LMFDB - Embedded newform 3969.2.a.w.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3969,2,Mod(1,3969)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3969, 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("3969.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3969 = 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3969.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:	$31.6926245622$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.14013.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} + 6x + 3 $ x^4 - x^3 - 6x^2 + 6x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 567)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.53652$ of defining polynomial
Character	$\chi$	$=$	3969.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.53652 q^{2} +0.360904 q^{4} -3.15761 q^{5} -2.51851 q^{8} +O(q^{10})$	$q+1.53652 q^{2} +0.360904 q^{4} -3.15761 q^{5} -2.51851 q^{8} -4.85173 q^{10} -5.74916 q^{11} +0.360904 q^{13} -4.59156 q^{16} -2.77684 q^{17} +7.23065 q^{19} -1.13959 q^{20} -8.83372 q^{22} -0.824381 q^{23} +4.97047 q^{25} +0.554537 q^{26} -4.27819 q^{29} +4.98199 q^{31} -2.01801 q^{32} -4.26668 q^{34} +7.49083 q^{37} +11.1101 q^{38} +7.95246 q^{40} +3.33138 q^{41} -7.86975 q^{43} -2.07489 q^{44} -1.26668 q^{46} +3.48149 q^{47} +7.63725 q^{50} +0.130252 q^{52} -2.91544 q^{53} +18.1536 q^{55} -6.57354 q^{58} -2.39878 q^{59} -3.20113 q^{61} +7.65494 q^{62} +6.08239 q^{64} -1.13959 q^{65} +1.89927 q^{67} -1.00217 q^{68} +1.60957 q^{71} +15.4138 q^{73} +11.5098 q^{74} +2.60957 q^{76} +5.47282 q^{79} +14.4983 q^{80} +5.11874 q^{82} +13.0348 q^{83} +8.76817 q^{85} -12.0921 q^{86} +14.4793 q^{88} +14.2677 q^{89} -0.297522 q^{92} +5.34939 q^{94} -22.8315 q^{95} -16.0053 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + 5 q^{4} - 2 q^{5} - 3 q^{8}+O(q^{10}) $ 4 * q + q^2 + 5 * q^4 - 2 * q^5 - 3 * q^8	$ 4 q + q^{2} + 5 q^{4} - 2 q^{5} - 3 q^{8} + 7 q^{10} + 5 q^{11} + 5 q^{13} - q^{16} - 6 q^{17} + 8 q^{19} + 8 q^{20} - 7 q^{22} - 12 q^{23} + 8 q^{25} - q^{26} - 10 q^{29} + 18 q^{31} - 10 q^{32} + 20 q^{38} + 18 q^{40} + 5 q^{41} - 7 q^{43} + 13 q^{44} + 12 q^{46} + 21 q^{47} + 38 q^{50} + 25 q^{52} - 12 q^{53} + 26 q^{55} - 7 q^{58} - 6 q^{59} + 20 q^{61} - 18 q^{62} - 23 q^{64} + 8 q^{65} - 5 q^{67} - 51 q^{68} - 9 q^{71} + 6 q^{73} - 5 q^{76} - 10 q^{79} + 2 q^{80} + 35 q^{82} - 9 q^{83} - 9 q^{85} - 22 q^{86} + 18 q^{88} + 22 q^{89} - 36 q^{92} + 15 q^{94} - 16 q^{95} + 9 q^{97}+O(q^{100}) $ 4 * q + q^2 + 5 * q^4 - 2 * q^5 - 3 * q^8 + 7 * q^10 + 5 * q^11 + 5 * q^13 - q^16 - 6 * q^17 + 8 * q^19 + 8 * q^20 - 7 * q^22 - 12 * q^23 + 8 * q^25 - q^26 - 10 * q^29 + 18 * q^31 - 10 * q^32 + 20 * q^38 + 18 * q^40 + 5 * q^41 - 7 * q^43 + 13 * q^44 + 12 * q^46 + 21 * q^47 + 38 * q^50 + 25 * q^52 - 12 * q^53 + 26 * q^55 - 7 * q^58 - 6 * q^59 + 20 * q^61 - 18 * q^62 - 23 * q^64 + 8 * q^65 - 5 * q^67 - 51 * q^68 - 9 * q^71 + 6 * q^73 - 5 * q^76 - 10 * q^79 + 2 * q^80 + 35 * q^82 - 9 * q^83 - 9 * q^85 - 22 * q^86 + 18 * q^88 + 22 * q^89 - 36 * q^92 + 15 * q^94 - 16 * q^95 + 9 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.53652	1.08649	0.543243	−	0.839576i	$-0.317196\pi$
$2$	1.53652	1.08649	0.543243	+	0.839576i	$0.317196\pi$
$3$	0	0
$3$	0	0
$4$	0.360904	0.180452
$5$	−3.15761	−1.41212	−0.706062	−	0.708150i	$-0.749530\pi$
$5$	−3.15761	−1.41212	−0.706062	+	0.708150i	$0.749530\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−2.51851	−0.890428
$9$	0	0
$10$	−4.85173	−1.53425
$11$	−5.74916	−1.73344	−0.866719	−	0.498797i	$-0.833775\pi$
$11$	−5.74916	−1.73344	−0.866719	+	0.498797i	$0.833775\pi$
$12$	0	0
$13$	0.360904	0.100097	0.0500484	−	0.998747i	$-0.484062\pi$
$13$	0.360904	0.100097	0.0500484	+	0.998747i	$0.484062\pi$
$14$	0	0
$15$	0	0
$16$	−4.59156	−1.14789
$17$	−2.77684	−0.673483	−0.336741	−	0.941597i	$-0.609325\pi$
$17$	−2.77684	−0.673483	−0.336741	+	0.941597i	$0.609325\pi$
$18$	0	0
$19$	7.23065	1.65883	0.829413	−	0.558636i	$-0.188675\pi$
$19$	7.23065	1.65883	0.829413	+	0.558636i	$0.188675\pi$
$20$	−1.13959	−0.254820
$21$	0	0
$22$	−8.83372	−1.88336
$23$	−0.824381	−0.171895	−0.0859476	−	0.996300i	$-0.527392\pi$
$23$	−0.824381	−0.171895	−0.0859476	+	0.996300i	$0.527392\pi$
$24$	0	0
$25$	4.97047	0.994095
$26$	0.554537	0.108754
$27$	0	0
$28$	0	0
$29$	−4.27819	−0.794440	−0.397220	−	0.917723i	$-0.630025\pi$
$29$	−4.27819	−0.794440	−0.397220	+	0.917723i	$0.630025\pi$
$30$	0	0
$31$	4.98199	0.894791	0.447396	−	0.894336i	$-0.352352\pi$
$31$	4.98199	0.894791	0.447396	+	0.894336i	$0.352352\pi$
$32$	−2.01801	−0.356738
$33$	0	0
$34$	−4.26668	−0.731730
$35$	0	0
$36$	0	0
$37$	7.49083	1.23149	0.615743	−	0.787947i	$-0.288856\pi$
$37$	7.49083	1.23149	0.615743	+	0.787947i	$0.288856\pi$
$38$	11.1101	1.80229
$39$	0	0
$40$	7.95246	1.25739
$41$	3.33138	0.520274	0.260137	−	0.965572i	$-0.416232\pi$
$41$	3.33138	0.520274	0.260137	+	0.965572i	$0.416232\pi$
$42$	0	0
$43$	−7.86975	−1.20013	−0.600063	−	0.799953i	$-0.704858\pi$
$43$	−7.86975	−1.20013	−0.600063	+	0.799953i	$0.704858\pi$
$44$	−2.07489	−0.312802
$45$	0	0
$46$	−1.26668	−0.186762
$47$	3.48149	0.507828	0.253914	−	0.967227i	$-0.418282\pi$
$47$	3.48149	0.507828	0.253914	+	0.967227i	$0.418282\pi$
$48$	0	0
$49$	0	0
$50$	7.63725	1.08007
$51$	0	0
$52$	0.130252	0.0180626
$53$	−2.91544	−0.400467	−0.200233	−	0.979748i	$-0.564170\pi$
$53$	−2.91544	−0.400467	−0.200233	+	0.979748i	$0.564170\pi$
$54$	0	0
$55$	18.1536	2.44783
$56$	0	0
$57$	0	0
$58$	−6.57354	−0.863148
$59$	−2.39878	−0.312294	−0.156147	−	0.987734i	$-0.549907\pi$
$59$	−2.39878	−0.312294	−0.156147	+	0.987734i	$0.549907\pi$
$60$	0	0
$61$	−3.20113	−0.409862	−0.204931	−	0.978776i	$-0.565697\pi$
$61$	−3.20113	−0.409862	−0.204931	+	0.978776i	$0.565697\pi$
$62$	7.65494	0.972178
$63$	0	0
$64$	6.08239	0.760298
$65$	−1.13959	−0.141349
$66$	0	0
$67$	1.89927	0.232033	0.116017	−	0.993247i	$-0.462987\pi$
$67$	1.89927	0.232033	0.116017	+	0.993247i	$0.462987\pi$
$68$	−1.00217	−0.121531
$69$	0	0
$70$	0	0
$71$	1.60957	0.191021	0.0955104	−	0.995428i	$-0.469552\pi$
$71$	1.60957	0.191021	0.0955104	+	0.995428i	$0.469552\pi$
$72$	0	0
$73$	15.4138	1.80404	0.902022	−	0.431689i	$-0.142082\pi$
$73$	15.4138	1.80404	0.902022	+	0.431689i	$0.142082\pi$
$74$	11.5098	1.33799
$75$	0	0
$76$	2.60957	0.299338
$77$	0	0
$78$	0	0
$79$	5.47282	0.615740	0.307870	−	0.951428i	$-0.400384\pi$
$79$	5.47282	0.615740	0.307870	+	0.951428i	$0.400384\pi$
$80$	14.4983	1.62096
$81$	0	0
$82$	5.11874	0.565270
$83$	13.0348	1.43076	0.715380	−	0.698735i	$-0.246254\pi$
$83$	13.0348	1.43076	0.715380	+	0.698735i	$0.246254\pi$
$84$	0	0
$85$	8.76817	0.951041
$86$	−12.0921	−1.30392
$87$	0	0
$88$	14.4793	1.54350
$89$	14.2677	1.51237	0.756185	−	0.654358i	$-0.227061\pi$
$89$	14.2677	1.51237	0.756185	+	0.654358i	$0.227061\pi$
$90$	0	0
$91$	0	0
$92$	−0.297522	−0.0310188
$93$	0	0
$94$	5.34939	0.551748
$95$	−22.8315	−2.34247
$96$	0	0
$97$	−16.0053	−1.62509	−0.812547	−	0.582896i	$-0.801920\pi$
$97$	−16.0053	−1.62509	−0.812547	+	0.582896i	$0.801920\pi$
$98$	0	0
$99$	0	0
$100$	1.79386	0.179386
$101$	−3.00749	−0.299257	−0.149628	−	0.988742i	$-0.547808\pi$
$101$	−3.00749	−0.299257	−0.149628	+	0.988742i	$0.547808\pi$
$102$	0	0
$103$	−9.72063	−0.957802	−0.478901	−	0.877869i	$-0.658965\pi$
$103$	−9.72063	−0.957802	−0.478901	+	0.877869i	$0.658965\pi$
$104$	−0.908940	−0.0891289
$105$	0	0
$106$	−4.47964	−0.435101
$107$	10.4243	1.00775	0.503877	−	0.863775i	$-0.331907\pi$
$107$	10.4243	1.00775	0.503877	+	0.863775i	$0.331907\pi$
$108$	0	0
$109$	−4.67427	−0.447714	−0.223857	−	0.974622i	$-0.571865\pi$
$109$	−4.67427	−0.447714	−0.223857	+	0.974622i	$0.571865\pi$
$110$	27.8934	2.65953
$111$	0	0
$112$	0	0
$113$	4.68664	0.440882	0.220441	−	0.975400i	$-0.429250\pi$
$113$	4.68664	0.440882	0.220441	+	0.975400i	$0.429250\pi$
$114$	0	0
$115$	2.60307	0.242737
$116$	−1.54402	−0.143358
$117$	0	0
$118$	−3.68578	−0.339304
$119$	0	0
$120$	0	0
$121$	22.0529	2.00481
$122$	−4.91860	−0.445310
$123$	0	0
$124$	1.79802	0.161467
$125$	0.0932326	0.00833898
$126$	0	0
$127$	−9.15945	−0.812770	−0.406385	−	0.913702i	$-0.633211\pi$
$127$	−9.15945	−0.812770	−0.406385	+	0.913702i	$0.633211\pi$
$128$	13.3818	1.18279
$129$	0	0
$130$	−1.75101	−0.153574
$131$	−9.21165	−0.804825	−0.402413	−	0.915458i	$-0.631828\pi$
$131$	−9.21165	−0.804825	−0.402413	+	0.915458i	$0.631828\pi$
$132$	0	0
$133$	0	0
$134$	2.91828	0.252101
$135$	0	0
$136$	6.99350	0.599688
$137$	6.82438	0.583046	0.291523	−	0.956564i	$-0.405838\pi$
$137$	6.82438	0.583046	0.291523	+	0.956564i	$0.405838\pi$
$138$	0	0
$139$	5.68664	0.482334	0.241167	−	0.970484i	$-0.422470\pi$
$139$	5.68664	0.482334	0.241167	+	0.970484i	$0.422470\pi$
$140$	0	0
$141$	0	0
$142$	2.47314	0.207541
$143$	−2.07489	−0.173511
$144$	0	0
$145$	13.5088	1.12185
$146$	23.6836	1.96007
$147$	0	0
$148$	2.70347	0.222224
$149$	21.5885	1.76860	0.884301	−	0.466918i	$-0.154636\pi$
$149$	21.5885	1.76860	0.884301	+	0.466918i	$0.154636\pi$
$150$	0	0
$151$	−5.55553	−0.452102	−0.226051	−	0.974115i	$-0.572582\pi$
$151$	−5.55553	−0.452102	−0.226051	+	0.974115i	$0.572582\pi$
$152$	−18.2105	−1.47706
$153$	0	0
$154$	0	0
$155$	−15.7311	−1.26356
$156$	0	0
$157$	−6.07120	−0.484534	−0.242267	−	0.970210i	$-0.577891\pi$
$157$	−6.07120	−0.484534	−0.242267	+	0.970210i	$0.577891\pi$
$158$	8.40911	0.668993
$159$	0	0
$160$	6.37209	0.503758
$161$	0	0
$162$	0	0
$163$	−3.85259	−0.301758	−0.150879	−	0.988552i	$-0.548210\pi$
$163$	−3.85259	−0.301758	−0.150879	+	0.988552i	$0.548210\pi$
$164$	1.20231	0.0938844
$165$	0	0
$166$	20.0283	1.55450
$167$	3.53837	0.273807	0.136904	−	0.990584i	$-0.456285\pi$
$167$	3.53837	0.273807	0.136904	+	0.990584i	$0.456285\pi$
$168$	0	0
$169$	−12.8697	−0.989981
$170$	13.4725	1.03329
$171$	0	0
$172$	−2.84022	−0.216565
$173$	9.85358	0.749154	0.374577	−	0.927196i	$-0.377788\pi$
$173$	9.85358	0.749154	0.374577	+	0.927196i	$0.377788\pi$
$174$	0	0
$175$	0	0
$176$	26.3976	1.98979
$177$	0	0
$178$	21.9226	1.64317
$179$	−19.8971	−1.48718	−0.743590	−	0.668636i	$-0.766878\pi$
$179$	−19.8971	−1.48718	−0.743590	+	0.668636i	$0.766878\pi$
$180$	0	0
$181$	−12.0930	−0.898869	−0.449434	−	0.893313i	$-0.648374\pi$
$181$	−12.0930	−0.898869	−0.449434	+	0.893313i	$0.648374\pi$
$182$	0	0
$183$	0	0
$184$	2.07621	0.153060
$185$	−23.6531	−1.73901
$186$	0	0
$187$	15.9645	1.16744
$188$	1.25648	0.0916385
$189$	0	0
$190$	−35.0812	−2.54506
$191$	9.71964	0.703288	0.351644	−	0.936134i	$-0.385623\pi$
$191$	9.71964	0.703288	0.351644	+	0.936134i	$0.385623\pi$
$192$	0	0
$193$	−2.30185	−0.165691	−0.0828454	−	0.996562i	$-0.526401\pi$
$193$	−2.30185	−0.165691	−0.0828454	+	0.996562i	$0.526401\pi$
$194$	−24.5925	−1.76564
$195$	0	0
$196$	0	0
$197$	5.06470	0.360845	0.180422	−	0.983589i	$-0.442254\pi$
$197$	5.06470	0.360845	0.180422	+	0.983589i	$0.442254\pi$
$198$	0	0
$199$	9.95332	0.705572	0.352786	−	0.935704i	$-0.385234\pi$
$199$	9.95332	0.705572	0.352786	+	0.935704i	$0.385234\pi$
$200$	−12.5182	−0.885169
$201$	0	0
$202$	−4.62108	−0.325138
$203$	0	0
$204$	0	0
$205$	−10.5192	−0.734691
$206$	−14.9360	−1.04064
$207$	0	0
$208$	−1.65711	−0.114900
$209$	−41.5702	−2.87547
$210$	0	0
$211$	−23.3007	−1.60408	−0.802042	−	0.597267i	$-0.796253\pi$
$211$	−23.3007	−1.60408	−0.802042	+	0.597267i	$0.796253\pi$
$212$	−1.05219	−0.0722650
$213$	0	0
$214$	16.0172	1.09491
$215$	24.8496	1.69473
$216$	0	0
$217$	0	0
$218$	−7.18212	−0.486435
$219$	0	0
$220$	6.55170	0.441715
$221$	−1.00217	−0.0674134
$222$	0	0
$223$	11.8754	0.795235	0.397618	−	0.917551i	$-0.369837\pi$
$223$	11.8754	0.795235	0.397618	+	0.917551i	$0.369837\pi$
$224$	0	0
$225$	0	0
$226$	7.20113	0.479012
$227$	−5.96081	−0.395633	−0.197816	−	0.980239i	$-0.563385\pi$
$227$	−5.96081	−0.395633	−0.197816	+	0.980239i	$0.563385\pi$
$228$	0	0
$229$	23.7206	1.56750	0.783752	−	0.621074i	$-0.213304\pi$
$229$	23.7206	1.56750	0.783752	+	0.621074i	$0.213304\pi$
$230$	3.99968	0.263731
$231$	0	0
$232$	10.7747	0.707392
$233$	−13.8592	−0.907948	−0.453974	−	0.891015i	$-0.649994\pi$
$233$	−13.8592	−0.907948	−0.453974	+	0.891015i	$0.649994\pi$
$234$	0	0
$235$	−10.9932	−0.717116
$236$	−0.865728	−0.0563541
$237$	0	0
$238$	0	0
$239$	22.3426	1.44522	0.722610	−	0.691256i	$-0.242942\pi$
$239$	22.3426	1.44522	0.722610	+	0.691256i	$0.242942\pi$
$240$	0	0
$241$	−9.72063	−0.626161	−0.313080	−	0.949727i	$-0.601361\pi$
$241$	−9.72063	−0.626161	−0.313080	+	0.949727i	$0.601361\pi$
$242$	33.8847	2.17819
$243$	0	0
$244$	−1.15530	−0.0739604
$245$	0	0
$246$	0	0
$247$	2.60957	0.166043
$248$	−12.5472	−0.796747
$249$	0	0
$250$	0.143254	0.00906018
$251$	6.51950	0.411507	0.205754	−	0.978604i	$-0.434035\pi$
$251$	6.51950	0.411507	0.205754	+	0.978604i	$0.434035\pi$
$252$	0	0
$253$	4.73950	0.297970
$254$	−14.0737	−0.883063
$255$	0	0
$256$	8.39661	0.524788
$257$	−5.53002	−0.344953	−0.172477	−	0.985014i	$-0.555177\pi$
$257$	−5.53002	−0.344953	−0.172477	+	0.985014i	$0.555177\pi$
$258$	0	0
$259$	0	0
$260$	−0.411283	−0.0255067
$261$	0	0
$262$	−14.1539	−0.874431
$263$	8.69499	0.536156	0.268078	−	0.963397i	$-0.413612\pi$
$263$	8.69499	0.536156	0.268078	+	0.963397i	$0.413612\pi$
$264$	0	0
$265$	9.20581	0.565509
$266$	0	0
$267$	0	0
$268$	0.685455	0.0418709
$269$	3.77901	0.230410	0.115205	−	0.993342i	$-0.463247\pi$
$269$	3.77901	0.230410	0.115205	+	0.993342i	$0.463247\pi$
$270$	0	0
$271$	−6.61990	−0.402130	−0.201065	−	0.979578i	$-0.564440\pi$
$271$	−6.61990	−0.402130	−0.201065	+	0.979578i	$0.564440\pi$
$272$	12.7500	0.773083
$273$	0	0
$274$	10.4858	0.633472
$275$	−28.5761	−1.72320
$276$	0	0
$277$	−25.2658	−1.51808	−0.759038	−	0.651046i	$-0.774330\pi$
$277$	−25.2658	−1.51808	−0.759038	+	0.651046i	$0.774330\pi$
$278$	8.73765	0.524049
$279$	0	0
$280$	0	0
$281$	7.42442	0.442904	0.221452	−	0.975171i	$-0.428920\pi$
$281$	7.42442	0.442904	0.221452	+	0.975171i	$0.428920\pi$
$282$	0	0
$283$	−15.4203	−0.916640	−0.458320	−	0.888787i	$-0.651549\pi$
$283$	−15.4203	−0.916640	−0.458320	+	0.888787i	$0.651549\pi$
$284$	0.580900	0.0344701
$285$	0	0
$286$	−3.18812	−0.188518
$287$	0	0
$288$	0	0
$289$	−9.28916	−0.546421
$290$	20.7567	1.21887
$291$	0	0
$292$	5.56289	0.325543
$293$	−30.5797	−1.78649	−0.893243	−	0.449574i	$-0.851576\pi$
$293$	−30.5797	−1.78649	−0.893243	+	0.449574i	$0.851576\pi$
$294$	0	0
$295$	7.57440	0.440999
$296$	−18.8657	−1.09655
$297$	0	0
$298$	33.1713	1.92156
$299$	−0.297522	−0.0172061
$300$	0	0
$301$	0	0
$302$	−8.53620	−0.491203
$303$	0	0
$304$	−33.1999	−1.90415
$305$	10.1079	0.578776
$306$	0	0
$307$	28.7794	1.64252	0.821262	−	0.570551i	$-0.193270\pi$
$307$	28.7794	1.64252	0.821262	+	0.570551i	$0.193270\pi$
$308$	0	0
$309$	0	0
$310$	−24.1713	−1.37284
$311$	2.57255	0.145876	0.0729380	−	0.997336i	$-0.476762\pi$
$311$	2.57255	0.145876	0.0729380	+	0.997336i	$0.476762\pi$
$312$	0	0
$313$	18.5145	1.04650	0.523250	−	0.852179i	$-0.324719\pi$
$313$	18.5145	1.04650	0.523250	+	0.852179i	$0.324719\pi$
$314$	−9.32854	−0.526440
$315$	0	0
$316$	1.97516	0.111111
$317$	15.5182	0.871588	0.435794	−	0.900046i	$-0.356468\pi$
$317$	15.5182	0.871588	0.435794	+	0.900046i	$0.356468\pi$
$318$	0	0
$319$	24.5960	1.37711
$320$	−19.2058	−1.07364
$321$	0	0
$322$	0	0
$323$	−20.0784	−1.11719
$324$	0	0
$325$	1.79386	0.0995056
$326$	−5.91960	−0.327856
$327$	0	0
$328$	−8.39011	−0.463266
$329$	0	0
$330$	0	0
$331$	31.5904	1.73636	0.868182	−	0.496246i	$-0.165289\pi$
$331$	31.5904	1.73636	0.868182	+	0.496246i	$0.165289\pi$
$332$	4.70433	0.258183
$333$	0	0
$334$	5.43679	0.297488
$335$	−5.99716	−0.327660
$336$	0	0
$337$	−11.4081	−0.621440	−0.310720	−	0.950502i	$-0.600570\pi$
$337$	−11.4081	−0.621440	−0.310720	+	0.950502i	$0.600570\pi$
$338$	−19.7747	−1.07560
$339$	0	0
$340$	3.16446	0.171617
$341$	−28.6422	−1.55106
$342$	0	0
$343$	0	0
$344$	19.8200	1.06862
$345$	0	0
$346$	15.1403	0.813945
$347$	−4.08140	−0.219101	−0.109550	−	0.993981i	$-0.534941\pi$
$347$	−4.08140	−0.219101	−0.109550	+	0.993981i	$0.534941\pi$
$348$	0	0
$349$	24.2779	1.29956	0.649782	−	0.760120i	$-0.274860\pi$
$349$	24.2779	1.29956	0.649782	+	0.760120i	$0.274860\pi$
$350$	0	0
$351$	0	0
$352$	11.6019	0.618383
$353$	34.0790	1.81384	0.906922	−	0.421299i	$-0.138426\pi$
$353$	34.0790	1.81384	0.906922	+	0.421299i	$0.138426\pi$
$354$	0	0
$355$	−5.08239	−0.269745
$356$	5.14926	0.272910
$357$	0	0
$358$	−30.5724	−1.61580
$359$	5.48931	0.289715	0.144857	−	0.989453i	$-0.453728\pi$
$359$	5.48931	0.289715	0.144857	+	0.989453i	$0.453728\pi$
$360$	0	0
$361$	33.2823	1.75170
$362$	−18.5812	−0.976608
$363$	0	0
$364$	0	0
$365$	−48.6706	−2.54754
$366$	0	0
$367$	13.6391	0.711955	0.355978	−	0.934495i	$-0.384148\pi$
$367$	13.6391	0.711955	0.355978	+	0.934495i	$0.384148\pi$
$368$	3.78519	0.197317
$369$	0	0
$370$	−36.3435	−1.88941
$371$	0	0
$372$	0	0
$373$	8.92379	0.462056	0.231028	−	0.972947i	$-0.425791\pi$
$373$	8.92379	0.462056	0.231028	+	0.972947i	$0.425791\pi$
$374$	24.5298	1.26841
$375$	0	0
$376$	−8.76817	−0.452184
$377$	−1.54402	−0.0795209
$378$	0	0
$379$	29.7035	1.52576	0.762882	−	0.646537i	$-0.223784\pi$
$379$	29.7035	1.52576	0.762882	+	0.646537i	$0.223784\pi$
$380$	−8.23999	−0.422703
$381$	0	0
$382$	14.9344	0.764113
$383$	17.7101	0.904944	0.452472	−	0.891779i	$-0.350542\pi$
$383$	17.7101	0.904944	0.452472	+	0.891779i	$0.350542\pi$
$384$	0	0
$385$	0	0
$386$	−3.53685	−0.180021
$387$	0	0
$388$	−5.77638	−0.293251
$389$	21.3255	1.08125	0.540624	−	0.841264i	$-0.318188\pi$
$389$	21.3255	1.08125	0.540624	+	0.841264i	$0.318188\pi$
$390$	0	0
$391$	2.28917	0.115768
$392$	0	0
$393$	0	0
$394$	7.78203	0.392053
$395$	−17.2810	−0.869501
$396$	0	0
$397$	2.23566	0.112205	0.0561024	−	0.998425i	$-0.482133\pi$
$397$	2.23566	0.112205	0.0561024	+	0.998425i	$0.482133\pi$
$398$	15.2935	0.766594
$399$	0	0
$400$	−22.8222	−1.14111
$401$	8.73702	0.436306	0.218153	−	0.975915i	$-0.429997\pi$
$401$	8.73702	0.436306	0.218153	+	0.975915i	$0.429997\pi$
$402$	0	0
$403$	1.79802	0.0895656
$404$	−1.08542	−0.0540014
$405$	0	0
$406$	0	0
$407$	−43.0660	−2.13470
$408$	0	0
$409$	28.6920	1.41873	0.709363	−	0.704843i	$-0.248983\pi$
$409$	28.6920	1.41873	0.709363	+	0.704843i	$0.248983\pi$
$410$	−16.1630	−0.798232
$411$	0	0
$412$	−3.50821	−0.172837
$413$	0	0
$414$	0	0
$415$	−41.1589	−2.02041
$416$	−0.728309	−0.0357083
$417$	0	0
$418$	−63.8736	−3.12416
$419$	−8.64442	−0.422307	−0.211154	−	0.977453i	$-0.567722\pi$
$419$	−8.64442	−0.422307	−0.211154	+	0.977453i	$0.567722\pi$
$420$	0	0
$421$	18.4669	0.900024	0.450012	−	0.893022i	$-0.351420\pi$
$421$	18.4669	0.900024	0.450012	+	0.893022i	$0.351420\pi$
$422$	−35.8020	−1.74282
$423$	0	0
$424$	7.34257	0.356587
$425$	−13.8022	−0.669506
$426$	0	0
$427$	0	0
$428$	3.76216	0.181851
$429$	0	0
$430$	38.1819	1.84130
$431$	−5.80736	−0.279731	−0.139865	−	0.990171i	$-0.544667\pi$
$431$	−5.80736	−0.279731	−0.139865	+	0.990171i	$0.544667\pi$
$432$	0	0
$433$	−3.63877	−0.174868	−0.0874341	−	0.996170i	$-0.527867\pi$
$433$	−3.63877	−0.174868	−0.0874341	+	0.996170i	$0.527867\pi$
$434$	0	0
$435$	0	0
$436$	−1.68696	−0.0807908
$437$	−5.96081	−0.285144
$438$	0	0
$439$	18.6619	0.890684	0.445342	−	0.895361i	$-0.353082\pi$
$439$	18.6619	0.890684	0.445342	+	0.895361i	$0.353082\pi$
$440$	−45.7200	−2.17961
$441$	0	0
$442$	−1.53986	−0.0732437
$443$	−26.0911	−1.23962	−0.619812	−	0.784750i	$-0.712791\pi$
$443$	−26.0911	−1.23962	−0.619812	+	0.784750i	$0.712791\pi$
$444$	0	0
$445$	−45.0517	−2.13565
$446$	18.2468	0.864012
$447$	0	0
$448$	0	0
$449$	−5.63824	−0.266085	−0.133042	−	0.991110i	$-0.542475\pi$
$449$	−5.63824	−0.266085	−0.133042	+	0.991110i	$0.542475\pi$
$450$	0	0
$451$	−19.1526	−0.901862
$452$	1.69142	0.0795579
$453$	0	0
$454$	−9.15892	−0.429849
$455$	0	0
$456$	0	0
$457$	−19.9923	−0.935201	−0.467601	−	0.883940i	$-0.654881\pi$
$457$	−19.9923	−0.935201	−0.467601	+	0.883940i	$0.654881\pi$
$458$	36.4473	1.70307
$459$	0	0
$460$	0.939457	0.0438024
$461$	−37.4495	−1.74420	−0.872098	−	0.489332i	$-0.837241\pi$
$461$	−37.4495	−1.74420	−0.872098	+	0.489332i	$0.837241\pi$
$462$	0	0
$463$	2.45513	0.114099	0.0570497	−	0.998371i	$-0.481831\pi$
$463$	2.45513	0.114099	0.0570497	+	0.998371i	$0.481831\pi$
$464$	19.6436	0.911929
$465$	0	0
$466$	−21.2950	−0.986473
$467$	−33.4107	−1.54606	−0.773032	−	0.634367i	$-0.781261\pi$
$467$	−33.4107	−1.54606	−0.773032	+	0.634367i	$0.781261\pi$
$468$	0	0
$469$	0	0
$470$	−16.8913	−0.779136
$471$	0	0
$472$	6.04135	0.278076
$473$	45.2445	2.08034
$474$	0	0
$475$	35.9398	1.64903
$476$	0	0
$477$	0	0
$478$	34.3299	1.57021
$479$	−34.6936	−1.58519	−0.792595	−	0.609748i	$-0.791271\pi$
$479$	−34.6936	−1.58519	−0.792595	+	0.609748i	$0.791271\pi$
$480$	0	0
$481$	2.70347	0.123268
$482$	−14.9360	−0.680315
$483$	0	0
$484$	7.95896	0.361771
$485$	50.5385	2.29483
$486$	0	0
$487$	0.959817	0.0434935	0.0217467	−	0.999764i	$-0.493077\pi$
$487$	0.959817	0.0434935	0.0217467	+	0.999764i	$0.493077\pi$
$488$	8.06207	0.364953
$489$	0	0
$490$	0	0
$491$	15.7738	0.711862	0.355931	−	0.934512i	$-0.384164\pi$
$491$	15.7738	0.711862	0.355931	+	0.934512i	$0.384164\pi$
$492$	0	0
$493$	11.8799	0.535042
$494$	4.00966	0.180403
$495$	0	0
$496$	−22.8751	−1.02712
$497$	0	0
$498$	0	0
$499$	−19.1287	−0.856320	−0.428160	−	0.903703i	$-0.640838\pi$
$499$	−19.1287	−0.856320	−0.428160	+	0.903703i	$0.640838\pi$
$500$	0.0336480	0.00150478
$501$	0	0
$502$	10.0174	0.447097
$503$	33.3898	1.48878	0.744388	−	0.667747i	$-0.232741\pi$
$503$	33.3898	1.48878	0.744388	+	0.667747i	$0.232741\pi$
$504$	0	0
$505$	9.49648	0.422588
$506$	7.28235	0.323740
$507$	0	0
$508$	−3.30568	−0.146666
$509$	−33.5176	−1.48564	−0.742822	−	0.669489i	$-0.766513\pi$
$509$	−33.5176	−1.48564	−0.742822	+	0.669489i	$0.766513\pi$
$510$	0	0
$511$	0	0
$512$	−13.8619	−0.612617
$513$	0	0
$514$	−8.49701	−0.374787
$515$	30.6939	1.35254
$516$	0	0
$517$	−20.0157	−0.880287
$518$	0	0
$519$	0	0
$520$	2.87007	0.125861
$521$	26.7243	1.17081	0.585407	−	0.810740i	$-0.300935\pi$
$521$	26.7243	1.17081	0.585407	+	0.810740i	$0.300935\pi$
$522$	0	0
$523$	−17.0644	−0.746173	−0.373086	−	0.927797i	$-0.621701\pi$
$523$	−17.0644	−0.746173	−0.373086	+	0.927797i	$0.621701\pi$
$524$	−3.32452	−0.145232
$525$	0	0
$526$	13.3600	0.582526
$527$	−13.8342	−0.602626
$528$	0	0
$529$	−22.3204	−0.970452
$530$	14.1449	0.614417
$531$	0	0
$532$	0	0
$533$	1.20231	0.0520777
$534$	0	0
$535$	−32.9158	−1.42307
$536$	−4.78334	−0.206609
$537$	0	0
$538$	5.80654	0.250338
$539$	0	0
$540$	0	0
$541$	12.3426	0.530648	0.265324	−	0.964159i	$-0.414521\pi$
$541$	12.3426	0.530648	0.265324	+	0.964159i	$0.414521\pi$
$542$	−10.1716	−0.436909
$543$	0	0
$544$	5.60370	0.240257
$545$	14.7595	0.632227
$546$	0	0
$547$	23.4424	1.00233	0.501163	−	0.865353i	$-0.332906\pi$
$547$	23.4424	1.00233	0.501163	+	0.865353i	$0.332906\pi$
$548$	2.46294	0.105212
$549$	0	0
$550$	−43.9078	−1.87223
$551$	−30.9341	−1.31784
$552$	0	0
$553$	0	0
$554$	−38.8215	−1.64937
$555$	0	0
$556$	2.05233	0.0870381
$557$	−24.5115	−1.03858	−0.519292	−	0.854597i	$-0.673804\pi$
$557$	−24.5115	−1.03858	−0.519292	+	0.854597i	$0.673804\pi$
$558$	0	0
$559$	−2.84022	−0.120129
$560$	0	0
$561$	0	0
$562$	11.4078	0.481209
$563$	13.3714	0.563538	0.281769	−	0.959482i	$-0.409079\pi$
$563$	13.3714	0.563538	0.281769	+	0.959482i	$0.409079\pi$
$564$	0	0
$565$	−14.7985	−0.622580
$566$	−23.6936	−0.995916
$567$	0	0
$568$	−4.05372	−0.170090
$569$	−14.2488	−0.597341	−0.298670	−	0.954356i	$-0.596543\pi$
$569$	−14.2488	−0.597341	−0.298670	+	0.954356i	$0.596543\pi$
$570$	0	0
$571$	−29.3237	−1.22716	−0.613579	−	0.789633i	$-0.710271\pi$
$571$	−29.3237	−1.22716	−0.613579	+	0.789633i	$0.710271\pi$
$572$	−0.748837	−0.0313105
$573$	0	0
$574$	0	0
$575$	−4.09756	−0.170880
$576$	0	0
$577$	18.1001	0.753516	0.376758	−	0.926312i	$-0.377039\pi$
$577$	18.1001	0.753516	0.376758	+	0.926312i	$0.377039\pi$
$578$	−14.2730	−0.593679
$579$	0	0
$580$	4.87539	0.202440
$581$	0	0
$582$	0	0
$583$	16.7613	0.694184
$584$	−38.8197	−1.60637
$585$	0	0
$586$	−46.9865	−1.94099
$587$	−6.52804	−0.269441	−0.134721	−	0.990884i	$-0.543014\pi$
$587$	−6.52804	−0.269441	−0.134721	+	0.990884i	$0.543014\pi$
$588$	0	0
$589$	36.0230	1.48430
$590$	11.6382	0.479139
$591$	0	0
$592$	−34.3946	−1.41361
$593$	25.4405	1.04471	0.522357	−	0.852727i	$-0.325053\pi$
$593$	25.4405	1.04471	0.522357	+	0.852727i	$0.325053\pi$
$594$	0	0
$595$	0	0
$596$	7.79138	0.319147
$597$	0	0
$598$	−0.457150	−0.0186942
$599$	6.17931	0.252480	0.126240	−	0.992000i	$-0.459709\pi$
$599$	6.17931	0.252480	0.126240	+	0.992000i	$0.459709\pi$
$600$	0	0
$601$	12.9344	0.527607	0.263804	−	0.964576i	$-0.415023\pi$
$601$	12.9344	0.527607	0.263804	+	0.964576i	$0.415023\pi$
$602$	0	0
$603$	0	0
$604$	−2.00501	−0.0815828
$605$	−69.6342	−2.83103
$606$	0	0
$607$	23.0756	0.936608	0.468304	−	0.883567i	$-0.344865\pi$
$607$	23.0756	0.936608	0.468304	+	0.883567i	$0.344865\pi$
$608$	−14.5916	−0.591766
$609$	0	0
$610$	15.5310	0.628832
$611$	1.25648	0.0508319
$612$	0	0
$613$	−17.6273	−0.711958	−0.355979	−	0.934494i	$-0.615853\pi$
$613$	−17.6273	−0.711958	−0.355979	+	0.934494i	$0.615853\pi$
$614$	44.2201	1.78458
$615$	0	0
$616$	0	0
$617$	21.6441	0.871358	0.435679	−	0.900102i	$-0.356508\pi$
$617$	21.6441	0.871358	0.435679	+	0.900102i	$0.356508\pi$
$618$	0	0
$619$	9.57941	0.385029	0.192514	−	0.981294i	$-0.438336\pi$
$619$	9.57941	0.385029	0.192514	+	0.981294i	$0.438336\pi$
$620$	−5.67743	−0.228011
$621$	0	0
$622$	3.95278	0.158492
$623$	0	0
$624$	0	0
$625$	−25.1468	−1.00587
$626$	28.4479	1.13701
$627$	0	0
$628$	−2.19112	−0.0874352
$629$	−20.8008	−0.829384
$630$	0	0
$631$	−31.1742	−1.24103	−0.620514	−	0.784196i	$-0.713076\pi$
$631$	−31.1742	−1.24103	−0.620514	+	0.784196i	$0.713076\pi$
$632$	−13.7833	−0.548272
$633$	0	0
$634$	23.8441	0.946968
$635$	28.9219	1.14773
$636$	0	0
$637$	0	0
$638$	37.7924	1.49621
$639$	0	0
$640$	−42.2543	−1.67025
$641$	−8.25214	−0.325940	−0.162970	−	0.986631i	$-0.552107\pi$
$641$	−8.25214	−0.325940	−0.162970	+	0.986631i	$0.552107\pi$
$642$	0	0
$643$	−24.4318	−0.963495	−0.481748	−	0.876310i	$-0.659998\pi$
$643$	−24.4318	−0.963495	−0.481748	+	0.876310i	$0.659998\pi$
$644$	0	0
$645$	0	0
$646$	−30.8509	−1.21381
$647$	−38.6864	−1.52092	−0.760461	−	0.649384i	$-0.775027\pi$
$647$	−38.6864	−1.52092	−0.760461	+	0.649384i	$0.775027\pi$
$648$	0	0
$649$	13.7910	0.541343
$650$	2.75631	0.108111
$651$	0	0
$652$	−1.39041	−0.0544528
$653$	7.65430	0.299536	0.149768	−	0.988721i	$-0.452147\pi$
$653$	7.65430	0.299536	0.149768	+	0.988721i	$0.452147\pi$
$654$	0	0
$655$	29.0867	1.13651
$656$	−15.2962	−0.597217
$657$	0	0
$658$	0	0
$659$	38.5145	1.50031	0.750156	−	0.661261i	$-0.229978\pi$
$659$	38.5145	1.50031	0.750156	+	0.661261i	$0.229978\pi$
$660$	0	0
$661$	−32.2132	−1.25295	−0.626474	−	0.779443i	$-0.715502\pi$
$661$	−32.2132	−1.25295	−0.626474	+	0.779443i	$0.715502\pi$
$662$	48.5393	1.88654
$663$	0	0
$664$	−32.8284	−1.27399
$665$	0	0
$666$	0	0
$667$	3.52686	0.136561
$668$	1.27701	0.0494091
$669$	0	0
$670$	−9.21478	−0.355998
$671$	18.4038	0.710470
$672$	0	0
$673$	1.26136	0.0486218	0.0243109	−	0.999704i	$-0.492261\pi$
$673$	1.26136	0.0486218	0.0243109	+	0.999704i	$0.492261\pi$
$674$	−17.5288	−0.675186
$675$	0	0
$676$	−4.64474	−0.178644
$677$	−14.8318	−0.570031	−0.285016	−	0.958523i	$-0.591999\pi$
$677$	−14.8318	−0.570031	−0.285016	+	0.958523i	$0.591999\pi$
$678$	0	0
$679$	0	0
$680$	−22.0827	−0.846833
$681$	0	0
$682$	−44.0095	−1.68521
$683$	−5.53120	−0.211646	−0.105823	−	0.994385i	$-0.533748\pi$
$683$	−5.53120	−0.211646	−0.105823	+	0.994385i	$0.533748\pi$
$684$	0	0
$685$	−21.5487	−0.823334
$686$	0	0
$687$	0	0
$688$	36.1344	1.37761
$689$	−1.05219	−0.0400854
$690$	0	0
$691$	−8.63792	−0.328602	−0.164301	−	0.986410i	$-0.552537\pi$
$691$	−8.63792	−0.328602	−0.164301	+	0.986410i	$0.552537\pi$
$692$	3.55620	0.135186
$693$	0	0
$694$	−6.27116	−0.238050
$695$	−17.9562	−0.681116
$696$	0	0
$697$	−9.25070	−0.350395
$698$	37.3035	1.41196
$699$	0	0
$700$	0	0
$701$	26.5897	1.00428	0.502140	−	0.864786i	$-0.332546\pi$
$701$	26.5897	1.00428	0.502140	+	0.864786i	$0.332546\pi$
$702$	0	0
$703$	54.1636	2.04282
$704$	−34.9686	−1.31793
$705$	0	0
$706$	52.3632	1.97072
$707$	0	0
$708$	0	0
$709$	10.1261	0.380294	0.190147	−	0.981756i	$-0.439104\pi$
$709$	10.1261	0.380294	0.190147	+	0.981756i	$0.439104\pi$
$710$	−7.80921	−0.293074
$711$	0	0
$712$	−35.9333	−1.34666
$713$	−4.10705	−0.153810
$714$	0	0
$715$	6.55170	0.245020
$716$	−7.18094	−0.268364
$717$	0	0
$718$	8.43445	0.314771
$719$	−32.3876	−1.20785	−0.603927	−	0.797040i	$-0.706398\pi$
$719$	−32.3876	−1.20785	−0.603927	+	0.797040i	$0.706398\pi$
$720$	0	0
$721$	0	0
$722$	51.1391	1.90320
$723$	0	0
$724$	−4.36443	−0.162203
$725$	−21.2646	−0.789749
$726$	0	0
$727$	−25.2348	−0.935907	−0.467953	−	0.883753i	$-0.655008\pi$
$727$	−25.2348	−0.935907	−0.467953	+	0.883753i	$0.655008\pi$
$728$	0	0
$729$	0	0
$730$	−74.7835	−2.76786
$731$	21.8530	0.808264
$732$	0	0
$733$	−2.84672	−0.105146	−0.0525731	−	0.998617i	$-0.516742\pi$
$733$	−2.84672	−0.105146	−0.0525731	+	0.998617i	$0.516742\pi$
$734$	20.9568	0.773529
$735$	0	0
$736$	1.66361	0.0613215
$737$	−10.9192	−0.402215
$738$	0	0
$739$	24.5417	0.902779	0.451390	−	0.892327i	$-0.350928\pi$
$739$	24.5417	0.902779	0.451390	+	0.892327i	$0.350928\pi$
$740$	−8.53649	−0.313808
$741$	0	0
$742$	0	0
$743$	4.59070	0.168416	0.0842082	−	0.996448i	$-0.473164\pi$
$743$	4.59070	0.168416	0.0842082	+	0.996448i	$0.473164\pi$
$744$	0	0
$745$	−68.1681	−2.49748
$746$	13.7116	0.502018
$747$	0	0
$748$	5.76165	0.210667
$749$	0	0
$750$	0	0
$751$	31.4356	1.14710	0.573551	−	0.819170i	$-0.305566\pi$
$751$	31.4356	1.14710	0.573551	+	0.819170i	$0.305566\pi$
$752$	−15.9855	−0.582930
$753$	0	0
$754$	−2.37242	−0.0863983
$755$	17.5422	0.638425
$756$	0	0
$757$	29.5432	1.07376	0.536882	−	0.843657i	$-0.319602\pi$
$757$	29.5432	1.07376	0.536882	+	0.843657i	$0.319602\pi$
$758$	45.6401	1.65772
$759$	0	0
$760$	57.5015	2.08580
$761$	46.4873	1.68516	0.842582	−	0.538567i	$-0.181034\pi$
$761$	46.4873	1.68516	0.842582	+	0.538567i	$0.181034\pi$
$762$	0	0
$763$	0	0
$764$	3.50785	0.126910
$765$	0	0
$766$	27.2120	0.983209
$767$	−0.865728	−0.0312596
$768$	0	0
$769$	−14.1706	−0.511006	−0.255503	−	0.966808i	$-0.582241\pi$
$769$	−14.1706	−0.511006	−0.255503	+	0.966808i	$0.582241\pi$
$770$	0	0
$771$	0	0
$772$	−0.830747	−0.0298992
$773$	34.0085	1.22320	0.611600	−	0.791167i	$-0.290526\pi$
$773$	34.0085	1.22320	0.611600	+	0.791167i	$0.290526\pi$
$774$	0	0
$775$	24.7628	0.889507
$776$	40.3096	1.44703
$777$	0	0
$778$	32.7672	1.17476
$779$	24.0880	0.863043
$780$	0	0
$781$	−9.25368	−0.331123
$782$	3.51737	0.125781
$783$	0	0
$784$	0	0
$785$	19.1705	0.684223
$786$	0	0
$787$	29.1059	1.03751	0.518757	−	0.854921i	$-0.326395\pi$
$787$	29.1059	1.03751	0.518757	+	0.854921i	$0.326395\pi$
$788$	1.82787	0.0651151
$789$	0	0
$790$	−26.5527	−0.944701
$791$	0	0
$792$	0	0
$793$	−1.15530	−0.0410259
$794$	3.43515	0.121909
$795$	0	0
$796$	3.59219	0.127322
$797$	18.5038	0.655439	0.327720	−	0.944775i	$-0.393720\pi$
$797$	18.5038	0.655439	0.327720	+	0.944775i	$0.393720\pi$
$798$	0	0
$799$	−9.66754	−0.342013
$800$	−10.0305	−0.354631
$801$	0	0
$802$	13.4246	0.474040
$803$	−88.6162	−3.12720
$804$	0	0
$805$	0	0
$806$	2.76270	0.0973118
$807$	0	0
$808$	7.57440	0.266466
$809$	41.0813	1.44434	0.722172	−	0.691714i	$-0.243144\pi$
$809$	41.0813	1.44434	0.722172	+	0.691714i	$0.243144\pi$
$810$	0	0
$811$	43.1361	1.51471	0.757357	−	0.653001i	$-0.226490\pi$
$811$	43.1361	1.51471	0.757357	+	0.653001i	$0.226490\pi$
$812$	0	0
$813$	0	0
$814$	−66.1719	−2.31932
$815$	12.1650	0.426120
$816$	0	0
$817$	−56.9034	−1.99080
$818$	44.0859	1.54143
$819$	0	0
$820$	−3.79641	−0.132576
$821$	13.1748	0.459802	0.229901	−	0.973214i	$-0.426160\pi$
$821$	13.1748	0.459802	0.229901	+	0.973214i	$0.426160\pi$
$822$	0	0
$823$	−11.9817	−0.417654	−0.208827	−	0.977953i	$-0.566965\pi$
$823$	−11.9817	−0.417654	−0.208827	+	0.977953i	$0.566965\pi$
$824$	24.4815	0.852853
$825$	0	0
$826$	0	0
$827$	29.9879	1.04278	0.521391	−	0.853318i	$-0.325413\pi$
$827$	29.9879	1.04278	0.521391	+	0.853318i	$0.325413\pi$
$828$	0	0
$829$	−26.9238	−0.935102	−0.467551	−	0.883966i	$-0.654864\pi$
$829$	−26.9238	−0.935102	−0.467551	+	0.883966i	$0.654864\pi$
$830$	−63.2416	−2.19515
$831$	0	0
$832$	2.19516	0.0761034
$833$	0	0
$834$	0	0
$835$	−11.1728	−0.386650
$836$	−15.0028	−0.518884
$837$	0	0
$838$	−13.2823	−0.458831
$839$	11.2238	0.387490	0.193745	−	0.981052i	$-0.437937\pi$
$839$	11.2238	0.387490	0.193745	+	0.981052i	$0.437937\pi$
$840$	0	0
$841$	−10.6971	−0.368864
$842$	28.3749	0.977864
$843$	0	0
$844$	−8.40930	−0.289460
$845$	40.6376	1.39798
$846$	0	0
$847$	0	0
$848$	13.3864	0.459691
$849$	0	0
$850$	−21.2074	−0.727408
$851$	−6.17529	−0.211686
$852$	0	0
$853$	12.1934	0.417496	0.208748	−	0.977970i	$-0.433061\pi$
$853$	12.1934	0.417496	0.208748	+	0.977970i	$0.433061\pi$
$854$	0	0
$855$	0	0
$856$	−26.2537	−0.897332
$857$	37.4382	1.27887	0.639433	−	0.768847i	$-0.279169\pi$
$857$	37.4382	1.27887	0.639433	+	0.768847i	$0.279169\pi$
$858$	0	0
$859$	−0.757734	−0.0258535	−0.0129268	−	0.999916i	$-0.504115\pi$
$859$	−0.757734	−0.0258535	−0.0129268	+	0.999916i	$0.504115\pi$
$860$	8.96830	0.305817
$861$	0	0
$862$	−8.92314	−0.303923
$863$	23.5592	0.801965	0.400983	−	0.916086i	$-0.368669\pi$
$863$	23.5592	0.801965	0.400983	+	0.916086i	$0.368669\pi$
$864$	0	0
$865$	−31.1137	−1.05790
$866$	−5.59106	−0.189992
$867$	0	0
$868$	0	0
$869$	−31.4641	−1.06735
$870$	0	0
$871$	0.685455	0.0232258
$872$	11.7722	0.398657
$873$	0	0
$874$	−9.15892	−0.309805
$875$	0	0
$876$	0	0
$877$	36.6739	1.23839	0.619196	−	0.785237i	$-0.287459\pi$
$877$	36.6739	1.23839	0.619196	+	0.785237i	$0.287459\pi$
$878$	28.6744	0.967716
$879$	0	0
$880$	−83.3532	−2.80984
$881$	39.4357	1.32862	0.664311	−	0.747456i	$-0.268725\pi$
$881$	39.4357	1.32862	0.664311	+	0.747456i	$0.268725\pi$
$882$	0	0
$883$	−8.76912	−0.295105	−0.147552	−	0.989054i	$-0.547139\pi$
$883$	−8.76912	−0.295105	−0.147552	+	0.989054i	$0.547139\pi$
$884$	−0.361688	−0.0121649
$885$	0	0
$886$	−40.0895	−1.34683
$887$	−7.82601	−0.262772	−0.131386	−	0.991331i	$-0.541943\pi$
$887$	−7.82601	−0.262772	−0.131386	+	0.991331i	$0.541943\pi$
$888$	0	0
$889$	0	0
$890$	−69.2230	−2.32036
$891$	0	0
$892$	4.28587	0.143502
$893$	25.1734	0.842397
$894$	0	0
$895$	62.8272	2.10008
$896$	0	0
$897$	0	0
$898$	−8.66329	−0.289098
$899$	−21.3139	−0.710858
$900$	0	0
$901$	8.09571	0.269707
$902$	−29.4285	−0.979860
$903$	0	0
$904$	−11.8033	−0.392573
$905$	38.1851	1.26931
$906$	0	0
$907$	51.9332	1.72442	0.862208	−	0.506555i	$-0.169081\pi$
$907$	51.9332	1.72442	0.862208	+	0.506555i	$0.169081\pi$
$908$	−2.15128	−0.0713927
$909$	0	0
$910$	0	0
$911$	36.1524	1.19778	0.598891	−	0.800830i	$-0.295608\pi$
$911$	36.1524	1.19778	0.598891	+	0.800830i	$0.295608\pi$
$912$	0	0
$913$	−74.9394	−2.48013
$914$	−30.7187	−1.01608
$915$	0	0
$916$	8.56086	0.282859
$917$	0	0
$918$	0	0
$919$	−11.1768	−0.368690	−0.184345	−	0.982862i	$-0.559016\pi$
$919$	−11.1768	−0.368690	−0.184345	+	0.982862i	$0.559016\pi$
$920$	−6.55585	−0.216140
$921$	0	0
$922$	−57.5420	−1.89504
$923$	0.580900	0.0191206
$924$	0	0
$925$	37.2330	1.22421
$926$	3.77236	0.123967
$927$	0	0
$928$	8.63345	0.283407
$929$	−32.8384	−1.07739	−0.538696	−	0.842500i	$-0.681083\pi$
$929$	−32.8384	−1.07739	−0.538696	+	0.842500i	$0.681083\pi$
$930$	0	0
$931$	0	0
$932$	−5.00185	−0.163841
$933$	0	0
$934$	−51.3364	−1.67978
$935$	−50.4096	−1.64857
$936$	0	0
$937$	25.9566	0.847965	0.423983	−	0.905670i	$-0.360632\pi$
$937$	25.9566	0.847965	0.423983	+	0.905670i	$0.360632\pi$
$938$	0	0
$939$	0	0
$940$	−3.96748	−0.129405
$941$	−29.4679	−0.960627	−0.480314	−	0.877097i	$-0.659477\pi$
$941$	−29.4679	−0.960627	−0.480314	+	0.877097i	$0.659477\pi$
$942$	0	0
$943$	−2.74632	−0.0894326
$944$	11.0141	0.358479
$945$	0	0
$946$	69.5192	2.26026
$947$	−11.5227	−0.374439	−0.187219	−	0.982318i	$-0.559948\pi$
$947$	−11.5227	−0.374439	−0.187219	+	0.982318i	$0.559948\pi$
$948$	0	0
$949$	5.56289	0.180579
$950$	55.2223	1.79165
$951$	0	0
$952$	0	0
$953$	29.2912	0.948835	0.474417	−	0.880300i	$-0.342659\pi$
$953$	29.2912	0.948835	0.474417	+	0.880300i	$0.342659\pi$
$954$	0	0
$955$	−30.6908	−0.993130
$956$	8.06352	0.260793
$957$	0	0
$958$	−53.3075	−1.72229
$959$	0	0
$960$	0	0
$961$	−6.17981	−0.199349
$962$	4.15394	0.133929
$963$	0	0
$964$	−3.50821	−0.112992
$965$	7.26834	0.233976
$966$	0	0
$967$	42.0803	1.35321	0.676606	−	0.736345i	$-0.263450\pi$
$967$	42.0803	1.35321	0.676606	+	0.736345i	$0.263450\pi$
$968$	−55.5403	−1.78513
$969$	0	0
$970$	77.6536	2.49331
$971$	35.1549	1.12817	0.564087	−	0.825715i	$-0.309228\pi$
$971$	35.1549	1.12817	0.564087	+	0.825715i	$0.309228\pi$
$972$	0	0
$973$	0	0
$974$	1.47478	0.0472551
$975$	0	0
$976$	14.6981	0.470476
$977$	−4.43025	−0.141736	−0.0708682	−	0.997486i	$-0.522577\pi$
$977$	−4.43025	−0.141736	−0.0708682	+	0.997486i	$0.522577\pi$
$978$	0	0
$979$	−82.0271	−2.62160
$980$	0	0
$981$	0	0
$982$	24.2368	0.773428
$983$	−22.1601	−0.706798	−0.353399	−	0.935473i	$-0.614974\pi$
$983$	−22.1601	−0.706798	−0.353399	+	0.935473i	$0.614974\pi$
$984$	0	0
$985$	−15.9923	−0.509558
$986$	18.2537	0.581316
$987$	0	0
$988$	0.941804	0.0299628
$989$	6.48767	0.206296
$990$	0	0
$991$	−36.7203	−1.16646	−0.583229	−	0.812307i	$-0.698211\pi$
$991$	−36.7203	−1.16646	−0.583229	+	0.812307i	$0.698211\pi$
$992$	−10.0537	−0.319206
$993$	0	0
$994$	0	0
$995$	−31.4286	−0.996355
$996$	0	0
$997$	39.9031	1.26374	0.631872	−	0.775073i	$-0.282287\pi$
$997$	39.9031	1.26374	0.631872	+	0.775073i	$0.282287\pi$
$998$	−29.3918	−0.930380
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.w.1.3		4
3.2	odd	2		3969.2.a.t.1.2		4
7.3	odd	6		567.2.e.c.163.2	✓	8
7.5	odd	6		567.2.e.c.487.2	yes	8
7.6	odd	2		3969.2.a.x.1.3		4
21.5	even	6		567.2.e.d.487.3	yes	8
21.17	even	6		567.2.e.d.163.3	yes	8
21.20	even	2		3969.2.a.s.1.2		4
63.5	even	6		567.2.h.j.298.2		8
63.31	odd	6		567.2.g.j.541.2		8
63.38	even	6		567.2.h.j.352.2		8
63.40	odd	6		567.2.h.k.298.3		8
63.47	even	6		567.2.g.k.109.3		8
63.52	odd	6		567.2.h.k.352.3		8
63.59	even	6		567.2.g.k.541.3		8
63.61	odd	6		567.2.g.j.109.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
567.2.e.c.163.2	✓	8	7.3	odd	6
567.2.e.c.487.2	yes	8	7.5	odd	6
567.2.e.d.163.3	yes	8	21.17	even	6
567.2.e.d.487.3	yes	8	21.5	even	6
567.2.g.j.109.2		8	63.61	odd	6
567.2.g.j.541.2		8	63.31	odd	6
567.2.g.k.109.3		8	63.47	even	6
567.2.g.k.541.3		8	63.59	even	6
567.2.h.j.298.2		8	63.5	even	6
567.2.h.j.352.2		8	63.38	even	6
567.2.h.k.298.3		8	63.40	odd	6
567.2.h.k.352.3		8	63.52	odd	6
3969.2.a.s.1.2		4	21.20	even	2
3969.2.a.t.1.2		4	3.2	odd	2
3969.2.a.w.1.3		4	1.1	even	1	trivial
3969.2.a.x.1.3		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:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

\(f(q)\)	\(=\)	\(q+1.53652 q^{2} +0.360904 q^{4} -3.15761 q^{5} -2.51851 q^{8} +O(q^{10})\)	\(q+1.53652 q^{2} +0.360904 q^{4} -3.15761 q^{5} -2.51851 q^{8} -4.85173 q^{10} -5.74916 q^{11} +0.360904 q^{13} -4.59156 q^{16} -2.77684 q^{17} +7.23065 q^{19} -1.13959 q^{20} -8.83372 q^{22} -0.824381 q^{23} +4.97047 q^{25} +0.554537 q^{26} -4.27819 q^{29} +4.98199 q^{31} -2.01801 q^{32} -4.26668 q^{34} +7.49083 q^{37} +11.1101 q^{38} +7.95246 q^{40} +3.33138 q^{41} -7.86975 q^{43} -2.07489 q^{44} -1.26668 q^{46} +3.48149 q^{47} +7.63725 q^{50} +0.130252 q^{52} -2.91544 q^{53} +18.1536 q^{55} -6.57354 q^{58} -2.39878 q^{59} -3.20113 q^{61} +7.65494 q^{62} +6.08239 q^{64} -1.13959 q^{65} +1.89927 q^{67} -1.00217 q^{68} +1.60957 q^{71} +15.4138 q^{73} +11.5098 q^{74} +2.60957 q^{76} +5.47282 q^{79} +14.4983 q^{80} +5.11874 q^{82} +13.0348 q^{83} +8.76817 q^{85} -12.0921 q^{86} +14.4793 q^{88} +14.2677 q^{89} -0.297522 q^{92} +5.34939 q^{94} -22.8315 q^{95} -16.0053 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + 5 q^{4} - 2 q^{5} - 3 q^{8}+O(q^{10}) \) 4 * q + q^2 + 5 * q^4 - 2 * q^5 - 3 * q^8	\( 4 q + q^{2} + 5 q^{4} - 2 q^{5} - 3 q^{8} + 7 q^{10} + 5 q^{11} + 5 q^{13} - q^{16} - 6 q^{17} + 8 q^{19} + 8 q^{20} - 7 q^{22} - 12 q^{23} + 8 q^{25} - q^{26} - 10 q^{29} + 18 q^{31} - 10 q^{32} + 20 q^{38} + 18 q^{40} + 5 q^{41} - 7 q^{43} + 13 q^{44} + 12 q^{46} + 21 q^{47} + 38 q^{50} + 25 q^{52} - 12 q^{53} + 26 q^{55} - 7 q^{58} - 6 q^{59} + 20 q^{61} - 18 q^{62} - 23 q^{64} + 8 q^{65} - 5 q^{67} - 51 q^{68} - 9 q^{71} + 6 q^{73} - 5 q^{76} - 10 q^{79} + 2 q^{80} + 35 q^{82} - 9 q^{83} - 9 q^{85} - 22 q^{86} + 18 q^{88} + 22 q^{89} - 36 q^{92} + 15 q^{94} - 16 q^{95} + 9 q^{97}+O(q^{100}) \) 4 * q + q^2 + 5 * q^4 - 2 * q^5 - 3 * q^8 + 7 * q^10 + 5 * q^11 + 5 * q^13 - q^16 - 6 * q^17 + 8 * q^19 + 8 * q^20 - 7 * q^22 - 12 * q^23 + 8 * q^25 - q^26 - 10 * q^29 + 18 * q^31 - 10 * q^32 + 20 * q^38 + 18 * q^40 + 5 * q^41 - 7 * q^43 + 13 * q^44 + 12 * q^46 + 21 * q^47 + 38 * q^50 + 25 * q^52 - 12 * q^53 + 26 * q^55 - 7 * q^58 - 6 * q^59 + 20 * q^61 - 18 * q^62 - 23 * q^64 + 8 * q^65 - 5 * q^67 - 51 * q^68 - 9 * q^71 + 6 * q^73 - 5 * q^76 - 10 * q^79 + 2 * q^80 + 35 * q^82 - 9 * q^83 - 9 * q^85 - 22 * q^86 + 18 * q^88 + 22 * q^89 - 36 * q^92 + 15 * q^94 - 16 * q^95 + 9 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more