LMFDB - Embedded newform 3920.2.a.bt.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3920,2,Mod(1,3920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3920, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3920.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3920 = 2^{4} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3920.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:	$31.3013575923$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{33}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 8 $ x^2 - x - 8
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 280)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.37228$ of defining polynomial
Character	$\chi$	$=$	3920.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.37228 q^{3} +1.00000 q^{5} +2.62772 q^{9} +O(q^{10})$	$q+2.37228 q^{3} +1.00000 q^{5} +2.62772 q^{9} -6.37228 q^{11} -4.37228 q^{13} +2.37228 q^{15} +0.372281 q^{17} -4.74456 q^{19} +4.74456 q^{23} +1.00000 q^{25} -0.883156 q^{27} -4.37228 q^{29} -8.00000 q^{31} -15.1168 q^{33} -2.00000 q^{37} -10.3723 q^{39} -6.74456 q^{41} +8.74456 q^{43} +2.62772 q^{45} -7.11684 q^{47} +0.883156 q^{51} +10.7446 q^{53} -6.37228 q^{55} -11.2554 q^{57} +8.00000 q^{59} +2.74456 q^{61} -4.37228 q^{65} +4.00000 q^{67} +11.2554 q^{69} -8.00000 q^{71} +6.00000 q^{73} +2.37228 q^{75} -15.1168 q^{79} -9.97825 q^{81} -9.48913 q^{83} +0.372281 q^{85} -10.3723 q^{87} -14.7446 q^{89} -18.9783 q^{93} -4.74456 q^{95} +9.86141 q^{97} -16.7446 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} + 2 q^{5} + 11 q^{9}+O(q^{10}) $ 2 * q - q^3 + 2 * q^5 + 11 * q^9	$ 2 q - q^{3} + 2 q^{5} + 11 q^{9} - 7 q^{11} - 3 q^{13} - q^{15} - 5 q^{17} + 2 q^{19} - 2 q^{23} + 2 q^{25} - 19 q^{27} - 3 q^{29} - 16 q^{31} - 13 q^{33} - 4 q^{37} - 15 q^{39} - 2 q^{41} + 6 q^{43} + 11 q^{45} + 3 q^{47} + 19 q^{51} + 10 q^{53} - 7 q^{55} - 34 q^{57} + 16 q^{59} - 6 q^{61} - 3 q^{65} + 8 q^{67} + 34 q^{69} - 16 q^{71} + 12 q^{73} - q^{75} - 13 q^{79} + 26 q^{81} + 4 q^{83} - 5 q^{85} - 15 q^{87} - 18 q^{89} + 8 q^{93} + 2 q^{95} - 9 q^{97} - 22 q^{99}+O(q^{100}) $ 2 * q - q^3 + 2 * q^5 + 11 * q^9 - 7 * q^11 - 3 * q^13 - q^15 - 5 * q^17 + 2 * q^19 - 2 * q^23 + 2 * q^25 - 19 * q^27 - 3 * q^29 - 16 * q^31 - 13 * q^33 - 4 * q^37 - 15 * q^39 - 2 * q^41 + 6 * q^43 + 11 * q^45 + 3 * q^47 + 19 * q^51 + 10 * q^53 - 7 * q^55 - 34 * q^57 + 16 * q^59 - 6 * q^61 - 3 * q^65 + 8 * q^67 + 34 * q^69 - 16 * q^71 + 12 * q^73 - q^75 - 13 * q^79 + 26 * q^81 + 4 * q^83 - 5 * q^85 - 15 * q^87 - 18 * q^89 + 8 * q^93 + 2 * q^95 - 9 * q^97 - 22 * q^99

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 $

$2$	0	0
$2$	0	0
$3$	2.37228	1.36964	0.684819	−	0.728714i	$-0.259881\pi$
$3$	2.37228	1.36964	0.684819	+	0.728714i	$0.259881\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.62772	0.875906
$10$	0	0
$11$	−6.37228	−1.92132	−0.960658	−	0.277736i	$-0.910416\pi$
$11$	−6.37228	−1.92132	−0.960658	+	0.277736i	$0.910416\pi$
$12$	0	0
$13$	−4.37228	−1.21265	−0.606326	−	0.795216i	$-0.707357\pi$
$13$	−4.37228	−1.21265	−0.606326	+	0.795216i	$0.707357\pi$
$14$	0	0
$15$	2.37228	0.612520
$16$	0	0
$17$	0.372281	0.0902915	0.0451457	−	0.998980i	$-0.485625\pi$
$17$	0.372281	0.0902915	0.0451457	+	0.998980i	$0.485625\pi$
$18$	0	0
$19$	−4.74456	−1.08848	−0.544239	−	0.838930i	$-0.683181\pi$
$19$	−4.74456	−1.08848	−0.544239	+	0.838930i	$0.683181\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.74456	0.989310	0.494655	−	0.869090i	$-0.335294\pi$
$23$	4.74456	0.989310	0.494655	+	0.869090i	$0.335294\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−0.883156	−0.169963
$28$	0	0
$29$	−4.37228	−0.811912	−0.405956	−	0.913893i	$-0.633061\pi$
$29$	−4.37228	−0.811912	−0.405956	+	0.913893i	$0.633061\pi$
$30$	0	0
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	0	0
$33$	−15.1168	−2.63150
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	−10.3723	−1.66089
$40$	0	0
$41$	−6.74456	−1.05332	−0.526662	−	0.850075i	$-0.676557\pi$
$41$	−6.74456	−1.05332	−0.526662	+	0.850075i	$0.676557\pi$
$42$	0	0
$43$	8.74456	1.33353	0.666767	−	0.745267i	$-0.267678\pi$
$43$	8.74456	1.33353	0.666767	+	0.745267i	$0.267678\pi$
$44$	0	0
$45$	2.62772	0.391717
$46$	0	0
$47$	−7.11684	−1.03810	−0.519049	−	0.854744i	$-0.673714\pi$
$47$	−7.11684	−1.03810	−0.519049	+	0.854744i	$0.673714\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0.883156	0.123667
$52$	0	0
$53$	10.7446	1.47588	0.737940	−	0.674867i	$-0.235799\pi$
$53$	10.7446	1.47588	0.737940	+	0.674867i	$0.235799\pi$
$54$	0	0
$55$	−6.37228	−0.859238
$56$	0	0
$57$	−11.2554	−1.49082
$58$	0	0
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	2.74456	0.351405	0.175703	−	0.984443i	$-0.443780\pi$
$61$	2.74456	0.351405	0.175703	+	0.984443i	$0.443780\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−4.37228	−0.542315
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	11.2554	1.35500
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	2.37228	0.273927
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−15.1168	−1.70078	−0.850389	−	0.526155i	$-0.823633\pi$
$79$	−15.1168	−1.70078	−0.850389	+	0.526155i	$0.823633\pi$
$80$	0	0
$81$	−9.97825	−1.10869
$82$	0	0
$83$	−9.48913	−1.04157	−0.520783	−	0.853689i	$-0.674360\pi$
$83$	−9.48913	−1.04157	−0.520783	+	0.853689i	$0.674360\pi$
$84$	0	0
$85$	0.372281	0.0403796
$86$	0	0
$87$	−10.3723	−1.11203
$88$	0	0
$89$	−14.7446	−1.56292	−0.781460	−	0.623955i	$-0.785525\pi$
$89$	−14.7446	−1.56292	−0.781460	+	0.623955i	$0.785525\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−18.9783	−1.96795
$94$	0	0
$95$	−4.74456	−0.486782
$96$	0	0
$97$	9.86141	1.00127	0.500637	−	0.865657i	$-0.333099\pi$
$97$	9.86141	1.00127	0.500637	+	0.865657i	$0.333099\pi$
$98$	0	0
$99$	−16.7446	−1.68289
$100$	0	0
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	0	0
$103$	−5.62772	−0.554516	−0.277258	−	0.960796i	$-0.589426\pi$
$103$	−5.62772	−0.554516	−0.277258	+	0.960796i	$0.589426\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−8.74456	−0.845369	−0.422684	−	0.906277i	$-0.638912\pi$
$107$	−8.74456	−0.845369	−0.422684	+	0.906277i	$0.638912\pi$
$108$	0	0
$109$	0.372281	0.0356581	0.0178290	−	0.999841i	$-0.494325\pi$
$109$	0.372281	0.0356581	0.0178290	+	0.999841i	$0.494325\pi$
$110$	0	0
$111$	−4.74456	−0.450334
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	4.74456	0.442433
$116$	0	0
$117$	−11.4891	−1.06217
$118$	0	0
$119$	0	0
$120$	0	0
$121$	29.6060	2.69145
$122$	0	0
$123$	−16.0000	−1.44267
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	0	0
$129$	20.7446	1.82646
$130$	0	0
$131$	−4.74456	−0.414534	−0.207267	−	0.978284i	$-0.566457\pi$
$131$	−4.74456	−0.414534	−0.207267	+	0.978284i	$0.566457\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−0.883156	−0.0760100
$136$	0	0
$137$	14.7446	1.25971	0.629857	−	0.776712i	$-0.283114\pi$
$137$	14.7446	1.25971	0.629857	+	0.776712i	$0.283114\pi$
$138$	0	0
$139$	−4.74456	−0.402429	−0.201214	−	0.979547i	$-0.564489\pi$
$139$	−4.74456	−0.402429	−0.201214	+	0.979547i	$0.564489\pi$
$140$	0	0
$141$	−16.8832	−1.42182
$142$	0	0
$143$	27.8614	2.32989
$144$	0	0
$145$	−4.37228	−0.363098
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.4891	1.26892	0.634459	−	0.772956i	$-0.281223\pi$
$149$	15.4891	1.26892	0.634459	+	0.772956i	$0.281223\pi$
$150$	0	0
$151$	−15.1168	−1.23019	−0.615096	−	0.788452i	$-0.710883\pi$
$151$	−15.1168	−1.23019	−0.615096	+	0.788452i	$0.710883\pi$
$152$	0	0
$153$	0.978251	0.0790869
$154$	0	0
$155$	−8.00000	−0.642575
$156$	0	0
$157$	15.4891	1.23617	0.618083	−	0.786113i	$-0.287909\pi$
$157$	15.4891	1.23617	0.618083	+	0.786113i	$0.287909\pi$
$158$	0	0
$159$	25.4891	2.02142
$160$	0	0
$161$	0	0
$162$	0	0
$163$	16.7446	1.31154	0.655768	−	0.754963i	$-0.272345\pi$
$163$	16.7446	1.31154	0.655768	+	0.754963i	$0.272345\pi$
$164$	0	0
$165$	−15.1168	−1.17684
$166$	0	0
$167$	−5.62772	−0.435486	−0.217743	−	0.976006i	$-0.569869\pi$
$167$	−5.62772	−0.435486	−0.217743	+	0.976006i	$0.569869\pi$
$168$	0	0
$169$	6.11684	0.470526
$170$	0	0
$171$	−12.4674	−0.953404
$172$	0	0
$173$	0.372281	0.0283040	0.0141520	−	0.999900i	$-0.495495\pi$
$173$	0.372281	0.0283040	0.0141520	+	0.999900i	$0.495495\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	18.9783	1.42649
$178$	0	0
$179$	22.9783	1.71748	0.858738	−	0.512416i	$-0.171249\pi$
$179$	22.9783	1.71748	0.858738	+	0.512416i	$0.171249\pi$
$180$	0	0
$181$	−16.2337	−1.20664	−0.603320	−	0.797499i	$-0.706156\pi$
$181$	−16.2337	−1.20664	−0.603320	+	0.797499i	$0.706156\pi$
$182$	0	0
$183$	6.51087	0.481298
$184$	0	0
$185$	−2.00000	−0.147043
$186$	0	0
$187$	−2.37228	−0.173478
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.86141	−0.279402	−0.139701	−	0.990194i	$-0.544614\pi$
$191$	−3.86141	−0.279402	−0.139701	+	0.990194i	$0.544614\pi$
$192$	0	0
$193$	6.74456	0.485484	0.242742	−	0.970091i	$-0.421953\pi$
$193$	6.74456	0.485484	0.242742	+	0.970091i	$0.421953\pi$
$194$	0	0
$195$	−10.3723	−0.742774
$196$	0	0
$197$	−8.23369	−0.586626	−0.293313	−	0.956016i	$-0.594758\pi$
$197$	−8.23369	−0.586626	−0.293313	+	0.956016i	$0.594758\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	0	0
$201$	9.48913	0.669311
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−6.74456	−0.471061
$206$	0	0
$207$	12.4674	0.866543
$208$	0	0
$209$	30.2337	2.09131
$210$	0	0
$211$	14.3723	0.989429	0.494714	−	0.869056i	$-0.335273\pi$
$211$	14.3723	0.989429	0.494714	+	0.869056i	$0.335273\pi$
$212$	0	0
$213$	−18.9783	−1.30037
$214$	0	0
$215$	8.74456	0.596374
$216$	0	0
$217$	0	0
$218$	0	0
$219$	14.2337	0.961823
$220$	0	0
$221$	−1.62772	−0.109492
$222$	0	0
$223$	−5.62772	−0.376860	−0.188430	−	0.982087i	$-0.560340\pi$
$223$	−5.62772	−0.376860	−0.188430	+	0.982087i	$0.560340\pi$
$224$	0	0
$225$	2.62772	0.175181
$226$	0	0
$227$	19.8614	1.31825	0.659124	−	0.752034i	$-0.270927\pi$
$227$	19.8614	1.31825	0.659124	+	0.752034i	$0.270927\pi$
$228$	0	0
$229$	12.2337	0.808425	0.404212	−	0.914665i	$-0.367546\pi$
$229$	12.2337	0.808425	0.404212	+	0.914665i	$0.367546\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.25544	−0.0822464	−0.0411232	−	0.999154i	$-0.513094\pi$
$233$	−1.25544	−0.0822464	−0.0411232	+	0.999154i	$0.513094\pi$
$234$	0	0
$235$	−7.11684	−0.464252
$236$	0	0
$237$	−35.8614	−2.32945
$238$	0	0
$239$	−13.6277	−0.881504	−0.440752	−	0.897629i	$-0.645288\pi$
$239$	−13.6277	−0.881504	−0.440752	+	0.897629i	$0.645288\pi$
$240$	0	0
$241$	−26.0000	−1.67481	−0.837404	−	0.546585i	$-0.815928\pi$
$241$	−26.0000	−1.67481	−0.837404	+	0.546585i	$0.815928\pi$
$242$	0	0
$243$	−21.0217	−1.34855
$244$	0	0
$245$	0	0
$246$	0	0
$247$	20.7446	1.31994
$248$	0	0
$249$	−22.5109	−1.42657
$250$	0	0
$251$	4.74456	0.299474	0.149737	−	0.988726i	$-0.452157\pi$
$251$	4.74456	0.299474	0.149737	+	0.988726i	$0.452157\pi$
$252$	0	0
$253$	−30.2337	−1.90078
$254$	0	0
$255$	0.883156	0.0553054
$256$	0	0
$257$	23.4891	1.46521	0.732606	−	0.680653i	$-0.238304\pi$
$257$	23.4891	1.46521	0.732606	+	0.680653i	$0.238304\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−11.4891	−0.711159
$262$	0	0
$263$	−22.2337	−1.37099	−0.685494	−	0.728078i	$-0.740414\pi$
$263$	−22.2337	−1.37099	−0.685494	+	0.728078i	$0.740414\pi$
$264$	0	0
$265$	10.7446	0.660033
$266$	0	0
$267$	−34.9783	−2.14063
$268$	0	0
$269$	6.00000	0.365826	0.182913	−	0.983129i	$-0.441447\pi$
$269$	6.00000	0.365826	0.182913	+	0.983129i	$0.441447\pi$
$270$	0	0
$271$	9.48913	0.576423	0.288212	−	0.957567i	$-0.406939\pi$
$271$	9.48913	0.576423	0.288212	+	0.957567i	$0.406939\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−6.37228	−0.384263
$276$	0	0
$277$	24.9783	1.50080	0.750399	−	0.660985i	$-0.229861\pi$
$277$	24.9783	1.50080	0.750399	+	0.660985i	$0.229861\pi$
$278$	0	0
$279$	−21.0217	−1.25854
$280$	0	0
$281$	18.6060	1.10994	0.554970	−	0.831871i	$-0.312730\pi$
$281$	18.6060	1.10994	0.554970	+	0.831871i	$0.312730\pi$
$282$	0	0
$283$	−8.88316	−0.528049	−0.264024	−	0.964516i	$-0.585050\pi$
$283$	−8.88316	−0.528049	−0.264024	+	0.964516i	$0.585050\pi$
$284$	0	0
$285$	−11.2554	−0.666715
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−16.8614	−0.991847
$290$	0	0
$291$	23.3940	1.37138
$292$	0	0
$293$	−25.1168	−1.46734	−0.733671	−	0.679505i	$-0.762195\pi$
$293$	−25.1168	−1.46734	−0.733671	+	0.679505i	$0.762195\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	0	0
$297$	5.62772	0.326553
$298$	0	0
$299$	−20.7446	−1.19969
$300$	0	0
$301$	0	0
$302$	0	0
$303$	14.2337	0.817704
$304$	0	0
$305$	2.74456	0.157153
$306$	0	0
$307$	31.1168	1.77593	0.887966	−	0.459909i	$-0.152118\pi$
$307$	31.1168	1.77593	0.887966	+	0.459909i	$0.152118\pi$
$308$	0	0
$309$	−13.3505	−0.759485
$310$	0	0
$311$	−12.7446	−0.722678	−0.361339	−	0.932435i	$-0.617680\pi$
$311$	−12.7446	−0.722678	−0.361339	+	0.932435i	$0.617680\pi$
$312$	0	0
$313$	−2.88316	−0.162966	−0.0814828	−	0.996675i	$-0.525966\pi$
$313$	−2.88316	−0.162966	−0.0814828	+	0.996675i	$0.525966\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	14.0000	0.786318	0.393159	−	0.919470i	$-0.371382\pi$
$317$	14.0000	0.786318	0.393159	+	0.919470i	$0.371382\pi$
$318$	0	0
$319$	27.8614	1.55994
$320$	0	0
$321$	−20.7446	−1.15785
$322$	0	0
$323$	−1.76631	−0.0982802
$324$	0	0
$325$	−4.37228	−0.242531
$326$	0	0
$327$	0.883156	0.0488386
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	0	0
$333$	−5.25544	−0.287996
$334$	0	0
$335$	4.00000	0.218543
$336$	0	0
$337$	−7.48913	−0.407959	−0.203979	−	0.978975i	$-0.565388\pi$
$337$	−7.48913	−0.407959	−0.203979	+	0.978975i	$0.565388\pi$
$338$	0	0
$339$	4.74456	0.257689
$340$	0	0
$341$	50.9783	2.76063
$342$	0	0
$343$	0	0
$344$	0	0
$345$	11.2554	0.605972
$346$	0	0
$347$	24.7446	1.32836	0.664179	−	0.747574i	$-0.268781\pi$
$347$	24.7446	1.32836	0.664179	+	0.747574i	$0.268781\pi$
$348$	0	0
$349$	−19.4891	−1.04323	−0.521614	−	0.853181i	$-0.674670\pi$
$349$	−19.4891	−1.04323	−0.521614	+	0.853181i	$0.674670\pi$
$350$	0	0
$351$	3.86141	0.206107
$352$	0	0
$353$	1.86141	0.0990727	0.0495363	−	0.998772i	$-0.484226\pi$
$353$	1.86141	0.0990727	0.0495363	+	0.998772i	$0.484226\pi$
$354$	0	0
$355$	−8.00000	−0.424596
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	3.51087	0.184783
$362$	0	0
$363$	70.2337	3.68631
$364$	0	0
$365$	6.00000	0.314054
$366$	0	0
$367$	19.8614	1.03676	0.518378	−	0.855151i	$-0.326536\pi$
$367$	19.8614	1.03676	0.518378	+	0.855151i	$0.326536\pi$
$368$	0	0
$369$	−17.7228	−0.922613
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−30.7446	−1.59189	−0.795947	−	0.605367i	$-0.793026\pi$
$373$	−30.7446	−1.59189	−0.795947	+	0.605367i	$0.793026\pi$
$374$	0	0
$375$	2.37228	0.122504
$376$	0	0
$377$	19.1168	0.984568
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	0	0
$381$	−18.9783	−0.972285
$382$	0	0
$383$	−17.4891	−0.893653	−0.446826	−	0.894621i	$-0.647446\pi$
$383$	−17.4891	−0.893653	−0.446826	+	0.894621i	$0.647446\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	22.9783	1.16805
$388$	0	0
$389$	17.8614	0.905609	0.452805	−	0.891610i	$-0.350424\pi$
$389$	17.8614	0.905609	0.452805	+	0.891610i	$0.350424\pi$
$390$	0	0
$391$	1.76631	0.0893262
$392$	0	0
$393$	−11.2554	−0.567762
$394$	0	0
$395$	−15.1168	−0.760611
$396$	0	0
$397$	−31.6277	−1.58735	−0.793675	−	0.608342i	$-0.791835\pi$
$397$	−31.6277	−1.58735	−0.793675	+	0.608342i	$0.791835\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−38.6060	−1.92789	−0.963945	−	0.266101i	$-0.914264\pi$
$401$	−38.6060	−1.92789	−0.963945	+	0.266101i	$0.914264\pi$
$402$	0	0
$403$	34.9783	1.74239
$404$	0	0
$405$	−9.97825	−0.495823
$406$	0	0
$407$	12.7446	0.631725
$408$	0	0
$409$	−11.4891	−0.568101	−0.284050	−	0.958809i	$-0.591678\pi$
$409$	−11.4891	−0.568101	−0.284050	+	0.958809i	$0.591678\pi$
$410$	0	0
$411$	34.9783	1.72535
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−9.48913	−0.465803
$416$	0	0
$417$	−11.2554	−0.551181
$418$	0	0
$419$	−14.5109	−0.708903	−0.354451	−	0.935074i	$-0.615332\pi$
$419$	−14.5109	−0.708903	−0.354451	+	0.935074i	$0.615332\pi$
$420$	0	0
$421$	−18.6060	−0.906799	−0.453400	−	0.891307i	$-0.649789\pi$
$421$	−18.6060	−0.906799	−0.453400	+	0.891307i	$0.649789\pi$
$422$	0	0
$423$	−18.7011	−0.909277
$424$	0	0
$425$	0.372281	0.0180583
$426$	0	0
$427$	0	0
$428$	0	0
$429$	66.0951	3.19110
$430$	0	0
$431$	−18.3723	−0.884962	−0.442481	−	0.896778i	$-0.645902\pi$
$431$	−18.3723	−0.884962	−0.442481	+	0.896778i	$0.645902\pi$
$432$	0	0
$433$	−28.9783	−1.39261	−0.696303	−	0.717748i	$-0.745173\pi$
$433$	−28.9783	−1.39261	−0.696303	+	0.717748i	$0.745173\pi$
$434$	0	0
$435$	−10.3723	−0.497313
$436$	0	0
$437$	−22.5109	−1.07684
$438$	0	0
$439$	−6.23369	−0.297518	−0.148759	−	0.988874i	$-0.547528\pi$
$439$	−6.23369	−0.297518	−0.148759	+	0.988874i	$0.547528\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	16.7446	0.795558	0.397779	−	0.917481i	$-0.369781\pi$
$443$	16.7446	0.795558	0.397779	+	0.917481i	$0.369781\pi$
$444$	0	0
$445$	−14.7446	−0.698959
$446$	0	0
$447$	36.7446	1.73796
$448$	0	0
$449$	17.1168	0.807794	0.403897	−	0.914805i	$-0.367655\pi$
$449$	17.1168	0.807794	0.403897	+	0.914805i	$0.367655\pi$
$450$	0	0
$451$	42.9783	2.02377
$452$	0	0
$453$	−35.8614	−1.68492
$454$	0	0
$455$	0	0
$456$	0	0
$457$	5.25544	0.245839	0.122919	−	0.992417i	$-0.460774\pi$
$457$	5.25544	0.245839	0.122919	+	0.992417i	$0.460774\pi$
$458$	0	0
$459$	−0.328782	−0.0153463
$460$	0	0
$461$	1.25544	0.0584715	0.0292358	−	0.999573i	$-0.490693\pi$
$461$	1.25544	0.0584715	0.0292358	+	0.999573i	$0.490693\pi$
$462$	0	0
$463$	6.51087	0.302586	0.151293	−	0.988489i	$-0.451656\pi$
$463$	6.51087	0.302586	0.151293	+	0.988489i	$0.451656\pi$
$464$	0	0
$465$	−18.9783	−0.880095
$466$	0	0
$467$	8.60597	0.398237	0.199118	−	0.979975i	$-0.436192\pi$
$467$	8.60597	0.398237	0.199118	+	0.979975i	$0.436192\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	36.7446	1.69310
$472$	0	0
$473$	−55.7228	−2.56214
$474$	0	0
$475$	−4.74456	−0.217695
$476$	0	0
$477$	28.2337	1.29273
$478$	0	0
$479$	22.2337	1.01588	0.507942	−	0.861392i	$-0.330407\pi$
$479$	22.2337	1.01588	0.507942	+	0.861392i	$0.330407\pi$
$480$	0	0
$481$	8.74456	0.398718
$482$	0	0
$483$	0	0
$484$	0	0
$485$	9.86141	0.447783
$486$	0	0
$487$	−30.2337	−1.37002	−0.685010	−	0.728534i	$-0.740202\pi$
$487$	−30.2337	−1.37002	−0.685010	+	0.728534i	$0.740202\pi$
$488$	0	0
$489$	39.7228	1.79633
$490$	0	0
$491$	−35.1168	−1.58480	−0.792400	−	0.610001i	$-0.791169\pi$
$491$	−35.1168	−1.58480	−0.792400	+	0.610001i	$0.791169\pi$
$492$	0	0
$493$	−1.62772	−0.0733088
$494$	0	0
$495$	−16.7446	−0.752612
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−31.8614	−1.42631	−0.713156	−	0.701005i	$-0.752735\pi$
$499$	−31.8614	−1.42631	−0.713156	+	0.701005i	$0.752735\pi$
$500$	0	0
$501$	−13.3505	−0.596458
$502$	0	0
$503$	−18.3723	−0.819180	−0.409590	−	0.912270i	$-0.634328\pi$
$503$	−18.3723	−0.819180	−0.409590	+	0.912270i	$0.634328\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	0	0
$507$	14.5109	0.644451
$508$	0	0
$509$	40.9783	1.81633	0.908165	−	0.418613i	$-0.137484\pi$
$509$	40.9783	1.81633	0.908165	+	0.418613i	$0.137484\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	4.19019	0.185001
$514$	0	0
$515$	−5.62772	−0.247987
$516$	0	0
$517$	45.3505	1.99451
$518$	0	0
$519$	0.883156	0.0387662
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	−36.4674	−1.59461	−0.797304	−	0.603579i	$-0.793741\pi$
$523$	−36.4674	−1.59461	−0.797304	+	0.603579i	$0.793741\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.97825	−0.129735
$528$	0	0
$529$	−0.489125	−0.0212663
$530$	0	0
$531$	21.0217	0.912266
$532$	0	0
$533$	29.4891	1.27732
$534$	0	0
$535$	−8.74456	−0.378060
$536$	0	0
$537$	54.5109	2.35232
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−31.3505	−1.34786	−0.673932	−	0.738793i	$-0.735396\pi$
$541$	−31.3505	−1.34786	−0.673932	+	0.738793i	$0.735396\pi$
$542$	0	0
$543$	−38.5109	−1.65266
$544$	0	0
$545$	0.372281	0.0159468
$546$	0	0
$547$	30.9783	1.32453	0.662267	−	0.749268i	$-0.269594\pi$
$547$	30.9783	1.32453	0.662267	+	0.749268i	$0.269594\pi$
$548$	0	0
$549$	7.21194	0.307798
$550$	0	0
$551$	20.7446	0.883748
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−4.74456	−0.201395
$556$	0	0
$557$	−3.76631	−0.159584	−0.0797919	−	0.996812i	$-0.525426\pi$
$557$	−3.76631	−0.159584	−0.0797919	+	0.996812i	$0.525426\pi$
$558$	0	0
$559$	−38.2337	−1.61711
$560$	0	0
$561$	−5.62772	−0.237602
$562$	0	0
$563$	17.4891	0.737079	0.368539	−	0.929612i	$-0.379858\pi$
$563$	17.4891	0.737079	0.368539	+	0.929612i	$0.379858\pi$
$564$	0	0
$565$	2.00000	0.0841406
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−24.9783	−1.04714	−0.523571	−	0.851982i	$-0.675401\pi$
$569$	−24.9783	−1.04714	−0.523571	+	0.851982i	$0.675401\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	−9.16034	−0.382679
$574$	0	0
$575$	4.74456	0.197862
$576$	0	0
$577$	22.6060	0.941099	0.470549	−	0.882374i	$-0.344056\pi$
$577$	22.6060	0.941099	0.470549	+	0.882374i	$0.344056\pi$
$578$	0	0
$579$	16.0000	0.664937
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−68.4674	−2.83563
$584$	0	0
$585$	−11.4891	−0.475017
$586$	0	0
$587$	−34.9783	−1.44371	−0.721853	−	0.692046i	$-0.756710\pi$
$587$	−34.9783	−1.44371	−0.721853	+	0.692046i	$0.756710\pi$
$588$	0	0
$589$	37.9565	1.56397
$590$	0	0
$591$	−19.5326	−0.803465
$592$	0	0
$593$	19.6277	0.806014	0.403007	−	0.915197i	$-0.367965\pi$
$593$	19.6277	0.806014	0.403007	+	0.915197i	$0.367965\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−37.9565	−1.55346
$598$	0	0
$599$	32.6060	1.33224	0.666122	−	0.745843i	$-0.267953\pi$
$599$	32.6060	1.33224	0.666122	+	0.745843i	$0.267953\pi$
$600$	0	0
$601$	−16.5109	−0.673493	−0.336746	−	0.941595i	$-0.609326\pi$
$601$	−16.5109	−0.673493	−0.336746	+	0.941595i	$0.609326\pi$
$602$	0	0
$603$	10.5109	0.428036
$604$	0	0
$605$	29.6060	1.20365
$606$	0	0
$607$	24.6060	0.998725	0.499363	−	0.866393i	$-0.333568\pi$
$607$	24.6060	0.998725	0.499363	+	0.866393i	$0.333568\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	31.1168	1.25885
$612$	0	0
$613$	−8.51087	−0.343751	−0.171875	−	0.985119i	$-0.554983\pi$
$613$	−8.51087	−0.343751	−0.171875	+	0.985119i	$0.554983\pi$
$614$	0	0
$615$	−16.0000	−0.645182
$616$	0	0
$617$	−6.00000	−0.241551	−0.120775	−	0.992680i	$-0.538538\pi$
$617$	−6.00000	−0.241551	−0.120775	+	0.992680i	$0.538538\pi$
$618$	0	0
$619$	−12.7446	−0.512247	−0.256124	−	0.966644i	$-0.582445\pi$
$619$	−12.7446	−0.512247	−0.256124	+	0.966644i	$0.582445\pi$
$620$	0	0
$621$	−4.19019	−0.168146
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	71.7228	2.86433
$628$	0	0
$629$	−0.744563	−0.0296877
$630$	0	0
$631$	−2.37228	−0.0944390	−0.0472195	−	0.998885i	$-0.515036\pi$
$631$	−2.37228	−0.0944390	−0.0472195	+	0.998885i	$0.515036\pi$
$632$	0	0
$633$	34.0951	1.35516
$634$	0	0
$635$	−8.00000	−0.317470
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−21.0217	−0.831608
$640$	0	0
$641$	2.00000	0.0789953	0.0394976	−	0.999220i	$-0.487424\pi$
$641$	2.00000	0.0789953	0.0394976	+	0.999220i	$0.487424\pi$
$642$	0	0
$643$	18.3723	0.724532	0.362266	−	0.932075i	$-0.382003\pi$
$643$	18.3723	0.724532	0.362266	+	0.932075i	$0.382003\pi$
$644$	0	0
$645$	20.7446	0.816816
$646$	0	0
$647$	−16.0000	−0.629025	−0.314512	−	0.949253i	$-0.601841\pi$
$647$	−16.0000	−0.629025	−0.314512	+	0.949253i	$0.601841\pi$
$648$	0	0
$649$	−50.9783	−2.00107
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	−4.74456	−0.185385
$656$	0	0
$657$	15.7663	0.615102
$658$	0	0
$659$	11.1168	0.433051	0.216525	−	0.976277i	$-0.430528\pi$
$659$	11.1168	0.433051	0.216525	+	0.976277i	$0.430528\pi$
$660$	0	0
$661$	−14.7446	−0.573497	−0.286749	−	0.958006i	$-0.592574\pi$
$661$	−14.7446	−0.573497	−0.286749	+	0.958006i	$0.592574\pi$
$662$	0	0
$663$	−3.86141	−0.149965
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−20.7446	−0.803233
$668$	0	0
$669$	−13.3505	−0.516161
$670$	0	0
$671$	−17.4891	−0.675160
$672$	0	0
$673$	25.7228	0.991542	0.495771	−	0.868453i	$-0.334886\pi$
$673$	25.7228	0.991542	0.495771	+	0.868453i	$0.334886\pi$
$674$	0	0
$675$	−0.883156	−0.0339927
$676$	0	0
$677$	−15.3505	−0.589969	−0.294984	−	0.955502i	$-0.595314\pi$
$677$	−15.3505	−0.589969	−0.294984	+	0.955502i	$0.595314\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	47.1168	1.80552
$682$	0	0
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	0	0
$685$	14.7446	0.563361
$686$	0	0
$687$	29.0217	1.10725
$688$	0	0
$689$	−46.9783	−1.78973
$690$	0	0
$691$	−9.48913	−0.360983	−0.180492	−	0.983577i	$-0.557769\pi$
$691$	−9.48913	−0.360983	−0.180492	+	0.983577i	$0.557769\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.74456	−0.179972
$696$	0	0
$697$	−2.51087	−0.0951062
$698$	0	0
$699$	−2.97825	−0.112648
$700$	0	0
$701$	2.13859	0.0807736	0.0403868	−	0.999184i	$-0.487141\pi$
$701$	2.13859	0.0807736	0.0403868	+	0.999184i	$0.487141\pi$
$702$	0	0
$703$	9.48913	0.357889
$704$	0	0
$705$	−16.8832	−0.635856
$706$	0	0
$707$	0	0
$708$	0	0
$709$	6.60597	0.248092	0.124046	−	0.992276i	$-0.460413\pi$
$709$	6.60597	0.248092	0.124046	+	0.992276i	$0.460413\pi$
$710$	0	0
$711$	−39.7228	−1.48972
$712$	0	0
$713$	−37.9565	−1.42148
$714$	0	0
$715$	27.8614	1.04196
$716$	0	0
$717$	−32.3288	−1.20734
$718$	0	0
$719$	3.25544	0.121407	0.0607037	−	0.998156i	$-0.480666\pi$
$719$	3.25544	0.121407	0.0607037	+	0.998156i	$0.480666\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−61.6793	−2.29388
$724$	0	0
$725$	−4.37228	−0.162382
$726$	0	0
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	0	0
$729$	−19.9348	−0.738324
$730$	0	0
$731$	3.25544	0.120407
$732$	0	0
$733$	10.1386	0.374477	0.187239	−	0.982314i	$-0.440046\pi$
$733$	10.1386	0.374477	0.187239	+	0.982314i	$0.440046\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−25.4891	−0.938904
$738$	0	0
$739$	20.6060	0.758003	0.379001	−	0.925396i	$-0.376268\pi$
$739$	20.6060	0.758003	0.379001	+	0.925396i	$0.376268\pi$
$740$	0	0
$741$	49.2119	1.80785
$742$	0	0
$743$	6.51087	0.238861	0.119430	−	0.992843i	$-0.461893\pi$
$743$	6.51087	0.238861	0.119430	+	0.992843i	$0.461893\pi$
$744$	0	0
$745$	15.4891	0.567478
$746$	0	0
$747$	−24.9348	−0.912315
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−20.1386	−0.734868	−0.367434	−	0.930050i	$-0.619764\pi$
$751$	−20.1386	−0.734868	−0.367434	+	0.930050i	$0.619764\pi$
$752$	0	0
$753$	11.2554	0.410171
$754$	0	0
$755$	−15.1168	−0.550158
$756$	0	0
$757$	−3.76631	−0.136889	−0.0684445	−	0.997655i	$-0.521804\pi$
$757$	−3.76631	−0.136889	−0.0684445	+	0.997655i	$0.521804\pi$
$758$	0	0
$759$	−71.7228	−2.60337
$760$	0	0
$761$	−4.97825	−0.180461	−0.0902307	−	0.995921i	$-0.528760\pi$
$761$	−4.97825	−0.180461	−0.0902307	+	0.995921i	$0.528760\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0.978251	0.0353687
$766$	0	0
$767$	−34.9783	−1.26299
$768$	0	0
$769$	−3.48913	−0.125821	−0.0629105	−	0.998019i	$-0.520038\pi$
$769$	−3.48913	−0.125821	−0.0629105	+	0.998019i	$0.520038\pi$
$770$	0	0
$771$	55.7228	2.00681
$772$	0	0
$773$	−4.37228	−0.157260	−0.0786300	−	0.996904i	$-0.525055\pi$
$773$	−4.37228	−0.157260	−0.0786300	+	0.996904i	$0.525055\pi$
$774$	0	0
$775$	−8.00000	−0.287368
$776$	0	0
$777$	0	0
$778$	0	0
$779$	32.0000	1.14652
$780$	0	0
$781$	50.9783	1.82415
$782$	0	0
$783$	3.86141	0.137995
$784$	0	0
$785$	15.4891	0.552831
$786$	0	0
$787$	−31.1168	−1.10920	−0.554598	−	0.832119i	$-0.687128\pi$
$787$	−31.1168	−1.10920	−0.554598	+	0.832119i	$0.687128\pi$
$788$	0	0
$789$	−52.7446	−1.87776
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−12.0000	−0.426132
$794$	0	0
$795$	25.4891	0.904006
$796$	0	0
$797$	−15.6277	−0.553562	−0.276781	−	0.960933i	$-0.589268\pi$
$797$	−15.6277	−0.553562	−0.276781	+	0.960933i	$0.589268\pi$
$798$	0	0
$799$	−2.64947	−0.0937314
$800$	0	0
$801$	−38.7446	−1.36897
$802$	0	0
$803$	−38.2337	−1.34924
$804$	0	0
$805$	0	0
$806$	0	0
$807$	14.2337	0.501050
$808$	0	0
$809$	20.3723	0.716251	0.358126	−	0.933673i	$-0.383416\pi$
$809$	20.3723	0.716251	0.358126	+	0.933673i	$0.383416\pi$
$810$	0	0
$811$	12.7446	0.447522	0.223761	−	0.974644i	$-0.428166\pi$
$811$	12.7446	0.447522	0.223761	+	0.974644i	$0.428166\pi$
$812$	0	0
$813$	22.5109	0.789491
$814$	0	0
$815$	16.7446	0.586536
$816$	0	0
$817$	−41.4891	−1.45152
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−25.1168	−0.876584	−0.438292	−	0.898833i	$-0.644416\pi$
$821$	−25.1168	−0.876584	−0.438292	+	0.898833i	$0.644416\pi$
$822$	0	0
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	0	0
$825$	−15.1168	−0.526301
$826$	0	0
$827$	−24.7446	−0.860453	−0.430226	−	0.902721i	$-0.641566\pi$
$827$	−24.7446	−0.860453	−0.430226	+	0.902721i	$0.641566\pi$
$828$	0	0
$829$	12.2337	0.424894	0.212447	−	0.977173i	$-0.431857\pi$
$829$	12.2337	0.424894	0.212447	+	0.977173i	$0.431857\pi$
$830$	0	0
$831$	59.2554	2.05555
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−5.62772	−0.194755
$836$	0	0
$837$	7.06525	0.244211
$838$	0	0
$839$	−11.2554	−0.388581	−0.194290	−	0.980944i	$-0.562240\pi$
$839$	−11.2554	−0.388581	−0.194290	+	0.980944i	$0.562240\pi$
$840$	0	0
$841$	−9.88316	−0.340798
$842$	0	0
$843$	44.1386	1.52021
$844$	0	0
$845$	6.11684	0.210426
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−21.0733	−0.723235
$850$	0	0
$851$	−9.48913	−0.325283
$852$	0	0
$853$	39.4891	1.35208	0.676041	−	0.736864i	$-0.263694\pi$
$853$	39.4891	1.35208	0.676041	+	0.736864i	$0.263694\pi$
$854$	0	0
$855$	−12.4674	−0.426375
$856$	0	0
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	44.4674	1.51721	0.758604	−	0.651552i	$-0.225882\pi$
$859$	44.4674	1.51721	0.758604	+	0.651552i	$0.225882\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−25.4891	−0.867660	−0.433830	−	0.900995i	$-0.642838\pi$
$863$	−25.4891	−0.867660	−0.433830	+	0.900995i	$0.642838\pi$
$864$	0	0
$865$	0.372281	0.0126579
$866$	0	0
$867$	−40.0000	−1.35847
$868$	0	0
$869$	96.3288	3.26773
$870$	0	0
$871$	−17.4891	−0.592596
$872$	0	0
$873$	25.9130	0.877022
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−7.02175	−0.237108	−0.118554	−	0.992948i	$-0.537826\pi$
$877$	−7.02175	−0.237108	−0.118554	+	0.992948i	$0.537826\pi$
$878$	0	0
$879$	−59.5842	−2.00973
$880$	0	0
$881$	25.2554	0.850877	0.425439	−	0.904987i	$-0.360120\pi$
$881$	25.2554	0.850877	0.425439	+	0.904987i	$0.360120\pi$
$882$	0	0
$883$	10.5109	0.353719	0.176860	−	0.984236i	$-0.443406\pi$
$883$	10.5109	0.353719	0.176860	+	0.984236i	$0.443406\pi$
$884$	0	0
$885$	18.9783	0.637947
$886$	0	0
$887$	−13.0217	−0.437228	−0.218614	−	0.975811i	$-0.570153\pi$
$887$	−13.0217	−0.437228	−0.218614	+	0.975811i	$0.570153\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	63.5842	2.13015
$892$	0	0
$893$	33.7663	1.12995
$894$	0	0
$895$	22.9783	0.768078
$896$	0	0
$897$	−49.2119	−1.64314
$898$	0	0
$899$	34.9783	1.16659
$900$	0	0
$901$	4.00000	0.133259
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−16.2337	−0.539626
$906$	0	0
$907$	−11.7228	−0.389250	−0.194625	−	0.980878i	$-0.562349\pi$
$907$	−11.7228	−0.389250	−0.194625	+	0.980878i	$0.562349\pi$
$908$	0	0
$909$	15.7663	0.522936
$910$	0	0
$911$	45.9565	1.52261	0.761303	−	0.648396i	$-0.224560\pi$
$911$	45.9565	1.52261	0.761303	+	0.648396i	$0.224560\pi$
$912$	0	0
$913$	60.4674	2.00118
$914$	0	0
$915$	6.51087	0.215243
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−13.6277	−0.449537	−0.224768	−	0.974412i	$-0.572163\pi$
$919$	−13.6277	−0.449537	−0.224768	+	0.974412i	$0.572163\pi$
$920$	0	0
$921$	73.8179	2.43238
$922$	0	0
$923$	34.9783	1.15132
$924$	0	0
$925$	−2.00000	−0.0657596
$926$	0	0
$927$	−14.7881	−0.485704
$928$	0	0
$929$	7.76631	0.254804	0.127402	−	0.991851i	$-0.459336\pi$
$929$	7.76631	0.254804	0.127402	+	0.991851i	$0.459336\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−30.2337	−0.989807
$934$	0	0
$935$	−2.37228	−0.0775819
$936$	0	0
$937$	−28.0951	−0.917827	−0.458913	−	0.888481i	$-0.651761\pi$
$937$	−28.0951	−0.917827	−0.458913	+	0.888481i	$0.651761\pi$
$938$	0	0
$939$	−6.83966	−0.223204
$940$	0	0
$941$	−32.2337	−1.05079	−0.525394	−	0.850859i	$-0.676082\pi$
$941$	−32.2337	−1.05079	−0.525394	+	0.850859i	$0.676082\pi$
$942$	0	0
$943$	−32.0000	−1.04206
$944$	0	0
$945$	0	0
$946$	0	0
$947$	28.0000	0.909878	0.454939	−	0.890523i	$-0.349661\pi$
$947$	28.0000	0.909878	0.454939	+	0.890523i	$0.349661\pi$
$948$	0	0
$949$	−26.2337	−0.851582
$950$	0	0
$951$	33.2119	1.07697
$952$	0	0
$953$	37.2554	1.20682	0.603411	−	0.797430i	$-0.293808\pi$
$953$	37.2554	1.20682	0.603411	+	0.797430i	$0.293808\pi$
$954$	0	0
$955$	−3.86141	−0.124952
$956$	0	0
$957$	66.0951	2.13655
$958$	0	0
$959$	0	0
$960$	0	0
$961$	33.0000	1.06452
$962$	0	0
$963$	−22.9783	−0.740464
$964$	0	0
$965$	6.74456	0.217115
$966$	0	0
$967$	−1.76631	−0.0568008	−0.0284004	−	0.999597i	$-0.509041\pi$
$967$	−1.76631	−0.0568008	−0.0284004	+	0.999597i	$0.509041\pi$
$968$	0	0
$969$	−4.19019	−0.134608
$970$	0	0
$971$	33.4891	1.07472	0.537359	−	0.843354i	$-0.319422\pi$
$971$	33.4891	1.07472	0.537359	+	0.843354i	$0.319422\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−10.3723	−0.332179
$976$	0	0
$977$	52.9783	1.69492	0.847462	−	0.530856i	$-0.178129\pi$
$977$	52.9783	1.69492	0.847462	+	0.530856i	$0.178129\pi$
$978$	0	0
$979$	93.9565	3.00286
$980$	0	0
$981$	0.978251	0.0312331
$982$	0	0
$983$	−10.3723	−0.330824	−0.165412	−	0.986225i	$-0.552895\pi$
$983$	−10.3723	−0.330824	−0.165412	+	0.986225i	$0.552895\pi$
$984$	0	0
$985$	−8.23369	−0.262347
$986$	0	0
$987$	0	0
$988$	0	0
$989$	41.4891	1.31928
$990$	0	0
$991$	37.9565	1.20573	0.602864	−	0.797844i	$-0.294026\pi$
$991$	37.9565	1.20573	0.602864	+	0.797844i	$0.294026\pi$
$992$	0	0
$993$	−28.4674	−0.903385
$994$	0	0
$995$	−16.0000	−0.507234
$996$	0	0
$997$	6.88316	0.217992	0.108996	−	0.994042i	$-0.465236\pi$
$997$	6.88316	0.217992	0.108996	+	0.994042i	$0.465236\pi$
$998$	0	0
$999$	1.76631	0.0558836

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3920.2.a.bt.1.2		2
4.3	odd	2		1960.2.a.s.1.1		2
7.6	odd	2		560.2.a.h.1.1		2
20.19	odd	2		9800.2.a.bu.1.2		2
21.20	even	2		5040.2.a.by.1.2		2
28.3	even	6		1960.2.q.t.961.1		4
28.11	odd	6		1960.2.q.r.961.2		4
28.19	even	6		1960.2.q.t.361.1		4
28.23	odd	6		1960.2.q.r.361.2		4
28.27	even	2		280.2.a.c.1.2	✓	2
35.13	even	4		2800.2.g.r.449.2		4
35.27	even	4		2800.2.g.r.449.3		4
35.34	odd	2		2800.2.a.bk.1.2		2
56.13	odd	2		2240.2.a.bg.1.2		2
56.27	even	2		2240.2.a.bk.1.1		2
84.83	odd	2		2520.2.a.x.1.1		2
140.27	odd	4		1400.2.g.i.449.2		4
140.83	odd	4		1400.2.g.i.449.3		4
140.139	even	2		1400.2.a.r.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.2.a.c.1.2	✓	2	28.27	even	2
560.2.a.h.1.1		2	7.6	odd	2
1400.2.a.r.1.1		2	140.139	even	2
1400.2.g.i.449.2		4	140.27	odd	4
1400.2.g.i.449.3		4	140.83	odd	4
1960.2.a.s.1.1		2	4.3	odd	2
1960.2.q.r.361.2		4	28.23	odd	6
1960.2.q.r.961.2		4	28.11	odd	6
1960.2.q.t.361.1		4	28.19	even	6
1960.2.q.t.961.1		4	28.3	even	6
2240.2.a.bg.1.2		2	56.13	odd	2
2240.2.a.bk.1.1		2	56.27	even	2
2520.2.a.x.1.1		2	84.83	odd	2
2800.2.a.bk.1.2		2	35.34	odd	2
2800.2.g.r.449.2		4	35.13	even	4
2800.2.g.r.449.3		4	35.27	even	4
3920.2.a.bt.1.2		2	1.1	even	1	trivial
5040.2.a.by.1.2		2	21.20	even	2
9800.2.a.bu.1.2		2	20.19	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3920.a (trivial)

\(f(q)\)	\(=\)	\(q+2.37228 q^{3} +1.00000 q^{5} +2.62772 q^{9} +O(q^{10})\)	\(q+2.37228 q^{3} +1.00000 q^{5} +2.62772 q^{9} -6.37228 q^{11} -4.37228 q^{13} +2.37228 q^{15} +0.372281 q^{17} -4.74456 q^{19} +4.74456 q^{23} +1.00000 q^{25} -0.883156 q^{27} -4.37228 q^{29} -8.00000 q^{31} -15.1168 q^{33} -2.00000 q^{37} -10.3723 q^{39} -6.74456 q^{41} +8.74456 q^{43} +2.62772 q^{45} -7.11684 q^{47} +0.883156 q^{51} +10.7446 q^{53} -6.37228 q^{55} -11.2554 q^{57} +8.00000 q^{59} +2.74456 q^{61} -4.37228 q^{65} +4.00000 q^{67} +11.2554 q^{69} -8.00000 q^{71} +6.00000 q^{73} +2.37228 q^{75} -15.1168 q^{79} -9.97825 q^{81} -9.48913 q^{83} +0.372281 q^{85} -10.3723 q^{87} -14.7446 q^{89} -18.9783 q^{93} -4.74456 q^{95} +9.86141 q^{97} -16.7446 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} + 2 q^{5} + 11 q^{9}+O(q^{10}) \) 2 * q - q^3 + 2 * q^5 + 11 * q^9	\( 2 q - q^{3} + 2 q^{5} + 11 q^{9} - 7 q^{11} - 3 q^{13} - q^{15} - 5 q^{17} + 2 q^{19} - 2 q^{23} + 2 q^{25} - 19 q^{27} - 3 q^{29} - 16 q^{31} - 13 q^{33} - 4 q^{37} - 15 q^{39} - 2 q^{41} + 6 q^{43} + 11 q^{45} + 3 q^{47} + 19 q^{51} + 10 q^{53} - 7 q^{55} - 34 q^{57} + 16 q^{59} - 6 q^{61} - 3 q^{65} + 8 q^{67} + 34 q^{69} - 16 q^{71} + 12 q^{73} - q^{75} - 13 q^{79} + 26 q^{81} + 4 q^{83} - 5 q^{85} - 15 q^{87} - 18 q^{89} + 8 q^{93} + 2 q^{95} - 9 q^{97} - 22 q^{99}+O(q^{100}) \) 2 * q - q^3 + 2 * q^5 + 11 * q^9 - 7 * q^11 - 3 * q^13 - q^15 - 5 * q^17 + 2 * q^19 - 2 * q^23 + 2 * q^25 - 19 * q^27 - 3 * q^29 - 16 * q^31 - 13 * q^33 - 4 * q^37 - 15 * q^39 - 2 * q^41 + 6 * q^43 + 11 * q^45 + 3 * q^47 + 19 * q^51 + 10 * q^53 - 7 * q^55 - 34 * q^57 + 16 * q^59 - 6 * q^61 - 3 * q^65 + 8 * q^67 + 34 * q^69 - 16 * q^71 + 12 * q^73 - q^75 - 13 * q^79 + 26 * q^81 + 4 * q^83 - 5 * q^85 - 15 * q^87 - 18 * q^89 + 8 * q^93 + 2 * q^95 - 9 * q^97 - 22 * q^99

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