LMFDB - Embedded newform 6012.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6012,2,Mod(1,6012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6012, 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("6012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6012 = 2^{2} \cdot 3^{2} \cdot 167 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6012.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:	$48.0060616952$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 668)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.30278$ of defining polynomial
Character	$\chi$	$=$	6012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000 q^{5} -0.302776 q^{7} +O(q^{10})$	$q+3.00000 q^{5} -0.302776 q^{7} +2.69722 q^{13} -2.30278 q^{17} +2.00000 q^{19} +2.30278 q^{23} +4.00000 q^{25} -7.60555 q^{29} +6.60555 q^{31} -0.908327 q^{35} +0.394449 q^{37} +6.21110 q^{41} +9.60555 q^{43} -1.60555 q^{47} -6.90833 q^{49} +4.60555 q^{53} -7.81665 q^{59} +6.81665 q^{61} +8.09167 q^{65} +8.21110 q^{67} -3.90833 q^{71} -3.09167 q^{73} +6.39445 q^{79} +1.60555 q^{83} -6.90833 q^{85} +13.8167 q^{89} -0.816654 q^{91} +6.00000 q^{95} +4.51388 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 6 q^{5} + 3 q^{7}+O(q^{10}) $ 2 * q + 6 * q^5 + 3 * q^7	$ 2 q + 6 q^{5} + 3 q^{7} + 9 q^{13} - q^{17} + 4 q^{19} + q^{23} + 8 q^{25} - 8 q^{29} + 6 q^{31} + 9 q^{35} + 8 q^{37} - 2 q^{41} + 12 q^{43} + 4 q^{47} - 3 q^{49} + 2 q^{53} + 6 q^{59} - 8 q^{61} + 27 q^{65} + 2 q^{67} + 3 q^{71} - 17 q^{73} + 20 q^{79} - 4 q^{83} - 3 q^{85} + 6 q^{89} + 20 q^{91} + 12 q^{95} - 9 q^{97}+O(q^{100}) $ 2 * q + 6 * q^5 + 3 * q^7 + 9 * q^13 - q^17 + 4 * q^19 + q^23 + 8 * q^25 - 8 * q^29 + 6 * q^31 + 9 * q^35 + 8 * q^37 - 2 * q^41 + 12 * q^43 + 4 * q^47 - 3 * q^49 + 2 * q^53 + 6 * q^59 - 8 * q^61 + 27 * q^65 + 2 * q^67 + 3 * q^71 - 17 * q^73 + 20 * q^79 - 4 * q^83 - 3 * q^85 + 6 * q^89 + 20 * q^91 + 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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	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
$9$	0	0
$10$	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
$15$	0	0
$16$	0	0
$17$	−2.30278	−0.558505	−0.279253	−	0.960218i	$-0.590087\pi$
$17$	−2.30278	−0.558505	−0.279253	+	0.960218i	$0.590087\pi$
$18$	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	0
$22$	0	0
$23$	2.30278	0.480162	0.240081	−	0.970753i	$-0.422826\pi$
$23$	2.30278	0.480162	0.240081	+	0.970753i	$0.422826\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.60555	−1.41232	−0.706158	−	0.708055i	$-0.749573\pi$
$29$	−7.60555	−1.41232	−0.706158	+	0.708055i	$0.749573\pi$
$30$	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
$39$	0	0
$40$	0	0
$41$	6.21110	0.970011	0.485006	−	0.874511i	$-0.338818\pi$
$41$	6.21110	0.970011	0.485006	+	0.874511i	$0.338818\pi$
$42$	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
$45$	0	0
$46$	0	0
$47$	−1.60555	−0.234194	−0.117097	−	0.993120i	$-0.537359\pi$
$47$	−1.60555	−0.234194	−0.117097	+	0.993120i	$0.537359\pi$
$48$	0	0
$49$	−6.90833	−0.986904
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.60555	0.632621	0.316311	−	0.948656i	$-0.397556\pi$
$53$	4.60555	0.632621	0.316311	+	0.948656i	$0.397556\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.81665	−1.01764	−0.508821	−	0.860872i	$-0.669918\pi$
$59$	−7.81665	−1.01764	−0.508821	+	0.860872i	$0.669918\pi$
$60$	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	0
$64$	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
$69$	0	0
$70$	0	0
$71$	−3.90833	−0.463833	−0.231917	−	0.972736i	$-0.574500\pi$
$71$	−3.90833	−0.463833	−0.231917	+	0.972736i	$0.574500\pi$
$72$	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
$75$	0	0
$76$	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
$81$	0	0
$82$	0	0
$83$	1.60555	0.176232	0.0881161	−	0.996110i	$-0.471915\pi$
$83$	1.60555	0.176232	0.0881161	+	0.996110i	$0.471915\pi$
$84$	0	0
$85$	−6.90833	−0.749313
$86$	0	0
$87$	0	0
$88$	0	0
$89$	13.8167	1.46456	0.732281	−	0.681002i	$-0.238456\pi$
$89$	13.8167	1.46456	0.732281	+	0.681002i	$0.238456\pi$
$90$	0	0
$91$	−0.816654	−0.0856086
$92$	0	0
$93$	0	0
$94$	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	$-0.319124\pi$
$101$	10.8167	1.07630	0.538149	+	0.842850i	$0.319124\pi$
$102$	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
$105$	0	0
$106$	0	0
$107$	−7.60555	−0.735256	−0.367628	−	0.929973i	$-0.619830\pi$
$107$	−7.60555	−0.735256	−0.367628	+	0.929973i	$0.619830\pi$
$108$	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	0
$112$	0	0
$113$	−9.42221	−0.886366	−0.443183	−	0.896431i	$-0.646151\pi$
$113$	−9.42221	−0.886366	−0.443183	+	0.896431i	$0.646151\pi$
$114$	0	0
$115$	6.90833	0.644205
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.697224	0.0639145
$120$	0	0
$121$	−11.0000	−1.00000
$122$	0	0
$123$	0	0
$124$	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
$129$	0	0
$130$	0	0
$131$	20.0278	1.74983	0.874917	−	0.484274i	$-0.160916\pi$
$131$	20.0278	1.74983	0.874917	+	0.484274i	$0.160916\pi$
$132$	0	0
$133$	−0.605551	−0.0525080
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.78890	0.238272	0.119136	−	0.992878i	$-0.461988\pi$
$137$	2.78890	0.238272	0.119136	+	0.992878i	$0.461988\pi$
$138$	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
$141$	0	0
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−22.8167	−1.89482
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−17.5139	−1.43479	−0.717396	−	0.696665i	$-0.754666\pi$
$149$	−17.5139	−1.43479	−0.717396	+	0.696665i	$0.754666\pi$
$150$	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
$153$	0	0
$154$	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
$159$	0	0
$160$	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
$171$	0	0
$172$	0	0
$173$	−5.78890	−0.440122	−0.220061	−	0.975486i	$-0.570626\pi$
$173$	−5.78890	−0.440122	−0.220061	+	0.975486i	$0.570626\pi$
$174$	0	0
$175$	−1.21110	−0.0915507
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−0.908327	−0.0678915	−0.0339458	−	0.999424i	$-0.510807\pi$
$179$	−0.908327	−0.0678915	−0.0339458	+	0.999424i	$0.510807\pi$
$180$	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
$183$	0	0
$184$	0	0
$185$	1.18335	0.0870013
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.3028	0.817840	0.408920	−	0.912570i	$-0.365905\pi$
$191$	11.3028	0.817840	0.408920	+	0.912570i	$0.365905\pi$
$192$	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
$195$	0	0
$196$	0	0
$197$	−3.69722	−0.263416	−0.131708	−	0.991289i	$-0.542046\pi$
$197$	−3.69722	−0.263416	−0.131708	+	0.991289i	$0.542046\pi$
$198$	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
$201$	0	0
$202$	0	0
$203$	2.30278	0.161623
$204$	0	0
$205$	18.6333	1.30141
$206$	0	0
$207$	0	0
$208$	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
$213$	0	0
$214$	0	0
$215$	28.8167	1.96528
$216$	0	0
$217$	−2.00000	−0.135769
$218$	0	0
$219$	0	0
$220$	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
$225$	0	0
$226$	0	0
$227$	−1.18335	−0.0785414	−0.0392707	−	0.999229i	$-0.512503\pi$
$227$	−1.18335	−0.0785414	−0.0392707	+	0.999229i	$0.512503\pi$
$228$	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	$-0.572660\pi$
$233$	−6.90833	−0.452580	−0.226290	+	0.974060i	$0.572660\pi$
$234$	0	0
$235$	−4.81665	−0.314204
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−11.0917	−0.717461	−0.358730	−	0.933441i	$-0.616790\pi$
$239$	−11.0917	−0.717461	−0.358730	+	0.933441i	$0.616790\pi$
$240$	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
$243$	0	0
$244$	0	0
$245$	−20.7250	−1.32407
$246$	0	0
$247$	5.39445	0.343241
$248$	0	0
$249$	0	0
$250$	0	0
$251$	10.3944	0.656092	0.328046	−	0.944662i	$-0.393610\pi$
$251$	10.3944	0.656092	0.328046	+	0.944662i	$0.393610\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−19.1194	−1.19264	−0.596319	−	0.802748i	$-0.703371\pi$
$257$	−19.1194	−1.19264	−0.596319	+	0.802748i	$0.703371\pi$
$258$	0	0
$259$	−0.119429	−0.00742099
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−12.2111	−0.752969	−0.376484	−	0.926423i	$-0.622867\pi$
$263$	−12.2111	−0.752969	−0.376484	+	0.926423i	$0.622867\pi$
$264$	0	0
$265$	13.8167	0.848750
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−5.30278	−0.323316	−0.161658	−	0.986847i	$-0.551684\pi$
$269$	−5.30278	−0.323316	−0.161658	+	0.986847i	$0.551684\pi$
$270$	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
$273$	0	0
$274$	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
$279$	0	0
$280$	0	0
$281$	15.4222	0.920012	0.460006	−	0.887916i	$-0.347847\pi$
$281$	15.4222	0.920012	0.460006	+	0.887916i	$0.347847\pi$
$282$	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
$285$	0	0
$286$	0	0
$287$	−1.88057	−0.111007
$288$	0	0
$289$	−11.6972	−0.688072
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	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
$303$	0	0
$304$	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
$309$	0	0
$310$	0	0
$311$	−6.21110	−0.352199	−0.176100	−	0.984372i	$-0.556348\pi$
$311$	−6.21110	−0.352199	−0.176100	+	0.984372i	$0.556348\pi$
$312$	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
$315$	0	0
$316$	0	0
$317$	−28.5416	−1.60306	−0.801529	−	0.597956i	$-0.795980\pi$
$317$	−28.5416	−1.60306	−0.801529	+	0.597956i	$0.795980\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−4.60555	−0.256260
$324$	0	0
$325$	10.7889	0.598460
$326$	0	0
$327$	0	0
$328$	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	0
$334$	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
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	4.21110	0.227378
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−21.9083	−1.17610	−0.588050	−	0.808824i	$-0.700104\pi$
$347$	−21.9083	−1.17610	−0.588050	+	0.808824i	$0.700104\pi$
$348$	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
$351$	0	0
$352$	0	0
$353$	4.18335	0.222657	0.111329	−	0.993784i	$-0.464489\pi$
$353$	4.18335	0.222657	0.111329	+	0.993784i	$0.464489\pi$
$354$	0	0
$355$	−11.7250	−0.622297
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1.18335	−0.0624546	−0.0312273	−	0.999512i	$-0.509942\pi$
$359$	−1.18335	−0.0624546	−0.0312273	+	0.999512i	$0.509942\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	0	0
$364$	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
$369$	0	0
$370$	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
$375$	0	0
$376$	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
$381$	0	0
$382$	0	0
$383$	27.8444	1.42278	0.711391	−	0.702796i	$-0.248065\pi$
$383$	27.8444	1.42278	0.711391	+	0.702796i	$0.248065\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4.81665	0.244214	0.122107	−	0.992517i	$-0.461035\pi$
$389$	4.81665	0.244214	0.122107	+	0.992517i	$0.461035\pi$
$390$	0	0
$391$	−5.30278	−0.268173
$392$	0	0
$393$	0	0
$394$	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	0
$400$	0	0
$401$	21.6972	1.08351	0.541754	−	0.840537i	$-0.317760\pi$
$401$	21.6972	1.08351	0.541754	+	0.840537i	$0.317760\pi$
$402$	0	0
$403$	17.8167	0.887511
$404$	0	0
$405$	0	0
$406$	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
$411$	0	0
$412$	0	0
$413$	2.36669	0.116457
$414$	0	0
$415$	4.81665	0.236440
$416$	0	0
$417$	0	0
$418$	0	0
$419$	12.2111	0.596551	0.298276	−	0.954480i	$-0.403589\pi$
$419$	12.2111	0.596551	0.298276	+	0.954480i	$0.403589\pi$
$420$	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
$423$	0	0
$424$	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	$-0.634506\pi$
$431$	−17.0278	−0.820198	−0.410099	+	0.912041i	$0.634506\pi$
$432$	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
$435$	0	0
$436$	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
$441$	0	0
$442$	0	0
$443$	−0.275019	−0.0130666	−0.00653328	−	0.999979i	$-0.502080\pi$
$443$	−0.275019	−0.0130666	−0.00653328	+	0.999979i	$0.502080\pi$
$444$	0	0
$445$	41.4500	1.96492
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−3.97224	−0.187462	−0.0937309	−	0.995598i	$-0.529879\pi$
$449$	−3.97224	−0.187462	−0.0937309	+	0.995598i	$0.529879\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	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
$459$	0	0
$460$	0	0
$461$	21.6972	1.01054	0.505270	−	0.862961i	$-0.331393\pi$
$461$	21.6972	1.01054	0.505270	+	0.862961i	$0.331393\pi$
$462$	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
$465$	0	0
$466$	0	0
$467$	13.1194	0.607095	0.303547	−	0.952816i	$-0.401829\pi$
$467$	13.1194	0.607095	0.303547	+	0.952816i	$0.401829\pi$
$468$	0	0
$469$	−2.48612	−0.114798
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	8.00000	0.367065
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3.63331	−0.166010	−0.0830050	−	0.996549i	$-0.526452\pi$
$479$	−3.63331	−0.166010	−0.0830050	+	0.996549i	$0.526452\pi$
$480$	0	0
$481$	1.06392	0.0485104
$482$	0	0
$483$	0	0
$484$	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
$489$	0	0
$490$	0	0
$491$	−0.633308	−0.0285808	−0.0142904	−	0.999898i	$-0.504549\pi$
$491$	−0.633308	−0.0285808	−0.0142904	+	0.999898i	$0.504549\pi$
$492$	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
$501$	0	0
$502$	0	0
$503$	0.486122	0.0216751	0.0108376	−	0.999941i	$-0.496550\pi$
$503$	0.486122	0.0216751	0.0108376	+	0.999941i	$0.496550\pi$
$504$	0	0
$505$	32.4500	1.44400
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−24.9083	−1.10404	−0.552021	−	0.833830i	$-0.686143\pi$
$509$	−24.9083	−1.10404	−0.552021	+	0.833830i	$0.686143\pi$
$510$	0	0
$511$	0.936083	0.0414099
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−9.90833	−0.436613
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	43.8167	1.91964	0.959821	−	0.280612i	$-0.0905375\pi$
$521$	43.8167	1.91964	0.959821	+	0.280612i	$0.0905375\pi$
$522$	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
$525$	0	0
$526$	0	0
$527$	−15.2111	−0.662606
$528$	0	0
$529$	−17.6972	−0.769445
$530$	0	0
$531$	0	0
$532$	0	0
$533$	16.7527	0.725642
$534$	0	0
$535$	−22.8167	−0.986450
$536$	0	0
$537$	0	0
$538$	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
$543$	0	0
$544$	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
$549$	0	0
$550$	0	0
$551$	−15.2111	−0.648015
$552$	0	0
$553$	−1.93608	−0.0823306
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10.8167	−0.458316	−0.229158	−	0.973389i	$-0.573597\pi$
$557$	−10.8167	−0.458316	−0.229158	+	0.973389i	$0.573597\pi$
$558$	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	$-0.665014\pi$
$563$	−23.5139	−0.990992	−0.495496	+	0.868610i	$0.665014\pi$
$564$	0	0
$565$	−28.2666	−1.18919
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−36.2111	−1.51805	−0.759024	−	0.651062i	$-0.774324\pi$
$569$	−36.2111	−1.51805	−0.759024	+	0.651062i	$0.774324\pi$
$570$	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
$573$	0	0
$574$	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
$579$	0	0
$580$	0	0
$581$	−0.486122	−0.0201677
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	42.9083	1.77102	0.885508	−	0.464624i	$-0.153810\pi$
$587$	42.9083	1.77102	0.885508	+	0.464624i	$0.153810\pi$
$588$	0	0
$589$	13.2111	0.544354
$590$	0	0
$591$	0	0
$592$	0	0
$593$	28.3944	1.16602	0.583010	−	0.812465i	$-0.301875\pi$
$593$	28.3944	1.16602	0.583010	+	0.812465i	$0.301875\pi$
$594$	0	0
$595$	2.09167	0.0857502
$596$	0	0
$597$	0	0
$598$	0	0
$599$	26.0917	1.06608	0.533038	−	0.846091i	$-0.321050\pi$
$599$	26.0917	1.06608	0.533038	+	0.846091i	$0.321050\pi$
$600$	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
$603$	0	0
$604$	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
$609$	0	0
$610$	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
$615$	0	0
$616$	0	0
$617$	38.4500	1.54794	0.773969	−	0.633224i	$-0.218269\pi$
$617$	38.4500	1.54794	0.773969	+	0.633224i	$0.218269\pi$
$618$	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
$621$	0	0
$622$	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
$633$	0	0
$634$	0	0
$635$	42.6333	1.69185
$636$	0	0
$637$	−18.6333	−0.738279
$638$	0	0
$639$	0	0
$640$	0	0
$641$	11.3028	0.446433	0.223216	−	0.974769i	$-0.428344\pi$
$641$	11.3028	0.446433	0.223216	+	0.974769i	$0.428344\pi$
$642$	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
$645$	0	0
$646$	0	0
$647$	35.9361	1.41279	0.706397	−	0.707816i	$-0.250320\pi$
$647$	35.9361	1.41279	0.706397	+	0.707816i	$0.250320\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	0	0
$653$	13.1194	0.513403	0.256701	−	0.966491i	$-0.417364\pi$
$653$	13.1194	0.513403	0.256701	+	0.966491i	$0.417364\pi$
$654$	0	0
$655$	60.0833	2.34765
$656$	0	0
$657$	0	0
$658$	0	0
$659$	19.8167	0.771947	0.385974	−	0.922510i	$-0.373866\pi$
$659$	19.8167	0.771947	0.385974	+	0.922510i	$0.373866\pi$
$660$	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
$663$	0	0
$664$	0	0
$665$	−1.81665	−0.0704468
$666$	0	0
$667$	−17.5139	−0.678140
$668$	0	0
$669$	0	0
$670$	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
$675$	0	0
$676$	0	0
$677$	13.8167	0.531017	0.265509	−	0.964108i	$-0.414460\pi$
$677$	13.8167	0.531017	0.265509	+	0.964108i	$0.414460\pi$
$678$	0	0
$679$	−1.36669	−0.0524488
$680$	0	0
$681$	0	0
$682$	0	0
$683$	49.5416	1.89566	0.947829	−	0.318779i	$-0.103273\pi$
$683$	49.5416	1.89566	0.947829	+	0.318779i	$0.103273\pi$
$684$	0	0
$685$	8.36669	0.319675
$686$	0	0
$687$	0	0
$688$	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
$699$	0	0
$700$	0	0
$701$	−8.02776	−0.303204	−0.151602	−	0.988442i	$-0.548443\pi$
$701$	−8.02776	−0.303204	−0.151602	+	0.988442i	$0.548443\pi$
$702$	0	0
$703$	0.788897	0.0297538
$704$	0	0
$705$	0	0
$706$	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
$711$	0	0
$712$	0	0
$713$	15.2111	0.569660
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	22.8806	0.853301	0.426651	−	0.904417i	$-0.359693\pi$
$719$	22.8806	0.853301	0.426651	+	0.904417i	$0.359693\pi$
$720$	0	0
$721$	1.00000	0.0372419
$722$	0	0
$723$	0	0
$724$	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
$729$	0	0
$730$	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
$735$	0	0
$736$	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
$741$	0	0
$742$	0	0
$743$	−52.1194	−1.91208	−0.956038	−	0.293242i	$-0.905266\pi$
$743$	−52.1194	−1.91208	−0.956038	+	0.293242i	$0.905266\pi$
$744$	0	0
$745$	−52.5416	−1.92498
$746$	0	0
$747$	0	0
$748$	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
$753$	0	0
$754$	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	$-0.478653\pi$
$761$	3.69722	0.134024	0.0670121	+	0.997752i	$0.478653\pi$
$762$	0	0
$763$	−0.605551	−0.0219224
$764$	0	0
$765$	0	0
$766$	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
$771$	0	0
$772$	0	0
$773$	−10.6056	−0.381455	−0.190728	−	0.981643i	$-0.561085\pi$
$773$	−10.6056	−0.381455	−0.190728	+	0.981643i	$0.561085\pi$
$774$	0	0
$775$	26.4222	0.949114
$776$	0	0
$777$	0	0
$778$	0	0
$779$	12.4222	0.445072
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	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
$789$	0	0
$790$	0	0
$791$	2.85281	0.101434
$792$	0	0
$793$	18.3860	0.652908
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−36.6333	−1.29762	−0.648809	−	0.760951i	$-0.724733\pi$
$797$	−36.6333	−1.29762	−0.648809	+	0.760951i	$0.724733\pi$
$798$	0	0
$799$	3.69722	0.130798
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	−2.09167	−0.0737218
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−23.7250	−0.834126	−0.417063	−	0.908878i	$-0.636941\pi$
$809$	−23.7250	−0.834126	−0.417063	+	0.908878i	$0.636941\pi$
$810$	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
$813$	0	0
$814$	0	0
$815$	−16.1833	−0.566878
$816$	0	0
$817$	19.2111	0.672111
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−14.7250	−0.513905	−0.256953	−	0.966424i	$-0.582718\pi$
$821$	−14.7250	−0.513905	−0.256953	+	0.966424i	$0.582718\pi$
$822$	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	$-0.508072\pi$
$827$	−1.45837	−0.0507123	−0.0253562	+	0.999678i	$0.508072\pi$
$828$	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
$831$	0	0
$832$	0	0
$833$	15.9083	0.551191
$834$	0	0
$835$	3.00000	0.103819
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−40.3944	−1.39457	−0.697286	−	0.716793i	$-0.745609\pi$
$839$	−40.3944	−1.39457	−0.697286	+	0.716793i	$0.745609\pi$
$840$	0	0
$841$	28.8444	0.994635
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−17.1749	−0.590836
$846$	0	0
$847$	3.33053	0.114438
$848$	0	0
$849$	0	0
$850$	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
$855$	0	0
$856$	0	0
$857$	−25.5416	−0.872486	−0.436243	−	0.899829i	$-0.643691\pi$
$857$	−25.5416	−0.872486	−0.436243	+	0.899829i	$0.643691\pi$
$858$	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
$861$	0	0
$862$	0	0
$863$	11.5139	0.391937	0.195968	−	0.980610i	$-0.437215\pi$
$863$	11.5139	0.391937	0.195968	+	0.980610i	$0.437215\pi$
$864$	0	0
$865$	−17.3667	−0.590485
$866$	0	0
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	22.1472	0.750429
$872$	0	0
$873$	0	0
$874$	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
$879$	0	0
$880$	0	0
$881$	14.7889	0.498251	0.249125	−	0.968471i	$-0.419857\pi$
$881$	14.7889	0.498251	0.249125	+	0.968471i	$0.419857\pi$
$882$	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
$885$	0	0
$886$	0	0
$887$	−10.8167	−0.363188	−0.181594	−	0.983374i	$-0.558126\pi$
$887$	−10.8167	−0.363188	−0.181594	+	0.983374i	$0.558126\pi$
$888$	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
$897$	0	0
$898$	0	0
$899$	−50.2389	−1.67556
$900$	0	0
$901$	−10.6056	−0.353322
$902$	0	0
$903$	0	0
$904$	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
$909$	0	0
$910$	0	0
$911$	−7.18335	−0.237995	−0.118997	−	0.992895i	$-0.537968\pi$
$911$	−7.18335	−0.237995	−0.118997	+	0.992895i	$0.537968\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	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
$921$	0	0
$922$	0	0
$923$	−10.5416	−0.346982
$924$	0	0
$925$	1.57779	0.0518776
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−15.4861	−0.508083	−0.254042	−	0.967193i	$-0.581760\pi$
$929$	−15.4861	−0.508083	−0.254042	+	0.967193i	$0.581760\pi$
$930$	0	0
$931$	−13.8167	−0.452823
$932$	0	0
$933$	0	0
$934$	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
$939$	0	0
$940$	0	0
$941$	−59.2389	−1.93113	−0.965566	−	0.260159i	$-0.916225\pi$
$941$	−59.2389	−1.93113	−0.965566	+	0.260159i	$0.916225\pi$
$942$	0	0
$943$	14.3028	0.465762
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−14.0917	−0.457918	−0.228959	−	0.973436i	$-0.573532\pi$
$947$	−14.0917	−0.457918	−0.228959	+	0.973436i	$0.573532\pi$
$948$	0	0
$949$	−8.33894	−0.270693
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−42.3583	−1.37212	−0.686060	−	0.727545i	$-0.740661\pi$
$953$	−42.3583	−1.37212	−0.686060	+	0.727545i	$0.740661\pi$
$954$	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
$963$	0	0
$964$	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
$969$	0	0
$970$	0	0
$971$	−49.4777	−1.58782	−0.793908	−	0.608038i	$-0.791957\pi$
$971$	−49.4777	−1.58782	−0.793908	+	0.608038i	$0.791957\pi$
$972$	0	0
$973$	−1.78890	−0.0573494
$974$	0	0
$975$	0	0
$976$	0	0
$977$	42.5694	1.36192	0.680958	−	0.732323i	$-0.261564\pi$
$977$	42.5694	1.36192	0.680958	+	0.732323i	$0.261564\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	0	0
$982$	0	0
$983$	31.2666	0.997250	0.498625	−	0.866818i	$-0.333838\pi$
$983$	31.2666	0.997250	0.498625	+	0.866818i	$0.333838\pi$
$984$	0	0
$985$	−11.0917	−0.353410
$986$	0	0
$987$	0	0
$988$	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
$993$	0	0
$994$	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
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6012.2.a.a.1.1		2
3.2	odd	2		668.2.a.a.1.1	✓	2
12.11	even	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	3.2	odd	2
2672.2.a.c.1.2		2	12.11	even	2
6012.2.a.a.1.1		2	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:	\( N \)	\(=\)	\( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6012.a (trivial)

\(f(q)\)	\(=\)	\(q+3.00000 q^{5} -0.302776 q^{7} +O(q^{10})\)	\(q+3.00000 q^{5} -0.302776 q^{7} +2.69722 q^{13} -2.30278 q^{17} +2.00000 q^{19} +2.30278 q^{23} +4.00000 q^{25} -7.60555 q^{29} +6.60555 q^{31} -0.908327 q^{35} +0.394449 q^{37} +6.21110 q^{41} +9.60555 q^{43} -1.60555 q^{47} -6.90833 q^{49} +4.60555 q^{53} -7.81665 q^{59} +6.81665 q^{61} +8.09167 q^{65} +8.21110 q^{67} -3.90833 q^{71} -3.09167 q^{73} +6.39445 q^{79} +1.60555 q^{83} -6.90833 q^{85} +13.8167 q^{89} -0.816654 q^{91} +6.00000 q^{95} +4.51388 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 6 q^{5} + 3 q^{7}+O(q^{10}) \) 2 * q + 6 * q^5 + 3 * q^7	\( 2 q + 6 q^{5} + 3 q^{7} + 9 q^{13} - q^{17} + 4 q^{19} + q^{23} + 8 q^{25} - 8 q^{29} + 6 q^{31} + 9 q^{35} + 8 q^{37} - 2 q^{41} + 12 q^{43} + 4 q^{47} - 3 q^{49} + 2 q^{53} + 6 q^{59} - 8 q^{61} + 27 q^{65} + 2 q^{67} + 3 q^{71} - 17 q^{73} + 20 q^{79} - 4 q^{83} - 3 q^{85} + 6 q^{89} + 20 q^{91} + 12 q^{95} - 9 q^{97}+O(q^{100}) \) 2 * q + 6 * q^5 + 3 * q^7 + 9 * q^13 - q^17 + 4 * q^19 + q^23 + 8 * q^25 - 8 * q^29 + 6 * q^31 + 9 * q^35 + 8 * q^37 - 2 * q^41 + 12 * q^43 + 4 * q^47 - 3 * q^49 + 2 * q^53 + 6 * q^59 - 8 * q^61 + 27 * q^65 + 2 * q^67 + 3 * q^71 - 17 * q^73 + 20 * q^79 - 4 * q^83 - 3 * q^85 + 6 * q^89 + 20 * q^91 + 12 * q^95 - 9 * q^97

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