LMFDB - Embedded newform 1288.2.a.n.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1288,2,Mod(1,1288)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1288, 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("1288.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1288 = 2^{3} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1288.a (trivial)

Newform invariants

comment: select newform

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

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

Self dual:	yes
Analytic conductor:	$10.2847317803$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.8468.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 3x + 4 $ x^4 - x^3 - 5x^2 + 3x + 4
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			1.3
Root			$1.31743$ of defining polynomial
Character	$\chi$	$=$	1288.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.317431 q^{3} -2.71878 q^{5} +1.00000 q^{7} -2.89924 q^{9} +O(q^{10})$	$q+0.317431 q^{3} -2.71878 q^{5} +1.00000 q^{7} -2.89924 q^{9} +2.30059 q^{11} +2.93011 q^{13} -0.863025 q^{15} -2.44483 q^{17} +0.802704 q^{19} +0.317431 q^{21} -1.00000 q^{23} +2.39178 q^{25} -1.87260 q^{27} -8.42799 q^{29} -0.211323 q^{31} +0.730278 q^{33} -2.71878 q^{35} -0.106108 q^{37} +0.930105 q^{39} -10.8964 q^{41} -12.7690 q^{43} +7.88240 q^{45} -2.31743 q^{47} +1.00000 q^{49} -0.776064 q^{51} -0.730278 q^{53} -6.25480 q^{55} +0.254803 q^{57} +1.24754 q^{59} -0.612674 q^{61} -2.89924 q^{63} -7.96632 q^{65} -8.02664 q^{67} -0.317431 q^{69} -5.76649 q^{71} +9.67107 q^{73} +0.759224 q^{75} +2.30059 q^{77} -2.30059 q^{79} +8.10329 q^{81} -15.7115 q^{83} +6.64697 q^{85} -2.67530 q^{87} -5.98850 q^{89} +2.93011 q^{91} -0.0670803 q^{93} -2.18238 q^{95} +7.52149 q^{97} -6.66996 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{3} + 4 q^{7} + q^{9}+O(q^{10}) $ 4 * q - 3 * q^3 + 4 * q^7 + q^9	$ 4 q - 3 q^{3} + 4 q^{7} + q^{9} - 10 q^{11} - 3 q^{13} - 6 q^{15} - 4 q^{17} - 10 q^{19} - 3 q^{21} - 4 q^{23} + 4 q^{25} - 9 q^{27} - 13 q^{29} + 3 q^{31} + 20 q^{33} - 11 q^{39} + q^{41} - 8 q^{43} + 4 q^{45} - 5 q^{47} + 4 q^{49} - 4 q^{51} - 20 q^{53} - 22 q^{55} - 2 q^{57} - 14 q^{59} + 8 q^{61} + q^{63} - 2 q^{65} - 18 q^{67} + 3 q^{69} - 25 q^{71} + 15 q^{73} - 11 q^{75} - 10 q^{77} + 10 q^{79} - 36 q^{83} - 28 q^{85} + q^{87} + 4 q^{89} - 3 q^{91} + 17 q^{93} - 36 q^{95} + 6 q^{97} - 28 q^{99}+O(q^{100}) $ 4 * q - 3 * q^3 + 4 * q^7 + q^9 - 10 * q^11 - 3 * q^13 - 6 * q^15 - 4 * q^17 - 10 * q^19 - 3 * q^21 - 4 * q^23 + 4 * q^25 - 9 * q^27 - 13 * q^29 + 3 * q^31 + 20 * q^33 - 11 * q^39 + q^41 - 8 * q^43 + 4 * q^45 - 5 * q^47 + 4 * q^49 - 4 * q^51 - 20 * q^53 - 22 * q^55 - 2 * q^57 - 14 * q^59 + 8 * q^61 + q^63 - 2 * q^65 - 18 * q^67 + 3 * q^69 - 25 * q^71 + 15 * q^73 - 11 * q^75 - 10 * q^77 + 10 * q^79 - 36 * q^83 - 28 * q^85 + q^87 + 4 * q^89 - 3 * q^91 + 17 * q^93 - 36 * q^95 + 6 * q^97 - 28 * 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
$3$	0.317431	0.183269	0.0916343	−	0.995793i	$-0.470791\pi$
$3$	0.317431	0.183269	0.0916343	+	0.995793i	$0.470791\pi$
$4$	0	0
$5$	−2.71878	−1.21588	−0.607938	−	0.793984i	$-0.708003\pi$
$5$	−2.71878	−1.21588	−0.607938	+	0.793984i	$0.708003\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−2.89924	−0.966413
$10$	0	0
$11$	2.30059	0.693654	0.346827	−	0.937929i	$-0.387259\pi$
$11$	2.30059	0.693654	0.346827	+	0.937929i	$0.387259\pi$
$12$	0	0
$13$	2.93011	0.812665	0.406332	−	0.913725i	$-0.366807\pi$
$13$	2.93011	0.812665	0.406332	+	0.913725i	$0.366807\pi$
$14$	0	0
$15$	−0.863025	−0.222832
$16$	0	0
$17$	−2.44483	−0.592959	−0.296479	−	0.955039i	$-0.595813\pi$
$17$	−2.44483	−0.592959	−0.296479	+	0.955039i	$0.595813\pi$
$18$	0	0
$19$	0.802704	0.184153	0.0920764	−	0.995752i	$-0.470650\pi$
$19$	0.802704	0.184153	0.0920764	+	0.995752i	$0.470650\pi$
$20$	0	0
$21$	0.317431	0.0692690
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	2.39178	0.478356
$26$	0	0
$27$	−1.87260	−0.360382
$28$	0	0
$29$	−8.42799	−1.56504	−0.782519	−	0.622626i	$-0.786066\pi$
$29$	−8.42799	−1.56504	−0.782519	+	0.622626i	$0.786066\pi$
$30$	0	0
$31$	−0.211323	−0.0379547	−0.0189773	−	0.999820i	$-0.506041\pi$
$31$	−0.211323	−0.0379547	−0.0189773	+	0.999820i	$0.506041\pi$
$32$	0	0
$33$	0.730278	0.127125
$34$	0	0
$35$	−2.71878	−0.459558
$36$	0	0
$37$	−0.106108	−0.0174441	−0.00872203	−	0.999962i	$-0.502776\pi$
$37$	−0.106108	−0.0174441	−0.00872203	+	0.999962i	$0.502776\pi$
$38$	0	0
$39$	0.930105	0.148936
$40$	0	0
$41$	−10.8964	−1.70174	−0.850868	−	0.525380i	$-0.823923\pi$
$41$	−10.8964	−1.70174	−0.850868	+	0.525380i	$0.823923\pi$
$42$	0	0
$43$	−12.7690	−1.94726	−0.973629	−	0.228138i	$-0.926736\pi$
$43$	−12.7690	−1.94726	−0.973629	+	0.228138i	$0.926736\pi$
$44$	0	0
$45$	7.88240	1.17504
$46$	0	0
$47$	−2.31743	−0.338032	−0.169016	−	0.985613i	$-0.554059\pi$
$47$	−2.31743	−0.338032	−0.169016	+	0.985613i	$0.554059\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.776064	−0.108671
$52$	0	0
$53$	−0.730278	−0.100311	−0.0501557	−	0.998741i	$-0.515972\pi$
$53$	−0.730278	−0.100311	−0.0501557	+	0.998741i	$0.515972\pi$
$54$	0	0
$55$	−6.25480	−0.843397
$56$	0	0
$57$	0.254803	0.0337494
$58$	0	0
$59$	1.24754	0.162415	0.0812077	−	0.996697i	$-0.474122\pi$
$59$	1.24754	0.162415	0.0812077	+	0.996697i	$0.474122\pi$
$60$	0	0
$61$	−0.612674	−0.0784449	−0.0392225	−	0.999231i	$-0.512488\pi$
$61$	−0.612674	−0.0784449	−0.0392225	+	0.999231i	$0.512488\pi$
$62$	0	0
$63$	−2.89924	−0.365270
$64$	0	0
$65$	−7.96632	−0.988100
$66$	0	0
$67$	−8.02664	−0.980610	−0.490305	−	0.871551i	$-0.663115\pi$
$67$	−8.02664	−0.980610	−0.490305	+	0.871551i	$0.663115\pi$
$68$	0	0
$69$	−0.317431	−0.0382142
$70$	0	0
$71$	−5.76649	−0.684357	−0.342178	−	0.939635i	$-0.611165\pi$
$71$	−5.76649	−0.684357	−0.342178	+	0.939635i	$0.611165\pi$
$72$	0	0
$73$	9.67107	1.13191	0.565957	−	0.824435i	$-0.308507\pi$
$73$	9.67107	1.13191	0.565957	+	0.824435i	$0.308507\pi$
$74$	0	0
$75$	0.759224	0.0876676
$76$	0	0
$77$	2.30059	0.262177
$78$	0	0
$79$	−2.30059	−0.258837	−0.129418	−	0.991590i	$-0.541311\pi$
$79$	−2.30059	−0.258837	−0.129418	+	0.991590i	$0.541311\pi$
$80$	0	0
$81$	8.10329	0.900366
$82$	0	0
$83$	−15.7115	−1.72456	−0.862281	−	0.506429i	$-0.830965\pi$
$83$	−15.7115	−1.72456	−0.862281	+	0.506429i	$0.830965\pi$
$84$	0	0
$85$	6.64697	0.720965
$86$	0	0
$87$	−2.67530	−0.286823
$88$	0	0
$89$	−5.98850	−0.634780	−0.317390	−	0.948295i	$-0.602806\pi$
$89$	−5.98850	−0.634780	−0.317390	+	0.948295i	$0.602806\pi$
$90$	0	0
$91$	2.93011	0.307158
$92$	0	0
$93$	−0.0670803	−0.00695590
$94$	0	0
$95$	−2.18238	−0.223907
$96$	0	0
$97$	7.52149	0.763691	0.381846	−	0.924226i	$-0.375289\pi$
$97$	7.52149	0.763691	0.381846	+	0.924226i	$0.375289\pi$
$98$	0	0
$99$	−6.66996	−0.670356
$100$	0	0
$101$	18.8223	1.87289	0.936444	−	0.350816i	$-0.114096\pi$
$101$	18.8223	1.87289	0.936444	+	0.350816i	$0.114096\pi$
$102$	0	0
$103$	10.3272	1.01757	0.508786	−	0.860893i	$-0.330094\pi$
$103$	10.3272	1.01757	0.508786	+	0.860893i	$0.330094\pi$
$104$	0	0
$105$	−0.863025	−0.0842226
$106$	0	0
$107$	−1.63909	−0.158457	−0.0792284	−	0.996856i	$-0.525246\pi$
$107$	−1.63909	−0.158457	−0.0792284	+	0.996856i	$0.525246\pi$
$108$	0	0
$109$	−12.0281	−1.15208	−0.576039	−	0.817422i	$-0.695402\pi$
$109$	−12.0281	−1.15208	−0.576039	+	0.817422i	$0.695402\pi$
$110$	0	0
$111$	−0.0336819	−0.00319695
$112$	0	0
$113$	14.1874	1.33464	0.667321	−	0.744770i	$-0.267441\pi$
$113$	14.1874	1.33464	0.667321	+	0.744770i	$0.267441\pi$
$114$	0	0
$115$	2.71878	0.253528
$116$	0	0
$117$	−8.49507	−0.785370
$118$	0	0
$119$	−2.44483	−0.224117
$120$	0	0
$121$	−5.70729	−0.518844
$122$	0	0
$123$	−3.45886	−0.311875
$124$	0	0
$125$	7.09119	0.634255
$126$	0	0
$127$	6.19983	0.550146	0.275073	−	0.961423i	$-0.411298\pi$
$127$	6.19983	0.550146	0.275073	+	0.961423i	$0.411298\pi$
$128$	0	0
$129$	−4.05328	−0.356871
$130$	0	0
$131$	2.28375	0.199532	0.0997660	−	0.995011i	$-0.468191\pi$
$131$	2.28375	0.199532	0.0997660	+	0.995011i	$0.468191\pi$
$132$	0	0
$133$	0.802704	0.0696032
$134$	0	0
$135$	5.09119	0.438180
$136$	0	0
$137$	1.96632	0.167994	0.0839969	−	0.996466i	$-0.473231\pi$
$137$	1.96632	0.167994	0.0839969	+	0.996466i	$0.473231\pi$
$138$	0	0
$139$	−18.1883	−1.54271	−0.771357	−	0.636403i	$-0.780422\pi$
$139$	−18.1883	−1.54271	−0.771357	+	0.636403i	$0.780422\pi$
$140$	0	0
$141$	−0.735623	−0.0619507
$142$	0	0
$143$	6.74097	0.563708
$144$	0	0
$145$	22.9139	1.90289
$146$	0	0
$147$	0.317431	0.0261812
$148$	0	0
$149$	−9.38474	−0.768827	−0.384414	−	0.923161i	$-0.625596\pi$
$149$	−9.38474	−0.768827	−0.384414	+	0.923161i	$0.625596\pi$
$150$	0	0
$151$	4.12740	0.335883	0.167942	−	0.985797i	$-0.446288\pi$
$151$	4.12740	0.335883	0.167942	+	0.985797i	$0.446288\pi$
$152$	0	0
$153$	7.08815	0.573043
$154$	0	0
$155$	0.574540	0.0461482
$156$	0	0
$157$	8.93100	0.712771	0.356386	−	0.934339i	$-0.384009\pi$
$157$	8.93100	0.712771	0.356386	+	0.934339i	$0.384009\pi$
$158$	0	0
$159$	−0.231812	−0.0183839
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	−5.01707	−0.392967	−0.196483	−	0.980507i	$-0.562952\pi$
$163$	−5.01707	−0.392967	−0.196483	+	0.980507i	$0.562952\pi$
$164$	0	0
$165$	−1.98547	−0.154568
$166$	0	0
$167$	1.34295	0.103921	0.0519604	−	0.998649i	$-0.483453\pi$
$167$	1.34295	0.103921	0.0519604	+	0.998649i	$0.483453\pi$
$168$	0	0
$169$	−4.41448	−0.339576
$170$	0	0
$171$	−2.32723	−0.177968
$172$	0	0
$173$	−12.1150	−0.921087	−0.460544	−	0.887637i	$-0.652346\pi$
$173$	−12.1150	−0.921087	−0.460544	+	0.887637i	$0.652346\pi$
$174$	0	0
$175$	2.39178	0.180801
$176$	0	0
$177$	0.396006	0.0297656
$178$	0	0
$179$	−25.5420	−1.90910	−0.954548	−	0.298056i	$-0.903662\pi$
$179$	−25.5420	−1.90910	−0.954548	+	0.298056i	$0.903662\pi$
$180$	0	0
$181$	11.4154	0.848498	0.424249	−	0.905545i	$-0.360538\pi$
$181$	11.4154	0.848498	0.424249	+	0.905545i	$0.360538\pi$
$182$	0	0
$183$	−0.194482	−0.0143765
$184$	0	0
$185$	0.288485	0.0212098
$186$	0	0
$187$	−5.62456	−0.411308
$188$	0	0
$189$	−1.87260	−0.136212
$190$	0	0
$191$	19.8990	1.43984	0.719919	−	0.694058i	$-0.244179\pi$
$191$	19.8990	1.43984	0.719919	+	0.694058i	$0.244179\pi$
$192$	0	0
$193$	2.79594	0.201256	0.100628	−	0.994924i	$-0.467915\pi$
$193$	2.79594	0.201256	0.100628	+	0.994924i	$0.467915\pi$
$194$	0	0
$195$	−2.52875	−0.181088
$196$	0	0
$197$	−15.7665	−1.12332	−0.561658	−	0.827370i	$-0.689836\pi$
$197$	−15.7665	−1.12332	−0.561658	+	0.827370i	$0.689836\pi$
$198$	0	0
$199$	22.9177	1.62459	0.812297	−	0.583244i	$-0.198217\pi$
$199$	22.9177	1.62459	0.812297	+	0.583244i	$0.198217\pi$
$200$	0	0
$201$	−2.54790	−0.179715
$202$	0	0
$203$	−8.42799	−0.591529
$204$	0	0
$205$	29.6250	2.06910
$206$	0	0
$207$	2.89924	0.201511
$208$	0	0
$209$	1.84669	0.127738
$210$	0	0
$211$	9.10471	0.626794	0.313397	−	0.949622i	$-0.398533\pi$
$211$	9.10471	0.626794	0.313397	+	0.949622i	$0.398533\pi$
$212$	0	0
$213$	−1.83046	−0.125421
$214$	0	0
$215$	34.7162	2.36762
$216$	0	0
$217$	−0.211323	−0.0143455
$218$	0	0
$219$	3.06989	0.207444
$220$	0	0
$221$	−7.16361	−0.481877
$222$	0	0
$223$	−8.52571	−0.570924	−0.285462	−	0.958390i	$-0.592147\pi$
$223$	−8.52571	−0.570924	−0.285462	+	0.958390i	$0.592147\pi$
$224$	0	0
$225$	−6.93433	−0.462289
$226$	0	0
$227$	−9.87090	−0.655155	−0.327577	−	0.944824i	$-0.606232\pi$
$227$	−9.87090	−0.655155	−0.327577	+	0.944824i	$0.606232\pi$
$228$	0	0
$229$	20.2714	1.33957	0.669785	−	0.742555i	$-0.266386\pi$
$229$	20.2714	1.33957	0.669785	+	0.742555i	$0.266386\pi$
$230$	0	0
$231$	0.730278	0.0480487
$232$	0	0
$233$	−12.3854	−0.811395	−0.405697	−	0.914007i	$-0.632971\pi$
$233$	−12.3854	−0.811395	−0.405697	+	0.914007i	$0.632971\pi$
$234$	0	0
$235$	6.30059	0.411005
$236$	0	0
$237$	−0.730278	−0.0474366
$238$	0	0
$239$	10.7903	0.697967	0.348984	−	0.937129i	$-0.386527\pi$
$239$	10.7903	0.697967	0.348984	+	0.937129i	$0.386527\pi$
$240$	0	0
$241$	21.4069	1.37894	0.689471	−	0.724314i	$-0.257843\pi$
$241$	21.4069	1.37894	0.689471	+	0.724314i	$0.257843\pi$
$242$	0	0
$243$	8.19003	0.525391
$244$	0	0
$245$	−2.71878	−0.173697
$246$	0	0
$247$	2.35201	0.149655
$248$	0	0
$249$	−4.98732	−0.316058
$250$	0	0
$251$	−28.9228	−1.82559	−0.912795	−	0.408418i	$-0.866080\pi$
$251$	−28.9228	−1.82559	−0.912795	+	0.408418i	$0.866080\pi$
$252$	0	0
$253$	−2.30059	−0.144637
$254$	0	0
$255$	2.10995	0.132130
$256$	0	0
$257$	−2.75157	−0.171638	−0.0858191	−	0.996311i	$-0.527351\pi$
$257$	−2.75157	−0.171638	−0.0858191	+	0.996311i	$0.527351\pi$
$258$	0	0
$259$	−0.106108	−0.00659323
$260$	0	0
$261$	24.4348	1.51247
$262$	0	0
$263$	7.90600	0.487505	0.243752	−	0.969838i	$-0.421622\pi$
$263$	7.90600	0.487505	0.243752	+	0.969838i	$0.421622\pi$
$264$	0	0
$265$	1.98547	0.121966
$266$	0	0
$267$	−1.90093	−0.116335
$268$	0	0
$269$	3.89782	0.237655	0.118827	−	0.992915i	$-0.462087\pi$
$269$	3.89782	0.237655	0.118827	+	0.992915i	$0.462087\pi$
$270$	0	0
$271$	−20.8001	−1.26352	−0.631758	−	0.775165i	$-0.717667\pi$
$271$	−20.8001	−1.26352	−0.631758	+	0.775165i	$0.717667\pi$
$272$	0	0
$273$	0.930105	0.0562925
$274$	0	0
$275$	5.50250	0.331813
$276$	0	0
$277$	−30.0867	−1.80773	−0.903867	−	0.427814i	$-0.859284\pi$
$277$	−30.0867	−1.80773	−0.903867	+	0.427814i	$0.859284\pi$
$278$	0	0
$279$	0.612674	0.0366799
$280$	0	0
$281$	−16.2403	−0.968813	−0.484407	−	0.874843i	$-0.660964\pi$
$281$	−16.2403	−0.968813	−0.484407	+	0.874843i	$0.660964\pi$
$282$	0	0
$283$	−3.79848	−0.225796	−0.112898	−	0.993607i	$-0.536013\pi$
$283$	−3.79848	−0.225796	−0.112898	+	0.993607i	$0.536013\pi$
$284$	0	0
$285$	−0.692753	−0.0410352
$286$	0	0
$287$	−10.8964	−0.643196
$288$	0	0
$289$	−11.0228	−0.648400
$290$	0	0
$291$	2.38755	0.139961
$292$	0	0
$293$	23.3396	1.36351	0.681756	−	0.731580i	$-0.261217\pi$
$293$	23.3396	1.36351	0.681756	+	0.731580i	$0.261217\pi$
$294$	0	0
$295$	−3.39178	−0.197477
$296$	0	0
$297$	−4.30808	−0.249980
$298$	0	0
$299$	−2.93011	−0.169452
$300$	0	0
$301$	−12.7690	−0.735994
$302$	0	0
$303$	5.97477	0.343242
$304$	0	0
$305$	1.66573	0.0953793
$306$	0	0
$307$	13.5411	0.772830	0.386415	−	0.922325i	$-0.373713\pi$
$307$	13.5411	0.772830	0.386415	+	0.922325i	$0.373713\pi$
$308$	0	0
$309$	3.27818	0.186489
$310$	0	0
$311$	6.59138	0.373763	0.186881	−	0.982382i	$-0.440162\pi$
$311$	6.59138	0.373763	0.186881	+	0.982382i	$0.440162\pi$
$312$	0	0
$313$	5.06055	0.286039	0.143019	−	0.989720i	$-0.454319\pi$
$313$	5.06055	0.286039	0.143019	+	0.989720i	$0.454319\pi$
$314$	0	0
$315$	7.88240	0.444123
$316$	0	0
$317$	8.98873	0.504857	0.252429	−	0.967616i	$-0.418771\pi$
$317$	8.98873	0.504857	0.252429	+	0.967616i	$0.418771\pi$
$318$	0	0
$319$	−19.3893	−1.08560
$320$	0	0
$321$	−0.520297	−0.0290402
$322$	0	0
$323$	−1.96248	−0.109195
$324$	0	0
$325$	7.00816	0.388743
$326$	0	0
$327$	−3.81807	−0.211140
$328$	0	0
$329$	−2.31743	−0.127764
$330$	0	0
$331$	16.1381	0.887030	0.443515	−	0.896267i	$-0.353731\pi$
$331$	16.1381	0.887030	0.443515	+	0.896267i	$0.353731\pi$
$332$	0	0
$333$	0.307632	0.0168582
$334$	0	0
$335$	21.8227	1.19230
$336$	0	0
$337$	24.1729	1.31678	0.658391	−	0.752676i	$-0.271237\pi$
$337$	24.1729	1.31678	0.658391	+	0.752676i	$0.271237\pi$
$338$	0	0
$339$	4.50353	0.244598
$340$	0	0
$341$	−0.486167	−0.0263274
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	0.863025	0.0464637
$346$	0	0
$347$	20.7017	1.11132	0.555662	−	0.831409i	$-0.312465\pi$
$347$	20.7017	1.11132	0.555662	+	0.831409i	$0.312465\pi$
$348$	0	0
$349$	3.96801	0.212403	0.106201	−	0.994345i	$-0.466131\pi$
$349$	3.96801	0.212403	0.106201	+	0.994345i	$0.466131\pi$
$350$	0	0
$351$	−5.48691	−0.292870
$352$	0	0
$353$	7.60934	0.405004	0.202502	−	0.979282i	$-0.435093\pi$
$353$	7.60934	0.405004	0.202502	+	0.979282i	$0.435093\pi$
$354$	0	0
$355$	15.6778	0.832093
$356$	0	0
$357$	−0.776064	−0.0410737
$358$	0	0
$359$	−11.7471	−0.619986	−0.309993	−	0.950739i	$-0.600327\pi$
$359$	−11.7471	−0.619986	−0.309993	+	0.950739i	$0.600327\pi$
$360$	0	0
$361$	−18.3557	−0.966088
$362$	0	0
$363$	−1.81167	−0.0950879
$364$	0	0
$365$	−26.2935	−1.37627
$366$	0	0
$367$	9.77042	0.510012	0.255006	−	0.966939i	$-0.417923\pi$
$367$	9.77042	0.510012	0.255006	+	0.966939i	$0.417923\pi$
$368$	0	0
$369$	31.5913	1.64458
$370$	0	0
$371$	−0.730278	−0.0379141
$372$	0	0
$373$	2.76012	0.142913	0.0714567	−	0.997444i	$-0.477235\pi$
$373$	2.76012	0.142913	0.0714567	+	0.997444i	$0.477235\pi$
$374$	0	0
$375$	2.25096	0.116239
$376$	0	0
$377$	−24.6949	−1.27185
$378$	0	0
$379$	2.70024	0.138702	0.0693511	−	0.997592i	$-0.477907\pi$
$379$	2.70024	0.138702	0.0693511	+	0.997592i	$0.477907\pi$
$380$	0	0
$381$	1.96801	0.100824
$382$	0	0
$383$	−15.4319	−0.788535	−0.394268	−	0.918996i	$-0.629002\pi$
$383$	−15.4319	−0.788535	−0.394268	+	0.918996i	$0.629002\pi$
$384$	0	0
$385$	−6.25480	−0.318774
$386$	0	0
$387$	37.0204	1.88185
$388$	0	0
$389$	30.7123	1.55718	0.778589	−	0.627534i	$-0.215936\pi$
$389$	30.7123	1.55718	0.778589	+	0.627534i	$0.215936\pi$
$390$	0	0
$391$	2.44483	0.123640
$392$	0	0
$393$	0.724932	0.0365680
$394$	0	0
$395$	6.25480	0.314713
$396$	0	0
$397$	−18.3396	−0.920439	−0.460220	−	0.887805i	$-0.652229\pi$
$397$	−18.3396	−0.920439	−0.460220	+	0.887805i	$0.652229\pi$
$398$	0	0
$399$	0.254803	0.0127561
$400$	0	0
$401$	17.1103	0.854449	0.427225	−	0.904145i	$-0.359491\pi$
$401$	17.1103	0.854449	0.427225	+	0.904145i	$0.359491\pi$
$402$	0	0
$403$	−0.619197	−0.0308444
$404$	0	0
$405$	−22.0311	−1.09473
$406$	0	0
$407$	−0.244111	−0.0121001
$408$	0	0
$409$	8.59948	0.425217	0.212609	−	0.977137i	$-0.431804\pi$
$409$	8.59948	0.425217	0.212609	+	0.977137i	$0.431804\pi$
$410$	0	0
$411$	0.624170	0.0307880
$412$	0	0
$413$	1.24754	0.0613872
$414$	0	0
$415$	42.7162	2.09686
$416$	0	0
$417$	−5.77353	−0.282731
$418$	0	0
$419$	10.5394	0.514886	0.257443	−	0.966294i	$-0.417120\pi$
$419$	10.5394	0.514886	0.257443	+	0.966294i	$0.417120\pi$
$420$	0	0
$421$	6.08088	0.296364	0.148182	−	0.988960i	$-0.452658\pi$
$421$	6.08088	0.296364	0.148182	+	0.988960i	$0.452658\pi$
$422$	0	0
$423$	6.71878	0.326678
$424$	0	0
$425$	−5.84750	−0.283645
$426$	0	0
$427$	−0.612674	−0.0296494
$428$	0	0
$429$	2.13979	0.103310
$430$	0	0
$431$	23.5296	1.13338	0.566690	−	0.823931i	$-0.308224\pi$
$431$	23.5296	1.13338	0.566690	+	0.823931i	$0.308224\pi$
$432$	0	0
$433$	29.0993	1.39842	0.699211	−	0.714915i	$-0.253535\pi$
$433$	29.0993	1.39842	0.699211	+	0.714915i	$0.253535\pi$
$434$	0	0
$435$	7.27356	0.348741
$436$	0	0
$437$	−0.802704	−0.0383985
$438$	0	0
$439$	7.42598	0.354423	0.177211	−	0.984173i	$-0.443292\pi$
$439$	7.42598	0.354423	0.177211	+	0.984173i	$0.443292\pi$
$440$	0	0
$441$	−2.89924	−0.138059
$442$	0	0
$443$	33.3919	1.58650	0.793250	−	0.608897i	$-0.208388\pi$
$443$	33.3919	1.58650	0.793250	+	0.608897i	$0.208388\pi$
$444$	0	0
$445$	16.2814	0.771814
$446$	0	0
$447$	−2.97900	−0.140902
$448$	0	0
$449$	−19.8779	−0.938098	−0.469049	−	0.883172i	$-0.655403\pi$
$449$	−19.8779	−0.938098	−0.469049	+	0.883172i	$0.655403\pi$
$450$	0	0
$451$	−25.0682	−1.18042
$452$	0	0
$453$	1.31016	0.0615569
$454$	0	0
$455$	−7.96632	−0.373467
$456$	0	0
$457$	−14.5203	−0.679231	−0.339615	−	0.940564i	$-0.610297\pi$
$457$	−14.5203	−0.679231	−0.339615	+	0.940564i	$0.610297\pi$
$458$	0	0
$459$	4.57819	0.213692
$460$	0	0
$461$	1.11710	0.0520283	0.0260142	−	0.999662i	$-0.491719\pi$
$461$	1.11710	0.0520283	0.0260142	+	0.999662i	$0.491719\pi$
$462$	0	0
$463$	−5.37020	−0.249574	−0.124787	−	0.992184i	$-0.539825\pi$
$463$	−5.37020	−0.249574	−0.124787	+	0.992184i	$0.539825\pi$
$464$	0	0
$465$	0.182377	0.00845751
$466$	0	0
$467$	−26.8504	−1.24249	−0.621243	−	0.783618i	$-0.713372\pi$
$467$	−26.8504	−1.24249	−0.621243	+	0.783618i	$0.713372\pi$
$468$	0	0
$469$	−8.02664	−0.370636
$470$	0	0
$471$	2.83497	0.130629
$472$	0	0
$473$	−29.3763	−1.35072
$474$	0	0
$475$	1.91989	0.0880905
$476$	0	0
$477$	2.11725	0.0969421
$478$	0	0
$479$	9.87090	0.451013	0.225507	−	0.974242i	$-0.427596\pi$
$479$	9.87090	0.451013	0.225507	+	0.974242i	$0.427596\pi$
$480$	0	0
$481$	−0.310908	−0.0141762
$482$	0	0
$483$	−0.317431	−0.0144436
$484$	0	0
$485$	−20.4493	−0.928554
$486$	0	0
$487$	−29.7631	−1.34869	−0.674347	−	0.738414i	$-0.735575\pi$
$487$	−29.7631	−1.34869	−0.674347	+	0.738414i	$0.735575\pi$
$488$	0	0
$489$	−1.59257	−0.0720185
$490$	0	0
$491$	10.3144	0.465482	0.232741	−	0.972539i	$-0.425231\pi$
$491$	10.3144	0.465482	0.232741	+	0.972539i	$0.425231\pi$
$492$	0	0
$493$	20.6050	0.928004
$494$	0	0
$495$	18.1342	0.815070
$496$	0	0
$497$	−5.76649	−0.258662
$498$	0	0
$499$	−38.5214	−1.72446	−0.862228	−	0.506520i	$-0.830931\pi$
$499$	−38.5214	−1.72446	−0.862228	+	0.506520i	$0.830931\pi$
$500$	0	0
$501$	0.426294	0.0190454
$502$	0	0
$503$	−16.7325	−0.746066	−0.373033	−	0.927818i	$-0.621682\pi$
$503$	−16.7325	−0.746066	−0.373033	+	0.927818i	$0.621682\pi$
$504$	0	0
$505$	−51.1737	−2.27720
$506$	0	0
$507$	−1.40129	−0.0622336
$508$	0	0
$509$	−26.3592	−1.16835	−0.584176	−	0.811627i	$-0.698582\pi$
$509$	−26.3592	−1.16835	−0.584176	+	0.811627i	$0.698582\pi$
$510$	0	0
$511$	9.67107	0.427823
$512$	0	0
$513$	−1.50314	−0.0663653
$514$	0	0
$515$	−28.0775	−1.23724
$516$	0	0
$517$	−5.33146	−0.234477
$518$	0	0
$519$	−3.84568	−0.168806
$520$	0	0
$521$	12.4051	0.543476	0.271738	−	0.962371i	$-0.412402\pi$
$521$	12.4051	0.543476	0.271738	+	0.962371i	$0.412402\pi$
$522$	0	0
$523$	−3.39882	−0.148620	−0.0743100	−	0.997235i	$-0.523675\pi$
$523$	−3.39882	−0.148620	−0.0743100	+	0.997235i	$0.523675\pi$
$524$	0	0
$525$	0.759224	0.0331352
$526$	0	0
$527$	0.516648	0.0225055
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−3.61690	−0.156960
$532$	0	0
$533$	−31.9277	−1.38294
$534$	0	0
$535$	4.45633	0.192664
$536$	0	0
$537$	−8.10780	−0.349878
$538$	0	0
$539$	2.30059	0.0990934
$540$	0	0
$541$	24.9511	1.07273	0.536366	−	0.843985i	$-0.319797\pi$
$541$	24.9511	1.07273	0.536366	+	0.843985i	$0.319797\pi$
$542$	0	0
$543$	3.62359	0.155503
$544$	0	0
$545$	32.7017	1.40079
$546$	0	0
$547$	−27.3887	−1.17105	−0.585527	−	0.810653i	$-0.699113\pi$
$547$	−27.3887	−1.17105	−0.585527	+	0.810653i	$0.699113\pi$
$548$	0	0
$549$	1.77629	0.0758102
$550$	0	0
$551$	−6.76518	−0.288206
$552$	0	0
$553$	−2.30059	−0.0978310
$554$	0	0
$555$	0.0915739	0.00388709
$556$	0	0
$557$	11.9668	0.507048	0.253524	−	0.967329i	$-0.418410\pi$
$557$	11.9668	0.507048	0.253524	+	0.967329i	$0.418410\pi$
$558$	0	0
$559$	−37.4146	−1.58247
$560$	0	0
$561$	−1.78541	−0.0753799
$562$	0	0
$563$	−10.3974	−0.438199	−0.219099	−	0.975703i	$-0.570312\pi$
$563$	−10.3974	−0.438199	−0.219099	+	0.975703i	$0.570312\pi$
$564$	0	0
$565$	−38.5726	−1.62276
$566$	0	0
$567$	8.10329	0.340306
$568$	0	0
$569$	−26.8364	−1.12504	−0.562520	−	0.826784i	$-0.690168\pi$
$569$	−26.8364	−1.12504	−0.562520	+	0.826784i	$0.690168\pi$
$570$	0	0
$571$	−32.8465	−1.37458	−0.687292	−	0.726381i	$-0.741201\pi$
$571$	−32.8465	−1.37458	−0.687292	+	0.726381i	$0.741201\pi$
$572$	0	0
$573$	6.31654	0.263877
$574$	0	0
$575$	−2.39178	−0.0997440
$576$	0	0
$577$	4.30978	0.179418	0.0897092	−	0.995968i	$-0.471406\pi$
$577$	4.30978	0.179418	0.0897092	+	0.995968i	$0.471406\pi$
$578$	0	0
$579$	0.887518	0.0368840
$580$	0	0
$581$	−15.7115	−0.651823
$582$	0	0
$583$	−1.68007	−0.0695813
$584$	0	0
$585$	23.0963	0.954912
$586$	0	0
$587$	−38.9910	−1.60933	−0.804666	−	0.593728i	$-0.797656\pi$
$587$	−38.9910	−1.60933	−0.804666	+	0.593728i	$0.797656\pi$
$588$	0	0
$589$	−0.169629	−0.00698946
$590$	0	0
$591$	−5.00477	−0.205869
$592$	0	0
$593$	−9.39882	−0.385963	−0.192982	−	0.981202i	$-0.561816\pi$
$593$	−9.39882	−0.385963	−0.192982	+	0.981202i	$0.561816\pi$
$594$	0	0
$595$	6.64697	0.272499
$596$	0	0
$597$	7.27478	0.297737
$598$	0	0
$599$	−29.7805	−1.21680	−0.608400	−	0.793630i	$-0.708188\pi$
$599$	−29.7805	−1.21680	−0.608400	+	0.793630i	$0.708188\pi$
$600$	0	0
$601$	−7.07496	−0.288594	−0.144297	−	0.989534i	$-0.546092\pi$
$601$	−7.07496	−0.288594	−0.144297	+	0.989534i	$0.546092\pi$
$602$	0	0
$603$	23.2711	0.947674
$604$	0	0
$605$	15.5169	0.630851
$606$	0	0
$607$	−3.21385	−0.130446	−0.0652231	−	0.997871i	$-0.520776\pi$
$607$	−3.21385	−0.130446	−0.0652231	+	0.997871i	$0.520776\pi$
$608$	0	0
$609$	−2.67530	−0.108409
$610$	0	0
$611$	−6.79032	−0.274707
$612$	0	0
$613$	−31.0480	−1.25402	−0.627009	−	0.779012i	$-0.715721\pi$
$613$	−31.0480	−1.25402	−0.627009	+	0.779012i	$0.715721\pi$
$614$	0	0
$615$	9.40388	0.379201
$616$	0	0
$617$	−20.4418	−0.822956	−0.411478	−	0.911420i	$-0.634987\pi$
$617$	−20.4418	−0.822956	−0.411478	+	0.911420i	$0.634987\pi$
$618$	0	0
$619$	23.9415	0.962292	0.481146	−	0.876641i	$-0.340221\pi$
$619$	23.9415	0.962292	0.481146	+	0.876641i	$0.340221\pi$
$620$	0	0
$621$	1.87260	0.0751448
$622$	0	0
$623$	−5.98850	−0.239924
$624$	0	0
$625$	−31.2383	−1.24953
$626$	0	0
$627$	0.586196	0.0234104
$628$	0	0
$629$	0.259416	0.0103436
$630$	0	0
$631$	44.8685	1.78619	0.893094	−	0.449870i	$-0.148530\pi$
$631$	44.8685	1.78619	0.893094	+	0.449870i	$0.148530\pi$
$632$	0	0
$633$	2.89011	0.114872
$634$	0	0
$635$	−16.8560	−0.668909
$636$	0	0
$637$	2.93011	0.116095
$638$	0	0
$639$	16.7184	0.661371
$640$	0	0
$641$	17.6306	0.696368	0.348184	−	0.937426i	$-0.386798\pi$
$641$	17.6306	0.696368	0.348184	+	0.937426i	$0.386798\pi$
$642$	0	0
$643$	16.7886	0.662078	0.331039	−	0.943617i	$-0.392601\pi$
$643$	16.7886	0.662078	0.331039	+	0.943617i	$0.392601\pi$
$644$	0	0
$645$	11.0200	0.433911
$646$	0	0
$647$	3.24360	0.127519	0.0637596	−	0.997965i	$-0.479691\pi$
$647$	3.24360	0.127519	0.0637596	+	0.997965i	$0.479691\pi$
$648$	0	0
$649$	2.87007	0.112660
$650$	0	0
$651$	−0.0670803	−0.00262908
$652$	0	0
$653$	−1.35838	−0.0531575	−0.0265788	−	0.999647i	$-0.508461\pi$
$653$	−1.35838	−0.0531575	−0.0265788	+	0.999647i	$0.508461\pi$
$654$	0	0
$655$	−6.20902	−0.242606
$656$	0	0
$657$	−28.0387	−1.09390
$658$	0	0
$659$	14.7864	0.575996	0.287998	−	0.957631i	$-0.407010\pi$
$659$	14.7864	0.575996	0.287998	+	0.957631i	$0.407010\pi$
$660$	0	0
$661$	−28.5312	−1.10974	−0.554868	−	0.831938i	$-0.687231\pi$
$661$	−28.5312	−1.10974	−0.554868	+	0.831938i	$0.687231\pi$
$662$	0	0
$663$	−2.27395	−0.0883129
$664$	0	0
$665$	−2.18238	−0.0846289
$666$	0	0
$667$	8.42799	0.326333
$668$	0	0
$669$	−2.70632	−0.104632
$670$	0	0
$671$	−1.40951	−0.0544136
$672$	0	0
$673$	−6.01988	−0.232049	−0.116025	−	0.993246i	$-0.537015\pi$
$673$	−6.01988	−0.232049	−0.116025	+	0.993246i	$0.537015\pi$
$674$	0	0
$675$	−4.47884	−0.172391
$676$	0	0
$677$	−23.7852	−0.914139	−0.457070	−	0.889431i	$-0.651101\pi$
$677$	−23.7852	−0.914139	−0.457070	+	0.889431i	$0.651101\pi$
$678$	0	0
$679$	7.52149	0.288648
$680$	0	0
$681$	−3.13333	−0.120069
$682$	0	0
$683$	−27.5061	−1.05249	−0.526245	−	0.850333i	$-0.676401\pi$
$683$	−27.5061	−1.05249	−0.526245	+	0.850333i	$0.676401\pi$
$684$	0	0
$685$	−5.34599	−0.204260
$686$	0	0
$687$	6.43475	0.245501
$688$	0	0
$689$	−2.13979	−0.0815195
$690$	0	0
$691$	15.3878	0.585379	0.292689	−	0.956208i	$-0.405450\pi$
$691$	15.3878	0.585379	0.292689	+	0.956208i	$0.405450\pi$
$692$	0	0
$693$	−6.66996	−0.253371
$694$	0	0
$695$	49.4501	1.87575
$696$	0	0
$697$	26.6399	1.00906
$698$	0	0
$699$	−3.93151	−0.148703
$700$	0	0
$701$	−41.3724	−1.56262	−0.781308	−	0.624146i	$-0.785447\pi$
$701$	−41.3724	−1.56262	−0.781308	+	0.624146i	$0.785447\pi$
$702$	0	0
$703$	−0.0851733	−0.00321237
$704$	0	0
$705$	2.00000	0.0753244
$706$	0	0
$707$	18.8223	0.707885
$708$	0	0
$709$	10.6562	0.400204	0.200102	−	0.979775i	$-0.435873\pi$
$709$	10.6562	0.400204	0.200102	+	0.979775i	$0.435873\pi$
$710$	0	0
$711$	6.66996	0.250143
$712$	0	0
$713$	0.211323	0.00791409
$714$	0	0
$715$	−18.3272	−0.685399
$716$	0	0
$717$	3.42518	0.127916
$718$	0	0
$719$	27.9066	1.04074	0.520370	−	0.853941i	$-0.325794\pi$
$719$	27.9066	1.04074	0.520370	+	0.853941i	$0.325794\pi$
$720$	0	0
$721$	10.3272	0.384606
$722$	0	0
$723$	6.79521	0.252717
$724$	0	0
$725$	−20.1579	−0.748645
$726$	0	0
$727$	−29.8602	−1.10745	−0.553727	−	0.832698i	$-0.686795\pi$
$727$	−29.8602	−1.10745	−0.553727	+	0.832698i	$0.686795\pi$
$728$	0	0
$729$	−21.7101	−0.804078
$730$	0	0
$731$	31.2181	1.15464
$732$	0	0
$733$	11.4210	0.421845	0.210922	−	0.977503i	$-0.432353\pi$
$733$	11.4210	0.421845	0.210922	+	0.977503i	$0.432353\pi$
$734$	0	0
$735$	−0.863025	−0.0318331
$736$	0	0
$737$	−18.4660	−0.680204
$738$	0	0
$739$	−34.3784	−1.26463	−0.632314	−	0.774712i	$-0.717895\pi$
$739$	−34.3784	−1.26463	−0.632314	+	0.774712i	$0.717895\pi$
$740$	0	0
$741$	0.746599	0.0274270
$742$	0	0
$743$	−18.7676	−0.688517	−0.344258	−	0.938875i	$-0.611870\pi$
$743$	−18.7676	−0.688517	−0.344258	+	0.938875i	$0.611870\pi$
$744$	0	0
$745$	25.5151	0.934799
$746$	0	0
$747$	45.5514	1.66664
$748$	0	0
$749$	−1.63909	−0.0598910
$750$	0	0
$751$	−16.9068	−0.616939	−0.308470	−	0.951234i	$-0.599817\pi$
$751$	−16.9068	−0.616939	−0.308470	+	0.951234i	$0.599817\pi$
$752$	0	0
$753$	−9.18098	−0.334573
$754$	0	0
$755$	−11.2215	−0.408392
$756$	0	0
$757$	−9.37583	−0.340770	−0.170385	−	0.985378i	$-0.554501\pi$
$757$	−9.37583	−0.340770	−0.170385	+	0.985378i	$0.554501\pi$
$758$	0	0
$759$	−0.730278	−0.0265074
$760$	0	0
$761$	49.3455	1.78877	0.894387	−	0.447294i	$-0.147612\pi$
$761$	49.3455	1.78877	0.894387	+	0.447294i	$0.147612\pi$
$762$	0	0
$763$	−12.0281	−0.435445
$764$	0	0
$765$	−19.2711	−0.696749
$766$	0	0
$767$	3.65541	0.131989
$768$	0	0
$769$	14.1428	0.510003	0.255002	−	0.966941i	$-0.417924\pi$
$769$	14.1428	0.510003	0.255002	+	0.966941i	$0.417924\pi$
$770$	0	0
$771$	−0.873433	−0.0314559
$772$	0	0
$773$	−47.9813	−1.72577	−0.862883	−	0.505403i	$-0.831344\pi$
$773$	−47.9813	−1.72577	−0.862883	+	0.505403i	$0.831344\pi$
$774$	0	0
$775$	−0.505437	−0.0181558
$776$	0	0
$777$	−0.0336819	−0.00120833
$778$	0	0
$779$	−8.74660	−0.313379
$780$	0	0
$781$	−13.2663	−0.474707
$782$	0	0
$783$	15.7822	0.564011
$784$	0	0
$785$	−24.2814	−0.866642
$786$	0	0
$787$	50.7634	1.80952	0.904760	−	0.425922i	$-0.140050\pi$
$787$	50.7634	1.80952	0.904760	+	0.425922i	$0.140050\pi$
$788$	0	0
$789$	2.50961	0.0893443
$790$	0	0
$791$	14.1874	0.504447
$792$	0	0
$793$	−1.79520	−0.0637494
$794$	0	0
$795$	0.630248	0.0223526
$796$	0	0
$797$	27.0797	0.959212	0.479606	−	0.877484i	$-0.340780\pi$
$797$	27.0797	0.959212	0.479606	+	0.877484i	$0.340780\pi$
$798$	0	0
$799$	5.66573	0.200439
$800$	0	0
$801$	17.3621	0.613460
$802$	0	0
$803$	22.2492	0.785156
$804$	0	0
$805$	2.71878	0.0958245
$806$	0	0
$807$	1.23729	0.0435546
$808$	0	0
$809$	17.7648	0.624577	0.312288	−	0.949987i	$-0.398904\pi$
$809$	17.7648	0.624577	0.312288	+	0.949987i	$0.398904\pi$
$810$	0	0
$811$	1.34971	0.0473948	0.0236974	−	0.999719i	$-0.492456\pi$
$811$	1.34971	0.0473948	0.0236974	+	0.999719i	$0.492456\pi$
$812$	0	0
$813$	−6.60259	−0.231563
$814$	0	0
$815$	13.6403	0.477799
$816$	0	0
$817$	−10.2497	−0.358593
$818$	0	0
$819$	−8.49507	−0.296842
$820$	0	0
$821$	34.9326	1.21916	0.609579	−	0.792725i	$-0.291339\pi$
$821$	34.9326	1.21916	0.609579	+	0.792725i	$0.291339\pi$
$822$	0	0
$823$	−22.2228	−0.774639	−0.387319	−	0.921946i	$-0.626599\pi$
$823$	−22.2228	−0.774639	−0.387319	+	0.921946i	$0.626599\pi$
$824$	0	0
$825$	1.74666	0.0608110
$826$	0	0
$827$	−19.1725	−0.666694	−0.333347	−	0.942804i	$-0.608178\pi$
$827$	−19.1725	−0.666694	−0.333347	+	0.942804i	$0.608178\pi$
$828$	0	0
$829$	37.9568	1.31830	0.659148	−	0.752013i	$-0.270917\pi$
$829$	37.9568	1.31830	0.659148	+	0.752013i	$0.270917\pi$
$830$	0	0
$831$	−9.55043	−0.331301
$832$	0	0
$833$	−2.44483	−0.0847084
$834$	0	0
$835$	−3.65119	−0.126355
$836$	0	0
$837$	0.395722	0.0136782
$838$	0	0
$839$	−31.6833	−1.09383	−0.546915	−	0.837188i	$-0.684198\pi$
$839$	−31.6833	−1.09383	−0.546915	+	0.837188i	$0.684198\pi$
$840$	0	0
$841$	42.0310	1.44935
$842$	0	0
$843$	−5.15516	−0.177553
$844$	0	0
$845$	12.0020	0.412882
$846$	0	0
$847$	−5.70729	−0.196105
$848$	0	0
$849$	−1.20575	−0.0413813
$850$	0	0
$851$	0.106108	0.00363734
$852$	0	0
$853$	−12.9054	−0.441873	−0.220937	−	0.975288i	$-0.570911\pi$
$853$	−12.9054	−0.441873	−0.220937	+	0.975288i	$0.570911\pi$
$854$	0	0
$855$	6.32723	0.216387
$856$	0	0
$857$	−53.2798	−1.82000	−0.910001	−	0.414606i	$-0.863919\pi$
$857$	−53.2798	−1.82000	−0.910001	+	0.414606i	$0.863919\pi$
$858$	0	0
$859$	2.90125	0.0989893	0.0494947	−	0.998774i	$-0.484239\pi$
$859$	2.90125	0.0989893	0.0494947	+	0.998774i	$0.484239\pi$
$860$	0	0
$861$	−3.45886	−0.117878
$862$	0	0
$863$	−48.3025	−1.64424	−0.822119	−	0.569316i	$-0.807208\pi$
$863$	−48.3025	−1.64424	−0.822119	+	0.569316i	$0.807208\pi$
$864$	0	0
$865$	32.9381	1.11993
$866$	0	0
$867$	−3.49897	−0.118831
$868$	0	0
$869$	−5.29271	−0.179543
$870$	0	0
$871$	−23.5189	−0.796907
$872$	0	0
$873$	−21.8066	−0.738041
$874$	0	0
$875$	7.09119	0.239726
$876$	0	0
$877$	−14.1136	−0.476582	−0.238291	−	0.971194i	$-0.576587\pi$
$877$	−14.1136	−0.476582	−0.238291	+	0.971194i	$0.576587\pi$
$878$	0	0
$879$	7.40869	0.249889
$880$	0	0
$881$	−39.4148	−1.32792	−0.663959	−	0.747769i	$-0.731125\pi$
$881$	−39.4148	−1.32792	−0.663959	+	0.747769i	$0.731125\pi$
$882$	0	0
$883$	−16.6343	−0.559789	−0.279894	−	0.960031i	$-0.590299\pi$
$883$	−16.6343	−0.559789	−0.279894	+	0.960031i	$0.590299\pi$
$884$	0	0
$885$	−1.07665	−0.0361913
$886$	0	0
$887$	48.7698	1.63753	0.818765	−	0.574129i	$-0.194659\pi$
$887$	48.7698	1.63753	0.818765	+	0.574129i	$0.194659\pi$
$888$	0	0
$889$	6.19983	0.207936
$890$	0	0
$891$	18.6424	0.624542
$892$	0	0
$893$	−1.86021	−0.0622496
$894$	0	0
$895$	69.4431	2.32123
$896$	0	0
$897$	−0.930105	−0.0310553
$898$	0	0
$899$	1.78102	0.0594005
$900$	0	0
$901$	1.78541	0.0594805
$902$	0	0
$903$	−4.05328	−0.134885
$904$	0	0
$905$	−31.0359	−1.03167
$906$	0	0
$907$	−52.8802	−1.75586	−0.877929	−	0.478791i	$-0.841075\pi$
$907$	−52.8802	−1.75586	−0.877929	+	0.478791i	$0.841075\pi$
$908$	0	0
$909$	−54.5703	−1.80998
$910$	0	0
$911$	38.4314	1.27329	0.636644	−	0.771158i	$-0.280322\pi$
$911$	38.4314	1.27329	0.636644	+	0.771158i	$0.280322\pi$
$912$	0	0
$913$	−36.1457	−1.19625
$914$	0	0
$915$	0.528753	0.0174800
$916$	0	0
$917$	2.28375	0.0754160
$918$	0	0
$919$	2.67886	0.0883675	0.0441837	−	0.999023i	$-0.485931\pi$
$919$	2.67886	0.0883675	0.0441837	+	0.999023i	$0.485931\pi$
$920$	0	0
$921$	4.29835	0.141636
$922$	0	0
$923$	−16.8964	−0.556153
$924$	0	0
$925$	−0.253787	−0.00834446
$926$	0	0
$927$	−29.9411	−0.983394
$928$	0	0
$929$	−18.6551	−0.612055	−0.306028	−	0.952023i	$-0.599000\pi$
$929$	−18.6551	−0.612055	−0.306028	+	0.952023i	$0.599000\pi$
$930$	0	0
$931$	0.802704	0.0263075
$932$	0	0
$933$	2.09231	0.0684990
$934$	0	0
$935$	15.2919	0.500100
$936$	0	0
$937$	−2.28038	−0.0744969	−0.0372484	−	0.999306i	$-0.511859\pi$
$937$	−2.28038	−0.0744969	−0.0372484	+	0.999306i	$0.511859\pi$
$938$	0	0
$939$	1.60637	0.0524220
$940$	0	0
$941$	−13.2385	−0.431563	−0.215782	−	0.976442i	$-0.569230\pi$
$941$	−13.2385	−0.431563	−0.215782	+	0.976442i	$0.569230\pi$
$942$	0	0
$943$	10.8964	0.354836
$944$	0	0
$945$	5.09119	0.165616
$946$	0	0
$947$	−56.1482	−1.82457	−0.912286	−	0.409554i	$-0.865684\pi$
$947$	−56.1482	−1.82457	−0.912286	+	0.409554i	$0.865684\pi$
$948$	0	0
$949$	28.3373	0.919866
$950$	0	0
$951$	2.85330	0.0925245
$952$	0	0
$953$	47.1883	1.52858	0.764289	−	0.644874i	$-0.223090\pi$
$953$	47.1883	1.52858	0.764289	+	0.644874i	$0.223090\pi$
$954$	0	0
$955$	−54.1009	−1.75066
$956$	0	0
$957$	−6.15477	−0.198956
$958$	0	0
$959$	1.96632	0.0634957
$960$	0	0
$961$	−30.9553	−0.998559
$962$	0	0
$963$	4.75211	0.153135
$964$	0	0
$965$	−7.60156	−0.244703
$966$	0	0
$967$	8.12919	0.261417	0.130709	−	0.991421i	$-0.458275\pi$
$967$	8.12919	0.261417	0.130709	+	0.991421i	$0.458275\pi$
$968$	0	0
$969$	−0.622950	−0.0200120
$970$	0	0
$971$	−24.6236	−0.790209	−0.395105	−	0.918636i	$-0.629292\pi$
$971$	−24.6236	−0.790209	−0.395105	+	0.918636i	$0.629292\pi$
$972$	0	0
$973$	−18.1883	−0.583091
$974$	0	0
$975$	2.22460	0.0712444
$976$	0	0
$977$	46.2453	1.47952	0.739760	−	0.672871i	$-0.234939\pi$
$977$	46.2453	1.47952	0.739760	+	0.672871i	$0.234939\pi$
$978$	0	0
$979$	−13.7771	−0.440318
$980$	0	0
$981$	34.8722	1.11338
$982$	0	0
$983$	16.4469	0.524573	0.262287	−	0.964990i	$-0.415523\pi$
$983$	16.4469	0.524573	0.262287	+	0.964990i	$0.415523\pi$
$984$	0	0
$985$	42.8657	1.36581
$986$	0	0
$987$	−0.735623	−0.0234152
$988$	0	0
$989$	12.7690	0.406031
$990$	0	0
$991$	49.6829	1.57823	0.789115	−	0.614246i	$-0.210540\pi$
$991$	49.6829	1.57823	0.789115	+	0.614246i	$0.210540\pi$
$992$	0	0
$993$	5.12272	0.162565
$994$	0	0
$995$	−62.3083	−1.97531
$996$	0	0
$997$	38.6254	1.22328	0.611639	−	0.791137i	$-0.290510\pi$
$997$	38.6254	1.22328	0.611639	+	0.791137i	$0.290510\pi$
$998$	0	0
$999$	0.198698	0.00628652

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1288.2.a.n.1.3	✓	4
4.3	odd	2		2576.2.a.ba.1.2		4
7.6	odd	2		9016.2.a.bf.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1288.2.a.n.1.3	✓	4	1.1	even	1	trivial
2576.2.a.ba.1.2		4	4.3	odd	2
9016.2.a.bf.1.2		4	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:	\( N \)	\(=\)	\( 1288 = 2^{3} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1288.a (trivial)

\(f(q)\)	\(=\)	\(q+0.317431 q^{3} -2.71878 q^{5} +1.00000 q^{7} -2.89924 q^{9} +O(q^{10})\)	\(q+0.317431 q^{3} -2.71878 q^{5} +1.00000 q^{7} -2.89924 q^{9} +2.30059 q^{11} +2.93011 q^{13} -0.863025 q^{15} -2.44483 q^{17} +0.802704 q^{19} +0.317431 q^{21} -1.00000 q^{23} +2.39178 q^{25} -1.87260 q^{27} -8.42799 q^{29} -0.211323 q^{31} +0.730278 q^{33} -2.71878 q^{35} -0.106108 q^{37} +0.930105 q^{39} -10.8964 q^{41} -12.7690 q^{43} +7.88240 q^{45} -2.31743 q^{47} +1.00000 q^{49} -0.776064 q^{51} -0.730278 q^{53} -6.25480 q^{55} +0.254803 q^{57} +1.24754 q^{59} -0.612674 q^{61} -2.89924 q^{63} -7.96632 q^{65} -8.02664 q^{67} -0.317431 q^{69} -5.76649 q^{71} +9.67107 q^{73} +0.759224 q^{75} +2.30059 q^{77} -2.30059 q^{79} +8.10329 q^{81} -15.7115 q^{83} +6.64697 q^{85} -2.67530 q^{87} -5.98850 q^{89} +2.93011 q^{91} -0.0670803 q^{93} -2.18238 q^{95} +7.52149 q^{97} -6.66996 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{3} + 4 q^{7} + q^{9}+O(q^{10}) \) 4 * q - 3 * q^3 + 4 * q^7 + q^9	\( 4 q - 3 q^{3} + 4 q^{7} + q^{9} - 10 q^{11} - 3 q^{13} - 6 q^{15} - 4 q^{17} - 10 q^{19} - 3 q^{21} - 4 q^{23} + 4 q^{25} - 9 q^{27} - 13 q^{29} + 3 q^{31} + 20 q^{33} - 11 q^{39} + q^{41} - 8 q^{43} + 4 q^{45} - 5 q^{47} + 4 q^{49} - 4 q^{51} - 20 q^{53} - 22 q^{55} - 2 q^{57} - 14 q^{59} + 8 q^{61} + q^{63} - 2 q^{65} - 18 q^{67} + 3 q^{69} - 25 q^{71} + 15 q^{73} - 11 q^{75} - 10 q^{77} + 10 q^{79} - 36 q^{83} - 28 q^{85} + q^{87} + 4 q^{89} - 3 q^{91} + 17 q^{93} - 36 q^{95} + 6 q^{97} - 28 q^{99}+O(q^{100}) \) 4 * q - 3 * q^3 + 4 * q^7 + q^9 - 10 * q^11 - 3 * q^13 - 6 * q^15 - 4 * q^17 - 10 * q^19 - 3 * q^21 - 4 * q^23 + 4 * q^25 - 9 * q^27 - 13 * q^29 + 3 * q^31 + 20 * q^33 - 11 * q^39 + q^41 - 8 * q^43 + 4 * q^45 - 5 * q^47 + 4 * q^49 - 4 * q^51 - 20 * q^53 - 22 * q^55 - 2 * q^57 - 14 * q^59 + 8 * q^61 + q^63 - 2 * q^65 - 18 * q^67 + 3 * q^69 - 25 * q^71 + 15 * q^73 - 11 * q^75 - 10 * q^77 + 10 * q^79 - 36 * q^83 - 28 * q^85 + q^87 + 4 * q^89 - 3 * q^91 + 17 * q^93 - 36 * q^95 + 6 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more