LMFDB - Embedded newform 6762.2.a.bz.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6762,2,Mod(1,6762)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6762, 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("6762.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6762.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:	$53.9948418468$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 966)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	6762.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{12} -4.47214 q^{13} +2.00000 q^{15} +1.00000 q^{16} +4.47214 q^{17} +1.00000 q^{18} +6.47214 q^{19} -2.00000 q^{20} -1.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} -4.47214 q^{26} -1.00000 q^{27} -2.00000 q^{29} +2.00000 q^{30} -6.47214 q^{31} +1.00000 q^{32} +4.47214 q^{34} +1.00000 q^{36} -10.9443 q^{37} +6.47214 q^{38} +4.47214 q^{39} -2.00000 q^{40} +6.00000 q^{41} +12.9443 q^{43} -2.00000 q^{45} -1.00000 q^{46} -6.47214 q^{47} -1.00000 q^{48} -1.00000 q^{50} -4.47214 q^{51} -4.47214 q^{52} +6.94427 q^{53} -1.00000 q^{54} -6.47214 q^{57} -2.00000 q^{58} -4.00000 q^{59} +2.00000 q^{60} +6.00000 q^{61} -6.47214 q^{62} +1.00000 q^{64} +8.94427 q^{65} -12.9443 q^{67} +4.47214 q^{68} +1.00000 q^{69} -12.9443 q^{71} +1.00000 q^{72} +2.94427 q^{73} -10.9443 q^{74} +1.00000 q^{75} +6.47214 q^{76} +4.47214 q^{78} +12.9443 q^{79} -2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +6.47214 q^{83} -8.94427 q^{85} +12.9443 q^{86} +2.00000 q^{87} +17.4164 q^{89} -2.00000 q^{90} -1.00000 q^{92} +6.47214 q^{93} -6.47214 q^{94} -12.9443 q^{95} -1.00000 q^{96} -8.47214 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 4 * q^5 - 2 * q^6 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9} - 4 q^{10} - 2 q^{12} + 4 q^{15} + 2 q^{16} + 2 q^{18} + 4 q^{19} - 4 q^{20} - 2 q^{23} - 2 q^{24} - 2 q^{25} - 2 q^{27} - 4 q^{29} + 4 q^{30} - 4 q^{31} + 2 q^{32} + 2 q^{36} - 4 q^{37} + 4 q^{38} - 4 q^{40} + 12 q^{41} + 8 q^{43} - 4 q^{45} - 2 q^{46} - 4 q^{47} - 2 q^{48} - 2 q^{50} - 4 q^{53} - 2 q^{54} - 4 q^{57} - 4 q^{58} - 8 q^{59} + 4 q^{60} + 12 q^{61} - 4 q^{62} + 2 q^{64} - 8 q^{67} + 2 q^{69} - 8 q^{71} + 2 q^{72} - 12 q^{73} - 4 q^{74} + 2 q^{75} + 4 q^{76} + 8 q^{79} - 4 q^{80} + 2 q^{81} + 12 q^{82} + 4 q^{83} + 8 q^{86} + 4 q^{87} + 8 q^{89} - 4 q^{90} - 2 q^{92} + 4 q^{93} - 4 q^{94} - 8 q^{95} - 2 q^{96} - 8 q^{97}+O(q^{100}) $ 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 4 * q^5 - 2 * q^6 + 2 * q^8 + 2 * q^9 - 4 * q^10 - 2 * q^12 + 4 * q^15 + 2 * q^16 + 2 * q^18 + 4 * q^19 - 4 * q^20 - 2 * q^23 - 2 * q^24 - 2 * q^25 - 2 * q^27 - 4 * q^29 + 4 * q^30 - 4 * q^31 + 2 * q^32 + 2 * q^36 - 4 * q^37 + 4 * q^38 - 4 * q^40 + 12 * q^41 + 8 * q^43 - 4 * q^45 - 2 * q^46 - 4 * q^47 - 2 * q^48 - 2 * q^50 - 4 * q^53 - 2 * q^54 - 4 * q^57 - 4 * q^58 - 8 * q^59 + 4 * q^60 + 12 * q^61 - 4 * q^62 + 2 * q^64 - 8 * q^67 + 2 * q^69 - 8 * q^71 + 2 * q^72 - 12 * q^73 - 4 * q^74 + 2 * q^75 + 4 * q^76 + 8 * q^79 - 4 * q^80 + 2 * q^81 + 12 * q^82 + 4 * q^83 + 8 * q^86 + 4 * q^87 + 8 * q^89 - 4 * q^90 - 2 * q^92 + 4 * q^93 - 4 * q^94 - 8 * q^95 - 2 * q^96 - 8 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	1.00000	0.500000
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−2.00000	−0.632456
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	−1.00000	−0.288675
$13$	−4.47214	−1.24035	−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214	−1.24035	−0.620174	+	0.784465i	$0.712938\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	4.47214	1.08465	0.542326	−	0.840168i	$-0.317544\pi$
$17$	4.47214	1.08465	0.542326	+	0.840168i	$0.317544\pi$
$18$	1.00000	0.235702
$19$	6.47214	1.48481	0.742405	−	0.669951i	$-0.233685\pi$
$19$	6.47214	1.48481	0.742405	+	0.669951i	$0.233685\pi$
$20$	−2.00000	−0.447214
$21$	0	0
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	−1.00000	−0.200000
$26$	−4.47214	−0.877058
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	2.00000	0.365148
$31$	−6.47214	−1.16243	−0.581215	−	0.813750i	$-0.697422\pi$
$31$	−6.47214	−1.16243	−0.581215	+	0.813750i	$0.697422\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	4.47214	0.766965
$35$	0	0
$36$	1.00000	0.166667
$37$	−10.9443	−1.79923	−0.899614	−	0.436687i	$-0.856152\pi$
$37$	−10.9443	−1.79923	−0.899614	+	0.436687i	$0.856152\pi$
$38$	6.47214	1.04992
$39$	4.47214	0.716115
$40$	−2.00000	−0.316228
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	12.9443	1.97398	0.986991	−	0.160773i	$-0.0513986\pi$
$43$	12.9443	1.97398	0.986991	+	0.160773i	$0.0513986\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	−1.00000	−0.147442
$47$	−6.47214	−0.944058	−0.472029	−	0.881583i	$-0.656478\pi$
$47$	−6.47214	−0.944058	−0.472029	+	0.881583i	$0.656478\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	−1.00000	−0.141421
$51$	−4.47214	−0.626224
$52$	−4.47214	−0.620174
$53$	6.94427	0.953869	0.476935	−	0.878939i	$-0.341748\pi$
$53$	6.94427	0.953869	0.476935	+	0.878939i	$0.341748\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	−6.47214	−0.857255
$58$	−2.00000	−0.262613
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	2.00000	0.258199
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	−6.47214	−0.821962
$63$	0	0
$64$	1.00000	0.125000
$65$	8.94427	1.10940
$66$	0	0
$67$	−12.9443	−1.58139	−0.790697	−	0.612207i	$-0.790282\pi$
$67$	−12.9443	−1.58139	−0.790697	+	0.612207i	$0.790282\pi$
$68$	4.47214	0.542326
$69$	1.00000	0.120386
$70$	0	0
$71$	−12.9443	−1.53620	−0.768101	−	0.640328i	$-0.778798\pi$
$71$	−12.9443	−1.53620	−0.768101	+	0.640328i	$0.778798\pi$
$72$	1.00000	0.117851
$73$	2.94427	0.344601	0.172300	−	0.985044i	$-0.444880\pi$
$73$	2.94427	0.344601	0.172300	+	0.985044i	$0.444880\pi$
$74$	−10.9443	−1.27225
$75$	1.00000	0.115470
$76$	6.47214	0.742405
$77$	0	0
$78$	4.47214	0.506370
$79$	12.9443	1.45634	0.728172	−	0.685394i	$-0.240370\pi$
$79$	12.9443	1.45634	0.728172	+	0.685394i	$0.240370\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	6.47214	0.710409	0.355205	−	0.934789i	$-0.384411\pi$
$83$	6.47214	0.710409	0.355205	+	0.934789i	$0.384411\pi$
$84$	0	0
$85$	−8.94427	−0.970143
$86$	12.9443	1.39582
$87$	2.00000	0.214423
$88$	0	0
$89$	17.4164	1.84614	0.923068	−	0.384637i	$-0.125673\pi$
$89$	17.4164	1.84614	0.923068	+	0.384637i	$0.125673\pi$
$90$	−2.00000	−0.210819
$91$	0	0
$92$	−1.00000	−0.104257
$93$	6.47214	0.671129
$94$	−6.47214	−0.667550
$95$	−12.9443	−1.32805
$96$	−1.00000	−0.102062
$97$	−8.47214	−0.860215	−0.430108	−	0.902778i	$-0.641524\pi$
$97$	−8.47214	−0.860215	−0.430108	+	0.902778i	$0.641524\pi$
$98$	0	0
$99$	0	0
$100$	−1.00000	−0.100000
$101$	−17.4164	−1.73300	−0.866499	−	0.499179i	$-0.833635\pi$
$101$	−17.4164	−1.73300	−0.866499	+	0.499179i	$0.833635\pi$
$102$	−4.47214	−0.442807
$103$	−12.9443	−1.27544	−0.637719	−	0.770270i	$-0.720122\pi$
$103$	−12.9443	−1.27544	−0.637719	+	0.770270i	$0.720122\pi$
$104$	−4.47214	−0.438529
$105$	0	0
$106$	6.94427	0.674487
$107$	−8.00000	−0.773389	−0.386695	−	0.922208i	$-0.626383\pi$
$107$	−8.00000	−0.773389	−0.386695	+	0.922208i	$0.626383\pi$
$108$	−1.00000	−0.0962250
$109$	14.9443	1.43140	0.715701	−	0.698407i	$-0.246107\pi$
$109$	14.9443	1.43140	0.715701	+	0.698407i	$0.246107\pi$
$110$	0	0
$111$	10.9443	1.03878
$112$	0	0
$113$	−1.05573	−0.0993145	−0.0496573	−	0.998766i	$-0.515813\pi$
$113$	−1.05573	−0.0993145	−0.0496573	+	0.998766i	$0.515813\pi$
$114$	−6.47214	−0.606171
$115$	2.00000	0.186501
$116$	−2.00000	−0.185695
$117$	−4.47214	−0.413449
$118$	−4.00000	−0.368230
$119$	0	0
$120$	2.00000	0.182574
$121$	−11.0000	−1.00000
$122$	6.00000	0.543214
$123$	−6.00000	−0.541002
$124$	−6.47214	−0.581215
$125$	12.0000	1.07331
$126$	0	0
$127$	3.05573	0.271152	0.135576	−	0.990767i	$-0.456712\pi$
$127$	3.05573	0.271152	0.135576	+	0.990767i	$0.456712\pi$
$128$	1.00000	0.0883883
$129$	−12.9443	−1.13968
$130$	8.94427	0.784465
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	0	0
$133$	0	0
$134$	−12.9443	−1.11821
$135$	2.00000	0.172133
$136$	4.47214	0.383482
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	1.00000	0.0851257
$139$	8.94427	0.758643	0.379322	−	0.925265i	$-0.376157\pi$
$139$	8.94427	0.758643	0.379322	+	0.925265i	$0.376157\pi$
$140$	0	0
$141$	6.47214	0.545052
$142$	−12.9443	−1.08626
$143$	0	0
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	2.94427	0.243670
$147$	0	0
$148$	−10.9443	−0.899614
$149$	−18.9443	−1.55198	−0.775988	−	0.630748i	$-0.782748\pi$
$149$	−18.9443	−1.55198	−0.775988	+	0.630748i	$0.782748\pi$
$150$	1.00000	0.0816497
$151$	−12.9443	−1.05339	−0.526695	−	0.850054i	$-0.676569\pi$
$151$	−12.9443	−1.05339	−0.526695	+	0.850054i	$0.676569\pi$
$152$	6.47214	0.524960
$153$	4.47214	0.361551
$154$	0	0
$155$	12.9443	1.03971
$156$	4.47214	0.358057
$157$	−14.9443	−1.19268	−0.596341	−	0.802731i	$-0.703380\pi$
$157$	−14.9443	−1.19268	−0.596341	+	0.802731i	$0.703380\pi$
$158$	12.9443	1.02979
$159$	−6.94427	−0.550717
$160$	−2.00000	−0.158114
$161$	0	0
$162$	1.00000	0.0785674
$163$	8.94427	0.700569	0.350285	−	0.936643i	$-0.386085\pi$
$163$	8.94427	0.700569	0.350285	+	0.936643i	$0.386085\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	6.47214	0.502335
$167$	−11.4164	−0.883428	−0.441714	−	0.897156i	$-0.645629\pi$
$167$	−11.4164	−0.883428	−0.441714	+	0.897156i	$0.645629\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	−8.94427	−0.685994
$171$	6.47214	0.494937
$172$	12.9443	0.986991
$173$	8.47214	0.644125	0.322062	−	0.946718i	$-0.395624\pi$
$173$	8.47214	0.644125	0.322062	+	0.946718i	$0.395624\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	0	0
$177$	4.00000	0.300658
$178$	17.4164	1.30541
$179$	−20.0000	−1.49487	−0.747435	−	0.664335i	$-0.768715\pi$
$179$	−20.0000	−1.49487	−0.747435	+	0.664335i	$0.768715\pi$
$180$	−2.00000	−0.149071
$181$	−2.00000	−0.148659	−0.0743294	−	0.997234i	$-0.523682\pi$
$181$	−2.00000	−0.148659	−0.0743294	+	0.997234i	$0.523682\pi$
$182$	0	0
$183$	−6.00000	−0.443533
$184$	−1.00000	−0.0737210
$185$	21.8885	1.60928
$186$	6.47214	0.474560
$187$	0	0
$188$	−6.47214	−0.472029
$189$	0	0
$190$	−12.9443	−0.939076
$191$	−20.9443	−1.51547	−0.757737	−	0.652560i	$-0.773695\pi$
$191$	−20.9443	−1.51547	−0.757737	+	0.652560i	$0.773695\pi$
$192$	−1.00000	−0.0721688
$193$	−23.8885	−1.71954	−0.859768	−	0.510686i	$-0.829392\pi$
$193$	−23.8885	−1.71954	−0.859768	+	0.510686i	$0.829392\pi$
$194$	−8.47214	−0.608264
$195$	−8.94427	−0.640513
$196$	0	0
$197$	2.94427	0.209771	0.104885	−	0.994484i	$-0.466552\pi$
$197$	2.94427	0.209771	0.104885	+	0.994484i	$0.466552\pi$
$198$	0	0
$199$	−20.9443	−1.48470	−0.742350	−	0.670012i	$-0.766289\pi$
$199$	−20.9443	−1.48470	−0.742350	+	0.670012i	$0.766289\pi$
$200$	−1.00000	−0.0707107
$201$	12.9443	0.913019
$202$	−17.4164	−1.22541
$203$	0	0
$204$	−4.47214	−0.313112
$205$	−12.0000	−0.838116
$206$	−12.9443	−0.901870
$207$	−1.00000	−0.0695048
$208$	−4.47214	−0.310087
$209$	0	0
$210$	0	0
$211$	16.9443	1.16649	0.583246	−	0.812296i	$-0.301782\pi$
$211$	16.9443	1.16649	0.583246	+	0.812296i	$0.301782\pi$
$212$	6.94427	0.476935
$213$	12.9443	0.886927
$214$	−8.00000	−0.546869
$215$	−25.8885	−1.76558
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	14.9443	1.01215
$219$	−2.94427	−0.198955
$220$	0	0
$221$	−20.0000	−1.34535
$222$	10.9443	0.734531
$223$	−9.52786	−0.638033	−0.319016	−	0.947749i	$-0.603353\pi$
$223$	−9.52786	−0.638033	−0.319016	+	0.947749i	$0.603353\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	−1.05573	−0.0702260
$227$	−14.4721	−0.960549	−0.480275	−	0.877118i	$-0.659463\pi$
$227$	−14.4721	−0.960549	−0.480275	+	0.877118i	$0.659463\pi$
$228$	−6.47214	−0.428628
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	2.00000	0.131876
$231$	0	0
$232$	−2.00000	−0.131306
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	−4.47214	−0.292353
$235$	12.9443	0.844391
$236$	−4.00000	−0.260378
$237$	−12.9443	−0.840821
$238$	0	0
$239$	3.05573	0.197659	0.0988293	−	0.995104i	$-0.468490\pi$
$239$	3.05573	0.197659	0.0988293	+	0.995104i	$0.468490\pi$
$240$	2.00000	0.129099
$241$	−11.5279	−0.742575	−0.371288	−	0.928518i	$-0.621084\pi$
$241$	−11.5279	−0.742575	−0.371288	+	0.928518i	$0.621084\pi$
$242$	−11.0000	−0.707107
$243$	−1.00000	−0.0641500
$244$	6.00000	0.384111
$245$	0	0
$246$	−6.00000	−0.382546
$247$	−28.9443	−1.84168
$248$	−6.47214	−0.410981
$249$	−6.47214	−0.410155
$250$	12.0000	0.758947
$251$	14.4721	0.913473	0.456737	−	0.889602i	$-0.349018\pi$
$251$	14.4721	0.913473	0.456737	+	0.889602i	$0.349018\pi$
$252$	0	0
$253$	0	0
$254$	3.05573	0.191733
$255$	8.94427	0.560112
$256$	1.00000	0.0625000
$257$	−14.9443	−0.932198	−0.466099	−	0.884733i	$-0.654341\pi$
$257$	−14.9443	−0.932198	−0.466099	+	0.884733i	$0.654341\pi$
$258$	−12.9443	−0.805875
$259$	0	0
$260$	8.94427	0.554700
$261$	−2.00000	−0.123797
$262$	−12.0000	−0.741362
$263$	−24.0000	−1.47990	−0.739952	−	0.672660i	$-0.765152\pi$
$263$	−24.0000	−1.47990	−0.739952	+	0.672660i	$0.765152\pi$
$264$	0	0
$265$	−13.8885	−0.853166
$266$	0	0
$267$	−17.4164	−1.06587
$268$	−12.9443	−0.790697
$269$	−7.52786	−0.458982	−0.229491	−	0.973311i	$-0.573706\pi$
$269$	−7.52786	−0.458982	−0.229491	+	0.973311i	$0.573706\pi$
$270$	2.00000	0.121716
$271$	11.4164	0.693497	0.346749	−	0.937958i	$-0.387286\pi$
$271$	11.4164	0.693497	0.346749	+	0.937958i	$0.387286\pi$
$272$	4.47214	0.271163
$273$	0	0
$274$	−14.0000	−0.845771
$275$	0	0
$276$	1.00000	0.0601929
$277$	23.8885	1.43532	0.717662	−	0.696392i	$-0.245212\pi$
$277$	23.8885	1.43532	0.717662	+	0.696392i	$0.245212\pi$
$278$	8.94427	0.536442
$279$	−6.47214	−0.387477
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	6.47214	0.385410
$283$	−9.52786	−0.566373	−0.283186	−	0.959065i	$-0.591391\pi$
$283$	−9.52786	−0.566373	−0.283186	+	0.959065i	$0.591391\pi$
$284$	−12.9443	−0.768101
$285$	12.9443	0.766752
$286$	0	0
$287$	0	0
$288$	1.00000	0.0589256
$289$	3.00000	0.176471
$290$	4.00000	0.234888
$291$	8.47214	0.496645
$292$	2.94427	0.172300
$293$	7.88854	0.460854	0.230427	−	0.973090i	$-0.425988\pi$
$293$	7.88854	0.460854	0.230427	+	0.973090i	$0.425988\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	−10.9443	−0.636123
$297$	0	0
$298$	−18.9443	−1.09741
$299$	4.47214	0.258630
$300$	1.00000	0.0577350
$301$	0	0
$302$	−12.9443	−0.744859
$303$	17.4164	1.00055
$304$	6.47214	0.371202
$305$	−12.0000	−0.687118
$306$	4.47214	0.255655
$307$	−7.05573	−0.402692	−0.201346	−	0.979520i	$-0.564532\pi$
$307$	−7.05573	−0.402692	−0.201346	+	0.979520i	$0.564532\pi$
$308$	0	0
$309$	12.9443	0.736374
$310$	12.9443	0.735185
$311$	−6.47214	−0.367001	−0.183501	−	0.983020i	$-0.558743\pi$
$311$	−6.47214	−0.367001	−0.183501	+	0.983020i	$0.558743\pi$
$312$	4.47214	0.253185
$313$	−13.4164	−0.758340	−0.379170	−	0.925327i	$-0.623790\pi$
$313$	−13.4164	−0.758340	−0.379170	+	0.925327i	$0.623790\pi$
$314$	−14.9443	−0.843354
$315$	0	0
$316$	12.9443	0.728172
$317$	−30.9443	−1.73800	−0.869002	−	0.494809i	$-0.835238\pi$
$317$	−30.9443	−1.73800	−0.869002	+	0.494809i	$0.835238\pi$
$318$	−6.94427	−0.389415
$319$	0	0
$320$	−2.00000	−0.111803
$321$	8.00000	0.446516
$322$	0	0
$323$	28.9443	1.61050
$324$	1.00000	0.0555556
$325$	4.47214	0.248069
$326$	8.94427	0.495377
$327$	−14.9443	−0.826420
$328$	6.00000	0.331295
$329$	0	0
$330$	0	0
$331$	21.8885	1.20310	0.601552	−	0.798834i	$-0.294549\pi$
$331$	21.8885	1.20310	0.601552	+	0.798834i	$0.294549\pi$
$332$	6.47214	0.355205
$333$	−10.9443	−0.599742
$334$	−11.4164	−0.624678
$335$	25.8885	1.41444
$336$	0	0
$337$	14.9443	0.814066	0.407033	−	0.913413i	$-0.366563\pi$
$337$	14.9443	0.814066	0.407033	+	0.913413i	$0.366563\pi$
$338$	7.00000	0.380750
$339$	1.05573	0.0573393
$340$	−8.94427	−0.485071
$341$	0	0
$342$	6.47214	0.349973
$343$	0	0
$344$	12.9443	0.697908
$345$	−2.00000	−0.107676
$346$	8.47214	0.455465
$347$	16.9443	0.909616	0.454808	−	0.890589i	$-0.349708\pi$
$347$	16.9443	0.909616	0.454808	+	0.890589i	$0.349708\pi$
$348$	2.00000	0.107211
$349$	−15.5279	−0.831188	−0.415594	−	0.909550i	$-0.636426\pi$
$349$	−15.5279	−0.831188	−0.415594	+	0.909550i	$0.636426\pi$
$350$	0	0
$351$	4.47214	0.238705
$352$	0	0
$353$	−2.00000	−0.106449	−0.0532246	−	0.998583i	$-0.516950\pi$
$353$	−2.00000	−0.106449	−0.0532246	+	0.998583i	$0.516950\pi$
$354$	4.00000	0.212598
$355$	25.8885	1.37402
$356$	17.4164	0.923068
$357$	0	0
$358$	−20.0000	−1.05703
$359$	22.8328	1.20507	0.602535	−	0.798092i	$-0.294157\pi$
$359$	22.8328	1.20507	0.602535	+	0.798092i	$0.294157\pi$
$360$	−2.00000	−0.105409
$361$	22.8885	1.20466
$362$	−2.00000	−0.105118
$363$	11.0000	0.577350
$364$	0	0
$365$	−5.88854	−0.308220
$366$	−6.00000	−0.313625
$367$	22.8328	1.19186	0.595932	−	0.803035i	$-0.296783\pi$
$367$	22.8328	1.19186	0.595932	+	0.803035i	$0.296783\pi$
$368$	−1.00000	−0.0521286
$369$	6.00000	0.312348
$370$	21.8885	1.13793
$371$	0	0
$372$	6.47214	0.335565
$373$	−10.9443	−0.566673	−0.283336	−	0.959021i	$-0.591441\pi$
$373$	−10.9443	−0.566673	−0.283336	+	0.959021i	$0.591441\pi$
$374$	0	0
$375$	−12.0000	−0.619677
$376$	−6.47214	−0.333775
$377$	8.94427	0.460653
$378$	0	0
$379$	24.0000	1.23280	0.616399	−	0.787434i	$-0.288591\pi$
$379$	24.0000	1.23280	0.616399	+	0.787434i	$0.288591\pi$
$380$	−12.9443	−0.664027
$381$	−3.05573	−0.156550
$382$	−20.9443	−1.07160
$383$	−16.0000	−0.817562	−0.408781	−	0.912633i	$-0.634046\pi$
$383$	−16.0000	−0.817562	−0.408781	+	0.912633i	$0.634046\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−23.8885	−1.21589
$387$	12.9443	0.657994
$388$	−8.47214	−0.430108
$389$	−1.05573	−0.0535275	−0.0267638	−	0.999642i	$-0.508520\pi$
$389$	−1.05573	−0.0535275	−0.0267638	+	0.999642i	$0.508520\pi$
$390$	−8.94427	−0.452911
$391$	−4.47214	−0.226166
$392$	0	0
$393$	12.0000	0.605320
$394$	2.94427	0.148330
$395$	−25.8885	−1.30259
$396$	0	0
$397$	27.5279	1.38158	0.690792	−	0.723054i	$-0.257262\pi$
$397$	27.5279	1.38158	0.690792	+	0.723054i	$0.257262\pi$
$398$	−20.9443	−1.04984
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−22.0000	−1.09863	−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000	−1.09863	−0.549314	+	0.835616i	$0.685111\pi$
$402$	12.9443	0.645602
$403$	28.9443	1.44182
$404$	−17.4164	−0.866499
$405$	−2.00000	−0.0993808
$406$	0	0
$407$	0	0
$408$	−4.47214	−0.221404
$409$	−19.8885	−0.983425	−0.491713	−	0.870758i	$-0.663629\pi$
$409$	−19.8885	−0.983425	−0.491713	+	0.870758i	$0.663629\pi$
$410$	−12.0000	−0.592638
$411$	14.0000	0.690569
$412$	−12.9443	−0.637719
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	−12.9443	−0.635409
$416$	−4.47214	−0.219265
$417$	−8.94427	−0.438003
$418$	0	0
$419$	−19.4164	−0.948554	−0.474277	−	0.880376i	$-0.657290\pi$
$419$	−19.4164	−0.948554	−0.474277	+	0.880376i	$0.657290\pi$
$420$	0	0
$421$	−20.8328	−1.01533	−0.507665	−	0.861555i	$-0.669491\pi$
$421$	−20.8328	−1.01533	−0.507665	+	0.861555i	$0.669491\pi$
$422$	16.9443	0.824834
$423$	−6.47214	−0.314686
$424$	6.94427	0.337244
$425$	−4.47214	−0.216930
$426$	12.9443	0.627152
$427$	0	0
$428$	−8.00000	−0.386695
$429$	0	0
$430$	−25.8885	−1.24846
$431$	−17.8885	−0.861661	−0.430830	−	0.902433i	$-0.641779\pi$
$431$	−17.8885	−0.861661	−0.430830	+	0.902433i	$0.641779\pi$
$432$	−1.00000	−0.0481125
$433$	1.41641	0.0680682	0.0340341	−	0.999421i	$-0.489165\pi$
$433$	1.41641	0.0680682	0.0340341	+	0.999421i	$0.489165\pi$
$434$	0	0
$435$	−4.00000	−0.191785
$436$	14.9443	0.715701
$437$	−6.47214	−0.309604
$438$	−2.94427	−0.140683
$439$	16.3607	0.780853	0.390426	−	0.920634i	$-0.372328\pi$
$439$	16.3607	0.780853	0.390426	+	0.920634i	$0.372328\pi$
$440$	0	0
$441$	0	0
$442$	−20.0000	−0.951303
$443$	−18.8328	−0.894774	−0.447387	−	0.894340i	$-0.647645\pi$
$443$	−18.8328	−0.894774	−0.447387	+	0.894340i	$0.647645\pi$
$444$	10.9443	0.519392
$445$	−34.8328	−1.65123
$446$	−9.52786	−0.451157
$447$	18.9443	0.896033
$448$	0	0
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	−1.00000	−0.0471405
$451$	0	0
$452$	−1.05573	−0.0496573
$453$	12.9443	0.608175
$454$	−14.4721	−0.679211
$455$	0	0
$456$	−6.47214	−0.303086
$457$	2.00000	0.0935561	0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000	0.0935561	0.0467780	+	0.998905i	$0.485105\pi$
$458$	14.0000	0.654177
$459$	−4.47214	−0.208741
$460$	2.00000	0.0932505
$461$	−33.4164	−1.55636	−0.778179	−	0.628043i	$-0.783856\pi$
$461$	−33.4164	−1.55636	−0.778179	+	0.628043i	$0.783856\pi$
$462$	0	0
$463$	41.8885	1.94673	0.973363	−	0.229270i	$-0.0736339\pi$
$463$	41.8885	1.94673	0.973363	+	0.229270i	$0.0736339\pi$
$464$	−2.00000	−0.0928477
$465$	−12.9443	−0.600276
$466$	−6.00000	−0.277945
$467$	29.3050	1.35607	0.678036	−	0.735029i	$-0.262831\pi$
$467$	29.3050	1.35607	0.678036	+	0.735029i	$0.262831\pi$
$468$	−4.47214	−0.206725
$469$	0	0
$470$	12.9443	0.597075
$471$	14.9443	0.688596
$472$	−4.00000	−0.184115
$473$	0	0
$474$	−12.9443	−0.594550
$475$	−6.47214	−0.296962
$476$	0	0
$477$	6.94427	0.317956
$478$	3.05573	0.139766
$479$	−12.9443	−0.591439	−0.295719	−	0.955275i	$-0.595559\pi$
$479$	−12.9443	−0.591439	−0.295719	+	0.955275i	$0.595559\pi$
$480$	2.00000	0.0912871
$481$	48.9443	2.23167
$482$	−11.5279	−0.525080
$483$	0	0
$484$	−11.0000	−0.500000
$485$	16.9443	0.769400
$486$	−1.00000	−0.0453609
$487$	24.0000	1.08754	0.543772	−	0.839233i	$-0.316996\pi$
$487$	24.0000	1.08754	0.543772	+	0.839233i	$0.316996\pi$
$488$	6.00000	0.271607
$489$	−8.94427	−0.404474
$490$	0	0
$491$	23.0557	1.04049	0.520245	−	0.854017i	$-0.325841\pi$
$491$	23.0557	1.04049	0.520245	+	0.854017i	$0.325841\pi$
$492$	−6.00000	−0.270501
$493$	−8.94427	−0.402830
$494$	−28.9443	−1.30226
$495$	0	0
$496$	−6.47214	−0.290607
$497$	0	0
$498$	−6.47214	−0.290023
$499$	15.0557	0.673987	0.336993	−	0.941507i	$-0.390590\pi$
$499$	15.0557	0.673987	0.336993	+	0.941507i	$0.390590\pi$
$500$	12.0000	0.536656
$501$	11.4164	0.510047
$502$	14.4721	0.645923
$503$	−25.8885	−1.15431	−0.577157	−	0.816634i	$-0.695838\pi$
$503$	−25.8885	−1.15431	−0.577157	+	0.816634i	$0.695838\pi$
$504$	0	0
$505$	34.8328	1.55004
$506$	0	0
$507$	−7.00000	−0.310881
$508$	3.05573	0.135576
$509$	−33.4164	−1.48116	−0.740578	−	0.671970i	$-0.765448\pi$
$509$	−33.4164	−1.48116	−0.740578	+	0.671970i	$0.765448\pi$
$510$	8.94427	0.396059
$511$	0	0
$512$	1.00000	0.0441942
$513$	−6.47214	−0.285752
$514$	−14.9443	−0.659164
$515$	25.8885	1.14079
$516$	−12.9443	−0.569840
$517$	0	0
$518$	0	0
$519$	−8.47214	−0.371885
$520$	8.94427	0.392232
$521$	−6.58359	−0.288432	−0.144216	−	0.989546i	$-0.546066\pi$
$521$	−6.58359	−0.288432	−0.144216	+	0.989546i	$0.546066\pi$
$522$	−2.00000	−0.0875376
$523$	−37.3050	−1.63123	−0.815616	−	0.578594i	$-0.803602\pi$
$523$	−37.3050	−1.63123	−0.815616	+	0.578594i	$0.803602\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−24.0000	−1.04645
$527$	−28.9443	−1.26083
$528$	0	0
$529$	1.00000	0.0434783
$530$	−13.8885	−0.603280
$531$	−4.00000	−0.173585
$532$	0	0
$533$	−26.8328	−1.16226
$534$	−17.4164	−0.753682
$535$	16.0000	0.691740
$536$	−12.9443	−0.559107
$537$	20.0000	0.863064
$538$	−7.52786	−0.324549
$539$	0	0
$540$	2.00000	0.0860663
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	11.4164	0.490377
$543$	2.00000	0.0858282
$544$	4.47214	0.191741
$545$	−29.8885	−1.28028
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	−14.0000	−0.598050
$549$	6.00000	0.256074
$550$	0	0
$551$	−12.9443	−0.551445
$552$	1.00000	0.0425628
$553$	0	0
$554$	23.8885	1.01493
$555$	−21.8885	−0.929117
$556$	8.94427	0.379322
$557$	−34.9443	−1.48064	−0.740318	−	0.672257i	$-0.765325\pi$
$557$	−34.9443	−1.48064	−0.740318	+	0.672257i	$0.765325\pi$
$558$	−6.47214	−0.273987
$559$	−57.8885	−2.44842
$560$	0	0
$561$	0	0
$562$	10.0000	0.421825
$563$	−17.5279	−0.738711	−0.369356	−	0.929288i	$-0.620422\pi$
$563$	−17.5279	−0.738711	−0.369356	+	0.929288i	$0.620422\pi$
$564$	6.47214	0.272526
$565$	2.11146	0.0888296
$566$	−9.52786	−0.400486
$567$	0	0
$568$	−12.9443	−0.543130
$569$	27.8885	1.16915	0.584574	−	0.811340i	$-0.301262\pi$
$569$	27.8885	1.16915	0.584574	+	0.811340i	$0.301262\pi$
$570$	12.9443	0.542176
$571$	4.94427	0.206911	0.103456	−	0.994634i	$-0.467010\pi$
$571$	4.94427	0.206911	0.103456	+	0.994634i	$0.467010\pi$
$572$	0	0
$573$	20.9443	0.874960
$574$	0	0
$575$	1.00000	0.0417029
$576$	1.00000	0.0416667
$577$	−14.9443	−0.622138	−0.311069	−	0.950387i	$-0.600687\pi$
$577$	−14.9443	−0.622138	−0.311069	+	0.950387i	$0.600687\pi$
$578$	3.00000	0.124784
$579$	23.8885	0.992774
$580$	4.00000	0.166091
$581$	0	0
$582$	8.47214	0.351181
$583$	0	0
$584$	2.94427	0.121835
$585$	8.94427	0.369800
$586$	7.88854	0.325873
$587$	−16.9443	−0.699365	−0.349682	−	0.936868i	$-0.613711\pi$
$587$	−16.9443	−0.699365	−0.349682	+	0.936868i	$0.613711\pi$
$588$	0	0
$589$	−41.8885	−1.72599
$590$	8.00000	0.329355
$591$	−2.94427	−0.121111
$592$	−10.9443	−0.449807
$593$	−24.8328	−1.01976	−0.509881	−	0.860245i	$-0.670310\pi$
$593$	−24.8328	−1.01976	−0.509881	+	0.860245i	$0.670310\pi$
$594$	0	0
$595$	0	0
$596$	−18.9443	−0.775988
$597$	20.9443	0.857192
$598$	4.47214	0.182879
$599$	−1.88854	−0.0771638	−0.0385819	−	0.999255i	$-0.512284\pi$
$599$	−1.88854	−0.0771638	−0.0385819	+	0.999255i	$0.512284\pi$
$600$	1.00000	0.0408248
$601$	12.8328	0.523461	0.261731	−	0.965141i	$-0.415707\pi$
$601$	12.8328	0.523461	0.261731	+	0.965141i	$0.415707\pi$
$602$	0	0
$603$	−12.9443	−0.527132
$604$	−12.9443	−0.526695
$605$	22.0000	0.894427
$606$	17.4164	0.707493
$607$	−16.3607	−0.664060	−0.332030	−	0.943269i	$-0.607733\pi$
$607$	−16.3607	−0.664060	−0.332030	+	0.943269i	$0.607733\pi$
$608$	6.47214	0.262480
$609$	0	0
$610$	−12.0000	−0.485866
$611$	28.9443	1.17096
$612$	4.47214	0.180775
$613$	−12.8328	−0.518313	−0.259156	−	0.965835i	$-0.583444\pi$
$613$	−12.8328	−0.518313	−0.259156	+	0.965835i	$0.583444\pi$
$614$	−7.05573	−0.284746
$615$	12.0000	0.483887
$616$	0	0
$617$	26.0000	1.04672	0.523360	−	0.852111i	$-0.324678\pi$
$617$	26.0000	1.04672	0.523360	+	0.852111i	$0.324678\pi$
$618$	12.9443	0.520695
$619$	25.5279	1.02605	0.513026	−	0.858373i	$-0.328525\pi$
$619$	25.5279	1.02605	0.513026	+	0.858373i	$0.328525\pi$
$620$	12.9443	0.519854
$621$	1.00000	0.0401286
$622$	−6.47214	−0.259509
$623$	0	0
$624$	4.47214	0.179029
$625$	−19.0000	−0.760000
$626$	−13.4164	−0.536228
$627$	0	0
$628$	−14.9443	−0.596341
$629$	−48.9443	−1.95154
$630$	0	0
$631$	−28.9443	−1.15225	−0.576127	−	0.817360i	$-0.695436\pi$
$631$	−28.9443	−1.15225	−0.576127	+	0.817360i	$0.695436\pi$
$632$	12.9443	0.514895
$633$	−16.9443	−0.673474
$634$	−30.9443	−1.22895
$635$	−6.11146	−0.242526
$636$	−6.94427	−0.275358
$637$	0	0
$638$	0	0
$639$	−12.9443	−0.512067
$640$	−2.00000	−0.0790569
$641$	10.0000	0.394976	0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000	0.394976	0.197488	+	0.980305i	$0.436722\pi$
$642$	8.00000	0.315735
$643$	1.52786	0.0602531	0.0301265	−	0.999546i	$-0.490409\pi$
$643$	1.52786	0.0602531	0.0301265	+	0.999546i	$0.490409\pi$
$644$	0	0
$645$	25.8885	1.01936
$646$	28.9443	1.13880
$647$	32.3607	1.27223	0.636115	−	0.771594i	$-0.280540\pi$
$647$	32.3607	1.27223	0.636115	+	0.771594i	$0.280540\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	4.47214	0.175412
$651$	0	0
$652$	8.94427	0.350285
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	−14.9443	−0.584367
$655$	24.0000	0.937758
$656$	6.00000	0.234261
$657$	2.94427	0.114867
$658$	0	0
$659$	−6.83282	−0.266169	−0.133084	−	0.991105i	$-0.542488\pi$
$659$	−6.83282	−0.266169	−0.133084	+	0.991105i	$0.542488\pi$
$660$	0	0
$661$	30.0000	1.16686	0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000	1.16686	0.583432	+	0.812162i	$0.301709\pi$
$662$	21.8885	0.850722
$663$	20.0000	0.776736
$664$	6.47214	0.251168
$665$	0	0
$666$	−10.9443	−0.424082
$667$	2.00000	0.0774403
$668$	−11.4164	−0.441714
$669$	9.52786	0.368369
$670$	25.8885	1.00016
$671$	0	0
$672$	0	0
$673$	−30.0000	−1.15642	−0.578208	−	0.815890i	$-0.696248\pi$
$673$	−30.0000	−1.15642	−0.578208	+	0.815890i	$0.696248\pi$
$674$	14.9443	0.575632
$675$	1.00000	0.0384900
$676$	7.00000	0.269231
$677$	39.8885	1.53304	0.766521	−	0.642220i	$-0.221986\pi$
$677$	39.8885	1.53304	0.766521	+	0.642220i	$0.221986\pi$
$678$	1.05573	0.0405450
$679$	0	0
$680$	−8.94427	−0.342997
$681$	14.4721	0.554573
$682$	0	0
$683$	−0.944272	−0.0361316	−0.0180658	−	0.999837i	$-0.505751\pi$
$683$	−0.944272	−0.0361316	−0.0180658	+	0.999837i	$0.505751\pi$
$684$	6.47214	0.247468
$685$	28.0000	1.06983
$686$	0	0
$687$	−14.0000	−0.534133
$688$	12.9443	0.493496
$689$	−31.0557	−1.18313
$690$	−2.00000	−0.0761387
$691$	32.9443	1.25326	0.626630	−	0.779317i	$-0.284434\pi$
$691$	32.9443	1.25326	0.626630	+	0.779317i	$0.284434\pi$
$692$	8.47214	0.322062
$693$	0	0
$694$	16.9443	0.643196
$695$	−17.8885	−0.678551
$696$	2.00000	0.0758098
$697$	26.8328	1.01637
$698$	−15.5279	−0.587738
$699$	6.00000	0.226941
$700$	0	0
$701$	32.8328	1.24008	0.620039	−	0.784571i	$-0.287117\pi$
$701$	32.8328	1.24008	0.620039	+	0.784571i	$0.287117\pi$
$702$	4.47214	0.168790
$703$	−70.8328	−2.67151
$704$	0	0
$705$	−12.9443	−0.487509
$706$	−2.00000	−0.0752710
$707$	0	0
$708$	4.00000	0.150329
$709$	−10.9443	−0.411021	−0.205510	−	0.978655i	$-0.565885\pi$
$709$	−10.9443	−0.411021	−0.205510	+	0.978655i	$0.565885\pi$
$710$	25.8885	0.971580
$711$	12.9443	0.485448
$712$	17.4164	0.652707
$713$	6.47214	0.242383
$714$	0	0
$715$	0	0
$716$	−20.0000	−0.747435
$717$	−3.05573	−0.114118
$718$	22.8328	0.852113
$719$	14.4721	0.539720	0.269860	−	0.962900i	$-0.413023\pi$
$719$	14.4721	0.539720	0.269860	+	0.962900i	$0.413023\pi$
$720$	−2.00000	−0.0745356
$721$	0	0
$722$	22.8885	0.851823
$723$	11.5279	0.428726
$724$	−2.00000	−0.0743294
$725$	2.00000	0.0742781
$726$	11.0000	0.408248
$727$	−20.9443	−0.776780	−0.388390	−	0.921495i	$-0.626969\pi$
$727$	−20.9443	−0.776780	−0.388390	+	0.921495i	$0.626969\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−5.88854	−0.217945
$731$	57.8885	2.14109
$732$	−6.00000	−0.221766
$733$	33.0557	1.22094	0.610471	−	0.792039i	$-0.290980\pi$
$733$	33.0557	1.22094	0.610471	+	0.792039i	$0.290980\pi$
$734$	22.8328	0.842775
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	0	0
$738$	6.00000	0.220863
$739$	8.94427	0.329020	0.164510	−	0.986375i	$-0.447396\pi$
$739$	8.94427	0.329020	0.164510	+	0.986375i	$0.447396\pi$
$740$	21.8885	0.804639
$741$	28.9443	1.06329
$742$	0	0
$743$	30.8328	1.13115	0.565573	−	0.824698i	$-0.308655\pi$
$743$	30.8328	1.13115	0.565573	+	0.824698i	$0.308655\pi$
$744$	6.47214	0.237280
$745$	37.8885	1.38813
$746$	−10.9443	−0.400698
$747$	6.47214	0.236803
$748$	0	0
$749$	0	0
$750$	−12.0000	−0.438178
$751$	−17.8885	−0.652762	−0.326381	−	0.945238i	$-0.605829\pi$
$751$	−17.8885	−0.652762	−0.326381	+	0.945238i	$0.605829\pi$
$752$	−6.47214	−0.236015
$753$	−14.4721	−0.527394
$754$	8.94427	0.325731
$755$	25.8885	0.942181
$756$	0	0
$757$	−25.0557	−0.910666	−0.455333	−	0.890321i	$-0.650480\pi$
$757$	−25.0557	−0.910666	−0.455333	+	0.890321i	$0.650480\pi$
$758$	24.0000	0.871719
$759$	0	0
$760$	−12.9443	−0.469538
$761$	−38.9443	−1.41173	−0.705864	−	0.708347i	$-0.749441\pi$
$761$	−38.9443	−1.41173	−0.705864	+	0.708347i	$0.749441\pi$
$762$	−3.05573	−0.110697
$763$	0	0
$764$	−20.9443	−0.757737
$765$	−8.94427	−0.323381
$766$	−16.0000	−0.578103
$767$	17.8885	0.645918
$768$	−1.00000	−0.0360844
$769$	14.3607	0.517859	0.258930	−	0.965896i	$-0.416630\pi$
$769$	14.3607	0.517859	0.258930	+	0.965896i	$0.416630\pi$
$770$	0	0
$771$	14.9443	0.538205
$772$	−23.8885	−0.859768
$773$	−22.9443	−0.825248	−0.412624	−	0.910901i	$-0.635388\pi$
$773$	−22.9443	−0.825248	−0.412624	+	0.910901i	$0.635388\pi$
$774$	12.9443	0.465272
$775$	6.47214	0.232486
$776$	−8.47214	−0.304132
$777$	0	0
$778$	−1.05573	−0.0378497
$779$	38.8328	1.39133
$780$	−8.94427	−0.320256
$781$	0	0
$782$	−4.47214	−0.159923
$783$	2.00000	0.0714742
$784$	0	0
$785$	29.8885	1.06677
$786$	12.0000	0.428026
$787$	27.4164	0.977289	0.488645	−	0.872483i	$-0.337491\pi$
$787$	27.4164	0.977289	0.488645	+	0.872483i	$0.337491\pi$
$788$	2.94427	0.104885
$789$	24.0000	0.854423
$790$	−25.8885	−0.921073
$791$	0	0
$792$	0	0
$793$	−26.8328	−0.952861
$794$	27.5279	0.976927
$795$	13.8885	0.492576
$796$	−20.9443	−0.742350
$797$	−24.8328	−0.879623	−0.439812	−	0.898090i	$-0.644955\pi$
$797$	−24.8328	−0.879623	−0.439812	+	0.898090i	$0.644955\pi$
$798$	0	0
$799$	−28.9443	−1.02397
$800$	−1.00000	−0.0353553
$801$	17.4164	0.615379
$802$	−22.0000	−0.776847
$803$	0	0
$804$	12.9443	0.456509
$805$	0	0
$806$	28.9443	1.01952
$807$	7.52786	0.264993
$808$	−17.4164	−0.612707
$809$	0.111456	0.00391859	0.00195930	−	0.999998i	$-0.499376\pi$
$809$	0.111456	0.00391859	0.00195930	+	0.999998i	$0.499376\pi$
$810$	−2.00000	−0.0702728
$811$	2.11146	0.0741433	0.0370716	−	0.999313i	$-0.488197\pi$
$811$	2.11146	0.0741433	0.0370716	+	0.999313i	$0.488197\pi$
$812$	0	0
$813$	−11.4164	−0.400391
$814$	0	0
$815$	−17.8885	−0.626608
$816$	−4.47214	−0.156556
$817$	83.7771	2.93099
$818$	−19.8885	−0.695387
$819$	0	0
$820$	−12.0000	−0.419058
$821$	36.8328	1.28547	0.642737	−	0.766087i	$-0.277799\pi$
$821$	36.8328	1.28547	0.642737	+	0.766087i	$0.277799\pi$
$822$	14.0000	0.488306
$823$	3.05573	0.106516	0.0532580	−	0.998581i	$-0.483039\pi$
$823$	3.05573	0.106516	0.0532580	+	0.998581i	$0.483039\pi$
$824$	−12.9443	−0.450935
$825$	0	0
$826$	0	0
$827$	−1.88854	−0.0656711	−0.0328356	−	0.999461i	$-0.510454\pi$
$827$	−1.88854	−0.0656711	−0.0328356	+	0.999461i	$0.510454\pi$
$828$	−1.00000	−0.0347524
$829$	13.4164	0.465971	0.232986	−	0.972480i	$-0.425151\pi$
$829$	13.4164	0.465971	0.232986	+	0.972480i	$0.425151\pi$
$830$	−12.9443	−0.449302
$831$	−23.8885	−0.828684
$832$	−4.47214	−0.155043
$833$	0	0
$834$	−8.94427	−0.309715
$835$	22.8328	0.790162
$836$	0	0
$837$	6.47214	0.223710
$838$	−19.4164	−0.670729
$839$	24.0000	0.828572	0.414286	−	0.910147i	$-0.364031\pi$
$839$	24.0000	0.828572	0.414286	+	0.910147i	$0.364031\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−20.8328	−0.717946
$843$	−10.0000	−0.344418
$844$	16.9443	0.583246
$845$	−14.0000	−0.481615
$846$	−6.47214	−0.222517
$847$	0	0
$848$	6.94427	0.238467
$849$	9.52786	0.326995
$850$	−4.47214	−0.153393
$851$	10.9443	0.375165
$852$	12.9443	0.443463
$853$	−38.3607	−1.31344	−0.656722	−	0.754132i	$-0.728058\pi$
$853$	−38.3607	−1.31344	−0.656722	+	0.754132i	$0.728058\pi$
$854$	0	0
$855$	−12.9443	−0.442685
$856$	−8.00000	−0.273434
$857$	12.1115	0.413719	0.206860	−	0.978371i	$-0.433676\pi$
$857$	12.1115	0.413719	0.206860	+	0.978371i	$0.433676\pi$
$858$	0	0
$859$	−15.0557	−0.513695	−0.256847	−	0.966452i	$-0.582684\pi$
$859$	−15.0557	−0.513695	−0.256847	+	0.966452i	$0.582684\pi$
$860$	−25.8885	−0.882792
$861$	0	0
$862$	−17.8885	−0.609286
$863$	3.05573	0.104018	0.0520091	−	0.998647i	$-0.483438\pi$
$863$	3.05573	0.104018	0.0520091	+	0.998647i	$0.483438\pi$
$864$	−1.00000	−0.0340207
$865$	−16.9443	−0.576123
$866$	1.41641	0.0481315
$867$	−3.00000	−0.101885
$868$	0	0
$869$	0	0
$870$	−4.00000	−0.135613
$871$	57.8885	1.96148
$872$	14.9443	0.506077
$873$	−8.47214	−0.286738
$874$	−6.47214	−0.218923
$875$	0	0
$876$	−2.94427	−0.0994777
$877$	54.7214	1.84781	0.923905	−	0.382623i	$-0.124979\pi$
$877$	54.7214	1.84781	0.923905	+	0.382623i	$0.124979\pi$
$878$	16.3607	0.552146
$879$	−7.88854	−0.266074
$880$	0	0
$881$	22.3607	0.753350	0.376675	−	0.926345i	$-0.377067\pi$
$881$	22.3607	0.753350	0.376675	+	0.926345i	$0.377067\pi$
$882$	0	0
$883$	−50.8328	−1.71066	−0.855330	−	0.518083i	$-0.826646\pi$
$883$	−50.8328	−1.71066	−0.855330	+	0.518083i	$0.826646\pi$
$884$	−20.0000	−0.672673
$885$	−8.00000	−0.268917
$886$	−18.8328	−0.632701
$887$	14.4721	0.485927	0.242963	−	0.970035i	$-0.421881\pi$
$887$	14.4721	0.485927	0.242963	+	0.970035i	$0.421881\pi$
$888$	10.9443	0.367266
$889$	0	0
$890$	−34.8328	−1.16760
$891$	0	0
$892$	−9.52786	−0.319016
$893$	−41.8885	−1.40175
$894$	18.9443	0.633591
$895$	40.0000	1.33705
$896$	0	0
$897$	−4.47214	−0.149320
$898$	18.0000	0.600668
$899$	12.9443	0.431716
$900$	−1.00000	−0.0333333
$901$	31.0557	1.03462
$902$	0	0
$903$	0	0
$904$	−1.05573	−0.0351130
$905$	4.00000	0.132964
$906$	12.9443	0.430045
$907$	−6.11146	−0.202928	−0.101464	−	0.994839i	$-0.532353\pi$
$907$	−6.11146	−0.202928	−0.101464	+	0.994839i	$0.532353\pi$
$908$	−14.4721	−0.480275
$909$	−17.4164	−0.577666
$910$	0	0
$911$	40.7214	1.34916	0.674579	−	0.738202i	$-0.264325\pi$
$911$	40.7214	1.34916	0.674579	+	0.738202i	$0.264325\pi$
$912$	−6.47214	−0.214314
$913$	0	0
$914$	2.00000	0.0661541
$915$	12.0000	0.396708
$916$	14.0000	0.462573
$917$	0	0
$918$	−4.47214	−0.147602
$919$	52.9443	1.74647	0.873235	−	0.487299i	$-0.162018\pi$
$919$	52.9443	1.74647	0.873235	+	0.487299i	$0.162018\pi$
$920$	2.00000	0.0659380
$921$	7.05573	0.232494
$922$	−33.4164	−1.10051
$923$	57.8885	1.90542
$924$	0	0
$925$	10.9443	0.359845
$926$	41.8885	1.37654
$927$	−12.9443	−0.425146
$928$	−2.00000	−0.0656532
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	−12.9443	−0.424459
$931$	0	0
$932$	−6.00000	−0.196537
$933$	6.47214	0.211888
$934$	29.3050	0.958887
$935$	0	0
$936$	−4.47214	−0.146176
$937$	40.2492	1.31488	0.657442	−	0.753505i	$-0.271638\pi$
$937$	40.2492	1.31488	0.657442	+	0.753505i	$0.271638\pi$
$938$	0	0
$939$	13.4164	0.437828
$940$	12.9443	0.422196
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	14.9443	0.486911
$943$	−6.00000	−0.195387
$944$	−4.00000	−0.130189
$945$	0	0
$946$	0	0
$947$	−16.9443	−0.550615	−0.275307	−	0.961356i	$-0.588780\pi$
$947$	−16.9443	−0.550615	−0.275307	+	0.961356i	$0.588780\pi$
$948$	−12.9443	−0.420410
$949$	−13.1672	−0.427425
$950$	−6.47214	−0.209984
$951$	30.9443	1.00344
$952$	0	0
$953$	−26.9443	−0.872811	−0.436405	−	0.899750i	$-0.643749\pi$
$953$	−26.9443	−0.872811	−0.436405	+	0.899750i	$0.643749\pi$
$954$	6.94427	0.224829
$955$	41.8885	1.35548
$956$	3.05573	0.0988293
$957$	0	0
$958$	−12.9443	−0.418210
$959$	0	0
$960$	2.00000	0.0645497
$961$	10.8885	0.351243
$962$	48.9443	1.57803
$963$	−8.00000	−0.257796
$964$	−11.5279	−0.371288
$965$	47.7771	1.53800
$966$	0	0
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	−11.0000	−0.353553
$969$	−28.9443	−0.929824
$970$	16.9443	0.544048
$971$	−3.41641	−0.109638	−0.0548189	−	0.998496i	$-0.517458\pi$
$971$	−3.41641	−0.109638	−0.0548189	+	0.998496i	$0.517458\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	24.0000	0.769010
$975$	−4.47214	−0.143223
$976$	6.00000	0.192055
$977$	−6.00000	−0.191957	−0.0959785	−	0.995383i	$-0.530598\pi$
$977$	−6.00000	−0.191957	−0.0959785	+	0.995383i	$0.530598\pi$
$978$	−8.94427	−0.286006
$979$	0	0
$980$	0	0
$981$	14.9443	0.477134
$982$	23.0557	0.735738
$983$	57.8885	1.84636	0.923179	−	0.384371i	$-0.125581\pi$
$983$	57.8885	1.84636	0.923179	+	0.384371i	$0.125581\pi$
$984$	−6.00000	−0.191273
$985$	−5.88854	−0.187625
$986$	−8.94427	−0.284844
$987$	0	0
$988$	−28.9443	−0.920840
$989$	−12.9443	−0.411604
$990$	0	0
$991$	43.0557	1.36771	0.683855	−	0.729618i	$-0.260302\pi$
$991$	43.0557	1.36771	0.683855	+	0.729618i	$0.260302\pi$
$992$	−6.47214	−0.205491
$993$	−21.8885	−0.694612
$994$	0	0
$995$	41.8885	1.32796
$996$	−6.47214	−0.205077
$997$	1.63932	0.0519178	0.0259589	−	0.999663i	$-0.491736\pi$
$997$	1.63932	0.0519178	0.0259589	+	0.999663i	$0.491736\pi$
$998$	15.0557	0.476581
$999$	10.9443	0.346261

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6762.2.a.bz.1.1		2
7.6	odd	2		966.2.a.p.1.2	✓	2
21.20	even	2		2898.2.a.v.1.2		2
28.27	even	2		7728.2.a.bd.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
966.2.a.p.1.2	✓	2	7.6	odd	2
2898.2.a.v.1.2		2	21.20	even	2
6762.2.a.bz.1.1		2	1.1	even	1	trivial
7728.2.a.bd.1.2		2	28.27	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:	\( N \)	\(=\)	\( 6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6762.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{12} -4.47214 q^{13} +2.00000 q^{15} +1.00000 q^{16} +4.47214 q^{17} +1.00000 q^{18} +6.47214 q^{19} -2.00000 q^{20} -1.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} -4.47214 q^{26} -1.00000 q^{27} -2.00000 q^{29} +2.00000 q^{30} -6.47214 q^{31} +1.00000 q^{32} +4.47214 q^{34} +1.00000 q^{36} -10.9443 q^{37} +6.47214 q^{38} +4.47214 q^{39} -2.00000 q^{40} +6.00000 q^{41} +12.9443 q^{43} -2.00000 q^{45} -1.00000 q^{46} -6.47214 q^{47} -1.00000 q^{48} -1.00000 q^{50} -4.47214 q^{51} -4.47214 q^{52} +6.94427 q^{53} -1.00000 q^{54} -6.47214 q^{57} -2.00000 q^{58} -4.00000 q^{59} +2.00000 q^{60} +6.00000 q^{61} -6.47214 q^{62} +1.00000 q^{64} +8.94427 q^{65} -12.9443 q^{67} +4.47214 q^{68} +1.00000 q^{69} -12.9443 q^{71} +1.00000 q^{72} +2.94427 q^{73} -10.9443 q^{74} +1.00000 q^{75} +6.47214 q^{76} +4.47214 q^{78} +12.9443 q^{79} -2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} +6.47214 q^{83} -8.94427 q^{85} +12.9443 q^{86} +2.00000 q^{87} +17.4164 q^{89} -2.00000 q^{90} -1.00000 q^{92} +6.47214 q^{93} -6.47214 q^{94} -12.9443 q^{95} -1.00000 q^{96} -8.47214 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 4 * q^5 - 2 * q^6 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9} - 4 q^{10} - 2 q^{12} + 4 q^{15} + 2 q^{16} + 2 q^{18} + 4 q^{19} - 4 q^{20} - 2 q^{23} - 2 q^{24} - 2 q^{25} - 2 q^{27} - 4 q^{29} + 4 q^{30} - 4 q^{31} + 2 q^{32} + 2 q^{36} - 4 q^{37} + 4 q^{38} - 4 q^{40} + 12 q^{41} + 8 q^{43} - 4 q^{45} - 2 q^{46} - 4 q^{47} - 2 q^{48} - 2 q^{50} - 4 q^{53} - 2 q^{54} - 4 q^{57} - 4 q^{58} - 8 q^{59} + 4 q^{60} + 12 q^{61} - 4 q^{62} + 2 q^{64} - 8 q^{67} + 2 q^{69} - 8 q^{71} + 2 q^{72} - 12 q^{73} - 4 q^{74} + 2 q^{75} + 4 q^{76} + 8 q^{79} - 4 q^{80} + 2 q^{81} + 12 q^{82} + 4 q^{83} + 8 q^{86} + 4 q^{87} + 8 q^{89} - 4 q^{90} - 2 q^{92} + 4 q^{93} - 4 q^{94} - 8 q^{95} - 2 q^{96} - 8 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 4 * q^5 - 2 * q^6 + 2 * q^8 + 2 * q^9 - 4 * q^10 - 2 * q^12 + 4 * q^15 + 2 * q^16 + 2 * q^18 + 4 * q^19 - 4 * q^20 - 2 * q^23 - 2 * q^24 - 2 * q^25 - 2 * q^27 - 4 * q^29 + 4 * q^30 - 4 * q^31 + 2 * q^32 + 2 * q^36 - 4 * q^37 + 4 * q^38 - 4 * q^40 + 12 * q^41 + 8 * q^43 - 4 * q^45 - 2 * q^46 - 4 * q^47 - 2 * q^48 - 2 * q^50 - 4 * q^53 - 2 * q^54 - 4 * q^57 - 4 * q^58 - 8 * q^59 + 4 * q^60 + 12 * q^61 - 4 * q^62 + 2 * q^64 - 8 * q^67 + 2 * q^69 - 8 * q^71 + 2 * q^72 - 12 * q^73 - 4 * q^74 + 2 * q^75 + 4 * q^76 + 8 * q^79 - 4 * q^80 + 2 * q^81 + 12 * q^82 + 4 * q^83 + 8 * q^86 + 4 * q^87 + 8 * q^89 - 4 * q^90 - 2 * q^92 + 4 * q^93 - 4 * q^94 - 8 * q^95 - 2 * q^96 - 8 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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