	−17.9483	−0.874745	−0.437372	−	0.899281i	$-0.644091\pi$
$421$	−17.9483	−0.874745	−0.437372	+	0.899281i	$0.644091\pi$
$422$	−6.22279	−0.302921
$423$	0	0
$424$	14.9654	0.726786
$425$	4.29812	0.208490
$426$	0	0
$427$	41.4586	2.00632
$428$	26.7903	1.29496
$429$	0	0
$430$	7.37196	0.355507
$431$	2.70538	0.130314	0.0651569	−	0.997875i	$-0.479245\pi$
$431$	2.70538	0.130314	0.0651569	+	0.997875i	$0.479245\pi$
$432$	0	0
$433$	30.0233	1.44283	0.721415	−	0.692503i	$-0.243492\pi$
$433$	30.0233	1.44283	0.721415	+	0.692503i	$0.243492\pi$
$434$	−20.2372	−0.971416
$435$	0	0
$436$	−1.34541	−0.0644335
$437$	−30.2015	−1.44473
$438$	0	0
$439$	−0.941071	−0.0449149	−0.0224574	−	0.999748i	$-0.507149\pi$
$439$	−0.941071	−0.0449149	−0.0224574	+	0.999748i	$0.507149\pi$
$440$	−11.4704	−0.546831
$441$	0	0
$442$	−8.22780	−0.391356
$443$	−5.00333	−0.237715	−0.118858	−	0.992911i	$-0.537923\pi$
$443$	−5.00333	−0.237715	−0.118858	+	0.992911i	$0.537923\pi$
$444$	0	0
$445$	−5.47595	−0.259585
$446$	1.57071	0.0743755
$447$	0	0
$448$	−0.538005	−0.0254183
$449$	5.34829	0.252401	0.126201	−	0.992005i	$-0.459722\pi$
$449$	5.34829	0.252401	0.126201	+	0.992005i	$0.459722\pi$
$450$	0	0
$451$	−5.71831	−0.269265
$452$	−9.53685	−0.448576
$453$	0	0
$454$	13.0207	0.611094
$455$	43.7534	2.05119
$456$	0	0
$457$	−31.1015	−1.45487	−0.727434	−	0.686178i	$-0.759287\pi$
$457$	−31.1015	−1.45487	−0.727434	+	0.686178i	$0.759287\pi$
$458$	−7.07603	−0.330641
$459$	0	0
$460$	13.9049	0.648321
$461$	−7.48123	−0.348436	−0.174218	−	0.984707i	$-0.555740\pi$
$461$	−7.48123	−0.348436	−0.174218	+	0.984707i	$0.555740\pi$
$462$	0	0
$463$	5.06565	0.235421	0.117710	−	0.993048i	$-0.462445\pi$
$463$	5.06565	0.235421	0.117710	+	0.993048i	$0.462445\pi$
$464$	−7.08860	−0.329080
$465$	0	0
$466$	−4.04455	−0.187360
$467$	10.7910	0.499347	0.249673	−	0.968330i	$-0.419677\pi$
$467$	10.7910	0.499347	0.249673	+	0.968330i	$0.419677\pi$
$468$	0	0
$469$	−9.52119	−0.439648
$470$	10.2371	0.472203
$471$	0	0
$472$	22.2965	1.02628
$473$	21.1106	0.970667
$474$	0	0
$475$	−12.6764	−0.581635
$476$	12.7246	0.583229
$477$	0	0
$478$	5.91663	0.270620
$479$	−37.2246	−1.70084	−0.850418	−	0.526107i	$-0.823651\pi$
$479$	−37.2246	−1.70084	−0.850418	+	0.526107i	$0.823651\pi$
$480$	0	0
$481$	−24.2256	−1.10459
$482$	−3.64569	−0.166056
$483$	0	0
$484$	3.07330	0.139695
$485$	9.90084	0.449574
$486$	0	0
$487$	−1.19893	−0.0543288	−0.0271644	−	0.999631i	$-0.508648\pi$
$487$	−1.19893	−0.0543288	−0.0271644	+	0.999631i	$0.508648\pi$
$488$	−23.7198	−1.07375
$489$	0	0
$490$	8.90990	0.402508
$491$	−10.7598	−0.485584	−0.242792	−	0.970078i	$-0.578063\pi$
$491$	−10.7598	−0.485584	−0.242792	+	0.970078i	$0.578063\pi$
$492$	0	0
$493$	7.80510	0.351524
$494$	24.2662	1.09179
$495$	0	0
$496$	−15.0656	−0.676467
$497$	61.4964	2.75849
$498$	0	0
$499$	32.5779	1.45839	0.729194	−	0.684307i	$-0.239895\pi$
$499$	32.5779	1.45839	0.729194	+	0.684307i	$0.239895\pi$
$500$	19.4651	0.870508
$501$	0	0
$502$	−0.622810	−0.0277974
$503$	−0.566614	−0.0252641	−0.0126320	−	0.999920i	$-0.504021\pi$
$503$	−0.566614	−0.0252641	−0.0126320	+	0.999920i	$0.504021\pi$
$504$	0	0
$505$	−11.3832	−0.506546
$506$	−9.58085	−0.425921
$507$	0	0
$508$	16.9802	0.753373
$509$	6.97468	0.309147	0.154574	−	0.987981i	$-0.450600\pi$
$509$	6.97468	0.309147	0.154574	+	0.987981i	$0.450600\pi$
$510$	0	0
$511$	42.6368	1.88614
$512$	18.4755	0.816509
$513$	0	0
$514$	12.4177	0.547723
$515$	−3.17869	−0.140070
$516$	0	0
$517$	29.3154	1.28929
$518$	−9.01471	−0.396083
$519$	0	0
$520$	−25.0328	−1.09776
$521$	15.1642	0.664355	0.332177	−	0.943217i	$-0.392217\pi$
$521$	15.1642	0.664355	0.332177	+	0.943217i	$0.392217\pi$
$522$	0	0
$523$	−18.2532	−0.798158	−0.399079	−	0.916917i	$-0.630670\pi$
$523$	−18.2532	−0.798158	−0.399079	+	0.916917i	$0.630670\pi$
$524$	−27.9351	−1.22035
$525$	0	0
$526$	−19.0170	−0.829180
$527$	16.5884	0.722604
$528$	0	0
$529$	3.02320	0.131443
$530$	−7.00525	−0.304289
$531$	0	0
$532$	−37.5285	−1.62707
$533$	−12.4795	−0.540548
$534$	0	0
$535$	−28.0982	−1.21479
$536$	5.44739	0.235291
$537$	0	0
$538$	3.27677	0.141271
$539$	25.5147	1.09900
$540$	0	0
$541$	−0.347914	−0.0149580	−0.00747899	−	0.999972i	$-0.502381\pi$
$541$	−0.347914	−0.0149580	−0.00747899	+	0.999972i	$0.502381\pi$
$542$	11.5573	0.496427
$543$	0	0
$544$	−11.3111	−0.484960
$545$	1.41109	0.0604446
$546$	0	0
$547$	29.1298	1.24550	0.622749	−	0.782422i	$-0.286016\pi$
$547$	29.1298	1.24550	0.622749	+	0.782422i	$0.286016\pi$
$548$	22.3588	0.955119
$549$	0	0
$550$	−4.02136	−0.171471
$551$	−23.0196	−0.980666
$552$	0	0
$553$	25.1816	1.07083
$554$	6.65402	0.282702
$555$	0	0
$556$	31.7165	1.34508
$557$	20.2253	0.856973	0.428487	−	0.903548i	$-0.359047\pi$
$557$	20.2253	0.856973	0.428487	+	0.903548i	$0.359047\pi$
$558$	0	0
$559$	46.0713	1.94861
$560$	12.1207	0.512191
$561$	0	0
$562$	13.7897	0.581685
$563$	3.71784	0.156688	0.0783440	−	0.996926i	$-0.475037\pi$
$563$	3.71784	0.156688	0.0783440	+	0.996926i	$0.475037\pi$
$564$	0	0
$565$	10.0024	0.420806
$566$	−1.40563	−0.0590831
$567$	0	0
$568$	−35.1841	−1.47629
$569$	−33.5532	−1.40662	−0.703311	−	0.710883i	$-0.748296\pi$
$569$	−33.5532	−1.40662	−0.703311	+	0.710883i	$0.748296\pi$
$570$	0	0
$571$	−25.5578	−1.06956	−0.534780	−	0.844992i	$-0.679605\pi$
$571$	−25.5578	−1.06956	−0.534780	+	0.844992i	$0.679605\pi$
$572$	−31.9934	−1.33771
$573$	0	0
$574$	−4.64380	−0.193829
$575$	10.9227	0.455507
$576$	0	0
$577$	13.3016	0.553753	0.276876	−	0.960906i	$-0.410701\pi$
$577$	13.3016	0.553753	0.276876	+	0.960906i	$0.410701\pi$
$578$	−8.07811	−0.336005
$579$	0	0
$580$	10.5983	0.440072
$581$	24.4950	1.01623
$582$	0	0
$583$	−20.0605	−0.830820
$584$	−24.3939	−1.00943
$585$	0	0
$586$	−11.3074	−0.467104
$587$	−32.4672	−1.34006	−0.670032	−	0.742332i	$-0.733720\pi$
$587$	−32.4672	−1.34006	−0.670032	+	0.742332i	$0.733720\pi$
$588$	0	0
$589$	−48.9242	−2.01589
$590$	−10.4369	−0.429680
$591$	0	0
$592$	−6.71103	−0.275821
$593$	5.79919	0.238144	0.119072	−	0.992886i	$-0.462008\pi$
$593$	5.79919	0.238144	0.119072	+	0.992886i	$0.462008\pi$
$594$	0	0
$595$	−13.3458	−0.547123
$596$	−0.0483058	−0.00197868
$597$	0	0
$598$	−20.9090	−0.855034
$599$	3.07138	0.125493	0.0627465	−	0.998029i	$-0.480014\pi$
$599$	3.07138	0.125493	0.0627465	+	0.998029i	$0.480014\pi$
$600$	0	0
$601$	23.0372	0.939707	0.469854	−	0.882744i	$-0.344307\pi$
$601$	23.0372	0.939707	0.469854	+	0.882744i	$0.344307\pi$
$602$	17.1438	0.698729
$603$	0	0
$604$	0.745580	0.0303372
$605$	−3.22334	−0.131047
$606$	0	0
$607$	−48.7294	−1.97786	−0.988932	−	0.148368i	$-0.952598\pi$
$607$	−48.7294	−1.97786	−0.988932	+	0.148368i	$0.952598\pi$
$608$	33.3598	1.35292
$609$	0	0
$610$	11.1031	0.449553
$611$	63.9773	2.58824
$612$	0	0
$613$	−1.69828	−0.0685929	−0.0342964	−	0.999412i	$-0.510919\pi$
$613$	−1.69828	−0.0685929	−0.0342964	+	0.999412i	$0.510919\pi$
$614$	2.40208	0.0969402
$615$	0	0
$616$	−26.6749	−1.07476
$617$	42.6700	1.71783	0.858914	−	0.512119i	$-0.171139\pi$
$617$	42.6700	1.71783	0.858914	+	0.512119i	$0.171139\pi$
$618$	0	0
$619$	−31.9703	−1.28500	−0.642498	−	0.766288i	$-0.722102\pi$
$619$	−31.9703	−1.28500	−0.642498	+	0.766288i	$0.722102\pi$
$620$	22.5250	0.904626
$621$	0	0
$622$	9.06861	0.363618
$623$	−12.7345	−0.510199
$624$	0	0
$625$	−9.70964	−0.388386
$626$	0.786157	0.0314212
$627$	0	0
$628$	9.56473	0.381674
$629$	7.38936	0.294633
$630$	0	0
$631$	−5.18897	−0.206570	−0.103285	−	0.994652i	$-0.532935\pi$
$631$	−5.18897	−0.206570	−0.103285	+	0.994652i	$0.532935\pi$
$632$	−14.4072	−0.573089
$633$	0	0
$634$	−2.62546	−0.104270
$635$	−17.8091	−0.706734
$636$	0	0
$637$	55.6827	2.20623
$638$	−7.30251	−0.289109
$639$	0	0
$640$	−19.1987	−0.758896
$641$	−19.3638	−0.764822	−0.382411	−	0.923992i	$-0.624906\pi$
$641$	−19.3638	−0.764822	−0.382411	+	0.923992i	$0.624906\pi$
$642$	0	0
$643$	21.5916	0.851489	0.425744	−	0.904844i	$-0.360012\pi$
$643$	21.5916	0.851489	0.425744	+	0.904844i	$0.360012\pi$
$644$	32.3365	1.27424
$645$	0	0
$646$	−7.40174	−0.291218
$647$	40.6062	1.59639	0.798196	−	0.602397i	$-0.205788\pi$
$647$	40.6062	1.59639	0.798196	+	0.602397i	$0.205788\pi$
$648$	0	0
$649$	−29.8875	−1.17319
$650$	−8.77613	−0.344228
$651$	0	0
$652$	8.61678	0.337459
$653$	−8.37874	−0.327886	−0.163943	−	0.986470i	$-0.552421\pi$
$653$	−8.37874	−0.327886	−0.163943	+	0.986470i	$0.552421\pi$
$654$	0	0
$655$	29.2989	1.14480
$656$	−3.45709	−0.134977
$657$	0	0
$658$	23.8069	0.928088
$659$	−2.74672	−0.106997	−0.0534985	−	0.998568i	$-0.517037\pi$
$659$	−2.74672	−0.106997	−0.0534985	+	0.998568i	$0.517037\pi$
$660$	0	0
$661$	2.29053	0.0890912	0.0445456	−	0.999007i	$-0.485816\pi$
$661$	2.29053	0.0890912	0.0445456	+	0.999007i	$0.485816\pi$
$662$	−12.6322	−0.490964
$663$	0	0
$664$	−14.0144	−0.543866
$665$	39.3607	1.52634
$666$	0	0
$667$	19.8348	0.768008
$668$	−26.8282	−1.03802
$669$	0	0
$670$	−2.54990	−0.0985111
$671$	31.7954	1.22745
$672$	0	0
$673$	3.55591	0.137070	0.0685352	−	0.997649i	$-0.478167\pi$
$673$	3.55591	0.137070	0.0685352	+	0.997649i	$0.478167\pi$
$674$	21.2948	0.820246
$675$	0	0
$676$	−48.8643	−1.87940
$677$	49.2272	1.89195	0.945977	−	0.324232i	$-0.105106\pi$
$677$	49.2272	1.89195	0.945977	+	0.324232i	$0.105106\pi$
$678$	0	0
$679$	23.0248	0.883611
$680$	7.63556	0.292810
$681$	0	0
$682$	−15.5203	−0.594302
$683$	2.75471	0.105406	0.0527030	−	0.998610i	$-0.483216\pi$
$683$	2.75471	0.105406	0.0527030	+	0.998610i	$0.483216\pi$
$684$	0	0
$685$	−23.4503	−0.895990
$686$	3.57790	0.136605
$687$	0	0
$688$	12.7628	0.486575
$689$	−43.7796	−1.66787
$690$	0	0
$691$	−4.74172	−0.180384	−0.0901918	−	0.995924i	$-0.528748\pi$
$691$	−4.74172	−0.180384	−0.0901918	+	0.995924i	$0.528748\pi$
$692$	−30.1649	−1.14670
$693$	0	0
$694$	−8.67700	−0.329374
$695$	−33.2649	−1.26181
$696$	0	0
$697$	3.80653	0.144183
$698$	−5.24466	−0.198513
$699$	0	0
$700$	13.5726	0.512995
$701$	−28.7187	−1.08469	−0.542345	−	0.840156i	$-0.682464\pi$
$701$	−28.7187	−1.08469	−0.542345	+	0.840156i	$0.682464\pi$
$702$	0	0
$703$	−21.7934	−0.821954
$704$	−0.412606	−0.0155507
$705$	0	0
$706$	3.23612	0.121793
$707$	−26.4721	−0.995587
$708$	0	0
$709$	−13.3937	−0.503009	−0.251505	−	0.967856i	$-0.580925\pi$
$709$	−13.3937	−0.503009	−0.251505	+	0.967856i	$0.580925\pi$
$710$	16.4695	0.618090
$711$	0	0
$712$	7.28586	0.273049
$713$	42.1557	1.57874
$714$	0	0
$715$	33.5553	1.25490
$716$	8.87849	0.331805
$717$	0	0
$718$	−22.6466	−0.845164
$719$	−51.9716	−1.93821	−0.969107	−	0.246640i	$-0.920674\pi$
$719$	−51.9716	−1.93821	−0.969107	+	0.246640i	$0.920674\pi$
$720$	0	0
$721$	−7.39217	−0.275299
$722$	9.99655	0.372033
$723$	0	0
$724$	−35.9843	−1.33735
$725$	8.32525	0.309192
$726$	0	0
$727$	−5.26831	−0.195391	−0.0976954	−	0.995216i	$-0.531147\pi$
$727$	−5.26831	−0.195391	−0.0976954	+	0.995216i	$0.531147\pi$
$728$	−58.2148	−2.15758
$729$	0	0
$730$	11.4187	0.422624
$731$	−14.0528	−0.519761
$732$	0	0
$733$	26.1445	0.965669	0.482835	−	0.875712i	$-0.339607\pi$
$733$	26.1445	0.965669	0.482835	+	0.875712i	$0.339607\pi$
$734$	−3.03891	−0.112168
$735$	0	0
$736$	−28.7446	−1.05954
$737$	−7.30198	−0.268972
$738$	0	0
$739$	11.7217	0.431189	0.215594	−	0.976483i	$-0.430831\pi$
$739$	11.7217	0.431189	0.215594	+	0.976483i	$0.430831\pi$
$740$	10.0338	0.368850
$741$	0	0
$742$	−16.2910	−0.598061
$743$	−18.3220	−0.672167	−0.336084	−	0.941832i	$-0.609102\pi$
$743$	−18.3220	−0.672167	−0.336084	+	0.941832i	$0.609102\pi$
$744$	0	0
$745$	0.0506641	0.00185619
$746$	−10.1981	−0.373380
$747$	0	0
$748$	9.75870	0.356813
$749$	−65.3436	−2.38760
$750$	0	0
$751$	23.5394	0.858965	0.429482	−	0.903075i	$-0.358696\pi$
$751$	23.5394	0.858965	0.429482	+	0.903075i	$0.358696\pi$
$752$	17.7231	0.646294
$753$	0	0
$754$	−15.9368	−0.580386
$755$	−0.781979	−0.0284591
$756$	0	0
$757$	−35.5799	−1.29317	−0.646587	−	0.762840i	$-0.723804\pi$
$757$	−35.5799	−1.29317	−0.646587	+	0.762840i	$0.723804\pi$
$758$	−7.74552	−0.281330
$759$	0	0
$760$	−22.5195	−0.816870
$761$	12.1130	0.439097	0.219549	−	0.975602i	$-0.429542\pi$
$761$	12.1130	0.439097	0.219549	+	0.975602i	$0.429542\pi$
$762$	0	0
$763$	3.28156	0.118800
$764$	15.2504	0.551742
$765$	0	0
$766$	9.99245	0.361042
$767$	−65.2258	−2.35517
$768$	0	0
$769$	10.1118	0.364640	0.182320	−	0.983239i	$-0.441639\pi$
$769$	10.1118	0.364640	0.182320	+	0.983239i	$0.441639\pi$
$770$	12.4864	0.449979
$771$	0	0
$772$	21.5754	0.776516
$773$	6.19658	0.222875	0.111438	−	0.993771i	$-0.464454\pi$
$773$	6.19658	0.222875	0.111438	+	0.993771i	$0.464454\pi$
$774$	0	0
$775$	17.6939	0.635585
$776$	−13.1733	−0.472893
$777$	0	0
$778$	−5.15200	−0.184708
$779$	−11.2266	−0.402234
$780$	0	0
$781$	47.1627	1.68761
$782$	6.37772	0.228067
$783$	0	0
$784$	15.4253	0.550905
$785$	−10.0317	−0.358046
$786$	0	0
$787$	−12.0316	−0.428881	−0.214441	−	0.976737i	$-0.568793\pi$
$787$	−12.0316	−0.428881	−0.214441	+	0.976737i	$0.568793\pi$
$788$	9.41184	0.335283
$789$	0	0
$790$	6.74396	0.239939
$791$	23.2611	0.827069
$792$	0	0
$793$	69.3895	2.46409
$794$	−16.5150	−0.586094
$795$	0	0
$796$	18.0716	0.640532
$797$	−5.35261	−0.189599	−0.0947996	−	0.995496i	$-0.530221\pi$
$797$	−5.35261	−0.189599	−0.0947996	+	0.995496i	$0.530221\pi$
$798$	0	0
$799$	−19.5145	−0.690373
$800$	−12.0649	−0.426560
$801$	0	0
$802$	−14.5920	−0.515260
$803$	32.6989	1.15392
$804$	0	0
$805$	−33.9152	−1.19535
$806$	−33.8711	−1.19306
$807$	0	0
$808$	15.1456	0.532820
$809$	−29.2497	−1.02836	−0.514182	−	0.857681i	$-0.671904\pi$
$809$	−29.2497	−1.02836	−0.514182	+	0.857681i	$0.671904\pi$
$810$	0	0
$811$	−30.7220	−1.07879	−0.539397	−	0.842052i	$-0.681348\pi$
$811$	−30.7220	−1.07879	−0.539397	+	0.842052i	$0.681348\pi$
$812$	24.6469	0.864935
$813$	0	0
$814$	−6.91355	−0.242320
$815$	−9.03745	−0.316568
$816$	0	0
$817$	41.4458	1.45001
$818$	−5.36611	−0.187622
$819$	0	0
$820$	5.16878	0.180502
$821$	10.6846	0.372894	0.186447	−	0.982465i	$-0.440303\pi$
$821$	10.6846	0.372894	0.186447	+	0.982465i	$0.440303\pi$
$822$	0	0
$823$	−31.9198	−1.11266	−0.556328	−	0.830963i	$-0.687790\pi$
$823$	−31.9198	−1.11266	−0.556328	+	0.830963i	$0.687790\pi$
$824$	4.22931	0.147335
$825$	0	0
$826$	−24.2714	−0.844512
$827$	−32.2088	−1.12001	−0.560005	−	0.828489i	$-0.689201\pi$
$827$	−32.2088	−1.12001	−0.560005	+	0.828489i	$0.689201\pi$
$828$	0	0
$829$	4.54003	0.157682	0.0788408	−	0.996887i	$-0.474878\pi$
$829$	4.54003	0.157682	0.0788408	+	0.996887i	$0.474878\pi$
$830$	6.56009	0.227704
$831$	0	0
$832$	−0.900462	−0.0312179
$833$	−16.9845	−0.588477
$834$	0	0
$835$	28.1380	0.973755
$836$	−28.7813	−0.995423
$837$	0	0
$838$	7.61786	0.263155
$839$	3.51756	0.121440	0.0607198	−	0.998155i	$-0.480660\pi$
$839$	3.51756	0.121440	0.0607198	+	0.998155i	$0.480660\pi$
$840$	0	0
$841$	−13.8819	−0.478686
$842$	−11.1784	−0.385232
$843$	0	0
$844$	16.1073	0.554437
$845$	51.2499	1.76305
$846$	0	0
$847$	−7.49600	−0.257566
$848$	−12.1279	−0.416473
$849$	0	0
$850$	2.67691	0.0918174
$851$	18.7783	0.643713
$852$	0	0
$853$	−41.4217	−1.41825	−0.709127	−	0.705081i	$-0.750911\pi$
$853$	−41.4217	−1.41825	−0.709127	+	0.705081i	$0.750911\pi$
$854$	25.8208	0.883570
$855$	0	0
$856$	37.3853	1.27780
$857$	−6.59439	−0.225260	−0.112630	−	0.993637i	$-0.535927\pi$
$857$	−6.59439	−0.225260	−0.112630	+	0.993637i	$0.535927\pi$
$858$	0	0
$859$	−54.3859	−1.85562	−0.927811	−	0.373050i	$-0.878312\pi$
$859$	−54.3859	−1.85562	−0.927811	+	0.373050i	$0.878312\pi$
$860$	−19.0819	−0.650687
$861$	0	0
$862$	1.68494	0.0573893
$863$	32.2610	1.09818	0.549089	−	0.835764i	$-0.314975\pi$
$863$	32.2610	1.09818	0.549089	+	0.835764i	$0.314975\pi$
$864$	0	0
$865$	31.6376	1.07571
$866$	18.6988	0.635412
$867$	0	0
$868$	52.3828	1.77799
$869$	19.3122	0.655123
$870$	0	0
$871$	−15.9357	−0.539960
$872$	−1.87749	−0.0635798
$873$	0	0
$874$	−18.8098	−0.636251
$875$	−47.4769	−1.60501
$876$	0	0
$877$	25.0725	0.846637	0.423318	−	0.905981i	$-0.360865\pi$
$877$	25.0725	0.846637	0.423318	+	0.905981i	$0.360865\pi$
$878$	−0.586108	−0.0197802
$879$	0	0
$880$	9.29555	0.313353
$881$	39.8888	1.34389	0.671944	−	0.740602i	$-0.265460\pi$
$881$	39.8888	1.34389	0.671944	+	0.740602i	$0.265460\pi$
$882$	0	0
$883$	−28.2464	−0.950566	−0.475283	−	0.879833i	$-0.657654\pi$
$883$	−28.2464	−0.950566	−0.475283	+	0.879833i	$0.657654\pi$
$884$	21.2972	0.716302
$885$	0	0
$886$	−3.11612	−0.104688
$887$	−1.88012	−0.0631283	−0.0315642	−	0.999502i	$-0.510049\pi$
$887$	−1.88012	−0.0631283	−0.0315642	+	0.999502i	$0.510049\pi$
$888$	0	0
$889$	−41.4159	−1.38904
$890$	−3.41048	−0.114319
$891$	0	0
$892$	−4.06570	−0.136130
$893$	57.5540	1.92597
$894$	0	0
$895$	−9.31194	−0.311264
$896$	−44.6474	−1.49157
$897$	0	0
$898$	3.33097	0.111156
$899$	32.1310	1.07163
$900$	0	0
$901$	13.3537	0.444877
$902$	−3.56142	−0.118582
$903$	0	0
$904$	−13.3084	−0.442632
$905$	37.7410	1.25455
$906$	0	0
$907$	−12.9333	−0.429443	−0.214722	−	0.976675i	$-0.568884\pi$
$907$	−12.9333	−0.429443	−0.214722	+	0.976675i	$0.568884\pi$
$908$	−33.7034	−1.11849
$909$	0	0
$910$	27.2501	0.903331
$911$	20.9733	0.694875	0.347438	−	0.937703i	$-0.387052\pi$
$911$	20.9733	0.694875	0.347438	+	0.937703i	$0.387052\pi$
$912$	0	0
$913$	18.7857	0.621716
$914$	−19.3703	−0.640714
$915$	0	0
$916$	18.3159	0.605174
$917$	68.1359	2.25004
$918$	0	0
$919$	−59.2829	−1.95556	−0.977781	−	0.209630i	$-0.932774\pi$
$919$	−59.2829	−1.95556	−0.977781	+	0.209630i	$0.932774\pi$
$920$	19.4040	0.639731
$921$	0	0
$922$	−4.65939	−0.153449
$923$	102.927	3.38788
$924$	0	0
$925$	7.88181	0.259152
$926$	3.15494	0.103678
$927$	0	0
$928$	−21.9091	−0.719201
$929$	52.9155	1.73610	0.868050	−	0.496476i	$-0.165373\pi$
$929$	52.9155	1.73610	0.868050	+	0.496476i	$0.165373\pi$
$930$	0	0
$931$	50.0923	1.64171
$932$	10.4691	0.342926
$933$	0	0
$934$	6.72073	0.219909
$935$	−10.2351	−0.334724
$936$	0	0
$937$	−14.9951	−0.489869	−0.244934	−	0.969540i	$-0.578766\pi$
$937$	−14.9951	−0.489869	−0.244934	+	0.969540i	$0.578766\pi$
$938$	−5.92989	−0.193618
$939$	0	0
$940$	−26.4982	−0.864276
$941$	−1.41283	−0.0460568	−0.0230284	−	0.999735i	$-0.507331\pi$
$941$	−1.41283	−0.0460568	−0.0230284	+	0.999735i	$0.507331\pi$
$942$	0	0
$943$	9.67341	0.315010
$944$	−18.0690	−0.588094
$945$	0	0
$946$	13.1479	0.427475
$947$	1.95072	0.0633898	0.0316949	−	0.999498i	$-0.489910\pi$
$947$	1.95072	0.0633898	0.0316949	+	0.999498i	$0.489910\pi$
$948$	0	0
$949$	71.3614	2.31649
$950$	−7.89502	−0.256148
$951$	0	0
$952$	17.7568	0.575502
$953$	−4.60766	−0.149257	−0.0746284	−	0.997211i	$-0.523777\pi$
$953$	−4.60766	−0.149257	−0.0746284	+	0.997211i	$0.523777\pi$
$954$	0	0
$955$	−15.9950	−0.517585
$956$	−15.3149	−0.495318
$957$	0	0
$958$	−23.1839	−0.749037
$959$	−54.5347	−1.76102
$960$	0	0
$961$	37.2891	1.20287
$962$	−15.0880	−0.486456
$963$	0	0
$964$	9.43664	0.303934
$965$	−22.6287	−0.728445
$966$	0	0
$967$	−21.2940	−0.684768	−0.342384	−	0.939560i	$-0.611234\pi$
$967$	−21.2940	−0.684768	−0.342384	+	0.939560i	$0.611234\pi$
$968$	4.28871	0.137844
$969$	0	0
$970$	6.16634	0.197989
$971$	−28.6035	−0.917930	−0.458965	−	0.888454i	$-0.651780\pi$
$971$	−28.6035	−0.917930	−0.458965	+	0.888454i	$0.651780\pi$
$972$	0	0
$973$	−77.3589	−2.48001
$974$	−0.746707	−0.0239260
$975$	0	0
$976$	19.2224	0.615294
$977$	−33.4519	−1.07022	−0.535110	−	0.844783i	$-0.679730\pi$
$977$	−33.4519	−1.07022	−0.535110	+	0.844783i	$0.679730\pi$
$978$	0	0
$979$	−9.76636	−0.312134
$980$	−23.0628	−0.736713
$981$	0	0
$982$	−6.70133	−0.213848
$983$	−45.8996	−1.46397	−0.731985	−	0.681321i	$-0.761406\pi$
$983$	−45.8996	−1.46397	−0.731985	+	0.681321i	$0.761406\pi$
$984$	0	0
$985$	−9.87133	−0.314527
$986$	4.86109	0.154809
$987$	0	0
$988$	−62.8117	−1.99831
$989$	−35.7119	−1.13557
$990$	0	0
$991$	−15.4162	−0.489713	−0.244856	−	0.969559i	$-0.578741\pi$
$991$	−15.4162	−0.489713	−0.244856	+	0.969559i	$0.578741\pi$
$992$	−46.5641	−1.47841
$993$	0	0
$994$	38.3006	1.21482
$995$	−18.9539	−0.600879
$996$	0	0
$997$	39.3302	1.24560	0.622800	−	0.782381i	$-0.285995\pi$
$997$	39.3302	1.24560	0.622800	+	0.782381i	$0.285995\pi$
$998$	20.2899	0.642264
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2259.2.a.k.1.10		17
3.2	odd	2		251.2.a.b.1.8	✓	17
12.11	even	2		4016.2.a.k.1.13		17
15.14	odd	2		6275.2.a.e.1.10		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
251.2.a.b.1.8	✓	17	3.2	odd	2
2259.2.a.k.1.10		17	1.1	even	1	trivial
4016.2.a.k.1.13		17	12.11	even	2
6275.2.a.e.1.10		17	15.14	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 2259 = 3^{2} \cdot 251 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2259.a (trivial)

\(f(q)\)	\(=\)	\(q+0.622810 q^{2} -1.61211 q^{4} +1.69081 q^{5} +3.93205 q^{7} -2.24966 q^{8} +O(q^{10})\)	\(q+0.622810 q^{2} -1.61211 q^{4} +1.69081 q^{5} +3.93205 q^{7} -2.24966 q^{8} +1.05305 q^{10} +3.01556 q^{11} +6.58110 q^{13} +2.44892 q^{14} +1.82311 q^{16} -2.00738 q^{17} +5.92036 q^{19} -2.72577 q^{20} +1.87812 q^{22} -5.10129 q^{23} -2.14116 q^{25} +4.09877 q^{26} -6.33889 q^{28} -3.88820 q^{29} -8.26372 q^{31} +5.63476 q^{32} -1.25022 q^{34} +6.64835 q^{35} -3.68109 q^{37} +3.68726 q^{38} -3.80374 q^{40} -1.89627 q^{41} +7.00055 q^{43} -4.86141 q^{44} -3.17714 q^{46} +9.72137 q^{47} +8.46101 q^{49} -1.33354 q^{50} -10.6094 q^{52} -6.65232 q^{53} +5.09874 q^{55} -8.84576 q^{56} -2.42161 q^{58} -9.91108 q^{59} +10.5438 q^{61} -5.14673 q^{62} -0.136826 q^{64} +11.1274 q^{65} -2.42143 q^{67} +3.23611 q^{68} +4.14066 q^{70} +15.6398 q^{71} +10.8434 q^{73} -2.29262 q^{74} -9.54426 q^{76} +11.8573 q^{77} +6.40419 q^{79} +3.08253 q^{80} -1.18101 q^{82} +6.22959 q^{83} -3.39410 q^{85} +4.36002 q^{86} -6.78398 q^{88} -3.23865 q^{89} +25.8772 q^{91} +8.22383 q^{92} +6.05457 q^{94} +10.0102 q^{95} +5.85568 q^{97} +5.26960 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - 2 q^{2} + 26 q^{4} - 3 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) \) 17 * q - 2 * q^2 + 26 * q^4 - 3 * q^5 + 3 * q^7 - 6 * q^8	\( 17 q - 2 q^{2} + 26 q^{4} - 3 q^{5} + 3 q^{7} - 6 q^{8} + 7 q^{10} + q^{11} + 22 q^{13} + 7 q^{14} + 40 q^{16} + q^{17} + 13 q^{19} + 14 q^{20} + 4 q^{22} + 2 q^{23} + 32 q^{25} + 9 q^{26} - 10 q^{28} - 28 q^{29} + 12 q^{31} - 4 q^{32} - 21 q^{34} + 15 q^{35} + 27 q^{37} + 37 q^{38} - 7 q^{40} + q^{41} + 9 q^{43} + 43 q^{44} + 4 q^{46} + 20 q^{47} + 32 q^{49} + 28 q^{50} - q^{52} - q^{53} - 11 q^{55} + 61 q^{56} - 46 q^{58} + 20 q^{59} + 59 q^{61} + 73 q^{62} + 54 q^{64} + 14 q^{65} + 15 q^{67} + 20 q^{68} - 11 q^{70} + 26 q^{71} + 8 q^{73} - 2 q^{74} + 38 q^{76} + 33 q^{79} + 29 q^{80} + 10 q^{82} + 67 q^{85} - 11 q^{86} + 27 q^{88} - 11 q^{89} - 2 q^{91} - 28 q^{92} + 29 q^{94} + 8 q^{95} - 10 q^{97} - 22 q^{98}+O(q^{100}) \) 17 * q - 2 * q^2 + 26 * q^4 - 3 * q^5 + 3 * q^7 - 6 * q^8 + 7 * q^10 + q^11 + 22 * q^13 + 7 * q^14 + 40 * q^16 + q^17 + 13 * q^19 + 14 * q^20 + 4 * q^22 + 2 * q^23 + 32 * q^25 + 9 * q^26 - 10 * q^28 - 28 * q^29 + 12 * q^31 - 4 * q^32 - 21 * q^34 + 15 * q^35 + 27 * q^37 + 37 * q^38 - 7 * q^40 + q^41 + 9 * q^43 + 43 * q^44 + 4 * q^46 + 20 * q^47 + 32 * q^49 + 28 * q^50 - q^52 - q^53 - 11 * q^55 + 61 * q^56 - 46 * q^58 + 20 * q^59 + 59 * q^61 + 73 * q^62 + 54 * q^64 + 14 * q^65 + 15 * q^67 + 20 * q^68 - 11 * q^70 + 26 * q^71 + 8 * q^73 - 2 * q^74 + 38 * q^76 + 33 * q^79 + 29 * q^80 + 10 * q^82 + 67 * q^85 - 11 * q^86 + 27 * q^88 - 11 * q^89 - 2 * q^91 - 28 * q^92 + 29 * q^94 + 8 * q^95 - 10 * q^97 - 22 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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