LMFDB - Embedded newform 3969.2.a.p.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:	$3$
Coefficient field:	3.3.321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 63)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.239123$ 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)

$f(q)$	$=$	$q+0.239123 q^{2} -1.94282 q^{4} +1.18194 q^{5} -0.942820 q^{8} +O(q^{10})$	$q+0.239123 q^{2} -1.94282 q^{4} +1.18194 q^{5} -0.942820 q^{8} +0.282630 q^{10} +3.70370 q^{11} -1.00000 q^{13} +3.66019 q^{16} -6.94282 q^{17} -1.94282 q^{19} -2.29630 q^{20} +0.885640 q^{22} +5.60301 q^{23} -3.60301 q^{25} -0.239123 q^{26} -0.239123 q^{29} -1.66019 q^{31} +2.76088 q^{32} -1.66019 q^{34} -9.54583 q^{37} -0.464574 q^{38} -1.11436 q^{40} -10.1819 q^{41} +2.22545 q^{43} -7.19562 q^{44} +1.33981 q^{46} +5.82846 q^{47} -0.861564 q^{50} +1.94282 q^{52} +11.6030 q^{53} +4.37756 q^{55} -0.0571799 q^{58} +2.60301 q^{59} +7.60301 q^{61} -0.396990 q^{62} -6.66019 q^{64} -1.18194 q^{65} +3.50808 q^{67} +13.4887 q^{68} -8.60301 q^{71} -15.1488 q^{73} -2.28263 q^{74} +3.77455 q^{76} +7.37756 q^{79} +4.32614 q^{80} -2.43474 q^{82} -6.94282 q^{83} -8.20602 q^{85} +0.532157 q^{86} -3.49192 q^{88} +2.74720 q^{89} -10.8856 q^{92} +1.39372 q^{94} -2.29630 q^{95} -7.16827 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 3 q^{4} - 5 q^{5} + 6 q^{8}+O(q^{10}) $ 3 * q + q^2 + 3 * q^4 - 5 * q^5 + 6 * q^8	$ 3 q + q^{2} + 3 q^{4} - 5 q^{5} + 6 q^{8} + 2 q^{11} - 3 q^{13} + 3 q^{16} - 12 q^{17} + 3 q^{19} - 16 q^{20} - 15 q^{22} + 6 q^{25} - q^{26} - q^{29} + 3 q^{31} + 8 q^{32} + 3 q^{34} - 3 q^{37} + 8 q^{38} - 21 q^{40} - 22 q^{41} - 3 q^{43} - 23 q^{44} + 12 q^{46} - 9 q^{47} - 10 q^{50} - 3 q^{52} + 18 q^{53} + 6 q^{55} - 9 q^{58} - 9 q^{59} + 6 q^{61} - 18 q^{62} - 12 q^{64} + 5 q^{65} + 6 q^{68} - 9 q^{71} - 3 q^{73} - 6 q^{74} + 21 q^{76} + 15 q^{79} + 11 q^{80} - 9 q^{82} - 12 q^{83} + 9 q^{85} - 34 q^{86} - 21 q^{88} - 2 q^{89} - 15 q^{92} - 24 q^{94} - 16 q^{95} - 3 q^{97}+O(q^{100}) $ 3 * q + q^2 + 3 * q^4 - 5 * q^5 + 6 * q^8 + 2 * q^11 - 3 * q^13 + 3 * q^16 - 12 * q^17 + 3 * q^19 - 16 * q^20 - 15 * q^22 + 6 * q^25 - q^26 - q^29 + 3 * q^31 + 8 * q^32 + 3 * q^34 - 3 * q^37 + 8 * q^38 - 21 * q^40 - 22 * q^41 - 3 * q^43 - 23 * q^44 + 12 * q^46 - 9 * q^47 - 10 * q^50 - 3 * q^52 + 18 * q^53 + 6 * q^55 - 9 * q^58 - 9 * q^59 + 6 * q^61 - 18 * q^62 - 12 * q^64 + 5 * q^65 + 6 * q^68 - 9 * q^71 - 3 * q^73 - 6 * q^74 + 21 * q^76 + 15 * q^79 + 11 * q^80 - 9 * q^82 - 12 * q^83 + 9 * q^85 - 34 * q^86 - 21 * q^88 - 2 * q^89 - 15 * q^92 - 24 * q^94 - 16 * q^95 - 3 * 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 $

$2$	0.239123	0.169086	0.0845428	−	0.996420i	$-0.473057\pi$
$2$	0.239123	0.169086	0.0845428	+	0.996420i	$0.473057\pi$
$3$	0	0
$3$	0	0
$4$	−1.94282	−0.971410
$5$	1.18194	0.528581	0.264291	−	0.964443i	$-0.414862\pi$
$5$	1.18194	0.528581	0.264291	+	0.964443i	$0.414862\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−0.942820	−0.333337
$9$	0	0
$10$	0.282630	0.0893755
$11$	3.70370	1.11671	0.558353	−	0.829603i	$-0.311433\pi$
$11$	3.70370	1.11671	0.558353	+	0.829603i	$0.311433\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	0	0
$16$	3.66019	0.915047
$17$	−6.94282	−1.68388	−0.841941	−	0.539570i	$-0.818587\pi$
$17$	−6.94282	−1.68388	−0.841941	+	0.539570i	$0.818587\pi$
$18$	0	0
$19$	−1.94282	−0.445713	−0.222857	−	0.974851i	$-0.571538\pi$
$19$	−1.94282	−0.445713	−0.222857	+	0.974851i	$0.571538\pi$
$20$	−2.29630	−0.513469
$21$	0	0
$22$	0.885640	0.188819
$23$	5.60301	1.16831	0.584154	−	0.811643i	$-0.301426\pi$
$23$	5.60301	1.16831	0.584154	+	0.811643i	$0.301426\pi$
$24$	0	0
$25$	−3.60301	−0.720602
$26$	−0.239123	−0.0468959
$27$	0	0
$28$	0	0
$29$	−0.239123	−0.0444041	−0.0222020	−	0.999754i	$-0.507068\pi$
$29$	−0.239123	−0.0444041	−0.0222020	+	0.999754i	$0.507068\pi$
$30$	0	0
$31$	−1.66019	−0.298179	−0.149089	−	0.988824i	$-0.547634\pi$
$31$	−1.66019	−0.298179	−0.149089	+	0.988824i	$0.547634\pi$
$32$	2.76088	0.488059
$33$	0	0
$34$	−1.66019	−0.284720
$35$	0	0
$36$	0	0
$37$	−9.54583	−1.56932	−0.784662	−	0.619923i	$-0.787164\pi$
$37$	−9.54583	−1.56932	−0.784662	+	0.619923i	$0.787164\pi$
$38$	−0.464574	−0.0753638
$39$	0	0
$40$	−1.11436	−0.176196
$41$	−10.1819	−1.59015	−0.795076	−	0.606510i	$-0.792569\pi$
$41$	−10.1819	−1.59015	−0.795076	+	0.606510i	$0.792569\pi$
$42$	0	0
$43$	2.22545	0.339378	0.169689	−	0.985498i	$-0.445724\pi$
$43$	2.22545	0.339378	0.169689	+	0.985498i	$0.445724\pi$
$44$	−7.19562	−1.08478
$45$	0	0
$46$	1.33981	0.197544
$47$	5.82846	0.850168	0.425084	−	0.905154i	$-0.360245\pi$
$47$	5.82846	0.850168	0.425084	+	0.905154i	$0.360245\pi$
$48$	0	0
$49$	0	0
$50$	−0.861564	−0.121843
$51$	0	0
$52$	1.94282	0.269421
$53$	11.6030	1.59380	0.796898	−	0.604114i	$-0.206473\pi$
$53$	11.6030	1.59380	0.796898	+	0.604114i	$0.206473\pi$
$54$	0	0
$55$	4.37756	0.590270
$56$	0	0
$57$	0	0
$58$	−0.0571799	−0.00750809
$59$	2.60301	0.338883	0.169442	−	0.985540i	$-0.445804\pi$
$59$	2.60301	0.338883	0.169442	+	0.985540i	$0.445804\pi$
$60$	0	0
$61$	7.60301	0.973466	0.486733	−	0.873551i	$-0.338189\pi$
$61$	7.60301	0.973466	0.486733	+	0.873551i	$0.338189\pi$
$62$	−0.396990	−0.0504178
$63$	0	0
$64$	−6.66019	−0.832524
$65$	−1.18194	−0.146602
$66$	0	0
$67$	3.50808	0.428580	0.214290	−	0.976770i	$-0.431256\pi$
$67$	3.50808	0.428580	0.214290	+	0.976770i	$0.431256\pi$
$68$	13.4887	1.63574
$69$	0	0
$70$	0	0
$71$	−8.60301	−1.02099	−0.510495	−	0.859881i	$-0.670538\pi$
$71$	−8.60301	−1.02099	−0.510495	+	0.859881i	$0.670538\pi$
$72$	0	0
$73$	−15.1488	−1.77304	−0.886519	−	0.462693i	$-0.846883\pi$
$73$	−15.1488	−1.77304	−0.886519	+	0.462693i	$0.846883\pi$
$74$	−2.28263	−0.265350
$75$	0	0
$76$	3.77455	0.432971
$77$	0	0
$78$	0	0
$79$	7.37756	0.830040	0.415020	−	0.909812i	$-0.363775\pi$
$79$	7.37756	0.830040	0.415020	+	0.909812i	$0.363775\pi$
$80$	4.32614	0.483677
$81$	0	0
$82$	−2.43474	−0.268872
$83$	−6.94282	−0.762074	−0.381037	−	0.924560i	$-0.624433\pi$
$83$	−6.94282	−0.762074	−0.381037	+	0.924560i	$0.624433\pi$
$84$	0	0
$85$	−8.20602	−0.890068
$86$	0.532157	0.0573840
$87$	0	0
$88$	−3.49192	−0.372240
$89$	2.74720	0.291203	0.145602	−	0.989343i	$-0.453488\pi$
$89$	2.74720	0.291203	0.145602	+	0.989343i	$0.453488\pi$
$90$	0	0
$91$	0	0
$92$	−10.8856	−1.13491
$93$	0	0
$94$	1.39372	0.143751
$95$	−2.29630	−0.235596
$96$	0	0
$97$	−7.16827	−0.727828	−0.363914	−	0.931433i	$-0.618560\pi$
$97$	−7.16827	−0.727828	−0.363914	+	0.931433i	$0.618560\pi$
$98$	0	0
$99$	0	0
$100$	7.00000	0.700000
$101$	−12.7850	−1.27215	−0.636075	−	0.771627i	$-0.719443\pi$
$101$	−12.7850	−1.27215	−0.636075	+	0.771627i	$0.719443\pi$
$102$	0	0
$103$	4.39699	0.433248	0.216624	−	0.976255i	$-0.430495\pi$
$103$	4.39699	0.433248	0.216624	+	0.976255i	$0.430495\pi$
$104$	0.942820	0.0924511
$105$	0	0
$106$	2.77455	0.269488
$107$	−13.7278	−1.32711	−0.663557	−	0.748126i	$-0.730954\pi$
$107$	−13.7278	−1.32711	−0.663557	+	0.748126i	$0.730954\pi$
$108$	0	0
$109$	1.26320	0.120993	0.0604963	−	0.998168i	$-0.480732\pi$
$109$	1.26320	0.120993	0.0604963	+	0.998168i	$0.480732\pi$
$110$	1.04678	0.0998062
$111$	0	0
$112$	0	0
$113$	−12.1625	−1.14415	−0.572076	−	0.820200i	$-0.693862\pi$
$113$	−12.1625	−1.14415	−0.572076	+	0.820200i	$0.693862\pi$
$114$	0	0
$115$	6.62244	0.617546
$116$	0.464574	0.0431346
$117$	0	0
$118$	0.622440	0.0573003
$119$	0	0
$120$	0	0
$121$	2.71737	0.247034
$122$	1.81806	0.164599
$123$	0	0
$124$	3.22545	0.289654
$125$	−10.1683	−0.909478
$126$	0	0
$127$	1.33981	0.118889	0.0594445	−	0.998232i	$-0.481067\pi$
$127$	1.33981	0.118889	0.0594445	+	0.998232i	$0.481067\pi$
$128$	−7.11436	−0.628827
$129$	0	0
$130$	−0.282630	−0.0247883
$131$	−4.96690	−0.433960	−0.216980	−	0.976176i	$-0.569621\pi$
$131$	−4.96690	−0.433960	−0.216980	+	0.976176i	$0.569621\pi$
$132$	0	0
$133$	0	0
$134$	0.838864	0.0724668
$135$	0	0
$136$	6.54583	0.561300
$137$	4.33981	0.370775	0.185387	−	0.982665i	$-0.440646\pi$
$137$	4.33981	0.370775	0.185387	+	0.982665i	$0.440646\pi$
$138$	0	0
$139$	−3.94282	−0.334426	−0.167213	−	0.985921i	$-0.553477\pi$
$139$	−3.94282	−0.334426	−0.167213	+	0.985921i	$0.553477\pi$
$140$	0	0
$141$	0	0
$142$	−2.05718	−0.172635
$143$	−3.70370	−0.309719
$144$	0	0
$145$	−0.282630	−0.0234712
$146$	−3.62244	−0.299795
$147$	0	0
$148$	18.5458	1.52446
$149$	−11.1111	−0.910256	−0.455128	−	0.890426i	$-0.650406\pi$
$149$	−11.1111	−0.910256	−0.455128	+	0.890426i	$0.650406\pi$
$150$	0	0
$151$	13.9234	1.13307	0.566535	−	0.824038i	$-0.308284\pi$
$151$	13.9234	1.13307	0.566535	+	0.824038i	$0.308284\pi$
$152$	1.83173	0.148573
$153$	0	0
$154$	0	0
$155$	−1.96225	−0.157612
$156$	0	0
$157$	−0.0571799	−0.00456346	−0.00228173	−	0.999997i	$-0.500726\pi$
$157$	−0.0571799	−0.00456346	−0.00228173	+	0.999997i	$0.500726\pi$
$158$	1.76415	0.140348
$159$	0	0
$160$	3.26320	0.257979
$161$	0	0
$162$	0	0
$163$	−1.50808	−0.118122	−0.0590610	−	0.998254i	$-0.518811\pi$
$163$	−1.50808	−0.118122	−0.0590610	+	0.998254i	$0.518811\pi$
$164$	19.7817	1.54469
$165$	0	0
$166$	−1.66019	−0.128856
$167$	−14.6843	−1.13630	−0.568151	−	0.822924i	$-0.692341\pi$
$167$	−14.6843	−1.13630	−0.568151	+	0.822924i	$0.692341\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	−1.96225	−0.150498
$171$	0	0
$172$	−4.32365	−0.329675
$173$	0.252796	0.0192197	0.00960987	−	0.999954i	$-0.496941\pi$
$173$	0.252796	0.0192197	0.00960987	+	0.999954i	$0.496941\pi$
$174$	0	0
$175$	0	0
$176$	13.5562	1.02184
$177$	0	0
$178$	0.656920	0.0492383
$179$	−14.1923	−1.06079	−0.530393	−	0.847752i	$-0.677956\pi$
$179$	−14.1923	−1.06079	−0.530393	+	0.847752i	$0.677956\pi$
$180$	0	0
$181$	1.43147	0.106400	0.0532002	−	0.998584i	$-0.483058\pi$
$181$	1.43147	0.106400	0.0532002	+	0.998584i	$0.483058\pi$
$182$	0	0
$183$	0	0
$184$	−5.28263	−0.389441
$185$	−11.2826	−0.829515
$186$	0	0
$187$	−25.7141	−1.88040
$188$	−11.3236	−0.825862
$189$	0	0
$190$	−0.549100	−0.0398359
$191$	−15.0676	−1.09025	−0.545126	−	0.838354i	$-0.683518\pi$
$191$	−15.0676	−1.09025	−0.545126	+	0.838354i	$0.683518\pi$
$192$	0	0
$193$	−7.84789	−0.564904	−0.282452	−	0.959282i	$-0.591148\pi$
$193$	−7.84789	−0.564904	−0.282452	+	0.959282i	$0.591148\pi$
$194$	−1.71410	−0.123065
$195$	0	0
$196$	0	0
$197$	−6.69002	−0.476644	−0.238322	−	0.971186i	$-0.576597\pi$
$197$	−6.69002	−0.476644	−0.238322	+	0.971186i	$0.576597\pi$
$198$	0	0
$199$	19.9396	1.41348	0.706739	−	0.707475i	$-0.250166\pi$
$199$	19.9396	1.41348	0.706739	+	0.707475i	$0.250166\pi$
$200$	3.39699	0.240203
$201$	0	0
$202$	−3.05718	−0.215102
$203$	0	0
$204$	0	0
$205$	−12.0345	−0.840525
$206$	1.05142	0.0732561
$207$	0	0
$208$	−3.66019	−0.253789
$209$	−7.19562	−0.497731
$210$	0	0
$211$	−18.0917	−1.24548	−0.622741	−	0.782428i	$-0.713981\pi$
$211$	−18.0917	−1.24548	−0.622741	+	0.782428i	$0.713981\pi$
$212$	−22.5426	−1.54823
$213$	0	0
$214$	−3.28263	−0.224396
$215$	2.63036	0.179389
$216$	0	0
$217$	0	0
$218$	0.302060	0.0204581
$219$	0	0
$220$	−8.50481	−0.573394
$221$	6.94282	0.467025
$222$	0	0
$223$	22.6569	1.51722	0.758610	−	0.651545i	$-0.225879\pi$
$223$	22.6569	1.51722	0.758610	+	0.651545i	$0.225879\pi$
$224$	0	0
$225$	0	0
$226$	−2.90834	−0.193460
$227$	−5.28263	−0.350620	−0.175310	−	0.984513i	$-0.556093\pi$
$227$	−5.28263	−0.350620	−0.175310	+	0.984513i	$0.556093\pi$
$228$	0	0
$229$	−19.3365	−1.27779	−0.638897	−	0.769292i	$-0.720609\pi$
$229$	−19.3365	−1.27779	−0.638897	+	0.769292i	$0.720609\pi$
$230$	1.58358	0.104418
$231$	0	0
$232$	0.225450	0.0148015
$233$	16.9806	1.11243	0.556217	−	0.831037i	$-0.312252\pi$
$233$	16.9806	1.11243	0.556217	+	0.831037i	$0.312252\pi$
$234$	0	0
$235$	6.88891	0.449383
$236$	−5.05718	−0.329194
$237$	0	0
$238$	0	0
$239$	−16.8856	−1.09224	−0.546121	−	0.837707i	$-0.683896\pi$
$239$	−16.8856	−1.09224	−0.546121	+	0.837707i	$0.683896\pi$
$240$	0	0
$241$	−27.1456	−1.74860	−0.874300	−	0.485386i	$-0.838679\pi$
$241$	−27.1456	−1.74860	−0.874300	+	0.485386i	$0.838679\pi$
$242$	0.649786	0.0417699
$243$	0	0
$244$	−14.7713	−0.945634
$245$	0	0
$246$	0	0
$247$	1.94282	0.123619
$248$	1.56526	0.0993941
$249$	0	0
$250$	−2.43147	−0.153780
$251$	−19.0780	−1.20419	−0.602096	−	0.798424i	$-0.705668\pi$
$251$	−19.0780	−1.20419	−0.602096	+	0.798424i	$0.705668\pi$
$252$	0	0
$253$	20.7518	1.30466
$254$	0.320380	0.0201024
$255$	0	0
$256$	11.6192	0.726198
$257$	−14.8421	−0.925827	−0.462913	−	0.886404i	$-0.653196\pi$
$257$	−14.8421	−0.925827	−0.462913	+	0.886404i	$0.653196\pi$
$258$	0	0
$259$	0	0
$260$	2.29630	0.142411
$261$	0	0
$262$	−1.18770	−0.0733764
$263$	7.74145	0.477358	0.238679	−	0.971099i	$-0.423286\pi$
$263$	7.74145	0.477358	0.238679	+	0.971099i	$0.423286\pi$
$264$	0	0
$265$	13.7141	0.842450
$266$	0	0
$267$	0	0
$268$	−6.81557	−0.416327
$269$	−1.51135	−0.0921486	−0.0460743	−	0.998938i	$-0.514671\pi$
$269$	−1.51135	−0.0921486	−0.0460743	+	0.998938i	$0.514671\pi$
$270$	0	0
$271$	21.9806	1.33522	0.667612	−	0.744509i	$-0.267316\pi$
$271$	21.9806	1.33522	0.667612	+	0.744509i	$0.267316\pi$
$272$	−25.4120	−1.54083
$273$	0	0
$274$	1.03775	0.0626927
$275$	−13.3445	−0.804701
$276$	0	0
$277$	−10.8285	−0.650619	−0.325310	−	0.945608i	$-0.605469\pi$
$277$	−10.8285	−0.650619	−0.325310	+	0.945608i	$0.605469\pi$
$278$	−0.942820	−0.0565466
$279$	0	0
$280$	0	0
$281$	16.8766	1.00677	0.503387	−	0.864061i	$-0.332087\pi$
$281$	16.8766	1.00677	0.503387	+	0.864061i	$0.332087\pi$
$282$	0	0
$283$	15.3171	0.910508	0.455254	−	0.890362i	$-0.349549\pi$
$283$	15.3171	0.910508	0.455254	+	0.890362i	$0.349549\pi$
$284$	16.7141	0.991799
$285$	0	0
$286$	−0.885640	−0.0523690
$287$	0	0
$288$	0	0
$289$	31.2028	1.83546
$290$	−0.0675835	−0.00396864
$291$	0	0
$292$	29.4315	1.72235
$293$	−9.36964	−0.547380	−0.273690	−	0.961818i	$-0.588244\pi$
$293$	−9.36964	−0.547380	−0.273690	+	0.961818i	$0.588244\pi$
$294$	0	0
$295$	3.07661	0.179127
$296$	9.00000	0.523114
$297$	0	0
$298$	−2.65692	−0.153911
$299$	−5.60301	−0.324030
$300$	0	0
$301$	0	0
$302$	3.32941	0.191586
$303$	0	0
$304$	−7.11109	−0.407849
$305$	8.98633	0.514556
$306$	0	0
$307$	−2.71410	−0.154902	−0.0774509	−	0.996996i	$-0.524678\pi$
$307$	−2.71410	−0.154902	−0.0774509	+	0.996996i	$0.524678\pi$
$308$	0	0
$309$	0	0
$310$	−0.469220	−0.0266499
$311$	−13.9806	−0.792765	−0.396383	−	0.918085i	$-0.629735\pi$
$311$	−13.9806	−0.792765	−0.396383	+	0.918085i	$0.629735\pi$
$312$	0	0
$313$	19.0539	1.07699	0.538495	−	0.842628i	$-0.318993\pi$
$313$	19.0539	1.07699	0.538495	+	0.842628i	$0.318993\pi$
$314$	−0.0136731	−0.000771615 0
$315$	0	0
$316$	−14.3333	−0.806309
$317$	4.01943	0.225754	0.112877	−	0.993609i	$-0.463993\pi$
$317$	4.01943	0.225754	0.112877	+	0.993609i	$0.463993\pi$
$318$	0	0
$319$	−0.885640	−0.0495863
$320$	−7.87197	−0.440056
$321$	0	0
$322$	0	0
$323$	13.4887	0.750529
$324$	0	0
$325$	3.60301	0.199859
$326$	−0.360617	−0.0199727
$327$	0	0
$328$	9.59974	0.530057
$329$	0	0
$330$	0	0
$331$	−12.3776	−0.680332	−0.340166	−	0.940365i	$-0.610483\pi$
$331$	−12.3776	−0.680332	−0.340166	+	0.940365i	$0.610483\pi$
$332$	13.4887	0.740286
$333$	0	0
$334$	−3.51135	−0.192133
$335$	4.14635	0.226539
$336$	0	0
$337$	12.2599	0.667841	0.333920	−	0.942601i	$-0.391628\pi$
$337$	12.2599	0.667841	0.333920	+	0.942601i	$0.391628\pi$
$338$	−2.86948	−0.156079
$339$	0	0
$340$	15.9428	0.864621
$341$	−6.14884	−0.332978
$342$	0	0
$343$	0	0
$344$	−2.09820	−0.113127
$345$	0	0
$346$	0.0604495	0.00324978
$347$	−6.64979	−0.356979	−0.178490	−	0.983942i	$-0.557121\pi$
$347$	−6.64979	−0.356979	−0.178490	+	0.983942i	$0.557121\pi$
$348$	0	0
$349$	−11.4347	−0.612088	−0.306044	−	0.952017i	$-0.599005\pi$
$349$	−11.4347	−0.612088	−0.306044	+	0.952017i	$0.599005\pi$
$350$	0	0
$351$	0	0
$352$	10.2255	0.545018
$353$	−22.1956	−1.18135	−0.590677	−	0.806908i	$-0.701139\pi$
$353$	−22.1956	−1.18135	−0.590677	+	0.806908i	$0.701139\pi$
$354$	0	0
$355$	−10.1683	−0.539676
$356$	−5.33732	−0.282878
$357$	0	0
$358$	−3.39372	−0.179364
$359$	7.55623	0.398803	0.199401	−	0.979918i	$-0.436100\pi$
$359$	7.55623	0.398803	0.199401	+	0.979918i	$0.436100\pi$
$360$	0	0
$361$	−15.2255	−0.801339
$362$	0.342298	0.0179908
$363$	0	0
$364$	0	0
$365$	−17.9051	−0.937194
$366$	0	0
$367$	18.5231	0.966900	0.483450	−	0.875372i	$-0.339384\pi$
$367$	18.5231	0.966900	0.483450	+	0.875372i	$0.339384\pi$
$368$	20.5081	1.06906
$369$	0	0
$370$	−2.69794	−0.140259
$371$	0	0
$372$	0	0
$373$	15.6602	0.810854	0.405427	−	0.914127i	$-0.367123\pi$
$373$	15.6602	0.810854	0.405427	+	0.914127i	$0.367123\pi$
$374$	−6.14884	−0.317949
$375$	0	0
$376$	−5.49519	−0.283393
$377$	0.239123	0.0123155
$378$	0	0
$379$	4.03775	0.207405	0.103703	−	0.994608i	$-0.466931\pi$
$379$	4.03775	0.207405	0.103703	+	0.994608i	$0.466931\pi$
$380$	4.46130	0.228860
$381$	0	0
$382$	−3.60301	−0.184346
$383$	0.225450	0.0115200	0.00575998	−	0.999983i	$-0.498167\pi$
$383$	0.225450	0.0115200	0.00575998	+	0.999983i	$0.498167\pi$
$384$	0	0
$385$	0	0
$386$	−1.87661	−0.0955171
$387$	0	0
$388$	13.9267	0.707019
$389$	−25.2632	−1.28090	−0.640448	−	0.768002i	$-0.721251\pi$
$389$	−25.2632	−1.28090	−0.640448	+	0.768002i	$0.721251\pi$
$390$	0	0
$391$	−38.9007	−1.96729
$392$	0	0
$393$	0	0
$394$	−1.59974	−0.0805938
$395$	8.71986	0.438744
$396$	0	0
$397$	20.3009	1.01888	0.509438	−	0.860508i	$-0.329853\pi$
$397$	20.3009	1.01888	0.509438	+	0.860508i	$0.329853\pi$
$398$	4.76801	0.238999
$399$	0	0
$400$	−13.1877	−0.659385
$401$	15.2255	0.760323	0.380161	−	0.924920i	$-0.375868\pi$
$401$	15.2255	0.760323	0.380161	+	0.924920i	$0.375868\pi$
$402$	0	0
$403$	1.66019	0.0826999
$404$	24.8389	1.23578
$405$	0	0
$406$	0	0
$407$	−35.3549	−1.75248
$408$	0	0
$409$	−1.65692	−0.0819294	−0.0409647	−	0.999161i	$-0.513043\pi$
$409$	−1.65692	−0.0819294	−0.0409647	+	0.999161i	$0.513043\pi$
$410$	−2.87772	−0.142121
$411$	0	0
$412$	−8.54256	−0.420862
$413$	0	0
$414$	0	0
$415$	−8.20602	−0.402818
$416$	−2.76088	−0.135363
$417$	0	0
$418$	−1.72064	−0.0841592
$419$	33.3743	1.63044	0.815220	−	0.579151i	$-0.196616\pi$
$419$	33.3743	1.63044	0.815220	+	0.579151i	$0.196616\pi$
$420$	0	0
$421$	18.2405	0.888988	0.444494	−	0.895782i	$-0.353384\pi$
$421$	18.2405	0.888988	0.444494	+	0.895782i	$0.353384\pi$
$422$	−4.32614	−0.210593
$423$	0	0
$424$	−10.9396	−0.531272
$425$	25.0150	1.21341
$426$	0	0
$427$	0	0
$428$	26.6706	1.28917
$429$	0	0
$430$	0.628979	0.0303321
$431$	−29.2826	−1.41049	−0.705247	−	0.708961i	$-0.749164\pi$
$431$	−29.2826	−1.41049	−0.705247	+	0.708961i	$0.749164\pi$
$432$	0	0
$433$	12.2449	0.588451	0.294226	−	0.955736i	$-0.404938\pi$
$433$	12.2449	0.588451	0.294226	+	0.955736i	$0.404938\pi$
$434$	0	0
$435$	0	0
$436$	−2.45417	−0.117533
$437$	−10.8856	−0.520731
$438$	0	0
$439$	4.83173	0.230606	0.115303	−	0.993330i	$-0.463216\pi$
$439$	4.83173	0.230606	0.115303	+	0.993330i	$0.463216\pi$
$440$	−4.12725	−0.196759
$441$	0	0
$442$	1.66019	0.0789672
$443$	−1.24488	−0.0591461	−0.0295730	−	0.999563i	$-0.509415\pi$
$443$	−1.24488	−0.0591461	−0.0295730	+	0.999563i	$0.509415\pi$
$444$	0	0
$445$	3.24704	0.153924
$446$	5.41780	0.256540
$447$	0	0
$448$	0	0
$449$	−8.82846	−0.416641	−0.208320	−	0.978061i	$-0.566800\pi$
$449$	−8.82846	−0.416641	−0.208320	+	0.978061i	$0.566800\pi$
$450$	0	0
$451$	−37.7108	−1.77573
$452$	23.6296	1.11144
$453$	0	0
$454$	−1.26320	−0.0592849
$455$	0	0
$456$	0	0
$457$	−10.5081	−0.491547	−0.245774	−	0.969327i	$-0.579042\pi$
$457$	−10.5081	−0.491547	−0.245774	+	0.969327i	$0.579042\pi$
$458$	−4.62382	−0.216057
$459$	0	0
$460$	−12.8662	−0.599890
$461$	22.5516	1.05033	0.525166	−	0.851000i	$-0.324003\pi$
$461$	22.5516	1.05033	0.525166	+	0.851000i	$0.324003\pi$
$462$	0	0
$463$	10.3970	0.483189	0.241595	−	0.970377i	$-0.422330\pi$
$463$	10.3970	0.483189	0.241595	+	0.970377i	$0.422330\pi$
$464$	−0.875237	−0.0406318
$465$	0	0
$466$	4.06045	0.188097
$467$	13.3171	0.616242	0.308121	−	0.951347i	$-0.400300\pi$
$467$	13.3171	0.616242	0.308121	+	0.951347i	$0.400300\pi$
$468$	0	0
$469$	0	0
$470$	1.64730	0.0759842
$471$	0	0
$472$	−2.45417	−0.112962
$473$	8.24239	0.378986
$474$	0	0
$475$	7.00000	0.321182
$476$	0	0
$477$	0	0
$478$	−4.03775	−0.184682
$479$	14.5354	0.664141	0.332070	−	0.943255i	$-0.392253\pi$
$479$	14.5354	0.664141	0.332070	+	0.943255i	$0.392253\pi$
$480$	0	0
$481$	9.54583	0.435252
$482$	−6.49114	−0.295663
$483$	0	0
$484$	−5.27936	−0.239971
$485$	−8.47249	−0.384716
$486$	0	0
$487$	13.0539	0.591529	0.295765	−	0.955261i	$-0.404426\pi$
$487$	13.0539	0.591529	0.295765	+	0.955261i	$0.404426\pi$
$488$	−7.16827	−0.324492
$489$	0	0
$490$	0	0
$491$	−19.3445	−0.873003	−0.436502	−	0.899704i	$-0.643783\pi$
$491$	−19.3445	−0.873003	−0.436502	+	0.899704i	$0.643783\pi$
$492$	0	0
$493$	1.66019	0.0747712
$494$	0.464574	0.0209022
$495$	0	0
$496$	−6.07661	−0.272848
$497$	0	0
$498$	0	0
$499$	−36.2222	−1.62153	−0.810764	−	0.585374i	$-0.800948\pi$
$499$	−36.2222	−1.62153	−0.810764	+	0.585374i	$0.800948\pi$
$500$	19.7551	0.883476
$501$	0	0
$502$	−4.56199	−0.203612
$503$	15.6764	0.698974	0.349487	−	0.936941i	$-0.386356\pi$
$503$	15.6764	0.698974	0.349487	+	0.936941i	$0.386356\pi$
$504$	0	0
$505$	−15.1111	−0.672435
$506$	4.96225	0.220599
$507$	0	0
$508$	−2.60301	−0.115490
$509$	34.3034	1.52047	0.760237	−	0.649646i	$-0.225083\pi$
$509$	34.3034	1.52047	0.760237	+	0.649646i	$0.225083\pi$
$510$	0	0
$511$	0	0
$512$	17.0071	0.751616
$513$	0	0
$514$	−3.54910	−0.156544
$515$	5.19699	0.229007
$516$	0	0
$517$	21.5868	0.949389
$518$	0	0
$519$	0	0
$520$	1.11436	0.0488679
$521$	−10.2449	−0.448836	−0.224418	−	0.974493i	$-0.572048\pi$
$521$	−10.2449	−0.448836	−0.224418	+	0.974493i	$0.572048\pi$
$522$	0	0
$523$	−30.6030	−1.33818	−0.669088	−	0.743183i	$-0.733315\pi$
$523$	−30.6030	−1.33818	−0.669088	+	0.743183i	$0.733315\pi$
$524$	9.64979	0.421553
$525$	0	0
$526$	1.85116	0.0807144
$527$	11.5264	0.502098
$528$	0	0
$529$	8.39372	0.364944
$530$	3.27936	0.142446
$531$	0	0
$532$	0	0
$533$	10.1819	0.441029
$534$	0	0
$535$	−16.2255	−0.701487
$536$	−3.30749	−0.142862
$537$	0	0
$538$	−0.361399	−0.0155810
$539$	0	0
$540$	0	0
$541$	−26.0917	−1.12177	−0.560884	−	0.827894i	$-0.689539\pi$
$541$	−26.0917	−1.12177	−0.560884	+	0.827894i	$0.689539\pi$
$542$	5.25607	0.225767
$543$	0	0
$544$	−19.1683	−0.821833
$545$	1.49303	0.0639544
$546$	0	0
$547$	−10.9234	−0.467050	−0.233525	−	0.972351i	$-0.575026\pi$
$547$	−10.9234	−0.467050	−0.233525	+	0.972351i	$0.575026\pi$
$548$	−8.43147	−0.360175
$549$	0	0
$550$	−3.19097	−0.136063
$551$	0.464574	0.0197915
$552$	0	0
$553$	0	0
$554$	−2.58934	−0.110010
$555$	0	0
$556$	7.66019	0.324864
$557$	13.9442	0.590835	0.295417	−	0.955368i	$-0.404541\pi$
$557$	13.9442	0.590835	0.295417	+	0.955368i	$0.404541\pi$
$558$	0	0
$559$	−2.22545	−0.0941265
$560$	0	0
$561$	0	0
$562$	4.03559	0.170231
$563$	30.2574	1.27520	0.637600	−	0.770368i	$-0.279927\pi$
$563$	30.2574	1.27520	0.637600	+	0.770368i	$0.279927\pi$
$564$	0	0
$565$	−14.3754	−0.604778
$566$	3.66268	0.153954
$567$	0	0
$568$	8.11109	0.340334
$569$	21.1352	0.886032	0.443016	−	0.896514i	$-0.353908\pi$
$569$	21.1352	0.886032	0.443016	+	0.896514i	$0.353908\pi$
$570$	0	0
$571$	−32.7863	−1.37207	−0.686033	−	0.727571i	$-0.740649\pi$
$571$	−32.7863	−1.37207	−0.686033	+	0.727571i	$0.740649\pi$
$572$	7.19562	0.300864
$573$	0	0
$574$	0	0
$575$	−20.1877	−0.841885
$576$	0	0
$577$	17.3743	0.723301	0.361651	−	0.932314i	$-0.382213\pi$
$577$	17.3743	0.723301	0.361651	+	0.932314i	$0.382213\pi$
$578$	7.46130	0.310349
$579$	0	0
$580$	0.549100	0.0228001
$581$	0	0
$582$	0	0
$583$	42.9740	1.77980
$584$	14.2826	0.591019
$585$	0	0
$586$	−2.24050	−0.0925542
$587$	16.9759	0.700671	0.350336	−	0.936624i	$-0.386068\pi$
$587$	16.9759	0.700671	0.350336	+	0.936624i	$0.386068\pi$
$588$	0	0
$589$	3.22545	0.132902
$590$	0.735689	0.0302878
$591$	0	0
$592$	−34.9396	−1.43601
$593$	−13.0733	−0.536858	−0.268429	−	0.963300i	$-0.586504\pi$
$593$	−13.0733	−0.536858	−0.268429	+	0.963300i	$0.586504\pi$
$594$	0	0
$595$	0	0
$596$	21.5868	0.884232
$597$	0	0
$598$	−1.33981	−0.0547889
$599$	−29.2060	−1.19333	−0.596663	−	0.802492i	$-0.703507\pi$
$599$	−29.2060	−1.19333	−0.596663	+	0.802492i	$0.703507\pi$
$600$	0	0
$601$	−7.79071	−0.317790	−0.158895	−	0.987296i	$-0.550793\pi$
$601$	−7.79071	−0.317790	−0.158895	+	0.987296i	$0.550793\pi$
$602$	0	0
$603$	0	0
$604$	−27.0506	−1.10067
$605$	3.21178	0.130577
$606$	0	0
$607$	19.6408	0.797194	0.398597	−	0.917126i	$-0.369497\pi$
$607$	19.6408	0.797194	0.398597	+	0.917126i	$0.369497\pi$
$608$	−5.36389	−0.217534
$609$	0	0
$610$	2.14884	0.0870040
$611$	−5.82846	−0.235794
$612$	0	0
$613$	23.5653	0.951792	0.475896	−	0.879502i	$-0.342124\pi$
$613$	23.5653	0.951792	0.475896	+	0.879502i	$0.342124\pi$
$614$	−0.649005	−0.0261917
$615$	0	0
$616$	0	0
$617$	10.6602	0.429163	0.214582	−	0.976706i	$-0.431161\pi$
$617$	10.6602	0.429163	0.214582	+	0.976706i	$0.431161\pi$
$618$	0	0
$619$	18.0150	0.724086	0.362043	−	0.932161i	$-0.382079\pi$
$619$	18.0150	0.724086	0.362043	+	0.932161i	$0.382079\pi$
$620$	3.81230	0.153106
$621$	0	0
$622$	−3.34308	−0.134045
$623$	0	0
$624$	0	0
$625$	5.99673	0.239869
$626$	4.55623	0.182104
$627$	0	0
$628$	0.111090	0.00443299
$629$	66.2750	2.64256
$630$	0	0
$631$	12.4703	0.496436	0.248218	−	0.968704i	$-0.420155\pi$
$631$	12.4703	0.496436	0.248218	+	0.968704i	$0.420155\pi$
$632$	−6.95571	−0.276683
$633$	0	0
$634$	0.961139	0.0381717
$635$	1.58358	0.0628424
$636$	0	0
$637$	0	0
$638$	−0.211777	−0.00838434
$639$	0	0
$640$	−8.40877	−0.332386
$641$	−19.1456	−0.756205	−0.378102	−	0.925764i	$-0.623423\pi$
$641$	−19.1456	−0.756205	−0.378102	+	0.925764i	$0.623423\pi$
$642$	0	0
$643$	−6.48865	−0.255887	−0.127944	−	0.991781i	$-0.540838\pi$
$643$	−6.48865	−0.255887	−0.127944	+	0.991781i	$0.540838\pi$
$644$	0	0
$645$	0	0
$646$	3.22545	0.126904
$647$	−48.0988	−1.89096	−0.945479	−	0.325682i	$-0.894406\pi$
$647$	−48.0988	−1.89096	−0.945479	+	0.325682i	$0.894406\pi$
$648$	0	0
$649$	9.64076	0.378433
$650$	0.861564	0.0337933
$651$	0	0
$652$	2.92993	0.114745
$653$	43.2405	1.69213	0.846066	−	0.533079i	$-0.178965\pi$
$653$	43.2405	1.69213	0.846066	+	0.533079i	$0.178965\pi$
$654$	0	0
$655$	−5.87059	−0.229383
$656$	−37.2678	−1.45506
$657$	0	0
$658$	0	0
$659$	2.50808	0.0977009	0.0488505	−	0.998806i	$-0.484444\pi$
$659$	2.50808	0.0977009	0.0488505	+	0.998806i	$0.484444\pi$
$660$	0	0
$661$	42.3354	1.64666	0.823329	−	0.567565i	$-0.192114\pi$
$661$	42.3354	1.64666	0.823329	+	0.567565i	$0.192114\pi$
$662$	−2.95976	−0.115034
$663$	0	0
$664$	6.54583	0.254027
$665$	0	0
$666$	0	0
$667$	−1.33981	−0.0518777
$668$	28.5289	1.10382
$669$	0	0
$670$	0.991489	0.0383046
$671$	28.1592	1.08708
$672$	0	0
$673$	13.4153	0.517122	0.258561	−	0.965995i	$-0.416752\pi$
$673$	13.4153	0.517122	0.258561	+	0.965995i	$0.416752\pi$
$674$	2.93163	0.112922
$675$	0	0
$676$	23.3138	0.896686
$677$	1.96225	0.0754154	0.0377077	−	0.999289i	$-0.487994\pi$
$677$	1.96225	0.0754154	0.0377077	+	0.999289i	$0.487994\pi$
$678$	0	0
$679$	0	0
$680$	7.73680	0.296693
$681$	0	0
$682$	−1.47033	−0.0563019
$683$	−27.1672	−1.03952	−0.519761	−	0.854312i	$-0.673979\pi$
$683$	−27.1672	−1.03952	−0.519761	+	0.854312i	$0.673979\pi$
$684$	0	0
$685$	5.12941	0.195985
$686$	0	0
$687$	0	0
$688$	8.14557	0.310547
$689$	−11.6030	−0.442039
$690$	0	0
$691$	50.3171	1.91415	0.957077	−	0.289835i	$-0.0936005\pi$
$691$	50.3171	1.91415	0.957077	+	0.289835i	$0.0936005\pi$
$692$	−0.491138	−0.0186703
$693$	0	0
$694$	−1.59012	−0.0603601
$695$	−4.66019	−0.176771
$696$	0	0
$697$	70.6914	2.67763
$698$	−2.73431	−0.103495
$699$	0	0
$700$	0	0
$701$	−45.1672	−1.70594	−0.852970	−	0.521960i	$-0.825201\pi$
$701$	−45.1672	−1.70594	−0.852970	+	0.521960i	$0.825201\pi$
$702$	0	0
$703$	18.5458	0.699469
$704$	−24.6673	−0.929685
$705$	0	0
$706$	−5.30749	−0.199750
$707$	0	0
$708$	0	0
$709$	39.6181	1.48789	0.743944	−	0.668242i	$-0.232953\pi$
$709$	39.6181	1.48789	0.743944	+	0.668242i	$0.232953\pi$
$710$	−2.43147	−0.0912514
$711$	0	0
$712$	−2.59012	−0.0970688
$713$	−9.30206	−0.348365
$714$	0	0
$715$	−4.37756	−0.163711
$716$	27.5732	1.03046
$717$	0	0
$718$	1.80687	0.0674318
$719$	−22.0377	−0.821869	−0.410935	−	0.911665i	$-0.634798\pi$
$719$	−22.0377	−0.821869	−0.410935	+	0.911665i	$0.634798\pi$
$720$	0	0
$721$	0	0
$722$	−3.64076	−0.135495
$723$	0	0
$724$	−2.78109	−0.103358
$725$	0.861564	0.0319977
$726$	0	0
$727$	−28.1111	−1.04258	−0.521291	−	0.853379i	$-0.674550\pi$
$727$	−28.1111	−1.04258	−0.521291	+	0.853379i	$0.674550\pi$
$728$	0	0
$729$	0	0
$730$	−4.28152	−0.158466
$731$	−15.4509	−0.571472
$732$	0	0
$733$	−11.8695	−0.438409	−0.219205	−	0.975679i	$-0.570346\pi$
$733$	−11.8695	−0.438409	−0.219205	+	0.975679i	$0.570346\pi$
$734$	4.42931	0.163489
$735$	0	0
$736$	15.4692	0.570203
$737$	12.9929	0.478598
$738$	0	0
$739$	−12.1844	−0.448212	−0.224106	−	0.974565i	$-0.571946\pi$
$739$	−12.1844	−0.448212	−0.224106	+	0.974565i	$0.571946\pi$
$740$	21.9201	0.805800
$741$	0	0
$742$	0	0
$743$	44.4854	1.63201	0.816005	−	0.578045i	$-0.196184\pi$
$743$	44.4854	1.63201	0.816005	+	0.578045i	$0.196184\pi$
$744$	0	0
$745$	−13.1327	−0.481144
$746$	3.74472	0.137104
$747$	0	0
$748$	49.9579	1.82664
$749$	0	0
$750$	0	0
$751$	42.8058	1.56200	0.781002	−	0.624528i	$-0.214709\pi$
$751$	42.8058	1.56200	0.781002	+	0.624528i	$0.214709\pi$
$752$	21.3333	0.777944
$753$	0	0
$754$	0.0571799	0.00208237
$755$	16.4567	0.598919
$756$	0	0
$757$	−22.4919	−0.817483	−0.408741	−	0.912650i	$-0.634032\pi$
$757$	−22.4919	−0.817483	−0.408741	+	0.912650i	$0.634032\pi$
$758$	0.965520	0.0350693
$759$	0	0
$760$	2.16500	0.0785328
$761$	−14.3365	−0.519699	−0.259850	−	0.965649i	$-0.583673\pi$
$761$	−14.3365	−0.519699	−0.259850	+	0.965649i	$0.583673\pi$
$762$	0	0
$763$	0	0
$764$	29.2736	1.05908
$765$	0	0
$766$	0.0539104	0.00194786
$767$	−2.60301	−0.0939892
$768$	0	0
$769$	31.2211	1.12586	0.562930	−	0.826504i	$-0.309674\pi$
$769$	31.2211	1.12586	0.562930	+	0.826504i	$0.309674\pi$
$770$	0	0
$771$	0	0
$772$	15.2470	0.548753
$773$	4.38005	0.157539	0.0787697	−	0.996893i	$-0.474901\pi$
$773$	4.38005	0.157539	0.0787697	+	0.996893i	$0.474901\pi$
$774$	0	0
$775$	5.98168	0.214868
$776$	6.75839	0.242612
$777$	0	0
$778$	−6.04102	−0.216581
$779$	19.7817	0.708752
$780$	0	0
$781$	−31.8629	−1.14015
$782$	−9.30206	−0.332641
$783$	0	0
$784$	0	0
$785$	−0.0675835	−0.00241216
$786$	0	0
$787$	−27.6213	−0.984594	−0.492297	−	0.870427i	$-0.663843\pi$
$787$	−27.6213	−0.984594	−0.492297	+	0.870427i	$0.663843\pi$
$788$	12.9975	0.463017
$789$	0	0
$790$	2.08512	0.0741853
$791$	0	0
$792$	0	0
$793$	−7.60301	−0.269991
$794$	4.85443	0.172277
$795$	0	0
$796$	−38.7390	−1.37307
$797$	−2.96363	−0.104977	−0.0524885	−	0.998622i	$-0.516715\pi$
$797$	−2.96363	−0.104977	−0.0524885	+	0.998622i	$0.516715\pi$
$798$	0	0
$799$	−40.4660	−1.43158
$800$	−9.94747	−0.351696
$801$	0	0
$802$	3.64076	0.128560
$803$	−56.1067	−1.97996
$804$	0	0
$805$	0	0
$806$	0.396990	0.0139834
$807$	0	0
$808$	12.0539	0.424055
$809$	24.7896	0.871556	0.435778	−	0.900054i	$-0.356473\pi$
$809$	24.7896	0.871556	0.435778	+	0.900054i	$0.356473\pi$
$810$	0	0
$811$	−8.24377	−0.289478	−0.144739	−	0.989470i	$-0.546234\pi$
$811$	−8.24377	−0.289478	−0.144739	+	0.989470i	$0.546234\pi$
$812$	0	0
$813$	0	0
$814$	−8.45417	−0.296319
$815$	−1.78247	−0.0624370
$816$	0	0
$817$	−4.32365	−0.151265
$818$	−0.396208	−0.0138531
$819$	0	0
$820$	23.3808	0.816494
$821$	28.8993	1.00859	0.504296	−	0.863531i	$-0.331752\pi$
$821$	28.8993	1.00859	0.504296	+	0.863531i	$0.331752\pi$
$822$	0	0
$823$	−36.0000	−1.25488	−0.627441	−	0.778664i	$-0.715897\pi$
$823$	−36.0000	−1.25488	−0.627441	+	0.778664i	$0.715897\pi$
$824$	−4.14557	−0.144418
$825$	0	0
$826$	0	0
$827$	50.7108	1.76339	0.881694	−	0.471821i	$-0.156403\pi$
$827$	50.7108	1.76339	0.881694	+	0.471821i	$0.156403\pi$
$828$	0	0
$829$	−14.8123	−0.514452	−0.257226	−	0.966351i	$-0.582809\pi$
$829$	−14.8123	−0.514452	−0.257226	+	0.966351i	$0.582809\pi$
$830$	−1.96225	−0.0681107
$831$	0	0
$832$	6.66019	0.230901
$833$	0	0
$834$	0	0
$835$	−17.3560	−0.600628
$836$	13.9798	0.483501
$837$	0	0
$838$	7.98057	0.275684
$839$	33.7212	1.16419	0.582093	−	0.813122i	$-0.302234\pi$
$839$	33.7212	1.16419	0.582093	+	0.813122i	$0.302234\pi$
$840$	0	0
$841$	−28.9428	−0.998028
$842$	4.36173	0.150315
$843$	0	0
$844$	35.1488	1.20987
$845$	−14.1833	−0.487921
$846$	0	0
$847$	0	0
$848$	42.4692	1.45840
$849$	0	0
$850$	5.98168	0.205170
$851$	−53.4854	−1.83346
$852$	0	0
$853$	−11.7896	−0.403668	−0.201834	−	0.979420i	$-0.564690\pi$
$853$	−11.7896	−0.403668	−0.201834	+	0.979420i	$0.564690\pi$
$854$	0	0
$855$	0	0
$856$	12.9428	0.442376
$857$	−31.3261	−1.07008	−0.535040	−	0.844827i	$-0.679704\pi$
$857$	−31.3261	−1.07008	−0.535040	+	0.844827i	$0.679704\pi$
$858$	0	0
$859$	50.3893	1.71926	0.859631	−	0.510915i	$-0.170693\pi$
$859$	50.3893	1.71926	0.859631	+	0.510915i	$0.170693\pi$
$860$	−5.11031	−0.174260
$861$	0	0
$862$	−7.00216	−0.238494
$863$	−1.13268	−0.0385568	−0.0192784	−	0.999814i	$-0.506137\pi$
$863$	−1.13268	−0.0385568	−0.0192784	+	0.999814i	$0.506137\pi$
$864$	0	0
$865$	0.298791	0.0101592
$866$	2.92804	0.0994987
$867$	0	0
$868$	0	0
$869$	27.3242	0.926911
$870$	0	0
$871$	−3.50808	−0.118867
$872$	−1.19097	−0.0403313
$873$	0	0
$874$	−2.60301	−0.0880481
$875$	0	0
$876$	0	0
$877$	−27.3937	−0.925020	−0.462510	−	0.886614i	$-0.653051\pi$
$877$	−27.3937	−0.925020	−0.462510	+	0.886614i	$0.653051\pi$
$878$	1.15538	0.0389922
$879$	0	0
$880$	16.0227	0.540125
$881$	1.20929	0.0407420	0.0203710	−	0.999792i	$-0.493515\pi$
$881$	1.20929	0.0407420	0.0203710	+	0.999792i	$0.493515\pi$
$882$	0	0
$883$	−51.0884	−1.71926	−0.859631	−	0.510916i	$-0.829306\pi$
$883$	−51.0884	−1.71926	−0.859631	+	0.510916i	$0.829306\pi$
$884$	−13.4887	−0.453672
$885$	0	0
$886$	−0.297680	−0.0100008
$887$	−41.5757	−1.39597	−0.697987	−	0.716110i	$-0.745921\pi$
$887$	−41.5757	−1.39597	−0.697987	+	0.716110i	$0.745921\pi$
$888$	0	0
$889$	0	0
$890$	0.776443	0.0260264
$891$	0	0
$892$	−44.0183	−1.47384
$893$	−11.3236	−0.378931
$894$	0	0
$895$	−16.7745	−0.560711
$896$	0	0
$897$	0	0
$898$	−2.11109	−0.0704480
$899$	0.396990	0.0132404
$900$	0	0
$901$	−80.5576	−2.68376
$902$	−9.01754	−0.300251
$903$	0	0
$904$	11.4671	0.381389
$905$	1.69192	0.0562412
$906$	0	0
$907$	35.4509	1.17713	0.588564	−	0.808451i	$-0.299694\pi$
$907$	35.4509	1.17713	0.588564	+	0.808451i	$0.299694\pi$
$908$	10.2632	0.340596
$909$	0	0
$910$	0	0
$911$	20.7108	0.686180	0.343090	−	0.939302i	$-0.388526\pi$
$911$	20.7108	0.686180	0.343090	+	0.939302i	$0.388526\pi$
$912$	0	0
$913$	−25.7141	−0.851013
$914$	−2.51273	−0.0831136
$915$	0	0
$916$	37.5674	1.24126
$917$	0	0
$918$	0	0
$919$	14.3926	0.474768	0.237384	−	0.971416i	$-0.423710\pi$
$919$	14.3926	0.474768	0.237384	+	0.971416i	$0.423710\pi$
$920$	−6.24377	−0.205851
$921$	0	0
$922$	5.39261	0.177596
$923$	8.60301	0.283172
$924$	0	0
$925$	34.3937	1.13086
$926$	2.48616	0.0817004
$927$	0	0
$928$	−0.660190	−0.0216718
$929$	−41.7428	−1.36954	−0.684769	−	0.728760i	$-0.740097\pi$
$929$	−41.7428	−1.36954	−0.684769	+	0.728760i	$0.740097\pi$
$930$	0	0
$931$	0	0
$932$	−32.9902	−1.08063
$933$	0	0
$934$	3.18443	0.104198
$935$	−30.3926	−0.993945
$936$	0	0
$937$	−3.17154	−0.103610	−0.0518048	−	0.998657i	$-0.516497\pi$
$937$	−3.17154	−0.103610	−0.0518048	+	0.998657i	$0.516497\pi$
$938$	0	0
$939$	0	0
$940$	−13.3839	−0.436535
$941$	−3.22080	−0.104995	−0.0524976	−	0.998621i	$-0.516718\pi$
$941$	−3.22080	−0.104995	−0.0524976	+	0.998621i	$0.516718\pi$
$942$	0	0
$943$	−57.0495	−1.85779
$944$	9.52751	0.310094
$945$	0	0
$946$	1.97095	0.0640810
$947$	45.3469	1.47358	0.736789	−	0.676123i	$-0.236341\pi$
$947$	45.3469	1.47358	0.736789	+	0.676123i	$0.236341\pi$
$948$	0	0
$949$	15.1488	0.491752
$950$	1.67386	0.0543073
$951$	0	0
$952$	0	0
$953$	54.2703	1.75799	0.878994	−	0.476832i	$-0.158215\pi$
$953$	54.2703	1.75799	0.878994	+	0.476832i	$0.158215\pi$
$954$	0	0
$955$	−17.8090	−0.576287
$956$	32.8058	1.06101
$957$	0	0
$958$	3.47576	0.112297
$959$	0	0
$960$	0	0
$961$	−28.2438	−0.911089
$962$	2.28263	0.0735950
$963$	0	0
$964$	52.7390	1.69861
$965$	−9.27576	−0.298597
$966$	0	0
$967$	25.6591	0.825140	0.412570	−	0.910926i	$-0.364631\pi$
$967$	25.6591	0.825140	0.412570	+	0.910926i	$0.364631\pi$
$968$	−2.56199	−0.0823455
$969$	0	0
$970$	−2.02597	−0.0650500
$971$	−21.0183	−0.674510	−0.337255	−	0.941413i	$-0.609498\pi$
$971$	−21.0183	−0.674510	−0.337255	+	0.941413i	$0.609498\pi$
$972$	0	0
$973$	0	0
$974$	3.12149	0.100019
$975$	0	0
$976$	27.8285	0.890767
$977$	2.09820	0.0671273	0.0335637	−	0.999437i	$-0.489314\pi$
$977$	2.09820	0.0671273	0.0335637	+	0.999437i	$0.489314\pi$
$978$	0	0
$979$	10.1748	0.325188
$980$	0	0
$981$	0	0
$982$	−4.62571	−0.147612
$983$	42.9923	1.37124	0.685622	−	0.727958i	$-0.259531\pi$
$983$	42.9923	1.37124	0.685622	+	0.727958i	$0.259531\pi$
$984$	0	0
$985$	−7.90723	−0.251945
$986$	0.396990	0.0126427
$987$	0	0
$988$	−3.77455	−0.120084
$989$	12.4692	0.396498
$990$	0	0
$991$	−17.2632	−0.548384	−0.274192	−	0.961675i	$-0.588410\pi$
$991$	−17.2632	−0.548384	−0.274192	+	0.961675i	$0.588410\pi$
$992$	−4.58358	−0.145529
$993$	0	0
$994$	0	0
$995$	23.5674	0.747137
$996$	0	0
$997$	38.9018	1.23203	0.616016	−	0.787733i	$-0.288746\pi$
$997$	38.9018	1.23203	0.616016	+	0.787733i	$0.288746\pi$
$998$	−8.66157	−0.274177
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3969.2.a.p.1.2		3
3.2	odd	2		3969.2.a.m.1.2		3
7.6	odd	2		567.2.a.g.1.2		3
9.2	odd	6		441.2.f.d.148.2		6
9.4	even	3		1323.2.f.c.883.2		6
9.5	odd	6		441.2.f.d.295.2		6
9.7	even	3		1323.2.f.c.442.2		6
21.20	even	2		567.2.a.d.1.2		3
28.27	even	2		9072.2.a.cd.1.1		3
63.2	odd	6		441.2.g.d.67.2		6
63.4	even	3		1323.2.g.b.667.2		6
63.5	even	6		441.2.h.c.214.2		6
63.11	odd	6		441.2.h.b.373.2		6
63.13	odd	6		189.2.f.a.127.2		6
63.16	even	3		1323.2.g.b.361.2		6
63.20	even	6		63.2.f.b.22.2	✓	6
63.23	odd	6		441.2.h.b.214.2		6
63.25	even	3		1323.2.h.e.226.2		6
63.31	odd	6		1323.2.g.c.667.2		6
63.32	odd	6		441.2.g.d.79.2		6
63.34	odd	6		189.2.f.a.64.2		6
63.38	even	6		441.2.h.c.373.2		6
63.40	odd	6		1323.2.h.d.802.2		6
63.41	even	6		63.2.f.b.43.2	yes	6
63.47	even	6		441.2.g.e.67.2		6
63.52	odd	6		1323.2.h.d.226.2		6
63.58	even	3		1323.2.h.e.802.2		6
63.59	even	6		441.2.g.e.79.2		6
63.61	odd	6		1323.2.g.c.361.2		6
84.83	odd	2		9072.2.a.bq.1.3		3
252.83	odd	6		1008.2.r.k.337.2		6
252.139	even	6		3024.2.r.g.2017.3		6
252.167	odd	6		1008.2.r.k.673.2		6
252.223	even	6		3024.2.r.g.1009.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
63.2.f.b.22.2	✓	6	63.20	even	6
63.2.f.b.43.2	yes	6	63.41	even	6
189.2.f.a.64.2		6	63.34	odd	6
189.2.f.a.127.2		6	63.13	odd	6
441.2.f.d.148.2		6	9.2	odd	6
441.2.f.d.295.2		6	9.5	odd	6
441.2.g.d.67.2		6	63.2	odd	6
441.2.g.d.79.2		6	63.32	odd	6
441.2.g.e.67.2		6	63.47	even	6
441.2.g.e.79.2		6	63.59	even	6
441.2.h.b.214.2		6	63.23	odd	6
441.2.h.b.373.2		6	63.11	odd	6
441.2.h.c.214.2		6	63.5	even	6
441.2.h.c.373.2		6	63.38	even	6
567.2.a.d.1.2		3	21.20	even	2
567.2.a.g.1.2		3	7.6	odd	2
1008.2.r.k.337.2		6	252.83	odd	6
1008.2.r.k.673.2		6	252.167	odd	6
1323.2.f.c.442.2		6	9.7	even	3
1323.2.f.c.883.2		6	9.4	even	3
1323.2.g.b.361.2		6	63.16	even	3
1323.2.g.b.667.2		6	63.4	even	3
1323.2.g.c.361.2		6	63.61	odd	6
1323.2.g.c.667.2		6	63.31	odd	6
1323.2.h.d.226.2		6	63.52	odd	6
1323.2.h.d.802.2		6	63.40	odd	6
1323.2.h.e.226.2		6	63.25	even	3
1323.2.h.e.802.2		6	63.58	even	3
3024.2.r.g.1009.3		6	252.223	even	6
3024.2.r.g.2017.3		6	252.139	even	6
3969.2.a.m.1.2		3	3.2	odd	2
3969.2.a.p.1.2		3	1.1	even	1	trivial
9072.2.a.bq.1.3		3	84.83	odd	2
9072.2.a.cd.1.1		3	28.27	even	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)

\(f(q)\)	\(=\)	\(q+0.239123 q^{2} -1.94282 q^{4} +1.18194 q^{5} -0.942820 q^{8} +O(q^{10})\)	\(q+0.239123 q^{2} -1.94282 q^{4} +1.18194 q^{5} -0.942820 q^{8} +0.282630 q^{10} +3.70370 q^{11} -1.00000 q^{13} +3.66019 q^{16} -6.94282 q^{17} -1.94282 q^{19} -2.29630 q^{20} +0.885640 q^{22} +5.60301 q^{23} -3.60301 q^{25} -0.239123 q^{26} -0.239123 q^{29} -1.66019 q^{31} +2.76088 q^{32} -1.66019 q^{34} -9.54583 q^{37} -0.464574 q^{38} -1.11436 q^{40} -10.1819 q^{41} +2.22545 q^{43} -7.19562 q^{44} +1.33981 q^{46} +5.82846 q^{47} -0.861564 q^{50} +1.94282 q^{52} +11.6030 q^{53} +4.37756 q^{55} -0.0571799 q^{58} +2.60301 q^{59} +7.60301 q^{61} -0.396990 q^{62} -6.66019 q^{64} -1.18194 q^{65} +3.50808 q^{67} +13.4887 q^{68} -8.60301 q^{71} -15.1488 q^{73} -2.28263 q^{74} +3.77455 q^{76} +7.37756 q^{79} +4.32614 q^{80} -2.43474 q^{82} -6.94282 q^{83} -8.20602 q^{85} +0.532157 q^{86} -3.49192 q^{88} +2.74720 q^{89} -10.8856 q^{92} +1.39372 q^{94} -2.29630 q^{95} -7.16827 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 3 q^{4} - 5 q^{5} + 6 q^{8}+O(q^{10}) \) 3 * q + q^2 + 3 * q^4 - 5 * q^5 + 6 * q^8	\( 3 q + q^{2} + 3 q^{4} - 5 q^{5} + 6 q^{8} + 2 q^{11} - 3 q^{13} + 3 q^{16} - 12 q^{17} + 3 q^{19} - 16 q^{20} - 15 q^{22} + 6 q^{25} - q^{26} - q^{29} + 3 q^{31} + 8 q^{32} + 3 q^{34} - 3 q^{37} + 8 q^{38} - 21 q^{40} - 22 q^{41} - 3 q^{43} - 23 q^{44} + 12 q^{46} - 9 q^{47} - 10 q^{50} - 3 q^{52} + 18 q^{53} + 6 q^{55} - 9 q^{58} - 9 q^{59} + 6 q^{61} - 18 q^{62} - 12 q^{64} + 5 q^{65} + 6 q^{68} - 9 q^{71} - 3 q^{73} - 6 q^{74} + 21 q^{76} + 15 q^{79} + 11 q^{80} - 9 q^{82} - 12 q^{83} + 9 q^{85} - 34 q^{86} - 21 q^{88} - 2 q^{89} - 15 q^{92} - 24 q^{94} - 16 q^{95} - 3 q^{97}+O(q^{100}) \) 3 * q + q^2 + 3 * q^4 - 5 * q^5 + 6 * q^8 + 2 * q^11 - 3 * q^13 + 3 * q^16 - 12 * q^17 + 3 * q^19 - 16 * q^20 - 15 * q^22 + 6 * q^25 - q^26 - q^29 + 3 * q^31 + 8 * q^32 + 3 * q^34 - 3 * q^37 + 8 * q^38 - 21 * q^40 - 22 * q^41 - 3 * q^43 - 23 * q^44 + 12 * q^46 - 9 * q^47 - 10 * q^50 - 3 * q^52 + 18 * q^53 + 6 * q^55 - 9 * q^58 - 9 * q^59 + 6 * q^61 - 18 * q^62 - 12 * q^64 + 5 * q^65 + 6 * q^68 - 9 * q^71 - 3 * q^73 - 6 * q^74 + 21 * q^76 + 15 * q^79 + 11 * q^80 - 9 * q^82 - 12 * q^83 + 9 * q^85 - 34 * q^86 - 21 * q^88 - 2 * q^89 - 15 * q^92 - 24 * q^94 - 16 * q^95 - 3 * q^97

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