LMFDB - Embedded newform 6579.2.a.t.1.18

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6579,2,Mod(1,6579)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6579, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6579.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6579 = 3^{2} \cdot 17 \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6579.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:	$52.5335794898$
Analytic rank:	$0$
Dimension:	$19$
Coefficient field:	$\mathbb{Q}[x]/(x^{19} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{19} - 2 x^{18} - 30 x^{17} + 62 x^{16} + 365 x^{15} - 786 x^{14} - 2295 x^{13} + 5233 x^{12} + \cdots + 64 $ x^19 - 2x^18 - 30x^17 + 62x^16 + 365x^15 - 786x^14 - 2295x^13 + 5233x^12 + 7816x^11 - 19517x^10 - 13527x^9 + 40173x^8 + 8942x^7 - 41911x^6 + 1140x^5 + 18520x^4 - 1520x^3 - 2640x^2 - 64x + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 731)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.18
Root			$-2.48303$ of defining polynomial
Character	$\chi$	$=$	6579.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.48303 q^{2} +4.16544 q^{4} +3.73374 q^{5} -2.89905 q^{7} +5.37685 q^{8} +O(q^{10})$	\(q+2.48303 q^{2} +4.16544 q^{4} +3.73374 q^{5} -2.89905 q^{7} +5.37685 q^{8} +9.27098 q^{10} +5.24858 q^{11} -2.10553 q^{13} -7.19842 q^{14} +5.02000 q^{16} -1.00000 q^{17} +4.79318 q^{19} +15.5526 q^{20} +13.0324 q^{22} -5.89086 q^{23} +8.94078 q^{25} -5.22810 q^{26} -12.0758 q^{28} -7.20505 q^{29} +4.25412 q^{31} +1.71111 q^{32} -2.48303 q^{34} -10.8243 q^{35} +10.0416 q^{37} +11.9016 q^{38} +20.0757 q^{40} +7.38412 q^{41} -1.00000 q^{43} +21.8626 q^{44} -14.6272 q^{46} +11.0405 q^{47} +1.40448 q^{49} +22.2002 q^{50} -8.77047 q^{52} +9.45938 q^{53} +19.5968 q^{55} -15.5877 q^{56} -17.8904 q^{58} -3.18824 q^{59} +4.67100 q^{61} +10.5631 q^{62} -5.79125 q^{64} -7.86151 q^{65} -3.98228 q^{67} -4.16544 q^{68} -26.8770 q^{70} -6.94579 q^{71} -2.14240 q^{73} +24.9336 q^{74} +19.9657 q^{76} -15.2159 q^{77} +6.32969 q^{79} +18.7434 q^{80} +18.3350 q^{82} +7.79387 q^{83} -3.73374 q^{85} -2.48303 q^{86} +28.2208 q^{88} -13.1696 q^{89} +6.10404 q^{91} -24.5380 q^{92} +27.4139 q^{94} +17.8965 q^{95} -17.0924 q^{97} +3.48736 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 19 q - 2 q^{2} + 26 q^{4} - 11 q^{5} + 7 q^{7} + 6 q^{8}+O(q^{10}) $ 19 * q - 2 * q^2 + 26 * q^4 - 11 * q^5 + 7 * q^7 + 6 * q^8	$ 19 q - 2 q^{2} + 26 q^{4} - 11 q^{5} + 7 q^{7} + 6 q^{8} - 2 q^{10} - 4 q^{11} + 14 q^{13} - 5 q^{14} + 32 q^{16} - 19 q^{17} + 12 q^{19} - 23 q^{20} + 36 q^{22} + q^{23} + 30 q^{25} + 21 q^{26} + 5 q^{28} - 41 q^{29} - 8 q^{31} + 20 q^{32} + 2 q^{34} - 3 q^{35} + 50 q^{37} + 29 q^{38} - 15 q^{40} - 6 q^{41} - 19 q^{43} - 16 q^{44} + 38 q^{46} + 21 q^{47} + 46 q^{49} + 36 q^{50} + 39 q^{52} + 9 q^{53} + 10 q^{55} + 12 q^{56} - 45 q^{58} + 4 q^{59} + 68 q^{61} + 25 q^{62} - 14 q^{64} - 22 q^{65} - 26 q^{68} - 37 q^{70} - 23 q^{71} - q^{73} + 30 q^{74} + 47 q^{76} + 19 q^{77} + 16 q^{79} - 28 q^{80} - 13 q^{82} + 32 q^{83} + 11 q^{85} + 2 q^{86} + 108 q^{88} - 11 q^{89} + 52 q^{91} + 23 q^{92} + 47 q^{94} + 25 q^{95} + 36 q^{97} + 100 q^{98}+O(q^{100}) $ 19 * q - 2 * q^2 + 26 * q^4 - 11 * q^5 + 7 * q^7 + 6 * q^8 - 2 * q^10 - 4 * q^11 + 14 * q^13 - 5 * q^14 + 32 * q^16 - 19 * q^17 + 12 * q^19 - 23 * q^20 + 36 * q^22 + q^23 + 30 * q^25 + 21 * q^26 + 5 * q^28 - 41 * q^29 - 8 * q^31 + 20 * q^32 + 2 * q^34 - 3 * q^35 + 50 * q^37 + 29 * q^38 - 15 * q^40 - 6 * q^41 - 19 * q^43 - 16 * q^44 + 38 * q^46 + 21 * q^47 + 46 * q^49 + 36 * q^50 + 39 * q^52 + 9 * q^53 + 10 * q^55 + 12 * q^56 - 45 * q^58 + 4 * q^59 + 68 * q^61 + 25 * q^62 - 14 * q^64 - 22 * q^65 - 26 * q^68 - 37 * q^70 - 23 * q^71 - q^73 + 30 * q^74 + 47 * q^76 + 19 * q^77 + 16 * q^79 - 28 * q^80 - 13 * q^82 + 32 * q^83 + 11 * q^85 + 2 * q^86 + 108 * q^88 - 11 * q^89 + 52 * q^91 + 23 * q^92 + 47 * q^94 + 25 * q^95 + 36 * q^97 + 100 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.48303	1.75577	0.877884	−	0.478874i	$-0.158955\pi$
$2$	2.48303	1.75577	0.877884	+	0.478874i	$0.158955\pi$
$3$	0	0
$3$	0	0
$4$	4.16544	2.08272
$5$	3.73374	1.66978	0.834889	−	0.550419i	$-0.185532\pi$
$5$	3.73374	1.66978	0.834889	+	0.550419i	$0.185532\pi$
$6$	0	0
$7$	−2.89905	−1.09574	−0.547869	−	0.836564i	$-0.684561\pi$
$7$	−2.89905	−1.09574	−0.547869	+	0.836564i	$0.684561\pi$
$8$	5.37685	1.90100
$9$	0	0
$10$	9.27098	2.93174
$11$	5.24858	1.58251	0.791253	−	0.611489i	$-0.209429\pi$
$11$	5.24858	1.58251	0.791253	+	0.611489i	$0.209429\pi$
$12$	0	0
$13$	−2.10553	−0.583970	−0.291985	−	0.956423i	$-0.594316\pi$
$13$	−2.10553	−0.583970	−0.291985	+	0.956423i	$0.594316\pi$
$14$	−7.19842	−1.92386
$15$	0	0
$16$	5.02000	1.25500
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	0	0
$19$	4.79318	1.09963	0.549815	−	0.835286i	$-0.314698\pi$
$19$	4.79318	1.09963	0.549815	+	0.835286i	$0.314698\pi$
$20$	15.5526	3.47768
$21$	0	0
$22$	13.0324	2.77851
$23$	−5.89086	−1.22833	−0.614165	−	0.789178i	$-0.710507\pi$
$23$	−5.89086	−1.22833	−0.614165	+	0.789178i	$0.710507\pi$
$24$	0	0
$25$	8.94078	1.78816
$26$	−5.22810	−1.02532
$27$	0	0
$28$	−12.0758	−2.28211
$29$	−7.20505	−1.33794	−0.668972	−	0.743288i	$-0.733265\pi$
$29$	−7.20505	−1.33794	−0.668972	+	0.743288i	$0.733265\pi$
$30$	0	0
$31$	4.25412	0.764062	0.382031	−	0.924149i	$-0.375225\pi$
$31$	4.25412	0.764062	0.382031	+	0.924149i	$0.375225\pi$
$32$	1.71111	0.302485
$33$	0	0
$34$	−2.48303	−0.425836
$35$	−10.8243	−1.82964
$36$	0	0
$37$	10.0416	1.65083	0.825416	−	0.564525i	$-0.190941\pi$
$37$	10.0416	1.65083	0.825416	+	0.564525i	$0.190941\pi$
$38$	11.9016	1.93070
$39$	0	0
$40$	20.0757	3.17425
$41$	7.38412	1.15321	0.576603	−	0.817024i	$-0.304378\pi$
$41$	7.38412	1.15321	0.576603	+	0.817024i	$0.304378\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	21.8626	3.29591
$45$	0	0
$46$	−14.6272	−2.15666
$47$	11.0405	1.61043	0.805213	−	0.592986i	$-0.202051\pi$
$47$	11.0405	1.61043	0.805213	+	0.592986i	$0.202051\pi$
$48$	0	0
$49$	1.40448	0.200640
$50$	22.2002	3.13959
$51$	0	0
$52$	−8.77047	−1.21625
$53$	9.45938	1.29935	0.649673	−	0.760214i	$-0.274906\pi$
$53$	9.45938	1.29935	0.649673	+	0.760214i	$0.274906\pi$
$54$	0	0
$55$	19.5968	2.64243
$56$	−15.5877	−2.08300
$57$	0	0
$58$	−17.8904	−2.34912
$59$	−3.18824	−0.415074	−0.207537	−	0.978227i	$-0.566545\pi$
$59$	−3.18824	−0.415074	−0.207537	+	0.978227i	$0.566545\pi$
$60$	0	0
$61$	4.67100	0.598060	0.299030	−	0.954244i	$-0.403337\pi$
$61$	4.67100	0.598060	0.299030	+	0.954244i	$0.403337\pi$
$62$	10.5631	1.34152
$63$	0	0
$64$	−5.79125	−0.723907
$65$	−7.86151	−0.975100
$66$	0	0
$67$	−3.98228	−0.486513	−0.243256	−	0.969962i	$-0.578216\pi$
$67$	−3.98228	−0.486513	−0.243256	+	0.969962i	$0.578216\pi$
$68$	−4.16544	−0.505134
$69$	0	0
$70$	−26.8770	−3.21242
$71$	−6.94579	−0.824314	−0.412157	−	0.911113i	$-0.635224\pi$
$71$	−6.94579	−0.824314	−0.412157	+	0.911113i	$0.635224\pi$
$72$	0	0
$73$	−2.14240	−0.250749	−0.125375	−	0.992109i	$-0.540013\pi$
$73$	−2.14240	−0.250749	−0.125375	+	0.992109i	$0.540013\pi$
$74$	24.9336	2.89848
$75$	0	0
$76$	19.9657	2.29022
$77$	−15.2159	−1.73401
$78$	0	0
$79$	6.32969	0.712146	0.356073	−	0.934458i	$-0.384115\pi$
$79$	6.32969	0.712146	0.356073	+	0.934458i	$0.384115\pi$
$80$	18.7434	2.09557
$81$	0	0
$82$	18.3350	2.02476
$83$	7.79387	0.855489	0.427744	−	0.903900i	$-0.359308\pi$
$83$	7.79387	0.855489	0.427744	+	0.903900i	$0.359308\pi$
$84$	0	0
$85$	−3.73374	−0.404980
$86$	−2.48303	−0.267752
$87$	0	0
$88$	28.2208	3.00835
$89$	−13.1696	−1.39597	−0.697987	−	0.716110i	$-0.745921\pi$
$89$	−13.1696	−1.39597	−0.697987	+	0.716110i	$0.745921\pi$
$90$	0	0
$91$	6.10404	0.639878
$92$	−24.5380	−2.55827
$93$	0	0
$94$	27.4139	2.82753
$95$	17.8965	1.83614
$96$	0	0
$97$	−17.0924	−1.73547	−0.867737	−	0.497023i	$-0.834426\pi$
$97$	−17.0924	−1.73547	−0.867737	+	0.497023i	$0.834426\pi$
$98$	3.48736	0.352277
$99$	0	0
$100$	37.2423	3.72423
$101$	6.46694	0.643485	0.321742	−	0.946827i	$-0.395732\pi$
$101$	6.46694	0.643485	0.321742	+	0.946827i	$0.395732\pi$
$102$	0	0
$103$	0.333796	0.0328899	0.0164449	−	0.999865i	$-0.494765\pi$
$103$	0.333796	0.0328899	0.0164449	+	0.999865i	$0.494765\pi$
$104$	−11.3211	−1.11013
$105$	0	0
$106$	23.4879	2.28135
$107$	−16.1810	−1.56428	−0.782138	−	0.623105i	$-0.785871\pi$
$107$	−16.1810	−1.56428	−0.782138	+	0.623105i	$0.785871\pi$
$108$	0	0
$109$	2.79226	0.267450	0.133725	−	0.991018i	$-0.457306\pi$
$109$	2.79226	0.267450	0.133725	+	0.991018i	$0.457306\pi$
$110$	48.6594	4.63950
$111$	0	0
$112$	−14.5532	−1.37515
$113$	5.23619	0.492580	0.246290	−	0.969196i	$-0.420789\pi$
$113$	5.23619	0.492580	0.246290	+	0.969196i	$0.420789\pi$
$114$	0	0
$115$	−21.9949	−2.05104
$116$	−30.0122	−2.78656
$117$	0	0
$118$	−7.91651	−0.728774
$119$	2.89905	0.265755
$120$	0	0
$121$	16.5476	1.50432
$122$	11.5982	1.05005
$123$	0	0
$124$	17.7203	1.59133
$125$	14.7138	1.31605
$126$	0	0
$127$	−6.36680	−0.564962	−0.282481	−	0.959273i	$-0.591157\pi$
$127$	−6.36680	−0.564962	−0.282481	+	0.959273i	$0.591157\pi$
$128$	−17.8021	−1.57350
$129$	0	0
$130$	−19.5204	−1.71205
$131$	−6.58609	−0.575429	−0.287715	−	0.957716i	$-0.592895\pi$
$131$	−6.58609	−0.575429	−0.287715	+	0.957716i	$0.592895\pi$
$132$	0	0
$133$	−13.8957	−1.20491
$134$	−9.88811	−0.854203
$135$	0	0
$136$	−5.37685	−0.461061
$137$	−18.9424	−1.61836	−0.809179	−	0.587562i	$-0.800088\pi$
$137$	−18.9424	−1.61836	−0.809179	+	0.587562i	$0.800088\pi$
$138$	0	0
$139$	11.1156	0.942812	0.471406	−	0.881916i	$-0.343747\pi$
$139$	11.1156	0.942812	0.471406	+	0.881916i	$0.343747\pi$
$140$	−45.0879	−3.81062
$141$	0	0
$142$	−17.2466	−1.44730
$143$	−11.0511	−0.924136
$144$	0	0
$145$	−26.9017	−2.23407
$146$	−5.31965	−0.440257
$147$	0	0
$148$	41.8277	3.43822
$149$	5.33805	0.437310	0.218655	−	0.975802i	$-0.429833\pi$
$149$	5.33805	0.437310	0.218655	+	0.975802i	$0.429833\pi$
$150$	0	0
$151$	−3.89448	−0.316928	−0.158464	−	0.987365i	$-0.550654\pi$
$151$	−3.89448	−0.316928	−0.158464	+	0.987365i	$0.550654\pi$
$152$	25.7722	2.09040
$153$	0	0
$154$	−37.7815	−3.04452
$155$	15.8838	1.27581
$156$	0	0
$157$	−18.1031	−1.44479	−0.722393	−	0.691483i	$-0.756958\pi$
$157$	−18.1031	−1.44479	−0.722393	+	0.691483i	$0.756958\pi$
$158$	15.7168	1.25036
$159$	0	0
$160$	6.38884	0.505082
$161$	17.0779	1.34593
$162$	0	0
$163$	−9.90070	−0.775483	−0.387741	−	0.921768i	$-0.626745\pi$
$163$	−9.90070	−0.775483	−0.387741	+	0.921768i	$0.626745\pi$
$164$	30.7581	2.40180
$165$	0	0
$166$	19.3524	1.50204
$167$	−5.07573	−0.392772	−0.196386	−	0.980527i	$-0.562921\pi$
$167$	−5.07573	−0.392772	−0.196386	+	0.980527i	$0.562921\pi$
$168$	0	0
$169$	−8.56673	−0.658979
$170$	−9.27098	−0.711052
$171$	0	0
$172$	−4.16544	−0.317612
$173$	−11.4702	−0.872065	−0.436033	−	0.899931i	$-0.643617\pi$
$173$	−11.4702	−0.872065	−0.436033	+	0.899931i	$0.643617\pi$
$174$	0	0
$175$	−25.9198	−1.95935
$176$	26.3479	1.98604
$177$	0	0
$178$	−32.7005	−2.45101
$179$	7.95513	0.594594	0.297297	−	0.954785i	$-0.403915\pi$
$179$	7.95513	0.594594	0.297297	+	0.954785i	$0.403915\pi$
$180$	0	0
$181$	22.5233	1.67414	0.837071	−	0.547095i	$-0.184266\pi$
$181$	22.5233	1.67414	0.837071	+	0.547095i	$0.184266\pi$
$182$	15.1565	1.12348
$183$	0	0
$184$	−31.6743	−2.33506
$185$	37.4927	2.75652
$186$	0	0
$187$	−5.24858	−0.383814
$188$	45.9886	3.35406
$189$	0	0
$190$	44.4375	3.22383
$191$	−15.9366	−1.15313	−0.576566	−	0.817050i	$-0.695608\pi$
$191$	−15.9366	−1.15313	−0.576566	+	0.817050i	$0.695608\pi$
$192$	0	0
$193$	7.48700	0.538926	0.269463	−	0.963011i	$-0.413154\pi$
$193$	7.48700	0.538926	0.269463	+	0.963011i	$0.413154\pi$
$194$	−42.4410	−3.04709
$195$	0	0
$196$	5.85027	0.417876
$197$	−24.4292	−1.74051	−0.870255	−	0.492602i	$-0.836046\pi$
$197$	−24.4292	−1.74051	−0.870255	+	0.492602i	$0.836046\pi$
$198$	0	0
$199$	−4.41253	−0.312796	−0.156398	−	0.987694i	$-0.549988\pi$
$199$	−4.41253	−0.312796	−0.156398	+	0.987694i	$0.549988\pi$
$200$	48.0732	3.39929
$201$	0	0
$202$	16.0576	1.12981
$203$	20.8878	1.46603
$204$	0	0
$205$	27.5704	1.92560
$206$	0.828826	0.0577470
$207$	0	0
$208$	−10.5698	−0.732882
$209$	25.1574	1.74017
$210$	0	0
$211$	7.20284	0.495864	0.247932	−	0.968777i	$-0.420249\pi$
$211$	7.20284	0.495864	0.247932	+	0.968777i	$0.420249\pi$
$212$	39.4025	2.70617
$213$	0	0
$214$	−40.1779	−2.74651
$215$	−3.73374	−0.254639
$216$	0	0
$217$	−12.3329	−0.837211
$218$	6.93327	0.469580
$219$	0	0
$220$	81.6293	5.50344
$221$	2.10553	0.141634
$222$	0	0
$223$	−13.5125	−0.904865	−0.452432	−	0.891799i	$-0.649444\pi$
$223$	−13.5125	−0.904865	−0.452432	+	0.891799i	$0.649444\pi$
$224$	−4.96060	−0.331444
$225$	0	0
$226$	13.0016	0.864855
$227$	−9.40679	−0.624350	−0.312175	−	0.950025i	$-0.601058\pi$
$227$	−9.40679	−0.624350	−0.312175	+	0.950025i	$0.601058\pi$
$228$	0	0
$229$	26.5418	1.75393	0.876965	−	0.480554i	$-0.159565\pi$
$229$	26.5418	1.75393	0.876965	+	0.480554i	$0.159565\pi$
$230$	−54.6141	−3.60114
$231$	0	0
$232$	−38.7405	−2.54344
$233$	14.1518	0.927113	0.463557	−	0.886067i	$-0.346573\pi$
$233$	14.1518	0.927113	0.463557	+	0.886067i	$0.346573\pi$
$234$	0	0
$235$	41.2224	2.68905
$236$	−13.2804	−0.864483
$237$	0	0
$238$	7.19842	0.466604
$239$	−21.2848	−1.37680	−0.688400	−	0.725331i	$-0.741687\pi$
$239$	−21.2848	−1.37680	−0.688400	+	0.725331i	$0.741687\pi$
$240$	0	0
$241$	−5.21334	−0.335821	−0.167910	−	0.985802i	$-0.553702\pi$
$241$	−5.21334	−0.335821	−0.167910	+	0.985802i	$0.553702\pi$
$242$	41.0881	2.64124
$243$	0	0
$244$	19.4568	1.24559
$245$	5.24395	0.335024
$246$	0	0
$247$	−10.0922	−0.642151
$248$	22.8738	1.45248
$249$	0	0
$250$	36.5349	2.31067
$251$	−13.9416	−0.879986	−0.439993	−	0.898001i	$-0.645019\pi$
$251$	−13.9416	−0.879986	−0.439993	+	0.898001i	$0.645019\pi$
$252$	0	0
$253$	−30.9187	−1.94384
$254$	−15.8089	−0.991942
$255$	0	0
$256$	−32.6206	−2.03879
$257$	20.9233	1.30516	0.652581	−	0.757719i	$-0.273686\pi$
$257$	20.9233	1.30516	0.652581	+	0.757719i	$0.273686\pi$
$258$	0	0
$259$	−29.1111	−1.80888
$260$	−32.7466	−2.03086
$261$	0	0
$262$	−16.3535	−1.01032
$263$	2.71526	0.167430	0.0837149	−	0.996490i	$-0.473321\pi$
$263$	2.71526	0.167430	0.0837149	+	0.996490i	$0.473321\pi$
$264$	0	0
$265$	35.3188	2.16962
$266$	−34.5033	−2.11553
$267$	0	0
$268$	−16.5879	−1.01327
$269$	−13.5472	−0.825990	−0.412995	−	0.910733i	$-0.635517\pi$
$269$	−13.5472	−0.825990	−0.412995	+	0.910733i	$0.635517\pi$
$270$	0	0
$271$	13.4674	0.818087	0.409044	−	0.912515i	$-0.365862\pi$
$271$	13.4674	0.818087	0.409044	+	0.912515i	$0.365862\pi$
$272$	−5.02000	−0.304382
$273$	0	0
$274$	−47.0346	−2.84146
$275$	46.9264	2.82977
$276$	0	0
$277$	31.1590	1.87216	0.936082	−	0.351782i	$-0.114424\pi$
$277$	31.1590	1.87216	0.936082	+	0.351782i	$0.114424\pi$
$278$	27.6004	1.65536
$279$	0	0
$280$	−58.2005	−3.47815
$281$	16.7994	1.00217	0.501084	−	0.865399i	$-0.332935\pi$
$281$	16.7994	1.00217	0.501084	+	0.865399i	$0.332935\pi$
$282$	0	0
$283$	17.2067	1.02283	0.511417	−	0.859333i	$-0.329121\pi$
$283$	17.2067	1.02283	0.511417	+	0.859333i	$0.329121\pi$
$284$	−28.9323	−1.71681
$285$	0	0
$286$	−27.4401	−1.62257
$287$	−21.4069	−1.26361
$288$	0	0
$289$	1.00000	0.0588235
$290$	−66.7978	−3.92250
$291$	0	0
$292$	−8.92405	−0.522240
$293$	−2.17110	−0.126837	−0.0634185	−	0.997987i	$-0.520200\pi$
$293$	−2.17110	−0.126837	−0.0634185	+	0.997987i	$0.520200\pi$
$294$	0	0
$295$	−11.9041	−0.693081
$296$	53.9923	3.13824
$297$	0	0
$298$	13.2545	0.767815
$299$	12.4034	0.717308
$300$	0	0
$301$	2.89905	0.167098
$302$	−9.67011	−0.556453
$303$	0	0
$304$	24.0618	1.38004
$305$	17.4403	0.998627
$306$	0	0
$307$	−18.5525	−1.05885	−0.529424	−	0.848357i	$-0.677592\pi$
$307$	−18.5525	−1.05885	−0.529424	+	0.848357i	$0.677592\pi$
$308$	−63.3808	−3.61146
$309$	0	0
$310$	39.4398	2.24003
$311$	−24.1577	−1.36986	−0.684928	−	0.728611i	$-0.740166\pi$
$311$	−24.1577	−1.36986	−0.684928	+	0.728611i	$0.740166\pi$
$312$	0	0
$313$	6.56000	0.370793	0.185397	−	0.982664i	$-0.440643\pi$
$313$	6.56000	0.370793	0.185397	+	0.982664i	$0.440643\pi$
$314$	−44.9506	−2.53671
$315$	0	0
$316$	26.3659	1.48320
$317$	−15.9186	−0.894076	−0.447038	−	0.894515i	$-0.647521\pi$
$317$	−15.9186	−0.894076	−0.447038	+	0.894515i	$0.647521\pi$
$318$	0	0
$319$	−37.8163	−2.11730
$320$	−21.6230	−1.20876
$321$	0	0
$322$	42.4049	2.36313
$323$	−4.79318	−0.266700
$324$	0	0
$325$	−18.8251	−1.04423
$326$	−24.5837	−1.36157
$327$	0	0
$328$	39.7033	2.19225
$329$	−32.0070	−1.76460
$330$	0	0
$331$	−1.30472	−0.0717140	−0.0358570	−	0.999357i	$-0.511416\pi$
$331$	−1.30472	−0.0717140	−0.0358570	+	0.999357i	$0.511416\pi$
$332$	32.4649	1.78174
$333$	0	0
$334$	−12.6032	−0.689616
$335$	−14.8688	−0.812368
$336$	0	0
$337$	−18.9400	−1.03173	−0.515864	−	0.856671i	$-0.672529\pi$
$337$	−18.9400	−1.03173	−0.515864	+	0.856671i	$0.672529\pi$
$338$	−21.2714	−1.15701
$339$	0	0
$340$	−15.5526	−0.843461
$341$	22.3281	1.20913
$342$	0	0
$343$	16.2217	0.875889
$344$	−5.37685	−0.289900
$345$	0	0
$346$	−28.4809	−1.53114
$347$	−5.92811	−0.318237	−0.159119	−	0.987259i	$-0.550865\pi$
$347$	−5.92811	−0.318237	−0.159119	+	0.987259i	$0.550865\pi$
$348$	0	0
$349$	1.25124	0.0669773	0.0334886	−	0.999439i	$-0.489338\pi$
$349$	1.25124	0.0669773	0.0334886	+	0.999439i	$0.489338\pi$
$350$	−64.3595	−3.44016
$351$	0	0
$352$	8.98091	0.478684
$353$	−8.22754	−0.437908	−0.218954	−	0.975735i	$-0.570264\pi$
$353$	−8.22754	−0.437908	−0.218954	+	0.975735i	$0.570264\pi$
$354$	0	0
$355$	−25.9338	−1.37642
$356$	−54.8571	−2.90742
$357$	0	0
$358$	19.7528	1.04397
$359$	5.56939	0.293941	0.146971	−	0.989141i	$-0.453048\pi$
$359$	5.56939	0.293941	0.146971	+	0.989141i	$0.453048\pi$
$360$	0	0
$361$	3.97456	0.209187
$362$	55.9260	2.93940
$363$	0	0
$364$	25.4260	1.33269
$365$	−7.99917	−0.418695
$366$	0	0
$367$	−21.9324	−1.14486	−0.572430	−	0.819954i	$-0.693999\pi$
$367$	−21.9324	−1.14486	−0.572430	+	0.819954i	$0.693999\pi$
$368$	−29.5721	−1.54155
$369$	0	0
$370$	93.0956	4.83981
$371$	−27.4232	−1.42374
$372$	0	0
$373$	15.9409	0.825391	0.412695	−	0.910869i	$-0.364587\pi$
$373$	15.9409	0.825391	0.412695	+	0.910869i	$0.364587\pi$
$374$	−13.0324	−0.673888
$375$	0	0
$376$	59.3632	3.06142
$377$	15.1705	0.781319
$378$	0	0
$379$	−5.55551	−0.285368	−0.142684	−	0.989768i	$-0.545573\pi$
$379$	−5.55551	−0.285368	−0.142684	+	0.989768i	$0.545573\pi$
$380$	74.5466	3.82416
$381$	0	0
$382$	−39.5711	−2.02463
$383$	−6.46880	−0.330540	−0.165270	−	0.986248i	$-0.552850\pi$
$383$	−6.46880	−0.330540	−0.165270	+	0.986248i	$0.552850\pi$
$384$	0	0
$385$	−56.8121	−2.89541
$386$	18.5904	0.946229
$387$	0	0
$388$	−71.1975	−3.61451
$389$	1.44264	0.0731446	0.0365723	−	0.999331i	$-0.488356\pi$
$389$	1.44264	0.0731446	0.0365723	+	0.999331i	$0.488356\pi$
$390$	0	0
$391$	5.89086	0.297914
$392$	7.55167	0.381417
$393$	0	0
$394$	−60.6585	−3.05593
$395$	23.6334	1.18912
$396$	0	0
$397$	−21.7105	−1.08962	−0.544810	−	0.838559i	$-0.683398\pi$
$397$	−21.7105	−1.08962	−0.544810	+	0.838559i	$0.683398\pi$
$398$	−10.9565	−0.549197
$399$	0	0
$400$	44.8827	2.24414
$401$	−20.1290	−1.00519	−0.502597	−	0.864521i	$-0.667622\pi$
$401$	−20.1290	−1.00519	−0.502597	+	0.864521i	$0.667622\pi$
$402$	0	0
$403$	−8.95719	−0.446190
$404$	26.9376	1.34020
$405$	0	0
$406$	51.8650	2.57402
$407$	52.7042	2.61245
$408$	0	0
$409$	10.4642	0.517423	0.258712	−	0.965955i	$-0.416702\pi$
$409$	10.4642	0.517423	0.258712	+	0.965955i	$0.416702\pi$
$410$	68.4580	3.38090
$411$	0	0
$412$	1.39041	0.0685004
$413$	9.24287	0.454812
$414$	0	0
$415$	29.1003	1.42848
$416$	−3.60281	−0.176642
$417$	0	0
$418$	62.4665	3.05534
$419$	0.222710	0.0108801	0.00544005	−	0.999985i	$-0.498268\pi$
$419$	0.222710	0.0108801	0.00544005	+	0.999985i	$0.498268\pi$
$420$	0	0
$421$	21.0554	1.02618	0.513089	−	0.858335i	$-0.328501\pi$
$421$	21.0554	1.02618	0.513089	+	0.858335i	$0.328501\pi$
$422$	17.8849	0.870622
$423$	0	0
$424$	50.8616	2.47006
$425$	−8.94078	−0.433692
$426$	0	0
$427$	−13.5414	−0.655317
$428$	−67.4010	−3.25795
$429$	0	0
$430$	−9.27098	−0.447086
$431$	23.3186	1.12322	0.561609	−	0.827403i	$-0.310183\pi$
$431$	23.3186	1.12322	0.561609	+	0.827403i	$0.310183\pi$
$432$	0	0
$433$	10.9526	0.526348	0.263174	−	0.964748i	$-0.415231\pi$
$433$	10.9526	0.526348	0.263174	+	0.964748i	$0.415231\pi$
$434$	−30.6229	−1.46995
$435$	0	0
$436$	11.6310	0.557023
$437$	−28.2360	−1.35071
$438$	0	0
$439$	−35.1971	−1.67987	−0.839934	−	0.542688i	$-0.817406\pi$
$439$	−35.1971	−1.67987	−0.839934	+	0.542688i	$0.817406\pi$
$440$	105.369	5.02327
$441$	0	0
$442$	5.22810	0.248676
$443$	33.0032	1.56803	0.784013	−	0.620744i	$-0.213169\pi$
$443$	33.0032	1.56803	0.784013	+	0.620744i	$0.213169\pi$
$444$	0	0
$445$	−49.1718	−2.33097
$446$	−33.5520	−1.58873
$447$	0	0
$448$	16.7891	0.793212
$449$	13.9175	0.656806	0.328403	−	0.944538i	$-0.393490\pi$
$449$	13.9175	0.656806	0.328403	+	0.944538i	$0.393490\pi$
$450$	0	0
$451$	38.7561	1.82496
$452$	21.8110	1.02590
$453$	0	0
$454$	−23.3573	−1.09621
$455$	22.7909	1.06845
$456$	0	0
$457$	29.0436	1.35860	0.679300	−	0.733861i	$-0.262283\pi$
$457$	29.0436	1.35860	0.679300	+	0.733861i	$0.262283\pi$
$458$	65.9040	3.07949
$459$	0	0
$460$	−91.6185	−4.27174
$461$	−9.21552	−0.429209	−0.214605	−	0.976701i	$-0.568846\pi$
$461$	−9.21552	−0.429209	−0.214605	+	0.976701i	$0.568846\pi$
$462$	0	0
$463$	10.8082	0.502299	0.251149	−	0.967948i	$-0.419191\pi$
$463$	10.8082	0.502299	0.251149	+	0.967948i	$0.419191\pi$
$464$	−36.1693	−1.67912
$465$	0	0
$466$	35.1393	1.62780
$467$	11.1493	0.515926	0.257963	−	0.966155i	$-0.416949\pi$
$467$	11.1493	0.515926	0.257963	+	0.966155i	$0.416949\pi$
$468$	0	0
$469$	11.5448	0.533090
$470$	102.356	4.72135
$471$	0	0
$472$	−17.1427	−0.789057
$473$	−5.24858	−0.241330
$474$	0	0
$475$	42.8548	1.96631
$476$	12.0758	0.553494
$477$	0	0
$478$	−52.8509	−2.41734
$479$	−21.1676	−0.967174	−0.483587	−	0.875296i	$-0.660666\pi$
$479$	−21.1676	−0.967174	−0.483587	+	0.875296i	$0.660666\pi$
$480$	0	0
$481$	−21.1430	−0.964036
$482$	−12.9449	−0.589623
$483$	0	0
$484$	68.9278	3.13308
$485$	−63.8186	−2.89786
$486$	0	0
$487$	−23.0411	−1.04409	−0.522046	−	0.852918i	$-0.674831\pi$
$487$	−23.0411	−1.04409	−0.522046	+	0.852918i	$0.674831\pi$
$488$	25.1152	1.13691
$489$	0	0
$490$	13.0209	0.588224
$491$	10.7845	0.486699	0.243349	−	0.969939i	$-0.421754\pi$
$491$	10.7845	0.486699	0.243349	+	0.969939i	$0.421754\pi$
$492$	0	0
$493$	7.20505	0.324499
$494$	−25.0592	−1.12747
$495$	0	0
$496$	21.3557	0.958898
$497$	20.1362	0.903231
$498$	0	0
$499$	−41.1827	−1.84359	−0.921795	−	0.387678i	$-0.873277\pi$
$499$	−41.1827	−1.84359	−0.921795	+	0.387678i	$0.873277\pi$
$500$	61.2896	2.74095
$501$	0	0
$502$	−34.6174	−1.54505
$503$	−19.3374	−0.862212	−0.431106	−	0.902301i	$-0.641877\pi$
$503$	−19.3374	−0.862212	−0.431106	+	0.902301i	$0.641877\pi$
$504$	0	0
$505$	24.1458	1.07448
$506$	−76.7719	−3.41293
$507$	0	0
$508$	−26.5205	−1.17666
$509$	−2.50526	−0.111044	−0.0555219	−	0.998457i	$-0.517682\pi$
$509$	−2.50526	−0.111044	−0.0555219	+	0.998457i	$0.517682\pi$
$510$	0	0
$511$	6.21093	0.274755
$512$	−45.3938	−2.00614
$513$	0	0
$514$	51.9533	2.29156
$515$	1.24631	0.0549188
$516$	0	0
$517$	57.9470	2.54851
$518$	−72.2838	−3.17597
$519$	0	0
$520$	−42.2701	−1.85367
$521$	−3.63655	−0.159320	−0.0796600	−	0.996822i	$-0.525383\pi$
$521$	−3.63655	−0.159320	−0.0796600	+	0.996822i	$0.525383\pi$
$522$	0	0
$523$	−4.42182	−0.193353	−0.0966763	−	0.995316i	$-0.530821\pi$
$523$	−4.42182	−0.193353	−0.0966763	+	0.995316i	$0.530821\pi$
$524$	−27.4339	−1.19846
$525$	0	0
$526$	6.74206	0.293968
$527$	−4.25412	−0.185312
$528$	0	0
$529$	11.7023	0.508795
$530$	87.6977	3.80934
$531$	0	0
$532$	−57.8815	−2.50948
$533$	−15.5475	−0.673438
$534$	0	0
$535$	−60.4156	−2.61199
$536$	−21.4121	−0.924862
$537$	0	0
$538$	−33.6382	−1.45025
$539$	7.37151	0.317513
$540$	0	0
$541$	38.8114	1.66863	0.834317	−	0.551285i	$-0.185862\pi$
$541$	38.8114	1.66863	0.834317	+	0.551285i	$0.185862\pi$
$542$	33.4400	1.43637
$543$	0	0
$544$	−1.71111	−0.0733634
$545$	10.4256	0.446582
$546$	0	0
$547$	30.0079	1.28304	0.641522	−	0.767105i	$-0.278303\pi$
$547$	30.0079	1.28304	0.641522	+	0.767105i	$0.278303\pi$
$548$	−78.9034	−3.37059
$549$	0	0
$550$	116.520	4.96841
$551$	−34.5351	−1.47124
$552$	0	0
$553$	−18.3501	−0.780324
$554$	77.3688	3.28708
$555$	0	0
$556$	46.3013	1.96361
$557$	−1.26983	−0.0538045	−0.0269022	−	0.999638i	$-0.508564\pi$
$557$	−1.26983	−0.0538045	−0.0269022	+	0.999638i	$0.508564\pi$
$558$	0	0
$559$	2.10553	0.0890546
$560$	−54.3379	−2.29619
$561$	0	0
$562$	41.7134	1.75957
$563$	−12.9251	−0.544727	−0.272363	−	0.962194i	$-0.587805\pi$
$563$	−12.9251	−0.544727	−0.272363	+	0.962194i	$0.587805\pi$
$564$	0	0
$565$	19.5506	0.822498
$566$	42.7248	1.79586
$567$	0	0
$568$	−37.3465	−1.56702
$569$	−31.7808	−1.33232	−0.666160	−	0.745809i	$-0.732063\pi$
$569$	−31.7808	−1.33232	−0.666160	+	0.745809i	$0.732063\pi$
$570$	0	0
$571$	−29.0170	−1.21432	−0.607162	−	0.794578i	$-0.707692\pi$
$571$	−29.0170	−1.21432	−0.607162	+	0.794578i	$0.707692\pi$
$572$	−46.0325	−1.92472
$573$	0	0
$574$	−53.1540	−2.21861
$575$	−52.6689	−2.19645
$576$	0	0
$577$	19.3667	0.806245	0.403123	−	0.915146i	$-0.367925\pi$
$577$	19.3667	0.806245	0.403123	+	0.915146i	$0.367925\pi$
$578$	2.48303	0.103280
$579$	0	0
$580$	−112.058	−4.65294
$581$	−22.5948	−0.937391
$582$	0	0
$583$	49.6483	2.05622
$584$	−11.5194	−0.476675
$585$	0	0
$586$	−5.39090	−0.222696
$587$	10.5817	0.436755	0.218378	−	0.975864i	$-0.429924\pi$
$587$	10.5817	0.436755	0.218378	+	0.975864i	$0.429924\pi$
$588$	0	0
$589$	20.3908	0.840186
$590$	−29.5581	−1.21689
$591$	0	0
$592$	50.4089	2.07179
$593$	−24.7891	−1.01797	−0.508983	−	0.860777i	$-0.669978\pi$
$593$	−24.7891	−1.01797	−0.508983	+	0.860777i	$0.669978\pi$
$594$	0	0
$595$	10.8243	0.443752
$596$	22.2353	0.910794
$597$	0	0
$598$	30.7980	1.25943
$599$	36.9287	1.50887	0.754433	−	0.656377i	$-0.227912\pi$
$599$	36.9287	1.50887	0.754433	+	0.656377i	$0.227912\pi$
$600$	0	0
$601$	32.7020	1.33394	0.666971	−	0.745083i	$-0.267590\pi$
$601$	32.7020	1.33394	0.666971	+	0.745083i	$0.267590\pi$
$602$	7.19842	0.293386
$603$	0	0
$604$	−16.2222	−0.660073
$605$	61.7842	2.51189
$606$	0	0
$607$	−10.9211	−0.443273	−0.221636	−	0.975129i	$-0.571140\pi$
$607$	−10.9211	−0.443273	−0.221636	+	0.975129i	$0.571140\pi$
$608$	8.20167	0.332622
$609$	0	0
$610$	43.3047	1.75336
$611$	−23.2462	−0.940440
$612$	0	0
$613$	34.5229	1.39437	0.697184	−	0.716892i	$-0.254436\pi$
$613$	34.5229	1.39437	0.697184	+	0.716892i	$0.254436\pi$
$614$	−46.0665	−1.85909
$615$	0	0
$616$	−81.8135	−3.29636
$617$	−29.6912	−1.19532	−0.597662	−	0.801748i	$-0.703903\pi$
$617$	−29.6912	−1.19532	−0.597662	+	0.801748i	$0.703903\pi$
$618$	0	0
$619$	−33.7639	−1.35709	−0.678544	−	0.734560i	$-0.737389\pi$
$619$	−33.7639	−1.35709	−0.678544	+	0.734560i	$0.737389\pi$
$620$	66.1628	2.65716
$621$	0	0
$622$	−59.9842	−2.40515
$623$	38.1793	1.52962
$624$	0	0
$625$	10.2337	0.409347
$626$	16.2887	0.651027
$627$	0	0
$628$	−75.4074	−3.00908
$629$	−10.0416	−0.400385
$630$	0	0
$631$	−11.5560	−0.460038	−0.230019	−	0.973186i	$-0.573879\pi$
$631$	−11.5560	−0.460038	−0.230019	+	0.973186i	$0.573879\pi$
$632$	34.0338	1.35379
$633$	0	0
$634$	−39.5263	−1.56979
$635$	−23.7719	−0.943361
$636$	0	0
$637$	−2.95718	−0.117168
$638$	−93.8989	−3.71749
$639$	0	0
$640$	−66.4683	−2.62739
$641$	9.48585	0.374669	0.187334	−	0.982296i	$-0.440015\pi$
$641$	9.48585	0.374669	0.187334	+	0.982296i	$0.440015\pi$
$642$	0	0
$643$	−48.3241	−1.90572	−0.952858	−	0.303416i	$-0.901873\pi$
$643$	−48.3241	−1.90572	−0.952858	+	0.303416i	$0.901873\pi$
$644$	71.1369	2.80319
$645$	0	0
$646$	−11.9016	−0.468262
$647$	−37.2131	−1.46300	−0.731500	−	0.681842i	$-0.761179\pi$
$647$	−37.2131	−1.46300	−0.731500	+	0.681842i	$0.761179\pi$
$648$	0	0
$649$	−16.7337	−0.656857
$650$	−46.7433	−1.83342
$651$	0	0
$652$	−41.2408	−1.61511
$653$	−4.28947	−0.167860	−0.0839299	−	0.996472i	$-0.526747\pi$
$653$	−4.28947	−0.167860	−0.0839299	+	0.996472i	$0.526747\pi$
$654$	0	0
$655$	−24.5907	−0.960839
$656$	37.0683	1.44727
$657$	0	0
$658$	−79.4743	−3.09823
$659$	23.2327	0.905018	0.452509	−	0.891760i	$-0.350529\pi$
$659$	23.2327	0.905018	0.452509	+	0.891760i	$0.350529\pi$
$660$	0	0
$661$	−32.1230	−1.24944	−0.624721	−	0.780848i	$-0.714787\pi$
$661$	−32.1230	−1.24944	−0.624721	+	0.780848i	$0.714787\pi$
$662$	−3.23966	−0.125913
$663$	0	0
$664$	41.9065	1.62629
$665$	−51.8827	−2.01192
$666$	0	0
$667$	42.4440	1.64344
$668$	−21.1426	−0.818033
$669$	0	0
$670$	−36.9196	−1.42633
$671$	24.5161	0.946433
$672$	0	0
$673$	−28.9871	−1.11737	−0.558685	−	0.829380i	$-0.688694\pi$
$673$	−28.9871	−1.11737	−0.558685	+	0.829380i	$0.688694\pi$
$674$	−47.0286	−1.81147
$675$	0	0
$676$	−35.6842	−1.37247
$677$	12.4469	0.478374	0.239187	−	0.970974i	$-0.423119\pi$
$677$	12.4469	0.478374	0.239187	+	0.970974i	$0.423119\pi$
$678$	0	0
$679$	49.5518	1.90162
$680$	−20.0757	−0.769869
$681$	0	0
$682$	55.4413	2.12296
$683$	34.8683	1.33420	0.667100	−	0.744968i	$-0.267536\pi$
$683$	34.8683	1.33420	0.667100	+	0.744968i	$0.267536\pi$
$684$	0	0
$685$	−70.7259	−2.70230
$686$	40.2789	1.53786
$687$	0	0
$688$	−5.02000	−0.191386
$689$	−19.9170	−0.758779
$690$	0	0
$691$	2.64606	0.100661	0.0503305	−	0.998733i	$-0.483973\pi$
$691$	2.64606	0.100661	0.0503305	+	0.998733i	$0.483973\pi$
$692$	−47.7785	−1.81627
$693$	0	0
$694$	−14.7197	−0.558751
$695$	41.5027	1.57429
$696$	0	0
$697$	−7.38412	−0.279694
$698$	3.10686	0.117596
$699$	0	0
$700$	−107.967	−4.08077
$701$	−25.9422	−0.979822	−0.489911	−	0.871772i	$-0.662971\pi$
$701$	−25.9422	−0.979822	−0.489911	+	0.871772i	$0.662971\pi$
$702$	0	0
$703$	48.1313	1.81530
$704$	−30.3958	−1.14559
$705$	0	0
$706$	−20.4292	−0.768865
$707$	−18.7480	−0.705090
$708$	0	0
$709$	32.4709	1.21947	0.609735	−	0.792605i	$-0.291276\pi$
$709$	32.4709	1.21947	0.609735	+	0.792605i	$0.291276\pi$
$710$	−64.3943	−2.41667
$711$	0	0
$712$	−70.8109	−2.65375
$713$	−25.0604	−0.938521
$714$	0	0
$715$	−41.2617	−1.54310
$716$	33.1366	1.23837
$717$	0	0
$718$	13.8290	0.516092
$719$	39.2066	1.46216	0.731080	−	0.682292i	$-0.239017\pi$
$719$	39.2066	1.46216	0.731080	+	0.682292i	$0.239017\pi$
$720$	0	0
$721$	−0.967691	−0.0360387
$722$	9.86896	0.367284
$723$	0	0
$724$	93.8193	3.48677
$725$	−64.4188	−2.39245
$726$	0	0
$727$	−16.1489	−0.598931	−0.299466	−	0.954107i	$-0.596808\pi$
$727$	−16.1489	−0.598931	−0.299466	+	0.954107i	$0.596808\pi$
$728$	32.8205	1.21641
$729$	0	0
$730$	−19.8622	−0.735132
$731$	1.00000	0.0369863
$732$	0	0
$733$	52.1551	1.92639	0.963196	−	0.268798i	$-0.0866265\pi$
$733$	52.1551	1.92639	0.963196	+	0.268798i	$0.0866265\pi$
$734$	−54.4587	−2.01011
$735$	0	0
$736$	−10.0799	−0.371551
$737$	−20.9013	−0.769909
$738$	0	0
$739$	18.9727	0.697923	0.348961	−	0.937137i	$-0.386534\pi$
$739$	18.9727	0.697923	0.348961	+	0.937137i	$0.386534\pi$
$740$	156.174	5.74106
$741$	0	0
$742$	−68.0926	−2.49976
$743$	15.1312	0.555111	0.277555	−	0.960710i	$-0.410476\pi$
$743$	15.1312	0.555111	0.277555	+	0.960710i	$0.410476\pi$
$744$	0	0
$745$	19.9309	0.730211
$746$	39.5818	1.44919
$747$	0	0
$748$	−21.8626	−0.799377
$749$	46.9095	1.71404
$750$	0	0
$751$	−6.07538	−0.221694	−0.110847	−	0.993837i	$-0.535356\pi$
$751$	−6.07538	−0.221694	−0.110847	+	0.993837i	$0.535356\pi$
$752$	55.4234	2.02108
$753$	0	0
$754$	37.6687	1.37181
$755$	−14.5410	−0.529200
$756$	0	0
$757$	−20.9641	−0.761955	−0.380977	−	0.924584i	$-0.624412\pi$
$757$	−20.9641	−0.761955	−0.380977	+	0.924584i	$0.624412\pi$
$758$	−13.7945	−0.501039
$759$	0	0
$760$	96.2266	3.49050
$761$	−1.78017	−0.0645312	−0.0322656	−	0.999479i	$-0.510272\pi$
$761$	−1.78017	−0.0645312	−0.0322656	+	0.999479i	$0.510272\pi$
$762$	0	0
$763$	−8.09490	−0.293055
$764$	−66.3830	−2.40165
$765$	0	0
$766$	−16.0622	−0.580352
$767$	6.71296	0.242391
$768$	0	0
$769$	1.10944	0.0400075	0.0200038	−	0.999800i	$-0.493632\pi$
$769$	1.10944	0.0400075	0.0200038	+	0.999800i	$0.493632\pi$
$770$	−141.066	−5.08367
$771$	0	0
$772$	31.1866	1.12243
$773$	−9.68588	−0.348377	−0.174188	−	0.984712i	$-0.555730\pi$
$773$	−9.68588	−0.348377	−0.174188	+	0.984712i	$0.555730\pi$
$774$	0	0
$775$	38.0352	1.36626
$776$	−91.9035	−3.29914
$777$	0	0
$778$	3.58211	0.128425
$779$	35.3934	1.26810
$780$	0	0
$781$	−36.4555	−1.30448
$782$	14.6272	0.523067
$783$	0	0
$784$	7.05048	0.251803
$785$	−67.5922	−2.41247
$786$	0	0
$787$	−20.3631	−0.725865	−0.362932	−	0.931815i	$-0.618224\pi$
$787$	−20.3631	−0.725865	−0.362932	+	0.931815i	$0.618224\pi$
$788$	−101.758	−3.62499
$789$	0	0
$790$	58.6824	2.08783
$791$	−15.1800	−0.539738
$792$	0	0
$793$	−9.83494	−0.349249
$794$	−53.9079	−1.91312
$795$	0	0
$796$	−18.3801	−0.651467
$797$	20.3068	0.719303	0.359651	−	0.933087i	$-0.382896\pi$
$797$	20.3068	0.719303	0.359651	+	0.933087i	$0.382896\pi$
$798$	0	0
$799$	−11.0405	−0.390586
$800$	15.2987	0.540890
$801$	0	0
$802$	−49.9809	−1.76489
$803$	−11.2446	−0.396812
$804$	0	0
$805$	63.7643	2.24740
$806$	−22.2410	−0.783405
$807$	0	0
$808$	34.7718	1.22327
$809$	14.5663	0.512123	0.256062	−	0.966660i	$-0.417575\pi$
$809$	14.5663	0.512123	0.256062	+	0.966660i	$0.417575\pi$
$810$	0	0
$811$	14.0121	0.492033	0.246016	−	0.969266i	$-0.420878\pi$
$811$	14.0121	0.492033	0.246016	+	0.969266i	$0.420878\pi$
$812$	87.0068	3.05334
$813$	0	0
$814$	130.866	4.58685
$815$	−36.9666	−1.29488
$816$	0	0
$817$	−4.79318	−0.167692
$818$	25.9830	0.908475
$819$	0	0
$820$	114.843	4.01048
$821$	−6.26833	−0.218766	−0.109383	−	0.994000i	$-0.534888\pi$
$821$	−6.26833	−0.218766	−0.109383	+	0.994000i	$0.534888\pi$
$822$	0	0
$823$	17.7633	0.619189	0.309595	−	0.950869i	$-0.399807\pi$
$823$	17.7633	0.619189	0.309595	+	0.950869i	$0.399807\pi$
$824$	1.79477	0.0625238
$825$	0	0
$826$	22.9503	0.798544
$827$	−13.3481	−0.464160	−0.232080	−	0.972697i	$-0.574553\pi$
$827$	−13.3481	−0.464160	−0.232080	+	0.972697i	$0.574553\pi$
$828$	0	0
$829$	27.8583	0.967558	0.483779	−	0.875190i	$-0.339264\pi$
$829$	27.8583	0.967558	0.483779	+	0.875190i	$0.339264\pi$
$830$	72.2568	2.50807
$831$	0	0
$832$	12.1937	0.422740
$833$	−1.40448	−0.0486623
$834$	0	0
$835$	−18.9514	−0.655841
$836$	104.791	3.62429
$837$	0	0
$838$	0.552996	0.0191029
$839$	5.04719	0.174248	0.0871241	−	0.996197i	$-0.472232\pi$
$839$	5.04719	0.174248	0.0871241	+	0.996197i	$0.472232\pi$
$840$	0	0
$841$	22.9127	0.790094
$842$	52.2813	1.80173
$843$	0	0
$844$	30.0030	1.03275
$845$	−31.9859	−1.10035
$846$	0	0
$847$	−47.9722	−1.64834
$848$	47.4861	1.63068
$849$	0	0
$850$	−22.2002	−0.761462
$851$	−59.1538	−2.02777
$852$	0	0
$853$	−49.6200	−1.69896	−0.849478	−	0.527624i	$-0.823083\pi$
$853$	−49.6200	−1.69896	−0.849478	+	0.527624i	$0.823083\pi$
$854$	−33.6238	−1.15058
$855$	0	0
$856$	−87.0028	−2.97369
$857$	5.03551	0.172010	0.0860049	−	0.996295i	$-0.472590\pi$
$857$	5.03551	0.172010	0.0860049	+	0.996295i	$0.472590\pi$
$858$	0	0
$859$	22.3089	0.761171	0.380586	−	0.924746i	$-0.375722\pi$
$859$	22.3089	0.761171	0.380586	+	0.924746i	$0.375722\pi$
$860$	−15.5526	−0.530341
$861$	0	0
$862$	57.9008	1.97211
$863$	33.9134	1.15443	0.577214	−	0.816593i	$-0.304140\pi$
$863$	33.9134	1.15443	0.577214	+	0.816593i	$0.304140\pi$
$864$	0	0
$865$	−42.8268	−1.45616
$866$	27.1956	0.924145
$867$	0	0
$868$	−51.3719	−1.74368
$869$	33.2219	1.12697
$870$	0	0
$871$	8.38482	0.284109
$872$	15.0136	0.508423
$873$	0	0
$874$	−70.1107	−2.37153
$875$	−42.6561	−1.44204
$876$	0	0
$877$	44.7321	1.51049	0.755247	−	0.655440i	$-0.227517\pi$
$877$	44.7321	1.51049	0.755247	+	0.655440i	$0.227517\pi$
$878$	−87.3956	−2.94946
$879$	0	0
$880$	98.3759	3.31625
$881$	13.4180	0.452064	0.226032	−	0.974120i	$-0.427425\pi$
$881$	13.4180	0.452064	0.226032	+	0.974120i	$0.427425\pi$
$882$	0	0
$883$	25.8417	0.869642	0.434821	−	0.900517i	$-0.356812\pi$
$883$	25.8417	0.869642	0.434821	+	0.900517i	$0.356812\pi$
$884$	8.77047	0.294983
$885$	0	0
$886$	81.9478	2.75309
$887$	−5.45803	−0.183263	−0.0916314	−	0.995793i	$-0.529208\pi$
$887$	−5.45803	−0.183263	−0.0916314	+	0.995793i	$0.529208\pi$
$888$	0	0
$889$	18.4576	0.619050
$890$	−122.095	−4.09263
$891$	0	0
$892$	−56.2855	−1.88458
$893$	52.9192	1.77087
$894$	0	0
$895$	29.7024	0.992840
$896$	51.6091	1.72414
$897$	0	0
$898$	34.5575	1.15320
$899$	−30.6511	−1.02227
$900$	0	0
$901$	−9.45938	−0.315138
$902$	96.2327	3.20420
$903$	0	0
$904$	28.1542	0.936395
$905$	84.0959	2.79544
$906$	0	0
$907$	2.56975	0.0853272	0.0426636	−	0.999089i	$-0.486416\pi$
$907$	2.56975	0.0853272	0.0426636	+	0.999089i	$0.486416\pi$
$908$	−39.1834	−1.30035
$909$	0	0
$910$	56.5904	1.87596
$911$	−0.704453	−0.0233396	−0.0116698	−	0.999932i	$-0.503715\pi$
$911$	−0.704453	−0.0233396	−0.0116698	+	0.999932i	$0.503715\pi$
$912$	0	0
$913$	40.9067	1.35382
$914$	72.1160	2.38539
$915$	0	0
$916$	110.558	3.65294
$917$	19.0934	0.630519
$918$	0	0
$919$	32.0951	1.05872	0.529360	−	0.848397i	$-0.322432\pi$
$919$	32.0951	1.05872	0.529360	+	0.848397i	$0.322432\pi$
$920$	−118.263	−3.89903
$921$	0	0
$922$	−22.8824	−0.753592
$923$	14.6246	0.481375
$924$	0	0
$925$	89.7799	2.95194
$926$	26.8370	0.881920
$927$	0	0
$928$	−12.3286	−0.404708
$929$	12.8948	0.423066	0.211533	−	0.977371i	$-0.432154\pi$
$929$	12.8948	0.423066	0.211533	+	0.977371i	$0.432154\pi$
$930$	0	0
$931$	6.73191	0.220630
$932$	58.9483	1.93092
$933$	0	0
$934$	27.6839	0.905846
$935$	−19.5968	−0.640884
$936$	0	0
$937$	41.9355	1.36997	0.684987	−	0.728555i	$-0.259808\pi$
$937$	41.9355	1.36997	0.684987	+	0.728555i	$0.259808\pi$
$938$	28.6661	0.935982
$939$	0	0
$940$	171.709	5.60054
$941$	1.22427	0.0399102	0.0199551	−	0.999801i	$-0.493648\pi$
$941$	1.22427	0.0399102	0.0199551	+	0.999801i	$0.493648\pi$
$942$	0	0
$943$	−43.4989	−1.41652
$944$	−16.0050	−0.520918
$945$	0	0
$946$	−13.0324	−0.423719
$947$	−28.3500	−0.921250	−0.460625	−	0.887595i	$-0.652375\pi$
$947$	−28.3500	−0.921250	−0.460625	+	0.887595i	$0.652375\pi$
$948$	0	0
$949$	4.51090	0.146430
$950$	106.410	3.45239
$951$	0	0
$952$	15.5877	0.505202
$953$	46.0640	1.49216	0.746079	−	0.665857i	$-0.231934\pi$
$953$	46.0640	1.49216	0.746079	+	0.665857i	$0.231934\pi$
$954$	0	0
$955$	−59.5031	−1.92547
$956$	−88.6606	−2.86749
$957$	0	0
$958$	−52.5599	−1.69813
$959$	54.9149	1.77330
$960$	0	0
$961$	−12.9025	−0.416209
$962$	−52.4986	−1.69262
$963$	0	0
$964$	−21.7158	−0.699420
$965$	27.9545	0.899886
$966$	0	0
$967$	−28.1322	−0.904672	−0.452336	−	0.891848i	$-0.649409\pi$
$967$	−28.1322	−0.904672	−0.452336	+	0.891848i	$0.649409\pi$
$968$	88.9737	2.85972
$969$	0	0
$970$	−158.464	−5.08796
$971$	47.2547	1.51647	0.758237	−	0.651979i	$-0.226061\pi$
$971$	47.2547	1.51647	0.758237	+	0.651979i	$0.226061\pi$
$972$	0	0
$973$	−32.2246	−1.03307
$974$	−57.2117	−1.83318
$975$	0	0
$976$	23.4484	0.750565
$977$	−11.1535	−0.356833	−0.178416	−	0.983955i	$-0.557097\pi$
$977$	−11.1535	−0.356833	−0.178416	+	0.983955i	$0.557097\pi$
$978$	0	0
$979$	−69.1216	−2.20914
$980$	21.8433	0.697760
$981$	0	0
$982$	26.7783	0.854530
$983$	48.8894	1.55933	0.779665	−	0.626197i	$-0.215389\pi$
$983$	48.8894	1.55933	0.779665	+	0.626197i	$0.215389\pi$
$984$	0	0
$985$	−91.2122	−2.90626
$986$	17.8904	0.569745
$987$	0	0
$988$	−42.0384	−1.33742
$989$	5.89086	0.187319
$990$	0	0
$991$	37.0459	1.17680	0.588401	−	0.808570i	$-0.299758\pi$
$991$	37.0459	1.17680	0.588401	+	0.808570i	$0.299758\pi$
$992$	7.27928	0.231117
$993$	0	0
$994$	49.9987	1.58586
$995$	−16.4752	−0.522300
$996$	0	0
$997$	−32.7001	−1.03562	−0.517812	−	0.855495i	$-0.673253\pi$
$997$	−32.7001	−1.03562	−0.517812	+	0.855495i	$0.673253\pi$
$998$	−102.258	−3.23691
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6579.2.a.t.1.18		19
3.2	odd	2		731.2.a.e.1.2	✓	19

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
731.2.a.e.1.2	✓	19	3.2	odd	2
6579.2.a.t.1.18		19	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6579 = 3^{2} \cdot 17 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6579.a (trivial)

\(f(q)\)	\(=\)	\(q+2.48303 q^{2} +4.16544 q^{4} +3.73374 q^{5} -2.89905 q^{7} +5.37685 q^{8} +O(q^{10})\)	\(q+2.48303 q^{2} +4.16544 q^{4} +3.73374 q^{5} -2.89905 q^{7} +5.37685 q^{8} +9.27098 q^{10} +5.24858 q^{11} -2.10553 q^{13} -7.19842 q^{14} +5.02000 q^{16} -1.00000 q^{17} +4.79318 q^{19} +15.5526 q^{20} +13.0324 q^{22} -5.89086 q^{23} +8.94078 q^{25} -5.22810 q^{26} -12.0758 q^{28} -7.20505 q^{29} +4.25412 q^{31} +1.71111 q^{32} -2.48303 q^{34} -10.8243 q^{35} +10.0416 q^{37} +11.9016 q^{38} +20.0757 q^{40} +7.38412 q^{41} -1.00000 q^{43} +21.8626 q^{44} -14.6272 q^{46} +11.0405 q^{47} +1.40448 q^{49} +22.2002 q^{50} -8.77047 q^{52} +9.45938 q^{53} +19.5968 q^{55} -15.5877 q^{56} -17.8904 q^{58} -3.18824 q^{59} +4.67100 q^{61} +10.5631 q^{62} -5.79125 q^{64} -7.86151 q^{65} -3.98228 q^{67} -4.16544 q^{68} -26.8770 q^{70} -6.94579 q^{71} -2.14240 q^{73} +24.9336 q^{74} +19.9657 q^{76} -15.2159 q^{77} +6.32969 q^{79} +18.7434 q^{80} +18.3350 q^{82} +7.79387 q^{83} -3.73374 q^{85} -2.48303 q^{86} +28.2208 q^{88} -13.1696 q^{89} +6.10404 q^{91} -24.5380 q^{92} +27.4139 q^{94} +17.8965 q^{95} -17.0924 q^{97} +3.48736 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 19 q - 2 q^{2} + 26 q^{4} - 11 q^{5} + 7 q^{7} + 6 q^{8}+O(q^{10}) \) 19 * q - 2 * q^2 + 26 * q^4 - 11 * q^5 + 7 * q^7 + 6 * q^8	\( 19 q - 2 q^{2} + 26 q^{4} - 11 q^{5} + 7 q^{7} + 6 q^{8} - 2 q^{10} - 4 q^{11} + 14 q^{13} - 5 q^{14} + 32 q^{16} - 19 q^{17} + 12 q^{19} - 23 q^{20} + 36 q^{22} + q^{23} + 30 q^{25} + 21 q^{26} + 5 q^{28} - 41 q^{29} - 8 q^{31} + 20 q^{32} + 2 q^{34} - 3 q^{35} + 50 q^{37} + 29 q^{38} - 15 q^{40} - 6 q^{41} - 19 q^{43} - 16 q^{44} + 38 q^{46} + 21 q^{47} + 46 q^{49} + 36 q^{50} + 39 q^{52} + 9 q^{53} + 10 q^{55} + 12 q^{56} - 45 q^{58} + 4 q^{59} + 68 q^{61} + 25 q^{62} - 14 q^{64} - 22 q^{65} - 26 q^{68} - 37 q^{70} - 23 q^{71} - q^{73} + 30 q^{74} + 47 q^{76} + 19 q^{77} + 16 q^{79} - 28 q^{80} - 13 q^{82} + 32 q^{83} + 11 q^{85} + 2 q^{86} + 108 q^{88} - 11 q^{89} + 52 q^{91} + 23 q^{92} + 47 q^{94} + 25 q^{95} + 36 q^{97} + 100 q^{98}+O(q^{100}) \) 19 * q - 2 * q^2 + 26 * q^4 - 11 * q^5 + 7 * q^7 + 6 * q^8 - 2 * q^10 - 4 * q^11 + 14 * q^13 - 5 * q^14 + 32 * q^16 - 19 * q^17 + 12 * q^19 - 23 * q^20 + 36 * q^22 + q^23 + 30 * q^25 + 21 * q^26 + 5 * q^28 - 41 * q^29 - 8 * q^31 + 20 * q^32 + 2 * q^34 - 3 * q^35 + 50 * q^37 + 29 * q^38 - 15 * q^40 - 6 * q^41 - 19 * q^43 - 16 * q^44 + 38 * q^46 + 21 * q^47 + 46 * q^49 + 36 * q^50 + 39 * q^52 + 9 * q^53 + 10 * q^55 + 12 * q^56 - 45 * q^58 + 4 * q^59 + 68 * q^61 + 25 * q^62 - 14 * q^64 - 22 * q^65 - 26 * q^68 - 37 * q^70 - 23 * q^71 - q^73 + 30 * q^74 + 47 * q^76 + 19 * q^77 + 16 * q^79 - 28 * q^80 - 13 * q^82 + 32 * q^83 + 11 * q^85 + 2 * q^86 + 108 * q^88 - 11 * q^89 + 52 * q^91 + 23 * q^92 + 47 * q^94 + 25 * q^95 + 36 * q^97 + 100 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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