LMFDB - Embedded newform 6027.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6027,2,Mod(1,6027)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6027, 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("6027.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6027 = 3 \cdot 7^{2} \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6027.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:	$48.1258372982$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 123)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	6027.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{2} -1.00000 q^{3} -3.41421 q^{5} +1.41421 q^{6} +2.82843 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.41421 q^{2} -1.00000 q^{3} -3.41421 q^{5} +1.41421 q^{6} +2.82843 q^{8} +1.00000 q^{9} +4.82843 q^{10} +2.41421 q^{11} -6.24264 q^{13} +3.41421 q^{15} -4.00000 q^{16} +0.414214 q^{17} -1.41421 q^{18} +5.41421 q^{19} -3.41421 q^{22} -1.41421 q^{23} -2.82843 q^{24} +6.65685 q^{25} +8.82843 q^{26} -1.00000 q^{27} -6.07107 q^{29} -4.82843 q^{30} +3.00000 q^{31} -2.41421 q^{33} -0.585786 q^{34} -9.48528 q^{37} -7.65685 q^{38} +6.24264 q^{39} -9.65685 q^{40} +1.00000 q^{41} -5.00000 q^{43} -3.41421 q^{45} +2.00000 q^{46} -10.4142 q^{47} +4.00000 q^{48} -9.41421 q^{50} -0.414214 q^{51} +6.82843 q^{53} +1.41421 q^{54} -8.24264 q^{55} -5.41421 q^{57} +8.58579 q^{58} +8.48528 q^{59} +4.65685 q^{61} -4.24264 q^{62} +8.00000 q^{64} +21.3137 q^{65} +3.41421 q^{66} +10.4853 q^{67} +1.41421 q^{69} +10.0711 q^{71} +2.82843 q^{72} +10.3137 q^{73} +13.4142 q^{74} -6.65685 q^{75} -8.82843 q^{78} -7.65685 q^{79} +13.6569 q^{80} +1.00000 q^{81} -1.41421 q^{82} -1.07107 q^{83} -1.41421 q^{85} +7.07107 q^{86} +6.07107 q^{87} +6.82843 q^{88} +11.6569 q^{89} +4.82843 q^{90} -3.00000 q^{93} +14.7279 q^{94} -18.4853 q^{95} -7.75736 q^{97} +2.41421 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9	$ 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9} + 4 q^{10} + 2 q^{11} - 4 q^{13} + 4 q^{15} - 8 q^{16} - 2 q^{17} + 8 q^{19} - 4 q^{22} + 2 q^{25} + 12 q^{26} - 2 q^{27} + 2 q^{29} - 4 q^{30} + 6 q^{31} - 2 q^{33} - 4 q^{34} - 2 q^{37} - 4 q^{38} + 4 q^{39} - 8 q^{40} + 2 q^{41} - 10 q^{43} - 4 q^{45} + 4 q^{46} - 18 q^{47} + 8 q^{48} - 16 q^{50} + 2 q^{51} + 8 q^{53} - 8 q^{55} - 8 q^{57} + 20 q^{58} - 2 q^{61} + 16 q^{64} + 20 q^{65} + 4 q^{66} + 4 q^{67} + 6 q^{71} - 2 q^{73} + 24 q^{74} - 2 q^{75} - 12 q^{78} - 4 q^{79} + 16 q^{80} + 2 q^{81} + 12 q^{83} - 2 q^{87} + 8 q^{88} + 12 q^{89} + 4 q^{90} - 6 q^{93} + 4 q^{94} - 20 q^{95} - 24 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9 + 4 * q^10 + 2 * q^11 - 4 * q^13 + 4 * q^15 - 8 * q^16 - 2 * q^17 + 8 * q^19 - 4 * q^22 + 2 * q^25 + 12 * q^26 - 2 * q^27 + 2 * q^29 - 4 * q^30 + 6 * q^31 - 2 * q^33 - 4 * q^34 - 2 * q^37 - 4 * q^38 + 4 * q^39 - 8 * q^40 + 2 * q^41 - 10 * q^43 - 4 * q^45 + 4 * q^46 - 18 * q^47 + 8 * q^48 - 16 * q^50 + 2 * q^51 + 8 * q^53 - 8 * q^55 - 8 * q^57 + 20 * q^58 - 2 * q^61 + 16 * q^64 + 20 * q^65 + 4 * q^66 + 4 * q^67 + 6 * q^71 - 2 * q^73 + 24 * q^74 - 2 * q^75 - 12 * q^78 - 4 * q^79 + 16 * q^80 + 2 * q^81 + 12 * q^83 - 2 * q^87 + 8 * q^88 + 12 * q^89 + 4 * q^90 - 6 * q^93 + 4 * q^94 - 20 * q^95 - 24 * q^97 + 2 * 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 $

$2$	−1.41421	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$2$	−1.41421	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−3.41421	−1.52688	−0.763441	−	0.645877i	$-0.776492\pi$
$5$	−3.41421	−1.52688	−0.763441	+	0.645877i	$0.776492\pi$
$6$	1.41421	0.577350
$7$	0	0
$7$	0	0
$8$	2.82843	1.00000
$9$	1.00000	0.333333
$10$	4.82843	1.52688
$11$	2.41421	0.727913	0.363956	−	0.931416i	$-0.381426\pi$
$11$	2.41421	0.727913	0.363956	+	0.931416i	$0.381426\pi$
$12$	0	0
$13$	−6.24264	−1.73140	−0.865699	−	0.500566i	$-0.833125\pi$
$13$	−6.24264	−1.73140	−0.865699	+	0.500566i	$0.833125\pi$
$14$	0	0
$15$	3.41421	0.881546
$16$	−4.00000	−1.00000
$17$	0.414214	0.100462	0.0502308	−	0.998738i	$-0.484004\pi$
$17$	0.414214	0.100462	0.0502308	+	0.998738i	$0.484004\pi$
$18$	−1.41421	−0.333333
$19$	5.41421	1.24211	0.621053	−	0.783769i	$-0.286705\pi$
$19$	5.41421	1.24211	0.621053	+	0.783769i	$0.286705\pi$
$20$	0	0
$21$	0	0
$22$	−3.41421	−0.727913
$23$	−1.41421	−0.294884	−0.147442	−	0.989071i	$-0.547104\pi$
$23$	−1.41421	−0.294884	−0.147442	+	0.989071i	$0.547104\pi$
$24$	−2.82843	−0.577350
$25$	6.65685	1.33137
$26$	8.82843	1.73140
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.07107	−1.12737	−0.563684	−	0.825990i	$-0.690617\pi$
$29$	−6.07107	−1.12737	−0.563684	+	0.825990i	$0.690617\pi$
$30$	−4.82843	−0.881546
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	0	0
$33$	−2.41421	−0.420261
$34$	−0.585786	−0.100462
$35$	0	0
$36$	0	0
$37$	−9.48528	−1.55937	−0.779685	−	0.626172i	$-0.784621\pi$
$37$	−9.48528	−1.55937	−0.779685	+	0.626172i	$0.784621\pi$
$38$	−7.65685	−1.24211
$39$	6.24264	0.999623
$40$	−9.65685	−1.52688
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	−5.00000	−0.762493	−0.381246	−	0.924473i	$-0.624505\pi$
$43$	−5.00000	−0.762493	−0.381246	+	0.924473i	$0.624505\pi$
$44$	0	0
$45$	−3.41421	−0.508961
$46$	2.00000	0.294884
$47$	−10.4142	−1.51907	−0.759535	−	0.650467i	$-0.774573\pi$
$47$	−10.4142	−1.51907	−0.759535	+	0.650467i	$0.774573\pi$
$48$	4.00000	0.577350
$49$	0	0
$50$	−9.41421	−1.33137
$51$	−0.414214	−0.0580015
$52$	0	0
$53$	6.82843	0.937957	0.468978	−	0.883210i	$-0.344622\pi$
$53$	6.82843	0.937957	0.468978	+	0.883210i	$0.344622\pi$
$54$	1.41421	0.192450
$55$	−8.24264	−1.11144
$56$	0	0
$57$	−5.41421	−0.717130
$58$	8.58579	1.12737
$59$	8.48528	1.10469	0.552345	−	0.833616i	$-0.313733\pi$
$59$	8.48528	1.10469	0.552345	+	0.833616i	$0.313733\pi$
$60$	0	0
$61$	4.65685	0.596249	0.298125	−	0.954527i	$-0.403639\pi$
$61$	4.65685	0.596249	0.298125	+	0.954527i	$0.403639\pi$
$62$	−4.24264	−0.538816
$63$	0	0
$64$	8.00000	1.00000
$65$	21.3137	2.64364
$66$	3.41421	0.420261
$67$	10.4853	1.28098	0.640490	−	0.767966i	$-0.278731\pi$
$67$	10.4853	1.28098	0.640490	+	0.767966i	$0.278731\pi$
$68$	0	0
$69$	1.41421	0.170251
$70$	0	0
$71$	10.0711	1.19522	0.597608	−	0.801788i	$-0.296118\pi$
$71$	10.0711	1.19522	0.597608	+	0.801788i	$0.296118\pi$
$72$	2.82843	0.333333
$73$	10.3137	1.20713	0.603564	−	0.797314i	$-0.293747\pi$
$73$	10.3137	1.20713	0.603564	+	0.797314i	$0.293747\pi$
$74$	13.4142	1.55937
$75$	−6.65685	−0.768667
$76$	0	0
$77$	0	0
$78$	−8.82843	−0.999623
$79$	−7.65685	−0.861463	−0.430732	−	0.902480i	$-0.641744\pi$
$79$	−7.65685	−0.861463	−0.430732	+	0.902480i	$0.641744\pi$
$80$	13.6569	1.52688
$81$	1.00000	0.111111
$82$	−1.41421	−0.156174
$83$	−1.07107	−0.117565	−0.0587825	−	0.998271i	$-0.518722\pi$
$83$	−1.07107	−0.117565	−0.0587825	+	0.998271i	$0.518722\pi$
$84$	0	0
$85$	−1.41421	−0.153393
$86$	7.07107	0.762493
$87$	6.07107	0.650887
$88$	6.82843	0.727913
$89$	11.6569	1.23562	0.617812	−	0.786326i	$-0.288019\pi$
$89$	11.6569	1.23562	0.617812	+	0.786326i	$0.288019\pi$
$90$	4.82843	0.508961
$91$	0	0
$92$	0	0
$93$	−3.00000	−0.311086
$94$	14.7279	1.51907
$95$	−18.4853	−1.89655
$96$	0	0
$97$	−7.75736	−0.787641	−0.393820	−	0.919187i	$-0.628847\pi$
$97$	−7.75736	−0.787641	−0.393820	+	0.919187i	$0.628847\pi$
$98$	0	0
$99$	2.41421	0.242638
$100$	0	0
$101$	0.414214	0.0412158	0.0206079	−	0.999788i	$-0.493440\pi$
$101$	0.414214	0.0412158	0.0206079	+	0.999788i	$0.493440\pi$
$102$	0.585786	0.0580015
$103$	−6.31371	−0.622108	−0.311054	−	0.950392i	$-0.600682\pi$
$103$	−6.31371	−0.622108	−0.311054	+	0.950392i	$0.600682\pi$
$104$	−17.6569	−1.73140
$105$	0	0
$106$	−9.65685	−0.937957
$107$	−0.343146	−0.0331732	−0.0165866	−	0.999862i	$-0.505280\pi$
$107$	−0.343146	−0.0331732	−0.0165866	+	0.999862i	$0.505280\pi$
$108$	0	0
$109$	7.89949	0.756634	0.378317	−	0.925676i	$-0.376503\pi$
$109$	7.89949	0.756634	0.378317	+	0.925676i	$0.376503\pi$
$110$	11.6569	1.11144
$111$	9.48528	0.900303
$112$	0	0
$113$	6.72792	0.632910	0.316455	−	0.948608i	$-0.397507\pi$
$113$	6.72792	0.632910	0.316455	+	0.948608i	$0.397507\pi$
$114$	7.65685	0.717130
$115$	4.82843	0.450253
$116$	0	0
$117$	−6.24264	−0.577132
$118$	−12.0000	−1.10469
$119$	0	0
$120$	9.65685	0.881546
$121$	−5.17157	−0.470143
$122$	−6.58579	−0.596249
$123$	−1.00000	−0.0901670
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	9.31371	0.826458	0.413229	−	0.910627i	$-0.364401\pi$
$127$	9.31371	0.826458	0.413229	+	0.910627i	$0.364401\pi$
$128$	−11.3137	−1.00000
$129$	5.00000	0.440225
$130$	−30.1421	−2.64364
$131$	10.5858	0.924884	0.462442	−	0.886649i	$-0.346973\pi$
$131$	10.5858	0.924884	0.462442	+	0.886649i	$0.346973\pi$
$132$	0	0
$133$	0	0
$134$	−14.8284	−1.28098
$135$	3.41421	0.293849
$136$	1.17157	0.100462
$137$	3.92893	0.335671	0.167836	−	0.985815i	$-0.446322\pi$
$137$	3.92893	0.335671	0.167836	+	0.985815i	$0.446322\pi$
$138$	−2.00000	−0.170251
$139$	−3.65685	−0.310170	−0.155085	−	0.987901i	$-0.549565\pi$
$139$	−3.65685	−0.310170	−0.155085	+	0.987901i	$0.549565\pi$
$140$	0	0
$141$	10.4142	0.877035
$142$	−14.2426	−1.19522
$143$	−15.0711	−1.26031
$144$	−4.00000	−0.333333
$145$	20.7279	1.72136
$146$	−14.5858	−1.20713
$147$	0	0
$148$	0	0
$149$	17.3137	1.41839	0.709197	−	0.705010i	$-0.249058\pi$
$149$	17.3137	1.41839	0.709197	+	0.705010i	$0.249058\pi$
$150$	9.41421	0.768667
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	15.3137	1.24211
$153$	0.414214	0.0334872
$154$	0	0
$155$	−10.2426	−0.822709
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	10.8284	0.861463
$159$	−6.82843	−0.541529
$160$	0	0
$161$	0	0
$162$	−1.41421	−0.111111
$163$	−9.48528	−0.742945	−0.371472	−	0.928444i	$-0.621147\pi$
$163$	−9.48528	−0.742945	−0.371472	+	0.928444i	$0.621147\pi$
$164$	0	0
$165$	8.24264	0.641689
$166$	1.51472	0.117565
$167$	3.65685	0.282976	0.141488	−	0.989940i	$-0.454811\pi$
$167$	3.65685	0.282976	0.141488	+	0.989940i	$0.454811\pi$
$168$	0	0
$169$	25.9706	1.99774
$170$	2.00000	0.153393
$171$	5.41421	0.414035
$172$	0	0
$173$	9.07107	0.689661	0.344830	−	0.938665i	$-0.387936\pi$
$173$	9.07107	0.689661	0.344830	+	0.938665i	$0.387936\pi$
$174$	−8.58579	−0.650887
$175$	0	0
$176$	−9.65685	−0.727913
$177$	−8.48528	−0.637793
$178$	−16.4853	−1.23562
$179$	−4.89949	−0.366205	−0.183103	−	0.983094i	$-0.558614\pi$
$179$	−4.89949	−0.366205	−0.183103	+	0.983094i	$0.558614\pi$
$180$	0	0
$181$	7.31371	0.543624	0.271812	−	0.962350i	$-0.412377\pi$
$181$	7.31371	0.543624	0.271812	+	0.962350i	$0.412377\pi$
$182$	0	0
$183$	−4.65685	−0.344245
$184$	−4.00000	−0.294884
$185$	32.3848	2.38098
$186$	4.24264	0.311086
$187$	1.00000	0.0731272
$188$	0	0
$189$	0	0
$190$	26.1421	1.89655
$191$	18.8284	1.36238	0.681189	−	0.732108i	$-0.261463\pi$
$191$	18.8284	1.36238	0.681189	+	0.732108i	$0.261463\pi$
$192$	−8.00000	−0.577350
$193$	0.585786	0.0421658	0.0210829	−	0.999778i	$-0.493289\pi$
$193$	0.585786	0.0421658	0.0210829	+	0.999778i	$0.493289\pi$
$194$	10.9706	0.787641
$195$	−21.3137	−1.52631
$196$	0	0
$197$	22.2426	1.58472	0.792361	−	0.610052i	$-0.208852\pi$
$197$	22.2426	1.58472	0.792361	+	0.610052i	$0.208852\pi$
$198$	−3.41421	−0.242638
$199$	16.2426	1.15141	0.575705	−	0.817657i	$-0.304728\pi$
$199$	16.2426	1.15141	0.575705	+	0.817657i	$0.304728\pi$
$200$	18.8284	1.33137
$201$	−10.4853	−0.739575
$202$	−0.585786	−0.0412158
$203$	0	0
$204$	0	0
$205$	−3.41421	−0.238459
$206$	8.92893	0.622108
$207$	−1.41421	−0.0982946
$208$	24.9706	1.73140
$209$	13.0711	0.904145
$210$	0	0
$211$	−24.4853	−1.68564	−0.842818	−	0.538198i	$-0.819105\pi$
$211$	−24.4853	−1.68564	−0.842818	+	0.538198i	$0.819105\pi$
$212$	0	0
$213$	−10.0711	−0.690058
$214$	0.485281	0.0331732
$215$	17.0711	1.16424
$216$	−2.82843	−0.192450
$217$	0	0
$218$	−11.1716	−0.756634
$219$	−10.3137	−0.696936
$220$	0	0
$221$	−2.58579	−0.173939
$222$	−13.4142	−0.900303
$223$	−10.3431	−0.692628	−0.346314	−	0.938119i	$-0.612567\pi$
$223$	−10.3431	−0.692628	−0.346314	+	0.938119i	$0.612567\pi$
$224$	0	0
$225$	6.65685	0.443790
$226$	−9.51472	−0.632910
$227$	−1.10051	−0.0730431	−0.0365215	−	0.999333i	$-0.511628\pi$
$227$	−1.10051	−0.0730431	−0.0365215	+	0.999333i	$0.511628\pi$
$228$	0	0
$229$	0.242641	0.0160341	0.00801707	−	0.999968i	$-0.497448\pi$
$229$	0.242641	0.0160341	0.00801707	+	0.999968i	$0.497448\pi$
$230$	−6.82843	−0.450253
$231$	0	0
$232$	−17.1716	−1.12737
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	8.82843	0.577132
$235$	35.5563	2.31944
$236$	0	0
$237$	7.65685	0.497366
$238$	0	0
$239$	18.1421	1.17352	0.586759	−	0.809762i	$-0.300404\pi$
$239$	18.1421	1.17352	0.586759	+	0.809762i	$0.300404\pi$
$240$	−13.6569	−0.881546
$241$	−9.48528	−0.611001	−0.305500	−	0.952192i	$-0.598824\pi$
$241$	−9.48528	−0.611001	−0.305500	+	0.952192i	$0.598824\pi$
$242$	7.31371	0.470143
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0	0
$246$	1.41421	0.0901670
$247$	−33.7990	−2.15058
$248$	8.48528	0.538816
$249$	1.07107	0.0678762
$250$	8.00000	0.505964
$251$	−8.82843	−0.557245	−0.278623	−	0.960401i	$-0.589878\pi$
$251$	−8.82843	−0.557245	−0.278623	+	0.960401i	$0.589878\pi$
$252$	0	0
$253$	−3.41421	−0.214650
$254$	−13.1716	−0.826458
$255$	1.41421	0.0885615
$256$	0	0
$257$	5.10051	0.318161	0.159080	−	0.987266i	$-0.449147\pi$
$257$	5.10051	0.318161	0.159080	+	0.987266i	$0.449147\pi$
$258$	−7.07107	−0.440225
$259$	0	0
$260$	0	0
$261$	−6.07107	−0.375790
$262$	−14.9706	−0.924884
$263$	2.41421	0.148867	0.0744334	−	0.997226i	$-0.476285\pi$
$263$	2.41421	0.148867	0.0744334	+	0.997226i	$0.476285\pi$
$264$	−6.82843	−0.420261
$265$	−23.3137	−1.43215
$266$	0	0
$267$	−11.6569	−0.713388
$268$	0	0
$269$	8.48528	0.517357	0.258678	−	0.965964i	$-0.416713\pi$
$269$	8.48528	0.517357	0.258678	+	0.965964i	$0.416713\pi$
$270$	−4.82843	−0.293849
$271$	−18.7990	−1.14196	−0.570979	−	0.820965i	$-0.693436\pi$
$271$	−18.7990	−1.14196	−0.570979	+	0.820965i	$0.693436\pi$
$272$	−1.65685	−0.100462
$273$	0	0
$274$	−5.55635	−0.335671
$275$	16.0711	0.969122
$276$	0	0
$277$	−27.2843	−1.63935	−0.819676	−	0.572827i	$-0.805847\pi$
$277$	−27.2843	−1.63935	−0.819676	+	0.572827i	$0.805847\pi$
$278$	5.17157	0.310170
$279$	3.00000	0.179605
$280$	0	0
$281$	17.3848	1.03709	0.518544	−	0.855051i	$-0.326474\pi$
$281$	17.3848	1.03709	0.518544	+	0.855051i	$0.326474\pi$
$282$	−14.7279	−0.877035
$283$	−16.1716	−0.961300	−0.480650	−	0.876912i	$-0.659599\pi$
$283$	−16.1716	−0.961300	−0.480650	+	0.876912i	$0.659599\pi$
$284$	0	0
$285$	18.4853	1.09497
$286$	21.3137	1.26031
$287$	0	0
$288$	0	0
$289$	−16.8284	−0.989907
$290$	−29.3137	−1.72136
$291$	7.75736	0.454744
$292$	0	0
$293$	−9.58579	−0.560008	−0.280004	−	0.959999i	$-0.590336\pi$
$293$	−9.58579	−0.560008	−0.280004	+	0.959999i	$0.590336\pi$
$294$	0	0
$295$	−28.9706	−1.68673
$296$	−26.8284	−1.55937
$297$	−2.41421	−0.140087
$298$	−24.4853	−1.41839
$299$	8.82843	0.510561
$300$	0	0
$301$	0	0
$302$	22.6274	1.30206
$303$	−0.414214	−0.0237959
$304$	−21.6569	−1.24211
$305$	−15.8995	−0.910402
$306$	−0.585786	−0.0334872
$307$	0.313708	0.0179043	0.00895214	−	0.999960i	$-0.497150\pi$
$307$	0.313708	0.0179043	0.00895214	+	0.999960i	$0.497150\pi$
$308$	0	0
$309$	6.31371	0.359174
$310$	14.4853	0.822709
$311$	−26.8284	−1.52130	−0.760650	−	0.649162i	$-0.775120\pi$
$311$	−26.8284	−1.52130	−0.760650	+	0.649162i	$0.775120\pi$
$312$	17.6569	0.999623
$313$	28.8701	1.63183	0.815916	−	0.578170i	$-0.196233\pi$
$313$	28.8701	1.63183	0.815916	+	0.578170i	$0.196233\pi$
$314$	25.4558	1.43656
$315$	0	0
$316$	0	0
$317$	−12.0711	−0.677979	−0.338989	−	0.940790i	$-0.610085\pi$
$317$	−12.0711	−0.677979	−0.338989	+	0.940790i	$0.610085\pi$
$318$	9.65685	0.541529
$319$	−14.6569	−0.820626
$320$	−27.3137	−1.52688
$321$	0.343146	0.0191525
$322$	0	0
$323$	2.24264	0.124784
$324$	0	0
$325$	−41.5563	−2.30513
$326$	13.4142	0.742945
$327$	−7.89949	−0.436843
$328$	2.82843	0.156174
$329$	0	0
$330$	−11.6569	−0.641689
$331$	16.2426	0.892776	0.446388	−	0.894839i	$-0.352710\pi$
$331$	16.2426	0.892776	0.446388	+	0.894839i	$0.352710\pi$
$332$	0	0
$333$	−9.48528	−0.519790
$334$	−5.17157	−0.282976
$335$	−35.7990	−1.95591
$336$	0	0
$337$	19.9706	1.08787	0.543933	−	0.839129i	$-0.316935\pi$
$337$	19.9706	1.08787	0.543933	+	0.839129i	$0.316935\pi$
$338$	−36.7279	−1.99774
$339$	−6.72792	−0.365411
$340$	0	0
$341$	7.24264	0.392211
$342$	−7.65685	−0.414035
$343$	0	0
$344$	−14.1421	−0.762493
$345$	−4.82843	−0.259954
$346$	−12.8284	−0.689661
$347$	0.899495	0.0482874	0.0241437	−	0.999708i	$-0.492314\pi$
$347$	0.899495	0.0482874	0.0241437	+	0.999708i	$0.492314\pi$
$348$	0	0
$349$	−9.14214	−0.489367	−0.244684	−	0.969603i	$-0.578684\pi$
$349$	−9.14214	−0.489367	−0.244684	+	0.969603i	$0.578684\pi$
$350$	0	0
$351$	6.24264	0.333208
$352$	0	0
$353$	34.9706	1.86130	0.930648	−	0.365917i	$-0.119244\pi$
$353$	34.9706	1.86130	0.930648	+	0.365917i	$0.119244\pi$
$354$	12.0000	0.637793
$355$	−34.3848	−1.82495
$356$	0	0
$357$	0	0
$358$	6.92893	0.366205
$359$	16.2426	0.857254	0.428627	−	0.903482i	$-0.358997\pi$
$359$	16.2426	0.857254	0.428627	+	0.903482i	$0.358997\pi$
$360$	−9.65685	−0.508961
$361$	10.3137	0.542827
$362$	−10.3431	−0.543624
$363$	5.17157	0.271437
$364$	0	0
$365$	−35.2132	−1.84314
$366$	6.58579	0.344245
$367$	6.65685	0.347485	0.173742	−	0.984791i	$-0.444414\pi$
$367$	6.65685	0.347485	0.173742	+	0.984791i	$0.444414\pi$
$368$	5.65685	0.294884
$369$	1.00000	0.0520579
$370$	−45.7990	−2.38098
$371$	0	0
$372$	0	0
$373$	33.1421	1.71603	0.858017	−	0.513621i	$-0.171696\pi$
$373$	33.1421	1.71603	0.858017	+	0.513621i	$0.171696\pi$
$374$	−1.41421	−0.0731272
$375$	5.65685	0.292119
$376$	−29.4558	−1.51907
$377$	37.8995	1.95192
$378$	0	0
$379$	26.9706	1.38538	0.692692	−	0.721233i	$-0.256424\pi$
$379$	26.9706	1.38538	0.692692	+	0.721233i	$0.256424\pi$
$380$	0	0
$381$	−9.31371	−0.477156
$382$	−26.6274	−1.36238
$383$	−17.1005	−0.873795	−0.436897	−	0.899511i	$-0.643923\pi$
$383$	−17.1005	−0.873795	−0.436897	+	0.899511i	$0.643923\pi$
$384$	11.3137	0.577350
$385$	0	0
$386$	−0.828427	−0.0421658
$387$	−5.00000	−0.254164
$388$	0	0
$389$	−22.2426	−1.12775	−0.563873	−	0.825861i	$-0.690689\pi$
$389$	−22.2426	−1.12775	−0.563873	+	0.825861i	$0.690689\pi$
$390$	30.1421	1.52631
$391$	−0.585786	−0.0296245
$392$	0	0
$393$	−10.5858	−0.533982
$394$	−31.4558	−1.58472
$395$	26.1421	1.31535
$396$	0	0
$397$	−30.5269	−1.53210	−0.766051	−	0.642780i	$-0.777781\pi$
$397$	−30.5269	−1.53210	−0.766051	+	0.642780i	$0.777781\pi$
$398$	−22.9706	−1.15141
$399$	0	0
$400$	−26.6274	−1.33137
$401$	10.6274	0.530708	0.265354	−	0.964151i	$-0.414511\pi$
$401$	10.6274	0.530708	0.265354	+	0.964151i	$0.414511\pi$
$402$	14.8284	0.739575
$403$	−18.7279	−0.932904
$404$	0	0
$405$	−3.41421	−0.169654
$406$	0	0
$407$	−22.8995	−1.13509
$408$	−1.17157	−0.0580015
$409$	−16.1716	−0.799633	−0.399816	−	0.916595i	$-0.630926\pi$
$409$	−16.1716	−0.799633	−0.399816	+	0.916595i	$0.630926\pi$
$410$	4.82843	0.238459
$411$	−3.92893	−0.193800
$412$	0	0
$413$	0	0
$414$	2.00000	0.0982946
$415$	3.65685	0.179508
$416$	0	0
$417$	3.65685	0.179077
$418$	−18.4853	−0.904145
$419$	−15.0711	−0.736270	−0.368135	−	0.929772i	$-0.620004\pi$
$419$	−15.0711	−0.736270	−0.368135	+	0.929772i	$0.620004\pi$
$420$	0	0
$421$	−8.97056	−0.437198	−0.218599	−	0.975815i	$-0.570149\pi$
$421$	−8.97056	−0.437198	−0.218599	+	0.975815i	$0.570149\pi$
$422$	34.6274	1.68564
$423$	−10.4142	−0.506356
$424$	19.3137	0.937957
$425$	2.75736	0.133752
$426$	14.2426	0.690058
$427$	0	0
$428$	0	0
$429$	15.0711	0.727638
$430$	−24.1421	−1.16424
$431$	24.9289	1.20078	0.600392	−	0.799706i	$-0.295011\pi$
$431$	24.9289	1.20078	0.600392	+	0.799706i	$0.295011\pi$
$432$	4.00000	0.192450
$433$	−5.48528	−0.263606	−0.131803	−	0.991276i	$-0.542077\pi$
$433$	−5.48528	−0.263606	−0.131803	+	0.991276i	$0.542077\pi$
$434$	0	0
$435$	−20.7279	−0.993828
$436$	0	0
$437$	−7.65685	−0.366277
$438$	14.5858	0.696936
$439$	−2.97056	−0.141777	−0.0708886	−	0.997484i	$-0.522583\pi$
$439$	−2.97056	−0.141777	−0.0708886	+	0.997484i	$0.522583\pi$
$440$	−23.3137	−1.11144
$441$	0	0
$442$	3.65685	0.173939
$443$	−36.2843	−1.72392	−0.861959	−	0.506978i	$-0.830762\pi$
$443$	−36.2843	−1.72392	−0.861959	+	0.506978i	$0.830762\pi$
$444$	0	0
$445$	−39.7990	−1.88665
$446$	14.6274	0.692628
$447$	−17.3137	−0.818910
$448$	0	0
$449$	3.51472	0.165870	0.0829349	−	0.996555i	$-0.473571\pi$
$449$	3.51472	0.165870	0.0829349	+	0.996555i	$0.473571\pi$
$450$	−9.41421	−0.443790
$451$	2.41421	0.113681
$452$	0	0
$453$	16.0000	0.751746
$454$	1.55635	0.0730431
$455$	0	0
$456$	−15.3137	−0.717130
$457$	−4.00000	−0.187112	−0.0935561	−	0.995614i	$-0.529823\pi$
$457$	−4.00000	−0.187112	−0.0935561	+	0.995614i	$0.529823\pi$
$458$	−0.343146	−0.0160341
$459$	−0.414214	−0.0193338
$460$	0	0
$461$	−28.6274	−1.33331	−0.666656	−	0.745366i	$-0.732275\pi$
$461$	−28.6274	−1.33331	−0.666656	+	0.745366i	$0.732275\pi$
$462$	0	0
$463$	−25.5563	−1.18770	−0.593852	−	0.804574i	$-0.702394\pi$
$463$	−25.5563	−1.18770	−0.593852	+	0.804574i	$0.702394\pi$
$464$	24.2843	1.12737
$465$	10.2426	0.474991
$466$	25.4558	1.17922
$467$	1.31371	0.0607912	0.0303956	−	0.999538i	$-0.490323\pi$
$467$	1.31371	0.0607912	0.0303956	+	0.999538i	$0.490323\pi$
$468$	0	0
$469$	0	0
$470$	−50.2843	−2.31944
$471$	18.0000	0.829396
$472$	24.0000	1.10469
$473$	−12.0711	−0.555028
$474$	−10.8284	−0.497366
$475$	36.0416	1.65370
$476$	0	0
$477$	6.82843	0.312652
$478$	−25.6569	−1.17352
$479$	−6.89949	−0.315246	−0.157623	−	0.987499i	$-0.550383\pi$
$479$	−6.89949	−0.315246	−0.157623	+	0.987499i	$0.550383\pi$
$480$	0	0
$481$	59.2132	2.69989
$482$	13.4142	0.611001
$483$	0	0
$484$	0	0
$485$	26.4853	1.20263
$486$	1.41421	0.0641500
$487$	1.97056	0.0892947	0.0446474	−	0.999003i	$-0.485784\pi$
$487$	1.97056	0.0892947	0.0446474	+	0.999003i	$0.485784\pi$
$488$	13.1716	0.596249
$489$	9.48528	0.428939
$490$	0	0
$491$	−13.0711	−0.589889	−0.294945	−	0.955514i	$-0.595301\pi$
$491$	−13.0711	−0.589889	−0.294945	+	0.955514i	$0.595301\pi$
$492$	0	0
$493$	−2.51472	−0.113257
$494$	47.7990	2.15058
$495$	−8.24264	−0.370479
$496$	−12.0000	−0.538816
$497$	0	0
$498$	−1.51472	−0.0678762
$499$	−19.5147	−0.873599	−0.436799	−	0.899559i	$-0.643888\pi$
$499$	−19.5147	−0.873599	−0.436799	+	0.899559i	$0.643888\pi$
$500$	0	0
$501$	−3.65685	−0.163376
$502$	12.4853	0.557245
$503$	−1.38478	−0.0617441	−0.0308721	−	0.999523i	$-0.509828\pi$
$503$	−1.38478	−0.0617441	−0.0308721	+	0.999523i	$0.509828\pi$
$504$	0	0
$505$	−1.41421	−0.0629317
$506$	4.82843	0.214650
$507$	−25.9706	−1.15339
$508$	0	0
$509$	−19.8701	−0.880725	−0.440362	−	0.897820i	$-0.645150\pi$
$509$	−19.8701	−0.880725	−0.440362	+	0.897820i	$0.645150\pi$
$510$	−2.00000	−0.0885615
$511$	0	0
$512$	22.6274	1.00000
$513$	−5.41421	−0.239043
$514$	−7.21320	−0.318161
$515$	21.5563	0.949886
$516$	0	0
$517$	−25.1421	−1.10575
$518$	0	0
$519$	−9.07107	−0.398176
$520$	60.2843	2.64364
$521$	−31.2426	−1.36876	−0.684382	−	0.729123i	$-0.739928\pi$
$521$	−31.2426	−1.36876	−0.684382	+	0.729123i	$0.739928\pi$
$522$	8.58579	0.375790
$523$	−34.0000	−1.48672	−0.743358	−	0.668894i	$-0.766768\pi$
$523$	−34.0000	−1.48672	−0.743358	+	0.668894i	$0.766768\pi$
$524$	0	0
$525$	0	0
$526$	−3.41421	−0.148867
$527$	1.24264	0.0541303
$528$	9.65685	0.420261
$529$	−21.0000	−0.913043
$530$	32.9706	1.43215
$531$	8.48528	0.368230
$532$	0	0
$533$	−6.24264	−0.270399
$534$	16.4853	0.713388
$535$	1.17157	0.0506515
$536$	29.6569	1.28098
$537$	4.89949	0.211429
$538$	−12.0000	−0.517357
$539$	0	0
$540$	0	0
$541$	−14.9706	−0.643635	−0.321817	−	0.946802i	$-0.604294\pi$
$541$	−14.9706	−0.643635	−0.321817	+	0.946802i	$0.604294\pi$
$542$	26.5858	1.14196
$543$	−7.31371	−0.313861
$544$	0	0
$545$	−26.9706	−1.15529
$546$	0	0
$547$	−24.4853	−1.04692	−0.523458	−	0.852052i	$-0.675358\pi$
$547$	−24.4853	−1.04692	−0.523458	+	0.852052i	$0.675358\pi$
$548$	0	0
$549$	4.65685	0.198750
$550$	−22.7279	−0.969122
$551$	−32.8701	−1.40031
$552$	4.00000	0.170251
$553$	0	0
$554$	38.5858	1.63935
$555$	−32.3848	−1.37466
$556$	0	0
$557$	14.2721	0.604727	0.302364	−	0.953193i	$-0.402224\pi$
$557$	14.2721	0.604727	0.302364	+	0.953193i	$0.402224\pi$
$558$	−4.24264	−0.179605
$559$	31.2132	1.32018
$560$	0	0
$561$	−1.00000	−0.0422200
$562$	−24.5858	−1.03709
$563$	−6.89949	−0.290779	−0.145390	−	0.989374i	$-0.546444\pi$
$563$	−6.89949	−0.290779	−0.145390	+	0.989374i	$0.546444\pi$
$564$	0	0
$565$	−22.9706	−0.966379
$566$	22.8701	0.961300
$567$	0	0
$568$	28.4853	1.19522
$569$	−4.44365	−0.186288	−0.0931438	−	0.995653i	$-0.529692\pi$
$569$	−4.44365	−0.186288	−0.0931438	+	0.995653i	$0.529692\pi$
$570$	−26.1421	−1.09497
$571$	−2.62742	−0.109954	−0.0549770	−	0.998488i	$-0.517509\pi$
$571$	−2.62742	−0.109954	−0.0549770	+	0.998488i	$0.517509\pi$
$572$	0	0
$573$	−18.8284	−0.786569
$574$	0	0
$575$	−9.41421	−0.392600
$576$	8.00000	0.333333
$577$	36.4853	1.51890	0.759451	−	0.650564i	$-0.225468\pi$
$577$	36.4853	1.51890	0.759451	+	0.650564i	$0.225468\pi$
$578$	23.7990	0.989907
$579$	−0.585786	−0.0243445
$580$	0	0
$581$	0	0
$582$	−10.9706	−0.454744
$583$	16.4853	0.682751
$584$	29.1716	1.20713
$585$	21.3137	0.881213
$586$	13.5563	0.560008
$587$	29.3848	1.21284	0.606420	−	0.795145i	$-0.292605\pi$
$587$	29.3848	1.21284	0.606420	+	0.795145i	$0.292605\pi$
$588$	0	0
$589$	16.2426	0.669266
$590$	40.9706	1.68673
$591$	−22.2426	−0.914940
$592$	37.9411	1.55937
$593$	−1.58579	−0.0651204	−0.0325602	−	0.999470i	$-0.510366\pi$
$593$	−1.58579	−0.0651204	−0.0325602	+	0.999470i	$0.510366\pi$
$594$	3.41421	0.140087
$595$	0	0
$596$	0	0
$597$	−16.2426	−0.664767
$598$	−12.4853	−0.510561
$599$	−35.3553	−1.44458	−0.722290	−	0.691590i	$-0.756910\pi$
$599$	−35.3553	−1.44458	−0.722290	+	0.691590i	$0.756910\pi$
$600$	−18.8284	−0.768667
$601$	26.4853	1.08036	0.540179	−	0.841550i	$-0.318357\pi$
$601$	26.4853	1.08036	0.540179	+	0.841550i	$0.318357\pi$
$602$	0	0
$603$	10.4853	0.426994
$604$	0	0
$605$	17.6569	0.717853
$606$	0.585786	0.0237959
$607$	34.4853	1.39971	0.699857	−	0.714283i	$-0.253247\pi$
$607$	34.4853	1.39971	0.699857	+	0.714283i	$0.253247\pi$
$608$	0	0
$609$	0	0
$610$	22.4853	0.910402
$611$	65.0122	2.63011
$612$	0	0
$613$	19.3137	0.780073	0.390037	−	0.920799i	$-0.372462\pi$
$613$	19.3137	0.780073	0.390037	+	0.920799i	$0.372462\pi$
$614$	−0.443651	−0.0179043
$615$	3.41421	0.137674
$616$	0	0
$617$	−15.2132	−0.612461	−0.306230	−	0.951957i	$-0.599068\pi$
$617$	−15.2132	−0.612461	−0.306230	+	0.951957i	$0.599068\pi$
$618$	−8.92893	−0.359174
$619$	10.3137	0.414543	0.207271	−	0.978283i	$-0.433542\pi$
$619$	10.3137	0.414543	0.207271	+	0.978283i	$0.433542\pi$
$620$	0	0
$621$	1.41421	0.0567504
$622$	37.9411	1.52130
$623$	0	0
$624$	−24.9706	−0.999623
$625$	−13.9706	−0.558823
$626$	−40.8284	−1.63183
$627$	−13.0711	−0.522008
$628$	0	0
$629$	−3.92893	−0.156657
$630$	0	0
$631$	−28.4558	−1.13281	−0.566405	−	0.824127i	$-0.691666\pi$
$631$	−28.4558	−1.13281	−0.566405	+	0.824127i	$0.691666\pi$
$632$	−21.6569	−0.861463
$633$	24.4853	0.973203
$634$	17.0711	0.677979
$635$	−31.7990	−1.26190
$636$	0	0
$637$	0	0
$638$	20.7279	0.820626
$639$	10.0711	0.398405
$640$	38.6274	1.52688
$641$	−8.55635	−0.337955	−0.168978	−	0.985620i	$-0.554047\pi$
$641$	−8.55635	−0.337955	−0.168978	+	0.985620i	$0.554047\pi$
$642$	−0.485281	−0.0191525
$643$	−37.4558	−1.47711	−0.738557	−	0.674191i	$-0.764493\pi$
$643$	−37.4558	−1.47711	−0.738557	+	0.674191i	$0.764493\pi$
$644$	0	0
$645$	−17.0711	−0.672173
$646$	−3.17157	−0.124784
$647$	23.5563	0.926096	0.463048	−	0.886333i	$-0.346756\pi$
$647$	23.5563	0.926096	0.463048	+	0.886333i	$0.346756\pi$
$648$	2.82843	0.111111
$649$	20.4853	0.804118
$650$	58.7696	2.30513
$651$	0	0
$652$	0	0
$653$	−6.89949	−0.269998	−0.134999	−	0.990846i	$-0.543103\pi$
$653$	−6.89949	−0.269998	−0.134999	+	0.990846i	$0.543103\pi$
$654$	11.1716	0.436843
$655$	−36.1421	−1.41219
$656$	−4.00000	−0.156174
$657$	10.3137	0.402376
$658$	0	0
$659$	−34.8284	−1.35672	−0.678361	−	0.734728i	$-0.737310\pi$
$659$	−34.8284	−1.35672	−0.678361	+	0.734728i	$0.737310\pi$
$660$	0	0
$661$	−13.4558	−0.523372	−0.261686	−	0.965153i	$-0.584278\pi$
$661$	−13.4558	−0.523372	−0.261686	+	0.965153i	$0.584278\pi$
$662$	−22.9706	−0.892776
$663$	2.58579	0.100424
$664$	−3.02944	−0.117565
$665$	0	0
$666$	13.4142	0.519790
$667$	8.58579	0.332443
$668$	0	0
$669$	10.3431	0.399889
$670$	50.6274	1.95591
$671$	11.2426	0.434017
$672$	0	0
$673$	21.7990	0.840289	0.420145	−	0.907457i	$-0.361979\pi$
$673$	21.7990	0.840289	0.420145	+	0.907457i	$0.361979\pi$
$674$	−28.2426	−1.08787
$675$	−6.65685	−0.256222
$676$	0	0
$677$	−33.5563	−1.28968	−0.644838	−	0.764320i	$-0.723075\pi$
$677$	−33.5563	−1.28968	−0.644838	+	0.764320i	$0.723075\pi$
$678$	9.51472	0.365411
$679$	0	0
$680$	−4.00000	−0.153393
$681$	1.10051	0.0421714
$682$	−10.2426	−0.392211
$683$	−6.28427	−0.240461	−0.120230	−	0.992746i	$-0.538363\pi$
$683$	−6.28427	−0.240461	−0.120230	+	0.992746i	$0.538363\pi$
$684$	0	0
$685$	−13.4142	−0.512531
$686$	0	0
$687$	−0.242641	−0.00925732
$688$	20.0000	0.762493
$689$	−42.6274	−1.62398
$690$	6.82843	0.259954
$691$	−10.7279	−0.408109	−0.204055	−	0.978959i	$-0.565412\pi$
$691$	−10.7279	−0.408109	−0.204055	+	0.978959i	$0.565412\pi$
$692$	0	0
$693$	0	0
$694$	−1.27208	−0.0482874
$695$	12.4853	0.473594
$696$	17.1716	0.650887
$697$	0.414214	0.0156895
$698$	12.9289	0.489367
$699$	18.0000	0.680823
$700$	0	0
$701$	−45.7990	−1.72980	−0.864902	−	0.501941i	$-0.832620\pi$
$701$	−45.7990	−1.72980	−0.864902	+	0.501941i	$0.832620\pi$
$702$	−8.82843	−0.333208
$703$	−51.3553	−1.93690
$704$	19.3137	0.727913
$705$	−35.5563	−1.33913
$706$	−49.4558	−1.86130
$707$	0	0
$708$	0	0
$709$	−30.4853	−1.14490	−0.572449	−	0.819940i	$-0.694007\pi$
$709$	−30.4853	−1.14490	−0.572449	+	0.819940i	$0.694007\pi$
$710$	48.6274	1.82495
$711$	−7.65685	−0.287154
$712$	32.9706	1.23562
$713$	−4.24264	−0.158888
$714$	0	0
$715$	51.4558	1.92434
$716$	0	0
$717$	−18.1421	−0.677530
$718$	−22.9706	−0.857254
$719$	−40.5563	−1.51250	−0.756248	−	0.654285i	$-0.772970\pi$
$719$	−40.5563	−1.51250	−0.756248	+	0.654285i	$0.772970\pi$
$720$	13.6569	0.508961
$721$	0	0
$722$	−14.5858	−0.542827
$723$	9.48528	0.352761
$724$	0	0
$725$	−40.4142	−1.50095
$726$	−7.31371	−0.271437
$727$	32.7279	1.21381	0.606906	−	0.794774i	$-0.292411\pi$
$727$	32.7279	1.21381	0.606906	+	0.794774i	$0.292411\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	49.7990	1.84314
$731$	−2.07107	−0.0766012
$732$	0	0
$733$	31.1421	1.15026	0.575131	−	0.818062i	$-0.304951\pi$
$733$	31.1421	1.15026	0.575131	+	0.818062i	$0.304951\pi$
$734$	−9.41421	−0.347485
$735$	0	0
$736$	0	0
$737$	25.3137	0.932442
$738$	−1.41421	−0.0520579
$739$	22.6569	0.833446	0.416723	−	0.909034i	$-0.363179\pi$
$739$	22.6569	0.833446	0.416723	+	0.909034i	$0.363179\pi$
$740$	0	0
$741$	33.7990	1.24164
$742$	0	0
$743$	16.6274	0.610001	0.305000	−	0.952352i	$-0.401343\pi$
$743$	16.6274	0.610001	0.305000	+	0.952352i	$0.401343\pi$
$744$	−8.48528	−0.311086
$745$	−59.1127	−2.16572
$746$	−46.8701	−1.71603
$747$	−1.07107	−0.0391883
$748$	0	0
$749$	0	0
$750$	−8.00000	−0.292119
$751$	34.9706	1.27609	0.638047	−	0.769997i	$-0.279743\pi$
$751$	34.9706	1.27609	0.638047	+	0.769997i	$0.279743\pi$
$752$	41.6569	1.51907
$753$	8.82843	0.321726
$754$	−53.5980	−1.95192
$755$	54.6274	1.98810
$756$	0	0
$757$	8.24264	0.299584	0.149792	−	0.988718i	$-0.452140\pi$
$757$	8.24264	0.299584	0.149792	+	0.988718i	$0.452140\pi$
$758$	−38.1421	−1.38538
$759$	3.41421	0.123928
$760$	−52.2843	−1.89655
$761$	−14.8284	−0.537530	−0.268765	−	0.963206i	$-0.586616\pi$
$761$	−14.8284	−0.537530	−0.268765	+	0.963206i	$0.586616\pi$
$762$	13.1716	0.477156
$763$	0	0
$764$	0	0
$765$	−1.41421	−0.0511310
$766$	24.1838	0.873795
$767$	−52.9706	−1.91266
$768$	0	0
$769$	33.4853	1.20751	0.603755	−	0.797170i	$-0.293670\pi$
$769$	33.4853	1.20751	0.603755	+	0.797170i	$0.293670\pi$
$770$	0	0
$771$	−5.10051	−0.183690
$772$	0	0
$773$	−50.4142	−1.81327	−0.906637	−	0.421912i	$-0.861359\pi$
$773$	−50.4142	−1.81327	−0.906637	+	0.421912i	$0.861359\pi$
$774$	7.07107	0.254164
$775$	19.9706	0.717364
$776$	−21.9411	−0.787641
$777$	0	0
$778$	31.4558	1.12775
$779$	5.41421	0.193984
$780$	0	0
$781$	24.3137	0.870013
$782$	0.828427	0.0296245
$783$	6.07107	0.216962
$784$	0	0
$785$	61.4558	2.19345
$786$	14.9706	0.533982
$787$	13.9706	0.497997	0.248998	−	0.968504i	$-0.419899\pi$
$787$	13.9706	0.497997	0.248998	+	0.968504i	$0.419899\pi$
$788$	0	0
$789$	−2.41421	−0.0859483
$790$	−36.9706	−1.31535
$791$	0	0
$792$	6.82843	0.242638
$793$	−29.0711	−1.03234
$794$	43.1716	1.53210
$795$	23.3137	0.826852
$796$	0	0
$797$	44.1421	1.56359	0.781797	−	0.623533i	$-0.214303\pi$
$797$	44.1421	1.56359	0.781797	+	0.623533i	$0.214303\pi$
$798$	0	0
$799$	−4.31371	−0.152608
$800$	0	0
$801$	11.6569	0.411875
$802$	−15.0294	−0.530708
$803$	24.8995	0.878684
$804$	0	0
$805$	0	0
$806$	26.4853	0.932904
$807$	−8.48528	−0.298696
$808$	1.17157	0.0412158
$809$	25.1127	0.882915	0.441458	−	0.897282i	$-0.354462\pi$
$809$	25.1127	0.882915	0.441458	+	0.897282i	$0.354462\pi$
$810$	4.82843	0.169654
$811$	17.9706	0.631032	0.315516	−	0.948920i	$-0.397822\pi$
$811$	17.9706	0.631032	0.315516	+	0.948920i	$0.397822\pi$
$812$	0	0
$813$	18.7990	0.659309
$814$	32.3848	1.13509
$815$	32.3848	1.13439
$816$	1.65685	0.0580015
$817$	−27.0711	−0.947097
$818$	22.8701	0.799633
$819$	0	0
$820$	0	0
$821$	−38.8701	−1.35657	−0.678287	−	0.734797i	$-0.737277\pi$
$821$	−38.8701	−1.35657	−0.678287	+	0.734797i	$0.737277\pi$
$822$	5.55635	0.193800
$823$	32.4264	1.13031	0.565157	−	0.824984i	$-0.308816\pi$
$823$	32.4264	1.13031	0.565157	+	0.824984i	$0.308816\pi$
$824$	−17.8579	−0.622108
$825$	−16.0711	−0.559523
$826$	0	0
$827$	7.92893	0.275716	0.137858	−	0.990452i	$-0.455978\pi$
$827$	7.92893	0.275716	0.137858	+	0.990452i	$0.455978\pi$
$828$	0	0
$829$	24.9411	0.866241	0.433121	−	0.901336i	$-0.357412\pi$
$829$	24.9411	0.866241	0.433121	+	0.901336i	$0.357412\pi$
$830$	−5.17157	−0.179508
$831$	27.2843	0.946481
$832$	−49.9411	−1.73140
$833$	0	0
$834$	−5.17157	−0.179077
$835$	−12.4853	−0.432071
$836$	0	0
$837$	−3.00000	−0.103695
$838$	21.3137	0.736270
$839$	22.8284	0.788125	0.394062	−	0.919084i	$-0.371069\pi$
$839$	22.8284	0.788125	0.394062	+	0.919084i	$0.371069\pi$
$840$	0	0
$841$	7.85786	0.270961
$842$	12.6863	0.437198
$843$	−17.3848	−0.598764
$844$	0	0
$845$	−88.6690	−3.05031
$846$	14.7279	0.506356
$847$	0	0
$848$	−27.3137	−0.937957
$849$	16.1716	0.555007
$850$	−3.89949	−0.133752
$851$	13.4142	0.459833
$852$	0	0
$853$	0	0		−	1.00000i	$-0.5\pi$
$853$	0	0			1.00000i	$0.5\pi$
$854$	0	0
$855$	−18.4853	−0.632183
$856$	−0.970563	−0.0331732
$857$	−51.7990	−1.76942	−0.884710	−	0.466142i	$-0.845644\pi$
$857$	−51.7990	−1.76942	−0.884710	+	0.466142i	$0.845644\pi$
$858$	−21.3137	−0.727638
$859$	−16.6569	−0.568325	−0.284162	−	0.958776i	$-0.591715\pi$
$859$	−16.6569	−0.568325	−0.284162	+	0.958776i	$0.591715\pi$
$860$	0	0
$861$	0	0
$862$	−35.2548	−1.20078
$863$	25.0711	0.853429	0.426715	−	0.904386i	$-0.359671\pi$
$863$	25.0711	0.853429	0.426715	+	0.904386i	$0.359671\pi$
$864$	0	0
$865$	−30.9706	−1.05303
$866$	7.75736	0.263606
$867$	16.8284	0.571523
$868$	0	0
$869$	−18.4853	−0.627070
$870$	29.3137	0.993828
$871$	−65.4558	−2.21789
$872$	22.3431	0.756634
$873$	−7.75736	−0.262547
$874$	10.8284	0.366277
$875$	0	0
$876$	0	0
$877$	−27.1421	−0.916525	−0.458262	−	0.888817i	$-0.651528\pi$
$877$	−27.1421	−0.916525	−0.458262	+	0.888817i	$0.651528\pi$
$878$	4.20101	0.141777
$879$	9.58579	0.323321
$880$	32.9706	1.11144
$881$	42.6274	1.43615	0.718077	−	0.695964i	$-0.245023\pi$
$881$	42.6274	1.43615	0.718077	+	0.695964i	$0.245023\pi$
$882$	0	0
$883$	−9.41421	−0.316814	−0.158407	−	0.987374i	$-0.550636\pi$
$883$	−9.41421	−0.316814	−0.158407	+	0.987374i	$0.550636\pi$
$884$	0	0
$885$	28.9706	0.973835
$886$	51.3137	1.72392
$887$	−36.0122	−1.20917	−0.604586	−	0.796540i	$-0.706661\pi$
$887$	−36.0122	−1.20917	−0.604586	+	0.796540i	$0.706661\pi$
$888$	26.8284	0.900303
$889$	0	0
$890$	56.2843	1.88665
$891$	2.41421	0.0808792
$892$	0	0
$893$	−56.3848	−1.88684
$894$	24.4853	0.818910
$895$	16.7279	0.559153
$896$	0	0
$897$	−8.82843	−0.294773
$898$	−4.97056	−0.165870
$899$	−18.2132	−0.607444
$900$	0	0
$901$	2.82843	0.0942286
$902$	−3.41421	−0.113681
$903$	0	0
$904$	19.0294	0.632910
$905$	−24.9706	−0.830050
$906$	−22.6274	−0.751746
$907$	15.6569	0.519877	0.259939	−	0.965625i	$-0.416298\pi$
$907$	15.6569	0.519877	0.259939	+	0.965625i	$0.416298\pi$
$908$	0	0
$909$	0.414214	0.0137386
$910$	0	0
$911$	34.1005	1.12980	0.564900	−	0.825159i	$-0.308915\pi$
$911$	34.1005	1.12980	0.564900	+	0.825159i	$0.308915\pi$
$912$	21.6569	0.717130
$913$	−2.58579	−0.0855770
$914$	5.65685	0.187112
$915$	15.8995	0.525621
$916$	0	0
$917$	0	0
$918$	0.585786	0.0193338
$919$	−15.4142	−0.508468	−0.254234	−	0.967143i	$-0.581823\pi$
$919$	−15.4142	−0.508468	−0.254234	+	0.967143i	$0.581823\pi$
$920$	13.6569	0.450253
$921$	−0.313708	−0.0103370
$922$	40.4853	1.33331
$923$	−62.8701	−2.06939
$924$	0	0
$925$	−63.1421	−2.07610
$926$	36.1421	1.18770
$927$	−6.31371	−0.207369
$928$	0	0
$929$	7.87006	0.258208	0.129104	−	0.991631i	$-0.458790\pi$
$929$	7.87006	0.258208	0.129104	+	0.991631i	$0.458790\pi$
$930$	−14.4853	−0.474991
$931$	0	0
$932$	0	0
$933$	26.8284	0.878323
$934$	−1.85786	−0.0607912
$935$	−3.41421	−0.111657
$936$	−17.6569	−0.577132
$937$	−52.9706	−1.73047	−0.865236	−	0.501364i	$-0.832832\pi$
$937$	−52.9706	−1.73047	−0.865236	+	0.501364i	$0.832832\pi$
$938$	0	0
$939$	−28.8701	−0.942139
$940$	0	0
$941$	52.6274	1.71560	0.857802	−	0.513980i	$-0.171829\pi$
$941$	52.6274	1.71560	0.857802	+	0.513980i	$0.171829\pi$
$942$	−25.4558	−0.829396
$943$	−1.41421	−0.0460531
$944$	−33.9411	−1.10469
$945$	0	0
$946$	17.0711	0.555028
$947$	21.5980	0.701840	0.350920	−	0.936405i	$-0.385869\pi$
$947$	21.5980	0.701840	0.350920	+	0.936405i	$0.385869\pi$
$948$	0	0
$949$	−64.3848	−2.09002
$950$	−50.9706	−1.65370
$951$	12.0711	0.391431
$952$	0	0
$953$	−41.6985	−1.35075	−0.675373	−	0.737476i	$-0.736017\pi$
$953$	−41.6985	−1.35075	−0.675373	+	0.737476i	$0.736017\pi$
$954$	−9.65685	−0.312652
$955$	−64.2843	−2.08019
$956$	0	0
$957$	14.6569	0.473789
$958$	9.75736	0.315246
$959$	0	0
$960$	27.3137	0.881546
$961$	−22.0000	−0.709677
$962$	−83.7401	−2.69989
$963$	−0.343146	−0.0110577
$964$	0	0
$965$	−2.00000	−0.0643823
$966$	0	0
$967$	−1.02944	−0.0331045	−0.0165522	−	0.999863i	$-0.505269\pi$
$967$	−1.02944	−0.0331045	−0.0165522	+	0.999863i	$0.505269\pi$
$968$	−14.6274	−0.470143
$969$	−2.24264	−0.0720440
$970$	−37.4558	−1.20263
$971$	−40.5563	−1.30152	−0.650758	−	0.759286i	$-0.725549\pi$
$971$	−40.5563	−1.30152	−0.650758	+	0.759286i	$0.725549\pi$
$972$	0	0
$973$	0	0
$974$	−2.78680	−0.0892947
$975$	41.5563	1.33087
$976$	−18.6274	−0.596249
$977$	7.58579	0.242691	0.121345	−	0.992610i	$-0.461279\pi$
$977$	7.58579	0.242691	0.121345	+	0.992610i	$0.461279\pi$
$978$	−13.4142	−0.428939
$979$	28.1421	0.899427
$980$	0	0
$981$	7.89949	0.252211
$982$	18.4853	0.589889
$983$	22.9289	0.731319	0.365660	−	0.930749i	$-0.380843\pi$
$983$	22.9289	0.731319	0.365660	+	0.930749i	$0.380843\pi$
$984$	−2.82843	−0.0901670
$985$	−75.9411	−2.41969
$986$	3.55635	0.113257
$987$	0	0
$988$	0	0
$989$	7.07107	0.224847
$990$	11.6569	0.370479
$991$	−28.4853	−0.904865	−0.452432	−	0.891799i	$-0.649444\pi$
$991$	−28.4853	−0.904865	−0.452432	+	0.891799i	$0.649444\pi$
$992$	0	0
$993$	−16.2426	−0.515445
$994$	0	0
$995$	−55.4558	−1.75807
$996$	0	0
$997$	−15.4142	−0.488173	−0.244087	−	0.969753i	$-0.578488\pi$
$997$	−15.4142	−0.488173	−0.244087	+	0.969753i	$0.578488\pi$
$998$	27.5980	0.873599
$999$	9.48528	0.300101

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6027.2.a.m.1.1		2
7.6	odd	2		123.2.a.c.1.1	✓	2
21.20	even	2		369.2.a.c.1.2		2
28.27	even	2		1968.2.a.r.1.2		2
35.34	odd	2		3075.2.a.p.1.2		2
56.13	odd	2		7872.2.a.bk.1.1		2
56.27	even	2		7872.2.a.bo.1.1		2
84.83	odd	2		5904.2.a.w.1.1		2
105.104	even	2		9225.2.a.bm.1.1		2
287.286	odd	2		5043.2.a.e.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
123.2.a.c.1.1	✓	2	7.6	odd	2
369.2.a.c.1.2		2	21.20	even	2
1968.2.a.r.1.2		2	28.27	even	2
3075.2.a.p.1.2		2	35.34	odd	2
5043.2.a.e.1.1		2	287.286	odd	2
5904.2.a.w.1.1		2	84.83	odd	2
6027.2.a.m.1.1		2	1.1	even	1	trivial
7872.2.a.bk.1.1		2	56.13	odd	2
7872.2.a.bo.1.1		2	56.27	even	2
9225.2.a.bm.1.1		2	105.104	even	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:	\( N \)	\(=\)	\( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6027.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} -1.00000 q^{3} -3.41421 q^{5} +1.41421 q^{6} +2.82843 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{2} -1.00000 q^{3} -3.41421 q^{5} +1.41421 q^{6} +2.82843 q^{8} +1.00000 q^{9} +4.82843 q^{10} +2.41421 q^{11} -6.24264 q^{13} +3.41421 q^{15} -4.00000 q^{16} +0.414214 q^{17} -1.41421 q^{18} +5.41421 q^{19} -3.41421 q^{22} -1.41421 q^{23} -2.82843 q^{24} +6.65685 q^{25} +8.82843 q^{26} -1.00000 q^{27} -6.07107 q^{29} -4.82843 q^{30} +3.00000 q^{31} -2.41421 q^{33} -0.585786 q^{34} -9.48528 q^{37} -7.65685 q^{38} +6.24264 q^{39} -9.65685 q^{40} +1.00000 q^{41} -5.00000 q^{43} -3.41421 q^{45} +2.00000 q^{46} -10.4142 q^{47} +4.00000 q^{48} -9.41421 q^{50} -0.414214 q^{51} +6.82843 q^{53} +1.41421 q^{54} -8.24264 q^{55} -5.41421 q^{57} +8.58579 q^{58} +8.48528 q^{59} +4.65685 q^{61} -4.24264 q^{62} +8.00000 q^{64} +21.3137 q^{65} +3.41421 q^{66} +10.4853 q^{67} +1.41421 q^{69} +10.0711 q^{71} +2.82843 q^{72} +10.3137 q^{73} +13.4142 q^{74} -6.65685 q^{75} -8.82843 q^{78} -7.65685 q^{79} +13.6569 q^{80} +1.00000 q^{81} -1.41421 q^{82} -1.07107 q^{83} -1.41421 q^{85} +7.07107 q^{86} +6.07107 q^{87} +6.82843 q^{88} +11.6569 q^{89} +4.82843 q^{90} -3.00000 q^{93} +14.7279 q^{94} -18.4853 q^{95} -7.75736 q^{97} +2.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9	\( 2 q - 2 q^{3} - 4 q^{5} + 2 q^{9} + 4 q^{10} + 2 q^{11} - 4 q^{13} + 4 q^{15} - 8 q^{16} - 2 q^{17} + 8 q^{19} - 4 q^{22} + 2 q^{25} + 12 q^{26} - 2 q^{27} + 2 q^{29} - 4 q^{30} + 6 q^{31} - 2 q^{33} - 4 q^{34} - 2 q^{37} - 4 q^{38} + 4 q^{39} - 8 q^{40} + 2 q^{41} - 10 q^{43} - 4 q^{45} + 4 q^{46} - 18 q^{47} + 8 q^{48} - 16 q^{50} + 2 q^{51} + 8 q^{53} - 8 q^{55} - 8 q^{57} + 20 q^{58} - 2 q^{61} + 16 q^{64} + 20 q^{65} + 4 q^{66} + 4 q^{67} + 6 q^{71} - 2 q^{73} + 24 q^{74} - 2 q^{75} - 12 q^{78} - 4 q^{79} + 16 q^{80} + 2 q^{81} + 12 q^{83} - 2 q^{87} + 8 q^{88} + 12 q^{89} + 4 q^{90} - 6 q^{93} + 4 q^{94} - 20 q^{95} - 24 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 4 * q^5 + 2 * q^9 + 4 * q^10 + 2 * q^11 - 4 * q^13 + 4 * q^15 - 8 * q^16 - 2 * q^17 + 8 * q^19 - 4 * q^22 + 2 * q^25 + 12 * q^26 - 2 * q^27 + 2 * q^29 - 4 * q^30 + 6 * q^31 - 2 * q^33 - 4 * q^34 - 2 * q^37 - 4 * q^38 + 4 * q^39 - 8 * q^40 + 2 * q^41 - 10 * q^43 - 4 * q^45 + 4 * q^46 - 18 * q^47 + 8 * q^48 - 16 * q^50 + 2 * q^51 + 8 * q^53 - 8 * q^55 - 8 * q^57 + 20 * q^58 - 2 * q^61 + 16 * q^64 + 20 * q^65 + 4 * q^66 + 4 * q^67 + 6 * q^71 - 2 * q^73 + 24 * q^74 - 2 * q^75 - 12 * q^78 - 4 * q^79 + 16 * q^80 + 2 * q^81 + 12 * q^83 - 2 * q^87 + 8 * q^88 + 12 * q^89 + 4 * q^90 - 6 * q^93 + 4 * q^94 - 20 * q^95 - 24 * q^97 + 2 * q^99

Introduction

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