LMFDB - Embedded newform 39.3.d.a.38.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [39,3,Mod(38,39)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(39, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("39.38");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 39 = 3 \cdot 13 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	39.d (of order $2$, degree $1$, minimal)

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:	$1.06267303101$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 3 $ x^2 - x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			38.2
Root			$-1.30278$ of defining polynomial
Character	$\chi$	$=$	39.38

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.60555 q^{2} -3.00000 q^{3} +9.00000 q^{4} -7.21110 q^{5} -10.8167 q^{6} +18.0278 q^{8} +9.00000 q^{9} +O(q^{10})$	$q+3.60555 q^{2} -3.00000 q^{3} +9.00000 q^{4} -7.21110 q^{5} -10.8167 q^{6} +18.0278 q^{8} +9.00000 q^{9} -26.0000 q^{10} -7.21110 q^{11} -27.0000 q^{12} +13.0000 q^{13} +21.6333 q^{15} +29.0000 q^{16} +32.4500 q^{18} -64.8999 q^{20} -26.0000 q^{22} -54.0833 q^{24} +27.0000 q^{25} +46.8722 q^{26} -27.0000 q^{27} +78.0000 q^{30} +32.4500 q^{32} +21.6333 q^{33} +81.0000 q^{36} -39.0000 q^{39} -130.000 q^{40} +79.3221 q^{41} -70.0000 q^{43} -64.8999 q^{44} -64.8999 q^{45} -93.7443 q^{47} -87.0000 q^{48} +49.0000 q^{49} +97.3499 q^{50} +117.000 q^{52} -97.3499 q^{54} +52.0000 q^{55} -7.21110 q^{59} +194.700 q^{60} -70.0000 q^{61} +1.00000 q^{64} -93.7443 q^{65} +78.0000 q^{66} +79.3221 q^{71} +162.250 q^{72} -81.0000 q^{75} -140.616 q^{78} +50.0000 q^{79} -209.122 q^{80} +81.0000 q^{81} +286.000 q^{82} +165.855 q^{83} -252.389 q^{86} -130.000 q^{88} +79.3221 q^{89} -234.000 q^{90} -338.000 q^{94} -97.3499 q^{96} +176.672 q^{98} -64.8999 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 18 q^{4} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 + 18 * q^4 + 18 * q^9	$ 2 q - 6 q^{3} + 18 q^{4} + 18 q^{9} - 52 q^{10} - 54 q^{12} + 26 q^{13} + 58 q^{16} - 52 q^{22} + 54 q^{25} - 54 q^{27} + 156 q^{30} + 162 q^{36} - 78 q^{39} - 260 q^{40} - 140 q^{43} - 174 q^{48} + 98 q^{49} + 234 q^{52} + 104 q^{55} - 140 q^{61} + 2 q^{64} + 156 q^{66} - 162 q^{75} + 100 q^{79} + 162 q^{81} + 572 q^{82} - 260 q^{88} - 468 q^{90} - 676 q^{94}+O(q^{100}) $ 2 * q - 6 * q^3 + 18 * q^4 + 18 * q^9 - 52 * q^10 - 54 * q^12 + 26 * q^13 + 58 * q^16 - 52 * q^22 + 54 * q^25 - 54 * q^27 + 156 * q^30 + 162 * q^36 - 78 * q^39 - 260 * q^40 - 140 * q^43 - 174 * q^48 + 98 * q^49 + 234 * q^52 + 104 * q^55 - 140 * q^61 + 2 * q^64 + 156 * q^66 - 162 * q^75 + 100 * q^79 + 162 * q^81 + 572 * q^82 - 260 * q^88 - 468 * q^90 - 676 * q^94

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/39\mathbb{Z}\right)^\times$.

$n$	$14$	$28$
$\chi(n)$	$-1$	$-1$

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 $

$2$	3.60555	1.80278	0.901388	−	0.433013i	$-0.142549\pi$
$2$	3.60555	1.80278	0.901388	+	0.433013i	$0.142549\pi$
$3$	−3.00000	−1.00000
$3$	−3.00000	−1.00000
$4$	9.00000	2.25000
$5$	−7.21110	−1.44222	−0.721110	−	0.692820i	$-0.756368\pi$
$5$	−7.21110	−1.44222	−0.721110	+	0.692820i	$0.756368\pi$
$6$	−10.8167	−1.80278
$7$	0	0	1.00000			$0$
$7$	0	0	−1.00000			$\pi$
$8$	18.0278	2.25347
$9$	9.00000	1.00000
$10$	−26.0000	−2.60000
$11$	−7.21110	−0.655555	−0.327777	−	0.944755i	$-0.606300\pi$
$11$	−7.21110	−0.655555	−0.327777	+	0.944755i	$0.606300\pi$
$12$	−27.0000	−2.25000
$13$	13.0000	1.00000
$13$	13.0000	1.00000
$14$	0	0
$15$	21.6333	1.44222
$16$	29.0000	1.81250
$17$	0	0	1.00000			$0$
$17$	0	0	−1.00000			$\pi$
$18$	32.4500	1.80278
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	−64.8999	−3.24500
$21$	0	0
$22$	−26.0000	−1.18182
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	−54.0833	−2.25347
$25$	27.0000	1.08000
$26$	46.8722	1.80278
$27$	−27.0000	−1.00000
$28$	0	0
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	78.0000	2.60000
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	32.4500	1.01406
$33$	21.6333	0.655555
$34$	0	0
$35$	0	0
$36$	81.0000	2.25000
$37$	0	0	1.00000			$0$
$37$	0	0	−1.00000			$\pi$
$38$	0	0
$39$	−39.0000	−1.00000
$40$	−130.000	−3.25000
$41$	79.3221	1.93469	0.967343	−	0.253471i	$-0.0815722\pi$
$41$	79.3221	1.93469	0.967343	+	0.253471i	$0.0815722\pi$
$42$	0	0
$43$	−70.0000	−1.62791	−0.813953	−	0.580930i	$-0.802689\pi$
$43$	−70.0000	−1.62791	−0.813953	+	0.580930i	$0.802689\pi$
$44$	−64.8999	−1.47500
$45$	−64.8999	−1.44222
$46$	0	0
$47$	−93.7443	−1.99456	−0.997280	−	0.0737043i	$-0.976518\pi$
$47$	−93.7443	−1.99456	−0.997280	+	0.0737043i	$0.976518\pi$
$48$	−87.0000	−1.81250
$49$	49.0000	1.00000
$50$	97.3499	1.94700
$51$	0	0
$52$	117.000	2.25000
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	−97.3499	−1.80278
$55$	52.0000	0.945455
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−7.21110	−0.122222	−0.0611110	−	0.998131i	$-0.519464\pi$
$59$	−7.21110	−0.122222	−0.0611110	+	0.998131i	$0.519464\pi$
$60$	194.700	3.24500
$61$	−70.0000	−1.14754	−0.573770	−	0.819016i	$-0.694520\pi$
$61$	−70.0000	−1.14754	−0.573770	+	0.819016i	$0.694520\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	0.0156250
$65$	−93.7443	−1.44222
$66$	78.0000	1.18182
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	79.3221	1.11721	0.558607	−	0.829433i	$-0.311336\pi$
$71$	79.3221	1.11721	0.558607	+	0.829433i	$0.311336\pi$
$72$	162.250	2.25347
$73$	0	0	1.00000			$0$
$73$	0	0	−1.00000			$\pi$
$74$	0	0
$75$	−81.0000	−1.08000
$76$	0	0
$77$	0	0
$78$	−140.616	−1.80278
$79$	50.0000	0.632911	0.316456	−	0.948607i	$-0.397507\pi$
$79$	50.0000	0.632911	0.316456	+	0.948607i	$0.397507\pi$
$80$	−209.122	−2.61402
$81$	81.0000	1.00000
$82$	286.000	3.48780
$83$	165.855	1.99826	0.999129	−	0.0417362i	$-0.0132889\pi$
$83$	165.855	1.99826	0.999129	+	0.0417362i	$0.0132889\pi$
$84$	0	0
$85$	0	0
$86$	−252.389	−2.93475
$87$	0	0
$88$	−130.000	−1.47727
$89$	79.3221	0.891260	0.445630	−	0.895217i	$-0.352980\pi$
$89$	79.3221	0.891260	0.445630	+	0.895217i	$0.352980\pi$
$90$	−234.000	−2.60000
$91$	0	0
$92$	0	0
$93$	0	0
$94$	−338.000	−3.59574
$95$	0	0
$96$	−97.3499	−1.01406
$97$	0	0	1.00000			$0$
$97$	0	0	−1.00000			$\pi$
$98$	176.672	1.80278
$99$	−64.8999	−0.655555
$100$	243.000	2.43000
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	50.0000	0.485437	0.242718	−	0.970097i	$-0.421961\pi$
$103$	50.0000	0.485437	0.242718	+	0.970097i	$0.421961\pi$
$104$	234.361	2.25347
$105$	0	0
$106$	0	0
$107$	0	0	1.00000			$0$
$107$	0	0	−1.00000			$\pi$
$108$	−243.000	−2.25000
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	187.489	1.70444
$111$	0	0
$112$	0	0
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	117.000	1.00000
$118$	−26.0000	−0.220339
$119$	0	0
$120$	390.000	3.25000
$121$	−69.0000	−0.570248
$122$	−252.389	−2.06876
$123$	−237.966	−1.93469
$124$	0	0
$125$	−14.4222	−0.115378
$126$	0	0
$127$	−46.0000	−0.362205	−0.181102	−	0.983464i	$-0.557967\pi$
$127$	−46.0000	−0.362205	−0.181102	+	0.983464i	$0.557967\pi$
$128$	−126.194	−0.985893
$129$	210.000	1.62791
$130$	−338.000	−2.60000
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	194.700	1.47500
$133$	0	0
$134$	0	0
$135$	194.700	1.44222
$136$	0	0
$137$	−266.811	−1.94752	−0.973762	−	0.227569i	$-0.926922\pi$
$137$	−266.811	−1.94752	−0.973762	+	0.227569i	$0.926922\pi$
$138$	0	0
$139$	122.000	0.877698	0.438849	−	0.898561i	$-0.355386\pi$
$139$	122.000	0.877698	0.438849	+	0.898561i	$0.355386\pi$
$140$	0	0
$141$	281.233	1.99456
$142$	286.000	2.01408
$143$	−93.7443	−0.655555
$144$	261.000	1.81250
$145$	0	0
$146$	0	0
$147$	−147.000	−1.00000
$148$	0	0
$149$	−7.21110	−0.0483967	−0.0241983	−	0.999707i	$-0.507703\pi$
$149$	−7.21110	−0.0483967	−0.0241983	+	0.999707i	$0.507703\pi$
$150$	−292.050	−1.94700
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	−351.000	−2.25000
$157$	−310.000	−1.97452	−0.987261	−	0.159108i	$-0.949138\pi$
$157$	−310.000	−1.97452	−0.987261	+	0.159108i	$0.949138\pi$
$158$	180.278	1.14100
$159$	0	0
$160$	−234.000	−1.46250
$161$	0	0
$162$	292.050	1.80278
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	713.899	4.35304
$165$	−156.000	−0.945455
$166$	598.000	3.60241
$167$	−266.811	−1.59767	−0.798835	−	0.601551i	$-0.794550\pi$
$167$	−266.811	−1.59767	−0.798835	+	0.601551i	$0.794550\pi$
$168$	0	0
$169$	169.000	1.00000
$170$	0	0
$171$	0	0
$172$	−630.000	−3.66279
$173$	0	0	1.00000			$0$
$173$	0	0	−1.00000			$\pi$
$174$	0	0
$175$	0	0
$176$	−209.122	−1.18819
$177$	21.6333	0.122222
$178$	286.000	1.60674
$179$	0	0	1.00000			$0$
$179$	0	0	−1.00000			$\pi$
$180$	−584.099	−3.24500
$181$	−262.000	−1.44751	−0.723757	−	0.690055i	$-0.757586\pi$
$181$	−262.000	−1.44751	−0.723757	+	0.690055i	$0.757586\pi$
$182$	0	0
$183$	210.000	1.14754
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	−843.699	−4.48776
$189$	0	0
$190$	0	0
$191$	0	0	1.00000			$0$
$191$	0	0	−1.00000			$\pi$
$192$	−3.00000	−0.0156250
$193$	0	0	1.00000			$0$
$193$	0	0	−1.00000			$\pi$
$194$	0	0
$195$	281.233	1.44222
$196$	441.000	2.25000
$197$	338.922	1.72042	0.860208	−	0.509944i	$-0.170334\pi$
$197$	338.922	1.72042	0.860208	+	0.509944i	$0.170334\pi$
$198$	−234.000	−1.18182
$199$	−190.000	−0.954774	−0.477387	−	0.878693i	$-0.658416\pi$
$199$	−190.000	−0.954774	−0.477387	+	0.878693i	$0.658416\pi$
$200$	486.749	2.43375
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−572.000	−2.79024
$206$	180.278	0.875134
$207$	0	0
$208$	377.000	1.81250
$209$	0	0
$210$	0	0
$211$	410.000	1.94313	0.971564	−	0.236777i	$-0.0760911\pi$
$211$	410.000	1.94313	0.971564	+	0.236777i	$0.0760911\pi$
$212$	0	0
$213$	−237.966	−1.11721
$214$	0	0
$215$	504.777	2.34780
$216$	−486.749	−2.25347
$217$	0	0
$218$	0	0
$219$	0	0
$220$	468.000	2.12727
$221$	0	0
$222$	0	0
$223$	0	0	1.00000			$0$
$223$	0	0	−1.00000			$\pi$
$224$	0	0
$225$	243.000	1.08000
$226$	0	0
$227$	165.855	0.730640	0.365320	−	0.930882i	$-0.380960\pi$
$227$	165.855	0.730640	0.365320	+	0.930882i	$0.380960\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	0	0	1.00000			$0$
$233$	0	0	−1.00000			$\pi$
$234$	421.849	1.80278
$235$	676.000	2.87660
$236$	−64.8999	−0.275000
$237$	−150.000	−0.632911
$238$	0	0
$239$	−439.877	−1.84049	−0.920245	−	0.391342i	$-0.872011\pi$
$239$	−439.877	−1.84049	−0.920245	+	0.391342i	$0.872011\pi$
$240$	627.366	2.61402
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	−248.783	−1.02803
$243$	−243.000	−1.00000
$244$	−630.000	−2.58197
$245$	−353.344	−1.44222
$246$	−858.000	−3.48780
$247$	0	0
$248$	0	0
$249$	−497.566	−1.99826
$250$	−52.0000	−0.208000
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	0	0
$253$	0	0
$254$	−165.855	−0.652974
$255$	0	0
$256$	−459.000	−1.79297
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	757.166	2.93475
$259$	0	0
$260$	−843.699	−3.24500
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	390.000	1.47727
$265$	0	0
$266$	0	0
$267$	−237.966	−0.891260
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	702.000	2.60000
$271$	0	0	1.00000			$0$
$271$	0	0	−1.00000			$\pi$
$272$	0	0
$273$	0	0
$274$	−962.000	−3.51095
$275$	−194.700	−0.707999
$276$	0	0
$277$	−70.0000	−0.252708	−0.126354	−	0.991985i	$-0.540327\pi$
$277$	−70.0000	−0.252708	−0.126354	+	0.991985i	$0.540327\pi$
$278$	439.877	1.58229
$279$	0	0
$280$	0	0
$281$	425.455	1.51407	0.757037	−	0.653371i	$-0.226646\pi$
$281$	425.455	1.51407	0.757037	+	0.653371i	$0.226646\pi$
$282$	1014.00	3.59574
$283$	266.000	0.939929	0.469965	−	0.882685i	$-0.344267\pi$
$283$	266.000	0.939929	0.469965	+	0.882685i	$0.344267\pi$
$284$	713.899	2.51373
$285$	0	0
$286$	−338.000	−1.18182
$287$	0	0
$288$	292.050	1.01406
$289$	289.000	1.00000
$290$	0	0
$291$	0	0
$292$	0	0
$293$	338.922	1.15673	0.578365	−	0.815778i	$-0.303691\pi$
$293$	338.922	1.15673	0.578365	+	0.815778i	$0.303691\pi$
$294$	−530.016	−1.80278
$295$	52.0000	0.176271
$296$	0	0
$297$	194.700	0.655555
$298$	−26.0000	−0.0872483
$299$	0	0
$300$	−729.000	−2.43000
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	504.777	1.65501
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	−150.000	−0.485437
$310$	0	0
$311$	0	0	1.00000			$0$
$311$	0	0	−1.00000			$\pi$
$312$	−703.082	−2.25347
$313$	−574.000	−1.83387	−0.916933	−	0.399041i	$-0.869343\pi$
$313$	−574.000	−1.83387	−0.916933	+	0.399041i	$0.869343\pi$
$314$	−1117.72	−3.55962
$315$	0	0
$316$	450.000	1.42405
$317$	−526.410	−1.66060	−0.830300	−	0.557316i	$-0.811831\pi$
$317$	−526.410	−1.66060	−0.830300	+	0.557316i	$0.811831\pi$
$318$	0	0
$319$	0	0
$320$	−7.21110	−0.0225347
$321$	0	0
$322$	0	0
$323$	0	0
$324$	729.000	2.25000
$325$	351.000	1.08000
$326$	0	0
$327$	0	0
$328$	1430.00	4.35976
$329$	0	0
$330$	−562.466	−1.70444
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	1492.70	4.49608
$333$	0	0
$334$	−962.000	−2.88024
$335$	0	0
$336$	0	0
$337$	626.000	1.85757	0.928783	−	0.370623i	$-0.120856\pi$
$337$	626.000	1.85757	0.928783	+	0.370623i	$0.120856\pi$
$338$	609.338	1.80278
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	0	0
$344$	−1261.94	−3.66844
$345$	0	0
$346$	0	0
$347$	0	0	1.00000			$0$
$347$	0	0	−1.00000			$\pi$
$348$	0	0
$349$	0	0	1.00000			$0$
$349$	0	0	−1.00000			$\pi$
$350$	0	0
$351$	−351.000	−1.00000
$352$	−234.000	−0.664773
$353$	−93.7443	−0.265565	−0.132782	−	0.991145i	$-0.542391\pi$
$353$	−93.7443	−0.265565	−0.132782	+	0.991145i	$0.542391\pi$
$354$	78.0000	0.220339
$355$	−572.000	−1.61127
$356$	713.899	2.00533
$357$	0	0
$358$	0	0
$359$	79.3221	0.220953	0.110477	−	0.993879i	$-0.464762\pi$
$359$	79.3221	0.220953	0.110477	+	0.993879i	$0.464762\pi$
$360$	−1170.00	−3.25000
$361$	361.000	1.00000
$362$	−944.654	−2.60954
$363$	207.000	0.570248
$364$	0	0
$365$	0	0
$366$	757.166	2.06876
$367$	−670.000	−1.82561	−0.912807	−	0.408392i	$-0.866090\pi$
$367$	−670.000	−1.82561	−0.912807	+	0.408392i	$0.866090\pi$
$368$	0	0
$369$	713.899	1.93469
$370$	0	0
$371$	0	0
$372$	0	0
$373$	554.000	1.48525	0.742627	−	0.669705i	$-0.233579\pi$
$373$	554.000	1.48525	0.742627	+	0.669705i	$0.233579\pi$
$374$	0	0
$375$	43.2666	0.115378
$376$	−1690.00	−4.49468
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	138.000	0.362205
$382$	0	0
$383$	598.522	1.56272	0.781360	−	0.624081i	$-0.214526\pi$
$383$	598.522	1.56272	0.781360	+	0.624081i	$0.214526\pi$
$384$	378.583	0.985893
$385$	0	0
$386$	0	0
$387$	−630.000	−1.62791
$388$	0	0
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	1014.00	2.60000
$391$	0	0
$392$	883.360	2.25347
$393$	0	0
$394$	1222.00	3.10152
$395$	−360.555	−0.912798
$396$	−584.099	−1.47500
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	−685.055	−1.72124
$399$	0	0
$400$	783.000	1.95750
$401$	−786.010	−1.96013	−0.980063	−	0.198689i	$-0.936332\pi$
$401$	−786.010	−1.96013	−0.980063	+	0.198689i	$0.936332\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	−584.099	−1.44222
$406$	0	0
$407$	0	0
$408$	0	0
$409$	0	0	1.00000			$0$
$409$	0	0	−1.00000			$\pi$
$410$	−2062.38	−5.03018
$411$	800.432	1.94752
$412$	450.000	1.09223
$413$	0	0
$414$	0	0
$415$	−1196.00	−2.88193
$416$	421.849	1.01406
$417$	−366.000	−0.877698
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	1478.28	3.50302
$423$	−843.699	−1.99456
$424$	0	0
$425$	0	0
$426$	−858.000	−2.01408
$427$	0	0
$428$	0	0
$429$	281.233	0.655555
$430$	1820.00	4.23256
$431$	−439.877	−1.02060	−0.510298	−	0.859997i	$-0.670465\pi$
$431$	−439.877	−1.02060	−0.510298	+	0.859997i	$0.670465\pi$
$432$	−783.000	−1.81250
$433$	434.000	1.00231	0.501155	−	0.865358i	$-0.332909\pi$
$433$	434.000	1.00231	0.501155	+	0.865358i	$0.332909\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	722.000	1.64465	0.822323	−	0.569020i	$-0.192677\pi$
$439$	722.000	1.64465	0.822323	+	0.569020i	$0.192677\pi$
$440$	937.443	2.13055
$441$	441.000	1.00000
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0	0
$445$	−572.000	−1.28539
$446$	0	0
$447$	21.6333	0.0483967
$448$	0	0
$449$	−439.877	−0.979682	−0.489841	−	0.871812i	$-0.662945\pi$
$449$	−439.877	−0.979682	−0.489841	+	0.871812i	$0.662945\pi$
$450$	876.149	1.94700
$451$	−572.000	−1.26829
$452$	0	0
$453$	0	0
$454$	598.000	1.31718
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−872.543	−1.89272	−0.946359	−	0.323116i	$-0.895270\pi$
$461$	−872.543	−1.89272	−0.946359	+	0.323116i	$0.895270\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0	1.00000			$0$
$467$	0	0	−1.00000			$\pi$
$468$	1053.00	2.25000
$469$	0	0
$470$	2437.35	5.18586
$471$	930.000	1.97452
$472$	−130.000	−0.275424
$473$	504.777	1.06718
$474$	−540.833	−1.14100
$475$	0	0
$476$	0	0
$477$	0	0
$478$	−1586.00	−3.31799
$479$	944.654	1.97214	0.986069	−	0.166335i	$-0.0531932\pi$
$479$	944.654	1.97214	0.986069	+	0.166335i	$0.0531932\pi$
$480$	702.000	1.46250
$481$	0	0
$482$	0	0
$483$	0	0
$484$	−621.000	−1.28306
$485$	0	0
$486$	−876.149	−1.80278
$487$	0	0	1.00000			$0$
$487$	0	0	−1.00000			$\pi$
$488$	−1261.94	−2.58595
$489$	0	0
$490$	−1274.00	−2.60000
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	−2141.70	−4.35304
$493$	0	0
$494$	0	0
$495$	468.000	0.945455
$496$	0	0
$497$	0	0
$498$	−1794.00	−3.60241
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	−129.800	−0.259600
$501$	800.432	1.59767
$502$	0	0
$503$	0	0	1.00000			$0$
$503$	0	0	−1.00000			$\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−507.000	−1.00000
$508$	−414.000	−0.814961
$509$	511.988	1.00587	0.502935	−	0.864324i	$-0.332253\pi$
$509$	511.988	1.00587	0.502935	+	0.864324i	$0.332253\pi$
$510$	0	0
$511$	0	0
$512$	−1150.17	−2.24643
$513$	0	0
$514$	0	0
$515$	−360.555	−0.700107
$516$	1890.00	3.66279
$517$	676.000	1.30754
$518$	0	0
$519$	0	0
$520$	−1690.00	−3.25000
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	890.000	1.70172	0.850860	−	0.525392i	$-0.176081\pi$
$523$	890.000	1.70172	0.850860	+	0.525392i	$0.176081\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	627.366	1.18819
$529$	529.000	1.00000
$530$	0	0
$531$	−64.8999	−0.122222
$532$	0	0
$533$	1031.19	1.93469
$534$	−858.000	−1.60674
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−353.344	−0.655555
$540$	1752.30	3.24500
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	786.000	1.44751
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−310.000	−0.566728	−0.283364	−	0.959012i	$-0.591450\pi$
$547$	−310.000	−0.566728	−0.283364	+	0.959012i	$0.591450\pi$
$548$	−2401.30	−4.38193
$549$	−630.000	−1.14754
$550$	−702.000	−1.27636
$551$	0	0
$552$	0	0
$553$	0	0
$554$	−252.389	−0.455575
$555$	0	0
$556$	1098.00	1.97482
$557$	165.855	0.297765	0.148883	−	0.988855i	$-0.452432\pi$
$557$	165.855	0.297765	0.148883	+	0.988855i	$0.452432\pi$
$558$	0	0
$559$	−910.000	−1.62791
$560$	0	0
$561$	0	0
$562$	1534.00	2.72954
$563$	0	0	1.00000			$0$
$563$	0	0	−1.00000			$\pi$
$564$	2531.10	4.48776
$565$	0	0
$566$	959.077	1.69448
$567$	0	0
$568$	1430.00	2.51761
$569$	0	0	1.00000			$0$
$569$	0	0	−1.00000			$\pi$
$570$	0	0
$571$	−310.000	−0.542907	−0.271454	−	0.962452i	$-0.587504\pi$
$571$	−310.000	−0.542907	−0.271454	+	0.962452i	$0.587504\pi$
$572$	−843.699	−1.47500
$573$	0	0
$574$	0	0
$575$	0	0
$576$	9.00000	0.0156250
$577$	0	0	1.00000			$0$
$577$	0	0	−1.00000			$\pi$
$578$	1042.00	1.80278
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	−843.699	−1.44222
$586$	1222.00	2.08532
$587$	1031.19	1.75671	0.878354	−	0.478011i	$-0.158642\pi$
$587$	1031.19	1.75671	0.878354	+	0.478011i	$0.158642\pi$
$588$	−1323.00	−2.25000
$589$	0	0
$590$	187.489	0.317777
$591$	−1016.77	−1.72042
$592$	0	0
$593$	−1132.14	−1.90918	−0.954589	−	0.297924i	$-0.903706\pi$
$593$	−1132.14	−1.90918	−0.954589	+	0.297924i	$0.903706\pi$
$594$	702.000	1.18182
$595$	0	0
$596$	−64.8999	−0.108892
$597$	570.000	0.954774
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	−1460.25	−2.43375
$601$	−1150.00	−1.91348	−0.956739	−	0.290948i	$-0.906029\pi$
$601$	−1150.00	−1.91348	−0.956739	+	0.290948i	$0.906029\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	497.566	0.822423
$606$	0	0
$607$	−190.000	−0.313015	−0.156507	−	0.987677i	$-0.550024\pi$
$607$	−190.000	−0.313015	−0.156507	+	0.987677i	$0.550024\pi$
$608$	0	0
$609$	0	0
$610$	1820.00	2.98361
$611$	−1218.68	−1.99456
$612$	0	0
$613$	0	0	1.00000			$0$
$613$	0	0	−1.00000			$\pi$
$614$	0	0
$615$	1716.00	2.79024
$616$	0	0
$617$	771.588	1.25055	0.625274	−	0.780405i	$-0.284987\pi$
$617$	771.588	1.25055	0.625274	+	0.780405i	$0.284987\pi$
$618$	−540.833	−0.875134
$619$	0	0	1.00000			$0$
$619$	0	0	−1.00000			$\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	−1131.00	−1.81250
$625$	−571.000	−0.913600
$626$	−2069.59	−3.30605
$627$	0	0
$628$	−2790.00	−4.44268
$629$	0	0
$630$	0	0
$631$	0	0	1.00000			$0$
$631$	0	0	−1.00000			$\pi$
$632$	901.388	1.42625
$633$	−1230.00	−1.94313
$634$	−1898.00	−2.99369
$635$	331.711	0.522379
$636$	0	0
$637$	637.000	1.00000
$638$	0	0
$639$	713.899	1.11721
$640$	910.000	1.42188
$641$	0	0	1.00000			$0$
$641$	0	0	−1.00000			$\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	−1514.33	−2.34780
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	1460.25	2.25347
$649$	52.0000	0.0801233
$650$	1265.55	1.94700
$651$	0	0
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	2300.34	3.50662
$657$	0	0
$658$	0	0
$659$	0	0	1.00000			$0$
$659$	0	0	−1.00000			$\pi$
$660$	−1404.00	−2.12727
$661$	0	0	1.00000			$0$
$661$	0	0	−1.00000			$\pi$
$662$	0	0
$663$	0	0
$664$	2990.00	4.50301
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−2401.30	−3.59476
$669$	0	0
$670$	0	0
$671$	504.777	0.752276
$672$	0	0
$673$	−1150.00	−1.70877	−0.854383	−	0.519643i	$-0.826065\pi$
$673$	−1150.00	−1.70877	−0.854383	+	0.519643i	$0.826065\pi$
$674$	2257.08	3.34878
$675$	−729.000	−1.08000
$676$	1521.00	2.25000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−497.566	−0.730640
$682$	0	0
$683$	338.922	0.496225	0.248113	−	0.968731i	$-0.420190\pi$
$683$	338.922	0.496225	0.248113	+	0.968731i	$0.420190\pi$
$684$	0	0
$685$	1924.00	2.80876
$686$	0	0
$687$	0	0
$688$	−2030.00	−2.95058
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−879.755	−1.26583
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	0	0	1.00000			$0$
$701$	0	0	−1.00000			$\pi$
$702$	−1265.55	−1.80278
$703$	0	0
$704$	−7.21110	−0.0102430
$705$	−2028.00	−2.87660
$706$	−338.000	−0.478754
$707$	0	0
$708$	194.700	0.275000
$709$	0	0	1.00000			$0$
$709$	0	0	−1.00000			$\pi$
$710$	−2062.38	−2.90475
$711$	450.000	0.632911
$712$	1430.00	2.00843
$713$	0	0
$714$	0	0
$715$	676.000	0.945455
$716$	0	0
$717$	1319.63	1.84049
$718$	286.000	0.398329
$719$	0	0	1.00000			$0$
$719$	0	0	−1.00000			$\pi$
$720$	−1882.10	−2.61402
$721$	0	0
$722$	1301.60	1.80278
$723$	0	0
$724$	−2358.00	−3.25691
$725$	0	0
$726$	746.349	1.02803
$727$	−1246.00	−1.71389	−0.856946	−	0.515406i	$-0.827641\pi$
$727$	−1246.00	−1.71389	−0.856946	+	0.515406i	$0.827641\pi$
$728$	0	0
$729$	729.000	1.00000
$730$	0	0
$731$	0	0
$732$	1890.00	2.58197
$733$	0	0	1.00000			$0$
$733$	0	0	−1.00000			$\pi$
$734$	−2415.72	−3.29117
$735$	1060.03	1.44222
$736$	0	0
$737$	0	0
$738$	2574.00	3.48780
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−266.811	−0.359099	−0.179550	−	0.983749i	$-0.557464\pi$
$743$	−266.811	−0.359099	−0.179550	+	0.983749i	$0.557464\pi$
$744$	0	0
$745$	52.0000	0.0697987
$746$	1997.48	2.67758
$747$	1492.70	1.99826
$748$	0	0
$749$	0	0
$750$	156.000	0.208000
$751$	98.0000	0.130493	0.0652463	−	0.997869i	$-0.479217\pi$
$751$	98.0000	0.130493	0.0652463	+	0.997869i	$0.479217\pi$
$752$	−2718.59	−3.61514
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−214.000	−0.282695	−0.141347	−	0.989960i	$-0.545143\pi$
$757$	−214.000	−0.282695	−0.141347	+	0.989960i	$0.545143\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1305.21	−1.71512	−0.857562	−	0.514380i	$-0.828022\pi$
$761$	−1305.21	−1.71512	−0.857562	+	0.514380i	$0.828022\pi$
$762$	497.566	0.652974
$763$	0	0
$764$	0	0
$765$	0	0
$766$	2158.00	2.81723
$767$	−93.7443	−0.122222
$768$	1377.00	1.79297
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−699.477	−0.904886	−0.452443	−	0.891793i	$-0.649447\pi$
$773$	−699.477	−0.904886	−0.452443	+	0.891793i	$0.649447\pi$
$774$	−2271.50	−2.93475
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	2531.10	3.24500
$781$	−572.000	−0.732394
$782$	0	0
$783$	0	0
$784$	1421.00	1.81250
$785$	2235.44	2.84770
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	3050.30	3.87093
$789$	0	0
$790$	−1300.00	−1.64557
$791$	0	0
$792$	−1170.00	−1.47727
$793$	−910.000	−1.14754
$794$	0	0
$795$	0	0
$796$	−1710.00	−2.14824
$797$	0	0	1.00000			$0$
$797$	0	0	−1.00000			$\pi$
$798$	0	0
$799$	0	0
$800$	876.149	1.09519
$801$	713.899	0.891260
$802$	−2834.00	−3.53367
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	−2106.00	−2.60000
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	−5148.00	−6.27805
$821$	−7.21110	−0.00878332	−0.00439166	−	0.999990i	$-0.501398\pi$
$821$	−7.21110	−0.00878332	−0.00439166	+	0.999990i	$0.501398\pi$
$822$	2886.00	3.51095
$823$	1490.00	1.81045	0.905225	−	0.424933i	$-0.139702\pi$
$823$	1490.00	1.81045	0.905225	+	0.424933i	$0.139702\pi$
$824$	901.388	1.09392
$825$	584.099	0.707999
$826$	0	0
$827$	−1391.74	−1.68288	−0.841441	−	0.540350i	$-0.818292\pi$
$827$	−1391.74	−1.68288	−0.841441	+	0.540350i	$0.818292\pi$
$828$	0	0
$829$	890.000	1.07358	0.536791	−	0.843715i	$-0.319636\pi$
$829$	890.000	1.07358	0.536791	+	0.843715i	$0.319636\pi$
$830$	−4312.24	−5.19547
$831$	210.000	0.252708
$832$	13.0000	0.0156250
$833$	0	0
$834$	−1319.63	−1.58229
$835$	1924.00	2.30419
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−1651.34	−1.96823	−0.984114	−	0.177540i	$-0.943186\pi$
$839$	−1651.34	−1.96823	−0.984114	+	0.177540i	$0.943186\pi$
$840$	0	0
$841$	841.000	1.00000
$842$	0	0
$843$	−1276.37	−1.51407
$844$	3690.00	4.37204
$845$	−1218.68	−1.44222
$846$	−3042.00	−3.59574
$847$	0	0
$848$	0	0
$849$	−798.000	−0.939929
$850$	0	0
$851$	0	0
$852$	−2141.70	−2.51373
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	1014.00	1.18182
$859$	1610.00	1.87427	0.937136	−	0.348964i	$-0.113466\pi$
$859$	1610.00	1.87427	0.937136	+	0.348964i	$0.113466\pi$
$860$	4542.99	5.28255
$861$	0	0
$862$	−1586.00	−1.83991
$863$	1636.92	1.89678	0.948390	−	0.317108i	$-0.102712\pi$
$863$	1636.92	1.89678	0.948390	+	0.317108i	$0.102712\pi$
$864$	−876.149	−1.01406
$865$	0	0
$866$	1564.81	1.80694
$867$	−867.000	−1.00000
$868$	0	0
$869$	−360.555	−0.414908
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	2603.21	2.96493
$879$	−1016.77	−1.15673
$880$	1508.00	1.71364
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	1590.05	1.80278
$883$	−934.000	−1.05776	−0.528879	−	0.848697i	$-0.677387\pi$
$883$	−934.000	−1.05776	−0.528879	+	0.848697i	$0.677387\pi$
$884$	0	0
$885$	−156.000	−0.176271
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	0	0
$890$	−2062.38	−2.31728
$891$	−584.099	−0.655555
$892$	0	0
$893$	0	0
$894$	78.0000	0.0872483
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−1586.00	−1.76615
$899$	0	0
$900$	2187.00	2.43000
$901$	0	0
$902$	−2062.38	−2.28645
$903$	0	0
$904$	0	0
$905$	1889.31	2.08763
$906$	0	0
$907$	1514.00	1.66924	0.834620	−	0.550827i	$-0.185687\pi$
$907$	1514.00	1.66924	0.834620	+	0.550827i	$0.185687\pi$
$908$	1492.70	1.64394
$909$	0	0
$910$	0	0
$911$	0	0	1.00000			$0$
$911$	0	0	−1.00000			$\pi$
$912$	0	0
$913$	−1196.00	−1.30997
$914$	0	0
$915$	−1514.33	−1.65501
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−1630.00	−1.77367	−0.886834	−	0.462089i	$-0.847100\pi$
$919$	−1630.00	−1.77367	−0.886834	+	0.462089i	$0.847100\pi$
$920$	0	0
$921$	0	0
$922$	−3146.00	−3.41215
$923$	1031.19	1.11721
$924$	0	0
$925$	0	0
$926$	0	0
$927$	450.000	0.485437
$928$	0	0
$929$	−786.010	−0.846082	−0.423041	−	0.906111i	$-0.639037\pi$
$929$	−786.010	−0.846082	−0.423041	+	0.906111i	$0.639037\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	2109.25	2.25347
$937$	674.000	0.719317	0.359658	−	0.933084i	$-0.382893\pi$
$937$	674.000	0.719317	0.359658	+	0.933084i	$0.382893\pi$
$938$	0	0
$939$	1722.00	1.83387
$940$	6084.00	6.47234
$941$	−1218.68	−1.29509	−0.647543	−	0.762029i	$-0.724203\pi$
$941$	−1218.68	−1.29509	−0.647543	+	0.762029i	$0.724203\pi$
$942$	3353.16	3.55962
$943$	0	0
$944$	−209.122	−0.221528
$945$	0	0
$946$	1820.00	1.92389
$947$	1204.25	1.27165	0.635826	−	0.771833i	$-0.280660\pi$
$947$	1204.25	1.27165	0.635826	+	0.771833i	$0.280660\pi$
$948$	−1350.00	−1.42405
$949$	0	0
$950$	0	0
$951$	1579.23	1.66060
$952$	0	0
$953$	0	0	1.00000			$0$
$953$	0	0	−1.00000			$\pi$
$954$	0	0
$955$	0	0
$956$	−3958.90	−4.14110
$957$	0	0
$958$	3406.00	3.55532
$959$	0	0
$960$	21.6333	0.0225347
$961$	961.000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	−1243.92	−1.28504
$969$	0	0
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	−2187.00	−2.25000
$973$	0	0
$974$	0	0
$975$	−1053.00	−1.08000
$976$	−2030.00	−2.07992
$977$	−1824.41	−1.86736	−0.933679	−	0.358111i	$-0.883421\pi$
$977$	−1824.41	−1.86736	−0.933679	+	0.358111i	$0.883421\pi$
$978$	0	0
$979$	−572.000	−0.584270
$980$	−3180.10	−3.24500
$981$	0	0
$982$	0	0
$983$	771.588	0.784932	0.392466	−	0.919767i	$-0.371622\pi$
$983$	771.588	0.784932	0.392466	+	0.919767i	$0.371622\pi$
$984$	−4290.00	−4.35976
$985$	−2444.00	−2.48122
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	1687.40	1.70444
$991$	−1918.00	−1.93542	−0.967709	−	0.252069i	$-0.918889\pi$
$991$	−1918.00	−1.93542	−0.967709	+	0.252069i	$0.918889\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1370.11	1.37699
$996$	−4478.09	−4.49608
$997$	1370.00	1.37412	0.687061	−	0.726600i	$-0.258900\pi$
$997$	1370.00	1.37412	0.687061	+	0.726600i	$0.258900\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.3.d.a.38.2	yes	2
3.2	odd	2	inner	39.3.d.a.38.1	✓	2
4.3	odd	2		624.3.l.c.545.1		2
12.11	even	2		624.3.l.c.545.2		2
13.5	odd	4		507.3.c.c.170.1		2
13.8	odd	4		507.3.c.c.170.2		2
13.12	even	2	inner	39.3.d.a.38.1	✓	2
39.5	even	4		507.3.c.c.170.2		2
39.8	even	4		507.3.c.c.170.1		2
39.38	odd	2	CM	39.3.d.a.38.2	yes	2
52.51	odd	2		624.3.l.c.545.2		2
156.155	even	2		624.3.l.c.545.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.3.d.a.38.1	✓	2	3.2	odd	2	inner
39.3.d.a.38.1	✓	2	13.12	even	2	inner
39.3.d.a.38.2	yes	2	1.1	even	1	trivial
39.3.d.a.38.2	yes	2	39.38	odd	2	CM
507.3.c.c.170.1		2	13.5	odd	4
507.3.c.c.170.1		2	39.8	even	4
507.3.c.c.170.2		2	13.8	odd	4
507.3.c.c.170.2		2	39.5	even	4
624.3.l.c.545.1		2	4.3	odd	2
624.3.l.c.545.1		2	156.155	even	2
624.3.l.c.545.2		2	12.11	even	2
624.3.l.c.545.2		2	52.51	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:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	39.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.60555 q^{2} -3.00000 q^{3} +9.00000 q^{4} -7.21110 q^{5} -10.8167 q^{6} +18.0278 q^{8} +9.00000 q^{9} +O(q^{10})\)	\(q+3.60555 q^{2} -3.00000 q^{3} +9.00000 q^{4} -7.21110 q^{5} -10.8167 q^{6} +18.0278 q^{8} +9.00000 q^{9} -26.0000 q^{10} -7.21110 q^{11} -27.0000 q^{12} +13.0000 q^{13} +21.6333 q^{15} +29.0000 q^{16} +32.4500 q^{18} -64.8999 q^{20} -26.0000 q^{22} -54.0833 q^{24} +27.0000 q^{25} +46.8722 q^{26} -27.0000 q^{27} +78.0000 q^{30} +32.4500 q^{32} +21.6333 q^{33} +81.0000 q^{36} -39.0000 q^{39} -130.000 q^{40} +79.3221 q^{41} -70.0000 q^{43} -64.8999 q^{44} -64.8999 q^{45} -93.7443 q^{47} -87.0000 q^{48} +49.0000 q^{49} +97.3499 q^{50} +117.000 q^{52} -97.3499 q^{54} +52.0000 q^{55} -7.21110 q^{59} +194.700 q^{60} -70.0000 q^{61} +1.00000 q^{64} -93.7443 q^{65} +78.0000 q^{66} +79.3221 q^{71} +162.250 q^{72} -81.0000 q^{75} -140.616 q^{78} +50.0000 q^{79} -209.122 q^{80} +81.0000 q^{81} +286.000 q^{82} +165.855 q^{83} -252.389 q^{86} -130.000 q^{88} +79.3221 q^{89} -234.000 q^{90} -338.000 q^{94} -97.3499 q^{96} +176.672 q^{98} -64.8999 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 18 q^{4} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 + 18 * q^4 + 18 * q^9	\( 2 q - 6 q^{3} + 18 q^{4} + 18 q^{9} - 52 q^{10} - 54 q^{12} + 26 q^{13} + 58 q^{16} - 52 q^{22} + 54 q^{25} - 54 q^{27} + 156 q^{30} + 162 q^{36} - 78 q^{39} - 260 q^{40} - 140 q^{43} - 174 q^{48} + 98 q^{49} + 234 q^{52} + 104 q^{55} - 140 q^{61} + 2 q^{64} + 156 q^{66} - 162 q^{75} + 100 q^{79} + 162 q^{81} + 572 q^{82} - 260 q^{88} - 468 q^{90} - 676 q^{94}+O(q^{100}) \) 2 * q - 6 * q^3 + 18 * q^4 + 18 * q^9 - 52 * q^10 - 54 * q^12 + 26 * q^13 + 58 * q^16 - 52 * q^22 + 54 * q^25 - 54 * q^27 + 156 * q^30 + 162 * q^36 - 78 * q^39 - 260 * q^40 - 140 * q^43 - 174 * q^48 + 98 * q^49 + 234 * q^52 + 104 * q^55 - 140 * q^61 + 2 * q^64 + 156 * q^66 - 162 * q^75 + 100 * q^79 + 162 * q^81 + 572 * q^82 - 260 * q^88 - 468 * q^90 - 676 * q^94

Introduction

L-functions

Modular forms

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