LMFDB - Embedded newform 6080.2.a.be.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6080,2,Mod(1,6080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6080, 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("6080.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6080 = 2^{6} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6080.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:	$48.5490444289$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3040)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	6080.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} +2.82843 q^{7} -3.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{5} +2.82843 q^{7} -3.00000 q^{9} +4.00000 q^{11} +0.828427 q^{13} -3.65685 q^{17} -1.00000 q^{19} +2.82843 q^{23} +1.00000 q^{25} -7.65685 q^{29} -5.65685 q^{31} -2.82843 q^{35} -10.4853 q^{37} +7.65685 q^{41} -8.48528 q^{43} +3.00000 q^{45} +10.8284 q^{47} +1.00000 q^{49} -12.8284 q^{53} -4.00000 q^{55} +1.65685 q^{59} -6.00000 q^{61} -8.48528 q^{63} -0.828427 q^{65} +11.3137 q^{67} -5.65685 q^{71} +4.34315 q^{73} +11.3137 q^{77} +5.65685 q^{79} +9.00000 q^{81} -10.8284 q^{83} +3.65685 q^{85} -0.343146 q^{89} +2.34315 q^{91} +1.00000 q^{95} +12.8284 q^{97} -12.0000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^5 - 6 * q^9	$ 2 q - 2 q^{5} - 6 q^{9} + 8 q^{11} - 4 q^{13} + 4 q^{17} - 2 q^{19} + 2 q^{25} - 4 q^{29} - 4 q^{37} + 4 q^{41} + 6 q^{45} + 16 q^{47} + 2 q^{49} - 20 q^{53} - 8 q^{55} - 8 q^{59} - 12 q^{61} + 4 q^{65} + 20 q^{73} + 18 q^{81} - 16 q^{83} - 4 q^{85} - 12 q^{89} + 16 q^{91} + 2 q^{95} + 20 q^{97} - 24 q^{99}+O(q^{100}) $ 2 * q - 2 * q^5 - 6 * q^9 + 8 * q^11 - 4 * q^13 + 4 * q^17 - 2 * q^19 + 2 * q^25 - 4 * q^29 - 4 * q^37 + 4 * q^41 + 6 * q^45 + 16 * q^47 + 2 * q^49 - 20 * q^53 - 8 * q^55 - 8 * q^59 - 12 * q^61 + 4 * q^65 + 20 * q^73 + 18 * q^81 - 16 * q^83 - 4 * q^85 - 12 * q^89 + 16 * q^91 + 2 * q^95 + 20 * q^97 - 24 * 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
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.82843	1.06904	0.534522	−	0.845154i	$-0.320491\pi$
$7$	2.82843	1.06904	0.534522	+	0.845154i	$0.320491\pi$
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	0.828427	0.229764	0.114882	−	0.993379i	$-0.463351\pi$
$13$	0.828427	0.229764	0.114882	+	0.993379i	$0.463351\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.65685	−0.886917	−0.443459	−	0.896295i	$-0.646249\pi$
$17$	−3.65685	−0.886917	−0.443459	+	0.896295i	$0.646249\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.82843	0.589768	0.294884	−	0.955533i	$-0.404719\pi$
$23$	2.82843	0.589768	0.294884	+	0.955533i	$0.404719\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.65685	−1.42184	−0.710921	−	0.703272i	$-0.751722\pi$
$29$	−7.65685	−1.42184	−0.710921	+	0.703272i	$0.751722\pi$
$30$	0	0
$31$	−5.65685	−1.01600	−0.508001	−	0.861357i	$-0.669615\pi$
$31$	−5.65685	−1.01600	−0.508001	+	0.861357i	$0.669615\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.82843	−0.478091
$36$	0	0
$37$	−10.4853	−1.72377	−0.861885	−	0.507104i	$-0.830716\pi$
$37$	−10.4853	−1.72377	−0.861885	+	0.507104i	$0.830716\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	7.65685	1.19580	0.597900	−	0.801571i	$-0.296002\pi$
$41$	7.65685	1.19580	0.597900	+	0.801571i	$0.296002\pi$
$42$	0	0
$43$	−8.48528	−1.29399	−0.646997	−	0.762493i	$-0.723975\pi$
$43$	−8.48528	−1.29399	−0.646997	+	0.762493i	$0.723975\pi$
$44$	0	0
$45$	3.00000	0.447214
$46$	0	0
$47$	10.8284	1.57949	0.789744	−	0.613436i	$-0.210213\pi$
$47$	10.8284	1.57949	0.789744	+	0.613436i	$0.210213\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−12.8284	−1.76212	−0.881060	−	0.473005i	$-0.843169\pi$
$53$	−12.8284	−1.76212	−0.881060	+	0.473005i	$0.843169\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.65685	0.215704	0.107852	−	0.994167i	$-0.465603\pi$
$59$	1.65685	0.215704	0.107852	+	0.994167i	$0.465603\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	0	0
$63$	−8.48528	−1.06904
$64$	0	0
$65$	−0.828427	−0.102754
$66$	0	0
$67$	11.3137	1.38219	0.691095	−	0.722764i	$-0.257129\pi$
$67$	11.3137	1.38219	0.691095	+	0.722764i	$0.257129\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.65685	−0.671345	−0.335673	−	0.941979i	$-0.608964\pi$
$71$	−5.65685	−0.671345	−0.335673	+	0.941979i	$0.608964\pi$
$72$	0	0
$73$	4.34315	0.508327	0.254163	−	0.967161i	$-0.418200\pi$
$73$	4.34315	0.508327	0.254163	+	0.967161i	$0.418200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	11.3137	1.28932
$78$	0	0
$79$	5.65685	0.636446	0.318223	−	0.948016i	$-0.396914\pi$
$79$	5.65685	0.636446	0.318223	+	0.948016i	$0.396914\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	−10.8284	−1.18857	−0.594287	−	0.804253i	$-0.702566\pi$
$83$	−10.8284	−1.18857	−0.594287	+	0.804253i	$0.702566\pi$
$84$	0	0
$85$	3.65685	0.396642
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.343146	−0.0363734	−0.0181867	−	0.999835i	$-0.505789\pi$
$89$	−0.343146	−0.0363734	−0.0181867	+	0.999835i	$0.505789\pi$
$90$	0	0
$91$	2.34315	0.245628
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	12.8284	1.30253	0.651265	−	0.758851i	$-0.274239\pi$
$97$	12.8284	1.30253	0.651265	+	0.758851i	$0.274239\pi$
$98$	0	0
$99$	−12.0000	−1.20605
$100$	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
$103$	−6.34315	−0.625009	−0.312504	−	0.949916i	$-0.601168\pi$
$103$	−6.34315	−0.625009	−0.312504	+	0.949916i	$0.601168\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−2.34315	−0.226520	−0.113260	−	0.993565i	$-0.536129\pi$
$107$	−2.34315	−0.226520	−0.113260	+	0.993565i	$0.536129\pi$
$108$	0	0
$109$	17.3137	1.65835	0.829176	−	0.558987i	$-0.188810\pi$
$109$	17.3137	1.65835	0.829176	+	0.558987i	$0.188810\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−6.48528	−0.610084	−0.305042	−	0.952339i	$-0.598670\pi$
$113$	−6.48528	−0.610084	−0.305042	+	0.952339i	$0.598670\pi$
$114$	0	0
$115$	−2.82843	−0.263752
$116$	0	0
$117$	−2.48528	−0.229764
$118$	0	0
$119$	−10.3431	−0.948155
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−20.9706	−1.86084	−0.930418	−	0.366499i	$-0.880556\pi$
$127$	−20.9706	−1.86084	−0.930418	+	0.366499i	$0.880556\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	1.65685	0.144760	0.0723800	−	0.997377i	$-0.476941\pi$
$131$	1.65685	0.144760	0.0723800	+	0.997377i	$0.476941\pi$
$132$	0	0
$133$	−2.82843	−0.245256
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.34315	0.371060	0.185530	−	0.982639i	$-0.440600\pi$
$137$	4.34315	0.371060	0.185530	+	0.982639i	$0.440600\pi$
$138$	0	0
$139$	20.9706	1.77870	0.889350	−	0.457227i	$-0.151157\pi$
$139$	20.9706	1.77870	0.889350	+	0.457227i	$0.151157\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	3.31371	0.277106
$144$	0	0
$145$	7.65685	0.635867
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−19.3137	−1.57173	−0.785864	−	0.618400i	$-0.787781\pi$
$151$	−19.3137	−1.57173	−0.785864	+	0.618400i	$0.787781\pi$
$152$	0	0
$153$	10.9706	0.886917
$154$	0	0
$155$	5.65685	0.454369
$156$	0	0
$157$	4.34315	0.346621	0.173310	−	0.984867i	$-0.444554\pi$
$157$	4.34315	0.346621	0.173310	+	0.984867i	$0.444554\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	8.00000	0.630488
$162$	0	0
$163$	−13.1716	−1.03168	−0.515839	−	0.856686i	$-0.672520\pi$
$163$	−13.1716	−1.03168	−0.515839	+	0.856686i	$0.672520\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−0.686292	−0.0531068	−0.0265534	−	0.999647i	$-0.508453\pi$
$167$	−0.686292	−0.0531068	−0.0265534	+	0.999647i	$0.508453\pi$
$168$	0	0
$169$	−12.3137	−0.947208
$170$	0	0
$171$	3.00000	0.229416
$172$	0	0
$173$	−16.1421	−1.22726	−0.613632	−	0.789592i	$-0.710292\pi$
$173$	−16.1421	−1.22726	−0.613632	+	0.789592i	$0.710292\pi$
$174$	0	0
$175$	2.82843	0.213809
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−20.9706	−1.56741	−0.783707	−	0.621131i	$-0.786673\pi$
$179$	−20.9706	−1.56741	−0.783707	+	0.621131i	$0.786673\pi$
$180$	0	0
$181$	9.31371	0.692283	0.346141	−	0.938182i	$-0.387492\pi$
$181$	9.31371	0.692283	0.346141	+	0.938182i	$0.387492\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	10.4853	0.770893
$186$	0	0
$187$	−14.6274	−1.06966
$188$	0	0
$189$	0	0
$190$	0	0
$191$	20.9706	1.51738	0.758688	−	0.651454i	$-0.225841\pi$
$191$	20.9706	1.51738	0.758688	+	0.651454i	$0.225841\pi$
$192$	0	0
$193$	−3.17157	−0.228295	−0.114147	−	0.993464i	$-0.536414\pi$
$193$	−3.17157	−0.228295	−0.114147	+	0.993464i	$0.536414\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	21.3137	1.51854	0.759269	−	0.650776i	$-0.225556\pi$
$197$	21.3137	1.51854	0.759269	+	0.650776i	$0.225556\pi$
$198$	0	0
$199$	−15.3137	−1.08556	−0.542780	−	0.839875i	$-0.682628\pi$
$199$	−15.3137	−1.08556	−0.542780	+	0.839875i	$0.682628\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−21.6569	−1.52001
$204$	0	0
$205$	−7.65685	−0.534778
$206$	0	0
$207$	−8.48528	−0.589768
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8.48528	0.578691
$216$	0	0
$217$	−16.0000	−1.08615
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−3.02944	−0.203782
$222$	0	0
$223$	−6.34315	−0.424768	−0.212384	−	0.977186i	$-0.568123\pi$
$223$	−6.34315	−0.424768	−0.212384	+	0.977186i	$0.568123\pi$
$224$	0	0
$225$	−3.00000	−0.200000
$226$	0	0
$227$	−5.65685	−0.375459	−0.187729	−	0.982221i	$-0.560113\pi$
$227$	−5.65685	−0.375459	−0.187729	+	0.982221i	$0.560113\pi$
$228$	0	0
$229$	−25.3137	−1.67278	−0.836388	−	0.548137i	$-0.815337\pi$
$229$	−25.3137	−1.67278	−0.836388	+	0.548137i	$0.815337\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	21.3137	1.39631	0.698154	−	0.715948i	$-0.254005\pi$
$233$	21.3137	1.39631	0.698154	+	0.715948i	$0.254005\pi$
$234$	0	0
$235$	−10.8284	−0.706369
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	26.9706	1.73733	0.868663	−	0.495403i	$-0.164980\pi$
$241$	26.9706	1.73733	0.868663	+	0.495403i	$0.164980\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1.00000	−0.0638877
$246$	0	0
$247$	−0.828427	−0.0527116
$248$	0	0
$249$	0	0
$250$	0	0
$251$	12.9706	0.818695	0.409347	−	0.912379i	$-0.365756\pi$
$251$	12.9706	0.818695	0.409347	+	0.912379i	$0.365756\pi$
$252$	0	0
$253$	11.3137	0.711287
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−16.8284	−1.04973	−0.524864	−	0.851186i	$-0.675884\pi$
$257$	−16.8284	−1.04973	−0.524864	+	0.851186i	$0.675884\pi$
$258$	0	0
$259$	−29.6569	−1.84279
$260$	0	0
$261$	22.9706	1.42184
$262$	0	0
$263$	−8.48528	−0.523225	−0.261612	−	0.965173i	$-0.584254\pi$
$263$	−8.48528	−0.523225	−0.261612	+	0.965173i	$0.584254\pi$
$264$	0	0
$265$	12.8284	0.788044
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	−14.9706	−0.899494	−0.449747	−	0.893156i	$-0.648486\pi$
$277$	−14.9706	−0.899494	−0.449747	+	0.893156i	$0.648486\pi$
$278$	0	0
$279$	16.9706	1.01600
$280$	0	0
$281$	4.34315	0.259090	0.129545	−	0.991574i	$-0.458648\pi$
$281$	4.34315	0.259090	0.129545	+	0.991574i	$0.458648\pi$
$282$	0	0
$283$	24.4853	1.45550	0.727749	−	0.685843i	$-0.240566\pi$
$283$	24.4853	1.45550	0.727749	+	0.685843i	$0.240566\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	21.6569	1.27836
$288$	0	0
$289$	−3.62742	−0.213377
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.48528	0.378874	0.189437	−	0.981893i	$-0.439334\pi$
$293$	6.48528	0.378874	0.189437	+	0.981893i	$0.439334\pi$
$294$	0	0
$295$	−1.65685	−0.0964658
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.34315	0.135508
$300$	0	0
$301$	−24.0000	−1.38334
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.00000	0.343559
$306$	0	0
$307$	−5.65685	−0.322854	−0.161427	−	0.986885i	$-0.551610\pi$
$307$	−5.65685	−0.322854	−0.161427	+	0.986885i	$0.551610\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	20.0000	1.13410	0.567048	−	0.823685i	$-0.308085\pi$
$311$	20.0000	1.13410	0.567048	+	0.823685i	$0.308085\pi$
$312$	0	0
$313$	−28.6274	−1.61812	−0.809059	−	0.587728i	$-0.800023\pi$
$313$	−28.6274	−1.61812	−0.809059	+	0.587728i	$0.800023\pi$
$314$	0	0
$315$	8.48528	0.478091
$316$	0	0
$317$	−7.17157	−0.402796	−0.201398	−	0.979510i	$-0.564548\pi$
$317$	−7.17157	−0.402796	−0.201398	+	0.979510i	$0.564548\pi$
$318$	0	0
$319$	−30.6274	−1.71481
$320$	0	0
$321$	0	0
$322$	0	0
$323$	3.65685	0.203473
$324$	0	0
$325$	0.828427	0.0459529
$326$	0	0
$327$	0	0
$328$	0	0
$329$	30.6274	1.68854
$330$	0	0
$331$	−31.3137	−1.72116	−0.860579	−	0.509318i	$-0.829898\pi$
$331$	−31.3137	−1.72116	−0.860579	+	0.509318i	$0.829898\pi$
$332$	0	0
$333$	31.4558	1.72377
$334$	0	0
$335$	−11.3137	−0.618134
$336$	0	0
$337$	−11.1716	−0.608554	−0.304277	−	0.952584i	$-0.598415\pi$
$337$	−11.1716	−0.608554	−0.304277	+	0.952584i	$0.598415\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−22.6274	−1.22534
$342$	0	0
$343$	−16.9706	−0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	34.8284	1.86969	0.934844	−	0.355059i	$-0.115539\pi$
$347$	34.8284	1.86969	0.934844	+	0.355059i	$0.115539\pi$
$348$	0	0
$349$	−31.9411	−1.70977	−0.854885	−	0.518818i	$-0.826372\pi$
$349$	−31.9411	−1.70977	−0.854885	+	0.518818i	$0.826372\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8.34315	−0.444061	−0.222030	−	0.975040i	$-0.571268\pi$
$353$	−8.34315	−0.444061	−0.222030	+	0.975040i	$0.571268\pi$
$354$	0	0
$355$	5.65685	0.300235
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−14.3431	−0.757002	−0.378501	−	0.925601i	$-0.623560\pi$
$359$	−14.3431	−0.757002	−0.378501	+	0.925601i	$0.623560\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.34315	−0.227331
$366$	0	0
$367$	17.4558	0.911188	0.455594	−	0.890188i	$-0.349427\pi$
$367$	17.4558	0.911188	0.455594	+	0.890188i	$0.349427\pi$
$368$	0	0
$369$	−22.9706	−1.19580
$370$	0	0
$371$	−36.2843	−1.88379
$372$	0	0
$373$	−16.1421	−0.835808	−0.417904	−	0.908491i	$-0.637235\pi$
$373$	−16.1421	−0.835808	−0.417904	+	0.908491i	$0.637235\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−6.34315	−0.326689
$378$	0	0
$379$	−4.00000	−0.205466	−0.102733	−	0.994709i	$-0.532759\pi$
$379$	−4.00000	−0.205466	−0.102733	+	0.994709i	$0.532759\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4.00000	−0.204390	−0.102195	−	0.994764i	$-0.532587\pi$
$383$	−4.00000	−0.204390	−0.102195	+	0.994764i	$0.532587\pi$
$384$	0	0
$385$	−11.3137	−0.576600
$386$	0	0
$387$	25.4558	1.29399
$388$	0	0
$389$	32.6274	1.65428	0.827138	−	0.561999i	$-0.189968\pi$
$389$	32.6274	1.65428	0.827138	+	0.561999i	$0.189968\pi$
$390$	0	0
$391$	−10.3431	−0.523075
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−5.65685	−0.284627
$396$	0	0
$397$	−6.97056	−0.349843	−0.174921	−	0.984582i	$-0.555967\pi$
$397$	−6.97056	−0.349843	−0.174921	+	0.984582i	$0.555967\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	26.0000	1.29838	0.649189	−	0.760627i	$-0.275108\pi$
$401$	26.0000	1.29838	0.649189	+	0.760627i	$0.275108\pi$
$402$	0	0
$403$	−4.68629	−0.233441
$404$	0	0
$405$	−9.00000	−0.447214
$406$	0	0
$407$	−41.9411	−2.07894
$408$	0	0
$409$	−9.31371	−0.460533	−0.230267	−	0.973128i	$-0.573960\pi$
$409$	−9.31371	−0.460533	−0.230267	+	0.973128i	$0.573960\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.68629	0.230597
$414$	0	0
$415$	10.8284	0.531547
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−23.3137	−1.13895	−0.569475	−	0.822009i	$-0.692853\pi$
$419$	−23.3137	−1.13895	−0.569475	+	0.822009i	$0.692853\pi$
$420$	0	0
$421$	−10.9706	−0.534673	−0.267336	−	0.963603i	$-0.586143\pi$
$421$	−10.9706	−0.534673	−0.267336	+	0.963603i	$0.586143\pi$
$422$	0	0
$423$	−32.4853	−1.57949
$424$	0	0
$425$	−3.65685	−0.177383
$426$	0	0
$427$	−16.9706	−0.821263
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−12.6863	−0.611077	−0.305539	−	0.952180i	$-0.598836\pi$
$431$	−12.6863	−0.611077	−0.305539	+	0.952180i	$0.598836\pi$
$432$	0	0
$433$	−3.17157	−0.152416	−0.0762080	−	0.997092i	$-0.524281\pi$
$433$	−3.17157	−0.152416	−0.0762080	+	0.997092i	$0.524281\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−2.82843	−0.135302
$438$	0	0
$439$	−8.97056	−0.428142	−0.214071	−	0.976818i	$-0.568672\pi$
$439$	−8.97056	−0.428142	−0.214071	+	0.976818i	$0.568672\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	16.4853	0.783239	0.391620	−	0.920127i	$-0.371915\pi$
$443$	16.4853	0.783239	0.391620	+	0.920127i	$0.371915\pi$
$444$	0	0
$445$	0.343146	0.0162667
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−10.6863	−0.504317	−0.252159	−	0.967686i	$-0.581140\pi$
$449$	−10.6863	−0.504317	−0.252159	+	0.967686i	$0.581140\pi$
$450$	0	0
$451$	30.6274	1.44219
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.34315	−0.109848
$456$	0	0
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	2.00000	0.0931493	0.0465746	−	0.998915i	$-0.485169\pi$
$461$	2.00000	0.0931493	0.0465746	+	0.998915i	$0.485169\pi$
$462$	0	0
$463$	22.1421	1.02903	0.514516	−	0.857481i	$-0.327972\pi$
$463$	22.1421	1.02903	0.514516	+	0.857481i	$0.327972\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14.1421	0.654420	0.327210	−	0.944952i	$-0.393892\pi$
$467$	14.1421	0.654420	0.327210	+	0.944952i	$0.393892\pi$
$468$	0	0
$469$	32.0000	1.47762
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−33.9411	−1.56061
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	38.4853	1.76212
$478$	0	0
$479$	−12.9706	−0.592640	−0.296320	−	0.955089i	$-0.595760\pi$
$479$	−12.9706	−0.592640	−0.296320	+	0.955089i	$0.595760\pi$
$480$	0	0
$481$	−8.68629	−0.396061
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−12.8284	−0.582509
$486$	0	0
$487$	26.6274	1.20660	0.603302	−	0.797513i	$-0.293851\pi$
$487$	26.6274	1.20660	0.603302	+	0.797513i	$0.293851\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	4.97056	0.224318	0.112159	−	0.993690i	$-0.464223\pi$
$491$	4.97056	0.224318	0.112159	+	0.993690i	$0.464223\pi$
$492$	0	0
$493$	28.0000	1.26106
$494$	0	0
$495$	12.0000	0.539360
$496$	0	0
$497$	−16.0000	−0.717698
$498$	0	0
$499$	16.2843	0.728984	0.364492	−	0.931207i	$-0.381243\pi$
$499$	16.2843	0.728984	0.364492	+	0.931207i	$0.381243\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−0.485281	−0.0216376	−0.0108188	−	0.999941i	$-0.503444\pi$
$503$	−0.485281	−0.0216376	−0.0108188	+	0.999941i	$0.503444\pi$
$504$	0	0
$505$	14.0000	0.622992
$506$	0	0
$507$	0	0
$508$	0	0
$509$	22.9706	1.01815	0.509076	−	0.860721i	$-0.329987\pi$
$509$	22.9706	1.01815	0.509076	+	0.860721i	$0.329987\pi$
$510$	0	0
$511$	12.2843	0.543424
$512$	0	0
$513$	0	0
$514$	0	0
$515$	6.34315	0.279512
$516$	0	0
$517$	43.3137	1.90493
$518$	0	0
$519$	0	0
$520$	0	0
$521$	29.3137	1.28426	0.642128	−	0.766597i	$-0.278052\pi$
$521$	29.3137	1.28426	0.642128	+	0.766597i	$0.278052\pi$
$522$	0	0
$523$	−7.02944	−0.307376	−0.153688	−	0.988119i	$-0.549115\pi$
$523$	−7.02944	−0.307376	−0.153688	+	0.988119i	$0.549115\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	20.6863	0.901109
$528$	0	0
$529$	−15.0000	−0.652174
$530$	0	0
$531$	−4.97056	−0.215704
$532$	0	0
$533$	6.34315	0.274752
$534$	0	0
$535$	2.34315	0.101303
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4.00000	0.172292
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−17.3137	−0.741638
$546$	0	0
$547$	−20.2843	−0.867293	−0.433646	−	0.901083i	$-0.642773\pi$
$547$	−20.2843	−0.867293	−0.433646	+	0.901083i	$0.642773\pi$
$548$	0	0
$549$	18.0000	0.768221
$550$	0	0
$551$	7.65685	0.326193
$552$	0	0
$553$	16.0000	0.680389
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−22.9706	−0.973294	−0.486647	−	0.873599i	$-0.661780\pi$
$557$	−22.9706	−0.973294	−0.486647	+	0.873599i	$0.661780\pi$
$558$	0	0
$559$	−7.02944	−0.297314
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−10.3431	−0.435912	−0.217956	−	0.975959i	$-0.569939\pi$
$563$	−10.3431	−0.435912	−0.217956	+	0.975959i	$0.569939\pi$
$564$	0	0
$565$	6.48528	0.272838
$566$	0	0
$567$	25.4558	1.06904
$568$	0	0
$569$	35.9411	1.50673	0.753365	−	0.657602i	$-0.228429\pi$
$569$	35.9411	1.50673	0.753365	+	0.657602i	$0.228429\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	2.82843	0.117954
$576$	0	0
$577$	−36.6274	−1.52482	−0.762410	−	0.647095i	$-0.775984\pi$
$577$	−36.6274	−1.52482	−0.762410	+	0.647095i	$0.775984\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−30.6274	−1.27064
$582$	0	0
$583$	−51.3137	−2.12520
$584$	0	0
$585$	2.48528	0.102754
$586$	0	0
$587$	22.1421	0.913904	0.456952	−	0.889491i	$-0.348941\pi$
$587$	22.1421	0.913904	0.456952	+	0.889491i	$0.348941\pi$
$588$	0	0
$589$	5.65685	0.233087
$590$	0	0
$591$	0	0
$592$	0	0
$593$	8.62742	0.354286	0.177143	−	0.984185i	$-0.443315\pi$
$593$	8.62742	0.354286	0.177143	+	0.984185i	$0.443315\pi$
$594$	0	0
$595$	10.3431	0.424028
$596$	0	0
$597$	0	0
$598$	0	0
$599$	5.65685	0.231133	0.115566	−	0.993300i	$-0.463132\pi$
$599$	5.65685	0.231133	0.115566	+	0.993300i	$0.463132\pi$
$600$	0	0
$601$	−38.9706	−1.58964	−0.794821	−	0.606844i	$-0.792435\pi$
$601$	−38.9706	−1.58964	−0.794821	+	0.606844i	$0.792435\pi$
$602$	0	0
$603$	−33.9411	−1.38219
$604$	0	0
$605$	−5.00000	−0.203279
$606$	0	0
$607$	36.0000	1.46119	0.730597	−	0.682808i	$-0.239242\pi$
$607$	36.0000	1.46119	0.730597	+	0.682808i	$0.239242\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	8.97056	0.362910
$612$	0	0
$613$	34.9706	1.41245	0.706224	−	0.707989i	$-0.250397\pi$
$613$	34.9706	1.41245	0.706224	+	0.707989i	$0.250397\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−45.5980	−1.83571	−0.917853	−	0.396921i	$-0.870079\pi$
$617$	−45.5980	−1.83571	−0.917853	+	0.396921i	$0.870079\pi$
$618$	0	0
$619$	−43.5980	−1.75235	−0.876175	−	0.481992i	$-0.839913\pi$
$619$	−43.5980	−1.75235	−0.876175	+	0.481992i	$0.839913\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−0.970563	−0.0388848
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	38.3431	1.52884
$630$	0	0
$631$	26.6274	1.06002	0.530010	−	0.847991i	$-0.322188\pi$
$631$	26.6274	1.06002	0.530010	+	0.847991i	$0.322188\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	20.9706	0.832191
$636$	0	0
$637$	0.828427	0.0328235
$638$	0	0
$639$	16.9706	0.671345
$640$	0	0
$641$	20.3431	0.803506	0.401753	−	0.915748i	$-0.368401\pi$
$641$	20.3431	0.803506	0.401753	+	0.915748i	$0.368401\pi$
$642$	0	0
$643$	15.1127	0.595987	0.297993	−	0.954568i	$-0.403683\pi$
$643$	15.1127	0.595987	0.297993	+	0.954568i	$0.403683\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.14214	−0.241472	−0.120736	−	0.992685i	$-0.538525\pi$
$647$	−6.14214	−0.241472	−0.120736	+	0.992685i	$0.538525\pi$
$648$	0	0
$649$	6.62742	0.260149
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−11.6569	−0.456168	−0.228084	−	0.973641i	$-0.573246\pi$
$653$	−11.6569	−0.456168	−0.228084	+	0.973641i	$0.573246\pi$
$654$	0	0
$655$	−1.65685	−0.0647387
$656$	0	0
$657$	−13.0294	−0.508327
$658$	0	0
$659$	−9.65685	−0.376178	−0.188089	−	0.982152i	$-0.560229\pi$
$659$	−9.65685	−0.376178	−0.188089	+	0.982152i	$0.560229\pi$
$660$	0	0
$661$	−12.3431	−0.480093	−0.240046	−	0.970761i	$-0.577163\pi$
$661$	−12.3431	−0.480093	−0.240046	+	0.970761i	$0.577163\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.82843	0.109682
$666$	0	0
$667$	−21.6569	−0.838557
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−24.0000	−0.926510
$672$	0	0
$673$	25.1127	0.968023	0.484012	−	0.875062i	$-0.339179\pi$
$673$	25.1127	0.968023	0.484012	+	0.875062i	$0.339179\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	40.4264	1.55371	0.776857	−	0.629678i	$-0.216813\pi$
$677$	40.4264	1.55371	0.776857	+	0.629678i	$0.216813\pi$
$678$	0	0
$679$	36.2843	1.39246
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−12.2843	−0.470045	−0.235022	−	0.971990i	$-0.575516\pi$
$683$	−12.2843	−0.470045	−0.235022	+	0.971990i	$0.575516\pi$
$684$	0	0
$685$	−4.34315	−0.165943
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−10.6274	−0.404872
$690$	0	0
$691$	−39.3137	−1.49556	−0.747782	−	0.663944i	$-0.768881\pi$
$691$	−39.3137	−1.49556	−0.747782	+	0.663944i	$0.768881\pi$
$692$	0	0
$693$	−33.9411	−1.28932
$694$	0	0
$695$	−20.9706	−0.795459
$696$	0	0
$697$	−28.0000	−1.06058
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−18.6863	−0.705771	−0.352886	−	0.935666i	$-0.614800\pi$
$701$	−18.6863	−0.705771	−0.352886	+	0.935666i	$0.614800\pi$
$702$	0	0
$703$	10.4853	0.395460
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−39.5980	−1.48924
$708$	0	0
$709$	43.9411	1.65024	0.825122	−	0.564955i	$-0.191106\pi$
$709$	43.9411	1.65024	0.825122	+	0.564955i	$0.191106\pi$
$710$	0	0
$711$	−16.9706	−0.636446
$712$	0	0
$713$	−16.0000	−0.599205
$714$	0	0
$715$	−3.31371	−0.123926
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−16.2843	−0.607301	−0.303650	−	0.952784i	$-0.598205\pi$
$719$	−16.2843	−0.607301	−0.303650	+	0.952784i	$0.598205\pi$
$720$	0	0
$721$	−17.9411	−0.668162
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−7.65685	−0.284368
$726$	0	0
$727$	−32.4853	−1.20481	−0.602406	−	0.798190i	$-0.705791\pi$
$727$	−32.4853	−1.20481	−0.602406	+	0.798190i	$0.705791\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	31.0294	1.14767
$732$	0	0
$733$	29.3137	1.08273	0.541363	−	0.840789i	$-0.317908\pi$
$733$	29.3137	1.08273	0.541363	+	0.840789i	$0.317908\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	45.2548	1.66698
$738$	0	0
$739$	−31.3137	−1.15189	−0.575947	−	0.817487i	$-0.695366\pi$
$739$	−31.3137	−1.15189	−0.575947	+	0.817487i	$0.695366\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	50.6274	1.85734	0.928670	−	0.370907i	$-0.120953\pi$
$743$	50.6274	1.85734	0.928670	+	0.370907i	$0.120953\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	0	0
$747$	32.4853	1.18857
$748$	0	0
$749$	−6.62742	−0.242161
$750$	0	0
$751$	−22.6274	−0.825686	−0.412843	−	0.910802i	$-0.635464\pi$
$751$	−22.6274	−0.825686	−0.412843	+	0.910802i	$0.635464\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	19.3137	0.702898
$756$	0	0
$757$	21.3137	0.774660	0.387330	−	0.921941i	$-0.373397\pi$
$757$	21.3137	0.774660	0.387330	+	0.921941i	$0.373397\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−20.6274	−0.747743	−0.373872	−	0.927480i	$-0.621970\pi$
$761$	−20.6274	−0.747743	−0.373872	+	0.927480i	$0.621970\pi$
$762$	0	0
$763$	48.9706	1.77285
$764$	0	0
$765$	−10.9706	−0.396642
$766$	0	0
$767$	1.37258	0.0495611
$768$	0	0
$769$	39.2548	1.41557	0.707783	−	0.706430i	$-0.249696\pi$
$769$	39.2548	1.41557	0.707783	+	0.706430i	$0.249696\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−42.4853	−1.52809	−0.764045	−	0.645163i	$-0.776789\pi$
$773$	−42.4853	−1.52809	−0.764045	+	0.645163i	$0.776789\pi$
$774$	0	0
$775$	−5.65685	−0.203200
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−7.65685	−0.274335
$780$	0	0
$781$	−22.6274	−0.809673
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−4.34315	−0.155014
$786$	0	0
$787$	12.2843	0.437887	0.218943	−	0.975738i	$-0.429739\pi$
$787$	12.2843	0.437887	0.218943	+	0.975738i	$0.429739\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−18.3431	−0.652207
$792$	0	0
$793$	−4.97056	−0.176510
$794$	0	0
$795$	0	0
$796$	0	0
$797$	11.1716	0.395717	0.197859	−	0.980231i	$-0.436601\pi$
$797$	11.1716	0.395717	0.197859	+	0.980231i	$0.436601\pi$
$798$	0	0
$799$	−39.5980	−1.40088
$800$	0	0
$801$	1.02944	0.0363734
$802$	0	0
$803$	17.3726	0.613065
$804$	0	0
$805$	−8.00000	−0.281963
$806$	0	0
$807$	0	0
$808$	0	0
$809$	3.37258	0.118574	0.0592869	−	0.998241i	$-0.481117\pi$
$809$	3.37258	0.118574	0.0592869	+	0.998241i	$0.481117\pi$
$810$	0	0
$811$	33.6569	1.18185	0.590926	−	0.806726i	$-0.298763\pi$
$811$	33.6569	1.18185	0.590926	+	0.806726i	$0.298763\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	13.1716	0.461380
$816$	0	0
$817$	8.48528	0.296862
$818$	0	0
$819$	−7.02944	−0.245628
$820$	0	0
$821$	13.3137	0.464652	0.232326	−	0.972638i	$-0.425366\pi$
$821$	13.3137	0.464652	0.232326	+	0.972638i	$0.425366\pi$
$822$	0	0
$823$	3.79899	0.132424	0.0662122	−	0.997806i	$-0.478909\pi$
$823$	3.79899	0.132424	0.0662122	+	0.997806i	$0.478909\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	16.0000	0.556375	0.278187	−	0.960527i	$-0.410266\pi$
$827$	16.0000	0.556375	0.278187	+	0.960527i	$0.410266\pi$
$828$	0	0
$829$	−39.2548	−1.36338	−0.681688	−	0.731643i	$-0.738754\pi$
$829$	−39.2548	−1.36338	−0.681688	+	0.731643i	$0.738754\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−3.65685	−0.126702
$834$	0	0
$835$	0.686292	0.0237501
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−26.3431	−0.909466	−0.454733	−	0.890628i	$-0.650265\pi$
$839$	−26.3431	−0.909466	−0.454733	+	0.890628i	$0.650265\pi$
$840$	0	0
$841$	29.6274	1.02164
$842$	0	0
$843$	0	0
$844$	0	0
$845$	12.3137	0.423604
$846$	0	0
$847$	14.1421	0.485930
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−29.6569	−1.01662
$852$	0	0
$853$	0.627417	0.0214823	0.0107412	−	0.999942i	$-0.496581\pi$
$853$	0.627417	0.0214823	0.0107412	+	0.999942i	$0.496581\pi$
$854$	0	0
$855$	−3.00000	−0.102598
$856$	0	0
$857$	21.7990	0.744639	0.372320	−	0.928105i	$-0.378563\pi$
$857$	21.7990	0.744639	0.372320	+	0.928105i	$0.378563\pi$
$858$	0	0
$859$	−0.686292	−0.0234160	−0.0117080	−	0.999931i	$-0.503727\pi$
$859$	−0.686292	−0.0234160	−0.0117080	+	0.999931i	$0.503727\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	52.0000	1.77010	0.885050	−	0.465495i	$-0.154124\pi$
$863$	52.0000	1.77010	0.885050	+	0.465495i	$0.154124\pi$
$864$	0	0
$865$	16.1421	0.548849
$866$	0	0
$867$	0	0
$868$	0	0
$869$	22.6274	0.767583
$870$	0	0
$871$	9.37258	0.317578
$872$	0	0
$873$	−38.4853	−1.30253
$874$	0	0
$875$	−2.82843	−0.0956183
$876$	0	0
$877$	9.79899	0.330888	0.165444	−	0.986219i	$-0.447094\pi$
$877$	9.79899	0.330888	0.165444	+	0.986219i	$0.447094\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	29.3137	0.987604	0.493802	−	0.869574i	$-0.335607\pi$
$881$	29.3137	0.987604	0.493802	+	0.869574i	$0.335607\pi$
$882$	0	0
$883$	10.8284	0.364406	0.182203	−	0.983261i	$-0.441677\pi$
$883$	10.8284	0.364406	0.182203	+	0.983261i	$0.441677\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	25.6569	0.861473	0.430736	−	0.902478i	$-0.358254\pi$
$887$	25.6569	0.861473	0.430736	+	0.902478i	$0.358254\pi$
$888$	0	0
$889$	−59.3137	−1.98932
$890$	0	0
$891$	36.0000	1.20605
$892$	0	0
$893$	−10.8284	−0.362359
$894$	0	0
$895$	20.9706	0.700969
$896$	0	0
$897$	0	0
$898$	0	0
$899$	43.3137	1.44459
$900$	0	0
$901$	46.9117	1.56285
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−9.31371	−0.309598
$906$	0	0
$907$	12.2843	0.407893	0.203946	−	0.978982i	$-0.434623\pi$
$907$	12.2843	0.407893	0.203946	+	0.978982i	$0.434623\pi$
$908$	0	0
$909$	42.0000	1.39305
$910$	0	0
$911$	−45.2548	−1.49936	−0.749680	−	0.661801i	$-0.769792\pi$
$911$	−45.2548	−1.49936	−0.749680	+	0.661801i	$0.769792\pi$
$912$	0	0
$913$	−43.3137	−1.43347
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.68629	0.154755
$918$	0	0
$919$	−13.3726	−0.441121	−0.220560	−	0.975373i	$-0.570789\pi$
$919$	−13.3726	−0.441121	−0.220560	+	0.975373i	$0.570789\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−4.68629	−0.154251
$924$	0	0
$925$	−10.4853	−0.344754
$926$	0	0
$927$	19.0294	0.625009
$928$	0	0
$929$	−38.0000	−1.24674	−0.623370	−	0.781927i	$-0.714237\pi$
$929$	−38.0000	−1.24674	−0.623370	+	0.781927i	$0.714237\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	14.6274	0.478368
$936$	0	0
$937$	−43.6569	−1.42621	−0.713104	−	0.701059i	$-0.752711\pi$
$937$	−43.6569	−1.42621	−0.713104	+	0.701059i	$0.752711\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	0	0
$943$	21.6569	0.705244
$944$	0	0
$945$	0	0
$946$	0	0
$947$	42.8284	1.39174	0.695868	−	0.718169i	$-0.255020\pi$
$947$	42.8284	1.39174	0.695868	+	0.718169i	$0.255020\pi$
$948$	0	0
$949$	3.59798	0.116795
$950$	0	0
$951$	0	0
$952$	0	0
$953$	37.7990	1.22443	0.612215	−	0.790692i	$-0.290279\pi$
$953$	37.7990	1.22443	0.612215	+	0.790692i	$0.290279\pi$
$954$	0	0
$955$	−20.9706	−0.678591
$956$	0	0
$957$	0	0
$958$	0	0
$959$	12.2843	0.396680
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	7.02944	0.226520
$964$	0	0
$965$	3.17157	0.102097
$966$	0	0
$967$	23.1127	0.743254	0.371627	−	0.928382i	$-0.378800\pi$
$967$	23.1127	0.743254	0.371627	+	0.928382i	$0.378800\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	4.97056	0.159513	0.0797565	−	0.996814i	$-0.474586\pi$
$971$	4.97056	0.159513	0.0797565	+	0.996814i	$0.474586\pi$
$972$	0	0
$973$	59.3137	1.90151
$974$	0	0
$975$	0	0
$976$	0	0
$977$	45.7990	1.46524	0.732620	−	0.680638i	$-0.238297\pi$
$977$	45.7990	1.46524	0.732620	+	0.680638i	$0.238297\pi$
$978$	0	0
$979$	−1.37258	−0.0438679
$980$	0	0
$981$	−51.9411	−1.65835
$982$	0	0
$983$	−7.31371	−0.233271	−0.116636	−	0.993175i	$-0.537211\pi$
$983$	−7.31371	−0.233271	−0.116636	+	0.993175i	$0.537211\pi$
$984$	0	0
$985$	−21.3137	−0.679111
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−24.0000	−0.763156
$990$	0	0
$991$	−8.00000	−0.254128	−0.127064	−	0.991894i	$-0.540555\pi$
$991$	−8.00000	−0.254128	−0.127064	+	0.991894i	$0.540555\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	15.3137	0.485477
$996$	0	0
$997$	23.6569	0.749220	0.374610	−	0.927182i	$-0.377777\pi$
$997$	23.6569	0.749220	0.374610	+	0.927182i	$0.377777\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6080.2.a.be.1.2		2
4.3	odd	2		6080.2.a.bd.1.1		2
8.3	odd	2		3040.2.a.g.1.1	yes	2
8.5	even	2		3040.2.a.f.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.f.1.2	✓	2	8.5	even	2
3040.2.a.g.1.1	yes	2	8.3	odd	2
6080.2.a.bd.1.1		2	4.3	odd	2
6080.2.a.be.1.2		2	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6080.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +2.82843 q^{7} -3.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{5} +2.82843 q^{7} -3.00000 q^{9} +4.00000 q^{11} +0.828427 q^{13} -3.65685 q^{17} -1.00000 q^{19} +2.82843 q^{23} +1.00000 q^{25} -7.65685 q^{29} -5.65685 q^{31} -2.82843 q^{35} -10.4853 q^{37} +7.65685 q^{41} -8.48528 q^{43} +3.00000 q^{45} +10.8284 q^{47} +1.00000 q^{49} -12.8284 q^{53} -4.00000 q^{55} +1.65685 q^{59} -6.00000 q^{61} -8.48528 q^{63} -0.828427 q^{65} +11.3137 q^{67} -5.65685 q^{71} +4.34315 q^{73} +11.3137 q^{77} +5.65685 q^{79} +9.00000 q^{81} -10.8284 q^{83} +3.65685 q^{85} -0.343146 q^{89} +2.34315 q^{91} +1.00000 q^{95} +12.8284 q^{97} -12.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^5 - 6 * q^9	\( 2 q - 2 q^{5} - 6 q^{9} + 8 q^{11} - 4 q^{13} + 4 q^{17} - 2 q^{19} + 2 q^{25} - 4 q^{29} - 4 q^{37} + 4 q^{41} + 6 q^{45} + 16 q^{47} + 2 q^{49} - 20 q^{53} - 8 q^{55} - 8 q^{59} - 12 q^{61} + 4 q^{65} + 20 q^{73} + 18 q^{81} - 16 q^{83} - 4 q^{85} - 12 q^{89} + 16 q^{91} + 2 q^{95} + 20 q^{97} - 24 q^{99}+O(q^{100}) \) 2 * q - 2 * q^5 - 6 * q^9 + 8 * q^11 - 4 * q^13 + 4 * q^17 - 2 * q^19 + 2 * q^25 - 4 * q^29 - 4 * q^37 + 4 * q^41 + 6 * q^45 + 16 * q^47 + 2 * q^49 - 20 * q^53 - 8 * q^55 - 8 * q^59 - 12 * q^61 + 4 * q^65 + 20 * q^73 + 18 * q^81 - 16 * q^83 - 4 * q^85 - 12 * q^89 + 16 * q^91 + 2 * q^95 + 20 * q^97 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more