LMFDB - Embedded newform 8379.2.a.bl.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [8379,2,Mod(1,8379)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8379.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8379, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 8379 = 3^{2} \cdot 7^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8379.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,2,0,2,2,0,0,6,0,2,-2,0,-4,0,0,6,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$66.9066518536$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 133)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	8379.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} +1.58579 q^{8} -0.414214 q^{10} -2.41421 q^{11} -6.24264 q^{13} +3.00000 q^{16} -4.00000 q^{17} +1.00000 q^{19} -1.82843 q^{20} +1.00000 q^{22} +8.41421 q^{23} -4.00000 q^{25} +2.58579 q^{26} +9.41421 q^{29} -2.24264 q^{31} -4.41421 q^{32} +1.65685 q^{34} -5.07107 q^{37} -0.414214 q^{38} +1.58579 q^{40} -8.82843 q^{41} -6.07107 q^{43} +4.41421 q^{44} -3.48528 q^{46} -1.58579 q^{47} +1.65685 q^{50} +11.4142 q^{52} -4.24264 q^{53} -2.41421 q^{55} -3.89949 q^{58} +1.17157 q^{59} -6.17157 q^{61} +0.928932 q^{62} -4.17157 q^{64} -6.24264 q^{65} +6.82843 q^{67} +7.31371 q^{68} +5.75736 q^{71} +13.1421 q^{73} +2.10051 q^{74} -1.82843 q^{76} -1.41421 q^{79} +3.00000 q^{80} +3.65685 q^{82} +0.757359 q^{83} -4.00000 q^{85} +2.51472 q^{86} -3.82843 q^{88} -8.58579 q^{89} -15.3848 q^{92} +0.656854 q^{94} +1.00000 q^{95} +8.82843 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 6 q^{8} + 2 q^{10} - 2 q^{11} - 4 q^{13} + 6 q^{16} - 8 q^{17} + 2 q^{19} + 2 q^{20} + 2 q^{22} + 14 q^{23} - 8 q^{25} + 8 q^{26} + 16 q^{29} + 4 q^{31} - 6 q^{32} - 8 q^{34}+ \cdots + 12 q^{97}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 6 * q^8 + 2 * q^10 - 2 * q^11 - 4 * q^13 + 6 * q^16 - 8 * q^17 + 2 * q^19 + 2 * q^20 + 2 * q^22 + 14 * q^23 - 8 * q^25 + 8 * q^26 + 16 * q^29 + 4 * q^31 - 6 * q^32 - 8 * q^34 + 4 * q^37 + 2 * q^38 + 6 * q^40 - 12 * q^41 + 2 * q^43 + 6 * q^44 + 10 * q^46 - 6 * q^47 - 8 * q^50 + 20 * q^52 - 2 * q^55 + 12 * q^58 + 8 * q^59 - 18 * q^61 + 16 * q^62 - 14 * q^64 - 4 * q^65 + 8 * q^67 - 8 * q^68 + 20 * q^71 - 2 * q^73 + 24 * q^74 + 2 * q^76 + 6 * q^80 - 4 * q^82 + 10 * q^83 - 8 * q^85 + 22 * q^86 - 2 * q^88 - 20 * q^89 + 6 * q^92 - 10 * q^94 + 2 * q^95 + 12 * 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 $

$2$	−0.414214	−0.292893	−0.146447	−	0.989219i	$-0.546784\pi$
$2$	−0.414214	−0.292893	−0.146447	+	0.989219i	$0.546784\pi$
$3$	0	0
$3$	0	0
$4$	−1.82843	−0.914214
$5$	1.00000	0.447214	0.223607	−	0.974679i	$-0.428217\pi$
$5$	1.00000	0.447214	0.223607	+	0.974679i	$0.428217\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	1.58579	0.560660
$9$	0	0
$10$	−0.414214	−0.130986
$11$	−2.41421	−0.727913	−0.363956	−	0.931416i	$-0.618574\pi$
$11$	−2.41421	−0.727913	−0.363956	+	0.931416i	$0.618574\pi$
$12$	0	0
$13$	−6.24264	−1.73140	−0.865699	−	0.500566i	$-0.833125\pi$
$13$	−6.24264	−1.73140	−0.865699	+	0.500566i	$0.833125\pi$
$14$	0	0
$15$	0	0
$16$	3.00000	0.750000
$17$	−4.00000	−0.970143	−0.485071	−	0.874475i	$-0.661206\pi$
$17$	−4.00000	−0.970143	−0.485071	+	0.874475i	$0.661206\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−1.82843	−0.408849
$21$	0	0
$22$	1.00000	0.213201
$23$	8.41421	1.75448	0.877242	−	0.480048i	$-0.159381\pi$
$23$	8.41421	1.75448	0.877242	+	0.480048i	$0.159381\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	2.58579	0.507114
$27$	0	0
$28$	0	0
$29$	9.41421	1.74818	0.874088	−	0.485768i	$-0.161460\pi$
$29$	9.41421	1.74818	0.874088	+	0.485768i	$0.161460\pi$
$30$	0	0
$31$	−2.24264	−0.402790	−0.201395	−	0.979510i	$-0.564548\pi$
$31$	−2.24264	−0.402790	−0.201395	+	0.979510i	$0.564548\pi$
$32$	−4.41421	−0.780330
$33$	0	0
$34$	1.65685	0.284148
$35$	0	0
$36$	0	0
$37$	−5.07107	−0.833678	−0.416839	−	0.908980i	$-0.636862\pi$
$37$	−5.07107	−0.833678	−0.416839	+	0.908980i	$0.636862\pi$
$38$	−0.414214	−0.0671943
$39$	0	0
$40$	1.58579	0.250735
$41$	−8.82843	−1.37877	−0.689384	−	0.724396i	$-0.742119\pi$
$41$	−8.82843	−1.37877	−0.689384	+	0.724396i	$0.742119\pi$
$42$	0	0
$43$	−6.07107	−0.925829	−0.462915	−	0.886403i	$-0.653196\pi$
$43$	−6.07107	−0.925829	−0.462915	+	0.886403i	$0.653196\pi$
$44$	4.41421	0.665468
$45$	0	0
$46$	−3.48528	−0.513877
$47$	−1.58579	−0.231311	−0.115655	−	0.993289i	$-0.536897\pi$
$47$	−1.58579	−0.231311	−0.115655	+	0.993289i	$0.536897\pi$
$48$	0	0
$49$	0	0
$50$	1.65685	0.234315
$51$	0	0
$52$	11.4142	1.58287
$53$	−4.24264	−0.582772	−0.291386	−	0.956606i	$-0.594116\pi$
$53$	−4.24264	−0.582772	−0.291386	+	0.956606i	$0.594116\pi$
$54$	0	0
$55$	−2.41421	−0.325532
$56$	0	0
$57$	0	0
$58$	−3.89949	−0.512029
$59$	1.17157	0.152526	0.0762629	−	0.997088i	$-0.475701\pi$
$59$	1.17157	0.152526	0.0762629	+	0.997088i	$0.475701\pi$
$60$	0	0
$61$	−6.17157	−0.790189	−0.395094	−	0.918640i	$-0.629288\pi$
$61$	−6.17157	−0.790189	−0.395094	+	0.918640i	$0.629288\pi$
$62$	0.928932	0.117975
$63$	0	0
$64$	−4.17157	−0.521447
$65$	−6.24264	−0.774304
$66$	0	0
$67$	6.82843	0.834225	0.417113	−	0.908855i	$-0.363042\pi$
$67$	6.82843	0.834225	0.417113	+	0.908855i	$0.363042\pi$
$68$	7.31371	0.886917
$69$	0	0
$70$	0	0
$71$	5.75736	0.683273	0.341636	−	0.939832i	$-0.389019\pi$
$71$	5.75736	0.683273	0.341636	+	0.939832i	$0.389019\pi$
$72$	0	0
$73$	13.1421	1.53817	0.769085	−	0.639146i	$-0.220712\pi$
$73$	13.1421	1.53817	0.769085	+	0.639146i	$0.220712\pi$
$74$	2.10051	0.244179
$75$	0	0
$76$	−1.82843	−0.209735
$77$	0	0
$78$	0	0
$79$	−1.41421	−0.159111	−0.0795557	−	0.996830i	$-0.525350\pi$
$79$	−1.41421	−0.159111	−0.0795557	+	0.996830i	$0.525350\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	3.65685	0.403832
$83$	0.757359	0.0831310	0.0415655	−	0.999136i	$-0.486765\pi$
$83$	0.757359	0.0831310	0.0415655	+	0.999136i	$0.486765\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	2.51472	0.271169
$87$	0	0
$88$	−3.82843	−0.408112
$89$	−8.58579	−0.910092	−0.455046	−	0.890468i	$-0.650377\pi$
$89$	−8.58579	−0.910092	−0.455046	+	0.890468i	$0.650377\pi$
$90$	0	0
$91$	0	0
$92$	−15.3848	−1.60397
$93$	0	0
$94$	0.656854	0.0677493
$95$	1.00000	0.102598
$96$	0	0
$97$	8.82843	0.896391	0.448195	−	0.893936i	$-0.352067\pi$
$97$	8.82843	0.896391	0.448195	+	0.893936i	$0.352067\pi$
$98$	0	0
$99$	0	0
$100$	7.31371	0.731371
$101$	−4.31371	−0.429230	−0.214615	−	0.976699i	$-0.568850\pi$
$101$	−4.31371	−0.429230	−0.214615	+	0.976699i	$0.568850\pi$
$102$	0	0
$103$	6.00000	0.591198	0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000	0.591198	0.295599	+	0.955312i	$0.404481\pi$
$104$	−9.89949	−0.970725
$105$	0	0
$106$	1.75736	0.170690
$107$	−5.89949	−0.570326	−0.285163	−	0.958479i	$-0.592048\pi$
$107$	−5.89949	−0.570326	−0.285163	+	0.958479i	$0.592048\pi$
$108$	0	0
$109$	1.65685	0.158698	0.0793489	−	0.996847i	$-0.474716\pi$
$109$	1.65685	0.158698	0.0793489	+	0.996847i	$0.474716\pi$
$110$	1.00000	0.0953463
$111$	0	0
$112$	0	0
$113$	−8.24264	−0.775402	−0.387701	−	0.921785i	$-0.626731\pi$
$113$	−8.24264	−0.775402	−0.387701	+	0.921785i	$0.626731\pi$
$114$	0	0
$115$	8.41421	0.784629
$116$	−17.2132	−1.59821
$117$	0	0
$118$	−0.485281	−0.0446738
$119$	0	0
$120$	0	0
$121$	−5.17157	−0.470143
$122$	2.55635	0.231441
$123$	0	0
$124$	4.10051	0.368236
$125$	−9.00000	−0.804984
$126$	0	0
$127$	−4.24264	−0.376473	−0.188237	−	0.982124i	$-0.560277\pi$
$127$	−4.24264	−0.376473	−0.188237	+	0.982124i	$0.560277\pi$
$128$	10.5563	0.933058
$129$	0	0
$130$	2.58579	0.226788
$131$	20.4853	1.78981	0.894904	−	0.446259i	$-0.147244\pi$
$131$	20.4853	1.78981	0.894904	+	0.446259i	$0.147244\pi$
$132$	0	0
$133$	0	0
$134$	−2.82843	−0.244339
$135$	0	0
$136$	−6.34315	−0.543920
$137$	14.1716	1.21076	0.605380	−	0.795937i	$-0.293021\pi$
$137$	14.1716	1.21076	0.605380	+	0.795937i	$0.293021\pi$
$138$	0	0
$139$	−8.07107	−0.684579	−0.342290	−	0.939595i	$-0.611202\pi$
$139$	−8.07107	−0.684579	−0.342290	+	0.939595i	$0.611202\pi$
$140$	0	0
$141$	0	0
$142$	−2.38478	−0.200126
$143$	15.0711	1.26031
$144$	0	0
$145$	9.41421	0.781808
$146$	−5.44365	−0.450520
$147$	0	0
$148$	9.27208	0.762160
$149$	5.82843	0.477483	0.238742	−	0.971083i	$-0.423265\pi$
$149$	5.82843	0.477483	0.238742	+	0.971083i	$0.423265\pi$
$150$	0	0
$151$	5.31371	0.432423	0.216212	−	0.976346i	$-0.430630\pi$
$151$	5.31371	0.432423	0.216212	+	0.976346i	$0.430630\pi$
$152$	1.58579	0.128624
$153$	0	0
$154$	0	0
$155$	−2.24264	−0.180133
$156$	0	0
$157$	−11.8284	−0.944011	−0.472006	−	0.881596i	$-0.656470\pi$
$157$	−11.8284	−0.944011	−0.472006	+	0.881596i	$0.656470\pi$
$158$	0.585786	0.0466027
$159$	0	0
$160$	−4.41421	−0.348974
$161$	0	0
$162$	0	0
$163$	18.0711	1.41544	0.707718	−	0.706495i	$-0.249725\pi$
$163$	18.0711	1.41544	0.707718	+	0.706495i	$0.249725\pi$
$164$	16.1421	1.26049
$165$	0	0
$166$	−0.313708	−0.0243485
$167$	−1.07107	−0.0828817	−0.0414409	−	0.999141i	$-0.513195\pi$
$167$	−1.07107	−0.0828817	−0.0414409	+	0.999141i	$0.513195\pi$
$168$	0	0
$169$	25.9706	1.99774
$170$	1.65685	0.127075
$171$	0	0
$172$	11.1005	0.846406
$173$	15.3137	1.16428	0.582140	−	0.813089i	$-0.302216\pi$
$173$	15.3137	1.16428	0.582140	+	0.813089i	$0.302216\pi$
$174$	0	0
$175$	0	0
$176$	−7.24264	−0.545935
$177$	0	0
$178$	3.55635	0.266560
$179$	21.7990	1.62933	0.814667	−	0.579930i	$-0.196920\pi$
$179$	21.7990	1.62933	0.814667	+	0.579930i	$0.196920\pi$
$180$	0	0
$181$	9.89949	0.735824	0.367912	−	0.929861i	$-0.380073\pi$
$181$	9.89949	0.735824	0.367912	+	0.929861i	$0.380073\pi$
$182$	0	0
$183$	0	0
$184$	13.3431	0.983670
$185$	−5.07107	−0.372832
$186$	0	0
$187$	9.65685	0.706179
$188$	2.89949	0.211467
$189$	0	0
$190$	−0.414214	−0.0300502
$191$	−5.58579	−0.404173	−0.202087	−	0.979368i	$-0.564772\pi$
$191$	−5.58579	−0.404173	−0.202087	+	0.979368i	$0.564772\pi$
$192$	0	0
$193$	−11.8995	−0.856544	−0.428272	−	0.903650i	$-0.640878\pi$
$193$	−11.8995	−0.856544	−0.428272	+	0.903650i	$0.640878\pi$
$194$	−3.65685	−0.262547
$195$	0	0
$196$	0	0
$197$	−21.4853	−1.53076	−0.765381	−	0.643577i	$-0.777450\pi$
$197$	−21.4853	−1.53076	−0.765381	+	0.643577i	$0.777450\pi$
$198$	0	0
$199$	−1.92893	−0.136738	−0.0683692	−	0.997660i	$-0.521780\pi$
$199$	−1.92893	−0.136738	−0.0683692	+	0.997660i	$0.521780\pi$
$200$	−6.34315	−0.448528
$201$	0	0
$202$	1.78680	0.125719
$203$	0	0
$204$	0	0
$205$	−8.82843	−0.616604
$206$	−2.48528	−0.173158
$207$	0	0
$208$	−18.7279	−1.29855
$209$	−2.41421	−0.166995
$210$	0	0
$211$	−6.82843	−0.470088	−0.235044	−	0.971985i	$-0.575523\pi$
$211$	−6.82843	−0.470088	−0.235044	+	0.971985i	$0.575523\pi$
$212$	7.75736	0.532778
$213$	0	0
$214$	2.44365	0.167045
$215$	−6.07107	−0.414043
$216$	0	0
$217$	0	0
$218$	−0.686292	−0.0464815
$219$	0	0
$220$	4.41421	0.297606
$221$	24.9706	1.67970
$222$	0	0
$223$	6.24264	0.418038	0.209019	−	0.977912i	$-0.432973\pi$
$223$	6.24264	0.418038	0.209019	+	0.977912i	$0.432973\pi$
$224$	0	0
$225$	0	0
$226$	3.41421	0.227110
$227$	−11.7574	−0.780363	−0.390182	−	0.920738i	$-0.627588\pi$
$227$	−11.7574	−0.780363	−0.390182	+	0.920738i	$0.627588\pi$
$228$	0	0
$229$	4.48528	0.296396	0.148198	−	0.988958i	$-0.452653\pi$
$229$	4.48528	0.296396	0.148198	+	0.988958i	$0.452653\pi$
$230$	−3.48528	−0.229813
$231$	0	0
$232$	14.9289	0.980132
$233$	16.4853	1.07999	0.539993	−	0.841669i	$-0.318427\pi$
$233$	16.4853	1.07999	0.539993	+	0.841669i	$0.318427\pi$
$234$	0	0
$235$	−1.58579	−0.103445
$236$	−2.14214	−0.139441
$237$	0	0
$238$	0	0
$239$	−2.00000	−0.129369	−0.0646846	−	0.997906i	$-0.520604\pi$
$239$	−2.00000	−0.129369	−0.0646846	+	0.997906i	$0.520604\pi$
$240$	0	0
$241$	−22.0416	−1.41983	−0.709913	−	0.704289i	$-0.751266\pi$
$241$	−22.0416	−1.41983	−0.709913	+	0.704289i	$0.751266\pi$
$242$	2.14214	0.137702
$243$	0	0
$244$	11.2843	0.722401
$245$	0	0
$246$	0	0
$247$	−6.24264	−0.397210
$248$	−3.55635	−0.225828
$249$	0	0
$250$	3.72792	0.235774
$251$	11.7279	0.740260	0.370130	−	0.928980i	$-0.379313\pi$
$251$	11.7279	0.740260	0.370130	+	0.928980i	$0.379313\pi$
$252$	0	0
$253$	−20.3137	−1.27711
$254$	1.75736	0.110267
$255$	0	0
$256$	3.97056	0.248160
$257$	−13.3137	−0.830486	−0.415243	−	0.909710i	$-0.636304\pi$
$257$	−13.3137	−0.830486	−0.415243	+	0.909710i	$0.636304\pi$
$258$	0	0
$259$	0	0
$260$	11.4142	0.707879
$261$	0	0
$262$	−8.48528	−0.524222
$263$	0.485281	0.0299237	0.0149619	−	0.999888i	$-0.495237\pi$
$263$	0.485281	0.0299237	0.0149619	+	0.999888i	$0.495237\pi$
$264$	0	0
$265$	−4.24264	−0.260623
$266$	0	0
$267$	0	0
$268$	−12.4853	−0.762660
$269$	−4.00000	−0.243884	−0.121942	−	0.992537i	$-0.538912\pi$
$269$	−4.00000	−0.243884	−0.121942	+	0.992537i	$0.538912\pi$
$270$	0	0
$271$	6.75736	0.410480	0.205240	−	0.978712i	$-0.434202\pi$
$271$	6.75736	0.410480	0.205240	+	0.978712i	$0.434202\pi$
$272$	−12.0000	−0.727607
$273$	0	0
$274$	−5.87006	−0.354623
$275$	9.65685	0.582330
$276$	0	0
$277$	28.4558	1.70975	0.854873	−	0.518837i	$-0.173635\pi$
$277$	28.4558	1.70975	0.854873	+	0.518837i	$0.173635\pi$
$278$	3.34315	0.200509
$279$	0	0
$280$	0	0
$281$	19.7574	1.17863	0.589313	−	0.807905i	$-0.299399\pi$
$281$	19.7574	1.17863	0.589313	+	0.807905i	$0.299399\pi$
$282$	0	0
$283$	−15.9289	−0.946877	−0.473438	−	0.880827i	$-0.656987\pi$
$283$	−15.9289	−0.946877	−0.473438	+	0.880827i	$0.656987\pi$
$284$	−10.5269	−0.624657
$285$	0	0
$286$	−6.24264	−0.369135
$287$	0	0
$288$	0	0
$289$	−1.00000	−0.0588235
$290$	−3.89949	−0.228986
$291$	0	0
$292$	−24.0294	−1.40622
$293$	8.82843	0.515762	0.257881	−	0.966177i	$-0.416976\pi$
$293$	8.82843	0.515762	0.257881	+	0.966177i	$0.416976\pi$
$294$	0	0
$295$	1.17157	0.0682116
$296$	−8.04163	−0.467410
$297$	0	0
$298$	−2.41421	−0.139852
$299$	−52.5269	−3.03771
$300$	0	0
$301$	0	0
$302$	−2.20101	−0.126654
$303$	0	0
$304$	3.00000	0.172062
$305$	−6.17157	−0.353383
$306$	0	0
$307$	−6.68629	−0.381607	−0.190803	−	0.981628i	$-0.561109\pi$
$307$	−6.68629	−0.381607	−0.190803	+	0.981628i	$0.561109\pi$
$308$	0	0
$309$	0	0
$310$	0.928932	0.0527598
$311$	15.7990	0.895879	0.447939	−	0.894064i	$-0.352158\pi$
$311$	15.7990	0.895879	0.447939	+	0.894064i	$0.352158\pi$
$312$	0	0
$313$	−4.65685	−0.263221	−0.131610	−	0.991302i	$-0.542015\pi$
$313$	−4.65685	−0.263221	−0.131610	+	0.991302i	$0.542015\pi$
$314$	4.89949	0.276494
$315$	0	0
$316$	2.58579	0.145462
$317$	11.7574	0.660359	0.330180	−	0.943918i	$-0.392891\pi$
$317$	11.7574	0.660359	0.330180	+	0.943918i	$0.392891\pi$
$318$	0	0
$319$	−22.7279	−1.27252
$320$	−4.17157	−0.233198
$321$	0	0
$322$	0	0
$323$	−4.00000	−0.222566
$324$	0	0
$325$	24.9706	1.38512
$326$	−7.48528	−0.414571
$327$	0	0
$328$	−14.0000	−0.773021
$329$	0	0
$330$	0	0
$331$	19.4558	1.06939	0.534695	−	0.845045i	$-0.320427\pi$
$331$	19.4558	1.06939	0.534695	+	0.845045i	$0.320427\pi$
$332$	−1.38478	−0.0759995
$333$	0	0
$334$	0.443651	0.0242755
$335$	6.82843	0.373077
$336$	0	0
$337$	−11.5563	−0.629514	−0.314757	−	0.949172i	$-0.601923\pi$
$337$	−11.5563	−0.629514	−0.314757	+	0.949172i	$0.601923\pi$
$338$	−10.7574	−0.585123
$339$	0	0
$340$	7.31371	0.396642
$341$	5.41421	0.293196
$342$	0	0
$343$	0	0
$344$	−9.62742	−0.519076
$345$	0	0
$346$	−6.34315	−0.341010
$347$	21.0416	1.12957	0.564787	−	0.825237i	$-0.308958\pi$
$347$	21.0416	1.12957	0.564787	+	0.825237i	$0.308958\pi$
$348$	0	0
$349$	12.4853	0.668322	0.334161	−	0.942516i	$-0.391547\pi$
$349$	12.4853	0.668322	0.334161	+	0.942516i	$0.391547\pi$
$350$	0	0
$351$	0	0
$352$	10.6569	0.568012
$353$	−16.4853	−0.877423	−0.438711	−	0.898628i	$-0.644565\pi$
$353$	−16.4853	−0.877423	−0.438711	+	0.898628i	$0.644565\pi$
$354$	0	0
$355$	5.75736	0.305569
$356$	15.6985	0.832018
$357$	0	0
$358$	−9.02944	−0.477221
$359$	3.24264	0.171140	0.0855700	−	0.996332i	$-0.472729\pi$
$359$	3.24264	0.171140	0.0855700	+	0.996332i	$0.472729\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−4.10051	−0.215518
$363$	0	0
$364$	0	0
$365$	13.1421	0.687891
$366$	0	0
$367$	35.6569	1.86127	0.930636	−	0.365945i	$-0.119254\pi$
$367$	35.6569	1.86127	0.930636	+	0.365945i	$0.119254\pi$
$368$	25.2426	1.31586
$369$	0	0
$370$	2.10051	0.109200
$371$	0	0
$372$	0	0
$373$	−11.3137	−0.585802	−0.292901	−	0.956143i	$-0.594621\pi$
$373$	−11.3137	−0.585802	−0.292901	+	0.956143i	$0.594621\pi$
$374$	−4.00000	−0.206835
$375$	0	0
$376$	−2.51472	−0.129687
$377$	−58.7696	−3.02679
$378$	0	0
$379$	−4.24264	−0.217930	−0.108965	−	0.994046i	$-0.534754\pi$
$379$	−4.24264	−0.217930	−0.108965	+	0.994046i	$0.534754\pi$
$380$	−1.82843	−0.0937963
$381$	0	0
$382$	2.31371	0.118380
$383$	28.2426	1.44313	0.721566	−	0.692346i	$-0.243423\pi$
$383$	28.2426	1.44313	0.721566	+	0.692346i	$0.243423\pi$
$384$	0	0
$385$	0	0
$386$	4.92893	0.250876
$387$	0	0
$388$	−16.1421	−0.819493
$389$	28.4853	1.44426	0.722131	−	0.691757i	$-0.243163\pi$
$389$	28.4853	1.44426	0.722131	+	0.691757i	$0.243163\pi$
$390$	0	0
$391$	−33.6569	−1.70210
$392$	0	0
$393$	0	0
$394$	8.89949	0.448350
$395$	−1.41421	−0.0711568
$396$	0	0
$397$	−36.6274	−1.83828	−0.919139	−	0.393934i	$-0.871114\pi$
$397$	−36.6274	−1.83828	−0.919139	+	0.393934i	$0.871114\pi$
$398$	0.798990	0.0400497
$399$	0	0
$400$	−12.0000	−0.600000
$401$	16.4853	0.823236	0.411618	−	0.911357i	$-0.364964\pi$
$401$	16.4853	0.823236	0.411618	+	0.911357i	$0.364964\pi$
$402$	0	0
$403$	14.0000	0.697390
$404$	7.88730	0.392408
$405$	0	0
$406$	0	0
$407$	12.2426	0.606845
$408$	0	0
$409$	25.3137	1.25168	0.625841	−	0.779951i	$-0.284756\pi$
$409$	25.3137	1.25168	0.625841	+	0.779951i	$0.284756\pi$
$410$	3.65685	0.180599
$411$	0	0
$412$	−10.9706	−0.540481
$413$	0	0
$414$	0	0
$415$	0.757359	0.0371773
$416$	27.5563	1.35106
$417$	0	0
$418$	1.00000	0.0489116
$419$	−8.07107	−0.394297	−0.197149	−	0.980374i	$-0.563168\pi$
$419$	−8.07107	−0.394297	−0.197149	+	0.980374i	$0.563168\pi$
$420$	0	0
$421$	5.21320	0.254076	0.127038	−	0.991898i	$-0.459453\pi$
$421$	5.21320	0.254076	0.127038	+	0.991898i	$0.459453\pi$
$422$	2.82843	0.137686
$423$	0	0
$424$	−6.72792	−0.326737
$425$	16.0000	0.776114
$426$	0	0
$427$	0	0
$428$	10.7868	0.521399
$429$	0	0
$430$	2.51472	0.121271
$431$	−38.0416	−1.83240	−0.916200	−	0.400720i	$-0.868760\pi$
$431$	−38.0416	−1.83240	−0.916200	+	0.400720i	$0.868760\pi$
$432$	0	0
$433$	−28.0000	−1.34559	−0.672797	−	0.739827i	$-0.734907\pi$
$433$	−28.0000	−1.34559	−0.672797	+	0.739827i	$0.734907\pi$
$434$	0	0
$435$	0	0
$436$	−3.02944	−0.145084
$437$	8.41421	0.402506
$438$	0	0
$439$	7.51472	0.358658	0.179329	−	0.983789i	$-0.442607\pi$
$439$	7.51472	0.358658	0.179329	+	0.983789i	$0.442607\pi$
$440$	−3.82843	−0.182513
$441$	0	0
$442$	−10.3431	−0.491973
$443$	−6.34315	−0.301372	−0.150686	−	0.988582i	$-0.548148\pi$
$443$	−6.34315	−0.301372	−0.150686	+	0.988582i	$0.548148\pi$
$444$	0	0
$445$	−8.58579	−0.407005
$446$	−2.58579	−0.122441
$447$	0	0
$448$	0	0
$449$	1.65685	0.0781918	0.0390959	−	0.999235i	$-0.487552\pi$
$449$	1.65685	0.0781918	0.0390959	+	0.999235i	$0.487552\pi$
$450$	0	0
$451$	21.3137	1.00362
$452$	15.0711	0.708883
$453$	0	0
$454$	4.87006	0.228563
$455$	0	0
$456$	0	0
$457$	−2.17157	−0.101582	−0.0507909	−	0.998709i	$-0.516174\pi$
$457$	−2.17157	−0.101582	−0.0507909	+	0.998709i	$0.516174\pi$
$458$	−1.85786	−0.0868123
$459$	0	0
$460$	−15.3848	−0.717319
$461$	−3.97056	−0.184928	−0.0924638	−	0.995716i	$-0.529474\pi$
$461$	−3.97056	−0.184928	−0.0924638	+	0.995716i	$0.529474\pi$
$462$	0	0
$463$	10.0711	0.468042	0.234021	−	0.972232i	$-0.424812\pi$
$463$	10.0711	0.468042	0.234021	+	0.972232i	$0.424812\pi$
$464$	28.2426	1.31113
$465$	0	0
$466$	−6.82843	−0.316321
$467$	38.8995	1.80005	0.900027	−	0.435834i	$-0.143547\pi$
$467$	38.8995	1.80005	0.900027	+	0.435834i	$0.143547\pi$
$468$	0	0
$469$	0	0
$470$	0.656854	0.0302984
$471$	0	0
$472$	1.85786	0.0855151
$473$	14.6569	0.673923
$474$	0	0
$475$	−4.00000	−0.183533
$476$	0	0
$477$	0	0
$478$	0.828427	0.0378914
$479$	11.3848	0.520184	0.260092	−	0.965584i	$-0.416247\pi$
$479$	11.3848	0.520184	0.260092	+	0.965584i	$0.416247\pi$
$480$	0	0
$481$	31.6569	1.44343
$482$	9.12994	0.415857
$483$	0	0
$484$	9.45584	0.429811
$485$	8.82843	0.400878
$486$	0	0
$487$	20.8284	0.943826	0.471913	−	0.881645i	$-0.343564\pi$
$487$	20.8284	0.943826	0.471913	+	0.881645i	$0.343564\pi$
$488$	−9.78680	−0.443027
$489$	0	0
$490$	0	0
$491$	32.5563	1.46925	0.734624	−	0.678475i	$-0.237359\pi$
$491$	32.5563	1.46925	0.734624	+	0.678475i	$0.237359\pi$
$492$	0	0
$493$	−37.6569	−1.69598
$494$	2.58579	0.116340
$495$	0	0
$496$	−6.72792	−0.302093
$497$	0	0
$498$	0	0
$499$	3.44365	0.154159	0.0770795	−	0.997025i	$-0.475440\pi$
$499$	3.44365	0.154159	0.0770795	+	0.997025i	$0.475440\pi$
$500$	16.4558	0.735928
$501$	0	0
$502$	−4.85786	−0.216817
$503$	19.0416	0.849024	0.424512	−	0.905422i	$-0.360446\pi$
$503$	19.0416	0.849024	0.424512	+	0.905422i	$0.360446\pi$
$504$	0	0
$505$	−4.31371	−0.191958
$506$	8.41421	0.374057
$507$	0	0
$508$	7.75736	0.344177
$509$	18.9706	0.840855	0.420428	−	0.907326i	$-0.361880\pi$
$509$	18.9706	0.840855	0.420428	+	0.907326i	$0.361880\pi$
$510$	0	0
$511$	0	0
$512$	−22.7574	−1.00574
$513$	0	0
$514$	5.51472	0.243244
$515$	6.00000	0.264392
$516$	0	0
$517$	3.82843	0.168374
$518$	0	0
$519$	0	0
$520$	−9.89949	−0.434122
$521$	10.4437	0.457545	0.228772	−	0.973480i	$-0.426529\pi$
$521$	10.4437	0.457545	0.228772	+	0.973480i	$0.426529\pi$
$522$	0	0
$523$	23.6569	1.03444	0.517221	−	0.855852i	$-0.326967\pi$
$523$	23.6569	1.03444	0.517221	+	0.855852i	$0.326967\pi$
$524$	−37.4558	−1.63627
$525$	0	0
$526$	−0.201010	−0.00876446
$527$	8.97056	0.390764
$528$	0	0
$529$	47.7990	2.07822
$530$	1.75736	0.0763348
$531$	0	0
$532$	0	0
$533$	55.1127	2.38720
$534$	0	0
$535$	−5.89949	−0.255057
$536$	10.8284	0.467717
$537$	0	0
$538$	1.65685	0.0714321
$539$	0	0
$540$	0	0
$541$	5.14214	0.221078	0.110539	−	0.993872i	$-0.464742\pi$
$541$	5.14214	0.221078	0.110539	+	0.993872i	$0.464742\pi$
$542$	−2.79899	−0.120227
$543$	0	0
$544$	17.6569	0.757031
$545$	1.65685	0.0709718
$546$	0	0
$547$	29.5563	1.26374	0.631869	−	0.775075i	$-0.282288\pi$
$547$	29.5563	1.26374	0.631869	+	0.775075i	$0.282288\pi$
$548$	−25.9117	−1.10689
$549$	0	0
$550$	−4.00000	−0.170561
$551$	9.41421	0.401059
$552$	0	0
$553$	0	0
$554$	−11.7868	−0.500773
$555$	0	0
$556$	14.7574	0.625851
$557$	−10.1716	−0.430983	−0.215492	−	0.976506i	$-0.569135\pi$
$557$	−10.1716	−0.430983	−0.215492	+	0.976506i	$0.569135\pi$
$558$	0	0
$559$	37.8995	1.60298
$560$	0	0
$561$	0	0
$562$	−8.18377	−0.345211
$563$	−13.8995	−0.585794	−0.292897	−	0.956144i	$-0.594619\pi$
$563$	−13.8995	−0.585794	−0.292897	+	0.956144i	$0.594619\pi$
$564$	0	0
$565$	−8.24264	−0.346770
$566$	6.59798	0.277334
$567$	0	0
$568$	9.12994	0.383084
$569$	42.9706	1.80142	0.900710	−	0.434421i	$-0.143047\pi$
$569$	42.9706	1.80142	0.900710	+	0.434421i	$0.143047\pi$
$570$	0	0
$571$	−18.8995	−0.790919	−0.395460	−	0.918483i	$-0.629415\pi$
$571$	−18.8995	−0.790919	−0.395460	+	0.918483i	$0.629415\pi$
$572$	−27.5563	−1.15219
$573$	0	0
$574$	0	0
$575$	−33.6569	−1.40359
$576$	0	0
$577$	−27.9706	−1.16443	−0.582215	−	0.813035i	$-0.697814\pi$
$577$	−27.9706	−1.16443	−0.582215	+	0.813035i	$0.697814\pi$
$578$	0.414214	0.0172290
$579$	0	0
$580$	−17.2132	−0.714739
$581$	0	0
$582$	0	0
$583$	10.2426	0.424207
$584$	20.8406	0.862391
$585$	0	0
$586$	−3.65685	−0.151063
$587$	14.0000	0.577842	0.288921	−	0.957353i	$-0.406704\pi$
$587$	14.0000	0.577842	0.288921	+	0.957353i	$0.406704\pi$
$588$	0	0
$589$	−2.24264	−0.0924064
$590$	−0.485281	−0.0199787
$591$	0	0
$592$	−15.2132	−0.625259
$593$	6.17157	0.253436	0.126718	−	0.991939i	$-0.459556\pi$
$593$	6.17157	0.253436	0.126718	+	0.991939i	$0.459556\pi$
$594$	0	0
$595$	0	0
$596$	−10.6569	−0.436522
$597$	0	0
$598$	21.7574	0.889725
$599$	31.6985	1.29516	0.647582	−	0.761995i	$-0.275780\pi$
$599$	31.6985	1.29516	0.647582	+	0.761995i	$0.275780\pi$
$600$	0	0
$601$	9.27208	0.378216	0.189108	−	0.981956i	$-0.439440\pi$
$601$	9.27208	0.378216	0.189108	+	0.981956i	$0.439440\pi$
$602$	0	0
$603$	0	0
$604$	−9.71573	−0.395327
$605$	−5.17157	−0.210254
$606$	0	0
$607$	−32.5269	−1.32023	−0.660113	−	0.751166i	$-0.729492\pi$
$607$	−32.5269	−1.32023	−0.660113	+	0.751166i	$0.729492\pi$
$608$	−4.41421	−0.179020
$609$	0	0
$610$	2.55635	0.103504
$611$	9.89949	0.400491
$612$	0	0
$613$	−8.68629	−0.350836	−0.175418	−	0.984494i	$-0.556128\pi$
$613$	−8.68629	−0.350836	−0.175418	+	0.984494i	$0.556128\pi$
$614$	2.76955	0.111770
$615$	0	0
$616$	0	0
$617$	42.4558	1.70921	0.854604	−	0.519280i	$-0.173800\pi$
$617$	42.4558	1.70921	0.854604	+	0.519280i	$0.173800\pi$
$618$	0	0
$619$	−3.44365	−0.138412	−0.0692060	−	0.997602i	$-0.522047\pi$
$619$	−3.44365	−0.138412	−0.0692060	+	0.997602i	$0.522047\pi$
$620$	4.10051	0.164680
$621$	0	0
$622$	−6.54416	−0.262397
$623$	0	0
$624$	0	0
$625$	11.0000	0.440000
$626$	1.92893	0.0770956
$627$	0	0
$628$	21.6274	0.863028
$629$	20.2843	0.808787
$630$	0	0
$631$	−5.44365	−0.216708	−0.108354	−	0.994112i	$-0.534558\pi$
$631$	−5.44365	−0.216708	−0.108354	+	0.994112i	$0.534558\pi$
$632$	−2.24264	−0.0892075
$633$	0	0
$634$	−4.87006	−0.193415
$635$	−4.24264	−0.168364
$636$	0	0
$637$	0	0
$638$	9.41421	0.372712
$639$	0	0
$640$	10.5563	0.417276
$641$	−3.79899	−0.150051	−0.0750255	−	0.997182i	$-0.523904\pi$
$641$	−3.79899	−0.150051	−0.0750255	+	0.997182i	$0.523904\pi$
$642$	0	0
$643$	−30.1421	−1.18869	−0.594345	−	0.804210i	$-0.702589\pi$
$643$	−30.1421	−1.18869	−0.594345	+	0.804210i	$0.702589\pi$
$644$	0	0
$645$	0	0
$646$	1.65685	0.0651881
$647$	6.21320	0.244266	0.122133	−	0.992514i	$-0.461027\pi$
$647$	6.21320	0.244266	0.122133	+	0.992514i	$0.461027\pi$
$648$	0	0
$649$	−2.82843	−0.111025
$650$	−10.3431	−0.405692
$651$	0	0
$652$	−33.0416	−1.29401
$653$	1.85786	0.0727039	0.0363519	−	0.999339i	$-0.488426\pi$
$653$	1.85786	0.0727039	0.0363519	+	0.999339i	$0.488426\pi$
$654$	0	0
$655$	20.4853	0.800426
$656$	−26.4853	−1.03408
$657$	0	0
$658$	0	0
$659$	38.0416	1.48189	0.740946	−	0.671565i	$-0.234378\pi$
$659$	38.0416	1.48189	0.740946	+	0.671565i	$0.234378\pi$
$660$	0	0
$661$	22.7279	0.884014	0.442007	−	0.897012i	$-0.354267\pi$
$661$	22.7279	0.884014	0.442007	+	0.897012i	$0.354267\pi$
$662$	−8.05887	−0.313217
$663$	0	0
$664$	1.20101	0.0466082
$665$	0	0
$666$	0	0
$667$	79.2132	3.06715
$668$	1.95837	0.0757716
$669$	0	0
$670$	−2.82843	−0.109272
$671$	14.8995	0.575189
$672$	0	0
$673$	35.7990	1.37995	0.689975	−	0.723833i	$-0.257622\pi$
$673$	35.7990	1.37995	0.689975	+	0.723833i	$0.257622\pi$
$674$	4.78680	0.184381
$675$	0	0
$676$	−47.4853	−1.82636
$677$	0.686292	0.0263763	0.0131882	−	0.999913i	$-0.495802\pi$
$677$	0.686292	0.0263763	0.0131882	+	0.999913i	$0.495802\pi$
$678$	0	0
$679$	0	0
$680$	−6.34315	−0.243249
$681$	0	0
$682$	−2.24264	−0.0858752
$683$	−10.4853	−0.401208	−0.200604	−	0.979672i	$-0.564290\pi$
$683$	−10.4853	−0.401208	−0.200604	+	0.979672i	$0.564290\pi$
$684$	0	0
$685$	14.1716	0.541468
$686$	0	0
$687$	0	0
$688$	−18.2132	−0.694372
$689$	26.4853	1.00901
$690$	0	0
$691$	2.97056	0.113006	0.0565028	−	0.998402i	$-0.482005\pi$
$691$	2.97056	0.113006	0.0565028	+	0.998402i	$0.482005\pi$
$692$	−28.0000	−1.06440
$693$	0	0
$694$	−8.71573	−0.330845
$695$	−8.07107	−0.306153
$696$	0	0
$697$	35.3137	1.33760
$698$	−5.17157	−0.195747
$699$	0	0
$700$	0	0
$701$	−3.82843	−0.144598	−0.0722988	−	0.997383i	$-0.523034\pi$
$701$	−3.82843	−0.144598	−0.0722988	+	0.997383i	$0.523034\pi$
$702$	0	0
$703$	−5.07107	−0.191259
$704$	10.0711	0.379568
$705$	0	0
$706$	6.82843	0.256991
$707$	0	0
$708$	0	0
$709$	24.9411	0.936684	0.468342	−	0.883547i	$-0.344852\pi$
$709$	24.9411	0.936684	0.468342	+	0.883547i	$0.344852\pi$
$710$	−2.38478	−0.0894991
$711$	0	0
$712$	−13.6152	−0.510252
$713$	−18.8701	−0.706689
$714$	0	0
$715$	15.0711	0.563626
$716$	−39.8579	−1.48956
$717$	0	0
$718$	−1.34315	−0.0501258
$719$	−44.0833	−1.64403	−0.822014	−	0.569467i	$-0.807150\pi$
$719$	−44.0833	−1.64403	−0.822014	+	0.569467i	$0.807150\pi$
$720$	0	0
$721$	0	0
$722$	−0.414214	−0.0154154
$723$	0	0
$724$	−18.1005	−0.672700
$725$	−37.6569	−1.39854
$726$	0	0
$727$	−5.92893	−0.219892	−0.109946	−	0.993938i	$-0.535068\pi$
$727$	−5.92893	−0.219892	−0.109946	+	0.993938i	$0.535068\pi$
$728$	0	0
$729$	0	0
$730$	−5.44365	−0.201479
$731$	24.2843	0.898186
$732$	0	0
$733$	29.4558	1.08798	0.543988	−	0.839093i	$-0.316914\pi$
$733$	29.4558	1.08798	0.543988	+	0.839093i	$0.316914\pi$
$734$	−14.7696	−0.545154
$735$	0	0
$736$	−37.1421	−1.36908
$737$	−16.4853	−0.607243
$738$	0	0
$739$	−19.6569	−0.723089	−0.361545	−	0.932355i	$-0.617750\pi$
$739$	−19.6569	−0.723089	−0.361545	+	0.932355i	$0.617750\pi$
$740$	9.27208	0.340848
$741$	0	0
$742$	0	0
$743$	−50.8701	−1.86624	−0.933121	−	0.359563i	$-0.882926\pi$
$743$	−50.8701	−1.86624	−0.933121	+	0.359563i	$0.882926\pi$
$744$	0	0
$745$	5.82843	0.213537
$746$	4.68629	0.171577
$747$	0	0
$748$	−17.6569	−0.645599
$749$	0	0
$750$	0	0
$751$	−41.4558	−1.51275	−0.756373	−	0.654141i	$-0.773030\pi$
$751$	−41.4558	−1.51275	−0.756373	+	0.654141i	$0.773030\pi$
$752$	−4.75736	−0.173483
$753$	0	0
$754$	24.3431	0.886525
$755$	5.31371	0.193386
$756$	0	0
$757$	−5.00000	−0.181728	−0.0908640	−	0.995863i	$-0.528963\pi$
$757$	−5.00000	−0.181728	−0.0908640	+	0.995863i	$0.528963\pi$
$758$	1.75736	0.0638302
$759$	0	0
$760$	1.58579	0.0575225
$761$	28.1127	1.01908	0.509542	−	0.860446i	$-0.329815\pi$
$761$	28.1127	1.01908	0.509542	+	0.860446i	$0.329815\pi$
$762$	0	0
$763$	0	0
$764$	10.2132	0.369501
$765$	0	0
$766$	−11.6985	−0.422683
$767$	−7.31371	−0.264083
$768$	0	0
$769$	−41.4264	−1.49387	−0.746937	−	0.664895i	$-0.768476\pi$
$769$	−41.4264	−1.49387	−0.746937	+	0.664895i	$0.768476\pi$
$770$	0	0
$771$	0	0
$772$	21.7574	0.783064
$773$	−25.7574	−0.926428	−0.463214	−	0.886247i	$-0.653304\pi$
$773$	−25.7574	−0.926428	−0.463214	+	0.886247i	$0.653304\pi$
$774$	0	0
$775$	8.97056	0.322232
$776$	14.0000	0.502571
$777$	0	0
$778$	−11.7990	−0.423014
$779$	−8.82843	−0.316311
$780$	0	0
$781$	−13.8995	−0.497363
$782$	13.9411	0.498534
$783$	0	0
$784$	0	0
$785$	−11.8284	−0.422175
$786$	0	0
$787$	41.0122	1.46193	0.730963	−	0.682417i	$-0.239071\pi$
$787$	41.0122	1.46193	0.730963	+	0.682417i	$0.239071\pi$
$788$	39.2843	1.39944
$789$	0	0
$790$	0.585786	0.0208413
$791$	0	0
$792$	0	0
$793$	38.5269	1.36813
$794$	15.1716	0.538419
$795$	0	0
$796$	3.52691	0.125008
$797$	−21.7574	−0.770685	−0.385343	−	0.922774i	$-0.625917\pi$
$797$	−21.7574	−0.770685	−0.385343	+	0.922774i	$0.625917\pi$
$798$	0	0
$799$	6.34315	0.224404
$800$	17.6569	0.624264
$801$	0	0
$802$	−6.82843	−0.241120
$803$	−31.7279	−1.11965
$804$	0	0
$805$	0	0
$806$	−5.79899	−0.204261
$807$	0	0
$808$	−6.84062	−0.240652
$809$	−2.85786	−0.100477	−0.0502386	−	0.998737i	$-0.515998\pi$
$809$	−2.85786	−0.100477	−0.0502386	+	0.998737i	$0.515998\pi$
$810$	0	0
$811$	51.4558	1.80686	0.903430	−	0.428737i	$-0.141041\pi$
$811$	51.4558	1.80686	0.903430	+	0.428737i	$0.141041\pi$
$812$	0	0
$813$	0	0
$814$	−5.07107	−0.177741
$815$	18.0711	0.633002
$816$	0	0
$817$	−6.07107	−0.212400
$818$	−10.4853	−0.366609
$819$	0	0
$820$	16.1421	0.563708
$821$	52.1127	1.81875	0.909373	−	0.415982i	$-0.136562\pi$
$821$	52.1127	1.81875	0.909373	+	0.415982i	$0.136562\pi$
$822$	0	0
$823$	36.2132	1.26231	0.631156	−	0.775656i	$-0.282581\pi$
$823$	36.2132	1.26231	0.631156	+	0.775656i	$0.282581\pi$
$824$	9.51472	0.331461
$825$	0	0
$826$	0	0
$827$	44.2843	1.53991	0.769957	−	0.638095i	$-0.220277\pi$
$827$	44.2843	1.53991	0.769957	+	0.638095i	$0.220277\pi$
$828$	0	0
$829$	−28.2843	−0.982353	−0.491177	−	0.871060i	$-0.663433\pi$
$829$	−28.2843	−0.982353	−0.491177	+	0.871060i	$0.663433\pi$
$830$	−0.313708	−0.0108890
$831$	0	0
$832$	26.0416	0.902831
$833$	0	0
$834$	0	0
$835$	−1.07107	−0.0370658
$836$	4.41421	0.152669
$837$	0	0
$838$	3.34315	0.115487
$839$	19.7990	0.683537	0.341769	−	0.939784i	$-0.388974\pi$
$839$	19.7990	0.683537	0.341769	+	0.939784i	$0.388974\pi$
$840$	0	0
$841$	59.6274	2.05612
$842$	−2.15938	−0.0744171
$843$	0	0
$844$	12.4853	0.429761
$845$	25.9706	0.893415
$846$	0	0
$847$	0	0
$848$	−12.7279	−0.437079
$849$	0	0
$850$	−6.62742	−0.227319
$851$	−42.6690	−1.46268
$852$	0	0
$853$	−10.0294	−0.343401	−0.171701	−	0.985149i	$-0.554926\pi$
$853$	−10.0294	−0.343401	−0.171701	+	0.985149i	$0.554926\pi$
$854$	0	0
$855$	0	0
$856$	−9.35534	−0.319759
$857$	20.9706	0.716341	0.358170	−	0.933656i	$-0.383401\pi$
$857$	20.9706	0.716341	0.358170	+	0.933656i	$0.383401\pi$
$858$	0	0
$859$	−2.07107	−0.0706639	−0.0353320	−	0.999376i	$-0.511249\pi$
$859$	−2.07107	−0.0706639	−0.0353320	+	0.999376i	$0.511249\pi$
$860$	11.1005	0.378524
$861$	0	0
$862$	15.7574	0.536698
$863$	−7.41421	−0.252383	−0.126191	−	0.992006i	$-0.540275\pi$
$863$	−7.41421	−0.252383	−0.126191	+	0.992006i	$0.540275\pi$
$864$	0	0
$865$	15.3137	0.520682
$866$	11.5980	0.394115
$867$	0	0
$868$	0	0
$869$	3.41421	0.115819
$870$	0	0
$871$	−42.6274	−1.44437
$872$	2.62742	0.0889756
$873$	0	0
$874$	−3.48528	−0.117891
$875$	0	0
$876$	0	0
$877$	−8.10051	−0.273535	−0.136767	−	0.990603i	$-0.543671\pi$
$877$	−8.10051	−0.273535	−0.136767	+	0.990603i	$0.543671\pi$
$878$	−3.11270	−0.105048
$879$	0	0
$880$	−7.24264	−0.244149
$881$	4.28427	0.144341	0.0721704	−	0.997392i	$-0.477007\pi$
$881$	4.28427	0.144341	0.0721704	+	0.997392i	$0.477007\pi$
$882$	0	0
$883$	5.65685	0.190368	0.0951842	−	0.995460i	$-0.469656\pi$
$883$	5.65685	0.190368	0.0951842	+	0.995460i	$0.469656\pi$
$884$	−45.6569	−1.53561
$885$	0	0
$886$	2.62742	0.0882698
$887$	1.31371	0.0441100	0.0220550	−	0.999757i	$-0.492979\pi$
$887$	1.31371	0.0441100	0.0220550	+	0.999757i	$0.492979\pi$
$888$	0	0
$889$	0	0
$890$	3.55635	0.119209
$891$	0	0
$892$	−11.4142	−0.382176
$893$	−1.58579	−0.0530663
$894$	0	0
$895$	21.7990	0.728660
$896$	0	0
$897$	0	0
$898$	−0.686292	−0.0229018
$899$	−21.1127	−0.704148
$900$	0	0
$901$	16.9706	0.565371
$902$	−8.82843	−0.293954
$903$	0	0
$904$	−13.0711	−0.434737
$905$	9.89949	0.329070
$906$	0	0
$907$	28.5858	0.949175	0.474588	−	0.880208i	$-0.342597\pi$
$907$	28.5858	0.949175	0.474588	+	0.880208i	$0.342597\pi$
$908$	21.4975	0.713419
$909$	0	0
$910$	0	0
$911$	−34.7279	−1.15059	−0.575294	−	0.817947i	$-0.695112\pi$
$911$	−34.7279	−1.15059	−0.575294	+	0.817947i	$0.695112\pi$
$912$	0	0
$913$	−1.82843	−0.0605121
$914$	0.899495	0.0297526
$915$	0	0
$916$	−8.20101	−0.270969
$917$	0	0
$918$	0	0
$919$	29.0416	0.957995	0.478997	−	0.877816i	$-0.341000\pi$
$919$	29.0416	0.957995	0.478997	+	0.877816i	$0.341000\pi$
$920$	13.3431	0.439910
$921$	0	0
$922$	1.64466	0.0541640
$923$	−35.9411	−1.18302
$924$	0	0
$925$	20.2843	0.666943
$926$	−4.17157	−0.137086
$927$	0	0
$928$	−41.5563	−1.36415
$929$	15.0000	0.492134	0.246067	−	0.969253i	$-0.420862\pi$
$929$	15.0000	0.492134	0.246067	+	0.969253i	$0.420862\pi$
$930$	0	0
$931$	0	0
$932$	−30.1421	−0.987338
$933$	0	0
$934$	−16.1127	−0.527224
$935$	9.65685	0.315813
$936$	0	0
$937$	56.3137	1.83969	0.919844	−	0.392284i	$-0.128315\pi$
$937$	56.3137	1.83969	0.919844	+	0.392284i	$0.128315\pi$
$938$	0	0
$939$	0	0
$940$	2.89949	0.0945711
$941$	−44.0416	−1.43572	−0.717858	−	0.696189i	$-0.754877\pi$
$941$	−44.0416	−1.43572	−0.717858	+	0.696189i	$0.754877\pi$
$942$	0	0
$943$	−74.2843	−2.41903
$944$	3.51472	0.114394
$945$	0	0
$946$	−6.07107	−0.197387
$947$	7.65685	0.248814	0.124407	−	0.992231i	$-0.460297\pi$
$947$	7.65685	0.248814	0.124407	+	0.992231i	$0.460297\pi$
$948$	0	0
$949$	−82.0416	−2.66318
$950$	1.65685	0.0537555
$951$	0	0
$952$	0	0
$953$	0.585786	0.0189755	0.00948774	−	0.999955i	$-0.496980\pi$
$953$	0.585786	0.0189755	0.00948774	+	0.999955i	$0.496980\pi$
$954$	0	0
$955$	−5.58579	−0.180752
$956$	3.65685	0.118271
$957$	0	0
$958$	−4.71573	−0.152358
$959$	0	0
$960$	0	0
$961$	−25.9706	−0.837760
$962$	−13.1127	−0.422770
$963$	0	0
$964$	40.3015	1.29802
$965$	−11.8995	−0.383058
$966$	0	0
$967$	20.2843	0.652298	0.326149	−	0.945318i	$-0.394249\pi$
$967$	20.2843	0.652298	0.326149	+	0.945318i	$0.394249\pi$
$968$	−8.20101	−0.263590
$969$	0	0
$970$	−3.65685	−0.117415
$971$	−26.8701	−0.862301	−0.431151	−	0.902280i	$-0.641892\pi$
$971$	−26.8701	−0.862301	−0.431151	+	0.902280i	$0.641892\pi$
$972$	0	0
$973$	0	0
$974$	−8.62742	−0.276440
$975$	0	0
$976$	−18.5147	−0.592642
$977$	−28.7696	−0.920420	−0.460210	−	0.887810i	$-0.652226\pi$
$977$	−28.7696	−0.920420	−0.460210	+	0.887810i	$0.652226\pi$
$978$	0	0
$979$	20.7279	0.662467
$980$	0	0
$981$	0	0
$982$	−13.4853	−0.430333
$983$	−46.0000	−1.46717	−0.733586	−	0.679597i	$-0.762155\pi$
$983$	−46.0000	−1.46717	−0.733586	+	0.679597i	$0.762155\pi$
$984$	0	0
$985$	−21.4853	−0.684578
$986$	15.5980	0.496741
$987$	0	0
$988$	11.4142	0.363135
$989$	−51.0833	−1.62435
$990$	0	0
$991$	−35.7990	−1.13719	−0.568596	−	0.822617i	$-0.692513\pi$
$991$	−35.7990	−1.13719	−0.568596	+	0.822617i	$0.692513\pi$
$992$	9.89949	0.314309
$993$	0	0
$994$	0	0
$995$	−1.92893	−0.0611513
$996$	0	0
$997$	−28.2843	−0.895772	−0.447886	−	0.894091i	$-0.647823\pi$
$997$	−28.2843	−0.895772	−0.447886	+	0.894091i	$0.647823\pi$
$998$	−1.42641	−0.0451521
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8379.2.a.bl.1.1		2
3.2	odd	2		931.2.a.e.1.2		2
7.3	odd	6		1197.2.j.e.856.2		4
7.5	odd	6		1197.2.j.e.172.2		4
7.6	odd	2		8379.2.a.bi.1.1		2
21.2	odd	6		931.2.f.i.704.1		4
21.5	even	6		133.2.f.c.39.1	✓	4
21.11	odd	6		931.2.f.i.324.1		4
21.17	even	6		133.2.f.c.58.1	yes	4
21.20	even	2		931.2.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
133.2.f.c.39.1	✓	4	21.5	even	6
133.2.f.c.58.1	yes	4	21.17	even	6
931.2.a.e.1.2		2	3.2	odd	2
931.2.a.f.1.2		2	21.20	even	2
931.2.f.i.324.1		4	21.11	odd	6
931.2.f.i.704.1		4	21.2	odd	6
1197.2.j.e.172.2		4	7.5	odd	6
1197.2.j.e.856.2		4	7.3	odd	6
8379.2.a.bi.1.1		2	7.6	odd	2
8379.2.a.bl.1.1		2	1.1	even	1	trivial

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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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 8379 = 3^{2} \cdot 7^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8379.a (trivial)

\(f(q)\)	\(=\)	\(q-0.414214 q^{2} -1.82843 q^{4} +1.00000 q^{5} +1.58579 q^{8} -0.414214 q^{10} -2.41421 q^{11} -6.24264 q^{13} +3.00000 q^{16} -4.00000 q^{17} +1.00000 q^{19} -1.82843 q^{20} +1.00000 q^{22} +8.41421 q^{23} -4.00000 q^{25} +2.58579 q^{26} +9.41421 q^{29} -2.24264 q^{31} -4.41421 q^{32} +1.65685 q^{34} -5.07107 q^{37} -0.414214 q^{38} +1.58579 q^{40} -8.82843 q^{41} -6.07107 q^{43} +4.41421 q^{44} -3.48528 q^{46} -1.58579 q^{47} +1.65685 q^{50} +11.4142 q^{52} -4.24264 q^{53} -2.41421 q^{55} -3.89949 q^{58} +1.17157 q^{59} -6.17157 q^{61} +0.928932 q^{62} -4.17157 q^{64} -6.24264 q^{65} +6.82843 q^{67} +7.31371 q^{68} +5.75736 q^{71} +13.1421 q^{73} +2.10051 q^{74} -1.82843 q^{76} -1.41421 q^{79} +3.00000 q^{80} +3.65685 q^{82} +0.757359 q^{83} -4.00000 q^{85} +2.51472 q^{86} -3.82843 q^{88} -8.58579 q^{89} -15.3848 q^{92} +0.656854 q^{94} +1.00000 q^{95} +8.82843 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} + 6 q^{8} + 2 q^{10} - 2 q^{11} - 4 q^{13} + 6 q^{16} - 8 q^{17} + 2 q^{19} + 2 q^{20} + 2 q^{22} + 14 q^{23} - 8 q^{25} + 8 q^{26} + 16 q^{29} + 4 q^{31} - 6 q^{32} - 8 q^{34}+ \cdots + 12 q^{97}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 + 6 * q^8 + 2 * q^10 - 2 * q^11 - 4 * q^13 + 6 * q^16 - 8 * q^17 + 2 * q^19 + 2 * q^20 + 2 * q^22 + 14 * q^23 - 8 * q^25 + 8 * q^26 + 16 * q^29 + 4 * q^31 - 6 * q^32 - 8 * q^34 + 4 * q^37 + 2 * q^38 + 6 * q^40 - 12 * q^41 + 2 * q^43 + 6 * q^44 + 10 * q^46 - 6 * q^47 - 8 * q^50 + 20 * q^52 - 2 * q^55 + 12 * q^58 + 8 * q^59 - 18 * q^61 + 16 * q^62 - 14 * q^64 - 4 * q^65 + 8 * q^67 - 8 * q^68 + 20 * q^71 - 2 * q^73 + 24 * q^74 + 2 * q^76 + 6 * q^80 - 4 * q^82 + 10 * q^83 - 8 * q^85 + 22 * q^86 - 2 * q^88 - 20 * q^89 + 6 * q^92 - 10 * q^94 + 2 * q^95 + 12 * q^97

Introduction

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