LMFDB - Embedded newform 1352.2.a.n.1.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1352,2,Mod(1,1352)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$10.7957743533$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.3728753.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} + 11x^{3} + 22x^{2} - 18x - 13 $ x^6 - x^5 - 10x^4 + 11x^3 + 22x^2 - 18x - 13
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			1.6
Root			$1.52958$ 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)

$f(q)$	$=$	$q+3.33152 q^{3} -0.932734 q^{5} +3.45729 q^{7} +8.09903 q^{9} +O(q^{10})$	$q+3.33152 q^{3} -0.932734 q^{5} +3.45729 q^{7} +8.09903 q^{9} -1.85844 q^{11} -3.10742 q^{15} +0.218412 q^{17} -5.10713 q^{19} +11.5180 q^{21} +7.49994 q^{23} -4.13001 q^{25} +16.9875 q^{27} -3.19676 q^{29} +7.91055 q^{31} -6.19143 q^{33} -3.22473 q^{35} -4.26852 q^{37} -3.27171 q^{41} -4.58448 q^{43} -7.55423 q^{45} +1.30969 q^{47} +4.95285 q^{49} +0.727644 q^{51} -2.68671 q^{53} +1.73343 q^{55} -17.0145 q^{57} -9.23569 q^{59} -2.22782 q^{61} +28.0007 q^{63} +4.43889 q^{67} +24.9862 q^{69} -2.60098 q^{71} +3.28199 q^{73} -13.7592 q^{75} -6.42517 q^{77} +7.33736 q^{79} +32.2971 q^{81} -3.58655 q^{83} -0.203720 q^{85} -10.6501 q^{87} +5.07548 q^{89} +26.3542 q^{93} +4.76359 q^{95} +13.4155 q^{97} -15.0516 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{3} + 2 q^{5} - 3 q^{7} + 9 q^{9}+O(q^{10}) $ 6 * q + 3 * q^3 + 2 * q^5 - 3 * q^7 + 9 * q^9	$ 6 q + 3 q^{3} + 2 q^{5} - 3 q^{7} + 9 q^{9} - 13 q^{11} + 14 q^{15} + 11 q^{17} - 15 q^{19} + 22 q^{21} + 15 q^{23} + 16 q^{25} + 21 q^{27} + 9 q^{29} + 3 q^{31} + 2 q^{33} + 14 q^{35} - 14 q^{37} + 20 q^{41} + 10 q^{43} + 9 q^{45} + 10 q^{47} + 27 q^{49} + 5 q^{51} + q^{53} + 14 q^{55} - 9 q^{57} - 50 q^{59} - 2 q^{63} - 6 q^{67} + 32 q^{69} - 9 q^{71} + 6 q^{73} + 40 q^{75} - 2 q^{77} + 13 q^{79} + 42 q^{81} - 36 q^{83} + 31 q^{85} + 22 q^{87} + 18 q^{89} + 12 q^{93} - 21 q^{95} + 14 q^{97} - 18 q^{99}+O(q^{100}) $ 6 * q + 3 * q^3 + 2 * q^5 - 3 * q^7 + 9 * q^9 - 13 * q^11 + 14 * q^15 + 11 * q^17 - 15 * q^19 + 22 * q^21 + 15 * q^23 + 16 * q^25 + 21 * q^27 + 9 * q^29 + 3 * q^31 + 2 * q^33 + 14 * q^35 - 14 * q^37 + 20 * q^41 + 10 * q^43 + 9 * q^45 + 10 * q^47 + 27 * q^49 + 5 * q^51 + q^53 + 14 * q^55 - 9 * q^57 - 50 * q^59 - 2 * q^63 - 6 * q^67 + 32 * q^69 - 9 * q^71 + 6 * q^73 + 40 * q^75 - 2 * q^77 + 13 * q^79 + 42 * q^81 - 36 * q^83 + 31 * q^85 + 22 * q^87 + 18 * q^89 + 12 * q^93 - 21 * q^95 + 14 * q^97 - 18 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	3.33152	1.92345	0.961727	−	0.274010i	$-0.0883500\pi$
$3$	3.33152	1.92345	0.961727	+	0.274010i	$0.0883500\pi$
$4$	0	0
$5$	−0.932734	−0.417131	−0.208566	−	0.978008i	$-0.566879\pi$
$5$	−0.932734	−0.417131	−0.208566	+	0.978008i	$0.566879\pi$
$6$	0	0
$7$	3.45729	1.30673	0.653366	−	0.757042i	$-0.273356\pi$
$7$	3.45729	1.30673	0.653366	+	0.757042i	$0.273356\pi$
$8$	0	0
$9$	8.09903	2.69968
$10$	0	0
$11$	−1.85844	−0.560341	−0.280170	−	0.959950i	$-0.590391\pi$
$11$	−1.85844	−0.560341	−0.280170	+	0.959950i	$0.590391\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−3.10742	−0.802333
$16$	0	0
$17$	0.218412	0.0529727	0.0264864	−	0.999649i	$-0.491568\pi$
$17$	0.218412	0.0529727	0.0264864	+	0.999649i	$0.491568\pi$
$18$	0	0
$19$	−5.10713	−1.17166	−0.585828	−	0.810436i	$-0.699230\pi$
$19$	−5.10713	−1.17166	−0.585828	+	0.810436i	$0.699230\pi$
$20$	0	0
$21$	11.5180	2.51344
$22$	0	0
$23$	7.49994	1.56385	0.781923	−	0.623375i	$-0.214239\pi$
$23$	7.49994	1.56385	0.781923	+	0.623375i	$0.214239\pi$
$24$	0	0
$25$	−4.13001	−0.826002
$26$	0	0
$27$	16.9875	3.26925
$28$	0	0
$29$	−3.19676	−0.593623	−0.296811	−	0.954936i	$-0.595923\pi$
$29$	−3.19676	−0.593623	−0.296811	+	0.954936i	$0.595923\pi$
$30$	0	0
$31$	7.91055	1.42078	0.710388	−	0.703810i	$-0.248519\pi$
$31$	7.91055	1.42078	0.710388	+	0.703810i	$0.248519\pi$
$32$	0	0
$33$	−6.19143	−1.07779
$34$	0	0
$35$	−3.22473	−0.545079
$36$	0	0
$37$	−4.26852	−0.701740	−0.350870	−	0.936424i	$-0.614114\pi$
$37$	−4.26852	−0.701740	−0.350870	+	0.936424i	$0.614114\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.27171	−0.510956	−0.255478	−	0.966815i	$-0.582233\pi$
$41$	−3.27171	−0.510956	−0.255478	+	0.966815i	$0.582233\pi$
$42$	0	0
$43$	−4.58448	−0.699127	−0.349563	−	0.936913i	$-0.613670\pi$
$43$	−4.58448	−0.699127	−0.349563	+	0.936913i	$0.613670\pi$
$44$	0	0
$45$	−7.55423	−1.12612
$46$	0	0
$47$	1.30969	0.191037	0.0955186	−	0.995428i	$-0.469549\pi$
$47$	1.30969	0.191037	0.0955186	+	0.995428i	$0.469549\pi$
$48$	0	0
$49$	4.95285	0.707551
$50$	0	0
$51$	0.727644	0.101891
$52$	0	0
$53$	−2.68671	−0.369048	−0.184524	−	0.982828i	$-0.559074\pi$
$53$	−2.68671	−0.369048	−0.184524	+	0.982828i	$0.559074\pi$
$54$	0	0
$55$	1.73343	0.233736
$56$	0	0
$57$	−17.0145	−2.25363
$58$	0	0
$59$	−9.23569	−1.20238	−0.601192	−	0.799105i	$-0.705307\pi$
$59$	−9.23569	−1.20238	−0.601192	+	0.799105i	$0.705307\pi$
$60$	0	0
$61$	−2.22782	−0.285243	−0.142622	−	0.989777i	$-0.545553\pi$
$61$	−2.22782	−0.285243	−0.142622	+	0.989777i	$0.545553\pi$
$62$	0	0
$63$	28.0007	3.52775
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.43889	0.542297	0.271148	−	0.962538i	$-0.412597\pi$
$67$	4.43889	0.542297	0.271148	+	0.962538i	$0.412597\pi$
$68$	0	0
$69$	24.9862	3.00798
$70$	0	0
$71$	−2.60098	−0.308679	−0.154340	−	0.988018i	$-0.549325\pi$
$71$	−2.60098	−0.308679	−0.154340	+	0.988018i	$0.549325\pi$
$72$	0	0
$73$	3.28199	0.384128	0.192064	−	0.981382i	$-0.438482\pi$
$73$	3.28199	0.384128	0.192064	+	0.981382i	$0.438482\pi$
$74$	0	0
$75$	−13.7592	−1.58878
$76$	0	0
$77$	−6.42517	−0.732216
$78$	0	0
$79$	7.33736	0.825517	0.412759	−	0.910840i	$-0.364565\pi$
$79$	7.33736	0.825517	0.412759	+	0.910840i	$0.364565\pi$
$80$	0	0
$81$	32.2971	3.58857
$82$	0	0
$83$	−3.58655	−0.393675	−0.196838	−	0.980436i	$-0.563067\pi$
$83$	−3.58655	−0.393675	−0.196838	+	0.980436i	$0.563067\pi$
$84$	0	0
$85$	−0.203720	−0.0220966
$86$	0	0
$87$	−10.6501	−1.14181
$88$	0	0
$89$	5.07548	0.538000	0.269000	−	0.963140i	$-0.413307\pi$
$89$	5.07548	0.538000	0.269000	+	0.963140i	$0.413307\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	26.3542	2.73280
$94$	0	0
$95$	4.76359	0.488734
$96$	0	0
$97$	13.4155	1.36214	0.681070	−	0.732219i	$-0.261515\pi$
$97$	13.4155	1.36214	0.681070	+	0.732219i	$0.261515\pi$
$98$	0	0
$99$	−15.0516	−1.51274
$100$	0	0
$101$	−15.1252	−1.50502	−0.752508	−	0.658583i	$-0.771156\pi$
$101$	−15.1252	−1.50502	−0.752508	+	0.658583i	$0.771156\pi$
$102$	0	0
$103$	−3.28988	−0.324161	−0.162081	−	0.986778i	$-0.551820\pi$
$103$	−3.28988	−0.324161	−0.162081	+	0.986778i	$0.551820\pi$
$104$	0	0
$105$	−10.7433	−1.04843
$106$	0	0
$107$	−14.6282	−1.41417	−0.707083	−	0.707131i	$-0.749989\pi$
$107$	−14.6282	−1.41417	−0.707083	+	0.707131i	$0.749989\pi$
$108$	0	0
$109$	14.2353	1.36350	0.681749	−	0.731586i	$-0.261220\pi$
$109$	14.2353	1.36350	0.681749	+	0.731586i	$0.261220\pi$
$110$	0	0
$111$	−14.2207	−1.34977
$112$	0	0
$113$	−5.68139	−0.534460	−0.267230	−	0.963633i	$-0.586108\pi$
$113$	−5.68139	−0.534460	−0.267230	+	0.963633i	$0.586108\pi$
$114$	0	0
$115$	−6.99545	−0.652329
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0.755114	0.0692212
$120$	0	0
$121$	−7.54620	−0.686018
$122$	0	0
$123$	−10.8998	−0.982799
$124$	0	0
$125$	8.51587	0.761682
$126$	0	0
$127$	13.3900	1.18817	0.594085	−	0.804402i	$-0.297514\pi$
$127$	13.3900	1.18817	0.594085	+	0.804402i	$0.297514\pi$
$128$	0	0
$129$	−15.2733	−1.34474
$130$	0	0
$131$	−10.9600	−0.957579	−0.478790	−	0.877930i	$-0.658924\pi$
$131$	−10.9600	−0.957579	−0.478790	+	0.877930i	$0.658924\pi$
$132$	0	0
$133$	−17.6568	−1.53104
$134$	0	0
$135$	−15.8448	−1.36370
$136$	0	0
$137$	−23.0733	−1.97128	−0.985641	−	0.168853i	$-0.945994\pi$
$137$	−23.0733	−1.97128	−0.985641	+	0.168853i	$0.945994\pi$
$138$	0	0
$139$	−7.17759	−0.608795	−0.304398	−	0.952545i	$-0.598455\pi$
$139$	−7.17759	−0.608795	−0.304398	+	0.952545i	$0.598455\pi$
$140$	0	0
$141$	4.36324	0.367451
$142$	0	0
$143$	0	0
$144$	0	0
$145$	2.98172	0.247619
$146$	0	0
$147$	16.5005	1.36094
$148$	0	0
$149$	3.53065	0.289242	0.144621	−	0.989487i	$-0.453804\pi$
$149$	3.53065	0.289242	0.144621	+	0.989487i	$0.453804\pi$
$150$	0	0
$151$	17.2588	1.40450	0.702250	−	0.711930i	$-0.252179\pi$
$151$	17.2588	1.40450	0.702250	+	0.711930i	$0.252179\pi$
$152$	0	0
$153$	1.76893	0.143009
$154$	0	0
$155$	−7.37844	−0.592650
$156$	0	0
$157$	16.5103	1.31767	0.658833	−	0.752289i	$-0.271050\pi$
$157$	16.5103	1.31767	0.658833	+	0.752289i	$0.271050\pi$
$158$	0	0
$159$	−8.95082	−0.709847
$160$	0	0
$161$	25.9295	2.04353
$162$	0	0
$163$	5.36629	0.420320	0.210160	−	0.977667i	$-0.432601\pi$
$163$	5.36629	0.420320	0.210160	+	0.977667i	$0.432601\pi$
$164$	0	0
$165$	5.77496	0.449580
$166$	0	0
$167$	−18.8702	−1.46022	−0.730110	−	0.683329i	$-0.760531\pi$
$167$	−18.8702	−1.46022	−0.730110	+	0.683329i	$0.760531\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−41.3628	−3.16309
$172$	0	0
$173$	−18.7237	−1.42354	−0.711769	−	0.702414i	$-0.752106\pi$
$173$	−18.7237	−1.42354	−0.711769	+	0.702414i	$0.752106\pi$
$174$	0	0
$175$	−14.2786	−1.07936
$176$	0	0
$177$	−30.7689	−2.31273
$178$	0	0
$179$	8.82566	0.659661	0.329830	−	0.944040i	$-0.393008\pi$
$179$	8.82566	0.659661	0.329830	+	0.944040i	$0.393008\pi$
$180$	0	0
$181$	15.1087	1.12302	0.561512	−	0.827469i	$-0.310220\pi$
$181$	15.1087	1.12302	0.561512	+	0.827469i	$0.310220\pi$
$182$	0	0
$183$	−7.42203	−0.548652
$184$	0	0
$185$	3.98139	0.292718
$186$	0	0
$187$	−0.405906	−0.0296828
$188$	0	0
$189$	58.7307	4.27203
$190$	0	0
$191$	−4.09143	−0.296045	−0.148023	−	0.988984i	$-0.547291\pi$
$191$	−4.09143	−0.296045	−0.148023	+	0.988984i	$0.547291\pi$
$192$	0	0
$193$	−11.1688	−0.803948	−0.401974	−	0.915651i	$-0.631676\pi$
$193$	−11.1688	−0.803948	−0.401974	+	0.915651i	$0.631676\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.1264	0.935217	0.467609	−	0.883936i	$-0.345116\pi$
$197$	13.1264	0.935217	0.467609	+	0.883936i	$0.345116\pi$
$198$	0	0
$199$	−15.5056	−1.09916	−0.549581	−	0.835440i	$-0.685213\pi$
$199$	−15.5056	−1.09916	−0.549581	+	0.835440i	$0.685213\pi$
$200$	0	0
$201$	14.7883	1.04308
$202$	0	0
$203$	−11.0521	−0.775707
$204$	0	0
$205$	3.05164	0.213135
$206$	0	0
$207$	60.7422	4.22187
$208$	0	0
$209$	9.49129	0.656526
$210$	0	0
$211$	−1.12641	−0.0775451	−0.0387725	−	0.999248i	$-0.512345\pi$
$211$	−1.12641	−0.0775451	−0.0387725	+	0.999248i	$0.512345\pi$
$212$	0	0
$213$	−8.66521	−0.593730
$214$	0	0
$215$	4.27610	0.291628
$216$	0	0
$217$	27.3491	1.85658
$218$	0	0
$219$	10.9340	0.738853
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−22.4415	−1.50280	−0.751398	−	0.659849i	$-0.770620\pi$
$223$	−22.4415	−1.50280	−0.751398	+	0.659849i	$0.770620\pi$
$224$	0	0
$225$	−33.4490	−2.22994
$226$	0	0
$227$	7.40764	0.491662	0.245831	−	0.969313i	$-0.420939\pi$
$227$	7.40764	0.491662	0.245831	+	0.969313i	$0.420939\pi$
$228$	0	0
$229$	16.7592	1.10748	0.553740	−	0.832690i	$-0.313200\pi$
$229$	16.7592	1.10748	0.553740	+	0.832690i	$0.313200\pi$
$230$	0	0
$231$	−21.4056	−1.40838
$232$	0	0
$233$	−11.7255	−0.768165	−0.384082	−	0.923299i	$-0.625482\pi$
$233$	−11.7255	−0.768165	−0.384082	+	0.923299i	$0.625482\pi$
$234$	0	0
$235$	−1.22159	−0.0796876
$236$	0	0
$237$	24.4445	1.58784
$238$	0	0
$239$	−25.8853	−1.67438	−0.837191	−	0.546911i	$-0.815803\pi$
$239$	−25.8853	−1.67438	−0.837191	+	0.546911i	$0.815803\pi$
$240$	0	0
$241$	13.2725	0.854958	0.427479	−	0.904025i	$-0.359402\pi$
$241$	13.2725	0.854958	0.427479	+	0.904025i	$0.359402\pi$
$242$	0	0
$243$	56.6360	3.63320
$244$	0	0
$245$	−4.61969	−0.295141
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−11.9487	−0.757216
$250$	0	0
$251$	18.1539	1.14586	0.572932	−	0.819603i	$-0.305806\pi$
$251$	18.1539	1.14586	0.572932	+	0.819603i	$0.305806\pi$
$252$	0	0
$253$	−13.9382	−0.876287
$254$	0	0
$255$	−0.678699	−0.0425018
$256$	0	0
$257$	22.2850	1.39010	0.695051	−	0.718961i	$-0.255382\pi$
$257$	22.2850	1.39010	0.695051	+	0.718961i	$0.255382\pi$
$258$	0	0
$259$	−14.7575	−0.916987
$260$	0	0
$261$	−25.8906	−1.60259
$262$	0	0
$263$	−8.38041	−0.516758	−0.258379	−	0.966044i	$-0.583188\pi$
$263$	−8.38041	−0.516758	−0.258379	+	0.966044i	$0.583188\pi$
$264$	0	0
$265$	2.50598	0.153941
$266$	0	0
$267$	16.9091	1.03482
$268$	0	0
$269$	−17.6019	−1.07321	−0.536604	−	0.843834i	$-0.680293\pi$
$269$	−17.6019	−1.07321	−0.536604	+	0.843834i	$0.680293\pi$
$270$	0	0
$271$	−28.9167	−1.75656	−0.878281	−	0.478144i	$-0.841310\pi$
$271$	−28.9167	−1.75656	−0.878281	+	0.478144i	$0.841310\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	7.67537	0.462842
$276$	0	0
$277$	−1.11756	−0.0671479	−0.0335740	−	0.999436i	$-0.510689\pi$
$277$	−1.11756	−0.0671479	−0.0335740	+	0.999436i	$0.510689\pi$
$278$	0	0
$279$	64.0677	3.83564
$280$	0	0
$281$	6.32322	0.377212	0.188606	−	0.982053i	$-0.439603\pi$
$281$	6.32322	0.377212	0.188606	+	0.982053i	$0.439603\pi$
$282$	0	0
$283$	13.3202	0.791801	0.395901	−	0.918293i	$-0.370432\pi$
$283$	13.3202	0.791801	0.395901	+	0.918293i	$0.370432\pi$
$284$	0	0
$285$	15.8700	0.940057
$286$	0	0
$287$	−11.3113	−0.667682
$288$	0	0
$289$	−16.9523	−0.997194
$290$	0	0
$291$	44.6941	2.62001
$292$	0	0
$293$	8.09418	0.472867	0.236433	−	0.971648i	$-0.424021\pi$
$293$	8.09418	0.472867	0.236433	+	0.971648i	$0.424021\pi$
$294$	0	0
$295$	8.61444	0.501552
$296$	0	0
$297$	−31.5703	−1.83189
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−15.8499	−0.913572
$302$	0	0
$303$	−50.3900	−2.89483
$304$	0	0
$305$	2.07796	0.118984
$306$	0	0
$307$	5.26113	0.300269	0.150134	−	0.988666i	$-0.452029\pi$
$307$	5.26113	0.300269	0.150134	+	0.988666i	$0.452029\pi$
$308$	0	0
$309$	−10.9603	−0.623509
$310$	0	0
$311$	−1.62402	−0.0920895	−0.0460448	−	0.998939i	$-0.514662\pi$
$311$	−1.62402	−0.0920895	−0.0460448	+	0.998939i	$0.514662\pi$
$312$	0	0
$313$	−18.8923	−1.06785	−0.533927	−	0.845531i	$-0.679284\pi$
$313$	−18.8923	−1.06785	−0.533927	+	0.845531i	$0.679284\pi$
$314$	0	0
$315$	−26.1172	−1.47154
$316$	0	0
$317$	24.0494	1.35075	0.675375	−	0.737475i	$-0.263982\pi$
$317$	24.0494	1.35075	0.675375	+	0.737475i	$0.263982\pi$
$318$	0	0
$319$	5.94098	0.332631
$320$	0	0
$321$	−48.7343	−2.72008
$322$	0	0
$323$	−1.11546	−0.0620658
$324$	0	0
$325$	0	0
$326$	0	0
$327$	47.4253	2.62263
$328$	0	0
$329$	4.52796	0.249635
$330$	0	0
$331$	−18.8085	−1.03381	−0.516903	−	0.856044i	$-0.672915\pi$
$331$	−18.8085	−1.03381	−0.516903	+	0.856044i	$0.672915\pi$
$332$	0	0
$333$	−34.5709	−1.89447
$334$	0	0
$335$	−4.14030	−0.226209
$336$	0	0
$337$	−1.37821	−0.0750758	−0.0375379	−	0.999295i	$-0.511951\pi$
$337$	−1.37821	−0.0750758	−0.0375379	+	0.999295i	$0.511951\pi$
$338$	0	0
$339$	−18.9277	−1.02801
$340$	0	0
$341$	−14.7013	−0.796119
$342$	0	0
$343$	−7.07758	−0.382153
$344$	0	0
$345$	−23.3055	−1.25472
$346$	0	0
$347$	−27.4601	−1.47414	−0.737069	−	0.675818i	$-0.763791\pi$
$347$	−27.4601	−1.47414	−0.737069	+	0.675818i	$0.763791\pi$
$348$	0	0
$349$	−18.0165	−0.964402	−0.482201	−	0.876061i	$-0.660162\pi$
$349$	−18.0165	−0.964402	−0.482201	+	0.876061i	$0.660162\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	19.5820	1.04224	0.521121	−	0.853483i	$-0.325514\pi$
$353$	19.5820	1.04224	0.521121	+	0.853483i	$0.325514\pi$
$354$	0	0
$355$	2.42602	0.128760
$356$	0	0
$357$	2.51568	0.133144
$358$	0	0
$359$	9.14213	0.482503	0.241252	−	0.970463i	$-0.422442\pi$
$359$	9.14213	0.482503	0.241252	+	0.970463i	$0.422442\pi$
$360$	0	0
$361$	7.08275	0.372776
$362$	0	0
$363$	−25.1403	−1.31952
$364$	0	0
$365$	−3.06123	−0.160232
$366$	0	0
$367$	6.61003	0.345041	0.172520	−	0.985006i	$-0.444809\pi$
$367$	6.61003	0.345041	0.172520	+	0.985006i	$0.444809\pi$
$368$	0	0
$369$	−26.4977	−1.37941
$370$	0	0
$371$	−9.28873	−0.482247
$372$	0	0
$373$	35.0063	1.81256	0.906280	−	0.422678i	$-0.138910\pi$
$373$	35.0063	1.81256	0.906280	+	0.422678i	$0.138910\pi$
$374$	0	0
$375$	28.3708	1.46506
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−14.2255	−0.730717	−0.365358	−	0.930867i	$-0.619054\pi$
$379$	−14.2255	−0.730717	−0.365358	+	0.930867i	$0.619054\pi$
$380$	0	0
$381$	44.6090	2.28539
$382$	0	0
$383$	−29.3638	−1.50042	−0.750210	−	0.661199i	$-0.770048\pi$
$383$	−29.3638	−1.50042	−0.750210	+	0.661199i	$0.770048\pi$
$384$	0	0
$385$	5.99297	0.305430
$386$	0	0
$387$	−37.1298	−1.88741
$388$	0	0
$389$	13.8196	0.700680	0.350340	−	0.936623i	$-0.386066\pi$
$389$	13.8196	0.700680	0.350340	+	0.936623i	$0.386066\pi$
$390$	0	0
$391$	1.63808	0.0828412
$392$	0	0
$393$	−36.5134	−1.84186
$394$	0	0
$395$	−6.84380	−0.344349
$396$	0	0
$397$	−27.8004	−1.39526	−0.697630	−	0.716458i	$-0.745762\pi$
$397$	−27.8004	−1.39526	−0.697630	+	0.716458i	$0.745762\pi$
$398$	0	0
$399$	−58.8240	−2.94489
$400$	0	0
$401$	31.2370	1.55990	0.779951	−	0.625841i	$-0.215244\pi$
$401$	31.2370	1.55990	0.779951	+	0.625841i	$0.215244\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−30.1246	−1.49690
$406$	0	0
$407$	7.93279	0.393214
$408$	0	0
$409$	20.9729	1.03704	0.518522	−	0.855065i	$-0.326483\pi$
$409$	20.9729	1.03704	0.518522	+	0.855065i	$0.326483\pi$
$410$	0	0
$411$	−76.8691	−3.79167
$412$	0	0
$413$	−31.9304	−1.57119
$414$	0	0
$415$	3.34530	0.164214
$416$	0	0
$417$	−23.9123	−1.17099
$418$	0	0
$419$	31.2949	1.52885	0.764427	−	0.644711i	$-0.223022\pi$
$419$	31.2949	1.52885	0.764427	+	0.644711i	$0.223022\pi$
$420$	0	0
$421$	−25.4264	−1.23921	−0.619603	−	0.784915i	$-0.712706\pi$
$421$	−25.4264	−1.23921	−0.619603	+	0.784915i	$0.712706\pi$
$422$	0	0
$423$	10.6072	0.515739
$424$	0	0
$425$	−0.902044	−0.0437556
$426$	0	0
$427$	−7.70222	−0.372737
$428$	0	0
$429$	0	0
$430$	0	0
$431$	7.24871	0.349158	0.174579	−	0.984643i	$-0.444144\pi$
$431$	7.24871	0.349158	0.174579	+	0.984643i	$0.444144\pi$
$432$	0	0
$433$	5.16009	0.247978	0.123989	−	0.992284i	$-0.460431\pi$
$433$	5.16009	0.247978	0.123989	+	0.992284i	$0.460431\pi$
$434$	0	0
$435$	9.93367	0.476283
$436$	0	0
$437$	−38.3031	−1.83229
$438$	0	0
$439$	20.4486	0.975958	0.487979	−	0.872855i	$-0.337734\pi$
$439$	20.4486	0.975958	0.487979	+	0.872855i	$0.337734\pi$
$440$	0	0
$441$	40.1133	1.91016
$442$	0	0
$443$	−17.8307	−0.847161	−0.423581	−	0.905858i	$-0.639227\pi$
$443$	−17.8307	−0.847161	−0.423581	+	0.905858i	$0.639227\pi$
$444$	0	0
$445$	−4.73407	−0.224417
$446$	0	0
$447$	11.7624	0.556344
$448$	0	0
$449$	−1.29579	−0.0611521	−0.0305760	−	0.999532i	$-0.509734\pi$
$449$	−1.29579	−0.0611521	−0.0305760	+	0.999532i	$0.509734\pi$
$450$	0	0
$451$	6.08028	0.286309
$452$	0	0
$453$	57.4980	2.70149
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−38.5698	−1.80422	−0.902110	−	0.431506i	$-0.857982\pi$
$457$	−38.5698	−1.80422	−0.902110	+	0.431506i	$0.857982\pi$
$458$	0	0
$459$	3.71028	0.173181
$460$	0	0
$461$	35.8260	1.66858	0.834291	−	0.551325i	$-0.185878\pi$
$461$	35.8260	1.66858	0.834291	+	0.551325i	$0.185878\pi$
$462$	0	0
$463$	−4.21294	−0.195792	−0.0978959	−	0.995197i	$-0.531211\pi$
$463$	−4.21294	−0.195792	−0.0978959	+	0.995197i	$0.531211\pi$
$464$	0	0
$465$	−24.5814	−1.13994
$466$	0	0
$467$	−5.00238	−0.231482	−0.115741	−	0.993279i	$-0.536924\pi$
$467$	−5.00238	−0.231482	−0.115741	+	0.993279i	$0.536924\pi$
$468$	0	0
$469$	15.3465	0.708637
$470$	0	0
$471$	55.0044	2.53447
$472$	0	0
$473$	8.51998	0.391749
$474$	0	0
$475$	21.0925	0.967789
$476$	0	0
$477$	−21.7597	−0.996309
$478$	0	0
$479$	32.6977	1.49400	0.746998	−	0.664827i	$-0.231495\pi$
$479$	32.6977	1.49400	0.746998	+	0.664827i	$0.231495\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	86.3845	3.93063
$484$	0	0
$485$	−12.5131	−0.568191
$486$	0	0
$487$	18.1107	0.820676	0.410338	−	0.911933i	$-0.365411\pi$
$487$	18.1107	0.820676	0.410338	+	0.911933i	$0.365411\pi$
$488$	0	0
$489$	17.8779	0.808467
$490$	0	0
$491$	−12.9720	−0.585419	−0.292709	−	0.956201i	$-0.594557\pi$
$491$	−12.9720	−0.585419	−0.292709	+	0.956201i	$0.594557\pi$
$492$	0	0
$493$	−0.698211	−0.0314458
$494$	0	0
$495$	14.0391	0.631010
$496$	0	0
$497$	−8.99234	−0.403361
$498$	0	0
$499$	17.3094	0.774877	0.387438	−	0.921896i	$-0.373360\pi$
$499$	17.3094	0.774877	0.387438	+	0.921896i	$0.373360\pi$
$500$	0	0
$501$	−62.8665	−2.80867
$502$	0	0
$503$	−8.97374	−0.400119	−0.200060	−	0.979784i	$-0.564114\pi$
$503$	−8.97374	−0.400119	−0.200060	+	0.979784i	$0.564114\pi$
$504$	0	0
$505$	14.1078	0.627789
$506$	0	0
$507$	0	0
$508$	0	0
$509$	0.191900	0.00850581	0.00425291	−	0.999991i	$-0.498646\pi$
$509$	0.191900	0.00850581	0.00425291	+	0.999991i	$0.498646\pi$
$510$	0	0
$511$	11.3468	0.501953
$512$	0	0
$513$	−86.7573	−3.83043
$514$	0	0
$515$	3.06858	0.135218
$516$	0	0
$517$	−2.43397	−0.107046
$518$	0	0
$519$	−62.3784	−2.73811
$520$	0	0
$521$	30.7590	1.34757	0.673787	−	0.738925i	$-0.264666\pi$
$521$	30.7590	1.34757	0.673787	+	0.738925i	$0.264666\pi$
$522$	0	0
$523$	6.70505	0.293191	0.146596	−	0.989197i	$-0.453168\pi$
$523$	6.70505	0.293191	0.146596	+	0.989197i	$0.453168\pi$
$524$	0	0
$525$	−47.5696	−2.07611
$526$	0	0
$527$	1.72776	0.0752624
$528$	0	0
$529$	33.2491	1.44561
$530$	0	0
$531$	−74.8001	−3.24605
$532$	0	0
$533$	0	0
$534$	0	0
$535$	13.6443	0.589893
$536$	0	0
$537$	29.4029	1.26883
$538$	0	0
$539$	−9.20458	−0.396470
$540$	0	0
$541$	37.7951	1.62494	0.812469	−	0.583004i	$-0.198123\pi$
$541$	37.7951	1.62494	0.812469	+	0.583004i	$0.198123\pi$
$542$	0	0
$543$	50.3351	2.16008
$544$	0	0
$545$	−13.2778	−0.568758
$546$	0	0
$547$	19.9942	0.854890	0.427445	−	0.904041i	$-0.359414\pi$
$547$	19.9942	0.854890	0.427445	+	0.904041i	$0.359414\pi$
$548$	0	0
$549$	−18.0432	−0.770064
$550$	0	0
$551$	16.3262	0.695522
$552$	0	0
$553$	25.3674	1.07873
$554$	0	0
$555$	13.2641	0.563029
$556$	0	0
$557$	−11.3709	−0.481802	−0.240901	−	0.970550i	$-0.577443\pi$
$557$	−11.3709	−0.481802	−0.240901	+	0.970550i	$0.577443\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−1.35228	−0.0570935
$562$	0	0
$563$	−6.76238	−0.285000	−0.142500	−	0.989795i	$-0.545514\pi$
$563$	−6.76238	−0.285000	−0.142500	+	0.989795i	$0.545514\pi$
$564$	0	0
$565$	5.29922	0.222940
$566$	0	0
$567$	111.661	4.68930
$568$	0	0
$569$	−19.4414	−0.815027	−0.407514	−	0.913199i	$-0.633604\pi$
$569$	−19.4414	−0.815027	−0.407514	+	0.913199i	$0.633604\pi$
$570$	0	0
$571$	35.8153	1.49882	0.749412	−	0.662104i	$-0.230336\pi$
$571$	35.8153	1.49882	0.749412	+	0.662104i	$0.230336\pi$
$572$	0	0
$573$	−13.6307	−0.569430
$574$	0	0
$575$	−30.9748	−1.29174
$576$	0	0
$577$	−5.53268	−0.230329	−0.115164	−	0.993346i	$-0.536739\pi$
$577$	−5.53268	−0.230329	−0.115164	+	0.993346i	$0.536739\pi$
$578$	0	0
$579$	−37.2091	−1.54636
$580$	0	0
$581$	−12.3997	−0.514428
$582$	0	0
$583$	4.99309	0.206793
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−28.6314	−1.18174	−0.590872	−	0.806765i	$-0.701216\pi$
$587$	−28.6314	−1.18174	−0.590872	+	0.806765i	$0.701216\pi$
$588$	0	0
$589$	−40.4002	−1.66466
$590$	0	0
$591$	43.7309	1.79885
$592$	0	0
$593$	3.53610	0.145210	0.0726051	−	0.997361i	$-0.476869\pi$
$593$	3.53610	0.145210	0.0726051	+	0.997361i	$0.476869\pi$
$594$	0	0
$595$	−0.704320	−0.0288743
$596$	0	0
$597$	−51.6572	−2.11419
$598$	0	0
$599$	−31.7899	−1.29890	−0.649450	−	0.760404i	$-0.725001\pi$
$599$	−31.7899	−1.29890	−0.649450	+	0.760404i	$0.725001\pi$
$600$	0	0
$601$	−1.82658	−0.0745076	−0.0372538	−	0.999306i	$-0.511861\pi$
$601$	−1.82658	−0.0745076	−0.0372538	+	0.999306i	$0.511861\pi$
$602$	0	0
$603$	35.9507	1.46403
$604$	0	0
$605$	7.03859	0.286160
$606$	0	0
$607$	16.9798	0.689190	0.344595	−	0.938751i	$-0.388016\pi$
$607$	16.9798	0.689190	0.344595	+	0.938751i	$0.388016\pi$
$608$	0	0
$609$	−36.8204	−1.49204
$610$	0	0
$611$	0	0
$612$	0	0
$613$	1.90736	0.0770377	0.0385189	−	0.999258i	$-0.487736\pi$
$613$	1.90736	0.0770377	0.0385189	+	0.999258i	$0.487736\pi$
$614$	0	0
$615$	10.1666	0.409956
$616$	0	0
$617$	−14.4962	−0.583596	−0.291798	−	0.956480i	$-0.594254\pi$
$617$	−14.4962	−0.583596	−0.291798	+	0.956480i	$0.594254\pi$
$618$	0	0
$619$	13.7192	0.551422	0.275711	−	0.961241i	$-0.411087\pi$
$619$	13.7192	0.551422	0.275711	+	0.961241i	$0.411087\pi$
$620$	0	0
$621$	127.405	5.11260
$622$	0	0
$623$	17.5474	0.703023
$624$	0	0
$625$	12.7070	0.508280
$626$	0	0
$627$	31.6204	1.26280
$628$	0	0
$629$	−0.932297	−0.0371731
$630$	0	0
$631$	38.9533	1.55071	0.775354	−	0.631527i	$-0.217571\pi$
$631$	38.9533	1.55071	0.775354	+	0.631527i	$0.217571\pi$
$632$	0	0
$633$	−3.75265	−0.149154
$634$	0	0
$635$	−12.4893	−0.495623
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−21.0654	−0.833334
$640$	0	0
$641$	38.0061	1.50115	0.750575	−	0.660785i	$-0.229776\pi$
$641$	38.0061	1.50115	0.750575	+	0.660785i	$0.229776\pi$
$642$	0	0
$643$	−15.6026	−0.615308	−0.307654	−	0.951498i	$-0.599544\pi$
$643$	−15.6026	−0.615308	−0.307654	+	0.951498i	$0.599544\pi$
$644$	0	0
$645$	14.2459	0.560932
$646$	0	0
$647$	−11.0471	−0.434306	−0.217153	−	0.976138i	$-0.569677\pi$
$647$	−11.0471	−0.434306	−0.217153	+	0.976138i	$0.569677\pi$
$648$	0	0
$649$	17.1640	0.673745
$650$	0	0
$651$	91.1139	3.57104
$652$	0	0
$653$	36.7657	1.43875	0.719376	−	0.694621i	$-0.244428\pi$
$653$	36.7657	1.43875	0.719376	+	0.694621i	$0.244428\pi$
$654$	0	0
$655$	10.2228	0.399436
$656$	0	0
$657$	26.5809	1.03702
$658$	0	0
$659$	41.3963	1.61257	0.806285	−	0.591528i	$-0.201475\pi$
$659$	41.3963	1.61257	0.806285	+	0.591528i	$0.201475\pi$
$660$	0	0
$661$	−9.45915	−0.367918	−0.183959	−	0.982934i	$-0.558891\pi$
$661$	−9.45915	−0.367918	−0.183959	+	0.982934i	$0.558891\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	16.4691	0.638645
$666$	0	0
$667$	−23.9755	−0.928335
$668$	0	0
$669$	−74.7644	−2.89056
$670$	0	0
$671$	4.14027	0.159833
$672$	0	0
$673$	4.56339	0.175906	0.0879528	−	0.996125i	$-0.471968\pi$
$673$	4.56339	0.175906	0.0879528	+	0.996125i	$0.471968\pi$
$674$	0	0
$675$	−70.1585	−2.70040
$676$	0	0
$677$	41.2034	1.58358	0.791788	−	0.610796i	$-0.209150\pi$
$677$	41.2034	1.58358	0.791788	+	0.610796i	$0.209150\pi$
$678$	0	0
$679$	46.3813	1.77995
$680$	0	0
$681$	24.6787	0.945689
$682$	0	0
$683$	−40.6939	−1.55711	−0.778555	−	0.627577i	$-0.784047\pi$
$683$	−40.6939	−1.55711	−0.778555	+	0.627577i	$0.784047\pi$
$684$	0	0
$685$	21.5212	0.822283
$686$	0	0
$687$	55.8337	2.13019
$688$	0	0
$689$	0	0
$690$	0	0
$691$	14.0852	0.535826	0.267913	−	0.963443i	$-0.413666\pi$
$691$	14.0852	0.535826	0.267913	+	0.963443i	$0.413666\pi$
$692$	0	0
$693$	−52.0376	−1.97674
$694$	0	0
$695$	6.69478	0.253948
$696$	0	0
$697$	−0.714582	−0.0270667
$698$	0	0
$699$	−39.0638	−1.47753
$700$	0	0
$701$	24.4122	0.922035	0.461018	−	0.887391i	$-0.347485\pi$
$701$	24.4122	0.922035	0.461018	+	0.887391i	$0.347485\pi$
$702$	0	0
$703$	21.7999	0.822198
$704$	0	0
$705$	−4.06974	−0.153275
$706$	0	0
$707$	−52.2923	−1.96665
$708$	0	0
$709$	2.10926	0.0792150	0.0396075	−	0.999215i	$-0.487389\pi$
$709$	2.10926	0.0792150	0.0396075	+	0.999215i	$0.487389\pi$
$710$	0	0
$711$	59.4254	2.22863
$712$	0	0
$713$	59.3286	2.22188
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−86.2374	−3.22059
$718$	0	0
$719$	20.1666	0.752087	0.376044	−	0.926602i	$-0.377284\pi$
$719$	20.1666	0.752087	0.376044	+	0.926602i	$0.377284\pi$
$720$	0	0
$721$	−11.3741	−0.423592
$722$	0	0
$723$	44.2177	1.64447
$724$	0	0
$725$	13.2026	0.490334
$726$	0	0
$727$	16.7786	0.622282	0.311141	−	0.950364i	$-0.399289\pi$
$727$	16.7786	0.622282	0.311141	+	0.950364i	$0.399289\pi$
$728$	0	0
$729$	91.7927	3.39973
$730$	0	0
$731$	−1.00131	−0.0370347
$732$	0	0
$733$	−10.2265	−0.377726	−0.188863	−	0.982003i	$-0.560480\pi$
$733$	−10.2265	−0.377726	−0.188863	+	0.982003i	$0.560480\pi$
$734$	0	0
$735$	−15.3906	−0.567691
$736$	0	0
$737$	−8.24941	−0.303871
$738$	0	0
$739$	−12.4262	−0.457106	−0.228553	−	0.973531i	$-0.573399\pi$
$739$	−12.4262	−0.457106	−0.228553	+	0.973531i	$0.573399\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−6.78280	−0.248837	−0.124418	−	0.992230i	$-0.539706\pi$
$743$	−6.78280	−0.248837	−0.124418	+	0.992230i	$0.539706\pi$
$744$	0	0
$745$	−3.29316	−0.120652
$746$	0	0
$747$	−29.0476	−1.06279
$748$	0	0
$749$	−50.5741	−1.84794
$750$	0	0
$751$	21.0479	0.768049	0.384024	−	0.923323i	$-0.374538\pi$
$751$	21.0479	0.768049	0.384024	+	0.923323i	$0.374538\pi$
$752$	0	0
$753$	60.4801	2.20402
$754$	0	0
$755$	−16.0979	−0.585861
$756$	0	0
$757$	40.6837	1.47867	0.739337	−	0.673336i	$-0.235139\pi$
$757$	40.6837	1.47867	0.739337	+	0.673336i	$0.235139\pi$
$758$	0	0
$759$	−46.4354	−1.68550
$760$	0	0
$761$	54.5284	1.97665	0.988327	−	0.152346i	$-0.0486828\pi$
$761$	54.5284	1.97665	0.988327	+	0.152346i	$0.0486828\pi$
$762$	0	0
$763$	49.2157	1.78173
$764$	0	0
$765$	−1.64994	−0.0596536
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−36.8009	−1.32707	−0.663536	−	0.748144i	$-0.730945\pi$
$769$	−36.8009	−1.32707	−0.663536	+	0.748144i	$0.730945\pi$
$770$	0	0
$771$	74.2430	2.67380
$772$	0	0
$773$	16.8191	0.604941	0.302471	−	0.953159i	$-0.402189\pi$
$773$	16.8191	0.604941	0.302471	+	0.953159i	$0.402189\pi$
$774$	0	0
$775$	−32.6706	−1.17356
$776$	0	0
$777$	−49.1649	−1.76378
$778$	0	0
$779$	16.7090	0.598664
$780$	0	0
$781$	4.83376	0.172966
$782$	0	0
$783$	−54.3049	−1.94070
$784$	0	0
$785$	−15.3997	−0.549639
$786$	0	0
$787$	38.5289	1.37341	0.686703	−	0.726938i	$-0.259057\pi$
$787$	38.5289	1.37341	0.686703	+	0.726938i	$0.259057\pi$
$788$	0	0
$789$	−27.9195	−0.993960
$790$	0	0
$791$	−19.6422	−0.698397
$792$	0	0
$793$	0	0
$794$	0	0
$795$	8.34874	0.296099
$796$	0	0
$797$	1.09737	0.0388709	0.0194354	−	0.999811i	$-0.493813\pi$
$797$	1.09737	0.0388709	0.0194354	+	0.999811i	$0.493813\pi$
$798$	0	0
$799$	0.286051	0.0101198
$800$	0	0
$801$	41.1065	1.45243
$802$	0	0
$803$	−6.09939	−0.215243
$804$	0	0
$805$	−24.1853	−0.852419
$806$	0	0
$807$	−58.6412	−2.06427
$808$	0	0
$809$	13.7000	0.481667	0.240834	−	0.970566i	$-0.422579\pi$
$809$	13.7000	0.481667	0.240834	+	0.970566i	$0.422579\pi$
$810$	0	0
$811$	−32.0498	−1.12542	−0.562711	−	0.826654i	$-0.690242\pi$
$811$	−32.0498	−1.12542	−0.562711	+	0.826654i	$0.690242\pi$
$812$	0	0
$813$	−96.3365	−3.37867
$814$	0	0
$815$	−5.00532	−0.175329
$816$	0	0
$817$	23.4135	0.819135
$818$	0	0
$819$	0	0
$820$	0	0
$821$	33.7727	1.17867	0.589337	−	0.807887i	$-0.299389\pi$
$821$	33.7727	1.17867	0.589337	+	0.807887i	$0.299389\pi$
$822$	0	0
$823$	−27.1260	−0.945555	−0.472777	−	0.881182i	$-0.656748\pi$
$823$	−27.1260	−0.945555	−0.472777	+	0.881182i	$0.656748\pi$
$824$	0	0
$825$	25.5707	0.890256
$826$	0	0
$827$	5.21060	0.181190	0.0905952	−	0.995888i	$-0.471123\pi$
$827$	5.21060	0.181190	0.0905952	+	0.995888i	$0.471123\pi$
$828$	0	0
$829$	−28.8248	−1.00113	−0.500564	−	0.865700i	$-0.666874\pi$
$829$	−28.8248	−1.00113	−0.500564	+	0.865700i	$0.666874\pi$
$830$	0	0
$831$	−3.72319	−0.129156
$832$	0	0
$833$	1.08176	0.0374809
$834$	0	0
$835$	17.6009	0.609104
$836$	0	0
$837$	134.380	4.64487
$838$	0	0
$839$	0.196498	0.00678385	0.00339193	−	0.999994i	$-0.498920\pi$
$839$	0.196498	0.00678385	0.00339193	+	0.999994i	$0.498920\pi$
$840$	0	0
$841$	−18.7807	−0.647612
$842$	0	0
$843$	21.0659	0.725550
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−26.0894	−0.896442
$848$	0	0
$849$	44.3764	1.52299
$850$	0	0
$851$	−32.0136	−1.09741
$852$	0	0
$853$	−18.3578	−0.628559	−0.314280	−	0.949330i	$-0.601763\pi$
$853$	−18.3578	−0.628559	−0.314280	+	0.949330i	$0.601763\pi$
$854$	0	0
$855$	38.5804	1.31942
$856$	0	0
$857$	−20.0212	−0.683910	−0.341955	−	0.939716i	$-0.611089\pi$
$857$	−20.0212	−0.683910	−0.341955	+	0.939716i	$0.611089\pi$
$858$	0	0
$859$	14.5650	0.496951	0.248476	−	0.968638i	$-0.420070\pi$
$859$	14.5650	0.496951	0.248476	+	0.968638i	$0.420070\pi$
$860$	0	0
$861$	−37.6837	−1.28426
$862$	0	0
$863$	−40.2185	−1.36905	−0.684526	−	0.728988i	$-0.739991\pi$
$863$	−40.2185	−1.36905	−0.684526	+	0.728988i	$0.739991\pi$
$864$	0	0
$865$	17.4642	0.593802
$866$	0	0
$867$	−56.4769	−1.91806
$868$	0	0
$869$	−13.6360	−0.462571
$870$	0	0
$871$	0	0
$872$	0	0
$873$	108.653	3.67733
$874$	0	0
$875$	29.4418	0.995315
$876$	0	0
$877$	−3.77692	−0.127538	−0.0637688	−	0.997965i	$-0.520312\pi$
$877$	−3.77692	−0.127538	−0.0637688	+	0.997965i	$0.520312\pi$
$878$	0	0
$879$	26.9659	0.909538
$880$	0	0
$881$	29.8489	1.00563	0.502817	−	0.864393i	$-0.332297\pi$
$881$	29.8489	1.00563	0.502817	+	0.864393i	$0.332297\pi$
$882$	0	0
$883$	−40.4518	−1.36131	−0.680655	−	0.732604i	$-0.738305\pi$
$883$	−40.4518	−1.36131	−0.680655	+	0.732604i	$0.738305\pi$
$884$	0	0
$885$	28.6992	0.964712
$886$	0	0
$887$	27.2244	0.914106	0.457053	−	0.889439i	$-0.348905\pi$
$887$	27.2244	0.914106	0.457053	+	0.889439i	$0.348905\pi$
$888$	0	0
$889$	46.2931	1.55262
$890$	0	0
$891$	−60.0223	−2.01082
$892$	0	0
$893$	−6.68873	−0.223830
$894$	0	0
$895$	−8.23199	−0.275165
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−25.2881	−0.843406
$900$	0	0
$901$	−0.586810	−0.0195495
$902$	0	0
$903$	−52.8042	−1.75721
$904$	0	0
$905$	−14.0924	−0.468448
$906$	0	0
$907$	26.5996	0.883225	0.441613	−	0.897206i	$-0.354407\pi$
$907$	26.5996	0.883225	0.441613	+	0.897206i	$0.354407\pi$
$908$	0	0
$909$	−122.500	−4.06306
$910$	0	0
$911$	−22.0346	−0.730040	−0.365020	−	0.931000i	$-0.618938\pi$
$911$	−22.0346	−0.730040	−0.365020	+	0.931000i	$0.618938\pi$
$912$	0	0
$913$	6.66539	0.220592
$914$	0	0
$915$	6.92278	0.228860
$916$	0	0
$917$	−37.8919	−1.25130
$918$	0	0
$919$	−47.9821	−1.58278	−0.791392	−	0.611309i	$-0.790643\pi$
$919$	−47.9821	−1.58278	−0.791392	+	0.611309i	$0.790643\pi$
$920$	0	0
$921$	17.5276	0.577553
$922$	0	0
$923$	0	0
$924$	0	0
$925$	17.6290	0.579639
$926$	0	0
$927$	−26.6448	−0.875130
$928$	0	0
$929$	28.0227	0.919395	0.459697	−	0.888076i	$-0.347958\pi$
$929$	28.0227	0.919395	0.459697	+	0.888076i	$0.347958\pi$
$930$	0	0
$931$	−25.2949	−0.829005
$932$	0	0
$933$	−5.41044	−0.177130
$934$	0	0
$935$	0.378602	0.0123816
$936$	0	0
$937$	2.99297	0.0977761	0.0488880	−	0.998804i	$-0.484432\pi$
$937$	2.99297	0.0977761	0.0488880	+	0.998804i	$0.484432\pi$
$938$	0	0
$939$	−62.9400	−2.05397
$940$	0	0
$941$	−25.7144	−0.838267	−0.419134	−	0.907925i	$-0.637666\pi$
$941$	−25.7144	−0.838267	−0.419134	+	0.907925i	$0.637666\pi$
$942$	0	0
$943$	−24.5376	−0.799056
$944$	0	0
$945$	−54.7801	−1.78200
$946$	0	0
$947$	13.8061	0.448637	0.224318	−	0.974516i	$-0.427984\pi$
$947$	13.8061	0.448637	0.224318	+	0.974516i	$0.427984\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	80.1211	2.59810
$952$	0	0
$953$	−23.8053	−0.771129	−0.385565	−	0.922681i	$-0.625993\pi$
$953$	−23.8053	−0.771129	−0.385565	+	0.922681i	$0.625993\pi$
$954$	0	0
$955$	3.81621	0.123490
$956$	0	0
$957$	19.7925	0.639801
$958$	0	0
$959$	−79.7710	−2.57594
$960$	0	0
$961$	31.5768	1.01861
$962$	0	0
$963$	−118.474	−3.81779
$964$	0	0
$965$	10.4175	0.335352
$966$	0	0
$967$	12.4663	0.400890	0.200445	−	0.979705i	$-0.435761\pi$
$967$	12.4663	0.400890	0.200445	+	0.979705i	$0.435761\pi$
$968$	0	0
$969$	−3.71617	−0.119381
$970$	0	0
$971$	12.4370	0.399123	0.199562	−	0.979885i	$-0.436048\pi$
$971$	12.4370	0.399123	0.199562	+	0.979885i	$0.436048\pi$
$972$	0	0
$973$	−24.8150	−0.795533
$974$	0	0
$975$	0	0
$976$	0	0
$977$	25.3036	0.809534	0.404767	−	0.914420i	$-0.367353\pi$
$977$	25.3036	0.809534	0.404767	+	0.914420i	$0.367353\pi$
$978$	0	0
$979$	−9.43248	−0.301464
$980$	0	0
$981$	115.292	3.68100
$982$	0	0
$983$	30.0391	0.958099	0.479050	−	0.877788i	$-0.340981\pi$
$983$	30.0391	0.958099	0.479050	+	0.877788i	$0.340981\pi$
$984$	0	0
$985$	−12.2434	−0.390108
$986$	0	0
$987$	15.0850	0.480161
$988$	0	0
$989$	−34.3833	−1.09333
$990$	0	0
$991$	10.5289	0.334463	0.167231	−	0.985918i	$-0.446517\pi$
$991$	10.5289	0.334463	0.167231	+	0.985918i	$0.446517\pi$
$992$	0	0
$993$	−62.6607	−1.98848
$994$	0	0
$995$	14.4626	0.458495
$996$	0	0
$997$	1.29608	0.0410472	0.0205236	−	0.999789i	$-0.493467\pi$
$997$	1.29608	0.0410472	0.0205236	+	0.999789i	$0.493467\pi$
$998$	0	0
$999$	−72.5115	−2.29416

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1352.2.a.n.1.6	yes	6
4.3	odd	2		2704.2.a.bg.1.1		6
13.2	odd	12		1352.2.o.h.1161.2		24
13.3	even	3		1352.2.i.n.529.1		12
13.4	even	6		1352.2.i.m.1329.1		12
13.5	odd	4		1352.2.f.g.337.12		12
13.6	odd	12		1352.2.o.h.361.1		24
13.7	odd	12		1352.2.o.h.361.2		24
13.8	odd	4		1352.2.f.g.337.11		12
13.9	even	3		1352.2.i.n.1329.1		12
13.10	even	6		1352.2.i.m.529.1		12
13.11	odd	12		1352.2.o.h.1161.1		24
13.12	even	2		1352.2.a.m.1.6	✓	6
52.31	even	4		2704.2.f.r.337.2		12
52.47	even	4		2704.2.f.r.337.1		12
52.51	odd	2		2704.2.a.bf.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1352.2.a.m.1.6	✓	6	13.12	even	2
1352.2.a.n.1.6	yes	6	1.1	even	1	trivial
1352.2.f.g.337.11		12	13.8	odd	4
1352.2.f.g.337.12		12	13.5	odd	4
1352.2.i.m.529.1		12	13.10	even	6
1352.2.i.m.1329.1		12	13.4	even	6
1352.2.i.n.529.1		12	13.3	even	3
1352.2.i.n.1329.1		12	13.9	even	3
1352.2.o.h.361.1		24	13.6	odd	12
1352.2.o.h.361.2		24	13.7	odd	12
1352.2.o.h.1161.1		24	13.11	odd	12
1352.2.o.h.1161.2		24	13.2	odd	12
2704.2.a.bf.1.1		6	52.51	odd	2
2704.2.a.bg.1.1		6	4.3	odd	2
2704.2.f.r.337.1		12	52.47	even	4
2704.2.f.r.337.2		12	52.31	even	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+3.33152 q^{3} -0.932734 q^{5} +3.45729 q^{7} +8.09903 q^{9} +O(q^{10})\)	\(q+3.33152 q^{3} -0.932734 q^{5} +3.45729 q^{7} +8.09903 q^{9} -1.85844 q^{11} -3.10742 q^{15} +0.218412 q^{17} -5.10713 q^{19} +11.5180 q^{21} +7.49994 q^{23} -4.13001 q^{25} +16.9875 q^{27} -3.19676 q^{29} +7.91055 q^{31} -6.19143 q^{33} -3.22473 q^{35} -4.26852 q^{37} -3.27171 q^{41} -4.58448 q^{43} -7.55423 q^{45} +1.30969 q^{47} +4.95285 q^{49} +0.727644 q^{51} -2.68671 q^{53} +1.73343 q^{55} -17.0145 q^{57} -9.23569 q^{59} -2.22782 q^{61} +28.0007 q^{63} +4.43889 q^{67} +24.9862 q^{69} -2.60098 q^{71} +3.28199 q^{73} -13.7592 q^{75} -6.42517 q^{77} +7.33736 q^{79} +32.2971 q^{81} -3.58655 q^{83} -0.203720 q^{85} -10.6501 q^{87} +5.07548 q^{89} +26.3542 q^{93} +4.76359 q^{95} +13.4155 q^{97} -15.0516 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{3} + 2 q^{5} - 3 q^{7} + 9 q^{9}+O(q^{10}) \) 6 * q + 3 * q^3 + 2 * q^5 - 3 * q^7 + 9 * q^9	\( 6 q + 3 q^{3} + 2 q^{5} - 3 q^{7} + 9 q^{9} - 13 q^{11} + 14 q^{15} + 11 q^{17} - 15 q^{19} + 22 q^{21} + 15 q^{23} + 16 q^{25} + 21 q^{27} + 9 q^{29} + 3 q^{31} + 2 q^{33} + 14 q^{35} - 14 q^{37} + 20 q^{41} + 10 q^{43} + 9 q^{45} + 10 q^{47} + 27 q^{49} + 5 q^{51} + q^{53} + 14 q^{55} - 9 q^{57} - 50 q^{59} - 2 q^{63} - 6 q^{67} + 32 q^{69} - 9 q^{71} + 6 q^{73} + 40 q^{75} - 2 q^{77} + 13 q^{79} + 42 q^{81} - 36 q^{83} + 31 q^{85} + 22 q^{87} + 18 q^{89} + 12 q^{93} - 21 q^{95} + 14 q^{97} - 18 q^{99}+O(q^{100}) \) 6 * q + 3 * q^3 + 2 * q^5 - 3 * q^7 + 9 * q^9 - 13 * q^11 + 14 * q^15 + 11 * q^17 - 15 * q^19 + 22 * q^21 + 15 * q^23 + 16 * q^25 + 21 * q^27 + 9 * q^29 + 3 * q^31 + 2 * q^33 + 14 * q^35 - 14 * q^37 + 20 * q^41 + 10 * q^43 + 9 * q^45 + 10 * q^47 + 27 * q^49 + 5 * q^51 + q^53 + 14 * q^55 - 9 * q^57 - 50 * q^59 - 2 * q^63 - 6 * q^67 + 32 * q^69 - 9 * q^71 + 6 * q^73 + 40 * q^75 - 2 * q^77 + 13 * q^79 + 42 * q^81 - 36 * q^83 + 31 * q^85 + 22 * q^87 + 18 * q^89 + 12 * q^93 - 21 * q^95 + 14 * q^97 - 18 * q^99

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