LMFDB - Embedded newform 3720.2.a.v.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3720,2,Mod(1,3720)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3720.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3720, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3720.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,0,5,0,-5,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$29.7043495519$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.2294036.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 8x^{3} + 8x^{2} + 6x - 4 $ x^5 - x^4 - 8x^3 + 8x^2 + 6*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.36670$ of defining polynomial
Character	$\chi$	$=$	3720.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +2.13213 q^{7} +1.00000 q^{9} -3.64738 q^{11} +4.85276 q^{13} -1.00000 q^{15} +1.80665 q^{17} +4.72063 q^{19} +2.13213 q^{21} +6.38079 q^{23} +1.00000 q^{25} +1.00000 q^{27} -4.65940 q^{29} +1.00000 q^{31} -3.64738 q^{33} -2.13213 q^{35} -0.588501 q^{37} +4.85276 q^{39} -8.90806 q^{41} -6.18743 q^{43} -1.00000 q^{45} -2.73340 q^{47} -2.45403 q^{49} +1.80665 q^{51} +10.9208 q^{53} +3.64738 q^{55} +4.72063 q^{57} -5.58616 q^{59} +4.92676 q^{61} +2.13213 q^{63} -4.85276 q^{65} +13.9088 q^{67} +6.38079 q^{69} -14.5710 q^{71} -2.67218 q^{73} +1.00000 q^{75} -7.77669 q^{77} +14.4845 q^{79} +1.00000 q^{81} +10.0282 q^{83} -1.80665 q^{85} -4.65940 q^{87} +0.747764 q^{89} +10.3467 q^{91} +1.00000 q^{93} -4.72063 q^{95} +17.1723 q^{97} -3.64738 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} - 5 q^{5} + 3 q^{7} + 5 q^{9} + 5 q^{11} + 2 q^{13} - 5 q^{15} + 6 q^{17} + 9 q^{19} + 3 q^{21} - 3 q^{23} + 5 q^{25} + 5 q^{27} + 2 q^{29} + 5 q^{31} + 5 q^{33} - 3 q^{35} + 4 q^{37} + 2 q^{39}+ \cdots + 5 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 - 5 * q^5 + 3 * q^7 + 5 * q^9 + 5 * q^11 + 2 * q^13 - 5 * q^15 + 6 * q^17 + 9 * q^19 + 3 * q^21 - 3 * q^23 + 5 * q^25 + 5 * q^27 + 2 * q^29 + 5 * q^31 + 5 * q^33 - 3 * q^35 + 4 * q^37 + 2 * q^39 + 8 * q^41 + 7 * q^43 - 5 * q^45 - 2 * q^47 + 14 * q^49 + 6 * q^51 + 5 * q^53 - 5 * q^55 + 9 * q^57 + 6 * q^59 + 16 * q^61 + 3 * q^63 - 2 * q^65 + 22 * q^67 - 3 * q^69 - 9 * q^71 + 9 * q^73 + 5 * q^75 + q^77 + 15 * q^79 + 5 * q^81 - 8 * q^83 - 6 * q^85 + 2 * q^87 + 17 * q^89 + 34 * q^91 + 5 * q^93 - 9 * q^95 + 18 * q^97 + 5 * 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
$7$	2.13213	0.805869	0.402934	−	0.915229i	$-0.367990\pi$
$7$	2.13213	0.805869	0.402934	+	0.915229i	$0.367990\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.64738	−1.09973	−0.549864	−	0.835254i	$-0.685320\pi$
$11$	−3.64738	−1.09973	−0.549864	+	0.835254i	$0.685320\pi$
$12$	0	0
$13$	4.85276	1.34591	0.672956	−	0.739682i	$-0.265024\pi$
$13$	4.85276	1.34591	0.672956	+	0.739682i	$0.265024\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	1.80665	0.438176	0.219088	−	0.975705i	$-0.429692\pi$
$17$	1.80665	0.438176	0.219088	+	0.975705i	$0.429692\pi$
$18$	0	0
$19$	4.72063	1.08299	0.541493	−	0.840705i	$-0.317859\pi$
$19$	4.72063	1.08299	0.541493	+	0.840705i	$0.317859\pi$
$20$	0	0
$21$	2.13213	0.465268
$22$	0	0
$23$	6.38079	1.33049	0.665243	−	0.746627i	$-0.268328\pi$
$23$	6.38079	1.33049	0.665243	+	0.746627i	$0.268328\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−4.65940	−0.865230	−0.432615	−	0.901579i	$-0.642409\pi$
$29$	−4.65940	−0.865230	−0.432615	+	0.901579i	$0.642409\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	−3.64738	−0.634928
$34$	0	0
$35$	−2.13213	−0.360395
$36$	0	0
$37$	−0.588501	−0.0967490	−0.0483745	−	0.998829i	$-0.515404\pi$
$37$	−0.588501	−0.0967490	−0.0483745	+	0.998829i	$0.515404\pi$
$38$	0	0
$39$	4.85276	0.777063
$40$	0	0
$41$	−8.90806	−1.39121	−0.695603	−	0.718427i	$-0.744863\pi$
$41$	−8.90806	−1.39121	−0.695603	+	0.718427i	$0.744863\pi$
$42$	0	0
$43$	−6.18743	−0.943575	−0.471787	−	0.881712i	$-0.656391\pi$
$43$	−6.18743	−0.943575	−0.471787	+	0.881712i	$0.656391\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−2.73340	−0.398708	−0.199354	−	0.979928i	$-0.563884\pi$
$47$	−2.73340	−0.398708	−0.199354	+	0.979928i	$0.563884\pi$
$48$	0	0
$49$	−2.45403	−0.350576
$50$	0	0
$51$	1.80665	0.252981
$52$	0	0
$53$	10.9208	1.50009	0.750046	−	0.661386i	$-0.230031\pi$
$53$	10.9208	1.50009	0.750046	+	0.661386i	$0.230031\pi$
$54$	0	0
$55$	3.64738	0.491813
$56$	0	0
$57$	4.72063	0.625263
$58$	0	0
$59$	−5.58616	−0.727256	−0.363628	−	0.931544i	$-0.618462\pi$
$59$	−5.58616	−0.727256	−0.363628	+	0.931544i	$0.618462\pi$
$60$	0	0
$61$	4.92676	0.630806	0.315403	−	0.948958i	$-0.397860\pi$
$61$	4.92676	0.630806	0.315403	+	0.948958i	$0.397860\pi$
$62$	0	0
$63$	2.13213	0.268623
$64$	0	0
$65$	−4.85276	−0.601910
$66$	0	0
$67$	13.9088	1.69923	0.849616	−	0.527401i	$-0.176834\pi$
$67$	13.9088	1.69923	0.849616	+	0.527401i	$0.176834\pi$
$68$	0	0
$69$	6.38079	0.768156
$70$	0	0
$71$	−14.5710	−1.72926	−0.864632	−	0.502405i	$-0.832449\pi$
$71$	−14.5710	−1.72926	−0.864632	+	0.502405i	$0.832449\pi$
$72$	0	0
$73$	−2.67218	−0.312755	−0.156377	−	0.987697i	$-0.549982\pi$
$73$	−2.67218	−0.312755	−0.156377	+	0.987697i	$0.549982\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−7.77669	−0.886236
$78$	0	0
$79$	14.4845	1.62964	0.814819	−	0.579715i	$-0.196836\pi$
$79$	14.4845	1.62964	0.814819	+	0.579715i	$0.196836\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	10.0282	1.10073	0.550367	−	0.834923i	$-0.314488\pi$
$83$	10.0282	1.10073	0.550367	+	0.834923i	$0.314488\pi$
$84$	0	0
$85$	−1.80665	−0.195958
$86$	0	0
$87$	−4.65940	−0.499541
$88$	0	0
$89$	0.747764	0.0792629	0.0396314	−	0.999214i	$-0.487382\pi$
$89$	0.747764	0.0792629	0.0396314	+	0.999214i	$0.487382\pi$
$90$	0	0
$91$	10.3467	1.08463
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	−4.72063	−0.484326
$96$	0	0
$97$	17.1723	1.74358	0.871792	−	0.489876i	$-0.162958\pi$
$97$	17.1723	1.74358	0.871792	+	0.489876i	$0.162958\pi$
$98$	0	0
$99$	−3.64738	−0.366576
$100$	0	0
$101$	17.8929	1.78041	0.890207	−	0.455556i	$-0.150559\pi$
$101$	17.8929	1.78041	0.890207	+	0.455556i	$0.150559\pi$
$102$	0	0
$103$	6.85276	0.675222	0.337611	−	0.941286i	$-0.390381\pi$
$103$	6.85276	0.675222	0.337611	+	0.941286i	$0.390381\pi$
$104$	0	0
$105$	−2.13213	−0.208074
$106$	0	0
$107$	18.0978	1.74958	0.874792	−	0.484499i	$-0.160998\pi$
$107$	18.0978	1.74958	0.874792	+	0.484499i	$0.160998\pi$
$108$	0	0
$109$	11.7639	1.12678	0.563389	−	0.826192i	$-0.309497\pi$
$109$	11.7639	1.12678	0.563389	+	0.826192i	$0.309497\pi$
$110$	0	0
$111$	−0.588501	−0.0558581
$112$	0	0
$113$	9.75114	0.917310	0.458655	−	0.888614i	$-0.348331\pi$
$113$	9.75114	0.917310	0.458655	+	0.888614i	$0.348331\pi$
$114$	0	0
$115$	−6.38079	−0.595011
$116$	0	0
$117$	4.85276	0.448638
$118$	0	0
$119$	3.85200	0.353113
$120$	0	0
$121$	2.30341	0.209401
$122$	0	0
$123$	−8.90806	−0.803213
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−16.7429	−1.48569	−0.742845	−	0.669463i	$-0.766524\pi$
$127$	−16.7429	−1.48569	−0.742845	+	0.669463i	$0.766524\pi$
$128$	0	0
$129$	−6.18743	−0.544773
$130$	0	0
$131$	−5.58616	−0.488065	−0.244032	−	0.969767i	$-0.578470\pi$
$131$	−5.58616	−0.488065	−0.244032	+	0.969767i	$0.578470\pi$
$132$	0	0
$133$	10.0650	0.872745
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−2.07090	−0.176929	−0.0884646	−	0.996079i	$-0.528196\pi$
$137$	−2.07090	−0.176929	−0.0884646	+	0.996079i	$0.528196\pi$
$138$	0	0
$139$	−15.5403	−1.31811	−0.659057	−	0.752093i	$-0.729044\pi$
$139$	−15.5403	−1.31811	−0.659057	+	0.752093i	$0.729044\pi$
$140$	0	0
$141$	−2.73340	−0.230194
$142$	0	0
$143$	−17.6999	−1.48014
$144$	0	0
$145$	4.65940	0.386942
$146$	0	0
$147$	−2.45403	−0.202405
$148$	0	0
$149$	−13.8929	−1.13815	−0.569077	−	0.822284i	$-0.692699\pi$
$149$	−13.8929	−1.13815	−0.569077	+	0.822284i	$0.692699\pi$
$150$	0	0
$151$	−18.1700	−1.47865	−0.739326	−	0.673348i	$-0.764856\pi$
$151$	−18.1700	−1.47865	−0.739326	+	0.673348i	$0.764856\pi$
$152$	0	0
$153$	1.80665	0.146059
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	0	0
$157$	−7.80073	−0.622566	−0.311283	−	0.950317i	$-0.600759\pi$
$157$	−7.80073	−0.622566	−0.311283	+	0.950317i	$0.600759\pi$
$158$	0	0
$159$	10.9208	0.866079
$160$	0	0
$161$	13.6047	1.07220
$162$	0	0
$163$	24.1971	1.89526	0.947632	−	0.319364i	$-0.103469\pi$
$163$	24.1971	1.89526	0.947632	+	0.319364i	$0.103469\pi$
$164$	0	0
$165$	3.64738	0.283948
$166$	0	0
$167$	22.1687	1.71547	0.857734	−	0.514093i	$-0.171871\pi$
$167$	22.1687	1.71547	0.857734	+	0.514093i	$0.171871\pi$
$168$	0	0
$169$	10.5492	0.811480
$170$	0	0
$171$	4.72063	0.360996
$172$	0	0
$173$	−5.61329	−0.426771	−0.213385	−	0.976968i	$-0.568449\pi$
$173$	−5.61329	−0.426771	−0.213385	+	0.976968i	$0.568449\pi$
$174$	0	0
$175$	2.13213	0.161174
$176$	0	0
$177$	−5.58616	−0.419881
$178$	0	0
$179$	18.0991	1.35279	0.676394	−	0.736540i	$-0.263542\pi$
$179$	18.0991	1.35279	0.676394	+	0.736540i	$0.263542\pi$
$180$	0	0
$181$	−14.9662	−1.11243	−0.556214	−	0.831039i	$-0.687747\pi$
$181$	−14.9662	−1.11243	−0.556214	+	0.831039i	$0.687747\pi$
$182$	0	0
$183$	4.92676	0.364196
$184$	0	0
$185$	0.588501	0.0432675
$186$	0	0
$187$	−6.58954	−0.481875
$188$	0	0
$189$	2.13213	0.155089
$190$	0	0
$191$	−3.85510	−0.278945	−0.139473	−	0.990226i	$-0.544541\pi$
$191$	−3.85510	−0.278945	−0.139473	+	0.990226i	$0.544541\pi$
$192$	0	0
$193$	−12.6438	−0.910121	−0.455061	−	0.890460i	$-0.650382\pi$
$193$	−12.6438	−0.910121	−0.455061	+	0.890460i	$0.650382\pi$
$194$	0	0
$195$	−4.85276	−0.347513
$196$	0	0
$197$	−8.41488	−0.599535	−0.299768	−	0.954012i	$-0.596909\pi$
$197$	−8.41488	−0.599535	−0.299768	+	0.954012i	$0.596909\pi$
$198$	0	0
$199$	−5.57180	−0.394974	−0.197487	−	0.980305i	$-0.563278\pi$
$199$	−5.57180	−0.394974	−0.197487	+	0.980305i	$0.563278\pi$
$200$	0	0
$201$	13.9088	0.981052
$202$	0	0
$203$	−9.93444	−0.697261
$204$	0	0
$205$	8.90806	0.622166
$206$	0	0
$207$	6.38079	0.443495
$208$	0	0
$209$	−17.2179	−1.19099
$210$	0	0
$211$	18.5741	1.27870	0.639348	−	0.768917i	$-0.279204\pi$
$211$	18.5741	1.27870	0.639348	+	0.768917i	$0.279204\pi$
$212$	0	0
$213$	−14.5710	−0.998391
$214$	0	0
$215$	6.18743	0.421979
$216$	0	0
$217$	2.13213	0.144738
$218$	0	0
$219$	−2.67218	−0.180569
$220$	0	0
$221$	8.76722	0.589747
$222$	0	0
$223$	27.1536	1.81834	0.909171	−	0.416424i	$-0.136717\pi$
$223$	27.1536	1.81834	0.909171	+	0.416424i	$0.136717\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	30.0436	1.99406	0.997031	−	0.0770037i	$-0.0245353\pi$
$227$	30.0436	1.99406	0.997031	+	0.0770037i	$0.0245353\pi$
$228$	0	0
$229$	−2.03409	−0.134416	−0.0672082	−	0.997739i	$-0.521409\pi$
$229$	−2.03409	−0.134416	−0.0672082	+	0.997739i	$0.521409\pi$
$230$	0	0
$231$	−7.77669	−0.511669
$232$	0	0
$233$	−22.2156	−1.45539	−0.727696	−	0.685899i	$-0.759409\pi$
$233$	−22.2156	−1.45539	−0.727696	+	0.685899i	$0.759409\pi$
$234$	0	0
$235$	2.73340	0.178308
$236$	0	0
$237$	14.4845	0.940872
$238$	0	0
$239$	−23.3798	−1.51232	−0.756158	−	0.654389i	$-0.772926\pi$
$239$	−23.3798	−1.51232	−0.756158	+	0.654389i	$0.772926\pi$
$240$	0	0
$241$	11.1227	0.716478	0.358239	−	0.933630i	$-0.383377\pi$
$241$	11.1227	0.716478	0.358239	+	0.933630i	$0.383377\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	2.45403	0.156782
$246$	0	0
$247$	22.9081	1.45761
$248$	0	0
$249$	10.0282	0.635509
$250$	0	0
$251$	15.3073	0.966186	0.483093	−	0.875569i	$-0.339513\pi$
$251$	15.3073	0.966186	0.483093	+	0.875569i	$0.339513\pi$
$252$	0	0
$253$	−23.2732	−1.46317
$254$	0	0
$255$	−1.80665	−0.113137
$256$	0	0
$257$	−23.0115	−1.43542	−0.717710	−	0.696342i	$-0.754810\pi$
$257$	−23.0115	−1.43542	−0.717710	+	0.696342i	$0.754810\pi$
$258$	0	0
$259$	−1.25476	−0.0779670
$260$	0	0
$261$	−4.65940	−0.288410
$262$	0	0
$263$	2.48578	0.153280	0.0766398	−	0.997059i	$-0.475581\pi$
$263$	2.48578	0.153280	0.0766398	+	0.997059i	$0.475581\pi$
$264$	0	0
$265$	−10.9208	−0.670862
$266$	0	0
$267$	0.747764	0.0457624
$268$	0	0
$269$	13.1312	0.800622	0.400311	−	0.916379i	$-0.368902\pi$
$269$	13.1312	0.800622	0.400311	+	0.916379i	$0.368902\pi$
$270$	0	0
$271$	19.5062	1.18492	0.592460	−	0.805600i	$-0.298157\pi$
$271$	19.5062	1.18492	0.592460	+	0.805600i	$0.298157\pi$
$272$	0	0
$273$	10.3467	0.626211
$274$	0	0
$275$	−3.64738	−0.219946
$276$	0	0
$277$	−22.1659	−1.33182	−0.665910	−	0.746032i	$-0.731957\pi$
$277$	−22.1659	−1.33182	−0.665910	+	0.746032i	$0.731957\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	31.4007	1.87321	0.936604	−	0.350389i	$-0.113951\pi$
$281$	31.4007	1.87321	0.936604	+	0.350389i	$0.113951\pi$
$282$	0	0
$283$	−16.9138	−1.00542	−0.502710	−	0.864455i	$-0.667664\pi$
$283$	−16.9138	−1.00542	−0.502710	+	0.864455i	$0.667664\pi$
$284$	0	0
$285$	−4.72063	−0.279626
$286$	0	0
$287$	−18.9931	−1.12113
$288$	0	0
$289$	−13.7360	−0.808002
$290$	0	0
$291$	17.1723	1.00666
$292$	0	0
$293$	−6.70936	−0.391965	−0.195983	−	0.980607i	$-0.562790\pi$
$293$	−6.70936	−0.391965	−0.195983	+	0.980607i	$0.562790\pi$
$294$	0	0
$295$	5.58616	0.325239
$296$	0	0
$297$	−3.64738	−0.211643
$298$	0	0
$299$	30.9644	1.79072
$300$	0	0
$301$	−13.1924	−0.760397
$302$	0	0
$303$	17.8929	1.02792
$304$	0	0
$305$	−4.92676	−0.282105
$306$	0	0
$307$	21.3860	1.22056	0.610280	−	0.792186i	$-0.291057\pi$
$307$	21.3860	1.22056	0.610280	+	0.792186i	$0.291057\pi$
$308$	0	0
$309$	6.85276	0.389840
$310$	0	0
$311$	−14.2371	−0.807313	−0.403657	−	0.914911i	$-0.632261\pi$
$311$	−14.2371	−0.807313	−0.403657	+	0.914911i	$0.632261\pi$
$312$	0	0
$313$	−23.6826	−1.33862	−0.669310	−	0.742983i	$-0.733410\pi$
$313$	−23.6826	−1.33862	−0.669310	+	0.742983i	$0.733410\pi$
$314$	0	0
$315$	−2.13213	−0.120132
$316$	0	0
$317$	31.5754	1.77345	0.886724	−	0.462299i	$-0.152975\pi$
$317$	31.5754	1.77345	0.886724	+	0.462299i	$0.152975\pi$
$318$	0	0
$319$	16.9946	0.951517
$320$	0	0
$321$	18.0978	1.01012
$322$	0	0
$323$	8.52851	0.474539
$324$	0	0
$325$	4.85276	0.269183
$326$	0	0
$327$	11.7639	0.650546
$328$	0	0
$329$	−5.82796	−0.321306
$330$	0	0
$331$	−16.7643	−0.921449	−0.460725	−	0.887543i	$-0.652410\pi$
$331$	−16.7643	−0.921449	−0.460725	+	0.887543i	$0.652410\pi$
$332$	0	0
$333$	−0.588501	−0.0322497
$334$	0	0
$335$	−13.9088	−0.759920
$336$	0	0
$337$	−14.6994	−0.800728	−0.400364	−	0.916356i	$-0.631116\pi$
$337$	−14.6994	−0.800728	−0.400364	+	0.916356i	$0.631116\pi$
$338$	0	0
$339$	9.75114	0.529609
$340$	0	0
$341$	−3.64738	−0.197517
$342$	0	0
$343$	−20.1572	−1.08839
$344$	0	0
$345$	−6.38079	−0.343530
$346$	0	0
$347$	−21.1364	−1.13466	−0.567332	−	0.823489i	$-0.692024\pi$
$347$	−21.1364	−1.13466	−0.567332	+	0.823489i	$0.692024\pi$
$348$	0	0
$349$	−2.73491	−0.146396	−0.0731982	−	0.997317i	$-0.523321\pi$
$349$	−2.73491	−0.146396	−0.0731982	+	0.997317i	$0.523321\pi$
$350$	0	0
$351$	4.85276	0.259021
$352$	0	0
$353$	−29.1921	−1.55374	−0.776870	−	0.629661i	$-0.783194\pi$
$353$	−29.1921	−1.55374	−0.776870	+	0.629661i	$0.783194\pi$
$354$	0	0
$355$	14.5710	0.773351
$356$	0	0
$357$	3.85200	0.203870
$358$	0	0
$359$	0.743080	0.0392183	0.0196091	−	0.999808i	$-0.493758\pi$
$359$	0.743080	0.0392183	0.0196091	+	0.999808i	$0.493758\pi$
$360$	0	0
$361$	3.28434	0.172860
$362$	0	0
$363$	2.30341	0.120898
$364$	0	0
$365$	2.67218	0.139868
$366$	0	0
$367$	20.8653	1.08916	0.544581	−	0.838708i	$-0.316689\pi$
$367$	20.8653	1.08916	0.544581	+	0.838708i	$0.316689\pi$
$368$	0	0
$369$	−8.90806	−0.463735
$370$	0	0
$371$	23.2846	1.20888
$372$	0	0
$373$	−28.5741	−1.47951	−0.739756	−	0.672875i	$-0.765059\pi$
$373$	−28.5741	−1.47951	−0.739756	+	0.672875i	$0.765059\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	−22.6110	−1.16452
$378$	0	0
$379$	14.4579	0.742651	0.371326	−	0.928503i	$-0.378903\pi$
$379$	14.4579	0.742651	0.371326	+	0.928503i	$0.378903\pi$
$380$	0	0
$381$	−16.7429	−0.857764
$382$	0	0
$383$	14.6835	0.750293	0.375146	−	0.926966i	$-0.377592\pi$
$383$	14.6835	0.750293	0.375146	+	0.926966i	$0.377592\pi$
$384$	0	0
$385$	7.77669	0.396337
$386$	0	0
$387$	−6.18743	−0.314525
$388$	0	0
$389$	−18.1007	−0.917740	−0.458870	−	0.888503i	$-0.651746\pi$
$389$	−18.1007	−0.917740	−0.458870	+	0.888503i	$0.651746\pi$
$390$	0	0
$391$	11.5278	0.582987
$392$	0	0
$393$	−5.58616	−0.281784
$394$	0	0
$395$	−14.4845	−0.728797
$396$	0	0
$397$	23.3853	1.17367	0.586837	−	0.809705i	$-0.300373\pi$
$397$	23.3853	1.17367	0.586837	+	0.809705i	$0.300373\pi$
$398$	0	0
$399$	10.0650	0.503879
$400$	0	0
$401$	−14.8343	−0.740791	−0.370396	−	0.928874i	$-0.620778\pi$
$401$	−14.8343	−0.740791	−0.370396	+	0.928874i	$0.620778\pi$
$402$	0	0
$403$	4.85276	0.241733
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	2.14649	0.106398
$408$	0	0
$409$	−6.81584	−0.337022	−0.168511	−	0.985700i	$-0.553896\pi$
$409$	−6.81584	−0.337022	−0.168511	+	0.985700i	$0.553896\pi$
$410$	0	0
$411$	−2.07090	−0.102150
$412$	0	0
$413$	−11.9104	−0.586073
$414$	0	0
$415$	−10.0282	−0.492264
$416$	0	0
$417$	−15.5403	−0.761013
$418$	0	0
$419$	−27.5247	−1.34467	−0.672336	−	0.740246i	$-0.734709\pi$
$419$	−27.5247	−1.34467	−0.672336	+	0.740246i	$0.734709\pi$
$420$	0	0
$421$	−18.1204	−0.883134	−0.441567	−	0.897228i	$-0.645577\pi$
$421$	−18.1204	−0.883134	−0.441567	+	0.897228i	$0.645577\pi$
$422$	0	0
$423$	−2.73340	−0.132903
$424$	0	0
$425$	1.80665	0.0876353
$426$	0	0
$427$	10.5045	0.508347
$428$	0	0
$429$	−17.6999	−0.854558
$430$	0	0
$431$	−10.8794	−0.524043	−0.262022	−	0.965062i	$-0.584389\pi$
$431$	−10.8794	−0.524043	−0.262022	+	0.965062i	$0.584389\pi$
$432$	0	0
$433$	−3.76411	−0.180892	−0.0904459	−	0.995901i	$-0.528829\pi$
$433$	−3.76411	−0.180892	−0.0904459	+	0.995901i	$0.528829\pi$
$434$	0	0
$435$	4.65940	0.223401
$436$	0	0
$437$	30.1213	1.44090
$438$	0	0
$439$	−20.3749	−0.972439	−0.486220	−	0.873837i	$-0.661625\pi$
$439$	−20.3749	−0.972439	−0.486220	+	0.873837i	$0.661625\pi$
$440$	0	0
$441$	−2.45403	−0.116859
$442$	0	0
$443$	−32.5111	−1.54465	−0.772323	−	0.635230i	$-0.780905\pi$
$443$	−32.5111	−1.54465	−0.772323	+	0.635230i	$0.780905\pi$
$444$	0	0
$445$	−0.747764	−0.0354474
$446$	0	0
$447$	−13.8929	−0.657114
$448$	0	0
$449$	8.44463	0.398527	0.199263	−	0.979946i	$-0.436145\pi$
$449$	8.44463	0.398527	0.199263	+	0.979946i	$0.436145\pi$
$450$	0	0
$451$	32.4911	1.52995
$452$	0	0
$453$	−18.1700	−0.853700
$454$	0	0
$455$	−10.3467	−0.485061
$456$	0	0
$457$	32.5292	1.52165	0.760827	−	0.648955i	$-0.224794\pi$
$457$	32.5292	1.52165	0.760827	+	0.648955i	$0.224794\pi$
$458$	0	0
$459$	1.80665	0.0843271
$460$	0	0
$461$	−2.66091	−0.123931	−0.0619655	−	0.998078i	$-0.519737\pi$
$461$	−2.66091	−0.123931	−0.0619655	+	0.998078i	$0.519737\pi$
$462$	0	0
$463$	−17.8604	−0.830042	−0.415021	−	0.909812i	$-0.636226\pi$
$463$	−17.8604	−0.830042	−0.415021	+	0.909812i	$0.636226\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	0	0
$467$	−3.49498	−0.161728	−0.0808641	−	0.996725i	$-0.525768\pi$
$467$	−3.49498	−0.161728	−0.0808641	+	0.996725i	$0.525768\pi$
$468$	0	0
$469$	29.6554	1.36936
$470$	0	0
$471$	−7.80073	−0.359439
$472$	0	0
$473$	22.5679	1.03768
$474$	0	0
$475$	4.72063	0.216597
$476$	0	0
$477$	10.9208	0.500031
$478$	0	0
$479$	−14.0425	−0.641620	−0.320810	−	0.947144i	$-0.603955\pi$
$479$	−14.0425	−0.641620	−0.320810	+	0.947144i	$0.603955\pi$
$480$	0	0
$481$	−2.85585	−0.130216
$482$	0	0
$483$	13.6047	0.619033
$484$	0	0
$485$	−17.1723	−0.779755
$486$	0	0
$487$	8.25175	0.373923	0.186961	−	0.982367i	$-0.440136\pi$
$487$	8.25175	0.373923	0.186961	+	0.982367i	$0.440136\pi$
$488$	0	0
$489$	24.1971	1.09423
$490$	0	0
$491$	8.46323	0.381940	0.190970	−	0.981596i	$-0.438837\pi$
$491$	8.46323	0.381940	0.190970	+	0.981596i	$0.438837\pi$
$492$	0	0
$493$	−8.41790	−0.379123
$494$	0	0
$495$	3.64738	0.163938
$496$	0	0
$497$	−31.0673	−1.39356
$498$	0	0
$499$	36.0944	1.61581	0.807903	−	0.589315i	$-0.200602\pi$
$499$	36.0944	1.61581	0.807903	+	0.589315i	$0.200602\pi$
$500$	0	0
$501$	22.1687	0.990426
$502$	0	0
$503$	−21.7352	−0.969124	−0.484562	−	0.874757i	$-0.661021\pi$
$503$	−21.7352	−0.969124	−0.484562	+	0.874757i	$0.661021\pi$
$504$	0	0
$505$	−17.8929	−0.796226
$506$	0	0
$507$	10.5492	0.468508
$508$	0	0
$509$	7.45121	0.330269	0.165134	−	0.986271i	$-0.447194\pi$
$509$	7.45121	0.330269	0.165134	+	0.986271i	$0.447194\pi$
$510$	0	0
$511$	−5.69742	−0.252039
$512$	0	0
$513$	4.72063	0.208421
$514$	0	0
$515$	−6.85276	−0.301969
$516$	0	0
$517$	9.96977	0.438470
$518$	0	0
$519$	−5.61329	−0.246396
$520$	0	0
$521$	24.8825	1.09012	0.545061	−	0.838396i	$-0.316506\pi$
$521$	24.8825	1.09012	0.545061	+	0.838396i	$0.316506\pi$
$522$	0	0
$523$	−3.87606	−0.169488	−0.0847442	−	0.996403i	$-0.527007\pi$
$523$	−3.87606	−0.169488	−0.0847442	+	0.996403i	$0.527007\pi$
$524$	0	0
$525$	2.13213	0.0930537
$526$	0	0
$527$	1.80665	0.0786988
$528$	0	0
$529$	17.7144	0.770193
$530$	0	0
$531$	−5.58616	−0.242419
$532$	0	0
$533$	−43.2287	−1.87244
$534$	0	0
$535$	−18.0978	−0.782438
$536$	0	0
$537$	18.0991	0.781033
$538$	0	0
$539$	8.95079	0.385538
$540$	0	0
$541$	−26.9075	−1.15684	−0.578422	−	0.815738i	$-0.696331\pi$
$541$	−26.9075	−1.15684	−0.578422	+	0.815738i	$0.696331\pi$
$542$	0	0
$543$	−14.9662	−0.642261
$544$	0	0
$545$	−11.7639	−0.503911
$546$	0	0
$547$	37.6128	1.60821	0.804104	−	0.594489i	$-0.202645\pi$
$547$	37.6128	1.60821	0.804104	+	0.594489i	$0.202645\pi$
$548$	0	0
$549$	4.92676	0.210269
$550$	0	0
$551$	−21.9953	−0.937032
$552$	0	0
$553$	30.8829	1.31327
$554$	0	0
$555$	0.588501	0.0249805
$556$	0	0
$557$	−38.4481	−1.62910	−0.814550	−	0.580094i	$-0.803016\pi$
$557$	−38.4481	−1.62910	−0.814550	+	0.580094i	$0.803016\pi$
$558$	0	0
$559$	−30.0261	−1.26997
$560$	0	0
$561$	−6.58954	−0.278210
$562$	0	0
$563$	−28.1516	−1.18645	−0.593224	−	0.805038i	$-0.702145\pi$
$563$	−28.1516	−1.18645	−0.593224	+	0.805038i	$0.702145\pi$
$564$	0	0
$565$	−9.75114	−0.410234
$566$	0	0
$567$	2.13213	0.0895409
$568$	0	0
$569$	−21.8587	−0.916363	−0.458181	−	0.888859i	$-0.651499\pi$
$569$	−21.8587	−0.916363	−0.458181	+	0.888859i	$0.651499\pi$
$570$	0	0
$571$	32.0688	1.34204	0.671020	−	0.741440i	$-0.265857\pi$
$571$	32.0688	1.34204	0.671020	+	0.741440i	$0.265857\pi$
$572$	0	0
$573$	−3.85510	−0.161049
$574$	0	0
$575$	6.38079	0.266097
$576$	0	0
$577$	−19.6342	−0.817381	−0.408690	−	0.912673i	$-0.634014\pi$
$577$	−19.6342	−0.817381	−0.408690	+	0.912673i	$0.634014\pi$
$578$	0	0
$579$	−12.6438	−0.525459
$580$	0	0
$581$	21.3813	0.887047
$582$	0	0
$583$	−39.8325	−1.64969
$584$	0	0
$585$	−4.85276	−0.200637
$586$	0	0
$587$	12.2586	0.505967	0.252983	−	0.967471i	$-0.418588\pi$
$587$	12.2586	0.505967	0.252983	+	0.967471i	$0.418588\pi$
$588$	0	0
$589$	4.72063	0.194510
$590$	0	0
$591$	−8.41488	−0.346142
$592$	0	0
$593$	45.0807	1.85124	0.925620	−	0.378453i	$-0.123544\pi$
$593$	45.0807	1.85124	0.925620	+	0.378453i	$0.123544\pi$
$594$	0	0
$595$	−3.85200	−0.157917
$596$	0	0
$597$	−5.57180	−0.228038
$598$	0	0
$599$	−24.3678	−0.995642	−0.497821	−	0.867280i	$-0.665866\pi$
$599$	−24.3678	−0.995642	−0.497821	+	0.867280i	$0.665866\pi$
$600$	0	0
$601$	−17.3141	−0.706258	−0.353129	−	0.935575i	$-0.614882\pi$
$601$	−17.3141	−0.706258	−0.353129	+	0.935575i	$0.614882\pi$
$602$	0	0
$603$	13.9088	0.566411
$604$	0	0
$605$	−2.30341	−0.0936470
$606$	0	0
$607$	25.3044	1.02708	0.513538	−	0.858067i	$-0.328335\pi$
$607$	25.3044	1.02708	0.513538	+	0.858067i	$0.328335\pi$
$608$	0	0
$609$	−9.93444	−0.402564
$610$	0	0
$611$	−13.2645	−0.536626
$612$	0	0
$613$	8.23259	0.332511	0.166256	−	0.986083i	$-0.446832\pi$
$613$	8.23259	0.332511	0.166256	+	0.986083i	$0.446832\pi$
$614$	0	0
$615$	8.90806	0.359208
$616$	0	0
$617$	15.4878	0.623517	0.311759	−	0.950161i	$-0.399082\pi$
$617$	15.4878	0.623517	0.311759	+	0.950161i	$0.399082\pi$
$618$	0	0
$619$	31.4795	1.26527	0.632633	−	0.774452i	$-0.281974\pi$
$619$	31.4795	1.26527	0.632633	+	0.774452i	$0.281974\pi$
$620$	0	0
$621$	6.38079	0.256052
$622$	0	0
$623$	1.59433	0.0638754
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−17.2179	−0.687619
$628$	0	0
$629$	−1.06321	−0.0423931
$630$	0	0
$631$	−16.0092	−0.637316	−0.318658	−	0.947870i	$-0.603232\pi$
$631$	−16.0092	−0.637316	−0.318658	+	0.947870i	$0.603232\pi$
$632$	0	0
$633$	18.5741	0.738256
$634$	0	0
$635$	16.7429	0.664421
$636$	0	0
$637$	−11.9088	−0.471844
$638$	0	0
$639$	−14.5710	−0.576422
$640$	0	0
$641$	9.56212	0.377681	0.188841	−	0.982008i	$-0.439527\pi$
$641$	9.56212	0.377681	0.188841	+	0.982008i	$0.439527\pi$
$642$	0	0
$643$	−46.5249	−1.83476	−0.917382	−	0.398008i	$-0.869702\pi$
$643$	−46.5249	−1.83476	−0.917382	+	0.398008i	$0.869702\pi$
$644$	0	0
$645$	6.18743	0.243630
$646$	0	0
$647$	−2.25421	−0.0886220	−0.0443110	−	0.999018i	$-0.514109\pi$
$647$	−2.25421	−0.0886220	−0.0443110	+	0.999018i	$0.514109\pi$
$648$	0	0
$649$	20.3749	0.799784
$650$	0	0
$651$	2.13213	0.0835647
$652$	0	0
$653$	−41.2189	−1.61302	−0.806510	−	0.591221i	$-0.798646\pi$
$653$	−41.2189	−1.61302	−0.806510	+	0.591221i	$0.798646\pi$
$654$	0	0
$655$	5.58616	0.218269
$656$	0	0
$657$	−2.67218	−0.104252
$658$	0	0
$659$	−2.40200	−0.0935687	−0.0467844	−	0.998905i	$-0.514897\pi$
$659$	−2.40200	−0.0935687	−0.0467844	+	0.998905i	$0.514897\pi$
$660$	0	0
$661$	−19.4721	−0.757375	−0.378688	−	0.925525i	$-0.623624\pi$
$661$	−19.4721	−0.757375	−0.378688	+	0.925525i	$0.623624\pi$
$662$	0	0
$663$	8.76722	0.340491
$664$	0	0
$665$	−10.0650	−0.390303
$666$	0	0
$667$	−29.7307	−1.15118
$668$	0	0
$669$	27.1536	1.04982
$670$	0	0
$671$	−17.9698	−0.693715
$672$	0	0
$673$	−36.2937	−1.39902	−0.699510	−	0.714623i	$-0.746598\pi$
$673$	−36.2937	−1.39902	−0.699510	+	0.714623i	$0.746598\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−23.4469	−0.901137	−0.450568	−	0.892742i	$-0.648779\pi$
$677$	−23.4469	−0.901137	−0.450568	+	0.892742i	$0.648779\pi$
$678$	0	0
$679$	36.6136	1.40510
$680$	0	0
$681$	30.0436	1.15127
$682$	0	0
$683$	−20.1362	−0.770490	−0.385245	−	0.922814i	$-0.625883\pi$
$683$	−20.1362	−0.770490	−0.385245	+	0.922814i	$0.625883\pi$
$684$	0	0
$685$	2.07090	0.0791251
$686$	0	0
$687$	−2.03409	−0.0776054
$688$	0	0
$689$	52.9962	2.01899
$690$	0	0
$691$	−33.5928	−1.27793	−0.638965	−	0.769235i	$-0.720637\pi$
$691$	−33.5928	−1.27793	−0.638965	+	0.769235i	$0.720637\pi$
$692$	0	0
$693$	−7.77669	−0.295412
$694$	0	0
$695$	15.5403	0.589478
$696$	0	0
$697$	−16.0937	−0.609593
$698$	0	0
$699$	−22.2156	−0.840271
$700$	0	0
$701$	2.67800	0.101147	0.0505733	−	0.998720i	$-0.483895\pi$
$701$	2.67800	0.101147	0.0505733	+	0.998720i	$0.483895\pi$
$702$	0	0
$703$	−2.77810	−0.104778
$704$	0	0
$705$	2.73340	0.102946
$706$	0	0
$707$	38.1500	1.43478
$708$	0	0
$709$	−19.0229	−0.714421	−0.357211	−	0.934024i	$-0.616272\pi$
$709$	−19.0229	−0.714421	−0.357211	+	0.934024i	$0.616272\pi$
$710$	0	0
$711$	14.4845	0.543213
$712$	0	0
$713$	6.38079	0.238962
$714$	0	0
$715$	17.6999	0.661937
$716$	0	0
$717$	−23.3798	−0.873136
$718$	0	0
$719$	23.6721	0.882821	0.441410	−	0.897305i	$-0.354478\pi$
$719$	23.6721	0.882821	0.441410	+	0.897305i	$0.354478\pi$
$720$	0	0
$721$	14.6110	0.544140
$722$	0	0
$723$	11.1227	0.413659
$724$	0	0
$725$	−4.65940	−0.173046
$726$	0	0
$727$	−4.75039	−0.176182	−0.0880911	−	0.996112i	$-0.528077\pi$
$727$	−4.75039	−0.176182	−0.0880911	+	0.996112i	$0.528077\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−11.1785	−0.413452
$732$	0	0
$733$	−14.9152	−0.550906	−0.275453	−	0.961315i	$-0.588828\pi$
$733$	−14.9152	−0.550906	−0.275453	+	0.961315i	$0.588828\pi$
$734$	0	0
$735$	2.45403	0.0905183
$736$	0	0
$737$	−50.7308	−1.86869
$738$	0	0
$739$	51.3965	1.89065	0.945325	−	0.326130i	$-0.105745\pi$
$739$	51.3965	1.89065	0.945325	+	0.326130i	$0.105745\pi$
$740$	0	0
$741$	22.9081	0.841549
$742$	0	0
$743$	12.3408	0.452739	0.226370	−	0.974041i	$-0.427314\pi$
$743$	12.3408	0.452739	0.226370	+	0.974041i	$0.427314\pi$
$744$	0	0
$745$	13.8929	0.508998
$746$	0	0
$747$	10.0282	0.366912
$748$	0	0
$749$	38.5869	1.40993
$750$	0	0
$751$	6.90806	0.252079	0.126039	−	0.992025i	$-0.459773\pi$
$751$	6.90806	0.252079	0.126039	+	0.992025i	$0.459773\pi$
$752$	0	0
$753$	15.3073	0.557828
$754$	0	0
$755$	18.1700	0.661273
$756$	0	0
$757$	−30.5209	−1.10930	−0.554650	−	0.832084i	$-0.687148\pi$
$757$	−30.5209	−1.10930	−0.554650	+	0.832084i	$0.687148\pi$
$758$	0	0
$759$	−23.2732	−0.844763
$760$	0	0
$761$	4.83005	0.175089	0.0875446	−	0.996161i	$-0.472098\pi$
$761$	4.83005	0.175089	0.0875446	+	0.996161i	$0.472098\pi$
$762$	0	0
$763$	25.0822	0.908036
$764$	0	0
$765$	−1.80665	−0.0653195
$766$	0	0
$767$	−27.1083	−0.978823
$768$	0	0
$769$	2.95822	0.106676	0.0533381	−	0.998577i	$-0.483014\pi$
$769$	2.95822	0.106676	0.0533381	+	0.998577i	$0.483014\pi$
$770$	0	0
$771$	−23.0115	−0.828741
$772$	0	0
$773$	50.1208	1.80272	0.901359	−	0.433072i	$-0.142570\pi$
$773$	50.1208	1.80272	0.901359	+	0.433072i	$0.142570\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	−1.25476	−0.0450142
$778$	0	0
$779$	−42.0517	−1.50666
$780$	0	0
$781$	53.1462	1.90172
$782$	0	0
$783$	−4.65940	−0.166514
$784$	0	0
$785$	7.80073	0.278420
$786$	0	0
$787$	44.2447	1.57715	0.788577	−	0.614936i	$-0.210818\pi$
$787$	44.2447	1.57715	0.788577	+	0.614936i	$0.210818\pi$
$788$	0	0
$789$	2.48578	0.0884961
$790$	0	0
$791$	20.7907	0.739231
$792$	0	0
$793$	23.9083	0.849010
$794$	0	0
$795$	−10.9208	−0.387322
$796$	0	0
$797$	−18.4333	−0.652940	−0.326470	−	0.945208i	$-0.605859\pi$
$797$	−18.4333	−0.652940	−0.326470	+	0.945208i	$0.605859\pi$
$798$	0	0
$799$	−4.93829	−0.174704
$800$	0	0
$801$	0.747764	0.0264210
$802$	0	0
$803$	9.74646	0.343945
$804$	0	0
$805$	−13.6047	−0.479501
$806$	0	0
$807$	13.1312	0.462239
$808$	0	0
$809$	−37.0372	−1.30216	−0.651079	−	0.759010i	$-0.725683\pi$
$809$	−37.0372	−1.30216	−0.651079	+	0.759010i	$0.725683\pi$
$810$	0	0
$811$	−19.6287	−0.689257	−0.344628	−	0.938739i	$-0.611995\pi$
$811$	−19.6287	−0.689257	−0.344628	+	0.938739i	$0.611995\pi$
$812$	0	0
$813$	19.5062	0.684114
$814$	0	0
$815$	−24.1971	−0.847588
$816$	0	0
$817$	−29.2086	−1.02188
$818$	0	0
$819$	10.3467	0.361543
$820$	0	0
$821$	−30.0888	−1.05011	−0.525054	−	0.851069i	$-0.675955\pi$
$821$	−30.0888	−1.05011	−0.525054	+	0.851069i	$0.675955\pi$
$822$	0	0
$823$	27.1931	0.947891	0.473946	−	0.880554i	$-0.342829\pi$
$823$	27.1931	0.947891	0.473946	+	0.880554i	$0.342829\pi$
$824$	0	0
$825$	−3.64738	−0.126986
$826$	0	0
$827$	22.7986	0.792784	0.396392	−	0.918081i	$-0.370262\pi$
$827$	22.7986	0.792784	0.396392	+	0.918081i	$0.370262\pi$
$828$	0	0
$829$	−11.9045	−0.413460	−0.206730	−	0.978398i	$-0.566282\pi$
$829$	−11.9045	−0.413460	−0.206730	+	0.978398i	$0.566282\pi$
$830$	0	0
$831$	−22.1659	−0.768927
$832$	0	0
$833$	−4.43357	−0.153614
$834$	0	0
$835$	−22.1687	−0.767181
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	0.691986	0.0238900	0.0119450	−	0.999929i	$-0.496198\pi$
$839$	0.691986	0.0238900	0.0119450	+	0.999929i	$0.496198\pi$
$840$	0	0
$841$	−7.28996	−0.251378
$842$	0	0
$843$	31.4007	1.08150
$844$	0	0
$845$	−10.5492	−0.362905
$846$	0	0
$847$	4.91117	0.168750
$848$	0	0
$849$	−16.9138	−0.580479
$850$	0	0
$851$	−3.75510	−0.128723
$852$	0	0
$853$	−30.6664	−1.05000	−0.524998	−	0.851103i	$-0.675934\pi$
$853$	−30.6664	−1.05000	−0.524998	+	0.851103i	$0.675934\pi$
$854$	0	0
$855$	−4.72063	−0.161442
$856$	0	0
$857$	22.0533	0.753327	0.376664	−	0.926350i	$-0.377071\pi$
$857$	22.0533	0.753327	0.376664	+	0.926350i	$0.377071\pi$
$858$	0	0
$859$	1.26932	0.0433087	0.0216543	−	0.999766i	$-0.493107\pi$
$859$	1.26932	0.0433087	0.0216543	+	0.999766i	$0.493107\pi$
$860$	0	0
$861$	−18.9931	−0.647284
$862$	0	0
$863$	−15.1925	−0.517159	−0.258579	−	0.965990i	$-0.583254\pi$
$863$	−15.1925	−0.517159	−0.258579	+	0.965990i	$0.583254\pi$
$864$	0	0
$865$	5.61329	0.190858
$866$	0	0
$867$	−13.7360	−0.466500
$868$	0	0
$869$	−52.8307	−1.79216
$870$	0	0
$871$	67.4961	2.28702
$872$	0	0
$873$	17.1723	0.581195
$874$	0	0
$875$	−2.13213	−0.0720791
$876$	0	0
$877$	3.17081	0.107071	0.0535353	−	0.998566i	$-0.482951\pi$
$877$	3.17081	0.107071	0.0535353	+	0.998566i	$0.482951\pi$
$878$	0	0
$879$	−6.70936	−0.226301
$880$	0	0
$881$	37.3767	1.25925	0.629627	−	0.776897i	$-0.283208\pi$
$881$	37.3767	1.25925	0.629627	+	0.776897i	$0.283208\pi$
$882$	0	0
$883$	−26.9178	−0.905857	−0.452928	−	0.891547i	$-0.649621\pi$
$883$	−26.9178	−0.905857	−0.452928	+	0.891547i	$0.649621\pi$
$884$	0	0
$885$	5.58616	0.187777
$886$	0	0
$887$	19.3214	0.648750	0.324375	−	0.945929i	$-0.394846\pi$
$887$	19.3214	0.648750	0.324375	+	0.945929i	$0.394846\pi$
$888$	0	0
$889$	−35.6980	−1.19727
$890$	0	0
$891$	−3.64738	−0.122192
$892$	0	0
$893$	−12.9034	−0.431795
$894$	0	0
$895$	−18.0991	−0.604985
$896$	0	0
$897$	30.9644	1.03387
$898$	0	0
$899$	−4.65940	−0.155400
$900$	0	0
$901$	19.7301	0.657305
$902$	0	0
$903$	−13.1924	−0.439016
$904$	0	0
$905$	14.9662	0.497493
$906$	0	0
$907$	52.7763	1.75241	0.876204	−	0.481940i	$-0.160068\pi$
$907$	52.7763	1.75241	0.876204	+	0.481940i	$0.160068\pi$
$908$	0	0
$909$	17.8929	0.593472
$910$	0	0
$911$	−48.3026	−1.60034	−0.800168	−	0.599776i	$-0.795256\pi$
$911$	−48.3026	−1.60034	−0.800168	+	0.599776i	$0.795256\pi$
$912$	0	0
$913$	−36.5766	−1.21051
$914$	0	0
$915$	−4.92676	−0.162874
$916$	0	0
$917$	−11.9104	−0.393316
$918$	0	0
$919$	−33.1984	−1.09512	−0.547558	−	0.836768i	$-0.684442\pi$
$919$	−33.1984	−1.09512	−0.547558	+	0.836768i	$0.684442\pi$
$920$	0	0
$921$	21.3860	0.704691
$922$	0	0
$923$	−70.7097	−2.32744
$924$	0	0
$925$	−0.588501	−0.0193498
$926$	0	0
$927$	6.85276	0.225074
$928$	0	0
$929$	−28.1325	−0.922999	−0.461499	−	0.887141i	$-0.652688\pi$
$929$	−28.1325	−0.922999	−0.461499	+	0.887141i	$0.652688\pi$
$930$	0	0
$931$	−11.5846	−0.379669
$932$	0	0
$933$	−14.2371	−0.466102
$934$	0	0
$935$	6.58954	0.215501
$936$	0	0
$937$	16.8538	0.550589	0.275295	−	0.961360i	$-0.411225\pi$
$937$	16.8538	0.550589	0.275295	+	0.961360i	$0.411225\pi$
$938$	0	0
$939$	−23.6826	−0.772853
$940$	0	0
$941$	3.10506	0.101222	0.0506111	−	0.998718i	$-0.483883\pi$
$941$	3.10506	0.101222	0.0506111	+	0.998718i	$0.483883\pi$
$942$	0	0
$943$	−56.8404	−1.85098
$944$	0	0
$945$	−2.13213	−0.0693581
$946$	0	0
$947$	−25.2718	−0.821225	−0.410612	−	0.911810i	$-0.634685\pi$
$947$	−25.2718	−0.821225	−0.410612	+	0.911810i	$0.634685\pi$
$948$	0	0
$949$	−12.9674	−0.420940
$950$	0	0
$951$	31.5754	1.02390
$952$	0	0
$953$	−0.695354	−0.0225247	−0.0112624	−	0.999937i	$-0.503585\pi$
$953$	−0.695354	−0.0225247	−0.0112624	+	0.999937i	$0.503585\pi$
$954$	0	0
$955$	3.85510	0.124748
$956$	0	0
$957$	16.9946	0.549359
$958$	0	0
$959$	−4.41543	−0.142582
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	18.0978	0.583195
$964$	0	0
$965$	12.6438	0.407019
$966$	0	0
$967$	−23.7734	−0.764501	−0.382250	−	0.924059i	$-0.624851\pi$
$967$	−23.7734	−0.764501	−0.382250	+	0.924059i	$0.624851\pi$
$968$	0	0
$969$	8.52851	0.273975
$970$	0	0
$971$	21.8874	0.702400	0.351200	−	0.936300i	$-0.385774\pi$
$971$	21.8874	0.702400	0.351200	+	0.936300i	$0.385774\pi$
$972$	0	0
$973$	−33.1340	−1.06223
$974$	0	0
$975$	4.85276	0.155413
$976$	0	0
$977$	24.1228	0.771757	0.385879	−	0.922550i	$-0.373898\pi$
$977$	24.1228	0.771757	0.385879	+	0.922550i	$0.373898\pi$
$978$	0	0
$979$	−2.72738	−0.0871676
$980$	0	0
$981$	11.7639	0.375593
$982$	0	0
$983$	−20.2829	−0.646926	−0.323463	−	0.946241i	$-0.604847\pi$
$983$	−20.2829	−0.646926	−0.323463	+	0.946241i	$0.604847\pi$
$984$	0	0
$985$	8.41488	0.268120
$986$	0	0
$987$	−5.82796	−0.185506
$988$	0	0
$989$	−39.4807	−1.25541
$990$	0	0
$991$	−11.4812	−0.364714	−0.182357	−	0.983232i	$-0.558373\pi$
$991$	−11.4812	−0.364714	−0.182357	+	0.983232i	$0.558373\pi$
$992$	0	0
$993$	−16.7643	−0.531999
$994$	0	0
$995$	5.57180	0.176638
$996$	0	0
$997$	−11.3562	−0.359654	−0.179827	−	0.983698i	$-0.557554\pi$
$997$	−11.3562	−0.359654	−0.179827	+	0.983698i	$0.557554\pi$
$998$	0	0
$999$	−0.588501	−0.0186194

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3720.2.a.v.1.3	✓	5
4.3	odd	2		7440.2.a.cd.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3720.2.a.v.1.3	✓	5	1.1	even	1	trivial
7440.2.a.cd.1.3		5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3720.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +2.13213 q^{7} +1.00000 q^{9} -3.64738 q^{11} +4.85276 q^{13} -1.00000 q^{15} +1.80665 q^{17} +4.72063 q^{19} +2.13213 q^{21} +6.38079 q^{23} +1.00000 q^{25} +1.00000 q^{27} -4.65940 q^{29} +1.00000 q^{31} -3.64738 q^{33} -2.13213 q^{35} -0.588501 q^{37} +4.85276 q^{39} -8.90806 q^{41} -6.18743 q^{43} -1.00000 q^{45} -2.73340 q^{47} -2.45403 q^{49} +1.80665 q^{51} +10.9208 q^{53} +3.64738 q^{55} +4.72063 q^{57} -5.58616 q^{59} +4.92676 q^{61} +2.13213 q^{63} -4.85276 q^{65} +13.9088 q^{67} +6.38079 q^{69} -14.5710 q^{71} -2.67218 q^{73} +1.00000 q^{75} -7.77669 q^{77} +14.4845 q^{79} +1.00000 q^{81} +10.0282 q^{83} -1.80665 q^{85} -4.65940 q^{87} +0.747764 q^{89} +10.3467 q^{91} +1.00000 q^{93} -4.72063 q^{95} +17.1723 q^{97} -3.64738 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} - 5 q^{5} + 3 q^{7} + 5 q^{9} + 5 q^{11} + 2 q^{13} - 5 q^{15} + 6 q^{17} + 9 q^{19} + 3 q^{21} - 3 q^{23} + 5 q^{25} + 5 q^{27} + 2 q^{29} + 5 q^{31} + 5 q^{33} - 3 q^{35} + 4 q^{37} + 2 q^{39}+ \cdots + 5 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 - 5 * q^5 + 3 * q^7 + 5 * q^9 + 5 * q^11 + 2 * q^13 - 5 * q^15 + 6 * q^17 + 9 * q^19 + 3 * q^21 - 3 * q^23 + 5 * q^25 + 5 * q^27 + 2 * q^29 + 5 * q^31 + 5 * q^33 - 3 * q^35 + 4 * q^37 + 2 * q^39 + 8 * q^41 + 7 * q^43 - 5 * q^45 - 2 * q^47 + 14 * q^49 + 6 * q^51 + 5 * q^53 - 5 * q^55 + 9 * q^57 + 6 * q^59 + 16 * q^61 + 3 * q^63 - 2 * q^65 + 22 * q^67 - 3 * q^69 - 9 * q^71 + 9 * q^73 + 5 * q^75 + q^77 + 15 * q^79 + 5 * q^81 - 8 * q^83 - 6 * q^85 + 2 * q^87 + 17 * q^89 + 34 * q^91 + 5 * q^93 - 9 * q^95 + 18 * q^97 + 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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