LMFDB - Embedded newform 7728.2.a.bn.1.1

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			1.1
Root			$-0.618034$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -0.618034 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.618034 q^{5} +1.00000 q^{7} +1.00000 q^{9} -2.23607 q^{11} -2.38197 q^{13} -0.618034 q^{15} +6.70820 q^{17} +3.47214 q^{19} +1.00000 q^{21} +1.00000 q^{23} -4.61803 q^{25} +1.00000 q^{27} -8.23607 q^{29} -6.70820 q^{31} -2.23607 q^{33} -0.618034 q^{35} -11.0000 q^{37} -2.38197 q^{39} -1.47214 q^{41} +1.61803 q^{43} -0.618034 q^{45} +7.23607 q^{47} +1.00000 q^{49} +6.70820 q^{51} -13.0902 q^{53} +1.38197 q^{55} +3.47214 q^{57} -9.38197 q^{59} -4.85410 q^{61} +1.00000 q^{63} +1.47214 q^{65} +5.09017 q^{67} +1.00000 q^{69} -4.38197 q^{71} -12.7082 q^{73} -4.61803 q^{75} -2.23607 q^{77} +9.47214 q^{79} +1.00000 q^{81} +9.18034 q^{83} -4.14590 q^{85} -8.23607 q^{87} +11.6180 q^{89} -2.38197 q^{91} -6.70820 q^{93} -2.14590 q^{95} +10.4164 q^{97} -2.23607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−0.618034	−0.276393	−0.138197	−	0.990405i	$-0.544131\pi$
$5$	−0.618034	−0.276393	−0.138197	+	0.990405i	$0.544131\pi$
$6$	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	$-0.609446\pi$
$11$	−2.23607	−0.674200	−0.337100	+	0.941469i	$0.609446\pi$
$12$	0	0
$13$	−2.38197	−0.660639	−0.330319	−	0.943869i	$-0.607156\pi$
$13$	−2.38197	−0.660639	−0.330319	+	0.943869i	$0.607156\pi$
$14$	0	0
$15$	−0.618034	−0.159576
$16$	0	0
$17$	6.70820	1.62698	0.813489	−	0.581580i	$-0.197565\pi$
$17$	6.70820	1.62698	0.813489	+	0.581580i	$0.197565\pi$
$18$	0	0
$19$	3.47214	0.796563	0.398281	−	0.917263i	$-0.369607\pi$
$19$	3.47214	0.796563	0.398281	+	0.917263i	$0.369607\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	−4.61803	−0.923607
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−8.23607	−1.52940	−0.764700	−	0.644387i	$-0.777113\pi$
$29$	−8.23607	−1.52940	−0.764700	+	0.644387i	$0.777113\pi$
$30$	0	0
$31$	−6.70820	−1.20483	−0.602414	−	0.798183i	$-0.705795\pi$
$31$	−6.70820	−1.20483	−0.602414	+	0.798183i	$0.705795\pi$
$32$	0	0
$33$	−2.23607	−0.389249
$34$	0	0
$35$	−0.618034	−0.104467
$36$	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
$39$	−2.38197	−0.381420
$40$	0	0
$41$	−1.47214	−0.229909	−0.114955	−	0.993371i	$-0.536672\pi$
$41$	−1.47214	−0.229909	−0.114955	+	0.993371i	$0.536672\pi$
$42$	0	0
$43$	1.61803	0.246748	0.123374	−	0.992360i	$-0.460629\pi$
$43$	1.61803	0.246748	0.123374	+	0.992360i	$0.460629\pi$
$44$	0	0
$45$	−0.618034	−0.0921311
$46$	0	0
$47$	7.23607	1.05549	0.527744	−	0.849403i	$-0.323038\pi$
$47$	7.23607	1.05549	0.527744	+	0.849403i	$0.323038\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.70820	0.939336
$52$	0	0
$53$	−13.0902	−1.79807	−0.899037	−	0.437874i	$-0.855732\pi$
$53$	−13.0902	−1.79807	−0.899037	+	0.437874i	$0.855732\pi$
$54$	0	0
$55$	1.38197	0.186344
$56$	0	0
$57$	3.47214	0.459896
$58$	0	0
$59$	−9.38197	−1.22143	−0.610714	−	0.791851i	$-0.709117\pi$
$59$	−9.38197	−1.22143	−0.610714	+	0.791851i	$0.709117\pi$
$60$	0	0
$61$	−4.85410	−0.621504	−0.310752	−	0.950491i	$-0.600581\pi$
$61$	−4.85410	−0.621504	−0.310752	+	0.950491i	$0.600581\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	1.47214	0.182596
$66$	0	0
$67$	5.09017	0.621863	0.310932	−	0.950432i	$-0.399359\pi$
$67$	5.09017	0.621863	0.310932	+	0.950432i	$0.399359\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−4.38197	−0.520044	−0.260022	−	0.965603i	$-0.583730\pi$
$71$	−4.38197	−0.520044	−0.260022	+	0.965603i	$0.583730\pi$
$72$	0	0
$73$	−12.7082	−1.48738	−0.743691	−	0.668523i	$-0.766927\pi$
$73$	−12.7082	−1.48738	−0.743691	+	0.668523i	$0.766927\pi$
$74$	0	0
$75$	−4.61803	−0.533245
$76$	0	0
$77$	−2.23607	−0.254824
$78$	0	0
$79$	9.47214	1.06570	0.532849	−	0.846210i	$-0.321121\pi$
$79$	9.47214	1.06570	0.532849	+	0.846210i	$0.321121\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.18034	1.00767	0.503837	−	0.863799i	$-0.331921\pi$
$83$	9.18034	1.00767	0.503837	+	0.863799i	$0.331921\pi$
$84$	0	0
$85$	−4.14590	−0.449686
$86$	0	0
$87$	−8.23607	−0.882999
$88$	0	0
$89$	11.6180	1.23151	0.615755	−	0.787938i	$-0.288851\pi$
$89$	11.6180	1.23151	0.615755	+	0.787938i	$0.288851\pi$
$90$	0	0
$91$	−2.38197	−0.249698
$92$	0	0
$93$	−6.70820	−0.695608
$94$	0	0
$95$	−2.14590	−0.220164
$96$	0	0
$97$	10.4164	1.05763	0.528813	−	0.848738i	$-0.322637\pi$
$97$	10.4164	1.05763	0.528813	+	0.848738i	$0.322637\pi$
$98$	0	0
$99$	−2.23607	−0.224733
$100$	0	0
$101$	4.14590	0.412532	0.206266	−	0.978496i	$-0.433869\pi$
$101$	4.14590	0.412532	0.206266	+	0.978496i	$0.433869\pi$
$102$	0	0
$103$	7.41641	0.730760	0.365380	−	0.930858i	$-0.380939\pi$
$103$	7.41641	0.730760	0.365380	+	0.930858i	$0.380939\pi$
$104$	0	0
$105$	−0.618034	−0.0603139
$106$	0	0
$107$	−3.32624	−0.321560	−0.160780	−	0.986990i	$-0.551401\pi$
$107$	−3.32624	−0.321560	−0.160780	+	0.986990i	$0.551401\pi$
$108$	0	0
$109$	19.2705	1.84578	0.922890	−	0.385064i	$-0.125820\pi$
$109$	19.2705	1.84578	0.922890	+	0.385064i	$0.125820\pi$
$110$	0	0
$111$	−11.0000	−1.04407
$112$	0	0
$113$	−0.909830	−0.0855896	−0.0427948	−	0.999084i	$-0.513626\pi$
$113$	−0.909830	−0.0855896	−0.0427948	+	0.999084i	$0.513626\pi$
$114$	0	0
$115$	−0.618034	−0.0576320
$116$	0	0
$117$	−2.38197	−0.220213
$118$	0	0
$119$	6.70820	0.614940
$120$	0	0
$121$	−6.00000	−0.545455
$122$	0	0
$123$	−1.47214	−0.132738
$124$	0	0
$125$	5.94427	0.531672
$126$	0	0
$127$	−22.2705	−1.97619	−0.988094	−	0.153851i	$-0.950832\pi$
$127$	−22.2705	−1.97619	−0.988094	+	0.153851i	$0.950832\pi$
$128$	0	0
$129$	1.61803	0.142460
$130$	0	0
$131$	−15.1803	−1.32631	−0.663156	−	0.748481i	$-0.730784\pi$
$131$	−15.1803	−1.32631	−0.663156	+	0.748481i	$0.730784\pi$
$132$	0	0
$133$	3.47214	0.301072
$134$	0	0
$135$	−0.618034	−0.0531919
$136$	0	0
$137$	−16.4164	−1.40255	−0.701274	−	0.712892i	$-0.747385\pi$
$137$	−16.4164	−1.40255	−0.701274	+	0.712892i	$0.747385\pi$
$138$	0	0
$139$	11.3820	0.965406	0.482703	−	0.875784i	$-0.339655\pi$
$139$	11.3820	0.965406	0.482703	+	0.875784i	$0.339655\pi$
$140$	0	0
$141$	7.23607	0.609387
$142$	0	0
$143$	5.32624	0.445402
$144$	0	0
$145$	5.09017	0.422716
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−19.2361	−1.57588	−0.787940	−	0.615752i	$-0.788852\pi$
$149$	−19.2361	−1.57588	−0.787940	+	0.615752i	$0.788852\pi$
$150$	0	0
$151$	15.2361	1.23989	0.619947	−	0.784644i	$-0.287154\pi$
$151$	15.2361	1.23989	0.619947	+	0.784644i	$0.287154\pi$
$152$	0	0
$153$	6.70820	0.542326
$154$	0	0
$155$	4.14590	0.333007
$156$	0	0
$157$	−0.291796	−0.0232879	−0.0116439	−	0.999932i	$-0.503706\pi$
$157$	−0.291796	−0.0232879	−0.0116439	+	0.999932i	$0.503706\pi$
$158$	0	0
$159$	−13.0902	−1.03812
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	0.618034	0.0484082	0.0242041	−	0.999707i	$-0.492295\pi$
$163$	0.618034	0.0484082	0.0242041	+	0.999707i	$0.492295\pi$
$164$	0	0
$165$	1.38197	0.107586
$166$	0	0
$167$	7.18034	0.555631	0.277816	−	0.960634i	$-0.410390\pi$
$167$	7.18034	0.555631	0.277816	+	0.960634i	$0.410390\pi$
$168$	0	0
$169$	−7.32624	−0.563557
$170$	0	0
$171$	3.47214	0.265521
$172$	0	0
$173$	−3.47214	−0.263982	−0.131991	−	0.991251i	$-0.542137\pi$
$173$	−3.47214	−0.263982	−0.131991	+	0.991251i	$0.542137\pi$
$174$	0	0
$175$	−4.61803	−0.349091
$176$	0	0
$177$	−9.38197	−0.705192
$178$	0	0
$179$	−1.85410	−0.138582	−0.0692910	−	0.997596i	$-0.522074\pi$
$179$	−1.85410	−0.138582	−0.0692910	+	0.997596i	$0.522074\pi$
$180$	0	0
$181$	−1.94427	−0.144517	−0.0722583	−	0.997386i	$-0.523021\pi$
$181$	−1.94427	−0.144517	−0.0722583	+	0.997386i	$0.523021\pi$
$182$	0	0
$183$	−4.85410	−0.358826
$184$	0	0
$185$	6.79837	0.499826
$186$	0	0
$187$	−15.0000	−1.09691
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−16.1803	−1.17077	−0.585384	−	0.810756i	$-0.699056\pi$
$191$	−16.1803	−1.17077	−0.585384	+	0.810756i	$0.699056\pi$
$192$	0	0
$193$	8.29180	0.596857	0.298428	−	0.954432i	$-0.403538\pi$
$193$	8.29180	0.596857	0.298428	+	0.954432i	$0.403538\pi$
$194$	0	0
$195$	1.47214	0.105422
$196$	0	0
$197$	−25.5066	−1.81727	−0.908634	−	0.417593i	$-0.862874\pi$
$197$	−25.5066	−1.81727	−0.908634	+	0.417593i	$0.862874\pi$
$198$	0	0
$199$	6.90983	0.489825	0.244912	−	0.969545i	$-0.421241\pi$
$199$	6.90983	0.489825	0.244912	+	0.969545i	$0.421241\pi$
$200$	0	0
$201$	5.09017	0.359033
$202$	0	0
$203$	−8.23607	−0.578059
$204$	0	0
$205$	0.909830	0.0635453
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−7.76393	−0.537042
$210$	0	0
$211$	−10.4164	−0.717095	−0.358548	−	0.933511i	$-0.616728\pi$
$211$	−10.4164	−0.717095	−0.358548	+	0.933511i	$0.616728\pi$
$212$	0	0
$213$	−4.38197	−0.300247
$214$	0	0
$215$	−1.00000	−0.0681994
$216$	0	0
$217$	−6.70820	−0.455383
$218$	0	0
$219$	−12.7082	−0.858741
$220$	0	0
$221$	−15.9787	−1.07484
$222$	0	0
$223$	17.8541	1.19560	0.597800	−	0.801646i	$-0.296042\pi$
$223$	17.8541	1.19560	0.597800	+	0.801646i	$0.296042\pi$
$224$	0	0
$225$	−4.61803	−0.307869
$226$	0	0
$227$	−24.3262	−1.61459	−0.807295	−	0.590149i	$-0.799069\pi$
$227$	−24.3262	−1.61459	−0.807295	+	0.590149i	$0.799069\pi$
$228$	0	0
$229$	−16.3262	−1.07887	−0.539434	−	0.842028i	$-0.681362\pi$
$229$	−16.3262	−1.07887	−0.539434	+	0.842028i	$0.681362\pi$
$230$	0	0
$231$	−2.23607	−0.147122
$232$	0	0
$233$	−11.0902	−0.726541	−0.363271	−	0.931684i	$-0.618340\pi$
$233$	−11.0902	−0.726541	−0.363271	+	0.931684i	$0.618340\pi$
$234$	0	0
$235$	−4.47214	−0.291730
$236$	0	0
$237$	9.47214	0.615281
$238$	0	0
$239$	4.79837	0.310381	0.155191	−	0.987885i	$-0.450401\pi$
$239$	4.79837	0.310381	0.155191	+	0.987885i	$0.450401\pi$
$240$	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
$245$	−0.618034	−0.0394847
$246$	0	0
$247$	−8.27051	−0.526240
$248$	0	0
$249$	9.18034	0.581780
$250$	0	0
$251$	−23.1246	−1.45961	−0.729806	−	0.683654i	$-0.760390\pi$
$251$	−23.1246	−1.45961	−0.729806	+	0.683654i	$0.760390\pi$
$252$	0	0
$253$	−2.23607	−0.140580
$254$	0	0
$255$	−4.14590	−0.259626
$256$	0	0
$257$	−3.23607	−0.201860	−0.100930	−	0.994894i	$-0.532182\pi$
$257$	−3.23607	−0.201860	−0.100930	+	0.994894i	$0.532182\pi$
$258$	0	0
$259$	−11.0000	−0.683507
$260$	0	0
$261$	−8.23607	−0.509800
$262$	0	0
$263$	0.0557281	0.00343634	0.00171817	−	0.999999i	$-0.499453\pi$
$263$	0.0557281	0.00343634	0.00171817	+	0.999999i	$0.499453\pi$
$264$	0	0
$265$	8.09017	0.496975
$266$	0	0
$267$	11.6180	0.711012
$268$	0	0
$269$	−32.0902	−1.95657	−0.978286	−	0.207259i	$-0.933546\pi$
$269$	−32.0902	−1.95657	−0.978286	+	0.207259i	$0.933546\pi$
$270$	0	0
$271$	−21.9443	−1.33302	−0.666510	−	0.745496i	$-0.732213\pi$
$271$	−21.9443	−1.33302	−0.666510	+	0.745496i	$0.732213\pi$
$272$	0	0
$273$	−2.38197	−0.144163
$274$	0	0
$275$	10.3262	0.622696
$276$	0	0
$277$	4.27051	0.256590	0.128295	−	0.991736i	$-0.459050\pi$
$277$	4.27051	0.256590	0.128295	+	0.991736i	$0.459050\pi$
$278$	0	0
$279$	−6.70820	−0.401610
$280$	0	0
$281$	−3.70820	−0.221213	−0.110606	−	0.993864i	$-0.535279\pi$
$281$	−3.70820	−0.221213	−0.110606	+	0.993864i	$0.535279\pi$
$282$	0	0
$283$	26.2148	1.55831	0.779154	−	0.626833i	$-0.215649\pi$
$283$	26.2148	1.55831	0.779154	+	0.626833i	$0.215649\pi$
$284$	0	0
$285$	−2.14590	−0.127112
$286$	0	0
$287$	−1.47214	−0.0868974
$288$	0	0
$289$	28.0000	1.64706
$290$	0	0
$291$	10.4164	0.610621
$292$	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
$295$	5.79837	0.337594
$296$	0	0
$297$	−2.23607	−0.129750
$298$	0	0
$299$	−2.38197	−0.137753
$300$	0	0
$301$	1.61803	0.0932619
$302$	0	0
$303$	4.14590	0.238176
$304$	0	0
$305$	3.00000	0.171780
$306$	0	0
$307$	−16.1246	−0.920280	−0.460140	−	0.887846i	$-0.652201\pi$
$307$	−16.1246	−0.920280	−0.460140	+	0.887846i	$0.652201\pi$
$308$	0	0
$309$	7.41641	0.421905
$310$	0	0
$311$	−15.3262	−0.869071	−0.434536	−	0.900655i	$-0.643087\pi$
$311$	−15.3262	−0.869071	−0.434536	+	0.900655i	$0.643087\pi$
$312$	0	0
$313$	−2.47214	−0.139733	−0.0698667	−	0.997556i	$-0.522257\pi$
$313$	−2.47214	−0.139733	−0.0698667	+	0.997556i	$0.522257\pi$
$314$	0	0
$315$	−0.618034	−0.0348223
$316$	0	0
$317$	14.5066	0.814771	0.407385	−	0.913256i	$-0.366441\pi$
$317$	14.5066	0.814771	0.407385	+	0.913256i	$0.366441\pi$
$318$	0	0
$319$	18.4164	1.03112
$320$	0	0
$321$	−3.32624	−0.185652
$322$	0	0
$323$	23.2918	1.29599
$324$	0	0
$325$	11.0000	0.610170
$326$	0	0
$327$	19.2705	1.06566
$328$	0	0
$329$	7.23607	0.398937
$330$	0	0
$331$	13.4164	0.737432	0.368716	−	0.929542i	$-0.379797\pi$
$331$	13.4164	0.737432	0.368716	+	0.929542i	$0.379797\pi$
$332$	0	0
$333$	−11.0000	−0.602796
$334$	0	0
$335$	−3.14590	−0.171879
$336$	0	0
$337$	−7.50658	−0.408909	−0.204455	−	0.978876i	$-0.565542\pi$
$337$	−7.50658	−0.408909	−0.204455	+	0.978876i	$0.565542\pi$
$338$	0	0
$339$	−0.909830	−0.0494152
$340$	0	0
$341$	15.0000	0.812296
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−0.618034	−0.0332738
$346$	0	0
$347$	−5.18034	−0.278095	−0.139048	−	0.990286i	$-0.544404\pi$
$347$	−5.18034	−0.278095	−0.139048	+	0.990286i	$0.544404\pi$
$348$	0	0
$349$	−11.8541	−0.634536	−0.317268	−	0.948336i	$-0.602765\pi$
$349$	−11.8541	−0.634536	−0.317268	+	0.948336i	$0.602765\pi$
$350$	0	0
$351$	−2.38197	−0.127140
$352$	0	0
$353$	30.3050	1.61297	0.806485	−	0.591255i	$-0.201367\pi$
$353$	30.3050	1.61297	0.806485	+	0.591255i	$0.201367\pi$
$354$	0	0
$355$	2.70820	0.143737
$356$	0	0
$357$	6.70820	0.355036
$358$	0	0
$359$	−23.0902	−1.21865	−0.609326	−	0.792920i	$-0.708560\pi$
$359$	−23.0902	−1.21865	−0.609326	+	0.792920i	$0.708560\pi$
$360$	0	0
$361$	−6.94427	−0.365488
$362$	0	0
$363$	−6.00000	−0.314918
$364$	0	0
$365$	7.85410	0.411102
$366$	0	0
$367$	−10.8541	−0.566580	−0.283290	−	0.959034i	$-0.591426\pi$
$367$	−10.8541	−0.566580	−0.283290	+	0.959034i	$0.591426\pi$
$368$	0	0
$369$	−1.47214	−0.0766363
$370$	0	0
$371$	−13.0902	−0.679608
$372$	0	0
$373$	24.4164	1.26423	0.632117	−	0.774873i	$-0.282186\pi$
$373$	24.4164	1.26423	0.632117	+	0.774873i	$0.282186\pi$
$374$	0	0
$375$	5.94427	0.306961
$376$	0	0
$377$	19.6180	1.01038
$378$	0	0
$379$	−7.41641	−0.380955	−0.190478	−	0.981692i	$-0.561004\pi$
$379$	−7.41641	−0.380955	−0.190478	+	0.981692i	$0.561004\pi$
$380$	0	0
$381$	−22.2705	−1.14095
$382$	0	0
$383$	0.708204	0.0361875	0.0180938	−	0.999836i	$-0.494240\pi$
$383$	0.708204	0.0361875	0.0180938	+	0.999836i	$0.494240\pi$
$384$	0	0
$385$	1.38197	0.0704315
$386$	0	0
$387$	1.61803	0.0822493
$388$	0	0
$389$	5.29180	0.268305	0.134152	−	0.990961i	$-0.457169\pi$
$389$	5.29180	0.268305	0.134152	+	0.990961i	$0.457169\pi$
$390$	0	0
$391$	6.70820	0.339248
$392$	0	0
$393$	−15.1803	−0.765747
$394$	0	0
$395$	−5.85410	−0.294552
$396$	0	0
$397$	22.0689	1.10761	0.553803	−	0.832648i	$-0.313176\pi$
$397$	22.0689	1.10761	0.553803	+	0.832648i	$0.313176\pi$
$398$	0	0
$399$	3.47214	0.173824
$400$	0	0
$401$	33.1803	1.65695	0.828474	−	0.560028i	$-0.189210\pi$
$401$	33.1803	1.65695	0.828474	+	0.560028i	$0.189210\pi$
$402$	0	0
$403$	15.9787	0.795956
$404$	0	0
$405$	−0.618034	−0.0307104
$406$	0	0
$407$	24.5967	1.21922
$408$	0	0
$409$	6.41641	0.317271	0.158635	−	0.987337i	$-0.449291\pi$
$409$	6.41641	0.317271	0.158635	+	0.987337i	$0.449291\pi$
$410$	0	0
$411$	−16.4164	−0.809762
$412$	0	0
$413$	−9.38197	−0.461656
$414$	0	0
$415$	−5.67376	−0.278514
$416$	0	0
$417$	11.3820	0.557377
$418$	0	0
$419$	−15.6738	−0.765713	−0.382857	−	0.923808i	$-0.625060\pi$
$419$	−15.6738	−0.765713	−0.382857	+	0.923808i	$0.625060\pi$
$420$	0	0
$421$	−10.2705	−0.500554	−0.250277	−	0.968174i	$-0.580522\pi$
$421$	−10.2705	−0.500554	−0.250277	+	0.968174i	$0.580522\pi$
$422$	0	0
$423$	7.23607	0.351830
$424$	0	0
$425$	−30.9787	−1.50269
$426$	0	0
$427$	−4.85410	−0.234906
$428$	0	0
$429$	5.32624	0.257153
$430$	0	0
$431$	−0.673762	−0.0324540	−0.0162270	−	0.999868i	$-0.505165\pi$
$431$	−0.673762	−0.0324540	−0.0162270	+	0.999868i	$0.505165\pi$
$432$	0	0
$433$	19.1246	0.919070	0.459535	−	0.888160i	$-0.348016\pi$
$433$	19.1246	0.919070	0.459535	+	0.888160i	$0.348016\pi$
$434$	0	0
$435$	5.09017	0.244055
$436$	0	0
$437$	3.47214	0.166095
$438$	0	0
$439$	−23.6525	−1.12887	−0.564436	−	0.825477i	$-0.690906\pi$
$439$	−23.6525	−1.12887	−0.564436	+	0.825477i	$0.690906\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	9.52786	0.452682	0.226341	−	0.974048i	$-0.427324\pi$
$443$	9.52786	0.452682	0.226341	+	0.974048i	$0.427324\pi$
$444$	0	0
$445$	−7.18034	−0.340381
$446$	0	0
$447$	−19.2361	−0.909835
$448$	0	0
$449$	−18.4377	−0.870129	−0.435064	−	0.900399i	$-0.643274\pi$
$449$	−18.4377	−0.870129	−0.435064	+	0.900399i	$0.643274\pi$
$450$	0	0
$451$	3.29180	0.155005
$452$	0	0
$453$	15.2361	0.715853
$454$	0	0
$455$	1.47214	0.0690148
$456$	0	0
$457$	−14.3262	−0.670153	−0.335077	−	0.942191i	$-0.608762\pi$
$457$	−14.3262	−0.670153	−0.335077	+	0.942191i	$0.608762\pi$
$458$	0	0
$459$	6.70820	0.313112
$460$	0	0
$461$	25.4508	1.18536	0.592682	−	0.805436i	$-0.298069\pi$
$461$	25.4508	1.18536	0.592682	+	0.805436i	$0.298069\pi$
$462$	0	0
$463$	−17.2918	−0.803618	−0.401809	−	0.915724i	$-0.631618\pi$
$463$	−17.2918	−0.803618	−0.401809	+	0.915724i	$0.631618\pi$
$464$	0	0
$465$	4.14590	0.192261
$466$	0	0
$467$	15.1803	0.702462	0.351231	−	0.936289i	$-0.385763\pi$
$467$	15.1803	0.702462	0.351231	+	0.936289i	$0.385763\pi$
$468$	0	0
$469$	5.09017	0.235042
$470$	0	0
$471$	−0.291796	−0.0134453
$472$	0	0
$473$	−3.61803	−0.166357
$474$	0	0
$475$	−16.0344	−0.735711
$476$	0	0
$477$	−13.0902	−0.599358
$478$	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
$481$	26.2016	1.19469
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	−6.43769	−0.292321
$486$	0	0
$487$	29.2918	1.32734	0.663669	−	0.748026i	$-0.268998\pi$
$487$	29.2918	1.32734	0.663669	+	0.748026i	$0.268998\pi$
$488$	0	0
$489$	0.618034	0.0279485
$490$	0	0
$491$	−9.79837	−0.442194	−0.221097	−	0.975252i	$-0.570964\pi$
$491$	−9.79837	−0.442194	−0.221097	+	0.975252i	$0.570964\pi$
$492$	0	0
$493$	−55.2492	−2.48830
$494$	0	0
$495$	1.38197	0.0621148
$496$	0	0
$497$	−4.38197	−0.196558
$498$	0	0
$499$	38.3951	1.71880	0.859401	−	0.511302i	$-0.170837\pi$
$499$	38.3951	1.71880	0.859401	+	0.511302i	$0.170837\pi$
$500$	0	0
$501$	7.18034	0.320794
$502$	0	0
$503$	−36.3262	−1.61971	−0.809853	−	0.586632i	$-0.800453\pi$
$503$	−36.3262	−1.61971	−0.809853	+	0.586632i	$0.800453\pi$
$504$	0	0
$505$	−2.56231	−0.114021
$506$	0	0
$507$	−7.32624	−0.325370
$508$	0	0
$509$	−15.5967	−0.691314	−0.345657	−	0.938361i	$-0.612344\pi$
$509$	−15.5967	−0.691314	−0.345657	+	0.938361i	$0.612344\pi$
$510$	0	0
$511$	−12.7082	−0.562178
$512$	0	0
$513$	3.47214	0.153299
$514$	0	0
$515$	−4.58359	−0.201977
$516$	0	0
$517$	−16.1803	−0.711611
$518$	0	0
$519$	−3.47214	−0.152410
$520$	0	0
$521$	3.52786	0.154559	0.0772793	−	0.997009i	$-0.475377\pi$
$521$	3.52786	0.154559	0.0772793	+	0.997009i	$0.475377\pi$
$522$	0	0
$523$	−21.4164	−0.936474	−0.468237	−	0.883603i	$-0.655111\pi$
$523$	−21.4164	−0.936474	−0.468237	+	0.883603i	$0.655111\pi$
$524$	0	0
$525$	−4.61803	−0.201548
$526$	0	0
$527$	−45.0000	−1.96023
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−9.38197	−0.407143
$532$	0	0
$533$	3.50658	0.151887
$534$	0	0
$535$	2.05573	0.0888769
$536$	0	0
$537$	−1.85410	−0.0800104
$538$	0	0
$539$	−2.23607	−0.0963143
$540$	0	0
$541$	−5.12461	−0.220324	−0.110162	−	0.993914i	$-0.535137\pi$
$541$	−5.12461	−0.220324	−0.110162	+	0.993914i	$0.535137\pi$
$542$	0	0
$543$	−1.94427	−0.0834367
$544$	0	0
$545$	−11.9098	−0.510161
$546$	0	0
$547$	−12.7984	−0.547219	−0.273609	−	0.961841i	$-0.588218\pi$
$547$	−12.7984	−0.547219	−0.273609	+	0.961841i	$0.588218\pi$
$548$	0	0
$549$	−4.85410	−0.207168
$550$	0	0
$551$	−28.5967	−1.21826
$552$	0	0
$553$	9.47214	0.402796
$554$	0	0
$555$	6.79837	0.288575
$556$	0	0
$557$	−5.81966	−0.246587	−0.123293	−	0.992370i	$-0.539346\pi$
$557$	−5.81966	−0.246587	−0.123293	+	0.992370i	$0.539346\pi$
$558$	0	0
$559$	−3.85410	−0.163011
$560$	0	0
$561$	−15.0000	−0.633300
$562$	0	0
$563$	23.5066	0.990684	0.495342	−	0.868698i	$-0.335043\pi$
$563$	23.5066	0.990684	0.495342	+	0.868698i	$0.335043\pi$
$564$	0	0
$565$	0.562306	0.0236564
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−16.3607	−0.685875	−0.342938	−	0.939358i	$-0.611422\pi$
$569$	−16.3607	−0.685875	−0.342938	+	0.939358i	$0.611422\pi$
$570$	0	0
$571$	44.4164	1.85877	0.929384	−	0.369113i	$-0.120339\pi$
$571$	44.4164	1.85877	0.929384	+	0.369113i	$0.120339\pi$
$572$	0	0
$573$	−16.1803	−0.675943
$574$	0	0
$575$	−4.61803	−0.192585
$576$	0	0
$577$	−36.8328	−1.53337	−0.766685	−	0.642023i	$-0.778095\pi$
$577$	−36.8328	−1.53337	−0.766685	+	0.642023i	$0.778095\pi$
$578$	0	0
$579$	8.29180	0.344595
$580$	0	0
$581$	9.18034	0.380865
$582$	0	0
$583$	29.2705	1.21226
$584$	0	0
$585$	1.47214	0.0608653
$586$	0	0
$587$	31.0344	1.28093	0.640464	−	0.767988i	$-0.278742\pi$
$587$	31.0344	1.28093	0.640464	+	0.767988i	$0.278742\pi$
$588$	0	0
$589$	−23.2918	−0.959722
$590$	0	0
$591$	−25.5066	−1.04920
$592$	0	0
$593$	−32.0689	−1.31691	−0.658456	−	0.752620i	$-0.728790\pi$
$593$	−32.0689	−1.31691	−0.658456	+	0.752620i	$0.728790\pi$
$594$	0	0
$595$	−4.14590	−0.169965
$596$	0	0
$597$	6.90983	0.282801
$598$	0	0
$599$	15.5066	0.633582	0.316791	−	0.948495i	$-0.397395\pi$
$599$	15.5066	0.633582	0.316791	+	0.948495i	$0.397395\pi$
$600$	0	0
$601$	0.909830	0.0371127	0.0185564	−	0.999828i	$-0.494093\pi$
$601$	0.909830	0.0371127	0.0185564	+	0.999828i	$0.494093\pi$
$602$	0	0
$603$	5.09017	0.207288
$604$	0	0
$605$	3.70820	0.150760
$606$	0	0
$607$	33.7984	1.37183	0.685917	−	0.727680i	$-0.259401\pi$
$607$	33.7984	1.37183	0.685917	+	0.727680i	$0.259401\pi$
$608$	0	0
$609$	−8.23607	−0.333742
$610$	0	0
$611$	−17.2361	−0.697297
$612$	0	0
$613$	−11.6525	−0.470639	−0.235320	−	0.971918i	$-0.575614\pi$
$613$	−11.6525	−0.470639	−0.235320	+	0.971918i	$0.575614\pi$
$614$	0	0
$615$	0.909830	0.0366879
$616$	0	0
$617$	−2.67376	−0.107642	−0.0538208	−	0.998551i	$-0.517140\pi$
$617$	−2.67376	−0.107642	−0.0538208	+	0.998551i	$0.517140\pi$
$618$	0	0
$619$	19.6180	0.788515	0.394258	−	0.919000i	$-0.371002\pi$
$619$	19.6180	0.788515	0.394258	+	0.919000i	$0.371002\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	11.6180	0.465467
$624$	0	0
$625$	19.4164	0.776656
$626$	0	0
$627$	−7.76393	−0.310062
$628$	0	0
$629$	−73.7902	−2.94221
$630$	0	0
$631$	−16.1246	−0.641911	−0.320955	−	0.947094i	$-0.604004\pi$
$631$	−16.1246	−0.641911	−0.320955	+	0.947094i	$0.604004\pi$
$632$	0	0
$633$	−10.4164	−0.414015
$634$	0	0
$635$	13.7639	0.546205
$636$	0	0
$637$	−2.38197	−0.0943769
$638$	0	0
$639$	−4.38197	−0.173348
$640$	0	0
$641$	17.7984	0.702994	0.351497	−	0.936189i	$-0.385673\pi$
$641$	17.7984	0.702994	0.351497	+	0.936189i	$0.385673\pi$
$642$	0	0
$643$	11.9787	0.472394	0.236197	−	0.971705i	$-0.424099\pi$
$643$	11.9787	0.472394	0.236197	+	0.971705i	$0.424099\pi$
$644$	0	0
$645$	−1.00000	−0.0393750
$646$	0	0
$647$	24.0344	0.944891	0.472446	−	0.881360i	$-0.343371\pi$
$647$	24.0344	0.944891	0.472446	+	0.881360i	$0.343371\pi$
$648$	0	0
$649$	20.9787	0.823487
$650$	0	0
$651$	−6.70820	−0.262915
$652$	0	0
$653$	27.7426	1.08565	0.542827	−	0.839845i	$-0.317354\pi$
$653$	27.7426	1.08565	0.542827	+	0.839845i	$0.317354\pi$
$654$	0	0
$655$	9.38197	0.366584
$656$	0	0
$657$	−12.7082	−0.495794
$658$	0	0
$659$	−9.65248	−0.376007	−0.188004	−	0.982168i	$-0.560202\pi$
$659$	−9.65248	−0.376007	−0.188004	+	0.982168i	$0.560202\pi$
$660$	0	0
$661$	14.4164	0.560733	0.280367	−	0.959893i	$-0.409544\pi$
$661$	14.4164	0.560733	0.280367	+	0.959893i	$0.409544\pi$
$662$	0	0
$663$	−15.9787	−0.620562
$664$	0	0
$665$	−2.14590	−0.0832144
$666$	0	0
$667$	−8.23607	−0.318902
$668$	0	0
$669$	17.8541	0.690279
$670$	0	0
$671$	10.8541	0.419018
$672$	0	0
$673$	12.4164	0.478617	0.239309	−	0.970944i	$-0.423079\pi$
$673$	12.4164	0.478617	0.239309	+	0.970944i	$0.423079\pi$
$674$	0	0
$675$	−4.61803	−0.177748
$676$	0	0
$677$	−1.25735	−0.0483240	−0.0241620	−	0.999708i	$-0.507692\pi$
$677$	−1.25735	−0.0483240	−0.0241620	+	0.999708i	$0.507692\pi$
$678$	0	0
$679$	10.4164	0.399745
$680$	0	0
$681$	−24.3262	−0.932183
$682$	0	0
$683$	−26.0132	−0.995366	−0.497683	−	0.867359i	$-0.665816\pi$
$683$	−26.0132	−0.995366	−0.497683	+	0.867359i	$0.665816\pi$
$684$	0	0
$685$	10.1459	0.387655
$686$	0	0
$687$	−16.3262	−0.622885
$688$	0	0
$689$	31.1803	1.18788
$690$	0	0
$691$	7.72949	0.294044	0.147022	−	0.989133i	$-0.453031\pi$
$691$	7.72949	0.294044	0.147022	+	0.989133i	$0.453031\pi$
$692$	0	0
$693$	−2.23607	−0.0849412
$694$	0	0
$695$	−7.03444	−0.266832
$696$	0	0
$697$	−9.87539	−0.374057
$698$	0	0
$699$	−11.0902	−0.419469
$700$	0	0
$701$	37.4508	1.41450	0.707250	−	0.706964i	$-0.249936\pi$
$701$	37.4508	1.41450	0.707250	+	0.706964i	$0.249936\pi$
$702$	0	0
$703$	−38.1935	−1.44049
$704$	0	0
$705$	−4.47214	−0.168430
$706$	0	0
$707$	4.14590	0.155923
$708$	0	0
$709$	−2.56231	−0.0962294	−0.0481147	−	0.998842i	$-0.515321\pi$
$709$	−2.56231	−0.0962294	−0.0481147	+	0.998842i	$0.515321\pi$
$710$	0	0
$711$	9.47214	0.355233
$712$	0	0
$713$	−6.70820	−0.251224
$714$	0	0
$715$	−3.29180	−0.123106
$716$	0	0
$717$	4.79837	0.179199
$718$	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
$721$	7.41641	0.276201
$722$	0	0
$723$	−11.0000	−0.409094
$724$	0	0
$725$	38.0344	1.41256
$726$	0	0
$727$	−18.8885	−0.700537	−0.350269	−	0.936649i	$-0.613910\pi$
$727$	−18.8885	−0.700537	−0.350269	+	0.936649i	$0.613910\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	10.8541	0.401453
$732$	0	0
$733$	25.5967	0.945437	0.472719	−	0.881213i	$-0.343273\pi$
$733$	25.5967	0.945437	0.472719	+	0.881213i	$0.343273\pi$
$734$	0	0
$735$	−0.618034	−0.0227965
$736$	0	0
$737$	−11.3820	−0.419260
$738$	0	0
$739$	−45.2492	−1.66452	−0.832260	−	0.554386i	$-0.812953\pi$
$739$	−45.2492	−1.66452	−0.832260	+	0.554386i	$0.812953\pi$
$740$	0	0
$741$	−8.27051	−0.303825
$742$	0	0
$743$	−11.6738	−0.428269	−0.214134	−	0.976804i	$-0.568693\pi$
$743$	−11.6738	−0.428269	−0.214134	+	0.976804i	$0.568693\pi$
$744$	0	0
$745$	11.8885	0.435563
$746$	0	0
$747$	9.18034	0.335891
$748$	0	0
$749$	−3.32624	−0.121538
$750$	0	0
$751$	−32.9787	−1.20341	−0.601705	−	0.798718i	$-0.705512\pi$
$751$	−32.9787	−1.20341	−0.601705	+	0.798718i	$0.705512\pi$
$752$	0	0
$753$	−23.1246	−0.842708
$754$	0	0
$755$	−9.41641	−0.342698
$756$	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
$761$	−14.4721	−0.524615	−0.262307	−	0.964984i	$-0.584483\pi$
$761$	−14.4721	−0.524615	−0.262307	+	0.964984i	$0.584483\pi$
$762$	0	0
$763$	19.2705	0.697639
$764$	0	0
$765$	−4.14590	−0.149895
$766$	0	0
$767$	22.3475	0.806922
$768$	0	0
$769$	33.2361	1.19852	0.599262	−	0.800553i	$-0.295461\pi$
$769$	33.2361	1.19852	0.599262	+	0.800553i	$0.295461\pi$
$770$	0	0
$771$	−3.23607	−0.116544
$772$	0	0
$773$	−19.9443	−0.717346	−0.358673	−	0.933463i	$-0.616771\pi$
$773$	−19.9443	−0.717346	−0.358673	+	0.933463i	$0.616771\pi$
$774$	0	0
$775$	30.9787	1.11279
$776$	0	0
$777$	−11.0000	−0.394623
$778$	0	0
$779$	−5.11146	−0.183137
$780$	0	0
$781$	9.79837	0.350613
$782$	0	0
$783$	−8.23607	−0.294333
$784$	0	0
$785$	0.180340	0.00643661
$786$	0	0
$787$	48.5755	1.73153	0.865764	−	0.500452i	$-0.166833\pi$
$787$	48.5755	1.73153	0.865764	+	0.500452i	$0.166833\pi$
$788$	0	0
$789$	0.0557281	0.00198397
$790$	0	0
$791$	−0.909830	−0.0323498
$792$	0	0
$793$	11.5623	0.410590
$794$	0	0
$795$	8.09017	0.286929
$796$	0	0
$797$	44.1803	1.56495	0.782474	−	0.622683i	$-0.213957\pi$
$797$	44.1803	1.56495	0.782474	+	0.622683i	$0.213957\pi$
$798$	0	0
$799$	48.5410	1.71726
$800$	0	0
$801$	11.6180	0.410503
$802$	0	0
$803$	28.4164	1.00279
$804$	0	0
$805$	−0.618034	−0.0217828
$806$	0	0
$807$	−32.0902	−1.12963
$808$	0	0
$809$	−9.43769	−0.331812	−0.165906	−	0.986142i	$-0.553055\pi$
$809$	−9.43769	−0.331812	−0.165906	+	0.986142i	$0.553055\pi$
$810$	0	0
$811$	−8.52786	−0.299454	−0.149727	−	0.988727i	$-0.547839\pi$
$811$	−8.52786	−0.299454	−0.149727	+	0.988727i	$0.547839\pi$
$812$	0	0
$813$	−21.9443	−0.769619
$814$	0	0
$815$	−0.381966	−0.0133797
$816$	0	0
$817$	5.61803	0.196550
$818$	0	0
$819$	−2.38197	−0.0832326
$820$	0	0
$821$	−29.1246	−1.01646	−0.508228	−	0.861223i	$-0.669699\pi$
$821$	−29.1246	−1.01646	−0.508228	+	0.861223i	$0.669699\pi$
$822$	0	0
$823$	39.8541	1.38923	0.694613	−	0.719383i	$-0.255575\pi$
$823$	39.8541	1.38923	0.694613	+	0.719383i	$0.255575\pi$
$824$	0	0
$825$	10.3262	0.359513
$826$	0	0
$827$	28.2148	0.981124	0.490562	−	0.871406i	$-0.336792\pi$
$827$	28.2148	0.981124	0.490562	+	0.871406i	$0.336792\pi$
$828$	0	0
$829$	−30.4164	−1.05641	−0.528203	−	0.849118i	$-0.677134\pi$
$829$	−30.4164	−1.05641	−0.528203	+	0.849118i	$0.677134\pi$
$830$	0	0
$831$	4.27051	0.148142
$832$	0	0
$833$	6.70820	0.232425
$834$	0	0
$835$	−4.43769	−0.153573
$836$	0	0
$837$	−6.70820	−0.231869
$838$	0	0
$839$	−16.7426	−0.578020	−0.289010	−	0.957326i	$-0.593326\pi$
$839$	−16.7426	−0.578020	−0.289010	+	0.957326i	$0.593326\pi$
$840$	0	0
$841$	38.8328	1.33906
$842$	0	0
$843$	−3.70820	−0.127717
$844$	0	0
$845$	4.52786	0.155763
$846$	0	0
$847$	−6.00000	−0.206162
$848$	0	0
$849$	26.2148	0.899689
$850$	0	0
$851$	−11.0000	−0.377075
$852$	0	0
$853$	−47.1246	−1.61352	−0.806758	−	0.590882i	$-0.798780\pi$
$853$	−47.1246	−1.61352	−0.806758	+	0.590882i	$0.798780\pi$
$854$	0	0
$855$	−2.14590	−0.0733882
$856$	0	0
$857$	39.7082	1.35641	0.678203	−	0.734874i	$-0.262759\pi$
$857$	39.7082	1.35641	0.678203	+	0.734874i	$0.262759\pi$
$858$	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
$861$	−1.47214	−0.0501703
$862$	0	0
$863$	8.06888	0.274668	0.137334	−	0.990525i	$-0.456147\pi$
$863$	8.06888	0.274668	0.137334	+	0.990525i	$0.456147\pi$
$864$	0	0
$865$	2.14590	0.0729627
$866$	0	0
$867$	28.0000	0.950930
$868$	0	0
$869$	−21.1803	−0.718494
$870$	0	0
$871$	−12.1246	−0.410827
$872$	0	0
$873$	10.4164	0.352542
$874$	0	0
$875$	5.94427	0.200953
$876$	0	0
$877$	16.5836	0.559988	0.279994	−	0.960002i	$-0.409667\pi$
$877$	16.5836	0.559988	0.279994	+	0.960002i	$0.409667\pi$
$878$	0	0
$879$	12.0000	0.404750
$880$	0	0
$881$	1.81966	0.0613059	0.0306530	−	0.999530i	$-0.490241\pi$
$881$	1.81966	0.0613059	0.0306530	+	0.999530i	$0.490241\pi$
$882$	0	0
$883$	3.90983	0.131576	0.0657881	−	0.997834i	$-0.479044\pi$
$883$	3.90983	0.131576	0.0657881	+	0.997834i	$0.479044\pi$
$884$	0	0
$885$	5.79837	0.194910
$886$	0	0
$887$	−9.79837	−0.328997	−0.164499	−	0.986377i	$-0.552601\pi$
$887$	−9.79837	−0.328997	−0.164499	+	0.986377i	$0.552601\pi$
$888$	0	0
$889$	−22.2705	−0.746929
$890$	0	0
$891$	−2.23607	−0.0749111
$892$	0	0
$893$	25.1246	0.840763
$894$	0	0
$895$	1.14590	0.0383031
$896$	0	0
$897$	−2.38197	−0.0795315
$898$	0	0
$899$	55.2492	1.84266
$900$	0	0
$901$	−87.8115	−2.92543
$902$	0	0
$903$	1.61803	0.0538448
$904$	0	0
$905$	1.20163	0.0399434
$906$	0	0
$907$	−48.9787	−1.62631	−0.813156	−	0.582046i	$-0.802252\pi$
$907$	−48.9787	−1.62631	−0.813156	+	0.582046i	$0.802252\pi$
$908$	0	0
$909$	4.14590	0.137511
$910$	0	0
$911$	7.59675	0.251691	0.125846	−	0.992050i	$-0.459836\pi$
$911$	7.59675	0.251691	0.125846	+	0.992050i	$0.459836\pi$
$912$	0	0
$913$	−20.5279	−0.679373
$914$	0	0
$915$	3.00000	0.0991769
$916$	0	0
$917$	−15.1803	−0.501299
$918$	0	0
$919$	50.1246	1.65346	0.826729	−	0.562600i	$-0.190199\pi$
$919$	50.1246	1.65346	0.826729	+	0.562600i	$0.190199\pi$
$920$	0	0
$921$	−16.1246	−0.531324
$922$	0	0
$923$	10.4377	0.343561
$924$	0	0
$925$	50.7984	1.67024
$926$	0	0
$927$	7.41641	0.243587
$928$	0	0
$929$	28.9098	0.948501	0.474250	−	0.880390i	$-0.342719\pi$
$929$	28.9098	0.948501	0.474250	+	0.880390i	$0.342719\pi$
$930$	0	0
$931$	3.47214	0.113795
$932$	0	0
$933$	−15.3262	−0.501759
$934$	0	0
$935$	9.27051	0.303178
$936$	0	0
$937$	2.70820	0.0884732	0.0442366	−	0.999021i	$-0.485914\pi$
$937$	2.70820	0.0884732	0.0442366	+	0.999021i	$0.485914\pi$
$938$	0	0
$939$	−2.47214	−0.0806751
$940$	0	0
$941$	−15.0557	−0.490803	−0.245401	−	0.969422i	$-0.578920\pi$
$941$	−15.0557	−0.490803	−0.245401	+	0.969422i	$0.578920\pi$
$942$	0	0
$943$	−1.47214	−0.0479393
$944$	0	0
$945$	−0.618034	−0.0201046
$946$	0	0
$947$	18.9443	0.615606	0.307803	−	0.951450i	$-0.400406\pi$
$947$	18.9443	0.615606	0.307803	+	0.951450i	$0.400406\pi$
$948$	0	0
$949$	30.2705	0.982622
$950$	0	0
$951$	14.5066	0.470408
$952$	0	0
$953$	12.1591	0.393870	0.196935	−	0.980417i	$-0.436901\pi$
$953$	12.1591	0.393870	0.196935	+	0.980417i	$0.436901\pi$
$954$	0	0
$955$	10.0000	0.323592
$956$	0	0
$957$	18.4164	0.595318
$958$	0	0
$959$	−16.4164	−0.530113
$960$	0	0
$961$	14.0000	0.451613
$962$	0	0
$963$	−3.32624	−0.107187
$964$	0	0
$965$	−5.12461	−0.164967
$966$	0	0
$967$	−50.3607	−1.61949	−0.809745	−	0.586782i	$-0.800395\pi$
$967$	−50.3607	−1.61949	−0.809745	+	0.586782i	$0.800395\pi$
$968$	0	0
$969$	23.2918	0.748240
$970$	0	0
$971$	19.6869	0.631783	0.315892	−	0.948795i	$-0.397696\pi$
$971$	19.6869	0.631783	0.315892	+	0.948795i	$0.397696\pi$
$972$	0	0
$973$	11.3820	0.364889
$974$	0	0
$975$	11.0000	0.352282
$976$	0	0
$977$	49.7984	1.59319	0.796596	−	0.604513i	$-0.206632\pi$
$977$	49.7984	1.59319	0.796596	+	0.604513i	$0.206632\pi$
$978$	0	0
$979$	−25.9787	−0.830283
$980$	0	0
$981$	19.2705	0.615260
$982$	0	0
$983$	7.40325	0.236127	0.118064	−	0.993006i	$-0.462331\pi$
$983$	7.40325	0.236127	0.118064	+	0.993006i	$0.462331\pi$
$984$	0	0
$985$	15.7639	0.502281
$986$	0	0
$987$	7.23607	0.230327
$988$	0	0
$989$	1.61803	0.0514505
$990$	0	0
$991$	35.5755	1.13009	0.565046	−	0.825059i	$-0.308858\pi$
$991$	35.5755	1.13009	0.565046	+	0.825059i	$0.308858\pi$
$992$	0	0
$993$	13.4164	0.425757
$994$	0	0
$995$	−4.27051	−0.135384
$996$	0	0
$997$	57.7214	1.82805	0.914027	−	0.405654i	$-0.132956\pi$
$997$	57.7214	1.82805	0.914027	+	0.405654i	$0.132956\pi$
$998$	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.1.1		2
4.3	odd	2		483.2.a.d.1.1	✓	2
12.11	even	2		1449.2.a.h.1.2		2
28.27	even	2		3381.2.a.r.1.1		2

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.618034 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.618034 q^{5} +1.00000 q^{7} +1.00000 q^{9} -2.23607 q^{11} -2.38197 q^{13} -0.618034 q^{15} +6.70820 q^{17} +3.47214 q^{19} +1.00000 q^{21} +1.00000 q^{23} -4.61803 q^{25} +1.00000 q^{27} -8.23607 q^{29} -6.70820 q^{31} -2.23607 q^{33} -0.618034 q^{35} -11.0000 q^{37} -2.38197 q^{39} -1.47214 q^{41} +1.61803 q^{43} -0.618034 q^{45} +7.23607 q^{47} +1.00000 q^{49} +6.70820 q^{51} -13.0902 q^{53} +1.38197 q^{55} +3.47214 q^{57} -9.38197 q^{59} -4.85410 q^{61} +1.00000 q^{63} +1.47214 q^{65} +5.09017 q^{67} +1.00000 q^{69} -4.38197 q^{71} -12.7082 q^{73} -4.61803 q^{75} -2.23607 q^{77} +9.47214 q^{79} +1.00000 q^{81} +9.18034 q^{83} -4.14590 q^{85} -8.23607 q^{87} +11.6180 q^{89} -2.38197 q^{91} -6.70820 q^{93} -2.14590 q^{95} +10.4164 q^{97} -2.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 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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