LMFDB - Embedded newform 9405.2.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9405,2,Mod(1,9405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9405, 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("9405.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9405.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:	$75.0993031010$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.905177.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 7x^{4} + 9x^{3} + 7x^{2} - 9x - 1 $ x^6 - x^5 - 7x^4 + 9x^3 + 7x^2 - 9x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 3135)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.04416$ of defining polynomial
Character	$\chi$	$=$	9405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.63929 q^{2} +0.687261 q^{4} -1.00000 q^{5} +1.85467 q^{7} +2.15196 q^{8} +O(q^{10})$	\(q-1.63929 q^{2} +0.687261 q^{4} -1.00000 q^{5} +1.85467 q^{7} +2.15196 q^{8} +1.63929 q^{10} +1.00000 q^{11} -5.04034 q^{13} -3.04034 q^{14} -4.90219 q^{16} -1.69834 q^{17} +1.00000 q^{19} -0.687261 q^{20} -1.63929 q^{22} +4.82051 q^{23} +1.00000 q^{25} +8.26256 q^{26} +1.27464 q^{28} -2.11506 q^{29} -4.16741 q^{31} +3.73219 q^{32} +2.78406 q^{34} -1.85467 q^{35} -6.48240 q^{37} -1.63929 q^{38} -2.15196 q^{40} +7.38560 q^{41} -0.542144 q^{43} +0.687261 q^{44} -7.90219 q^{46} -1.52395 q^{47} -3.56019 q^{49} -1.63929 q^{50} -3.46403 q^{52} -3.72042 q^{53} -1.00000 q^{55} +3.99117 q^{56} +3.46718 q^{58} +1.48340 q^{59} -4.02948 q^{61} +6.83158 q^{62} +3.68625 q^{64} +5.04034 q^{65} +6.00618 q^{67} -1.16720 q^{68} +3.04034 q^{70} +7.47665 q^{71} +2.27836 q^{73} +10.6265 q^{74} +0.687261 q^{76} +1.85467 q^{77} -8.33010 q^{79} +4.90219 q^{80} -12.1071 q^{82} -1.75111 q^{83} +1.69834 q^{85} +0.888729 q^{86} +2.15196 q^{88} +2.84163 q^{89} -9.34818 q^{91} +3.31295 q^{92} +2.49820 q^{94} -1.00000 q^{95} -0.585347 q^{97} +5.83617 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{2} + 5 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) $ 6 * q + q^2 + 5 * q^4 - 6 * q^5 + 3 * q^7 + 6 * q^8	$ 6 q + q^{2} + 5 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8} - q^{10} + 6 q^{11} - 4 q^{13} + 8 q^{14} - q^{16} + 4 q^{17} + 6 q^{19} - 5 q^{20} + q^{22} + 6 q^{25} + 9 q^{26} + 5 q^{28} + 21 q^{29} - 16 q^{31} + 8 q^{32} - 7 q^{34} - 3 q^{35} - q^{37} + q^{38} - 6 q^{40} + 31 q^{41} + 6 q^{43} + 5 q^{44} - 19 q^{46} + 4 q^{47} - 5 q^{49} + q^{50} + 17 q^{52} + 9 q^{53} - 6 q^{55} + 11 q^{56} + 19 q^{58} + 24 q^{59} - 11 q^{61} - 3 q^{62} - 30 q^{64} + 4 q^{65} - 11 q^{67} - 8 q^{70} + 23 q^{71} - 6 q^{73} - 17 q^{74} + 5 q^{76} + 3 q^{77} + 4 q^{79} + q^{80} + 20 q^{82} - 17 q^{83} - 4 q^{85} + 9 q^{86} + 6 q^{88} + 37 q^{89} + 3 q^{91} + 17 q^{92} - 2 q^{94} - 6 q^{95} - 26 q^{97} + 2 q^{98}+O(q^{100}) $ 6 * q + q^2 + 5 * q^4 - 6 * q^5 + 3 * q^7 + 6 * q^8 - q^10 + 6 * q^11 - 4 * q^13 + 8 * q^14 - q^16 + 4 * q^17 + 6 * q^19 - 5 * q^20 + q^22 + 6 * q^25 + 9 * q^26 + 5 * q^28 + 21 * q^29 - 16 * q^31 + 8 * q^32 - 7 * q^34 - 3 * q^35 - q^37 + q^38 - 6 * q^40 + 31 * q^41 + 6 * q^43 + 5 * q^44 - 19 * q^46 + 4 * q^47 - 5 * q^49 + q^50 + 17 * q^52 + 9 * q^53 - 6 * q^55 + 11 * q^56 + 19 * q^58 + 24 * q^59 - 11 * q^61 - 3 * q^62 - 30 * q^64 + 4 * q^65 - 11 * q^67 - 8 * q^70 + 23 * q^71 - 6 * q^73 - 17 * q^74 + 5 * q^76 + 3 * q^77 + 4 * q^79 + q^80 + 20 * q^82 - 17 * q^83 - 4 * q^85 + 9 * q^86 + 6 * q^88 + 37 * q^89 + 3 * q^91 + 17 * q^92 - 2 * q^94 - 6 * q^95 - 26 * 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 $

$2$	−1.63929	−1.15915	−0.579575	−	0.814919i	$-0.696782\pi$
$2$	−1.63929	−1.15915	−0.579575	+	0.814919i	$0.696782\pi$
$3$	0	0
$3$	0	0
$4$	0.687261	0.343631
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.85467	0.701000	0.350500	−	0.936563i	$-0.386012\pi$
$7$	1.85467	0.701000	0.350500	+	0.936563i	$0.386012\pi$
$8$	2.15196	0.760831
$9$	0	0
$10$	1.63929	0.518388
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.04034	−1.39794	−0.698969	−	0.715152i	$-0.746358\pi$
$13$	−5.04034	−1.39794	−0.698969	+	0.715152i	$0.746358\pi$
$14$	−3.04034	−0.812565
$15$	0	0
$16$	−4.90219	−1.22555
$17$	−1.69834	−0.411907	−0.205953	−	0.978562i	$-0.566030\pi$
$17$	−1.69834	−0.411907	−0.205953	+	0.978562i	$0.566030\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−0.687261	−0.153676
$21$	0	0
$22$	−1.63929	−0.349497
$23$	4.82051	1.00515	0.502573	−	0.864535i	$-0.332387\pi$
$23$	4.82051	1.00515	0.502573	+	0.864535i	$0.332387\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	8.26256	1.62042
$27$	0	0
$28$	1.27464	0.240885
$29$	−2.11506	−0.392756	−0.196378	−	0.980528i	$-0.562918\pi$
$29$	−2.11506	−0.392756	−0.196378	+	0.980528i	$0.562918\pi$
$30$	0	0
$31$	−4.16741	−0.748489	−0.374245	−	0.927330i	$-0.622098\pi$
$31$	−4.16741	−0.748489	−0.374245	+	0.927330i	$0.622098\pi$
$32$	3.73219	0.659765
$33$	0	0
$34$	2.78406	0.477462
$35$	−1.85467	−0.313497
$36$	0	0
$37$	−6.48240	−1.06570	−0.532850	−	0.846210i	$-0.678879\pi$
$37$	−6.48240	−1.06570	−0.532850	+	0.846210i	$0.678879\pi$
$38$	−1.63929	−0.265927
$39$	0	0
$40$	−2.15196	−0.340254
$41$	7.38560	1.15344	0.576718	−	0.816943i	$-0.304333\pi$
$41$	7.38560	1.15344	0.576718	+	0.816943i	$0.304333\pi$
$42$	0	0
$43$	−0.542144	−0.0826762	−0.0413381	−	0.999145i	$-0.513162\pi$
$43$	−0.542144	−0.0826762	−0.0413381	+	0.999145i	$0.513162\pi$
$44$	0.687261	0.103609
$45$	0	0
$46$	−7.90219	−1.16512
$47$	−1.52395	−0.222291	−0.111146	−	0.993804i	$-0.535452\pi$
$47$	−1.52395	−0.222291	−0.111146	+	0.993804i	$0.535452\pi$
$48$	0	0
$49$	−3.56019	−0.508599
$50$	−1.63929	−0.231830
$51$	0	0
$52$	−3.46403	−0.480375
$53$	−3.72042	−0.511039	−0.255519	−	0.966804i	$-0.582246\pi$
$53$	−3.72042	−0.511039	−0.255519	+	0.966804i	$0.582246\pi$
$54$	0	0
$55$	−1.00000	−0.134840
$56$	3.99117	0.533343
$57$	0	0
$58$	3.46718	0.455263
$59$	1.48340	0.193123	0.0965613	−	0.995327i	$-0.469216\pi$
$59$	1.48340	0.193123	0.0965613	+	0.995327i	$0.469216\pi$
$60$	0	0
$61$	−4.02948	−0.515921	−0.257961	−	0.966155i	$-0.583051\pi$
$61$	−4.02948	−0.515921	−0.257961	+	0.966155i	$0.583051\pi$
$62$	6.83158	0.867612
$63$	0	0
$64$	3.68625	0.460782
$65$	5.04034	0.625177
$66$	0	0
$67$	6.00618	0.733771	0.366886	−	0.930266i	$-0.380424\pi$
$67$	6.00618	0.733771	0.366886	+	0.930266i	$0.380424\pi$
$68$	−1.16720	−0.141544
$69$	0	0
$70$	3.04034	0.363390
$71$	7.47665	0.887315	0.443657	−	0.896196i	$-0.353681\pi$
$71$	7.47665	0.887315	0.443657	+	0.896196i	$0.353681\pi$
$72$	0	0
$73$	2.27836	0.266662	0.133331	−	0.991072i	$-0.457433\pi$
$73$	2.27836	0.266662	0.133331	+	0.991072i	$0.457433\pi$
$74$	10.6265	1.23531
$75$	0	0
$76$	0.687261	0.0788343
$77$	1.85467	0.211360
$78$	0	0
$79$	−8.33010	−0.937209	−0.468605	−	0.883408i	$-0.655243\pi$
$79$	−8.33010	−0.937209	−0.468605	+	0.883408i	$0.655243\pi$
$80$	4.90219	0.548082
$81$	0	0
$82$	−12.1071	−1.33701
$83$	−1.75111	−0.192210	−0.0961048	−	0.995371i	$-0.530638\pi$
$83$	−1.75111	−0.192210	−0.0961048	+	0.995371i	$0.530638\pi$
$84$	0	0
$85$	1.69834	0.184210
$86$	0.888729	0.0958342
$87$	0	0
$88$	2.15196	0.229399
$89$	2.84163	0.301212	0.150606	−	0.988594i	$-0.451878\pi$
$89$	2.84163	0.301212	0.150606	+	0.988594i	$0.451878\pi$
$90$	0	0
$91$	−9.34818	−0.979955
$92$	3.31295	0.345399
$93$	0	0
$94$	2.49820	0.257669
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−0.585347	−0.0594330	−0.0297165	−	0.999558i	$-0.509460\pi$
$97$	−0.585347	−0.0594330	−0.0297165	+	0.999558i	$0.509460\pi$
$98$	5.83617	0.589543
$99$	0	0
$100$	0.687261	0.0687261
$101$	−2.89501	−0.288064	−0.144032	−	0.989573i	$-0.546007\pi$
$101$	−2.89501	−0.288064	−0.144032	+	0.989573i	$0.546007\pi$
$102$	0	0
$103$	0.963762	0.0949623	0.0474812	−	0.998872i	$-0.484881\pi$
$103$	0.963762	0.0949623	0.0474812	+	0.998872i	$0.484881\pi$
$104$	−10.8466	−1.06360
$105$	0	0
$106$	6.09883	0.592371
$107$	2.94040	0.284259	0.142129	−	0.989848i	$-0.454605\pi$
$107$	2.94040	0.284259	0.142129	+	0.989848i	$0.454605\pi$
$108$	0	0
$109$	−2.71620	−0.260165	−0.130083	−	0.991503i	$-0.541524\pi$
$109$	−2.71620	−0.260165	−0.130083	+	0.991503i	$0.541524\pi$
$110$	1.63929	0.156300
$111$	0	0
$112$	−9.09196	−0.859110
$113$	9.21403	0.866783	0.433392	−	0.901206i	$-0.357317\pi$
$113$	9.21403	0.866783	0.433392	+	0.901206i	$0.357317\pi$
$114$	0	0
$115$	−4.82051	−0.449515
$116$	−1.45360	−0.134963
$117$	0	0
$118$	−2.43172	−0.223858
$119$	−3.14986	−0.288747
$120$	0	0
$121$	1.00000	0.0909091
$122$	6.60547	0.598031
$123$	0	0
$124$	−2.86410	−0.257204
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−10.8555	−0.963272	−0.481636	−	0.876371i	$-0.659957\pi$
$127$	−10.8555	−0.963272	−0.481636	+	0.876371i	$0.659957\pi$
$128$	−13.5072	−1.19388
$129$	0	0
$130$	−8.26256	−0.724675
$131$	16.4910	1.44083	0.720413	−	0.693546i	$-0.243952\pi$
$131$	16.4910	1.44083	0.720413	+	0.693546i	$0.243952\pi$
$132$	0	0
$133$	1.85467	0.160820
$134$	−9.84584	−0.850551
$135$	0	0
$136$	−3.65474	−0.313392
$137$	−1.38963	−0.118724	−0.0593622	−	0.998237i	$-0.518907\pi$
$137$	−1.38963	−0.118724	−0.0593622	+	0.998237i	$0.518907\pi$
$138$	0	0
$139$	2.09823	0.177969	0.0889846	−	0.996033i	$-0.471638\pi$
$139$	2.09823	0.177969	0.0889846	+	0.996033i	$0.471638\pi$
$140$	−1.27464	−0.107727
$141$	0	0
$142$	−12.2564	−1.02853
$143$	−5.04034	−0.421494
$144$	0	0
$145$	2.11506	0.175646
$146$	−3.73489	−0.309102
$147$	0	0
$148$	−4.45510	−0.366207
$149$	22.8670	1.87334	0.936668	−	0.350218i	$-0.113892\pi$
$149$	22.8670	1.87334	0.936668	+	0.350218i	$0.113892\pi$
$150$	0	0
$151$	12.1356	0.987584	0.493792	−	0.869580i	$-0.335610\pi$
$151$	12.1356	0.987584	0.493792	+	0.869580i	$0.335610\pi$
$152$	2.15196	0.174547
$153$	0	0
$154$	−3.04034	−0.244998
$155$	4.16741	0.334735
$156$	0	0
$157$	19.2179	1.53376	0.766879	−	0.641792i	$-0.221809\pi$
$157$	19.2179	1.53376	0.766879	+	0.641792i	$0.221809\pi$
$158$	13.6554	1.08637
$159$	0	0
$160$	−3.73219	−0.295056
$161$	8.94046	0.704607
$162$	0	0
$163$	−2.83648	−0.222170	−0.111085	−	0.993811i	$-0.535433\pi$
$163$	−2.83648	−0.222170	−0.111085	+	0.993811i	$0.535433\pi$
$164$	5.07584	0.396356
$165$	0	0
$166$	2.87058	0.222800
$167$	10.2869	0.796024	0.398012	−	0.917380i	$-0.369700\pi$
$167$	10.2869	0.796024	0.398012	+	0.917380i	$0.369700\pi$
$168$	0	0
$169$	12.4050	0.954233
$170$	−2.78406	−0.213528
$171$	0	0
$172$	−0.372595	−0.0284101
$173$	10.1153	0.769050	0.384525	−	0.923115i	$-0.374365\pi$
$173$	10.1153	0.769050	0.384525	+	0.923115i	$0.374365\pi$
$174$	0	0
$175$	1.85467	0.140200
$176$	−4.90219	−0.369517
$177$	0	0
$178$	−4.65824	−0.349150
$179$	−5.83622	−0.436219	−0.218110	−	0.975924i	$-0.569989\pi$
$179$	−5.83622	−0.436219	−0.218110	+	0.975924i	$0.569989\pi$
$180$	0	0
$181$	−0.927249	−0.0689219	−0.0344609	−	0.999406i	$-0.510971\pi$
$181$	−0.927249	−0.0689219	−0.0344609	+	0.999406i	$0.510971\pi$
$182$	15.3243	1.13592
$183$	0	0
$184$	10.3735	0.764746
$185$	6.48240	0.476595
$186$	0	0
$187$	−1.69834	−0.124195
$188$	−1.04735	−0.0763861
$189$	0	0
$190$	1.63929	0.118926
$191$	−15.5055	−1.12194	−0.560971	−	0.827836i	$-0.689572\pi$
$191$	−15.5055	−1.12194	−0.560971	+	0.827836i	$0.689572\pi$
$192$	0	0
$193$	8.58558	0.618004	0.309002	−	0.951061i	$-0.400005\pi$
$193$	8.58558	0.618004	0.309002	+	0.951061i	$0.400005\pi$
$194$	0.959552	0.0688918
$195$	0	0
$196$	−2.44678	−0.174770
$197$	3.22355	0.229669	0.114834	−	0.993385i	$-0.463366\pi$
$197$	3.22355	0.229669	0.114834	+	0.993385i	$0.463366\pi$
$198$	0	0
$199$	3.13980	0.222574	0.111287	−	0.993788i	$-0.464503\pi$
$199$	3.13980	0.222574	0.111287	+	0.993788i	$0.464503\pi$
$200$	2.15196	0.152166
$201$	0	0
$202$	4.74576	0.333910
$203$	−3.92273	−0.275322
$204$	0	0
$205$	−7.38560	−0.515832
$206$	−1.57988	−0.110076
$207$	0	0
$208$	24.7087	1.71324
$209$	1.00000	0.0691714
$210$	0	0
$211$	−8.72743	−0.600821	−0.300410	−	0.953810i	$-0.597124\pi$
$211$	−8.72743	−0.600821	−0.300410	+	0.953810i	$0.597124\pi$
$212$	−2.55690	−0.175609
$213$	0	0
$214$	−4.82015	−0.329499
$215$	0.542144	0.0369739
$216$	0	0
$217$	−7.72918	−0.524691
$218$	4.45264	0.301571
$219$	0	0
$220$	−0.687261	−0.0463352
$221$	8.56019	0.575821
$222$	0	0
$223$	−23.8601	−1.59779	−0.798895	−	0.601471i	$-0.794582\pi$
$223$	−23.8601	−1.59779	−0.798895	+	0.601471i	$0.794582\pi$
$224$	6.92200	0.462495
$225$	0	0
$226$	−15.1044	−1.00473
$227$	−26.3877	−1.75141	−0.875706	−	0.482844i	$-0.839604\pi$
$227$	−26.3877	−1.75141	−0.875706	+	0.482844i	$0.839604\pi$
$228$	0	0
$229$	−5.21073	−0.344335	−0.172167	−	0.985068i	$-0.555077\pi$
$229$	−5.21073	−0.344335	−0.172167	+	0.985068i	$0.555077\pi$
$230$	7.90219	0.521055
$231$	0	0
$232$	−4.55150	−0.298821
$233$	18.3469	1.20194	0.600971	−	0.799271i	$-0.294781\pi$
$233$	18.3469	1.20194	0.600971	+	0.799271i	$0.294781\pi$
$234$	0	0
$235$	1.52395	0.0994117
$236$	1.01949	0.0663629
$237$	0	0
$238$	5.16352	0.334701
$239$	−22.6042	−1.46214	−0.731072	−	0.682300i	$-0.760980\pi$
$239$	−22.6042	−1.46214	−0.731072	+	0.682300i	$0.760980\pi$
$240$	0	0
$241$	−20.0385	−1.29080	−0.645398	−	0.763847i	$-0.723308\pi$
$241$	−20.0385	−1.29080	−0.645398	+	0.763847i	$0.723308\pi$
$242$	−1.63929	−0.105377
$243$	0	0
$244$	−2.76930	−0.177286
$245$	3.56019	0.227452
$246$	0	0
$247$	−5.04034	−0.320709
$248$	−8.96808	−0.569474
$249$	0	0
$250$	1.63929	0.103678
$251$	27.7333	1.75051	0.875257	−	0.483659i	$-0.160692\pi$
$251$	27.7333	1.75051	0.875257	+	0.483659i	$0.160692\pi$
$252$	0	0
$253$	4.82051	0.303063
$254$	17.7953	1.11658
$255$	0	0
$256$	14.7697	0.923106
$257$	−14.2166	−0.886805	−0.443402	−	0.896323i	$-0.646229\pi$
$257$	−14.2166	−0.886805	−0.443402	+	0.896323i	$0.646229\pi$
$258$	0	0
$259$	−12.0227	−0.747055
$260$	3.46403	0.214830
$261$	0	0
$262$	−27.0335	−1.67013
$263$	−23.0874	−1.42363	−0.711815	−	0.702367i	$-0.752126\pi$
$263$	−23.0874	−1.42363	−0.711815	+	0.702367i	$0.752126\pi$
$264$	0	0
$265$	3.72042	0.228544
$266$	−3.04034	−0.186415
$267$	0	0
$268$	4.12781	0.252146
$269$	11.7459	0.716162	0.358081	−	0.933690i	$-0.383431\pi$
$269$	11.7459	0.716162	0.358081	+	0.933690i	$0.383431\pi$
$270$	0	0
$271$	19.0195	1.15535	0.577677	−	0.816265i	$-0.303959\pi$
$271$	19.0195	1.15535	0.577677	+	0.816265i	$0.303959\pi$
$272$	8.32557	0.504812
$273$	0	0
$274$	2.27801	0.137620
$275$	1.00000	0.0603023
$276$	0	0
$277$	−6.58956	−0.395928	−0.197964	−	0.980209i	$-0.563433\pi$
$277$	−6.58956	−0.395928	−0.197964	+	0.980209i	$0.563433\pi$
$278$	−3.43959	−0.206293
$279$	0	0
$280$	−3.99117	−0.238518
$281$	−13.8947	−0.828891	−0.414446	−	0.910074i	$-0.636025\pi$
$281$	−13.8947	−0.828891	−0.414446	+	0.910074i	$0.636025\pi$
$282$	0	0
$283$	22.7610	1.35300	0.676501	−	0.736442i	$-0.263496\pi$
$283$	22.7610	1.35300	0.676501	+	0.736442i	$0.263496\pi$
$284$	5.13841	0.304909
$285$	0	0
$286$	8.26256	0.488576
$287$	13.6979	0.808559
$288$	0	0
$289$	−14.1157	−0.830333
$290$	−3.46718	−0.203600
$291$	0	0
$292$	1.56583	0.0916333
$293$	−11.0103	−0.643228	−0.321614	−	0.946871i	$-0.604225\pi$
$293$	−11.0103	−0.643228	−0.321614	+	0.946871i	$0.604225\pi$
$294$	0	0
$295$	−1.48340	−0.0863670
$296$	−13.9498	−0.810817
$297$	0	0
$298$	−37.4855	−2.17148
$299$	−24.2970	−1.40513
$300$	0	0
$301$	−1.00550	−0.0579560
$302$	−19.8938	−1.14476
$303$	0	0
$304$	−4.90219	−0.281160
$305$	4.02948	0.230727
$306$	0	0
$307$	−6.55796	−0.374283	−0.187141	−	0.982333i	$-0.559922\pi$
$307$	−6.55796	−0.374283	−0.187141	+	0.982333i	$0.559922\pi$
$308$	1.27464	0.0726296
$309$	0	0
$310$	−6.83158	−0.388008
$311$	−10.7525	−0.609719	−0.304859	−	0.952397i	$-0.598609\pi$
$311$	−10.7525	−0.609719	−0.304859	+	0.952397i	$0.598609\pi$
$312$	0	0
$313$	−4.76748	−0.269474	−0.134737	−	0.990881i	$-0.543019\pi$
$313$	−4.76748	−0.269474	−0.134737	+	0.990881i	$0.543019\pi$
$314$	−31.5037	−1.77786
$315$	0	0
$316$	−5.72495	−0.322054
$317$	−2.10430	−0.118190	−0.0590948	−	0.998252i	$-0.518821\pi$
$317$	−2.10430	−0.118190	−0.0590948	+	0.998252i	$0.518821\pi$
$318$	0	0
$319$	−2.11506	−0.118420
$320$	−3.68625	−0.206068
$321$	0	0
$322$	−14.6560	−0.816746
$323$	−1.69834	−0.0944979
$324$	0	0
$325$	−5.04034	−0.279588
$326$	4.64981	0.257529
$327$	0	0
$328$	15.8935	0.877570
$329$	−2.82643	−0.155826
$330$	0	0
$331$	11.2501	0.618363	0.309181	−	0.951003i	$-0.399945\pi$
$331$	11.2501	0.618363	0.309181	+	0.951003i	$0.399945\pi$
$332$	−1.20347	−0.0660491
$333$	0	0
$334$	−16.8632	−0.922711
$335$	−6.00618	−0.328152
$336$	0	0
$337$	7.14424	0.389172	0.194586	−	0.980885i	$-0.437664\pi$
$337$	7.14424	0.389172	0.194586	+	0.980885i	$0.437664\pi$
$338$	−20.3354	−1.10610
$339$	0	0
$340$	1.16720	0.0633003
$341$	−4.16741	−0.225678
$342$	0	0
$343$	−19.5857	−1.05753
$344$	−1.16667	−0.0629026
$345$	0	0
$346$	−16.5818	−0.891445
$347$	17.7076	0.950596	0.475298	−	0.879825i	$-0.342340\pi$
$347$	17.7076	0.950596	0.475298	+	0.879825i	$0.342340\pi$
$348$	0	0
$349$	33.5327	1.79496	0.897482	−	0.441052i	$-0.145395\pi$
$349$	33.5327	1.79496	0.897482	+	0.441052i	$0.145395\pi$
$350$	−3.04034	−0.162513
$351$	0	0
$352$	3.73219	0.198927
$353$	1.97575	0.105158	0.0525792	−	0.998617i	$-0.483256\pi$
$353$	1.97575	0.105158	0.0525792	+	0.998617i	$0.483256\pi$
$354$	0	0
$355$	−7.47665	−0.396819
$356$	1.95294	0.103506
$357$	0	0
$358$	9.56724	0.505644
$359$	−15.5778	−0.822163	−0.411081	−	0.911599i	$-0.634849\pi$
$359$	−15.5778	−0.822163	−0.411081	+	0.911599i	$0.634849\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	1.52003	0.0798909
$363$	0	0
$364$	−6.42464	−0.336743
$365$	−2.27836	−0.119255
$366$	0	0
$367$	9.11481	0.475789	0.237895	−	0.971291i	$-0.423543\pi$
$367$	9.11481	0.475789	0.237895	+	0.971291i	$0.423543\pi$
$368$	−23.6311	−1.23185
$369$	0	0
$370$	−10.6265	−0.552446
$371$	−6.90016	−0.358238
$372$	0	0
$373$	2.71091	0.140366	0.0701829	−	0.997534i	$-0.477642\pi$
$373$	2.71091	0.140366	0.0701829	+	0.997534i	$0.477642\pi$
$374$	2.78406	0.143960
$375$	0	0
$376$	−3.27948	−0.169126
$377$	10.6606	0.549049
$378$	0	0
$379$	27.9866	1.43758	0.718788	−	0.695230i	$-0.244697\pi$
$379$	27.9866	1.43758	0.718788	+	0.695230i	$0.244697\pi$
$380$	−0.687261	−0.0352558
$381$	0	0
$382$	25.4180	1.30050
$383$	−13.9290	−0.711741	−0.355871	−	0.934535i	$-0.615816\pi$
$383$	−13.9290	−0.711741	−0.355871	+	0.934535i	$0.615816\pi$
$384$	0	0
$385$	−1.85467	−0.0945229
$386$	−14.0742	−0.716359
$387$	0	0
$388$	−0.402287	−0.0204230
$389$	−0.478930	−0.0242827	−0.0121414	−	0.999926i	$-0.503865\pi$
$389$	−0.478930	−0.0242827	−0.0121414	+	0.999926i	$0.503865\pi$
$390$	0	0
$391$	−8.18684	−0.414026
$392$	−7.66137	−0.386958
$393$	0	0
$394$	−5.28433	−0.266220
$395$	8.33010	0.419133
$396$	0	0
$397$	32.6045	1.63637	0.818187	−	0.574952i	$-0.194979\pi$
$397$	32.6045	1.63637	0.818187	+	0.574952i	$0.194979\pi$
$398$	−5.14703	−0.257997
$399$	0	0
$400$	−4.90219	−0.245110
$401$	16.2164	0.809810	0.404905	−	0.914359i	$-0.367305\pi$
$401$	16.2164	0.809810	0.404905	+	0.914359i	$0.367305\pi$
$402$	0	0
$403$	21.0052	1.04634
$404$	−1.98963	−0.0989878
$405$	0	0
$406$	6.43049	0.319140
$407$	−6.48240	−0.321320
$408$	0	0
$409$	−16.3636	−0.809129	−0.404565	−	0.914509i	$-0.632577\pi$
$409$	−16.3636	−0.809129	−0.404565	+	0.914509i	$0.632577\pi$
$410$	12.1071	0.597928
$411$	0	0
$412$	0.662357	0.0326320
$413$	2.75123	0.135379
$414$	0	0
$415$	1.75111	0.0859587
$416$	−18.8115	−0.922311
$417$	0	0
$418$	−1.63929	−0.0801801
$419$	24.9036	1.21662	0.608311	−	0.793698i	$-0.291847\pi$
$419$	24.9036	1.21662	0.608311	+	0.793698i	$0.291847\pi$
$420$	0	0
$421$	27.2339	1.32730	0.663649	−	0.748044i	$-0.269007\pi$
$421$	27.2339	1.32730	0.663649	+	0.748044i	$0.269007\pi$
$422$	14.3068	0.696442
$423$	0	0
$424$	−8.00618	−0.388814
$425$	−1.69834	−0.0823814
$426$	0	0
$427$	−7.47336	−0.361661
$428$	2.02082	0.0976801
$429$	0	0
$430$	−0.888729	−0.0428583
$431$	−14.1597	−0.682048	−0.341024	−	0.940055i	$-0.610774\pi$
$431$	−14.1597	−0.682048	−0.341024	+	0.940055i	$0.610774\pi$
$432$	0	0
$433$	5.25993	0.252776	0.126388	−	0.991981i	$-0.459662\pi$
$433$	5.25993	0.252776	0.126388	+	0.991981i	$0.459662\pi$
$434$	12.6703	0.608196
$435$	0	0
$436$	−1.86674	−0.0894007
$437$	4.82051	0.230596
$438$	0	0
$439$	16.4662	0.785888	0.392944	−	0.919562i	$-0.371457\pi$
$439$	16.4662	0.785888	0.392944	+	0.919562i	$0.371457\pi$
$440$	−2.15196	−0.102590
$441$	0	0
$442$	−14.0326	−0.667463
$443$	0.846120	0.0402004	0.0201002	−	0.999798i	$-0.493601\pi$
$443$	0.846120	0.0402004	0.0201002	+	0.999798i	$0.493601\pi$
$444$	0	0
$445$	−2.84163	−0.134706
$446$	39.1135	1.85208
$447$	0	0
$448$	6.83679	0.323008
$449$	14.8884	0.702628	0.351314	−	0.936258i	$-0.385735\pi$
$449$	14.8884	0.702628	0.351314	+	0.936258i	$0.385735\pi$
$450$	0	0
$451$	7.38560	0.347774
$452$	6.33245	0.297853
$453$	0	0
$454$	43.2570	2.03015
$455$	9.34818	0.438249
$456$	0	0
$457$	−22.2238	−1.03959	−0.519793	−	0.854292i	$-0.673991\pi$
$457$	−22.2238	−1.03959	−0.519793	+	0.854292i	$0.673991\pi$
$458$	8.54188	0.399136
$459$	0	0
$460$	−3.31295	−0.154467
$461$	1.69186	0.0787978	0.0393989	−	0.999224i	$-0.487456\pi$
$461$	1.69186	0.0787978	0.0393989	+	0.999224i	$0.487456\pi$
$462$	0	0
$463$	10.9509	0.508932	0.254466	−	0.967082i	$-0.418100\pi$
$463$	10.9509	0.508932	0.254466	+	0.967082i	$0.418100\pi$
$464$	10.3684	0.481341
$465$	0	0
$466$	−30.0758	−1.39323
$467$	−38.3629	−1.77522	−0.887611	−	0.460593i	$-0.847637\pi$
$467$	−38.3629	−1.77522	−0.887611	+	0.460593i	$0.847637\pi$
$468$	0	0
$469$	11.1395	0.514374
$470$	−2.49820	−0.115233
$471$	0	0
$472$	3.19222	0.146934
$473$	−0.542144	−0.0249278
$474$	0	0
$475$	1.00000	0.0458831
$476$	−2.16477	−0.0992223
$477$	0	0
$478$	37.0548	1.69485
$479$	34.2559	1.56519	0.782596	−	0.622530i	$-0.213895\pi$
$479$	34.2559	1.56519	0.782596	+	0.622530i	$0.213895\pi$
$480$	0	0
$481$	32.6735	1.48978
$482$	32.8489	1.49623
$483$	0	0
$484$	0.687261	0.0312392
$485$	0.585347	0.0265793
$486$	0	0
$487$	−19.5996	−0.888141	−0.444070	−	0.895992i	$-0.646466\pi$
$487$	−19.5996	−0.888141	−0.444070	+	0.895992i	$0.646466\pi$
$488$	−8.67125	−0.392529
$489$	0	0
$490$	−5.83617	−0.263651
$491$	29.4373	1.32849	0.664244	−	0.747515i	$-0.268753\pi$
$491$	29.4373	1.32849	0.664244	+	0.747515i	$0.268753\pi$
$492$	0	0
$493$	3.59207	0.161779
$494$	8.26256	0.371750
$495$	0	0
$496$	20.4295	0.917310
$497$	13.8667	0.622008
$498$	0	0
$499$	−8.59227	−0.384643	−0.192321	−	0.981332i	$-0.561602\pi$
$499$	−8.59227	−0.384643	−0.192321	+	0.981332i	$0.561602\pi$
$500$	−0.687261	−0.0307353
$501$	0	0
$502$	−45.4629	−2.02911
$503$	−36.6391	−1.63366	−0.816829	−	0.576880i	$-0.804270\pi$
$503$	−36.6391	−1.63366	−0.816829	+	0.576880i	$0.804270\pi$
$504$	0	0
$505$	2.89501	0.128826
$506$	−7.90219	−0.351295
$507$	0	0
$508$	−7.46058	−0.331010
$509$	−12.3644	−0.548044	−0.274022	−	0.961723i	$-0.588354\pi$
$509$	−12.3644	−0.548044	−0.274022	+	0.961723i	$0.588354\pi$
$510$	0	0
$511$	4.22562	0.186930
$512$	2.80267	0.123862
$513$	0	0
$514$	23.3050	1.02794
$515$	−0.963762	−0.0424684
$516$	0	0
$517$	−1.52395	−0.0670234
$518$	19.7087	0.865950
$519$	0	0
$520$	10.8466	0.475654
$521$	27.0650	1.18574	0.592870	−	0.805298i	$-0.297995\pi$
$521$	27.0650	1.18574	0.592870	+	0.805298i	$0.297995\pi$
$522$	0	0
$523$	27.6678	1.20983	0.604915	−	0.796290i	$-0.293207\pi$
$523$	27.6678	1.20983	0.604915	+	0.796290i	$0.293207\pi$
$524$	11.3336	0.495112
$525$	0	0
$526$	37.8469	1.65020
$527$	7.07766	0.308308
$528$	0	0
$529$	0.237292	0.0103171
$530$	−6.09883	−0.264916
$531$	0	0
$532$	1.27464	0.0552629
$533$	−37.2259	−1.61243
$534$	0	0
$535$	−2.94040	−0.127124
$536$	12.9250	0.558276
$537$	0	0
$538$	−19.2550	−0.830140
$539$	−3.56019	−0.153348
$540$	0	0
$541$	−1.13268	−0.0486978	−0.0243489	−	0.999704i	$-0.507751\pi$
$541$	−1.13268	−0.0486978	−0.0243489	+	0.999704i	$0.507751\pi$
$542$	−31.1785	−1.33923
$543$	0	0
$544$	−6.33852	−0.271762
$545$	2.71620	0.116349
$546$	0	0
$547$	−3.17754	−0.135862	−0.0679310	−	0.997690i	$-0.521640\pi$
$547$	−3.17754	−0.135862	−0.0679310	+	0.997690i	$0.521640\pi$
$548$	−0.955042	−0.0407974
$549$	0	0
$550$	−1.63929	−0.0698994
$551$	−2.11506	−0.0901044
$552$	0	0
$553$	−15.4496	−0.656984
$554$	10.8022	0.458941
$555$	0	0
$556$	1.44203	0.0611557
$557$	−31.5754	−1.33789	−0.668945	−	0.743312i	$-0.733254\pi$
$557$	−31.5754	−1.33789	−0.668945	+	0.743312i	$0.733254\pi$
$558$	0	0
$559$	2.73259	0.115576
$560$	9.09196	0.384206
$561$	0	0
$562$	22.7775	0.960810
$563$	21.1420	0.891031	0.445515	−	0.895274i	$-0.353020\pi$
$563$	21.1420	0.891031	0.445515	+	0.895274i	$0.353020\pi$
$564$	0	0
$565$	−9.21403	−0.387637
$566$	−37.3118	−1.56833
$567$	0	0
$568$	16.0894	0.675097
$569$	−29.9993	−1.25763	−0.628817	−	0.777553i	$-0.716461\pi$
$569$	−29.9993	−1.25763	−0.628817	+	0.777553i	$0.716461\pi$
$570$	0	0
$571$	33.7683	1.41316	0.706580	−	0.707633i	$-0.250237\pi$
$571$	33.7683	1.41316	0.706580	+	0.707633i	$0.250237\pi$
$572$	−3.46403	−0.144838
$573$	0	0
$574$	−22.4547	−0.937242
$575$	4.82051	0.201029
$576$	0	0
$577$	7.78804	0.324221	0.162110	−	0.986773i	$-0.448170\pi$
$577$	7.78804	0.324221	0.162110	+	0.986773i	$0.448170\pi$
$578$	23.1396	0.962481
$579$	0	0
$580$	1.45360	0.0603573
$581$	−3.24774	−0.134739
$582$	0	0
$583$	−3.72042	−0.154084
$584$	4.90294	0.202885
$585$	0	0
$586$	18.0490	0.745598
$587$	−4.47878	−0.184859	−0.0924295	−	0.995719i	$-0.529463\pi$
$587$	−4.47878	−0.184859	−0.0924295	+	0.995719i	$0.529463\pi$
$588$	0	0
$589$	−4.16741	−0.171715
$590$	2.43172	0.100112
$591$	0	0
$592$	31.7780	1.30607
$593$	31.2806	1.28454	0.642271	−	0.766478i	$-0.277992\pi$
$593$	31.2806	1.28454	0.642271	+	0.766478i	$0.277992\pi$
$594$	0	0
$595$	3.14986	0.129132
$596$	15.7156	0.643736
$597$	0	0
$598$	39.8297	1.62876
$599$	18.4847	0.755265	0.377633	−	0.925955i	$-0.376738\pi$
$599$	18.4847	0.755265	0.377633	+	0.925955i	$0.376738\pi$
$600$	0	0
$601$	35.7960	1.46015	0.730075	−	0.683367i	$-0.239485\pi$
$601$	35.7960	1.46015	0.730075	+	0.683367i	$0.239485\pi$
$602$	1.64830	0.0671798
$603$	0	0
$604$	8.34035	0.339364
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−5.90733	−0.239771	−0.119886	−	0.992788i	$-0.538253\pi$
$607$	−5.90733	−0.239771	−0.119886	+	0.992788i	$0.538253\pi$
$608$	3.73219	0.151360
$609$	0	0
$610$	−6.60547	−0.267448
$611$	7.68124	0.310750
$612$	0	0
$613$	22.0504	0.890606	0.445303	−	0.895380i	$-0.353096\pi$
$613$	22.0504	0.890606	0.445303	+	0.895380i	$0.353096\pi$
$614$	10.7504	0.433850
$615$	0	0
$616$	3.99117	0.160809
$617$	3.46953	0.139678	0.0698391	−	0.997558i	$-0.477751\pi$
$617$	3.46953	0.139678	0.0698391	+	0.997558i	$0.477751\pi$
$618$	0	0
$619$	18.2464	0.733384	0.366692	−	0.930342i	$-0.380490\pi$
$619$	18.2464	0.733384	0.366692	+	0.930342i	$0.380490\pi$
$620$	2.86410	0.115025
$621$	0	0
$622$	17.6264	0.706756
$623$	5.27029	0.211150
$624$	0	0
$625$	1.00000	0.0400000
$626$	7.81527	0.312361
$627$	0	0
$628$	13.2077	0.527046
$629$	11.0093	0.438969
$630$	0	0
$631$	−27.5412	−1.09640	−0.548199	−	0.836348i	$-0.684686\pi$
$631$	−27.5412	−1.09640	−0.548199	+	0.836348i	$0.684686\pi$
$632$	−17.9260	−0.713058
$633$	0	0
$634$	3.44956	0.136999
$635$	10.8555	0.430788
$636$	0	0
$637$	17.9446	0.710990
$638$	3.46718	0.137267
$639$	0	0
$640$	13.5072	0.533920
$641$	49.1116	1.93979	0.969895	−	0.243523i	$-0.0783031\pi$
$641$	49.1116	1.93979	0.969895	+	0.243523i	$0.0783031\pi$
$642$	0	0
$643$	33.1494	1.30728	0.653642	−	0.756804i	$-0.273240\pi$
$643$	33.1494	1.30728	0.653642	+	0.756804i	$0.273240\pi$
$644$	6.14443	0.242125
$645$	0	0
$646$	2.78406	0.109537
$647$	−28.8413	−1.13387	−0.566933	−	0.823764i	$-0.691870\pi$
$647$	−28.8413	−1.13387	−0.566933	+	0.823764i	$0.691870\pi$
$648$	0	0
$649$	1.48340	0.0582287
$650$	8.26256	0.324084
$651$	0	0
$652$	−1.94940	−0.0763446
$653$	−0.888179	−0.0347571	−0.0173786	−	0.999849i	$-0.505532\pi$
$653$	−0.888179	−0.0347571	−0.0173786	+	0.999849i	$0.505532\pi$
$654$	0	0
$655$	−16.4910	−0.644357
$656$	−36.2056	−1.41359
$657$	0	0
$658$	4.63334	0.180626
$659$	35.0627	1.36585	0.682924	−	0.730489i	$-0.260708\pi$
$659$	35.0627	1.36585	0.682924	+	0.730489i	$0.260708\pi$
$660$	0	0
$661$	−1.11734	−0.0434594	−0.0217297	−	0.999764i	$-0.506917\pi$
$661$	−1.11734	−0.0434594	−0.0217297	+	0.999764i	$0.506917\pi$
$662$	−18.4422	−0.716776
$663$	0	0
$664$	−3.76831	−0.146239
$665$	−1.85467	−0.0719211
$666$	0	0
$667$	−10.1956	−0.394777
$668$	7.06978	0.273538
$669$	0	0
$670$	9.84584	0.380378
$671$	−4.02948	−0.155556
$672$	0	0
$673$	14.2325	0.548624	0.274312	−	0.961641i	$-0.411550\pi$
$673$	14.2325	0.548624	0.274312	+	0.961641i	$0.411550\pi$
$674$	−11.7115	−0.451109
$675$	0	0
$676$	8.52550	0.327904
$677$	39.6699	1.52464	0.762319	−	0.647202i	$-0.224061\pi$
$677$	39.6699	1.52464	0.762319	+	0.647202i	$0.224061\pi$
$678$	0	0
$679$	−1.08563	−0.0416626
$680$	3.65474	0.140153
$681$	0	0
$682$	6.83158	0.261595
$683$	17.5612	0.671961	0.335981	−	0.941869i	$-0.390932\pi$
$683$	17.5612	0.671961	0.335981	+	0.941869i	$0.390932\pi$
$684$	0	0
$685$	1.38963	0.0530952
$686$	32.1066	1.22583
$687$	0	0
$688$	2.65770	0.101324
$689$	18.7522	0.714401
$690$	0	0
$691$	26.6517	1.01388	0.506940	−	0.861981i	$-0.330777\pi$
$691$	26.6517	1.01388	0.506940	+	0.861981i	$0.330777\pi$
$692$	6.95184	0.264269
$693$	0	0
$694$	−29.0279	−1.10188
$695$	−2.09823	−0.0795902
$696$	0	0
$697$	−12.5432	−0.475109
$698$	−54.9697	−2.08063
$699$	0	0
$700$	1.27464	0.0481770
$701$	6.40959	0.242087	0.121043	−	0.992647i	$-0.461376\pi$
$701$	6.40959	0.242087	0.121043	+	0.992647i	$0.461376\pi$
$702$	0	0
$703$	−6.48240	−0.244488
$704$	3.68625	0.138931
$705$	0	0
$706$	−3.23882	−0.121894
$707$	−5.36930	−0.201933
$708$	0	0
$709$	9.24288	0.347123	0.173562	−	0.984823i	$-0.444472\pi$
$709$	9.24288	0.347123	0.173562	+	0.984823i	$0.444472\pi$
$710$	12.2564	0.459973
$711$	0	0
$712$	6.11506	0.229171
$713$	−20.0890	−0.752340
$714$	0	0
$715$	5.04034	0.188498
$716$	−4.01101	−0.149898
$717$	0	0
$718$	25.5364	0.953011
$719$	−15.8392	−0.590702	−0.295351	−	0.955389i	$-0.595437\pi$
$719$	−15.8392	−0.590702	−0.295351	+	0.955389i	$0.595437\pi$
$720$	0	0
$721$	1.78746	0.0665686
$722$	−1.63929	−0.0610079
$723$	0	0
$724$	−0.637262	−0.0236837
$725$	−2.11506	−0.0785512
$726$	0	0
$727$	−25.1063	−0.931141	−0.465570	−	0.885011i	$-0.654151\pi$
$727$	−25.1063	−0.931141	−0.465570	+	0.885011i	$0.654151\pi$
$728$	−20.1169	−0.745581
$729$	0	0
$730$	3.73489	0.138235
$731$	0.920743	0.0340549
$732$	0	0
$733$	39.1019	1.44426	0.722132	−	0.691756i	$-0.243163\pi$
$733$	39.1019	1.44426	0.722132	+	0.691756i	$0.243163\pi$
$734$	−14.9418	−0.551512
$735$	0	0
$736$	17.9911	0.663159
$737$	6.00618	0.221240
$738$	0	0
$739$	14.5427	0.534963	0.267482	−	0.963563i	$-0.413809\pi$
$739$	14.5427	0.534963	0.267482	+	0.963563i	$0.413809\pi$
$740$	4.45510	0.163773
$741$	0	0
$742$	11.3113	0.415252
$743$	1.46126	0.0536083	0.0268041	−	0.999641i	$-0.491467\pi$
$743$	1.46126	0.0536083	0.0268041	+	0.999641i	$0.491467\pi$
$744$	0	0
$745$	−22.8670	−0.837782
$746$	−4.44397	−0.162705
$747$	0	0
$748$	−1.16720	−0.0426771
$749$	5.45347	0.199266
$750$	0	0
$751$	−9.26538	−0.338099	−0.169049	−	0.985608i	$-0.554070\pi$
$751$	−9.26538	−0.338099	−0.169049	+	0.985608i	$0.554070\pi$
$752$	7.47071	0.272429
$753$	0	0
$754$	−17.4758	−0.636430
$755$	−12.1356	−0.441661
$756$	0	0
$757$	24.8627	0.903649	0.451824	−	0.892107i	$-0.350773\pi$
$757$	24.8627	0.903649	0.451824	+	0.892107i	$0.350773\pi$
$758$	−45.8781	−1.66637
$759$	0	0
$760$	−2.15196	−0.0780596
$761$	49.2950	1.78694	0.893471	−	0.449122i	$-0.148263\pi$
$761$	49.2950	1.78694	0.893471	+	0.449122i	$0.148263\pi$
$762$	0	0
$763$	−5.03767	−0.182376
$764$	−10.6564	−0.385534
$765$	0	0
$766$	22.8337	0.825015
$767$	−7.47685	−0.269974
$768$	0	0
$769$	−45.4467	−1.63885	−0.819426	−	0.573186i	$-0.805707\pi$
$769$	−45.4467	−1.63885	−0.819426	+	0.573186i	$0.805707\pi$
$770$	3.04034	0.109566
$771$	0	0
$772$	5.90054	0.212365
$773$	36.9140	1.32770	0.663851	−	0.747864i	$-0.268921\pi$
$773$	36.9140	1.32770	0.663851	+	0.747864i	$0.268921\pi$
$774$	0	0
$775$	−4.16741	−0.149698
$776$	−1.25964	−0.0452185
$777$	0	0
$778$	0.785104	0.0281473
$779$	7.38560	0.264616
$780$	0	0
$781$	7.47665	0.267535
$782$	13.4206	0.479919
$783$	0	0
$784$	17.4527	0.623312
$785$	−19.2179	−0.685917
$786$	0	0
$787$	−39.8533	−1.42062	−0.710309	−	0.703890i	$-0.751445\pi$
$787$	−39.8533	−1.42062	−0.710309	+	0.703890i	$0.751445\pi$
$788$	2.21542	0.0789212
$789$	0	0
$790$	−13.6554	−0.485838
$791$	17.0890	0.607615
$792$	0	0
$793$	20.3099	0.721227
$794$	−53.4482	−1.89680
$795$	0	0
$796$	2.15786	0.0764834
$797$	11.3599	0.402387	0.201193	−	0.979552i	$-0.435518\pi$
$797$	11.3599	0.402387	0.201193	+	0.979552i	$0.435518\pi$
$798$	0	0
$799$	2.58818	0.0915634
$800$	3.73219	0.131953
$801$	0	0
$802$	−26.5834	−0.938692
$803$	2.27836	0.0804017
$804$	0	0
$805$	−8.94046	−0.315110
$806$	−34.4335	−1.21287
$807$	0	0
$808$	−6.22994	−0.219168
$809$	37.7496	1.32721	0.663603	−	0.748085i	$-0.269026\pi$
$809$	37.7496	1.32721	0.663603	+	0.748085i	$0.269026\pi$
$810$	0	0
$811$	46.5412	1.63428	0.817141	−	0.576437i	$-0.195558\pi$
$811$	46.5412	1.63428	0.817141	+	0.576437i	$0.195558\pi$
$812$	−2.69594	−0.0946091
$813$	0	0
$814$	10.6265	0.372459
$815$	2.83648	0.0993576
$816$	0	0
$817$	−0.542144	−0.0189672
$818$	26.8247	0.937903
$819$	0	0
$820$	−5.07584	−0.177256
$821$	20.2161	0.705547	0.352774	−	0.935709i	$-0.385239\pi$
$821$	20.2161	0.705547	0.352774	+	0.935709i	$0.385239\pi$
$822$	0	0
$823$	2.10501	0.0733760	0.0366880	−	0.999327i	$-0.488319\pi$
$823$	2.10501	0.0733760	0.0366880	+	0.999327i	$0.488319\pi$
$824$	2.07397	0.0722503
$825$	0	0
$826$	−4.51005	−0.156925
$827$	−14.4835	−0.503642	−0.251821	−	0.967774i	$-0.581029\pi$
$827$	−14.4835	−0.503642	−0.251821	+	0.967774i	$0.581029\pi$
$828$	0	0
$829$	−27.4850	−0.954593	−0.477296	−	0.878742i	$-0.658383\pi$
$829$	−27.4850	−0.954593	−0.477296	+	0.878742i	$0.658383\pi$
$830$	−2.87058	−0.0996391
$831$	0	0
$832$	−18.5800	−0.644145
$833$	6.04640	0.209495
$834$	0	0
$835$	−10.2869	−0.355993
$836$	0.687261	0.0237694
$837$	0	0
$838$	−40.8242	−1.41025
$839$	24.1194	0.832696	0.416348	−	0.909205i	$-0.363310\pi$
$839$	24.1194	0.832696	0.416348	+	0.909205i	$0.363310\pi$
$840$	0	0
$841$	−24.5265	−0.845743
$842$	−44.6441	−1.53854
$843$	0	0
$844$	−5.99802	−0.206461
$845$	−12.4050	−0.426746
$846$	0	0
$847$	1.85467	0.0637273
$848$	18.2382	0.626303
$849$	0	0
$850$	2.78406	0.0954925
$851$	−31.2484	−1.07118
$852$	0	0
$853$	7.81329	0.267522	0.133761	−	0.991014i	$-0.457295\pi$
$853$	7.81329	0.267522	0.133761	+	0.991014i	$0.457295\pi$
$854$	12.2510	0.419220
$855$	0	0
$856$	6.32760	0.216273
$857$	17.1582	0.586114	0.293057	−	0.956095i	$-0.405327\pi$
$857$	17.1582	0.586114	0.293057	+	0.956095i	$0.405327\pi$
$858$	0	0
$859$	−14.7301	−0.502584	−0.251292	−	0.967911i	$-0.580855\pi$
$859$	−14.7301	−0.502584	−0.251292	+	0.967911i	$0.580855\pi$
$860$	0.372595	0.0127054
$861$	0	0
$862$	23.2118	0.790596
$863$	24.6653	0.839616	0.419808	−	0.907613i	$-0.362097\pi$
$863$	24.6653	0.839616	0.419808	+	0.907613i	$0.362097\pi$
$864$	0	0
$865$	−10.1153	−0.343930
$866$	−8.62253	−0.293005
$867$	0	0
$868$	−5.31197	−0.180300
$869$	−8.33010	−0.282579
$870$	0	0
$871$	−30.2732	−1.02577
$872$	−5.84515	−0.197942
$873$	0	0
$874$	−7.90219	−0.267296
$875$	−1.85467	−0.0626994
$876$	0	0
$877$	−31.8972	−1.07709	−0.538547	−	0.842596i	$-0.681026\pi$
$877$	−31.8972	−1.07709	−0.538547	+	0.842596i	$0.681026\pi$
$878$	−26.9928	−0.910962
$879$	0	0
$880$	4.90219	0.165253
$881$	27.3744	0.922268	0.461134	−	0.887330i	$-0.347443\pi$
$881$	27.3744	0.922268	0.461134	+	0.887330i	$0.347443\pi$
$882$	0	0
$883$	−15.5817	−0.524365	−0.262182	−	0.965018i	$-0.584442\pi$
$883$	−15.5817	−0.524365	−0.262182	+	0.965018i	$0.584442\pi$
$884$	5.88309	0.197870
$885$	0	0
$886$	−1.38703	−0.0465983
$887$	−35.8772	−1.20464	−0.602319	−	0.798255i	$-0.705757\pi$
$887$	−35.8772	−1.20464	−0.602319	+	0.798255i	$0.705757\pi$
$888$	0	0
$889$	−20.1334	−0.675254
$890$	4.65824	0.156145
$891$	0	0
$892$	−16.3981	−0.549050
$893$	−1.52395	−0.0509971
$894$	0	0
$895$	5.83622	0.195083
$896$	−25.0515	−0.836910
$897$	0	0
$898$	−24.4064	−0.814452
$899$	8.81430	0.293973
$900$	0	0
$901$	6.31852	0.210500
$902$	−12.1071	−0.403123
$903$	0	0
$904$	19.8282	0.659476
$905$	0.927249	0.0308228
$906$	0	0
$907$	−24.2089	−0.803843	−0.401922	−	0.915674i	$-0.631658\pi$
$907$	−24.2089	−0.803843	−0.401922	+	0.915674i	$0.631658\pi$
$908$	−18.1352	−0.601839
$909$	0	0
$910$	−15.3243	−0.507997
$911$	34.1836	1.13255	0.566276	−	0.824216i	$-0.308384\pi$
$911$	34.1836	1.13255	0.566276	+	0.824216i	$0.308384\pi$
$912$	0	0
$913$	−1.75111	−0.0579534
$914$	36.4312	1.20504
$915$	0	0
$916$	−3.58113	−0.118324
$917$	30.5854	1.01002
$918$	0	0
$919$	27.4375	0.905080	0.452540	−	0.891744i	$-0.350518\pi$
$919$	27.4375	0.905080	0.452540	+	0.891744i	$0.350518\pi$
$920$	−10.3735	−0.342005
$921$	0	0
$922$	−2.77344	−0.0913385
$923$	−37.6848	−1.24041
$924$	0	0
$925$	−6.48240	−0.213140
$926$	−17.9517	−0.589929
$927$	0	0
$928$	−7.89379	−0.259126
$929$	−13.6344	−0.447330	−0.223665	−	0.974666i	$-0.571802\pi$
$929$	−13.6344	−0.447330	−0.223665	+	0.974666i	$0.571802\pi$
$930$	0	0
$931$	−3.56019	−0.116681
$932$	12.6091	0.413024
$933$	0	0
$934$	62.8878	2.05775
$935$	1.69834	0.0555415
$936$	0	0
$937$	−12.9313	−0.422448	−0.211224	−	0.977438i	$-0.567745\pi$
$937$	−12.9313	−0.422448	−0.211224	+	0.977438i	$0.567745\pi$
$938$	−18.2608	−0.596237
$939$	0	0
$940$	1.04735	0.0341609
$941$	42.3415	1.38029	0.690147	−	0.723669i	$-0.257546\pi$
$941$	42.3415	1.38029	0.690147	+	0.723669i	$0.257546\pi$
$942$	0	0
$943$	35.6023	1.15937
$944$	−7.27193	−0.236681
$945$	0	0
$946$	0.888729	0.0288951
$947$	−22.1576	−0.720026	−0.360013	−	0.932947i	$-0.617228\pi$
$947$	−22.1576	−0.720026	−0.360013	+	0.932947i	$0.617228\pi$
$948$	0	0
$949$	−11.4837	−0.372778
$950$	−1.63929	−0.0531855
$951$	0	0
$952$	−6.77835	−0.219688
$953$	15.1161	0.489658	0.244829	−	0.969566i	$-0.421268\pi$
$953$	15.1161	0.489658	0.244829	+	0.969566i	$0.421268\pi$
$954$	0	0
$955$	15.5055	0.501748
$956$	−15.5350	−0.502438
$957$	0	0
$958$	−56.1552	−1.81429
$959$	−2.57732	−0.0832259
$960$	0	0
$961$	−13.6327	−0.439764
$962$	−53.5612	−1.72688
$963$	0	0
$964$	−13.7717	−0.443557
$965$	−8.58558	−0.276380
$966$	0	0
$967$	−21.2309	−0.682741	−0.341370	−	0.939929i	$-0.610891\pi$
$967$	−21.2309	−0.682741	−0.341370	+	0.939929i	$0.610891\pi$
$968$	2.15196	0.0691665
$969$	0	0
$970$	−0.959552	−0.0308094
$971$	49.6913	1.59467	0.797335	−	0.603537i	$-0.206243\pi$
$971$	49.6913	1.59467	0.797335	+	0.603537i	$0.206243\pi$
$972$	0	0
$973$	3.89152	0.124756
$974$	32.1293	1.02949
$975$	0	0
$976$	19.7533	0.632287
$977$	29.5793	0.946326	0.473163	−	0.880975i	$-0.343112\pi$
$977$	29.5793	0.946326	0.473163	+	0.880975i	$0.343112\pi$
$978$	0	0
$979$	2.84163	0.0908188
$980$	2.44678	0.0781596
$981$	0	0
$982$	−48.2563	−1.53992
$983$	−37.8228	−1.20636	−0.603180	−	0.797605i	$-0.706100\pi$
$983$	−37.8228	−1.20636	−0.603180	+	0.797605i	$0.706100\pi$
$984$	0	0
$985$	−3.22355	−0.102711
$986$	−5.88844	−0.187526
$987$	0	0
$988$	−3.46403	−0.110206
$989$	−2.61341	−0.0831016
$990$	0	0
$991$	6.65600	0.211435	0.105717	−	0.994396i	$-0.466286\pi$
$991$	6.65600	0.211435	0.105717	+	0.994396i	$0.466286\pi$
$992$	−15.5536	−0.493827
$993$	0	0
$994$	−22.7315	−0.721001
$995$	−3.13980	−0.0995383
$996$	0	0
$997$	−8.68050	−0.274914	−0.137457	−	0.990508i	$-0.543893\pi$
$997$	−8.68050	−0.274914	−0.137457	+	0.990508i	$0.543893\pi$
$998$	14.0852	0.445859
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9405.2.a.x.1.2		6
3.2	odd	2		3135.2.a.p.1.5	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3135.2.a.p.1.5	✓	6	3.2	odd	2
9405.2.a.x.1.2		6	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:	\( N \)	\(=\)	\( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9405.a (trivial)

\(f(q)\)	\(=\)	\(q-1.63929 q^{2} +0.687261 q^{4} -1.00000 q^{5} +1.85467 q^{7} +2.15196 q^{8} +O(q^{10})\)	\(q-1.63929 q^{2} +0.687261 q^{4} -1.00000 q^{5} +1.85467 q^{7} +2.15196 q^{8} +1.63929 q^{10} +1.00000 q^{11} -5.04034 q^{13} -3.04034 q^{14} -4.90219 q^{16} -1.69834 q^{17} +1.00000 q^{19} -0.687261 q^{20} -1.63929 q^{22} +4.82051 q^{23} +1.00000 q^{25} +8.26256 q^{26} +1.27464 q^{28} -2.11506 q^{29} -4.16741 q^{31} +3.73219 q^{32} +2.78406 q^{34} -1.85467 q^{35} -6.48240 q^{37} -1.63929 q^{38} -2.15196 q^{40} +7.38560 q^{41} -0.542144 q^{43} +0.687261 q^{44} -7.90219 q^{46} -1.52395 q^{47} -3.56019 q^{49} -1.63929 q^{50} -3.46403 q^{52} -3.72042 q^{53} -1.00000 q^{55} +3.99117 q^{56} +3.46718 q^{58} +1.48340 q^{59} -4.02948 q^{61} +6.83158 q^{62} +3.68625 q^{64} +5.04034 q^{65} +6.00618 q^{67} -1.16720 q^{68} +3.04034 q^{70} +7.47665 q^{71} +2.27836 q^{73} +10.6265 q^{74} +0.687261 q^{76} +1.85467 q^{77} -8.33010 q^{79} +4.90219 q^{80} -12.1071 q^{82} -1.75111 q^{83} +1.69834 q^{85} +0.888729 q^{86} +2.15196 q^{88} +2.84163 q^{89} -9.34818 q^{91} +3.31295 q^{92} +2.49820 q^{94} -1.00000 q^{95} -0.585347 q^{97} +5.83617 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} + 5 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8}+O(q^{10}) \) 6 * q + q^2 + 5 * q^4 - 6 * q^5 + 3 * q^7 + 6 * q^8	\( 6 q + q^{2} + 5 q^{4} - 6 q^{5} + 3 q^{7} + 6 q^{8} - q^{10} + 6 q^{11} - 4 q^{13} + 8 q^{14} - q^{16} + 4 q^{17} + 6 q^{19} - 5 q^{20} + q^{22} + 6 q^{25} + 9 q^{26} + 5 q^{28} + 21 q^{29} - 16 q^{31} + 8 q^{32} - 7 q^{34} - 3 q^{35} - q^{37} + q^{38} - 6 q^{40} + 31 q^{41} + 6 q^{43} + 5 q^{44} - 19 q^{46} + 4 q^{47} - 5 q^{49} + q^{50} + 17 q^{52} + 9 q^{53} - 6 q^{55} + 11 q^{56} + 19 q^{58} + 24 q^{59} - 11 q^{61} - 3 q^{62} - 30 q^{64} + 4 q^{65} - 11 q^{67} - 8 q^{70} + 23 q^{71} - 6 q^{73} - 17 q^{74} + 5 q^{76} + 3 q^{77} + 4 q^{79} + q^{80} + 20 q^{82} - 17 q^{83} - 4 q^{85} + 9 q^{86} + 6 q^{88} + 37 q^{89} + 3 q^{91} + 17 q^{92} - 2 q^{94} - 6 q^{95} - 26 q^{97} + 2 q^{98}+O(q^{100}) \) 6 * q + q^2 + 5 * q^4 - 6 * q^5 + 3 * q^7 + 6 * q^8 - q^10 + 6 * q^11 - 4 * q^13 + 8 * q^14 - q^16 + 4 * q^17 + 6 * q^19 - 5 * q^20 + q^22 + 6 * q^25 + 9 * q^26 + 5 * q^28 + 21 * q^29 - 16 * q^31 + 8 * q^32 - 7 * q^34 - 3 * q^35 - q^37 + q^38 - 6 * q^40 + 31 * q^41 + 6 * q^43 + 5 * q^44 - 19 * q^46 + 4 * q^47 - 5 * q^49 + q^50 + 17 * q^52 + 9 * q^53 - 6 * q^55 + 11 * q^56 + 19 * q^58 + 24 * q^59 - 11 * q^61 - 3 * q^62 - 30 * q^64 + 4 * q^65 - 11 * q^67 - 8 * q^70 + 23 * q^71 - 6 * q^73 - 17 * q^74 + 5 * q^76 + 3 * q^77 + 4 * q^79 + q^80 + 20 * q^82 - 17 * q^83 - 4 * q^85 + 9 * q^86 + 6 * q^88 + 37 * q^89 + 3 * q^91 + 17 * q^92 - 2 * q^94 - 6 * q^95 - 26 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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