LMFDB - Embedded newform 1148.2.a.c.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1148,2,Mod(1,1148)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1148, 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("1148.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1148 = 2^{2} \cdot 7 \cdot 41 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1148.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:	$9.16682615204$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.470117.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 8x^{2} + 7x + 1 $ x^5 - 2x^4 - 6x^3 + 8x^2 + 7x + 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-2.05768$ of defining polynomial
Character	$\chi$	$=$	1148.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.05768 q^{3} -2.77770 q^{5} -1.00000 q^{7} +1.23404 q^{9} +O(q^{10})$	$q+2.05768 q^{3} -2.77770 q^{5} -1.00000 q^{7} +1.23404 q^{9} -0.650081 q^{11} -3.20159 q^{13} -5.71561 q^{15} +3.01173 q^{17} -6.99947 q^{19} -2.05768 q^{21} -0.230150 q^{23} +2.71561 q^{25} -3.63378 q^{27} -7.95514 q^{29} +7.36112 q^{31} -1.33766 q^{33} +2.77770 q^{35} -7.01958 q^{37} -6.58784 q^{39} +1.00000 q^{41} +0.887550 q^{43} -3.42778 q^{45} +1.56384 q^{47} +1.00000 q^{49} +6.19718 q^{51} -9.53360 q^{53} +1.80573 q^{55} -14.4027 q^{57} -2.25033 q^{59} -8.08129 q^{61} -1.23404 q^{63} +8.89305 q^{65} +1.72514 q^{67} -0.473574 q^{69} +3.79347 q^{71} +5.34109 q^{73} +5.58784 q^{75} +0.650081 q^{77} -4.14795 q^{79} -11.1793 q^{81} -3.08586 q^{83} -8.36569 q^{85} -16.3691 q^{87} +5.26991 q^{89} +3.20159 q^{91} +15.1468 q^{93} +19.4424 q^{95} +19.4644 q^{97} -0.802223 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{3} - q^{5} - 5 q^{7} + q^{9}+O(q^{10}) $ 5 * q - 2 * q^3 - q^5 - 5 * q^7 + q^9	$ 5 q - 2 q^{3} - q^{5} - 5 q^{7} + q^{9} - 2 q^{11} - q^{13} - 9 q^{15} - 3 q^{17} - 4 q^{19} + 2 q^{21} - 8 q^{23} - 6 q^{25} - 8 q^{27} - 9 q^{29} - 11 q^{31} + 5 q^{33} + q^{35} - 11 q^{37} - 17 q^{39} + 5 q^{41} - 27 q^{43} - 3 q^{45} - 3 q^{47} + 5 q^{49} - 3 q^{51} - 19 q^{53} - 13 q^{55} - 11 q^{57} - 15 q^{59} - q^{63} + 7 q^{65} - 21 q^{67} + 14 q^{69} - 16 q^{71} - 10 q^{73} + 12 q^{75} + 2 q^{77} - 14 q^{79} - 7 q^{81} - 2 q^{83} - 21 q^{85} - 36 q^{87} + 6 q^{89} + q^{91} + 17 q^{93} + 9 q^{95} + 20 q^{97} - 17 q^{99}+O(q^{100}) $ 5 * q - 2 * q^3 - q^5 - 5 * q^7 + q^9 - 2 * q^11 - q^13 - 9 * q^15 - 3 * q^17 - 4 * q^19 + 2 * q^21 - 8 * q^23 - 6 * q^25 - 8 * q^27 - 9 * q^29 - 11 * q^31 + 5 * q^33 + q^35 - 11 * q^37 - 17 * q^39 + 5 * q^41 - 27 * q^43 - 3 * q^45 - 3 * q^47 + 5 * q^49 - 3 * q^51 - 19 * q^53 - 13 * q^55 - 11 * q^57 - 15 * q^59 - q^63 + 7 * q^65 - 21 * q^67 + 14 * q^69 - 16 * q^71 - 10 * q^73 + 12 * q^75 + 2 * q^77 - 14 * q^79 - 7 * q^81 - 2 * q^83 - 21 * q^85 - 36 * q^87 + 6 * q^89 + q^91 + 17 * q^93 + 9 * q^95 + 20 * q^97 - 17 * 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
$3$	2.05768	1.18800	0.594000	−	0.804465i	$-0.297548\pi$
$3$	2.05768	1.18800	0.594000	+	0.804465i	$0.297548\pi$
$4$	0	0
$5$	−2.77770	−1.24222	−0.621112	−	0.783722i	$-0.713319\pi$
$5$	−2.77770	−1.24222	−0.621112	+	0.783722i	$0.713319\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.23404	0.411345
$10$	0	0
$11$	−0.650081	−0.196007	−0.0980034	−	0.995186i	$-0.531246\pi$
$11$	−0.650081	−0.196007	−0.0980034	+	0.995186i	$0.531246\pi$
$12$	0	0
$13$	−3.20159	−0.887962	−0.443981	−	0.896036i	$-0.646434\pi$
$13$	−3.20159	−0.887962	−0.443981	+	0.896036i	$0.646434\pi$
$14$	0	0
$15$	−5.71561	−1.47576
$16$	0	0
$17$	3.01173	0.730453	0.365226	−	0.930919i	$-0.380992\pi$
$17$	3.01173	0.730453	0.365226	+	0.930919i	$0.380992\pi$
$18$	0	0
$19$	−6.99947	−1.60579	−0.802894	−	0.596121i	$-0.796708\pi$
$19$	−6.99947	−1.60579	−0.802894	+	0.596121i	$0.796708\pi$
$20$	0	0
$21$	−2.05768	−0.449022
$22$	0	0
$23$	−0.230150	−0.0479896	−0.0239948	−	0.999712i	$-0.507639\pi$
$23$	−0.230150	−0.0479896	−0.0239948	+	0.999712i	$0.507639\pi$
$24$	0	0
$25$	2.71561	0.543121
$26$	0	0
$27$	−3.63378	−0.699322
$28$	0	0
$29$	−7.95514	−1.47723	−0.738617	−	0.674126i	$-0.764521\pi$
$29$	−7.95514	−1.47723	−0.738617	+	0.674126i	$0.764521\pi$
$30$	0	0
$31$	7.36112	1.32210	0.661048	−	0.750343i	$-0.270112\pi$
$31$	7.36112	1.32210	0.661048	+	0.750343i	$0.270112\pi$
$32$	0	0
$33$	−1.33766	−0.232856
$34$	0	0
$35$	2.77770	0.469517
$36$	0	0
$37$	−7.01958	−1.15401	−0.577006	−	0.816740i	$-0.695779\pi$
$37$	−7.01958	−1.15401	−0.577006	+	0.816740i	$0.695779\pi$
$38$	0	0
$39$	−6.58784	−1.05490
$40$	0	0
$41$	1.00000	0.156174
$41$	1.00000	0.156174
$42$	0	0
$43$	0.887550	0.135350	0.0676750	−	0.997707i	$-0.478442\pi$
$43$	0.887550	0.135350	0.0676750	+	0.997707i	$0.478442\pi$
$44$	0	0
$45$	−3.42778	−0.510983
$46$	0	0
$47$	1.56384	0.228110	0.114055	−	0.993474i	$-0.463616\pi$
$47$	1.56384	0.228110	0.114055	+	0.993474i	$0.463616\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.19718	0.867778
$52$	0	0
$53$	−9.53360	−1.30954	−0.654770	−	0.755828i	$-0.727235\pi$
$53$	−9.53360	−1.30954	−0.654770	+	0.755828i	$0.727235\pi$
$54$	0	0
$55$	1.80573	0.243484
$56$	0	0
$57$	−14.4027	−1.90768
$58$	0	0
$59$	−2.25033	−0.292968	−0.146484	−	0.989213i	$-0.546796\pi$
$59$	−2.25033	−0.292968	−0.146484	+	0.989213i	$0.546796\pi$
$60$	0	0
$61$	−8.08129	−1.03470	−0.517352	−	0.855773i	$-0.673082\pi$
$61$	−8.08129	−1.03470	−0.517352	+	0.855773i	$0.673082\pi$
$62$	0	0
$63$	−1.23404	−0.155474
$64$	0	0
$65$	8.89305	1.10305
$66$	0	0
$67$	1.72514	0.210760	0.105380	−	0.994432i	$-0.466394\pi$
$67$	1.72514	0.210760	0.105380	+	0.994432i	$0.466394\pi$
$68$	0	0
$69$	−0.473574	−0.0570116
$70$	0	0
$71$	3.79347	0.450202	0.225101	−	0.974335i	$-0.427729\pi$
$71$	3.79347	0.450202	0.225101	+	0.974335i	$0.427729\pi$
$72$	0	0
$73$	5.34109	0.625127	0.312564	−	0.949897i	$-0.398812\pi$
$73$	5.34109	0.625127	0.312564	+	0.949897i	$0.398812\pi$
$74$	0	0
$75$	5.58784	0.645228
$76$	0	0
$77$	0.650081	0.0740836
$78$	0	0
$79$	−4.14795	−0.466681	−0.233340	−	0.972395i	$-0.574966\pi$
$79$	−4.14795	−0.466681	−0.233340	+	0.972395i	$0.574966\pi$
$80$	0	0
$81$	−11.1793	−1.24214
$82$	0	0
$83$	−3.08586	−0.338717	−0.169358	−	0.985555i	$-0.554170\pi$
$83$	−3.08586	−0.338717	−0.169358	+	0.985555i	$0.554170\pi$
$84$	0	0
$85$	−8.36569	−0.907386
$86$	0	0
$87$	−16.3691	−1.75495
$88$	0	0
$89$	5.26991	0.558610	0.279305	−	0.960202i	$-0.409896\pi$
$89$	5.26991	0.558610	0.279305	+	0.960202i	$0.409896\pi$
$90$	0	0
$91$	3.20159	0.335618
$92$	0	0
$93$	15.1468	1.57065
$94$	0	0
$95$	19.4424	1.99475
$96$	0	0
$97$	19.4644	1.97631	0.988153	−	0.153470i	$-0.0490447\pi$
$97$	19.4644	1.97631	0.988153	+	0.153470i	$0.0490447\pi$
$98$	0	0
$99$	−0.802223	−0.0806265
$100$	0	0
$101$	7.59304	0.755536	0.377768	−	0.925900i	$-0.376692\pi$
$101$	7.59304	0.755536	0.377768	+	0.925900i	$0.376692\pi$
$102$	0	0
$103$	−0.992004	−0.0977451	−0.0488725	−	0.998805i	$-0.515563\pi$
$103$	−0.992004	−0.0977451	−0.0488725	+	0.998805i	$0.515563\pi$
$104$	0	0
$105$	5.71561	0.557786
$106$	0	0
$107$	18.7077	1.80854	0.904272	−	0.426958i	$-0.140415\pi$
$107$	18.7077	1.80854	0.904272	+	0.426958i	$0.140415\pi$
$108$	0	0
$109$	−3.11992	−0.298834	−0.149417	−	0.988774i	$-0.547740\pi$
$109$	−3.11992	−0.298834	−0.149417	+	0.988774i	$0.547740\pi$
$110$	0	0
$111$	−14.4440	−1.37097
$112$	0	0
$113$	−4.36583	−0.410703	−0.205352	−	0.978688i	$-0.565834\pi$
$113$	−4.36583	−0.410703	−0.205352	+	0.978688i	$0.565834\pi$
$114$	0	0
$115$	0.639287	0.0596138
$116$	0	0
$117$	−3.95088	−0.365259
$118$	0	0
$119$	−3.01173	−0.276085
$120$	0	0
$121$	−10.5774	−0.961581
$122$	0	0
$123$	2.05768	0.185535
$124$	0	0
$125$	6.34536	0.567546
$126$	0	0
$127$	−16.0571	−1.42484	−0.712421	−	0.701753i	$-0.752401\pi$
$127$	−16.0571	−1.42484	−0.712421	+	0.701753i	$0.752401\pi$
$128$	0	0
$129$	1.82629	0.160796
$130$	0	0
$131$	11.8057	1.03147	0.515736	−	0.856748i	$-0.327519\pi$
$131$	11.8057	1.03147	0.515736	+	0.856748i	$0.327519\pi$
$132$	0	0
$133$	6.99947	0.606931
$134$	0	0
$135$	10.0936	0.868715
$136$	0	0
$137$	18.6525	1.59359	0.796793	−	0.604252i	$-0.206528\pi$
$137$	18.6525	1.59359	0.796793	+	0.604252i	$0.206528\pi$
$138$	0	0
$139$	−7.70481	−0.653514	−0.326757	−	0.945108i	$-0.605956\pi$
$139$	−7.70481	−0.653514	−0.326757	+	0.945108i	$0.605956\pi$
$140$	0	0
$141$	3.21789	0.270995
$142$	0	0
$143$	2.08129	0.174046
$144$	0	0
$145$	22.0970	1.83506
$146$	0	0
$147$	2.05768	0.169714
$148$	0	0
$149$	10.9709	0.898772	0.449386	−	0.893338i	$-0.351643\pi$
$149$	10.9709	0.898772	0.449386	+	0.893338i	$0.351643\pi$
$150$	0	0
$151$	−9.44075	−0.768277	−0.384139	−	0.923275i	$-0.625502\pi$
$151$	−9.44075	−0.768277	−0.384139	+	0.923275i	$0.625502\pi$
$152$	0	0
$153$	3.71659	0.300468
$154$	0	0
$155$	−20.4470	−1.64234
$156$	0	0
$157$	−2.53413	−0.202245	−0.101123	−	0.994874i	$-0.532243\pi$
$157$	−2.53413	−0.202245	−0.101123	+	0.994874i	$0.532243\pi$
$158$	0	0
$159$	−19.6171	−1.55573
$160$	0	0
$161$	0.230150	0.0181384
$162$	0	0
$163$	−8.81968	−0.690811	−0.345405	−	0.938454i	$-0.612259\pi$
$163$	−8.81968	−0.690811	−0.345405	+	0.938454i	$0.612259\pi$
$164$	0	0
$165$	3.71561	0.289260
$166$	0	0
$167$	−1.44751	−0.112012	−0.0560058	−	0.998430i	$-0.517837\pi$
$167$	−1.44751	−0.112012	−0.0560058	+	0.998430i	$0.517837\pi$
$168$	0	0
$169$	−2.74982	−0.211524
$170$	0	0
$171$	−8.63760	−0.660534
$172$	0	0
$173$	−18.3600	−1.39588	−0.697942	−	0.716154i	$-0.745901\pi$
$173$	−18.3600	−1.39588	−0.697942	+	0.716154i	$0.745901\pi$
$174$	0	0
$175$	−2.71561	−0.205281
$176$	0	0
$177$	−4.63046	−0.348047
$178$	0	0
$179$	12.2089	0.912537	0.456268	−	0.889842i	$-0.349186\pi$
$179$	12.2089	0.912537	0.456268	+	0.889842i	$0.349186\pi$
$180$	0	0
$181$	−15.4906	−1.15140	−0.575702	−	0.817660i	$-0.695271\pi$
$181$	−15.4906	−1.15140	−0.575702	+	0.817660i	$0.695271\pi$
$182$	0	0
$183$	−16.6287	−1.22923
$184$	0	0
$185$	19.4983	1.43354
$186$	0	0
$187$	−1.95787	−0.143174
$188$	0	0
$189$	3.63378	0.264319
$190$	0	0
$191$	1.79629	0.129975	0.0649874	−	0.997886i	$-0.479299\pi$
$191$	1.79629	0.129975	0.0649874	+	0.997886i	$0.479299\pi$
$192$	0	0
$193$	−13.0364	−0.938378	−0.469189	−	0.883098i	$-0.655454\pi$
$193$	−13.0364	−0.938378	−0.469189	+	0.883098i	$0.655454\pi$
$194$	0	0
$195$	18.2990	1.31042
$196$	0	0
$197$	7.86295	0.560212	0.280106	−	0.959969i	$-0.409630\pi$
$197$	7.86295	0.560212	0.280106	+	0.959969i	$0.409630\pi$
$198$	0	0
$199$	−6.42879	−0.455725	−0.227863	−	0.973693i	$-0.573174\pi$
$199$	−6.42879	−0.455725	−0.227863	+	0.973693i	$0.573174\pi$
$200$	0	0
$201$	3.54979	0.250383
$202$	0	0
$203$	7.95514	0.558342
$204$	0	0
$205$	−2.77770	−0.194003
$206$	0	0
$207$	−0.284013	−0.0197403
$208$	0	0
$209$	4.55022	0.314745
$210$	0	0
$211$	−20.0269	−1.37871	−0.689354	−	0.724424i	$-0.742106\pi$
$211$	−20.0269	−1.37871	−0.689354	+	0.724424i	$0.742106\pi$
$212$	0	0
$213$	7.80573	0.534840
$214$	0	0
$215$	−2.46535	−0.168135
$216$	0	0
$217$	−7.36112	−0.499706
$218$	0	0
$219$	10.9902	0.742652
$220$	0	0
$221$	−9.64234	−0.648614
$222$	0	0
$223$	−20.6042	−1.37976	−0.689879	−	0.723925i	$-0.742336\pi$
$223$	−20.6042	−1.37976	−0.689879	+	0.723925i	$0.742336\pi$
$224$	0	0
$225$	3.35116	0.223410
$226$	0	0
$227$	−0.370025	−0.0245595	−0.0122797	−	0.999925i	$-0.503909\pi$
$227$	−0.370025	−0.0245595	−0.0122797	+	0.999925i	$0.503909\pi$
$228$	0	0
$229$	25.9103	1.71220	0.856102	−	0.516807i	$-0.172880\pi$
$229$	25.9103	1.71220	0.856102	+	0.516807i	$0.172880\pi$
$230$	0	0
$231$	1.33766	0.0880113
$232$	0	0
$233$	−4.41439	−0.289196	−0.144598	−	0.989490i	$-0.546189\pi$
$233$	−4.41439	−0.289196	−0.144598	+	0.989490i	$0.546189\pi$
$234$	0	0
$235$	−4.34389	−0.283364
$236$	0	0
$237$	−8.53514	−0.554417
$238$	0	0
$239$	1.26883	0.0820735	0.0410368	−	0.999158i	$-0.486934\pi$
$239$	1.26883	0.0820735	0.0410368	+	0.999158i	$0.486934\pi$
$240$	0	0
$241$	20.3831	1.31299	0.656494	−	0.754331i	$-0.272039\pi$
$241$	20.3831	1.31299	0.656494	+	0.754331i	$0.272039\pi$
$242$	0	0
$243$	−12.1020	−0.776341
$244$	0	0
$245$	−2.77770	−0.177461
$246$	0	0
$247$	22.4094	1.42588
$248$	0	0
$249$	−6.34970	−0.402396
$250$	0	0
$251$	27.4118	1.73022	0.865108	−	0.501585i	$-0.167250\pi$
$251$	27.4118	1.73022	0.865108	+	0.501585i	$0.167250\pi$
$252$	0	0
$253$	0.149616	0.00940628
$254$	0	0
$255$	−17.2139	−1.07798
$256$	0	0
$257$	−0.989482	−0.0617222	−0.0308611	−	0.999524i	$-0.509825\pi$
$257$	−0.989482	−0.0617222	−0.0308611	+	0.999524i	$0.509825\pi$
$258$	0	0
$259$	7.01958	0.436176
$260$	0	0
$261$	−9.81693	−0.607653
$262$	0	0
$263$	−10.9961	−0.678051	−0.339026	−	0.940777i	$-0.610097\pi$
$263$	−10.9961	−0.678051	−0.339026	+	0.940777i	$0.610097\pi$
$264$	0	0
$265$	26.4815	1.62674
$266$	0	0
$267$	10.8438	0.663629
$268$	0	0
$269$	−24.6103	−1.50052	−0.750259	−	0.661144i	$-0.770071\pi$
$269$	−24.6103	−1.50052	−0.750259	+	0.661144i	$0.770071\pi$
$270$	0	0
$271$	1.08909	0.0661574	0.0330787	−	0.999453i	$-0.489469\pi$
$271$	1.08909	0.0661574	0.0330787	+	0.999453i	$0.489469\pi$
$272$	0	0
$273$	6.58784	0.398714
$274$	0	0
$275$	−1.76536	−0.106455
$276$	0	0
$277$	13.1719	0.791423	0.395711	−	0.918375i	$-0.370498\pi$
$277$	13.1719	0.791423	0.395711	+	0.918375i	$0.370498\pi$
$278$	0	0
$279$	9.08389	0.543838
$280$	0	0
$281$	13.3499	0.796389	0.398195	−	0.917301i	$-0.369637\pi$
$281$	13.3499	0.796389	0.398195	+	0.917301i	$0.369637\pi$
$282$	0	0
$283$	−23.8824	−1.41966	−0.709830	−	0.704373i	$-0.751228\pi$
$283$	−23.8824	−1.41966	−0.709830	+	0.704373i	$0.751228\pi$
$284$	0	0
$285$	40.0062	2.36976
$286$	0	0
$287$	−1.00000	−0.0590281
$288$	0	0
$289$	−7.92946	−0.466439
$290$	0	0
$291$	40.0514	2.34785
$292$	0	0
$293$	19.8922	1.16212	0.581058	−	0.813862i	$-0.302639\pi$
$293$	19.8922	1.16212	0.581058	+	0.813862i	$0.302639\pi$
$294$	0	0
$295$	6.25074	0.363932
$296$	0	0
$297$	2.36225	0.137072
$298$	0	0
$299$	0.736846	0.0426129
$300$	0	0
$301$	−0.887550	−0.0511575
$302$	0	0
$303$	15.6240	0.897577
$304$	0	0
$305$	22.4474	1.28533
$306$	0	0
$307$	−26.9305	−1.53701	−0.768504	−	0.639845i	$-0.778998\pi$
$307$	−26.9305	−1.53701	−0.768504	+	0.639845i	$0.778998\pi$
$308$	0	0
$309$	−2.04122	−0.116121
$310$	0	0
$311$	−20.5399	−1.16471	−0.582355	−	0.812935i	$-0.697869\pi$
$311$	−20.5399	−1.16471	−0.582355	+	0.812935i	$0.697869\pi$
$312$	0	0
$313$	10.4785	0.592278	0.296139	−	0.955145i	$-0.404301\pi$
$313$	10.4785	0.592278	0.296139	+	0.955145i	$0.404301\pi$
$314$	0	0
$315$	3.42778	0.193133
$316$	0	0
$317$	27.5360	1.54657	0.773287	−	0.634056i	$-0.218611\pi$
$317$	27.5360	1.54657	0.773287	+	0.634056i	$0.218611\pi$
$318$	0	0
$319$	5.17149	0.289548
$320$	0	0
$321$	38.4944	2.14855
$322$	0	0
$323$	−21.0805	−1.17295
$324$	0	0
$325$	−8.69426	−0.482271
$326$	0	0
$327$	−6.41978	−0.355015
$328$	0	0
$329$	−1.56384	−0.0862175
$330$	0	0
$331$	4.97091	0.273226	0.136613	−	0.990624i	$-0.456378\pi$
$331$	4.97091	0.273226	0.136613	+	0.990624i	$0.456378\pi$
$332$	0	0
$333$	−8.66242	−0.474697
$334$	0	0
$335$	−4.79193	−0.261811
$336$	0	0
$337$	−10.6607	−0.580727	−0.290363	−	0.956916i	$-0.593776\pi$
$337$	−10.6607	−0.580727	−0.290363	+	0.956916i	$0.593776\pi$
$338$	0	0
$339$	−8.98348	−0.487916
$340$	0	0
$341$	−4.78533	−0.259140
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	1.31545	0.0708212
$346$	0	0
$347$	−12.1949	−0.654656	−0.327328	−	0.944911i	$-0.606148\pi$
$347$	−12.1949	−0.654656	−0.327328	+	0.944911i	$0.606148\pi$
$348$	0	0
$349$	30.9064	1.65438	0.827192	−	0.561920i	$-0.189937\pi$
$349$	30.9064	1.65438	0.827192	+	0.561920i	$0.189937\pi$
$350$	0	0
$351$	11.6339	0.620971
$352$	0	0
$353$	11.6821	0.621773	0.310886	−	0.950447i	$-0.399374\pi$
$353$	11.6821	0.621773	0.310886	+	0.950447i	$0.399374\pi$
$354$	0	0
$355$	−10.5371	−0.559251
$356$	0	0
$357$	−6.19718	−0.327989
$358$	0	0
$359$	−31.8729	−1.68219	−0.841093	−	0.540890i	$-0.818087\pi$
$359$	−31.8729	−1.68219	−0.841093	+	0.540890i	$0.818087\pi$
$360$	0	0
$361$	29.9926	1.57856
$362$	0	0
$363$	−21.7649	−1.14236
$364$	0	0
$365$	−14.8359	−0.776548
$366$	0	0
$367$	28.9799	1.51274	0.756368	−	0.654146i	$-0.226972\pi$
$367$	28.9799	1.51274	0.756368	+	0.654146i	$0.226972\pi$
$368$	0	0
$369$	1.23404	0.0642413
$370$	0	0
$371$	9.53360	0.494960
$372$	0	0
$373$	6.19997	0.321022	0.160511	−	0.987034i	$-0.448686\pi$
$373$	6.19997	0.321022	0.160511	+	0.987034i	$0.448686\pi$
$374$	0	0
$375$	13.0567	0.674245
$376$	0	0
$377$	25.4691	1.31173
$378$	0	0
$379$	−1.59353	−0.0818543	−0.0409271	−	0.999162i	$-0.513031\pi$
$379$	−1.59353	−0.0818543	−0.0409271	+	0.999162i	$0.513031\pi$
$380$	0	0
$381$	−33.0404	−1.69271
$382$	0	0
$383$	−2.56955	−0.131298	−0.0656489	−	0.997843i	$-0.520912\pi$
$383$	−2.56955	−0.131298	−0.0656489	+	0.997843i	$0.520912\pi$
$384$	0	0
$385$	−1.80573	−0.0920284
$386$	0	0
$387$	1.09527	0.0556756
$388$	0	0
$389$	−15.0487	−0.763001	−0.381501	−	0.924369i	$-0.624593\pi$
$389$	−15.0487	−0.763001	−0.381501	+	0.924369i	$0.624593\pi$
$390$	0	0
$391$	−0.693150	−0.0350541
$392$	0	0
$393$	24.2924	1.22539
$394$	0	0
$395$	11.5217	0.579722
$396$	0	0
$397$	−0.530236	−0.0266118	−0.0133059	−	0.999911i	$-0.504236\pi$
$397$	−0.530236	−0.0266118	−0.0133059	+	0.999911i	$0.504236\pi$
$398$	0	0
$399$	14.4027	0.721035
$400$	0	0
$401$	−35.8246	−1.78899	−0.894497	−	0.447073i	$-0.852466\pi$
$401$	−35.8246	−1.78899	−0.894497	+	0.447073i	$0.852466\pi$
$402$	0	0
$403$	−23.5673	−1.17397
$404$	0	0
$405$	31.0526	1.54302
$406$	0	0
$407$	4.56330	0.226194
$408$	0	0
$409$	−19.9916	−0.988520	−0.494260	−	0.869314i	$-0.664561\pi$
$409$	−19.9916	−0.988520	−0.494260	+	0.869314i	$0.664561\pi$
$410$	0	0
$411$	38.3807	1.89318
$412$	0	0
$413$	2.25033	0.110732
$414$	0	0
$415$	8.57157	0.420762
$416$	0	0
$417$	−15.8540	−0.776375
$418$	0	0
$419$	−27.0806	−1.32297	−0.661486	−	0.749957i	$-0.730074\pi$
$419$	−27.0806	−1.32297	−0.661486	+	0.749957i	$0.730074\pi$
$420$	0	0
$421$	14.6977	0.716323	0.358162	−	0.933660i	$-0.383404\pi$
$421$	14.6977	0.716323	0.358162	+	0.933660i	$0.383404\pi$
$422$	0	0
$423$	1.92984	0.0938320
$424$	0	0
$425$	8.17868	0.396724
$426$	0	0
$427$	8.08129	0.391081
$428$	0	0
$429$	4.28263	0.206767
$430$	0	0
$431$	13.9935	0.674045	0.337023	−	0.941497i	$-0.390580\pi$
$431$	13.9935	0.674045	0.337023	+	0.941497i	$0.390580\pi$
$432$	0	0
$433$	8.20114	0.394122	0.197061	−	0.980391i	$-0.436860\pi$
$433$	8.20114	0.394122	0.197061	+	0.980391i	$0.436860\pi$
$434$	0	0
$435$	45.4685	2.18005
$436$	0	0
$437$	1.61093	0.0770611
$438$	0	0
$439$	3.02985	0.144607	0.0723033	−	0.997383i	$-0.476965\pi$
$439$	3.02985	0.144607	0.0723033	+	0.997383i	$0.476965\pi$
$440$	0	0
$441$	1.23404	0.0587636
$442$	0	0
$443$	−1.23293	−0.0585781	−0.0292891	−	0.999571i	$-0.509324\pi$
$443$	−1.23293	−0.0585781	−0.0292891	+	0.999571i	$0.509324\pi$
$444$	0	0
$445$	−14.6382	−0.693919
$446$	0	0
$447$	22.5746	1.06774
$448$	0	0
$449$	4.17031	0.196809	0.0984045	−	0.995147i	$-0.468626\pi$
$449$	4.17031	0.196809	0.0984045	+	0.995147i	$0.468626\pi$
$450$	0	0
$451$	−0.650081	−0.0306111
$452$	0	0
$453$	−19.4260	−0.912714
$454$	0	0
$455$	−8.89305	−0.416913
$456$	0	0
$457$	−26.3524	−1.23271	−0.616356	−	0.787468i	$-0.711392\pi$
$457$	−26.3524	−1.23271	−0.616356	+	0.787468i	$0.711392\pi$
$458$	0	0
$459$	−10.9440	−0.510822
$460$	0	0
$461$	16.8303	0.783866	0.391933	−	0.919994i	$-0.371806\pi$
$461$	16.8303	0.783866	0.391933	+	0.919994i	$0.371806\pi$
$462$	0	0
$463$	−36.4971	−1.69617	−0.848083	−	0.529863i	$-0.822243\pi$
$463$	−36.4971	−1.69617	−0.848083	+	0.529863i	$0.822243\pi$
$464$	0	0
$465$	−42.0733	−1.95110
$466$	0	0
$467$	19.0522	0.881631	0.440815	−	0.897598i	$-0.354689\pi$
$467$	19.0522	0.881631	0.440815	+	0.897598i	$0.354689\pi$
$468$	0	0
$469$	−1.72514	−0.0796597
$470$	0	0
$471$	−5.21441	−0.240267
$472$	0	0
$473$	−0.576979	−0.0265295
$474$	0	0
$475$	−19.0078	−0.872138
$476$	0	0
$477$	−11.7648	−0.538673
$478$	0	0
$479$	−19.2346	−0.878853	−0.439426	−	0.898279i	$-0.644818\pi$
$479$	−19.2346	−0.878853	−0.439426	+	0.898279i	$0.644818\pi$
$480$	0	0
$481$	22.4738	1.02472
$482$	0	0
$483$	0.473574	0.0215484
$484$	0	0
$485$	−54.0661	−2.45502
$486$	0	0
$487$	−33.0271	−1.49660	−0.748300	−	0.663361i	$-0.769130\pi$
$487$	−33.0271	−1.49660	−0.748300	+	0.663361i	$0.769130\pi$
$488$	0	0
$489$	−18.1481	−0.820683
$490$	0	0
$491$	−23.8861	−1.07796	−0.538981	−	0.842318i	$-0.681191\pi$
$491$	−23.8861	−1.07796	−0.538981	+	0.842318i	$0.681191\pi$
$492$	0	0
$493$	−23.9588	−1.07905
$494$	0	0
$495$	2.22833	0.100156
$496$	0	0
$497$	−3.79347	−0.170160
$498$	0	0
$499$	−21.2438	−0.951003	−0.475502	−	0.879715i	$-0.657733\pi$
$499$	−21.2438	−0.951003	−0.475502	+	0.879715i	$0.657733\pi$
$500$	0	0
$501$	−2.97851	−0.133070
$502$	0	0
$503$	27.2308	1.21416	0.607081	−	0.794640i	$-0.292340\pi$
$503$	27.2308	1.21416	0.607081	+	0.794640i	$0.292340\pi$
$504$	0	0
$505$	−21.0912	−0.938545
$506$	0	0
$507$	−5.65823	−0.251291
$508$	0	0
$509$	−9.50329	−0.421226	−0.210613	−	0.977570i	$-0.567546\pi$
$509$	−9.50329	−0.421226	−0.210613	+	0.977570i	$0.567546\pi$
$510$	0	0
$511$	−5.34109	−0.236276
$512$	0	0
$513$	25.4346	1.12296
$514$	0	0
$515$	2.75549	0.121421
$516$	0	0
$517$	−1.01663	−0.0447111
$518$	0	0
$519$	−37.7789	−1.65831
$520$	0	0
$521$	−11.5555	−0.506257	−0.253128	−	0.967433i	$-0.581459\pi$
$521$	−11.5555	−0.506257	−0.253128	+	0.967433i	$0.581459\pi$
$522$	0	0
$523$	−1.48247	−0.0648238	−0.0324119	−	0.999475i	$-0.510319\pi$
$523$	−1.48247	−0.0648238	−0.0324119	+	0.999475i	$0.510319\pi$
$524$	0	0
$525$	−5.58784	−0.243873
$526$	0	0
$527$	22.1697	0.965729
$528$	0	0
$529$	−22.9470	−0.997697
$530$	0	0
$531$	−2.77699	−0.120511
$532$	0	0
$533$	−3.20159	−0.138676
$534$	0	0
$535$	−51.9644	−2.24662
$536$	0	0
$537$	25.1220	1.08409
$538$	0	0
$539$	−0.650081	−0.0280010
$540$	0	0
$541$	4.96794	0.213588	0.106794	−	0.994281i	$-0.465941\pi$
$541$	4.96794	0.213588	0.106794	+	0.994281i	$0.465941\pi$
$542$	0	0
$543$	−31.8746	−1.36787
$544$	0	0
$545$	8.66619	0.371219
$546$	0	0
$547$	2.24010	0.0957798	0.0478899	−	0.998853i	$-0.484750\pi$
$547$	2.24010	0.0957798	0.0478899	+	0.998853i	$0.484750\pi$
$548$	0	0
$549$	−9.97261	−0.425620
$550$	0	0
$551$	55.6818	2.37212
$552$	0	0
$553$	4.14795	0.176389
$554$	0	0
$555$	40.1212	1.70305
$556$	0	0
$557$	−37.1135	−1.57255	−0.786274	−	0.617878i	$-0.787993\pi$
$557$	−37.1135	−1.57255	−0.786274	+	0.617878i	$0.787993\pi$
$558$	0	0
$559$	−2.84157	−0.120186
$560$	0	0
$561$	−4.02867	−0.170090
$562$	0	0
$563$	14.0640	0.592728	0.296364	−	0.955075i	$-0.404226\pi$
$563$	14.0640	0.592728	0.296364	+	0.955075i	$0.404226\pi$
$564$	0	0
$565$	12.1270	0.510185
$566$	0	0
$567$	11.1793	0.469485
$568$	0	0
$569$	−23.2493	−0.974659	−0.487330	−	0.873218i	$-0.662029\pi$
$569$	−23.2493	−0.974659	−0.487330	+	0.873218i	$0.662029\pi$
$570$	0	0
$571$	−35.7638	−1.49667	−0.748333	−	0.663323i	$-0.769146\pi$
$571$	−35.7638	−1.49667	−0.748333	+	0.663323i	$0.769146\pi$
$572$	0	0
$573$	3.69618	0.154410
$574$	0	0
$575$	−0.624997	−0.0260642
$576$	0	0
$577$	−7.29391	−0.303649	−0.151825	−	0.988407i	$-0.548515\pi$
$577$	−7.29391	−0.303649	−0.151825	+	0.988407i	$0.548515\pi$
$578$	0	0
$579$	−26.8246	−1.11479
$580$	0	0
$581$	3.08586	0.128023
$582$	0	0
$583$	6.19761	0.256679
$584$	0	0
$585$	10.9743	0.453733
$586$	0	0
$587$	3.23634	0.133578	0.0667890	−	0.997767i	$-0.478725\pi$
$587$	3.23634	0.133578	0.0667890	+	0.997767i	$0.478725\pi$
$588$	0	0
$589$	−51.5240	−2.12301
$590$	0	0
$591$	16.1794	0.665532
$592$	0	0
$593$	−7.75949	−0.318644	−0.159322	−	0.987227i	$-0.550931\pi$
$593$	−7.75949	−0.318644	−0.159322	+	0.987227i	$0.550931\pi$
$594$	0	0
$595$	8.36569	0.342960
$596$	0	0
$597$	−13.2284	−0.541402
$598$	0	0
$599$	6.54344	0.267358	0.133679	−	0.991025i	$-0.457321\pi$
$599$	6.54344	0.267358	0.133679	+	0.991025i	$0.457321\pi$
$600$	0	0
$601$	−17.6484	−0.719891	−0.359946	−	0.932973i	$-0.617205\pi$
$601$	−17.6484	−0.719891	−0.359946	+	0.932973i	$0.617205\pi$
$602$	0	0
$603$	2.12889	0.0866950
$604$	0	0
$605$	29.3808	1.19450
$606$	0	0
$607$	38.5586	1.56504	0.782522	−	0.622623i	$-0.213933\pi$
$607$	38.5586	1.56504	0.782522	+	0.622623i	$0.213933\pi$
$608$	0	0
$609$	16.3691	0.663310
$610$	0	0
$611$	−5.00679	−0.202553
$612$	0	0
$613$	−20.8305	−0.841335	−0.420667	−	0.907215i	$-0.638204\pi$
$613$	−20.8305	−0.841335	−0.420667	+	0.907215i	$0.638204\pi$
$614$	0	0
$615$	−5.71561	−0.230475
$616$	0	0
$617$	−48.4441	−1.95028	−0.975142	−	0.221580i	$-0.928879\pi$
$617$	−48.4441	−1.95028	−0.975142	+	0.221580i	$0.928879\pi$
$618$	0	0
$619$	35.6461	1.43274	0.716369	−	0.697721i	$-0.245803\pi$
$619$	35.6461	1.43274	0.716369	+	0.697721i	$0.245803\pi$
$620$	0	0
$621$	0.836315	0.0335602
$622$	0	0
$623$	−5.26991	−0.211135
$624$	0	0
$625$	−31.2035	−1.24814
$626$	0	0
$627$	9.36289	0.373918
$628$	0	0
$629$	−21.1411	−0.842951
$630$	0	0
$631$	−5.10970	−0.203414	−0.101707	−	0.994814i	$-0.532430\pi$
$631$	−5.10970	−0.203414	−0.101707	+	0.994814i	$0.532430\pi$
$632$	0	0
$633$	−41.2089	−1.63791
$634$	0	0
$635$	44.6019	1.76997
$636$	0	0
$637$	−3.20159	−0.126852
$638$	0	0
$639$	4.68127	0.185188
$640$	0	0
$641$	17.1406	0.677012	0.338506	−	0.940964i	$-0.390079\pi$
$641$	17.1406	0.677012	0.338506	+	0.940964i	$0.390079\pi$
$642$	0	0
$643$	−47.2815	−1.86460	−0.932301	−	0.361683i	$-0.882202\pi$
$643$	−47.2815	−1.86460	−0.932301	+	0.361683i	$0.882202\pi$
$644$	0	0
$645$	−5.07289	−0.199745
$646$	0	0
$647$	−32.9021	−1.29351	−0.646757	−	0.762696i	$-0.723875\pi$
$647$	−32.9021	−1.29351	−0.646757	+	0.762696i	$0.723875\pi$
$648$	0	0
$649$	1.46290	0.0574238
$650$	0	0
$651$	−15.1468	−0.593651
$652$	0	0
$653$	30.1228	1.17880	0.589399	−	0.807842i	$-0.299365\pi$
$653$	30.1228	1.17880	0.589399	+	0.807842i	$0.299365\pi$
$654$	0	0
$655$	−32.7927	−1.28132
$656$	0	0
$657$	6.59110	0.257143
$658$	0	0
$659$	27.1319	1.05691	0.528454	−	0.848962i	$-0.322772\pi$
$659$	27.1319	1.05691	0.528454	+	0.848962i	$0.322772\pi$
$660$	0	0
$661$	34.9848	1.36075	0.680375	−	0.732864i	$-0.261817\pi$
$661$	34.9848	1.36075	0.680375	+	0.732864i	$0.261817\pi$
$662$	0	0
$663$	−19.8408	−0.770554
$664$	0	0
$665$	−19.4424	−0.753945
$666$	0	0
$667$	1.83088	0.0708918
$668$	0	0
$669$	−42.3967	−1.63915
$670$	0	0
$671$	5.25349	0.202809
$672$	0	0
$673$	21.8089	0.840670	0.420335	−	0.907369i	$-0.361913\pi$
$673$	21.8089	0.840670	0.420335	+	0.907369i	$0.361913\pi$
$674$	0	0
$675$	−9.86793	−0.379817
$676$	0	0
$677$	19.3947	0.745399	0.372699	−	0.927952i	$-0.378432\pi$
$677$	19.3947	0.745399	0.372699	+	0.927952i	$0.378432\pi$
$678$	0	0
$679$	−19.4644	−0.746974
$680$	0	0
$681$	−0.761393	−0.0291766
$682$	0	0
$683$	7.62321	0.291694	0.145847	−	0.989307i	$-0.453409\pi$
$683$	7.62321	0.291694	0.145847	+	0.989307i	$0.453409\pi$
$684$	0	0
$685$	−51.8109	−1.97959
$686$	0	0
$687$	53.3151	2.03410
$688$	0	0
$689$	30.5227	1.16282
$690$	0	0
$691$	32.0467	1.21911	0.609556	−	0.792743i	$-0.291348\pi$
$691$	32.0467	1.21911	0.609556	+	0.792743i	$0.291348\pi$
$692$	0	0
$693$	0.802223	0.0304739
$694$	0	0
$695$	21.4016	0.811811
$696$	0	0
$697$	3.01173	0.114078
$698$	0	0
$699$	−9.08338	−0.343565
$700$	0	0
$701$	−37.5575	−1.41853	−0.709264	−	0.704943i	$-0.750972\pi$
$701$	−37.5575	−1.41853	−0.709264	+	0.704943i	$0.750972\pi$
$702$	0	0
$703$	49.1334	1.85310
$704$	0	0
$705$	−8.93832	−0.336637
$706$	0	0
$707$	−7.59304	−0.285566
$708$	0	0
$709$	11.6662	0.438132	0.219066	−	0.975710i	$-0.429699\pi$
$709$	11.6662	0.438132	0.219066	+	0.975710i	$0.429699\pi$
$710$	0	0
$711$	−5.11872	−0.191967
$712$	0	0
$713$	−1.69416	−0.0634469
$714$	0	0
$715$	−5.78120	−0.216205
$716$	0	0
$717$	2.61083	0.0975034
$718$	0	0
$719$	−47.3583	−1.76617	−0.883083	−	0.469216i	$-0.844537\pi$
$719$	−47.3583	−1.76617	−0.883083	+	0.469216i	$0.844537\pi$
$720$	0	0
$721$	0.992004	0.0369442
$722$	0	0
$723$	41.9417	1.55983
$724$	0	0
$725$	−21.6030	−0.802317
$726$	0	0
$727$	−37.5129	−1.39128	−0.695638	−	0.718393i	$-0.744878\pi$
$727$	−37.5129	−1.39128	−0.695638	+	0.718393i	$0.744878\pi$
$728$	0	0
$729$	8.63586	0.319847
$730$	0	0
$731$	2.67306	0.0988668
$732$	0	0
$733$	−16.9727	−0.626900	−0.313450	−	0.949605i	$-0.601485\pi$
$733$	−16.9727	−0.626900	−0.313450	+	0.949605i	$0.601485\pi$
$734$	0	0
$735$	−5.71561	−0.210823
$736$	0	0
$737$	−1.12148	−0.0413103
$738$	0	0
$739$	29.0306	1.06791	0.533955	−	0.845513i	$-0.320705\pi$
$739$	29.0306	1.06791	0.533955	+	0.845513i	$0.320705\pi$
$740$	0	0
$741$	46.1114	1.69394
$742$	0	0
$743$	−27.5161	−1.00947	−0.504734	−	0.863275i	$-0.668409\pi$
$743$	−27.5161	−1.00947	−0.504734	+	0.863275i	$0.668409\pi$
$744$	0	0
$745$	−30.4739	−1.11648
$746$	0	0
$747$	−3.80806	−0.139329
$748$	0	0
$749$	−18.7077	−0.683565
$750$	0	0
$751$	−2.13796	−0.0780154	−0.0390077	−	0.999239i	$-0.512420\pi$
$751$	−2.13796	−0.0780154	−0.0390077	+	0.999239i	$0.512420\pi$
$752$	0	0
$753$	56.4046	2.05550
$754$	0	0
$755$	26.2235	0.954373
$756$	0	0
$757$	52.5711	1.91073	0.955365	−	0.295428i	$-0.0954622\pi$
$757$	52.5711	1.91073	0.955365	+	0.295428i	$0.0954622\pi$
$758$	0	0
$759$	0.307862	0.0111747
$760$	0	0
$761$	−52.9946	−1.92105	−0.960526	−	0.278189i	$-0.910266\pi$
$761$	−52.9946	−1.92105	−0.960526	+	0.278189i	$0.910266\pi$
$762$	0	0
$763$	3.11992	0.112949
$764$	0	0
$765$	−10.3236	−0.373249
$766$	0	0
$767$	7.20464	0.260145
$768$	0	0
$769$	53.9262	1.94463	0.972314	−	0.233677i	$-0.0750759\pi$
$769$	53.9262	1.94463	0.972314	+	0.233677i	$0.0750759\pi$
$770$	0	0
$771$	−2.03603	−0.0733260
$772$	0	0
$773$	28.2936	1.01765	0.508826	−	0.860869i	$-0.330080\pi$
$773$	28.2936	1.01765	0.508826	+	0.860869i	$0.330080\pi$
$774$	0	0
$775$	19.9899	0.718059
$776$	0	0
$777$	14.4440	0.518177
$778$	0	0
$779$	−6.99947	−0.250782
$780$	0	0
$781$	−2.46606	−0.0882425
$782$	0	0
$783$	28.9073	1.03306
$784$	0	0
$785$	7.03904	0.251234
$786$	0	0
$787$	−9.69789	−0.345692	−0.172846	−	0.984949i	$-0.555296\pi$
$787$	−9.69789	−0.345692	−0.172846	+	0.984949i	$0.555296\pi$
$788$	0	0
$789$	−22.6265	−0.805525
$790$	0	0
$791$	4.36583	0.155231
$792$	0	0
$793$	25.8730	0.918777
$794$	0	0
$795$	54.4903	1.93257
$796$	0	0
$797$	−30.7203	−1.08817	−0.544085	−	0.839030i	$-0.683123\pi$
$797$	−30.7203	−1.08817	−0.544085	+	0.839030i	$0.683123\pi$
$798$	0	0
$799$	4.70988	0.166624
$800$	0	0
$801$	6.50326	0.229781
$802$	0	0
$803$	−3.47214	−0.122529
$804$	0	0
$805$	−0.639287	−0.0225319
$806$	0	0
$807$	−50.6401	−1.78262
$808$	0	0
$809$	−26.8979	−0.945679	−0.472840	−	0.881148i	$-0.656771\pi$
$809$	−26.8979	−0.945679	−0.472840	+	0.881148i	$0.656771\pi$
$810$	0	0
$811$	−35.3077	−1.23982	−0.619911	−	0.784672i	$-0.712831\pi$
$811$	−35.3077	−1.23982	−0.619911	+	0.784672i	$0.712831\pi$
$812$	0	0
$813$	2.24099	0.0785951
$814$	0	0
$815$	24.4984	0.858142
$816$	0	0
$817$	−6.21238	−0.217344
$818$	0	0
$819$	3.95088	0.138055
$820$	0	0
$821$	29.2280	1.02006	0.510032	−	0.860155i	$-0.329633\pi$
$821$	29.2280	1.02006	0.510032	+	0.860155i	$0.329633\pi$
$822$	0	0
$823$	−8.43332	−0.293967	−0.146983	−	0.989139i	$-0.546956\pi$
$823$	−8.43332	−0.293967	−0.146983	+	0.989139i	$0.546956\pi$
$824$	0	0
$825$	−3.63255	−0.126469
$826$	0	0
$827$	−40.2571	−1.39988	−0.699938	−	0.714203i	$-0.746789\pi$
$827$	−40.2571	−1.39988	−0.699938	+	0.714203i	$0.746789\pi$
$828$	0	0
$829$	46.5302	1.61606	0.808030	−	0.589141i	$-0.200534\pi$
$829$	46.5302	1.61606	0.808030	+	0.589141i	$0.200534\pi$
$830$	0	0
$831$	27.1035	0.940211
$832$	0	0
$833$	3.01173	0.104350
$834$	0	0
$835$	4.02074	0.139144
$836$	0	0
$837$	−26.7487	−0.924572
$838$	0	0
$839$	41.3731	1.42836	0.714179	−	0.699963i	$-0.246800\pi$
$839$	41.3731	1.42836	0.714179	+	0.699963i	$0.246800\pi$
$840$	0	0
$841$	34.2843	1.18222
$842$	0	0
$843$	27.4698	0.946111
$844$	0	0
$845$	7.63816	0.262761
$846$	0	0
$847$	10.5774	0.363444
$848$	0	0
$849$	−49.1422	−1.68656
$850$	0	0
$851$	1.61556	0.0553806
$852$	0	0
$853$	17.6793	0.605327	0.302664	−	0.953097i	$-0.402124\pi$
$853$	17.6793	0.605327	0.302664	+	0.953097i	$0.402124\pi$
$854$	0	0
$855$	23.9926	0.820531
$856$	0	0
$857$	28.2236	0.964100	0.482050	−	0.876144i	$-0.339892\pi$
$857$	28.2236	0.964100	0.482050	+	0.876144i	$0.339892\pi$
$858$	0	0
$859$	−3.41810	−0.116624	−0.0583120	−	0.998298i	$-0.518572\pi$
$859$	−3.41810	−0.116624	−0.0583120	+	0.998298i	$0.518572\pi$
$860$	0	0
$861$	−2.05768	−0.0701255
$862$	0	0
$863$	−17.1112	−0.582472	−0.291236	−	0.956651i	$-0.594067\pi$
$863$	−17.1112	−0.582472	−0.291236	+	0.956651i	$0.594067\pi$
$864$	0	0
$865$	50.9985	1.73400
$866$	0	0
$867$	−16.3163	−0.554130
$868$	0	0
$869$	2.69650	0.0914725
$870$	0	0
$871$	−5.52320	−0.187147
$872$	0	0
$873$	24.0197	0.812944
$874$	0	0
$875$	−6.34536	−0.214512
$876$	0	0
$877$	−29.0707	−0.981648	−0.490824	−	0.871259i	$-0.663304\pi$
$877$	−29.0707	−0.981648	−0.490824	+	0.871259i	$0.663304\pi$
$878$	0	0
$879$	40.9318	1.38059
$880$	0	0
$881$	−40.5892	−1.36749	−0.683743	−	0.729723i	$-0.739649\pi$
$881$	−40.5892	−1.36749	−0.683743	+	0.729723i	$0.739649\pi$
$882$	0	0
$883$	−20.0747	−0.675567	−0.337784	−	0.941224i	$-0.609677\pi$
$883$	−20.0747	−0.675567	−0.337784	+	0.941224i	$0.609677\pi$
$884$	0	0
$885$	12.8620	0.432352
$886$	0	0
$887$	20.1877	0.677837	0.338918	−	0.940816i	$-0.389939\pi$
$887$	20.1877	0.677837	0.338918	+	0.940816i	$0.389939\pi$
$888$	0	0
$889$	16.0571	0.538539
$890$	0	0
$891$	7.26742	0.243468
$892$	0	0
$893$	−10.9461	−0.366297
$894$	0	0
$895$	−33.9127	−1.13358
$896$	0	0
$897$	1.51619	0.0506241
$898$	0	0
$899$	−58.5588	−1.95305
$900$	0	0
$901$	−28.7127	−0.956557
$902$	0	0
$903$	−1.82629	−0.0607752
$904$	0	0
$905$	43.0281	1.43030
$906$	0	0
$907$	−29.0566	−0.964808	−0.482404	−	0.875949i	$-0.660236\pi$
$907$	−29.0566	−0.964808	−0.482404	+	0.875949i	$0.660236\pi$
$908$	0	0
$909$	9.37008	0.310786
$910$	0	0
$911$	25.7618	0.853526	0.426763	−	0.904364i	$-0.359654\pi$
$911$	25.7618	0.853526	0.426763	+	0.904364i	$0.359654\pi$
$912$	0	0
$913$	2.00606	0.0663908
$914$	0	0
$915$	46.1895	1.52698
$916$	0	0
$917$	−11.8057	−0.389859
$918$	0	0
$919$	−2.01560	−0.0664885	−0.0332442	−	0.999447i	$-0.510584\pi$
$919$	−2.01560	−0.0664885	−0.0332442	+	0.999447i	$0.510584\pi$
$920$	0	0
$921$	−55.4144	−1.82597
$922$	0	0
$923$	−12.1451	−0.399762
$924$	0	0
$925$	−19.0624	−0.626768
$926$	0	0
$927$	−1.22417	−0.0402070
$928$	0	0
$929$	−19.9533	−0.654646	−0.327323	−	0.944913i	$-0.606146\pi$
$929$	−19.9533	−0.654646	−0.327323	+	0.944913i	$0.606146\pi$
$930$	0	0
$931$	−6.99947	−0.229398
$932$	0	0
$933$	−42.2644	−1.38368
$934$	0	0
$935$	5.43837	0.177854
$936$	0	0
$937$	−19.8678	−0.649052	−0.324526	−	0.945877i	$-0.605205\pi$
$937$	−19.8678	−0.649052	−0.324526	+	0.945877i	$0.605205\pi$
$938$	0	0
$939$	21.5613	0.703627
$940$	0	0
$941$	42.0530	1.37089	0.685444	−	0.728126i	$-0.259608\pi$
$941$	42.0530	1.37089	0.685444	+	0.728126i	$0.259608\pi$
$942$	0	0
$943$	−0.230150	−0.00749471
$944$	0	0
$945$	−10.0936	−0.328343
$946$	0	0
$947$	−35.0910	−1.14031	−0.570153	−	0.821539i	$-0.693116\pi$
$947$	−35.0910	−1.14031	−0.570153	+	0.821539i	$0.693116\pi$
$948$	0	0
$949$	−17.1000	−0.555089
$950$	0	0
$951$	56.6601	1.83733
$952$	0	0
$953$	20.5115	0.664432	0.332216	−	0.943203i	$-0.392204\pi$
$953$	20.5115	0.664432	0.332216	+	0.943203i	$0.392204\pi$
$954$	0	0
$955$	−4.98955	−0.161458
$956$	0	0
$957$	10.6413	0.343983
$958$	0	0
$959$	−18.6525	−0.602319
$960$	0	0
$961$	23.1861	0.747940
$962$	0	0
$963$	23.0860	0.743936
$964$	0	0
$965$	36.2111	1.16568
$966$	0	0
$967$	−32.6418	−1.04969	−0.524845	−	0.851198i	$-0.675877\pi$
$967$	−32.6418	−1.04969	−0.524845	+	0.851198i	$0.675877\pi$
$968$	0	0
$969$	−43.3770	−1.39347
$970$	0	0
$971$	−3.56624	−0.114446	−0.0572231	−	0.998361i	$-0.518225\pi$
$971$	−3.56624	−0.114446	−0.0572231	+	0.998361i	$0.518225\pi$
$972$	0	0
$973$	7.70481	0.247005
$974$	0	0
$975$	−17.8900	−0.572938
$976$	0	0
$977$	57.9920	1.85533	0.927665	−	0.373413i	$-0.121813\pi$
$977$	57.9920	1.85533	0.927665	+	0.373413i	$0.121813\pi$
$978$	0	0
$979$	−3.42587	−0.109491
$980$	0	0
$981$	−3.85009	−0.122924
$982$	0	0
$983$	2.96228	0.0944821	0.0472410	−	0.998884i	$-0.484957\pi$
$983$	2.96228	0.0944821	0.0472410	+	0.998884i	$0.484957\pi$
$984$	0	0
$985$	−21.8409	−0.695909
$986$	0	0
$987$	−3.21789	−0.102426
$988$	0	0
$989$	−0.204270	−0.00649539
$990$	0	0
$991$	14.5840	0.463276	0.231638	−	0.972802i	$-0.425592\pi$
$991$	14.5840	0.463276	0.231638	+	0.972802i	$0.425592\pi$
$992$	0	0
$993$	10.2285	0.324593
$994$	0	0
$995$	17.8573	0.566113
$996$	0	0
$997$	−9.52531	−0.301670	−0.150835	−	0.988559i	$-0.548196\pi$
$997$	−9.52531	−0.301670	−0.150835	+	0.988559i	$0.548196\pi$
$998$	0	0
$999$	25.5076	0.807026

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1148.2.a.c.1.5	✓	5
4.3	odd	2		4592.2.a.be.1.1		5
7.6	odd	2		8036.2.a.l.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1148.2.a.c.1.5	✓	5	1.1	even	1	trivial
4592.2.a.be.1.1		5	4.3	odd	2
8036.2.a.l.1.1		5	7.6	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1148 = 2^{2} \cdot 7 \cdot 41 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1148.a (trivial)

\(f(q)\)	\(=\)	\(q+2.05768 q^{3} -2.77770 q^{5} -1.00000 q^{7} +1.23404 q^{9} +O(q^{10})\)	\(q+2.05768 q^{3} -2.77770 q^{5} -1.00000 q^{7} +1.23404 q^{9} -0.650081 q^{11} -3.20159 q^{13} -5.71561 q^{15} +3.01173 q^{17} -6.99947 q^{19} -2.05768 q^{21} -0.230150 q^{23} +2.71561 q^{25} -3.63378 q^{27} -7.95514 q^{29} +7.36112 q^{31} -1.33766 q^{33} +2.77770 q^{35} -7.01958 q^{37} -6.58784 q^{39} +1.00000 q^{41} +0.887550 q^{43} -3.42778 q^{45} +1.56384 q^{47} +1.00000 q^{49} +6.19718 q^{51} -9.53360 q^{53} +1.80573 q^{55} -14.4027 q^{57} -2.25033 q^{59} -8.08129 q^{61} -1.23404 q^{63} +8.89305 q^{65} +1.72514 q^{67} -0.473574 q^{69} +3.79347 q^{71} +5.34109 q^{73} +5.58784 q^{75} +0.650081 q^{77} -4.14795 q^{79} -11.1793 q^{81} -3.08586 q^{83} -8.36569 q^{85} -16.3691 q^{87} +5.26991 q^{89} +3.20159 q^{91} +15.1468 q^{93} +19.4424 q^{95} +19.4644 q^{97} -0.802223 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{3} - q^{5} - 5 q^{7} + q^{9}+O(q^{10}) \) 5 * q - 2 * q^3 - q^5 - 5 * q^7 + q^9	\( 5 q - 2 q^{3} - q^{5} - 5 q^{7} + q^{9} - 2 q^{11} - q^{13} - 9 q^{15} - 3 q^{17} - 4 q^{19} + 2 q^{21} - 8 q^{23} - 6 q^{25} - 8 q^{27} - 9 q^{29} - 11 q^{31} + 5 q^{33} + q^{35} - 11 q^{37} - 17 q^{39} + 5 q^{41} - 27 q^{43} - 3 q^{45} - 3 q^{47} + 5 q^{49} - 3 q^{51} - 19 q^{53} - 13 q^{55} - 11 q^{57} - 15 q^{59} - q^{63} + 7 q^{65} - 21 q^{67} + 14 q^{69} - 16 q^{71} - 10 q^{73} + 12 q^{75} + 2 q^{77} - 14 q^{79} - 7 q^{81} - 2 q^{83} - 21 q^{85} - 36 q^{87} + 6 q^{89} + q^{91} + 17 q^{93} + 9 q^{95} + 20 q^{97} - 17 q^{99}+O(q^{100}) \) 5 * q - 2 * q^3 - q^5 - 5 * q^7 + q^9 - 2 * q^11 - q^13 - 9 * q^15 - 3 * q^17 - 4 * q^19 + 2 * q^21 - 8 * q^23 - 6 * q^25 - 8 * q^27 - 9 * q^29 - 11 * q^31 + 5 * q^33 + q^35 - 11 * q^37 - 17 * q^39 + 5 * q^41 - 27 * q^43 - 3 * q^45 - 3 * q^47 + 5 * q^49 - 3 * q^51 - 19 * q^53 - 13 * q^55 - 11 * q^57 - 15 * q^59 - q^63 + 7 * q^65 - 21 * q^67 + 14 * q^69 - 16 * q^71 - 10 * q^73 + 12 * q^75 + 2 * q^77 - 14 * q^79 - 7 * q^81 - 2 * q^83 - 21 * q^85 - 36 * q^87 + 6 * q^89 + q^91 + 17 * q^93 + 9 * q^95 + 20 * q^97 - 17 * q^99

Introduction

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