LMFDB - Embedded newform 2550.2.a.bl.1.2

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:	$ 2 $
Twist minimal:	no (minimal twist has level 510)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.89898 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.89898 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} -6.89898 q^{13} +4.89898 q^{14} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +4.89898 q^{21} +4.00000 q^{23} +1.00000 q^{24} -6.89898 q^{26} +1.00000 q^{27} +4.89898 q^{28} +6.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} +1.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} +4.00000 q^{38} -6.89898 q^{39} -2.89898 q^{41} +4.89898 q^{42} -8.89898 q^{43} +4.00000 q^{46} -9.79796 q^{47} +1.00000 q^{48} +17.0000 q^{49} +1.00000 q^{51} -6.89898 q^{52} +7.79796 q^{53} +1.00000 q^{54} +4.89898 q^{56} +4.00000 q^{57} +6.00000 q^{58} +4.89898 q^{59} +11.7980 q^{61} +4.00000 q^{62} +4.89898 q^{63} +1.00000 q^{64} -0.898979 q^{67} +1.00000 q^{68} +4.00000 q^{69} -8.89898 q^{71} +1.00000 q^{72} +10.8990 q^{73} -6.00000 q^{74} +4.00000 q^{76} -6.89898 q^{78} -5.79796 q^{79} +1.00000 q^{81} -2.89898 q^{82} -13.7980 q^{83} +4.89898 q^{84} -8.89898 q^{86} +6.00000 q^{87} -7.79796 q^{89} -33.7980 q^{91} +4.00000 q^{92} +4.00000 q^{93} -9.79796 q^{94} +1.00000 q^{96} +12.6969 q^{97} +17.0000 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^8 + 2 * q^9	$ 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9} + 2 q^{12} - 4 q^{13} + 2 q^{16} + 2 q^{17} + 2 q^{18} + 8 q^{19} + 8 q^{23} + 2 q^{24} - 4 q^{26} + 2 q^{27} + 12 q^{29} + 8 q^{31} + 2 q^{32} + 2 q^{34} + 2 q^{36} - 12 q^{37} + 8 q^{38} - 4 q^{39} + 4 q^{41} - 8 q^{43} + 8 q^{46} + 2 q^{48} + 34 q^{49} + 2 q^{51} - 4 q^{52} - 4 q^{53} + 2 q^{54} + 8 q^{57} + 12 q^{58} + 4 q^{61} + 8 q^{62} + 2 q^{64} + 8 q^{67} + 2 q^{68} + 8 q^{69} - 8 q^{71} + 2 q^{72} + 12 q^{73} - 12 q^{74} + 8 q^{76} - 4 q^{78} + 8 q^{79} + 2 q^{81} + 4 q^{82} - 8 q^{83} - 8 q^{86} + 12 q^{87} + 4 q^{89} - 48 q^{91} + 8 q^{92} + 8 q^{93} + 2 q^{96} - 4 q^{97} + 34 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 - 4 * q^13 + 2 * q^16 + 2 * q^17 + 2 * q^18 + 8 * q^19 + 8 * q^23 + 2 * q^24 - 4 * q^26 + 2 * q^27 + 12 * q^29 + 8 * q^31 + 2 * q^32 + 2 * q^34 + 2 * q^36 - 12 * q^37 + 8 * q^38 - 4 * q^39 + 4 * q^41 - 8 * q^43 + 8 * q^46 + 2 * q^48 + 34 * q^49 + 2 * q^51 - 4 * q^52 - 4 * q^53 + 2 * q^54 + 8 * q^57 + 12 * q^58 + 4 * q^61 + 8 * q^62 + 2 * q^64 + 8 * q^67 + 2 * q^68 + 8 * q^69 - 8 * q^71 + 2 * q^72 + 12 * q^73 - 12 * q^74 + 8 * q^76 - 4 * q^78 + 8 * q^79 + 2 * q^81 + 4 * q^82 - 8 * q^83 - 8 * q^86 + 12 * q^87 + 4 * q^89 - 48 * q^91 + 8 * q^92 + 8 * q^93 + 2 * q^96 - 4 * q^97 + 34 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	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
$7$	4.89898	1.85164	0.925820	−	0.377964i	$-0.123376\pi$
$7$	4.89898	1.85164	0.925820	+	0.377964i	$0.123376\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000	0.288675
$13$	−6.89898	−1.91343	−0.956716	−	0.291022i	$-0.906005\pi$
$13$	−6.89898	−1.91343	−0.956716	+	0.291022i	$0.906005\pi$
$14$	4.89898	1.30931
$15$	0	0
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	1.00000	0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	4.89898	1.06904
$22$	0	0
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	1.00000	0.204124
$25$	0	0
$26$	−6.89898	−1.35300
$27$	1.00000	0.192450
$28$	4.89898	0.925820
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	1.00000	0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	4.00000	0.648886
$39$	−6.89898	−1.10472
$40$	0	0
$41$	−2.89898	−0.452745	−0.226372	−	0.974041i	$-0.572687\pi$
$41$	−2.89898	−0.452745	−0.226372	+	0.974041i	$0.572687\pi$
$42$	4.89898	0.755929
$43$	−8.89898	−1.35708	−0.678541	−	0.734563i	$-0.737387\pi$
$43$	−8.89898	−1.35708	−0.678541	+	0.734563i	$0.737387\pi$
$44$	0	0
$45$	0	0
$46$	4.00000	0.589768
$47$	−9.79796	−1.42918	−0.714590	−	0.699544i	$-0.753387\pi$
$47$	−9.79796	−1.42918	−0.714590	+	0.699544i	$0.753387\pi$
$48$	1.00000	0.144338
$49$	17.0000	2.42857
$50$	0	0
$51$	1.00000	0.140028
$52$	−6.89898	−0.956716
$53$	7.79796	1.07113	0.535566	−	0.844493i	$-0.320098\pi$
$53$	7.79796	1.07113	0.535566	+	0.844493i	$0.320098\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	4.89898	0.654654
$57$	4.00000	0.529813
$58$	6.00000	0.787839
$59$	4.89898	0.637793	0.318896	−	0.947790i	$-0.396688\pi$
$59$	4.89898	0.637793	0.318896	+	0.947790i	$0.396688\pi$
$60$	0	0
$61$	11.7980	1.51057	0.755287	−	0.655394i	$-0.227498\pi$
$61$	11.7980	1.51057	0.755287	+	0.655394i	$0.227498\pi$
$62$	4.00000	0.508001
$63$	4.89898	0.617213
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−0.898979	−0.109828	−0.0549139	−	0.998491i	$-0.517488\pi$
$67$	−0.898979	−0.109828	−0.0549139	+	0.998491i	$0.517488\pi$
$68$	1.00000	0.121268
$69$	4.00000	0.481543
$70$	0	0
$71$	−8.89898	−1.05611	−0.528057	−	0.849209i	$-0.677079\pi$
$71$	−8.89898	−1.05611	−0.528057	+	0.849209i	$0.677079\pi$
$72$	1.00000	0.117851
$73$	10.8990	1.27563	0.637815	−	0.770190i	$-0.279839\pi$
$73$	10.8990	1.27563	0.637815	+	0.770190i	$0.279839\pi$
$74$	−6.00000	−0.697486
$75$	0	0
$76$	4.00000	0.458831
$77$	0	0
$78$	−6.89898	−0.781156
$79$	−5.79796	−0.652321	−0.326161	−	0.945314i	$-0.605755\pi$
$79$	−5.79796	−0.652321	−0.326161	+	0.945314i	$0.605755\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−2.89898	−0.320139
$83$	−13.7980	−1.51452	−0.757261	−	0.653112i	$-0.773463\pi$
$83$	−13.7980	−1.51452	−0.757261	+	0.653112i	$0.773463\pi$
$84$	4.89898	0.534522
$85$	0	0
$86$	−8.89898	−0.959602
$87$	6.00000	0.643268
$88$	0	0
$89$	−7.79796	−0.826582	−0.413291	−	0.910599i	$-0.635621\pi$
$89$	−7.79796	−0.826582	−0.413291	+	0.910599i	$0.635621\pi$
$90$	0	0
$91$	−33.7980	−3.54299
$92$	4.00000	0.417029
$93$	4.00000	0.414781
$94$	−9.79796	−1.01058
$95$	0	0
$96$	1.00000	0.102062
$97$	12.6969	1.28918	0.644589	−	0.764529i	$-0.277028\pi$
$97$	12.6969	1.28918	0.644589	+	0.764529i	$0.277028\pi$
$98$	17.0000	1.71726
$99$	0	0
$100$	0	0
$101$	−18.8990	−1.88052	−0.940259	−	0.340459i	$-0.889418\pi$
$101$	−18.8990	−1.88052	−0.940259	+	0.340459i	$0.889418\pi$
$102$	1.00000	0.0990148
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−6.89898	−0.676501
$105$	0	0
$106$	7.79796	0.757405
$107$	−5.79796	−0.560510	−0.280255	−	0.959926i	$-0.590419\pi$
$107$	−5.79796	−0.560510	−0.280255	+	0.959926i	$0.590419\pi$
$108$	1.00000	0.0962250
$109$	11.7980	1.13004	0.565020	−	0.825077i	$-0.308869\pi$
$109$	11.7980	1.13004	0.565020	+	0.825077i	$0.308869\pi$
$110$	0	0
$111$	−6.00000	−0.569495
$112$	4.89898	0.462910
$113$	−7.79796	−0.733570	−0.366785	−	0.930306i	$-0.619542\pi$
$113$	−7.79796	−0.733570	−0.366785	+	0.930306i	$0.619542\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	6.00000	0.557086
$117$	−6.89898	−0.637811
$118$	4.89898	0.450988
$119$	4.89898	0.449089
$120$	0	0
$121$	−11.0000	−1.00000
$122$	11.7980	1.06814
$123$	−2.89898	−0.261392
$124$	4.00000	0.359211
$125$	0	0
$126$	4.89898	0.436436
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	1.00000	0.0883883
$129$	−8.89898	−0.783511
$130$	0	0
$131$	9.79796	0.856052	0.428026	−	0.903767i	$-0.359209\pi$
$131$	9.79796	0.856052	0.428026	+	0.903767i	$0.359209\pi$
$132$	0	0
$133$	19.5959	1.69918
$134$	−0.898979	−0.0776600
$135$	0	0
$136$	1.00000	0.0857493
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	4.00000	0.340503
$139$	13.7980	1.17033	0.585164	−	0.810915i	$-0.301030\pi$
$139$	13.7980	1.17033	0.585164	+	0.810915i	$0.301030\pi$
$140$	0	0
$141$	−9.79796	−0.825137
$142$	−8.89898	−0.746786
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	10.8990	0.902006
$147$	17.0000	1.40214
$148$	−6.00000	−0.493197
$149$	−18.8990	−1.54826	−0.774132	−	0.633024i	$-0.781814\pi$
$149$	−18.8990	−1.54826	−0.774132	+	0.633024i	$0.781814\pi$
$150$	0	0
$151$	−9.79796	−0.797347	−0.398673	−	0.917093i	$-0.630529\pi$
$151$	−9.79796	−0.797347	−0.398673	+	0.917093i	$0.630529\pi$
$152$	4.00000	0.324443
$153$	1.00000	0.0808452
$154$	0	0
$155$	0	0
$156$	−6.89898	−0.552360
$157$	1.10102	0.0878710	0.0439355	−	0.999034i	$-0.486010\pi$
$157$	1.10102	0.0878710	0.0439355	+	0.999034i	$0.486010\pi$
$158$	−5.79796	−0.461261
$159$	7.79796	0.618418
$160$	0	0
$161$	19.5959	1.54437
$162$	1.00000	0.0785674
$163$	2.20204	0.172477	0.0862386	−	0.996275i	$-0.472515\pi$
$163$	2.20204	0.172477	0.0862386	+	0.996275i	$0.472515\pi$
$164$	−2.89898	−0.226372
$165$	0	0
$166$	−13.7980	−1.07093
$167$	−2.20204	−0.170399	−0.0851995	−	0.996364i	$-0.527153\pi$
$167$	−2.20204	−0.170399	−0.0851995	+	0.996364i	$0.527153\pi$
$168$	4.89898	0.377964
$169$	34.5959	2.66122
$170$	0	0
$171$	4.00000	0.305888
$172$	−8.89898	−0.678541
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	0	0
$177$	4.89898	0.368230
$178$	−7.79796	−0.584482
$179$	−4.89898	−0.366167	−0.183083	−	0.983097i	$-0.558608\pi$
$179$	−4.89898	−0.366167	−0.183083	+	0.983097i	$0.558608\pi$
$180$	0	0
$181$	−4.20204	−0.312335	−0.156168	−	0.987731i	$-0.549914\pi$
$181$	−4.20204	−0.312335	−0.156168	+	0.987731i	$0.549914\pi$
$182$	−33.7980	−2.50527
$183$	11.7980	0.872130
$184$	4.00000	0.294884
$185$	0	0
$186$	4.00000	0.293294
$187$	0	0
$188$	−9.79796	−0.714590
$189$	4.89898	0.356348
$190$	0	0
$191$	−9.79796	−0.708955	−0.354478	−	0.935064i	$-0.615341\pi$
$191$	−9.79796	−0.708955	−0.354478	+	0.935064i	$0.615341\pi$
$192$	1.00000	0.0721688
$193$	1.10102	0.0792532	0.0396266	−	0.999215i	$-0.487383\pi$
$193$	1.10102	0.0792532	0.0396266	+	0.999215i	$0.487383\pi$
$194$	12.6969	0.911587
$195$	0	0
$196$	17.0000	1.21429
$197$	13.5959	0.968669	0.484335	−	0.874883i	$-0.339062\pi$
$197$	13.5959	0.968669	0.484335	+	0.874883i	$0.339062\pi$
$198$	0	0
$199$	15.5959	1.10557	0.552783	−	0.833325i	$-0.313566\pi$
$199$	15.5959	1.10557	0.552783	+	0.833325i	$0.313566\pi$
$200$	0	0
$201$	−0.898979	−0.0634091
$202$	−18.8990	−1.32973
$203$	29.3939	2.06305
$204$	1.00000	0.0700140
$205$	0	0
$206$	−4.00000	−0.278693
$207$	4.00000	0.278019
$208$	−6.89898	−0.478358
$209$	0	0
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	7.79796	0.535566
$213$	−8.89898	−0.609748
$214$	−5.79796	−0.396340
$215$	0	0
$216$	1.00000	0.0680414
$217$	19.5959	1.33026
$218$	11.7980	0.799059
$219$	10.8990	0.736485
$220$	0	0
$221$	−6.89898	−0.464076
$222$	−6.00000	−0.402694
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	4.89898	0.327327
$225$	0	0
$226$	−7.79796	−0.518713
$227$	21.7980	1.44678	0.723391	−	0.690439i	$-0.242583\pi$
$227$	21.7980	1.44678	0.723391	+	0.690439i	$0.242583\pi$
$228$	4.00000	0.264906
$229$	−13.5959	−0.898444	−0.449222	−	0.893420i	$-0.648299\pi$
$229$	−13.5959	−0.898444	−0.449222	+	0.893420i	$0.648299\pi$
$230$	0	0
$231$	0	0
$232$	6.00000	0.393919
$233$	−14.0000	−0.917170	−0.458585	−	0.888650i	$-0.651644\pi$
$233$	−14.0000	−0.917170	−0.458585	+	0.888650i	$0.651644\pi$
$234$	−6.89898	−0.451000
$235$	0	0
$236$	4.89898	0.318896
$237$	−5.79796	−0.376618
$238$	4.89898	0.317554
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	13.5959	0.875790	0.437895	−	0.899026i	$-0.355724\pi$
$241$	13.5959	0.875790	0.437895	+	0.899026i	$0.355724\pi$
$242$	−11.0000	−0.707107
$243$	1.00000	0.0641500
$244$	11.7980	0.755287
$245$	0	0
$246$	−2.89898	−0.184832
$247$	−27.5959	−1.75589
$248$	4.00000	0.254000
$249$	−13.7980	−0.874410
$250$	0	0
$251$	4.89898	0.309221	0.154610	−	0.987976i	$-0.450588\pi$
$251$	4.89898	0.309221	0.154610	+	0.987976i	$0.450588\pi$
$252$	4.89898	0.308607
$253$	0	0
$254$	12.0000	0.752947
$255$	0	0
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	−8.89898	−0.554026
$259$	−29.3939	−1.82645
$260$	0	0
$261$	6.00000	0.371391
$262$	9.79796	0.605320
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	0	0
$265$	0	0
$266$	19.5959	1.20150
$267$	−7.79796	−0.477227
$268$	−0.898979	−0.0549139
$269$	9.59592	0.585073	0.292537	−	0.956254i	$-0.405501\pi$
$269$	9.59592	0.585073	0.292537	+	0.956254i	$0.405501\pi$
$270$	0	0
$271$	−1.79796	−0.109218	−0.0546091	−	0.998508i	$-0.517391\pi$
$271$	−1.79796	−0.109218	−0.0546091	+	0.998508i	$0.517391\pi$
$272$	1.00000	0.0606339
$273$	−33.7980	−2.04555
$274$	−14.0000	−0.845771
$275$	0	0
$276$	4.00000	0.240772
$277$	−7.79796	−0.468534	−0.234267	−	0.972172i	$-0.575269\pi$
$277$	−7.79796	−0.468534	−0.234267	+	0.972172i	$0.575269\pi$
$278$	13.7980	0.827547
$279$	4.00000	0.239474
$280$	0	0
$281$	27.7980	1.65829	0.829144	−	0.559036i	$-0.188829\pi$
$281$	27.7980	1.65829	0.829144	+	0.559036i	$0.188829\pi$
$282$	−9.79796	−0.583460
$283$	−23.5959	−1.40263	−0.701316	−	0.712851i	$-0.747404\pi$
$283$	−23.5959	−1.40263	−0.701316	+	0.712851i	$0.747404\pi$
$284$	−8.89898	−0.528057
$285$	0	0
$286$	0	0
$287$	−14.2020	−0.838320
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	0	0
$291$	12.6969	0.744308
$292$	10.8990	0.637815
$293$	−13.5959	−0.794282	−0.397141	−	0.917758i	$-0.629998\pi$
$293$	−13.5959	−0.794282	−0.397141	+	0.917758i	$0.629998\pi$
$294$	17.0000	0.991460
$295$	0	0
$296$	−6.00000	−0.348743
$297$	0	0
$298$	−18.8990	−1.09479
$299$	−27.5959	−1.59591
$300$	0	0
$301$	−43.5959	−2.51283
$302$	−9.79796	−0.563809
$303$	−18.8990	−1.08572
$304$	4.00000	0.229416
$305$	0	0
$306$	1.00000	0.0571662
$307$	−0.898979	−0.0513075	−0.0256537	−	0.999671i	$-0.508167\pi$
$307$	−0.898979	−0.0513075	−0.0256537	+	0.999671i	$0.508167\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−7.10102	−0.402662	−0.201331	−	0.979523i	$-0.564527\pi$
$311$	−7.10102	−0.402662	−0.201331	+	0.979523i	$0.564527\pi$
$312$	−6.89898	−0.390578
$313$	−6.89898	−0.389953	−0.194977	−	0.980808i	$-0.562463\pi$
$313$	−6.89898	−0.389953	−0.194977	+	0.980808i	$0.562463\pi$
$314$	1.10102	0.0621342
$315$	0	0
$316$	−5.79796	−0.326161
$317$	−14.0000	−0.786318	−0.393159	−	0.919470i	$-0.628618\pi$
$317$	−14.0000	−0.786318	−0.393159	+	0.919470i	$0.628618\pi$
$318$	7.79796	0.437288
$319$	0	0
$320$	0	0
$321$	−5.79796	−0.323611
$322$	19.5959	1.09204
$323$	4.00000	0.222566
$324$	1.00000	0.0555556
$325$	0	0
$326$	2.20204	0.121960
$327$	11.7980	0.652429
$328$	−2.89898	−0.160069
$329$	−48.0000	−2.64633
$330$	0	0
$331$	−5.79796	−0.318685	−0.159342	−	0.987223i	$-0.550937\pi$
$331$	−5.79796	−0.318685	−0.159342	+	0.987223i	$0.550937\pi$
$332$	−13.7980	−0.757261
$333$	−6.00000	−0.328798
$334$	−2.20204	−0.120490
$335$	0	0
$336$	4.89898	0.267261
$337$	−32.6969	−1.78112	−0.890558	−	0.454870i	$-0.849686\pi$
$337$	−32.6969	−1.78112	−0.890558	+	0.454870i	$0.849686\pi$
$338$	34.5959	1.88177
$339$	−7.79796	−0.423527
$340$	0	0
$341$	0	0
$342$	4.00000	0.216295
$343$	48.9898	2.64520
$344$	−8.89898	−0.479801
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−21.7980	−1.17018	−0.585088	−	0.810970i	$-0.698940\pi$
$347$	−21.7980	−1.17018	−0.585088	+	0.810970i	$0.698940\pi$
$348$	6.00000	0.321634
$349$	4.20204	0.224930	0.112465	−	0.993656i	$-0.464125\pi$
$349$	4.20204	0.224930	0.112465	+	0.993656i	$0.464125\pi$
$350$	0	0
$351$	−6.89898	−0.368240
$352$	0	0
$353$	26.0000	1.38384	0.691920	−	0.721974i	$-0.256765\pi$
$353$	26.0000	1.38384	0.691920	+	0.721974i	$0.256765\pi$
$354$	4.89898	0.260378
$355$	0	0
$356$	−7.79796	−0.413291
$357$	4.89898	0.259281
$358$	−4.89898	−0.258919
$359$	37.3939	1.97357	0.986787	−	0.162025i	$-0.0518025\pi$
$359$	37.3939	1.97357	0.986787	+	0.162025i	$0.0518025\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−4.20204	−0.220854
$363$	−11.0000	−0.577350
$364$	−33.7980	−1.77149
$365$	0	0
$366$	11.7980	0.616689
$367$	27.1010	1.41466	0.707331	−	0.706883i	$-0.249899\pi$
$367$	27.1010	1.41466	0.707331	+	0.706883i	$0.249899\pi$
$368$	4.00000	0.208514
$369$	−2.89898	−0.150915
$370$	0	0
$371$	38.2020	1.98335
$372$	4.00000	0.207390
$373$	−24.6969	−1.27876	−0.639380	−	0.768891i	$-0.720809\pi$
$373$	−24.6969	−1.27876	−0.639380	+	0.768891i	$0.720809\pi$
$374$	0	0
$375$	0	0
$376$	−9.79796	−0.505291
$377$	−41.3939	−2.13189
$378$	4.89898	0.251976
$379$	−29.7980	−1.53062	−0.765309	−	0.643663i	$-0.777414\pi$
$379$	−29.7980	−1.53062	−0.765309	+	0.643663i	$0.777414\pi$
$380$	0	0
$381$	12.0000	0.614779
$382$	−9.79796	−0.501307
$383$	−8.00000	−0.408781	−0.204390	−	0.978889i	$-0.565521\pi$
$383$	−8.00000	−0.408781	−0.204390	+	0.978889i	$0.565521\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	1.10102	0.0560405
$387$	−8.89898	−0.452361
$388$	12.6969	0.644589
$389$	−38.4949	−1.95177	−0.975884	−	0.218288i	$-0.929953\pi$
$389$	−38.4949	−1.95177	−0.975884	+	0.218288i	$0.929953\pi$
$390$	0	0
$391$	4.00000	0.202289
$392$	17.0000	0.858630
$393$	9.79796	0.494242
$394$	13.5959	0.684952
$395$	0	0
$396$	0	0
$397$	−1.59592	−0.0800968	−0.0400484	−	0.999198i	$-0.512751\pi$
$397$	−1.59592	−0.0800968	−0.0400484	+	0.999198i	$0.512751\pi$
$398$	15.5959	0.781753
$399$	19.5959	0.981023
$400$	0	0
$401$	22.8990	1.14352	0.571760	−	0.820421i	$-0.306261\pi$
$401$	22.8990	1.14352	0.571760	+	0.820421i	$0.306261\pi$
$402$	−0.898979	−0.0448370
$403$	−27.5959	−1.37465
$404$	−18.8990	−0.940259
$405$	0	0
$406$	29.3939	1.45879
$407$	0	0
$408$	1.00000	0.0495074
$409$	−17.5959	−0.870062	−0.435031	−	0.900415i	$-0.643263\pi$
$409$	−17.5959	−0.870062	−0.435031	+	0.900415i	$0.643263\pi$
$410$	0	0
$411$	−14.0000	−0.690569
$412$	−4.00000	−0.197066
$413$	24.0000	1.18096
$414$	4.00000	0.196589
$415$	0	0
$416$	−6.89898	−0.338250
$417$	13.7980	0.675689
$418$	0	0
$419$	−17.7980	−0.869487	−0.434744	−	0.900554i	$-0.643161\pi$
$419$	−17.7980	−0.869487	−0.434744	+	0.900554i	$0.643161\pi$
$420$	0	0
$421$	14.0000	0.682318	0.341159	−	0.940006i	$-0.389181\pi$
$421$	14.0000	0.682318	0.341159	+	0.940006i	$0.389181\pi$
$422$	−12.0000	−0.584151
$423$	−9.79796	−0.476393
$424$	7.79796	0.378702
$425$	0	0
$426$	−8.89898	−0.431157
$427$	57.7980	2.79704
$428$	−5.79796	−0.280255
$429$	0	0
$430$	0	0
$431$	−23.1010	−1.11274	−0.556369	−	0.830936i	$-0.687806\pi$
$431$	−23.1010	−1.11274	−0.556369	+	0.830936i	$0.687806\pi$
$432$	1.00000	0.0481125
$433$	35.3939	1.70092	0.850461	−	0.526039i	$-0.176323\pi$
$433$	35.3939	1.70092	0.850461	+	0.526039i	$0.176323\pi$
$434$	19.5959	0.940634
$435$	0	0
$436$	11.7980	0.565020
$437$	16.0000	0.765384
$438$	10.8990	0.520773
$439$	5.79796	0.276721	0.138361	−	0.990382i	$-0.455817\pi$
$439$	5.79796	0.276721	0.138361	+	0.990382i	$0.455817\pi$
$440$	0	0
$441$	17.0000	0.809524
$442$	−6.89898	−0.328151
$443$	−37.7980	−1.79584	−0.897918	−	0.440164i	$-0.854920\pi$
$443$	−37.7980	−1.79584	−0.897918	+	0.440164i	$0.854920\pi$
$444$	−6.00000	−0.284747
$445$	0	0
$446$	4.00000	0.189405
$447$	−18.8990	−0.893891
$448$	4.89898	0.231455
$449$	6.89898	0.325583	0.162791	−	0.986660i	$-0.447950\pi$
$449$	6.89898	0.325583	0.162791	+	0.986660i	$0.447950\pi$
$450$	0	0
$451$	0	0
$452$	−7.79796	−0.366785
$453$	−9.79796	−0.460348
$454$	21.7980	1.02303
$455$	0	0
$456$	4.00000	0.187317
$457$	−16.2020	−0.757900	−0.378950	−	0.925417i	$-0.623715\pi$
$457$	−16.2020	−0.757900	−0.378950	+	0.925417i	$0.623715\pi$
$458$	−13.5959	−0.635296
$459$	1.00000	0.0466760
$460$	0	0
$461$	−28.6969	−1.33655	−0.668275	−	0.743914i	$-0.732967\pi$
$461$	−28.6969	−1.33655	−0.668275	+	0.743914i	$0.732967\pi$
$462$	0	0
$463$	−7.59592	−0.353012	−0.176506	−	0.984300i	$-0.556480\pi$
$463$	−7.59592	−0.353012	−0.176506	+	0.984300i	$0.556480\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−14.0000	−0.648537
$467$	7.59592	0.351497	0.175749	−	0.984435i	$-0.443765\pi$
$467$	7.59592	0.351497	0.175749	+	0.984435i	$0.443765\pi$
$468$	−6.89898	−0.318905
$469$	−4.40408	−0.203362
$470$	0	0
$471$	1.10102	0.0507323
$472$	4.89898	0.225494
$473$	0	0
$474$	−5.79796	−0.266309
$475$	0	0
$476$	4.89898	0.224544
$477$	7.79796	0.357044
$478$	0	0
$479$	−32.8990	−1.50319	−0.751596	−	0.659623i	$-0.770716\pi$
$479$	−32.8990	−1.50319	−0.751596	+	0.659623i	$0.770716\pi$
$480$	0	0
$481$	41.3939	1.88740
$482$	13.5959	0.619277
$483$	19.5959	0.891645
$484$	−11.0000	−0.500000
$485$	0	0
$486$	1.00000	0.0453609
$487$	20.8990	0.947023	0.473512	−	0.880788i	$-0.342986\pi$
$487$	20.8990	0.947023	0.473512	+	0.880788i	$0.342986\pi$
$488$	11.7980	0.534069
$489$	2.20204	0.0995797
$490$	0	0
$491$	−1.30306	−0.0588063	−0.0294032	−	0.999568i	$-0.509361\pi$
$491$	−1.30306	−0.0588063	−0.0294032	+	0.999568i	$0.509361\pi$
$492$	−2.89898	−0.130696
$493$	6.00000	0.270226
$494$	−27.5959	−1.24160
$495$	0	0
$496$	4.00000	0.179605
$497$	−43.5959	−1.95554
$498$	−13.7980	−0.618301
$499$	−39.5959	−1.77256	−0.886278	−	0.463153i	$-0.846718\pi$
$499$	−39.5959	−1.77256	−0.886278	+	0.463153i	$0.846718\pi$
$500$	0	0
$501$	−2.20204	−0.0983799
$502$	4.89898	0.218652
$503$	−4.00000	−0.178351	−0.0891756	−	0.996016i	$-0.528423\pi$
$503$	−4.00000	−0.178351	−0.0891756	+	0.996016i	$0.528423\pi$
$504$	4.89898	0.218218
$505$	0	0
$506$	0	0
$507$	34.5959	1.53646
$508$	12.0000	0.532414
$509$	−15.3031	−0.678296	−0.339148	−	0.940733i	$-0.610139\pi$
$509$	−15.3031	−0.678296	−0.339148	+	0.940733i	$0.610139\pi$
$510$	0	0
$511$	53.3939	2.36201
$512$	1.00000	0.0441942
$513$	4.00000	0.176604
$514$	2.00000	0.0882162
$515$	0	0
$516$	−8.89898	−0.391756
$517$	0	0
$518$	−29.3939	−1.29149
$519$	−6.00000	−0.263371
$520$	0	0
$521$	−40.2929	−1.76526	−0.882631	−	0.470066i	$-0.844230\pi$
$521$	−40.2929	−1.76526	−0.882631	+	0.470066i	$0.844230\pi$
$522$	6.00000	0.262613
$523$	−18.6969	−0.817560	−0.408780	−	0.912633i	$-0.634046\pi$
$523$	−18.6969	−0.817560	−0.408780	+	0.912633i	$0.634046\pi$
$524$	9.79796	0.428026
$525$	0	0
$526$	−8.00000	−0.348817
$527$	4.00000	0.174243
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	4.89898	0.212598
$532$	19.5959	0.849591
$533$	20.0000	0.866296
$534$	−7.79796	−0.337451
$535$	0	0
$536$	−0.898979	−0.0388300
$537$	−4.89898	−0.211407
$538$	9.59592	0.413709
$539$	0	0
$540$	0	0
$541$	−7.79796	−0.335260	−0.167630	−	0.985850i	$-0.553611\pi$
$541$	−7.79796	−0.335260	−0.167630	+	0.985850i	$0.553611\pi$
$542$	−1.79796	−0.0772290
$543$	−4.20204	−0.180327
$544$	1.00000	0.0428746
$545$	0	0
$546$	−33.7980	−1.44642
$547$	−0.404082	−0.0172773	−0.00863865	−	0.999963i	$-0.502750\pi$
$547$	−0.404082	−0.0172773	−0.00863865	+	0.999963i	$0.502750\pi$
$548$	−14.0000	−0.598050
$549$	11.7980	0.503525
$550$	0	0
$551$	24.0000	1.02243
$552$	4.00000	0.170251
$553$	−28.4041	−1.20786
$554$	−7.79796	−0.331304
$555$	0	0
$556$	13.7980	0.585164
$557$	−19.7980	−0.838866	−0.419433	−	0.907786i	$-0.637771\pi$
$557$	−19.7980	−0.838866	−0.419433	+	0.907786i	$0.637771\pi$
$558$	4.00000	0.169334
$559$	61.3939	2.59668
$560$	0	0
$561$	0	0
$562$	27.7980	1.17259
$563$	−21.7980	−0.918674	−0.459337	−	0.888262i	$-0.651913\pi$
$563$	−21.7980	−0.918674	−0.459337	+	0.888262i	$0.651913\pi$
$564$	−9.79796	−0.412568
$565$	0	0
$566$	−23.5959	−0.991810
$567$	4.89898	0.205738
$568$	−8.89898	−0.373393
$569$	34.0000	1.42535	0.712677	−	0.701492i	$-0.247483\pi$
$569$	34.0000	1.42535	0.712677	+	0.701492i	$0.247483\pi$
$570$	0	0
$571$	29.7980	1.24701	0.623503	−	0.781821i	$-0.285709\pi$
$571$	29.7980	1.24701	0.623503	+	0.781821i	$0.285709\pi$
$572$	0	0
$573$	−9.79796	−0.409316
$574$	−14.2020	−0.592782
$575$	0	0
$576$	1.00000	0.0416667
$577$	27.3939	1.14042	0.570211	−	0.821498i	$-0.306861\pi$
$577$	27.3939	1.14042	0.570211	+	0.821498i	$0.306861\pi$
$578$	1.00000	0.0415945
$579$	1.10102	0.0457569
$580$	0	0
$581$	−67.5959	−2.80435
$582$	12.6969	0.526305
$583$	0	0
$584$	10.8990	0.451003
$585$	0	0
$586$	−13.5959	−0.561642
$587$	41.3939	1.70851	0.854254	−	0.519856i	$-0.174014\pi$
$587$	41.3939	1.70851	0.854254	+	0.519856i	$0.174014\pi$
$588$	17.0000	0.701068
$589$	16.0000	0.659269
$590$	0	0
$591$	13.5959	0.559261
$592$	−6.00000	−0.246598
$593$	−1.59592	−0.0655365	−0.0327682	−	0.999463i	$-0.510432\pi$
$593$	−1.59592	−0.0655365	−0.0327682	+	0.999463i	$0.510432\pi$
$594$	0	0
$595$	0	0
$596$	−18.8990	−0.774132
$597$	15.5959	0.638298
$598$	−27.5959	−1.12848
$599$	21.3939	0.874130	0.437065	−	0.899430i	$-0.356018\pi$
$599$	21.3939	0.874130	0.437065	+	0.899430i	$0.356018\pi$
$600$	0	0
$601$	16.2020	0.660895	0.330448	−	0.943824i	$-0.392800\pi$
$601$	16.2020	0.660895	0.330448	+	0.943824i	$0.392800\pi$
$602$	−43.5959	−1.77684
$603$	−0.898979	−0.0366093
$604$	−9.79796	−0.398673
$605$	0	0
$606$	−18.8990	−0.767719
$607$	32.4949	1.31893	0.659464	−	0.751736i	$-0.270783\pi$
$607$	32.4949	1.31893	0.659464	+	0.751736i	$0.270783\pi$
$608$	4.00000	0.162221
$609$	29.3939	1.19110
$610$	0	0
$611$	67.5959	2.73464
$612$	1.00000	0.0404226
$613$	14.4949	0.585443	0.292722	−	0.956198i	$-0.405439\pi$
$613$	14.4949	0.585443	0.292722	+	0.956198i	$0.405439\pi$
$614$	−0.898979	−0.0362799
$615$	0	0
$616$	0	0
$617$	37.5959	1.51355	0.756777	−	0.653673i	$-0.226773\pi$
$617$	37.5959	1.51355	0.756777	+	0.653673i	$0.226773\pi$
$618$	−4.00000	−0.160904
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	0	0
$621$	4.00000	0.160514
$622$	−7.10102	−0.284725
$623$	−38.2020	−1.53053
$624$	−6.89898	−0.276180
$625$	0	0
$626$	−6.89898	−0.275739
$627$	0	0
$628$	1.10102	0.0439355
$629$	−6.00000	−0.239236
$630$	0	0
$631$	−40.0000	−1.59237	−0.796187	−	0.605050i	$-0.793153\pi$
$631$	−40.0000	−1.59237	−0.796187	+	0.605050i	$0.793153\pi$
$632$	−5.79796	−0.230630
$633$	−12.0000	−0.476957
$634$	−14.0000	−0.556011
$635$	0	0
$636$	7.79796	0.309209
$637$	−117.283	−4.64691
$638$	0	0
$639$	−8.89898	−0.352038
$640$	0	0
$641$	−20.6969	−0.817480	−0.408740	−	0.912651i	$-0.634032\pi$
$641$	−20.6969	−0.817480	−0.408740	+	0.912651i	$0.634032\pi$
$642$	−5.79796	−0.228827
$643$	−29.7980	−1.17512	−0.587558	−	0.809182i	$-0.699911\pi$
$643$	−29.7980	−1.17512	−0.587558	+	0.809182i	$0.699911\pi$
$644$	19.5959	0.772187
$645$	0	0
$646$	4.00000	0.157378
$647$	32.0000	1.25805	0.629025	−	0.777385i	$-0.283454\pi$
$647$	32.0000	1.25805	0.629025	+	0.777385i	$0.283454\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	0	0
$651$	19.5959	0.768025
$652$	2.20204	0.0862386
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	11.7980	0.461337
$655$	0	0
$656$	−2.89898	−0.113186
$657$	10.8990	0.425210
$658$	−48.0000	−1.87123
$659$	30.6969	1.19578	0.597891	−	0.801577i	$-0.296005\pi$
$659$	30.6969	1.19578	0.597891	+	0.801577i	$0.296005\pi$
$660$	0	0
$661$	−19.7980	−0.770051	−0.385026	−	0.922906i	$-0.625807\pi$
$661$	−19.7980	−0.770051	−0.385026	+	0.922906i	$0.625807\pi$
$662$	−5.79796	−0.225344
$663$	−6.89898	−0.267934
$664$	−13.7980	−0.535465
$665$	0	0
$666$	−6.00000	−0.232495
$667$	24.0000	0.929284
$668$	−2.20204	−0.0851995
$669$	4.00000	0.154649
$670$	0	0
$671$	0	0
$672$	4.89898	0.188982
$673$	33.1010	1.27595	0.637975	−	0.770057i	$-0.279772\pi$
$673$	33.1010	1.27595	0.637975	+	0.770057i	$0.279772\pi$
$674$	−32.6969	−1.25944
$675$	0	0
$676$	34.5959	1.33061
$677$	13.5959	0.522534	0.261267	−	0.965267i	$-0.415860\pi$
$677$	13.5959	0.522534	0.261267	+	0.965267i	$0.415860\pi$
$678$	−7.79796	−0.299479
$679$	62.2020	2.38710
$680$	0	0
$681$	21.7980	0.835300
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	4.00000	0.152944
$685$	0	0
$686$	48.9898	1.87044
$687$	−13.5959	−0.518717
$688$	−8.89898	−0.339270
$689$	−53.7980	−2.04954
$690$	0	0
$691$	−27.1918	−1.03443	−0.517213	−	0.855857i	$-0.673031\pi$
$691$	−27.1918	−1.03443	−0.517213	+	0.855857i	$0.673031\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	−21.7980	−0.827439
$695$	0	0
$696$	6.00000	0.227429
$697$	−2.89898	−0.109807
$698$	4.20204	0.159050
$699$	−14.0000	−0.529529
$700$	0	0
$701$	14.8990	0.562727	0.281363	−	0.959601i	$-0.409213\pi$
$701$	14.8990	0.562727	0.281363	+	0.959601i	$0.409213\pi$
$702$	−6.89898	−0.260385
$703$	−24.0000	−0.905177
$704$	0	0
$705$	0	0
$706$	26.0000	0.978523
$707$	−92.5857	−3.48204
$708$	4.89898	0.184115
$709$	−31.7980	−1.19420	−0.597099	−	0.802168i	$-0.703680\pi$
$709$	−31.7980	−1.19420	−0.597099	+	0.802168i	$0.703680\pi$
$710$	0	0
$711$	−5.79796	−0.217440
$712$	−7.79796	−0.292241
$713$	16.0000	0.599205
$714$	4.89898	0.183340
$715$	0	0
$716$	−4.89898	−0.183083
$717$	0	0
$718$	37.3939	1.39553
$719$	−12.4949	−0.465981	−0.232991	−	0.972479i	$-0.574851\pi$
$719$	−12.4949	−0.465981	−0.232991	+	0.972479i	$0.574851\pi$
$720$	0	0
$721$	−19.5959	−0.729790
$722$	−3.00000	−0.111648
$723$	13.5959	0.505638
$724$	−4.20204	−0.156168
$725$	0	0
$726$	−11.0000	−0.408248
$727$	−0.404082	−0.0149866	−0.00749329	−	0.999972i	$-0.502385\pi$
$727$	−0.404082	−0.0149866	−0.00749329	+	0.999972i	$0.502385\pi$
$728$	−33.7980	−1.25264
$729$	1.00000	0.0370370
$730$	0	0
$731$	−8.89898	−0.329141
$732$	11.7980	0.436065
$733$	46.4949	1.71733	0.858664	−	0.512539i	$-0.171295\pi$
$733$	46.4949	1.71733	0.858664	+	0.512539i	$0.171295\pi$
$734$	27.1010	1.00032
$735$	0	0
$736$	4.00000	0.147442
$737$	0	0
$738$	−2.89898	−0.106713
$739$	−9.39388	−0.345559	−0.172780	−	0.984960i	$-0.555275\pi$
$739$	−9.39388	−0.345559	−0.172780	+	0.984960i	$0.555275\pi$
$740$	0	0
$741$	−27.5959	−1.01376
$742$	38.2020	1.40244
$743$	9.39388	0.344628	0.172314	−	0.985042i	$-0.444876\pi$
$743$	9.39388	0.344628	0.172314	+	0.985042i	$0.444876\pi$
$744$	4.00000	0.146647
$745$	0	0
$746$	−24.6969	−0.904219
$747$	−13.7980	−0.504841
$748$	0	0
$749$	−28.4041	−1.03786
$750$	0	0
$751$	21.7980	0.795419	0.397709	−	0.917511i	$-0.369805\pi$
$751$	21.7980	0.795419	0.397709	+	0.917511i	$0.369805\pi$
$752$	−9.79796	−0.357295
$753$	4.89898	0.178529
$754$	−41.3939	−1.50748
$755$	0	0
$756$	4.89898	0.178174
$757$	4.69694	0.170713	0.0853566	−	0.996350i	$-0.472797\pi$
$757$	4.69694	0.170713	0.0853566	+	0.996350i	$0.472797\pi$
$758$	−29.7980	−1.08231
$759$	0	0
$760$	0	0
$761$	−1.59592	−0.0578520	−0.0289260	−	0.999582i	$-0.509209\pi$
$761$	−1.59592	−0.0578520	−0.0289260	+	0.999582i	$0.509209\pi$
$762$	12.0000	0.434714
$763$	57.7980	2.09243
$764$	−9.79796	−0.354478
$765$	0	0
$766$	−8.00000	−0.289052
$767$	−33.7980	−1.22037
$768$	1.00000	0.0360844
$769$	26.0000	0.937584	0.468792	−	0.883309i	$-0.344689\pi$
$769$	26.0000	0.937584	0.468792	+	0.883309i	$0.344689\pi$
$770$	0	0
$771$	2.00000	0.0720282
$772$	1.10102	0.0396266
$773$	35.3939	1.27303	0.636515	−	0.771265i	$-0.280375\pi$
$773$	35.3939	1.27303	0.636515	+	0.771265i	$0.280375\pi$
$774$	−8.89898	−0.319867
$775$	0	0
$776$	12.6969	0.455794
$777$	−29.3939	−1.05450
$778$	−38.4949	−1.38011
$779$	−11.5959	−0.415467
$780$	0	0
$781$	0	0
$782$	4.00000	0.143040
$783$	6.00000	0.214423
$784$	17.0000	0.607143
$785$	0	0
$786$	9.79796	0.349482
$787$	45.7980	1.63252	0.816260	−	0.577684i	$-0.196043\pi$
$787$	45.7980	1.63252	0.816260	+	0.577684i	$0.196043\pi$
$788$	13.5959	0.484335
$789$	−8.00000	−0.284808
$790$	0	0
$791$	−38.2020	−1.35831
$792$	0	0
$793$	−81.3939	−2.89038
$794$	−1.59592	−0.0566370
$795$	0	0
$796$	15.5959	0.552783
$797$	−26.0000	−0.920967	−0.460484	−	0.887668i	$-0.652324\pi$
$797$	−26.0000	−0.920967	−0.460484	+	0.887668i	$0.652324\pi$
$798$	19.5959	0.693688
$799$	−9.79796	−0.346627
$800$	0	0
$801$	−7.79796	−0.275527
$802$	22.8990	0.808591
$803$	0	0
$804$	−0.898979	−0.0317046
$805$	0	0
$806$	−27.5959	−0.972025
$807$	9.59592	0.337792
$808$	−18.8990	−0.664864
$809$	32.6969	1.14956	0.574782	−	0.818307i	$-0.305087\pi$
$809$	32.6969	1.14956	0.574782	+	0.818307i	$0.305087\pi$
$810$	0	0
$811$	25.3939	0.891700	0.445850	−	0.895108i	$-0.352902\pi$
$811$	25.3939	0.891700	0.445850	+	0.895108i	$0.352902\pi$
$812$	29.3939	1.03152
$813$	−1.79796	−0.0630572
$814$	0	0
$815$	0	0
$816$	1.00000	0.0350070
$817$	−35.5959	−1.24534
$818$	−17.5959	−0.615227
$819$	−33.7980	−1.18100
$820$	0	0
$821$	15.7980	0.551353	0.275676	−	0.961251i	$-0.411098\pi$
$821$	15.7980	0.551353	0.275676	+	0.961251i	$0.411098\pi$
$822$	−14.0000	−0.488306
$823$	−20.8990	−0.728493	−0.364246	−	0.931303i	$-0.618673\pi$
$823$	−20.8990	−0.728493	−0.364246	+	0.931303i	$0.618673\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	24.0000	0.835067
$827$	4.00000	0.139094	0.0695468	−	0.997579i	$-0.477845\pi$
$827$	4.00000	0.139094	0.0695468	+	0.997579i	$0.477845\pi$
$828$	4.00000	0.139010
$829$	3.39388	0.117874	0.0589371	−	0.998262i	$-0.481229\pi$
$829$	3.39388	0.117874	0.0589371	+	0.998262i	$0.481229\pi$
$830$	0	0
$831$	−7.79796	−0.270508
$832$	−6.89898	−0.239179
$833$	17.0000	0.589015
$834$	13.7980	0.477784
$835$	0	0
$836$	0	0
$837$	4.00000	0.138260
$838$	−17.7980	−0.614820
$839$	−7.10102	−0.245154	−0.122577	−	0.992459i	$-0.539116\pi$
$839$	−7.10102	−0.245154	−0.122577	+	0.992459i	$0.539116\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	14.0000	0.482472
$843$	27.7980	0.957413
$844$	−12.0000	−0.413057
$845$	0	0
$846$	−9.79796	−0.336861
$847$	−53.8888	−1.85164
$848$	7.79796	0.267783
$849$	−23.5959	−0.809810
$850$	0	0
$851$	−24.0000	−0.822709
$852$	−8.89898	−0.304874
$853$	−11.3939	−0.390119	−0.195059	−	0.980791i	$-0.562490\pi$
$853$	−11.3939	−0.390119	−0.195059	+	0.980791i	$0.562490\pi$
$854$	57.7980	1.97781
$855$	0	0
$856$	−5.79796	−0.198170
$857$	10.0000	0.341593	0.170797	−	0.985306i	$-0.445366\pi$
$857$	10.0000	0.341593	0.170797	+	0.985306i	$0.445366\pi$
$858$	0	0
$859$	5.79796	0.197824	0.0989119	−	0.995096i	$-0.468464\pi$
$859$	5.79796	0.197824	0.0989119	+	0.995096i	$0.468464\pi$
$860$	0	0
$861$	−14.2020	−0.484004
$862$	−23.1010	−0.786824
$863$	45.3939	1.54523	0.772613	−	0.634878i	$-0.218949\pi$
$863$	45.3939	1.54523	0.772613	+	0.634878i	$0.218949\pi$
$864$	1.00000	0.0340207
$865$	0	0
$866$	35.3939	1.20273
$867$	1.00000	0.0339618
$868$	19.5959	0.665129
$869$	0	0
$870$	0	0
$871$	6.20204	0.210148
$872$	11.7980	0.399529
$873$	12.6969	0.429726
$874$	16.0000	0.541208
$875$	0	0
$876$	10.8990	0.368242
$877$	31.3939	1.06010	0.530048	−	0.847968i	$-0.322174\pi$
$877$	31.3939	1.06010	0.530048	+	0.847968i	$0.322174\pi$
$878$	5.79796	0.195672
$879$	−13.5959	−0.458579
$880$	0	0
$881$	34.4949	1.16216	0.581081	−	0.813846i	$-0.302630\pi$
$881$	34.4949	1.16216	0.581081	+	0.813846i	$0.302630\pi$
$882$	17.0000	0.572420
$883$	−26.6969	−0.898424	−0.449212	−	0.893425i	$-0.648295\pi$
$883$	−26.6969	−0.898424	−0.449212	+	0.893425i	$0.648295\pi$
$884$	−6.89898	−0.232038
$885$	0	0
$886$	−37.7980	−1.26985
$887$	12.0000	0.402921	0.201460	−	0.979497i	$-0.435431\pi$
$887$	12.0000	0.402921	0.201460	+	0.979497i	$0.435431\pi$
$888$	−6.00000	−0.201347
$889$	58.7878	1.97168
$890$	0	0
$891$	0	0
$892$	4.00000	0.133930
$893$	−39.1918	−1.31150
$894$	−18.8990	−0.632076
$895$	0	0
$896$	4.89898	0.163663
$897$	−27.5959	−0.921401
$898$	6.89898	0.230222
$899$	24.0000	0.800445
$900$	0	0
$901$	7.79796	0.259788
$902$	0	0
$903$	−43.5959	−1.45078
$904$	−7.79796	−0.259356
$905$	0	0
$906$	−9.79796	−0.325515
$907$	33.3939	1.10883	0.554413	−	0.832242i	$-0.312943\pi$
$907$	33.3939	1.10883	0.554413	+	0.832242i	$0.312943\pi$
$908$	21.7980	0.723391
$909$	−18.8990	−0.626840
$910$	0	0
$911$	23.1010	0.765371	0.382685	−	0.923879i	$-0.374999\pi$
$911$	23.1010	0.765371	0.382685	+	0.923879i	$0.374999\pi$
$912$	4.00000	0.132453
$913$	0	0
$914$	−16.2020	−0.535916
$915$	0	0
$916$	−13.5959	−0.449222
$917$	48.0000	1.58510
$918$	1.00000	0.0330049
$919$	1.79796	0.0593092	0.0296546	−	0.999560i	$-0.490559\pi$
$919$	1.79796	0.0593092	0.0296546	+	0.999560i	$0.490559\pi$
$920$	0	0
$921$	−0.898979	−0.0296224
$922$	−28.6969	−0.945083
$923$	61.3939	2.02080
$924$	0	0
$925$	0	0
$926$	−7.59592	−0.249617
$927$	−4.00000	−0.131377
$928$	6.00000	0.196960
$929$	36.2929	1.19073	0.595365	−	0.803455i	$-0.297007\pi$
$929$	36.2929	1.19073	0.595365	+	0.803455i	$0.297007\pi$
$930$	0	0
$931$	68.0000	2.22861
$932$	−14.0000	−0.458585
$933$	−7.10102	−0.232477
$934$	7.59592	0.248546
$935$	0	0
$936$	−6.89898	−0.225500
$937$	−39.3939	−1.28694	−0.643471	−	0.765471i	$-0.722506\pi$
$937$	−39.3939	−1.28694	−0.643471	+	0.765471i	$0.722506\pi$
$938$	−4.40408	−0.143798
$939$	−6.89898	−0.225140
$940$	0	0
$941$	−16.2020	−0.528171	−0.264086	−	0.964499i	$-0.585070\pi$
$941$	−16.2020	−0.528171	−0.264086	+	0.964499i	$0.585070\pi$
$942$	1.10102	0.0358732
$943$	−11.5959	−0.377615
$944$	4.89898	0.159448
$945$	0	0
$946$	0	0
$947$	−21.7980	−0.708338	−0.354169	−	0.935181i	$-0.615236\pi$
$947$	−21.7980	−0.708338	−0.354169	+	0.935181i	$0.615236\pi$
$948$	−5.79796	−0.188309
$949$	−75.1918	−2.44083
$950$	0	0
$951$	−14.0000	−0.453981
$952$	4.89898	0.158777
$953$	41.1918	1.33433	0.667167	−	0.744908i	$-0.267507\pi$
$953$	41.1918	1.33433	0.667167	+	0.744908i	$0.267507\pi$
$954$	7.79796	0.252468
$955$	0	0
$956$	0	0
$957$	0	0
$958$	−32.8990	−1.06292
$959$	−68.5857	−2.21475
$960$	0	0
$961$	−15.0000	−0.483871
$962$	41.3939	1.33459
$963$	−5.79796	−0.186837
$964$	13.5959	0.437895
$965$	0	0
$966$	19.5959	0.630488
$967$	41.3939	1.33114	0.665569	−	0.746337i	$-0.268189\pi$
$967$	41.3939	1.33114	0.665569	+	0.746337i	$0.268189\pi$
$968$	−11.0000	−0.353553
$969$	4.00000	0.128499
$970$	0	0
$971$	38.6969	1.24184	0.620922	−	0.783872i	$-0.286758\pi$
$971$	38.6969	1.24184	0.620922	+	0.783872i	$0.286758\pi$
$972$	1.00000	0.0320750
$973$	67.5959	2.16703
$974$	20.8990	0.669646
$975$	0	0
$976$	11.7980	0.377643
$977$	29.5959	0.946857	0.473429	−	0.880832i	$-0.343016\pi$
$977$	29.5959	0.946857	0.473429	+	0.880832i	$0.343016\pi$
$978$	2.20204	0.0704135
$979$	0	0
$980$	0	0
$981$	11.7980	0.376680
$982$	−1.30306	−0.0415824
$983$	−9.39388	−0.299618	−0.149809	−	0.988715i	$-0.547866\pi$
$983$	−9.39388	−0.299618	−0.149809	+	0.988715i	$0.547866\pi$
$984$	−2.89898	−0.0924161
$985$	0	0
$986$	6.00000	0.191079
$987$	−48.0000	−1.52786
$988$	−27.5959	−0.877943
$989$	−35.5959	−1.13188
$990$	0	0
$991$	−15.5959	−0.495421	−0.247710	−	0.968834i	$-0.579678\pi$
$991$	−15.5959	−0.495421	−0.247710	+	0.968834i	$0.579678\pi$
$992$	4.00000	0.127000
$993$	−5.79796	−0.183993
$994$	−43.5959	−1.38278
$995$	0	0
$996$	−13.7980	−0.437205
$997$	21.5959	0.683950	0.341975	−	0.939709i	$-0.388904\pi$
$997$	21.5959	0.683950	0.341975	+	0.939709i	$0.388904\pi$
$998$	−39.5959	−1.25339
$999$	−6.00000	−0.189832

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2550.2.a.bl.1.2		2
3.2	odd	2		7650.2.a.cu.1.2		2
5.2	odd	4		2550.2.d.u.2449.4		4
5.3	odd	4		2550.2.d.u.2449.1		4
5.4	even	2		510.2.a.h.1.1	✓	2
15.14	odd	2		1530.2.a.s.1.1		2
20.19	odd	2		4080.2.a.bq.1.2		2
85.84	even	2		8670.2.a.be.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.a.h.1.1	✓	2	5.4	even	2
1530.2.a.s.1.1		2	15.14	odd	2
2550.2.a.bl.1.2		2	1.1	even	1	trivial
2550.2.d.u.2449.1		4	5.3	odd	4
2550.2.d.u.2449.4		4	5.2	odd	4
4080.2.a.bq.1.2		2	20.19	odd	2
7650.2.a.cu.1.2		2	3.2	odd	2
8670.2.a.be.1.2		2	85.84	even	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.89898 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +4.89898 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} -6.89898 q^{13} +4.89898 q^{14} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} +4.89898 q^{21} +4.00000 q^{23} +1.00000 q^{24} -6.89898 q^{26} +1.00000 q^{27} +4.89898 q^{28} +6.00000 q^{29} +4.00000 q^{31} +1.00000 q^{32} +1.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} +4.00000 q^{38} -6.89898 q^{39} -2.89898 q^{41} +4.89898 q^{42} -8.89898 q^{43} +4.00000 q^{46} -9.79796 q^{47} +1.00000 q^{48} +17.0000 q^{49} +1.00000 q^{51} -6.89898 q^{52} +7.79796 q^{53} +1.00000 q^{54} +4.89898 q^{56} +4.00000 q^{57} +6.00000 q^{58} +4.89898 q^{59} +11.7980 q^{61} +4.00000 q^{62} +4.89898 q^{63} +1.00000 q^{64} -0.898979 q^{67} +1.00000 q^{68} +4.00000 q^{69} -8.89898 q^{71} +1.00000 q^{72} +10.8990 q^{73} -6.00000 q^{74} +4.00000 q^{76} -6.89898 q^{78} -5.79796 q^{79} +1.00000 q^{81} -2.89898 q^{82} -13.7980 q^{83} +4.89898 q^{84} -8.89898 q^{86} +6.00000 q^{87} -7.79796 q^{89} -33.7980 q^{91} +4.00000 q^{92} +4.00000 q^{93} -9.79796 q^{94} +1.00000 q^{96} +12.6969 q^{97} +17.0000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{6} + 2 q^{8} + 2 q^{9} + 2 q^{12} - 4 q^{13} + 2 q^{16} + 2 q^{17} + 2 q^{18} + 8 q^{19} + 8 q^{23} + 2 q^{24} - 4 q^{26} + 2 q^{27} + 12 q^{29} + 8 q^{31} + 2 q^{32} + 2 q^{34} + 2 q^{36} - 12 q^{37} + 8 q^{38} - 4 q^{39} + 4 q^{41} - 8 q^{43} + 8 q^{46} + 2 q^{48} + 34 q^{49} + 2 q^{51} - 4 q^{52} - 4 q^{53} + 2 q^{54} + 8 q^{57} + 12 q^{58} + 4 q^{61} + 8 q^{62} + 2 q^{64} + 8 q^{67} + 2 q^{68} + 8 q^{69} - 8 q^{71} + 2 q^{72} + 12 q^{73} - 12 q^{74} + 8 q^{76} - 4 q^{78} + 8 q^{79} + 2 q^{81} + 4 q^{82} - 8 q^{83} - 8 q^{86} + 12 q^{87} + 4 q^{89} - 48 q^{91} + 8 q^{92} + 8 q^{93} + 2 q^{96} - 4 q^{97} + 34 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^3 + 2 * q^4 + 2 * q^6 + 2 * q^8 + 2 * q^9 + 2 * q^12 - 4 * q^13 + 2 * q^16 + 2 * q^17 + 2 * q^18 + 8 * q^19 + 8 * q^23 + 2 * q^24 - 4 * q^26 + 2 * q^27 + 12 * q^29 + 8 * q^31 + 2 * q^32 + 2 * q^34 + 2 * q^36 - 12 * q^37 + 8 * q^38 - 4 * q^39 + 4 * q^41 - 8 * q^43 + 8 * q^46 + 2 * q^48 + 34 * q^49 + 2 * q^51 - 4 * q^52 - 4 * q^53 + 2 * q^54 + 8 * q^57 + 12 * q^58 + 4 * q^61 + 8 * q^62 + 2 * q^64 + 8 * q^67 + 2 * q^68 + 8 * q^69 - 8 * q^71 + 2 * q^72 + 12 * q^73 - 12 * q^74 + 8 * q^76 - 4 * q^78 + 8 * q^79 + 2 * q^81 + 4 * q^82 - 8 * q^83 - 8 * q^86 + 12 * q^87 + 4 * q^89 - 48 * q^91 + 8 * q^92 + 8 * q^93 + 2 * q^96 - 4 * q^97 + 34 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more