LMFDB - Embedded newform 2259.2.a.k.1.6

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			1.6
Root			$1.37907$ 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)

$f(q)$	$=$	$q-1.37907 q^{2} -0.0981670 q^{4} -1.31548 q^{5} -3.48956 q^{7} +2.89352 q^{8} +O(q^{10})$	\(q-1.37907 q^{2} -0.0981670 q^{4} -1.31548 q^{5} -3.48956 q^{7} +2.89352 q^{8} +1.81414 q^{10} -3.66893 q^{11} +4.41851 q^{13} +4.81235 q^{14} -3.79403 q^{16} +7.90905 q^{17} -4.04690 q^{19} +0.129137 q^{20} +5.05972 q^{22} -0.625539 q^{23} -3.26950 q^{25} -6.09344 q^{26} +0.342560 q^{28} -10.4241 q^{29} -4.37191 q^{31} -0.554806 q^{32} -10.9071 q^{34} +4.59046 q^{35} -3.01090 q^{37} +5.58096 q^{38} -3.80637 q^{40} -12.0639 q^{41} +0.164454 q^{43} +0.360168 q^{44} +0.862661 q^{46} +0.235969 q^{47} +5.17706 q^{49} +4.50887 q^{50} -0.433752 q^{52} +11.4830 q^{53} +4.82642 q^{55} -10.0971 q^{56} +14.3756 q^{58} -7.18778 q^{59} +7.92880 q^{61} +6.02917 q^{62} +8.35318 q^{64} -5.81248 q^{65} +0.478919 q^{67} -0.776408 q^{68} -6.33057 q^{70} -2.65624 q^{71} +9.29721 q^{73} +4.15224 q^{74} +0.397272 q^{76} +12.8030 q^{77} +6.33824 q^{79} +4.99098 q^{80} +16.6370 q^{82} +14.2551 q^{83} -10.4042 q^{85} -0.226794 q^{86} -10.6161 q^{88} +9.55758 q^{89} -15.4187 q^{91} +0.0614073 q^{92} -0.325418 q^{94} +5.32363 q^{95} +3.65366 q^{97} -7.13952 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.37907	−0.975149	−0.487575	−	0.873081i	$-0.662118\pi$
$2$	−1.37907	−0.975149	−0.487575	+	0.873081i	$0.662118\pi$
$3$	0	0
$3$	0	0
$4$	−0.0981670	−0.0490835
$5$	−1.31548	−0.588302	−0.294151	−	0.955759i	$-0.595037\pi$
$5$	−1.31548	−0.588302	−0.294151	+	0.955759i	$0.595037\pi$
$6$	0	0
$7$	−3.48956	−1.31893	−0.659466	−	0.751735i	$-0.729217\pi$
$7$	−3.48956	−1.31893	−0.659466	+	0.751735i	$0.729217\pi$
$8$	2.89352	1.02301
$9$	0	0
$10$	1.81414	0.573682
$11$	−3.66893	−1.10623	−0.553113	−	0.833106i	$-0.686560\pi$
$11$	−3.66893	−1.10623	−0.553113	+	0.833106i	$0.686560\pi$
$12$	0	0
$13$	4.41851	1.22547	0.612737	−	0.790287i	$-0.290068\pi$
$13$	4.41851	1.22547	0.612737	+	0.790287i	$0.290068\pi$
$14$	4.81235	1.28615
$15$	0	0
$16$	−3.79403	−0.948507
$17$	7.90905	1.91823	0.959113	−	0.283022i	$-0.0913371\pi$
$17$	7.90905	1.91823	0.959113	+	0.283022i	$0.0913371\pi$
$18$	0	0
$19$	−4.04690	−0.928423	−0.464211	−	0.885724i	$-0.653662\pi$
$19$	−4.04690	−0.928423	−0.464211	+	0.885724i	$0.653662\pi$
$20$	0.129137	0.0288759
$21$	0	0
$22$	5.05972	1.07874
$23$	−0.625539	−0.130434	−0.0652169	−	0.997871i	$-0.520774\pi$
$23$	−0.625539	−0.130434	−0.0652169	+	0.997871i	$0.520774\pi$
$24$	0	0
$25$	−3.26950	−0.653901
$26$	−6.09344	−1.19502
$27$	0	0
$28$	0.342560	0.0647378
$29$	−10.4241	−1.93571	−0.967855	−	0.251509i	$-0.919073\pi$
$29$	−10.4241	−1.93571	−0.967855	+	0.251509i	$0.919073\pi$
$30$	0	0
$31$	−4.37191	−0.785218	−0.392609	−	0.919705i	$-0.628428\pi$
$31$	−4.37191	−0.785218	−0.392609	+	0.919705i	$0.628428\pi$
$32$	−0.554806	−0.0980768
$33$	0	0
$34$	−10.9071	−1.87056
$35$	4.59046	0.775930
$36$	0	0
$37$	−3.01090	−0.494989	−0.247495	−	0.968889i	$-0.579607\pi$
$37$	−3.01090	−0.494989	−0.247495	+	0.968889i	$0.579607\pi$
$38$	5.58096	0.905351
$39$	0	0
$40$	−3.80637	−0.601841
$41$	−12.0639	−1.88406	−0.942032	−	0.335523i	$-0.891087\pi$
$41$	−12.0639	−1.88406	−0.942032	+	0.335523i	$0.891087\pi$
$42$	0	0
$43$	0.164454	0.0250790	0.0125395	−	0.999921i	$-0.496008\pi$
$43$	0.164454	0.0250790	0.0125395	+	0.999921i	$0.496008\pi$
$44$	0.360168	0.0542974
$45$	0	0
$46$	0.862661	0.127192
$47$	0.235969	0.0344196	0.0172098	−	0.999852i	$-0.494522\pi$
$47$	0.235969	0.0344196	0.0172098	+	0.999852i	$0.494522\pi$
$48$	0	0
$49$	5.17706	0.739579
$50$	4.50887	0.637651
$51$	0	0
$52$	−0.433752	−0.0601506
$53$	11.4830	1.57731	0.788656	−	0.614835i	$-0.210777\pi$
$53$	11.4830	1.57731	0.788656	+	0.614835i	$0.210777\pi$
$54$	0	0
$55$	4.82642	0.650794
$56$	−10.0971	−1.34928
$57$	0	0
$58$	14.3756	1.88761
$59$	−7.18778	−0.935769	−0.467885	−	0.883790i	$-0.654984\pi$
$59$	−7.18778	−0.935769	−0.467885	+	0.883790i	$0.654984\pi$
$60$	0	0
$61$	7.92880	1.01518	0.507589	−	0.861599i	$-0.330537\pi$
$61$	7.92880	1.01518	0.507589	+	0.861599i	$0.330537\pi$
$62$	6.02917	0.765705
$63$	0	0
$64$	8.35318	1.04415
$65$	−5.81248	−0.720949
$66$	0	0
$67$	0.478919	0.0585093	0.0292547	−	0.999572i	$-0.490687\pi$
$67$	0.478919	0.0585093	0.0292547	+	0.999572i	$0.490687\pi$
$68$	−0.776408	−0.0941533
$69$	0	0
$70$	−6.33057	−0.756647
$71$	−2.65624	−0.315238	−0.157619	−	0.987500i	$-0.550382\pi$
$71$	−2.65624	−0.315238	−0.157619	+	0.987500i	$0.550382\pi$
$72$	0	0
$73$	9.29721	1.08816	0.544078	−	0.839035i	$-0.316879\pi$
$73$	9.29721	1.08816	0.544078	+	0.839035i	$0.316879\pi$
$74$	4.15224	0.482688
$75$	0	0
$76$	0.397272	0.0455703
$77$	12.8030	1.45903
$78$	0	0
$79$	6.33824	0.713107	0.356554	−	0.934275i	$-0.383952\pi$
$79$	6.33824	0.713107	0.356554	+	0.934275i	$0.383952\pi$
$80$	4.99098	0.558009
$81$	0	0
$82$	16.6370	1.83724
$83$	14.2551	1.56470	0.782349	−	0.622840i	$-0.214021\pi$
$83$	14.2551	1.56470	0.782349	+	0.622840i	$0.214021\pi$
$84$	0	0
$85$	−10.4042	−1.12850
$86$	−0.226794	−0.0244558
$87$	0	0
$88$	−10.6161	−1.13168
$89$	9.55758	1.01310	0.506551	−	0.862210i	$-0.330920\pi$
$89$	9.55758	1.01310	0.506551	+	0.862210i	$0.330920\pi$
$90$	0	0
$91$	−15.4187	−1.61632
$92$	0.0614073	0.00640215
$93$	0	0
$94$	−0.325418	−0.0335643
$95$	5.32363	0.546193
$96$	0	0
$97$	3.65366	0.370973	0.185487	−	0.982647i	$-0.440614\pi$
$97$	3.65366	0.370973	0.185487	+	0.982647i	$0.440614\pi$
$98$	−7.13952	−0.721200
$99$	0	0
$100$	0.320958	0.0320958
$101$	−7.04820	−0.701322	−0.350661	−	0.936502i	$-0.614043\pi$
$101$	−7.04820	−0.701322	−0.350661	+	0.936502i	$0.614043\pi$
$102$	0	0
$103$	8.53341	0.840822	0.420411	−	0.907334i	$-0.361886\pi$
$103$	8.53341	0.840822	0.420411	+	0.907334i	$0.361886\pi$
$104$	12.7850	1.25368
$105$	0	0
$106$	−15.8359	−1.53811
$107$	−7.02362	−0.678999	−0.339500	−	0.940606i	$-0.610258\pi$
$107$	−7.02362	−0.678999	−0.339500	+	0.940606i	$0.610258\pi$
$108$	0	0
$109$	10.9647	1.05023	0.525116	−	0.851031i	$-0.324022\pi$
$109$	10.9647	1.05023	0.525116	+	0.851031i	$0.324022\pi$
$110$	−6.65597	−0.634622
$111$	0	0
$112$	13.2395	1.25102
$113$	−3.26571	−0.307212	−0.153606	−	0.988132i	$-0.549089\pi$
$113$	−3.26571	−0.307212	−0.153606	+	0.988132i	$0.549089\pi$
$114$	0	0
$115$	0.822885	0.0767345
$116$	1.02330	0.0950114
$117$	0	0
$118$	9.91245	0.912515
$119$	−27.5991	−2.53001
$120$	0	0
$121$	2.46108	0.223734
$122$	−10.9344	−0.989951
$123$	0	0
$124$	0.429178	0.0385413
$125$	10.8784	0.972993
$126$	0	0
$127$	−1.19962	−0.106449	−0.0532244	−	0.998583i	$-0.516950\pi$
$127$	−1.19962	−0.106449	−0.0532244	+	0.998583i	$0.516950\pi$
$128$	−10.4100	−0.920122
$129$	0	0
$130$	8.01581	0.703033
$131$	8.44030	0.737432	0.368716	−	0.929542i	$-0.379797\pi$
$131$	8.44030	0.737432	0.368716	+	0.929542i	$0.379797\pi$
$132$	0	0
$133$	14.1219	1.22453
$134$	−0.660463	−0.0570553
$135$	0	0
$136$	22.8850	1.96237
$137$	13.0671	1.11640	0.558199	−	0.829707i	$-0.311493\pi$
$137$	13.0671	1.11640	0.558199	+	0.829707i	$0.311493\pi$
$138$	0	0
$139$	−5.12717	−0.434881	−0.217441	−	0.976074i	$-0.569771\pi$
$139$	−5.12717	−0.434881	−0.217441	+	0.976074i	$0.569771\pi$
$140$	−0.450632	−0.0380854
$141$	0	0
$142$	3.66314	0.307404
$143$	−16.2112	−1.35565
$144$	0	0
$145$	13.7127	1.13878
$146$	−12.8215	−1.06112
$147$	0	0
$148$	0.295571	0.0242958
$149$	1.68186	0.137784	0.0688919	−	0.997624i	$-0.478054\pi$
$149$	1.68186	0.137784	0.0688919	+	0.997624i	$0.478054\pi$
$150$	0	0
$151$	4.85361	0.394981	0.197491	−	0.980305i	$-0.436721\pi$
$151$	4.85361	0.394981	0.197491	+	0.980305i	$0.436721\pi$
$152$	−11.7098	−0.949789
$153$	0	0
$154$	−17.6562	−1.42278
$155$	5.75117	0.461945
$156$	0	0
$157$	14.9123	1.19013	0.595064	−	0.803678i	$-0.297127\pi$
$157$	14.9123	1.19013	0.595064	+	0.803678i	$0.297127\pi$
$158$	−8.74087	−0.695386
$159$	0	0
$160$	0.729838	0.0576988
$161$	2.18286	0.172033
$162$	0	0
$163$	−22.4041	−1.75483	−0.877413	−	0.479735i	$-0.840733\pi$
$163$	−22.4041	−1.75483	−0.877413	+	0.479735i	$0.840733\pi$
$164$	1.18428	0.0924765
$165$	0	0
$166$	−19.6587	−1.52581
$167$	2.57870	0.199546	0.0997729	−	0.995010i	$-0.468188\pi$
$167$	2.57870	0.199546	0.0997729	+	0.995010i	$0.468188\pi$
$168$	0	0
$169$	6.52325	0.501788
$170$	14.3481	1.10045
$171$	0	0
$172$	−0.0161440	−0.00123097
$173$	2.69452	0.204861	0.102430	−	0.994740i	$-0.467338\pi$
$173$	2.69452	0.204861	0.102430	+	0.994740i	$0.467338\pi$
$174$	0	0
$175$	11.4091	0.862450
$176$	13.9200	1.04926
$177$	0	0
$178$	−13.1806	−0.987925
$179$	24.9712	1.86644	0.933219	−	0.359309i	$-0.116988\pi$
$179$	24.9712	1.86644	0.933219	+	0.359309i	$0.116988\pi$
$180$	0	0
$181$	2.85397	0.212134	0.106067	−	0.994359i	$-0.466174\pi$
$181$	2.85397	0.212134	0.106067	+	0.994359i	$0.466174\pi$
$182$	21.2634	1.57615
$183$	0	0
$184$	−1.81001	−0.133436
$185$	3.96079	0.291203
$186$	0	0
$187$	−29.0178	−2.12199
$188$	−0.0231644	−0.00168944
$189$	0	0
$190$	−7.34166	−0.532620
$191$	−3.43102	−0.248260	−0.124130	−	0.992266i	$-0.539614\pi$
$191$	−3.43102	−0.248260	−0.124130	+	0.992266i	$0.539614\pi$
$192$	0	0
$193$	10.7258	0.772060	0.386030	−	0.922486i	$-0.373846\pi$
$193$	10.7258	0.772060	0.386030	+	0.922486i	$0.373846\pi$
$194$	−5.03866	−0.361755
$195$	0	0
$196$	−0.508216	−0.0363012
$197$	−22.1350	−1.57705	−0.788527	−	0.615001i	$-0.789156\pi$
$197$	−22.1350	−1.57705	−0.788527	+	0.615001i	$0.789156\pi$
$198$	0	0
$199$	−16.1670	−1.14605	−0.573024	−	0.819538i	$-0.694230\pi$
$199$	−16.1670	−1.14605	−0.573024	+	0.819538i	$0.694230\pi$
$200$	−9.46037	−0.668949
$201$	0	0
$202$	9.71995	0.683894
$203$	36.3756	2.55307
$204$	0	0
$205$	15.8699	1.10840
$206$	−11.7682	−0.819927
$207$	0	0
$208$	−16.7640	−1.16237
$209$	14.8478	1.02704
$210$	0	0
$211$	7.10640	0.489225	0.244612	−	0.969621i	$-0.421339\pi$
$211$	7.10640	0.489225	0.244612	+	0.969621i	$0.421339\pi$
$212$	−1.12725	−0.0774200
$213$	0	0
$214$	9.68607	0.662126
$215$	−0.216337	−0.0147540
$216$	0	0
$217$	15.2561	1.03565
$218$	−15.1212	−1.02413
$219$	0	0
$220$	−0.473795	−0.0319433
$221$	34.9462	2.35074
$222$	0	0
$223$	−2.08114	−0.139364	−0.0696818	−	0.997569i	$-0.522198\pi$
$223$	−2.08114	−0.139364	−0.0696818	+	0.997569i	$0.522198\pi$
$224$	1.93603	0.129357
$225$	0	0
$226$	4.50364	0.299578
$227$	18.1848	1.20697	0.603485	−	0.797374i	$-0.293778\pi$
$227$	18.1848	1.20697	0.603485	+	0.797374i	$0.293778\pi$
$228$	0	0
$229$	0.908662	0.0600461	0.0300230	−	0.999549i	$-0.490442\pi$
$229$	0.908662	0.0600461	0.0300230	+	0.999549i	$0.490442\pi$
$230$	−1.13482	−0.0748276
$231$	0	0
$232$	−30.1624	−1.98026
$233$	−12.2053	−0.799594	−0.399797	−	0.916604i	$-0.630919\pi$
$233$	−12.2053	−0.799594	−0.399797	+	0.916604i	$0.630919\pi$
$234$	0	0
$235$	−0.310413	−0.0202491
$236$	0.705603	0.0459308
$237$	0	0
$238$	38.0611	2.46714
$239$	−9.40536	−0.608382	−0.304191	−	0.952611i	$-0.598386\pi$
$239$	−9.40536	−0.608382	−0.304191	+	0.952611i	$0.598386\pi$
$240$	0	0
$241$	−7.45266	−0.480068	−0.240034	−	0.970764i	$-0.577159\pi$
$241$	−7.45266	−0.480068	−0.240034	+	0.970764i	$0.577159\pi$
$242$	−3.39400	−0.218174
$243$	0	0
$244$	−0.778346	−0.0498285
$245$	−6.81033	−0.435096
$246$	0	0
$247$	−17.8813	−1.13776
$248$	−12.6502	−0.803289
$249$	0	0
$250$	−15.0021	−0.948814
$251$	−1.00000	−0.0631194
$251$	−1.00000	−0.0631194
$252$	0	0
$253$	2.29506	0.144289
$254$	1.65435	0.103803
$255$	0	0
$256$	−2.35024	−0.146890
$257$	8.59388	0.536071	0.268036	−	0.963409i	$-0.413625\pi$
$257$	8.59388	0.536071	0.268036	+	0.963409i	$0.413625\pi$
$258$	0	0
$259$	10.5067	0.652857
$260$	0.570594	0.0353867
$261$	0	0
$262$	−11.6398	−0.719107
$263$	−5.00613	−0.308691	−0.154346	−	0.988017i	$-0.549327\pi$
$263$	−5.00613	−0.308691	−0.154346	+	0.988017i	$0.549327\pi$
$264$	0	0
$265$	−15.1057	−0.927935
$266$	−19.4751	−1.19410
$267$	0	0
$268$	−0.0470141	−0.00287184
$269$	5.42980	0.331061	0.165530	−	0.986205i	$-0.447066\pi$
$269$	5.42980	0.331061	0.165530	+	0.986205i	$0.447066\pi$
$270$	0	0
$271$	−18.0972	−1.09933	−0.549664	−	0.835386i	$-0.685244\pi$
$271$	−18.0972	−1.09933	−0.549664	+	0.835386i	$0.685244\pi$
$272$	−30.0072	−1.81945
$273$	0	0
$274$	−18.0204	−1.08865
$275$	11.9956	0.723362
$276$	0	0
$277$	16.4601	0.988994	0.494497	−	0.869179i	$-0.335352\pi$
$277$	16.4601	0.988994	0.494497	+	0.869179i	$0.335352\pi$
$278$	7.07073	0.424074
$279$	0	0
$280$	13.2826	0.793786
$281$	3.66646	0.218722	0.109361	−	0.994002i	$-0.465119\pi$
$281$	3.66646	0.218722	0.109361	+	0.994002i	$0.465119\pi$
$282$	0	0
$283$	11.3048	0.672000	0.336000	−	0.941862i	$-0.390926\pi$
$283$	11.3048	0.672000	0.336000	+	0.941862i	$0.390926\pi$
$284$	0.260755	0.0154730
$285$	0	0
$286$	22.3564	1.32196
$287$	42.0977	2.48495
$288$	0	0
$289$	45.5531	2.67959
$290$	−18.9108	−1.11048
$291$	0	0
$292$	−0.912680	−0.0534105
$293$	27.8052	1.62440	0.812199	−	0.583380i	$-0.198270\pi$
$293$	27.8052	1.62440	0.812199	+	0.583380i	$0.198270\pi$
$294$	0	0
$295$	9.45540	0.550515
$296$	−8.71210	−0.506380
$297$	0	0
$298$	−2.31941	−0.134360
$299$	−2.76395	−0.159843
$300$	0	0
$301$	−0.573873	−0.0330775
$302$	−6.69347	−0.385166
$303$	0	0
$304$	15.3541	0.880616
$305$	−10.4302	−0.597231
$306$	0	0
$307$	−18.2217	−1.03997	−0.519985	−	0.854176i	$-0.674062\pi$
$307$	−18.2217	−1.03997	−0.519985	+	0.854176i	$0.674062\pi$
$308$	−1.25683	−0.0716146
$309$	0	0
$310$	−7.93127	−0.450466
$311$	16.7179	0.947987	0.473994	−	0.880528i	$-0.342812\pi$
$311$	16.7179	0.947987	0.473994	+	0.880528i	$0.342812\pi$
$312$	0	0
$313$	22.1319	1.25097	0.625483	−	0.780238i	$-0.284902\pi$
$313$	22.1319	1.25097	0.625483	+	0.780238i	$0.284902\pi$
$314$	−20.5650	−1.16055
$315$	0	0
$316$	−0.622206	−0.0350018
$317$	2.71767	0.152639	0.0763197	−	0.997083i	$-0.475683\pi$
$317$	2.71767	0.152639	0.0763197	+	0.997083i	$0.475683\pi$
$318$	0	0
$319$	38.2454	2.14133
$320$	−10.9885	−0.614274
$321$	0	0
$322$	−3.01031	−0.167758
$323$	−32.0072	−1.78093
$324$	0	0
$325$	−14.4463	−0.801339
$326$	30.8968	1.71122
$327$	0	0
$328$	−34.9071	−1.92742
$329$	−0.823429	−0.0453971
$330$	0	0
$331$	−32.1394	−1.76654	−0.883270	−	0.468864i	$-0.844663\pi$
$331$	−32.1394	−1.76654	−0.883270	+	0.468864i	$0.844663\pi$
$332$	−1.39938	−0.0768009
$333$	0	0
$334$	−3.55621	−0.194587
$335$	−0.630010	−0.0344211
$336$	0	0
$337$	−26.4970	−1.44338	−0.721692	−	0.692215i	$-0.756635\pi$
$337$	−26.4970	−1.44338	−0.721692	+	0.692215i	$0.756635\pi$
$338$	−8.99601	−0.489319
$339$	0	0
$340$	1.02135	0.0553906
$341$	16.0403	0.868629
$342$	0	0
$343$	6.36128	0.343477
$344$	0.475851	0.0256562
$345$	0	0
$346$	−3.71593	−0.199770
$347$	22.2777	1.19593	0.597964	−	0.801523i	$-0.295977\pi$
$347$	22.2777	1.19593	0.597964	+	0.801523i	$0.295977\pi$
$348$	0	0
$349$	13.3137	0.712667	0.356333	−	0.934359i	$-0.384027\pi$
$349$	13.3137	0.712667	0.356333	+	0.934359i	$0.384027\pi$
$350$	−15.7340	−0.841018
$351$	0	0
$352$	2.03555	0.108495
$353$	20.7420	1.10399	0.551994	−	0.833848i	$-0.313867\pi$
$353$	20.7420	1.10399	0.551994	+	0.833848i	$0.313867\pi$
$354$	0	0
$355$	3.49424	0.185455
$356$	−0.938239	−0.0497266
$357$	0	0
$358$	−34.4371	−1.82006
$359$	8.64717	0.456380	0.228190	−	0.973617i	$-0.426719\pi$
$359$	8.64717	0.456380	0.228190	+	0.973617i	$0.426719\pi$
$360$	0	0
$361$	−2.62259	−0.138031
$362$	−3.93582	−0.206862
$363$	0	0
$364$	1.51361	0.0793345
$365$	−12.2303	−0.640164
$366$	0	0
$367$	25.1717	1.31395	0.656976	−	0.753912i	$-0.271835\pi$
$367$	25.1717	1.31395	0.656976	+	0.753912i	$0.271835\pi$
$368$	2.37331	0.123717
$369$	0	0
$370$	−5.46221	−0.283967
$371$	−40.0707	−2.08037
$372$	0	0
$373$	−0.893363	−0.0462566	−0.0231283	−	0.999733i	$-0.507363\pi$
$373$	−0.893363	−0.0462566	−0.0231283	+	0.999733i	$0.507363\pi$
$374$	40.0175	2.06926
$375$	0	0
$376$	0.682780	0.0352117
$377$	−46.0591	−2.37216
$378$	0	0
$379$	−2.52089	−0.129489	−0.0647447	−	0.997902i	$-0.520623\pi$
$379$	−2.52089	−0.129489	−0.0647447	+	0.997902i	$0.520623\pi$
$380$	−0.522605	−0.0268091
$381$	0	0
$382$	4.73161	0.242090
$383$	−3.74890	−0.191560	−0.0957800	−	0.995403i	$-0.530535\pi$
$383$	−3.74890	−0.191560	−0.0957800	+	0.995403i	$0.530535\pi$
$384$	0	0
$385$	−16.8421	−0.858353
$386$	−14.7916	−0.752874
$387$	0	0
$388$	−0.358669	−0.0182087
$389$	−0.193533	−0.00981252	−0.00490626	−	0.999988i	$-0.501562\pi$
$389$	−0.193533	−0.00981252	−0.00490626	+	0.999988i	$0.501562\pi$
$390$	0	0
$391$	−4.94742	−0.250202
$392$	14.9799	0.756599
$393$	0	0
$394$	30.5257	1.53786
$395$	−8.33784	−0.419522
$396$	0	0
$397$	15.8009	0.793025	0.396513	−	0.918029i	$-0.370220\pi$
$397$	15.8009	0.793025	0.396513	+	0.918029i	$0.370220\pi$
$398$	22.2954	1.11757
$399$	0	0
$400$	12.4046	0.620230
$401$	−15.2558	−0.761839	−0.380920	−	0.924608i	$-0.624393\pi$
$401$	−15.2558	−0.761839	−0.380920	+	0.924608i	$0.624393\pi$
$402$	0	0
$403$	−19.3173	−0.962265
$404$	0.691901	0.0344233
$405$	0	0
$406$	−50.1645	−2.48962
$407$	11.0468	0.547570
$408$	0	0
$409$	12.0065	0.593683	0.296842	−	0.954927i	$-0.404067\pi$
$409$	12.0065	0.593683	0.296842	+	0.954927i	$0.404067\pi$
$410$	−21.8856	−1.08085
$411$	0	0
$412$	−0.837700	−0.0412705
$413$	25.0822	1.23421
$414$	0	0
$415$	−18.7523	−0.920515
$416$	−2.45142	−0.120191
$417$	0	0
$418$	−20.4762	−1.00152
$419$	15.9744	0.780402	0.390201	−	0.920730i	$-0.372405\pi$
$419$	15.9744	0.780402	0.390201	+	0.920730i	$0.372405\pi$
$420$	0	0
$421$	25.2569	1.23095	0.615474	−	0.788157i	$-0.288965\pi$
$421$	25.2569	1.23095	0.615474	+	0.788157i	$0.288965\pi$
$422$	−9.80022	−0.477067
$423$	0	0
$424$	33.2263	1.61361
$425$	−25.8587	−1.25433
$426$	0	0
$427$	−27.6680	−1.33895
$428$	0.689488	0.0333277
$429$	0	0
$430$	0.298343	0.0143874
$431$	1.29490	0.0623733	0.0311867	−	0.999514i	$-0.490071\pi$
$431$	1.29490	0.0623733	0.0311867	+	0.999514i	$0.490071\pi$
$432$	0	0
$433$	23.8563	1.14646	0.573231	−	0.819394i	$-0.305690\pi$
$433$	23.8563	1.14646	0.573231	+	0.819394i	$0.305690\pi$
$434$	−21.0392	−1.00991
$435$	0	0
$436$	−1.07638	−0.0515491
$437$	2.53149	0.121098
$438$	0	0
$439$	13.5767	0.647980	0.323990	−	0.946060i	$-0.394976\pi$
$439$	13.5767	0.647980	0.323990	+	0.946060i	$0.394976\pi$
$440$	13.9653	0.665771
$441$	0	0
$442$	−48.1933	−2.29232
$443$	−22.4586	−1.06704	−0.533520	−	0.845787i	$-0.679131\pi$
$443$	−22.4586	−1.06704	−0.533520	+	0.845787i	$0.679131\pi$
$444$	0	0
$445$	−12.5728	−0.596009
$446$	2.87004	0.135900
$447$	0	0
$448$	−29.1489	−1.37716
$449$	−29.9262	−1.41230	−0.706152	−	0.708060i	$-0.749571\pi$
$449$	−29.9262	−1.41230	−0.706152	+	0.708060i	$0.749571\pi$
$450$	0	0
$451$	44.2616	2.08420
$452$	0.320585	0.0150790
$453$	0	0
$454$	−25.0781	−1.17698
$455$	20.2830	0.950882
$456$	0	0
$457$	−31.2410	−1.46139	−0.730696	−	0.682703i	$-0.760804\pi$
$457$	−31.2410	−1.46139	−0.730696	+	0.682703i	$0.760804\pi$
$458$	−1.25311	−0.0585539
$459$	0	0
$460$	−0.0807802	−0.00376640
$461$	24.9115	1.16025	0.580123	−	0.814529i	$-0.303005\pi$
$461$	24.9115	1.16025	0.580123	+	0.814529i	$0.303005\pi$
$462$	0	0
$463$	−16.2898	−0.757050	−0.378525	−	0.925591i	$-0.623569\pi$
$463$	−16.2898	−0.757050	−0.378525	+	0.925591i	$0.623569\pi$
$464$	39.5494	1.83603
$465$	0	0
$466$	16.8319	0.779724
$467$	27.7607	1.28461	0.642305	−	0.766449i	$-0.277978\pi$
$467$	27.7607	1.28461	0.642305	+	0.766449i	$0.277978\pi$
$468$	0	0
$469$	−1.67122	−0.0771698
$470$	0.428081	0.0197459
$471$	0	0
$472$	−20.7980	−0.957304
$473$	−0.603372	−0.0277431
$474$	0	0
$475$	13.2314	0.607097
$476$	2.70933	0.124182
$477$	0	0
$478$	12.9706	0.593263
$479$	33.4396	1.52789	0.763947	−	0.645279i	$-0.223259\pi$
$479$	33.4396	1.52789	0.763947	+	0.645279i	$0.223259\pi$
$480$	0	0
$481$	−13.3037	−0.606597
$482$	10.2777	0.468138
$483$	0	0
$484$	−0.241597	−0.0109817
$485$	−4.80633	−0.218244
$486$	0	0
$487$	−5.85497	−0.265314	−0.132657	−	0.991162i	$-0.542351\pi$
$487$	−5.85497	−0.265314	−0.132657	+	0.991162i	$0.542351\pi$
$488$	22.9421	1.03854
$489$	0	0
$490$	9.39192	0.424284
$491$	−27.9694	−1.26224	−0.631121	−	0.775684i	$-0.717405\pi$
$491$	−27.9694	−1.26224	−0.631121	+	0.775684i	$0.717405\pi$
$492$	0	0
$493$	−82.4449	−3.71313
$494$	24.6595	1.10948
$495$	0	0
$496$	16.5872	0.744785
$497$	9.26912	0.415777
$498$	0	0
$499$	28.6178	1.28111	0.640553	−	0.767914i	$-0.278705\pi$
$499$	28.6178	1.28111	0.640553	+	0.767914i	$0.278705\pi$
$500$	−1.06790	−0.0477579
$501$	0	0
$502$	1.37907	0.0615509
$503$	−4.06882	−0.181420	−0.0907098	−	0.995877i	$-0.528914\pi$
$503$	−4.06882	−0.181420	−0.0907098	+	0.995877i	$0.528914\pi$
$504$	0	0
$505$	9.27178	0.412589
$506$	−3.16505	−0.140704
$507$	0	0
$508$	0.117763	0.00522488
$509$	3.27751	0.145273	0.0726364	−	0.997358i	$-0.476859\pi$
$509$	3.27751	0.145273	0.0726364	+	0.997358i	$0.476859\pi$
$510$	0	0
$511$	−32.4432	−1.43520
$512$	24.0611	1.06336
$513$	0	0
$514$	−11.8516	−0.522750
$515$	−11.2256	−0.494657
$516$	0	0
$517$	−0.865754	−0.0380758
$518$	−14.4895	−0.636633
$519$	0	0
$520$	−16.8185	−0.737540
$521$	−19.6775	−0.862085	−0.431043	−	0.902332i	$-0.641854\pi$
$521$	−19.6775	−0.862085	−0.431043	+	0.902332i	$0.641854\pi$
$522$	0	0
$523$	35.9586	1.57236	0.786181	−	0.617996i	$-0.212055\pi$
$523$	35.9586	1.57236	0.786181	+	0.617996i	$0.212055\pi$
$524$	−0.828559	−0.0361958
$525$	0	0
$526$	6.90380	0.301020
$527$	−34.5777	−1.50623
$528$	0	0
$529$	−22.6087	−0.982987
$530$	20.8318	0.904876
$531$	0	0
$532$	−1.38631	−0.0601040
$533$	−53.3045	−2.30887
$534$	0	0
$535$	9.23946	0.399457
$536$	1.38576	0.0598558
$537$	0	0
$538$	−7.48807	−0.322833
$539$	−18.9943	−0.818141
$540$	0	0
$541$	−14.0585	−0.604421	−0.302211	−	0.953241i	$-0.597725\pi$
$541$	−14.0585	−0.604421	−0.302211	+	0.953241i	$0.597725\pi$
$542$	24.9573	1.07201
$543$	0	0
$544$	−4.38799	−0.188134
$545$	−14.4239	−0.617854
$546$	0	0
$547$	18.0040	0.769795	0.384898	−	0.922959i	$-0.374237\pi$
$547$	18.0040	0.769795	0.384898	+	0.922959i	$0.374237\pi$
$548$	−1.28276	−0.0547967
$549$	0	0
$550$	−16.5428	−0.705386
$551$	42.1854	1.79716
$552$	0	0
$553$	−22.1177	−0.940540
$554$	−22.6997	−0.964417
$555$	0	0
$556$	0.503319	0.0213455
$557$	−7.94515	−0.336647	−0.168323	−	0.985732i	$-0.553835\pi$
$557$	−7.94515	−0.336647	−0.168323	+	0.985732i	$0.553835\pi$
$558$	0	0
$559$	0.726643	0.0307337
$560$	−17.4163	−0.735975
$561$	0	0
$562$	−5.05630	−0.213287
$563$	−17.1327	−0.722059	−0.361029	−	0.932554i	$-0.617575\pi$
$563$	−17.1327	−0.722059	−0.361029	+	0.932554i	$0.617575\pi$
$564$	0	0
$565$	4.29598	0.180733
$566$	−15.5901	−0.655301
$567$	0	0
$568$	−7.68588	−0.322492
$569$	−22.1683	−0.929343	−0.464672	−	0.885483i	$-0.653828\pi$
$569$	−22.1683	−0.929343	−0.464672	+	0.885483i	$0.653828\pi$
$570$	0	0
$571$	6.04758	0.253083	0.126542	−	0.991961i	$-0.459612\pi$
$571$	6.04758	0.253083	0.126542	+	0.991961i	$0.459612\pi$
$572$	1.59141	0.0665401
$573$	0	0
$574$	−58.0557	−2.42320
$575$	2.04520	0.0852908
$576$	0	0
$577$	23.3358	0.971483	0.485742	−	0.874102i	$-0.338550\pi$
$577$	23.3358	0.971483	0.485742	+	0.874102i	$0.338550\pi$
$578$	−62.8209	−2.61300
$579$	0	0
$580$	−1.34614	−0.0558954
$581$	−49.7440	−2.06373
$582$	0	0
$583$	−42.1304	−1.74486
$584$	26.9017	1.11320
$585$	0	0
$586$	−38.3453	−1.58403
$587$	−5.01691	−0.207070	−0.103535	−	0.994626i	$-0.533015\pi$
$587$	−5.01691	−0.207070	−0.103535	+	0.994626i	$0.533015\pi$
$588$	0	0
$589$	17.6927	0.729015
$590$	−13.0397	−0.536834
$591$	0	0
$592$	11.4234	0.469501
$593$	−14.9574	−0.614228	−0.307114	−	0.951673i	$-0.599363\pi$
$593$	−14.9574	−0.614228	−0.307114	+	0.951673i	$0.599363\pi$
$594$	0	0
$595$	36.3062	1.48841
$596$	−0.165104	−0.00676291
$597$	0	0
$598$	3.81168	0.155871
$599$	37.6200	1.53711	0.768555	−	0.639784i	$-0.220976\pi$
$599$	37.6200	1.53711	0.768555	+	0.639784i	$0.220976\pi$
$600$	0	0
$601$	−25.2201	−1.02875	−0.514374	−	0.857566i	$-0.671976\pi$
$601$	−25.2201	−1.02875	−0.514374	+	0.857566i	$0.671976\pi$
$602$	0.791411	0.0322555
$603$	0	0
$604$	−0.476464	−0.0193871
$605$	−3.23751	−0.131623
$606$	0	0
$607$	−27.6339	−1.12162	−0.560812	−	0.827943i	$-0.689511\pi$
$607$	−27.6339	−1.12162	−0.560812	+	0.827943i	$0.689511\pi$
$608$	2.24525	0.0910568
$609$	0	0
$610$	14.3840	0.582390
$611$	1.04263	0.0421804
$612$	0	0
$613$	−33.9869	−1.37272	−0.686359	−	0.727263i	$-0.740792\pi$
$613$	−33.9869	−1.37272	−0.686359	+	0.727263i	$0.740792\pi$
$614$	25.1290	1.01413
$615$	0	0
$616$	37.0457	1.49261
$617$	−33.4920	−1.34834	−0.674169	−	0.738577i	$-0.735498\pi$
$617$	−33.4920	−1.34834	−0.674169	+	0.738577i	$0.735498\pi$
$618$	0	0
$619$	26.5574	1.06743	0.533717	−	0.845663i	$-0.320795\pi$
$619$	26.5574	1.06743	0.533717	+	0.845663i	$0.320795\pi$
$620$	−0.564576	−0.0226739
$621$	0	0
$622$	−23.0552	−0.924429
$623$	−33.3518	−1.33621
$624$	0	0
$625$	2.03718	0.0814873
$626$	−30.5214	−1.21988
$627$	0	0
$628$	−1.46389	−0.0584157
$629$	−23.8134	−0.949501
$630$	0	0
$631$	−0.456653	−0.0181791	−0.00908953	−	0.999959i	$-0.502893\pi$
$631$	−0.456653	−0.0181791	−0.00908953	+	0.999959i	$0.502893\pi$
$632$	18.3398	0.729518
$633$	0	0
$634$	−3.74785	−0.148846
$635$	1.57808	0.0626240
$636$	0	0
$637$	22.8749	0.906336
$638$	−52.7431	−2.08812
$639$	0	0
$640$	13.6942	0.541310
$641$	−33.0081	−1.30374	−0.651871	−	0.758330i	$-0.726016\pi$
$641$	−33.0081	−1.30374	−0.651871	+	0.758330i	$0.726016\pi$
$642$	0	0
$643$	−28.9957	−1.14348	−0.571739	−	0.820436i	$-0.693731\pi$
$643$	−28.9957	−1.14348	−0.571739	+	0.820436i	$0.693731\pi$
$644$	−0.214285	−0.00844400
$645$	0	0
$646$	44.1401	1.73667
$647$	−5.00136	−0.196624	−0.0983119	−	0.995156i	$-0.531344\pi$
$647$	−5.00136	−0.196624	−0.0983119	+	0.995156i	$0.531344\pi$
$648$	0	0
$649$	26.3715	1.03517
$650$	19.9225	0.781425
$651$	0	0
$652$	2.19935	0.0861331
$653$	−5.31388	−0.207948	−0.103974	−	0.994580i	$-0.533156\pi$
$653$	−5.31388	−0.207948	−0.103974	+	0.994580i	$0.533156\pi$
$654$	0	0
$655$	−11.1031	−0.433833
$656$	45.7708	1.78705
$657$	0	0
$658$	1.13557	0.0442689
$659$	19.1561	0.746214	0.373107	−	0.927788i	$-0.378292\pi$
$659$	19.1561	0.746214	0.373107	+	0.927788i	$0.378292\pi$
$660$	0	0
$661$	22.9498	0.892643	0.446322	−	0.894873i	$-0.352734\pi$
$661$	22.9498	0.892643	0.446322	+	0.894873i	$0.352734\pi$
$662$	44.3224	1.72264
$663$	0	0
$664$	41.2473	1.60071
$665$	−18.5771	−0.720391
$666$	0	0
$667$	6.52069	0.252482
$668$	−0.253143	−0.00979441
$669$	0	0
$670$	0.868828	0.0335658
$671$	−29.0902	−1.12302
$672$	0	0
$673$	19.6759	0.758450	0.379225	−	0.925305i	$-0.376191\pi$
$673$	19.6759	0.758450	0.379225	+	0.925305i	$0.376191\pi$
$674$	36.5412	1.40751
$675$	0	0
$676$	−0.640368	−0.0246295
$677$	32.7409	1.25834	0.629168	−	0.777269i	$-0.283396\pi$
$677$	32.7409	1.25834	0.629168	+	0.777269i	$0.283396\pi$
$678$	0	0
$679$	−12.7497	−0.489288
$680$	−30.1048	−1.15447
$681$	0	0
$682$	−22.1206	−0.847043
$683$	−4.74368	−0.181512	−0.0907560	−	0.995873i	$-0.528928\pi$
$683$	−4.74368	−0.181512	−0.0907560	+	0.995873i	$0.528928\pi$
$684$	0	0
$685$	−17.1896	−0.656779
$686$	−8.77265	−0.334941
$687$	0	0
$688$	−0.623944	−0.0237876
$689$	50.7378	1.93296
$690$	0	0
$691$	−17.4515	−0.663886	−0.331943	−	0.943300i	$-0.607704\pi$
$691$	−17.4515	−0.663886	−0.331943	+	0.943300i	$0.607704\pi$
$692$	−0.264513	−0.0100553
$693$	0	0
$694$	−30.7224	−1.16621
$695$	6.74471	0.255841
$696$	0	0
$697$	−95.4140	−3.61406
$698$	−18.3605	−0.694957
$699$	0	0
$700$	−1.12000	−0.0423321
$701$	−45.8717	−1.73255	−0.866274	−	0.499569i	$-0.833492\pi$
$701$	−45.8717	−1.73255	−0.866274	+	0.499569i	$0.833492\pi$
$702$	0	0
$703$	12.1848	0.459559
$704$	−30.6472	−1.15506
$705$	0	0
$706$	−28.6047	−1.07655
$707$	24.5951	0.924995
$708$	0	0
$709$	−20.8242	−0.782069	−0.391035	−	0.920376i	$-0.627883\pi$
$709$	−20.8242	−0.782069	−0.391035	+	0.920376i	$0.627883\pi$
$710$	−4.81880	−0.180846
$711$	0	0
$712$	27.6550	1.03642
$713$	2.73480	0.102419
$714$	0	0
$715$	21.3256	0.797532
$716$	−2.45135	−0.0916113
$717$	0	0
$718$	−11.9251	−0.445039
$719$	−38.1880	−1.42417	−0.712086	−	0.702092i	$-0.752249\pi$
$719$	−38.1880	−1.42417	−0.712086	+	0.702092i	$0.752249\pi$
$720$	0	0
$721$	−29.7779	−1.10899
$722$	3.61673	0.134601
$723$	0	0
$724$	−0.280165	−0.0104123
$725$	34.0817	1.26576
$726$	0	0
$727$	37.1384	1.37739	0.688693	−	0.725053i	$-0.258185\pi$
$727$	37.1384	1.37739	0.688693	+	0.725053i	$0.258185\pi$
$728$	−44.6142	−1.65351
$729$	0	0
$730$	16.8665	0.624256
$731$	1.30068	0.0481073
$732$	0	0
$733$	27.7780	1.02601	0.513003	−	0.858387i	$-0.328533\pi$
$733$	27.7780	1.02601	0.513003	+	0.858387i	$0.328533\pi$
$734$	−34.7135	−1.28130
$735$	0	0
$736$	0.347053	0.0127925
$737$	−1.75712	−0.0647245
$738$	0	0
$739$	24.3250	0.894809	0.447405	−	0.894332i	$-0.352348\pi$
$739$	24.3250	0.894809	0.447405	+	0.894332i	$0.352348\pi$
$740$	−0.388819	−0.0142933
$741$	0	0
$742$	55.2602	2.02867
$743$	−22.6948	−0.832590	−0.416295	−	0.909230i	$-0.636672\pi$
$743$	−22.6948	−0.832590	−0.416295	+	0.909230i	$0.636672\pi$
$744$	0	0
$745$	−2.21246	−0.0810584
$746$	1.23201	0.0451071
$747$	0	0
$748$	2.84859	0.104155
$749$	24.5094	0.895553
$750$	0	0
$751$	53.8859	1.96632	0.983162	−	0.182738i	$-0.0584960\pi$
$751$	53.8859	1.96632	0.983162	+	0.182738i	$0.0584960\pi$
$752$	−0.895273	−0.0326472
$753$	0	0
$754$	63.5187	2.31321
$755$	−6.38484	−0.232368
$756$	0	0
$757$	38.8162	1.41080	0.705400	−	0.708810i	$-0.250768\pi$
$757$	38.8162	1.41080	0.705400	+	0.708810i	$0.250768\pi$
$758$	3.47648	0.126272
$759$	0	0
$760$	15.4040	0.558763
$761$	28.5923	1.03647	0.518235	−	0.855238i	$-0.326589\pi$
$761$	28.5923	1.03647	0.518235	+	0.855238i	$0.326589\pi$
$762$	0	0
$763$	−38.2622	−1.38518
$764$	0.336813	0.0121855
$765$	0	0
$766$	5.17000	0.186800
$767$	−31.7593	−1.14676
$768$	0	0
$769$	−36.2609	−1.30760	−0.653800	−	0.756667i	$-0.726826\pi$
$769$	−36.2609	−1.30760	−0.653800	+	0.756667i	$0.726826\pi$
$770$	23.2264	0.837022
$771$	0	0
$772$	−1.05292	−0.0378954
$773$	41.4690	1.49154	0.745768	−	0.666205i	$-0.232083\pi$
$773$	41.4690	1.49154	0.745768	+	0.666205i	$0.232083\pi$
$774$	0	0
$775$	14.2940	0.513455
$776$	10.5719	0.379511
$777$	0	0
$778$	0.266896	0.00956867
$779$	48.8214	1.74921
$780$	0	0
$781$	9.74557	0.348724
$782$	6.82283	0.243984
$783$	0	0
$784$	−19.6419	−0.701496
$785$	−19.6168	−0.700155
$786$	0	0
$787$	−46.6878	−1.66424	−0.832119	−	0.554597i	$-0.812873\pi$
$787$	−46.6878	−1.66424	−0.832119	+	0.554597i	$0.812873\pi$
$788$	2.17293	0.0774073
$789$	0	0
$790$	11.4985	0.409097
$791$	11.3959	0.405191
$792$	0	0
$793$	35.0335	1.24408
$794$	−21.7906	−0.773318
$795$	0	0
$796$	1.58707	0.0562521
$797$	45.0698	1.59646	0.798228	−	0.602356i	$-0.205771\pi$
$797$	45.0698	1.59646	0.798228	+	0.602356i	$0.205771\pi$
$798$	0	0
$799$	1.86629	0.0660246
$800$	1.81394	0.0641325
$801$	0	0
$802$	21.0388	0.742907
$803$	−34.1109	−1.20375
$804$	0	0
$805$	−2.87151	−0.101207
$806$	26.6400	0.938353
$807$	0	0
$808$	−20.3941	−0.717462
$809$	−0.551761	−0.0193989	−0.00969944	−	0.999953i	$-0.503087\pi$
$809$	−0.551761	−0.0193989	−0.00969944	+	0.999953i	$0.503087\pi$
$810$	0	0
$811$	4.33715	0.152298	0.0761490	−	0.997096i	$-0.475738\pi$
$811$	4.33715	0.152298	0.0761490	+	0.997096i	$0.475738\pi$
$812$	−3.57089	−0.125314
$813$	0	0
$814$	−15.2343	−0.533962
$815$	29.4722	1.03237
$816$	0	0
$817$	−0.665530	−0.0232840
$818$	−16.5578	−0.578930
$819$	0	0
$820$	−1.55790	−0.0544041
$821$	−7.79694	−0.272115	−0.136058	−	0.990701i	$-0.543443\pi$
$821$	−7.79694	−0.272115	−0.136058	+	0.990701i	$0.543443\pi$
$822$	0	0
$823$	23.2012	0.808741	0.404371	−	0.914595i	$-0.367491\pi$
$823$	23.2012	0.808741	0.404371	+	0.914595i	$0.367491\pi$
$824$	24.6916	0.860172
$825$	0	0
$826$	−34.5901	−1.20354
$827$	−40.8965	−1.42211	−0.711055	−	0.703136i	$-0.751783\pi$
$827$	−40.8965	−1.42211	−0.711055	+	0.703136i	$0.751783\pi$
$828$	0	0
$829$	13.7145	0.476324	0.238162	−	0.971225i	$-0.423455\pi$
$829$	13.7145	0.476324	0.238162	+	0.971225i	$0.423455\pi$
$830$	25.8607	0.897639
$831$	0	0
$832$	36.9086	1.27958
$833$	40.9456	1.41868
$834$	0	0
$835$	−3.39224	−0.117393
$836$	−1.45757	−0.0504110
$837$	0	0
$838$	−22.0299	−0.761009
$839$	32.4698	1.12098	0.560491	−	0.828161i	$-0.310613\pi$
$839$	32.4698	1.12098	0.560491	+	0.828161i	$0.310613\pi$
$840$	0	0
$841$	79.6622	2.74697
$842$	−34.8311	−1.20036
$843$	0	0
$844$	−0.697614	−0.0240129
$845$	−8.58122	−0.295203
$846$	0	0
$847$	−8.58809	−0.295090
$848$	−43.5668	−1.49609
$849$	0	0
$850$	35.6609	1.22316
$851$	1.88344	0.0645633
$852$	0	0
$853$	−0.229410	−0.00785485	−0.00392743	−	0.999992i	$-0.501250\pi$
$853$	−0.229410	−0.00785485	−0.00392743	+	0.999992i	$0.501250\pi$
$854$	38.1562	1.30568
$855$	0	0
$856$	−20.3230	−0.694625
$857$	33.4957	1.14419	0.572096	−	0.820186i	$-0.306130\pi$
$857$	33.4957	1.14419	0.572096	+	0.820186i	$0.306130\pi$
$858$	0	0
$859$	−18.0814	−0.616931	−0.308465	−	0.951236i	$-0.599815\pi$
$859$	−18.0814	−0.616931	−0.308465	+	0.951236i	$0.599815\pi$
$860$	0.0212371	0.000724180 0
$861$	0	0
$862$	−1.78576	−0.0608233
$863$	−10.3333	−0.351751	−0.175875	−	0.984412i	$-0.556276\pi$
$863$	−10.3333	−0.351751	−0.175875	+	0.984412i	$0.556276\pi$
$864$	0	0
$865$	−3.54460	−0.120520
$866$	−32.8995	−1.11797
$867$	0	0
$868$	−1.49764	−0.0508333
$869$	−23.2546	−0.788857
$870$	0	0
$871$	2.11611	0.0717017
$872$	31.7267	1.07440
$873$	0	0
$874$	−3.49111	−0.118088
$875$	−37.9608	−1.28331
$876$	0	0
$877$	32.9398	1.11230	0.556149	−	0.831083i	$-0.312278\pi$
$877$	32.9398	1.11230	0.556149	+	0.831083i	$0.312278\pi$
$878$	−18.7232	−0.631878
$879$	0	0
$880$	−18.3116	−0.617283
$881$	−14.5017	−0.488573	−0.244287	−	0.969703i	$-0.578554\pi$
$881$	−14.5017	−0.488573	−0.244287	+	0.969703i	$0.578554\pi$
$882$	0	0
$883$	28.1025	0.945725	0.472862	−	0.881136i	$-0.343221\pi$
$883$	28.1025	0.945725	0.472862	+	0.881136i	$0.343221\pi$
$884$	−3.43057	−0.115382
$885$	0	0
$886$	30.9720	1.04052
$887$	−10.2546	−0.344314	−0.172157	−	0.985070i	$-0.555074\pi$
$887$	−10.2546	−0.344314	−0.172157	+	0.985070i	$0.555074\pi$
$888$	0	0
$889$	4.18614	0.140399
$890$	17.3388	0.581198
$891$	0	0
$892$	0.204300	0.00684046
$893$	−0.954943	−0.0319559
$894$	0	0
$895$	−32.8492	−1.09803
$896$	36.3263	1.21358
$897$	0	0
$898$	41.2703	1.37721
$899$	45.5733	1.51996
$900$	0	0
$901$	90.8196	3.02564
$902$	−61.0399	−2.03241
$903$	0	0
$904$	−9.44939	−0.314282
$905$	−3.75434	−0.124799
$906$	0	0
$907$	34.6173	1.14945	0.574725	−	0.818347i	$-0.305109\pi$
$907$	34.6173	1.14945	0.574725	+	0.818347i	$0.305109\pi$
$908$	−1.78515	−0.0592423
$909$	0	0
$910$	−27.9717	−0.927252
$911$	−19.5034	−0.646178	−0.323089	−	0.946369i	$-0.604721\pi$
$911$	−19.5034	−0.646178	−0.323089	+	0.946369i	$0.604721\pi$
$912$	0	0
$913$	−52.3009	−1.73091
$914$	43.0835	1.42508
$915$	0	0
$916$	−0.0892007	−0.00294727
$917$	−29.4530	−0.972622
$918$	0	0
$919$	−26.3765	−0.870081	−0.435041	−	0.900411i	$-0.643266\pi$
$919$	−26.3765	−0.870081	−0.435041	+	0.900411i	$0.643266\pi$
$920$	2.38103	0.0785004
$921$	0	0
$922$	−34.3547	−1.13141
$923$	−11.7366	−0.386316
$924$	0	0
$925$	9.84416	0.323674
$926$	22.4647	0.738237
$927$	0	0
$928$	5.78337	0.189848
$929$	28.6371	0.939553	0.469776	−	0.882785i	$-0.344335\pi$
$929$	28.6371	0.939553	0.469776	+	0.882785i	$0.344335\pi$
$930$	0	0
$931$	−20.9510	−0.686642
$932$	1.19816	0.0392469
$933$	0	0
$934$	−38.2839	−1.25269
$935$	38.1724	1.24837
$936$	0	0
$937$	29.0180	0.947977	0.473989	−	0.880531i	$-0.342814\pi$
$937$	29.0180	0.947977	0.473989	+	0.880531i	$0.342814\pi$
$938$	2.30473	0.0752520
$939$	0	0
$940$	0.0304723	0.000993898 0
$941$	−28.8763	−0.941339	−0.470670	−	0.882310i	$-0.655988\pi$
$941$	−28.8763	−0.941339	−0.470670	+	0.882310i	$0.655988\pi$
$942$	0	0
$943$	7.54643	0.245746
$944$	27.2706	0.887584
$945$	0	0
$946$	0.832092	0.0270536
$947$	−20.2389	−0.657676	−0.328838	−	0.944386i	$-0.606657\pi$
$947$	−20.2389	−0.657676	−0.328838	+	0.944386i	$0.606657\pi$
$948$	0	0
$949$	41.0798	1.33351
$950$	−18.2470	−0.592010
$951$	0	0
$952$	−79.8586	−2.58823
$953$	12.2731	0.397564	0.198782	−	0.980044i	$-0.436301\pi$
$953$	12.2731	0.397564	0.198782	+	0.980044i	$0.436301\pi$
$954$	0	0
$955$	4.51345	0.146052
$956$	0.923296	0.0298615
$957$	0	0
$958$	−46.1155	−1.48993
$959$	−45.5985	−1.47245
$960$	0	0
$961$	−11.8864	−0.383432
$962$	18.3467	0.591522
$963$	0	0
$964$	0.731606	0.0235634
$965$	−14.1096	−0.454204
$966$	0	0
$967$	−7.18637	−0.231098	−0.115549	−	0.993302i	$-0.536863\pi$
$967$	−7.18637	−0.231098	−0.115549	+	0.993302i	$0.536863\pi$
$968$	7.12118	0.228883
$969$	0	0
$970$	6.62827	0.212821
$971$	−27.1179	−0.870256	−0.435128	−	0.900369i	$-0.643297\pi$
$971$	−27.1179	−0.870256	−0.435128	+	0.900369i	$0.643297\pi$
$972$	0	0
$973$	17.8916	0.573578
$974$	8.07441	0.258721
$975$	0	0
$976$	−30.0821	−0.962904
$977$	29.1247	0.931782	0.465891	−	0.884842i	$-0.345734\pi$
$977$	29.1247	0.931782	0.465891	+	0.884842i	$0.345734\pi$
$978$	0	0
$979$	−35.0661	−1.12072
$980$	0.668550	0.0213560
$981$	0	0
$982$	38.5718	1.23088
$983$	47.7824	1.52402	0.762010	−	0.647565i	$-0.224212\pi$
$983$	47.7824	1.52402	0.762010	+	0.647565i	$0.224212\pi$
$984$	0	0
$985$	29.1182	0.927783
$986$	113.697	3.62086
$987$	0	0
$988$	1.75535	0.0558452
$989$	−0.102872	−0.00327115
$990$	0	0
$991$	42.7324	1.35744	0.678720	−	0.734397i	$-0.262535\pi$
$991$	42.7324	1.35744	0.678720	+	0.734397i	$0.262535\pi$
$992$	2.42556	0.0770118
$993$	0	0
$994$	−12.7828	−0.405445
$995$	21.2674	0.674223
$996$	0	0
$997$	−20.8160	−0.659250	−0.329625	−	0.944112i	$-0.606922\pi$
$997$	−20.8160	−0.659250	−0.329625	+	0.944112i	$0.606922\pi$
$998$	−39.4659	−1.24927
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2259.2.a.k.1.6		17
3.2	odd	2		251.2.a.b.1.12	✓	17
12.11	even	2		4016.2.a.k.1.3		17
15.14	odd	2		6275.2.a.e.1.6		17

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
251.2.a.b.1.12	✓	17	3.2	odd	2
2259.2.a.k.1.6		17	1.1	even	1	trivial
4016.2.a.k.1.3		17	12.11	even	2
6275.2.a.e.1.6		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)

\(f(q)\)	\(=\)	\(q-1.37907 q^{2} -0.0981670 q^{4} -1.31548 q^{5} -3.48956 q^{7} +2.89352 q^{8} +O(q^{10})\)	\(q-1.37907 q^{2} -0.0981670 q^{4} -1.31548 q^{5} -3.48956 q^{7} +2.89352 q^{8} +1.81414 q^{10} -3.66893 q^{11} +4.41851 q^{13} +4.81235 q^{14} -3.79403 q^{16} +7.90905 q^{17} -4.04690 q^{19} +0.129137 q^{20} +5.05972 q^{22} -0.625539 q^{23} -3.26950 q^{25} -6.09344 q^{26} +0.342560 q^{28} -10.4241 q^{29} -4.37191 q^{31} -0.554806 q^{32} -10.9071 q^{34} +4.59046 q^{35} -3.01090 q^{37} +5.58096 q^{38} -3.80637 q^{40} -12.0639 q^{41} +0.164454 q^{43} +0.360168 q^{44} +0.862661 q^{46} +0.235969 q^{47} +5.17706 q^{49} +4.50887 q^{50} -0.433752 q^{52} +11.4830 q^{53} +4.82642 q^{55} -10.0971 q^{56} +14.3756 q^{58} -7.18778 q^{59} +7.92880 q^{61} +6.02917 q^{62} +8.35318 q^{64} -5.81248 q^{65} +0.478919 q^{67} -0.776408 q^{68} -6.33057 q^{70} -2.65624 q^{71} +9.29721 q^{73} +4.15224 q^{74} +0.397272 q^{76} +12.8030 q^{77} +6.33824 q^{79} +4.99098 q^{80} +16.6370 q^{82} +14.2551 q^{83} -10.4042 q^{85} -0.226794 q^{86} -10.6161 q^{88} +9.55758 q^{89} -15.4187 q^{91} +0.0614073 q^{92} -0.325418 q^{94} +5.32363 q^{95} +3.65366 q^{97} -7.13952 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - 2 q^{2} + 26 q^{4} - 3 q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) \) 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

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