LMFDB - Embedded newform 3450.2.a.bp.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3450,2,Mod(1,3450)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3450, 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("3450.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3450.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:	$27.5483886973$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.3368.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 15x + 11 $ x^3 - x^2 - 15*x + 11
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$4.03932$ of defining polynomial
Character	$\chi$	$=$	3450.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.03932 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.03932 q^{7} -1.00000 q^{8} +1.00000 q^{9} -5.69736 q^{11} +1.00000 q^{12} +0.381275 q^{13} -4.03932 q^{14} +1.00000 q^{16} -3.65804 q^{17} -1.00000 q^{18} +0.381275 q^{19} +4.03932 q^{21} +5.69736 q^{22} -1.00000 q^{23} -1.00000 q^{24} -0.381275 q^{26} +1.00000 q^{27} +4.03932 q^{28} -1.00000 q^{29} +6.00000 q^{31} -1.00000 q^{32} -5.69736 q^{33} +3.65804 q^{34} +1.00000 q^{36} +3.65804 q^{37} -0.381275 q^{38} +0.381275 q^{39} +5.61873 q^{41} -4.03932 q^{42} +3.61873 q^{43} -5.69736 q^{44} +1.00000 q^{46} -3.38127 q^{47} +1.00000 q^{48} +9.31608 q^{49} -3.65804 q^{51} +0.381275 q^{52} +9.31608 q^{53} -1.00000 q^{54} -4.03932 q^{56} +0.381275 q^{57} +1.00000 q^{58} +13.3947 q^{59} +14.0786 q^{61} -6.00000 q^{62} +4.03932 q^{63} +1.00000 q^{64} +5.69736 q^{66} -2.76255 q^{67} -3.65804 q^{68} -1.00000 q^{69} +10.6974 q^{71} -1.00000 q^{72} -7.07863 q^{73} -3.65804 q^{74} +0.381275 q^{76} -23.0134 q^{77} -0.381275 q^{78} +5.69736 q^{79} +1.00000 q^{81} -5.61873 q^{82} +16.1180 q^{83} +4.03932 q^{84} -3.61873 q^{86} -1.00000 q^{87} +5.69736 q^{88} +5.65804 q^{89} +1.54009 q^{91} -1.00000 q^{92} +6.00000 q^{93} +3.38127 q^{94} -1.00000 q^{96} -4.07863 q^{97} -9.31608 q^{98} -5.69736 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{6} + q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^6 + q^7 - 3 * q^8 + 3 * q^9	$ 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{6} + q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{11} + 3 q^{12} - q^{13} - q^{14} + 3 q^{16} - 2 q^{17} - 3 q^{18} - q^{19} + q^{21} - 3 q^{22} - 3 q^{23} - 3 q^{24} + q^{26} + 3 q^{27} + q^{28} - 3 q^{29} + 18 q^{31} - 3 q^{32} + 3 q^{33} + 2 q^{34} + 3 q^{36} + 2 q^{37} + q^{38} - q^{39} + 19 q^{41} - q^{42} + 13 q^{43} + 3 q^{44} + 3 q^{46} - 8 q^{47} + 3 q^{48} + 10 q^{49} - 2 q^{51} - q^{52} + 10 q^{53} - 3 q^{54} - q^{56} - q^{57} + 3 q^{58} + 20 q^{61} - 18 q^{62} + q^{63} + 3 q^{64} - 3 q^{66} - 4 q^{67} - 2 q^{68} - 3 q^{69} + 12 q^{71} - 3 q^{72} + q^{73} - 2 q^{74} - q^{76} - 31 q^{77} + q^{78} - 3 q^{79} + 3 q^{81} - 19 q^{82} + 15 q^{83} + q^{84} - 13 q^{86} - 3 q^{87} - 3 q^{88} + 8 q^{89} + 29 q^{91} - 3 q^{92} + 18 q^{93} + 8 q^{94} - 3 q^{96} + 10 q^{97} - 10 q^{98} + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^6 + q^7 - 3 * q^8 + 3 * q^9 + 3 * q^11 + 3 * q^12 - q^13 - q^14 + 3 * q^16 - 2 * q^17 - 3 * q^18 - q^19 + q^21 - 3 * q^22 - 3 * q^23 - 3 * q^24 + q^26 + 3 * q^27 + q^28 - 3 * q^29 + 18 * q^31 - 3 * q^32 + 3 * q^33 + 2 * q^34 + 3 * q^36 + 2 * q^37 + q^38 - q^39 + 19 * q^41 - q^42 + 13 * q^43 + 3 * q^44 + 3 * q^46 - 8 * q^47 + 3 * q^48 + 10 * q^49 - 2 * q^51 - q^52 + 10 * q^53 - 3 * q^54 - q^56 - q^57 + 3 * q^58 + 20 * q^61 - 18 * q^62 + q^63 + 3 * q^64 - 3 * q^66 - 4 * q^67 - 2 * q^68 - 3 * q^69 + 12 * q^71 - 3 * q^72 + q^73 - 2 * q^74 - q^76 - 31 * q^77 + q^78 - 3 * q^79 + 3 * q^81 - 19 * q^82 + 15 * q^83 + q^84 - 13 * q^86 - 3 * q^87 - 3 * q^88 + 8 * q^89 + 29 * q^91 - 3 * q^92 + 18 * q^93 + 8 * q^94 - 3 * q^96 + 10 * q^97 - 10 * q^98 + 3 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.00000	−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
$7$	4.03932	1.52672	0.763359	−	0.645974i	$-0.223549\pi$
$7$	4.03932	1.52672	0.763359	+	0.645974i	$0.223549\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−5.69736	−1.71782	−0.858909	−	0.512128i	$-0.828857\pi$
$11$	−5.69736	−1.71782	−0.858909	+	0.512128i	$0.828857\pi$
$12$	1.00000	0.288675
$13$	0.381275	0.105747	0.0528733	−	0.998601i	$-0.483162\pi$
$13$	0.381275	0.105747	0.0528733	+	0.998601i	$0.483162\pi$
$14$	−4.03932	−1.07955
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.65804	−0.887206	−0.443603	−	0.896223i	$-0.646300\pi$
$17$	−3.65804	−0.887206	−0.443603	+	0.896223i	$0.646300\pi$
$18$	−1.00000	−0.235702
$19$	0.381275	0.0874705	0.0437352	−	0.999043i	$-0.486074\pi$
$19$	0.381275	0.0874705	0.0437352	+	0.999043i	$0.486074\pi$
$20$	0	0
$21$	4.03932	0.881451
$22$	5.69736	1.21468
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	−1.00000	−0.204124
$25$	0	0
$26$	−0.381275	−0.0747742
$27$	1.00000	0.192450
$28$	4.03932	0.763359
$29$	−1.00000	−0.185695	−0.0928477	−	0.995680i	$-0.529597\pi$
$29$	−1.00000	−0.185695	−0.0928477	+	0.995680i	$0.529597\pi$
$30$	0	0
$31$	6.00000	1.07763	0.538816	−	0.842424i	$-0.318872\pi$
$31$	6.00000	1.07763	0.538816	+	0.842424i	$0.318872\pi$
$32$	−1.00000	−0.176777
$33$	−5.69736	−0.991783
$34$	3.65804	0.627349
$35$	0	0
$36$	1.00000	0.166667
$37$	3.65804	0.601378	0.300689	−	0.953722i	$-0.402783\pi$
$37$	3.65804	0.601378	0.300689	+	0.953722i	$0.402783\pi$
$38$	−0.381275	−0.0618510
$39$	0.381275	0.0610528
$40$	0	0
$41$	5.61873	0.877497	0.438749	−	0.898610i	$-0.355422\pi$
$41$	5.61873	0.877497	0.438749	+	0.898610i	$0.355422\pi$
$42$	−4.03932	−0.623280
$43$	3.61873	0.551850	0.275925	−	0.961179i	$-0.411016\pi$
$43$	3.61873	0.551850	0.275925	+	0.961179i	$0.411016\pi$
$44$	−5.69736	−0.858909
$45$	0	0
$46$	1.00000	0.147442
$47$	−3.38127	−0.493210	−0.246605	−	0.969116i	$-0.579315\pi$
$47$	−3.38127	−0.493210	−0.246605	+	0.969116i	$0.579315\pi$
$48$	1.00000	0.144338
$49$	9.31608	1.33087
$50$	0	0
$51$	−3.65804	−0.512228
$52$	0.381275	0.0528733
$53$	9.31608	1.27966	0.639831	−	0.768515i	$-0.279004\pi$
$53$	9.31608	1.27966	0.639831	+	0.768515i	$0.279004\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−4.03932	−0.539776
$57$	0.381275	0.0505011
$58$	1.00000	0.131306
$59$	13.3947	1.74384	0.871922	−	0.489645i	$-0.162874\pi$
$59$	13.3947	1.74384	0.871922	+	0.489645i	$0.162874\pi$
$60$	0	0
$61$	14.0786	1.80258	0.901292	−	0.433212i	$-0.142620\pi$
$61$	14.0786	1.80258	0.901292	+	0.433212i	$0.142620\pi$
$62$	−6.00000	−0.762001
$63$	4.03932	0.508906
$64$	1.00000	0.125000
$65$	0	0
$66$	5.69736	0.701297
$67$	−2.76255	−0.337499	−0.168750	−	0.985659i	$-0.553973\pi$
$67$	−2.76255	−0.337499	−0.168750	+	0.985659i	$0.553973\pi$
$68$	−3.65804	−0.443603
$69$	−1.00000	−0.120386
$70$	0	0
$71$	10.6974	1.26954	0.634772	−	0.772700i	$-0.281094\pi$
$71$	10.6974	1.26954	0.634772	+	0.772700i	$0.281094\pi$
$72$	−1.00000	−0.117851
$73$	−7.07863	−0.828492	−0.414246	−	0.910165i	$-0.635955\pi$
$73$	−7.07863	−0.828492	−0.414246	+	0.910165i	$0.635955\pi$
$74$	−3.65804	−0.425239
$75$	0	0
$76$	0.381275	0.0437352
$77$	−23.0134	−2.62263
$78$	−0.381275	−0.0431709
$79$	5.69736	0.641003	0.320502	−	0.947248i	$-0.396149\pi$
$79$	5.69736	0.641003	0.320502	+	0.947248i	$0.396149\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−5.61873	−0.620484
$83$	16.1180	1.76918	0.884588	−	0.466374i	$-0.154440\pi$
$83$	16.1180	1.76918	0.884588	+	0.466374i	$0.154440\pi$
$84$	4.03932	0.440726
$85$	0	0
$86$	−3.61873	−0.390217
$87$	−1.00000	−0.107211
$88$	5.69736	0.607341
$89$	5.65804	0.599751	0.299876	−	0.953978i	$-0.403055\pi$
$89$	5.65804	0.599751	0.299876	+	0.953978i	$0.403055\pi$
$90$	0	0
$91$	1.54009	0.161445
$92$	−1.00000	−0.104257
$93$	6.00000	0.622171
$94$	3.38127	0.348752
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−4.07863	−0.414123	−0.207061	−	0.978328i	$-0.566390\pi$
$97$	−4.07863	−0.414123	−0.207061	+	0.978328i	$0.566390\pi$
$98$	−9.31608	−0.941067
$99$	−5.69736	−0.572606
$100$	0	0
$101$	16.7760	1.66927	0.834637	−	0.550801i	$-0.185677\pi$
$101$	16.7760	1.66927	0.834637	+	0.550801i	$0.185677\pi$
$102$	3.65804	0.362200
$103$	13.3554	1.31595	0.657973	−	0.753041i	$-0.271414\pi$
$103$	13.3554	1.31595	0.657973	+	0.753041i	$0.271414\pi$
$104$	−0.381275	−0.0373871
$105$	0	0
$106$	−9.31608	−0.904858
$107$	−6.76255	−0.653760	−0.326880	−	0.945066i	$-0.605997\pi$
$107$	−6.76255	−0.653760	−0.326880	+	0.945066i	$0.605997\pi$
$108$	1.00000	0.0962250
$109$	−13.0528	−1.25023	−0.625114	−	0.780534i	$-0.714948\pi$
$109$	−13.0528	−1.25023	−0.625114	+	0.780534i	$0.714948\pi$
$110$	0	0
$111$	3.65804	0.347206
$112$	4.03932	0.381680
$113$	−1.65804	−0.155976	−0.0779878	−	0.996954i	$-0.524850\pi$
$113$	−1.65804	−0.155976	−0.0779878	+	0.996954i	$0.524850\pi$
$114$	−0.381275	−0.0357097
$115$	0	0
$116$	−1.00000	−0.0928477
$117$	0.381275	0.0352489
$118$	−13.3947	−1.23308
$119$	−14.7760	−1.35451
$120$	0	0
$121$	21.4599	1.95090
$122$	−14.0786	−1.27462
$123$	5.61873	0.506623
$124$	6.00000	0.538816
$125$	0	0
$126$	−4.03932	−0.359851
$127$	−18.6322	−1.65334	−0.826669	−	0.562689i	$-0.809767\pi$
$127$	−18.6322	−1.65334	−0.826669	+	0.562689i	$0.809767\pi$
$128$	−1.00000	−0.0883883
$129$	3.61873	0.318611
$130$	0	0
$131$	−17.4734	−1.52665	−0.763327	−	0.646012i	$-0.776435\pi$
$131$	−17.4734	−1.52665	−0.763327	+	0.646012i	$0.776435\pi$
$132$	−5.69736	−0.495892
$133$	1.54009	0.133543
$134$	2.76255	0.238648
$135$	0	0
$136$	3.65804	0.313675
$137$	−14.3420	−1.22532	−0.612658	−	0.790348i	$-0.709900\pi$
$137$	−14.3420	−1.22532	−0.612658	+	0.790348i	$0.709900\pi$
$138$	1.00000	0.0851257
$139$	−8.01344	−0.679692	−0.339846	−	0.940481i	$-0.610375\pi$
$139$	−8.01344	−0.679692	−0.339846	+	0.940481i	$0.610375\pi$
$140$	0	0
$141$	−3.38127	−0.284755
$142$	−10.6974	−0.897702
$143$	−2.17226	−0.181654
$144$	1.00000	0.0833333
$145$	0	0
$146$	7.07863	0.585832
$147$	9.31608	0.768378
$148$	3.65804	0.300689
$149$	3.23745	0.265222	0.132611	−	0.991168i	$-0.457664\pi$
$149$	3.23745	0.265222	0.132611	+	0.991168i	$0.457664\pi$
$150$	0	0
$151$	−9.31608	−0.758132	−0.379066	−	0.925370i	$-0.623755\pi$
$151$	−9.31608	−0.758132	−0.379066	+	0.925370i	$0.623755\pi$
$152$	−0.381275	−0.0309255
$153$	−3.65804	−0.295735
$154$	23.0134	1.85448
$155$	0	0
$156$	0.381275	0.0305264
$157$	−6.84118	−0.545986	−0.272993	−	0.962016i	$-0.588014\pi$
$157$	−6.84118	−0.545986	−0.272993	+	0.962016i	$0.588014\pi$
$158$	−5.69736	−0.453258
$159$	9.31608	0.738814
$160$	0	0
$161$	−4.03932	−0.318343
$162$	−1.00000	−0.0785674
$163$	4.76255	0.373032	0.186516	−	0.982452i	$-0.440280\pi$
$163$	4.76255	0.373032	0.186516	+	0.982452i	$0.440280\pi$
$164$	5.61873	0.438749
$165$	0	0
$166$	−16.1180	−1.25100
$167$	14.8546	1.14949	0.574743	−	0.818334i	$-0.305102\pi$
$167$	14.8546	1.14949	0.574743	+	0.818334i	$0.305102\pi$
$168$	−4.03932	−0.311640
$169$	−12.8546	−0.988818
$170$	0	0
$171$	0.381275	0.0291568
$172$	3.61873	0.275925
$173$	9.69736	0.737277	0.368638	−	0.929573i	$-0.379824\pi$
$173$	9.69736	0.737277	0.368638	+	0.929573i	$0.379824\pi$
$174$	1.00000	0.0758098
$175$	0	0
$176$	−5.69736	−0.429455
$177$	13.3947	1.00681
$178$	−5.65804	−0.424088
$179$	−5.31608	−0.397343	−0.198671	−	0.980066i	$-0.563663\pi$
$179$	−5.31608	−0.397343	−0.198671	+	0.980066i	$0.563663\pi$
$180$	0	0
$181$	24.3688	1.81132	0.905661	−	0.424002i	$-0.139375\pi$
$181$	24.3688	1.81132	0.905661	+	0.424002i	$0.139375\pi$
$182$	−1.54009	−0.114159
$183$	14.0786	1.04072
$184$	1.00000	0.0737210
$185$	0	0
$186$	−6.00000	−0.439941
$187$	20.8412	1.52406
$188$	−3.38127	−0.246605
$189$	4.03932	0.293817
$190$	0	0
$191$	10.4599	0.756853	0.378426	−	0.925631i	$-0.376465\pi$
$191$	10.4599	0.756853	0.378426	+	0.925631i	$0.376465\pi$
$192$	1.00000	0.0721688
$193$	−14.0134	−1.00871	−0.504355	−	0.863496i	$-0.668270\pi$
$193$	−14.0134	−1.00871	−0.504355	+	0.863496i	$0.668270\pi$
$194$	4.07863	0.292829
$195$	0	0
$196$	9.31608	0.665435
$197$	8.47335	0.603702	0.301851	−	0.953355i	$-0.402396\pi$
$197$	8.47335	0.603702	0.301851	+	0.953355i	$0.402396\pi$
$198$	5.69736	0.404894
$199$	1.19813	0.0849334	0.0424667	−	0.999098i	$-0.486478\pi$
$199$	1.19813	0.0849334	0.0424667	+	0.999098i	$0.486478\pi$
$200$	0	0
$201$	−2.76255	−0.194855
$202$	−16.7760	−1.18035
$203$	−4.03932	−0.283505
$204$	−3.65804	−0.256114
$205$	0	0
$206$	−13.3554	−0.930515
$207$	−1.00000	−0.0695048
$208$	0.381275	0.0264367
$209$	−2.17226	−0.150258
$210$	0	0
$211$	−16.2225	−1.11680	−0.558400	−	0.829572i	$-0.688585\pi$
$211$	−16.2225	−1.11680	−0.558400	+	0.829572i	$0.688585\pi$
$212$	9.31608	0.639831
$213$	10.6974	0.732971
$214$	6.76255	0.462278
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	24.2359	1.64524
$218$	13.0528	0.884045
$219$	−7.07863	−0.478330
$220$	0	0
$221$	−1.39472	−0.0938190
$222$	−3.65804	−0.245512
$223$	−13.6037	−0.910973	−0.455487	−	0.890243i	$-0.650535\pi$
$223$	−13.6037	−0.910973	−0.455487	+	0.890243i	$0.650535\pi$
$224$	−4.03932	−0.269888
$225$	0	0
$226$	1.65804	0.110291
$227$	−12.4992	−0.829603	−0.414801	−	0.909912i	$-0.636149\pi$
$227$	−12.4992	−0.829603	−0.414801	+	0.909912i	$0.636149\pi$
$228$	0.381275	0.0252505
$229$	21.4734	1.41900	0.709500	−	0.704706i	$-0.248921\pi$
$229$	21.4734	1.41900	0.709500	+	0.704706i	$0.248921\pi$
$230$	0	0
$231$	−23.0134	−1.51417
$232$	1.00000	0.0656532
$233$	−11.1438	−0.730056	−0.365028	−	0.930996i	$-0.618941\pi$
$233$	−11.1438	−0.730056	−0.365028	+	0.930996i	$0.618941\pi$
$234$	−0.381275	−0.0249247
$235$	0	0
$236$	13.3947	0.871922
$237$	5.69736	0.370083
$238$	14.7760	0.957785
$239$	22.1707	1.43410	0.717052	−	0.697020i	$-0.245491\pi$
$239$	22.1707	1.43410	0.717052	+	0.697020i	$0.245491\pi$
$240$	0	0
$241$	−13.8696	−0.893421	−0.446710	−	0.894679i	$-0.647405\pi$
$241$	−13.8696	−0.893421	−0.446710	+	0.894679i	$0.647405\pi$
$242$	−21.4599	−1.37950
$243$	1.00000	0.0641500
$244$	14.0786	0.901292
$245$	0	0
$246$	−5.61873	−0.358237
$247$	0.145371	0.00924971
$248$	−6.00000	−0.381000
$249$	16.1180	1.02143
$250$	0	0
$251$	5.73668	0.362096	0.181048	−	0.983474i	$-0.442051\pi$
$251$	5.73668	0.362096	0.181048	+	0.983474i	$0.442051\pi$
$252$	4.03932	0.254453
$253$	5.69736	0.358190
$254$	18.6322	1.16909
$255$	0	0
$256$	1.00000	0.0625000
$257$	−20.6322	−1.28700	−0.643500	−	0.765446i	$-0.722518\pi$
$257$	−20.6322	−1.28700	−0.643500	+	0.765446i	$0.722518\pi$
$258$	−3.61873	−0.225292
$259$	14.7760	0.918136
$260$	0	0
$261$	−1.00000	−0.0618984
$262$	17.4734	1.07951
$263$	19.8696	1.22521	0.612607	−	0.790388i	$-0.290121\pi$
$263$	19.8696	1.22521	0.612607	+	0.790388i	$0.290121\pi$
$264$	5.69736	0.350648
$265$	0	0
$266$	−1.54009	−0.0944290
$267$	5.65804	0.346267
$268$	−2.76255	−0.168750
$269$	−8.17226	−0.498272	−0.249136	−	0.968469i	$-0.580147\pi$
$269$	−8.17226	−0.498272	−0.249136	+	0.968469i	$0.580147\pi$
$270$	0	0
$271$	−11.3947	−0.692180	−0.346090	−	0.938201i	$-0.612491\pi$
$271$	−11.3947	−0.692180	−0.346090	+	0.938201i	$0.612491\pi$
$272$	−3.65804	−0.221801
$273$	1.54009	0.0932105
$274$	14.3420	0.866429
$275$	0	0
$276$	−1.00000	−0.0601929
$277$	−4.93481	−0.296504	−0.148252	−	0.988950i	$-0.547365\pi$
$277$	−4.93481	−0.296504	−0.148252	+	0.988950i	$0.547365\pi$
$278$	8.01344	0.480614
$279$	6.00000	0.359211
$280$	0	0
$281$	14.8955	0.888591	0.444295	−	0.895880i	$-0.353454\pi$
$281$	14.8955	0.888591	0.444295	+	0.895880i	$0.353454\pi$
$282$	3.38127	0.201352
$283$	−14.6322	−0.869792	−0.434896	−	0.900481i	$-0.643215\pi$
$283$	−14.6322	−0.869792	−0.434896	+	0.900481i	$0.643215\pi$
$284$	10.6974	0.634772
$285$	0	0
$286$	2.17226	0.128448
$287$	22.6958	1.33969
$288$	−1.00000	−0.0589256
$289$	−3.61873	−0.212866
$290$	0	0
$291$	−4.07863	−0.239094
$292$	−7.07863	−0.414246
$293$	−9.39472	−0.548845	−0.274423	−	0.961609i	$-0.588487\pi$
$293$	−9.39472	−0.548845	−0.274423	+	0.961609i	$0.588487\pi$
$294$	−9.31608	−0.543325
$295$	0	0
$296$	−3.65804	−0.212619
$297$	−5.69736	−0.330594
$298$	−3.23745	−0.187540
$299$	−0.381275	−0.0220497
$300$	0	0
$301$	14.6172	0.842520
$302$	9.31608	0.536080
$303$	16.7760	0.963756
$304$	0.381275	0.0218676
$305$	0	0
$306$	3.65804	0.209116
$307$	9.22401	0.526442	0.263221	−	0.964736i	$-0.415215\pi$
$307$	9.22401	0.526442	0.263221	+	0.964736i	$0.415215\pi$
$308$	−23.0134	−1.31131
$309$	13.3554	0.759762
$310$	0	0
$311$	11.3295	0.642439	0.321219	−	0.947005i	$-0.395907\pi$
$311$	11.3295	0.642439	0.321219	+	0.947005i	$0.395907\pi$
$312$	−0.381275	−0.0215854
$313$	15.6037	0.881975	0.440988	−	0.897513i	$-0.354628\pi$
$313$	15.6037	0.881975	0.440988	+	0.897513i	$0.354628\pi$
$314$	6.84118	0.386070
$315$	0	0
$316$	5.69736	0.320502
$317$	−11.0000	−0.617822	−0.308911	−	0.951091i	$-0.599964\pi$
$317$	−11.0000	−0.617822	−0.308911	+	0.951091i	$0.599964\pi$
$318$	−9.31608	−0.522420
$319$	5.69736	0.318991
$320$	0	0
$321$	−6.76255	−0.377449
$322$	4.03932	0.225102
$323$	−1.39472	−0.0776043
$324$	1.00000	0.0555556
$325$	0	0
$326$	−4.76255	−0.263773
$327$	−13.0528	−0.721819
$328$	−5.61873	−0.310242
$329$	−13.6580	−0.752992
$330$	0	0
$331$	2.69736	0.148260	0.0741302	−	0.997249i	$-0.476382\pi$
$331$	2.69736	0.148260	0.0741302	+	0.997249i	$0.476382\pi$
$332$	16.1180	0.884588
$333$	3.65804	0.200459
$334$	−14.8546	−0.812809
$335$	0	0
$336$	4.03932	0.220363
$337$	−17.3161	−0.943267	−0.471634	−	0.881795i	$-0.656336\pi$
$337$	−17.3161	−0.943267	−0.471634	+	0.881795i	$0.656336\pi$
$338$	12.8546	0.699200
$339$	−1.65804	−0.0900525
$340$	0	0
$341$	−34.1842	−1.85118
$342$	−0.381275	−0.0206170
$343$	9.35540	0.505144
$344$	−3.61873	−0.195109
$345$	0	0
$346$	−9.69736	−0.521333
$347$	−3.47335	−0.186459	−0.0932297	−	0.995645i	$-0.529719\pi$
$347$	−3.47335	−0.186459	−0.0932297	+	0.995645i	$0.529719\pi$
$348$	−1.00000	−0.0536056
$349$	−13.1438	−0.703573	−0.351786	−	0.936080i	$-0.614426\pi$
$349$	−13.1438	−0.703573	−0.351786	+	0.936080i	$0.614426\pi$
$350$	0	0
$351$	0.381275	0.0203509
$352$	5.69736	0.303670
$353$	−13.8546	−0.737408	−0.368704	−	0.929547i	$-0.620198\pi$
$353$	−13.8546	−0.737408	−0.368704	+	0.929547i	$0.620198\pi$
$354$	−13.3947	−0.711921
$355$	0	0
$356$	5.65804	0.299876
$357$	−14.7760	−0.782029
$358$	5.31608	0.280964
$359$	14.9348	0.788229	0.394115	−	0.919061i	$-0.371051\pi$
$359$	14.9348	0.788229	0.394115	+	0.919061i	$0.371051\pi$
$360$	0	0
$361$	−18.8546	−0.992349
$362$	−24.3688	−1.28080
$363$	21.4599	1.12635
$364$	1.54009	0.0807227
$365$	0	0
$366$	−14.0786	−0.735902
$367$	28.9348	1.51038	0.755192	−	0.655503i	$-0.227543\pi$
$367$	28.9348	1.51038	0.755192	+	0.655503i	$0.227543\pi$
$368$	−1.00000	−0.0521286
$369$	5.61873	0.292499
$370$	0	0
$371$	37.6306	1.95368
$372$	6.00000	0.311086
$373$	26.4475	1.36940	0.684699	−	0.728826i	$-0.259934\pi$
$373$	26.4475	1.36940	0.684699	+	0.728826i	$0.259934\pi$
$374$	−20.8412	−1.07767
$375$	0	0
$376$	3.38127	0.174376
$377$	−0.381275	−0.0196367
$378$	−4.03932	−0.207760
$379$	−26.2359	−1.34765	−0.673824	−	0.738892i	$-0.735349\pi$
$379$	−26.2359	−1.34765	−0.673824	+	0.738892i	$0.735349\pi$
$380$	0	0
$381$	−18.6322	−0.954555
$382$	−10.4599	−0.535176
$383$	−29.7242	−1.51884	−0.759419	−	0.650602i	$-0.774517\pi$
$383$	−29.7242	−1.51884	−0.759419	+	0.650602i	$0.774517\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	14.0134	0.713266
$387$	3.61873	0.183950
$388$	−4.07863	−0.207061
$389$	8.47490	0.429695	0.214847	−	0.976648i	$-0.431075\pi$
$389$	8.47490	0.429695	0.214847	+	0.976648i	$0.431075\pi$
$390$	0	0
$391$	3.65804	0.184995
$392$	−9.31608	−0.470533
$393$	−17.4734	−0.881414
$394$	−8.47335	−0.426881
$395$	0	0
$396$	−5.69736	−0.286303
$397$	21.8696	1.09760	0.548802	−	0.835952i	$-0.315084\pi$
$397$	21.8696	1.09760	0.548802	+	0.835952i	$0.315084\pi$
$398$	−1.19813	−0.0600570
$399$	1.54009	0.0771010
$400$	0	0
$401$	−4.71080	−0.235246	−0.117623	−	0.993058i	$-0.537527\pi$
$401$	−4.71080	−0.235246	−0.117623	+	0.993058i	$0.537527\pi$
$402$	2.76255	0.137783
$403$	2.28765	0.113956
$404$	16.7760	0.834637
$405$	0	0
$406$	4.03932	0.200468
$407$	−20.8412	−1.03306
$408$	3.65804	0.181100
$409$	35.1055	1.73586	0.867928	−	0.496690i	$-0.165451\pi$
$409$	35.1055	1.73586	0.867928	+	0.496690i	$0.165451\pi$
$410$	0	0
$411$	−14.3420	−0.707437
$412$	13.3554	0.657973
$413$	54.1055	2.66236
$414$	1.00000	0.0491473
$415$	0	0
$416$	−0.381275	−0.0186935
$417$	−8.01344	−0.392420
$418$	2.17226	0.106249
$419$	−24.7232	−1.20781	−0.603904	−	0.797057i	$-0.706389\pi$
$419$	−24.7232	−1.20781	−0.603904	+	0.797057i	$0.706389\pi$
$420$	0	0
$421$	17.4734	0.851599	0.425800	−	0.904817i	$-0.359993\pi$
$421$	17.4734	0.851599	0.425800	+	0.904817i	$0.359993\pi$
$422$	16.2225	0.789697
$423$	−3.38127	−0.164403
$424$	−9.31608	−0.452429
$425$	0	0
$426$	−10.6974	−0.518289
$427$	56.8681	2.75204
$428$	−6.76255	−0.326880
$429$	−2.17226	−0.104878
$430$	0	0
$431$	4.84118	0.233192	0.116596	−	0.993179i	$-0.462802\pi$
$431$	4.84118	0.233192	0.116596	+	0.993179i	$0.462802\pi$
$432$	1.00000	0.0481125
$433$	16.8412	0.809336	0.404668	−	0.914464i	$-0.367387\pi$
$433$	16.8412	0.809336	0.404668	+	0.914464i	$0.367387\pi$
$434$	−24.2359	−1.16336
$435$	0	0
$436$	−13.0528	−0.625114
$437$	−0.381275	−0.0182389
$438$	7.07863	0.338230
$439$	−20.7108	−0.988473	−0.494236	−	0.869328i	$-0.664552\pi$
$439$	−20.7108	−0.988473	−0.494236	+	0.869328i	$0.664552\pi$
$440$	0	0
$441$	9.31608	0.443623
$442$	1.39472	0.0663401
$443$	−14.8412	−0.705126	−0.352563	−	0.935788i	$-0.614690\pi$
$443$	−14.8412	−0.705126	−0.352563	+	0.935788i	$0.614690\pi$
$444$	3.65804	0.173603
$445$	0	0
$446$	13.6037	0.644155
$447$	3.23745	0.153126
$448$	4.03932	0.190840
$449$	−30.1573	−1.42321	−0.711605	−	0.702580i	$-0.752031\pi$
$449$	−30.1573	−1.42321	−0.711605	+	0.702580i	$0.752031\pi$
$450$	0	0
$451$	−32.0119	−1.50738
$452$	−1.65804	−0.0779878
$453$	−9.31608	−0.437708
$454$	12.4992	0.586618
$455$	0	0
$456$	−0.381275	−0.0178548
$457$	12.7625	0.597007	0.298503	−	0.954409i	$-0.403513\pi$
$457$	12.7625	0.597007	0.298503	+	0.954409i	$0.403513\pi$
$458$	−21.4734	−1.00338
$459$	−3.65804	−0.170743
$460$	0	0
$461$	17.2359	0.802756	0.401378	−	0.915912i	$-0.368531\pi$
$461$	17.2359	0.802756	0.401378	+	0.915912i	$0.368531\pi$
$462$	23.0134	1.07068
$463$	−12.0000	−0.557687	−0.278844	−	0.960337i	$-0.589951\pi$
$463$	−12.0000	−0.557687	−0.278844	+	0.960337i	$0.589951\pi$
$464$	−1.00000	−0.0464238
$465$	0	0
$466$	11.1438	0.516228
$467$	−5.88205	−0.272189	−0.136094	−	0.990696i	$-0.543455\pi$
$467$	−5.88205	−0.272189	−0.136094	+	0.990696i	$0.543455\pi$
$468$	0.381275	0.0176244
$469$	−11.1588	−0.515266
$470$	0	0
$471$	−6.84118	−0.315225
$472$	−13.3947	−0.616542
$473$	−20.6172	−0.947979
$474$	−5.69736	−0.261688
$475$	0	0
$476$	−14.7760	−0.677257
$477$	9.31608	0.426554
$478$	−22.1707	−1.01406
$479$	−28.4868	−1.30160	−0.650798	−	0.759251i	$-0.725565\pi$
$479$	−28.4868	−1.30160	−0.650798	+	0.759251i	$0.725565\pi$
$480$	0	0
$481$	1.39472	0.0635937
$482$	13.8696	0.631744
$483$	−4.03932	−0.183795
$484$	21.4599	0.975450
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−25.6306	−1.16143	−0.580717	−	0.814105i	$-0.697228\pi$
$487$	−25.6306	−1.16143	−0.580717	+	0.814105i	$0.697228\pi$
$488$	−14.0786	−0.637310
$489$	4.76255	0.215370
$490$	0	0
$491$	33.9483	1.53206	0.766032	−	0.642803i	$-0.222229\pi$
$491$	33.9483	1.53206	0.766032	+	0.642803i	$0.222229\pi$
$492$	5.61873	0.253312
$493$	3.65804	0.164750
$494$	−0.145371	−0.00654053
$495$	0	0
$496$	6.00000	0.269408
$497$	43.2100	1.93823
$498$	−16.1180	−0.722263
$499$	18.0134	0.806393	0.403196	−	0.915114i	$-0.367899\pi$
$499$	18.0134	0.806393	0.403196	+	0.915114i	$0.367899\pi$
$500$	0	0
$501$	14.8546	0.663656
$502$	−5.73668	−0.256040
$503$	14.3813	0.641229	0.320615	−	0.947210i	$-0.396111\pi$
$503$	14.3813	0.641229	0.320615	+	0.947210i	$0.396111\pi$
$504$	−4.03932	−0.179925
$505$	0	0
$506$	−5.69736	−0.253279
$507$	−12.8546	−0.570894
$508$	−18.6322	−0.826669
$509$	32.3797	1.43521	0.717603	−	0.696452i	$-0.245239\pi$
$509$	32.3797	1.43521	0.717603	+	0.696452i	$0.245239\pi$
$510$	0	0
$511$	−28.5929	−1.26487
$512$	−1.00000	−0.0441942
$513$	0.381275	0.0168337
$514$	20.6322	0.910046
$515$	0	0
$516$	3.61873	0.159305
$517$	19.2643	0.847245
$518$	−14.7760	−0.649220
$519$	9.69736	0.425667
$520$	0	0
$521$	24.3420	1.06644	0.533220	−	0.845976i	$-0.320982\pi$
$521$	24.3420	1.06644	0.533220	+	0.845976i	$0.320982\pi$
$522$	1.00000	0.0437688
$523$	3.54009	0.154797	0.0773987	−	0.997000i	$-0.475339\pi$
$523$	3.54009	0.154797	0.0773987	+	0.997000i	$0.475339\pi$
$524$	−17.4734	−0.763327
$525$	0	0
$526$	−19.8696	−0.866357
$527$	−21.9483	−0.956081
$528$	−5.69736	−0.247946
$529$	1.00000	0.0434783
$530$	0	0
$531$	13.3947	0.581281
$532$	1.54009	0.0667714
$533$	2.14228	0.0927924
$534$	−5.65804	−0.244847
$535$	0	0
$536$	2.76255	0.119324
$537$	−5.31608	−0.229406
$538$	8.17226	0.352331
$539$	−53.0771	−2.28619
$540$	0	0
$541$	−23.2493	−0.999568	−0.499784	−	0.866150i	$-0.666587\pi$
$541$	−23.2493	−0.999568	−0.499784	+	0.866150i	$0.666587\pi$
$542$	11.3947	0.489445
$543$	24.3688	1.04577
$544$	3.65804	0.156837
$545$	0	0
$546$	−1.54009	−0.0659098
$547$	32.2225	1.37773	0.688866	−	0.724888i	$-0.258109\pi$
$547$	32.2225	1.37773	0.688866	+	0.724888i	$0.258109\pi$
$548$	−14.3420	−0.612658
$549$	14.0786	0.600861
$550$	0	0
$551$	−0.381275	−0.0162429
$552$	1.00000	0.0425628
$553$	23.0134	0.978631
$554$	4.93481	0.209660
$555$	0	0
$556$	−8.01344	−0.339846
$557$	−44.9985	−1.90665	−0.953323	−	0.301953i	$-0.902361\pi$
$557$	−44.9985	−1.90665	−0.953323	+	0.301953i	$0.902361\pi$
$558$	−6.00000	−0.254000
$559$	1.37973	0.0583563
$560$	0	0
$561$	20.8412	0.879916
$562$	−14.8955	−0.628328
$563$	−13.0921	−0.551765	−0.275883	−	0.961191i	$-0.588970\pi$
$563$	−13.0921	−0.551765	−0.275883	+	0.961191i	$0.588970\pi$
$564$	−3.38127	−0.142377
$565$	0	0
$566$	14.6322	0.615036
$567$	4.03932	0.169635
$568$	−10.6974	−0.448851
$569$	−34.0786	−1.42865	−0.714325	−	0.699814i	$-0.753266\pi$
$569$	−34.0786	−1.42865	−0.714325	+	0.699814i	$0.753266\pi$
$570$	0	0
$571$	23.4734	0.982329	0.491165	−	0.871067i	$-0.336571\pi$
$571$	23.4734	0.982329	0.491165	+	0.871067i	$0.336571\pi$
$572$	−2.17226	−0.0908268
$573$	10.4599	0.436969
$574$	−22.6958	−0.947305
$575$	0	0
$576$	1.00000	0.0416667
$577$	24.9348	1.03805	0.519025	−	0.854759i	$-0.326295\pi$
$577$	24.9348	1.03805	0.519025	+	0.854759i	$0.326295\pi$
$578$	3.61873	0.150519
$579$	−14.0134	−0.582379
$580$	0	0
$581$	65.1055	2.70103
$582$	4.07863	0.169065
$583$	−53.0771	−2.19823
$584$	7.07863	0.292916
$585$	0	0
$586$	9.39472	0.388092
$587$	−30.0786	−1.24148	−0.620739	−	0.784017i	$-0.713167\pi$
$587$	−30.0786	−1.24148	−0.620739	+	0.784017i	$0.713167\pi$
$588$	9.31608	0.384189
$589$	2.28765	0.0942610
$590$	0	0
$591$	8.47335	0.348547
$592$	3.65804	0.150345
$593$	16.4599	0.675927	0.337964	−	0.941159i	$-0.390262\pi$
$593$	16.4599	0.675927	0.337964	+	0.941159i	$0.390262\pi$
$594$	5.69736	0.233766
$595$	0	0
$596$	3.23745	0.132611
$597$	1.19813	0.0490363
$598$	0.381275	0.0155915
$599$	−26.0269	−1.06343	−0.531715	−	0.846923i	$-0.678452\pi$
$599$	−26.0269	−1.06343	−0.531715	+	0.846923i	$0.678452\pi$
$600$	0	0
$601$	−30.8546	−1.25859	−0.629293	−	0.777168i	$-0.716656\pi$
$601$	−30.8546	−1.25859	−0.629293	+	0.777168i	$0.716656\pi$
$602$	−14.6172	−0.595752
$603$	−2.76255	−0.112500
$604$	−9.31608	−0.379066
$605$	0	0
$606$	−16.7760	−0.681478
$607$	5.05020	0.204981	0.102491	−	0.994734i	$-0.467319\pi$
$607$	5.05020	0.204981	0.102491	+	0.994734i	$0.467319\pi$
$608$	−0.381275	−0.0154627
$609$	−4.03932	−0.163681
$610$	0	0
$611$	−1.28920	−0.0521553
$612$	−3.65804	−0.147868
$613$	48.5261	1.95995	0.979976	−	0.199117i	$-0.0638074\pi$
$613$	48.5261	1.95995	0.979976	+	0.199117i	$0.0638074\pi$
$614$	−9.22401	−0.372251
$615$	0	0
$616$	23.0134	0.927238
$617$	−20.7108	−0.833786	−0.416893	−	0.908956i	$-0.636881\pi$
$617$	−20.7108	−0.833786	−0.416893	+	0.908956i	$0.636881\pi$
$618$	−13.3554	−0.537233
$619$	42.2359	1.69760	0.848802	−	0.528711i	$-0.177324\pi$
$619$	42.2359	1.69760	0.848802	+	0.528711i	$0.177324\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	−11.3295	−0.454273
$623$	22.8546	0.915651
$624$	0.381275	0.0152632
$625$	0	0
$626$	−15.6037	−0.623651
$627$	−2.17226	−0.0867517
$628$	−6.84118	−0.272993
$629$	−13.3813	−0.533546
$630$	0	0
$631$	25.5913	1.01877	0.509387	−	0.860538i	$-0.329872\pi$
$631$	25.5913	1.01877	0.509387	+	0.860538i	$0.329872\pi$
$632$	−5.69736	−0.226629
$633$	−16.2225	−0.644785
$634$	11.0000	0.436866
$635$	0	0
$636$	9.31608	0.369407
$637$	3.55199	0.140735
$638$	−5.69736	−0.225561
$639$	10.6974	0.423181
$640$	0	0
$641$	−36.6834	−1.44891	−0.724453	−	0.689324i	$-0.757908\pi$
$641$	−36.6834	−1.44891	−0.724453	+	0.689324i	$0.757908\pi$
$642$	6.76255	0.266897
$643$	13.6974	0.540171	0.270086	−	0.962836i	$-0.412948\pi$
$643$	13.6974	0.540171	0.270086	+	0.962836i	$0.412948\pi$
$644$	−4.03932	−0.159171
$645$	0	0
$646$	1.39472	0.0548745
$647$	24.0652	0.946100	0.473050	−	0.881036i	$-0.343153\pi$
$647$	24.0652	0.946100	0.473050	+	0.881036i	$0.343153\pi$
$648$	−1.00000	−0.0392837
$649$	−76.3145	−2.99561
$650$	0	0
$651$	24.2359	0.949880
$652$	4.76255	0.186516
$653$	−14.9214	−0.583918	−0.291959	−	0.956431i	$-0.594307\pi$
$653$	−14.9214	−0.583918	−0.291959	+	0.956431i	$0.594307\pi$
$654$	13.0528	0.510403
$655$	0	0
$656$	5.61873	0.219374
$657$	−7.07863	−0.276164
$658$	13.6580	0.532446
$659$	40.8288	1.59046	0.795231	−	0.606306i	$-0.207349\pi$
$659$	40.8288	1.59046	0.795231	+	0.606306i	$0.207349\pi$
$660$	0	0
$661$	−29.1045	−1.13203	−0.566017	−	0.824394i	$-0.691516\pi$
$661$	−29.1045	−1.13203	−0.566017	+	0.824394i	$0.691516\pi$
$662$	−2.69736	−0.104836
$663$	−1.39472	−0.0541664
$664$	−16.1180	−0.625498
$665$	0	0
$666$	−3.65804	−0.141746
$667$	1.00000	0.0387202
$668$	14.8546	0.574743
$669$	−13.6037	−0.525951
$670$	0	0
$671$	−80.2110	−3.09651
$672$	−4.03932	−0.155820
$673$	−41.9198	−1.61589	−0.807945	−	0.589258i	$-0.799420\pi$
$673$	−41.9198	−1.61589	−0.807945	+	0.589258i	$0.799420\pi$
$674$	17.3161	0.666991
$675$	0	0
$676$	−12.8546	−0.494409
$677$	−24.3145	−0.934484	−0.467242	−	0.884130i	$-0.654752\pi$
$677$	−24.3145	−0.934484	−0.467242	+	0.884130i	$0.654752\pi$
$678$	1.65804	0.0636767
$679$	−16.4749	−0.632249
$680$	0	0
$681$	−12.4992	−0.478971
$682$	34.1842	1.30898
$683$	44.2628	1.69367	0.846834	−	0.531857i	$-0.178506\pi$
$683$	44.2628	1.69367	0.846834	+	0.531857i	$0.178506\pi$
$684$	0.381275	0.0145784
$685$	0	0
$686$	−9.35540	−0.357191
$687$	21.4734	0.819260
$688$	3.61873	0.137963
$689$	3.55199	0.135320
$690$	0	0
$691$	−37.1423	−1.41296	−0.706479	−	0.707734i	$-0.749718\pi$
$691$	−37.1423	−1.41296	−0.706479	+	0.707734i	$0.749718\pi$
$692$	9.69736	0.368638
$693$	−23.0134	−0.874208
$694$	3.47335	0.131847
$695$	0	0
$696$	1.00000	0.0379049
$697$	−20.5535	−0.778521
$698$	13.1438	0.497501
$699$	−11.1438	−0.421498
$700$	0	0
$701$	−0.130380	−0.00492438	−0.00246219	−	0.999997i	$-0.500784\pi$
$701$	−0.130380	−0.00492438	−0.00246219	+	0.999997i	$0.500784\pi$
$702$	−0.381275	−0.0143903
$703$	1.39472	0.0526029
$704$	−5.69736	−0.214727
$705$	0	0
$706$	13.8546	0.521426
$707$	67.7636	2.54851
$708$	13.3947	0.503404
$709$	−21.2618	−0.798503	−0.399251	−	0.916841i	$-0.630730\pi$
$709$	−21.2618	−0.798503	−0.399251	+	0.916841i	$0.630730\pi$
$710$	0	0
$711$	5.69736	0.213668
$712$	−5.65804	−0.212044
$713$	−6.00000	−0.224702
$714$	14.7760	0.552978
$715$	0	0
$716$	−5.31608	−0.198671
$717$	22.1707	0.827981
$718$	−14.9348	−0.557362
$719$	−23.3026	−0.869042	−0.434521	−	0.900662i	$-0.643082\pi$
$719$	−23.3026	−0.869042	−0.434521	+	0.900662i	$0.643082\pi$
$720$	0	0
$721$	53.9467	2.00908
$722$	18.8546	0.701697
$723$	−13.8696	−0.515817
$724$	24.3688	0.905661
$725$	0	0
$726$	−21.4599	−0.796452
$727$	14.6053	0.541680	0.270840	−	0.962624i	$-0.412699\pi$
$727$	14.6053	0.541680	0.270840	+	0.962624i	$0.412699\pi$
$728$	−1.54009	−0.0570795
$729$	1.00000	0.0370370
$730$	0	0
$731$	−13.2375	−0.489605
$732$	14.0786	0.520361
$733$	26.6565	0.984580	0.492290	−	0.870431i	$-0.336160\pi$
$733$	26.6565	0.984580	0.492290	+	0.870431i	$0.336160\pi$
$734$	−28.9348	−1.06800
$735$	0	0
$736$	1.00000	0.0368605
$737$	15.7392	0.579762
$738$	−5.61873	−0.206828
$739$	49.1992	1.80982	0.904910	−	0.425603i	$-0.139938\pi$
$739$	49.1992	1.80982	0.904910	+	0.425603i	$0.139938\pi$
$740$	0	0
$741$	0.145371	0.00534032
$742$	−37.6306	−1.38146
$743$	−17.1438	−0.628946	−0.314473	−	0.949266i	$-0.601828\pi$
$743$	−17.1438	−0.628946	−0.314473	+	0.949266i	$0.601828\pi$
$744$	−6.00000	−0.219971
$745$	0	0
$746$	−26.4475	−0.968311
$747$	16.1180	0.589725
$748$	20.8412	0.762029
$749$	−27.3161	−0.998108
$750$	0	0
$751$	42.2752	1.54264	0.771322	−	0.636445i	$-0.219596\pi$
$751$	42.2752	1.54264	0.771322	+	0.636445i	$0.219596\pi$
$752$	−3.38127	−0.123302
$753$	5.73668	0.209056
$754$	0.381275	0.0138852
$755$	0	0
$756$	4.03932	0.146909
$757$	−53.8939	−1.95881	−0.979404	−	0.201908i	$-0.935286\pi$
$757$	−53.8939	−1.95881	−0.979404	+	0.201908i	$0.935286\pi$
$758$	26.2359	0.952931
$759$	5.69736	0.206801
$760$	0	0
$761$	−41.6456	−1.50965	−0.754826	−	0.655925i	$-0.772279\pi$
$761$	−41.6456	−1.50965	−0.754826	+	0.655925i	$0.772279\pi$
$762$	18.6322	0.674972
$763$	−52.7242	−1.90875
$764$	10.4599	0.378426
$765$	0	0
$766$	29.7242	1.07398
$767$	5.10707	0.184406
$768$	1.00000	0.0360844
$769$	−12.5266	−0.451722	−0.225861	−	0.974159i	$-0.572520\pi$
$769$	−12.5266	−0.451722	−0.225861	+	0.974159i	$0.572520\pi$
$770$	0	0
$771$	−20.6322	−0.743049
$772$	−14.0134	−0.504355
$773$	18.3663	0.660589	0.330295	−	0.943878i	$-0.392852\pi$
$773$	18.3663	0.660589	0.330295	+	0.943878i	$0.392852\pi$
$774$	−3.61873	−0.130072
$775$	0	0
$776$	4.07863	0.146414
$777$	14.7760	0.530086
$778$	−8.47490	−0.303840
$779$	2.14228	0.0767551
$780$	0	0
$781$	−60.9467	−2.18084
$782$	−3.65804	−0.130811
$783$	−1.00000	−0.0357371
$784$	9.31608	0.332717
$785$	0	0
$786$	17.4734	0.623254
$787$	−32.6172	−1.16268	−0.581338	−	0.813662i	$-0.697471\pi$
$787$	−32.6172	−1.16268	−0.581338	+	0.813662i	$0.697471\pi$
$788$	8.47335	0.301851
$789$	19.8696	0.707377
$790$	0	0
$791$	−6.69736	−0.238131
$792$	5.69736	0.202447
$793$	5.36783	0.190617
$794$	−21.8696	−0.776124
$795$	0	0
$796$	1.19813	0.0424667
$797$	−21.7910	−0.771876	−0.385938	−	0.922525i	$-0.626122\pi$
$797$	−21.7910	−0.771876	−0.385938	+	0.922525i	$0.626122\pi$
$798$	−1.54009	−0.0545186
$799$	12.3688	0.437578
$800$	0	0
$801$	5.65804	0.199917
$802$	4.71080	0.166344
$803$	40.3295	1.42320
$804$	−2.76255	−0.0974276
$805$	0	0
$806$	−2.28765	−0.0805790
$807$	−8.17226	−0.287677
$808$	−16.7760	−0.590177
$809$	8.17226	0.287321	0.143661	−	0.989627i	$-0.454113\pi$
$809$	8.17226	0.287321	0.143661	+	0.989627i	$0.454113\pi$
$810$	0	0
$811$	−29.4216	−1.03313	−0.516566	−	0.856247i	$-0.672790\pi$
$811$	−29.4216	−1.03313	−0.516566	+	0.856247i	$0.672790\pi$
$812$	−4.03932	−0.141752
$813$	−11.3947	−0.399630
$814$	20.8412	0.730483
$815$	0	0
$816$	−3.65804	−0.128057
$817$	1.37973	0.0482706
$818$	−35.1055	−1.22744
$819$	1.54009	0.0538151
$820$	0	0
$821$	41.6974	1.45525	0.727624	−	0.685976i	$-0.240625\pi$
$821$	41.6974	1.45525	0.727624	+	0.685976i	$0.240625\pi$
$822$	14.3420	0.500233
$823$	−26.2359	−0.914526	−0.457263	−	0.889331i	$-0.651170\pi$
$823$	−26.2359	−0.914526	−0.457263	+	0.889331i	$0.651170\pi$
$824$	−13.3554	−0.465257
$825$	0	0
$826$	−54.1055	−1.88257
$827$	−41.9358	−1.45825	−0.729126	−	0.684380i	$-0.760073\pi$
$827$	−41.9358	−1.45825	−0.729126	+	0.684380i	$0.760073\pi$
$828$	−1.00000	−0.0347524
$829$	−27.7760	−0.964700	−0.482350	−	0.875979i	$-0.660217\pi$
$829$	−27.7760	−0.964700	−0.482350	+	0.875979i	$0.660217\pi$
$830$	0	0
$831$	−4.93481	−0.171187
$832$	0.381275	0.0132183
$833$	−34.0786	−1.18075
$834$	8.01344	0.277483
$835$	0	0
$836$	−2.17226	−0.0751292
$837$	6.00000	0.207390
$838$	24.7232	0.854050
$839$	31.3011	1.08063	0.540317	−	0.841462i	$-0.318304\pi$
$839$	31.3011	1.08063	0.540317	+	0.841462i	$0.318304\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	−17.4734	−0.602172
$843$	14.8955	0.513028
$844$	−16.2225	−0.558400
$845$	0	0
$846$	3.38127	0.116251
$847$	86.6834	2.97848
$848$	9.31608	0.319916
$849$	−14.6322	−0.502175
$850$	0	0
$851$	−3.65804	−0.125396
$852$	10.6974	0.366486
$853$	−20.6172	−0.705919	−0.352959	−	0.935639i	$-0.614825\pi$
$853$	−20.6172	−0.705919	−0.352959	+	0.935639i	$0.614825\pi$
$854$	−56.8681	−1.94599
$855$	0	0
$856$	6.76255	0.231139
$857$	0.235904	0.00805834	0.00402917	−	0.999992i	$-0.498717\pi$
$857$	0.235904	0.00805834	0.00402917	+	0.999992i	$0.498717\pi$
$858$	2.17226	0.0741597
$859$	−23.7392	−0.809972	−0.404986	−	0.914323i	$-0.632724\pi$
$859$	−23.7392	−0.809972	−0.404986	+	0.914323i	$0.632724\pi$
$860$	0	0
$861$	22.6958	0.773471
$862$	−4.84118	−0.164891
$863$	−20.2225	−0.688381	−0.344190	−	0.938900i	$-0.611847\pi$
$863$	−20.2225	−0.688381	−0.344190	+	0.938900i	$0.611847\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−16.8412	−0.572287
$867$	−3.61873	−0.122898
$868$	24.2359	0.822620
$869$	−32.4599	−1.10113
$870$	0	0
$871$	−1.05329	−0.0356894
$872$	13.0528	0.442022
$873$	−4.07863	−0.138041
$874$	0.381275	0.0128968
$875$	0	0
$876$	−7.07863	−0.239165
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	20.7108	0.698956
$879$	−9.39472	−0.316876
$880$	0	0
$881$	26.0786	0.878612	0.439306	−	0.898338i	$-0.355224\pi$
$881$	26.0786	0.878612	0.439306	+	0.898338i	$0.355224\pi$
$882$	−9.31608	−0.313689
$883$	4.40971	0.148399	0.0741993	−	0.997243i	$-0.476360\pi$
$883$	4.40971	0.148399	0.0741993	+	0.997243i	$0.476360\pi$
$884$	−1.39472	−0.0469095
$885$	0	0
$886$	14.8412	0.498599
$887$	22.6456	0.760365	0.380183	−	0.924911i	$-0.375861\pi$
$887$	22.6456	0.760365	0.380183	+	0.924911i	$0.375861\pi$
$888$	−3.65804	−0.122756
$889$	−75.2612	−2.52418
$890$	0	0
$891$	−5.69736	−0.190869
$892$	−13.6037	−0.455487
$893$	−1.28920	−0.0431413
$894$	−3.23745	−0.108277
$895$	0	0
$896$	−4.03932	−0.134944
$897$	−0.381275	−0.0127304
$898$	30.1573	1.00636
$899$	−6.00000	−0.200111
$900$	0	0
$901$	−34.0786	−1.13532
$902$	32.0119	1.06588
$903$	14.6172	0.486429
$904$	1.65804	0.0551457
$905$	0	0
$906$	9.31608	0.309506
$907$	16.2240	0.538709	0.269355	−	0.963041i	$-0.413190\pi$
$907$	16.2240	0.538709	0.269355	+	0.963041i	$0.413190\pi$
$908$	−12.4992	−0.414801
$909$	16.7760	0.556425
$910$	0	0
$911$	−1.77599	−0.0588413	−0.0294207	−	0.999567i	$-0.509366\pi$
$911$	−1.77599	−0.0588413	−0.0294207	+	0.999567i	$0.509366\pi$
$912$	0.381275	0.0126253
$913$	−91.8298	−3.03912
$914$	−12.7625	−0.422148
$915$	0	0
$916$	21.4734	0.709500
$917$	−70.5804	−2.33077
$918$	3.65804	0.120733
$919$	−27.6580	−0.912355	−0.456177	−	0.889889i	$-0.650782\pi$
$919$	−27.6580	−0.912355	−0.456177	+	0.889889i	$0.650782\pi$
$920$	0	0
$921$	9.22401	0.303941
$922$	−17.2359	−0.567634
$923$	4.07863	0.134250
$924$	−23.0134	−0.757087
$925$	0	0
$926$	12.0000	0.394344
$927$	13.3554	0.438649
$928$	1.00000	0.0328266
$929$	0.777541	0.0255103	0.0127551	−	0.999919i	$-0.495940\pi$
$929$	0.777541	0.0255103	0.0127551	+	0.999919i	$0.495940\pi$
$930$	0	0
$931$	3.55199	0.116412
$932$	−11.1438	−0.365028
$933$	11.3295	0.370912
$934$	5.88205	0.192466
$935$	0	0
$936$	−0.381275	−0.0124624
$937$	7.52510	0.245834	0.122917	−	0.992417i	$-0.460775\pi$
$937$	7.52510	0.245834	0.122917	+	0.992417i	$0.460775\pi$
$938$	11.1588	0.364348
$939$	15.6037	0.509209
$940$	0	0
$941$	−40.6322	−1.32457	−0.662285	−	0.749252i	$-0.730413\pi$
$941$	−40.6322	−1.32457	−0.662285	+	0.749252i	$0.730413\pi$
$942$	6.84118	0.222898
$943$	−5.61873	−0.182971
$944$	13.3947	0.435961
$945$	0	0
$946$	20.6172	0.670322
$947$	23.3947	0.760226	0.380113	−	0.924940i	$-0.375885\pi$
$947$	23.3947	0.760226	0.380113	+	0.924940i	$0.375885\pi$
$948$	5.69736	0.185042
$949$	−2.69891	−0.0876102
$950$	0	0
$951$	−11.0000	−0.356699
$952$	14.7760	0.478893
$953$	−34.2902	−1.11077	−0.555384	−	0.831594i	$-0.687429\pi$
$953$	−34.2902	−1.11077	−0.555384	+	0.831594i	$0.687429\pi$
$954$	−9.31608	−0.301619
$955$	0	0
$956$	22.1707	0.717052
$957$	5.69736	0.184169
$958$	28.4868	0.920367
$959$	−57.9317	−1.87071
$960$	0	0
$961$	5.00000	0.161290
$962$	−1.39472	−0.0449676
$963$	−6.76255	−0.217920
$964$	−13.8696	−0.446710
$965$	0	0
$966$	4.03932	0.129963
$967$	−5.39472	−0.173482	−0.0867412	−	0.996231i	$-0.527645\pi$
$967$	−5.39472	−0.173482	−0.0867412	+	0.996231i	$0.527645\pi$
$968$	−21.4599	−0.689748
$969$	−1.39472	−0.0448049
$970$	0	0
$971$	58.3807	1.87353	0.936764	−	0.349963i	$-0.113806\pi$
$971$	58.3807	1.87353	0.936764	+	0.349963i	$0.113806\pi$
$972$	1.00000	0.0320750
$973$	−32.3688	−1.03770
$974$	25.6306	0.821258
$975$	0	0
$976$	14.0786	0.450646
$977$	−33.4190	−1.06917	−0.534585	−	0.845115i	$-0.679532\pi$
$977$	−33.4190	−1.06917	−0.534585	+	0.845115i	$0.679532\pi$
$978$	−4.76255	−0.152290
$979$	−32.2359	−1.03026
$980$	0	0
$981$	−13.0528	−0.416743
$982$	−33.9483	−1.08333
$983$	−42.4868	−1.35512	−0.677559	−	0.735468i	$-0.736962\pi$
$983$	−42.4868	−1.35512	−0.677559	+	0.735468i	$0.736962\pi$
$984$	−5.61873	−0.179118
$985$	0	0
$986$	−3.65804	−0.116496
$987$	−13.6580	−0.434740
$988$	0.145371	0.00462485
$989$	−3.61873	−0.115069
$990$	0	0
$991$	16.7625	0.532480	0.266240	−	0.963907i	$-0.414219\pi$
$991$	16.7625	0.532480	0.266240	+	0.963907i	$0.414219\pi$
$992$	−6.00000	−0.190500
$993$	2.69736	0.0855981
$994$	−43.2100	−1.37054
$995$	0	0
$996$	16.1180	0.510717
$997$	15.7760	0.499631	0.249815	−	0.968293i	$-0.419630\pi$
$997$	15.7760	0.499631	0.249815	+	0.968293i	$0.419630\pi$
$998$	−18.0134	−0.570206
$999$	3.65804	0.115735

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3450.2.a.bp.1.3	✓	3
5.2	odd	4		3450.2.d.ba.2899.3		6
5.3	odd	4		3450.2.d.ba.2899.4		6
5.4	even	2		3450.2.a.bs.1.1	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3450.2.a.bp.1.3	✓	3	1.1	even	1	trivial
3450.2.a.bs.1.1	yes	3	5.4	even	2
3450.2.d.ba.2899.3		6	5.2	odd	4
3450.2.d.ba.2899.4		6	5.3	odd	4

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:	\( N \)	\(=\)	\( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3450.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.03932 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +4.03932 q^{7} -1.00000 q^{8} +1.00000 q^{9} -5.69736 q^{11} +1.00000 q^{12} +0.381275 q^{13} -4.03932 q^{14} +1.00000 q^{16} -3.65804 q^{17} -1.00000 q^{18} +0.381275 q^{19} +4.03932 q^{21} +5.69736 q^{22} -1.00000 q^{23} -1.00000 q^{24} -0.381275 q^{26} +1.00000 q^{27} +4.03932 q^{28} -1.00000 q^{29} +6.00000 q^{31} -1.00000 q^{32} -5.69736 q^{33} +3.65804 q^{34} +1.00000 q^{36} +3.65804 q^{37} -0.381275 q^{38} +0.381275 q^{39} +5.61873 q^{41} -4.03932 q^{42} +3.61873 q^{43} -5.69736 q^{44} +1.00000 q^{46} -3.38127 q^{47} +1.00000 q^{48} +9.31608 q^{49} -3.65804 q^{51} +0.381275 q^{52} +9.31608 q^{53} -1.00000 q^{54} -4.03932 q^{56} +0.381275 q^{57} +1.00000 q^{58} +13.3947 q^{59} +14.0786 q^{61} -6.00000 q^{62} +4.03932 q^{63} +1.00000 q^{64} +5.69736 q^{66} -2.76255 q^{67} -3.65804 q^{68} -1.00000 q^{69} +10.6974 q^{71} -1.00000 q^{72} -7.07863 q^{73} -3.65804 q^{74} +0.381275 q^{76} -23.0134 q^{77} -0.381275 q^{78} +5.69736 q^{79} +1.00000 q^{81} -5.61873 q^{82} +16.1180 q^{83} +4.03932 q^{84} -3.61873 q^{86} -1.00000 q^{87} +5.69736 q^{88} +5.65804 q^{89} +1.54009 q^{91} -1.00000 q^{92} +6.00000 q^{93} +3.38127 q^{94} -1.00000 q^{96} -4.07863 q^{97} -9.31608 q^{98} -5.69736 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{6} + q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^6 + q^7 - 3 * q^8 + 3 * q^9	\( 3 q - 3 q^{2} + 3 q^{3} + 3 q^{4} - 3 q^{6} + q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{11} + 3 q^{12} - q^{13} - q^{14} + 3 q^{16} - 2 q^{17} - 3 q^{18} - q^{19} + q^{21} - 3 q^{22} - 3 q^{23} - 3 q^{24} + q^{26} + 3 q^{27} + q^{28} - 3 q^{29} + 18 q^{31} - 3 q^{32} + 3 q^{33} + 2 q^{34} + 3 q^{36} + 2 q^{37} + q^{38} - q^{39} + 19 q^{41} - q^{42} + 13 q^{43} + 3 q^{44} + 3 q^{46} - 8 q^{47} + 3 q^{48} + 10 q^{49} - 2 q^{51} - q^{52} + 10 q^{53} - 3 q^{54} - q^{56} - q^{57} + 3 q^{58} + 20 q^{61} - 18 q^{62} + q^{63} + 3 q^{64} - 3 q^{66} - 4 q^{67} - 2 q^{68} - 3 q^{69} + 12 q^{71} - 3 q^{72} + q^{73} - 2 q^{74} - q^{76} - 31 q^{77} + q^{78} - 3 q^{79} + 3 q^{81} - 19 q^{82} + 15 q^{83} + q^{84} - 13 q^{86} - 3 q^{87} - 3 q^{88} + 8 q^{89} + 29 q^{91} - 3 q^{92} + 18 q^{93} + 8 q^{94} - 3 q^{96} + 10 q^{97} - 10 q^{98} + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 + 3 * q^3 + 3 * q^4 - 3 * q^6 + q^7 - 3 * q^8 + 3 * q^9 + 3 * q^11 + 3 * q^12 - q^13 - q^14 + 3 * q^16 - 2 * q^17 - 3 * q^18 - q^19 + q^21 - 3 * q^22 - 3 * q^23 - 3 * q^24 + q^26 + 3 * q^27 + q^28 - 3 * q^29 + 18 * q^31 - 3 * q^32 + 3 * q^33 + 2 * q^34 + 3 * q^36 + 2 * q^37 + q^38 - q^39 + 19 * q^41 - q^42 + 13 * q^43 + 3 * q^44 + 3 * q^46 - 8 * q^47 + 3 * q^48 + 10 * q^49 - 2 * q^51 - q^52 + 10 * q^53 - 3 * q^54 - q^56 - q^57 + 3 * q^58 + 20 * q^61 - 18 * q^62 + q^63 + 3 * q^64 - 3 * q^66 - 4 * q^67 - 2 * q^68 - 3 * q^69 + 12 * q^71 - 3 * q^72 + q^73 - 2 * q^74 - q^76 - 31 * q^77 + q^78 - 3 * q^79 + 3 * q^81 - 19 * q^82 + 15 * q^83 + q^84 - 13 * q^86 - 3 * q^87 - 3 * q^88 + 8 * q^89 + 29 * q^91 - 3 * q^92 + 18 * q^93 + 8 * q^94 - 3 * q^96 + 10 * q^97 - 10 * q^98 + 3 * q^99

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