LMFDB - Embedded newform 1305.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,2,Mod(1,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, 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("1305.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1305.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:	$10.4204774638$
Analytic rank:	$0$
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 145)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	1305.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.414214 q^{2} -1.82843 q^{4} -1.00000 q^{5} -4.82843 q^{7} +1.58579 q^{8} +0.414214 q^{10} -0.828427 q^{11} -2.00000 q^{13} +2.00000 q^{14} +3.00000 q^{16} -2.82843 q^{17} -4.82843 q^{19} +1.82843 q^{20} +0.343146 q^{22} +3.17157 q^{23} +1.00000 q^{25} +0.828427 q^{26} +8.82843 q^{28} -1.00000 q^{29} +6.48528 q^{31} -4.41421 q^{32} +1.17157 q^{34} +4.82843 q^{35} -8.48528 q^{37} +2.00000 q^{38} -1.58579 q^{40} +6.00000 q^{41} -6.00000 q^{43} +1.51472 q^{44} -1.31371 q^{46} +11.6569 q^{47} +16.3137 q^{49} -0.414214 q^{50} +3.65685 q^{52} +3.65685 q^{53} +0.828427 q^{55} -7.65685 q^{56} +0.414214 q^{58} -3.65685 q^{61} -2.68629 q^{62} -4.17157 q^{64} +2.00000 q^{65} +6.48528 q^{67} +5.17157 q^{68} -2.00000 q^{70} +15.3137 q^{71} +8.48528 q^{73} +3.51472 q^{74} +8.82843 q^{76} +4.00000 q^{77} -2.48528 q^{79} -3.00000 q^{80} -2.48528 q^{82} -7.17157 q^{83} +2.82843 q^{85} +2.48528 q^{86} -1.31371 q^{88} +7.65685 q^{89} +9.65685 q^{91} -5.79899 q^{92} -4.82843 q^{94} +4.82843 q^{95} -12.4853 q^{97} -6.75736 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{7} + 6 q^{8} - 2 q^{10} + 4 q^{11} - 4 q^{13} + 4 q^{14} + 6 q^{16} - 4 q^{19} - 2 q^{20} + 12 q^{22} + 12 q^{23} + 2 q^{25} - 4 q^{26} + 12 q^{28} - 2 q^{29} - 4 q^{31}+ \cdots - 22 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 4 * q^7 + 6 * q^8 - 2 * q^10 + 4 * q^11 - 4 * q^13 + 4 * q^14 + 6 * q^16 - 4 * q^19 - 2 * q^20 + 12 * q^22 + 12 * q^23 + 2 * q^25 - 4 * q^26 + 12 * q^28 - 2 * q^29 - 4 * q^31 - 6 * q^32 + 8 * q^34 + 4 * q^35 + 4 * q^38 - 6 * q^40 + 12 * q^41 - 12 * q^43 + 20 * q^44 + 20 * q^46 + 12 * q^47 + 10 * q^49 + 2 * q^50 - 4 * q^52 - 4 * q^53 - 4 * q^55 - 4 * q^56 - 2 * q^58 + 4 * q^61 - 28 * q^62 - 14 * q^64 + 4 * q^65 - 4 * q^67 + 16 * q^68 - 4 * q^70 + 8 * q^71 + 24 * q^74 + 12 * q^76 + 8 * q^77 + 12 * q^79 - 6 * q^80 + 12 * q^82 - 20 * q^83 - 12 * q^86 + 20 * q^88 + 4 * q^89 + 8 * q^91 + 28 * q^92 - 4 * q^94 + 4 * q^95 - 8 * q^97 - 22 * q^98

Coefficient data

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

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

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

$2$	−0.414214	−0.292893	−0.146447	−	0.989219i	$-0.546784\pi$
$2$	−0.414214	−0.292893	−0.146447	+	0.989219i	$0.546784\pi$
$3$	0	0
$3$	0	0
$4$	−1.82843	−0.914214
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.82843	−1.82497	−0.912487	−	0.409106i	$-0.865841\pi$
$7$	−4.82843	−1.82497	−0.912487	+	0.409106i	$0.865841\pi$
$8$	1.58579	0.560660
$9$	0	0
$10$	0.414214	0.130986
$11$	−0.828427	−0.249780	−0.124890	−	0.992171i	$-0.539858\pi$
$11$	−0.828427	−0.249780	−0.124890	+	0.992171i	$0.539858\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	3.00000	0.750000
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	0	0
$19$	−4.82843	−1.10772	−0.553859	−	0.832611i	$-0.686845\pi$
$19$	−4.82843	−1.10772	−0.553859	+	0.832611i	$0.686845\pi$
$20$	1.82843	0.408849
$21$	0	0
$22$	0.343146	0.0731589
$23$	3.17157	0.661319	0.330659	−	0.943750i	$-0.392729\pi$
$23$	3.17157	0.661319	0.330659	+	0.943750i	$0.392729\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.828427	0.162468
$27$	0	0
$28$	8.82843	1.66842
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	6.48528	1.16479	0.582395	−	0.812906i	$-0.302116\pi$
$31$	6.48528	1.16479	0.582395	+	0.812906i	$0.302116\pi$
$32$	−4.41421	−0.780330
$33$	0	0
$34$	1.17157	0.200923
$35$	4.82843	0.816153
$36$	0	0
$37$	−8.48528	−1.39497	−0.697486	−	0.716599i	$-0.745698\pi$
$37$	−8.48528	−1.39497	−0.697486	+	0.716599i	$0.745698\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	−1.58579	−0.250735
$41$	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
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	1.51472	0.228352
$45$	0	0
$46$	−1.31371	−0.193696
$47$	11.6569	1.70033	0.850163	−	0.526519i	$-0.176503\pi$
$47$	11.6569	1.70033	0.850163	+	0.526519i	$0.176503\pi$
$48$	0	0
$49$	16.3137	2.33053
$50$	−0.414214	−0.0585786
$51$	0	0
$52$	3.65685	0.507114
$53$	3.65685	0.502308	0.251154	−	0.967947i	$-0.419190\pi$
$53$	3.65685	0.502308	0.251154	+	0.967947i	$0.419190\pi$
$54$	0	0
$55$	0.828427	0.111705
$56$	−7.65685	−1.02319
$57$	0	0
$58$	0.414214	0.0543889
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−3.65685	−0.468212	−0.234106	−	0.972211i	$-0.575216\pi$
$61$	−3.65685	−0.468212	−0.234106	+	0.972211i	$0.575216\pi$
$62$	−2.68629	−0.341159
$63$	0	0
$64$	−4.17157	−0.521447
$65$	2.00000	0.248069
$66$	0	0
$67$	6.48528	0.792303	0.396152	−	0.918185i	$-0.370345\pi$
$67$	6.48528	0.792303	0.396152	+	0.918185i	$0.370345\pi$
$68$	5.17157	0.627145
$69$	0	0
$70$	−2.00000	−0.239046
$71$	15.3137	1.81740	0.908701	−	0.417447i	$-0.137075\pi$
$71$	15.3137	1.81740	0.908701	+	0.417447i	$0.137075\pi$
$72$	0	0
$73$	8.48528	0.993127	0.496564	−	0.868000i	$-0.334595\pi$
$73$	8.48528	0.993127	0.496564	+	0.868000i	$0.334595\pi$
$74$	3.51472	0.408578
$75$	0	0
$76$	8.82843	1.01269
$77$	4.00000	0.455842
$78$	0	0
$79$	−2.48528	−0.279616	−0.139808	−	0.990179i	$-0.544649\pi$
$79$	−2.48528	−0.279616	−0.139808	+	0.990179i	$0.544649\pi$
$80$	−3.00000	−0.335410
$81$	0	0
$82$	−2.48528	−0.274453
$83$	−7.17157	−0.787182	−0.393591	−	0.919286i	$-0.628767\pi$
$83$	−7.17157	−0.787182	−0.393591	+	0.919286i	$0.628767\pi$
$84$	0	0
$85$	2.82843	0.306786
$86$	2.48528	0.267995
$87$	0	0
$88$	−1.31371	−0.140042
$89$	7.65685	0.811625	0.405812	−	0.913956i	$-0.366989\pi$
$89$	7.65685	0.811625	0.405812	+	0.913956i	$0.366989\pi$
$90$	0	0
$91$	9.65685	1.01231
$92$	−5.79899	−0.604586
$93$	0	0
$94$	−4.82843	−0.498014
$95$	4.82843	0.495386
$96$	0	0
$97$	−12.4853	−1.26769	−0.633844	−	0.773461i	$-0.718524\pi$
$97$	−12.4853	−1.26769	−0.633844	+	0.773461i	$0.718524\pi$
$98$	−6.75736	−0.682596
$99$	0	0
$100$	−1.82843	−0.182843
$101$	−15.6569	−1.55792	−0.778958	−	0.627077i	$-0.784251\pi$
$101$	−15.6569	−1.55792	−0.778958	+	0.627077i	$0.784251\pi$
$102$	0	0
$103$	16.1421	1.59053	0.795266	−	0.606261i	$-0.207331\pi$
$103$	16.1421	1.59053	0.795266	+	0.606261i	$0.207331\pi$
$104$	−3.17157	−0.310998
$105$	0	0
$106$	−1.51472	−0.147122
$107$	−20.1421	−1.94721	−0.973607	−	0.228232i	$-0.926706\pi$
$107$	−20.1421	−1.94721	−0.973607	+	0.228232i	$0.926706\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	−0.343146	−0.0327177
$111$	0	0
$112$	−14.4853	−1.36873
$113$	2.82843	0.266076	0.133038	−	0.991111i	$-0.457527\pi$
$113$	2.82843	0.266076	0.133038	+	0.991111i	$0.457527\pi$
$114$	0	0
$115$	−3.17157	−0.295751
$116$	1.82843	0.169765
$117$	0	0
$118$	0	0
$119$	13.6569	1.25192
$120$	0	0
$121$	−10.3137	−0.937610
$122$	1.51472	0.137136
$123$	0	0
$124$	−11.8579	−1.06487
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	6.00000	0.532414	0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000	0.532414	0.266207	+	0.963916i	$0.414230\pi$
$128$	10.5563	0.933058
$129$	0	0
$130$	−0.828427	−0.0726579
$131$	12.1421	1.06086	0.530432	−	0.847728i	$-0.322030\pi$
$131$	12.1421	1.06086	0.530432	+	0.847728i	$0.322030\pi$
$132$	0	0
$133$	23.3137	2.02155
$134$	−2.68629	−0.232060
$135$	0	0
$136$	−4.48528	−0.384610
$137$	5.17157	0.441837	0.220919	−	0.975292i	$-0.429094\pi$
$137$	5.17157	0.441837	0.220919	+	0.975292i	$0.429094\pi$
$138$	0	0
$139$	21.6569	1.83691	0.918455	−	0.395525i	$-0.129437\pi$
$139$	21.6569	1.83691	0.918455	+	0.395525i	$0.129437\pi$
$140$	−8.82843	−0.746138
$141$	0	0
$142$	−6.34315	−0.532305
$143$	1.65685	0.138553
$144$	0	0
$145$	1.00000	0.0830455
$146$	−3.51472	−0.290880
$147$	0	0
$148$	15.5147	1.27530
$149$	−9.31371	−0.763009	−0.381504	−	0.924367i	$-0.624594\pi$
$149$	−9.31371	−0.763009	−0.381504	+	0.924367i	$0.624594\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	−7.65685	−0.621053
$153$	0	0
$154$	−1.65685	−0.133513
$155$	−6.48528	−0.520910
$156$	0	0
$157$	0.485281	0.0387297	0.0193648	−	0.999812i	$-0.493836\pi$
$157$	0.485281	0.0387297	0.0193648	+	0.999812i	$0.493836\pi$
$158$	1.02944	0.0818976
$159$	0	0
$160$	4.41421	0.348974
$161$	−15.3137	−1.20689
$162$	0	0
$163$	−8.34315	−0.653486	−0.326743	−	0.945113i	$-0.605951\pi$
$163$	−8.34315	−0.653486	−0.326743	+	0.945113i	$0.605951\pi$
$164$	−10.9706	−0.856657
$165$	0	0
$166$	2.97056	0.230560
$167$	2.48528	0.192317	0.0961584	−	0.995366i	$-0.469344\pi$
$167$	2.48528	0.192317	0.0961584	+	0.995366i	$0.469344\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−1.17157	−0.0898555
$171$	0	0
$172$	10.9706	0.836498
$173$	−17.3137	−1.31634	−0.658168	−	0.752871i	$-0.728669\pi$
$173$	−17.3137	−1.31634	−0.658168	+	0.752871i	$0.728669\pi$
$174$	0	0
$175$	−4.82843	−0.364995
$176$	−2.48528	−0.187335
$177$	0	0
$178$	−3.17157	−0.237719
$179$	23.3137	1.74255	0.871274	−	0.490797i	$-0.163294\pi$
$179$	23.3137	1.74255	0.871274	+	0.490797i	$0.163294\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	−4.00000	−0.296500
$183$	0	0
$184$	5.02944	0.370775
$185$	8.48528	0.623850
$186$	0	0
$187$	2.34315	0.171348
$188$	−21.3137	−1.55446
$189$	0	0
$190$	−2.00000	−0.145095
$191$	20.8284	1.50709	0.753546	−	0.657395i	$-0.228342\pi$
$191$	20.8284	1.50709	0.753546	+	0.657395i	$0.228342\pi$
$192$	0	0
$193$	4.48528	0.322858	0.161429	−	0.986884i	$-0.448390\pi$
$193$	4.48528	0.322858	0.161429	+	0.986884i	$0.448390\pi$
$194$	5.17157	0.371297
$195$	0	0
$196$	−29.8284	−2.13060
$197$	19.6569	1.40049	0.700246	−	0.713901i	$-0.253073\pi$
$197$	19.6569	1.40049	0.700246	+	0.713901i	$0.253073\pi$
$198$	0	0
$199$	12.0000	0.850657	0.425329	−	0.905039i	$-0.360158\pi$
$199$	12.0000	0.850657	0.425329	+	0.905039i	$0.360158\pi$
$200$	1.58579	0.112132
$201$	0	0
$202$	6.48528	0.456303
$203$	4.82843	0.338889
$204$	0	0
$205$	−6.00000	−0.419058
$206$	−6.68629	−0.465856
$207$	0	0
$208$	−6.00000	−0.416025
$209$	4.00000	0.276686
$210$	0	0
$211$	−0.828427	−0.0570313	−0.0285156	−	0.999593i	$-0.509078\pi$
$211$	−0.828427	−0.0570313	−0.0285156	+	0.999593i	$0.509078\pi$
$212$	−6.68629	−0.459216
$213$	0	0
$214$	8.34315	0.570326
$215$	6.00000	0.409197
$216$	0	0
$217$	−31.3137	−2.12571
$218$	−0.828427	−0.0561082
$219$	0	0
$220$	−1.51472	−0.102122
$221$	5.65685	0.380521
$222$	0	0
$223$	−17.7990	−1.19191	−0.595954	−	0.803018i	$-0.703226\pi$
$223$	−17.7990	−1.19191	−0.595954	+	0.803018i	$0.703226\pi$
$224$	21.3137	1.42408
$225$	0	0
$226$	−1.17157	−0.0779319
$227$	−20.1421	−1.33688	−0.668440	−	0.743766i	$-0.733038\pi$
$227$	−20.1421	−1.33688	−0.668440	+	0.743766i	$0.733038\pi$
$228$	0	0
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	1.31371	0.0866234
$231$	0	0
$232$	−1.58579	−0.104112
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	−11.6569	−0.760409
$236$	0	0
$237$	0	0
$238$	−5.65685	−0.366679
$239$	0.686292	0.0443925	0.0221963	−	0.999754i	$-0.492934\pi$
$239$	0.686292	0.0443925	0.0221963	+	0.999754i	$0.492934\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	4.27208	0.274620
$243$	0	0
$244$	6.68629	0.428046
$245$	−16.3137	−1.04224
$246$	0	0
$247$	9.65685	0.614451
$248$	10.2843	0.653052
$249$	0	0
$250$	0.414214	0.0261972
$251$	−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
$253$	−2.62742	−0.165184
$254$	−2.48528	−0.155940
$255$	0	0
$256$	3.97056	0.248160
$257$	−6.68629	−0.417079	−0.208540	−	0.978014i	$-0.566871\pi$
$257$	−6.68629	−0.417079	−0.208540	+	0.978014i	$0.566871\pi$
$258$	0	0
$259$	40.9706	2.54579
$260$	−3.65685	−0.226788
$261$	0	0
$262$	−5.02944	−0.310720
$263$	19.6569	1.21209	0.606047	−	0.795429i	$-0.292754\pi$
$263$	19.6569	1.21209	0.606047	+	0.795429i	$0.292754\pi$
$264$	0	0
$265$	−3.65685	−0.224639
$266$	−9.65685	−0.592100
$267$	0	0
$268$	−11.8579	−0.724334
$269$	21.3137	1.29952	0.649760	−	0.760140i	$-0.274869\pi$
$269$	21.3137	1.29952	0.649760	+	0.760140i	$0.274869\pi$
$270$	0	0
$271$	−9.79899	−0.595246	−0.297623	−	0.954683i	$-0.596194\pi$
$271$	−9.79899	−0.595246	−0.297623	+	0.954683i	$0.596194\pi$
$272$	−8.48528	−0.514496
$273$	0	0
$274$	−2.14214	−0.129411
$275$	−0.828427	−0.0499560
$276$	0	0
$277$	−3.65685	−0.219719	−0.109860	−	0.993947i	$-0.535040\pi$
$277$	−3.65685	−0.219719	−0.109860	+	0.993947i	$0.535040\pi$
$278$	−8.97056	−0.538019
$279$	0	0
$280$	7.65685	0.457585
$281$	29.3137	1.74871	0.874355	−	0.485288i	$-0.161285\pi$
$281$	29.3137	1.74871	0.874355	+	0.485288i	$0.161285\pi$
$282$	0	0
$283$	4.82843	0.287020	0.143510	−	0.989649i	$-0.454161\pi$
$283$	4.82843	0.287020	0.143510	+	0.989649i	$0.454161\pi$
$284$	−28.0000	−1.66149
$285$	0	0
$286$	−0.686292	−0.0405813
$287$	−28.9706	−1.71008
$288$	0	0
$289$	−9.00000	−0.529412
$290$	−0.414214	−0.0243235
$291$	0	0
$292$	−15.5147	−0.907930
$293$	−8.48528	−0.495715	−0.247858	−	0.968796i	$-0.579727\pi$
$293$	−8.48528	−0.495715	−0.247858	+	0.968796i	$0.579727\pi$
$294$	0	0
$295$	0	0
$296$	−13.4558	−0.782105
$297$	0	0
$298$	3.85786	0.223480
$299$	−6.34315	−0.366834
$300$	0	0
$301$	28.9706	1.66984
$302$	4.97056	0.286024
$303$	0	0
$304$	−14.4853	−0.830788
$305$	3.65685	0.209391
$306$	0	0
$307$	−22.9706	−1.31100	−0.655500	−	0.755195i	$-0.727542\pi$
$307$	−22.9706	−1.31100	−0.655500	+	0.755195i	$0.727542\pi$
$308$	−7.31371	−0.416737
$309$	0	0
$310$	2.68629	0.152571
$311$	−14.4853	−0.821385	−0.410692	−	0.911774i	$-0.634713\pi$
$311$	−14.4853	−0.821385	−0.410692	+	0.911774i	$0.634713\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	−0.201010	−0.0113437
$315$	0	0
$316$	4.54416	0.255629
$317$	−2.82843	−0.158860	−0.0794301	−	0.996840i	$-0.525310\pi$
$317$	−2.82843	−0.158860	−0.0794301	+	0.996840i	$0.525310\pi$
$318$	0	0
$319$	0.828427	0.0463830
$320$	4.17157	0.233198
$321$	0	0
$322$	6.34315	0.353490
$323$	13.6569	0.759888
$324$	0	0
$325$	−2.00000	−0.110940
$326$	3.45584	0.191402
$327$	0	0
$328$	9.51472	0.525362
$329$	−56.2843	−3.10305
$330$	0	0
$331$	21.7990	1.19818	0.599090	−	0.800681i	$-0.295529\pi$
$331$	21.7990	1.19818	0.599090	+	0.800681i	$0.295529\pi$
$332$	13.1127	0.719653
$333$	0	0
$334$	−1.02944	−0.0563283
$335$	−6.48528	−0.354329
$336$	0	0
$337$	−1.17157	−0.0638196	−0.0319098	−	0.999491i	$-0.510159\pi$
$337$	−1.17157	−0.0638196	−0.0319098	+	0.999491i	$0.510159\pi$
$338$	3.72792	0.202772
$339$	0	0
$340$	−5.17157	−0.280468
$341$	−5.37258	−0.290942
$342$	0	0
$343$	−44.9706	−2.42818
$344$	−9.51472	−0.512999
$345$	0	0
$346$	7.17157	0.385546
$347$	8.14214	0.437093	0.218546	−	0.975827i	$-0.429869\pi$
$347$	8.14214	0.437093	0.218546	+	0.975827i	$0.429869\pi$
$348$	0	0
$349$	20.6274	1.10416	0.552080	−	0.833791i	$-0.313834\pi$
$349$	20.6274	1.10416	0.552080	+	0.833791i	$0.313834\pi$
$350$	2.00000	0.106904
$351$	0	0
$352$	3.65685	0.194911
$353$	4.34315	0.231162	0.115581	−	0.993298i	$-0.463127\pi$
$353$	4.34315	0.231162	0.115581	+	0.993298i	$0.463127\pi$
$354$	0	0
$355$	−15.3137	−0.812767
$356$	−14.0000	−0.741999
$357$	0	0
$358$	−9.65685	−0.510381
$359$	3.85786	0.203610	0.101805	−	0.994804i	$-0.467538\pi$
$359$	3.85786	0.203610	0.101805	+	0.994804i	$0.467538\pi$
$360$	0	0
$361$	4.31371	0.227037
$362$	2.48528	0.130623
$363$	0	0
$364$	−17.6569	−0.925471
$365$	−8.48528	−0.444140
$366$	0	0
$367$	18.0000	0.939592	0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000	0.939592	0.469796	+	0.882775i	$0.344327\pi$
$368$	9.51472	0.495989
$369$	0	0
$370$	−3.51472	−0.182722
$371$	−17.6569	−0.916698
$372$	0	0
$373$	−6.97056	−0.360922	−0.180461	−	0.983582i	$-0.557759\pi$
$373$	−6.97056	−0.360922	−0.180461	+	0.983582i	$0.557759\pi$
$374$	−0.970563	−0.0501866
$375$	0	0
$376$	18.4853	0.953306
$377$	2.00000	0.103005
$378$	0	0
$379$	22.4853	1.15499	0.577496	−	0.816394i	$-0.304030\pi$
$379$	22.4853	1.15499	0.577496	+	0.816394i	$0.304030\pi$
$380$	−8.82843	−0.452889
$381$	0	0
$382$	−8.62742	−0.441417
$383$	−2.48528	−0.126992	−0.0634960	−	0.997982i	$-0.520225\pi$
$383$	−2.48528	−0.126992	−0.0634960	+	0.997982i	$0.520225\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	−1.85786	−0.0945628
$387$	0	0
$388$	22.8284	1.15894
$389$	29.3137	1.48626	0.743132	−	0.669145i	$-0.233339\pi$
$389$	29.3137	1.48626	0.743132	+	0.669145i	$0.233339\pi$
$390$	0	0
$391$	−8.97056	−0.453661
$392$	25.8701	1.30664
$393$	0	0
$394$	−8.14214	−0.410195
$395$	2.48528	0.125048
$396$	0	0
$397$	−19.6569	−0.986549	−0.493275	−	0.869874i	$-0.664200\pi$
$397$	−19.6569	−0.986549	−0.493275	+	0.869874i	$0.664200\pi$
$398$	−4.97056	−0.249152
$399$	0	0
$400$	3.00000	0.150000
$401$	6.68629	0.333897	0.166949	−	0.985966i	$-0.446609\pi$
$401$	6.68629	0.333897	0.166949	+	0.985966i	$0.446609\pi$
$402$	0	0
$403$	−12.9706	−0.646110
$404$	28.6274	1.42427
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	7.02944	0.348436
$408$	0	0
$409$	−2.97056	−0.146885	−0.0734424	−	0.997299i	$-0.523399\pi$
$409$	−2.97056	−0.146885	−0.0734424	+	0.997299i	$0.523399\pi$
$410$	2.48528	0.122739
$411$	0	0
$412$	−29.5147	−1.45409
$413$	0	0
$414$	0	0
$415$	7.17157	0.352039
$416$	8.82843	0.432849
$417$	0	0
$418$	−1.65685	−0.0810394
$419$	−28.9706	−1.41530	−0.707652	−	0.706561i	$-0.750246\pi$
$419$	−28.9706	−1.41530	−0.707652	+	0.706561i	$0.750246\pi$
$420$	0	0
$421$	18.9706	0.924569	0.462284	−	0.886732i	$-0.347030\pi$
$421$	18.9706	0.924569	0.462284	+	0.886732i	$0.347030\pi$
$422$	0.343146	0.0167041
$423$	0	0
$424$	5.79899	0.281624
$425$	−2.82843	−0.137199
$426$	0	0
$427$	17.6569	0.854475
$428$	36.8284	1.78017
$429$	0	0
$430$	−2.48528	−0.119851
$431$	−3.31371	−0.159616	−0.0798079	−	0.996810i	$-0.525431\pi$
$431$	−3.31371	−0.159616	−0.0798079	+	0.996810i	$0.525431\pi$
$432$	0	0
$433$	−29.1716	−1.40190	−0.700948	−	0.713212i	$-0.747240\pi$
$433$	−29.1716	−1.40190	−0.700948	+	0.713212i	$0.747240\pi$
$434$	12.9706	0.622607
$435$	0	0
$436$	−3.65685	−0.175132
$437$	−15.3137	−0.732554
$438$	0	0
$439$	−10.3431	−0.493651	−0.246826	−	0.969060i	$-0.579388\pi$
$439$	−10.3431	−0.493651	−0.246826	+	0.969060i	$0.579388\pi$
$440$	1.31371	0.0626286
$441$	0	0
$442$	−2.34315	−0.111452
$443$	7.65685	0.363788	0.181894	−	0.983318i	$-0.441777\pi$
$443$	7.65685	0.363788	0.181894	+	0.983318i	$0.441777\pi$
$444$	0	0
$445$	−7.65685	−0.362970
$446$	7.37258	0.349102
$447$	0	0
$448$	20.1421	0.951626
$449$	−11.6569	−0.550121	−0.275060	−	0.961427i	$-0.588698\pi$
$449$	−11.6569	−0.550121	−0.275060	+	0.961427i	$0.588698\pi$
$450$	0	0
$451$	−4.97056	−0.234055
$452$	−5.17157	−0.243250
$453$	0	0
$454$	8.34315	0.391563
$455$	−9.65685	−0.452720
$456$	0	0
$457$	19.6569	0.919509	0.459754	−	0.888046i	$-0.347937\pi$
$457$	19.6569	0.919509	0.459754	+	0.888046i	$0.347937\pi$
$458$	0.828427	0.0387099
$459$	0	0
$460$	5.79899	0.270379
$461$	35.6569	1.66071	0.830353	−	0.557238i	$-0.188139\pi$
$461$	35.6569	1.66071	0.830353	+	0.557238i	$0.188139\pi$
$462$	0	0
$463$	21.7990	1.01308	0.506542	−	0.862215i	$-0.330923\pi$
$463$	21.7990	1.01308	0.506542	+	0.862215i	$0.330923\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	7.45584	0.345385
$467$	−10.9706	−0.507657	−0.253829	−	0.967249i	$-0.581690\pi$
$467$	−10.9706	−0.507657	−0.253829	+	0.967249i	$0.581690\pi$
$468$	0	0
$469$	−31.3137	−1.44593
$470$	4.82843	0.222719
$471$	0	0
$472$	0	0
$473$	4.97056	0.228547
$474$	0	0
$475$	−4.82843	−0.221543
$476$	−24.9706	−1.14452
$477$	0	0
$478$	−0.284271	−0.0130023
$479$	−7.17157	−0.327678	−0.163839	−	0.986487i	$-0.552388\pi$
$479$	−7.17157	−0.327678	−0.163839	+	0.986487i	$0.552388\pi$
$480$	0	0
$481$	16.9706	0.773791
$482$	−4.14214	−0.188669
$483$	0	0
$484$	18.8579	0.857176
$485$	12.4853	0.566927
$486$	0	0
$487$	9.79899	0.444035	0.222017	−	0.975043i	$-0.428736\pi$
$487$	9.79899	0.444035	0.222017	+	0.975043i	$0.428736\pi$
$488$	−5.79899	−0.262508
$489$	0	0
$490$	6.75736	0.305266
$491$	7.45584	0.336478	0.168239	−	0.985746i	$-0.446192\pi$
$491$	7.45584	0.336478	0.168239	+	0.985746i	$0.446192\pi$
$492$	0	0
$493$	2.82843	0.127386
$494$	−4.00000	−0.179969
$495$	0	0
$496$	19.4558	0.873593
$497$	−73.9411	−3.31671
$498$	0	0
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	1.82843	0.0817697
$501$	0	0
$502$	3.65685	0.163213
$503$	30.0000	1.33763	0.668817	−	0.743427i	$-0.266801\pi$
$503$	30.0000	1.33763	0.668817	+	0.743427i	$0.266801\pi$
$504$	0	0
$505$	15.6569	0.696721
$506$	1.08831	0.0483814
$507$	0	0
$508$	−10.9706	−0.486740
$509$	−0.627417	−0.0278098	−0.0139049	−	0.999903i	$-0.504426\pi$
$509$	−0.627417	−0.0278098	−0.0139049	+	0.999903i	$0.504426\pi$
$510$	0	0
$511$	−40.9706	−1.81243
$512$	−22.7574	−1.00574
$513$	0	0
$514$	2.76955	0.122160
$515$	−16.1421	−0.711307
$516$	0	0
$517$	−9.65685	−0.424708
$518$	−16.9706	−0.745644
$519$	0	0
$520$	3.17157	0.139083
$521$	21.3137	0.933771	0.466885	−	0.884318i	$-0.345376\pi$
$521$	21.3137	0.933771	0.466885	+	0.884318i	$0.345376\pi$
$522$	0	0
$523$	2.48528	0.108674	0.0543369	−	0.998523i	$-0.482696\pi$
$523$	2.48528	0.108674	0.0543369	+	0.998523i	$0.482696\pi$
$524$	−22.2010	−0.969856
$525$	0	0
$526$	−8.14214	−0.355014
$527$	−18.3431	−0.799040
$528$	0	0
$529$	−12.9411	−0.562658
$530$	1.51472	0.0657952
$531$	0	0
$532$	−42.6274	−1.84813
$533$	−12.0000	−0.519778
$534$	0	0
$535$	20.1421	0.870820
$536$	10.2843	0.444213
$537$	0	0
$538$	−8.82843	−0.380621
$539$	−13.5147	−0.582120
$540$	0	0
$541$	−5.02944	−0.216232	−0.108116	−	0.994138i	$-0.534482\pi$
$541$	−5.02944	−0.216232	−0.108116	+	0.994138i	$0.534482\pi$
$542$	4.05887	0.174344
$543$	0	0
$544$	12.4853	0.535302
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	−2.48528	−0.106263	−0.0531315	−	0.998588i	$-0.516920\pi$
$547$	−2.48528	−0.106263	−0.0531315	+	0.998588i	$0.516920\pi$
$548$	−9.45584	−0.403934
$549$	0	0
$550$	0.343146	0.0146318
$551$	4.82843	0.205698
$552$	0	0
$553$	12.0000	0.510292
$554$	1.51472	0.0643542
$555$	0	0
$556$	−39.5980	−1.67933
$557$	27.9411	1.18390	0.591952	−	0.805973i	$-0.298358\pi$
$557$	27.9411	1.18390	0.591952	+	0.805973i	$0.298358\pi$
$558$	0	0
$559$	12.0000	0.507546
$560$	14.4853	0.612115
$561$	0	0
$562$	−12.1421	−0.512185
$563$	−7.65685	−0.322698	−0.161349	−	0.986897i	$-0.551584\pi$
$563$	−7.65685	−0.322698	−0.161349	+	0.986897i	$0.551584\pi$
$564$	0	0
$565$	−2.82843	−0.118993
$566$	−2.00000	−0.0840663
$567$	0	0
$568$	24.2843	1.01895
$569$	−27.6569	−1.15944	−0.579718	−	0.814817i	$-0.696837\pi$
$569$	−27.6569	−1.15944	−0.579718	+	0.814817i	$0.696837\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	−3.02944	−0.126667
$573$	0	0
$574$	12.0000	0.500870
$575$	3.17157	0.132264
$576$	0	0
$577$	23.7990	0.990765	0.495382	−	0.868675i	$-0.335028\pi$
$577$	23.7990	0.990765	0.495382	+	0.868675i	$0.335028\pi$
$578$	3.72792	0.155061
$579$	0	0
$580$	−1.82843	−0.0759213
$581$	34.6274	1.43659
$582$	0	0
$583$	−3.02944	−0.125466
$584$	13.4558	0.556807
$585$	0	0
$586$	3.51472	0.145192
$587$	29.7990	1.22994	0.614968	−	0.788552i	$-0.289169\pi$
$587$	29.7990	1.22994	0.614968	+	0.788552i	$0.289169\pi$
$588$	0	0
$589$	−31.3137	−1.29026
$590$	0	0
$591$	0	0
$592$	−25.4558	−1.04623
$593$	7.65685	0.314429	0.157215	−	0.987564i	$-0.449749\pi$
$593$	7.65685	0.314429	0.157215	+	0.987564i	$0.449749\pi$
$594$	0	0
$595$	−13.6569	−0.559876
$596$	17.0294	0.697553
$597$	0	0
$598$	2.62742	0.107443
$599$	37.7990	1.54442	0.772212	−	0.635364i	$-0.219150\pi$
$599$	37.7990	1.54442	0.772212	+	0.635364i	$0.219150\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	−12.0000	−0.489083
$603$	0	0
$604$	21.9411	0.892772
$605$	10.3137	0.419312
$606$	0	0
$607$	9.02944	0.366494	0.183247	−	0.983067i	$-0.441339\pi$
$607$	9.02944	0.366494	0.183247	+	0.983067i	$0.441339\pi$
$608$	21.3137	0.864385
$609$	0	0
$610$	−1.51472	−0.0613292
$611$	−23.3137	−0.943172
$612$	0	0
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	9.51472	0.383983
$615$	0	0
$616$	6.34315	0.255573
$617$	9.17157	0.369234	0.184617	−	0.982811i	$-0.440896\pi$
$617$	9.17157	0.369234	0.184617	+	0.982811i	$0.440896\pi$
$618$	0	0
$619$	−9.79899	−0.393855	−0.196927	−	0.980418i	$-0.563096\pi$
$619$	−9.79899	−0.393855	−0.196927	+	0.980418i	$0.563096\pi$
$620$	11.8579	0.476223
$621$	0	0
$622$	6.00000	0.240578
$623$	−36.9706	−1.48119
$624$	0	0
$625$	1.00000	0.0400000
$626$	2.48528	0.0993318
$627$	0	0
$628$	−0.887302	−0.0354072
$629$	24.0000	0.956943
$630$	0	0
$631$	36.9706	1.47177	0.735887	−	0.677104i	$-0.236765\pi$
$631$	36.9706	1.47177	0.735887	+	0.677104i	$0.236765\pi$
$632$	−3.94113	−0.156770
$633$	0	0
$634$	1.17157	0.0465291
$635$	−6.00000	−0.238103
$636$	0	0
$637$	−32.6274	−1.29275
$638$	−0.343146	−0.0135853
$639$	0	0
$640$	−10.5563	−0.417276
$641$	−0.627417	−0.0247815	−0.0123907	−	0.999923i	$-0.503944\pi$
$641$	−0.627417	−0.0247815	−0.0123907	+	0.999923i	$0.503944\pi$
$642$	0	0
$643$	19.4558	0.767264	0.383632	−	0.923486i	$-0.374673\pi$
$643$	19.4558	0.767264	0.383632	+	0.923486i	$0.374673\pi$
$644$	28.0000	1.10335
$645$	0	0
$646$	−5.65685	−0.222566
$647$	41.1127	1.61631	0.808153	−	0.588972i	$-0.200467\pi$
$647$	41.1127	1.61631	0.808153	+	0.588972i	$0.200467\pi$
$648$	0	0
$649$	0	0
$650$	0.828427	0.0324936
$651$	0	0
$652$	15.2548	0.597425
$653$	17.1716	0.671976	0.335988	−	0.941866i	$-0.390930\pi$
$653$	17.1716	0.671976	0.335988	+	0.941866i	$0.390930\pi$
$654$	0	0
$655$	−12.1421	−0.474432
$656$	18.0000	0.702782
$657$	0	0
$658$	23.3137	0.908863
$659$	−1.79899	−0.0700787	−0.0350393	−	0.999386i	$-0.511156\pi$
$659$	−1.79899	−0.0700787	−0.0350393	+	0.999386i	$0.511156\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	−9.02944	−0.350939
$663$	0	0
$664$	−11.3726	−0.441342
$665$	−23.3137	−0.904067
$666$	0	0
$667$	−3.17157	−0.122804
$668$	−4.54416	−0.175819
$669$	0	0
$670$	2.68629	0.103780
$671$	3.02944	0.116950
$672$	0	0
$673$	22.9706	0.885450	0.442725	−	0.896657i	$-0.354012\pi$
$673$	22.9706	0.885450	0.442725	+	0.896657i	$0.354012\pi$
$674$	0.485281	0.0186923
$675$	0	0
$676$	16.4558	0.632917
$677$	36.7696	1.41317	0.706584	−	0.707629i	$-0.250235\pi$
$677$	36.7696	1.41317	0.706584	+	0.707629i	$0.250235\pi$
$678$	0	0
$679$	60.2843	2.31350
$680$	4.48528	0.172003
$681$	0	0
$682$	2.22540	0.0852148
$683$	−11.8579	−0.453729	−0.226864	−	0.973926i	$-0.572847\pi$
$683$	−11.8579	−0.453729	−0.226864	+	0.973926i	$0.572847\pi$
$684$	0	0
$685$	−5.17157	−0.197596
$686$	18.6274	0.711198
$687$	0	0
$688$	−18.0000	−0.686244
$689$	−7.31371	−0.278630
$690$	0	0
$691$	−44.9706	−1.71076	−0.855380	−	0.518000i	$-0.826677\pi$
$691$	−44.9706	−1.71076	−0.855380	+	0.518000i	$0.826677\pi$
$692$	31.6569	1.20341
$693$	0	0
$694$	−3.37258	−0.128022
$695$	−21.6569	−0.821491
$696$	0	0
$697$	−16.9706	−0.642806
$698$	−8.54416	−0.323401
$699$	0	0
$700$	8.82843	0.333683
$701$	−6.68629	−0.252538	−0.126269	−	0.991996i	$-0.540300\pi$
$701$	−6.68629	−0.252538	−0.126269	+	0.991996i	$0.540300\pi$
$702$	0	0
$703$	40.9706	1.54523
$704$	3.45584	0.130247
$705$	0	0
$706$	−1.79899	−0.0677059
$707$	75.5980	2.84315
$708$	0	0
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	6.34315	0.238054
$711$	0	0
$712$	12.1421	0.455046
$713$	20.5685	0.770298
$714$	0	0
$715$	−1.65685	−0.0619628
$716$	−42.6274	−1.59306
$717$	0	0
$718$	−1.59798	−0.0596361
$719$	34.6274	1.29138	0.645692	−	0.763598i	$-0.276569\pi$
$719$	34.6274	1.29138	0.645692	+	0.763598i	$0.276569\pi$
$720$	0	0
$721$	−77.9411	−2.90268
$722$	−1.78680	−0.0664977
$723$	0	0
$724$	10.9706	0.407718
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	−23.9411	−0.887927	−0.443964	−	0.896045i	$-0.646428\pi$
$727$	−23.9411	−0.887927	−0.443964	+	0.896045i	$0.646428\pi$
$728$	15.3137	0.567564
$729$	0	0
$730$	3.51472	0.130086
$731$	16.9706	0.627679
$732$	0	0
$733$	−22.8284	−0.843187	−0.421594	−	0.906785i	$-0.638529\pi$
$733$	−22.8284	−0.843187	−0.421594	+	0.906785i	$0.638529\pi$
$734$	−7.45584	−0.275200
$735$	0	0
$736$	−14.0000	−0.516047
$737$	−5.37258	−0.197902
$738$	0	0
$739$	14.4853	0.532850	0.266425	−	0.963856i	$-0.414158\pi$
$739$	14.4853	0.532850	0.266425	+	0.963856i	$0.414158\pi$
$740$	−15.5147	−0.570332
$741$	0	0
$742$	7.31371	0.268495
$743$	52.6274	1.93071	0.965356	−	0.260935i	$-0.0840309\pi$
$743$	52.6274	1.93071	0.965356	+	0.260935i	$0.0840309\pi$
$744$	0	0
$745$	9.31371	0.341228
$746$	2.88730	0.105712
$747$	0	0
$748$	−4.28427	−0.156648
$749$	97.2548	3.55361
$750$	0	0
$751$	16.1421	0.589035	0.294517	−	0.955646i	$-0.404841\pi$
$751$	16.1421	0.589035	0.294517	+	0.955646i	$0.404841\pi$
$752$	34.9706	1.27525
$753$	0	0
$754$	−0.828427	−0.0301695
$755$	12.0000	0.436725
$756$	0	0
$757$	19.5147	0.709275	0.354637	−	0.935004i	$-0.384604\pi$
$757$	19.5147	0.709275	0.354637	+	0.935004i	$0.384604\pi$
$758$	−9.31371	−0.338289
$759$	0	0
$760$	7.65685	0.277743
$761$	−8.62742	−0.312744	−0.156372	−	0.987698i	$-0.549980\pi$
$761$	−8.62742	−0.312744	−0.156372	+	0.987698i	$0.549980\pi$
$762$	0	0
$763$	−9.65685	−0.349602
$764$	−38.0833	−1.37780
$765$	0	0
$766$	1.02944	0.0371951
$767$	0	0
$768$	0	0
$769$	−15.6569	−0.564601	−0.282300	−	0.959326i	$-0.591097\pi$
$769$	−15.6569	−0.564601	−0.282300	+	0.959326i	$0.591097\pi$
$770$	1.65685	0.0597089
$771$	0	0
$772$	−8.20101	−0.295161
$773$	8.48528	0.305194	0.152597	−	0.988288i	$-0.451236\pi$
$773$	8.48528	0.305194	0.152597	+	0.988288i	$0.451236\pi$
$774$	0	0
$775$	6.48528	0.232958
$776$	−19.7990	−0.710742
$777$	0	0
$778$	−12.1421	−0.435317
$779$	−28.9706	−1.03798
$780$	0	0
$781$	−12.6863	−0.453951
$782$	3.71573	0.132874
$783$	0	0
$784$	48.9411	1.74790
$785$	−0.485281	−0.0173204
$786$	0	0
$787$	17.7990	0.634465	0.317233	−	0.948348i	$-0.397246\pi$
$787$	17.7990	0.634465	0.317233	+	0.948348i	$0.397246\pi$
$788$	−35.9411	−1.28035
$789$	0	0
$790$	−1.02944	−0.0366257
$791$	−13.6569	−0.485582
$792$	0	0
$793$	7.31371	0.259717
$794$	8.14214	0.288954
$795$	0	0
$796$	−21.9411	−0.777683
$797$	5.85786	0.207496	0.103748	−	0.994604i	$-0.466916\pi$
$797$	5.85786	0.207496	0.103748	+	0.994604i	$0.466916\pi$
$798$	0	0
$799$	−32.9706	−1.16641
$800$	−4.41421	−0.156066
$801$	0	0
$802$	−2.76955	−0.0977963
$803$	−7.02944	−0.248063
$804$	0	0
$805$	15.3137	0.539737
$806$	5.37258	0.189241
$807$	0	0
$808$	−24.8284	−0.873461
$809$	−42.2843	−1.48664	−0.743318	−	0.668938i	$-0.766749\pi$
$809$	−42.2843	−1.48664	−0.743318	+	0.668938i	$0.766749\pi$
$810$	0	0
$811$	37.6569	1.32231	0.661155	−	0.750249i	$-0.270066\pi$
$811$	37.6569	1.32231	0.661155	+	0.750249i	$0.270066\pi$
$812$	−8.82843	−0.309817
$813$	0	0
$814$	−2.91169	−0.102055
$815$	8.34315	0.292248
$816$	0	0
$817$	28.9706	1.01355
$818$	1.23045	0.0430216
$819$	0	0
$820$	10.9706	0.383109
$821$	22.6863	0.791757	0.395879	−	0.918303i	$-0.370440\pi$
$821$	22.6863	0.791757	0.395879	+	0.918303i	$0.370440\pi$
$822$	0	0
$823$	−30.9706	−1.07957	−0.539783	−	0.841804i	$-0.681494\pi$
$823$	−30.9706	−1.07957	−0.539783	+	0.841804i	$0.681494\pi$
$824$	25.5980	0.891748
$825$	0	0
$826$	0	0
$827$	−17.3137	−0.602057	−0.301028	−	0.953615i	$-0.597330\pi$
$827$	−17.3137	−0.602057	−0.301028	+	0.953615i	$0.597330\pi$
$828$	0	0
$829$	20.6274	0.716420	0.358210	−	0.933641i	$-0.383387\pi$
$829$	20.6274	0.716420	0.358210	+	0.933641i	$0.383387\pi$
$830$	−2.97056	−0.103110
$831$	0	0
$832$	8.34315	0.289247
$833$	−46.1421	−1.59873
$834$	0	0
$835$	−2.48528	−0.0860067
$836$	−7.31371	−0.252950
$837$	0	0
$838$	12.0000	0.414533
$839$	−2.48528	−0.0858014	−0.0429007	−	0.999079i	$-0.513660\pi$
$839$	−2.48528	−0.0858014	−0.0429007	+	0.999079i	$0.513660\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−7.85786	−0.270800
$843$	0	0
$844$	1.51472	0.0521388
$845$	9.00000	0.309609
$846$	0	0
$847$	49.7990	1.71111
$848$	10.9706	0.376731
$849$	0	0
$850$	1.17157	0.0401846
$851$	−26.9117	−0.922521
$852$	0	0
$853$	51.1127	1.75007	0.875033	−	0.484064i	$-0.160840\pi$
$853$	51.1127	1.75007	0.875033	+	0.484064i	$0.160840\pi$
$854$	−7.31371	−0.250270
$855$	0	0
$856$	−31.9411	−1.09173
$857$	−3.37258	−0.115205	−0.0576026	−	0.998340i	$-0.518346\pi$
$857$	−3.37258	−0.115205	−0.0576026	+	0.998340i	$0.518346\pi$
$858$	0	0
$859$	−56.4264	−1.92524	−0.962622	−	0.270848i	$-0.912696\pi$
$859$	−56.4264	−1.92524	−0.962622	+	0.270848i	$0.912696\pi$
$860$	−10.9706	−0.374093
$861$	0	0
$862$	1.37258	0.0467504
$863$	36.1421	1.23029	0.615146	−	0.788413i	$-0.289097\pi$
$863$	36.1421	1.23029	0.615146	+	0.788413i	$0.289097\pi$
$864$	0	0
$865$	17.3137	0.588684
$866$	12.0833	0.410606
$867$	0	0
$868$	57.2548	1.94336
$869$	2.05887	0.0698425
$870$	0	0
$871$	−12.9706	−0.439491
$872$	3.17157	0.107403
$873$	0	0
$874$	6.34315	0.214560
$875$	4.82843	0.163231
$876$	0	0
$877$	38.2843	1.29277	0.646384	−	0.763012i	$-0.276280\pi$
$877$	38.2843	1.29277	0.646384	+	0.763012i	$0.276280\pi$
$878$	4.28427	0.144587
$879$	0	0
$880$	2.48528	0.0837788
$881$	−29.3137	−0.987604	−0.493802	−	0.869574i	$-0.664393\pi$
$881$	−29.3137	−0.987604	−0.493802	+	0.869574i	$0.664393\pi$
$882$	0	0
$883$	−14.4853	−0.487469	−0.243734	−	0.969842i	$-0.578372\pi$
$883$	−14.4853	−0.487469	−0.243734	+	0.969842i	$0.578372\pi$
$884$	−10.3431	−0.347878
$885$	0	0
$886$	−3.17157	−0.106551
$887$	6.68629	0.224504	0.112252	−	0.993680i	$-0.464194\pi$
$887$	6.68629	0.224504	0.112252	+	0.993680i	$0.464194\pi$
$888$	0	0
$889$	−28.9706	−0.971641
$890$	3.17157	0.106311
$891$	0	0
$892$	32.5442	1.08966
$893$	−56.2843	−1.88348
$894$	0	0
$895$	−23.3137	−0.779291
$896$	−50.9706	−1.70281
$897$	0	0
$898$	4.82843	0.161127
$899$	−6.48528	−0.216296
$900$	0	0
$901$	−10.3431	−0.344580
$902$	2.05887	0.0685530
$903$	0	0
$904$	4.48528	0.149178
$905$	6.00000	0.199447
$906$	0	0
$907$	−10.0000	−0.332045	−0.166022	−	0.986122i	$-0.553092\pi$
$907$	−10.0000	−0.332045	−0.166022	+	0.986122i	$0.553092\pi$
$908$	36.8284	1.22219
$909$	0	0
$910$	4.00000	0.132599
$911$	−32.1421	−1.06492	−0.532458	−	0.846456i	$-0.678732\pi$
$911$	−32.1421	−1.06492	−0.532458	+	0.846456i	$0.678732\pi$
$912$	0	0
$913$	5.94113	0.196623
$914$	−8.14214	−0.269318
$915$	0	0
$916$	3.65685	0.120826
$917$	−58.6274	−1.93605
$918$	0	0
$919$	−36.0000	−1.18753	−0.593765	−	0.804638i	$-0.702359\pi$
$919$	−36.0000	−1.18753	−0.593765	+	0.804638i	$0.702359\pi$
$920$	−5.02944	−0.165816
$921$	0	0
$922$	−14.7696	−0.486409
$923$	−30.6274	−1.00811
$924$	0	0
$925$	−8.48528	−0.278994
$926$	−9.02944	−0.296726
$927$	0	0
$928$	4.41421	0.144904
$929$	4.62742	0.151821	0.0759103	−	0.997115i	$-0.475814\pi$
$929$	4.62742	0.151821	0.0759103	+	0.997115i	$0.475814\pi$
$930$	0	0
$931$	−78.7696	−2.58157
$932$	32.9117	1.07806
$933$	0	0
$934$	4.54416	0.148689
$935$	−2.34315	−0.0766291
$936$	0	0
$937$	19.6569	0.642161	0.321081	−	0.947052i	$-0.395954\pi$
$937$	19.6569	0.642161	0.321081	+	0.947052i	$0.395954\pi$
$938$	12.9706	0.423504
$939$	0	0
$940$	21.3137	0.695177
$941$	27.9411	0.910855	0.455427	−	0.890273i	$-0.349486\pi$
$941$	27.9411	0.910855	0.455427	+	0.890273i	$0.349486\pi$
$942$	0	0
$943$	19.0294	0.619684
$944$	0	0
$945$	0	0
$946$	−2.05887	−0.0669398
$947$	−44.9117	−1.45943	−0.729717	−	0.683749i	$-0.760348\pi$
$947$	−44.9117	−1.45943	−0.729717	+	0.683749i	$0.760348\pi$
$948$	0	0
$949$	−16.9706	−0.550888
$950$	2.00000	0.0648886
$951$	0	0
$952$	21.6569	0.701903
$953$	−29.3137	−0.949564	−0.474782	−	0.880103i	$-0.657473\pi$
$953$	−29.3137	−0.949564	−0.474782	+	0.880103i	$0.657473\pi$
$954$	0	0
$955$	−20.8284	−0.673992
$956$	−1.25483	−0.0405842
$957$	0	0
$958$	2.97056	0.0959745
$959$	−24.9706	−0.806342
$960$	0	0
$961$	11.0589	0.356738
$962$	−7.02944	−0.226638
$963$	0	0
$964$	−18.2843	−0.588897
$965$	−4.48528	−0.144386
$966$	0	0
$967$	14.9706	0.481421	0.240710	−	0.970597i	$-0.422620\pi$
$967$	14.9706	0.481421	0.240710	+	0.970597i	$0.422620\pi$
$968$	−16.3553	−0.525681
$969$	0	0
$970$	−5.17157	−0.166049
$971$	−28.1421	−0.903124	−0.451562	−	0.892240i	$-0.649133\pi$
$971$	−28.1421	−0.903124	−0.451562	+	0.892240i	$0.649133\pi$
$972$	0	0
$973$	−104.569	−3.35231
$974$	−4.05887	−0.130055
$975$	0	0
$976$	−10.9706	−0.351159
$977$	2.68629	0.0859421	0.0429710	−	0.999076i	$-0.486318\pi$
$977$	2.68629	0.0859421	0.0429710	+	0.999076i	$0.486318\pi$
$978$	0	0
$979$	−6.34315	−0.202728
$980$	29.8284	0.952834
$981$	0	0
$982$	−3.08831	−0.0985520
$983$	9.31371	0.297061	0.148531	−	0.988908i	$-0.452546\pi$
$983$	9.31371	0.297061	0.148531	+	0.988908i	$0.452546\pi$
$984$	0	0
$985$	−19.6569	−0.626319
$986$	−1.17157	−0.0373105
$987$	0	0
$988$	−17.6569	−0.561739
$989$	−19.0294	−0.605101
$990$	0	0
$991$	52.0000	1.65183	0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000	1.65183	0.825917	+	0.563791i	$0.190658\pi$
$992$	−28.6274	−0.908921
$993$	0	0
$994$	30.6274	0.971443
$995$	−12.0000	−0.380426
$996$	0	0
$997$	6.82843	0.216258	0.108129	−	0.994137i	$-0.465514\pi$
$997$	6.82843	0.216258	0.108129	+	0.994137i	$0.465514\pi$
$998$	−14.9117	−0.472021
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.2.a.n.1.1		2
3.2	odd	2		145.2.a.b.1.2	✓	2
5.4	even	2		6525.2.a.p.1.2		2
12.11	even	2		2320.2.a.k.1.2		2
15.2	even	4		725.2.b.c.349.3		4
15.8	even	4		725.2.b.c.349.2		4
15.14	odd	2		725.2.a.c.1.1		2
21.20	even	2		7105.2.a.e.1.2		2
24.5	odd	2		9280.2.a.be.1.1		2
24.11	even	2		9280.2.a.w.1.2		2
87.86	odd	2		4205.2.a.d.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.a.b.1.2	✓	2	3.2	odd	2
725.2.a.c.1.1		2	15.14	odd	2
725.2.b.c.349.2		4	15.8	even	4
725.2.b.c.349.3		4	15.2	even	4
1305.2.a.n.1.1		2	1.1	even	1	trivial
2320.2.a.k.1.2		2	12.11	even	2
4205.2.a.d.1.1		2	87.86	odd	2
6525.2.a.p.1.2		2	5.4	even	2
7105.2.a.e.1.2		2	21.20	even	2
9280.2.a.w.1.2		2	24.11	even	2
9280.2.a.be.1.1		2	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1305.a (trivial)

\(f(q)\)	\(=\)	\(q-0.414214 q^{2} -1.82843 q^{4} -1.00000 q^{5} -4.82843 q^{7} +1.58579 q^{8} +0.414214 q^{10} -0.828427 q^{11} -2.00000 q^{13} +2.00000 q^{14} +3.00000 q^{16} -2.82843 q^{17} -4.82843 q^{19} +1.82843 q^{20} +0.343146 q^{22} +3.17157 q^{23} +1.00000 q^{25} +0.828427 q^{26} +8.82843 q^{28} -1.00000 q^{29} +6.48528 q^{31} -4.41421 q^{32} +1.17157 q^{34} +4.82843 q^{35} -8.48528 q^{37} +2.00000 q^{38} -1.58579 q^{40} +6.00000 q^{41} -6.00000 q^{43} +1.51472 q^{44} -1.31371 q^{46} +11.6569 q^{47} +16.3137 q^{49} -0.414214 q^{50} +3.65685 q^{52} +3.65685 q^{53} +0.828427 q^{55} -7.65685 q^{56} +0.414214 q^{58} -3.65685 q^{61} -2.68629 q^{62} -4.17157 q^{64} +2.00000 q^{65} +6.48528 q^{67} +5.17157 q^{68} -2.00000 q^{70} +15.3137 q^{71} +8.48528 q^{73} +3.51472 q^{74} +8.82843 q^{76} +4.00000 q^{77} -2.48528 q^{79} -3.00000 q^{80} -2.48528 q^{82} -7.17157 q^{83} +2.82843 q^{85} +2.48528 q^{86} -1.31371 q^{88} +7.65685 q^{89} +9.65685 q^{91} -5.79899 q^{92} -4.82843 q^{94} +4.82843 q^{95} -12.4853 q^{97} -6.75736 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{7} + 6 q^{8} - 2 q^{10} + 4 q^{11} - 4 q^{13} + 4 q^{14} + 6 q^{16} - 4 q^{19} - 2 q^{20} + 12 q^{22} + 12 q^{23} + 2 q^{25} - 4 q^{26} + 12 q^{28} - 2 q^{29} - 4 q^{31}+ \cdots - 22 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 4 * q^7 + 6 * q^8 - 2 * q^10 + 4 * q^11 - 4 * q^13 + 4 * q^14 + 6 * q^16 - 4 * q^19 - 2 * q^20 + 12 * q^22 + 12 * q^23 + 2 * q^25 - 4 * q^26 + 12 * q^28 - 2 * q^29 - 4 * q^31 - 6 * q^32 + 8 * q^34 + 4 * q^35 + 4 * q^38 - 6 * q^40 + 12 * q^41 - 12 * q^43 + 20 * q^44 + 20 * q^46 + 12 * q^47 + 10 * q^49 + 2 * q^50 - 4 * q^52 - 4 * q^53 - 4 * q^55 - 4 * q^56 - 2 * q^58 + 4 * q^61 - 28 * q^62 - 14 * q^64 + 4 * q^65 - 4 * q^67 + 16 * q^68 - 4 * q^70 + 8 * q^71 + 24 * q^74 + 12 * q^76 + 8 * q^77 + 12 * q^79 - 6 * q^80 + 12 * q^82 - 20 * q^83 - 12 * q^86 + 20 * q^88 + 4 * q^89 + 8 * q^91 + 28 * q^92 - 4 * q^94 + 4 * q^95 - 8 * q^97 - 22 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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