LMFDB - Embedded newform 1352.2.a.d.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1352,2,Mod(1,1352)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1352, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1352.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1352 = 2^{3} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1352.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-1,0,-1,0,5] 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:	$10.7957743533$
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:	$\Z[a_1, a_2, a_3]$
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			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	1352.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155 q^{3} +1.56155 q^{5} +0.438447 q^{7} -0.561553 q^{9} +5.12311 q^{11} +2.43845 q^{15} +3.56155 q^{17} -2.00000 q^{19} +0.684658 q^{21} +3.12311 q^{23} -2.56155 q^{25} -5.56155 q^{27} -5.12311 q^{29} +5.12311 q^{31} +8.00000 q^{33} +0.684658 q^{35} +9.56155 q^{37} +8.00000 q^{41} -9.56155 q^{43} -0.876894 q^{45} -7.56155 q^{47} -6.80776 q^{49} +5.56155 q^{51} +12.2462 q^{53} +8.00000 q^{55} -3.12311 q^{57} +10.0000 q^{59} +2.87689 q^{61} -0.246211 q^{63} +9.12311 q^{67} +4.87689 q^{69} -6.68466 q^{71} -9.36932 q^{73} -4.00000 q^{75} +2.24621 q^{77} -11.1231 q^{79} -7.00000 q^{81} -8.24621 q^{83} +5.56155 q^{85} -8.00000 q^{87} -3.12311 q^{89} +8.00000 q^{93} -3.12311 q^{95} -6.24621 q^{97} -2.87689 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - q^{5} + 5 q^{7} + 3 q^{9} + 2 q^{11} + 9 q^{15} + 3 q^{17} - 4 q^{19} - 11 q^{21} - 2 q^{23} - q^{25} - 7 q^{27} - 2 q^{29} + 2 q^{31} + 16 q^{33} - 11 q^{35} + 15 q^{37} + 16 q^{41} - 15 q^{43}+ \cdots - 14 q^{99}+O(q^{100}) $ 2 * q - q^3 - q^5 + 5 * q^7 + 3 * q^9 + 2 * q^11 + 9 * q^15 + 3 * q^17 - 4 * q^19 - 11 * q^21 - 2 * q^23 - q^25 - 7 * q^27 - 2 * q^29 + 2 * q^31 + 16 * q^33 - 11 * q^35 + 15 * q^37 + 16 * q^41 - 15 * q^43 - 10 * q^45 - 11 * q^47 + 7 * q^49 + 7 * q^51 + 8 * q^53 + 16 * q^55 + 2 * q^57 + 20 * q^59 + 14 * q^61 + 16 * q^63 + 10 * q^67 + 18 * q^69 - q^71 + 6 * q^73 - 8 * q^75 - 12 * q^77 - 14 * q^79 - 14 * q^81 + 7 * q^85 - 16 * q^87 + 2 * q^89 + 16 * q^93 + 2 * q^95 + 4 * q^97 - 14 * 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
$3$	1.56155	0.901563	0.450781	−	0.892634i	$-0.351145\pi$
$3$	1.56155	0.901563	0.450781	+	0.892634i	$0.351145\pi$
$4$	0	0
$5$	1.56155	0.698348	0.349174	−	0.937058i	$-0.386462\pi$
$5$	1.56155	0.698348	0.349174	+	0.937058i	$0.386462\pi$
$6$	0	0
$7$	0.438447	0.165717	0.0828587	−	0.996561i	$-0.473595\pi$
$7$	0.438447	0.165717	0.0828587	+	0.996561i	$0.473595\pi$
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	5.12311	1.54467	0.772337	−	0.635213i	$-0.219088\pi$
$11$	5.12311	1.54467	0.772337	+	0.635213i	$0.219088\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	2.43845	0.629604
$16$	0	0
$17$	3.56155	0.863803	0.431902	−	0.901921i	$-0.357843\pi$
$17$	3.56155	0.863803	0.431902	+	0.901921i	$0.357843\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	0.684658	0.149405
$22$	0	0
$23$	3.12311	0.651213	0.325606	−	0.945505i	$-0.394432\pi$
$23$	3.12311	0.651213	0.325606	+	0.945505i	$0.394432\pi$
$24$	0	0
$25$	−2.56155	−0.512311
$26$	0	0
$27$	−5.56155	−1.07032
$28$	0	0
$29$	−5.12311	−0.951337	−0.475668	−	0.879625i	$-0.657794\pi$
$29$	−5.12311	−0.951337	−0.475668	+	0.879625i	$0.657794\pi$
$30$	0	0
$31$	5.12311	0.920137	0.460068	−	0.887883i	$-0.347825\pi$
$31$	5.12311	0.920137	0.460068	+	0.887883i	$0.347825\pi$
$32$	0	0
$33$	8.00000	1.39262
$34$	0	0
$35$	0.684658	0.115728
$36$	0	0
$37$	9.56155	1.57191	0.785955	−	0.618284i	$-0.212172\pi$
$37$	9.56155	1.57191	0.785955	+	0.618284i	$0.212172\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	8.00000	1.24939	0.624695	−	0.780869i	$-0.285223\pi$
$41$	8.00000	1.24939	0.624695	+	0.780869i	$0.285223\pi$
$42$	0	0
$43$	−9.56155	−1.45812	−0.729062	−	0.684448i	$-0.760043\pi$
$43$	−9.56155	−1.45812	−0.729062	+	0.684448i	$0.760043\pi$
$44$	0	0
$45$	−0.876894	−0.130720
$46$	0	0
$47$	−7.56155	−1.10297	−0.551483	−	0.834186i	$-0.685938\pi$
$47$	−7.56155	−1.10297	−0.551483	+	0.834186i	$0.685938\pi$
$48$	0	0
$49$	−6.80776	−0.972538
$50$	0	0
$51$	5.56155	0.778773
$52$	0	0
$53$	12.2462	1.68215	0.841073	−	0.540921i	$-0.181924\pi$
$53$	12.2462	1.68215	0.841073	+	0.540921i	$0.181924\pi$
$54$	0	0
$55$	8.00000	1.07872
$56$	0	0
$57$	−3.12311	−0.413665
$58$	0	0
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	0	0
$61$	2.87689	0.368349	0.184174	−	0.982894i	$-0.441039\pi$
$61$	2.87689	0.368349	0.184174	+	0.982894i	$0.441039\pi$
$62$	0	0
$63$	−0.246211	−0.0310197
$64$	0	0
$65$	0	0
$66$	0	0
$67$	9.12311	1.11456	0.557282	−	0.830323i	$-0.311844\pi$
$67$	9.12311	1.11456	0.557282	+	0.830323i	$0.311844\pi$
$68$	0	0
$69$	4.87689	0.587109
$70$	0	0
$71$	−6.68466	−0.793323	−0.396662	−	0.917965i	$-0.629831\pi$
$71$	−6.68466	−0.793323	−0.396662	+	0.917965i	$0.629831\pi$
$72$	0	0
$73$	−9.36932	−1.09660	−0.548298	−	0.836283i	$-0.684724\pi$
$73$	−9.36932	−1.09660	−0.548298	+	0.836283i	$0.684724\pi$
$74$	0	0
$75$	−4.00000	−0.461880
$76$	0	0
$77$	2.24621	0.255980
$78$	0	0
$79$	−11.1231	−1.25145	−0.625724	−	0.780045i	$-0.715196\pi$
$79$	−11.1231	−1.25145	−0.625724	+	0.780045i	$0.715196\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−8.24621	−0.905139	−0.452570	−	0.891729i	$-0.649493\pi$
$83$	−8.24621	−0.905139	−0.452570	+	0.891729i	$0.649493\pi$
$84$	0	0
$85$	5.56155	0.603235
$86$	0	0
$87$	−8.00000	−0.857690
$88$	0	0
$89$	−3.12311	−0.331049	−0.165524	−	0.986206i	$-0.552932\pi$
$89$	−3.12311	−0.331049	−0.165524	+	0.986206i	$0.552932\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	8.00000	0.829561
$94$	0	0
$95$	−3.12311	−0.320424
$96$	0	0
$97$	−6.24621	−0.634207	−0.317103	−	0.948391i	$-0.602710\pi$
$97$	−6.24621	−0.634207	−0.317103	+	0.948391i	$0.602710\pi$
$98$	0	0
$99$	−2.87689	−0.289139
$100$	0	0
$101$	11.3693	1.13129	0.565645	−	0.824649i	$-0.308627\pi$
$101$	11.3693	1.13129	0.565645	+	0.824649i	$0.308627\pi$
$102$	0	0
$103$	−3.12311	−0.307729	−0.153864	−	0.988092i	$-0.549172\pi$
$103$	−3.12311	−0.307729	−0.153864	+	0.988092i	$0.549172\pi$
$104$	0	0
$105$	1.06913	0.104336
$106$	0	0
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	0	0
$109$	3.31534	0.317552	0.158776	−	0.987315i	$-0.449245\pi$
$109$	3.31534	0.317552	0.158776	+	0.987315i	$0.449245\pi$
$110$	0	0
$111$	14.9309	1.41718
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	0	0
$115$	4.87689	0.454773
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1.56155	0.143147
$120$	0	0
$121$	15.2462	1.38602
$122$	0	0
$123$	12.4924	1.12640
$124$	0	0
$125$	−11.8078	−1.05612
$126$	0	0
$127$	4.87689	0.432754	0.216377	−	0.976310i	$-0.430576\pi$
$127$	4.87689	0.432754	0.216377	+	0.976310i	$0.430576\pi$
$128$	0	0
$129$	−14.9309	−1.31459
$130$	0	0
$131$	−3.31534	−0.289663	−0.144831	−	0.989456i	$-0.546264\pi$
$131$	−3.31534	−0.289663	−0.144831	+	0.989456i	$0.546264\pi$
$132$	0	0
$133$	−0.876894	−0.0760364
$134$	0	0
$135$	−8.68466	−0.747456
$136$	0	0
$137$	−3.12311	−0.266825	−0.133412	−	0.991061i	$-0.542594\pi$
$137$	−3.12311	−0.266825	−0.133412	+	0.991061i	$0.542594\pi$
$138$	0	0
$139$	−7.80776	−0.662246	−0.331123	−	0.943588i	$-0.607427\pi$
$139$	−7.80776	−0.662246	−0.331123	+	0.943588i	$0.607427\pi$
$140$	0	0
$141$	−11.8078	−0.994393
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−8.00000	−0.664364
$146$	0	0
$147$	−10.6307	−0.876804
$148$	0	0
$149$	−7.12311	−0.583548	−0.291774	−	0.956487i	$-0.594245\pi$
$149$	−7.12311	−0.583548	−0.291774	+	0.956487i	$0.594245\pi$
$150$	0	0
$151$	11.5616	0.940866	0.470433	−	0.882436i	$-0.344098\pi$
$151$	11.5616	0.940866	0.470433	+	0.882436i	$0.344098\pi$
$152$	0	0
$153$	−2.00000	−0.161690
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	12.2462	0.977354	0.488677	−	0.872465i	$-0.337480\pi$
$157$	12.2462	0.977354	0.488677	+	0.872465i	$0.337480\pi$
$158$	0	0
$159$	19.1231	1.51656
$160$	0	0
$161$	1.36932	0.107917
$162$	0	0
$163$	−20.2462	−1.58581	−0.792903	−	0.609348i	$-0.791431\pi$
$163$	−20.2462	−1.58581	−0.792903	+	0.609348i	$0.791431\pi$
$164$	0	0
$165$	12.4924	0.972534
$166$	0	0
$167$	−6.87689	−0.532150	−0.266075	−	0.963952i	$-0.585727\pi$
$167$	−6.87689	−0.532150	−0.266075	+	0.963952i	$0.585727\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	1.12311	0.0858860
$172$	0	0
$173$	−4.24621	−0.322833	−0.161417	−	0.986886i	$-0.551606\pi$
$173$	−4.24621	−0.322833	−0.161417	+	0.986886i	$0.551606\pi$
$174$	0	0
$175$	−1.12311	−0.0848988
$176$	0	0
$177$	15.6155	1.17373
$178$	0	0
$179$	−17.5616	−1.31261	−0.656306	−	0.754495i	$-0.727882\pi$
$179$	−17.5616	−1.31261	−0.656306	+	0.754495i	$0.727882\pi$
$180$	0	0
$181$	8.24621	0.612936	0.306468	−	0.951881i	$-0.400853\pi$
$181$	8.24621	0.612936	0.306468	+	0.951881i	$0.400853\pi$
$182$	0	0
$183$	4.49242	0.332089
$184$	0	0
$185$	14.9309	1.09774
$186$	0	0
$187$	18.2462	1.33430
$188$	0	0
$189$	−2.43845	−0.177371
$190$	0	0
$191$	−22.2462	−1.60968	−0.804840	−	0.593492i	$-0.797749\pi$
$191$	−22.2462	−1.60968	−0.804840	+	0.593492i	$0.797749\pi$
$192$	0	0
$193$	23.6155	1.69988	0.849941	−	0.526877i	$-0.176637\pi$
$193$	23.6155	1.69988	0.849941	+	0.526877i	$0.176637\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−9.56155	−0.681232	−0.340616	−	0.940202i	$-0.610636\pi$
$197$	−9.56155	−0.681232	−0.340616	+	0.940202i	$0.610636\pi$
$198$	0	0
$199$	14.2462	1.00989	0.504944	−	0.863152i	$-0.331513\pi$
$199$	14.2462	1.00989	0.504944	+	0.863152i	$0.331513\pi$
$200$	0	0
$201$	14.2462	1.00485
$202$	0	0
$203$	−2.24621	−0.157653
$204$	0	0
$205$	12.4924	0.872509
$206$	0	0
$207$	−1.75379	−0.121897
$208$	0	0
$209$	−10.2462	−0.708745
$210$	0	0
$211$	−22.0540	−1.51826	−0.759129	−	0.650940i	$-0.774375\pi$
$211$	−22.0540	−1.51826	−0.759129	+	0.650940i	$0.774375\pi$
$212$	0	0
$213$	−10.4384	−0.715231
$214$	0	0
$215$	−14.9309	−1.01828
$216$	0	0
$217$	2.24621	0.152483
$218$	0	0
$219$	−14.6307	−0.988650
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−27.1771	−1.81991	−0.909956	−	0.414705i	$-0.863885\pi$
$223$	−27.1771	−1.81991	−0.909956	+	0.414705i	$0.863885\pi$
$224$	0	0
$225$	1.43845	0.0958965
$226$	0	0
$227$	−3.36932	−0.223629	−0.111815	−	0.993729i	$-0.535666\pi$
$227$	−3.36932	−0.223629	−0.111815	+	0.993729i	$0.535666\pi$
$228$	0	0
$229$	20.6847	1.36688	0.683440	−	0.730006i	$-0.260483\pi$
$229$	20.6847	1.36688	0.683440	+	0.730006i	$0.260483\pi$
$230$	0	0
$231$	3.50758	0.230782
$232$	0	0
$233$	3.56155	0.233325	0.116663	−	0.993172i	$-0.462780\pi$
$233$	3.56155	0.233325	0.116663	+	0.993172i	$0.462780\pi$
$234$	0	0
$235$	−11.8078	−0.770254
$236$	0	0
$237$	−17.3693	−1.12826
$238$	0	0
$239$	8.93087	0.577690	0.288845	−	0.957376i	$-0.406729\pi$
$239$	8.93087	0.577690	0.288845	+	0.957376i	$0.406729\pi$
$240$	0	0
$241$	−27.1231	−1.74715	−0.873576	−	0.486687i	$-0.838205\pi$
$241$	−27.1231	−1.74715	−0.873576	+	0.486687i	$0.838205\pi$
$242$	0	0
$243$	5.75379	0.369106
$244$	0	0
$245$	−10.6307	−0.679169
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−12.8769	−0.816040
$250$	0	0
$251$	−24.4924	−1.54595	−0.772974	−	0.634438i	$-0.781232\pi$
$251$	−24.4924	−1.54595	−0.772974	+	0.634438i	$0.781232\pi$
$252$	0	0
$253$	16.0000	1.00591
$254$	0	0
$255$	8.68466	0.543854
$256$	0	0
$257$	−0.930870	−0.0580661	−0.0290330	−	0.999578i	$-0.509243\pi$
$257$	−0.930870	−0.0580661	−0.0290330	+	0.999578i	$0.509243\pi$
$258$	0	0
$259$	4.19224	0.260493
$260$	0	0
$261$	2.87689	0.178075
$262$	0	0
$263$	−4.49242	−0.277015	−0.138507	−	0.990361i	$-0.544230\pi$
$263$	−4.49242	−0.277015	−0.138507	+	0.990361i	$0.544230\pi$
$264$	0	0
$265$	19.1231	1.17472
$266$	0	0
$267$	−4.87689	−0.298461
$268$	0	0
$269$	10.8769	0.663176	0.331588	−	0.943424i	$-0.392416\pi$
$269$	10.8769	0.663176	0.331588	+	0.943424i	$0.392416\pi$
$270$	0	0
$271$	14.6847	0.892029	0.446015	−	0.895026i	$-0.352843\pi$
$271$	14.6847	0.892029	0.446015	+	0.895026i	$0.352843\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−13.1231	−0.791353
$276$	0	0
$277$	−7.36932	−0.442779	−0.221390	−	0.975185i	$-0.571059\pi$
$277$	−7.36932	−0.442779	−0.221390	+	0.975185i	$0.571059\pi$
$278$	0	0
$279$	−2.87689	−0.172235
$280$	0	0
$281$	9.36932	0.558927	0.279463	−	0.960156i	$-0.409843\pi$
$281$	9.36932	0.558927	0.279463	+	0.960156i	$0.409843\pi$
$282$	0	0
$283$	−12.0000	−0.713326	−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000	−0.713326	−0.356663	+	0.934233i	$0.616086\pi$
$284$	0	0
$285$	−4.87689	−0.288882
$286$	0	0
$287$	3.50758	0.207046
$288$	0	0
$289$	−4.31534	−0.253844
$290$	0	0
$291$	−9.75379	−0.571777
$292$	0	0
$293$	−12.6847	−0.741046	−0.370523	−	0.928823i	$-0.620821\pi$
$293$	−12.6847	−0.741046	−0.370523	+	0.928823i	$0.620821\pi$
$294$	0	0
$295$	15.6155	0.909171
$296$	0	0
$297$	−28.4924	−1.65330
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−4.19224	−0.241636
$302$	0	0
$303$	17.7538	1.01993
$304$	0	0
$305$	4.49242	0.257235
$306$	0	0
$307$	18.0000	1.02731	0.513657	−	0.857996i	$-0.328290\pi$
$307$	18.0000	1.02731	0.513657	+	0.857996i	$0.328290\pi$
$308$	0	0
$309$	−4.87689	−0.277437
$310$	0	0
$311$	−19.1231	−1.08437	−0.542186	−	0.840259i	$-0.682403\pi$
$311$	−19.1231	−1.08437	−0.542186	+	0.840259i	$0.682403\pi$
$312$	0	0
$313$	4.43845	0.250876	0.125438	−	0.992101i	$-0.459966\pi$
$313$	4.43845	0.250876	0.125438	+	0.992101i	$0.459966\pi$
$314$	0	0
$315$	−0.384472	−0.0216625
$316$	0	0
$317$	−15.1231	−0.849398	−0.424699	−	0.905335i	$-0.639620\pi$
$317$	−15.1231	−0.849398	−0.424699	+	0.905335i	$0.639620\pi$
$318$	0	0
$319$	−26.2462	−1.46951
$320$	0	0
$321$	−6.24621	−0.348630
$322$	0	0
$323$	−7.12311	−0.396340
$324$	0	0
$325$	0	0
$326$	0	0
$327$	5.17708	0.286293
$328$	0	0
$329$	−3.31534	−0.182781
$330$	0	0
$331$	−0.246211	−0.0135330	−0.00676650	−	0.999977i	$-0.502154\pi$
$331$	−0.246211	−0.0135330	−0.00676650	+	0.999977i	$0.502154\pi$
$332$	0	0
$333$	−5.36932	−0.294237
$334$	0	0
$335$	14.2462	0.778354
$336$	0	0
$337$	−1.31534	−0.0716512	−0.0358256	−	0.999358i	$-0.511406\pi$
$337$	−1.31534	−0.0716512	−0.0358256	+	0.999358i	$0.511406\pi$
$338$	0	0
$339$	15.6155	0.848119
$340$	0	0
$341$	26.2462	1.42131
$342$	0	0
$343$	−6.05398	−0.326884
$344$	0	0
$345$	7.61553	0.410006
$346$	0	0
$347$	−1.56155	−0.0838285	−0.0419143	−	0.999121i	$-0.513346\pi$
$347$	−1.56155	−0.0838285	−0.0419143	+	0.999121i	$0.513346\pi$
$348$	0	0
$349$	−10.9309	−0.585116	−0.292558	−	0.956248i	$-0.594506\pi$
$349$	−10.9309	−0.585116	−0.292558	+	0.956248i	$0.594506\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−7.61553	−0.405334	−0.202667	−	0.979248i	$-0.564961\pi$
$353$	−7.61553	−0.405334	−0.202667	+	0.979248i	$0.564961\pi$
$354$	0	0
$355$	−10.4384	−0.554015
$356$	0	0
$357$	2.43845	0.129056
$358$	0	0
$359$	−13.1231	−0.692611	−0.346306	−	0.938122i	$-0.612564\pi$
$359$	−13.1231	−0.692611	−0.346306	+	0.938122i	$0.612564\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	0	0
$363$	23.8078	1.24958
$364$	0	0
$365$	−14.6307	−0.765805
$366$	0	0
$367$	28.4924	1.48729	0.743646	−	0.668573i	$-0.233095\pi$
$367$	28.4924	1.48729	0.743646	+	0.668573i	$0.233095\pi$
$368$	0	0
$369$	−4.49242	−0.233866
$370$	0	0
$371$	5.36932	0.278761
$372$	0	0
$373$	−35.3693	−1.83135	−0.915677	−	0.401915i	$-0.868345\pi$
$373$	−35.3693	−1.83135	−0.915677	+	0.401915i	$0.868345\pi$
$374$	0	0
$375$	−18.4384	−0.952157
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−26.0000	−1.33553	−0.667765	−	0.744372i	$-0.732749\pi$
$379$	−26.0000	−1.33553	−0.667765	+	0.744372i	$0.732749\pi$
$380$	0	0
$381$	7.61553	0.390155
$382$	0	0
$383$	29.8078	1.52311	0.761553	−	0.648103i	$-0.224437\pi$
$383$	29.8078	1.52311	0.761553	+	0.648103i	$0.224437\pi$
$384$	0	0
$385$	3.50758	0.178763
$386$	0	0
$387$	5.36932	0.272938
$388$	0	0
$389$	11.7538	0.595941	0.297970	−	0.954575i	$-0.403690\pi$
$389$	11.7538	0.595941	0.297970	+	0.954575i	$0.403690\pi$
$390$	0	0
$391$	11.1231	0.562520
$392$	0	0
$393$	−5.17708	−0.261149
$394$	0	0
$395$	−17.3693	−0.873945
$396$	0	0
$397$	−0.876894	−0.0440101	−0.0220050	−	0.999758i	$-0.507005\pi$
$397$	−0.876894	−0.0440101	−0.0220050	+	0.999758i	$0.507005\pi$
$398$	0	0
$399$	−1.36932	−0.0685516
$400$	0	0
$401$	−17.3693	−0.867382	−0.433691	−	0.901062i	$-0.642789\pi$
$401$	−17.3693	−0.867382	−0.433691	+	0.901062i	$0.642789\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−10.9309	−0.543159
$406$	0	0
$407$	48.9848	2.42809
$408$	0	0
$409$	−25.3693	−1.25443	−0.627216	−	0.778845i	$-0.715806\pi$
$409$	−25.3693	−1.25443	−0.627216	+	0.778845i	$0.715806\pi$
$410$	0	0
$411$	−4.87689	−0.240559
$412$	0	0
$413$	4.38447	0.215746
$414$	0	0
$415$	−12.8769	−0.632102
$416$	0	0
$417$	−12.1922	−0.597056
$418$	0	0
$419$	−30.4384	−1.48702	−0.743508	−	0.668727i	$-0.766839\pi$
$419$	−30.4384	−1.48702	−0.743508	+	0.668727i	$0.766839\pi$
$420$	0	0
$421$	23.8078	1.16032	0.580160	−	0.814503i	$-0.302990\pi$
$421$	23.8078	1.16032	0.580160	+	0.814503i	$0.302990\pi$
$422$	0	0
$423$	4.24621	0.206458
$424$	0	0
$425$	−9.12311	−0.442536
$426$	0	0
$427$	1.26137	0.0610418
$428$	0	0
$429$	0	0
$430$	0	0
$431$	37.4233	1.80262	0.901308	−	0.433178i	$-0.142608\pi$
$431$	37.4233	1.80262	0.901308	+	0.433178i	$0.142608\pi$
$432$	0	0
$433$	−9.31534	−0.447667	−0.223833	−	0.974627i	$-0.571857\pi$
$433$	−9.31534	−0.447667	−0.223833	+	0.974627i	$0.571857\pi$
$434$	0	0
$435$	−12.4924	−0.598966
$436$	0	0
$437$	−6.24621	−0.298797
$438$	0	0
$439$	37.8617	1.80704	0.903521	−	0.428544i	$-0.140973\pi$
$439$	37.8617	1.80704	0.903521	+	0.428544i	$0.140973\pi$
$440$	0	0
$441$	3.82292	0.182044
$442$	0	0
$443$	−9.17708	−0.436016	−0.218008	−	0.975947i	$-0.569956\pi$
$443$	−9.17708	−0.436016	−0.218008	+	0.975947i	$0.569956\pi$
$444$	0	0
$445$	−4.87689	−0.231187
$446$	0	0
$447$	−11.1231	−0.526105
$448$	0	0
$449$	16.0000	0.755087	0.377543	−	0.925992i	$-0.376769\pi$
$449$	16.0000	0.755087	0.377543	+	0.925992i	$0.376769\pi$
$450$	0	0
$451$	40.9848	1.92990
$452$	0	0
$453$	18.0540	0.848250
$454$	0	0
$455$	0	0
$456$	0	0
$457$	15.6155	0.730464	0.365232	−	0.930917i	$-0.380990\pi$
$457$	15.6155	0.730464	0.365232	+	0.930917i	$0.380990\pi$
$458$	0	0
$459$	−19.8078	−0.924547
$460$	0	0
$461$	−9.17708	−0.427419	−0.213710	−	0.976897i	$-0.568555\pi$
$461$	−9.17708	−0.427419	−0.213710	+	0.976897i	$0.568555\pi$
$462$	0	0
$463$	−1.61553	−0.0750800	−0.0375400	−	0.999295i	$-0.511952\pi$
$463$	−1.61553	−0.0750800	−0.0375400	+	0.999295i	$0.511952\pi$
$464$	0	0
$465$	12.4924	0.579322
$466$	0	0
$467$	−16.4924	−0.763178	−0.381589	−	0.924332i	$-0.624623\pi$
$467$	−16.4924	−0.763178	−0.381589	+	0.924332i	$0.624623\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	0	0
$471$	19.1231	0.881146
$472$	0	0
$473$	−48.9848	−2.25233
$474$	0	0
$475$	5.12311	0.235064
$476$	0	0
$477$	−6.87689	−0.314871
$478$	0	0
$479$	−29.8078	−1.36195	−0.680976	−	0.732306i	$-0.738444\pi$
$479$	−29.8078	−1.36195	−0.680976	+	0.732306i	$0.738444\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	2.13826	0.0972942
$484$	0	0
$485$	−9.75379	−0.442897
$486$	0	0
$487$	15.8617	0.718764	0.359382	−	0.933191i	$-0.382988\pi$
$487$	15.8617	0.718764	0.359382	+	0.933191i	$0.382988\pi$
$488$	0	0
$489$	−31.6155	−1.42970
$490$	0	0
$491$	−31.8078	−1.43546	−0.717732	−	0.696319i	$-0.754820\pi$
$491$	−31.8078	−1.43546	−0.717732	+	0.696319i	$0.754820\pi$
$492$	0	0
$493$	−18.2462	−0.821768
$494$	0	0
$495$	−4.49242	−0.201919
$496$	0	0
$497$	−2.93087	−0.131467
$498$	0	0
$499$	25.1231	1.12466	0.562332	−	0.826911i	$-0.309904\pi$
$499$	25.1231	1.12466	0.562332	+	0.826911i	$0.309904\pi$
$500$	0	0
$501$	−10.7386	−0.479767
$502$	0	0
$503$	9.36932	0.417757	0.208879	−	0.977942i	$-0.433019\pi$
$503$	9.36932	0.417757	0.208879	+	0.977942i	$0.433019\pi$
$504$	0	0
$505$	17.7538	0.790033
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−2.63068	−0.116603	−0.0583015	−	0.998299i	$-0.518568\pi$
$509$	−2.63068	−0.116603	−0.0583015	+	0.998299i	$0.518568\pi$
$510$	0	0
$511$	−4.10795	−0.181725
$512$	0	0
$513$	11.1231	0.491097
$514$	0	0
$515$	−4.87689	−0.214902
$516$	0	0
$517$	−38.7386	−1.70372
$518$	0	0
$519$	−6.63068	−0.291055
$520$	0	0
$521$	0.930870	0.0407821	0.0203911	−	0.999792i	$-0.493509\pi$
$521$	0.930870	0.0407821	0.0203911	+	0.999792i	$0.493509\pi$
$522$	0	0
$523$	−13.7538	−0.601411	−0.300706	−	0.953717i	$-0.597222\pi$
$523$	−13.7538	−0.601411	−0.300706	+	0.953717i	$0.597222\pi$
$524$	0	0
$525$	−1.75379	−0.0765416
$526$	0	0
$527$	18.2462	0.794817
$528$	0	0
$529$	−13.2462	−0.575922
$530$	0	0
$531$	−5.61553	−0.243693
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−6.24621	−0.270047
$536$	0	0
$537$	−27.4233	−1.18340
$538$	0	0
$539$	−34.8769	−1.50225
$540$	0	0
$541$	25.1771	1.08245	0.541224	−	0.840879i	$-0.317961\pi$
$541$	25.1771	1.08245	0.541224	+	0.840879i	$0.317961\pi$
$542$	0	0
$543$	12.8769	0.552600
$544$	0	0
$545$	5.17708	0.221762
$546$	0	0
$547$	23.8078	1.01795	0.508973	−	0.860782i	$-0.330025\pi$
$547$	23.8078	1.01795	0.508973	+	0.860782i	$0.330025\pi$
$548$	0	0
$549$	−1.61553	−0.0689491
$550$	0	0
$551$	10.2462	0.436503
$552$	0	0
$553$	−4.87689	−0.207387
$554$	0	0
$555$	23.3153	0.989681
$556$	0	0
$557$	18.9309	0.802127	0.401063	−	0.916050i	$-0.368641\pi$
$557$	18.9309	0.802127	0.401063	+	0.916050i	$0.368641\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	28.4924	1.20295
$562$	0	0
$563$	26.9309	1.13500	0.567500	−	0.823373i	$-0.307911\pi$
$563$	26.9309	1.13500	0.567500	+	0.823373i	$0.307911\pi$
$564$	0	0
$565$	15.6155	0.656950
$566$	0	0
$567$	−3.06913	−0.128891
$568$	0	0
$569$	2.19224	0.0919033	0.0459517	−	0.998944i	$-0.485368\pi$
$569$	2.19224	0.0919033	0.0459517	+	0.998944i	$0.485368\pi$
$570$	0	0
$571$	14.0540	0.588141	0.294070	−	0.955784i	$-0.404990\pi$
$571$	14.0540	0.588141	0.294070	+	0.955784i	$0.404990\pi$
$572$	0	0
$573$	−34.7386	−1.45123
$574$	0	0
$575$	−8.00000	−0.333623
$576$	0	0
$577$	29.8617	1.24316	0.621580	−	0.783351i	$-0.286491\pi$
$577$	29.8617	1.24316	0.621580	+	0.783351i	$0.286491\pi$
$578$	0	0
$579$	36.8769	1.53255
$580$	0	0
$581$	−3.61553	−0.149997
$582$	0	0
$583$	62.7386	2.59837
$584$	0	0
$585$	0	0
$586$	0	0
$587$	9.50758	0.392420	0.196210	−	0.980562i	$-0.437137\pi$
$587$	9.50758	0.392420	0.196210	+	0.980562i	$0.437137\pi$
$588$	0	0
$589$	−10.2462	−0.422188
$590$	0	0
$591$	−14.9309	−0.614174
$592$	0	0
$593$	−25.7538	−1.05758	−0.528791	−	0.848752i	$-0.677354\pi$
$593$	−25.7538	−1.05758	−0.528791	+	0.848752i	$0.677354\pi$
$594$	0	0
$595$	2.43845	0.0999666
$596$	0	0
$597$	22.2462	0.910477
$598$	0	0
$599$	41.3693	1.69030	0.845152	−	0.534526i	$-0.179510\pi$
$599$	41.3693	1.69030	0.845152	+	0.534526i	$0.179510\pi$
$600$	0	0
$601$	21.8078	0.889557	0.444779	−	0.895641i	$-0.353282\pi$
$601$	21.8078	0.889557	0.444779	+	0.895641i	$0.353282\pi$
$602$	0	0
$603$	−5.12311	−0.208629
$604$	0	0
$605$	23.8078	0.967923
$606$	0	0
$607$	−7.61553	−0.309105	−0.154552	−	0.987985i	$-0.549394\pi$
$607$	−7.61553	−0.309105	−0.154552	+	0.987985i	$0.549394\pi$
$608$	0	0
$609$	−3.50758	−0.142134
$610$	0	0
$611$	0	0
$612$	0	0
$613$	18.6307	0.752486	0.376243	−	0.926521i	$-0.377216\pi$
$613$	18.6307	0.752486	0.376243	+	0.926521i	$0.377216\pi$
$614$	0	0
$615$	19.5076	0.786621
$616$	0	0
$617$	1.75379	0.0706049	0.0353024	−	0.999377i	$-0.488761\pi$
$617$	1.75379	0.0706049	0.0353024	+	0.999377i	$0.488761\pi$
$618$	0	0
$619$	17.6155	0.708028	0.354014	−	0.935240i	$-0.384817\pi$
$619$	17.6155	0.708028	0.354014	+	0.935240i	$0.384817\pi$
$620$	0	0
$621$	−17.3693	−0.697007
$622$	0	0
$623$	−1.36932	−0.0548605
$624$	0	0
$625$	−5.63068	−0.225227
$626$	0	0
$627$	−16.0000	−0.638978
$628$	0	0
$629$	34.0540	1.35782
$630$	0	0
$631$	11.5616	0.460258	0.230129	−	0.973160i	$-0.426085\pi$
$631$	11.5616	0.460258	0.230129	+	0.973160i	$0.426085\pi$
$632$	0	0
$633$	−34.4384	−1.36881
$634$	0	0
$635$	7.61553	0.302213
$636$	0	0
$637$	0	0
$638$	0	0
$639$	3.75379	0.148498
$640$	0	0
$641$	−3.75379	−0.148266	−0.0741329	−	0.997248i	$-0.523619\pi$
$641$	−3.75379	−0.148266	−0.0741329	+	0.997248i	$0.523619\pi$
$642$	0	0
$643$	25.1231	0.990759	0.495379	−	0.868677i	$-0.335029\pi$
$643$	25.1231	0.990759	0.495379	+	0.868677i	$0.335029\pi$
$644$	0	0
$645$	−23.3153	−0.918041
$646$	0	0
$647$	−3.12311	−0.122782	−0.0613910	−	0.998114i	$-0.519554\pi$
$647$	−3.12311	−0.122782	−0.0613910	+	0.998114i	$0.519554\pi$
$648$	0	0
$649$	51.2311	2.01099
$650$	0	0
$651$	3.50758	0.137473
$652$	0	0
$653$	13.6155	0.532817	0.266408	−	0.963860i	$-0.414163\pi$
$653$	13.6155	0.532817	0.266408	+	0.963860i	$0.414163\pi$
$654$	0	0
$655$	−5.17708	−0.202285
$656$	0	0
$657$	5.26137	0.205265
$658$	0	0
$659$	6.73863	0.262500	0.131250	−	0.991349i	$-0.458101\pi$
$659$	6.73863	0.262500	0.131250	+	0.991349i	$0.458101\pi$
$660$	0	0
$661$	32.8769	1.27876	0.639381	−	0.768890i	$-0.279190\pi$
$661$	32.8769	1.27876	0.639381	+	0.768890i	$0.279190\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−1.36932	−0.0530998
$666$	0	0
$667$	−16.0000	−0.619522
$668$	0	0
$669$	−42.4384	−1.64077
$670$	0	0
$671$	14.7386	0.568979
$672$	0	0
$673$	22.3002	0.859609	0.429805	−	0.902922i	$-0.358582\pi$
$673$	22.3002	0.859609	0.429805	+	0.902922i	$0.358582\pi$
$674$	0	0
$675$	14.2462	0.548337
$676$	0	0
$677$	−28.7386	−1.10452	−0.552258	−	0.833673i	$-0.686234\pi$
$677$	−28.7386	−1.10452	−0.552258	+	0.833673i	$0.686234\pi$
$678$	0	0
$679$	−2.73863	−0.105099
$680$	0	0
$681$	−5.26137	−0.201616
$682$	0	0
$683$	−12.2462	−0.468588	−0.234294	−	0.972166i	$-0.575278\pi$
$683$	−12.2462	−0.468588	−0.234294	+	0.972166i	$0.575278\pi$
$684$	0	0
$685$	−4.87689	−0.186337
$686$	0	0
$687$	32.3002	1.23233
$688$	0	0
$689$	0	0
$690$	0	0
$691$	27.8617	1.05991	0.529955	−	0.848026i	$-0.322209\pi$
$691$	27.8617	1.05991	0.529955	+	0.848026i	$0.322209\pi$
$692$	0	0
$693$	−1.26137	−0.0479153
$694$	0	0
$695$	−12.1922	−0.462478
$696$	0	0
$697$	28.4924	1.07923
$698$	0	0
$699$	5.56155	0.210357
$700$	0	0
$701$	−37.2311	−1.40620	−0.703099	−	0.711092i	$-0.748201\pi$
$701$	−37.2311	−1.40620	−0.703099	+	0.711092i	$0.748201\pi$
$702$	0	0
$703$	−19.1231	−0.721242
$704$	0	0
$705$	−18.4384	−0.694432
$706$	0	0
$707$	4.98485	0.187474
$708$	0	0
$709$	7.12311	0.267514	0.133757	−	0.991014i	$-0.457296\pi$
$709$	7.12311	0.267514	0.133757	+	0.991014i	$0.457296\pi$
$710$	0	0
$711$	6.24621	0.234251
$712$	0	0
$713$	16.0000	0.599205
$714$	0	0
$715$	0	0
$716$	0	0
$717$	13.9460	0.520824
$718$	0	0
$719$	16.0000	0.596699	0.298350	−	0.954457i	$-0.403564\pi$
$719$	16.0000	0.596699	0.298350	+	0.954457i	$0.403564\pi$
$720$	0	0
$721$	−1.36932	−0.0509960
$722$	0	0
$723$	−42.3542	−1.57517
$724$	0	0
$725$	13.1231	0.487380
$726$	0	0
$727$	−39.2311	−1.45500	−0.727500	−	0.686108i	$-0.759318\pi$
$727$	−39.2311	−1.45500	−0.727500	+	0.686108i	$0.759318\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	−34.0540	−1.25953
$732$	0	0
$733$	−4.30019	−0.158831	−0.0794155	−	0.996842i	$-0.525305\pi$
$733$	−4.30019	−0.158831	−0.0794155	+	0.996842i	$0.525305\pi$
$734$	0	0
$735$	−16.6004	−0.612314
$736$	0	0
$737$	46.7386	1.72164
$738$	0	0
$739$	15.8617	0.583484	0.291742	−	0.956497i	$-0.405765\pi$
$739$	15.8617	0.583484	0.291742	+	0.956497i	$0.405765\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	7.94602	0.291511	0.145756	−	0.989321i	$-0.453439\pi$
$743$	7.94602	0.291511	0.145756	+	0.989321i	$0.453439\pi$
$744$	0	0
$745$	−11.1231	−0.407519
$746$	0	0
$747$	4.63068	0.169428
$748$	0	0
$749$	−1.75379	−0.0640820
$750$	0	0
$751$	3.50758	0.127993	0.0639967	−	0.997950i	$-0.479615\pi$
$751$	3.50758	0.127993	0.0639967	+	0.997950i	$0.479615\pi$
$752$	0	0
$753$	−38.2462	−1.39377
$754$	0	0
$755$	18.0540	0.657051
$756$	0	0
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	0	0
$759$	24.9848	0.906892
$760$	0	0
$761$	44.8769	1.62679	0.813393	−	0.581714i	$-0.197618\pi$
$761$	44.8769	1.62679	0.813393	+	0.581714i	$0.197618\pi$
$762$	0	0
$763$	1.45360	0.0526239
$764$	0	0
$765$	−3.12311	−0.112916
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−28.4924	−1.02746	−0.513732	−	0.857951i	$-0.671737\pi$
$769$	−28.4924	−1.02746	−0.513732	+	0.857951i	$0.671737\pi$
$770$	0	0
$771$	−1.45360	−0.0523502
$772$	0	0
$773$	14.0540	0.505486	0.252743	−	0.967533i	$-0.418667\pi$
$773$	14.0540	0.505486	0.252743	+	0.967533i	$0.418667\pi$
$774$	0	0
$775$	−13.1231	−0.471396
$776$	0	0
$777$	6.54640	0.234851
$778$	0	0
$779$	−16.0000	−0.573259
$780$	0	0
$781$	−34.2462	−1.22543
$782$	0	0
$783$	28.4924	1.01824
$784$	0	0
$785$	19.1231	0.682533
$786$	0	0
$787$	3.36932	0.120103	0.0600516	−	0.998195i	$-0.480873\pi$
$787$	3.36932	0.120103	0.0600516	+	0.998195i	$0.480873\pi$
$788$	0	0
$789$	−7.01515	−0.249746
$790$	0	0
$791$	4.38447	0.155894
$792$	0	0
$793$	0	0
$794$	0	0
$795$	29.8617	1.05909
$796$	0	0
$797$	−12.2462	−0.433783	−0.216892	−	0.976196i	$-0.569592\pi$
$797$	−12.2462	−0.433783	−0.216892	+	0.976196i	$0.569592\pi$
$798$	0	0
$799$	−26.9309	−0.952746
$800$	0	0
$801$	1.75379	0.0619671
$802$	0	0
$803$	−48.0000	−1.69388
$804$	0	0
$805$	2.13826	0.0753638
$806$	0	0
$807$	16.9848	0.597895
$808$	0	0
$809$	42.6847	1.50071	0.750356	−	0.661034i	$-0.229882\pi$
$809$	42.6847	1.50071	0.750356	+	0.661034i	$0.229882\pi$
$810$	0	0
$811$	23.3693	0.820608	0.410304	−	0.911949i	$-0.365423\pi$
$811$	23.3693	0.820608	0.410304	+	0.911949i	$0.365423\pi$
$812$	0	0
$813$	22.9309	0.804221
$814$	0	0
$815$	−31.6155	−1.10744
$816$	0	0
$817$	19.1231	0.669033
$818$	0	0
$819$	0	0
$820$	0	0
$821$	42.5464	1.48488	0.742440	−	0.669913i	$-0.233669\pi$
$821$	42.5464	1.48488	0.742440	+	0.669913i	$0.233669\pi$
$822$	0	0
$823$	50.3542	1.75524	0.877618	−	0.479361i	$-0.159131\pi$
$823$	50.3542	1.75524	0.877618	+	0.479361i	$0.159131\pi$
$824$	0	0
$825$	−20.4924	−0.713455
$826$	0	0
$827$	−22.0000	−0.765015	−0.382507	−	0.923952i	$-0.624939\pi$
$827$	−22.0000	−0.765015	−0.382507	+	0.923952i	$0.624939\pi$
$828$	0	0
$829$	6.87689	0.238844	0.119422	−	0.992844i	$-0.461896\pi$
$829$	6.87689	0.238844	0.119422	+	0.992844i	$0.461896\pi$
$830$	0	0
$831$	−11.5076	−0.399193
$832$	0	0
$833$	−24.2462	−0.840081
$834$	0	0
$835$	−10.7386	−0.371626
$836$	0	0
$837$	−28.4924	−0.984842
$838$	0	0
$839$	43.8617	1.51428	0.757138	−	0.653255i	$-0.226597\pi$
$839$	43.8617	1.51428	0.757138	+	0.653255i	$0.226597\pi$
$840$	0	0
$841$	−2.75379	−0.0949582
$842$	0	0
$843$	14.6307	0.503907
$844$	0	0
$845$	0	0
$846$	0	0
$847$	6.68466	0.229688
$848$	0	0
$849$	−18.7386	−0.643108
$850$	0	0
$851$	29.8617	1.02365
$852$	0	0
$853$	4.68466	0.160400	0.0801998	−	0.996779i	$-0.474444\pi$
$853$	4.68466	0.160400	0.0801998	+	0.996779i	$0.474444\pi$
$854$	0	0
$855$	1.75379	0.0599783
$856$	0	0
$857$	50.4924	1.72479	0.862394	−	0.506237i	$-0.168964\pi$
$857$	50.4924	1.72479	0.862394	+	0.506237i	$0.168964\pi$
$858$	0	0
$859$	−10.2462	−0.349596	−0.174798	−	0.984604i	$-0.555927\pi$
$859$	−10.2462	−0.349596	−0.174798	+	0.984604i	$0.555927\pi$
$860$	0	0
$861$	5.47727	0.186665
$862$	0	0
$863$	−48.0540	−1.63578	−0.817888	−	0.575377i	$-0.804855\pi$
$863$	−48.0540	−1.63578	−0.817888	+	0.575377i	$0.804855\pi$
$864$	0	0
$865$	−6.63068	−0.225450
$866$	0	0
$867$	−6.73863	−0.228856
$868$	0	0
$869$	−56.9848	−1.93308
$870$	0	0
$871$	0	0
$872$	0	0
$873$	3.50758	0.118714
$874$	0	0
$875$	−5.17708	−0.175017
$876$	0	0
$877$	7.42329	0.250667	0.125333	−	0.992115i	$-0.460000\pi$
$877$	7.42329	0.250667	0.125333	+	0.992115i	$0.460000\pi$
$878$	0	0
$879$	−19.8078	−0.668099
$880$	0	0
$881$	42.7926	1.44172	0.720860	−	0.693081i	$-0.243747\pi$
$881$	42.7926	1.44172	0.720860	+	0.693081i	$0.243747\pi$
$882$	0	0
$883$	31.8078	1.07042	0.535208	−	0.844720i	$-0.320233\pi$
$883$	31.8078	1.07042	0.535208	+	0.844720i	$0.320233\pi$
$884$	0	0
$885$	24.3845	0.819675
$886$	0	0
$887$	−5.26137	−0.176659	−0.0883297	−	0.996091i	$-0.528153\pi$
$887$	−5.26137	−0.176659	−0.0883297	+	0.996091i	$0.528153\pi$
$888$	0	0
$889$	2.13826	0.0717150
$890$	0	0
$891$	−35.8617	−1.20141
$892$	0	0
$893$	15.1231	0.506075
$894$	0	0
$895$	−27.4233	−0.916660
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−26.2462	−0.875360
$900$	0	0
$901$	43.6155	1.45304
$902$	0	0
$903$	−6.54640	−0.217850
$904$	0	0
$905$	12.8769	0.428042
$906$	0	0
$907$	22.4384	0.745056	0.372528	−	0.928021i	$-0.378491\pi$
$907$	22.4384	0.745056	0.372528	+	0.928021i	$0.378491\pi$
$908$	0	0
$909$	−6.38447	−0.211760
$910$	0	0
$911$	25.7538	0.853261	0.426631	−	0.904426i	$-0.359700\pi$
$911$	25.7538	0.853261	0.426631	+	0.904426i	$0.359700\pi$
$912$	0	0
$913$	−42.2462	−1.39815
$914$	0	0
$915$	7.01515	0.231914
$916$	0	0
$917$	−1.45360	−0.0480022
$918$	0	0
$919$	1.75379	0.0578522	0.0289261	−	0.999582i	$-0.490791\pi$
$919$	1.75379	0.0578522	0.0289261	+	0.999582i	$0.490791\pi$
$920$	0	0
$921$	28.1080	0.926188
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−24.4924	−0.805306
$926$	0	0
$927$	1.75379	0.0576020
$928$	0	0
$929$	18.7386	0.614795	0.307397	−	0.951581i	$-0.400542\pi$
$929$	18.7386	0.614795	0.307397	+	0.951581i	$0.400542\pi$
$930$	0	0
$931$	13.6155	0.446231
$932$	0	0
$933$	−29.8617	−0.977629
$934$	0	0
$935$	28.4924	0.931802
$936$	0	0
$937$	19.7538	0.645328	0.322664	−	0.946514i	$-0.395422\pi$
$937$	19.7538	0.645328	0.322664	+	0.946514i	$0.395422\pi$
$938$	0	0
$939$	6.93087	0.226180
$940$	0	0
$941$	−42.5464	−1.38697	−0.693486	−	0.720470i	$-0.743926\pi$
$941$	−42.5464	−1.38697	−0.693486	+	0.720470i	$0.743926\pi$
$942$	0	0
$943$	24.9848	0.813618
$944$	0	0
$945$	−3.80776	−0.123867
$946$	0	0
$947$	−15.3693	−0.499436	−0.249718	−	0.968319i	$-0.580338\pi$
$947$	−15.3693	−0.499436	−0.249718	+	0.968319i	$0.580338\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	−23.6155	−0.765786
$952$	0	0
$953$	−20.0540	−0.649612	−0.324806	−	0.945781i	$-0.605299\pi$
$953$	−20.0540	−0.649612	−0.324806	+	0.945781i	$0.605299\pi$
$954$	0	0
$955$	−34.7386	−1.12412
$956$	0	0
$957$	−40.9848	−1.32485
$958$	0	0
$959$	−1.36932	−0.0442175
$960$	0	0
$961$	−4.75379	−0.153348
$962$	0	0
$963$	2.24621	0.0723831
$964$	0	0
$965$	36.8769	1.18711
$966$	0	0
$967$	15.5616	0.500426	0.250213	−	0.968191i	$-0.419499\pi$
$967$	15.5616	0.500426	0.250213	+	0.968191i	$0.419499\pi$
$968$	0	0
$969$	−11.1231	−0.357326
$970$	0	0
$971$	−57.1771	−1.83490	−0.917450	−	0.397851i	$-0.869756\pi$
$971$	−57.1771	−1.83490	−0.917450	+	0.397851i	$0.869756\pi$
$972$	0	0
$973$	−3.42329	−0.109746
$974$	0	0
$975$	0	0
$976$	0	0
$977$	47.2311	1.51106	0.755528	−	0.655117i	$-0.227381\pi$
$977$	47.2311	1.51106	0.755528	+	0.655117i	$0.227381\pi$
$978$	0	0
$979$	−16.0000	−0.511362
$980$	0	0
$981$	−1.86174	−0.0594408
$982$	0	0
$983$	10.1922	0.325082	0.162541	−	0.986702i	$-0.448031\pi$
$983$	10.1922	0.325082	0.162541	+	0.986702i	$0.448031\pi$
$984$	0	0
$985$	−14.9309	−0.475737
$986$	0	0
$987$	−5.17708	−0.164788
$988$	0	0
$989$	−29.8617	−0.949548
$990$	0	0
$991$	−7.61553	−0.241915	−0.120958	−	0.992658i	$-0.538597\pi$
$991$	−7.61553	−0.241915	−0.120958	+	0.992658i	$0.538597\pi$
$992$	0	0
$993$	−0.384472	−0.0122008
$994$	0	0
$995$	22.2462	0.705252
$996$	0	0
$997$	−1.61553	−0.0511643	−0.0255821	−	0.999673i	$-0.508144\pi$
$997$	−1.61553	−0.0511643	−0.0255821	+	0.999673i	$0.508144\pi$
$998$	0	0
$999$	−53.1771	−1.68245

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1352.2.a.d.1.2		2
4.3	odd	2		2704.2.a.s.1.1		2
13.2	odd	12		1352.2.o.e.1161.1		8
13.3	even	3		1352.2.i.g.529.1		4
13.4	even	6		1352.2.i.h.1329.1		4
13.5	odd	4		104.2.f.a.25.3	✓	4
13.6	odd	12		1352.2.o.e.361.1		8
13.7	odd	12		1352.2.o.e.361.2		8
13.8	odd	4		104.2.f.a.25.4	yes	4
13.9	even	3		1352.2.i.g.1329.1		4
13.10	even	6		1352.2.i.h.529.1		4
13.11	odd	12		1352.2.o.e.1161.2		8
13.12	even	2		1352.2.a.e.1.2		2
39.5	even	4		936.2.c.d.649.3		4
39.8	even	4		936.2.c.d.649.2		4
52.31	even	4		208.2.f.b.129.1		4
52.47	even	4		208.2.f.b.129.2		4
52.51	odd	2		2704.2.a.t.1.1		2
65.8	even	4		2600.2.f.a.649.3		4
65.18	even	4		2600.2.f.b.649.3		4
65.34	odd	4		2600.2.k.a.2001.2		4
65.44	odd	4		2600.2.k.a.2001.1		4
65.47	even	4		2600.2.f.b.649.2		4
65.57	even	4		2600.2.f.a.649.2		4
104.5	odd	4		832.2.f.i.129.2		4
104.21	odd	4		832.2.f.i.129.1		4
104.83	even	4		832.2.f.f.129.4		4
104.99	even	4		832.2.f.f.129.3		4
156.47	odd	4		1872.2.c.j.1585.2		4
156.83	odd	4		1872.2.c.j.1585.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.f.a.25.3	✓	4	13.5	odd	4
104.2.f.a.25.4	yes	4	13.8	odd	4
208.2.f.b.129.1		4	52.31	even	4
208.2.f.b.129.2		4	52.47	even	4
832.2.f.f.129.3		4	104.99	even	4
832.2.f.f.129.4		4	104.83	even	4
832.2.f.i.129.1		4	104.21	odd	4
832.2.f.i.129.2		4	104.5	odd	4
936.2.c.d.649.2		4	39.8	even	4
936.2.c.d.649.3		4	39.5	even	4
1352.2.a.d.1.2		2	1.1	even	1	trivial
1352.2.a.e.1.2		2	13.12	even	2
1352.2.i.g.529.1		4	13.3	even	3
1352.2.i.g.1329.1		4	13.9	even	3
1352.2.i.h.529.1		4	13.10	even	6
1352.2.i.h.1329.1		4	13.4	even	6
1352.2.o.e.361.1		8	13.6	odd	12
1352.2.o.e.361.2		8	13.7	odd	12
1352.2.o.e.1161.1		8	13.2	odd	12
1352.2.o.e.1161.2		8	13.11	odd	12
1872.2.c.j.1585.2		4	156.47	odd	4
1872.2.c.j.1585.3		4	156.83	odd	4
2600.2.f.a.649.2		4	65.57	even	4
2600.2.f.a.649.3		4	65.8	even	4
2600.2.f.b.649.2		4	65.47	even	4
2600.2.f.b.649.3		4	65.18	even	4
2600.2.k.a.2001.1		4	65.44	odd	4
2600.2.k.a.2001.2		4	65.34	odd	4
2704.2.a.s.1.1		2	4.3	odd	2
2704.2.a.t.1.1		2	52.51	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1352 = 2^{3} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1352.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56155 q^{3} +1.56155 q^{5} +0.438447 q^{7} -0.561553 q^{9} +5.12311 q^{11} +2.43845 q^{15} +3.56155 q^{17} -2.00000 q^{19} +0.684658 q^{21} +3.12311 q^{23} -2.56155 q^{25} -5.56155 q^{27} -5.12311 q^{29} +5.12311 q^{31} +8.00000 q^{33} +0.684658 q^{35} +9.56155 q^{37} +8.00000 q^{41} -9.56155 q^{43} -0.876894 q^{45} -7.56155 q^{47} -6.80776 q^{49} +5.56155 q^{51} +12.2462 q^{53} +8.00000 q^{55} -3.12311 q^{57} +10.0000 q^{59} +2.87689 q^{61} -0.246211 q^{63} +9.12311 q^{67} +4.87689 q^{69} -6.68466 q^{71} -9.36932 q^{73} -4.00000 q^{75} +2.24621 q^{77} -11.1231 q^{79} -7.00000 q^{81} -8.24621 q^{83} +5.56155 q^{85} -8.00000 q^{87} -3.12311 q^{89} +8.00000 q^{93} -3.12311 q^{95} -6.24621 q^{97} -2.87689 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - q^{5} + 5 q^{7} + 3 q^{9} + 2 q^{11} + 9 q^{15} + 3 q^{17} - 4 q^{19} - 11 q^{21} - 2 q^{23} - q^{25} - 7 q^{27} - 2 q^{29} + 2 q^{31} + 16 q^{33} - 11 q^{35} + 15 q^{37} + 16 q^{41} - 15 q^{43}+ \cdots - 14 q^{99}+O(q^{100}) \) 2 * q - q^3 - q^5 + 5 * q^7 + 3 * q^9 + 2 * q^11 + 9 * q^15 + 3 * q^17 - 4 * q^19 - 11 * q^21 - 2 * q^23 - q^25 - 7 * q^27 - 2 * q^29 + 2 * q^31 + 16 * q^33 - 11 * q^35 + 15 * q^37 + 16 * q^41 - 15 * q^43 - 10 * q^45 - 11 * q^47 + 7 * q^49 + 7 * q^51 + 8 * q^53 + 16 * q^55 + 2 * q^57 + 20 * q^59 + 14 * q^61 + 16 * q^63 + 10 * q^67 + 18 * q^69 - q^71 + 6 * q^73 - 8 * q^75 - 12 * q^77 - 14 * q^79 - 14 * q^81 + 7 * q^85 - 16 * q^87 + 2 * q^89 + 16 * q^93 + 2 * q^95 + 4 * q^97 - 14 * q^99

Introduction

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