LMFDB - Embedded newform 2550.2.a.bi.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2550,2,Mod(1,2550)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2550, 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("2550.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2550.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:	$20.3618525154$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 510)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	2550.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.44949 q^{7} -1.00000 q^{8} +1.00000 q^{9} -4.89898 q^{11} +1.00000 q^{12} +6.00000 q^{13} +2.44949 q^{14} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +6.89898 q^{19} -2.44949 q^{21} +4.89898 q^{22} +6.44949 q^{23} -1.00000 q^{24} -6.00000 q^{26} +1.00000 q^{27} -2.44949 q^{28} -9.34847 q^{29} -6.44949 q^{31} -1.00000 q^{32} -4.89898 q^{33} -1.00000 q^{34} +1.00000 q^{36} -0.449490 q^{37} -6.89898 q^{38} +6.00000 q^{39} -1.10102 q^{41} +2.44949 q^{42} -2.89898 q^{43} -4.89898 q^{44} -6.44949 q^{46} +4.89898 q^{47} +1.00000 q^{48} -1.00000 q^{49} +1.00000 q^{51} +6.00000 q^{52} -1.10102 q^{53} -1.00000 q^{54} +2.44949 q^{56} +6.89898 q^{57} +9.34847 q^{58} +5.79796 q^{59} +13.3485 q^{61} +6.44949 q^{62} -2.44949 q^{63} +1.00000 q^{64} +4.89898 q^{66} +4.00000 q^{67} +1.00000 q^{68} +6.44949 q^{69} +2.44949 q^{71} -1.00000 q^{72} +14.8990 q^{73} +0.449490 q^{74} +6.89898 q^{76} +12.0000 q^{77} -6.00000 q^{78} -1.55051 q^{79} +1.00000 q^{81} +1.10102 q^{82} +2.89898 q^{83} -2.44949 q^{84} +2.89898 q^{86} -9.34847 q^{87} +4.89898 q^{88} -1.79796 q^{89} -14.6969 q^{91} +6.44949 q^{92} -6.44949 q^{93} -4.89898 q^{94} -1.00000 q^{96} -3.79796 q^{97} +1.00000 q^{98} -4.89898 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{12} + 12 q^{13} + 2 q^{16} + 2 q^{17} - 2 q^{18} + 4 q^{19} + 8 q^{23} - 2 q^{24} - 12 q^{26} + 2 q^{27} - 4 q^{29} - 8 q^{31} - 2 q^{32}+ \cdots + 2 q^{98}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^8 + 2 * q^9 + 2 * q^12 + 12 * q^13 + 2 * q^16 + 2 * q^17 - 2 * q^18 + 4 * q^19 + 8 * q^23 - 2 * q^24 - 12 * q^26 + 2 * q^27 - 4 * q^29 - 8 * q^31 - 2 * q^32 - 2 * q^34 + 2 * q^36 + 4 * q^37 - 4 * q^38 + 12 * q^39 - 12 * q^41 + 4 * q^43 - 8 * q^46 + 2 * q^48 - 2 * q^49 + 2 * q^51 + 12 * q^52 - 12 * q^53 - 2 * q^54 + 4 * q^57 + 4 * q^58 - 8 * q^59 + 12 * q^61 + 8 * q^62 + 2 * q^64 + 8 * q^67 + 2 * q^68 + 8 * q^69 - 2 * q^72 + 20 * q^73 - 4 * q^74 + 4 * q^76 + 24 * q^77 - 12 * q^78 - 8 * q^79 + 2 * q^81 + 12 * q^82 - 4 * q^83 - 4 * q^86 - 4 * q^87 + 16 * q^89 + 8 * q^92 - 8 * q^93 - 2 * q^96 + 12 * q^97 + 2 * q^98

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	−2.44949	−0.925820	−0.462910	−	0.886405i	$-0.653195\pi$
$7$	−2.44949	−0.925820	−0.462910	+	0.886405i	$0.653195\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.89898	−1.47710	−0.738549	−	0.674200i	$-0.764489\pi$
$11$	−4.89898	−1.47710	−0.738549	+	0.674200i	$0.764489\pi$
$12$	1.00000	0.288675
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	2.44949	0.654654
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	−1.00000	−0.235702
$19$	6.89898	1.58273	0.791367	−	0.611341i	$-0.209370\pi$
$19$	6.89898	1.58273	0.791367	+	0.611341i	$0.209370\pi$
$20$	0	0
$21$	−2.44949	−0.534522
$22$	4.89898	1.04447
$23$	6.44949	1.34481	0.672406	−	0.740183i	$-0.265261\pi$
$23$	6.44949	1.34481	0.672406	+	0.740183i	$0.265261\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−6.00000	−1.17670
$27$	1.00000	0.192450
$28$	−2.44949	−0.462910
$29$	−9.34847	−1.73597	−0.867984	−	0.496593i	$-0.834584\pi$
$29$	−9.34847	−1.73597	−0.867984	+	0.496593i	$0.834584\pi$
$30$	0	0
$31$	−6.44949	−1.15836	−0.579181	−	0.815199i	$-0.696628\pi$
$31$	−6.44949	−1.15836	−0.579181	+	0.815199i	$0.696628\pi$
$32$	−1.00000	−0.176777
$33$	−4.89898	−0.852803
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	−0.449490	−0.0738957	−0.0369478	−	0.999317i	$-0.511764\pi$
$37$	−0.449490	−0.0738957	−0.0369478	+	0.999317i	$0.511764\pi$
$38$	−6.89898	−1.11916
$39$	6.00000	0.960769
$40$	0	0
$41$	−1.10102	−0.171951	−0.0859753	−	0.996297i	$-0.527401\pi$
$41$	−1.10102	−0.171951	−0.0859753	+	0.996297i	$0.527401\pi$
$42$	2.44949	0.377964
$43$	−2.89898	−0.442090	−0.221045	−	0.975264i	$-0.570947\pi$
$43$	−2.89898	−0.442090	−0.221045	+	0.975264i	$0.570947\pi$
$44$	−4.89898	−0.738549
$45$	0	0
$46$	−6.44949	−0.950925
$47$	4.89898	0.714590	0.357295	−	0.933992i	$-0.383699\pi$
$47$	4.89898	0.714590	0.357295	+	0.933992i	$0.383699\pi$
$48$	1.00000	0.144338
$49$	−1.00000	−0.142857
$50$	0	0
$51$	1.00000	0.140028
$52$	6.00000	0.832050
$53$	−1.10102	−0.151237	−0.0756184	−	0.997137i	$-0.524093\pi$
$53$	−1.10102	−0.151237	−0.0756184	+	0.997137i	$0.524093\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	2.44949	0.327327
$57$	6.89898	0.913792
$58$	9.34847	1.22751
$59$	5.79796	0.754830	0.377415	−	0.926044i	$-0.376813\pi$
$59$	5.79796	0.754830	0.377415	+	0.926044i	$0.376813\pi$
$60$	0	0
$61$	13.3485	1.70910	0.854548	−	0.519372i	$-0.173834\pi$
$61$	13.3485	1.70910	0.854548	+	0.519372i	$0.173834\pi$
$62$	6.44949	0.819086
$63$	−2.44949	−0.308607
$64$	1.00000	0.125000
$65$	0	0
$66$	4.89898	0.603023
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	1.00000	0.121268
$69$	6.44949	0.776427
$70$	0	0
$71$	2.44949	0.290701	0.145350	−	0.989380i	$-0.453569\pi$
$71$	2.44949	0.290701	0.145350	+	0.989380i	$0.453569\pi$
$72$	−1.00000	−0.117851
$73$	14.8990	1.74379	0.871897	−	0.489690i	$-0.162890\pi$
$73$	14.8990	1.74379	0.871897	+	0.489690i	$0.162890\pi$
$74$	0.449490	0.0522521
$75$	0	0
$76$	6.89898	0.791367
$77$	12.0000	1.36753
$78$	−6.00000	−0.679366
$79$	−1.55051	−0.174446	−0.0872230	−	0.996189i	$-0.527799\pi$
$79$	−1.55051	−0.174446	−0.0872230	+	0.996189i	$0.527799\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	1.10102	0.121587
$83$	2.89898	0.318204	0.159102	−	0.987262i	$-0.449140\pi$
$83$	2.89898	0.318204	0.159102	+	0.987262i	$0.449140\pi$
$84$	−2.44949	−0.267261
$85$	0	0
$86$	2.89898	0.312605
$87$	−9.34847	−1.00226
$88$	4.89898	0.522233
$89$	−1.79796	−0.190583	−0.0952916	−	0.995449i	$-0.530378\pi$
$89$	−1.79796	−0.190583	−0.0952916	+	0.995449i	$0.530378\pi$
$90$	0	0
$91$	−14.6969	−1.54066
$92$	6.44949	0.672406
$93$	−6.44949	−0.668781
$94$	−4.89898	−0.505291
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−3.79796	−0.385624	−0.192812	−	0.981236i	$-0.561761\pi$
$97$	−3.79796	−0.385624	−0.192812	+	0.981236i	$0.561761\pi$
$98$	1.00000	0.101015
$99$	−4.89898	−0.492366
$100$	0	0
$101$	3.79796	0.377911	0.188956	−	0.981986i	$-0.439490\pi$
$101$	3.79796	0.377911	0.188956	+	0.981986i	$0.439490\pi$
$102$	−1.00000	−0.0990148
$103$	16.8990	1.66511	0.832553	−	0.553945i	$-0.186878\pi$
$103$	16.8990	1.66511	0.832553	+	0.553945i	$0.186878\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	1.10102	0.106941
$107$	3.10102	0.299787	0.149893	−	0.988702i	$-0.452107\pi$
$107$	3.10102	0.299787	0.149893	+	0.988702i	$0.452107\pi$
$108$	1.00000	0.0962250
$109$	18.2474	1.74779	0.873894	−	0.486116i	$-0.161587\pi$
$109$	18.2474	1.74779	0.873894	+	0.486116i	$0.161587\pi$
$110$	0	0
$111$	−0.449490	−0.0426637
$112$	−2.44949	−0.231455
$113$	16.6969	1.57072	0.785358	−	0.619042i	$-0.212479\pi$
$113$	16.6969	1.57072	0.785358	+	0.619042i	$0.212479\pi$
$114$	−6.89898	−0.646149
$115$	0	0
$116$	−9.34847	−0.867984
$117$	6.00000	0.554700
$118$	−5.79796	−0.533745
$119$	−2.44949	−0.224544
$120$	0	0
$121$	13.0000	1.18182
$122$	−13.3485	−1.20851
$123$	−1.10102	−0.0992757
$124$	−6.44949	−0.579181
$125$	0	0
$126$	2.44949	0.218218
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	−1.00000	−0.0883883
$129$	−2.89898	−0.255241
$130$	0	0
$131$	4.89898	0.428026	0.214013	−	0.976831i	$-0.431347\pi$
$131$	4.89898	0.428026	0.214013	+	0.976831i	$0.431347\pi$
$132$	−4.89898	−0.426401
$133$	−16.8990	−1.46533
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	12.0000	1.02523	0.512615	−	0.858619i	$-0.328677\pi$
$137$	12.0000	1.02523	0.512615	+	0.858619i	$0.328677\pi$
$138$	−6.44949	−0.549017
$139$	−16.8990	−1.43335	−0.716676	−	0.697406i	$-0.754338\pi$
$139$	−16.8990	−1.43335	−0.716676	+	0.697406i	$0.754338\pi$
$140$	0	0
$141$	4.89898	0.412568
$142$	−2.44949	−0.205557
$143$	−29.3939	−2.45804
$144$	1.00000	0.0833333
$145$	0	0
$146$	−14.8990	−1.23305
$147$	−1.00000	−0.0824786
$148$	−0.449490	−0.0369478
$149$	12.6969	1.04017	0.520087	−	0.854113i	$-0.325900\pi$
$149$	12.6969	1.04017	0.520087	+	0.854113i	$0.325900\pi$
$150$	0	0
$151$	13.7980	1.12286	0.561431	−	0.827524i	$-0.310251\pi$
$151$	13.7980	1.12286	0.561431	+	0.827524i	$0.310251\pi$
$152$	−6.89898	−0.559581
$153$	1.00000	0.0808452
$154$	−12.0000	−0.966988
$155$	0	0
$156$	6.00000	0.480384
$157$	16.6969	1.33256	0.666280	−	0.745701i	$-0.267885\pi$
$157$	16.6969	1.33256	0.666280	+	0.745701i	$0.267885\pi$
$158$	1.55051	0.123352
$159$	−1.10102	−0.0873166
$160$	0	0
$161$	−15.7980	−1.24505
$162$	−1.00000	−0.0785674
$163$	5.79796	0.454131	0.227066	−	0.973879i	$-0.427087\pi$
$163$	5.79796	0.454131	0.227066	+	0.973879i	$0.427087\pi$
$164$	−1.10102	−0.0859753
$165$	0	0
$166$	−2.89898	−0.225004
$167$	4.65153	0.359946	0.179973	−	0.983672i	$-0.442399\pi$
$167$	4.65153	0.359946	0.179973	+	0.983672i	$0.442399\pi$
$168$	2.44949	0.188982
$169$	23.0000	1.76923
$170$	0	0
$171$	6.89898	0.527578
$172$	−2.89898	−0.221045
$173$	−13.3485	−1.01487	−0.507433	−	0.861691i	$-0.669405\pi$
$173$	−13.3485	−1.01487	−0.507433	+	0.861691i	$0.669405\pi$
$174$	9.34847	0.708706
$175$	0	0
$176$	−4.89898	−0.369274
$177$	5.79796	0.435801
$178$	1.79796	0.134763
$179$	−20.6969	−1.54696	−0.773481	−	0.633820i	$-0.781486\pi$
$179$	−20.6969	−1.54696	−0.773481	+	0.633820i	$0.781486\pi$
$180$	0	0
$181$	4.44949	0.330728	0.165364	−	0.986233i	$-0.447120\pi$
$181$	4.44949	0.330728	0.165364	+	0.986233i	$0.447120\pi$
$182$	14.6969	1.08941
$183$	13.3485	0.986747
$184$	−6.44949	−0.475463
$185$	0	0
$186$	6.44949	0.472900
$187$	−4.89898	−0.358249
$188$	4.89898	0.357295
$189$	−2.44949	−0.178174
$190$	0	0
$191$	−16.8990	−1.22277	−0.611384	−	0.791334i	$-0.709387\pi$
$191$	−16.8990	−1.22277	−0.611384	+	0.791334i	$0.709387\pi$
$192$	1.00000	0.0721688
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	3.79796	0.272678
$195$	0	0
$196$	−1.00000	−0.0714286
$197$	19.1464	1.36413	0.682063	−	0.731293i	$-0.261083\pi$
$197$	19.1464	1.36413	0.682063	+	0.731293i	$0.261083\pi$
$198$	4.89898	0.348155
$199$	−18.0454	−1.27921	−0.639603	−	0.768706i	$-0.720901\pi$
$199$	−18.0454	−1.27921	−0.639603	+	0.768706i	$0.720901\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	−3.79796	−0.267223
$203$	22.8990	1.60719
$204$	1.00000	0.0700140
$205$	0	0
$206$	−16.8990	−1.17741
$207$	6.44949	0.448271
$208$	6.00000	0.416025
$209$	−33.7980	−2.33785
$210$	0	0
$211$	−3.10102	−0.213483	−0.106742	−	0.994287i	$-0.534042\pi$
$211$	−3.10102	−0.213483	−0.106742	+	0.994287i	$0.534042\pi$
$212$	−1.10102	−0.0756184
$213$	2.44949	0.167836
$214$	−3.10102	−0.211981
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	15.7980	1.07244
$218$	−18.2474	−1.23587
$219$	14.8990	1.00678
$220$	0	0
$221$	6.00000	0.403604
$222$	0.449490	0.0301678
$223$	4.89898	0.328060	0.164030	−	0.986455i	$-0.447551\pi$
$223$	4.89898	0.328060	0.164030	+	0.986455i	$0.447551\pi$
$224$	2.44949	0.163663
$225$	0	0
$226$	−16.6969	−1.11066
$227$	−19.5959	−1.30063	−0.650313	−	0.759666i	$-0.725362\pi$
$227$	−19.5959	−1.30063	−0.650313	+	0.759666i	$0.725362\pi$
$228$	6.89898	0.456896
$229$	18.0000	1.18947	0.594737	−	0.803921i	$-0.297256\pi$
$229$	18.0000	1.18947	0.594737	+	0.803921i	$0.297256\pi$
$230$	0	0
$231$	12.0000	0.789542
$232$	9.34847	0.613757
$233$	−1.10102	−0.0721303	−0.0360651	−	0.999349i	$-0.511482\pi$
$233$	−1.10102	−0.0721303	−0.0360651	+	0.999349i	$0.511482\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	5.79796	0.377415
$237$	−1.55051	−0.100716
$238$	2.44949	0.158777
$239$	8.89898	0.575627	0.287814	−	0.957686i	$-0.407072\pi$
$239$	8.89898	0.575627	0.287814	+	0.957686i	$0.407072\pi$
$240$	0	0
$241$	−6.89898	−0.444402	−0.222201	−	0.975001i	$-0.571324\pi$
$241$	−6.89898	−0.444402	−0.222201	+	0.975001i	$0.571324\pi$
$242$	−13.0000	−0.835672
$243$	1.00000	0.0641500
$244$	13.3485	0.854548
$245$	0	0
$246$	1.10102	0.0701985
$247$	41.3939	2.63383
$248$	6.44949	0.409543
$249$	2.89898	0.183715
$250$	0	0
$251$	−20.6969	−1.30638	−0.653190	−	0.757194i	$-0.726570\pi$
$251$	−20.6969	−1.30638	−0.653190	+	0.757194i	$0.726570\pi$
$252$	−2.44949	−0.154303
$253$	−31.5959	−1.98642
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−20.0000	−1.24757	−0.623783	−	0.781598i	$-0.714405\pi$
$257$	−20.0000	−1.24757	−0.623783	+	0.781598i	$0.714405\pi$
$258$	2.89898	0.180483
$259$	1.10102	0.0684141
$260$	0	0
$261$	−9.34847	−0.578656
$262$	−4.89898	−0.302660
$263$	−26.6969	−1.64620	−0.823102	−	0.567894i	$-0.807758\pi$
$263$	−26.6969	−1.64620	−0.823102	+	0.567894i	$0.807758\pi$
$264$	4.89898	0.301511
$265$	0	0
$266$	16.8990	1.03614
$267$	−1.79796	−0.110033
$268$	4.00000	0.244339
$269$	23.5505	1.43590	0.717950	−	0.696095i	$-0.245081\pi$
$269$	23.5505	1.43590	0.717950	+	0.696095i	$0.245081\pi$
$270$	0	0
$271$	−7.59592	−0.461419	−0.230710	−	0.973023i	$-0.574105\pi$
$271$	−7.59592	−0.461419	−0.230710	+	0.973023i	$0.574105\pi$
$272$	1.00000	0.0606339
$273$	−14.6969	−0.889499
$274$	−12.0000	−0.724947
$275$	0	0
$276$	6.44949	0.388214
$277$	−28.4495	−1.70936	−0.854682	−	0.519152i	$-0.826248\pi$
$277$	−28.4495	−1.70936	−0.854682	+	0.519152i	$0.826248\pi$
$278$	16.8990	1.01353
$279$	−6.44949	−0.386121
$280$	0	0
$281$	9.79796	0.584497	0.292249	−	0.956342i	$-0.405597\pi$
$281$	9.79796	0.584497	0.292249	+	0.956342i	$0.405597\pi$
$282$	−4.89898	−0.291730
$283$	−6.69694	−0.398092	−0.199046	−	0.979990i	$-0.563784\pi$
$283$	−6.69694	−0.398092	−0.199046	+	0.979990i	$0.563784\pi$
$284$	2.44949	0.145350
$285$	0	0
$286$	29.3939	1.73810
$287$	2.69694	0.159195
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	0	0
$291$	−3.79796	−0.222640
$292$	14.8990	0.871897
$293$	−10.0000	−0.584206	−0.292103	−	0.956387i	$-0.594355\pi$
$293$	−10.0000	−0.584206	−0.292103	+	0.956387i	$0.594355\pi$
$294$	1.00000	0.0583212
$295$	0	0
$296$	0.449490	0.0261261
$297$	−4.89898	−0.284268
$298$	−12.6969	−0.735514
$299$	38.6969	2.23790
$300$	0	0
$301$	7.10102	0.409296
$302$	−13.7980	−0.793983
$303$	3.79796	0.218187
$304$	6.89898	0.395684
$305$	0	0
$306$	−1.00000	−0.0571662
$307$	10.8990	0.622038	0.311019	−	0.950404i	$-0.399330\pi$
$307$	10.8990	0.622038	0.311019	+	0.950404i	$0.399330\pi$
$308$	12.0000	0.683763
$309$	16.8990	0.961349
$310$	0	0
$311$	−23.3485	−1.32397	−0.661985	−	0.749517i	$-0.730286\pi$
$311$	−23.3485	−1.32397	−0.661985	+	0.749517i	$0.730286\pi$
$312$	−6.00000	−0.339683
$313$	2.00000	0.113047	0.0565233	−	0.998401i	$-0.481998\pi$
$313$	2.00000	0.113047	0.0565233	+	0.998401i	$0.481998\pi$
$314$	−16.6969	−0.942263
$315$	0	0
$316$	−1.55051	−0.0872230
$317$	−8.44949	−0.474571	−0.237285	−	0.971440i	$-0.576258\pi$
$317$	−8.44949	−0.474571	−0.237285	+	0.971440i	$0.576258\pi$
$318$	1.10102	0.0617422
$319$	45.7980	2.56419
$320$	0	0
$321$	3.10102	0.173082
$322$	15.7980	0.880386
$323$	6.89898	0.383869
$324$	1.00000	0.0555556
$325$	0	0
$326$	−5.79796	−0.321119
$327$	18.2474	1.00909
$328$	1.10102	0.0607937
$329$	−12.0000	−0.661581
$330$	0	0
$331$	33.3939	1.83549	0.917747	−	0.397166i	$-0.130006\pi$
$331$	33.3939	1.83549	0.917747	+	0.397166i	$0.130006\pi$
$332$	2.89898	0.159102
$333$	−0.449490	−0.0246319
$334$	−4.65153	−0.254520
$335$	0	0
$336$	−2.44949	−0.133631
$337$	2.89898	0.157917	0.0789587	−	0.996878i	$-0.474840\pi$
$337$	2.89898	0.157917	0.0789587	+	0.996878i	$0.474840\pi$
$338$	−23.0000	−1.25104
$339$	16.6969	0.906853
$340$	0	0
$341$	31.5959	1.71101
$342$	−6.89898	−0.373054
$343$	19.5959	1.05808
$344$	2.89898	0.156302
$345$	0	0
$346$	13.3485	0.717618
$347$	−13.7980	−0.740713	−0.370357	−	0.928890i	$-0.620765\pi$
$347$	−13.7980	−0.740713	−0.370357	+	0.928890i	$0.620765\pi$
$348$	−9.34847	−0.501131
$349$	7.30306	0.390924	0.195462	−	0.980711i	$-0.437379\pi$
$349$	7.30306	0.390924	0.195462	+	0.980711i	$0.437379\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	4.89898	0.261116
$353$	10.0000	0.532246	0.266123	−	0.963939i	$-0.414257\pi$
$353$	10.0000	0.532246	0.266123	+	0.963939i	$0.414257\pi$
$354$	−5.79796	−0.308158
$355$	0	0
$356$	−1.79796	−0.0952916
$357$	−2.44949	−0.129641
$358$	20.6969	1.09387
$359$	−6.20204	−0.327331	−0.163666	−	0.986516i	$-0.552332\pi$
$359$	−6.20204	−0.327331	−0.163666	+	0.986516i	$0.552332\pi$
$360$	0	0
$361$	28.5959	1.50505
$362$	−4.44949	−0.233860
$363$	13.0000	0.682323
$364$	−14.6969	−0.770329
$365$	0	0
$366$	−13.3485	−0.697736
$367$	−14.0454	−0.733164	−0.366582	−	0.930386i	$-0.619472\pi$
$367$	−14.0454	−0.733164	−0.366582	+	0.930386i	$0.619472\pi$
$368$	6.44949	0.336203
$369$	−1.10102	−0.0573168
$370$	0	0
$371$	2.69694	0.140018
$372$	−6.44949	−0.334390
$373$	−11.7980	−0.610875	−0.305438	−	0.952212i	$-0.598803\pi$
$373$	−11.7980	−0.610875	−0.305438	+	0.952212i	$0.598803\pi$
$374$	4.89898	0.253320
$375$	0	0
$376$	−4.89898	−0.252646
$377$	−56.0908	−2.88882
$378$	2.44949	0.125988
$379$	−10.2020	−0.524044	−0.262022	−	0.965062i	$-0.584389\pi$
$379$	−10.2020	−0.524044	−0.262022	+	0.965062i	$0.584389\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	16.8990	0.864627
$383$	16.8990	0.863498	0.431749	−	0.901994i	$-0.357897\pi$
$383$	16.8990	0.863498	0.431749	+	0.901994i	$0.357897\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	−2.89898	−0.147363
$388$	−3.79796	−0.192812
$389$	3.30306	0.167472	0.0837359	−	0.996488i	$-0.473315\pi$
$389$	3.30306	0.167472	0.0837359	+	0.996488i	$0.473315\pi$
$390$	0	0
$391$	6.44949	0.326165
$392$	1.00000	0.0505076
$393$	4.89898	0.247121
$394$	−19.1464	−0.964583
$395$	0	0
$396$	−4.89898	−0.246183
$397$	−36.0454	−1.80907	−0.904534	−	0.426402i	$-0.859781\pi$
$397$	−36.0454	−1.80907	−0.904534	+	0.426402i	$0.859781\pi$
$398$	18.0454	0.904535
$399$	−16.8990	−0.846007
$400$	0	0
$401$	−0.696938	−0.0348034	−0.0174017	−	0.999849i	$-0.505539\pi$
$401$	−0.696938	−0.0348034	−0.0174017	+	0.999849i	$0.505539\pi$
$402$	−4.00000	−0.199502
$403$	−38.6969	−1.92763
$404$	3.79796	0.188956
$405$	0	0
$406$	−22.8990	−1.13646
$407$	2.20204	0.109151
$408$	−1.00000	−0.0495074
$409$	15.5959	0.771169	0.385584	−	0.922673i	$-0.374000\pi$
$409$	15.5959	0.771169	0.385584	+	0.922673i	$0.374000\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	16.8990	0.832553
$413$	−14.2020	−0.698837
$414$	−6.44949	−0.316975
$415$	0	0
$416$	−6.00000	−0.294174
$417$	−16.8990	−0.827547
$418$	33.7980	1.65311
$419$	7.59592	0.371085	0.185542	−	0.982636i	$-0.440596\pi$
$419$	7.59592	0.371085	0.185542	+	0.982636i	$0.440596\pi$
$420$	0	0
$421$	−0.696938	−0.0339667	−0.0169834	−	0.999856i	$-0.505406\pi$
$421$	−0.696938	−0.0339667	−0.0169834	+	0.999856i	$0.505406\pi$
$422$	3.10102	0.150955
$423$	4.89898	0.238197
$424$	1.10102	0.0534703
$425$	0	0
$426$	−2.44949	−0.118678
$427$	−32.6969	−1.58232
$428$	3.10102	0.149893
$429$	−29.3939	−1.41915
$430$	0	0
$431$	25.1464	1.21126	0.605630	−	0.795746i	$-0.292921\pi$
$431$	25.1464	1.21126	0.605630	+	0.795746i	$0.292921\pi$
$432$	1.00000	0.0481125
$433$	−10.0000	−0.480569	−0.240285	−	0.970702i	$-0.577241\pi$
$433$	−10.0000	−0.480569	−0.240285	+	0.970702i	$0.577241\pi$
$434$	−15.7980	−0.758326
$435$	0	0
$436$	18.2474	0.873894
$437$	44.4949	2.12848
$438$	−14.8990	−0.711901
$439$	−27.3485	−1.30527	−0.652636	−	0.757672i	$-0.726337\pi$
$439$	−27.3485	−1.30527	−0.652636	+	0.757672i	$0.726337\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	−6.00000	−0.285391
$443$	−10.2020	−0.484714	−0.242357	−	0.970187i	$-0.577920\pi$
$443$	−10.2020	−0.484714	−0.242357	+	0.970187i	$0.577920\pi$
$444$	−0.449490	−0.0213318
$445$	0	0
$446$	−4.89898	−0.231973
$447$	12.6969	0.600545
$448$	−2.44949	−0.115728
$449$	−32.6969	−1.54306	−0.771532	−	0.636191i	$-0.780509\pi$
$449$	−32.6969	−1.54306	−0.771532	+	0.636191i	$0.780509\pi$
$450$	0	0
$451$	5.39388	0.253988
$452$	16.6969	0.785358
$453$	13.7980	0.648285
$454$	19.5959	0.919682
$455$	0	0
$456$	−6.89898	−0.323074
$457$	1.59592	0.0746539	0.0373269	−	0.999303i	$-0.488116\pi$
$457$	1.59592	0.0746539	0.0373269	+	0.999303i	$0.488116\pi$
$458$	−18.0000	−0.841085
$459$	1.00000	0.0466760
$460$	0	0
$461$	−21.5959	−1.00582	−0.502911	−	0.864338i	$-0.667738\pi$
$461$	−21.5959	−1.00582	−0.502911	+	0.864338i	$0.667738\pi$
$462$	−12.0000	−0.558291
$463$	−5.30306	−0.246454	−0.123227	−	0.992378i	$-0.539324\pi$
$463$	−5.30306	−0.246454	−0.123227	+	0.992378i	$0.539324\pi$
$464$	−9.34847	−0.433992
$465$	0	0
$466$	1.10102	0.0510038
$467$	−0.404082	−0.0186987	−0.00934934	−	0.999956i	$-0.502976\pi$
$467$	−0.404082	−0.0186987	−0.00934934	+	0.999956i	$0.502976\pi$
$468$	6.00000	0.277350
$469$	−9.79796	−0.452428
$470$	0	0
$471$	16.6969	0.769354
$472$	−5.79796	−0.266873
$473$	14.2020	0.653011
$474$	1.55051	0.0712173
$475$	0	0
$476$	−2.44949	−0.112272
$477$	−1.10102	−0.0504123
$478$	−8.89898	−0.407030
$479$	15.3485	0.701289	0.350645	−	0.936509i	$-0.385962\pi$
$479$	15.3485	0.701289	0.350645	+	0.936509i	$0.385962\pi$
$480$	0	0
$481$	−2.69694	−0.122970
$482$	6.89898	0.314240
$483$	−15.7980	−0.718832
$484$	13.0000	0.590909
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−15.3485	−0.695506	−0.347753	−	0.937586i	$-0.613055\pi$
$487$	−15.3485	−0.695506	−0.347753	+	0.937586i	$0.613055\pi$
$488$	−13.3485	−0.604257
$489$	5.79796	0.262193
$490$	0	0
$491$	−14.8990	−0.672382	−0.336191	−	0.941794i	$-0.609139\pi$
$491$	−14.8990	−0.672382	−0.336191	+	0.941794i	$0.609139\pi$
$492$	−1.10102	−0.0496378
$493$	−9.34847	−0.421034
$494$	−41.3939	−1.86240
$495$	0	0
$496$	−6.44949	−0.289591
$497$	−6.00000	−0.269137
$498$	−2.89898	−0.129906
$499$	0.898979	0.0402438	0.0201219	−	0.999798i	$-0.493595\pi$
$499$	0.898979	0.0402438	0.0201219	+	0.999798i	$0.493595\pi$
$500$	0	0
$501$	4.65153	0.207815
$502$	20.6969	0.923750
$503$	33.5505	1.49594	0.747972	−	0.663731i	$-0.231028\pi$
$503$	33.5505	1.49594	0.747972	+	0.663731i	$0.231028\pi$
$504$	2.44949	0.109109
$505$	0	0
$506$	31.5959	1.40461
$507$	23.0000	1.02147
$508$	−8.00000	−0.354943
$509$	−10.8990	−0.483089	−0.241544	−	0.970390i	$-0.577654\pi$
$509$	−10.8990	−0.483089	−0.241544	+	0.970390i	$0.577654\pi$
$510$	0	0
$511$	−36.4949	−1.61444
$512$	−1.00000	−0.0441942
$513$	6.89898	0.304597
$514$	20.0000	0.882162
$515$	0	0
$516$	−2.89898	−0.127620
$517$	−24.0000	−1.05552
$518$	−1.10102	−0.0483761
$519$	−13.3485	−0.585933
$520$	0	0
$521$	−16.6969	−0.731506	−0.365753	−	0.930712i	$-0.619189\pi$
$521$	−16.6969	−0.731506	−0.365753	+	0.930712i	$0.619189\pi$
$522$	9.34847	0.409171
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	4.89898	0.214013
$525$	0	0
$526$	26.6969	1.16404
$527$	−6.44949	−0.280944
$528$	−4.89898	−0.213201
$529$	18.5959	0.808518
$530$	0	0
$531$	5.79796	0.251610
$532$	−16.8990	−0.732664
$533$	−6.60612	−0.286143
$534$	1.79796	0.0778053
$535$	0	0
$536$	−4.00000	−0.172774
$537$	−20.6969	−0.893139
$538$	−23.5505	−1.01533
$539$	4.89898	0.211014
$540$	0	0
$541$	31.1464	1.33909	0.669545	−	0.742772i	$-0.266489\pi$
$541$	31.1464	1.33909	0.669545	+	0.742772i	$0.266489\pi$
$542$	7.59592	0.326273
$543$	4.44949	0.190946
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	14.6969	0.628971
$547$	18.6969	0.799423	0.399712	−	0.916641i	$-0.369110\pi$
$547$	18.6969	0.799423	0.399712	+	0.916641i	$0.369110\pi$
$548$	12.0000	0.512615
$549$	13.3485	0.569699
$550$	0	0
$551$	−64.4949	−2.74758
$552$	−6.44949	−0.274509
$553$	3.79796	0.161506
$554$	28.4495	1.20870
$555$	0	0
$556$	−16.8990	−0.716676
$557$	−2.89898	−0.122834	−0.0614169	−	0.998112i	$-0.519562\pi$
$557$	−2.89898	−0.122834	−0.0614169	+	0.998112i	$0.519562\pi$
$558$	6.44949	0.273029
$559$	−17.3939	−0.735683
$560$	0	0
$561$	−4.89898	−0.206835
$562$	−9.79796	−0.413302
$563$	6.89898	0.290757	0.145379	−	0.989376i	$-0.453560\pi$
$563$	6.89898	0.290757	0.145379	+	0.989376i	$0.453560\pi$
$564$	4.89898	0.206284
$565$	0	0
$566$	6.69694	0.281493
$567$	−2.44949	−0.102869
$568$	−2.44949	−0.102778
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	−21.3939	−0.895306	−0.447653	−	0.894207i	$-0.647740\pi$
$571$	−21.3939	−0.895306	−0.447653	+	0.894207i	$0.647740\pi$
$572$	−29.3939	−1.22902
$573$	−16.8990	−0.705965
$574$	−2.69694	−0.112568
$575$	0	0
$576$	1.00000	0.0416667
$577$	−10.2020	−0.424717	−0.212358	−	0.977192i	$-0.568114\pi$
$577$	−10.2020	−0.424717	−0.212358	+	0.977192i	$0.568114\pi$
$578$	−1.00000	−0.0415945
$579$	14.0000	0.581820
$580$	0	0
$581$	−7.10102	−0.294600
$582$	3.79796	0.157430
$583$	5.39388	0.223392
$584$	−14.8990	−0.616524
$585$	0	0
$586$	10.0000	0.413096
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	−1.00000	−0.0412393
$589$	−44.4949	−1.83338
$590$	0	0
$591$	19.1464	0.787579
$592$	−0.449490	−0.0184739
$593$	1.79796	0.0738333	0.0369167	−	0.999318i	$-0.488246\pi$
$593$	1.79796	0.0738333	0.0369167	+	0.999318i	$0.488246\pi$
$594$	4.89898	0.201008
$595$	0	0
$596$	12.6969	0.520087
$597$	−18.0454	−0.738549
$598$	−38.6969	−1.58244
$599$	13.7980	0.563769	0.281885	−	0.959448i	$-0.409040\pi$
$599$	13.7980	0.563769	0.281885	+	0.959448i	$0.409040\pi$
$600$	0	0
$601$	0.202041	0.00824143	0.00412071	−	0.999992i	$-0.498688\pi$
$601$	0.202041	0.00824143	0.00412071	+	0.999992i	$0.498688\pi$
$602$	−7.10102	−0.289416
$603$	4.00000	0.162893
$604$	13.7980	0.561431
$605$	0	0
$606$	−3.79796	−0.154282
$607$	−26.4495	−1.07355	−0.536776	−	0.843725i	$-0.680358\pi$
$607$	−26.4495	−1.07355	−0.536776	+	0.843725i	$0.680358\pi$
$608$	−6.89898	−0.279791
$609$	22.8990	0.927913
$610$	0	0
$611$	29.3939	1.18915
$612$	1.00000	0.0404226
$613$	−48.2929	−1.95053	−0.975265	−	0.221039i	$-0.929055\pi$
$613$	−48.2929	−1.95053	−0.975265	+	0.221039i	$0.929055\pi$
$614$	−10.8990	−0.439847
$615$	0	0
$616$	−12.0000	−0.483494
$617$	−26.4949	−1.06664	−0.533322	−	0.845912i	$-0.679057\pi$
$617$	−26.4949	−1.06664	−0.533322	+	0.845912i	$0.679057\pi$
$618$	−16.8990	−0.679777
$619$	−16.0000	−0.643094	−0.321547	−	0.946894i	$-0.604203\pi$
$619$	−16.0000	−0.643094	−0.321547	+	0.946894i	$0.604203\pi$
$620$	0	0
$621$	6.44949	0.258809
$622$	23.3485	0.936188
$623$	4.40408	0.176446
$624$	6.00000	0.240192
$625$	0	0
$626$	−2.00000	−0.0799361
$627$	−33.7980	−1.34976
$628$	16.6969	0.666280
$629$	−0.449490	−0.0179223
$630$	0	0
$631$	24.4949	0.975126	0.487563	−	0.873088i	$-0.337886\pi$
$631$	24.4949	0.975126	0.487563	+	0.873088i	$0.337886\pi$
$632$	1.55051	0.0616760
$633$	−3.10102	−0.123255
$634$	8.44949	0.335572
$635$	0	0
$636$	−1.10102	−0.0436583
$637$	−6.00000	−0.237729
$638$	−45.7980	−1.81316
$639$	2.44949	0.0969003
$640$	0	0
$641$	8.69694	0.343508	0.171754	−	0.985140i	$-0.445057\pi$
$641$	8.69694	0.343508	0.171754	+	0.985140i	$0.445057\pi$
$642$	−3.10102	−0.122388
$643$	−8.00000	−0.315489	−0.157745	−	0.987480i	$-0.550422\pi$
$643$	−8.00000	−0.315489	−0.157745	+	0.987480i	$0.550422\pi$
$644$	−15.7980	−0.622527
$645$	0	0
$646$	−6.89898	−0.271437
$647$	30.2929	1.19094	0.595468	−	0.803379i	$-0.296967\pi$
$647$	30.2929	1.19094	0.595468	+	0.803379i	$0.296967\pi$
$648$	−1.00000	−0.0392837
$649$	−28.4041	−1.11496
$650$	0	0
$651$	15.7980	0.619171
$652$	5.79796	0.227066
$653$	20.4495	0.800250	0.400125	−	0.916460i	$-0.368967\pi$
$653$	20.4495	0.800250	0.400125	+	0.916460i	$0.368967\pi$
$654$	−18.2474	−0.713532
$655$	0	0
$656$	−1.10102	−0.0429876
$657$	14.8990	0.581265
$658$	12.0000	0.467809
$659$	−15.5959	−0.607531	−0.303765	−	0.952747i	$-0.598244\pi$
$659$	−15.5959	−0.607531	−0.303765	+	0.952747i	$0.598244\pi$
$660$	0	0
$661$	−46.4949	−1.80844	−0.904221	−	0.427065i	$-0.859548\pi$
$661$	−46.4949	−1.80844	−0.904221	+	0.427065i	$0.859548\pi$
$662$	−33.3939	−1.29789
$663$	6.00000	0.233021
$664$	−2.89898	−0.112502
$665$	0	0
$666$	0.449490	0.0174174
$667$	−60.2929	−2.33455
$668$	4.65153	0.179973
$669$	4.89898	0.189405
$670$	0	0
$671$	−65.3939	−2.52450
$672$	2.44949	0.0944911
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	−2.89898	−0.111665
$675$	0	0
$676$	23.0000	0.884615
$677$	47.6413	1.83100	0.915502	−	0.402312i	$-0.131793\pi$
$677$	47.6413	1.83100	0.915502	+	0.402312i	$0.131793\pi$
$678$	−16.6969	−0.641242
$679$	9.30306	0.357019
$680$	0	0
$681$	−19.5959	−0.750917
$682$	−31.5959	−1.20987
$683$	−9.79796	−0.374908	−0.187454	−	0.982273i	$-0.560024\pi$
$683$	−9.79796	−0.374908	−0.187454	+	0.982273i	$0.560024\pi$
$684$	6.89898	0.263789
$685$	0	0
$686$	−19.5959	−0.748176
$687$	18.0000	0.686743
$688$	−2.89898	−0.110523
$689$	−6.60612	−0.251673
$690$	0	0
$691$	−50.2929	−1.91323	−0.956615	−	0.291354i	$-0.905894\pi$
$691$	−50.2929	−1.91323	−0.956615	+	0.291354i	$0.905894\pi$
$692$	−13.3485	−0.507433
$693$	12.0000	0.455842
$694$	13.7980	0.523763
$695$	0	0
$696$	9.34847	0.354353
$697$	−1.10102	−0.0417041
$698$	−7.30306	−0.276425
$699$	−1.10102	−0.0416444
$700$	0	0
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	−6.00000	−0.226455
$703$	−3.10102	−0.116957
$704$	−4.89898	−0.184637
$705$	0	0
$706$	−10.0000	−0.376355
$707$	−9.30306	−0.349878
$708$	5.79796	0.217901
$709$	−24.4495	−0.918220	−0.459110	−	0.888379i	$-0.651832\pi$
$709$	−24.4495	−0.918220	−0.459110	+	0.888379i	$0.651832\pi$
$710$	0	0
$711$	−1.55051	−0.0581487
$712$	1.79796	0.0673814
$713$	−41.5959	−1.55778
$714$	2.44949	0.0916698
$715$	0	0
$716$	−20.6969	−0.773481
$717$	8.89898	0.332338
$718$	6.20204	0.231458
$719$	12.2474	0.456753	0.228376	−	0.973573i	$-0.426658\pi$
$719$	12.2474	0.456753	0.228376	+	0.973573i	$0.426658\pi$
$720$	0	0
$721$	−41.3939	−1.54159
$722$	−28.5959	−1.06423
$723$	−6.89898	−0.256576
$724$	4.44949	0.165364
$725$	0	0
$726$	−13.0000	−0.482475
$727$	−32.0000	−1.18681	−0.593407	−	0.804902i	$-0.702218\pi$
$727$	−32.0000	−1.18681	−0.593407	+	0.804902i	$0.702218\pi$
$728$	14.6969	0.544705
$729$	1.00000	0.0370370
$730$	0	0
$731$	−2.89898	−0.107223
$732$	13.3485	0.493374
$733$	19.7980	0.731254	0.365627	−	0.930761i	$-0.380855\pi$
$733$	19.7980	0.731254	0.365627	+	0.930761i	$0.380855\pi$
$734$	14.0454	0.518425
$735$	0	0
$736$	−6.44949	−0.237731
$737$	−19.5959	−0.721825
$738$	1.10102	0.0405291
$739$	−17.1010	−0.629071	−0.314536	−	0.949246i	$-0.601849\pi$
$739$	−17.1010	−0.629071	−0.314536	+	0.949246i	$0.601849\pi$
$740$	0	0
$741$	41.3939	1.52064
$742$	−2.69694	−0.0990077
$743$	14.4495	0.530100	0.265050	−	0.964235i	$-0.414611\pi$
$743$	14.4495	0.530100	0.265050	+	0.964235i	$0.414611\pi$
$744$	6.44949	0.236450
$745$	0	0
$746$	11.7980	0.431954
$747$	2.89898	0.106068
$748$	−4.89898	−0.179124
$749$	−7.59592	−0.277549
$750$	0	0
$751$	8.24745	0.300954	0.150477	−	0.988614i	$-0.451919\pi$
$751$	8.24745	0.300954	0.150477	+	0.988614i	$0.451919\pi$
$752$	4.89898	0.178647
$753$	−20.6969	−0.754238
$754$	56.0908	2.04271
$755$	0	0
$756$	−2.44949	−0.0890871
$757$	5.50510	0.200086	0.100043	−	0.994983i	$-0.468102\pi$
$757$	5.50510	0.200086	0.100043	+	0.994983i	$0.468102\pi$
$758$	10.2020	0.370555
$759$	−31.5959	−1.14686
$760$	0	0
$761$	9.59592	0.347852	0.173926	−	0.984759i	$-0.444355\pi$
$761$	9.59592	0.347852	0.173926	+	0.984759i	$0.444355\pi$
$762$	8.00000	0.289809
$763$	−44.6969	−1.61814
$764$	−16.8990	−0.611384
$765$	0	0
$766$	−16.8990	−0.610585
$767$	34.7878	1.25611
$768$	1.00000	0.0360844
$769$	27.3939	0.987848	0.493924	−	0.869505i	$-0.335562\pi$
$769$	27.3939	0.987848	0.493924	+	0.869505i	$0.335562\pi$
$770$	0	0
$771$	−20.0000	−0.720282
$772$	14.0000	0.503871
$773$	42.0000	1.51064	0.755318	−	0.655359i	$-0.227483\pi$
$773$	42.0000	1.51064	0.755318	+	0.655359i	$0.227483\pi$
$774$	2.89898	0.104202
$775$	0	0
$776$	3.79796	0.136339
$777$	1.10102	0.0394989
$778$	−3.30306	−0.118420
$779$	−7.59592	−0.272152
$780$	0	0
$781$	−12.0000	−0.429394
$782$	−6.44949	−0.230633
$783$	−9.34847	−0.334087
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	−4.89898	−0.174741
$787$	−17.3031	−0.616788	−0.308394	−	0.951259i	$-0.599791\pi$
$787$	−17.3031	−0.616788	−0.308394	+	0.951259i	$0.599791\pi$
$788$	19.1464	0.682063
$789$	−26.6969	−0.950436
$790$	0	0
$791$	−40.8990	−1.45420
$792$	4.89898	0.174078
$793$	80.0908	2.84411
$794$	36.0454	1.27920
$795$	0	0
$796$	−18.0454	−0.639603
$797$	−22.8990	−0.811123	−0.405562	−	0.914068i	$-0.632924\pi$
$797$	−22.8990	−0.811123	−0.405562	+	0.914068i	$0.632924\pi$
$798$	16.8990	0.598217
$799$	4.89898	0.173313
$800$	0	0
$801$	−1.79796	−0.0635278
$802$	0.696938	0.0246098
$803$	−72.9898	−2.57575
$804$	4.00000	0.141069
$805$	0	0
$806$	38.6969	1.36304
$807$	23.5505	0.829017
$808$	−3.79796	−0.133612
$809$	42.0908	1.47983	0.739917	−	0.672698i	$-0.234865\pi$
$809$	42.0908	1.47983	0.739917	+	0.672698i	$0.234865\pi$
$810$	0	0
$811$	1.79796	0.0631349	0.0315674	−	0.999502i	$-0.489950\pi$
$811$	1.79796	0.0631349	0.0315674	+	0.999502i	$0.489950\pi$
$812$	22.8990	0.803597
$813$	−7.59592	−0.266400
$814$	−2.20204	−0.0771815
$815$	0	0
$816$	1.00000	0.0350070
$817$	−20.0000	−0.699711
$818$	−15.5959	−0.545298
$819$	−14.6969	−0.513553
$820$	0	0
$821$	35.6413	1.24389	0.621945	−	0.783061i	$-0.286343\pi$
$821$	35.6413	1.24389	0.621945	+	0.783061i	$0.286343\pi$
$822$	−12.0000	−0.418548
$823$	−26.4495	−0.921971	−0.460986	−	0.887408i	$-0.652504\pi$
$823$	−26.4495	−0.921971	−0.460986	+	0.887408i	$0.652504\pi$
$824$	−16.8990	−0.588704
$825$	0	0
$826$	14.2020	0.494152
$827$	16.4949	0.573584	0.286792	−	0.957993i	$-0.407411\pi$
$827$	16.4949	0.573584	0.286792	+	0.957993i	$0.407411\pi$
$828$	6.44949	0.224135
$829$	8.20204	0.284869	0.142434	−	0.989804i	$-0.454507\pi$
$829$	8.20204	0.284869	0.142434	+	0.989804i	$0.454507\pi$
$830$	0	0
$831$	−28.4495	−0.986902
$832$	6.00000	0.208013
$833$	−1.00000	−0.0346479
$834$	16.8990	0.585164
$835$	0	0
$836$	−33.7980	−1.16893
$837$	−6.44949	−0.222927
$838$	−7.59592	−0.262397
$839$	−39.3485	−1.35846	−0.679230	−	0.733925i	$-0.737686\pi$
$839$	−39.3485	−1.35846	−0.679230	+	0.733925i	$0.737686\pi$
$840$	0	0
$841$	58.3939	2.01358
$842$	0.696938	0.0240181
$843$	9.79796	0.337460
$844$	−3.10102	−0.106742
$845$	0	0
$846$	−4.89898	−0.168430
$847$	−31.8434	−1.09415
$848$	−1.10102	−0.0378092
$849$	−6.69694	−0.229838
$850$	0	0
$851$	−2.89898	−0.0993757
$852$	2.44949	0.0839181
$853$	10.6515	0.364701	0.182351	−	0.983234i	$-0.441629\pi$
$853$	10.6515	0.364701	0.182351	+	0.983234i	$0.441629\pi$
$854$	32.6969	1.11887
$855$	0	0
$856$	−3.10102	−0.105991
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	29.3939	1.00349
$859$	−26.2020	−0.894002	−0.447001	−	0.894533i	$-0.647508\pi$
$859$	−26.2020	−0.894002	−0.447001	+	0.894533i	$0.647508\pi$
$860$	0	0
$861$	2.69694	0.0919114
$862$	−25.1464	−0.856491
$863$	23.5959	0.803214	0.401607	−	0.915812i	$-0.368452\pi$
$863$	23.5959	0.803214	0.401607	+	0.915812i	$0.368452\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	10.0000	0.339814
$867$	1.00000	0.0339618
$868$	15.7980	0.536218
$869$	7.59592	0.257674
$870$	0	0
$871$	24.0000	0.813209
$872$	−18.2474	−0.617937
$873$	−3.79796	−0.128541
$874$	−44.4949	−1.50506
$875$	0	0
$876$	14.8990	0.503390
$877$	18.6515	0.629817	0.314909	−	0.949122i	$-0.398026\pi$
$877$	18.6515	0.629817	0.314909	+	0.949122i	$0.398026\pi$
$878$	27.3485	0.922966
$879$	−10.0000	−0.337292
$880$	0	0
$881$	−37.1918	−1.25302	−0.626512	−	0.779411i	$-0.715518\pi$
$881$	−37.1918	−1.25302	−0.626512	+	0.779411i	$0.715518\pi$
$882$	1.00000	0.0336718
$883$	−10.2020	−0.343326	−0.171663	−	0.985156i	$-0.554914\pi$
$883$	−10.2020	−0.343326	−0.171663	+	0.985156i	$0.554914\pi$
$884$	6.00000	0.201802
$885$	0	0
$886$	10.2020	0.342744
$887$	40.2474	1.35138	0.675689	−	0.737187i	$-0.263846\pi$
$887$	40.2474	1.35138	0.675689	+	0.737187i	$0.263846\pi$
$888$	0.449490	0.0150839
$889$	19.5959	0.657226
$890$	0	0
$891$	−4.89898	−0.164122
$892$	4.89898	0.164030
$893$	33.7980	1.13101
$894$	−12.6969	−0.424649
$895$	0	0
$896$	2.44949	0.0818317
$897$	38.6969	1.29205
$898$	32.6969	1.09111
$899$	60.2929	2.01088
$900$	0	0
$901$	−1.10102	−0.0366803
$902$	−5.39388	−0.179596
$903$	7.10102	0.236307
$904$	−16.6969	−0.555332
$905$	0	0
$906$	−13.7980	−0.458406
$907$	50.6969	1.68336	0.841682	−	0.539973i	$-0.181566\pi$
$907$	50.6969	1.68336	0.841682	+	0.539973i	$0.181566\pi$
$908$	−19.5959	−0.650313
$909$	3.79796	0.125970
$910$	0	0
$911$	−32.6515	−1.08179	−0.540897	−	0.841089i	$-0.681915\pi$
$911$	−32.6515	−1.08179	−0.540897	+	0.841089i	$0.681915\pi$
$912$	6.89898	0.228448
$913$	−14.2020	−0.470019
$914$	−1.59592	−0.0527883
$915$	0	0
$916$	18.0000	0.594737
$917$	−12.0000	−0.396275
$918$	−1.00000	−0.0330049
$919$	−24.0000	−0.791687	−0.395843	−	0.918318i	$-0.629548\pi$
$919$	−24.0000	−0.791687	−0.395843	+	0.918318i	$0.629548\pi$
$920$	0	0
$921$	10.8990	0.359134
$922$	21.5959	0.711224
$923$	14.6969	0.483756
$924$	12.0000	0.394771
$925$	0	0
$926$	5.30306	0.174269
$927$	16.8990	0.555035
$928$	9.34847	0.306879
$929$	−36.6969	−1.20399	−0.601994	−	0.798501i	$-0.705627\pi$
$929$	−36.6969	−1.20399	−0.601994	+	0.798501i	$0.705627\pi$
$930$	0	0
$931$	−6.89898	−0.226105
$932$	−1.10102	−0.0360651
$933$	−23.3485	−0.764395
$934$	0.404082	0.0132220
$935$	0	0
$936$	−6.00000	−0.196116
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	9.79796	0.319915
$939$	2.00000	0.0652675
$940$	0	0
$941$	9.75255	0.317924	0.158962	−	0.987285i	$-0.449185\pi$
$941$	9.75255	0.317924	0.158962	+	0.987285i	$0.449185\pi$
$942$	−16.6969	−0.544016
$943$	−7.10102	−0.231241
$944$	5.79796	0.188707
$945$	0	0
$946$	−14.2020	−0.461748
$947$	31.1010	1.01065	0.505324	−	0.862930i	$-0.331373\pi$
$947$	31.1010	1.01065	0.505324	+	0.862930i	$0.331373\pi$
$948$	−1.55051	−0.0503582
$949$	89.3939	2.90185
$950$	0	0
$951$	−8.44949	−0.273993
$952$	2.44949	0.0793884
$953$	−33.5959	−1.08828	−0.544139	−	0.838995i	$-0.683144\pi$
$953$	−33.5959	−1.08828	−0.544139	+	0.838995i	$0.683144\pi$
$954$	1.10102	0.0356469
$955$	0	0
$956$	8.89898	0.287814
$957$	45.7980	1.48044
$958$	−15.3485	−0.495887
$959$	−29.3939	−0.949178
$960$	0	0
$961$	10.5959	0.341804
$962$	2.69694	0.0869528
$963$	3.10102	0.0999290
$964$	−6.89898	−0.222201
$965$	0	0
$966$	15.7980	0.508291
$967$	−42.6969	−1.37304	−0.686520	−	0.727110i	$-0.740863\pi$
$967$	−42.6969	−1.37304	−0.686520	+	0.727110i	$0.740863\pi$
$968$	−13.0000	−0.417836
$969$	6.89898	0.221627
$970$	0	0
$971$	−44.0000	−1.41203	−0.706014	−	0.708198i	$-0.749508\pi$
$971$	−44.0000	−1.41203	−0.706014	+	0.708198i	$0.749508\pi$
$972$	1.00000	0.0320750
$973$	41.3939	1.32703
$974$	15.3485	0.491797
$975$	0	0
$976$	13.3485	0.427274
$977$	0.404082	0.0129277	0.00646387	−	0.999979i	$-0.497942\pi$
$977$	0.404082	0.0129277	0.00646387	+	0.999979i	$0.497942\pi$
$978$	−5.79796	−0.185398
$979$	8.80816	0.281510
$980$	0	0
$981$	18.2474	0.582596
$982$	14.8990	0.475446
$983$	18.8536	0.601336	0.300668	−	0.953729i	$-0.402790\pi$
$983$	18.8536	0.601336	0.300668	+	0.953729i	$0.402790\pi$
$984$	1.10102	0.0350993
$985$	0	0
$986$	9.34847	0.297716
$987$	−12.0000	−0.381964
$988$	41.3939	1.31691
$989$	−18.6969	−0.594528
$990$	0	0
$991$	32.2474	1.02437	0.512187	−	0.858874i	$-0.328835\pi$
$991$	32.2474	1.02437	0.512187	+	0.858874i	$0.328835\pi$
$992$	6.44949	0.204772
$993$	33.3939	1.05972
$994$	6.00000	0.190308
$995$	0	0
$996$	2.89898	0.0918577
$997$	46.7423	1.48034	0.740172	−	0.672417i	$-0.234744\pi$
$997$	46.7423	1.48034	0.740172	+	0.672417i	$0.234744\pi$
$998$	−0.898979	−0.0284567
$999$	−0.449490	−0.0142212

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2550.2.a.bi.1.1		2
3.2	odd	2		7650.2.a.dg.1.1		2
5.2	odd	4		510.2.d.c.409.1	✓	4
5.3	odd	4		510.2.d.c.409.3	yes	4
5.4	even	2		2550.2.a.bj.1.2		2
15.2	even	4		1530.2.d.e.919.4		4
15.8	even	4		1530.2.d.e.919.2		4
15.14	odd	2		7650.2.a.ct.1.2		2
20.3	even	4		4080.2.m.o.2449.1		4
20.7	even	4		4080.2.m.o.2449.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.d.c.409.1	✓	4	5.2	odd	4
510.2.d.c.409.3	yes	4	5.3	odd	4
1530.2.d.e.919.2		4	15.8	even	4
1530.2.d.e.919.4		4	15.2	even	4
2550.2.a.bi.1.1		2	1.1	even	1	trivial
2550.2.a.bj.1.2		2	5.4	even	2
4080.2.m.o.2449.1		4	20.3	even	4
4080.2.m.o.2449.3		4	20.7	even	4
7650.2.a.ct.1.2		2	15.14	odd	2
7650.2.a.dg.1.1		2	3.2	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:	\( N \)	\(=\)	\( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2550.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.44949 q^{7} -1.00000 q^{8} +1.00000 q^{9} -4.89898 q^{11} +1.00000 q^{12} +6.00000 q^{13} +2.44949 q^{14} +1.00000 q^{16} +1.00000 q^{17} -1.00000 q^{18} +6.89898 q^{19} -2.44949 q^{21} +4.89898 q^{22} +6.44949 q^{23} -1.00000 q^{24} -6.00000 q^{26} +1.00000 q^{27} -2.44949 q^{28} -9.34847 q^{29} -6.44949 q^{31} -1.00000 q^{32} -4.89898 q^{33} -1.00000 q^{34} +1.00000 q^{36} -0.449490 q^{37} -6.89898 q^{38} +6.00000 q^{39} -1.10102 q^{41} +2.44949 q^{42} -2.89898 q^{43} -4.89898 q^{44} -6.44949 q^{46} +4.89898 q^{47} +1.00000 q^{48} -1.00000 q^{49} +1.00000 q^{51} +6.00000 q^{52} -1.10102 q^{53} -1.00000 q^{54} +2.44949 q^{56} +6.89898 q^{57} +9.34847 q^{58} +5.79796 q^{59} +13.3485 q^{61} +6.44949 q^{62} -2.44949 q^{63} +1.00000 q^{64} +4.89898 q^{66} +4.00000 q^{67} +1.00000 q^{68} +6.44949 q^{69} +2.44949 q^{71} -1.00000 q^{72} +14.8990 q^{73} +0.449490 q^{74} +6.89898 q^{76} +12.0000 q^{77} -6.00000 q^{78} -1.55051 q^{79} +1.00000 q^{81} +1.10102 q^{82} +2.89898 q^{83} -2.44949 q^{84} +2.89898 q^{86} -9.34847 q^{87} +4.89898 q^{88} -1.79796 q^{89} -14.6969 q^{91} +6.44949 q^{92} -6.44949 q^{93} -4.89898 q^{94} -1.00000 q^{96} -3.79796 q^{97} +1.00000 q^{98} -4.89898 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - 2 q^{8} + 2 q^{9} + 2 q^{12} + 12 q^{13} + 2 q^{16} + 2 q^{17} - 2 q^{18} + 4 q^{19} + 8 q^{23} - 2 q^{24} - 12 q^{26} + 2 q^{27} - 4 q^{29} - 8 q^{31} - 2 q^{32}+ \cdots + 2 q^{98}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^3 + 2 * q^4 - 2 * q^6 - 2 * q^8 + 2 * q^9 + 2 * q^12 + 12 * q^13 + 2 * q^16 + 2 * q^17 - 2 * q^18 + 4 * q^19 + 8 * q^23 - 2 * q^24 - 12 * q^26 + 2 * q^27 - 4 * q^29 - 8 * q^31 - 2 * q^32 - 2 * q^34 + 2 * q^36 + 4 * q^37 - 4 * q^38 + 12 * q^39 - 12 * q^41 + 4 * q^43 - 8 * q^46 + 2 * q^48 - 2 * q^49 + 2 * q^51 + 12 * q^52 - 12 * q^53 - 2 * q^54 + 4 * q^57 + 4 * q^58 - 8 * q^59 + 12 * q^61 + 8 * q^62 + 2 * q^64 + 8 * q^67 + 2 * q^68 + 8 * q^69 - 2 * q^72 + 20 * q^73 - 4 * q^74 + 4 * q^76 + 24 * q^77 - 12 * q^78 - 8 * q^79 + 2 * q^81 + 12 * q^82 - 4 * q^83 - 4 * q^86 - 4 * q^87 + 16 * q^89 + 8 * q^92 - 8 * q^93 - 2 * q^96 + 12 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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