LMFDB - Embedded newform 6762.2.a.bz.1.2

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			1.2
Root			$-0.618034$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{12} +4.47214 q^{13} +2.00000 q^{15} +1.00000 q^{16} -4.47214 q^{17} +1.00000 q^{18} -2.47214 q^{19} -2.00000 q^{20} -1.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} +4.47214 q^{26} -1.00000 q^{27} -2.00000 q^{29} +2.00000 q^{30} +2.47214 q^{31} +1.00000 q^{32} -4.47214 q^{34} +1.00000 q^{36} +6.94427 q^{37} -2.47214 q^{38} -4.47214 q^{39} -2.00000 q^{40} +6.00000 q^{41} -4.94427 q^{43} -2.00000 q^{45} -1.00000 q^{46} +2.47214 q^{47} -1.00000 q^{48} -1.00000 q^{50} +4.47214 q^{51} +4.47214 q^{52} -10.9443 q^{53} -1.00000 q^{54} +2.47214 q^{57} -2.00000 q^{58} -4.00000 q^{59} +2.00000 q^{60} +6.00000 q^{61} +2.47214 q^{62} +1.00000 q^{64} -8.94427 q^{65} +4.94427 q^{67} -4.47214 q^{68} +1.00000 q^{69} +4.94427 q^{71} +1.00000 q^{72} -14.9443 q^{73} +6.94427 q^{74} +1.00000 q^{75} -2.47214 q^{76} -4.47214 q^{78} -4.94427 q^{79} -2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -2.47214 q^{83} +8.94427 q^{85} -4.94427 q^{86} +2.00000 q^{87} -9.41641 q^{89} -2.00000 q^{90} -1.00000 q^{92} -2.47214 q^{93} +2.47214 q^{94} +4.94427 q^{95} -1.00000 q^{96} +0.472136 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.287062\pi$
$13$	4.47214	1.24035	0.620174	+	0.784465i	$0.287062\pi$
$14$	0	0
$15$	2.00000	0.516398
$16$	1.00000	0.250000
$17$	−4.47214	−1.08465	−0.542326	−	0.840168i	$-0.682456\pi$
$17$	−4.47214	−1.08465	−0.542326	+	0.840168i	$0.682456\pi$
$18$	1.00000	0.235702
$19$	−2.47214	−0.567147	−0.283573	−	0.958951i	$-0.591520\pi$
$19$	−2.47214	−0.567147	−0.283573	+	0.958951i	$0.591520\pi$
$20$	−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
$31$	2.47214	0.444009	0.222004	−	0.975046i	$-0.428740\pi$
$31$	2.47214	0.444009	0.222004	+	0.975046i	$0.428740\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−4.47214	−0.766965
$35$	0	0
$36$	1.00000	0.166667
$37$	6.94427	1.14163	0.570816	−	0.821078i	$-0.306627\pi$
$37$	6.94427	1.14163	0.570816	+	0.821078i	$0.306627\pi$
$38$	−2.47214	−0.401033
$39$	−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
$43$	−4.94427	−0.753994	−0.376997	−	0.926214i	$-0.623043\pi$
$43$	−4.94427	−0.753994	−0.376997	+	0.926214i	$0.623043\pi$
$44$	0	0
$45$	−2.00000	−0.298142
$46$	−1.00000	−0.147442
$47$	2.47214	0.360598	0.180299	−	0.983612i	$-0.442293\pi$
$47$	2.47214	0.360598	0.180299	+	0.983612i	$0.442293\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	−1.00000	−0.141421
$51$	4.47214	0.626224
$52$	4.47214	0.620174
$53$	−10.9443	−1.50331	−0.751656	−	0.659556i	$-0.770744\pi$
$53$	−10.9443	−1.50331	−0.751656	+	0.659556i	$0.770744\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	2.47214	0.327442
$58$	−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	$0.374508\pi$
$62$	2.47214	0.313962
$63$	0	0
$64$	1.00000	0.125000
$65$	−8.94427	−1.10940
$66$	0	0
$67$	4.94427	0.604039	0.302019	−	0.953302i	$-0.402339\pi$
$67$	4.94427	0.604039	0.302019	+	0.953302i	$0.402339\pi$
$68$	−4.47214	−0.542326
$69$	1.00000	0.120386
$70$	0	0
$71$	4.94427	0.586777	0.293389	−	0.955993i	$-0.405217\pi$
$71$	4.94427	0.586777	0.293389	+	0.955993i	$0.405217\pi$
$72$	1.00000	0.117851
$73$	−14.9443	−1.74909	−0.874547	−	0.484940i	$-0.838841\pi$
$73$	−14.9443	−1.74909	−0.874547	+	0.484940i	$0.838841\pi$
$74$	6.94427	0.807255
$75$	1.00000	0.115470
$76$	−2.47214	−0.283573
$77$	0	0
$78$	−4.47214	−0.506370
$79$	−4.94427	−0.556274	−0.278137	−	0.960541i	$-0.589717\pi$
$79$	−4.94427	−0.556274	−0.278137	+	0.960541i	$0.589717\pi$
$80$	−2.00000	−0.223607
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	−2.47214	−0.271352	−0.135676	−	0.990753i	$-0.543321\pi$
$83$	−2.47214	−0.271352	−0.135676	+	0.990753i	$0.543321\pi$
$84$	0	0
$85$	8.94427	0.970143
$86$	−4.94427	−0.533155
$87$	2.00000	0.214423
$88$	0	0
$89$	−9.41641	−0.998137	−0.499069	−	0.866562i	$-0.666324\pi$
$89$	−9.41641	−0.998137	−0.499069	+	0.866562i	$0.666324\pi$
$90$	−2.00000	−0.210819
$91$	0	0
$92$	−1.00000	−0.104257
$93$	−2.47214	−0.256349
$94$	2.47214	0.254981
$95$	4.94427	0.507272
$96$	−1.00000	−0.102062
$97$	0.472136	0.0479381	0.0239691	−	0.999713i	$-0.492370\pi$
$97$	0.472136	0.0479381	0.0239691	+	0.999713i	$0.492370\pi$
$98$	0	0
$99$	0	0
$100$	−1.00000	−0.100000
$101$	9.41641	0.936968	0.468484	−	0.883472i	$-0.344800\pi$
$101$	9.41641	0.936968	0.468484	+	0.883472i	$0.344800\pi$
$102$	4.47214	0.442807
$103$	4.94427	0.487174	0.243587	−	0.969879i	$-0.421676\pi$
$103$	4.94427	0.487174	0.243587	+	0.969879i	$0.421676\pi$
$104$	4.47214	0.438529
$105$	0	0
$106$	−10.9443	−1.06300
$107$	−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
$109$	−2.94427	−0.282010	−0.141005	−	0.990009i	$-0.545033\pi$
$109$	−2.94427	−0.282010	−0.141005	+	0.990009i	$0.545033\pi$
$110$	0	0
$111$	−6.94427	−0.659121
$112$	0	0
$113$	−18.9443	−1.78213	−0.891064	−	0.453878i	$-0.850040\pi$
$113$	−18.9443	−1.78213	−0.891064	+	0.453878i	$0.850040\pi$
$114$	2.47214	0.231537
$115$	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
$124$	2.47214	0.222004
$125$	12.0000	1.07331
$126$	0	0
$127$	20.9443	1.85850	0.929252	−	0.369447i	$-0.120453\pi$
$127$	20.9443	1.85850	0.929252	+	0.369447i	$0.120453\pi$
$128$	1.00000	0.0883883
$129$	4.94427	0.435319
$130$	−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
$134$	4.94427	0.427120
$135$	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	$-0.623843\pi$
$139$	−8.94427	−0.758643	−0.379322	+	0.925265i	$0.623843\pi$
$140$	0	0
$141$	−2.47214	−0.208191
$142$	4.94427	0.414914
$143$	0	0
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	−14.9443	−1.23680
$147$	0	0
$148$	6.94427	0.570816
$149$	−1.05573	−0.0864886	−0.0432443	−	0.999065i	$-0.513769\pi$
$149$	−1.05573	−0.0864886	−0.0432443	+	0.999065i	$0.513769\pi$
$150$	1.00000	0.0816497
$151$	4.94427	0.402359	0.201180	−	0.979554i	$-0.435523\pi$
$151$	4.94427	0.402359	0.201180	+	0.979554i	$0.435523\pi$
$152$	−2.47214	−0.200517
$153$	−4.47214	−0.361551
$154$	0	0
$155$	−4.94427	−0.397133
$156$	−4.47214	−0.358057
$157$	2.94427	0.234978	0.117489	−	0.993074i	$-0.462515\pi$
$157$	2.94427	0.234978	0.117489	+	0.993074i	$0.462515\pi$
$158$	−4.94427	−0.393345
$159$	10.9443	0.867937
$160$	−2.00000	−0.158114
$161$	0	0
$162$	1.00000	0.0785674
$163$	−8.94427	−0.700569	−0.350285	−	0.936643i	$-0.613915\pi$
$163$	−8.94427	−0.700569	−0.350285	+	0.936643i	$0.613915\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−2.47214	−0.191875
$167$	15.4164	1.19296	0.596479	−	0.802629i	$-0.296566\pi$
$167$	15.4164	1.19296	0.596479	+	0.802629i	$0.296566\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	8.94427	0.685994
$171$	−2.47214	−0.189049
$172$	−4.94427	−0.376997
$173$	−0.472136	−0.0358958	−0.0179479	−	0.999839i	$-0.505713\pi$
$173$	−0.472136	−0.0358958	−0.0179479	+	0.999839i	$0.505713\pi$
$174$	2.00000	0.151620
$175$	0	0
$176$	0	0
$177$	4.00000	0.300658
$178$	−9.41641	−0.705790
$179$	−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
$185$	−13.8885	−1.02111
$186$	−2.47214	−0.181266
$187$	0	0
$188$	2.47214	0.180299
$189$	0	0
$190$	4.94427	0.358695
$191$	−3.05573	−0.221105	−0.110552	−	0.993870i	$-0.535262\pi$
$191$	−3.05573	−0.221105	−0.110552	+	0.993870i	$0.535262\pi$
$192$	−1.00000	−0.0721688
$193$	11.8885	0.855756	0.427878	−	0.903836i	$-0.359261\pi$
$193$	11.8885	0.855756	0.427878	+	0.903836i	$0.359261\pi$
$194$	0.472136	0.0338974
$195$	8.94427	0.640513
$196$	0	0
$197$	−14.9443	−1.06474	−0.532368	−	0.846513i	$-0.678698\pi$
$197$	−14.9443	−1.06474	−0.532368	+	0.846513i	$0.678698\pi$
$198$	0	0
$199$	−3.05573	−0.216615	−0.108307	−	0.994117i	$-0.534543\pi$
$199$	−3.05573	−0.216615	−0.108307	+	0.994117i	$0.534543\pi$
$200$	−1.00000	−0.0707107
$201$	−4.94427	−0.348742
$202$	9.41641	0.662536
$203$	0	0
$204$	4.47214	0.313112
$205$	−12.0000	−0.838116
$206$	4.94427	0.344484
$207$	−1.00000	−0.0695048
$208$	4.47214	0.310087
$209$	0	0
$210$	0	0
$211$	−0.944272	−0.0650064	−0.0325032	−	0.999472i	$-0.510348\pi$
$211$	−0.944272	−0.0650064	−0.0325032	+	0.999472i	$0.510348\pi$
$212$	−10.9443	−0.751656
$213$	−4.94427	−0.338776
$214$	−8.00000	−0.546869
$215$	9.88854	0.674393
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−2.94427	−0.199411
$219$	14.9443	1.00984
$220$	0	0
$221$	−20.0000	−1.34535
$222$	−6.94427	−0.466069
$223$	−18.4721	−1.23699	−0.618493	−	0.785790i	$-0.712256\pi$
$223$	−18.4721	−1.23699	−0.618493	+	0.785790i	$0.712256\pi$
$224$	0	0
$225$	−1.00000	−0.0666667
$226$	−18.9443	−1.26015
$227$	−5.52786	−0.366897	−0.183449	−	0.983029i	$-0.558726\pi$
$227$	−5.52786	−0.366897	−0.183449	+	0.983029i	$0.558726\pi$
$228$	2.47214	0.163721
$229$	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
$235$	−4.94427	−0.322529
$236$	−4.00000	−0.260378
$237$	4.94427	0.321165
$238$	0	0
$239$	20.9443	1.35477	0.677386	−	0.735628i	$-0.263113\pi$
$239$	20.9443	1.35477	0.677386	+	0.735628i	$0.263113\pi$
$240$	2.00000	0.129099
$241$	−20.4721	−1.31873	−0.659363	−	0.751825i	$-0.729174\pi$
$241$	−20.4721	−1.31873	−0.659363	+	0.751825i	$0.729174\pi$
$242$	−11.0000	−0.707107
$243$	−1.00000	−0.0641500
$244$	6.00000	0.384111
$245$	0	0
$246$	−6.00000	−0.382546
$247$	−11.0557	−0.703459
$248$	2.47214	0.156981
$249$	2.47214	0.156665
$250$	12.0000	0.758947
$251$	5.52786	0.348916	0.174458	−	0.984665i	$-0.444183\pi$
$251$	5.52786	0.348916	0.174458	+	0.984665i	$0.444183\pi$
$252$	0	0
$253$	0	0
$254$	20.9443	1.31416
$255$	−8.94427	−0.560112
$256$	1.00000	0.0625000
$257$	2.94427	0.183659	0.0918293	−	0.995775i	$-0.470729\pi$
$257$	2.94427	0.183659	0.0918293	+	0.995775i	$0.470729\pi$
$258$	4.94427	0.307817
$259$	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
$265$	21.8885	1.34460
$266$	0	0
$267$	9.41641	0.576275
$268$	4.94427	0.302019
$269$	−16.4721	−1.00432	−0.502162	−	0.864774i	$-0.667462\pi$
$269$	−16.4721	−1.00432	−0.502162	+	0.864774i	$0.667462\pi$
$270$	2.00000	0.121716
$271$	−15.4164	−0.936480	−0.468240	−	0.883601i	$-0.655112\pi$
$271$	−15.4164	−0.936480	−0.468240	+	0.883601i	$0.655112\pi$
$272$	−4.47214	−0.271163
$273$	0	0
$274$	−14.0000	−0.845771
$275$	0	0
$276$	1.00000	0.0601929
$277$	−11.8885	−0.714313	−0.357157	−	0.934044i	$-0.616254\pi$
$277$	−11.8885	−0.714313	−0.357157	+	0.934044i	$0.616254\pi$
$278$	−8.94427	−0.536442
$279$	2.47214	0.148003
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	−2.47214	−0.147214
$283$	−18.4721	−1.09805	−0.549027	−	0.835804i	$-0.685002\pi$
$283$	−18.4721	−1.09805	−0.549027	+	0.835804i	$0.685002\pi$
$284$	4.94427	0.293389
$285$	−4.94427	−0.292873
$286$	0	0
$287$	0	0
$288$	1.00000	0.0589256
$289$	3.00000	0.176471
$290$	4.00000	0.234888
$291$	−0.472136	−0.0276771
$292$	−14.9443	−0.874547
$293$	−27.8885	−1.62927	−0.814633	−	0.579977i	$-0.803062\pi$
$293$	−27.8885	−1.62927	−0.814633	+	0.579977i	$0.803062\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	6.94427	0.403628
$297$	0	0
$298$	−1.05573	−0.0611567
$299$	−4.47214	−0.258630
$300$	1.00000	0.0577350
$301$	0	0
$302$	4.94427	0.284511
$303$	−9.41641	−0.540958
$304$	−2.47214	−0.141787
$305$	−12.0000	−0.687118
$306$	−4.47214	−0.255655
$307$	−24.9443	−1.42364	−0.711822	−	0.702360i	$-0.752130\pi$
$307$	−24.9443	−1.42364	−0.711822	+	0.702360i	$0.752130\pi$
$308$	0	0
$309$	−4.94427	−0.281270
$310$	−4.94427	−0.280816
$311$	2.47214	0.140182	0.0700910	−	0.997541i	$-0.477671\pi$
$311$	2.47214	0.140182	0.0700910	+	0.997541i	$0.477671\pi$
$312$	−4.47214	−0.253185
$313$	13.4164	0.758340	0.379170	−	0.925327i	$-0.376210\pi$
$313$	13.4164	0.758340	0.379170	+	0.925327i	$0.376210\pi$
$314$	2.94427	0.166155
$315$	0	0
$316$	−4.94427	−0.278137
$317$	−13.0557	−0.733283	−0.366641	−	0.930362i	$-0.619492\pi$
$317$	−13.0557	−0.733283	−0.366641	+	0.930362i	$0.619492\pi$
$318$	10.9443	0.613724
$319$	0	0
$320$	−2.00000	−0.111803
$321$	8.00000	0.446516
$322$	0	0
$323$	11.0557	0.615157
$324$	1.00000	0.0555556
$325$	−4.47214	−0.248069
$326$	−8.94427	−0.495377
$327$	2.94427	0.162819
$328$	6.00000	0.331295
$329$	0	0
$330$	0	0
$331$	−13.8885	−0.763383	−0.381692	−	0.924290i	$-0.624658\pi$
$331$	−13.8885	−0.763383	−0.381692	+	0.924290i	$0.624658\pi$
$332$	−2.47214	−0.135676
$333$	6.94427	0.380544
$334$	15.4164	0.843548
$335$	−9.88854	−0.540269
$336$	0	0
$337$	−2.94427	−0.160385	−0.0801924	−	0.996779i	$-0.525553\pi$
$337$	−2.94427	−0.160385	−0.0801924	+	0.996779i	$0.525553\pi$
$338$	7.00000	0.380750
$339$	18.9443	1.02891
$340$	8.94427	0.485071
$341$	0	0
$342$	−2.47214	−0.133678
$343$	0	0
$344$	−4.94427	−0.266577
$345$	−2.00000	−0.107676
$346$	−0.472136	−0.0253822
$347$	−0.944272	−0.0506912	−0.0253456	−	0.999679i	$-0.508069\pi$
$347$	−0.944272	−0.0506912	−0.0253456	+	0.999679i	$0.508069\pi$
$348$	2.00000	0.107211
$349$	−24.4721	−1.30996	−0.654982	−	0.755645i	$-0.727324\pi$
$349$	−24.4721	−1.30996	−0.654982	+	0.755645i	$0.727324\pi$
$350$	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
$355$	−9.88854	−0.524829
$356$	−9.41641	−0.499069
$357$	0	0
$358$	−20.0000	−1.05703
$359$	−30.8328	−1.62729	−0.813647	−	0.581359i	$-0.802521\pi$
$359$	−30.8328	−1.62729	−0.813647	+	0.581359i	$0.802521\pi$
$360$	−2.00000	−0.105409
$361$	−12.8885	−0.678344
$362$	−2.00000	−0.105118
$363$	11.0000	0.577350
$364$	0	0
$365$	29.8885	1.56444
$366$	−6.00000	−0.313625
$367$	−30.8328	−1.60946	−0.804730	−	0.593641i	$-0.797690\pi$
$367$	−30.8328	−1.60946	−0.804730	+	0.593641i	$0.797690\pi$
$368$	−1.00000	−0.0521286
$369$	6.00000	0.312348
$370$	−13.8885	−0.722031
$371$	0	0
$372$	−2.47214	−0.128174
$373$	6.94427	0.359561	0.179780	−	0.983707i	$-0.442461\pi$
$373$	6.94427	0.359561	0.179780	+	0.983707i	$0.442461\pi$
$374$	0	0
$375$	−12.0000	−0.619677
$376$	2.47214	0.127491
$377$	−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	$0.288591\pi$
$380$	4.94427	0.253636
$381$	−20.9443	−1.07301
$382$	−3.05573	−0.156345
$383$	−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
$386$	11.8885	0.605111
$387$	−4.94427	−0.251331
$388$	0.472136	0.0239691
$389$	−18.9443	−0.960513	−0.480256	−	0.877128i	$-0.659456\pi$
$389$	−18.9443	−0.960513	−0.480256	+	0.877128i	$0.659456\pi$
$390$	8.94427	0.452911
$391$	4.47214	0.226166
$392$	0	0
$393$	12.0000	0.605320
$394$	−14.9443	−0.752882
$395$	9.88854	0.497547
$396$	0	0
$397$	36.4721	1.83048	0.915242	−	0.402905i	$-0.131999\pi$
$397$	36.4721	1.83048	0.915242	+	0.402905i	$0.131999\pi$
$398$	−3.05573	−0.153170
$399$	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	$0.685111\pi$
$402$	−4.94427	−0.246598
$403$	11.0557	0.550725
$404$	9.41641	0.468484
$405$	−2.00000	−0.0993808
$406$	0	0
$407$	0	0
$408$	4.47214	0.221404
$409$	15.8885	0.785638	0.392819	−	0.919616i	$-0.371500\pi$
$409$	15.8885	0.785638	0.392819	+	0.919616i	$0.371500\pi$
$410$	−12.0000	−0.592638
$411$	14.0000	0.690569
$412$	4.94427	0.243587
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	4.94427	0.242705
$416$	4.47214	0.219265
$417$	8.94427	0.438003
$418$	0	0
$419$	7.41641	0.362315	0.181158	−	0.983454i	$-0.442016\pi$
$419$	7.41641	0.362315	0.181158	+	0.983454i	$0.442016\pi$
$420$	0	0
$421$	32.8328	1.60017	0.800087	−	0.599884i	$-0.204787\pi$
$421$	32.8328	1.60017	0.800087	+	0.599884i	$0.204787\pi$
$422$	−0.944272	−0.0459664
$423$	2.47214	0.120199
$424$	−10.9443	−0.531501
$425$	4.47214	0.216930
$426$	−4.94427	−0.239551
$427$	0	0
$428$	−8.00000	−0.386695
$429$	0	0
$430$	9.88854	0.476868
$431$	17.8885	0.861661	0.430830	−	0.902433i	$-0.358221\pi$
$431$	17.8885	0.861661	0.430830	+	0.902433i	$0.358221\pi$
$432$	−1.00000	−0.0481125
$433$	−25.4164	−1.22143	−0.610717	−	0.791849i	$-0.709119\pi$
$433$	−25.4164	−1.22143	−0.610717	+	0.791849i	$0.709119\pi$
$434$	0	0
$435$	−4.00000	−0.191785
$436$	−2.94427	−0.141005
$437$	2.47214	0.118258
$438$	14.9443	0.714065
$439$	−28.3607	−1.35358	−0.676791	−	0.736175i	$-0.736630\pi$
$439$	−28.3607	−1.35358	−0.676791	+	0.736175i	$0.736630\pi$
$440$	0	0
$441$	0	0
$442$	−20.0000	−0.951303
$443$	34.8328	1.65496	0.827479	−	0.561497i	$-0.189775\pi$
$443$	34.8328	1.65496	0.827479	+	0.561497i	$0.189775\pi$
$444$	−6.94427	−0.329561
$445$	18.8328	0.892761
$446$	−18.4721	−0.874681
$447$	1.05573	0.0499342
$448$	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
$452$	−18.9443	−0.891064
$453$	−4.94427	−0.232302
$454$	−5.52786	−0.259436
$455$	0	0
$456$	2.47214	0.115768
$457$	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
$461$	−6.58359	−0.306628	−0.153314	−	0.988177i	$-0.548995\pi$
$461$	−6.58359	−0.306628	−0.153314	+	0.988177i	$0.548995\pi$
$462$	0	0
$463$	6.11146	0.284023	0.142012	−	0.989865i	$-0.454643\pi$
$463$	6.11146	0.284023	0.142012	+	0.989865i	$0.454643\pi$
$464$	−2.00000	−0.0928477
$465$	4.94427	0.229285
$466$	−6.00000	−0.277945
$467$	−33.3050	−1.54117	−0.770585	−	0.637338i	$-0.780036\pi$
$467$	−33.3050	−1.54117	−0.770585	+	0.637338i	$0.780036\pi$
$468$	4.47214	0.206725
$469$	0	0
$470$	−4.94427	−0.228062
$471$	−2.94427	−0.135665
$472$	−4.00000	−0.184115
$473$	0	0
$474$	4.94427	0.227098
$475$	2.47214	0.113429
$476$	0	0
$477$	−10.9443	−0.501104
$478$	20.9443	0.957969
$479$	4.94427	0.225910	0.112955	−	0.993600i	$-0.463968\pi$
$479$	4.94427	0.225910	0.112955	+	0.993600i	$0.463968\pi$
$480$	2.00000	0.0912871
$481$	31.0557	1.41602
$482$	−20.4721	−0.932480
$483$	0	0
$484$	−11.0000	−0.500000
$485$	−0.944272	−0.0428772
$486$	−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
$491$	40.9443	1.84779	0.923895	−	0.382647i	$-0.124987\pi$
$491$	40.9443	1.84779	0.923895	+	0.382647i	$0.124987\pi$
$492$	−6.00000	−0.270501
$493$	8.94427	0.402830
$494$	−11.0557	−0.497421
$495$	0	0
$496$	2.47214	0.111002
$497$	0	0
$498$	2.47214	0.110779
$499$	32.9443	1.47479	0.737394	−	0.675463i	$-0.236056\pi$
$499$	32.9443	1.47479	0.737394	+	0.675463i	$0.236056\pi$
$500$	12.0000	0.536656
$501$	−15.4164	−0.688754
$502$	5.52786	0.246721
$503$	9.88854	0.440908	0.220454	−	0.975397i	$-0.429246\pi$
$503$	9.88854	0.440908	0.220454	+	0.975397i	$0.429246\pi$
$504$	0	0
$505$	−18.8328	−0.838049
$506$	0	0
$507$	−7.00000	−0.310881
$508$	20.9443	0.929252
$509$	−6.58359	−0.291813	−0.145906	−	0.989298i	$-0.546610\pi$
$509$	−6.58359	−0.291813	−0.145906	+	0.989298i	$0.546610\pi$
$510$	−8.94427	−0.396059
$511$	0	0
$512$	1.00000	0.0441942
$513$	2.47214	0.109147
$514$	2.94427	0.129866
$515$	−9.88854	−0.435741
$516$	4.94427	0.217659
$517$	0	0
$518$	0	0
$519$	0.472136	0.0207245
$520$	−8.94427	−0.392232
$521$	−33.4164	−1.46400	−0.732000	−	0.681305i	$-0.761413\pi$
$521$	−33.4164	−1.46400	−0.732000	+	0.681305i	$0.761413\pi$
$522$	−2.00000	−0.0875376
$523$	25.3050	1.10651	0.553254	−	0.833013i	$-0.313386\pi$
$523$	25.3050	1.10651	0.553254	+	0.833013i	$0.313386\pi$
$524$	−12.0000	−0.524222
$525$	0	0
$526$	−24.0000	−1.04645
$527$	−11.0557	−0.481595
$528$	0	0
$529$	1.00000	0.0434783
$530$	21.8885	0.950778
$531$	−4.00000	−0.173585
$532$	0	0
$533$	26.8328	1.16226
$534$	9.41641	0.407488
$535$	16.0000	0.691740
$536$	4.94427	0.213560
$537$	20.0000	0.863064
$538$	−16.4721	−0.710164
$539$	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	$0.568964\pi$
$542$	−15.4164	−0.662191
$543$	2.00000	0.0858282
$544$	−4.47214	−0.191741
$545$	5.88854	0.252238
$546$	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
$551$	4.94427	0.210633
$552$	1.00000	0.0425628
$553$	0	0
$554$	−11.8885	−0.505096
$555$	13.8885	0.589536
$556$	−8.94427	−0.379322
$557$	−17.0557	−0.722674	−0.361337	−	0.932435i	$-0.617680\pi$
$557$	−17.0557	−0.722674	−0.361337	+	0.932435i	$0.617680\pi$
$558$	2.47214	0.104654
$559$	−22.1115	−0.935215
$560$	0	0
$561$	0	0
$562$	10.0000	0.421825
$563$	−26.4721	−1.11567	−0.557834	−	0.829953i	$-0.688367\pi$
$563$	−26.4721	−1.11567	−0.557834	+	0.829953i	$0.688367\pi$
$564$	−2.47214	−0.104096
$565$	37.8885	1.59398
$566$	−18.4721	−0.776442
$567$	0	0
$568$	4.94427	0.207457
$569$	−7.88854	−0.330705	−0.165352	−	0.986235i	$-0.552876\pi$
$569$	−7.88854	−0.330705	−0.165352	+	0.986235i	$0.552876\pi$
$570$	−4.94427	−0.207093
$571$	−12.9443	−0.541701	−0.270850	−	0.962621i	$-0.587305\pi$
$571$	−12.9443	−0.541701	−0.270850	+	0.962621i	$0.587305\pi$
$572$	0	0
$573$	3.05573	0.127655
$574$	0	0
$575$	1.00000	0.0417029
$576$	1.00000	0.0416667
$577$	2.94427	0.122572	0.0612858	−	0.998120i	$-0.480480\pi$
$577$	2.94427	0.122572	0.0612858	+	0.998120i	$0.480480\pi$
$578$	3.00000	0.124784
$579$	−11.8885	−0.494071
$580$	4.00000	0.166091
$581$	0	0
$582$	−0.472136	−0.0195707
$583$	0	0
$584$	−14.9443	−0.618398
$585$	−8.94427	−0.369800
$586$	−27.8885	−1.15207
$587$	0.944272	0.0389743	0.0194871	−	0.999810i	$-0.493797\pi$
$587$	0.944272	0.0389743	0.0194871	+	0.999810i	$0.493797\pi$
$588$	0	0
$589$	−6.11146	−0.251818
$590$	8.00000	0.329355
$591$	14.9443	0.614725
$592$	6.94427	0.285408
$593$	28.8328	1.18402	0.592011	−	0.805930i	$-0.298334\pi$
$593$	28.8328	1.18402	0.592011	+	0.805930i	$0.298334\pi$
$594$	0	0
$595$	0	0
$596$	−1.05573	−0.0432443
$597$	3.05573	0.125063
$598$	−4.47214	−0.182879
$599$	33.8885	1.38465	0.692324	−	0.721587i	$-0.256587\pi$
$599$	33.8885	1.38465	0.692324	+	0.721587i	$0.256587\pi$
$600$	1.00000	0.0408248
$601$	−40.8328	−1.66561	−0.832803	−	0.553570i	$-0.813265\pi$
$601$	−40.8328	−1.66561	−0.832803	+	0.553570i	$0.813265\pi$
$602$	0	0
$603$	4.94427	0.201346
$604$	4.94427	0.201180
$605$	22.0000	0.894427
$606$	−9.41641	−0.382515
$607$	28.3607	1.15112	0.575562	−	0.817758i	$-0.304783\pi$
$607$	28.3607	1.15112	0.575562	+	0.817758i	$0.304783\pi$
$608$	−2.47214	−0.100258
$609$	0	0
$610$	−12.0000	−0.485866
$611$	11.0557	0.447267
$612$	−4.47214	−0.180775
$613$	40.8328	1.64922	0.824611	−	0.565700i	$-0.191394\pi$
$613$	40.8328	1.64922	0.824611	+	0.565700i	$0.191394\pi$
$614$	−24.9443	−1.00667
$615$	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	$0.324678\pi$
$618$	−4.94427	−0.198888
$619$	34.4721	1.38555	0.692776	−	0.721153i	$-0.256387\pi$
$619$	34.4721	1.38555	0.692776	+	0.721153i	$0.256387\pi$
$620$	−4.94427	−0.198567
$621$	1.00000	0.0401286
$622$	2.47214	0.0991236
$623$	0	0
$624$	−4.47214	−0.179029
$625$	−19.0000	−0.760000
$626$	13.4164	0.536228
$627$	0	0
$628$	2.94427	0.117489
$629$	−31.0557	−1.23827
$630$	0	0
$631$	−11.0557	−0.440122	−0.220061	−	0.975486i	$-0.570626\pi$
$631$	−11.0557	−0.440122	−0.220061	+	0.975486i	$0.570626\pi$
$632$	−4.94427	−0.196673
$633$	0.944272	0.0375314
$634$	−13.0557	−0.518509
$635$	−41.8885	−1.66230
$636$	10.9443	0.433969
$637$	0	0
$638$	0	0
$639$	4.94427	0.195592
$640$	−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
$643$	10.4721	0.412981	0.206490	−	0.978449i	$-0.433796\pi$
$643$	10.4721	0.412981	0.206490	+	0.978449i	$0.433796\pi$
$644$	0	0
$645$	−9.88854	−0.389361
$646$	11.0557	0.434982
$647$	−12.3607	−0.485948	−0.242974	−	0.970033i	$-0.578123\pi$
$647$	−12.3607	−0.485948	−0.242974	+	0.970033i	$0.578123\pi$
$648$	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	$0.462544\pi$
$654$	2.94427	0.115130
$655$	24.0000	0.937758
$656$	6.00000	0.234261
$657$	−14.9443	−0.583032
$658$	0	0
$659$	46.8328	1.82435	0.912174	−	0.409804i	$-0.134403\pi$
$659$	46.8328	1.82435	0.912174	+	0.409804i	$0.134403\pi$
$660$	0	0
$661$	30.0000	1.16686	0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000	1.16686	0.583432	+	0.812162i	$0.301709\pi$
$662$	−13.8885	−0.539794
$663$	20.0000	0.776736
$664$	−2.47214	−0.0959375
$665$	0	0
$666$	6.94427	0.269085
$667$	2.00000	0.0774403
$668$	15.4164	0.596479
$669$	18.4721	0.714174
$670$	−9.88854	−0.382028
$671$	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	$0.696248\pi$
$674$	−2.94427	−0.113409
$675$	1.00000	0.0384900
$676$	7.00000	0.269231
$677$	4.11146	0.158016	0.0790080	−	0.996874i	$-0.474825\pi$
$677$	4.11146	0.158016	0.0790080	+	0.996874i	$0.474825\pi$
$678$	18.9443	0.727550
$679$	0	0
$680$	8.94427	0.342997
$681$	5.52786	0.211828
$682$	0	0
$683$	16.9443	0.648355	0.324177	−	0.945996i	$-0.394913\pi$
$683$	16.9443	0.648355	0.324177	+	0.945996i	$0.394913\pi$
$684$	−2.47214	−0.0945245
$685$	28.0000	1.06983
$686$	0	0
$687$	−14.0000	−0.534133
$688$	−4.94427	−0.188499
$689$	−48.9443	−1.86463
$690$	−2.00000	−0.0761387
$691$	15.0557	0.572747	0.286373	−	0.958118i	$-0.407550\pi$
$691$	15.0557	0.572747	0.286373	+	0.958118i	$0.407550\pi$
$692$	−0.472136	−0.0179479
$693$	0	0
$694$	−0.944272	−0.0358441
$695$	17.8885	0.678551
$696$	2.00000	0.0758098
$697$	−26.8328	−1.01637
$698$	−24.4721	−0.926284
$699$	6.00000	0.226941
$700$	0	0
$701$	−20.8328	−0.786845	−0.393422	−	0.919358i	$-0.628709\pi$
$701$	−20.8328	−0.786845	−0.393422	+	0.919358i	$0.628709\pi$
$702$	−4.47214	−0.168790
$703$	−17.1672	−0.647473
$704$	0	0
$705$	4.94427	0.186212
$706$	−2.00000	−0.0752710
$707$	0	0
$708$	4.00000	0.150329
$709$	6.94427	0.260798	0.130399	−	0.991462i	$-0.458374\pi$
$709$	6.94427	0.260798	0.130399	+	0.991462i	$0.458374\pi$
$710$	−9.88854	−0.371110
$711$	−4.94427	−0.185425
$712$	−9.41641	−0.352895
$713$	−2.47214	−0.0925822
$714$	0	0
$715$	0	0
$716$	−20.0000	−0.747435
$717$	−20.9443	−0.782178
$718$	−30.8328	−1.15067
$719$	5.52786	0.206155	0.103077	−	0.994673i	$-0.467131\pi$
$719$	5.52786	0.206155	0.103077	+	0.994673i	$0.467131\pi$
$720$	−2.00000	−0.0745356
$721$	0	0
$722$	−12.8885	−0.479662
$723$	20.4721	0.761367
$724$	−2.00000	−0.0743294
$725$	2.00000	0.0742781
$726$	11.0000	0.408248
$727$	−3.05573	−0.113331	−0.0566653	−	0.998393i	$-0.518047\pi$
$727$	−3.05573	−0.113331	−0.0566653	+	0.998393i	$0.518047\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	29.8885	1.10622
$731$	22.1115	0.817822
$732$	−6.00000	−0.221766
$733$	50.9443	1.88167	0.940835	−	0.338866i	$-0.110043\pi$
$733$	50.9443	1.88167	0.940835	+	0.338866i	$0.110043\pi$
$734$	−30.8328	−1.13806
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	0	0
$738$	6.00000	0.220863
$739$	−8.94427	−0.329020	−0.164510	−	0.986375i	$-0.552604\pi$
$739$	−8.94427	−0.329020	−0.164510	+	0.986375i	$0.552604\pi$
$740$	−13.8885	−0.510553
$741$	11.0557	0.406142
$742$	0	0
$743$	−22.8328	−0.837655	−0.418827	−	0.908066i	$-0.637559\pi$
$743$	−22.8328	−0.837655	−0.418827	+	0.908066i	$0.637559\pi$
$744$	−2.47214	−0.0906329
$745$	2.11146	0.0773578
$746$	6.94427	0.254248
$747$	−2.47214	−0.0904507
$748$	0	0
$749$	0	0
$750$	−12.0000	−0.438178
$751$	17.8885	0.652762	0.326381	−	0.945238i	$-0.394171\pi$
$751$	17.8885	0.652762	0.326381	+	0.945238i	$0.394171\pi$
$752$	2.47214	0.0901495
$753$	−5.52786	−0.201447
$754$	−8.94427	−0.325731
$755$	−9.88854	−0.359881
$756$	0	0
$757$	−42.9443	−1.56084	−0.780418	−	0.625258i	$-0.784994\pi$
$757$	−42.9443	−1.56084	−0.780418	+	0.625258i	$0.784994\pi$
$758$	24.0000	0.871719
$759$	0	0
$760$	4.94427	0.179348
$761$	−21.0557	−0.763270	−0.381635	−	0.924313i	$-0.624639\pi$
$761$	−21.0557	−0.763270	−0.381635	+	0.924313i	$0.624639\pi$
$762$	−20.9443	−0.758731
$763$	0	0
$764$	−3.05573	−0.110552
$765$	8.94427	0.323381
$766$	−16.0000	−0.578103
$767$	−17.8885	−0.645918
$768$	−1.00000	−0.0360844
$769$	−30.3607	−1.09483	−0.547417	−	0.836860i	$-0.684389\pi$
$769$	−30.3607	−1.09483	−0.547417	+	0.836860i	$0.684389\pi$
$770$	0	0
$771$	−2.94427	−0.106035
$772$	11.8885	0.427878
$773$	−5.05573	−0.181842	−0.0909210	−	0.995858i	$-0.528981\pi$
$773$	−5.05573	−0.181842	−0.0909210	+	0.995858i	$0.528981\pi$
$774$	−4.94427	−0.177718
$775$	−2.47214	−0.0888017
$776$	0.472136	0.0169487
$777$	0	0
$778$	−18.9443	−0.679185
$779$	−14.8328	−0.531441
$780$	8.94427	0.320256
$781$	0	0
$782$	4.47214	0.159923
$783$	2.00000	0.0714742
$784$	0	0
$785$	−5.88854	−0.210171
$786$	12.0000	0.428026
$787$	0.583592	0.0208028	0.0104014	−	0.999946i	$-0.496689\pi$
$787$	0.583592	0.0208028	0.0104014	+	0.999946i	$0.496689\pi$
$788$	−14.9443	−0.532368
$789$	24.0000	0.854423
$790$	9.88854	0.351819
$791$	0	0
$792$	0	0
$793$	26.8328	0.952861
$794$	36.4721	1.29435
$795$	−21.8885	−0.776307
$796$	−3.05573	−0.108307
$797$	28.8328	1.02131	0.510655	−	0.859785i	$-0.329403\pi$
$797$	28.8328	1.02131	0.510655	+	0.859785i	$0.329403\pi$
$798$	0	0
$799$	−11.0557	−0.391124
$800$	−1.00000	−0.0353553
$801$	−9.41641	−0.332712
$802$	−22.0000	−0.776847
$803$	0	0
$804$	−4.94427	−0.174371
$805$	0	0
$806$	11.0557	0.389421
$807$	16.4721	0.579847
$808$	9.41641	0.331268
$809$	35.8885	1.26177	0.630887	−	0.775875i	$-0.282691\pi$
$809$	35.8885	1.26177	0.630887	+	0.775875i	$0.282691\pi$
$810$	−2.00000	−0.0702728
$811$	37.8885	1.33045	0.665223	−	0.746644i	$-0.268336\pi$
$811$	37.8885	1.33045	0.665223	+	0.746644i	$0.268336\pi$
$812$	0	0
$813$	15.4164	0.540677
$814$	0	0
$815$	17.8885	0.626608
$816$	4.47214	0.156556
$817$	12.2229	0.427626
$818$	15.8885	0.555530
$819$	0	0
$820$	−12.0000	−0.419058
$821$	−16.8328	−0.587469	−0.293735	−	0.955887i	$-0.594898\pi$
$821$	−16.8328	−0.587469	−0.293735	+	0.955887i	$0.594898\pi$
$822$	14.0000	0.488306
$823$	20.9443	0.730071	0.365036	−	0.930994i	$-0.381057\pi$
$823$	20.9443	0.730071	0.365036	+	0.930994i	$0.381057\pi$
$824$	4.94427	0.172242
$825$	0	0
$826$	0	0
$827$	33.8885	1.17842	0.589210	−	0.807980i	$-0.299439\pi$
$827$	33.8885	1.17842	0.589210	+	0.807980i	$0.299439\pi$
$828$	−1.00000	−0.0347524
$829$	−13.4164	−0.465971	−0.232986	−	0.972480i	$-0.574849\pi$
$829$	−13.4164	−0.465971	−0.232986	+	0.972480i	$0.574849\pi$
$830$	4.94427	0.171618
$831$	11.8885	0.412409
$832$	4.47214	0.155043
$833$	0	0
$834$	8.94427	0.309715
$835$	−30.8328	−1.06701
$836$	0	0
$837$	−2.47214	−0.0854495
$838$	7.41641	0.256196
$839$	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
$842$	32.8328	1.13149
$843$	−10.0000	−0.344418
$844$	−0.944272	−0.0325032
$845$	−14.0000	−0.481615
$846$	2.47214	0.0849938
$847$	0	0
$848$	−10.9443	−0.375828
$849$	18.4721	0.633962
$850$	4.47214	0.153393
$851$	−6.94427	−0.238047
$852$	−4.94427	−0.169388
$853$	6.36068	0.217786	0.108893	−	0.994054i	$-0.465269\pi$
$853$	6.36068	0.217786	0.108893	+	0.994054i	$0.465269\pi$
$854$	0	0
$855$	4.94427	0.169091
$856$	−8.00000	−0.273434
$857$	47.8885	1.63584	0.817921	−	0.575331i	$-0.195127\pi$
$857$	47.8885	1.63584	0.817921	+	0.575331i	$0.195127\pi$
$858$	0	0
$859$	−32.9443	−1.12404	−0.562022	−	0.827122i	$-0.689976\pi$
$859$	−32.9443	−1.12404	−0.562022	+	0.827122i	$0.689976\pi$
$860$	9.88854	0.337197
$861$	0	0
$862$	17.8885	0.609286
$863$	20.9443	0.712951	0.356476	−	0.934305i	$-0.383978\pi$
$863$	20.9443	0.712951	0.356476	+	0.934305i	$0.383978\pi$
$864$	−1.00000	−0.0340207
$865$	0.944272	0.0321062
$866$	−25.4164	−0.863685
$867$	−3.00000	−0.101885
$868$	0	0
$869$	0	0
$870$	−4.00000	−0.135613
$871$	22.1115	0.749218
$872$	−2.94427	−0.0997056
$873$	0.472136	0.0159794
$874$	2.47214	0.0836212
$875$	0	0
$876$	14.9443	0.504920
$877$	−34.7214	−1.17246	−0.586229	−	0.810146i	$-0.699388\pi$
$877$	−34.7214	−1.17246	−0.586229	+	0.810146i	$0.699388\pi$
$878$	−28.3607	−0.957127
$879$	27.8885	0.940657
$880$	0	0
$881$	−22.3607	−0.753350	−0.376675	−	0.926345i	$-0.622933\pi$
$881$	−22.3607	−0.753350	−0.376675	+	0.926345i	$0.622933\pi$
$882$	0	0
$883$	2.83282	0.0953318	0.0476659	−	0.998863i	$-0.484822\pi$
$883$	2.83282	0.0953318	0.0476659	+	0.998863i	$0.484822\pi$
$884$	−20.0000	−0.672673
$885$	−8.00000	−0.268917
$886$	34.8328	1.17023
$887$	5.52786	0.185608	0.0928038	−	0.995684i	$-0.470417\pi$
$887$	5.52786	0.185608	0.0928038	+	0.995684i	$0.470417\pi$
$888$	−6.94427	−0.233035
$889$	0	0
$890$	18.8328	0.631277
$891$	0	0
$892$	−18.4721	−0.618493
$893$	−6.11146	−0.204512
$894$	1.05573	0.0353088
$895$	40.0000	1.33705
$896$	0	0
$897$	4.47214	0.149320
$898$	18.0000	0.600668
$899$	−4.94427	−0.164901
$900$	−1.00000	−0.0333333
$901$	48.9443	1.63057
$902$	0	0
$903$	0	0
$904$	−18.9443	−0.630077
$905$	4.00000	0.132964
$906$	−4.94427	−0.164262
$907$	−41.8885	−1.39089	−0.695443	−	0.718581i	$-0.744792\pi$
$907$	−41.8885	−1.39089	−0.695443	+	0.718581i	$0.744792\pi$
$908$	−5.52786	−0.183449
$909$	9.41641	0.312323
$910$	0	0
$911$	−48.7214	−1.61421	−0.807105	−	0.590407i	$-0.798967\pi$
$911$	−48.7214	−1.61421	−0.807105	+	0.590407i	$0.798967\pi$
$912$	2.47214	0.0818606
$913$	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
$919$	35.0557	1.15638	0.578191	−	0.815902i	$-0.303759\pi$
$919$	35.0557	1.15638	0.578191	+	0.815902i	$0.303759\pi$
$920$	2.00000	0.0659380
$921$	24.9443	0.821942
$922$	−6.58359	−0.216819
$923$	22.1115	0.727807
$924$	0	0
$925$	−6.94427	−0.228326
$926$	6.11146	0.200835
$927$	4.94427	0.162391
$928$	−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	$0.595413\pi$
$930$	4.94427	0.162129
$931$	0	0
$932$	−6.00000	−0.196537
$933$	−2.47214	−0.0809341
$934$	−33.3050	−1.08977
$935$	0	0
$936$	4.47214	0.146176
$937$	−40.2492	−1.31488	−0.657442	−	0.753505i	$-0.728362\pi$
$937$	−40.2492	−1.31488	−0.657442	+	0.753505i	$0.728362\pi$
$938$	0	0
$939$	−13.4164	−0.437828
$940$	−4.94427	−0.161264
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	−2.94427	−0.0959296
$943$	−6.00000	−0.195387
$944$	−4.00000	−0.130189
$945$	0	0
$946$	0	0
$947$	0.944272	0.0306847	0.0153424	−	0.999882i	$-0.495116\pi$
$947$	0.944272	0.0306847	0.0153424	+	0.999882i	$0.495116\pi$
$948$	4.94427	0.160582
$949$	−66.8328	−2.16949
$950$	2.47214	0.0802067
$951$	13.0557	0.423361
$952$	0	0
$953$	−9.05573	−0.293344	−0.146672	−	0.989185i	$-0.546856\pi$
$953$	−9.05573	−0.293344	−0.146672	+	0.989185i	$0.546856\pi$
$954$	−10.9443	−0.354334
$955$	6.11146	0.197762
$956$	20.9443	0.677386
$957$	0	0
$958$	4.94427	0.159742
$959$	0	0
$960$	2.00000	0.0645497
$961$	−24.8885	−0.802856
$962$	31.0557	1.00128
$963$	−8.00000	−0.257796
$964$	−20.4721	−0.659363
$965$	−23.7771	−0.765412
$966$	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
$969$	−11.0557	−0.355161
$970$	−0.944272	−0.0303187
$971$	23.4164	0.751468	0.375734	−	0.926727i	$-0.377391\pi$
$971$	23.4164	0.751468	0.375734	+	0.926727i	$0.377391\pi$
$972$	−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
$981$	−2.94427	−0.0940034
$982$	40.9443	1.30658
$983$	22.1115	0.705246	0.352623	−	0.935765i	$-0.385290\pi$
$983$	22.1115	0.705246	0.352623	+	0.935765i	$0.385290\pi$
$984$	−6.00000	−0.191273
$985$	29.8885	0.952328
$986$	8.94427	0.284844
$987$	0	0
$988$	−11.0557	−0.351730
$989$	4.94427	0.157219
$990$	0	0
$991$	60.9443	1.93596	0.967979	−	0.251030i	$-0.0807693\pi$
$991$	60.9443	1.93596	0.967979	+	0.251030i	$0.0807693\pi$
$992$	2.47214	0.0784904
$993$	13.8885	0.440740
$994$	0	0
$995$	6.11146	0.193746
$996$	2.47214	0.0783326
$997$	46.3607	1.46826	0.734129	−	0.679010i	$-0.237591\pi$
$997$	46.3607	1.46826	0.734129	+	0.679010i	$0.237591\pi$
$998$	32.9443	1.04283
$999$	−6.94427	−0.219707

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{12} +4.47214 q^{13} +2.00000 q^{15} +1.00000 q^{16} -4.47214 q^{17} +1.00000 q^{18} -2.47214 q^{19} -2.00000 q^{20} -1.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} +4.47214 q^{26} -1.00000 q^{27} -2.00000 q^{29} +2.00000 q^{30} +2.47214 q^{31} +1.00000 q^{32} -4.47214 q^{34} +1.00000 q^{36} +6.94427 q^{37} -2.47214 q^{38} -4.47214 q^{39} -2.00000 q^{40} +6.00000 q^{41} -4.94427 q^{43} -2.00000 q^{45} -1.00000 q^{46} +2.47214 q^{47} -1.00000 q^{48} -1.00000 q^{50} +4.47214 q^{51} +4.47214 q^{52} -10.9443 q^{53} -1.00000 q^{54} +2.47214 q^{57} -2.00000 q^{58} -4.00000 q^{59} +2.00000 q^{60} +6.00000 q^{61} +2.47214 q^{62} +1.00000 q^{64} -8.94427 q^{65} +4.94427 q^{67} -4.47214 q^{68} +1.00000 q^{69} +4.94427 q^{71} +1.00000 q^{72} -14.9443 q^{73} +6.94427 q^{74} +1.00000 q^{75} -2.47214 q^{76} -4.47214 q^{78} -4.94427 q^{79} -2.00000 q^{80} +1.00000 q^{81} +6.00000 q^{82} -2.47214 q^{83} +8.94427 q^{85} -4.94427 q^{86} +2.00000 q^{87} -9.41641 q^{89} -2.00000 q^{90} -1.00000 q^{92} -2.47214 q^{93} +2.47214 q^{94} +4.94427 q^{95} -1.00000 q^{96} +0.472136 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 4 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 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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