LMFDB - Embedded newform 668.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [668,2,Mod(1,668)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(668, 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("668.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 668 = 2^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	668.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:	$5.33400685502$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
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			1.1
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	668.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.30278 q^{3} -3.00000 q^{5} -0.302776 q^{7} -1.30278 q^{9} +O(q^{10})$	$q-1.30278 q^{3} -3.00000 q^{5} -0.302776 q^{7} -1.30278 q^{9} +2.69722 q^{13} +3.90833 q^{15} +2.30278 q^{17} +2.00000 q^{19} +0.394449 q^{21} -2.30278 q^{23} +4.00000 q^{25} +5.60555 q^{27} +7.60555 q^{29} +6.60555 q^{31} +0.908327 q^{35} +0.394449 q^{37} -3.51388 q^{39} -6.21110 q^{41} +9.60555 q^{43} +3.90833 q^{45} +1.60555 q^{47} -6.90833 q^{49} -3.00000 q^{51} -4.60555 q^{53} -2.60555 q^{57} +7.81665 q^{59} +6.81665 q^{61} +0.394449 q^{63} -8.09167 q^{65} +8.21110 q^{67} +3.00000 q^{69} +3.90833 q^{71} -3.09167 q^{73} -5.21110 q^{75} +6.39445 q^{79} -3.39445 q^{81} -1.60555 q^{83} -6.90833 q^{85} -9.90833 q^{87} -13.8167 q^{89} -0.816654 q^{91} -8.60555 q^{93} -6.00000 q^{95} +4.51388 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 6 q^{5} + 3 q^{7} + q^{9}+O(q^{10}) $ 2 * q + q^3 - 6 * q^5 + 3 * q^7 + q^9	$ 2 q + q^{3} - 6 q^{5} + 3 q^{7} + q^{9} + 9 q^{13} - 3 q^{15} + q^{17} + 4 q^{19} + 8 q^{21} - q^{23} + 8 q^{25} + 4 q^{27} + 8 q^{29} + 6 q^{31} - 9 q^{35} + 8 q^{37} + 11 q^{39} + 2 q^{41} + 12 q^{43} - 3 q^{45} - 4 q^{47} - 3 q^{49} - 6 q^{51} - 2 q^{53} + 2 q^{57} - 6 q^{59} - 8 q^{61} + 8 q^{63} - 27 q^{65} + 2 q^{67} + 6 q^{69} - 3 q^{71} - 17 q^{73} + 4 q^{75} + 20 q^{79} - 14 q^{81} + 4 q^{83} - 3 q^{85} - 9 q^{87} - 6 q^{89} + 20 q^{91} - 10 q^{93} - 12 q^{95} - 9 q^{97}+O(q^{100}) $ 2 * q + q^3 - 6 * q^5 + 3 * q^7 + q^9 + 9 * q^13 - 3 * q^15 + q^17 + 4 * q^19 + 8 * q^21 - q^23 + 8 * q^25 + 4 * q^27 + 8 * q^29 + 6 * q^31 - 9 * q^35 + 8 * q^37 + 11 * q^39 + 2 * q^41 + 12 * q^43 - 3 * q^45 - 4 * q^47 - 3 * q^49 - 6 * q^51 - 2 * q^53 + 2 * q^57 - 6 * q^59 - 8 * q^61 + 8 * q^63 - 27 * q^65 + 2 * q^67 + 6 * q^69 - 3 * q^71 - 17 * q^73 + 4 * q^75 + 20 * q^79 - 14 * q^81 + 4 * q^83 - 3 * q^85 - 9 * q^87 - 6 * q^89 + 20 * q^91 - 10 * q^93 - 12 * q^95 - 9 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.30278	−0.752158	−0.376079	−	0.926588i	$-0.622728\pi$
$3$	−1.30278	−0.752158	−0.376079	+	0.926588i	$0.622728\pi$
$4$	0	0
$5$	−3.00000	−1.34164	−0.670820	−	0.741620i	$-0.734058\pi$
$5$	−3.00000	−1.34164	−0.670820	+	0.741620i	$0.734058\pi$
$6$	0	0
$7$	−0.302776	−0.114438	−0.0572192	−	0.998362i	$-0.518223\pi$
$7$	−0.302776	−0.114438	−0.0572192	+	0.998362i	$0.518223\pi$
$8$	0	0
$9$	−1.30278	−0.434259
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	0	0
$13$	2.69722	0.748075	0.374038	−	0.927413i	$-0.377973\pi$
$13$	2.69722	0.748075	0.374038	+	0.927413i	$0.377973\pi$
$14$	0	0
$15$	3.90833	1.00913
$16$	0	0
$17$	2.30278	0.558505	0.279253	−	0.960218i	$-0.409913\pi$
$17$	2.30278	0.558505	0.279253	+	0.960218i	$0.409913\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0.394449	0.0860758
$22$	0	0
$23$	−2.30278	−0.480162	−0.240081	−	0.970753i	$-0.577174\pi$
$23$	−2.30278	−0.480162	−0.240081	+	0.970753i	$0.577174\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	5.60555	1.07879
$28$	0	0
$29$	7.60555	1.41232	0.706158	−	0.708055i	$-0.250427\pi$
$29$	7.60555	1.41232	0.706158	+	0.708055i	$0.250427\pi$
$30$	0	0
$31$	6.60555	1.18639	0.593196	−	0.805058i	$-0.297866\pi$
$31$	6.60555	1.18639	0.593196	+	0.805058i	$0.297866\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.908327	0.153535
$36$	0	0
$37$	0.394449	0.0648470	0.0324235	−	0.999474i	$-0.489677\pi$
$37$	0.394449	0.0648470	0.0324235	+	0.999474i	$0.489677\pi$
$38$	0	0
$39$	−3.51388	−0.562671
$40$	0	0
$41$	−6.21110	−0.970011	−0.485006	−	0.874511i	$-0.661182\pi$
$41$	−6.21110	−0.970011	−0.485006	+	0.874511i	$0.661182\pi$
$42$	0	0
$43$	9.60555	1.46483	0.732416	−	0.680857i	$-0.238392\pi$
$43$	9.60555	1.46483	0.732416	+	0.680857i	$0.238392\pi$
$44$	0	0
$45$	3.90833	0.582619
$46$	0	0
$47$	1.60555	0.234194	0.117097	−	0.993120i	$-0.462641\pi$
$47$	1.60555	0.234194	0.117097	+	0.993120i	$0.462641\pi$
$48$	0	0
$49$	−6.90833	−0.986904
$50$	0	0
$51$	−3.00000	−0.420084
$52$	0	0
$53$	−4.60555	−0.632621	−0.316311	−	0.948656i	$-0.602444\pi$
$53$	−4.60555	−0.632621	−0.316311	+	0.948656i	$0.602444\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−2.60555	−0.345114
$58$	0	0
$59$	7.81665	1.01764	0.508821	−	0.860872i	$-0.330082\pi$
$59$	7.81665	1.01764	0.508821	+	0.860872i	$0.330082\pi$
$60$	0	0
$61$	6.81665	0.872783	0.436392	−	0.899757i	$-0.356256\pi$
$61$	6.81665	0.872783	0.436392	+	0.899757i	$0.356256\pi$
$62$	0	0
$63$	0.394449	0.0496959
$64$	0	0
$65$	−8.09167	−1.00365
$66$	0	0
$67$	8.21110	1.00315	0.501573	−	0.865115i	$-0.332755\pi$
$67$	8.21110	1.00315	0.501573	+	0.865115i	$0.332755\pi$
$68$	0	0
$69$	3.00000	0.361158
$70$	0	0
$71$	3.90833	0.463833	0.231917	−	0.972736i	$-0.425500\pi$
$71$	3.90833	0.463833	0.231917	+	0.972736i	$0.425500\pi$
$72$	0	0
$73$	−3.09167	−0.361853	−0.180926	−	0.983497i	$-0.557910\pi$
$73$	−3.09167	−0.361853	−0.180926	+	0.983497i	$0.557910\pi$
$74$	0	0
$75$	−5.21110	−0.601726
$76$	0	0
$77$	0	0
$78$	0	0
$79$	6.39445	0.719432	0.359716	−	0.933062i	$-0.382874\pi$
$79$	6.39445	0.719432	0.359716	+	0.933062i	$0.382874\pi$
$80$	0	0
$81$	−3.39445	−0.377161
$82$	0	0
$83$	−1.60555	−0.176232	−0.0881161	−	0.996110i	$-0.528085\pi$
$83$	−1.60555	−0.176232	−0.0881161	+	0.996110i	$0.528085\pi$
$84$	0	0
$85$	−6.90833	−0.749313
$86$	0	0
$87$	−9.90833	−1.06228
$88$	0	0
$89$	−13.8167	−1.46456	−0.732281	−	0.681002i	$-0.761544\pi$
$89$	−13.8167	−1.46456	−0.732281	+	0.681002i	$0.761544\pi$
$90$	0	0
$91$	−0.816654	−0.0856086
$92$	0	0
$93$	−8.60555	−0.892354
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	4.51388	0.458315	0.229157	−	0.973389i	$-0.426403\pi$
$97$	4.51388	0.458315	0.229157	+	0.973389i	$0.426403\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.8167	−1.07630	−0.538149	−	0.842850i	$-0.680876\pi$
$101$	−10.8167	−1.07630	−0.538149	+	0.842850i	$0.680876\pi$
$102$	0	0
$103$	−3.30278	−0.325432	−0.162716	−	0.986673i	$-0.552025\pi$
$103$	−3.30278	−0.325432	−0.162716	+	0.986673i	$0.552025\pi$
$104$	0	0
$105$	−1.18335	−0.115483
$106$	0	0
$107$	7.60555	0.735256	0.367628	−	0.929973i	$-0.380170\pi$
$107$	7.60555	0.735256	0.367628	+	0.929973i	$0.380170\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−0.513878	−0.0487752
$112$	0	0
$113$	9.42221	0.886366	0.443183	−	0.896431i	$-0.353849\pi$
$113$	9.42221	0.886366	0.443183	+	0.896431i	$0.353849\pi$
$114$	0	0
$115$	6.90833	0.644205
$116$	0	0
$117$	−3.51388	−0.324858
$118$	0	0
$119$	−0.697224	−0.0639145
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	8.09167	0.729602
$124$	0	0
$125$	3.00000	0.268328
$126$	0	0
$127$	14.2111	1.26103	0.630516	−	0.776176i	$-0.282843\pi$
$127$	14.2111	1.26103	0.630516	+	0.776176i	$0.282843\pi$
$128$	0	0
$129$	−12.5139	−1.10179
$130$	0	0
$131$	−20.0278	−1.74983	−0.874917	−	0.484274i	$-0.839084\pi$
$131$	−20.0278	−1.74983	−0.874917	+	0.484274i	$0.839084\pi$
$132$	0	0
$133$	−0.605551	−0.0525080
$134$	0	0
$135$	−16.8167	−1.44735
$136$	0	0
$137$	−2.78890	−0.238272	−0.119136	−	0.992878i	$-0.538012\pi$
$137$	−2.78890	−0.238272	−0.119136	+	0.992878i	$0.538012\pi$
$138$	0	0
$139$	5.90833	0.501138	0.250569	−	0.968099i	$-0.419382\pi$
$139$	5.90833	0.501138	0.250569	+	0.968099i	$0.419382\pi$
$140$	0	0
$141$	−2.09167	−0.176151
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−22.8167	−1.89482
$146$	0	0
$147$	9.00000	0.742307
$148$	0	0
$149$	17.5139	1.43479	0.717396	−	0.696665i	$-0.245334\pi$
$149$	17.5139	1.43479	0.717396	+	0.696665i	$0.245334\pi$
$150$	0	0
$151$	19.5139	1.58802	0.794008	−	0.607907i	$-0.207991\pi$
$151$	19.5139	1.58802	0.794008	+	0.607907i	$0.207991\pi$
$152$	0	0
$153$	−3.00000	−0.242536
$154$	0	0
$155$	−19.8167	−1.59171
$156$	0	0
$157$	0.816654	0.0651761	0.0325880	−	0.999469i	$-0.489625\pi$
$157$	0.816654	0.0651761	0.0325880	+	0.999469i	$0.489625\pi$
$158$	0	0
$159$	6.00000	0.475831
$160$	0	0
$161$	0.697224	0.0549490
$162$	0	0
$163$	−5.39445	−0.422526	−0.211263	−	0.977429i	$-0.567758\pi$
$163$	−5.39445	−0.422526	−0.211263	+	0.977429i	$0.567758\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−1.00000	−0.0773823
$167$	−1.00000	−0.0773823
$168$	0	0
$169$	−5.72498	−0.440383
$170$	0	0
$171$	−2.60555	−0.199251
$172$	0	0
$173$	5.78890	0.440122	0.220061	−	0.975486i	$-0.429374\pi$
$173$	5.78890	0.440122	0.220061	+	0.975486i	$0.429374\pi$
$174$	0	0
$175$	−1.21110	−0.0915507
$176$	0	0
$177$	−10.1833	−0.765427
$178$	0	0
$179$	0.908327	0.0678915	0.0339458	−	0.999424i	$-0.489193\pi$
$179$	0.908327	0.0678915	0.0339458	+	0.999424i	$0.489193\pi$
$180$	0	0
$181$	−7.21110	−0.535997	−0.267999	−	0.963419i	$-0.586362\pi$
$181$	−7.21110	−0.535997	−0.267999	+	0.963419i	$0.586362\pi$
$182$	0	0
$183$	−8.88057	−0.656471
$184$	0	0
$185$	−1.18335	−0.0870013
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−1.69722	−0.123455
$190$	0	0
$191$	−11.3028	−0.817840	−0.408920	−	0.912570i	$-0.634095\pi$
$191$	−11.3028	−0.817840	−0.408920	+	0.912570i	$0.634095\pi$
$192$	0	0
$193$	5.90833	0.425291	0.212645	−	0.977129i	$-0.431792\pi$
$193$	5.90833	0.425291	0.212645	+	0.977129i	$0.431792\pi$
$194$	0	0
$195$	10.5416	0.754902
$196$	0	0
$197$	3.69722	0.263416	0.131708	−	0.991289i	$-0.457954\pi$
$197$	3.69722	0.263416	0.131708	+	0.991289i	$0.457954\pi$
$198$	0	0
$199$	−10.6972	−0.758306	−0.379153	−	0.925334i	$-0.623785\pi$
$199$	−10.6972	−0.758306	−0.379153	+	0.925334i	$0.623785\pi$
$200$	0	0
$201$	−10.6972	−0.754524
$202$	0	0
$203$	−2.30278	−0.161623
$204$	0	0
$205$	18.6333	1.30141
$206$	0	0
$207$	3.00000	0.208514
$208$	0	0
$209$	0	0
$210$	0	0
$211$	13.5139	0.930334	0.465167	−	0.885223i	$-0.345994\pi$
$211$	13.5139	0.930334	0.465167	+	0.885223i	$0.345994\pi$
$212$	0	0
$213$	−5.09167	−0.348876
$214$	0	0
$215$	−28.8167	−1.96528
$216$	0	0
$217$	−2.00000	−0.135769
$218$	0	0
$219$	4.02776	0.272171
$220$	0	0
$221$	6.21110	0.417804
$222$	0	0
$223$	−0.513878	−0.0344118	−0.0172059	−	0.999852i	$-0.505477\pi$
$223$	−0.513878	−0.0344118	−0.0172059	+	0.999852i	$0.505477\pi$
$224$	0	0
$225$	−5.21110	−0.347407
$226$	0	0
$227$	1.18335	0.0785414	0.0392707	−	0.999229i	$-0.487497\pi$
$227$	1.18335	0.0785414	0.0392707	+	0.999229i	$0.487497\pi$
$228$	0	0
$229$	8.69722	0.574729	0.287364	−	0.957821i	$-0.407221\pi$
$229$	8.69722	0.574729	0.287364	+	0.957821i	$0.407221\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.90833	0.452580	0.226290	−	0.974060i	$-0.427340\pi$
$233$	6.90833	0.452580	0.226290	+	0.974060i	$0.427340\pi$
$234$	0	0
$235$	−4.81665	−0.314204
$236$	0	0
$237$	−8.33053	−0.541126
$238$	0	0
$239$	11.0917	0.717461	0.358730	−	0.933441i	$-0.383210\pi$
$239$	11.0917	0.717461	0.358730	+	0.933441i	$0.383210\pi$
$240$	0	0
$241$	2.69722	0.173743	0.0868717	−	0.996220i	$-0.472313\pi$
$241$	2.69722	0.173743	0.0868717	+	0.996220i	$0.472313\pi$
$242$	0	0
$243$	−12.3944	−0.795104
$244$	0	0
$245$	20.7250	1.32407
$246$	0	0
$247$	5.39445	0.343241
$248$	0	0
$249$	2.09167	0.132554
$250$	0	0
$251$	−10.3944	−0.656092	−0.328046	−	0.944662i	$-0.606390\pi$
$251$	−10.3944	−0.656092	−0.328046	+	0.944662i	$0.606390\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	9.00000	0.563602
$256$	0	0
$257$	19.1194	1.19264	0.596319	−	0.802748i	$-0.296629\pi$
$257$	19.1194	1.19264	0.596319	+	0.802748i	$0.296629\pi$
$258$	0	0
$259$	−0.119429	−0.00742099
$260$	0	0
$261$	−9.90833	−0.613310
$262$	0	0
$263$	12.2111	0.752969	0.376484	−	0.926423i	$-0.377133\pi$
$263$	12.2111	0.752969	0.376484	+	0.926423i	$0.377133\pi$
$264$	0	0
$265$	13.8167	0.848750
$266$	0	0
$267$	18.0000	1.10158
$268$	0	0
$269$	5.30278	0.323316	0.161658	−	0.986847i	$-0.448316\pi$
$269$	5.30278	0.323316	0.161658	+	0.986847i	$0.448316\pi$
$270$	0	0
$271$	14.4222	0.876087	0.438043	−	0.898954i	$-0.355672\pi$
$271$	14.4222	0.876087	0.438043	+	0.898954i	$0.355672\pi$
$272$	0	0
$273$	1.06392	0.0643912
$274$	0	0
$275$	0	0
$276$	0	0
$277$	25.5139	1.53298	0.766490	−	0.642256i	$-0.222001\pi$
$277$	25.5139	1.53298	0.766490	+	0.642256i	$0.222001\pi$
$278$	0	0
$279$	−8.60555	−0.515201
$280$	0	0
$281$	−15.4222	−0.920012	−0.460006	−	0.887916i	$-0.652153\pi$
$281$	−15.4222	−0.920012	−0.460006	+	0.887916i	$0.652153\pi$
$282$	0	0
$283$	−10.4222	−0.619536	−0.309768	−	0.950812i	$-0.600251\pi$
$283$	−10.4222	−0.619536	−0.309768	+	0.950812i	$0.600251\pi$
$284$	0	0
$285$	7.81665	0.463019
$286$	0	0
$287$	1.88057	0.111007
$288$	0	0
$289$	−11.6972	−0.688072
$290$	0	0
$291$	−5.88057	−0.344725
$292$	0	0
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	0	0
$295$	−23.4500	−1.36531
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−6.21110	−0.359197
$300$	0	0
$301$	−2.90833	−0.167633
$302$	0	0
$303$	14.0917	0.809545
$304$	0	0
$305$	−20.4500	−1.17096
$306$	0	0
$307$	25.9361	1.48025	0.740125	−	0.672469i	$-0.234766\pi$
$307$	25.9361	1.48025	0.740125	+	0.672469i	$0.234766\pi$
$308$	0	0
$309$	4.30278	0.244776
$310$	0	0
$311$	6.21110	0.352199	0.176100	−	0.984372i	$-0.443652\pi$
$311$	6.21110	0.352199	0.176100	+	0.984372i	$0.443652\pi$
$312$	0	0
$313$	10.5139	0.594280	0.297140	−	0.954834i	$-0.403967\pi$
$313$	10.5139	0.594280	0.297140	+	0.954834i	$0.403967\pi$
$314$	0	0
$315$	−1.18335	−0.0666740
$316$	0	0
$317$	28.5416	1.60306	0.801529	−	0.597956i	$-0.204020\pi$
$317$	28.5416	1.60306	0.801529	+	0.597956i	$0.204020\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−9.90833	−0.553029
$322$	0	0
$323$	4.60555	0.256260
$324$	0	0
$325$	10.7889	0.598460
$326$	0	0
$327$	−2.60555	−0.144087
$328$	0	0
$329$	−0.486122	−0.0268008
$330$	0	0
$331$	9.18335	0.504762	0.252381	−	0.967628i	$-0.418786\pi$
$331$	9.18335	0.504762	0.252381	+	0.967628i	$0.418786\pi$
$332$	0	0
$333$	−0.513878	−0.0281604
$334$	0	0
$335$	−24.6333	−1.34586
$336$	0	0
$337$	27.6056	1.50377	0.751885	−	0.659294i	$-0.229145\pi$
$337$	27.6056	1.50377	0.751885	+	0.659294i	$0.229145\pi$
$338$	0	0
$339$	−12.2750	−0.666688
$340$	0	0
$341$	0	0
$342$	0	0
$343$	4.21110	0.227378
$344$	0	0
$345$	−9.00000	−0.484544
$346$	0	0
$347$	21.9083	1.17610	0.588050	−	0.808824i	$-0.299896\pi$
$347$	21.9083	1.17610	0.588050	+	0.808824i	$0.299896\pi$
$348$	0	0
$349$	−36.4500	−1.95112	−0.975561	−	0.219729i	$-0.929483\pi$
$349$	−36.4500	−1.95112	−0.975561	+	0.219729i	$0.929483\pi$
$350$	0	0
$351$	15.1194	0.807015
$352$	0	0
$353$	−4.18335	−0.222657	−0.111329	−	0.993784i	$-0.535511\pi$
$353$	−4.18335	−0.222657	−0.111329	+	0.993784i	$0.535511\pi$
$354$	0	0
$355$	−11.7250	−0.622297
$356$	0	0
$357$	0.908327	0.0480738
$358$	0	0
$359$	1.18335	0.0624546	0.0312273	−	0.999512i	$-0.490058\pi$
$359$	1.18335	0.0624546	0.0312273	+	0.999512i	$0.490058\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	14.3305	0.752158
$364$	0	0
$365$	9.27502	0.485477
$366$	0	0
$367$	18.8167	0.982221	0.491111	−	0.871097i	$-0.336591\pi$
$367$	18.8167	0.982221	0.491111	+	0.871097i	$0.336591\pi$
$368$	0	0
$369$	8.09167	0.421236
$370$	0	0
$371$	1.39445	0.0723962
$372$	0	0
$373$	−16.4861	−0.853619	−0.426810	−	0.904342i	$-0.640363\pi$
$373$	−16.4861	−0.853619	−0.426810	+	0.904342i	$0.640363\pi$
$374$	0	0
$375$	−3.90833	−0.201825
$376$	0	0
$377$	20.5139	1.05652
$378$	0	0
$379$	−0.0277564	−0.00142575	−0.000712875	−	1.00000i	$-0.500227\pi$
$379$	−0.0277564	−0.00142575	−0.000712875		1.00000i	$0.500227\pi$
$380$	0	0
$381$	−18.5139	−0.948495
$382$	0	0
$383$	−27.8444	−1.42278	−0.711391	−	0.702796i	$-0.751935\pi$
$383$	−27.8444	−1.42278	−0.711391	+	0.702796i	$0.751935\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−12.5139	−0.636116
$388$	0	0
$389$	−4.81665	−0.244214	−0.122107	−	0.992517i	$-0.538965\pi$
$389$	−4.81665	−0.244214	−0.122107	+	0.992517i	$0.538965\pi$
$390$	0	0
$391$	−5.30278	−0.268173
$392$	0	0
$393$	26.0917	1.31615
$394$	0	0
$395$	−19.1833	−0.965219
$396$	0	0
$397$	3.11943	0.156560	0.0782798	−	0.996931i	$-0.475057\pi$
$397$	3.11943	0.156560	0.0782798	+	0.996931i	$0.475057\pi$
$398$	0	0
$399$	0.788897	0.0394943
$400$	0	0
$401$	−21.6972	−1.08351	−0.541754	−	0.840537i	$-0.682240\pi$
$401$	−21.6972	−1.08351	−0.541754	+	0.840537i	$0.682240\pi$
$402$	0	0
$403$	17.8167	0.887511
$404$	0	0
$405$	10.1833	0.506015
$406$	0	0
$407$	0	0
$408$	0	0
$409$	19.0917	0.944022	0.472011	−	0.881593i	$-0.343528\pi$
$409$	19.0917	0.944022	0.472011	+	0.881593i	$0.343528\pi$
$410$	0	0
$411$	3.63331	0.179218
$412$	0	0
$413$	−2.36669	−0.116457
$414$	0	0
$415$	4.81665	0.236440
$416$	0	0
$417$	−7.69722	−0.376935
$418$	0	0
$419$	−12.2111	−0.596551	−0.298276	−	0.954480i	$-0.596411\pi$
$419$	−12.2111	−0.596551	−0.298276	+	0.954480i	$0.596411\pi$
$420$	0	0
$421$	12.6056	0.614357	0.307178	−	0.951652i	$-0.400615\pi$
$421$	12.6056	0.614357	0.307178	+	0.951652i	$0.400615\pi$
$422$	0	0
$423$	−2.09167	−0.101701
$424$	0	0
$425$	9.21110	0.446804
$426$	0	0
$427$	−2.06392	−0.0998799
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17.0278	0.820198	0.410099	−	0.912041i	$-0.365494\pi$
$431$	17.0278	0.820198	0.410099	+	0.912041i	$0.365494\pi$
$432$	0	0
$433$	−29.3305	−1.40954	−0.704768	−	0.709438i	$-0.748949\pi$
$433$	−29.3305	−1.40954	−0.704768	+	0.709438i	$0.748949\pi$
$434$	0	0
$435$	29.7250	1.42520
$436$	0	0
$437$	−4.60555	−0.220313
$438$	0	0
$439$	−22.2111	−1.06008	−0.530039	−	0.847973i	$-0.677823\pi$
$439$	−22.2111	−1.06008	−0.530039	+	0.847973i	$0.677823\pi$
$440$	0	0
$441$	9.00000	0.428571
$442$	0	0
$443$	0.275019	0.0130666	0.00653328	−	0.999979i	$-0.497920\pi$
$443$	0.275019	0.0130666	0.00653328	+	0.999979i	$0.497920\pi$
$444$	0	0
$445$	41.4500	1.96492
$446$	0	0
$447$	−22.8167	−1.07919
$448$	0	0
$449$	3.97224	0.187462	0.0937309	−	0.995598i	$-0.470121\pi$
$449$	3.97224	0.187462	0.0937309	+	0.995598i	$0.470121\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−25.4222	−1.19444
$454$	0	0
$455$	2.44996	0.114856
$456$	0	0
$457$	6.88057	0.321860	0.160930	−	0.986966i	$-0.448551\pi$
$457$	6.88057	0.321860	0.160930	+	0.986966i	$0.448551\pi$
$458$	0	0
$459$	12.9083	0.602509
$460$	0	0
$461$	−21.6972	−1.01054	−0.505270	−	0.862961i	$-0.668607\pi$
$461$	−21.6972	−1.01054	−0.505270	+	0.862961i	$0.668607\pi$
$462$	0	0
$463$	1.78890	0.0831371	0.0415686	−	0.999136i	$-0.486765\pi$
$463$	1.78890	0.0831371	0.0415686	+	0.999136i	$0.486765\pi$
$464$	0	0
$465$	25.8167	1.19722
$466$	0	0
$467$	−13.1194	−0.607095	−0.303547	−	0.952816i	$-0.598171\pi$
$467$	−13.1194	−0.607095	−0.303547	+	0.952816i	$0.598171\pi$
$468$	0	0
$469$	−2.48612	−0.114798
$470$	0	0
$471$	−1.06392	−0.0490227
$472$	0	0
$473$	0	0
$474$	0	0
$475$	8.00000	0.367065
$476$	0	0
$477$	6.00000	0.274721
$478$	0	0
$479$	3.63331	0.166010	0.0830050	−	0.996549i	$-0.473548\pi$
$479$	3.63331	0.166010	0.0830050	+	0.996549i	$0.473548\pi$
$480$	0	0
$481$	1.06392	0.0485104
$482$	0	0
$483$	−0.908327	−0.0413303
$484$	0	0
$485$	−13.5416	−0.614894
$486$	0	0
$487$	30.8167	1.39644	0.698218	−	0.715885i	$-0.253977\pi$
$487$	30.8167	1.39644	0.698218	+	0.715885i	$0.253977\pi$
$488$	0	0
$489$	7.02776	0.317806
$490$	0	0
$491$	0.633308	0.0285808	0.0142904	−	0.999898i	$-0.495451\pi$
$491$	0.633308	0.0285808	0.0142904	+	0.999898i	$0.495451\pi$
$492$	0	0
$493$	17.5139	0.788785
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1.18335	−0.0530803
$498$	0	0
$499$	0.669468	0.0299695	0.0149848	−	0.999888i	$-0.495230\pi$
$499$	0.669468	0.0299695	0.0149848	+	0.999888i	$0.495230\pi$
$500$	0	0
$501$	1.30278	0.0582037
$502$	0	0
$503$	−0.486122	−0.0216751	−0.0108376	−	0.999941i	$-0.503450\pi$
$503$	−0.486122	−0.0216751	−0.0108376	+	0.999941i	$0.503450\pi$
$504$	0	0
$505$	32.4500	1.44400
$506$	0	0
$507$	7.45837	0.331238
$508$	0	0
$509$	24.9083	1.10404	0.552021	−	0.833830i	$-0.313857\pi$
$509$	24.9083	1.10404	0.552021	+	0.833830i	$0.313857\pi$
$510$	0	0
$511$	0.936083	0.0414099
$512$	0	0
$513$	11.2111	0.494982
$514$	0	0
$515$	9.90833	0.436613
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−7.54163	−0.331041
$520$	0	0
$521$	−43.8167	−1.91964	−0.959821	−	0.280612i	$-0.909463\pi$
$521$	−43.8167	−1.91964	−0.959821	+	0.280612i	$0.909463\pi$
$522$	0	0
$523$	−8.81665	−0.385525	−0.192763	−	0.981245i	$-0.561745\pi$
$523$	−8.81665	−0.385525	−0.192763	+	0.981245i	$0.561745\pi$
$524$	0	0
$525$	1.57779	0.0688606
$526$	0	0
$527$	15.2111	0.662606
$528$	0	0
$529$	−17.6972	−0.769445
$530$	0	0
$531$	−10.1833	−0.441920
$532$	0	0
$533$	−16.7527	−0.725642
$534$	0	0
$535$	−22.8167	−0.986450
$536$	0	0
$537$	−1.18335	−0.0510652
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−35.6056	−1.53080	−0.765401	−	0.643554i	$-0.777459\pi$
$541$	−35.6056	−1.53080	−0.765401	+	0.643554i	$0.777459\pi$
$542$	0	0
$543$	9.39445	0.403154
$544$	0	0
$545$	−6.00000	−0.257012
$546$	0	0
$547$	−34.3583	−1.46905	−0.734527	−	0.678579i	$-0.762596\pi$
$547$	−34.3583	−1.46905	−0.734527	+	0.678579i	$0.762596\pi$
$548$	0	0
$549$	−8.88057	−0.379014
$550$	0	0
$551$	15.2111	0.648015
$552$	0	0
$553$	−1.93608	−0.0823306
$554$	0	0
$555$	1.54163	0.0654387
$556$	0	0
$557$	10.8167	0.458316	0.229158	−	0.973389i	$-0.426403\pi$
$557$	10.8167	0.458316	0.229158	+	0.973389i	$0.426403\pi$
$558$	0	0
$559$	25.9083	1.09581
$560$	0	0
$561$	0	0
$562$	0	0
$563$	23.5139	0.990992	0.495496	−	0.868610i	$-0.334986\pi$
$563$	23.5139	0.990992	0.495496	+	0.868610i	$0.334986\pi$
$564$	0	0
$565$	−28.2666	−1.18919
$566$	0	0
$567$	1.02776	0.0431617
$568$	0	0
$569$	36.2111	1.51805	0.759024	−	0.651062i	$-0.225676\pi$
$569$	36.2111	1.51805	0.759024	+	0.651062i	$0.225676\pi$
$570$	0	0
$571$	−14.5416	−0.608548	−0.304274	−	0.952584i	$-0.598414\pi$
$571$	−14.5416	−0.608548	−0.304274	+	0.952584i	$0.598414\pi$
$572$	0	0
$573$	14.7250	0.615145
$574$	0	0
$575$	−9.21110	−0.384130
$576$	0	0
$577$	−24.5139	−1.02053	−0.510263	−	0.860018i	$-0.670452\pi$
$577$	−24.5139	−1.02053	−0.510263	+	0.860018i	$0.670452\pi$
$578$	0	0
$579$	−7.69722	−0.319886
$580$	0	0
$581$	0.486122	0.0201677
$582$	0	0
$583$	0	0
$584$	0	0
$585$	10.5416	0.435843
$586$	0	0
$587$	−42.9083	−1.77102	−0.885508	−	0.464624i	$-0.846190\pi$
$587$	−42.9083	−1.77102	−0.885508	+	0.464624i	$0.846190\pi$
$588$	0	0
$589$	13.2111	0.544354
$590$	0	0
$591$	−4.81665	−0.198131
$592$	0	0
$593$	−28.3944	−1.16602	−0.583010	−	0.812465i	$-0.698125\pi$
$593$	−28.3944	−1.16602	−0.583010	+	0.812465i	$0.698125\pi$
$594$	0	0
$595$	2.09167	0.0857502
$596$	0	0
$597$	13.9361	0.570366
$598$	0	0
$599$	−26.0917	−1.06608	−0.533038	−	0.846091i	$-0.678950\pi$
$599$	−26.0917	−1.06608	−0.533038	+	0.846091i	$0.678950\pi$
$600$	0	0
$601$	24.5416	1.00107	0.500537	−	0.865715i	$-0.333136\pi$
$601$	24.5416	1.00107	0.500537	+	0.865715i	$0.333136\pi$
$602$	0	0
$603$	−10.6972	−0.435625
$604$	0	0
$605$	33.0000	1.34164
$606$	0	0
$607$	13.7250	0.557080	0.278540	−	0.960425i	$-0.410150\pi$
$607$	13.7250	0.557080	0.278540	+	0.960425i	$0.410150\pi$
$608$	0	0
$609$	3.00000	0.121566
$610$	0	0
$611$	4.33053	0.175195
$612$	0	0
$613$	−38.3305	−1.54816	−0.774078	−	0.633090i	$-0.781786\pi$
$613$	−38.3305	−1.54816	−0.774078	+	0.633090i	$0.781786\pi$
$614$	0	0
$615$	−24.2750	−0.978863
$616$	0	0
$617$	−38.4500	−1.54794	−0.773969	−	0.633224i	$-0.781731\pi$
$617$	−38.4500	−1.54794	−0.773969	+	0.633224i	$0.781731\pi$
$618$	0	0
$619$	25.0278	1.00595	0.502975	−	0.864301i	$-0.332239\pi$
$619$	25.0278	1.00595	0.502975	+	0.864301i	$0.332239\pi$
$620$	0	0
$621$	−12.9083	−0.517993
$622$	0	0
$623$	4.18335	0.167602
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	0.908327	0.0362174
$630$	0	0
$631$	−33.9361	−1.35097	−0.675487	−	0.737372i	$-0.736067\pi$
$631$	−33.9361	−1.35097	−0.675487	+	0.737372i	$0.736067\pi$
$632$	0	0
$633$	−17.6056	−0.699758
$634$	0	0
$635$	−42.6333	−1.69185
$636$	0	0
$637$	−18.6333	−0.738279
$638$	0	0
$639$	−5.09167	−0.201423
$640$	0	0
$641$	−11.3028	−0.446433	−0.223216	−	0.974769i	$-0.571656\pi$
$641$	−11.3028	−0.446433	−0.223216	+	0.974769i	$0.571656\pi$
$642$	0	0
$643$	−12.5139	−0.493499	−0.246750	−	0.969079i	$-0.579363\pi$
$643$	−12.5139	−0.493499	−0.246750	+	0.969079i	$0.579363\pi$
$644$	0	0
$645$	37.5416	1.47820
$646$	0	0
$647$	−35.9361	−1.41279	−0.706397	−	0.707816i	$-0.749680\pi$
$647$	−35.9361	−1.41279	−0.706397	+	0.707816i	$0.749680\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	2.60555	0.102120
$652$	0	0
$653$	−13.1194	−0.513403	−0.256701	−	0.966491i	$-0.582636\pi$
$653$	−13.1194	−0.513403	−0.256701	+	0.966491i	$0.582636\pi$
$654$	0	0
$655$	60.0833	2.34765
$656$	0	0
$657$	4.02776	0.157138
$658$	0	0
$659$	−19.8167	−0.771947	−0.385974	−	0.922510i	$-0.626134\pi$
$659$	−19.8167	−0.771947	−0.385974	+	0.922510i	$0.626134\pi$
$660$	0	0
$661$	−38.1194	−1.48267	−0.741337	−	0.671133i	$-0.765808\pi$
$661$	−38.1194	−1.48267	−0.741337	+	0.671133i	$0.765808\pi$
$662$	0	0
$663$	−8.09167	−0.314255
$664$	0	0
$665$	1.81665	0.0704468
$666$	0	0
$667$	−17.5139	−0.678140
$668$	0	0
$669$	0.669468	0.0258831
$670$	0	0
$671$	0	0
$672$	0	0
$673$	41.6333	1.60485	0.802423	−	0.596756i	$-0.203544\pi$
$673$	41.6333	1.60485	0.802423	+	0.596756i	$0.203544\pi$
$674$	0	0
$675$	22.4222	0.863031
$676$	0	0
$677$	−13.8167	−0.531017	−0.265509	−	0.964108i	$-0.585540\pi$
$677$	−13.8167	−0.531017	−0.265509	+	0.964108i	$0.585540\pi$
$678$	0	0
$679$	−1.36669	−0.0524488
$680$	0	0
$681$	−1.54163	−0.0590756
$682$	0	0
$683$	−49.5416	−1.89566	−0.947829	−	0.318779i	$-0.896727\pi$
$683$	−49.5416	−1.89566	−0.947829	+	0.318779i	$0.896727\pi$
$684$	0	0
$685$	8.36669	0.319675
$686$	0	0
$687$	−11.3305	−0.432287
$688$	0	0
$689$	−12.4222	−0.473248
$690$	0	0
$691$	−42.4500	−1.61487	−0.807436	−	0.589955i	$-0.799146\pi$
$691$	−42.4500	−1.61487	−0.807436	+	0.589955i	$0.799146\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−17.7250	−0.672347
$696$	0	0
$697$	−14.3028	−0.541756
$698$	0	0
$699$	−9.00000	−0.340411
$700$	0	0
$701$	8.02776	0.303204	0.151602	−	0.988442i	$-0.451557\pi$
$701$	8.02776	0.303204	0.151602	+	0.988442i	$0.451557\pi$
$702$	0	0
$703$	0.788897	0.0297538
$704$	0	0
$705$	6.27502	0.236331
$706$	0	0
$707$	3.27502	0.123170
$708$	0	0
$709$	50.1472	1.88332	0.941659	−	0.336570i	$-0.109267\pi$
$709$	50.1472	1.88332	0.941659	+	0.336570i	$0.109267\pi$
$710$	0	0
$711$	−8.33053	−0.312419
$712$	0	0
$713$	−15.2111	−0.569660
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−14.4500	−0.539644
$718$	0	0
$719$	−22.8806	−0.853301	−0.426651	−	0.904417i	$-0.640307\pi$
$719$	−22.8806	−0.853301	−0.426651	+	0.904417i	$0.640307\pi$
$720$	0	0
$721$	1.00000	0.0372419
$722$	0	0
$723$	−3.51388	−0.130683
$724$	0	0
$725$	30.4222	1.12985
$726$	0	0
$727$	28.7250	1.06535	0.532675	−	0.846320i	$-0.321187\pi$
$727$	28.7250	1.06535	0.532675	+	0.846320i	$0.321187\pi$
$728$	0	0
$729$	26.3305	0.975205
$730$	0	0
$731$	22.1194	0.818117
$732$	0	0
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	0	0
$735$	−27.0000	−0.995910
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−3.93608	−0.144791	−0.0723956	−	0.997376i	$-0.523064\pi$
$739$	−3.93608	−0.144791	−0.0723956	+	0.997376i	$0.523064\pi$
$740$	0	0
$741$	−7.02776	−0.258171
$742$	0	0
$743$	52.1194	1.91208	0.956038	−	0.293242i	$-0.0947342\pi$
$743$	52.1194	1.91208	0.956038	+	0.293242i	$0.0947342\pi$
$744$	0	0
$745$	−52.5416	−1.92498
$746$	0	0
$747$	2.09167	0.0765303
$748$	0	0
$749$	−2.30278	−0.0841416
$750$	0	0
$751$	−23.6056	−0.861379	−0.430689	−	0.902500i	$-0.641730\pi$
$751$	−23.6056	−0.861379	−0.430689	+	0.902500i	$0.641730\pi$
$752$	0	0
$753$	13.5416	0.493485
$754$	0	0
$755$	−58.5416	−2.13055
$756$	0	0
$757$	22.7250	0.825953	0.412977	−	0.910742i	$-0.364489\pi$
$757$	22.7250	0.825953	0.412977	+	0.910742i	$0.364489\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−3.69722	−0.134024	−0.0670121	−	0.997752i	$-0.521347\pi$
$761$	−3.69722	−0.134024	−0.0670121	+	0.997752i	$0.521347\pi$
$762$	0	0
$763$	−0.605551	−0.0219224
$764$	0	0
$765$	9.00000	0.325396
$766$	0	0
$767$	21.0833	0.761273
$768$	0	0
$769$	−14.8167	−0.534302	−0.267151	−	0.963655i	$-0.586082\pi$
$769$	−14.8167	−0.534302	−0.267151	+	0.963655i	$0.586082\pi$
$770$	0	0
$771$	−24.9083	−0.897051
$772$	0	0
$773$	10.6056	0.381455	0.190728	−	0.981643i	$-0.438915\pi$
$773$	10.6056	0.381455	0.190728	+	0.981643i	$0.438915\pi$
$774$	0	0
$775$	26.4222	0.949114
$776$	0	0
$777$	0.155590	0.00558175
$778$	0	0
$779$	−12.4222	−0.445072
$780$	0	0
$781$	0	0
$782$	0	0
$783$	42.6333	1.52359
$784$	0	0
$785$	−2.44996	−0.0874429
$786$	0	0
$787$	43.4500	1.54882	0.774412	−	0.632682i	$-0.218046\pi$
$787$	43.4500	1.54882	0.774412	+	0.632682i	$0.218046\pi$
$788$	0	0
$789$	−15.9083	−0.566351
$790$	0	0
$791$	−2.85281	−0.101434
$792$	0	0
$793$	18.3860	0.652908
$794$	0	0
$795$	−18.0000	−0.638394
$796$	0	0
$797$	36.6333	1.29762	0.648809	−	0.760951i	$-0.275267\pi$
$797$	36.6333	1.29762	0.648809	+	0.760951i	$0.275267\pi$
$798$	0	0
$799$	3.69722	0.130798
$800$	0	0
$801$	18.0000	0.635999
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−2.09167	−0.0737218
$806$	0	0
$807$	−6.90833	−0.243185
$808$	0	0
$809$	23.7250	0.834126	0.417063	−	0.908878i	$-0.363059\pi$
$809$	23.7250	0.834126	0.417063	+	0.908878i	$0.363059\pi$
$810$	0	0
$811$	−37.8444	−1.32890	−0.664448	−	0.747334i	$-0.731333\pi$
$811$	−37.8444	−1.32890	−0.664448	+	0.747334i	$0.731333\pi$
$812$	0	0
$813$	−18.7889	−0.658955
$814$	0	0
$815$	16.1833	0.566878
$816$	0	0
$817$	19.2111	0.672111
$818$	0	0
$819$	1.06392	0.0371763
$820$	0	0
$821$	14.7250	0.513905	0.256953	−	0.966424i	$-0.417282\pi$
$821$	14.7250	0.513905	0.256953	+	0.966424i	$0.417282\pi$
$822$	0	0
$823$	−21.3028	−0.742568	−0.371284	−	0.928519i	$-0.621082\pi$
$823$	−21.3028	−0.742568	−0.371284	+	0.928519i	$0.621082\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.45837	0.0507123	0.0253562	−	0.999678i	$-0.491928\pi$
$827$	1.45837	0.0507123	0.0253562	+	0.999678i	$0.491928\pi$
$828$	0	0
$829$	37.8722	1.31535	0.657677	−	0.753300i	$-0.271539\pi$
$829$	37.8722	1.31535	0.657677	+	0.753300i	$0.271539\pi$
$830$	0	0
$831$	−33.2389	−1.15304
$832$	0	0
$833$	−15.9083	−0.551191
$834$	0	0
$835$	3.00000	0.103819
$836$	0	0
$837$	37.0278	1.27987
$838$	0	0
$839$	40.3944	1.39457	0.697286	−	0.716793i	$-0.254391\pi$
$839$	40.3944	1.39457	0.697286	+	0.716793i	$0.254391\pi$
$840$	0	0
$841$	28.8444	0.994635
$842$	0	0
$843$	20.0917	0.691994
$844$	0	0
$845$	17.1749	0.590836
$846$	0	0
$847$	3.33053	0.114438
$848$	0	0
$849$	13.5778	0.465989
$850$	0	0
$851$	−0.908327	−0.0311370
$852$	0	0
$853$	40.7250	1.39440	0.697198	−	0.716878i	$-0.254430\pi$
$853$	40.7250	1.39440	0.697198	+	0.716878i	$0.254430\pi$
$854$	0	0
$855$	7.81665	0.267324
$856$	0	0
$857$	25.5416	0.872486	0.436243	−	0.899829i	$-0.356309\pi$
$857$	25.5416	0.872486	0.436243	+	0.899829i	$0.356309\pi$
$858$	0	0
$859$	−41.3944	−1.41236	−0.706180	−	0.708032i	$-0.749583\pi$
$859$	−41.3944	−1.41236	−0.706180	+	0.708032i	$0.749583\pi$
$860$	0	0
$861$	−2.44996	−0.0834945
$862$	0	0
$863$	−11.5139	−0.391937	−0.195968	−	0.980610i	$-0.562785\pi$
$863$	−11.5139	−0.391937	−0.195968	+	0.980610i	$0.562785\pi$
$864$	0	0
$865$	−17.3667	−0.590485
$866$	0	0
$867$	15.2389	0.517539
$868$	0	0
$869$	0	0
$870$	0	0
$871$	22.1472	0.750429
$872$	0	0
$873$	−5.88057	−0.199027
$874$	0	0
$875$	−0.908327	−0.0307071
$876$	0	0
$877$	19.5139	0.658937	0.329468	−	0.944167i	$-0.393130\pi$
$877$	19.5139	0.658937	0.329468	+	0.944167i	$0.393130\pi$
$878$	0	0
$879$	−23.4500	−0.790948
$880$	0	0
$881$	−14.7889	−0.498251	−0.249125	−	0.968471i	$-0.580143\pi$
$881$	−14.7889	−0.498251	−0.249125	+	0.968471i	$0.580143\pi$
$882$	0	0
$883$	−27.9361	−0.940124	−0.470062	−	0.882633i	$-0.655768\pi$
$883$	−27.9361	−0.940124	−0.470062	+	0.882633i	$0.655768\pi$
$884$	0	0
$885$	30.5500	1.02693
$886$	0	0
$887$	10.8167	0.363188	0.181594	−	0.983374i	$-0.441874\pi$
$887$	10.8167	0.363188	0.181594	+	0.983374i	$0.441874\pi$
$888$	0	0
$889$	−4.30278	−0.144310
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.21110	0.107455
$894$	0	0
$895$	−2.72498	−0.0910861
$896$	0	0
$897$	8.09167	0.270173
$898$	0	0
$899$	50.2389	1.67556
$900$	0	0
$901$	−10.6056	−0.353322
$902$	0	0
$903$	3.78890	0.126087
$904$	0	0
$905$	21.6333	0.719115
$906$	0	0
$907$	−4.63331	−0.153846	−0.0769232	−	0.997037i	$-0.524510\pi$
$907$	−4.63331	−0.153846	−0.0769232	+	0.997037i	$0.524510\pi$
$908$	0	0
$909$	14.0917	0.467391
$910$	0	0
$911$	7.18335	0.237995	0.118997	−	0.992895i	$-0.462032\pi$
$911$	7.18335	0.237995	0.118997	+	0.992895i	$0.462032\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	26.6417	0.880748
$916$	0	0
$917$	6.06392	0.200248
$918$	0	0
$919$	−0.0916731	−0.00302402	−0.00151201	−	0.999999i	$-0.500481\pi$
$919$	−0.0916731	−0.00302402	−0.00151201	+	0.999999i	$0.500481\pi$
$920$	0	0
$921$	−33.7889	−1.11338
$922$	0	0
$923$	10.5416	0.346982
$924$	0	0
$925$	1.57779	0.0518776
$926$	0	0
$927$	4.30278	0.141322
$928$	0	0
$929$	15.4861	0.508083	0.254042	−	0.967193i	$-0.418240\pi$
$929$	15.4861	0.508083	0.254042	+	0.967193i	$0.418240\pi$
$930$	0	0
$931$	−13.8167	−0.452823
$932$	0	0
$933$	−8.09167	−0.264909
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−51.4500	−1.68080	−0.840398	−	0.541969i	$-0.817679\pi$
$937$	−51.4500	−1.68080	−0.840398	+	0.541969i	$0.817679\pi$
$938$	0	0
$939$	−13.6972	−0.446992
$940$	0	0
$941$	59.2389	1.93113	0.965566	−	0.260159i	$-0.0837750\pi$
$941$	59.2389	1.93113	0.965566	+	0.260159i	$0.0837750\pi$
$942$	0	0
$943$	14.3028	0.465762
$944$	0	0
$945$	5.09167	0.165632
$946$	0	0
$947$	14.0917	0.457918	0.228959	−	0.973436i	$-0.426468\pi$
$947$	14.0917	0.457918	0.228959	+	0.973436i	$0.426468\pi$
$948$	0	0
$949$	−8.33894	−0.270693
$950$	0	0
$951$	−37.1833	−1.20575
$952$	0	0
$953$	42.3583	1.37212	0.686060	−	0.727545i	$-0.259339\pi$
$953$	42.3583	1.37212	0.686060	+	0.727545i	$0.259339\pi$
$954$	0	0
$955$	33.9083	1.09725
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0.844410	0.0272674
$960$	0	0
$961$	12.6333	0.407526
$962$	0	0
$963$	−9.90833	−0.319291
$964$	0	0
$965$	−17.7250	−0.570587
$966$	0	0
$967$	−25.4861	−0.819578	−0.409789	−	0.912180i	$-0.634398\pi$
$967$	−25.4861	−0.819578	−0.409789	+	0.912180i	$0.634398\pi$
$968$	0	0
$969$	−6.00000	−0.192748
$970$	0	0
$971$	49.4777	1.58782	0.793908	−	0.608038i	$-0.208043\pi$
$971$	49.4777	1.58782	0.793908	+	0.608038i	$0.208043\pi$
$972$	0	0
$973$	−1.78890	−0.0573494
$974$	0	0
$975$	−14.0555	−0.450137
$976$	0	0
$977$	−42.5694	−1.36192	−0.680958	−	0.732323i	$-0.738436\pi$
$977$	−42.5694	−1.36192	−0.680958	+	0.732323i	$0.738436\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	−2.60555	−0.0831888
$982$	0	0
$983$	−31.2666	−0.997250	−0.498625	−	0.866818i	$-0.666162\pi$
$983$	−31.2666	−0.997250	−0.498625	+	0.866818i	$0.666162\pi$
$984$	0	0
$985$	−11.0917	−0.353410
$986$	0	0
$987$	0.633308	0.0201584
$988$	0	0
$989$	−22.1194	−0.703357
$990$	0	0
$991$	−19.6333	−0.623673	−0.311836	−	0.950136i	$-0.600944\pi$
$991$	−19.6333	−0.623673	−0.311836	+	0.950136i	$0.600944\pi$
$992$	0	0
$993$	−11.9638	−0.379661
$994$	0	0
$995$	32.0917	1.01737
$996$	0	0
$997$	−24.7889	−0.785072	−0.392536	−	0.919737i	$-0.628402\pi$
$997$	−24.7889	−0.785072	−0.392536	+	0.919737i	$0.628402\pi$
$998$	0	0
$999$	2.21110	0.0699562

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	668.2.a.a.1.1	✓	2
3.2	odd	2		6012.2.a.a.1.1		2
4.3	odd	2		2672.2.a.c.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
668.2.a.a.1.1	✓	2	1.1	even	1	trivial
2672.2.a.c.1.2		2	4.3	odd	2
6012.2.a.a.1.1		2	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 668 = 2^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	668.a (trivial)

\(f(q)\)	\(=\)	\(q-1.30278 q^{3} -3.00000 q^{5} -0.302776 q^{7} -1.30278 q^{9} +O(q^{10})\)	\(q-1.30278 q^{3} -3.00000 q^{5} -0.302776 q^{7} -1.30278 q^{9} +2.69722 q^{13} +3.90833 q^{15} +2.30278 q^{17} +2.00000 q^{19} +0.394449 q^{21} -2.30278 q^{23} +4.00000 q^{25} +5.60555 q^{27} +7.60555 q^{29} +6.60555 q^{31} +0.908327 q^{35} +0.394449 q^{37} -3.51388 q^{39} -6.21110 q^{41} +9.60555 q^{43} +3.90833 q^{45} +1.60555 q^{47} -6.90833 q^{49} -3.00000 q^{51} -4.60555 q^{53} -2.60555 q^{57} +7.81665 q^{59} +6.81665 q^{61} +0.394449 q^{63} -8.09167 q^{65} +8.21110 q^{67} +3.00000 q^{69} +3.90833 q^{71} -3.09167 q^{73} -5.21110 q^{75} +6.39445 q^{79} -3.39445 q^{81} -1.60555 q^{83} -6.90833 q^{85} -9.90833 q^{87} -13.8167 q^{89} -0.816654 q^{91} -8.60555 q^{93} -6.00000 q^{95} +4.51388 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 6 q^{5} + 3 q^{7} + q^{9}+O(q^{10}) \) 2 * q + q^3 - 6 * q^5 + 3 * q^7 + q^9	\( 2 q + q^{3} - 6 q^{5} + 3 q^{7} + q^{9} + 9 q^{13} - 3 q^{15} + q^{17} + 4 q^{19} + 8 q^{21} - q^{23} + 8 q^{25} + 4 q^{27} + 8 q^{29} + 6 q^{31} - 9 q^{35} + 8 q^{37} + 11 q^{39} + 2 q^{41} + 12 q^{43} - 3 q^{45} - 4 q^{47} - 3 q^{49} - 6 q^{51} - 2 q^{53} + 2 q^{57} - 6 q^{59} - 8 q^{61} + 8 q^{63} - 27 q^{65} + 2 q^{67} + 6 q^{69} - 3 q^{71} - 17 q^{73} + 4 q^{75} + 20 q^{79} - 14 q^{81} + 4 q^{83} - 3 q^{85} - 9 q^{87} - 6 q^{89} + 20 q^{91} - 10 q^{93} - 12 q^{95} - 9 q^{97}+O(q^{100}) \) 2 * q + q^3 - 6 * q^5 + 3 * q^7 + q^9 + 9 * q^13 - 3 * q^15 + q^17 + 4 * q^19 + 8 * q^21 - q^23 + 8 * q^25 + 4 * q^27 + 8 * q^29 + 6 * q^31 - 9 * q^35 + 8 * q^37 + 11 * q^39 + 2 * q^41 + 12 * q^43 - 3 * q^45 - 4 * q^47 - 3 * q^49 - 6 * q^51 - 2 * q^53 + 2 * q^57 - 6 * q^59 - 8 * q^61 + 8 * q^63 - 27 * q^65 + 2 * q^67 + 6 * q^69 - 3 * q^71 - 17 * q^73 + 4 * q^75 + 20 * q^79 - 14 * q^81 + 4 * q^83 - 3 * q^85 - 9 * q^87 - 6 * q^89 + 20 * q^91 - 10 * q^93 - 12 * q^95 - 9 * q^97

Introduction

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