LMFDB - Embedded newform 1725.2.a.y.1.1

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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
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			1.1
Root			$-2.44949$ 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)

$f(q)$	$=$	$q-2.44949 q^{2} -1.00000 q^{3} +4.00000 q^{4} +2.44949 q^{6} +1.00000 q^{7} -4.89898 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.44949 q^{2} -1.00000 q^{3} +4.00000 q^{4} +2.44949 q^{6} +1.00000 q^{7} -4.89898 q^{8} +1.00000 q^{9} -2.44949 q^{11} -4.00000 q^{12} +0.449490 q^{13} -2.44949 q^{14} +4.00000 q^{16} +0.550510 q^{17} -2.44949 q^{18} -0.449490 q^{19} -1.00000 q^{21} +6.00000 q^{22} +1.00000 q^{23} +4.89898 q^{24} -1.10102 q^{26} -1.00000 q^{27} +4.00000 q^{28} -4.34847 q^{29} +9.89898 q^{31} +2.44949 q^{33} -1.34847 q^{34} +4.00000 q^{36} +5.89898 q^{37} +1.10102 q^{38} -0.449490 q^{39} +0.550510 q^{41} +2.44949 q^{42} -2.00000 q^{43} -9.79796 q^{44} -2.44949 q^{46} -3.55051 q^{47} -4.00000 q^{48} -6.00000 q^{49} -0.550510 q^{51} +1.79796 q^{52} -5.44949 q^{53} +2.44949 q^{54} -4.89898 q^{56} +0.449490 q^{57} +10.6515 q^{58} -4.34847 q^{59} +15.3485 q^{61} -24.2474 q^{62} +1.00000 q^{63} -8.00000 q^{64} -6.00000 q^{66} +7.00000 q^{67} +2.20204 q^{68} -1.00000 q^{69} +10.3485 q^{71} -4.89898 q^{72} -9.34847 q^{73} -14.4495 q^{74} -1.79796 q^{76} -2.44949 q^{77} +1.10102 q^{78} -4.00000 q^{79} +1.00000 q^{81} -1.34847 q^{82} +9.24745 q^{83} -4.00000 q^{84} +4.89898 q^{86} +4.34847 q^{87} +12.0000 q^{88} -7.10102 q^{89} +0.449490 q^{91} +4.00000 q^{92} -9.89898 q^{93} +8.69694 q^{94} -12.8990 q^{97} +14.6969 q^{98} -2.44949 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} + 8 q^{4} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 + 8 * q^4 + 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} + 8 q^{4} + 2 q^{7} + 2 q^{9} - 8 q^{12} - 4 q^{13} + 8 q^{16} + 6 q^{17} + 4 q^{19} - 2 q^{21} + 12 q^{22} + 2 q^{23} - 12 q^{26} - 2 q^{27} + 8 q^{28} + 6 q^{29} + 10 q^{31} + 12 q^{34} + 8 q^{36} + 2 q^{37} + 12 q^{38} + 4 q^{39} + 6 q^{41} - 4 q^{43} - 12 q^{47} - 8 q^{48} - 12 q^{49} - 6 q^{51} - 16 q^{52} - 6 q^{53} - 4 q^{57} + 36 q^{58} + 6 q^{59} + 16 q^{61} - 24 q^{62} + 2 q^{63} - 16 q^{64} - 12 q^{66} + 14 q^{67} + 24 q^{68} - 2 q^{69} + 6 q^{71} - 4 q^{73} - 24 q^{74} + 16 q^{76} + 12 q^{78} - 8 q^{79} + 2 q^{81} + 12 q^{82} - 6 q^{83} - 8 q^{84} - 6 q^{87} + 24 q^{88} - 24 q^{89} - 4 q^{91} + 8 q^{92} - 10 q^{93} - 12 q^{94} - 16 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 + 8 * q^4 + 2 * q^7 + 2 * q^9 - 8 * q^12 - 4 * q^13 + 8 * q^16 + 6 * q^17 + 4 * q^19 - 2 * q^21 + 12 * q^22 + 2 * q^23 - 12 * q^26 - 2 * q^27 + 8 * q^28 + 6 * q^29 + 10 * q^31 + 12 * q^34 + 8 * q^36 + 2 * q^37 + 12 * q^38 + 4 * q^39 + 6 * q^41 - 4 * q^43 - 12 * q^47 - 8 * q^48 - 12 * q^49 - 6 * q^51 - 16 * q^52 - 6 * q^53 - 4 * q^57 + 36 * q^58 + 6 * q^59 + 16 * q^61 - 24 * q^62 + 2 * q^63 - 16 * q^64 - 12 * q^66 + 14 * q^67 + 24 * q^68 - 2 * q^69 + 6 * q^71 - 4 * q^73 - 24 * q^74 + 16 * q^76 + 12 * q^78 - 8 * q^79 + 2 * q^81 + 12 * q^82 - 6 * q^83 - 8 * q^84 - 6 * q^87 + 24 * q^88 - 24 * q^89 - 4 * q^91 + 8 * q^92 - 10 * q^93 - 12 * q^94 - 16 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−2.44949	−1.73205	−0.866025	−	0.500000i	$-0.833333\pi$
$2$	−2.44949	−1.73205	−0.866025	+	0.500000i	$0.833333\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	4.00000	2.00000
$5$	0	0
$5$	0	0
$6$	2.44949	1.00000
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	−4.89898	−1.73205
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.44949	−0.738549	−0.369274	−	0.929320i	$-0.620394\pi$
$11$	−2.44949	−0.738549	−0.369274	+	0.929320i	$0.620394\pi$
$12$	−4.00000	−1.15470
$13$	0.449490	0.124666	0.0623330	−	0.998055i	$-0.480146\pi$
$13$	0.449490	0.124666	0.0623330	+	0.998055i	$0.480146\pi$
$14$	−2.44949	−0.654654
$15$	0	0
$16$	4.00000	1.00000
$17$	0.550510	0.133518	0.0667592	−	0.997769i	$-0.478734\pi$
$17$	0.550510	0.133518	0.0667592	+	0.997769i	$0.478734\pi$
$18$	−2.44949	−0.577350
$19$	−0.449490	−0.103120	−0.0515600	−	0.998670i	$-0.516419\pi$
$19$	−0.449490	−0.103120	−0.0515600	+	0.998670i	$0.516419\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	6.00000	1.27920
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	4.89898	1.00000
$25$	0	0
$26$	−1.10102	−0.215928
$27$	−1.00000	−0.192450
$28$	4.00000	0.755929
$29$	−4.34847	−0.807490	−0.403745	−	0.914871i	$-0.632292\pi$
$29$	−4.34847	−0.807490	−0.403745	+	0.914871i	$0.632292\pi$
$30$	0	0
$31$	9.89898	1.77791	0.888955	−	0.457995i	$-0.151432\pi$
$31$	9.89898	1.77791	0.888955	+	0.457995i	$0.151432\pi$
$32$	0	0
$33$	2.44949	0.426401
$34$	−1.34847	−0.231261
$35$	0	0
$36$	4.00000	0.666667
$37$	5.89898	0.969786	0.484893	−	0.874573i	$-0.338858\pi$
$37$	5.89898	0.969786	0.484893	+	0.874573i	$0.338858\pi$
$38$	1.10102	0.178609
$39$	−0.449490	−0.0719760
$40$	0	0
$41$	0.550510	0.0859753	0.0429876	−	0.999076i	$-0.486312\pi$
$41$	0.550510	0.0859753	0.0429876	+	0.999076i	$0.486312\pi$
$42$	2.44949	0.377964
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	−9.79796	−1.47710
$45$	0	0
$46$	−2.44949	−0.361158
$47$	−3.55051	−0.517895	−0.258948	−	0.965891i	$-0.583376\pi$
$47$	−3.55051	−0.517895	−0.258948	+	0.965891i	$0.583376\pi$
$48$	−4.00000	−0.577350
$49$	−6.00000	−0.857143
$50$	0	0
$51$	−0.550510	−0.0770869
$52$	1.79796	0.249332
$53$	−5.44949	−0.748545	−0.374272	−	0.927319i	$-0.622108\pi$
$53$	−5.44949	−0.748545	−0.374272	+	0.927319i	$0.622108\pi$
$54$	2.44949	0.333333
$55$	0	0
$56$	−4.89898	−0.654654
$57$	0.449490	0.0595364
$58$	10.6515	1.39861
$59$	−4.34847	−0.566122	−0.283061	−	0.959102i	$-0.591350\pi$
$59$	−4.34847	−0.566122	−0.283061	+	0.959102i	$0.591350\pi$
$60$	0	0
$61$	15.3485	1.96517	0.982585	−	0.185813i	$-0.0594920\pi$
$61$	15.3485	1.96517	0.982585	+	0.185813i	$0.0594920\pi$
$62$	−24.2474	−3.07943
$63$	1.00000	0.125988
$64$	−8.00000	−1.00000
$65$	0	0
$66$	−6.00000	−0.738549
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	2.20204	0.267037
$69$	−1.00000	−0.120386
$70$	0	0
$71$	10.3485	1.22814	0.614069	−	0.789253i	$-0.289532\pi$
$71$	10.3485	1.22814	0.614069	+	0.789253i	$0.289532\pi$
$72$	−4.89898	−0.577350
$73$	−9.34847	−1.09416	−0.547078	−	0.837082i	$-0.684260\pi$
$73$	−9.34847	−1.09416	−0.547078	+	0.837082i	$0.684260\pi$
$74$	−14.4495	−1.67972
$75$	0	0
$76$	−1.79796	−0.206240
$77$	−2.44949	−0.279145
$78$	1.10102	0.124666
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−1.34847	−0.148914
$83$	9.24745	1.01504	0.507520	−	0.861640i	$-0.330562\pi$
$83$	9.24745	1.01504	0.507520	+	0.861640i	$0.330562\pi$
$84$	−4.00000	−0.436436
$85$	0	0
$86$	4.89898	0.528271
$87$	4.34847	0.466205
$88$	12.0000	1.27920
$89$	−7.10102	−0.752707	−0.376353	−	0.926476i	$-0.622822\pi$
$89$	−7.10102	−0.752707	−0.376353	+	0.926476i	$0.622822\pi$
$90$	0	0
$91$	0.449490	0.0471193
$92$	4.00000	0.417029
$93$	−9.89898	−1.02648
$94$	8.69694	0.897021
$95$	0	0
$96$	0	0
$97$	−12.8990	−1.30969	−0.654846	−	0.755762i	$-0.727267\pi$
$97$	−12.8990	−1.30969	−0.654846	+	0.755762i	$0.727267\pi$
$98$	14.6969	1.48461
$99$	−2.44949	−0.246183
$100$	0	0
$101$	−17.4495	−1.73629	−0.868145	−	0.496311i	$-0.834687\pi$
$101$	−17.4495	−1.73629	−0.868145	+	0.496311i	$0.834687\pi$
$102$	1.34847	0.133518
$103$	−16.6969	−1.64520	−0.822599	−	0.568622i	$-0.807477\pi$
$103$	−16.6969	−1.64520	−0.822599	+	0.568622i	$0.807477\pi$
$104$	−2.20204	−0.215928
$105$	0	0
$106$	13.3485	1.29652
$107$	0.550510	0.0532198	0.0266099	−	0.999646i	$-0.491529\pi$
$107$	0.550510	0.0532198	0.0266099	+	0.999646i	$0.491529\pi$
$108$	−4.00000	−0.384900
$109$	16.4495	1.57558	0.787788	−	0.615947i	$-0.211226\pi$
$109$	16.4495	1.57558	0.787788	+	0.615947i	$0.211226\pi$
$110$	0	0
$111$	−5.89898	−0.559906
$112$	4.00000	0.377964
$113$	16.3485	1.53793	0.768967	−	0.639288i	$-0.220771\pi$
$113$	16.3485	1.53793	0.768967	+	0.639288i	$0.220771\pi$
$114$	−1.10102	−0.103120
$115$	0	0
$116$	−17.3939	−1.61498
$117$	0.449490	0.0415553
$118$	10.6515	0.980553
$119$	0.550510	0.0504652
$120$	0	0
$121$	−5.00000	−0.454545
$122$	−37.5959	−3.40377
$123$	−0.550510	−0.0496378
$124$	39.5959	3.55582
$125$	0	0
$126$	−2.44949	−0.218218
$127$	6.44949	0.572300	0.286150	−	0.958185i	$-0.407624\pi$
$127$	6.44949	0.572300	0.286150	+	0.958185i	$0.407624\pi$
$128$	19.5959	1.73205
$129$	2.00000	0.176090
$130$	0	0
$131$	19.5959	1.71210	0.856052	−	0.516890i	$-0.172910\pi$
$131$	19.5959	1.71210	0.856052	+	0.516890i	$0.172910\pi$
$132$	9.79796	0.852803
$133$	−0.449490	−0.0389757
$134$	−17.1464	−1.48123
$135$	0	0
$136$	−2.69694	−0.231261
$137$	7.10102	0.606681	0.303341	−	0.952882i	$-0.401898\pi$
$137$	7.10102	0.606681	0.303341	+	0.952882i	$0.401898\pi$
$138$	2.44949	0.208514
$139$	14.7980	1.25515	0.627573	−	0.778558i	$-0.284048\pi$
$139$	14.7980	1.25515	0.627573	+	0.778558i	$0.284048\pi$
$140$	0	0
$141$	3.55051	0.299007
$142$	−25.3485	−2.12720
$143$	−1.10102	−0.0920720
$144$	4.00000	0.333333
$145$	0	0
$146$	22.8990	1.89513
$147$	6.00000	0.494872
$148$	23.5959	1.93957
$149$	13.3485	1.09355	0.546775	−	0.837280i	$-0.315855\pi$
$149$	13.3485	1.09355	0.546775	+	0.837280i	$0.315855\pi$
$150$	0	0
$151$	14.0000	1.13930	0.569652	−	0.821886i	$-0.307078\pi$
$151$	14.0000	1.13930	0.569652	+	0.821886i	$0.307078\pi$
$152$	2.20204	0.178609
$153$	0.550510	0.0445061
$154$	6.00000	0.483494
$155$	0	0
$156$	−1.79796	−0.143952
$157$	20.5959	1.64373	0.821867	−	0.569680i	$-0.192933\pi$
$157$	20.5959	1.64373	0.821867	+	0.569680i	$0.192933\pi$
$158$	9.79796	0.779484
$159$	5.44949	0.432173
$160$	0	0
$161$	1.00000	0.0788110
$162$	−2.44949	−0.192450
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	2.20204	0.171951
$165$	0	0
$166$	−22.6515	−1.75810
$167$	1.34847	0.104348	0.0521738	−	0.998638i	$-0.483385\pi$
$167$	1.34847	0.104348	0.0521738	+	0.998638i	$0.483385\pi$
$168$	4.89898	0.377964
$169$	−12.7980	−0.984458
$170$	0	0
$171$	−0.449490	−0.0343733
$172$	−8.00000	−0.609994
$173$	19.5959	1.48985	0.744925	−	0.667148i	$-0.232485\pi$
$173$	19.5959	1.48985	0.744925	+	0.667148i	$0.232485\pi$
$174$	−10.6515	−0.807490
$175$	0	0
$176$	−9.79796	−0.738549
$177$	4.34847	0.326851
$178$	17.3939	1.30373
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	0	0
$181$	0.898979	0.0668206	0.0334103	−	0.999442i	$-0.489363\pi$
$181$	0.898979	0.0668206	0.0334103	+	0.999442i	$0.489363\pi$
$182$	−1.10102	−0.0816131
$183$	−15.3485	−1.13459
$184$	−4.89898	−0.361158
$185$	0	0
$186$	24.2474	1.77791
$187$	−1.34847	−0.0986098
$188$	−14.2020	−1.03579
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	6.24745	0.452050	0.226025	−	0.974122i	$-0.427427\pi$
$191$	6.24745	0.452050	0.226025	+	0.974122i	$0.427427\pi$
$192$	8.00000	0.577350
$193$	−22.6969	−1.63376	−0.816881	−	0.576807i	$-0.804299\pi$
$193$	−22.6969	−1.63376	−0.816881	+	0.576807i	$0.804299\pi$
$194$	31.5959	2.26845
$195$	0	0
$196$	−24.0000	−1.71429
$197$	13.1010	0.933409	0.466705	−	0.884413i	$-0.345441\pi$
$197$	13.1010	0.933409	0.466705	+	0.884413i	$0.345441\pi$
$198$	6.00000	0.426401
$199$	6.89898	0.489056	0.244528	−	0.969642i	$-0.421367\pi$
$199$	6.89898	0.489056	0.244528	+	0.969642i	$0.421367\pi$
$200$	0	0
$201$	−7.00000	−0.493742
$202$	42.7423	3.00734
$203$	−4.34847	−0.305203
$204$	−2.20204	−0.154174
$205$	0	0
$206$	40.8990	2.84957
$207$	1.00000	0.0695048
$208$	1.79796	0.124666
$209$	1.10102	0.0761592
$210$	0	0
$211$	11.0000	0.757271	0.378636	−	0.925546i	$-0.376393\pi$
$211$	11.0000	0.757271	0.378636	+	0.925546i	$0.376393\pi$
$212$	−21.7980	−1.49709
$213$	−10.3485	−0.709065
$214$	−1.34847	−0.0921795
$215$	0	0
$216$	4.89898	0.333333
$217$	9.89898	0.671987
$218$	−40.2929	−2.72898
$219$	9.34847	0.631711
$220$	0	0
$221$	0.247449	0.0166452
$222$	14.4495	0.969786
$223$	17.5959	1.17831	0.589155	−	0.808020i	$-0.299461\pi$
$223$	17.5959	1.17831	0.589155	+	0.808020i	$0.299461\pi$
$224$	0	0
$225$	0	0
$226$	−40.0454	−2.66378
$227$	−2.20204	−0.146155	−0.0730773	−	0.997326i	$-0.523282\pi$
$227$	−2.20204	−0.146155	−0.0730773	+	0.997326i	$0.523282\pi$
$228$	1.79796	0.119073
$229$	20.4949	1.35434	0.677170	−	0.735826i	$-0.263206\pi$
$229$	20.4949	1.35434	0.677170	+	0.735826i	$0.263206\pi$
$230$	0	0
$231$	2.44949	0.161165
$232$	21.3031	1.39861
$233$	−2.20204	−0.144261	−0.0721303	−	0.997395i	$-0.522980\pi$
$233$	−2.20204	−0.144261	−0.0721303	+	0.997395i	$0.522980\pi$
$234$	−1.10102	−0.0719760
$235$	0	0
$236$	−17.3939	−1.13224
$237$	4.00000	0.259828
$238$	−1.34847	−0.0874083
$239$	11.4495	0.740606	0.370303	−	0.928911i	$-0.379254\pi$
$239$	11.4495	0.740606	0.370303	+	0.928911i	$0.379254\pi$
$240$	0	0
$241$	−14.6515	−0.943788	−0.471894	−	0.881655i	$-0.656430\pi$
$241$	−14.6515	−0.943788	−0.471894	+	0.881655i	$0.656430\pi$
$242$	12.2474	0.787296
$243$	−1.00000	−0.0641500
$244$	61.3939	3.93034
$245$	0	0
$246$	1.34847	0.0859753
$247$	−0.202041	−0.0128556
$248$	−48.4949	−3.07943
$249$	−9.24745	−0.586033
$250$	0	0
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	4.00000	0.251976
$253$	−2.44949	−0.153998
$254$	−15.7980	−0.991252
$255$	0	0
$256$	−32.0000	−2.00000
$257$	10.6515	0.664424	0.332212	−	0.943205i	$-0.392205\pi$
$257$	10.6515	0.664424	0.332212	+	0.943205i	$0.392205\pi$
$258$	−4.89898	−0.304997
$259$	5.89898	0.366545
$260$	0	0
$261$	−4.34847	−0.269163
$262$	−48.0000	−2.96545
$263$	−2.75255	−0.169730	−0.0848648	−	0.996392i	$-0.527046\pi$
$263$	−2.75255	−0.169730	−0.0848648	+	0.996392i	$0.527046\pi$
$264$	−12.0000	−0.738549
$265$	0	0
$266$	1.10102	0.0675079
$267$	7.10102	0.434575
$268$	28.0000	1.71037
$269$	−0.550510	−0.0335652	−0.0167826	−	0.999859i	$-0.505342\pi$
$269$	−0.550510	−0.0335652	−0.0167826	+	0.999859i	$0.505342\pi$
$270$	0	0
$271$	2.30306	0.139901	0.0699505	−	0.997550i	$-0.477716\pi$
$271$	2.30306	0.139901	0.0699505	+	0.997550i	$0.477716\pi$
$272$	2.20204	0.133518
$273$	−0.449490	−0.0272044
$274$	−17.3939	−1.05080
$275$	0	0
$276$	−4.00000	−0.240772
$277$	10.0000	0.600842	0.300421	−	0.953807i	$-0.402873\pi$
$277$	10.0000	0.600842	0.300421	+	0.953807i	$0.402873\pi$
$278$	−36.2474	−2.17398
$279$	9.89898	0.592636
$280$	0	0
$281$	−4.65153	−0.277487	−0.138744	−	0.990328i	$-0.544306\pi$
$281$	−4.65153	−0.277487	−0.138744	+	0.990328i	$0.544306\pi$
$282$	−8.69694	−0.517895
$283$	11.8990	0.707321	0.353660	−	0.935374i	$-0.384937\pi$
$283$	11.8990	0.707321	0.353660	+	0.935374i	$0.384937\pi$
$284$	41.3939	2.45627
$285$	0	0
$286$	2.69694	0.159473
$287$	0.550510	0.0324956
$288$	0	0
$289$	−16.6969	−0.982173
$290$	0	0
$291$	12.8990	0.756152
$292$	−37.3939	−2.18831
$293$	−3.24745	−0.189718	−0.0948590	−	0.995491i	$-0.530240\pi$
$293$	−3.24745	−0.189718	−0.0948590	+	0.995491i	$0.530240\pi$
$294$	−14.6969	−0.857143
$295$	0	0
$296$	−28.8990	−1.67972
$297$	2.44949	0.142134
$298$	−32.6969	−1.89408
$299$	0.449490	0.0259947
$300$	0	0
$301$	−2.00000	−0.115278
$302$	−34.2929	−1.97333
$303$	17.4495	1.00245
$304$	−1.79796	−0.103120
$305$	0	0
$306$	−1.34847	−0.0770869
$307$	17.3485	0.990129	0.495065	−	0.868856i	$-0.335144\pi$
$307$	17.3485	0.990129	0.495065	+	0.868856i	$0.335144\pi$
$308$	−9.79796	−0.558291
$309$	16.6969	0.949856
$310$	0	0
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	2.20204	0.124666
$313$	9.69694	0.548103	0.274052	−	0.961715i	$-0.411636\pi$
$313$	9.69694	0.548103	0.274052	+	0.961715i	$0.411636\pi$
$314$	−50.4495	−2.84703
$315$	0	0
$316$	−16.0000	−0.900070
$317$	18.2474	1.02488	0.512439	−	0.858723i	$-0.328742\pi$
$317$	18.2474	1.02488	0.512439	+	0.858723i	$0.328742\pi$
$318$	−13.3485	−0.748545
$319$	10.6515	0.596371
$320$	0	0
$321$	−0.550510	−0.0307265
$322$	−2.44949	−0.136505
$323$	−0.247449	−0.0137684
$324$	4.00000	0.222222
$325$	0	0
$326$	−24.4949	−1.35665
$327$	−16.4495	−0.909659
$328$	−2.69694	−0.148914
$329$	−3.55051	−0.195746
$330$	0	0
$331$	−14.5959	−0.802264	−0.401132	−	0.916020i	$-0.631383\pi$
$331$	−14.5959	−0.802264	−0.401132	+	0.916020i	$0.631383\pi$
$332$	36.9898	2.03008
$333$	5.89898	0.323262
$334$	−3.30306	−0.180735
$335$	0	0
$336$	−4.00000	−0.218218
$337$	0.202041	0.0110059	0.00550294	−	0.999985i	$-0.498248\pi$
$337$	0.202041	0.0110059	0.00550294	+	0.999985i	$0.498248\pi$
$338$	31.3485	1.70513
$339$	−16.3485	−0.887927
$340$	0	0
$341$	−24.2474	−1.31307
$342$	1.10102	0.0595364
$343$	−13.0000	−0.701934
$344$	9.79796	0.528271
$345$	0	0
$346$	−48.0000	−2.58050
$347$	19.1010	1.02540	0.512698	−	0.858569i	$-0.328646\pi$
$347$	19.1010	1.02540	0.512698	+	0.858569i	$0.328646\pi$
$348$	17.3939	0.932410
$349$	−25.4949	−1.36471	−0.682355	−	0.731021i	$-0.739044\pi$
$349$	−25.4949	−1.36471	−0.682355	+	0.731021i	$0.739044\pi$
$350$	0	0
$351$	−0.449490	−0.0239920
$352$	0	0
$353$	20.4495	1.08842	0.544208	−	0.838950i	$-0.316830\pi$
$353$	20.4495	1.08842	0.544208	+	0.838950i	$0.316830\pi$
$354$	−10.6515	−0.566122
$355$	0	0
$356$	−28.4041	−1.50541
$357$	−0.550510	−0.0291361
$358$	14.6969	0.776757
$359$	−8.44949	−0.445947	−0.222974	−	0.974825i	$-0.571576\pi$
$359$	−8.44949	−0.445947	−0.222974	+	0.974825i	$0.571576\pi$
$360$	0	0
$361$	−18.7980	−0.989366
$362$	−2.20204	−0.115737
$363$	5.00000	0.262432
$364$	1.79796	0.0942387
$365$	0	0
$366$	37.5959	1.96517
$367$	15.6969	0.819374	0.409687	−	0.912226i	$-0.365638\pi$
$367$	15.6969	0.819374	0.409687	+	0.912226i	$0.365638\pi$
$368$	4.00000	0.208514
$369$	0.550510	0.0286584
$370$	0	0
$371$	−5.44949	−0.282923
$372$	−39.5959	−2.05295
$373$	11.5959	0.600414	0.300207	−	0.953874i	$-0.402944\pi$
$373$	11.5959	0.600414	0.300207	+	0.953874i	$0.402944\pi$
$374$	3.30306	0.170797
$375$	0	0
$376$	17.3939	0.897021
$377$	−1.95459	−0.100667
$378$	2.44949	0.125988
$379$	−33.3939	−1.71533	−0.857664	−	0.514210i	$-0.828085\pi$
$379$	−33.3939	−1.71533	−0.857664	+	0.514210i	$0.828085\pi$
$380$	0	0
$381$	−6.44949	−0.330417
$382$	−15.3031	−0.782973
$383$	−16.3485	−0.835368	−0.417684	−	0.908592i	$-0.637158\pi$
$383$	−16.3485	−0.835368	−0.417684	+	0.908592i	$0.637158\pi$
$384$	−19.5959	−1.00000
$385$	0	0
$386$	55.5959	2.82976
$387$	−2.00000	−0.101666
$388$	−51.5959	−2.61939
$389$	−32.6969	−1.65780	−0.828900	−	0.559396i	$-0.811033\pi$
$389$	−32.6969	−1.65780	−0.828900	+	0.559396i	$0.811033\pi$
$390$	0	0
$391$	0.550510	0.0278405
$392$	29.3939	1.48461
$393$	−19.5959	−0.988483
$394$	−32.0908	−1.61671
$395$	0	0
$396$	−9.79796	−0.492366
$397$	−5.79796	−0.290991	−0.145496	−	0.989359i	$-0.546478\pi$
$397$	−5.79796	−0.290991	−0.145496	+	0.989359i	$0.546478\pi$
$398$	−16.8990	−0.847069
$399$	0.449490	0.0225026
$400$	0	0
$401$	−22.8990	−1.14352	−0.571760	−	0.820421i	$-0.693739\pi$
$401$	−22.8990	−1.14352	−0.571760	+	0.820421i	$0.693739\pi$
$402$	17.1464	0.855186
$403$	4.44949	0.221645
$404$	−69.7980	−3.47258
$405$	0	0
$406$	10.6515	0.528627
$407$	−14.4495	−0.716235
$408$	2.69694	0.133518
$409$	12.1010	0.598357	0.299178	−	0.954197i	$-0.403287\pi$
$409$	12.1010	0.598357	0.299178	+	0.954197i	$0.403287\pi$
$410$	0	0
$411$	−7.10102	−0.350268
$412$	−66.7878	−3.29040
$413$	−4.34847	−0.213974
$414$	−2.44949	−0.120386
$415$	0	0
$416$	0	0
$417$	−14.7980	−0.724659
$418$	−2.69694	−0.131912
$419$	38.4495	1.87838	0.939190	−	0.343397i	$-0.111578\pi$
$419$	38.4495	1.87838	0.939190	+	0.343397i	$0.111578\pi$
$420$	0	0
$421$	2.24745	0.109534	0.0547670	−	0.998499i	$-0.482558\pi$
$421$	2.24745	0.109534	0.0547670	+	0.998499i	$0.482558\pi$
$422$	−26.9444	−1.31163
$423$	−3.55051	−0.172632
$424$	26.6969	1.29652
$425$	0	0
$426$	25.3485	1.22814
$427$	15.3485	0.742764
$428$	2.20204	0.106440
$429$	1.10102	0.0531578
$430$	0	0
$431$	24.4949	1.17988	0.589939	−	0.807448i	$-0.299152\pi$
$431$	24.4949	1.17988	0.589939	+	0.807448i	$0.299152\pi$
$432$	−4.00000	−0.192450
$433$	16.7980	0.807258	0.403629	−	0.914923i	$-0.367749\pi$
$433$	16.7980	0.807258	0.403629	+	0.914923i	$0.367749\pi$
$434$	−24.2474	−1.16391
$435$	0	0
$436$	65.7980	3.15115
$437$	−0.449490	−0.0215020
$438$	−22.8990	−1.09416
$439$	−7.30306	−0.348556	−0.174278	−	0.984696i	$-0.555759\pi$
$439$	−7.30306	−0.348556	−0.174278	+	0.984696i	$0.555759\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	−0.606123	−0.0288303
$443$	15.5505	0.738827	0.369414	−	0.929265i	$-0.379559\pi$
$443$	15.5505	0.738827	0.369414	+	0.929265i	$0.379559\pi$
$444$	−23.5959	−1.11981
$445$	0	0
$446$	−43.1010	−2.04089
$447$	−13.3485	−0.631361
$448$	−8.00000	−0.377964
$449$	3.24745	0.153257	0.0766283	−	0.997060i	$-0.475585\pi$
$449$	3.24745	0.153257	0.0766283	+	0.997060i	$0.475585\pi$
$450$	0	0
$451$	−1.34847	−0.0634969
$452$	65.3939	3.07587
$453$	−14.0000	−0.657777
$454$	5.39388	0.253147
$455$	0	0
$456$	−2.20204	−0.103120
$457$	−9.89898	−0.463055	−0.231527	−	0.972828i	$-0.574372\pi$
$457$	−9.89898	−0.463055	−0.231527	+	0.972828i	$0.574372\pi$
$458$	−50.2020	−2.34579
$459$	−0.550510	−0.0256956
$460$	0	0
$461$	30.4949	1.42029	0.710144	−	0.704056i	$-0.248630\pi$
$461$	30.4949	1.42029	0.710144	+	0.704056i	$0.248630\pi$
$462$	−6.00000	−0.279145
$463$	−28.9444	−1.34516	−0.672580	−	0.740025i	$-0.734814\pi$
$463$	−28.9444	−1.34516	−0.672580	+	0.740025i	$0.734814\pi$
$464$	−17.3939	−0.807490
$465$	0	0
$466$	5.39388	0.249867
$467$	−28.8434	−1.33471	−0.667356	−	0.744739i	$-0.732574\pi$
$467$	−28.8434	−1.33471	−0.667356	+	0.744739i	$0.732574\pi$
$468$	1.79796	0.0831107
$469$	7.00000	0.323230
$470$	0	0
$471$	−20.5959	−0.949010
$472$	21.3031	0.980553
$473$	4.89898	0.225255
$474$	−9.79796	−0.450035
$475$	0	0
$476$	2.20204	0.100930
$477$	−5.44949	−0.249515
$478$	−28.0454	−1.28277
$479$	−20.9444	−0.956973	−0.478487	−	0.878095i	$-0.658815\pi$
$479$	−20.9444	−0.956973	−0.478487	+	0.878095i	$0.658815\pi$
$480$	0	0
$481$	2.65153	0.120899
$482$	35.8888	1.63469
$483$	−1.00000	−0.0455016
$484$	−20.0000	−0.909091
$485$	0	0
$486$	2.44949	0.111111
$487$	−34.9444	−1.58348	−0.791741	−	0.610857i	$-0.790825\pi$
$487$	−34.9444	−1.58348	−0.791741	+	0.610857i	$0.790825\pi$
$488$	−75.1918	−3.40377
$489$	−10.0000	−0.452216
$490$	0	0
$491$	17.4495	0.787484	0.393742	−	0.919221i	$-0.371180\pi$
$491$	17.4495	0.787484	0.393742	+	0.919221i	$0.371180\pi$
$492$	−2.20204	−0.0992757
$493$	−2.39388	−0.107815
$494$	0.494897	0.0222665
$495$	0	0
$496$	39.5959	1.77791
$497$	10.3485	0.464192
$498$	22.6515	1.01504
$499$	−1.00000	−0.0447661	−0.0223831	−	0.999749i	$-0.507125\pi$
$499$	−1.00000	−0.0447661	−0.0223831	+	0.999749i	$0.507125\pi$
$500$	0	0
$501$	−1.34847	−0.0602452
$502$	−44.0908	−1.96787
$503$	37.0454	1.65177	0.825887	−	0.563836i	$-0.190675\pi$
$503$	37.0454	1.65177	0.825887	+	0.563836i	$0.190675\pi$
$504$	−4.89898	−0.218218
$505$	0	0
$506$	6.00000	0.266733
$507$	12.7980	0.568377
$508$	25.7980	1.14460
$509$	19.1010	0.846638	0.423319	−	0.905981i	$-0.360865\pi$
$509$	19.1010	0.846638	0.423319	+	0.905981i	$0.360865\pi$
$510$	0	0
$511$	−9.34847	−0.413552
$512$	39.1918	1.73205
$513$	0.449490	0.0198455
$514$	−26.0908	−1.15082
$515$	0	0
$516$	8.00000	0.352180
$517$	8.69694	0.382491
$518$	−14.4495	−0.634874
$519$	−19.5959	−0.860165
$520$	0	0
$521$	19.8434	0.869354	0.434677	−	0.900587i	$-0.356863\pi$
$521$	19.8434	0.869354	0.434677	+	0.900587i	$0.356863\pi$
$522$	10.6515	0.466205
$523$	−25.3939	−1.11040	−0.555198	−	0.831718i	$-0.687358\pi$
$523$	−25.3939	−1.11040	−0.555198	+	0.831718i	$0.687358\pi$
$524$	78.3837	3.42421
$525$	0	0
$526$	6.74235	0.293980
$527$	5.44949	0.237384
$528$	9.79796	0.426401
$529$	1.00000	0.0434783
$530$	0	0
$531$	−4.34847	−0.188707
$532$	−1.79796	−0.0779514
$533$	0.247449	0.0107182
$534$	−17.3939	−0.752707
$535$	0	0
$536$	−34.2929	−1.48123
$537$	6.00000	0.258919
$538$	1.34847	0.0581366
$539$	14.6969	0.633042
$540$	0	0
$541$	3.10102	0.133323	0.0666616	−	0.997776i	$-0.478765\pi$
$541$	3.10102	0.133323	0.0666616	+	0.997776i	$0.478765\pi$
$542$	−5.64133	−0.242316
$543$	−0.898979	−0.0385789
$544$	0	0
$545$	0	0
$546$	1.10102	0.0471193
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	28.4041	1.21336
$549$	15.3485	0.655057
$550$	0	0
$551$	1.95459	0.0832684
$552$	4.89898	0.208514
$553$	−4.00000	−0.170097
$554$	−24.4949	−1.04069
$555$	0	0
$556$	59.1918	2.51029
$557$	−25.0454	−1.06121	−0.530604	−	0.847620i	$-0.678035\pi$
$557$	−25.0454	−1.06121	−0.530604	+	0.847620i	$0.678035\pi$
$558$	−24.2474	−1.02648
$559$	−0.898979	−0.0380228
$560$	0	0
$561$	1.34847	0.0569324
$562$	11.3939	0.480622
$563$	−0.550510	−0.0232012	−0.0116006	−	0.999933i	$-0.503693\pi$
$563$	−0.550510	−0.0232012	−0.0116006	+	0.999933i	$0.503693\pi$
$564$	14.2020	0.598014
$565$	0	0
$566$	−29.1464	−1.22512
$567$	1.00000	0.0419961
$568$	−50.6969	−2.12720
$569$	40.2929	1.68916	0.844582	−	0.535426i	$-0.179849\pi$
$569$	40.2929	1.68916	0.844582	+	0.535426i	$0.179849\pi$
$570$	0	0
$571$	−21.1464	−0.884950	−0.442475	−	0.896781i	$-0.645900\pi$
$571$	−21.1464	−0.884950	−0.442475	+	0.896781i	$0.645900\pi$
$572$	−4.40408	−0.184144
$573$	−6.24745	−0.260991
$574$	−1.34847	−0.0562840
$575$	0	0
$576$	−8.00000	−0.333333
$577$	28.0000	1.16566	0.582828	−	0.812596i	$-0.301946\pi$
$577$	28.0000	1.16566	0.582828	+	0.812596i	$0.301946\pi$
$578$	40.8990	1.70117
$579$	22.6969	0.943253
$580$	0	0
$581$	9.24745	0.383649
$582$	−31.5959	−1.30969
$583$	13.3485	0.552837
$584$	45.7980	1.89513
$585$	0	0
$586$	7.95459	0.328601
$587$	6.00000	0.247647	0.123823	−	0.992304i	$-0.460484\pi$
$587$	6.00000	0.247647	0.123823	+	0.992304i	$0.460484\pi$
$588$	24.0000	0.989743
$589$	−4.44949	−0.183338
$590$	0	0
$591$	−13.1010	−0.538904
$592$	23.5959	0.969786
$593$	−32.4495	−1.33254	−0.666270	−	0.745710i	$-0.732110\pi$
$593$	−32.4495	−1.33254	−0.666270	+	0.745710i	$0.732110\pi$
$594$	−6.00000	−0.246183
$595$	0	0
$596$	53.3939	2.18710
$597$	−6.89898	−0.282356
$598$	−1.10102	−0.0450241
$599$	26.6969	1.09081	0.545404	−	0.838174i	$-0.316376\pi$
$599$	26.6969	1.09081	0.545404	+	0.838174i	$0.316376\pi$
$600$	0	0
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	4.89898	0.199667
$603$	7.00000	0.285062
$604$	56.0000	2.27861
$605$	0	0
$606$	−42.7423	−1.73629
$607$	−8.24745	−0.334754	−0.167377	−	0.985893i	$-0.553530\pi$
$607$	−8.24745	−0.334754	−0.167377	+	0.985893i	$0.553530\pi$
$608$	0	0
$609$	4.34847	0.176209
$610$	0	0
$611$	−1.59592	−0.0645639
$612$	2.20204	0.0890122
$613$	−14.0000	−0.565455	−0.282727	−	0.959200i	$-0.591239\pi$
$613$	−14.0000	−0.565455	−0.282727	+	0.959200i	$0.591239\pi$
$614$	−42.4949	−1.71495
$615$	0	0
$616$	12.0000	0.483494
$617$	−17.4495	−0.702490	−0.351245	−	0.936284i	$-0.614242\pi$
$617$	−17.4495	−0.702490	−0.351245	+	0.936284i	$0.614242\pi$
$618$	−40.8990	−1.64520
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	−29.3939	−1.17859
$623$	−7.10102	−0.284496
$624$	−1.79796	−0.0719760
$625$	0	0
$626$	−23.7526	−0.949343
$627$	−1.10102	−0.0439705
$628$	82.3837	3.28747
$629$	3.24745	0.129484
$630$	0	0
$631$	34.4495	1.37141	0.685706	−	0.727878i	$-0.259493\pi$
$631$	34.4495	1.37141	0.685706	+	0.727878i	$0.259493\pi$
$632$	19.5959	0.779484
$633$	−11.0000	−0.437211
$634$	−44.6969	−1.77514
$635$	0	0
$636$	21.7980	0.864345
$637$	−2.69694	−0.106857
$638$	−26.0908	−1.03295
$639$	10.3485	0.409379
$640$	0	0
$641$	−26.9444	−1.06424	−0.532120	−	0.846669i	$-0.678604\pi$
$641$	−26.9444	−1.06424	−0.532120	+	0.846669i	$0.678604\pi$
$642$	1.34847	0.0532198
$643$	−19.6969	−0.776771	−0.388386	−	0.921497i	$-0.626967\pi$
$643$	−19.6969	−0.776771	−0.388386	+	0.921497i	$0.626967\pi$
$644$	4.00000	0.157622
$645$	0	0
$646$	0.606123	0.0238476
$647$	−46.0454	−1.81023	−0.905116	−	0.425165i	$-0.860216\pi$
$647$	−46.0454	−1.81023	−0.905116	+	0.425165i	$0.860216\pi$
$648$	−4.89898	−0.192450
$649$	10.6515	0.418109
$650$	0	0
$651$	−9.89898	−0.387972
$652$	40.0000	1.56652
$653$	19.8434	0.776531	0.388265	−	0.921548i	$-0.373074\pi$
$653$	19.8434	0.776531	0.388265	+	0.921548i	$0.373074\pi$
$654$	40.2929	1.57558
$655$	0	0
$656$	2.20204	0.0859753
$657$	−9.34847	−0.364719
$658$	8.69694	0.339042
$659$	29.1464	1.13538	0.567692	−	0.823241i	$-0.307837\pi$
$659$	29.1464	1.13538	0.567692	+	0.823241i	$0.307837\pi$
$660$	0	0
$661$	−42.0908	−1.63714	−0.818571	−	0.574405i	$-0.805234\pi$
$661$	−42.0908	−1.63714	−0.818571	+	0.574405i	$0.805234\pi$
$662$	35.7526	1.38956
$663$	−0.247449	−0.00961011
$664$	−45.3031	−1.75810
$665$	0	0
$666$	−14.4495	−0.559906
$667$	−4.34847	−0.168373
$668$	5.39388	0.208695
$669$	−17.5959	−0.680297
$670$	0	0
$671$	−37.5959	−1.45137
$672$	0	0
$673$	−17.5505	−0.676522	−0.338261	−	0.941052i	$-0.609839\pi$
$673$	−17.5505	−0.676522	−0.338261	+	0.941052i	$0.609839\pi$
$674$	−0.494897	−0.0190627
$675$	0	0
$676$	−51.1918	−1.96892
$677$	17.4495	0.670638	0.335319	−	0.942105i	$-0.391156\pi$
$677$	17.4495	0.670638	0.335319	+	0.942105i	$0.391156\pi$
$678$	40.0454	1.53793
$679$	−12.8990	−0.495017
$680$	0	0
$681$	2.20204	0.0843824
$682$	59.3939	2.27431
$683$	6.24745	0.239052	0.119526	−	0.992831i	$-0.461863\pi$
$683$	6.24745	0.239052	0.119526	+	0.992831i	$0.461863\pi$
$684$	−1.79796	−0.0687467
$685$	0	0
$686$	31.8434	1.21579
$687$	−20.4949	−0.781929
$688$	−8.00000	−0.304997
$689$	−2.44949	−0.0933181
$690$	0	0
$691$	35.7980	1.36182	0.680909	−	0.732368i	$-0.261585\pi$
$691$	35.7980	1.36182	0.680909	+	0.732368i	$0.261585\pi$
$692$	78.3837	2.97970
$693$	−2.44949	−0.0930484
$694$	−46.7878	−1.77604
$695$	0	0
$696$	−21.3031	−0.807490
$697$	0.303062	0.0114793
$698$	62.4495	2.36375
$699$	2.20204	0.0832888
$700$	0	0
$701$	−7.95459	−0.300441	−0.150220	−	0.988653i	$-0.547998\pi$
$701$	−7.95459	−0.300441	−0.150220	+	0.988653i	$0.547998\pi$
$702$	1.10102	0.0415553
$703$	−2.65153	−0.100004
$704$	19.5959	0.738549
$705$	0	0
$706$	−50.0908	−1.88519
$707$	−17.4495	−0.656256
$708$	17.3939	0.653702
$709$	44.7423	1.68033	0.840167	−	0.542328i	$-0.182457\pi$
$709$	44.7423	1.68033	0.840167	+	0.542328i	$0.182457\pi$
$710$	0	0
$711$	−4.00000	−0.150012
$712$	34.7878	1.30373
$713$	9.89898	0.370720
$714$	1.34847	0.0504652
$715$	0	0
$716$	−24.0000	−0.896922
$717$	−11.4495	−0.427589
$718$	20.6969	0.772403
$719$	41.9444	1.56426	0.782131	−	0.623114i	$-0.214133\pi$
$719$	41.9444	1.56426	0.782131	+	0.623114i	$0.214133\pi$
$720$	0	0
$721$	−16.6969	−0.621826
$722$	46.0454	1.71363
$723$	14.6515	0.544896
$724$	3.59592	0.133641
$725$	0	0
$726$	−12.2474	−0.454545
$727$	45.6969	1.69481	0.847403	−	0.530951i	$-0.178165\pi$
$727$	45.6969	1.69481	0.847403	+	0.530951i	$0.178165\pi$
$728$	−2.20204	−0.0816131
$729$	1.00000	0.0370370
$730$	0	0
$731$	−1.10102	−0.0407227
$732$	−61.3939	−2.26918
$733$	−30.5959	−1.13009	−0.565043	−	0.825061i	$-0.691140\pi$
$733$	−30.5959	−1.13009	−0.565043	+	0.825061i	$0.691140\pi$
$734$	−38.4495	−1.41920
$735$	0	0
$736$	0	0
$737$	−17.1464	−0.631597
$738$	−1.34847	−0.0496378
$739$	−28.7980	−1.05935	−0.529675	−	0.848201i	$-0.677686\pi$
$739$	−28.7980	−1.05935	−0.529675	+	0.848201i	$0.677686\pi$
$740$	0	0
$741$	0.202041	0.00742216
$742$	13.3485	0.490038
$743$	2.20204	0.0807851	0.0403925	−	0.999184i	$-0.487139\pi$
$743$	2.20204	0.0807851	0.0403925	+	0.999184i	$0.487139\pi$
$744$	48.4949	1.77791
$745$	0	0
$746$	−28.4041	−1.03995
$747$	9.24745	0.338346
$748$	−5.39388	−0.197220
$749$	0.550510	0.0201152
$750$	0	0
$751$	45.3485	1.65479	0.827395	−	0.561621i	$-0.189822\pi$
$751$	45.3485	1.65479	0.827395	+	0.561621i	$0.189822\pi$
$752$	−14.2020	−0.517895
$753$	−18.0000	−0.655956
$754$	4.78775	0.174360
$755$	0	0
$756$	−4.00000	−0.145479
$757$	−25.6969	−0.933971	−0.466986	−	0.884265i	$-0.654660\pi$
$757$	−25.6969	−0.933971	−0.466986	+	0.884265i	$0.654660\pi$
$758$	81.7980	2.97104
$759$	2.44949	0.0889108
$760$	0	0
$761$	−20.1464	−0.730307	−0.365154	−	0.930947i	$-0.618984\pi$
$761$	−20.1464	−0.730307	−0.365154	+	0.930947i	$0.618984\pi$
$762$	15.7980	0.572300
$763$	16.4495	0.595512
$764$	24.9898	0.904099
$765$	0	0
$766$	40.0454	1.44690
$767$	−1.95459	−0.0705762
$768$	32.0000	1.15470
$769$	5.55051	0.200157	0.100078	−	0.994980i	$-0.468091\pi$
$769$	5.55051	0.200157	0.100078	+	0.994980i	$0.468091\pi$
$770$	0	0
$771$	−10.6515	−0.383606
$772$	−90.7878	−3.26752
$773$	−52.2929	−1.88084	−0.940422	−	0.340010i	$-0.889569\pi$
$773$	−52.2929	−1.88084	−0.940422	+	0.340010i	$0.889569\pi$
$774$	4.89898	0.176090
$775$	0	0
$776$	63.1918	2.26845
$777$	−5.89898	−0.211625
$778$	80.0908	2.87139
$779$	−0.247449	−0.00886577
$780$	0	0
$781$	−25.3485	−0.907040
$782$	−1.34847	−0.0482212
$783$	4.34847	0.155402
$784$	−24.0000	−0.857143
$785$	0	0
$786$	48.0000	1.71210
$787$	−7.69694	−0.274366	−0.137183	−	0.990546i	$-0.543805\pi$
$787$	−7.69694	−0.274366	−0.137183	+	0.990546i	$0.543805\pi$
$788$	52.4041	1.86682
$789$	2.75255	0.0979934
$790$	0	0
$791$	16.3485	0.581285
$792$	12.0000	0.426401
$793$	6.89898	0.244990
$794$	14.2020	0.504012
$795$	0	0
$796$	27.5959	0.978111
$797$	15.2474	0.540092	0.270046	−	0.962847i	$-0.412961\pi$
$797$	15.2474	0.540092	0.270046	+	0.962847i	$0.412961\pi$
$798$	−1.10102	−0.0389757
$799$	−1.95459	−0.0691485
$800$	0	0
$801$	−7.10102	−0.250902
$802$	56.0908	1.98064
$803$	22.8990	0.808087
$804$	−28.0000	−0.987484
$805$	0	0
$806$	−10.8990	−0.383900
$807$	0.550510	0.0193789
$808$	85.4847	3.00734
$809$	11.4495	0.402543	0.201271	−	0.979536i	$-0.435493\pi$
$809$	11.4495	0.402543	0.201271	+	0.979536i	$0.435493\pi$
$810$	0	0
$811$	16.3939	0.575667	0.287833	−	0.957680i	$-0.407065\pi$
$811$	16.3939	0.575667	0.287833	+	0.957680i	$0.407065\pi$
$812$	−17.3939	−0.610405
$813$	−2.30306	−0.0807719
$814$	35.3939	1.24055
$815$	0	0
$816$	−2.20204	−0.0770869
$817$	0.898979	0.0314513
$818$	−29.6413	−1.03638
$819$	0.449490	0.0157064
$820$	0	0
$821$	50.6969	1.76934	0.884668	−	0.466222i	$-0.154385\pi$
$821$	50.6969	1.76934	0.884668	+	0.466222i	$0.154385\pi$
$822$	17.3939	0.606681
$823$	35.5959	1.24080	0.620398	−	0.784287i	$-0.286971\pi$
$823$	35.5959	1.24080	0.620398	+	0.784287i	$0.286971\pi$
$824$	81.7980	2.84957
$825$	0	0
$826$	10.6515	0.370614
$827$	2.14643	0.0746386	0.0373193	−	0.999303i	$-0.488118\pi$
$827$	2.14643	0.0746386	0.0373193	+	0.999303i	$0.488118\pi$
$828$	4.00000	0.139010
$829$	1.69694	0.0589371	0.0294686	−	0.999566i	$-0.490619\pi$
$829$	1.69694	0.0589371	0.0294686	+	0.999566i	$0.490619\pi$
$830$	0	0
$831$	−10.0000	−0.346896
$832$	−3.59592	−0.124666
$833$	−3.30306	−0.114444
$834$	36.2474	1.25515
$835$	0	0
$836$	4.40408	0.152318
$837$	−9.89898	−0.342159
$838$	−94.1816	−3.25345
$839$	33.1918	1.14591	0.572955	−	0.819587i	$-0.305797\pi$
$839$	33.1918	1.14591	0.572955	+	0.819587i	$0.305797\pi$
$840$	0	0
$841$	−10.0908	−0.347959
$842$	−5.50510	−0.189718
$843$	4.65153	0.160207
$844$	44.0000	1.51454
$845$	0	0
$846$	8.69694	0.299007
$847$	−5.00000	−0.171802
$848$	−21.7980	−0.748545
$849$	−11.8990	−0.408372
$850$	0	0
$851$	5.89898	0.202214
$852$	−41.3939	−1.41813
$853$	42.0908	1.44116	0.720581	−	0.693371i	$-0.243875\pi$
$853$	42.0908	1.44116	0.720581	+	0.693371i	$0.243875\pi$
$854$	−37.5959	−1.28651
$855$	0	0
$856$	−2.69694	−0.0921795
$857$	−25.5959	−0.874340	−0.437170	−	0.899379i	$-0.644019\pi$
$857$	−25.5959	−0.874340	−0.437170	+	0.899379i	$0.644019\pi$
$858$	−2.69694	−0.0920720
$859$	−40.1918	−1.37133	−0.685664	−	0.727918i	$-0.740488\pi$
$859$	−40.1918	−1.37133	−0.685664	+	0.727918i	$0.740488\pi$
$860$	0	0
$861$	−0.550510	−0.0187613
$862$	−60.0000	−2.04361
$863$	42.4949	1.44654	0.723272	−	0.690564i	$-0.242637\pi$
$863$	42.4949	1.44654	0.723272	+	0.690564i	$0.242637\pi$
$864$	0	0
$865$	0	0
$866$	−41.1464	−1.39821
$867$	16.6969	0.567058
$868$	39.5959	1.34397
$869$	9.79796	0.332373
$870$	0	0
$871$	3.14643	0.106613
$872$	−80.5857	−2.72898
$873$	−12.8990	−0.436564
$874$	1.10102	0.0372426
$875$	0	0
$876$	37.3939	1.26342
$877$	−38.4949	−1.29988	−0.649940	−	0.759985i	$-0.725206\pi$
$877$	−38.4949	−1.29988	−0.649940	+	0.759985i	$0.725206\pi$
$878$	17.8888	0.603717
$879$	3.24745	0.109534
$880$	0	0
$881$	3.55051	0.119620	0.0598099	−	0.998210i	$-0.480951\pi$
$881$	3.55051	0.119620	0.0598099	+	0.998210i	$0.480951\pi$
$882$	14.6969	0.494872
$883$	42.4495	1.42854	0.714270	−	0.699871i	$-0.246759\pi$
$883$	42.4495	1.42854	0.714270	+	0.699871i	$0.246759\pi$
$884$	0.989795	0.0332904
$885$	0	0
$886$	−38.0908	−1.27969
$887$	−42.4949	−1.42684	−0.713420	−	0.700737i	$-0.752855\pi$
$887$	−42.4949	−1.42684	−0.713420	+	0.700737i	$0.752855\pi$
$888$	28.8990	0.969786
$889$	6.44949	0.216309
$890$	0	0
$891$	−2.44949	−0.0820610
$892$	70.3837	2.35662
$893$	1.59592	0.0534054
$894$	32.6969	1.09355
$895$	0	0
$896$	19.5959	0.654654
$897$	−0.449490	−0.0150080
$898$	−7.95459	−0.265448
$899$	−43.0454	−1.43564
$900$	0	0
$901$	−3.00000	−0.0999445
$902$	3.30306	0.109980
$903$	2.00000	0.0665558
$904$	−80.0908	−2.66378
$905$	0	0
$906$	34.2929	1.13930
$907$	−19.2020	−0.637593	−0.318797	−	0.947823i	$-0.603279\pi$
$907$	−19.2020	−0.637593	−0.318797	+	0.947823i	$0.603279\pi$
$908$	−8.80816	−0.292309
$909$	−17.4495	−0.578763
$910$	0	0
$911$	−33.7980	−1.11978	−0.559888	−	0.828568i	$-0.689156\pi$
$911$	−33.7980	−1.11978	−0.559888	+	0.828568i	$0.689156\pi$
$912$	1.79796	0.0595364
$913$	−22.6515	−0.749656
$914$	24.2474	0.802034
$915$	0	0
$916$	81.9796	2.70868
$917$	19.5959	0.647114
$918$	1.34847	0.0445061
$919$	13.3939	0.441823	0.220912	−	0.975294i	$-0.429097\pi$
$919$	13.3939	0.441823	0.220912	+	0.975294i	$0.429097\pi$
$920$	0	0
$921$	−17.3485	−0.571651
$922$	−74.6969	−2.46001
$923$	4.65153	0.153107
$924$	9.79796	0.322329
$925$	0	0
$926$	70.8990	2.32989
$927$	−16.6969	−0.548399
$928$	0	0
$929$	−40.8434	−1.34003	−0.670014	−	0.742349i	$-0.733712\pi$
$929$	−40.8434	−1.34003	−0.670014	+	0.742349i	$0.733712\pi$
$930$	0	0
$931$	2.69694	0.0883886
$932$	−8.80816	−0.288521
$933$	−12.0000	−0.392862
$934$	70.6515	2.31179
$935$	0	0
$936$	−2.20204	−0.0719760
$937$	−0.898979	−0.0293684	−0.0146842	−	0.999892i	$-0.504674\pi$
$937$	−0.898979	−0.0293684	−0.0146842	+	0.999892i	$0.504674\pi$
$938$	−17.1464	−0.559851
$939$	−9.69694	−0.316448
$940$	0	0
$941$	−59.1464	−1.92812	−0.964059	−	0.265687i	$-0.914401\pi$
$941$	−59.1464	−1.92812	−0.964059	+	0.265687i	$0.914401\pi$
$942$	50.4495	1.64373
$943$	0.550510	0.0179271
$944$	−17.3939	−0.566122
$945$	0	0
$946$	−12.0000	−0.390154
$947$	−48.4949	−1.57587	−0.787936	−	0.615757i	$-0.788850\pi$
$947$	−48.4949	−1.57587	−0.787936	+	0.615757i	$0.788850\pi$
$948$	16.0000	0.519656
$949$	−4.20204	−0.136404
$950$	0	0
$951$	−18.2474	−0.591714
$952$	−2.69694	−0.0874083
$953$	−49.5959	−1.60657	−0.803285	−	0.595595i	$-0.796916\pi$
$953$	−49.5959	−1.60657	−0.803285	+	0.595595i	$0.796916\pi$
$954$	13.3485	0.432173
$955$	0	0
$956$	45.7980	1.48121
$957$	−10.6515	−0.344315
$958$	51.3031	1.65753
$959$	7.10102	0.229304
$960$	0	0
$961$	66.9898	2.16096
$962$	−6.49490	−0.209404
$963$	0.550510	0.0177399
$964$	−58.6061	−1.88758
$965$	0	0
$966$	2.44949	0.0788110
$967$	−9.95459	−0.320118	−0.160059	−	0.987107i	$-0.551168\pi$
$967$	−9.95459	−0.320118	−0.160059	+	0.987107i	$0.551168\pi$
$968$	24.4949	0.787296
$969$	0.247449	0.00794920
$970$	0	0
$971$	−46.2929	−1.48561	−0.742804	−	0.669509i	$-0.766505\pi$
$971$	−46.2929	−1.48561	−0.742804	+	0.669509i	$0.766505\pi$
$972$	−4.00000	−0.128300
$973$	14.7980	0.474401
$974$	85.5959	2.74267
$975$	0	0
$976$	61.3939	1.96517
$977$	−34.3485	−1.09890	−0.549452	−	0.835525i	$-0.685164\pi$
$977$	−34.3485	−1.09890	−0.549452	+	0.835525i	$0.685164\pi$
$978$	24.4949	0.783260
$979$	17.3939	0.555911
$980$	0	0
$981$	16.4495	0.525192
$982$	−42.7423	−1.36396
$983$	−39.7423	−1.26758	−0.633792	−	0.773504i	$-0.718502\pi$
$983$	−39.7423	−1.26758	−0.633792	+	0.773504i	$0.718502\pi$
$984$	2.69694	0.0859753
$985$	0	0
$986$	5.86378	0.186741
$987$	3.55051	0.113014
$988$	−0.808164	−0.0257111
$989$	−2.00000	−0.0635963
$990$	0	0
$991$	−59.7878	−1.89922	−0.949610	−	0.313433i	$-0.898521\pi$
$991$	−59.7878	−1.89922	−0.949610	+	0.313433i	$0.898521\pi$
$992$	0	0
$993$	14.5959	0.463187
$994$	−25.3485	−0.804005
$995$	0	0
$996$	−36.9898	−1.17207
$997$	48.0908	1.52305	0.761526	−	0.648135i	$-0.224451\pi$
$997$	48.0908	1.52305	0.761526	+	0.648135i	$0.224451\pi$
$998$	2.44949	0.0775372
$999$	−5.89898	−0.186635

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1725.2.a.y.1.1		2
3.2	odd	2		5175.2.a.bl.1.2		2
5.2	odd	4		1725.2.b.m.1174.2		4
5.3	odd	4		1725.2.b.m.1174.3		4
5.4	even	2		345.2.a.i.1.2	✓	2
15.14	odd	2		1035.2.a.k.1.1		2
20.19	odd	2		5520.2.a.bi.1.2		2
115.114	odd	2		7935.2.a.t.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
345.2.a.i.1.2	✓	2	5.4	even	2
1035.2.a.k.1.1		2	15.14	odd	2
1725.2.a.y.1.1		2	1.1	even	1	trivial
1725.2.b.m.1174.2		4	5.2	odd	4
1725.2.b.m.1174.3		4	5.3	odd	4
5175.2.a.bl.1.2		2	3.2	odd	2
5520.2.a.bi.1.2		2	20.19	odd	2
7935.2.a.t.1.2		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)

\(f(q)\)	\(=\)	\(q-2.44949 q^{2} -1.00000 q^{3} +4.00000 q^{4} +2.44949 q^{6} +1.00000 q^{7} -4.89898 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.44949 q^{2} -1.00000 q^{3} +4.00000 q^{4} +2.44949 q^{6} +1.00000 q^{7} -4.89898 q^{8} +1.00000 q^{9} -2.44949 q^{11} -4.00000 q^{12} +0.449490 q^{13} -2.44949 q^{14} +4.00000 q^{16} +0.550510 q^{17} -2.44949 q^{18} -0.449490 q^{19} -1.00000 q^{21} +6.00000 q^{22} +1.00000 q^{23} +4.89898 q^{24} -1.10102 q^{26} -1.00000 q^{27} +4.00000 q^{28} -4.34847 q^{29} +9.89898 q^{31} +2.44949 q^{33} -1.34847 q^{34} +4.00000 q^{36} +5.89898 q^{37} +1.10102 q^{38} -0.449490 q^{39} +0.550510 q^{41} +2.44949 q^{42} -2.00000 q^{43} -9.79796 q^{44} -2.44949 q^{46} -3.55051 q^{47} -4.00000 q^{48} -6.00000 q^{49} -0.550510 q^{51} +1.79796 q^{52} -5.44949 q^{53} +2.44949 q^{54} -4.89898 q^{56} +0.449490 q^{57} +10.6515 q^{58} -4.34847 q^{59} +15.3485 q^{61} -24.2474 q^{62} +1.00000 q^{63} -8.00000 q^{64} -6.00000 q^{66} +7.00000 q^{67} +2.20204 q^{68} -1.00000 q^{69} +10.3485 q^{71} -4.89898 q^{72} -9.34847 q^{73} -14.4495 q^{74} -1.79796 q^{76} -2.44949 q^{77} +1.10102 q^{78} -4.00000 q^{79} +1.00000 q^{81} -1.34847 q^{82} +9.24745 q^{83} -4.00000 q^{84} +4.89898 q^{86} +4.34847 q^{87} +12.0000 q^{88} -7.10102 q^{89} +0.449490 q^{91} +4.00000 q^{92} -9.89898 q^{93} +8.69694 q^{94} -12.8990 q^{97} +14.6969 q^{98} -2.44949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 8 q^{4} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 8 * q^4 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 8 q^{4} + 2 q^{7} + 2 q^{9} - 8 q^{12} - 4 q^{13} + 8 q^{16} + 6 q^{17} + 4 q^{19} - 2 q^{21} + 12 q^{22} + 2 q^{23} - 12 q^{26} - 2 q^{27} + 8 q^{28} + 6 q^{29} + 10 q^{31} + 12 q^{34} + 8 q^{36} + 2 q^{37} + 12 q^{38} + 4 q^{39} + 6 q^{41} - 4 q^{43} - 12 q^{47} - 8 q^{48} - 12 q^{49} - 6 q^{51} - 16 q^{52} - 6 q^{53} - 4 q^{57} + 36 q^{58} + 6 q^{59} + 16 q^{61} - 24 q^{62} + 2 q^{63} - 16 q^{64} - 12 q^{66} + 14 q^{67} + 24 q^{68} - 2 q^{69} + 6 q^{71} - 4 q^{73} - 24 q^{74} + 16 q^{76} + 12 q^{78} - 8 q^{79} + 2 q^{81} + 12 q^{82} - 6 q^{83} - 8 q^{84} - 6 q^{87} + 24 q^{88} - 24 q^{89} - 4 q^{91} + 8 q^{92} - 10 q^{93} - 12 q^{94} - 16 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 + 8 * q^4 + 2 * q^7 + 2 * q^9 - 8 * q^12 - 4 * q^13 + 8 * q^16 + 6 * q^17 + 4 * q^19 - 2 * q^21 + 12 * q^22 + 2 * q^23 - 12 * q^26 - 2 * q^27 + 8 * q^28 + 6 * q^29 + 10 * q^31 + 12 * q^34 + 8 * q^36 + 2 * q^37 + 12 * q^38 + 4 * q^39 + 6 * q^41 - 4 * q^43 - 12 * q^47 - 8 * q^48 - 12 * q^49 - 6 * q^51 - 16 * q^52 - 6 * q^53 - 4 * q^57 + 36 * q^58 + 6 * q^59 + 16 * q^61 - 24 * q^62 + 2 * q^63 - 16 * q^64 - 12 * q^66 + 14 * q^67 + 24 * q^68 - 2 * q^69 + 6 * q^71 - 4 * q^73 - 24 * q^74 + 16 * q^76 + 12 * q^78 - 8 * q^79 + 2 * q^81 + 12 * q^82 - 6 * q^83 - 8 * q^84 - 6 * q^87 + 24 * q^88 - 24 * q^89 - 4 * q^91 + 8 * q^92 - 10 * q^93 - 12 * q^94 - 16 * q^97

Introduction

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