LMFDB - Embedded newform 6003.2.a.r.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6003,2,Mod(1,6003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, 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("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.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:	$47.9341963334$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{15} - 28 x^{14} + 27 x^{13} + 316 x^{12} - 295 x^{11} - 1835 x^{10} + 1665 x^{9} + \cdots - 192 $ x^16 - x^15 - 28x^14 + 27x^13 + 316x^12 - 295x^11 - 1835x^10 + 1665x^9 + 5778x^8 - 5124x^7 - 9405x^6 + 8288x^5 + 6405x^4 - 6032x^3 - 400x^2 + 1088x - 192
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.510814$ of defining polynomial
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.510814 q^{2} -1.73907 q^{4} -2.52081 q^{5} +1.21289 q^{7} +1.90997 q^{8} +O(q^{10})$	\(q-0.510814 q^{2} -1.73907 q^{4} -2.52081 q^{5} +1.21289 q^{7} +1.90997 q^{8} +1.28766 q^{10} +0.502071 q^{11} +6.57584 q^{13} -0.619559 q^{14} +2.50250 q^{16} +5.32759 q^{17} +2.19688 q^{19} +4.38386 q^{20} -0.256465 q^{22} +1.00000 q^{23} +1.35448 q^{25} -3.35903 q^{26} -2.10929 q^{28} +1.00000 q^{29} +9.31204 q^{31} -5.09825 q^{32} -2.72141 q^{34} -3.05746 q^{35} -1.80268 q^{37} -1.12220 q^{38} -4.81466 q^{40} -0.987595 q^{41} +7.62698 q^{43} -0.873136 q^{44} -0.510814 q^{46} +6.90467 q^{47} -5.52891 q^{49} -0.691885 q^{50} -11.4358 q^{52} -13.2261 q^{53} -1.26562 q^{55} +2.31658 q^{56} -0.510814 q^{58} +11.6425 q^{59} -5.76654 q^{61} -4.75672 q^{62} -2.40075 q^{64} -16.5764 q^{65} +7.64146 q^{67} -9.26505 q^{68} +1.56179 q^{70} -0.291616 q^{71} +7.58236 q^{73} +0.920831 q^{74} -3.82053 q^{76} +0.608955 q^{77} +6.30526 q^{79} -6.30833 q^{80} +0.504477 q^{82} +1.76118 q^{83} -13.4298 q^{85} -3.89597 q^{86} +0.958939 q^{88} -16.6702 q^{89} +7.97576 q^{91} -1.73907 q^{92} -3.52700 q^{94} -5.53792 q^{95} -7.03740 q^{97} +2.82424 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7}+O(q^{10}) $ 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7	$ 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7} + 11 q^{10} - 8 q^{11} + 19 q^{13} - 16 q^{14} + 31 q^{16} + 4 q^{17} + 19 q^{19} - 16 q^{20} + 6 q^{22} + 16 q^{23} + 23 q^{25} + 15 q^{26} + 18 q^{28} + 16 q^{29} + 24 q^{31} + 21 q^{32} - 9 q^{34} + 13 q^{35} + 26 q^{37} - 22 q^{40} + 15 q^{41} + 33 q^{43} - 6 q^{44} + q^{46} - 13 q^{47} + 41 q^{49} - 13 q^{50} - 26 q^{52} - 5 q^{53} + 9 q^{55} - 40 q^{56} + q^{58} - 2 q^{59} + 29 q^{61} + 32 q^{62} + 28 q^{64} - 18 q^{65} + 32 q^{67} + 26 q^{68} + 18 q^{70} - 29 q^{71} + 19 q^{73} + 16 q^{74} + 64 q^{76} + 21 q^{77} + 56 q^{79} + 14 q^{82} - 5 q^{83} + 16 q^{85} + 20 q^{86} + q^{88} - 7 q^{89} - 6 q^{91} + 25 q^{92} - 11 q^{94} - 39 q^{95} + 35 q^{97} + 109 q^{98}+O(q^{100}) $ 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7 + 11 * q^10 - 8 * q^11 + 19 * q^13 - 16 * q^14 + 31 * q^16 + 4 * q^17 + 19 * q^19 - 16 * q^20 + 6 * q^22 + 16 * q^23 + 23 * q^25 + 15 * q^26 + 18 * q^28 + 16 * q^29 + 24 * q^31 + 21 * q^32 - 9 * q^34 + 13 * q^35 + 26 * q^37 - 22 * q^40 + 15 * q^41 + 33 * q^43 - 6 * q^44 + q^46 - 13 * q^47 + 41 * q^49 - 13 * q^50 - 26 * q^52 - 5 * q^53 + 9 * q^55 - 40 * q^56 + q^58 - 2 * q^59 + 29 * q^61 + 32 * q^62 + 28 * q^64 - 18 * q^65 + 32 * q^67 + 26 * q^68 + 18 * q^70 - 29 * q^71 + 19 * q^73 + 16 * q^74 + 64 * q^76 + 21 * q^77 + 56 * q^79 + 14 * q^82 - 5 * q^83 + 16 * q^85 + 20 * q^86 + q^88 - 7 * q^89 - 6 * q^91 + 25 * q^92 - 11 * q^94 - 39 * q^95 + 35 * q^97 + 109 * 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 $

$2$	−0.510814	−0.361200	−0.180600	−	0.983557i	$-0.557804\pi$
$2$	−0.510814	−0.361200	−0.180600	+	0.983557i	$0.557804\pi$
$3$	0	0
$3$	0	0
$4$	−1.73907	−0.869535
$5$	−2.52081	−1.12734	−0.563670	−	0.826000i	$-0.690611\pi$
$5$	−2.52081	−1.12734	−0.563670	+	0.826000i	$0.690611\pi$
$6$	0	0
$7$	1.21289	0.458428	0.229214	−	0.973376i	$-0.426384\pi$
$7$	1.21289	0.458428	0.229214	+	0.973376i	$0.426384\pi$
$8$	1.90997	0.675276
$9$	0	0
$10$	1.28766	0.407195
$11$	0.502071	0.151380	0.0756900	−	0.997131i	$-0.475884\pi$
$11$	0.502071	0.151380	0.0756900	+	0.997131i	$0.475884\pi$
$12$	0	0
$13$	6.57584	1.82381	0.911906	−	0.410400i	$-0.134611\pi$
$13$	6.57584	1.82381	0.911906	+	0.410400i	$0.134611\pi$
$14$	−0.619559	−0.165584
$15$	0	0
$16$	2.50250	0.625625
$17$	5.32759	1.29213	0.646066	−	0.763282i	$-0.276413\pi$
$17$	5.32759	1.29213	0.646066	+	0.763282i	$0.276413\pi$
$18$	0	0
$19$	2.19688	0.504000	0.252000	−	0.967727i	$-0.418912\pi$
$19$	2.19688	0.504000	0.252000	+	0.967727i	$0.418912\pi$
$20$	4.38386	0.980261
$21$	0	0
$22$	−0.256465	−0.0546784
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.35448	0.270895
$26$	−3.35903	−0.658760
$27$	0	0
$28$	−2.10929	−0.398619
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	9.31204	1.67249	0.836246	−	0.548355i	$-0.184746\pi$
$31$	9.31204	1.67249	0.836246	+	0.548355i	$0.184746\pi$
$32$	−5.09825	−0.901251
$33$	0	0
$34$	−2.72141	−0.466717
$35$	−3.05746	−0.516804
$36$	0	0
$37$	−1.80268	−0.296358	−0.148179	−	0.988961i	$-0.547341\pi$
$37$	−1.80268	−0.296358	−0.148179	+	0.988961i	$0.547341\pi$
$38$	−1.12220	−0.182045
$39$	0	0
$40$	−4.81466	−0.761265
$41$	−0.987595	−0.154236	−0.0771182	−	0.997022i	$-0.524572\pi$
$41$	−0.987595	−0.154236	−0.0771182	+	0.997022i	$0.524572\pi$
$42$	0	0
$43$	7.62698	1.16310	0.581552	−	0.813509i	$-0.302446\pi$
$43$	7.62698	1.16310	0.581552	+	0.813509i	$0.302446\pi$
$44$	−0.873136	−0.131630
$45$	0	0
$46$	−0.510814	−0.0753154
$47$	6.90467	1.00715	0.503575	−	0.863952i	$-0.332018\pi$
$47$	6.90467	1.00715	0.503575	+	0.863952i	$0.332018\pi$
$48$	0	0
$49$	−5.52891	−0.789844
$50$	−0.691885	−0.0978473
$51$	0	0
$52$	−11.4358	−1.58587
$53$	−13.2261	−1.81674	−0.908371	−	0.418165i	$-0.862673\pi$
$53$	−13.2261	−1.81674	−0.908371	+	0.418165i	$0.862673\pi$
$54$	0	0
$55$	−1.26562	−0.170657
$56$	2.31658	0.309565
$57$	0	0
$58$	−0.510814	−0.0670731
$59$	11.6425	1.51572	0.757860	−	0.652417i	$-0.226245\pi$
$59$	11.6425	1.51572	0.757860	+	0.652417i	$0.226245\pi$
$60$	0	0
$61$	−5.76654	−0.738330	−0.369165	−	0.929364i	$-0.620356\pi$
$61$	−5.76654	−0.738330	−0.369165	+	0.929364i	$0.620356\pi$
$62$	−4.75672	−0.604104
$63$	0	0
$64$	−2.40075	−0.300094
$65$	−16.5764	−2.05605
$66$	0	0
$67$	7.64146	0.933553	0.466776	−	0.884375i	$-0.345415\pi$
$67$	7.64146	0.933553	0.466776	+	0.884375i	$0.345415\pi$
$68$	−9.26505	−1.12355
$69$	0	0
$70$	1.56179	0.186670
$71$	−0.291616	−0.0346085	−0.0173042	−	0.999850i	$-0.505508\pi$
$71$	−0.291616	−0.0346085	−0.0173042	+	0.999850i	$0.505508\pi$
$72$	0	0
$73$	7.58236	0.887448	0.443724	−	0.896164i	$-0.353657\pi$
$73$	7.58236	0.887448	0.443724	+	0.896164i	$0.353657\pi$
$74$	0.920831	0.107044
$75$	0	0
$76$	−3.82053	−0.438245
$77$	0.608955	0.0693969
$78$	0	0
$79$	6.30526	0.709397	0.354698	−	0.934981i	$-0.384584\pi$
$79$	6.30526	0.709397	0.354698	+	0.934981i	$0.384584\pi$
$80$	−6.30833	−0.705292
$81$	0	0
$82$	0.504477	0.0557101
$83$	1.76118	0.193315	0.0966575	−	0.995318i	$-0.469185\pi$
$83$	1.76118	0.193315	0.0966575	+	0.995318i	$0.469185\pi$
$84$	0	0
$85$	−13.4298	−1.45667
$86$	−3.89597	−0.420113
$87$	0	0
$88$	0.958939	0.102223
$89$	−16.6702	−1.76704	−0.883521	−	0.468392i	$-0.844834\pi$
$89$	−16.6702	−1.76704	−0.883521	+	0.468392i	$0.844834\pi$
$90$	0	0
$91$	7.97576	0.836086
$92$	−1.73907	−0.181311
$93$	0	0
$94$	−3.52700	−0.363782
$95$	−5.53792	−0.568179
$96$	0	0
$97$	−7.03740	−0.714540	−0.357270	−	0.934001i	$-0.616292\pi$
$97$	−7.03740	−0.714540	−0.357270	+	0.934001i	$0.616292\pi$
$98$	2.82424	0.285291
$99$	0	0
$100$	−2.35553	−0.235553
$101$	−12.3857	−1.23242	−0.616209	−	0.787582i	$-0.711332\pi$
$101$	−12.3857	−1.23242	−0.616209	+	0.787582i	$0.711332\pi$
$102$	0	0
$103$	−9.26533	−0.912940	−0.456470	−	0.889739i	$-0.650886\pi$
$103$	−9.26533	−0.912940	−0.456470	+	0.889739i	$0.650886\pi$
$104$	12.5597	1.23158
$105$	0	0
$106$	6.75606	0.656207
$107$	−1.85332	−0.179167	−0.0895837	−	0.995979i	$-0.528554\pi$
$107$	−1.85332	−0.179167	−0.0895837	+	0.995979i	$0.528554\pi$
$108$	0	0
$109$	8.89625	0.852106	0.426053	−	0.904698i	$-0.359904\pi$
$109$	8.89625	0.852106	0.426053	+	0.904698i	$0.359904\pi$
$110$	0.646498	0.0616412
$111$	0	0
$112$	3.03525	0.286804
$113$	4.93101	0.463871	0.231935	−	0.972731i	$-0.425494\pi$
$113$	4.93101	0.463871	0.231935	+	0.972731i	$0.425494\pi$
$114$	0	0
$115$	−2.52081	−0.235067
$116$	−1.73907	−0.161469
$117$	0	0
$118$	−5.94713	−0.547478
$119$	6.46177	0.592349
$120$	0	0
$121$	−10.7479	−0.977084
$122$	2.94563	0.266685
$123$	0	0
$124$	−16.1943	−1.45429
$125$	9.18967	0.821949
$126$	0	0
$127$	0.0259951	0.00230669	0.00115335	−	0.999999i	$-0.499633\pi$
$127$	0.0259951	0.00230669	0.00115335	+	0.999999i	$0.499633\pi$
$128$	11.4228	1.00965
$129$	0	0
$130$	8.46747	0.742647
$131$	−10.3177	−0.901460	−0.450730	−	0.892660i	$-0.648836\pi$
$131$	−10.3177	−0.901460	−0.450730	+	0.892660i	$0.648836\pi$
$132$	0	0
$133$	2.66457	0.231048
$134$	−3.90336	−0.337199
$135$	0	0
$136$	10.1755	0.872544
$137$	1.23202	0.105258	0.0526291	−	0.998614i	$-0.483240\pi$
$137$	1.23202	0.105258	0.0526291	+	0.998614i	$0.483240\pi$
$138$	0	0
$139$	8.44793	0.716544	0.358272	−	0.933617i	$-0.383366\pi$
$139$	8.44793	0.716544	0.358272	+	0.933617i	$0.383366\pi$
$140$	5.31713	0.449379
$141$	0	0
$142$	0.148962	0.0125006
$143$	3.30154	0.276088
$144$	0	0
$145$	−2.52081	−0.209342
$146$	−3.87317	−0.320546
$147$	0	0
$148$	3.13498	0.257694
$149$	9.61316	0.787541	0.393770	−	0.919209i	$-0.371170\pi$
$149$	9.61316	0.787541	0.393770	+	0.919209i	$0.371170\pi$
$150$	0	0
$151$	−2.56226	−0.208514	−0.104257	−	0.994550i	$-0.533246\pi$
$151$	−2.56226	−0.208514	−0.104257	+	0.994550i	$0.533246\pi$
$152$	4.19598	0.340339
$153$	0	0
$154$	−0.311062	−0.0250661
$155$	−23.4739	−1.88547
$156$	0	0
$157$	−9.73470	−0.776914	−0.388457	−	0.921467i	$-0.626992\pi$
$157$	−9.73470	−0.776914	−0.388457	+	0.921467i	$0.626992\pi$
$158$	−3.22081	−0.256234
$159$	0	0
$160$	12.8517	1.01602
$161$	1.21289	0.0955889
$162$	0	0
$163$	3.98394	0.312046	0.156023	−	0.987753i	$-0.450133\pi$
$163$	3.98394	0.312046	0.156023	+	0.987753i	$0.450133\pi$
$164$	1.71750	0.134114
$165$	0	0
$166$	−0.899636	−0.0698253
$167$	11.7573	0.909805	0.454902	−	0.890541i	$-0.349674\pi$
$167$	11.7573	0.909805	0.454902	+	0.890541i	$0.349674\pi$
$168$	0	0
$169$	30.2417	2.32629
$170$	6.86015	0.526149
$171$	0	0
$172$	−13.2639	−1.01136
$173$	21.8207	1.65900	0.829498	−	0.558510i	$-0.188627\pi$
$173$	21.8207	1.65900	0.829498	+	0.558510i	$0.188627\pi$
$174$	0	0
$175$	1.64283	0.124186
$176$	1.25643	0.0947072
$177$	0	0
$178$	8.51538	0.638255
$179$	−6.05354	−0.452463	−0.226231	−	0.974074i	$-0.572641\pi$
$179$	−6.05354	−0.452463	−0.226231	+	0.974074i	$0.572641\pi$
$180$	0	0
$181$	−8.79738	−0.653904	−0.326952	−	0.945041i	$-0.606022\pi$
$181$	−8.79738	−0.653904	−0.326952	+	0.945041i	$0.606022\pi$
$182$	−4.07413	−0.301994
$183$	0	0
$184$	1.90997	0.140805
$185$	4.54420	0.334096
$186$	0	0
$187$	2.67483	0.195603
$188$	−12.0077	−0.875752
$189$	0	0
$190$	2.82885	0.205226
$191$	8.17121	0.591248	0.295624	−	0.955304i	$-0.404472\pi$
$191$	8.17121	0.591248	0.295624	+	0.955304i	$0.404472\pi$
$192$	0	0
$193$	−14.0888	−1.01413	−0.507065	−	0.861908i	$-0.669270\pi$
$193$	−14.0888	−1.01413	−0.507065	+	0.861908i	$0.669270\pi$
$194$	3.59480	0.258092
$195$	0	0
$196$	9.61515	0.686796
$197$	−5.62762	−0.400952	−0.200476	−	0.979699i	$-0.564249\pi$
$197$	−5.62762	−0.400952	−0.200476	+	0.979699i	$0.564249\pi$
$198$	0	0
$199$	−7.30901	−0.518122	−0.259061	−	0.965861i	$-0.583413\pi$
$199$	−7.30901	−0.518122	−0.259061	+	0.965861i	$0.583413\pi$
$200$	2.58701	0.182929
$201$	0	0
$202$	6.32676	0.445149
$203$	1.21289	0.0851280
$204$	0	0
$205$	2.48954	0.173877
$206$	4.73286	0.329754
$207$	0	0
$208$	16.4561	1.14102
$209$	1.10299	0.0762955
$210$	0	0
$211$	20.2921	1.39697	0.698484	−	0.715626i	$-0.253858\pi$
$211$	20.2921	1.39697	0.698484	+	0.715626i	$0.253858\pi$
$212$	23.0011	1.57972
$213$	0	0
$214$	0.946702	0.0647152
$215$	−19.2262	−1.31121
$216$	0	0
$217$	11.2945	0.766717
$218$	−4.54433	−0.307781
$219$	0	0
$220$	2.20101	0.148392
$221$	35.0334	2.35660
$222$	0	0
$223$	10.3132	0.690621	0.345311	−	0.938488i	$-0.387774\pi$
$223$	10.3132	0.690621	0.345311	+	0.938488i	$0.387774\pi$
$224$	−6.18360	−0.413159
$225$	0	0
$226$	−2.51883	−0.167550
$227$	−10.6159	−0.704602	−0.352301	−	0.935887i	$-0.614601\pi$
$227$	−10.6159	−0.704602	−0.352301	+	0.935887i	$0.614601\pi$
$228$	0	0
$229$	12.2748	0.811139	0.405569	−	0.914064i	$-0.367073\pi$
$229$	12.2748	0.811139	0.405569	+	0.914064i	$0.367073\pi$
$230$	1.28766	0.0849060
$231$	0	0
$232$	1.90997	0.125396
$233$	−1.08278	−0.0709356	−0.0354678	−	0.999371i	$-0.511292\pi$
$233$	−1.08278	−0.0709356	−0.0354678	+	0.999371i	$0.511292\pi$
$234$	0	0
$235$	−17.4054	−1.13540
$236$	−20.2471	−1.31797
$237$	0	0
$238$	−3.30076	−0.213956
$239$	22.4858	1.45449	0.727243	−	0.686380i	$-0.240802\pi$
$239$	22.4858	1.45449	0.727243	+	0.686380i	$0.240802\pi$
$240$	0	0
$241$	−14.7491	−0.950073	−0.475036	−	0.879966i	$-0.657565\pi$
$241$	−14.7491	−0.950073	−0.475036	+	0.879966i	$0.657565\pi$
$242$	5.49019	0.352923
$243$	0	0
$244$	10.0284	0.642004
$245$	13.9373	0.890422
$246$	0	0
$247$	14.4464	0.919200
$248$	17.7857	1.12939
$249$	0	0
$250$	−4.69421	−0.296888
$251$	−3.10594	−0.196045	−0.0980226	−	0.995184i	$-0.531252\pi$
$251$	−3.10594	−0.196045	−0.0980226	+	0.995184i	$0.531252\pi$
$252$	0	0
$253$	0.502071	0.0315649
$254$	−0.0132786	−0.000833176 0
$255$	0	0
$256$	−1.03344	−0.0645901
$257$	23.2603	1.45094	0.725470	−	0.688254i	$-0.241623\pi$
$257$	23.2603	1.45094	0.725470	+	0.688254i	$0.241623\pi$
$258$	0	0
$259$	−2.18644	−0.135859
$260$	28.8276	1.78781
$261$	0	0
$262$	5.27041	0.325607
$263$	−14.4964	−0.893884	−0.446942	−	0.894563i	$-0.647487\pi$
$263$	−14.4964	−0.893884	−0.446942	+	0.894563i	$0.647487\pi$
$264$	0	0
$265$	33.3404	2.04809
$266$	−1.36110	−0.0834544
$267$	0	0
$268$	−13.2890	−0.811757
$269$	18.7400	1.14260	0.571299	−	0.820742i	$-0.306440\pi$
$269$	18.7400	1.14260	0.571299	+	0.820742i	$0.306440\pi$
$270$	0	0
$271$	−12.6693	−0.769605	−0.384802	−	0.922999i	$-0.625730\pi$
$271$	−12.6693	−0.769605	−0.384802	+	0.922999i	$0.625730\pi$
$272$	13.3323	0.808390
$273$	0	0
$274$	−0.629330	−0.0380192
$275$	0.680043	0.0410081
$276$	0	0
$277$	−28.8115	−1.73111	−0.865557	−	0.500811i	$-0.833035\pi$
$277$	−28.8115	−1.73111	−0.865557	+	0.500811i	$0.833035\pi$
$278$	−4.31532	−0.258816
$279$	0	0
$280$	−5.83964	−0.348985
$281$	25.8074	1.53954	0.769770	−	0.638321i	$-0.220371\pi$
$281$	25.8074	1.53954	0.769770	+	0.638321i	$0.220371\pi$
$282$	0	0
$283$	−5.95438	−0.353951	−0.176975	−	0.984215i	$-0.556631\pi$
$283$	−5.95438	−0.353951	−0.176975	+	0.984215i	$0.556631\pi$
$284$	0.507141	0.0300933
$285$	0	0
$286$	−1.68647	−0.0997231
$287$	−1.19784	−0.0707063
$288$	0	0
$289$	11.3832	0.669603
$290$	1.28766	0.0756142
$291$	0	0
$292$	−13.1862	−0.771667
$293$	−8.47294	−0.494994	−0.247497	−	0.968889i	$-0.579608\pi$
$293$	−8.47294	−0.494994	−0.247497	+	0.968889i	$0.579608\pi$
$294$	0	0
$295$	−29.3484	−1.70873
$296$	−3.44305	−0.200123
$297$	0	0
$298$	−4.91053	−0.284460
$299$	6.57584	0.380291
$300$	0	0
$301$	9.25067	0.533200
$302$	1.30884	0.0753151
$303$	0	0
$304$	5.49771	0.315315
$305$	14.5363	0.832349
$306$	0	0
$307$	16.9797	0.969084	0.484542	−	0.874768i	$-0.338986\pi$
$307$	16.9797	0.969084	0.484542	+	0.874768i	$0.338986\pi$
$308$	−1.05901	−0.0603430
$309$	0	0
$310$	11.9908	0.681030
$311$	−7.85324	−0.445317	−0.222658	−	0.974897i	$-0.571473\pi$
$311$	−7.85324	−0.445317	−0.222658	+	0.974897i	$0.571473\pi$
$312$	0	0
$313$	−21.4281	−1.21119	−0.605593	−	0.795775i	$-0.707064\pi$
$313$	−21.4281	−1.21119	−0.605593	+	0.795775i	$0.707064\pi$
$314$	4.97262	0.280621
$315$	0	0
$316$	−10.9653	−0.616845
$317$	−0.861567	−0.0483904	−0.0241952	−	0.999707i	$-0.507702\pi$
$317$	−0.861567	−0.0483904	−0.0241952	+	0.999707i	$0.507702\pi$
$318$	0	0
$319$	0.502071	0.0281106
$320$	6.05183	0.338307
$321$	0	0
$322$	−0.619559	−0.0345267
$323$	11.7041	0.651234
$324$	0	0
$325$	8.90683	0.494062
$326$	−2.03505	−0.112711
$327$	0	0
$328$	−1.88627	−0.104152
$329$	8.37458	0.461706
$330$	0	0
$331$	−13.6652	−0.751105	−0.375552	−	0.926801i	$-0.622547\pi$
$331$	−13.6652	−0.751105	−0.375552	+	0.926801i	$0.622547\pi$
$332$	−3.06282	−0.168094
$333$	0	0
$334$	−6.00577	−0.328621
$335$	−19.2627	−1.05243
$336$	0	0
$337$	23.5815	1.28456	0.642282	−	0.766468i	$-0.277988\pi$
$337$	23.5815	1.28456	0.642282	+	0.766468i	$0.277988\pi$
$338$	−15.4479	−0.840254
$339$	0	0
$340$	23.3554	1.26663
$341$	4.67530	0.253182
$342$	0	0
$343$	−15.1961	−0.820515
$344$	14.5673	0.785416
$345$	0	0
$346$	−11.1463	−0.599229
$347$	4.28468	0.230013	0.115007	−	0.993365i	$-0.463311\pi$
$347$	4.28468	0.230013	0.115007	+	0.993365i	$0.463311\pi$
$348$	0	0
$349$	−7.92294	−0.424105	−0.212053	−	0.977258i	$-0.568015\pi$
$349$	−7.92294	−0.424105	−0.212053	+	0.977258i	$0.568015\pi$
$350$	−0.839178	−0.0448560
$351$	0	0
$352$	−2.55968	−0.136431
$353$	12.7525	0.678749	0.339374	−	0.940651i	$-0.389785\pi$
$353$	12.7525	0.678749	0.339374	+	0.940651i	$0.389785\pi$
$354$	0	0
$355$	0.735109	0.0390155
$356$	28.9907	1.53650
$357$	0	0
$358$	3.09223	0.163430
$359$	13.4404	0.709359	0.354680	−	0.934988i	$-0.384590\pi$
$359$	13.4404	0.709359	0.354680	+	0.934988i	$0.384590\pi$
$360$	0	0
$361$	−14.1737	−0.745984
$362$	4.49382	0.236190
$363$	0	0
$364$	−13.8704	−0.727006
$365$	−19.1137	−1.00046
$366$	0	0
$367$	3.47897	0.181601	0.0908003	−	0.995869i	$-0.471058\pi$
$367$	3.47897	0.181601	0.0908003	+	0.995869i	$0.471058\pi$
$368$	2.50250	0.130452
$369$	0	0
$370$	−2.32124	−0.120676
$371$	−16.0417	−0.832846
$372$	0	0
$373$	25.7726	1.33445	0.667227	−	0.744854i	$-0.267481\pi$
$373$	25.7726	1.33445	0.667227	+	0.744854i	$0.267481\pi$
$374$	−1.36634	−0.0706517
$375$	0	0
$376$	13.1877	0.680104
$377$	6.57584	0.338673
$378$	0	0
$379$	−15.3700	−0.789505	−0.394753	−	0.918787i	$-0.629170\pi$
$379$	−15.3700	−0.789505	−0.394753	+	0.918787i	$0.629170\pi$
$380$	9.63084	0.494051
$381$	0	0
$382$	−4.17397	−0.213559
$383$	10.7152	0.547522	0.273761	−	0.961798i	$-0.411732\pi$
$383$	10.7152	0.547522	0.273761	+	0.961798i	$0.411732\pi$
$384$	0	0
$385$	−1.53506	−0.0782338
$386$	7.19673	0.366304
$387$	0	0
$388$	12.2385	0.621317
$389$	9.32168	0.472628	0.236314	−	0.971677i	$-0.424061\pi$
$389$	9.32168	0.472628	0.236314	+	0.971677i	$0.424061\pi$
$390$	0	0
$391$	5.32759	0.269428
$392$	−10.5600	−0.533362
$393$	0	0
$394$	2.87467	0.144824
$395$	−15.8943	−0.799731
$396$	0	0
$397$	5.71651	0.286903	0.143452	−	0.989657i	$-0.454180\pi$
$397$	5.71651	0.286903	0.143452	+	0.989657i	$0.454180\pi$
$398$	3.73354	0.187146
$399$	0	0
$400$	3.38958	0.169479
$401$	16.7193	0.834923	0.417462	−	0.908695i	$-0.362920\pi$
$401$	16.7193	0.834923	0.417462	+	0.908695i	$0.362920\pi$
$402$	0	0
$403$	61.2345	3.05031
$404$	21.5395	1.07163
$405$	0	0
$406$	−0.619559	−0.0307482
$407$	−0.905071	−0.0448627
$408$	0	0
$409$	4.89630	0.242106	0.121053	−	0.992646i	$-0.461373\pi$
$409$	4.89630	0.242106	0.121053	+	0.992646i	$0.461373\pi$
$410$	−1.27169	−0.0628043
$411$	0	0
$412$	16.1131	0.793833
$413$	14.1210	0.694849
$414$	0	0
$415$	−4.43961	−0.217932
$416$	−33.5253	−1.64371
$417$	0	0
$418$	−0.563423	−0.0275579
$419$	9.53235	0.465686	0.232843	−	0.972514i	$-0.425197\pi$
$419$	9.53235	0.465686	0.232843	+	0.972514i	$0.425197\pi$
$420$	0	0
$421$	13.9000	0.677446	0.338723	−	0.940886i	$-0.390005\pi$
$421$	13.9000	0.677446	0.338723	+	0.940886i	$0.390005\pi$
$422$	−10.3655	−0.504584
$423$	0	0
$424$	−25.2614	−1.22680
$425$	7.21610	0.350032
$426$	0	0
$427$	−6.99416	−0.338471
$428$	3.22305	0.155792
$429$	0	0
$430$	9.82099	0.473610
$431$	34.2413	1.64935	0.824673	−	0.565609i	$-0.191359\pi$
$431$	34.2413	1.64935	0.824673	+	0.565609i	$0.191359\pi$
$432$	0	0
$433$	−14.3103	−0.687709	−0.343854	−	0.939023i	$-0.611733\pi$
$433$	−14.3103	−0.687709	−0.343854	+	0.939023i	$0.611733\pi$
$434$	−5.76936	−0.276938
$435$	0	0
$436$	−15.4712	−0.740936
$437$	2.19688	0.105091
$438$	0	0
$439$	−24.6983	−1.17879	−0.589393	−	0.807846i	$-0.700633\pi$
$439$	−24.6983	−1.17879	−0.589393	+	0.807846i	$0.700633\pi$
$440$	−2.41730	−0.115240
$441$	0	0
$442$	−17.8955	−0.851204
$443$	2.83321	0.134610	0.0673048	−	0.997732i	$-0.478560\pi$
$443$	2.83321	0.134610	0.0673048	+	0.997732i	$0.478560\pi$
$444$	0	0
$445$	42.0225	1.99206
$446$	−5.26811	−0.249452
$447$	0	0
$448$	−2.91184	−0.137571
$449$	16.8933	0.797245	0.398623	−	0.917115i	$-0.369488\pi$
$449$	16.8933	0.797245	0.398623	+	0.917115i	$0.369488\pi$
$450$	0	0
$451$	−0.495842	−0.0233483
$452$	−8.57537	−0.403352
$453$	0	0
$454$	5.42275	0.254502
$455$	−20.1054	−0.942554
$456$	0	0
$457$	−26.4008	−1.23498	−0.617490	−	0.786579i	$-0.711850\pi$
$457$	−26.4008	−1.23498	−0.617490	+	0.786579i	$0.711850\pi$
$458$	−6.27011	−0.292983
$459$	0	0
$460$	4.38386	0.204399
$461$	−0.0827947	−0.00385614	−0.00192807	−	0.999998i	$-0.500614\pi$
$461$	−0.0827947	−0.00385614	−0.00192807	+	0.999998i	$0.500614\pi$
$462$	0	0
$463$	6.85070	0.318379	0.159189	−	0.987248i	$-0.449112\pi$
$463$	6.85070	0.318379	0.159189	+	0.987248i	$0.449112\pi$
$464$	2.50250	0.116176
$465$	0	0
$466$	0.553101	0.0256219
$467$	−28.9311	−1.33877	−0.669386	−	0.742914i	$-0.733443\pi$
$467$	−28.9311	−1.33877	−0.669386	+	0.742914i	$0.733443\pi$
$468$	0	0
$469$	9.26823	0.427967
$470$	8.89089	0.410106
$471$	0	0
$472$	22.2367	1.02353
$473$	3.82928	0.176071
$474$	0	0
$475$	2.97563	0.136531
$476$	−11.2375	−0.515068
$477$	0	0
$478$	−11.4861	−0.525360
$479$	10.8301	0.494841	0.247421	−	0.968908i	$-0.420417\pi$
$479$	10.8301	0.494841	0.247421	+	0.968908i	$0.420417\pi$
$480$	0	0
$481$	−11.8541	−0.540501
$482$	7.53404	0.343166
$483$	0	0
$484$	18.6914	0.849609
$485$	17.7399	0.805529
$486$	0	0
$487$	−25.0320	−1.13431	−0.567154	−	0.823612i	$-0.691956\pi$
$487$	−25.0320	−1.13431	−0.567154	+	0.823612i	$0.691956\pi$
$488$	−11.0139	−0.498576
$489$	0	0
$490$	−7.11937	−0.321620
$491$	−20.1985	−0.911543	−0.455772	−	0.890097i	$-0.650637\pi$
$491$	−20.1985	−0.911543	−0.455772	+	0.890097i	$0.650637\pi$
$492$	0	0
$493$	5.32759	0.239943
$494$	−7.37940	−0.332015
$495$	0	0
$496$	23.3034	1.04635
$497$	−0.353698	−0.0158655
$498$	0	0
$499$	38.6070	1.72829	0.864143	−	0.503247i	$-0.167861\pi$
$499$	38.6070	1.72829	0.864143	+	0.503247i	$0.167861\pi$
$500$	−15.9815	−0.714713
$501$	0	0
$502$	1.58656	0.0708115
$503$	−18.7133	−0.834385	−0.417193	−	0.908818i	$-0.636986\pi$
$503$	−18.7133	−0.834385	−0.417193	+	0.908818i	$0.636986\pi$
$504$	0	0
$505$	31.2219	1.38935
$506$	−0.256465	−0.0114012
$507$	0	0
$508$	−0.0452073	−0.00200575
$509$	−1.89925	−0.0841826	−0.0420913	−	0.999114i	$-0.513402\pi$
$509$	−1.89925	−0.0841826	−0.0420913	+	0.999114i	$0.513402\pi$
$510$	0	0
$511$	9.19654	0.406831
$512$	−22.3178	−0.986315
$513$	0	0
$514$	−11.8817	−0.524079
$515$	23.3561	1.02919
$516$	0	0
$517$	3.46663	0.152462
$518$	1.11686	0.0490722
$519$	0	0
$520$	−31.6605	−1.38840
$521$	−6.18015	−0.270757	−0.135379	−	0.990794i	$-0.543225\pi$
$521$	−6.18015	−0.270757	−0.135379	+	0.990794i	$0.543225\pi$
$522$	0	0
$523$	−27.8680	−1.21858	−0.609291	−	0.792947i	$-0.708546\pi$
$523$	−27.8680	−1.21858	−0.609291	+	0.792947i	$0.708546\pi$
$524$	17.9432	0.783851
$525$	0	0
$526$	7.40494	0.322871
$527$	49.6107	2.16108
$528$	0	0
$529$	1.00000	0.0434783
$530$	−17.0307	−0.739768
$531$	0	0
$532$	−4.63388	−0.200904
$533$	−6.49427	−0.281298
$534$	0	0
$535$	4.67187	0.201983
$536$	14.5949	0.630405
$537$	0	0
$538$	−9.57264	−0.412706
$539$	−2.77590	−0.119567
$540$	0	0
$541$	−14.8798	−0.639733	−0.319866	−	0.947463i	$-0.603638\pi$
$541$	−14.8798	−0.639733	−0.319866	+	0.947463i	$0.603638\pi$
$542$	6.47165	0.277981
$543$	0	0
$544$	−27.1614	−1.16453
$545$	−22.4257	−0.960614
$546$	0	0
$547$	−26.2488	−1.12232	−0.561159	−	0.827708i	$-0.689644\pi$
$547$	−26.2488	−1.12232	−0.561159	+	0.827708i	$0.689644\pi$
$548$	−2.14256	−0.0915256
$549$	0	0
$550$	−0.347375	−0.0148121
$551$	2.19688	0.0935904
$552$	0	0
$553$	7.64756	0.325207
$554$	14.7173	0.625278
$555$	0	0
$556$	−14.6915	−0.623060
$557$	−28.2623	−1.19751	−0.598756	−	0.800931i	$-0.704338\pi$
$557$	−28.2623	−1.19751	−0.598756	+	0.800931i	$0.704338\pi$
$558$	0	0
$559$	50.1539	2.12128
$560$	−7.65129	−0.323326
$561$	0	0
$562$	−13.1828	−0.556082
$563$	−10.4549	−0.440623	−0.220311	−	0.975430i	$-0.570707\pi$
$563$	−10.4549	−0.440623	−0.220311	+	0.975430i	$0.570707\pi$
$564$	0	0
$565$	−12.4301	−0.522940
$566$	3.04158	0.127847
$567$	0	0
$568$	−0.556978	−0.0233703
$569$	21.3535	0.895185	0.447593	−	0.894238i	$-0.352281\pi$
$569$	21.3535	0.895185	0.447593	+	0.894238i	$0.352281\pi$
$570$	0	0
$571$	−21.1597	−0.885508	−0.442754	−	0.896643i	$-0.645998\pi$
$571$	−21.1597	−0.885508	−0.442754	+	0.896643i	$0.645998\pi$
$572$	−5.74160	−0.240069
$573$	0	0
$574$	0.611873	0.0255391
$575$	1.35448	0.0564856
$576$	0	0
$577$	14.3012	0.595368	0.297684	−	0.954664i	$-0.403786\pi$
$577$	14.3012	0.595368	0.297684	+	0.954664i	$0.403786\pi$
$578$	−5.81472	−0.241860
$579$	0	0
$580$	4.38386	0.182030
$581$	2.13612	0.0886210
$582$	0	0
$583$	−6.64043	−0.275018
$584$	14.4821	0.599272
$585$	0	0
$586$	4.32809	0.178792
$587$	−25.4131	−1.04891	−0.524455	−	0.851438i	$-0.675731\pi$
$587$	−25.4131	−1.04891	−0.524455	+	0.851438i	$0.675731\pi$
$588$	0	0
$589$	20.4575	0.842935
$590$	14.9916	0.617193
$591$	0	0
$592$	−4.51120	−0.185409
$593$	−11.0315	−0.453008	−0.226504	−	0.974010i	$-0.572730\pi$
$593$	−11.0315	−0.453008	−0.226504	+	0.974010i	$0.572730\pi$
$594$	0	0
$595$	−16.2889	−0.667779
$596$	−16.7180	−0.684794
$597$	0	0
$598$	−3.35903	−0.137361
$599$	48.1605	1.96779	0.983893	−	0.178761i	$-0.0572087\pi$
$599$	48.1605	1.96779	0.983893	+	0.178761i	$0.0572087\pi$
$600$	0	0
$601$	−27.1335	−1.10680	−0.553400	−	0.832915i	$-0.686670\pi$
$601$	−27.1335	−1.10680	−0.553400	+	0.832915i	$0.686670\pi$
$602$	−4.72537	−0.192592
$603$	0	0
$604$	4.45594	0.181310
$605$	27.0935	1.10151
$606$	0	0
$607$	−20.5726	−0.835015	−0.417507	−	0.908674i	$-0.637096\pi$
$607$	−20.5726	−0.835015	−0.417507	+	0.908674i	$0.637096\pi$
$608$	−11.2003	−0.454230
$609$	0	0
$610$	−7.42537	−0.300644
$611$	45.4040	1.83685
$612$	0	0
$613$	19.5402	0.789221	0.394611	−	0.918848i	$-0.370880\pi$
$613$	19.5402	0.789221	0.394611	+	0.918848i	$0.370880\pi$
$614$	−8.67348	−0.350033
$615$	0	0
$616$	1.16308	0.0468620
$617$	−4.95291	−0.199396	−0.0996982	−	0.995018i	$-0.531788\pi$
$617$	−4.95291	−0.199396	−0.0996982	+	0.995018i	$0.531788\pi$
$618$	0	0
$619$	40.8807	1.64313	0.821567	−	0.570111i	$-0.193100\pi$
$619$	40.8807	1.64313	0.821567	+	0.570111i	$0.193100\pi$
$620$	40.8227	1.63948
$621$	0	0
$622$	4.01154	0.160848
$623$	−20.2191	−0.810062
$624$	0	0
$625$	−29.9378	−1.19751
$626$	10.9457	0.437480
$627$	0	0
$628$	16.9293	0.675554
$629$	−9.60392	−0.382933
$630$	0	0
$631$	0.835217	0.0332494	0.0166247	−	0.999862i	$-0.494708\pi$
$631$	0.835217	0.0332494	0.0166247	+	0.999862i	$0.494708\pi$
$632$	12.0428	0.479038
$633$	0	0
$634$	0.440100	0.0174786
$635$	−0.0655286	−0.00260042
$636$	0	0
$637$	−36.3572	−1.44053
$638$	−0.256465	−0.0101535
$639$	0	0
$640$	−28.7948	−1.13821
$641$	−27.4004	−1.08225	−0.541126	−	0.840942i	$-0.682002\pi$
$641$	−27.4004	−1.08225	−0.541126	+	0.840942i	$0.682002\pi$
$642$	0	0
$643$	41.9630	1.65486	0.827430	−	0.561569i	$-0.189802\pi$
$643$	41.9630	1.65486	0.827430	+	0.561569i	$0.189802\pi$
$644$	−2.10929	−0.0831179
$645$	0	0
$646$	−5.97862	−0.235226
$647$	16.1202	0.633750	0.316875	−	0.948467i	$-0.397367\pi$
$647$	16.1202	0.633750	0.316875	+	0.948467i	$0.397367\pi$
$648$	0	0
$649$	5.84534	0.229450
$650$	−4.54973	−0.178455
$651$	0	0
$652$	−6.92834	−0.271335
$653$	8.66870	0.339232	0.169616	−	0.985510i	$-0.445747\pi$
$653$	8.66870	0.339232	0.169616	+	0.985510i	$0.445747\pi$
$654$	0	0
$655$	26.0089	1.01625
$656$	−2.47146	−0.0964942
$657$	0	0
$658$	−4.27785	−0.166768
$659$	−15.8552	−0.617631	−0.308815	−	0.951122i	$-0.599933\pi$
$659$	−15.8552	−0.617631	−0.308815	+	0.951122i	$0.599933\pi$
$660$	0	0
$661$	41.2840	1.60576	0.802881	−	0.596140i	$-0.203300\pi$
$661$	41.2840	1.60576	0.802881	+	0.596140i	$0.203300\pi$
$662$	6.98035	0.271299
$663$	0	0
$664$	3.36380	0.130541
$665$	−6.71688	−0.260469
$666$	0	0
$667$	1.00000	0.0387202
$668$	−20.4467	−0.791107
$669$	0	0
$670$	9.83963	0.380138
$671$	−2.89521	−0.111768
$672$	0	0
$673$	−12.8807	−0.496515	−0.248258	−	0.968694i	$-0.579858\pi$
$673$	−12.8807	−0.496515	−0.248258	+	0.968694i	$0.579858\pi$
$674$	−12.0457	−0.463984
$675$	0	0
$676$	−52.5925	−2.02279
$677$	−4.01480	−0.154301	−0.0771506	−	0.997019i	$-0.524582\pi$
$677$	−4.01480	−0.154301	−0.0771506	+	0.997019i	$0.524582\pi$
$678$	0	0
$679$	−8.53557	−0.327565
$680$	−25.6506	−0.983654
$681$	0	0
$682$	−2.38821	−0.0914492
$683$	−12.8052	−0.489979	−0.244990	−	0.969526i	$-0.578785\pi$
$683$	−12.8052	−0.489979	−0.244990	+	0.969526i	$0.578785\pi$
$684$	0	0
$685$	−3.10567	−0.118662
$686$	7.76240	0.296370
$687$	0	0
$688$	19.0865	0.727667
$689$	−86.9726	−3.31339
$690$	0	0
$691$	−50.1919	−1.90939	−0.954694	−	0.297589i	$-0.903818\pi$
$691$	−50.1919	−1.90939	−0.954694	+	0.297589i	$0.903818\pi$
$692$	−37.9477	−1.44255
$693$	0	0
$694$	−2.18867	−0.0830808
$695$	−21.2956	−0.807789
$696$	0	0
$697$	−5.26150	−0.199294
$698$	4.04715	0.153187
$699$	0	0
$700$	−2.85699	−0.107984
$701$	47.5567	1.79619	0.898095	−	0.439802i	$-0.144951\pi$
$701$	47.5567	1.79619	0.898095	+	0.439802i	$0.144951\pi$
$702$	0	0
$703$	−3.96027	−0.149364
$704$	−1.20535	−0.0454282
$705$	0	0
$706$	−6.51417	−0.245164
$707$	−15.0224	−0.564975
$708$	0	0
$709$	27.2576	1.02368	0.511841	−	0.859080i	$-0.328964\pi$
$709$	27.2576	1.02368	0.511841	+	0.859080i	$0.328964\pi$
$710$	−0.375504	−0.0140924
$711$	0	0
$712$	−31.8396	−1.19324
$713$	9.31204	0.348739
$714$	0	0
$715$	−8.32255	−0.311246
$716$	10.5275	0.393432
$717$	0	0
$718$	−6.86556	−0.256220
$719$	−30.0570	−1.12094	−0.560468	−	0.828176i	$-0.689379\pi$
$719$	−30.0570	−1.12094	−0.560468	+	0.828176i	$0.689379\pi$
$720$	0	0
$721$	−11.2378	−0.418517
$722$	7.24012	0.269449
$723$	0	0
$724$	15.2993	0.568593
$725$	1.35448	0.0503040
$726$	0	0
$727$	−30.8267	−1.14330	−0.571650	−	0.820497i	$-0.693697\pi$
$727$	−30.8267	−1.14330	−0.571650	+	0.820497i	$0.693697\pi$
$728$	15.2334	0.564589
$729$	0	0
$730$	9.76352	0.361364
$731$	40.6335	1.50288
$732$	0	0
$733$	−44.0671	−1.62765	−0.813827	−	0.581107i	$-0.802620\pi$
$733$	−44.0671	−1.62765	−0.813827	+	0.581107i	$0.802620\pi$
$734$	−1.77710	−0.0655941
$735$	0	0
$736$	−5.09825	−0.187924
$737$	3.83655	0.141321
$738$	0	0
$739$	−3.94561	−0.145142	−0.0725708	−	0.997363i	$-0.523120\pi$
$739$	−3.94561	−0.145142	−0.0725708	+	0.997363i	$0.523120\pi$
$740$	−7.90268	−0.290508
$741$	0	0
$742$	8.19434	0.300824
$743$	−41.0117	−1.50457	−0.752287	−	0.658836i	$-0.771049\pi$
$743$	−41.0117	−1.50457	−0.752287	+	0.658836i	$0.771049\pi$
$744$	0	0
$745$	−24.2329	−0.887826
$746$	−13.1650	−0.482005
$747$	0	0
$748$	−4.65171	−0.170083
$749$	−2.24787	−0.0821354
$750$	0	0
$751$	40.2066	1.46716	0.733581	−	0.679602i	$-0.237848\pi$
$751$	40.2066	1.46716	0.733581	+	0.679602i	$0.237848\pi$
$752$	17.2789	0.630098
$753$	0	0
$754$	−3.35903	−0.122329
$755$	6.45896	0.235066
$756$	0	0
$757$	−26.5726	−0.965796	−0.482898	−	0.875677i	$-0.660416\pi$
$757$	−26.5726	−0.965796	−0.482898	+	0.875677i	$0.660416\pi$
$758$	7.85122	0.285169
$759$	0	0
$760$	−10.5773	−0.383677
$761$	32.4080	1.17479	0.587395	−	0.809301i	$-0.300154\pi$
$761$	32.4080	1.17479	0.587395	+	0.809301i	$0.300154\pi$
$762$	0	0
$763$	10.7901	0.390630
$764$	−14.2103	−0.514111
$765$	0	0
$766$	−5.47348	−0.197765
$767$	76.5590	2.76439
$768$	0	0
$769$	−16.4485	−0.593149	−0.296575	−	0.955010i	$-0.595844\pi$
$769$	−16.4485	−0.593149	−0.296575	+	0.955010i	$0.595844\pi$
$770$	0.784129	0.0282580
$771$	0	0
$772$	24.5013	0.881822
$773$	33.8571	1.21775	0.608877	−	0.793265i	$-0.291620\pi$
$773$	33.8571	1.21775	0.608877	+	0.793265i	$0.291620\pi$
$774$	0	0
$775$	12.6129	0.453070
$776$	−13.4412	−0.482511
$777$	0	0
$778$	−4.76164	−0.170713
$779$	−2.16963	−0.0777351
$780$	0	0
$781$	−0.146412	−0.00523903
$782$	−2.72141	−0.0973173
$783$	0	0
$784$	−13.8361	−0.494146
$785$	24.5393	0.875846
$786$	0	0
$787$	36.1190	1.28750	0.643751	−	0.765235i	$-0.277377\pi$
$787$	36.1190	1.28750	0.643751	+	0.765235i	$0.277377\pi$
$788$	9.78683	0.348641
$789$	0	0
$790$	8.11905	0.288863
$791$	5.98076	0.212651
$792$	0	0
$793$	−37.9199	−1.34657
$794$	−2.92007	−0.103629
$795$	0	0
$796$	12.7109	0.450525
$797$	−24.0715	−0.852656	−0.426328	−	0.904569i	$-0.640193\pi$
$797$	−24.0715	−0.852656	−0.426328	+	0.904569i	$0.640193\pi$
$798$	0	0
$799$	36.7853	1.30137
$800$	−6.90546	−0.244145
$801$	0	0
$802$	−8.54046	−0.301574
$803$	3.80688	0.134342
$804$	0	0
$805$	−3.05746	−0.107761
$806$	−31.2794	−1.10177
$807$	0	0
$808$	−23.6562	−0.832222
$809$	−19.3808	−0.681393	−0.340696	−	0.940173i	$-0.610663\pi$
$809$	−19.3808	−0.681393	−0.340696	+	0.940173i	$0.610663\pi$
$810$	0	0
$811$	53.5687	1.88105	0.940525	−	0.339723i	$-0.110333\pi$
$811$	53.5687	1.88105	0.940525	+	0.339723i	$0.110333\pi$
$812$	−2.10929	−0.0740217
$813$	0	0
$814$	0.462322	0.0162044
$815$	−10.0427	−0.351782
$816$	0	0
$817$	16.7556	0.586204
$818$	−2.50109	−0.0874487
$819$	0	0
$820$	−4.32948	−0.151192
$821$	−24.7395	−0.863416	−0.431708	−	0.902013i	$-0.642089\pi$
$821$	−24.7395	−0.863416	−0.431708	+	0.902013i	$0.642089\pi$
$822$	0	0
$823$	−0.0903977	−0.00315106	−0.00157553	−	0.999999i	$-0.500502\pi$
$823$	−0.0903977	−0.00315106	−0.00157553	+	0.999999i	$0.500502\pi$
$824$	−17.6965	−0.616486
$825$	0	0
$826$	−7.21320	−0.250979
$827$	−10.1907	−0.354364	−0.177182	−	0.984178i	$-0.556698\pi$
$827$	−10.1907	−0.354364	−0.177182	+	0.984178i	$0.556698\pi$
$828$	0	0
$829$	−16.6333	−0.577698	−0.288849	−	0.957375i	$-0.593273\pi$
$829$	−16.6333	−0.577698	−0.288849	+	0.957375i	$0.593273\pi$
$830$	2.26781	0.0787169
$831$	0	0
$832$	−15.7869	−0.547314
$833$	−29.4558	−1.02058
$834$	0	0
$835$	−29.6378	−1.02566
$836$	−1.91818	−0.0663416
$837$	0	0
$838$	−4.86925	−0.168206
$839$	−36.8517	−1.27226	−0.636131	−	0.771581i	$-0.719466\pi$
$839$	−36.8517	−1.27226	−0.636131	+	0.771581i	$0.719466\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−7.10033	−0.244693
$843$	0	0
$844$	−35.2894	−1.21471
$845$	−76.2336	−2.62252
$846$	0	0
$847$	−13.0360	−0.447923
$848$	−33.0983	−1.13660
$849$	0	0
$850$	−3.68608	−0.126432
$851$	−1.80268	−0.0617949
$852$	0	0
$853$	30.6934	1.05092	0.525461	−	0.850818i	$-0.323893\pi$
$853$	30.6934	1.05092	0.525461	+	0.850818i	$0.323893\pi$
$854$	3.57271	0.122256
$855$	0	0
$856$	−3.53978	−0.120987
$857$	38.6243	1.31938	0.659690	−	0.751538i	$-0.270688\pi$
$857$	38.6243	1.31938	0.659690	+	0.751538i	$0.270688\pi$
$858$	0	0
$859$	49.7993	1.69913	0.849564	−	0.527485i	$-0.176865\pi$
$859$	49.7993	1.69913	0.849564	+	0.527485i	$0.176865\pi$
$860$	33.4356	1.14015
$861$	0	0
$862$	−17.4909	−0.595744
$863$	−50.3080	−1.71250	−0.856252	−	0.516559i	$-0.827213\pi$
$863$	−50.3080	−1.71250	−0.856252	+	0.516559i	$0.827213\pi$
$864$	0	0
$865$	−55.0058	−1.87025
$866$	7.30989	0.248400
$867$	0	0
$868$	−19.6418	−0.666687
$869$	3.16568	0.107388
$870$	0	0
$871$	50.2490	1.70262
$872$	16.9916	0.575407
$873$	0	0
$874$	−1.12220	−0.0379589
$875$	11.1460	0.376805
$876$	0	0
$877$	−26.6720	−0.900650	−0.450325	−	0.892865i	$-0.648692\pi$
$877$	−26.6720	−0.900650	−0.450325	+	0.892865i	$0.648692\pi$
$878$	12.6162	0.425777
$879$	0	0
$880$	−3.16723	−0.106767
$881$	18.4999	0.623277	0.311638	−	0.950201i	$-0.399122\pi$
$881$	18.4999	0.623277	0.311638	+	0.950201i	$0.399122\pi$
$882$	0	0
$883$	43.6866	1.47017	0.735085	−	0.677975i	$-0.237142\pi$
$883$	43.6866	1.47017	0.735085	+	0.677975i	$0.237142\pi$
$884$	−60.9255	−2.04915
$885$	0	0
$886$	−1.44724	−0.0486210
$887$	−5.09759	−0.171160	−0.0855802	−	0.996331i	$-0.527274\pi$
$887$	−5.09759	−0.171160	−0.0855802	+	0.996331i	$0.527274\pi$
$888$	0	0
$889$	0.0315291	0.00105745
$890$	−21.4657	−0.719530
$891$	0	0
$892$	−17.9353	−0.600519
$893$	15.1688	0.507603
$894$	0	0
$895$	15.2598	0.510080
$896$	13.8546	0.462850
$897$	0	0
$898$	−8.62934	−0.287965
$899$	9.31204	0.310574
$900$	0	0
$901$	−70.4632	−2.34747
$902$	0.253283	0.00843340
$903$	0	0
$904$	9.41808	0.313241
$905$	22.1765	0.737173
$906$	0	0
$907$	1.46127	0.0485207	0.0242604	−	0.999706i	$-0.492277\pi$
$907$	1.46127	0.0485207	0.0242604	+	0.999706i	$0.492277\pi$
$908$	18.4618	0.612676
$909$	0	0
$910$	10.2701	0.340450
$911$	2.61553	0.0866562	0.0433281	−	0.999061i	$-0.486204\pi$
$911$	2.61553	0.0866562	0.0433281	+	0.999061i	$0.486204\pi$
$912$	0	0
$913$	0.884238	0.0292640
$914$	13.4859	0.446074
$915$	0	0
$916$	−21.3467	−0.705313
$917$	−12.5142	−0.413255
$918$	0	0
$919$	−8.94338	−0.295015	−0.147508	−	0.989061i	$-0.547125\pi$
$919$	−8.94338	−0.295015	−0.147508	+	0.989061i	$0.547125\pi$
$920$	−4.81466	−0.158735
$921$	0	0
$922$	0.0422927	0.00139284
$923$	−1.91762	−0.0631194
$924$	0	0
$925$	−2.44168	−0.0802820
$926$	−3.49943	−0.114998
$927$	0	0
$928$	−5.09825	−0.167358
$929$	−2.35870	−0.0773866	−0.0386933	−	0.999251i	$-0.512320\pi$
$929$	−2.35870	−0.0773866	−0.0386933	+	0.999251i	$0.512320\pi$
$930$	0	0
$931$	−12.1464	−0.398081
$932$	1.88304	0.0616809
$933$	0	0
$934$	14.7784	0.483564
$935$	−6.74273	−0.220511
$936$	0	0
$937$	−2.96754	−0.0969453	−0.0484726	−	0.998825i	$-0.515435\pi$
$937$	−2.96754	−0.0969453	−0.0484726	+	0.998825i	$0.515435\pi$
$938$	−4.73434	−0.154582
$939$	0	0
$940$	30.2691	0.987270
$941$	−9.55520	−0.311490	−0.155745	−	0.987797i	$-0.549778\pi$
$941$	−9.55520	−0.311490	−0.155745	+	0.987797i	$0.549778\pi$
$942$	0	0
$943$	−0.987595	−0.0321605
$944$	29.1353	0.948273
$945$	0	0
$946$	−1.95605	−0.0635967
$947$	32.3691	1.05185	0.525927	−	0.850529i	$-0.323718\pi$
$947$	32.3691	1.05185	0.525927	+	0.850529i	$0.323718\pi$
$948$	0	0
$949$	49.8604	1.61854
$950$	−1.51999	−0.0493150
$951$	0	0
$952$	12.3418	0.399999
$953$	3.76001	0.121799	0.0608993	−	0.998144i	$-0.480603\pi$
$953$	3.76001	0.121799	0.0608993	+	0.998144i	$0.480603\pi$
$954$	0	0
$955$	−20.5981	−0.666538
$956$	−39.1044	−1.26473
$957$	0	0
$958$	−5.53218	−0.178737
$959$	1.49430	0.0482533
$960$	0	0
$961$	55.7141	1.79723
$962$	6.05524	0.195229
$963$	0	0
$964$	25.6497	0.826121
$965$	35.5151	1.14327
$966$	0	0
$967$	57.6853	1.85504	0.927518	−	0.373779i	$-0.121938\pi$
$967$	57.6853	1.85504	0.927518	+	0.373779i	$0.121938\pi$
$968$	−20.5282	−0.659801
$969$	0	0
$970$	−9.06180	−0.290957
$971$	−42.5620	−1.36588	−0.682940	−	0.730475i	$-0.739299\pi$
$971$	−42.5620	−1.36588	−0.682940	+	0.730475i	$0.739299\pi$
$972$	0	0
$973$	10.2464	0.328484
$974$	12.7867	0.409712
$975$	0	0
$976$	−14.4308	−0.461918
$977$	25.8972	0.828524	0.414262	−	0.910158i	$-0.364040\pi$
$977$	25.8972	0.828524	0.414262	+	0.910158i	$0.364040\pi$
$978$	0	0
$979$	−8.36964	−0.267495
$980$	−24.2380	−0.774253
$981$	0	0
$982$	10.3176	0.329249
$983$	−36.3230	−1.15852	−0.579262	−	0.815142i	$-0.696659\pi$
$983$	−36.3230	−1.15852	−0.579262	+	0.815142i	$0.696659\pi$
$984$	0	0
$985$	14.1862	0.452009
$986$	−2.72141	−0.0866673
$987$	0	0
$988$	−25.1232	−0.799277
$989$	7.62698	0.242524
$990$	0	0
$991$	−45.8075	−1.45512	−0.727562	−	0.686042i	$-0.759347\pi$
$991$	−45.8075	−1.45512	−0.727562	+	0.686042i	$0.759347\pi$
$992$	−47.4751	−1.50734
$993$	0	0
$994$	0.180674	0.00573062
$995$	18.4246	0.584099
$996$	0	0
$997$	49.2764	1.56060	0.780300	−	0.625405i	$-0.215066\pi$
$997$	49.2764	1.56060	0.780300	+	0.625405i	$0.215066\pi$
$998$	−19.7210	−0.624256
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.r.1.7		16
3.2	odd	2		2001.2.a.n.1.10	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.n.1.10	✓	16	3.2	odd	2
6003.2.a.r.1.7		16	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q-0.510814 q^{2} -1.73907 q^{4} -2.52081 q^{5} +1.21289 q^{7} +1.90997 q^{8} +O(q^{10})\)	\(q-0.510814 q^{2} -1.73907 q^{4} -2.52081 q^{5} +1.21289 q^{7} +1.90997 q^{8} +1.28766 q^{10} +0.502071 q^{11} +6.57584 q^{13} -0.619559 q^{14} +2.50250 q^{16} +5.32759 q^{17} +2.19688 q^{19} +4.38386 q^{20} -0.256465 q^{22} +1.00000 q^{23} +1.35448 q^{25} -3.35903 q^{26} -2.10929 q^{28} +1.00000 q^{29} +9.31204 q^{31} -5.09825 q^{32} -2.72141 q^{34} -3.05746 q^{35} -1.80268 q^{37} -1.12220 q^{38} -4.81466 q^{40} -0.987595 q^{41} +7.62698 q^{43} -0.873136 q^{44} -0.510814 q^{46} +6.90467 q^{47} -5.52891 q^{49} -0.691885 q^{50} -11.4358 q^{52} -13.2261 q^{53} -1.26562 q^{55} +2.31658 q^{56} -0.510814 q^{58} +11.6425 q^{59} -5.76654 q^{61} -4.75672 q^{62} -2.40075 q^{64} -16.5764 q^{65} +7.64146 q^{67} -9.26505 q^{68} +1.56179 q^{70} -0.291616 q^{71} +7.58236 q^{73} +0.920831 q^{74} -3.82053 q^{76} +0.608955 q^{77} +6.30526 q^{79} -6.30833 q^{80} +0.504477 q^{82} +1.76118 q^{83} -13.4298 q^{85} -3.89597 q^{86} +0.958939 q^{88} -16.6702 q^{89} +7.97576 q^{91} -1.73907 q^{92} -3.52700 q^{94} -5.53792 q^{95} -7.03740 q^{97} +2.82424 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7}+O(q^{10}) \) 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7	\( 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7} + 11 q^{10} - 8 q^{11} + 19 q^{13} - 16 q^{14} + 31 q^{16} + 4 q^{17} + 19 q^{19} - 16 q^{20} + 6 q^{22} + 16 q^{23} + 23 q^{25} + 15 q^{26} + 18 q^{28} + 16 q^{29} + 24 q^{31} + 21 q^{32} - 9 q^{34} + 13 q^{35} + 26 q^{37} - 22 q^{40} + 15 q^{41} + 33 q^{43} - 6 q^{44} + q^{46} - 13 q^{47} + 41 q^{49} - 13 q^{50} - 26 q^{52} - 5 q^{53} + 9 q^{55} - 40 q^{56} + q^{58} - 2 q^{59} + 29 q^{61} + 32 q^{62} + 28 q^{64} - 18 q^{65} + 32 q^{67} + 26 q^{68} + 18 q^{70} - 29 q^{71} + 19 q^{73} + 16 q^{74} + 64 q^{76} + 21 q^{77} + 56 q^{79} + 14 q^{82} - 5 q^{83} + 16 q^{85} + 20 q^{86} + q^{88} - 7 q^{89} - 6 q^{91} + 25 q^{92} - 11 q^{94} - 39 q^{95} + 35 q^{97} + 109 q^{98}+O(q^{100}) \) 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7 + 11 * q^10 - 8 * q^11 + 19 * q^13 - 16 * q^14 + 31 * q^16 + 4 * q^17 + 19 * q^19 - 16 * q^20 + 6 * q^22 + 16 * q^23 + 23 * q^25 + 15 * q^26 + 18 * q^28 + 16 * q^29 + 24 * q^31 + 21 * q^32 - 9 * q^34 + 13 * q^35 + 26 * q^37 - 22 * q^40 + 15 * q^41 + 33 * q^43 - 6 * q^44 + q^46 - 13 * q^47 + 41 * q^49 - 13 * q^50 - 26 * q^52 - 5 * q^53 + 9 * q^55 - 40 * q^56 + q^58 - 2 * q^59 + 29 * q^61 + 32 * q^62 + 28 * q^64 - 18 * q^65 + 32 * q^67 + 26 * q^68 + 18 * q^70 - 29 * q^71 + 19 * q^73 + 16 * q^74 + 64 * q^76 + 21 * q^77 + 56 * q^79 + 14 * q^82 - 5 * q^83 + 16 * q^85 + 20 * q^86 + q^88 - 7 * q^89 - 6 * q^91 + 25 * q^92 - 11 * q^94 - 39 * q^95 + 35 * q^97 + 109 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more