LMFDB - Embedded newform 7938.2.a.bm.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7938,2,Mod(1,7938)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7938, 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("7938.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7938 = 2 \cdot 3^{4} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7938.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:	$63.3852491245$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 126)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.44949$ of defining polynomial
Character	$\chi$	$=$	7938.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -1.44949 q^{5} -1.00000 q^{8} +1.44949 q^{10} -2.00000 q^{11} -4.89898 q^{13} +1.00000 q^{16} +2.00000 q^{17} -2.55051 q^{19} -1.44949 q^{20} +2.00000 q^{22} +1.00000 q^{23} -2.89898 q^{25} +4.89898 q^{26} +6.89898 q^{29} -6.00000 q^{31} -1.00000 q^{32} -2.00000 q^{34} +11.7980 q^{37} +2.55051 q^{38} +1.44949 q^{40} +9.79796 q^{41} +6.89898 q^{43} -2.00000 q^{44} -1.00000 q^{46} +9.79796 q^{47} +2.89898 q^{50} -4.89898 q^{52} +10.8990 q^{53} +2.89898 q^{55} -6.89898 q^{58} -2.00000 q^{59} -6.55051 q^{61} +6.00000 q^{62} +1.00000 q^{64} +7.10102 q^{65} -12.8990 q^{67} +2.00000 q^{68} -0.101021 q^{71} +6.89898 q^{73} -11.7980 q^{74} -2.55051 q^{76} -1.89898 q^{79} -1.44949 q^{80} -9.79796 q^{82} +2.00000 q^{83} -2.89898 q^{85} -6.89898 q^{86} +2.00000 q^{88} -16.8990 q^{89} +1.00000 q^{92} -9.79796 q^{94} +3.69694 q^{95} -2.89898 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8} - 2 q^{10} - 4 q^{11} + 2 q^{16} + 4 q^{17} - 10 q^{19} + 2 q^{20} + 4 q^{22} + 2 q^{23} + 4 q^{25} + 4 q^{29} - 12 q^{31} - 2 q^{32} - 4 q^{34} + 4 q^{37} + 10 q^{38}+ \cdots + 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^8 - 2 * q^10 - 4 * q^11 + 2 * q^16 + 4 * q^17 - 10 * q^19 + 2 * q^20 + 4 * q^22 + 2 * q^23 + 4 * q^25 + 4 * q^29 - 12 * q^31 - 2 * q^32 - 4 * q^34 + 4 * q^37 + 10 * q^38 - 2 * q^40 + 4 * q^43 - 4 * q^44 - 2 * q^46 - 4 * q^50 + 12 * q^53 - 4 * q^55 - 4 * q^58 - 4 * q^59 - 18 * q^61 + 12 * q^62 + 2 * q^64 + 24 * q^65 - 16 * q^67 + 4 * q^68 - 10 * q^71 + 4 * q^73 - 4 * q^74 - 10 * q^76 + 6 * q^79 + 2 * q^80 + 4 * q^83 + 4 * q^85 - 4 * q^86 + 4 * q^88 - 24 * q^89 + 2 * q^92 - 22 * q^95 + 4 * q^97

Coefficient data

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

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

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

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.44949	−0.648232	−0.324116	−	0.946017i	$-0.605067\pi$
$5$	−1.44949	−0.648232	−0.324116	+	0.946017i	$0.605067\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.44949	0.458369
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	0	0
$13$	−4.89898	−1.35873	−0.679366	−	0.733799i	$-0.737745\pi$
$13$	−4.89898	−1.35873	−0.679366	+	0.733799i	$0.737745\pi$
$14$	0	0
$15$	0	0
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	−2.55051	−0.585127	−0.292564	−	0.956246i	$-0.594508\pi$
$19$	−2.55051	−0.585127	−0.292564	+	0.956246i	$0.594508\pi$
$20$	−1.44949	−0.324116
$21$	0	0
$22$	2.00000	0.426401
$23$	1.00000	0.208514	0.104257	−	0.994550i	$-0.466753\pi$
$23$	1.00000	0.208514	0.104257	+	0.994550i	$0.466753\pi$
$24$	0	0
$25$	−2.89898	−0.579796
$26$	4.89898	0.960769
$27$	0	0
$28$	0	0
$29$	6.89898	1.28111	0.640554	−	0.767913i	$-0.278705\pi$
$29$	6.89898	1.28111	0.640554	+	0.767913i	$0.278705\pi$
$30$	0	0
$31$	−6.00000	−1.07763	−0.538816	−	0.842424i	$-0.681128\pi$
$31$	−6.00000	−1.07763	−0.538816	+	0.842424i	$0.681128\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−2.00000	−0.342997
$35$	0	0
$36$	0	0
$37$	11.7980	1.93957	0.969786	−	0.243956i	$-0.0784453\pi$
$37$	11.7980	1.93957	0.969786	+	0.243956i	$0.0784453\pi$
$38$	2.55051	0.413747
$39$	0	0
$40$	1.44949	0.229184
$41$	9.79796	1.53018	0.765092	−	0.643921i	$-0.222693\pi$
$41$	9.79796	1.53018	0.765092	+	0.643921i	$0.222693\pi$
$42$	0	0
$43$	6.89898	1.05208	0.526042	−	0.850458i	$-0.323675\pi$
$43$	6.89898	1.05208	0.526042	+	0.850458i	$0.323675\pi$
$44$	−2.00000	−0.301511
$45$	0	0
$46$	−1.00000	−0.147442
$47$	9.79796	1.42918	0.714590	−	0.699544i	$-0.246613\pi$
$47$	9.79796	1.42918	0.714590	+	0.699544i	$0.246613\pi$
$48$	0	0
$49$	0	0
$50$	2.89898	0.409978
$51$	0	0
$52$	−4.89898	−0.679366
$53$	10.8990	1.49709	0.748545	−	0.663084i	$-0.230753\pi$
$53$	10.8990	1.49709	0.748545	+	0.663084i	$0.230753\pi$
$54$	0	0
$55$	2.89898	0.390898
$56$	0	0
$57$	0	0
$58$	−6.89898	−0.905880
$59$	−2.00000	−0.260378	−0.130189	−	0.991489i	$-0.541558\pi$
$59$	−2.00000	−0.260378	−0.130189	+	0.991489i	$0.541558\pi$
$60$	0	0
$61$	−6.55051	−0.838707	−0.419353	−	0.907823i	$-0.637743\pi$
$61$	−6.55051	−0.838707	−0.419353	+	0.907823i	$0.637743\pi$
$62$	6.00000	0.762001
$63$	0	0
$64$	1.00000	0.125000
$65$	7.10102	0.880773
$66$	0	0
$67$	−12.8990	−1.57586	−0.787931	−	0.615764i	$-0.788847\pi$
$67$	−12.8990	−1.57586	−0.787931	+	0.615764i	$0.788847\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	0	0
$71$	−0.101021	−0.0119889	−0.00599446	−	0.999982i	$-0.501908\pi$
$71$	−0.101021	−0.0119889	−0.00599446	+	0.999982i	$0.501908\pi$
$72$	0	0
$73$	6.89898	0.807464	0.403732	−	0.914877i	$-0.367713\pi$
$73$	6.89898	0.807464	0.403732	+	0.914877i	$0.367713\pi$
$74$	−11.7980	−1.37148
$75$	0	0
$76$	−2.55051	−0.292564
$77$	0	0
$78$	0	0
$79$	−1.89898	−0.213652	−0.106826	−	0.994278i	$-0.534069\pi$
$79$	−1.89898	−0.213652	−0.106826	+	0.994278i	$0.534069\pi$
$80$	−1.44949	−0.162058
$81$	0	0
$82$	−9.79796	−1.08200
$83$	2.00000	0.219529	0.109764	−	0.993958i	$-0.464990\pi$
$83$	2.00000	0.219529	0.109764	+	0.993958i	$0.464990\pi$
$84$	0	0
$85$	−2.89898	−0.314438
$86$	−6.89898	−0.743936
$87$	0	0
$88$	2.00000	0.213201
$89$	−16.8990	−1.79129	−0.895644	−	0.444771i	$-0.853285\pi$
$89$	−16.8990	−1.79129	−0.895644	+	0.444771i	$0.853285\pi$
$90$	0	0
$91$	0	0
$92$	1.00000	0.104257
$93$	0	0
$94$	−9.79796	−1.01058
$95$	3.69694	0.379298
$96$	0	0
$97$	−2.89898	−0.294347	−0.147173	−	0.989111i	$-0.547018\pi$
$97$	−2.89898	−0.294347	−0.147173	+	0.989111i	$0.547018\pi$
$98$	0	0
$99$	0	0
$100$	−2.89898	−0.289898
$101$	−17.2474	−1.71619	−0.858093	−	0.513495i	$-0.828351\pi$
$101$	−17.2474	−1.71619	−0.858093	+	0.513495i	$0.828351\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	4.89898	0.480384
$105$	0	0
$106$	−10.8990	−1.05860
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	12.6969	1.21615	0.608073	−	0.793881i	$-0.291943\pi$
$109$	12.6969	1.21615	0.608073	+	0.793881i	$0.291943\pi$
$110$	−2.89898	−0.276407
$111$	0	0
$112$	0	0
$113$	6.10102	0.573936	0.286968	−	0.957940i	$-0.407353\pi$
$113$	6.10102	0.573936	0.286968	+	0.957940i	$0.407353\pi$
$114$	0	0
$115$	−1.44949	−0.135166
$116$	6.89898	0.640554
$117$	0	0
$118$	2.00000	0.184115
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	6.55051	0.593055
$123$	0	0
$124$	−6.00000	−0.538816
$125$	11.4495	1.02407
$126$	0	0
$127$	−3.00000	−0.266207	−0.133103	−	0.991102i	$-0.542494\pi$
$127$	−3.00000	−0.266207	−0.133103	+	0.991102i	$0.542494\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−7.10102	−0.622801
$131$	−8.55051	−0.747062	−0.373531	−	0.927618i	$-0.621853\pi$
$131$	−8.55051	−0.747062	−0.373531	+	0.927618i	$0.621853\pi$
$132$	0	0
$133$	0	0
$134$	12.8990	1.11430
$135$	0	0
$136$	−2.00000	−0.171499
$137$	−7.79796	−0.666225	−0.333112	−	0.942887i	$-0.608099\pi$
$137$	−7.79796	−0.666225	−0.333112	+	0.942887i	$0.608099\pi$
$138$	0	0
$139$	−4.55051	−0.385969	−0.192985	−	0.981202i	$-0.561817\pi$
$139$	−4.55051	−0.385969	−0.192985	+	0.981202i	$0.561817\pi$
$140$	0	0
$141$	0	0
$142$	0.101021	0.00847745
$143$	9.79796	0.819346
$144$	0	0
$145$	−10.0000	−0.830455
$146$	−6.89898	−0.570964
$147$	0	0
$148$	11.7980	0.969786
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−5.00000	−0.406894	−0.203447	−	0.979086i	$-0.565214\pi$
$151$	−5.00000	−0.406894	−0.203447	+	0.979086i	$0.565214\pi$
$152$	2.55051	0.206874
$153$	0	0
$154$	0	0
$155$	8.69694	0.698555
$156$	0	0
$157$	8.34847	0.666280	0.333140	−	0.942877i	$-0.391892\pi$
$157$	8.34847	0.666280	0.333140	+	0.942877i	$0.391892\pi$
$158$	1.89898	0.151075
$159$	0	0
$160$	1.44949	0.114592
$161$	0	0
$162$	0	0
$163$	−19.7980	−1.55070	−0.775348	−	0.631534i	$-0.782425\pi$
$163$	−19.7980	−1.55070	−0.775348	+	0.631534i	$0.782425\pi$
$164$	9.79796	0.765092
$165$	0	0
$166$	−2.00000	−0.155230
$167$	−10.6969	−0.827754	−0.413877	−	0.910333i	$-0.635826\pi$
$167$	−10.6969	−0.827754	−0.413877	+	0.910333i	$0.635826\pi$
$168$	0	0
$169$	11.0000	0.846154
$170$	2.89898	0.222342
$171$	0	0
$172$	6.89898	0.526042
$173$	−3.10102	−0.235766	−0.117883	−	0.993027i	$-0.537611\pi$
$173$	−3.10102	−0.235766	−0.117883	+	0.993027i	$0.537611\pi$
$174$	0	0
$175$	0	0
$176$	−2.00000	−0.150756
$177$	0	0
$178$	16.8990	1.26663
$179$	−20.6969	−1.54696	−0.773481	−	0.633820i	$-0.781486\pi$
$179$	−20.6969	−1.54696	−0.773481	+	0.633820i	$0.781486\pi$
$180$	0	0
$181$	10.3485	0.769196	0.384598	−	0.923084i	$-0.374340\pi$
$181$	10.3485	0.769196	0.384598	+	0.923084i	$0.374340\pi$
$182$	0	0
$183$	0	0
$184$	−1.00000	−0.0737210
$185$	−17.1010	−1.25729
$186$	0	0
$187$	−4.00000	−0.292509
$188$	9.79796	0.714590
$189$	0	0
$190$	−3.69694	−0.268204
$191$	−4.10102	−0.296739	−0.148370	−	0.988932i	$-0.547403\pi$
$191$	−4.10102	−0.296739	−0.148370	+	0.988932i	$0.547403\pi$
$192$	0	0
$193$	−17.8990	−1.28840	−0.644198	−	0.764858i	$-0.722809\pi$
$193$	−17.8990	−1.28840	−0.644198	+	0.764858i	$0.722809\pi$
$194$	2.89898	0.208135
$195$	0	0
$196$	0	0
$197$	−16.6969	−1.18961	−0.594804	−	0.803871i	$-0.702770\pi$
$197$	−16.6969	−1.18961	−0.594804	+	0.803871i	$0.702770\pi$
$198$	0	0
$199$	2.89898	0.205503	0.102752	−	0.994707i	$-0.467235\pi$
$199$	2.89898	0.205503	0.102752	+	0.994707i	$0.467235\pi$
$200$	2.89898	0.204989
$201$	0	0
$202$	17.2474	1.21353
$203$	0	0
$204$	0	0
$205$	−14.2020	−0.991914
$206$	14.0000	0.975426
$207$	0	0
$208$	−4.89898	−0.339683
$209$	5.10102	0.352845
$210$	0	0
$211$	12.8990	0.888002	0.444001	−	0.896026i	$-0.353559\pi$
$211$	12.8990	0.888002	0.444001	+	0.896026i	$0.353559\pi$
$212$	10.8990	0.748545
$213$	0	0
$214$	−12.0000	−0.820303
$215$	−10.0000	−0.681994
$216$	0	0
$217$	0	0
$218$	−12.6969	−0.859945
$219$	0	0
$220$	2.89898	0.195449
$221$	−9.79796	−0.659082
$222$	0	0
$223$	11.1010	0.743379	0.371690	−	0.928357i	$-0.378779\pi$
$223$	11.1010	0.743379	0.371690	+	0.928357i	$0.378779\pi$
$224$	0	0
$225$	0	0
$226$	−6.10102	−0.405834
$227$	−5.44949	−0.361695	−0.180848	−	0.983511i	$-0.557884\pi$
$227$	−5.44949	−0.361695	−0.180848	+	0.983511i	$0.557884\pi$
$228$	0	0
$229$	−1.24745	−0.0824337	−0.0412169	−	0.999150i	$-0.513123\pi$
$229$	−1.24745	−0.0824337	−0.0412169	+	0.999150i	$0.513123\pi$
$230$	1.44949	0.0955765
$231$	0	0
$232$	−6.89898	−0.452940
$233$	7.00000	0.458585	0.229293	−	0.973358i	$-0.426359\pi$
$233$	7.00000	0.458585	0.229293	+	0.973358i	$0.426359\pi$
$234$	0	0
$235$	−14.2020	−0.926439
$236$	−2.00000	−0.130189
$237$	0	0
$238$	0	0
$239$	−6.79796	−0.439723	−0.219862	−	0.975531i	$-0.570561\pi$
$239$	−6.79796	−0.439723	−0.219862	+	0.975531i	$0.570561\pi$
$240$	0	0
$241$	−0.898979	−0.0579084	−0.0289542	−	0.999581i	$-0.509218\pi$
$241$	−0.898979	−0.0579084	−0.0289542	+	0.999581i	$0.509218\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	−6.55051	−0.419353
$245$	0	0
$246$	0	0
$247$	12.4949	0.795031
$248$	6.00000	0.381000
$249$	0	0
$250$	−11.4495	−0.724129
$251$	17.4495	1.10140	0.550701	−	0.834703i	$-0.314360\pi$
$251$	17.4495	1.10140	0.550701	+	0.834703i	$0.314360\pi$
$252$	0	0
$253$	−2.00000	−0.125739
$254$	3.00000	0.188237
$255$	0	0
$256$	1.00000	0.0625000
$257$	8.20204	0.511629	0.255815	−	0.966726i	$-0.417656\pi$
$257$	8.20204	0.511629	0.255815	+	0.966726i	$0.417656\pi$
$258$	0	0
$259$	0	0
$260$	7.10102	0.440387
$261$	0	0
$262$	8.55051	0.528252
$263$	−25.8990	−1.59700	−0.798500	−	0.601995i	$-0.794373\pi$
$263$	−25.8990	−1.59700	−0.798500	+	0.601995i	$0.794373\pi$
$264$	0	0
$265$	−15.7980	−0.970461
$266$	0	0
$267$	0	0
$268$	−12.8990	−0.787931
$269$	−18.3485	−1.11873	−0.559363	−	0.828923i	$-0.688954\pi$
$269$	−18.3485	−1.11873	−0.559363	+	0.828923i	$0.688954\pi$
$270$	0	0
$271$	−7.10102	−0.431356	−0.215678	−	0.976465i	$-0.569196\pi$
$271$	−7.10102	−0.431356	−0.215678	+	0.976465i	$0.569196\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	7.79796	0.471092
$275$	5.79796	0.349630
$276$	0	0
$277$	−18.6969	−1.12339	−0.561695	−	0.827344i	$-0.689851\pi$
$277$	−18.6969	−1.12339	−0.561695	+	0.827344i	$0.689851\pi$
$278$	4.55051	0.272921
$279$	0	0
$280$	0	0
$281$	19.0000	1.13344	0.566722	−	0.823909i	$-0.308211\pi$
$281$	19.0000	1.13344	0.566722	+	0.823909i	$0.308211\pi$
$282$	0	0
$283$	25.4495	1.51282	0.756408	−	0.654101i	$-0.226953\pi$
$283$	25.4495	1.51282	0.756408	+	0.654101i	$0.226953\pi$
$284$	−0.101021	−0.00599446
$285$	0	0
$286$	−9.79796	−0.579365
$287$	0	0
$288$	0	0
$289$	−13.0000	−0.764706
$290$	10.0000	0.587220
$291$	0	0
$292$	6.89898	0.403732
$293$	2.75255	0.160806	0.0804029	−	0.996762i	$-0.474379\pi$
$293$	2.75255	0.160806	0.0804029	+	0.996762i	$0.474379\pi$
$294$	0	0
$295$	2.89898	0.168785
$296$	−11.7980	−0.685742
$297$	0	0
$298$	−6.00000	−0.347571
$299$	−4.89898	−0.283315
$300$	0	0
$301$	0	0
$302$	5.00000	0.287718
$303$	0	0
$304$	−2.55051	−0.146282
$305$	9.49490	0.543676
$306$	0	0
$307$	−25.2474	−1.44095	−0.720474	−	0.693482i	$-0.756076\pi$
$307$	−25.2474	−1.44095	−0.720474	+	0.693482i	$0.756076\pi$
$308$	0	0
$309$	0	0
$310$	−8.69694	−0.493953
$311$	30.6969	1.74066	0.870332	−	0.492466i	$-0.163904\pi$
$311$	30.6969	1.74066	0.870332	+	0.492466i	$0.163904\pi$
$312$	0	0
$313$	4.69694	0.265487	0.132743	−	0.991150i	$-0.457621\pi$
$313$	4.69694	0.265487	0.132743	+	0.991150i	$0.457621\pi$
$314$	−8.34847	−0.471131
$315$	0	0
$316$	−1.89898	−0.106826
$317$	−20.6969	−1.16246	−0.581228	−	0.813741i	$-0.697428\pi$
$317$	−20.6969	−1.16246	−0.581228	+	0.813741i	$0.697428\pi$
$318$	0	0
$319$	−13.7980	−0.772537
$320$	−1.44949	−0.0810289
$321$	0	0
$322$	0	0
$323$	−5.10102	−0.283828
$324$	0	0
$325$	14.2020	0.787787
$326$	19.7980	1.09651
$327$	0	0
$328$	−9.79796	−0.541002
$329$	0	0
$330$	0	0
$331$	4.69694	0.258167	0.129084	−	0.991634i	$-0.458796\pi$
$331$	4.69694	0.258167	0.129084	+	0.991634i	$0.458796\pi$
$332$	2.00000	0.109764
$333$	0	0
$334$	10.6969	0.585310
$335$	18.6969	1.02152
$336$	0	0
$337$	−23.3939	−1.27435	−0.637173	−	0.770721i	$-0.719896\pi$
$337$	−23.3939	−1.27435	−0.637173	+	0.770721i	$0.719896\pi$
$338$	−11.0000	−0.598321
$339$	0	0
$340$	−2.89898	−0.157219
$341$	12.0000	0.649836
$342$	0	0
$343$	0	0
$344$	−6.89898	−0.371968
$345$	0	0
$346$	3.10102	0.166712
$347$	−19.5959	−1.05196	−0.525982	−	0.850496i	$-0.676302\pi$
$347$	−19.5959	−1.05196	−0.525982	+	0.850496i	$0.676302\pi$
$348$	0	0
$349$	−11.1010	−0.594224	−0.297112	−	0.954843i	$-0.596023\pi$
$349$	−11.1010	−0.594224	−0.297112	+	0.954843i	$0.596023\pi$
$350$	0	0
$351$	0	0
$352$	2.00000	0.106600
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	0.146428	0.00777160
$356$	−16.8990	−0.895644
$357$	0	0
$358$	20.6969	1.09387
$359$	8.79796	0.464339	0.232169	−	0.972675i	$-0.425418\pi$
$359$	8.79796	0.464339	0.232169	+	0.972675i	$0.425418\pi$
$360$	0	0
$361$	−12.4949	−0.657626
$362$	−10.3485	−0.543903
$363$	0	0
$364$	0	0
$365$	−10.0000	−0.523424
$366$	0	0
$367$	13.7980	0.720248	0.360124	−	0.932905i	$-0.382734\pi$
$367$	13.7980	0.720248	0.360124	+	0.932905i	$0.382734\pi$
$368$	1.00000	0.0521286
$369$	0	0
$370$	17.1010	0.889040
$371$	0	0
$372$	0	0
$373$	−6.89898	−0.357216	−0.178608	−	0.983920i	$-0.557159\pi$
$373$	−6.89898	−0.357216	−0.178608	+	0.983920i	$0.557159\pi$
$374$	4.00000	0.206835
$375$	0	0
$376$	−9.79796	−0.505291
$377$	−33.7980	−1.74068
$378$	0	0
$379$	22.4949	1.15549	0.577743	−	0.816219i	$-0.303934\pi$
$379$	22.4949	1.15549	0.577743	+	0.816219i	$0.303934\pi$
$380$	3.69694	0.189649
$381$	0	0
$382$	4.10102	0.209826
$383$	−2.89898	−0.148131	−0.0740655	−	0.997253i	$-0.523597\pi$
$383$	−2.89898	−0.148131	−0.0740655	+	0.997253i	$0.523597\pi$
$384$	0	0
$385$	0	0
$386$	17.8990	0.911034
$387$	0	0
$388$	−2.89898	−0.147173
$389$	24.8990	1.26243	0.631214	−	0.775609i	$-0.282557\pi$
$389$	24.8990	1.26243	0.631214	+	0.775609i	$0.282557\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	0	0
$393$	0	0
$394$	16.6969	0.841180
$395$	2.75255	0.138496
$396$	0	0
$397$	−38.6969	−1.94214	−0.971072	−	0.238788i	$-0.923250\pi$
$397$	−38.6969	−1.94214	−0.971072	+	0.238788i	$0.923250\pi$
$398$	−2.89898	−0.145313
$399$	0	0
$400$	−2.89898	−0.144949
$401$	19.8990	0.993708	0.496854	−	0.867834i	$-0.334489\pi$
$401$	19.8990	0.993708	0.496854	+	0.867834i	$0.334489\pi$
$402$	0	0
$403$	29.3939	1.46421
$404$	−17.2474	−0.858093
$405$	0	0
$406$	0	0
$407$	−23.5959	−1.16961
$408$	0	0
$409$	13.7980	0.682265	0.341133	−	0.940015i	$-0.389189\pi$
$409$	13.7980	0.682265	0.341133	+	0.940015i	$0.389189\pi$
$410$	14.2020	0.701389
$411$	0	0
$412$	−14.0000	−0.689730
$413$	0	0
$414$	0	0
$415$	−2.89898	−0.142305
$416$	4.89898	0.240192
$417$	0	0
$418$	−5.10102	−0.249499
$419$	29.4495	1.43870	0.719351	−	0.694647i	$-0.244439\pi$
$419$	29.4495	1.43870	0.719351	+	0.694647i	$0.244439\pi$
$420$	0	0
$421$	22.8990	1.11603	0.558014	−	0.829832i	$-0.311564\pi$
$421$	22.8990	1.11603	0.558014	+	0.829832i	$0.311564\pi$
$422$	−12.8990	−0.627912
$423$	0	0
$424$	−10.8990	−0.529301
$425$	−5.79796	−0.281242
$426$	0	0
$427$	0	0
$428$	12.0000	0.580042
$429$	0	0
$430$	10.0000	0.482243
$431$	−31.5959	−1.52192	−0.760961	−	0.648798i	$-0.775272\pi$
$431$	−31.5959	−1.52192	−0.760961	+	0.648798i	$0.775272\pi$
$432$	0	0
$433$	7.79796	0.374746	0.187373	−	0.982289i	$-0.440003\pi$
$433$	7.79796	0.374746	0.187373	+	0.982289i	$0.440003\pi$
$434$	0	0
$435$	0	0
$436$	12.6969	0.608073
$437$	−2.55051	−0.122007
$438$	0	0
$439$	−2.20204	−0.105098	−0.0525488	−	0.998618i	$-0.516735\pi$
$439$	−2.20204	−0.105098	−0.0525488	+	0.998618i	$0.516735\pi$
$440$	−2.89898	−0.138203
$441$	0	0
$442$	9.79796	0.466041
$443$	14.8990	0.707872	0.353936	−	0.935270i	$-0.384843\pi$
$443$	14.8990	0.707872	0.353936	+	0.935270i	$0.384843\pi$
$444$	0	0
$445$	24.4949	1.16117
$446$	−11.1010	−0.525649
$447$	0	0
$448$	0	0
$449$	−20.5959	−0.971981	−0.485991	−	0.873964i	$-0.661541\pi$
$449$	−20.5959	−0.971981	−0.485991	+	0.873964i	$0.661541\pi$
$450$	0	0
$451$	−19.5959	−0.922736
$452$	6.10102	0.286968
$453$	0	0
$454$	5.44949	0.255757
$455$	0	0
$456$	0	0
$457$	−17.4949	−0.818377	−0.409188	−	0.912450i	$-0.634188\pi$
$457$	−17.4949	−0.818377	−0.409188	+	0.912450i	$0.634188\pi$
$458$	1.24745	0.0582895
$459$	0	0
$460$	−1.44949	−0.0675828
$461$	5.65153	0.263218	0.131609	−	0.991302i	$-0.457986\pi$
$461$	5.65153	0.263218	0.131609	+	0.991302i	$0.457986\pi$
$462$	0	0
$463$	3.69694	0.171811	0.0859057	−	0.996303i	$-0.472622\pi$
$463$	3.69694	0.171811	0.0859057	+	0.996303i	$0.472622\pi$
$464$	6.89898	0.320277
$465$	0	0
$466$	−7.00000	−0.324269
$467$	10.0000	0.462745	0.231372	−	0.972865i	$-0.425678\pi$
$467$	10.0000	0.462745	0.231372	+	0.972865i	$0.425678\pi$
$468$	0	0
$469$	0	0
$470$	14.2020	0.655091
$471$	0	0
$472$	2.00000	0.0920575
$473$	−13.7980	−0.634431
$474$	0	0
$475$	7.39388	0.339254
$476$	0	0
$477$	0	0
$478$	6.79796	0.310931
$479$	9.59592	0.438449	0.219224	−	0.975674i	$-0.429647\pi$
$479$	9.59592	0.438449	0.219224	+	0.975674i	$0.429647\pi$
$480$	0	0
$481$	−57.7980	−2.63536
$482$	0.898979	0.0409474
$483$	0	0
$484$	−7.00000	−0.318182
$485$	4.20204	0.190805
$486$	0	0
$487$	−36.3939	−1.64916	−0.824582	−	0.565742i	$-0.808590\pi$
$487$	−36.3939	−1.64916	−0.824582	+	0.565742i	$0.808590\pi$
$488$	6.55051	0.296528
$489$	0	0
$490$	0	0
$491$	−15.7980	−0.712952	−0.356476	−	0.934304i	$-0.616022\pi$
$491$	−15.7980	−0.712952	−0.356476	+	0.934304i	$0.616022\pi$
$492$	0	0
$493$	13.7980	0.621429
$494$	−12.4949	−0.562172
$495$	0	0
$496$	−6.00000	−0.269408
$497$	0	0
$498$	0	0
$499$	−25.3939	−1.13679	−0.568393	−	0.822757i	$-0.692435\pi$
$499$	−25.3939	−1.13679	−0.568393	+	0.822757i	$0.692435\pi$
$500$	11.4495	0.512037
$501$	0	0
$502$	−17.4495	−0.778809
$503$	−24.4949	−1.09217	−0.546087	−	0.837729i	$-0.683883\pi$
$503$	−24.4949	−1.09217	−0.546087	+	0.837729i	$0.683883\pi$
$504$	0	0
$505$	25.0000	1.11249
$506$	2.00000	0.0889108
$507$	0	0
$508$	−3.00000	−0.133103
$509$	−7.10102	−0.314747	−0.157374	−	0.987539i	$-0.550303\pi$
$509$	−7.10102	−0.314747	−0.157374	+	0.987539i	$0.550303\pi$
$510$	0	0
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−8.20204	−0.361777
$515$	20.2929	0.894210
$516$	0	0
$517$	−19.5959	−0.861827
$518$	0	0
$519$	0	0
$520$	−7.10102	−0.311400
$521$	−9.30306	−0.407575	−0.203787	−	0.979015i	$-0.565325\pi$
$521$	−9.30306	−0.407575	−0.203787	+	0.979015i	$0.565325\pi$
$522$	0	0
$523$	14.3485	0.627415	0.313707	−	0.949520i	$-0.398429\pi$
$523$	14.3485	0.627415	0.313707	+	0.949520i	$0.398429\pi$
$524$	−8.55051	−0.373531
$525$	0	0
$526$	25.8990	1.12925
$527$	−12.0000	−0.522728
$528$	0	0
$529$	−22.0000	−0.956522
$530$	15.7980	0.686219
$531$	0	0
$532$	0	0
$533$	−48.0000	−2.07911
$534$	0	0
$535$	−17.3939	−0.752003
$536$	12.8990	0.557151
$537$	0	0
$538$	18.3485	0.791059
$539$	0	0
$540$	0	0
$541$	−18.4949	−0.795158	−0.397579	−	0.917568i	$-0.630149\pi$
$541$	−18.4949	−0.795158	−0.397579	+	0.917568i	$0.630149\pi$
$542$	7.10102	0.305015
$543$	0	0
$544$	−2.00000	−0.0857493
$545$	−18.4041	−0.788344
$546$	0	0
$547$	−7.59592	−0.324778	−0.162389	−	0.986727i	$-0.551920\pi$
$547$	−7.59592	−0.324778	−0.162389	+	0.986727i	$0.551920\pi$
$548$	−7.79796	−0.333112
$549$	0	0
$550$	−5.79796	−0.247226
$551$	−17.5959	−0.749611
$552$	0	0
$553$	0	0
$554$	18.6969	0.794357
$555$	0	0
$556$	−4.55051	−0.192985
$557$	12.8990	0.546547	0.273274	−	0.961936i	$-0.411894\pi$
$557$	12.8990	0.546547	0.273274	+	0.961936i	$0.411894\pi$
$558$	0	0
$559$	−33.7980	−1.42950
$560$	0	0
$561$	0	0
$562$	−19.0000	−0.801467
$563$	−39.9444	−1.68346	−0.841728	−	0.539902i	$-0.818461\pi$
$563$	−39.9444	−1.68346	−0.841728	+	0.539902i	$0.818461\pi$
$564$	0	0
$565$	−8.84337	−0.372043
$566$	−25.4495	−1.06972
$567$	0	0
$568$	0.101021	0.00423873
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	0	0
$571$	33.7980	1.41440	0.707200	−	0.707013i	$-0.249958\pi$
$571$	33.7980	1.41440	0.707200	+	0.707013i	$0.249958\pi$
$572$	9.79796	0.409673
$573$	0	0
$574$	0	0
$575$	−2.89898	−0.120896
$576$	0	0
$577$	15.5959	0.649267	0.324633	−	0.945840i	$-0.394759\pi$
$577$	15.5959	0.649267	0.324633	+	0.945840i	$0.394759\pi$
$578$	13.0000	0.540729
$579$	0	0
$580$	−10.0000	−0.415227
$581$	0	0
$582$	0	0
$583$	−21.7980	−0.902779
$584$	−6.89898	−0.285482
$585$	0	0
$586$	−2.75255	−0.113707
$587$	16.1464	0.666434	0.333217	−	0.942850i	$-0.391866\pi$
$587$	16.1464	0.666434	0.333217	+	0.942850i	$0.391866\pi$
$588$	0	0
$589$	15.3031	0.630552
$590$	−2.89898	−0.119349
$591$	0	0
$592$	11.7980	0.484893
$593$	14.6969	0.603531	0.301765	−	0.953382i	$-0.402424\pi$
$593$	14.6969	0.603531	0.301765	+	0.953382i	$0.402424\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	0	0
$598$	4.89898	0.200334
$599$	33.7980	1.38095	0.690474	−	0.723358i	$-0.257402\pi$
$599$	33.7980	1.38095	0.690474	+	0.723358i	$0.257402\pi$
$600$	0	0
$601$	−16.6969	−0.681082	−0.340541	−	0.940230i	$-0.610610\pi$
$601$	−16.6969	−0.681082	−0.340541	+	0.940230i	$0.610610\pi$
$602$	0	0
$603$	0	0
$604$	−5.00000	−0.203447
$605$	10.1464	0.412511
$606$	0	0
$607$	−20.6969	−0.840063	−0.420031	−	0.907510i	$-0.637981\pi$
$607$	−20.6969	−0.840063	−0.420031	+	0.907510i	$0.637981\pi$
$608$	2.55051	0.103437
$609$	0	0
$610$	−9.49490	−0.384437
$611$	−48.0000	−1.94187
$612$	0	0
$613$	−14.6969	−0.593604	−0.296802	−	0.954939i	$-0.595920\pi$
$613$	−14.6969	−0.593604	−0.296802	+	0.954939i	$0.595920\pi$
$614$	25.2474	1.01890
$615$	0	0
$616$	0	0
$617$	15.3939	0.619734	0.309867	−	0.950780i	$-0.399715\pi$
$617$	15.3939	0.619734	0.309867	+	0.950780i	$0.399715\pi$
$618$	0	0
$619$	−30.1464	−1.21169	−0.605844	−	0.795584i	$-0.707164\pi$
$619$	−30.1464	−1.21169	−0.605844	+	0.795584i	$0.707164\pi$
$620$	8.69694	0.349277
$621$	0	0
$622$	−30.6969	−1.23084
$623$	0	0
$624$	0	0
$625$	−2.10102	−0.0840408
$626$	−4.69694	−0.187727
$627$	0	0
$628$	8.34847	0.333140
$629$	23.5959	0.940831
$630$	0	0
$631$	27.8990	1.11064	0.555320	−	0.831636i	$-0.312596\pi$
$631$	27.8990	1.11064	0.555320	+	0.831636i	$0.312596\pi$
$632$	1.89898	0.0755373
$633$	0	0
$634$	20.6969	0.821980
$635$	4.34847	0.172564
$636$	0	0
$637$	0	0
$638$	13.7980	0.546266
$639$	0	0
$640$	1.44949	0.0572961
$641$	7.49490	0.296031	0.148015	−	0.988985i	$-0.452712\pi$
$641$	7.49490	0.296031	0.148015	+	0.988985i	$0.452712\pi$
$642$	0	0
$643$	39.3939	1.55354	0.776771	−	0.629783i	$-0.216856\pi$
$643$	39.3939	1.55354	0.776771	+	0.629783i	$0.216856\pi$
$644$	0	0
$645$	0	0
$646$	5.10102	0.200697
$647$	−50.6969	−1.99310	−0.996551	−	0.0829807i	$-0.973556\pi$
$647$	−50.6969	−1.99310	−0.996551	+	0.0829807i	$0.973556\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	−14.2020	−0.557050
$651$	0	0
$652$	−19.7980	−0.775348
$653$	−9.79796	−0.383424	−0.191712	−	0.981451i	$-0.561404\pi$
$653$	−9.79796	−0.383424	−0.191712	+	0.981451i	$0.561404\pi$
$654$	0	0
$655$	12.3939	0.484269
$656$	9.79796	0.382546
$657$	0	0
$658$	0	0
$659$	24.6969	0.962056	0.481028	−	0.876705i	$-0.340264\pi$
$659$	24.6969	0.962056	0.481028	+	0.876705i	$0.340264\pi$
$660$	0	0
$661$	−4.55051	−0.176994	−0.0884972	−	0.996076i	$-0.528206\pi$
$661$	−4.55051	−0.176994	−0.0884972	+	0.996076i	$0.528206\pi$
$662$	−4.69694	−0.182552
$663$	0	0
$664$	−2.00000	−0.0776151
$665$	0	0
$666$	0	0
$667$	6.89898	0.267130
$668$	−10.6969	−0.413877
$669$	0	0
$670$	−18.6969	−0.722326
$671$	13.1010	0.505759
$672$	0	0
$673$	−8.59592	−0.331348	−0.165674	−	0.986181i	$-0.552980\pi$
$673$	−8.59592	−0.331348	−0.165674	+	0.986181i	$0.552980\pi$
$674$	23.3939	0.901098
$675$	0	0
$676$	11.0000	0.423077
$677$	14.6969	0.564849	0.282425	−	0.959289i	$-0.408861\pi$
$677$	14.6969	0.564849	0.282425	+	0.959289i	$0.408861\pi$
$678$	0	0
$679$	0	0
$680$	2.89898	0.111171
$681$	0	0
$682$	−12.0000	−0.459504
$683$	−51.7980	−1.98199	−0.990997	−	0.133885i	$-0.957255\pi$
$683$	−51.7980	−1.98199	−0.990997	+	0.133885i	$0.957255\pi$
$684$	0	0
$685$	11.3031	0.431868
$686$	0	0
$687$	0	0
$688$	6.89898	0.263021
$689$	−53.3939	−2.03414
$690$	0	0
$691$	−51.0454	−1.94186	−0.970929	−	0.239366i	$-0.923060\pi$
$691$	−51.0454	−1.94186	−0.970929	+	0.239366i	$0.923060\pi$
$692$	−3.10102	−0.117883
$693$	0	0
$694$	19.5959	0.743851
$695$	6.59592	0.250197
$696$	0	0
$697$	19.5959	0.742248
$698$	11.1010	0.420180
$699$	0	0
$700$	0	0
$701$	7.39388	0.279263	0.139631	−	0.990204i	$-0.455408\pi$
$701$	7.39388	0.279263	0.139631	+	0.990204i	$0.455408\pi$
$702$	0	0
$703$	−30.0908	−1.13490
$704$	−2.00000	−0.0753778
$705$	0	0
$706$	6.00000	0.225813
$707$	0	0
$708$	0	0
$709$	27.5959	1.03639	0.518193	−	0.855264i	$-0.326605\pi$
$709$	27.5959	1.03639	0.518193	+	0.855264i	$0.326605\pi$
$710$	−0.146428	−0.00549535
$711$	0	0
$712$	16.8990	0.633316
$713$	−6.00000	−0.224702
$714$	0	0
$715$	−14.2020	−0.531126
$716$	−20.6969	−0.773481
$717$	0	0
$718$	−8.79796	−0.328337
$719$	9.79796	0.365402	0.182701	−	0.983169i	$-0.441516\pi$
$719$	9.79796	0.365402	0.182701	+	0.983169i	$0.441516\pi$
$720$	0	0
$721$	0	0
$722$	12.4949	0.465012
$723$	0	0
$724$	10.3485	0.384598
$725$	−20.0000	−0.742781
$726$	0	0
$727$	8.49490	0.315058	0.157529	−	0.987514i	$-0.449647\pi$
$727$	8.49490	0.315058	0.157529	+	0.987514i	$0.449647\pi$
$728$	0	0
$729$	0	0
$730$	10.0000	0.370117
$731$	13.7980	0.510336
$732$	0	0
$733$	17.4495	0.644512	0.322256	−	0.946653i	$-0.395559\pi$
$733$	17.4495	0.644512	0.322256	+	0.946653i	$0.395559\pi$
$734$	−13.7980	−0.509292
$735$	0	0
$736$	−1.00000	−0.0368605
$737$	25.7980	0.950280
$738$	0	0
$739$	13.5959	0.500134	0.250067	−	0.968229i	$-0.419547\pi$
$739$	13.5959	0.500134	0.250067	+	0.968229i	$0.419547\pi$
$740$	−17.1010	−0.628646
$741$	0	0
$742$	0	0
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	−8.69694	−0.318631
$746$	6.89898	0.252590
$747$	0	0
$748$	−4.00000	−0.146254
$749$	0	0
$750$	0	0
$751$	1.40408	0.0512357	0.0256178	−	0.999672i	$-0.491845\pi$
$751$	1.40408	0.0512357	0.0256178	+	0.999672i	$0.491845\pi$
$752$	9.79796	0.357295
$753$	0	0
$754$	33.7980	1.23085
$755$	7.24745	0.263762
$756$	0	0
$757$	−35.3939	−1.28641	−0.643206	−	0.765693i	$-0.722396\pi$
$757$	−35.3939	−1.28641	−0.643206	+	0.765693i	$0.722396\pi$
$758$	−22.4949	−0.817051
$759$	0	0
$760$	−3.69694	−0.134102
$761$	2.00000	0.0724999	0.0362500	−	0.999343i	$-0.488459\pi$
$761$	2.00000	0.0724999	0.0362500	+	0.999343i	$0.488459\pi$
$762$	0	0
$763$	0	0
$764$	−4.10102	−0.148370
$765$	0	0
$766$	2.89898	0.104744
$767$	9.79796	0.353784
$768$	0	0
$769$	34.0908	1.22935	0.614673	−	0.788782i	$-0.289288\pi$
$769$	34.0908	1.22935	0.614673	+	0.788782i	$0.289288\pi$
$770$	0	0
$771$	0	0
$772$	−17.8990	−0.644198
$773$	33.9444	1.22089	0.610447	−	0.792057i	$-0.290990\pi$
$773$	33.9444	1.22089	0.610447	+	0.792057i	$0.290990\pi$
$774$	0	0
$775$	17.3939	0.624807
$776$	2.89898	0.104067
$777$	0	0
$778$	−24.8990	−0.892672
$779$	−24.9898	−0.895352
$780$	0	0
$781$	0.202041	0.00722960
$782$	−2.00000	−0.0715199
$783$	0	0
$784$	0	0
$785$	−12.1010	−0.431904
$786$	0	0
$787$	11.3939	0.406148	0.203074	−	0.979163i	$-0.434907\pi$
$787$	11.3939	0.406148	0.203074	+	0.979163i	$0.434907\pi$
$788$	−16.6969	−0.594804
$789$	0	0
$790$	−2.75255	−0.0979314
$791$	0	0
$792$	0	0
$793$	32.0908	1.13958
$794$	38.6969	1.37330
$795$	0	0
$796$	2.89898	0.102752
$797$	−17.9444	−0.635623	−0.317811	−	0.948154i	$-0.602948\pi$
$797$	−17.9444	−0.635623	−0.317811	+	0.948154i	$0.602948\pi$
$798$	0	0
$799$	19.5959	0.693254
$800$	2.89898	0.102494
$801$	0	0
$802$	−19.8990	−0.702657
$803$	−13.7980	−0.486919
$804$	0	0
$805$	0	0
$806$	−29.3939	−1.03536
$807$	0	0
$808$	17.2474	0.606763
$809$	16.2020	0.569633	0.284817	−	0.958582i	$-0.408067\pi$
$809$	16.2020	0.569633	0.284817	+	0.958582i	$0.408067\pi$
$810$	0	0
$811$	2.00000	0.0702295	0.0351147	−	0.999383i	$-0.488820\pi$
$811$	2.00000	0.0702295	0.0351147	+	0.999383i	$0.488820\pi$
$812$	0	0
$813$	0	0
$814$	23.5959	0.827036
$815$	28.6969	1.00521
$816$	0	0
$817$	−17.5959	−0.615603
$818$	−13.7980	−0.482434
$819$	0	0
$820$	−14.2020	−0.495957
$821$	−0.404082	−0.0141026	−0.00705128	−	0.999975i	$-0.502245\pi$
$821$	−0.404082	−0.0141026	−0.00705128	+	0.999975i	$0.502245\pi$
$822$	0	0
$823$	−13.3939	−0.466881	−0.233441	−	0.972371i	$-0.574998\pi$
$823$	−13.3939	−0.466881	−0.233441	+	0.972371i	$0.574998\pi$
$824$	14.0000	0.487713
$825$	0	0
$826$	0	0
$827$	36.4949	1.26905	0.634526	−	0.772902i	$-0.281195\pi$
$827$	36.4949	1.26905	0.634526	+	0.772902i	$0.281195\pi$
$828$	0	0
$829$	−1.30306	−0.0452572	−0.0226286	−	0.999744i	$-0.507204\pi$
$829$	−1.30306	−0.0452572	−0.0226286	+	0.999744i	$0.507204\pi$
$830$	2.89898	0.100625
$831$	0	0
$832$	−4.89898	−0.169842
$833$	0	0
$834$	0	0
$835$	15.5051	0.536576
$836$	5.10102	0.176422
$837$	0	0
$838$	−29.4495	−1.01732
$839$	−35.1010	−1.21182	−0.605911	−	0.795533i	$-0.707191\pi$
$839$	−35.1010	−1.21182	−0.605911	+	0.795533i	$0.707191\pi$
$840$	0	0
$841$	18.5959	0.641239
$842$	−22.8990	−0.789151
$843$	0	0
$844$	12.8990	0.444001
$845$	−15.9444	−0.548504
$846$	0	0
$847$	0	0
$848$	10.8990	0.374272
$849$	0	0
$850$	5.79796	0.198868
$851$	11.7980	0.404429
$852$	0	0
$853$	−24.8434	−0.850621	−0.425310	−	0.905048i	$-0.639835\pi$
$853$	−24.8434	−0.850621	−0.425310	+	0.905048i	$0.639835\pi$
$854$	0	0
$855$	0	0
$856$	−12.0000	−0.410152
$857$	−34.8990	−1.19213	−0.596063	−	0.802938i	$-0.703269\pi$
$857$	−34.8990	−1.19213	−0.596063	+	0.802938i	$0.703269\pi$
$858$	0	0
$859$	10.0000	0.341196	0.170598	−	0.985341i	$-0.445430\pi$
$859$	10.0000	0.341196	0.170598	+	0.985341i	$0.445430\pi$
$860$	−10.0000	−0.340997
$861$	0	0
$862$	31.5959	1.07616
$863$	−11.8990	−0.405046	−0.202523	−	0.979278i	$-0.564914\pi$
$863$	−11.8990	−0.405046	−0.202523	+	0.979278i	$0.564914\pi$
$864$	0	0
$865$	4.49490	0.152831
$866$	−7.79796	−0.264985
$867$	0	0
$868$	0	0
$869$	3.79796	0.128837
$870$	0	0
$871$	63.1918	2.14117
$872$	−12.6969	−0.429973
$873$	0	0
$874$	2.55051	0.0862723
$875$	0	0
$876$	0	0
$877$	22.4949	0.759599	0.379799	−	0.925069i	$-0.375993\pi$
$877$	22.4949	0.759599	0.379799	+	0.925069i	$0.375993\pi$
$878$	2.20204	0.0743153
$879$	0	0
$880$	2.89898	0.0977246
$881$	19.5959	0.660203	0.330102	−	0.943945i	$-0.392917\pi$
$881$	19.5959	0.660203	0.330102	+	0.943945i	$0.392917\pi$
$882$	0	0
$883$	−19.7980	−0.666254	−0.333127	−	0.942882i	$-0.608104\pi$
$883$	−19.7980	−0.666254	−0.333127	+	0.942882i	$0.608104\pi$
$884$	−9.79796	−0.329541
$885$	0	0
$886$	−14.8990	−0.500541
$887$	−14.2020	−0.476858	−0.238429	−	0.971160i	$-0.576632\pi$
$887$	−14.2020	−0.476858	−0.238429	+	0.971160i	$0.576632\pi$
$888$	0	0
$889$	0	0
$890$	−24.4949	−0.821071
$891$	0	0
$892$	11.1010	0.371690
$893$	−24.9898	−0.836252
$894$	0	0
$895$	30.0000	1.00279
$896$	0	0
$897$	0	0
$898$	20.5959	0.687295
$899$	−41.3939	−1.38056
$900$	0	0
$901$	21.7980	0.726195
$902$	19.5959	0.652473
$903$	0	0
$904$	−6.10102	−0.202917
$905$	−15.0000	−0.498617
$906$	0	0
$907$	2.69694	0.0895504	0.0447752	−	0.998997i	$-0.485743\pi$
$907$	2.69694	0.0895504	0.0447752	+	0.998997i	$0.485743\pi$
$908$	−5.44949	−0.180848
$909$	0	0
$910$	0	0
$911$	−51.9898	−1.72250	−0.861249	−	0.508183i	$-0.830318\pi$
$911$	−51.9898	−1.72250	−0.861249	+	0.508183i	$0.830318\pi$
$912$	0	0
$913$	−4.00000	−0.132381
$914$	17.4949	0.578680
$915$	0	0
$916$	−1.24745	−0.0412169
$917$	0	0
$918$	0	0
$919$	−25.6969	−0.847664	−0.423832	−	0.905741i	$-0.639315\pi$
$919$	−25.6969	−0.847664	−0.423832	+	0.905741i	$0.639315\pi$
$920$	1.44949	0.0477883
$921$	0	0
$922$	−5.65153	−0.186123
$923$	0.494897	0.0162897
$924$	0	0
$925$	−34.2020	−1.12456
$926$	−3.69694	−0.121489
$927$	0	0
$928$	−6.89898	−0.226470
$929$	34.2929	1.12511	0.562556	−	0.826759i	$-0.309818\pi$
$929$	34.2929	1.12511	0.562556	+	0.826759i	$0.309818\pi$
$930$	0	0
$931$	0	0
$932$	7.00000	0.229293
$933$	0	0
$934$	−10.0000	−0.327210
$935$	5.79796	0.189614
$936$	0	0
$937$	−45.5959	−1.48955	−0.744777	−	0.667314i	$-0.767444\pi$
$937$	−45.5959	−1.48955	−0.744777	+	0.667314i	$0.767444\pi$
$938$	0	0
$939$	0	0
$940$	−14.2020	−0.463220
$941$	−1.44949	−0.0472520	−0.0236260	−	0.999721i	$-0.507521\pi$
$941$	−1.44949	−0.0472520	−0.0236260	+	0.999721i	$0.507521\pi$
$942$	0	0
$943$	9.79796	0.319065
$944$	−2.00000	−0.0650945
$945$	0	0
$946$	13.7980	0.448610
$947$	−52.4949	−1.70585	−0.852927	−	0.522029i	$-0.825175\pi$
$947$	−52.4949	−1.70585	−0.852927	+	0.522029i	$0.825175\pi$
$948$	0	0
$949$	−33.7980	−1.09713
$950$	−7.39388	−0.239889
$951$	0	0
$952$	0	0
$953$	3.39388	0.109938	0.0549692	−	0.998488i	$-0.482494\pi$
$953$	3.39388	0.109938	0.0549692	+	0.998488i	$0.482494\pi$
$954$	0	0
$955$	5.94439	0.192356
$956$	−6.79796	−0.219862
$957$	0	0
$958$	−9.59592	−0.310030
$959$	0	0
$960$	0	0
$961$	5.00000	0.161290
$962$	57.7980	1.86348
$963$	0	0
$964$	−0.898979	−0.0289542
$965$	25.9444	0.835179
$966$	0	0
$967$	24.5959	0.790951	0.395476	−	0.918476i	$-0.370580\pi$
$967$	24.5959	0.790951	0.395476	+	0.918476i	$0.370580\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	−4.20204	−0.134919
$971$	−0.0556128	−0.00178470	−0.000892350	−	1.00000i	$-0.500284\pi$
$971$	−0.0556128	−0.00178470	−0.000892350		1.00000i	$0.500284\pi$
$972$	0	0
$973$	0	0
$974$	36.3939	1.16614
$975$	0	0
$976$	−6.55051	−0.209677
$977$	37.5959	1.20280	0.601400	−	0.798948i	$-0.294610\pi$
$977$	37.5959	1.20280	0.601400	+	0.798948i	$0.294610\pi$
$978$	0	0
$979$	33.7980	1.08019
$980$	0	0
$981$	0	0
$982$	15.7980	0.504133
$983$	−33.1918	−1.05866	−0.529328	−	0.848418i	$-0.677556\pi$
$983$	−33.1918	−1.05866	−0.529328	+	0.848418i	$0.677556\pi$
$984$	0	0
$985$	24.2020	0.771141
$986$	−13.7980	−0.439417
$987$	0	0
$988$	12.4949	0.397516
$989$	6.89898	0.219375
$990$	0	0
$991$	−1.79796	−0.0571140	−0.0285570	−	0.999592i	$-0.509091\pi$
$991$	−1.79796	−0.0571140	−0.0285570	+	0.999592i	$0.509091\pi$
$992$	6.00000	0.190500
$993$	0	0
$994$	0	0
$995$	−4.20204	−0.133214
$996$	0	0
$997$	−52.1464	−1.65149	−0.825747	−	0.564041i	$-0.809246\pi$
$997$	−52.1464	−1.65149	−0.825747	+	0.564041i	$0.809246\pi$
$998$	25.3939	0.803829
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7938.2.a.bm.1.1		2
3.2	odd	2		7938.2.a.bn.1.2		2
7.6	odd	2		1134.2.a.i.1.2		2
9.2	odd	6		882.2.f.j.589.2		4
9.4	even	3		2646.2.f.k.883.2		4
9.5	odd	6		882.2.f.j.295.1		4
9.7	even	3		2646.2.f.k.1765.2		4
21.20	even	2		1134.2.a.p.1.1		2
28.27	even	2		9072.2.a.bd.1.2		2
63.2	odd	6		882.2.h.l.67.1		4
63.4	even	3		2646.2.h.n.667.1		4
63.5	even	6		882.2.e.m.655.1		4
63.11	odd	6		882.2.e.n.373.2		4
63.13	odd	6		378.2.f.d.127.1		4
63.16	even	3		2646.2.h.n.361.1		4
63.20	even	6		126.2.f.c.85.1	yes	4
63.23	odd	6		882.2.e.n.655.2		4
63.25	even	3		2646.2.e.k.1549.2		4
63.31	odd	6		2646.2.h.m.667.2		4
63.32	odd	6		882.2.h.l.79.1		4
63.34	odd	6		378.2.f.d.253.1		4
63.38	even	6		882.2.e.m.373.1		4
63.40	odd	6		2646.2.e.l.2125.1		4
63.41	even	6		126.2.f.c.43.2	✓	4
63.47	even	6		882.2.h.k.67.2		4
63.52	odd	6		2646.2.e.l.1549.1		4
63.58	even	3		2646.2.e.k.2125.2		4
63.59	even	6		882.2.h.k.79.2		4
63.61	odd	6		2646.2.h.m.361.2		4
84.83	odd	2		9072.2.a.bk.1.1		2
252.83	odd	6		1008.2.r.e.337.2		4
252.139	even	6		3024.2.r.e.2017.1		4
252.167	odd	6		1008.2.r.e.673.1		4
252.223	even	6		3024.2.r.e.1009.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
126.2.f.c.43.2	✓	4	63.41	even	6
126.2.f.c.85.1	yes	4	63.20	even	6
378.2.f.d.127.1		4	63.13	odd	6
378.2.f.d.253.1		4	63.34	odd	6
882.2.e.m.373.1		4	63.38	even	6
882.2.e.m.655.1		4	63.5	even	6
882.2.e.n.373.2		4	63.11	odd	6
882.2.e.n.655.2		4	63.23	odd	6
882.2.f.j.295.1		4	9.5	odd	6
882.2.f.j.589.2		4	9.2	odd	6
882.2.h.k.67.2		4	63.47	even	6
882.2.h.k.79.2		4	63.59	even	6
882.2.h.l.67.1		4	63.2	odd	6
882.2.h.l.79.1		4	63.32	odd	6
1008.2.r.e.337.2		4	252.83	odd	6
1008.2.r.e.673.1		4	252.167	odd	6
1134.2.a.i.1.2		2	7.6	odd	2
1134.2.a.p.1.1		2	21.20	even	2
2646.2.e.k.1549.2		4	63.25	even	3
2646.2.e.k.2125.2		4	63.58	even	3
2646.2.e.l.1549.1		4	63.52	odd	6
2646.2.e.l.2125.1		4	63.40	odd	6
2646.2.f.k.883.2		4	9.4	even	3
2646.2.f.k.1765.2		4	9.7	even	3
2646.2.h.m.361.2		4	63.61	odd	6
2646.2.h.m.667.2		4	63.31	odd	6
2646.2.h.n.361.1		4	63.16	even	3
2646.2.h.n.667.1		4	63.4	even	3
3024.2.r.e.1009.1		4	252.223	even	6
3024.2.r.e.2017.1		4	252.139	even	6
7938.2.a.bm.1.1		2	1.1	even	1	trivial
7938.2.a.bn.1.2		2	3.2	odd	2
9072.2.a.bd.1.2		2	28.27	even	2
9072.2.a.bk.1.1		2	84.83	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:	\( N \)	\(=\)	\( 7938 = 2 \cdot 3^{4} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7938.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -1.44949 q^{5} -1.00000 q^{8} +1.44949 q^{10} -2.00000 q^{11} -4.89898 q^{13} +1.00000 q^{16} +2.00000 q^{17} -2.55051 q^{19} -1.44949 q^{20} +2.00000 q^{22} +1.00000 q^{23} -2.89898 q^{25} +4.89898 q^{26} +6.89898 q^{29} -6.00000 q^{31} -1.00000 q^{32} -2.00000 q^{34} +11.7980 q^{37} +2.55051 q^{38} +1.44949 q^{40} +9.79796 q^{41} +6.89898 q^{43} -2.00000 q^{44} -1.00000 q^{46} +9.79796 q^{47} +2.89898 q^{50} -4.89898 q^{52} +10.8990 q^{53} +2.89898 q^{55} -6.89898 q^{58} -2.00000 q^{59} -6.55051 q^{61} +6.00000 q^{62} +1.00000 q^{64} +7.10102 q^{65} -12.8990 q^{67} +2.00000 q^{68} -0.101021 q^{71} +6.89898 q^{73} -11.7980 q^{74} -2.55051 q^{76} -1.89898 q^{79} -1.44949 q^{80} -9.79796 q^{82} +2.00000 q^{83} -2.89898 q^{85} -6.89898 q^{86} +2.00000 q^{88} -16.8990 q^{89} +1.00000 q^{92} -9.79796 q^{94} +3.69694 q^{95} -2.89898 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} + 2 q^{5} - 2 q^{8} - 2 q^{10} - 4 q^{11} + 2 q^{16} + 4 q^{17} - 10 q^{19} + 2 q^{20} + 4 q^{22} + 2 q^{23} + 4 q^{25} + 4 q^{29} - 12 q^{31} - 2 q^{32} - 4 q^{34} + 4 q^{37} + 10 q^{38}+ \cdots + 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 + 2 * q^5 - 2 * q^8 - 2 * q^10 - 4 * q^11 + 2 * q^16 + 4 * q^17 - 10 * q^19 + 2 * q^20 + 4 * q^22 + 2 * q^23 + 4 * q^25 + 4 * q^29 - 12 * q^31 - 2 * q^32 - 4 * q^34 + 4 * q^37 + 10 * q^38 - 2 * q^40 + 4 * q^43 - 4 * q^44 - 2 * q^46 - 4 * q^50 + 12 * q^53 - 4 * q^55 - 4 * q^58 - 4 * q^59 - 18 * q^61 + 12 * q^62 + 2 * q^64 + 24 * q^65 - 16 * q^67 + 4 * q^68 - 10 * q^71 + 4 * q^73 - 4 * q^74 - 10 * q^76 + 6 * q^79 + 2 * q^80 + 4 * q^83 + 4 * q^85 - 4 * q^86 + 4 * q^88 - 24 * q^89 + 2 * q^92 - 22 * q^95 + 4 * q^97

Introduction

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