LMFDB - Embedded newform 8046.2.a.o.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, 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("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 29 x^{10} + 76 x^{9} + 320 x^{8} - 724 x^{7} - 1643 x^{6} + 3265 x^{5} + 3921 x^{4} + \cdots + 423 $ x^12 - 3x^11 - 29x^10 + 76x^9 + 320x^8 - 724x^7 - 1643x^6 + 3265x^5 + 3921x^4 - 6927x^3 - 3639x^2 + 5508*x + 423
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.46524$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -1.46524 q^{5} -3.53999 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -1.46524 q^{5} -3.53999 q^{7} +1.00000 q^{8} -1.46524 q^{10} +4.22810 q^{11} -0.227772 q^{13} -3.53999 q^{14} +1.00000 q^{16} -6.27991 q^{17} +2.57419 q^{19} -1.46524 q^{20} +4.22810 q^{22} -3.30009 q^{23} -2.85307 q^{25} -0.227772 q^{26} -3.53999 q^{28} +3.57821 q^{29} +7.37510 q^{31} +1.00000 q^{32} -6.27991 q^{34} +5.18694 q^{35} -4.39823 q^{37} +2.57419 q^{38} -1.46524 q^{40} +4.53941 q^{41} -1.36273 q^{43} +4.22810 q^{44} -3.30009 q^{46} -7.53590 q^{47} +5.53153 q^{49} -2.85307 q^{50} -0.227772 q^{52} +10.3373 q^{53} -6.19518 q^{55} -3.53999 q^{56} +3.57821 q^{58} +1.84299 q^{59} -0.0848386 q^{61} +7.37510 q^{62} +1.00000 q^{64} +0.333741 q^{65} +4.55903 q^{67} -6.27991 q^{68} +5.18694 q^{70} -2.24203 q^{71} -6.79670 q^{73} -4.39823 q^{74} +2.57419 q^{76} -14.9674 q^{77} +3.14110 q^{79} -1.46524 q^{80} +4.53941 q^{82} -1.74078 q^{83} +9.20158 q^{85} -1.36273 q^{86} +4.22810 q^{88} +5.71960 q^{89} +0.806312 q^{91} -3.30009 q^{92} -7.53590 q^{94} -3.77181 q^{95} +19.1029 q^{97} +5.53153 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 12 q^{2} + 12 q^{4} + 3 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) $ 12 * q + 12 * q^2 + 12 * q^4 + 3 * q^5 + 6 * q^7 + 12 * q^8	$ 12 q + 12 q^{2} + 12 q^{4} + 3 q^{5} + 6 q^{7} + 12 q^{8} + 3 q^{10} + 10 q^{11} + 5 q^{13} + 6 q^{14} + 12 q^{16} + 8 q^{17} + 2 q^{19} + 3 q^{20} + 10 q^{22} + 9 q^{23} + 7 q^{25} + 5 q^{26} + 6 q^{28} + 19 q^{29} + 10 q^{31} + 12 q^{32} + 8 q^{34} + 20 q^{35} + 11 q^{37} + 2 q^{38} + 3 q^{40} + 8 q^{41} + 13 q^{43} + 10 q^{44} + 9 q^{46} + 11 q^{47} + 2 q^{49} + 7 q^{50} + 5 q^{52} + 24 q^{53} + 3 q^{55} + 6 q^{56} + 19 q^{58} + 10 q^{59} + 10 q^{62} + 12 q^{64} + 28 q^{65} + 21 q^{67} + 8 q^{68} + 20 q^{70} + 37 q^{71} - 2 q^{73} + 11 q^{74} + 2 q^{76} + 2 q^{77} + 7 q^{79} + 3 q^{80} + 8 q^{82} + 22 q^{83} + 15 q^{85} + 13 q^{86} + 10 q^{88} + 40 q^{89} + q^{91} + 9 q^{92} + 11 q^{94} + 11 q^{95} + 7 q^{97} + 2 q^{98}+O(q^{100}) $ 12 * q + 12 * q^2 + 12 * q^4 + 3 * q^5 + 6 * q^7 + 12 * q^8 + 3 * q^10 + 10 * q^11 + 5 * q^13 + 6 * q^14 + 12 * q^16 + 8 * q^17 + 2 * q^19 + 3 * q^20 + 10 * q^22 + 9 * q^23 + 7 * q^25 + 5 * q^26 + 6 * q^28 + 19 * q^29 + 10 * q^31 + 12 * q^32 + 8 * q^34 + 20 * q^35 + 11 * q^37 + 2 * q^38 + 3 * q^40 + 8 * q^41 + 13 * q^43 + 10 * q^44 + 9 * q^46 + 11 * q^47 + 2 * q^49 + 7 * q^50 + 5 * q^52 + 24 * q^53 + 3 * q^55 + 6 * q^56 + 19 * q^58 + 10 * q^59 + 10 * q^62 + 12 * q^64 + 28 * q^65 + 21 * q^67 + 8 * q^68 + 20 * q^70 + 37 * q^71 - 2 * q^73 + 11 * q^74 + 2 * q^76 + 2 * q^77 + 7 * q^79 + 3 * q^80 + 8 * q^82 + 22 * q^83 + 15 * q^85 + 13 * q^86 + 10 * q^88 + 40 * q^89 + q^91 + 9 * q^92 + 11 * q^94 + 11 * q^95 + 7 * q^97 + 2 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.46524	−0.655276	−0.327638	−	0.944803i	$-0.606253\pi$
$5$	−1.46524	−0.655276	−0.327638	+	0.944803i	$0.606253\pi$
$6$	0	0
$7$	−3.53999	−1.33799	−0.668995	−	0.743267i	$-0.733275\pi$
$7$	−3.53999	−1.33799	−0.668995	+	0.743267i	$0.733275\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.46524	−0.463350
$11$	4.22810	1.27482	0.637410	−	0.770525i	$-0.280006\pi$
$11$	4.22810	1.27482	0.637410	+	0.770525i	$0.280006\pi$
$12$	0	0
$13$	−0.227772	−0.0631727	−0.0315863	−	0.999501i	$-0.510056\pi$
$13$	−0.227772	−0.0631727	−0.0315863	+	0.999501i	$0.510056\pi$
$14$	−3.53999	−0.946102
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.27991	−1.52310	−0.761551	−	0.648105i	$-0.775562\pi$
$17$	−6.27991	−1.52310	−0.761551	+	0.648105i	$0.775562\pi$
$18$	0	0
$19$	2.57419	0.590560	0.295280	−	0.955411i	$-0.404587\pi$
$19$	2.57419	0.590560	0.295280	+	0.955411i	$0.404587\pi$
$20$	−1.46524	−0.327638
$21$	0	0
$22$	4.22810	0.901433
$23$	−3.30009	−0.688116	−0.344058	−	0.938948i	$-0.611802\pi$
$23$	−3.30009	−0.688116	−0.344058	+	0.938948i	$0.611802\pi$
$24$	0	0
$25$	−2.85307	−0.570614
$26$	−0.227772	−0.0446698
$27$	0	0
$28$	−3.53999	−0.668995
$29$	3.57821	0.664456	0.332228	−	0.943199i	$-0.392200\pi$
$29$	3.57821	0.664456	0.332228	+	0.943199i	$0.392200\pi$
$30$	0	0
$31$	7.37510	1.32461	0.662304	−	0.749236i	$-0.269579\pi$
$31$	7.37510	1.32461	0.662304	+	0.749236i	$0.269579\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−6.27991	−1.07700
$35$	5.18694	0.876753
$36$	0	0
$37$	−4.39823	−0.723065	−0.361532	−	0.932360i	$-0.617746\pi$
$37$	−4.39823	−0.723065	−0.361532	+	0.932360i	$0.617746\pi$
$38$	2.57419	0.417589
$39$	0	0
$40$	−1.46524	−0.231675
$41$	4.53941	0.708937	0.354469	−	0.935068i	$-0.384662\pi$
$41$	4.53941	0.708937	0.354469	+	0.935068i	$0.384662\pi$
$42$	0	0
$43$	−1.36273	−0.207814	−0.103907	−	0.994587i	$-0.533135\pi$
$43$	−1.36273	−0.207814	−0.103907	+	0.994587i	$0.533135\pi$
$44$	4.22810	0.637410
$45$	0	0
$46$	−3.30009	−0.486572
$47$	−7.53590	−1.09922	−0.549612	−	0.835420i	$-0.685224\pi$
$47$	−7.53590	−1.09922	−0.549612	+	0.835420i	$0.685224\pi$
$48$	0	0
$49$	5.53153	0.790219
$50$	−2.85307	−0.403485
$51$	0	0
$52$	−0.227772	−0.0315863
$53$	10.3373	1.41994	0.709972	−	0.704230i	$-0.248708\pi$
$53$	10.3373	1.41994	0.709972	+	0.704230i	$0.248708\pi$
$54$	0	0
$55$	−6.19518	−0.835358
$56$	−3.53999	−0.473051
$57$	0	0
$58$	3.57821	0.469842
$59$	1.84299	0.239937	0.119968	−	0.992778i	$-0.461721\pi$
$59$	1.84299	0.239937	0.119968	+	0.992778i	$0.461721\pi$
$60$	0	0
$61$	−0.0848386	−0.0108625	−0.00543123	−	0.999985i	$-0.501729\pi$
$61$	−0.0848386	−0.0108625	−0.00543123	+	0.999985i	$0.501729\pi$
$62$	7.37510	0.936639
$63$	0	0
$64$	1.00000	0.125000
$65$	0.333741	0.0413955
$66$	0	0
$67$	4.55903	0.556974	0.278487	−	0.960440i	$-0.410167\pi$
$67$	4.55903	0.556974	0.278487	+	0.960440i	$0.410167\pi$
$68$	−6.27991	−0.761551
$69$	0	0
$70$	5.18694	0.619958
$71$	−2.24203	−0.266080	−0.133040	−	0.991111i	$-0.542474\pi$
$71$	−2.24203	−0.266080	−0.133040	+	0.991111i	$0.542474\pi$
$72$	0	0
$73$	−6.79670	−0.795494	−0.397747	−	0.917495i	$-0.630208\pi$
$73$	−6.79670	−0.795494	−0.397747	+	0.917495i	$0.630208\pi$
$74$	−4.39823	−0.511284
$75$	0	0
$76$	2.57419	0.295280
$77$	−14.9674	−1.70570
$78$	0	0
$79$	3.14110	0.353401	0.176701	−	0.984265i	$-0.443458\pi$
$79$	3.14110	0.353401	0.176701	+	0.984265i	$0.443458\pi$
$80$	−1.46524	−0.163819
$81$	0	0
$82$	4.53941	0.501294
$83$	−1.74078	−0.191076	−0.0955378	−	0.995426i	$-0.530457\pi$
$83$	−1.74078	−0.191076	−0.0955378	+	0.995426i	$0.530457\pi$
$84$	0	0
$85$	9.20158	0.998051
$86$	−1.36273	−0.146947
$87$	0	0
$88$	4.22810	0.450717
$89$	5.71960	0.606276	0.303138	−	0.952947i	$-0.401966\pi$
$89$	5.71960	0.606276	0.303138	+	0.952947i	$0.401966\pi$
$90$	0	0
$91$	0.806312	0.0845245
$92$	−3.30009	−0.344058
$93$	0	0
$94$	−7.53590	−0.777268
$95$	−3.77181	−0.386980
$96$	0	0
$97$	19.1029	1.93961	0.969804	−	0.243884i	$-0.0784217\pi$
$97$	19.1029	1.93961	0.969804	+	0.243884i	$0.0784217\pi$
$98$	5.53153	0.558769
$99$	0	0
$100$	−2.85307	−0.285307
$101$	3.67787	0.365962	0.182981	−	0.983116i	$-0.441425\pi$
$101$	3.67787	0.365962	0.182981	+	0.983116i	$0.441425\pi$
$102$	0	0
$103$	1.13416	0.111752	0.0558759	−	0.998438i	$-0.482205\pi$
$103$	1.13416	0.111752	0.0558759	+	0.998438i	$0.482205\pi$
$104$	−0.227772	−0.0223349
$105$	0	0
$106$	10.3373	1.00405
$107$	7.11155	0.687500	0.343750	−	0.939061i	$-0.388303\pi$
$107$	7.11155	0.687500	0.343750	+	0.939061i	$0.388303\pi$
$108$	0	0
$109$	16.1888	1.55061	0.775305	−	0.631587i	$-0.217596\pi$
$109$	16.1888	1.55061	0.775305	+	0.631587i	$0.217596\pi$
$110$	−6.19518	−0.590687
$111$	0	0
$112$	−3.53999	−0.334498
$113$	1.96757	0.185093	0.0925466	−	0.995708i	$-0.470499\pi$
$113$	1.96757	0.185093	0.0925466	+	0.995708i	$0.470499\pi$
$114$	0	0
$115$	4.83543	0.450906
$116$	3.57821	0.332228
$117$	0	0
$118$	1.84299	0.169661
$119$	22.2308	2.03790
$120$	0	0
$121$	6.87680	0.625164
$122$	−0.0848386	−0.00768092
$123$	0	0
$124$	7.37510	0.662304
$125$	11.5066	1.02919
$126$	0	0
$127$	2.56617	0.227710	0.113855	−	0.993497i	$-0.463680\pi$
$127$	2.56617	0.227710	0.113855	+	0.993497i	$0.463680\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0.333741	0.0292711
$131$	−21.1716	−1.84978	−0.924888	−	0.380240i	$-0.875841\pi$
$131$	−21.1716	−1.84978	−0.924888	+	0.380240i	$0.875841\pi$
$132$	0	0
$133$	−9.11262	−0.790164
$134$	4.55903	0.393840
$135$	0	0
$136$	−6.27991	−0.538498
$137$	17.0465	1.45638	0.728192	−	0.685373i	$-0.240361\pi$
$137$	17.0465	1.45638	0.728192	+	0.685373i	$0.240361\pi$
$138$	0	0
$139$	−11.5556	−0.980132	−0.490066	−	0.871685i	$-0.663027\pi$
$139$	−11.5556	−0.980132	−0.490066	+	0.871685i	$0.663027\pi$
$140$	5.18694	0.438376
$141$	0	0
$142$	−2.24203	−0.188147
$143$	−0.963044	−0.0805337
$144$	0	0
$145$	−5.24293	−0.435402
$146$	−6.79670	−0.562499
$147$	0	0
$148$	−4.39823	−0.361532
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	5.85207	0.476235	0.238117	−	0.971236i	$-0.423470\pi$
$151$	5.85207	0.476235	0.238117	+	0.971236i	$0.423470\pi$
$152$	2.57419	0.208795
$153$	0	0
$154$	−14.9674	−1.20611
$155$	−10.8063	−0.867983
$156$	0	0
$157$	−7.89904	−0.630412	−0.315206	−	0.949023i	$-0.602074\pi$
$157$	−7.89904	−0.630412	−0.315206	+	0.949023i	$0.602074\pi$
$158$	3.14110	0.249892
$159$	0	0
$160$	−1.46524	−0.115837
$161$	11.6823	0.920693
$162$	0	0
$163$	2.40642	0.188485	0.0942427	−	0.995549i	$-0.469957\pi$
$163$	2.40642	0.188485	0.0942427	+	0.995549i	$0.469957\pi$
$164$	4.53941	0.354469
$165$	0	0
$166$	−1.74078	−0.135111
$167$	4.87364	0.377133	0.188567	−	0.982060i	$-0.439616\pi$
$167$	4.87364	0.377133	0.188567	+	0.982060i	$0.439616\pi$
$168$	0	0
$169$	−12.9481	−0.996009
$170$	9.20158	0.705729
$171$	0	0
$172$	−1.36273	−0.103907
$173$	12.8982	0.980633	0.490317	−	0.871544i	$-0.336881\pi$
$173$	12.8982	0.980633	0.490317	+	0.871544i	$0.336881\pi$
$174$	0	0
$175$	10.0998	0.763476
$176$	4.22810	0.318705
$177$	0	0
$178$	5.71960	0.428702
$179$	−8.33812	−0.623221	−0.311610	−	0.950210i	$-0.600868\pi$
$179$	−8.33812	−0.623221	−0.311610	+	0.950210i	$0.600868\pi$
$180$	0	0
$181$	25.7248	1.91211	0.956055	−	0.293187i	$-0.0947158\pi$
$181$	25.7248	1.91211	0.956055	+	0.293187i	$0.0947158\pi$
$182$	0.806312	0.0597678
$183$	0	0
$184$	−3.30009	−0.243286
$185$	6.44447	0.473807
$186$	0	0
$187$	−26.5521	−1.94168
$188$	−7.53590	−0.549612
$189$	0	0
$190$	−3.77181	−0.273636
$191$	21.3232	1.54289	0.771446	−	0.636295i	$-0.219534\pi$
$191$	21.3232	1.54289	0.771446	+	0.636295i	$0.219534\pi$
$192$	0	0
$193$	9.44168	0.679627	0.339813	−	0.940493i	$-0.389636\pi$
$193$	9.44168	0.679627	0.339813	+	0.940493i	$0.389636\pi$
$194$	19.1029	1.37151
$195$	0	0
$196$	5.53153	0.395110
$197$	13.7269	0.978004	0.489002	−	0.872283i	$-0.337361\pi$
$197$	13.7269	0.978004	0.489002	+	0.872283i	$0.337361\pi$
$198$	0	0
$199$	−25.9142	−1.83701	−0.918504	−	0.395412i	$-0.870602\pi$
$199$	−25.9142	−1.83701	−0.918504	+	0.395412i	$0.870602\pi$
$200$	−2.85307	−0.201742
$201$	0	0
$202$	3.67787	0.258774
$203$	−12.6668	−0.889036
$204$	0	0
$205$	−6.65134	−0.464549
$206$	1.13416	0.0790204
$207$	0	0
$208$	−0.227772	−0.0157932
$209$	10.8839	0.752858
$210$	0	0
$211$	4.68939	0.322831	0.161416	−	0.986887i	$-0.448394\pi$
$211$	4.68939	0.322831	0.161416	+	0.986887i	$0.448394\pi$
$212$	10.3373	0.709972
$213$	0	0
$214$	7.11155	0.486136
$215$	1.99673	0.136176
$216$	0	0
$217$	−26.1078	−1.77231
$218$	16.1888	1.09645
$219$	0	0
$220$	−6.19518	−0.417679
$221$	1.43039	0.0962184
$222$	0	0
$223$	21.3356	1.42874	0.714369	−	0.699769i	$-0.246714\pi$
$223$	21.3356	1.42874	0.714369	+	0.699769i	$0.246714\pi$
$224$	−3.53999	−0.236526
$225$	0	0
$226$	1.96757	0.130881
$227$	14.9565	0.992696	0.496348	−	0.868124i	$-0.334674\pi$
$227$	14.9565	0.992696	0.496348	+	0.868124i	$0.334674\pi$
$228$	0	0
$229$	−10.3500	−0.683945	−0.341973	−	0.939710i	$-0.611095\pi$
$229$	−10.3500	−0.683945	−0.341973	+	0.939710i	$0.611095\pi$
$230$	4.83543	0.318839
$231$	0	0
$232$	3.57821	0.234921
$233$	3.89733	0.255323	0.127661	−	0.991818i	$-0.459253\pi$
$233$	3.89733	0.255323	0.127661	+	0.991818i	$0.459253\pi$
$234$	0	0
$235$	11.0419	0.720294
$236$	1.84299	0.119968
$237$	0	0
$238$	22.2308	1.44101
$239$	28.9563	1.87303	0.936515	−	0.350628i	$-0.114032\pi$
$239$	28.9563	1.87303	0.936515	+	0.350628i	$0.114032\pi$
$240$	0	0
$241$	−14.6773	−0.945449	−0.472725	−	0.881210i	$-0.656729\pi$
$241$	−14.6773	−0.945449	−0.472725	+	0.881210i	$0.656729\pi$
$242$	6.87680	0.442057
$243$	0	0
$244$	−0.0848386	−0.00543123
$245$	−8.10503	−0.517811
$246$	0	0
$247$	−0.586330	−0.0373073
$248$	7.37510	0.468319
$249$	0	0
$250$	11.5066	0.727744
$251$	1.07802	0.0680438	0.0340219	−	0.999421i	$-0.489168\pi$
$251$	1.07802	0.0680438	0.0340219	+	0.999421i	$0.489168\pi$
$252$	0	0
$253$	−13.9531	−0.877224
$254$	2.56617	0.161016
$255$	0	0
$256$	1.00000	0.0625000
$257$	−13.6721	−0.852839	−0.426420	−	0.904525i	$-0.640225\pi$
$257$	−13.6721	−0.852839	−0.426420	+	0.904525i	$0.640225\pi$
$258$	0	0
$259$	15.5697	0.967454
$260$	0.333741	0.0206978
$261$	0	0
$262$	−21.1716	−1.30799
$263$	11.7549	0.724839	0.362419	−	0.932015i	$-0.381951\pi$
$263$	11.7549	0.724839	0.362419	+	0.932015i	$0.381951\pi$
$264$	0	0
$265$	−15.1467	−0.930455
$266$	−9.11262	−0.558730
$267$	0	0
$268$	4.55903	0.278487
$269$	18.2194	1.11086	0.555428	−	0.831565i	$-0.312554\pi$
$269$	18.2194	1.11086	0.555428	+	0.831565i	$0.312554\pi$
$270$	0	0
$271$	−15.5134	−0.942372	−0.471186	−	0.882034i	$-0.656174\pi$
$271$	−15.5134	−0.942372	−0.471186	+	0.882034i	$0.656174\pi$
$272$	−6.27991	−0.380775
$273$	0	0
$274$	17.0465	1.02982
$275$	−12.0630	−0.727429
$276$	0	0
$277$	−32.5931	−1.95833	−0.979164	−	0.203073i	$-0.934907\pi$
$277$	−32.5931	−1.95833	−0.979164	+	0.203073i	$0.934907\pi$
$278$	−11.5556	−0.693058
$279$	0	0
$280$	5.18694	0.309979
$281$	27.5743	1.64495	0.822473	−	0.568805i	$-0.192594\pi$
$281$	27.5743	1.64495	0.822473	+	0.568805i	$0.192594\pi$
$282$	0	0
$283$	−0.606670	−0.0360628	−0.0180314	−	0.999837i	$-0.505740\pi$
$283$	−0.606670	−0.0360628	−0.0180314	+	0.999837i	$0.505740\pi$
$284$	−2.24203	−0.133040
$285$	0	0
$286$	−0.963044	−0.0569460
$287$	−16.0695	−0.948552
$288$	0	0
$289$	22.4373	1.31984
$290$	−5.24293	−0.307876
$291$	0	0
$292$	−6.79670	−0.397747
$293$	−9.36340	−0.547016	−0.273508	−	0.961870i	$-0.588184\pi$
$293$	−9.36340	−0.547016	−0.273508	+	0.961870i	$0.588184\pi$
$294$	0	0
$295$	−2.70042	−0.157225
$296$	−4.39823	−0.255642
$297$	0	0
$298$	1.00000	0.0579284
$299$	0.751669	0.0434702
$300$	0	0
$301$	4.82405	0.278054
$302$	5.85207	0.336749
$303$	0	0
$304$	2.57419	0.147640
$305$	0.124309	0.00711791
$306$	0	0
$307$	13.5000	0.770486	0.385243	−	0.922815i	$-0.374118\pi$
$307$	13.5000	0.770486	0.385243	+	0.922815i	$0.374118\pi$
$308$	−14.9674	−0.852848
$309$	0	0
$310$	−10.8063	−0.613757
$311$	27.1871	1.54164	0.770821	−	0.637052i	$-0.219847\pi$
$311$	27.1871	1.54164	0.770821	+	0.637052i	$0.219847\pi$
$312$	0	0
$313$	−11.7246	−0.662711	−0.331356	−	0.943506i	$-0.607506\pi$
$313$	−11.7246	−0.662711	−0.331356	+	0.943506i	$0.607506\pi$
$314$	−7.89904	−0.445769
$315$	0	0
$316$	3.14110	0.176701
$317$	13.8607	0.778493	0.389247	−	0.921134i	$-0.372735\pi$
$317$	13.8607	0.778493	0.389247	+	0.921134i	$0.372735\pi$
$318$	0	0
$319$	15.1290	0.847061
$320$	−1.46524	−0.0819095
$321$	0	0
$322$	11.6823	0.651028
$323$	−16.1657	−0.899483
$324$	0	0
$325$	0.649850	0.0360472
$326$	2.40642	0.133279
$327$	0	0
$328$	4.53941	0.250647
$329$	26.6770	1.47075
$330$	0	0
$331$	9.46657	0.520330	0.260165	−	0.965564i	$-0.416223\pi$
$331$	9.46657	0.520330	0.260165	+	0.965564i	$0.416223\pi$
$332$	−1.74078	−0.0955378
$333$	0	0
$334$	4.87364	0.266674
$335$	−6.68007	−0.364971
$336$	0	0
$337$	−3.81960	−0.208067	−0.104033	−	0.994574i	$-0.533175\pi$
$337$	−3.81960	−0.208067	−0.104033	+	0.994574i	$0.533175\pi$
$338$	−12.9481	−0.704285
$339$	0	0
$340$	9.20158	0.499026
$341$	31.1826	1.68863
$342$	0	0
$343$	5.19836	0.280685
$344$	−1.36273	−0.0734735
$345$	0	0
$346$	12.8982	0.693413
$347$	−23.0564	−1.23773	−0.618866	−	0.785497i	$-0.712408\pi$
$347$	−23.0564	−1.23773	−0.618866	+	0.785497i	$0.712408\pi$
$348$	0	0
$349$	31.8302	1.70383	0.851916	−	0.523678i	$-0.175441\pi$
$349$	31.8302	1.70383	0.851916	+	0.523678i	$0.175441\pi$
$350$	10.0998	0.539859
$351$	0	0
$352$	4.22810	0.225358
$353$	18.8310	1.00227	0.501137	−	0.865368i	$-0.332915\pi$
$353$	18.8310	1.00227	0.501137	+	0.865368i	$0.332915\pi$
$354$	0	0
$355$	3.28511	0.174355
$356$	5.71960	0.303138
$357$	0	0
$358$	−8.33812	−0.440684
$359$	7.29303	0.384911	0.192456	−	0.981306i	$-0.438355\pi$
$359$	7.29303	0.384911	0.192456	+	0.981306i	$0.438355\pi$
$360$	0	0
$361$	−12.3735	−0.651239
$362$	25.7248	1.35207
$363$	0	0
$364$	0.806312	0.0422622
$365$	9.95881	0.521268
$366$	0	0
$367$	24.6833	1.28846	0.644229	−	0.764833i	$-0.277178\pi$
$367$	24.6833	1.28846	0.644229	+	0.764833i	$0.277178\pi$
$368$	−3.30009	−0.172029
$369$	0	0
$370$	6.44447	0.335032
$371$	−36.5941	−1.89987
$372$	0	0
$373$	9.75225	0.504952	0.252476	−	0.967603i	$-0.418755\pi$
$373$	9.75225	0.504952	0.252476	+	0.967603i	$0.418755\pi$
$374$	−26.5521	−1.37297
$375$	0	0
$376$	−7.53590	−0.388634
$377$	−0.815017	−0.0419755
$378$	0	0
$379$	−11.0298	−0.566565	−0.283282	−	0.959037i	$-0.591423\pi$
$379$	−11.0298	−0.566565	−0.283282	+	0.959037i	$0.591423\pi$
$380$	−3.77181	−0.193490
$381$	0	0
$382$	21.3232	1.09099
$383$	12.3380	0.630441	0.315220	−	0.949018i	$-0.397922\pi$
$383$	12.3380	0.630441	0.315220	+	0.949018i	$0.397922\pi$
$384$	0	0
$385$	21.9309	1.11770
$386$	9.44168	0.480569
$387$	0	0
$388$	19.1029	0.969804
$389$	−8.33709	−0.422707	−0.211354	−	0.977410i	$-0.567787\pi$
$389$	−8.33709	−0.422707	−0.211354	+	0.977410i	$0.567787\pi$
$390$	0	0
$391$	20.7243	1.04807
$392$	5.53153	0.279385
$393$	0	0
$394$	13.7269	0.691553
$395$	−4.60246	−0.231575
$396$	0	0
$397$	−23.3671	−1.17276	−0.586382	−	0.810035i	$-0.699448\pi$
$397$	−23.3671	−1.17276	−0.586382	+	0.810035i	$0.699448\pi$
$398$	−25.9142	−1.29896
$399$	0	0
$400$	−2.85307	−0.142653
$401$	21.0592	1.05165	0.525824	−	0.850593i	$-0.323757\pi$
$401$	21.0592	1.05165	0.525824	+	0.850593i	$0.323757\pi$
$402$	0	0
$403$	−1.67984	−0.0836790
$404$	3.67787	0.182981
$405$	0	0
$406$	−12.6668	−0.628644
$407$	−18.5962	−0.921777
$408$	0	0
$409$	−2.26707	−0.112099	−0.0560497	−	0.998428i	$-0.517851\pi$
$409$	−2.26707	−0.112099	−0.0560497	+	0.998428i	$0.517851\pi$
$410$	−6.65134	−0.328486
$411$	0	0
$412$	1.13416	0.0558759
$413$	−6.52416	−0.321033
$414$	0	0
$415$	2.55066	0.125207
$416$	−0.227772	−0.0111675
$417$	0	0
$418$	10.8839	0.532351
$419$	−34.4011	−1.68060	−0.840302	−	0.542119i	$-0.817622\pi$
$419$	−34.4011	−1.68060	−0.840302	+	0.542119i	$0.817622\pi$
$420$	0	0
$421$	20.5050	0.999355	0.499677	−	0.866212i	$-0.333452\pi$
$421$	20.5050	0.999355	0.499677	+	0.866212i	$0.333452\pi$
$422$	4.68939	0.228276
$423$	0	0
$424$	10.3373	0.502026
$425$	17.9170	0.869103
$426$	0	0
$427$	0.300328	0.0145339
$428$	7.11155	0.343750
$429$	0	0
$430$	1.99673	0.0962908
$431$	−15.7752	−0.759864	−0.379932	−	0.925014i	$-0.624053\pi$
$431$	−15.7752	−0.759864	−0.379932	+	0.925014i	$0.624053\pi$
$432$	0	0
$433$	−22.9521	−1.10301	−0.551504	−	0.834172i	$-0.685946\pi$
$433$	−22.9521	−1.10301	−0.551504	+	0.834172i	$0.685946\pi$
$434$	−26.1078	−1.25321
$435$	0	0
$436$	16.1888	0.775305
$437$	−8.49507	−0.406374
$438$	0	0
$439$	15.0417	0.717899	0.358950	−	0.933357i	$-0.383135\pi$
$439$	15.0417	0.717899	0.358950	+	0.933357i	$0.383135\pi$
$440$	−6.19518	−0.295344
$441$	0	0
$442$	1.43039	0.0680367
$443$	11.3602	0.539738	0.269869	−	0.962897i	$-0.413020\pi$
$443$	11.3602	0.539738	0.269869	+	0.962897i	$0.413020\pi$
$444$	0	0
$445$	−8.38059	−0.397278
$446$	21.3356	1.01027
$447$	0	0
$448$	−3.53999	−0.167249
$449$	14.1274	0.666712	0.333356	−	0.942801i	$-0.391819\pi$
$449$	14.1274	0.666712	0.333356	+	0.942801i	$0.391819\pi$
$450$	0	0
$451$	19.1931	0.903767
$452$	1.96757	0.0925466
$453$	0	0
$454$	14.9565	0.701942
$455$	−1.18144	−0.0553868
$456$	0	0
$457$	−29.3781	−1.37425	−0.687124	−	0.726540i	$-0.741127\pi$
$457$	−29.3781	−1.37425	−0.687124	+	0.726540i	$0.741127\pi$
$458$	−10.3500	−0.483622
$459$	0	0
$460$	4.83543	0.225453
$461$	14.3347	0.667632	0.333816	−	0.942638i	$-0.391664\pi$
$461$	14.3347	0.667632	0.333816	+	0.942638i	$0.391664\pi$
$462$	0	0
$463$	10.6011	0.492675	0.246338	−	0.969184i	$-0.420773\pi$
$463$	10.6011	0.492675	0.246338	+	0.969184i	$0.420773\pi$
$464$	3.57821	0.166114
$465$	0	0
$466$	3.89733	0.180540
$467$	−6.22349	−0.287989	−0.143994	−	0.989579i	$-0.545995\pi$
$467$	−6.22349	−0.287989	−0.143994	+	0.989579i	$0.545995\pi$
$468$	0	0
$469$	−16.1389	−0.745226
$470$	11.0419	0.509325
$471$	0	0
$472$	1.84299	0.0848305
$473$	−5.76176	−0.264926
$474$	0	0
$475$	−7.34435	−0.336982
$476$	22.2308	1.01895
$477$	0	0
$478$	28.9563	1.32443
$479$	20.7476	0.947983	0.473991	−	0.880529i	$-0.342813\pi$
$479$	20.7476	0.947983	0.473991	+	0.880529i	$0.342813\pi$
$480$	0	0
$481$	1.00180	0.0456780
$482$	−14.6773	−0.668534
$483$	0	0
$484$	6.87680	0.312582
$485$	−27.9904	−1.27098
$486$	0	0
$487$	7.42424	0.336425	0.168212	−	0.985751i	$-0.446201\pi$
$487$	7.42424	0.336425	0.168212	+	0.985751i	$0.446201\pi$
$488$	−0.0848386	−0.00384046
$489$	0	0
$490$	−8.10503	−0.366148
$491$	11.2139	0.506078	0.253039	−	0.967456i	$-0.418570\pi$
$491$	11.2139	0.506078	0.253039	+	0.967456i	$0.418570\pi$
$492$	0	0
$493$	−22.4708	−1.01203
$494$	−0.586330	−0.0263802
$495$	0	0
$496$	7.37510	0.331152
$497$	7.93675	0.356012
$498$	0	0
$499$	−26.6372	−1.19244	−0.596222	−	0.802820i	$-0.703332\pi$
$499$	−26.6372	−1.19244	−0.596222	+	0.802820i	$0.703332\pi$
$500$	11.5066	0.514593
$501$	0	0
$502$	1.07802	0.0481143
$503$	43.8172	1.95371	0.976856	−	0.213896i	$-0.0686155\pi$
$503$	43.8172	1.95371	0.976856	+	0.213896i	$0.0686155\pi$
$504$	0	0
$505$	−5.38897	−0.239806
$506$	−13.9531	−0.620291
$507$	0	0
$508$	2.56617	0.113855
$509$	−12.7752	−0.566252	−0.283126	−	0.959083i	$-0.591371\pi$
$509$	−12.7752	−0.566252	−0.283126	+	0.959083i	$0.591371\pi$
$510$	0	0
$511$	24.0603	1.06436
$512$	1.00000	0.0441942
$513$	0	0
$514$	−13.6721	−0.603048
$515$	−1.66181	−0.0732282
$516$	0	0
$517$	−31.8625	−1.40131
$518$	15.5697	0.684093
$519$	0	0
$520$	0.333741	0.0146355
$521$	10.0510	0.440341	0.220171	−	0.975461i	$-0.429339\pi$
$521$	10.0510	0.440341	0.220171	+	0.975461i	$0.429339\pi$
$522$	0	0
$523$	−5.79781	−0.253521	−0.126760	−	0.991933i	$-0.540458\pi$
$523$	−5.79781	−0.253521	−0.126760	+	0.991933i	$0.540458\pi$
$524$	−21.1716	−0.924888
$525$	0	0
$526$	11.7549	0.512539
$527$	−46.3150	−2.01751
$528$	0	0
$529$	−12.1094	−0.526496
$530$	−15.1467	−0.657931
$531$	0	0
$532$	−9.11262	−0.395082
$533$	−1.03395	−0.0447855
$534$	0	0
$535$	−10.4201	−0.450502
$536$	4.55903	0.196920
$537$	0	0
$538$	18.2194	0.785494
$539$	23.3879	1.00739
$540$	0	0
$541$	−14.8299	−0.637586	−0.318793	−	0.947824i	$-0.603277\pi$
$541$	−14.8299	−0.637586	−0.318793	+	0.947824i	$0.603277\pi$
$542$	−15.5134	−0.666358
$543$	0	0
$544$	−6.27991	−0.269249
$545$	−23.7205	−1.01608
$546$	0	0
$547$	−1.80078	−0.0769958	−0.0384979	−	0.999259i	$-0.512257\pi$
$547$	−1.80078	−0.0769958	−0.0384979	+	0.999259i	$0.512257\pi$
$548$	17.0465	0.728192
$549$	0	0
$550$	−12.0630	−0.514370
$551$	9.21099	0.392401
$552$	0	0
$553$	−11.1195	−0.472847
$554$	−32.5931	−1.38475
$555$	0	0
$556$	−11.5556	−0.490066
$557$	−17.8340	−0.755651	−0.377825	−	0.925877i	$-0.623328\pi$
$557$	−17.8340	−0.755651	−0.377825	+	0.925877i	$0.623328\pi$
$558$	0	0
$559$	0.310392	0.0131282
$560$	5.18694	0.219188
$561$	0	0
$562$	27.5743	1.16315
$563$	43.5290	1.83453	0.917264	−	0.398279i	$-0.130393\pi$
$563$	43.5290	1.83453	0.917264	+	0.398279i	$0.130393\pi$
$564$	0	0
$565$	−2.88296	−0.121287
$566$	−0.606670	−0.0255002
$567$	0	0
$568$	−2.24203	−0.0940733
$569$	−23.8279	−0.998919	−0.499460	−	0.866337i	$-0.666468\pi$
$569$	−23.8279	−0.998919	−0.499460	+	0.866337i	$0.666468\pi$
$570$	0	0
$571$	−16.8475	−0.705044	−0.352522	−	0.935804i	$-0.614676\pi$
$571$	−16.8475	−0.705044	−0.352522	+	0.935804i	$0.614676\pi$
$572$	−0.963044	−0.0402669
$573$	0	0
$574$	−16.0695	−0.670727
$575$	9.41538	0.392649
$576$	0	0
$577$	−14.9952	−0.624260	−0.312130	−	0.950039i	$-0.601042\pi$
$577$	−14.9952	−0.624260	−0.312130	+	0.950039i	$0.601042\pi$
$578$	22.4373	0.933267
$579$	0	0
$580$	−5.24293	−0.217701
$581$	6.16235	0.255657
$582$	0	0
$583$	43.7073	1.81017
$584$	−6.79670	−0.281249
$585$	0	0
$586$	−9.36340	−0.386799
$587$	−33.7563	−1.39327	−0.696636	−	0.717425i	$-0.745321\pi$
$587$	−33.7563	−1.39327	−0.696636	+	0.717425i	$0.745321\pi$
$588$	0	0
$589$	18.9849	0.782260
$590$	−2.70042	−0.111175
$591$	0	0
$592$	−4.39823	−0.180766
$593$	8.47497	0.348025	0.174013	−	0.984743i	$-0.444327\pi$
$593$	8.47497	0.348025	0.174013	+	0.984743i	$0.444327\pi$
$594$	0	0
$595$	−32.5735	−1.33538
$596$	1.00000	0.0409616
$597$	0	0
$598$	0.751669	0.0307380
$599$	28.5142	1.16506	0.582530	−	0.812809i	$-0.302063\pi$
$599$	28.5142	1.16506	0.582530	+	0.812809i	$0.302063\pi$
$600$	0	0
$601$	−45.4417	−1.85361	−0.926803	−	0.375549i	$-0.877454\pi$
$601$	−45.4417	−1.85361	−0.926803	+	0.375549i	$0.877454\pi$
$602$	4.82405	0.196614
$603$	0	0
$604$	5.85207	0.238117
$605$	−10.0762	−0.409654
$606$	0	0
$607$	−21.2507	−0.862541	−0.431270	−	0.902223i	$-0.641934\pi$
$607$	−21.2507	−0.862541	−0.431270	+	0.902223i	$0.641934\pi$
$608$	2.57419	0.104397
$609$	0	0
$610$	0.124309	0.00503312
$611$	1.71647	0.0694409
$612$	0	0
$613$	−11.6344	−0.469909	−0.234955	−	0.972006i	$-0.575494\pi$
$613$	−11.6344	−0.469909	−0.234955	+	0.972006i	$0.575494\pi$
$614$	13.5000	0.544816
$615$	0	0
$616$	−14.9674	−0.603055
$617$	−22.9162	−0.922571	−0.461286	−	0.887252i	$-0.652612\pi$
$617$	−22.9162	−0.922571	−0.461286	+	0.887252i	$0.652612\pi$
$618$	0	0
$619$	46.4773	1.86808	0.934041	−	0.357167i	$-0.116257\pi$
$619$	46.4773	1.86808	0.934041	+	0.357167i	$0.116257\pi$
$620$	−10.8063	−0.433991
$621$	0	0
$622$	27.1871	1.09010
$623$	−20.2473	−0.811191
$624$	0	0
$625$	−2.59465	−0.103786
$626$	−11.7246	−0.468608
$627$	0	0
$628$	−7.89904	−0.315206
$629$	27.6205	1.10130
$630$	0	0
$631$	−6.74494	−0.268512	−0.134256	−	0.990947i	$-0.542864\pi$
$631$	−6.74494	−0.268512	−0.134256	+	0.990947i	$0.542864\pi$
$632$	3.14110	0.124946
$633$	0	0
$634$	13.8607	0.550478
$635$	−3.76005	−0.149213
$636$	0	0
$637$	−1.25993	−0.0499203
$638$	15.1290	0.598963
$639$	0	0
$640$	−1.46524	−0.0579187
$641$	−3.15648	−0.124673	−0.0623367	−	0.998055i	$-0.519855\pi$
$641$	−3.15648	−0.124673	−0.0623367	+	0.998055i	$0.519855\pi$
$642$	0	0
$643$	11.0283	0.434913	0.217457	−	0.976070i	$-0.430224\pi$
$643$	11.0283	0.434913	0.217457	+	0.976070i	$0.430224\pi$
$644$	11.6823	0.460347
$645$	0	0
$646$	−16.1657	−0.636031
$647$	−17.4386	−0.685584	−0.342792	−	0.939411i	$-0.611373\pi$
$647$	−17.4386	−0.685584	−0.342792	+	0.939411i	$0.611373\pi$
$648$	0	0
$649$	7.79234	0.305876
$650$	0.649850	0.0254892
$651$	0	0
$652$	2.40642	0.0942427
$653$	−13.2528	−0.518621	−0.259310	−	0.965794i	$-0.583495\pi$
$653$	−13.2528	−0.518621	−0.259310	+	0.965794i	$0.583495\pi$
$654$	0	0
$655$	31.0216	1.21211
$656$	4.53941	0.177234
$657$	0	0
$658$	26.6770	1.03998
$659$	34.7215	1.35256	0.676279	−	0.736646i	$-0.263591\pi$
$659$	34.7215	1.35256	0.676279	+	0.736646i	$0.263591\pi$
$660$	0	0
$661$	−33.8202	−1.31545	−0.657726	−	0.753257i	$-0.728482\pi$
$661$	−33.8202	−1.31545	−0.657726	+	0.753257i	$0.728482\pi$
$662$	9.46657	0.367929
$663$	0	0
$664$	−1.74078	−0.0675554
$665$	13.3522	0.517775
$666$	0	0
$667$	−11.8084	−0.457223
$668$	4.87364	0.188567
$669$	0	0
$670$	−6.68007	−0.258074
$671$	−0.358706	−0.0138477
$672$	0	0
$673$	26.3295	1.01493	0.507464	−	0.861673i	$-0.330583\pi$
$673$	26.3295	1.01493	0.507464	+	0.861673i	$0.330583\pi$
$674$	−3.81960	−0.147125
$675$	0	0
$676$	−12.9481	−0.498005
$677$	−16.3556	−0.628596	−0.314298	−	0.949324i	$-0.601769\pi$
$677$	−16.3556	−0.628596	−0.314298	+	0.949324i	$0.601769\pi$
$678$	0	0
$679$	−67.6242	−2.59518
$680$	9.20158	0.352864
$681$	0	0
$682$	31.1826	1.19404
$683$	−45.4585	−1.73942	−0.869711	−	0.493561i	$-0.835695\pi$
$683$	−45.4585	−1.73942	−0.869711	+	0.493561i	$0.835695\pi$
$684$	0	0
$685$	−24.9773	−0.954333
$686$	5.19836	0.198474
$687$	0	0
$688$	−1.36273	−0.0519536
$689$	−2.35456	−0.0897017
$690$	0	0
$691$	−6.61901	−0.251799	−0.125899	−	0.992043i	$-0.540182\pi$
$691$	−6.61901	−0.251799	−0.125899	+	0.992043i	$0.540182\pi$
$692$	12.8982	0.490317
$693$	0	0
$694$	−23.0564	−0.875208
$695$	16.9317	0.642257
$696$	0	0
$697$	−28.5071	−1.07978
$698$	31.8302	1.20479
$699$	0	0
$700$	10.0998	0.381738
$701$	−35.8627	−1.35452	−0.677258	−	0.735745i	$-0.736832\pi$
$701$	−35.8627	−1.35452	−0.677258	+	0.735745i	$0.736832\pi$
$702$	0	0
$703$	−11.3219	−0.427013
$704$	4.22810	0.159352
$705$	0	0
$706$	18.8310	0.708714
$707$	−13.0196	−0.489653
$708$	0	0
$709$	−1.55359	−0.0583463	−0.0291732	−	0.999574i	$-0.509287\pi$
$709$	−1.55359	−0.0583463	−0.0291732	+	0.999574i	$0.509287\pi$
$710$	3.28511	0.123288
$711$	0	0
$712$	5.71960	0.214351
$713$	−24.3385	−0.911484
$714$	0	0
$715$	1.41109	0.0527718
$716$	−8.33812	−0.311610
$717$	0	0
$718$	7.29303	0.272173
$719$	2.47572	0.0923288	0.0461644	−	0.998934i	$-0.485300\pi$
$719$	2.47572	0.0923288	0.0461644	+	0.998934i	$0.485300\pi$
$720$	0	0
$721$	−4.01490	−0.149523
$722$	−12.3735	−0.460495
$723$	0	0
$724$	25.7248	0.956055
$725$	−10.2089	−0.379148
$726$	0	0
$727$	−15.0691	−0.558884	−0.279442	−	0.960163i	$-0.590149\pi$
$727$	−15.0691	−0.558884	−0.279442	+	0.960163i	$0.590149\pi$
$728$	0.806312	0.0298839
$729$	0	0
$730$	9.95881	0.368592
$731$	8.55782	0.316523
$732$	0	0
$733$	1.22863	0.0453805	0.0226902	−	0.999743i	$-0.492777\pi$
$733$	1.22863	0.0453805	0.0226902	+	0.999743i	$0.492777\pi$
$734$	24.6833	0.911077
$735$	0	0
$736$	−3.30009	−0.121643
$737$	19.2760	0.710041
$738$	0	0
$739$	23.7074	0.872090	0.436045	−	0.899925i	$-0.356379\pi$
$739$	23.7074	0.872090	0.436045	+	0.899925i	$0.356379\pi$
$740$	6.44447	0.236903
$741$	0	0
$742$	−36.5941	−1.34341
$743$	47.1223	1.72875	0.864375	−	0.502848i	$-0.167714\pi$
$743$	47.1223	1.72875	0.864375	+	0.502848i	$0.167714\pi$
$744$	0	0
$745$	−1.46524	−0.0536823
$746$	9.75225	0.357055
$747$	0	0
$748$	−26.5521	−0.970839
$749$	−25.1748	−0.919868
$750$	0	0
$751$	−38.3517	−1.39947	−0.699737	−	0.714401i	$-0.746699\pi$
$751$	−38.3517	−1.39947	−0.699737	+	0.714401i	$0.746699\pi$
$752$	−7.53590	−0.274806
$753$	0	0
$754$	−0.815017	−0.0296812
$755$	−8.57469	−0.312065
$756$	0	0
$757$	2.13944	0.0777591	0.0388796	−	0.999244i	$-0.487621\pi$
$757$	2.13944	0.0777591	0.0388796	+	0.999244i	$0.487621\pi$
$758$	−11.0298	−0.400622
$759$	0	0
$760$	−3.77181	−0.136818
$761$	24.7239	0.896241	0.448121	−	0.893973i	$-0.352094\pi$
$761$	24.7239	0.896241	0.448121	+	0.893973i	$0.352094\pi$
$762$	0	0
$763$	−57.3083	−2.07470
$764$	21.3232	0.771446
$765$	0	0
$766$	12.3380	0.445789
$767$	−0.419782	−0.0151575
$768$	0	0
$769$	−42.4880	−1.53216	−0.766079	−	0.642746i	$-0.777795\pi$
$769$	−42.4880	−1.53216	−0.766079	+	0.642746i	$0.777795\pi$
$770$	21.9309	0.790334
$771$	0	0
$772$	9.44168	0.339813
$773$	−32.6589	−1.17466	−0.587330	−	0.809348i	$-0.699821\pi$
$773$	−32.6589	−1.17466	−0.587330	+	0.809348i	$0.699821\pi$
$774$	0	0
$775$	−21.0417	−0.755839
$776$	19.1029	0.685755
$777$	0	0
$778$	−8.33709	−0.298899
$779$	11.6853	0.418670
$780$	0	0
$781$	−9.47950	−0.339203
$782$	20.7243	0.741098
$783$	0	0
$784$	5.53153	0.197555
$785$	11.5740	0.413094
$786$	0	0
$787$	−49.4554	−1.76289	−0.881446	−	0.472284i	$-0.843430\pi$
$787$	−49.4554	−1.76289	−0.881446	+	0.472284i	$0.843430\pi$
$788$	13.7269	0.489002
$789$	0	0
$790$	−4.60246	−0.163748
$791$	−6.96517	−0.247653
$792$	0	0
$793$	0.0193239	0.000686211 0
$794$	−23.3671	−0.829269
$795$	0	0
$796$	−25.9142	−0.918504
$797$	−32.3598	−1.14624	−0.573121	−	0.819471i	$-0.694267\pi$
$797$	−32.3598	−1.14624	−0.573121	+	0.819471i	$0.694267\pi$
$798$	0	0
$799$	47.3247	1.67423
$800$	−2.85307	−0.100871
$801$	0	0
$802$	21.0592	0.743627
$803$	−28.7371	−1.01411
$804$	0	0
$805$	−17.1174	−0.603308
$806$	−1.67984	−0.0591700
$807$	0	0
$808$	3.67787	0.129387
$809$	24.6466	0.866528	0.433264	−	0.901267i	$-0.357362\pi$
$809$	24.6466	0.866528	0.433264	+	0.901267i	$0.357362\pi$
$810$	0	0
$811$	29.9169	1.05052	0.525262	−	0.850941i	$-0.323967\pi$
$811$	29.9169	1.05052	0.525262	+	0.850941i	$0.323967\pi$
$812$	−12.6668	−0.444518
$813$	0	0
$814$	−18.5962	−0.651795
$815$	−3.52599	−0.123510
$816$	0	0
$817$	−3.50793	−0.122727
$818$	−2.26707	−0.0792662
$819$	0	0
$820$	−6.65134	−0.232275
$821$	−4.48913	−0.156672	−0.0783359	−	0.996927i	$-0.524961\pi$
$821$	−4.48913	−0.156672	−0.0783359	+	0.996927i	$0.524961\pi$
$822$	0	0
$823$	−19.3547	−0.674664	−0.337332	−	0.941386i	$-0.609524\pi$
$823$	−19.3547	−0.674664	−0.337332	+	0.941386i	$0.609524\pi$
$824$	1.13416	0.0395102
$825$	0	0
$826$	−6.52416	−0.227005
$827$	−18.4040	−0.639970	−0.319985	−	0.947423i	$-0.603678\pi$
$827$	−18.4040	−0.639970	−0.319985	+	0.947423i	$0.603678\pi$
$828$	0	0
$829$	−24.2823	−0.843361	−0.421680	−	0.906745i	$-0.638560\pi$
$829$	−24.2823	−0.843361	−0.421680	+	0.906745i	$0.638560\pi$
$830$	2.55066	0.0885348
$831$	0	0
$832$	−0.227772	−0.00789659
$833$	−34.7375	−1.20358
$834$	0	0
$835$	−7.14105	−0.247126
$836$	10.8839	0.376429
$837$	0	0
$838$	−34.4011	−1.18837
$839$	34.6031	1.19463	0.597316	−	0.802006i	$-0.296234\pi$
$839$	34.6031	1.19463	0.597316	+	0.802006i	$0.296234\pi$
$840$	0	0
$841$	−16.1964	−0.558498
$842$	20.5050	0.706650
$843$	0	0
$844$	4.68939	0.161416
$845$	18.9721	0.652661
$846$	0	0
$847$	−24.3438	−0.836463
$848$	10.3373	0.354986
$849$	0	0
$850$	17.9170	0.614548
$851$	14.5146	0.497553
$852$	0	0
$853$	2.09858	0.0718541	0.0359271	−	0.999354i	$-0.488562\pi$
$853$	2.09858	0.0718541	0.0359271	+	0.999354i	$0.488562\pi$
$854$	0.300328	0.0102770
$855$	0	0
$856$	7.11155	0.243068
$857$	34.2049	1.16842	0.584209	−	0.811603i	$-0.301405\pi$
$857$	34.2049	1.16842	0.584209	+	0.811603i	$0.301405\pi$
$858$	0	0
$859$	5.92208	0.202059	0.101029	−	0.994883i	$-0.467786\pi$
$859$	5.92208	0.202059	0.101029	+	0.994883i	$0.467786\pi$
$860$	1.99673	0.0680879
$861$	0	0
$862$	−15.7752	−0.537305
$863$	7.96972	0.271292	0.135646	−	0.990757i	$-0.456689\pi$
$863$	7.96972	0.271292	0.135646	+	0.990757i	$0.456689\pi$
$864$	0	0
$865$	−18.8990	−0.642585
$866$	−22.9521	−0.779944
$867$	0	0
$868$	−26.1078	−0.886156
$869$	13.2809	0.450522
$870$	0	0
$871$	−1.03842	−0.0351855
$872$	16.1888	0.548223
$873$	0	0
$874$	−8.49507	−0.287350
$875$	−40.7334	−1.37704
$876$	0	0
$877$	0.249393	0.00842141	0.00421071	−	0.999991i	$-0.498660\pi$
$877$	0.249393	0.00842141	0.00421071	+	0.999991i	$0.498660\pi$
$878$	15.0417	0.507631
$879$	0	0
$880$	−6.19518	−0.208839
$881$	−44.4729	−1.49833	−0.749165	−	0.662383i	$-0.769545\pi$
$881$	−44.4729	−1.49833	−0.749165	+	0.662383i	$0.769545\pi$
$882$	0	0
$883$	36.3933	1.22473	0.612366	−	0.790575i	$-0.290218\pi$
$883$	36.3933	1.22473	0.612366	+	0.790575i	$0.290218\pi$
$884$	1.43039	0.0481092
$885$	0	0
$886$	11.3602	0.381653
$887$	−57.1344	−1.91838	−0.959192	−	0.282756i	$-0.908751\pi$
$887$	−57.1344	−1.91838	−0.959192	+	0.282756i	$0.908751\pi$
$888$	0	0
$889$	−9.08420	−0.304674
$890$	−8.38059	−0.280918
$891$	0	0
$892$	21.3356	0.714369
$893$	−19.3988	−0.649158
$894$	0	0
$895$	12.2174	0.408381
$896$	−3.53999	−0.118263
$897$	0	0
$898$	14.1274	0.471437
$899$	26.3896	0.880143
$900$	0	0
$901$	−64.9176	−2.16272
$902$	19.1931	0.639060
$903$	0	0
$904$	1.96757	0.0654404
$905$	−37.6931	−1.25296
$906$	0	0
$907$	32.9692	1.09472	0.547362	−	0.836896i	$-0.315632\pi$
$907$	32.9692	1.09472	0.547362	+	0.836896i	$0.315632\pi$
$908$	14.9565	0.496348
$909$	0	0
$910$	−1.18144	−0.0391644
$911$	−17.3024	−0.573254	−0.286627	−	0.958042i	$-0.592534\pi$
$911$	−17.3024	−0.573254	−0.286627	+	0.958042i	$0.592534\pi$
$912$	0	0
$913$	−7.36019	−0.243587
$914$	−29.3781	−0.971741
$915$	0	0
$916$	−10.3500	−0.341973
$917$	74.9474	2.47498
$918$	0	0
$919$	38.5986	1.27325	0.636626	−	0.771173i	$-0.280330\pi$
$919$	38.5986	1.27325	0.636626	+	0.771173i	$0.280330\pi$
$920$	4.83543	0.159419
$921$	0	0
$922$	14.3347	0.472087
$923$	0.510672	0.0168090
$924$	0	0
$925$	12.5485	0.412591
$926$	10.6011	0.348374
$927$	0	0
$928$	3.57821	0.117460
$929$	7.90865	0.259474	0.129737	−	0.991548i	$-0.458587\pi$
$929$	7.90865	0.259474	0.129737	+	0.991548i	$0.458587\pi$
$930$	0	0
$931$	14.2392	0.466672
$932$	3.89733	0.127661
$933$	0	0
$934$	−6.22349	−0.203639
$935$	38.9052	1.27233
$936$	0	0
$937$	−22.8586	−0.746759	−0.373379	−	0.927679i	$-0.621801\pi$
$937$	−22.8586	−0.746759	−0.373379	+	0.927679i	$0.621801\pi$
$938$	−16.1389	−0.526954
$939$	0	0
$940$	11.0419	0.360147
$941$	37.2088	1.21297	0.606486	−	0.795094i	$-0.292579\pi$
$941$	37.2088	1.21297	0.606486	+	0.795094i	$0.292579\pi$
$942$	0	0
$943$	−14.9805	−0.487831
$944$	1.84299	0.0599842
$945$	0	0
$946$	−5.76176	−0.187331
$947$	29.8932	0.971397	0.485699	−	0.874126i	$-0.338565\pi$
$947$	29.8932	0.971397	0.485699	+	0.874126i	$0.338565\pi$
$948$	0	0
$949$	1.54810	0.0502535
$950$	−7.34435	−0.238282
$951$	0	0
$952$	22.2308	0.720505
$953$	−18.1015	−0.586364	−0.293182	−	0.956057i	$-0.594714\pi$
$953$	−18.1015	−0.586364	−0.293182	+	0.956057i	$0.594714\pi$
$954$	0	0
$955$	−31.2436	−1.01102
$956$	28.9563	0.936515
$957$	0	0
$958$	20.7476	0.670325
$959$	−60.3446	−1.94863
$960$	0	0
$961$	23.3921	0.754584
$962$	1.00180	0.0322992
$963$	0	0
$964$	−14.6773	−0.472725
$965$	−13.8343	−0.445343
$966$	0	0
$967$	8.95620	0.288012	0.144006	−	0.989577i	$-0.454002\pi$
$967$	8.95620	0.288012	0.144006	+	0.989577i	$0.454002\pi$
$968$	6.87680	0.221029
$969$	0	0
$970$	−27.9904	−0.898718
$971$	56.3517	1.80841	0.904206	−	0.427096i	$-0.140463\pi$
$971$	56.3517	1.80841	0.904206	+	0.427096i	$0.140463\pi$
$972$	0	0
$973$	40.9067	1.31141
$974$	7.42424	0.237888
$975$	0	0
$976$	−0.0848386	−0.00271562
$977$	−36.6371	−1.17213	−0.586063	−	0.810266i	$-0.699323\pi$
$977$	−36.6371	−1.17213	−0.586063	+	0.810266i	$0.699323\pi$
$978$	0	0
$979$	24.1830	0.772892
$980$	−8.10503	−0.258906
$981$	0	0
$982$	11.2139	0.357851
$983$	−48.0649	−1.53303	−0.766516	−	0.642225i	$-0.778011\pi$
$983$	−48.0649	−1.53303	−0.766516	+	0.642225i	$0.778011\pi$
$984$	0	0
$985$	−20.1133	−0.640862
$986$	−22.4708	−0.715616
$987$	0	0
$988$	−0.586330	−0.0186536
$989$	4.49713	0.143001
$990$	0	0
$991$	−5.53139	−0.175710	−0.0878552	−	0.996133i	$-0.528001\pi$
$991$	−5.53139	−0.175710	−0.0878552	+	0.996133i	$0.528001\pi$
$992$	7.37510	0.234160
$993$	0	0
$994$	7.93675	0.251738
$995$	37.9705	1.20375
$996$	0	0
$997$	−37.3951	−1.18431	−0.592157	−	0.805823i	$-0.701723\pi$
$997$	−37.3951	−1.18431	−0.592157	+	0.805823i	$0.701723\pi$
$998$	−26.6372	−0.843185
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.o.1.5	yes	12
3.2	odd	2		8046.2.a.j.1.8	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.j.1.8	✓	12	3.2	odd	2
8046.2.a.o.1.5	yes	12	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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.46524 q^{5} -3.53999 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -1.46524 q^{5} -3.53999 q^{7} +1.00000 q^{8} -1.46524 q^{10} +4.22810 q^{11} -0.227772 q^{13} -3.53999 q^{14} +1.00000 q^{16} -6.27991 q^{17} +2.57419 q^{19} -1.46524 q^{20} +4.22810 q^{22} -3.30009 q^{23} -2.85307 q^{25} -0.227772 q^{26} -3.53999 q^{28} +3.57821 q^{29} +7.37510 q^{31} +1.00000 q^{32} -6.27991 q^{34} +5.18694 q^{35} -4.39823 q^{37} +2.57419 q^{38} -1.46524 q^{40} +4.53941 q^{41} -1.36273 q^{43} +4.22810 q^{44} -3.30009 q^{46} -7.53590 q^{47} +5.53153 q^{49} -2.85307 q^{50} -0.227772 q^{52} +10.3373 q^{53} -6.19518 q^{55} -3.53999 q^{56} +3.57821 q^{58} +1.84299 q^{59} -0.0848386 q^{61} +7.37510 q^{62} +1.00000 q^{64} +0.333741 q^{65} +4.55903 q^{67} -6.27991 q^{68} +5.18694 q^{70} -2.24203 q^{71} -6.79670 q^{73} -4.39823 q^{74} +2.57419 q^{76} -14.9674 q^{77} +3.14110 q^{79} -1.46524 q^{80} +4.53941 q^{82} -1.74078 q^{83} +9.20158 q^{85} -1.36273 q^{86} +4.22810 q^{88} +5.71960 q^{89} +0.806312 q^{91} -3.30009 q^{92} -7.53590 q^{94} -3.77181 q^{95} +19.1029 q^{97} +5.53153 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 12 q^{2} + 12 q^{4} + 3 q^{5} + 6 q^{7} + 12 q^{8}+O(q^{10}) \) 12 * q + 12 * q^2 + 12 * q^4 + 3 * q^5 + 6 * q^7 + 12 * q^8	\( 12 q + 12 q^{2} + 12 q^{4} + 3 q^{5} + 6 q^{7} + 12 q^{8} + 3 q^{10} + 10 q^{11} + 5 q^{13} + 6 q^{14} + 12 q^{16} + 8 q^{17} + 2 q^{19} + 3 q^{20} + 10 q^{22} + 9 q^{23} + 7 q^{25} + 5 q^{26} + 6 q^{28} + 19 q^{29} + 10 q^{31} + 12 q^{32} + 8 q^{34} + 20 q^{35} + 11 q^{37} + 2 q^{38} + 3 q^{40} + 8 q^{41} + 13 q^{43} + 10 q^{44} + 9 q^{46} + 11 q^{47} + 2 q^{49} + 7 q^{50} + 5 q^{52} + 24 q^{53} + 3 q^{55} + 6 q^{56} + 19 q^{58} + 10 q^{59} + 10 q^{62} + 12 q^{64} + 28 q^{65} + 21 q^{67} + 8 q^{68} + 20 q^{70} + 37 q^{71} - 2 q^{73} + 11 q^{74} + 2 q^{76} + 2 q^{77} + 7 q^{79} + 3 q^{80} + 8 q^{82} + 22 q^{83} + 15 q^{85} + 13 q^{86} + 10 q^{88} + 40 q^{89} + q^{91} + 9 q^{92} + 11 q^{94} + 11 q^{95} + 7 q^{97} + 2 q^{98}+O(q^{100}) \) 12 * q + 12 * q^2 + 12 * q^4 + 3 * q^5 + 6 * q^7 + 12 * q^8 + 3 * q^10 + 10 * q^11 + 5 * q^13 + 6 * q^14 + 12 * q^16 + 8 * q^17 + 2 * q^19 + 3 * q^20 + 10 * q^22 + 9 * q^23 + 7 * q^25 + 5 * q^26 + 6 * q^28 + 19 * q^29 + 10 * q^31 + 12 * q^32 + 8 * q^34 + 20 * q^35 + 11 * q^37 + 2 * q^38 + 3 * q^40 + 8 * q^41 + 13 * q^43 + 10 * q^44 + 9 * q^46 + 11 * q^47 + 2 * q^49 + 7 * q^50 + 5 * q^52 + 24 * q^53 + 3 * q^55 + 6 * q^56 + 19 * q^58 + 10 * q^59 + 10 * q^62 + 12 * q^64 + 28 * q^65 + 21 * q^67 + 8 * q^68 + 20 * q^70 + 37 * q^71 - 2 * q^73 + 11 * q^74 + 2 * q^76 + 2 * q^77 + 7 * q^79 + 3 * q^80 + 8 * q^82 + 22 * q^83 + 15 * q^85 + 13 * q^86 + 10 * q^88 + 40 * q^89 + q^91 + 9 * q^92 + 11 * q^94 + 11 * q^95 + 7 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more