LMFDB - Embedded newform 6080.2.a.ci.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [6080,2,Mod(1,6080)] 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("6080.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [5,0,-4,0,5,0,-4,0,7,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$48.5490444289$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.387268.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 7x^{3} + 4x^{2} + 12x - 2 $ x^5 - x^4 - 7x^3 + 4x^2 + 12*x - 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			1.3
Root			$2.42170$ 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)

$f(q)$	$=$	$q-1.44292 q^{3} +1.00000 q^{5} -2.92540 q^{7} -0.917992 q^{9} -4.55509 q^{11} +1.15462 q^{13} -1.44292 q^{15} +5.35892 q^{17} +1.00000 q^{19} +4.22111 q^{21} -6.10565 q^{23} +1.00000 q^{25} +5.65334 q^{27} +5.09212 q^{29} +1.54157 q^{31} +6.57262 q^{33} -2.92540 q^{35} +8.92953 q^{37} -1.66602 q^{39} -2.88583 q^{41} +7.86833 q^{43} -0.917992 q^{45} +7.58614 q^{47} +1.55797 q^{49} -7.73247 q^{51} -4.48008 q^{53} -4.55509 q^{55} -1.44292 q^{57} +2.09953 q^{59} -1.59013 q^{61} +2.68549 q^{63} +1.15462 q^{65} -3.73733 q^{67} +8.80994 q^{69} +3.73263 q^{71} -9.81735 q^{73} -1.44292 q^{75} +13.3255 q^{77} -4.60365 q^{79} -5.40331 q^{81} +9.89538 q^{83} +5.35892 q^{85} -7.34751 q^{87} +1.91688 q^{89} -3.37772 q^{91} -2.22435 q^{93} +1.00000 q^{95} -15.6391 q^{97} +4.18154 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 4 q^{3} + 5 q^{5} - 4 q^{7} + 7 q^{9} - 2 q^{11} + 4 q^{13} - 4 q^{15} - 12 q^{17} + 5 q^{19} + 10 q^{21} - 8 q^{23} + 5 q^{25} - 16 q^{27} + 6 q^{29} - 10 q^{31} - 18 q^{33} - 4 q^{35} + 6 q^{37}+ \cdots + 14 q^{99}+O(q^{100}) $ 5 * q - 4 * q^3 + 5 * q^5 - 4 * q^7 + 7 * q^9 - 2 * q^11 + 4 * q^13 - 4 * q^15 - 12 * q^17 + 5 * q^19 + 10 * q^21 - 8 * q^23 + 5 * q^25 - 16 * q^27 + 6 * q^29 - 10 * q^31 - 18 * q^33 - 4 * q^35 + 6 * q^37 - 18 * q^39 - 8 * q^41 - 12 * q^43 + 7 * q^45 - 16 * q^47 + 7 * q^49 + 14 * q^51 + 18 * q^53 - 2 * q^55 - 4 * q^57 - 8 * q^59 - 2 * q^61 - 36 * q^63 + 4 * q^65 - 10 * q^67 + 22 * q^69 + 18 * q^71 - 28 * q^73 - 4 * q^75 + 28 * q^77 - 14 * q^79 + 25 * q^81 - 8 * q^83 - 12 * q^85 - 24 * q^87 - 30 * q^89 - 28 * q^91 + 24 * q^93 + 5 * q^95 - 18 * q^97 + 14 * 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
$3$	−1.44292	−0.833068	−0.416534	−	0.909120i	$-0.636755\pi$
$3$	−1.44292	−0.833068	−0.416534	+	0.909120i	$0.636755\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.92540	−1.10570	−0.552849	−	0.833282i	$-0.686459\pi$
$7$	−2.92540	−1.10570	−0.552849	+	0.833282i	$0.686459\pi$
$8$	0	0
$9$	−0.917992	−0.305997
$10$	0	0
$11$	−4.55509	−1.37341	−0.686706	−	0.726935i	$-0.740944\pi$
$11$	−4.55509	−1.37341	−0.686706	+	0.726935i	$0.740944\pi$
$12$	0	0
$13$	1.15462	0.320233	0.160116	−	0.987098i	$-0.448813\pi$
$13$	1.15462	0.320233	0.160116	+	0.987098i	$0.448813\pi$
$14$	0	0
$15$	−1.44292	−0.372559
$16$	0	0
$17$	5.35892	1.29973	0.649864	−	0.760050i	$-0.274826\pi$
$17$	5.35892	1.29973	0.649864	+	0.760050i	$0.274826\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	4.22111	0.921121
$22$	0	0
$23$	−6.10565	−1.27312	−0.636558	−	0.771229i	$-0.719642\pi$
$23$	−6.10565	−1.27312	−0.636558	+	0.771229i	$0.719642\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.65334	1.08798
$28$	0	0
$29$	5.09212	0.945584	0.472792	−	0.881174i	$-0.343246\pi$
$29$	5.09212	0.945584	0.472792	+	0.881174i	$0.343246\pi$
$30$	0	0
$31$	1.54157	0.276874	0.138437	−	0.990371i	$-0.455792\pi$
$31$	1.54157	0.276874	0.138437	+	0.990371i	$0.455792\pi$
$32$	0	0
$33$	6.57262	1.14415
$34$	0	0
$35$	−2.92540	−0.494483
$36$	0	0
$37$	8.92953	1.46801	0.734003	−	0.679147i	$-0.237650\pi$
$37$	8.92953	1.46801	0.734003	+	0.679147i	$0.237650\pi$
$38$	0	0
$39$	−1.66602	−0.266776
$40$	0	0
$41$	−2.88583	−0.450691	−0.225346	−	0.974279i	$-0.572351\pi$
$41$	−2.88583	−0.450691	−0.225346	+	0.974279i	$0.572351\pi$
$42$	0	0
$43$	7.86833	1.19991	0.599954	−	0.800034i	$-0.295185\pi$
$43$	7.86833	1.19991	0.599954	+	0.800034i	$0.295185\pi$
$44$	0	0
$45$	−0.917992	−0.136846
$46$	0	0
$47$	7.58614	1.10655	0.553276	−	0.832998i	$-0.313377\pi$
$47$	7.58614	1.10655	0.553276	+	0.832998i	$0.313377\pi$
$48$	0	0
$49$	1.55797	0.222567
$50$	0	0
$51$	−7.73247	−1.08276
$52$	0	0
$53$	−4.48008	−0.615387	−0.307693	−	0.951486i	$-0.599557\pi$
$53$	−4.48008	−0.615387	−0.307693	+	0.951486i	$0.599557\pi$
$54$	0	0
$55$	−4.55509	−0.614209
$56$	0	0
$57$	−1.44292	−0.191119
$58$	0	0
$59$	2.09953	0.273336	0.136668	−	0.990617i	$-0.456361\pi$
$59$	2.09953	0.273336	0.136668	+	0.990617i	$0.456361\pi$
$60$	0	0
$61$	−1.59013	−0.203595	−0.101797	−	0.994805i	$-0.532459\pi$
$61$	−1.59013	−0.203595	−0.101797	+	0.994805i	$0.532459\pi$
$62$	0	0
$63$	2.68549	0.338341
$64$	0	0
$65$	1.15462	0.143213
$66$	0	0
$67$	−3.73733	−0.456588	−0.228294	−	0.973592i	$-0.573315\pi$
$67$	−3.73733	−0.456588	−0.228294	+	0.973592i	$0.573315\pi$
$68$	0	0
$69$	8.80994	1.06059
$70$	0	0
$71$	3.73263	0.442982	0.221491	−	0.975162i	$-0.428908\pi$
$71$	3.73263	0.442982	0.221491	+	0.975162i	$0.428908\pi$
$72$	0	0
$73$	−9.81735	−1.14903	−0.574517	−	0.818493i	$-0.694810\pi$
$73$	−9.81735	−1.14903	−0.574517	+	0.818493i	$0.694810\pi$
$74$	0	0
$75$	−1.44292	−0.166614
$76$	0	0
$77$	13.3255	1.51858
$78$	0	0
$79$	−4.60365	−0.517951	−0.258976	−	0.965884i	$-0.583385\pi$
$79$	−4.60365	−0.517951	−0.258976	+	0.965884i	$0.583385\pi$
$80$	0	0
$81$	−5.40331	−0.600368
$82$	0	0
$83$	9.89538	1.08616	0.543079	−	0.839681i	$-0.317258\pi$
$83$	9.89538	1.08616	0.543079	+	0.839681i	$0.317258\pi$
$84$	0	0
$85$	5.35892	0.581256
$86$	0	0
$87$	−7.34751	−0.787736
$88$	0	0
$89$	1.91688	0.203189	0.101595	−	0.994826i	$-0.467606\pi$
$89$	1.91688	0.203189	0.101595	+	0.994826i	$0.467606\pi$
$90$	0	0
$91$	−3.37772	−0.354081
$92$	0	0
$93$	−2.22435	−0.230655
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	−15.6391	−1.58791	−0.793955	−	0.607976i	$-0.791982\pi$
$97$	−15.6391	−1.58791	−0.793955	+	0.607976i	$0.791982\pi$
$98$	0	0
$99$	4.18154	0.420261
$100$	0	0
$101$	8.55509	0.851264	0.425632	−	0.904896i	$-0.360052\pi$
$101$	8.55509	0.851264	0.425632	+	0.904896i	$0.360052\pi$
$102$	0	0
$103$	−9.01793	−0.888563	−0.444282	−	0.895887i	$-0.646541\pi$
$103$	−9.01793	−0.888563	−0.444282	+	0.895887i	$0.646541\pi$
$104$	0	0
$105$	4.22111	0.411938
$106$	0	0
$107$	−19.4780	−1.88301	−0.941503	−	0.337004i	$-0.890586\pi$
$107$	−19.4780	−1.88301	−0.941503	+	0.337004i	$0.890586\pi$
$108$	0	0
$109$	−8.16328	−0.781900	−0.390950	−	0.920412i	$-0.627854\pi$
$109$	−8.16328	−0.781900	−0.390950	+	0.920412i	$0.627854\pi$
$110$	0	0
$111$	−12.8846	−1.22295
$112$	0	0
$113$	−14.5893	−1.37244	−0.686221	−	0.727393i	$-0.740732\pi$
$113$	−14.5893	−1.37244	−0.686221	+	0.727393i	$0.740732\pi$
$114$	0	0
$115$	−6.10565	−0.569355
$116$	0	0
$117$	−1.05993	−0.0979905
$118$	0	0
$119$	−15.6770	−1.43711
$120$	0	0
$121$	9.74887	0.886261
$122$	0	0
$123$	4.16402	0.375457
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−9.99514	−0.886925	−0.443462	−	0.896293i	$-0.646250\pi$
$127$	−9.99514	−0.886925	−0.443462	+	0.896293i	$0.646250\pi$
$128$	0	0
$129$	−11.3533	−0.999605
$130$	0	0
$131$	−4.11817	−0.359806	−0.179903	−	0.983684i	$-0.557578\pi$
$131$	−4.11817	−0.359806	−0.179903	+	0.983684i	$0.557578\pi$
$132$	0	0
$133$	−2.92540	−0.253664
$134$	0	0
$135$	5.65334	0.486562
$136$	0	0
$137$	−11.7513	−1.00398	−0.501989	−	0.864874i	$-0.667398\pi$
$137$	−11.7513	−1.00398	−0.501989	+	0.864874i	$0.667398\pi$
$138$	0	0
$139$	12.8455	1.08954	0.544772	−	0.838584i	$-0.316616\pi$
$139$	12.8455	1.08954	0.544772	+	0.838584i	$0.316616\pi$
$140$	0	0
$141$	−10.9462	−0.921834
$142$	0	0
$143$	−5.25939	−0.439812
$144$	0	0
$145$	5.09212	0.422878
$146$	0	0
$147$	−2.24801	−0.185413
$148$	0	0
$149$	22.9907	1.88348	0.941738	−	0.336348i	$-0.109192\pi$
$149$	22.9907	1.88348	0.941738	+	0.336348i	$0.109192\pi$
$150$	0	0
$151$	1.76768	0.143852	0.0719260	−	0.997410i	$-0.477085\pi$
$151$	1.76768	0.143852	0.0719260	+	0.997410i	$0.477085\pi$
$152$	0	0
$153$	−4.91945	−0.397714
$154$	0	0
$155$	1.54157	0.123822
$156$	0	0
$157$	6.57660	0.524870	0.262435	−	0.964950i	$-0.415475\pi$
$157$	6.57660	0.524870	0.262435	+	0.964950i	$0.415475\pi$
$158$	0	0
$159$	6.46439	0.512659
$160$	0	0
$161$	17.8615	1.40768
$162$	0	0
$163$	−3.01353	−0.236038	−0.118019	−	0.993011i	$-0.537654\pi$
$163$	−3.01353	−0.236038	−0.118019	+	0.993011i	$0.537654\pi$
$164$	0	0
$165$	6.57262	0.511678
$166$	0	0
$167$	6.45533	0.499529	0.249764	−	0.968307i	$-0.419647\pi$
$167$	6.45533	0.499529	0.249764	+	0.968307i	$0.419647\pi$
$168$	0	0
$169$	−11.6669	−0.897451
$170$	0	0
$171$	−0.917992	−0.0702006
$172$	0	0
$173$	11.6824	0.888195	0.444097	−	0.895979i	$-0.353524\pi$
$173$	11.6824	0.888195	0.444097	+	0.895979i	$0.353524\pi$
$174$	0	0
$175$	−2.92540	−0.221139
$176$	0	0
$177$	−3.02945	−0.227707
$178$	0	0
$179$	−15.5646	−1.16336	−0.581678	−	0.813419i	$-0.697604\pi$
$179$	−15.5646	−1.16336	−0.581678	+	0.813419i	$0.697604\pi$
$180$	0	0
$181$	0.603651	0.0448690	0.0224345	−	0.999748i	$-0.492858\pi$
$181$	0.603651	0.0448690	0.0224345	+	0.999748i	$0.492858\pi$
$182$	0	0
$183$	2.29442	0.169608
$184$	0	0
$185$	8.92953	0.656512
$186$	0	0
$187$	−24.4104	−1.78506
$188$	0	0
$189$	−16.5383	−1.20298
$190$	0	0
$191$	3.09611	0.224026	0.112013	−	0.993707i	$-0.464270\pi$
$191$	3.09611	0.224026	0.112013	+	0.993707i	$0.464270\pi$
$192$	0	0
$193$	−20.0144	−1.44067	−0.720335	−	0.693626i	$-0.756012\pi$
$193$	−20.0144	−1.44067	−0.720335	+	0.693626i	$0.756012\pi$
$194$	0	0
$195$	−1.66602	−0.119306
$196$	0	0
$197$	−15.9790	−1.13846	−0.569228	−	0.822180i	$-0.692758\pi$
$197$	−15.9790	−1.13846	−0.569228	+	0.822180i	$0.692758\pi$
$198$	0	0
$199$	−4.38254	−0.310670	−0.155335	−	0.987862i	$-0.549646\pi$
$199$	−4.38254	−0.310670	−0.155335	+	0.987862i	$0.549646\pi$
$200$	0	0
$201$	5.39266	0.380369
$202$	0	0
$203$	−14.8965	−1.04553
$204$	0	0
$205$	−2.88583	−0.201555
$206$	0	0
$207$	5.60494	0.389570
$208$	0	0
$209$	−4.55509	−0.315082
$210$	0	0
$211$	−0.759249	−0.0522689	−0.0261344	−	0.999658i	$-0.508320\pi$
$211$	−0.759249	−0.0522689	−0.0261344	+	0.999658i	$0.508320\pi$
$212$	0	0
$213$	−5.38588	−0.369034
$214$	0	0
$215$	7.86833	0.536615
$216$	0	0
$217$	−4.50970	−0.306138
$218$	0	0
$219$	14.1656	0.957224
$220$	0	0
$221$	6.18749	0.416216
$222$	0	0
$223$	12.9078	0.864368	0.432184	−	0.901785i	$-0.357743\pi$
$223$	12.9078	0.864368	0.432184	+	0.901785i	$0.357743\pi$
$224$	0	0
$225$	−0.917992	−0.0611995
$226$	0	0
$227$	−22.7792	−1.51191	−0.755955	−	0.654624i	$-0.772827\pi$
$227$	−22.7792	−1.51191	−0.755955	+	0.654624i	$0.772827\pi$
$228$	0	0
$229$	−10.4181	−0.688449	−0.344225	−	0.938887i	$-0.611858\pi$
$229$	−10.4181	−0.688449	−0.344225	+	0.938887i	$0.611858\pi$
$230$	0	0
$231$	−19.2275	−1.26508
$232$	0	0
$233$	−13.2432	−0.867589	−0.433794	−	0.901012i	$-0.642826\pi$
$233$	−13.2432	−0.867589	−0.433794	+	0.901012i	$0.642826\pi$
$234$	0	0
$235$	7.58614	0.494865
$236$	0	0
$237$	6.64268	0.431489
$238$	0	0
$239$	24.6568	1.59491	0.797457	−	0.603376i	$-0.206178\pi$
$239$	24.6568	1.59491	0.797457	+	0.603376i	$0.206178\pi$
$240$	0	0
$241$	24.1951	1.55854	0.779271	−	0.626687i	$-0.215589\pi$
$241$	24.1951	1.55854	0.779271	+	0.626687i	$0.215589\pi$
$242$	0	0
$243$	−9.16348	−0.587837
$244$	0	0
$245$	1.55797	0.0995348
$246$	0	0
$247$	1.15462	0.0734665
$248$	0	0
$249$	−14.2782	−0.904844
$250$	0	0
$251$	−23.0140	−1.45263	−0.726316	−	0.687361i	$-0.758769\pi$
$251$	−23.0140	−1.45263	−0.726316	+	0.687361i	$0.758769\pi$
$252$	0	0
$253$	27.8118	1.74851
$254$	0	0
$255$	−7.73247	−0.484226
$256$	0	0
$257$	9.58128	0.597664	0.298832	−	0.954306i	$-0.403403\pi$
$257$	9.58128	0.597664	0.298832	+	0.954306i	$0.403403\pi$
$258$	0	0
$259$	−26.1224	−1.62317
$260$	0	0
$261$	−4.67453	−0.289346
$262$	0	0
$263$	−20.5381	−1.26643	−0.633215	−	0.773976i	$-0.718265\pi$
$263$	−20.5381	−1.26643	−0.633215	+	0.773976i	$0.718265\pi$
$264$	0	0
$265$	−4.48008	−0.275209
$266$	0	0
$267$	−2.76590	−0.169270
$268$	0	0
$269$	27.0932	1.65190	0.825949	−	0.563744i	$-0.190640\pi$
$269$	27.0932	1.65190	0.825949	+	0.563744i	$0.190640\pi$
$270$	0	0
$271$	6.05357	0.367728	0.183864	−	0.982952i	$-0.441139\pi$
$271$	6.05357	0.367728	0.183864	+	0.982952i	$0.441139\pi$
$272$	0	0
$273$	4.87376	0.294973
$274$	0	0
$275$	−4.55509	−0.274682
$276$	0	0
$277$	−3.69253	−0.221863	−0.110931	−	0.993828i	$-0.535383\pi$
$277$	−3.69253	−0.221863	−0.110931	+	0.993828i	$0.535383\pi$
$278$	0	0
$279$	−1.41515	−0.0847226
$280$	0	0
$281$	−22.4212	−1.33754	−0.668768	−	0.743471i	$-0.733178\pi$
$281$	−22.4212	−1.33754	−0.668768	+	0.743471i	$0.733178\pi$
$282$	0	0
$283$	9.02152	0.536274	0.268137	−	0.963381i	$-0.413592\pi$
$283$	9.02152	0.536274	0.268137	+	0.963381i	$0.413592\pi$
$284$	0	0
$285$	−1.44292	−0.0854710
$286$	0	0
$287$	8.44222	0.498328
$288$	0	0
$289$	11.7180	0.689294
$290$	0	0
$291$	22.5659	1.32284
$292$	0	0
$293$	−8.55922	−0.500035	−0.250017	−	0.968241i	$-0.580436\pi$
$293$	−8.55922	−0.500035	−0.250017	+	0.968241i	$0.580436\pi$
$294$	0	0
$295$	2.09953	0.122240
$296$	0	0
$297$	−25.7515	−1.49425
$298$	0	0
$299$	−7.04968	−0.407694
$300$	0	0
$301$	−23.0180	−1.32674
$302$	0	0
$303$	−12.3443	−0.709161
$304$	0	0
$305$	−1.59013	−0.0910503
$306$	0	0
$307$	−1.07318	−0.0612498	−0.0306249	−	0.999531i	$-0.509750\pi$
$307$	−1.07318	−0.0612498	−0.0306249	+	0.999531i	$0.509750\pi$
$308$	0	0
$309$	13.0121	0.740234
$310$	0	0
$311$	9.99150	0.566566	0.283283	−	0.959036i	$-0.408576\pi$
$311$	9.99150	0.566566	0.283283	+	0.959036i	$0.408576\pi$
$312$	0	0
$313$	24.6935	1.39576	0.697878	−	0.716216i	$-0.254128\pi$
$313$	24.6935	1.39576	0.697878	+	0.716216i	$0.254128\pi$
$314$	0	0
$315$	2.68549	0.151310
$316$	0	0
$317$	9.23375	0.518619	0.259310	−	0.965794i	$-0.416505\pi$
$317$	9.23375	0.518619	0.259310	+	0.965794i	$0.416505\pi$
$318$	0	0
$319$	−23.1951	−1.29868
$320$	0	0
$321$	28.1051	1.56867
$322$	0	0
$323$	5.35892	0.298178
$324$	0	0
$325$	1.15462	0.0640466
$326$	0	0
$327$	11.7789	0.651376
$328$	0	0
$329$	−22.1925	−1.22351
$330$	0	0
$331$	8.22994	0.452358	0.226179	−	0.974086i	$-0.427377\pi$
$331$	8.22994	0.452358	0.226179	+	0.974086i	$0.427377\pi$
$332$	0	0
$333$	−8.19724	−0.449206
$334$	0	0
$335$	−3.73733	−0.204192
$336$	0	0
$337$	6.04627	0.329362	0.164681	−	0.986347i	$-0.447341\pi$
$337$	6.04627	0.329362	0.164681	+	0.986347i	$0.447341\pi$
$338$	0	0
$339$	21.0511	1.14334
$340$	0	0
$341$	−7.02198	−0.380262
$342$	0	0
$343$	15.9201	0.859606
$344$	0	0
$345$	8.80994	0.474311
$346$	0	0
$347$	−17.1938	−0.923011	−0.461506	−	0.887137i	$-0.652691\pi$
$347$	−17.1938	−0.923011	−0.461506	+	0.887137i	$0.652691\pi$
$348$	0	0
$349$	−7.79272	−0.417135	−0.208567	−	0.978008i	$-0.566880\pi$
$349$	−7.79272	−0.417135	−0.208567	+	0.978008i	$0.566880\pi$
$350$	0	0
$351$	6.52743	0.348409
$352$	0	0
$353$	−10.2758	−0.546925	−0.273462	−	0.961883i	$-0.588169\pi$
$353$	−10.2758	−0.546925	−0.273462	+	0.961883i	$0.588169\pi$
$354$	0	0
$355$	3.73263	0.198108
$356$	0	0
$357$	22.6206	1.19721
$358$	0	0
$359$	1.51424	0.0799183	0.0399591	−	0.999201i	$-0.487277\pi$
$359$	1.51424	0.0799183	0.0399591	+	0.999201i	$0.487277\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−14.0668	−0.738316
$364$	0	0
$365$	−9.81735	−0.513864
$366$	0	0
$367$	−2.02549	−0.105730	−0.0528648	−	0.998602i	$-0.516835\pi$
$367$	−2.02549	−0.105730	−0.0528648	+	0.998602i	$0.516835\pi$
$368$	0	0
$369$	2.64917	0.137910
$370$	0	0
$371$	13.1060	0.680432
$372$	0	0
$373$	−32.6834	−1.69228	−0.846142	−	0.532958i	$-0.821080\pi$
$373$	−32.6834	−1.69228	−0.846142	+	0.532958i	$0.821080\pi$
$374$	0	0
$375$	−1.44292	−0.0745119
$376$	0	0
$377$	5.87945	0.302807
$378$	0	0
$379$	5.64250	0.289836	0.144918	−	0.989444i	$-0.453708\pi$
$379$	5.64250	0.289836	0.144918	+	0.989444i	$0.453708\pi$
$380$	0	0
$381$	14.4221	0.738869
$382$	0	0
$383$	20.2873	1.03663	0.518317	−	0.855188i	$-0.326559\pi$
$383$	20.2873	1.03663	0.518317	+	0.855188i	$0.326559\pi$
$384$	0	0
$385$	13.3255	0.679129
$386$	0	0
$387$	−7.22306	−0.367169
$388$	0	0
$389$	−32.6268	−1.65424	−0.827121	−	0.562024i	$-0.810023\pi$
$389$	−32.6268	−1.65424	−0.827121	+	0.562024i	$0.810023\pi$
$390$	0	0
$391$	−32.7197	−1.65471
$392$	0	0
$393$	5.94217	0.299743
$394$	0	0
$395$	−4.60365	−0.231635
$396$	0	0
$397$	−6.23917	−0.313135	−0.156567	−	0.987667i	$-0.550043\pi$
$397$	−6.23917	−0.313135	−0.156567	+	0.987667i	$0.550043\pi$
$398$	0	0
$399$	4.22111	0.211320
$400$	0	0
$401$	−12.4252	−0.620484	−0.310242	−	0.950658i	$-0.600410\pi$
$401$	−12.4252	−0.620484	−0.310242	+	0.950658i	$0.600410\pi$
$402$	0	0
$403$	1.77992	0.0886641
$404$	0	0
$405$	−5.40331	−0.268493
$406$	0	0
$407$	−40.6748	−2.01618
$408$	0	0
$409$	35.4878	1.75476	0.877378	−	0.479799i	$-0.159291\pi$
$409$	35.4878	1.75476	0.877378	+	0.479799i	$0.159291\pi$
$410$	0	0
$411$	16.9561	0.836383
$412$	0	0
$413$	−6.14197	−0.302227
$414$	0	0
$415$	9.89538	0.485745
$416$	0	0
$417$	−18.5350	−0.907664
$418$	0	0
$419$	5.77992	0.282368	0.141184	−	0.989983i	$-0.454909\pi$
$419$	5.77992	0.282368	0.141184	+	0.989983i	$0.454909\pi$
$420$	0	0
$421$	5.31824	0.259195	0.129598	−	0.991567i	$-0.458631\pi$
$421$	5.31824	0.259195	0.129598	+	0.991567i	$0.458631\pi$
$422$	0	0
$423$	−6.96402	−0.338602
$424$	0	0
$425$	5.35892	0.259946
$426$	0	0
$427$	4.65175	0.225114
$428$	0	0
$429$	7.58885	0.366393
$430$	0	0
$431$	−34.6708	−1.67004	−0.835018	−	0.550223i	$-0.814543\pi$
$431$	−34.6708	−1.67004	−0.835018	+	0.550223i	$0.814543\pi$
$432$	0	0
$433$	35.4581	1.70401	0.852003	−	0.523537i	$-0.175388\pi$
$433$	35.4581	1.70401	0.852003	+	0.523537i	$0.175388\pi$
$434$	0	0
$435$	−7.34751	−0.352286
$436$	0	0
$437$	−6.10565	−0.292073
$438$	0	0
$439$	0.804934	0.0384174	0.0192087	−	0.999815i	$-0.493885\pi$
$439$	0.804934	0.0384174	0.0192087	+	0.999815i	$0.493885\pi$
$440$	0	0
$441$	−1.43020	−0.0681048
$442$	0	0
$443$	32.0901	1.52465	0.762323	−	0.647196i	$-0.224059\pi$
$443$	32.0901	1.52465	0.762323	+	0.647196i	$0.224059\pi$
$444$	0	0
$445$	1.91688	0.0908690
$446$	0	0
$447$	−33.1737	−1.56906
$448$	0	0
$449$	10.0871	0.476041	0.238021	−	0.971260i	$-0.423501\pi$
$449$	10.0871	0.476041	0.238021	+	0.971260i	$0.423501\pi$
$450$	0	0
$451$	13.1452	0.618985
$452$	0	0
$453$	−2.55062	−0.119839
$454$	0	0
$455$	−3.37772	−0.158350
$456$	0	0
$457$	−38.2058	−1.78719	−0.893595	−	0.448874i	$-0.851825\pi$
$457$	−38.2058	−1.78719	−0.893595	+	0.448874i	$0.851825\pi$
$458$	0	0
$459$	30.2958	1.41408
$460$	0	0
$461$	−35.1925	−1.63908	−0.819540	−	0.573023i	$-0.805771\pi$
$461$	−35.1925	−1.63908	−0.819540	+	0.573023i	$0.805771\pi$
$462$	0	0
$463$	18.3358	0.852138	0.426069	−	0.904691i	$-0.359898\pi$
$463$	18.3358	0.852138	0.426069	+	0.904691i	$0.359898\pi$
$464$	0	0
$465$	−2.22435	−0.103152
$466$	0	0
$467$	5.05941	0.234122	0.117061	−	0.993125i	$-0.462653\pi$
$467$	5.05941	0.234122	0.117061	+	0.993125i	$0.462653\pi$
$468$	0	0
$469$	10.9332	0.504848
$470$	0	0
$471$	−9.48948	−0.437252
$472$	0	0
$473$	−35.8409	−1.64797
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	4.11268	0.188307
$478$	0	0
$479$	−21.4868	−0.981757	−0.490879	−	0.871228i	$-0.663324\pi$
$479$	−21.4868	−0.981757	−0.490879	+	0.871228i	$0.663324\pi$
$480$	0	0
$481$	10.3102	0.470104
$482$	0	0
$483$	−25.7726	−1.17269
$484$	0	0
$485$	−15.6391	−0.710135
$486$	0	0
$487$	−27.7776	−1.25872	−0.629362	−	0.777112i	$-0.716684\pi$
$487$	−27.7776	−1.25872	−0.629362	+	0.777112i	$0.716684\pi$
$488$	0	0
$489$	4.34827	0.196635
$490$	0	0
$491$	18.4624	0.833199	0.416599	−	0.909090i	$-0.363222\pi$
$491$	18.4624	0.833199	0.416599	+	0.909090i	$0.363222\pi$
$492$	0	0
$493$	27.2883	1.22900
$494$	0	0
$495$	4.18154	0.187946
$496$	0	0
$497$	−10.9194	−0.489804
$498$	0	0
$499$	−37.0366	−1.65799	−0.828994	−	0.559258i	$-0.811086\pi$
$499$	−37.0366	−1.65799	−0.828994	+	0.559258i	$0.811086\pi$
$500$	0	0
$501$	−9.31451	−0.416141
$502$	0	0
$503$	−18.5003	−0.824886	−0.412443	−	0.910984i	$-0.635324\pi$
$503$	−18.5003	−0.824886	−0.412443	+	0.910984i	$0.635324\pi$
$504$	0	0
$505$	8.55509	0.380697
$506$	0	0
$507$	16.8343	0.747638
$508$	0	0
$509$	32.9570	1.46079	0.730397	−	0.683023i	$-0.239335\pi$
$509$	32.9570	1.46079	0.730397	+	0.683023i	$0.239335\pi$
$510$	0	0
$511$	28.7197	1.27048
$512$	0	0
$513$	5.65334	0.249601
$514$	0	0
$515$	−9.01793	−0.397378
$516$	0	0
$517$	−34.5556	−1.51975
$518$	0	0
$519$	−16.8567	−0.739927
$520$	0	0
$521$	−26.8526	−1.17643	−0.588216	−	0.808704i	$-0.700170\pi$
$521$	−26.8526	−1.17643	−0.588216	+	0.808704i	$0.700170\pi$
$522$	0	0
$523$	−38.2131	−1.67094	−0.835472	−	0.549534i	$-0.814805\pi$
$523$	−38.2131	−1.67094	−0.835472	+	0.549534i	$0.814805\pi$
$524$	0	0
$525$	4.22111	0.184224
$526$	0	0
$527$	8.26113	0.359861
$528$	0	0
$529$	14.2790	0.620825
$530$	0	0
$531$	−1.92735	−0.0836401
$532$	0	0
$533$	−3.33203	−0.144326
$534$	0	0
$535$	−19.4780	−0.842106
$536$	0	0
$537$	22.4585	0.969154
$538$	0	0
$539$	−7.09668	−0.305676
$540$	0	0
$541$	−10.4930	−0.451129	−0.225564	−	0.974228i	$-0.572423\pi$
$541$	−10.4930	−0.451129	−0.225564	+	0.974228i	$0.572423\pi$
$542$	0	0
$543$	−0.871017	−0.0373789
$544$	0	0
$545$	−8.16328	−0.349676
$546$	0	0
$547$	−38.0140	−1.62536	−0.812680	−	0.582710i	$-0.801992\pi$
$547$	−38.0140	−1.62536	−0.812680	+	0.582710i	$0.801992\pi$
$548$	0	0
$549$	1.45972	0.0622995
$550$	0	0
$551$	5.09212	0.216932
$552$	0	0
$553$	13.4675	0.572697
$554$	0	0
$555$	−12.8846	−0.546919
$556$	0	0
$557$	15.6925	0.664914	0.332457	−	0.943118i	$-0.392122\pi$
$557$	15.6925	0.664914	0.332457	+	0.943118i	$0.392122\pi$
$558$	0	0
$559$	9.08490	0.384250
$560$	0	0
$561$	35.2221	1.48708
$562$	0	0
$563$	−3.58794	−0.151213	−0.0756067	−	0.997138i	$-0.524089\pi$
$563$	−3.58794	−0.151213	−0.0756067	+	0.997138i	$0.524089\pi$
$564$	0	0
$565$	−14.5893	−0.613775
$566$	0	0
$567$	15.8069	0.663825
$568$	0	0
$569$	28.1242	1.17903	0.589514	−	0.807758i	$-0.299319\pi$
$569$	28.1242	1.17903	0.589514	+	0.807758i	$0.299319\pi$
$570$	0	0
$571$	−41.3658	−1.73111	−0.865553	−	0.500818i	$-0.833033\pi$
$571$	−41.3658	−1.73111	−0.865553	+	0.500818i	$0.833033\pi$
$572$	0	0
$573$	−4.46742	−0.186629
$574$	0	0
$575$	−6.10565	−0.254623
$576$	0	0
$577$	−27.6480	−1.15100	−0.575500	−	0.817802i	$-0.695192\pi$
$577$	−27.6480	−1.15100	−0.575500	+	0.817802i	$0.695192\pi$
$578$	0	0
$579$	28.8791	1.20018
$580$	0	0
$581$	−28.9479	−1.20096
$582$	0	0
$583$	20.4072	0.845180
$584$	0	0
$585$	−1.05993	−0.0438227
$586$	0	0
$587$	25.2812	1.04347	0.521733	−	0.853109i	$-0.325286\pi$
$587$	25.2812	1.04347	0.521733	+	0.853109i	$0.325286\pi$
$588$	0	0
$589$	1.54157	0.0635192
$590$	0	0
$591$	23.0563	0.948411
$592$	0	0
$593$	−29.9920	−1.23163	−0.615813	−	0.787893i	$-0.711172\pi$
$593$	−29.9920	−1.23163	−0.615813	+	0.787893i	$0.711172\pi$
$594$	0	0
$595$	−15.6770	−0.642693
$596$	0	0
$597$	6.32364	0.258809
$598$	0	0
$599$	23.4775	0.959266	0.479633	−	0.877469i	$-0.340770\pi$
$599$	23.4775	0.959266	0.479633	+	0.877469i	$0.340770\pi$
$600$	0	0
$601$	−0.213065	−0.00869108	−0.00434554	−	0.999991i	$-0.501383\pi$
$601$	−0.213065	−0.00869108	−0.00434554	+	0.999991i	$0.501383\pi$
$602$	0	0
$603$	3.43084	0.139715
$604$	0	0
$605$	9.74887	0.396348
$606$	0	0
$607$	−32.3246	−1.31202	−0.656008	−	0.754754i	$-0.727756\pi$
$607$	−32.3246	−1.31202	−0.656008	+	0.754754i	$0.727756\pi$
$608$	0	0
$609$	21.4944	0.870997
$610$	0	0
$611$	8.75909	0.354355
$612$	0	0
$613$	−39.5596	−1.59780	−0.798898	−	0.601466i	$-0.794583\pi$
$613$	−39.5596	−1.59780	−0.798898	+	0.601466i	$0.794583\pi$
$614$	0	0
$615$	4.16402	0.167909
$616$	0	0
$617$	−37.8781	−1.52492	−0.762458	−	0.647037i	$-0.776008\pi$
$617$	−37.8781	−1.52492	−0.762458	+	0.647037i	$0.776008\pi$
$618$	0	0
$619$	−6.77119	−0.272157	−0.136079	−	0.990698i	$-0.543450\pi$
$619$	−6.77119	−0.272157	−0.136079	+	0.990698i	$0.543450\pi$
$620$	0	0
$621$	−34.5173	−1.38513
$622$	0	0
$623$	−5.60765	−0.224666
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	6.57262	0.262485
$628$	0	0
$629$	47.8526	1.90801
$630$	0	0
$631$	−28.8072	−1.14680	−0.573398	−	0.819277i	$-0.694375\pi$
$631$	−28.8072	−1.14680	−0.573398	+	0.819277i	$0.694375\pi$
$632$	0	0
$633$	1.09553	0.0435435
$634$	0	0
$635$	−9.99514	−0.396645
$636$	0	0
$637$	1.79885	0.0712731
$638$	0	0
$639$	−3.42653	−0.135551
$640$	0	0
$641$	4.50030	0.177751	0.0888756	−	0.996043i	$-0.471673\pi$
$641$	4.50030	0.177751	0.0888756	+	0.996043i	$0.471673\pi$
$642$	0	0
$643$	−26.7345	−1.05431	−0.527154	−	0.849770i	$-0.676741\pi$
$643$	−26.7345	−1.05431	−0.527154	+	0.849770i	$0.676741\pi$
$644$	0	0
$645$	−11.3533	−0.447037
$646$	0	0
$647$	12.9066	0.507411	0.253705	−	0.967282i	$-0.418351\pi$
$647$	12.9066	0.507411	0.253705	+	0.967282i	$0.418351\pi$
$648$	0	0
$649$	−9.56357	−0.375403
$650$	0	0
$651$	6.50712	0.255034
$652$	0	0
$653$	−20.0704	−0.785414	−0.392707	−	0.919664i	$-0.628461\pi$
$653$	−20.0704	−0.785414	−0.392707	+	0.919664i	$0.628461\pi$
$654$	0	0
$655$	−4.11817	−0.160910
$656$	0	0
$657$	9.01225	0.351601
$658$	0	0
$659$	31.1534	1.21356	0.606782	−	0.794868i	$-0.292460\pi$
$659$	31.1534	1.21356	0.606782	+	0.794868i	$0.292460\pi$
$660$	0	0
$661$	−26.3854	−1.02627	−0.513137	−	0.858307i	$-0.671517\pi$
$661$	−26.3854	−1.02627	−0.513137	+	0.858307i	$0.671517\pi$
$662$	0	0
$663$	−8.92804	−0.346736
$664$	0	0
$665$	−2.92540	−0.113442
$666$	0	0
$667$	−31.0907	−1.20384
$668$	0	0
$669$	−18.6248	−0.720077
$670$	0	0
$671$	7.24317	0.279619
$672$	0	0
$673$	−19.0512	−0.734371	−0.367185	−	0.930148i	$-0.619679\pi$
$673$	−19.0512	−0.734371	−0.367185	+	0.930148i	$0.619679\pi$
$674$	0	0
$675$	5.65334	0.217597
$676$	0	0
$677$	22.5780	0.867744	0.433872	−	0.900975i	$-0.357147\pi$
$677$	22.5780	0.867744	0.433872	+	0.900975i	$0.357147\pi$
$678$	0	0
$679$	45.7506	1.75575
$680$	0	0
$681$	32.8685	1.25952
$682$	0	0
$683$	39.2754	1.50283	0.751415	−	0.659830i	$-0.229372\pi$
$683$	39.2754	1.50283	0.751415	+	0.659830i	$0.229372\pi$
$684$	0	0
$685$	−11.7513	−0.448993
$686$	0	0
$687$	15.0325	0.573525
$688$	0	0
$689$	−5.17278	−0.197067
$690$	0	0
$691$	−21.3498	−0.812185	−0.406093	−	0.913832i	$-0.633109\pi$
$691$	−21.3498	−0.812185	−0.406093	+	0.913832i	$0.633109\pi$
$692$	0	0
$693$	−12.2327	−0.464681
$694$	0	0
$695$	12.8455	0.487259
$696$	0	0
$697$	−15.4649	−0.585776
$698$	0	0
$699$	19.1088	0.722761
$700$	0	0
$701$	−45.8535	−1.73186	−0.865932	−	0.500161i	$-0.833274\pi$
$701$	−45.8535	−1.73186	−0.865932	+	0.500161i	$0.833274\pi$
$702$	0	0
$703$	8.92953	0.336783
$704$	0	0
$705$	−10.9462	−0.412257
$706$	0	0
$707$	−25.0271	−0.941240
$708$	0	0
$709$	−6.34908	−0.238445	−0.119222	−	0.992868i	$-0.538040\pi$
$709$	−6.34908	−0.238445	−0.119222	+	0.992868i	$0.538040\pi$
$710$	0	0
$711$	4.22612	0.158492
$712$	0	0
$713$	−9.41227	−0.352492
$714$	0	0
$715$	−5.25939	−0.196690
$716$	0	0
$717$	−35.5776	−1.32867
$718$	0	0
$719$	21.5313	0.802982	0.401491	−	0.915863i	$-0.368492\pi$
$719$	21.5313	0.802982	0.401491	+	0.915863i	$0.368492\pi$
$720$	0	0
$721$	26.3811	0.982482
$722$	0	0
$723$	−34.9115	−1.29837
$724$	0	0
$725$	5.09212	0.189117
$726$	0	0
$727$	−49.4459	−1.83385	−0.916924	−	0.399062i	$-0.869336\pi$
$727$	−49.4459	−1.83385	−0.916924	+	0.399062i	$0.869336\pi$
$728$	0	0
$729$	29.4321	1.09008
$730$	0	0
$731$	42.1657	1.55955
$732$	0	0
$733$	53.0404	1.95909	0.979546	−	0.201220i	$-0.0644906\pi$
$733$	53.0404	1.95909	0.979546	+	0.201220i	$0.0644906\pi$
$734$	0	0
$735$	−2.24801	−0.0829192
$736$	0	0
$737$	17.0239	0.627084
$738$	0	0
$739$	26.3898	0.970765	0.485382	−	0.874302i	$-0.338680\pi$
$739$	26.3898	0.970765	0.485382	+	0.874302i	$0.338680\pi$
$740$	0	0
$741$	−1.66602	−0.0612026
$742$	0	0
$743$	−14.2042	−0.521102	−0.260551	−	0.965460i	$-0.583904\pi$
$743$	−14.2042	−0.521102	−0.260551	+	0.965460i	$0.583904\pi$
$744$	0	0
$745$	22.9907	0.842316
$746$	0	0
$747$	−9.08388	−0.332362
$748$	0	0
$749$	56.9808	2.08203
$750$	0	0
$751$	31.4599	1.14799	0.573994	−	0.818860i	$-0.305393\pi$
$751$	31.4599	1.14799	0.573994	+	0.818860i	$0.305393\pi$
$752$	0	0
$753$	33.2073	1.21014
$754$	0	0
$755$	1.76768	0.0643326
$756$	0	0
$757$	−15.3981	−0.559654	−0.279827	−	0.960050i	$-0.590277\pi$
$757$	−15.3981	−0.559654	−0.279827	+	0.960050i	$0.590277\pi$
$758$	0	0
$759$	−40.1301	−1.45663
$760$	0	0
$761$	12.4838	0.452537	0.226269	−	0.974065i	$-0.427347\pi$
$761$	12.4838	0.452537	0.226269	+	0.974065i	$0.427347\pi$
$762$	0	0
$763$	23.8809	0.864545
$764$	0	0
$765$	−4.91945	−0.177863
$766$	0	0
$767$	2.42416	0.0875312
$768$	0	0
$769$	−23.4197	−0.844538	−0.422269	−	0.906471i	$-0.638766\pi$
$769$	−23.4197	−0.844538	−0.422269	+	0.906471i	$0.638766\pi$
$770$	0	0
$771$	−13.8250	−0.497895
$772$	0	0
$773$	−52.0914	−1.87360	−0.936799	−	0.349868i	$-0.886226\pi$
$773$	−52.0914	−1.87360	−0.936799	+	0.349868i	$0.886226\pi$
$774$	0	0
$775$	1.54157	0.0553747
$776$	0	0
$777$	37.6925	1.35221
$778$	0	0
$779$	−2.88583	−0.103396
$780$	0	0
$781$	−17.0025	−0.608397
$782$	0	0
$783$	28.7875	1.02878
$784$	0	0
$785$	6.57660	0.234729
$786$	0	0
$787$	8.88515	0.316721	0.158361	−	0.987381i	$-0.449379\pi$
$787$	8.88515	0.316721	0.158361	+	0.987381i	$0.449379\pi$
$788$	0	0
$789$	29.6347	1.05502
$790$	0	0
$791$	42.6794	1.51751
$792$	0	0
$793$	−1.83598	−0.0651977
$794$	0	0
$795$	6.46439	0.229268
$796$	0	0
$797$	23.0257	0.815610	0.407805	−	0.913069i	$-0.366294\pi$
$797$	23.0257	0.815610	0.407805	+	0.913069i	$0.366294\pi$
$798$	0	0
$799$	40.6535	1.43822
$800$	0	0
$801$	−1.75968	−0.0621754
$802$	0	0
$803$	44.7189	1.57810
$804$	0	0
$805$	17.8615	0.629534
$806$	0	0
$807$	−39.0932	−1.37614
$808$	0	0
$809$	32.9398	1.15810	0.579051	−	0.815291i	$-0.303423\pi$
$809$	32.9398	1.15810	0.579051	+	0.815291i	$0.303423\pi$
$810$	0	0
$811$	−28.7407	−1.00922	−0.504612	−	0.863346i	$-0.668364\pi$
$811$	−28.7407	−1.00922	−0.504612	+	0.863346i	$0.668364\pi$
$812$	0	0
$813$	−8.73479	−0.306342
$814$	0	0
$815$	−3.01353	−0.105559
$816$	0	0
$817$	7.86833	0.275278
$818$	0	0
$819$	3.10072	0.108348
$820$	0	0
$821$	−21.9542	−0.766205	−0.383103	−	0.923706i	$-0.625144\pi$
$821$	−21.9542	−0.766205	−0.383103	+	0.923706i	$0.625144\pi$
$822$	0	0
$823$	−31.2671	−1.08990	−0.544951	−	0.838468i	$-0.683452\pi$
$823$	−31.2671	−1.08990	−0.544951	+	0.838468i	$0.683452\pi$
$824$	0	0
$825$	6.57262	0.228829
$826$	0	0
$827$	16.9193	0.588343	0.294172	−	0.955753i	$-0.404956\pi$
$827$	16.9193	0.588343	0.294172	+	0.955753i	$0.404956\pi$
$828$	0	0
$829$	−2.71964	−0.0944571	−0.0472286	−	0.998884i	$-0.515039\pi$
$829$	−2.71964	−0.0944571	−0.0472286	+	0.998884i	$0.515039\pi$
$830$	0	0
$831$	5.32801	0.184827
$832$	0	0
$833$	8.34901	0.289276
$834$	0	0
$835$	6.45533	0.223396
$836$	0	0
$837$	8.71500	0.301234
$838$	0	0
$839$	35.7791	1.23523	0.617617	−	0.786479i	$-0.288098\pi$
$839$	35.7791	1.23523	0.617617	+	0.786479i	$0.288098\pi$
$840$	0	0
$841$	−3.07026	−0.105871
$842$	0	0
$843$	32.3519	1.11426
$844$	0	0
$845$	−11.6669	−0.401352
$846$	0	0
$847$	−28.5193	−0.979936
$848$	0	0
$849$	−13.0173	−0.446753
$850$	0	0
$851$	−54.5206	−1.86894
$852$	0	0
$853$	49.9150	1.70906	0.854528	−	0.519406i	$-0.173847\pi$
$853$	49.9150	1.70906	0.854528	+	0.519406i	$0.173847\pi$
$854$	0	0
$855$	−0.917992	−0.0313947
$856$	0	0
$857$	−21.1297	−0.721778	−0.360889	−	0.932609i	$-0.617527\pi$
$857$	−21.1297	−0.721778	−0.360889	+	0.932609i	$0.617527\pi$
$858$	0	0
$859$	34.5319	1.17821	0.589106	−	0.808056i	$-0.299480\pi$
$859$	34.5319	1.17821	0.589106	+	0.808056i	$0.299480\pi$
$860$	0	0
$861$	−12.1814	−0.415141
$862$	0	0
$863$	−29.2411	−0.995380	−0.497690	−	0.867355i	$-0.665818\pi$
$863$	−29.2411	−0.995380	−0.497690	+	0.867355i	$0.665818\pi$
$864$	0	0
$865$	11.6824	0.397213
$866$	0	0
$867$	−16.9081	−0.574229
$868$	0	0
$869$	20.9701	0.711360
$870$	0	0
$871$	−4.31519	−0.146215
$872$	0	0
$873$	14.3566	0.485897
$874$	0	0
$875$	−2.92540	−0.0988966
$876$	0	0
$877$	42.4819	1.43451	0.717256	−	0.696810i	$-0.245398\pi$
$877$	42.4819	1.43451	0.717256	+	0.696810i	$0.245398\pi$
$878$	0	0
$879$	12.3502	0.416563
$880$	0	0
$881$	27.2758	0.918946	0.459473	−	0.888192i	$-0.348038\pi$
$881$	27.2758	0.918946	0.459473	+	0.888192i	$0.348038\pi$
$882$	0	0
$883$	8.38201	0.282077	0.141039	−	0.990004i	$-0.454956\pi$
$883$	8.38201	0.282077	0.141039	+	0.990004i	$0.454956\pi$
$884$	0	0
$885$	−3.02945	−0.101834
$886$	0	0
$887$	−44.0775	−1.47998	−0.739988	−	0.672620i	$-0.765169\pi$
$887$	−44.0775	−1.47998	−0.739988	+	0.672620i	$0.765169\pi$
$888$	0	0
$889$	29.2398	0.980670
$890$	0	0
$891$	24.6126	0.824553
$892$	0	0
$893$	7.58614	0.253861
$894$	0	0
$895$	−15.5646	−0.520268
$896$	0	0
$897$	10.1721	0.339637
$898$	0	0
$899$	7.84985	0.261807
$900$	0	0
$901$	−24.0084	−0.799836
$902$	0	0
$903$	33.2130	1.10526
$904$	0	0
$905$	0.603651	0.0200660
$906$	0	0
$907$	24.4179	0.810784	0.405392	−	0.914143i	$-0.367135\pi$
$907$	24.4179	0.810784	0.405392	+	0.914143i	$0.367135\pi$
$908$	0	0
$909$	−7.85351	−0.260484
$910$	0	0
$911$	5.36495	0.177749	0.0888744	−	0.996043i	$-0.471673\pi$
$911$	5.36495	0.177749	0.0888744	+	0.996043i	$0.471673\pi$
$912$	0	0
$913$	−45.0744	−1.49174
$914$	0	0
$915$	2.29442	0.0758511
$916$	0	0
$917$	12.0473	0.397836
$918$	0	0
$919$	12.8165	0.422777	0.211389	−	0.977402i	$-0.432201\pi$
$919$	12.8165	0.422777	0.211389	+	0.977402i	$0.432201\pi$
$920$	0	0
$921$	1.54851	0.0510252
$922$	0	0
$923$	4.30976	0.141857
$924$	0	0
$925$	8.92953	0.293601
$926$	0	0
$927$	8.27839	0.271898
$928$	0	0
$929$	−42.4601	−1.39307	−0.696535	−	0.717523i	$-0.745276\pi$
$929$	−42.4601	−1.39307	−0.696535	+	0.717523i	$0.745276\pi$
$930$	0	0
$931$	1.55797	0.0510603
$932$	0	0
$933$	−14.4169	−0.471988
$934$	0	0
$935$	−24.4104	−0.798304
$936$	0	0
$937$	59.5062	1.94398	0.971991	−	0.235017i	$-0.0755146\pi$
$937$	59.5062	1.94398	0.971991	+	0.235017i	$0.0755146\pi$
$938$	0	0
$939$	−35.6306	−1.16276
$940$	0	0
$941$	52.9888	1.72739	0.863693	−	0.504018i	$-0.168146\pi$
$941$	52.9888	1.72739	0.863693	+	0.504018i	$0.168146\pi$
$942$	0	0
$943$	17.6199	0.573782
$944$	0	0
$945$	−16.5383	−0.537990
$946$	0	0
$947$	−40.7246	−1.32337	−0.661685	−	0.749782i	$-0.730158\pi$
$947$	−40.7246	−1.32337	−0.661685	+	0.749782i	$0.730158\pi$
$948$	0	0
$949$	−11.3353	−0.367959
$950$	0	0
$951$	−13.3235	−0.432045
$952$	0	0
$953$	−15.3515	−0.497284	−0.248642	−	0.968595i	$-0.579984\pi$
$953$	−15.3515	−0.497284	−0.248642	+	0.968595i	$0.579984\pi$
$954$	0	0
$955$	3.09611	0.100188
$956$	0	0
$957$	33.4686	1.08189
$958$	0	0
$959$	34.3772	1.11010
$960$	0	0
$961$	−28.6236	−0.923341
$962$	0	0
$963$	17.8806	0.576195
$964$	0	0
$965$	−20.0144	−0.644287
$966$	0	0
$967$	−4.61690	−0.148470	−0.0742348	−	0.997241i	$-0.523651\pi$
$967$	−4.61690	−0.148470	−0.0742348	+	0.997241i	$0.523651\pi$
$968$	0	0
$969$	−7.73247	−0.248403
$970$	0	0
$971$	−11.9619	−0.383876	−0.191938	−	0.981407i	$-0.561477\pi$
$971$	−11.9619	−0.383876	−0.191938	+	0.981407i	$0.561477\pi$
$972$	0	0
$973$	−37.5783	−1.20471
$974$	0	0
$975$	−1.66602	−0.0533552
$976$	0	0
$977$	−41.1459	−1.31637	−0.658187	−	0.752854i	$-0.728676\pi$
$977$	−41.1459	−1.31637	−0.658187	+	0.752854i	$0.728676\pi$
$978$	0	0
$979$	−8.73158	−0.279063
$980$	0	0
$981$	7.49383	0.239259
$982$	0	0
$983$	59.4969	1.89766	0.948828	−	0.315793i	$-0.102271\pi$
$983$	59.4969	1.89766	0.948828	+	0.315793i	$0.102271\pi$
$984$	0	0
$985$	−15.9790	−0.509133
$986$	0	0
$987$	32.0219	1.01927
$988$	0	0
$989$	−48.0412	−1.52762
$990$	0	0
$991$	52.4403	1.66582	0.832910	−	0.553408i	$-0.186673\pi$
$991$	52.4403	1.66582	0.832910	+	0.553408i	$0.186673\pi$
$992$	0	0
$993$	−11.8751	−0.376845
$994$	0	0
$995$	−4.38254	−0.138936
$996$	0	0
$997$	26.9834	0.854574	0.427287	−	0.904116i	$-0.359469\pi$
$997$	26.9834	0.854574	0.427287	+	0.904116i	$0.359469\pi$
$998$	0	0
$999$	50.4816	1.59717

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6080.2.a.ci.1.3		5
4.3	odd	2		6080.2.a.cl.1.3		5
8.3	odd	2		3040.2.a.u.1.3	✓	5
8.5	even	2		3040.2.a.x.1.3	yes	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3040.2.a.u.1.3	✓	5	8.3	odd	2
3040.2.a.x.1.3	yes	5	8.5	even	2
6080.2.a.ci.1.3		5	1.1	even	1	trivial
6080.2.a.cl.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:	\( N \)	\(=\)	\( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6080.a (trivial)

\(f(q)\)	\(=\)	\(q-1.44292 q^{3} +1.00000 q^{5} -2.92540 q^{7} -0.917992 q^{9} -4.55509 q^{11} +1.15462 q^{13} -1.44292 q^{15} +5.35892 q^{17} +1.00000 q^{19} +4.22111 q^{21} -6.10565 q^{23} +1.00000 q^{25} +5.65334 q^{27} +5.09212 q^{29} +1.54157 q^{31} +6.57262 q^{33} -2.92540 q^{35} +8.92953 q^{37} -1.66602 q^{39} -2.88583 q^{41} +7.86833 q^{43} -0.917992 q^{45} +7.58614 q^{47} +1.55797 q^{49} -7.73247 q^{51} -4.48008 q^{53} -4.55509 q^{55} -1.44292 q^{57} +2.09953 q^{59} -1.59013 q^{61} +2.68549 q^{63} +1.15462 q^{65} -3.73733 q^{67} +8.80994 q^{69} +3.73263 q^{71} -9.81735 q^{73} -1.44292 q^{75} +13.3255 q^{77} -4.60365 q^{79} -5.40331 q^{81} +9.89538 q^{83} +5.35892 q^{85} -7.34751 q^{87} +1.91688 q^{89} -3.37772 q^{91} -2.22435 q^{93} +1.00000 q^{95} -15.6391 q^{97} +4.18154 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 4 q^{3} + 5 q^{5} - 4 q^{7} + 7 q^{9} - 2 q^{11} + 4 q^{13} - 4 q^{15} - 12 q^{17} + 5 q^{19} + 10 q^{21} - 8 q^{23} + 5 q^{25} - 16 q^{27} + 6 q^{29} - 10 q^{31} - 18 q^{33} - 4 q^{35} + 6 q^{37}+ \cdots + 14 q^{99}+O(q^{100}) \) 5 * q - 4 * q^3 + 5 * q^5 - 4 * q^7 + 7 * q^9 - 2 * q^11 + 4 * q^13 - 4 * q^15 - 12 * q^17 + 5 * q^19 + 10 * q^21 - 8 * q^23 + 5 * q^25 - 16 * q^27 + 6 * q^29 - 10 * q^31 - 18 * q^33 - 4 * q^35 + 6 * q^37 - 18 * q^39 - 8 * q^41 - 12 * q^43 + 7 * q^45 - 16 * q^47 + 7 * q^49 + 14 * q^51 + 18 * q^53 - 2 * q^55 - 4 * q^57 - 8 * q^59 - 2 * q^61 - 36 * q^63 + 4 * q^65 - 10 * q^67 + 22 * q^69 + 18 * q^71 - 28 * q^73 - 4 * q^75 + 28 * q^77 - 14 * q^79 + 25 * q^81 - 8 * q^83 - 12 * q^85 - 24 * q^87 - 30 * q^89 - 28 * q^91 + 24 * q^93 + 5 * q^95 - 18 * q^97 + 14 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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