LMFDB - Embedded newform 5408.2.a.bs.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5408,2,Mod(1,5408)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5408, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("5408.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5408 = 2^{5} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5408.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:	$43.1830974131$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 2x^{8} - 14x^{7} + 23x^{6} + 63x^{5} - 85x^{4} - 99x^{3} + 98x^{2} + 35x - 7 $ x^9 - 2x^8 - 14x^7 + 23x^6 + 63x^5 - 85x^4 - 99x^3 + 98x^2 + 35x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-2.34228$ of defining polynomial
Character	$\chi$	$=$	5408.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.34228 q^{3} +3.54539 q^{5} +2.43267 q^{7} +8.17085 q^{9} +1.56000 q^{11} +11.8497 q^{15} -4.70875 q^{17} -5.85552 q^{19} +8.13068 q^{21} -1.89070 q^{23} +7.56979 q^{25} +17.2824 q^{27} -2.14833 q^{29} -3.62508 q^{31} +5.21397 q^{33} +8.62478 q^{35} -9.60085 q^{37} -1.62658 q^{41} +3.33570 q^{43} +28.9689 q^{45} +1.03301 q^{47} -1.08210 q^{49} -15.7380 q^{51} +10.9116 q^{53} +5.53082 q^{55} -19.5708 q^{57} +6.70920 q^{59} +1.04224 q^{61} +19.8770 q^{63} -7.98257 q^{67} -6.31927 q^{69} -1.07822 q^{71} -8.85642 q^{73} +25.3004 q^{75} +3.79498 q^{77} +8.62755 q^{79} +33.2503 q^{81} +3.37447 q^{83} -16.6944 q^{85} -7.18031 q^{87} -10.5809 q^{89} -12.1160 q^{93} -20.7601 q^{95} +6.08937 q^{97} +12.7466 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 7 q^{3} + 6 q^{5} + 10 q^{9} - q^{11} + 18 q^{15} - 7 q^{17} - 5 q^{19} + 12 q^{21} + 12 q^{23} + 11 q^{25} + 34 q^{27} - 8 q^{29} + 20 q^{31} - 12 q^{33} + 6 q^{35} + 6 q^{37} - 31 q^{41} + 33 q^{43}+ \cdots - 47 q^{99}+O(q^{100}) $ 9 * q + 7 * q^3 + 6 * q^5 + 10 * q^9 - q^11 + 18 * q^15 - 7 * q^17 - 5 * q^19 + 12 * q^21 + 12 * q^23 + 11 * q^25 + 34 * q^27 - 8 * q^29 + 20 * q^31 - 12 * q^33 + 6 * q^35 + 6 * q^37 - 31 * q^41 + 33 * q^43 + 22 * q^45 - 14 * q^47 + 3 * q^49 + 12 * q^51 + 14 * q^53 + 38 * q^55 - 10 * q^57 + 9 * q^59 + 8 * q^61 + 20 * q^63 + 3 * q^67 + 26 * q^69 - 26 * q^71 - 5 * q^73 + 25 * q^75 - 6 * q^77 + 44 * q^79 + 45 * q^81 + 17 * q^83 + 6 * q^85 + 24 * q^87 - 41 * q^89 + 26 * q^93 + 4 * q^95 - 15 * q^97 - 47 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	3.34228	1.92967	0.964834	−	0.262861i	$-0.0846660\pi$
$3$	3.34228	1.92967	0.964834	+	0.262861i	$0.0846660\pi$
$4$	0	0
$5$	3.54539	1.58555	0.792773	−	0.609516i	$-0.208636\pi$
$5$	3.54539	1.58555	0.792773	+	0.609516i	$0.208636\pi$
$6$	0	0
$7$	2.43267	0.919464	0.459732	−	0.888058i	$-0.347945\pi$
$7$	2.43267	0.919464	0.459732	+	0.888058i	$0.347945\pi$
$8$	0	0
$9$	8.17085	2.72362
$10$	0	0
$11$	1.56000	0.470359	0.235179	−	0.971952i	$-0.424432\pi$
$11$	1.56000	0.470359	0.235179	+	0.971952i	$0.424432\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	11.8497	3.05958
$16$	0	0
$17$	−4.70875	−1.14204	−0.571020	−	0.820936i	$-0.693452\pi$
$17$	−4.70875	−1.14204	−0.571020	+	0.820936i	$0.693452\pi$
$18$	0	0
$19$	−5.85552	−1.34335	−0.671674	−	0.740847i	$-0.734425\pi$
$19$	−5.85552	−1.34335	−0.671674	+	0.740847i	$0.734425\pi$
$20$	0	0
$21$	8.13068	1.77426
$22$	0	0
$23$	−1.89070	−0.394239	−0.197119	−	0.980379i	$-0.563159\pi$
$23$	−1.89070	−0.394239	−0.197119	+	0.980379i	$0.563159\pi$
$24$	0	0
$25$	7.56979	1.51396
$26$	0	0
$27$	17.2824	3.32601
$28$	0	0
$29$	−2.14833	−0.398934	−0.199467	−	0.979905i	$-0.563921\pi$
$29$	−2.14833	−0.398934	−0.199467	+	0.979905i	$0.563921\pi$
$30$	0	0
$31$	−3.62508	−0.651083	−0.325542	−	0.945528i	$-0.605547\pi$
$31$	−3.62508	−0.651083	−0.325542	+	0.945528i	$0.605547\pi$
$32$	0	0
$33$	5.21397	0.907636
$34$	0	0
$35$	8.62478	1.45785
$36$	0	0
$37$	−9.60085	−1.57837	−0.789185	−	0.614155i	$-0.789497\pi$
$37$	−9.60085	−1.57837	−0.789185	+	0.614155i	$0.789497\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.62658	−0.254030	−0.127015	−	0.991901i	$-0.540540\pi$
$41$	−1.62658	−0.254030	−0.127015	+	0.991901i	$0.540540\pi$
$42$	0	0
$43$	3.33570	0.508690	0.254345	−	0.967114i	$-0.418140\pi$
$43$	3.33570	0.508690	0.254345	+	0.967114i	$0.418140\pi$
$44$	0	0
$45$	28.9689	4.31842
$46$	0	0
$47$	1.03301	0.150681	0.0753403	−	0.997158i	$-0.475996\pi$
$47$	1.03301	0.150681	0.0753403	+	0.997158i	$0.475996\pi$
$48$	0	0
$49$	−1.08210	−0.154585
$50$	0	0
$51$	−15.7380	−2.20376
$52$	0	0
$53$	10.9116	1.49883	0.749415	−	0.662101i	$-0.230335\pi$
$53$	10.9116	1.49883	0.749415	+	0.662101i	$0.230335\pi$
$54$	0	0
$55$	5.53082	0.745776
$56$	0	0
$57$	−19.5708	−2.59222
$58$	0	0
$59$	6.70920	0.873463	0.436732	−	0.899592i	$-0.356136\pi$
$59$	6.70920	0.873463	0.436732	+	0.899592i	$0.356136\pi$
$60$	0	0
$61$	1.04224	0.133445	0.0667226	−	0.997772i	$-0.478746\pi$
$61$	1.04224	0.133445	0.0667226	+	0.997772i	$0.478746\pi$
$62$	0	0
$63$	19.8770	2.50427
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.98257	−0.975226	−0.487613	−	0.873060i	$-0.662132\pi$
$67$	−7.98257	−0.975226	−0.487613	+	0.873060i	$0.662132\pi$
$68$	0	0
$69$	−6.31927	−0.760750
$70$	0	0
$71$	−1.07822	−0.127961	−0.0639807	−	0.997951i	$-0.520380\pi$
$71$	−1.07822	−0.127961	−0.0639807	+	0.997951i	$0.520380\pi$
$72$	0	0
$73$	−8.85642	−1.03657	−0.518283	−	0.855209i	$-0.673429\pi$
$73$	−8.85642	−1.03657	−0.518283	+	0.855209i	$0.673429\pi$
$74$	0	0
$75$	25.3004	2.92144
$76$	0	0
$77$	3.79498	0.432478
$78$	0	0
$79$	8.62755	0.970675	0.485337	−	0.874327i	$-0.338697\pi$
$79$	8.62755	0.970675	0.485337	+	0.874327i	$0.338697\pi$
$80$	0	0
$81$	33.2503	3.69447
$82$	0	0
$83$	3.37447	0.370396	0.185198	−	0.982701i	$-0.440707\pi$
$83$	3.37447	0.370396	0.185198	+	0.982701i	$0.440707\pi$
$84$	0	0
$85$	−16.6944	−1.81076
$86$	0	0
$87$	−7.18031	−0.769810
$88$	0	0
$89$	−10.5809	−1.12157	−0.560785	−	0.827961i	$-0.689501\pi$
$89$	−10.5809	−1.12157	−0.560785	+	0.827961i	$0.689501\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−12.1160	−1.25637
$94$	0	0
$95$	−20.7601	−2.12994
$96$	0	0
$97$	6.08937	0.618282	0.309141	−	0.951016i	$-0.399959\pi$
$97$	6.08937	0.618282	0.309141	+	0.951016i	$0.399959\pi$
$98$	0	0
$99$	12.7466	1.28108
$100$	0	0
$101$	14.7273	1.46543	0.732713	−	0.680538i	$-0.238254\pi$
$101$	14.7273	1.46543	0.732713	+	0.680538i	$0.238254\pi$
$102$	0	0
$103$	15.8099	1.55780	0.778898	−	0.627151i	$-0.215779\pi$
$103$	15.8099	1.55780	0.778898	+	0.627151i	$0.215779\pi$
$104$	0	0
$105$	28.8264	2.81317
$106$	0	0
$107$	−13.1878	−1.27492	−0.637459	−	0.770485i	$-0.720014\pi$
$107$	−13.1878	−1.27492	−0.637459	+	0.770485i	$0.720014\pi$
$108$	0	0
$109$	4.33283	0.415010	0.207505	−	0.978234i	$-0.433466\pi$
$109$	4.33283	0.415010	0.207505	+	0.978234i	$0.433466\pi$
$110$	0	0
$111$	−32.0888	−3.04573
$112$	0	0
$113$	9.54353	0.897780	0.448890	−	0.893587i	$-0.351819\pi$
$113$	9.54353	0.897780	0.448890	+	0.893587i	$0.351819\pi$
$114$	0	0
$115$	−6.70328	−0.625084
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−11.4549	−1.05006
$120$	0	0
$121$	−8.56639	−0.778763
$122$	0	0
$123$	−5.43650	−0.490193
$124$	0	0
$125$	9.11092	0.814905
$126$	0	0
$127$	10.7995	0.958300	0.479150	−	0.877733i	$-0.340945\pi$
$127$	10.7995	0.958300	0.479150	+	0.877733i	$0.340945\pi$
$128$	0	0
$129$	11.1489	0.981603
$130$	0	0
$131$	−11.0540	−0.965792	−0.482896	−	0.875678i	$-0.660415\pi$
$131$	−11.0540	−0.965792	−0.482896	+	0.875678i	$0.660415\pi$
$132$	0	0
$133$	−14.2446	−1.23516
$134$	0	0
$135$	61.2730	5.27354
$136$	0	0
$137$	−12.8633	−1.09899	−0.549494	−	0.835498i	$-0.685179\pi$
$137$	−12.8633	−1.09899	−0.549494	+	0.835498i	$0.685179\pi$
$138$	0	0
$139$	−14.9531	−1.26831	−0.634154	−	0.773206i	$-0.718652\pi$
$139$	−14.9531	−1.26831	−0.634154	+	0.773206i	$0.718652\pi$
$140$	0	0
$141$	3.45262	0.290763
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−7.61666	−0.632529
$146$	0	0
$147$	−3.61668	−0.298298
$148$	0	0
$149$	−4.04297	−0.331213	−0.165607	−	0.986192i	$-0.552958\pi$
$149$	−4.04297	−0.331213	−0.165607	+	0.986192i	$0.552958\pi$
$150$	0	0
$151$	−9.20810	−0.749345	−0.374672	−	0.927157i	$-0.622245\pi$
$151$	−9.20810	−0.749345	−0.374672	+	0.927157i	$0.622245\pi$
$152$	0	0
$153$	−38.4745	−3.11048
$154$	0	0
$155$	−12.8523	−1.03232
$156$	0	0
$157$	−14.4979	−1.15706	−0.578529	−	0.815662i	$-0.696373\pi$
$157$	−14.4979	−1.15706	−0.578529	+	0.815662i	$0.696373\pi$
$158$	0	0
$159$	36.4698	2.89224
$160$	0	0
$161$	−4.59947	−0.362489
$162$	0	0
$163$	−9.46323	−0.741218	−0.370609	−	0.928789i	$-0.620851\pi$
$163$	−9.46323	−0.741218	−0.370609	+	0.928789i	$0.620851\pi$
$164$	0	0
$165$	18.4856	1.43910
$166$	0	0
$167$	15.1520	1.17249	0.586247	−	0.810132i	$-0.300605\pi$
$167$	15.1520	1.17249	0.586247	+	0.810132i	$0.300605\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−47.8446	−3.65877
$172$	0	0
$173$	−18.9107	−1.43775	−0.718876	−	0.695139i	$-0.755343\pi$
$173$	−18.9107	−1.43775	−0.718876	+	0.695139i	$0.755343\pi$
$174$	0	0
$175$	18.4148	1.39203
$176$	0	0
$177$	22.4240	1.68549
$178$	0	0
$179$	−9.57124	−0.715388	−0.357694	−	0.933839i	$-0.616437\pi$
$179$	−9.57124	−0.715388	−0.357694	+	0.933839i	$0.616437\pi$
$180$	0	0
$181$	26.0996	1.93997	0.969985	−	0.243166i	$-0.0781859\pi$
$181$	26.0996	1.93997	0.969985	+	0.243166i	$0.0781859\pi$
$182$	0	0
$183$	3.48346	0.257505
$184$	0	0
$185$	−34.0388	−2.50258
$186$	0	0
$187$	−7.34567	−0.537169
$188$	0	0
$189$	42.0426	3.05815
$190$	0	0
$191$	8.67623	0.627790	0.313895	−	0.949458i	$-0.398366\pi$
$191$	8.67623	0.627790	0.313895	+	0.949458i	$0.398366\pi$
$192$	0	0
$193$	14.7887	1.06451	0.532257	−	0.846583i	$-0.321344\pi$
$193$	14.7887	1.06451	0.532257	+	0.846583i	$0.321344\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1.89803	0.135229	0.0676145	−	0.997712i	$-0.478461\pi$
$197$	1.89803	0.135229	0.0676145	+	0.997712i	$0.478461\pi$
$198$	0	0
$199$	−9.55304	−0.677197	−0.338598	−	0.940931i	$-0.609953\pi$
$199$	−9.55304	−0.677197	−0.338598	+	0.940931i	$0.609953\pi$
$200$	0	0
$201$	−26.6800	−1.88186
$202$	0	0
$203$	−5.22618	−0.366806
$204$	0	0
$205$	−5.76688	−0.402776
$206$	0	0
$207$	−15.4487	−1.07376
$208$	0	0
$209$	−9.13464	−0.631856
$210$	0	0
$211$	8.56429	0.589590	0.294795	−	0.955560i	$-0.404749\pi$
$211$	8.56429	0.589590	0.294795	+	0.955560i	$0.404749\pi$
$212$	0	0
$213$	−3.60372	−0.246923
$214$	0	0
$215$	11.8264	0.806552
$216$	0	0
$217$	−8.81864	−0.598648
$218$	0	0
$219$	−29.6007	−2.00023
$220$	0	0
$221$	0	0
$222$	0	0
$223$	5.28598	0.353975	0.176988	−	0.984213i	$-0.443365\pi$
$223$	5.28598	0.353975	0.176988	+	0.984213i	$0.443365\pi$
$224$	0	0
$225$	61.8517	4.12344
$226$	0	0
$227$	−28.2562	−1.87543	−0.937715	−	0.347406i	$-0.887062\pi$
$227$	−28.2562	−1.87543	−0.937715	+	0.347406i	$0.887062\pi$
$228$	0	0
$229$	−4.21203	−0.278339	−0.139170	−	0.990269i	$-0.544443\pi$
$229$	−4.21203	−0.278339	−0.139170	+	0.990269i	$0.544443\pi$
$230$	0	0
$231$	12.6839	0.834539
$232$	0	0
$233$	−9.50448	−0.622659	−0.311330	−	0.950302i	$-0.600774\pi$
$233$	−9.50448	−0.622659	−0.311330	+	0.950302i	$0.600774\pi$
$234$	0	0
$235$	3.66244	0.238911
$236$	0	0
$237$	28.8357	1.87308
$238$	0	0
$239$	20.8800	1.35061	0.675307	−	0.737536i	$-0.264011\pi$
$239$	20.8800	1.35061	0.675307	+	0.737536i	$0.264011\pi$
$240$	0	0
$241$	−19.4449	−1.25255	−0.626277	−	0.779600i	$-0.715422\pi$
$241$	−19.4449	−1.25255	−0.626277	+	0.779600i	$0.715422\pi$
$242$	0	0
$243$	59.2844	3.80310
$244$	0	0
$245$	−3.83646	−0.245102
$246$	0	0
$247$	0	0
$248$	0	0
$249$	11.2784	0.714741
$250$	0	0
$251$	31.2802	1.97439	0.987196	−	0.159513i	$-0.0509924\pi$
$251$	31.2802	1.97439	0.987196	+	0.159513i	$0.0509924\pi$
$252$	0	0
$253$	−2.94950	−0.185434
$254$	0	0
$255$	−55.7973	−3.49416
$256$	0	0
$257$	−0.649810	−0.0405340	−0.0202670	−	0.999795i	$-0.506452\pi$
$257$	−0.649810	−0.0405340	−0.0202670	+	0.999795i	$0.506452\pi$
$258$	0	0
$259$	−23.3557	−1.45126
$260$	0	0
$261$	−17.5537	−1.08654
$262$	0	0
$263$	3.80983	0.234924	0.117462	−	0.993077i	$-0.462524\pi$
$263$	3.80983	0.234924	0.117462	+	0.993077i	$0.462524\pi$
$264$	0	0
$265$	38.6860	2.37646
$266$	0	0
$267$	−35.3643	−2.16426
$268$	0	0
$269$	−18.0883	−1.10287	−0.551433	−	0.834219i	$-0.685919\pi$
$269$	−18.0883	−1.10287	−0.551433	+	0.834219i	$0.685919\pi$
$270$	0	0
$271$	−17.6833	−1.07418	−0.537092	−	0.843524i	$-0.680477\pi$
$271$	−17.6833	−1.07418	−0.537092	+	0.843524i	$0.680477\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	11.8089	0.712104
$276$	0	0
$277$	−20.4411	−1.22819	−0.614093	−	0.789234i	$-0.710478\pi$
$277$	−20.4411	−1.22819	−0.614093	+	0.789234i	$0.710478\pi$
$278$	0	0
$279$	−29.6200	−1.77330
$280$	0	0
$281$	−4.13083	−0.246425	−0.123212	−	0.992380i	$-0.539320\pi$
$281$	−4.13083	−0.246425	−0.123212	+	0.992380i	$0.539320\pi$
$282$	0	0
$283$	11.9921	0.712854	0.356427	−	0.934323i	$-0.383995\pi$
$283$	11.9921	0.712854	0.356427	+	0.934323i	$0.383995\pi$
$284$	0	0
$285$	−69.3861	−4.11008
$286$	0	0
$287$	−3.95695	−0.233571
$288$	0	0
$289$	5.17234	0.304255
$290$	0	0
$291$	20.3524	1.19308
$292$	0	0
$293$	0.724538	0.0423279	0.0211640	−	0.999776i	$-0.493263\pi$
$293$	0.724538	0.0423279	0.0211640	+	0.999776i	$0.493263\pi$
$294$	0	0
$295$	23.7867	1.38492
$296$	0	0
$297$	26.9607	1.56442
$298$	0	0
$299$	0	0
$300$	0	0
$301$	8.11468	0.467722
$302$	0	0
$303$	49.2229	2.82778
$304$	0	0
$305$	3.69515	0.211584
$306$	0	0
$307$	2.76712	0.157928	0.0789639	−	0.996877i	$-0.474839\pi$
$307$	2.76712	0.157928	0.0789639	+	0.996877i	$0.474839\pi$
$308$	0	0
$309$	52.8411	3.00603
$310$	0	0
$311$	29.6292	1.68012	0.840059	−	0.542494i	$-0.182520\pi$
$311$	29.6292	1.68012	0.840059	+	0.542494i	$0.182520\pi$
$312$	0	0
$313$	10.7690	0.608700	0.304350	−	0.952560i	$-0.401561\pi$
$313$	10.7690	0.608700	0.304350	+	0.952560i	$0.401561\pi$
$314$	0	0
$315$	70.4718	3.97064
$316$	0	0
$317$	0.379391	0.0213087	0.0106544	−	0.999943i	$-0.496609\pi$
$317$	0.379391	0.0213087	0.0106544	+	0.999943i	$0.496609\pi$
$318$	0	0
$319$	−3.35140	−0.187642
$320$	0	0
$321$	−44.0775	−2.46017
$322$	0	0
$323$	27.5722	1.53416
$324$	0	0
$325$	0	0
$326$	0	0
$327$	14.4815	0.800831
$328$	0	0
$329$	2.51299	0.138545
$330$	0	0
$331$	24.3183	1.33665	0.668327	−	0.743868i	$-0.267011\pi$
$331$	24.3183	1.33665	0.668327	+	0.743868i	$0.267011\pi$
$332$	0	0
$333$	−78.4471	−4.29888
$334$	0	0
$335$	−28.3013	−1.54627
$336$	0	0
$337$	−14.9198	−0.812733	−0.406366	−	0.913710i	$-0.633204\pi$
$337$	−14.9198	−0.812733	−0.406366	+	0.913710i	$0.633204\pi$
$338$	0	0
$339$	31.8972	1.73242
$340$	0	0
$341$	−5.65514	−0.306243
$342$	0	0
$343$	−19.6611	−1.06160
$344$	0	0
$345$	−22.4043	−1.20620
$346$	0	0
$347$	21.4559	1.15181	0.575907	−	0.817515i	$-0.304649\pi$
$347$	21.4559	1.15181	0.575907	+	0.817515i	$0.304649\pi$
$348$	0	0
$349$	−1.71661	−0.0918882	−0.0459441	−	0.998944i	$-0.514630\pi$
$349$	−1.71661	−0.0918882	−0.0459441	+	0.998944i	$0.514630\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−23.3569	−1.24316	−0.621580	−	0.783351i	$-0.713509\pi$
$353$	−23.3569	−1.24316	−0.621580	+	0.783351i	$0.713509\pi$
$354$	0	0
$355$	−3.82272	−0.202889
$356$	0	0
$357$	−38.2854	−2.02628
$358$	0	0
$359$	31.3222	1.65312	0.826561	−	0.562847i	$-0.190294\pi$
$359$	31.3222	1.65312	0.826561	+	0.562847i	$0.190294\pi$
$360$	0	0
$361$	15.2871	0.804586
$362$	0	0
$363$	−28.6313	−1.50275
$364$	0	0
$365$	−31.3995	−1.64352
$366$	0	0
$367$	24.2134	1.26393	0.631964	−	0.774998i	$-0.282249\pi$
$367$	24.2134	1.26393	0.631964	+	0.774998i	$0.282249\pi$
$368$	0	0
$369$	−13.2906	−0.691880
$370$	0	0
$371$	26.5445	1.37812
$372$	0	0
$373$	−32.5412	−1.68492	−0.842460	−	0.538759i	$-0.818893\pi$
$373$	−32.5412	−1.68492	−0.842460	+	0.538759i	$0.818893\pi$
$374$	0	0
$375$	30.4513	1.57250
$376$	0	0
$377$	0	0
$378$	0	0
$379$	26.3195	1.35194	0.675972	−	0.736927i	$-0.263724\pi$
$379$	26.3195	1.35194	0.675972	+	0.736927i	$0.263724\pi$
$380$	0	0
$381$	36.0949	1.84920
$382$	0	0
$383$	−9.29522	−0.474964	−0.237482	−	0.971392i	$-0.576322\pi$
$383$	−9.29522	−0.474964	−0.237482	+	0.971392i	$0.576322\pi$
$384$	0	0
$385$	13.4547	0.685714
$386$	0	0
$387$	27.2555	1.38548
$388$	0	0
$389$	−6.76758	−0.343130	−0.171565	−	0.985173i	$-0.554882\pi$
$389$	−6.76758	−0.343130	−0.171565	+	0.985173i	$0.554882\pi$
$390$	0	0
$391$	8.90285	0.450237
$392$	0	0
$393$	−36.9456	−1.86366
$394$	0	0
$395$	30.5880	1.53905
$396$	0	0
$397$	16.2305	0.814586	0.407293	−	0.913298i	$-0.366473\pi$
$397$	16.2305	0.814586	0.407293	+	0.913298i	$0.366473\pi$
$398$	0	0
$399$	−47.6094	−2.38345
$400$	0	0
$401$	−8.12021	−0.405504	−0.202752	−	0.979230i	$-0.564988\pi$
$401$	−8.12021	−0.405504	−0.202752	+	0.979230i	$0.564988\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	117.885	5.85776
$406$	0	0
$407$	−14.9774	−0.742400
$408$	0	0
$409$	−14.6048	−0.722161	−0.361081	−	0.932535i	$-0.617592\pi$
$409$	−14.6048	−0.722161	−0.361081	+	0.932535i	$0.617592\pi$
$410$	0	0
$411$	−42.9928	−2.12068
$412$	0	0
$413$	16.3213	0.803118
$414$	0	0
$415$	11.9638	0.587280
$416$	0	0
$417$	−49.9776	−2.44741
$418$	0	0
$419$	−16.9755	−0.829307	−0.414654	−	0.909979i	$-0.636097\pi$
$419$	−16.9755	−0.829307	−0.414654	+	0.909979i	$0.636097\pi$
$420$	0	0
$421$	−10.2034	−0.497284	−0.248642	−	0.968595i	$-0.579984\pi$
$421$	−10.2034	−0.497284	−0.248642	+	0.968595i	$0.579984\pi$
$422$	0	0
$423$	8.44060	0.410396
$424$	0	0
$425$	−35.6443	−1.72900
$426$	0	0
$427$	2.53543	0.122698
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−6.33244	−0.305023	−0.152512	−	0.988302i	$-0.548736\pi$
$431$	−6.33244	−0.305023	−0.152512	+	0.988302i	$0.548736\pi$
$432$	0	0
$433$	−19.8673	−0.954761	−0.477381	−	0.878697i	$-0.658414\pi$
$433$	−19.8673	−0.954761	−0.477381	+	0.878697i	$0.658414\pi$
$434$	0	0
$435$	−25.4570	−1.22057
$436$	0	0
$437$	11.0711	0.529600
$438$	0	0
$439$	−12.3304	−0.588496	−0.294248	−	0.955729i	$-0.595069\pi$
$439$	−12.3304	−0.588496	−0.294248	+	0.955729i	$0.595069\pi$
$440$	0	0
$441$	−8.84166	−0.421031
$442$	0	0
$443$	1.24249	0.0590323	0.0295161	−	0.999564i	$-0.490603\pi$
$443$	1.24249	0.0590323	0.0295161	+	0.999564i	$0.490603\pi$
$444$	0	0
$445$	−37.5133	−1.77830
$446$	0	0
$447$	−13.5127	−0.639131
$448$	0	0
$449$	17.0128	0.802884	0.401442	−	0.915884i	$-0.368509\pi$
$449$	17.0128	0.802884	0.401442	+	0.915884i	$0.368509\pi$
$450$	0	0
$451$	−2.53748	−0.119485
$452$	0	0
$453$	−30.7761	−1.44599
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−18.9023	−0.884210	−0.442105	−	0.896963i	$-0.645768\pi$
$457$	−18.9023	−0.884210	−0.442105	+	0.896963i	$0.645768\pi$
$458$	0	0
$459$	−81.3787	−3.79843
$460$	0	0
$461$	9.90226	0.461194	0.230597	−	0.973049i	$-0.425932\pi$
$461$	9.90226	0.461194	0.230597	+	0.973049i	$0.425932\pi$
$462$	0	0
$463$	1.49093	0.0692896	0.0346448	−	0.999400i	$-0.488970\pi$
$463$	1.49093	0.0692896	0.0346448	+	0.999400i	$0.488970\pi$
$464$	0	0
$465$	−42.9561	−1.99204
$466$	0	0
$467$	19.8898	0.920392	0.460196	−	0.887817i	$-0.347779\pi$
$467$	19.8898	0.920392	0.460196	+	0.887817i	$0.347779\pi$
$468$	0	0
$469$	−19.4190	−0.896685
$470$	0	0
$471$	−48.4560	−2.23274
$472$	0	0
$473$	5.20371	0.239267
$474$	0	0
$475$	−44.3251	−2.03377
$476$	0	0
$477$	89.1574	4.08224
$478$	0	0
$479$	−40.1034	−1.83237	−0.916186	−	0.400754i	$-0.868748\pi$
$479$	−40.1034	−1.83237	−0.916186	+	0.400754i	$0.868748\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−15.3727	−0.699483
$484$	0	0
$485$	21.5892	0.980315
$486$	0	0
$487$	21.7252	0.984464	0.492232	−	0.870464i	$-0.336181\pi$
$487$	21.7252	0.984464	0.492232	+	0.870464i	$0.336181\pi$
$488$	0	0
$489$	−31.6288	−1.43030
$490$	0	0
$491$	9.29640	0.419541	0.209770	−	0.977751i	$-0.432728\pi$
$491$	9.29640	0.419541	0.209770	+	0.977751i	$0.432728\pi$
$492$	0	0
$493$	10.1159	0.455599
$494$	0	0
$495$	45.1915	2.03121
$496$	0	0
$497$	−2.62296	−0.117656
$498$	0	0
$499$	25.2366	1.12975	0.564874	−	0.825177i	$-0.308925\pi$
$499$	25.2366	1.12975	0.564874	+	0.825177i	$0.308925\pi$
$500$	0	0
$501$	50.6422	2.26253
$502$	0	0
$503$	32.2114	1.43624	0.718118	−	0.695921i	$-0.245004\pi$
$503$	32.2114	1.43624	0.718118	+	0.695921i	$0.245004\pi$
$504$	0	0
$505$	52.2142	2.32350
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−32.1572	−1.42534	−0.712672	−	0.701498i	$-0.752515\pi$
$509$	−32.1572	−1.42534	−0.712672	+	0.701498i	$0.752515\pi$
$510$	0	0
$511$	−21.5448	−0.953085
$512$	0	0
$513$	−101.198	−4.46799
$514$	0	0
$515$	56.0522	2.46996
$516$	0	0
$517$	1.61151	0.0708739
$518$	0	0
$519$	−63.2048	−2.77438
$520$	0	0
$521$	−14.3244	−0.627563	−0.313782	−	0.949495i	$-0.601596\pi$
$521$	−14.3244	−0.627563	−0.313782	+	0.949495i	$0.601596\pi$
$522$	0	0
$523$	−31.2863	−1.36805	−0.684026	−	0.729457i	$-0.739773\pi$
$523$	−31.2863	−1.36805	−0.684026	+	0.729457i	$0.739773\pi$
$524$	0	0
$525$	61.5476	2.68616
$526$	0	0
$527$	17.0696	0.743563
$528$	0	0
$529$	−19.4252	−0.844576
$530$	0	0
$531$	54.8199	2.37898
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−46.7561	−2.02144
$536$	0	0
$537$	−31.9898	−1.38046
$538$	0	0
$539$	−1.68808	−0.0727106
$540$	0	0
$541$	26.7368	1.14950	0.574752	−	0.818327i	$-0.305098\pi$
$541$	26.7368	1.14950	0.574752	+	0.818327i	$0.305098\pi$
$542$	0	0
$543$	87.2323	3.74350
$544$	0	0
$545$	15.3616	0.658017
$546$	0	0
$547$	11.0699	0.473316	0.236658	−	0.971593i	$-0.423948\pi$
$547$	11.0699	0.473316	0.236658	+	0.971593i	$0.423948\pi$
$548$	0	0
$549$	8.51599	0.363454
$550$	0	0
$551$	12.5796	0.535908
$552$	0	0
$553$	20.9880	0.892501
$554$	0	0
$555$	−113.767	−4.82915
$556$	0	0
$557$	41.7738	1.77001	0.885006	−	0.465579i	$-0.154154\pi$
$557$	41.7738	1.77001	0.885006	+	0.465579i	$0.154154\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−24.5513	−1.03656
$562$	0	0
$563$	26.9475	1.13570	0.567850	−	0.823132i	$-0.307775\pi$
$563$	26.9475	1.13570	0.567850	+	0.823132i	$0.307775\pi$
$564$	0	0
$565$	33.8356	1.42347
$566$	0	0
$567$	80.8870	3.39694
$568$	0	0
$569$	21.3611	0.895502	0.447751	−	0.894158i	$-0.352225\pi$
$569$	21.3611	0.895502	0.447751	+	0.894158i	$0.352225\pi$
$570$	0	0
$571$	−4.66374	−0.195171	−0.0975857	−	0.995227i	$-0.531112\pi$
$571$	−4.66374	−0.195171	−0.0975857	+	0.995227i	$0.531112\pi$
$572$	0	0
$573$	28.9984	1.21143
$574$	0	0
$575$	−14.3122	−0.596861
$576$	0	0
$577$	16.3810	0.681951	0.340976	−	0.940072i	$-0.389243\pi$
$577$	16.3810	0.681951	0.340976	+	0.940072i	$0.389243\pi$
$578$	0	0
$579$	49.4281	2.05416
$580$	0	0
$581$	8.20897	0.340566
$582$	0	0
$583$	17.0222	0.704988
$584$	0	0
$585$	0	0
$586$	0	0
$587$	7.63817	0.315261	0.157630	−	0.987498i	$-0.449615\pi$
$587$	7.63817	0.315261	0.157630	+	0.987498i	$0.449615\pi$
$588$	0	0
$589$	21.2267	0.874632
$590$	0	0
$591$	6.34375	0.260947
$592$	0	0
$593$	20.6572	0.848289	0.424145	−	0.905594i	$-0.360575\pi$
$593$	20.6572	0.848289	0.424145	+	0.905594i	$0.360575\pi$
$594$	0	0
$595$	−40.6119	−1.66493
$596$	0	0
$597$	−31.9289	−1.30676
$598$	0	0
$599$	−31.4017	−1.28304	−0.641519	−	0.767107i	$-0.721696\pi$
$599$	−31.4017	−1.28304	−0.641519	+	0.767107i	$0.721696\pi$
$600$	0	0
$601$	34.5110	1.40773	0.703867	−	0.710332i	$-0.251455\pi$
$601$	34.5110	1.40773	0.703867	+	0.710332i	$0.251455\pi$
$602$	0	0
$603$	−65.2244	−2.65614
$604$	0	0
$605$	−30.3712	−1.23476
$606$	0	0
$607$	−23.4675	−0.952515	−0.476257	−	0.879306i	$-0.658007\pi$
$607$	−23.4675	−0.952515	−0.476257	+	0.879306i	$0.658007\pi$
$608$	0	0
$609$	−17.4674	−0.707813
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−1.03159	−0.0416655	−0.0208328	−	0.999783i	$-0.506632\pi$
$613$	−1.03159	−0.0416655	−0.0208328	+	0.999783i	$0.506632\pi$
$614$	0	0
$615$	−19.2745	−0.777224
$616$	0	0
$617$	5.85958	0.235898	0.117949	−	0.993020i	$-0.462368\pi$
$617$	5.85958	0.235898	0.117949	+	0.993020i	$0.462368\pi$
$618$	0	0
$619$	20.2565	0.814176	0.407088	−	0.913389i	$-0.366544\pi$
$619$	20.2565	0.814176	0.407088	+	0.913389i	$0.366544\pi$
$620$	0	0
$621$	−32.6760	−1.31124
$622$	0	0
$623$	−25.7398	−1.03124
$624$	0	0
$625$	−5.54720	−0.221888
$626$	0	0
$627$	−30.5305	−1.21927
$628$	0	0
$629$	45.2080	1.80256
$630$	0	0
$631$	−27.7509	−1.10475	−0.552374	−	0.833597i	$-0.686278\pi$
$631$	−27.7509	−1.10475	−0.552374	+	0.833597i	$0.686278\pi$
$632$	0	0
$633$	28.6243	1.13771
$634$	0	0
$635$	38.2884	1.51943
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−8.80999	−0.348518
$640$	0	0
$641$	−10.3750	−0.409789	−0.204895	−	0.978784i	$-0.565685\pi$
$641$	−10.3750	−0.409789	−0.204895	+	0.978784i	$0.565685\pi$
$642$	0	0
$643$	44.0936	1.73888	0.869440	−	0.494038i	$-0.164480\pi$
$643$	44.0936	1.73888	0.869440	+	0.494038i	$0.164480\pi$
$644$	0	0
$645$	39.5271	1.55638
$646$	0	0
$647$	39.0961	1.53703	0.768513	−	0.639834i	$-0.220997\pi$
$647$	39.0961	1.53703	0.768513	+	0.639834i	$0.220997\pi$
$648$	0	0
$649$	10.4664	0.410841
$650$	0	0
$651$	−29.4744	−1.15519
$652$	0	0
$653$	14.2435	0.557393	0.278696	−	0.960379i	$-0.410098\pi$
$653$	14.2435	0.557393	0.278696	+	0.960379i	$0.410098\pi$
$654$	0	0
$655$	−39.1907	−1.53131
$656$	0	0
$657$	−72.3645	−2.82321
$658$	0	0
$659$	−10.1685	−0.396110	−0.198055	−	0.980191i	$-0.563462\pi$
$659$	−10.1685	−0.396110	−0.198055	+	0.980191i	$0.563462\pi$
$660$	0	0
$661$	26.9220	1.04714	0.523572	−	0.851982i	$-0.324599\pi$
$661$	26.9220	1.04714	0.523572	+	0.851982i	$0.324599\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−50.5026	−1.95841
$666$	0	0
$667$	4.06185	0.157275
$668$	0	0
$669$	17.6672	0.683054
$670$	0	0
$671$	1.62590	0.0627671
$672$	0	0
$673$	26.5902	1.02498	0.512488	−	0.858694i	$-0.328724\pi$
$673$	26.5902	1.02498	0.512488	+	0.858694i	$0.328724\pi$
$674$	0	0
$675$	130.825	5.03544
$676$	0	0
$677$	6.74621	0.259278	0.129639	−	0.991561i	$-0.458618\pi$
$677$	6.74621	0.259278	0.129639	+	0.991561i	$0.458618\pi$
$678$	0	0
$679$	14.8134	0.568488
$680$	0	0
$681$	−94.4402	−3.61896
$682$	0	0
$683$	−29.0411	−1.11123	−0.555614	−	0.831441i	$-0.687517\pi$
$683$	−29.0411	−1.11123	−0.555614	+	0.831441i	$0.687517\pi$
$684$	0	0
$685$	−45.6055	−1.74250
$686$	0	0
$687$	−14.0778	−0.537102
$688$	0	0
$689$	0	0
$690$	0	0
$691$	6.09637	0.231917	0.115958	−	0.993254i	$-0.463006\pi$
$691$	6.09637	0.231917	0.115958	+	0.993254i	$0.463006\pi$
$692$	0	0
$693$	31.0082	1.17791
$694$	0	0
$695$	−53.0147	−2.01096
$696$	0	0
$697$	7.65918	0.290112
$698$	0	0
$699$	−31.7667	−1.20153
$700$	0	0
$701$	−15.3975	−0.581555	−0.290778	−	0.956791i	$-0.593914\pi$
$701$	−15.3975	−0.581555	−0.290778	+	0.956791i	$0.593914\pi$
$702$	0	0
$703$	56.2180	2.12030
$704$	0	0
$705$	12.2409	0.461019
$706$	0	0
$707$	35.8268	1.34741
$708$	0	0
$709$	24.4324	0.917578	0.458789	−	0.888545i	$-0.348283\pi$
$709$	24.4324	0.917578	0.458789	+	0.888545i	$0.348283\pi$
$710$	0	0
$711$	70.4944	2.64375
$712$	0	0
$713$	6.85395	0.256682
$714$	0	0
$715$	0	0
$716$	0	0
$717$	69.7868	2.60624
$718$	0	0
$719$	−7.10505	−0.264974	−0.132487	−	0.991185i	$-0.542296\pi$
$719$	−7.10505	−0.264974	−0.132487	+	0.991185i	$0.542296\pi$
$720$	0	0
$721$	38.4603	1.43234
$722$	0	0
$723$	−64.9903	−2.41701
$724$	0	0
$725$	−16.2624	−0.603970
$726$	0	0
$727$	−0.395638	−0.0146734	−0.00733670	−	0.999973i	$-0.502335\pi$
$727$	−0.395638	−0.0146734	−0.00733670	+	0.999973i	$0.502335\pi$
$728$	0	0
$729$	98.3945	3.64424
$730$	0	0
$731$	−15.7070	−0.580944
$732$	0	0
$733$	−42.7691	−1.57971	−0.789857	−	0.613291i	$-0.789845\pi$
$733$	−42.7691	−1.57971	−0.789857	+	0.613291i	$0.789845\pi$
$734$	0	0
$735$	−12.8225	−0.472966
$736$	0	0
$737$	−12.4528	−0.458706
$738$	0	0
$739$	−46.0715	−1.69477	−0.847383	−	0.530982i	$-0.821823\pi$
$739$	−46.0715	−1.69477	−0.847383	+	0.530982i	$0.821823\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	11.9311	0.437708	0.218854	−	0.975758i	$-0.429768\pi$
$743$	11.9311	0.437708	0.218854	+	0.975758i	$0.429768\pi$
$744$	0	0
$745$	−14.3339	−0.525154
$746$	0	0
$747$	27.5723	1.00882
$748$	0	0
$749$	−32.0817	−1.17224
$750$	0	0
$751$	15.1934	0.554415	0.277208	−	0.960810i	$-0.410591\pi$
$751$	15.1934	0.554415	0.277208	+	0.960810i	$0.410591\pi$
$752$	0	0
$753$	104.547	3.80992
$754$	0	0
$755$	−32.6463	−1.18812
$756$	0	0
$757$	−0.527100	−0.0191578	−0.00957889	−	0.999954i	$-0.503049\pi$
$757$	−0.527100	−0.0191578	−0.00957889	+	0.999954i	$0.503049\pi$
$758$	0	0
$759$	−9.85808	−0.357826
$760$	0	0
$761$	13.2511	0.480352	0.240176	−	0.970729i	$-0.422795\pi$
$761$	13.2511	0.480352	0.240176	+	0.970729i	$0.422795\pi$
$762$	0	0
$763$	10.5404	0.381587
$764$	0	0
$765$	−136.407	−4.93181
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−23.8483	−0.859991	−0.429996	−	0.902831i	$-0.641485\pi$
$769$	−23.8483	−0.859991	−0.429996	+	0.902831i	$0.641485\pi$
$770$	0	0
$771$	−2.17185	−0.0782172
$772$	0	0
$773$	49.0214	1.76318	0.881589	−	0.472019i	$-0.156474\pi$
$773$	49.0214	1.76318	0.881589	+	0.472019i	$0.156474\pi$
$774$	0	0
$775$	−27.4411	−0.985713
$776$	0	0
$777$	−78.0615	−2.80044
$778$	0	0
$779$	9.52450	0.341251
$780$	0	0
$781$	−1.68203	−0.0601878
$782$	0	0
$783$	−37.1283	−1.32686
$784$	0	0
$785$	−51.4007	−1.83457
$786$	0	0
$787$	51.7724	1.84549	0.922744	−	0.385415i	$-0.125942\pi$
$787$	51.7724	1.84549	0.922744	+	0.385415i	$0.125942\pi$
$788$	0	0
$789$	12.7335	0.453325
$790$	0	0
$791$	23.2163	0.825477
$792$	0	0
$793$	0	0
$794$	0	0
$795$	129.300	4.58578
$796$	0	0
$797$	−43.7601	−1.55006	−0.775031	−	0.631924i	$-0.782266\pi$
$797$	−43.7601	−1.55006	−0.775031	+	0.631924i	$0.782266\pi$
$798$	0	0
$799$	−4.86421	−0.172083
$800$	0	0
$801$	−86.4547	−3.05473
$802$	0	0
$803$	−13.8161	−0.487558
$804$	0	0
$805$	−16.3069	−0.574743
$806$	0	0
$807$	−60.4563	−2.12816
$808$	0	0
$809$	3.86431	0.135862	0.0679309	−	0.997690i	$-0.478360\pi$
$809$	3.86431	0.135862	0.0679309	+	0.997690i	$0.478360\pi$
$810$	0	0
$811$	−23.4043	−0.821836	−0.410918	−	0.911672i	$-0.634792\pi$
$811$	−23.4043	−0.821836	−0.410918	+	0.911672i	$0.634792\pi$
$812$	0	0
$813$	−59.1026	−2.07282
$814$	0	0
$815$	−33.5509	−1.17524
$816$	0	0
$817$	−19.5323	−0.683348
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−40.0306	−1.39708	−0.698538	−	0.715573i	$-0.746166\pi$
$821$	−40.0306	−1.39708	−0.698538	+	0.715573i	$0.746166\pi$
$822$	0	0
$823$	32.9333	1.14798	0.573991	−	0.818862i	$-0.305394\pi$
$823$	32.9333	1.14798	0.573991	+	0.818862i	$0.305394\pi$
$824$	0	0
$825$	39.4687	1.37412
$826$	0	0
$827$	−30.2742	−1.05274	−0.526368	−	0.850257i	$-0.676447\pi$
$827$	−30.2742	−1.05274	−0.526368	+	0.850257i	$0.676447\pi$
$828$	0	0
$829$	4.32213	0.150114	0.0750570	−	0.997179i	$-0.476086\pi$
$829$	4.32213	0.150114	0.0750570	+	0.997179i	$0.476086\pi$
$830$	0	0
$831$	−68.3199	−2.36999
$832$	0	0
$833$	5.09533	0.176543
$834$	0	0
$835$	53.7197	1.85905
$836$	0	0
$837$	−62.6502	−2.16551
$838$	0	0
$839$	41.3956	1.42913	0.714567	−	0.699567i	$-0.246624\pi$
$839$	41.3956	1.42913	0.714567	+	0.699567i	$0.246624\pi$
$840$	0	0
$841$	−24.3847	−0.840852
$842$	0	0
$843$	−13.8064	−0.475518
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−20.8392	−0.716044
$848$	0	0
$849$	40.0809	1.37557
$850$	0	0
$851$	18.1524	0.622255
$852$	0	0
$853$	40.6391	1.39145	0.695727	−	0.718306i	$-0.255082\pi$
$853$	40.6391	1.39145	0.695727	+	0.718306i	$0.255082\pi$
$854$	0	0
$855$	−169.628	−5.80115
$856$	0	0
$857$	−37.0489	−1.26557	−0.632784	−	0.774328i	$-0.718088\pi$
$857$	−37.0489	−1.26557	−0.632784	+	0.774328i	$0.718088\pi$
$858$	0	0
$859$	52.6967	1.79799	0.898994	−	0.437961i	$-0.144299\pi$
$859$	52.6967	1.79799	0.898994	+	0.437961i	$0.144299\pi$
$860$	0	0
$861$	−13.2252	−0.450715
$862$	0	0
$863$	−53.1621	−1.80966	−0.904830	−	0.425773i	$-0.860002\pi$
$863$	−53.1621	−1.80966	−0.904830	+	0.425773i	$0.860002\pi$
$864$	0	0
$865$	−67.0457	−2.27962
$866$	0	0
$867$	17.2874	0.587111
$868$	0	0
$869$	13.4590	0.456566
$870$	0	0
$871$	0	0
$872$	0	0
$873$	49.7553	1.68396
$874$	0	0
$875$	22.1639	0.749276
$876$	0	0
$877$	37.4162	1.26345	0.631727	−	0.775191i	$-0.282346\pi$
$877$	37.4162	1.26345	0.631727	+	0.775191i	$0.282346\pi$
$878$	0	0
$879$	2.42161	0.0816789
$880$	0	0
$881$	−11.2014	−0.377384	−0.188692	−	0.982036i	$-0.560425\pi$
$881$	−11.2014	−0.377384	−0.188692	+	0.982036i	$0.560425\pi$
$882$	0	0
$883$	37.9278	1.27637	0.638186	−	0.769882i	$-0.279685\pi$
$883$	37.9278	1.27637	0.638186	+	0.769882i	$0.279685\pi$
$884$	0	0
$885$	79.5020	2.67243
$886$	0	0
$887$	22.6989	0.762154	0.381077	−	0.924543i	$-0.375553\pi$
$887$	22.6989	0.762154	0.381077	+	0.924543i	$0.375553\pi$
$888$	0	0
$889$	26.2716	0.881122
$890$	0	0
$891$	51.8705	1.73773
$892$	0	0
$893$	−6.04883	−0.202417
$894$	0	0
$895$	−33.9338	−1.13428
$896$	0	0
$897$	0	0
$898$	0	0
$899$	7.78785	0.259739
$900$	0	0
$901$	−51.3802	−1.71172
$902$	0	0
$903$	27.1216	0.902549
$904$	0	0
$905$	92.5333	3.07591
$906$	0	0
$907$	−3.58717	−0.119110	−0.0595550	−	0.998225i	$-0.518968\pi$
$907$	−3.58717	−0.119110	−0.0595550	+	0.998225i	$0.518968\pi$
$908$	0	0
$909$	120.335	3.99126
$910$	0	0
$911$	−21.7449	−0.720442	−0.360221	−	0.932867i	$-0.617299\pi$
$911$	−21.7449	−0.720442	−0.360221	+	0.932867i	$0.617299\pi$
$912$	0	0
$913$	5.26418	0.174219
$914$	0	0
$915$	12.3502	0.408286
$916$	0	0
$917$	−26.8908	−0.888011
$918$	0	0
$919$	50.1334	1.65375	0.826875	−	0.562386i	$-0.190116\pi$
$919$	50.1334	1.65375	0.826875	+	0.562386i	$0.190116\pi$
$920$	0	0
$921$	9.24849	0.304748
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−72.6765	−2.38959
$926$	0	0
$927$	129.180	4.24284
$928$	0	0
$929$	−8.39068	−0.275289	−0.137645	−	0.990482i	$-0.543953\pi$
$929$	−8.39068	−0.275289	−0.137645	+	0.990482i	$0.543953\pi$
$930$	0	0
$931$	6.33625	0.207662
$932$	0	0
$933$	99.0292	3.24207
$934$	0	0
$935$	−26.0433	−0.851706
$936$	0	0
$937$	−4.09505	−0.133780	−0.0668898	−	0.997760i	$-0.521308\pi$
$937$	−4.09505	−0.133780	−0.0668898	+	0.997760i	$0.521308\pi$
$938$	0	0
$939$	35.9930	1.17459
$940$	0	0
$941$	61.0652	1.99067	0.995334	−	0.0964892i	$-0.0307613\pi$
$941$	61.0652	1.99067	0.995334	+	0.0964892i	$0.0307613\pi$
$942$	0	0
$943$	3.07539	0.100148
$944$	0	0
$945$	149.057	4.84883
$946$	0	0
$947$	−32.4455	−1.05434	−0.527169	−	0.849760i	$-0.676747\pi$
$947$	−32.4455	−1.05434	−0.527169	+	0.849760i	$0.676747\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	1.26803	0.0411187
$952$	0	0
$953$	14.9299	0.483627	0.241814	−	0.970323i	$-0.422258\pi$
$953$	14.9299	0.483627	0.241814	+	0.970323i	$0.422258\pi$
$954$	0	0
$955$	30.7606	0.995390
$956$	0	0
$957$	−11.2013	−0.362087
$958$	0	0
$959$	−31.2923	−1.01048
$960$	0	0
$961$	−17.8588	−0.576090
$962$	0	0
$963$	−107.756	−3.47239
$964$	0	0
$965$	52.4318	1.68784
$966$	0	0
$967$	−50.3131	−1.61796	−0.808980	−	0.587836i	$-0.799980\pi$
$967$	−50.3131	−1.61796	−0.808980	+	0.587836i	$0.799980\pi$
$968$	0	0
$969$	92.1541	2.96041
$970$	0	0
$971$	10.8825	0.349237	0.174618	−	0.984636i	$-0.444131\pi$
$971$	10.8825	0.349237	0.174618	+	0.984636i	$0.444131\pi$
$972$	0	0
$973$	−36.3761	−1.16616
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−4.17393	−0.133536	−0.0667680	−	0.997769i	$-0.521269\pi$
$977$	−4.17393	−0.133536	−0.0667680	+	0.997769i	$0.521269\pi$
$978$	0	0
$979$	−16.5062	−0.527540
$980$	0	0
$981$	35.4029	1.13033
$982$	0	0
$983$	−10.7680	−0.343445	−0.171722	−	0.985145i	$-0.554933\pi$
$983$	−10.7680	−0.343445	−0.171722	+	0.985145i	$0.554933\pi$
$984$	0	0
$985$	6.72926	0.214412
$986$	0	0
$987$	8.39911	0.267347
$988$	0	0
$989$	−6.30683	−0.200545
$990$	0	0
$991$	−23.1226	−0.734515	−0.367257	−	0.930119i	$-0.619703\pi$
$991$	−23.1226	−0.734515	−0.367257	+	0.930119i	$0.619703\pi$
$992$	0	0
$993$	81.2786	2.57930
$994$	0	0
$995$	−33.8692	−1.07373
$996$	0	0
$997$	41.9826	1.32960	0.664801	−	0.747020i	$-0.268516\pi$
$997$	41.9826	1.32960	0.664801	+	0.747020i	$0.268516\pi$
$998$	0	0
$999$	−165.926	−5.24967

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5408.2.a.bs.1.9	yes	9
4.3	odd	2		5408.2.a.bq.1.1	yes	9
13.12	even	2		5408.2.a.br.1.9	yes	9
52.51	odd	2		5408.2.a.bp.1.1	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5408.2.a.bp.1.1	✓	9	52.51	odd	2
5408.2.a.bq.1.1	yes	9	4.3	odd	2
5408.2.a.br.1.9	yes	9	13.12	even	2
5408.2.a.bs.1.9	yes	9	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5408 = 2^{5} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5408.a (trivial)

\(f(q)\)	\(=\)	\(q+3.34228 q^{3} +3.54539 q^{5} +2.43267 q^{7} +8.17085 q^{9} +1.56000 q^{11} +11.8497 q^{15} -4.70875 q^{17} -5.85552 q^{19} +8.13068 q^{21} -1.89070 q^{23} +7.56979 q^{25} +17.2824 q^{27} -2.14833 q^{29} -3.62508 q^{31} +5.21397 q^{33} +8.62478 q^{35} -9.60085 q^{37} -1.62658 q^{41} +3.33570 q^{43} +28.9689 q^{45} +1.03301 q^{47} -1.08210 q^{49} -15.7380 q^{51} +10.9116 q^{53} +5.53082 q^{55} -19.5708 q^{57} +6.70920 q^{59} +1.04224 q^{61} +19.8770 q^{63} -7.98257 q^{67} -6.31927 q^{69} -1.07822 q^{71} -8.85642 q^{73} +25.3004 q^{75} +3.79498 q^{77} +8.62755 q^{79} +33.2503 q^{81} +3.37447 q^{83} -16.6944 q^{85} -7.18031 q^{87} -10.5809 q^{89} -12.1160 q^{93} -20.7601 q^{95} +6.08937 q^{97} +12.7466 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 7 q^{3} + 6 q^{5} + 10 q^{9} - q^{11} + 18 q^{15} - 7 q^{17} - 5 q^{19} + 12 q^{21} + 12 q^{23} + 11 q^{25} + 34 q^{27} - 8 q^{29} + 20 q^{31} - 12 q^{33} + 6 q^{35} + 6 q^{37} - 31 q^{41} + 33 q^{43}+ \cdots - 47 q^{99}+O(q^{100}) \) 9 * q + 7 * q^3 + 6 * q^5 + 10 * q^9 - q^11 + 18 * q^15 - 7 * q^17 - 5 * q^19 + 12 * q^21 + 12 * q^23 + 11 * q^25 + 34 * q^27 - 8 * q^29 + 20 * q^31 - 12 * q^33 + 6 * q^35 + 6 * q^37 - 31 * q^41 + 33 * q^43 + 22 * q^45 - 14 * q^47 + 3 * q^49 + 12 * q^51 + 14 * q^53 + 38 * q^55 - 10 * q^57 + 9 * q^59 + 8 * q^61 + 20 * q^63 + 3 * q^67 + 26 * q^69 - 26 * q^71 - 5 * q^73 + 25 * q^75 - 6 * q^77 + 44 * q^79 + 45 * q^81 + 17 * q^83 + 6 * q^85 + 24 * q^87 - 41 * q^89 + 26 * q^93 + 4 * q^95 - 15 * q^97 - 47 * q^99

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