LMFDB - Embedded newform 1488.2.a.r.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1488,2,Mod(1,1488)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1488, 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("1488.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1488 = 2^{4} \cdot 3 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1488.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:	$11.8817398208$
Analytic rank:	$0$
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 186)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	1488.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -0.561553 q^{5} -5.12311 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -0.561553 q^{5} -5.12311 q^{7} +1.00000 q^{9} +2.56155 q^{11} -0.561553 q^{13} -0.561553 q^{15} +5.68466 q^{17} +2.56155 q^{19} -5.12311 q^{21} +8.00000 q^{23} -4.68466 q^{25} +1.00000 q^{27} -7.12311 q^{29} +1.00000 q^{31} +2.56155 q^{33} +2.87689 q^{35} +8.24621 q^{37} -0.561553 q^{39} +4.24621 q^{41} +9.12311 q^{43} -0.561553 q^{45} +3.68466 q^{47} +19.2462 q^{49} +5.68466 q^{51} -2.00000 q^{53} -1.43845 q^{55} +2.56155 q^{57} +1.12311 q^{59} -0.561553 q^{61} -5.12311 q^{63} +0.315342 q^{65} -0.315342 q^{67} +8.00000 q^{69} +1.43845 q^{71} +10.0000 q^{73} -4.68466 q^{75} -13.1231 q^{77} +3.68466 q^{79} +1.00000 q^{81} -12.8078 q^{83} -3.19224 q^{85} -7.12311 q^{87} -0.246211 q^{89} +2.87689 q^{91} +1.00000 q^{93} -1.43845 q^{95} -14.8078 q^{97} +2.56155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 3 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 3 * q^5 - 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 3 q^{5} - 2 q^{7} + 2 q^{9} + q^{11} + 3 q^{13} + 3 q^{15} - q^{17} + q^{19} - 2 q^{21} + 16 q^{23} + 3 q^{25} + 2 q^{27} - 6 q^{29} + 2 q^{31} + q^{33} + 14 q^{35} + 3 q^{39} - 8 q^{41} + 10 q^{43} + 3 q^{45} - 5 q^{47} + 22 q^{49} - q^{51} - 4 q^{53} - 7 q^{55} + q^{57} - 6 q^{59} + 3 q^{61} - 2 q^{63} + 13 q^{65} - 13 q^{67} + 16 q^{69} + 7 q^{71} + 20 q^{73} + 3 q^{75} - 18 q^{77} - 5 q^{79} + 2 q^{81} - 5 q^{83} - 27 q^{85} - 6 q^{87} + 16 q^{89} + 14 q^{91} + 2 q^{93} - 7 q^{95} - 9 q^{97} + q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 3 * q^5 - 2 * q^7 + 2 * q^9 + q^11 + 3 * q^13 + 3 * q^15 - q^17 + q^19 - 2 * q^21 + 16 * q^23 + 3 * q^25 + 2 * q^27 - 6 * q^29 + 2 * q^31 + q^33 + 14 * q^35 + 3 * q^39 - 8 * q^41 + 10 * q^43 + 3 * q^45 - 5 * q^47 + 22 * q^49 - q^51 - 4 * q^53 - 7 * q^55 + q^57 - 6 * q^59 + 3 * q^61 - 2 * q^63 + 13 * q^65 - 13 * q^67 + 16 * q^69 + 7 * q^71 + 20 * q^73 + 3 * q^75 - 18 * q^77 - 5 * q^79 + 2 * q^81 - 5 * q^83 - 27 * q^85 - 6 * q^87 + 16 * q^89 + 14 * q^91 + 2 * q^93 - 7 * q^95 - 9 * q^97 + 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
$5$	−0.561553	−0.251134	−0.125567	−	0.992085i	$-0.540075\pi$
$5$	−0.561553	−0.251134	−0.125567	+	0.992085i	$0.540075\pi$
$6$	0	0
$7$	−5.12311	−1.93635	−0.968176	−	0.250270i	$-0.919480\pi$
$7$	−5.12311	−1.93635	−0.968176	+	0.250270i	$0.919480\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.56155	0.772337	0.386169	−	0.922428i	$-0.373798\pi$
$11$	2.56155	0.772337	0.386169	+	0.922428i	$0.373798\pi$
$12$	0	0
$13$	−0.561553	−0.155747	−0.0778734	−	0.996963i	$-0.524813\pi$
$13$	−0.561553	−0.155747	−0.0778734	+	0.996963i	$0.524813\pi$
$14$	0	0
$15$	−0.561553	−0.144992
$16$	0	0
$17$	5.68466	1.37873	0.689366	−	0.724413i	$-0.257889\pi$
$17$	5.68466	1.37873	0.689366	+	0.724413i	$0.257889\pi$
$18$	0	0
$19$	2.56155	0.587661	0.293830	−	0.955858i	$-0.405070\pi$
$19$	2.56155	0.587661	0.293830	+	0.955858i	$0.405070\pi$
$20$	0	0
$21$	−5.12311	−1.11795
$22$	0	0
$23$	8.00000	1.66812	0.834058	−	0.551677i	$-0.186012\pi$
$23$	8.00000	1.66812	0.834058	+	0.551677i	$0.186012\pi$
$24$	0	0
$25$	−4.68466	−0.936932
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−7.12311	−1.32273	−0.661364	−	0.750065i	$-0.730022\pi$
$29$	−7.12311	−1.32273	−0.661364	+	0.750065i	$0.730022\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	2.56155	0.445909
$34$	0	0
$35$	2.87689	0.486284
$36$	0	0
$37$	8.24621	1.35567	0.677834	−	0.735215i	$-0.262919\pi$
$37$	8.24621	1.35567	0.677834	+	0.735215i	$0.262919\pi$
$38$	0	0
$39$	−0.561553	−0.0899204
$40$	0	0
$41$	4.24621	0.663147	0.331573	−	0.943429i	$-0.392421\pi$
$41$	4.24621	0.663147	0.331573	+	0.943429i	$0.392421\pi$
$42$	0	0
$43$	9.12311	1.39126	0.695630	−	0.718400i	$-0.255125\pi$
$43$	9.12311	1.39126	0.695630	+	0.718400i	$0.255125\pi$
$44$	0	0
$45$	−0.561553	−0.0837114
$46$	0	0
$47$	3.68466	0.537463	0.268731	−	0.963215i	$-0.413396\pi$
$47$	3.68466	0.537463	0.268731	+	0.963215i	$0.413396\pi$
$48$	0	0
$49$	19.2462	2.74946
$50$	0	0
$51$	5.68466	0.796011
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	−1.43845	−0.193960
$56$	0	0
$57$	2.56155	0.339286
$58$	0	0
$59$	1.12311	0.146216	0.0731079	−	0.997324i	$-0.476708\pi$
$59$	1.12311	0.146216	0.0731079	+	0.997324i	$0.476708\pi$
$60$	0	0
$61$	−0.561553	−0.0718995	−0.0359497	−	0.999354i	$-0.511446\pi$
$61$	−0.561553	−0.0718995	−0.0359497	+	0.999354i	$0.511446\pi$
$62$	0	0
$63$	−5.12311	−0.645451
$64$	0	0
$65$	0.315342	0.0391133
$66$	0	0
$67$	−0.315342	−0.0385251	−0.0192626	−	0.999814i	$-0.506132\pi$
$67$	−0.315342	−0.0385251	−0.0192626	+	0.999814i	$0.506132\pi$
$68$	0	0
$69$	8.00000	0.963087
$70$	0	0
$71$	1.43845	0.170712	0.0853561	−	0.996351i	$-0.472797\pi$
$71$	1.43845	0.170712	0.0853561	+	0.996351i	$0.472797\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	−4.68466	−0.540938
$76$	0	0
$77$	−13.1231	−1.49552
$78$	0	0
$79$	3.68466	0.414556	0.207278	−	0.978282i	$-0.433539\pi$
$79$	3.68466	0.414556	0.207278	+	0.978282i	$0.433539\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.8078	−1.40583	−0.702917	−	0.711272i	$-0.748120\pi$
$83$	−12.8078	−1.40583	−0.702917	+	0.711272i	$0.748120\pi$
$84$	0	0
$85$	−3.19224	−0.346247
$86$	0	0
$87$	−7.12311	−0.763677
$88$	0	0
$89$	−0.246211	−0.0260983	−0.0130492	−	0.999915i	$-0.504154\pi$
$89$	−0.246211	−0.0260983	−0.0130492	+	0.999915i	$0.504154\pi$
$90$	0	0
$91$	2.87689	0.301580
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	−1.43845	−0.147582
$96$	0	0
$97$	−14.8078	−1.50350	−0.751750	−	0.659448i	$-0.770790\pi$
$97$	−14.8078	−1.50350	−0.751750	+	0.659448i	$0.770790\pi$
$98$	0	0
$99$	2.56155	0.257446
$100$	0	0
$101$	16.2462	1.61656	0.808279	−	0.588799i	$-0.200399\pi$
$101$	16.2462	1.61656	0.808279	+	0.588799i	$0.200399\pi$
$102$	0	0
$103$	5.12311	0.504795	0.252397	−	0.967624i	$-0.418781\pi$
$103$	5.12311	0.504795	0.252397	+	0.967624i	$0.418781\pi$
$104$	0	0
$105$	2.87689	0.280756
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	0	0
$111$	8.24621	0.782696
$112$	0	0
$113$	−14.0000	−1.31701	−0.658505	−	0.752577i	$-0.728811\pi$
$113$	−14.0000	−1.31701	−0.658505	+	0.752577i	$0.728811\pi$
$114$	0	0
$115$	−4.49242	−0.418921
$116$	0	0
$117$	−0.561553	−0.0519156
$118$	0	0
$119$	−29.1231	−2.66971
$120$	0	0
$121$	−4.43845	−0.403495
$122$	0	0
$123$	4.24621	0.382868
$124$	0	0
$125$	5.43845	0.486430
$126$	0	0
$127$	−10.2462	−0.909204	−0.454602	−	0.890695i	$-0.650219\pi$
$127$	−10.2462	−0.909204	−0.454602	+	0.890695i	$0.650219\pi$
$128$	0	0
$129$	9.12311	0.803245
$130$	0	0
$131$	−17.1231	−1.49605	−0.748026	−	0.663669i	$-0.768998\pi$
$131$	−17.1231	−1.49605	−0.748026	+	0.663669i	$0.768998\pi$
$132$	0	0
$133$	−13.1231	−1.13792
$134$	0	0
$135$	−0.561553	−0.0483308
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	3.68466	0.310304
$142$	0	0
$143$	−1.43845	−0.120289
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	19.2462	1.58740
$148$	0	0
$149$	9.68466	0.793398	0.396699	−	0.917949i	$-0.370156\pi$
$149$	9.68466	0.793398	0.396699	+	0.917949i	$0.370156\pi$
$150$	0	0
$151$	−5.93087	−0.482647	−0.241324	−	0.970445i	$-0.577582\pi$
$151$	−5.93087	−0.482647	−0.241324	+	0.970445i	$0.577582\pi$
$152$	0	0
$153$	5.68466	0.459577
$154$	0	0
$155$	−0.561553	−0.0451050
$156$	0	0
$157$	−9.36932	−0.747753	−0.373876	−	0.927479i	$-0.621972\pi$
$157$	−9.36932	−0.747753	−0.373876	+	0.927479i	$0.621972\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	0	0
$161$	−40.9848	−3.23006
$162$	0	0
$163$	23.0540	1.80573	0.902863	−	0.429928i	$-0.141461\pi$
$163$	23.0540	1.80573	0.902863	+	0.429928i	$0.141461\pi$
$164$	0	0
$165$	−1.43845	−0.111983
$166$	0	0
$167$	5.12311	0.396438	0.198219	−	0.980158i	$-0.436484\pi$
$167$	5.12311	0.396438	0.198219	+	0.980158i	$0.436484\pi$
$168$	0	0
$169$	−12.6847	−0.975743
$170$	0	0
$171$	2.56155	0.195887
$172$	0	0
$173$	4.56155	0.346808	0.173404	−	0.984851i	$-0.444523\pi$
$173$	4.56155	0.346808	0.173404	+	0.984851i	$0.444523\pi$
$174$	0	0
$175$	24.0000	1.81423
$176$	0	0
$177$	1.12311	0.0844178
$178$	0	0
$179$	−23.0540	−1.72314	−0.861568	−	0.507643i	$-0.830517\pi$
$179$	−23.0540	−1.72314	−0.861568	+	0.507643i	$0.830517\pi$
$180$	0	0
$181$	−12.2462	−0.910254	−0.455127	−	0.890427i	$-0.650406\pi$
$181$	−12.2462	−0.910254	−0.455127	+	0.890427i	$0.650406\pi$
$182$	0	0
$183$	−0.561553	−0.0415112
$184$	0	0
$185$	−4.63068	−0.340455
$186$	0	0
$187$	14.5616	1.06485
$188$	0	0
$189$	−5.12311	−0.372651
$190$	0	0
$191$	−16.0000	−1.15772	−0.578860	−	0.815427i	$-0.696502\pi$
$191$	−16.0000	−1.15772	−0.578860	+	0.815427i	$0.696502\pi$
$192$	0	0
$193$	24.5616	1.76798	0.883990	−	0.467507i	$-0.154848\pi$
$193$	24.5616	1.76798	0.883990	+	0.467507i	$0.154848\pi$
$194$	0	0
$195$	0.315342	0.0225821
$196$	0	0
$197$	8.24621	0.587518	0.293759	−	0.955879i	$-0.405094\pi$
$197$	8.24621	0.587518	0.293759	+	0.955879i	$0.405094\pi$
$198$	0	0
$199$	−4.31534	−0.305906	−0.152953	−	0.988233i	$-0.548878\pi$
$199$	−4.31534	−0.305906	−0.152953	+	0.988233i	$0.548878\pi$
$200$	0	0
$201$	−0.315342	−0.0222425
$202$	0	0
$203$	36.4924	2.56127
$204$	0	0
$205$	−2.38447	−0.166539
$206$	0	0
$207$	8.00000	0.556038
$208$	0	0
$209$	6.56155	0.453872
$210$	0	0
$211$	1.75379	0.120736	0.0603679	−	0.998176i	$-0.480773\pi$
$211$	1.75379	0.120736	0.0603679	+	0.998176i	$0.480773\pi$
$212$	0	0
$213$	1.43845	0.0985608
$214$	0	0
$215$	−5.12311	−0.349393
$216$	0	0
$217$	−5.12311	−0.347779
$218$	0	0
$219$	10.0000	0.675737
$220$	0	0
$221$	−3.19224	−0.214733
$222$	0	0
$223$	−19.6847	−1.31818	−0.659091	−	0.752063i	$-0.729059\pi$
$223$	−19.6847	−1.31818	−0.659091	+	0.752063i	$0.729059\pi$
$224$	0	0
$225$	−4.68466	−0.312311
$226$	0	0
$227$	19.3693	1.28559	0.642793	−	0.766040i	$-0.277775\pi$
$227$	19.3693	1.28559	0.642793	+	0.766040i	$0.277775\pi$
$228$	0	0
$229$	17.6847	1.16864	0.584318	−	0.811525i	$-0.301362\pi$
$229$	17.6847	1.16864	0.584318	+	0.811525i	$0.301362\pi$
$230$	0	0
$231$	−13.1231	−0.863437
$232$	0	0
$233$	11.6155	0.760959	0.380479	−	0.924789i	$-0.375759\pi$
$233$	11.6155	0.760959	0.380479	+	0.924789i	$0.375759\pi$
$234$	0	0
$235$	−2.06913	−0.134975
$236$	0	0
$237$	3.68466	0.239344
$238$	0	0
$239$	20.4924	1.32554	0.662772	−	0.748821i	$-0.269380\pi$
$239$	20.4924	1.32554	0.662772	+	0.748821i	$0.269380\pi$
$240$	0	0
$241$	7.75379	0.499465	0.249733	−	0.968315i	$-0.419657\pi$
$241$	7.75379	0.499465	0.249733	+	0.968315i	$0.419657\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−10.8078	−0.690483
$246$	0	0
$247$	−1.43845	−0.0915262
$248$	0	0
$249$	−12.8078	−0.811659
$250$	0	0
$251$	−20.0000	−1.26239	−0.631194	−	0.775625i	$-0.717435\pi$
$251$	−20.0000	−1.26239	−0.631194	+	0.775625i	$0.717435\pi$
$252$	0	0
$253$	20.4924	1.28835
$254$	0	0
$255$	−3.19224	−0.199906
$256$	0	0
$257$	−16.8769	−1.05275	−0.526376	−	0.850252i	$-0.676450\pi$
$257$	−16.8769	−1.05275	−0.526376	+	0.850252i	$0.676450\pi$
$258$	0	0
$259$	−42.2462	−2.62505
$260$	0	0
$261$	−7.12311	−0.440909
$262$	0	0
$263$	2.24621	0.138507	0.0692537	−	0.997599i	$-0.477938\pi$
$263$	2.24621	0.138507	0.0692537	+	0.997599i	$0.477938\pi$
$264$	0	0
$265$	1.12311	0.0689918
$266$	0	0
$267$	−0.246211	−0.0150679
$268$	0	0
$269$	−14.4924	−0.883619	−0.441809	−	0.897109i	$-0.645663\pi$
$269$	−14.4924	−0.883619	−0.441809	+	0.897109i	$0.645663\pi$
$270$	0	0
$271$	3.68466	0.223827	0.111914	−	0.993718i	$-0.464302\pi$
$271$	3.68466	0.223827	0.111914	+	0.993718i	$0.464302\pi$
$272$	0	0
$273$	2.87689	0.174118
$274$	0	0
$275$	−12.0000	−0.723627
$276$	0	0
$277$	7.43845	0.446933	0.223466	−	0.974712i	$-0.428263\pi$
$277$	7.43845	0.446933	0.223466	+	0.974712i	$0.428263\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	14.4924	0.864545	0.432273	−	0.901743i	$-0.357712\pi$
$281$	14.4924	0.864545	0.432273	+	0.901743i	$0.357712\pi$
$282$	0	0
$283$	3.19224	0.189759	0.0948794	−	0.995489i	$-0.469753\pi$
$283$	3.19224	0.189759	0.0948794	+	0.995489i	$0.469753\pi$
$284$	0	0
$285$	−1.43845	−0.0852063
$286$	0	0
$287$	−21.7538	−1.28409
$288$	0	0
$289$	15.3153	0.900902
$290$	0	0
$291$	−14.8078	−0.868047
$292$	0	0
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	−0.630683	−0.0367198
$296$	0	0
$297$	2.56155	0.148636
$298$	0	0
$299$	−4.49242	−0.259804
$300$	0	0
$301$	−46.7386	−2.69397
$302$	0	0
$303$	16.2462	0.933320
$304$	0	0
$305$	0.315342	0.0180564
$306$	0	0
$307$	1.75379	0.100094	0.0500470	−	0.998747i	$-0.484063\pi$
$307$	1.75379	0.100094	0.0500470	+	0.998747i	$0.484063\pi$
$308$	0	0
$309$	5.12311	0.291443
$310$	0	0
$311$	−21.9309	−1.24359	−0.621793	−	0.783182i	$-0.713595\pi$
$311$	−21.9309	−1.24359	−0.621793	+	0.783182i	$0.713595\pi$
$312$	0	0
$313$	−16.2462	−0.918290	−0.459145	−	0.888361i	$-0.651844\pi$
$313$	−16.2462	−0.918290	−0.459145	+	0.888361i	$0.651844\pi$
$314$	0	0
$315$	2.87689	0.162095
$316$	0	0
$317$	10.3153	0.579367	0.289684	−	0.957122i	$-0.406450\pi$
$317$	10.3153	0.579367	0.289684	+	0.957122i	$0.406450\pi$
$318$	0	0
$319$	−18.2462	−1.02159
$320$	0	0
$321$	4.00000	0.223258
$322$	0	0
$323$	14.5616	0.810226
$324$	0	0
$325$	2.63068	0.145924
$326$	0	0
$327$	14.0000	0.774202
$328$	0	0
$329$	−18.8769	−1.04072
$330$	0	0
$331$	12.0000	0.659580	0.329790	−	0.944054i	$-0.393022\pi$
$331$	12.0000	0.659580	0.329790	+	0.944054i	$0.393022\pi$
$332$	0	0
$333$	8.24621	0.451890
$334$	0	0
$335$	0.177081	0.00967497
$336$	0	0
$337$	−27.1231	−1.47749	−0.738745	−	0.673985i	$-0.764581\pi$
$337$	−27.1231	−1.47749	−0.738745	+	0.673985i	$0.764581\pi$
$338$	0	0
$339$	−14.0000	−0.760376
$340$	0	0
$341$	2.56155	0.138716
$342$	0	0
$343$	−62.7386	−3.38757
$344$	0	0
$345$	−4.49242	−0.241864
$346$	0	0
$347$	−12.1771	−0.653700	−0.326850	−	0.945076i	$-0.605987\pi$
$347$	−12.1771	−0.653700	−0.326850	+	0.945076i	$0.605987\pi$
$348$	0	0
$349$	15.6155	0.835880	0.417940	−	0.908475i	$-0.362752\pi$
$349$	15.6155	0.835880	0.417940	+	0.908475i	$0.362752\pi$
$350$	0	0
$351$	−0.561553	−0.0299735
$352$	0	0
$353$	18.8078	1.00104	0.500518	−	0.865726i	$-0.333143\pi$
$353$	18.8078	1.00104	0.500518	+	0.865726i	$0.333143\pi$
$354$	0	0
$355$	−0.807764	−0.0428717
$356$	0	0
$357$	−29.1231	−1.54136
$358$	0	0
$359$	−29.3002	−1.54640	−0.773202	−	0.634159i	$-0.781346\pi$
$359$	−29.3002	−1.54640	−0.773202	+	0.634159i	$0.781346\pi$
$360$	0	0
$361$	−12.4384	−0.654655
$362$	0	0
$363$	−4.43845	−0.232958
$364$	0	0
$365$	−5.61553	−0.293930
$366$	0	0
$367$	−32.8078	−1.71255	−0.856276	−	0.516519i	$-0.827228\pi$
$367$	−32.8078	−1.71255	−0.856276	+	0.516519i	$0.827228\pi$
$368$	0	0
$369$	4.24621	0.221049
$370$	0	0
$371$	10.2462	0.531957
$372$	0	0
$373$	3.12311	0.161708	0.0808541	−	0.996726i	$-0.474235\pi$
$373$	3.12311	0.161708	0.0808541	+	0.996726i	$0.474235\pi$
$374$	0	0
$375$	5.43845	0.280840
$376$	0	0
$377$	4.00000	0.206010
$378$	0	0
$379$	17.9309	0.921047	0.460523	−	0.887648i	$-0.347662\pi$
$379$	17.9309	0.921047	0.460523	+	0.887648i	$0.347662\pi$
$380$	0	0
$381$	−10.2462	−0.524929
$382$	0	0
$383$	17.6155	0.900111	0.450056	−	0.893000i	$-0.351404\pi$
$383$	17.6155	0.900111	0.450056	+	0.893000i	$0.351404\pi$
$384$	0	0
$385$	7.36932	0.375575
$386$	0	0
$387$	9.12311	0.463754
$388$	0	0
$389$	−12.2462	−0.620908	−0.310454	−	0.950588i	$-0.600481\pi$
$389$	−12.2462	−0.620908	−0.310454	+	0.950588i	$0.600481\pi$
$390$	0	0
$391$	45.4773	2.29988
$392$	0	0
$393$	−17.1231	−0.863746
$394$	0	0
$395$	−2.06913	−0.104109
$396$	0	0
$397$	12.7386	0.639334	0.319667	−	0.947530i	$-0.396429\pi$
$397$	12.7386	0.639334	0.319667	+	0.947530i	$0.396429\pi$
$398$	0	0
$399$	−13.1231	−0.656977
$400$	0	0
$401$	−27.9309	−1.39480	−0.697401	−	0.716682i	$-0.745660\pi$
$401$	−27.9309	−1.39480	−0.697401	+	0.716682i	$0.745660\pi$
$402$	0	0
$403$	−0.561553	−0.0279729
$404$	0	0
$405$	−0.561553	−0.0279038
$406$	0	0
$407$	21.1231	1.04703
$408$	0	0
$409$	4.24621	0.209962	0.104981	−	0.994474i	$-0.466522\pi$
$409$	4.24621	0.209962	0.104981	+	0.994474i	$0.466522\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	0	0
$413$	−5.75379	−0.283125
$414$	0	0
$415$	7.19224	0.353053
$416$	0	0
$417$	12.0000	0.587643
$418$	0	0
$419$	6.24621	0.305147	0.152574	−	0.988292i	$-0.451244\pi$
$419$	6.24621	0.305147	0.152574	+	0.988292i	$0.451244\pi$
$420$	0	0
$421$	−24.7386	−1.20569	−0.602844	−	0.797859i	$-0.705966\pi$
$421$	−24.7386	−1.20569	−0.602844	+	0.797859i	$0.705966\pi$
$422$	0	0
$423$	3.68466	0.179154
$424$	0	0
$425$	−26.6307	−1.29178
$426$	0	0
$427$	2.87689	0.139223
$428$	0	0
$429$	−1.43845	−0.0694489
$430$	0	0
$431$	−40.9848	−1.97417	−0.987085	−	0.160196i	$-0.948787\pi$
$431$	−40.9848	−1.97417	−0.987085	+	0.160196i	$0.948787\pi$
$432$	0	0
$433$	35.6155	1.71157	0.855787	−	0.517329i	$-0.173074\pi$
$433$	35.6155	1.71157	0.855787	+	0.517329i	$0.173074\pi$
$434$	0	0
$435$	4.00000	0.191785
$436$	0	0
$437$	20.4924	0.980286
$438$	0	0
$439$	38.7386	1.84889	0.924447	−	0.381310i	$-0.124527\pi$
$439$	38.7386	1.84889	0.924447	+	0.381310i	$0.124527\pi$
$440$	0	0
$441$	19.2462	0.916486
$442$	0	0
$443$	11.3693	0.540173	0.270086	−	0.962836i	$-0.412948\pi$
$443$	11.3693	0.540173	0.270086	+	0.962836i	$0.412948\pi$
$444$	0	0
$445$	0.138261	0.00655418
$446$	0	0
$447$	9.68466	0.458069
$448$	0	0
$449$	2.80776	0.132507	0.0662533	−	0.997803i	$-0.478895\pi$
$449$	2.80776	0.132507	0.0662533	+	0.997803i	$0.478895\pi$
$450$	0	0
$451$	10.8769	0.512173
$452$	0	0
$453$	−5.93087	−0.278657
$454$	0	0
$455$	−1.61553	−0.0757371
$456$	0	0
$457$	−0.246211	−0.0115173	−0.00575864	−	0.999983i	$-0.501833\pi$
$457$	−0.246211	−0.0115173	−0.00575864	+	0.999983i	$0.501833\pi$
$458$	0	0
$459$	5.68466	0.265337
$460$	0	0
$461$	32.2462	1.50186	0.750928	−	0.660384i	$-0.229607\pi$
$461$	32.2462	1.50186	0.750928	+	0.660384i	$0.229607\pi$
$462$	0	0
$463$	13.9309	0.647422	0.323711	−	0.946156i	$-0.395069\pi$
$463$	13.9309	0.647422	0.323711	+	0.946156i	$0.395069\pi$
$464$	0	0
$465$	−0.561553	−0.0260414
$466$	0	0
$467$	22.2462	1.02943	0.514716	−	0.857361i	$-0.327897\pi$
$467$	22.2462	1.02943	0.514716	+	0.857361i	$0.327897\pi$
$468$	0	0
$469$	1.61553	0.0745982
$470$	0	0
$471$	−9.36932	−0.431715
$472$	0	0
$473$	23.3693	1.07452
$474$	0	0
$475$	−12.0000	−0.550598
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	0	0
$479$	−29.9309	−1.36758	−0.683788	−	0.729681i	$-0.739669\pi$
$479$	−29.9309	−1.36758	−0.683788	+	0.729681i	$0.739669\pi$
$480$	0	0
$481$	−4.63068	−0.211141
$482$	0	0
$483$	−40.9848	−1.86488
$484$	0	0
$485$	8.31534	0.377580
$486$	0	0
$487$	11.6847	0.529482	0.264741	−	0.964319i	$-0.414713\pi$
$487$	11.6847	0.529482	0.264741	+	0.964319i	$0.414713\pi$
$488$	0	0
$489$	23.0540	1.04254
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	0	0
$493$	−40.4924	−1.82369
$494$	0	0
$495$	−1.43845	−0.0646534
$496$	0	0
$497$	−7.36932	−0.330559
$498$	0	0
$499$	−1.75379	−0.0785104	−0.0392552	−	0.999229i	$-0.512499\pi$
$499$	−1.75379	−0.0785104	−0.0392552	+	0.999229i	$0.512499\pi$
$500$	0	0
$501$	5.12311	0.228883
$502$	0	0
$503$	32.1771	1.43471	0.717353	−	0.696710i	$-0.245354\pi$
$503$	32.1771	1.43471	0.717353	+	0.696710i	$0.245354\pi$
$504$	0	0
$505$	−9.12311	−0.405973
$506$	0	0
$507$	−12.6847	−0.563345
$508$	0	0
$509$	−37.8617	−1.67819	−0.839096	−	0.543983i	$-0.816915\pi$
$509$	−37.8617	−1.67819	−0.839096	+	0.543983i	$0.816915\pi$
$510$	0	0
$511$	−51.2311	−2.26633
$512$	0	0
$513$	2.56155	0.113095
$514$	0	0
$515$	−2.87689	−0.126771
$516$	0	0
$517$	9.43845	0.415102
$518$	0	0
$519$	4.56155	0.200230
$520$	0	0
$521$	−23.6155	−1.03462	−0.517308	−	0.855800i	$-0.673066\pi$
$521$	−23.6155	−1.03462	−0.517308	+	0.855800i	$0.673066\pi$
$522$	0	0
$523$	25.1231	1.09856	0.549278	−	0.835639i	$-0.314903\pi$
$523$	25.1231	1.09856	0.549278	+	0.835639i	$0.314903\pi$
$524$	0	0
$525$	24.0000	1.04745
$526$	0	0
$527$	5.68466	0.247628
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	1.12311	0.0487386
$532$	0	0
$533$	−2.38447	−0.103283
$534$	0	0
$535$	−2.24621	−0.0971122
$536$	0	0
$537$	−23.0540	−0.994852
$538$	0	0
$539$	49.3002	2.12351
$540$	0	0
$541$	−41.3693	−1.77861	−0.889303	−	0.457319i	$-0.848810\pi$
$541$	−41.3693	−1.77861	−0.889303	+	0.457319i	$0.848810\pi$
$542$	0	0
$543$	−12.2462	−0.525535
$544$	0	0
$545$	−7.86174	−0.336760
$546$	0	0
$547$	26.7386	1.14326	0.571631	−	0.820511i	$-0.306311\pi$
$547$	26.7386	1.14326	0.571631	+	0.820511i	$0.306311\pi$
$548$	0	0
$549$	−0.561553	−0.0239665
$550$	0	0
$551$	−18.2462	−0.777315
$552$	0	0
$553$	−18.8769	−0.802727
$554$	0	0
$555$	−4.63068	−0.196562
$556$	0	0
$557$	−17.3693	−0.735962	−0.367981	−	0.929833i	$-0.619951\pi$
$557$	−17.3693	−0.735962	−0.367981	+	0.929833i	$0.619951\pi$
$558$	0	0
$559$	−5.12311	−0.216684
$560$	0	0
$561$	14.5616	0.614789
$562$	0	0
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	0	0
$565$	7.86174	0.330746
$566$	0	0
$567$	−5.12311	−0.215150
$568$	0	0
$569$	4.24621	0.178010	0.0890052	−	0.996031i	$-0.471631\pi$
$569$	4.24621	0.178010	0.0890052	+	0.996031i	$0.471631\pi$
$570$	0	0
$571$	−28.9848	−1.21298	−0.606489	−	0.795092i	$-0.707423\pi$
$571$	−28.9848	−1.21298	−0.606489	+	0.795092i	$0.707423\pi$
$572$	0	0
$573$	−16.0000	−0.668410
$574$	0	0
$575$	−37.4773	−1.56291
$576$	0	0
$577$	−7.43845	−0.309667	−0.154833	−	0.987941i	$-0.549484\pi$
$577$	−7.43845	−0.309667	−0.154833	+	0.987941i	$0.549484\pi$
$578$	0	0
$579$	24.5616	1.02074
$580$	0	0
$581$	65.6155	2.72219
$582$	0	0
$583$	−5.12311	−0.212177
$584$	0	0
$585$	0.315342	0.0130378
$586$	0	0
$587$	−25.3002	−1.04425	−0.522125	−	0.852869i	$-0.674861\pi$
$587$	−25.3002	−1.04425	−0.522125	+	0.852869i	$0.674861\pi$
$588$	0	0
$589$	2.56155	0.105547
$590$	0	0
$591$	8.24621	0.339204
$592$	0	0
$593$	−34.4924	−1.41643	−0.708217	−	0.705995i	$-0.750500\pi$
$593$	−34.4924	−1.41643	−0.708217	+	0.705995i	$0.750500\pi$
$594$	0	0
$595$	16.3542	0.670455
$596$	0	0
$597$	−4.31534	−0.176615
$598$	0	0
$599$	20.3153	0.830062	0.415031	−	0.909807i	$-0.363771\pi$
$599$	20.3153	0.830062	0.415031	+	0.909807i	$0.363771\pi$
$600$	0	0
$601$	−11.7538	−0.479447	−0.239724	−	0.970841i	$-0.577057\pi$
$601$	−11.7538	−0.479447	−0.239724	+	0.970841i	$0.577057\pi$
$602$	0	0
$603$	−0.315342	−0.0128417
$604$	0	0
$605$	2.49242	0.101331
$606$	0	0
$607$	−7.36932	−0.299111	−0.149556	−	0.988753i	$-0.547784\pi$
$607$	−7.36932	−0.299111	−0.149556	+	0.988753i	$0.547784\pi$
$608$	0	0
$609$	36.4924	1.47875
$610$	0	0
$611$	−2.06913	−0.0837081
$612$	0	0
$613$	13.1922	0.532829	0.266415	−	0.963859i	$-0.414161\pi$
$613$	13.1922	0.532829	0.266415	+	0.963859i	$0.414161\pi$
$614$	0	0
$615$	−2.38447	−0.0961512
$616$	0	0
$617$	−22.0000	−0.885687	−0.442843	−	0.896599i	$-0.646030\pi$
$617$	−22.0000	−0.885687	−0.442843	+	0.896599i	$0.646030\pi$
$618$	0	0
$619$	−9.75379	−0.392038	−0.196019	−	0.980600i	$-0.562801\pi$
$619$	−9.75379	−0.392038	−0.196019	+	0.980600i	$0.562801\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	0	0
$623$	1.26137	0.0505356
$624$	0	0
$625$	20.3693	0.814773
$626$	0	0
$627$	6.56155	0.262043
$628$	0	0
$629$	46.8769	1.86910
$630$	0	0
$631$	−8.00000	−0.318475	−0.159237	−	0.987240i	$-0.550904\pi$
$631$	−8.00000	−0.318475	−0.159237	+	0.987240i	$0.550904\pi$
$632$	0	0
$633$	1.75379	0.0697068
$634$	0	0
$635$	5.75379	0.228332
$636$	0	0
$637$	−10.8078	−0.428219
$638$	0	0
$639$	1.43845	0.0569041
$640$	0	0
$641$	−10.3153	−0.407431	−0.203716	−	0.979030i	$-0.565302\pi$
$641$	−10.3153	−0.407431	−0.203716	+	0.979030i	$0.565302\pi$
$642$	0	0
$643$	20.0000	0.788723	0.394362	−	0.918955i	$-0.370966\pi$
$643$	20.0000	0.788723	0.394362	+	0.918955i	$0.370966\pi$
$644$	0	0
$645$	−5.12311	−0.201722
$646$	0	0
$647$	35.8617	1.40987	0.704935	−	0.709272i	$-0.250976\pi$
$647$	35.8617	1.40987	0.704935	+	0.709272i	$0.250976\pi$
$648$	0	0
$649$	2.87689	0.112928
$650$	0	0
$651$	−5.12311	−0.200790
$652$	0	0
$653$	−33.5464	−1.31277	−0.656386	−	0.754425i	$-0.727916\pi$
$653$	−33.5464	−1.31277	−0.656386	+	0.754425i	$0.727916\pi$
$654$	0	0
$655$	9.61553	0.375710
$656$	0	0
$657$	10.0000	0.390137
$658$	0	0
$659$	19.3693	0.754521	0.377261	−	0.926107i	$-0.376866\pi$
$659$	19.3693	0.754521	0.377261	+	0.926107i	$0.376866\pi$
$660$	0	0
$661$	44.1080	1.71560	0.857800	−	0.513983i	$-0.171831\pi$
$661$	44.1080	1.71560	0.857800	+	0.513983i	$0.171831\pi$
$662$	0	0
$663$	−3.19224	−0.123976
$664$	0	0
$665$	7.36932	0.285770
$666$	0	0
$667$	−56.9848	−2.20646
$668$	0	0
$669$	−19.6847	−0.761053
$670$	0	0
$671$	−1.43845	−0.0555306
$672$	0	0
$673$	−0.876894	−0.0338018	−0.0169009	−	0.999857i	$-0.505380\pi$
$673$	−0.876894	−0.0338018	−0.0169009	+	0.999857i	$0.505380\pi$
$674$	0	0
$675$	−4.68466	−0.180313
$676$	0	0
$677$	0.876894	0.0337018	0.0168509	−	0.999858i	$-0.494636\pi$
$677$	0.876894	0.0337018	0.0168509	+	0.999858i	$0.494636\pi$
$678$	0	0
$679$	75.8617	2.91131
$680$	0	0
$681$	19.3693	0.742234
$682$	0	0
$683$	50.7386	1.94146	0.970730	−	0.240174i	$-0.0772044\pi$
$683$	50.7386	1.94146	0.970730	+	0.240174i	$0.0772044\pi$
$684$	0	0
$685$	3.36932	0.128735
$686$	0	0
$687$	17.6847	0.674712
$688$	0	0
$689$	1.12311	0.0427869
$690$	0	0
$691$	4.17708	0.158904	0.0794518	−	0.996839i	$-0.474683\pi$
$691$	4.17708	0.158904	0.0794518	+	0.996839i	$0.474683\pi$
$692$	0	0
$693$	−13.1231	−0.498506
$694$	0	0
$695$	−6.73863	−0.255611
$696$	0	0
$697$	24.1383	0.914302
$698$	0	0
$699$	11.6155	0.439340
$700$	0	0
$701$	−4.06913	−0.153689	−0.0768445	−	0.997043i	$-0.524484\pi$
$701$	−4.06913	−0.153689	−0.0768445	+	0.997043i	$0.524484\pi$
$702$	0	0
$703$	21.1231	0.796673
$704$	0	0
$705$	−2.06913	−0.0779280
$706$	0	0
$707$	−83.2311	−3.13023
$708$	0	0
$709$	−20.4233	−0.767013	−0.383506	−	0.923538i	$-0.625284\pi$
$709$	−20.4233	−0.767013	−0.383506	+	0.923538i	$0.625284\pi$
$710$	0	0
$711$	3.68466	0.138185
$712$	0	0
$713$	8.00000	0.299602
$714$	0	0
$715$	0.807764	0.0302087
$716$	0	0
$717$	20.4924	0.765304
$718$	0	0
$719$	−4.49242	−0.167539	−0.0837695	−	0.996485i	$-0.526696\pi$
$719$	−4.49242	−0.167539	−0.0837695	+	0.996485i	$0.526696\pi$
$720$	0	0
$721$	−26.2462	−0.977460
$722$	0	0
$723$	7.75379	0.288367
$724$	0	0
$725$	33.3693	1.23931
$726$	0	0
$727$	13.7538	0.510100	0.255050	−	0.966928i	$-0.417908\pi$
$727$	13.7538	0.510100	0.255050	+	0.966928i	$0.417908\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	51.8617	1.91818
$732$	0	0
$733$	8.24621	0.304581	0.152290	−	0.988336i	$-0.451335\pi$
$733$	8.24621	0.304581	0.152290	+	0.988336i	$0.451335\pi$
$734$	0	0
$735$	−10.8078	−0.398650
$736$	0	0
$737$	−0.807764	−0.0297544
$738$	0	0
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	0	0
$741$	−1.43845	−0.0528427
$742$	0	0
$743$	8.00000	0.293492	0.146746	−	0.989174i	$-0.453120\pi$
$743$	8.00000	0.293492	0.146746	+	0.989174i	$0.453120\pi$
$744$	0	0
$745$	−5.43845	−0.199249
$746$	0	0
$747$	−12.8078	−0.468612
$748$	0	0
$749$	−20.4924	−0.748777
$750$	0	0
$751$	18.8769	0.688828	0.344414	−	0.938818i	$-0.388078\pi$
$751$	18.8769	0.688828	0.344414	+	0.938818i	$0.388078\pi$
$752$	0	0
$753$	−20.0000	−0.728841
$754$	0	0
$755$	3.33050	0.121209
$756$	0	0
$757$	40.2462	1.46277	0.731387	−	0.681963i	$-0.238873\pi$
$757$	40.2462	1.46277	0.731387	+	0.681963i	$0.238873\pi$
$758$	0	0
$759$	20.4924	0.743828
$760$	0	0
$761$	−40.4233	−1.46534	−0.732672	−	0.680582i	$-0.761727\pi$
$761$	−40.4233	−1.46534	−0.732672	+	0.680582i	$0.761727\pi$
$762$	0	0
$763$	−71.7235	−2.59656
$764$	0	0
$765$	−3.19224	−0.115416
$766$	0	0
$767$	−0.630683	−0.0227726
$768$	0	0
$769$	7.75379	0.279609	0.139804	−	0.990179i	$-0.455353\pi$
$769$	7.75379	0.279609	0.139804	+	0.990179i	$0.455353\pi$
$770$	0	0
$771$	−16.8769	−0.607807
$772$	0	0
$773$	31.6155	1.13713	0.568566	−	0.822638i	$-0.307498\pi$
$773$	31.6155	1.13713	0.568566	+	0.822638i	$0.307498\pi$
$774$	0	0
$775$	−4.68466	−0.168278
$776$	0	0
$777$	−42.2462	−1.51557
$778$	0	0
$779$	10.8769	0.389705
$780$	0	0
$781$	3.68466	0.131847
$782$	0	0
$783$	−7.12311	−0.254559
$784$	0	0
$785$	5.26137	0.187786
$786$	0	0
$787$	15.5076	0.552785	0.276393	−	0.961045i	$-0.410861\pi$
$787$	15.5076	0.552785	0.276393	+	0.961045i	$0.410861\pi$
$788$	0	0
$789$	2.24621	0.0799672
$790$	0	0
$791$	71.7235	2.55019
$792$	0	0
$793$	0.315342	0.0111981
$794$	0	0
$795$	1.12311	0.0398325
$796$	0	0
$797$	54.3542	1.92532	0.962662	−	0.270708i	$-0.0872577\pi$
$797$	54.3542	1.92532	0.962662	+	0.270708i	$0.0872577\pi$
$798$	0	0
$799$	20.9460	0.741017
$800$	0	0
$801$	−0.246211	−0.00869945
$802$	0	0
$803$	25.6155	0.903952
$804$	0	0
$805$	23.0152	0.811178
$806$	0	0
$807$	−14.4924	−0.510157
$808$	0	0
$809$	−17.0540	−0.599586	−0.299793	−	0.954004i	$-0.596918\pi$
$809$	−17.0540	−0.599586	−0.299793	+	0.954004i	$0.596918\pi$
$810$	0	0
$811$	−16.4924	−0.579127	−0.289564	−	0.957159i	$-0.593510\pi$
$811$	−16.4924	−0.579127	−0.289564	+	0.957159i	$0.593510\pi$
$812$	0	0
$813$	3.68466	0.129227
$814$	0	0
$815$	−12.9460	−0.453479
$816$	0	0
$817$	23.3693	0.817589
$818$	0	0
$819$	2.87689	0.100527
$820$	0	0
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	0	0
$823$	−11.6847	−0.407302	−0.203651	−	0.979044i	$-0.565281\pi$
$823$	−11.6847	−0.407302	−0.203651	+	0.979044i	$0.565281\pi$
$824$	0	0
$825$	−12.0000	−0.417786
$826$	0	0
$827$	30.4233	1.05792	0.528961	−	0.848646i	$-0.322582\pi$
$827$	30.4233	1.05792	0.528961	+	0.848646i	$0.322582\pi$
$828$	0	0
$829$	−4.24621	−0.147477	−0.0737385	−	0.997278i	$-0.523493\pi$
$829$	−4.24621	−0.147477	−0.0737385	+	0.997278i	$0.523493\pi$
$830$	0	0
$831$	7.43845	0.258037
$832$	0	0
$833$	109.408	3.79077
$834$	0	0
$835$	−2.87689	−0.0995590
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	13.7538	0.474834	0.237417	−	0.971408i	$-0.423699\pi$
$839$	13.7538	0.474834	0.237417	+	0.971408i	$0.423699\pi$
$840$	0	0
$841$	21.7386	0.749608
$842$	0	0
$843$	14.4924	0.499146
$844$	0	0
$845$	7.12311	0.245042
$846$	0	0
$847$	22.7386	0.781309
$848$	0	0
$849$	3.19224	0.109557
$850$	0	0
$851$	65.9697	2.26141
$852$	0	0
$853$	−29.2311	−1.00085	−0.500426	−	0.865779i	$-0.666823\pi$
$853$	−29.2311	−1.00085	−0.500426	+	0.865779i	$0.666823\pi$
$854$	0	0
$855$	−1.43845	−0.0491939
$856$	0	0
$857$	−28.1080	−0.960149	−0.480075	−	0.877228i	$-0.659390\pi$
$857$	−28.1080	−0.960149	−0.480075	+	0.877228i	$0.659390\pi$
$858$	0	0
$859$	−58.1080	−1.98262	−0.991309	−	0.131555i	$-0.958003\pi$
$859$	−58.1080	−1.98262	−0.991309	+	0.131555i	$0.958003\pi$
$860$	0	0
$861$	−21.7538	−0.741367
$862$	0	0
$863$	−17.6155	−0.599640	−0.299820	−	0.953996i	$-0.596927\pi$
$863$	−17.6155	−0.599640	−0.299820	+	0.953996i	$0.596927\pi$
$864$	0	0
$865$	−2.56155	−0.0870954
$866$	0	0
$867$	15.3153	0.520136
$868$	0	0
$869$	9.43845	0.320177
$870$	0	0
$871$	0.177081	0.00600016
$872$	0	0
$873$	−14.8078	−0.501167
$874$	0	0
$875$	−27.8617	−0.941899
$876$	0	0
$877$	31.6155	1.06758	0.533790	−	0.845617i	$-0.320767\pi$
$877$	31.6155	1.06758	0.533790	+	0.845617i	$0.320767\pi$
$878$	0	0
$879$	6.00000	0.202375
$880$	0	0
$881$	47.9309	1.61483	0.807416	−	0.589983i	$-0.200865\pi$
$881$	47.9309	1.61483	0.807416	+	0.589983i	$0.200865\pi$
$882$	0	0
$883$	−14.8769	−0.500647	−0.250324	−	0.968162i	$-0.580537\pi$
$883$	−14.8769	−0.500647	−0.250324	+	0.968162i	$0.580537\pi$
$884$	0	0
$885$	−0.630683	−0.0212002
$886$	0	0
$887$	−28.4924	−0.956682	−0.478341	−	0.878174i	$-0.658762\pi$
$887$	−28.4924	−0.956682	−0.478341	+	0.878174i	$0.658762\pi$
$888$	0	0
$889$	52.4924	1.76054
$890$	0	0
$891$	2.56155	0.0858152
$892$	0	0
$893$	9.43845	0.315846
$894$	0	0
$895$	12.9460	0.432738
$896$	0	0
$897$	−4.49242	−0.149998
$898$	0	0
$899$	−7.12311	−0.237569
$900$	0	0
$901$	−11.3693	−0.378767
$902$	0	0
$903$	−46.7386	−1.55536
$904$	0	0
$905$	6.87689	0.228596
$906$	0	0
$907$	−5.43845	−0.180581	−0.0902903	−	0.995915i	$-0.528780\pi$
$907$	−5.43845	−0.180581	−0.0902903	+	0.995915i	$0.528780\pi$
$908$	0	0
$909$	16.2462	0.538853
$910$	0	0
$911$	16.0000	0.530104	0.265052	−	0.964234i	$-0.414611\pi$
$911$	16.0000	0.530104	0.265052	+	0.964234i	$0.414611\pi$
$912$	0	0
$913$	−32.8078	−1.08578
$914$	0	0
$915$	0.315342	0.0104249
$916$	0	0
$917$	87.7235	2.89688
$918$	0	0
$919$	−44.4924	−1.46767	−0.733835	−	0.679328i	$-0.762271\pi$
$919$	−44.4924	−1.46767	−0.733835	+	0.679328i	$0.762271\pi$
$920$	0	0
$921$	1.75379	0.0577893
$922$	0	0
$923$	−0.807764	−0.0265879
$924$	0	0
$925$	−38.6307	−1.27017
$926$	0	0
$927$	5.12311	0.168265
$928$	0	0
$929$	6.49242	0.213009	0.106505	−	0.994312i	$-0.466034\pi$
$929$	6.49242	0.213009	0.106505	+	0.994312i	$0.466034\pi$
$930$	0	0
$931$	49.3002	1.61575
$932$	0	0
$933$	−21.9309	−0.717984
$934$	0	0
$935$	−8.17708	−0.267419
$936$	0	0
$937$	−1.05398	−0.0344319	−0.0172159	−	0.999852i	$-0.505480\pi$
$937$	−1.05398	−0.0344319	−0.0172159	+	0.999852i	$0.505480\pi$
$938$	0	0
$939$	−16.2462	−0.530175
$940$	0	0
$941$	−2.63068	−0.0857578	−0.0428789	−	0.999080i	$-0.513653\pi$
$941$	−2.63068	−0.0857578	−0.0428789	+	0.999080i	$0.513653\pi$
$942$	0	0
$943$	33.9697	1.10621
$944$	0	0
$945$	2.87689	0.0935854
$946$	0	0
$947$	−34.5616	−1.12310	−0.561550	−	0.827443i	$-0.689795\pi$
$947$	−34.5616	−1.12310	−0.561550	+	0.827443i	$0.689795\pi$
$948$	0	0
$949$	−5.61553	−0.182288
$950$	0	0
$951$	10.3153	0.334498
$952$	0	0
$953$	−21.5464	−0.697956	−0.348978	−	0.937131i	$-0.613471\pi$
$953$	−21.5464	−0.697956	−0.348978	+	0.937131i	$0.613471\pi$
$954$	0	0
$955$	8.98485	0.290743
$956$	0	0
$957$	−18.2462	−0.589816
$958$	0	0
$959$	30.7386	0.992602
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	4.00000	0.128898
$964$	0	0
$965$	−13.7926	−0.444000
$966$	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
$969$	14.5616	0.467784
$970$	0	0
$971$	18.3845	0.589986	0.294993	−	0.955499i	$-0.404683\pi$
$971$	18.3845	0.589986	0.294993	+	0.955499i	$0.404683\pi$
$972$	0	0
$973$	−61.4773	−1.97087
$974$	0	0
$975$	2.63068	0.0842493
$976$	0	0
$977$	44.2462	1.41556	0.707781	−	0.706432i	$-0.249696\pi$
$977$	44.2462	1.41556	0.707781	+	0.706432i	$0.249696\pi$
$978$	0	0
$979$	−0.630683	−0.0201567
$980$	0	0
$981$	14.0000	0.446986
$982$	0	0
$983$	18.2462	0.581964	0.290982	−	0.956729i	$-0.406018\pi$
$983$	18.2462	0.581964	0.290982	+	0.956729i	$0.406018\pi$
$984$	0	0
$985$	−4.63068	−0.147546
$986$	0	0
$987$	−18.8769	−0.600858
$988$	0	0
$989$	72.9848	2.32078
$990$	0	0
$991$	−46.7386	−1.48470	−0.742351	−	0.670011i	$-0.766289\pi$
$991$	−46.7386	−1.48470	−0.742351	+	0.670011i	$0.766289\pi$
$992$	0	0
$993$	12.0000	0.380808
$994$	0	0
$995$	2.42329	0.0768235
$996$	0	0
$997$	−26.0000	−0.823428	−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000	−0.823428	−0.411714	+	0.911313i	$0.635070\pi$
$998$	0	0
$999$	8.24621	0.260899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1488.2.a.r.1.1		2
3.2	odd	2		4464.2.a.bb.1.2		2
4.3	odd	2		186.2.a.d.1.1	✓	2
8.3	odd	2		5952.2.a.bs.1.2		2
8.5	even	2		5952.2.a.bk.1.2		2
12.11	even	2		558.2.a.i.1.2		2
20.3	even	4		4650.2.d.bc.3349.1		4
20.7	even	4		4650.2.d.bc.3349.4		4
20.19	odd	2		4650.2.a.cd.1.1		2
28.27	even	2		9114.2.a.be.1.2		2
124.123	even	2		5766.2.a.v.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
186.2.a.d.1.1	✓	2	4.3	odd	2
558.2.a.i.1.2		2	12.11	even	2
1488.2.a.r.1.1		2	1.1	even	1	trivial
4464.2.a.bb.1.2		2	3.2	odd	2
4650.2.a.cd.1.1		2	20.19	odd	2
4650.2.d.bc.3349.1		4	20.3	even	4
4650.2.d.bc.3349.4		4	20.7	even	4
5766.2.a.v.1.1		2	124.123	even	2
5952.2.a.bk.1.2		2	8.5	even	2
5952.2.a.bs.1.2		2	8.3	odd	2
9114.2.a.be.1.2		2	28.27	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:	\( N \)	\(=\)	\( 1488 = 2^{4} \cdot 3 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1488.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -0.561553 q^{5} -5.12311 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -0.561553 q^{5} -5.12311 q^{7} +1.00000 q^{9} +2.56155 q^{11} -0.561553 q^{13} -0.561553 q^{15} +5.68466 q^{17} +2.56155 q^{19} -5.12311 q^{21} +8.00000 q^{23} -4.68466 q^{25} +1.00000 q^{27} -7.12311 q^{29} +1.00000 q^{31} +2.56155 q^{33} +2.87689 q^{35} +8.24621 q^{37} -0.561553 q^{39} +4.24621 q^{41} +9.12311 q^{43} -0.561553 q^{45} +3.68466 q^{47} +19.2462 q^{49} +5.68466 q^{51} -2.00000 q^{53} -1.43845 q^{55} +2.56155 q^{57} +1.12311 q^{59} -0.561553 q^{61} -5.12311 q^{63} +0.315342 q^{65} -0.315342 q^{67} +8.00000 q^{69} +1.43845 q^{71} +10.0000 q^{73} -4.68466 q^{75} -13.1231 q^{77} +3.68466 q^{79} +1.00000 q^{81} -12.8078 q^{83} -3.19224 q^{85} -7.12311 q^{87} -0.246211 q^{89} +2.87689 q^{91} +1.00000 q^{93} -1.43845 q^{95} -14.8078 q^{97} +2.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 3 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 3 * q^5 - 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 3 q^{5} - 2 q^{7} + 2 q^{9} + q^{11} + 3 q^{13} + 3 q^{15} - q^{17} + q^{19} - 2 q^{21} + 16 q^{23} + 3 q^{25} + 2 q^{27} - 6 q^{29} + 2 q^{31} + q^{33} + 14 q^{35} + 3 q^{39} - 8 q^{41} + 10 q^{43} + 3 q^{45} - 5 q^{47} + 22 q^{49} - q^{51} - 4 q^{53} - 7 q^{55} + q^{57} - 6 q^{59} + 3 q^{61} - 2 q^{63} + 13 q^{65} - 13 q^{67} + 16 q^{69} + 7 q^{71} + 20 q^{73} + 3 q^{75} - 18 q^{77} - 5 q^{79} + 2 q^{81} - 5 q^{83} - 27 q^{85} - 6 q^{87} + 16 q^{89} + 14 q^{91} + 2 q^{93} - 7 q^{95} - 9 q^{97} + q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 3 * q^5 - 2 * q^7 + 2 * q^9 + q^11 + 3 * q^13 + 3 * q^15 - q^17 + q^19 - 2 * q^21 + 16 * q^23 + 3 * q^25 + 2 * q^27 - 6 * q^29 + 2 * q^31 + q^33 + 14 * q^35 + 3 * q^39 - 8 * q^41 + 10 * q^43 + 3 * q^45 - 5 * q^47 + 22 * q^49 - q^51 - 4 * q^53 - 7 * q^55 + q^57 - 6 * q^59 + 3 * q^61 - 2 * q^63 + 13 * q^65 - 13 * q^67 + 16 * q^69 + 7 * q^71 + 20 * q^73 + 3 * q^75 - 18 * q^77 - 5 * q^79 + 2 * q^81 - 5 * q^83 - 27 * q^85 - 6 * q^87 + 16 * q^89 + 14 * q^91 + 2 * q^93 - 7 * q^95 - 9 * q^97 + q^99

Introduction

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