LMFDB - Embedded newform 2368.2.a.bi.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2368,2,Mod(1,2368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2368, 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("2368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2368 = 2^{6} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2368.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:	$18.9085751986$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.2225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 2x + 4 $ x^4 - x^3 - 5x^2 + 2x + 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1184)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.43828$ of defining polynomial
Character	$\chi$	$=$	2368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.43828 q^{3} -0.797785 q^{5} -2.50694 q^{7} -0.931341 q^{9} +O(q^{10})$	$q-1.43828 q^{3} -0.797785 q^{5} -2.50694 q^{7} -0.931341 q^{9} +6.18129 q^{11} +0.526911 q^{13} +1.14744 q^{15} -2.00000 q^{17} +0.986116 q^{19} +3.60569 q^{21} +0.486973 q^{23} -4.36354 q^{25} +5.65438 q^{27} +0.0825425 q^{29} -0.325649 q^{31} -8.89045 q^{33} +2.00000 q^{35} -1.00000 q^{37} -0.757847 q^{39} +6.56480 q^{41} +8.15756 q^{43} +0.743010 q^{45} +2.97908 q^{47} -0.715242 q^{49} +2.87657 q^{51} -5.74301 q^{53} -4.93134 q^{55} -1.41831 q^{57} +3.59557 q^{59} -14.9831 q^{61} +2.33482 q^{63} -0.420361 q^{65} -9.16037 q^{67} -0.700405 q^{69} -3.08863 q^{71} +5.82179 q^{73} +6.27601 q^{75} -15.4961 q^{77} -12.3774 q^{79} -5.33858 q^{81} -2.34186 q^{83} +1.59557 q^{85} -0.118720 q^{87} -8.49990 q^{89} -1.32093 q^{91} +0.468375 q^{93} -0.786708 q^{95} -3.52786 q^{97} -5.75689 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{3} - 3 q^{5} - 6 q^{7} + q^{9}+O(q^{10}) $ 4 * q + 3 * q^3 - 3 * q^5 - 6 * q^7 + q^9	$ 4 q + 3 q^{3} - 3 q^{5} - 6 q^{7} + q^{9} + 3 q^{11} - 3 q^{13} - 8 q^{15} - 8 q^{17} + 12 q^{19} - 8 q^{21} - q^{23} + 3 q^{25} + 6 q^{27} - 3 q^{29} - 19 q^{31} - 10 q^{33} + 8 q^{35} - 4 q^{37} - 5 q^{39} - 17 q^{41} - 2 q^{43} - 10 q^{45} - 10 q^{47} - 6 q^{49} - 6 q^{51} - 10 q^{53} - 15 q^{55} + 2 q^{57} + 14 q^{59} - 9 q^{61} - 18 q^{63} - 30 q^{65} + 7 q^{67} + 9 q^{69} - 16 q^{71} - 7 q^{73} + 14 q^{75} - 14 q^{77} - 21 q^{79} - 8 q^{81} - 12 q^{83} + 6 q^{85} + 19 q^{87} + 14 q^{91} - 30 q^{93} - 2 q^{95} - 32 q^{97} - 2 q^{99}+O(q^{100}) $ 4 * q + 3 * q^3 - 3 * q^5 - 6 * q^7 + q^9 + 3 * q^11 - 3 * q^13 - 8 * q^15 - 8 * q^17 + 12 * q^19 - 8 * q^21 - q^23 + 3 * q^25 + 6 * q^27 - 3 * q^29 - 19 * q^31 - 10 * q^33 + 8 * q^35 - 4 * q^37 - 5 * q^39 - 17 * q^41 - 2 * q^43 - 10 * q^45 - 10 * q^47 - 6 * q^49 - 6 * q^51 - 10 * q^53 - 15 * q^55 + 2 * q^57 + 14 * q^59 - 9 * q^61 - 18 * q^63 - 30 * q^65 + 7 * q^67 + 9 * q^69 - 16 * q^71 - 7 * q^73 + 14 * q^75 - 14 * q^77 - 21 * q^79 - 8 * q^81 - 12 * q^83 + 6 * q^85 + 19 * q^87 + 14 * q^91 - 30 * q^93 - 2 * q^95 - 32 * q^97 - 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.43828	−0.830393	−0.415197	−	0.909732i	$-0.636287\pi$
$3$	−1.43828	−0.830393	−0.415197	+	0.909732i	$0.636287\pi$
$4$	0	0
$5$	−0.797785	−0.356780	−0.178390	−	0.983960i	$-0.557089\pi$
$5$	−0.797785	−0.356780	−0.178390	+	0.983960i	$0.557089\pi$
$6$	0	0
$7$	−2.50694	−0.947535	−0.473767	−	0.880650i	$-0.657106\pi$
$7$	−2.50694	−0.947535	−0.473767	+	0.880650i	$0.657106\pi$
$8$	0	0
$9$	−0.931341	−0.310447
$10$	0	0
$11$	6.18129	1.86373	0.931865	−	0.362805i	$-0.118181\pi$
$11$	6.18129	1.86373	0.931865	+	0.362805i	$0.118181\pi$
$12$	0	0
$13$	0.526911	0.146139	0.0730694	−	0.997327i	$-0.476721\pi$
$13$	0.526911	0.146139	0.0730694	+	0.997327i	$0.476721\pi$
$14$	0	0
$15$	1.14744	0.296268
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	0.986116	0.226231	0.113115	−	0.993582i	$-0.463917\pi$
$19$	0.986116	0.226231	0.113115	+	0.993582i	$0.463917\pi$
$20$	0	0
$21$	3.60569	0.786827
$22$	0	0
$23$	0.486973	0.101541	0.0507704	−	0.998710i	$-0.483832\pi$
$23$	0.486973	0.101541	0.0507704	+	0.998710i	$0.483832\pi$
$24$	0	0
$25$	−4.36354	−0.872708
$26$	0	0
$27$	5.65438	1.08819
$28$	0	0
$29$	0.0825425	0.0153278	0.00766388	−	0.999971i	$-0.497560\pi$
$29$	0.0825425	0.0153278	0.00766388	+	0.999971i	$0.497560\pi$
$30$	0	0
$31$	−0.325649	−0.0584882	−0.0292441	−	0.999572i	$-0.509310\pi$
$31$	−0.325649	−0.0584882	−0.0292441	+	0.999572i	$0.509310\pi$
$32$	0	0
$33$	−8.89045	−1.54763
$34$	0	0
$35$	2.00000	0.338062
$36$	0	0
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	0	0
$39$	−0.757847	−0.121353
$40$	0	0
$41$	6.56480	1.02525	0.512625	−	0.858613i	$-0.328673\pi$
$41$	6.56480	1.02525	0.512625	+	0.858613i	$0.328673\pi$
$42$	0	0
$43$	8.15756	1.24402	0.622008	−	0.783011i	$-0.286317\pi$
$43$	8.15756	1.24402	0.622008	+	0.783011i	$0.286317\pi$
$44$	0	0
$45$	0.743010	0.110761
$46$	0	0
$47$	2.97908	0.434543	0.217272	−	0.976111i	$-0.430284\pi$
$47$	2.97908	0.434543	0.217272	+	0.976111i	$0.430284\pi$
$48$	0	0
$49$	−0.715242	−0.102177
$50$	0	0
$51$	2.87657	0.402800
$52$	0	0
$53$	−5.74301	−0.788863	−0.394432	−	0.918925i	$-0.629058\pi$
$53$	−5.74301	−0.788863	−0.394432	+	0.918925i	$0.629058\pi$
$54$	0	0
$55$	−4.93134	−0.664942
$56$	0	0
$57$	−1.41831	−0.187860
$58$	0	0
$59$	3.59557	0.468103	0.234052	−	0.972224i	$-0.424802\pi$
$59$	3.59557	0.468103	0.234052	+	0.972224i	$0.424802\pi$
$60$	0	0
$61$	−14.9831	−1.91839	−0.959196	−	0.282743i	$-0.908756\pi$
$61$	−14.9831	−1.91839	−0.959196	+	0.282743i	$0.908756\pi$
$62$	0	0
$63$	2.33482	0.294159
$64$	0	0
$65$	−0.420361	−0.0521394
$66$	0	0
$67$	−9.16037	−1.11912	−0.559559	−	0.828791i	$-0.689029\pi$
$67$	−9.16037	−1.11912	−0.559559	+	0.828791i	$0.689029\pi$
$68$	0	0
$69$	−0.700405	−0.0843189
$70$	0	0
$71$	−3.08863	−0.366553	−0.183276	−	0.983061i	$-0.558670\pi$
$71$	−3.08863	−0.366553	−0.183276	+	0.983061i	$0.558670\pi$
$72$	0	0
$73$	5.82179	0.681389	0.340695	−	0.940174i	$-0.389338\pi$
$73$	5.82179	0.681389	0.340695	+	0.940174i	$0.389338\pi$
$74$	0	0
$75$	6.27601	0.724691
$76$	0	0
$77$	−15.4961	−1.76595
$78$	0	0
$79$	−12.3774	−1.39257	−0.696284	−	0.717766i	$-0.745165\pi$
$79$	−12.3774	−1.39257	−0.696284	+	0.717766i	$0.745165\pi$
$80$	0	0
$81$	−5.33858	−0.593175
$82$	0	0
$83$	−2.34186	−0.257052	−0.128526	−	0.991706i	$-0.541025\pi$
$83$	−2.34186	−0.257052	−0.128526	+	0.991706i	$0.541025\pi$
$84$	0	0
$85$	1.59557	0.173064
$86$	0	0
$87$	−0.118720	−0.0127281
$88$	0	0
$89$	−8.49990	−0.900988	−0.450494	−	0.892779i	$-0.648752\pi$
$89$	−8.49990	−0.900988	−0.450494	+	0.892779i	$0.648752\pi$
$90$	0	0
$91$	−1.32093	−0.138472
$92$	0	0
$93$	0.468375	0.0485682
$94$	0	0
$95$	−0.786708	−0.0807146
$96$	0	0
$97$	−3.52786	−0.358200	−0.179100	−	0.983831i	$-0.557319\pi$
$97$	−3.52786	−0.358200	−0.179100	+	0.983831i	$0.557319\pi$
$98$	0	0
$99$	−5.75689	−0.578590
$100$	0	0
$101$	−13.2290	−1.31634	−0.658169	−	0.752870i	$-0.728669\pi$
$101$	−13.2290	−1.31634	−0.658169	+	0.752870i	$0.728669\pi$
$102$	0	0
$103$	−11.8319	−1.16583	−0.582917	−	0.812532i	$-0.698089\pi$
$103$	−11.8319	−1.16583	−0.582917	+	0.812532i	$0.698089\pi$
$104$	0	0
$105$	−2.87657	−0.280724
$106$	0	0
$107$	1.87752	0.181507	0.0907533	−	0.995873i	$-0.471073\pi$
$107$	1.87752	0.181507	0.0907533	+	0.995873i	$0.471073\pi$
$108$	0	0
$109$	10.7948	1.03395	0.516976	−	0.856000i	$-0.327057\pi$
$109$	10.7948	1.03395	0.516976	+	0.856000i	$0.327057\pi$
$110$	0	0
$111$	1.43828	0.136516
$112$	0	0
$113$	−6.20502	−0.583719	−0.291860	−	0.956461i	$-0.594274\pi$
$113$	−6.20502	−0.583719	−0.291860	+	0.956461i	$0.594274\pi$
$114$	0	0
$115$	−0.388500	−0.0362278
$116$	0	0
$117$	−0.490734	−0.0453684
$118$	0	0
$119$	5.01388	0.459622
$120$	0	0
$121$	27.2084	2.47349
$122$	0	0
$123$	−9.44204	−0.851360
$124$	0	0
$125$	7.47009	0.668145
$126$	0	0
$127$	−14.8695	−1.31946	−0.659729	−	0.751504i	$-0.729329\pi$
$127$	−14.8695	−1.31946	−0.659729	+	0.751504i	$0.729329\pi$
$128$	0	0
$129$	−11.7329	−1.03302
$130$	0	0
$131$	4.03994	0.352971	0.176486	−	0.984303i	$-0.443527\pi$
$131$	4.03994	0.352971	0.176486	+	0.984303i	$0.443527\pi$
$132$	0	0
$133$	−2.47214	−0.214361
$134$	0	0
$135$	−4.51098	−0.388243
$136$	0	0
$137$	−18.8857	−1.61352	−0.806759	−	0.590881i	$-0.798780\pi$
$137$	−18.8857	−1.61352	−0.806759	+	0.590881i	$0.798780\pi$
$138$	0	0
$139$	−2.01293	−0.170735	−0.0853673	−	0.996350i	$-0.527206\pi$
$139$	−2.01293	−0.170735	−0.0853673	+	0.996350i	$0.527206\pi$
$140$	0	0
$141$	−4.28476	−0.360842
$142$	0	0
$143$	3.25699	0.272363
$144$	0	0
$145$	−0.0658512	−0.00546864
$146$	0	0
$147$	1.02872	0.0848475
$148$	0	0
$149$	−8.44232	−0.691622	−0.345811	−	0.938304i	$-0.612396\pi$
$149$	−8.44232	−0.691622	−0.345811	+	0.938304i	$0.612396\pi$
$150$	0	0
$151$	4.43220	0.360687	0.180344	−	0.983604i	$-0.442279\pi$
$151$	4.43220	0.360687	0.180344	+	0.983604i	$0.442279\pi$
$152$	0	0
$153$	1.86268	0.150589
$154$	0	0
$155$	0.259798	0.0208674
$156$	0	0
$157$	−22.9822	−1.83418	−0.917088	−	0.398685i	$-0.869467\pi$
$157$	−22.9822	−1.83418	−0.917088	+	0.398685i	$0.869467\pi$
$158$	0	0
$159$	8.26007	0.655067
$160$	0	0
$161$	−1.22081	−0.0962136
$162$	0	0
$163$	−9.72537	−0.761749	−0.380875	−	0.924627i	$-0.624377\pi$
$163$	−9.72537	−0.761749	−0.380875	+	0.924627i	$0.624377\pi$
$164$	0	0
$165$	7.09267	0.552163
$166$	0	0
$167$	−9.35978	−0.724281	−0.362141	−	0.932123i	$-0.617954\pi$
$167$	−9.35978	−0.724281	−0.362141	+	0.932123i	$0.617954\pi$
$168$	0	0
$169$	−12.7224	−0.978643
$170$	0	0
$171$	−0.918411	−0.0702326
$172$	0	0
$173$	6.03789	0.459052	0.229526	−	0.973302i	$-0.426282\pi$
$173$	6.03789	0.459052	0.229526	+	0.973302i	$0.426282\pi$
$174$	0	0
$175$	10.9391	0.826921
$176$	0	0
$177$	−5.17145	−0.388710
$178$	0	0
$179$	3.17897	0.237607	0.118804	−	0.992918i	$-0.462094\pi$
$179$	3.17897	0.237607	0.118804	+	0.992918i	$0.462094\pi$
$180$	0	0
$181$	16.9619	1.26077	0.630385	−	0.776283i	$-0.282897\pi$
$181$	16.9619	1.26077	0.630385	+	0.776283i	$0.282897\pi$
$182$	0	0
$183$	21.5500	1.59302
$184$	0	0
$185$	0.797785	0.0586543
$186$	0	0
$187$	−12.3626	−0.904042
$188$	0	0
$189$	−14.1752	−1.03109
$190$	0	0
$191$	24.3793	1.76402	0.882011	−	0.471228i	$-0.156189\pi$
$191$	24.3793	1.76402	0.882011	+	0.471228i	$0.156189\pi$
$192$	0	0
$193$	−13.6835	−0.984961	−0.492481	−	0.870323i	$-0.663910\pi$
$193$	−13.6835	−0.984961	−0.492481	+	0.870323i	$0.663910\pi$
$194$	0	0
$195$	0.604599	0.0432962
$196$	0	0
$197$	16.4404	1.17133	0.585666	−	0.810553i	$-0.300833\pi$
$197$	16.4404	1.17133	0.585666	+	0.810553i	$0.300833\pi$
$198$	0	0
$199$	−1.66327	−0.117906	−0.0589532	−	0.998261i	$-0.518776\pi$
$199$	−1.66327	−0.117906	−0.0589532	+	0.998261i	$0.518776\pi$
$200$	0	0
$201$	13.1752	0.929308
$202$	0	0
$203$	−0.206929	−0.0145236
$204$	0	0
$205$	−5.23730	−0.365789
$206$	0	0
$207$	−0.453538	−0.0315231
$208$	0	0
$209$	6.09547	0.421633
$210$	0	0
$211$	25.5300	1.75756	0.878779	−	0.477230i	$-0.158359\pi$
$211$	25.5300	1.75756	0.878779	+	0.477230i	$0.158359\pi$
$212$	0	0
$213$	4.44232	0.304383
$214$	0	0
$215$	−6.50798	−0.443841
$216$	0	0
$217$	0.816383	0.0554197
$218$	0	0
$219$	−8.37339	−0.565821
$220$	0	0
$221$	−1.05382	−0.0708877
$222$	0	0
$223$	−13.1563	−0.881013	−0.440507	−	0.897749i	$-0.645201\pi$
$223$	−13.1563	−0.881013	−0.440507	+	0.897749i	$0.645201\pi$
$224$	0	0
$225$	4.06394	0.270930
$226$	0	0
$227$	−19.1296	−1.26968	−0.634838	−	0.772645i	$-0.718933\pi$
$227$	−19.1296	−1.26968	−0.634838	+	0.772645i	$0.718933\pi$
$228$	0	0
$229$	14.1677	0.936227	0.468114	−	0.883668i	$-0.344934\pi$
$229$	14.1677	0.936227	0.468114	+	0.883668i	$0.344934\pi$
$230$	0	0
$231$	22.2878	1.46643
$232$	0	0
$233$	−15.1167	−0.990326	−0.495163	−	0.868800i	$-0.664892\pi$
$233$	−15.1167	−0.990326	−0.495163	+	0.868800i	$0.664892\pi$
$234$	0	0
$235$	−2.37666	−0.155036
$236$	0	0
$237$	17.8022	1.15638
$238$	0	0
$239$	22.1465	1.43254	0.716269	−	0.697825i	$-0.245849\pi$
$239$	22.1465	1.43254	0.716269	+	0.697825i	$0.245849\pi$
$240$	0	0
$241$	7.36087	0.474155	0.237078	−	0.971491i	$-0.423810\pi$
$241$	7.36087	0.474155	0.237078	+	0.971491i	$0.423810\pi$
$242$	0	0
$243$	−9.28476	−0.595617
$244$	0	0
$245$	0.570609	0.0364549
$246$	0	0
$247$	0.519595	0.0330611
$248$	0	0
$249$	3.36825	0.213454
$250$	0	0
$251$	−22.1834	−1.40021	−0.700103	−	0.714042i	$-0.746862\pi$
$251$	−22.1834	−1.40021	−0.700103	+	0.714042i	$0.746862\pi$
$252$	0	0
$253$	3.01012	0.189245
$254$	0	0
$255$	−2.29488	−0.143711
$256$	0	0
$257$	−27.0897	−1.68981	−0.844903	−	0.534919i	$-0.820342\pi$
$257$	−27.0897	−1.68981	−0.844903	+	0.534919i	$0.820342\pi$
$258$	0	0
$259$	2.50694	0.155774
$260$	0	0
$261$	−0.0768752	−0.00475846
$262$	0	0
$263$	0.895775	0.0552358	0.0276179	−	0.999619i	$-0.491208\pi$
$263$	0.895775	0.0552358	0.0276179	+	0.999619i	$0.491208\pi$
$264$	0	0
$265$	4.58169	0.281451
$266$	0	0
$267$	12.2253	0.748174
$268$	0	0
$269$	−26.9923	−1.64575	−0.822874	−	0.568223i	$-0.807631\pi$
$269$	−26.9923	−1.64575	−0.822874	+	0.568223i	$0.807631\pi$
$270$	0	0
$271$	−13.0290	−0.791455	−0.395728	−	0.918368i	$-0.629508\pi$
$271$	−13.0290	−0.791455	−0.395728	+	0.918368i	$0.629508\pi$
$272$	0	0
$273$	1.89988	0.114986
$274$	0	0
$275$	−26.9723	−1.62649
$276$	0	0
$277$	2.51850	0.151322	0.0756610	−	0.997134i	$-0.475893\pi$
$277$	2.51850	0.151322	0.0756610	+	0.997134i	$0.475893\pi$
$278$	0	0
$279$	0.303290	0.0181575
$280$	0	0
$281$	13.7531	0.820443	0.410222	−	0.911986i	$-0.365451\pi$
$281$	13.7531	0.820443	0.410222	+	0.911986i	$0.365451\pi$
$282$	0	0
$283$	−20.8625	−1.24015	−0.620073	−	0.784544i	$-0.712897\pi$
$283$	−20.8625	−1.24015	−0.620073	+	0.784544i	$0.712897\pi$
$284$	0	0
$285$	1.13151	0.0670248
$286$	0	0
$287$	−16.4576	−0.971460
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	5.07407	0.297447
$292$	0	0
$293$	−10.9739	−0.641105	−0.320552	−	0.947231i	$-0.603869\pi$
$293$	−10.9739	−0.641105	−0.320552	+	0.947231i	$0.603869\pi$
$294$	0	0
$295$	−2.86849	−0.167010
$296$	0	0
$297$	34.9514	2.02809
$298$	0	0
$299$	0.256591	0.0148391
$300$	0	0
$301$	−20.4505	−1.17875
$302$	0	0
$303$	19.0271	1.09308
$304$	0	0
$305$	11.9533	0.684444
$306$	0	0
$307$	−1.39835	−0.0798078	−0.0399039	−	0.999204i	$-0.512705\pi$
$307$	−1.39835	−0.0798078	−0.0399039	+	0.999204i	$0.512705\pi$
$308$	0	0
$309$	17.0176	0.968100
$310$	0	0
$311$	−7.94332	−0.450424	−0.225212	−	0.974310i	$-0.572308\pi$
$311$	−7.94332	−0.450424	−0.225212	+	0.974310i	$0.572308\pi$
$312$	0	0
$313$	14.4524	0.816900	0.408450	−	0.912781i	$-0.366069\pi$
$313$	14.4524	0.816900	0.408450	+	0.912781i	$0.366069\pi$
$314$	0	0
$315$	−1.86268	−0.104950
$316$	0	0
$317$	3.21139	0.180369	0.0901847	−	0.995925i	$-0.471254\pi$
$317$	3.21139	0.180369	0.0901847	+	0.995925i	$0.471254\pi$
$318$	0	0
$319$	0.510219	0.0285668
$320$	0	0
$321$	−2.70041	−0.150722
$322$	0	0
$323$	−1.97223	−0.109738
$324$	0	0
$325$	−2.29920	−0.127536
$326$	0	0
$327$	−15.5260	−0.858587
$328$	0	0
$329$	−7.46838	−0.411745
$330$	0	0
$331$	28.5757	1.57066	0.785331	−	0.619076i	$-0.212493\pi$
$331$	28.5757	1.57066	0.785331	+	0.619076i	$0.212493\pi$
$332$	0	0
$333$	0.931341	0.0510372
$334$	0	0
$335$	7.30800	0.399279
$336$	0	0
$337$	18.7661	1.02225	0.511126	−	0.859506i	$-0.329228\pi$
$337$	18.7661	1.02225	0.511126	+	0.859506i	$0.329228\pi$
$338$	0	0
$339$	8.92458	0.484717
$340$	0	0
$341$	−2.01293	−0.109006
$342$	0	0
$343$	19.3417	1.04435
$344$	0	0
$345$	0.558773	0.0300833
$346$	0	0
$347$	−28.3030	−1.51938	−0.759691	−	0.650284i	$-0.774650\pi$
$347$	−28.3030	−1.51938	−0.759691	+	0.650284i	$0.774650\pi$
$348$	0	0
$349$	2.69931	0.144491	0.0722454	−	0.997387i	$-0.476984\pi$
$349$	2.69931	0.144491	0.0722454	+	0.997387i	$0.476984\pi$
$350$	0	0
$351$	2.97936	0.159026
$352$	0	0
$353$	15.7790	0.839831	0.419916	−	0.907563i	$-0.362060\pi$
$353$	15.7790	0.839831	0.419916	+	0.907563i	$0.362060\pi$
$354$	0	0
$355$	2.46406	0.130779
$356$	0	0
$357$	−7.21139	−0.381667
$358$	0	0
$359$	−3.97736	−0.209917	−0.104959	−	0.994477i	$-0.533471\pi$
$359$	−3.97736	−0.209917	−0.104959	+	0.994477i	$0.533471\pi$
$360$	0	0
$361$	−18.0276	−0.948820
$362$	0	0
$363$	−39.1334	−2.05397
$364$	0	0
$365$	−4.64454	−0.243106
$366$	0	0
$367$	−2.41195	−0.125903	−0.0629514	−	0.998017i	$-0.520051\pi$
$367$	−2.41195	−0.125903	−0.0629514	+	0.998017i	$0.520051\pi$
$368$	0	0
$369$	−6.11407	−0.318286
$370$	0	0
$371$	14.3974	0.747475
$372$	0	0
$373$	0.613407	0.0317610	0.0158805	−	0.999874i	$-0.494945\pi$
$373$	0.613407	0.0317610	0.0158805	+	0.999874i	$0.494945\pi$
$374$	0	0
$375$	−10.7441	−0.554823
$376$	0	0
$377$	0.0434925	0.00223998
$378$	0	0
$379$	17.5737	0.902700	0.451350	−	0.892347i	$-0.350943\pi$
$379$	17.5737	0.902700	0.451350	+	0.892347i	$0.350943\pi$
$380$	0	0
$381$	21.3866	1.09567
$382$	0	0
$383$	25.4785	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$383$	25.4785	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$384$	0	0
$385$	12.3626	0.630056
$386$	0	0
$387$	−7.59748	−0.386201
$388$	0	0
$389$	19.0889	0.967846	0.483923	−	0.875111i	$-0.339211\pi$
$389$	19.0889	0.967846	0.483923	+	0.875111i	$0.339211\pi$
$390$	0	0
$391$	−0.973946	−0.0492546
$392$	0	0
$393$	−5.81057	−0.293105
$394$	0	0
$395$	9.87452	0.496841
$396$	0	0
$397$	28.9206	1.45148	0.725742	−	0.687967i	$-0.241496\pi$
$397$	28.9206	1.45148	0.725742	+	0.687967i	$0.241496\pi$
$398$	0	0
$399$	3.55563	0.178004
$400$	0	0
$401$	−3.61582	−0.180565	−0.0902826	−	0.995916i	$-0.528777\pi$
$401$	−3.61582	−0.180565	−0.0902826	+	0.995916i	$0.528777\pi$
$402$	0	0
$403$	−0.171588	−0.00854740
$404$	0	0
$405$	4.25904	0.211633
$406$	0	0
$407$	−6.18129	−0.306395
$408$	0	0
$409$	15.9525	0.788802	0.394401	−	0.918938i	$-0.370952\pi$
$409$	15.9525	0.788802	0.394401	+	0.918938i	$0.370952\pi$
$410$	0	0
$411$	27.1630	1.33985
$412$	0	0
$413$	−9.01388	−0.443544
$414$	0	0
$415$	1.86830	0.0917111
$416$	0	0
$417$	2.89516	0.141777
$418$	0	0
$419$	−19.1218	−0.934161	−0.467081	−	0.884215i	$-0.654694\pi$
$419$	−19.1218	−0.934161	−0.467081	+	0.884215i	$0.654694\pi$
$420$	0	0
$421$	19.6428	0.957333	0.478666	−	0.877997i	$-0.341120\pi$
$421$	19.6428	0.957333	0.478666	+	0.877997i	$0.341120\pi$
$422$	0	0
$423$	−2.77454	−0.134903
$424$	0	0
$425$	8.72708	0.423326
$426$	0	0
$427$	37.5618	1.81774
$428$	0	0
$429$	−4.68447	−0.226169
$430$	0	0
$431$	−2.78109	−0.133961	−0.0669803	−	0.997754i	$-0.521336\pi$
$431$	−2.78109	−0.133961	−0.0669803	+	0.997754i	$0.521336\pi$
$432$	0	0
$433$	−10.0306	−0.482039	−0.241019	−	0.970520i	$-0.577482\pi$
$433$	−10.0306	−0.482039	−0.241019	+	0.970520i	$0.577482\pi$
$434$	0	0
$435$	0.0947126	0.00454112
$436$	0	0
$437$	0.480212	0.0229717
$438$	0	0
$439$	−2.34349	−0.111848	−0.0559242	−	0.998435i	$-0.517811\pi$
$439$	−2.34349	−0.111848	−0.0559242	+	0.998435i	$0.517811\pi$
$440$	0	0
$441$	0.666135	0.0317207
$442$	0	0
$443$	9.32497	0.443043	0.221521	−	0.975156i	$-0.428898\pi$
$443$	9.32497	0.443043	0.221521	+	0.975156i	$0.428898\pi$
$444$	0	0
$445$	6.78109	0.321455
$446$	0	0
$447$	12.1424	0.574318
$448$	0	0
$449$	−11.0938	−0.523547	−0.261773	−	0.965129i	$-0.584307\pi$
$449$	−11.0938	−0.523547	−0.261773	+	0.965129i	$0.584307\pi$
$450$	0	0
$451$	40.5790	1.91079
$452$	0	0
$453$	−6.37476	−0.299512
$454$	0	0
$455$	1.05382	0.0494039
$456$	0	0
$457$	3.76530	0.176133	0.0880667	−	0.996115i	$-0.471931\pi$
$457$	3.76530	0.176133	0.0880667	+	0.996115i	$0.471931\pi$
$458$	0	0
$459$	−11.3088	−0.527848
$460$	0	0
$461$	−35.5543	−1.65593	−0.827964	−	0.560781i	$-0.810501\pi$
$461$	−35.5543	−1.65593	−0.827964	+	0.560781i	$0.810501\pi$
$462$	0	0
$463$	−19.8079	−0.920552	−0.460276	−	0.887776i	$-0.652249\pi$
$463$	−19.8079	−0.920552	−0.460276	+	0.887776i	$0.652249\pi$
$464$	0	0
$465$	−0.373663	−0.0173282
$466$	0	0
$467$	−19.7113	−0.912130	−0.456065	−	0.889947i	$-0.650741\pi$
$467$	−19.7113	−0.912130	−0.456065	+	0.889947i	$0.650741\pi$
$468$	0	0
$469$	22.9645	1.06040
$470$	0	0
$471$	33.0549	1.52309
$472$	0	0
$473$	50.4243	2.31851
$474$	0	0
$475$	−4.30296	−0.197433
$476$	0	0
$477$	5.34870	0.244900
$478$	0	0
$479$	30.2457	1.38196	0.690981	−	0.722873i	$-0.257179\pi$
$479$	30.2457	1.38196	0.690981	+	0.722873i	$0.257179\pi$
$480$	0	0
$481$	−0.526911	−0.0240251
$482$	0	0
$483$	1.75588	0.0798951
$484$	0	0
$485$	2.81448	0.127799
$486$	0	0
$487$	19.4367	0.880759	0.440379	−	0.897812i	$-0.354844\pi$
$487$	19.4367	0.880759	0.440379	+	0.897812i	$0.354844\pi$
$488$	0	0
$489$	13.9878	0.632552
$490$	0	0
$491$	32.1907	1.45275	0.726374	−	0.687300i	$-0.241204\pi$
$491$	32.1907	1.45275	0.726374	+	0.687300i	$0.241204\pi$
$492$	0	0
$493$	−0.165085	−0.00743506
$494$	0	0
$495$	4.59276	0.206429
$496$	0	0
$497$	7.74301	0.347321
$498$	0	0
$499$	19.2613	0.862254	0.431127	−	0.902291i	$-0.358116\pi$
$499$	19.2613	0.862254	0.431127	+	0.902291i	$0.358116\pi$
$500$	0	0
$501$	13.4620	0.601438
$502$	0	0
$503$	−17.7920	−0.793305	−0.396653	−	0.917969i	$-0.629828\pi$
$503$	−17.7920	−0.793305	−0.396653	+	0.917969i	$0.629828\pi$
$504$	0	0
$505$	10.5539	0.469643
$506$	0	0
$507$	18.2984	0.812659
$508$	0	0
$509$	−30.5853	−1.35567	−0.677834	−	0.735215i	$-0.737081\pi$
$509$	−30.5853	−1.35567	−0.677834	+	0.735215i	$0.737081\pi$
$510$	0	0
$511$	−14.5949	−0.645640
$512$	0	0
$513$	5.57588	0.246181
$514$	0	0
$515$	9.43932	0.415946
$516$	0	0
$517$	18.4146	0.809871
$518$	0	0
$519$	−8.68420	−0.381194
$520$	0	0
$521$	14.3530	0.628817	0.314409	−	0.949288i	$-0.398194\pi$
$521$	14.3530	0.628817	0.314409	+	0.949288i	$0.398194\pi$
$522$	0	0
$523$	9.56780	0.418371	0.209185	−	0.977876i	$-0.432919\pi$
$523$	9.56780	0.418371	0.209185	+	0.977876i	$0.432919\pi$
$524$	0	0
$525$	−15.7336	−0.686670
$526$	0	0
$527$	0.651298	0.0283710
$528$	0	0
$529$	−22.7629	−0.989689
$530$	0	0
$531$	−3.34870	−0.145321
$532$	0	0
$533$	3.45906	0.149829
$534$	0	0
$535$	−1.49786	−0.0647580
$536$	0	0
$537$	−4.57226	−0.197307
$538$	0	0
$539$	−4.42112	−0.190431
$540$	0	0
$541$	10.4570	0.449580	0.224790	−	0.974407i	$-0.427830\pi$
$541$	10.4570	0.449580	0.224790	+	0.974407i	$0.427830\pi$
$542$	0	0
$543$	−24.3960	−1.04693
$544$	0	0
$545$	−8.61191	−0.368894
$546$	0	0
$547$	24.5676	1.05044	0.525218	−	0.850968i	$-0.323984\pi$
$547$	24.5676	1.05044	0.525218	+	0.850968i	$0.323984\pi$
$548$	0	0
$549$	13.9544	0.595559
$550$	0	0
$551$	0.0813965	0.00346761
$552$	0	0
$553$	31.0295	1.31951
$554$	0	0
$555$	−1.14744	−0.0487061
$556$	0	0
$557$	36.9837	1.56705	0.783524	−	0.621361i	$-0.213420\pi$
$557$	36.9837	1.56705	0.783524	+	0.621361i	$0.213420\pi$
$558$	0	0
$559$	4.29831	0.181799
$560$	0	0
$561$	17.7809	0.750710
$562$	0	0
$563$	−7.73289	−0.325902	−0.162951	−	0.986634i	$-0.552101\pi$
$563$	−7.73289	−0.325902	−0.162951	+	0.986634i	$0.552101\pi$
$564$	0	0
$565$	4.95027	0.208260
$566$	0	0
$567$	13.3835	0.562055
$568$	0	0
$569$	−24.0264	−1.00724	−0.503620	−	0.863925i	$-0.667999\pi$
$569$	−24.0264	−1.00724	−0.503620	+	0.863925i	$0.667999\pi$
$570$	0	0
$571$	−20.1216	−0.842063	−0.421032	−	0.907046i	$-0.638332\pi$
$571$	−20.1216	−0.842063	−0.421032	+	0.907046i	$0.638332\pi$
$572$	0	0
$573$	−35.0643	−1.46483
$574$	0	0
$575$	−2.12493	−0.0886155
$576$	0	0
$577$	−38.7576	−1.61350	−0.806750	−	0.590893i	$-0.798775\pi$
$577$	−38.7576	−1.61350	−0.806750	+	0.590893i	$0.798775\pi$
$578$	0	0
$579$	19.6808	0.817905
$580$	0	0
$581$	5.87090	0.243566
$582$	0	0
$583$	−35.4992	−1.47023
$584$	0	0
$585$	0.391500	0.0161865
$586$	0	0
$587$	−23.4644	−0.968480	−0.484240	−	0.874935i	$-0.660904\pi$
$587$	−23.4644	−0.968480	−0.484240	+	0.874935i	$0.660904\pi$
$588$	0	0
$589$	−0.321128	−0.0132318
$590$	0	0
$591$	−23.6460	−0.972665
$592$	0	0
$593$	−7.09814	−0.291486	−0.145743	−	0.989323i	$-0.546557\pi$
$593$	−7.09814	−0.291486	−0.145743	+	0.989323i	$0.546557\pi$
$594$	0	0
$595$	−4.00000	−0.163984
$596$	0	0
$597$	2.39226	0.0979087
$598$	0	0
$599$	−31.4038	−1.28312	−0.641561	−	0.767072i	$-0.721713\pi$
$599$	−31.4038	−1.28312	−0.641561	+	0.767072i	$0.721713\pi$
$600$	0	0
$601$	4.83963	0.197413	0.0987063	−	0.995117i	$-0.468530\pi$
$601$	4.83963	0.197413	0.0987063	+	0.995117i	$0.468530\pi$
$602$	0	0
$603$	8.53143	0.347427
$604$	0	0
$605$	−21.7064	−0.882492
$606$	0	0
$607$	−30.5905	−1.24163	−0.620815	−	0.783957i	$-0.713198\pi$
$607$	−30.5905	−1.24163	−0.620815	+	0.783957i	$0.713198\pi$
$608$	0	0
$609$	0.297623	0.0120603
$610$	0	0
$611$	1.56971	0.0635036
$612$	0	0
$613$	15.0564	0.608123	0.304062	−	0.952652i	$-0.401657\pi$
$613$	15.0564	0.608123	0.304062	+	0.952652i	$0.401657\pi$
$614$	0	0
$615$	7.53272	0.303749
$616$	0	0
$617$	−15.2641	−0.614510	−0.307255	−	0.951627i	$-0.599410\pi$
$617$	−15.2641	−0.614510	−0.307255	+	0.951627i	$0.599410\pi$
$618$	0	0
$619$	16.4243	0.660150	0.330075	−	0.943955i	$-0.392926\pi$
$619$	16.4243	0.660150	0.330075	+	0.943955i	$0.392926\pi$
$620$	0	0
$621$	2.75353	0.110495
$622$	0	0
$623$	21.3088	0.853718
$624$	0	0
$625$	15.8582	0.634327
$626$	0	0
$627$	−8.76702	−0.350121
$628$	0	0
$629$	2.00000	0.0797452
$630$	0	0
$631$	12.2881	0.489182	0.244591	−	0.969626i	$-0.421346\pi$
$631$	12.2881	0.489182	0.244591	+	0.969626i	$0.421346\pi$
$632$	0	0
$633$	−36.7194	−1.45946
$634$	0	0
$635$	11.8627	0.470756
$636$	0	0
$637$	−0.376869	−0.0149321
$638$	0	0
$639$	2.87657	0.113795
$640$	0	0
$641$	−25.9413	−1.02462	−0.512309	−	0.858801i	$-0.671210\pi$
$641$	−25.9413	−1.02462	−0.512309	+	0.858801i	$0.671210\pi$
$642$	0	0
$643$	7.22698	0.285004	0.142502	−	0.989794i	$-0.454485\pi$
$643$	7.22698	0.285004	0.142502	+	0.989794i	$0.454485\pi$
$644$	0	0
$645$	9.36032	0.368562
$646$	0	0
$647$	−18.8693	−0.741827	−0.370913	−	0.928667i	$-0.620955\pi$
$647$	−18.8693	−0.741827	−0.370913	+	0.928667i	$0.620955\pi$
$648$	0	0
$649$	22.2253	0.872418
$650$	0	0
$651$	−1.17419	−0.0460201
$652$	0	0
$653$	44.0849	1.72518	0.862588	−	0.505908i	$-0.168842\pi$
$653$	44.0849	1.72518	0.862588	+	0.505908i	$0.168842\pi$
$654$	0	0
$655$	−3.22300	−0.125933
$656$	0	0
$657$	−5.42208	−0.211535
$658$	0	0
$659$	−22.4122	−0.873056	−0.436528	−	0.899691i	$-0.643792\pi$
$659$	−22.4122	−0.873056	−0.436528	+	0.899691i	$0.643792\pi$
$660$	0	0
$661$	32.9173	1.28033	0.640167	−	0.768236i	$-0.278865\pi$
$661$	32.9173	1.28033	0.640167	+	0.768236i	$0.278865\pi$
$662$	0	0
$663$	1.51569	0.0588647
$664$	0	0
$665$	1.97223	0.0764799
$666$	0	0
$667$	0.0401960	0.00155639
$668$	0	0
$669$	18.9225	0.731587
$670$	0	0
$671$	−92.6150	−3.57536
$672$	0	0
$673$	21.4570	0.827105	0.413553	−	0.910480i	$-0.364288\pi$
$673$	21.4570	0.827105	0.413553	+	0.910480i	$0.364288\pi$
$674$	0	0
$675$	−24.6731	−0.949669
$676$	0	0
$677$	24.3586	0.936178	0.468089	−	0.883681i	$-0.344943\pi$
$677$	24.3586	0.936178	0.468089	+	0.883681i	$0.344943\pi$
$678$	0	0
$679$	8.84415	0.339407
$680$	0	0
$681$	27.5138	1.05433
$682$	0	0
$683$	−48.6453	−1.86136	−0.930680	−	0.365834i	$-0.880784\pi$
$683$	−48.6453	−1.86136	−0.930680	+	0.365834i	$0.880784\pi$
$684$	0	0
$685$	15.0668	0.575671
$686$	0	0
$687$	−20.3771	−0.777437
$688$	0	0
$689$	−3.02605	−0.115283
$690$	0	0
$691$	−19.0681	−0.725383	−0.362692	−	0.931909i	$-0.618142\pi$
$691$	−19.0681	−0.725383	−0.362692	+	0.931909i	$0.618142\pi$
$692$	0	0
$693$	14.4322	0.548234
$694$	0	0
$695$	1.60589	0.0609147
$696$	0	0
$697$	−13.1296	−0.497319
$698$	0	0
$699$	21.7421	0.822360
$700$	0	0
$701$	25.2300	0.952923	0.476462	−	0.879195i	$-0.341919\pi$
$701$	25.2300	0.952923	0.476462	+	0.879195i	$0.341919\pi$
$702$	0	0
$703$	−0.986116	−0.0371921
$704$	0	0
$705$	3.41831	0.128741
$706$	0	0
$707$	33.1644	1.24728
$708$	0	0
$709$	23.9708	0.900241	0.450120	−	0.892968i	$-0.351381\pi$
$709$	23.9708	0.900241	0.450120	+	0.892968i	$0.351381\pi$
$710$	0	0
$711$	11.5276	0.432319
$712$	0	0
$713$	−0.158582	−0.00593895
$714$	0	0
$715$	−2.59838	−0.0971738
$716$	0	0
$717$	−31.8529	−1.18957
$718$	0	0
$719$	−42.0834	−1.56945	−0.784723	−	0.619847i	$-0.787195\pi$
$719$	−42.0834	−1.56945	−0.784723	+	0.619847i	$0.787195\pi$
$720$	0	0
$721$	29.6619	1.10467
$722$	0	0
$723$	−10.5870	−0.393735
$724$	0	0
$725$	−0.360177	−0.0133767
$726$	0	0
$727$	−37.1388	−1.37740	−0.688700	−	0.725046i	$-0.741818\pi$
$727$	−37.1388	−1.37740	−0.688700	+	0.725046i	$0.741818\pi$
$728$	0	0
$729$	29.3698	1.08777
$730$	0	0
$731$	−16.3151	−0.603437
$732$	0	0
$733$	−11.3367	−0.418730	−0.209365	−	0.977838i	$-0.567140\pi$
$733$	−11.3367	−0.418730	−0.209365	+	0.977838i	$0.567140\pi$
$734$	0	0
$735$	−0.820698	−0.0302719
$736$	0	0
$737$	−56.6229	−2.08573
$738$	0	0
$739$	39.7397	1.46185	0.730924	−	0.682459i	$-0.239090\pi$
$739$	39.7397	1.46185	0.730924	+	0.682459i	$0.239090\pi$
$740$	0	0
$741$	−0.747325	−0.0274537
$742$	0	0
$743$	32.6071	1.19624	0.598119	−	0.801408i	$-0.295915\pi$
$743$	32.6071	1.19624	0.598119	+	0.801408i	$0.295915\pi$
$744$	0	0
$745$	6.73516	0.246757
$746$	0	0
$747$	2.18107	0.0798011
$748$	0	0
$749$	−4.70683	−0.171984
$750$	0	0
$751$	−11.1442	−0.406656	−0.203328	−	0.979111i	$-0.565176\pi$
$751$	−11.1442	−0.406656	−0.203328	+	0.979111i	$0.565176\pi$
$752$	0	0
$753$	31.9060	1.16272
$754$	0	0
$755$	−3.53594	−0.128686
$756$	0	0
$757$	−42.4631	−1.54335	−0.771674	−	0.636018i	$-0.780581\pi$
$757$	−42.4631	−1.54335	−0.771674	+	0.636018i	$0.780581\pi$
$758$	0	0
$759$	−4.32941	−0.157148
$760$	0	0
$761$	24.7779	0.898198	0.449099	−	0.893482i	$-0.351745\pi$
$761$	24.7779	0.898198	0.449099	+	0.893482i	$0.351745\pi$
$762$	0	0
$763$	−27.0619	−0.979706
$764$	0	0
$765$	−1.48602	−0.0537272
$766$	0	0
$767$	1.89454	0.0684080
$768$	0	0
$769$	34.3809	1.23981	0.619904	−	0.784678i	$-0.287172\pi$
$769$	34.3809	1.23981	0.619904	+	0.784678i	$0.287172\pi$
$770$	0	0
$771$	38.9626	1.40320
$772$	0	0
$773$	−31.3292	−1.12683	−0.563416	−	0.826174i	$-0.690513\pi$
$773$	−31.3292	−1.12683	−0.563416	+	0.826174i	$0.690513\pi$
$774$	0	0
$775$	1.42098	0.0510432
$776$	0	0
$777$	−3.60569	−0.129353
$778$	0	0
$779$	6.47366	0.231943
$780$	0	0
$781$	−19.0917	−0.683155
$782$	0	0
$783$	0.466727	0.0166795
$784$	0	0
$785$	18.3348	0.654398
$786$	0	0
$787$	−22.8377	−0.814075	−0.407037	−	0.913411i	$-0.633438\pi$
$787$	−22.8377	−0.814075	−0.407037	+	0.913411i	$0.633438\pi$
$788$	0	0
$789$	−1.28838	−0.0458675
$790$	0	0
$791$	15.5556	0.553095
$792$	0	0
$793$	−7.89477	−0.280351
$794$	0	0
$795$	−6.58976	−0.233715
$796$	0	0
$797$	−30.1710	−1.06871	−0.534357	−	0.845259i	$-0.679446\pi$
$797$	−30.1710	−1.06871	−0.534357	+	0.845259i	$0.679446\pi$
$798$	0	0
$799$	−5.95816	−0.210784
$800$	0	0
$801$	7.91631	0.279709
$802$	0	0
$803$	35.9862	1.26993
$804$	0	0
$805$	0.973946	0.0343271
$806$	0	0
$807$	38.8226	1.36662
$808$	0	0
$809$	−35.6117	−1.25204	−0.626021	−	0.779806i	$-0.715317\pi$
$809$	−35.6117	−1.25204	−0.626021	+	0.779806i	$0.715317\pi$
$810$	0	0
$811$	−1.92978	−0.0677637	−0.0338818	−	0.999426i	$-0.510787\pi$
$811$	−1.92978	−0.0677637	−0.0338818	+	0.999426i	$0.510787\pi$
$812$	0	0
$813$	18.7394	0.657219
$814$	0	0
$815$	7.75875	0.271777
$816$	0	0
$817$	8.04431	0.281435
$818$	0	0
$819$	1.23024	0.0429881
$820$	0	0
$821$	−32.7706	−1.14370	−0.571851	−	0.820358i	$-0.693774\pi$
$821$	−32.7706	−1.14370	−0.571851	+	0.820358i	$0.693774\pi$
$822$	0	0
$823$	42.6974	1.48834	0.744169	−	0.667991i	$-0.232846\pi$
$823$	42.6974	1.48834	0.744169	+	0.667991i	$0.232846\pi$
$824$	0	0
$825$	38.7938	1.35063
$826$	0	0
$827$	−29.1639	−1.01413	−0.507065	−	0.861908i	$-0.669269\pi$
$827$	−29.1639	−1.01413	−0.507065	+	0.861908i	$0.669269\pi$
$828$	0	0
$829$	−13.3217	−0.462682	−0.231341	−	0.972873i	$-0.574311\pi$
$829$	−13.3217	−0.462682	−0.231341	+	0.972873i	$0.574311\pi$
$830$	0	0
$831$	−3.62232	−0.125657
$832$	0	0
$833$	1.43048	0.0495633
$834$	0	0
$835$	7.46709	0.258409
$836$	0	0
$837$	−1.84134	−0.0636461
$838$	0	0
$839$	15.6638	0.540775	0.270388	−	0.962752i	$-0.412848\pi$
$839$	15.6638	0.540775	0.270388	+	0.962752i	$0.412848\pi$
$840$	0	0
$841$	−28.9932	−0.999765
$842$	0	0
$843$	−19.7809	−0.681290
$844$	0	0
$845$	10.1497	0.349161
$846$	0	0
$847$	−68.2098	−2.34372
$848$	0	0
$849$	30.0062	1.02981
$850$	0	0
$851$	−0.486973	−0.0166932
$852$	0	0
$853$	−14.1065	−0.482999	−0.241500	−	0.970401i	$-0.577639\pi$
$853$	−14.1065	−0.482999	−0.241500	+	0.970401i	$0.577639\pi$
$854$	0	0
$855$	0.732694	0.0250576
$856$	0	0
$857$	−49.9013	−1.70459	−0.852297	−	0.523057i	$-0.824791\pi$
$857$	−49.9013	−1.70459	−0.852297	+	0.523057i	$0.824791\pi$
$858$	0	0
$859$	25.7353	0.878079	0.439039	−	0.898468i	$-0.355319\pi$
$859$	25.7353	0.878079	0.439039	+	0.898468i	$0.355319\pi$
$860$	0	0
$861$	23.6707	0.806694
$862$	0	0
$863$	1.75695	0.0598071	0.0299036	−	0.999553i	$-0.490480\pi$
$863$	1.75695	0.0598071	0.0299036	+	0.999553i	$0.490480\pi$
$864$	0	0
$865$	−4.81694	−0.163781
$866$	0	0
$867$	18.6977	0.635007
$868$	0	0
$869$	−76.5085	−2.59537
$870$	0	0
$871$	−4.82670	−0.163546
$872$	0	0
$873$	3.28565	0.111202
$874$	0	0
$875$	−18.7271	−0.633091
$876$	0	0
$877$	−49.1630	−1.66012	−0.830058	−	0.557677i	$-0.811693\pi$
$877$	−49.1630	−1.66012	−0.830058	+	0.557677i	$0.811693\pi$
$878$	0	0
$879$	15.7836	0.532369
$880$	0	0
$881$	−22.5889	−0.761041	−0.380521	−	0.924772i	$-0.624255\pi$
$881$	−22.5889	−0.761041	−0.380521	+	0.924772i	$0.624255\pi$
$882$	0	0
$883$	11.9959	0.403694	0.201847	−	0.979417i	$-0.435306\pi$
$883$	11.9959	0.403694	0.201847	+	0.979417i	$0.435306\pi$
$884$	0	0
$885$	4.12570	0.138684
$886$	0	0
$887$	11.6643	0.391649	0.195825	−	0.980639i	$-0.437262\pi$
$887$	11.6643	0.391649	0.195825	+	0.980639i	$0.437262\pi$
$888$	0	0
$889$	37.2770	1.25023
$890$	0	0
$891$	−32.9993	−1.10552
$892$	0	0
$893$	2.93772	0.0983069
$894$	0	0
$895$	−2.53613	−0.0847736
$896$	0	0
$897$	−0.369051	−0.0123223
$898$	0	0
$899$	−0.0268799	−0.000896494 0
$900$	0	0
$901$	11.4860	0.382655
$902$	0	0
$903$	29.4137	0.978826
$904$	0	0
$905$	−13.5320	−0.449818
$906$	0	0
$907$	35.6139	1.18254	0.591270	−	0.806474i	$-0.298627\pi$
$907$	35.6139	1.18254	0.591270	+	0.806474i	$0.298627\pi$
$908$	0	0
$909$	12.3207	0.408653
$910$	0	0
$911$	17.0835	0.566001	0.283001	−	0.959120i	$-0.408670\pi$
$911$	17.0835	0.566001	0.283001	+	0.959120i	$0.408670\pi$
$912$	0	0
$913$	−14.4757	−0.479076
$914$	0	0
$915$	−17.1922	−0.568358
$916$	0	0
$917$	−10.1279	−0.334452
$918$	0	0
$919$	−45.0756	−1.48691	−0.743453	−	0.668788i	$-0.766813\pi$
$919$	−45.0756	−1.48691	−0.743453	+	0.668788i	$0.766813\pi$
$920$	0	0
$921$	2.01122	0.0662718
$922$	0	0
$923$	−1.62743	−0.0535676
$924$	0	0
$925$	4.36354	0.143472
$926$	0	0
$927$	11.0196	0.361930
$928$	0	0
$929$	−51.4533	−1.68813	−0.844065	−	0.536241i	$-0.819844\pi$
$929$	−51.4533	−1.68813	−0.844065	+	0.536241i	$0.819844\pi$
$930$	0	0
$931$	−0.705312	−0.0231157
$932$	0	0
$933$	11.4247	0.374029
$934$	0	0
$935$	9.86268	0.322544
$936$	0	0
$937$	21.3774	0.698368	0.349184	−	0.937054i	$-0.386459\pi$
$937$	21.3774	0.698368	0.349184	+	0.937054i	$0.386459\pi$
$938$	0	0
$939$	−20.7867	−0.678349
$940$	0	0
$941$	25.9460	0.845815	0.422907	−	0.906173i	$-0.361010\pi$
$941$	25.9460	0.845815	0.422907	+	0.906173i	$0.361010\pi$
$942$	0	0
$943$	3.19688	0.104105
$944$	0	0
$945$	11.3088	0.367874
$946$	0	0
$947$	−5.25323	−0.170707	−0.0853535	−	0.996351i	$-0.527202\pi$
$947$	−5.25323	−0.170707	−0.0853535	+	0.996351i	$0.527202\pi$
$948$	0	0
$949$	3.06756	0.0995774
$950$	0	0
$951$	−4.61888	−0.149777
$952$	0	0
$953$	6.17221	0.199937	0.0999687	−	0.994991i	$-0.468126\pi$
$953$	6.17221	0.199937	0.0999687	+	0.994991i	$0.468126\pi$
$954$	0	0
$955$	−19.4494	−0.629368
$956$	0	0
$957$	−0.733840	−0.0237217
$958$	0	0
$959$	47.3454	1.52886
$960$	0	0
$961$	−30.8940	−0.996579
$962$	0	0
$963$	−1.74861	−0.0563482
$964$	0	0
$965$	10.9165	0.351415
$966$	0	0
$967$	32.3652	1.04079	0.520397	−	0.853924i	$-0.325784\pi$
$967$	32.3652	1.04079	0.520397	+	0.853924i	$0.325784\pi$
$968$	0	0
$969$	2.83663	0.0911256
$970$	0	0
$971$	16.4907	0.529213	0.264606	−	0.964356i	$-0.414758\pi$
$971$	16.4907	0.529213	0.264606	+	0.964356i	$0.414758\pi$
$972$	0	0
$973$	5.04630	0.161777
$974$	0	0
$975$	3.30690	0.105905
$976$	0	0
$977$	10.4341	0.333817	0.166908	−	0.985972i	$-0.446622\pi$
$977$	10.4341	0.333817	0.166908	+	0.985972i	$0.446622\pi$
$978$	0	0
$979$	−52.5404	−1.67920
$980$	0	0
$981$	−10.0536	−0.320988
$982$	0	0
$983$	60.6011	1.93287	0.966437	−	0.256904i	$-0.0827025\pi$
$983$	60.6011	1.93287	0.966437	+	0.256904i	$0.0827025\pi$
$984$	0	0
$985$	−13.1159	−0.417908
$986$	0	0
$987$	10.7416	0.341910
$988$	0	0
$989$	3.97251	0.126319
$990$	0	0
$991$	15.5489	0.493926	0.246963	−	0.969025i	$-0.420567\pi$
$991$	15.5489	0.493926	0.246963	+	0.969025i	$0.420567\pi$
$992$	0	0
$993$	−41.0999	−1.30427
$994$	0	0
$995$	1.32694	0.0420667
$996$	0	0
$997$	−39.1493	−1.23987	−0.619935	−	0.784653i	$-0.712841\pi$
$997$	−39.1493	−1.23987	−0.619935	+	0.784653i	$0.712841\pi$
$998$	0	0
$999$	−5.65438	−0.178897

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2368.2.a.bi.1.1		4
4.3	odd	2		2368.2.a.bf.1.4		4
8.3	odd	2		1184.2.a.o.1.1	yes	4
8.5	even	2		1184.2.a.n.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1184.2.a.n.1.4	✓	4	8.5	even	2
1184.2.a.o.1.1	yes	4	8.3	odd	2
2368.2.a.bf.1.4		4	4.3	odd	2
2368.2.a.bi.1.1		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2368 = 2^{6} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2368.a (trivial)

\(f(q)\)	\(=\)	\(q-1.43828 q^{3} -0.797785 q^{5} -2.50694 q^{7} -0.931341 q^{9} +O(q^{10})\)	\(q-1.43828 q^{3} -0.797785 q^{5} -2.50694 q^{7} -0.931341 q^{9} +6.18129 q^{11} +0.526911 q^{13} +1.14744 q^{15} -2.00000 q^{17} +0.986116 q^{19} +3.60569 q^{21} +0.486973 q^{23} -4.36354 q^{25} +5.65438 q^{27} +0.0825425 q^{29} -0.325649 q^{31} -8.89045 q^{33} +2.00000 q^{35} -1.00000 q^{37} -0.757847 q^{39} +6.56480 q^{41} +8.15756 q^{43} +0.743010 q^{45} +2.97908 q^{47} -0.715242 q^{49} +2.87657 q^{51} -5.74301 q^{53} -4.93134 q^{55} -1.41831 q^{57} +3.59557 q^{59} -14.9831 q^{61} +2.33482 q^{63} -0.420361 q^{65} -9.16037 q^{67} -0.700405 q^{69} -3.08863 q^{71} +5.82179 q^{73} +6.27601 q^{75} -15.4961 q^{77} -12.3774 q^{79} -5.33858 q^{81} -2.34186 q^{83} +1.59557 q^{85} -0.118720 q^{87} -8.49990 q^{89} -1.32093 q^{91} +0.468375 q^{93} -0.786708 q^{95} -3.52786 q^{97} -5.75689 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{3} - 3 q^{5} - 6 q^{7} + q^{9}+O(q^{10}) \) 4 * q + 3 * q^3 - 3 * q^5 - 6 * q^7 + q^9	\( 4 q + 3 q^{3} - 3 q^{5} - 6 q^{7} + q^{9} + 3 q^{11} - 3 q^{13} - 8 q^{15} - 8 q^{17} + 12 q^{19} - 8 q^{21} - q^{23} + 3 q^{25} + 6 q^{27} - 3 q^{29} - 19 q^{31} - 10 q^{33} + 8 q^{35} - 4 q^{37} - 5 q^{39} - 17 q^{41} - 2 q^{43} - 10 q^{45} - 10 q^{47} - 6 q^{49} - 6 q^{51} - 10 q^{53} - 15 q^{55} + 2 q^{57} + 14 q^{59} - 9 q^{61} - 18 q^{63} - 30 q^{65} + 7 q^{67} + 9 q^{69} - 16 q^{71} - 7 q^{73} + 14 q^{75} - 14 q^{77} - 21 q^{79} - 8 q^{81} - 12 q^{83} + 6 q^{85} + 19 q^{87} + 14 q^{91} - 30 q^{93} - 2 q^{95} - 32 q^{97} - 2 q^{99}+O(q^{100}) \) 4 * q + 3 * q^3 - 3 * q^5 - 6 * q^7 + q^9 + 3 * q^11 - 3 * q^13 - 8 * q^15 - 8 * q^17 + 12 * q^19 - 8 * q^21 - q^23 + 3 * q^25 + 6 * q^27 - 3 * q^29 - 19 * q^31 - 10 * q^33 + 8 * q^35 - 4 * q^37 - 5 * q^39 - 17 * q^41 - 2 * q^43 - 10 * q^45 - 10 * q^47 - 6 * q^49 - 6 * q^51 - 10 * q^53 - 15 * q^55 + 2 * q^57 + 14 * q^59 - 9 * q^61 - 18 * q^63 - 30 * q^65 + 7 * q^67 + 9 * q^69 - 16 * q^71 - 7 * q^73 + 14 * q^75 - 14 * q^77 - 21 * q^79 - 8 * q^81 - 12 * q^83 + 6 * q^85 + 19 * q^87 + 14 * q^91 - 30 * q^93 - 2 * q^95 - 32 * q^97 - 2 * q^99

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