LMFDB - Embedded newform 1920.2.a.y.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1920,2,Mod(1,1920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1920 = 2^{7} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1920.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:	$15.3312771881$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	1920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} -3.12311 q^{7} +1.00000 q^{9} -2.00000 q^{11} +3.12311 q^{13} +1.00000 q^{15} +7.12311 q^{17} +3.12311 q^{19} +3.12311 q^{21} +3.12311 q^{23} +1.00000 q^{25} -1.00000 q^{27} -8.24621 q^{29} +1.12311 q^{31} +2.00000 q^{33} +3.12311 q^{35} -3.12311 q^{37} -3.12311 q^{39} -2.00000 q^{41} -10.2462 q^{43} -1.00000 q^{45} -4.87689 q^{47} +2.75379 q^{49} -7.12311 q^{51} -10.0000 q^{53} +2.00000 q^{55} -3.12311 q^{57} -6.00000 q^{59} +2.00000 q^{61} -3.12311 q^{63} -3.12311 q^{65} -10.2462 q^{67} -3.12311 q^{69} -8.00000 q^{71} +12.2462 q^{73} -1.00000 q^{75} +6.24621 q^{77} -13.1231 q^{79} +1.00000 q^{81} +4.00000 q^{83} -7.12311 q^{85} +8.24621 q^{87} -10.0000 q^{89} -9.75379 q^{91} -1.12311 q^{93} -3.12311 q^{95} +10.0000 q^{97} -2.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 4 q^{11} - 2 q^{13} + 2 q^{15} + 6 q^{17} - 2 q^{19} - 2 q^{21} - 2 q^{23} + 2 q^{25} - 2 q^{27} - 6 q^{31} + 4 q^{33} - 2 q^{35} + 2 q^{37} + 2 q^{39} - 4 q^{41}+ \cdots - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 - 4 * q^11 - 2 * q^13 + 2 * q^15 + 6 * q^17 - 2 * q^19 - 2 * q^21 - 2 * q^23 + 2 * q^25 - 2 * q^27 - 6 * q^31 + 4 * q^33 - 2 * q^35 + 2 * q^37 + 2 * q^39 - 4 * q^41 - 4 * q^43 - 2 * q^45 - 18 * q^47 + 22 * q^49 - 6 * q^51 - 20 * q^53 + 4 * q^55 + 2 * q^57 - 12 * q^59 + 4 * q^61 + 2 * q^63 + 2 * q^65 - 4 * q^67 + 2 * q^69 - 16 * q^71 + 8 * q^73 - 2 * q^75 - 4 * q^77 - 18 * q^79 + 2 * q^81 + 8 * q^83 - 6 * q^85 - 20 * q^89 - 36 * q^91 + 6 * q^93 + 2 * q^95 + 20 * q^97 - 4 * q^99

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−3.12311	−1.18042	−0.590211	−	0.807249i	$-0.700956\pi$
$7$	−3.12311	−1.18042	−0.590211	+	0.807249i	$0.700956\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	3.12311	0.866194	0.433097	−	0.901347i	$-0.357421\pi$
$13$	3.12311	0.866194	0.433097	+	0.901347i	$0.357421\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	7.12311	1.72761	0.863803	−	0.503829i	$-0.168076\pi$
$17$	7.12311	1.72761	0.863803	+	0.503829i	$0.168076\pi$
$18$	0	0
$19$	3.12311	0.716490	0.358245	−	0.933628i	$-0.383375\pi$
$19$	3.12311	0.716490	0.358245	+	0.933628i	$0.383375\pi$
$20$	0	0
$21$	3.12311	0.681518
$22$	0	0
$23$	3.12311	0.651213	0.325606	−	0.945505i	$-0.394432\pi$
$23$	3.12311	0.651213	0.325606	+	0.945505i	$0.394432\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−8.24621	−1.53128	−0.765641	−	0.643268i	$-0.777578\pi$
$29$	−8.24621	−1.53128	−0.765641	+	0.643268i	$0.777578\pi$
$30$	0	0
$31$	1.12311	0.201716	0.100858	−	0.994901i	$-0.467841\pi$
$31$	1.12311	0.201716	0.100858	+	0.994901i	$0.467841\pi$
$32$	0	0
$33$	2.00000	0.348155
$34$	0	0
$35$	3.12311	0.527901
$36$	0	0
$37$	−3.12311	−0.513435	−0.256718	−	0.966486i	$-0.582641\pi$
$37$	−3.12311	−0.513435	−0.256718	+	0.966486i	$0.582641\pi$
$38$	0	0
$39$	−3.12311	−0.500097
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−10.2462	−1.56253	−0.781266	−	0.624198i	$-0.785426\pi$
$43$	−10.2462	−1.56253	−0.781266	+	0.624198i	$0.785426\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−4.87689	−0.711368	−0.355684	−	0.934606i	$-0.615752\pi$
$47$	−4.87689	−0.711368	−0.355684	+	0.934606i	$0.615752\pi$
$48$	0	0
$49$	2.75379	0.393398
$50$	0	0
$51$	−7.12311	−0.997434
$52$	0	0
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	0	0
$55$	2.00000	0.269680
$56$	0	0
$57$	−3.12311	−0.413665
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	0	0
$63$	−3.12311	−0.393474
$64$	0	0
$65$	−3.12311	−0.387374
$66$	0	0
$67$	−10.2462	−1.25177	−0.625887	−	0.779914i	$-0.715263\pi$
$67$	−10.2462	−1.25177	−0.625887	+	0.779914i	$0.715263\pi$
$68$	0	0
$69$	−3.12311	−0.375978
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	12.2462	1.43331	0.716655	−	0.697428i	$-0.245672\pi$
$73$	12.2462	1.43331	0.716655	+	0.697428i	$0.245672\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	6.24621	0.711822
$78$	0	0
$79$	−13.1231	−1.47646	−0.738232	−	0.674546i	$-0.764339\pi$
$79$	−13.1231	−1.47646	−0.738232	+	0.674546i	$0.764339\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	−7.12311	−0.772609
$86$	0	0
$87$	8.24621	0.884087
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−9.75379	−1.02247
$92$	0	0
$93$	−1.12311	−0.116461
$94$	0	0
$95$	−3.12311	−0.320424
$96$	0	0
$97$	10.0000	1.01535	0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000	1.01535	0.507673	+	0.861550i	$0.330506\pi$
$98$	0	0
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−14.0000	−1.39305	−0.696526	−	0.717532i	$-0.745272\pi$
$101$	−14.0000	−1.39305	−0.696526	+	0.717532i	$0.745272\pi$
$102$	0	0
$103$	13.3693	1.31732	0.658659	−	0.752442i	$-0.271124\pi$
$103$	13.3693	1.31732	0.658659	+	0.752442i	$0.271124\pi$
$104$	0	0
$105$	−3.12311	−0.304784
$106$	0	0
$107$	14.2462	1.37723	0.688617	−	0.725126i	$-0.258218\pi$
$107$	14.2462	1.37723	0.688617	+	0.725126i	$0.258218\pi$
$108$	0	0
$109$	12.2462	1.17297	0.586487	−	0.809959i	$-0.300510\pi$
$109$	12.2462	1.17297	0.586487	+	0.809959i	$0.300510\pi$
$110$	0	0
$111$	3.12311	0.296432
$112$	0	0
$113$	15.1231	1.42266	0.711331	−	0.702857i	$-0.248093\pi$
$113$	15.1231	1.42266	0.711331	+	0.702857i	$0.248093\pi$
$114$	0	0
$115$	−3.12311	−0.291231
$116$	0	0
$117$	3.12311	0.288731
$118$	0	0
$119$	−22.2462	−2.03931
$120$	0	0
$121$	−7.00000	−0.636364
$122$	0	0
$123$	2.00000	0.180334
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−15.1231	−1.34196	−0.670979	−	0.741476i	$-0.734126\pi$
$127$	−15.1231	−1.34196	−0.670979	+	0.741476i	$0.734126\pi$
$128$	0	0
$129$	10.2462	0.902129
$130$	0	0
$131$	−8.24621	−0.720475	−0.360237	−	0.932861i	$-0.617304\pi$
$131$	−8.24621	−0.720475	−0.360237	+	0.932861i	$0.617304\pi$
$132$	0	0
$133$	−9.75379	−0.845761
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−11.1231	−0.950311	−0.475156	−	0.879902i	$-0.657608\pi$
$137$	−11.1231	−0.950311	−0.475156	+	0.879902i	$0.657608\pi$
$138$	0	0
$139$	9.36932	0.794695	0.397348	−	0.917668i	$-0.369931\pi$
$139$	9.36932	0.794695	0.397348	+	0.917668i	$0.369931\pi$
$140$	0	0
$141$	4.87689	0.410709
$142$	0	0
$143$	−6.24621	−0.522334
$144$	0	0
$145$	8.24621	0.684811
$146$	0	0
$147$	−2.75379	−0.227129
$148$	0	0
$149$	−3.75379	−0.307522	−0.153761	−	0.988108i	$-0.549139\pi$
$149$	−3.75379	−0.307522	−0.153761	+	0.988108i	$0.549139\pi$
$150$	0	0
$151$	−7.36932	−0.599707	−0.299853	−	0.953985i	$-0.596938\pi$
$151$	−7.36932	−0.599707	−0.299853	+	0.953985i	$0.596938\pi$
$152$	0	0
$153$	7.12311	0.575869
$154$	0	0
$155$	−1.12311	−0.0902100
$156$	0	0
$157$	−5.36932	−0.428518	−0.214259	−	0.976777i	$-0.568734\pi$
$157$	−5.36932	−0.428518	−0.214259	+	0.976777i	$0.568734\pi$
$158$	0	0
$159$	10.0000	0.793052
$160$	0	0
$161$	−9.75379	−0.768706
$162$	0	0
$163$	24.4924	1.91839	0.959197	−	0.282738i	$-0.0912426\pi$
$163$	24.4924	1.91839	0.959197	+	0.282738i	$0.0912426\pi$
$164$	0	0
$165$	−2.00000	−0.155700
$166$	0	0
$167$	−19.1231	−1.47979	−0.739895	−	0.672722i	$-0.765125\pi$
$167$	−19.1231	−1.47979	−0.739895	+	0.672722i	$0.765125\pi$
$168$	0	0
$169$	−3.24621	−0.249709
$170$	0	0
$171$	3.12311	0.238830
$172$	0	0
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	−3.12311	−0.236085
$176$	0	0
$177$	6.00000	0.450988
$178$	0	0
$179$	−4.24621	−0.317377	−0.158688	−	0.987329i	$-0.550727\pi$
$179$	−4.24621	−0.317377	−0.158688	+	0.987329i	$0.550727\pi$
$180$	0	0
$181$	−16.2462	−1.20757	−0.603786	−	0.797147i	$-0.706342\pi$
$181$	−16.2462	−1.20757	−0.603786	+	0.797147i	$0.706342\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	0	0
$185$	3.12311	0.229615
$186$	0	0
$187$	−14.2462	−1.04179
$188$	0	0
$189$	3.12311	0.227173
$190$	0	0
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	7.75379	0.558130	0.279065	−	0.960272i	$-0.409976\pi$
$193$	7.75379	0.558130	0.279065	+	0.960272i	$0.409976\pi$
$194$	0	0
$195$	3.12311	0.223650
$196$	0	0
$197$	−2.00000	−0.142494	−0.0712470	−	0.997459i	$-0.522698\pi$
$197$	−2.00000	−0.142494	−0.0712470	+	0.997459i	$0.522698\pi$
$198$	0	0
$199$	21.6155	1.53228	0.766142	−	0.642671i	$-0.222174\pi$
$199$	21.6155	1.53228	0.766142	+	0.642671i	$0.222174\pi$
$200$	0	0
$201$	10.2462	0.722712
$202$	0	0
$203$	25.7538	1.80756
$204$	0	0
$205$	2.00000	0.139686
$206$	0	0
$207$	3.12311	0.217071
$208$	0	0
$209$	−6.24621	−0.432059
$210$	0	0
$211$	−19.1231	−1.31649	−0.658244	−	0.752804i	$-0.728701\pi$
$211$	−19.1231	−1.31649	−0.658244	+	0.752804i	$0.728701\pi$
$212$	0	0
$213$	8.00000	0.548151
$214$	0	0
$215$	10.2462	0.698786
$216$	0	0
$217$	−3.50758	−0.238110
$218$	0	0
$219$	−12.2462	−0.827522
$220$	0	0
$221$	22.2462	1.49644
$222$	0	0
$223$	23.1231	1.54844	0.774219	−	0.632918i	$-0.218143\pi$
$223$	23.1231	1.54844	0.774219	+	0.632918i	$0.218143\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	0	0		−	1.00000i	$-0.5\pi$
$227$	0	0			1.00000i	$0.5\pi$
$228$	0	0
$229$	10.0000	0.660819	0.330409	−	0.943838i	$-0.392813\pi$
$229$	10.0000	0.660819	0.330409	+	0.943838i	$0.392813\pi$
$230$	0	0
$231$	−6.24621	−0.410971
$232$	0	0
$233$	−15.6155	−1.02301	−0.511504	−	0.859281i	$-0.670911\pi$
$233$	−15.6155	−1.02301	−0.511504	+	0.859281i	$0.670911\pi$
$234$	0	0
$235$	4.87689	0.318134
$236$	0	0
$237$	13.1231	0.852437
$238$	0	0
$239$	10.2462	0.662772	0.331386	−	0.943495i	$-0.392484\pi$
$239$	10.2462	0.662772	0.331386	+	0.943495i	$0.392484\pi$
$240$	0	0
$241$	−24.2462	−1.56184	−0.780918	−	0.624634i	$-0.785248\pi$
$241$	−24.2462	−1.56184	−0.780918	+	0.624634i	$0.785248\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−2.75379	−0.175933
$246$	0	0
$247$	9.75379	0.620619
$248$	0	0
$249$	−4.00000	−0.253490
$250$	0	0
$251$	−12.2462	−0.772974	−0.386487	−	0.922295i	$-0.626312\pi$
$251$	−12.2462	−0.772974	−0.386487	+	0.922295i	$0.626312\pi$
$252$	0	0
$253$	−6.24621	−0.392696
$254$	0	0
$255$	7.12311	0.446066
$256$	0	0
$257$	−25.8617	−1.61321	−0.806605	−	0.591090i	$-0.798698\pi$
$257$	−25.8617	−1.61321	−0.806605	+	0.591090i	$0.798698\pi$
$258$	0	0
$259$	9.75379	0.606071
$260$	0	0
$261$	−8.24621	−0.510428
$262$	0	0
$263$	−9.36932	−0.577737	−0.288868	−	0.957369i	$-0.593279\pi$
$263$	−9.36932	−0.577737	−0.288868	+	0.957369i	$0.593279\pi$
$264$	0	0
$265$	10.0000	0.614295
$266$	0	0
$267$	10.0000	0.611990
$268$	0	0
$269$	−8.24621	−0.502780	−0.251390	−	0.967886i	$-0.580888\pi$
$269$	−8.24621	−0.502780	−0.251390	+	0.967886i	$0.580888\pi$
$270$	0	0
$271$	1.12311	0.0682238	0.0341119	−	0.999418i	$-0.489140\pi$
$271$	1.12311	0.0682238	0.0341119	+	0.999418i	$0.489140\pi$
$272$	0	0
$273$	9.75379	0.590326
$274$	0	0
$275$	−2.00000	−0.120605
$276$	0	0
$277$	23.1231	1.38933	0.694666	−	0.719332i	$-0.255552\pi$
$277$	23.1231	1.38933	0.694666	+	0.719332i	$0.255552\pi$
$278$	0	0
$279$	1.12311	0.0672386
$280$	0	0
$281$	−16.2462	−0.969168	−0.484584	−	0.874745i	$-0.661029\pi$
$281$	−16.2462	−0.969168	−0.484584	+	0.874745i	$0.661029\pi$
$282$	0	0
$283$	−13.7538	−0.817578	−0.408789	−	0.912629i	$-0.634049\pi$
$283$	−13.7538	−0.817578	−0.408789	+	0.912629i	$0.634049\pi$
$284$	0	0
$285$	3.12311	0.184997
$286$	0	0
$287$	6.24621	0.368702
$288$	0	0
$289$	33.7386	1.98463
$290$	0	0
$291$	−10.0000	−0.586210
$292$	0	0
$293$	−20.7386	−1.21156	−0.605782	−	0.795631i	$-0.707140\pi$
$293$	−20.7386	−1.21156	−0.605782	+	0.795631i	$0.707140\pi$
$294$	0	0
$295$	6.00000	0.349334
$296$	0	0
$297$	2.00000	0.116052
$298$	0	0
$299$	9.75379	0.564076
$300$	0	0
$301$	32.0000	1.84445
$302$	0	0
$303$	14.0000	0.804279
$304$	0	0
$305$	−2.00000	−0.114520
$306$	0	0
$307$	2.24621	0.128198	0.0640990	−	0.997944i	$-0.479583\pi$
$307$	2.24621	0.128198	0.0640990	+	0.997944i	$0.479583\pi$
$308$	0	0
$309$	−13.3693	−0.760554
$310$	0	0
$311$	−13.7538	−0.779906	−0.389953	−	0.920835i	$-0.627509\pi$
$311$	−13.7538	−0.779906	−0.389953	+	0.920835i	$0.627509\pi$
$312$	0	0
$313$	−28.7386	−1.62440	−0.812202	−	0.583377i	$-0.801731\pi$
$313$	−28.7386	−1.62440	−0.812202	+	0.583377i	$0.801731\pi$
$314$	0	0
$315$	3.12311	0.175967
$316$	0	0
$317$	−2.00000	−0.112331	−0.0561656	−	0.998421i	$-0.517887\pi$
$317$	−2.00000	−0.112331	−0.0561656	+	0.998421i	$0.517887\pi$
$318$	0	0
$319$	16.4924	0.923398
$320$	0	0
$321$	−14.2462	−0.795146
$322$	0	0
$323$	22.2462	1.23781
$324$	0	0
$325$	3.12311	0.173239
$326$	0	0
$327$	−12.2462	−0.677217
$328$	0	0
$329$	15.2311	0.839715
$330$	0	0
$331$	−7.12311	−0.391521	−0.195761	−	0.980652i	$-0.562718\pi$
$331$	−7.12311	−0.391521	−0.195761	+	0.980652i	$0.562718\pi$
$332$	0	0
$333$	−3.12311	−0.171145
$334$	0	0
$335$	10.2462	0.559810
$336$	0	0
$337$	24.7386	1.34760	0.673800	−	0.738914i	$-0.264661\pi$
$337$	24.7386	1.34760	0.673800	+	0.738914i	$0.264661\pi$
$338$	0	0
$339$	−15.1231	−0.821374
$340$	0	0
$341$	−2.24621	−0.121639
$342$	0	0
$343$	13.2614	0.716046
$344$	0	0
$345$	3.12311	0.168142
$346$	0	0
$347$	6.24621	0.335314	0.167657	−	0.985845i	$-0.446380\pi$
$347$	6.24621	0.335314	0.167657	+	0.985845i	$0.446380\pi$
$348$	0	0
$349$	22.4924	1.20399	0.601996	−	0.798499i	$-0.294372\pi$
$349$	22.4924	1.20399	0.601996	+	0.798499i	$0.294372\pi$
$350$	0	0
$351$	−3.12311	−0.166699
$352$	0	0
$353$	29.8617	1.58938	0.794690	−	0.607015i	$-0.207633\pi$
$353$	29.8617	1.58938	0.794690	+	0.607015i	$0.207633\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	0	0
$357$	22.2462	1.17739
$358$	0	0
$359$	4.00000	0.211112	0.105556	−	0.994413i	$-0.466338\pi$
$359$	4.00000	0.211112	0.105556	+	0.994413i	$0.466338\pi$
$360$	0	0
$361$	−9.24621	−0.486643
$362$	0	0
$363$	7.00000	0.367405
$364$	0	0
$365$	−12.2462	−0.640996
$366$	0	0
$367$	−17.3693	−0.906671	−0.453335	−	0.891340i	$-0.649766\pi$
$367$	−17.3693	−0.906671	−0.453335	+	0.891340i	$0.649766\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	31.2311	1.62144
$372$	0	0
$373$	−8.87689	−0.459628	−0.229814	−	0.973235i	$-0.573812\pi$
$373$	−8.87689	−0.459628	−0.229814	+	0.973235i	$0.573812\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−25.7538	−1.32639
$378$	0	0
$379$	−21.3693	−1.09767	−0.548834	−	0.835931i	$-0.684928\pi$
$379$	−21.3693	−1.09767	−0.548834	+	0.835931i	$0.684928\pi$
$380$	0	0
$381$	15.1231	0.774780
$382$	0	0
$383$	−3.12311	−0.159583	−0.0797916	−	0.996812i	$-0.525425\pi$
$383$	−3.12311	−0.159583	−0.0797916	+	0.996812i	$0.525425\pi$
$384$	0	0
$385$	−6.24621	−0.318336
$386$	0	0
$387$	−10.2462	−0.520844
$388$	0	0
$389$	22.4924	1.14041	0.570206	−	0.821502i	$-0.306864\pi$
$389$	22.4924	1.14041	0.570206	+	0.821502i	$0.306864\pi$
$390$	0	0
$391$	22.2462	1.12504
$392$	0	0
$393$	8.24621	0.415966
$394$	0	0
$395$	13.1231	0.660295
$396$	0	0
$397$	−9.36932	−0.470233	−0.235116	−	0.971967i	$-0.575547\pi$
$397$	−9.36932	−0.470233	−0.235116	+	0.971967i	$0.575547\pi$
$398$	0	0
$399$	9.75379	0.488300
$400$	0	0
$401$	−28.2462	−1.41055	−0.705274	−	0.708935i	$-0.749176\pi$
$401$	−28.2462	−1.41055	−0.705274	+	0.708935i	$0.749176\pi$
$402$	0	0
$403$	3.50758	0.174725
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	6.24621	0.309613
$408$	0	0
$409$	−14.4924	−0.716604	−0.358302	−	0.933606i	$-0.616644\pi$
$409$	−14.4924	−0.716604	−0.358302	+	0.933606i	$0.616644\pi$
$410$	0	0
$411$	11.1231	0.548662
$412$	0	0
$413$	18.7386	0.922068
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	−9.36932	−0.458817
$418$	0	0
$419$	16.2462	0.793679	0.396840	−	0.917888i	$-0.370107\pi$
$419$	16.2462	0.793679	0.396840	+	0.917888i	$0.370107\pi$
$420$	0	0
$421$	34.9848	1.70506	0.852529	−	0.522681i	$-0.175068\pi$
$421$	34.9848	1.70506	0.852529	+	0.522681i	$0.175068\pi$
$422$	0	0
$423$	−4.87689	−0.237123
$424$	0	0
$425$	7.12311	0.345521
$426$	0	0
$427$	−6.24621	−0.302275
$428$	0	0
$429$	6.24621	0.301570
$430$	0	0
$431$	−16.4924	−0.794412	−0.397206	−	0.917729i	$-0.630020\pi$
$431$	−16.4924	−0.794412	−0.397206	+	0.917729i	$0.630020\pi$
$432$	0	0
$433$	−16.2462	−0.780743	−0.390372	−	0.920657i	$-0.627653\pi$
$433$	−16.2462	−0.780743	−0.390372	+	0.920657i	$0.627653\pi$
$434$	0	0
$435$	−8.24621	−0.395376
$436$	0	0
$437$	9.75379	0.466587
$438$	0	0
$439$	−27.3693	−1.30627	−0.653133	−	0.757243i	$-0.726546\pi$
$439$	−27.3693	−1.30627	−0.653133	+	0.757243i	$0.726546\pi$
$440$	0	0
$441$	2.75379	0.131133
$442$	0	0
$443$	36.4924	1.73381	0.866904	−	0.498476i	$-0.166107\pi$
$443$	36.4924	1.73381	0.866904	+	0.498476i	$0.166107\pi$
$444$	0	0
$445$	10.0000	0.474045
$446$	0	0
$447$	3.75379	0.177548
$448$	0	0
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	4.00000	0.188353
$452$	0	0
$453$	7.36932	0.346241
$454$	0	0
$455$	9.75379	0.457265
$456$	0	0
$457$	−26.4924	−1.23926	−0.619632	−	0.784893i	$-0.712718\pi$
$457$	−26.4924	−1.23926	−0.619632	+	0.784893i	$0.712718\pi$
$458$	0	0
$459$	−7.12311	−0.332478
$460$	0	0
$461$	23.7538	1.10632	0.553162	−	0.833074i	$-0.313421\pi$
$461$	23.7538	1.10632	0.553162	+	0.833074i	$0.313421\pi$
$462$	0	0
$463$	28.8769	1.34202	0.671012	−	0.741447i	$-0.265860\pi$
$463$	28.8769	1.34202	0.671012	+	0.741447i	$0.265860\pi$
$464$	0	0
$465$	1.12311	0.0520828
$466$	0	0
$467$	−30.7386	−1.42241	−0.711207	−	0.702982i	$-0.751851\pi$
$467$	−30.7386	−1.42241	−0.711207	+	0.702982i	$0.751851\pi$
$468$	0	0
$469$	32.0000	1.47762
$470$	0	0
$471$	5.36932	0.247405
$472$	0	0
$473$	20.4924	0.942243
$474$	0	0
$475$	3.12311	0.143298
$476$	0	0
$477$	−10.0000	−0.457869
$478$	0	0
$479$	3.50758	0.160265	0.0801327	−	0.996784i	$-0.474466\pi$
$479$	3.50758	0.160265	0.0801327	+	0.996784i	$0.474466\pi$
$480$	0	0
$481$	−9.75379	−0.444734
$482$	0	0
$483$	9.75379	0.443813
$484$	0	0
$485$	−10.0000	−0.454077
$486$	0	0
$487$	27.6155	1.25138	0.625690	−	0.780072i	$-0.284818\pi$
$487$	27.6155	1.25138	0.625690	+	0.780072i	$0.284818\pi$
$488$	0	0
$489$	−24.4924	−1.10759
$490$	0	0
$491$	34.4924	1.55662	0.778311	−	0.627879i	$-0.216077\pi$
$491$	34.4924	1.55662	0.778311	+	0.627879i	$0.216077\pi$
$492$	0	0
$493$	−58.7386	−2.64546
$494$	0	0
$495$	2.00000	0.0898933
$496$	0	0
$497$	24.9848	1.12072
$498$	0	0
$499$	29.3693	1.31475	0.657376	−	0.753563i	$-0.271667\pi$
$499$	29.3693	1.31475	0.657376	+	0.753563i	$0.271667\pi$
$500$	0	0
$501$	19.1231	0.854357
$502$	0	0
$503$	39.6155	1.76637	0.883185	−	0.469026i	$-0.155395\pi$
$503$	39.6155	1.76637	0.883185	+	0.469026i	$0.155395\pi$
$504$	0	0
$505$	14.0000	0.622992
$506$	0	0
$507$	3.24621	0.144169
$508$	0	0
$509$	−14.0000	−0.620539	−0.310270	−	0.950649i	$-0.600419\pi$
$509$	−14.0000	−0.620539	−0.310270	+	0.950649i	$0.600419\pi$
$510$	0	0
$511$	−38.2462	−1.69191
$512$	0	0
$513$	−3.12311	−0.137888
$514$	0	0
$515$	−13.3693	−0.589122
$516$	0	0
$517$	9.75379	0.428971
$518$	0	0
$519$	14.0000	0.614532
$520$	0	0
$521$	10.0000	0.438108	0.219054	−	0.975713i	$-0.429703\pi$
$521$	10.0000	0.438108	0.219054	+	0.975713i	$0.429703\pi$
$522$	0	0
$523$	30.7386	1.34411	0.672053	−	0.740503i	$-0.265413\pi$
$523$	30.7386	1.34411	0.672053	+	0.740503i	$0.265413\pi$
$524$	0	0
$525$	3.12311	0.136304
$526$	0	0
$527$	8.00000	0.348485
$528$	0	0
$529$	−13.2462	−0.575922
$530$	0	0
$531$	−6.00000	−0.260378
$532$	0	0
$533$	−6.24621	−0.270553
$534$	0	0
$535$	−14.2462	−0.615917
$536$	0	0
$537$	4.24621	0.183238
$538$	0	0
$539$	−5.50758	−0.237228
$540$	0	0
$541$	−8.24621	−0.354532	−0.177266	−	0.984163i	$-0.556725\pi$
$541$	−8.24621	−0.354532	−0.177266	+	0.984163i	$0.556725\pi$
$542$	0	0
$543$	16.2462	0.697192
$544$	0	0
$545$	−12.2462	−0.524570
$546$	0	0
$547$	26.2462	1.12221	0.561103	−	0.827746i	$-0.310377\pi$
$547$	26.2462	1.12221	0.561103	+	0.827746i	$0.310377\pi$
$548$	0	0
$549$	2.00000	0.0853579
$550$	0	0
$551$	−25.7538	−1.09715
$552$	0	0
$553$	40.9848	1.74285
$554$	0	0
$555$	−3.12311	−0.132568
$556$	0	0
$557$	−34.4924	−1.46149	−0.730745	−	0.682650i	$-0.760827\pi$
$557$	−34.4924	−1.46149	−0.730745	+	0.682650i	$0.760827\pi$
$558$	0	0
$559$	−32.0000	−1.35346
$560$	0	0
$561$	14.2462	0.601476
$562$	0	0
$563$	−26.7386	−1.12690	−0.563450	−	0.826150i	$-0.690526\pi$
$563$	−26.7386	−1.12690	−0.563450	+	0.826150i	$0.690526\pi$
$564$	0	0
$565$	−15.1231	−0.636234
$566$	0	0
$567$	−3.12311	−0.131158
$568$	0	0
$569$	−24.7386	−1.03710	−0.518549	−	0.855048i	$-0.673528\pi$
$569$	−24.7386	−1.03710	−0.518549	+	0.855048i	$0.673528\pi$
$570$	0	0
$571$	−43.6155	−1.82525	−0.912627	−	0.408794i	$-0.865949\pi$
$571$	−43.6155	−1.82525	−0.912627	+	0.408794i	$0.865949\pi$
$572$	0	0
$573$	4.00000	0.167102
$574$	0	0
$575$	3.12311	0.130243
$576$	0	0
$577$	−22.9848	−0.956872	−0.478436	−	0.878123i	$-0.658796\pi$
$577$	−22.9848	−0.956872	−0.478436	+	0.878123i	$0.658796\pi$
$578$	0	0
$579$	−7.75379	−0.322236
$580$	0	0
$581$	−12.4924	−0.518273
$582$	0	0
$583$	20.0000	0.828315
$584$	0	0
$585$	−3.12311	−0.129125
$586$	0	0
$587$	−2.73863	−0.113036	−0.0565178	−	0.998402i	$-0.518000\pi$
$587$	−2.73863	−0.113036	−0.0565178	+	0.998402i	$0.518000\pi$
$588$	0	0
$589$	3.50758	0.144527
$590$	0	0
$591$	2.00000	0.0822690
$592$	0	0
$593$	−17.8617	−0.733494	−0.366747	−	0.930321i	$-0.619528\pi$
$593$	−17.8617	−0.733494	−0.366747	+	0.930321i	$0.619528\pi$
$594$	0	0
$595$	22.2462	0.912006
$596$	0	0
$597$	−21.6155	−0.884665
$598$	0	0
$599$	8.49242	0.346991	0.173495	−	0.984835i	$-0.444494\pi$
$599$	8.49242	0.346991	0.173495	+	0.984835i	$0.444494\pi$
$600$	0	0
$601$	34.9848	1.42706	0.713531	−	0.700624i	$-0.247095\pi$
$601$	34.9848	1.42706	0.713531	+	0.700624i	$0.247095\pi$
$602$	0	0
$603$	−10.2462	−0.417258
$604$	0	0
$605$	7.00000	0.284590
$606$	0	0
$607$	−4.87689	−0.197947	−0.0989735	−	0.995090i	$-0.531556\pi$
$607$	−4.87689	−0.197947	−0.0989735	+	0.995090i	$0.531556\pi$
$608$	0	0
$609$	−25.7538	−1.04360
$610$	0	0
$611$	−15.2311	−0.616183
$612$	0	0
$613$	−17.3693	−0.701540	−0.350770	−	0.936462i	$-0.614080\pi$
$613$	−17.3693	−0.701540	−0.350770	+	0.936462i	$0.614080\pi$
$614$	0	0
$615$	−2.00000	−0.0806478
$616$	0	0
$617$	−31.6155	−1.27279	−0.636397	−	0.771362i	$-0.719576\pi$
$617$	−31.6155	−1.27279	−0.636397	+	0.771362i	$0.719576\pi$
$618$	0	0
$619$	12.8769	0.517566	0.258783	−	0.965935i	$-0.416679\pi$
$619$	12.8769	0.517566	0.258783	+	0.965935i	$0.416679\pi$
$620$	0	0
$621$	−3.12311	−0.125326
$622$	0	0
$623$	31.2311	1.25125
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	6.24621	0.249450
$628$	0	0
$629$	−22.2462	−0.887015
$630$	0	0
$631$	43.8617	1.74611	0.873054	−	0.487623i	$-0.162136\pi$
$631$	43.8617	1.74611	0.873054	+	0.487623i	$0.162136\pi$
$632$	0	0
$633$	19.1231	0.760075
$634$	0	0
$635$	15.1231	0.600142
$636$	0	0
$637$	8.60037	0.340759
$638$	0	0
$639$	−8.00000	−0.316475
$640$	0	0
$641$	−19.7538	−0.780228	−0.390114	−	0.920767i	$-0.627564\pi$
$641$	−19.7538	−0.780228	−0.390114	+	0.920767i	$0.627564\pi$
$642$	0	0
$643$	−0.492423	−0.0194192	−0.00970962	−	0.999953i	$-0.503091\pi$
$643$	−0.492423	−0.0194192	−0.00970962	+	0.999953i	$0.503091\pi$
$644$	0	0
$645$	−10.2462	−0.403444
$646$	0	0
$647$	4.87689	0.191731	0.0958653	−	0.995394i	$-0.469438\pi$
$647$	4.87689	0.191731	0.0958653	+	0.995394i	$0.469438\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	3.50758	0.137473
$652$	0	0
$653$	4.73863	0.185437	0.0927185	−	0.995692i	$-0.470444\pi$
$653$	4.73863	0.185437	0.0927185	+	0.995692i	$0.470444\pi$
$654$	0	0
$655$	8.24621	0.322206
$656$	0	0
$657$	12.2462	0.477770
$658$	0	0
$659$	−26.0000	−1.01282	−0.506408	−	0.862294i	$-0.669027\pi$
$659$	−26.0000	−1.01282	−0.506408	+	0.862294i	$0.669027\pi$
$660$	0	0
$661$	2.00000	0.0777910	0.0388955	−	0.999243i	$-0.487616\pi$
$661$	2.00000	0.0777910	0.0388955	+	0.999243i	$0.487616\pi$
$662$	0	0
$663$	−22.2462	−0.863971
$664$	0	0
$665$	9.75379	0.378236
$666$	0	0
$667$	−25.7538	−0.997191
$668$	0	0
$669$	−23.1231	−0.893991
$670$	0	0
$671$	−4.00000	−0.154418
$672$	0	0
$673$	−3.75379	−0.144698	−0.0723490	−	0.997379i	$-0.523050\pi$
$673$	−3.75379	−0.144698	−0.0723490	+	0.997379i	$0.523050\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	44.7386	1.71945	0.859723	−	0.510761i	$-0.170636\pi$
$677$	44.7386	1.71945	0.859723	+	0.510761i	$0.170636\pi$
$678$	0	0
$679$	−31.2311	−1.19854
$680$	0	0
$681$	0	0
$682$	0	0
$683$	26.2462	1.00428	0.502142	−	0.864785i	$-0.332546\pi$
$683$	26.2462	1.00428	0.502142	+	0.864785i	$0.332546\pi$
$684$	0	0
$685$	11.1231	0.424992
$686$	0	0
$687$	−10.0000	−0.381524
$688$	0	0
$689$	−31.2311	−1.18981
$690$	0	0
$691$	4.87689	0.185526	0.0927629	−	0.995688i	$-0.470430\pi$
$691$	4.87689	0.185526	0.0927629	+	0.995688i	$0.470430\pi$
$692$	0	0
$693$	6.24621	0.237274
$694$	0	0
$695$	−9.36932	−0.355398
$696$	0	0
$697$	−14.2462	−0.539614
$698$	0	0
$699$	15.6155	0.590634
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	−9.75379	−0.367871
$704$	0	0
$705$	−4.87689	−0.183674
$706$	0	0
$707$	43.7235	1.64439
$708$	0	0
$709$	−14.9848	−0.562768	−0.281384	−	0.959595i	$-0.590793\pi$
$709$	−14.9848	−0.562768	−0.281384	+	0.959595i	$0.590793\pi$
$710$	0	0
$711$	−13.1231	−0.492155
$712$	0	0
$713$	3.50758	0.131360
$714$	0	0
$715$	6.24621	0.233595
$716$	0	0
$717$	−10.2462	−0.382652
$718$	0	0
$719$	5.75379	0.214580	0.107290	−	0.994228i	$-0.465783\pi$
$719$	5.75379	0.214580	0.107290	+	0.994228i	$0.465783\pi$
$720$	0	0
$721$	−41.7538	−1.55499
$722$	0	0
$723$	24.2462	0.901726
$724$	0	0
$725$	−8.24621	−0.306257
$726$	0	0
$727$	11.6155	0.430796	0.215398	−	0.976526i	$-0.430895\pi$
$727$	11.6155	0.430796	0.215398	+	0.976526i	$0.430895\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−72.9848	−2.69944
$732$	0	0
$733$	3.61553	0.133543	0.0667713	−	0.997768i	$-0.478730\pi$
$733$	3.61553	0.133543	0.0667713	+	0.997768i	$0.478730\pi$
$734$	0	0
$735$	2.75379	0.101575
$736$	0	0
$737$	20.4924	0.754848
$738$	0	0
$739$	24.8769	0.915111	0.457556	−	0.889181i	$-0.348725\pi$
$739$	24.8769	0.915111	0.457556	+	0.889181i	$0.348725\pi$
$740$	0	0
$741$	−9.75379	−0.358314
$742$	0	0
$743$	−37.8617	−1.38901	−0.694506	−	0.719487i	$-0.744377\pi$
$743$	−37.8617	−1.38901	−0.694506	+	0.719487i	$0.744377\pi$
$744$	0	0
$745$	3.75379	0.137528
$746$	0	0
$747$	4.00000	0.146352
$748$	0	0
$749$	−44.4924	−1.62572
$750$	0	0
$751$	−27.3693	−0.998721	−0.499360	−	0.866394i	$-0.666432\pi$
$751$	−27.3693	−0.998721	−0.499360	+	0.866394i	$0.666432\pi$
$752$	0	0
$753$	12.2462	0.446277
$754$	0	0
$755$	7.36932	0.268197
$756$	0	0
$757$	27.6155	1.00370	0.501852	−	0.864954i	$-0.332652\pi$
$757$	27.6155	1.00370	0.501852	+	0.864954i	$0.332652\pi$
$758$	0	0
$759$	6.24621	0.226723
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	−38.2462	−1.38461
$764$	0	0
$765$	−7.12311	−0.257536
$766$	0	0
$767$	−18.7386	−0.676613
$768$	0	0
$769$	48.2462	1.73980	0.869901	−	0.493226i	$-0.164182\pi$
$769$	48.2462	1.73980	0.869901	+	0.493226i	$0.164182\pi$
$770$	0	0
$771$	25.8617	0.931388
$772$	0	0
$773$	51.4773	1.85151	0.925754	−	0.378126i	$-0.123431\pi$
$773$	51.4773	1.85151	0.925754	+	0.378126i	$0.123431\pi$
$774$	0	0
$775$	1.12311	0.0403431
$776$	0	0
$777$	−9.75379	−0.349915
$778$	0	0
$779$	−6.24621	−0.223794
$780$	0	0
$781$	16.0000	0.572525
$782$	0	0
$783$	8.24621	0.294696
$784$	0	0
$785$	5.36932	0.191639
$786$	0	0
$787$	−21.7538	−0.775439	−0.387719	−	0.921777i	$-0.626737\pi$
$787$	−21.7538	−0.775439	−0.387719	+	0.921777i	$0.626737\pi$
$788$	0	0
$789$	9.36932	0.333557
$790$	0	0
$791$	−47.2311	−1.67934
$792$	0	0
$793$	6.24621	0.221809
$794$	0	0
$795$	−10.0000	−0.354663
$796$	0	0
$797$	−28.2462	−1.00053	−0.500266	−	0.865872i	$-0.666764\pi$
$797$	−28.2462	−1.00053	−0.500266	+	0.865872i	$0.666764\pi$
$798$	0	0
$799$	−34.7386	−1.22896
$800$	0	0
$801$	−10.0000	−0.353333
$802$	0	0
$803$	−24.4924	−0.864319
$804$	0	0
$805$	9.75379	0.343776
$806$	0	0
$807$	8.24621	0.290280
$808$	0	0
$809$	4.24621	0.149289	0.0746444	−	0.997210i	$-0.476218\pi$
$809$	4.24621	0.149289	0.0746444	+	0.997210i	$0.476218\pi$
$810$	0	0
$811$	8.87689	0.311710	0.155855	−	0.987780i	$-0.450187\pi$
$811$	8.87689	0.311710	0.155855	+	0.987780i	$0.450187\pi$
$812$	0	0
$813$	−1.12311	−0.0393890
$814$	0	0
$815$	−24.4924	−0.857932
$816$	0	0
$817$	−32.0000	−1.11954
$818$	0	0
$819$	−9.75379	−0.340825
$820$	0	0
$821$	24.7386	0.863384	0.431692	−	0.902021i	$-0.357917\pi$
$821$	24.7386	0.863384	0.431692	+	0.902021i	$0.357917\pi$
$822$	0	0
$823$	45.3693	1.58147	0.790737	−	0.612155i	$-0.209697\pi$
$823$	45.3693	1.58147	0.790737	+	0.612155i	$0.209697\pi$
$824$	0	0
$825$	2.00000	0.0696311
$826$	0	0
$827$	−56.4924	−1.96443	−0.982217	−	0.187749i	$-0.939881\pi$
$827$	−56.4924	−1.96443	−0.982217	+	0.187749i	$0.939881\pi$
$828$	0	0
$829$	−28.7386	−0.998134	−0.499067	−	0.866563i	$-0.666324\pi$
$829$	−28.7386	−0.998134	−0.499067	+	0.866563i	$0.666324\pi$
$830$	0	0
$831$	−23.1231	−0.802132
$832$	0	0
$833$	19.6155	0.679638
$834$	0	0
$835$	19.1231	0.661782
$836$	0	0
$837$	−1.12311	−0.0388202
$838$	0	0
$839$	44.0000	1.51905	0.759524	−	0.650479i	$-0.225432\pi$
$839$	44.0000	1.51905	0.759524	+	0.650479i	$0.225432\pi$
$840$	0	0
$841$	39.0000	1.34483
$842$	0	0
$843$	16.2462	0.559549
$844$	0	0
$845$	3.24621	0.111673
$846$	0	0
$847$	21.8617	0.751178
$848$	0	0
$849$	13.7538	0.472029
$850$	0	0
$851$	−9.75379	−0.334356
$852$	0	0
$853$	−12.8769	−0.440896	−0.220448	−	0.975399i	$-0.570752\pi$
$853$	−12.8769	−0.440896	−0.220448	+	0.975399i	$0.570752\pi$
$854$	0	0
$855$	−3.12311	−0.106808
$856$	0	0
$857$	39.1231	1.33642	0.668210	−	0.743973i	$-0.267061\pi$
$857$	39.1231	1.33642	0.668210	+	0.743973i	$0.267061\pi$
$858$	0	0
$859$	−16.8769	−0.575832	−0.287916	−	0.957656i	$-0.592962\pi$
$859$	−16.8769	−0.575832	−0.287916	+	0.957656i	$0.592962\pi$
$860$	0	0
$861$	−6.24621	−0.212870
$862$	0	0
$863$	−22.6307	−0.770357	−0.385179	−	0.922842i	$-0.625860\pi$
$863$	−22.6307	−0.770357	−0.385179	+	0.922842i	$0.625860\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	0	0
$867$	−33.7386	−1.14582
$868$	0	0
$869$	26.2462	0.890342
$870$	0	0
$871$	−32.0000	−1.08428
$872$	0	0
$873$	10.0000	0.338449
$874$	0	0
$875$	3.12311	0.105580
$876$	0	0
$877$	−12.3845	−0.418194	−0.209097	−	0.977895i	$-0.567052\pi$
$877$	−12.3845	−0.418194	−0.209097	+	0.977895i	$0.567052\pi$
$878$	0	0
$879$	20.7386	0.699497
$880$	0	0
$881$	18.4924	0.623026	0.311513	−	0.950242i	$-0.399164\pi$
$881$	18.4924	0.623026	0.311513	+	0.950242i	$0.399164\pi$
$882$	0	0
$883$	−32.4924	−1.09346	−0.546729	−	0.837310i	$-0.684127\pi$
$883$	−32.4924	−1.09346	−0.546729	+	0.837310i	$0.684127\pi$
$884$	0	0
$885$	−6.00000	−0.201688
$886$	0	0
$887$	−47.6155	−1.59877	−0.799386	−	0.600817i	$-0.794842\pi$
$887$	−47.6155	−1.59877	−0.799386	+	0.600817i	$0.794842\pi$
$888$	0	0
$889$	47.2311	1.58408
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	0	0
$893$	−15.2311	−0.509688
$894$	0	0
$895$	4.24621	0.141935
$896$	0	0
$897$	−9.75379	−0.325670
$898$	0	0
$899$	−9.26137	−0.308884
$900$	0	0
$901$	−71.2311	−2.37305
$902$	0	0
$903$	−32.0000	−1.06489
$904$	0	0
$905$	16.2462	0.540042
$906$	0	0
$907$	36.9848	1.22806	0.614031	−	0.789282i	$-0.289547\pi$
$907$	36.9848	1.22806	0.614031	+	0.789282i	$0.289547\pi$
$908$	0	0
$909$	−14.0000	−0.464351
$910$	0	0
$911$	58.7386	1.94610	0.973049	−	0.230599i	$-0.0740685\pi$
$911$	58.7386	1.94610	0.973049	+	0.230599i	$0.0740685\pi$
$912$	0	0
$913$	−8.00000	−0.264761
$914$	0	0
$915$	2.00000	0.0661180
$916$	0	0
$917$	25.7538	0.850465
$918$	0	0
$919$	−30.8769	−1.01853	−0.509267	−	0.860608i	$-0.670084\pi$
$919$	−30.8769	−1.01853	−0.509267	+	0.860608i	$0.670084\pi$
$920$	0	0
$921$	−2.24621	−0.0740152
$922$	0	0
$923$	−24.9848	−0.822386
$924$	0	0
$925$	−3.12311	−0.102687
$926$	0	0
$927$	13.3693	0.439106
$928$	0	0
$929$	−18.4924	−0.606717	−0.303358	−	0.952877i	$-0.598108\pi$
$929$	−18.4924	−0.606717	−0.303358	+	0.952877i	$0.598108\pi$
$930$	0	0
$931$	8.60037	0.281866
$932$	0	0
$933$	13.7538	0.450279
$934$	0	0
$935$	14.2462	0.465901
$936$	0	0
$937$	−14.0000	−0.457360	−0.228680	−	0.973502i	$-0.573441\pi$
$937$	−14.0000	−0.457360	−0.228680	+	0.973502i	$0.573441\pi$
$938$	0	0
$939$	28.7386	0.937850
$940$	0	0
$941$	55.4773	1.80851	0.904254	−	0.426996i	$-0.140428\pi$
$941$	55.4773	1.80851	0.904254	+	0.426996i	$0.140428\pi$
$942$	0	0
$943$	−6.24621	−0.203405
$944$	0	0
$945$	−3.12311	−0.101595
$946$	0	0
$947$	−22.7386	−0.738906	−0.369453	−	0.929249i	$-0.620455\pi$
$947$	−22.7386	−0.738906	−0.369453	+	0.929249i	$0.620455\pi$
$948$	0	0
$949$	38.2462	1.24152
$950$	0	0
$951$	2.00000	0.0648544
$952$	0	0
$953$	36.8769	1.19456	0.597280	−	0.802033i	$-0.296248\pi$
$953$	36.8769	1.19456	0.597280	+	0.802033i	$0.296248\pi$
$954$	0	0
$955$	4.00000	0.129437
$956$	0	0
$957$	−16.4924	−0.533124
$958$	0	0
$959$	34.7386	1.12177
$960$	0	0
$961$	−29.7386	−0.959311
$962$	0	0
$963$	14.2462	0.459078
$964$	0	0
$965$	−7.75379	−0.249603
$966$	0	0
$967$	26.6307	0.856385	0.428193	−	0.903688i	$-0.359151\pi$
$967$	26.6307	0.856385	0.428193	+	0.903688i	$0.359151\pi$
$968$	0	0
$969$	−22.2462	−0.714651
$970$	0	0
$971$	−15.7538	−0.505563	−0.252782	−	0.967523i	$-0.581345\pi$
$971$	−15.7538	−0.505563	−0.252782	+	0.967523i	$0.581345\pi$
$972$	0	0
$973$	−29.2614	−0.938076
$974$	0	0
$975$	−3.12311	−0.100019
$976$	0	0
$977$	39.1231	1.25166	0.625830	−	0.779960i	$-0.284761\pi$
$977$	39.1231	1.25166	0.625830	+	0.779960i	$0.284761\pi$
$978$	0	0
$979$	20.0000	0.639203
$980$	0	0
$981$	12.2462	0.390991
$982$	0	0
$983$	22.6307	0.721807	0.360903	−	0.932603i	$-0.382468\pi$
$983$	22.6307	0.721807	0.360903	+	0.932603i	$0.382468\pi$
$984$	0	0
$985$	2.00000	0.0637253
$986$	0	0
$987$	−15.2311	−0.484810
$988$	0	0
$989$	−32.0000	−1.01754
$990$	0	0
$991$	6.87689	0.218452	0.109226	−	0.994017i	$-0.465163\pi$
$991$	6.87689	0.218452	0.109226	+	0.994017i	$0.465163\pi$
$992$	0	0
$993$	7.12311	0.226045
$994$	0	0
$995$	−21.6155	−0.685258
$996$	0	0
$997$	−35.6155	−1.12796	−0.563978	−	0.825790i	$-0.690730\pi$
$997$	−35.6155	−1.12796	−0.563978	+	0.825790i	$0.690730\pi$
$998$	0	0
$999$	3.12311	0.0988107

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1920.2.a.y.1.1	✓	2
3.2	odd	2		5760.2.a.cj.1.1		2
4.3	odd	2		1920.2.a.ba.1.2	yes	2
5.4	even	2		9600.2.a.cw.1.2		2
8.3	odd	2		1920.2.a.z.1.2	yes	2
8.5	even	2		1920.2.a.bb.1.1	yes	2
12.11	even	2		5760.2.a.cg.1.2		2
16.3	odd	4		3840.2.k.bd.1921.3		4
16.5	even	4		3840.2.k.bc.1921.4		4
16.11	odd	4		3840.2.k.bd.1921.1		4
16.13	even	4		3840.2.k.bc.1921.2		4
20.19	odd	2		9600.2.a.ct.1.1		2
24.5	odd	2		5760.2.a.cc.1.1		2
24.11	even	2		5760.2.a.bx.1.2		2
40.19	odd	2		9600.2.a.dg.1.1		2
40.29	even	2		9600.2.a.cl.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1920.2.a.y.1.1	✓	2	1.1	even	1	trivial
1920.2.a.z.1.2	yes	2	8.3	odd	2
1920.2.a.ba.1.2	yes	2	4.3	odd	2
1920.2.a.bb.1.1	yes	2	8.5	even	2
3840.2.k.bc.1921.2		4	16.13	even	4
3840.2.k.bc.1921.4		4	16.5	even	4
3840.2.k.bd.1921.1		4	16.11	odd	4
3840.2.k.bd.1921.3		4	16.3	odd	4
5760.2.a.bx.1.2		2	24.11	even	2
5760.2.a.cc.1.1		2	24.5	odd	2
5760.2.a.cg.1.2		2	12.11	even	2
5760.2.a.cj.1.1		2	3.2	odd	2
9600.2.a.cl.1.2		2	40.29	even	2
9600.2.a.ct.1.1		2	20.19	odd	2
9600.2.a.cw.1.2		2	5.4	even	2
9600.2.a.dg.1.1		2	40.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1920 = 2^{7} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1920.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} -3.12311 q^{7} +1.00000 q^{9} -2.00000 q^{11} +3.12311 q^{13} +1.00000 q^{15} +7.12311 q^{17} +3.12311 q^{19} +3.12311 q^{21} +3.12311 q^{23} +1.00000 q^{25} -1.00000 q^{27} -8.24621 q^{29} +1.12311 q^{31} +2.00000 q^{33} +3.12311 q^{35} -3.12311 q^{37} -3.12311 q^{39} -2.00000 q^{41} -10.2462 q^{43} -1.00000 q^{45} -4.87689 q^{47} +2.75379 q^{49} -7.12311 q^{51} -10.0000 q^{53} +2.00000 q^{55} -3.12311 q^{57} -6.00000 q^{59} +2.00000 q^{61} -3.12311 q^{63} -3.12311 q^{65} -10.2462 q^{67} -3.12311 q^{69} -8.00000 q^{71} +12.2462 q^{73} -1.00000 q^{75} +6.24621 q^{77} -13.1231 q^{79} +1.00000 q^{81} +4.00000 q^{83} -7.12311 q^{85} +8.24621 q^{87} -10.0000 q^{89} -9.75379 q^{91} -1.12311 q^{93} -3.12311 q^{95} +10.0000 q^{97} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} - 4 q^{11} - 2 q^{13} + 2 q^{15} + 6 q^{17} - 2 q^{19} - 2 q^{21} - 2 q^{23} + 2 q^{25} - 2 q^{27} - 6 q^{31} + 4 q^{33} - 2 q^{35} + 2 q^{37} + 2 q^{39} - 4 q^{41}+ \cdots - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 - 4 * q^11 - 2 * q^13 + 2 * q^15 + 6 * q^17 - 2 * q^19 - 2 * q^21 - 2 * q^23 + 2 * q^25 - 2 * q^27 - 6 * q^31 + 4 * q^33 - 2 * q^35 + 2 * q^37 + 2 * q^39 - 4 * q^41 - 4 * q^43 - 2 * q^45 - 18 * q^47 + 22 * q^49 - 6 * q^51 - 20 * q^53 + 4 * q^55 + 2 * q^57 - 12 * q^59 + 4 * q^61 + 2 * q^63 + 2 * q^65 - 4 * q^67 + 2 * q^69 - 16 * q^71 + 8 * q^73 - 2 * q^75 - 4 * q^77 - 18 * q^79 + 2 * q^81 + 8 * q^83 - 6 * q^85 - 20 * q^89 - 36 * q^91 + 6 * q^93 + 2 * q^95 + 20 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more