LMFDB - Embedded newform 2550.2.a.bj.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:	$1$
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)

$f(q)$	$=$	$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.44949 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.44949 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.89898 q^{11} -1.00000 q^{12} -6.00000 q^{13} -2.44949 q^{14} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} -2.89898 q^{19} +2.44949 q^{21} +4.89898 q^{22} -1.55051 q^{23} -1.00000 q^{24} -6.00000 q^{26} -1.00000 q^{27} -2.44949 q^{28} +5.34847 q^{29} -1.55051 q^{31} +1.00000 q^{32} -4.89898 q^{33} -1.00000 q^{34} +1.00000 q^{36} -4.44949 q^{37} -2.89898 q^{38} +6.00000 q^{39} -10.8990 q^{41} +2.44949 q^{42} -6.89898 q^{43} +4.89898 q^{44} -1.55051 q^{46} +4.89898 q^{47} -1.00000 q^{48} -1.00000 q^{49} +1.00000 q^{51} -6.00000 q^{52} +10.8990 q^{53} -1.00000 q^{54} -2.44949 q^{56} +2.89898 q^{57} +5.34847 q^{58} -13.7980 q^{59} -1.34847 q^{61} -1.55051 q^{62} -2.44949 q^{63} +1.00000 q^{64} -4.89898 q^{66} -4.00000 q^{67} -1.00000 q^{68} +1.55051 q^{69} -2.44949 q^{71} +1.00000 q^{72} -5.10102 q^{73} -4.44949 q^{74} -2.89898 q^{76} -12.0000 q^{77} +6.00000 q^{78} -6.44949 q^{79} +1.00000 q^{81} -10.8990 q^{82} +6.89898 q^{83} +2.44949 q^{84} -6.89898 q^{86} -5.34847 q^{87} +4.89898 q^{88} +17.7980 q^{89} +14.6969 q^{91} -1.55051 q^{92} +1.55051 q^{93} +4.89898 q^{94} -1.00000 q^{96} -15.7980 q^{97} -1.00000 q^{98} +4.89898 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{8} + 2 q^{9}+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} - 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}+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

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
$7$	−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	$-0.235511\pi$
$11$	4.89898	1.47710	0.738549	+	0.674200i	$0.235511\pi$
$12$	−1.00000	−0.288675
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	−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
$19$	−2.89898	−0.665072	−0.332536	−	0.943091i	$-0.607904\pi$
$19$	−2.89898	−0.665072	−0.332536	+	0.943091i	$0.607904\pi$
$20$	0	0
$21$	2.44949	0.534522
$22$	4.89898	1.04447
$23$	−1.55051	−0.323304	−0.161652	−	0.986848i	$-0.551682\pi$
$23$	−1.55051	−0.323304	−0.161652	+	0.986848i	$0.551682\pi$
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−6.00000	−1.17670
$27$	−1.00000	−0.192450
$28$	−2.44949	−0.462910
$29$	5.34847	0.993186	0.496593	−	0.867984i	$-0.334584\pi$
$29$	5.34847	0.993186	0.496593	+	0.867984i	$0.334584\pi$
$30$	0	0
$31$	−1.55051	−0.278480	−0.139240	−	0.990259i	$-0.544466\pi$
$31$	−1.55051	−0.278480	−0.139240	+	0.990259i	$0.544466\pi$
$32$	1.00000	0.176777
$33$	−4.89898	−0.852803
$34$	−1.00000	−0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	−4.44949	−0.731492	−0.365746	−	0.930715i	$-0.619186\pi$
$37$	−4.44949	−0.731492	−0.365746	+	0.930715i	$0.619186\pi$
$38$	−2.89898	−0.470277
$39$	6.00000	0.960769
$40$	0	0
$41$	−10.8990	−1.70213	−0.851067	−	0.525057i	$-0.824044\pi$
$41$	−10.8990	−1.70213	−0.851067	+	0.525057i	$0.824044\pi$
$42$	2.44949	0.377964
$43$	−6.89898	−1.05208	−0.526042	−	0.850458i	$-0.676325\pi$
$43$	−6.89898	−1.05208	−0.526042	+	0.850458i	$0.676325\pi$
$44$	4.89898	0.738549
$45$	0	0
$46$	−1.55051	−0.228610
$47$	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
$53$	10.8990	1.49709	0.748545	−	0.663084i	$-0.230753\pi$
$53$	10.8990	1.49709	0.748545	+	0.663084i	$0.230753\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−2.44949	−0.327327
$57$	2.89898	0.383979
$58$	5.34847	0.702288
$59$	−13.7980	−1.79634	−0.898171	−	0.439647i	$-0.855104\pi$
$59$	−13.7980	−1.79634	−0.898171	+	0.439647i	$0.855104\pi$
$60$	0	0
$61$	−1.34847	−0.172654	−0.0863269	−	0.996267i	$-0.527513\pi$
$61$	−1.34847	−0.172654	−0.0863269	+	0.996267i	$0.527513\pi$
$62$	−1.55051	−0.196915
$63$	−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	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−1.00000	−0.121268
$69$	1.55051	0.186660
$70$	0	0
$71$	−2.44949	−0.290701	−0.145350	−	0.989380i	$-0.546431\pi$
$71$	−2.44949	−0.290701	−0.145350	+	0.989380i	$0.546431\pi$
$72$	1.00000	0.117851
$73$	−5.10102	−0.597029	−0.298515	−	0.954405i	$-0.596491\pi$
$73$	−5.10102	−0.597029	−0.298515	+	0.954405i	$0.596491\pi$
$74$	−4.44949	−0.517243
$75$	0	0
$76$	−2.89898	−0.332536
$77$	−12.0000	−1.36753
$78$	6.00000	0.679366
$79$	−6.44949	−0.725624	−0.362812	−	0.931862i	$-0.618183\pi$
$79$	−6.44949	−0.725624	−0.362812	+	0.931862i	$0.618183\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−10.8990	−1.20359
$83$	6.89898	0.757261	0.378631	−	0.925548i	$-0.376395\pi$
$83$	6.89898	0.757261	0.378631	+	0.925548i	$0.376395\pi$
$84$	2.44949	0.267261
$85$	0	0
$86$	−6.89898	−0.743936
$87$	−5.34847	−0.573416
$88$	4.89898	0.522233
$89$	17.7980	1.88658	0.943290	−	0.331970i	$-0.107713\pi$
$89$	17.7980	1.88658	0.943290	+	0.331970i	$0.107713\pi$
$90$	0	0
$91$	14.6969	1.54066
$92$	−1.55051	−0.161652
$93$	1.55051	0.160780
$94$	4.89898	0.505291
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−15.7980	−1.60404	−0.802020	−	0.597297i	$-0.796241\pi$
$97$	−15.7980	−1.60404	−0.802020	+	0.597297i	$0.796241\pi$
$98$	−1.00000	−0.101015
$99$	4.89898	0.492366
$100$	0	0
$101$	−15.7980	−1.57196	−0.785978	−	0.618255i	$-0.787840\pi$
$101$	−15.7980	−1.57196	−0.785978	+	0.618255i	$0.787840\pi$
$102$	1.00000	0.0990148
$103$	−7.10102	−0.699684	−0.349842	−	0.936809i	$-0.613765\pi$
$103$	−7.10102	−0.699684	−0.349842	+	0.936809i	$0.613765\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	10.8990	1.05860
$107$	−12.8990	−1.24699	−0.623496	−	0.781827i	$-0.714288\pi$
$107$	−12.8990	−1.24699	−0.623496	+	0.781827i	$0.714288\pi$
$108$	−1.00000	−0.0962250
$109$	−6.24745	−0.598397	−0.299199	−	0.954191i	$-0.596719\pi$
$109$	−6.24745	−0.598397	−0.299199	+	0.954191i	$0.596719\pi$
$110$	0	0
$111$	4.44949	0.422327
$112$	−2.44949	−0.231455
$113$	12.6969	1.19443	0.597214	−	0.802082i	$-0.296274\pi$
$113$	12.6969	1.19443	0.597214	+	0.802082i	$0.296274\pi$
$114$	2.89898	0.271514
$115$	0	0
$116$	5.34847	0.496593
$117$	−6.00000	−0.554700
$118$	−13.7980	−1.27021
$119$	2.44949	0.224544
$120$	0	0
$121$	13.0000	1.18182
$122$	−1.34847	−0.122085
$123$	10.8990	0.982728
$124$	−1.55051	−0.139240
$125$	0	0
$126$	−2.44949	−0.218218
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	6.89898	0.607421
$130$	0	0
$131$	−4.89898	−0.428026	−0.214013	−	0.976831i	$-0.568653\pi$
$131$	−4.89898	−0.428026	−0.214013	+	0.976831i	$0.568653\pi$
$132$	−4.89898	−0.426401
$133$	7.10102	0.615737
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−1.00000	−0.0857493
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	1.55051	0.131988
$139$	−7.10102	−0.602301	−0.301150	−	0.953577i	$-0.597371\pi$
$139$	−7.10102	−0.602301	−0.301150	+	0.953577i	$0.597371\pi$
$140$	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
$146$	−5.10102	−0.422163
$147$	1.00000	0.0824786
$148$	−4.44949	−0.365746
$149$	−16.6969	−1.36787	−0.683933	−	0.729545i	$-0.739732\pi$
$149$	−16.6969	−1.36787	−0.683933	+	0.729545i	$0.739732\pi$
$150$	0	0
$151$	−5.79796	−0.471831	−0.235916	−	0.971774i	$-0.575809\pi$
$151$	−5.79796	−0.471831	−0.235916	+	0.971774i	$0.575809\pi$
$152$	−2.89898	−0.235138
$153$	−1.00000	−0.0808452
$154$	−12.0000	−0.966988
$155$	0	0
$156$	6.00000	0.480384
$157$	12.6969	1.01333	0.506663	−	0.862144i	$-0.330879\pi$
$157$	12.6969	1.01333	0.506663	+	0.862144i	$0.330879\pi$
$158$	−6.44949	−0.513094
$159$	−10.8990	−0.864345
$160$	0	0
$161$	3.79796	0.299321
$162$	1.00000	0.0785674
$163$	13.7980	1.08074	0.540370	−	0.841428i	$-0.318284\pi$
$163$	13.7980	1.08074	0.540370	+	0.841428i	$0.318284\pi$
$164$	−10.8990	−0.851067
$165$	0	0
$166$	6.89898	0.535465
$167$	−19.3485	−1.49723	−0.748615	−	0.663005i	$-0.769281\pi$
$167$	−19.3485	−1.49723	−0.748615	+	0.663005i	$0.769281\pi$
$168$	2.44949	0.188982
$169$	23.0000	1.76923
$170$	0	0
$171$	−2.89898	−0.221691
$172$	−6.89898	−0.526042
$173$	−1.34847	−0.102522	−0.0512611	−	0.998685i	$-0.516324\pi$
$173$	−1.34847	−0.102522	−0.0512611	+	0.998685i	$0.516324\pi$
$174$	−5.34847	−0.405466
$175$	0	0
$176$	4.89898	0.369274
$177$	13.7980	1.03712
$178$	17.7980	1.33401
$179$	8.69694	0.650040	0.325020	−	0.945707i	$-0.394629\pi$
$179$	8.69694	0.650040	0.325020	+	0.945707i	$0.394629\pi$
$180$	0	0
$181$	−0.449490	−0.0334103	−0.0167052	−	0.999860i	$-0.505318\pi$
$181$	−0.449490	−0.0334103	−0.0167052	+	0.999860i	$0.505318\pi$
$182$	14.6969	1.08941
$183$	1.34847	0.0996817
$184$	−1.55051	−0.114305
$185$	0	0
$186$	1.55051	0.113689
$187$	−4.89898	−0.358249
$188$	4.89898	0.357295
$189$	2.44949	0.178174
$190$	0	0
$191$	−7.10102	−0.513812	−0.256906	−	0.966436i	$-0.582703\pi$
$191$	−7.10102	−0.513812	−0.256906	+	0.966436i	$0.582703\pi$
$192$	−1.00000	−0.0721688
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	−15.7980	−1.13423
$195$	0	0
$196$	−1.00000	−0.0714286
$197$	15.1464	1.07914	0.539569	−	0.841941i	$-0.318587\pi$
$197$	15.1464	1.07914	0.539569	+	0.841941i	$0.318587\pi$
$198$	4.89898	0.348155
$199$	26.0454	1.84631	0.923155	−	0.384428i	$-0.125601\pi$
$199$	26.0454	1.84631	0.923155	+	0.384428i	$0.125601\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	−15.7980	−1.11154
$203$	−13.1010	−0.919511
$204$	1.00000	0.0700140
$205$	0	0
$206$	−7.10102	−0.494752
$207$	−1.55051	−0.107768
$208$	−6.00000	−0.416025
$209$	−14.2020	−0.982376
$210$	0	0
$211$	−12.8990	−0.888002	−0.444001	−	0.896026i	$-0.646441\pi$
$211$	−12.8990	−0.888002	−0.444001	+	0.896026i	$0.646441\pi$
$212$	10.8990	0.748545
$213$	2.44949	0.167836
$214$	−12.8990	−0.881756
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	3.79796	0.257822
$218$	−6.24745	−0.423131
$219$	5.10102	0.344695
$220$	0	0
$221$	6.00000	0.403604
$222$	4.44949	0.298630
$223$	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
$226$	12.6969	0.844588
$227$	−19.5959	−1.30063	−0.650313	−	0.759666i	$-0.725362\pi$
$227$	−19.5959	−1.30063	−0.650313	+	0.759666i	$0.725362\pi$
$228$	2.89898	0.191990
$229$	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
$232$	5.34847	0.351144
$233$	10.8990	0.714016	0.357008	−	0.934101i	$-0.383797\pi$
$233$	10.8990	0.714016	0.357008	+	0.934101i	$0.383797\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	−13.7980	−0.898171
$237$	6.44949	0.418939
$238$	2.44949	0.158777
$239$	−0.898979	−0.0581501	−0.0290751	−	0.999577i	$-0.509256\pi$
$239$	−0.898979	−0.0581501	−0.0290751	+	0.999577i	$0.509256\pi$
$240$	0	0
$241$	2.89898	0.186740	0.0933698	−	0.995631i	$-0.470236\pi$
$241$	2.89898	0.186740	0.0933698	+	0.995631i	$0.470236\pi$
$242$	13.0000	0.835672
$243$	−1.00000	−0.0641500
$244$	−1.34847	−0.0863269
$245$	0	0
$246$	10.8990	0.694894
$247$	17.3939	1.10675
$248$	−1.55051	−0.0984575
$249$	−6.89898	−0.437205
$250$	0	0
$251$	8.69694	0.548946	0.274473	−	0.961595i	$-0.411497\pi$
$251$	8.69694	0.548946	0.274473	+	0.961595i	$0.411497\pi$
$252$	−2.44949	−0.154303
$253$	−7.59592	−0.477551
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	20.0000	1.24757	0.623783	−	0.781598i	$-0.285595\pi$
$257$	20.0000	1.24757	0.623783	+	0.781598i	$0.285595\pi$
$258$	6.89898	0.429512
$259$	10.8990	0.677230
$260$	0	0
$261$	5.34847	0.331062
$262$	−4.89898	−0.302660
$263$	−2.69694	−0.166300	−0.0831502	−	0.996537i	$-0.526498\pi$
$263$	−2.69694	−0.166300	−0.0831502	+	0.996537i	$0.526498\pi$
$264$	−4.89898	−0.301511
$265$	0	0
$266$	7.10102	0.435392
$267$	−17.7980	−1.08922
$268$	−4.00000	−0.244339
$269$	28.4495	1.73460	0.867298	−	0.497789i	$-0.165855\pi$
$269$	28.4495	1.73460	0.867298	+	0.497789i	$0.165855\pi$
$270$	0	0
$271$	31.5959	1.91932	0.959658	−	0.281171i	$-0.0907228\pi$
$271$	31.5959	1.91932	0.959658	+	0.281171i	$0.0907228\pi$
$272$	−1.00000	−0.0606339
$273$	−14.6969	−0.889499
$274$	−12.0000	−0.724947
$275$	0	0
$276$	1.55051	0.0933298
$277$	23.5505	1.41501	0.707507	−	0.706707i	$-0.249820\pi$
$277$	23.5505	1.41501	0.707507	+	0.706707i	$0.249820\pi$
$278$	−7.10102	−0.425891
$279$	−1.55051	−0.0928266
$280$	0	0
$281$	−9.79796	−0.584497	−0.292249	−	0.956342i	$-0.594403\pi$
$281$	−9.79796	−0.584497	−0.292249	+	0.956342i	$0.594403\pi$
$282$	−4.89898	−0.291730
$283$	−22.6969	−1.34919	−0.674596	−	0.738187i	$-0.735682\pi$
$283$	−22.6969	−1.34919	−0.674596	+	0.738187i	$0.735682\pi$
$284$	−2.44949	−0.145350
$285$	0	0
$286$	−29.3939	−1.73810
$287$	26.6969	1.57587
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	0	0
$291$	15.7980	0.926093
$292$	−5.10102	−0.298515
$293$	10.0000	0.584206	0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000	0.584206	0.292103	+	0.956387i	$0.405645\pi$
$294$	1.00000	0.0583212
$295$	0	0
$296$	−4.44949	−0.258621
$297$	−4.89898	−0.284268
$298$	−16.6969	−0.967228
$299$	9.30306	0.538010
$300$	0	0
$301$	16.8990	0.974041
$302$	−5.79796	−0.333635
$303$	15.7980	0.907569
$304$	−2.89898	−0.166268
$305$	0	0
$306$	−1.00000	−0.0571662
$307$	−1.10102	−0.0628386	−0.0314193	−	0.999506i	$-0.510003\pi$
$307$	−1.10102	−0.0628386	−0.0314193	+	0.999506i	$0.510003\pi$
$308$	−12.0000	−0.683763
$309$	7.10102	0.403963
$310$	0	0
$311$	−8.65153	−0.490583	−0.245292	−	0.969449i	$-0.578884\pi$
$311$	−8.65153	−0.490583	−0.245292	+	0.969449i	$0.578884\pi$
$312$	6.00000	0.339683
$313$	−2.00000	−0.113047	−0.0565233	−	0.998401i	$-0.518002\pi$
$313$	−2.00000	−0.113047	−0.0565233	+	0.998401i	$0.518002\pi$
$314$	12.6969	0.716530
$315$	0	0
$316$	−6.44949	−0.362812
$317$	3.55051	0.199417	0.0997083	−	0.995017i	$-0.468209\pi$
$317$	3.55051	0.199417	0.0997083	+	0.995017i	$0.468209\pi$
$318$	−10.8990	−0.611184
$319$	26.2020	1.46703
$320$	0	0
$321$	12.8990	0.719951
$322$	3.79796	0.211652
$323$	2.89898	0.161304
$324$	1.00000	0.0555556
$325$	0	0
$326$	13.7980	0.764198
$327$	6.24745	0.345485
$328$	−10.8990	−0.601795
$329$	−12.0000	−0.661581
$330$	0	0
$331$	−25.3939	−1.39577	−0.697887	−	0.716208i	$-0.745876\pi$
$331$	−25.3939	−1.39577	−0.697887	+	0.716208i	$0.745876\pi$
$332$	6.89898	0.378631
$333$	−4.44949	−0.243831
$334$	−19.3485	−1.05870
$335$	0	0
$336$	2.44949	0.133631
$337$	6.89898	0.375811	0.187906	−	0.982187i	$-0.439830\pi$
$337$	6.89898	0.375811	0.187906	+	0.982187i	$0.439830\pi$
$338$	23.0000	1.25104
$339$	−12.6969	−0.689603
$340$	0	0
$341$	−7.59592	−0.411342
$342$	−2.89898	−0.156759
$343$	19.5959	1.05808
$344$	−6.89898	−0.371968
$345$	0	0
$346$	−1.34847	−0.0724942
$347$	−5.79796	−0.311251	−0.155625	−	0.987816i	$-0.549739\pi$
$347$	−5.79796	−0.311251	−0.155625	+	0.987816i	$0.549739\pi$
$348$	−5.34847	−0.286708
$349$	36.6969	1.96434	0.982171	−	0.187989i	$-0.0601971\pi$
$349$	36.6969	1.96434	0.982171	+	0.187989i	$0.0601971\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	4.89898	0.261116
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	13.7980	0.733353
$355$	0	0
$356$	17.7980	0.943290
$357$	−2.44949	−0.129641
$358$	8.69694	0.459647
$359$	−25.7980	−1.36156	−0.680782	−	0.732486i	$-0.738360\pi$
$359$	−25.7980	−1.36156	−0.680782	+	0.732486i	$0.738360\pi$
$360$	0	0
$361$	−10.5959	−0.557680
$362$	−0.449490	−0.0236247
$363$	−13.0000	−0.682323
$364$	14.6969	0.770329
$365$	0	0
$366$	1.34847	0.0704856
$367$	−30.0454	−1.56836	−0.784179	−	0.620535i	$-0.786915\pi$
$367$	−30.0454	−1.56836	−0.784179	+	0.620535i	$0.786915\pi$
$368$	−1.55051	−0.0808259
$369$	−10.8990	−0.567378
$370$	0	0
$371$	−26.6969	−1.38604
$372$	1.55051	0.0803902
$373$	−7.79796	−0.403763	−0.201882	−	0.979410i	$-0.564706\pi$
$373$	−7.79796	−0.403763	−0.201882	+	0.979410i	$0.564706\pi$
$374$	−4.89898	−0.253320
$375$	0	0
$376$	4.89898	0.252646
$377$	−32.0908	−1.65276
$378$	2.44949	0.125988
$379$	−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
$381$	−8.00000	−0.409852
$382$	−7.10102	−0.363320
$383$	−7.10102	−0.362845	−0.181423	−	0.983405i	$-0.558070\pi$
$383$	−7.10102	−0.362845	−0.181423	+	0.983405i	$0.558070\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	−6.89898	−0.350695
$388$	−15.7980	−0.802020
$389$	32.6969	1.65780	0.828900	−	0.559396i	$-0.188967\pi$
$389$	32.6969	1.65780	0.828900	+	0.559396i	$0.188967\pi$
$390$	0	0
$391$	1.55051	0.0784127
$392$	−1.00000	−0.0505076
$393$	4.89898	0.247121
$394$	15.1464	0.763066
$395$	0	0
$396$	4.89898	0.246183
$397$	−8.04541	−0.403787	−0.201894	−	0.979407i	$-0.564710\pi$
$397$	−8.04541	−0.403787	−0.201894	+	0.979407i	$0.564710\pi$
$398$	26.0454	1.30554
$399$	−7.10102	−0.355496
$400$	0	0
$401$	28.6969	1.43306	0.716528	−	0.697558i	$-0.245730\pi$
$401$	28.6969	1.43306	0.716528	+	0.697558i	$0.245730\pi$
$402$	4.00000	0.199502
$403$	9.30306	0.463419
$404$	−15.7980	−0.785978
$405$	0	0
$406$	−13.1010	−0.650193
$407$	−21.7980	−1.08048
$408$	1.00000	0.0495074
$409$	−23.5959	−1.16674	−0.583372	−	0.812205i	$-0.698267\pi$
$409$	−23.5959	−1.16674	−0.583372	+	0.812205i	$0.698267\pi$
$410$	0	0
$411$	12.0000	0.591916
$412$	−7.10102	−0.349842
$413$	33.7980	1.66309
$414$	−1.55051	−0.0762034
$415$	0	0
$416$	−6.00000	−0.294174
$417$	7.10102	0.347738
$418$	−14.2020	−0.694645
$419$	−31.5959	−1.54356	−0.771781	−	0.635889i	$-0.780634\pi$
$419$	−31.5959	−1.54356	−0.771781	+	0.635889i	$0.780634\pi$
$420$	0	0
$421$	28.6969	1.39860	0.699302	−	0.714827i	$-0.253494\pi$
$421$	28.6969	1.39860	0.699302	+	0.714827i	$0.253494\pi$
$422$	−12.8990	−0.627912
$423$	4.89898	0.238197
$424$	10.8990	0.529301
$425$	0	0
$426$	2.44949	0.118678
$427$	3.30306	0.159846
$428$	−12.8990	−0.623496
$429$	29.3939	1.41915
$430$	0	0
$431$	−9.14643	−0.440568	−0.220284	−	0.975436i	$-0.570698\pi$
$431$	−9.14643	−0.440568	−0.220284	+	0.975436i	$0.570698\pi$
$432$	−1.00000	−0.0481125
$433$	10.0000	0.480569	0.240285	−	0.970702i	$-0.422759\pi$
$433$	10.0000	0.480569	0.240285	+	0.970702i	$0.422759\pi$
$434$	3.79796	0.182308
$435$	0	0
$436$	−6.24745	−0.299199
$437$	4.49490	0.215020
$438$	5.10102	0.243736
$439$	−12.6515	−0.603825	−0.301912	−	0.953336i	$-0.597625\pi$
$439$	−12.6515	−0.603825	−0.301912	+	0.953336i	$0.597625\pi$
$440$	0	0
$441$	−1.00000	−0.0476190
$442$	6.00000	0.285391
$443$	29.7980	1.41574	0.707872	−	0.706341i	$-0.249655\pi$
$443$	29.7980	1.41574	0.707872	+	0.706341i	$0.249655\pi$
$444$	4.44949	0.211163
$445$	0	0
$446$	4.89898	0.231973
$447$	16.6969	0.789738
$448$	−2.44949	−0.115728
$449$	−3.30306	−0.155881	−0.0779406	−	0.996958i	$-0.524834\pi$
$449$	−3.30306	−0.155881	−0.0779406	+	0.996958i	$0.524834\pi$
$450$	0	0
$451$	−53.3939	−2.51422
$452$	12.6969	0.597214
$453$	5.79796	0.272412
$454$	−19.5959	−0.919682
$455$	0	0
$456$	2.89898	0.135757
$457$	37.5959	1.75866	0.879331	−	0.476210i	$-0.157990\pi$
$457$	37.5959	1.75866	0.879331	+	0.476210i	$0.157990\pi$
$458$	18.0000	0.841085
$459$	1.00000	0.0466760
$460$	0	0
$461$	17.5959	0.819524	0.409762	−	0.912193i	$-0.365612\pi$
$461$	17.5959	0.819524	0.409762	+	0.912193i	$0.365612\pi$
$462$	12.0000	0.558291
$463$	34.6969	1.61250	0.806252	−	0.591573i	$-0.201493\pi$
$463$	34.6969	1.61250	0.806252	+	0.591573i	$0.201493\pi$
$464$	5.34847	0.248296
$465$	0	0
$466$	10.8990	0.504885
$467$	39.5959	1.83228	0.916140	−	0.400858i	$-0.131288\pi$
$467$	39.5959	1.83228	0.916140	+	0.400858i	$0.131288\pi$
$468$	−6.00000	−0.277350
$469$	9.79796	0.452428
$470$	0	0
$471$	−12.6969	−0.585044
$472$	−13.7980	−0.635103
$473$	−33.7980	−1.55403
$474$	6.44949	0.296235
$475$	0	0
$476$	2.44949	0.112272
$477$	10.8990	0.499030
$478$	−0.898979	−0.0411184
$479$	0.651531	0.0297692	0.0148846	−	0.999889i	$-0.495262\pi$
$479$	0.651531	0.0297692	0.0148846	+	0.999889i	$0.495262\pi$
$480$	0	0
$481$	26.6969	1.21728
$482$	2.89898	0.132045
$483$	−3.79796	−0.172813
$484$	13.0000	0.590909
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	0.651531	0.0295237	0.0147618	−	0.999891i	$-0.495301\pi$
$487$	0.651531	0.0295237	0.0147618	+	0.999891i	$0.495301\pi$
$488$	−1.34847	−0.0610423
$489$	−13.7980	−0.623965
$490$	0	0
$491$	−5.10102	−0.230206	−0.115103	−	0.993354i	$-0.536720\pi$
$491$	−5.10102	−0.230206	−0.115103	+	0.993354i	$0.536720\pi$
$492$	10.8990	0.491364
$493$	−5.34847	−0.240883
$494$	17.3939	0.782588
$495$	0	0
$496$	−1.55051	−0.0696200
$497$	6.00000	0.269137
$498$	−6.89898	−0.309151
$499$	−8.89898	−0.398373	−0.199187	−	0.979962i	$-0.563830\pi$
$499$	−8.89898	−0.398373	−0.199187	+	0.979962i	$0.563830\pi$
$500$	0	0
$501$	19.3485	0.864426
$502$	8.69694	0.388163
$503$	−38.4495	−1.71438	−0.857189	−	0.515002i	$-0.827791\pi$
$503$	−38.4495	−1.71438	−0.857189	+	0.515002i	$0.827791\pi$
$504$	−2.44949	−0.109109
$505$	0	0
$506$	−7.59592	−0.337680
$507$	−23.0000	−1.02147
$508$	8.00000	0.354943
$509$	−1.10102	−0.0488019	−0.0244009	−	0.999702i	$-0.507768\pi$
$509$	−1.10102	−0.0488019	−0.0244009	+	0.999702i	$0.507768\pi$
$510$	0	0
$511$	12.4949	0.552742
$512$	1.00000	0.0441942
$513$	2.89898	0.127993
$514$	20.0000	0.882162
$515$	0	0
$516$	6.89898	0.303711
$517$	24.0000	1.05552
$518$	10.8990	0.478874
$519$	1.34847	0.0591912
$520$	0	0
$521$	12.6969	0.556263	0.278131	−	0.960543i	$-0.410285\pi$
$521$	12.6969	0.556263	0.278131	+	0.960543i	$0.410285\pi$
$522$	5.34847	0.234096
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	−4.89898	−0.214013
$525$	0	0
$526$	−2.69694	−0.117592
$527$	1.55051	0.0675413
$528$	−4.89898	−0.213201
$529$	−20.5959	−0.895475
$530$	0	0
$531$	−13.7980	−0.598780
$532$	7.10102	0.307868
$533$	65.3939	2.83252
$534$	−17.7980	−0.770193
$535$	0	0
$536$	−4.00000	−0.172774
$537$	−8.69694	−0.375301
$538$	28.4495	1.22654
$539$	−4.89898	−0.211014
$540$	0	0
$541$	−3.14643	−0.135276	−0.0676378	−	0.997710i	$-0.521546\pi$
$541$	−3.14643	−0.135276	−0.0676378	+	0.997710i	$0.521546\pi$
$542$	31.5959	1.35716
$543$	0.449490	0.0192895
$544$	−1.00000	−0.0428746
$545$	0	0
$546$	−14.6969	−0.628971
$547$	10.6969	0.457368	0.228684	−	0.973501i	$-0.426558\pi$
$547$	10.6969	0.457368	0.228684	+	0.973501i	$0.426558\pi$
$548$	−12.0000	−0.512615
$549$	−1.34847	−0.0575513
$550$	0	0
$551$	−15.5051	−0.660540
$552$	1.55051	0.0659941
$553$	15.7980	0.671798
$554$	23.5505	1.00057
$555$	0	0
$556$	−7.10102	−0.301150
$557$	−6.89898	−0.292319	−0.146160	−	0.989261i	$-0.546691\pi$
$557$	−6.89898	−0.292319	−0.146160	+	0.989261i	$0.546691\pi$
$558$	−1.55051	−0.0656383
$559$	41.3939	1.75077
$560$	0	0
$561$	4.89898	0.206835
$562$	−9.79796	−0.413302
$563$	2.89898	0.122177	0.0610887	−	0.998132i	$-0.480543\pi$
$563$	2.89898	0.122177	0.0610887	+	0.998132i	$0.480543\pi$
$564$	−4.89898	−0.206284
$565$	0	0
$566$	−22.6969	−0.954023
$567$	−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
$571$	37.3939	1.56489	0.782443	−	0.622723i	$-0.213974\pi$
$571$	37.3939	1.56489	0.782443	+	0.622723i	$0.213974\pi$
$572$	−29.3939	−1.22902
$573$	7.10102	0.296649
$574$	26.6969	1.11431
$575$	0	0
$576$	1.00000	0.0416667
$577$	29.7980	1.24051	0.620253	−	0.784402i	$-0.287030\pi$
$577$	29.7980	1.24051	0.620253	+	0.784402i	$0.287030\pi$
$578$	1.00000	0.0415945
$579$	14.0000	0.581820
$580$	0	0
$581$	−16.8990	−0.701088
$582$	15.7980	0.654846
$583$	53.3939	2.21135
$584$	−5.10102	−0.211082
$585$	0	0
$586$	10.0000	0.413096
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	1.00000	0.0412393
$589$	4.49490	0.185209
$590$	0	0
$591$	−15.1464	−0.623041
$592$	−4.44949	−0.182873
$593$	17.7980	0.730875	0.365437	−	0.930836i	$-0.380919\pi$
$593$	17.7980	0.730875	0.365437	+	0.930836i	$0.380919\pi$
$594$	−4.89898	−0.201008
$595$	0	0
$596$	−16.6969	−0.683933
$597$	−26.0454	−1.06597
$598$	9.30306	0.380430
$599$	−5.79796	−0.236898	−0.118449	−	0.992960i	$-0.537792\pi$
$599$	−5.79796	−0.236898	−0.118449	+	0.992960i	$0.537792\pi$
$600$	0	0
$601$	19.7980	0.807576	0.403788	−	0.914853i	$-0.367693\pi$
$601$	19.7980	0.807576	0.403788	+	0.914853i	$0.367693\pi$
$602$	16.8990	0.688751
$603$	−4.00000	−0.162893
$604$	−5.79796	−0.235916
$605$	0	0
$606$	15.7980	0.641748
$607$	21.5505	0.874708	0.437354	−	0.899289i	$-0.355916\pi$
$607$	21.5505	0.874708	0.437354	+	0.899289i	$0.355916\pi$
$608$	−2.89898	−0.117569
$609$	13.1010	0.530880
$610$	0	0
$611$	−29.3939	−1.18915
$612$	−1.00000	−0.0404226
$613$	−20.2929	−0.819621	−0.409810	−	0.912171i	$-0.634405\pi$
$613$	−20.2929	−0.819621	−0.409810	+	0.912171i	$0.634405\pi$
$614$	−1.10102	−0.0444336
$615$	0	0
$616$	−12.0000	−0.483494
$617$	−22.4949	−0.905610	−0.452805	−	0.891609i	$-0.649577\pi$
$617$	−22.4949	−0.905610	−0.452805	+	0.891609i	$0.649577\pi$
$618$	7.10102	0.285645
$619$	−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
$621$	1.55051	0.0622198
$622$	−8.65153	−0.346895
$623$	−43.5959	−1.74663
$624$	6.00000	0.240192
$625$	0	0
$626$	−2.00000	−0.0799361
$627$	14.2020	0.567175
$628$	12.6969	0.506663
$629$	4.44949	0.177413
$630$	0	0
$631$	−24.4949	−0.975126	−0.487563	−	0.873088i	$-0.662114\pi$
$631$	−24.4949	−0.975126	−0.487563	+	0.873088i	$0.662114\pi$
$632$	−6.44949	−0.256547
$633$	12.8990	0.512688
$634$	3.55051	0.141009
$635$	0	0
$636$	−10.8990	−0.432173
$637$	6.00000	0.237729
$638$	26.2020	1.03735
$639$	−2.44949	−0.0969003
$640$	0	0
$641$	−20.6969	−0.817480	−0.408740	−	0.912651i	$-0.634032\pi$
$641$	−20.6969	−0.817480	−0.408740	+	0.912651i	$0.634032\pi$
$642$	12.8990	0.509082
$643$	8.00000	0.315489	0.157745	−	0.987480i	$-0.449578\pi$
$643$	8.00000	0.315489	0.157745	+	0.987480i	$0.449578\pi$
$644$	3.79796	0.149661
$645$	0	0
$646$	2.89898	0.114059
$647$	38.2929	1.50545	0.752724	−	0.658336i	$-0.228740\pi$
$647$	38.2929	1.50545	0.752724	+	0.658336i	$0.228740\pi$
$648$	1.00000	0.0392837
$649$	−67.5959	−2.65337
$650$	0	0
$651$	−3.79796	−0.148854
$652$	13.7980	0.540370
$653$	−15.5505	−0.608538	−0.304269	−	0.952586i	$-0.598412\pi$
$653$	−15.5505	−0.608538	−0.304269	+	0.952586i	$0.598412\pi$
$654$	6.24745	0.244295
$655$	0	0
$656$	−10.8990	−0.425534
$657$	−5.10102	−0.199010
$658$	−12.0000	−0.467809
$659$	23.5959	0.919166	0.459583	−	0.888135i	$-0.347999\pi$
$659$	23.5959	0.919166	0.459583	+	0.888135i	$0.347999\pi$
$660$	0	0
$661$	2.49490	0.0970403	0.0485201	−	0.998822i	$-0.484549\pi$
$661$	2.49490	0.0970403	0.0485201	+	0.998822i	$0.484549\pi$
$662$	−25.3939	−0.986961
$663$	−6.00000	−0.233021
$664$	6.89898	0.267732
$665$	0	0
$666$	−4.44949	−0.172414
$667$	−8.29286	−0.321101
$668$	−19.3485	−0.748615
$669$	−4.89898	−0.189405
$670$	0	0
$671$	−6.60612	−0.255027
$672$	2.44949	0.0944911
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	6.89898	0.265739
$675$	0	0
$676$	23.0000	0.884615
$677$	35.6413	1.36981	0.684904	−	0.728634i	$-0.259844\pi$
$677$	35.6413	1.36981	0.684904	+	0.728634i	$0.259844\pi$
$678$	−12.6969	−0.487623
$679$	38.6969	1.48505
$680$	0	0
$681$	19.5959	0.750917
$682$	−7.59592	−0.290863
$683$	−9.79796	−0.374908	−0.187454	−	0.982273i	$-0.560024\pi$
$683$	−9.79796	−0.374908	−0.187454	+	0.982273i	$0.560024\pi$
$684$	−2.89898	−0.110845
$685$	0	0
$686$	19.5959	0.748176
$687$	−18.0000	−0.686743
$688$	−6.89898	−0.263021
$689$	−65.3939	−2.49131
$690$	0	0
$691$	18.2929	0.695893	0.347947	−	0.937514i	$-0.386879\pi$
$691$	18.2929	0.695893	0.347947	+	0.937514i	$0.386879\pi$
$692$	−1.34847	−0.0512611
$693$	−12.0000	−0.455842
$694$	−5.79796	−0.220088
$695$	0	0
$696$	−5.34847	−0.202733
$697$	10.8990	0.412828
$698$	36.6969	1.38900
$699$	−10.8990	−0.412237
$700$	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
$703$	12.8990	0.486494
$704$	4.89898	0.184637
$705$	0	0
$706$	−10.0000	−0.376355
$707$	38.6969	1.45535
$708$	13.7980	0.518559
$709$	−19.5505	−0.734235	−0.367117	−	0.930175i	$-0.619655\pi$
$709$	−19.5505	−0.734235	−0.367117	+	0.930175i	$0.619655\pi$
$710$	0	0
$711$	−6.44949	−0.241875
$712$	17.7980	0.667007
$713$	2.40408	0.0900336
$714$	−2.44949	−0.0916698
$715$	0	0
$716$	8.69694	0.325020
$717$	0.898979	0.0335730
$718$	−25.7980	−0.962771
$719$	−12.2474	−0.456753	−0.228376	−	0.973573i	$-0.573342\pi$
$719$	−12.2474	−0.456753	−0.228376	+	0.973573i	$0.573342\pi$
$720$	0	0
$721$	17.3939	0.647782
$722$	−10.5959	−0.394339
$723$	−2.89898	−0.107814
$724$	−0.449490	−0.0167052
$725$	0	0
$726$	−13.0000	−0.482475
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	14.6969	0.544705
$729$	1.00000	0.0370370
$730$	0	0
$731$	6.89898	0.255168
$732$	1.34847	0.0498409
$733$	−0.202041	−0.00746256	−0.00373128	−	0.999993i	$-0.501188\pi$
$733$	−0.202041	−0.00746256	−0.00373128	+	0.999993i	$0.501188\pi$
$734$	−30.0454	−1.10900
$735$	0	0
$736$	−1.55051	−0.0571526
$737$	−19.5959	−0.721825
$738$	−10.8990	−0.401197
$739$	−26.8990	−0.989495	−0.494747	−	0.869037i	$-0.664739\pi$
$739$	−26.8990	−0.989495	−0.494747	+	0.869037i	$0.664739\pi$
$740$	0	0
$741$	−17.3939	−0.638980
$742$	−26.6969	−0.980075
$743$	−9.55051	−0.350374	−0.175187	−	0.984535i	$-0.556053\pi$
$743$	−9.55051	−0.350374	−0.175187	+	0.984535i	$0.556053\pi$
$744$	1.55051	0.0568445
$745$	0	0
$746$	−7.79796	−0.285504
$747$	6.89898	0.252420
$748$	−4.89898	−0.179124
$749$	31.5959	1.15449
$750$	0	0
$751$	−16.2474	−0.592878	−0.296439	−	0.955052i	$-0.595799\pi$
$751$	−16.2474	−0.592878	−0.296439	+	0.955052i	$0.595799\pi$
$752$	4.89898	0.178647
$753$	−8.69694	−0.316934
$754$	−32.0908	−1.16868
$755$	0	0
$756$	2.44949	0.0890871
$757$	−54.4949	−1.98065	−0.990325	−	0.138765i	$-0.955687\pi$
$757$	−54.4949	−1.98065	−0.990325	+	0.138765i	$0.955687\pi$
$758$	−29.7980	−1.08231
$759$	7.59592	0.275714
$760$	0	0
$761$	−29.5959	−1.07285	−0.536426	−	0.843948i	$-0.680226\pi$
$761$	−29.5959	−1.07285	−0.536426	+	0.843948i	$0.680226\pi$
$762$	−8.00000	−0.289809
$763$	15.3031	0.554008
$764$	−7.10102	−0.256906
$765$	0	0
$766$	−7.10102	−0.256570
$767$	82.7878	2.98929
$768$	−1.00000	−0.0360844
$769$	−31.3939	−1.13209	−0.566046	−	0.824374i	$-0.691528\pi$
$769$	−31.3939	−1.13209	−0.566046	+	0.824374i	$0.691528\pi$
$770$	0	0
$771$	−20.0000	−0.720282
$772$	−14.0000	−0.503871
$773$	−42.0000	−1.51064	−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000	−1.51064	−0.755318	+	0.655359i	$0.772517\pi$
$774$	−6.89898	−0.247979
$775$	0	0
$776$	−15.7980	−0.567114
$777$	−10.8990	−0.390999
$778$	32.6969	1.17224
$779$	31.5959	1.13204
$780$	0	0
$781$	−12.0000	−0.429394
$782$	1.55051	0.0554461
$783$	−5.34847	−0.191139
$784$	−1.00000	−0.0357143
$785$	0	0
$786$	4.89898	0.174741
$787$	46.6969	1.66457	0.832283	−	0.554351i	$-0.187033\pi$
$787$	46.6969	1.66457	0.832283	+	0.554351i	$0.187033\pi$
$788$	15.1464	0.539569
$789$	2.69694	0.0960136
$790$	0	0
$791$	−31.1010	−1.10582
$792$	4.89898	0.174078
$793$	8.09082	0.287313
$794$	−8.04541	−0.285521
$795$	0	0
$796$	26.0454	0.923155
$797$	13.1010	0.464062	0.232031	−	0.972708i	$-0.425463\pi$
$797$	13.1010	0.464062	0.232031	+	0.972708i	$0.425463\pi$
$798$	−7.10102	−0.251373
$799$	−4.89898	−0.173313
$800$	0	0
$801$	17.7980	0.628860
$802$	28.6969	1.01332
$803$	−24.9898	−0.881871
$804$	4.00000	0.141069
$805$	0	0
$806$	9.30306	0.327686
$807$	−28.4495	−1.00147
$808$	−15.7980	−0.555770
$809$	−46.0908	−1.62047	−0.810233	−	0.586107i	$-0.800660\pi$
$809$	−46.0908	−1.62047	−0.810233	+	0.586107i	$0.800660\pi$
$810$	0	0
$811$	−17.7980	−0.624971	−0.312485	−	0.949923i	$-0.601162\pi$
$811$	−17.7980	−0.624971	−0.312485	+	0.949923i	$0.601162\pi$
$812$	−13.1010	−0.459756
$813$	−31.5959	−1.10812
$814$	−21.7980	−0.764018
$815$	0	0
$816$	1.00000	0.0350070
$817$	20.0000	0.699711
$818$	−23.5959	−0.825012
$819$	14.6969	0.513553
$820$	0	0
$821$	−47.6413	−1.66269	−0.831347	−	0.555754i	$-0.812430\pi$
$821$	−47.6413	−1.66269	−0.831347	+	0.555754i	$0.812430\pi$
$822$	12.0000	0.418548
$823$	21.5505	0.751204	0.375602	−	0.926781i	$-0.377436\pi$
$823$	21.5505	0.751204	0.375602	+	0.926781i	$0.377436\pi$
$824$	−7.10102	−0.247376
$825$	0	0
$826$	33.7980	1.17598
$827$	32.4949	1.12996	0.564979	−	0.825105i	$-0.308884\pi$
$827$	32.4949	1.12996	0.564979	+	0.825105i	$0.308884\pi$
$828$	−1.55051	−0.0538840
$829$	27.7980	0.965463	0.482732	−	0.875768i	$-0.339645\pi$
$829$	27.7980	0.965463	0.482732	+	0.875768i	$0.339645\pi$
$830$	0	0
$831$	−23.5505	−0.816958
$832$	−6.00000	−0.208013
$833$	1.00000	0.0346479
$834$	7.10102	0.245888
$835$	0	0
$836$	−14.2020	−0.491188
$837$	1.55051	0.0535935
$838$	−31.5959	−1.09146
$839$	−24.6515	−0.851065	−0.425533	−	0.904943i	$-0.639913\pi$
$839$	−24.6515	−0.851065	−0.425533	+	0.904943i	$0.639913\pi$
$840$	0	0
$841$	−0.393877	−0.0135820
$842$	28.6969	0.988962
$843$	9.79796	0.337460
$844$	−12.8990	−0.444001
$845$	0	0
$846$	4.89898	0.168430
$847$	−31.8434	−1.09415
$848$	10.8990	0.374272
$849$	22.6969	0.778957
$850$	0	0
$851$	6.89898	0.236494
$852$	2.44949	0.0839181
$853$	−25.3485	−0.867915	−0.433958	−	0.900933i	$-0.642883\pi$
$853$	−25.3485	−0.867915	−0.433958	+	0.900933i	$0.642883\pi$
$854$	3.30306	0.113028
$855$	0	0
$856$	−12.8990	−0.440878
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	29.3939	1.00349
$859$	−45.7980	−1.56261	−0.781303	−	0.624152i	$-0.785445\pi$
$859$	−45.7980	−1.56261	−0.781303	+	0.624152i	$0.785445\pi$
$860$	0	0
$861$	−26.6969	−0.909829
$862$	−9.14643	−0.311529
$863$	15.5959	0.530891	0.265446	−	0.964126i	$-0.414481\pi$
$863$	15.5959	0.530891	0.265446	+	0.964126i	$0.414481\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	10.0000	0.339814
$867$	−1.00000	−0.0339618
$868$	3.79796	0.128911
$869$	−31.5959	−1.07182
$870$	0	0
$871$	24.0000	0.813209
$872$	−6.24745	−0.211565
$873$	−15.7980	−0.534680
$874$	4.49490	0.152042
$875$	0	0
$876$	5.10102	0.172348
$877$	−33.3485	−1.12610	−0.563049	−	0.826424i	$-0.690372\pi$
$877$	−33.3485	−1.12610	−0.563049	+	0.826424i	$0.690372\pi$
$878$	−12.6515	−0.426968
$879$	−10.0000	−0.337292
$880$	0	0
$881$	41.1918	1.38779	0.693894	−	0.720077i	$-0.255894\pi$
$881$	41.1918	1.38779	0.693894	+	0.720077i	$0.255894\pi$
$882$	−1.00000	−0.0336718
$883$	29.7980	1.00278	0.501391	−	0.865221i	$-0.332822\pi$
$883$	29.7980	1.00278	0.501391	+	0.865221i	$0.332822\pi$
$884$	6.00000	0.201802
$885$	0	0
$886$	29.7980	1.00108
$887$	−15.7526	−0.528919	−0.264459	−	0.964397i	$-0.585194\pi$
$887$	−15.7526	−0.528919	−0.264459	+	0.964397i	$0.585194\pi$
$888$	4.44949	0.149315
$889$	−19.5959	−0.657226
$890$	0	0
$891$	4.89898	0.164122
$892$	4.89898	0.164030
$893$	−14.2020	−0.475253
$894$	16.6969	0.558429
$895$	0	0
$896$	−2.44949	−0.0818317
$897$	−9.30306	−0.310620
$898$	−3.30306	−0.110225
$899$	−8.29286	−0.276582
$900$	0	0
$901$	−10.8990	−0.363098
$902$	−53.3939	−1.77782
$903$	−16.8990	−0.562363
$904$	12.6969	0.422294
$905$	0	0
$906$	5.79796	0.192624
$907$	−21.3031	−0.707357	−0.353678	−	0.935367i	$-0.615069\pi$
$907$	−21.3031	−0.707357	−0.353678	+	0.935367i	$0.615069\pi$
$908$	−19.5959	−0.650313
$909$	−15.7980	−0.523985
$910$	0	0
$911$	−47.3485	−1.56872	−0.784362	−	0.620303i	$-0.787010\pi$
$911$	−47.3485	−1.56872	−0.784362	+	0.620303i	$0.787010\pi$
$912$	2.89898	0.0959948
$913$	33.7980	1.11855
$914$	37.5959	1.24356
$915$	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
$921$	1.10102	0.0362799
$922$	17.5959	0.579491
$923$	14.6969	0.483756
$924$	12.0000	0.394771
$925$	0	0
$926$	34.6969	1.14021
$927$	−7.10102	−0.233228
$928$	5.34847	0.175572
$929$	−7.30306	−0.239606	−0.119803	−	0.992798i	$-0.538226\pi$
$929$	−7.30306	−0.239606	−0.119803	+	0.992798i	$0.538226\pi$
$930$	0	0
$931$	2.89898	0.0950102
$932$	10.8990	0.357008
$933$	8.65153	0.283238
$934$	39.5959	1.29562
$935$	0	0
$936$	−6.00000	−0.196116
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	9.79796	0.319915
$939$	2.00000	0.0652675
$940$	0	0
$941$	34.2474	1.11643	0.558217	−	0.829695i	$-0.311485\pi$
$941$	34.2474	1.11643	0.558217	+	0.829695i	$0.311485\pi$
$942$	−12.6969	−0.413689
$943$	16.8990	0.550306
$944$	−13.7980	−0.449085
$945$	0	0
$946$	−33.7980	−1.09887
$947$	−40.8990	−1.32904	−0.664519	−	0.747271i	$-0.731364\pi$
$947$	−40.8990	−1.32904	−0.664519	+	0.747271i	$0.731364\pi$
$948$	6.44949	0.209470
$949$	30.6061	0.993517
$950$	0	0
$951$	−3.55051	−0.115133
$952$	2.44949	0.0793884
$953$	−5.59592	−0.181270	−0.0906348	−	0.995884i	$-0.528890\pi$
$953$	−5.59592	−0.181270	−0.0906348	+	0.995884i	$0.528890\pi$
$954$	10.8990	0.352867
$955$	0	0
$956$	−0.898979	−0.0290751
$957$	−26.2020	−0.846992
$958$	0.651531	0.0210500
$959$	29.3939	0.949178
$960$	0	0
$961$	−28.5959	−0.922449
$962$	26.6969	0.860744
$963$	−12.8990	−0.415664
$964$	2.89898	0.0933698
$965$	0	0
$966$	−3.79796	−0.122197
$967$	13.3031	0.427798	0.213899	−	0.976856i	$-0.431384\pi$
$967$	13.3031	0.427798	0.213899	+	0.976856i	$0.431384\pi$
$968$	13.0000	0.417836
$969$	−2.89898	−0.0931286
$970$	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
$973$	17.3939	0.557622
$974$	0.651531	0.0208764
$975$	0	0
$976$	−1.34847	−0.0431634
$977$	−39.5959	−1.26679	−0.633393	−	0.773830i	$-0.718338\pi$
$977$	−39.5959	−1.26679	−0.633393	+	0.773830i	$0.718338\pi$
$978$	−13.7980	−0.441210
$979$	87.1918	2.78666
$980$	0	0
$981$	−6.24745	−0.199466
$982$	−5.10102	−0.162780
$983$	−53.1464	−1.69511	−0.847554	−	0.530709i	$-0.821926\pi$
$983$	−53.1464	−1.69511	−0.847554	+	0.530709i	$0.821926\pi$
$984$	10.8990	0.347447
$985$	0	0
$986$	−5.34847	−0.170330
$987$	12.0000	0.381964
$988$	17.3939	0.553373
$989$	10.6969	0.340143
$990$	0	0
$991$	7.75255	0.246268	0.123134	−	0.992390i	$-0.460706\pi$
$991$	7.75255	0.246268	0.123134	+	0.992390i	$0.460706\pi$
$992$	−1.55051	−0.0492287
$993$	25.3939	0.805850
$994$	6.00000	0.190308
$995$	0	0
$996$	−6.89898	−0.218603
$997$	26.7423	0.846939	0.423469	−	0.905910i	$-0.360812\pi$
$997$	26.7423	0.846939	0.423469	+	0.905910i	$0.360812\pi$
$998$	−8.89898	−0.281692
$999$	4.44949	0.140776

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2550.2.a.bj.1.1		2
3.2	odd	2		7650.2.a.ct.1.1		2
5.2	odd	4		510.2.d.c.409.4	yes	4
5.3	odd	4		510.2.d.c.409.2	✓	4
5.4	even	2		2550.2.a.bi.1.2		2
15.2	even	4		1530.2.d.e.919.1		4
15.8	even	4		1530.2.d.e.919.3		4
15.14	odd	2		7650.2.a.dg.1.2		2
20.3	even	4		4080.2.m.o.2449.4		4
20.7	even	4		4080.2.m.o.2449.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
510.2.d.c.409.2	✓	4	5.3	odd	4
510.2.d.c.409.4	yes	4	5.2	odd	4
1530.2.d.e.919.1		4	15.2	even	4
1530.2.d.e.919.3		4	15.8	even	4
2550.2.a.bi.1.2		2	5.4	even	2
2550.2.a.bj.1.1		2	1.1	even	1	trivial
4080.2.m.o.2449.2		4	20.7	even	4
4080.2.m.o.2449.4		4	20.3	even	4
7650.2.a.ct.1.1		2	3.2	odd	2
7650.2.a.dg.1.2		2	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2550.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.44949 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.44949 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.89898 q^{11} -1.00000 q^{12} -6.00000 q^{13} -2.44949 q^{14} +1.00000 q^{16} -1.00000 q^{17} +1.00000 q^{18} -2.89898 q^{19} +2.44949 q^{21} +4.89898 q^{22} -1.55051 q^{23} -1.00000 q^{24} -6.00000 q^{26} -1.00000 q^{27} -2.44949 q^{28} +5.34847 q^{29} -1.55051 q^{31} +1.00000 q^{32} -4.89898 q^{33} -1.00000 q^{34} +1.00000 q^{36} -4.44949 q^{37} -2.89898 q^{38} +6.00000 q^{39} -10.8990 q^{41} +2.44949 q^{42} -6.89898 q^{43} +4.89898 q^{44} -1.55051 q^{46} +4.89898 q^{47} -1.00000 q^{48} -1.00000 q^{49} +1.00000 q^{51} -6.00000 q^{52} +10.8990 q^{53} -1.00000 q^{54} -2.44949 q^{56} +2.89898 q^{57} +5.34847 q^{58} -13.7980 q^{59} -1.34847 q^{61} -1.55051 q^{62} -2.44949 q^{63} +1.00000 q^{64} -4.89898 q^{66} -4.00000 q^{67} -1.00000 q^{68} +1.55051 q^{69} -2.44949 q^{71} +1.00000 q^{72} -5.10102 q^{73} -4.44949 q^{74} -2.89898 q^{76} -12.0000 q^{77} +6.00000 q^{78} -6.44949 q^{79} +1.00000 q^{81} -10.8990 q^{82} +6.89898 q^{83} +2.44949 q^{84} -6.89898 q^{86} -5.34847 q^{87} +4.89898 q^{88} +17.7980 q^{89} +14.6969 q^{91} -1.55051 q^{92} +1.55051 q^{93} +4.89898 q^{94} -1.00000 q^{96} -15.7980 q^{97} -1.00000 q^{98} +4.89898 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^2 - 2 * q^3 + 2 * q^4 - 2 * q^6 + 2 * q^8 + 2 * q^9	\( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{6} + 2 q^{8} + 2 q^{9} - 2 q^{12} - 12 q^{13} + 2 q^{16} - 2 q^{17} + 2 q^{18} + 4 q^{19} - 8 q^{23} - 2 q^{24} - 12 q^{26} - 2 q^{27} - 4 q^{29} - 8 q^{31} + 2 q^{32} - 2 q^{34} + 2 q^{36} - 4 q^{37} + 4 q^{38} + 12 q^{39} - 12 q^{41} - 4 q^{43} - 8 q^{46} - 2 q^{48} - 2 q^{49} + 2 q^{51} - 12 q^{52} + 12 q^{53} - 2 q^{54} - 4 q^{57} - 4 q^{58} - 8 q^{59} + 12 q^{61} - 8 q^{62} + 2 q^{64} - 8 q^{67} - 2 q^{68} + 8 q^{69} + 2 q^{72} - 20 q^{73} - 4 q^{74} + 4 q^{76} - 24 q^{77} + 12 q^{78} - 8 q^{79} + 2 q^{81} - 12 q^{82} + 4 q^{83} - 4 q^{86} + 4 q^{87} + 16 q^{89} - 8 q^{92} + 8 q^{93} - 2 q^{96} - 12 q^{97} - 2 q^{98}+O(q^{100}) \) 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

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