LMFDB - Embedded newform 7728.2.a.bn.1.2

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
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			1.2
Root			$1.61803$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.61803 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.61803 q^{5} +1.00000 q^{7} +1.00000 q^{9} +2.23607 q^{11} -4.61803 q^{13} +1.61803 q^{15} -6.70820 q^{17} -5.47214 q^{19} +1.00000 q^{21} +1.00000 q^{23} -2.38197 q^{25} +1.00000 q^{27} -3.76393 q^{29} +6.70820 q^{31} +2.23607 q^{33} +1.61803 q^{35} -11.0000 q^{37} -4.61803 q^{39} +7.47214 q^{41} -0.618034 q^{43} +1.61803 q^{45} +2.76393 q^{47} +1.00000 q^{49} -6.70820 q^{51} -1.90983 q^{53} +3.61803 q^{55} -5.47214 q^{57} -11.6180 q^{59} +1.85410 q^{61} +1.00000 q^{63} -7.47214 q^{65} -6.09017 q^{67} +1.00000 q^{69} -6.61803 q^{71} +0.708204 q^{73} -2.38197 q^{75} +2.23607 q^{77} +0.527864 q^{79} +1.00000 q^{81} -13.1803 q^{83} -10.8541 q^{85} -3.76393 q^{87} +9.38197 q^{89} -4.61803 q^{91} +6.70820 q^{93} -8.85410 q^{95} -16.4164 q^{97} +2.23607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} - 7 q^{13} + q^{15} - 2 q^{19} + 2 q^{21} + 2 q^{23} - 7 q^{25} + 2 q^{27} - 12 q^{29} + q^{35} - 22 q^{37} - 7 q^{39} + 6 q^{41} + q^{43} + q^{45} + 10 q^{47} + 2 q^{49} - 15 q^{53} + 5 q^{55} - 2 q^{57} - 21 q^{59} - 3 q^{61} + 2 q^{63} - 6 q^{65} - q^{67} + 2 q^{69} - 11 q^{71} - 12 q^{73} - 7 q^{75} + 10 q^{79} + 2 q^{81} - 4 q^{83} - 15 q^{85} - 12 q^{87} + 21 q^{89} - 7 q^{91} - 11 q^{95} - 6 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9 - 7 * q^13 + q^15 - 2 * q^19 + 2 * q^21 + 2 * q^23 - 7 * q^25 + 2 * q^27 - 12 * q^29 + q^35 - 22 * q^37 - 7 * q^39 + 6 * q^41 + q^43 + q^45 + 10 * q^47 + 2 * q^49 - 15 * q^53 + 5 * q^55 - 2 * q^57 - 21 * q^59 - 3 * q^61 + 2 * q^63 - 6 * q^65 - q^67 + 2 * q^69 - 11 * q^71 - 12 * q^73 - 7 * q^75 + 10 * q^79 + 2 * q^81 - 4 * q^83 - 15 * q^85 - 12 * q^87 + 21 * q^89 - 7 * q^91 - 11 * q^95 - 6 * q^97

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
$5$	1.61803	0.723607	0.361803	−	0.932254i	$-0.382161\pi$
$5$	1.61803	0.723607	0.361803	+	0.932254i	$0.382161\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.23607	0.674200	0.337100	−	0.941469i	$-0.390554\pi$
$11$	2.23607	0.674200	0.337100	+	0.941469i	$0.390554\pi$
$12$	0	0
$13$	−4.61803	−1.28081	−0.640406	−	0.768036i	$-0.721234\pi$
$13$	−4.61803	−1.28081	−0.640406	+	0.768036i	$0.721234\pi$
$14$	0	0
$15$	1.61803	0.417775
$16$	0	0
$17$	−6.70820	−1.62698	−0.813489	−	0.581580i	$-0.802435\pi$
$17$	−6.70820	−1.62698	−0.813489	+	0.581580i	$0.802435\pi$
$18$	0	0
$19$	−5.47214	−1.25539	−0.627697	−	0.778458i	$-0.716002\pi$
$19$	−5.47214	−1.25539	−0.627697	+	0.778458i	$0.716002\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−2.38197	−0.476393
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.76393	−0.698945	−0.349472	−	0.936947i	$-0.613639\pi$
$29$	−3.76393	−0.698945	−0.349472	+	0.936947i	$0.613639\pi$
$30$	0	0
$31$	6.70820	1.20483	0.602414	−	0.798183i	$-0.294205\pi$
$31$	6.70820	1.20483	0.602414	+	0.798183i	$0.294205\pi$
$32$	0	0
$33$	2.23607	0.389249
$34$	0	0
$35$	1.61803	0.273498
$36$	0	0
$37$	−11.0000	−1.80839	−0.904194	−	0.427121i	$-0.859528\pi$
$37$	−11.0000	−1.80839	−0.904194	+	0.427121i	$0.859528\pi$
$38$	0	0
$39$	−4.61803	−0.739477
$40$	0	0
$41$	7.47214	1.16695	0.583476	−	0.812131i	$-0.301692\pi$
$41$	7.47214	1.16695	0.583476	+	0.812131i	$0.301692\pi$
$42$	0	0
$43$	−0.618034	−0.0942493	−0.0471246	−	0.998889i	$-0.515006\pi$
$43$	−0.618034	−0.0942493	−0.0471246	+	0.998889i	$0.515006\pi$
$44$	0	0
$45$	1.61803	0.241202
$46$	0	0
$47$	2.76393	0.403161	0.201580	−	0.979472i	$-0.435392\pi$
$47$	2.76393	0.403161	0.201580	+	0.979472i	$0.435392\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−6.70820	−0.939336
$52$	0	0
$53$	−1.90983	−0.262335	−0.131168	−	0.991360i	$-0.541873\pi$
$53$	−1.90983	−0.262335	−0.131168	+	0.991360i	$0.541873\pi$
$54$	0	0
$55$	3.61803	0.487856
$56$	0	0
$57$	−5.47214	−0.724802
$58$	0	0
$59$	−11.6180	−1.51254	−0.756270	−	0.654260i	$-0.772980\pi$
$59$	−11.6180	−1.51254	−0.756270	+	0.654260i	$0.772980\pi$
$60$	0	0
$61$	1.85410	0.237393	0.118697	−	0.992931i	$-0.462128\pi$
$61$	1.85410	0.237393	0.118697	+	0.992931i	$0.462128\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−7.47214	−0.926804
$66$	0	0
$67$	−6.09017	−0.744033	−0.372016	−	0.928226i	$-0.621333\pi$
$67$	−6.09017	−0.744033	−0.372016	+	0.928226i	$0.621333\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−6.61803	−0.785416	−0.392708	−	0.919663i	$-0.628462\pi$
$71$	−6.61803	−0.785416	−0.392708	+	0.919663i	$0.628462\pi$
$72$	0	0
$73$	0.708204	0.0828890	0.0414445	−	0.999141i	$-0.486804\pi$
$73$	0.708204	0.0828890	0.0414445	+	0.999141i	$0.486804\pi$
$74$	0	0
$75$	−2.38197	−0.275046
$76$	0	0
$77$	2.23607	0.254824
$78$	0	0
$79$	0.527864	0.0593893	0.0296947	−	0.999559i	$-0.490547\pi$
$79$	0.527864	0.0593893	0.0296947	+	0.999559i	$0.490547\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−13.1803	−1.44673	−0.723365	−	0.690466i	$-0.757406\pi$
$83$	−13.1803	−1.44673	−0.723365	+	0.690466i	$0.757406\pi$
$84$	0	0
$85$	−10.8541	−1.17729
$86$	0	0
$87$	−3.76393	−0.403536
$88$	0	0
$89$	9.38197	0.994486	0.497243	−	0.867611i	$-0.334346\pi$
$89$	9.38197	0.994486	0.497243	+	0.867611i	$0.334346\pi$
$90$	0	0
$91$	−4.61803	−0.484102
$92$	0	0
$93$	6.70820	0.695608
$94$	0	0
$95$	−8.85410	−0.908412
$96$	0	0
$97$	−16.4164	−1.66683	−0.833417	−	0.552645i	$-0.813619\pi$
$97$	−16.4164	−1.66683	−0.833417	+	0.552645i	$0.813619\pi$
$98$	0	0
$99$	2.23607	0.224733
$100$	0	0
$101$	10.8541	1.08002	0.540012	−	0.841657i	$-0.318420\pi$
$101$	10.8541	1.08002	0.540012	+	0.841657i	$0.318420\pi$
$102$	0	0
$103$	−19.4164	−1.91316	−0.956578	−	0.291477i	$-0.905853\pi$
$103$	−19.4164	−1.91316	−0.956578	+	0.291477i	$0.905853\pi$
$104$	0	0
$105$	1.61803	0.157904
$106$	0	0
$107$	12.3262	1.19162	0.595811	−	0.803125i	$-0.296831\pi$
$107$	12.3262	1.19162	0.595811	+	0.803125i	$0.296831\pi$
$108$	0	0
$109$	−14.2705	−1.36687	−0.683433	−	0.730013i	$-0.739514\pi$
$109$	−14.2705	−1.36687	−0.683433	+	0.730013i	$0.739514\pi$
$110$	0	0
$111$	−11.0000	−1.04407
$112$	0	0
$113$	−12.0902	−1.13735	−0.568674	−	0.822563i	$-0.692543\pi$
$113$	−12.0902	−1.13735	−0.568674	+	0.822563i	$0.692543\pi$
$114$	0	0
$115$	1.61803	0.150882
$116$	0	0
$117$	−4.61803	−0.426937
$118$	0	0
$119$	−6.70820	−0.614940
$120$	0	0
$121$	−6.00000	−0.545455
$122$	0	0
$123$	7.47214	0.673740
$124$	0	0
$125$	−11.9443	−1.06833
$126$	0	0
$127$	11.2705	1.00010	0.500048	−	0.865998i	$-0.333316\pi$
$127$	11.2705	1.00010	0.500048	+	0.865998i	$0.333316\pi$
$128$	0	0
$129$	−0.618034	−0.0544149
$130$	0	0
$131$	7.18034	0.627349	0.313675	−	0.949531i	$-0.398440\pi$
$131$	7.18034	0.627349	0.313675	+	0.949531i	$0.398440\pi$
$132$	0	0
$133$	−5.47214	−0.474494
$134$	0	0
$135$	1.61803	0.139258
$136$	0	0
$137$	10.4164	0.889934	0.444967	−	0.895547i	$-0.353215\pi$
$137$	10.4164	0.889934	0.444967	+	0.895547i	$0.353215\pi$
$138$	0	0
$139$	13.6180	1.15507	0.577533	−	0.816367i	$-0.304015\pi$
$139$	13.6180	1.15507	0.577533	+	0.816367i	$0.304015\pi$
$140$	0	0
$141$	2.76393	0.232765
$142$	0	0
$143$	−10.3262	−0.863523
$144$	0	0
$145$	−6.09017	−0.505761
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−14.7639	−1.20951	−0.604754	−	0.796412i	$-0.706729\pi$
$149$	−14.7639	−1.20951	−0.604754	+	0.796412i	$0.706729\pi$
$150$	0	0
$151$	10.7639	0.875956	0.437978	−	0.898986i	$-0.355695\pi$
$151$	10.7639	0.875956	0.437978	+	0.898986i	$0.355695\pi$
$152$	0	0
$153$	−6.70820	−0.542326
$154$	0	0
$155$	10.8541	0.871822
$156$	0	0
$157$	−13.7082	−1.09403	−0.547017	−	0.837122i	$-0.684237\pi$
$157$	−13.7082	−1.09403	−0.547017	+	0.837122i	$0.684237\pi$
$158$	0	0
$159$	−1.90983	−0.151459
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	−1.61803	−0.126734	−0.0633671	−	0.997990i	$-0.520184\pi$
$163$	−1.61803	−0.126734	−0.0633671	+	0.997990i	$0.520184\pi$
$164$	0	0
$165$	3.61803	0.281664
$166$	0	0
$167$	−15.1803	−1.17469	−0.587345	−	0.809337i	$-0.699827\pi$
$167$	−15.1803	−1.17469	−0.587345	+	0.809337i	$0.699827\pi$
$168$	0	0
$169$	8.32624	0.640480
$170$	0	0
$171$	−5.47214	−0.418465
$172$	0	0
$173$	5.47214	0.416039	0.208019	−	0.978125i	$-0.433298\pi$
$173$	5.47214	0.416039	0.208019	+	0.978125i	$0.433298\pi$
$174$	0	0
$175$	−2.38197	−0.180060
$176$	0	0
$177$	−11.6180	−0.873265
$178$	0	0
$179$	4.85410	0.362813	0.181406	−	0.983408i	$-0.441935\pi$
$179$	4.85410	0.362813	0.181406	+	0.983408i	$0.441935\pi$
$180$	0	0
$181$	15.9443	1.18513	0.592564	−	0.805523i	$-0.298116\pi$
$181$	15.9443	1.18513	0.592564	+	0.805523i	$0.298116\pi$
$182$	0	0
$183$	1.85410	0.137059
$184$	0	0
$185$	−17.7984	−1.30856
$186$	0	0
$187$	−15.0000	−1.09691
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	6.18034	0.447194	0.223597	−	0.974682i	$-0.428220\pi$
$191$	6.18034	0.447194	0.223597	+	0.974682i	$0.428220\pi$
$192$	0	0
$193$	21.7082	1.56259	0.781295	−	0.624161i	$-0.214559\pi$
$193$	21.7082	1.56259	0.781295	+	0.624161i	$0.214559\pi$
$194$	0	0
$195$	−7.47214	−0.535091
$196$	0	0
$197$	12.5066	0.891057	0.445528	−	0.895268i	$-0.353016\pi$
$197$	12.5066	0.891057	0.445528	+	0.895268i	$0.353016\pi$
$198$	0	0
$199$	18.0902	1.28238	0.641189	−	0.767383i	$-0.278441\pi$
$199$	18.0902	1.28238	0.641189	+	0.767383i	$0.278441\pi$
$200$	0	0
$201$	−6.09017	−0.429567
$202$	0	0
$203$	−3.76393	−0.264176
$204$	0	0
$205$	12.0902	0.844414
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−12.2361	−0.846387
$210$	0	0
$211$	16.4164	1.13015	0.565076	−	0.825039i	$-0.308847\pi$
$211$	16.4164	1.13015	0.565076	+	0.825039i	$0.308847\pi$
$212$	0	0
$213$	−6.61803	−0.453460
$214$	0	0
$215$	−1.00000	−0.0681994
$216$	0	0
$217$	6.70820	0.455383
$218$	0	0
$219$	0.708204	0.0478560
$220$	0	0
$221$	30.9787	2.08385
$222$	0	0
$223$	11.1459	0.746385	0.373192	−	0.927754i	$-0.378263\pi$
$223$	11.1459	0.746385	0.373192	+	0.927754i	$0.378263\pi$
$224$	0	0
$225$	−2.38197	−0.158798
$226$	0	0
$227$	−8.67376	−0.575698	−0.287849	−	0.957676i	$-0.592940\pi$
$227$	−8.67376	−0.575698	−0.287849	+	0.957676i	$0.592940\pi$
$228$	0	0
$229$	−0.673762	−0.0445235	−0.0222617	−	0.999752i	$-0.507087\pi$
$229$	−0.673762	−0.0445235	−0.0222617	+	0.999752i	$0.507087\pi$
$230$	0	0
$231$	2.23607	0.147122
$232$	0	0
$233$	0.0901699	0.00590723	0.00295361	−	0.999996i	$-0.499060\pi$
$233$	0.0901699	0.00590723	0.00295361	+	0.999996i	$0.499060\pi$
$234$	0	0
$235$	4.47214	0.291730
$236$	0	0
$237$	0.527864	0.0342885
$238$	0	0
$239$	−19.7984	−1.28065	−0.640325	−	0.768104i	$-0.721200\pi$
$239$	−19.7984	−1.28065	−0.640325	+	0.768104i	$0.721200\pi$
$240$	0	0
$241$	−11.0000	−0.708572	−0.354286	−	0.935137i	$-0.615276\pi$
$241$	−11.0000	−0.708572	−0.354286	+	0.935137i	$0.615276\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	1.61803	0.103372
$246$	0	0
$247$	25.2705	1.60792
$248$	0	0
$249$	−13.1803	−0.835270
$250$	0	0
$251$	17.1246	1.08090	0.540448	−	0.841377i	$-0.318255\pi$
$251$	17.1246	1.08090	0.540448	+	0.841377i	$0.318255\pi$
$252$	0	0
$253$	2.23607	0.140580
$254$	0	0
$255$	−10.8541	−0.679710
$256$	0	0
$257$	1.23607	0.0771038	0.0385519	−	0.999257i	$-0.487726\pi$
$257$	1.23607	0.0771038	0.0385519	+	0.999257i	$0.487726\pi$
$258$	0	0
$259$	−11.0000	−0.683507
$260$	0	0
$261$	−3.76393	−0.232982
$262$	0	0
$263$	17.9443	1.10649	0.553246	−	0.833018i	$-0.313389\pi$
$263$	17.9443	1.10649	0.553246	+	0.833018i	$0.313389\pi$
$264$	0	0
$265$	−3.09017	−0.189828
$266$	0	0
$267$	9.38197	0.574167
$268$	0	0
$269$	−20.9098	−1.27489	−0.637447	−	0.770494i	$-0.720010\pi$
$269$	−20.9098	−1.27489	−0.637447	+	0.770494i	$0.720010\pi$
$270$	0	0
$271$	−4.05573	−0.246368	−0.123184	−	0.992384i	$-0.539311\pi$
$271$	−4.05573	−0.246368	−0.123184	+	0.992384i	$0.539311\pi$
$272$	0	0
$273$	−4.61803	−0.279496
$274$	0	0
$275$	−5.32624	−0.321184
$276$	0	0
$277$	−29.2705	−1.75869	−0.879347	−	0.476181i	$-0.842021\pi$
$277$	−29.2705	−1.75869	−0.879347	+	0.476181i	$0.842021\pi$
$278$	0	0
$279$	6.70820	0.401610
$280$	0	0
$281$	9.70820	0.579143	0.289571	−	0.957156i	$-0.406487\pi$
$281$	9.70820	0.579143	0.289571	+	0.957156i	$0.406487\pi$
$282$	0	0
$283$	−25.2148	−1.49886	−0.749432	−	0.662082i	$-0.769673\pi$
$283$	−25.2148	−1.49886	−0.749432	+	0.662082i	$0.769673\pi$
$284$	0	0
$285$	−8.85410	−0.524472
$286$	0	0
$287$	7.47214	0.441066
$288$	0	0
$289$	28.0000	1.64706
$290$	0	0
$291$	−16.4164	−0.962347
$292$	0	0
$293$	12.0000	0.701047	0.350524	−	0.936554i	$-0.386004\pi$
$293$	12.0000	0.701047	0.350524	+	0.936554i	$0.386004\pi$
$294$	0	0
$295$	−18.7984	−1.09448
$296$	0	0
$297$	2.23607	0.129750
$298$	0	0
$299$	−4.61803	−0.267068
$300$	0	0
$301$	−0.618034	−0.0356229
$302$	0	0
$303$	10.8541	0.623552
$304$	0	0
$305$	3.00000	0.171780
$306$	0	0
$307$	24.1246	1.37686	0.688432	−	0.725301i	$-0.258299\pi$
$307$	24.1246	1.37686	0.688432	+	0.725301i	$0.258299\pi$
$308$	0	0
$309$	−19.4164	−1.10456
$310$	0	0
$311$	0.326238	0.0184993	0.00924963	−	0.999957i	$-0.497056\pi$
$311$	0.326238	0.0184993	0.00924963	+	0.999957i	$0.497056\pi$
$312$	0	0
$313$	6.47214	0.365827	0.182913	−	0.983129i	$-0.441447\pi$
$313$	6.47214	0.365827	0.182913	+	0.983129i	$0.441447\pi$
$314$	0	0
$315$	1.61803	0.0911659
$316$	0	0
$317$	−23.5066	−1.32026	−0.660130	−	0.751151i	$-0.729499\pi$
$317$	−23.5066	−1.32026	−0.660130	+	0.751151i	$0.729499\pi$
$318$	0	0
$319$	−8.41641	−0.471228
$320$	0	0
$321$	12.3262	0.687984
$322$	0	0
$323$	36.7082	2.04250
$324$	0	0
$325$	11.0000	0.610170
$326$	0	0
$327$	−14.2705	−0.789161
$328$	0	0
$329$	2.76393	0.152381
$330$	0	0
$331$	−13.4164	−0.737432	−0.368716	−	0.929542i	$-0.620203\pi$
$331$	−13.4164	−0.737432	−0.368716	+	0.929542i	$0.620203\pi$
$332$	0	0
$333$	−11.0000	−0.602796
$334$	0	0
$335$	−9.85410	−0.538387
$336$	0	0
$337$	30.5066	1.66180	0.830900	−	0.556422i	$-0.187826\pi$
$337$	30.5066	1.66180	0.830900	+	0.556422i	$0.187826\pi$
$338$	0	0
$339$	−12.0902	−0.656648
$340$	0	0
$341$	15.0000	0.812296
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	1.61803	0.0871120
$346$	0	0
$347$	17.1803	0.922289	0.461144	−	0.887325i	$-0.347439\pi$
$347$	17.1803	0.922289	0.461144	+	0.887325i	$0.347439\pi$
$348$	0	0
$349$	−5.14590	−0.275454	−0.137727	−	0.990470i	$-0.543980\pi$
$349$	−5.14590	−0.275454	−0.137727	+	0.990470i	$0.543980\pi$
$350$	0	0
$351$	−4.61803	−0.246492
$352$	0	0
$353$	−32.3050	−1.71942	−0.859710	−	0.510783i	$-0.829355\pi$
$353$	−32.3050	−1.71942	−0.859710	+	0.510783i	$0.829355\pi$
$354$	0	0
$355$	−10.7082	−0.568332
$356$	0	0
$357$	−6.70820	−0.355036
$358$	0	0
$359$	−11.9098	−0.628577	−0.314288	−	0.949328i	$-0.601766\pi$
$359$	−11.9098	−0.628577	−0.314288	+	0.949328i	$0.601766\pi$
$360$	0	0
$361$	10.9443	0.576014
$362$	0	0
$363$	−6.00000	−0.314918
$364$	0	0
$365$	1.14590	0.0599790
$366$	0	0
$367$	−4.14590	−0.216414	−0.108207	−	0.994128i	$-0.534511\pi$
$367$	−4.14590	−0.216414	−0.108207	+	0.994128i	$0.534511\pi$
$368$	0	0
$369$	7.47214	0.388984
$370$	0	0
$371$	−1.90983	−0.0991534
$372$	0	0
$373$	−2.41641	−0.125117	−0.0625584	−	0.998041i	$-0.519926\pi$
$373$	−2.41641	−0.125117	−0.0625584	+	0.998041i	$0.519926\pi$
$374$	0	0
$375$	−11.9443	−0.616800
$376$	0	0
$377$	17.3820	0.895217
$378$	0	0
$379$	19.4164	0.997354	0.498677	−	0.866788i	$-0.333819\pi$
$379$	19.4164	0.997354	0.498677	+	0.866788i	$0.333819\pi$
$380$	0	0
$381$	11.2705	0.577406
$382$	0	0
$383$	−12.7082	−0.649359	−0.324679	−	0.945824i	$-0.605256\pi$
$383$	−12.7082	−0.649359	−0.324679	+	0.945824i	$0.605256\pi$
$384$	0	0
$385$	3.61803	0.184392
$386$	0	0
$387$	−0.618034	−0.0314164
$388$	0	0
$389$	18.7082	0.948544	0.474272	−	0.880378i	$-0.342711\pi$
$389$	18.7082	0.948544	0.474272	+	0.880378i	$0.342711\pi$
$390$	0	0
$391$	−6.70820	−0.339248
$392$	0	0
$393$	7.18034	0.362200
$394$	0	0
$395$	0.854102	0.0429745
$396$	0	0
$397$	−36.0689	−1.81025	−0.905123	−	0.425150i	$-0.860221\pi$
$397$	−36.0689	−1.81025	−0.905123	+	0.425150i	$0.860221\pi$
$398$	0	0
$399$	−5.47214	−0.273949
$400$	0	0
$401$	10.8197	0.540308	0.270154	−	0.962817i	$-0.412925\pi$
$401$	10.8197	0.540308	0.270154	+	0.962817i	$0.412925\pi$
$402$	0	0
$403$	−30.9787	−1.54316
$404$	0	0
$405$	1.61803	0.0804008
$406$	0	0
$407$	−24.5967	−1.21922
$408$	0	0
$409$	−20.4164	−1.00953	−0.504763	−	0.863258i	$-0.668420\pi$
$409$	−20.4164	−1.00953	−0.504763	+	0.863258i	$0.668420\pi$
$410$	0	0
$411$	10.4164	0.513804
$412$	0	0
$413$	−11.6180	−0.571686
$414$	0	0
$415$	−21.3262	−1.04686
$416$	0	0
$417$	13.6180	0.666878
$418$	0	0
$419$	−31.3262	−1.53039	−0.765193	−	0.643800i	$-0.777357\pi$
$419$	−31.3262	−1.53039	−0.765193	+	0.643800i	$0.777357\pi$
$420$	0	0
$421$	23.2705	1.13414	0.567068	−	0.823671i	$-0.308078\pi$
$421$	23.2705	1.13414	0.567068	+	0.823671i	$0.308078\pi$
$422$	0	0
$423$	2.76393	0.134387
$424$	0	0
$425$	15.9787	0.775081
$426$	0	0
$427$	1.85410	0.0897263
$428$	0	0
$429$	−10.3262	−0.498555
$430$	0	0
$431$	−16.3262	−0.786407	−0.393204	−	0.919451i	$-0.628633\pi$
$431$	−16.3262	−0.786407	−0.393204	+	0.919451i	$0.628633\pi$
$432$	0	0
$433$	−21.1246	−1.01518	−0.507592	−	0.861598i	$-0.669464\pi$
$433$	−21.1246	−1.01518	−0.507592	+	0.861598i	$0.669464\pi$
$434$	0	0
$435$	−6.09017	−0.292001
$436$	0	0
$437$	−5.47214	−0.261768
$438$	0	0
$439$	7.65248	0.365233	0.182616	−	0.983184i	$-0.441543\pi$
$439$	7.65248	0.365233	0.182616	+	0.983184i	$0.441543\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	18.4721	0.877638	0.438819	−	0.898576i	$-0.355397\pi$
$443$	18.4721	0.877638	0.438819	+	0.898576i	$0.355397\pi$
$444$	0	0
$445$	15.1803	0.719617
$446$	0	0
$447$	−14.7639	−0.698310
$448$	0	0
$449$	−38.5623	−1.81987	−0.909934	−	0.414753i	$-0.863868\pi$
$449$	−38.5623	−1.81987	−0.909934	+	0.414753i	$0.863868\pi$
$450$	0	0
$451$	16.7082	0.786759
$452$	0	0
$453$	10.7639	0.505734
$454$	0	0
$455$	−7.47214	−0.350299
$456$	0	0
$457$	1.32624	0.0620388	0.0310194	−	0.999519i	$-0.490125\pi$
$457$	1.32624	0.0620388	0.0310194	+	0.999519i	$0.490125\pi$
$458$	0	0
$459$	−6.70820	−0.313112
$460$	0	0
$461$	−30.4508	−1.41824	−0.709119	−	0.705089i	$-0.750907\pi$
$461$	−30.4508	−1.41824	−0.709119	+	0.705089i	$0.750907\pi$
$462$	0	0
$463$	−30.7082	−1.42713	−0.713566	−	0.700588i	$-0.752921\pi$
$463$	−30.7082	−1.42713	−0.713566	+	0.700588i	$0.752921\pi$
$464$	0	0
$465$	10.8541	0.503347
$466$	0	0
$467$	−7.18034	−0.332267	−0.166133	−	0.986103i	$-0.553128\pi$
$467$	−7.18034	−0.332267	−0.166133	+	0.986103i	$0.553128\pi$
$468$	0	0
$469$	−6.09017	−0.281218
$470$	0	0
$471$	−13.7082	−0.631641
$472$	0	0
$473$	−1.38197	−0.0635429
$474$	0	0
$475$	13.0344	0.598061
$476$	0	0
$477$	−1.90983	−0.0874451
$478$	0	0
$479$	−21.0000	−0.959514	−0.479757	−	0.877401i	$-0.659275\pi$
$479$	−21.0000	−0.959514	−0.479757	+	0.877401i	$0.659275\pi$
$480$	0	0
$481$	50.7984	2.31621
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	−26.5623	−1.20613
$486$	0	0
$487$	42.7082	1.93529	0.967647	−	0.252309i	$-0.0811899\pi$
$487$	42.7082	1.93529	0.967647	+	0.252309i	$0.0811899\pi$
$488$	0	0
$489$	−1.61803	−0.0731700
$490$	0	0
$491$	14.7984	0.667841	0.333921	−	0.942601i	$-0.391628\pi$
$491$	14.7984	0.667841	0.333921	+	0.942601i	$0.391628\pi$
$492$	0	0
$493$	25.2492	1.13717
$494$	0	0
$495$	3.61803	0.162619
$496$	0	0
$497$	−6.61803	−0.296859
$498$	0	0
$499$	−35.3951	−1.58450	−0.792252	−	0.610195i	$-0.791091\pi$
$499$	−35.3951	−1.58450	−0.792252	+	0.610195i	$0.791091\pi$
$500$	0	0
$501$	−15.1803	−0.678208
$502$	0	0
$503$	−20.6738	−0.921797	−0.460899	−	0.887453i	$-0.652473\pi$
$503$	−20.6738	−0.921797	−0.460899	+	0.887453i	$0.652473\pi$
$504$	0	0
$505$	17.5623	0.781512
$506$	0	0
$507$	8.32624	0.369781
$508$	0	0
$509$	33.5967	1.48915	0.744575	−	0.667539i	$-0.232652\pi$
$509$	33.5967	1.48915	0.744575	+	0.667539i	$0.232652\pi$
$510$	0	0
$511$	0.708204	0.0313291
$512$	0	0
$513$	−5.47214	−0.241601
$514$	0	0
$515$	−31.4164	−1.38437
$516$	0	0
$517$	6.18034	0.271811
$518$	0	0
$519$	5.47214	0.240200
$520$	0	0
$521$	12.4721	0.546414	0.273207	−	0.961955i	$-0.411916\pi$
$521$	12.4721	0.546414	0.273207	+	0.961955i	$0.411916\pi$
$522$	0	0
$523$	5.41641	0.236843	0.118421	−	0.992963i	$-0.462217\pi$
$523$	5.41641	0.236843	0.118421	+	0.992963i	$0.462217\pi$
$524$	0	0
$525$	−2.38197	−0.103958
$526$	0	0
$527$	−45.0000	−1.96023
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−11.6180	−0.504180
$532$	0	0
$533$	−34.5066	−1.49465
$534$	0	0
$535$	19.9443	0.862266
$536$	0	0
$537$	4.85410	0.209470
$538$	0	0
$539$	2.23607	0.0963143
$540$	0	0
$541$	35.1246	1.51013	0.755063	−	0.655653i	$-0.227606\pi$
$541$	35.1246	1.51013	0.755063	+	0.655653i	$0.227606\pi$
$542$	0	0
$543$	15.9443	0.684234
$544$	0	0
$545$	−23.0902	−0.989074
$546$	0	0
$547$	11.7984	0.504462	0.252231	−	0.967667i	$-0.418836\pi$
$547$	11.7984	0.504462	0.252231	+	0.967667i	$0.418836\pi$
$548$	0	0
$549$	1.85410	0.0791311
$550$	0	0
$551$	20.5967	0.877451
$552$	0	0
$553$	0.527864	0.0224471
$554$	0	0
$555$	−17.7984	−0.755499
$556$	0	0
$557$	−28.1803	−1.19404	−0.597020	−	0.802227i	$-0.703649\pi$
$557$	−28.1803	−1.19404	−0.597020	+	0.802227i	$0.703649\pi$
$558$	0	0
$559$	2.85410	0.120716
$560$	0	0
$561$	−15.0000	−0.633300
$562$	0	0
$563$	−14.5066	−0.611379	−0.305690	−	0.952131i	$-0.598887\pi$
$563$	−14.5066	−0.611379	−0.305690	+	0.952131i	$0.598887\pi$
$564$	0	0
$565$	−19.5623	−0.822992
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	28.3607	1.18894	0.594471	−	0.804117i	$-0.297362\pi$
$569$	28.3607	1.18894	0.594471	+	0.804117i	$0.297362\pi$
$570$	0	0
$571$	17.5836	0.735850	0.367925	−	0.929855i	$-0.380068\pi$
$571$	17.5836	0.735850	0.367925	+	0.929855i	$0.380068\pi$
$572$	0	0
$573$	6.18034	0.258187
$574$	0	0
$575$	−2.38197	−0.0993348
$576$	0	0
$577$	16.8328	0.700759	0.350380	−	0.936608i	$-0.386053\pi$
$577$	16.8328	0.700759	0.350380	+	0.936608i	$0.386053\pi$
$578$	0	0
$579$	21.7082	0.902162
$580$	0	0
$581$	−13.1803	−0.546813
$582$	0	0
$583$	−4.27051	−0.176866
$584$	0	0
$585$	−7.47214	−0.308935
$586$	0	0
$587$	1.96556	0.0811273	0.0405636	−	0.999177i	$-0.487085\pi$
$587$	1.96556	0.0811273	0.0405636	+	0.999177i	$0.487085\pi$
$588$	0	0
$589$	−36.7082	−1.51254
$590$	0	0
$591$	12.5066	0.514452
$592$	0	0
$593$	26.0689	1.07052	0.535260	−	0.844687i	$-0.320213\pi$
$593$	26.0689	1.07052	0.535260	+	0.844687i	$0.320213\pi$
$594$	0	0
$595$	−10.8541	−0.444975
$596$	0	0
$597$	18.0902	0.740381
$598$	0	0
$599$	−22.5066	−0.919594	−0.459797	−	0.888024i	$-0.652078\pi$
$599$	−22.5066	−0.919594	−0.459797	+	0.888024i	$0.652078\pi$
$600$	0	0
$601$	12.0902	0.493168	0.246584	−	0.969121i	$-0.420692\pi$
$601$	12.0902	0.493168	0.246584	+	0.969121i	$0.420692\pi$
$602$	0	0
$603$	−6.09017	−0.248011
$604$	0	0
$605$	−9.70820	−0.394695
$606$	0	0
$607$	9.20163	0.373482	0.186741	−	0.982409i	$-0.440207\pi$
$607$	9.20163	0.373482	0.186741	+	0.982409i	$0.440207\pi$
$608$	0	0
$609$	−3.76393	−0.152522
$610$	0	0
$611$	−12.7639	−0.516373
$612$	0	0
$613$	19.6525	0.793756	0.396878	−	0.917871i	$-0.370094\pi$
$613$	19.6525	0.793756	0.396878	+	0.917871i	$0.370094\pi$
$614$	0	0
$615$	12.0902	0.487523
$616$	0	0
$617$	−18.3262	−0.737787	−0.368893	−	0.929472i	$-0.620263\pi$
$617$	−18.3262	−0.737787	−0.368893	+	0.929472i	$0.620263\pi$
$618$	0	0
$619$	17.3820	0.698640	0.349320	−	0.937003i	$-0.386413\pi$
$619$	17.3820	0.698640	0.349320	+	0.937003i	$0.386413\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	9.38197	0.375881
$624$	0	0
$625$	−7.41641	−0.296656
$626$	0	0
$627$	−12.2361	−0.488661
$628$	0	0
$629$	73.7902	2.94221
$630$	0	0
$631$	24.1246	0.960386	0.480193	−	0.877163i	$-0.340567\pi$
$631$	24.1246	0.960386	0.480193	+	0.877163i	$0.340567\pi$
$632$	0	0
$633$	16.4164	0.652494
$634$	0	0
$635$	18.2361	0.723676
$636$	0	0
$637$	−4.61803	−0.182973
$638$	0	0
$639$	−6.61803	−0.261805
$640$	0	0
$641$	−6.79837	−0.268520	−0.134260	−	0.990946i	$-0.542866\pi$
$641$	−6.79837	−0.268520	−0.134260	+	0.990946i	$0.542866\pi$
$642$	0	0
$643$	−34.9787	−1.37943	−0.689713	−	0.724083i	$-0.742263\pi$
$643$	−34.9787	−1.37943	−0.689713	+	0.724083i	$0.742263\pi$
$644$	0	0
$645$	−1.00000	−0.0393750
$646$	0	0
$647$	−5.03444	−0.197924	−0.0989622	−	0.995091i	$-0.531552\pi$
$647$	−5.03444	−0.197924	−0.0989622	+	0.995091i	$0.531552\pi$
$648$	0	0
$649$	−25.9787	−1.01975
$650$	0	0
$651$	6.70820	0.262915
$652$	0	0
$653$	−14.7426	−0.576924	−0.288462	−	0.957491i	$-0.593144\pi$
$653$	−14.7426	−0.576924	−0.288462	+	0.957491i	$0.593144\pi$
$654$	0	0
$655$	11.6180	0.453954
$656$	0	0
$657$	0.708204	0.0276297
$658$	0	0
$659$	21.6525	0.843461	0.421730	−	0.906721i	$-0.361423\pi$
$659$	21.6525	0.843461	0.421730	+	0.906721i	$0.361423\pi$
$660$	0	0
$661$	−12.4164	−0.482942	−0.241471	−	0.970408i	$-0.577630\pi$
$661$	−12.4164	−0.482942	−0.241471	+	0.970408i	$0.577630\pi$
$662$	0	0
$663$	30.9787	1.20311
$664$	0	0
$665$	−8.85410	−0.343347
$666$	0	0
$667$	−3.76393	−0.145740
$668$	0	0
$669$	11.1459	0.430925
$670$	0	0
$671$	4.14590	0.160051
$672$	0	0
$673$	−14.4164	−0.555712	−0.277856	−	0.960623i	$-0.589624\pi$
$673$	−14.4164	−0.555712	−0.277856	+	0.960623i	$0.589624\pi$
$674$	0	0
$675$	−2.38197	−0.0916819
$676$	0	0
$677$	−43.7426	−1.68117	−0.840583	−	0.541682i	$-0.817788\pi$
$677$	−43.7426	−1.68117	−0.840583	+	0.541682i	$0.817788\pi$
$678$	0	0
$679$	−16.4164	−0.630004
$680$	0	0
$681$	−8.67376	−0.332379
$682$	0	0
$683$	50.0132	1.91370	0.956850	−	0.290582i	$-0.0938489\pi$
$683$	50.0132	1.91370	0.956850	+	0.290582i	$0.0938489\pi$
$684$	0	0
$685$	16.8541	0.643962
$686$	0	0
$687$	−0.673762	−0.0257056
$688$	0	0
$689$	8.81966	0.336002
$690$	0	0
$691$	41.2705	1.57000	0.785002	−	0.619493i	$-0.212662\pi$
$691$	41.2705	1.57000	0.785002	+	0.619493i	$0.212662\pi$
$692$	0	0
$693$	2.23607	0.0849412
$694$	0	0
$695$	22.0344	0.835814
$696$	0	0
$697$	−50.1246	−1.89861
$698$	0	0
$699$	0.0901699	0.00341054
$700$	0	0
$701$	−18.4508	−0.696879	−0.348439	−	0.937331i	$-0.613288\pi$
$701$	−18.4508	−0.696879	−0.348439	+	0.937331i	$0.613288\pi$
$702$	0	0
$703$	60.1935	2.27024
$704$	0	0
$705$	4.47214	0.168430
$706$	0	0
$707$	10.8541	0.408211
$708$	0	0
$709$	17.5623	0.659566	0.329783	−	0.944057i	$-0.393024\pi$
$709$	17.5623	0.659566	0.329783	+	0.944057i	$0.393024\pi$
$710$	0	0
$711$	0.527864	0.0197964
$712$	0	0
$713$	6.70820	0.251224
$714$	0	0
$715$	−16.7082	−0.624851
$716$	0	0
$717$	−19.7984	−0.739384
$718$	0	0
$719$	3.00000	0.111881	0.0559406	−	0.998434i	$-0.482184\pi$
$719$	3.00000	0.111881	0.0559406	+	0.998434i	$0.482184\pi$
$720$	0	0
$721$	−19.4164	−0.723105
$722$	0	0
$723$	−11.0000	−0.409094
$724$	0	0
$725$	8.96556	0.332972
$726$	0	0
$727$	16.8885	0.626361	0.313181	−	0.949694i	$-0.398605\pi$
$727$	16.8885	0.626361	0.313181	+	0.949694i	$0.398605\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.14590	0.153342
$732$	0	0
$733$	−23.5967	−0.871566	−0.435783	−	0.900052i	$-0.643528\pi$
$733$	−23.5967	−0.871566	−0.435783	+	0.900052i	$0.643528\pi$
$734$	0	0
$735$	1.61803	0.0596821
$736$	0	0
$737$	−13.6180	−0.501627
$738$	0	0
$739$	35.2492	1.29666	0.648332	−	0.761358i	$-0.275467\pi$
$739$	35.2492	1.29666	0.648332	+	0.761358i	$0.275467\pi$
$740$	0	0
$741$	25.2705	0.928335
$742$	0	0
$743$	−27.3262	−1.00250	−0.501251	−	0.865302i	$-0.667127\pi$
$743$	−27.3262	−1.00250	−0.501251	+	0.865302i	$0.667127\pi$
$744$	0	0
$745$	−23.8885	−0.875209
$746$	0	0
$747$	−13.1803	−0.482243
$748$	0	0
$749$	12.3262	0.450391
$750$	0	0
$751$	13.9787	0.510091	0.255045	−	0.966929i	$-0.417910\pi$
$751$	13.9787	0.510091	0.255045	+	0.966929i	$0.417910\pi$
$752$	0	0
$753$	17.1246	0.624056
$754$	0	0
$755$	17.4164	0.633848
$756$	0	0
$757$	−17.0000	−0.617876	−0.308938	−	0.951082i	$-0.599973\pi$
$757$	−17.0000	−0.617876	−0.308938	+	0.951082i	$0.599973\pi$
$758$	0	0
$759$	2.23607	0.0811641
$760$	0	0
$761$	−5.52786	−0.200385	−0.100192	−	0.994968i	$-0.531946\pi$
$761$	−5.52786	−0.200385	−0.100192	+	0.994968i	$0.531946\pi$
$762$	0	0
$763$	−14.2705	−0.516627
$764$	0	0
$765$	−10.8541	−0.392431
$766$	0	0
$767$	53.6525	1.93728
$768$	0	0
$769$	28.7639	1.03725	0.518627	−	0.855001i	$-0.326443\pi$
$769$	28.7639	1.03725	0.518627	+	0.855001i	$0.326443\pi$
$770$	0	0
$771$	1.23607	0.0445159
$772$	0	0
$773$	−2.05573	−0.0739394	−0.0369697	−	0.999316i	$-0.511771\pi$
$773$	−2.05573	−0.0739394	−0.0369697	+	0.999316i	$0.511771\pi$
$774$	0	0
$775$	−15.9787	−0.573972
$776$	0	0
$777$	−11.0000	−0.394623
$778$	0	0
$779$	−40.8885	−1.46498
$780$	0	0
$781$	−14.7984	−0.529527
$782$	0	0
$783$	−3.76393	−0.134512
$784$	0	0
$785$	−22.1803	−0.791650
$786$	0	0
$787$	−47.5755	−1.69588	−0.847941	−	0.530091i	$-0.822158\pi$
$787$	−47.5755	−1.69588	−0.847941	+	0.530091i	$0.822158\pi$
$788$	0	0
$789$	17.9443	0.638833
$790$	0	0
$791$	−12.0902	−0.429877
$792$	0	0
$793$	−8.56231	−0.304056
$794$	0	0
$795$	−3.09017	−0.109597
$796$	0	0
$797$	21.8197	0.772892	0.386446	−	0.922312i	$-0.373703\pi$
$797$	21.8197	0.772892	0.386446	+	0.922312i	$0.373703\pi$
$798$	0	0
$799$	−18.5410	−0.655934
$800$	0	0
$801$	9.38197	0.331495
$802$	0	0
$803$	1.58359	0.0558838
$804$	0	0
$805$	1.61803	0.0570282
$806$	0	0
$807$	−20.9098	−0.736061
$808$	0	0
$809$	−29.5623	−1.03936	−0.519678	−	0.854362i	$-0.673948\pi$
$809$	−29.5623	−1.03936	−0.519678	+	0.854362i	$0.673948\pi$
$810$	0	0
$811$	−17.4721	−0.613530	−0.306765	−	0.951785i	$-0.599247\pi$
$811$	−17.4721	−0.613530	−0.306765	+	0.951785i	$0.599247\pi$
$812$	0	0
$813$	−4.05573	−0.142241
$814$	0	0
$815$	−2.61803	−0.0917057
$816$	0	0
$817$	3.38197	0.118320
$818$	0	0
$819$	−4.61803	−0.161367
$820$	0	0
$821$	11.1246	0.388252	0.194126	−	0.980977i	$-0.437813\pi$
$821$	11.1246	0.388252	0.194126	+	0.980977i	$0.437813\pi$
$822$	0	0
$823$	33.1459	1.15539	0.577697	−	0.816252i	$-0.303952\pi$
$823$	33.1459	1.15539	0.577697	+	0.816252i	$0.303952\pi$
$824$	0	0
$825$	−5.32624	−0.185436
$826$	0	0
$827$	−23.2148	−0.807257	−0.403629	−	0.914923i	$-0.632251\pi$
$827$	−23.2148	−0.807257	−0.403629	+	0.914923i	$0.632251\pi$
$828$	0	0
$829$	−3.58359	−0.124463	−0.0622316	−	0.998062i	$-0.519822\pi$
$829$	−3.58359	−0.124463	−0.0622316	+	0.998062i	$0.519822\pi$
$830$	0	0
$831$	−29.2705	−1.01538
$832$	0	0
$833$	−6.70820	−0.232425
$834$	0	0
$835$	−24.5623	−0.850014
$836$	0	0
$837$	6.70820	0.231869
$838$	0	0
$839$	25.7426	0.888735	0.444367	−	0.895845i	$-0.353428\pi$
$839$	25.7426	0.888735	0.444367	+	0.895845i	$0.353428\pi$
$840$	0	0
$841$	−14.8328	−0.511476
$842$	0	0
$843$	9.70820	0.334368
$844$	0	0
$845$	13.4721	0.463456
$846$	0	0
$847$	−6.00000	−0.206162
$848$	0	0
$849$	−25.2148	−0.865369
$850$	0	0
$851$	−11.0000	−0.377075
$852$	0	0
$853$	−6.87539	−0.235409	−0.117704	−	0.993049i	$-0.537554\pi$
$853$	−6.87539	−0.235409	−0.117704	+	0.993049i	$0.537554\pi$
$854$	0	0
$855$	−8.85410	−0.302804
$856$	0	0
$857$	26.2918	0.898111	0.449055	−	0.893504i	$-0.351761\pi$
$857$	26.2918	0.898111	0.449055	+	0.893504i	$0.351761\pi$
$858$	0	0
$859$	14.0000	0.477674	0.238837	−	0.971060i	$-0.423234\pi$
$859$	14.0000	0.477674	0.238837	+	0.971060i	$0.423234\pi$
$860$	0	0
$861$	7.47214	0.254650
$862$	0	0
$863$	−50.0689	−1.70436	−0.852182	−	0.523245i	$-0.824721\pi$
$863$	−50.0689	−1.70436	−0.852182	+	0.523245i	$0.824721\pi$
$864$	0	0
$865$	8.85410	0.301048
$866$	0	0
$867$	28.0000	0.950930
$868$	0	0
$869$	1.18034	0.0400403
$870$	0	0
$871$	28.1246	0.952966
$872$	0	0
$873$	−16.4164	−0.555611
$874$	0	0
$875$	−11.9443	−0.403790
$876$	0	0
$877$	43.4164	1.46607	0.733034	−	0.680192i	$-0.238104\pi$
$877$	43.4164	1.46607	0.733034	+	0.680192i	$0.238104\pi$
$878$	0	0
$879$	12.0000	0.404750
$880$	0	0
$881$	24.1803	0.814656	0.407328	−	0.913282i	$-0.366461\pi$
$881$	24.1803	0.814656	0.407328	+	0.913282i	$0.366461\pi$
$882$	0	0
$883$	15.0902	0.507825	0.253912	−	0.967227i	$-0.418283\pi$
$883$	15.0902	0.507825	0.253912	+	0.967227i	$0.418283\pi$
$884$	0	0
$885$	−18.7984	−0.631900
$886$	0	0
$887$	14.7984	0.496881	0.248440	−	0.968647i	$-0.420082\pi$
$887$	14.7984	0.496881	0.248440	+	0.968647i	$0.420082\pi$
$888$	0	0
$889$	11.2705	0.378001
$890$	0	0
$891$	2.23607	0.0749111
$892$	0	0
$893$	−15.1246	−0.506126
$894$	0	0
$895$	7.85410	0.262534
$896$	0	0
$897$	−4.61803	−0.154192
$898$	0	0
$899$	−25.2492	−0.842109
$900$	0	0
$901$	12.8115	0.426814
$902$	0	0
$903$	−0.618034	−0.0205669
$904$	0	0
$905$	25.7984	0.857567
$906$	0	0
$907$	−2.02129	−0.0671157	−0.0335579	−	0.999437i	$-0.510684\pi$
$907$	−2.02129	−0.0671157	−0.0335579	+	0.999437i	$0.510684\pi$
$908$	0	0
$909$	10.8541	0.360008
$910$	0	0
$911$	−41.5967	−1.37816	−0.689081	−	0.724684i	$-0.741986\pi$
$911$	−41.5967	−1.37816	−0.689081	+	0.724684i	$0.741986\pi$
$912$	0	0
$913$	−29.4721	−0.975385
$914$	0	0
$915$	3.00000	0.0991769
$916$	0	0
$917$	7.18034	0.237116
$918$	0	0
$919$	9.87539	0.325759	0.162879	−	0.986646i	$-0.447922\pi$
$919$	9.87539	0.325759	0.162879	+	0.986646i	$0.447922\pi$
$920$	0	0
$921$	24.1246	0.794933
$922$	0	0
$923$	30.5623	1.00597
$924$	0	0
$925$	26.2016	0.861504
$926$	0	0
$927$	−19.4164	−0.637719
$928$	0	0
$929$	40.0902	1.31532	0.657658	−	0.753317i	$-0.271547\pi$
$929$	40.0902	1.31532	0.657658	+	0.753317i	$0.271547\pi$
$930$	0	0
$931$	−5.47214	−0.179342
$932$	0	0
$933$	0.326238	0.0106806
$934$	0	0
$935$	−24.2705	−0.793731
$936$	0	0
$937$	−10.7082	−0.349822	−0.174911	−	0.984584i	$-0.555964\pi$
$937$	−10.7082	−0.349822	−0.174911	+	0.984584i	$0.555964\pi$
$938$	0	0
$939$	6.47214	0.211210
$940$	0	0
$941$	−32.9443	−1.07395	−0.536976	−	0.843597i	$-0.680434\pi$
$941$	−32.9443	−1.07395	−0.536976	+	0.843597i	$0.680434\pi$
$942$	0	0
$943$	7.47214	0.243326
$944$	0	0
$945$	1.61803	0.0526346
$946$	0	0
$947$	1.05573	0.0343066	0.0171533	−	0.999853i	$-0.494540\pi$
$947$	1.05573	0.0343066	0.0171533	+	0.999853i	$0.494540\pi$
$948$	0	0
$949$	−3.27051	−0.106165
$950$	0	0
$951$	−23.5066	−0.762253
$952$	0	0
$953$	−57.1591	−1.85156	−0.925782	−	0.378059i	$-0.876592\pi$
$953$	−57.1591	−1.85156	−0.925782	+	0.378059i	$0.876592\pi$
$954$	0	0
$955$	10.0000	0.323592
$956$	0	0
$957$	−8.41641	−0.272064
$958$	0	0
$959$	10.4164	0.336363
$960$	0	0
$961$	14.0000	0.451613
$962$	0	0
$963$	12.3262	0.397207
$964$	0	0
$965$	35.1246	1.13070
$966$	0	0
$967$	−5.63932	−0.181348	−0.0906742	−	0.995881i	$-0.528902\pi$
$967$	−5.63932	−0.181348	−0.0906742	+	0.995881i	$0.528902\pi$
$968$	0	0
$969$	36.7082	1.17924
$970$	0	0
$971$	−40.6869	−1.30571	−0.652853	−	0.757485i	$-0.726428\pi$
$971$	−40.6869	−1.30571	−0.652853	+	0.757485i	$0.726428\pi$
$972$	0	0
$973$	13.6180	0.436574
$974$	0	0
$975$	11.0000	0.352282
$976$	0	0
$977$	25.2016	0.806271	0.403136	−	0.915140i	$-0.367920\pi$
$977$	25.2016	0.806271	0.403136	+	0.915140i	$0.367920\pi$
$978$	0	0
$979$	20.9787	0.670483
$980$	0	0
$981$	−14.2705	−0.455622
$982$	0	0
$983$	56.5967	1.80516	0.902578	−	0.430526i	$-0.141672\pi$
$983$	56.5967	1.80516	0.902578	+	0.430526i	$0.141672\pi$
$984$	0	0
$985$	20.2361	0.644775
$986$	0	0
$987$	2.76393	0.0879769
$988$	0	0
$989$	−0.618034	−0.0196523
$990$	0	0
$991$	−60.5755	−1.92424	−0.962121	−	0.272621i	$-0.912109\pi$
$991$	−60.5755	−1.92424	−0.962121	+	0.272621i	$0.912109\pi$
$992$	0	0
$993$	−13.4164	−0.425757
$994$	0	0
$995$	29.2705	0.927938
$996$	0	0
$997$	−31.7214	−1.00463	−0.502313	−	0.864686i	$-0.667517\pi$
$997$	−31.7214	−1.00463	−0.502313	+	0.864686i	$0.667517\pi$
$998$	0	0
$999$	−11.0000	−0.348025

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.bn.1.2		2
4.3	odd	2		483.2.a.d.1.2	✓	2
12.11	even	2		1449.2.a.h.1.1		2
28.27	even	2		3381.2.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
483.2.a.d.1.2	✓	2	4.3	odd	2
1449.2.a.h.1.1		2	12.11	even	2
3381.2.a.r.1.2		2	28.27	even	2
7728.2.a.bn.1.2		2	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.61803 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.61803 q^{5} +1.00000 q^{7} +1.00000 q^{9} +2.23607 q^{11} -4.61803 q^{13} +1.61803 q^{15} -6.70820 q^{17} -5.47214 q^{19} +1.00000 q^{21} +1.00000 q^{23} -2.38197 q^{25} +1.00000 q^{27} -3.76393 q^{29} +6.70820 q^{31} +2.23607 q^{33} +1.61803 q^{35} -11.0000 q^{37} -4.61803 q^{39} +7.47214 q^{41} -0.618034 q^{43} +1.61803 q^{45} +2.76393 q^{47} +1.00000 q^{49} -6.70820 q^{51} -1.90983 q^{53} +3.61803 q^{55} -5.47214 q^{57} -11.6180 q^{59} +1.85410 q^{61} +1.00000 q^{63} -7.47214 q^{65} -6.09017 q^{67} +1.00000 q^{69} -6.61803 q^{71} +0.708204 q^{73} -2.38197 q^{75} +2.23607 q^{77} +0.527864 q^{79} +1.00000 q^{81} -13.1803 q^{83} -10.8541 q^{85} -3.76393 q^{87} +9.38197 q^{89} -4.61803 q^{91} +6.70820 q^{93} -8.85410 q^{95} -16.4164 q^{97} +2.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9} - 7 q^{13} + q^{15} - 2 q^{19} + 2 q^{21} + 2 q^{23} - 7 q^{25} + 2 q^{27} - 12 q^{29} + q^{35} - 22 q^{37} - 7 q^{39} + 6 q^{41} + q^{43} + q^{45} + 10 q^{47} + 2 q^{49} - 15 q^{53} + 5 q^{55} - 2 q^{57} - 21 q^{59} - 3 q^{61} + 2 q^{63} - 6 q^{65} - q^{67} + 2 q^{69} - 11 q^{71} - 12 q^{73} - 7 q^{75} + 10 q^{79} + 2 q^{81} - 4 q^{83} - 15 q^{85} - 12 q^{87} + 21 q^{89} - 7 q^{91} - 11 q^{95} - 6 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 + q^5 + 2 * q^7 + 2 * q^9 - 7 * q^13 + q^15 - 2 * q^19 + 2 * q^21 + 2 * q^23 - 7 * q^25 + 2 * q^27 - 12 * q^29 + q^35 - 22 * q^37 - 7 * q^39 + 6 * q^41 + q^43 + q^45 + 10 * q^47 + 2 * q^49 - 15 * q^53 + 5 * q^55 - 2 * q^57 - 21 * q^59 - 3 * q^61 + 2 * q^63 - 6 * q^65 - q^67 + 2 * q^69 - 11 * q^71 - 12 * q^73 - 7 * q^75 + 10 * q^79 + 2 * q^81 - 4 * q^83 - 15 * q^85 - 12 * q^87 + 21 * q^89 - 7 * q^91 - 11 * q^95 - 6 * q^97

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