LMFDB - Embedded newform 2280.2.a.u.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2280,2,Mod(1,2280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("2280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2280.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:	$18.2058916609$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1772.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 12x + 8 $ x^3 - x^2 - 12*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.67370$ of defining polynomial
Character	$\chi$	$=$	2280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} +0.911179 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} +0.911179 q^{7} +1.00000 q^{9} +2.91118 q^{11} -6.25857 q^{13} +1.00000 q^{15} +4.00000 q^{17} +1.00000 q^{19} +0.911179 q^{21} +5.34740 q^{23} +1.00000 q^{25} +1.00000 q^{27} -0.911179 q^{29} -2.00000 q^{31} +2.91118 q^{33} +0.911179 q^{35} +4.43622 q^{37} -6.25857 q^{39} +6.43622 q^{41} +8.25857 q^{43} +1.00000 q^{45} -5.34740 q^{47} -6.16975 q^{49} +4.00000 q^{51} +3.34740 q^{53} +2.91118 q^{55} +1.00000 q^{57} +5.52504 q^{59} +13.1698 q^{61} +0.911179 q^{63} -6.25857 q^{65} +4.00000 q^{67} +5.34740 q^{69} -16.5171 q^{71} +6.00000 q^{73} +1.00000 q^{75} +2.65260 q^{77} -10.9921 q^{79} +1.00000 q^{81} -3.82236 q^{83} +4.00000 q^{85} -0.911179 q^{87} -2.43622 q^{89} -5.70268 q^{91} -2.00000 q^{93} +1.00000 q^{95} +16.4362 q^{97} +2.91118 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{5} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^5 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{5} + 3 q^{9} + 6 q^{11} + 4 q^{13} + 3 q^{15} + 12 q^{17} + 3 q^{19} - 4 q^{23} + 3 q^{25} + 3 q^{27} - 6 q^{31} + 6 q^{33} - 4 q^{37} + 4 q^{39} + 2 q^{41} + 2 q^{43} + 3 q^{45} + 4 q^{47} + 7 q^{49} + 12 q^{51} - 10 q^{53} + 6 q^{55} + 3 q^{57} + 2 q^{59} + 14 q^{61} + 4 q^{65} + 12 q^{67} - 4 q^{69} - 4 q^{71} + 18 q^{73} + 3 q^{75} + 28 q^{77} - 2 q^{79} + 3 q^{81} - 6 q^{83} + 12 q^{85} + 10 q^{89} - 8 q^{91} - 6 q^{93} + 3 q^{95} + 32 q^{97} + 6 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^5 + 3 * q^9 + 6 * q^11 + 4 * q^13 + 3 * q^15 + 12 * q^17 + 3 * q^19 - 4 * q^23 + 3 * q^25 + 3 * q^27 - 6 * q^31 + 6 * q^33 - 4 * q^37 + 4 * q^39 + 2 * q^41 + 2 * q^43 + 3 * q^45 + 4 * q^47 + 7 * q^49 + 12 * q^51 - 10 * q^53 + 6 * q^55 + 3 * q^57 + 2 * q^59 + 14 * q^61 + 4 * q^65 + 12 * q^67 - 4 * q^69 - 4 * q^71 + 18 * q^73 + 3 * q^75 + 28 * q^77 - 2 * q^79 + 3 * q^81 - 6 * q^83 + 12 * q^85 + 10 * q^89 - 8 * q^91 - 6 * q^93 + 3 * q^95 + 32 * q^97 + 6 * 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
$3$	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
$7$	0.911179	0.344393	0.172197	−	0.985063i	$-0.444914\pi$
$7$	0.911179	0.344393	0.172197	+	0.985063i	$0.444914\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.91118	0.877753	0.438877	−	0.898547i	$-0.355376\pi$
$11$	2.91118	0.877753	0.438877	+	0.898547i	$0.355376\pi$
$12$	0	0
$13$	−6.25857	−1.73582	−0.867908	−	0.496725i	$-0.834536\pi$
$13$	−6.25857	−1.73582	−0.867908	+	0.496725i	$0.834536\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	4.00000	0.970143	0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000	0.970143	0.485071	+	0.874475i	$0.338794\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	0.911179	0.198836
$22$	0	0
$23$	5.34740	1.11501	0.557505	−	0.830174i	$-0.311759\pi$
$23$	5.34740	1.11501	0.557505	+	0.830174i	$0.311759\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.911179	−0.169202	−0.0846008	−	0.996415i	$-0.526962\pi$
$29$	−0.911179	−0.169202	−0.0846008	+	0.996415i	$0.526962\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	0	0
$33$	2.91118	0.506771
$34$	0	0
$35$	0.911179	0.154017
$36$	0	0
$37$	4.43622	0.729310	0.364655	−	0.931143i	$-0.381187\pi$
$37$	4.43622	0.729310	0.364655	+	0.931143i	$0.381187\pi$
$38$	0	0
$39$	−6.25857	−1.00217
$40$	0	0
$41$	6.43622	1.00517	0.502584	−	0.864528i	$-0.332383\pi$
$41$	6.43622	1.00517	0.502584	+	0.864528i	$0.332383\pi$
$42$	0	0
$43$	8.25857	1.25942	0.629710	−	0.776830i	$-0.283174\pi$
$43$	8.25857	1.25942	0.629710	+	0.776830i	$0.283174\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−5.34740	−0.779998	−0.389999	−	0.920815i	$-0.627525\pi$
$47$	−5.34740	−0.779998	−0.389999	+	0.920815i	$0.627525\pi$
$48$	0	0
$49$	−6.16975	−0.881393
$50$	0	0
$51$	4.00000	0.560112
$52$	0	0
$53$	3.34740	0.459800	0.229900	−	0.973214i	$-0.426160\pi$
$53$	3.34740	0.459800	0.229900	+	0.973214i	$0.426160\pi$
$54$	0	0
$55$	2.91118	0.392543
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	5.52504	0.719299	0.359649	−	0.933088i	$-0.382896\pi$
$59$	5.52504	0.719299	0.359649	+	0.933088i	$0.382896\pi$
$60$	0	0
$61$	13.1698	1.68621	0.843107	−	0.537746i	$-0.180724\pi$
$61$	13.1698	1.68621	0.843107	+	0.537746i	$0.180724\pi$
$62$	0	0
$63$	0.911179	0.114798
$64$	0	0
$65$	−6.25857	−0.776281
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	5.34740	0.643751
$70$	0	0
$71$	−16.5171	−1.96022	−0.980112	−	0.198443i	$-0.936412\pi$
$71$	−16.5171	−1.96022	−0.980112	+	0.198443i	$0.936412\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	2.65260	0.302292
$78$	0	0
$79$	−10.9921	−1.23671	−0.618355	−	0.785899i	$-0.712200\pi$
$79$	−10.9921	−1.23671	−0.618355	+	0.785899i	$0.712200\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−3.82236	−0.419558	−0.209779	−	0.977749i	$-0.567274\pi$
$83$	−3.82236	−0.419558	−0.209779	+	0.977749i	$0.567274\pi$
$84$	0	0
$85$	4.00000	0.433861
$86$	0	0
$87$	−0.911179	−0.0976886
$88$	0	0
$89$	−2.43622	−0.258238	−0.129119	−	0.991629i	$-0.541215\pi$
$89$	−2.43622	−0.258238	−0.129119	+	0.991629i	$0.541215\pi$
$90$	0	0
$91$	−5.70268	−0.597803
$92$	0	0
$93$	−2.00000	−0.207390
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	16.4362	1.66884	0.834422	−	0.551125i	$-0.185801\pi$
$97$	16.4362	1.66884	0.834422	+	0.551125i	$0.185801\pi$
$98$	0	0
$99$	2.91118	0.292584
$100$	0	0
$101$	−0.177642	−0.0176761	−0.00883804	−	0.999961i	$-0.502813\pi$
$101$	−0.177642	−0.0176761	−0.00883804	+	0.999961i	$0.502813\pi$
$102$	0	0
$103$	−3.64472	−0.359124	−0.179562	−	0.983747i	$-0.557468\pi$
$103$	−3.64472	−0.359124	−0.179562	+	0.983747i	$0.557468\pi$
$104$	0	0
$105$	0.911179	0.0889219
$106$	0	0
$107$	−3.64472	−0.352348	−0.176174	−	0.984359i	$-0.556372\pi$
$107$	−3.64472	−0.352348	−0.176174	+	0.984359i	$0.556372\pi$
$108$	0	0
$109$	−3.52504	−0.337637	−0.168819	−	0.985647i	$-0.553995\pi$
$109$	−3.52504	−0.337637	−0.168819	+	0.985647i	$0.553995\pi$
$110$	0	0
$111$	4.43622	0.421067
$112$	0	0
$113$	−13.5250	−1.27233	−0.636164	−	0.771554i	$-0.719480\pi$
$113$	−13.5250	−1.27233	−0.636164	+	0.771554i	$0.719480\pi$
$114$	0	0
$115$	5.34740	0.498647
$116$	0	0
$117$	−6.25857	−0.578605
$118$	0	0
$119$	3.64472	0.334110
$120$	0	0
$121$	−2.52504	−0.229549
$122$	0	0
$123$	6.43622	0.580334
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	22.3395	1.98231	0.991155	−	0.132707i	$-0.0423669\pi$
$127$	22.3395	1.98231	0.991155	+	0.132707i	$0.0423669\pi$
$128$	0	0
$129$	8.25857	0.727127
$130$	0	0
$131$	−8.73354	−0.763053	−0.381526	−	0.924358i	$-0.624601\pi$
$131$	−8.73354	−0.763053	−0.381526	+	0.924358i	$0.624601\pi$
$132$	0	0
$133$	0.911179	0.0790092
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	13.3474	1.14034	0.570172	−	0.821525i	$-0.306876\pi$
$137$	13.3474	1.14034	0.570172	+	0.821525i	$0.306876\pi$
$138$	0	0
$139$	1.82236	0.154570	0.0772852	−	0.997009i	$-0.475375\pi$
$139$	1.82236	0.154570	0.0772852	+	0.997009i	$0.475375\pi$
$140$	0	0
$141$	−5.34740	−0.450332
$142$	0	0
$143$	−18.2198	−1.52362
$144$	0	0
$145$	−0.911179	−0.0756693
$146$	0	0
$147$	−6.16975	−0.508873
$148$	0	0
$149$	4.69479	0.384612	0.192306	−	0.981335i	$-0.438403\pi$
$149$	4.69479	0.384612	0.192306	+	0.981335i	$0.438403\pi$
$150$	0	0
$151$	4.69479	0.382057	0.191028	−	0.981585i	$-0.438818\pi$
$151$	4.69479	0.382057	0.191028	+	0.981585i	$0.438818\pi$
$152$	0	0
$153$	4.00000	0.323381
$154$	0	0
$155$	−2.00000	−0.160644
$156$	0	0
$157$	−3.16975	−0.252974	−0.126487	−	0.991968i	$-0.540370\pi$
$157$	−3.16975	−0.252974	−0.126487	+	0.991968i	$0.540370\pi$
$158$	0	0
$159$	3.34740	0.265466
$160$	0	0
$161$	4.87243	0.384002
$162$	0	0
$163$	−6.43622	−0.504123	−0.252062	−	0.967711i	$-0.581109\pi$
$163$	−6.43622	−0.504123	−0.252062	+	0.967711i	$0.581109\pi$
$164$	0	0
$165$	2.91118	0.226635
$166$	0	0
$167$	3.16975	0.245283	0.122641	−	0.992451i	$-0.460863\pi$
$167$	3.16975	0.245283	0.122641	+	0.992451i	$0.460863\pi$
$168$	0	0
$169$	26.1698	2.01306
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−9.52504	−0.724175	−0.362088	−	0.932144i	$-0.617936\pi$
$173$	−9.52504	−0.724175	−0.362088	+	0.932144i	$0.617936\pi$
$174$	0	0
$175$	0.911179	0.0688786
$176$	0	0
$177$	5.52504	0.415287
$178$	0	0
$179$	5.16975	0.386405	0.193203	−	0.981159i	$-0.438112\pi$
$179$	5.16975	0.386405	0.193203	+	0.981159i	$0.438112\pi$
$180$	0	0
$181$	18.8145	1.39847	0.699234	−	0.714893i	$-0.253524\pi$
$181$	18.8145	1.39847	0.699234	+	0.714893i	$0.253524\pi$
$182$	0	0
$183$	13.1698	0.973536
$184$	0	0
$185$	4.43622	0.326157
$186$	0	0
$187$	11.6447	0.851546
$188$	0	0
$189$	0.911179	0.0662785
$190$	0	0
$191$	7.78361	0.563202	0.281601	−	0.959532i	$-0.409134\pi$
$191$	7.78361	0.563202	0.281601	+	0.959532i	$0.409134\pi$
$192$	0	0
$193$	−15.7256	−1.13196	−0.565978	−	0.824420i	$-0.691501\pi$
$193$	−15.7256	−1.13196	−0.565978	+	0.824420i	$0.691501\pi$
$194$	0	0
$195$	−6.25857	−0.448186
$196$	0	0
$197$	−7.70268	−0.548793	−0.274397	−	0.961617i	$-0.588478\pi$
$197$	−7.70268	−0.548793	−0.274397	+	0.961617i	$0.588478\pi$
$198$	0	0
$199$	−20.5171	−1.45442	−0.727211	−	0.686414i	$-0.759184\pi$
$199$	−20.5171	−1.45442	−0.727211	+	0.686414i	$0.759184\pi$
$200$	0	0
$201$	4.00000	0.282138
$202$	0	0
$203$	−0.830247	−0.0582719
$204$	0	0
$205$	6.43622	0.449525
$206$	0	0
$207$	5.34740	0.371670
$208$	0	0
$209$	2.91118	0.201370
$210$	0	0
$211$	−22.3395	−1.53792	−0.768958	−	0.639300i	$-0.779224\pi$
$211$	−22.3395	−1.53792	−0.768958	+	0.639300i	$0.779224\pi$
$212$	0	0
$213$	−16.5171	−1.13174
$214$	0	0
$215$	8.25857	0.563230
$216$	0	0
$217$	−1.82236	−0.123710
$218$	0	0
$219$	6.00000	0.405442
$220$	0	0
$221$	−25.0343	−1.68399
$222$	0	0
$223$	−18.3395	−1.22810	−0.614052	−	0.789266i	$-0.710462\pi$
$223$	−18.3395	−1.22810	−0.614052	+	0.789266i	$0.710462\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−17.8645	−1.18571	−0.592856	−	0.805309i	$-0.702000\pi$
$227$	−17.8645	−1.18571	−0.592856	+	0.805309i	$0.702000\pi$
$228$	0	0
$229$	25.1698	1.66326	0.831632	−	0.555327i	$-0.187407\pi$
$229$	25.1698	1.66326	0.831632	+	0.555327i	$0.187407\pi$
$230$	0	0
$231$	2.65260	0.174529
$232$	0	0
$233$	15.6869	1.02768	0.513842	−	0.857885i	$-0.328222\pi$
$233$	15.6869	1.02768	0.513842	+	0.857885i	$0.328222\pi$
$234$	0	0
$235$	−5.34740	−0.348826
$236$	0	0
$237$	−10.9921	−0.714014
$238$	0	0
$239$	3.42833	0.221760	0.110880	−	0.993834i	$-0.464633\pi$
$239$	3.42833	0.221760	0.110880	+	0.993834i	$0.464633\pi$
$240$	0	0
$241$	−23.0343	−1.48377	−0.741885	−	0.670527i	$-0.766068\pi$
$241$	−23.0343	−1.48377	−0.741885	+	0.670527i	$0.766068\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−6.16975	−0.394171
$246$	0	0
$247$	−6.25857	−0.398224
$248$	0	0
$249$	−3.82236	−0.242232
$250$	0	0
$251$	19.0730	1.20388	0.601940	−	0.798541i	$-0.294395\pi$
$251$	19.0730	1.20388	0.601940	+	0.798541i	$0.294395\pi$
$252$	0	0
$253$	15.5672	0.978703
$254$	0	0
$255$	4.00000	0.250490
$256$	0	0
$257$	−15.8645	−0.989603	−0.494802	−	0.869006i	$-0.664759\pi$
$257$	−15.8645	−0.989603	−0.494802	+	0.869006i	$0.664759\pi$
$258$	0	0
$259$	4.04219	0.251169
$260$	0	0
$261$	−0.911179	−0.0564006
$262$	0	0
$263$	−3.16975	−0.195455	−0.0977277	−	0.995213i	$-0.531157\pi$
$263$	−3.16975	−0.195455	−0.0977277	+	0.995213i	$0.531157\pi$
$264$	0	0
$265$	3.34740	0.205629
$266$	0	0
$267$	−2.43622	−0.149094
$268$	0	0
$269$	−13.7836	−0.840402	−0.420201	−	0.907431i	$-0.638040\pi$
$269$	−13.7836	−0.840402	−0.420201	+	0.907431i	$0.638040\pi$
$270$	0	0
$271$	−17.8224	−1.08263	−0.541316	−	0.840820i	$-0.682074\pi$
$271$	−17.8224	−1.08263	−0.541316	+	0.840820i	$0.682074\pi$
$272$	0	0
$273$	−5.70268	−0.345142
$274$	0	0
$275$	2.91118	0.175551
$276$	0	0
$277$	21.3474	1.28264	0.641320	−	0.767273i	$-0.278387\pi$
$277$	21.3474	1.28264	0.641320	+	0.767273i	$0.278387\pi$
$278$	0	0
$279$	−2.00000	−0.119737
$280$	0	0
$281$	14.4362	0.861192	0.430596	−	0.902545i	$-0.358303\pi$
$281$	14.4362	0.861192	0.430596	+	0.902545i	$0.358303\pi$
$282$	0	0
$283$	−6.08093	−0.361474	−0.180737	−	0.983531i	$-0.557848\pi$
$283$	−6.08093	−0.361474	−0.180737	+	0.983531i	$0.557848\pi$
$284$	0	0
$285$	1.00000	0.0592349
$286$	0	0
$287$	5.86454	0.346173
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	0	0
$291$	16.4362	0.963508
$292$	0	0
$293$	−25.1698	−1.47043	−0.735216	−	0.677833i	$-0.762919\pi$
$293$	−25.1698	−1.47043	−0.735216	+	0.677833i	$0.762919\pi$
$294$	0	0
$295$	5.52504	0.321680
$296$	0	0
$297$	2.91118	0.168924
$298$	0	0
$299$	−33.4671	−1.93545
$300$	0	0
$301$	7.52504	0.433736
$302$	0	0
$303$	−0.177642	−0.0102053
$304$	0	0
$305$	13.1698	0.754098
$306$	0	0
$307$	22.1776	1.26574	0.632872	−	0.774256i	$-0.281876\pi$
$307$	22.1776	1.26574	0.632872	+	0.774256i	$0.281876\pi$
$308$	0	0
$309$	−3.64472	−0.207341
$310$	0	0
$311$	−23.7836	−1.34864	−0.674322	−	0.738437i	$-0.735564\pi$
$311$	−23.7836	−1.34864	−0.674322	+	0.738437i	$0.735564\pi$
$312$	0	0
$313$	15.4671	0.874251	0.437125	−	0.899401i	$-0.355997\pi$
$313$	15.4671	0.874251	0.437125	+	0.899401i	$0.355997\pi$
$314$	0	0
$315$	0.911179	0.0513391
$316$	0	0
$317$	32.3817	1.81874	0.909369	−	0.415991i	$-0.136565\pi$
$317$	32.3817	1.81874	0.909369	+	0.415991i	$0.136565\pi$
$318$	0	0
$319$	−2.65260	−0.148517
$320$	0	0
$321$	−3.64472	−0.203428
$322$	0	0
$323$	4.00000	0.222566
$324$	0	0
$325$	−6.25857	−0.347163
$326$	0	0
$327$	−3.52504	−0.194935
$328$	0	0
$329$	−4.87243	−0.268626
$330$	0	0
$331$	−20.0422	−1.10162	−0.550809	−	0.834631i	$-0.685681\pi$
$331$	−20.0422	−1.10162	−0.550809	+	0.834631i	$0.685681\pi$
$332$	0	0
$333$	4.43622	0.243103
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	13.9033	0.757360	0.378680	−	0.925528i	$-0.376378\pi$
$337$	13.9033	0.757360	0.378680	+	0.925528i	$0.376378\pi$
$338$	0	0
$339$	−13.5250	−0.734579
$340$	0	0
$341$	−5.82236	−0.315298
$342$	0	0
$343$	−12.0000	−0.647939
$344$	0	0
$345$	5.34740	0.287894
$346$	0	0
$347$	7.46707	0.400853	0.200427	−	0.979709i	$-0.435767\pi$
$347$	7.46707	0.400853	0.200427	+	0.979709i	$0.435767\pi$
$348$	0	0
$349$	16.6948	0.893652	0.446826	−	0.894621i	$-0.352554\pi$
$349$	16.6948	0.893652	0.446826	+	0.894621i	$0.352554\pi$
$350$	0	0
$351$	−6.25857	−0.334058
$352$	0	0
$353$	−9.70268	−0.516422	−0.258211	−	0.966089i	$-0.583133\pi$
$353$	−9.70268	−0.516422	−0.258211	+	0.966089i	$0.583133\pi$
$354$	0	0
$355$	−16.5171	−0.876639
$356$	0	0
$357$	3.64472	0.192899
$358$	0	0
$359$	−29.6060	−1.56254	−0.781272	−	0.624191i	$-0.785429\pi$
$359$	−29.6060	−1.56254	−0.781272	+	0.624191i	$0.785429\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−2.52504	−0.132530
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	−13.7836	−0.719499	−0.359749	−	0.933049i	$-0.617138\pi$
$367$	−13.7836	−0.719499	−0.359749	+	0.933049i	$0.617138\pi$
$368$	0	0
$369$	6.43622	0.335056
$370$	0	0
$371$	3.05008	0.158352
$372$	0	0
$373$	−2.25857	−0.116945	−0.0584723	−	0.998289i	$-0.518623\pi$
$373$	−2.25857	−0.116945	−0.0584723	+	0.998289i	$0.518623\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	5.70268	0.293703
$378$	0	0
$379$	−17.7027	−0.909326	−0.454663	−	0.890664i	$-0.650240\pi$
$379$	−17.7027	−0.909326	−0.454663	+	0.890664i	$0.650240\pi$
$380$	0	0
$381$	22.3395	1.14449
$382$	0	0
$383$	15.0501	0.769023	0.384511	−	0.923120i	$-0.374370\pi$
$383$	15.0501	0.769023	0.384511	+	0.923120i	$0.374370\pi$
$384$	0	0
$385$	2.65260	0.135189
$386$	0	0
$387$	8.25857	0.419807
$388$	0	0
$389$	−14.0000	−0.709828	−0.354914	−	0.934899i	$-0.615490\pi$
$389$	−14.0000	−0.709828	−0.354914	+	0.934899i	$0.615490\pi$
$390$	0	0
$391$	21.3896	1.08172
$392$	0	0
$393$	−8.73354	−0.440549
$394$	0	0
$395$	−10.9921	−0.553073
$396$	0	0
$397$	0.313098	0.0157139	0.00785697	−	0.999969i	$-0.497499\pi$
$397$	0.313098	0.0157139	0.00785697	+	0.999969i	$0.497499\pi$
$398$	0	0
$399$	0.911179	0.0456160
$400$	0	0
$401$	−7.03086	−0.351104	−0.175552	−	0.984470i	$-0.556171\pi$
$401$	−7.03086	−0.351104	−0.175552	+	0.984470i	$0.556171\pi$
$402$	0	0
$403$	12.5171	0.623524
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	12.9146	0.640154
$408$	0	0
$409$	20.1776	0.997720	0.498860	−	0.866683i	$-0.333752\pi$
$409$	20.1776	0.997720	0.498860	+	0.866683i	$0.333752\pi$
$410$	0	0
$411$	13.3474	0.658378
$412$	0	0
$413$	5.03430	0.247722
$414$	0	0
$415$	−3.82236	−0.187632
$416$	0	0
$417$	1.82236	0.0892412
$418$	0	0
$419$	−15.4283	−0.753723	−0.376862	−	0.926270i	$-0.622997\pi$
$419$	−15.4283	−0.753723	−0.376862	+	0.926270i	$0.622997\pi$
$420$	0	0
$421$	27.3316	1.33206	0.666031	−	0.745924i	$-0.267992\pi$
$421$	27.3316	1.33206	0.666031	+	0.745924i	$0.267992\pi$
$422$	0	0
$423$	−5.34740	−0.259999
$424$	0	0
$425$	4.00000	0.194029
$426$	0	0
$427$	12.0000	0.580721
$428$	0	0
$429$	−18.2198	−0.879662
$430$	0	0
$431$	−1.22772	−0.0591371	−0.0295686	−	0.999563i	$-0.509413\pi$
$431$	−1.22772	−0.0591371	−0.0295686	+	0.999563i	$0.509413\pi$
$432$	0	0
$433$	−24.4362	−1.17433	−0.587165	−	0.809467i	$-0.699756\pi$
$433$	−24.4362	−1.17433	−0.587165	+	0.809467i	$0.699756\pi$
$434$	0	0
$435$	−0.911179	−0.0436877
$436$	0	0
$437$	5.34740	0.255801
$438$	0	0
$439$	−34.9921	−1.67008	−0.835041	−	0.550187i	$-0.814556\pi$
$439$	−34.9921	−1.67008	−0.835041	+	0.550187i	$0.814556\pi$
$440$	0	0
$441$	−6.16975	−0.293798
$442$	0	0
$443$	−31.3896	−1.49136	−0.745682	−	0.666302i	$-0.767876\pi$
$443$	−31.3896	−1.49136	−0.745682	+	0.666302i	$0.767876\pi$
$444$	0	0
$445$	−2.43622	−0.115488
$446$	0	0
$447$	4.69479	0.222056
$448$	0	0
$449$	−12.2586	−0.578518	−0.289259	−	0.957251i	$-0.593409\pi$
$449$	−12.2586	−0.578518	−0.289259	+	0.957251i	$0.593409\pi$
$450$	0	0
$451$	18.7370	0.882290
$452$	0	0
$453$	4.69479	0.220581
$454$	0	0
$455$	−5.70268	−0.267346
$456$	0	0
$457$	−17.2119	−0.805141	−0.402570	−	0.915389i	$-0.631883\pi$
$457$	−17.2119	−0.805141	−0.402570	+	0.915389i	$0.631883\pi$
$458$	0	0
$459$	4.00000	0.186704
$460$	0	0
$461$	−28.6948	−1.33645	−0.668225	−	0.743959i	$-0.732946\pi$
$461$	−28.6948	−1.33645	−0.668225	+	0.743959i	$0.732946\pi$
$462$	0	0
$463$	−7.08882	−0.329445	−0.164723	−	0.986340i	$-0.552673\pi$
$463$	−7.08882	−0.329445	−0.164723	+	0.986340i	$0.552673\pi$
$464$	0	0
$465$	−2.00000	−0.0927478
$466$	0	0
$467$	−26.6790	−1.23456	−0.617279	−	0.786745i	$-0.711765\pi$
$467$	−26.6790	−1.23456	−0.617279	+	0.786745i	$0.711765\pi$
$468$	0	0
$469$	3.64472	0.168297
$470$	0	0
$471$	−3.16975	−0.146055
$472$	0	0
$473$	24.0422	1.10546
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	3.34740	0.153267
$478$	0	0
$479$	−29.0888	−1.32910	−0.664551	−	0.747243i	$-0.731377\pi$
$479$	−29.0888	−1.32910	−0.664551	+	0.747243i	$0.731377\pi$
$480$	0	0
$481$	−27.7644	−1.26595
$482$	0	0
$483$	4.87243	0.221703
$484$	0	0
$485$	16.4362	0.746330
$486$	0	0
$487$	23.8066	1.07878	0.539390	−	0.842056i	$-0.318655\pi$
$487$	23.8066	1.07878	0.539390	+	0.842056i	$0.318655\pi$
$488$	0	0
$489$	−6.43622	−0.291056
$490$	0	0
$491$	−13.6060	−0.614029	−0.307014	−	0.951705i	$-0.599330\pi$
$491$	−13.6060	−0.614029	−0.307014	+	0.951705i	$0.599330\pi$
$492$	0	0
$493$	−3.64472	−0.164150
$494$	0	0
$495$	2.91118	0.130848
$496$	0	0
$497$	−15.0501	−0.675088
$498$	0	0
$499$	−6.17764	−0.276549	−0.138275	−	0.990394i	$-0.544156\pi$
$499$	−6.17764	−0.276549	−0.138275	+	0.990394i	$0.544156\pi$
$500$	0	0
$501$	3.16975	0.141614
$502$	0	0
$503$	2.81447	0.125491	0.0627455	−	0.998030i	$-0.480014\pi$
$503$	2.81447	0.125491	0.0627455	+	0.998030i	$0.480014\pi$
$504$	0	0
$505$	−0.177642	−0.00790498
$506$	0	0
$507$	26.1698	1.16224
$508$	0	0
$509$	−39.7678	−1.76268	−0.881339	−	0.472484i	$-0.843357\pi$
$509$	−39.7678	−1.76268	−0.881339	+	0.472484i	$0.843357\pi$
$510$	0	0
$511$	5.46707	0.241849
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	−3.64472	−0.160605
$516$	0	0
$517$	−15.5672	−0.684646
$518$	0	0
$519$	−9.52504	−0.418103
$520$	0	0
$521$	−38.5981	−1.69101	−0.845506	−	0.533965i	$-0.820701\pi$
$521$	−38.5981	−1.69101	−0.845506	+	0.533965i	$0.820701\pi$
$522$	0	0
$523$	38.5014	1.68355	0.841774	−	0.539831i	$-0.181512\pi$
$523$	38.5014	1.68355	0.841774	+	0.539831i	$0.181512\pi$
$524$	0	0
$525$	0.911179	0.0397671
$526$	0	0
$527$	−8.00000	−0.348485
$528$	0	0
$529$	5.59464	0.243245
$530$	0	0
$531$	5.52504	0.239766
$532$	0	0
$533$	−40.2815	−1.74479
$534$	0	0
$535$	−3.64472	−0.157575
$536$	0	0
$537$	5.16975	0.223091
$538$	0	0
$539$	−17.9613	−0.773646
$540$	0	0
$541$	−10.0000	−0.429934	−0.214967	−	0.976621i	$-0.568964\pi$
$541$	−10.0000	−0.429934	−0.214967	+	0.976621i	$0.568964\pi$
$542$	0	0
$543$	18.8145	0.807406
$544$	0	0
$545$	−3.52504	−0.150996
$546$	0	0
$547$	14.1776	0.606192	0.303096	−	0.952960i	$-0.401980\pi$
$547$	14.1776	0.606192	0.303096	+	0.952960i	$0.401980\pi$
$548$	0	0
$549$	13.1698	0.562071
$550$	0	0
$551$	−0.911179	−0.0388175
$552$	0	0
$553$	−10.0158	−0.425914
$554$	0	0
$555$	4.43622	0.188307
$556$	0	0
$557$	6.00000	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$558$	0	0
$559$	−51.6869	−2.18612
$560$	0	0
$561$	11.6447	0.491640
$562$	0	0
$563$	28.8302	1.21505	0.607525	−	0.794301i	$-0.292162\pi$
$563$	28.8302	1.21505	0.607525	+	0.794301i	$0.292162\pi$
$564$	0	0
$565$	−13.5250	−0.569003
$566$	0	0
$567$	0.911179	0.0382659
$568$	0	0
$569$	−28.2586	−1.18466	−0.592331	−	0.805695i	$-0.701792\pi$
$569$	−28.2586	−1.18466	−0.592331	+	0.805695i	$0.701792\pi$
$570$	0	0
$571$	0.872434	0.0365102	0.0182551	−	0.999833i	$-0.494189\pi$
$571$	0.872434	0.0365102	0.0182551	+	0.999833i	$0.494189\pi$
$572$	0	0
$573$	7.78361	0.325165
$574$	0	0
$575$	5.34740	0.223002
$576$	0	0
$577$	25.8066	1.07434	0.537171	−	0.843473i	$-0.319493\pi$
$577$	25.8066	1.07434	0.537171	+	0.843473i	$0.319493\pi$
$578$	0	0
$579$	−15.7256	−0.653536
$580$	0	0
$581$	−3.48285	−0.144493
$582$	0	0
$583$	9.74487	0.403591
$584$	0	0
$585$	−6.25857	−0.258760
$586$	0	0
$587$	16.6948	0.689068	0.344534	−	0.938774i	$-0.388037\pi$
$587$	16.6948	0.689068	0.344534	+	0.938774i	$0.388037\pi$
$588$	0	0
$589$	−2.00000	−0.0824086
$590$	0	0
$591$	−7.70268	−0.316846
$592$	0	0
$593$	−20.6368	−0.847453	−0.423726	−	0.905790i	$-0.639278\pi$
$593$	−20.6368	−0.847453	−0.423726	+	0.905790i	$0.639278\pi$
$594$	0	0
$595$	3.64472	0.149419
$596$	0	0
$597$	−20.5171	−0.839711
$598$	0	0
$599$	−14.9341	−0.610193	−0.305096	−	0.952321i	$-0.598689\pi$
$599$	−14.9341	−0.610193	−0.305096	+	0.952321i	$0.598689\pi$
$600$	0	0
$601$	−6.51715	−0.265840	−0.132920	−	0.991127i	$-0.542435\pi$
$601$	−6.51715	−0.265840	−0.132920	+	0.991127i	$0.542435\pi$
$602$	0	0
$603$	4.00000	0.162893
$604$	0	0
$605$	−2.52504	−0.102657
$606$	0	0
$607$	−34.5014	−1.40037	−0.700184	−	0.713963i	$-0.746899\pi$
$607$	−34.5014	−1.40037	−0.700184	+	0.713963i	$0.746899\pi$
$608$	0	0
$609$	−0.830247	−0.0336433
$610$	0	0
$611$	33.4671	1.35393
$612$	0	0
$613$	−28.5593	−1.15350	−0.576750	−	0.816920i	$-0.695679\pi$
$613$	−28.5593	−1.15350	−0.576750	+	0.816920i	$0.695679\pi$
$614$	0	0
$615$	6.43622	0.259533
$616$	0	0
$617$	25.9842	1.04609	0.523043	−	0.852306i	$-0.324797\pi$
$617$	25.9842	1.04609	0.523043	+	0.852306i	$0.324797\pi$
$618$	0	0
$619$	5.22772	0.210120	0.105060	−	0.994466i	$-0.466497\pi$
$619$	5.22772	0.210120	0.105060	+	0.994466i	$0.466497\pi$
$620$	0	0
$621$	5.34740	0.214584
$622$	0	0
$623$	−2.21983	−0.0889356
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	2.91118	0.116261
$628$	0	0
$629$	17.7449	0.707534
$630$	0	0
$631$	−45.3896	−1.80693	−0.903465	−	0.428661i	$-0.858985\pi$
$631$	−45.3896	−1.80693	−0.903465	+	0.428661i	$0.858985\pi$
$632$	0	0
$633$	−22.3395	−0.887916
$634$	0	0
$635$	22.3395	0.886516
$636$	0	0
$637$	38.6139	1.52994
$638$	0	0
$639$	−16.5171	−0.653408
$640$	0	0
$641$	11.5480	0.456119	0.228059	−	0.973647i	$-0.426762\pi$
$641$	11.5480	0.456119	0.228059	+	0.973647i	$0.426762\pi$
$642$	0	0
$643$	15.7414	0.620781	0.310391	−	0.950609i	$-0.399540\pi$
$643$	15.7414	0.620781	0.310391	+	0.950609i	$0.399540\pi$
$644$	0	0
$645$	8.25857	0.325181
$646$	0	0
$647$	−17.8645	−0.702328	−0.351164	−	0.936314i	$-0.614214\pi$
$647$	−17.8645	−0.702328	−0.351164	+	0.936314i	$0.614214\pi$
$648$	0	0
$649$	16.0844	0.631367
$650$	0	0
$651$	−1.82236	−0.0714238
$652$	0	0
$653$	−42.9921	−1.68241	−0.841206	−	0.540715i	$-0.818154\pi$
$653$	−42.9921	−1.68241	−0.841206	+	0.540715i	$0.818154\pi$
$654$	0	0
$655$	−8.73354	−0.341248
$656$	0	0
$657$	6.00000	0.234082
$658$	0	0
$659$	48.3817	1.88468	0.942342	−	0.334653i	$-0.108619\pi$
$659$	48.3817	1.88468	0.942342	+	0.334653i	$0.108619\pi$
$660$	0	0
$661$	−25.8645	−1.00601	−0.503007	−	0.864282i	$-0.667773\pi$
$661$	−25.8645	−1.00601	−0.503007	+	0.864282i	$0.667773\pi$
$662$	0	0
$663$	−25.0343	−0.972252
$664$	0	0
$665$	0.911179	0.0353340
$666$	0	0
$667$	−4.87243	−0.188661
$668$	0	0
$669$	−18.3395	−0.709046
$670$	0	0
$671$	38.3395	1.48008
$672$	0	0
$673$	42.9376	1.65512	0.827561	−	0.561376i	$-0.189728\pi$
$673$	42.9376	1.65512	0.827561	+	0.561376i	$0.189728\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	10.0422	0.385953	0.192976	−	0.981203i	$-0.438186\pi$
$677$	10.0422	0.385953	0.192976	+	0.981203i	$0.438186\pi$
$678$	0	0
$679$	14.9763	0.574739
$680$	0	0
$681$	−17.8645	−0.684571
$682$	0	0
$683$	30.6948	1.17450	0.587252	−	0.809404i	$-0.300210\pi$
$683$	30.6948	1.17450	0.587252	+	0.809404i	$0.300210\pi$
$684$	0	0
$685$	13.3474	0.509978
$686$	0	0
$687$	25.1698	0.960286
$688$	0	0
$689$	−20.9499	−0.798129
$690$	0	0
$691$	−3.48285	−0.132494	−0.0662470	−	0.997803i	$-0.521103\pi$
$691$	−3.48285	−0.132494	−0.0662470	+	0.997803i	$0.521103\pi$
$692$	0	0
$693$	2.65260	0.100764
$694$	0	0
$695$	1.82236	0.0691260
$696$	0	0
$697$	25.7449	0.975156
$698$	0	0
$699$	15.6869	0.593333
$700$	0	0
$701$	−23.3896	−0.883412	−0.441706	−	0.897160i	$-0.645627\pi$
$701$	−23.3896	−0.883412	−0.441706	+	0.897160i	$0.645627\pi$
$702$	0	0
$703$	4.43622	0.167315
$704$	0	0
$705$	−5.34740	−0.201395
$706$	0	0
$707$	−0.161864	−0.00608752
$708$	0	0
$709$	−20.4592	−0.768361	−0.384180	−	0.923258i	$-0.625516\pi$
$709$	−20.4592	−0.768361	−0.384180	+	0.923258i	$0.625516\pi$
$710$	0	0
$711$	−10.9921	−0.412236
$712$	0	0
$713$	−10.6948	−0.400523
$714$	0	0
$715$	−18.2198	−0.681383
$716$	0	0
$717$	3.42833	0.128033
$718$	0	0
$719$	12.3783	0.461631	0.230815	−	0.972998i	$-0.425861\pi$
$719$	12.3783	0.461631	0.230815	+	0.972998i	$0.425861\pi$
$720$	0	0
$721$	−3.32099	−0.123680
$722$	0	0
$723$	−23.0343	−0.856655
$724$	0	0
$725$	−0.911179	−0.0338403
$726$	0	0
$727$	2.49418	0.0925041	0.0462520	−	0.998930i	$-0.485272\pi$
$727$	2.49418	0.0925041	0.0462520	+	0.998930i	$0.485272\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	33.0343	1.22182
$732$	0	0
$733$	14.6526	0.541206	0.270603	−	0.962691i	$-0.412777\pi$
$733$	14.6526	0.541206	0.270603	+	0.962691i	$0.412777\pi$
$734$	0	0
$735$	−6.16975	−0.227575
$736$	0	0
$737$	11.6447	0.428939
$738$	0	0
$739$	21.0343	0.773759	0.386880	−	0.922130i	$-0.373553\pi$
$739$	21.0343	0.773759	0.386880	+	0.922130i	$0.373553\pi$
$740$	0	0
$741$	−6.25857	−0.229914
$742$	0	0
$743$	−0.949924	−0.0348493	−0.0174247	−	0.999848i	$-0.505547\pi$
$743$	−0.949924	−0.0348493	−0.0174247	+	0.999848i	$0.505547\pi$
$744$	0	0
$745$	4.69479	0.172004
$746$	0	0
$747$	−3.82236	−0.139853
$748$	0	0
$749$	−3.32099	−0.121346
$750$	0	0
$751$	−20.6948	−0.755164	−0.377582	−	0.925976i	$-0.623244\pi$
$751$	−20.6948	−0.755164	−0.377582	+	0.925976i	$0.623244\pi$
$752$	0	0
$753$	19.0730	0.695060
$754$	0	0
$755$	4.69479	0.170861
$756$	0	0
$757$	23.8803	0.867945	0.433973	−	0.900926i	$-0.357111\pi$
$757$	23.8803	0.867945	0.433973	+	0.900926i	$0.357111\pi$
$758$	0	0
$759$	15.5672	0.565054
$760$	0	0
$761$	53.3738	1.93480	0.967399	−	0.253255i	$-0.0815013\pi$
$761$	53.3738	1.93480	0.967399	+	0.253255i	$0.0815013\pi$
$762$	0	0
$763$	−3.21194	−0.116280
$764$	0	0
$765$	4.00000	0.144620
$766$	0	0
$767$	−34.5789	−1.24857
$768$	0	0
$769$	29.0501	1.04757	0.523786	−	0.851850i	$-0.324519\pi$
$769$	29.0501	1.04757	0.523786	+	0.851850i	$0.324519\pi$
$770$	0	0
$771$	−15.8645	−0.571348
$772$	0	0
$773$	−31.3474	−1.12749	−0.563744	−	0.825950i	$-0.690639\pi$
$773$	−31.3474	−1.12749	−0.563744	+	0.825950i	$0.690639\pi$
$774$	0	0
$775$	−2.00000	−0.0718421
$776$	0	0
$777$	4.04219	0.145013
$778$	0	0
$779$	6.43622	0.230601
$780$	0	0
$781$	−48.0844	−1.72059
$782$	0	0
$783$	−0.911179	−0.0325629
$784$	0	0
$785$	−3.16975	−0.113133
$786$	0	0
$787$	23.2119	0.827416	0.413708	−	0.910410i	$-0.364233\pi$
$787$	23.2119	0.827416	0.413708	+	0.910410i	$0.364233\pi$
$788$	0	0
$789$	−3.16975	−0.112846
$790$	0	0
$791$	−12.3237	−0.438181
$792$	0	0
$793$	−82.4239	−2.92696
$794$	0	0
$795$	3.34740	0.118720
$796$	0	0
$797$	−35.8645	−1.27039	−0.635194	−	0.772353i	$-0.719080\pi$
$797$	−35.8645	−1.27039	−0.635194	+	0.772353i	$0.719080\pi$
$798$	0	0
$799$	−21.3896	−0.756709
$800$	0	0
$801$	−2.43622	−0.0860795
$802$	0	0
$803$	17.4671	0.616400
$804$	0	0
$805$	4.87243	0.171731
$806$	0	0
$807$	−13.7836	−0.485206
$808$	0	0
$809$	2.35528	0.0828074	0.0414037	−	0.999142i	$-0.486817\pi$
$809$	2.35528	0.0828074	0.0414037	+	0.999142i	$0.486817\pi$
$810$	0	0
$811$	−43.0185	−1.51058	−0.755292	−	0.655388i	$-0.772505\pi$
$811$	−43.0185	−1.51058	−0.755292	+	0.655388i	$0.772505\pi$
$812$	0	0
$813$	−17.8224	−0.625057
$814$	0	0
$815$	−6.43622	−0.225451
$816$	0	0
$817$	8.25857	0.288931
$818$	0	0
$819$	−5.70268	−0.199268
$820$	0	0
$821$	47.9067	1.67196	0.835978	−	0.548763i	$-0.184901\pi$
$821$	47.9067	1.67196	0.835978	+	0.548763i	$0.184901\pi$
$822$	0	0
$823$	−12.3940	−0.432029	−0.216014	−	0.976390i	$-0.569306\pi$
$823$	−12.3940	−0.432029	−0.216014	+	0.976390i	$0.569306\pi$
$824$	0	0
$825$	2.91118	0.101354
$826$	0	0
$827$	11.1698	0.388410	0.194205	−	0.980961i	$-0.437787\pi$
$827$	11.1698	0.388410	0.194205	+	0.980961i	$0.437787\pi$
$828$	0	0
$829$	−0.830247	−0.0288357	−0.0144178	−	0.999896i	$-0.504589\pi$
$829$	−0.830247	−0.0288357	−0.0144178	+	0.999896i	$0.504589\pi$
$830$	0	0
$831$	21.3474	0.740533
$832$	0	0
$833$	−24.6790	−0.855077
$834$	0	0
$835$	3.16975	0.109694
$836$	0	0
$837$	−2.00000	−0.0691301
$838$	0	0
$839$	−12.8724	−0.444406	−0.222203	−	0.975000i	$-0.571325\pi$
$839$	−12.8724	−0.444406	−0.222203	+	0.975000i	$0.571325\pi$
$840$	0	0
$841$	−28.1698	−0.971371
$842$	0	0
$843$	14.4362	0.497210
$844$	0	0
$845$	26.1698	0.900267
$846$	0	0
$847$	−2.30076	−0.0790551
$848$	0	0
$849$	−6.08093	−0.208697
$850$	0	0
$851$	23.7222	0.813187
$852$	0	0
$853$	12.8302	0.439299	0.219650	−	0.975579i	$-0.429509\pi$
$853$	12.8302	0.439299	0.219650	+	0.975579i	$0.429509\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	0	0
$857$	−35.0765	−1.19819	−0.599095	−	0.800678i	$-0.704473\pi$
$857$	−35.0765	−1.19819	−0.599095	+	0.800678i	$0.704473\pi$
$858$	0	0
$859$	−22.6948	−0.774336	−0.387168	−	0.922009i	$-0.626547\pi$
$859$	−22.6948	−0.774336	−0.387168	+	0.922009i	$0.626547\pi$
$860$	0	0
$861$	5.86454	0.199863
$862$	0	0
$863$	−18.6948	−0.636378	−0.318189	−	0.948027i	$-0.603075\pi$
$863$	−18.6948	−0.636378	−0.318189	+	0.948027i	$0.603075\pi$
$864$	0	0
$865$	−9.52504	−0.323861
$866$	0	0
$867$	−1.00000	−0.0339618
$868$	0	0
$869$	−32.0000	−1.08553
$870$	0	0
$871$	−25.0343	−0.848255
$872$	0	0
$873$	16.4362	0.556282
$874$	0	0
$875$	0.911179	0.0308035
$876$	0	0
$877$	−10.7757	−0.363870	−0.181935	−	0.983311i	$-0.558236\pi$
$877$	−10.7757	−0.363870	−0.181935	+	0.983311i	$0.558236\pi$
$878$	0	0
$879$	−25.1698	−0.848955
$880$	0	0
$881$	1.56722	0.0528011	0.0264006	−	0.999651i	$-0.491595\pi$
$881$	1.56722	0.0528011	0.0264006	+	0.999651i	$0.491595\pi$
$882$	0	0
$883$	−5.13101	−0.172672	−0.0863361	−	0.996266i	$-0.527516\pi$
$883$	−5.13101	−0.172672	−0.0863361	+	0.996266i	$0.527516\pi$
$884$	0	0
$885$	5.52504	0.185722
$886$	0	0
$887$	−9.74487	−0.327201	−0.163600	−	0.986527i	$-0.552311\pi$
$887$	−9.74487	−0.327201	−0.163600	+	0.986527i	$0.552311\pi$
$888$	0	0
$889$	20.3553	0.682694
$890$	0	0
$891$	2.91118	0.0975282
$892$	0	0
$893$	−5.34740	−0.178944
$894$	0	0
$895$	5.16975	0.172806
$896$	0	0
$897$	−33.4671	−1.11743
$898$	0	0
$899$	1.82236	0.0607790
$900$	0	0
$901$	13.3896	0.446072
$902$	0	0
$903$	7.52504	0.250418
$904$	0	0
$905$	18.8145	0.625414
$906$	0	0
$907$	−4.51715	−0.149989	−0.0749947	−	0.997184i	$-0.523894\pi$
$907$	−4.51715	−0.149989	−0.0749947	+	0.997184i	$0.523894\pi$
$908$	0	0
$909$	−0.177642	−0.00589203
$910$	0	0
$911$	21.0343	0.696897	0.348449	−	0.937328i	$-0.386709\pi$
$911$	21.0343	0.696897	0.348449	+	0.937328i	$0.386709\pi$
$912$	0	0
$913$	−11.1276	−0.368269
$914$	0	0
$915$	13.1698	0.435379
$916$	0	0
$917$	−7.95781	−0.262790
$918$	0	0
$919$	−0.949924	−0.0313351	−0.0156676	−	0.999877i	$-0.504987\pi$
$919$	−0.949924	−0.0313351	−0.0156676	+	0.999877i	$0.504987\pi$
$920$	0	0
$921$	22.1776	0.730778
$922$	0	0
$923$	103.374	3.40259
$924$	0	0
$925$	4.43622	0.145862
$926$	0	0
$927$	−3.64472	−0.119708
$928$	0	0
$929$	29.4513	0.966266	0.483133	−	0.875547i	$-0.339499\pi$
$929$	29.4513	0.966266	0.483133	+	0.875547i	$0.339499\pi$
$930$	0	0
$931$	−6.16975	−0.202205
$932$	0	0
$933$	−23.7836	−0.778641
$934$	0	0
$935$	11.6447	0.380823
$936$	0	0
$937$	−54.6015	−1.78375	−0.891877	−	0.452278i	$-0.850611\pi$
$937$	−54.6015	−1.78375	−0.891877	+	0.452278i	$0.850611\pi$
$938$	0	0
$939$	15.4671	0.504749
$940$	0	0
$941$	32.2006	1.04971	0.524855	−	0.851192i	$-0.324120\pi$
$941$	32.2006	1.04971	0.524855	+	0.851192i	$0.324120\pi$
$942$	0	0
$943$	34.4170	1.12077
$944$	0	0
$945$	0.911179	0.0296406
$946$	0	0
$947$	42.5171	1.38162	0.690811	−	0.723036i	$-0.257254\pi$
$947$	42.5171	1.38162	0.690811	+	0.723036i	$0.257254\pi$
$948$	0	0
$949$	−37.5514	−1.21897
$950$	0	0
$951$	32.3817	1.05005
$952$	0	0
$953$	28.1039	0.910375	0.455187	−	0.890396i	$-0.349572\pi$
$953$	28.1039	0.910375	0.455187	+	0.890396i	$0.349572\pi$
$954$	0	0
$955$	7.78361	0.251872
$956$	0	0
$957$	−2.65260	−0.0857465
$958$	0	0
$959$	12.1619	0.392727
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	−3.64472	−0.117449
$964$	0	0
$965$	−15.7256	−0.506226
$966$	0	0
$967$	42.4626	1.36551	0.682753	−	0.730649i	$-0.260783\pi$
$967$	42.4626	1.36551	0.682753	+	0.730649i	$0.260783\pi$
$968$	0	0
$969$	4.00000	0.128499
$970$	0	0
$971$	−22.0422	−0.707367	−0.353684	−	0.935365i	$-0.615071\pi$
$971$	−22.0422	−0.707367	−0.353684	+	0.935365i	$0.615071\pi$
$972$	0	0
$973$	1.66049	0.0532330
$974$	0	0
$975$	−6.25857	−0.200435
$976$	0	0
$977$	−15.3474	−0.491007	−0.245503	−	0.969396i	$-0.578953\pi$
$977$	−15.3474	−0.491007	−0.245503	+	0.969396i	$0.578953\pi$
$978$	0	0
$979$	−7.09226	−0.226670
$980$	0	0
$981$	−3.52504	−0.112546
$982$	0	0
$983$	−5.78017	−0.184359	−0.0921794	−	0.995742i	$-0.529383\pi$
$983$	−5.78017	−0.184359	−0.0921794	+	0.995742i	$0.529383\pi$
$984$	0	0
$985$	−7.70268	−0.245428
$986$	0	0
$987$	−4.87243	−0.155091
$988$	0	0
$989$	44.1619	1.40427
$990$	0	0
$991$	60.7370	1.92937	0.964687	−	0.263399i	$-0.0848437\pi$
$991$	60.7370	1.92937	0.964687	+	0.263399i	$0.0848437\pi$
$992$	0	0
$993$	−20.0422	−0.636020
$994$	0	0
$995$	−20.5171	−0.650437
$996$	0	0
$997$	41.5093	1.31461	0.657306	−	0.753624i	$-0.271696\pi$
$997$	41.5093	1.31461	0.657306	+	0.753624i	$0.271696\pi$
$998$	0	0
$999$	4.43622	0.140356

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2280.2.a.u.1.2	✓	3
3.2	odd	2		6840.2.a.bh.1.2		3
4.3	odd	2		4560.2.a.bs.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.2.a.u.1.2	✓	3	1.1	even	1	trivial
4560.2.a.bs.1.2		3	4.3	odd	2
6840.2.a.bh.1.2		3	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2280 = 2^{3} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2280.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} +0.911179 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} +0.911179 q^{7} +1.00000 q^{9} +2.91118 q^{11} -6.25857 q^{13} +1.00000 q^{15} +4.00000 q^{17} +1.00000 q^{19} +0.911179 q^{21} +5.34740 q^{23} +1.00000 q^{25} +1.00000 q^{27} -0.911179 q^{29} -2.00000 q^{31} +2.91118 q^{33} +0.911179 q^{35} +4.43622 q^{37} -6.25857 q^{39} +6.43622 q^{41} +8.25857 q^{43} +1.00000 q^{45} -5.34740 q^{47} -6.16975 q^{49} +4.00000 q^{51} +3.34740 q^{53} +2.91118 q^{55} +1.00000 q^{57} +5.52504 q^{59} +13.1698 q^{61} +0.911179 q^{63} -6.25857 q^{65} +4.00000 q^{67} +5.34740 q^{69} -16.5171 q^{71} +6.00000 q^{73} +1.00000 q^{75} +2.65260 q^{77} -10.9921 q^{79} +1.00000 q^{81} -3.82236 q^{83} +4.00000 q^{85} -0.911179 q^{87} -2.43622 q^{89} -5.70268 q^{91} -2.00000 q^{93} +1.00000 q^{95} +16.4362 q^{97} +2.91118 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{5} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^5 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{5} + 3 q^{9} + 6 q^{11} + 4 q^{13} + 3 q^{15} + 12 q^{17} + 3 q^{19} - 4 q^{23} + 3 q^{25} + 3 q^{27} - 6 q^{31} + 6 q^{33} - 4 q^{37} + 4 q^{39} + 2 q^{41} + 2 q^{43} + 3 q^{45} + 4 q^{47} + 7 q^{49} + 12 q^{51} - 10 q^{53} + 6 q^{55} + 3 q^{57} + 2 q^{59} + 14 q^{61} + 4 q^{65} + 12 q^{67} - 4 q^{69} - 4 q^{71} + 18 q^{73} + 3 q^{75} + 28 q^{77} - 2 q^{79} + 3 q^{81} - 6 q^{83} + 12 q^{85} + 10 q^{89} - 8 q^{91} - 6 q^{93} + 3 q^{95} + 32 q^{97} + 6 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^5 + 3 * q^9 + 6 * q^11 + 4 * q^13 + 3 * q^15 + 12 * q^17 + 3 * q^19 - 4 * q^23 + 3 * q^25 + 3 * q^27 - 6 * q^31 + 6 * q^33 - 4 * q^37 + 4 * q^39 + 2 * q^41 + 2 * q^43 + 3 * q^45 + 4 * q^47 + 7 * q^49 + 12 * q^51 - 10 * q^53 + 6 * q^55 + 3 * q^57 + 2 * q^59 + 14 * q^61 + 4 * q^65 + 12 * q^67 - 4 * q^69 - 4 * q^71 + 18 * q^73 + 3 * q^75 + 28 * q^77 - 2 * q^79 + 3 * q^81 - 6 * q^83 + 12 * q^85 + 10 * q^89 - 8 * q^91 - 6 * q^93 + 3 * q^95 + 32 * q^97 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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