LMFDB - Embedded newform 4160.2.a.bs.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4160,2,Mod(1,4160)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4160, 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("4160.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4160 = 2^{6} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4160.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:	$33.2177672409$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4752.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 3x^{2} + 4x + 1 $ x^4 - 2x^3 - 3x^2 + 4*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 2080)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.49551$ of defining polynomial
Character	$\chi$	$=$	4160.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.73205 q^{3} -1.00000 q^{5} -2.99102 q^{7} +4.46410 q^{9} +O(q^{10})$	$q-2.73205 q^{3} -1.00000 q^{5} -2.99102 q^{7} +4.46410 q^{9} -2.06939 q^{11} +1.00000 q^{13} +2.73205 q^{15} -0.189575 q^{17} -7.91264 q^{19} +8.17161 q^{21} +4.92163 q^{23} +1.00000 q^{25} -4.00000 q^{27} -5.84325 q^{29} -7.18717 q^{31} +5.65368 q^{33} +2.99102 q^{35} -2.32835 q^{37} -2.73205 q^{39} -5.79246 q^{41} -12.7141 q^{43} -4.46410 q^{45} +0.0628121 q^{47} +1.94617 q^{49} +0.517929 q^{51} -0.189575 q^{53} +2.06939 q^{55} +21.6177 q^{57} +7.91264 q^{59} -7.44613 q^{61} -13.3522 q^{63} -1.00000 q^{65} -6.47309 q^{67} -13.4461 q^{69} +4.63811 q^{71} -4.59985 q^{73} -2.73205 q^{75} +6.18958 q^{77} -12.7715 q^{79} -2.46410 q^{81} -6.99102 q^{83} +0.189575 q^{85} +15.9641 q^{87} -9.30735 q^{89} -2.99102 q^{91} +19.6357 q^{93} +7.91264 q^{95} +11.9820 q^{97} -9.23796 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 4 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 4 * q^5 + 4 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} - 4 q^{5} + 4 q^{7} + 4 q^{9} - 4 q^{11} + 4 q^{13} + 4 q^{15} + 4 q^{17} - 4 q^{19} - 4 q^{21} + 8 q^{23} + 4 q^{25} - 16 q^{27} + 8 q^{31} + 4 q^{33} - 4 q^{35} + 4 q^{37} - 4 q^{39} + 4 q^{41} - 12 q^{43} - 4 q^{45} + 12 q^{47} + 12 q^{49} - 16 q^{51} + 4 q^{53} + 4 q^{55} + 4 q^{57} + 4 q^{59} + 16 q^{61} + 4 q^{63} - 4 q^{65} - 28 q^{67} - 8 q^{69} - 4 q^{73} - 4 q^{75} + 20 q^{77} + 4 q^{81} - 12 q^{83} - 4 q^{85} + 4 q^{91} + 28 q^{93} + 4 q^{95} + 16 q^{97} - 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 4 * q^5 + 4 * q^7 + 4 * q^9 - 4 * q^11 + 4 * q^13 + 4 * q^15 + 4 * q^17 - 4 * q^19 - 4 * q^21 + 8 * q^23 + 4 * q^25 - 16 * q^27 + 8 * q^31 + 4 * q^33 - 4 * q^35 + 4 * q^37 - 4 * q^39 + 4 * q^41 - 12 * q^43 - 4 * q^45 + 12 * q^47 + 12 * q^49 - 16 * q^51 + 4 * q^53 + 4 * q^55 + 4 * q^57 + 4 * q^59 + 16 * q^61 + 4 * q^63 - 4 * q^65 - 28 * q^67 - 8 * q^69 - 4 * q^73 - 4 * q^75 + 20 * q^77 + 4 * q^81 - 12 * q^83 - 4 * q^85 + 4 * q^91 + 28 * q^93 + 4 * q^95 + 16 * q^97 - 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.73205	−1.57735	−0.788675	−	0.614810i	$-0.789233\pi$
$3$	−2.73205	−1.57735	−0.788675	+	0.614810i	$0.789233\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−2.99102	−1.13050	−0.565249	−	0.824921i	$-0.691220\pi$
$7$	−2.99102	−1.13050	−0.565249	+	0.824921i	$0.691220\pi$
$8$	0	0
$9$	4.46410	1.48803
$10$	0	0
$11$	−2.06939	−0.623944	−0.311972	−	0.950091i	$-0.600990\pi$
$11$	−2.06939	−0.623944	−0.311972	+	0.950091i	$0.600990\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	2.73205	0.705412
$16$	0	0
$17$	−0.189575	−0.0459787	−0.0229894	−	0.999736i	$-0.507318\pi$
$17$	−0.189575	−0.0459787	−0.0229894	+	0.999736i	$0.507318\pi$
$18$	0	0
$19$	−7.91264	−1.81528	−0.907642	−	0.419745i	$-0.862120\pi$
$19$	−7.91264	−1.81528	−0.907642	+	0.419745i	$0.862120\pi$
$20$	0	0
$21$	8.17161	1.78319
$22$	0	0
$23$	4.92163	1.02623	0.513115	−	0.858320i	$-0.328491\pi$
$23$	4.92163	1.02623	0.513115	+	0.858320i	$0.328491\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−4.00000	−0.769800
$28$	0	0
$29$	−5.84325	−1.08506	−0.542532	−	0.840035i	$-0.682534\pi$
$29$	−5.84325	−1.08506	−0.542532	+	0.840035i	$0.682534\pi$
$30$	0	0
$31$	−7.18717	−1.29085	−0.645427	−	0.763822i	$-0.723320\pi$
$31$	−7.18717	−1.29085	−0.645427	+	0.763822i	$0.723320\pi$
$32$	0	0
$33$	5.65368	0.984179
$34$	0	0
$35$	2.99102	0.505574
$36$	0	0
$37$	−2.32835	−0.382779	−0.191390	−	0.981514i	$-0.561299\pi$
$37$	−2.32835	−0.382779	−0.191390	+	0.981514i	$0.561299\pi$
$38$	0	0
$39$	−2.73205	−0.437478
$40$	0	0
$41$	−5.79246	−0.904630	−0.452315	−	0.891858i	$-0.649402\pi$
$41$	−5.79246	−0.904630	−0.452315	+	0.891858i	$0.649402\pi$
$42$	0	0
$43$	−12.7141	−1.93888	−0.969440	−	0.245330i	$-0.921104\pi$
$43$	−12.7141	−1.93888	−0.969440	+	0.245330i	$0.921104\pi$
$44$	0	0
$45$	−4.46410	−0.665469
$46$	0	0
$47$	0.0628121	0.00916208	0.00458104	−	0.999990i	$-0.498542\pi$
$47$	0.0628121	0.00916208	0.00458104	+	0.999990i	$0.498542\pi$
$48$	0	0
$49$	1.94617	0.278025
$50$	0	0
$51$	0.517929	0.0725246
$52$	0	0
$53$	−0.189575	−0.0260402	−0.0130201	−	0.999915i	$-0.504145\pi$
$53$	−0.189575	−0.0260402	−0.0130201	+	0.999915i	$0.504145\pi$
$54$	0	0
$55$	2.06939	0.279036
$56$	0	0
$57$	21.6177	2.86334
$58$	0	0
$59$	7.91264	1.03014	0.515069	−	0.857149i	$-0.327766\pi$
$59$	7.91264	1.03014	0.515069	+	0.857149i	$0.327766\pi$
$60$	0	0
$61$	−7.44613	−0.953380	−0.476690	−	0.879072i	$-0.658163\pi$
$61$	−7.44613	−0.953380	−0.476690	+	0.879072i	$0.658163\pi$
$62$	0	0
$63$	−13.3522	−1.68222
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−6.47309	−0.790813	−0.395407	−	0.918506i	$-0.629396\pi$
$67$	−6.47309	−0.790813	−0.395407	+	0.918506i	$0.629396\pi$
$68$	0	0
$69$	−13.4461	−1.61872
$70$	0	0
$71$	4.63811	0.550443	0.275221	−	0.961381i	$-0.411249\pi$
$71$	4.63811	0.550443	0.275221	+	0.961381i	$0.411249\pi$
$72$	0	0
$73$	−4.59985	−0.538372	−0.269186	−	0.963088i	$-0.586755\pi$
$73$	−4.59985	−0.538372	−0.269186	+	0.963088i	$0.586755\pi$
$74$	0	0
$75$	−2.73205	−0.315470
$76$	0	0
$77$	6.18958	0.705368
$78$	0	0
$79$	−12.7715	−1.43690	−0.718450	−	0.695578i	$-0.755148\pi$
$79$	−12.7715	−1.43690	−0.718450	+	0.695578i	$0.755148\pi$
$80$	0	0
$81$	−2.46410	−0.273789
$82$	0	0
$83$	−6.99102	−0.767364	−0.383682	−	0.923465i	$-0.625344\pi$
$83$	−6.99102	−0.767364	−0.383682	+	0.923465i	$0.625344\pi$
$84$	0	0
$85$	0.189575	0.0205623
$86$	0	0
$87$	15.9641	1.71153
$88$	0	0
$89$	−9.30735	−0.986578	−0.493289	−	0.869866i	$-0.664205\pi$
$89$	−9.30735	−0.986578	−0.493289	+	0.869866i	$0.664205\pi$
$90$	0	0
$91$	−2.99102	−0.313544
$92$	0	0
$93$	19.6357	2.03613
$94$	0	0
$95$	7.91264	0.811820
$96$	0	0
$97$	11.9820	1.21659	0.608295	−	0.793711i	$-0.291854\pi$
$97$	11.9820	1.21659	0.608295	+	0.793711i	$0.291854\pi$
$98$	0	0
$99$	−9.23796	−0.928450
$100$	0	0
$101$	13.5849	1.35175	0.675875	−	0.737017i	$-0.263766\pi$
$101$	13.5849	1.35175	0.675875	+	0.737017i	$0.263766\pi$
$102$	0	0
$103$	17.3139	1.70599	0.852996	−	0.521917i	$-0.174783\pi$
$103$	17.3139	1.70599	0.852996	+	0.521917i	$0.174783\pi$
$104$	0	0
$105$	−8.17161	−0.797467
$106$	0	0
$107$	−11.1112	−1.07416	−0.537080	−	0.843531i	$-0.680473\pi$
$107$	−11.1112	−1.07416	−0.537080	+	0.843531i	$0.680473\pi$
$108$	0	0
$109$	6.37915	0.611012	0.305506	−	0.952190i	$-0.401174\pi$
$109$	6.37915	0.611012	0.305506	+	0.952190i	$0.401174\pi$
$110$	0	0
$111$	6.36118	0.603777
$112$	0	0
$113$	−7.11778	−0.669584	−0.334792	−	0.942292i	$-0.608666\pi$
$113$	−7.11778	−0.669584	−0.334792	+	0.942292i	$0.608666\pi$
$114$	0	0
$115$	−4.92163	−0.458944
$116$	0	0
$117$	4.46410	0.412706
$118$	0	0
$119$	0.567022	0.0519788
$120$	0	0
$121$	−6.71763	−0.610693
$122$	0	0
$123$	15.8253	1.42692
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	1.61427	0.143244	0.0716218	−	0.997432i	$-0.477183\pi$
$127$	1.61427	0.143244	0.0716218	+	0.997432i	$0.477183\pi$
$128$	0	0
$129$	34.7355	3.05829
$130$	0	0
$131$	−2.01797	−0.176311	−0.0881554	−	0.996107i	$-0.528097\pi$
$131$	−2.01797	−0.176311	−0.0881554	+	0.996107i	$0.528097\pi$
$132$	0	0
$133$	23.6668	2.05217
$134$	0	0
$135$	4.00000	0.344265
$136$	0	0
$137$	−19.5669	−1.67172	−0.835858	−	0.548945i	$-0.815030\pi$
$137$	−19.5669	−1.67172	−0.835858	+	0.548945i	$0.815030\pi$
$138$	0	0
$139$	2.53590	0.215092	0.107546	−	0.994200i	$-0.465701\pi$
$139$	2.53590	0.215092	0.107546	+	0.994200i	$0.465701\pi$
$140$	0	0
$141$	−0.171606	−0.0144518
$142$	0	0
$143$	−2.06939	−0.173051
$144$	0	0
$145$	5.84325	0.485256
$146$	0	0
$147$	−5.31704	−0.438542
$148$	0	0
$149$	−12.9282	−1.05912	−0.529560	−	0.848273i	$-0.677643\pi$
$149$	−12.9282	−1.05912	−0.529560	+	0.848273i	$0.677643\pi$
$150$	0	0
$151$	−11.9455	−0.972108	−0.486054	−	0.873929i	$-0.661564\pi$
$151$	−11.9455	−0.972108	−0.486054	+	0.873929i	$0.661564\pi$
$152$	0	0
$153$	−0.846283	−0.0684179
$154$	0	0
$155$	7.18717	0.577287
$156$	0	0
$157$	9.62085	0.767827	0.383914	−	0.923369i	$-0.374576\pi$
$157$	9.62085	0.767827	0.383914	+	0.923369i	$0.374576\pi$
$158$	0	0
$159$	0.517929	0.0410744
$160$	0	0
$161$	−14.7207	−1.16015
$162$	0	0
$163$	24.7086	1.93533	0.967665	−	0.252238i	$-0.0811667\pi$
$163$	24.7086	1.93533	0.967665	+	0.252238i	$0.0811667\pi$
$164$	0	0
$165$	−5.65368	−0.440138
$166$	0	0
$167$	12.3163	0.953067	0.476533	−	0.879156i	$-0.341893\pi$
$167$	12.3163	0.953067	0.476533	+	0.879156i	$0.341893\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−35.3228	−2.70120
$172$	0	0
$173$	23.4969	1.78644	0.893219	−	0.449621i	$-0.148441\pi$
$173$	23.4969	1.78644	0.893219	+	0.449621i	$0.148441\pi$
$174$	0	0
$175$	−2.99102	−0.226100
$176$	0	0
$177$	−21.6177	−1.62489
$178$	0	0
$179$	−5.22373	−0.390440	−0.195220	−	0.980759i	$-0.562542\pi$
$179$	−5.22373	−0.390440	−0.195220	+	0.980759i	$0.562542\pi$
$180$	0	0
$181$	6.53590	0.485810	0.242905	−	0.970050i	$-0.421900\pi$
$181$	6.53590	0.485810	0.242905	+	0.970050i	$0.421900\pi$
$182$	0	0
$183$	20.3432	1.50381
$184$	0	0
$185$	2.32835	0.171184
$186$	0	0
$187$	0.392305	0.0286882
$188$	0	0
$189$	11.9641	0.870257
$190$	0	0
$191$	−24.3432	−1.76141	−0.880706	−	0.473662i	$-0.842932\pi$
$191$	−24.3432	−1.76141	−0.880706	+	0.473662i	$0.842932\pi$
$192$	0	0
$193$	−17.8384	−1.28404	−0.642019	−	0.766688i	$-0.721903\pi$
$193$	−17.8384	−1.28404	−0.642019	+	0.766688i	$0.721903\pi$
$194$	0	0
$195$	2.73205	0.195646
$196$	0	0
$197$	15.4773	1.10271	0.551354	−	0.834271i	$-0.314111\pi$
$197$	15.4773	1.10271	0.551354	+	0.834271i	$0.314111\pi$
$198$	0	0
$199$	5.70447	0.404379	0.202190	−	0.979346i	$-0.435194\pi$
$199$	5.70447	0.404379	0.202190	+	0.979346i	$0.435194\pi$
$200$	0	0
$201$	17.6848	1.24739
$202$	0	0
$203$	17.4773	1.22666
$204$	0	0
$205$	5.79246	0.404563
$206$	0	0
$207$	21.9706	1.52706
$208$	0	0
$209$	16.3743	1.13264
$210$	0	0
$211$	7.71054	0.530815	0.265408	−	0.964136i	$-0.414493\pi$
$211$	7.71054	0.530815	0.265408	+	0.964136i	$0.414493\pi$
$212$	0	0
$213$	−12.6716	−0.868241
$214$	0	0
$215$	12.7141	0.867093
$216$	0	0
$217$	21.4969	1.45931
$218$	0	0
$219$	12.5670	0.849201
$220$	0	0
$221$	−0.189575	−0.0127522
$222$	0	0
$223$	8.69549	0.582293	0.291146	−	0.956678i	$-0.405963\pi$
$223$	8.69549	0.582293	0.291146	+	0.956678i	$0.405963\pi$
$224$	0	0
$225$	4.46410	0.297607
$226$	0	0
$227$	−8.76861	−0.581993	−0.290997	−	0.956724i	$-0.593987\pi$
$227$	−8.76861	−0.581993	−0.290997	+	0.956724i	$0.593987\pi$
$228$	0	0
$229$	13.4790	0.890715	0.445357	−	0.895353i	$-0.353077\pi$
$229$	13.4790	0.890715	0.445357	+	0.895353i	$0.353077\pi$
$230$	0	0
$231$	−16.9102	−1.11261
$232$	0	0
$233$	25.5849	1.67612	0.838062	−	0.545576i	$-0.183689\pi$
$233$	25.5849	1.67612	0.838062	+	0.545576i	$0.183689\pi$
$234$	0	0
$235$	−0.0628121	−0.00409741
$236$	0	0
$237$	34.8923	2.26650
$238$	0	0
$239$	−8.27212	−0.535079	−0.267539	−	0.963547i	$-0.586211\pi$
$239$	−8.27212	−0.535079	−0.267539	+	0.963547i	$0.586211\pi$
$240$	0	0
$241$	−0.858190	−0.0552809	−0.0276405	−	0.999618i	$-0.508799\pi$
$241$	−0.858190	−0.0552809	−0.0276405	+	0.999618i	$0.508799\pi$
$242$	0	0
$243$	18.7321	1.20166
$244$	0	0
$245$	−1.94617	−0.124336
$246$	0	0
$247$	−7.91264	−0.503469
$248$	0	0
$249$	19.0998	1.21040
$250$	0	0
$251$	−15.0670	−0.951019	−0.475510	−	0.879710i	$-0.657736\pi$
$251$	−15.0670	−0.951019	−0.475510	+	0.879710i	$0.657736\pi$
$252$	0	0
$253$	−10.1848	−0.640310
$254$	0	0
$255$	−0.517929	−0.0324340
$256$	0	0
$257$	4.54905	0.283762	0.141881	−	0.989884i	$-0.454685\pi$
$257$	4.54905	0.283762	0.141881	+	0.989884i	$0.454685\pi$
$258$	0	0
$259$	6.96414	0.432731
$260$	0	0
$261$	−26.0849	−1.61461
$262$	0	0
$263$	6.14536	0.378939	0.189469	−	0.981887i	$-0.439323\pi$
$263$	6.14536	0.378939	0.189469	+	0.981887i	$0.439323\pi$
$264$	0	0
$265$	0.189575	0.0116455
$266$	0	0
$267$	25.4282	1.55618
$268$	0	0
$269$	3.70447	0.225866	0.112933	−	0.993603i	$-0.463975\pi$
$269$	3.70447	0.225866	0.112933	+	0.993603i	$0.463975\pi$
$270$	0	0
$271$	8.08425	0.491083	0.245542	−	0.969386i	$-0.421034\pi$
$271$	8.08425	0.491083	0.245542	+	0.969386i	$0.421034\pi$
$272$	0	0
$273$	8.17161	0.494568
$274$	0	0
$275$	−2.06939	−0.124789
$276$	0	0
$277$	−25.3533	−1.52333	−0.761667	−	0.647968i	$-0.775619\pi$
$277$	−25.3533	−1.52333	−0.761667	+	0.647968i	$0.775619\pi$
$278$	0	0
$279$	−32.0842	−1.92083
$280$	0	0
$281$	−5.03415	−0.300312	−0.150156	−	0.988662i	$-0.547978\pi$
$281$	−5.03415	−0.300312	−0.150156	+	0.988662i	$0.547978\pi$
$282$	0	0
$283$	24.7272	1.46988	0.734941	−	0.678131i	$-0.237210\pi$
$283$	24.7272	1.46988	0.734941	+	0.678131i	$0.237210\pi$
$284$	0	0
$285$	−21.6177	−1.28052
$286$	0	0
$287$	17.3253	1.02268
$288$	0	0
$289$	−16.9641	−0.997886
$290$	0	0
$291$	−32.7355	−1.91899
$292$	0	0
$293$	16.5639	0.967674	0.483837	−	0.875158i	$-0.339243\pi$
$293$	16.5639	0.967674	0.483837	+	0.875158i	$0.339243\pi$
$294$	0	0
$295$	−7.91264	−0.460692
$296$	0	0
$297$	8.27756	0.480313
$298$	0	0
$299$	4.92163	0.284625
$300$	0	0
$301$	38.0280	2.19190
$302$	0	0
$303$	−37.1147	−2.13218
$304$	0	0
$305$	7.44613	0.426364
$306$	0	0
$307$	−23.9012	−1.36412	−0.682058	−	0.731298i	$-0.738915\pi$
$307$	−23.9012	−1.36412	−0.682058	+	0.731298i	$0.738915\pi$
$308$	0	0
$309$	−47.3025	−2.69095
$310$	0	0
$311$	20.0132	1.13484	0.567421	−	0.823428i	$-0.307941\pi$
$311$	20.0132	1.13484	0.567421	+	0.823428i	$0.307941\pi$
$312$	0	0
$313$	30.4251	1.71973	0.859865	−	0.510521i	$-0.170548\pi$
$313$	30.4251	1.71973	0.859865	+	0.510521i	$0.170548\pi$
$314$	0	0
$315$	13.3522	0.752311
$316$	0	0
$317$	7.08665	0.398026	0.199013	−	0.979997i	$-0.436226\pi$
$317$	7.08665	0.398026	0.199013	+	0.979997i	$0.436226\pi$
$318$	0	0
$319$	12.0920	0.677020
$320$	0	0
$321$	30.3564	1.69433
$322$	0	0
$323$	1.50004	0.0834645
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	−17.4282	−0.963780
$328$	0	0
$329$	−0.187872	−0.0103577
$330$	0	0
$331$	15.1412	0.832235	0.416117	−	0.909311i	$-0.363391\pi$
$331$	15.1412	0.832235	0.416117	+	0.909311i	$0.363391\pi$
$332$	0	0
$333$	−10.3940	−0.569588
$334$	0	0
$335$	6.47309	0.353662
$336$	0	0
$337$	−0.846283	−0.0461000	−0.0230500	−	0.999734i	$-0.507338\pi$
$337$	−0.846283	−0.0461000	−0.0230500	+	0.999734i	$0.507338\pi$
$338$	0	0
$339$	19.4461	1.05617
$340$	0	0
$341$	14.8730	0.805421
$342$	0	0
$343$	15.1161	0.816191
$344$	0	0
$345$	13.4461	0.723915
$346$	0	0
$347$	−4.29775	−0.230715	−0.115358	−	0.993324i	$-0.536801\pi$
$347$	−4.29775	−0.230715	−0.115358	+	0.993324i	$0.536801\pi$
$348$	0	0
$349$	−14.9299	−0.799180	−0.399590	−	0.916694i	$-0.630847\pi$
$349$	−14.9299	−0.799180	−0.399590	+	0.916694i	$0.630847\pi$
$350$	0	0
$351$	−4.00000	−0.213504
$352$	0	0
$353$	−27.8713	−1.48344	−0.741719	−	0.670711i	$-0.765989\pi$
$353$	−27.8713	−1.48344	−0.741719	+	0.670711i	$0.765989\pi$
$354$	0	0
$355$	−4.63811	−0.246166
$356$	0	0
$357$	−1.54913	−0.0819888
$358$	0	0
$359$	32.0615	1.69214	0.846070	−	0.533072i	$-0.178963\pi$
$359$	32.0615	1.69214	0.846070	+	0.533072i	$0.178963\pi$
$360$	0	0
$361$	43.6099	2.29526
$362$	0	0
$363$	18.3529	0.963277
$364$	0	0
$365$	4.59985	0.240767
$366$	0	0
$367$	9.43955	0.492741	0.246370	−	0.969176i	$-0.420762\pi$
$367$	9.43955	0.492741	0.246370	+	0.969176i	$0.420762\pi$
$368$	0	0
$369$	−25.8581	−1.34612
$370$	0	0
$371$	0.567022	0.0294383
$372$	0	0
$373$	7.72850	0.400167	0.200083	−	0.979779i	$-0.435879\pi$
$373$	7.72850	0.400167	0.200083	+	0.979779i	$0.435879\pi$
$374$	0	0
$375$	2.73205	0.141082
$376$	0	0
$377$	−5.84325	−0.300943
$378$	0	0
$379$	19.9126	1.02284	0.511422	−	0.859330i	$-0.329119\pi$
$379$	19.9126	1.02284	0.511422	+	0.859330i	$0.329119\pi$
$380$	0	0
$381$	−4.41027	−0.225945
$382$	0	0
$383$	4.59390	0.234737	0.117369	−	0.993088i	$-0.462554\pi$
$383$	4.59390	0.234737	0.117369	+	0.993088i	$0.462554\pi$
$384$	0	0
$385$	−6.18958	−0.315450
$386$	0	0
$387$	−56.7570	−2.88512
$388$	0	0
$389$	−25.5609	−1.29599	−0.647994	−	0.761645i	$-0.724392\pi$
$389$	−25.5609	−1.29599	−0.647994	+	0.761645i	$0.724392\pi$
$390$	0	0
$391$	−0.933018	−0.0471848
$392$	0	0
$393$	5.51319	0.278104
$394$	0	0
$395$	12.7715	0.642602
$396$	0	0
$397$	−21.3222	−1.07013	−0.535066	−	0.844811i	$-0.679713\pi$
$397$	−21.3222	−1.07013	−0.535066	+	0.844811i	$0.679713\pi$
$398$	0	0
$399$	−64.6590	−3.23700
$400$	0	0
$401$	−1.93435	−0.0965966	−0.0482983	−	0.998833i	$-0.515380\pi$
$401$	−1.93435	−0.0965966	−0.0482983	+	0.998833i	$0.515380\pi$
$402$	0	0
$403$	−7.18717	−0.358018
$404$	0	0
$405$	2.46410	0.122442
$406$	0	0
$407$	4.81827	0.238833
$408$	0	0
$409$	−22.1357	−1.09454	−0.547269	−	0.836957i	$-0.684333\pi$
$409$	−22.1357	−1.09454	−0.547269	+	0.836957i	$0.684333\pi$
$410$	0	0
$411$	53.4579	2.63688
$412$	0	0
$413$	−23.6668	−1.16457
$414$	0	0
$415$	6.99102	0.343175
$416$	0	0
$417$	−6.92820	−0.339276
$418$	0	0
$419$	8.51793	0.416128	0.208064	−	0.978115i	$-0.433284\pi$
$419$	8.51793	0.416128	0.208064	+	0.978115i	$0.433284\pi$
$420$	0	0
$421$	16.8266	0.820079	0.410039	−	0.912068i	$-0.365515\pi$
$421$	16.8266	0.820079	0.410039	+	0.912068i	$0.365515\pi$
$422$	0	0
$423$	0.280400	0.0136335
$424$	0	0
$425$	−0.189575	−0.00919575
$426$	0	0
$427$	22.2715	1.07779
$428$	0	0
$429$	5.65368	0.272962
$430$	0	0
$431$	−38.0794	−1.83422	−0.917111	−	0.398632i	$-0.869485\pi$
$431$	−38.0794	−1.83422	−0.917111	+	0.398632i	$0.869485\pi$
$432$	0	0
$433$	−9.20576	−0.442401	−0.221200	−	0.975228i	$-0.570997\pi$
$433$	−9.20576	−0.442401	−0.221200	+	0.975228i	$0.570997\pi$
$434$	0	0
$435$	−15.9641	−0.765418
$436$	0	0
$437$	−38.9431	−1.86290
$438$	0	0
$439$	12.1279	0.578833	0.289417	−	0.957203i	$-0.406539\pi$
$439$	12.1279	0.578833	0.289417	+	0.957203i	$0.406539\pi$
$440$	0	0
$441$	8.68791	0.413710
$442$	0	0
$443$	17.5096	0.831905	0.415952	−	0.909386i	$-0.363448\pi$
$443$	17.5096	0.831905	0.415952	+	0.909386i	$0.363448\pi$
$444$	0	0
$445$	9.30735	0.441211
$446$	0	0
$447$	35.3205	1.67060
$448$	0	0
$449$	−39.2075	−1.85031	−0.925157	−	0.379584i	$-0.876067\pi$
$449$	−39.2075	−1.85031	−0.925157	+	0.379584i	$0.876067\pi$
$450$	0	0
$451$	11.9868	0.564439
$452$	0	0
$453$	32.6356	1.53336
$454$	0	0
$455$	2.99102	0.140221
$456$	0	0
$457$	−28.8682	−1.35040	−0.675199	−	0.737635i	$-0.735942\pi$
$457$	−28.8682	−1.35040	−0.675199	+	0.737635i	$0.735942\pi$
$458$	0	0
$459$	0.758301	0.0353944
$460$	0	0
$461$	−28.7863	−1.34071	−0.670356	−	0.742040i	$-0.733859\pi$
$461$	−28.7863	−1.34071	−0.670356	+	0.742040i	$0.733859\pi$
$462$	0	0
$463$	−17.7804	−0.826327	−0.413164	−	0.910657i	$-0.635576\pi$
$463$	−17.7804	−0.826327	−0.413164	+	0.910657i	$0.635576\pi$
$464$	0	0
$465$	−19.6357	−0.910584
$466$	0	0
$467$	17.1064	0.791589	0.395795	−	0.918339i	$-0.370469\pi$
$467$	17.1064	0.791589	0.395795	+	0.918339i	$0.370469\pi$
$468$	0	0
$469$	19.3611	0.894013
$470$	0	0
$471$	−26.2847	−1.21113
$472$	0	0
$473$	26.3104	1.20975
$474$	0	0
$475$	−7.91264	−0.363057
$476$	0	0
$477$	−0.846283	−0.0387486
$478$	0	0
$479$	−19.8307	−0.906089	−0.453044	−	0.891488i	$-0.649662\pi$
$479$	−19.8307	−0.906089	−0.453044	+	0.891488i	$0.649662\pi$
$480$	0	0
$481$	−2.32835	−0.106164
$482$	0	0
$483$	40.2176	1.82996
$484$	0	0
$485$	−11.9820	−0.544076
$486$	0	0
$487$	−37.2638	−1.68858	−0.844291	−	0.535885i	$-0.819978\pi$
$487$	−37.2638	−1.68858	−0.844291	+	0.535885i	$0.819978\pi$
$488$	0	0
$489$	−67.5053	−3.05269
$490$	0	0
$491$	−25.2762	−1.14070	−0.570350	−	0.821402i	$-0.693192\pi$
$491$	−25.2762	−1.14070	−0.570350	+	0.821402i	$0.693192\pi$
$492$	0	0
$493$	1.10774	0.0498899
$494$	0	0
$495$	9.23796	0.415216
$496$	0	0
$497$	−13.8727	−0.622274
$498$	0	0
$499$	36.5585	1.63658	0.818291	−	0.574804i	$-0.194922\pi$
$499$	36.5585	1.63658	0.818291	+	0.574804i	$0.194922\pi$
$500$	0	0
$501$	−33.6489	−1.50332
$502$	0	0
$503$	29.5315	1.31675	0.658373	−	0.752692i	$-0.271245\pi$
$503$	29.5315	1.31675	0.658373	+	0.752692i	$0.271245\pi$
$504$	0	0
$505$	−13.5849	−0.604521
$506$	0	0
$507$	−2.73205	−0.121335
$508$	0	0
$509$	−12.2793	−0.544269	−0.272134	−	0.962259i	$-0.587730\pi$
$509$	−12.2793	−0.544269	−0.272134	+	0.962259i	$0.587730\pi$
$510$	0	0
$511$	13.7582	0.608628
$512$	0	0
$513$	31.6506	1.39741
$514$	0	0
$515$	−17.3139	−0.762943
$516$	0	0
$517$	−0.129983	−0.00571663
$518$	0	0
$519$	−64.1948	−2.81784
$520$	0	0
$521$	29.1387	1.27659	0.638295	−	0.769792i	$-0.279640\pi$
$521$	29.1387	1.27659	0.638295	+	0.769792i	$0.279640\pi$
$522$	0	0
$523$	19.8827	0.869408	0.434704	−	0.900573i	$-0.356853\pi$
$523$	19.8827	0.869408	0.434704	+	0.900573i	$0.356853\pi$
$524$	0	0
$525$	8.17161	0.356638
$526$	0	0
$527$	1.36251	0.0593518
$528$	0	0
$529$	1.22240	0.0531479
$530$	0	0
$531$	35.3228	1.53288
$532$	0	0
$533$	−5.79246	−0.250899
$534$	0	0
$535$	11.1112	0.480379
$536$	0	0
$537$	14.2715	0.615860
$538$	0	0
$539$	−4.02739	−0.173472
$540$	0	0
$541$	27.3354	1.17524	0.587620	−	0.809137i	$-0.300065\pi$
$541$	27.3354	1.17524	0.587620	+	0.809137i	$0.300065\pi$
$542$	0	0
$543$	−17.8564	−0.766292
$544$	0	0
$545$	−6.37915	−0.273253
$546$	0	0
$547$	−18.6305	−0.796581	−0.398290	−	0.917259i	$-0.630396\pi$
$547$	−18.6305	−0.796581	−0.398290	+	0.917259i	$0.630396\pi$
$548$	0	0
$549$	−33.2403	−1.41866
$550$	0	0
$551$	46.2356	1.96970
$552$	0	0
$553$	38.1996	1.62441
$554$	0	0
$555$	−6.36118	−0.270017
$556$	0	0
$557$	12.5376	0.531235	0.265618	−	0.964078i	$-0.414424\pi$
$557$	12.5376	0.531235	0.265618	+	0.964078i	$0.414424\pi$
$558$	0	0
$559$	−12.7141	−0.537748
$560$	0	0
$561$	−1.07180	−0.0452513
$562$	0	0
$563$	−35.7439	−1.50642	−0.753212	−	0.657777i	$-0.771497\pi$
$563$	−35.7439	−1.50642	−0.753212	+	0.657777i	$0.771497\pi$
$564$	0	0
$565$	7.11778	0.299447
$566$	0	0
$567$	7.37017	0.309518
$568$	0	0
$569$	27.9413	1.17136	0.585680	−	0.810543i	$-0.300828\pi$
$569$	27.9413	1.17136	0.585680	+	0.810543i	$0.300828\pi$
$570$	0	0
$571$	31.6997	1.32659	0.663295	−	0.748358i	$-0.269158\pi$
$571$	31.6997	1.32659	0.663295	+	0.748358i	$0.269158\pi$
$572$	0	0
$573$	66.5069	2.77837
$574$	0	0
$575$	4.92163	0.205246
$576$	0	0
$577$	−16.5639	−0.689565	−0.344782	−	0.938683i	$-0.612047\pi$
$577$	−16.5639	−0.689565	−0.344782	+	0.938683i	$0.612047\pi$
$578$	0	0
$579$	48.7355	2.02538
$580$	0	0
$581$	20.9102	0.867503
$582$	0	0
$583$	0.392305	0.0162476
$584$	0	0
$585$	−4.46410	−0.184568
$586$	0	0
$587$	−5.42532	−0.223927	−0.111963	−	0.993712i	$-0.535714\pi$
$587$	−5.42532	−0.223927	−0.111963	+	0.993712i	$0.535714\pi$
$588$	0	0
$589$	56.8695	2.34327
$590$	0	0
$591$	−42.2847	−1.73936
$592$	0	0
$593$	3.56694	0.146477	0.0732384	−	0.997314i	$-0.476667\pi$
$593$	3.56694	0.146477	0.0732384	+	0.997314i	$0.476667\pi$
$594$	0	0
$595$	−0.567022	−0.0232456
$596$	0	0
$597$	−15.5849	−0.637848
$598$	0	0
$599$	34.4640	1.40816	0.704081	−	0.710120i	$-0.251359\pi$
$599$	34.4640	1.40816	0.704081	+	0.710120i	$0.251359\pi$
$600$	0	0
$601$	11.1615	0.455286	0.227643	−	0.973745i	$-0.426898\pi$
$601$	11.1615	0.455286	0.227643	+	0.973745i	$0.426898\pi$
$602$	0	0
$603$	−28.8965	−1.17676
$604$	0	0
$605$	6.71763	0.273110
$606$	0	0
$607$	3.87386	0.157235	0.0786176	−	0.996905i	$-0.474949\pi$
$607$	3.87386	0.157235	0.0786176	+	0.996905i	$0.474949\pi$
$608$	0	0
$609$	−47.7488	−1.93488
$610$	0	0
$611$	0.0628121	0.00254110
$612$	0	0
$613$	−17.2058	−0.694934	−0.347467	−	0.937692i	$-0.612958\pi$
$613$	−17.2058	−0.694934	−0.347467	+	0.937692i	$0.612958\pi$
$614$	0	0
$615$	−15.8253	−0.638137
$616$	0	0
$617$	36.4531	1.46755	0.733774	−	0.679393i	$-0.237757\pi$
$617$	36.4531	1.46755	0.733774	+	0.679393i	$0.237757\pi$
$618$	0	0
$619$	−23.4210	−0.941370	−0.470685	−	0.882301i	$-0.655993\pi$
$619$	−23.4210	−0.941370	−0.470685	+	0.882301i	$0.655993\pi$
$620$	0	0
$621$	−19.6865	−0.789992
$622$	0	0
$623$	27.8384	1.11532
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−44.7355	−1.78656
$628$	0	0
$629$	0.441398	0.0175997
$630$	0	0
$631$	−12.0842	−0.481066	−0.240533	−	0.970641i	$-0.577322\pi$
$631$	−12.0842	−0.481066	−0.240533	+	0.970641i	$0.577322\pi$
$632$	0	0
$633$	−21.0656	−0.837281
$634$	0	0
$635$	−1.61427	−0.0640604
$636$	0	0
$637$	1.94617	0.0771102
$638$	0	0
$639$	20.7050	0.819078
$640$	0	0
$641$	−13.4330	−0.530571	−0.265285	−	0.964170i	$-0.585466\pi$
$641$	−13.4330	−0.530571	−0.265285	+	0.964170i	$0.585466\pi$
$642$	0	0
$643$	−24.1188	−0.951154	−0.475577	−	0.879674i	$-0.657761\pi$
$643$	−24.1188	−0.951154	−0.475577	+	0.879674i	$0.657761\pi$
$644$	0	0
$645$	−34.7355	−1.36771
$646$	0	0
$647$	−35.4776	−1.39477	−0.697384	−	0.716697i	$-0.745653\pi$
$647$	−35.4776	−1.39477	−0.697384	+	0.716697i	$0.745653\pi$
$648$	0	0
$649$	−16.3743	−0.642749
$650$	0	0
$651$	−58.7307	−2.30184
$652$	0	0
$653$	−17.2714	−0.675883	−0.337941	−	0.941167i	$-0.609731\pi$
$653$	−17.2714	−0.675883	−0.337941	+	0.941167i	$0.609731\pi$
$654$	0	0
$655$	2.01797	0.0788486
$656$	0	0
$657$	−20.5342	−0.801115
$658$	0	0
$659$	−20.4820	−0.797865	−0.398933	−	0.916980i	$-0.630619\pi$
$659$	−20.4820	−0.797865	−0.398933	+	0.916980i	$0.630619\pi$
$660$	0	0
$661$	27.4773	1.06874	0.534371	−	0.845250i	$-0.320549\pi$
$661$	27.4773	1.06874	0.534371	+	0.845250i	$0.320549\pi$
$662$	0	0
$663$	0.517929	0.0201147
$664$	0	0
$665$	−23.6668	−0.917760
$666$	0	0
$667$	−28.7583	−1.11353
$668$	0	0
$669$	−23.7565	−0.918480
$670$	0	0
$671$	15.4089	0.594856
$672$	0	0
$673$	19.3594	0.746250	0.373125	−	0.927781i	$-0.378286\pi$
$673$	19.3594	0.746250	0.373125	+	0.927781i	$0.378286\pi$
$674$	0	0
$675$	−4.00000	−0.153960
$676$	0	0
$677$	−42.7683	−1.64372	−0.821860	−	0.569689i	$-0.807064\pi$
$677$	−42.7683	−1.64372	−0.821860	+	0.569689i	$0.807064\pi$
$678$	0	0
$679$	−35.8384	−1.37535
$680$	0	0
$681$	23.9563	0.918007
$682$	0	0
$683$	30.5268	1.16808	0.584038	−	0.811726i	$-0.301472\pi$
$683$	30.5268	1.16808	0.584038	+	0.811726i	$0.301472\pi$
$684$	0	0
$685$	19.5669	0.747614
$686$	0	0
$687$	−36.8252	−1.40497
$688$	0	0
$689$	−0.189575	−0.00722224
$690$	0	0
$691$	39.5107	1.50306	0.751529	−	0.659700i	$-0.229317\pi$
$691$	39.5107	1.50306	0.751529	+	0.659700i	$0.229317\pi$
$692$	0	0
$693$	27.6309	1.04961
$694$	0	0
$695$	−2.53590	−0.0961921
$696$	0	0
$697$	1.09811	0.0415937
$698$	0	0
$699$	−69.8993	−2.64383
$700$	0	0
$701$	38.8384	1.46690	0.733452	−	0.679741i	$-0.237908\pi$
$701$	38.8384	1.46690	0.733452	+	0.679741i	$0.237908\pi$
$702$	0	0
$703$	18.4234	0.694853
$704$	0	0
$705$	0.171606	0.00646305
$706$	0	0
$707$	−40.6327	−1.52815
$708$	0	0
$709$	−35.2075	−1.32224	−0.661122	−	0.750278i	$-0.729919\pi$
$709$	−35.2075	−1.32224	−0.661122	+	0.750278i	$0.729919\pi$
$710$	0	0
$711$	−57.0131	−2.13816
$712$	0	0
$713$	−35.3726	−1.32471
$714$	0	0
$715$	2.06939	0.0773908
$716$	0	0
$717$	22.5998	0.844007
$718$	0	0
$719$	0.818271	0.0305163	0.0152582	−	0.999884i	$-0.495143\pi$
$719$	0.818271	0.0305163	0.0152582	+	0.999884i	$0.495143\pi$
$720$	0	0
$721$	−51.7862	−1.92862
$722$	0	0
$723$	2.34462	0.0871974
$724$	0	0
$725$	−5.84325	−0.217013
$726$	0	0
$727$	−46.5783	−1.72749	−0.863746	−	0.503928i	$-0.831888\pi$
$727$	−46.5783	−1.72749	−0.863746	+	0.503928i	$0.831888\pi$
$728$	0	0
$729$	−43.7846	−1.62165
$730$	0	0
$731$	2.41027	0.0891472
$732$	0	0
$733$	−22.4711	−0.829990	−0.414995	−	0.909824i	$-0.636217\pi$
$733$	−22.4711	−0.829990	−0.414995	+	0.909824i	$0.636217\pi$
$734$	0	0
$735$	5.31704	0.196122
$736$	0	0
$737$	13.3953	0.493424
$738$	0	0
$739$	25.5024	0.938120	0.469060	−	0.883166i	$-0.344593\pi$
$739$	25.5024	0.938120	0.469060	+	0.883166i	$0.344593\pi$
$740$	0	0
$741$	21.6177	0.794147
$742$	0	0
$743$	−8.98495	−0.329626	−0.164813	−	0.986325i	$-0.552702\pi$
$743$	−8.98495	−0.329626	−0.164813	+	0.986325i	$0.552702\pi$
$744$	0	0
$745$	12.9282	0.473653
$746$	0	0
$747$	−31.2086	−1.14186
$748$	0	0
$749$	33.2338	1.21434
$750$	0	0
$751$	−10.8910	−0.397419	−0.198709	−	0.980058i	$-0.563675\pi$
$751$	−10.8910	−0.397419	−0.198709	+	0.980058i	$0.563675\pi$
$752$	0	0
$753$	41.1638	1.50009
$754$	0	0
$755$	11.9455	0.434740
$756$	0	0
$757$	54.2517	1.97181	0.985905	−	0.167306i	$-0.0535066\pi$
$757$	54.2517	1.97181	0.985905	+	0.167306i	$0.0535066\pi$
$758$	0	0
$759$	27.8253	1.00999
$760$	0	0
$761$	−24.5788	−0.890980	−0.445490	−	0.895287i	$-0.646970\pi$
$761$	−24.5788	−0.890980	−0.445490	+	0.895287i	$0.646970\pi$
$762$	0	0
$763$	−19.0801	−0.690747
$764$	0	0
$765$	0.846283	0.0305974
$766$	0	0
$767$	7.91264	0.285709
$768$	0	0
$769$	−2.79424	−0.100763	−0.0503814	−	0.998730i	$-0.516044\pi$
$769$	−2.79424	−0.100763	−0.0503814	+	0.998730i	$0.516044\pi$
$770$	0	0
$771$	−12.4282	−0.447592
$772$	0	0
$773$	−46.6356	−1.67737	−0.838683	−	0.544619i	$-0.816674\pi$
$773$	−46.6356	−1.67737	−0.838683	+	0.544619i	$0.816674\pi$
$774$	0	0
$775$	−7.18717	−0.258171
$776$	0	0
$777$	−19.0264	−0.682568
$778$	0	0
$779$	45.8336	1.64216
$780$	0	0
$781$	−9.59807	−0.343446
$782$	0	0
$783$	23.3730	0.835283
$784$	0	0
$785$	−9.62085	−0.343383
$786$	0	0
$787$	20.3404	0.725056	0.362528	−	0.931973i	$-0.381914\pi$
$787$	20.3404	0.725056	0.362528	+	0.931973i	$0.381914\pi$
$788$	0	0
$789$	−16.7894	−0.597719
$790$	0	0
$791$	21.2894	0.756963
$792$	0	0
$793$	−7.44613	−0.264420
$794$	0	0
$795$	−0.517929	−0.0183690
$796$	0	0
$797$	−32.1996	−1.14057	−0.570284	−	0.821447i	$-0.693167\pi$
$797$	−32.1996	−1.14057	−0.570284	+	0.821447i	$0.693167\pi$
$798$	0	0
$799$	−0.0119076	−0.000421261 0
$800$	0	0
$801$	−41.5490	−1.46806
$802$	0	0
$803$	9.51888	0.335914
$804$	0	0
$805$	14.7207	0.518835
$806$	0	0
$807$	−10.1208	−0.356269
$808$	0	0
$809$	−5.58973	−0.196524	−0.0982621	−	0.995161i	$-0.531328\pi$
$809$	−5.58973	−0.196524	−0.0982621	+	0.995161i	$0.531328\pi$
$810$	0	0
$811$	−29.2380	−1.02668	−0.513342	−	0.858184i	$-0.671593\pi$
$811$	−29.2380	−1.02668	−0.513342	+	0.858184i	$0.671593\pi$
$812$	0	0
$813$	−22.0866	−0.774610
$814$	0	0
$815$	−24.7086	−0.865506
$816$	0	0
$817$	100.602	3.51962
$818$	0	0
$819$	−13.3522	−0.466563
$820$	0	0
$821$	−21.6865	−0.756864	−0.378432	−	0.925629i	$-0.623537\pi$
$821$	−21.6865	−0.756864	−0.378432	+	0.925629i	$0.623537\pi$
$822$	0	0
$823$	−1.64058	−0.0571871	−0.0285935	−	0.999591i	$-0.509103\pi$
$823$	−1.64058	−0.0571871	−0.0285935	+	0.999591i	$0.509103\pi$
$824$	0	0
$825$	5.65368	0.196836
$826$	0	0
$827$	−23.2686	−0.809128	−0.404564	−	0.914510i	$-0.632577\pi$
$827$	−23.2686	−0.809128	−0.404564	+	0.914510i	$0.632577\pi$
$828$	0	0
$829$	−20.4523	−0.710337	−0.355168	−	0.934802i	$-0.615576\pi$
$829$	−20.4523	−0.710337	−0.355168	+	0.934802i	$0.615576\pi$
$830$	0	0
$831$	69.2666	2.40283
$832$	0	0
$833$	−0.368946	−0.0127832
$834$	0	0
$835$	−12.3163	−0.426224
$836$	0	0
$837$	28.7487	0.993699
$838$	0	0
$839$	53.0208	1.83048	0.915240	−	0.402908i	$-0.132001\pi$
$839$	53.0208	1.83048	0.915240	+	0.402908i	$0.132001\pi$
$840$	0	0
$841$	5.14359	0.177365
$842$	0	0
$843$	13.7536	0.473698
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	20.0925	0.690387
$848$	0	0
$849$	−67.5561	−2.31852
$850$	0	0
$851$	−11.4593	−0.392819
$852$	0	0
$853$	−36.8055	−1.26020	−0.630099	−	0.776515i	$-0.716986\pi$
$853$	−36.8055	−1.26020	−0.630099	+	0.776515i	$0.716986\pi$
$854$	0	0
$855$	35.3228	1.20802
$856$	0	0
$857$	−24.4772	−0.836124	−0.418062	−	0.908418i	$-0.637291\pi$
$857$	−24.4772	−0.836124	−0.418062	+	0.908418i	$0.637291\pi$
$858$	0	0
$859$	24.3804	0.831848	0.415924	−	0.909399i	$-0.363458\pi$
$859$	24.3804	0.831848	0.415924	+	0.909399i	$0.363458\pi$
$860$	0	0
$861$	−47.3337	−1.61313
$862$	0	0
$863$	13.0090	0.442831	0.221415	−	0.975180i	$-0.428932\pi$
$863$	13.0090	0.442831	0.221415	+	0.975180i	$0.428932\pi$
$864$	0	0
$865$	−23.4969	−0.798920
$866$	0	0
$867$	46.3467	1.57402
$868$	0	0
$869$	26.4291	0.896546
$870$	0	0
$871$	−6.47309	−0.219332
$872$	0	0
$873$	53.4890	1.81033
$874$	0	0
$875$	2.99102	0.101115
$876$	0	0
$877$	−25.7128	−0.868260	−0.434130	−	0.900850i	$-0.642944\pi$
$877$	−25.7128	−0.868260	−0.434130	+	0.900850i	$0.642944\pi$
$878$	0	0
$879$	−45.2534	−1.52636
$880$	0	0
$881$	−33.0227	−1.11256	−0.556282	−	0.830994i	$-0.687772\pi$
$881$	−33.0227	−1.11256	−0.556282	+	0.830994i	$0.687772\pi$
$882$	0	0
$883$	17.4855	0.588435	0.294218	−	0.955738i	$-0.404941\pi$
$883$	17.4855	0.588435	0.294218	+	0.955738i	$0.404941\pi$
$884$	0	0
$885$	21.6177	0.726672
$886$	0	0
$887$	−7.68827	−0.258147	−0.129073	−	0.991635i	$-0.541200\pi$
$887$	−7.68827	−0.258147	−0.129073	+	0.991635i	$0.541200\pi$
$888$	0	0
$889$	−4.82831	−0.161936
$890$	0	0
$891$	5.09919	0.170829
$892$	0	0
$893$	−0.497009	−0.0166318
$894$	0	0
$895$	5.22373	0.174610
$896$	0	0
$897$	−13.4461	−0.448953
$898$	0	0
$899$	41.9964	1.40066
$900$	0	0
$901$	0.0359387	0.00119729
$902$	0	0
$903$	−103.894	−3.45739
$904$	0	0
$905$	−6.53590	−0.217261
$906$	0	0
$907$	29.0704	0.965268	0.482634	−	0.875822i	$-0.339680\pi$
$907$	29.0704	0.965268	0.482634	+	0.875822i	$0.339680\pi$
$908$	0	0
$909$	60.6444	2.01145
$910$	0	0
$911$	−8.97589	−0.297384	−0.148692	−	0.988884i	$-0.547506\pi$
$911$	−8.97589	−0.297384	−0.148692	+	0.988884i	$0.547506\pi$
$912$	0	0
$913$	14.4671	0.478792
$914$	0	0
$915$	−20.3432	−0.672526
$916$	0	0
$917$	6.03578	0.199319
$918$	0	0
$919$	−33.7057	−1.11185	−0.555925	−	0.831233i	$-0.687636\pi$
$919$	−33.7057	−1.11185	−0.555925	+	0.831233i	$0.687636\pi$
$920$	0	0
$921$	65.2994	2.15169
$922$	0	0
$923$	4.63811	0.152665
$924$	0	0
$925$	−2.32835	−0.0765558
$926$	0	0
$927$	77.2911	2.53857
$928$	0	0
$929$	13.4430	0.441051	0.220526	−	0.975381i	$-0.429223\pi$
$929$	13.4430	0.441051	0.220526	+	0.975381i	$0.429223\pi$
$930$	0	0
$931$	−15.3994	−0.504694
$932$	0	0
$933$	−54.6770	−1.79004
$934$	0	0
$935$	−0.392305	−0.0128297
$936$	0	0
$937$	−51.7845	−1.69173	−0.845863	−	0.533400i	$-0.820914\pi$
$937$	−51.7845	−1.69173	−0.845863	+	0.533400i	$0.820914\pi$
$938$	0	0
$939$	−83.1230	−2.71262
$940$	0	0
$941$	31.9658	1.04205	0.521027	−	0.853540i	$-0.325549\pi$
$941$	31.9658	1.04205	0.521027	+	0.853540i	$0.325549\pi$
$942$	0	0
$943$	−28.5083	−0.928358
$944$	0	0
$945$	−11.9641	−0.389191
$946$	0	0
$947$	−24.8115	−0.806265	−0.403132	−	0.915142i	$-0.632079\pi$
$947$	−24.8115	−0.806265	−0.403132	+	0.915142i	$0.632079\pi$
$948$	0	0
$949$	−4.59985	−0.149317
$950$	0	0
$951$	−19.3611	−0.627827
$952$	0	0
$953$	22.2776	0.721641	0.360820	−	0.932635i	$-0.382497\pi$
$953$	22.2776	0.721641	0.360820	+	0.932635i	$0.382497\pi$
$954$	0	0
$955$	24.3432	0.787728
$956$	0	0
$957$	−33.0359	−1.06790
$958$	0	0
$959$	58.5250	1.88987
$960$	0	0
$961$	20.6554	0.666303
$962$	0	0
$963$	−49.6015	−1.59839
$964$	0	0
$965$	17.8384	0.574240
$966$	0	0
$967$	45.4550	1.46174	0.730868	−	0.682519i	$-0.239116\pi$
$967$	45.4550	1.46174	0.730868	+	0.682519i	$0.239116\pi$
$968$	0	0
$969$	−4.09819	−0.131653
$970$	0	0
$971$	−56.4233	−1.81071	−0.905354	−	0.424657i	$-0.860395\pi$
$971$	−56.4233	−1.81071	−0.905354	+	0.424657i	$0.860395\pi$
$972$	0	0
$973$	−7.58491	−0.243161
$974$	0	0
$975$	−2.73205	−0.0874957
$976$	0	0
$977$	1.25656	0.0402008	0.0201004	−	0.999798i	$-0.493601\pi$
$977$	1.25656	0.0402008	0.0201004	+	0.999798i	$0.493601\pi$
$978$	0	0
$979$	19.2605	0.615569
$980$	0	0
$981$	28.4772	0.909206
$982$	0	0
$983$	−19.0939	−0.608999	−0.304500	−	0.952512i	$-0.598489\pi$
$983$	−19.0939	−0.608999	−0.304500	+	0.952512i	$0.598489\pi$
$984$	0	0
$985$	−15.4773	−0.493146
$986$	0	0
$987$	0.513276	0.0163377
$988$	0	0
$989$	−62.5740	−1.98974
$990$	0	0
$991$	22.6410	0.719216	0.359608	−	0.933104i	$-0.382910\pi$
$991$	22.6410	0.719216	0.359608	+	0.933104i	$0.382910\pi$
$992$	0	0
$993$	−41.3665	−1.31273
$994$	0	0
$995$	−5.70447	−0.180844
$996$	0	0
$997$	6.33317	0.200573	0.100287	−	0.994959i	$-0.468024\pi$
$997$	6.33317	0.200573	0.100287	+	0.994959i	$0.468024\pi$
$998$	0	0
$999$	9.31342	0.294663

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4160.2.a.bs.1.1		4
4.3	odd	2		4160.2.a.bx.1.4		4
8.3	odd	2		2080.2.a.o.1.2	✓	4
8.5	even	2		2080.2.a.t.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2080.2.a.o.1.2	✓	4	8.3	odd	2
2080.2.a.t.1.3	yes	4	8.5	even	2
4160.2.a.bs.1.1		4	1.1	even	1	trivial
4160.2.a.bx.1.4		4	4.3	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:	\( N \)	\(=\)	\( 4160 = 2^{6} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4160.a (trivial)

\(f(q)\)	\(=\)	\(q-2.73205 q^{3} -1.00000 q^{5} -2.99102 q^{7} +4.46410 q^{9} +O(q^{10})\)	\(q-2.73205 q^{3} -1.00000 q^{5} -2.99102 q^{7} +4.46410 q^{9} -2.06939 q^{11} +1.00000 q^{13} +2.73205 q^{15} -0.189575 q^{17} -7.91264 q^{19} +8.17161 q^{21} +4.92163 q^{23} +1.00000 q^{25} -4.00000 q^{27} -5.84325 q^{29} -7.18717 q^{31} +5.65368 q^{33} +2.99102 q^{35} -2.32835 q^{37} -2.73205 q^{39} -5.79246 q^{41} -12.7141 q^{43} -4.46410 q^{45} +0.0628121 q^{47} +1.94617 q^{49} +0.517929 q^{51} -0.189575 q^{53} +2.06939 q^{55} +21.6177 q^{57} +7.91264 q^{59} -7.44613 q^{61} -13.3522 q^{63} -1.00000 q^{65} -6.47309 q^{67} -13.4461 q^{69} +4.63811 q^{71} -4.59985 q^{73} -2.73205 q^{75} +6.18958 q^{77} -12.7715 q^{79} -2.46410 q^{81} -6.99102 q^{83} +0.189575 q^{85} +15.9641 q^{87} -9.30735 q^{89} -2.99102 q^{91} +19.6357 q^{93} +7.91264 q^{95} +11.9820 q^{97} -9.23796 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 4 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 4 * q^5 + 4 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} - 4 q^{5} + 4 q^{7} + 4 q^{9} - 4 q^{11} + 4 q^{13} + 4 q^{15} + 4 q^{17} - 4 q^{19} - 4 q^{21} + 8 q^{23} + 4 q^{25} - 16 q^{27} + 8 q^{31} + 4 q^{33} - 4 q^{35} + 4 q^{37} - 4 q^{39} + 4 q^{41} - 12 q^{43} - 4 q^{45} + 12 q^{47} + 12 q^{49} - 16 q^{51} + 4 q^{53} + 4 q^{55} + 4 q^{57} + 4 q^{59} + 16 q^{61} + 4 q^{63} - 4 q^{65} - 28 q^{67} - 8 q^{69} - 4 q^{73} - 4 q^{75} + 20 q^{77} + 4 q^{81} - 12 q^{83} - 4 q^{85} + 4 q^{91} + 28 q^{93} + 4 q^{95} + 16 q^{97} - 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 4 * q^5 + 4 * q^7 + 4 * q^9 - 4 * q^11 + 4 * q^13 + 4 * q^15 + 4 * q^17 - 4 * q^19 - 4 * q^21 + 8 * q^23 + 4 * q^25 - 16 * q^27 + 8 * q^31 + 4 * q^33 - 4 * q^35 + 4 * q^37 - 4 * q^39 + 4 * q^41 - 12 * q^43 - 4 * q^45 + 12 * q^47 + 12 * q^49 - 16 * q^51 + 4 * q^53 + 4 * q^55 + 4 * q^57 + 4 * q^59 + 16 * q^61 + 4 * q^63 - 4 * q^65 - 28 * q^67 - 8 * q^69 - 4 * q^73 - 4 * q^75 + 20 * q^77 + 4 * q^81 - 12 * q^83 - 4 * q^85 + 4 * q^91 + 28 * q^93 + 4 * q^95 + 16 * q^97 - 4 * q^99

Introduction

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