LMFDB - Embedded newform 1666.2.a.w.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1666,2,Mod(1,1666)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1666, 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("1666.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1666 = 2 \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1666.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:	$13.3030769767$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4352.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} - 4x + 2 $ x^4 - 6x^2 - 4x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.68554$ of defining polynomial
Character	$\chi$	$=$	1666.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -1.27133 q^{3} +1.00000 q^{4} -0.526602 q^{5} +1.27133 q^{6} -1.00000 q^{8} -1.38372 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -1.27133 q^{3} +1.00000 q^{4} -0.526602 q^{5} +1.27133 q^{6} -1.00000 q^{8} -1.38372 q^{9} +0.526602 q^{10} -5.51397 q^{11} -1.27133 q^{12} +1.63899 q^{13} +0.669485 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.38372 q^{18} -7.01008 q^{19} -0.526602 q^{20} +5.51397 q^{22} +8.62636 q^{23} +1.27133 q^{24} -4.72269 q^{25} -1.63899 q^{26} +5.57316 q^{27} -5.45298 q^{29} -0.669485 q^{30} +2.74473 q^{31} -1.00000 q^{32} +7.01008 q^{33} -1.00000 q^{34} -1.38372 q^{36} -9.02614 q^{37} +7.01008 q^{38} -2.08370 q^{39} +0.526602 q^{40} +10.8511 q^{41} +4.60365 q^{43} -5.51397 q^{44} +0.728670 q^{45} -8.62636 q^{46} +8.78530 q^{47} -1.27133 q^{48} +4.72269 q^{50} -1.27133 q^{51} +1.63899 q^{52} -2.09311 q^{53} -5.57316 q^{54} +2.90367 q^{55} +8.91213 q^{57} +5.45298 q^{58} +6.58579 q^{59} +0.669485 q^{60} -13.0440 q^{61} -2.74473 q^{62} +1.00000 q^{64} -0.863096 q^{65} -7.01008 q^{66} -10.5174 q^{67} +1.00000 q^{68} -10.9670 q^{69} +7.79538 q^{71} +1.38372 q^{72} +2.03212 q^{73} +9.02614 q^{74} +6.00410 q^{75} -7.01008 q^{76} +2.08370 q^{78} -1.33897 q^{79} -0.526602 q^{80} -2.93416 q^{81} -10.8511 q^{82} +3.21470 q^{83} -0.526602 q^{85} -4.60365 q^{86} +6.93254 q^{87} +5.51397 q^{88} +1.81835 q^{89} -0.728670 q^{90} +8.62636 q^{92} -3.48946 q^{93} -8.78530 q^{94} +3.69152 q^{95} +1.27133 q^{96} -3.69152 q^{97} +7.62979 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 4 q^{4} + 4 q^{5} - 4 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 4 * q^4 + 4 * q^5 - 4 * q^8	$ 4 q - 4 q^{2} + 4 q^{4} + 4 q^{5} - 4 q^{8} - 4 q^{10} - 4 q^{15} + 4 q^{16} + 4 q^{17} + 4 q^{20} + 12 q^{23} + 12 q^{27} + 4 q^{30} + 12 q^{31} - 4 q^{32} - 4 q^{34} - 4 q^{37} + 4 q^{39} - 4 q^{40} + 20 q^{41} + 8 q^{43} + 8 q^{45} - 12 q^{46} + 8 q^{47} - 12 q^{54} + 8 q^{55} + 12 q^{57} + 32 q^{59} - 4 q^{60} - 4 q^{61} - 12 q^{62} + 4 q^{64} - 28 q^{65} + 4 q^{68} - 24 q^{71} + 4 q^{74} - 28 q^{75} - 4 q^{78} + 8 q^{79} + 4 q^{80} - 8 q^{81} - 20 q^{82} + 40 q^{83} + 4 q^{85} - 8 q^{86} + 32 q^{87} + 24 q^{89} - 8 q^{90} + 12 q^{92} - 16 q^{93} - 8 q^{94} + 28 q^{95} - 28 q^{97} - 12 q^{99}+O(q^{100}) $ 4 * q - 4 * q^2 + 4 * q^4 + 4 * q^5 - 4 * q^8 - 4 * q^10 - 4 * q^15 + 4 * q^16 + 4 * q^17 + 4 * q^20 + 12 * q^23 + 12 * q^27 + 4 * q^30 + 12 * q^31 - 4 * q^32 - 4 * q^34 - 4 * q^37 + 4 * q^39 - 4 * q^40 + 20 * q^41 + 8 * q^43 + 8 * q^45 - 12 * q^46 + 8 * q^47 - 12 * q^54 + 8 * q^55 + 12 * q^57 + 32 * q^59 - 4 * q^60 - 4 * q^61 - 12 * q^62 + 4 * q^64 - 28 * q^65 + 4 * q^68 - 24 * q^71 + 4 * q^74 - 28 * q^75 - 4 * q^78 + 8 * q^79 + 4 * q^80 - 8 * q^81 - 20 * q^82 + 40 * q^83 + 4 * q^85 - 8 * q^86 + 32 * q^87 + 24 * q^89 - 8 * q^90 + 12 * q^92 - 16 * q^93 - 8 * q^94 + 28 * q^95 - 28 * q^97 - 12 * q^99

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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−1.27133	−0.734003	−0.367001	−	0.930220i	$-0.619616\pi$
$3$	−1.27133	−0.734003	−0.367001	+	0.930220i	$0.619616\pi$
$4$	1.00000	0.500000
$5$	−0.526602	−0.235504	−0.117752	−	0.993043i	$-0.537569\pi$
$5$	−0.526602	−0.235504	−0.117752	+	0.993043i	$0.537569\pi$
$6$	1.27133	0.519018
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	−1.38372	−0.461240
$10$	0.526602	0.166526
$11$	−5.51397	−1.66252	−0.831262	−	0.555880i	$-0.812381\pi$
$11$	−5.51397	−1.66252	−0.831262	+	0.555880i	$0.812381\pi$
$12$	−1.27133	−0.367001
$13$	1.63899	0.454574	0.227287	−	0.973828i	$-0.427014\pi$
$13$	1.63899	0.454574	0.227287	+	0.973828i	$0.427014\pi$
$14$	0	0
$15$	0.669485	0.172860
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	1.38372	0.326146
$19$	−7.01008	−1.60822	−0.804111	−	0.594479i	$-0.797358\pi$
$19$	−7.01008	−1.60822	−0.804111	+	0.594479i	$0.797358\pi$
$20$	−0.526602	−0.117752
$21$	0	0
$22$	5.51397	1.17558
$23$	8.62636	1.79872	0.899360	−	0.437208i	$-0.144033\pi$
$23$	8.62636	1.79872	0.899360	+	0.437208i	$0.144033\pi$
$24$	1.27133	0.259509
$25$	−4.72269	−0.944538
$26$	−1.63899	−0.321433
$27$	5.57316	1.07255
$28$	0	0
$29$	−5.45298	−1.01259	−0.506297	−	0.862359i	$-0.668986\pi$
$29$	−5.45298	−1.01259	−0.506297	+	0.862359i	$0.668986\pi$
$30$	−0.669485	−0.122231
$31$	2.74473	0.492968	0.246484	−	0.969147i	$-0.420725\pi$
$31$	2.74473	0.492968	0.246484	+	0.969147i	$0.420725\pi$
$32$	−1.00000	−0.176777
$33$	7.01008	1.22030
$34$	−1.00000	−0.171499
$35$	0	0
$36$	−1.38372	−0.230620
$37$	−9.02614	−1.48389	−0.741944	−	0.670462i	$-0.766096\pi$
$37$	−9.02614	−1.48389	−0.741944	+	0.670462i	$0.766096\pi$
$38$	7.01008	1.13718
$39$	−2.08370	−0.333659
$40$	0.526602	0.0832631
$41$	10.8511	1.69466	0.847331	−	0.531064i	$-0.178208\pi$
$41$	10.8511	1.69466	0.847331	+	0.531064i	$0.178208\pi$
$42$	0	0
$43$	4.60365	0.702050	0.351025	−	0.936366i	$-0.385833\pi$
$43$	4.60365	0.702050	0.351025	+	0.936366i	$0.385833\pi$
$44$	−5.51397	−0.831262
$45$	0.728670	0.108624
$46$	−8.62636	−1.27189
$47$	8.78530	1.28147	0.640734	−	0.767763i	$-0.278630\pi$
$47$	8.78530	1.28147	0.640734	+	0.767763i	$0.278630\pi$
$48$	−1.27133	−0.183501
$49$	0	0
$50$	4.72269	0.667889
$51$	−1.27133	−0.178022
$52$	1.63899	0.227287
$53$	−2.09311	−0.287510	−0.143755	−	0.989613i	$-0.545918\pi$
$53$	−2.09311	−0.287510	−0.143755	+	0.989613i	$0.545918\pi$
$54$	−5.57316	−0.758410
$55$	2.90367	0.391531
$56$	0	0
$57$	8.91213	1.18044
$58$	5.45298	0.716012
$59$	6.58579	0.857396	0.428698	−	0.903448i	$-0.358972\pi$
$59$	6.58579	0.857396	0.428698	+	0.903448i	$0.358972\pi$
$60$	0.669485	0.0864302
$61$	−13.0440	−1.67011	−0.835057	−	0.550164i	$-0.814565\pi$
$61$	−13.0440	−1.67011	−0.835057	+	0.550164i	$0.814565\pi$
$62$	−2.74473	−0.348581
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.863096	−0.107054
$66$	−7.01008	−0.862881
$67$	−10.5174	−1.28490	−0.642452	−	0.766326i	$-0.722083\pi$
$67$	−10.5174	−1.28490	−0.642452	+	0.766326i	$0.722083\pi$
$68$	1.00000	0.121268
$69$	−10.9670	−1.32027
$70$	0	0
$71$	7.79538	0.925141	0.462571	−	0.886582i	$-0.346927\pi$
$71$	7.79538	0.925141	0.462571	+	0.886582i	$0.346927\pi$
$72$	1.38372	0.163073
$73$	2.03212	0.237841	0.118921	−	0.992904i	$-0.462057\pi$
$73$	2.03212	0.237841	0.118921	+	0.992904i	$0.462057\pi$
$74$	9.02614	1.04927
$75$	6.00410	0.693294
$76$	−7.01008	−0.804111
$77$	0	0
$78$	2.08370	0.235932
$79$	−1.33897	−0.150646	−0.0753230	−	0.997159i	$-0.523999\pi$
$79$	−1.33897	−0.150646	−0.0753230	+	0.997159i	$0.523999\pi$
$80$	−0.526602	−0.0588759
$81$	−2.93416	−0.326018
$82$	−10.8511	−1.19831
$83$	3.21470	0.352859	0.176430	−	0.984313i	$-0.443545\pi$
$83$	3.21470	0.352859	0.176430	+	0.984313i	$0.443545\pi$
$84$	0	0
$85$	−0.526602	−0.0571180
$86$	−4.60365	−0.496424
$87$	6.93254	0.743246
$88$	5.51397	0.587791
$89$	1.81835	0.192745	0.0963723	−	0.995345i	$-0.469276\pi$
$89$	1.81835	0.192745	0.0963723	+	0.995345i	$0.469276\pi$
$90$	−0.728670	−0.0768085
$91$	0	0
$92$	8.62636	0.899360
$93$	−3.48946	−0.361840
$94$	−8.78530	−0.906135
$95$	3.69152	0.378742
$96$	1.27133	0.129755
$97$	−3.69152	−0.374817	−0.187409	−	0.982282i	$-0.560009\pi$
$97$	−3.69152	−0.374817	−0.187409	+	0.982282i	$0.560009\pi$
$98$	0	0
$99$	7.62979	0.766822
$100$	−4.72269	−0.472269
$101$	8.84268	0.879880	0.439940	−	0.898027i	$-0.355000\pi$
$101$	8.84268	0.879880	0.439940	+	0.898027i	$0.355000\pi$
$102$	1.27133	0.125880
$103$	14.1706	1.39627	0.698137	−	0.715964i	$-0.254012\pi$
$103$	14.1706	1.39627	0.698137	+	0.715964i	$0.254012\pi$
$104$	−1.63899	−0.160716
$105$	0	0
$106$	2.09311	0.203300
$107$	−5.19609	−0.502325	−0.251162	−	0.967945i	$-0.580813\pi$
$107$	−5.19609	−0.502325	−0.251162	+	0.967945i	$0.580813\pi$
$108$	5.57316	0.536277
$109$	8.86235	0.848859	0.424430	−	0.905461i	$-0.360475\pi$
$109$	8.86235	0.848859	0.424430	+	0.905461i	$0.360475\pi$
$110$	−2.90367	−0.276854
$111$	11.4752	1.08918
$112$	0	0
$113$	2.79015	0.262475	0.131238	−	0.991351i	$-0.458105\pi$
$113$	2.79015	0.262475	0.131238	+	0.991351i	$0.458105\pi$
$114$	−8.91213	−0.834697
$115$	−4.54266	−0.423605
$116$	−5.45298	−0.506297
$117$	−2.26790	−0.209668
$118$	−6.58579	−0.606271
$119$	0	0
$120$	−0.669485	−0.0611154
$121$	19.4039	1.76399
$122$	13.0440	1.18095
$123$	−13.7954	−1.24389
$124$	2.74473	0.246484
$125$	5.11999	0.457946
$126$	0	0
$127$	0.608497	0.0539953	0.0269977	−	0.999635i	$-0.491405\pi$
$127$	0.608497	0.0539953	0.0269977	+	0.999635i	$0.491405\pi$
$128$	−1.00000	−0.0883883
$129$	−5.85276	−0.515307
$130$	0.863096	0.0756986
$131$	16.9188	1.47820	0.739100	−	0.673595i	$-0.235251\pi$
$131$	16.9188	1.47820	0.739100	+	0.673595i	$0.235251\pi$
$132$	7.01008	0.610149
$133$	0	0
$134$	10.5174	0.908565
$135$	−2.93484	−0.252590
$136$	−1.00000	−0.0857493
$137$	−0.530959	−0.0453629	−0.0226815	−	0.999743i	$-0.507220\pi$
$137$	−0.530959	−0.0453629	−0.0226815	+	0.999743i	$0.507220\pi$
$138$	10.9670	0.933569
$139$	12.2289	1.03724	0.518620	−	0.855005i	$-0.326446\pi$
$139$	12.2289	1.03724	0.518620	+	0.855005i	$0.326446\pi$
$140$	0	0
$141$	−11.1690	−0.940601
$142$	−7.79538	−0.654174
$143$	−9.03735	−0.755741
$144$	−1.38372	−0.115310
$145$	2.87155	0.238469
$146$	−2.03212	−0.168179
$147$	0	0
$148$	−9.02614	−0.741944
$149$	20.2316	1.65744	0.828720	−	0.559664i	$-0.189070\pi$
$149$	20.2316	1.65744	0.828720	+	0.559664i	$0.189070\pi$
$150$	−6.00410	−0.490233
$151$	6.17910	0.502848	0.251424	−	0.967877i	$-0.419101\pi$
$151$	6.17910	0.502848	0.251424	+	0.967877i	$0.419101\pi$
$152$	7.01008	0.568592
$153$	−1.38372	−0.111867
$154$	0	0
$155$	−1.44538	−0.116096
$156$	−2.08370	−0.166829
$157$	3.55274	0.283539	0.141770	−	0.989900i	$-0.454721\pi$
$157$	3.55274	0.283539	0.141770	+	0.989900i	$0.454721\pi$
$158$	1.33897	0.106523
$159$	2.66103	0.211033
$160$	0.526602	0.0416316
$161$	0	0
$162$	2.93416	0.230530
$163$	19.6846	1.54182	0.770909	−	0.636945i	$-0.219802\pi$
$163$	19.6846	1.54182	0.770909	+	0.636945i	$0.219802\pi$
$164$	10.8511	0.847331
$165$	−3.69152	−0.287385
$166$	−3.21470	−0.249509
$167$	10.2952	0.796664	0.398332	−	0.917241i	$-0.369589\pi$
$167$	10.2952	0.796664	0.398332	+	0.917241i	$0.369589\pi$
$168$	0	0
$169$	−10.3137	−0.793362
$170$	0.526602	0.0403886
$171$	9.69998	0.741776
$172$	4.60365	0.351025
$173$	−13.4456	−1.02225	−0.511124	−	0.859507i	$-0.670771\pi$
$173$	−13.4456	−1.02225	−0.511124	+	0.859507i	$0.670771\pi$
$174$	−6.93254	−0.525555
$175$	0	0
$176$	−5.51397	−0.415631
$177$	−8.37271	−0.629331
$178$	−1.81835	−0.136291
$179$	23.4234	1.75074	0.875372	−	0.483450i	$-0.160616\pi$
$179$	23.4234	1.75074	0.875372	+	0.483450i	$0.160616\pi$
$180$	0.728670	0.0543118
$181$	−10.2289	−0.760306	−0.380153	−	0.924924i	$-0.624129\pi$
$181$	−10.2289	−0.760306	−0.380153	+	0.924924i	$0.624129\pi$
$182$	0	0
$183$	16.5832	1.22587
$184$	−8.62636	−0.635944
$185$	4.75318	0.349461
$186$	3.48946	0.255859
$187$	−5.51397	−0.403221
$188$	8.78530	0.640734
$189$	0	0
$190$	−3.69152	−0.267811
$191$	−13.4486	−0.973108	−0.486554	−	0.873650i	$-0.661746\pi$
$191$	−13.4486	−0.973108	−0.486554	+	0.873650i	$0.661746\pi$
$192$	−1.27133	−0.0917504
$193$	11.3484	0.816874	0.408437	−	0.912786i	$-0.366074\pi$
$193$	11.3484	0.816874	0.408437	+	0.912786i	$0.366074\pi$
$194$	3.69152	0.265036
$195$	1.09728	0.0785779
$196$	0	0
$197$	−7.56880	−0.539255	−0.269627	−	0.962965i	$-0.586900\pi$
$197$	−7.56880	−0.539255	−0.269627	+	0.962965i	$0.586900\pi$
$198$	−7.62979	−0.542225
$199$	−6.29262	−0.446072	−0.223036	−	0.974810i	$-0.571597\pi$
$199$	−6.29262	−0.446072	−0.223036	+	0.974810i	$0.571597\pi$
$200$	4.72269	0.333945
$201$	13.3711	0.943124
$202$	−8.84268	−0.622169
$203$	0	0
$204$	−1.27133	−0.0890109
$205$	−5.71423	−0.399099
$206$	−14.1706	−0.987315
$207$	−11.9365	−0.829641
$208$	1.63899	0.113644
$209$	38.6534	2.67371
$210$	0	0
$211$	12.5062	0.860961	0.430481	−	0.902600i	$-0.358344\pi$
$211$	12.5062	0.860961	0.430481	+	0.902600i	$0.358344\pi$
$212$	−2.09311	−0.143755
$213$	−9.91050	−0.679057
$214$	5.19609	0.355197
$215$	−2.42429	−0.165335
$216$	−5.57316	−0.379205
$217$	0	0
$218$	−8.86235	−0.600234
$219$	−2.58349	−0.174576
$220$	2.90367	0.195765
$221$	1.63899	0.110250
$222$	−11.4752	−0.770165
$223$	−27.7591	−1.85889	−0.929444	−	0.368964i	$-0.879713\pi$
$223$	−27.7591	−1.85889	−0.929444	+	0.368964i	$0.879713\pi$
$224$	0	0
$225$	6.53488	0.435658
$226$	−2.79015	−0.185598
$227$	17.2208	1.14299	0.571493	−	0.820607i	$-0.306365\pi$
$227$	17.2208	1.14299	0.571493	+	0.820607i	$0.306365\pi$
$228$	8.91213	0.590220
$229$	24.7146	1.63319	0.816594	−	0.577212i	$-0.195860\pi$
$229$	24.7146	1.63319	0.816594	+	0.577212i	$0.195860\pi$
$230$	4.54266	0.299534
$231$	0	0
$232$	5.45298	0.358006
$233$	9.66371	0.633091	0.316545	−	0.948577i	$-0.397477\pi$
$233$	9.66371	0.633091	0.316545	+	0.948577i	$0.397477\pi$
$234$	2.26790	0.148257
$235$	−4.62636	−0.301790
$236$	6.58579	0.428698
$237$	1.70227	0.110575
$238$	0	0
$239$	−29.6770	−1.91965	−0.959823	−	0.280606i	$-0.909465\pi$
$239$	−29.6770	−1.91965	−0.959823	+	0.280606i	$0.909465\pi$
$240$	0.669485	0.0432151
$241$	25.1866	1.62241	0.811207	−	0.584760i	$-0.198811\pi$
$241$	25.1866	1.62241	0.811207	+	0.584760i	$0.198811\pi$
$242$	−19.4039	−1.24733
$243$	−12.9892	−0.833256
$244$	−13.0440	−0.835057
$245$	0	0
$246$	13.7954	0.879561
$247$	−11.4895	−0.731057
$248$	−2.74473	−0.174290
$249$	−4.08694	−0.259000
$250$	−5.11999	−0.323817
$251$	−29.3229	−1.85084	−0.925421	−	0.378940i	$-0.876289\pi$
$251$	−29.3229	−1.85084	−0.925421	+	0.378940i	$0.876289\pi$
$252$	0	0
$253$	−47.5655	−2.99042
$254$	−0.608497	−0.0381805
$255$	0.669485	0.0419248
$256$	1.00000	0.0625000
$257$	−14.0523	−0.876557	−0.438278	−	0.898839i	$-0.644412\pi$
$257$	−14.0523	−0.876557	−0.438278	+	0.898839i	$0.644412\pi$
$258$	5.85276	0.364377
$259$	0	0
$260$	−0.863096	−0.0535270
$261$	7.54540	0.467048
$262$	−16.9188	−1.04525
$263$	−11.8612	−0.731394	−0.365697	−	0.930734i	$-0.619169\pi$
$263$	−11.8612	−0.731394	−0.365697	+	0.930734i	$0.619169\pi$
$264$	−7.01008	−0.431441
$265$	1.10223	0.0677097
$266$	0	0
$267$	−2.31172	−0.141475
$268$	−10.5174	−0.642452
$269$	−23.3071	−1.42106	−0.710528	−	0.703669i	$-0.751544\pi$
$269$	−23.3071	−1.42106	−0.710528	+	0.703669i	$0.751544\pi$
$270$	2.93484	0.178608
$271$	−0.599090	−0.0363921	−0.0181961	−	0.999834i	$-0.505792\pi$
$271$	−0.599090	−0.0363921	−0.0181961	+	0.999834i	$0.505792\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	0.530959	0.0320764
$275$	26.0408	1.57032
$276$	−10.9670	−0.660133
$277$	8.19447	0.492358	0.246179	−	0.969224i	$-0.420825\pi$
$277$	8.19447	0.492358	0.246179	+	0.969224i	$0.420825\pi$
$278$	−12.2289	−0.733439
$279$	−3.79793	−0.227376
$280$	0	0
$281$	−16.4396	−0.980705	−0.490352	−	0.871524i	$-0.663132\pi$
$281$	−16.4396	−0.980705	−0.490352	+	0.871524i	$0.663132\pi$
$282$	11.1690	0.665105
$283$	8.85134	0.526158	0.263079	−	0.964774i	$-0.415262\pi$
$283$	8.85134	0.526158	0.263079	+	0.964774i	$0.415262\pi$
$284$	7.79538	0.462571
$285$	−4.69315	−0.277998
$286$	9.03735	0.534390
$287$	0	0
$288$	1.38372	0.0815364
$289$	1.00000	0.0588235
$290$	−2.87155	−0.168623
$291$	4.69315	0.275117
$292$	2.03212	0.118921
$293$	−7.32796	−0.428104	−0.214052	−	0.976822i	$-0.568666\pi$
$293$	−7.32796	−0.428104	−0.214052	+	0.976822i	$0.568666\pi$
$294$	0	0
$295$	−3.46809	−0.201920
$296$	9.02614	0.524634
$297$	−30.7302	−1.78315
$298$	−20.2316	−1.17199
$299$	14.1385	0.817652
$300$	6.00410	0.346647
$301$	0	0
$302$	−6.17910	−0.355567
$303$	−11.2420	−0.645834
$304$	−7.01008	−0.402056
$305$	6.86900	0.393318
$306$	1.38372	0.0791020
$307$	12.0380	0.687046	0.343523	−	0.939144i	$-0.388380\pi$
$307$	12.0380	0.687046	0.343523	+	0.939144i	$0.388380\pi$
$308$	0	0
$309$	−18.0156	−1.02487
$310$	1.44538	0.0820921
$311$	13.4486	0.762602	0.381301	−	0.924451i	$-0.375476\pi$
$311$	13.4486	0.762602	0.381301	+	0.924451i	$0.375476\pi$
$312$	2.08370	0.117966
$313$	−10.8475	−0.613139	−0.306569	−	0.951848i	$-0.599181\pi$
$313$	−10.8475	−0.613139	−0.306569	+	0.951848i	$0.599181\pi$
$314$	−3.55274	−0.200493
$315$	0	0
$316$	−1.33897	−0.0753230
$317$	31.1910	1.75186	0.875930	−	0.482438i	$-0.160249\pi$
$317$	31.1910	1.75186	0.875930	+	0.482438i	$0.160249\pi$
$318$	−2.66103	−0.149223
$319$	30.0676	1.68346
$320$	−0.526602	−0.0294380
$321$	6.60594	0.368708
$322$	0	0
$323$	−7.01008	−0.390051
$324$	−2.93416	−0.163009
$325$	−7.74045	−0.429363
$326$	−19.6846	−1.09023
$327$	−11.2670	−0.623065
$328$	−10.8511	−0.599154
$329$	0	0
$330$	3.69152	0.203212
$331$	−3.92571	−0.215776	−0.107888	−	0.994163i	$-0.534409\pi$
$331$	−3.92571	−0.215776	−0.107888	+	0.994163i	$0.534409\pi$
$332$	3.21470	0.176430
$333$	12.4896	0.684428
$334$	−10.2952	−0.563327
$335$	5.53849	0.302600
$336$	0	0
$337$	3.42591	0.186621	0.0933107	−	0.995637i	$-0.470255\pi$
$337$	3.42591	0.186621	0.0933107	+	0.995637i	$0.470255\pi$
$338$	10.3137	0.560992
$339$	−3.54720	−0.192657
$340$	−0.526602	−0.0285590
$341$	−15.1344	−0.819571
$342$	−9.69998	−0.524515
$343$	0	0
$344$	−4.60365	−0.248212
$345$	5.77522	0.310928
$346$	13.4456	0.722839
$347$	−22.9435	−1.23167	−0.615836	−	0.787875i	$-0.711181\pi$
$347$	−22.9435	−1.23167	−0.615836	+	0.787875i	$0.711181\pi$
$348$	6.93254	0.371623
$349$	−7.27208	−0.389265	−0.194633	−	0.980876i	$-0.562351\pi$
$349$	−7.27208	−0.389265	−0.194633	+	0.980876i	$0.562351\pi$
$350$	0	0
$351$	9.13435	0.487556
$352$	5.51397	0.293896
$353$	9.45223	0.503092	0.251546	−	0.967845i	$-0.419061\pi$
$353$	9.45223	0.503092	0.251546	+	0.967845i	$0.419061\pi$
$354$	8.37271	0.445005
$355$	−4.10507	−0.217874
$356$	1.81835	0.0963723
$357$	0	0
$358$	−23.4234	−1.23796
$359$	1.70738	0.0901120	0.0450560	−	0.998984i	$-0.485653\pi$
$359$	1.70738	0.0901120	0.0450560	+	0.998984i	$0.485653\pi$
$360$	−0.728670	−0.0384043
$361$	30.1412	1.58638
$362$	10.2289	0.537618
$363$	−24.6687	−1.29477
$364$	0	0
$365$	−1.07012	−0.0560125
$366$	−16.5832	−0.866820
$367$	−1.00162	−0.0522843	−0.0261421	−	0.999658i	$-0.508322\pi$
$367$	−1.00162	−0.0522843	−0.0261421	+	0.999658i	$0.508322\pi$
$368$	8.62636	0.449680
$369$	−15.0149	−0.781646
$370$	−4.75318	−0.247106
$371$	0	0
$372$	−3.48946	−0.180920
$373$	−8.39218	−0.434530	−0.217265	−	0.976113i	$-0.569714\pi$
$373$	−8.39218	−0.434530	−0.217265	+	0.976113i	$0.569714\pi$
$374$	5.51397	0.285121
$375$	−6.50920	−0.336134
$376$	−8.78530	−0.453067
$377$	−8.93739	−0.460299
$378$	0	0
$379$	14.4679	0.743167	0.371583	−	0.928400i	$-0.378815\pi$
$379$	14.4679	0.743167	0.371583	+	0.928400i	$0.378815\pi$
$380$	3.69152	0.189371
$381$	−0.773600	−0.0396327
$382$	13.4486	0.688092
$383$	4.71006	0.240673	0.120336	−	0.992733i	$-0.461603\pi$
$383$	4.71006	0.240673	0.120336	+	0.992733i	$0.461603\pi$
$384$	1.27133	0.0648773
$385$	0	0
$386$	−11.3484	−0.577617
$387$	−6.37016	−0.323813
$388$	−3.69152	−0.187409
$389$	−27.7633	−1.40765	−0.703827	−	0.710372i	$-0.748527\pi$
$389$	−27.7633	−1.40765	−0.703827	+	0.710372i	$0.748527\pi$
$390$	−1.09728	−0.0555630
$391$	8.62636	0.436254
$392$	0	0
$393$	−21.5094	−1.08500
$394$	7.56880	0.381311
$395$	0.705105	0.0354777
$396$	7.62979	0.383411
$397$	9.68946	0.486300	0.243150	−	0.969989i	$-0.421819\pi$
$397$	9.68946	0.486300	0.243150	+	0.969989i	$0.421819\pi$
$398$	6.29262	0.315421
$399$	0	0
$400$	−4.72269	−0.236135
$401$	4.53686	0.226560	0.113280	−	0.993563i	$-0.463864\pi$
$401$	4.53686	0.226560	0.113280	+	0.993563i	$0.463864\pi$
$402$	−13.3711	−0.666889
$403$	4.49858	0.224090
$404$	8.84268	0.439940
$405$	1.54514	0.0767785
$406$	0	0
$407$	49.7699	2.46700
$408$	1.27133	0.0629402
$409$	−26.1839	−1.29471	−0.647356	−	0.762187i	$-0.724125\pi$
$409$	−26.1839	−1.29471	−0.647356	+	0.762187i	$0.724125\pi$
$410$	5.71423	0.282206
$411$	0.675025	0.0332965
$412$	14.1706	0.698137
$413$	0	0
$414$	11.9365	0.586645
$415$	−1.69287	−0.0830996
$416$	−1.63899	−0.0803581
$417$	−15.5469	−0.761337
$418$	−38.6534	−1.89060
$419$	20.2227	0.987944	0.493972	−	0.869478i	$-0.335544\pi$
$419$	20.2227	0.987944	0.493972	+	0.869478i	$0.335544\pi$
$420$	0	0
$421$	−11.9206	−0.580975	−0.290487	−	0.956879i	$-0.593817\pi$
$421$	−11.9206	−0.580975	−0.290487	+	0.956879i	$0.593817\pi$
$422$	−12.5062	−0.608792
$423$	−12.1564	−0.591064
$424$	2.09311	0.101650
$425$	−4.72269	−0.229084
$426$	9.91050	0.480165
$427$	0	0
$428$	−5.19609	−0.251162
$429$	11.4895	0.554716
$430$	2.42429	0.116910
$431$	6.60259	0.318036	0.159018	−	0.987276i	$-0.449167\pi$
$431$	6.60259	0.318036	0.159018	+	0.987276i	$0.449167\pi$
$432$	5.57316	0.268139
$433$	−36.1733	−1.73838	−0.869189	−	0.494480i	$-0.835359\pi$
$433$	−36.1733	−1.73838	−0.869189	+	0.494480i	$0.835359\pi$
$434$	0	0
$435$	−3.65069	−0.175037
$436$	8.86235	0.424430
$437$	−60.4715	−2.89274
$438$	2.58349	0.123444
$439$	−5.97219	−0.285037	−0.142518	−	0.989792i	$-0.545520\pi$
$439$	−5.97219	−0.285037	−0.142518	+	0.989792i	$0.545520\pi$
$440$	−2.90367	−0.138427
$441$	0	0
$442$	−1.63899	−0.0779589
$443$	−4.76058	−0.226182	−0.113091	−	0.993585i	$-0.536075\pi$
$443$	−4.76058	−0.226182	−0.113091	+	0.993585i	$0.536075\pi$
$444$	11.4752	0.544589
$445$	−0.957546	−0.0453920
$446$	27.7591	1.31443
$447$	−25.7211	−1.21657
$448$	0	0
$449$	5.79177	0.273331	0.136665	−	0.990617i	$-0.456362\pi$
$449$	5.79177	0.273331	0.136665	+	0.990617i	$0.456362\pi$
$450$	−6.53488	−0.308057
$451$	−59.8329	−2.81742
$452$	2.79015	0.131238
$453$	−7.85568	−0.369092
$454$	−17.2208	−0.808212
$455$	0	0
$456$	−8.91213	−0.417349
$457$	−28.6779	−1.34150	−0.670749	−	0.741685i	$-0.734027\pi$
$457$	−28.6779	−1.34150	−0.670749	+	0.741685i	$0.734027\pi$
$458$	−24.7146	−1.15484
$459$	5.57316	0.260133
$460$	−4.54266	−0.211803
$461$	−6.40132	−0.298139	−0.149070	−	0.988827i	$-0.547628\pi$
$461$	−6.40132	−0.298139	−0.149070	+	0.988827i	$0.547628\pi$
$462$	0	0
$463$	18.9667	0.881457	0.440729	−	0.897640i	$-0.354720\pi$
$463$	18.9667	0.881457	0.440729	+	0.897640i	$0.354720\pi$
$464$	−5.45298	−0.253148
$465$	1.83756	0.0852146
$466$	−9.66371	−0.447663
$467$	−31.6390	−1.46408	−0.732039	−	0.681263i	$-0.761431\pi$
$467$	−31.6390	−1.46408	−0.732039	+	0.681263i	$0.761431\pi$
$468$	−2.26790	−0.104834
$469$	0	0
$470$	4.62636	0.213398
$471$	−4.51671	−0.208119
$472$	−6.58579	−0.303135
$473$	−25.3844	−1.16718
$474$	−1.70227	−0.0781881
$475$	33.1064	1.51903
$476$	0	0
$477$	2.89627	0.132611
$478$	29.6770	1.35739
$479$	30.6141	1.39879	0.699397	−	0.714733i	$-0.253452\pi$
$479$	30.6141	1.39879	0.699397	+	0.714733i	$0.253452\pi$
$480$	−0.669485	−0.0305577
$481$	−14.7938	−0.674537
$482$	−25.1866	−1.14722
$483$	0	0
$484$	19.4039	0.881994
$485$	1.94396	0.0882709
$486$	12.9892	0.589201
$487$	4.41715	0.200160	0.100080	−	0.994979i	$-0.468090\pi$
$487$	4.41715	0.200160	0.100080	+	0.994979i	$0.468090\pi$
$488$	13.0440	0.590474
$489$	−25.0256	−1.13170
$490$	0	0
$491$	−27.4940	−1.24079	−0.620394	−	0.784290i	$-0.713027\pi$
$491$	−27.4940	−1.24079	−0.620394	+	0.784290i	$0.713027\pi$
$492$	−13.7954	−0.621944
$493$	−5.45298	−0.245590
$494$	11.4895	0.516935
$495$	−4.01786	−0.180590
$496$	2.74473	0.123242
$497$	0	0
$498$	4.08694	0.183140
$499$	33.2963	1.49055	0.745274	−	0.666758i	$-0.232319\pi$
$499$	33.2963	1.49055	0.745274	+	0.666758i	$0.232319\pi$
$500$	5.11999	0.228973
$501$	−13.0886	−0.584754
$502$	29.3229	1.30874
$503$	6.43045	0.286720	0.143360	−	0.989671i	$-0.454209\pi$
$503$	6.43045	0.286720	0.143360	+	0.989671i	$0.454209\pi$
$504$	0	0
$505$	−4.65658	−0.207215
$506$	47.5655	2.11454
$507$	13.1121	0.582330
$508$	0.608497	0.0269977
$509$	24.2054	1.07289	0.536443	−	0.843936i	$-0.319768\pi$
$509$	24.2054	1.07289	0.536443	+	0.843936i	$0.319768\pi$
$510$	−0.669485	−0.0296453
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	−39.0683	−1.72491
$514$	14.0523	0.619819
$515$	−7.46229	−0.328828
$516$	−5.85276	−0.257653
$517$	−48.4419	−2.13047
$518$	0	0
$519$	17.0938	0.750334
$520$	0.863096	0.0378493
$521$	20.9132	0.916223	0.458112	−	0.888895i	$-0.348526\pi$
$521$	20.9132	0.916223	0.458112	+	0.888895i	$0.348526\pi$
$522$	−7.54540	−0.330253
$523$	25.6628	1.12215	0.561077	−	0.827764i	$-0.310387\pi$
$523$	25.6628	1.12215	0.561077	+	0.827764i	$0.310387\pi$
$524$	16.9188	0.739100
$525$	0	0
$526$	11.8612	0.517174
$527$	2.74473	0.119562
$528$	7.01008	0.305075
$529$	51.4141	2.23539
$530$	−1.10223	−0.0478780
$531$	−9.11288	−0.395465
$532$	0	0
$533$	17.7849	0.770350
$534$	2.31172	0.100038
$535$	2.73627	0.118299
$536$	10.5174	0.454282
$537$	−29.7788	−1.28505
$538$	23.3071	1.00484
$539$	0	0
$540$	−2.93484	−0.126295
$541$	2.69240	0.115755	0.0578776	−	0.998324i	$-0.481567\pi$
$541$	2.69240	0.115755	0.0578776	+	0.998324i	$0.481567\pi$
$542$	0.599090	0.0257331
$543$	13.0043	0.558067
$544$	−1.00000	−0.0428746
$545$	−4.66693	−0.199909
$546$	0	0
$547$	7.84261	0.335326	0.167663	−	0.985844i	$-0.446378\pi$
$547$	7.84261	0.335326	0.167663	+	0.985844i	$0.446378\pi$
$548$	−0.530959	−0.0226815
$549$	18.0492	0.770322
$550$	−26.0408	−1.11038
$551$	38.2258	1.62848
$552$	10.9670	0.466785
$553$	0	0
$554$	−8.19447	−0.348149
$555$	−6.04287	−0.256505
$556$	12.2289	0.518620
$557$	−18.0880	−0.766413	−0.383207	−	0.923663i	$-0.625180\pi$
$557$	−18.0880	−0.766413	−0.383207	+	0.923663i	$0.625180\pi$
$558$	3.79793	0.160779
$559$	7.54534	0.319134
$560$	0	0
$561$	7.01008	0.295966
$562$	16.4396	0.693463
$563$	13.9284	0.587011	0.293505	−	0.955957i	$-0.405178\pi$
$563$	13.9284	0.587011	0.293505	+	0.955957i	$0.405178\pi$
$564$	−11.1690	−0.470301
$565$	−1.46930	−0.0618139
$566$	−8.85134	−0.372050
$567$	0	0
$568$	−7.79538	−0.327087
$569$	10.6968	0.448431	0.224216	−	0.974540i	$-0.428018\pi$
$569$	10.6968	0.448431	0.224216	+	0.974540i	$0.428018\pi$
$570$	4.69315	0.196574
$571$	−16.1018	−0.673838	−0.336919	−	0.941534i	$-0.609385\pi$
$571$	−16.1018	−0.673838	−0.336919	+	0.941534i	$0.609385\pi$
$572$	−9.03735	−0.377871
$573$	17.0976	0.714264
$574$	0	0
$575$	−40.7396	−1.69896
$576$	−1.38372	−0.0576550
$577$	−45.5334	−1.89558	−0.947790	−	0.318896i	$-0.896688\pi$
$577$	−45.5334	−1.89558	−0.947790	+	0.318896i	$0.896688\pi$
$578$	−1.00000	−0.0415945
$579$	−14.4275	−0.599588
$580$	2.87155	0.119235
$581$	0	0
$582$	−4.69315	−0.194537
$583$	11.5413	0.477993
$584$	−2.03212	−0.0840896
$585$	1.19428	0.0493775
$586$	7.32796	0.302715
$587$	24.5641	1.01387	0.506935	−	0.861984i	$-0.330778\pi$
$587$	24.5641	1.01387	0.506935	+	0.861984i	$0.330778\pi$
$588$	0	0
$589$	−19.2408	−0.792802
$590$	3.46809	0.142779
$591$	9.62244	0.395814
$592$	−9.02614	−0.370972
$593$	17.7775	0.730035	0.365018	−	0.931001i	$-0.381063\pi$
$593$	17.7775	0.730035	0.365018	+	0.931001i	$0.381063\pi$
$594$	30.7302	1.26088
$595$	0	0
$596$	20.2316	0.828720
$597$	8.00000	0.327418
$598$	−14.1385	−0.578167
$599$	−8.28416	−0.338482	−0.169241	−	0.985575i	$-0.554132\pi$
$599$	−8.28416	−0.338482	−0.169241	+	0.985575i	$0.554132\pi$
$600$	−6.00410	−0.245116
$601$	−14.6231	−0.596489	−0.298245	−	0.954489i	$-0.596401\pi$
$601$	−14.6231	−0.596489	−0.298245	+	0.954489i	$0.596401\pi$
$602$	0	0
$603$	14.5531	0.592649
$604$	6.17910	0.251424
$605$	−10.2181	−0.415426
$606$	11.2420	0.456674
$607$	25.5655	1.03767	0.518836	−	0.854874i	$-0.326366\pi$
$607$	25.5655	1.03767	0.518836	+	0.854874i	$0.326366\pi$
$608$	7.01008	0.284296
$609$	0	0
$610$	−6.86900	−0.278118
$611$	14.3990	0.582522
$612$	−1.38372	−0.0559335
$613$	25.3696	1.02467	0.512334	−	0.858786i	$-0.328781\pi$
$613$	25.3696	1.02467	0.512334	+	0.858786i	$0.328781\pi$
$614$	−12.0380	−0.485815
$615$	7.26468	0.292940
$616$	0	0
$617$	12.6465	0.509130	0.254565	−	0.967056i	$-0.418068\pi$
$617$	12.6465	0.509130	0.254565	+	0.967056i	$0.418068\pi$
$618$	18.0156	0.724692
$619$	6.68325	0.268622	0.134311	−	0.990939i	$-0.457118\pi$
$619$	6.68325	0.268622	0.134311	+	0.990939i	$0.457118\pi$
$620$	−1.44538	−0.0580479
$621$	48.0760	1.92923
$622$	−13.4486	−0.539241
$623$	0	0
$624$	−2.08370	−0.0834147
$625$	20.9173	0.836690
$626$	10.8475	0.433554
$627$	−49.1412	−1.96251
$628$	3.55274	0.141770
$629$	−9.02614	−0.359896
$630$	0	0
$631$	−8.72526	−0.347347	−0.173674	−	0.984803i	$-0.555564\pi$
$631$	−8.72526	−0.347347	−0.173674	+	0.984803i	$0.555564\pi$
$632$	1.33897	0.0532614
$633$	−15.8995	−0.631948
$634$	−31.1910	−1.23875
$635$	−0.320436	−0.0127161
$636$	2.66103	0.105517
$637$	0	0
$638$	−30.0676	−1.19039
$639$	−10.7866	−0.426712
$640$	0.526602	0.0208158
$641$	−6.59424	−0.260457	−0.130228	−	0.991484i	$-0.541571\pi$
$641$	−6.59424	−0.260457	−0.130228	+	0.991484i	$0.541571\pi$
$642$	−6.60594	−0.260716
$643$	−21.8500	−0.861680	−0.430840	−	0.902428i	$-0.641783\pi$
$643$	−21.8500	−0.861680	−0.430840	+	0.902428i	$0.641783\pi$
$644$	0	0
$645$	3.08208	0.121357
$646$	7.01008	0.275808
$647$	−4.38117	−0.172241	−0.0861207	−	0.996285i	$-0.527447\pi$
$647$	−4.38117	−0.172241	−0.0861207	+	0.996285i	$0.527447\pi$
$648$	2.93416	0.115265
$649$	−36.3138	−1.42544
$650$	7.74045	0.303605
$651$	0	0
$652$	19.6846	0.770909
$653$	−32.2633	−1.26256	−0.631280	−	0.775555i	$-0.717470\pi$
$653$	−32.2633	−1.26256	−0.631280	+	0.775555i	$0.717470\pi$
$654$	11.2670	0.440574
$655$	−8.90947	−0.348122
$656$	10.8511	0.423666
$657$	−2.81188	−0.109702
$658$	0	0
$659$	31.1820	1.21468	0.607340	−	0.794442i	$-0.292237\pi$
$659$	31.1820	1.21468	0.607340	+	0.794442i	$0.292237\pi$
$660$	−3.69152	−0.143692
$661$	−6.62151	−0.257547	−0.128774	−	0.991674i	$-0.541104\pi$
$661$	−6.62151	−0.257547	−0.128774	+	0.991674i	$0.541104\pi$
$662$	3.92571	0.152577
$663$	−2.08370	−0.0809242
$664$	−3.21470	−0.124754
$665$	0	0
$666$	−12.4896	−0.483964
$667$	−47.0394	−1.82137
$668$	10.2952	0.398332
$669$	35.2910	1.36443
$670$	−5.53849	−0.213970
$671$	71.9242	2.77660
$672$	0	0
$673$	−12.7101	−0.489937	−0.244968	−	0.969531i	$-0.578778\pi$
$673$	−12.7101	−0.489937	−0.244968	+	0.969531i	$0.578778\pi$
$674$	−3.42591	−0.131961
$675$	−26.3203	−1.01307
$676$	−10.3137	−0.396681
$677$	−17.2129	−0.661545	−0.330773	−	0.943710i	$-0.607309\pi$
$677$	−17.2129	−0.661545	−0.330773	+	0.943710i	$0.607309\pi$
$678$	3.54720	0.136229
$679$	0	0
$680$	0.526602	0.0201943
$681$	−21.8933	−0.838954
$682$	15.1344	0.579524
$683$	23.4757	0.898272	0.449136	−	0.893463i	$-0.351732\pi$
$683$	23.4757	0.898272	0.449136	+	0.893463i	$0.351732\pi$
$684$	9.69998	0.370888
$685$	0.279604	0.0106831
$686$	0	0
$687$	−31.4204	−1.19876
$688$	4.60365	0.175512
$689$	−3.43058	−0.130695
$690$	−5.77522	−0.219859
$691$	−33.4953	−1.27422	−0.637111	−	0.770772i	$-0.719871\pi$
$691$	−33.4953	−1.27422	−0.637111	+	0.770772i	$0.719871\pi$
$692$	−13.4456	−0.511124
$693$	0	0
$694$	22.9435	0.870923
$695$	−6.43975	−0.244274
$696$	−6.93254	−0.262777
$697$	10.8511	0.411016
$698$	7.27208	0.275252
$699$	−12.2858	−0.464690
$700$	0	0
$701$	27.8862	1.05325	0.526624	−	0.850098i	$-0.323458\pi$
$701$	27.8862	1.05325	0.526624	+	0.850098i	$0.323458\pi$
$702$	−9.13435	−0.344754
$703$	63.2739	2.38642
$704$	−5.51397	−0.207816
$705$	5.88163	0.221515
$706$	−9.45223	−0.355740
$707$	0	0
$708$	−8.37271	−0.314666
$709$	−43.4671	−1.63244	−0.816220	−	0.577742i	$-0.803934\pi$
$709$	−43.4671	−1.63244	−0.816220	+	0.577742i	$0.803934\pi$
$710$	4.10507	0.154060
$711$	1.85276	0.0694839
$712$	−1.81835	−0.0681455
$713$	23.6770	0.886711
$714$	0	0
$715$	4.75909	0.177980
$716$	23.4234	0.875372
$717$	37.7293	1.40903
$718$	−1.70738	−0.0637188
$719$	30.9514	1.15429	0.577146	−	0.816641i	$-0.304166\pi$
$719$	30.9514	1.15429	0.577146	+	0.816641i	$0.304166\pi$
$720$	0.728670	0.0271559
$721$	0	0
$722$	−30.1412	−1.12174
$723$	−32.0205	−1.19086
$724$	−10.2289	−0.380153
$725$	25.7527	0.956433
$726$	24.6687	0.915543
$727$	−12.3884	−0.459460	−0.229730	−	0.973254i	$-0.573784\pi$
$727$	−12.3884	−0.459460	−0.229730	+	0.973254i	$0.573784\pi$
$728$	0	0
$729$	25.3160	0.937630
$730$	1.07012	0.0396068
$731$	4.60365	0.170272
$732$	16.5832	0.612934
$733$	15.7908	0.583247	0.291624	−	0.956533i	$-0.405805\pi$
$733$	15.7908	0.583247	0.291624	+	0.956533i	$0.405805\pi$
$734$	1.00162	0.0369706
$735$	0	0
$736$	−8.62636	−0.317972
$737$	57.9926	2.13619
$738$	15.0149	0.552707
$739$	4.56643	0.167979	0.0839894	−	0.996467i	$-0.473234\pi$
$739$	4.56643	0.167979	0.0839894	+	0.996467i	$0.473234\pi$
$740$	4.75318	0.174731
$741$	14.6069	0.536598
$742$	0	0
$743$	38.5281	1.41346	0.706730	−	0.707483i	$-0.250169\pi$
$743$	38.5281	1.41346	0.706730	+	0.707483i	$0.250169\pi$
$744$	3.48946	0.127930
$745$	−10.6540	−0.390333
$746$	8.39218	0.307259
$747$	−4.44824	−0.162753
$748$	−5.51397	−0.201611
$749$	0	0
$750$	6.50920	0.237682
$751$	−43.7001	−1.59464	−0.797320	−	0.603557i	$-0.793750\pi$
$751$	−43.7001	−1.59464	−0.797320	+	0.603557i	$0.793750\pi$
$752$	8.78530	0.320367
$753$	37.2790	1.35852
$754$	8.93739	0.325480
$755$	−3.25393	−0.118423
$756$	0	0
$757$	9.31371	0.338512	0.169256	−	0.985572i	$-0.445863\pi$
$757$	9.31371	0.338512	0.169256	+	0.985572i	$0.445863\pi$
$758$	−14.4679	−0.525498
$759$	60.4715	2.19498
$760$	−3.69152	−0.133906
$761$	19.0853	0.691842	0.345921	−	0.938264i	$-0.387566\pi$
$761$	19.0853	0.691842	0.345921	+	0.938264i	$0.387566\pi$
$762$	0.773600	0.0280246
$763$	0	0
$764$	−13.4486	−0.486554
$765$	0.728670	0.0263451
$766$	−4.71006	−0.170181
$767$	10.7940	0.389750
$768$	−1.27133	−0.0458752
$769$	25.2949	0.912157	0.456079	−	0.889939i	$-0.349254\pi$
$769$	25.2949	0.912157	0.456079	+	0.889939i	$0.349254\pi$
$770$	0	0
$771$	17.8651	0.643395
$772$	11.3484	0.408437
$773$	44.2100	1.59012	0.795062	−	0.606529i	$-0.207438\pi$
$773$	44.2100	1.59012	0.795062	+	0.606529i	$0.207438\pi$
$774$	6.37016	0.228971
$775$	−12.9625	−0.465627
$776$	3.69152	0.132518
$777$	0	0
$778$	27.7633	0.995361
$779$	−76.0673	−2.72539
$780$	1.09728	0.0392890
$781$	−42.9835	−1.53807
$782$	−8.62636	−0.308478
$783$	−30.3903	−1.08606
$784$	0	0
$785$	−1.87088	−0.0667746
$786$	21.5094	0.767213
$787$	25.0158	0.891716	0.445858	−	0.895104i	$-0.352899\pi$
$787$	25.0158	0.891716	0.445858	+	0.895104i	$0.352899\pi$
$788$	−7.56880	−0.269627
$789$	15.0795	0.536845
$790$	−0.705105	−0.0250865
$791$	0	0
$792$	−7.62979	−0.271113
$793$	−21.3790	−0.759191
$794$	−9.68946	−0.343866
$795$	−1.40130	−0.0496991
$796$	−6.29262	−0.223036
$797$	−41.3229	−1.46373	−0.731865	−	0.681449i	$-0.761350\pi$
$797$	−41.3229	−1.46373	−0.731865	+	0.681449i	$0.761350\pi$
$798$	0	0
$799$	8.78530	0.310802
$800$	4.72269	0.166972
$801$	−2.51608	−0.0889014
$802$	−4.53686	−0.160202
$803$	−11.2050	−0.395417
$804$	13.3711	0.471562
$805$	0	0
$806$	−4.49858	−0.158456
$807$	29.6310	1.04306
$808$	−8.84268	−0.311084
$809$	37.5593	1.32052	0.660258	−	0.751039i	$-0.270447\pi$
$809$	37.5593	1.32052	0.660258	+	0.751039i	$0.270447\pi$
$810$	−1.54514	−0.0542906
$811$	33.6490	1.18158	0.590788	−	0.806827i	$-0.298817\pi$
$811$	33.6490	1.18158	0.590788	+	0.806827i	$0.298817\pi$
$812$	0	0
$813$	0.761641	0.0267119
$814$	−49.7699	−1.74443
$815$	−10.3660	−0.363104
$816$	−1.27133	−0.0445055
$817$	−32.2719	−1.12905
$818$	26.1839	0.915500
$819$	0	0
$820$	−5.71423	−0.199550
$821$	10.8371	0.378217	0.189108	−	0.981956i	$-0.439440\pi$
$821$	10.8371	0.378217	0.189108	+	0.981956i	$0.439440\pi$
$822$	−0.675025	−0.0235442
$823$	−16.4307	−0.572740	−0.286370	−	0.958119i	$-0.592449\pi$
$823$	−16.4307	−0.572740	−0.286370	+	0.958119i	$0.592449\pi$
$824$	−14.1706	−0.493658
$825$	−33.1064	−1.15262
$826$	0	0
$827$	23.6056	0.820847	0.410423	−	0.911895i	$-0.365381\pi$
$827$	23.6056	0.820847	0.410423	+	0.911895i	$0.365381\pi$
$828$	−11.9365	−0.414821
$829$	33.2334	1.15424	0.577121	−	0.816659i	$-0.304176\pi$
$829$	33.2334	1.15424	0.577121	+	0.816659i	$0.304176\pi$
$830$	1.69287	0.0587603
$831$	−10.4179	−0.361392
$832$	1.63899	0.0568218
$833$	0	0
$834$	15.5469	0.538347
$835$	−5.42146	−0.187617
$836$	38.6534	1.33685
$837$	15.2968	0.528735
$838$	−20.2227	−0.698582
$839$	21.3053	0.735540	0.367770	−	0.929917i	$-0.380121\pi$
$839$	21.3053	0.735540	0.367770	+	0.929917i	$0.380121\pi$
$840$	0	0
$841$	0.735014	0.0253453
$842$	11.9206	0.410811
$843$	20.9002	0.719840
$844$	12.5062	0.430481
$845$	5.43122	0.186840
$846$	12.1564	0.417945
$847$	0	0
$848$	−2.09311	−0.0718776
$849$	−11.2530	−0.386201
$850$	4.72269	0.161987
$851$	−77.8627	−2.66910
$852$	−9.91050	−0.339528
$853$	2.57790	0.0882655	0.0441328	−	0.999026i	$-0.485948\pi$
$853$	2.57790	0.0882655	0.0441328	+	0.999026i	$0.485948\pi$
$854$	0	0
$855$	−5.10803	−0.174691
$856$	5.19609	0.177599
$857$	−30.1818	−1.03099	−0.515495	−	0.856893i	$-0.672392\pi$
$857$	−30.1818	−1.03099	−0.515495	+	0.856893i	$0.672392\pi$
$858$	−11.4895	−0.392244
$859$	−25.4881	−0.869644	−0.434822	−	0.900516i	$-0.643189\pi$
$859$	−25.4881	−0.869644	−0.434822	+	0.900516i	$0.643189\pi$
$860$	−2.42429	−0.0826677
$861$	0	0
$862$	−6.60259	−0.224885
$863$	9.90850	0.337289	0.168645	−	0.985677i	$-0.446061\pi$
$863$	9.90850	0.337289	0.168645	+	0.985677i	$0.446061\pi$
$864$	−5.57316	−0.189603
$865$	7.08047	0.240743
$866$	36.1733	1.22922
$867$	−1.27133	−0.0431766
$868$	0	0
$869$	7.38305	0.250453
$870$	3.65069	0.123770
$871$	−17.2379	−0.584085
$872$	−8.86235	−0.300117
$873$	5.10803	0.172881
$874$	60.4715	2.04548
$875$	0	0
$876$	−2.58349	−0.0872881
$877$	−13.5699	−0.458221	−0.229111	−	0.973400i	$-0.573582\pi$
$877$	−13.5699	−0.458221	−0.229111	+	0.973400i	$0.573582\pi$
$878$	5.97219	0.201551
$879$	9.31626	0.314230
$880$	2.90367	0.0978827
$881$	47.4951	1.60015	0.800075	−	0.599900i	$-0.204793\pi$
$881$	47.4951	1.60015	0.800075	+	0.599900i	$0.204793\pi$
$882$	0	0
$883$	13.6027	0.457768	0.228884	−	0.973454i	$-0.426492\pi$
$883$	13.6027	0.457768	0.228884	+	0.973454i	$0.426492\pi$
$884$	1.63899	0.0551252
$885$	4.40909	0.148210
$886$	4.76058	0.159935
$887$	−40.5613	−1.36192	−0.680958	−	0.732322i	$-0.738436\pi$
$887$	−40.5613	−1.36192	−0.680958	+	0.732322i	$0.738436\pi$
$888$	−11.4752	−0.385083
$889$	0	0
$890$	0.957546	0.0320970
$891$	16.1789	0.542013
$892$	−27.7591	−0.929444
$893$	−61.5857	−2.06089
$894$	25.7211	0.860242
$895$	−12.3348	−0.412307
$896$	0	0
$897$	−17.9747	−0.600159
$898$	−5.79177	−0.193274
$899$	−14.9670	−0.499176
$900$	6.53488	0.217829
$901$	−2.09311	−0.0697315
$902$	59.8329	1.99222
$903$	0	0
$904$	−2.79015	−0.0927990
$905$	5.38655	0.179055
$906$	7.85568	0.260987
$907$	16.0617	0.533321	0.266661	−	0.963790i	$-0.414080\pi$
$907$	16.0617	0.533321	0.266661	+	0.963790i	$0.414080\pi$
$908$	17.2208	0.571493
$909$	−12.2358	−0.405835
$910$	0	0
$911$	4.48138	0.148475	0.0742375	−	0.997241i	$-0.476348\pi$
$911$	4.48138	0.148475	0.0742375	+	0.997241i	$0.476348\pi$
$912$	8.91213	0.295110
$913$	−17.7258	−0.586637
$914$	28.6779	0.948582
$915$	−8.73277	−0.288696
$916$	24.7146	0.816594
$917$	0	0
$918$	−5.57316	−0.183942
$919$	−48.3453	−1.59476	−0.797381	−	0.603476i	$-0.793782\pi$
$919$	−48.3453	−1.59476	−0.797381	+	0.603476i	$0.793782\pi$
$920$	4.54266	0.149767
$921$	−15.3043	−0.504294
$922$	6.40132	0.210816
$923$	12.7766	0.420546
$924$	0	0
$925$	42.6276	1.40159
$926$	−18.9667	−0.623284
$927$	−19.6082	−0.644017
$928$	5.45298	0.179003
$929$	1.98670	0.0651814	0.0325907	−	0.999469i	$-0.489624\pi$
$929$	1.98670	0.0651814	0.0325907	+	0.999469i	$0.489624\pi$
$930$	−1.83756	−0.0602558
$931$	0	0
$932$	9.66371	0.316545
$933$	−17.0976	−0.559752
$934$	31.6390	1.03526
$935$	2.90367	0.0949602
$936$	2.26790	0.0741287
$937$	−14.7509	−0.481891	−0.240945	−	0.970539i	$-0.577457\pi$
$937$	−14.7509	−0.481891	−0.240945	+	0.970539i	$0.577457\pi$
$938$	0	0
$939$	13.7908	0.450045
$940$	−4.62636	−0.150895
$941$	39.0999	1.27462	0.637310	−	0.770608i	$-0.280047\pi$
$941$	39.0999	1.27462	0.637310	+	0.770608i	$0.280047\pi$
$942$	4.51671	0.147162
$943$	93.6058	3.04822
$944$	6.58579	0.214349
$945$	0	0
$946$	25.3844	0.825318
$947$	35.2783	1.14639	0.573195	−	0.819419i	$-0.305704\pi$
$947$	35.2783	1.14639	0.573195	+	0.819419i	$0.305704\pi$
$948$	1.70227	0.0552873
$949$	3.33062	0.108117
$950$	−33.1064	−1.07411
$951$	−39.6540	−1.28587
$952$	0	0
$953$	−52.8078	−1.71061	−0.855305	−	0.518124i	$-0.826631\pi$
$953$	−52.8078	−1.71061	−0.855305	+	0.518124i	$0.826631\pi$
$954$	−2.89627	−0.0937702
$955$	7.08208	0.229171
$956$	−29.6770	−0.959823
$957$	−38.2258	−1.23567
$958$	−30.6141	−0.989097
$959$	0	0
$960$	0.669485	0.0216076
$961$	−23.4665	−0.756983
$962$	14.7938	0.476970
$963$	7.18993	0.231692
$964$	25.1866	0.811207
$965$	−5.97608	−0.192377
$966$	0	0
$967$	2.51642	0.0809226	0.0404613	−	0.999181i	$-0.487117\pi$
$967$	2.51642	0.0809226	0.0404613	+	0.999181i	$0.487117\pi$
$968$	−19.4039	−0.623664
$969$	8.91213	0.286299
$970$	−1.94396	−0.0624169
$971$	−30.0192	−0.963362	−0.481681	−	0.876347i	$-0.659974\pi$
$971$	−30.0192	−0.963362	−0.481681	+	0.876347i	$0.659974\pi$
$972$	−12.9892	−0.416628
$973$	0	0
$974$	−4.41715	−0.141535
$975$	9.84066	0.315153
$976$	−13.0440	−0.417528
$977$	24.2867	0.777002	0.388501	−	0.921448i	$-0.372993\pi$
$977$	24.2867	0.777002	0.388501	+	0.921448i	$0.372993\pi$
$978$	25.0256	0.800232
$979$	−10.0263	−0.320443
$980$	0	0
$981$	−12.2630	−0.391527
$982$	27.4940	0.877370
$983$	14.0982	0.449663	0.224832	−	0.974398i	$-0.427817\pi$
$983$	14.0982	0.449663	0.224832	+	0.974398i	$0.427817\pi$
$984$	13.7954	0.439781
$985$	3.98575	0.126996
$986$	5.45298	0.173658
$987$	0	0
$988$	−11.4895	−0.365528
$989$	39.7127	1.26279
$990$	4.01786	0.127696
$991$	0.711919	0.0226148	0.0113074	−	0.999936i	$-0.496401\pi$
$991$	0.711919	0.0226148	0.0113074	+	0.999936i	$0.496401\pi$
$992$	−2.74473	−0.0871452
$993$	4.99087	0.158381
$994$	0	0
$995$	3.31371	0.105052
$996$	−4.08694	−0.129500
$997$	5.29051	0.167552	0.0837760	−	0.996485i	$-0.473302\pi$
$997$	5.29051	0.167552	0.0837760	+	0.996485i	$0.473302\pi$
$998$	−33.2963	−1.05398
$999$	−50.3041	−1.59155

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1666.2.a.w.1.2	yes	4
7.6	odd	2		1666.2.a.v.1.3	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1666.2.a.v.1.3	✓	4	7.6	odd	2
1666.2.a.w.1.2	yes	4	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:	\( N \)	\(=\)	\( 1666 = 2 \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1666.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -1.27133 q^{3} +1.00000 q^{4} -0.526602 q^{5} +1.27133 q^{6} -1.00000 q^{8} -1.38372 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -1.27133 q^{3} +1.00000 q^{4} -0.526602 q^{5} +1.27133 q^{6} -1.00000 q^{8} -1.38372 q^{9} +0.526602 q^{10} -5.51397 q^{11} -1.27133 q^{12} +1.63899 q^{13} +0.669485 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.38372 q^{18} -7.01008 q^{19} -0.526602 q^{20} +5.51397 q^{22} +8.62636 q^{23} +1.27133 q^{24} -4.72269 q^{25} -1.63899 q^{26} +5.57316 q^{27} -5.45298 q^{29} -0.669485 q^{30} +2.74473 q^{31} -1.00000 q^{32} +7.01008 q^{33} -1.00000 q^{34} -1.38372 q^{36} -9.02614 q^{37} +7.01008 q^{38} -2.08370 q^{39} +0.526602 q^{40} +10.8511 q^{41} +4.60365 q^{43} -5.51397 q^{44} +0.728670 q^{45} -8.62636 q^{46} +8.78530 q^{47} -1.27133 q^{48} +4.72269 q^{50} -1.27133 q^{51} +1.63899 q^{52} -2.09311 q^{53} -5.57316 q^{54} +2.90367 q^{55} +8.91213 q^{57} +5.45298 q^{58} +6.58579 q^{59} +0.669485 q^{60} -13.0440 q^{61} -2.74473 q^{62} +1.00000 q^{64} -0.863096 q^{65} -7.01008 q^{66} -10.5174 q^{67} +1.00000 q^{68} -10.9670 q^{69} +7.79538 q^{71} +1.38372 q^{72} +2.03212 q^{73} +9.02614 q^{74} +6.00410 q^{75} -7.01008 q^{76} +2.08370 q^{78} -1.33897 q^{79} -0.526602 q^{80} -2.93416 q^{81} -10.8511 q^{82} +3.21470 q^{83} -0.526602 q^{85} -4.60365 q^{86} +6.93254 q^{87} +5.51397 q^{88} +1.81835 q^{89} -0.728670 q^{90} +8.62636 q^{92} -3.48946 q^{93} -8.78530 q^{94} +3.69152 q^{95} +1.27133 q^{96} -3.69152 q^{97} +7.62979 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 4 q^{4} + 4 q^{5} - 4 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 4 * q^4 + 4 * q^5 - 4 * q^8	\( 4 q - 4 q^{2} + 4 q^{4} + 4 q^{5} - 4 q^{8} - 4 q^{10} - 4 q^{15} + 4 q^{16} + 4 q^{17} + 4 q^{20} + 12 q^{23} + 12 q^{27} + 4 q^{30} + 12 q^{31} - 4 q^{32} - 4 q^{34} - 4 q^{37} + 4 q^{39} - 4 q^{40} + 20 q^{41} + 8 q^{43} + 8 q^{45} - 12 q^{46} + 8 q^{47} - 12 q^{54} + 8 q^{55} + 12 q^{57} + 32 q^{59} - 4 q^{60} - 4 q^{61} - 12 q^{62} + 4 q^{64} - 28 q^{65} + 4 q^{68} - 24 q^{71} + 4 q^{74} - 28 q^{75} - 4 q^{78} + 8 q^{79} + 4 q^{80} - 8 q^{81} - 20 q^{82} + 40 q^{83} + 4 q^{85} - 8 q^{86} + 32 q^{87} + 24 q^{89} - 8 q^{90} + 12 q^{92} - 16 q^{93} - 8 q^{94} + 28 q^{95} - 28 q^{97} - 12 q^{99}+O(q^{100}) \) 4 * q - 4 * q^2 + 4 * q^4 + 4 * q^5 - 4 * q^8 - 4 * q^10 - 4 * q^15 + 4 * q^16 + 4 * q^17 + 4 * q^20 + 12 * q^23 + 12 * q^27 + 4 * q^30 + 12 * q^31 - 4 * q^32 - 4 * q^34 - 4 * q^37 + 4 * q^39 - 4 * q^40 + 20 * q^41 + 8 * q^43 + 8 * q^45 - 12 * q^46 + 8 * q^47 - 12 * q^54 + 8 * q^55 + 12 * q^57 + 32 * q^59 - 4 * q^60 - 4 * q^61 - 12 * q^62 + 4 * q^64 - 28 * q^65 + 4 * q^68 - 24 * q^71 + 4 * q^74 - 28 * q^75 - 4 * q^78 + 8 * q^79 + 4 * q^80 - 8 * q^81 - 20 * q^82 + 40 * q^83 + 4 * q^85 - 8 * q^86 + 32 * q^87 + 24 * q^89 - 8 * q^90 + 12 * q^92 - 16 * q^93 - 8 * q^94 + 28 * q^95 - 28 * q^97 - 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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