LMFDB - Embedded newform 2448.2.a.v.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2448,2,Mod(1,2448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2448, 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("2448.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2448 = 2^{4} \cdot 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2448.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:	$19.5473784148$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 51)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	2448.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.561553 q^{5} +O(q^{10})$	$q+0.561553 q^{5} -2.56155 q^{11} +4.56155 q^{13} -1.00000 q^{17} -7.68466 q^{19} -6.56155 q^{23} -4.68466 q^{25} -8.24621 q^{29} +5.12311 q^{31} +3.12311 q^{37} -0.561553 q^{41} +7.68466 q^{43} -2.87689 q^{47} -7.00000 q^{49} +4.24621 q^{53} -1.43845 q^{55} -1.12311 q^{59} +0.876894 q^{61} +2.56155 q^{65} -4.00000 q^{67} +10.2462 q^{71} +4.24621 q^{73} -15.3693 q^{79} -9.12311 q^{83} -0.561553 q^{85} -7.12311 q^{89} -4.31534 q^{95} -11.1231 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{5}+O(q^{10}) $ 2 * q - 3 * q^5	$ 2 q - 3 q^{5} - q^{11} + 5 q^{13} - 2 q^{17} - 3 q^{19} - 9 q^{23} + 3 q^{25} + 2 q^{31} - 2 q^{37} + 3 q^{41} + 3 q^{43} - 14 q^{47} - 14 q^{49} - 8 q^{53} - 7 q^{55} + 6 q^{59} + 10 q^{61} + q^{65} - 8 q^{67} + 4 q^{71} - 8 q^{73} - 6 q^{79} - 10 q^{83} + 3 q^{85} - 6 q^{89} - 21 q^{95} - 14 q^{97}+O(q^{100}) $ 2 * q - 3 * q^5 - q^11 + 5 * q^13 - 2 * q^17 - 3 * q^19 - 9 * q^23 + 3 * q^25 + 2 * q^31 - 2 * q^37 + 3 * q^41 + 3 * q^43 - 14 * q^47 - 14 * q^49 - 8 * q^53 - 7 * q^55 + 6 * q^59 + 10 * q^61 + q^65 - 8 * q^67 + 4 * q^71 - 8 * q^73 - 6 * q^79 - 10 * q^83 + 3 * q^85 - 6 * q^89 - 21 * q^95 - 14 * q^97

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
$3$	0	0
$4$	0	0
$5$	0.561553	0.251134	0.125567	−	0.992085i	$-0.459925\pi$
$5$	0.561553	0.251134	0.125567	+	0.992085i	$0.459925\pi$
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.56155	−0.772337	−0.386169	−	0.922428i	$-0.626202\pi$
$11$	−2.56155	−0.772337	−0.386169	+	0.922428i	$0.626202\pi$
$12$	0	0
$13$	4.56155	1.26515	0.632574	−	0.774500i	$-0.281999\pi$
$13$	4.56155	1.26515	0.632574	+	0.774500i	$0.281999\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	−7.68466	−1.76298	−0.881491	−	0.472201i	$-0.843460\pi$
$19$	−7.68466	−1.76298	−0.881491	+	0.472201i	$0.843460\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−6.56155	−1.36818	−0.684089	−	0.729398i	$-0.739800\pi$
$23$	−6.56155	−1.36818	−0.684089	+	0.729398i	$0.739800\pi$
$24$	0	0
$25$	−4.68466	−0.936932
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8.24621	−1.53128	−0.765641	−	0.643268i	$-0.777578\pi$
$29$	−8.24621	−1.53128	−0.765641	+	0.643268i	$0.777578\pi$
$30$	0	0
$31$	5.12311	0.920137	0.460068	−	0.887883i	$-0.347825\pi$
$31$	5.12311	0.920137	0.460068	+	0.887883i	$0.347825\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.12311	0.513435	0.256718	−	0.966486i	$-0.417359\pi$
$37$	3.12311	0.513435	0.256718	+	0.966486i	$0.417359\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−0.561553	−0.0876998	−0.0438499	−	0.999038i	$-0.513962\pi$
$41$	−0.561553	−0.0876998	−0.0438499	+	0.999038i	$0.513962\pi$
$42$	0	0
$43$	7.68466	1.17190	0.585950	−	0.810347i	$-0.300722\pi$
$43$	7.68466	1.17190	0.585950	+	0.810347i	$0.300722\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.87689	−0.419638	−0.209819	−	0.977740i	$-0.567288\pi$
$47$	−2.87689	−0.419638	−0.209819	+	0.977740i	$0.567288\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.24621	0.583262	0.291631	−	0.956531i	$-0.405802\pi$
$53$	4.24621	0.583262	0.291631	+	0.956531i	$0.405802\pi$
$54$	0	0
$55$	−1.43845	−0.193960
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.12311	−0.146216	−0.0731079	−	0.997324i	$-0.523292\pi$
$59$	−1.12311	−0.146216	−0.0731079	+	0.997324i	$0.523292\pi$
$60$	0	0
$61$	0.876894	0.112275	0.0561374	−	0.998423i	$-0.482122\pi$
$61$	0.876894	0.112275	0.0561374	+	0.998423i	$0.482122\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.56155	0.317722
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.2462	1.21600	0.608001	−	0.793936i	$-0.291972\pi$
$71$	10.2462	1.21600	0.608001	+	0.793936i	$0.291972\pi$
$72$	0	0
$73$	4.24621	0.496981	0.248491	−	0.968634i	$-0.420065\pi$
$73$	4.24621	0.496981	0.248491	+	0.968634i	$0.420065\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−15.3693	−1.72918	−0.864592	−	0.502475i	$-0.832423\pi$
$79$	−15.3693	−1.72918	−0.864592	+	0.502475i	$0.832423\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−9.12311	−1.00139	−0.500695	−	0.865624i	$-0.666922\pi$
$83$	−9.12311	−1.00139	−0.500695	+	0.865624i	$0.666922\pi$
$84$	0	0
$85$	−0.561553	−0.0609090
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−7.12311	−0.755048	−0.377524	−	0.926000i	$-0.623224\pi$
$89$	−7.12311	−0.755048	−0.377524	+	0.926000i	$0.623224\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.31534	−0.442745
$96$	0	0
$97$	−11.1231	−1.12938	−0.564690	−	0.825303i	$-0.691004\pi$
$97$	−11.1231	−1.12938	−0.564690	+	0.825303i	$0.691004\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−19.1231	−1.90282	−0.951410	−	0.307927i	$-0.900365\pi$
$101$	−19.1231	−1.90282	−0.951410	+	0.307927i	$0.900365\pi$
$102$	0	0
$103$	−4.31534	−0.425203	−0.212602	−	0.977139i	$-0.568194\pi$
$103$	−4.31534	−0.425203	−0.212602	+	0.977139i	$0.568194\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7.68466	0.742904	0.371452	−	0.928452i	$-0.378860\pi$
$107$	7.68466	0.742904	0.371452	+	0.928452i	$0.378860\pi$
$108$	0	0
$109$	−15.1231	−1.44853	−0.724265	−	0.689521i	$-0.757821\pi$
$109$	−15.1231	−1.44853	−0.724265	+	0.689521i	$0.757821\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	4.56155	0.429115	0.214557	−	0.976711i	$-0.431169\pi$
$113$	4.56155	0.429115	0.214557	+	0.976711i	$0.431169\pi$
$114$	0	0
$115$	−3.68466	−0.343596
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−4.43845	−0.403495
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−5.43845	−0.486430
$126$	0	0
$127$	0.807764	0.0716775	0.0358387	−	0.999358i	$-0.488590\pi$
$127$	0.807764	0.0716775	0.0358387	+	0.999358i	$0.488590\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	18.5616	1.62173	0.810865	−	0.585233i	$-0.198997\pi$
$131$	18.5616	1.62173	0.810865	+	0.585233i	$0.198997\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	16.2462	1.38801	0.694004	−	0.719971i	$-0.255845\pi$
$137$	16.2462	1.38801	0.694004	+	0.719971i	$0.255845\pi$
$138$	0	0
$139$	9.12311	0.773812	0.386906	−	0.922119i	$-0.373544\pi$
$139$	9.12311	0.773812	0.386906	+	0.922119i	$0.373544\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−11.6847	−0.977120
$144$	0	0
$145$	−4.63068	−0.384557
$146$	0	0
$147$	0	0
$148$	0	0
$149$	4.24621	0.347863	0.173932	−	0.984758i	$-0.444353\pi$
$149$	4.24621	0.347863	0.173932	+	0.984758i	$0.444353\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2.87689	0.231078
$156$	0	0
$157$	−5.68466	−0.453685	−0.226843	−	0.973931i	$-0.572840\pi$
$157$	−5.68466	−0.453685	−0.226843	+	0.973931i	$0.572840\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	6.87689	0.538640	0.269320	−	0.963051i	$-0.413201\pi$
$163$	6.87689	0.538640	0.269320	+	0.963051i	$0.413201\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−0.807764	−0.0625067	−0.0312533	−	0.999511i	$-0.509950\pi$
$167$	−0.807764	−0.0625067	−0.0312533	+	0.999511i	$0.509950\pi$
$168$	0	0
$169$	7.80776	0.600597
$170$	0	0
$171$	0	0
$172$	0	0
$173$	18.8078	1.42993	0.714964	−	0.699161i	$-0.246443\pi$
$173$	18.8078	1.42993	0.714964	+	0.699161i	$0.246443\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−9.12311	−0.681893	−0.340946	−	0.940083i	$-0.610747\pi$
$179$	−9.12311	−0.681893	−0.340946	+	0.940083i	$0.610747\pi$
$180$	0	0
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.75379	0.128941
$186$	0	0
$187$	2.56155	0.187319
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.1231	−0.949555	−0.474777	−	0.880106i	$-0.657471\pi$
$191$	−13.1231	−0.949555	−0.474777	+	0.880106i	$0.657471\pi$
$192$	0	0
$193$	−24.2462	−1.74528	−0.872640	−	0.488364i	$-0.837594\pi$
$193$	−24.2462	−1.74528	−0.872640	+	0.488364i	$0.837594\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−19.9309	−1.42002	−0.710008	−	0.704194i	$-0.751309\pi$
$197$	−19.9309	−1.42002	−0.710008	+	0.704194i	$0.751309\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−0.315342	−0.0220244
$206$	0	0
$207$	0	0
$208$	0	0
$209$	19.6847	1.36162
$210$	0	0
$211$	11.3693	0.782696	0.391348	−	0.920243i	$-0.372009\pi$
$211$	11.3693	0.782696	0.391348	+	0.920243i	$0.372009\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.31534	0.294304
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.56155	−0.306843
$222$	0	0
$223$	−13.9309	−0.932880	−0.466440	−	0.884553i	$-0.654464\pi$
$223$	−13.9309	−0.932880	−0.466440	+	0.884553i	$0.654464\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	23.0540	1.53015	0.765073	−	0.643944i	$-0.222703\pi$
$227$	23.0540	1.53015	0.765073	+	0.643944i	$0.222703\pi$
$228$	0	0
$229$	6.00000	0.396491	0.198246	−	0.980152i	$-0.436476\pi$
$229$	6.00000	0.396491	0.198246	+	0.980152i	$0.436476\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−0.561553	−0.0367885	−0.0183943	−	0.999831i	$-0.505855\pi$
$233$	−0.561553	−0.0367885	−0.0183943	+	0.999831i	$0.505855\pi$
$234$	0	0
$235$	−1.61553	−0.105385
$236$	0	0
$237$	0	0
$238$	0	0
$239$	10.2462	0.662772	0.331386	−	0.943495i	$-0.392484\pi$
$239$	10.2462	0.662772	0.331386	+	0.943495i	$0.392484\pi$
$240$	0	0
$241$	−21.3693	−1.37652	−0.688259	−	0.725465i	$-0.741625\pi$
$241$	−21.3693	−1.37652	−0.688259	+	0.725465i	$0.741625\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−3.93087	−0.251134
$246$	0	0
$247$	−35.0540	−2.23043
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−24.4924	−1.54595	−0.772974	−	0.634438i	$-0.781232\pi$
$251$	−24.4924	−1.54595	−0.772974	+	0.634438i	$0.781232\pi$
$252$	0	0
$253$	16.8078	1.05670
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−9.36932	−0.584442	−0.292221	−	0.956351i	$-0.594394\pi$
$257$	−9.36932	−0.584442	−0.292221	+	0.956351i	$0.594394\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	12.4924	0.770316	0.385158	−	0.922851i	$-0.374147\pi$
$263$	12.4924	0.770316	0.385158	+	0.922851i	$0.374147\pi$
$264$	0	0
$265$	2.38447	0.146477
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−20.5616	−1.25366	−0.626830	−	0.779156i	$-0.715648\pi$
$269$	−20.5616	−1.25366	−0.626830	+	0.779156i	$0.715648\pi$
$270$	0	0
$271$	0.807764	0.0490682	0.0245341	−	0.999699i	$-0.492190\pi$
$271$	0.807764	0.0490682	0.0245341	+	0.999699i	$0.492190\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	12.0000	0.723627
$276$	0	0
$277$	6.00000	0.360505	0.180253	−	0.983620i	$-0.442309\pi$
$277$	6.00000	0.360505	0.180253	+	0.983620i	$0.442309\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	19.1231	1.14079	0.570394	−	0.821371i	$-0.306790\pi$
$281$	19.1231	1.14079	0.570394	+	0.821371i	$0.306790\pi$
$282$	0	0
$283$	3.36932	0.200285	0.100143	−	0.994973i	$-0.468070\pi$
$283$	3.36932	0.200285	0.100143	+	0.994973i	$0.468070\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	0	0
$292$	0	0
$293$	7.12311	0.416136	0.208068	−	0.978114i	$-0.433282\pi$
$293$	7.12311	0.416136	0.208068	+	0.978114i	$0.433282\pi$
$294$	0	0
$295$	−0.630683	−0.0367198
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−29.9309	−1.73095
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0.492423	0.0281960
$306$	0	0
$307$	0.492423	0.0281040	0.0140520	−	0.999901i	$-0.495527\pi$
$307$	0.492423	0.0281040	0.0140520	+	0.999901i	$0.495527\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−7.61553	−0.430455	−0.215228	−	0.976564i	$-0.569049\pi$
$313$	−7.61553	−0.430455	−0.215228	+	0.976564i	$0.569049\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	21.1231	1.18267
$320$	0	0
$321$	0	0
$322$	0	0
$323$	7.68466	0.427586
$324$	0	0
$325$	−21.3693	−1.18536
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	6.06913	0.333590	0.166795	−	0.985992i	$-0.446658\pi$
$331$	6.06913	0.333590	0.166795	+	0.985992i	$0.446658\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−2.24621	−0.122724
$336$	0	0
$337$	32.7386	1.78339	0.891694	−	0.452640i	$-0.149518\pi$
$337$	32.7386	1.78339	0.891694	+	0.452640i	$0.149518\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−13.1231	−0.710656
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	24.4924	1.31482	0.657411	−	0.753532i	$-0.271652\pi$
$347$	24.4924	1.31482	0.657411	+	0.753532i	$0.271652\pi$
$348$	0	0
$349$	7.43845	0.398171	0.199085	−	0.979982i	$-0.436203\pi$
$349$	7.43845	0.398171	0.199085	+	0.979982i	$0.436203\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−22.4924	−1.19715	−0.598575	−	0.801066i	$-0.704266\pi$
$353$	−22.4924	−1.19715	−0.598575	+	0.801066i	$0.704266\pi$
$354$	0	0
$355$	5.75379	0.305379
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−2.24621	−0.118550	−0.0592752	−	0.998242i	$-0.518879\pi$
$359$	−2.24621	−0.118550	−0.0592752	+	0.998242i	$0.518879\pi$
$360$	0	0
$361$	40.0540	2.10810
$362$	0	0
$363$	0	0
$364$	0	0
$365$	2.38447	0.124809
$366$	0	0
$367$	18.2462	0.952444	0.476222	−	0.879325i	$-0.342006\pi$
$367$	18.2462	0.952444	0.476222	+	0.879325i	$0.342006\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	16.2462	0.841197	0.420598	−	0.907247i	$-0.361820\pi$
$373$	16.2462	0.841197	0.420598	+	0.907247i	$0.361820\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−37.6155	−1.93730
$378$	0	0
$379$	12.0000	0.616399	0.308199	−	0.951322i	$-0.400274\pi$
$379$	12.0000	0.616399	0.308199	+	0.951322i	$0.400274\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−10.2462	−0.523557	−0.261778	−	0.965128i	$-0.584309\pi$
$383$	−10.2462	−0.523557	−0.261778	+	0.965128i	$0.584309\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	21.8617	1.10843	0.554217	−	0.832372i	$-0.313018\pi$
$389$	21.8617	1.10843	0.554217	+	0.832372i	$0.313018\pi$
$390$	0	0
$391$	6.56155	0.331832
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−8.63068	−0.434257
$396$	0	0
$397$	5.36932	0.269478	0.134739	−	0.990881i	$-0.456980\pi$
$397$	5.36932	0.269478	0.134739	+	0.990881i	$0.456980\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	6.17708	0.308469	0.154234	−	0.988034i	$-0.450709\pi$
$401$	6.17708	0.308469	0.154234	+	0.988034i	$0.450709\pi$
$402$	0	0
$403$	23.3693	1.16411
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−8.00000	−0.396545
$408$	0	0
$409$	−2.31534	−0.114486	−0.0572431	−	0.998360i	$-0.518231\pi$
$409$	−2.31534	−0.114486	−0.0572431	+	0.998360i	$0.518231\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−5.12311	−0.251483
$416$	0	0
$417$	0	0
$418$	0	0
$419$	32.4924	1.58736	0.793679	−	0.608336i	$-0.208163\pi$
$419$	32.4924	1.58736	0.793679	+	0.608336i	$0.208163\pi$
$420$	0	0
$421$	28.5616	1.39200	0.696002	−	0.718039i	$-0.254960\pi$
$421$	28.5616	1.39200	0.696002	+	0.718039i	$0.254960\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.68466	0.227239
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	14.3153	0.687951	0.343976	−	0.938979i	$-0.388226\pi$
$433$	14.3153	0.687951	0.343976	+	0.938979i	$0.388226\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	50.4233	2.41207
$438$	0	0
$439$	5.75379	0.274613	0.137307	−	0.990529i	$-0.456155\pi$
$439$	5.75379	0.274613	0.137307	+	0.990529i	$0.456155\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−22.8769	−1.08691	−0.543457	−	0.839437i	$-0.682885\pi$
$443$	−22.8769	−1.08691	−0.543457	+	0.839437i	$0.682885\pi$
$444$	0	0
$445$	−4.00000	−0.189618
$446$	0	0
$447$	0	0
$448$	0	0
$449$	12.7386	0.601173	0.300587	−	0.953755i	$-0.402818\pi$
$449$	12.7386	0.601173	0.300587	+	0.953755i	$0.402818\pi$
$450$	0	0
$451$	1.43845	0.0677338
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−6.80776	−0.318454	−0.159227	−	0.987242i	$-0.550900\pi$
$457$	−6.80776	−0.318454	−0.159227	+	0.987242i	$0.550900\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−8.24621	−0.384064	−0.192032	−	0.981389i	$-0.561508\pi$
$461$	−8.24621	−0.384064	−0.192032	+	0.981389i	$0.561508\pi$
$462$	0	0
$463$	−24.9848	−1.16114	−0.580572	−	0.814209i	$-0.697171\pi$
$463$	−24.9848	−1.16114	−0.580572	+	0.814209i	$0.697171\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.36932	−0.155913	−0.0779567	−	0.996957i	$-0.524840\pi$
$467$	−3.36932	−0.155913	−0.0779567	+	0.996957i	$0.524840\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−19.6847	−0.905102
$474$	0	0
$475$	36.0000	1.65179
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−29.3002	−1.33876	−0.669380	−	0.742920i	$-0.733440\pi$
$479$	−29.3002	−1.33876	−0.669380	+	0.742920i	$0.733440\pi$
$480$	0	0
$481$	14.2462	0.649571
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−6.24621	−0.283626
$486$	0	0
$487$	7.36932	0.333936	0.166968	−	0.985962i	$-0.446602\pi$
$487$	7.36932	0.333936	0.166968	+	0.985962i	$0.446602\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3.36932	0.152055	0.0760276	−	0.997106i	$-0.475776\pi$
$491$	3.36932	0.152055	0.0760276	+	0.997106i	$0.475776\pi$
$492$	0	0
$493$	8.24621	0.371391
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	11.3693	0.508961	0.254480	−	0.967078i	$-0.418096\pi$
$499$	11.3693	0.508961	0.254480	+	0.967078i	$0.418096\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−25.4384	−1.13424	−0.567122	−	0.823634i	$-0.691943\pi$
$503$	−25.4384	−1.13424	−0.567122	+	0.823634i	$0.691943\pi$
$504$	0	0
$505$	−10.7386	−0.477863
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−16.8769	−0.748055	−0.374028	−	0.927418i	$-0.622023\pi$
$509$	−16.8769	−0.748055	−0.374028	+	0.927418i	$0.622023\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−2.42329	−0.106783
$516$	0	0
$517$	7.36932	0.324102
$518$	0	0
$519$	0	0
$520$	0	0
$521$	31.4384	1.37734	0.688672	−	0.725073i	$-0.258194\pi$
$521$	31.4384	1.37734	0.688672	+	0.725073i	$0.258194\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−5.12311	−0.223166
$528$	0	0
$529$	20.0540	0.871912
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−2.56155	−0.110953
$534$	0	0
$535$	4.31534	0.186568
$536$	0	0
$537$	0	0
$538$	0	0
$539$	17.9309	0.772337
$540$	0	0
$541$	−40.1080	−1.72438	−0.862188	−	0.506589i	$-0.830906\pi$
$541$	−40.1080	−1.72438	−0.862188	+	0.506589i	$0.830906\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−8.49242	−0.363775
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	63.3693	2.69962
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	6.49242	0.275093	0.137546	−	0.990495i	$-0.456078\pi$
$557$	6.49242	0.275093	0.137546	+	0.990495i	$0.456078\pi$
$558$	0	0
$559$	35.0540	1.48263
$560$	0	0
$561$	0	0
$562$	0	0
$563$	22.8769	0.964146	0.482073	−	0.876131i	$-0.339884\pi$
$563$	22.8769	0.964146	0.482073	+	0.876131i	$0.339884\pi$
$564$	0	0
$565$	2.56155	0.107765
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−12.8769	−0.539827	−0.269914	−	0.962885i	$-0.586995\pi$
$569$	−12.8769	−0.539827	−0.269914	+	0.962885i	$0.586995\pi$
$570$	0	0
$571$	−18.7386	−0.784187	−0.392094	−	0.919925i	$-0.628249\pi$
$571$	−18.7386	−0.784187	−0.392094	+	0.919925i	$0.628249\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	30.7386	1.28189
$576$	0	0
$577$	−41.0540	−1.70910	−0.854550	−	0.519370i	$-0.826167\pi$
$577$	−41.0540	−1.70910	−0.854550	+	0.519370i	$0.826167\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−10.8769	−0.450475
$584$	0	0
$585$	0	0
$586$	0	0
$587$	36.9848	1.52653	0.763264	−	0.646087i	$-0.223596\pi$
$587$	36.9848	1.52653	0.763264	+	0.646087i	$0.223596\pi$
$588$	0	0
$589$	−39.3693	−1.62218
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−44.2462	−1.81697	−0.908487	−	0.417913i	$-0.862762\pi$
$593$	−44.2462	−1.81697	−0.908487	+	0.417913i	$0.862762\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−41.6155	−1.70036	−0.850182	−	0.526489i	$-0.823508\pi$
$599$	−41.6155	−1.70036	−0.850182	+	0.526489i	$0.823508\pi$
$600$	0	0
$601$	34.9848	1.42706	0.713531	−	0.700624i	$-0.247095\pi$
$601$	34.9848	1.42706	0.713531	+	0.700624i	$0.247095\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−2.49242	−0.101331
$606$	0	0
$607$	−15.3693	−0.623821	−0.311911	−	0.950111i	$-0.600969\pi$
$607$	−15.3693	−0.623821	−0.311911	+	0.950111i	$0.600969\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−13.1231	−0.530904
$612$	0	0
$613$	2.31534	0.0935158	0.0467579	−	0.998906i	$-0.485111\pi$
$613$	2.31534	0.0935158	0.0467579	+	0.998906i	$0.485111\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	27.7538	1.11733	0.558663	−	0.829395i	$-0.311315\pi$
$617$	27.7538	1.11733	0.558663	+	0.829395i	$0.311315\pi$
$618$	0	0
$619$	19.3693	0.778519	0.389259	−	0.921128i	$-0.372731\pi$
$619$	19.3693	0.778519	0.389259	+	0.921128i	$0.372731\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	20.3693	0.814773
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3.12311	−0.124526
$630$	0	0
$631$	−11.6847	−0.465159	−0.232579	−	0.972577i	$-0.574717\pi$
$631$	−11.6847	−0.465159	−0.232579	+	0.972577i	$0.574717\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0.453602	0.0180007
$636$	0	0
$637$	−31.9309	−1.26515
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0.0691303	0.00273048	0.00136524	−	0.999999i	$-0.499565\pi$
$641$	0.0691303	0.00273048	0.00136524	+	0.999999i	$0.499565\pi$
$642$	0	0
$643$	30.2462	1.19279	0.596397	−	0.802690i	$-0.296598\pi$
$643$	30.2462	1.19279	0.596397	+	0.802690i	$0.296598\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	15.3693	0.604230	0.302115	−	0.953271i	$-0.402307\pi$
$647$	15.3693	0.604230	0.302115	+	0.953271i	$0.402307\pi$
$648$	0	0
$649$	2.87689	0.112928
$650$	0	0
$651$	0	0
$652$	0	0
$653$	4.06913	0.159237	0.0796187	−	0.996825i	$-0.474630\pi$
$653$	4.06913	0.159237	0.0796187	+	0.996825i	$0.474630\pi$
$654$	0	0
$655$	10.4233	0.407272
$656$	0	0
$657$	0	0
$658$	0	0
$659$	47.8617	1.86443	0.932214	−	0.361907i	$-0.117874\pi$
$659$	47.8617	1.86443	0.932214	+	0.361907i	$0.117874\pi$
$660$	0	0
$661$	25.6847	0.999017	0.499509	−	0.866309i	$-0.333514\pi$
$661$	25.6847	0.999017	0.499509	+	0.866309i	$0.333514\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	54.1080	2.09507
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−2.24621	−0.0867140
$672$	0	0
$673$	48.7386	1.87874	0.939368	−	0.342910i	$-0.111413\pi$
$673$	48.7386	1.87874	0.939368	+	0.342910i	$0.111413\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13.6847	0.525944	0.262972	−	0.964803i	$-0.415297\pi$
$677$	13.6847	0.525944	0.262972	+	0.964803i	$0.415297\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−5.43845	−0.208096	−0.104048	−	0.994572i	$-0.533180\pi$
$683$	−5.43845	−0.208096	−0.104048	+	0.994572i	$0.533180\pi$
$684$	0	0
$685$	9.12311	0.348576
$686$	0	0
$687$	0	0
$688$	0	0
$689$	19.3693	0.737912
$690$	0	0
$691$	−36.9848	−1.40697	−0.703485	−	0.710710i	$-0.748374\pi$
$691$	−36.9848	−1.40697	−0.703485	+	0.710710i	$0.748374\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.12311	0.194330
$696$	0	0
$697$	0.561553	0.0212703
$698$	0	0
$699$	0	0
$700$	0	0
$701$	9.36932	0.353874	0.176937	−	0.984222i	$-0.443381\pi$
$701$	9.36932	0.353874	0.176937	+	0.984222i	$0.443381\pi$
$702$	0	0
$703$	−24.0000	−0.905177
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	4.73863	0.177963	0.0889816	−	0.996033i	$-0.471639\pi$
$709$	4.73863	0.177963	0.0889816	+	0.996033i	$0.471639\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−33.6155	−1.25891
$714$	0	0
$715$	−6.56155	−0.245388
$716$	0	0
$717$	0	0
$718$	0	0
$719$	8.80776	0.328474	0.164237	−	0.986421i	$-0.447484\pi$
$719$	8.80776	0.328474	0.164237	+	0.986421i	$0.447484\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	38.6307	1.43471
$726$	0	0
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−7.68466	−0.284227
$732$	0	0
$733$	−28.2462	−1.04330	−0.521649	−	0.853160i	$-0.674683\pi$
$733$	−28.2462	−1.04330	−0.521649	+	0.853160i	$0.674683\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	10.2462	0.377424
$738$	0	0
$739$	8.31534	0.305885	0.152942	−	0.988235i	$-0.451125\pi$
$739$	8.31534	0.305885	0.152942	+	0.988235i	$0.451125\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−4.49242	−0.164811	−0.0824055	−	0.996599i	$-0.526260\pi$
$743$	−4.49242	−0.164811	−0.0824055	+	0.996599i	$0.526260\pi$
$744$	0	0
$745$	2.38447	0.0873603
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	0.630683	0.0230140	0.0115070	−	0.999934i	$-0.496337\pi$
$751$	0.630683	0.0230140	0.0115070	+	0.999934i	$0.496337\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−4.49242	−0.163496
$756$	0	0
$757$	−21.0540	−0.765220	−0.382610	−	0.923910i	$-0.624975\pi$
$757$	−21.0540	−0.765220	−0.382610	+	0.923910i	$0.624975\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	32.2462	1.16892	0.584462	−	0.811421i	$-0.301306\pi$
$761$	32.2462	1.16892	0.584462	+	0.811421i	$0.301306\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−5.12311	−0.184985
$768$	0	0
$769$	−29.5464	−1.06547	−0.532735	−	0.846282i	$-0.678836\pi$
$769$	−29.5464	−1.06547	−0.532735	+	0.846282i	$0.678836\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	33.3693	1.20021	0.600105	−	0.799921i	$-0.295125\pi$
$773$	33.3693	1.20021	0.600105	+	0.799921i	$0.295125\pi$
$774$	0	0
$775$	−24.0000	−0.862105
$776$	0	0
$777$	0	0
$778$	0	0
$779$	4.31534	0.154613
$780$	0	0
$781$	−26.2462	−0.939163
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−3.19224	−0.113936
$786$	0	0
$787$	−6.24621	−0.222653	−0.111327	−	0.993784i	$-0.535510\pi$
$787$	−6.24621	−0.222653	−0.111327	+	0.993784i	$0.535510\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	4.00000	0.142044
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−31.6155	−1.11988	−0.559940	−	0.828533i	$-0.689176\pi$
$797$	−31.6155	−1.11988	−0.559940	+	0.828533i	$0.689176\pi$
$798$	0	0
$799$	2.87689	0.101777
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−10.8769	−0.383837
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−53.0540	−1.86528	−0.932639	−	0.360810i	$-0.882500\pi$
$809$	−53.0540	−1.86528	−0.932639	+	0.360810i	$0.882500\pi$
$810$	0	0
$811$	20.6307	0.724441	0.362221	−	0.932092i	$-0.382019\pi$
$811$	20.6307	0.724441	0.362221	+	0.932092i	$0.382019\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	3.86174	0.135271
$816$	0	0
$817$	−59.0540	−2.06604
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16.5616	0.578002	0.289001	−	0.957329i	$-0.406677\pi$
$821$	16.5616	0.578002	0.289001	+	0.957329i	$0.406677\pi$
$822$	0	0
$823$	−36.4924	−1.27205	−0.636023	−	0.771670i	$-0.719422\pi$
$823$	−36.4924	−1.27205	−0.636023	+	0.771670i	$0.719422\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−14.4233	−0.501547	−0.250774	−	0.968046i	$-0.580685\pi$
$827$	−14.4233	−0.501547	−0.250774	+	0.968046i	$0.580685\pi$
$828$	0	0
$829$	50.4924	1.75367	0.876837	−	0.480787i	$-0.159649\pi$
$829$	50.4924	1.75367	0.876837	+	0.480787i	$0.159649\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	7.00000	0.242536
$834$	0	0
$835$	−0.453602	−0.0156976
$836$	0	0
$837$	0	0
$838$	0	0
$839$	11.0540	0.381626	0.190813	−	0.981626i	$-0.438888\pi$
$839$	11.0540	0.381626	0.190813	+	0.981626i	$0.438888\pi$
$840$	0	0
$841$	39.0000	1.34483
$842$	0	0
$843$	0	0
$844$	0	0
$845$	4.38447	0.150830
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−20.4924	−0.702471
$852$	0	0
$853$	20.7386	0.710077	0.355039	−	0.934852i	$-0.384468\pi$
$853$	20.7386	0.710077	0.355039	+	0.934852i	$0.384468\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	−12.0000	−0.409435	−0.204717	−	0.978821i	$-0.565628\pi$
$859$	−12.0000	−0.409435	−0.204717	+	0.978821i	$0.565628\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−26.2462	−0.893431	−0.446716	−	0.894676i	$-0.647406\pi$
$863$	−26.2462	−0.893431	−0.446716	+	0.894676i	$0.647406\pi$
$864$	0	0
$865$	10.5616	0.359104
$866$	0	0
$867$	0	0
$868$	0	0
$869$	39.3693	1.33551
$870$	0	0
$871$	−18.2462	−0.618249
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−34.0000	−1.14810	−0.574049	−	0.818821i	$-0.694628\pi$
$877$	−34.0000	−1.14810	−0.574049	+	0.818821i	$0.694628\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−23.7538	−0.800285	−0.400143	−	0.916453i	$-0.631039\pi$
$881$	−23.7538	−0.800285	−0.400143	+	0.916453i	$0.631039\pi$
$882$	0	0
$883$	−38.4233	−1.29305	−0.646523	−	0.762894i	$-0.723778\pi$
$883$	−38.4233	−1.29305	−0.646523	+	0.762894i	$0.723778\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	22.5616	0.757543	0.378771	−	0.925490i	$-0.376347\pi$
$887$	22.5616	0.757543	0.378771	+	0.925490i	$0.376347\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	22.1080	0.739814
$894$	0	0
$895$	−5.12311	−0.171247
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−42.2462	−1.40899
$900$	0	0
$901$	−4.24621	−0.141462
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.36932	0.112000
$906$	0	0
$907$	−47.8617	−1.58922	−0.794611	−	0.607118i	$-0.792325\pi$
$907$	−47.8617	−1.58922	−0.794611	+	0.607118i	$0.792325\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−29.3002	−0.970758	−0.485379	−	0.874304i	$-0.661318\pi$
$911$	−29.3002	−0.970758	−0.485379	+	0.874304i	$0.661318\pi$
$912$	0	0
$913$	23.3693	0.773412
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	4.31534	0.142350	0.0711750	−	0.997464i	$-0.477325\pi$
$919$	4.31534	0.142350	0.0711750	+	0.997464i	$0.477325\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	46.7386	1.53842
$924$	0	0
$925$	−14.6307	−0.481054
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−31.9309	−1.04762	−0.523809	−	0.851836i	$-0.675489\pi$
$929$	−31.9309	−1.04762	−0.523809	+	0.851836i	$0.675489\pi$
$930$	0	0
$931$	53.7926	1.76298
$932$	0	0
$933$	0	0
$934$	0	0
$935$	1.43845	0.0470423
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−30.0000	−0.977972	−0.488986	−	0.872292i	$-0.662633\pi$
$941$	−30.0000	−0.977972	−0.488986	+	0.872292i	$0.662633\pi$
$942$	0	0
$943$	3.68466	0.119989
$944$	0	0
$945$	0	0
$946$	0	0
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	0	0
$949$	19.3693	0.628755
$950$	0	0
$951$	0	0
$952$	0	0
$953$	54.3542	1.76070	0.880352	−	0.474321i	$-0.157306\pi$
$953$	54.3542	1.76070	0.880352	+	0.474321i	$0.157306\pi$
$954$	0	0
$955$	−7.36932	−0.238465
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−4.75379	−0.153348
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−13.6155	−0.438299
$966$	0	0
$967$	−46.5616	−1.49732	−0.748659	−	0.662955i	$-0.769302\pi$
$967$	−46.5616	−1.49732	−0.748659	+	0.662955i	$0.769302\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−2.38447	−0.0765213	−0.0382607	−	0.999268i	$-0.512182\pi$
$971$	−2.38447	−0.0765213	−0.0382607	+	0.999268i	$0.512182\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	8.24621	0.263820	0.131910	−	0.991262i	$-0.457889\pi$
$977$	8.24621	0.263820	0.131910	+	0.991262i	$0.457889\pi$
$978$	0	0
$979$	18.2462	0.583151
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.06913	0.0659950	0.0329975	−	0.999455i	$-0.489495\pi$
$983$	2.06913	0.0659950	0.0329975	+	0.999455i	$0.489495\pi$
$984$	0	0
$985$	−11.1922	−0.356614
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−50.4233	−1.60337
$990$	0	0
$991$	6.73863	0.214060	0.107030	−	0.994256i	$-0.465866\pi$
$991$	6.73863	0.214060	0.107030	+	0.994256i	$0.465866\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−8.98485	−0.284839
$996$	0	0
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2448.2.a.v.1.2		2
3.2	odd	2		816.2.a.m.1.1		2
4.3	odd	2		153.2.a.e.1.1		2
8.3	odd	2		9792.2.a.cy.1.1		2
8.5	even	2		9792.2.a.cz.1.1		2
12.11	even	2		51.2.a.b.1.2	✓	2
20.19	odd	2		3825.2.a.s.1.2		2
24.5	odd	2		3264.2.a.bg.1.2		2
24.11	even	2		3264.2.a.bl.1.2		2
28.27	even	2		7497.2.a.v.1.1		2
60.23	odd	4		1275.2.b.d.1174.2		4
60.47	odd	4		1275.2.b.d.1174.3		4
60.59	even	2		1275.2.a.n.1.1		2
68.67	odd	2		2601.2.a.t.1.1		2
84.83	odd	2		2499.2.a.o.1.2		2
132.131	odd	2		6171.2.a.p.1.1		2
156.155	even	2		8619.2.a.q.1.1		2
204.11	odd	16		867.2.h.j.733.4		16
204.23	odd	16		867.2.h.j.733.3		16
204.47	even	4		867.2.d.c.577.1		4
204.59	even	8		867.2.e.f.829.2		8
204.71	odd	16		867.2.h.j.757.3		16
204.83	even	8		867.2.e.f.616.4		8
204.95	odd	16		867.2.h.j.712.1		16
204.107	odd	16		867.2.h.j.688.2		16
204.131	odd	16		867.2.h.j.688.1		16
204.143	odd	16		867.2.h.j.712.2		16
204.155	even	8		867.2.e.f.616.3		8
204.167	odd	16		867.2.h.j.757.4		16
204.179	even	8		867.2.e.f.829.1		8
204.191	even	4		867.2.d.c.577.2		4
204.203	even	2		867.2.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
51.2.a.b.1.2	✓	2	12.11	even	2
153.2.a.e.1.1		2	4.3	odd	2
816.2.a.m.1.1		2	3.2	odd	2
867.2.a.f.1.2		2	204.203	even	2
867.2.d.c.577.1		4	204.47	even	4
867.2.d.c.577.2		4	204.191	even	4
867.2.e.f.616.3		8	204.155	even	8
867.2.e.f.616.4		8	204.83	even	8
867.2.e.f.829.1		8	204.179	even	8
867.2.e.f.829.2		8	204.59	even	8
867.2.h.j.688.1		16	204.131	odd	16
867.2.h.j.688.2		16	204.107	odd	16
867.2.h.j.712.1		16	204.95	odd	16
867.2.h.j.712.2		16	204.143	odd	16
867.2.h.j.733.3		16	204.23	odd	16
867.2.h.j.733.4		16	204.11	odd	16
867.2.h.j.757.3		16	204.71	odd	16
867.2.h.j.757.4		16	204.167	odd	16
1275.2.a.n.1.1		2	60.59	even	2
1275.2.b.d.1174.2		4	60.23	odd	4
1275.2.b.d.1174.3		4	60.47	odd	4
2448.2.a.v.1.2		2	1.1	even	1	trivial
2499.2.a.o.1.2		2	84.83	odd	2
2601.2.a.t.1.1		2	68.67	odd	2
3264.2.a.bg.1.2		2	24.5	odd	2
3264.2.a.bl.1.2		2	24.11	even	2
3825.2.a.s.1.2		2	20.19	odd	2
6171.2.a.p.1.1		2	132.131	odd	2
7497.2.a.v.1.1		2	28.27	even	2
8619.2.a.q.1.1		2	156.155	even	2
9792.2.a.cy.1.1		2	8.3	odd	2
9792.2.a.cz.1.1		2	8.5	even	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:	\( N \)	\(=\)	\( 2448 = 2^{4} \cdot 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2448.a (trivial)

\(f(q)\)	\(=\)	\(q+0.561553 q^{5} +O(q^{10})\)	\(q+0.561553 q^{5} -2.56155 q^{11} +4.56155 q^{13} -1.00000 q^{17} -7.68466 q^{19} -6.56155 q^{23} -4.68466 q^{25} -8.24621 q^{29} +5.12311 q^{31} +3.12311 q^{37} -0.561553 q^{41} +7.68466 q^{43} -2.87689 q^{47} -7.00000 q^{49} +4.24621 q^{53} -1.43845 q^{55} -1.12311 q^{59} +0.876894 q^{61} +2.56155 q^{65} -4.00000 q^{67} +10.2462 q^{71} +4.24621 q^{73} -15.3693 q^{79} -9.12311 q^{83} -0.561553 q^{85} -7.12311 q^{89} -4.31534 q^{95} -11.1231 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{5}+O(q^{10}) \) 2 * q - 3 * q^5	\( 2 q - 3 q^{5} - q^{11} + 5 q^{13} - 2 q^{17} - 3 q^{19} - 9 q^{23} + 3 q^{25} + 2 q^{31} - 2 q^{37} + 3 q^{41} + 3 q^{43} - 14 q^{47} - 14 q^{49} - 8 q^{53} - 7 q^{55} + 6 q^{59} + 10 q^{61} + q^{65} - 8 q^{67} + 4 q^{71} - 8 q^{73} - 6 q^{79} - 10 q^{83} + 3 q^{85} - 6 q^{89} - 21 q^{95} - 14 q^{97}+O(q^{100}) \) 2 * q - 3 * q^5 - q^11 + 5 * q^13 - 2 * q^17 - 3 * q^19 - 9 * q^23 + 3 * q^25 + 2 * q^31 - 2 * q^37 + 3 * q^41 + 3 * q^43 - 14 * q^47 - 14 * q^49 - 8 * q^53 - 7 * q^55 + 6 * q^59 + 10 * q^61 + q^65 - 8 * q^67 + 4 * q^71 - 8 * q^73 - 6 * q^79 - 10 * q^83 + 3 * q^85 - 6 * q^89 - 21 * q^95 - 14 * q^97

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