LMFDB - Embedded newform 7728.2.a.ce.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.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:	$61.7083906820$
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:	$\Z[a_1, \ldots, a_{11}]$
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			$2.69353$ of defining polynomial
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.51256 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.51256 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.74252 q^{11} +6.33805 q^{13} +3.51256 q^{15} -5.94860 q^{17} -1.74252 q^{19} +1.00000 q^{21} -1.00000 q^{23} +7.33805 q^{25} +1.00000 q^{27} -5.68466 q^{29} -7.94860 q^{31} +1.74252 q^{33} +3.51256 q^{35} +1.53644 q^{37} +6.33805 q^{39} +12.1547 q^{41} -6.43605 q^{43} +3.51256 q^{45} -3.59313 q^{47} +1.00000 q^{49} -5.94860 q^{51} +12.9397 q^{53} +6.12071 q^{55} -1.74252 q^{57} +8.69353 q^{59} +8.47618 q^{61} +1.00000 q^{63} +22.2628 q^{65} +4.46971 q^{67} -1.00000 q^{69} +13.4612 q^{71} +4.29761 q^{73} +7.33805 q^{75} +1.74252 q^{77} +7.42071 q^{79} +1.00000 q^{81} +7.75262 q^{83} -20.8948 q^{85} -5.68466 q^{87} +4.18503 q^{89} +6.33805 q^{91} -7.94860 q^{93} -6.12071 q^{95} +5.53644 q^{97} +1.74252 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 5 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 5 * q^5 + 4 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 5 q^{5} + 4 q^{7} + 4 q^{9} - q^{11} + 7 q^{13} + 5 q^{15} + 2 q^{17} + q^{19} + 4 q^{21} - 4 q^{23} + 11 q^{25} + 4 q^{27} + 2 q^{29} - 6 q^{31} - q^{33} + 5 q^{35} + 16 q^{37} + 7 q^{39} + 5 q^{41} - 9 q^{43} + 5 q^{45} + 21 q^{47} + 4 q^{49} + 2 q^{51} + 10 q^{53} - 17 q^{55} + q^{57} + 26 q^{59} + 2 q^{61} + 4 q^{63} + 26 q^{65} - 5 q^{67} - 4 q^{69} + 19 q^{71} + 10 q^{73} + 11 q^{75} - q^{77} + 6 q^{79} + 4 q^{81} + 2 q^{83} - 7 q^{85} + 2 q^{87} - 17 q^{89} + 7 q^{91} - 6 q^{93} + 17 q^{95} + 32 q^{97} - q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 5 * q^5 + 4 * q^7 + 4 * q^9 - q^11 + 7 * q^13 + 5 * q^15 + 2 * q^17 + q^19 + 4 * q^21 - 4 * q^23 + 11 * q^25 + 4 * q^27 + 2 * q^29 - 6 * q^31 - q^33 + 5 * q^35 + 16 * q^37 + 7 * q^39 + 5 * q^41 - 9 * q^43 + 5 * q^45 + 21 * q^47 + 4 * q^49 + 2 * q^51 + 10 * q^53 - 17 * q^55 + q^57 + 26 * q^59 + 2 * q^61 + 4 * q^63 + 26 * q^65 - 5 * q^67 - 4 * q^69 + 19 * q^71 + 10 * q^73 + 11 * q^75 - q^77 + 6 * q^79 + 4 * q^81 + 2 * q^83 - 7 * q^85 + 2 * q^87 - 17 * q^89 + 7 * q^91 - 6 * q^93 + 17 * q^95 + 32 * q^97 - 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 $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.51256	1.57086	0.785432	−	0.618949i	$-0.212441\pi$
$5$	3.51256	1.57086	0.785432	+	0.618949i	$0.212441\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	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	$0.158258\pi$
$14$	0	0
$15$	3.51256	0.906938
$16$	0	0
$17$	−5.94860	−1.44275	−0.721374	−	0.692546i	$-0.756489\pi$
$17$	−5.94860	−1.44275	−0.721374	+	0.692546i	$0.756489\pi$
$18$	0	0
$19$	−1.74252	−0.399762	−0.199881	−	0.979820i	$-0.564056\pi$
$19$	−1.74252	−0.399762	−0.199881	+	0.979820i	$0.564056\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	7.33805	1.46761
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−5.68466	−1.05561	−0.527807	−	0.849364i	$-0.676986\pi$
$29$	−5.68466	−1.05561	−0.527807	+	0.849364i	$0.676986\pi$
$30$	0	0
$31$	−7.94860	−1.42761	−0.713806	−	0.700344i	$-0.753030\pi$
$31$	−7.94860	−1.42761	−0.713806	+	0.700344i	$0.753030\pi$
$32$	0	0
$33$	1.74252	0.303334
$34$	0	0
$35$	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	$0.459692\pi$
$38$	0	0
$39$	6.33805	1.01490
$40$	0	0
$41$	12.1547	1.89824	0.949121	−	0.314910i	$-0.101974\pi$
$41$	12.1547	1.89824	0.949121	+	0.314910i	$0.101974\pi$
$42$	0	0
$43$	−6.43605	−0.981488	−0.490744	−	0.871304i	$-0.663275\pi$
$43$	−6.43605	−0.981488	−0.490744	+	0.871304i	$0.663275\pi$
$44$	0	0
$45$	3.51256	0.523621
$46$	0	0
$47$	−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
$50$	0	0
$51$	−5.94860	−0.832971
$52$	0	0
$53$	12.9397	1.77741	0.888705	−	0.458480i	$-0.151606\pi$
$53$	12.9397	1.77741	0.888705	+	0.458480i	$0.151606\pi$
$54$	0	0
$55$	6.12071	0.825316
$56$	0	0
$57$	−1.74252	−0.230803
$58$	0	0
$59$	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	$0.317428\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	22.2628	2.76136
$66$	0	0
$67$	4.46971	0.546062	0.273031	−	0.962005i	$-0.411974\pi$
$67$	4.46971	0.546062	0.273031	+	0.962005i	$0.411974\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	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	$0.419077\pi$
$74$	0	0
$75$	7.33805	0.847326
$76$	0	0
$77$	1.74252	0.198579
$78$	0	0
$79$	7.42071	0.834896	0.417448	−	0.908701i	$-0.362925\pi$
$79$	7.42071	0.834896	0.417448	+	0.908701i	$0.362925\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	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
$86$	0	0
$87$	−5.68466	−0.609459
$88$	0	0
$89$	4.18503	0.443613	0.221806	−	0.975091i	$-0.428805\pi$
$89$	4.18503	0.443613	0.221806	+	0.975091i	$0.428805\pi$
$90$	0	0
$91$	6.33805	0.664409
$92$	0	0
$93$	−7.94860	−0.824232
$94$	0	0
$95$	−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	$0.409311\pi$
$98$	0	0
$99$	1.74252	0.175130
$100$	0	0
$101$	−15.9778	−1.58985	−0.794924	−	0.606709i	$-0.792490\pi$
$101$	−15.9778	−1.58985	−0.794924	+	0.606709i	$0.792490\pi$
$102$	0	0
$103$	−8.96725	−0.883569	−0.441785	−	0.897121i	$-0.645654\pi$
$103$	−8.96725	−0.883569	−0.441785	+	0.897121i	$0.645654\pi$
$104$	0	0
$105$	3.51256	0.342790
$106$	0	0
$107$	−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	$0.550774\pi$
$110$	0	0
$111$	1.53644	0.145832
$112$	0	0
$113$	16.1207	1.51651	0.758254	−	0.651959i	$-0.226053\pi$
$113$	16.1207	1.51651	0.758254	+	0.651959i	$0.226053\pi$
$114$	0	0
$115$	−3.51256	−0.327548
$116$	0	0
$117$	6.33805	0.585953
$118$	0	0
$119$	−5.94860	−0.545308
$120$	0	0
$121$	−7.96362	−0.723965
$122$	0	0
$123$	12.1547	1.09595
$124$	0	0
$125$	8.21255	0.734553
$126$	0	0
$127$	−0.480977	−0.0426798	−0.0213399	−	0.999772i	$-0.506793\pi$
$127$	−0.480977	−0.0426798	−0.0213399	+	0.999772i	$0.506793\pi$
$128$	0	0
$129$	−6.43605	−0.566662
$130$	0	0
$131$	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
$134$	0	0
$135$	3.51256	0.302313
$136$	0	0
$137$	−2.96362	−0.253199	−0.126600	−	0.991954i	$-0.540406\pi$
$137$	−2.96362	−0.253199	−0.126600	+	0.991954i	$0.540406\pi$
$138$	0	0
$139$	8.80161	0.746543	0.373272	−	0.927722i	$-0.378236\pi$
$139$	8.80161	0.746543	0.373272	+	0.927722i	$0.378236\pi$
$140$	0	0
$141$	−3.59313	−0.302596
$142$	0	0
$143$	11.0442	0.923562
$144$	0	0
$145$	−19.9677	−1.65823
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−11.2441	−0.921155	−0.460577	−	0.887620i	$-0.652358\pi$
$149$	−11.2441	−0.921155	−0.460577	+	0.887620i	$0.652358\pi$
$150$	0	0
$151$	−5.01502	−0.408116	−0.204058	−	0.978959i	$-0.565413\pi$
$151$	−5.01502	−0.408116	−0.204058	+	0.978959i	$0.565413\pi$
$152$	0	0
$153$	−5.94860	−0.480916
$154$	0	0
$155$	−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	$0.298598\pi$
$158$	0	0
$159$	12.9397	1.02619
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−2.11541	−0.165692	−0.0828458	−	0.996562i	$-0.526401\pi$
$163$	−2.11541	−0.165692	−0.0828458	+	0.996562i	$0.526401\pi$
$164$	0	0
$165$	6.12071	0.476496
$166$	0	0
$167$	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
$170$	0	0
$171$	−1.74252	−0.133254
$172$	0	0
$173$	−9.73969	−0.740495	−0.370247	−	0.928933i	$-0.620727\pi$
$173$	−9.73969	−0.740495	−0.370247	+	0.928933i	$0.620727\pi$
$174$	0	0
$175$	7.33805	0.554705
$176$	0	0
$177$	8.69353	0.653445
$178$	0	0
$179$	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	$0.810074\pi$
$182$	0	0
$183$	8.47618	0.626577
$184$	0	0
$185$	5.39683	0.396783
$186$	0	0
$187$	−10.3656	−0.758005
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−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	$0.442580\pi$
$194$	0	0
$195$	22.2628	1.59427
$196$	0	0
$197$	3.41573	0.243361	0.121681	−	0.992569i	$-0.461172\pi$
$197$	3.41573	0.243361	0.121681	+	0.992569i	$0.461172\pi$
$198$	0	0
$199$	3.91462	0.277500	0.138750	−	0.990327i	$-0.455692\pi$
$199$	3.91462	0.277500	0.138750	+	0.990327i	$0.455692\pi$
$200$	0	0
$201$	4.46971	0.315269
$202$	0	0
$203$	−5.68466	−0.398985
$204$	0	0
$205$	42.6940	2.98188
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−3.03638	−0.210031
$210$	0	0
$211$	1.59430	0.109756	0.0548782	−	0.998493i	$-0.482523\pi$
$211$	1.59430	0.109756	0.0548782	+	0.998493i	$0.482523\pi$
$212$	0	0
$213$	13.4612	0.922343
$214$	0	0
$215$	−22.6070	−1.54178
$216$	0	0
$217$	−7.94860	−0.539586
$218$	0	0
$219$	4.29761	0.290405
$220$	0	0
$221$	−37.7026	−2.53615
$222$	0	0
$223$	−2.88667	−0.193306	−0.0966530	−	0.995318i	$-0.530814\pi$
$223$	−2.88667	−0.193306	−0.0966530	+	0.995318i	$0.530814\pi$
$224$	0	0
$225$	7.33805	0.489204
$226$	0	0
$227$	−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	$0.654627\pi$
$230$	0	0
$231$	1.74252	0.114649
$232$	0	0
$233$	−2.88187	−0.188798	−0.0943989	−	0.995534i	$-0.530093\pi$
$233$	−2.88187	−0.188798	−0.0943989	+	0.995534i	$0.530093\pi$
$234$	0	0
$235$	−12.6211	−0.823309
$236$	0	0
$237$	7.42071	0.482027
$238$	0	0
$239$	−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	$0.696867\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.51256	0.224409
$246$	0	0
$247$	−11.0442	−0.702725
$248$	0	0
$249$	7.75262	0.491302
$250$	0	0
$251$	−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	0
$255$	−20.8948	−1.30848
$256$	0	0
$257$	−30.4045	−1.89658	−0.948291	−	0.317402i	$-0.897190\pi$
$257$	−30.4045	−1.89658	−0.948291	+	0.317402i	$0.897190\pi$
$258$	0	0
$259$	1.53644	0.0954697
$260$	0	0
$261$	−5.68466	−0.351872
$262$	0	0
$263$	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
$266$	0	0
$267$	4.18503	0.256120
$268$	0	0
$269$	7.65714	0.466864	0.233432	−	0.972373i	$-0.425004\pi$
$269$	7.65714	0.466864	0.233432	+	0.972373i	$0.425004\pi$
$270$	0	0
$271$	−10.2628	−0.623419	−0.311710	−	0.950177i	$-0.600902\pi$
$271$	−10.2628	−0.623419	−0.311710	+	0.950177i	$0.600902\pi$
$272$	0	0
$273$	6.33805	0.383596
$274$	0	0
$275$	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	$0.424384\pi$
$278$	0	0
$279$	−7.94860	−0.475870
$280$	0	0
$281$	−2.34420	−0.139843	−0.0699217	−	0.997552i	$-0.522275\pi$
$281$	−2.34420	−0.139843	−0.0699217	+	0.997552i	$0.522275\pi$
$282$	0	0
$283$	−0.517476	−0.0307607	−0.0153804	−	0.999882i	$-0.504896\pi$
$283$	−0.517476	−0.0307607	−0.0153804	+	0.999882i	$0.504896\pi$
$284$	0	0
$285$	−6.12071	−0.362559
$286$	0	0
$287$	12.1547	0.717468
$288$	0	0
$289$	18.3859	1.08152
$290$	0	0
$291$	5.53644	0.324552
$292$	0	0
$293$	−9.15314	−0.534732	−0.267366	−	0.963595i	$-0.586153\pi$
$293$	−9.15314	−0.534732	−0.267366	+	0.963595i	$0.586153\pi$
$294$	0	0
$295$	30.5365	1.77790
$296$	0	0
$297$	1.74252	0.101111
$298$	0	0
$299$	−6.33805	−0.366539
$300$	0	0
$301$	−6.43605	−0.370968
$302$	0	0
$303$	−15.9778	−0.917900
$304$	0	0
$305$	29.7730	1.70480
$306$	0	0
$307$	33.0492	1.88622	0.943108	−	0.332487i	$-0.107888\pi$
$307$	33.0492	1.88622	0.943108	+	0.332487i	$0.107888\pi$
$308$	0	0
$309$	−8.96725	−0.510129
$310$	0	0
$311$	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	$0.499187\pi$
$314$	0	0
$315$	3.51256	0.197910
$316$	0	0
$317$	−13.1029	−0.735930	−0.367965	−	0.929840i	$-0.619945\pi$
$317$	−13.1029	−0.735930	−0.367965	+	0.929840i	$0.619945\pi$
$318$	0	0
$319$	−9.90564	−0.554609
$320$	0	0
$321$	−20.2353	−1.12942
$322$	0	0
$323$	10.3656	0.576756
$324$	0	0
$325$	46.5090	2.57985
$326$	0	0
$327$	−3.31657	−0.183407
$328$	0	0
$329$	−3.59313	−0.198096
$330$	0	0
$331$	−4.04013	−0.222066	−0.111033	−	0.993817i	$-0.535416\pi$
$331$	−4.04013	−0.222066	−0.111033	+	0.993817i	$0.535416\pi$
$332$	0	0
$333$	1.53644	0.0841964
$334$	0	0
$335$	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	$0.519474\pi$
$338$	0	0
$339$	16.1207	0.875557
$340$	0	0
$341$	−13.8506	−0.750053
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−3.51256	−0.189110
$346$	0	0
$347$	−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	$0.786154\pi$
$350$	0	0
$351$	6.33805	0.338300
$352$	0	0
$353$	10.6595	0.567350	0.283675	−	0.958920i	$-0.408446\pi$
$353$	10.6595	0.567350	0.283675	+	0.958920i	$0.408446\pi$
$354$	0	0
$355$	47.2831	2.50953
$356$	0	0
$357$	−5.94860	−0.314833
$358$	0	0
$359$	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
$362$	0	0
$363$	−7.96362	−0.417982
$364$	0	0
$365$	15.0956	0.790139
$366$	0	0
$367$	−11.4640	−0.598416	−0.299208	−	0.954188i	$-0.596722\pi$
$367$	−11.4640	−0.598416	−0.299208	+	0.954188i	$0.596722\pi$
$368$	0	0
$369$	12.1547	0.632748
$370$	0	0
$371$	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	$0.757942\pi$
$374$	0	0
$375$	8.21255	0.424094
$376$	0	0
$377$	−36.0297	−1.85562
$378$	0	0
$379$	−11.1604	−0.573271	−0.286636	−	0.958040i	$-0.592537\pi$
$379$	−11.1604	−0.573271	−0.286636	+	0.958040i	$0.592537\pi$
$380$	0	0
$381$	−0.480977	−0.0246412
$382$	0	0
$383$	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
$386$	0	0
$387$	−6.43605	−0.327163
$388$	0	0
$389$	27.8158	1.41032	0.705158	−	0.709050i	$-0.250876\pi$
$389$	27.8158	1.41032	0.705158	+	0.709050i	$0.250876\pi$
$390$	0	0
$391$	5.94860	0.300834
$392$	0	0
$393$	6.71741	0.338848
$394$	0	0
$395$	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	$0.867120\pi$
$398$	0	0
$399$	−1.74252	−0.0872352
$400$	0	0
$401$	−1.41605	−0.0707142	−0.0353571	−	0.999375i	$-0.511257\pi$
$401$	−1.41605	−0.0707142	−0.0353571	+	0.999375i	$0.511257\pi$
$402$	0	0
$403$	−50.3787	−2.50954
$404$	0	0
$405$	3.51256	0.174540
$406$	0	0
$407$	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	$0.502673\pi$
$410$	0	0
$411$	−2.96362	−0.146185
$412$	0	0
$413$	8.69353	0.427780
$414$	0	0
$415$	27.2315	1.33674
$416$	0	0
$417$	8.80161	0.431017
$418$	0	0
$419$	−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	$0.340880\pi$
$422$	0	0
$423$	−3.59313	−0.174704
$424$	0	0
$425$	−43.6512	−2.11739
$426$	0	0
$427$	8.47618	0.410191
$428$	0	0
$429$	11.0442	0.533219
$430$	0	0
$431$	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	$0.733250\pi$
$434$	0	0
$435$	−19.9677	−0.957377
$436$	0	0
$437$	1.74252	0.0833561
$438$	0	0
$439$	28.7859	1.37387	0.686937	−	0.726717i	$-0.258955\pi$
$439$	28.7859	1.37387	0.686937	+	0.726717i	$0.258955\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	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
$446$	0	0
$447$	−11.2441	−0.531829
$448$	0	0
$449$	0.359419	0.0169620	0.00848101	−	0.999964i	$-0.497300\pi$
$449$	0.359419	0.0169620	0.00848101	+	0.999964i	$0.497300\pi$
$450$	0	0
$451$	21.1798	0.997318
$452$	0	0
$453$	−5.01502	−0.235626
$454$	0	0
$455$	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	$0.721108\pi$
$458$	0	0
$459$	−5.94860	−0.277657
$460$	0	0
$461$	−7.43230	−0.346157	−0.173078	−	0.984908i	$-0.555371\pi$
$461$	−7.43230	−0.346157	−0.173078	+	0.984908i	$0.555371\pi$
$462$	0	0
$463$	−6.45346	−0.299918	−0.149959	−	0.988692i	$-0.547914\pi$
$463$	−6.45346	−0.299918	−0.149959	+	0.988692i	$0.547914\pi$
$464$	0	0
$465$	−27.9199	−1.29476
$466$	0	0
$467$	−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
$470$	0	0
$471$	14.8190	0.682825
$472$	0	0
$473$	−11.2149	−0.515664
$474$	0	0
$475$	−12.7867	−0.586695
$476$	0	0
$477$	12.9397	0.592470
$478$	0	0
$479$	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
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	19.4471	0.883045
$486$	0	0
$487$	−7.21180	−0.326798	−0.163399	−	0.986560i	$-0.552246\pi$
$487$	−7.21180	−0.326798	−0.163399	+	0.986560i	$0.552246\pi$
$488$	0	0
$489$	−2.11541	−0.0956621
$490$	0	0
$491$	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
$494$	0	0
$495$	6.12071	0.275105
$496$	0	0
$497$	13.4612	0.603816
$498$	0	0
$499$	29.8147	1.33469	0.667344	−	0.744750i	$-0.267431\pi$
$499$	29.8147	1.33469	0.667344	+	0.744750i	$0.267431\pi$
$500$	0	0
$501$	14.4186	0.644177
$502$	0	0
$503$	−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
$506$	0	0
$507$	27.1709	1.20670
$508$	0	0
$509$	19.4401	0.861668	0.430834	−	0.902431i	$-0.358219\pi$
$509$	19.4401	0.861668	0.430834	+	0.902431i	$0.358219\pi$
$510$	0	0
$511$	4.29761	0.190115
$512$	0	0
$513$	−1.74252	−0.0769342
$514$	0	0
$515$	−31.4980	−1.38797
$516$	0	0
$517$	−6.26111	−0.275363
$518$	0	0
$519$	−9.73969	−0.427525
$520$	0	0
$521$	−9.59843	−0.420515	−0.210257	−	0.977646i	$-0.567430\pi$
$521$	−9.59843	−0.420515	−0.210257	+	0.977646i	$0.567430\pi$
$522$	0	0
$523$	−14.6384	−0.640094	−0.320047	−	0.947402i	$-0.603699\pi$
$523$	−14.6384	−0.640094	−0.320047	+	0.947402i	$0.603699\pi$
$524$	0	0
$525$	7.33805	0.320259
$526$	0	0
$527$	47.2831	2.05968
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	8.69353	0.377267
$532$	0	0
$533$	77.0371	3.33685
$534$	0	0
$535$	−71.0775	−3.07295
$536$	0	0
$537$	0.376519	0.0162480
$538$	0	0
$539$	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	$0.462278\pi$
$542$	0	0
$543$	−22.2580	−0.955181
$544$	0	0
$545$	−11.6496	−0.499016
$546$	0	0
$547$	−6.59357	−0.281921	−0.140960	−	0.990015i	$-0.545019\pi$
$547$	−6.59357	−0.281921	−0.140960	+	0.990015i	$0.545019\pi$
$548$	0	0
$549$	8.47618	0.361754
$550$	0	0
$551$	9.90564	0.421994
$552$	0	0
$553$	7.42071	0.315561
$554$	0	0
$555$	5.39683	0.229083
$556$	0	0
$557$	−5.45021	−0.230933	−0.115466	−	0.993311i	$-0.536836\pi$
$557$	−5.45021	−0.230933	−0.115466	+	0.993311i	$0.536836\pi$
$558$	0	0
$559$	−40.7920	−1.72532
$560$	0	0
$561$	−10.3656	−0.437635
$562$	0	0
$563$	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	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−20.2263	−0.847930	−0.423965	−	0.905679i	$-0.639362\pi$
$569$	−20.2263	−0.847930	−0.423965	+	0.905679i	$0.639362\pi$
$570$	0	0
$571$	−34.6328	−1.44934	−0.724670	−	0.689096i	$-0.758008\pi$
$571$	−34.6328	−1.44934	−0.724670	+	0.689096i	$0.758008\pi$
$572$	0	0
$573$	−2.98498	−0.124699
$574$	0	0
$575$	−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	$0.777391\pi$
$578$	0	0
$579$	4.98498	0.207169
$580$	0	0
$581$	7.75262	0.321633
$582$	0	0
$583$	22.5478	0.933833
$584$	0	0
$585$	22.2628	0.920452
$586$	0	0
$587$	−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
$590$	0	0
$591$	3.41573	0.140505
$592$	0	0
$593$	−23.5701	−0.967908	−0.483954	−	0.875093i	$-0.660800\pi$
$593$	−23.5701	−0.967908	−0.483954	+	0.875093i	$0.660800\pi$
$594$	0	0
$595$	−20.8948	−0.856604
$596$	0	0
$597$	3.91462	0.160215
$598$	0	0
$599$	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	$0.249548\pi$
$602$	0	0
$603$	4.46971	0.182021
$604$	0	0
$605$	−27.9727	−1.13725
$606$	0	0
$607$	5.60938	0.227678	0.113839	−	0.993499i	$-0.463685\pi$
$607$	5.60938	0.227678	0.113839	+	0.993499i	$0.463685\pi$
$608$	0	0
$609$	−5.68466	−0.230354
$610$	0	0
$611$	−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	$0.407961\pi$
$614$	0	0
$615$	42.6940	1.72159
$616$	0	0
$617$	37.0094	1.48994	0.744970	−	0.667098i	$-0.232464\pi$
$617$	37.0094	1.48994	0.744970	+	0.667098i	$0.232464\pi$
$618$	0	0
$619$	−42.2855	−1.69960	−0.849799	−	0.527107i	$-0.823277\pi$
$619$	−42.2855	−1.69960	−0.849799	+	0.527107i	$0.823277\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	4.18503	0.167670
$624$	0	0
$625$	−7.84323	−0.313729
$626$	0	0
$627$	−3.03638	−0.121261
$628$	0	0
$629$	−9.13967	−0.364422
$630$	0	0
$631$	−31.1770	−1.24114	−0.620568	−	0.784153i	$-0.713098\pi$
$631$	−31.1770	−1.24114	−0.620568	+	0.784153i	$0.713098\pi$
$632$	0	0
$633$	1.59430	0.0633678
$634$	0	0
$635$	−1.68946	−0.0670442
$636$	0	0
$637$	6.33805	0.251123
$638$	0	0
$639$	13.4612	0.532515
$640$	0	0
$641$	35.9135	1.41850	0.709248	−	0.704959i	$-0.249035\pi$
$641$	35.9135	1.41850	0.709248	+	0.704959i	$0.249035\pi$
$642$	0	0
$643$	23.3808	0.922047	0.461024	−	0.887388i	$-0.347482\pi$
$643$	23.3808	0.922047	0.461024	+	0.887388i	$0.347482\pi$
$644$	0	0
$645$	−22.6070	−0.890149
$646$	0	0
$647$	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
$650$	0	0
$651$	−7.94860	−0.311530
$652$	0	0
$653$	31.9927	1.25197	0.625985	−	0.779835i	$-0.284697\pi$
$653$	31.9927	1.25197	0.625985	+	0.779835i	$0.284697\pi$
$654$	0	0
$655$	23.5953	0.921944
$656$	0	0
$657$	4.29761	0.167666
$658$	0	0
$659$	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	$0.619229\pi$
$662$	0	0
$663$	−37.7026	−1.46425
$664$	0	0
$665$	−6.12071	−0.237351
$666$	0	0
$667$	5.68466	0.220111
$668$	0	0
$669$	−2.88667	−0.111605
$670$	0	0
$671$	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	$0.733065\pi$
$674$	0	0
$675$	7.33805	0.282442
$676$	0	0
$677$	12.9841	0.499021	0.249510	−	0.968372i	$-0.419730\pi$
$677$	12.9841	0.499021	0.249510	+	0.968372i	$0.419730\pi$
$678$	0	0
$679$	5.53644	0.212469
$680$	0	0
$681$	−2.14324	−0.0821291
$682$	0	0
$683$	−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
$686$	0	0
$687$	−14.1308	−0.539124
$688$	0	0
$689$	82.0128	3.12444
$690$	0	0
$691$	−37.8312	−1.43917	−0.719584	−	0.694406i	$-0.755667\pi$
$691$	−37.8312	−1.43917	−0.719584	+	0.694406i	$0.755667\pi$
$692$	0	0
$693$	1.74252	0.0661929
$694$	0	0
$695$	30.9162	1.17272
$696$	0	0
$697$	−72.3034	−2.73869
$698$	0	0
$699$	−2.88187	−0.109002
$700$	0	0
$701$	7.93899	0.299851	0.149926	−	0.988697i	$-0.452097\pi$
$701$	7.93899	0.299851	0.149926	+	0.988697i	$0.452097\pi$
$702$	0	0
$703$	−2.67728	−0.100975
$704$	0	0
$705$	−12.6211	−0.475337
$706$	0	0
$707$	−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	$0.492006\pi$
$710$	0	0
$711$	7.42071	0.278299
$712$	0	0
$713$	7.94860	0.297678
$714$	0	0
$715$	38.7934	1.45079
$716$	0	0
$717$	−16.4956	−0.616039
$718$	0	0
$719$	−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
$722$	0	0
$723$	−18.0017	−0.669489
$724$	0	0
$725$	−41.7143	−1.54923
$726$	0	0
$727$	−13.1675	−0.488356	−0.244178	−	0.969730i	$-0.578518\pi$
$727$	−13.1675	−0.488356	−0.244178	+	0.969730i	$0.578518\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	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	$0.719462\pi$
$734$	0	0
$735$	3.51256	0.129563
$736$	0	0
$737$	7.78856	0.286895
$738$	0	0
$739$	7.05398	0.259485	0.129742	−	0.991548i	$-0.458585\pi$
$739$	7.05398	0.259485	0.129742	+	0.991548i	$0.458585\pi$
$740$	0	0
$741$	−11.0442	−0.405719
$742$	0	0
$743$	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
$746$	0	0
$747$	7.75262	0.283653
$748$	0	0
$749$	−20.2353	−0.739380
$750$	0	0
$751$	−2.31669	−0.0845372	−0.0422686	−	0.999106i	$-0.513459\pi$
$751$	−2.31669	−0.0845372	−0.0422686	+	0.999106i	$0.513459\pi$
$752$	0	0
$753$	−11.4373	−0.416798
$754$	0	0
$755$	−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	$0.563286\pi$
$758$	0	0
$759$	−1.74252	−0.0632495
$760$	0	0
$761$	−14.1760	−0.513881	−0.256941	−	0.966427i	$-0.582715\pi$
$761$	−14.1760	−0.513881	−0.256941	+	0.966427i	$0.582715\pi$
$762$	0	0
$763$	−3.31657	−0.120068
$764$	0	0
$765$	−20.8948	−0.755453
$766$	0	0
$767$	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	$0.435896\pi$
$770$	0	0
$771$	−30.4045	−1.09499
$772$	0	0
$773$	26.3956	0.949384	0.474692	−	0.880152i	$-0.342559\pi$
$773$	26.3956	0.949384	0.474692	+	0.880152i	$0.342559\pi$
$774$	0	0
$775$	−58.3273	−2.09518
$776$	0	0
$777$	1.53644	0.0551195
$778$	0	0
$779$	−21.1798	−0.758845
$780$	0	0
$781$	23.4564	0.839335
$782$	0	0
$783$	−5.68466	−0.203153
$784$	0	0
$785$	52.0527	1.85784
$786$	0	0
$787$	−21.0254	−0.749476	−0.374738	−	0.927131i	$-0.622267\pi$
$787$	−21.0254	−0.749476	−0.374738	+	0.927131i	$0.622267\pi$
$788$	0	0
$789$	18.2226	0.648743
$790$	0	0
$791$	16.1207	0.573186
$792$	0	0
$793$	53.7225	1.90774
$794$	0	0
$795$	45.4516	1.61200
$796$	0	0
$797$	−3.60323	−0.127633	−0.0638165	−	0.997962i	$-0.520327\pi$
$797$	−3.60323	−0.127633	−0.0638165	+	0.997962i	$0.520327\pi$
$798$	0	0
$799$	21.3741	0.756162
$800$	0	0
$801$	4.18503	0.147871
$802$	0	0
$803$	7.48867	0.264270
$804$	0	0
$805$	−3.51256	−0.123801
$806$	0	0
$807$	7.65714	0.269544
$808$	0	0
$809$	−46.2734	−1.62689	−0.813443	−	0.581644i	$-0.802410\pi$
$809$	−46.2734	−1.62689	−0.813443	+	0.581644i	$0.802410\pi$
$810$	0	0
$811$	−23.3657	−0.820480	−0.410240	−	0.911978i	$-0.634555\pi$
$811$	−23.3657	−0.820480	−0.410240	+	0.911978i	$0.634555\pi$
$812$	0	0
$813$	−10.2628	−0.359931
$814$	0	0
$815$	−7.43049	−0.260279
$816$	0	0
$817$	11.2149	0.392361
$818$	0	0
$819$	6.33805	0.221470
$820$	0	0
$821$	−29.6333	−1.03421	−0.517104	−	0.855923i	$-0.672990\pi$
$821$	−29.6333	−1.03421	−0.517104	+	0.855923i	$0.672990\pi$
$822$	0	0
$823$	−3.93716	−0.137241	−0.0686203	−	0.997643i	$-0.521860\pi$
$823$	−3.93716	−0.137241	−0.0686203	+	0.997643i	$0.521860\pi$
$824$	0	0
$825$	12.7867	0.445176
$826$	0	0
$827$	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	$0.383777\pi$
$830$	0	0
$831$	7.83319	0.271730
$832$	0	0
$833$	−5.94860	−0.206107
$834$	0	0
$835$	50.6463	1.75269
$836$	0	0
$837$	−7.94860	−0.274744
$838$	0	0
$839$	−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
$842$	0	0
$843$	−2.34420	−0.0807387
$844$	0	0
$845$	95.4394	3.28322
$846$	0	0
$847$	−7.96362	−0.273633
$848$	0	0
$849$	−0.517476	−0.0177597
$850$	0	0
$851$	−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	$0.467015\pi$
$854$	0	0
$855$	−6.12071	−0.209324
$856$	0	0
$857$	7.86521	0.268670	0.134335	−	0.990936i	$-0.457110\pi$
$857$	7.86521	0.268670	0.134335	+	0.990936i	$0.457110\pi$
$858$	0	0
$859$	54.5758	1.86210	0.931051	−	0.364890i	$-0.118893\pi$
$859$	54.5758	1.86210	0.931051	+	0.364890i	$0.118893\pi$
$860$	0	0
$861$	12.1547	0.414231
$862$	0	0
$863$	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
$866$	0	0
$867$	18.3859	0.624417
$868$	0	0
$869$	12.9308	0.438646
$870$	0	0
$871$	28.3293	0.959900
$872$	0	0
$873$	5.53644	0.187380
$874$	0	0
$875$	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	$0.375418\pi$
$878$	0	0
$879$	−9.15314	−0.308728
$880$	0	0
$881$	−32.4203	−1.09227	−0.546134	−	0.837698i	$-0.683901\pi$
$881$	−32.4203	−1.09227	−0.546134	+	0.837698i	$0.683901\pi$
$882$	0	0
$883$	−46.9261	−1.57919	−0.789595	−	0.613628i	$-0.789709\pi$
$883$	−46.9261	−1.57919	−0.789595	+	0.613628i	$0.789709\pi$
$884$	0	0
$885$	30.5365	1.02647
$886$	0	0
$887$	−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
$890$	0	0
$891$	1.74252	0.0583767
$892$	0	0
$893$	6.26111	0.209520
$894$	0	0
$895$	1.32254	0.0442078
$896$	0	0
$897$	−6.33805	−0.211621
$898$	0	0
$899$	45.1851	1.50701
$900$	0	0
$901$	−76.9734	−2.56435
$902$	0	0
$903$	−6.43605	−0.214178
$904$	0	0
$905$	−78.1824	−2.59887
$906$	0	0
$907$	44.4382	1.47555	0.737773	−	0.675049i	$-0.235877\pi$
$907$	44.4382	1.47555	0.737773	+	0.675049i	$0.235877\pi$
$908$	0	0
$909$	−15.9778	−0.529950
$910$	0	0
$911$	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
$914$	0	0
$915$	29.7730	0.984267
$916$	0	0
$917$	6.71741	0.221828
$918$	0	0
$919$	41.8741	1.38130	0.690650	−	0.723189i	$-0.257324\pi$
$919$	41.8741	1.38130	0.690650	+	0.723189i	$0.257324\pi$
$920$	0	0
$921$	33.0492	1.08901
$922$	0	0
$923$	85.3176	2.80826
$924$	0	0
$925$	11.2745	0.370703
$926$	0	0
$927$	−8.96725	−0.294523
$928$	0	0
$929$	21.7717	0.714306	0.357153	−	0.934046i	$-0.383747\pi$
$929$	21.7717	0.714306	0.357153	+	0.934046i	$0.383747\pi$
$930$	0	0
$931$	−1.74252	−0.0571088
$932$	0	0
$933$	28.4591	0.931708
$934$	0	0
$935$	−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	$0.183911\pi$
$938$	0	0
$939$	0.0903560	0.00294866
$940$	0	0
$941$	−50.8018	−1.65609	−0.828046	−	0.560661i	$-0.810547\pi$
$941$	−50.8018	−1.65609	−0.828046	+	0.560661i	$0.810547\pi$
$942$	0	0
$943$	−12.1547	−0.395811
$944$	0	0
$945$	3.51256	0.114263
$946$	0	0
$947$	−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
$950$	0	0
$951$	−13.1029	−0.424889
$952$	0	0
$953$	−6.05417	−0.196114	−0.0980570	−	0.995181i	$-0.531263\pi$
$953$	−6.05417	−0.196114	−0.0980570	+	0.995181i	$0.531263\pi$
$954$	0	0
$955$	−10.4849	−0.339284
$956$	0	0
$957$	−9.90564	−0.320204
$958$	0	0
$959$	−2.96362	−0.0957003
$960$	0	0
$961$	32.1803	1.03807
$962$	0	0
$963$	−20.2353	−0.652072
$964$	0	0
$965$	17.5100	0.563668
$966$	0	0
$967$	−40.4901	−1.30207	−0.651037	−	0.759046i	$-0.725666\pi$
$967$	−40.4901	−1.30207	−0.651037	+	0.759046i	$0.725666\pi$
$968$	0	0
$969$	10.3656	0.332990
$970$	0	0
$971$	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
$974$	0	0
$975$	46.5090	1.48948
$976$	0	0
$977$	15.5013	0.495930	0.247965	−	0.968769i	$-0.420238\pi$
$977$	15.5013	0.495930	0.247965	+	0.968769i	$0.420238\pi$
$978$	0	0
$979$	7.29251	0.233070
$980$	0	0
$981$	−3.31657	−0.105890
$982$	0	0
$983$	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
$986$	0	0
$987$	−3.59313	−0.114371
$988$	0	0
$989$	6.43605	0.204654
$990$	0	0
$991$	3.26567	0.103737	0.0518687	−	0.998654i	$-0.483482\pi$
$991$	3.26567	0.103737	0.0518687	+	0.998654i	$0.483482\pi$
$992$	0	0
$993$	−4.04013	−0.128210
$994$	0	0
$995$	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	$0.585058\pi$
$998$	0	0
$999$	1.53644	0.0486108

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.ce.1.4		4
4.3	odd	2		483.2.a.j.1.1	✓	4
12.11	even	2		1449.2.a.o.1.4		4
28.27	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	4.3	odd	2
1449.2.a.o.1.4		4	12.11	even	2
3381.2.a.x.1.1		4	28.27	even	2
7728.2.a.ce.1.4		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.51256 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.51256 q^{5} +1.00000 q^{7} +1.00000 q^{9} +1.74252 q^{11} +6.33805 q^{13} +3.51256 q^{15} -5.94860 q^{17} -1.74252 q^{19} +1.00000 q^{21} -1.00000 q^{23} +7.33805 q^{25} +1.00000 q^{27} -5.68466 q^{29} -7.94860 q^{31} +1.74252 q^{33} +3.51256 q^{35} +1.53644 q^{37} +6.33805 q^{39} +12.1547 q^{41} -6.43605 q^{43} +3.51256 q^{45} -3.59313 q^{47} +1.00000 q^{49} -5.94860 q^{51} +12.9397 q^{53} +6.12071 q^{55} -1.74252 q^{57} +8.69353 q^{59} +8.47618 q^{61} +1.00000 q^{63} +22.2628 q^{65} +4.46971 q^{67} -1.00000 q^{69} +13.4612 q^{71} +4.29761 q^{73} +7.33805 q^{75} +1.74252 q^{77} +7.42071 q^{79} +1.00000 q^{81} +7.75262 q^{83} -20.8948 q^{85} -5.68466 q^{87} +4.18503 q^{89} +6.33805 q^{91} -7.94860 q^{93} -6.12071 q^{95} +5.53644 q^{97} +1.74252 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 5 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 5 * q^5 + 4 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 5 q^{5} + 4 q^{7} + 4 q^{9} - q^{11} + 7 q^{13} + 5 q^{15} + 2 q^{17} + q^{19} + 4 q^{21} - 4 q^{23} + 11 q^{25} + 4 q^{27} + 2 q^{29} - 6 q^{31} - q^{33} + 5 q^{35} + 16 q^{37} + 7 q^{39} + 5 q^{41} - 9 q^{43} + 5 q^{45} + 21 q^{47} + 4 q^{49} + 2 q^{51} + 10 q^{53} - 17 q^{55} + q^{57} + 26 q^{59} + 2 q^{61} + 4 q^{63} + 26 q^{65} - 5 q^{67} - 4 q^{69} + 19 q^{71} + 10 q^{73} + 11 q^{75} - q^{77} + 6 q^{79} + 4 q^{81} + 2 q^{83} - 7 q^{85} + 2 q^{87} - 17 q^{89} + 7 q^{91} - 6 q^{93} + 17 q^{95} + 32 q^{97} - q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 5 * q^5 + 4 * q^7 + 4 * q^9 - q^11 + 7 * q^13 + 5 * q^15 + 2 * q^17 + q^19 + 4 * q^21 - 4 * q^23 + 11 * q^25 + 4 * q^27 + 2 * q^29 - 6 * q^31 - q^33 + 5 * q^35 + 16 * q^37 + 7 * q^39 + 5 * q^41 - 9 * q^43 + 5 * q^45 + 21 * q^47 + 4 * q^49 + 2 * q^51 + 10 * q^53 - 17 * q^55 + q^57 + 26 * q^59 + 2 * q^61 + 4 * q^63 + 26 * q^65 - 5 * q^67 - 4 * q^69 + 19 * q^71 + 10 * q^73 + 11 * q^75 - q^77 + 6 * q^79 + 4 * q^81 + 2 * q^83 - 7 * q^85 + 2 * q^87 - 17 * q^89 + 7 * q^91 - 6 * q^93 + 17 * q^95 + 32 * q^97 - q^99

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