LMFDB - Embedded newform 1449.2.a.o.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1449,2,Mod(1,1449)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1449, 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("1449.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1449 = 3^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1449.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:	$11.5703232529$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.15317.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 2 $ x^4 - 2x^3 - 4x^2 + 5*x + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 483)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.69353$ of defining polynomial
Character	$\chi$	$=$	1449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.69353 q^{2} +0.868028 q^{4} -3.51256 q^{5} -1.00000 q^{7} -1.91702 q^{8} +O(q^{10})$	\(q+1.69353 q^{2} +0.868028 q^{4} -3.51256 q^{5} -1.00000 q^{7} -1.91702 q^{8} -5.94860 q^{10} +1.74252 q^{11} +6.33805 q^{13} -1.69353 q^{14} -4.98258 q^{16} +5.94860 q^{17} +1.74252 q^{19} -3.04900 q^{20} +2.95100 q^{22} -1.00000 q^{23} +7.33805 q^{25} +10.7337 q^{26} -0.868028 q^{28} +5.68466 q^{29} +7.94860 q^{31} -4.60408 q^{32} +10.0741 q^{34} +3.51256 q^{35} +1.53644 q^{37} +2.95100 q^{38} +6.73365 q^{40} -12.1547 q^{41} +6.43605 q^{43} +1.51256 q^{44} -1.69353 q^{46} -3.59313 q^{47} +1.00000 q^{49} +12.4272 q^{50} +5.50161 q^{52} -12.9397 q^{53} -6.12071 q^{55} +1.91702 q^{56} +9.62711 q^{58} +8.69353 q^{59} +8.47618 q^{61} +13.4612 q^{62} +2.16804 q^{64} -22.2628 q^{65} -4.46971 q^{67} +5.16355 q^{68} +5.94860 q^{70} +13.4612 q^{71} +4.29761 q^{73} +2.60200 q^{74} +1.51256 q^{76} -1.74252 q^{77} -7.42071 q^{79} +17.5016 q^{80} -20.5843 q^{82} +7.75262 q^{83} -20.8948 q^{85} +10.8996 q^{86} -3.34045 q^{88} -4.18503 q^{89} -6.33805 q^{91} -0.868028 q^{92} -6.08506 q^{94} -6.12071 q^{95} +5.53644 q^{97} +1.69353 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 4 q^{4} - 5 q^{5} - 4 q^{7} - 9 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 + 4 * q^4 - 5 * q^5 - 4 * q^7 - 9 * q^8	$ 4 q - 2 q^{2} + 4 q^{4} - 5 q^{5} - 4 q^{7} - 9 q^{8} + 2 q^{10} - q^{11} + 7 q^{13} + 2 q^{14} + 8 q^{16} - 2 q^{17} - q^{19} - 13 q^{20} + 11 q^{22} - 4 q^{23} + 11 q^{25} + 19 q^{26} - 4 q^{28} - 2 q^{29} + 6 q^{31} - 20 q^{32} + 23 q^{34} + 5 q^{35} + 16 q^{37} + 11 q^{38} + 3 q^{40} - 5 q^{41} + 9 q^{43} - 3 q^{44} + 2 q^{46} + 21 q^{47} + 4 q^{49} + 17 q^{50} - 24 q^{52} - 10 q^{53} + 17 q^{55} + 9 q^{56} + q^{58} + 26 q^{59} + 2 q^{61} + 19 q^{62} + 27 q^{64} - 26 q^{65} + 5 q^{67} - 7 q^{68} - 2 q^{70} + 19 q^{71} + 10 q^{73} - 9 q^{74} - 3 q^{76} + q^{77} - 6 q^{79} + 24 q^{80} - 31 q^{82} + 2 q^{83} - 7 q^{85} + 17 q^{86} - 20 q^{88} + 17 q^{89} - 7 q^{91} - 4 q^{92} - 44 q^{94} + 17 q^{95} + 32 q^{97} - 2 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 4 * q^4 - 5 * q^5 - 4 * q^7 - 9 * q^8 + 2 * q^10 - q^11 + 7 * q^13 + 2 * q^14 + 8 * q^16 - 2 * q^17 - q^19 - 13 * q^20 + 11 * q^22 - 4 * q^23 + 11 * q^25 + 19 * q^26 - 4 * q^28 - 2 * q^29 + 6 * q^31 - 20 * q^32 + 23 * q^34 + 5 * q^35 + 16 * q^37 + 11 * q^38 + 3 * q^40 - 5 * q^41 + 9 * q^43 - 3 * q^44 + 2 * q^46 + 21 * q^47 + 4 * q^49 + 17 * q^50 - 24 * q^52 - 10 * q^53 + 17 * q^55 + 9 * q^56 + q^58 + 26 * q^59 + 2 * q^61 + 19 * q^62 + 27 * q^64 - 26 * q^65 + 5 * q^67 - 7 * q^68 - 2 * q^70 + 19 * q^71 + 10 * q^73 - 9 * q^74 - 3 * q^76 + q^77 - 6 * q^79 + 24 * q^80 - 31 * q^82 + 2 * q^83 - 7 * q^85 + 17 * q^86 - 20 * q^88 + 17 * q^89 - 7 * q^91 - 4 * q^92 - 44 * q^94 + 17 * q^95 + 32 * 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 $

$2$	1.69353	1.19750	0.598752	−	0.800935i	$-0.295664\pi$
$2$	1.69353	1.19750	0.598752	+	0.800935i	$0.295664\pi$
$3$	0	0
$3$	0	0
$4$	0.868028	0.434014
$5$	−3.51256	−1.57086	−0.785432	−	0.618949i	$-0.787559\pi$
$5$	−3.51256	−1.57086	−0.785432	+	0.618949i	$0.787559\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.91702	−0.677770
$9$	0	0
$10$	−5.94860	−1.88111
$11$	1.74252	0.525390	0.262695	−	0.964879i	$-0.415389\pi$
$11$	1.74252	0.525390	0.262695	+	0.964879i	$0.415389\pi$
$12$	0	0
$13$	6.33805	1.75786	0.878930	−	0.476951i	$-0.158258\pi$
$13$	6.33805	1.75786	0.878930	+	0.476951i	$0.158258\pi$
$14$	−1.69353	−0.452614
$15$	0	0
$16$	−4.98258	−1.24565
$17$	5.94860	1.44275	0.721374	−	0.692546i	$-0.243511\pi$
$17$	5.94860	1.44275	0.721374	+	0.692546i	$0.243511\pi$
$18$	0	0
$19$	1.74252	0.399762	0.199881	−	0.979820i	$-0.435944\pi$
$19$	1.74252	0.399762	0.199881	+	0.979820i	$0.435944\pi$
$20$	−3.04900	−0.681776
$21$	0	0
$22$	2.95100	0.629156
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	7.33805	1.46761
$26$	10.7337	2.10504
$27$	0	0
$28$	−0.868028	−0.164042
$29$	5.68466	1.05561	0.527807	−	0.849364i	$-0.323014\pi$
$29$	5.68466	1.05561	0.527807	+	0.849364i	$0.323014\pi$
$30$	0	0
$31$	7.94860	1.42761	0.713806	−	0.700344i	$-0.246970\pi$
$31$	7.94860	1.42761	0.713806	+	0.700344i	$0.246970\pi$
$32$	−4.60408	−0.813895
$33$	0	0
$34$	10.0741	1.72770
$35$	3.51256	0.593730
$36$	0	0
$37$	1.53644	0.252589	0.126295	−	0.991993i	$-0.459692\pi$
$37$	1.53644	0.252589	0.126295	+	0.991993i	$0.459692\pi$
$38$	2.95100	0.478716
$39$	0	0
$40$	6.73365	1.06468
$41$	−12.1547	−1.89824	−0.949121	−	0.314910i	$-0.898026\pi$
$41$	−12.1547	−1.89824	−0.949121	+	0.314910i	$0.898026\pi$
$42$	0	0
$43$	6.43605	0.981488	0.490744	−	0.871304i	$-0.336725\pi$
$43$	6.43605	0.981488	0.490744	+	0.871304i	$0.336725\pi$
$44$	1.51256	0.228026
$45$	0	0
$46$	−1.69353	−0.249697
$47$	−3.59313	−0.524112	−0.262056	−	0.965053i	$-0.584401\pi$
$47$	−3.59313	−0.524112	−0.262056	+	0.965053i	$0.584401\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	12.4272	1.75747
$51$	0	0
$52$	5.50161	0.762935
$53$	−12.9397	−1.77741	−0.888705	−	0.458480i	$-0.848394\pi$
$53$	−12.9397	−1.77741	−0.888705	+	0.458480i	$0.848394\pi$
$54$	0	0
$55$	−6.12071	−0.825316
$56$	1.91702	0.256173
$57$	0	0
$58$	9.62711	1.26410
$59$	8.69353	1.13180	0.565900	−	0.824474i	$-0.308529\pi$
$59$	8.69353	1.13180	0.565900	+	0.824474i	$0.308529\pi$
$60$	0	0
$61$	8.47618	1.08526	0.542632	−	0.839971i	$-0.317428\pi$
$61$	8.47618	1.08526	0.542632	+	0.839971i	$0.317428\pi$
$62$	13.4612	1.70957
$63$	0	0
$64$	2.16804	0.271005
$65$	−22.2628	−2.76136
$66$	0	0
$67$	−4.46971	−0.546062	−0.273031	−	0.962005i	$-0.588026\pi$
$67$	−4.46971	−0.546062	−0.273031	+	0.962005i	$0.588026\pi$
$68$	5.16355	0.626173
$69$	0	0
$70$	5.94860	0.710994
$71$	13.4612	1.59755	0.798773	−	0.601633i	$-0.205483\pi$
$71$	13.4612	1.59755	0.798773	+	0.601633i	$0.205483\pi$
$72$	0	0
$73$	4.29761	0.502997	0.251498	−	0.967858i	$-0.419077\pi$
$73$	4.29761	0.502997	0.251498	+	0.967858i	$0.419077\pi$
$74$	2.60200	0.302476
$75$	0	0
$76$	1.51256	0.173502
$77$	−1.74252	−0.198579
$78$	0	0
$79$	−7.42071	−0.834896	−0.417448	−	0.908701i	$-0.637075\pi$
$79$	−7.42071	−0.834896	−0.417448	+	0.908701i	$0.637075\pi$
$80$	17.5016	1.95674
$81$	0	0
$82$	−20.5843	−2.27315
$83$	7.75262	0.850960	0.425480	−	0.904968i	$-0.360105\pi$
$83$	7.75262	0.850960	0.425480	+	0.904968i	$0.360105\pi$
$84$	0	0
$85$	−20.8948	−2.26636
$86$	10.8996	1.17533
$87$	0	0
$88$	−3.34045	−0.356094
$89$	−4.18503	−0.443613	−0.221806	−	0.975091i	$-0.571195\pi$
$89$	−4.18503	−0.443613	−0.221806	+	0.975091i	$0.571195\pi$
$90$	0	0
$91$	−6.33805	−0.664409
$92$	−0.868028	−0.0904981
$93$	0	0
$94$	−6.08506	−0.627626
$95$	−6.12071	−0.627971
$96$	0	0
$97$	5.53644	0.562140	0.281070	−	0.959687i	$-0.409311\pi$
$97$	5.53644	0.562140	0.281070	+	0.959687i	$0.409311\pi$
$98$	1.69353	0.171072
$99$	0	0
$100$	6.36963	0.636963
$101$	15.9778	1.58985	0.794924	−	0.606709i	$-0.207510\pi$
$101$	15.9778	1.58985	0.794924	+	0.606709i	$0.207510\pi$
$102$	0	0
$103$	8.96725	0.883569	0.441785	−	0.897121i	$-0.354346\pi$
$103$	8.96725	0.883569	0.441785	+	0.897121i	$0.354346\pi$
$104$	−12.1502	−1.19143
$105$	0	0
$106$	−21.9138	−2.12845
$107$	−20.2353	−1.95622	−0.978108	−	0.208097i	$-0.933273\pi$
$107$	−20.2353	−1.95622	−0.978108	+	0.208097i	$0.933273\pi$
$108$	0	0
$109$	−3.31657	−0.317670	−0.158835	−	0.987305i	$-0.550774\pi$
$109$	−3.31657	−0.317670	−0.158835	+	0.987305i	$0.550774\pi$
$110$	−10.3656	−0.988318
$111$	0	0
$112$	4.98258	0.470810
$113$	−16.1207	−1.51651	−0.758254	−	0.651959i	$-0.773947\pi$
$113$	−16.1207	−1.51651	−0.758254	+	0.651959i	$0.773947\pi$
$114$	0	0
$115$	3.51256	0.327548
$116$	4.93444	0.458151
$117$	0	0
$118$	14.7227	1.35533
$119$	−5.94860	−0.545308
$120$	0	0
$121$	−7.96362	−0.723965
$122$	14.3546	1.29961
$123$	0	0
$124$	6.89961	0.619603
$125$	−8.21255	−0.734553
$126$	0	0
$127$	0.480977	0.0426798	0.0213399	−	0.999772i	$-0.493207\pi$
$127$	0.480977	0.0426798	0.0213399	+	0.999772i	$0.493207\pi$
$128$	12.8798	1.13842
$129$	0	0
$130$	−37.7026	−3.30673
$131$	6.71741	0.586903	0.293451	−	0.955974i	$-0.405196\pi$
$131$	6.71741	0.586903	0.293451	+	0.955974i	$0.405196\pi$
$132$	0	0
$133$	−1.74252	−0.151096
$134$	−7.56957	−0.653911
$135$	0	0
$136$	−11.4036	−0.977852
$137$	2.96362	0.253199	0.126600	−	0.991954i	$-0.459594\pi$
$137$	2.96362	0.253199	0.126600	+	0.991954i	$0.459594\pi$
$138$	0	0
$139$	−8.80161	−0.746543	−0.373272	−	0.927722i	$-0.621764\pi$
$139$	−8.80161	−0.746543	−0.373272	+	0.927722i	$0.621764\pi$
$140$	3.04900	0.257687
$141$	0	0
$142$	22.7968	1.91307
$143$	11.0442	0.923562
$144$	0	0
$145$	−19.9677	−1.65823
$146$	7.27811	0.602340
$147$	0	0
$148$	1.33367	0.109627
$149$	11.2441	0.921155	0.460577	−	0.887620i	$-0.347642\pi$
$149$	11.2441	0.921155	0.460577	+	0.887620i	$0.347642\pi$
$150$	0	0
$151$	5.01502	0.408116	0.204058	−	0.978959i	$-0.434587\pi$
$151$	5.01502	0.408116	0.204058	+	0.978959i	$0.434587\pi$
$152$	−3.34045	−0.270947
$153$	0	0
$154$	−2.95100	−0.237799
$155$	−27.9199	−2.24258
$156$	0	0
$157$	14.8190	1.18269	0.591344	−	0.806420i	$-0.298598\pi$
$157$	14.8190	1.18269	0.591344	+	0.806420i	$0.298598\pi$
$158$	−12.5672	−0.999790
$159$	0	0
$160$	16.1721	1.27852
$161$	1.00000	0.0788110
$162$	0	0
$163$	2.11541	0.165692	0.0828458	−	0.996562i	$-0.473599\pi$
$163$	2.11541	0.165692	0.0828458	+	0.996562i	$0.473599\pi$
$164$	−10.5506	−0.823864
$165$	0	0
$166$	13.1293	1.01903
$167$	14.4186	1.11575	0.557874	−	0.829926i	$-0.311617\pi$
$167$	14.4186	1.11575	0.557874	+	0.829926i	$0.311617\pi$
$168$	0	0
$169$	27.1709	2.09007
$170$	−35.3859	−2.71397
$171$	0	0
$172$	5.58667	0.425979
$173$	9.73969	0.740495	0.370247	−	0.928933i	$-0.379273\pi$
$173$	9.73969	0.740495	0.370247	+	0.928933i	$0.379273\pi$
$174$	0	0
$175$	−7.33805	−0.554705
$176$	−8.68226	−0.654450
$177$	0	0
$178$	−7.08746	−0.531228
$179$	0.376519	0.0281423	0.0140712	−	0.999901i	$-0.495521\pi$
$179$	0.376519	0.0281423	0.0140712	+	0.999901i	$0.495521\pi$
$180$	0	0
$181$	−22.2580	−1.65442	−0.827211	−	0.561891i	$-0.810074\pi$
$181$	−22.2580	−1.65442	−0.827211	+	0.561891i	$0.810074\pi$
$182$	−10.7337	−0.795631
$183$	0	0
$184$	1.91702	0.141325
$185$	−5.39683	−0.396783
$186$	0	0
$187$	10.3656	0.758005
$188$	−3.11894	−0.227472
$189$	0	0
$190$	−10.3656	−0.751997
$191$	−2.98498	−0.215986	−0.107993	−	0.994152i	$-0.534442\pi$
$191$	−2.98498	−0.215986	−0.107993	+	0.994152i	$0.534442\pi$
$192$	0	0
$193$	4.98498	0.358827	0.179413	−	0.983774i	$-0.442580\pi$
$193$	4.98498	0.358827	0.179413	+	0.983774i	$0.442580\pi$
$194$	9.37610	0.673165
$195$	0	0
$196$	0.868028	0.0620020
$197$	−3.41573	−0.243361	−0.121681	−	0.992569i	$-0.538828\pi$
$197$	−3.41573	−0.243361	−0.121681	+	0.992569i	$0.538828\pi$
$198$	0	0
$199$	−3.91462	−0.277500	−0.138750	−	0.990327i	$-0.544308\pi$
$199$	−3.91462	−0.277500	−0.138750	+	0.990327i	$0.544308\pi$
$200$	−14.0672	−0.994703
$201$	0	0
$202$	27.0588	1.90385
$203$	−5.68466	−0.398985
$204$	0	0
$205$	42.6940	2.98188
$206$	15.1863	1.05808
$207$	0	0
$208$	−31.5799	−2.18967
$209$	3.03638	0.210031
$210$	0	0
$211$	−1.59430	−0.109756	−0.0548782	−	0.998493i	$-0.517477\pi$
$211$	−1.59430	−0.109756	−0.0548782	+	0.998493i	$0.517477\pi$
$212$	−11.2320	−0.771420
$213$	0	0
$214$	−34.2689	−2.34258
$215$	−22.6070	−1.54178
$216$	0	0
$217$	−7.94860	−0.539586
$218$	−5.61670	−0.380411
$219$	0	0
$220$	−5.31294	−0.358198
$221$	37.7026	2.53615
$222$	0	0
$223$	2.88667	0.193306	0.0966530	−	0.995318i	$-0.469186\pi$
$223$	2.88667	0.193306	0.0966530	+	0.995318i	$0.469186\pi$
$224$	4.60408	0.307623
$225$	0	0
$226$	−27.3008	−1.81602
$227$	−2.14324	−0.142252	−0.0711259	−	0.997467i	$-0.522659\pi$
$227$	−2.14324	−0.142252	−0.0711259	+	0.997467i	$0.522659\pi$
$228$	0	0
$229$	−14.1308	−0.933790	−0.466895	−	0.884313i	$-0.654627\pi$
$229$	−14.1308	−0.933790	−0.466895	+	0.884313i	$0.654627\pi$
$230$	5.94860	0.392239
$231$	0	0
$232$	−10.8976	−0.715464
$233$	2.88187	0.188798	0.0943989	−	0.995534i	$-0.469907\pi$
$233$	2.88187	0.188798	0.0943989	+	0.995534i	$0.469907\pi$
$234$	0	0
$235$	12.6211	0.823309
$236$	7.54622	0.491217
$237$	0	0
$238$	−10.0741	−0.653008
$239$	−16.4956	−1.06701	−0.533505	−	0.845797i	$-0.679125\pi$
$239$	−16.4956	−1.06701	−0.533505	+	0.845797i	$0.679125\pi$
$240$	0	0
$241$	−18.0017	−1.15959	−0.579795	−	0.814763i	$-0.696867\pi$
$241$	−18.0017	−1.15959	−0.579795	+	0.814763i	$0.696867\pi$
$242$	−13.4866	−0.866951
$243$	0	0
$244$	7.35755	0.471019
$245$	−3.51256	−0.224409
$246$	0	0
$247$	11.0442	0.702725
$248$	−15.2377	−0.967592
$249$	0	0
$250$	−13.9082	−0.879629
$251$	−11.4373	−0.721915	−0.360957	−	0.932582i	$-0.617550\pi$
$251$	−11.4373	−0.721915	−0.360957	+	0.932582i	$0.617550\pi$
$252$	0	0
$253$	−1.74252	−0.109551
$254$	0.814547	0.0511092
$255$	0	0
$256$	17.4762	1.09226
$257$	30.4045	1.89658	0.948291	−	0.317402i	$-0.102810\pi$
$257$	30.4045	1.89658	0.948291	+	0.317402i	$0.102810\pi$
$258$	0	0
$259$	−1.53644	−0.0954697
$260$	−19.3247	−1.19847
$261$	0	0
$262$	11.3761	0.702818
$263$	18.2226	1.12366	0.561828	−	0.827254i	$-0.310098\pi$
$263$	18.2226	1.12366	0.561828	+	0.827254i	$0.310098\pi$
$264$	0	0
$265$	45.4516	2.79207
$266$	−2.95100	−0.180938
$267$	0	0
$268$	−3.87983	−0.236998
$269$	−7.65714	−0.466864	−0.233432	−	0.972373i	$-0.574996\pi$
$269$	−7.65714	−0.466864	−0.233432	+	0.972373i	$0.574996\pi$
$270$	0	0
$271$	10.2628	0.623419	0.311710	−	0.950177i	$-0.399098\pi$
$271$	10.2628	0.623419	0.311710	+	0.950177i	$0.399098\pi$
$272$	−29.6394	−1.79715
$273$	0	0
$274$	5.01896	0.303207
$275$	12.7867	0.771068
$276$	0	0
$277$	7.83319	0.470651	0.235326	−	0.971917i	$-0.424384\pi$
$277$	7.83319	0.470651	0.235326	+	0.971917i	$0.424384\pi$
$278$	−14.9058	−0.893988
$279$	0	0
$280$	−6.73365	−0.402413
$281$	2.34420	0.139843	0.0699217	−	0.997552i	$-0.477725\pi$
$281$	2.34420	0.139843	0.0699217	+	0.997552i	$0.477725\pi$
$282$	0	0
$283$	0.517476	0.0307607	0.0153804	−	0.999882i	$-0.495104\pi$
$283$	0.517476	0.0307607	0.0153804	+	0.999882i	$0.495104\pi$
$284$	11.6847	0.693357
$285$	0	0
$286$	18.7036	1.10597
$287$	12.1547	0.717468
$288$	0	0
$289$	18.3859	1.08152
$290$	−33.8158	−1.98573
$291$	0	0
$292$	3.73044	0.218308
$293$	9.15314	0.534732	0.267366	−	0.963595i	$-0.413847\pi$
$293$	9.15314	0.534732	0.267366	+	0.963595i	$0.413847\pi$
$294$	0	0
$295$	−30.5365	−1.77790
$296$	−2.94539	−0.171197
$297$	0	0
$298$	19.0422	1.10309
$299$	−6.33805	−0.366539
$300$	0	0
$301$	−6.43605	−0.370968
$302$	8.49306	0.488720
$303$	0	0
$304$	−8.68226	−0.497962
$305$	−29.7730	−1.70480
$306$	0	0
$307$	−33.0492	−1.88622	−0.943108	−	0.332487i	$-0.892112\pi$
$307$	−33.0492	−1.88622	−0.943108	+	0.332487i	$0.892112\pi$
$308$	−1.51256	−0.0861859
$309$	0	0
$310$	−47.2831	−2.68550
$311$	28.4591	1.61377	0.806883	−	0.590711i	$-0.201153\pi$
$311$	28.4591	1.61377	0.806883	+	0.590711i	$0.201153\pi$
$312$	0	0
$313$	0.0903560	0.00510722	0.00255361	−	0.999997i	$-0.499187\pi$
$313$	0.0903560	0.00510722	0.00255361	+	0.999997i	$0.499187\pi$
$314$	25.0964	1.41627
$315$	0	0
$316$	−6.44138	−0.362356
$317$	13.1029	0.735930	0.367965	−	0.929840i	$-0.380055\pi$
$317$	13.1029	0.735930	0.367965	+	0.929840i	$0.380055\pi$
$318$	0	0
$319$	9.90564	0.554609
$320$	−7.61535	−0.425711
$321$	0	0
$322$	1.69353	0.0943765
$323$	10.3656	0.576756
$324$	0	0
$325$	46.5090	2.57985
$326$	3.58250	0.198416
$327$	0	0
$328$	23.3008	1.28657
$329$	3.59313	0.198096
$330$	0	0
$331$	4.04013	0.222066	0.111033	−	0.993817i	$-0.464584\pi$
$331$	4.04013	0.222066	0.111033	+	0.993817i	$0.464584\pi$
$332$	6.72949	0.369329
$333$	0	0
$334$	24.4183	1.33611
$335$	15.7001	0.857789
$336$	0	0
$337$	−2.24486	−0.122285	−0.0611427	−	0.998129i	$-0.519474\pi$
$337$	−2.24486	−0.122285	−0.0611427	+	0.998129i	$0.519474\pi$
$338$	46.0147	2.50287
$339$	0	0
$340$	−18.1373	−0.983631
$341$	13.8506	0.750053
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	−12.3381	−0.665223
$345$	0	0
$346$	16.4944	0.886745
$347$	−14.4458	−0.775493	−0.387746	−	0.921766i	$-0.626746\pi$
$347$	−14.4458	−0.775493	−0.387746	+	0.921766i	$0.626746\pi$
$348$	0	0
$349$	−29.2438	−1.56539	−0.782693	−	0.622408i	$-0.786154\pi$
$349$	−29.2438	−1.56539	−0.782693	+	0.622408i	$0.786154\pi$
$350$	−12.4272	−0.664261
$351$	0	0
$352$	−8.02271	−0.427612
$353$	−10.6595	−0.567350	−0.283675	−	0.958920i	$-0.591554\pi$
$353$	−10.6595	−0.567350	−0.283675	+	0.958920i	$0.591554\pi$
$354$	0	0
$355$	−47.2831	−2.50953
$356$	−3.63273	−0.192534
$357$	0	0
$358$	0.637644	0.0337005
$359$	5.44460	0.287355	0.143677	−	0.989625i	$-0.454107\pi$
$359$	5.44460	0.287355	0.143677	+	0.989625i	$0.454107\pi$
$360$	0	0
$361$	−15.9636	−0.840190
$362$	−37.6944	−1.98118
$363$	0	0
$364$	−5.50161	−0.288363
$365$	−15.0956	−0.790139
$366$	0	0
$367$	11.4640	0.598416	0.299208	−	0.954188i	$-0.403278\pi$
$367$	11.4640	0.598416	0.299208	+	0.954188i	$0.403278\pi$
$368$	4.98258	0.259735
$369$	0	0
$370$	−9.13967	−0.475149
$371$	12.9397	0.671798
$372$	0	0
$373$	−27.9859	−1.44905	−0.724527	−	0.689246i	$-0.757942\pi$
$373$	−27.9859	−1.44905	−0.724527	+	0.689246i	$0.757942\pi$
$374$	17.5544	0.907714
$375$	0	0
$376$	6.88812	0.355228
$377$	36.0297	1.85562
$378$	0	0
$379$	11.1604	0.573271	0.286636	−	0.958040i	$-0.407463\pi$
$379$	11.1604	0.573271	0.286636	+	0.958040i	$0.407463\pi$
$380$	−5.31294	−0.272548
$381$	0	0
$382$	−5.05515	−0.258644
$383$	17.2677	0.882338	0.441169	−	0.897424i	$-0.354564\pi$
$383$	17.2677	0.882338	0.441169	+	0.897424i	$0.354564\pi$
$384$	0	0
$385$	6.12071	0.311940
$386$	8.44220	0.429696
$387$	0	0
$388$	4.80578	0.243977
$389$	−27.8158	−1.41032	−0.705158	−	0.709050i	$-0.749124\pi$
$389$	−27.8158	−1.41032	−0.705158	+	0.709050i	$0.749124\pi$
$390$	0	0
$391$	−5.94860	−0.300834
$392$	−1.91702	−0.0968243
$393$	0	0
$394$	−5.78463	−0.291426
$395$	26.0657	1.31151
$396$	0	0
$397$	−36.4276	−1.82825	−0.914123	−	0.405436i	$-0.867120\pi$
$397$	−36.4276	−1.82825	−0.914123	+	0.405436i	$0.867120\pi$
$398$	−6.62951	−0.332307
$399$	0	0
$400$	−36.5625	−1.82812
$401$	1.41605	0.0707142	0.0353571	−	0.999375i	$-0.488743\pi$
$401$	1.41605	0.0707142	0.0353571	+	0.999375i	$0.488743\pi$
$402$	0	0
$403$	50.3787	2.50954
$404$	13.8692	0.690016
$405$	0	0
$406$	−9.62711	−0.477786
$407$	2.67728	0.132708
$408$	0	0
$409$	−0.339704	−0.0167973	−0.00839863	−	0.999965i	$-0.502673\pi$
$409$	−0.339704	−0.0167973	−0.00839863	+	0.999965i	$0.502673\pi$
$410$	72.3034	3.57081
$411$	0	0
$412$	7.78382	0.383481
$413$	−8.69353	−0.427780
$414$	0	0
$415$	−27.2315	−1.33674
$416$	−29.1809	−1.43071
$417$	0	0
$418$	5.14219	0.251513
$419$	−36.7313	−1.79444	−0.897221	−	0.441582i	$-0.854418\pi$
$419$	−36.7313	−1.79444	−0.897221	+	0.441582i	$0.854418\pi$
$420$	0	0
$421$	19.6701	0.958661	0.479330	−	0.877635i	$-0.340880\pi$
$421$	19.6701	0.958661	0.479330	+	0.877635i	$0.340880\pi$
$422$	−2.69999	−0.131434
$423$	0	0
$424$	24.8058	1.20468
$425$	43.6512	2.11739
$426$	0	0
$427$	−8.47618	−0.410191
$428$	−17.5648	−0.849025
$429$	0	0
$430$	−38.2855	−1.84629
$431$	2.51827	0.121301	0.0606504	−	0.998159i	$-0.480683\pi$
$431$	2.51827	0.121301	0.0606504	+	0.998159i	$0.480683\pi$
$432$	0	0
$433$	−27.8393	−1.33787	−0.668937	−	0.743319i	$-0.733250\pi$
$433$	−27.8393	−1.33787	−0.668937	+	0.743319i	$0.733250\pi$
$434$	−13.4612	−0.646156
$435$	0	0
$436$	−2.87888	−0.137873
$437$	−1.74252	−0.0833561
$438$	0	0
$439$	−28.7859	−1.37387	−0.686937	−	0.726717i	$-0.741045\pi$
$439$	−28.7859	−1.37387	−0.686937	+	0.726717i	$0.741045\pi$
$440$	11.7335	0.559374
$441$	0	0
$442$	63.8503	3.03705
$443$	12.3920	0.588760	0.294380	−	0.955688i	$-0.404887\pi$
$443$	12.3920	0.588760	0.294380	+	0.955688i	$0.404887\pi$
$444$	0	0
$445$	14.7002	0.696855
$446$	4.88866	0.231485
$447$	0	0
$448$	−2.16804	−0.102430
$449$	−0.359419	−0.0169620	−0.00848101	−	0.999964i	$-0.502700\pi$
$449$	−0.359419	−0.0169620	−0.00848101	+	0.999964i	$0.502700\pi$
$450$	0	0
$451$	−21.1798	−0.997318
$452$	−13.9932	−0.658186
$453$	0	0
$454$	−3.62963	−0.170347
$455$	22.2628	1.04369
$456$	0	0
$457$	−27.3677	−1.28021	−0.640103	−	0.768289i	$-0.721108\pi$
$457$	−27.3677	−1.28021	−0.640103	+	0.768289i	$0.721108\pi$
$458$	−23.9309	−1.11822
$459$	0	0
$460$	3.04900	0.142160
$461$	7.43230	0.346157	0.173078	−	0.984908i	$-0.444629\pi$
$461$	7.43230	0.346157	0.173078	+	0.984908i	$0.444629\pi$
$462$	0	0
$463$	6.45346	0.299918	0.149959	−	0.988692i	$-0.452086\pi$
$463$	6.45346	0.299918	0.149959	+	0.988692i	$0.452086\pi$
$464$	−28.3243	−1.31492
$465$	0	0
$466$	4.88053	0.226086
$467$	−42.2953	−1.95719	−0.978596	−	0.205792i	$-0.934023\pi$
$467$	−42.2953	−1.95719	−0.978596	+	0.205792i	$0.934023\pi$
$468$	0	0
$469$	4.46971	0.206392
$470$	21.3741	0.985915
$471$	0	0
$472$	−16.6657	−0.767101
$473$	11.2149	0.515664
$474$	0	0
$475$	12.7867	0.586695
$476$	−5.16355	−0.236671
$477$	0	0
$478$	−27.9357	−1.27775
$479$	21.6417	0.988834	0.494417	−	0.869225i	$-0.335382\pi$
$479$	21.6417	0.988834	0.494417	+	0.869225i	$0.335382\pi$
$480$	0	0
$481$	9.73804	0.444016
$482$	−30.4863	−1.38861
$483$	0	0
$484$	−6.91264	−0.314211
$485$	−19.4471	−0.883045
$486$	0	0
$487$	7.21180	0.326798	0.163399	−	0.986560i	$-0.447754\pi$
$487$	7.21180	0.326798	0.163399	+	0.986560i	$0.447754\pi$
$488$	−16.2490	−0.735559
$489$	0	0
$490$	−5.94860	−0.268731
$491$	7.42387	0.335034	0.167517	−	0.985869i	$-0.446425\pi$
$491$	7.42387	0.335034	0.167517	+	0.985869i	$0.446425\pi$
$492$	0	0
$493$	33.8158	1.52299
$494$	18.7036	0.841516
$495$	0	0
$496$	−39.6046	−1.77830
$497$	−13.4612	−0.603816
$498$	0	0
$499$	−29.8147	−1.33469	−0.667344	−	0.744750i	$-0.732569\pi$
$499$	−29.8147	−1.33469	−0.667344	+	0.744750i	$0.732569\pi$
$500$	−7.12872	−0.318806
$501$	0	0
$502$	−19.3693	−0.864495
$503$	−32.1036	−1.43143	−0.715715	−	0.698393i	$-0.753899\pi$
$503$	−32.1036	−1.43143	−0.715715	+	0.698393i	$0.753899\pi$
$504$	0	0
$505$	−56.1229	−2.49743
$506$	−2.95100	−0.131188
$507$	0	0
$508$	0.417501	0.0185236
$509$	−19.4401	−0.861668	−0.430834	−	0.902431i	$-0.641781\pi$
$509$	−19.4401	−0.861668	−0.430834	+	0.902431i	$0.641781\pi$
$510$	0	0
$511$	−4.29761	−0.190115
$512$	3.83676	0.169563
$513$	0	0
$514$	51.4908	2.27116
$515$	−31.4980	−1.38797
$516$	0	0
$517$	−6.26111	−0.275363
$518$	−2.60200	−0.114325
$519$	0	0
$520$	42.6783	1.87157
$521$	9.59843	0.420515	0.210257	−	0.977646i	$-0.432570\pi$
$521$	9.59843	0.420515	0.210257	+	0.977646i	$0.432570\pi$
$522$	0	0
$523$	14.6384	0.640094	0.320047	−	0.947402i	$-0.396301\pi$
$523$	14.6384	0.640094	0.320047	+	0.947402i	$0.396301\pi$
$524$	5.83090	0.254724
$525$	0	0
$526$	30.8605	1.34558
$527$	47.2831	2.05968
$528$	0	0
$529$	1.00000	0.0434783
$530$	76.9734	3.34351
$531$	0	0
$532$	−1.51256	−0.0655776
$533$	−77.0371	−3.33685
$534$	0	0
$535$	71.0775	3.07295
$536$	8.56854	0.370105
$537$	0	0
$538$	−12.9676	−0.559072
$539$	1.74252	0.0750557
$540$	0	0
$541$	5.49994	0.236461	0.118230	−	0.992986i	$-0.462278\pi$
$541$	5.49994	0.236461	0.118230	+	0.992986i	$0.462278\pi$
$542$	17.3803	0.746546
$543$	0	0
$544$	−27.3879	−1.17424
$545$	11.6496	0.499016
$546$	0	0
$547$	6.59357	0.281921	0.140960	−	0.990015i	$-0.454981\pi$
$547$	6.59357	0.281921	0.140960	+	0.990015i	$0.454981\pi$
$548$	2.57250	0.109892
$549$	0	0
$550$	21.6546	0.923356
$551$	9.90564	0.421994
$552$	0	0
$553$	7.42071	0.315561
$554$	13.2657	0.563606
$555$	0	0
$556$	−7.64004	−0.324010
$557$	5.45021	0.230933	0.115466	−	0.993311i	$-0.463164\pi$
$557$	5.45021	0.230933	0.115466	+	0.993311i	$0.463164\pi$
$558$	0	0
$559$	40.7920	1.72532
$560$	−17.5016	−0.739578
$561$	0	0
$562$	3.96997	0.167463
$563$	19.3154	0.814047	0.407024	−	0.913418i	$-0.366567\pi$
$563$	19.3154	0.814047	0.407024	+	0.913418i	$0.366567\pi$
$564$	0	0
$565$	56.6249	2.38223
$566$	0.876358	0.0368361
$567$	0	0
$568$	−25.8054	−1.08277
$569$	20.2263	0.847930	0.423965	−	0.905679i	$-0.360638\pi$
$569$	20.2263	0.847930	0.423965	+	0.905679i	$0.360638\pi$
$570$	0	0
$571$	34.6328	1.44934	0.724670	−	0.689096i	$-0.241992\pi$
$571$	34.6328	1.44934	0.724670	+	0.689096i	$0.241992\pi$
$572$	9.58667	0.400839
$573$	0	0
$574$	20.5843	0.859171
$575$	−7.33805	−0.306018
$576$	0	0
$577$	−36.7645	−1.53053	−0.765263	−	0.643718i	$-0.777391\pi$
$577$	−36.7645	−1.53053	−0.765263	+	0.643718i	$0.777391\pi$
$578$	31.1370	1.29513
$579$	0	0
$580$	−17.3325	−0.719693
$581$	−7.75262	−0.321633
$582$	0	0
$583$	−22.5478	−0.933833
$584$	−8.23862	−0.340916
$585$	0	0
$586$	15.5011	0.640343
$587$	−4.37818	−0.180707	−0.0903535	−	0.995910i	$-0.528800\pi$
$587$	−4.37818	−0.180707	−0.0903535	+	0.995910i	$0.528800\pi$
$588$	0	0
$589$	13.8506	0.570704
$590$	−51.7143	−2.12905
$591$	0	0
$592$	−7.65544	−0.314637
$593$	23.5701	0.967908	0.483954	−	0.875093i	$-0.339200\pi$
$593$	23.5701	0.967908	0.483954	+	0.875093i	$0.339200\pi$
$594$	0	0
$595$	20.8948	0.856604
$596$	9.76021	0.399794
$597$	0	0
$598$	−10.7337	−0.438932
$599$	22.1721	0.905928	0.452964	−	0.891529i	$-0.350367\pi$
$599$	22.1721	0.905928	0.452964	+	0.891529i	$0.350367\pi$
$600$	0	0
$601$	34.7190	1.41622	0.708109	−	0.706103i	$-0.249548\pi$
$601$	34.7190	1.41622	0.708109	+	0.706103i	$0.249548\pi$
$602$	−10.8996	−0.444235
$603$	0	0
$604$	4.35317	0.177128
$605$	27.9727	1.13725
$606$	0	0
$607$	−5.60938	−0.227678	−0.113839	−	0.993499i	$-0.536315\pi$
$607$	−5.60938	−0.227678	−0.113839	+	0.993499i	$0.536315\pi$
$608$	−8.02271	−0.325364
$609$	0	0
$610$	−50.4214	−2.04150
$611$	−22.7735	−0.921316
$612$	0	0
$613$	14.1194	0.570275	0.285138	−	0.958487i	$-0.407961\pi$
$613$	14.1194	0.570275	0.285138	+	0.958487i	$0.407961\pi$
$614$	−55.9696	−2.25875
$615$	0	0
$616$	3.34045	0.134591
$617$	−37.0094	−1.48994	−0.744970	−	0.667098i	$-0.767536\pi$
$617$	−37.0094	−1.48994	−0.744970	+	0.667098i	$0.767536\pi$
$618$	0	0
$619$	42.2855	1.69960	0.849799	−	0.527107i	$-0.176723\pi$
$619$	42.2855	1.69960	0.849799	+	0.527107i	$0.176723\pi$
$620$	−24.2353	−0.973311
$621$	0	0
$622$	48.1962	1.93249
$623$	4.18503	0.167670
$624$	0	0
$625$	−7.84323	−0.313729
$626$	0.153020	0.00611592
$627$	0	0
$628$	12.8633	0.513303
$629$	9.13967	0.364422
$630$	0	0
$631$	31.1770	1.24114	0.620568	−	0.784153i	$-0.286902\pi$
$631$	31.1770	1.24114	0.620568	+	0.784153i	$0.286902\pi$
$632$	14.2257	0.565867
$633$	0	0
$634$	22.1900	0.881278
$635$	−1.68946	−0.0670442
$636$	0	0
$637$	6.33805	0.251123
$638$	16.7754	0.664146
$639$	0	0
$640$	−45.2410	−1.78831
$641$	−35.9135	−1.41850	−0.709248	−	0.704959i	$-0.750965\pi$
$641$	−35.9135	−1.41850	−0.709248	+	0.704959i	$0.750965\pi$
$642$	0	0
$643$	−23.3808	−0.922047	−0.461024	−	0.887388i	$-0.652518\pi$
$643$	−23.3808	−0.922047	−0.461024	+	0.887388i	$0.652518\pi$
$644$	0.868028	0.0342051
$645$	0	0
$646$	17.5544	0.690667
$647$	3.09559	0.121700	0.0608501	−	0.998147i	$-0.480619\pi$
$647$	3.09559	0.121700	0.0608501	+	0.998147i	$0.480619\pi$
$648$	0	0
$649$	15.1487	0.594637
$650$	78.7641	3.08938
$651$	0	0
$652$	1.83623	0.0719125
$653$	−31.9927	−1.25197	−0.625985	−	0.779835i	$-0.715303\pi$
$653$	−31.9927	−1.25197	−0.625985	+	0.779835i	$0.715303\pi$
$654$	0	0
$655$	−23.5953	−0.921944
$656$	60.5617	2.36454
$657$	0	0
$658$	6.08506	0.237220
$659$	4.45396	0.173502	0.0867508	−	0.996230i	$-0.472352\pi$
$659$	4.45396	0.173502	0.0867508	+	0.996230i	$0.472352\pi$
$660$	0	0
$661$	−18.8131	−0.731743	−0.365872	−	0.930665i	$-0.619229\pi$
$661$	−18.8131	−0.731743	−0.365872	+	0.930665i	$0.619229\pi$
$662$	6.84206	0.265924
$663$	0	0
$664$	−14.8620	−0.576756
$665$	6.12071	0.237351
$666$	0	0
$667$	−5.68466	−0.220111
$668$	12.5158	0.484250
$669$	0	0
$670$	26.5885	1.02720
$671$	14.7699	0.570186
$672$	0	0
$673$	−34.6850	−1.33701	−0.668505	−	0.743708i	$-0.733065\pi$
$673$	−34.6850	−1.33701	−0.668505	+	0.743708i	$0.733065\pi$
$674$	−3.80173	−0.146437
$675$	0	0
$676$	23.5851	0.907120
$677$	−12.9841	−0.499021	−0.249510	−	0.968372i	$-0.580270\pi$
$677$	−12.9841	−0.499021	−0.249510	+	0.968372i	$0.580270\pi$
$678$	0	0
$679$	−5.53644	−0.212469
$680$	40.0558	1.53607
$681$	0	0
$682$	23.4564	0.898190
$683$	−8.41814	−0.322111	−0.161055	−	0.986945i	$-0.551490\pi$
$683$	−8.41814	−0.322111	−0.161055	+	0.986945i	$0.551490\pi$
$684$	0	0
$685$	−10.4099	−0.397741
$686$	−1.69353	−0.0646591
$687$	0	0
$688$	−32.0681	−1.22259
$689$	−82.0128	−3.12444
$690$	0	0
$691$	37.8312	1.43917	0.719584	−	0.694406i	$-0.244333\pi$
$691$	37.8312	1.43917	0.719584	+	0.694406i	$0.244333\pi$
$692$	8.45432	0.321385
$693$	0	0
$694$	−24.4644	−0.928655
$695$	30.9162	1.17272
$696$	0	0
$697$	−72.3034	−2.73869
$698$	−49.5251	−1.87455
$699$	0	0
$700$	−6.36963	−0.240750
$701$	−7.93899	−0.299851	−0.149926	−	0.988697i	$-0.547903\pi$
$701$	−7.93899	−0.299851	−0.149926	+	0.988697i	$0.547903\pi$
$702$	0	0
$703$	2.67728	0.100975
$704$	3.77785	0.142383
$705$	0	0
$706$	−18.0522	−0.679404
$707$	−15.9778	−0.600906
$708$	0	0
$709$	1.33730	0.0502235	0.0251117	−	0.999685i	$-0.492006\pi$
$709$	1.33730	0.0502235	0.0251117	+	0.999685i	$0.492006\pi$
$710$	−80.0751	−3.00516
$711$	0	0
$712$	8.02281	0.300668
$713$	−7.94860	−0.297678
$714$	0	0
$715$	−38.7934	−1.45079
$716$	0.326829	0.0122142
$717$	0	0
$718$	9.22056	0.344108
$719$	−43.6532	−1.62799	−0.813994	−	0.580873i	$-0.802711\pi$
$719$	−43.6532	−1.62799	−0.813994	+	0.580873i	$0.802711\pi$
$720$	0	0
$721$	−8.96725	−0.333958
$722$	−27.0348	−1.00613
$723$	0	0
$724$	−19.3205	−0.718042
$725$	41.7143	1.54923
$726$	0	0
$727$	13.1675	0.488356	0.244178	−	0.969730i	$-0.421482\pi$
$727$	13.1675	0.488356	0.244178	+	0.969730i	$0.421482\pi$
$728$	12.1502	0.450316
$729$	0	0
$730$	−25.5648	−0.946194
$731$	38.2855	1.41604
$732$	0	0
$733$	−34.4447	−1.27224	−0.636121	−	0.771589i	$-0.719462\pi$
$733$	−34.4447	−1.27224	−0.636121	+	0.771589i	$0.719462\pi$
$734$	19.4146	0.716605
$735$	0	0
$736$	4.60408	0.169709
$737$	−7.78856	−0.286895
$738$	0	0
$739$	−7.05398	−0.259485	−0.129742	−	0.991548i	$-0.541415\pi$
$739$	−7.05398	−0.259485	−0.129742	+	0.991548i	$0.541415\pi$
$740$	−4.68460	−0.172209
$741$	0	0
$742$	21.9138	0.804480
$743$	6.49211	0.238172	0.119086	−	0.992884i	$-0.462004\pi$
$743$	6.49211	0.238172	0.119086	+	0.992884i	$0.462004\pi$
$744$	0	0
$745$	−39.4956	−1.44701
$746$	−47.3948	−1.73525
$747$	0	0
$748$	8.99760	0.328985
$749$	20.2353	0.739380
$750$	0	0
$751$	2.31669	0.0845372	0.0422686	−	0.999106i	$-0.486541\pi$
$751$	2.31669	0.0845372	0.0422686	+	0.999106i	$0.486541\pi$
$752$	17.9031	0.652858
$753$	0	0
$754$	61.0172	2.22211
$755$	−17.6155	−0.641095
$756$	0	0
$757$	−10.8685	−0.395021	−0.197510	−	0.980301i	$-0.563286\pi$
$757$	−10.8685	−0.395021	−0.197510	+	0.980301i	$0.563286\pi$
$758$	18.9004	0.686494
$759$	0	0
$760$	11.7335	0.425620
$761$	14.1760	0.513881	0.256941	−	0.966427i	$-0.417285\pi$
$761$	14.1760	0.513881	0.256941	+	0.966427i	$0.417285\pi$
$762$	0	0
$763$	3.31657	0.120068
$764$	−2.59105	−0.0937408
$765$	0	0
$766$	29.2433	1.05660
$767$	55.1000	1.98955
$768$	0	0
$769$	11.0939	0.400058	0.200029	−	0.979790i	$-0.435896\pi$
$769$	11.0939	0.400058	0.200029	+	0.979790i	$0.435896\pi$
$770$	10.3656	0.373549
$771$	0	0
$772$	4.32710	0.155736
$773$	−26.3956	−0.949384	−0.474692	−	0.880152i	$-0.657441\pi$
$773$	−26.3956	−0.949384	−0.474692	+	0.880152i	$0.657441\pi$
$774$	0	0
$775$	58.3273	2.09518
$776$	−10.6135	−0.381002
$777$	0	0
$778$	−47.1067	−1.68886
$779$	−21.1798	−0.758845
$780$	0	0
$781$	23.4564	0.839335
$782$	−10.0741	−0.360249
$783$	0	0
$784$	−4.98258	−0.177949
$785$	−52.0527	−1.85784
$786$	0	0
$787$	21.0254	0.749476	0.374738	−	0.927131i	$-0.377733\pi$
$787$	21.0254	0.749476	0.374738	+	0.927131i	$0.377733\pi$
$788$	−2.96495	−0.105622
$789$	0	0
$790$	44.1429	1.57053
$791$	16.1207	0.573186
$792$	0	0
$793$	53.7225	1.90774
$794$	−61.6910	−2.18933
$795$	0	0
$796$	−3.39800	−0.120439
$797$	3.60323	0.127633	0.0638165	−	0.997962i	$-0.479673\pi$
$797$	3.60323	0.127633	0.0638165	+	0.997962i	$0.479673\pi$
$798$	0	0
$799$	−21.3741	−0.756162
$800$	−33.7850	−1.19448
$801$	0	0
$802$	2.39812	0.0846805
$803$	7.48867	0.264270
$804$	0	0
$805$	−3.51256	−0.123801
$806$	85.3176	3.00518
$807$	0	0
$808$	−30.6298	−1.07755
$809$	46.2734	1.62689	0.813443	−	0.581644i	$-0.197590\pi$
$809$	46.2734	1.62689	0.813443	+	0.581644i	$0.197590\pi$
$810$	0	0
$811$	23.3657	0.820480	0.410240	−	0.911978i	$-0.365445\pi$
$811$	23.3657	0.820480	0.410240	+	0.911978i	$0.365445\pi$
$812$	−4.93444	−0.173165
$813$	0	0
$814$	4.53404	0.158918
$815$	−7.43049	−0.260279
$816$	0	0
$817$	11.2149	0.392361
$818$	−0.575297	−0.0201148
$819$	0	0
$820$	37.0596	1.29418
$821$	29.6333	1.03421	0.517104	−	0.855923i	$-0.327010\pi$
$821$	29.6333	1.03421	0.517104	+	0.855923i	$0.327010\pi$
$822$	0	0
$823$	3.93716	0.137241	0.0686203	−	0.997643i	$-0.478140\pi$
$823$	3.93716	0.137241	0.0686203	+	0.997643i	$0.478140\pi$
$824$	−17.1904	−0.598857
$825$	0	0
$826$	−14.7227	−0.512268
$827$	21.6408	0.752526	0.376263	−	0.926513i	$-0.377209\pi$
$827$	21.6408	0.752526	0.376263	+	0.926513i	$0.377209\pi$
$828$	0	0
$829$	20.5616	0.714132	0.357066	−	0.934079i	$-0.383777\pi$
$829$	20.5616	0.714132	0.357066	+	0.934079i	$0.383777\pi$
$830$	−46.1173	−1.60075
$831$	0	0
$832$	13.7411	0.476388
$833$	5.94860	0.206107
$834$	0	0
$835$	−50.6463	−1.75269
$836$	2.63566	0.0911563
$837$	0	0
$838$	−62.2054	−2.14885
$839$	−14.3203	−0.494392	−0.247196	−	0.968965i	$-0.579509\pi$
$839$	−14.3203	−0.494392	−0.247196	+	0.968965i	$0.579509\pi$
$840$	0	0
$841$	3.31534	0.114322
$842$	33.3118	1.14800
$843$	0	0
$844$	−1.38390	−0.0476358
$845$	−95.4394	−3.28322
$846$	0	0
$847$	7.96362	0.273633
$848$	64.4733	2.21402
$849$	0	0
$850$	73.9244	2.53558
$851$	−1.53644	−0.0526685
$852$	0	0
$853$	6.04210	0.206877	0.103439	−	0.994636i	$-0.467015\pi$
$853$	6.04210	0.206877	0.103439	+	0.994636i	$0.467015\pi$
$854$	−14.3546	−0.491205
$855$	0	0
$856$	38.7915	1.32587
$857$	−7.86521	−0.268670	−0.134335	−	0.990936i	$-0.542890\pi$
$857$	−7.86521	−0.268670	−0.134335	+	0.990936i	$0.542890\pi$
$858$	0	0
$859$	−54.5758	−1.86210	−0.931051	−	0.364890i	$-0.881107\pi$
$859$	−54.5758	−1.86210	−0.931051	+	0.364890i	$0.881107\pi$
$860$	−19.6235	−0.669155
$861$	0	0
$862$	4.26476	0.145258
$863$	41.3174	1.40646	0.703230	−	0.710962i	$-0.251740\pi$
$863$	41.3174	1.40646	0.703230	+	0.710962i	$0.251740\pi$
$864$	0	0
$865$	−34.2112	−1.16322
$866$	−47.1466	−1.60211
$867$	0	0
$868$	−6.89961	−0.234188
$869$	−12.9308	−0.438646
$870$	0	0
$871$	−28.3293	−0.959900
$872$	6.35795	0.215307
$873$	0	0
$874$	−2.95100	−0.0998192
$875$	8.21255	0.277635
$876$	0	0
$877$	22.5939	0.762941	0.381471	−	0.924381i	$-0.375418\pi$
$877$	22.5939	0.762941	0.381471	+	0.924381i	$0.375418\pi$
$878$	−48.7496	−1.64522
$879$	0	0
$880$	30.4969	1.02805
$881$	32.4203	1.09227	0.546134	−	0.837698i	$-0.316099\pi$
$881$	32.4203	1.09227	0.546134	+	0.837698i	$0.316099\pi$
$882$	0	0
$883$	46.9261	1.57919	0.789595	−	0.613628i	$-0.210291\pi$
$883$	46.9261	1.57919	0.789595	+	0.613628i	$0.210291\pi$
$884$	32.7269	1.10072
$885$	0	0
$886$	20.9861	0.705042
$887$	−37.9121	−1.27296	−0.636482	−	0.771291i	$-0.719611\pi$
$887$	−37.9121	−1.27296	−0.636482	+	0.771291i	$0.719611\pi$
$888$	0	0
$889$	−0.480977	−0.0161315
$890$	24.8951	0.834486
$891$	0	0
$892$	2.50571	0.0838975
$893$	−6.26111	−0.209520
$894$	0	0
$895$	−1.32254	−0.0442078
$896$	−12.8798	−0.430284
$897$	0	0
$898$	−0.608685	−0.0203121
$899$	45.1851	1.50701
$900$	0	0
$901$	−76.9734	−2.56435
$902$	−35.8685	−1.19429
$903$	0	0
$904$	30.9038	1.02784
$905$	78.1824	2.59887
$906$	0	0
$907$	−44.4382	−1.47555	−0.737773	−	0.675049i	$-0.764123\pi$
$907$	−44.4382	−1.47555	−0.737773	+	0.675049i	$0.764123\pi$
$908$	−1.86039	−0.0617393
$909$	0	0
$910$	37.7026	1.24983
$911$	32.9725	1.09243	0.546215	−	0.837645i	$-0.316068\pi$
$911$	32.9725	1.09243	0.546215	+	0.837645i	$0.316068\pi$
$912$	0	0
$913$	13.5091	0.447086
$914$	−46.3478	−1.53305
$915$	0	0
$916$	−12.2659	−0.405278
$917$	−6.71741	−0.221828
$918$	0	0
$919$	−41.8741	−1.38130	−0.690650	−	0.723189i	$-0.742676\pi$
$919$	−41.8741	−1.38130	−0.690650	+	0.723189i	$0.742676\pi$
$920$	−6.73365	−0.222002
$921$	0	0
$922$	12.5868	0.414524
$923$	85.3176	2.80826
$924$	0	0
$925$	11.2745	0.370703
$926$	10.9291	0.359153
$927$	0	0
$928$	−26.1726	−0.859159
$929$	−21.7717	−0.714306	−0.357153	−	0.934046i	$-0.616253\pi$
$929$	−21.7717	−0.714306	−0.357153	+	0.934046i	$0.616253\pi$
$930$	0	0
$931$	1.74252	0.0571088
$932$	2.50155	0.0819409
$933$	0	0
$934$	−71.6281	−2.34374
$935$	−36.4096	−1.19072
$936$	0	0
$937$	51.2836	1.67536	0.837681	−	0.546160i	$-0.183911\pi$
$937$	51.2836	1.67536	0.837681	+	0.546160i	$0.183911\pi$
$938$	7.56957	0.247155
$939$	0	0
$940$	10.9554	0.357327
$941$	50.8018	1.65609	0.828046	−	0.560661i	$-0.189453\pi$
$941$	50.8018	1.65609	0.828046	+	0.560661i	$0.189453\pi$
$942$	0	0
$943$	12.1547	0.395811
$944$	−43.3162	−1.40982
$945$	0	0
$946$	18.9928	0.617509
$947$	−40.5255	−1.31690	−0.658452	−	0.752623i	$-0.728788\pi$
$947$	−40.5255	−1.31690	−0.658452	+	0.752623i	$0.728788\pi$
$948$	0	0
$949$	27.2385	0.884198
$950$	21.6546	0.702569
$951$	0	0
$952$	11.4036	0.369593
$953$	6.05417	0.196114	0.0980570	−	0.995181i	$-0.468737\pi$
$953$	6.05417	0.196114	0.0980570	+	0.995181i	$0.468737\pi$
$954$	0	0
$955$	10.4849	0.339284
$956$	−14.3186	−0.463097
$957$	0	0
$958$	36.6508	1.18413
$959$	−2.96362	−0.0957003
$960$	0	0
$961$	32.1803	1.03807
$962$	16.4916	0.531711
$963$	0	0
$964$	−15.6259	−0.503278
$965$	−17.5100	−0.563668
$966$	0	0
$967$	40.4901	1.30207	0.651037	−	0.759046i	$-0.274334\pi$
$967$	40.4901	1.30207	0.651037	+	0.759046i	$0.274334\pi$
$968$	15.2664	0.490682
$969$	0	0
$970$	−32.9341	−1.05745
$971$	17.9470	0.575945	0.287973	−	0.957639i	$-0.407019\pi$
$971$	17.9470	0.575945	0.287973	+	0.957639i	$0.407019\pi$
$972$	0	0
$973$	8.80161	0.282167
$974$	12.2134	0.391341
$975$	0	0
$976$	−42.2333	−1.35185
$977$	−15.5013	−0.495930	−0.247965	−	0.968769i	$-0.579762\pi$
$977$	−15.5013	−0.495930	−0.247965	+	0.968769i	$0.579762\pi$
$978$	0	0
$979$	−7.29251	−0.233070
$980$	−3.04900	−0.0973966
$981$	0	0
$982$	12.5725	0.401205
$983$	24.1422	0.770018	0.385009	−	0.922913i	$-0.374198\pi$
$983$	24.1422	0.770018	0.385009	+	0.922913i	$0.374198\pi$
$984$	0	0
$985$	11.9980	0.382287
$986$	57.2679	1.82378
$987$	0	0
$988$	9.58667	0.304992
$989$	−6.43605	−0.204654
$990$	0	0
$991$	−3.26567	−0.103737	−0.0518687	−	0.998654i	$-0.516518\pi$
$991$	−3.26567	−0.103737	−0.0518687	+	0.998654i	$0.516518\pi$
$992$	−36.5960	−1.16193
$993$	0	0
$994$	−22.7968	−0.723071
$995$	13.7503	0.435915
$996$	0	0
$997$	−16.6749	−0.528101	−0.264050	−	0.964509i	$-0.585058\pi$
$997$	−16.6749	−0.528101	−0.264050	+	0.964509i	$0.585058\pi$
$998$	−50.4919	−1.59829
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1449.2.a.o.1.4		4
3.2	odd	2		483.2.a.j.1.1	✓	4
12.11	even	2		7728.2.a.ce.1.4		4
21.20	even	2		3381.2.a.x.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.j.1.1	✓	4	3.2	odd	2
1449.2.a.o.1.4		4	1.1	even	1	trivial
3381.2.a.x.1.1		4	21.20	even	2
7728.2.a.ce.1.4		4	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1449 = 3^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1449.a (trivial)

\(f(q)\)	\(=\)	\(q+1.69353 q^{2} +0.868028 q^{4} -3.51256 q^{5} -1.00000 q^{7} -1.91702 q^{8} +O(q^{10})\)	\(q+1.69353 q^{2} +0.868028 q^{4} -3.51256 q^{5} -1.00000 q^{7} -1.91702 q^{8} -5.94860 q^{10} +1.74252 q^{11} +6.33805 q^{13} -1.69353 q^{14} -4.98258 q^{16} +5.94860 q^{17} +1.74252 q^{19} -3.04900 q^{20} +2.95100 q^{22} -1.00000 q^{23} +7.33805 q^{25} +10.7337 q^{26} -0.868028 q^{28} +5.68466 q^{29} +7.94860 q^{31} -4.60408 q^{32} +10.0741 q^{34} +3.51256 q^{35} +1.53644 q^{37} +2.95100 q^{38} +6.73365 q^{40} -12.1547 q^{41} +6.43605 q^{43} +1.51256 q^{44} -1.69353 q^{46} -3.59313 q^{47} +1.00000 q^{49} +12.4272 q^{50} +5.50161 q^{52} -12.9397 q^{53} -6.12071 q^{55} +1.91702 q^{56} +9.62711 q^{58} +8.69353 q^{59} +8.47618 q^{61} +13.4612 q^{62} +2.16804 q^{64} -22.2628 q^{65} -4.46971 q^{67} +5.16355 q^{68} +5.94860 q^{70} +13.4612 q^{71} +4.29761 q^{73} +2.60200 q^{74} +1.51256 q^{76} -1.74252 q^{77} -7.42071 q^{79} +17.5016 q^{80} -20.5843 q^{82} +7.75262 q^{83} -20.8948 q^{85} +10.8996 q^{86} -3.34045 q^{88} -4.18503 q^{89} -6.33805 q^{91} -0.868028 q^{92} -6.08506 q^{94} -6.12071 q^{95} +5.53644 q^{97} +1.69353 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 4 q^{4} - 5 q^{5} - 4 q^{7} - 9 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 + 4 * q^4 - 5 * q^5 - 4 * q^7 - 9 * q^8	\( 4 q - 2 q^{2} + 4 q^{4} - 5 q^{5} - 4 q^{7} - 9 q^{8} + 2 q^{10} - q^{11} + 7 q^{13} + 2 q^{14} + 8 q^{16} - 2 q^{17} - q^{19} - 13 q^{20} + 11 q^{22} - 4 q^{23} + 11 q^{25} + 19 q^{26} - 4 q^{28} - 2 q^{29} + 6 q^{31} - 20 q^{32} + 23 q^{34} + 5 q^{35} + 16 q^{37} + 11 q^{38} + 3 q^{40} - 5 q^{41} + 9 q^{43} - 3 q^{44} + 2 q^{46} + 21 q^{47} + 4 q^{49} + 17 q^{50} - 24 q^{52} - 10 q^{53} + 17 q^{55} + 9 q^{56} + q^{58} + 26 q^{59} + 2 q^{61} + 19 q^{62} + 27 q^{64} - 26 q^{65} + 5 q^{67} - 7 q^{68} - 2 q^{70} + 19 q^{71} + 10 q^{73} - 9 q^{74} - 3 q^{76} + q^{77} - 6 q^{79} + 24 q^{80} - 31 q^{82} + 2 q^{83} - 7 q^{85} + 17 q^{86} - 20 q^{88} + 17 q^{89} - 7 q^{91} - 4 q^{92} - 44 q^{94} + 17 q^{95} + 32 q^{97} - 2 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 4 * q^4 - 5 * q^5 - 4 * q^7 - 9 * q^8 + 2 * q^10 - q^11 + 7 * q^13 + 2 * q^14 + 8 * q^16 - 2 * q^17 - q^19 - 13 * q^20 + 11 * q^22 - 4 * q^23 + 11 * q^25 + 19 * q^26 - 4 * q^28 - 2 * q^29 + 6 * q^31 - 20 * q^32 + 23 * q^34 + 5 * q^35 + 16 * q^37 + 11 * q^38 + 3 * q^40 - 5 * q^41 + 9 * q^43 - 3 * q^44 + 2 * q^46 + 21 * q^47 + 4 * q^49 + 17 * q^50 - 24 * q^52 - 10 * q^53 + 17 * q^55 + 9 * q^56 + q^58 + 26 * q^59 + 2 * q^61 + 19 * q^62 + 27 * q^64 - 26 * q^65 + 5 * q^67 - 7 * q^68 - 2 * q^70 + 19 * q^71 + 10 * q^73 - 9 * q^74 - 3 * q^76 + q^77 - 6 * q^79 + 24 * q^80 - 31 * q^82 + 2 * q^83 - 7 * q^85 + 17 * q^86 - 20 * q^88 + 17 * q^89 - 7 * q^91 - 4 * q^92 - 44 * q^94 + 17 * q^95 + 32 * q^97 - 2 * q^98

Introduction

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