LMFDB - Embedded newform 3969.2.a.w.1.1

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			1.1
Root			$-2.37167$ 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)

$f(q)$	$=$	$q-2.37167 q^{2} +3.62484 q^{4} -1.22875 q^{5} -3.85358 q^{8} +O(q^{10})$	$q-2.37167 q^{2} +3.62484 q^{4} -1.22875 q^{5} -3.85358 q^{8} +2.91418 q^{10} +2.66102 q^{11} +3.62484 q^{13} +1.88977 q^{16} -6.73158 q^{17} -2.51460 q^{19} -4.45400 q^{20} -6.31107 q^{22} -7.99651 q^{23} -3.49018 q^{25} -8.59693 q^{26} +2.24967 q^{29} +10.2253 q^{31} +3.22526 q^{32} +15.9651 q^{34} -3.53902 q^{37} +5.96382 q^{38} +4.73507 q^{40} -1.86535 q^{41} +5.13944 q^{43} +9.64576 q^{44} +18.9651 q^{46} +2.14642 q^{47} +8.27758 q^{50} +13.1394 q^{52} +2.97209 q^{53} -3.26972 q^{55} -5.33549 q^{58} +8.72809 q^{59} +15.0048 q^{61} -24.2510 q^{62} -11.4288 q^{64} -4.45400 q^{65} -2.64925 q^{67} -24.4009 q^{68} -10.1150 q^{71} -7.29413 q^{73} +8.39340 q^{74} -9.11502 q^{76} -0.313764 q^{79} -2.32204 q^{80} +4.42400 q^{82} -7.69371 q^{83} +8.27140 q^{85} -12.1891 q^{86} -10.2545 q^{88} +7.19256 q^{89} -28.9860 q^{92} -5.09060 q^{94} +3.08981 q^{95} +13.1839 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.37167	−1.67703	−0.838513	−	0.544881i	$-0.816575\pi$
$2$	−2.37167	−1.67703	−0.838513	+	0.544881i	$0.816575\pi$
$3$	0	0
$3$	0	0
$4$	3.62484	1.81242
$5$	−1.22875	−0.549512	−0.274756	−	0.961514i	$-0.588597\pi$
$5$	−1.22875	−0.549512	−0.274756	+	0.961514i	$0.588597\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−3.85358	−1.36245
$9$	0	0
$10$	2.91418	0.921546
$11$	2.66102	0.802328	0.401164	−	0.916006i	$-0.368606\pi$
$11$	2.66102	0.802328	0.401164	+	0.916006i	$0.368606\pi$
$12$	0	0
$13$	3.62484	1.00535	0.502674	−	0.864476i	$-0.332349\pi$
$13$	3.62484	1.00535	0.502674	+	0.864476i	$0.332349\pi$
$14$	0	0
$15$	0	0
$16$	1.88977	0.472441
$17$	−6.73158	−1.63265	−0.816324	−	0.577594i	$-0.803992\pi$
$17$	−6.73158	−1.63265	−0.816324	+	0.577594i	$0.803992\pi$
$18$	0	0
$19$	−2.51460	−0.576889	−0.288445	−	0.957497i	$-0.593138\pi$
$19$	−2.51460	−0.576889	−0.288445	+	0.957497i	$0.593138\pi$
$20$	−4.45400	−0.995945
$21$	0	0
$22$	−6.31107	−1.34553
$23$	−7.99651	−1.66739	−0.833694	−	0.552227i	$-0.813778\pi$
$23$	−7.99651	−1.66739	−0.833694	+	0.552227i	$0.813778\pi$
$24$	0	0
$25$	−3.49018	−0.698037
$26$	−8.59693	−1.68600
$27$	0	0
$28$	0	0
$29$	2.24967	0.417754	0.208877	−	0.977942i	$-0.433019\pi$
$29$	2.24967	0.417754	0.208877	+	0.977942i	$0.433019\pi$
$30$	0	0
$31$	10.2253	1.83651	0.918255	−	0.395989i	$-0.129598\pi$
$31$	10.2253	1.83651	0.918255	+	0.395989i	$0.129598\pi$
$32$	3.22526	0.570150
$33$	0	0
$34$	15.9651	2.73799
$35$	0	0
$36$	0	0
$37$	−3.53902	−0.581811	−0.290906	−	0.956752i	$-0.593957\pi$
$37$	−3.53902	−0.581811	−0.290906	+	0.956752i	$0.593957\pi$
$38$	5.96382	0.967459
$39$	0	0
$40$	4.73507	0.748680
$41$	−1.86535	−0.291319	−0.145659	−	0.989335i	$-0.546530\pi$
$41$	−1.86535	−0.291319	−0.145659	+	0.989335i	$0.546530\pi$
$42$	0	0
$43$	5.13944	0.783757	0.391879	−	0.920017i	$-0.371825\pi$
$43$	5.13944	0.783757	0.391879	+	0.920017i	$0.371825\pi$
$44$	9.64576	1.45415
$45$	0	0
$46$	18.9651	2.79625
$47$	2.14642	0.313087	0.156544	−	0.987671i	$-0.449965\pi$
$47$	2.14642	0.313087	0.156544	+	0.987671i	$0.449965\pi$
$48$	0	0
$49$	0	0
$50$	8.27758	1.17063
$51$	0	0
$52$	13.1394	1.82211
$53$	2.97209	0.408248	0.204124	−	0.978945i	$-0.434565\pi$
$53$	2.97209	0.408248	0.204124	+	0.978945i	$0.434565\pi$
$54$	0	0
$55$	−3.26972	−0.440888
$56$	0	0
$57$	0	0
$58$	−5.33549	−0.700584
$59$	8.72809	1.13630	0.568150	−	0.822925i	$-0.307659\pi$
$59$	8.72809	1.13630	0.568150	+	0.822925i	$0.307659\pi$
$60$	0	0
$61$	15.0048	1.92117	0.960583	−	0.277993i	$-0.0896693\pi$
$61$	15.0048	1.92117	0.960583	+	0.277993i	$0.0896693\pi$
$62$	−24.2510	−3.07988
$63$	0	0
$64$	−11.4288	−1.42860
$65$	−4.45400	−0.552451
$66$	0	0
$67$	−2.64925	−0.323658	−0.161829	−	0.986819i	$-0.551739\pi$
$67$	−2.64925	−0.323658	−0.161829	+	0.986819i	$0.551739\pi$
$68$	−24.4009	−2.95904
$69$	0	0
$70$	0	0
$71$	−10.1150	−1.20043	−0.600216	−	0.799838i	$-0.704919\pi$
$71$	−10.1150	−1.20043	−0.600216	+	0.799838i	$0.704919\pi$
$72$	0	0
$73$	−7.29413	−0.853714	−0.426857	−	0.904319i	$-0.640379\pi$
$73$	−7.29413	−0.853714	−0.426857	+	0.904319i	$0.640379\pi$
$74$	8.39340	0.975713
$75$	0	0
$76$	−9.11502	−1.04556
$77$	0	0
$78$	0	0
$79$	−0.313764	−0.0353012	−0.0176506	−	0.999844i	$-0.505619\pi$
$79$	−0.313764	−0.0353012	−0.0176506	+	0.999844i	$0.505619\pi$
$80$	−2.32204	−0.259612
$81$	0	0
$82$	4.42400	0.488549
$83$	−7.69371	−0.844495	−0.422247	−	0.906481i	$-0.638759\pi$
$83$	−7.69371	−0.844495	−0.422247	+	0.906481i	$0.638759\pi$
$84$	0	0
$85$	8.27140	0.897159
$86$	−12.1891	−1.31438
$87$	0	0
$88$	−10.2545	−1.09313
$89$	7.19256	0.762410	0.381205	−	0.924491i	$-0.375509\pi$
$89$	7.19256	0.762410	0.381205	+	0.924491i	$0.375509\pi$
$90$	0	0
$91$	0	0
$92$	−28.9860	−3.02200
$93$	0	0
$94$	−5.09060	−0.525056
$95$	3.08981	0.317007
$96$	0	0
$97$	13.1839	1.33862	0.669311	−	0.742982i	$-0.266589\pi$
$97$	13.1839	1.33862	0.669311	+	0.742982i	$0.266589\pi$
$98$	0	0
$99$	0	0
$100$	−12.6513	−1.26513
$101$	2.78302	0.276921	0.138460	−	0.990368i	$-0.455785\pi$
$101$	2.78302	0.276921	0.138460	+	0.990368i	$0.455785\pi$
$102$	0	0
$103$	−6.00647	−0.591835	−0.295917	−	0.955214i	$-0.595625\pi$
$103$	−6.00647	−0.591835	−0.295917	+	0.955214i	$0.595625\pi$
$104$	−13.9686	−1.36973
$105$	0	0
$106$	−7.04883	−0.684643
$107$	−11.7364	−1.13460	−0.567299	−	0.823512i	$-0.692012\pi$
$107$	−11.7364	−1.13460	−0.567299	+	0.823512i	$0.692012\pi$
$108$	0	0
$109$	−7.98474	−0.764800	−0.382400	−	0.923997i	$-0.624902\pi$
$109$	−7.98474	−0.764800	−0.382400	+	0.923997i	$0.624902\pi$
$110$	7.75470	0.739382
$111$	0	0
$112$	0	0
$113$	4.64009	0.436503	0.218252	−	0.975893i	$-0.429965\pi$
$113$	4.64009	0.436503	0.218252	+	0.975893i	$0.429965\pi$
$114$	0	0
$115$	9.82567	0.916249
$116$	8.15470	0.757144
$117$	0	0
$118$	−20.7002	−1.90561
$119$	0	0
$120$	0	0
$121$	−3.91897	−0.356270
$122$	−35.5865	−3.22185
$123$	0	0
$124$	37.0649	3.32852
$125$	10.4323	0.933091
$126$	0	0
$127$	−3.32633	−0.295164	−0.147582	−	0.989050i	$-0.547149\pi$
$127$	−3.32633	−0.295164	−0.147582	+	0.989050i	$0.547149\pi$
$128$	20.6548	1.82565
$129$	0	0
$130$	10.5634	0.926475
$131$	8.44702	0.738020	0.369010	−	0.929425i	$-0.379697\pi$
$131$	8.44702	0.738020	0.369010	+	0.929425i	$0.379697\pi$
$132$	0	0
$133$	0	0
$134$	6.28317	0.542783
$135$	0	0
$136$	25.9407	2.22440
$137$	13.9965	1.19580	0.597901	−	0.801570i	$-0.296001\pi$
$137$	13.9965	1.19580	0.597901	+	0.801570i	$0.296001\pi$
$138$	0	0
$139$	5.64009	0.478386	0.239193	−	0.970972i	$-0.423117\pi$
$139$	5.64009	0.478386	0.239193	+	0.970972i	$0.423117\pi$
$140$	0	0
$141$	0	0
$142$	23.9895	2.01316
$143$	9.64576	0.806619
$144$	0	0
$145$	−2.76427	−0.229561
$146$	17.2993	1.43170
$147$	0	0
$148$	−12.8284	−1.05449
$149$	8.76945	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$149$	8.76945	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$150$	0	0
$151$	−9.56075	−0.778042	−0.389021	−	0.921229i	$-0.627187\pi$
$151$	−9.56075	−0.778042	−0.389021	+	0.921229i	$0.627187\pi$
$152$	9.69022	0.785981
$153$	0	0
$154$	0	0
$155$	−12.5642	−1.00918
$156$	0	0
$157$	−2.15907	−0.172312	−0.0861562	−	0.996282i	$-0.527458\pi$
$157$	−2.15907	−0.172312	−0.0861562	+	0.996282i	$0.527458\pi$
$158$	0.744146	0.0592011
$159$	0	0
$160$	−3.96302	−0.313304
$161$	0	0
$162$	0	0
$163$	20.9743	1.64283	0.821416	−	0.570330i	$-0.193185\pi$
$163$	20.9743	1.64283	0.821416	+	0.570330i	$0.193185\pi$
$164$	−6.76158	−0.527991
$165$	0	0
$166$	18.2470	1.41624
$167$	−4.27409	−0.330739	−0.165370	−	0.986232i	$-0.552882\pi$
$167$	−4.27409	−0.330739	−0.165370	+	0.986232i	$0.552882\pi$
$168$	0	0
$169$	0.139438	0.0107260
$170$	−19.6171	−1.50456
$171$	0	0
$172$	18.6296	1.42050
$173$	−1.81660	−0.138113	−0.0690567	−	0.997613i	$-0.521999\pi$
$173$	−1.81660	−0.138113	−0.0690567	+	0.997613i	$0.521999\pi$
$174$	0	0
$175$	0	0
$176$	5.02871	0.379053
$177$	0	0
$178$	−17.0584	−1.27858
$179$	8.05013	0.601695	0.300848	−	0.953672i	$-0.402730\pi$
$179$	8.05013	0.601695	0.300848	+	0.953672i	$0.402730\pi$
$180$	0	0
$181$	−25.3467	−1.88401	−0.942004	−	0.335601i	$-0.891061\pi$
$181$	−25.3467	−1.88401	−0.942004	+	0.335601i	$0.891061\pi$
$182$	0	0
$183$	0	0
$184$	30.8152	2.27173
$185$	4.34855	0.319712
$186$	0	0
$187$	−17.9129	−1.30992
$188$	7.78042	0.567445
$189$	0	0
$190$	−7.32801	−0.531630
$191$	−7.15121	−0.517443	−0.258722	−	0.965952i	$-0.583301\pi$
$191$	−7.15121	−0.517443	−0.258722	+	0.965952i	$0.583301\pi$
$192$	0	0
$193$	11.3555	0.817389	0.408695	−	0.912671i	$-0.365984\pi$
$193$	11.3555	0.817389	0.408695	+	0.912671i	$0.365984\pi$
$194$	−31.2679	−2.24490
$195$	0	0
$196$	0	0
$197$	20.0998	1.43205	0.716024	−	0.698075i	$-0.245960\pi$
$197$	20.0998	1.43205	0.716024	+	0.698075i	$0.245960\pi$
$198$	0	0
$199$	−10.3250	−0.731921	−0.365961	−	0.930630i	$-0.619260\pi$
$199$	−10.3250	−0.731921	−0.365961	+	0.930630i	$0.619260\pi$
$200$	13.4497	0.951038
$201$	0	0
$202$	−6.60042	−0.464404
$203$	0	0
$204$	0	0
$205$	2.29204	0.160083
$206$	14.2454	0.992523
$207$	0	0
$208$	6.85009	0.474968
$209$	−6.69141	−0.462854
$210$	0	0
$211$	0.598737	0.0412188	0.0206094	−	0.999788i	$-0.493439\pi$
$211$	0.598737	0.0412188	0.0206094	+	0.999788i	$0.493439\pi$
$212$	10.7734	0.739917
$213$	0	0
$214$	27.8348	1.90275
$215$	−6.31506	−0.430684
$216$	0	0
$217$	0	0
$218$	18.9372	1.28259
$219$	0	0
$220$	−11.8522	−0.799074
$221$	−24.4009	−1.64138
$222$	0	0
$223$	−3.02004	−0.202237	−0.101119	−	0.994874i	$-0.532242\pi$
$223$	−3.02004	−0.202237	−0.101119	+	0.994874i	$0.532242\pi$
$224$	0	0
$225$	0	0
$226$	−11.0048	−0.732028
$227$	20.1080	1.33462	0.667309	−	0.744781i	$-0.267446\pi$
$227$	20.1080	1.33462	0.667309	+	0.744781i	$0.267446\pi$
$228$	0	0
$229$	20.0065	1.32206	0.661032	−	0.750357i	$-0.270119\pi$
$229$	20.0065	1.32206	0.661032	+	0.750357i	$0.270119\pi$
$230$	−23.3033	−1.53657
$231$	0	0
$232$	−8.66930	−0.569167
$233$	−0.302795	−0.0198368	−0.00991839	−	0.999951i	$-0.503157\pi$
$233$	−0.302795	−0.0198368	−0.00991839	+	0.999951i	$0.503157\pi$
$234$	0	0
$235$	−2.63740	−0.172045
$236$	31.6379	2.05945
$237$	0	0
$238$	0	0
$239$	3.54680	0.229423	0.114712	−	0.993399i	$-0.463406\pi$
$239$	3.54680	0.229423	0.114712	+	0.993399i	$0.463406\pi$
$240$	0	0
$241$	−6.00647	−0.386911	−0.193455	−	0.981109i	$-0.561969\pi$
$241$	−6.00647	−0.386911	−0.193455	+	0.981109i	$0.561969\pi$
$242$	9.29452	0.597474
$243$	0	0
$244$	54.3899	3.48196
$245$	0	0
$246$	0	0
$247$	−9.11502	−0.579975
$248$	−39.4039	−2.50215
$249$	0	0
$250$	−24.7420	−1.56482
$251$	21.0113	1.32622	0.663109	−	0.748523i	$-0.269236\pi$
$251$	21.0113	1.32622	0.663109	+	0.748523i	$0.269236\pi$
$252$	0	0
$253$	−21.2789	−1.33779
$254$	7.88897	0.494998
$255$	0	0
$256$	−26.1290	−1.63306
$257$	−20.5690	−1.28306	−0.641530	−	0.767098i	$-0.721700\pi$
$257$	−20.5690	−1.28306	−0.641530	+	0.767098i	$0.721700\pi$
$258$	0	0
$259$	0	0
$260$	−16.1450	−1.00127
$261$	0	0
$262$	−20.0336	−1.23768
$263$	−14.2030	−0.875796	−0.437898	−	0.899025i	$-0.644277\pi$
$263$	−14.2030	−0.875796	−0.437898	+	0.899025i	$0.644277\pi$
$264$	0	0
$265$	−3.65195	−0.224337
$266$	0	0
$267$	0	0
$268$	−9.60311	−0.586603
$269$	31.1325	1.89818	0.949090	−	0.315006i	$-0.102006\pi$
$269$	31.1325	1.89818	0.949090	+	0.315006i	$0.102006\pi$
$270$	0	0
$271$	1.64278	0.0997921	0.0498960	−	0.998754i	$-0.484111\pi$
$271$	1.64278	0.0997921	0.0498960	+	0.998754i	$0.484111\pi$
$272$	−12.7211	−0.771331
$273$	0	0
$274$	−33.1952	−2.00539
$275$	−9.28745	−0.560055
$276$	0	0
$277$	−22.0950	−1.32756	−0.663779	−	0.747929i	$-0.731049\pi$
$277$	−22.0950	−1.32756	−0.663779	+	0.747929i	$0.731049\pi$
$278$	−13.3765	−0.802267
$279$	0	0
$280$	0	0
$281$	15.4814	0.923543	0.461771	−	0.886999i	$-0.347214\pi$
$281$	15.4814	0.923543	0.461771	+	0.886999i	$0.347214\pi$
$282$	0	0
$283$	26.2348	1.55950	0.779749	−	0.626092i	$-0.215346\pi$
$283$	26.2348	1.55950	0.779749	+	0.626092i	$0.215346\pi$
$284$	−36.6653	−2.17569
$285$	0	0
$286$	−22.8766	−1.35272
$287$	0	0
$288$	0	0
$289$	28.3142	1.66554
$290$	6.55596	0.384979
$291$	0	0
$292$	−26.4400	−1.54729
$293$	16.9085	0.987805	0.493903	−	0.869517i	$-0.335570\pi$
$293$	16.9085	0.987805	0.493903	+	0.869517i	$0.335570\pi$
$294$	0	0
$295$	−10.7246	−0.624410
$296$	13.6379	0.792687
$297$	0	0
$298$	−20.7983	−1.20481
$299$	−28.9860	−1.67631
$300$	0	0
$301$	0	0
$302$	22.6750	1.30480
$303$	0	0
$304$	−4.75201	−0.272546
$305$	−18.4371	−1.05570
$306$	0	0
$307$	14.6835	0.838034	0.419017	−	0.907978i	$-0.362375\pi$
$307$	14.6835	0.838034	0.419017	+	0.907978i	$0.362375\pi$
$308$	0	0
$309$	0	0
$310$	29.7983	1.69243
$311$	−11.8222	−0.670375	−0.335187	−	0.942152i	$-0.608800\pi$
$311$	−11.8222	−0.670375	−0.335187	+	0.942152i	$0.608800\pi$
$312$	0	0
$313$	0.355120	0.0200726	0.0100363	−	0.999950i	$-0.496805\pi$
$313$	0.355120	0.0200726	0.0100363	+	0.999950i	$0.496805\pi$
$314$	5.12061	0.288973
$315$	0	0
$316$	−1.13734	−0.0639805
$317$	−10.4497	−0.586914	−0.293457	−	0.955972i	$-0.594806\pi$
$317$	−10.4497	−0.586914	−0.293457	+	0.955972i	$0.594806\pi$
$318$	0	0
$319$	5.98643	0.335175
$320$	14.0431	0.785031
$321$	0	0
$322$	0	0
$323$	16.9272	0.941857
$324$	0	0
$325$	−12.6513	−0.701771
$326$	−49.7441	−2.75507
$327$	0	0
$328$	7.18827	0.396906
$329$	0	0
$330$	0	0
$331$	14.8670	0.817166	0.408583	−	0.912721i	$-0.366023\pi$
$331$	14.8670	0.817166	0.408583	+	0.912721i	$0.366023\pi$
$332$	−27.8885	−1.53058
$333$	0	0
$334$	10.1367	0.554658
$335$	3.25526	0.177854
$336$	0	0
$337$	9.41353	0.512788	0.256394	−	0.966572i	$-0.417466\pi$
$337$	9.41353	0.512788	0.256394	+	0.966572i	$0.417466\pi$
$338$	−0.330703	−0.0179878
$339$	0	0
$340$	29.9825	1.62603
$341$	27.2096	1.47348
$342$	0	0
$343$	0	0
$344$	−19.8052	−1.06783
$345$	0	0
$346$	4.30838	0.231620
$347$	26.5865	1.42724	0.713618	−	0.700535i	$-0.247055\pi$
$347$	26.5865	1.42724	0.713618	+	0.700535i	$0.247055\pi$
$348$	0	0
$349$	−9.55297	−0.511359	−0.255679	−	0.966762i	$-0.582299\pi$
$349$	−9.55297	−0.511359	−0.255679	+	0.966762i	$0.582299\pi$
$350$	0	0
$351$	0	0
$352$	8.58247	0.457447
$353$	−17.0729	−0.908697	−0.454349	−	0.890824i	$-0.650128\pi$
$353$	−17.0729	−0.908697	−0.454349	+	0.890824i	$0.650128\pi$
$354$	0	0
$355$	12.4288	0.659651
$356$	26.0719	1.38181
$357$	0	0
$358$	−19.0923	−1.00906
$359$	25.6667	1.35464	0.677318	−	0.735690i	$-0.263142\pi$
$359$	25.6667	1.35464	0.677318	+	0.735690i	$0.263142\pi$
$360$	0	0
$361$	−12.6768	−0.667199
$362$	60.1142	3.15953
$363$	0	0
$364$	0	0
$365$	8.96263	0.469126
$366$	0	0
$367$	10.3752	0.541579	0.270790	−	0.962639i	$-0.412715\pi$
$367$	10.3752	0.541579	0.270790	+	0.962639i	$0.412715\pi$
$368$	−15.1115	−0.787743
$369$	0	0
$370$	−10.3134	−0.536166
$371$	0	0
$372$	0	0
$373$	−19.8152	−1.02599	−0.512996	−	0.858391i	$-0.671465\pi$
$373$	−19.8152	−1.02599	−0.512996	+	0.858391i	$0.671465\pi$
$374$	42.4835	2.19677
$375$	0	0
$376$	−8.27140	−0.426565
$377$	8.15470	0.419988
$378$	0	0
$379$	14.1716	0.727948	0.363974	−	0.931409i	$-0.381420\pi$
$379$	14.1716	0.727948	0.363974	+	0.931409i	$0.381420\pi$
$380$	11.2000	0.574550
$381$	0	0
$382$	16.9603	0.867766
$383$	13.4487	0.687197	0.343598	−	0.939117i	$-0.388354\pi$
$383$	13.4487	0.687197	0.343598	+	0.939117i	$0.388354\pi$
$384$	0	0
$385$	0	0
$386$	−26.9316	−1.37078
$387$	0	0
$388$	47.7895	2.42614
$389$	20.9297	1.06118	0.530590	−	0.847629i	$-0.321971\pi$
$389$	20.9297	1.06118	0.530590	+	0.847629i	$0.321971\pi$
$390$	0	0
$391$	53.8292	2.72226
$392$	0	0
$393$	0	0
$394$	−47.6701	−2.40158
$395$	0.385536	0.0193984
$396$	0	0
$397$	25.1415	1.26182	0.630909	−	0.775857i	$-0.282682\pi$
$397$	25.1415	1.26182	0.630909	+	0.775857i	$0.282682\pi$
$398$	24.4876	1.22745
$399$	0	0
$400$	−6.59563	−0.329782
$401$	21.1603	1.05669	0.528347	−	0.849029i	$-0.322812\pi$
$401$	21.1603	1.05669	0.528347	+	0.849029i	$0.322812\pi$
$402$	0	0
$403$	37.0649	1.84633
$404$	10.0880	0.501897
$405$	0	0
$406$	0	0
$407$	−9.41740	−0.466803
$408$	0	0
$409$	−0.543806	−0.0268895	−0.0134447	−	0.999910i	$-0.504280\pi$
$409$	−0.543806	−0.0268895	−0.0134447	+	0.999910i	$0.504280\pi$
$410$	−5.43597	−0.268463
$411$	0	0
$412$	−21.7725	−1.07265
$413$	0	0
$414$	0	0
$415$	9.45362	0.464060
$416$	11.6910	0.573200
$417$	0	0
$418$	15.8698	0.776219
$419$	23.8087	1.16313	0.581566	−	0.813499i	$-0.302440\pi$
$419$	23.8087	1.16313	0.581566	+	0.813499i	$0.302440\pi$
$420$	0	0
$421$	−2.90981	−0.141815	−0.0709077	−	0.997483i	$-0.522590\pi$
$421$	−2.90981	−0.141815	−0.0709077	+	0.997483i	$0.522590\pi$
$422$	−1.42001	−0.0691250
$423$	0	0
$424$	−11.4532	−0.556217
$425$	23.4945	1.13965
$426$	0	0
$427$	0	0
$428$	−42.5424	−2.05637
$429$	0	0
$430$	14.9773	0.722268
$431$	−31.3794	−1.51149	−0.755747	−	0.654863i	$-0.772726\pi$
$431$	−31.3794	−1.51149	−0.755747	+	0.654863i	$0.772726\pi$
$432$	0	0
$433$	26.9281	1.29408	0.647042	−	0.762455i	$-0.276006\pi$
$433$	26.9281	1.29408	0.647042	+	0.762455i	$0.276006\pi$
$434$	0	0
$435$	0	0
$436$	−28.9434	−1.38614
$437$	20.1080	0.961898
$438$	0	0
$439$	25.3294	1.20891	0.604453	−	0.796641i	$-0.293392\pi$
$439$	25.3294	1.20891	0.604453	+	0.796641i	$0.293392\pi$
$440$	12.6001	0.600687
$441$	0	0
$442$	57.8709	2.75264
$443$	−13.0314	−0.619140	−0.309570	−	0.950877i	$-0.600185\pi$
$443$	−13.0314	−0.619140	−0.309570	+	0.950877i	$0.600185\pi$
$444$	0	0
$445$	−8.83783	−0.418953
$446$	7.16256	0.339157
$447$	0	0
$448$	0	0
$449$	−19.4353	−0.917206	−0.458603	−	0.888641i	$-0.651650\pi$
$449$	−19.4353	−0.917206	−0.458603	+	0.888641i	$0.651650\pi$
$450$	0	0
$451$	−4.96373	−0.233733
$452$	16.8196	0.791126
$453$	0	0
$454$	−47.6897	−2.23819
$455$	0	0
$456$	0	0
$457$	−28.6975	−1.34241	−0.671206	−	0.741271i	$-0.734223\pi$
$457$	−28.6975	−1.34241	−0.671206	+	0.741271i	$0.734223\pi$
$458$	−47.4488	−2.21714
$459$	0	0
$460$	35.6165	1.66063
$461$	23.0479	1.07345	0.536725	−	0.843757i	$-0.319661\pi$
$461$	23.0479	1.07345	0.536725	+	0.843757i	$0.319661\pi$
$462$	0	0
$463$	29.2148	1.35773	0.678863	−	0.734265i	$-0.262473\pi$
$463$	29.2148	1.35773	0.678863	+	0.734265i	$0.262473\pi$
$464$	4.25135	0.197364
$465$	0	0
$466$	0.718132	0.0332668
$467$	−4.36508	−0.201992	−0.100996	−	0.994887i	$-0.532203\pi$
$467$	−4.36508	−0.201992	−0.100996	+	0.994887i	$0.532203\pi$
$468$	0	0
$469$	0	0
$470$	6.25506	0.288524
$471$	0	0
$472$	−33.6344	−1.54815
$473$	13.6762	0.628830
$474$	0	0
$475$	8.77643	0.402690
$476$	0	0
$477$	0	0
$478$	−8.41185	−0.384749
$479$	15.9229	0.727535	0.363767	−	0.931490i	$-0.381490\pi$
$479$	15.9229	0.727535	0.363767	+	0.931490i	$0.381490\pi$
$480$	0	0
$481$	−12.8284	−0.584923
$482$	14.2454	0.648859
$483$	0	0
$484$	−14.2056	−0.645710
$485$	−16.1997	−0.735588
$486$	0	0
$487$	−38.2657	−1.73399	−0.866993	−	0.498321i	$-0.833950\pi$
$487$	−38.2657	−1.73399	−0.866993	+	0.498321i	$0.833950\pi$
$488$	−57.8222	−2.61749
$489$	0	0
$490$	0	0
$491$	13.3908	0.604318	0.302159	−	0.953258i	$-0.402293\pi$
$491$	13.3908	0.604318	0.302159	+	0.953258i	$0.402293\pi$
$492$	0	0
$493$	−15.1439	−0.682045
$494$	21.6179	0.972633
$495$	0	0
$496$	19.3233	0.867643
$497$	0	0
$498$	0	0
$499$	5.40706	0.242053	0.121027	−	0.992649i	$-0.461381\pi$
$499$	5.40706	0.242053	0.121027	+	0.992649i	$0.461381\pi$
$500$	37.8153	1.69115
$501$	0	0
$502$	−49.8318	−2.22410
$503$	−9.49157	−0.423208	−0.211604	−	0.977355i	$-0.567869\pi$
$503$	−9.49157	−0.423208	−0.211604	+	0.977355i	$0.567869\pi$
$504$	0	0
$505$	−3.41962	−0.152171
$506$	50.4666	2.24351
$507$	0	0
$508$	−12.0574	−0.534961
$509$	37.2297	1.65018	0.825088	−	0.565005i	$-0.191126\pi$
$509$	37.2297	1.65018	0.825088	+	0.565005i	$0.191126\pi$
$510$	0	0
$511$	0	0
$512$	20.6597	0.913039
$513$	0	0
$514$	48.7830	2.15173
$515$	7.38042	0.325220
$516$	0	0
$517$	5.71166	0.251199
$518$	0	0
$519$	0	0
$520$	17.1639	0.752685
$521$	15.2016	0.665996	0.332998	−	0.942928i	$-0.391940\pi$
$521$	15.2016	0.665996	0.332998	+	0.942928i	$0.391940\pi$
$522$	0	0
$523$	−4.79647	−0.209735	−0.104867	−	0.994486i	$-0.533442\pi$
$523$	−4.79647	−0.209735	−0.104867	+	0.994486i	$0.533442\pi$
$524$	30.6191	1.33760
$525$	0	0
$526$	33.6849	1.46873
$527$	−68.8321	−2.99837
$528$	0	0
$529$	40.9442	1.78018
$530$	8.66122	0.376219
$531$	0	0
$532$	0	0
$533$	−6.76158	−0.292877
$534$	0	0
$535$	14.4210	0.623475
$536$	10.2091	0.440967
$537$	0	0
$538$	−73.8360	−3.18330
$539$	0	0
$540$	0	0
$541$	−6.45320	−0.277445	−0.138722	−	0.990331i	$-0.544300\pi$
$541$	−6.45320	−0.277445	−0.138722	+	0.990331i	$0.544300\pi$
$542$	−3.89615	−0.167354
$543$	0	0
$544$	−21.7111	−0.930854
$545$	9.81122	0.420266
$546$	0	0
$547$	26.2561	1.12263	0.561316	−	0.827602i	$-0.310295\pi$
$547$	26.2561	1.12263	0.561316	+	0.827602i	$0.310295\pi$
$548$	50.7351	2.16729
$549$	0	0
$550$	22.0268	0.939226
$551$	−5.65703	−0.240998
$552$	0	0
$553$	0	0
$554$	52.4021	2.22635
$555$	0	0
$556$	20.4444	0.867036
$557$	−0.0143322	−0.000607275 0	−0.000303638	−	1.00000i	$-0.500097\pi$
$557$	−0.0143322	−0.000607275 0	−0.000303638		1.00000i	$0.500097\pi$
$558$	0	0
$559$	18.6296	0.787949
$560$	0	0
$561$	0	0
$562$	−36.7168	−1.54881
$563$	17.1826	0.724160	0.362080	−	0.932147i	$-0.382067\pi$
$563$	17.1826	0.724160	0.362080	+	0.932147i	$0.382067\pi$
$564$	0	0
$565$	−5.70149	−0.239864
$566$	−62.2205	−2.61532
$567$	0	0
$568$	38.9791	1.63553
$569$	−29.4779	−1.23578	−0.617889	−	0.786265i	$-0.712012\pi$
$569$	−29.4779	−1.23578	−0.617889	+	0.786265i	$0.712012\pi$
$570$	0	0
$571$	−32.8321	−1.37398	−0.686991	−	0.726666i	$-0.741069\pi$
$571$	−32.8321	−1.37398	−0.686991	+	0.726666i	$0.741069\pi$
$572$	34.9643	1.46193
$573$	0	0
$574$	0	0
$575$	27.9093	1.16390
$576$	0	0
$577$	−31.9573	−1.33040	−0.665201	−	0.746665i	$-0.731654\pi$
$577$	−31.9573	−1.33040	−0.665201	+	0.746665i	$0.731654\pi$
$578$	−67.1520	−2.79315
$579$	0	0
$580$	−10.0200	−0.416060
$581$	0	0
$582$	0	0
$583$	7.90880	0.327549
$584$	28.1085	1.16314
$585$	0	0
$586$	−40.1014	−1.65658
$587$	4.74633	0.195902	0.0979509	−	0.995191i	$-0.468771\pi$
$587$	4.74633	0.195902	0.0979509	+	0.995191i	$0.468771\pi$
$588$	0	0
$589$	−25.7124	−1.05946
$590$	25.4353	1.04715
$591$	0	0
$592$	−6.68792	−0.274872
$593$	1.94079	0.0796988	0.0398494	−	0.999206i	$-0.487312\pi$
$593$	1.94079	0.0796988	0.0398494	+	0.999206i	$0.487312\pi$
$594$	0	0
$595$	0	0
$596$	31.7878	1.30208
$597$	0	0
$598$	68.7454	2.81121
$599$	−8.80134	−0.359613	−0.179807	−	0.983702i	$-0.557547\pi$
$599$	−8.80134	−0.359613	−0.179807	+	0.983702i	$0.557547\pi$
$600$	0	0
$601$	14.9603	0.610244	0.305122	−	0.952313i	$-0.401303\pi$
$601$	14.9603	0.610244	0.305122	+	0.952313i	$0.401303\pi$
$602$	0	0
$603$	0	0
$604$	−34.6561	−1.41014
$605$	4.81542	0.195775
$606$	0	0
$607$	−2.79138	−0.113299	−0.0566494	−	0.998394i	$-0.518042\pi$
$607$	−2.79138	−0.113299	−0.0566494	+	0.998394i	$0.518042\pi$
$608$	−8.11023	−0.328913
$609$	0	0
$610$	43.7267	1.77044
$611$	7.78042	0.314762
$612$	0	0
$613$	26.6436	1.07612	0.538062	−	0.842905i	$-0.319157\pi$
$613$	26.6436	1.07612	0.538062	+	0.842905i	$0.319157\pi$
$614$	−34.8246	−1.40541
$615$	0	0
$616$	0	0
$617$	−38.1120	−1.53433	−0.767166	−	0.641448i	$-0.778334\pi$
$617$	−38.1120	−1.53433	−0.767166	+	0.641448i	$0.778334\pi$
$618$	0	0
$619$	23.9315	0.961890	0.480945	−	0.876751i	$-0.340294\pi$
$619$	23.9315	0.961890	0.480945	+	0.876751i	$0.340294\pi$
$620$	−45.5433	−1.82906
$621$	0	0
$622$	28.0384	1.12424
$623$	0	0
$624$	0	0
$625$	4.63231	0.185293
$626$	−0.842228	−0.0336622
$627$	0	0
$628$	−7.82627	−0.312302
$629$	23.8232	0.949893
$630$	0	0
$631$	−16.6748	−0.663812	−0.331906	−	0.943313i	$-0.607692\pi$
$631$	−16.6748	−0.663812	−0.331906	+	0.943313i	$0.607692\pi$
$632$	1.20912	0.0480960
$633$	0	0
$634$	24.7833	0.984271
$635$	4.08721	0.162196
$636$	0	0
$637$	0	0
$638$	−14.1978	−0.562098
$639$	0	0
$640$	−25.3795	−1.00321
$641$	32.0213	1.26477	0.632383	−	0.774656i	$-0.282077\pi$
$641$	32.0213	1.26477	0.632383	+	0.774656i	$0.282077\pi$
$642$	0	0
$643$	3.51939	0.138791	0.0693956	−	0.997589i	$-0.477893\pi$
$643$	3.51939	0.138791	0.0693956	+	0.997589i	$0.477893\pi$
$644$	0	0
$645$	0	0
$646$	−40.1459	−1.57952
$647$	−41.5546	−1.63368	−0.816839	−	0.576865i	$-0.804276\pi$
$647$	−41.5546	−1.63368	−0.816839	+	0.576865i	$0.804276\pi$
$648$	0	0
$649$	23.2256	0.911686
$650$	30.0049	1.17689
$651$	0	0
$652$	76.0283	2.97750
$653$	10.2858	0.402513	0.201257	−	0.979539i	$-0.435497\pi$
$653$	10.2858	0.402513	0.201257	+	0.979539i	$0.435497\pi$
$654$	0	0
$655$	−10.3792	−0.405550
$656$	−3.52507	−0.137631
$657$	0	0
$658$	0	0
$659$	20.3551	0.792923	0.396461	−	0.918051i	$-0.370238\pi$
$659$	20.3551	0.792923	0.396461	+	0.918051i	$0.370238\pi$
$660$	0	0
$661$	16.6527	0.647716	0.323858	−	0.946106i	$-0.395020\pi$
$661$	16.6527	0.647716	0.323858	+	0.946106i	$0.395020\pi$
$662$	−35.2597	−1.37041
$663$	0	0
$664$	29.6484	1.15058
$665$	0	0
$666$	0	0
$667$	−17.9895	−0.696557
$668$	−15.4929	−0.599437
$669$	0	0
$670$	−7.72041	−0.298265
$671$	39.9280	1.54141
$672$	0	0
$673$	10.2188	0.393905	0.196953	−	0.980413i	$-0.436895\pi$
$673$	10.2188	0.393905	0.196953	+	0.980413i	$0.436895\pi$
$674$	−22.3258	−0.859958
$675$	0	0
$676$	0.505442	0.0194401
$677$	−6.38685	−0.245466	−0.122733	−	0.992440i	$-0.539166\pi$
$677$	−6.38685	−0.245466	−0.122733	+	0.992440i	$0.539166\pi$
$678$	0	0
$679$	0	0
$680$	−31.8745	−1.22233
$681$	0	0
$682$	−64.5323	−2.47107
$683$	−30.8122	−1.17900	−0.589499	−	0.807769i	$-0.700675\pi$
$683$	−30.8122	−1.17900	−0.589499	+	0.807769i	$0.700675\pi$
$684$	0	0
$685$	−17.1981	−0.657107
$686$	0	0
$687$	0	0
$688$	9.71234	0.370279
$689$	10.7734	0.410432
$690$	0	0
$691$	4.86804	0.185189	0.0925945	−	0.995704i	$-0.470484\pi$
$691$	4.86804	0.185189	0.0925945	+	0.995704i	$0.470484\pi$
$692$	−6.58487	−0.250319
$693$	0	0
$694$	−63.0544	−2.39351
$695$	−6.93024	−0.262879
$696$	0	0
$697$	12.5567	0.475621
$698$	22.6565	0.857562
$699$	0	0
$700$	0	0
$701$	44.4038	1.67711	0.838554	−	0.544819i	$-0.183402\pi$
$701$	44.4038	1.67711	0.838554	+	0.544819i	$0.183402\pi$
$702$	0	0
$703$	8.89923	0.335641
$704$	−30.4122	−1.14620
$705$	0	0
$706$	40.4913	1.52391
$707$	0	0
$708$	0	0
$709$	−26.5768	−0.998112	−0.499056	−	0.866570i	$-0.666320\pi$
$709$	−26.5768	−0.998112	−0.499056	+	0.866570i	$0.666320\pi$
$710$	−29.4770	−1.10625
$711$	0	0
$712$	−27.7171	−1.03874
$713$	−81.7664	−3.06217
$714$	0	0
$715$	−11.8522	−0.443247
$716$	29.1804	1.09052
$717$	0	0
$718$	−60.8730	−2.27176
$719$	33.8924	1.26397	0.631987	−	0.774979i	$-0.282239\pi$
$719$	33.8924	1.26397	0.631987	+	0.774979i	$0.282239\pi$
$720$	0	0
$721$	0	0
$722$	30.0652	1.11891
$723$	0	0
$724$	−91.8778	−3.41461
$725$	−7.85177	−0.291608
$726$	0	0
$727$	23.9417	0.887949	0.443974	−	0.896039i	$-0.353568\pi$
$727$	23.9417	0.887949	0.443974	+	0.896039i	$0.353568\pi$
$728$	0	0
$729$	0	0
$730$	−21.2564	−0.786736
$731$	−34.5965	−1.27960
$732$	0	0
$733$	37.5703	1.38769	0.693846	−	0.720124i	$-0.255915\pi$
$733$	37.5703	1.38769	0.693846	+	0.720124i	$0.255915\pi$
$734$	−24.6065	−0.908243
$735$	0	0
$736$	−25.7908	−0.950661
$737$	−7.04972	−0.259680
$738$	0	0
$739$	−5.64110	−0.207511	−0.103756	−	0.994603i	$-0.533086\pi$
$739$	−5.64110	−0.207511	−0.103756	+	0.994603i	$0.533086\pi$
$740$	15.7628	0.579452
$741$	0	0
$742$	0	0
$743$	15.1703	0.556545	0.278273	−	0.960502i	$-0.410238\pi$
$743$	15.1703	0.556545	0.278273	+	0.960502i	$0.410238\pi$
$744$	0	0
$745$	−10.7754	−0.394781
$746$	46.9952	1.72062
$747$	0	0
$748$	−64.9312	−2.37412
$749$	0	0
$750$	0	0
$751$	25.8935	0.944869	0.472434	−	0.881366i	$-0.343375\pi$
$751$	25.8935	0.944869	0.472434	+	0.881366i	$0.343375\pi$
$752$	4.05623	0.147915
$753$	0	0
$754$	−19.3403	−0.704331
$755$	11.7477	0.427543
$756$	0	0
$757$	36.9054	1.34135	0.670675	−	0.741752i	$-0.266005\pi$
$757$	36.9054	1.34135	0.670675	+	0.741752i	$0.266005\pi$
$758$	−33.6105	−1.22079
$759$	0	0
$760$	−11.9068	−0.431906
$761$	−28.4009	−1.02953	−0.514765	−	0.857331i	$-0.672121\pi$
$761$	−28.4009	−1.02953	−0.514765	+	0.857331i	$0.672121\pi$
$762$	0	0
$763$	0	0
$764$	−25.9219	−0.937823
$765$	0	0
$766$	−31.8959	−1.15245
$767$	31.6379	1.14238
$768$	0	0
$769$	5.26152	0.189735	0.0948677	−	0.995490i	$-0.469757\pi$
$769$	5.26152	0.189735	0.0948677	+	0.995490i	$0.469757\pi$
$770$	0	0
$771$	0	0
$772$	41.1620	1.48145
$773$	41.3747	1.48814	0.744071	−	0.668100i	$-0.232892\pi$
$773$	41.3747	1.48814	0.744071	+	0.668100i	$0.232892\pi$
$774$	0	0
$775$	−35.6880	−1.28195
$776$	−50.8052	−1.82380
$777$	0	0
$778$	−49.6385	−1.77963
$779$	4.69061	0.168059
$780$	0	0
$781$	−26.9163	−0.963140
$782$	−127.665	−4.56530
$783$	0	0
$784$	0	0
$785$	2.65295	0.0946877
$786$	0	0
$787$	−5.36129	−0.191109	−0.0955547	−	0.995424i	$-0.530462\pi$
$787$	−5.36129	−0.191109	−0.0955547	+	0.995424i	$0.530462\pi$
$788$	72.8584	2.59547
$789$	0	0
$790$	−0.914366	−0.0325317
$791$	0	0
$792$	0	0
$793$	54.3899	1.93144
$794$	−59.6275	−2.11610
$795$	0	0
$796$	−37.4265	−1.32655
$797$	−30.4204	−1.07755	−0.538773	−	0.842451i	$-0.681112\pi$
$797$	−30.4204	−1.07755	−0.538773	+	0.842451i	$0.681112\pi$
$798$	0	0
$799$	−14.4488	−0.511161
$800$	−11.2567	−0.397986
$801$	0	0
$802$	−50.1853	−1.77210
$803$	−19.4098	−0.684958
$804$	0	0
$805$	0	0
$806$	−87.9058	−3.09635
$807$	0	0
$808$	−10.7246	−0.377290
$809$	43.5458	1.53099	0.765494	−	0.643443i	$-0.222495\pi$
$809$	43.5458	1.53099	0.765494	+	0.643443i	$0.222495\pi$
$810$	0	0
$811$	−17.4078	−0.611272	−0.305636	−	0.952148i	$-0.598869\pi$
$811$	−17.4078	−0.611272	−0.305636	+	0.952148i	$0.598869\pi$
$812$	0	0
$813$	0	0
$814$	22.3350	0.782842
$815$	−25.7720	−0.902755
$816$	0	0
$817$	−12.9236	−0.452141
$818$	1.28973	0.0450944
$819$	0	0
$820$	8.30826	0.290137
$821$	23.0636	0.804925	0.402462	−	0.915437i	$-0.368154\pi$
$821$	23.0636	0.804925	0.402462	+	0.915437i	$0.368154\pi$
$822$	0	0
$823$	10.0780	0.351298	0.175649	−	0.984453i	$-0.443798\pi$
$823$	10.0780	0.351298	0.175649	+	0.984453i	$0.443798\pi$
$824$	23.1464	0.806344
$825$	0	0
$826$	0	0
$827$	−28.4954	−0.990882	−0.495441	−	0.868642i	$-0.664993\pi$
$827$	−28.4954	−0.990882	−0.495441	+	0.868642i	$0.664993\pi$
$828$	0	0
$829$	1.81520	0.0630447	0.0315223	−	0.999503i	$-0.489964\pi$
$829$	1.81520	0.0630447	0.0315223	+	0.999503i	$0.489964\pi$
$830$	−22.4209	−0.778241
$831$	0	0
$832$	−41.4275	−1.43624
$833$	0	0
$834$	0	0
$835$	5.25177	0.181745
$836$	−24.2553	−0.838886
$837$	0	0
$838$	−56.4666	−1.95060
$839$	−6.87720	−0.237427	−0.118714	−	0.992929i	$-0.537877\pi$
$839$	−6.87720	−0.237427	−0.118714	+	0.992929i	$0.537877\pi$
$840$	0	0
$841$	−23.9390	−0.825482
$842$	6.90112	0.237828
$843$	0	0
$844$	2.17032	0.0747057
$845$	−0.171334	−0.00589408
$846$	0	0
$847$	0	0
$848$	5.61656	0.192873
$849$	0	0
$850$	−55.7212	−1.91122
$851$	28.2998	0.970105
$852$	0	0
$853$	2.69270	0.0921965	0.0460982	−	0.998937i	$-0.485321\pi$
$853$	2.69270	0.0921965	0.0460982	+	0.998937i	$0.485321\pi$
$854$	0	0
$855$	0	0
$856$	45.2271	1.54583
$857$	23.6722	0.808625	0.404313	−	0.914621i	$-0.367511\pi$
$857$	23.6722	0.808625	0.404313	+	0.914621i	$0.367511\pi$
$858$	0	0
$859$	13.0275	0.444492	0.222246	−	0.974991i	$-0.428661\pi$
$859$	13.0275	0.444492	0.222246	+	0.974991i	$0.428661\pi$
$860$	−22.8911	−0.780579
$861$	0	0
$862$	74.4218	2.53482
$863$	19.7559	0.672499	0.336250	−	0.941773i	$-0.390841\pi$
$863$	19.7559	0.672499	0.336250	+	0.941773i	$0.390841\pi$
$864$	0	0
$865$	2.23214	0.0758950
$866$	−63.8647	−2.17021
$867$	0	0
$868$	0	0
$869$	−0.834932	−0.0283231
$870$	0	0
$871$	−9.60311	−0.325389
$872$	30.7699	1.04200
$873$	0	0
$874$	−47.6897	−1.61313
$875$	0	0
$876$	0	0
$877$	12.6814	0.428222	0.214111	−	0.976809i	$-0.431315\pi$
$877$	12.6814	0.428222	0.214111	+	0.976809i	$0.431315\pi$
$878$	−60.0731	−2.02737
$879$	0	0
$880$	−6.17900	−0.208294
$881$	43.7202	1.47297	0.736485	−	0.676454i	$-0.236484\pi$
$881$	43.7202	1.47297	0.736485	+	0.676454i	$0.236484\pi$
$882$	0	0
$883$	−1.03795	−0.0349298	−0.0174649	−	0.999847i	$-0.505560\pi$
$883$	−1.03795	−0.0349298	−0.0174649	+	0.999847i	$0.505560\pi$
$884$	−88.4492	−2.97487
$885$	0	0
$886$	30.9062	1.03831
$887$	6.38256	0.214305	0.107153	−	0.994243i	$-0.465827\pi$
$887$	6.38256	0.214305	0.107153	+	0.994243i	$0.465827\pi$
$888$	0	0
$889$	0	0
$890$	20.9604	0.702596
$891$	0	0
$892$	−10.9472	−0.366538
$893$	−5.39739	−0.180617
$894$	0	0
$895$	−9.89156	−0.330638
$896$	0	0
$897$	0	0
$898$	46.0941	1.53818
$899$	23.0035	0.767209
$900$	0	0
$901$	−20.0069	−0.666526
$902$	11.7723	0.391976
$903$	0	0
$904$	−17.8810	−0.594712
$905$	31.1447	1.03528
$906$	0	0
$907$	−45.4262	−1.50835	−0.754176	−	0.656672i	$-0.771964\pi$
$907$	−45.4262	−1.50835	−0.754176	+	0.656672i	$0.771964\pi$
$908$	72.8884	2.41889
$909$	0	0
$910$	0	0
$911$	4.48708	0.148664	0.0743318	−	0.997234i	$-0.476318\pi$
$911$	4.48708	0.148664	0.0743318	+	0.997234i	$0.476318\pi$
$912$	0	0
$913$	−20.4731	−0.677562
$914$	68.0611	2.25126
$915$	0	0
$916$	72.5202	2.39613
$917$	0	0
$918$	0	0
$919$	−34.6378	−1.14260	−0.571298	−	0.820743i	$-0.693560\pi$
$919$	−34.6378	−1.14260	−0.571298	+	0.820743i	$0.693560\pi$
$920$	−37.8640	−1.24834
$921$	0	0
$922$	−54.6622	−1.80020
$923$	−36.6653	−1.20685
$924$	0	0
$925$	12.3518	0.406126
$926$	−69.2879	−2.27694
$927$	0	0
$928$	7.25577	0.238182
$929$	−15.2728	−0.501084	−0.250542	−	0.968106i	$-0.580609\pi$
$929$	−15.2728	−0.501084	−0.250542	+	0.968106i	$0.580609\pi$
$930$	0	0
$931$	0	0
$932$	−1.09758	−0.0359526
$933$	0	0
$934$	10.3525	0.338746
$935$	22.0104	0.719816
$936$	0	0
$937$	−16.6920	−0.545305	−0.272652	−	0.962113i	$-0.587901\pi$
$937$	−16.6920	−0.545305	−0.272652	+	0.962113i	$0.587901\pi$
$938$	0	0
$939$	0	0
$940$	−9.56015	−0.311818
$941$	−21.2479	−0.692661	−0.346330	−	0.938113i	$-0.612572\pi$
$941$	−21.2479	−0.692661	−0.346330	+	0.938113i	$0.612572\pi$
$942$	0	0
$943$	14.9163	0.485741
$944$	16.4940	0.536836
$945$	0	0
$946$	−32.4354	−1.05456
$947$	39.3146	1.27755	0.638777	−	0.769392i	$-0.279441\pi$
$947$	39.3146	1.27755	0.638777	+	0.769392i	$0.279441\pi$
$948$	0	0
$949$	−26.4400	−0.858280
$950$	−20.8148	−0.675322
$951$	0	0
$952$	0	0
$953$	−15.1311	−0.490143	−0.245072	−	0.969505i	$-0.578811\pi$
$953$	−15.1311	−0.490143	−0.245072	+	0.969505i	$0.578811\pi$
$954$	0	0
$955$	8.78701	0.284341
$956$	12.8566	0.415811
$957$	0	0
$958$	−37.7639	−1.22009
$959$	0	0
$960$	0	0
$961$	73.5558	2.37277
$962$	30.4247	0.980932
$963$	0	0
$964$	−21.7725	−0.701244
$965$	−13.9531	−0.449165
$966$	0	0
$967$	31.3881	1.00937	0.504687	−	0.863303i	$-0.331608\pi$
$967$	31.3881	1.00937	0.504687	+	0.863303i	$0.331608\pi$
$968$	15.1021	0.485399
$969$	0	0
$970$	38.4203	1.23360
$971$	−34.9521	−1.12167	−0.560833	−	0.827929i	$-0.689519\pi$
$971$	−34.9521	−1.12167	−0.560833	+	0.827929i	$0.689519\pi$
$972$	0	0
$973$	0	0
$974$	90.7538	2.90794
$975$	0	0
$976$	28.3555	0.907639
$977$	−7.68632	−0.245907	−0.122954	−	0.992412i	$-0.539237\pi$
$977$	−7.68632	−0.245907	−0.122954	+	0.992412i	$0.539237\pi$
$978$	0	0
$979$	19.1396	0.611703
$980$	0	0
$981$	0	0
$982$	−31.7586	−1.01346
$983$	−2.18071	−0.0695538	−0.0347769	−	0.999395i	$-0.511072\pi$
$983$	−2.18071	−0.0695538	−0.0347769	+	0.999395i	$0.511072\pi$
$984$	0	0
$985$	−24.6975	−0.786928
$986$	35.9163	1.14381
$987$	0	0
$988$	−33.0405	−1.05116
$989$	−41.0976	−1.30683
$990$	0	0
$991$	−5.70317	−0.181167	−0.0905837	−	0.995889i	$-0.528873\pi$
$991$	−5.70317	−0.181167	−0.0905837	+	0.995889i	$0.528873\pi$
$992$	32.9791	1.04709
$993$	0	0
$994$	0	0
$995$	12.6868	0.402199
$996$	0	0
$997$	−31.3797	−0.993804	−0.496902	−	0.867807i	$-0.665529\pi$
$997$	−31.3797	−0.993804	−0.496902	+	0.867807i	$0.665529\pi$
$998$	−12.8238	−0.405930
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.w.1.1		4
3.2	odd	2		3969.2.a.t.1.4		4
7.3	odd	6		567.2.e.c.163.4	✓	8
7.5	odd	6		567.2.e.c.487.4	yes	8
7.6	odd	2		3969.2.a.x.1.1		4
21.5	even	6		567.2.e.d.487.1	yes	8
21.17	even	6		567.2.e.d.163.1	yes	8
21.20	even	2		3969.2.a.s.1.4		4
63.5	even	6		567.2.h.j.298.4		8
63.31	odd	6		567.2.g.j.541.4		8
63.38	even	6		567.2.h.j.352.4		8
63.40	odd	6		567.2.h.k.298.1		8
63.47	even	6		567.2.g.k.109.1		8
63.52	odd	6		567.2.h.k.352.1		8
63.59	even	6		567.2.g.k.541.1		8
63.61	odd	6		567.2.g.j.109.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
567.2.e.c.163.4	✓	8	7.3	odd	6
567.2.e.c.487.4	yes	8	7.5	odd	6
567.2.e.d.163.1	yes	8	21.17	even	6
567.2.e.d.487.1	yes	8	21.5	even	6
567.2.g.j.109.4		8	63.61	odd	6
567.2.g.j.541.4		8	63.31	odd	6
567.2.g.k.109.1		8	63.47	even	6
567.2.g.k.541.1		8	63.59	even	6
567.2.h.j.298.4		8	63.5	even	6
567.2.h.j.352.4		8	63.38	even	6
567.2.h.k.298.1		8	63.40	odd	6
567.2.h.k.352.1		8	63.52	odd	6
3969.2.a.s.1.4		4	21.20	even	2
3969.2.a.t.1.4		4	3.2	odd	2
3969.2.a.w.1.1		4	1.1	even	1	trivial
3969.2.a.x.1.1		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)

\(f(q)\)	\(=\)	\(q-2.37167 q^{2} +3.62484 q^{4} -1.22875 q^{5} -3.85358 q^{8} +O(q^{10})\)	\(q-2.37167 q^{2} +3.62484 q^{4} -1.22875 q^{5} -3.85358 q^{8} +2.91418 q^{10} +2.66102 q^{11} +3.62484 q^{13} +1.88977 q^{16} -6.73158 q^{17} -2.51460 q^{19} -4.45400 q^{20} -6.31107 q^{22} -7.99651 q^{23} -3.49018 q^{25} -8.59693 q^{26} +2.24967 q^{29} +10.2253 q^{31} +3.22526 q^{32} +15.9651 q^{34} -3.53902 q^{37} +5.96382 q^{38} +4.73507 q^{40} -1.86535 q^{41} +5.13944 q^{43} +9.64576 q^{44} +18.9651 q^{46} +2.14642 q^{47} +8.27758 q^{50} +13.1394 q^{52} +2.97209 q^{53} -3.26972 q^{55} -5.33549 q^{58} +8.72809 q^{59} +15.0048 q^{61} -24.2510 q^{62} -11.4288 q^{64} -4.45400 q^{65} -2.64925 q^{67} -24.4009 q^{68} -10.1150 q^{71} -7.29413 q^{73} +8.39340 q^{74} -9.11502 q^{76} -0.313764 q^{79} -2.32204 q^{80} +4.42400 q^{82} -7.69371 q^{83} +8.27140 q^{85} -12.1891 q^{86} -10.2545 q^{88} +7.19256 q^{89} -28.9860 q^{92} -5.09060 q^{94} +3.08981 q^{95} +13.1839 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + 5 q^{4} - 2 q^{5} - 3 q^{8}+O(q^{10}) \) 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

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