LMFDB - Embedded newform 2704.2.a.bd.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2704,2,Mod(1,2704)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2704.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2704, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2704 = 2^{4} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2704.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$21.5915487066$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.13968.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 7x^{2} + 8x + 4 $ x^4 - 2x^3 - 7x^2 + 8*x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.38651$ of defining polynomial
Character	$\chi$	$=$	2704.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.38651 q^{3} -3.44247 q^{5} -3.17452 q^{7} -1.07759 q^{9} -3.17452 q^{11} -4.77302 q^{15} +1.77302 q^{17} +6.40150 q^{19} -4.40150 q^{21} -1.38651 q^{23} +6.85061 q^{25} -5.65362 q^{27} +5.11192 q^{29} +1.35218 q^{31} -4.40150 q^{33} +10.9282 q^{35} -11.3081 q^{37} +2.26795 q^{41} +4.27145 q^{43} +3.70958 q^{45} -1.57603 q^{47} +3.07759 q^{49} +2.45831 q^{51} +3.73869 q^{53} +10.9282 q^{55} +8.87574 q^{57} +13.5236 q^{59} -10.8131 q^{61} +3.42084 q^{63} +1.01934 q^{67} -1.92241 q^{69} +13.3297 q^{71} -5.33055 q^{73} +9.49843 q^{75} +10.0776 q^{77} +11.1521 q^{79} -4.60602 q^{81} +14.3490 q^{83} -6.10356 q^{85} +7.08773 q^{87} +4.62535 q^{89} +1.87481 q^{93} -22.0370 q^{95} -11.8223 q^{97} +3.42084 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{3} - 4 q^{5} + 4 q^{7} + 6 q^{9} + 4 q^{11} - 12 q^{15} + 16 q^{19} - 8 q^{21} - 2 q^{23} + 10 q^{25} + 14 q^{27} + 8 q^{29} + 4 q^{31} - 8 q^{33} + 16 q^{35} - 12 q^{37} + 16 q^{41} - 6 q^{43}+ \cdots + 8 q^{99}+O(q^{100}) $ 4 * q + 2 * q^3 - 4 * q^5 + 4 * q^7 + 6 * q^9 + 4 * q^11 - 12 * q^15 + 16 * q^19 - 8 * q^21 - 2 * q^23 + 10 * q^25 + 14 * q^27 + 8 * q^29 + 4 * q^31 - 8 * q^33 + 16 * q^35 - 12 * q^37 + 16 * q^41 - 6 * q^43 - 28 * q^45 + 20 * q^47 + 2 * q^49 + 34 * q^51 + 10 * q^53 + 16 * q^55 - 16 * q^57 + 4 * q^59 + 4 * q^61 + 8 * q^63 + 8 * q^67 - 18 * q^69 + 16 * q^71 - 24 * q^73 + 22 * q^75 + 30 * q^77 - 8 * q^79 + 20 * q^81 + 24 * q^83 - 20 * q^85 - 10 * q^87 - 16 * q^89 + 16 * q^93 - 16 * q^95 - 32 * q^97 + 8 * q^99

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.38651	0.800501	0.400251	−	0.916406i	$-0.368923\pi$
$3$	1.38651	0.800501	0.400251	+	0.916406i	$0.368923\pi$
$4$	0	0
$5$	−3.44247	−1.53952	−0.769760	−	0.638333i	$-0.779624\pi$
$5$	−3.44247	−1.53952	−0.769760	+	0.638333i	$0.779624\pi$
$6$	0	0
$7$	−3.17452	−1.19986	−0.599928	−	0.800054i	$-0.704804\pi$
$7$	−3.17452	−1.19986	−0.599928	+	0.800054i	$0.704804\pi$
$8$	0	0
$9$	−1.07759	−0.359198
$10$	0	0
$11$	−3.17452	−0.957155	−0.478577	−	0.878045i	$-0.658847\pi$
$11$	−3.17452	−0.957155	−0.478577	+	0.878045i	$0.658847\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−4.77302	−1.23239
$16$	0	0
$17$	1.77302	0.430020	0.215010	−	0.976612i	$-0.431022\pi$
$17$	1.77302	0.430020	0.215010	+	0.976612i	$0.431022\pi$
$18$	0	0
$19$	6.40150	1.46861	0.734303	−	0.678822i	$-0.237509\pi$
$19$	6.40150	1.46861	0.734303	+	0.678822i	$0.237509\pi$
$20$	0	0
$21$	−4.40150	−0.960487
$22$	0	0
$23$	−1.38651	−0.289107	−0.144554	−	0.989497i	$-0.546175\pi$
$23$	−1.38651	−0.289107	−0.144554	+	0.989497i	$0.546175\pi$
$24$	0	0
$25$	6.85061	1.37012
$26$	0	0
$27$	−5.65362	−1.08804
$28$	0	0
$29$	5.11192	0.949261	0.474630	−	0.880185i	$-0.342582\pi$
$29$	5.11192	0.949261	0.474630	+	0.880185i	$0.342582\pi$
$30$	0	0
$31$	1.35218	0.242858	0.121429	−	0.992600i	$-0.461252\pi$
$31$	1.35218	0.242858	0.121429	+	0.992600i	$0.461252\pi$
$32$	0	0
$33$	−4.40150	−0.766204
$34$	0	0
$35$	10.9282	1.84720
$36$	0	0
$37$	−11.3081	−1.85904	−0.929518	−	0.368776i	$-0.879777\pi$
$37$	−11.3081	−1.85904	−0.929518	+	0.368776i	$0.879777\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	2.26795	0.354194	0.177097	−	0.984193i	$-0.443329\pi$
$41$	2.26795	0.354194	0.177097	+	0.984193i	$0.443329\pi$
$42$	0	0
$43$	4.27145	0.651390	0.325695	−	0.945475i	$-0.394402\pi$
$43$	4.27145	0.651390	0.325695	+	0.945475i	$0.394402\pi$
$44$	0	0
$45$	3.70958	0.552992
$46$	0	0
$47$	−1.57603	−0.229887	−0.114944	−	0.993372i	$-0.536669\pi$
$47$	−1.57603	−0.229887	−0.114944	+	0.993372i	$0.536669\pi$
$48$	0	0
$49$	3.07759	0.439656
$50$	0	0
$51$	2.45831	0.344232
$52$	0	0
$53$	3.73869	0.513548	0.256774	−	0.966471i	$-0.417340\pi$
$53$	3.73869	0.513548	0.256774	+	0.966471i	$0.417340\pi$
$54$	0	0
$55$	10.9282	1.47356
$56$	0	0
$57$	8.87574	1.17562
$58$	0	0
$59$	13.5236	1.76062	0.880309	−	0.474400i	$-0.157335\pi$
$59$	13.5236	1.76062	0.880309	+	0.474400i	$0.157335\pi$
$60$	0	0
$61$	−10.8131	−1.38448	−0.692241	−	0.721667i	$-0.743376\pi$
$61$	−10.8131	−1.38448	−0.692241	+	0.721667i	$0.743376\pi$
$62$	0	0
$63$	3.42084	0.430986
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.01934	0.124532	0.0622659	−	0.998060i	$-0.480167\pi$
$67$	1.01934	0.124532	0.0622659	+	0.998060i	$0.480167\pi$
$68$	0	0
$69$	−1.92241	−0.231431
$70$	0	0
$71$	13.3297	1.58195	0.790973	−	0.611852i	$-0.209575\pi$
$71$	13.3297	1.58195	0.790973	+	0.611852i	$0.209575\pi$
$72$	0	0
$73$	−5.33055	−0.623893	−0.311947	−	0.950100i	$-0.600981\pi$
$73$	−5.33055	−0.623893	−0.311947	+	0.950100i	$0.600981\pi$
$74$	0	0
$75$	9.49843	1.09678
$76$	0	0
$77$	10.0776	1.14845
$78$	0	0
$79$	11.1521	1.25470	0.627352	−	0.778736i	$-0.284139\pi$
$79$	11.1521	1.25470	0.627352	+	0.778736i	$0.284139\pi$
$80$	0	0
$81$	−4.60602	−0.511780
$82$	0	0
$83$	14.3490	1.57501	0.787506	−	0.616307i	$-0.211372\pi$
$83$	14.3490	1.57501	0.787506	+	0.616307i	$0.211372\pi$
$84$	0	0
$85$	−6.10356	−0.662025
$86$	0	0
$87$	7.08773	0.759884
$88$	0	0
$89$	4.62535	0.490287	0.245143	−	0.969487i	$-0.421165\pi$
$89$	4.62535	0.490287	0.245143	+	0.969487i	$0.421165\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	1.87481	0.194408
$94$	0	0
$95$	−22.0370	−2.26095
$96$	0	0
$97$	−11.8223	−1.20038	−0.600189	−	0.799858i	$-0.704908\pi$
$97$	−11.8223	−1.20038	−0.600189	+	0.799858i	$0.704908\pi$
$98$	0	0
$99$	3.42084	0.343808
$100$	0	0
$101$	6.33891	0.630745	0.315372	−	0.948968i	$-0.397871\pi$
$101$	6.33891	0.630745	0.315372	+	0.948968i	$0.397871\pi$
$102$	0	0
$103$	6.88494	0.678394	0.339197	−	0.940715i	$-0.389845\pi$
$103$	6.88494	0.678394	0.339197	+	0.940715i	$0.389845\pi$
$104$	0	0
$105$	15.1521	1.47869
$106$	0	0
$107$	−0.159527	−0.0154221	−0.00771104	−	0.999970i	$-0.502455\pi$
$107$	−0.159527	−0.0154221	−0.00771104	+	0.999970i	$0.502455\pi$
$108$	0	0
$109$	10.9282	1.04673	0.523366	−	0.852108i	$-0.324676\pi$
$109$	10.9282	1.04673	0.523366	+	0.852108i	$0.324676\pi$
$110$	0	0
$111$	−15.6788	−1.48816
$112$	0	0
$113$	11.9969	1.12857	0.564285	−	0.825580i	$-0.309152\pi$
$113$	11.9969	1.12857	0.564285	+	0.825580i	$0.309152\pi$
$114$	0	0
$115$	4.77302	0.445086
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−5.62849	−0.515962
$120$	0	0
$121$	−0.922407	−0.0838552
$122$	0	0
$123$	3.14453	0.283533
$124$	0	0
$125$	−6.37067	−0.569810
$126$	0	0
$127$	−12.5386	−1.11262	−0.556309	−	0.830976i	$-0.687783\pi$
$127$	−12.5386	−1.11262	−0.556309	+	0.830976i	$0.687783\pi$
$128$	0	0
$129$	5.92241	0.521439
$130$	0	0
$131$	9.33891	0.815944	0.407972	−	0.912994i	$-0.366236\pi$
$131$	9.33891	0.815944	0.407972	+	0.912994i	$0.366236\pi$
$132$	0	0
$133$	−20.3217	−1.76212
$134$	0	0
$135$	19.4624	1.67506
$136$	0	0
$137$	5.15289	0.440241	0.220121	−	0.975473i	$-0.429355\pi$
$137$	5.15289	0.440241	0.220121	+	0.975473i	$0.429355\pi$
$138$	0	0
$139$	−3.19966	−0.271391	−0.135696	−	0.990751i	$-0.543327\pi$
$139$	−3.19966	−0.271391	−0.135696	+	0.990751i	$0.543327\pi$
$140$	0	0
$141$	−2.18518	−0.184025
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−17.5977	−1.46141
$146$	0	0
$147$	4.26711	0.351945
$148$	0	0
$149$	7.62013	0.624265	0.312133	−	0.950039i	$-0.398957\pi$
$149$	7.62013	0.624265	0.312133	+	0.950039i	$0.398957\pi$
$150$	0	0
$151$	12.5042	1.01758	0.508790	−	0.860891i	$-0.330093\pi$
$151$	12.5042	1.01758	0.508790	+	0.860891i	$0.330093\pi$
$152$	0	0
$153$	−1.91059	−0.154462
$154$	0	0
$155$	−4.65483	−0.373885
$156$	0	0
$157$	1.80735	0.144242	0.0721211	−	0.997396i	$-0.477023\pi$
$157$	1.80735	0.144242	0.0721211	+	0.997396i	$0.477023\pi$
$158$	0	0
$159$	5.18372	0.411096
$160$	0	0
$161$	4.40150	0.346887
$162$	0	0
$163$	11.7537	0.920619	0.460310	−	0.887758i	$-0.347738\pi$
$163$	11.7537	0.920619	0.460310	+	0.887758i	$0.347738\pi$
$164$	0	0
$165$	15.1521	1.17959
$166$	0	0
$167$	−17.0309	−1.31789	−0.658946	−	0.752190i	$-0.728998\pi$
$167$	−17.0309	−1.31789	−0.658946	+	0.752190i	$0.728998\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−6.89821	−0.527520
$172$	0	0
$173$	−15.3249	−1.16513	−0.582563	−	0.812785i	$-0.697950\pi$
$173$	−15.3249	−1.16513	−0.582563	+	0.812785i	$0.697950\pi$
$174$	0	0
$175$	−21.7474	−1.64395
$176$	0	0
$177$	18.7505	1.40938
$178$	0	0
$179$	1.27458	0.0952669	0.0476334	−	0.998865i	$-0.484832\pi$
$179$	1.27458	0.0952669	0.0476334	+	0.998865i	$0.484832\pi$
$180$	0	0
$181$	2.41650	0.179617	0.0898085	−	0.995959i	$-0.471375\pi$
$181$	2.41650	0.179617	0.0898085	+	0.995959i	$0.471375\pi$
$182$	0	0
$183$	−14.9925	−1.10828
$184$	0	0
$185$	38.9277	2.86202
$186$	0	0
$187$	−5.62849	−0.411596
$188$	0	0
$189$	17.9475	1.30549
$190$	0	0
$191$	−6.70870	−0.485424	−0.242712	−	0.970098i	$-0.578037\pi$
$191$	−6.70870	−0.485424	−0.242712	+	0.970098i	$0.578037\pi$
$192$	0	0
$193$	−5.36002	−0.385823	−0.192911	−	0.981216i	$-0.561793\pi$
$193$	−5.36002	−0.385823	−0.192911	+	0.981216i	$0.561793\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.62535	−0.614531	−0.307266	−	0.951624i	$-0.599414\pi$
$197$	−8.62535	−0.614531	−0.307266	+	0.951624i	$0.599414\pi$
$198$	0	0
$199$	−10.8639	−0.770120	−0.385060	−	0.922892i	$-0.625819\pi$
$199$	−10.8639	−0.770120	−0.385060	+	0.922892i	$0.625819\pi$
$200$	0	0
$201$	1.41332	0.0996879
$202$	0	0
$203$	−16.2279	−1.13898
$204$	0	0
$205$	−7.80735	−0.545289
$206$	0	0
$207$	1.49409	0.103847
$208$	0	0
$209$	−20.3217	−1.40568
$210$	0	0
$211$	−10.8373	−0.746073	−0.373037	−	0.927817i	$-0.621683\pi$
$211$	−10.8373	−0.746073	−0.373037	+	0.927817i	$0.621683\pi$
$212$	0	0
$213$	18.4818	1.26635
$214$	0	0
$215$	−14.7044	−1.00283
$216$	0	0
$217$	−4.29251	−0.291395
$218$	0	0
$219$	−7.39085	−0.499427
$220$	0	0
$221$	0	0
$222$	0	0
$223$	7.24945	0.485459	0.242730	−	0.970094i	$-0.421957\pi$
$223$	7.24945	0.485459	0.242730	+	0.970094i	$0.421957\pi$
$224$	0	0
$225$	−7.38217	−0.492144
$226$	0	0
$227$	−9.59850	−0.637075	−0.318537	−	0.947910i	$-0.603192\pi$
$227$	−9.59850	−0.637075	−0.318537	+	0.947910i	$0.603192\pi$
$228$	0	0
$229$	2.92820	0.193501	0.0967506	−	0.995309i	$-0.469155\pi$
$229$	2.92820	0.193501	0.0967506	+	0.995309i	$0.469155\pi$
$230$	0	0
$231$	13.9727	0.919334
$232$	0	0
$233$	−13.1521	−0.861620	−0.430810	−	0.902443i	$-0.641772\pi$
$233$	−13.1521	−0.861620	−0.430810	+	0.902443i	$0.641772\pi$
$234$	0	0
$235$	5.42543	0.353916
$236$	0	0
$237$	15.4624	1.00439
$238$	0	0
$239$	−9.02686	−0.583899	−0.291949	−	0.956434i	$-0.594304\pi$
$239$	−9.02686	−0.583899	−0.291949	+	0.956434i	$0.594304\pi$
$240$	0	0
$241$	−16.8408	−1.08481	−0.542407	−	0.840116i	$-0.682487\pi$
$241$	−16.8408	−1.08481	−0.542407	+	0.840116i	$0.682487\pi$
$242$	0	0
$243$	10.5746	0.678359
$244$	0	0
$245$	−10.5945	−0.676859
$246$	0	0
$247$	0	0
$248$	0	0
$249$	19.8951	1.26080
$250$	0	0
$251$	29.9207	1.88858	0.944290	−	0.329115i	$-0.106750\pi$
$251$	29.9207	1.88858	0.944290	+	0.329115i	$0.106750\pi$
$252$	0	0
$253$	4.40150	0.276720
$254$	0	0
$255$	−8.46265	−0.529952
$256$	0	0
$257$	4.15205	0.258998	0.129499	−	0.991580i	$-0.458663\pi$
$257$	4.15205	0.258998	0.129499	+	0.991580i	$0.458663\pi$
$258$	0	0
$259$	35.8977	2.23058
$260$	0	0
$261$	−5.50857	−0.340972
$262$	0	0
$263$	−9.80880	−0.604837	−0.302418	−	0.953175i	$-0.597794\pi$
$263$	−9.80880	−0.604837	−0.302418	+	0.953175i	$0.597794\pi$
$264$	0	0
$265$	−12.8703	−0.790617
$266$	0	0
$267$	6.41310	0.392475
$268$	0	0
$269$	−6.91928	−0.421876	−0.210938	−	0.977499i	$-0.567652\pi$
$269$	−6.91928	−0.421876	−0.210938	+	0.977499i	$0.567652\pi$
$270$	0	0
$271$	−12.7206	−0.772719	−0.386359	−	0.922348i	$-0.626268\pi$
$271$	−12.7206	−0.772719	−0.386359	+	0.922348i	$0.626268\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−21.7474	−1.31142
$276$	0	0
$277$	8.19531	0.492409	0.246204	−	0.969218i	$-0.420817\pi$
$277$	8.19531	0.492409	0.246204	+	0.969218i	$0.420817\pi$
$278$	0	0
$279$	−1.45710	−0.0872340
$280$	0	0
$281$	8.00836	0.477739	0.238869	−	0.971052i	$-0.423223\pi$
$281$	8.00836	0.477739	0.238869	+	0.971052i	$0.423223\pi$
$282$	0	0
$283$	22.0327	1.30971	0.654853	−	0.755757i	$-0.272731\pi$
$283$	22.0327	1.30971	0.654853	+	0.755757i	$0.272731\pi$
$284$	0	0
$285$	−30.5545	−1.80989
$286$	0	0
$287$	−7.19966	−0.424982
$288$	0	0
$289$	−13.8564	−0.815083
$290$	0	0
$291$	−16.3918	−0.960904
$292$	0	0
$293$	−8.05111	−0.470351	−0.235175	−	0.971953i	$-0.575566\pi$
$293$	−8.05111	−0.470351	−0.235175	+	0.971953i	$0.575566\pi$
$294$	0	0
$295$	−46.5545	−2.71051
$296$	0	0
$297$	17.9475	1.04142
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−13.5598	−0.781575
$302$	0	0
$303$	8.78895	0.504912
$304$	0	0
$305$	37.2239	2.13144
$306$	0	0
$307$	−16.7281	−0.954722	−0.477361	−	0.878707i	$-0.658407\pi$
$307$	−16.7281	−0.954722	−0.477361	+	0.878707i	$0.658407\pi$
$308$	0	0
$309$	9.54604	0.543055
$310$	0	0
$311$	−3.79287	−0.215074	−0.107537	−	0.994201i	$-0.534296\pi$
$311$	−3.79287	−0.215074	−0.107537	+	0.994201i	$0.534296\pi$
$312$	0	0
$313$	−26.6981	−1.50907	−0.754533	−	0.656263i	$-0.772136\pi$
$313$	−26.6981	−1.50907	−0.754533	+	0.656263i	$0.772136\pi$
$314$	0	0
$315$	−11.7762	−0.663511
$316$	0	0
$317$	−18.6378	−1.04680	−0.523401	−	0.852086i	$-0.675337\pi$
$317$	−18.6378	−1.04680	−0.523401	+	0.852086i	$0.675337\pi$
$318$	0	0
$319$	−16.2279	−0.908589
$320$	0	0
$321$	−0.221186	−0.0123454
$322$	0	0
$323$	11.3500	0.631530
$324$	0	0
$325$	0	0
$326$	0	0
$327$	15.1521	0.837910
$328$	0	0
$329$	5.00313	0.275832
$330$	0	0
$331$	−13.2997	−0.731019	−0.365509	−	0.930808i	$-0.619105\pi$
$331$	−13.2997	−0.731019	−0.365509	+	0.930808i	$0.619105\pi$
$332$	0	0
$333$	12.1855	0.667761
$334$	0	0
$335$	−3.50904	−0.191719
$336$	0	0
$337$	20.6174	1.12310	0.561550	−	0.827443i	$-0.310205\pi$
$337$	20.6174	1.12310	0.561550	+	0.827443i	$0.310205\pi$
$338$	0	0
$339$	16.6338	0.903422
$340$	0	0
$341$	−4.29251	−0.232453
$342$	0	0
$343$	12.4518	0.672332
$344$	0	0
$345$	6.61783	0.356292
$346$	0	0
$347$	10.6135	0.569762	0.284881	−	0.958563i	$-0.408046\pi$
$347$	10.6135	0.569762	0.284881	+	0.958563i	$0.408046\pi$
$348$	0	0
$349$	12.0278	0.643833	0.321917	−	0.946768i	$-0.395673\pi$
$349$	12.0278	0.643833	0.321917	+	0.946768i	$0.395673\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−23.9329	−1.27382	−0.636910	−	0.770938i	$-0.719788\pi$
$353$	−23.9329	−1.27382	−0.636910	+	0.770938i	$0.719788\pi$
$354$	0	0
$355$	−45.8871	−2.43544
$356$	0	0
$357$	−7.80395	−0.413029
$358$	0	0
$359$	−0.104919	−0.00553740	−0.00276870	−	0.999996i	$-0.500881\pi$
$359$	−0.104919	−0.00553740	−0.00276870	+	0.999996i	$0.500881\pi$
$360$	0	0
$361$	21.9793	1.15680
$362$	0	0
$363$	−1.27893	−0.0671262
$364$	0	0
$365$	18.3503	0.960496
$366$	0	0
$367$	12.1508	0.634269	0.317134	−	0.948381i	$-0.397279\pi$
$367$	12.1508	0.634269	0.317134	+	0.948381i	$0.397279\pi$
$368$	0	0
$369$	−2.44393	−0.127226
$370$	0	0
$371$	−11.8685	−0.616184
$372$	0	0
$373$	34.1204	1.76669	0.883343	−	0.468727i	$-0.155287\pi$
$373$	34.1204	1.76669	0.883343	+	0.468727i	$0.155287\pi$
$374$	0	0
$375$	−8.83300	−0.456134
$376$	0	0
$377$	0	0
$378$	0	0
$379$	32.6456	1.67689	0.838447	−	0.544984i	$-0.183464\pi$
$379$	32.6456	1.67689	0.838447	+	0.544984i	$0.183464\pi$
$380$	0	0
$381$	−17.3848	−0.890652
$382$	0	0
$383$	30.8008	1.57385	0.786924	−	0.617050i	$-0.211672\pi$
$383$	30.8008	1.57385	0.786924	+	0.617050i	$0.211672\pi$
$384$	0	0
$385$	−34.6918	−1.76806
$386$	0	0
$387$	−4.60289	−0.233978
$388$	0	0
$389$	−1.41963	−0.0719782	−0.0359891	−	0.999352i	$-0.511458\pi$
$389$	−1.41963	−0.0719782	−0.0359891	+	0.999352i	$0.511458\pi$
$390$	0	0
$391$	−2.45831	−0.124322
$392$	0	0
$393$	12.9485	0.653164
$394$	0	0
$395$	−38.3906	−1.93164
$396$	0	0
$397$	20.8308	1.04547	0.522734	−	0.852496i	$-0.324912\pi$
$397$	20.8308	1.04547	0.522734	+	0.852496i	$0.324912\pi$
$398$	0	0
$399$	−28.1762	−1.41058
$400$	0	0
$401$	8.39314	0.419134	0.209567	−	0.977794i	$-0.432795\pi$
$401$	8.39314	0.419134	0.209567	+	0.977794i	$0.432795\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	15.8561	0.787895
$406$	0	0
$407$	35.8977	1.77939
$408$	0	0
$409$	−5.44654	−0.269314	−0.134657	−	0.990892i	$-0.542993\pi$
$409$	−5.44654	−0.269314	−0.134657	+	0.990892i	$0.542993\pi$
$410$	0	0
$411$	7.14453	0.352414
$412$	0	0
$413$	−42.9309	−2.11249
$414$	0	0
$415$	−49.3962	−2.42476
$416$	0	0
$417$	−4.43635	−0.217249
$418$	0	0
$419$	12.4520	0.608322	0.304161	−	0.952621i	$-0.401624\pi$
$419$	12.4520	0.608322	0.304161	+	0.952621i	$0.401624\pi$
$420$	0	0
$421$	18.7584	0.914228	0.457114	−	0.889408i	$-0.348883\pi$
$421$	18.7584	0.914228	0.457114	+	0.889408i	$0.348883\pi$
$422$	0	0
$423$	1.69831	0.0825749
$424$	0	0
$425$	12.1463	0.589180
$426$	0	0
$427$	34.3266	1.66118
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17.2548	0.831133	0.415567	−	0.909563i	$-0.363583\pi$
$431$	17.2548	0.831133	0.415567	+	0.909563i	$0.363583\pi$
$432$	0	0
$433$	17.9219	0.861273	0.430637	−	0.902525i	$-0.358289\pi$
$433$	17.9219	0.861273	0.430637	+	0.902525i	$0.358289\pi$
$434$	0	0
$435$	−24.3993	−1.16986
$436$	0	0
$437$	−8.87574	−0.424584
$438$	0	0
$439$	−27.5100	−1.31298	−0.656491	−	0.754334i	$-0.727960\pi$
$439$	−27.5100	−1.31298	−0.656491	+	0.754334i	$0.727960\pi$
$440$	0	0
$441$	−3.31639	−0.157923
$442$	0	0
$443$	0.223850	0.0106354	0.00531772	−	0.999986i	$-0.498307\pi$
$443$	0.223850	0.0106354	0.00531772	+	0.999986i	$0.498307\pi$
$444$	0	0
$445$	−15.9227	−0.754806
$446$	0	0
$447$	10.5654	0.499725
$448$	0	0
$449$	28.1327	1.32766	0.663832	−	0.747881i	$-0.268929\pi$
$449$	28.1327	1.32766	0.663832	+	0.747881i	$0.268929\pi$
$450$	0	0
$451$	−7.19966	−0.339019
$452$	0	0
$453$	17.3372	0.814574
$454$	0	0
$455$	0	0
$456$	0	0
$457$	8.34821	0.390513	0.195256	−	0.980752i	$-0.437446\pi$
$457$	8.34821	0.390513	0.195256	+	0.980752i	$0.437446\pi$
$458$	0	0
$459$	−10.0240	−0.467879
$460$	0	0
$461$	12.1543	0.566084	0.283042	−	0.959107i	$-0.408656\pi$
$461$	12.1543	0.566084	0.283042	+	0.959107i	$0.408656\pi$
$462$	0	0
$463$	30.3490	1.41044	0.705220	−	0.708989i	$-0.250848\pi$
$463$	30.3490	1.41044	0.705220	+	0.708989i	$0.250848\pi$
$464$	0	0
$465$	−6.45396	−0.299295
$466$	0	0
$467$	35.7902	1.65617	0.828086	−	0.560602i	$-0.189430\pi$
$467$	35.7902	1.65617	0.828086	+	0.560602i	$0.189430\pi$
$468$	0	0
$469$	−3.23591	−0.149420
$470$	0	0
$471$	2.50591	0.115466
$472$	0	0
$473$	−13.5598	−0.623481
$474$	0	0
$475$	43.8542	2.01217
$476$	0	0
$477$	−4.02878	−0.184465
$478$	0	0
$479$	16.9744	0.775580	0.387790	−	0.921748i	$-0.373239\pi$
$479$	16.9744	0.775580	0.387790	+	0.921748i	$0.373239\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	6.10273	0.277684
$484$	0	0
$485$	40.6981	1.84801
$486$	0	0
$487$	8.30630	0.376394	0.188197	−	0.982131i	$-0.439736\pi$
$487$	8.30630	0.376394	0.188197	+	0.982131i	$0.439736\pi$
$488$	0	0
$489$	16.2966	0.736957
$490$	0	0
$491$	−39.0498	−1.76229	−0.881146	−	0.472844i	$-0.843228\pi$
$491$	−39.0498	−1.76229	−0.881146	+	0.472844i	$0.843228\pi$
$492$	0	0
$493$	9.06354	0.408201
$494$	0	0
$495$	−11.7762	−0.529299
$496$	0	0
$497$	−42.3155	−1.89811
$498$	0	0
$499$	−32.7546	−1.46630	−0.733149	−	0.680068i	$-0.761950\pi$
$499$	−32.7546	−1.46630	−0.733149	+	0.680068i	$0.761950\pi$
$500$	0	0
$501$	−23.6135	−1.05498
$502$	0	0
$503$	10.1365	0.451966	0.225983	−	0.974131i	$-0.427441\pi$
$503$	10.1365	0.451966	0.225983	+	0.974131i	$0.427441\pi$
$504$	0	0
$505$	−21.8215	−0.971044
$506$	0	0
$507$	0	0
$508$	0	0
$509$	41.5089	1.83985	0.919926	−	0.392093i	$-0.128249\pi$
$509$	41.5089	1.83985	0.919926	+	0.392093i	$0.128249\pi$
$510$	0	0
$511$	16.9219	0.748583
$512$	0	0
$513$	−36.1917	−1.59790
$514$	0	0
$515$	−23.7012	−1.04440
$516$	0	0
$517$	5.00313	0.220037
$518$	0	0
$519$	−21.2480	−0.932686
$520$	0	0
$521$	38.8591	1.70245	0.851223	−	0.524804i	$-0.175861\pi$
$521$	38.8591	1.70245	0.851223	+	0.524804i	$0.175861\pi$
$522$	0	0
$523$	−28.3401	−1.23923	−0.619613	−	0.784907i	$-0.712711\pi$
$523$	−28.3401	−1.23923	−0.619613	+	0.784907i	$0.712711\pi$
$524$	0	0
$525$	−30.1530	−1.31598
$526$	0	0
$527$	2.39743	0.104434
$528$	0	0
$529$	−21.0776	−0.916417
$530$	0	0
$531$	−14.5729	−0.632410
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0.549168	0.0237426
$536$	0	0
$537$	1.76722	0.0762613
$538$	0	0
$539$	−9.76989	−0.420819
$540$	0	0
$541$	34.4879	1.48275	0.741376	−	0.671090i	$-0.234174\pi$
$541$	34.4879	1.48275	0.741376	+	0.671090i	$0.234174\pi$
$542$	0	0
$543$	3.35050	0.143784
$544$	0	0
$545$	−37.6200	−1.61147
$546$	0	0
$547$	27.7469	1.18637	0.593186	−	0.805065i	$-0.297870\pi$
$547$	27.7469	1.18637	0.593186	+	0.805065i	$0.297870\pi$
$548$	0	0
$549$	11.6522	0.497302
$550$	0	0
$551$	32.7240	1.39409
$552$	0	0
$553$	−35.4024	−1.50547
$554$	0	0
$555$	53.9737	2.29105
$556$	0	0
$557$	14.0308	0.594505	0.297253	−	0.954799i	$-0.403930\pi$
$557$	14.0308	0.594505	0.297253	+	0.954799i	$0.403930\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−7.80395	−0.329483
$562$	0	0
$563$	41.6536	1.75549	0.877745	−	0.479127i	$-0.159047\pi$
$563$	41.6536	1.75549	0.877745	+	0.479127i	$0.159047\pi$
$564$	0	0
$565$	−41.2989	−1.73746
$566$	0	0
$567$	14.6219	0.614062
$568$	0	0
$569$	−36.1603	−1.51592	−0.757959	−	0.652303i	$-0.773803\pi$
$569$	−36.1603	−1.51592	−0.757959	+	0.652303i	$0.773803\pi$
$570$	0	0
$571$	−23.1786	−0.969994	−0.484997	−	0.874516i	$-0.661179\pi$
$571$	−23.1786	−0.969994	−0.484997	+	0.874516i	$0.661179\pi$
$572$	0	0
$573$	−9.30167	−0.388583
$574$	0	0
$575$	−9.49843	−0.396112
$576$	0	0
$577$	−2.75597	−0.114733	−0.0573663	−	0.998353i	$-0.518270\pi$
$577$	−2.75597	−0.114733	−0.0573663	+	0.998353i	$0.518270\pi$
$578$	0	0
$579$	−7.43172	−0.308852
$580$	0	0
$581$	−45.5514	−1.88979
$582$	0	0
$583$	−11.8685	−0.491545
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−16.1414	−0.666227	−0.333113	−	0.942887i	$-0.608099\pi$
$587$	−16.1414	−0.666227	−0.333113	+	0.942887i	$0.608099\pi$
$588$	0	0
$589$	8.65596	0.356663
$590$	0	0
$591$	−11.9591	−0.491933
$592$	0	0
$593$	−13.0028	−0.533961	−0.266981	−	0.963702i	$-0.586026\pi$
$593$	−13.0028	−0.533961	−0.266981	+	0.963702i	$0.586026\pi$
$594$	0	0
$595$	19.3759	0.794335
$596$	0	0
$597$	−15.0629	−0.616482
$598$	0	0
$599$	9.58930	0.391808	0.195904	−	0.980623i	$-0.437236\pi$
$599$	9.58930	0.391808	0.195904	+	0.980623i	$0.437236\pi$
$600$	0	0
$601$	−20.9516	−0.854634	−0.427317	−	0.904102i	$-0.640541\pi$
$601$	−20.9516	−0.854634	−0.427317	+	0.904102i	$0.640541\pi$
$602$	0	0
$603$	−1.09843	−0.0447315
$604$	0	0
$605$	3.17536	0.129097
$606$	0	0
$607$	7.53543	0.305854	0.152927	−	0.988238i	$-0.451130\pi$
$607$	7.53543	0.305854	0.152927	+	0.988238i	$0.451130\pi$
$608$	0	0
$609$	−22.5002	−0.911752
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−7.81744	−0.315743	−0.157872	−	0.987460i	$-0.550463\pi$
$613$	−7.81744	−0.315743	−0.157872	+	0.987460i	$0.550463\pi$
$614$	0	0
$615$	−10.8250	−0.436505
$616$	0	0
$617$	10.6973	0.430655	0.215328	−	0.976542i	$-0.430918\pi$
$617$	10.6973	0.430655	0.215328	+	0.976542i	$0.430918\pi$
$618$	0	0
$619$	2.18518	0.0878296	0.0439148	−	0.999035i	$-0.486017\pi$
$619$	2.18518	0.0878296	0.0439148	+	0.999035i	$0.486017\pi$
$620$	0	0
$621$	7.83879	0.314560
$622$	0	0
$623$	−14.6833	−0.588274
$624$	0	0
$625$	−12.3222	−0.492887
$626$	0	0
$627$	−28.1762	−1.12525
$628$	0	0
$629$	−20.0494	−0.799423
$630$	0	0
$631$	13.0493	0.519486	0.259743	−	0.965678i	$-0.416362\pi$
$631$	13.0493	0.519486	0.259743	+	0.965678i	$0.416362\pi$
$632$	0	0
$633$	−15.0261	−0.597233
$634$	0	0
$635$	43.1636	1.71290
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−14.3640	−0.568231
$640$	0	0
$641$	−2.86509	−0.113164	−0.0565821	−	0.998398i	$-0.518020\pi$
$641$	−2.86509	−0.113164	−0.0565821	+	0.998398i	$0.518020\pi$
$642$	0	0
$643$	34.0162	1.34147	0.670734	−	0.741698i	$-0.265979\pi$
$643$	34.0162	1.34147	0.670734	+	0.741698i	$0.265979\pi$
$644$	0	0
$645$	−20.3877	−0.802766
$646$	0	0
$647$	9.16799	0.360431	0.180215	−	0.983627i	$-0.442321\pi$
$647$	9.16799	0.360431	0.180215	+	0.983627i	$0.442321\pi$
$648$	0	0
$649$	−42.9309	−1.68518
$650$	0	0
$651$	−5.95161	−0.233262
$652$	0	0
$653$	12.9309	0.506024	0.253012	−	0.967463i	$-0.418579\pi$
$653$	12.9309	0.506024	0.253012	+	0.967463i	$0.418579\pi$
$654$	0	0
$655$	−32.1489	−1.25616
$656$	0	0
$657$	5.74416	0.224101
$658$	0	0
$659$	45.2112	1.76118	0.880590	−	0.473879i	$-0.157147\pi$
$659$	45.2112	1.76118	0.880590	+	0.473879i	$0.157147\pi$
$660$	0	0
$661$	17.8643	0.694839	0.347419	−	0.937710i	$-0.387058\pi$
$661$	17.8643	0.694839	0.347419	+	0.937710i	$0.387058\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	69.9569	2.71281
$666$	0	0
$667$	−7.08773	−0.274438
$668$	0	0
$669$	10.0514	0.388611
$670$	0	0
$671$	34.3266	1.32516
$672$	0	0
$673$	−31.8330	−1.22707	−0.613536	−	0.789667i	$-0.710253\pi$
$673$	−31.8330	−1.22707	−0.613536	+	0.789667i	$0.710253\pi$
$674$	0	0
$675$	−38.7307	−1.49075
$676$	0	0
$677$	13.6123	0.523162	0.261581	−	0.965181i	$-0.415756\pi$
$677$	13.6123	0.523162	0.261581	+	0.965181i	$0.415756\pi$
$678$	0	0
$679$	37.5303	1.44028
$680$	0	0
$681$	−13.3084	−0.509979
$682$	0	0
$683$	−8.80950	−0.337086	−0.168543	−	0.985694i	$-0.553906\pi$
$683$	−8.80950	−0.337086	−0.168543	+	0.985694i	$0.553906\pi$
$684$	0	0
$685$	−17.7387	−0.677760
$686$	0	0
$687$	4.05998	0.154898
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−20.0781	−0.763806	−0.381903	−	0.924203i	$-0.624731\pi$
$691$	−20.0781	−0.763806	−0.381903	+	0.924203i	$0.624731\pi$
$692$	0	0
$693$	−10.8595	−0.412520
$694$	0	0
$695$	11.0147	0.417812
$696$	0	0
$697$	4.02112	0.152311
$698$	0	0
$699$	−18.2354	−0.689728
$700$	0	0
$701$	48.6918	1.83906	0.919532	−	0.393014i	$-0.128568\pi$
$701$	48.6918	1.83906	0.919532	+	0.393014i	$0.128568\pi$
$702$	0	0
$703$	−72.3887	−2.73019
$704$	0	0
$705$	7.52240	0.283310
$706$	0	0
$707$	−20.1230	−0.756803
$708$	0	0
$709$	1.70206	0.0639222	0.0319611	−	0.999489i	$-0.489825\pi$
$709$	1.70206	0.0639222	0.0319611	+	0.999489i	$0.489825\pi$
$710$	0	0
$711$	−12.0174	−0.450687
$712$	0	0
$713$	−1.87481	−0.0702120
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−12.5158	−0.467412
$718$	0	0
$719$	−11.3548	−0.423464	−0.211732	−	0.977328i	$-0.567910\pi$
$719$	−11.3548	−0.423464	−0.211732	+	0.977328i	$0.567910\pi$
$720$	0	0
$721$	−21.8564	−0.813975
$722$	0	0
$723$	−23.3500	−0.868395
$724$	0	0
$725$	35.0198	1.30060
$726$	0	0
$727$	−14.9882	−0.555881	−0.277940	−	0.960598i	$-0.589652\pi$
$727$	−14.9882	−0.555881	−0.277940	+	0.960598i	$0.589652\pi$
$728$	0	0
$729$	28.4798	1.05481
$730$	0	0
$731$	7.57336	0.280111
$732$	0	0
$733$	−35.0688	−1.29529	−0.647647	−	0.761940i	$-0.724247\pi$
$733$	−35.0688	−1.29529	−0.647647	+	0.761940i	$0.724247\pi$
$734$	0	0
$735$	−14.6894	−0.541827
$736$	0	0
$737$	−3.23591	−0.119196
$738$	0	0
$739$	32.1414	1.18234	0.591170	−	0.806547i	$-0.298666\pi$
$739$	32.1414	1.18234	0.591170	+	0.806547i	$0.298666\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	19.8992	0.730029	0.365015	−	0.931002i	$-0.381064\pi$
$743$	19.8992	0.730029	0.365015	+	0.931002i	$0.381064\pi$
$744$	0	0
$745$	−26.2321	−0.961069
$746$	0	0
$747$	−15.4624	−0.565740
$748$	0	0
$749$	0.506423	0.0185043
$750$	0	0
$751$	−24.1216	−0.880210	−0.440105	−	0.897946i	$-0.645059\pi$
$751$	−24.1216	−0.880210	−0.440105	+	0.897946i	$0.645059\pi$
$752$	0	0
$753$	41.4854	1.51181
$754$	0	0
$755$	−43.0455	−1.56658
$756$	0	0
$757$	2.76409	0.100463	0.0502313	−	0.998738i	$-0.484004\pi$
$757$	2.76409	0.100463	0.0502313	+	0.998738i	$0.484004\pi$
$758$	0	0
$759$	6.10273	0.221515
$760$	0	0
$761$	4.46045	0.161691	0.0808457	−	0.996727i	$-0.474238\pi$
$761$	4.46045	0.161691	0.0808457	+	0.996727i	$0.474238\pi$
$762$	0	0
$763$	−34.6918	−1.25593
$764$	0	0
$765$	6.57716	0.237798
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−32.0278	−1.15495	−0.577476	−	0.816408i	$-0.695962\pi$
$769$	−32.0278	−1.15495	−0.577476	+	0.816408i	$0.695962\pi$
$770$	0	0
$771$	5.75686	0.207328
$772$	0	0
$773$	−25.2045	−0.906543	−0.453272	−	0.891372i	$-0.649743\pi$
$773$	−25.2045	−0.906543	−0.453272	+	0.891372i	$0.649743\pi$
$774$	0	0
$775$	9.26324	0.332745
$776$	0	0
$777$	49.7726	1.78558
$778$	0	0
$779$	14.5183	0.520172
$780$	0	0
$781$	−42.3155	−1.51417
$782$	0	0
$783$	−28.9009	−1.03283
$784$	0	0
$785$	−6.22175	−0.222064
$786$	0	0
$787$	38.1327	1.35928	0.679642	−	0.733544i	$-0.262135\pi$
$787$	38.1327	1.35928	0.679642	+	0.733544i	$0.262135\pi$
$788$	0	0
$789$	−13.6000	−0.484173
$790$	0	0
$791$	−38.0843	−1.35412
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−17.8448	−0.632890
$796$	0	0
$797$	5.31326	0.188205	0.0941026	−	0.995563i	$-0.470002\pi$
$797$	5.31326	0.188205	0.0941026	+	0.995563i	$0.470002\pi$
$798$	0	0
$799$	−2.79432	−0.0988561
$800$	0	0
$801$	−4.98425	−0.176110
$802$	0	0
$803$	16.9219	0.597162
$804$	0	0
$805$	−15.1521	−0.534040
$806$	0	0
$807$	−9.59364	−0.337712
$808$	0	0
$809$	−17.1755	−0.603857	−0.301929	−	0.953331i	$-0.597630\pi$
$809$	−17.1755	−0.603857	−0.301929	+	0.953331i	$0.597630\pi$
$810$	0	0
$811$	12.7281	0.446943	0.223472	−	0.974710i	$-0.428261\pi$
$811$	12.7281	0.446943	0.223472	+	0.974710i	$0.428261\pi$
$812$	0	0
$813$	−17.6372	−0.618563
$814$	0	0
$815$	−40.4617	−1.41731
$816$	0	0
$817$	27.3437	0.956636
$818$	0	0
$819$	0	0
$820$	0	0
$821$	19.0875	0.666157	0.333078	−	0.942899i	$-0.391913\pi$
$821$	19.0875	0.666157	0.333078	+	0.942899i	$0.391913\pi$
$822$	0	0
$823$	−44.8743	−1.56422	−0.782111	−	0.623140i	$-0.785857\pi$
$823$	−44.8743	−1.56422	−0.782111	+	0.623140i	$0.785857\pi$
$824$	0	0
$825$	−30.1530	−1.04979
$826$	0	0
$827$	29.7249	1.03364	0.516819	−	0.856095i	$-0.327116\pi$
$827$	29.7249	1.03364	0.516819	+	0.856095i	$0.327116\pi$
$828$	0	0
$829$	46.9910	1.63207	0.816033	−	0.578005i	$-0.196169\pi$
$829$	46.9910	1.63207	0.816033	+	0.578005i	$0.196169\pi$
$830$	0	0
$831$	11.3629	0.394174
$832$	0	0
$833$	5.45663	0.189061
$834$	0	0
$835$	58.6285	2.02892
$836$	0	0
$837$	−7.64469	−0.264239
$838$	0	0
$839$	−45.7774	−1.58041	−0.790206	−	0.612842i	$-0.790026\pi$
$839$	−45.7774	−1.58041	−0.790206	+	0.612842i	$0.790026\pi$
$840$	0	0
$841$	−2.86822	−0.0989042
$842$	0	0
$843$	11.1037	0.382431
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.92820	0.100614
$848$	0	0
$849$	30.5485	1.04842
$850$	0	0
$851$	15.6788	0.537461
$852$	0	0
$853$	−35.3624	−1.21079	−0.605393	−	0.795927i	$-0.706984\pi$
$853$	−35.3624	−1.21079	−0.605393	+	0.795927i	$0.706984\pi$
$854$	0	0
$855$	23.7469	0.812127
$856$	0	0
$857$	43.3068	1.47933	0.739666	−	0.672975i	$-0.234984\pi$
$857$	43.3068	1.47933	0.739666	+	0.672975i	$0.234984\pi$
$858$	0	0
$859$	−10.2009	−0.348049	−0.174025	−	0.984741i	$-0.555677\pi$
$859$	−10.2009	−0.348049	−0.174025	+	0.984741i	$0.555677\pi$
$860$	0	0
$861$	−9.98239	−0.340199
$862$	0	0
$863$	−4.26879	−0.145311	−0.0726556	−	0.997357i	$-0.523147\pi$
$863$	−4.26879	−0.145311	−0.0726556	+	0.997357i	$0.523147\pi$
$864$	0	0
$865$	52.7554	1.79374
$866$	0	0
$867$	−19.2120	−0.652475
$868$	0	0
$869$	−35.4024	−1.20095
$870$	0	0
$871$	0	0
$872$	0	0
$873$	12.7397	0.431173
$874$	0	0
$875$	20.2238	0.683691
$876$	0	0
$877$	−8.48311	−0.286454	−0.143227	−	0.989690i	$-0.545748\pi$
$877$	−8.48311	−0.286454	−0.143227	+	0.989690i	$0.545748\pi$
$878$	0	0
$879$	−11.1629	−0.376516
$880$	0	0
$881$	−51.1005	−1.72162	−0.860810	−	0.508926i	$-0.830043\pi$
$881$	−51.1005	−1.72162	−0.860810	+	0.508926i	$0.830043\pi$
$882$	0	0
$883$	−21.5628	−0.725645	−0.362822	−	0.931858i	$-0.618187\pi$
$883$	−21.5628	−0.725645	−0.362822	+	0.931858i	$0.618187\pi$
$884$	0	0
$885$	−64.5482	−2.16977
$886$	0	0
$887$	−7.07101	−0.237421	−0.118711	−	0.992929i	$-0.537876\pi$
$887$	−7.07101	−0.237421	−0.118711	+	0.992929i	$0.537876\pi$
$888$	0	0
$889$	39.8039	1.33498
$890$	0	0
$891$	14.6219	0.489852
$892$	0	0
$893$	−10.0889	−0.337614
$894$	0	0
$895$	−4.38772	−0.146665
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.91223	0.230536
$900$	0	0
$901$	6.62876	0.220836
$902$	0	0
$903$	−18.8008	−0.625652
$904$	0	0
$905$	−8.31873	−0.276524
$906$	0	0
$907$	40.4406	1.34281	0.671405	−	0.741091i	$-0.265691\pi$
$907$	40.4406	1.34281	0.671405	+	0.741091i	$0.265691\pi$
$908$	0	0
$909$	−6.83076	−0.226562
$910$	0	0
$911$	5.91974	0.196130	0.0980649	−	0.995180i	$-0.468735\pi$
$911$	5.91974	0.196130	0.0980649	+	0.995180i	$0.468735\pi$
$912$	0	0
$913$	−45.5514	−1.50753
$914$	0	0
$915$	51.6113	1.70622
$916$	0	0
$917$	−29.6466	−0.979016
$918$	0	0
$919$	50.4177	1.66313	0.831563	−	0.555431i	$-0.187447\pi$
$919$	50.4177	1.66313	0.831563	+	0.555431i	$0.187447\pi$
$920$	0	0
$921$	−23.1936	−0.764256
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−77.4672	−2.54711
$926$	0	0
$927$	−7.41916	−0.243677
$928$	0	0
$929$	41.4449	1.35976	0.679881	−	0.733323i	$-0.262031\pi$
$929$	41.4449	1.35976	0.679881	+	0.733323i	$0.262031\pi$
$930$	0	0
$931$	19.7012	0.645681
$932$	0	0
$933$	−5.25885	−0.172167
$934$	0	0
$935$	19.3759	0.633660
$936$	0	0
$937$	5.67515	0.185399	0.0926995	−	0.995694i	$-0.470450\pi$
$937$	5.67515	0.185399	0.0926995	+	0.995694i	$0.470450\pi$
$938$	0	0
$939$	−37.0171	−1.20801
$940$	0	0
$941$	−18.7044	−0.609744	−0.304872	−	0.952393i	$-0.598614\pi$
$941$	−18.7044	−0.609744	−0.304872	+	0.952393i	$0.598614\pi$
$942$	0	0
$943$	−3.14453	−0.102400
$944$	0	0
$945$	−61.7839	−2.00983
$946$	0	0
$947$	−36.7830	−1.19529	−0.597643	−	0.801762i	$-0.703896\pi$
$947$	−36.7830	−1.19529	−0.597643	+	0.801762i	$0.703896\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−25.8415	−0.837967
$952$	0	0
$953$	−18.3396	−0.594077	−0.297039	−	0.954865i	$-0.595999\pi$
$953$	−18.3396	−0.594077	−0.297039	+	0.954865i	$0.595999\pi$
$954$	0	0
$955$	23.0945	0.747320
$956$	0	0
$957$	−22.5002	−0.727327
$958$	0	0
$959$	−16.3580	−0.528226
$960$	0	0
$961$	−29.1716	−0.941020
$962$	0	0
$963$	0.171905	0.00553957
$964$	0	0
$965$	18.4517	0.593982
$966$	0	0
$967$	−17.0558	−0.548478	−0.274239	−	0.961662i	$-0.588426\pi$
$967$	−17.0558	−0.548478	−0.274239	+	0.961662i	$0.588426\pi$
$968$	0	0
$969$	15.7369	0.505541
$970$	0	0
$971$	−18.6454	−0.598358	−0.299179	−	0.954197i	$-0.596713\pi$
$971$	−18.6454	−0.598358	−0.299179	+	0.954197i	$0.596713\pi$
$972$	0	0
$973$	10.1574	0.325631
$974$	0	0
$975$	0	0
$976$	0	0
$977$	16.1752	0.517489	0.258745	−	0.965946i	$-0.416691\pi$
$977$	16.1752	0.517489	0.258745	+	0.965946i	$0.416691\pi$
$978$	0	0
$979$	−14.6833	−0.469280
$980$	0	0
$981$	−11.7762	−0.375984
$982$	0	0
$983$	−55.0587	−1.75610	−0.878050	−	0.478568i	$-0.841156\pi$
$983$	−55.0587	−1.75610	−0.878050	+	0.478568i	$0.841156\pi$
$984$	0	0
$985$	29.6925	0.946083
$986$	0	0
$987$	6.93689	0.220804
$988$	0	0
$989$	−5.92241	−0.188322
$990$	0	0
$991$	37.0728	1.17766	0.588828	−	0.808258i	$-0.299590\pi$
$991$	37.0728	1.17766	0.588828	+	0.808258i	$0.299590\pi$
$992$	0	0
$993$	−18.4402	−0.585181
$994$	0	0
$995$	37.3986	1.18562
$996$	0	0
$997$	48.6633	1.54118	0.770591	−	0.637330i	$-0.219961\pi$
$997$	48.6633	1.54118	0.770591	+	0.637330i	$0.219961\pi$
$998$	0	0
$999$	63.9316	2.02271

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2704.2.a.bd.1.3		4
4.3	odd	2		1352.2.a.k.1.2		4
13.5	odd	4		2704.2.f.q.337.6		8
13.6	odd	12		208.2.w.c.49.2		8
13.8	odd	4		2704.2.f.q.337.5		8
13.11	odd	12		208.2.w.c.17.2		8
13.12	even	2		2704.2.a.be.1.3		4
39.11	even	12		1872.2.by.n.433.4		8
39.32	even	12		1872.2.by.n.1297.1		8
52.3	odd	6		1352.2.i.k.529.3		8
52.7	even	12		1352.2.o.f.361.3		8
52.11	even	12		104.2.o.a.17.3	✓	8
52.15	even	12		1352.2.o.f.1161.3		8
52.19	even	12		104.2.o.a.49.3	yes	8
52.23	odd	6		1352.2.i.l.529.3		8
52.31	even	4		1352.2.f.f.337.4		8
52.35	odd	6		1352.2.i.k.1329.3		8
52.43	odd	6		1352.2.i.l.1329.3		8
52.47	even	4		1352.2.f.f.337.3		8
52.51	odd	2		1352.2.a.l.1.2		4
104.11	even	12		832.2.w.g.641.2		8
104.19	even	12		832.2.w.g.257.2		8
104.37	odd	12		832.2.w.i.641.3		8
104.45	odd	12		832.2.w.i.257.3		8
156.11	odd	12		936.2.bi.b.433.4		8
156.71	odd	12		936.2.bi.b.361.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.o.a.17.3	✓	8	52.11	even	12
104.2.o.a.49.3	yes	8	52.19	even	12
208.2.w.c.17.2		8	13.11	odd	12
208.2.w.c.49.2		8	13.6	odd	12
832.2.w.g.257.2		8	104.19	even	12
832.2.w.g.641.2		8	104.11	even	12
832.2.w.i.257.3		8	104.45	odd	12
832.2.w.i.641.3		8	104.37	odd	12
936.2.bi.b.361.1		8	156.71	odd	12
936.2.bi.b.433.4		8	156.11	odd	12
1352.2.a.k.1.2		4	4.3	odd	2
1352.2.a.l.1.2		4	52.51	odd	2
1352.2.f.f.337.3		8	52.47	even	4
1352.2.f.f.337.4		8	52.31	even	4
1352.2.i.k.529.3		8	52.3	odd	6
1352.2.i.k.1329.3		8	52.35	odd	6
1352.2.i.l.529.3		8	52.23	odd	6
1352.2.i.l.1329.3		8	52.43	odd	6
1352.2.o.f.361.3		8	52.7	even	12
1352.2.o.f.1161.3		8	52.15	even	12
1872.2.by.n.433.4		8	39.11	even	12
1872.2.by.n.1297.1		8	39.32	even	12
2704.2.a.bd.1.3		4	1.1	even	1	trivial
2704.2.a.be.1.3		4	13.12	even	2
2704.2.f.q.337.5		8	13.8	odd	4
2704.2.f.q.337.6		8	13.5	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2704 = 2^{4} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2704.a (trivial)

\(f(q)\)	\(=\)	\(q+1.38651 q^{3} -3.44247 q^{5} -3.17452 q^{7} -1.07759 q^{9} -3.17452 q^{11} -4.77302 q^{15} +1.77302 q^{17} +6.40150 q^{19} -4.40150 q^{21} -1.38651 q^{23} +6.85061 q^{25} -5.65362 q^{27} +5.11192 q^{29} +1.35218 q^{31} -4.40150 q^{33} +10.9282 q^{35} -11.3081 q^{37} +2.26795 q^{41} +4.27145 q^{43} +3.70958 q^{45} -1.57603 q^{47} +3.07759 q^{49} +2.45831 q^{51} +3.73869 q^{53} +10.9282 q^{55} +8.87574 q^{57} +13.5236 q^{59} -10.8131 q^{61} +3.42084 q^{63} +1.01934 q^{67} -1.92241 q^{69} +13.3297 q^{71} -5.33055 q^{73} +9.49843 q^{75} +10.0776 q^{77} +11.1521 q^{79} -4.60602 q^{81} +14.3490 q^{83} -6.10356 q^{85} +7.08773 q^{87} +4.62535 q^{89} +1.87481 q^{93} -22.0370 q^{95} -11.8223 q^{97} +3.42084 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{3} - 4 q^{5} + 4 q^{7} + 6 q^{9} + 4 q^{11} - 12 q^{15} + 16 q^{19} - 8 q^{21} - 2 q^{23} + 10 q^{25} + 14 q^{27} + 8 q^{29} + 4 q^{31} - 8 q^{33} + 16 q^{35} - 12 q^{37} + 16 q^{41} - 6 q^{43}+ \cdots + 8 q^{99}+O(q^{100}) \) 4 * q + 2 * q^3 - 4 * q^5 + 4 * q^7 + 6 * q^9 + 4 * q^11 - 12 * q^15 + 16 * q^19 - 8 * q^21 - 2 * q^23 + 10 * q^25 + 14 * q^27 + 8 * q^29 + 4 * q^31 - 8 * q^33 + 16 * q^35 - 12 * q^37 + 16 * q^41 - 6 * q^43 - 28 * q^45 + 20 * q^47 + 2 * q^49 + 34 * q^51 + 10 * q^53 + 16 * q^55 - 16 * q^57 + 4 * q^59 + 4 * q^61 + 8 * q^63 + 8 * q^67 - 18 * q^69 + 16 * q^71 - 24 * q^73 + 22 * q^75 + 30 * q^77 - 8 * q^79 + 20 * q^81 + 24 * q^83 - 20 * q^85 - 10 * q^87 - 16 * q^89 + 16 * q^93 - 16 * q^95 - 32 * q^97 + 8 * q^99

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