LMFDB - Embedded newform 6032.2.a.ba.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6032,2,Mod(1,6032)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6032, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6032.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6032 = 2^{4} \cdot 13 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6032.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:	$48.1657624992$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 21x^{8} + 40x^{7} + 138x^{6} - 243x^{5} - 318x^{4} + 448x^{3} + 312x^{2} - 240x - 128 $ x^10 - 2x^9 - 21x^8 + 40x^7 + 138x^6 - 243x^5 - 318x^4 + 448x^3 + 312x^2 - 240*x - 128
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 3016)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-2.71446$ of defining polynomial
Character	$\chi$	$=$	6032.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.71446 q^{3} -0.957650 q^{5} -1.45512 q^{7} +4.36832 q^{9} +O(q^{10})$	$q+2.71446 q^{3} -0.957650 q^{5} -1.45512 q^{7} +4.36832 q^{9} +4.83356 q^{11} +1.00000 q^{13} -2.59951 q^{15} +7.18535 q^{17} +4.48094 q^{19} -3.94986 q^{21} +1.00431 q^{23} -4.08291 q^{25} +3.71424 q^{27} -1.00000 q^{29} -2.69838 q^{31} +13.1205 q^{33} +1.39349 q^{35} -4.54623 q^{37} +2.71446 q^{39} +1.33262 q^{41} -2.40853 q^{43} -4.18332 q^{45} +2.47227 q^{47} -4.88263 q^{49} +19.5044 q^{51} +3.16124 q^{53} -4.62885 q^{55} +12.1634 q^{57} +4.04941 q^{59} +0.0266504 q^{61} -6.35641 q^{63} -0.957650 q^{65} +4.16633 q^{67} +2.72616 q^{69} +4.29890 q^{71} +0.151929 q^{73} -11.0829 q^{75} -7.03339 q^{77} +0.682554 q^{79} -3.02277 q^{81} +6.76374 q^{83} -6.88105 q^{85} -2.71446 q^{87} +5.62684 q^{89} -1.45512 q^{91} -7.32466 q^{93} -4.29117 q^{95} +15.7797 q^{97} +21.1145 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{3} + 5 q^{5} + 2 q^{7} + 16 q^{9}+O(q^{10}) $ 10 * q - 2 * q^3 + 5 * q^5 + 2 * q^7 + 16 * q^9	$ 10 q - 2 q^{3} + 5 q^{5} + 2 q^{7} + 16 q^{9} - 4 q^{11} + 10 q^{13} - 8 q^{15} + 10 q^{17} + q^{19} + q^{21} - 23 q^{23} + 25 q^{25} - 2 q^{27} - 10 q^{29} + 13 q^{31} + 15 q^{33} + 12 q^{35} - 7 q^{37} - 2 q^{39} + 16 q^{41} + 12 q^{43} + 55 q^{45} - 11 q^{47} + 25 q^{51} + 11 q^{53} - 22 q^{55} - 6 q^{57} + 11 q^{59} + 34 q^{61} - 37 q^{63} + 5 q^{65} + 23 q^{67} + 2 q^{69} + 4 q^{71} + 39 q^{73} - 11 q^{75} + 32 q^{77} - 5 q^{79} + 38 q^{81} - 6 q^{83} + 45 q^{85} + 2 q^{87} - 24 q^{89} + 2 q^{91} + 13 q^{93} - 33 q^{95} + 19 q^{97}+O(q^{100}) $ 10 * q - 2 * q^3 + 5 * q^5 + 2 * q^7 + 16 * q^9 - 4 * q^11 + 10 * q^13 - 8 * q^15 + 10 * q^17 + q^19 + q^21 - 23 * q^23 + 25 * q^25 - 2 * q^27 - 10 * q^29 + 13 * q^31 + 15 * q^33 + 12 * q^35 - 7 * q^37 - 2 * q^39 + 16 * q^41 + 12 * q^43 + 55 * q^45 - 11 * q^47 + 25 * q^51 + 11 * q^53 - 22 * q^55 - 6 * q^57 + 11 * q^59 + 34 * q^61 - 37 * q^63 + 5 * q^65 + 23 * q^67 + 2 * q^69 + 4 * q^71 + 39 * q^73 - 11 * q^75 + 32 * q^77 - 5 * q^79 + 38 * q^81 - 6 * q^83 + 45 * q^85 + 2 * q^87 - 24 * q^89 + 2 * q^91 + 13 * q^93 - 33 * q^95 + 19 * q^97

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
$3$	2.71446	1.56720	0.783598	−	0.621268i	$-0.213382\pi$
$3$	2.71446	1.56720	0.783598	+	0.621268i	$0.213382\pi$
$4$	0	0
$5$	−0.957650	−0.428274	−0.214137	−	0.976804i	$-0.568694\pi$
$5$	−0.957650	−0.428274	−0.214137	+	0.976804i	$0.568694\pi$
$6$	0	0
$7$	−1.45512	−0.549983	−0.274991	−	0.961447i	$-0.588675\pi$
$7$	−1.45512	−0.549983	−0.274991	+	0.961447i	$0.588675\pi$
$8$	0	0
$9$	4.36832	1.45611
$10$	0	0
$11$	4.83356	1.45737	0.728686	−	0.684848i	$-0.240131\pi$
$11$	4.83356	1.45737	0.728686	+	0.684848i	$0.240131\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−2.59951	−0.671190
$16$	0	0
$17$	7.18535	1.74270	0.871351	−	0.490660i	$-0.163244\pi$
$17$	7.18535	1.74270	0.871351	+	0.490660i	$0.163244\pi$
$18$	0	0
$19$	4.48094	1.02800	0.513999	−	0.857791i	$-0.328163\pi$
$19$	4.48094	1.02800	0.513999	+	0.857791i	$0.328163\pi$
$20$	0	0
$21$	−3.94986	−0.861931
$22$	0	0
$23$	1.00431	0.209413	0.104707	−	0.994503i	$-0.466610\pi$
$23$	1.00431	0.209413	0.104707	+	0.994503i	$0.466610\pi$
$24$	0	0
$25$	−4.08291	−0.816581
$26$	0	0
$27$	3.71424	0.714806
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−2.69838	−0.484643	−0.242322	−	0.970196i	$-0.577909\pi$
$31$	−2.69838	−0.484643	−0.242322	+	0.970196i	$0.577909\pi$
$32$	0	0
$33$	13.1205	2.28399
$34$	0	0
$35$	1.39349	0.235543
$36$	0	0
$37$	−4.54623	−0.747395	−0.373698	−	0.927551i	$-0.621910\pi$
$37$	−4.54623	−0.747395	−0.373698	+	0.927551i	$0.621910\pi$
$38$	0	0
$39$	2.71446	0.434662
$40$	0	0
$41$	1.33262	0.208120	0.104060	−	0.994571i	$-0.466817\pi$
$41$	1.33262	0.208120	0.104060	+	0.994571i	$0.466817\pi$
$42$	0	0
$43$	−2.40853	−0.367298	−0.183649	−	0.982992i	$-0.558791\pi$
$43$	−2.40853	−0.367298	−0.183649	+	0.982992i	$0.558791\pi$
$44$	0	0
$45$	−4.18332	−0.623612
$46$	0	0
$47$	2.47227	0.360618	0.180309	−	0.983610i	$-0.442290\pi$
$47$	2.47227	0.360618	0.180309	+	0.983610i	$0.442290\pi$
$48$	0	0
$49$	−4.88263	−0.697519
$50$	0	0
$51$	19.5044	2.73116
$52$	0	0
$53$	3.16124	0.434230	0.217115	−	0.976146i	$-0.430335\pi$
$53$	3.16124	0.434230	0.217115	+	0.976146i	$0.430335\pi$
$54$	0	0
$55$	−4.62885	−0.624155
$56$	0	0
$57$	12.1634	1.61108
$58$	0	0
$59$	4.04941	0.527188	0.263594	−	0.964634i	$-0.415092\pi$
$59$	4.04941	0.527188	0.263594	+	0.964634i	$0.415092\pi$
$60$	0	0
$61$	0.0266504	0.00341224	0.00170612	−	0.999999i	$-0.499457\pi$
$61$	0.0266504	0.00341224	0.00170612	+	0.999999i	$0.499457\pi$
$62$	0	0
$63$	−6.35641	−0.800832
$64$	0	0
$65$	−0.957650	−0.118782
$66$	0	0
$67$	4.16633	0.508998	0.254499	−	0.967073i	$-0.418089\pi$
$67$	4.16633	0.508998	0.254499	+	0.967073i	$0.418089\pi$
$68$	0	0
$69$	2.72616	0.328192
$70$	0	0
$71$	4.29890	0.510186	0.255093	−	0.966917i	$-0.417894\pi$
$71$	4.29890	0.510186	0.255093	+	0.966917i	$0.417894\pi$
$72$	0	0
$73$	0.151929	0.0177820	0.00889100	−	0.999960i	$-0.497170\pi$
$73$	0.151929	0.0177820	0.00889100	+	0.999960i	$0.497170\pi$
$74$	0	0
$75$	−11.0829	−1.27974
$76$	0	0
$77$	−7.03339	−0.801529
$78$	0	0
$79$	0.682554	0.0767933	0.0383966	−	0.999263i	$-0.487775\pi$
$79$	0.682554	0.0767933	0.0383966	+	0.999263i	$0.487775\pi$
$80$	0	0
$81$	−3.02277	−0.335863
$82$	0	0
$83$	6.76374	0.742417	0.371209	−	0.928549i	$-0.378943\pi$
$83$	6.76374	0.742417	0.371209	+	0.928549i	$0.378943\pi$
$84$	0	0
$85$	−6.88105	−0.746354
$86$	0	0
$87$	−2.71446	−0.291021
$88$	0	0
$89$	5.62684	0.596444	0.298222	−	0.954497i	$-0.403606\pi$
$89$	5.62684	0.596444	0.298222	+	0.954497i	$0.403606\pi$
$90$	0	0
$91$	−1.45512	−0.152538
$92$	0	0
$93$	−7.32466	−0.759532
$94$	0	0
$95$	−4.29117	−0.440265
$96$	0	0
$97$	15.7797	1.60218	0.801092	−	0.598541i	$-0.204253\pi$
$97$	15.7797	1.60218	0.801092	+	0.598541i	$0.204253\pi$
$98$	0	0
$99$	21.1145	2.12209
$100$	0	0
$101$	16.7889	1.67056	0.835279	−	0.549826i	$-0.185306\pi$
$101$	16.7889	1.67056	0.835279	+	0.549826i	$0.185306\pi$
$102$	0	0
$103$	7.49640	0.738642	0.369321	−	0.929302i	$-0.379590\pi$
$103$	7.49640	0.738642	0.369321	+	0.929302i	$0.379590\pi$
$104$	0	0
$105$	3.78259	0.369143
$106$	0	0
$107$	−16.1921	−1.56535	−0.782673	−	0.622434i	$-0.786144\pi$
$107$	−16.1921	−1.56535	−0.782673	+	0.622434i	$0.786144\pi$
$108$	0	0
$109$	−12.9087	−1.23643	−0.618214	−	0.786010i	$-0.712143\pi$
$109$	−12.9087	−1.23643	−0.618214	+	0.786010i	$0.712143\pi$
$110$	0	0
$111$	−12.3406	−1.17132
$112$	0	0
$113$	−4.11400	−0.387013	−0.193506	−	0.981099i	$-0.561986\pi$
$113$	−4.11400	−0.387013	−0.193506	+	0.981099i	$0.561986\pi$
$114$	0	0
$115$	−0.961778	−0.0896862
$116$	0	0
$117$	4.36832	0.403851
$118$	0	0
$119$	−10.4555	−0.958456
$120$	0	0
$121$	12.3633	1.12393
$122$	0	0
$123$	3.61734	0.326165
$124$	0	0
$125$	8.69824	0.777995
$126$	0	0
$127$	−14.7756	−1.31112	−0.655560	−	0.755143i	$-0.727567\pi$
$127$	−14.7756	−1.31112	−0.655560	+	0.755143i	$0.727567\pi$
$128$	0	0
$129$	−6.53788	−0.575628
$130$	0	0
$131$	−2.41966	−0.211406	−0.105703	−	0.994398i	$-0.533709\pi$
$131$	−2.41966	−0.211406	−0.105703	+	0.994398i	$0.533709\pi$
$132$	0	0
$133$	−6.52029	−0.565381
$134$	0	0
$135$	−3.55694	−0.306133
$136$	0	0
$137$	16.7248	1.42889	0.714447	−	0.699689i	$-0.246678\pi$
$137$	16.7248	1.42889	0.714447	+	0.699689i	$0.246678\pi$
$138$	0	0
$139$	−11.1508	−0.945800	−0.472900	−	0.881116i	$-0.656793\pi$
$139$	−11.1508	−0.945800	−0.472900	+	0.881116i	$0.656793\pi$
$140$	0	0
$141$	6.71090	0.565159
$142$	0	0
$143$	4.83356	0.404202
$144$	0	0
$145$	0.957650	0.0795285
$146$	0	0
$147$	−13.2537	−1.09315
$148$	0	0
$149$	7.85137	0.643209	0.321605	−	0.946874i	$-0.395778\pi$
$149$	7.85137	0.643209	0.321605	+	0.946874i	$0.395778\pi$
$150$	0	0
$151$	−1.48065	−0.120494	−0.0602470	−	0.998184i	$-0.519189\pi$
$151$	−1.48065	−0.120494	−0.0602470	+	0.998184i	$0.519189\pi$
$152$	0	0
$153$	31.3879	2.53756
$154$	0	0
$155$	2.58410	0.207560
$156$	0	0
$157$	7.67008	0.612139	0.306070	−	0.952009i	$-0.400986\pi$
$157$	7.67008	0.612139	0.306070	+	0.952009i	$0.400986\pi$
$158$	0	0
$159$	8.58107	0.680523
$160$	0	0
$161$	−1.46139	−0.115174
$162$	0	0
$163$	−9.85502	−0.771904	−0.385952	−	0.922519i	$-0.626127\pi$
$163$	−9.85502	−0.771904	−0.385952	+	0.922519i	$0.626127\pi$
$164$	0	0
$165$	−12.5649	−0.978173
$166$	0	0
$167$	7.61916	0.589589	0.294794	−	0.955561i	$-0.404749\pi$
$167$	7.61916	0.589589	0.294794	+	0.955561i	$0.404749\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	19.5742	1.49687
$172$	0	0
$173$	1.98351	0.150804	0.0754019	−	0.997153i	$-0.475976\pi$
$173$	1.98351	0.150804	0.0754019	+	0.997153i	$0.475976\pi$
$174$	0	0
$175$	5.94111	0.449106
$176$	0	0
$177$	10.9920	0.826207
$178$	0	0
$179$	1.59223	0.119009	0.0595046	−	0.998228i	$-0.481048\pi$
$179$	1.59223	0.119009	0.0595046	+	0.998228i	$0.481048\pi$
$180$	0	0
$181$	5.68887	0.422851	0.211425	−	0.977394i	$-0.432189\pi$
$181$	5.68887	0.422851	0.211425	+	0.977394i	$0.432189\pi$
$182$	0	0
$183$	0.0723416	0.00534764
$184$	0	0
$185$	4.35370	0.320090
$186$	0	0
$187$	34.7308	2.53977
$188$	0	0
$189$	−5.40466	−0.393131
$190$	0	0
$191$	−19.0628	−1.37934	−0.689668	−	0.724126i	$-0.742244\pi$
$191$	−19.0628	−1.37934	−0.689668	+	0.724126i	$0.742244\pi$
$192$	0	0
$193$	−5.57434	−0.401250	−0.200625	−	0.979668i	$-0.564297\pi$
$193$	−5.57434	−0.401250	−0.200625	+	0.979668i	$0.564297\pi$
$194$	0	0
$195$	−2.59951	−0.186154
$196$	0	0
$197$	−1.00512	−0.0716121	−0.0358061	−	0.999359i	$-0.511400\pi$
$197$	−1.00512	−0.0716121	−0.0358061	+	0.999359i	$0.511400\pi$
$198$	0	0
$199$	1.32520	0.0939408	0.0469704	−	0.998896i	$-0.485043\pi$
$199$	1.32520	0.0939408	0.0469704	+	0.998896i	$0.485043\pi$
$200$	0	0
$201$	11.3094	0.797701
$202$	0	0
$203$	1.45512	0.102129
$204$	0	0
$205$	−1.27618	−0.0891324
$206$	0	0
$207$	4.38714	0.304928
$208$	0	0
$209$	21.6589	1.49818
$210$	0	0
$211$	−6.98899	−0.481142	−0.240571	−	0.970632i	$-0.577335\pi$
$211$	−6.98899	−0.481142	−0.240571	+	0.970632i	$0.577335\pi$
$212$	0	0
$213$	11.6692	0.799562
$214$	0	0
$215$	2.30653	0.157304
$216$	0	0
$217$	3.92646	0.266545
$218$	0	0
$219$	0.412407	0.0278679
$220$	0	0
$221$	7.18535	0.483339
$222$	0	0
$223$	10.1084	0.676908	0.338454	−	0.940983i	$-0.390096\pi$
$223$	10.1084	0.676908	0.338454	+	0.940983i	$0.390096\pi$
$224$	0	0
$225$	−17.8354	−1.18903
$226$	0	0
$227$	−19.7743	−1.31247	−0.656233	−	0.754558i	$-0.727851\pi$
$227$	−19.7743	−1.31247	−0.656233	+	0.754558i	$0.727851\pi$
$228$	0	0
$229$	3.94345	0.260590	0.130295	−	0.991475i	$-0.458408\pi$
$229$	3.94345	0.260590	0.130295	+	0.991475i	$0.458408\pi$
$230$	0	0
$231$	−19.0919	−1.25615
$232$	0	0
$233$	19.0555	1.24837	0.624184	−	0.781277i	$-0.285431\pi$
$233$	19.0555	1.24837	0.624184	+	0.781277i	$0.285431\pi$
$234$	0	0
$235$	−2.36757	−0.154443
$236$	0	0
$237$	1.85277	0.120350
$238$	0	0
$239$	2.85476	0.184659	0.0923296	−	0.995728i	$-0.470569\pi$
$239$	2.85476	0.184659	0.0923296	+	0.995728i	$0.470569\pi$
$240$	0	0
$241$	−4.75999	−0.306618	−0.153309	−	0.988178i	$-0.548993\pi$
$241$	−4.75999	−0.306618	−0.153309	+	0.988178i	$0.548993\pi$
$242$	0	0
$243$	−19.3479	−1.24117
$244$	0	0
$245$	4.67585	0.298729
$246$	0	0
$247$	4.48094	0.285115
$248$	0	0
$249$	18.3599	1.16351
$250$	0	0
$251$	−6.07425	−0.383403	−0.191702	−	0.981453i	$-0.561401\pi$
$251$	−6.07425	−0.383403	−0.191702	+	0.981453i	$0.561401\pi$
$252$	0	0
$253$	4.85439	0.305193
$254$	0	0
$255$	−18.6784	−1.16968
$256$	0	0
$257$	12.9616	0.808524	0.404262	−	0.914643i	$-0.367528\pi$
$257$	12.9616	0.808524	0.404262	+	0.914643i	$0.367528\pi$
$258$	0	0
$259$	6.61530	0.411054
$260$	0	0
$261$	−4.36832	−0.270392
$262$	0	0
$263$	−22.4112	−1.38193	−0.690966	−	0.722887i	$-0.742815\pi$
$263$	−22.4112	−1.38193	−0.690966	+	0.722887i	$0.742815\pi$
$264$	0	0
$265$	−3.02736	−0.185969
$266$	0	0
$267$	15.2739	0.934745
$268$	0	0
$269$	16.8814	1.02928	0.514638	−	0.857408i	$-0.327926\pi$
$269$	16.8814	1.02928	0.514638	+	0.857408i	$0.327926\pi$
$270$	0	0
$271$	26.0123	1.58013	0.790067	−	0.613021i	$-0.210046\pi$
$271$	26.0123	1.58013	0.790067	+	0.613021i	$0.210046\pi$
$272$	0	0
$273$	−3.94986	−0.239057
$274$	0	0
$275$	−19.7350	−1.19006
$276$	0	0
$277$	12.9730	0.779473	0.389736	−	0.920926i	$-0.372566\pi$
$277$	12.9730	0.779473	0.389736	+	0.920926i	$0.372566\pi$
$278$	0	0
$279$	−11.7874	−0.705692
$280$	0	0
$281$	24.0323	1.43364	0.716822	−	0.697256i	$-0.245596\pi$
$281$	24.0323	1.43364	0.716822	+	0.697256i	$0.245596\pi$
$282$	0	0
$283$	−11.6780	−0.694188	−0.347094	−	0.937830i	$-0.612832\pi$
$283$	−11.6780	−0.694188	−0.347094	+	0.937830i	$0.612832\pi$
$284$	0	0
$285$	−11.6482	−0.689982
$286$	0	0
$287$	−1.93912	−0.114462
$288$	0	0
$289$	34.6292	2.03701
$290$	0	0
$291$	42.8334	2.51094
$292$	0	0
$293$	−8.62418	−0.503830	−0.251915	−	0.967749i	$-0.581060\pi$
$293$	−8.62418	−0.503830	−0.251915	+	0.967749i	$0.581060\pi$
$294$	0	0
$295$	−3.87792	−0.225781
$296$	0	0
$297$	17.9530	1.04174
$298$	0	0
$299$	1.00431	0.0580808
$300$	0	0
$301$	3.50470	0.202008
$302$	0	0
$303$	45.5729	2.61809
$304$	0	0
$305$	−0.0255218	−0.00146137
$306$	0	0
$307$	4.34207	0.247815	0.123907	−	0.992294i	$-0.460457\pi$
$307$	4.34207	0.247815	0.123907	+	0.992294i	$0.460457\pi$
$308$	0	0
$309$	20.3487	1.15760
$310$	0	0
$311$	3.64278	0.206563	0.103282	−	0.994652i	$-0.467066\pi$
$311$	3.64278	0.206563	0.103282	+	0.994652i	$0.467066\pi$
$312$	0	0
$313$	−26.7805	−1.51372	−0.756860	−	0.653577i	$-0.773268\pi$
$313$	−26.7805	−1.51372	−0.756860	+	0.653577i	$0.773268\pi$
$314$	0	0
$315$	6.08722	0.342976
$316$	0	0
$317$	22.0363	1.23768	0.618841	−	0.785516i	$-0.287602\pi$
$317$	22.0363	1.23768	0.618841	+	0.785516i	$0.287602\pi$
$318$	0	0
$319$	−4.83356	−0.270627
$320$	0	0
$321$	−43.9528	−2.45320
$322$	0	0
$323$	32.1971	1.79150
$324$	0	0
$325$	−4.08291	−0.226479
$326$	0	0
$327$	−35.0401	−1.93772
$328$	0	0
$329$	−3.59745	−0.198334
$330$	0	0
$331$	6.32622	0.347720	0.173860	−	0.984770i	$-0.444376\pi$
$331$	6.32622	0.347720	0.173860	+	0.984770i	$0.444376\pi$
$332$	0	0
$333$	−19.8594	−1.08829
$334$	0	0
$335$	−3.98989	−0.217991
$336$	0	0
$337$	7.37126	0.401538	0.200769	−	0.979639i	$-0.435656\pi$
$337$	7.37126	0.401538	0.200769	+	0.979639i	$0.435656\pi$
$338$	0	0
$339$	−11.1673	−0.606525
$340$	0	0
$341$	−13.0428	−0.706306
$342$	0	0
$343$	17.2906	0.933606
$344$	0	0
$345$	−2.61071	−0.140556
$346$	0	0
$347$	−1.10093	−0.0591008	−0.0295504	−	0.999563i	$-0.509408\pi$
$347$	−1.10093	−0.0591008	−0.0295504	+	0.999563i	$0.509408\pi$
$348$	0	0
$349$	28.8578	1.54472	0.772362	−	0.635183i	$-0.219075\pi$
$349$	28.8578	1.54472	0.772362	+	0.635183i	$0.219075\pi$
$350$	0	0
$351$	3.71424	0.198252
$352$	0	0
$353$	−19.2459	−1.02436	−0.512178	−	0.858879i	$-0.671161\pi$
$353$	−19.2459	−1.02436	−0.512178	+	0.858879i	$0.671161\pi$
$354$	0	0
$355$	−4.11685	−0.218499
$356$	0	0
$357$	−28.3811	−1.50209
$358$	0	0
$359$	−16.6578	−0.879167	−0.439584	−	0.898202i	$-0.644874\pi$
$359$	−16.6578	−0.879167	−0.439584	+	0.898202i	$0.644874\pi$
$360$	0	0
$361$	1.07883	0.0567807
$362$	0	0
$363$	33.5597	1.76143
$364$	0	0
$365$	−0.145495	−0.00761556
$366$	0	0
$367$	25.0636	1.30831	0.654154	−	0.756361i	$-0.273025\pi$
$367$	25.0636	1.30831	0.654154	+	0.756361i	$0.273025\pi$
$368$	0	0
$369$	5.82130	0.303045
$370$	0	0
$371$	−4.59998	−0.238819
$372$	0	0
$373$	−27.7441	−1.43654	−0.718268	−	0.695766i	$-0.755065\pi$
$373$	−27.7441	−1.43654	−0.718268	+	0.695766i	$0.755065\pi$
$374$	0	0
$375$	23.6111	1.21927
$376$	0	0
$377$	−1.00000	−0.0515026
$378$	0	0
$379$	−23.3777	−1.20083	−0.600415	−	0.799688i	$-0.704998\pi$
$379$	−23.3777	−1.20083	−0.600415	+	0.799688i	$0.704998\pi$
$380$	0	0
$381$	−40.1078	−2.05478
$382$	0	0
$383$	3.48051	0.177846	0.0889229	−	0.996039i	$-0.471658\pi$
$383$	3.48051	0.177846	0.0889229	+	0.996039i	$0.471658\pi$
$384$	0	0
$385$	6.73553	0.343274
$386$	0	0
$387$	−10.5212	−0.534825
$388$	0	0
$389$	−19.7247	−1.00008	−0.500041	−	0.866002i	$-0.666682\pi$
$389$	−19.7247	−1.00008	−0.500041	+	0.866002i	$0.666682\pi$
$390$	0	0
$391$	7.21632	0.364945
$392$	0	0
$393$	−6.56807	−0.331316
$394$	0	0
$395$	−0.653647	−0.0328886
$396$	0	0
$397$	−25.8846	−1.29911	−0.649554	−	0.760315i	$-0.725044\pi$
$397$	−25.8846	−1.29911	−0.649554	+	0.760315i	$0.725044\pi$
$398$	0	0
$399$	−17.6991	−0.886064
$400$	0	0
$401$	−28.8917	−1.44278	−0.721392	−	0.692527i	$-0.756497\pi$
$401$	−28.8917	−1.44278	−0.721392	+	0.692527i	$0.756497\pi$
$402$	0	0
$403$	−2.69838	−0.134416
$404$	0	0
$405$	2.89475	0.143841
$406$	0	0
$407$	−21.9745	−1.08923
$408$	0	0
$409$	13.1075	0.648126	0.324063	−	0.946036i	$-0.394951\pi$
$409$	13.1075	0.648126	0.324063	+	0.946036i	$0.394951\pi$
$410$	0	0
$411$	45.3988	2.23936
$412$	0	0
$413$	−5.89236	−0.289944
$414$	0	0
$415$	−6.47730	−0.317958
$416$	0	0
$417$	−30.2685	−1.48225
$418$	0	0
$419$	16.5829	0.810127	0.405064	−	0.914289i	$-0.367249\pi$
$419$	16.5829	0.810127	0.405064	+	0.914289i	$0.367249\pi$
$420$	0	0
$421$	6.59648	0.321493	0.160746	−	0.986996i	$-0.448610\pi$
$421$	6.59648	0.321493	0.160746	+	0.986996i	$0.448610\pi$
$422$	0	0
$423$	10.7997	0.525098
$424$	0	0
$425$	−29.3371	−1.42306
$426$	0	0
$427$	−0.0387795	−0.00187667
$428$	0	0
$429$	13.1205	0.633465
$430$	0	0
$431$	−29.1663	−1.40489	−0.702445	−	0.711738i	$-0.747908\pi$
$431$	−29.1663	−1.40489	−0.702445	+	0.711738i	$0.747908\pi$
$432$	0	0
$433$	−14.7050	−0.706676	−0.353338	−	0.935496i	$-0.614953\pi$
$433$	−14.7050	−0.706676	−0.353338	+	0.935496i	$0.614953\pi$
$434$	0	0
$435$	2.59951	0.124637
$436$	0	0
$437$	4.50026	0.215276
$438$	0	0
$439$	9.38070	0.447716	0.223858	−	0.974622i	$-0.428135\pi$
$439$	9.38070	0.447716	0.223858	+	0.974622i	$0.428135\pi$
$440$	0	0
$441$	−21.3289	−1.01566
$442$	0	0
$443$	−7.53940	−0.358208	−0.179104	−	0.983830i	$-0.557320\pi$
$443$	−7.53940	−0.358208	−0.179104	+	0.983830i	$0.557320\pi$
$444$	0	0
$445$	−5.38854	−0.255441
$446$	0	0
$447$	21.3123	1.00804
$448$	0	0
$449$	15.7830	0.744844	0.372422	−	0.928064i	$-0.378527\pi$
$449$	15.7830	0.744844	0.372422	+	0.928064i	$0.378527\pi$
$450$	0	0
$451$	6.44129	0.303308
$452$	0	0
$453$	−4.01918	−0.188838
$454$	0	0
$455$	1.39349	0.0653279
$456$	0	0
$457$	−2.95133	−0.138057	−0.0690286	−	0.997615i	$-0.521990\pi$
$457$	−2.95133	−0.138057	−0.0690286	+	0.997615i	$0.521990\pi$
$458$	0	0
$459$	26.6881	1.24569
$460$	0	0
$461$	27.8790	1.29845	0.649226	−	0.760595i	$-0.275093\pi$
$461$	27.8790	1.29845	0.649226	+	0.760595i	$0.275093\pi$
$462$	0	0
$463$	−35.1234	−1.63232	−0.816162	−	0.577823i	$-0.803902\pi$
$463$	−35.1234	−1.63232	−0.816162	+	0.577823i	$0.803902\pi$
$464$	0	0
$465$	7.01446	0.325288
$466$	0	0
$467$	29.3446	1.35791	0.678953	−	0.734181i	$-0.262434\pi$
$467$	29.3446	1.35791	0.678953	+	0.734181i	$0.262434\pi$
$468$	0	0
$469$	−6.06250	−0.279940
$470$	0	0
$471$	20.8202	0.959343
$472$	0	0
$473$	−11.6418	−0.535290
$474$	0	0
$475$	−18.2953	−0.839444
$476$	0	0
$477$	13.8093	0.632284
$478$	0	0
$479$	−35.1611	−1.60655	−0.803275	−	0.595608i	$-0.796911\pi$
$479$	−35.1611	−1.60655	−0.803275	+	0.595608i	$0.796911\pi$
$480$	0	0
$481$	−4.54623	−0.207290
$482$	0	0
$483$	−3.96689	−0.180500
$484$	0	0
$485$	−15.1114	−0.686174
$486$	0	0
$487$	29.1296	1.31999	0.659994	−	0.751271i	$-0.270559\pi$
$487$	29.1296	1.31999	0.659994	+	0.751271i	$0.270559\pi$
$488$	0	0
$489$	−26.7511	−1.20973
$490$	0	0
$491$	−13.2518	−0.598045	−0.299023	−	0.954246i	$-0.596661\pi$
$491$	−13.2518	−0.598045	−0.299023	+	0.954246i	$0.596661\pi$
$492$	0	0
$493$	−7.18535	−0.323612
$494$	0	0
$495$	−20.2203	−0.908835
$496$	0	0
$497$	−6.25541	−0.280593
$498$	0	0
$499$	19.3093	0.864401	0.432201	−	0.901778i	$-0.357737\pi$
$499$	19.3093	0.864401	0.432201	+	0.901778i	$0.357737\pi$
$500$	0	0
$501$	20.6819	0.924001
$502$	0	0
$503$	22.8757	1.01998	0.509989	−	0.860181i	$-0.329650\pi$
$503$	22.8757	1.01998	0.509989	+	0.860181i	$0.329650\pi$
$504$	0	0
$505$	−16.0779	−0.715457
$506$	0	0
$507$	2.71446	0.120554
$508$	0	0
$509$	−5.56220	−0.246540	−0.123270	−	0.992373i	$-0.539338\pi$
$509$	−5.56220	−0.246540	−0.123270	+	0.992373i	$0.539338\pi$
$510$	0	0
$511$	−0.221075	−0.00977979
$512$	0	0
$513$	16.6433	0.734820
$514$	0	0
$515$	−7.17892	−0.316341
$516$	0	0
$517$	11.9499	0.525555
$518$	0	0
$519$	5.38418	0.236339
$520$	0	0
$521$	−2.33273	−0.102199	−0.0510994	−	0.998694i	$-0.516273\pi$
$521$	−2.33273	−0.102199	−0.0510994	+	0.998694i	$0.516273\pi$
$522$	0	0
$523$	−43.5967	−1.90635	−0.953174	−	0.302422i	$-0.902205\pi$
$523$	−43.5967	−1.90635	−0.953174	+	0.302422i	$0.902205\pi$
$524$	0	0
$525$	16.1269	0.703837
$526$	0	0
$527$	−19.3888	−0.844589
$528$	0	0
$529$	−21.9914	−0.956146
$530$	0	0
$531$	17.6891	0.767641
$532$	0	0
$533$	1.33262	0.0577221
$534$	0	0
$535$	15.5063	0.670397
$536$	0	0
$537$	4.32206	0.186511
$538$	0	0
$539$	−23.6005	−1.01655
$540$	0	0
$541$	−20.1743	−0.867359	−0.433680	−	0.901067i	$-0.642785\pi$
$541$	−20.1743	−0.867359	−0.433680	+	0.901067i	$0.642785\pi$
$542$	0	0
$543$	15.4422	0.662690
$544$	0	0
$545$	12.3620	0.529530
$546$	0	0
$547$	−41.6394	−1.78037	−0.890186	−	0.455597i	$-0.849426\pi$
$547$	−41.6394	−1.78037	−0.890186	+	0.455597i	$0.849426\pi$
$548$	0	0
$549$	0.116417	0.00496857
$550$	0	0
$551$	−4.48094	−0.190895
$552$	0	0
$553$	−0.993195	−0.0422350
$554$	0	0
$555$	11.8180	0.501644
$556$	0	0
$557$	−4.71694	−0.199863	−0.0999317	−	0.994994i	$-0.531862\pi$
$557$	−4.71694	−0.199863	−0.0999317	+	0.994994i	$0.531862\pi$
$558$	0	0
$559$	−2.40853	−0.101870
$560$	0	0
$561$	94.2755	3.98031
$562$	0	0
$563$	20.8677	0.879468	0.439734	−	0.898128i	$-0.355073\pi$
$563$	20.8677	0.879468	0.439734	+	0.898128i	$0.355073\pi$
$564$	0	0
$565$	3.93977	0.165747
$566$	0	0
$567$	4.39848	0.184719
$568$	0	0
$569$	8.98465	0.376656	0.188328	−	0.982106i	$-0.439693\pi$
$569$	8.98465	0.376656	0.188328	+	0.982106i	$0.439693\pi$
$570$	0	0
$571$	−34.3479	−1.43742	−0.718708	−	0.695312i	$-0.755266\pi$
$571$	−34.3479	−1.43742	−0.718708	+	0.695312i	$0.755266\pi$
$572$	0	0
$573$	−51.7453	−2.16169
$574$	0	0
$575$	−4.10051	−0.171003
$576$	0	0
$577$	−26.7846	−1.11506	−0.557528	−	0.830158i	$-0.688250\pi$
$577$	−26.7846	−1.11506	−0.557528	+	0.830158i	$0.688250\pi$
$578$	0	0
$579$	−15.1314	−0.628837
$580$	0	0
$581$	−9.84204	−0.408317
$582$	0	0
$583$	15.2800	0.632834
$584$	0	0
$585$	−4.18332	−0.172959
$586$	0	0
$587$	−5.87201	−0.242364	−0.121182	−	0.992630i	$-0.538668\pi$
$587$	−5.87201	−0.242364	−0.121182	+	0.992630i	$0.538668\pi$
$588$	0	0
$589$	−12.0913	−0.498213
$590$	0	0
$591$	−2.72837	−0.112230
$592$	0	0
$593$	−43.0777	−1.76899	−0.884495	−	0.466550i	$-0.845497\pi$
$593$	−43.0777	−1.76899	−0.884495	+	0.466550i	$0.845497\pi$
$594$	0	0
$595$	10.0127	0.410482
$596$	0	0
$597$	3.59720	0.147224
$598$	0	0
$599$	−10.1662	−0.415378	−0.207689	−	0.978195i	$-0.566594\pi$
$599$	−10.1662	−0.415378	−0.207689	+	0.978195i	$0.566594\pi$
$600$	0	0
$601$	−5.81407	−0.237161	−0.118580	−	0.992944i	$-0.537834\pi$
$601$	−5.81407	−0.237161	−0.118580	+	0.992944i	$0.537834\pi$
$602$	0	0
$603$	18.1999	0.741155
$604$	0	0
$605$	−11.8397	−0.481352
$606$	0	0
$607$	34.5618	1.40282	0.701411	−	0.712757i	$-0.252554\pi$
$607$	34.5618	1.40282	0.701411	+	0.712757i	$0.252554\pi$
$608$	0	0
$609$	3.94986	0.160057
$610$	0	0
$611$	2.47227	0.100017
$612$	0	0
$613$	−35.2755	−1.42476	−0.712381	−	0.701792i	$-0.752383\pi$
$613$	−35.2755	−1.42476	−0.712381	+	0.701792i	$0.752383\pi$
$614$	0	0
$615$	−3.46415	−0.139688
$616$	0	0
$617$	42.3322	1.70423	0.852116	−	0.523354i	$-0.175319\pi$
$617$	42.3322	1.70423	0.852116	+	0.523354i	$0.175319\pi$
$618$	0	0
$619$	−22.8629	−0.918936	−0.459468	−	0.888194i	$-0.651960\pi$
$619$	−22.8629	−0.918936	−0.459468	+	0.888194i	$0.651960\pi$
$620$	0	0
$621$	3.73025	0.149690
$622$	0	0
$623$	−8.18771	−0.328034
$624$	0	0
$625$	12.0847	0.483386
$626$	0	0
$627$	58.7923	2.34794
$628$	0	0
$629$	−32.6662	−1.30249
$630$	0	0
$631$	−29.0089	−1.15483	−0.577413	−	0.816452i	$-0.695938\pi$
$631$	−29.0089	−1.15483	−0.577413	+	0.816452i	$0.695938\pi$
$632$	0	0
$633$	−18.9714	−0.754044
$634$	0	0
$635$	14.1498	0.561519
$636$	0	0
$637$	−4.88263	−0.193457
$638$	0	0
$639$	18.7790	0.742885
$640$	0	0
$641$	−25.6099	−1.01153	−0.505765	−	0.862671i	$-0.668790\pi$
$641$	−25.6099	−1.01153	−0.505765	+	0.862671i	$0.668790\pi$
$642$	0	0
$643$	11.6849	0.460809	0.230404	−	0.973095i	$-0.425995\pi$
$643$	11.6849	0.460809	0.230404	+	0.973095i	$0.425995\pi$
$644$	0	0
$645$	6.26100	0.246527
$646$	0	0
$647$	−33.3092	−1.30952	−0.654760	−	0.755837i	$-0.727230\pi$
$647$	−33.3092	−1.30952	−0.654760	+	0.755837i	$0.727230\pi$
$648$	0	0
$649$	19.5730	0.768309
$650$	0	0
$651$	10.6582	0.417729
$652$	0	0
$653$	−23.0950	−0.903776	−0.451888	−	0.892075i	$-0.649249\pi$
$653$	−23.0950	−0.903776	−0.451888	+	0.892075i	$0.649249\pi$
$654$	0	0
$655$	2.31718	0.0905399
$656$	0	0
$657$	0.663676	0.0258924
$658$	0	0
$659$	−3.02142	−0.117698	−0.0588489	−	0.998267i	$-0.518743\pi$
$659$	−3.02142	−0.117698	−0.0588489	+	0.998267i	$0.518743\pi$
$660$	0	0
$661$	8.63115	0.335713	0.167856	−	0.985811i	$-0.446316\pi$
$661$	8.63115	0.335713	0.167856	+	0.985811i	$0.446316\pi$
$662$	0	0
$663$	19.5044	0.757487
$664$	0	0
$665$	6.24416	0.242138
$666$	0	0
$667$	−1.00431	−0.0388871
$668$	0	0
$669$	27.4389	1.06085
$670$	0	0
$671$	0.128816	0.00497290
$672$	0	0
$673$	46.0102	1.77356	0.886782	−	0.462187i	$-0.152935\pi$
$673$	46.0102	1.77356	0.886782	+	0.462187i	$0.152935\pi$
$674$	0	0
$675$	−15.1649	−0.583698
$676$	0	0
$677$	−27.1052	−1.04174	−0.520868	−	0.853637i	$-0.674392\pi$
$677$	−27.1052	−1.04174	−0.520868	+	0.853637i	$0.674392\pi$
$678$	0	0
$679$	−22.9613	−0.881173
$680$	0	0
$681$	−53.6766	−2.05689
$682$	0	0
$683$	−27.1352	−1.03830	−0.519150	−	0.854683i	$-0.673751\pi$
$683$	−27.1352	−1.03830	−0.519150	+	0.854683i	$0.673751\pi$
$684$	0	0
$685$	−16.0165	−0.611958
$686$	0	0
$687$	10.7043	0.408396
$688$	0	0
$689$	3.16124	0.120434
$690$	0	0
$691$	−12.6056	−0.479538	−0.239769	−	0.970830i	$-0.577072\pi$
$691$	−12.6056	−0.479538	−0.239769	+	0.970830i	$0.577072\pi$
$692$	0	0
$693$	−30.7241	−1.16711
$694$	0	0
$695$	10.6786	0.405061
$696$	0	0
$697$	9.57532	0.362691
$698$	0	0
$699$	51.7255	1.95644
$700$	0	0
$701$	−37.5880	−1.41968	−0.709840	−	0.704362i	$-0.751233\pi$
$701$	−37.5880	−1.41968	−0.709840	+	0.704362i	$0.751233\pi$
$702$	0	0
$703$	−20.3714	−0.768321
$704$	0	0
$705$	−6.42669	−0.242043
$706$	0	0
$707$	−24.4298	−0.918778
$708$	0	0
$709$	−14.2488	−0.535125	−0.267562	−	0.963541i	$-0.586218\pi$
$709$	−14.2488	−0.535125	−0.267562	+	0.963541i	$0.586218\pi$
$710$	0	0
$711$	2.98161	0.111819
$712$	0	0
$713$	−2.71001	−0.101491
$714$	0	0
$715$	−4.62885	−0.173109
$716$	0	0
$717$	7.74915	0.289397
$718$	0	0
$719$	−41.5044	−1.54785	−0.773927	−	0.633275i	$-0.781710\pi$
$719$	−41.5044	−1.54785	−0.773927	+	0.633275i	$0.781710\pi$
$720$	0	0
$721$	−10.9081	−0.406240
$722$	0	0
$723$	−12.9208	−0.480531
$724$	0	0
$725$	4.08291	0.151635
$726$	0	0
$727$	−43.2505	−1.60407	−0.802037	−	0.597275i	$-0.796250\pi$
$727$	−43.2505	−1.60407	−0.802037	+	0.597275i	$0.796250\pi$
$728$	0	0
$729$	−43.4509	−1.60929
$730$	0	0
$731$	−17.3062	−0.640091
$732$	0	0
$733$	3.48787	0.128828	0.0644138	−	0.997923i	$-0.479482\pi$
$733$	3.48787	0.128828	0.0644138	+	0.997923i	$0.479482\pi$
$734$	0	0
$735$	12.6924	0.468168
$736$	0	0
$737$	20.1382	0.741800
$738$	0	0
$739$	−12.6446	−0.465140	−0.232570	−	0.972580i	$-0.574713\pi$
$739$	−12.6446	−0.465140	−0.232570	+	0.972580i	$0.574713\pi$
$740$	0	0
$741$	12.1634	0.446832
$742$	0	0
$743$	42.2790	1.55107	0.775533	−	0.631307i	$-0.217481\pi$
$743$	42.2790	1.55107	0.775533	+	0.631307i	$0.217481\pi$
$744$	0	0
$745$	−7.51886	−0.275470
$746$	0	0
$747$	29.5462	1.08104
$748$	0	0
$749$	23.5613	0.860913
$750$	0	0
$751$	−30.0406	−1.09620	−0.548099	−	0.836413i	$-0.684648\pi$
$751$	−30.0406	−1.09620	−0.548099	+	0.836413i	$0.684648\pi$
$752$	0	0
$753$	−16.4883	−0.600868
$754$	0	0
$755$	1.41795	0.0516044
$756$	0	0
$757$	9.34562	0.339672	0.169836	−	0.985472i	$-0.445676\pi$
$757$	9.34562	0.339672	0.169836	+	0.985472i	$0.445676\pi$
$758$	0	0
$759$	13.1771	0.478297
$760$	0	0
$761$	−27.8775	−1.01056	−0.505280	−	0.862956i	$-0.668611\pi$
$761$	−27.8775	−1.01056	−0.505280	+	0.862956i	$0.668611\pi$
$762$	0	0
$763$	18.7836	0.680013
$764$	0	0
$765$	−30.0586	−1.08677
$766$	0	0
$767$	4.04941	0.146216
$768$	0	0
$769$	30.9911	1.11757	0.558783	−	0.829314i	$-0.311268\pi$
$769$	30.9911	1.11757	0.558783	+	0.829314i	$0.311268\pi$
$770$	0	0
$771$	35.1839	1.26712
$772$	0	0
$773$	−29.0247	−1.04395	−0.521973	−	0.852962i	$-0.674804\pi$
$773$	−29.0247	−1.04395	−0.521973	+	0.852962i	$0.674804\pi$
$774$	0	0
$775$	11.0172	0.395751
$776$	0	0
$777$	17.9570	0.644203
$778$	0	0
$779$	5.97138	0.213947
$780$	0	0
$781$	20.7790	0.743531
$782$	0	0
$783$	−3.71424	−0.132736
$784$	0	0
$785$	−7.34526	−0.262163
$786$	0	0
$787$	−7.92560	−0.282517	−0.141259	−	0.989973i	$-0.545115\pi$
$787$	−7.92560	−0.282517	−0.141259	+	0.989973i	$0.545115\pi$
$788$	0	0
$789$	−60.8343	−2.16576
$790$	0	0
$791$	5.98635	0.212850
$792$	0	0
$793$	0.0266504	0.000946384 0
$794$	0	0
$795$	−8.21766	−0.291450
$796$	0	0
$797$	8.48546	0.300571	0.150285	−	0.988643i	$-0.451981\pi$
$797$	8.48546	0.300571	0.150285	+	0.988643i	$0.451981\pi$
$798$	0	0
$799$	17.7641	0.628450
$800$	0	0
$801$	24.5798	0.868485
$802$	0	0
$803$	0.734359	0.0259150
$804$	0	0
$805$	1.39950	0.0493259
$806$	0	0
$807$	45.8239	1.61308
$808$	0	0
$809$	−54.0567	−1.90053	−0.950266	−	0.311441i	$-0.899188\pi$
$809$	−54.0567	−1.90053	−0.950266	+	0.311441i	$0.899188\pi$
$810$	0	0
$811$	37.9541	1.33275	0.666375	−	0.745617i	$-0.267845\pi$
$811$	37.9541	1.33275	0.666375	+	0.745617i	$0.267845\pi$
$812$	0	0
$813$	70.6094	2.47638
$814$	0	0
$815$	9.43765	0.330587
$816$	0	0
$817$	−10.7925	−0.377582
$818$	0	0
$819$	−6.35641	−0.222111
$820$	0	0
$821$	13.4996	0.471138	0.235569	−	0.971858i	$-0.424305\pi$
$821$	13.4996	0.471138	0.235569	+	0.971858i	$0.424305\pi$
$822$	0	0
$823$	18.5710	0.647346	0.323673	−	0.946169i	$-0.395082\pi$
$823$	18.5710	0.647346	0.323673	+	0.946169i	$0.395082\pi$
$824$	0	0
$825$	−53.5698	−1.86506
$826$	0	0
$827$	−3.27680	−0.113946	−0.0569728	−	0.998376i	$-0.518145\pi$
$827$	−3.27680	−0.113946	−0.0569728	+	0.998376i	$0.518145\pi$
$828$	0	0
$829$	3.66795	0.127393	0.0636966	−	0.997969i	$-0.479711\pi$
$829$	3.66795	0.127393	0.0636966	+	0.997969i	$0.479711\pi$
$830$	0	0
$831$	35.2148	1.22159
$832$	0	0
$833$	−35.0834	−1.21557
$834$	0	0
$835$	−7.29649	−0.252505
$836$	0	0
$837$	−10.0224	−0.346426
$838$	0	0
$839$	15.7301	0.543064	0.271532	−	0.962429i	$-0.412470\pi$
$839$	15.7301	0.543064	0.271532	+	0.962429i	$0.412470\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	65.2347	2.24680
$844$	0	0
$845$	−0.957650	−0.0329442
$846$	0	0
$847$	−17.9900	−0.618144
$848$	0	0
$849$	−31.6996	−1.08793
$850$	0	0
$851$	−4.56583	−0.156514
$852$	0	0
$853$	54.9197	1.88041	0.940207	−	0.340602i	$-0.110631\pi$
$853$	54.9197	1.88041	0.940207	+	0.340602i	$0.110631\pi$
$854$	0	0
$855$	−18.7452	−0.641072
$856$	0	0
$857$	34.9148	1.19267	0.596334	−	0.802737i	$-0.296623\pi$
$857$	34.9148	1.19267	0.596334	+	0.802737i	$0.296623\pi$
$858$	0	0
$859$	22.7982	0.777865	0.388932	−	0.921266i	$-0.372844\pi$
$859$	22.7982	0.777865	0.388932	+	0.921266i	$0.372844\pi$
$860$	0	0
$861$	−5.26366	−0.179385
$862$	0	0
$863$	−40.5057	−1.37883	−0.689414	−	0.724367i	$-0.742132\pi$
$863$	−40.5057	−1.37883	−0.689414	+	0.724367i	$0.742132\pi$
$864$	0	0
$865$	−1.89951	−0.0645853
$866$	0	0
$867$	93.9997	3.19240
$868$	0	0
$869$	3.29916	0.111916
$870$	0	0
$871$	4.16633	0.141171
$872$	0	0
$873$	68.9306	2.33295
$874$	0	0
$875$	−12.6570	−0.427883
$876$	0	0
$877$	8.38959	0.283296	0.141648	−	0.989917i	$-0.454760\pi$
$877$	8.38959	0.283296	0.141648	+	0.989917i	$0.454760\pi$
$878$	0	0
$879$	−23.4100	−0.789601
$880$	0	0
$881$	44.1726	1.48821	0.744107	−	0.668061i	$-0.232875\pi$
$881$	44.1726	1.48821	0.744107	+	0.668061i	$0.232875\pi$
$882$	0	0
$883$	45.5972	1.53447	0.767234	−	0.641367i	$-0.221632\pi$
$883$	45.5972	1.53447	0.767234	+	0.641367i	$0.221632\pi$
$884$	0	0
$885$	−10.5265	−0.353843
$886$	0	0
$887$	−5.27881	−0.177245	−0.0886226	−	0.996065i	$-0.528247\pi$
$887$	−5.27881	−0.177245	−0.0886226	+	0.996065i	$0.528247\pi$
$888$	0	0
$889$	21.5002	0.721093
$890$	0	0
$891$	−14.6107	−0.489477
$892$	0	0
$893$	11.0781	0.370715
$894$	0	0
$895$	−1.52480	−0.0509685
$896$	0	0
$897$	2.72616	0.0910240
$898$	0	0
$899$	2.69838	0.0899960
$900$	0	0
$901$	22.7146	0.756733
$902$	0	0
$903$	9.51338	0.316585
$904$	0	0
$905$	−5.44795	−0.181096
$906$	0	0
$907$	40.2743	1.33728	0.668642	−	0.743584i	$-0.266876\pi$
$907$	40.2743	1.33728	0.668642	+	0.743584i	$0.266876\pi$
$908$	0	0
$909$	73.3392	2.43251
$910$	0	0
$911$	−46.0561	−1.52591	−0.762953	−	0.646454i	$-0.776251\pi$
$911$	−46.0561	−1.52591	−0.762953	+	0.646454i	$0.776251\pi$
$912$	0	0
$913$	32.6929	1.08198
$914$	0	0
$915$	−0.0692779	−0.00229026
$916$	0	0
$917$	3.52089	0.116270
$918$	0	0
$919$	−19.4058	−0.640139	−0.320069	−	0.947394i	$-0.603706\pi$
$919$	−19.4058	−0.640139	−0.320069	+	0.947394i	$0.603706\pi$
$920$	0	0
$921$	11.7864	0.388375
$922$	0	0
$923$	4.29890	0.141500
$924$	0	0
$925$	18.5618	0.610309
$926$	0	0
$927$	32.7466	1.07554
$928$	0	0
$929$	46.8086	1.53574	0.767870	−	0.640606i	$-0.221317\pi$
$929$	46.8086	1.53574	0.767870	+	0.640606i	$0.221317\pi$
$930$	0	0
$931$	−21.8788	−0.717049
$932$	0	0
$933$	9.88820	0.323725
$934$	0	0
$935$	−33.2599	−1.08772
$936$	0	0
$937$	−18.4550	−0.602897	−0.301449	−	0.953482i	$-0.597470\pi$
$937$	−18.4550	−0.602897	−0.301449	+	0.953482i	$0.597470\pi$
$938$	0	0
$939$	−72.6946	−2.37230
$940$	0	0
$941$	38.1411	1.24336	0.621681	−	0.783270i	$-0.286450\pi$
$941$	38.1411	1.24336	0.621681	+	0.783270i	$0.286450\pi$
$942$	0	0
$943$	1.33836	0.0435831
$944$	0	0
$945$	5.17577	0.168368
$946$	0	0
$947$	29.9185	0.972221	0.486110	−	0.873897i	$-0.338415\pi$
$947$	29.9185	0.972221	0.486110	+	0.873897i	$0.338415\pi$
$948$	0	0
$949$	0.151929	0.00493184
$950$	0	0
$951$	59.8168	1.93969
$952$	0	0
$953$	−20.6000	−0.667299	−0.333649	−	0.942697i	$-0.608280\pi$
$953$	−20.6000	−0.667299	−0.333649	+	0.942697i	$0.608280\pi$
$954$	0	0
$955$	18.2555	0.590734
$956$	0	0
$957$	−13.1205	−0.424126
$958$	0	0
$959$	−24.3365	−0.785867
$960$	0	0
$961$	−23.7187	−0.765121
$962$	0	0
$963$	−70.7320	−2.27931
$964$	0	0
$965$	5.33827	0.171845
$966$	0	0
$967$	−54.9190	−1.76607	−0.883037	−	0.469302i	$-0.844505\pi$
$967$	−54.9190	−1.76607	−0.883037	+	0.469302i	$0.844505\pi$
$968$	0	0
$969$	87.3979	2.80763
$970$	0	0
$971$	−54.0866	−1.73572	−0.867860	−	0.496808i	$-0.834505\pi$
$971$	−54.0866	−1.73572	−0.867860	+	0.496808i	$0.834505\pi$
$972$	0	0
$973$	16.2257	0.520173
$974$	0	0
$975$	−11.0829	−0.354937
$976$	0	0
$977$	−11.4829	−0.367371	−0.183685	−	0.982985i	$-0.558803\pi$
$977$	−11.4829	−0.367371	−0.183685	+	0.982985i	$0.558803\pi$
$978$	0	0
$979$	27.1976	0.869241
$980$	0	0
$981$	−56.3892	−1.80037
$982$	0	0
$983$	−4.10574	−0.130953	−0.0654764	−	0.997854i	$-0.520857\pi$
$983$	−4.10574	−0.130953	−0.0654764	+	0.997854i	$0.520857\pi$
$984$	0	0
$985$	0.962557	0.0306696
$986$	0	0
$987$	−9.76514	−0.310828
$988$	0	0
$989$	−2.41892	−0.0769171
$990$	0	0
$991$	−51.1201	−1.62388	−0.811941	−	0.583740i	$-0.801589\pi$
$991$	−51.1201	−1.62388	−0.811941	+	0.583740i	$0.801589\pi$
$992$	0	0
$993$	17.1723	0.544946
$994$	0	0
$995$	−1.26908	−0.0402324
$996$	0	0
$997$	40.8734	1.29447	0.647237	−	0.762288i	$-0.275924\pi$
$997$	40.8734	1.29447	0.647237	+	0.762288i	$0.275924\pi$
$998$	0	0
$999$	−16.8858	−0.534243

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6032.2.a.ba.1.9		10
4.3	odd	2		3016.2.a.g.1.2	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3016.2.a.g.1.2	✓	10	4.3	odd	2
6032.2.a.ba.1.9		10	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:	\( N \)	\(=\)	\( 6032 = 2^{4} \cdot 13 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6032.a (trivial)

\(f(q)\)	\(=\)	\(q+2.71446 q^{3} -0.957650 q^{5} -1.45512 q^{7} +4.36832 q^{9} +O(q^{10})\)	\(q+2.71446 q^{3} -0.957650 q^{5} -1.45512 q^{7} +4.36832 q^{9} +4.83356 q^{11} +1.00000 q^{13} -2.59951 q^{15} +7.18535 q^{17} +4.48094 q^{19} -3.94986 q^{21} +1.00431 q^{23} -4.08291 q^{25} +3.71424 q^{27} -1.00000 q^{29} -2.69838 q^{31} +13.1205 q^{33} +1.39349 q^{35} -4.54623 q^{37} +2.71446 q^{39} +1.33262 q^{41} -2.40853 q^{43} -4.18332 q^{45} +2.47227 q^{47} -4.88263 q^{49} +19.5044 q^{51} +3.16124 q^{53} -4.62885 q^{55} +12.1634 q^{57} +4.04941 q^{59} +0.0266504 q^{61} -6.35641 q^{63} -0.957650 q^{65} +4.16633 q^{67} +2.72616 q^{69} +4.29890 q^{71} +0.151929 q^{73} -11.0829 q^{75} -7.03339 q^{77} +0.682554 q^{79} -3.02277 q^{81} +6.76374 q^{83} -6.88105 q^{85} -2.71446 q^{87} +5.62684 q^{89} -1.45512 q^{91} -7.32466 q^{93} -4.29117 q^{95} +15.7797 q^{97} +21.1145 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{3} + 5 q^{5} + 2 q^{7} + 16 q^{9}+O(q^{10}) \) 10 * q - 2 * q^3 + 5 * q^5 + 2 * q^7 + 16 * q^9	\( 10 q - 2 q^{3} + 5 q^{5} + 2 q^{7} + 16 q^{9} - 4 q^{11} + 10 q^{13} - 8 q^{15} + 10 q^{17} + q^{19} + q^{21} - 23 q^{23} + 25 q^{25} - 2 q^{27} - 10 q^{29} + 13 q^{31} + 15 q^{33} + 12 q^{35} - 7 q^{37} - 2 q^{39} + 16 q^{41} + 12 q^{43} + 55 q^{45} - 11 q^{47} + 25 q^{51} + 11 q^{53} - 22 q^{55} - 6 q^{57} + 11 q^{59} + 34 q^{61} - 37 q^{63} + 5 q^{65} + 23 q^{67} + 2 q^{69} + 4 q^{71} + 39 q^{73} - 11 q^{75} + 32 q^{77} - 5 q^{79} + 38 q^{81} - 6 q^{83} + 45 q^{85} + 2 q^{87} - 24 q^{89} + 2 q^{91} + 13 q^{93} - 33 q^{95} + 19 q^{97}+O(q^{100}) \) 10 * q - 2 * q^3 + 5 * q^5 + 2 * q^7 + 16 * q^9 - 4 * q^11 + 10 * q^13 - 8 * q^15 + 10 * q^17 + q^19 + q^21 - 23 * q^23 + 25 * q^25 - 2 * q^27 - 10 * q^29 + 13 * q^31 + 15 * q^33 + 12 * q^35 - 7 * q^37 - 2 * q^39 + 16 * q^41 + 12 * q^43 + 55 * q^45 - 11 * q^47 + 25 * q^51 + 11 * q^53 - 22 * q^55 - 6 * q^57 + 11 * q^59 + 34 * q^61 - 37 * q^63 + 5 * q^65 + 23 * q^67 + 2 * q^69 + 4 * q^71 + 39 * q^73 - 11 * q^75 + 32 * q^77 - 5 * q^79 + 38 * q^81 - 6 * q^83 + 45 * q^85 + 2 * q^87 - 24 * q^89 + 2 * q^91 + 13 * q^93 - 33 * q^95 + 19 * q^97

Introduction

L-functions

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