LMFDB - Embedded newform 3969.2.a.y.1.2

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:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{2}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 6x^{2} + 4 $ x^4 - 6*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.28825$ 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)

$f(q)$	$=$	$q-0.618034 q^{2} -1.61803 q^{4} +2.28825 q^{5} +2.23607 q^{8} +O(q^{10})$	$q-0.618034 q^{2} -1.61803 q^{4} +2.28825 q^{5} +2.23607 q^{8} -1.41421 q^{10} +1.00000 q^{11} +4.57649 q^{13} +1.85410 q^{16} -3.16228 q^{17} -4.03631 q^{19} -3.70246 q^{20} -0.618034 q^{22} -7.23607 q^{23} +0.236068 q^{25} -2.82843 q^{26} +1.23607 q^{29} -5.99070 q^{31} -5.61803 q^{32} +1.95440 q^{34} -10.7082 q^{37} +2.49458 q^{38} +5.11667 q^{40} -5.78437 q^{41} +5.94427 q^{43} -1.61803 q^{44} +4.47214 q^{46} -3.03476 q^{47} -0.145898 q^{50} -7.40492 q^{52} +10.7082 q^{53} +2.28825 q^{55} -0.763932 q^{58} -14.4760 q^{59} +8.94665 q^{61} +3.70246 q^{62} -0.236068 q^{64} +10.4721 q^{65} +1.47214 q^{67} +5.11667 q^{68} +5.47214 q^{71} -6.73722 q^{73} +6.61803 q^{74} +6.53089 q^{76} -7.76393 q^{79} +4.24264 q^{80} +3.57494 q^{82} -8.61280 q^{83} -7.23607 q^{85} -3.67376 q^{86} +2.23607 q^{88} +6.32456 q^{89} +11.7082 q^{92} +1.87558 q^{94} -9.23607 q^{95} +6.19704 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} - 2 q^{4}+O(q^{10}) $ 4 * q + 2 * q^2 - 2 * q^4	$ 4 q + 2 q^{2} - 2 q^{4} + 4 q^{11} - 6 q^{16} + 2 q^{22} - 20 q^{23} - 8 q^{25} - 4 q^{29} - 18 q^{32} - 16 q^{37} - 12 q^{43} - 2 q^{44} - 14 q^{50} + 16 q^{53} - 12 q^{58} + 8 q^{64} + 24 q^{65} - 12 q^{67} + 4 q^{71} + 22 q^{74} - 40 q^{79} - 20 q^{85} - 46 q^{86} + 20 q^{92} - 28 q^{95}+O(q^{100}) $ 4 * q + 2 * q^2 - 2 * q^4 + 4 * q^11 - 6 * q^16 + 2 * q^22 - 20 * q^23 - 8 * q^25 - 4 * q^29 - 18 * q^32 - 16 * q^37 - 12 * q^43 - 2 * q^44 - 14 * q^50 + 16 * q^53 - 12 * q^58 + 8 * q^64 + 24 * q^65 - 12 * q^67 + 4 * q^71 + 22 * q^74 - 40 * q^79 - 20 * q^85 - 46 * q^86 + 20 * q^92 - 28 * q^95

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 $

$2$	−0.618034	−0.437016	−0.218508	−	0.975835i	$-0.570119\pi$
$2$	−0.618034	−0.437016	−0.218508	+	0.975835i	$0.570119\pi$
$3$	0	0
$3$	0	0
$4$	−1.61803	−0.809017
$5$	2.28825	1.02333	0.511667	−	0.859184i	$-0.329028\pi$
$5$	2.28825	1.02333	0.511667	+	0.859184i	$0.329028\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	2.23607	0.790569
$9$	0	0
$10$	−1.41421	−0.447214
$11$	1.00000	0.301511	0.150756	−	0.988571i	$-0.451829\pi$
$11$	1.00000	0.301511	0.150756	+	0.988571i	$0.451829\pi$
$12$	0	0
$13$	4.57649	1.26929	0.634645	−	0.772804i	$-0.281146\pi$
$13$	4.57649	1.26929	0.634645	+	0.772804i	$0.281146\pi$
$14$	0	0
$15$	0	0
$16$	1.85410	0.463525
$17$	−3.16228	−0.766965	−0.383482	−	0.923548i	$-0.625275\pi$
$17$	−3.16228	−0.766965	−0.383482	+	0.923548i	$0.625275\pi$
$18$	0	0
$19$	−4.03631	−0.925993	−0.462996	−	0.886360i	$-0.653226\pi$
$19$	−4.03631	−0.925993	−0.462996	+	0.886360i	$0.653226\pi$
$20$	−3.70246	−0.827895
$21$	0	0
$22$	−0.618034	−0.131765
$23$	−7.23607	−1.50882	−0.754412	−	0.656401i	$-0.772078\pi$
$23$	−7.23607	−1.50882	−0.754412	+	0.656401i	$0.772078\pi$
$24$	0	0
$25$	0.236068	0.0472136
$26$	−2.82843	−0.554700
$27$	0	0
$28$	0	0
$29$	1.23607	0.229532	0.114766	−	0.993393i	$-0.463388\pi$
$29$	1.23607	0.229532	0.114766	+	0.993393i	$0.463388\pi$
$30$	0	0
$31$	−5.99070	−1.07596	−0.537981	−	0.842957i	$-0.680813\pi$
$31$	−5.99070	−1.07596	−0.537981	+	0.842957i	$0.680813\pi$
$32$	−5.61803	−0.993137
$33$	0	0
$34$	1.95440	0.335176
$35$	0	0
$36$	0	0
$37$	−10.7082	−1.76042	−0.880209	−	0.474586i	$-0.842598\pi$
$37$	−10.7082	−1.76042	−0.880209	+	0.474586i	$0.842598\pi$
$38$	2.49458	0.404674
$39$	0	0
$40$	5.11667	0.809017
$41$	−5.78437	−0.903367	−0.451684	−	0.892178i	$-0.649176\pi$
$41$	−5.78437	−0.903367	−0.451684	+	0.892178i	$0.649176\pi$
$42$	0	0
$43$	5.94427	0.906493	0.453246	−	0.891385i	$-0.350266\pi$
$43$	5.94427	0.906493	0.453246	+	0.891385i	$0.350266\pi$
$44$	−1.61803	−0.243928
$45$	0	0
$46$	4.47214	0.659380
$47$	−3.03476	−0.442665	−0.221332	−	0.975198i	$-0.571041\pi$
$47$	−3.03476	−0.442665	−0.221332	+	0.975198i	$0.571041\pi$
$48$	0	0
$49$	0	0
$50$	−0.145898	−0.0206331
$51$	0	0
$52$	−7.40492	−1.02688
$53$	10.7082	1.47088	0.735442	−	0.677587i	$-0.236974\pi$
$53$	10.7082	1.47088	0.735442	+	0.677587i	$0.236974\pi$
$54$	0	0
$55$	2.28825	0.308547
$56$	0	0
$57$	0	0
$58$	−0.763932	−0.100309
$59$	−14.4760	−1.88461	−0.942306	−	0.334752i	$-0.891348\pi$
$59$	−14.4760	−1.88461	−0.942306	+	0.334752i	$0.891348\pi$
$60$	0	0
$61$	8.94665	1.14550	0.572751	−	0.819730i	$-0.305876\pi$
$61$	8.94665	1.14550	0.572751	+	0.819730i	$0.305876\pi$
$62$	3.70246	0.470213
$63$	0	0
$64$	−0.236068	−0.0295085
$65$	10.4721	1.29891
$66$	0	0
$67$	1.47214	0.179850	0.0899250	−	0.995949i	$-0.471337\pi$
$67$	1.47214	0.179850	0.0899250	+	0.995949i	$0.471337\pi$
$68$	5.11667	0.620488
$69$	0	0
$70$	0	0
$71$	5.47214	0.649423	0.324712	−	0.945813i	$-0.394733\pi$
$71$	5.47214	0.649423	0.324712	+	0.945813i	$0.394733\pi$
$72$	0	0
$73$	−6.73722	−0.788532	−0.394266	−	0.918996i	$-0.629001\pi$
$73$	−6.73722	−0.788532	−0.394266	+	0.918996i	$0.629001\pi$
$74$	6.61803	0.769331
$75$	0	0
$76$	6.53089	0.749144
$77$	0	0
$78$	0	0
$79$	−7.76393	−0.873511	−0.436755	−	0.899580i	$-0.643872\pi$
$79$	−7.76393	−0.873511	−0.436755	+	0.899580i	$0.643872\pi$
$80$	4.24264	0.474342
$81$	0	0
$82$	3.57494	0.394786
$83$	−8.61280	−0.945378	−0.472689	−	0.881229i	$-0.656717\pi$
$83$	−8.61280	−0.945378	−0.472689	+	0.881229i	$0.656717\pi$
$84$	0	0
$85$	−7.23607	−0.784862
$86$	−3.67376	−0.396152
$87$	0	0
$88$	2.23607	0.238366
$89$	6.32456	0.670402	0.335201	−	0.942147i	$-0.391196\pi$
$89$	6.32456	0.670402	0.335201	+	0.942147i	$0.391196\pi$
$90$	0	0
$91$	0	0
$92$	11.7082	1.22066
$93$	0	0
$94$	1.87558	0.193452
$95$	−9.23607	−0.947601
$96$	0	0
$97$	6.19704	0.629214	0.314607	−	0.949222i	$-0.398127\pi$
$97$	6.19704	0.629214	0.314607	+	0.949222i	$0.398127\pi$
$98$	0	0
$99$	0	0
$100$	−0.381966	−0.0381966
$101$	18.7186	1.86257	0.931286	−	0.364288i	$-0.118687\pi$
$101$	18.7186	1.86257	0.931286	+	0.364288i	$0.118687\pi$
$102$	0	0
$103$	−5.78437	−0.569951	−0.284976	−	0.958535i	$-0.591986\pi$
$103$	−5.78437	−0.569951	−0.284976	+	0.958535i	$0.591986\pi$
$104$	10.2333	1.00346
$105$	0	0
$106$	−6.61803	−0.642800
$107$	11.9443	1.15470	0.577348	−	0.816498i	$-0.304088\pi$
$107$	11.9443	1.15470	0.577348	+	0.816498i	$0.304088\pi$
$108$	0	0
$109$	−14.9443	−1.43140	−0.715701	−	0.698407i	$-0.753893\pi$
$109$	−14.9443	−1.43140	−0.715701	+	0.698407i	$0.753893\pi$
$110$	−1.41421	−0.134840
$111$	0	0
$112$	0	0
$113$	−3.76393	−0.354081	−0.177040	−	0.984204i	$-0.556652\pi$
$113$	−3.76393	−0.354081	−0.177040	+	0.984204i	$0.556652\pi$
$114$	0	0
$115$	−16.5579	−1.54403
$116$	−2.00000	−0.185695
$117$	0	0
$118$	8.94665	0.823606
$119$	0	0
$120$	0	0
$121$	−10.0000	−0.909091
$122$	−5.52933	−0.500602
$123$	0	0
$124$	9.69316	0.870472
$125$	−10.9010	−0.975019
$126$	0	0
$127$	−17.6525	−1.56640	−0.783202	−	0.621767i	$-0.786415\pi$
$127$	−17.6525	−1.56640	−0.783202	+	0.621767i	$0.786415\pi$
$128$	11.3820	1.00603
$129$	0	0
$130$	−6.47214	−0.567644
$131$	20.2604	1.77016	0.885078	−	0.465443i	$-0.154105\pi$
$131$	20.2604	1.77016	0.885078	+	0.465443i	$0.154105\pi$
$132$	0	0
$133$	0	0
$134$	−0.909830	−0.0785973
$135$	0	0
$136$	−7.07107	−0.606339
$137$	8.23607	0.703655	0.351827	−	0.936065i	$-0.385560\pi$
$137$	8.23607	0.703655	0.351827	+	0.936065i	$0.385560\pi$
$138$	0	0
$139$	−9.69316	−0.822163	−0.411082	−	0.911598i	$-0.634849\pi$
$139$	−9.69316	−0.822163	−0.411082	+	0.911598i	$0.634849\pi$
$140$	0	0
$141$	0	0
$142$	−3.38197	−0.283808
$143$	4.57649	0.382705
$144$	0	0
$145$	2.82843	0.234888
$146$	4.16383	0.344601
$147$	0	0
$148$	17.3262	1.42421
$149$	−10.5279	−0.862476	−0.431238	−	0.902238i	$-0.641923\pi$
$149$	−10.5279	−0.862476	−0.431238	+	0.902238i	$0.641923\pi$
$150$	0	0
$151$	−18.7082	−1.52245	−0.761226	−	0.648487i	$-0.775402\pi$
$151$	−18.7082	−1.52245	−0.761226	+	0.648487i	$0.775402\pi$
$152$	−9.02546	−0.732062
$153$	0	0
$154$	0	0
$155$	−13.7082	−1.10107
$156$	0	0
$157$	−14.8098	−1.18195	−0.590977	−	0.806689i	$-0.701258\pi$
$157$	−14.8098	−1.18195	−0.590977	+	0.806689i	$0.701258\pi$
$158$	4.79837	0.381738
$159$	0	0
$160$	−12.8554	−1.01631
$161$	0	0
$162$	0	0
$163$	−3.76393	−0.294814	−0.147407	−	0.989076i	$-0.547093\pi$
$163$	−3.76393	−0.294814	−0.147407	+	0.989076i	$0.547093\pi$
$164$	9.35931	0.730840
$165$	0	0
$166$	5.32300	0.413145
$167$	18.1784	1.40669	0.703345	−	0.710848i	$-0.251689\pi$
$167$	18.1784	1.40669	0.703345	+	0.710848i	$0.251689\pi$
$168$	0	0
$169$	7.94427	0.611098
$170$	4.47214	0.342997
$171$	0	0
$172$	−9.61803	−0.733368
$173$	−16.0965	−1.22380	−0.611898	−	0.790936i	$-0.709594\pi$
$173$	−16.0965	−1.22380	−0.611898	+	0.790936i	$0.709594\pi$
$174$	0	0
$175$	0	0
$176$	1.85410	0.139758
$177$	0	0
$178$	−3.90879	−0.292976
$179$	−24.3607	−1.82080	−0.910401	−	0.413726i	$-0.864227\pi$
$179$	−24.3607	−1.82080	−0.910401	+	0.413726i	$0.864227\pi$
$180$	0	0
$181$	16.6367	1.23660	0.618299	−	0.785943i	$-0.287822\pi$
$181$	16.6367	1.23660	0.618299	+	0.785943i	$0.287822\pi$
$182$	0	0
$183$	0	0
$184$	−16.1803	−1.19283
$185$	−24.5030	−1.80150
$186$	0	0
$187$	−3.16228	−0.231249
$188$	4.91034	0.358123
$189$	0	0
$190$	5.70820	0.414117
$191$	22.4164	1.62199	0.810997	−	0.585050i	$-0.198925\pi$
$191$	22.4164	1.62199	0.810997	+	0.585050i	$0.198925\pi$
$192$	0	0
$193$	−7.52786	−0.541868	−0.270934	−	0.962598i	$-0.587332\pi$
$193$	−7.52786	−0.541868	−0.270934	+	0.962598i	$0.587332\pi$
$194$	−3.82998	−0.274976
$195$	0	0
$196$	0	0
$197$	−8.41641	−0.599644	−0.299822	−	0.953995i	$-0.596927\pi$
$197$	−8.41641	−0.599644	−0.299822	+	0.953995i	$0.596927\pi$
$198$	0	0
$199$	4.37016	0.309792	0.154896	−	0.987931i	$-0.450496\pi$
$199$	4.37016	0.309792	0.154896	+	0.987931i	$0.450496\pi$
$200$	0.527864	0.0373256
$201$	0	0
$202$	−11.5687	−0.813974
$203$	0	0
$204$	0	0
$205$	−13.2361	−0.924447
$206$	3.57494	0.249078
$207$	0	0
$208$	8.48528	0.588348
$209$	−4.03631	−0.279197
$210$	0	0
$211$	−16.7082	−1.15024	−0.575120	−	0.818069i	$-0.695045\pi$
$211$	−16.7082	−1.15024	−0.575120	+	0.818069i	$0.695045\pi$
$212$	−17.3262	−1.18997
$213$	0	0
$214$	−7.38197	−0.504621
$215$	13.6020	0.927646
$216$	0	0
$217$	0	0
$218$	9.23607	0.625545
$219$	0	0
$220$	−3.70246	−0.249620
$221$	−14.4721	−0.973501
$222$	0	0
$223$	5.32300	0.356455	0.178227	−	0.983989i	$-0.442964\pi$
$223$	5.32300	0.356455	0.178227	+	0.983989i	$0.442964\pi$
$224$	0	0
$225$	0	0
$226$	2.32624	0.154739
$227$	0.952843	0.0632424	0.0316212	−	0.999500i	$-0.489933\pi$
$227$	0.952843	0.0632424	0.0316212	+	0.999500i	$0.489933\pi$
$228$	0	0
$229$	5.78437	0.382242	0.191121	−	0.981566i	$-0.438788\pi$
$229$	5.78437	0.382242	0.191121	+	0.981566i	$0.438788\pi$
$230$	10.2333	0.674767
$231$	0	0
$232$	2.76393	0.181461
$233$	5.05573	0.331212	0.165606	−	0.986192i	$-0.447042\pi$
$233$	5.05573	0.331212	0.165606	+	0.986192i	$0.447042\pi$
$234$	0	0
$235$	−6.94427	−0.452994
$236$	23.4226	1.52468
$237$	0	0
$238$	0	0
$239$	−3.94427	−0.255134	−0.127567	−	0.991830i	$-0.540717\pi$
$239$	−3.94427	−0.255134	−0.127567	+	0.991830i	$0.540717\pi$
$240$	0	0
$241$	16.3516	1.05330	0.526649	−	0.850083i	$-0.323448\pi$
$241$	16.3516	1.05330	0.526649	+	0.850083i	$0.323448\pi$
$242$	6.18034	0.397287
$243$	0	0
$244$	−14.4760	−0.926730
$245$	0	0
$246$	0	0
$247$	−18.4721	−1.17535
$248$	−13.3956	−0.850623
$249$	0	0
$250$	6.73722	0.426099
$251$	−18.5123	−1.16849	−0.584243	−	0.811579i	$-0.698608\pi$
$251$	−18.5123	−1.16849	−0.584243	+	0.811579i	$0.698608\pi$
$252$	0	0
$253$	−7.23607	−0.454928
$254$	10.9098	0.684544
$255$	0	0
$256$	−6.56231	−0.410144
$257$	9.56564	0.596689	0.298344	−	0.954458i	$-0.403566\pi$
$257$	9.56564	0.596689	0.298344	+	0.954458i	$0.403566\pi$
$258$	0	0
$259$	0	0
$260$	−16.9443	−1.05084
$261$	0	0
$262$	−12.5216	−0.773586
$263$	9.76393	0.602070	0.301035	−	0.953613i	$-0.402668\pi$
$263$	9.76393	0.602070	0.301035	+	0.953613i	$0.402668\pi$
$264$	0	0
$265$	24.5030	1.50521
$266$	0	0
$267$	0	0
$268$	−2.38197	−0.145502
$269$	4.16383	0.253873	0.126937	−	0.991911i	$-0.459486\pi$
$269$	4.16383	0.253873	0.126937	+	0.991911i	$0.459486\pi$
$270$	0	0
$271$	26.1723	1.58985	0.794926	−	0.606707i	$-0.207510\pi$
$271$	26.1723	1.58985	0.794926	+	0.606707i	$0.207510\pi$
$272$	−5.86319	−0.355508
$273$	0	0
$274$	−5.09017	−0.307508
$275$	0.236068	0.0142354
$276$	0	0
$277$	10.2361	0.615026	0.307513	−	0.951544i	$-0.400503\pi$
$277$	10.2361	0.615026	0.307513	+	0.951544i	$0.400503\pi$
$278$	5.99070	0.359299
$279$	0	0
$280$	0	0
$281$	−5.00000	−0.298275	−0.149137	−	0.988816i	$-0.547650\pi$
$281$	−5.00000	−0.298275	−0.149137	+	0.988816i	$0.547650\pi$
$282$	0	0
$283$	−16.3516	−0.972000	−0.486000	−	0.873959i	$-0.661544\pi$
$283$	−16.3516	−0.972000	−0.486000	+	0.873959i	$0.661544\pi$
$284$	−8.85410	−0.525394
$285$	0	0
$286$	−2.82843	−0.167248
$287$	0	0
$288$	0	0
$289$	−7.00000	−0.411765
$290$	−1.74806	−0.102650
$291$	0	0
$292$	10.9010	0.637935
$293$	13.5231	0.790030	0.395015	−	0.918675i	$-0.370739\pi$
$293$	13.5231	0.790030	0.395015	+	0.918675i	$0.370739\pi$
$294$	0	0
$295$	−33.1246	−1.92859
$296$	−23.9443	−1.39173
$297$	0	0
$298$	6.50658	0.376916
$299$	−33.1158	−1.91514
$300$	0	0
$301$	0	0
$302$	11.5623	0.665336
$303$	0	0
$304$	−7.48373	−0.429221
$305$	20.4721	1.17223
$306$	0	0
$307$	−12.9830	−0.740977	−0.370488	−	0.928837i	$-0.620810\pi$
$307$	−12.9830	−0.740977	−0.370488	+	0.928837i	$0.620810\pi$
$308$	0	0
$309$	0	0
$310$	8.47214	0.481185
$311$	−12.6004	−0.714503	−0.357252	−	0.934008i	$-0.616286\pi$
$311$	−12.6004	−0.714503	−0.357252	+	0.934008i	$0.616286\pi$
$312$	0	0
$313$	0.746512	0.0421954	0.0210977	−	0.999777i	$-0.493284\pi$
$313$	0.746512	0.0421954	0.0210977	+	0.999777i	$0.493284\pi$
$314$	9.15298	0.516533
$315$	0	0
$316$	12.5623	0.706685
$317$	−34.8328	−1.95641	−0.978203	−	0.207651i	$-0.933418\pi$
$317$	−34.8328	−1.95641	−0.978203	+	0.207651i	$0.933418\pi$
$318$	0	0
$319$	1.23607	0.0692065
$320$	−0.540182	−0.0301971
$321$	0	0
$322$	0	0
$323$	12.7639	0.710204
$324$	0	0
$325$	1.08036	0.0599278
$326$	2.32624	0.128838
$327$	0	0
$328$	−12.9343	−0.714175
$329$	0	0
$330$	0	0
$331$	2.00000	0.109930	0.0549650	−	0.998488i	$-0.482495\pi$
$331$	2.00000	0.109930	0.0549650	+	0.998488i	$0.482495\pi$
$332$	13.9358	0.764827
$333$	0	0
$334$	−11.2349	−0.614746
$335$	3.36861	0.184047
$336$	0	0
$337$	5.18034	0.282191	0.141096	−	0.989996i	$-0.454938\pi$
$337$	5.18034	0.282191	0.141096	+	0.989996i	$0.454938\pi$
$338$	−4.90983	−0.267060
$339$	0	0
$340$	11.7082	0.634967
$341$	−5.99070	−0.324415
$342$	0	0
$343$	0	0
$344$	13.2918	0.716646
$345$	0	0
$346$	9.94820	0.534819
$347$	4.05573	0.217723	0.108861	−	0.994057i	$-0.465280\pi$
$347$	4.05573	0.217723	0.108861	+	0.994057i	$0.465280\pi$
$348$	0	0
$349$	18.7673	1.00459	0.502296	−	0.864696i	$-0.332489\pi$
$349$	18.7673	1.00459	0.502296	+	0.864696i	$0.332489\pi$
$350$	0	0
$351$	0	0
$352$	−5.61803	−0.299442
$353$	33.2433	1.76936	0.884682	−	0.466195i	$-0.154376\pi$
$353$	33.2433	1.76936	0.884682	+	0.466195i	$0.154376\pi$
$354$	0	0
$355$	12.5216	0.664577
$356$	−10.2333	−0.542366
$357$	0	0
$358$	15.0557	0.795720
$359$	−5.18034	−0.273408	−0.136704	−	0.990612i	$-0.543651\pi$
$359$	−5.18034	−0.273408	−0.136704	+	0.990612i	$0.543651\pi$
$360$	0	0
$361$	−2.70820	−0.142537
$362$	−10.2821	−0.540413
$363$	0	0
$364$	0	0
$365$	−15.4164	−0.806932
$366$	0	0
$367$	−4.57649	−0.238891	−0.119445	−	0.992841i	$-0.538112\pi$
$367$	−4.57649	−0.238891	−0.119445	+	0.992841i	$0.538112\pi$
$368$	−13.4164	−0.699379
$369$	0	0
$370$	15.1437	0.787283
$371$	0	0
$372$	0	0
$373$	13.1803	0.682452	0.341226	−	0.939981i	$-0.389158\pi$
$373$	13.1803	0.682452	0.341226	+	0.939981i	$0.389158\pi$
$374$	1.95440	0.101059
$375$	0	0
$376$	−6.78593	−0.349957
$377$	5.65685	0.291343
$378$	0	0
$379$	−0.347524	−0.0178511	−0.00892556	−	0.999960i	$-0.502841\pi$
$379$	−0.347524	−0.0178511	−0.00892556	+	0.999960i	$0.502841\pi$
$380$	14.9443	0.766625
$381$	0	0
$382$	−13.8541	−0.708838
$383$	−24.6305	−1.25856	−0.629280	−	0.777178i	$-0.716650\pi$
$383$	−24.6305	−1.25856	−0.629280	+	0.777178i	$0.716650\pi$
$384$	0	0
$385$	0	0
$386$	4.65248	0.236805
$387$	0	0
$388$	−10.0270	−0.509045
$389$	−6.94427	−0.352089	−0.176044	−	0.984382i	$-0.556330\pi$
$389$	−6.94427	−0.352089	−0.176044	+	0.984382i	$0.556330\pi$
$390$	0	0
$391$	22.8825	1.15722
$392$	0	0
$393$	0	0
$394$	5.20163	0.262054
$395$	−17.7658	−0.893894
$396$	0	0
$397$	−31.9867	−1.60537	−0.802684	−	0.596405i	$-0.796595\pi$
$397$	−31.9867	−1.60537	−0.802684	+	0.596405i	$0.796595\pi$
$398$	−2.70091	−0.135384
$399$	0	0
$400$	0.437694	0.0218847
$401$	−21.0689	−1.05213	−0.526065	−	0.850444i	$-0.676333\pi$
$401$	−21.0689	−1.05213	−0.526065	+	0.850444i	$0.676333\pi$
$402$	0	0
$403$	−27.4164	−1.36571
$404$	−30.2874	−1.50685
$405$	0	0
$406$	0	0
$407$	−10.7082	−0.530786
$408$	0	0
$409$	19.9265	0.985302	0.492651	−	0.870227i	$-0.336028\pi$
$409$	19.9265	0.985302	0.492651	+	0.870227i	$0.336028\pi$
$410$	8.18034	0.403998
$411$	0	0
$412$	9.35931	0.461100
$413$	0	0
$414$	0	0
$415$	−19.7082	−0.967438
$416$	−25.7109	−1.26058
$417$	0	0
$418$	2.49458	0.122014
$419$	−23.0100	−1.12411	−0.562055	−	0.827100i	$-0.689989\pi$
$419$	−23.0100	−1.12411	−0.562055	+	0.827100i	$0.689989\pi$
$420$	0	0
$421$	−20.8885	−1.01805	−0.509023	−	0.860753i	$-0.669993\pi$
$421$	−20.8885	−1.01805	−0.509023	+	0.860753i	$0.669993\pi$
$422$	10.3262	0.502673
$423$	0	0
$424$	23.9443	1.16284
$425$	−0.746512	−0.0362112
$426$	0	0
$427$	0	0
$428$	−19.3262	−0.934169
$429$	0	0
$430$	−8.40647	−0.405396
$431$	3.41641	0.164563	0.0822813	−	0.996609i	$-0.473779\pi$
$431$	3.41641	0.164563	0.0822813	+	0.996609i	$0.473779\pi$
$432$	0	0
$433$	−33.2433	−1.59757	−0.798786	−	0.601615i	$-0.794524\pi$
$433$	−33.2433	−1.59757	−0.798786	+	0.601615i	$0.794524\pi$
$434$	0	0
$435$	0	0
$436$	24.1803	1.15803
$437$	29.2070	1.39716
$438$	0	0
$439$	−35.7379	−1.70568	−0.852838	−	0.522175i	$-0.825121\pi$
$439$	−35.7379	−1.70568	−0.852838	+	0.522175i	$0.825121\pi$
$440$	5.11667	0.243928
$441$	0	0
$442$	8.94427	0.425436
$443$	−15.7082	−0.746319	−0.373160	−	0.927767i	$-0.621726\pi$
$443$	−15.7082	−0.746319	−0.373160	+	0.927767i	$0.621726\pi$
$444$	0	0
$445$	14.4721	0.686045
$446$	−3.28980	−0.155776
$447$	0	0
$448$	0	0
$449$	22.5279	1.06316	0.531578	−	0.847009i	$-0.321599\pi$
$449$	22.5279	1.06316	0.531578	+	0.847009i	$0.321599\pi$
$450$	0	0
$451$	−5.78437	−0.272376
$452$	6.09017	0.286457
$453$	0	0
$454$	−0.588890	−0.0276380
$455$	0	0
$456$	0	0
$457$	−29.8328	−1.39552	−0.697760	−	0.716331i	$-0.745820\pi$
$457$	−29.8328	−1.39552	−0.697760	+	0.716331i	$0.745820\pi$
$458$	−3.57494	−0.167046
$459$	0	0
$460$	26.7912	1.24915
$461$	8.69161	0.404809	0.202404	−	0.979302i	$-0.435124\pi$
$461$	8.69161	0.404809	0.202404	+	0.979302i	$0.435124\pi$
$462$	0	0
$463$	−1.11146	−0.0516537	−0.0258269	−	0.999666i	$-0.508222\pi$
$463$	−1.11146	−0.0516537	−0.0258269	+	0.999666i	$0.508222\pi$
$464$	2.29180	0.106394
$465$	0	0
$466$	−3.12461	−0.144745
$467$	−13.4443	−0.622129	−0.311065	−	0.950389i	$-0.600686\pi$
$467$	−13.4443	−0.622129	−0.311065	+	0.950389i	$0.600686\pi$
$468$	0	0
$469$	0	0
$470$	4.29180	0.197966
$471$	0	0
$472$	−32.3693	−1.48992
$473$	5.94427	0.273318
$474$	0	0
$475$	−0.952843	−0.0437195
$476$	0	0
$477$	0	0
$478$	2.43769	0.111498
$479$	14.0633	0.642570	0.321285	−	0.946983i	$-0.395885\pi$
$479$	14.0633	0.642570	0.321285	+	0.946983i	$0.395885\pi$
$480$	0	0
$481$	−49.0060	−2.23448
$482$	−10.1058	−0.460308
$483$	0	0
$484$	16.1803	0.735470
$485$	14.1803	0.643896
$486$	0	0
$487$	3.94427	0.178732	0.0893660	−	0.995999i	$-0.471516\pi$
$487$	3.94427	0.178732	0.0893660	+	0.995999i	$0.471516\pi$
$488$	20.0053	0.905598
$489$	0	0
$490$	0	0
$491$	22.0000	0.992846	0.496423	−	0.868081i	$-0.334646\pi$
$491$	22.0000	0.992846	0.496423	+	0.868081i	$0.334646\pi$
$492$	0	0
$493$	−3.90879	−0.176043
$494$	11.4164	0.513648
$495$	0	0
$496$	−11.1074	−0.498736
$497$	0	0
$498$	0	0
$499$	21.8885	0.979866	0.489933	−	0.871760i	$-0.337021\pi$
$499$	21.8885	0.979866	0.489933	+	0.871760i	$0.337021\pi$
$500$	17.6383	0.788807
$501$	0	0
$502$	11.4412	0.510647
$503$	−6.48218	−0.289026	−0.144513	−	0.989503i	$-0.546162\pi$
$503$	−6.48218	−0.289026	−0.144513	+	0.989503i	$0.546162\pi$
$504$	0	0
$505$	42.8328	1.90604
$506$	4.47214	0.198811
$507$	0	0
$508$	28.5623	1.26725
$509$	21.4682	0.951563	0.475782	−	0.879563i	$-0.342165\pi$
$509$	21.4682	0.951563	0.475782	+	0.879563i	$0.342165\pi$
$510$	0	0
$511$	0	0
$512$	−18.7082	−0.826794
$513$	0	0
$514$	−5.91189	−0.260762
$515$	−13.2361	−0.583251
$516$	0	0
$517$	−3.03476	−0.133469
$518$	0	0
$519$	0	0
$520$	23.4164	1.02688
$521$	−9.23179	−0.404452	−0.202226	−	0.979339i	$-0.564818\pi$
$521$	−9.23179	−0.404452	−0.202226	+	0.979339i	$0.564818\pi$
$522$	0	0
$523$	−1.28669	−0.0562632	−0.0281316	−	0.999604i	$-0.508956\pi$
$523$	−1.28669	−0.0562632	−0.0281316	+	0.999604i	$0.508956\pi$
$524$	−32.7820	−1.43209
$525$	0	0
$526$	−6.03444	−0.263114
$527$	18.9443	0.825225
$528$	0	0
$529$	29.3607	1.27655
$530$	−15.1437	−0.657800
$531$	0	0
$532$	0	0
$533$	−26.4721	−1.14664
$534$	0	0
$535$	27.3314	1.18164
$536$	3.29180	0.142184
$537$	0	0
$538$	−2.57339	−0.110947
$539$	0	0
$540$	0	0
$541$	30.5967	1.31546	0.657728	−	0.753255i	$-0.271517\pi$
$541$	30.5967	1.31546	0.657728	+	0.753255i	$0.271517\pi$
$542$	−16.1753	−0.694790
$543$	0	0
$544$	17.7658	0.761702
$545$	−34.1962	−1.46480
$546$	0	0
$547$	10.7082	0.457850	0.228925	−	0.973444i	$-0.426479\pi$
$547$	10.7082	0.457850	0.228925	+	0.973444i	$0.426479\pi$
$548$	−13.3262	−0.569269
$549$	0	0
$550$	−0.145898	−0.00622111
$551$	−4.98915	−0.212545
$552$	0	0
$553$	0	0
$554$	−6.32624	−0.268776
$555$	0	0
$556$	15.6839	0.665144
$557$	7.29180	0.308963	0.154482	−	0.987996i	$-0.450629\pi$
$557$	7.29180	0.308963	0.154482	+	0.987996i	$0.450629\pi$
$558$	0	0
$559$	27.2039	1.15060
$560$	0	0
$561$	0	0
$562$	3.09017	0.130351
$563$	27.9504	1.17797	0.588985	−	0.808144i	$-0.299528\pi$
$563$	27.9504	1.17797	0.588985	+	0.808144i	$0.299528\pi$
$564$	0	0
$565$	−8.61280	−0.362343
$566$	10.1058	0.424780
$567$	0	0
$568$	12.2361	0.513414
$569$	−15.5836	−0.653298	−0.326649	−	0.945146i	$-0.605920\pi$
$569$	−15.5836	−0.653298	−0.326649	+	0.945146i	$0.605920\pi$
$570$	0	0
$571$	−17.8197	−0.745730	−0.372865	−	0.927886i	$-0.621624\pi$
$571$	−17.8197	−0.745730	−0.372865	+	0.927886i	$0.621624\pi$
$572$	−7.40492	−0.309615
$573$	0	0
$574$	0	0
$575$	−1.70820	−0.0712370
$576$	0	0
$577$	3.03476	0.126339	0.0631693	−	0.998003i	$-0.479879\pi$
$577$	3.03476	0.126339	0.0631693	+	0.998003i	$0.479879\pi$
$578$	4.32624	0.179948
$579$	0	0
$580$	−4.57649	−0.190028
$581$	0	0
$582$	0	0
$583$	10.7082	0.443488
$584$	−15.0649	−0.623389
$585$	0	0
$586$	−8.35776	−0.345256
$587$	28.2055	1.16416	0.582082	−	0.813130i	$-0.302238\pi$
$587$	28.2055	1.16416	0.582082	+	0.813130i	$0.302238\pi$
$588$	0	0
$589$	24.1803	0.996334
$590$	20.4721	0.842824
$591$	0	0
$592$	−19.8541	−0.815999
$593$	34.2750	1.40750	0.703752	−	0.710445i	$-0.251506\pi$
$593$	34.2750	1.40750	0.703752	+	0.710445i	$0.251506\pi$
$594$	0	0
$595$	0	0
$596$	17.0344	0.697758
$597$	0	0
$598$	20.4667	0.836945
$599$	−13.8328	−0.565194	−0.282597	−	0.959239i	$-0.591196\pi$
$599$	−13.8328	−0.565194	−0.282597	+	0.959239i	$0.591196\pi$
$600$	0	0
$601$	28.8245	1.17577	0.587887	−	0.808943i	$-0.299960\pi$
$601$	28.8245	1.17577	0.587887	+	0.808943i	$0.299960\pi$
$602$	0	0
$603$	0	0
$604$	30.2705	1.23169
$605$	−22.8825	−0.930304
$606$	0	0
$607$	2.20943	0.0896782	0.0448391	−	0.998994i	$-0.485722\pi$
$607$	2.20943	0.0896782	0.0448391	+	0.998994i	$0.485722\pi$
$608$	22.6761	0.919638
$609$	0	0
$610$	−12.6525	−0.512284
$611$	−13.8885	−0.561870
$612$	0	0
$613$	16.8197	0.679340	0.339670	−	0.940545i	$-0.389685\pi$
$613$	16.8197	0.679340	0.339670	+	0.940545i	$0.389685\pi$
$614$	8.02391	0.323819
$615$	0	0
$616$	0	0
$617$	−5.41641	−0.218056	−0.109028	−	0.994039i	$-0.534774\pi$
$617$	−5.41641	−0.218056	−0.109028	+	0.994039i	$0.534774\pi$
$618$	0	0
$619$	45.0485	1.81065	0.905326	−	0.424717i	$-0.139626\pi$
$619$	45.0485	1.81065	0.905326	+	0.424717i	$0.139626\pi$
$620$	22.1803	0.890784
$621$	0	0
$622$	7.78748	0.312249
$623$	0	0
$624$	0	0
$625$	−26.1246	−1.04498
$626$	−0.461370	−0.0184401
$627$	0	0
$628$	23.9628	0.956221
$629$	33.8623	1.35018
$630$	0	0
$631$	−42.6525	−1.69797	−0.848984	−	0.528418i	$-0.822785\pi$
$631$	−42.6525	−1.69797	−0.848984	+	0.528418i	$0.822785\pi$
$632$	−17.3607	−0.690571
$633$	0	0
$634$	21.5279	0.854981
$635$	−40.3932	−1.60296
$636$	0	0
$637$	0	0
$638$	−0.763932	−0.0302444
$639$	0	0
$640$	26.0447	1.02951
$641$	−21.4721	−0.848098	−0.424049	−	0.905639i	$-0.639392\pi$
$641$	−21.4721	−0.848098	−0.424049	+	0.905639i	$0.639392\pi$
$642$	0	0
$643$	3.08347	0.121600	0.0608000	−	0.998150i	$-0.480635\pi$
$643$	3.08347	0.121600	0.0608000	+	0.998150i	$0.480635\pi$
$644$	0	0
$645$	0	0
$646$	−7.88854	−0.310371
$647$	36.7394	1.44438	0.722188	−	0.691696i	$-0.243136\pi$
$647$	36.7394	1.44438	0.722188	+	0.691696i	$0.243136\pi$
$648$	0	0
$649$	−14.4760	−0.568232
$650$	−0.667701	−0.0261894
$651$	0	0
$652$	6.09017	0.238509
$653$	16.7082	0.653843	0.326921	−	0.945052i	$-0.393989\pi$
$653$	16.7082	0.653843	0.326921	+	0.945052i	$0.393989\pi$
$654$	0	0
$655$	46.3607	1.81146
$656$	−10.7248	−0.418734
$657$	0	0
$658$	0	0
$659$	−16.2361	−0.632467	−0.316234	−	0.948681i	$-0.602418\pi$
$659$	−16.2361	−0.632467	−0.316234	+	0.948681i	$0.602418\pi$
$660$	0	0
$661$	4.98915	0.194056	0.0970278	−	0.995282i	$-0.469066\pi$
$661$	4.98915	0.194056	0.0970278	+	0.995282i	$0.469066\pi$
$662$	−1.23607	−0.0480411
$663$	0	0
$664$	−19.2588	−0.747387
$665$	0	0
$666$	0	0
$667$	−8.94427	−0.346324
$668$	−29.4133	−1.13804
$669$	0	0
$670$	−2.08191	−0.0804314
$671$	8.94665	0.345382
$672$	0	0
$673$	−21.6525	−0.834642	−0.417321	−	0.908759i	$-0.637031\pi$
$673$	−21.6525	−0.834642	−0.417321	+	0.908759i	$0.637031\pi$
$674$	−3.20163	−0.123322
$675$	0	0
$676$	−12.8541	−0.494389
$677$	7.07107	0.271763	0.135882	−	0.990725i	$-0.456613\pi$
$677$	7.07107	0.271763	0.135882	+	0.990725i	$0.456613\pi$
$678$	0	0
$679$	0	0
$680$	−16.1803	−0.620488
$681$	0	0
$682$	3.70246	0.141774
$683$	−22.2361	−0.850839	−0.425420	−	0.904996i	$-0.639874\pi$
$683$	−22.2361	−0.850839	−0.425420	+	0.904996i	$0.639874\pi$
$684$	0	0
$685$	18.8461	0.720074
$686$	0	0
$687$	0	0
$688$	11.0213	0.420183
$689$	49.0060	1.86698
$690$	0	0
$691$	−37.7711	−1.43688	−0.718440	−	0.695589i	$-0.755144\pi$
$691$	−37.7711	−1.43688	−0.718440	+	0.695589i	$0.755144\pi$
$692$	26.0447	0.990072
$693$	0	0
$694$	−2.50658	−0.0951484
$695$	−22.1803	−0.841348
$696$	0	0
$697$	18.2918	0.692851
$698$	−11.5989	−0.439023
$699$	0	0
$700$	0	0
$701$	9.18034	0.346737	0.173368	−	0.984857i	$-0.444535\pi$
$701$	9.18034	0.346737	0.173368	+	0.984857i	$0.444535\pi$
$702$	0	0
$703$	43.2216	1.63013
$704$	−0.236068	−0.00889715
$705$	0	0
$706$	−20.5455	−0.773240
$707$	0	0
$708$	0	0
$709$	25.6525	0.963399	0.481699	−	0.876336i	$-0.340020\pi$
$709$	25.6525	0.963399	0.481699	+	0.876336i	$0.340020\pi$
$710$	−7.73877	−0.290431
$711$	0	0
$712$	14.1421	0.529999
$713$	43.3491	1.62344
$714$	0	0
$715$	10.4721	0.391636
$716$	39.4164	1.47306
$717$	0	0
$718$	3.20163	0.119484
$719$	51.7556	1.93016	0.965079	−	0.261958i	$-0.0843680\pi$
$719$	51.7556	1.93016	0.965079	+	0.261958i	$0.0843680\pi$
$720$	0	0
$721$	0	0
$722$	1.67376	0.0622910
$723$	0	0
$724$	−26.9188	−1.00043
$725$	0.291796	0.0108370
$726$	0	0
$727$	−48.1320	−1.78512	−0.892558	−	0.450933i	$-0.851091\pi$
$727$	−48.1320	−1.78512	−0.892558	+	0.450933i	$0.851091\pi$
$728$	0	0
$729$	0	0
$730$	9.52786	0.352642
$731$	−18.7974	−0.695248
$732$	0	0
$733$	−5.78437	−0.213651	−0.106825	−	0.994278i	$-0.534069\pi$
$733$	−5.78437	−0.213651	−0.106825	+	0.994278i	$0.534069\pi$
$734$	2.82843	0.104399
$735$	0	0
$736$	40.6525	1.49847
$737$	1.47214	0.0542268
$738$	0	0
$739$	11.3607	0.417909	0.208955	−	0.977925i	$-0.432994\pi$
$739$	11.3607	0.417909	0.208955	+	0.977925i	$0.432994\pi$
$740$	39.6467	1.45744
$741$	0	0
$742$	0	0
$743$	39.2492	1.43991	0.719957	−	0.694018i	$-0.244161\pi$
$743$	39.2492	1.43991	0.719957	+	0.694018i	$0.244161\pi$
$744$	0	0
$745$	−24.0903	−0.882602
$746$	−8.14590	−0.298243
$747$	0	0
$748$	5.11667	0.187084
$749$	0	0
$750$	0	0
$751$	−32.3050	−1.17882	−0.589412	−	0.807832i	$-0.700641\pi$
$751$	−32.3050	−1.17882	−0.589412	+	0.807832i	$0.700641\pi$
$752$	−5.62675	−0.205186
$753$	0	0
$754$	−3.49613	−0.127321
$755$	−42.8090	−1.55798
$756$	0	0
$757$	−42.0689	−1.52902	−0.764510	−	0.644612i	$-0.777019\pi$
$757$	−42.0689	−1.52902	−0.764510	+	0.644612i	$0.777019\pi$
$758$	0.214782	0.00780122
$759$	0	0
$760$	−20.6525	−0.749144
$761$	20.2604	0.734437	0.367219	−	0.930135i	$-0.380310\pi$
$761$	20.2604	0.734437	0.367219	+	0.930135i	$0.380310\pi$
$762$	0	0
$763$	0	0
$764$	−36.2705	−1.31222
$765$	0	0
$766$	15.2225	0.550011
$767$	−66.2492	−2.39212
$768$	0	0
$769$	−14.1120	−0.508893	−0.254446	−	0.967087i	$-0.581893\pi$
$769$	−14.1120	−0.508893	−0.254446	+	0.967087i	$0.581893\pi$
$770$	0	0
$771$	0	0
$772$	12.1803	0.438380
$773$	−42.6026	−1.53231	−0.766155	−	0.642656i	$-0.777833\pi$
$773$	−42.6026	−1.53231	−0.766155	+	0.642656i	$0.777833\pi$
$774$	0	0
$775$	−1.41421	−0.0508001
$776$	13.8570	0.497437
$777$	0	0
$778$	4.29180	0.153868
$779$	23.3475	0.836512
$780$	0	0
$781$	5.47214	0.195808
$782$	−14.1421	−0.505722
$783$	0	0
$784$	0	0
$785$	−33.8885	−1.20953
$786$	0	0
$787$	43.6343	1.55539	0.777697	−	0.628639i	$-0.216388\pi$
$787$	43.6343	1.55539	0.777697	+	0.628639i	$0.216388\pi$
$788$	13.6180	0.485122
$789$	0	0
$790$	10.9799	0.390646
$791$	0	0
$792$	0	0
$793$	40.9443	1.45397
$794$	19.7689	0.701572
$795$	0	0
$796$	−7.07107	−0.250627
$797$	−20.8794	−0.739585	−0.369792	−	0.929114i	$-0.620571\pi$
$797$	−20.8794	−0.739585	−0.369792	+	0.929114i	$0.620571\pi$
$798$	0	0
$799$	9.59675	0.339509
$800$	−1.32624	−0.0468896
$801$	0	0
$802$	13.0213	0.459798
$803$	−6.73722	−0.237751
$804$	0	0
$805$	0	0
$806$	16.9443	0.596837
$807$	0	0
$808$	41.8561	1.47249
$809$	−23.5410	−0.827658	−0.413829	−	0.910355i	$-0.635809\pi$
$809$	−23.5410	−0.827658	−0.413829	+	0.910355i	$0.635809\pi$
$810$	0	0
$811$	−50.4990	−1.77326	−0.886630	−	0.462479i	$-0.846960\pi$
$811$	−50.4990	−1.77326	−0.886630	+	0.462479i	$0.846960\pi$
$812$	0	0
$813$	0	0
$814$	6.61803	0.231962
$815$	−8.61280	−0.301693
$816$	0	0
$817$	−23.9929	−0.839406
$818$	−12.3153	−0.430593
$819$	0	0
$820$	21.4164	0.747893
$821$	−10.8197	−0.377609	−0.188804	−	0.982015i	$-0.560461\pi$
$821$	−10.8197	−0.377609	−0.188804	+	0.982015i	$0.560461\pi$
$822$	0	0
$823$	17.4164	0.607098	0.303549	−	0.952816i	$-0.401828\pi$
$823$	17.4164	0.607098	0.303549	+	0.952816i	$0.401828\pi$
$824$	−12.9343	−0.450586
$825$	0	0
$826$	0	0
$827$	9.65248	0.335649	0.167825	−	0.985817i	$-0.446326\pi$
$827$	9.65248	0.335649	0.167825	+	0.985817i	$0.446326\pi$
$828$	0	0
$829$	−3.24109	−0.112568	−0.0562838	−	0.998415i	$-0.517925\pi$
$829$	−3.24109	−0.112568	−0.0562838	+	0.998415i	$0.517925\pi$
$830$	12.1803	0.422786
$831$	0	0
$832$	−1.08036	−0.0374548
$833$	0	0
$834$	0	0
$835$	41.5967	1.43951
$836$	6.53089	0.225875
$837$	0	0
$838$	14.2209	0.491254
$839$	19.0038	0.656083	0.328041	−	0.944663i	$-0.393611\pi$
$839$	19.0038	0.656083	0.328041	+	0.944663i	$0.393611\pi$
$840$	0	0
$841$	−27.4721	−0.947315
$842$	12.9098	0.444902
$843$	0	0
$844$	27.0344	0.930564
$845$	18.1784	0.625358
$846$	0	0
$847$	0	0
$848$	19.8541	0.681793
$849$	0	0
$850$	0.461370	0.0158249
$851$	77.4853	2.65616
$852$	0	0
$853$	−37.7410	−1.29223	−0.646114	−	0.763241i	$-0.723607\pi$
$853$	−37.7410	−1.29223	−0.646114	+	0.763241i	$0.723607\pi$
$854$	0	0
$855$	0	0
$856$	26.7082	0.912868
$857$	−55.2517	−1.88736	−0.943682	−	0.330854i	$-0.892663\pi$
$857$	−55.2517	−1.88736	−0.943682	+	0.330854i	$0.892663\pi$
$858$	0	0
$859$	−41.3460	−1.41071	−0.705354	−	0.708855i	$-0.749212\pi$
$859$	−41.3460	−1.41071	−0.705354	+	0.708855i	$0.749212\pi$
$860$	−22.0084	−0.750481
$861$	0	0
$862$	−2.11146	−0.0719165
$863$	40.3050	1.37200	0.685998	−	0.727603i	$-0.259366\pi$
$863$	40.3050	1.37200	0.685998	+	0.727603i	$0.259366\pi$
$864$	0	0
$865$	−36.8328	−1.25235
$866$	20.5455	0.698165
$867$	0	0
$868$	0	0
$869$	−7.76393	−0.263373
$870$	0	0
$871$	6.73722	0.228282
$872$	−33.4164	−1.13162
$873$	0	0
$874$	−18.0509	−0.610582
$875$	0	0
$876$	0	0
$877$	14.2361	0.480718	0.240359	−	0.970684i	$-0.422735\pi$
$877$	14.2361	0.480718	0.240359	+	0.970684i	$0.422735\pi$
$878$	22.0872	0.745408
$879$	0	0
$880$	4.24264	0.143019
$881$	24.9157	0.839430	0.419715	−	0.907656i	$-0.362130\pi$
$881$	24.9157	0.839430	0.419715	+	0.907656i	$0.362130\pi$
$882$	0	0
$883$	10.3475	0.348222	0.174111	−	0.984726i	$-0.444295\pi$
$883$	10.3475	0.348222	0.174111	+	0.984726i	$0.444295\pi$
$884$	23.4164	0.787579
$885$	0	0
$886$	9.70820	0.326153
$887$	−14.1421	−0.474846	−0.237423	−	0.971406i	$-0.576303\pi$
$887$	−14.1421	−0.474846	−0.237423	+	0.971406i	$0.576303\pi$
$888$	0	0
$889$	0	0
$890$	−8.94427	−0.299813
$891$	0	0
$892$	−8.61280	−0.288378
$893$	12.2492	0.409905
$894$	0	0
$895$	−55.7432	−1.86329
$896$	0	0
$897$	0	0
$898$	−13.9230	−0.464616
$899$	−7.40492	−0.246968
$900$	0	0
$901$	−33.8623	−1.12812
$902$	3.57494	0.119032
$903$	0	0
$904$	−8.41641	−0.279926
$905$	38.0689	1.26545
$906$	0	0
$907$	−55.5410	−1.84421	−0.922105	−	0.386941i	$-0.873532\pi$
$907$	−55.5410	−1.84421	−0.922105	+	0.386941i	$0.873532\pi$
$908$	−1.54173	−0.0511642
$909$	0	0
$910$	0	0
$911$	40.2492	1.33352	0.666758	−	0.745274i	$-0.267681\pi$
$911$	40.2492	1.33352	0.666758	+	0.745274i	$0.267681\pi$
$912$	0	0
$913$	−8.61280	−0.285042
$914$	18.4377	0.609865
$915$	0	0
$916$	−9.35931	−0.309240
$917$	0	0
$918$	0	0
$919$	−3.18034	−0.104910	−0.0524549	−	0.998623i	$-0.516705\pi$
$919$	−3.18034	−0.104910	−0.0524549	+	0.998623i	$0.516705\pi$
$920$	−37.0246	−1.22066
$921$	0	0
$922$	−5.37171	−0.176908
$923$	25.0432	0.824306
$924$	0	0
$925$	−2.52786	−0.0831157
$926$	0.686918	0.0225735
$927$	0	0
$928$	−6.94427	−0.227957
$929$	32.0841	1.05265	0.526323	−	0.850284i	$-0.323570\pi$
$929$	32.0841	1.05265	0.526323	+	0.850284i	$0.323570\pi$
$930$	0	0
$931$	0	0
$932$	−8.18034	−0.267956
$933$	0	0
$934$	8.30905	0.271881
$935$	−7.23607	−0.236645
$936$	0	0
$937$	51.1667	1.67154	0.835772	−	0.549077i	$-0.185020\pi$
$937$	51.1667	1.67154	0.835772	+	0.549077i	$0.185020\pi$
$938$	0	0
$939$	0	0
$940$	11.2361	0.366480
$941$	21.2920	0.694100	0.347050	−	0.937847i	$-0.387183\pi$
$941$	21.2920	0.694100	0.347050	+	0.937847i	$0.387183\pi$
$942$	0	0
$943$	41.8561	1.36302
$944$	−26.8400	−0.873566
$945$	0	0
$946$	−3.67376	−0.119444
$947$	−42.7639	−1.38964	−0.694821	−	0.719183i	$-0.744516\pi$
$947$	−42.7639	−1.38964	−0.694821	+	0.719183i	$0.744516\pi$
$948$	0	0
$949$	−30.8328	−1.00088
$950$	0.588890	0.0191061
$951$	0	0
$952$	0	0
$953$	57.4164	1.85990	0.929950	−	0.367686i	$-0.119850\pi$
$953$	57.4164	1.85990	0.929950	+	0.367686i	$0.119850\pi$
$954$	0	0
$955$	51.2942	1.65984
$956$	6.38197	0.206408
$957$	0	0
$958$	−8.69161	−0.280813
$959$	0	0
$960$	0	0
$961$	4.88854	0.157695
$962$	30.2874	0.976504
$963$	0	0
$964$	−26.4574	−0.852135
$965$	−17.2256	−0.554512
$966$	0	0
$967$	−0.472136	−0.0151829	−0.00759143	−	0.999971i	$-0.502416\pi$
$967$	−0.472136	−0.0151829	−0.00759143	+	0.999971i	$0.502416\pi$
$968$	−22.3607	−0.718699
$969$	0	0
$970$	−8.76393	−0.281393
$971$	−5.99070	−0.192251	−0.0961254	−	0.995369i	$-0.530645\pi$
$971$	−5.99070	−0.192251	−0.0961254	+	0.995369i	$0.530645\pi$
$972$	0	0
$973$	0	0
$974$	−2.43769	−0.0781088
$975$	0	0
$976$	16.5880	0.530969
$977$	31.3050	1.00153	0.500767	−	0.865582i	$-0.333051\pi$
$977$	31.3050	1.00153	0.500767	+	0.865582i	$0.333051\pi$
$978$	0	0
$979$	6.32456	0.202134
$980$	0	0
$981$	0	0
$982$	−13.5967	−0.433890
$983$	2.36706	0.0754974	0.0377487	−	0.999287i	$-0.487981\pi$
$983$	2.36706	0.0754974	0.0377487	+	0.999287i	$0.487981\pi$
$984$	0	0
$985$	−19.2588	−0.613637
$986$	2.41577	0.0769336
$987$	0	0
$988$	29.8885	0.950881
$989$	−43.0132	−1.36774
$990$	0	0
$991$	20.4164	0.648549	0.324274	−	0.945963i	$-0.394880\pi$
$991$	20.4164	0.648549	0.324274	+	0.945963i	$0.394880\pi$
$992$	33.6560	1.06858
$993$	0	0
$994$	0	0
$995$	10.0000	0.317021
$996$	0	0
$997$	19.8778	0.629536	0.314768	−	0.949169i	$-0.398073\pi$
$997$	19.8778	0.629536	0.314768	+	0.949169i	$0.398073\pi$
$998$	−13.5279	−0.428217
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.y.1.2	yes	4
3.2	odd	2		3969.2.a.r.1.3	✓	4
7.6	odd	2	inner	3969.2.a.y.1.1	yes	4
21.20	even	2		3969.2.a.r.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3969.2.a.r.1.3	✓	4	3.2	odd	2
3969.2.a.r.1.4	yes	4	21.20	even	2
3969.2.a.y.1.1	yes	4	7.6	odd	2	inner
3969.2.a.y.1.2	yes	4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3969 = 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3969.a (trivial)

\(f(q)\)	\(=\)	\(q-0.618034 q^{2} -1.61803 q^{4} +2.28825 q^{5} +2.23607 q^{8} +O(q^{10})\)	\(q-0.618034 q^{2} -1.61803 q^{4} +2.28825 q^{5} +2.23607 q^{8} -1.41421 q^{10} +1.00000 q^{11} +4.57649 q^{13} +1.85410 q^{16} -3.16228 q^{17} -4.03631 q^{19} -3.70246 q^{20} -0.618034 q^{22} -7.23607 q^{23} +0.236068 q^{25} -2.82843 q^{26} +1.23607 q^{29} -5.99070 q^{31} -5.61803 q^{32} +1.95440 q^{34} -10.7082 q^{37} +2.49458 q^{38} +5.11667 q^{40} -5.78437 q^{41} +5.94427 q^{43} -1.61803 q^{44} +4.47214 q^{46} -3.03476 q^{47} -0.145898 q^{50} -7.40492 q^{52} +10.7082 q^{53} +2.28825 q^{55} -0.763932 q^{58} -14.4760 q^{59} +8.94665 q^{61} +3.70246 q^{62} -0.236068 q^{64} +10.4721 q^{65} +1.47214 q^{67} +5.11667 q^{68} +5.47214 q^{71} -6.73722 q^{73} +6.61803 q^{74} +6.53089 q^{76} -7.76393 q^{79} +4.24264 q^{80} +3.57494 q^{82} -8.61280 q^{83} -7.23607 q^{85} -3.67376 q^{86} +2.23607 q^{88} +6.32456 q^{89} +11.7082 q^{92} +1.87558 q^{94} -9.23607 q^{95} +6.19704 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} - 2 q^{4}+O(q^{10}) \) 4 * q + 2 * q^2 - 2 * q^4	\( 4 q + 2 q^{2} - 2 q^{4} + 4 q^{11} - 6 q^{16} + 2 q^{22} - 20 q^{23} - 8 q^{25} - 4 q^{29} - 18 q^{32} - 16 q^{37} - 12 q^{43} - 2 q^{44} - 14 q^{50} + 16 q^{53} - 12 q^{58} + 8 q^{64} + 24 q^{65} - 12 q^{67} + 4 q^{71} + 22 q^{74} - 40 q^{79} - 20 q^{85} - 46 q^{86} + 20 q^{92} - 28 q^{95}+O(q^{100}) \) 4 * q + 2 * q^2 - 2 * q^4 + 4 * q^11 - 6 * q^16 + 2 * q^22 - 20 * q^23 - 8 * q^25 - 4 * q^29 - 18 * q^32 - 16 * q^37 - 12 * q^43 - 2 * q^44 - 14 * q^50 + 16 * q^53 - 12 * q^58 + 8 * q^64 + 24 * q^65 - 12 * q^67 + 4 * q^71 + 22 * q^74 - 40 * q^79 - 20 * q^85 - 46 * q^86 + 20 * q^92 - 28 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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