LMFDB - Embedded newform 1755.2.a.v.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1755,2,Mod(1,1755)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1755, 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("1755.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1755 = 3^{3} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1755.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:	$14.0137455547$
Analytic rank:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 12x^{5} + 9x^{4} + 39x^{3} - 16x^{2} - 30x + 12 $ x^7 - x^6 - 12x^5 + 9x^4 + 39x^3 - 16x^2 - 30*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.18218$ of defining polynomial
Character	$\chi$	$=$	1755.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.18218 q^{2} -0.602453 q^{4} +1.00000 q^{5} -5.08623 q^{7} +3.07657 q^{8} +O(q^{10})$	\(q-1.18218 q^{2} -0.602453 q^{4} +1.00000 q^{5} -5.08623 q^{7} +3.07657 q^{8} -1.18218 q^{10} -3.05473 q^{11} +1.00000 q^{13} +6.01284 q^{14} -2.43214 q^{16} +1.22503 q^{17} +3.82969 q^{19} -0.602453 q^{20} +3.61123 q^{22} -6.12723 q^{23} +1.00000 q^{25} -1.18218 q^{26} +3.06422 q^{28} -9.62407 q^{29} -8.15873 q^{31} -3.27790 q^{32} -1.44821 q^{34} -5.08623 q^{35} +10.7602 q^{37} -4.52738 q^{38} +3.07657 q^{40} -1.43126 q^{41} +3.10658 q^{43} +1.84033 q^{44} +7.24348 q^{46} +11.9884 q^{47} +18.8698 q^{49} -1.18218 q^{50} -0.602453 q^{52} +3.61269 q^{53} -3.05473 q^{55} -15.6481 q^{56} +11.3774 q^{58} +7.99928 q^{59} +1.70123 q^{61} +9.64508 q^{62} +8.73935 q^{64} +1.00000 q^{65} -8.90383 q^{67} -0.738026 q^{68} +6.01284 q^{70} -13.1957 q^{71} -5.69892 q^{73} -12.7205 q^{74} -2.30721 q^{76} +15.5370 q^{77} +12.0225 q^{79} -2.43214 q^{80} +1.69200 q^{82} -1.37522 q^{83} +1.22503 q^{85} -3.67253 q^{86} -9.39806 q^{88} -9.68648 q^{89} -5.08623 q^{91} +3.69137 q^{92} -14.1725 q^{94} +3.82969 q^{95} +4.70970 q^{97} -22.3074 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + q^{2} + 11 q^{4} + 7 q^{5} + 2 q^{7} + 6 q^{8}+O(q^{10}) $ 7 * q + q^2 + 11 * q^4 + 7 * q^5 + 2 * q^7 + 6 * q^8	$ 7 q + q^{2} + 11 q^{4} + 7 q^{5} + 2 q^{7} + 6 q^{8} + q^{10} + q^{11} + 7 q^{13} - 12 q^{14} + 23 q^{16} + 11 q^{17} + 2 q^{19} + 11 q^{20} + 16 q^{22} - q^{23} + 7 q^{25} + q^{26} + 10 q^{28} - 4 q^{29} + 13 q^{32} + q^{34} + 2 q^{35} + 23 q^{37} - 15 q^{38} + 6 q^{40} + 2 q^{41} + 8 q^{43} + 10 q^{44} + 37 q^{46} + 2 q^{47} + 43 q^{49} + q^{50} + 11 q^{52} + 10 q^{53} + q^{55} - 68 q^{56} - 26 q^{58} - 13 q^{59} + 21 q^{61} + 9 q^{62} + 46 q^{64} + 7 q^{65} + 21 q^{67} + 53 q^{68} - 12 q^{70} - 10 q^{71} + 13 q^{73} - 68 q^{74} - 41 q^{76} + 6 q^{77} + 8 q^{79} + 23 q^{80} - 26 q^{82} - 4 q^{83} + 11 q^{85} - 12 q^{86} + 44 q^{88} - 27 q^{89} + 2 q^{91} - 9 q^{92} - 24 q^{94} + 2 q^{95} - 15 q^{97} + 7 q^{98}+O(q^{100}) $ 7 * q + q^2 + 11 * q^4 + 7 * q^5 + 2 * q^7 + 6 * q^8 + q^10 + q^11 + 7 * q^13 - 12 * q^14 + 23 * q^16 + 11 * q^17 + 2 * q^19 + 11 * q^20 + 16 * q^22 - q^23 + 7 * q^25 + q^26 + 10 * q^28 - 4 * q^29 + 13 * q^32 + q^34 + 2 * q^35 + 23 * q^37 - 15 * q^38 + 6 * q^40 + 2 * q^41 + 8 * q^43 + 10 * q^44 + 37 * q^46 + 2 * q^47 + 43 * q^49 + q^50 + 11 * q^52 + 10 * q^53 + q^55 - 68 * q^56 - 26 * q^58 - 13 * q^59 + 21 * q^61 + 9 * q^62 + 46 * q^64 + 7 * q^65 + 21 * q^67 + 53 * q^68 - 12 * q^70 - 10 * q^71 + 13 * q^73 - 68 * q^74 - 41 * q^76 + 6 * q^77 + 8 * q^79 + 23 * q^80 - 26 * q^82 - 4 * q^83 + 11 * q^85 - 12 * q^86 + 44 * q^88 - 27 * q^89 + 2 * q^91 - 9 * q^92 - 24 * q^94 + 2 * q^95 - 15 * q^97 + 7 * q^98

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 $

$2$	−1.18218	−0.835927	−0.417963	−	0.908464i	$-0.637256\pi$
$2$	−1.18218	−0.835927	−0.417963	+	0.908464i	$0.637256\pi$
$3$	0	0
$3$	0	0
$4$	−0.602453	−0.301227
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−5.08623	−1.92241	−0.961207	−	0.275827i	$-0.911048\pi$
$7$	−5.08623	−1.92241	−0.961207	+	0.275827i	$0.911048\pi$
$8$	3.07657	1.08773
$9$	0	0
$10$	−1.18218	−0.373838
$11$	−3.05473	−0.921034	−0.460517	−	0.887651i	$-0.652336\pi$
$11$	−3.05473	−0.921034	−0.460517	+	0.887651i	$0.652336\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	6.01284	1.60700
$15$	0	0
$16$	−2.43214	−0.608036
$17$	1.22503	0.297114	0.148557	−	0.988904i	$-0.452537\pi$
$17$	1.22503	0.297114	0.148557	+	0.988904i	$0.452537\pi$
$18$	0	0
$19$	3.82969	0.878591	0.439296	−	0.898343i	$-0.355228\pi$
$19$	3.82969	0.878591	0.439296	+	0.898343i	$0.355228\pi$
$20$	−0.602453	−0.134713
$21$	0	0
$22$	3.61123	0.769917
$23$	−6.12723	−1.27762	−0.638808	−	0.769366i	$-0.720572\pi$
$23$	−6.12723	−1.27762	−0.638808	+	0.769366i	$0.720572\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−1.18218	−0.231844
$27$	0	0
$28$	3.06422	0.579082
$29$	−9.62407	−1.78714	−0.893572	−	0.448919i	$-0.851809\pi$
$29$	−9.62407	−1.78714	−0.893572	+	0.448919i	$0.851809\pi$
$30$	0	0
$31$	−8.15873	−1.46535	−0.732676	−	0.680578i	$-0.761729\pi$
$31$	−8.15873	−1.46535	−0.732676	+	0.680578i	$0.761729\pi$
$32$	−3.27790	−0.579456
$33$	0	0
$34$	−1.44821	−0.248366
$35$	−5.08623	−0.859730
$36$	0	0
$37$	10.7602	1.76897	0.884485	−	0.466569i	$-0.154510\pi$
$37$	10.7602	1.76897	0.884485	+	0.466569i	$0.154510\pi$
$38$	−4.52738	−0.734438
$39$	0	0
$40$	3.07657	0.486448
$41$	−1.43126	−0.223525	−0.111762	−	0.993735i	$-0.535650\pi$
$41$	−1.43126	−0.223525	−0.111762	+	0.993735i	$0.535650\pi$
$42$	0	0
$43$	3.10658	0.473749	0.236874	−	0.971540i	$-0.423877\pi$
$43$	3.10658	0.473749	0.236874	+	0.971540i	$0.423877\pi$
$44$	1.84033	0.277440
$45$	0	0
$46$	7.24348	1.06799
$47$	11.9884	1.74869	0.874346	−	0.485304i	$-0.161291\pi$
$47$	11.9884	1.74869	0.874346	+	0.485304i	$0.161291\pi$
$48$	0	0
$49$	18.8698	2.69568
$50$	−1.18218	−0.167185
$51$	0	0
$52$	−0.602453	−0.0835452
$53$	3.61269	0.496241	0.248121	−	0.968729i	$-0.420187\pi$
$53$	3.61269	0.496241	0.248121	+	0.968729i	$0.420187\pi$
$54$	0	0
$55$	−3.05473	−0.411899
$56$	−15.6481	−2.09107
$57$	0	0
$58$	11.3774	1.49392
$59$	7.99928	1.04142	0.520709	−	0.853734i	$-0.325668\pi$
$59$	7.99928	1.04142	0.520709	+	0.853734i	$0.325668\pi$
$60$	0	0
$61$	1.70123	0.217820	0.108910	−	0.994052i	$-0.465264\pi$
$61$	1.70123	0.217820	0.108910	+	0.994052i	$0.465264\pi$
$62$	9.64508	1.22493
$63$	0	0
$64$	8.73935	1.09242
$65$	1.00000	0.124035
$66$	0	0
$67$	−8.90383	−1.08778	−0.543888	−	0.839158i	$-0.683048\pi$
$67$	−8.90383	−1.08778	−0.543888	+	0.839158i	$0.683048\pi$
$68$	−0.738026	−0.0894988
$69$	0	0
$70$	6.01284	0.718671
$71$	−13.1957	−1.56604	−0.783020	−	0.621997i	$-0.786322\pi$
$71$	−13.1957	−1.56604	−0.783020	+	0.621997i	$0.786322\pi$
$72$	0	0
$73$	−5.69892	−0.667008	−0.333504	−	0.942749i	$-0.608231\pi$
$73$	−5.69892	−0.667008	−0.333504	+	0.942749i	$0.608231\pi$
$74$	−12.7205	−1.47873
$75$	0	0
$76$	−2.30721	−0.264655
$77$	15.5370	1.77061
$78$	0	0
$79$	12.0225	1.35264	0.676319	−	0.736609i	$-0.263574\pi$
$79$	12.0225	1.35264	0.676319	+	0.736609i	$0.263574\pi$
$80$	−2.43214	−0.271922
$81$	0	0
$82$	1.69200	0.186850
$83$	−1.37522	−0.150949	−0.0754747	−	0.997148i	$-0.524047\pi$
$83$	−1.37522	−0.150949	−0.0754747	+	0.997148i	$0.524047\pi$
$84$	0	0
$85$	1.22503	0.132874
$86$	−3.67253	−0.396019
$87$	0	0
$88$	−9.39806	−1.00184
$89$	−9.68648	−1.02676	−0.513382	−	0.858160i	$-0.671608\pi$
$89$	−9.68648	−1.02676	−0.513382	+	0.858160i	$0.671608\pi$
$90$	0	0
$91$	−5.08623	−0.533182
$92$	3.69137	0.384852
$93$	0	0
$94$	−14.1725	−1.46178
$95$	3.82969	0.392918
$96$	0	0
$97$	4.70970	0.478198	0.239099	−	0.970995i	$-0.423148\pi$
$97$	4.70970	0.478198	0.239099	+	0.970995i	$0.423148\pi$
$98$	−22.3074	−2.25339
$99$	0	0
$100$	−0.602453	−0.0602453
$101$	10.5368	1.04845	0.524226	−	0.851579i	$-0.324355\pi$
$101$	10.5368	1.04845	0.524226	+	0.851579i	$0.324355\pi$
$102$	0	0
$103$	17.1933	1.69411	0.847055	−	0.531506i	$-0.178374\pi$
$103$	17.1933	1.69411	0.847055	+	0.531506i	$0.178374\pi$
$104$	3.07657	0.301682
$105$	0	0
$106$	−4.27085	−0.414821
$107$	−2.10666	−0.203658	−0.101829	−	0.994802i	$-0.532469\pi$
$107$	−2.10666	−0.203658	−0.101829	+	0.994802i	$0.532469\pi$
$108$	0	0
$109$	13.1796	1.26238	0.631188	−	0.775629i	$-0.282567\pi$
$109$	13.1796	1.26238	0.631188	+	0.775629i	$0.282567\pi$
$110$	3.61123	0.344317
$111$	0	0
$112$	12.3704	1.16890
$113$	16.7809	1.57861	0.789306	−	0.614000i	$-0.210441\pi$
$113$	16.7809	1.57861	0.789306	+	0.614000i	$0.210441\pi$
$114$	0	0
$115$	−6.12723	−0.571367
$116$	5.79805	0.538335
$117$	0	0
$118$	−9.45658	−0.870549
$119$	−6.23081	−0.571177
$120$	0	0
$121$	−1.66865	−0.151696
$122$	−2.01115	−0.182081
$123$	0	0
$124$	4.91525	0.441403
$125$	1.00000	0.0894427
$126$	0	0
$127$	20.0606	1.78009	0.890047	−	0.455870i	$-0.150672\pi$
$127$	20.0606	1.78009	0.890047	+	0.455870i	$0.150672\pi$
$128$	−3.77568	−0.333726
$129$	0	0
$130$	−1.18218	−0.103684
$131$	5.84342	0.510542	0.255271	−	0.966870i	$-0.417835\pi$
$131$	5.84342	0.510542	0.255271	+	0.966870i	$0.417835\pi$
$132$	0	0
$133$	−19.4787	−1.68902
$134$	10.5259	0.909301
$135$	0	0
$136$	3.76890	0.323180
$137$	5.91981	0.505763	0.252882	−	0.967497i	$-0.418622\pi$
$137$	5.91981	0.505763	0.252882	+	0.967497i	$0.418622\pi$
$138$	0	0
$139$	−16.5578	−1.40441	−0.702206	−	0.711974i	$-0.747801\pi$
$139$	−16.5578	−1.40441	−0.702206	+	0.711974i	$0.747801\pi$
$140$	3.06422	0.258974
$141$	0	0
$142$	15.5997	1.30909
$143$	−3.05473	−0.255449
$144$	0	0
$145$	−9.62407	−0.799235
$146$	6.73714	0.557570
$147$	0	0
$148$	−6.48253	−0.532861
$149$	−1.23641	−0.101291	−0.0506454	−	0.998717i	$-0.516128\pi$
$149$	−1.23641	−0.101291	−0.0506454	+	0.998717i	$0.516128\pi$
$150$	0	0
$151$	1.92780	0.156882	0.0784409	−	0.996919i	$-0.475006\pi$
$151$	1.92780	0.156882	0.0784409	+	0.996919i	$0.475006\pi$
$152$	11.7823	0.955670
$153$	0	0
$154$	−18.3676	−1.48010
$155$	−8.15873	−0.655325
$156$	0	0
$157$	10.6385	0.849046	0.424523	−	0.905417i	$-0.360442\pi$
$157$	10.6385	0.849046	0.424523	+	0.905417i	$0.360442\pi$
$158$	−14.2127	−1.13071
$159$	0	0
$160$	−3.27790	−0.259141
$161$	31.1645	2.45611
$162$	0	0
$163$	−6.70970	−0.525544	−0.262772	−	0.964858i	$-0.584637\pi$
$163$	−6.70970	−0.525544	−0.262772	+	0.964858i	$0.584637\pi$
$164$	0.862265	0.0673316
$165$	0	0
$166$	1.62575	0.126183
$167$	6.94448	0.537380	0.268690	−	0.963227i	$-0.413409\pi$
$167$	6.94448	0.537380	0.268690	+	0.963227i	$0.413409\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−1.44821	−0.111073
$171$	0	0
$172$	−1.87157	−0.142706
$173$	11.1572	0.848266	0.424133	−	0.905600i	$-0.360579\pi$
$173$	11.1572	0.848266	0.424133	+	0.905600i	$0.360579\pi$
$174$	0	0
$175$	−5.08623	−0.384483
$176$	7.42953	0.560022
$177$	0	0
$178$	11.4512	0.858300
$179$	−10.0455	−0.750833	−0.375416	−	0.926856i	$-0.622500\pi$
$179$	−10.0455	−0.750833	−0.375416	+	0.926856i	$0.622500\pi$
$180$	0	0
$181$	16.3078	1.21215	0.606074	−	0.795408i	$-0.292744\pi$
$181$	16.3078	1.21215	0.606074	+	0.795408i	$0.292744\pi$
$182$	6.01284	0.445701
$183$	0	0
$184$	−18.8508	−1.38970
$185$	10.7602	0.791107
$186$	0	0
$187$	−3.74214	−0.273653
$188$	−7.22246	−0.526752
$189$	0	0
$190$	−4.52738	−0.328451
$191$	3.85285	0.278783	0.139391	−	0.990237i	$-0.455485\pi$
$191$	3.85285	0.278783	0.139391	+	0.990237i	$0.455485\pi$
$192$	0	0
$193$	4.16267	0.299636	0.149818	−	0.988714i	$-0.452131\pi$
$193$	4.16267	0.299636	0.149818	+	0.988714i	$0.452131\pi$
$194$	−5.56771	−0.399738
$195$	0	0
$196$	−11.3681	−0.812010
$197$	−10.0115	−0.713290	−0.356645	−	0.934240i	$-0.616079\pi$
$197$	−10.0115	−0.713290	−0.356645	+	0.934240i	$0.616079\pi$
$198$	0	0
$199$	21.6408	1.53407	0.767036	−	0.641604i	$-0.221731\pi$
$199$	21.6408	1.53407	0.767036	+	0.641604i	$0.221731\pi$
$200$	3.07657	0.217546
$201$	0	0
$202$	−12.4564	−0.876430
$203$	48.9502	3.43563
$204$	0	0
$205$	−1.43126	−0.0999633
$206$	−20.3256	−1.41615
$207$	0	0
$208$	−2.43214	−0.168639
$209$	−11.6987	−0.809213
$210$	0	0
$211$	−4.74218	−0.326465	−0.163232	−	0.986588i	$-0.552192\pi$
$211$	−4.74218	−0.326465	−0.163232	+	0.986588i	$0.552192\pi$
$212$	−2.17648	−0.149481
$213$	0	0
$214$	2.49045	0.170243
$215$	3.10658	0.211867
$216$	0	0
$217$	41.4972	2.81701
$218$	−15.5806	−1.05525
$219$	0	0
$220$	1.84033	0.124075
$221$	1.22503	0.0824047
$222$	0	0
$223$	19.6327	1.31470	0.657351	−	0.753584i	$-0.271677\pi$
$223$	19.6327	1.31470	0.657351	+	0.753584i	$0.271677\pi$
$224$	16.6722	1.11396
$225$	0	0
$226$	−19.8380	−1.31960
$227$	1.10918	0.0736191	0.0368096	−	0.999322i	$-0.488281\pi$
$227$	1.10918	0.0736191	0.0368096	+	0.999322i	$0.488281\pi$
$228$	0	0
$229$	23.9164	1.58044	0.790222	−	0.612821i	$-0.209965\pi$
$229$	23.9164	1.58044	0.790222	+	0.612821i	$0.209965\pi$
$230$	7.24348	0.477621
$231$	0	0
$232$	−29.6091	−1.94393
$233$	−22.2796	−1.45958	−0.729791	−	0.683670i	$-0.760383\pi$
$233$	−22.2796	−1.45958	−0.729791	+	0.683670i	$0.760383\pi$
$234$	0	0
$235$	11.9884	0.782039
$236$	−4.81919	−0.313703
$237$	0	0
$238$	7.36593	0.477462
$239$	−19.4553	−1.25846	−0.629230	−	0.777219i	$-0.716630\pi$
$239$	−19.4553	−1.25846	−0.629230	+	0.777219i	$0.716630\pi$
$240$	0	0
$241$	9.75882	0.628621	0.314310	−	0.949320i	$-0.398227\pi$
$241$	9.75882	0.628621	0.314310	+	0.949320i	$0.398227\pi$
$242$	1.97265	0.126806
$243$	0	0
$244$	−1.02491	−0.0656130
$245$	18.8698	1.20554
$246$	0	0
$247$	3.82969	0.243677
$248$	−25.1009	−1.59391
$249$	0	0
$250$	−1.18218	−0.0747676
$251$	−24.1416	−1.52380	−0.761901	−	0.647694i	$-0.775734\pi$
$251$	−24.1416	−1.52380	−0.761901	+	0.647694i	$0.775734\pi$
$252$	0	0
$253$	18.7170	1.17673
$254$	−23.7153	−1.48803
$255$	0	0
$256$	−13.0152	−0.813449
$257$	3.23557	0.201830	0.100915	−	0.994895i	$-0.467823\pi$
$257$	3.23557	0.201830	0.100915	+	0.994895i	$0.467823\pi$
$258$	0	0
$259$	−54.7290	−3.40069
$260$	−0.602453	−0.0373626
$261$	0	0
$262$	−6.90797	−0.426776
$263$	0.158339	0.00976357	0.00488179	−	0.999988i	$-0.498446\pi$
$263$	0.158339	0.00976357	0.00488179	+	0.999988i	$0.498446\pi$
$264$	0	0
$265$	3.61269	0.221926
$266$	23.0273	1.41189
$267$	0	0
$268$	5.36414	0.327667
$269$	−6.12296	−0.373323	−0.186662	−	0.982424i	$-0.559767\pi$
$269$	−6.12296	−0.373323	−0.186662	+	0.982424i	$0.559767\pi$
$270$	0	0
$271$	−4.80749	−0.292034	−0.146017	−	0.989282i	$-0.546645\pi$
$271$	−4.80749	−0.292034	−0.146017	+	0.989282i	$0.546645\pi$
$272$	−2.97946	−0.180656
$273$	0	0
$274$	−6.99827	−0.422781
$275$	−3.05473	−0.184207
$276$	0	0
$277$	7.84342	0.471265	0.235633	−	0.971842i	$-0.424284\pi$
$277$	7.84342	0.471265	0.235633	+	0.971842i	$0.424284\pi$
$278$	19.5742	1.17399
$279$	0	0
$280$	−15.6481	−0.935154
$281$	−5.93820	−0.354243	−0.177122	−	0.984189i	$-0.556679\pi$
$281$	−5.93820	−0.354243	−0.177122	+	0.984189i	$0.556679\pi$
$282$	0	0
$283$	7.43554	0.441997	0.220998	−	0.975274i	$-0.429068\pi$
$283$	7.43554	0.441997	0.220998	+	0.975274i	$0.429068\pi$
$284$	7.94978	0.471733
$285$	0	0
$286$	3.61123	0.213537
$287$	7.27970	0.429707
$288$	0	0
$289$	−15.4993	−0.911723
$290$	11.3774	0.668102
$291$	0	0
$292$	3.43333	0.200921
$293$	10.1604	0.593578	0.296789	−	0.954943i	$-0.404084\pi$
$293$	10.1604	0.593578	0.296789	+	0.954943i	$0.404084\pi$
$294$	0	0
$295$	7.99928	0.465736
$296$	33.1045	1.92416
$297$	0	0
$298$	1.46166	0.0846717
$299$	−6.12723	−0.354347
$300$	0	0
$301$	−15.8008	−0.910742
$302$	−2.27900	−0.131142
$303$	0	0
$304$	−9.31436	−0.534215
$305$	1.70123	0.0974119
$306$	0	0
$307$	−28.2980	−1.61505	−0.807527	−	0.589830i	$-0.799195\pi$
$307$	−28.2980	−1.61505	−0.807527	+	0.589830i	$0.799195\pi$
$308$	−9.36034	−0.533355
$309$	0	0
$310$	9.64508	0.547804
$311$	0.949999	0.0538695	0.0269348	−	0.999637i	$-0.491425\pi$
$311$	0.949999	0.0538695	0.0269348	+	0.999637i	$0.491425\pi$
$312$	0	0
$313$	13.9865	0.790563	0.395282	−	0.918560i	$-0.370647\pi$
$313$	13.9865	0.790563	0.395282	+	0.918560i	$0.370647\pi$
$314$	−12.5766	−0.709740
$315$	0	0
$316$	−7.24299	−0.407450
$317$	−5.03267	−0.282663	−0.141331	−	0.989962i	$-0.545138\pi$
$317$	−5.03267	−0.282663	−0.141331	+	0.989962i	$0.545138\pi$
$318$	0	0
$319$	29.3989	1.64602
$320$	8.73935	0.488545
$321$	0	0
$322$	−36.8420	−2.05313
$323$	4.69150	0.261042
$324$	0	0
$325$	1.00000	0.0554700
$326$	7.93207	0.439317
$327$	0	0
$328$	−4.40335	−0.243135
$329$	−60.9759	−3.36171
$330$	0	0
$331$	−34.8862	−1.91752	−0.958758	−	0.284223i	$-0.908264\pi$
$331$	−34.8862	−1.91752	−0.958758	+	0.284223i	$0.908264\pi$
$332$	0.828503	0.0454700
$333$	0	0
$334$	−8.20962	−0.449210
$335$	−8.90383	−0.486468
$336$	0	0
$337$	3.67560	0.200223	0.100111	−	0.994976i	$-0.468080\pi$
$337$	3.67560	0.200223	0.100111	+	0.994976i	$0.468080\pi$
$338$	−1.18218	−0.0643021
$339$	0	0
$340$	−0.738026	−0.0400251
$341$	24.9227	1.34964
$342$	0	0
$343$	−60.3723	−3.25980
$344$	9.55759	0.515311
$345$	0	0
$346$	−13.1898	−0.709088
$347$	26.8662	1.44225	0.721127	−	0.692803i	$-0.243624\pi$
$347$	26.8662	1.44225	0.721127	+	0.692803i	$0.243624\pi$
$348$	0	0
$349$	8.40207	0.449752	0.224876	−	0.974387i	$-0.427802\pi$
$349$	8.40207	0.449752	0.224876	+	0.974387i	$0.427802\pi$
$350$	6.01284	0.321400
$351$	0	0
$352$	10.0131	0.533699
$353$	−13.3255	−0.709244	−0.354622	−	0.935010i	$-0.615390\pi$
$353$	−13.3255	−0.709244	−0.354622	+	0.935010i	$0.615390\pi$
$354$	0	0
$355$	−13.1957	−0.700354
$356$	5.83565	0.309289
$357$	0	0
$358$	11.8755	0.627641
$359$	7.52750	0.397286	0.198643	−	0.980072i	$-0.436347\pi$
$359$	7.52750	0.397286	0.198643	+	0.980072i	$0.436347\pi$
$360$	0	0
$361$	−4.33347	−0.228077
$362$	−19.2787	−1.01327
$363$	0	0
$364$	3.06422	0.160609
$365$	−5.69892	−0.298295
$366$	0	0
$367$	6.07819	0.317279	0.158639	−	0.987337i	$-0.449289\pi$
$367$	6.07819	0.317279	0.158639	+	0.987337i	$0.449289\pi$
$368$	14.9023	0.776836
$369$	0	0
$370$	−12.7205	−0.661308
$371$	−18.3750	−0.953981
$372$	0	0
$373$	3.50078	0.181263	0.0906317	−	0.995884i	$-0.471111\pi$
$373$	3.50078	0.181263	0.0906317	+	0.995884i	$0.471111\pi$
$374$	4.42388	0.228754
$375$	0	0
$376$	36.8832	1.90210
$377$	−9.62407	−0.495665
$378$	0	0
$379$	−9.71881	−0.499221	−0.249611	−	0.968346i	$-0.580303\pi$
$379$	−9.71881	−0.499221	−0.249611	+	0.968346i	$0.580303\pi$
$380$	−2.30721	−0.118357
$381$	0	0
$382$	−4.55476	−0.233042
$383$	15.2846	0.781005	0.390502	−	0.920602i	$-0.372301\pi$
$383$	15.2846	0.781005	0.390502	+	0.920602i	$0.372301\pi$
$384$	0	0
$385$	15.5370	0.791841
$386$	−4.92102	−0.250473
$387$	0	0
$388$	−2.83737	−0.144046
$389$	37.6557	1.90922	0.954610	−	0.297857i	$-0.0962720\pi$
$389$	37.6557	1.90922	0.954610	+	0.297857i	$0.0962720\pi$
$390$	0	0
$391$	−7.50606	−0.379598
$392$	58.0540	2.93217
$393$	0	0
$394$	11.8354	0.596258
$395$	12.0225	0.604918
$396$	0	0
$397$	−2.91967	−0.146534	−0.0732670	−	0.997312i	$-0.523343\pi$
$397$	−2.91967	−0.146534	−0.0732670	+	0.997312i	$0.523343\pi$
$398$	−25.5832	−1.28237
$399$	0	0
$400$	−2.43214	−0.121607
$401$	−15.6306	−0.780555	−0.390278	−	0.920697i	$-0.627621\pi$
$401$	−15.6306	−0.780555	−0.390278	+	0.920697i	$0.627621\pi$
$402$	0	0
$403$	−8.15873	−0.406415
$404$	−6.34794	−0.315822
$405$	0	0
$406$	−57.8679	−2.87194
$407$	−32.8695	−1.62928
$408$	0	0
$409$	−12.3252	−0.609443	−0.304721	−	0.952442i	$-0.598563\pi$
$409$	−12.3252	−0.609443	−0.304721	+	0.952442i	$0.598563\pi$
$410$	1.69200	0.0835620
$411$	0	0
$412$	−10.3582	−0.510311
$413$	−40.6862	−2.00204
$414$	0	0
$415$	−1.37522	−0.0675067
$416$	−3.27790	−0.160712
$417$	0	0
$418$	13.8299	0.676443
$419$	21.3750	1.04424	0.522118	−	0.852873i	$-0.325142\pi$
$419$	21.3750	1.04424	0.522118	+	0.852873i	$0.325142\pi$
$420$	0	0
$421$	−12.2414	−0.596611	−0.298306	−	0.954470i	$-0.596421\pi$
$421$	−12.2414	−0.596611	−0.298306	+	0.954470i	$0.596421\pi$
$422$	5.60610	0.272901
$423$	0	0
$424$	11.1147	0.539776
$425$	1.22503	0.0594229
$426$	0	0
$427$	−8.65282	−0.418740
$428$	1.26916	0.0613473
$429$	0	0
$430$	−3.67253	−0.177105
$431$	21.4653	1.03395	0.516973	−	0.856002i	$-0.327059\pi$
$431$	21.4653	1.03395	0.516973	+	0.856002i	$0.327059\pi$
$432$	0	0
$433$	1.96983	0.0946638	0.0473319	−	0.998879i	$-0.484928\pi$
$433$	1.96983	0.0946638	0.0473319	+	0.998879i	$0.484928\pi$
$434$	−49.0571	−2.35482
$435$	0	0
$436$	−7.94009	−0.380261
$437$	−23.4654	−1.12250
$438$	0	0
$439$	2.04964	0.0978241	0.0489120	−	0.998803i	$-0.484425\pi$
$439$	2.04964	0.0978241	0.0489120	+	0.998803i	$0.484425\pi$
$440$	−9.39806	−0.448035
$441$	0	0
$442$	−1.44821	−0.0688843
$443$	−32.4481	−1.54166	−0.770828	−	0.637044i	$-0.780157\pi$
$443$	−32.4481	−1.54166	−0.770828	+	0.637044i	$0.780157\pi$
$444$	0	0
$445$	−9.68648	−0.459183
$446$	−23.2094	−1.09900
$447$	0	0
$448$	−44.4504	−2.10008
$449$	−20.4178	−0.963577	−0.481788	−	0.876288i	$-0.660013\pi$
$449$	−20.4178	−0.963577	−0.481788	+	0.876288i	$0.660013\pi$
$450$	0	0
$451$	4.37210	0.205874
$452$	−10.1097	−0.475520
$453$	0	0
$454$	−1.31125	−0.0615402
$455$	−5.08623	−0.238446
$456$	0	0
$457$	−4.17535	−0.195315	−0.0976573	−	0.995220i	$-0.531135\pi$
$457$	−4.17535	−0.195315	−0.0976573	+	0.995220i	$0.531135\pi$
$458$	−28.2735	−1.32113
$459$	0	0
$460$	3.69137	0.172111
$461$	−23.5976	−1.09905	−0.549525	−	0.835477i	$-0.685191\pi$
$461$	−23.5976	−1.09905	−0.549525	+	0.835477i	$0.685191\pi$
$462$	0	0
$463$	0.955834	0.0444214	0.0222107	−	0.999753i	$-0.492930\pi$
$463$	0.955834	0.0444214	0.0222107	+	0.999753i	$0.492930\pi$
$464$	23.4071	1.08665
$465$	0	0
$466$	26.3384	1.22010
$467$	25.2304	1.16752	0.583761	−	0.811926i	$-0.301581\pi$
$467$	25.2304	1.16752	0.583761	+	0.811926i	$0.301581\pi$
$468$	0	0
$469$	45.2869	2.09116
$470$	−14.1725	−0.653727
$471$	0	0
$472$	24.6103	1.13278
$473$	−9.48975	−0.436339
$474$	0	0
$475$	3.82969	0.175718
$476$	3.75377	0.172054
$477$	0	0
$478$	22.9997	1.05198
$479$	−4.59150	−0.209791	−0.104895	−	0.994483i	$-0.533451\pi$
$479$	−4.59150	−0.209791	−0.104895	+	0.994483i	$0.533451\pi$
$480$	0	0
$481$	10.7602	0.490624
$482$	−11.5367	−0.525481
$483$	0	0
$484$	1.00529	0.0456948
$485$	4.70970	0.213856
$486$	0	0
$487$	−28.1620	−1.27614	−0.638071	−	0.769978i	$-0.720267\pi$
$487$	−28.1620	−1.27614	−0.638071	+	0.769978i	$0.720267\pi$
$488$	5.23393	0.236929
$489$	0	0
$490$	−22.3074	−1.00775
$491$	−37.3150	−1.68400	−0.842000	−	0.539477i	$-0.818622\pi$
$491$	−37.3150	−1.68400	−0.842000	+	0.539477i	$0.818622\pi$
$492$	0	0
$493$	−11.7898	−0.530986
$494$	−4.52738	−0.203696
$495$	0	0
$496$	19.8432	0.890987
$497$	67.1163	3.01058
$498$	0	0
$499$	1.64892	0.0738158	0.0369079	−	0.999319i	$-0.488249\pi$
$499$	1.64892	0.0738158	0.0369079	+	0.999319i	$0.488249\pi$
$500$	−0.602453	−0.0269425
$501$	0	0
$502$	28.5396	1.27379
$503$	−9.30661	−0.414961	−0.207481	−	0.978239i	$-0.566526\pi$
$503$	−9.30661	−0.414961	−0.207481	+	0.978239i	$0.566526\pi$
$504$	0	0
$505$	10.5368	0.468882
$506$	−22.1268	−0.983658
$507$	0	0
$508$	−12.0856	−0.536211
$509$	−10.6433	−0.471757	−0.235878	−	0.971783i	$-0.575797\pi$
$509$	−10.6433	−0.471757	−0.235878	+	0.971783i	$0.575797\pi$
$510$	0	0
$511$	28.9860	1.28227
$512$	22.9376	1.01371
$513$	0	0
$514$	−3.82503	−0.168715
$515$	17.1933	0.757629
$516$	0	0
$517$	−36.6213	−1.61060
$518$	64.6994	2.84273
$519$	0	0
$520$	3.07657	0.134916
$521$	16.1142	0.705976	0.352988	−	0.935628i	$-0.385166\pi$
$521$	16.1142	0.705976	0.352988	+	0.935628i	$0.385166\pi$
$522$	0	0
$523$	8.17546	0.357488	0.178744	−	0.983896i	$-0.442797\pi$
$523$	8.17546	0.357488	0.178744	+	0.983896i	$0.442797\pi$
$524$	−3.52039	−0.153789
$525$	0	0
$526$	−0.187185	−0.00816163
$527$	−9.99473	−0.435377
$528$	0	0
$529$	14.5429	0.632301
$530$	−4.27085	−0.185514
$531$	0	0
$532$	11.7350	0.508777
$533$	−1.43126	−0.0619946
$534$	0	0
$535$	−2.10666	−0.0910787
$536$	−27.3932	−1.18321
$537$	0	0
$538$	7.23843	0.312071
$539$	−57.6419	−2.48281
$540$	0	0
$541$	−2.31254	−0.0994238	−0.0497119	−	0.998764i	$-0.515830\pi$
$541$	−2.31254	−0.0994238	−0.0497119	+	0.998764i	$0.515830\pi$
$542$	5.68331	0.244119
$543$	0	0
$544$	−4.01554	−0.172165
$545$	13.1796	0.564552
$546$	0	0
$547$	−1.11920	−0.0478533	−0.0239267	−	0.999714i	$-0.507617\pi$
$547$	−1.11920	−0.0478533	−0.0239267	+	0.999714i	$0.507617\pi$
$548$	−3.56641	−0.152349
$549$	0	0
$550$	3.61123	0.153983
$551$	−36.8572	−1.57017
$552$	0	0
$553$	−61.1492	−2.60033
$554$	−9.27233	−0.393943
$555$	0	0
$556$	9.97528	0.423046
$557$	−5.19407	−0.220080	−0.110040	−	0.993927i	$-0.535098\pi$
$557$	−5.19407	−0.220080	−0.110040	+	0.993927i	$0.535098\pi$
$558$	0	0
$559$	3.10658	0.131394
$560$	12.3704	0.522747
$561$	0	0
$562$	7.02002	0.296122
$563$	16.2598	0.685267	0.342634	−	0.939469i	$-0.388681\pi$
$563$	16.2598	0.685267	0.342634	+	0.939469i	$0.388681\pi$
$564$	0	0
$565$	16.7809	0.705977
$566$	−8.79014	−0.369477
$567$	0	0
$568$	−40.5974	−1.70343
$569$	10.3662	0.434574	0.217287	−	0.976108i	$-0.430279\pi$
$569$	10.3662	0.434574	0.217287	+	0.976108i	$0.430279\pi$
$570$	0	0
$571$	−9.78220	−0.409372	−0.204686	−	0.978828i	$-0.565617\pi$
$571$	−9.78220	−0.409372	−0.204686	+	0.978828i	$0.565617\pi$
$572$	1.84033	0.0769480
$573$	0	0
$574$	−8.60591	−0.359204
$575$	−6.12723	−0.255523
$576$	0	0
$577$	−3.45820	−0.143967	−0.0719834	−	0.997406i	$-0.522933\pi$
$577$	−3.45820	−0.143967	−0.0719834	+	0.997406i	$0.522933\pi$
$578$	18.3229	0.762134
$579$	0	0
$580$	5.79805	0.240751
$581$	6.99466	0.290188
$582$	0	0
$583$	−11.0358	−0.457055
$584$	−17.5331	−0.725525
$585$	0	0
$586$	−12.0114	−0.496188
$587$	21.4313	0.884563	0.442281	−	0.896876i	$-0.354169\pi$
$587$	21.4313	0.884563	0.442281	+	0.896876i	$0.354169\pi$
$588$	0	0
$589$	−31.2454	−1.28745
$590$	−9.45658	−0.389321
$591$	0	0
$592$	−26.1704	−1.07560
$593$	14.4716	0.594276	0.297138	−	0.954835i	$-0.403968\pi$
$593$	14.4716	0.594276	0.297138	+	0.954835i	$0.403968\pi$
$594$	0	0
$595$	−6.23081	−0.255438
$596$	0.744880	0.0305115
$597$	0	0
$598$	7.24348	0.296208
$599$	3.81824	0.156009	0.0780046	−	0.996953i	$-0.475145\pi$
$599$	3.81824	0.156009	0.0780046	+	0.996953i	$0.475145\pi$
$600$	0	0
$601$	41.4200	1.68956	0.844780	−	0.535115i	$-0.179732\pi$
$601$	41.4200	1.68956	0.844780	+	0.535115i	$0.179732\pi$
$602$	18.6794	0.761313
$603$	0	0
$604$	−1.16141	−0.0472569
$605$	−1.66865	−0.0678404
$606$	0	0
$607$	8.85661	0.359479	0.179739	−	0.983714i	$-0.442475\pi$
$607$	8.85661	0.359479	0.179739	+	0.983714i	$0.442475\pi$
$608$	−12.5533	−0.509105
$609$	0	0
$610$	−2.01115	−0.0814292
$611$	11.9884	0.485000
$612$	0	0
$613$	24.6951	0.997427	0.498713	−	0.866767i	$-0.333806\pi$
$613$	24.6951	0.997427	0.498713	+	0.866767i	$0.333806\pi$
$614$	33.4533	1.35007
$615$	0	0
$616$	47.8007	1.92595
$617$	23.0744	0.928940	0.464470	−	0.885589i	$-0.346245\pi$
$617$	23.0744	0.928940	0.464470	+	0.885589i	$0.346245\pi$
$618$	0	0
$619$	33.7276	1.35563	0.677813	−	0.735235i	$-0.262928\pi$
$619$	33.7276	1.35563	0.677813	+	0.735235i	$0.262928\pi$
$620$	4.91525	0.197401
$621$	0	0
$622$	−1.12307	−0.0450310
$623$	49.2677	1.97387
$624$	0	0
$625$	1.00000	0.0400000
$626$	−16.5345	−0.660853
$627$	0	0
$628$	−6.40921	−0.255755
$629$	13.1816	0.525586
$630$	0	0
$631$	−7.28545	−0.290029	−0.145015	−	0.989430i	$-0.546323\pi$
$631$	−7.28545	−0.290029	−0.145015	+	0.989430i	$0.546323\pi$
$632$	36.9880	1.47130
$633$	0	0
$634$	5.94951	0.236285
$635$	20.0606	0.796082
$636$	0	0
$637$	18.8698	0.747647
$638$	−34.7547	−1.37595
$639$	0	0
$640$	−3.77568	−0.149247
$641$	−16.3481	−0.645713	−0.322856	−	0.946448i	$-0.604643\pi$
$641$	−16.3481	−0.645713	−0.322856	+	0.946448i	$0.604643\pi$
$642$	0	0
$643$	7.47645	0.294842	0.147421	−	0.989074i	$-0.452903\pi$
$643$	7.47645	0.294842	0.147421	+	0.989074i	$0.452903\pi$
$644$	−18.7751	−0.739845
$645$	0	0
$646$	−5.54620	−0.218212
$647$	43.5894	1.71368	0.856839	−	0.515584i	$-0.172425\pi$
$647$	43.5894	1.71368	0.856839	+	0.515584i	$0.172425\pi$
$648$	0	0
$649$	−24.4356	−0.959182
$650$	−1.18218	−0.0463689
$651$	0	0
$652$	4.04228	0.158308
$653$	−32.2421	−1.26173	−0.630866	−	0.775892i	$-0.717300\pi$
$653$	−32.2421	−1.26173	−0.630866	+	0.775892i	$0.717300\pi$
$654$	0	0
$655$	5.84342	0.228321
$656$	3.48102	0.135911
$657$	0	0
$658$	72.0844	2.81014
$659$	20.9167	0.814799	0.407400	−	0.913250i	$-0.366436\pi$
$659$	20.9167	0.814799	0.407400	+	0.913250i	$0.366436\pi$
$660$	0	0
$661$	−7.26420	−0.282545	−0.141272	−	0.989971i	$-0.545119\pi$
$661$	−7.26420	−0.282545	−0.141272	+	0.989971i	$0.545119\pi$
$662$	41.2417	1.60290
$663$	0	0
$664$	−4.23094	−0.164192
$665$	−19.4787	−0.755351
$666$	0	0
$667$	58.9689	2.28328
$668$	−4.18372	−0.161873
$669$	0	0
$670$	10.5259	0.406652
$671$	−5.19678	−0.200619
$672$	0	0
$673$	−0.793515	−0.0305877	−0.0152939	−	0.999883i	$-0.504868\pi$
$673$	−0.793515	−0.0305877	−0.0152939	+	0.999883i	$0.504868\pi$
$674$	−4.34522	−0.167372
$675$	0	0
$676$	−0.602453	−0.0231713
$677$	−15.2675	−0.586778	−0.293389	−	0.955993i	$-0.594783\pi$
$677$	−15.2675	−0.586778	−0.293389	+	0.955993i	$0.594783\pi$
$678$	0	0
$679$	−23.9546	−0.919294
$680$	3.76890	0.144531
$681$	0	0
$682$	−29.4631	−1.12820
$683$	−4.06694	−0.155617	−0.0778086	−	0.996968i	$-0.524792\pi$
$683$	−4.06694	−0.155617	−0.0778086	+	0.996968i	$0.524792\pi$
$684$	0	0
$685$	5.91981	0.226184
$686$	71.3709	2.72495
$687$	0	0
$688$	−7.55565	−0.288056
$689$	3.61269	0.137633
$690$	0	0
$691$	29.1051	1.10721	0.553606	−	0.832779i	$-0.313252\pi$
$691$	29.1051	1.10721	0.553606	+	0.832779i	$0.313252\pi$
$692$	−6.72169	−0.255520
$693$	0	0
$694$	−31.7607	−1.20562
$695$	−16.5578	−0.628072
$696$	0	0
$697$	−1.75334	−0.0664124
$698$	−9.93275	−0.375960
$699$	0	0
$700$	3.06422	0.115816
$701$	−13.0716	−0.493706	−0.246853	−	0.969053i	$-0.579396\pi$
$701$	−13.0716	−0.493706	−0.246853	+	0.969053i	$0.579396\pi$
$702$	0	0
$703$	41.2083	1.55420
$704$	−26.6963	−1.00616
$705$	0	0
$706$	15.7531	0.592876
$707$	−53.5927	−2.01556
$708$	0	0
$709$	−4.61239	−0.173222	−0.0866110	−	0.996242i	$-0.527604\pi$
$709$	−4.61239	−0.173222	−0.0866110	+	0.996242i	$0.527604\pi$
$710$	15.5997	0.585445
$711$	0	0
$712$	−29.8011	−1.11684
$713$	49.9904	1.87216
$714$	0	0
$715$	−3.05473	−0.114240
$716$	6.05192	0.226171
$717$	0	0
$718$	−8.89885	−0.332102
$719$	28.4059	1.05936	0.529681	−	0.848197i	$-0.322312\pi$
$719$	28.4059	1.05936	0.529681	+	0.848197i	$0.322312\pi$
$720$	0	0
$721$	−87.4493	−3.25678
$722$	5.12293	0.190656
$723$	0	0
$724$	−9.82467	−0.365131
$725$	−9.62407	−0.357429
$726$	0	0
$727$	−7.94370	−0.294616	−0.147308	−	0.989091i	$-0.547061\pi$
$727$	−7.94370	−0.294616	−0.147308	+	0.989091i	$0.547061\pi$
$728$	−15.6481	−0.579958
$729$	0	0
$730$	6.73714	0.249353
$731$	3.80567	0.140758
$732$	0	0
$733$	−30.3715	−1.12180	−0.560898	−	0.827885i	$-0.689544\pi$
$733$	−30.3715	−1.12180	−0.560898	+	0.827885i	$0.689544\pi$
$734$	−7.18550	−0.265222
$735$	0	0
$736$	20.0844	0.740322
$737$	27.1987	1.00188
$738$	0	0
$739$	25.1254	0.924253	0.462126	−	0.886814i	$-0.347087\pi$
$739$	25.1254	0.924253	0.462126	+	0.886814i	$0.347087\pi$
$740$	−6.48253	−0.238302
$741$	0	0
$742$	21.7225	0.797458
$743$	6.06286	0.222425	0.111212	−	0.993797i	$-0.464527\pi$
$743$	6.06286	0.222425	0.111212	+	0.993797i	$0.464527\pi$
$744$	0	0
$745$	−1.23641	−0.0452986
$746$	−4.13854	−0.151523
$747$	0	0
$748$	2.25447	0.0824314
$749$	10.7149	0.391516
$750$	0	0
$751$	−16.9422	−0.618230	−0.309115	−	0.951025i	$-0.600033\pi$
$751$	−16.9422	−0.618230	−0.309115	+	0.951025i	$0.600033\pi$
$752$	−29.1576	−1.06327
$753$	0	0
$754$	11.3774	0.414339
$755$	1.92780	0.0701596
$756$	0	0
$757$	−12.2308	−0.444535	−0.222268	−	0.974986i	$-0.571346\pi$
$757$	−12.2308	−0.444535	−0.222268	+	0.974986i	$0.571346\pi$
$758$	11.4894	0.417313
$759$	0	0
$760$	11.7823	0.427389
$761$	−5.05467	−0.183232	−0.0916159	−	0.995794i	$-0.529203\pi$
$761$	−5.05467	−0.183232	−0.0916159	+	0.995794i	$0.529203\pi$
$762$	0	0
$763$	−67.0345	−2.42681
$764$	−2.32116	−0.0839767
$765$	0	0
$766$	−18.0691	−0.652863
$767$	7.99928	0.288837
$768$	0	0
$769$	40.8717	1.47387	0.736936	−	0.675962i	$-0.236272\pi$
$769$	40.8717	1.47387	0.736936	+	0.675962i	$0.236272\pi$
$770$	−18.3676	−0.661921
$771$	0	0
$772$	−2.50781	−0.0902582
$773$	−42.0690	−1.51312	−0.756558	−	0.653927i	$-0.773120\pi$
$773$	−42.0690	−1.51312	−0.756558	+	0.653927i	$0.773120\pi$
$774$	0	0
$775$	−8.15873	−0.293070
$776$	14.4897	0.520150
$777$	0	0
$778$	−44.5158	−1.59597
$779$	−5.48127	−0.196387
$780$	0	0
$781$	40.3092	1.44238
$782$	8.87351	0.317316
$783$	0	0
$784$	−45.8940	−1.63907
$785$	10.6385	0.379705
$786$	0	0
$787$	−39.3176	−1.40152	−0.700761	−	0.713396i	$-0.747156\pi$
$787$	−39.3176	−1.40152	−0.700761	+	0.713396i	$0.747156\pi$
$788$	6.03146	0.214862
$789$	0	0
$790$	−14.2127	−0.505667
$791$	−85.3514	−3.03475
$792$	0	0
$793$	1.70123	0.0604123
$794$	3.45157	0.122492
$795$	0	0
$796$	−13.0375	−0.462103
$797$	−28.2363	−1.00018	−0.500090	−	0.865974i	$-0.666700\pi$
$797$	−28.2363	−1.00018	−0.500090	+	0.865974i	$0.666700\pi$
$798$	0	0
$799$	14.6862	0.519562
$800$	−3.27790	−0.115891
$801$	0	0
$802$	18.4782	0.652487
$803$	17.4086	0.614338
$804$	0	0
$805$	31.1645	1.09840
$806$	9.64508	0.339734
$807$	0	0
$808$	32.4172	1.14043
$809$	−25.2242	−0.886834	−0.443417	−	0.896315i	$-0.646234\pi$
$809$	−25.2242	−0.886834	−0.443417	+	0.896315i	$0.646234\pi$
$810$	0	0
$811$	−31.0755	−1.09121	−0.545604	−	0.838043i	$-0.683700\pi$
$811$	−31.0755	−1.09121	−0.545604	+	0.838043i	$0.683700\pi$
$812$	−29.4902	−1.03490
$813$	0	0
$814$	38.8577	1.36196
$815$	−6.70970	−0.235031
$816$	0	0
$817$	11.8972	0.416232
$818$	14.5706	0.509449
$819$	0	0
$820$	0.862265	0.0301116
$821$	−4.48786	−0.156627	−0.0783136	−	0.996929i	$-0.524954\pi$
$821$	−4.48786	−0.156627	−0.0783136	+	0.996929i	$0.524954\pi$
$822$	0	0
$823$	47.1277	1.64277	0.821383	−	0.570376i	$-0.193203\pi$
$823$	47.1277	1.64277	0.821383	+	0.570376i	$0.193203\pi$
$824$	52.8964	1.84273
$825$	0	0
$826$	48.0984	1.67356
$827$	−30.6145	−1.06457	−0.532286	−	0.846565i	$-0.678667\pi$
$827$	−30.6145	−1.06457	−0.532286	+	0.846565i	$0.678667\pi$
$828$	0	0
$829$	8.22031	0.285503	0.142752	−	0.989759i	$-0.454405\pi$
$829$	8.22031	0.285503	0.142752	+	0.989759i	$0.454405\pi$
$830$	1.62575	0.0564306
$831$	0	0
$832$	8.73935	0.302983
$833$	23.1161	0.800925
$834$	0	0
$835$	6.94448	0.240324
$836$	7.04789	0.243756
$837$	0	0
$838$	−25.2691	−0.872905
$839$	4.62444	0.159653	0.0798267	−	0.996809i	$-0.474563\pi$
$839$	4.62444	0.159653	0.0798267	+	0.996809i	$0.474563\pi$
$840$	0	0
$841$	63.6227	2.19389
$842$	14.4716	0.498723
$843$	0	0
$844$	2.85694	0.0983399
$845$	1.00000	0.0344010
$846$	0	0
$847$	8.48715	0.291622
$848$	−8.78658	−0.301732
$849$	0	0
$850$	−1.44821	−0.0496732
$851$	−65.9303	−2.26006
$852$	0	0
$853$	31.5258	1.07942	0.539711	−	0.841850i	$-0.318533\pi$
$853$	31.5258	1.07942	0.539711	+	0.841850i	$0.318533\pi$
$854$	10.2292	0.350036
$855$	0	0
$856$	−6.48127	−0.221525
$857$	−37.3535	−1.27597	−0.637986	−	0.770048i	$-0.720232\pi$
$857$	−37.3535	−1.27597	−0.637986	+	0.770048i	$0.720232\pi$
$858$	0	0
$859$	−17.8726	−0.609804	−0.304902	−	0.952384i	$-0.598624\pi$
$859$	−17.8726	−0.609804	−0.304902	+	0.952384i	$0.598624\pi$
$860$	−1.87157	−0.0638199
$861$	0	0
$862$	−25.3758	−0.864302
$863$	−18.2336	−0.620678	−0.310339	−	0.950626i	$-0.600442\pi$
$863$	−18.2336	−0.620678	−0.310339	+	0.950626i	$0.600442\pi$
$864$	0	0
$865$	11.1572	0.379356
$866$	−2.32869	−0.0791320
$867$	0	0
$868$	−25.0001	−0.848559
$869$	−36.7254	−1.24583
$870$	0	0
$871$	−8.90383	−0.301695
$872$	40.5479	1.37313
$873$	0	0
$874$	27.7403	0.938329
$875$	−5.08623	−0.171946
$876$	0	0
$877$	4.85882	0.164071	0.0820354	−	0.996629i	$-0.473858\pi$
$877$	4.85882	0.164071	0.0820354	+	0.996629i	$0.473858\pi$
$878$	−2.42304	−0.0817738
$879$	0	0
$880$	7.42953	0.250449
$881$	8.06212	0.271620	0.135810	−	0.990735i	$-0.456636\pi$
$881$	8.06212	0.271620	0.135810	+	0.990735i	$0.456636\pi$
$882$	0	0
$883$	−12.1686	−0.409506	−0.204753	−	0.978814i	$-0.565639\pi$
$883$	−12.1686	−0.409506	−0.204753	+	0.978814i	$0.565639\pi$
$884$	−0.738026	−0.0248225
$885$	0	0
$886$	38.3594	1.28871
$887$	56.8286	1.90812	0.954058	−	0.299621i	$-0.0968602\pi$
$887$	56.8286	1.90812	0.954058	+	0.299621i	$0.0968602\pi$
$888$	0	0
$889$	−102.033	−3.42208
$890$	11.4512	0.383844
$891$	0	0
$892$	−11.8278	−0.396023
$893$	45.9120	1.53639
$894$	0	0
$895$	−10.0455	−0.335783
$896$	19.2040	0.641560
$897$	0	0
$898$	24.1375	0.805480
$899$	78.5202	2.61880
$900$	0	0
$901$	4.42567	0.147440
$902$	−5.16860	−0.172096
$903$	0	0
$904$	51.6274	1.71710
$905$	16.3078	0.542089
$906$	0	0
$907$	12.4132	0.412172	0.206086	−	0.978534i	$-0.433927\pi$
$907$	12.4132	0.412172	0.206086	+	0.978534i	$0.433927\pi$
$908$	−0.668231	−0.0221760
$909$	0	0
$910$	6.01284	0.199324
$911$	42.3145	1.40194	0.700970	−	0.713191i	$-0.252751\pi$
$911$	42.3145	1.40194	0.700970	+	0.713191i	$0.252751\pi$
$912$	0	0
$913$	4.20090	0.139030
$914$	4.93601	0.163269
$915$	0	0
$916$	−14.4085	−0.476071
$917$	−29.7210	−0.981473
$918$	0	0
$919$	51.4568	1.69740	0.848701	−	0.528873i	$-0.177385\pi$
$919$	51.4568	1.69740	0.848701	+	0.528873i	$0.177385\pi$
$920$	−18.8508	−0.621493
$921$	0	0
$922$	27.8966	0.918726
$923$	−13.1957	−0.434341
$924$	0	0
$925$	10.7602	0.353794
$926$	−1.12997	−0.0371330
$927$	0	0
$928$	31.5467	1.03557
$929$	36.4085	1.19452	0.597262	−	0.802046i	$-0.296255\pi$
$929$	36.4085	1.19452	0.597262	+	0.802046i	$0.296255\pi$
$930$	0	0
$931$	72.2653	2.36840
$932$	13.4224	0.439665
$933$	0	0
$934$	−29.8268	−0.975963
$935$	−3.74214	−0.122381
$936$	0	0
$937$	8.30585	0.271340	0.135670	−	0.990754i	$-0.456681\pi$
$937$	8.30585	0.271340	0.135670	+	0.990754i	$0.456681\pi$
$938$	−53.5372	−1.74805
$939$	0	0
$940$	−7.22246	−0.235571
$941$	−11.2038	−0.365234	−0.182617	−	0.983184i	$-0.558457\pi$
$941$	−11.2038	−0.365234	−0.182617	+	0.983184i	$0.558457\pi$
$942$	0	0
$943$	8.76963	0.285579
$944$	−19.4554	−0.633220
$945$	0	0
$946$	11.2186	0.364747
$947$	45.4189	1.47592	0.737958	−	0.674847i	$-0.235790\pi$
$947$	45.4189	1.47592	0.737958	+	0.674847i	$0.235790\pi$
$948$	0	0
$949$	−5.69892	−0.184995
$950$	−4.52738	−0.146888
$951$	0	0
$952$	−19.1695	−0.621287
$953$	46.9578	1.52111	0.760557	−	0.649272i	$-0.224926\pi$
$953$	46.9578	1.52111	0.760557	+	0.649272i	$0.224926\pi$
$954$	0	0
$955$	3.85285	0.124675
$956$	11.7209	0.379081
$957$	0	0
$958$	5.42797	0.175370
$959$	−30.1095	−0.972287
$960$	0	0
$961$	35.5649	1.14726
$962$	−12.7205	−0.410126
$963$	0	0
$964$	−5.87923	−0.189357
$965$	4.16267	0.134001
$966$	0	0
$967$	30.0414	0.966065	0.483032	−	0.875602i	$-0.339535\pi$
$967$	30.0414	0.966065	0.483032	+	0.875602i	$0.339535\pi$
$968$	−5.13372	−0.165004
$969$	0	0
$970$	−5.56771	−0.178768
$971$	28.5868	0.917393	0.458696	−	0.888593i	$-0.348317\pi$
$971$	28.5868	0.917393	0.458696	+	0.888593i	$0.348317\pi$
$972$	0	0
$973$	84.2166	2.69986
$974$	33.2925	1.06676
$975$	0	0
$976$	−4.13762	−0.132442
$977$	10.0672	0.322077	0.161038	−	0.986948i	$-0.448516\pi$
$977$	10.0672	0.322077	0.161038	+	0.986948i	$0.448516\pi$
$978$	0	0
$979$	29.5895	0.945686
$980$	−11.3681	−0.363142
$981$	0	0
$982$	44.1130	1.40770
$983$	55.0757	1.75664	0.878321	−	0.478072i	$-0.158664\pi$
$983$	55.0757	1.75664	0.878321	+	0.478072i	$0.158664\pi$
$984$	0	0
$985$	−10.0115	−0.318993
$986$	13.9377	0.443866
$987$	0	0
$988$	−2.30721	−0.0734021
$989$	−19.0347	−0.605269
$990$	0	0
$991$	−0.195325	−0.00620470	−0.00310235	−	0.999995i	$-0.500988\pi$
$991$	−0.195325	−0.00620470	−0.00310235	+	0.999995i	$0.500988\pi$
$992$	26.7435	0.849108
$993$	0	0
$994$	−79.3435	−2.51662
$995$	21.6408	0.686058
$996$	0	0
$997$	−36.6474	−1.16063	−0.580317	−	0.814390i	$-0.697071\pi$
$997$	−36.6474	−1.16063	−0.580317	+	0.814390i	$0.697071\pi$
$998$	−1.94932	−0.0617046
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1755.2.a.v.1.3	yes	7
3.2	odd	2		1755.2.a.u.1.5	✓	7
5.4	even	2		8775.2.a.bw.1.5		7
15.14	odd	2		8775.2.a.bx.1.3		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1755.2.a.u.1.5	✓	7	3.2	odd	2
1755.2.a.v.1.3	yes	7	1.1	even	1	trivial
8775.2.a.bw.1.5		7	5.4	even	2
8775.2.a.bx.1.3		7	15.14	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:	\( N \)	\(=\)	\( 1755 = 3^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1755.a (trivial)

\(f(q)\)	\(=\)	\(q-1.18218 q^{2} -0.602453 q^{4} +1.00000 q^{5} -5.08623 q^{7} +3.07657 q^{8} +O(q^{10})\)	\(q-1.18218 q^{2} -0.602453 q^{4} +1.00000 q^{5} -5.08623 q^{7} +3.07657 q^{8} -1.18218 q^{10} -3.05473 q^{11} +1.00000 q^{13} +6.01284 q^{14} -2.43214 q^{16} +1.22503 q^{17} +3.82969 q^{19} -0.602453 q^{20} +3.61123 q^{22} -6.12723 q^{23} +1.00000 q^{25} -1.18218 q^{26} +3.06422 q^{28} -9.62407 q^{29} -8.15873 q^{31} -3.27790 q^{32} -1.44821 q^{34} -5.08623 q^{35} +10.7602 q^{37} -4.52738 q^{38} +3.07657 q^{40} -1.43126 q^{41} +3.10658 q^{43} +1.84033 q^{44} +7.24348 q^{46} +11.9884 q^{47} +18.8698 q^{49} -1.18218 q^{50} -0.602453 q^{52} +3.61269 q^{53} -3.05473 q^{55} -15.6481 q^{56} +11.3774 q^{58} +7.99928 q^{59} +1.70123 q^{61} +9.64508 q^{62} +8.73935 q^{64} +1.00000 q^{65} -8.90383 q^{67} -0.738026 q^{68} +6.01284 q^{70} -13.1957 q^{71} -5.69892 q^{73} -12.7205 q^{74} -2.30721 q^{76} +15.5370 q^{77} +12.0225 q^{79} -2.43214 q^{80} +1.69200 q^{82} -1.37522 q^{83} +1.22503 q^{85} -3.67253 q^{86} -9.39806 q^{88} -9.68648 q^{89} -5.08623 q^{91} +3.69137 q^{92} -14.1725 q^{94} +3.82969 q^{95} +4.70970 q^{97} -22.3074 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + q^{2} + 11 q^{4} + 7 q^{5} + 2 q^{7} + 6 q^{8}+O(q^{10}) \) 7 * q + q^2 + 11 * q^4 + 7 * q^5 + 2 * q^7 + 6 * q^8	\( 7 q + q^{2} + 11 q^{4} + 7 q^{5} + 2 q^{7} + 6 q^{8} + q^{10} + q^{11} + 7 q^{13} - 12 q^{14} + 23 q^{16} + 11 q^{17} + 2 q^{19} + 11 q^{20} + 16 q^{22} - q^{23} + 7 q^{25} + q^{26} + 10 q^{28} - 4 q^{29} + 13 q^{32} + q^{34} + 2 q^{35} + 23 q^{37} - 15 q^{38} + 6 q^{40} + 2 q^{41} + 8 q^{43} + 10 q^{44} + 37 q^{46} + 2 q^{47} + 43 q^{49} + q^{50} + 11 q^{52} + 10 q^{53} + q^{55} - 68 q^{56} - 26 q^{58} - 13 q^{59} + 21 q^{61} + 9 q^{62} + 46 q^{64} + 7 q^{65} + 21 q^{67} + 53 q^{68} - 12 q^{70} - 10 q^{71} + 13 q^{73} - 68 q^{74} - 41 q^{76} + 6 q^{77} + 8 q^{79} + 23 q^{80} - 26 q^{82} - 4 q^{83} + 11 q^{85} - 12 q^{86} + 44 q^{88} - 27 q^{89} + 2 q^{91} - 9 q^{92} - 24 q^{94} + 2 q^{95} - 15 q^{97} + 7 q^{98}+O(q^{100}) \) 7 * q + q^2 + 11 * q^4 + 7 * q^5 + 2 * q^7 + 6 * q^8 + q^10 + q^11 + 7 * q^13 - 12 * q^14 + 23 * q^16 + 11 * q^17 + 2 * q^19 + 11 * q^20 + 16 * q^22 - q^23 + 7 * q^25 + q^26 + 10 * q^28 - 4 * q^29 + 13 * q^32 + q^34 + 2 * q^35 + 23 * q^37 - 15 * q^38 + 6 * q^40 + 2 * q^41 + 8 * q^43 + 10 * q^44 + 37 * q^46 + 2 * q^47 + 43 * q^49 + q^50 + 11 * q^52 + 10 * q^53 + q^55 - 68 * q^56 - 26 * q^58 - 13 * q^59 + 21 * q^61 + 9 * q^62 + 46 * q^64 + 7 * q^65 + 21 * q^67 + 53 * q^68 - 12 * q^70 - 10 * q^71 + 13 * q^73 - 68 * q^74 - 41 * q^76 + 6 * q^77 + 8 * q^79 + 23 * q^80 - 26 * q^82 - 4 * q^83 + 11 * q^85 - 12 * q^86 + 44 * q^88 - 27 * q^89 + 2 * q^91 - 9 * q^92 - 24 * q^94 + 2 * q^95 - 15 * q^97 + 7 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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