LMFDB - Embedded newform 9801.2.a.by.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9801,2,Mod(1,9801)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9801, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9801.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9801 = 3^{4} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9801.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:	$78.2613790211$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.287107358976.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 15x^{6} + 74x^{4} - 147x^{2} + 100 $ x^8 - 15x^6 + 74x^4 - 147*x^2 + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.23950$ of defining polynomial
Character	$\chi$	$=$	9801.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.683225 q^{2} -1.53320 q^{4} +2.18081 q^{5} +3.85674 q^{7} +2.41397 q^{8} +O(q^{10})$	$q-0.683225 q^{2} -1.53320 q^{4} +2.18081 q^{5} +3.85674 q^{7} +2.41397 q^{8} -1.48998 q^{10} +5.99921 q^{13} -2.63502 q^{14} +1.41712 q^{16} +3.16675 q^{17} -0.530894 q^{19} -3.34363 q^{20} -7.02266 q^{23} -0.244069 q^{25} -4.09881 q^{26} -5.91317 q^{28} -7.62725 q^{29} +0.516471 q^{31} -5.79616 q^{32} -2.16361 q^{34} +8.41082 q^{35} -5.72760 q^{37} +0.362720 q^{38} +5.26441 q^{40} +3.61052 q^{41} +10.9656 q^{43} +4.79806 q^{46} +12.8275 q^{47} +7.87447 q^{49} +0.166754 q^{50} -9.19802 q^{52} -8.02009 q^{53} +9.31007 q^{56} +5.21113 q^{58} +10.6467 q^{59} +4.57377 q^{61} -0.352866 q^{62} +1.12584 q^{64} +13.0831 q^{65} -2.38633 q^{67} -4.85528 q^{68} -5.74648 q^{70} -2.94145 q^{71} +11.4664 q^{73} +3.91324 q^{74} +0.813969 q^{76} -0.254390 q^{79} +3.09047 q^{80} -2.46680 q^{82} -6.81919 q^{83} +6.90609 q^{85} -7.49198 q^{86} +15.5570 q^{89} +23.1374 q^{91} +10.7672 q^{92} -8.76407 q^{94} -1.15778 q^{95} -6.45274 q^{97} -5.38003 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{4} + 18 q^{8}+O(q^{10}) $ 8 * q + 12 * q^4 + 18 * q^8	$ 8 q + 12 q^{4} + 18 q^{8} + 52 q^{16} + 12 q^{17} + 28 q^{25} + 12 q^{29} + 12 q^{31} + 66 q^{32} + 38 q^{34} + 24 q^{35} - 8 q^{37} - 12 q^{41} + 2 q^{49} - 12 q^{50} - 4 q^{58} - 66 q^{62} + 70 q^{64} + 24 q^{65} - 18 q^{67} - 36 q^{68} + 50 q^{70} - 78 q^{74} - 44 q^{82} + 6 q^{83} + 36 q^{91} - 18 q^{95} - 2 q^{97} + 42 q^{98}+O(q^{100}) $ 8 * q + 12 * q^4 + 18 * q^8 + 52 * q^16 + 12 * q^17 + 28 * q^25 + 12 * q^29 + 12 * q^31 + 66 * q^32 + 38 * q^34 + 24 * q^35 - 8 * q^37 - 12 * q^41 + 2 * q^49 - 12 * q^50 - 4 * q^58 - 66 * q^62 + 70 * q^64 + 24 * q^65 - 18 * q^67 - 36 * q^68 + 50 * q^70 - 78 * q^74 - 44 * q^82 + 6 * q^83 + 36 * q^91 - 18 * q^95 - 2 * q^97 + 42 * q^98

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 $

$2$	−0.683225	−0.483113	−0.241556	−	0.970387i	$-0.577658\pi$
$2$	−0.683225	−0.483113	−0.241556	+	0.970387i	$0.577658\pi$
$3$	0	0
$3$	0	0
$4$	−1.53320	−0.766602
$5$	2.18081	0.975288	0.487644	−	0.873043i	$-0.337856\pi$
$5$	2.18081	0.975288	0.487644	+	0.873043i	$0.337856\pi$
$6$	0	0
$7$	3.85674	1.45771	0.728856	−	0.684667i	$-0.240052\pi$
$7$	3.85674	1.45771	0.728856	+	0.684667i	$0.240052\pi$
$8$	2.41397	0.853468
$9$	0	0
$10$	−1.48998	−0.471174
$11$	0	0
$11$	0	0
$12$	0	0
$13$	5.99921	1.66388	0.831941	−	0.554864i	$-0.187230\pi$
$13$	5.99921	1.66388	0.831941	+	0.554864i	$0.187230\pi$
$14$	−2.63502	−0.704239
$15$	0	0
$16$	1.41712	0.354280
$17$	3.16675	0.768051	0.384025	−	0.923323i	$-0.374538\pi$
$17$	3.16675	0.768051	0.384025	+	0.923323i	$0.374538\pi$
$18$	0	0
$19$	−0.530894	−0.121795	−0.0608977	−	0.998144i	$-0.519396\pi$
$19$	−0.530894	−0.121795	−0.0608977	+	0.998144i	$0.519396\pi$
$20$	−3.34363	−0.747657
$21$	0	0
$22$	0	0
$23$	−7.02266	−1.46433	−0.732163	−	0.681129i	$-0.761489\pi$
$23$	−7.02266	−1.46433	−0.732163	+	0.681129i	$0.761489\pi$
$24$	0	0
$25$	−0.244069	−0.0488139
$26$	−4.09881	−0.803843
$27$	0	0
$28$	−5.91317	−1.11748
$29$	−7.62725	−1.41635	−0.708173	−	0.706039i	$-0.750480\pi$
$29$	−7.62725	−1.41635	−0.708173	+	0.706039i	$0.750480\pi$
$30$	0	0
$31$	0.516471	0.0927609	0.0463804	−	0.998924i	$-0.485231\pi$
$31$	0.516471	0.0927609	0.0463804	+	0.998924i	$0.485231\pi$
$32$	−5.79616	−1.02463
$33$	0	0
$34$	−2.16361	−0.371055
$35$	8.41082	1.42169
$36$	0	0
$37$	−5.72760	−0.941611	−0.470806	−	0.882237i	$-0.656037\pi$
$37$	−5.72760	−0.941611	−0.470806	+	0.882237i	$0.656037\pi$
$38$	0.362720	0.0588410
$39$	0	0
$40$	5.26441	0.832377
$41$	3.61052	0.563868	0.281934	−	0.959434i	$-0.409024\pi$
$41$	3.61052	0.563868	0.281934	+	0.959434i	$0.409024\pi$
$42$	0	0
$43$	10.9656	1.67224	0.836121	−	0.548546i	$-0.184818\pi$
$43$	10.9656	1.67224	0.836121	+	0.548546i	$0.184818\pi$
$44$	0	0
$45$	0	0
$46$	4.79806	0.707435
$47$	12.8275	1.87108	0.935542	−	0.353216i	$-0.114912\pi$
$47$	12.8275	1.87108	0.935542	+	0.353216i	$0.114912\pi$
$48$	0	0
$49$	7.87447	1.12492
$50$	0.166754	0.0235826
$51$	0	0
$52$	−9.19802	−1.27554
$53$	−8.02009	−1.10164	−0.550822	−	0.834623i	$-0.685686\pi$
$53$	−8.02009	−1.10164	−0.550822	+	0.834623i	$0.685686\pi$
$54$	0	0
$55$	0	0
$56$	9.31007	1.24411
$57$	0	0
$58$	5.21113	0.684255
$59$	10.6467	1.38608	0.693041	−	0.720898i	$-0.256271\pi$
$59$	10.6467	1.38608	0.693041	+	0.720898i	$0.256271\pi$
$60$	0	0
$61$	4.57377	0.585611	0.292805	−	0.956172i	$-0.405411\pi$
$61$	4.57377	0.585611	0.292805	+	0.956172i	$0.405411\pi$
$62$	−0.352866	−0.0448140
$63$	0	0
$64$	1.12584	0.140729
$65$	13.0831	1.62276
$66$	0	0
$67$	−2.38633	−0.291537	−0.145768	−	0.989319i	$-0.546565\pi$
$67$	−2.38633	−0.291537	−0.145768	+	0.989319i	$0.546565\pi$
$68$	−4.85528	−0.588789
$69$	0	0
$70$	−5.74648	−0.686836
$71$	−2.94145	−0.349086	−0.174543	−	0.984650i	$-0.555845\pi$
$71$	−2.94145	−0.349086	−0.174543	+	0.984650i	$0.555845\pi$
$72$	0	0
$73$	11.4664	1.34204	0.671021	−	0.741438i	$-0.265856\pi$
$73$	11.4664	1.34204	0.671021	+	0.741438i	$0.265856\pi$
$74$	3.91324	0.454905
$75$	0	0
$76$	0.813969	0.0933687
$77$	0	0
$78$	0	0
$79$	−0.254390	−0.0286211	−0.0143105	−	0.999898i	$-0.504555\pi$
$79$	−0.254390	−0.0286211	−0.0143105	+	0.999898i	$0.504555\pi$
$80$	3.09047	0.345525
$81$	0	0
$82$	−2.46680	−0.272412
$83$	−6.81919	−0.748503	−0.374252	−	0.927327i	$-0.622100\pi$
$83$	−6.81919	−0.748503	−0.374252	+	0.927327i	$0.622100\pi$
$84$	0	0
$85$	6.90609	0.749070
$86$	−7.49198	−0.807881
$87$	0	0
$88$	0	0
$89$	15.5570	1.64904	0.824519	−	0.565835i	$-0.191446\pi$
$89$	15.5570	1.64904	0.824519	+	0.565835i	$0.191446\pi$
$90$	0	0
$91$	23.1374	2.42546
$92$	10.7672	1.12256
$93$	0	0
$94$	−8.76407	−0.903945
$95$	−1.15778	−0.118786
$96$	0	0
$97$	−6.45274	−0.655176	−0.327588	−	0.944821i	$-0.606236\pi$
$97$	−6.45274	−0.655176	−0.327588	+	0.944821i	$0.606236\pi$
$98$	−5.38003	−0.543465
$99$	0	0
$100$	0.374208	0.0374208
$101$	0.974817	0.0969979	0.0484990	−	0.998823i	$-0.484556\pi$
$101$	0.974817	0.0969979	0.0484990	+	0.998823i	$0.484556\pi$
$102$	0	0
$103$	1.33666	0.131705	0.0658524	−	0.997829i	$-0.479023\pi$
$103$	1.33666	0.131705	0.0658524	+	0.997829i	$0.479023\pi$
$104$	14.4819	1.42007
$105$	0	0
$106$	5.47952	0.532218
$107$	−3.00315	−0.290325	−0.145163	−	0.989408i	$-0.546371\pi$
$107$	−3.00315	−0.290325	−0.145163	+	0.989408i	$0.546371\pi$
$108$	0	0
$109$	−1.34628	−0.128950	−0.0644750	−	0.997919i	$-0.520537\pi$
$109$	−1.34628	−0.128950	−0.0644750	+	0.997919i	$0.520537\pi$
$110$	0	0
$111$	0	0
$112$	5.46547	0.516439
$113$	15.4625	1.45459	0.727296	−	0.686324i	$-0.240777\pi$
$113$	15.4625	1.45459	0.727296	+	0.686324i	$0.240777\pi$
$114$	0	0
$115$	−15.3151	−1.42814
$116$	11.6941	1.08577
$117$	0	0
$118$	−7.27409	−0.669634
$119$	12.2134	1.11960
$120$	0	0
$121$	0	0
$122$	−3.12491	−0.282916
$123$	0	0
$124$	−0.791855	−0.0711107
$125$	−11.4363	−1.02290
$126$	0	0
$127$	5.73937	0.509287	0.254643	−	0.967035i	$-0.418042\pi$
$127$	5.73937	0.509287	0.254643	+	0.967035i	$0.418042\pi$
$128$	10.8231	0.956637
$129$	0	0
$130$	−8.93873	−0.783978
$131$	−10.9916	−0.960336	−0.480168	−	0.877176i	$-0.659424\pi$
$131$	−10.9916	−0.960336	−0.480168	+	0.877176i	$0.659424\pi$
$132$	0	0
$133$	−2.04752	−0.177543
$134$	1.63040	0.140845
$135$	0	0
$136$	7.64446	0.655507
$137$	18.2126	1.55600	0.778002	−	0.628261i	$-0.216233\pi$
$137$	18.2126	1.55600	0.778002	+	0.628261i	$0.216233\pi$
$138$	0	0
$139$	20.6108	1.74819	0.874093	−	0.485759i	$-0.161457\pi$
$139$	20.6108	1.74819	0.874093	+	0.485759i	$0.161457\pi$
$140$	−12.8955	−1.08987
$141$	0	0
$142$	2.00967	0.168648
$143$	0	0
$144$	0	0
$145$	−16.6336	−1.38134
$146$	−7.83414	−0.648358
$147$	0	0
$148$	8.78158	0.721841
$149$	−4.28383	−0.350945	−0.175473	−	0.984484i	$-0.556145\pi$
$149$	−4.28383	−0.350945	−0.175473	+	0.984484i	$0.556145\pi$
$150$	0	0
$151$	−11.1954	−0.911066	−0.455533	−	0.890219i	$-0.650551\pi$
$151$	−11.1954	−0.911066	−0.455533	+	0.890219i	$0.650551\pi$
$152$	−1.28156	−0.103949
$153$	0	0
$154$	0	0
$155$	1.12632	0.0904685
$156$	0	0
$157$	7.31048	0.583440	0.291720	−	0.956504i	$-0.405773\pi$
$157$	7.31048	0.583440	0.291720	+	0.956504i	$0.405773\pi$
$158$	0.173805	0.0138272
$159$	0	0
$160$	−12.6403	−0.999305
$161$	−27.0846	−2.13457
$162$	0	0
$163$	−21.9125	−1.71632	−0.858162	−	0.513380i	$-0.828393\pi$
$163$	−21.9125	−1.71632	−0.858162	+	0.513380i	$0.828393\pi$
$164$	−5.53566	−0.432263
$165$	0	0
$166$	4.65904	0.361612
$167$	−7.22104	−0.558781	−0.279390	−	0.960178i	$-0.590132\pi$
$167$	−7.22104	−0.558781	−0.279390	+	0.960178i	$0.590132\pi$
$168$	0	0
$169$	22.9906	1.76850
$170$	−4.71841	−0.361886
$171$	0	0
$172$	−16.8125	−1.28194
$173$	2.40767	0.183052	0.0915260	−	0.995803i	$-0.470826\pi$
$173$	2.40767	0.183052	0.0915260	+	0.995803i	$0.470826\pi$
$174$	0	0
$175$	−0.941313	−0.0711565
$176$	0	0
$177$	0	0
$178$	−10.6289	−0.796671
$179$	−14.6200	−1.09275	−0.546375	−	0.837540i	$-0.683993\pi$
$179$	−14.6200	−1.09275	−0.546375	+	0.837540i	$0.683993\pi$
$180$	0	0
$181$	0.900651	0.0669449	0.0334724	−	0.999440i	$-0.489343\pi$
$181$	0.900651	0.0669449	0.0334724	+	0.999440i	$0.489343\pi$
$182$	−15.8081	−1.17177
$183$	0	0
$184$	−16.9525	−1.24976
$185$	−12.4908	−0.918342
$186$	0	0
$187$	0	0
$188$	−19.6672	−1.43438
$189$	0	0
$190$	0.791024	0.0573869
$191$	8.05453	0.582805	0.291403	−	0.956601i	$-0.405878\pi$
$191$	8.05453	0.582805	0.291403	+	0.956601i	$0.405878\pi$
$192$	0	0
$193$	1.20116	0.0864611	0.0432306	−	0.999065i	$-0.486235\pi$
$193$	1.20116	0.0864611	0.0432306	+	0.999065i	$0.486235\pi$
$194$	4.40867	0.316524
$195$	0	0
$196$	−12.0732	−0.862369
$197$	−25.9462	−1.84859	−0.924294	−	0.381680i	$-0.875345\pi$
$197$	−25.9462	−1.84859	−0.924294	+	0.381680i	$0.875345\pi$
$198$	0	0
$199$	−6.08775	−0.431549	−0.215775	−	0.976443i	$-0.569228\pi$
$199$	−6.08775	−0.431549	−0.215775	+	0.976443i	$0.569228\pi$
$200$	−0.589177	−0.0416611
$201$	0	0
$202$	−0.666019	−0.0468609
$203$	−29.4164	−2.06462
$204$	0	0
$205$	7.87385	0.549934
$206$	−0.913238	−0.0636283
$207$	0	0
$208$	8.50161	0.589481
$209$	0	0
$210$	0	0
$211$	−8.93130	−0.614856	−0.307428	−	0.951571i	$-0.599468\pi$
$211$	−8.93130	−0.614856	−0.307428	+	0.951571i	$0.599468\pi$
$212$	12.2964	0.844522
$213$	0	0
$214$	2.05183	0.140260
$215$	23.9139	1.63092
$216$	0	0
$217$	1.99189	0.135219
$218$	0.919810	0.0622974
$219$	0	0
$220$	0	0
$221$	18.9980	1.27795
$222$	0	0
$223$	26.5216	1.77602	0.888009	−	0.459826i	$-0.152088\pi$
$223$	26.5216	1.77602	0.888009	+	0.459826i	$0.152088\pi$
$224$	−22.3543	−1.49361
$225$	0	0
$226$	−10.5644	−0.702732
$227$	2.55624	0.169663	0.0848316	−	0.996395i	$-0.472965\pi$
$227$	2.55624	0.169663	0.0848316	+	0.996395i	$0.472965\pi$
$228$	0	0
$229$	22.2984	1.47352	0.736758	−	0.676156i	$-0.236356\pi$
$229$	22.2984	1.47352	0.736758	+	0.676156i	$0.236356\pi$
$230$	10.4636	0.689952
$231$	0	0
$232$	−18.4120	−1.20881
$233$	−18.4651	−1.20969	−0.604845	−	0.796343i	$-0.706765\pi$
$233$	−18.4651	−1.20969	−0.604845	+	0.796343i	$0.706765\pi$
$234$	0	0
$235$	27.9743	1.82485
$236$	−16.3236	−1.06257
$237$	0	0
$238$	−8.34447	−0.540892
$239$	6.57973	0.425607	0.212804	−	0.977095i	$-0.431741\pi$
$239$	6.57973	0.425607	0.212804	+	0.977095i	$0.431741\pi$
$240$	0	0
$241$	−6.95830	−0.448224	−0.224112	−	0.974563i	$-0.571948\pi$
$241$	−6.95830	−0.448224	−0.224112	+	0.974563i	$0.571948\pi$
$242$	0	0
$243$	0	0
$244$	−7.01251	−0.448930
$245$	17.1727	1.09712
$246$	0	0
$247$	−3.18495	−0.202653
$248$	1.24675	0.0791684
$249$	0	0
$250$	7.81358	0.494174
$251$	−0.0914931	−0.00577500	−0.00288750	−	0.999996i	$-0.500919\pi$
$251$	−0.0914931	−0.00577500	−0.00288750	+	0.999996i	$0.500919\pi$
$252$	0	0
$253$	0	0
$254$	−3.92128	−0.246043
$255$	0	0
$256$	−9.64629	−0.602893
$257$	−11.0833	−0.691358	−0.345679	−	0.938353i	$-0.612351\pi$
$257$	−11.0833	−0.691358	−0.345679	+	0.938353i	$0.612351\pi$
$258$	0	0
$259$	−22.0899	−1.37260
$260$	−20.0591	−1.24401
$261$	0	0
$262$	7.50970	0.463951
$263$	12.5347	0.772920	0.386460	−	0.922306i	$-0.373698\pi$
$263$	12.5347	0.772920	0.386460	+	0.922306i	$0.373698\pi$
$264$	0	0
$265$	−17.4903	−1.07442
$266$	1.39892	0.0857732
$267$	0	0
$268$	3.65873	0.223493
$269$	−17.3696	−1.05904	−0.529522	−	0.848296i	$-0.677629\pi$
$269$	−17.3696	−1.05904	−0.529522	+	0.848296i	$0.677629\pi$
$270$	0	0
$271$	19.5849	1.18970	0.594848	−	0.803838i	$-0.297212\pi$
$271$	19.5849	1.18970	0.594848	+	0.803838i	$0.297212\pi$
$272$	4.48768	0.272105
$273$	0	0
$274$	−12.4433	−0.751726
$275$	0	0
$276$	0	0
$277$	−28.2560	−1.69774	−0.848870	−	0.528601i	$-0.822717\pi$
$277$	−28.2560	−1.69774	−0.848870	+	0.528601i	$0.822717\pi$
$278$	−14.0818	−0.844571
$279$	0	0
$280$	20.3035	1.21337
$281$	−7.89750	−0.471125	−0.235563	−	0.971859i	$-0.575693\pi$
$281$	−7.89750	−0.471125	−0.235563	+	0.971859i	$0.575693\pi$
$282$	0	0
$283$	26.3792	1.56808	0.784039	−	0.620712i	$-0.213156\pi$
$283$	26.3792	1.56808	0.784039	+	0.620712i	$0.213156\pi$
$284$	4.50984	0.267610
$285$	0	0
$286$	0	0
$287$	13.9248	0.821958
$288$	0	0
$289$	−6.97167	−0.410098
$290$	11.3645	0.667345
$291$	0	0
$292$	−17.5804	−1.02881
$293$	14.8301	0.866384	0.433192	−	0.901302i	$-0.357387\pi$
$293$	14.8301	0.866384	0.433192	+	0.901302i	$0.357387\pi$
$294$	0	0
$295$	23.2184	1.35183
$296$	−13.8263	−0.803635
$297$	0	0
$298$	2.92682	0.169546
$299$	−42.1304	−2.43647
$300$	0	0
$301$	42.2916	2.43765
$302$	7.64895	0.440148
$303$	0	0
$304$	−0.752342	−0.0431498
$305$	9.97451	0.571139
$306$	0	0
$307$	4.86259	0.277523	0.138761	−	0.990326i	$-0.455688\pi$
$307$	4.86259	0.277523	0.138761	+	0.990326i	$0.455688\pi$
$308$	0	0
$309$	0	0
$310$	−0.769533	−0.0437065
$311$	15.2753	0.866182	0.433091	−	0.901350i	$-0.357423\pi$
$311$	15.2753	0.866182	0.433091	+	0.901350i	$0.357423\pi$
$312$	0	0
$313$	32.7228	1.84960	0.924800	−	0.380454i	$-0.124232\pi$
$313$	32.7228	1.84960	0.924800	+	0.380454i	$0.124232\pi$
$314$	−4.99470	−0.281867
$315$	0	0
$316$	0.390031	0.0219410
$317$	−3.08378	−0.173202	−0.0866012	−	0.996243i	$-0.527601\pi$
$317$	−3.08378	−0.173202	−0.0866012	+	0.996243i	$0.527601\pi$
$318$	0	0
$319$	0	0
$320$	2.45523	0.137252
$321$	0	0
$322$	18.5049	1.03124
$323$	−1.68121	−0.0935451
$324$	0	0
$325$	−1.46422	−0.0812205
$326$	14.9712	0.829178
$327$	0	0
$328$	8.71569	0.481244
$329$	49.4724	2.72750
$330$	0	0
$331$	27.1750	1.49367	0.746836	−	0.665008i	$-0.231572\pi$
$331$	27.1750	1.49367	0.746836	+	0.665008i	$0.231572\pi$
$332$	10.4552	0.573804
$333$	0	0
$334$	4.93359	0.269954
$335$	−5.20414	−0.284332
$336$	0	0
$337$	36.4605	1.98613	0.993066	−	0.117562i	$-0.0375080\pi$
$337$	36.4605	1.98613	0.993066	+	0.117562i	$0.0375080\pi$
$338$	−15.7077	−0.854387
$339$	0	0
$340$	−10.5884	−0.574239
$341$	0	0
$342$	0	0
$343$	3.37261	0.182104
$344$	26.4707	1.42720
$345$	0	0
$346$	−1.64498	−0.0884348
$347$	−18.6220	−0.999679	−0.499839	−	0.866118i	$-0.666608\pi$
$347$	−18.6220	−0.999679	−0.499839	+	0.866118i	$0.666608\pi$
$348$	0	0
$349$	26.6433	1.42618	0.713090	−	0.701072i	$-0.247295\pi$
$349$	26.6433	1.42618	0.713090	+	0.701072i	$0.247295\pi$
$350$	0.643128	0.0343766
$351$	0	0
$352$	0	0
$353$	−18.9338	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$353$	−18.9338	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$354$	0	0
$355$	−6.41474	−0.340459
$356$	−23.8520	−1.26415
$357$	0	0
$358$	9.98875	0.527922
$359$	−15.7915	−0.833446	−0.416723	−	0.909034i	$-0.636822\pi$
$359$	−15.7915	−0.833446	−0.416723	+	0.909034i	$0.636822\pi$
$360$	0	0
$361$	−18.7182	−0.985166
$362$	−0.615347	−0.0323419
$363$	0	0
$364$	−35.4744	−1.85936
$365$	25.0061	1.30888
$366$	0	0
$367$	4.63255	0.241817	0.120909	−	0.992664i	$-0.461419\pi$
$367$	4.63255	0.241817	0.120909	+	0.992664i	$0.461419\pi$
$368$	−9.95196	−0.518782
$369$	0	0
$370$	8.53403	0.443663
$371$	−30.9314	−1.60588
$372$	0	0
$373$	34.6192	1.79251	0.896257	−	0.443535i	$-0.146276\pi$
$373$	34.6192	1.79251	0.896257	+	0.443535i	$0.146276\pi$
$374$	0	0
$375$	0	0
$376$	30.9652	1.59691
$377$	−45.7575	−2.35663
$378$	0	0
$379$	−4.42388	−0.227240	−0.113620	−	0.993524i	$-0.536245\pi$
$379$	−4.42388	−0.227240	−0.113620	+	0.993524i	$0.536245\pi$
$380$	1.77511	0.0910613
$381$	0	0
$382$	−5.50305	−0.281561
$383$	27.5705	1.40879	0.704394	−	0.709809i	$-0.251219\pi$
$383$	27.5705	1.40879	0.704394	+	0.709809i	$0.251219\pi$
$384$	0	0
$385$	0	0
$386$	−0.820660	−0.0417705
$387$	0	0
$388$	9.89337	0.502260
$389$	20.0416	1.01615	0.508074	−	0.861313i	$-0.330358\pi$
$389$	20.0416	1.01615	0.508074	+	0.861313i	$0.330358\pi$
$390$	0	0
$391$	−22.2390	−1.12468
$392$	19.0088	0.960087
$393$	0	0
$394$	17.7271	0.893077
$395$	−0.554775	−0.0279138
$396$	0	0
$397$	−24.4787	−1.22855	−0.614275	−	0.789092i	$-0.710552\pi$
$397$	−24.4787	−1.22855	−0.614275	+	0.789092i	$0.710552\pi$
$398$	4.15930	0.208487
$399$	0	0
$400$	−0.345876	−0.0172938
$401$	−15.4281	−0.770442	−0.385221	−	0.922824i	$-0.625875\pi$
$401$	−15.4281	−0.770442	−0.385221	+	0.922824i	$0.625875\pi$
$402$	0	0
$403$	3.09842	0.154343
$404$	−1.49459	−0.0743588
$405$	0	0
$406$	20.0980	0.997446
$407$	0	0
$408$	0	0
$409$	−8.50534	−0.420562	−0.210281	−	0.977641i	$-0.567438\pi$
$409$	−8.50534	−0.420562	−0.210281	+	0.977641i	$0.567438\pi$
$410$	−5.37961	−0.265680
$411$	0	0
$412$	−2.04937	−0.100965
$413$	41.0616	2.02051
$414$	0	0
$415$	−14.8714	−0.730006
$416$	−34.7724	−1.70486
$417$	0	0
$418$	0	0
$419$	−11.2348	−0.548857	−0.274428	−	0.961608i	$-0.588489\pi$
$419$	−11.2348	−0.548857	−0.274428	+	0.961608i	$0.588489\pi$
$420$	0	0
$421$	−23.6185	−1.15110	−0.575548	−	0.817768i	$-0.695211\pi$
$421$	−23.6185	−1.15110	−0.575548	+	0.817768i	$0.695211\pi$
$422$	6.10209	0.297045
$423$	0	0
$424$	−19.3603	−0.940218
$425$	−0.772907	−0.0374915
$426$	0	0
$427$	17.6398	0.853652
$428$	4.60444	0.222564
$429$	0	0
$430$	−16.3386	−0.787917
$431$	5.55985	0.267808	0.133904	−	0.990994i	$-0.457249\pi$
$431$	5.55985	0.267808	0.133904	+	0.990994i	$0.457249\pi$
$432$	0	0
$433$	21.6137	1.03869	0.519343	−	0.854566i	$-0.326177\pi$
$433$	21.6137	1.03869	0.519343	+	0.854566i	$0.326177\pi$
$434$	−1.36091	−0.0653259
$435$	0	0
$436$	2.06412	0.0988532
$437$	3.72829	0.178348
$438$	0	0
$439$	−7.66239	−0.365706	−0.182853	−	0.983140i	$-0.558533\pi$
$439$	−7.66239	−0.365706	−0.182853	+	0.983140i	$0.558533\pi$
$440$	0	0
$441$	0	0
$442$	−12.9799	−0.617392
$443$	23.9011	1.13558	0.567788	−	0.823175i	$-0.307799\pi$
$443$	23.9011	1.13558	0.567788	+	0.823175i	$0.307799\pi$
$444$	0	0
$445$	33.9268	1.60829
$446$	−18.1202	−0.858017
$447$	0	0
$448$	4.34206	0.205143
$449$	33.7817	1.59426	0.797129	−	0.603810i	$-0.206351\pi$
$449$	33.7817	1.59426	0.797129	+	0.603810i	$0.206351\pi$
$450$	0	0
$451$	0	0
$452$	−23.7072	−1.11509
$453$	0	0
$454$	−1.74648	−0.0819665
$455$	50.4583	2.36552
$456$	0	0
$457$	−9.92967	−0.464491	−0.232245	−	0.972657i	$-0.574607\pi$
$457$	−9.92967	−0.464491	−0.232245	+	0.972657i	$0.574607\pi$
$458$	−15.2348	−0.711875
$459$	0	0
$460$	23.4811	1.09481
$461$	3.76269	0.175246	0.0876230	−	0.996154i	$-0.472073\pi$
$461$	3.76269	0.175246	0.0876230	+	0.996154i	$0.472073\pi$
$462$	0	0
$463$	29.4887	1.37045	0.685227	−	0.728329i	$-0.259703\pi$
$463$	29.4887	1.37045	0.685227	+	0.728329i	$0.259703\pi$
$464$	−10.8087	−0.501783
$465$	0	0
$466$	12.6158	0.584417
$467$	−12.2272	−0.565809	−0.282904	−	0.959148i	$-0.591298\pi$
$467$	−12.2272	−0.565809	−0.282904	+	0.959148i	$0.591298\pi$
$468$	0	0
$469$	−9.20347	−0.424977
$470$	−19.1128	−0.881606
$471$	0	0
$472$	25.7008	1.18298
$473$	0	0
$474$	0	0
$475$	0.129575	0.00594531
$476$	−18.7256	−0.858285
$477$	0	0
$478$	−4.49543	−0.205616
$479$	−31.0255	−1.41759	−0.708795	−	0.705414i	$-0.750761\pi$
$479$	−31.0255	−1.41759	−0.708795	+	0.705414i	$0.750761\pi$
$480$	0	0
$481$	−34.3611	−1.56673
$482$	4.75408	0.216543
$483$	0	0
$484$	0	0
$485$	−14.0722	−0.638986
$486$	0	0
$487$	13.7298	0.622154	0.311077	−	0.950385i	$-0.399310\pi$
$487$	13.7298	0.622154	0.311077	+	0.950385i	$0.399310\pi$
$488$	11.0409	0.499800
$489$	0	0
$490$	−11.7328	−0.530035
$491$	−8.43140	−0.380504	−0.190252	−	0.981735i	$-0.560930\pi$
$491$	−8.43140	−0.380504	−0.190252	+	0.981735i	$0.560930\pi$
$492$	0	0
$493$	−24.1536	−1.08782
$494$	2.17604	0.0979045
$495$	0	0
$496$	0.731902	0.0328634
$497$	−11.3444	−0.508867
$498$	0	0
$499$	−24.7395	−1.10749	−0.553746	−	0.832686i	$-0.686802\pi$
$499$	−24.7395	−1.10749	−0.553746	+	0.832686i	$0.686802\pi$
$500$	17.5342	0.784153
$501$	0	0
$502$	0.0625104	0.00278997
$503$	16.3439	0.728740	0.364370	−	0.931254i	$-0.381284\pi$
$503$	16.3439	0.728740	0.364370	+	0.931254i	$0.381284\pi$
$504$	0	0
$505$	2.12589	0.0946009
$506$	0	0
$507$	0	0
$508$	−8.79962	−0.390420
$509$	−20.4702	−0.907327	−0.453664	−	0.891173i	$-0.649883\pi$
$509$	−20.4702	−0.907327	−0.453664	+	0.891173i	$0.649883\pi$
$510$	0	0
$511$	44.2230	1.95631
$512$	−15.0556	−0.665372
$513$	0	0
$514$	7.57239	0.334004
$515$	2.91500	0.128450
$516$	0	0
$517$	0	0
$518$	15.0924	0.663120
$519$	0	0
$520$	31.5823	1.38498
$521$	−40.1151	−1.75747	−0.878737	−	0.477306i	$-0.841613\pi$
$521$	−40.1151	−1.75747	−0.878737	+	0.477306i	$0.841613\pi$
$522$	0	0
$523$	−17.6107	−0.770064	−0.385032	−	0.922903i	$-0.625810\pi$
$523$	−17.6107	−0.770064	−0.385032	+	0.922903i	$0.625810\pi$
$524$	16.8523	0.736196
$525$	0	0
$526$	−8.56399	−0.373408
$527$	1.63554	0.0712450
$528$	0	0
$529$	26.3178	1.14425
$530$	11.9498	0.519066
$531$	0	0
$532$	3.13927	0.136105
$533$	21.6603	0.938210
$534$	0	0
$535$	−6.54930	−0.283151
$536$	−5.76054	−0.248817
$537$	0	0
$538$	11.8674	0.511638
$539$	0	0
$540$	0	0
$541$	−20.5537	−0.883675	−0.441837	−	0.897095i	$-0.645673\pi$
$541$	−20.5537	−0.883675	−0.441837	+	0.897095i	$0.645673\pi$
$542$	−13.3809	−0.574757
$543$	0	0
$544$	−18.3550	−0.786964
$545$	−2.93597	−0.125763
$546$	0	0
$547$	−16.0069	−0.684407	−0.342204	−	0.939626i	$-0.611173\pi$
$547$	−16.0069	−0.684407	−0.342204	+	0.939626i	$0.611173\pi$
$548$	−27.9236	−1.19284
$549$	0	0
$550$	0	0
$551$	4.04926	0.172504
$552$	0	0
$553$	−0.981115	−0.0417213
$554$	19.3052	0.820200
$555$	0	0
$556$	−31.6006	−1.34016
$557$	−12.2300	−0.518202	−0.259101	−	0.965850i	$-0.583426\pi$
$557$	−12.2300	−0.518202	−0.259101	+	0.965850i	$0.583426\pi$
$558$	0	0
$559$	65.7851	2.78241
$560$	11.9192	0.503676
$561$	0	0
$562$	5.39577	0.227607
$563$	−31.6605	−1.33433	−0.667165	−	0.744910i	$-0.732493\pi$
$563$	−31.6605	−1.33433	−0.667165	+	0.744910i	$0.732493\pi$
$564$	0	0
$565$	33.7208	1.41865
$566$	−18.0229	−0.757559
$567$	0	0
$568$	−7.10058	−0.297934
$569$	−28.6418	−1.20073	−0.600364	−	0.799727i	$-0.704978\pi$
$569$	−28.6418	−1.20073	−0.600364	+	0.799727i	$0.704978\pi$
$570$	0	0
$571$	13.7649	0.576044	0.288022	−	0.957624i	$-0.407002\pi$
$571$	13.7649	0.576044	0.288022	+	0.957624i	$0.407002\pi$
$572$	0	0
$573$	0	0
$574$	−9.51380	−0.397098
$575$	1.71402	0.0714794
$576$	0	0
$577$	12.9997	0.541184	0.270592	−	0.962694i	$-0.412781\pi$
$577$	12.9997	0.541184	0.270592	+	0.962694i	$0.412781\pi$
$578$	4.76322	0.198124
$579$	0	0
$580$	25.5027	1.05894
$581$	−26.2999	−1.09110
$582$	0	0
$583$	0	0
$584$	27.6796	1.14539
$585$	0	0
$586$	−10.1323	−0.418561
$587$	−22.4701	−0.927439	−0.463720	−	0.885982i	$-0.653486\pi$
$587$	−22.4701	−0.927439	−0.463720	+	0.885982i	$0.653486\pi$
$588$	0	0
$589$	−0.274191	−0.0112979
$590$	−15.8634	−0.653086
$591$	0	0
$592$	−8.11670	−0.333594
$593$	−8.07538	−0.331616	−0.165808	−	0.986158i	$-0.553023\pi$
$593$	−8.07538	−0.331616	−0.165808	+	0.986158i	$0.553023\pi$
$594$	0	0
$595$	26.6350	1.09193
$596$	6.56799	0.269035
$597$	0	0
$598$	28.7846	1.17709
$599$	34.7973	1.42178	0.710889	−	0.703304i	$-0.248293\pi$
$599$	34.7973	1.42178	0.710889	+	0.703304i	$0.248293\pi$
$600$	0	0
$601$	−6.58485	−0.268602	−0.134301	−	0.990941i	$-0.542879\pi$
$601$	−6.58485	−0.268602	−0.134301	+	0.990941i	$0.542879\pi$
$602$	−28.8947	−1.17766
$603$	0	0
$604$	17.1648	0.698425
$605$	0	0
$606$	0	0
$607$	1.62390	0.0659119	0.0329560	−	0.999457i	$-0.489508\pi$
$607$	1.62390	0.0659119	0.0329560	+	0.999457i	$0.489508\pi$
$608$	3.07715	0.124795
$609$	0	0
$610$	−6.81483	−0.275925
$611$	76.9549	3.11326
$612$	0	0
$613$	−9.80435	−0.395994	−0.197997	−	0.980203i	$-0.563444\pi$
$613$	−9.80435	−0.395994	−0.197997	+	0.980203i	$0.563444\pi$
$614$	−3.32224	−0.134075
$615$	0	0
$616$	0	0
$617$	−25.7948	−1.03846	−0.519230	−	0.854635i	$-0.673781\pi$
$617$	−25.7948	−1.03846	−0.519230	+	0.854635i	$0.673781\pi$
$618$	0	0
$619$	27.9268	1.12247	0.561237	−	0.827655i	$-0.310326\pi$
$619$	27.9268	1.12247	0.561237	+	0.827655i	$0.310326\pi$
$620$	−1.72688	−0.0693534
$621$	0	0
$622$	−10.4365	−0.418464
$623$	59.9993	2.40382
$624$	0	0
$625$	−23.7201	−0.948803
$626$	−22.3570	−0.893566
$627$	0	0
$628$	−11.2085	−0.447266
$629$	−18.1379	−0.723205
$630$	0	0
$631$	−26.0594	−1.03741	−0.518705	−	0.854954i	$-0.673586\pi$
$631$	−26.0594	−1.03741	−0.518705	+	0.854954i	$0.673586\pi$
$632$	−0.614089	−0.0244272
$633$	0	0
$634$	2.10692	0.0836763
$635$	12.5165	0.496701
$636$	0	0
$637$	47.2406	1.87174
$638$	0	0
$639$	0	0
$640$	23.6032	0.932997
$641$	−22.9474	−0.906367	−0.453184	−	0.891417i	$-0.649712\pi$
$641$	−22.9474	−0.906367	−0.453184	+	0.891417i	$0.649712\pi$
$642$	0	0
$643$	−31.2843	−1.23373	−0.616866	−	0.787068i	$-0.711598\pi$
$643$	−31.2843	−1.23373	−0.616866	+	0.787068i	$0.711598\pi$
$644$	41.5262	1.63636
$645$	0	0
$646$	1.14865	0.0451929
$647$	−14.9853	−0.589131	−0.294566	−	0.955631i	$-0.595175\pi$
$647$	−14.9853	−0.589131	−0.294566	+	0.955631i	$0.595175\pi$
$648$	0	0
$649$	0	0
$650$	1.00039	0.0392387
$651$	0	0
$652$	33.5964	1.31574
$653$	−4.90689	−0.192021	−0.0960107	−	0.995380i	$-0.530608\pi$
$653$	−4.90689	−0.192021	−0.0960107	+	0.995380i	$0.530608\pi$
$654$	0	0
$655$	−23.9705	−0.936604
$656$	5.11654	0.199768
$657$	0	0
$658$	−33.8008	−1.31769
$659$	3.66895	0.142922	0.0714610	−	0.997443i	$-0.477234\pi$
$659$	3.66895	0.142922	0.0714610	+	0.997443i	$0.477234\pi$
$660$	0	0
$661$	22.1469	0.861413	0.430707	−	0.902492i	$-0.358264\pi$
$661$	22.1469	0.861413	0.430707	+	0.902492i	$0.358264\pi$
$662$	−18.5666	−0.721612
$663$	0	0
$664$	−16.4613	−0.638824
$665$	−4.46526	−0.173155
$666$	0	0
$667$	53.5636	2.07399
$668$	11.0713	0.428362
$669$	0	0
$670$	3.55559	0.137365
$671$	0	0
$672$	0	0
$673$	22.0420	0.849657	0.424828	−	0.905274i	$-0.360334\pi$
$673$	22.0420	0.849657	0.424828	+	0.905274i	$0.360334\pi$
$674$	−24.9107	−0.959526
$675$	0	0
$676$	−35.2492	−1.35574
$677$	28.7363	1.10443	0.552214	−	0.833703i	$-0.313783\pi$
$677$	28.7363	1.10443	0.552214	+	0.833703i	$0.313783\pi$
$678$	0	0
$679$	−24.8866	−0.955059
$680$	16.6711	0.639308
$681$	0	0
$682$	0	0
$683$	46.4334	1.77672	0.888362	−	0.459143i	$-0.151843\pi$
$683$	46.4334	1.77672	0.888362	+	0.459143i	$0.151843\pi$
$684$	0	0
$685$	39.7182	1.51755
$686$	−2.30425	−0.0879767
$687$	0	0
$688$	15.5396	0.592442
$689$	−48.1142	−1.83301
$690$	0	0
$691$	−7.83156	−0.297926	−0.148963	−	0.988843i	$-0.547594\pi$
$691$	−7.83156	−0.297926	−0.148963	+	0.988843i	$0.547594\pi$
$692$	−3.69146	−0.140328
$693$	0	0
$694$	12.7230	0.482958
$695$	44.9482	1.70498
$696$	0	0
$697$	11.4336	0.433079
$698$	−18.2033	−0.689006
$699$	0	0
$700$	1.44322	0.0545487
$701$	27.6808	1.04549	0.522746	−	0.852489i	$-0.324908\pi$
$701$	27.6808	1.04549	0.522746	+	0.852489i	$0.324908\pi$
$702$	0	0
$703$	3.04075	0.114684
$704$	0	0
$705$	0	0
$706$	12.9360	0.486853
$707$	3.75962	0.141395
$708$	0	0
$709$	−30.4153	−1.14227	−0.571135	−	0.820856i	$-0.693497\pi$
$709$	−30.4153	−1.14227	−0.571135	+	0.820856i	$0.693497\pi$
$710$	4.38271	0.164480
$711$	0	0
$712$	37.5541	1.40740
$713$	−3.62700	−0.135832
$714$	0	0
$715$	0	0
$716$	22.4155	0.837705
$717$	0	0
$718$	10.7892	0.402649
$719$	−37.3869	−1.39430	−0.697148	−	0.716927i	$-0.745548\pi$
$719$	−37.3869	−1.39430	−0.697148	+	0.716927i	$0.745548\pi$
$720$	0	0
$721$	5.15515	0.191988
$722$	12.7887	0.475946
$723$	0	0
$724$	−1.38088	−0.0513201
$725$	1.86158	0.0691373
$726$	0	0
$727$	16.3288	0.605603	0.302802	−	0.953054i	$-0.402078\pi$
$727$	16.3288	0.605603	0.302802	+	0.953054i	$0.402078\pi$
$728$	55.8531	2.07005
$729$	0	0
$730$	−17.0848	−0.632336
$731$	34.7254	1.28437
$732$	0	0
$733$	−22.5505	−0.832921	−0.416461	−	0.909154i	$-0.636730\pi$
$733$	−22.5505	−0.832921	−0.416461	+	0.909154i	$0.636730\pi$
$734$	−3.16508	−0.116825
$735$	0	0
$736$	40.7044	1.50039
$737$	0	0
$738$	0	0
$739$	−13.7185	−0.504641	−0.252321	−	0.967644i	$-0.581194\pi$
$739$	−13.7185	−0.504641	−0.252321	+	0.967644i	$0.581194\pi$
$740$	19.1509	0.704003
$741$	0	0
$742$	21.1331	0.775821
$743$	−41.0233	−1.50500	−0.752499	−	0.658594i	$-0.771152\pi$
$743$	−41.0233	−1.50500	−0.752499	+	0.658594i	$0.771152\pi$
$744$	0	0
$745$	−9.34223	−0.342273
$746$	−23.6527	−0.865987
$747$	0	0
$748$	0	0
$749$	−11.5824	−0.423211
$750$	0	0
$751$	37.9309	1.38412	0.692059	−	0.721841i	$-0.256704\pi$
$751$	37.9309	1.38412	0.692059	+	0.721841i	$0.256704\pi$
$752$	18.1781	0.662888
$753$	0	0
$754$	31.2627	1.13852
$755$	−24.4150	−0.888551
$756$	0	0
$757$	−22.7700	−0.827589	−0.413794	−	0.910370i	$-0.635797\pi$
$757$	−22.7700	−0.827589	−0.413794	+	0.910370i	$0.635797\pi$
$758$	3.02251	0.109782
$759$	0	0
$760$	−2.79485	−0.101380
$761$	−4.64031	−0.168211	−0.0841056	−	0.996457i	$-0.526803\pi$
$761$	−4.64031	−0.168211	−0.0841056	+	0.996457i	$0.526803\pi$
$762$	0	0
$763$	−5.19224	−0.187972
$764$	−12.3492	−0.446780
$765$	0	0
$766$	−18.8369	−0.680604
$767$	63.8718	2.30628
$768$	0	0
$769$	2.74051	0.0988253	0.0494127	−	0.998778i	$-0.484265\pi$
$769$	2.74051	0.0988253	0.0494127	+	0.998778i	$0.484265\pi$
$770$	0	0
$771$	0	0
$772$	−1.84162	−0.0662813
$773$	11.0421	0.397158	0.198579	−	0.980085i	$-0.436367\pi$
$773$	11.0421	0.397158	0.198579	+	0.980085i	$0.436367\pi$
$774$	0	0
$775$	−0.126055	−0.00452802
$776$	−15.5767	−0.559172
$777$	0	0
$778$	−13.6929	−0.490914
$779$	−1.91680	−0.0686766
$780$	0	0
$781$	0	0
$782$	15.1943	0.543346
$783$	0	0
$784$	11.1591	0.398539
$785$	15.9428	0.569021
$786$	0	0
$787$	29.3886	1.04759	0.523795	−	0.851844i	$-0.324516\pi$
$787$	29.3886	1.04759	0.523795	+	0.851844i	$0.324516\pi$
$788$	39.7808	1.41713
$789$	0	0
$790$	0.379036	0.0134855
$791$	59.6350	2.12038
$792$	0	0
$793$	27.4390	0.974387
$794$	16.7244	0.593529
$795$	0	0
$796$	9.33376	0.330826
$797$	−1.11623	−0.0395390	−0.0197695	−	0.999805i	$-0.506293\pi$
$797$	−1.11623	−0.0395390	−0.0197695	+	0.999805i	$0.506293\pi$
$798$	0	0
$799$	40.6216	1.43709
$800$	1.41466	0.0500159
$801$	0	0
$802$	10.5409	0.372210
$803$	0	0
$804$	0	0
$805$	−59.0664	−2.08182
$806$	−2.11692	−0.0745652
$807$	0	0
$808$	2.35318	0.0827846
$809$	−43.3724	−1.52489	−0.762445	−	0.647053i	$-0.776001\pi$
$809$	−43.3724	−1.52489	−0.762445	+	0.647053i	$0.776001\pi$
$810$	0	0
$811$	−33.0062	−1.15900	−0.579502	−	0.814971i	$-0.696753\pi$
$811$	−33.0062	−1.15900	−0.579502	+	0.814971i	$0.696753\pi$
$812$	45.1013	1.58274
$813$	0	0
$814$	0	0
$815$	−47.7871	−1.67391
$816$	0	0
$817$	−5.82158	−0.203671
$818$	5.81106	0.203179
$819$	0	0
$820$	−12.0722	−0.421580
$821$	27.0131	0.942762	0.471381	−	0.881930i	$-0.343756\pi$
$821$	27.0131	0.942762	0.471381	+	0.881930i	$0.343756\pi$
$822$	0	0
$823$	2.23678	0.0779694	0.0389847	−	0.999240i	$-0.487588\pi$
$823$	2.23678	0.0779694	0.0389847	+	0.999240i	$0.487588\pi$
$824$	3.22665	0.112406
$825$	0	0
$826$	−28.0543	−0.976133
$827$	24.0696	0.836980	0.418490	−	0.908221i	$-0.362559\pi$
$827$	24.0696	0.836980	0.418490	+	0.908221i	$0.362559\pi$
$828$	0	0
$829$	−7.42634	−0.257927	−0.128964	−	0.991649i	$-0.541165\pi$
$829$	−7.42634	−0.257927	−0.128964	+	0.991649i	$0.541165\pi$
$830$	10.1605	0.352675
$831$	0	0
$832$	6.75413	0.234157
$833$	24.9365	0.863999
$834$	0	0
$835$	−15.7477	−0.544972
$836$	0	0
$837$	0	0
$838$	7.67591	0.265160
$839$	11.3704	0.392550	0.196275	−	0.980549i	$-0.437115\pi$
$839$	11.3704	0.392550	0.196275	+	0.980549i	$0.437115\pi$
$840$	0	0
$841$	29.1750	1.00603
$842$	16.1367	0.556109
$843$	0	0
$844$	13.6935	0.471350
$845$	50.1380	1.72480
$846$	0	0
$847$	0	0
$848$	−11.3654	−0.390291
$849$	0	0
$850$	0.528070	0.0181126
$851$	40.2230	1.37883
$852$	0	0
$853$	3.35595	0.114905	0.0574527	−	0.998348i	$-0.481702\pi$
$853$	3.35595	0.114905	0.0574527	+	0.998348i	$0.481702\pi$
$854$	−12.0520	−0.412410
$855$	0	0
$856$	−7.24952	−0.247783
$857$	18.5664	0.634215	0.317107	−	0.948390i	$-0.397288\pi$
$857$	18.5664	0.634215	0.317107	+	0.948390i	$0.397288\pi$
$858$	0	0
$859$	14.1518	0.482853	0.241426	−	0.970419i	$-0.422385\pi$
$859$	14.1518	0.482853	0.241426	+	0.970419i	$0.422385\pi$
$860$	−36.6649	−1.25026
$861$	0	0
$862$	−3.79863	−0.129382
$863$	52.4715	1.78615	0.893076	−	0.449906i	$-0.148542\pi$
$863$	52.4715	1.78615	0.893076	+	0.449906i	$0.148542\pi$
$864$	0	0
$865$	5.25068	0.178528
$866$	−14.7670	−0.501803
$867$	0	0
$868$	−3.05398	−0.103659
$869$	0	0
$870$	0	0
$871$	−14.3161	−0.485083
$872$	−3.24987	−0.110055
$873$	0	0
$874$	−2.54726	−0.0861624
$875$	−44.1069	−1.49109
$876$	0	0
$877$	31.8136	1.07427	0.537135	−	0.843496i	$-0.319506\pi$
$877$	31.8136	1.07427	0.537135	+	0.843496i	$0.319506\pi$
$878$	5.23513	0.176677
$879$	0	0
$880$	0	0
$881$	−29.3806	−0.989858	−0.494929	−	0.868933i	$-0.664806\pi$
$881$	−29.3806	−0.989858	−0.494929	+	0.868933i	$0.664806\pi$
$882$	0	0
$883$	−0.606382	−0.0204064	−0.0102032	−	0.999948i	$-0.503248\pi$
$883$	−0.606382	−0.0204064	−0.0102032	+	0.999948i	$0.503248\pi$
$884$	−29.1279	−0.979676
$885$	0	0
$886$	−16.3298	−0.548611
$887$	7.56882	0.254136	0.127068	−	0.991894i	$-0.459443\pi$
$887$	7.56882	0.254136	0.127068	+	0.991894i	$0.459443\pi$
$888$	0	0
$889$	22.1353	0.742393
$890$	−23.1796	−0.776984
$891$	0	0
$892$	−40.6630	−1.36150
$893$	−6.81005	−0.227890
$894$	0	0
$895$	−31.8835	−1.06575
$896$	41.7420	1.39450
$897$	0	0
$898$	−23.0805	−0.770206
$899$	−3.93925	−0.131381
$900$	0	0
$901$	−25.3977	−0.846119
$902$	0	0
$903$	0	0
$904$	37.3261	1.24145
$905$	1.96415	0.0652905
$906$	0	0
$907$	−30.7730	−1.02180	−0.510900	−	0.859640i	$-0.670688\pi$
$907$	−30.7730	−1.02180	−0.510900	+	0.859640i	$0.670688\pi$
$908$	−3.91923	−0.130064
$909$	0	0
$910$	−34.4744	−1.14281
$911$	−11.6236	−0.385106	−0.192553	−	0.981287i	$-0.561677\pi$
$911$	−11.6236	−0.385106	−0.192553	+	0.981287i	$0.561677\pi$
$912$	0	0
$913$	0	0
$914$	6.78420	0.224401
$915$	0	0
$916$	−34.1879	−1.12960
$917$	−42.3916	−1.39989
$918$	0	0
$919$	−32.0118	−1.05597	−0.527986	−	0.849253i	$-0.677053\pi$
$919$	−32.0118	−1.05597	−0.527986	+	0.849253i	$0.677053\pi$
$920$	−36.9702	−1.21887
$921$	0	0
$922$	−2.57076	−0.0846636
$923$	−17.6464	−0.580838
$924$	0	0
$925$	1.39793	0.0459637
$926$	−20.1474	−0.662084
$927$	0	0
$928$	44.2088	1.45122
$929$	−16.0439	−0.526384	−0.263192	−	0.964744i	$-0.584775\pi$
$929$	−16.0439	−0.526384	−0.263192	+	0.964744i	$0.584775\pi$
$930$	0	0
$931$	−4.18051	−0.137011
$932$	28.3108	0.927350
$933$	0	0
$934$	8.35395	0.273350
$935$	0	0
$936$	0	0
$937$	28.7829	0.940296	0.470148	−	0.882588i	$-0.344201\pi$
$937$	28.7829	0.940296	0.470148	+	0.882588i	$0.344201\pi$
$938$	6.28804	0.205312
$939$	0	0
$940$	−42.8904	−1.39893
$941$	−32.0887	−1.04606	−0.523032	−	0.852313i	$-0.675199\pi$
$941$	−32.0887	−1.04606	−0.523032	+	0.852313i	$0.675199\pi$
$942$	0	0
$943$	−25.3554	−0.825687
$944$	15.0877	0.491062
$945$	0	0
$946$	0	0
$947$	31.8805	1.03598	0.517989	−	0.855387i	$-0.326681\pi$
$947$	31.8805	1.03598	0.517989	+	0.855387i	$0.326681\pi$
$948$	0	0
$949$	68.7895	2.23300
$950$	−0.0885288	−0.00287225
$951$	0	0
$952$	29.4827	0.955540
$953$	33.5509	1.08682	0.543410	−	0.839468i	$-0.317133\pi$
$953$	33.5509	1.08682	0.543410	+	0.839468i	$0.317133\pi$
$954$	0	0
$955$	17.5654	0.568403
$956$	−10.0881	−0.326271
$957$	0	0
$958$	21.1974	0.684856
$959$	70.2412	2.26821
$960$	0	0
$961$	−30.7333	−0.991395
$962$	23.4763	0.756908
$963$	0	0
$964$	10.6685	0.343609
$965$	2.61949	0.0843245
$966$	0	0
$967$	−17.3428	−0.557708	−0.278854	−	0.960333i	$-0.589955\pi$
$967$	−17.3428	−0.557708	−0.278854	+	0.960333i	$0.589955\pi$
$968$	0	0
$969$	0	0
$970$	9.61447	0.308702
$971$	42.3522	1.35915	0.679573	−	0.733608i	$-0.262165\pi$
$971$	42.3522	1.35915	0.679573	+	0.733608i	$0.262165\pi$
$972$	0	0
$973$	79.4906	2.54835
$974$	−9.38051	−0.300571
$975$	0	0
$976$	6.48158	0.207470
$977$	0.820225	0.0262413	0.0131207	−	0.999914i	$-0.495823\pi$
$977$	0.820225	0.0262413	0.0131207	+	0.999914i	$0.495823\pi$
$978$	0	0
$979$	0	0
$980$	−26.3293	−0.841058
$981$	0	0
$982$	5.76054	0.183826
$983$	31.2508	0.996747	0.498374	−	0.866962i	$-0.333931\pi$
$983$	31.2508	0.996747	0.498374	+	0.866962i	$0.333931\pi$
$984$	0	0
$985$	−56.5837	−1.80291
$986$	16.5024	0.525542
$987$	0	0
$988$	4.88317	0.155354
$989$	−77.0078	−2.44871
$990$	0	0
$991$	4.89850	0.155606	0.0778030	−	0.996969i	$-0.475209\pi$
$991$	4.89850	0.155606	0.0778030	+	0.996969i	$0.475209\pi$
$992$	−2.99355	−0.0950452
$993$	0	0
$994$	7.75079	0.245840
$995$	−13.2762	−0.420885
$996$	0	0
$997$	21.6834	0.686721	0.343360	−	0.939204i	$-0.388435\pi$
$997$	21.6834	0.686721	0.343360	+	0.939204i	$0.388435\pi$
$998$	16.9026	0.535044
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9801.2.a.by.1.4	yes	8
3.2	odd	2		9801.2.a.bx.1.5	✓	8
11.10	odd	2		9801.2.a.bx.1.6	yes	8
33.32	even	2	inner	9801.2.a.by.1.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9801.2.a.bx.1.5	✓	8	3.2	odd	2
9801.2.a.bx.1.6	yes	8	11.10	odd	2
9801.2.a.by.1.3	yes	8	33.32	even	2	inner
9801.2.a.by.1.4	yes	8	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:	\( N \)	\(=\)	\( 9801 = 3^{4} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9801.a (trivial)

\(f(q)\)	\(=\)	\(q-0.683225 q^{2} -1.53320 q^{4} +2.18081 q^{5} +3.85674 q^{7} +2.41397 q^{8} +O(q^{10})\)	\(q-0.683225 q^{2} -1.53320 q^{4} +2.18081 q^{5} +3.85674 q^{7} +2.41397 q^{8} -1.48998 q^{10} +5.99921 q^{13} -2.63502 q^{14} +1.41712 q^{16} +3.16675 q^{17} -0.530894 q^{19} -3.34363 q^{20} -7.02266 q^{23} -0.244069 q^{25} -4.09881 q^{26} -5.91317 q^{28} -7.62725 q^{29} +0.516471 q^{31} -5.79616 q^{32} -2.16361 q^{34} +8.41082 q^{35} -5.72760 q^{37} +0.362720 q^{38} +5.26441 q^{40} +3.61052 q^{41} +10.9656 q^{43} +4.79806 q^{46} +12.8275 q^{47} +7.87447 q^{49} +0.166754 q^{50} -9.19802 q^{52} -8.02009 q^{53} +9.31007 q^{56} +5.21113 q^{58} +10.6467 q^{59} +4.57377 q^{61} -0.352866 q^{62} +1.12584 q^{64} +13.0831 q^{65} -2.38633 q^{67} -4.85528 q^{68} -5.74648 q^{70} -2.94145 q^{71} +11.4664 q^{73} +3.91324 q^{74} +0.813969 q^{76} -0.254390 q^{79} +3.09047 q^{80} -2.46680 q^{82} -6.81919 q^{83} +6.90609 q^{85} -7.49198 q^{86} +15.5570 q^{89} +23.1374 q^{91} +10.7672 q^{92} -8.76407 q^{94} -1.15778 q^{95} -6.45274 q^{97} -5.38003 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{4} + 18 q^{8}+O(q^{10}) \) 8 * q + 12 * q^4 + 18 * q^8	\( 8 q + 12 q^{4} + 18 q^{8} + 52 q^{16} + 12 q^{17} + 28 q^{25} + 12 q^{29} + 12 q^{31} + 66 q^{32} + 38 q^{34} + 24 q^{35} - 8 q^{37} - 12 q^{41} + 2 q^{49} - 12 q^{50} - 4 q^{58} - 66 q^{62} + 70 q^{64} + 24 q^{65} - 18 q^{67} - 36 q^{68} + 50 q^{70} - 78 q^{74} - 44 q^{82} + 6 q^{83} + 36 q^{91} - 18 q^{95} - 2 q^{97} + 42 q^{98}+O(q^{100}) \) 8 * q + 12 * q^4 + 18 * q^8 + 52 * q^16 + 12 * q^17 + 28 * q^25 + 12 * q^29 + 12 * q^31 + 66 * q^32 + 38 * q^34 + 24 * q^35 - 8 * q^37 - 12 * q^41 + 2 * q^49 - 12 * q^50 - 4 * q^58 - 66 * q^62 + 70 * q^64 + 24 * q^65 - 18 * q^67 - 36 * q^68 + 50 * q^70 - 78 * q^74 - 44 * q^82 + 6 * q^83 + 36 * q^91 - 18 * q^95 - 2 * q^97 + 42 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more