LMFDB - Embedded newform 1725.2.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1725,2,Mod(1,1725)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1725, 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("1725.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1725 = 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1725.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:	$13.7741943487$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 345)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	1725.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +1.00000 q^{3} +1.41421 q^{6} -1.82843 q^{7} -2.82843 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.41421 q^{2} +1.00000 q^{3} +1.41421 q^{6} -1.82843 q^{7} -2.82843 q^{8} +1.00000 q^{9} -2.58579 q^{11} -3.41421 q^{13} -2.58579 q^{14} -4.00000 q^{16} +6.07107 q^{17} +1.41421 q^{18} -6.24264 q^{19} -1.82843 q^{21} -3.65685 q^{22} +1.00000 q^{23} -2.82843 q^{24} -4.82843 q^{26} +1.00000 q^{27} -3.58579 q^{29} -4.17157 q^{31} -2.58579 q^{33} +8.58579 q^{34} -3.00000 q^{37} -8.82843 q^{38} -3.41421 q^{39} -10.4142 q^{41} -2.58579 q^{42} -6.00000 q^{43} +1.41421 q^{46} +8.58579 q^{47} -4.00000 q^{48} -3.65685 q^{49} +6.07107 q^{51} -7.24264 q^{53} +1.41421 q^{54} +5.17157 q^{56} -6.24264 q^{57} -5.07107 q^{58} -6.89949 q^{59} +5.41421 q^{61} -5.89949 q^{62} -1.82843 q^{63} +8.00000 q^{64} -3.65685 q^{66} +11.4853 q^{67} +1.00000 q^{69} +8.89949 q^{71} -2.82843 q^{72} +10.2426 q^{73} -4.24264 q^{74} +4.72792 q^{77} -4.82843 q^{78} -15.3137 q^{79} +1.00000 q^{81} -14.7279 q^{82} -14.0711 q^{83} -8.48528 q^{86} -3.58579 q^{87} +7.31371 q^{88} +10.1421 q^{89} +6.24264 q^{91} -4.17157 q^{93} +12.1421 q^{94} +1.17157 q^{97} -5.17157 q^{98} -2.58579 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9} - 8 q^{11} - 4 q^{13} - 8 q^{14} - 8 q^{16} - 2 q^{17} - 4 q^{19} + 2 q^{21} + 4 q^{22} + 2 q^{23} - 4 q^{26} + 2 q^{27} - 10 q^{29} - 14 q^{31} - 8 q^{33} + 20 q^{34} - 6 q^{37} - 12 q^{38} - 4 q^{39} - 18 q^{41} - 8 q^{42} - 12 q^{43} + 20 q^{47} - 8 q^{48} + 4 q^{49} - 2 q^{51} - 6 q^{53} + 16 q^{56} - 4 q^{57} + 4 q^{58} + 6 q^{59} + 8 q^{61} + 8 q^{62} + 2 q^{63} + 16 q^{64} + 4 q^{66} + 6 q^{67} + 2 q^{69} - 2 q^{71} + 12 q^{73} - 16 q^{77} - 4 q^{78} - 8 q^{79} + 2 q^{81} - 4 q^{82} - 14 q^{83} - 10 q^{87} - 8 q^{88} - 8 q^{89} + 4 q^{91} - 14 q^{93} - 4 q^{94} + 8 q^{97} - 16 q^{98} - 8 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^9 - 8 * q^11 - 4 * q^13 - 8 * q^14 - 8 * q^16 - 2 * q^17 - 4 * q^19 + 2 * q^21 + 4 * q^22 + 2 * q^23 - 4 * q^26 + 2 * q^27 - 10 * q^29 - 14 * q^31 - 8 * q^33 + 20 * q^34 - 6 * q^37 - 12 * q^38 - 4 * q^39 - 18 * q^41 - 8 * q^42 - 12 * q^43 + 20 * q^47 - 8 * q^48 + 4 * q^49 - 2 * q^51 - 6 * q^53 + 16 * q^56 - 4 * q^57 + 4 * q^58 + 6 * q^59 + 8 * q^61 + 8 * q^62 + 2 * q^63 + 16 * q^64 + 4 * q^66 + 6 * q^67 + 2 * q^69 - 2 * q^71 + 12 * q^73 - 16 * q^77 - 4 * q^78 - 8 * q^79 + 2 * q^81 - 4 * q^82 - 14 * q^83 - 10 * q^87 - 8 * q^88 - 8 * q^89 + 4 * q^91 - 14 * q^93 - 4 * q^94 + 8 * q^97 - 16 * q^98 - 8 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.41421	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$2$	1.41421	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	0	0
$5$	0	0
$6$	1.41421	0.577350
$7$	−1.82843	−0.691080	−0.345540	−	0.938404i	$-0.612304\pi$
$7$	−1.82843	−0.691080	−0.345540	+	0.938404i	$0.612304\pi$
$8$	−2.82843	−1.00000
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.58579	−0.779644	−0.389822	−	0.920890i	$-0.627463\pi$
$11$	−2.58579	−0.779644	−0.389822	+	0.920890i	$0.627463\pi$
$12$	0	0
$13$	−3.41421	−0.946932	−0.473466	−	0.880812i	$-0.656997\pi$
$13$	−3.41421	−0.946932	−0.473466	+	0.880812i	$0.656997\pi$
$14$	−2.58579	−0.691080
$15$	0	0
$16$	−4.00000	−1.00000
$17$	6.07107	1.47245	0.736225	−	0.676737i	$-0.236606\pi$
$17$	6.07107	1.47245	0.736225	+	0.676737i	$0.236606\pi$
$18$	1.41421	0.333333
$19$	−6.24264	−1.43216	−0.716080	−	0.698018i	$-0.754065\pi$
$19$	−6.24264	−1.43216	−0.716080	+	0.698018i	$0.754065\pi$
$20$	0	0
$21$	−1.82843	−0.398996
$22$	−3.65685	−0.779644
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−2.82843	−0.577350
$25$	0	0
$26$	−4.82843	−0.946932
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.58579	−0.665864	−0.332932	−	0.942951i	$-0.608038\pi$
$29$	−3.58579	−0.665864	−0.332932	+	0.942951i	$0.608038\pi$
$30$	0	0
$31$	−4.17157	−0.749237	−0.374618	−	0.927179i	$-0.622226\pi$
$31$	−4.17157	−0.749237	−0.374618	+	0.927179i	$0.622226\pi$
$32$	0	0
$33$	−2.58579	−0.450128
$34$	8.58579	1.47245
$35$	0	0
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	−8.82843	−1.43216
$39$	−3.41421	−0.546712
$40$	0	0
$41$	−10.4142	−1.62643	−0.813213	−	0.581966i	$-0.802284\pi$
$41$	−10.4142	−1.62643	−0.813213	+	0.581966i	$0.802284\pi$
$42$	−2.58579	−0.398996
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	0	0
$45$	0	0
$46$	1.41421	0.208514
$47$	8.58579	1.25237	0.626183	−	0.779676i	$-0.284616\pi$
$47$	8.58579	1.25237	0.626183	+	0.779676i	$0.284616\pi$
$48$	−4.00000	−0.577350
$49$	−3.65685	−0.522408
$50$	0	0
$51$	6.07107	0.850120
$52$	0	0
$53$	−7.24264	−0.994853	−0.497427	−	0.867506i	$-0.665722\pi$
$53$	−7.24264	−0.994853	−0.497427	+	0.867506i	$0.665722\pi$
$54$	1.41421	0.192450
$55$	0	0
$56$	5.17157	0.691080
$57$	−6.24264	−0.826858
$58$	−5.07107	−0.665864
$59$	−6.89949	−0.898238	−0.449119	−	0.893472i	$-0.648262\pi$
$59$	−6.89949	−0.898238	−0.449119	+	0.893472i	$0.648262\pi$
$60$	0	0
$61$	5.41421	0.693219	0.346610	−	0.938010i	$-0.387333\pi$
$61$	5.41421	0.693219	0.346610	+	0.938010i	$0.387333\pi$
$62$	−5.89949	−0.749237
$63$	−1.82843	−0.230360
$64$	8.00000	1.00000
$65$	0	0
$66$	−3.65685	−0.450128
$67$	11.4853	1.40315	0.701575	−	0.712595i	$-0.252480\pi$
$67$	11.4853	1.40315	0.701575	+	0.712595i	$0.252480\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	8.89949	1.05618	0.528088	−	0.849190i	$-0.322909\pi$
$71$	8.89949	1.05618	0.528088	+	0.849190i	$0.322909\pi$
$72$	−2.82843	−0.333333
$73$	10.2426	1.19881	0.599405	−	0.800446i	$-0.295404\pi$
$73$	10.2426	1.19881	0.599405	+	0.800446i	$0.295404\pi$
$74$	−4.24264	−0.493197
$75$	0	0
$76$	0	0
$77$	4.72792	0.538797
$78$	−4.82843	−0.546712
$79$	−15.3137	−1.72293	−0.861463	−	0.507820i	$-0.830452\pi$
$79$	−15.3137	−1.72293	−0.861463	+	0.507820i	$0.830452\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−14.7279	−1.62643
$83$	−14.0711	−1.54450	−0.772250	−	0.635319i	$-0.780869\pi$
$83$	−14.0711	−1.54450	−0.772250	+	0.635319i	$0.780869\pi$
$84$	0	0
$85$	0	0
$86$	−8.48528	−0.914991
$87$	−3.58579	−0.384437
$88$	7.31371	0.779644
$89$	10.1421	1.07506	0.537532	−	0.843243i	$-0.319357\pi$
$89$	10.1421	1.07506	0.537532	+	0.843243i	$0.319357\pi$
$90$	0	0
$91$	6.24264	0.654407
$92$	0	0
$93$	−4.17157	−0.432572
$94$	12.1421	1.25237
$95$	0	0
$96$	0	0
$97$	1.17157	0.118955	0.0594776	−	0.998230i	$-0.481057\pi$
$97$	1.17157	0.118955	0.0594776	+	0.998230i	$0.481057\pi$
$98$	−5.17157	−0.522408
$99$	−2.58579	−0.259881
$100$	0	0
$101$	12.5563	1.24940	0.624702	−	0.780863i	$-0.285221\pi$
$101$	12.5563	1.24940	0.624702	+	0.780863i	$0.285221\pi$
$102$	8.58579	0.850120
$103$	−0.828427	−0.0816274	−0.0408137	−	0.999167i	$-0.512995\pi$
$103$	−0.828427	−0.0816274	−0.0408137	+	0.999167i	$0.512995\pi$
$104$	9.65685	0.946932
$105$	0	0
$106$	−10.2426	−0.994853
$107$	4.41421	0.426738	0.213369	−	0.976972i	$-0.431556\pi$
$107$	4.41421	0.426738	0.213369	+	0.976972i	$0.431556\pi$
$108$	0	0
$109$	14.2426	1.36420	0.682099	−	0.731260i	$-0.261067\pi$
$109$	14.2426	1.36420	0.682099	+	0.731260i	$0.261067\pi$
$110$	0	0
$111$	−3.00000	−0.284747
$112$	7.31371	0.691080
$113$	10.4142	0.979687	0.489843	−	0.871810i	$-0.337054\pi$
$113$	10.4142	0.979687	0.489843	+	0.871810i	$0.337054\pi$
$114$	−8.82843	−0.826858
$115$	0	0
$116$	0	0
$117$	−3.41421	−0.315644
$118$	−9.75736	−0.898238
$119$	−11.1005	−1.01758
$120$	0	0
$121$	−4.31371	−0.392155
$122$	7.65685	0.693219
$123$	−10.4142	−0.939018
$124$	0	0
$125$	0	0
$126$	−2.58579	−0.230360
$127$	0.242641	0.0215309	0.0107654	−	0.999942i	$-0.496573\pi$
$127$	0.242641	0.0215309	0.0107654	+	0.999942i	$0.496573\pi$
$128$	11.3137	1.00000
$129$	−6.00000	−0.528271
$130$	0	0
$131$	−13.6569	−1.19320	−0.596602	−	0.802537i	$-0.703483\pi$
$131$	−13.6569	−1.19320	−0.596602	+	0.802537i	$0.703483\pi$
$132$	0	0
$133$	11.4142	0.989738
$134$	16.2426	1.40315
$135$	0	0
$136$	−17.1716	−1.47245
$137$	−2.82843	−0.241649	−0.120824	−	0.992674i	$-0.538554\pi$
$137$	−2.82843	−0.241649	−0.120824	+	0.992674i	$0.538554\pi$
$138$	1.41421	0.120386
$139$	−11.0000	−0.933008	−0.466504	−	0.884519i	$-0.654487\pi$
$139$	−11.0000	−0.933008	−0.466504	+	0.884519i	$0.654487\pi$
$140$	0	0
$141$	8.58579	0.723054
$142$	12.5858	1.05618
$143$	8.82843	0.738270
$144$	−4.00000	−0.333333
$145$	0	0
$146$	14.4853	1.19881
$147$	−3.65685	−0.301612
$148$	0	0
$149$	9.07107	0.743131	0.371565	−	0.928407i	$-0.378821\pi$
$149$	9.07107	0.743131	0.371565	+	0.928407i	$0.378821\pi$
$150$	0	0
$151$	−9.31371	−0.757939	−0.378969	−	0.925409i	$-0.623721\pi$
$151$	−9.31371	−0.757939	−0.378969	+	0.925409i	$0.623721\pi$
$152$	17.6569	1.43216
$153$	6.07107	0.490817
$154$	6.68629	0.538797
$155$	0	0
$156$	0	0
$157$	1.48528	0.118538	0.0592692	−	0.998242i	$-0.481123\pi$
$157$	1.48528	0.118538	0.0592692	+	0.998242i	$0.481123\pi$
$158$	−21.6569	−1.72293
$159$	−7.24264	−0.574379
$160$	0	0
$161$	−1.82843	−0.144100
$162$	1.41421	0.111111
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	0	0
$165$	0	0
$166$	−19.8995	−1.54450
$167$	6.72792	0.520622	0.260311	−	0.965525i	$-0.416175\pi$
$167$	6.72792	0.520622	0.260311	+	0.965525i	$0.416175\pi$
$168$	5.17157	0.398996
$169$	−1.34315	−0.103319
$170$	0	0
$171$	−6.24264	−0.477387
$172$	0	0
$173$	−13.6569	−1.03831	−0.519156	−	0.854680i	$-0.673754\pi$
$173$	−13.6569	−1.03831	−0.519156	+	0.854680i	$0.673754\pi$
$174$	−5.07107	−0.384437
$175$	0	0
$176$	10.3431	0.779644
$177$	−6.89949	−0.518598
$178$	14.3431	1.07506
$179$	−7.65685	−0.572300	−0.286150	−	0.958185i	$-0.592376\pi$
$179$	−7.65685	−0.572300	−0.286150	+	0.958185i	$0.592376\pi$
$180$	0	0
$181$	−6.82843	−0.507553	−0.253776	−	0.967263i	$-0.581673\pi$
$181$	−6.82843	−0.507553	−0.253776	+	0.967263i	$0.581673\pi$
$182$	8.82843	0.654407
$183$	5.41421	0.400230
$184$	−2.82843	−0.208514
$185$	0	0
$186$	−5.89949	−0.432572
$187$	−15.6985	−1.14799
$188$	0	0
$189$	−1.82843	−0.132999
$190$	0	0
$191$	10.2426	0.741131	0.370566	−	0.928806i	$-0.379164\pi$
$191$	10.2426	0.741131	0.370566	+	0.928806i	$0.379164\pi$
$192$	8.00000	0.577350
$193$	8.48528	0.610784	0.305392	−	0.952227i	$-0.401213\pi$
$193$	8.48528	0.610784	0.305392	+	0.952227i	$0.401213\pi$
$194$	1.65685	0.118955
$195$	0	0
$196$	0	0
$197$	−16.8284	−1.19898	−0.599488	−	0.800384i	$-0.704629\pi$
$197$	−16.8284	−1.19898	−0.599488	+	0.800384i	$0.704629\pi$
$198$	−3.65685	−0.259881
$199$	10.4853	0.743282	0.371641	−	0.928377i	$-0.378795\pi$
$199$	10.4853	0.743282	0.371641	+	0.928377i	$0.378795\pi$
$200$	0	0
$201$	11.4853	0.810109
$202$	17.7574	1.24940
$203$	6.55635	0.460166
$204$	0	0
$205$	0	0
$206$	−1.17157	−0.0816274
$207$	1.00000	0.0695048
$208$	13.6569	0.946932
$209$	16.1421	1.11657
$210$	0	0
$211$	0.656854	0.0452197	0.0226099	−	0.999744i	$-0.492802\pi$
$211$	0.656854	0.0452197	0.0226099	+	0.999744i	$0.492802\pi$
$212$	0	0
$213$	8.89949	0.609783
$214$	6.24264	0.426738
$215$	0	0
$216$	−2.82843	−0.192450
$217$	7.62742	0.517783
$218$	20.1421	1.36420
$219$	10.2426	0.692134
$220$	0	0
$221$	−20.7279	−1.39431
$222$	−4.24264	−0.284747
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	0	0
$225$	0	0
$226$	14.7279	0.979687
$227$	−17.6569	−1.17193	−0.585963	−	0.810338i	$-0.699284\pi$
$227$	−17.6569	−1.17193	−0.585963	+	0.810338i	$0.699284\pi$
$228$	0	0
$229$	23.7990	1.57268	0.786341	−	0.617793i	$-0.211973\pi$
$229$	23.7990	1.57268	0.786341	+	0.617793i	$0.211973\pi$
$230$	0	0
$231$	4.72792	0.311074
$232$	10.1421	0.665864
$233$	−9.65685	−0.632642	−0.316321	−	0.948652i	$-0.602448\pi$
$233$	−9.65685	−0.632642	−0.316321	+	0.948652i	$0.602448\pi$
$234$	−4.82843	−0.315644
$235$	0	0
$236$	0	0
$237$	−15.3137	−0.994732
$238$	−15.6985	−1.01758
$239$	−1.58579	−0.102576	−0.0512880	−	0.998684i	$-0.516333\pi$
$239$	−1.58579	−0.102576	−0.0512880	+	0.998684i	$0.516333\pi$
$240$	0	0
$241$	23.4142	1.50824	0.754121	−	0.656735i	$-0.228063\pi$
$241$	23.4142	1.50824	0.754121	+	0.656735i	$0.228063\pi$
$242$	−6.10051	−0.392155
$243$	1.00000	0.0641500
$244$	0	0
$245$	0	0
$246$	−14.7279	−0.939018
$247$	21.3137	1.35616
$248$	11.7990	0.749237
$249$	−14.0711	−0.891718
$250$	0	0
$251$	−20.6274	−1.30199	−0.650996	−	0.759082i	$-0.725648\pi$
$251$	−20.6274	−1.30199	−0.650996	+	0.759082i	$0.725648\pi$
$252$	0	0
$253$	−2.58579	−0.162567
$254$	0.343146	0.0215309
$255$	0	0
$256$	0	0
$257$	−30.0416	−1.87395	−0.936973	−	0.349403i	$-0.886385\pi$
$257$	−30.0416	−1.87395	−0.936973	+	0.349403i	$0.886385\pi$
$258$	−8.48528	−0.528271
$259$	5.48528	0.340839
$260$	0	0
$261$	−3.58579	−0.221955
$262$	−19.3137	−1.19320
$263$	−3.72792	−0.229874	−0.114937	−	0.993373i	$-0.536667\pi$
$263$	−3.72792	−0.229874	−0.114937	+	0.993373i	$0.536667\pi$
$264$	7.31371	0.450128
$265$	0	0
$266$	16.1421	0.989738
$267$	10.1421	0.620689
$268$	0	0
$269$	−26.5563	−1.61917	−0.809585	−	0.587003i	$-0.800308\pi$
$269$	−26.5563	−1.61917	−0.809585	+	0.587003i	$0.800308\pi$
$270$	0	0
$271$	−6.51472	−0.395741	−0.197870	−	0.980228i	$-0.563403\pi$
$271$	−6.51472	−0.395741	−0.197870	+	0.980228i	$0.563403\pi$
$272$	−24.2843	−1.47245
$273$	6.24264	0.377822
$274$	−4.00000	−0.241649
$275$	0	0
$276$	0	0
$277$	−25.3137	−1.52095	−0.760477	−	0.649365i	$-0.775035\pi$
$277$	−25.3137	−1.52095	−0.760477	+	0.649365i	$0.775035\pi$
$278$	−15.5563	−0.933008
$279$	−4.17157	−0.249746
$280$	0	0
$281$	17.4142	1.03884	0.519422	−	0.854518i	$-0.326147\pi$
$281$	17.4142	1.03884	0.519422	+	0.854518i	$0.326147\pi$
$282$	12.1421	0.723054
$283$	28.6569	1.70347	0.851737	−	0.523970i	$-0.175550\pi$
$283$	28.6569	1.70347	0.851737	+	0.523970i	$0.175550\pi$
$284$	0	0
$285$	0	0
$286$	12.4853	0.738270
$287$	19.0416	1.12399
$288$	0	0
$289$	19.8579	1.16811
$290$	0	0
$291$	1.17157	0.0686788
$292$	0	0
$293$	3.10051	0.181133	0.0905667	−	0.995890i	$-0.471132\pi$
$293$	3.10051	0.181133	0.0905667	+	0.995890i	$0.471132\pi$
$294$	−5.17157	−0.301612
$295$	0	0
$296$	8.48528	0.493197
$297$	−2.58579	−0.150043
$298$	12.8284	0.743131
$299$	−3.41421	−0.197449
$300$	0	0
$301$	10.9706	0.632333
$302$	−13.1716	−0.757939
$303$	12.5563	0.721343
$304$	24.9706	1.43216
$305$	0	0
$306$	8.58579	0.490817
$307$	−27.2132	−1.55314	−0.776570	−	0.630031i	$-0.783042\pi$
$307$	−27.2132	−1.55314	−0.776570	+	0.630031i	$0.783042\pi$
$308$	0	0
$309$	−0.828427	−0.0471276
$310$	0	0
$311$	17.6569	1.00123	0.500614	−	0.865671i	$-0.333108\pi$
$311$	17.6569	1.00123	0.500614	+	0.865671i	$0.333108\pi$
$312$	9.65685	0.546712
$313$	−25.9706	−1.46794	−0.733971	−	0.679180i	$-0.762335\pi$
$313$	−25.9706	−1.46794	−0.733971	+	0.679180i	$0.762335\pi$
$314$	2.10051	0.118538
$315$	0	0
$316$	0	0
$317$	29.5563	1.66005	0.830025	−	0.557726i	$-0.188326\pi$
$317$	29.5563	1.66005	0.830025	+	0.557726i	$0.188326\pi$
$318$	−10.2426	−0.574379
$319$	9.27208	0.519137
$320$	0	0
$321$	4.41421	0.246377
$322$	−2.58579	−0.144100
$323$	−37.8995	−2.10878
$324$	0	0
$325$	0	0
$326$	2.82843	0.156652
$327$	14.2426	0.787620
$328$	29.4558	1.62643
$329$	−15.6985	−0.865485
$330$	0	0
$331$	1.00000	0.0549650	0.0274825	−	0.999622i	$-0.491251\pi$
$331$	1.00000	0.0549650	0.0274825	+	0.999622i	$0.491251\pi$
$332$	0	0
$333$	−3.00000	−0.164399
$334$	9.51472	0.520622
$335$	0	0
$336$	7.31371	0.398996
$337$	−30.9706	−1.68707	−0.843537	−	0.537071i	$-0.819531\pi$
$337$	−30.9706	−1.68707	−0.843537	+	0.537071i	$0.819531\pi$
$338$	−1.89949	−0.103319
$339$	10.4142	0.565622
$340$	0	0
$341$	10.7868	0.584138
$342$	−8.82843	−0.477387
$343$	19.4853	1.05211
$344$	16.9706	0.914991
$345$	0	0
$346$	−19.3137	−1.03831
$347$	−13.1716	−0.707087	−0.353544	−	0.935418i	$-0.615023\pi$
$347$	−13.1716	−0.707087	−0.353544	+	0.935418i	$0.615023\pi$
$348$	0	0
$349$	−32.1127	−1.71895	−0.859477	−	0.511175i	$-0.829210\pi$
$349$	−32.1127	−1.71895	−0.859477	+	0.511175i	$0.829210\pi$
$350$	0	0
$351$	−3.41421	−0.182237
$352$	0	0
$353$	34.5269	1.83768	0.918841	−	0.394628i	$-0.129126\pi$
$353$	34.5269	1.83768	0.918841	+	0.394628i	$0.129126\pi$
$354$	−9.75736	−0.518598
$355$	0	0
$356$	0	0
$357$	−11.1005	−0.587501
$358$	−10.8284	−0.572300
$359$	−10.9289	−0.576807	−0.288403	−	0.957509i	$-0.593124\pi$
$359$	−10.9289	−0.576807	−0.288403	+	0.957509i	$0.593124\pi$
$360$	0	0
$361$	19.9706	1.05108
$362$	−9.65685	−0.507553
$363$	−4.31371	−0.226411
$364$	0	0
$365$	0	0
$366$	7.65685	0.400230
$367$	13.9706	0.729257	0.364629	−	0.931153i	$-0.381196\pi$
$367$	13.9706	0.729257	0.364629	+	0.931153i	$0.381196\pi$
$368$	−4.00000	−0.208514
$369$	−10.4142	−0.542142
$370$	0	0
$371$	13.2426	0.687524
$372$	0	0
$373$	12.9706	0.671590	0.335795	−	0.941935i	$-0.390995\pi$
$373$	12.9706	0.671590	0.335795	+	0.941935i	$0.390995\pi$
$374$	−22.2010	−1.14799
$375$	0	0
$376$	−24.2843	−1.25237
$377$	12.2426	0.630528
$378$	−2.58579	−0.132999
$379$	28.9706	1.48812	0.744059	−	0.668114i	$-0.232898\pi$
$379$	28.9706	1.48812	0.744059	+	0.668114i	$0.232898\pi$
$380$	0	0
$381$	0.242641	0.0124309
$382$	14.4853	0.741131
$383$	−14.4142	−0.736532	−0.368266	−	0.929720i	$-0.620048\pi$
$383$	−14.4142	−0.736532	−0.368266	+	0.929720i	$0.620048\pi$
$384$	11.3137	0.577350
$385$	0	0
$386$	12.0000	0.610784
$387$	−6.00000	−0.304997
$388$	0	0
$389$	14.4853	0.734433	0.367216	−	0.930136i	$-0.380311\pi$
$389$	14.4853	0.734433	0.367216	+	0.930136i	$0.380311\pi$
$390$	0	0
$391$	6.07107	0.307027
$392$	10.3431	0.522408
$393$	−13.6569	−0.688897
$394$	−23.7990	−1.19898
$395$	0	0
$396$	0	0
$397$	15.3137	0.768573	0.384286	−	0.923214i	$-0.374447\pi$
$397$	15.3137	0.768573	0.384286	+	0.923214i	$0.374447\pi$
$398$	14.8284	0.743282
$399$	11.4142	0.571425
$400$	0	0
$401$	−32.1421	−1.60510	−0.802551	−	0.596584i	$-0.796524\pi$
$401$	−32.1421	−1.60510	−0.802551	+	0.596584i	$0.796524\pi$
$402$	16.2426	0.810109
$403$	14.2426	0.709476
$404$	0	0
$405$	0	0
$406$	9.27208	0.460166
$407$	7.75736	0.384518
$408$	−17.1716	−0.850120
$409$	1.48528	0.0734424	0.0367212	−	0.999326i	$-0.488309\pi$
$409$	1.48528	0.0734424	0.0367212	+	0.999326i	$0.488309\pi$
$410$	0	0
$411$	−2.82843	−0.139516
$412$	0	0
$413$	12.6152	0.620755
$414$	1.41421	0.0695048
$415$	0	0
$416$	0	0
$417$	−11.0000	−0.538672
$418$	22.8284	1.11657
$419$	−24.0416	−1.17451	−0.587255	−	0.809402i	$-0.699792\pi$
$419$	−24.0416	−1.17451	−0.587255	+	0.809402i	$0.699792\pi$
$420$	0	0
$421$	−1.75736	−0.0856485	−0.0428242	−	0.999083i	$-0.513636\pi$
$421$	−1.75736	−0.0856485	−0.0428242	+	0.999083i	$0.513636\pi$
$422$	0.928932	0.0452197
$423$	8.58579	0.417455
$424$	20.4853	0.994853
$425$	0	0
$426$	12.5858	0.609783
$427$	−9.89949	−0.479070
$428$	0	0
$429$	8.82843	0.426240
$430$	0	0
$431$	−27.7990	−1.33903	−0.669515	−	0.742798i	$-0.733498\pi$
$431$	−27.7990	−1.33903	−0.669515	+	0.742798i	$0.733498\pi$
$432$	−4.00000	−0.192450
$433$	1.82843	0.0878686	0.0439343	−	0.999034i	$-0.486011\pi$
$433$	1.82843	0.0878686	0.0439343	+	0.999034i	$0.486011\pi$
$434$	10.7868	0.517783
$435$	0	0
$436$	0	0
$437$	−6.24264	−0.298626
$438$	14.4853	0.692134
$439$	−18.4853	−0.882254	−0.441127	−	0.897445i	$-0.645421\pi$
$439$	−18.4853	−0.882254	−0.441127	+	0.897445i	$0.645421\pi$
$440$	0	0
$441$	−3.65685	−0.174136
$442$	−29.3137	−1.39431
$443$	−30.5269	−1.45038	−0.725189	−	0.688550i	$-0.758247\pi$
$443$	−30.5269	−1.45038	−0.725189	+	0.688550i	$0.758247\pi$
$444$	0	0
$445$	0	0
$446$	19.7990	0.937509
$447$	9.07107	0.429047
$448$	−14.6274	−0.691080
$449$	−13.2426	−0.624959	−0.312479	−	0.949925i	$-0.601160\pi$
$449$	−13.2426	−0.624959	−0.312479	+	0.949925i	$0.601160\pi$
$450$	0	0
$451$	26.9289	1.26803
$452$	0	0
$453$	−9.31371	−0.437596
$454$	−24.9706	−1.17193
$455$	0	0
$456$	17.6569	0.826858
$457$	31.0000	1.45012	0.725059	−	0.688686i	$-0.241812\pi$
$457$	31.0000	1.45012	0.725059	+	0.688686i	$0.241812\pi$
$458$	33.6569	1.57268
$459$	6.07107	0.283373
$460$	0	0
$461$	−9.51472	−0.443145	−0.221572	−	0.975144i	$-0.571119\pi$
$461$	−9.51472	−0.443145	−0.221572	+	0.975144i	$0.571119\pi$
$462$	6.68629	0.311074
$463$	−17.0711	−0.793360	−0.396680	−	0.917957i	$-0.629838\pi$
$463$	−17.0711	−0.793360	−0.396680	+	0.917957i	$0.629838\pi$
$464$	14.3431	0.665864
$465$	0	0
$466$	−13.6569	−0.632642
$467$	−3.58579	−0.165930	−0.0829652	−	0.996552i	$-0.526439\pi$
$467$	−3.58579	−0.165930	−0.0829652	+	0.996552i	$0.526439\pi$
$468$	0	0
$469$	−21.0000	−0.969690
$470$	0	0
$471$	1.48528	0.0684382
$472$	19.5147	0.898238
$473$	15.5147	0.713368
$474$	−21.6569	−0.994732
$475$	0	0
$476$	0	0
$477$	−7.24264	−0.331618
$478$	−2.24264	−0.102576
$479$	−18.7279	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$479$	−18.7279	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$480$	0	0
$481$	10.2426	0.467024
$482$	33.1127	1.50824
$483$	−1.82843	−0.0831963
$484$	0	0
$485$	0	0
$486$	1.41421	0.0641500
$487$	−10.3848	−0.470579	−0.235290	−	0.971925i	$-0.575604\pi$
$487$	−10.3848	−0.470579	−0.235290	+	0.971925i	$0.575604\pi$
$488$	−15.3137	−0.693219
$489$	2.00000	0.0904431
$490$	0	0
$491$	2.07107	0.0934660	0.0467330	−	0.998907i	$-0.485119\pi$
$491$	2.07107	0.0934660	0.0467330	+	0.998907i	$0.485119\pi$
$492$	0	0
$493$	−21.7696	−0.980451
$494$	30.1421	1.35616
$495$	0	0
$496$	16.6863	0.749237
$497$	−16.2721	−0.729902
$498$	−19.8995	−0.891718
$499$	−13.0000	−0.581960	−0.290980	−	0.956729i	$-0.593981\pi$
$499$	−13.0000	−0.581960	−0.290980	+	0.956729i	$0.593981\pi$
$500$	0	0
$501$	6.72792	0.300581
$502$	−29.1716	−1.30199
$503$	14.2721	0.636361	0.318180	−	0.948030i	$-0.396928\pi$
$503$	14.2721	0.636361	0.318180	+	0.948030i	$0.396928\pi$
$504$	5.17157	0.230360
$505$	0	0
$506$	−3.65685	−0.162567
$507$	−1.34315	−0.0596512
$508$	0	0
$509$	−31.7990	−1.40947	−0.704733	−	0.709473i	$-0.748933\pi$
$509$	−31.7990	−1.40947	−0.704733	+	0.709473i	$0.748933\pi$
$510$	0	0
$511$	−18.7279	−0.828474
$512$	−22.6274	−1.00000
$513$	−6.24264	−0.275619
$514$	−42.4853	−1.87395
$515$	0	0
$516$	0	0
$517$	−22.2010	−0.976399
$518$	7.75736	0.340839
$519$	−13.6569	−0.599469
$520$	0	0
$521$	−0.443651	−0.0194367	−0.00971835	−	0.999953i	$-0.503093\pi$
$521$	−0.443651	−0.0194367	−0.00971835	+	0.999953i	$0.503093\pi$
$522$	−5.07107	−0.221955
$523$	−12.0000	−0.524723	−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000	−0.524723	−0.262362	+	0.964970i	$0.584501\pi$
$524$	0	0
$525$	0	0
$526$	−5.27208	−0.229874
$527$	−25.3259	−1.10321
$528$	10.3431	0.450128
$529$	1.00000	0.0434783
$530$	0	0
$531$	−6.89949	−0.299413
$532$	0	0
$533$	35.5563	1.54012
$534$	14.3431	0.620689
$535$	0	0
$536$	−32.4853	−1.40315
$537$	−7.65685	−0.330418
$538$	−37.5563	−1.61917
$539$	9.45584	0.407292
$540$	0	0
$541$	34.1421	1.46789	0.733943	−	0.679212i	$-0.237678\pi$
$541$	34.1421	1.46789	0.733943	+	0.679212i	$0.237678\pi$
$542$	−9.21320	−0.395741
$543$	−6.82843	−0.293036
$544$	0	0
$545$	0	0
$546$	8.82843	0.377822
$547$	20.9706	0.896637	0.448318	−	0.893874i	$-0.352023\pi$
$547$	20.9706	0.896637	0.448318	+	0.893874i	$0.352023\pi$
$548$	0	0
$549$	5.41421	0.231073
$550$	0	0
$551$	22.3848	0.953624
$552$	−2.82843	−0.120386
$553$	28.0000	1.19068
$554$	−35.7990	−1.52095
$555$	0	0
$556$	0	0
$557$	−12.8995	−0.546569	−0.273285	−	0.961933i	$-0.588110\pi$
$557$	−12.8995	−0.546569	−0.273285	+	0.961933i	$0.588110\pi$
$558$	−5.89949	−0.249746
$559$	20.4853	0.866435
$560$	0	0
$561$	−15.6985	−0.662791
$562$	24.6274	1.03884
$563$	−1.10051	−0.0463808	−0.0231904	−	0.999731i	$-0.507382\pi$
$563$	−1.10051	−0.0463808	−0.0231904	+	0.999731i	$0.507382\pi$
$564$	0	0
$565$	0	0
$566$	40.5269	1.70347
$567$	−1.82843	−0.0767867
$568$	−25.1716	−1.05618
$569$	−16.1421	−0.676714	−0.338357	−	0.941018i	$-0.609871\pi$
$569$	−16.1421	−0.676714	−0.338357	+	0.941018i	$0.609871\pi$
$570$	0	0
$571$	3.27208	0.136932	0.0684661	−	0.997653i	$-0.478190\pi$
$571$	3.27208	0.136932	0.0684661	+	0.997653i	$0.478190\pi$
$572$	0	0
$573$	10.2426	0.427892
$574$	26.9289	1.12399
$575$	0	0
$576$	8.00000	0.333333
$577$	36.9706	1.53910	0.769552	−	0.638584i	$-0.220479\pi$
$577$	36.9706	1.53910	0.769552	+	0.638584i	$0.220479\pi$
$578$	28.0833	1.16811
$579$	8.48528	0.352636
$580$	0	0
$581$	25.7279	1.06737
$582$	1.65685	0.0686788
$583$	18.7279	0.775631
$584$	−28.9706	−1.19881
$585$	0	0
$586$	4.38478	0.181133
$587$	−40.6274	−1.67687	−0.838436	−	0.544999i	$-0.816530\pi$
$587$	−40.6274	−1.67687	−0.838436	+	0.544999i	$0.816530\pi$
$588$	0	0
$589$	26.0416	1.07303
$590$	0	0
$591$	−16.8284	−0.692229
$592$	12.0000	0.493197
$593$	2.44365	0.100349	0.0501744	−	0.998740i	$-0.484022\pi$
$593$	2.44365	0.100349	0.0501744	+	0.998740i	$0.484022\pi$
$594$	−3.65685	−0.150043
$595$	0	0
$596$	0	0
$597$	10.4853	0.429134
$598$	−4.82843	−0.197449
$599$	33.1716	1.35535	0.677677	−	0.735360i	$-0.262987\pi$
$599$	33.1716	1.35535	0.677677	+	0.735360i	$0.262987\pi$
$600$	0	0
$601$	−18.3137	−0.747032	−0.373516	−	0.927624i	$-0.621848\pi$
$601$	−18.3137	−0.747032	−0.373516	+	0.927624i	$0.621848\pi$
$602$	15.5147	0.632333
$603$	11.4853	0.467717
$604$	0	0
$605$	0	0
$606$	17.7574	0.721343
$607$	38.3848	1.55799	0.778995	−	0.627030i	$-0.215730\pi$
$607$	38.3848	1.55799	0.778995	+	0.627030i	$0.215730\pi$
$608$	0	0
$609$	6.55635	0.265677
$610$	0	0
$611$	−29.3137	−1.18591
$612$	0	0
$613$	32.6274	1.31781	0.658904	−	0.752227i	$-0.271020\pi$
$613$	32.6274	1.31781	0.658904	+	0.752227i	$0.271020\pi$
$614$	−38.4853	−1.55314
$615$	0	0
$616$	−13.3726	−0.538797
$617$	−4.89949	−0.197246	−0.0986231	−	0.995125i	$-0.531444\pi$
$617$	−4.89949	−0.197246	−0.0986231	+	0.995125i	$0.531444\pi$
$618$	−1.17157	−0.0471276
$619$	−48.2843	−1.94071	−0.970354	−	0.241687i	$-0.922299\pi$
$619$	−48.2843	−1.94071	−0.970354	+	0.241687i	$0.922299\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	24.9706	1.00123
$623$	−18.5442	−0.742956
$624$	13.6569	0.546712
$625$	0	0
$626$	−36.7279	−1.46794
$627$	16.1421	0.644655
$628$	0	0
$629$	−18.2132	−0.726208
$630$	0	0
$631$	33.2132	1.32220	0.661098	−	0.750299i	$-0.270091\pi$
$631$	33.2132	1.32220	0.661098	+	0.750299i	$0.270091\pi$
$632$	43.3137	1.72293
$633$	0.656854	0.0261076
$634$	41.7990	1.66005
$635$	0	0
$636$	0	0
$637$	12.4853	0.494685
$638$	13.1127	0.519137
$639$	8.89949	0.352059
$640$	0	0
$641$	4.92893	0.194681	0.0973406	−	0.995251i	$-0.468966\pi$
$641$	4.92893	0.194681	0.0973406	+	0.995251i	$0.468966\pi$
$642$	6.24264	0.246377
$643$	−33.0000	−1.30139	−0.650696	−	0.759338i	$-0.725523\pi$
$643$	−33.0000	−1.30139	−0.650696	+	0.759338i	$0.725523\pi$
$644$	0	0
$645$	0	0
$646$	−53.5980	−2.10878
$647$	−21.2132	−0.833977	−0.416989	−	0.908912i	$-0.636914\pi$
$647$	−21.2132	−0.833977	−0.416989	+	0.908912i	$0.636914\pi$
$648$	−2.82843	−0.111111
$649$	17.8406	0.700306
$650$	0	0
$651$	7.62742	0.298942
$652$	0	0
$653$	2.58579	0.101190	0.0505948	−	0.998719i	$-0.483888\pi$
$653$	2.58579	0.101190	0.0505948	+	0.998719i	$0.483888\pi$
$654$	20.1421	0.787620
$655$	0	0
$656$	41.6569	1.62643
$657$	10.2426	0.399603
$658$	−22.2010	−0.865485
$659$	2.38478	0.0928977	0.0464488	−	0.998921i	$-0.485210\pi$
$659$	2.38478	0.0928977	0.0464488	+	0.998921i	$0.485210\pi$
$660$	0	0
$661$	11.4558	0.445581	0.222790	−	0.974866i	$-0.428483\pi$
$661$	11.4558	0.445581	0.222790	+	0.974866i	$0.428483\pi$
$662$	1.41421	0.0549650
$663$	−20.7279	−0.805006
$664$	39.7990	1.54450
$665$	0	0
$666$	−4.24264	−0.164399
$667$	−3.58579	−0.138842
$668$	0	0
$669$	14.0000	0.541271
$670$	0	0
$671$	−14.0000	−0.540464
$672$	0	0
$673$	−41.6985	−1.60736	−0.803679	−	0.595063i	$-0.797127\pi$
$673$	−41.6985	−1.60736	−0.803679	+	0.595063i	$0.797127\pi$
$674$	−43.7990	−1.68707
$675$	0	0
$676$	0	0
$677$	19.2426	0.739555	0.369777	−	0.929120i	$-0.379434\pi$
$677$	19.2426	0.739555	0.369777	+	0.929120i	$0.379434\pi$
$678$	14.7279	0.565622
$679$	−2.14214	−0.0822076
$680$	0	0
$681$	−17.6569	−0.676612
$682$	15.2548	0.584138
$683$	32.1838	1.23148	0.615739	−	0.787950i	$-0.288858\pi$
$683$	32.1838	1.23148	0.615739	+	0.787950i	$0.288858\pi$
$684$	0	0
$685$	0	0
$686$	27.5563	1.05211
$687$	23.7990	0.907988
$688$	24.0000	0.914991
$689$	24.7279	0.942059
$690$	0	0
$691$	−15.6569	−0.595615	−0.297807	−	0.954626i	$-0.596255\pi$
$691$	−15.6569	−0.595615	−0.297807	+	0.954626i	$0.596255\pi$
$692$	0	0
$693$	4.72792	0.179599
$694$	−18.6274	−0.707087
$695$	0	0
$696$	10.1421	0.384437
$697$	−63.2254	−2.39483
$698$	−45.4142	−1.71895
$699$	−9.65685	−0.365256
$700$	0	0
$701$	−36.3848	−1.37423	−0.687117	−	0.726547i	$-0.741124\pi$
$701$	−36.3848	−1.37423	−0.687117	+	0.726547i	$0.741124\pi$
$702$	−4.82843	−0.182237
$703$	18.7279	0.706337
$704$	−20.6863	−0.779644
$705$	0	0
$706$	48.8284	1.83768
$707$	−22.9584	−0.863438
$708$	0	0
$709$	16.7279	0.628230	0.314115	−	0.949385i	$-0.398292\pi$
$709$	16.7279	0.628230	0.314115	+	0.949385i	$0.398292\pi$
$710$	0	0
$711$	−15.3137	−0.574309
$712$	−28.6863	−1.07506
$713$	−4.17157	−0.156227
$714$	−15.6985	−0.587501
$715$	0	0
$716$	0	0
$717$	−1.58579	−0.0592223
$718$	−15.4558	−0.576807
$719$	36.2132	1.35052	0.675262	−	0.737578i	$-0.264030\pi$
$719$	36.2132	1.35052	0.675262	+	0.737578i	$0.264030\pi$
$720$	0	0
$721$	1.51472	0.0564111
$722$	28.2426	1.05108
$723$	23.4142	0.870784
$724$	0	0
$725$	0	0
$726$	−6.10051	−0.226411
$727$	23.0000	0.853023	0.426511	−	0.904482i	$-0.359742\pi$
$727$	23.0000	0.853023	0.426511	+	0.904482i	$0.359742\pi$
$728$	−17.6569	−0.654407
$729$	1.00000	0.0370370
$730$	0	0
$731$	−36.4264	−1.34728
$732$	0	0
$733$	17.4853	0.645834	0.322917	−	0.946427i	$-0.395337\pi$
$733$	17.4853	0.645834	0.322917	+	0.946427i	$0.395337\pi$
$734$	19.7574	0.729257
$735$	0	0
$736$	0	0
$737$	−29.6985	−1.09396
$738$	−14.7279	−0.542142
$739$	−53.6274	−1.97272	−0.986358	−	0.164613i	$-0.947362\pi$
$739$	−53.6274	−1.97272	−0.986358	+	0.164613i	$0.947362\pi$
$740$	0	0
$741$	21.3137	0.782979
$742$	18.7279	0.687524
$743$	25.6569	0.941259	0.470629	−	0.882331i	$-0.344027\pi$
$743$	25.6569	0.941259	0.470629	+	0.882331i	$0.344027\pi$
$744$	11.7990	0.432572
$745$	0	0
$746$	18.3431	0.671590
$747$	−14.0711	−0.514833
$748$	0	0
$749$	−8.07107	−0.294910
$750$	0	0
$751$	−9.27208	−0.338343	−0.169171	−	0.985587i	$-0.554109\pi$
$751$	−9.27208	−0.338343	−0.169171	+	0.985587i	$0.554109\pi$
$752$	−34.3431	−1.25237
$753$	−20.6274	−0.751705
$754$	17.3137	0.630528
$755$	0	0
$756$	0	0
$757$	−27.0000	−0.981332	−0.490666	−	0.871348i	$-0.663246\pi$
$757$	−27.0000	−0.981332	−0.490666	+	0.871348i	$0.663246\pi$
$758$	40.9706	1.48812
$759$	−2.58579	−0.0938581
$760$	0	0
$761$	−19.9289	−0.722423	−0.361212	−	0.932484i	$-0.617637\pi$
$761$	−19.9289	−0.722423	−0.361212	+	0.932484i	$0.617637\pi$
$762$	0.343146	0.0124309
$763$	−26.0416	−0.942770
$764$	0	0
$765$	0	0
$766$	−20.3848	−0.736532
$767$	23.5563	0.850570
$768$	0	0
$769$	20.0416	0.722720	0.361360	−	0.932426i	$-0.382313\pi$
$769$	20.0416	0.722720	0.361360	+	0.932426i	$0.382313\pi$
$770$	0	0
$771$	−30.0416	−1.08192
$772$	0	0
$773$	3.17157	0.114074	0.0570368	−	0.998372i	$-0.481835\pi$
$773$	3.17157	0.114074	0.0570368	+	0.998372i	$0.481835\pi$
$774$	−8.48528	−0.304997
$775$	0	0
$776$	−3.31371	−0.118955
$777$	5.48528	0.196783
$778$	20.4853	0.734433
$779$	65.0122	2.32930
$780$	0	0
$781$	−23.0122	−0.823441
$782$	8.58579	0.307027
$783$	−3.58579	−0.128146
$784$	14.6274	0.522408
$785$	0	0
$786$	−19.3137	−0.688897
$787$	−3.62742	−0.129303	−0.0646517	−	0.997908i	$-0.520594\pi$
$787$	−3.62742	−0.129303	−0.0646517	+	0.997908i	$0.520594\pi$
$788$	0	0
$789$	−3.72792	−0.132718
$790$	0	0
$791$	−19.0416	−0.677042
$792$	7.31371	0.259881
$793$	−18.4853	−0.656432
$794$	21.6569	0.768573
$795$	0	0
$796$	0	0
$797$	−16.7574	−0.593576	−0.296788	−	0.954943i	$-0.595915\pi$
$797$	−16.7574	−0.593576	−0.296788	+	0.954943i	$0.595915\pi$
$798$	16.1421	0.571425
$799$	52.1249	1.84405
$800$	0	0
$801$	10.1421	0.358355
$802$	−45.4558	−1.60510
$803$	−26.4853	−0.934645
$804$	0	0
$805$	0	0
$806$	20.1421	0.709476
$807$	−26.5563	−0.934828
$808$	−35.5147	−1.24940
$809$	−5.87006	−0.206380	−0.103190	−	0.994662i	$-0.532905\pi$
$809$	−5.87006	−0.206380	−0.103190	+	0.994662i	$0.532905\pi$
$810$	0	0
$811$	39.2843	1.37946	0.689729	−	0.724068i	$-0.257730\pi$
$811$	39.2843	1.37946	0.689729	+	0.724068i	$0.257730\pi$
$812$	0	0
$813$	−6.51472	−0.228481
$814$	10.9706	0.384518
$815$	0	0
$816$	−24.2843	−0.850120
$817$	37.4558	1.31041
$818$	2.10051	0.0734424
$819$	6.24264	0.218136
$820$	0	0
$821$	−43.7990	−1.52860	−0.764298	−	0.644864i	$-0.776914\pi$
$821$	−43.7990	−1.52860	−0.764298	+	0.644864i	$0.776914\pi$
$822$	−4.00000	−0.139516
$823$	−45.6569	−1.59150	−0.795749	−	0.605627i	$-0.792923\pi$
$823$	−45.6569	−1.59150	−0.795749	+	0.605627i	$0.792923\pi$
$824$	2.34315	0.0816274
$825$	0	0
$826$	17.8406	0.620755
$827$	50.0122	1.73909	0.869547	−	0.493850i	$-0.164411\pi$
$827$	50.0122	1.73909	0.869547	+	0.493850i	$0.164411\pi$
$828$	0	0
$829$	23.4853	0.815678	0.407839	−	0.913054i	$-0.366283\pi$
$829$	23.4853	0.815678	0.407839	+	0.913054i	$0.366283\pi$
$830$	0	0
$831$	−25.3137	−0.878123
$832$	−27.3137	−0.946932
$833$	−22.2010	−0.769219
$834$	−15.5563	−0.538672
$835$	0	0
$836$	0	0
$837$	−4.17157	−0.144191
$838$	−34.0000	−1.17451
$839$	−4.62742	−0.159756	−0.0798781	−	0.996805i	$-0.525453\pi$
$839$	−4.62742	−0.159756	−0.0798781	+	0.996805i	$0.525453\pi$
$840$	0	0
$841$	−16.1421	−0.556625
$842$	−2.48528	−0.0856485
$843$	17.4142	0.599777
$844$	0	0
$845$	0	0
$846$	12.1421	0.417455
$847$	7.88730	0.271011
$848$	28.9706	0.994853
$849$	28.6569	0.983501
$850$	0	0
$851$	−3.00000	−0.102839
$852$	0	0
$853$	31.4558	1.07703	0.538514	−	0.842617i	$-0.318986\pi$
$853$	31.4558	1.07703	0.538514	+	0.842617i	$0.318986\pi$
$854$	−14.0000	−0.479070
$855$	0	0
$856$	−12.4853	−0.426738
$857$	−21.5980	−0.737773	−0.368886	−	0.929474i	$-0.620261\pi$
$857$	−21.5980	−0.737773	−0.368886	+	0.929474i	$0.620261\pi$
$858$	12.4853	0.426240
$859$	−57.0000	−1.94481	−0.972407	−	0.233289i	$-0.925051\pi$
$859$	−57.0000	−1.94481	−0.972407	+	0.233289i	$0.925051\pi$
$860$	0	0
$861$	19.0416	0.648937
$862$	−39.3137	−1.33903
$863$	−11.1716	−0.380285	−0.190142	−	0.981757i	$-0.560895\pi$
$863$	−11.1716	−0.380285	−0.190142	+	0.981757i	$0.560895\pi$
$864$	0	0
$865$	0	0
$866$	2.58579	0.0878686
$867$	19.8579	0.674408
$868$	0	0
$869$	39.5980	1.34327
$870$	0	0
$871$	−39.2132	−1.32869
$872$	−40.2843	−1.36420
$873$	1.17157	0.0396517
$874$	−8.82843	−0.298626
$875$	0	0
$876$	0	0
$877$	1.51472	0.0511484	0.0255742	−	0.999673i	$-0.491859\pi$
$877$	1.51472	0.0511484	0.0255742	+	0.999673i	$0.491859\pi$
$878$	−26.1421	−0.882254
$879$	3.10051	0.104577
$880$	0	0
$881$	19.4142	0.654081	0.327041	−	0.945010i	$-0.393949\pi$
$881$	19.4142	0.654081	0.327041	+	0.945010i	$0.393949\pi$
$882$	−5.17157	−0.174136
$883$	1.89949	0.0639231	0.0319615	−	0.999489i	$-0.489825\pi$
$883$	1.89949	0.0639231	0.0319615	+	0.999489i	$0.489825\pi$
$884$	0	0
$885$	0	0
$886$	−43.1716	−1.45038
$887$	−21.7990	−0.731938	−0.365969	−	0.930627i	$-0.619262\pi$
$887$	−21.7990	−0.731938	−0.365969	+	0.930627i	$0.619262\pi$
$888$	8.48528	0.284747
$889$	−0.443651	−0.0148796
$890$	0	0
$891$	−2.58579	−0.0866271
$892$	0	0
$893$	−53.5980	−1.79359
$894$	12.8284	0.429047
$895$	0	0
$896$	−20.6863	−0.691080
$897$	−3.41421	−0.113997
$898$	−18.7279	−0.624959
$899$	14.9584	0.498890
$900$	0	0
$901$	−43.9706	−1.46487
$902$	38.0833	1.26803
$903$	10.9706	0.365077
$904$	−29.4558	−0.979687
$905$	0	0
$906$	−13.1716	−0.437596
$907$	−31.1421	−1.03406	−0.517029	−	0.855968i	$-0.672962\pi$
$907$	−31.1421	−1.03406	−0.517029	+	0.855968i	$0.672962\pi$
$908$	0	0
$909$	12.5563	0.416468
$910$	0	0
$911$	−5.65685	−0.187420	−0.0937100	−	0.995600i	$-0.529873\pi$
$911$	−5.65685	−0.187420	−0.0937100	+	0.995600i	$0.529873\pi$
$912$	24.9706	0.826858
$913$	36.3848	1.20416
$914$	43.8406	1.45012
$915$	0	0
$916$	0	0
$917$	24.9706	0.824601
$918$	8.58579	0.283373
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	−27.2132	−0.896706
$922$	−13.4558	−0.443145
$923$	−30.3848	−1.00013
$924$	0	0
$925$	0	0
$926$	−24.1421	−0.793360
$927$	−0.828427	−0.0272091
$928$	0	0
$929$	41.8701	1.37371	0.686856	−	0.726794i	$-0.258990\pi$
$929$	41.8701	1.37371	0.686856	+	0.726794i	$0.258990\pi$
$930$	0	0
$931$	22.8284	0.748171
$932$	0	0
$933$	17.6569	0.578059
$934$	−5.07107	−0.165930
$935$	0	0
$936$	9.65685	0.315644
$937$	59.7990	1.95355	0.976774	−	0.214272i	$-0.0687381\pi$
$937$	59.7990	1.95355	0.976774	+	0.214272i	$0.0687381\pi$
$938$	−29.6985	−0.969690
$939$	−25.9706	−0.847517
$940$	0	0
$941$	−7.41421	−0.241696	−0.120848	−	0.992671i	$-0.538561\pi$
$941$	−7.41421	−0.241696	−0.120848	+	0.992671i	$0.538561\pi$
$942$	2.10051	0.0684382
$943$	−10.4142	−0.339133
$944$	27.5980	0.898238
$945$	0	0
$946$	21.9411	0.713368
$947$	3.79899	0.123451	0.0617253	−	0.998093i	$-0.480340\pi$
$947$	3.79899	0.123451	0.0617253	+	0.998093i	$0.480340\pi$
$948$	0	0
$949$	−34.9706	−1.13519
$950$	0	0
$951$	29.5563	0.958430
$952$	31.3970	1.01758
$953$	−7.37258	−0.238821	−0.119411	−	0.992845i	$-0.538101\pi$
$953$	−7.37258	−0.238821	−0.119411	+	0.992845i	$0.538101\pi$
$954$	−10.2426	−0.331618
$955$	0	0
$956$	0	0
$957$	9.27208	0.299724
$958$	−26.4853	−0.855701
$959$	5.17157	0.166999
$960$	0	0
$961$	−13.5980	−0.438645
$962$	14.4853	0.467024
$963$	4.41421	0.142246
$964$	0	0
$965$	0	0
$966$	−2.58579	−0.0831963
$967$	−0.443651	−0.0142668	−0.00713342	−	0.999975i	$-0.502271\pi$
$967$	−0.443651	−0.0142668	−0.00713342	+	0.999975i	$0.502271\pi$
$968$	12.2010	0.392155
$969$	−37.8995	−1.21751
$970$	0	0
$971$	30.4264	0.976430	0.488215	−	0.872723i	$-0.337648\pi$
$971$	30.4264	0.976430	0.488215	+	0.872723i	$0.337648\pi$
$972$	0	0
$973$	20.1127	0.644784
$974$	−14.6863	−0.470579
$975$	0	0
$976$	−21.6569	−0.693219
$977$	15.5858	0.498633	0.249317	−	0.968422i	$-0.419794\pi$
$977$	15.5858	0.498633	0.249317	+	0.968422i	$0.419794\pi$
$978$	2.82843	0.0904431
$979$	−26.2254	−0.838167
$980$	0	0
$981$	14.2426	0.454733
$982$	2.92893	0.0934660
$983$	−24.0122	−0.765870	−0.382935	−	0.923775i	$-0.625087\pi$
$983$	−24.0122	−0.765870	−0.382935	+	0.923775i	$0.625087\pi$
$984$	29.4558	0.939018
$985$	0	0
$986$	−30.7868	−0.980451
$987$	−15.6985	−0.499688
$988$	0	0
$989$	−6.00000	−0.190789
$990$	0	0
$991$	1.34315	0.0426664	0.0213332	−	0.999772i	$-0.493209\pi$
$991$	1.34315	0.0426664	0.0213332	+	0.999772i	$0.493209\pi$
$992$	0	0
$993$	1.00000	0.0317340
$994$	−23.0122	−0.729902
$995$	0	0
$996$	0	0
$997$	34.1421	1.08129	0.540646	−	0.841250i	$-0.318180\pi$
$997$	34.1421	1.08129	0.540646	+	0.841250i	$0.318180\pi$
$998$	−18.3848	−0.581960
$999$	−3.00000	−0.0949158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1725.2.a.z.1.2		2
3.2	odd	2		5175.2.a.bj.1.1		2
5.2	odd	4		1725.2.b.s.1174.3		4
5.3	odd	4		1725.2.b.s.1174.2		4
5.4	even	2		345.2.a.h.1.1	✓	2
15.14	odd	2		1035.2.a.j.1.2		2
20.19	odd	2		5520.2.a.bm.1.1		2
115.114	odd	2		7935.2.a.q.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
345.2.a.h.1.1	✓	2	5.4	even	2
1035.2.a.j.1.2		2	15.14	odd	2
1725.2.a.z.1.2		2	1.1	even	1	trivial
1725.2.b.s.1174.2		4	5.3	odd	4
1725.2.b.s.1174.3		4	5.2	odd	4
5175.2.a.bj.1.1		2	3.2	odd	2
5520.2.a.bm.1.1		2	20.19	odd	2
7935.2.a.q.1.1		2	115.114	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:	\( N \)	\(=\)	\( 1725 = 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1725.a (trivial)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +1.00000 q^{3} +1.41421 q^{6} -1.82843 q^{7} -2.82843 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.41421 q^{2} +1.00000 q^{3} +1.41421 q^{6} -1.82843 q^{7} -2.82843 q^{8} +1.00000 q^{9} -2.58579 q^{11} -3.41421 q^{13} -2.58579 q^{14} -4.00000 q^{16} +6.07107 q^{17} +1.41421 q^{18} -6.24264 q^{19} -1.82843 q^{21} -3.65685 q^{22} +1.00000 q^{23} -2.82843 q^{24} -4.82843 q^{26} +1.00000 q^{27} -3.58579 q^{29} -4.17157 q^{31} -2.58579 q^{33} +8.58579 q^{34} -3.00000 q^{37} -8.82843 q^{38} -3.41421 q^{39} -10.4142 q^{41} -2.58579 q^{42} -6.00000 q^{43} +1.41421 q^{46} +8.58579 q^{47} -4.00000 q^{48} -3.65685 q^{49} +6.07107 q^{51} -7.24264 q^{53} +1.41421 q^{54} +5.17157 q^{56} -6.24264 q^{57} -5.07107 q^{58} -6.89949 q^{59} +5.41421 q^{61} -5.89949 q^{62} -1.82843 q^{63} +8.00000 q^{64} -3.65685 q^{66} +11.4853 q^{67} +1.00000 q^{69} +8.89949 q^{71} -2.82843 q^{72} +10.2426 q^{73} -4.24264 q^{74} +4.72792 q^{77} -4.82843 q^{78} -15.3137 q^{79} +1.00000 q^{81} -14.7279 q^{82} -14.0711 q^{83} -8.48528 q^{86} -3.58579 q^{87} +7.31371 q^{88} +10.1421 q^{89} +6.24264 q^{91} -4.17157 q^{93} +12.1421 q^{94} +1.17157 q^{97} -5.17157 q^{98} -2.58579 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{7} + 2 q^{9} - 8 q^{11} - 4 q^{13} - 8 q^{14} - 8 q^{16} - 2 q^{17} - 4 q^{19} + 2 q^{21} + 4 q^{22} + 2 q^{23} - 4 q^{26} + 2 q^{27} - 10 q^{29} - 14 q^{31} - 8 q^{33} + 20 q^{34} - 6 q^{37} - 12 q^{38} - 4 q^{39} - 18 q^{41} - 8 q^{42} - 12 q^{43} + 20 q^{47} - 8 q^{48} + 4 q^{49} - 2 q^{51} - 6 q^{53} + 16 q^{56} - 4 q^{57} + 4 q^{58} + 6 q^{59} + 8 q^{61} + 8 q^{62} + 2 q^{63} + 16 q^{64} + 4 q^{66} + 6 q^{67} + 2 q^{69} - 2 q^{71} + 12 q^{73} - 16 q^{77} - 4 q^{78} - 8 q^{79} + 2 q^{81} - 4 q^{82} - 14 q^{83} - 10 q^{87} - 8 q^{88} - 8 q^{89} + 4 q^{91} - 14 q^{93} - 4 q^{94} + 8 q^{97} - 16 q^{98} - 8 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^7 + 2 * q^9 - 8 * q^11 - 4 * q^13 - 8 * q^14 - 8 * q^16 - 2 * q^17 - 4 * q^19 + 2 * q^21 + 4 * q^22 + 2 * q^23 - 4 * q^26 + 2 * q^27 - 10 * q^29 - 14 * q^31 - 8 * q^33 + 20 * q^34 - 6 * q^37 - 12 * q^38 - 4 * q^39 - 18 * q^41 - 8 * q^42 - 12 * q^43 + 20 * q^47 - 8 * q^48 + 4 * q^49 - 2 * q^51 - 6 * q^53 + 16 * q^56 - 4 * q^57 + 4 * q^58 + 6 * q^59 + 8 * q^61 + 8 * q^62 + 2 * q^63 + 16 * q^64 + 4 * q^66 + 6 * q^67 + 2 * q^69 - 2 * q^71 + 12 * q^73 - 16 * q^77 - 4 * q^78 - 8 * q^79 + 2 * q^81 - 4 * q^82 - 14 * q^83 - 10 * q^87 - 8 * q^88 - 8 * q^89 + 4 * q^91 - 14 * q^93 - 4 * q^94 + 8 * q^97 - 16 * q^98 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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