LMFDB - Embedded newform 9405.2.a.z.1.1

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:	6.6.7281497.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 7x^{3} + 6x^{2} - 2x - 1 $ x^6 - 2x^5 - 5x^4 + 7x^3 + 6x^2 - 2*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1045)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.326248$ 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)

$f(q)$	$=$	$q-1.73890 q^{2} +1.02379 q^{4} +1.00000 q^{5} +2.22757 q^{7} +1.69754 q^{8} +O(q^{10})$	\(q-1.73890 q^{2} +1.02379 q^{4} +1.00000 q^{5} +2.22757 q^{7} +1.69754 q^{8} -1.73890 q^{10} +1.00000 q^{11} -3.64023 q^{13} -3.87353 q^{14} -4.99943 q^{16} -0.644551 q^{17} -1.00000 q^{19} +1.02379 q^{20} -1.73890 q^{22} +6.15983 q^{23} +1.00000 q^{25} +6.33001 q^{26} +2.28056 q^{28} +1.88204 q^{29} -0.183675 q^{31} +5.29846 q^{32} +1.12081 q^{34} +2.22757 q^{35} -4.11428 q^{37} +1.73890 q^{38} +1.69754 q^{40} +1.53371 q^{41} -1.53577 q^{43} +1.02379 q^{44} -10.7113 q^{46} -1.75837 q^{47} -2.03793 q^{49} -1.73890 q^{50} -3.72682 q^{52} +2.81491 q^{53} +1.00000 q^{55} +3.78139 q^{56} -3.27269 q^{58} +3.90068 q^{59} +0.734057 q^{61} +0.319392 q^{62} +0.785358 q^{64} -3.64023 q^{65} -1.30264 q^{67} -0.659883 q^{68} -3.87353 q^{70} -10.5493 q^{71} -5.17599 q^{73} +7.15433 q^{74} -1.02379 q^{76} +2.22757 q^{77} +2.89974 q^{79} -4.99943 q^{80} -2.66698 q^{82} -13.8209 q^{83} -0.644551 q^{85} +2.67055 q^{86} +1.69754 q^{88} +6.39884 q^{89} -8.10886 q^{91} +6.30635 q^{92} +3.05763 q^{94} -1.00000 q^{95} +6.23352 q^{97} +3.54376 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 5 q^{7} + 12 q^{8}+O(q^{10}) $ 6 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 5 * q^7 + 12 * q^8	$ 6 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 5 q^{7} + 12 q^{8} + 2 q^{10} + 6 q^{11} - 5 q^{13} + 8 q^{14} + 4 q^{16} - q^{17} - 6 q^{19} + 4 q^{20} + 2 q^{22} - 4 q^{23} + 6 q^{25} + 14 q^{26} + 10 q^{28} + 9 q^{29} - 21 q^{31} + q^{32} + 5 q^{35} - 3 q^{37} - 2 q^{38} + 12 q^{40} + 23 q^{41} + 7 q^{43} + 4 q^{44} - 12 q^{46} + 18 q^{47} - 3 q^{49} + 2 q^{50} + 13 q^{52} + 17 q^{53} + 6 q^{55} + 2 q^{56} + 23 q^{58} + 29 q^{59} + 17 q^{61} - 2 q^{62} - 18 q^{64} - 5 q^{65} + 8 q^{67} + q^{68} + 8 q^{70} + 12 q^{71} + 2 q^{73} + 37 q^{74} - 4 q^{76} + 5 q^{77} + 3 q^{79} + 4 q^{80} + 24 q^{82} + 11 q^{83} - q^{85} + 12 q^{86} + 12 q^{88} + 22 q^{89} - 18 q^{91} + 15 q^{92} + 22 q^{94} - 6 q^{95} - 2 q^{97} + q^{98}+O(q^{100}) $ 6 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 5 * q^7 + 12 * q^8 + 2 * q^10 + 6 * q^11 - 5 * q^13 + 8 * q^14 + 4 * q^16 - q^17 - 6 * q^19 + 4 * q^20 + 2 * q^22 - 4 * q^23 + 6 * q^25 + 14 * q^26 + 10 * q^28 + 9 * q^29 - 21 * q^31 + q^32 + 5 * q^35 - 3 * q^37 - 2 * q^38 + 12 * q^40 + 23 * q^41 + 7 * q^43 + 4 * q^44 - 12 * q^46 + 18 * q^47 - 3 * q^49 + 2 * q^50 + 13 * q^52 + 17 * q^53 + 6 * q^55 + 2 * q^56 + 23 * q^58 + 29 * q^59 + 17 * q^61 - 2 * q^62 - 18 * q^64 - 5 * q^65 + 8 * q^67 + q^68 + 8 * q^70 + 12 * q^71 + 2 * q^73 + 37 * q^74 - 4 * q^76 + 5 * q^77 + 3 * q^79 + 4 * q^80 + 24 * q^82 + 11 * q^83 - q^85 + 12 * q^86 + 12 * q^88 + 22 * q^89 - 18 * q^91 + 15 * q^92 + 22 * q^94 - 6 * q^95 - 2 * q^97 + 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 $

$2$	−1.73890	−1.22959	−0.614795	−	0.788687i	$-0.710761\pi$
$2$	−1.73890	−1.22959	−0.614795	+	0.788687i	$0.710761\pi$
$3$	0	0
$3$	0	0
$4$	1.02379	0.511894
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	2.22757	0.841943	0.420971	−	0.907074i	$-0.361689\pi$
$7$	2.22757	0.841943	0.420971	+	0.907074i	$0.361689\pi$
$8$	1.69754	0.600171
$9$	0	0
$10$	−1.73890	−0.549890
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−3.64023	−1.00962	−0.504809	−	0.863231i	$-0.668437\pi$
$13$	−3.64023	−1.00962	−0.504809	+	0.863231i	$0.668437\pi$
$14$	−3.87353	−1.03525
$15$	0	0
$16$	−4.99943	−1.24986
$17$	−0.644551	−0.156327	−0.0781633	−	0.996941i	$-0.524906\pi$
$17$	−0.644551	−0.156327	−0.0781633	+	0.996941i	$0.524906\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	1.02379	0.228926
$21$	0	0
$22$	−1.73890	−0.370736
$23$	6.15983	1.28441	0.642206	−	0.766532i	$-0.278019\pi$
$23$	6.15983	1.28441	0.642206	+	0.766532i	$0.278019\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	6.33001	1.24142
$27$	0	0
$28$	2.28056	0.430985
$29$	1.88204	0.349487	0.174743	−	0.984614i	$-0.444090\pi$
$29$	1.88204	0.349487	0.174743	+	0.984614i	$0.444090\pi$
$30$	0	0
$31$	−0.183675	−0.0329889	−0.0164945	−	0.999864i	$-0.505251\pi$
$31$	−0.183675	−0.0329889	−0.0164945	+	0.999864i	$0.505251\pi$
$32$	5.29846	0.936644
$33$	0	0
$34$	1.12081	0.192218
$35$	2.22757	0.376528
$36$	0	0
$37$	−4.11428	−0.676383	−0.338192	−	0.941077i	$-0.609815\pi$
$37$	−4.11428	−0.676383	−0.338192	+	0.941077i	$0.609815\pi$
$38$	1.73890	0.282088
$39$	0	0
$40$	1.69754	0.268405
$41$	1.53371	0.239525	0.119763	−	0.992803i	$-0.461787\pi$
$41$	1.53371	0.239525	0.119763	+	0.992803i	$0.461787\pi$
$42$	0	0
$43$	−1.53577	−0.234202	−0.117101	−	0.993120i	$-0.537360\pi$
$43$	−1.53577	−0.234202	−0.117101	+	0.993120i	$0.537360\pi$
$44$	1.02379	0.154342
$45$	0	0
$46$	−10.7113	−1.57930
$47$	−1.75837	−0.256484	−0.128242	−	0.991743i	$-0.540933\pi$
$47$	−1.75837	−0.256484	−0.128242	+	0.991743i	$0.540933\pi$
$48$	0	0
$49$	−2.03793	−0.291133
$50$	−1.73890	−0.245918
$51$	0	0
$52$	−3.72682	−0.516817
$53$	2.81491	0.386658	0.193329	−	0.981134i	$-0.438071\pi$
$53$	2.81491	0.386658	0.193329	+	0.981134i	$0.438071\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	3.78139	0.505309
$57$	0	0
$58$	−3.27269	−0.429726
$59$	3.90068	0.507825	0.253913	−	0.967227i	$-0.418282\pi$
$59$	3.90068	0.507825	0.253913	+	0.967227i	$0.418282\pi$
$60$	0	0
$61$	0.734057	0.0939863	0.0469932	−	0.998895i	$-0.485036\pi$
$61$	0.734057	0.0939863	0.0469932	+	0.998895i	$0.485036\pi$
$62$	0.319392	0.0405629
$63$	0	0
$64$	0.785358	0.0981698
$65$	−3.64023	−0.451515
$66$	0	0
$67$	−1.30264	−0.159143	−0.0795717	−	0.996829i	$-0.525355\pi$
$67$	−1.30264	−0.159143	−0.0795717	+	0.996829i	$0.525355\pi$
$68$	−0.659883	−0.0800226
$69$	0	0
$70$	−3.87353	−0.462976
$71$	−10.5493	−1.25197	−0.625983	−	0.779837i	$-0.715302\pi$
$71$	−10.5493	−1.25197	−0.625983	+	0.779837i	$0.715302\pi$
$72$	0	0
$73$	−5.17599	−0.605804	−0.302902	−	0.953022i	$-0.597956\pi$
$73$	−5.17599	−0.605804	−0.302902	+	0.953022i	$0.597956\pi$
$74$	7.15433	0.831674
$75$	0	0
$76$	−1.02379	−0.117437
$77$	2.22757	0.253855
$78$	0	0
$79$	2.89974	0.326246	0.163123	−	0.986606i	$-0.447843\pi$
$79$	2.89974	0.326246	0.163123	+	0.986606i	$0.447843\pi$
$80$	−4.99943	−0.558954
$81$	0	0
$82$	−2.66698	−0.294518
$83$	−13.8209	−1.51704	−0.758522	−	0.651647i	$-0.774078\pi$
$83$	−13.8209	−1.51704	−0.758522	+	0.651647i	$0.774078\pi$
$84$	0	0
$85$	−0.644551	−0.0699114
$86$	2.67055	0.287973
$87$	0	0
$88$	1.69754	0.180958
$89$	6.39884	0.678275	0.339138	−	0.940737i	$-0.389865\pi$
$89$	6.39884	0.678275	0.339138	+	0.940737i	$0.389865\pi$
$90$	0	0
$91$	−8.10886	−0.850040
$92$	6.30635	0.657483
$93$	0	0
$94$	3.05763	0.315371
$95$	−1.00000	−0.102598
$96$	0	0
$97$	6.23352	0.632918	0.316459	−	0.948606i	$-0.397506\pi$
$97$	6.23352	0.632918	0.316459	+	0.948606i	$0.397506\pi$
$98$	3.54376	0.357974
$99$	0	0
$100$	1.02379	0.102379
$101$	16.3563	1.62751	0.813756	−	0.581206i	$-0.197419\pi$
$101$	16.3563	1.62751	0.813756	+	0.581206i	$0.197419\pi$
$102$	0	0
$103$	0.645519	0.0636049	0.0318025	−	0.999494i	$-0.489875\pi$
$103$	0.645519	0.0636049	0.0318025	+	0.999494i	$0.489875\pi$
$104$	−6.17943	−0.605943
$105$	0	0
$106$	−4.89487	−0.475432
$107$	17.5900	1.70049	0.850243	−	0.526390i	$-0.176455\pi$
$107$	17.5900	1.70049	0.850243	+	0.526390i	$0.176455\pi$
$108$	0	0
$109$	19.4359	1.86162	0.930808	−	0.365507i	$-0.119104\pi$
$109$	19.4359	1.86162	0.930808	+	0.365507i	$0.119104\pi$
$110$	−1.73890	−0.165798
$111$	0	0
$112$	−11.1366	−1.05231
$113$	5.88383	0.553504	0.276752	−	0.960941i	$-0.410742\pi$
$113$	5.88383	0.553504	0.276752	+	0.960941i	$0.410742\pi$
$114$	0	0
$115$	6.15983	0.574407
$116$	1.92681	0.178900
$117$	0	0
$118$	−6.78291	−0.624417
$119$	−1.43578	−0.131618
$120$	0	0
$121$	1.00000	0.0909091
$122$	−1.27645	−0.115565
$123$	0	0
$124$	−0.188044	−0.0168868
$125$	1.00000	0.0894427
$126$	0	0
$127$	15.9676	1.41690	0.708450	−	0.705762i	$-0.249395\pi$
$127$	15.9676	1.41690	0.708450	+	0.705762i	$0.249395\pi$
$128$	−11.9626	−1.05735
$129$	0	0
$130$	6.33001	0.555178
$131$	3.93173	0.343517	0.171759	−	0.985139i	$-0.445055\pi$
$131$	3.93173	0.343517	0.171759	+	0.985139i	$0.445055\pi$
$132$	0	0
$133$	−2.22757	−0.193155
$134$	2.26517	0.195681
$135$	0	0
$136$	−1.09415	−0.0938226
$137$	−15.5422	−1.32786	−0.663929	−	0.747796i	$-0.731112\pi$
$137$	−15.5422	−1.32786	−0.663929	+	0.747796i	$0.731112\pi$
$138$	0	0
$139$	−6.69900	−0.568202	−0.284101	−	0.958794i	$-0.591695\pi$
$139$	−6.69900	−0.568202	−0.284101	+	0.958794i	$0.591695\pi$
$140$	2.28056	0.192742
$141$	0	0
$142$	18.3441	1.53941
$143$	−3.64023	−0.304411
$144$	0	0
$145$	1.88204	0.156295
$146$	9.00056	0.744891
$147$	0	0
$148$	−4.21215	−0.346236
$149$	−0.783045	−0.0641496	−0.0320748	−	0.999485i	$-0.510211\pi$
$149$	−0.783045	−0.0641496	−0.0320748	+	0.999485i	$0.510211\pi$
$150$	0	0
$151$	8.46749	0.689075	0.344537	−	0.938773i	$-0.388036\pi$
$151$	8.46749	0.689075	0.344537	+	0.938773i	$0.388036\pi$
$152$	−1.69754	−0.137689
$153$	0	0
$154$	−3.87353	−0.312138
$155$	−0.183675	−0.0147531
$156$	0	0
$157$	8.27024	0.660037	0.330019	−	0.943974i	$-0.392945\pi$
$157$	8.27024	0.660037	0.330019	+	0.943974i	$0.392945\pi$
$158$	−5.04237	−0.401150
$159$	0	0
$160$	5.29846	0.418880
$161$	13.7215	1.08140
$162$	0	0
$163$	23.9030	1.87223	0.936113	−	0.351699i	$-0.114396\pi$
$163$	23.9030	1.87223	0.936113	+	0.351699i	$0.114396\pi$
$164$	1.57019	0.122612
$165$	0	0
$166$	24.0333	1.86534
$167$	19.4405	1.50435	0.752176	−	0.658962i	$-0.229004\pi$
$167$	19.4405	1.50435	0.752176	+	0.658962i	$0.229004\pi$
$168$	0	0
$169$	0.251253	0.0193272
$170$	1.12081	0.0859624
$171$	0	0
$172$	−1.57230	−0.119887
$173$	11.6888	0.888682	0.444341	−	0.895858i	$-0.353438\pi$
$173$	11.6888	0.888682	0.444341	+	0.895858i	$0.353438\pi$
$174$	0	0
$175$	2.22757	0.168389
$176$	−4.99943	−0.376847
$177$	0	0
$178$	−11.1270	−0.834001
$179$	6.77452	0.506351	0.253176	−	0.967420i	$-0.418525\pi$
$179$	6.77452	0.506351	0.253176	+	0.967420i	$0.418525\pi$
$180$	0	0
$181$	−4.34551	−0.322999	−0.161500	−	0.986873i	$-0.551633\pi$
$181$	−4.34551	−0.322999	−0.161500	+	0.986873i	$0.551633\pi$
$182$	14.1005	1.04520
$183$	0	0
$184$	10.4565	0.770867
$185$	−4.11428	−0.302488
$186$	0	0
$187$	−0.644551	−0.0471342
$188$	−1.80020	−0.131293
$189$	0	0
$190$	1.73890	0.126153
$191$	1.37612	0.0995728	0.0497864	−	0.998760i	$-0.484146\pi$
$191$	1.37612	0.0995728	0.0497864	+	0.998760i	$0.484146\pi$
$192$	0	0
$193$	5.15711	0.371217	0.185609	−	0.982624i	$-0.440574\pi$
$193$	5.15711	0.371217	0.185609	+	0.982624i	$0.440574\pi$
$194$	−10.8395	−0.778231
$195$	0	0
$196$	−2.08641	−0.149029
$197$	−12.0998	−0.862072	−0.431036	−	0.902335i	$-0.641852\pi$
$197$	−12.0998	−0.862072	−0.431036	+	0.902335i	$0.641852\pi$
$198$	0	0
$199$	−13.2948	−0.942445	−0.471223	−	0.882014i	$-0.656187\pi$
$199$	−13.2948	−0.942445	−0.471223	+	0.882014i	$0.656187\pi$
$200$	1.69754	0.120034
$201$	0	0
$202$	−28.4420	−2.00117
$203$	4.19239	0.294248
$204$	0	0
$205$	1.53371	0.107119
$206$	−1.12250	−0.0782080
$207$	0	0
$208$	18.1991	1.26188
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−26.0866	−1.79588	−0.897939	−	0.440120i	$-0.854936\pi$
$211$	−26.0866	−1.79588	−0.897939	+	0.440120i	$0.854936\pi$
$212$	2.88188	0.197928
$213$	0	0
$214$	−30.5873	−2.09090
$215$	−1.53577	−0.104738
$216$	0	0
$217$	−0.409148	−0.0277748
$218$	−33.7971	−2.28903
$219$	0	0
$220$	1.02379	0.0690238
$221$	2.34631	0.157830
$222$	0	0
$223$	13.8606	0.928175	0.464087	−	0.885789i	$-0.346382\pi$
$223$	13.8606	0.928175	0.464087	+	0.885789i	$0.346382\pi$
$224$	11.8027	0.788600
$225$	0	0
$226$	−10.2314	−0.680583
$227$	12.9104	0.856893	0.428446	−	0.903567i	$-0.359061\pi$
$227$	12.9104	0.856893	0.428446	+	0.903567i	$0.359061\pi$
$228$	0	0
$229$	7.93710	0.524499	0.262249	−	0.965000i	$-0.415536\pi$
$229$	7.93710	0.524499	0.262249	+	0.965000i	$0.415536\pi$
$230$	−10.7113	−0.706285
$231$	0	0
$232$	3.19484	0.209752
$233$	−17.6265	−1.15475	−0.577374	−	0.816480i	$-0.695922\pi$
$233$	−17.6265	−1.15475	−0.577374	+	0.816480i	$0.695922\pi$
$234$	0	0
$235$	−1.75837	−0.114703
$236$	3.99347	0.259953
$237$	0	0
$238$	2.49669	0.161836
$239$	18.6909	1.20901	0.604507	−	0.796600i	$-0.293370\pi$
$239$	18.6909	1.20901	0.604507	+	0.796600i	$0.293370\pi$
$240$	0	0
$241$	−0.877285	−0.0565109	−0.0282555	−	0.999601i	$-0.508995\pi$
$241$	−0.877285	−0.0565109	−0.0282555	+	0.999601i	$0.508995\pi$
$242$	−1.73890	−0.111781
$243$	0	0
$244$	0.751518	0.0481110
$245$	−2.03793	−0.130198
$246$	0	0
$247$	3.64023	0.231622
$248$	−0.311795	−0.0197990
$249$	0	0
$250$	−1.73890	−0.109978
$251$	0.448818	0.0283291	0.0141646	−	0.999900i	$-0.495491\pi$
$251$	0.448818	0.0283291	0.0141646	+	0.999900i	$0.495491\pi$
$252$	0	0
$253$	6.15983	0.387265
$254$	−27.7662	−1.74221
$255$	0	0
$256$	19.2311	1.20194
$257$	7.85809	0.490174	0.245087	−	0.969501i	$-0.421183\pi$
$257$	7.85809	0.490174	0.245087	+	0.969501i	$0.421183\pi$
$258$	0	0
$259$	−9.16484	−0.569476
$260$	−3.72682	−0.231128
$261$	0	0
$262$	−6.83691	−0.422386
$263$	−0.585215	−0.0360859	−0.0180429	−	0.999837i	$-0.505744\pi$
$263$	−0.585215	−0.0360859	−0.0180429	+	0.999837i	$0.505744\pi$
$264$	0	0
$265$	2.81491	0.172919
$266$	3.87353	0.237502
$267$	0	0
$268$	−1.33363	−0.0814645
$269$	3.59078	0.218934	0.109467	−	0.993990i	$-0.465086\pi$
$269$	3.59078	0.218934	0.109467	+	0.993990i	$0.465086\pi$
$270$	0	0
$271$	−23.4466	−1.42428	−0.712139	−	0.702038i	$-0.752274\pi$
$271$	−23.4466	−1.42428	−0.712139	+	0.702038i	$0.752274\pi$
$272$	3.22239	0.195386
$273$	0	0
$274$	27.0264	1.63272
$275$	1.00000	0.0603023
$276$	0	0
$277$	−19.2912	−1.15909	−0.579547	−	0.814939i	$-0.696771\pi$
$277$	−19.2912	−1.15909	−0.579547	+	0.814939i	$0.696771\pi$
$278$	11.6489	0.698656
$279$	0	0
$280$	3.78139	0.225981
$281$	5.82550	0.347520	0.173760	−	0.984788i	$-0.444408\pi$
$281$	5.82550	0.347520	0.173760	+	0.984788i	$0.444408\pi$
$282$	0	0
$283$	−15.5166	−0.922368	−0.461184	−	0.887305i	$-0.652575\pi$
$283$	−15.5166	−0.922368	−0.461184	+	0.887305i	$0.652575\pi$
$284$	−10.8002	−0.640874
$285$	0	0
$286$	6.33001	0.374301
$287$	3.41645	0.201667
$288$	0	0
$289$	−16.5846	−0.975562
$290$	−3.27269	−0.192179
$291$	0	0
$292$	−5.29912	−0.310107
$293$	12.8089	0.748305	0.374152	−	0.927367i	$-0.377934\pi$
$293$	12.8089	0.748305	0.374152	+	0.927367i	$0.377934\pi$
$294$	0	0
$295$	3.90068	0.227106
$296$	−6.98415	−0.405945
$297$	0	0
$298$	1.36164	0.0788777
$299$	−22.4232	−1.29677
$300$	0	0
$301$	−3.42103	−0.197185
$302$	−14.7242	−0.847280
$303$	0	0
$304$	4.99943	0.286737
$305$	0.734057	0.0420320
$306$	0	0
$307$	−14.0052	−0.799321	−0.399661	−	0.916663i	$-0.630872\pi$
$307$	−14.0052	−0.799321	−0.399661	+	0.916663i	$0.630872\pi$
$308$	2.28056	0.129947
$309$	0	0
$310$	0.319392	0.0181403
$311$	−4.10325	−0.232674	−0.116337	−	0.993210i	$-0.537115\pi$
$311$	−4.10325	−0.232674	−0.116337	+	0.993210i	$0.537115\pi$
$312$	0	0
$313$	21.5394	1.21748	0.608740	−	0.793370i	$-0.291675\pi$
$313$	21.5394	1.21748	0.608740	+	0.793370i	$0.291675\pi$
$314$	−14.3812	−0.811576
$315$	0	0
$316$	2.96872	0.167004
$317$	8.36917	0.470059	0.235030	−	0.971988i	$-0.424481\pi$
$317$	8.36917	0.470059	0.235030	+	0.971988i	$0.424481\pi$
$318$	0	0
$319$	1.88204	0.105374
$320$	0.785358	0.0439029
$321$	0	0
$322$	−23.8603	−1.32968
$323$	0.644551	0.0358638
$324$	0	0
$325$	−3.64023	−0.201923
$326$	−41.5650	−2.30207
$327$	0	0
$328$	2.60353	0.143756
$329$	−3.91689	−0.215945
$330$	0	0
$331$	−29.1194	−1.60055	−0.800274	−	0.599635i	$-0.795312\pi$
$331$	−29.1194	−1.60055	−0.800274	+	0.599635i	$0.795312\pi$
$332$	−14.1497	−0.776565
$333$	0	0
$334$	−33.8052	−1.84974
$335$	−1.30264	−0.0711711
$336$	0	0
$337$	3.36844	0.183491	0.0917453	−	0.995783i	$-0.470755\pi$
$337$	3.36844	0.183491	0.0917453	+	0.995783i	$0.470755\pi$
$338$	−0.436905	−0.0237645
$339$	0	0
$340$	−0.659883	−0.0357872
$341$	−0.183675	−0.00994653
$342$	0	0
$343$	−20.1326	−1.08706
$344$	−2.60702	−0.140561
$345$	0	0
$346$	−20.3257	−1.09272
$347$	−3.97272	−0.213267	−0.106633	−	0.994298i	$-0.534007\pi$
$347$	−3.97272	−0.213267	−0.106633	+	0.994298i	$0.534007\pi$
$348$	0	0
$349$	1.51616	0.0811581	0.0405790	−	0.999176i	$-0.487080\pi$
$349$	1.51616	0.0811581	0.0405790	+	0.999176i	$0.487080\pi$
$350$	−3.87353	−0.207049
$351$	0	0
$352$	5.29846	0.282409
$353$	−4.76876	−0.253816	−0.126908	−	0.991915i	$-0.540505\pi$
$353$	−4.76876	−0.253816	−0.126908	+	0.991915i	$0.540505\pi$
$354$	0	0
$355$	−10.5493	−0.559896
$356$	6.55105	0.347205
$357$	0	0
$358$	−11.7802	−0.622605
$359$	−0.542750	−0.0286453	−0.0143226	−	0.999897i	$-0.504559\pi$
$359$	−0.542750	−0.0286453	−0.0143226	+	0.999897i	$0.504559\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	7.55643	0.397157
$363$	0	0
$364$	−8.30176	−0.435130
$365$	−5.17599	−0.270924
$366$	0	0
$367$	−26.1792	−1.36654	−0.683271	−	0.730165i	$-0.739443\pi$
$367$	−26.1792	−1.36654	−0.683271	+	0.730165i	$0.739443\pi$
$368$	−30.7956	−1.60533
$369$	0	0
$370$	7.15433	0.371936
$371$	6.27042	0.325544
$372$	0	0
$373$	−12.7807	−0.661760	−0.330880	−	0.943673i	$-0.607345\pi$
$373$	−12.7807	−0.661760	−0.330880	+	0.943673i	$0.607345\pi$
$374$	1.12081	0.0579558
$375$	0	0
$376$	−2.98490	−0.153934
$377$	−6.85107	−0.352848
$378$	0	0
$379$	−26.0931	−1.34031	−0.670157	−	0.742219i	$-0.733773\pi$
$379$	−26.0931	−1.34031	−0.670157	+	0.742219i	$0.733773\pi$
$380$	−1.02379	−0.0525192
$381$	0	0
$382$	−2.39295	−0.122434
$383$	12.7812	0.653091	0.326545	−	0.945182i	$-0.394115\pi$
$383$	12.7812	0.653091	0.326545	+	0.945182i	$0.394115\pi$
$384$	0	0
$385$	2.22757	0.113528
$386$	−8.96773	−0.456445
$387$	0	0
$388$	6.38180	0.323987
$389$	8.97979	0.455293	0.227647	−	0.973744i	$-0.426897\pi$
$389$	8.97979	0.455293	0.227647	+	0.973744i	$0.426897\pi$
$390$	0	0
$391$	−3.97032	−0.200788
$392$	−3.45946	−0.174729
$393$	0	0
$394$	21.0403	1.06000
$395$	2.89974	0.145902
$396$	0	0
$397$	26.8417	1.34715	0.673573	−	0.739120i	$-0.264758\pi$
$397$	26.8417	1.34715	0.673573	+	0.739120i	$0.264758\pi$
$398$	23.1184	1.15882
$399$	0	0
$400$	−4.99943	−0.249972
$401$	9.25518	0.462182	0.231091	−	0.972932i	$-0.425771\pi$
$401$	9.25518	0.462182	0.231091	+	0.972932i	$0.425771\pi$
$402$	0	0
$403$	0.668617	0.0333062
$404$	16.7454	0.833114
$405$	0	0
$406$	−7.29016	−0.361804
$407$	−4.11428	−0.203937
$408$	0	0
$409$	26.0835	1.28975	0.644874	−	0.764289i	$-0.276910\pi$
$409$	26.0835	1.28975	0.644874	+	0.764289i	$0.276910\pi$
$410$	−2.66698	−0.131713
$411$	0	0
$412$	0.660875	0.0325590
$413$	8.68904	0.427560
$414$	0	0
$415$	−13.8209	−0.678443
$416$	−19.2876	−0.945652
$417$	0	0
$418$	1.73890	0.0850526
$419$	32.5181	1.58861	0.794307	−	0.607517i	$-0.207834\pi$
$419$	32.5181	1.58861	0.794307	+	0.607517i	$0.207834\pi$
$420$	0	0
$421$	3.16934	0.154464	0.0772320	−	0.997013i	$-0.475392\pi$
$421$	3.16934	0.154464	0.0772320	+	0.997013i	$0.475392\pi$
$422$	45.3621	2.20819
$423$	0	0
$424$	4.77843	0.232061
$425$	−0.644551	−0.0312653
$426$	0	0
$427$	1.63516	0.0791311
$428$	18.0084	0.870469
$429$	0	0
$430$	2.67055	0.128785
$431$	1.19700	0.0576575	0.0288288	−	0.999584i	$-0.490822\pi$
$431$	1.19700	0.0576575	0.0288288	+	0.999584i	$0.490822\pi$
$432$	0	0
$433$	23.4218	1.12558	0.562789	−	0.826601i	$-0.309728\pi$
$433$	23.4218	1.12558	0.562789	+	0.826601i	$0.309728\pi$
$434$	0.711469	0.0341516
$435$	0	0
$436$	19.8982	0.952950
$437$	−6.15983	−0.294664
$438$	0	0
$439$	−21.4961	−1.02595	−0.512975	−	0.858403i	$-0.671457\pi$
$439$	−21.4961	−1.02595	−0.512975	+	0.858403i	$0.671457\pi$
$440$	1.69754	0.0809270
$441$	0	0
$442$	−4.08001	−0.194066
$443$	−23.6133	−1.12190	−0.560950	−	0.827850i	$-0.689564\pi$
$443$	−23.6133	−1.12190	−0.560950	+	0.827850i	$0.689564\pi$
$444$	0	0
$445$	6.39884	0.303334
$446$	−24.1023	−1.14128
$447$	0	0
$448$	1.74944	0.0826533
$449$	5.35004	0.252484	0.126242	−	0.991999i	$-0.459708\pi$
$449$	5.35004	0.252484	0.126242	+	0.991999i	$0.459708\pi$
$450$	0	0
$451$	1.53371	0.0722196
$452$	6.02379	0.283335
$453$	0	0
$454$	−22.4499	−1.05363
$455$	−8.10886	−0.380149
$456$	0	0
$457$	39.7929	1.86143	0.930716	−	0.365742i	$-0.119185\pi$
$457$	39.7929	1.86143	0.930716	+	0.365742i	$0.119185\pi$
$458$	−13.8019	−0.644919
$459$	0	0
$460$	6.30635	0.294035
$461$	3.21566	0.149768	0.0748840	−	0.997192i	$-0.476141\pi$
$461$	3.21566	0.149768	0.0748840	+	0.997192i	$0.476141\pi$
$462$	0	0
$463$	16.7065	0.776416	0.388208	−	0.921572i	$-0.373094\pi$
$463$	16.7065	0.776416	0.388208	+	0.921572i	$0.373094\pi$
$464$	−9.40915	−0.436809
$465$	0	0
$466$	30.6507	1.41987
$467$	37.8565	1.75179	0.875894	−	0.482503i	$-0.160272\pi$
$467$	37.8565	1.75179	0.875894	+	0.482503i	$0.160272\pi$
$468$	0	0
$469$	−2.90173	−0.133990
$470$	3.05763	0.141038
$471$	0	0
$472$	6.62156	0.304782
$473$	−1.53577	−0.0706146
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	−1.46994	−0.0673744
$477$	0	0
$478$	−32.5017	−1.48659
$479$	16.6101	0.758936	0.379468	−	0.925205i	$-0.376107\pi$
$479$	16.6101	0.758936	0.379468	+	0.925205i	$0.376107\pi$
$480$	0	0
$481$	14.9769	0.682888
$482$	1.52552	0.0694853
$483$	0	0
$484$	1.02379	0.0465358
$485$	6.23352	0.283050
$486$	0	0
$487$	29.2975	1.32760	0.663798	−	0.747912i	$-0.268944\pi$
$487$	29.2975	1.32760	0.663798	+	0.747912i	$0.268944\pi$
$488$	1.24609	0.0564079
$489$	0	0
$490$	3.54376	0.160091
$491$	0.806717	0.0364066	0.0182033	−	0.999834i	$-0.494205\pi$
$491$	0.806717	0.0364066	0.0182033	+	0.999834i	$0.494205\pi$
$492$	0	0
$493$	−1.21307	−0.0546340
$494$	−6.33001	−0.284800
$495$	0	0
$496$	0.918269	0.0412315
$497$	−23.4992	−1.05408
$498$	0	0
$499$	−30.7212	−1.37527	−0.687634	−	0.726058i	$-0.741351\pi$
$499$	−30.7212	−1.37527	−0.687634	+	0.726058i	$0.741351\pi$
$500$	1.02379	0.0457852
$501$	0	0
$502$	−0.780451	−0.0348332
$503$	13.0527	0.581991	0.290995	−	0.956724i	$-0.406014\pi$
$503$	13.0527	0.581991	0.290995	+	0.956724i	$0.406014\pi$
$504$	0	0
$505$	16.3563	0.727846
$506$	−10.7113	−0.476178
$507$	0	0
$508$	16.3475	0.725302
$509$	22.6615	1.00445	0.502226	−	0.864736i	$-0.332515\pi$
$509$	22.6615	1.00445	0.502226	+	0.864736i	$0.332515\pi$
$510$	0	0
$511$	−11.5299	−0.510052
$512$	−9.51581	−0.420544
$513$	0	0
$514$	−13.6645	−0.602713
$515$	0.645519	0.0284450
$516$	0	0
$517$	−1.75837	−0.0773329
$518$	15.9368	0.700222
$519$	0	0
$520$	−6.17943	−0.270986
$521$	−19.0608	−0.835070	−0.417535	−	0.908661i	$-0.637106\pi$
$521$	−19.0608	−0.835070	−0.417535	+	0.908661i	$0.637106\pi$
$522$	0	0
$523$	15.1864	0.664054	0.332027	−	0.943270i	$-0.392268\pi$
$523$	15.1864	0.664054	0.332027	+	0.943270i	$0.392268\pi$
$524$	4.02526	0.175844
$525$	0	0
$526$	1.01763	0.0443709
$527$	0.118388	0.00515704
$528$	0	0
$529$	14.9435	0.649716
$530$	−4.89487	−0.212619
$531$	0	0
$532$	−2.28056	−0.0988748
$533$	−5.58305	−0.241829
$534$	0	0
$535$	17.5900	0.760481
$536$	−2.21129	−0.0955132
$537$	0	0
$538$	−6.24402	−0.269199
$539$	−2.03793	−0.0877798
$540$	0	0
$541$	41.1432	1.76888	0.884442	−	0.466650i	$-0.154539\pi$
$541$	41.1432	1.76888	0.884442	+	0.466650i	$0.154539\pi$
$542$	40.7714	1.75128
$543$	0	0
$544$	−3.41513	−0.146422
$545$	19.4359	0.832540
$546$	0	0
$547$	40.7714	1.74326	0.871630	−	0.490164i	$-0.163063\pi$
$547$	40.7714	1.74326	0.871630	+	0.490164i	$0.163063\pi$
$548$	−15.9119	−0.679722
$549$	0	0
$550$	−1.73890	−0.0741471
$551$	−1.88204	−0.0801778
$552$	0	0
$553$	6.45938	0.274681
$554$	33.5455	1.42521
$555$	0	0
$556$	−6.85835	−0.290859
$557$	14.2315	0.603009	0.301504	−	0.953465i	$-0.402511\pi$
$557$	14.2315	0.603009	0.301504	+	0.953465i	$0.402511\pi$
$558$	0	0
$559$	5.59054	0.236454
$560$	−11.1366	−0.470607
$561$	0	0
$562$	−10.1300	−0.427308
$563$	11.1511	0.469963	0.234982	−	0.972000i	$-0.424497\pi$
$563$	11.1511	0.469963	0.234982	+	0.972000i	$0.424497\pi$
$564$	0	0
$565$	5.88383	0.247534
$566$	26.9819	1.13414
$567$	0	0
$568$	−17.9078	−0.751393
$569$	−11.4022	−0.478005	−0.239003	−	0.971019i	$-0.576820\pi$
$569$	−11.4022	−0.478005	−0.239003	+	0.971019i	$0.576820\pi$
$570$	0	0
$571$	12.5111	0.523573	0.261787	−	0.965126i	$-0.415688\pi$
$571$	12.5111	0.523573	0.261787	+	0.965126i	$0.415688\pi$
$572$	−3.72682	−0.155826
$573$	0	0
$574$	−5.94088	−0.247967
$575$	6.15983	0.256883
$576$	0	0
$577$	7.95520	0.331179	0.165590	−	0.986195i	$-0.447047\pi$
$577$	7.95520	0.331179	0.165590	+	0.986195i	$0.447047\pi$
$578$	28.8390	1.19954
$579$	0	0
$580$	1.92681	0.0800066
$581$	−30.7871	−1.27726
$582$	0	0
$583$	2.81491	0.116582
$584$	−8.78645	−0.363586
$585$	0	0
$586$	−22.2735	−0.920109
$587$	−45.7680	−1.88905	−0.944524	−	0.328443i	$-0.893476\pi$
$587$	−45.7680	−1.88905	−0.944524	+	0.328443i	$0.893476\pi$
$588$	0	0
$589$	0.183675	0.00756818
$590$	−6.78291	−0.279248
$591$	0	0
$592$	20.5691	0.845383
$593$	26.3613	1.08253	0.541264	−	0.840853i	$-0.317946\pi$
$593$	26.3613	1.08253	0.541264	+	0.840853i	$0.317946\pi$
$594$	0	0
$595$	−1.43578	−0.0588614
$596$	−0.801672	−0.0328378
$597$	0	0
$598$	38.9917	1.59449
$599$	−17.3368	−0.708361	−0.354180	−	0.935177i	$-0.615240\pi$
$599$	−17.3368	−0.708361	−0.354180	+	0.935177i	$0.615240\pi$
$600$	0	0
$601$	−15.7946	−0.644274	−0.322137	−	0.946693i	$-0.604401\pi$
$601$	−15.7946	−0.644274	−0.322137	+	0.946693i	$0.604401\pi$
$602$	5.94884	0.242457
$603$	0	0
$604$	8.66891	0.352733
$605$	1.00000	0.0406558
$606$	0	0
$607$	−1.80754	−0.0733659	−0.0366829	−	0.999327i	$-0.511679\pi$
$607$	−1.80754	−0.0733659	−0.0366829	+	0.999327i	$0.511679\pi$
$608$	−5.29846	−0.214881
$609$	0	0
$610$	−1.27645	−0.0516821
$611$	6.40086	0.258951
$612$	0	0
$613$	10.9246	0.441239	0.220619	−	0.975360i	$-0.429192\pi$
$613$	10.9246	0.441239	0.220619	+	0.975360i	$0.429192\pi$
$614$	24.3538	0.982838
$615$	0	0
$616$	3.78139	0.152357
$617$	2.68742	0.108191	0.0540957	−	0.998536i	$-0.482772\pi$
$617$	2.68742	0.108191	0.0540957	+	0.998536i	$0.482772\pi$
$618$	0	0
$619$	18.2439	0.733284	0.366642	−	0.930362i	$-0.380507\pi$
$619$	18.2439	0.733284	0.366642	+	0.930362i	$0.380507\pi$
$620$	−0.188044	−0.00755202
$621$	0	0
$622$	7.13515	0.286094
$623$	14.2539	0.571069
$624$	0	0
$625$	1.00000	0.0400000
$626$	−37.4550	−1.49700
$627$	0	0
$628$	8.46697	0.337869
$629$	2.65186	0.105737
$630$	0	0
$631$	9.12012	0.363066	0.181533	−	0.983385i	$-0.441894\pi$
$631$	9.12012	0.363066	0.181533	+	0.983385i	$0.441894\pi$
$632$	4.92243	0.195804
$633$	0	0
$634$	−14.5532	−0.577981
$635$	15.9676	0.633657
$636$	0	0
$637$	7.41852	0.293932
$638$	−3.27269	−0.129567
$639$	0	0
$640$	−11.9626	−0.472862
$641$	2.90773	0.114848	0.0574241	−	0.998350i	$-0.481711\pi$
$641$	2.90773	0.114848	0.0574241	+	0.998350i	$0.481711\pi$
$642$	0	0
$643$	−3.35603	−0.132349	−0.0661745	−	0.997808i	$-0.521079\pi$
$643$	−3.35603	−0.132349	−0.0661745	+	0.997808i	$0.521079\pi$
$644$	14.0479	0.553563
$645$	0	0
$646$	−1.12081	−0.0440978
$647$	−26.5428	−1.04351	−0.521753	−	0.853096i	$-0.674722\pi$
$647$	−26.5428	−1.04351	−0.521753	+	0.853096i	$0.674722\pi$
$648$	0	0
$649$	3.90068	0.153115
$650$	6.33001	0.248283
$651$	0	0
$652$	24.4716	0.958381
$653$	−28.6844	−1.12251	−0.561253	−	0.827644i	$-0.689681\pi$
$653$	−28.6844	−1.12251	−0.561253	+	0.827644i	$0.689681\pi$
$654$	0	0
$655$	3.93173	0.153626
$656$	−7.66768	−0.299373
$657$	0	0
$658$	6.81110	0.265524
$659$	39.0719	1.52203	0.761013	−	0.648737i	$-0.224703\pi$
$659$	39.0719	1.52203	0.761013	+	0.648737i	$0.224703\pi$
$660$	0	0
$661$	−21.1901	−0.824199	−0.412099	−	0.911139i	$-0.635204\pi$
$661$	−21.1901	−0.824199	−0.412099	+	0.911139i	$0.635204\pi$
$662$	50.6358	1.96802
$663$	0	0
$664$	−23.4616	−0.910486
$665$	−2.22757	−0.0863815
$666$	0	0
$667$	11.5931	0.448885
$668$	19.9030	0.770069
$669$	0	0
$670$	2.26517	0.0875113
$671$	0.734057	0.0283379
$672$	0	0
$673$	−24.6931	−0.951849	−0.475924	−	0.879486i	$-0.657886\pi$
$673$	−24.6931	−0.951849	−0.475924	+	0.879486i	$0.657886\pi$
$674$	−5.85739	−0.225618
$675$	0	0
$676$	0.257230	0.00989345
$677$	−5.41516	−0.208122	−0.104061	−	0.994571i	$-0.533184\pi$
$677$	−5.41516	−0.208122	−0.104061	+	0.994571i	$0.533184\pi$
$678$	0	0
$679$	13.8856	0.532881
$680$	−1.09415	−0.0419588
$681$	0	0
$682$	0.319392	0.0122302
$683$	13.7885	0.527604	0.263802	−	0.964577i	$-0.415023\pi$
$683$	13.7885	0.527604	0.263802	+	0.964577i	$0.415023\pi$
$684$	0	0
$685$	−15.5422	−0.593836
$686$	35.0087	1.33664
$687$	0	0
$688$	7.67796	0.292719
$689$	−10.2469	−0.390377
$690$	0	0
$691$	−31.5750	−1.20117	−0.600586	−	0.799560i	$-0.705066\pi$
$691$	−31.5750	−1.20117	−0.600586	+	0.799560i	$0.705066\pi$
$692$	11.9668	0.454911
$693$	0	0
$694$	6.90818	0.262231
$695$	−6.69900	−0.254108
$696$	0	0
$697$	−0.988554	−0.0374442
$698$	−2.63645	−0.0997912
$699$	0	0
$700$	2.28056	0.0861971
$701$	17.4459	0.658924	0.329462	−	0.944169i	$-0.393133\pi$
$701$	17.4459	0.658924	0.329462	+	0.944169i	$0.393133\pi$
$702$	0	0
$703$	4.11428	0.155173
$704$	0.785358	0.0295993
$705$	0	0
$706$	8.29242	0.312089
$707$	36.4348	1.37027
$708$	0	0
$709$	−38.7266	−1.45441	−0.727205	−	0.686421i	$-0.759181\pi$
$709$	−38.7266	−1.45441	−0.727205	+	0.686421i	$0.759181\pi$
$710$	18.3441	0.688443
$711$	0	0
$712$	10.8623	0.407081
$713$	−1.13140	−0.0423714
$714$	0	0
$715$	−3.64023	−0.136137
$716$	6.93567	0.259198
$717$	0	0
$718$	0.943791	0.0352220
$719$	28.9402	1.07929	0.539644	−	0.841894i	$-0.318559\pi$
$719$	28.9402	1.07929	0.539644	+	0.841894i	$0.318559\pi$
$720$	0	0
$721$	1.43794	0.0535517
$722$	−1.73890	−0.0647153
$723$	0	0
$724$	−4.44888	−0.165341
$725$	1.88204	0.0698973
$726$	0	0
$727$	−0.331738	−0.0123035	−0.00615174	−	0.999981i	$-0.501958\pi$
$727$	−0.331738	−0.0123035	−0.00615174	+	0.999981i	$0.501958\pi$
$728$	−13.7651	−0.510169
$729$	0	0
$730$	9.00056	0.333126
$731$	0.989879	0.0366120
$732$	0	0
$733$	−30.5241	−1.12743	−0.563716	−	0.825969i	$-0.690629\pi$
$733$	−30.5241	−1.12743	−0.563716	+	0.825969i	$0.690629\pi$
$734$	45.5231	1.68029
$735$	0	0
$736$	32.6376	1.20304
$737$	−1.30264	−0.0479835
$738$	0	0
$739$	−8.65314	−0.318311	−0.159155	−	0.987254i	$-0.550877\pi$
$739$	−8.65314	−0.318311	−0.159155	+	0.987254i	$0.550877\pi$
$740$	−4.21215	−0.154842
$741$	0	0
$742$	−10.9037	−0.400286
$743$	−33.9919	−1.24704	−0.623521	−	0.781807i	$-0.714298\pi$
$743$	−33.9919	−1.24704	−0.623521	+	0.781807i	$0.714298\pi$
$744$	0	0
$745$	−0.783045	−0.0286886
$746$	22.2244	0.813694
$747$	0	0
$748$	−0.659883	−0.0241277
$749$	39.1829	1.43171
$750$	0	0
$751$	−50.2581	−1.83394	−0.916972	−	0.398951i	$-0.869374\pi$
$751$	−50.2581	−1.83394	−0.916972	+	0.398951i	$0.869374\pi$
$752$	8.79085	0.320569
$753$	0	0
$754$	11.9133	0.433859
$755$	8.46749	0.308164
$756$	0	0
$757$	10.7948	0.392344	0.196172	−	0.980570i	$-0.437149\pi$
$757$	10.7948	0.392344	0.196172	+	0.980570i	$0.437149\pi$
$758$	45.3734	1.64804
$759$	0	0
$760$	−1.69754	−0.0615762
$761$	−15.5661	−0.564272	−0.282136	−	0.959374i	$-0.591043\pi$
$761$	−15.5661	−0.564272	−0.282136	+	0.959374i	$0.591043\pi$
$762$	0	0
$763$	43.2947	1.56737
$764$	1.40886	0.0509707
$765$	0	0
$766$	−22.2254	−0.803035
$767$	−14.1994	−0.512709
$768$	0	0
$769$	19.6520	0.708671	0.354335	−	0.935118i	$-0.384707\pi$
$769$	19.6520	0.708671	0.354335	+	0.935118i	$0.384707\pi$
$770$	−3.87353	−0.139592
$771$	0	0
$772$	5.27979	0.190024
$773$	41.0208	1.47541	0.737707	−	0.675121i	$-0.235908\pi$
$773$	41.0208	1.47541	0.737707	+	0.675121i	$0.235908\pi$
$774$	0	0
$775$	−0.183675	−0.00659778
$776$	10.5817	0.379859
$777$	0	0
$778$	−15.6150	−0.559825
$779$	−1.53371	−0.0549509
$780$	0	0
$781$	−10.5493	−0.377482
$782$	6.90401	0.246887
$783$	0	0
$784$	10.1885	0.363874
$785$	8.27024	0.295178
$786$	0	0
$787$	25.2114	0.898688	0.449344	−	0.893359i	$-0.351658\pi$
$787$	25.2114	0.898688	0.449344	+	0.893359i	$0.351658\pi$
$788$	−12.3876	−0.441289
$789$	0	0
$790$	−5.04237	−0.179400
$791$	13.1066	0.466018
$792$	0	0
$793$	−2.67213	−0.0948902
$794$	−46.6752	−1.65644
$795$	0	0
$796$	−13.6111	−0.482432
$797$	23.4053	0.829057	0.414528	−	0.910036i	$-0.363947\pi$
$797$	23.4053	0.829057	0.414528	+	0.910036i	$0.363947\pi$
$798$	0	0
$799$	1.13336	0.0400953
$800$	5.29846	0.187329
$801$	0	0
$802$	−16.0939	−0.568294
$803$	−5.17599	−0.182657
$804$	0	0
$805$	13.7215	0.483618
$806$	−1.16266	−0.0409530
$807$	0	0
$808$	27.7655	0.976786
$809$	25.3235	0.890325	0.445163	−	0.895450i	$-0.353146\pi$
$809$	25.3235	0.890325	0.445163	+	0.895450i	$0.353146\pi$
$810$	0	0
$811$	−1.24610	−0.0437566	−0.0218783	−	0.999761i	$-0.506965\pi$
$811$	−1.24610	−0.0437566	−0.0218783	+	0.999761i	$0.506965\pi$
$812$	4.29211	0.150624
$813$	0	0
$814$	7.15433	0.250759
$815$	23.9030	0.837285
$816$	0	0
$817$	1.53577	0.0537296
$818$	−45.3567	−1.58586
$819$	0	0
$820$	1.57019	0.0548335
$821$	−20.4113	−0.712359	−0.356180	−	0.934418i	$-0.615921\pi$
$821$	−20.4113	−0.712359	−0.356180	+	0.934418i	$0.615921\pi$
$822$	0	0
$823$	−15.4655	−0.539093	−0.269546	−	0.962987i	$-0.586874\pi$
$823$	−15.4655	−0.539093	−0.269546	+	0.962987i	$0.586874\pi$
$824$	1.09579	0.0381738
$825$	0	0
$826$	−15.1094	−0.525723
$827$	46.2008	1.60656	0.803280	−	0.595602i	$-0.203087\pi$
$827$	46.2008	1.60656	0.803280	+	0.595602i	$0.203087\pi$
$828$	0	0
$829$	2.91356	0.101192	0.0505961	−	0.998719i	$-0.483888\pi$
$829$	2.91356	0.101192	0.0505961	+	0.998719i	$0.483888\pi$
$830$	24.0333	0.834207
$831$	0	0
$832$	−2.85888	−0.0991139
$833$	1.31355	0.0455117
$834$	0	0
$835$	19.4405	0.672767
$836$	−1.02379	−0.0354084
$837$	0	0
$838$	−56.5459	−1.95334
$839$	−28.2887	−0.976633	−0.488317	−	0.872667i	$-0.662389\pi$
$839$	−28.2887	−0.976633	−0.488317	+	0.872667i	$0.662389\pi$
$840$	0	0
$841$	−25.4579	−0.877859
$842$	−5.51117	−0.189928
$843$	0	0
$844$	−26.7072	−0.919299
$845$	0.251253	0.00864337
$846$	0	0
$847$	2.22757	0.0765402
$848$	−14.0730	−0.483268
$849$	0	0
$850$	1.12081	0.0384435
$851$	−25.3432	−0.868755
$852$	0	0
$853$	41.9234	1.43543	0.717715	−	0.696337i	$-0.245188\pi$
$853$	41.9234	1.43543	0.717715	+	0.696337i	$0.245188\pi$
$854$	−2.84339	−0.0972989
$855$	0	0
$856$	29.8597	1.02058
$857$	−52.4352	−1.79115	−0.895576	−	0.444909i	$-0.853236\pi$
$857$	−52.4352	−1.79115	−0.895576	+	0.444909i	$0.853236\pi$
$858$	0	0
$859$	−21.2305	−0.724374	−0.362187	−	0.932105i	$-0.617970\pi$
$859$	−21.2305	−0.724374	−0.362187	+	0.932105i	$0.617970\pi$
$860$	−1.57230	−0.0536149
$861$	0	0
$862$	−2.08147	−0.0708952
$863$	−1.75582	−0.0597690	−0.0298845	−	0.999553i	$-0.509514\pi$
$863$	−1.75582	−0.0597690	−0.0298845	+	0.999553i	$0.509514\pi$
$864$	0	0
$865$	11.6888	0.397431
$866$	−40.7282	−1.38400
$867$	0	0
$868$	−0.418881	−0.0142177
$869$	2.89974	0.0983670
$870$	0	0
$871$	4.74192	0.160674
$872$	32.9931	1.11729
$873$	0	0
$874$	10.7113	0.362317
$875$	2.22757	0.0753056
$876$	0	0
$877$	27.7873	0.938309	0.469155	−	0.883116i	$-0.344559\pi$
$877$	27.7873	0.938309	0.469155	+	0.883116i	$0.344559\pi$
$878$	37.3796	1.26150
$879$	0	0
$880$	−4.99943	−0.168531
$881$	−38.8712	−1.30960	−0.654802	−	0.755800i	$-0.727248\pi$
$881$	−38.8712	−1.30960	−0.654802	+	0.755800i	$0.727248\pi$
$882$	0	0
$883$	−1.20467	−0.0405404	−0.0202702	−	0.999795i	$-0.506453\pi$
$883$	−1.20467	−0.0405404	−0.0202702	+	0.999795i	$0.506453\pi$
$884$	2.40212	0.0807922
$885$	0	0
$886$	41.0612	1.37948
$887$	44.0947	1.48056	0.740278	−	0.672301i	$-0.234694\pi$
$887$	44.0947	1.48056	0.740278	+	0.672301i	$0.234694\pi$
$888$	0	0
$889$	35.5691	1.19295
$890$	−11.1270	−0.372977
$891$	0	0
$892$	14.1903	0.475127
$893$	1.75837	0.0588415
$894$	0	0
$895$	6.77452	0.226447
$896$	−26.6475	−0.890230
$897$	0	0
$898$	−9.30320	−0.310452
$899$	−0.345683	−0.0115292
$900$	0	0
$901$	−1.81436	−0.0604450
$902$	−2.66698	−0.0888006
$903$	0	0
$904$	9.98803	0.332197
$905$	−4.34551	−0.144450
$906$	0	0
$907$	36.2297	1.20299	0.601493	−	0.798878i	$-0.294573\pi$
$907$	36.2297	1.20299	0.601493	+	0.798878i	$0.294573\pi$
$908$	13.2175	0.438638
$909$	0	0
$910$	14.1005	0.467428
$911$	5.41417	0.179380	0.0896898	−	0.995970i	$-0.471412\pi$
$911$	5.41417	0.179380	0.0896898	+	0.995970i	$0.471412\pi$
$912$	0	0
$913$	−13.8209	−0.457406
$914$	−69.1960	−2.28880
$915$	0	0
$916$	8.12591	0.268488
$917$	8.75822	0.289222
$918$	0	0
$919$	54.0931	1.78437	0.892183	−	0.451675i	$-0.149173\pi$
$919$	54.0931	1.78437	0.892183	+	0.451675i	$0.149173\pi$
$920$	10.4565	0.344742
$921$	0	0
$922$	−5.59172	−0.184153
$923$	38.4017	1.26401
$924$	0	0
$925$	−4.11428	−0.135277
$926$	−29.0510	−0.954674
$927$	0	0
$928$	9.97193	0.327345
$929$	−52.0785	−1.70864	−0.854320	−	0.519747i	$-0.826026\pi$
$929$	−52.0785	−1.70864	−0.854320	+	0.519747i	$0.826026\pi$
$930$	0	0
$931$	2.03793	0.0667904
$932$	−18.0457	−0.591108
$933$	0	0
$934$	−65.8288	−2.15398
$935$	−0.644551	−0.0210791
$936$	0	0
$937$	35.9861	1.17561	0.587807	−	0.809002i	$-0.299992\pi$
$937$	35.9861	1.17561	0.587807	+	0.809002i	$0.299992\pi$
$938$	5.04584	0.164752
$939$	0	0
$940$	−1.80020	−0.0587159
$941$	−11.9972	−0.391098	−0.195549	−	0.980694i	$-0.562649\pi$
$941$	−11.9972	−0.391098	−0.195549	+	0.980694i	$0.562649\pi$
$942$	0	0
$943$	9.44739	0.307649
$944$	−19.5012	−0.634709
$945$	0	0
$946$	2.67055	0.0868271
$947$	−37.1663	−1.20774	−0.603871	−	0.797082i	$-0.706376\pi$
$947$	−37.1663	−1.20774	−0.603871	+	0.797082i	$0.706376\pi$
$948$	0	0
$949$	18.8418	0.611630
$950$	1.73890	0.0564175
$951$	0	0
$952$	−2.43730	−0.0789933
$953$	−16.5500	−0.536106	−0.268053	−	0.963404i	$-0.586380\pi$
$953$	−16.5500	−0.536106	−0.268053	+	0.963404i	$0.586380\pi$
$954$	0	0
$955$	1.37612	0.0445303
$956$	19.1355	0.618886
$957$	0	0
$958$	−28.8834	−0.933180
$959$	−34.6213	−1.11798
$960$	0	0
$961$	−30.9663	−0.998912
$962$	−26.0434	−0.839673
$963$	0	0
$964$	−0.898154	−0.0289276
$965$	5.15711	0.166013
$966$	0	0
$967$	37.7443	1.21377	0.606887	−	0.794788i	$-0.292418\pi$
$967$	37.7443	1.21377	0.606887	+	0.794788i	$0.292418\pi$
$968$	1.69754	0.0545610
$969$	0	0
$970$	−10.8395	−0.348035
$971$	40.2912	1.29301	0.646504	−	0.762911i	$-0.276231\pi$
$971$	40.2912	1.29301	0.646504	+	0.762911i	$0.276231\pi$
$972$	0	0
$973$	−14.9225	−0.478393
$974$	−50.9455	−1.63240
$975$	0	0
$976$	−3.66987	−0.117470
$977$	−1.24031	−0.0396812	−0.0198406	−	0.999803i	$-0.506316\pi$
$977$	−1.24031	−0.0396812	−0.0198406	+	0.999803i	$0.506316\pi$
$978$	0	0
$979$	6.39884	0.204508
$980$	−2.08641	−0.0666478
$981$	0	0
$982$	−1.40280	−0.0447653
$983$	−8.75471	−0.279232	−0.139616	−	0.990206i	$-0.544587\pi$
$983$	−8.75471	−0.279232	−0.139616	+	0.990206i	$0.544587\pi$
$984$	0	0
$985$	−12.0998	−0.385530
$986$	2.10942	0.0671775
$987$	0	0
$988$	3.72682	0.118566
$989$	−9.46005	−0.300812
$990$	0	0
$991$	49.7066	1.57898	0.789491	−	0.613763i	$-0.210345\pi$
$991$	49.7066	1.57898	0.789491	+	0.613763i	$0.210345\pi$
$992$	−0.973192	−0.0308989
$993$	0	0
$994$	40.8629	1.29609
$995$	−13.2948	−0.421474
$996$	0	0
$997$	46.3005	1.46635	0.733176	−	0.680039i	$-0.238037\pi$
$997$	46.3005	1.46635	0.733176	+	0.680039i	$0.238037\pi$
$998$	53.4211	1.69102
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9405.2.a.z.1.1		6
3.2	odd	2		1045.2.a.f.1.6	✓	6
15.14	odd	2		5225.2.a.l.1.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.f.1.6	✓	6	3.2	odd	2
5225.2.a.l.1.1		6	15.14	odd	2
9405.2.a.z.1.1		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)

\(f(q)\)	\(=\)	\(q-1.73890 q^{2} +1.02379 q^{4} +1.00000 q^{5} +2.22757 q^{7} +1.69754 q^{8} +O(q^{10})\)	\(q-1.73890 q^{2} +1.02379 q^{4} +1.00000 q^{5} +2.22757 q^{7} +1.69754 q^{8} -1.73890 q^{10} +1.00000 q^{11} -3.64023 q^{13} -3.87353 q^{14} -4.99943 q^{16} -0.644551 q^{17} -1.00000 q^{19} +1.02379 q^{20} -1.73890 q^{22} +6.15983 q^{23} +1.00000 q^{25} +6.33001 q^{26} +2.28056 q^{28} +1.88204 q^{29} -0.183675 q^{31} +5.29846 q^{32} +1.12081 q^{34} +2.22757 q^{35} -4.11428 q^{37} +1.73890 q^{38} +1.69754 q^{40} +1.53371 q^{41} -1.53577 q^{43} +1.02379 q^{44} -10.7113 q^{46} -1.75837 q^{47} -2.03793 q^{49} -1.73890 q^{50} -3.72682 q^{52} +2.81491 q^{53} +1.00000 q^{55} +3.78139 q^{56} -3.27269 q^{58} +3.90068 q^{59} +0.734057 q^{61} +0.319392 q^{62} +0.785358 q^{64} -3.64023 q^{65} -1.30264 q^{67} -0.659883 q^{68} -3.87353 q^{70} -10.5493 q^{71} -5.17599 q^{73} +7.15433 q^{74} -1.02379 q^{76} +2.22757 q^{77} +2.89974 q^{79} -4.99943 q^{80} -2.66698 q^{82} -13.8209 q^{83} -0.644551 q^{85} +2.67055 q^{86} +1.69754 q^{88} +6.39884 q^{89} -8.10886 q^{91} +6.30635 q^{92} +3.05763 q^{94} -1.00000 q^{95} +6.23352 q^{97} +3.54376 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 5 q^{7} + 12 q^{8}+O(q^{10}) \) 6 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 5 * q^7 + 12 * q^8	\( 6 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 5 q^{7} + 12 q^{8} + 2 q^{10} + 6 q^{11} - 5 q^{13} + 8 q^{14} + 4 q^{16} - q^{17} - 6 q^{19} + 4 q^{20} + 2 q^{22} - 4 q^{23} + 6 q^{25} + 14 q^{26} + 10 q^{28} + 9 q^{29} - 21 q^{31} + q^{32} + 5 q^{35} - 3 q^{37} - 2 q^{38} + 12 q^{40} + 23 q^{41} + 7 q^{43} + 4 q^{44} - 12 q^{46} + 18 q^{47} - 3 q^{49} + 2 q^{50} + 13 q^{52} + 17 q^{53} + 6 q^{55} + 2 q^{56} + 23 q^{58} + 29 q^{59} + 17 q^{61} - 2 q^{62} - 18 q^{64} - 5 q^{65} + 8 q^{67} + q^{68} + 8 q^{70} + 12 q^{71} + 2 q^{73} + 37 q^{74} - 4 q^{76} + 5 q^{77} + 3 q^{79} + 4 q^{80} + 24 q^{82} + 11 q^{83} - q^{85} + 12 q^{86} + 12 q^{88} + 22 q^{89} - 18 q^{91} + 15 q^{92} + 22 q^{94} - 6 q^{95} - 2 q^{97} + q^{98}+O(q^{100}) \) 6 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 5 * q^7 + 12 * q^8 + 2 * q^10 + 6 * q^11 - 5 * q^13 + 8 * q^14 + 4 * q^16 - q^17 - 6 * q^19 + 4 * q^20 + 2 * q^22 - 4 * q^23 + 6 * q^25 + 14 * q^26 + 10 * q^28 + 9 * q^29 - 21 * q^31 + q^32 + 5 * q^35 - 3 * q^37 - 2 * q^38 + 12 * q^40 + 23 * q^41 + 7 * q^43 + 4 * q^44 - 12 * q^46 + 18 * q^47 - 3 * q^49 + 2 * q^50 + 13 * q^52 + 17 * q^53 + 6 * q^55 + 2 * q^56 + 23 * q^58 + 29 * q^59 + 17 * q^61 - 2 * q^62 - 18 * q^64 - 5 * q^65 + 8 * q^67 + q^68 + 8 * q^70 + 12 * q^71 + 2 * q^73 + 37 * q^74 - 4 * q^76 + 5 * q^77 + 3 * q^79 + 4 * q^80 + 24 * q^82 + 11 * q^83 - q^85 + 12 * q^86 + 12 * q^88 + 22 * q^89 - 18 * q^91 + 15 * q^92 + 22 * q^94 - 6 * q^95 - 2 * q^97 + q^98

Introduction

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