LMFDB - Embedded newform 405.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [405,2,Mod(1,405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(405, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("405.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 405 = 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	405.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:	$3.23394128186$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 45)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.571993$ of defining polynomial
Character	$\chi$	$=$	405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.571993 q^{2} -1.67282 q^{4} +1.00000 q^{5} +1.42801 q^{7} -2.10083 q^{8} +O(q^{10})$	\(q+0.571993 q^{2} -1.67282 q^{4} +1.00000 q^{5} +1.42801 q^{7} -2.10083 q^{8} +0.571993 q^{10} +2.67282 q^{11} +4.67282 q^{13} +0.816810 q^{14} +2.14399 q^{16} +2.67282 q^{17} +4.67282 q^{19} -1.67282 q^{20} +1.52884 q^{22} -5.91764 q^{23} +1.00000 q^{25} +2.67282 q^{26} -2.38880 q^{28} -9.48963 q^{29} +6.96080 q^{31} +5.42801 q^{32} +1.52884 q^{34} +1.42801 q^{35} -1.81681 q^{37} +2.67282 q^{38} -2.10083 q^{40} -1.47116 q^{41} +0.471163 q^{43} -4.47116 q^{44} -3.38485 q^{46} +6.95684 q^{47} -4.96080 q^{49} +0.571993 q^{50} -7.81681 q^{52} -1.14399 q^{53} +2.67282 q^{55} -3.00000 q^{56} -5.42801 q^{58} -1.14399 q^{59} -2.52884 q^{61} +3.98153 q^{62} -1.18319 q^{64} +4.67282 q^{65} -6.59046 q^{67} -4.47116 q^{68} +0.816810 q^{70} -12.8745 q^{71} -1.71203 q^{73} -1.03920 q^{74} -7.81681 q^{76} +3.81681 q^{77} -0.287973 q^{79} +2.14399 q^{80} -0.841495 q^{82} -4.28402 q^{83} +2.67282 q^{85} +0.269502 q^{86} -5.61515 q^{88} -3.00000 q^{89} +6.67282 q^{91} +9.89917 q^{92} +3.97927 q^{94} +4.67282 q^{95} +7.83528 q^{97} -2.83754 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 5 q^{4} + 3 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + q^2 + 5 * q^4 + 3 * q^5 + 5 * q^7 + 3 * q^8	$ 3 q + q^{2} + 5 q^{4} + 3 q^{5} + 5 q^{7} + 3 q^{8} + q^{10} - 2 q^{11} + 4 q^{13} - 9 q^{14} + 5 q^{16} - 2 q^{17} + 4 q^{19} + 5 q^{20} - 4 q^{22} + 3 q^{23} + 3 q^{25} - 2 q^{26} + 5 q^{28} - 7 q^{29} + 8 q^{31} + 17 q^{32} - 4 q^{34} + 5 q^{35} + 6 q^{37} - 2 q^{38} + 3 q^{40} - 13 q^{41} + 10 q^{43} - 22 q^{44} - 3 q^{46} + 13 q^{47} - 2 q^{49} + q^{50} - 12 q^{52} - 2 q^{53} - 2 q^{55} - 9 q^{56} - 17 q^{58} - 2 q^{59} + q^{61} + 42 q^{62} - 15 q^{64} + 4 q^{65} + 11 q^{67} - 22 q^{68} - 9 q^{70} - 10 q^{71} - 8 q^{73} - 16 q^{74} - 12 q^{76} + 2 q^{79} + 5 q^{80} - 29 q^{82} - 15 q^{83} - 2 q^{85} + 28 q^{86} - 24 q^{88} - 9 q^{89} + 10 q^{91} + 39 q^{92} - 31 q^{94} + 4 q^{95} - 18 q^{97} - 40 q^{98}+O(q^{100}) $ 3 * q + q^2 + 5 * q^4 + 3 * q^5 + 5 * q^7 + 3 * q^8 + q^10 - 2 * q^11 + 4 * q^13 - 9 * q^14 + 5 * q^16 - 2 * q^17 + 4 * q^19 + 5 * q^20 - 4 * q^22 + 3 * q^23 + 3 * q^25 - 2 * q^26 + 5 * q^28 - 7 * q^29 + 8 * q^31 + 17 * q^32 - 4 * q^34 + 5 * q^35 + 6 * q^37 - 2 * q^38 + 3 * q^40 - 13 * q^41 + 10 * q^43 - 22 * q^44 - 3 * q^46 + 13 * q^47 - 2 * q^49 + q^50 - 12 * q^52 - 2 * q^53 - 2 * q^55 - 9 * q^56 - 17 * q^58 - 2 * q^59 + q^61 + 42 * q^62 - 15 * q^64 + 4 * q^65 + 11 * q^67 - 22 * q^68 - 9 * q^70 - 10 * q^71 - 8 * q^73 - 16 * q^74 - 12 * q^76 + 2 * q^79 + 5 * q^80 - 29 * q^82 - 15 * q^83 - 2 * q^85 + 28 * q^86 - 24 * q^88 - 9 * q^89 + 10 * q^91 + 39 * q^92 - 31 * q^94 + 4 * q^95 - 18 * q^97 - 40 * 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 $

$2$	0.571993	0.404460	0.202230	−	0.979338i	$-0.435181\pi$
$2$	0.571993	0.404460	0.202230	+	0.979338i	$0.435181\pi$
$3$	0	0
$3$	0	0
$4$	−1.67282	−0.836412
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.42801	0.539736	0.269868	−	0.962897i	$-0.413020\pi$
$7$	1.42801	0.539736	0.269868	+	0.962897i	$0.413020\pi$
$8$	−2.10083	−0.742756
$9$	0	0
$10$	0.571993	0.180880
$11$	2.67282	0.805887	0.402943	−	0.915225i	$-0.367987\pi$
$11$	2.67282	0.805887	0.402943	+	0.915225i	$0.367987\pi$
$12$	0	0
$13$	4.67282	1.29601	0.648004	−	0.761637i	$-0.275604\pi$
$13$	4.67282	1.29601	0.648004	+	0.761637i	$0.275604\pi$
$14$	0.816810	0.218302
$15$	0	0
$16$	2.14399	0.535997
$17$	2.67282	0.648255	0.324127	−	0.946013i	$-0.394929\pi$
$17$	2.67282	0.648255	0.324127	+	0.946013i	$0.394929\pi$
$18$	0	0
$19$	4.67282	1.07202	0.536010	−	0.844212i	$-0.319931\pi$
$19$	4.67282	1.07202	0.536010	+	0.844212i	$0.319931\pi$
$20$	−1.67282	−0.374055
$21$	0	0
$22$	1.52884	0.325949
$23$	−5.91764	−1.23391	−0.616957	−	0.786997i	$-0.711635\pi$
$23$	−5.91764	−1.23391	−0.616957	+	0.786997i	$0.711635\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.67282	0.524184
$27$	0	0
$28$	−2.38880	−0.451441
$29$	−9.48963	−1.76218	−0.881090	−	0.472948i	$-0.843190\pi$
$29$	−9.48963	−1.76218	−0.881090	+	0.472948i	$0.843190\pi$
$30$	0	0
$31$	6.96080	1.25020	0.625098	−	0.780546i	$-0.285059\pi$
$31$	6.96080	1.25020	0.625098	+	0.780546i	$0.285059\pi$
$32$	5.42801	0.959545
$33$	0	0
$34$	1.52884	0.262193
$35$	1.42801	0.241377
$36$	0	0
$37$	−1.81681	−0.298682	−0.149341	−	0.988786i	$-0.547715\pi$
$37$	−1.81681	−0.298682	−0.149341	+	0.988786i	$0.547715\pi$
$38$	2.67282	0.433589
$39$	0	0
$40$	−2.10083	−0.332170
$41$	−1.47116	−0.229757	−0.114879	−	0.993380i	$-0.536648\pi$
$41$	−1.47116	−0.229757	−0.114879	+	0.993380i	$0.536648\pi$
$42$	0	0
$43$	0.471163	0.0718517	0.0359258	−	0.999354i	$-0.488562\pi$
$43$	0.471163	0.0718517	0.0359258	+	0.999354i	$0.488562\pi$
$44$	−4.47116	−0.674053
$45$	0	0
$46$	−3.38485	−0.499069
$47$	6.95684	1.01476	0.507380	−	0.861722i	$-0.330614\pi$
$47$	6.95684	1.01476	0.507380	+	0.861722i	$0.330614\pi$
$48$	0	0
$49$	−4.96080	−0.708685
$50$	0.571993	0.0808921
$51$	0	0
$52$	−7.81681	−1.08400
$53$	−1.14399	−0.157139	−0.0785693	−	0.996909i	$-0.525035\pi$
$53$	−1.14399	−0.157139	−0.0785693	+	0.996909i	$0.525035\pi$
$54$	0	0
$55$	2.67282	0.360403
$56$	−3.00000	−0.400892
$57$	0	0
$58$	−5.42801	−0.712732
$59$	−1.14399	−0.148934	−0.0744672	−	0.997223i	$-0.523726\pi$
$59$	−1.14399	−0.148934	−0.0744672	+	0.997223i	$0.523726\pi$
$60$	0	0
$61$	−2.52884	−0.323784	−0.161892	−	0.986808i	$-0.551760\pi$
$61$	−2.52884	−0.323784	−0.161892	+	0.986808i	$0.551760\pi$
$62$	3.98153	0.505655
$63$	0	0
$64$	−1.18319	−0.147899
$65$	4.67282	0.579592
$66$	0	0
$67$	−6.59046	−0.805153	−0.402577	−	0.915386i	$-0.631885\pi$
$67$	−6.59046	−0.805153	−0.402577	+	0.915386i	$0.631885\pi$
$68$	−4.47116	−0.542208
$69$	0	0
$70$	0.816810	0.0976275
$71$	−12.8745	−1.52792	−0.763960	−	0.645263i	$-0.776748\pi$
$71$	−12.8745	−1.52792	−0.763960	+	0.645263i	$0.776748\pi$
$72$	0	0
$73$	−1.71203	−0.200378	−0.100189	−	0.994968i	$-0.531945\pi$
$73$	−1.71203	−0.200378	−0.100189	+	0.994968i	$0.531945\pi$
$74$	−1.03920	−0.120805
$75$	0	0
$76$	−7.81681	−0.896650
$77$	3.81681	0.434966
$78$	0	0
$79$	−0.287973	−0.0323995	−0.0161998	−	0.999869i	$-0.505157\pi$
$79$	−0.287973	−0.0323995	−0.0161998	+	0.999869i	$0.505157\pi$
$80$	2.14399	0.239705
$81$	0	0
$82$	−0.841495	−0.0929276
$83$	−4.28402	−0.470232	−0.235116	−	0.971967i	$-0.575547\pi$
$83$	−4.28402	−0.470232	−0.235116	+	0.971967i	$0.575547\pi$
$84$	0	0
$85$	2.67282	0.289908
$86$	0.269502	0.0290611
$87$	0	0
$88$	−5.61515	−0.598577
$89$	−3.00000	−0.317999	−0.159000	−	0.987279i	$-0.550827\pi$
$89$	−3.00000	−0.317999	−0.159000	+	0.987279i	$0.550827\pi$
$90$	0	0
$91$	6.67282	0.699502
$92$	9.89917	1.03206
$93$	0	0
$94$	3.97927	0.410430
$95$	4.67282	0.479422
$96$	0	0
$97$	7.83528	0.795552	0.397776	−	0.917483i	$-0.369782\pi$
$97$	7.83528	0.795552	0.397776	+	0.917483i	$0.369782\pi$
$98$	−2.83754	−0.286635
$99$	0	0
$100$	−1.67282	−0.167282
$101$	−4.20166	−0.418081	−0.209040	−	0.977907i	$-0.567034\pi$
$101$	−4.20166	−0.418081	−0.209040	+	0.977907i	$0.567034\pi$
$102$	0	0
$103$	−1.81681	−0.179016	−0.0895078	−	0.995986i	$-0.528529\pi$
$103$	−1.81681	−0.179016	−0.0895078	+	0.995986i	$0.528529\pi$
$104$	−9.81681	−0.962617
$105$	0	0
$106$	−0.654353	−0.0635563
$107$	11.9176	1.15212	0.576061	−	0.817407i	$-0.304589\pi$
$107$	11.9176	1.15212	0.576061	+	0.817407i	$0.304589\pi$
$108$	0	0
$109$	−16.6521	−1.59498	−0.797491	−	0.603331i	$-0.793840\pi$
$109$	−16.6521	−1.59498	−0.797491	+	0.603331i	$0.793840\pi$
$110$	1.52884	0.145769
$111$	0	0
$112$	3.06163	0.289297
$113$	20.1233	1.89304	0.946518	−	0.322650i	$-0.104574\pi$
$113$	20.1233	1.89304	0.946518	+	0.322650i	$0.104574\pi$
$114$	0	0
$115$	−5.91764	−0.551823
$116$	15.8745	1.47391
$117$	0	0
$118$	−0.654353	−0.0602380
$119$	3.81681	0.349886
$120$	0	0
$121$	−3.85601	−0.350547
$122$	−1.44648	−0.130958
$123$	0	0
$124$	−11.6442	−1.04568
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.18714	0.194078	0.0970388	−	0.995281i	$-0.469063\pi$
$127$	2.18714	0.194078	0.0970388	+	0.995281i	$0.469063\pi$
$128$	−11.5328	−1.01936
$129$	0	0
$130$	2.67282	0.234422
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	6.67282	0.578607
$134$	−3.76970	−0.325653
$135$	0	0
$136$	−5.61515	−0.481495
$137$	−10.2017	−0.871587	−0.435793	−	0.900047i	$-0.643532\pi$
$137$	−10.2017	−0.871587	−0.435793	+	0.900047i	$0.643532\pi$
$138$	0	0
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	−2.38880	−0.201891
$141$	0	0
$142$	−7.36412	−0.617983
$143$	12.4896	1.04444
$144$	0	0
$145$	−9.48963	−0.788071
$146$	−0.979268	−0.0810448
$147$	0	0
$148$	3.03920	0.249821
$149$	−20.0761	−1.64470	−0.822351	−	0.568981i	$-0.807338\pi$
$149$	−20.0761	−1.64470	−0.822351	+	0.568981i	$0.807338\pi$
$150$	0	0
$151$	3.03920	0.247327	0.123663	−	0.992324i	$-0.460536\pi$
$151$	3.03920	0.247327	0.123663	+	0.992324i	$0.460536\pi$
$152$	−9.81681	−0.796248
$153$	0	0
$154$	2.18319	0.175926
$155$	6.96080	0.559105
$156$	0	0
$157$	0.201661	0.0160943	0.00804714	−	0.999968i	$-0.497438\pi$
$157$	0.201661	0.0160943	0.00804714	+	0.999968i	$0.497438\pi$
$158$	−0.164719	−0.0131043
$159$	0	0
$160$	5.42801	0.429122
$161$	−8.45043	−0.665987
$162$	0	0
$163$	17.8168	1.39552	0.697760	−	0.716331i	$-0.254180\pi$
$163$	17.8168	1.39552	0.697760	+	0.716331i	$0.254180\pi$
$164$	2.46100	0.192172
$165$	0	0
$166$	−2.45043	−0.190190
$167$	14.1008	1.09116	0.545578	−	0.838060i	$-0.316310\pi$
$167$	14.1008	1.09116	0.545578	+	0.838060i	$0.316310\pi$
$168$	0	0
$169$	8.83528	0.679637
$170$	1.52884	0.117256
$171$	0	0
$172$	−0.788172	−0.0600976
$173$	4.36638	0.331970	0.165985	−	0.986128i	$-0.446920\pi$
$173$	4.36638	0.331970	0.165985	+	0.986128i	$0.446920\pi$
$174$	0	0
$175$	1.42801	0.107947
$176$	5.73050	0.431953
$177$	0	0
$178$	−1.71598	−0.128618
$179$	−15.1625	−1.13330	−0.566648	−	0.823960i	$-0.691760\pi$
$179$	−15.1625	−1.13330	−0.566648	+	0.823960i	$0.691760\pi$
$180$	0	0
$181$	3.20166	0.237978	0.118989	−	0.992896i	$-0.462035\pi$
$181$	3.20166	0.237978	0.118989	+	0.992896i	$0.462035\pi$
$182$	3.81681	0.282921
$183$	0	0
$184$	12.4320	0.916496
$185$	−1.81681	−0.133575
$186$	0	0
$187$	7.14399	0.522420
$188$	−11.6376	−0.848757
$189$	0	0
$190$	2.67282	0.193907
$191$	−2.83754	−0.205317	−0.102659	−	0.994717i	$-0.532735\pi$
$191$	−2.83754	−0.205317	−0.102659	+	0.994717i	$0.532735\pi$
$192$	0	0
$193$	−18.7882	−1.35240	−0.676201	−	0.736717i	$-0.736375\pi$
$193$	−18.7882	−1.35240	−0.676201	+	0.736717i	$0.736375\pi$
$194$	4.48173	0.321769
$195$	0	0
$196$	8.29854	0.592753
$197$	5.83528	0.415747	0.207873	−	0.978156i	$-0.433346\pi$
$197$	5.83528	0.415747	0.207873	+	0.978156i	$0.433346\pi$
$198$	0	0
$199$	13.0761	0.926943	0.463472	−	0.886112i	$-0.346604\pi$
$199$	13.0761	0.926943	0.463472	+	0.886112i	$0.346604\pi$
$200$	−2.10083	−0.148551
$201$	0	0
$202$	−2.40332	−0.169097
$203$	−13.5513	−0.951112
$204$	0	0
$205$	−1.47116	−0.102750
$206$	−1.03920	−0.0724047
$207$	0	0
$208$	10.0185	0.694656
$209$	12.4896	0.863926
$210$	0	0
$211$	8.38485	0.577237	0.288618	−	0.957444i	$-0.406804\pi$
$211$	8.38485	0.577237	0.288618	+	0.957444i	$0.406804\pi$
$212$	1.91369	0.131433
$213$	0	0
$214$	6.81681	0.465988
$215$	0.471163	0.0321330
$216$	0	0
$217$	9.94006	0.674776
$218$	−9.52488	−0.645107
$219$	0	0
$220$	−4.47116	−0.301446
$221$	12.4896	0.840144
$222$	0	0
$223$	9.16641	0.613828	0.306914	−	0.951737i	$-0.400704\pi$
$223$	9.16641	0.613828	0.306914	+	0.951737i	$0.400704\pi$
$224$	7.75123	0.517901
$225$	0	0
$226$	11.5104	0.765658
$227$	−2.67282	−0.177402	−0.0887008	−	0.996058i	$-0.528271\pi$
$227$	−2.67282	−0.177402	−0.0887008	+	0.996058i	$0.528271\pi$
$228$	0	0
$229$	2.54731	0.168331	0.0841654	−	0.996452i	$-0.473178\pi$
$229$	2.54731	0.168331	0.0841654	+	0.996452i	$0.473178\pi$
$230$	−3.38485	−0.223190
$231$	0	0
$232$	19.9361	1.30887
$233$	−6.22013	−0.407494	−0.203747	−	0.979024i	$-0.565312\pi$
$233$	−6.22013	−0.407494	−0.203747	+	0.979024i	$0.565312\pi$
$234$	0	0
$235$	6.95684	0.453814
$236$	1.91369	0.124570
$237$	0	0
$238$	2.18319	0.141515
$239$	8.12325	0.525450	0.262725	−	0.964871i	$-0.415379\pi$
$239$	8.12325	0.525450	0.262725	+	0.964871i	$0.415379\pi$
$240$	0	0
$241$	−26.3641	−1.69826	−0.849131	−	0.528182i	$-0.822874\pi$
$241$	−26.3641	−1.69826	−0.849131	+	0.528182i	$0.822874\pi$
$242$	−2.20561	−0.141782
$243$	0	0
$244$	4.23030	0.270817
$245$	−4.96080	−0.316934
$246$	0	0
$247$	21.8353	1.38935
$248$	−14.6235	−0.928590
$249$	0	0
$250$	0.571993	0.0361760
$251$	−0.549569	−0.0346885	−0.0173443	−	0.999850i	$-0.505521\pi$
$251$	−0.549569	−0.0346885	−0.0173443	+	0.999850i	$0.505521\pi$
$252$	0	0
$253$	−15.8168	−0.994394
$254$	1.25103	0.0784967
$255$	0	0
$256$	−4.23030	−0.264394
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	0	0
$259$	−2.59442	−0.161209
$260$	−7.81681	−0.484778
$261$	0	0
$262$	−3.43196	−0.212027
$263$	11.8952	0.733490	0.366745	−	0.930321i	$-0.380472\pi$
$263$	11.8952	0.733490	0.366745	+	0.930321i	$0.380472\pi$
$264$	0	0
$265$	−1.14399	−0.0702745
$266$	3.81681	0.234024
$267$	0	0
$268$	11.0247	0.673440
$269$	28.5737	1.74217	0.871084	−	0.491134i	$-0.163417\pi$
$269$	28.5737	1.74217	0.871084	+	0.491134i	$0.163417\pi$
$270$	0	0
$271$	−23.3641	−1.41927	−0.709635	−	0.704570i	$-0.751140\pi$
$271$	−23.3641	−1.41927	−0.709635	+	0.704570i	$0.751140\pi$
$272$	5.73050	0.347462
$273$	0	0
$274$	−5.83528	−0.352522
$275$	2.67282	0.161177
$276$	0	0
$277$	−15.0761	−0.905838	−0.452919	−	0.891552i	$-0.649617\pi$
$277$	−15.0761	−0.905838	−0.452919	+	0.891552i	$0.649617\pi$
$278$	4.57595	0.274447
$279$	0	0
$280$	−3.00000	−0.179284
$281$	6.65209	0.396831	0.198415	−	0.980118i	$-0.436421\pi$
$281$	6.65209	0.396831	0.198415	+	0.980118i	$0.436421\pi$
$282$	0	0
$283$	26.8969	1.59886	0.799428	−	0.600762i	$-0.205136\pi$
$283$	26.8969	1.59886	0.799428	+	0.600762i	$0.205136\pi$
$284$	21.5367	1.27797
$285$	0	0
$286$	7.14399	0.422433
$287$	−2.10083	−0.124008
$288$	0	0
$289$	−9.85601	−0.579765
$290$	−5.42801	−0.318744
$291$	0	0
$292$	2.86392	0.167598
$293$	12.3849	0.723531	0.361765	−	0.932269i	$-0.382174\pi$
$293$	12.3849	0.723531	0.361765	+	0.932269i	$0.382174\pi$
$294$	0	0
$295$	−1.14399	−0.0666055
$296$	3.81681	0.221848
$297$	0	0
$298$	−11.4834	−0.665217
$299$	−27.6521	−1.59916
$300$	0	0
$301$	0.672824	0.0387809
$302$	1.73840	0.100034
$303$	0	0
$304$	10.0185	0.574599
$305$	−2.52884	−0.144801
$306$	0	0
$307$	−2.49359	−0.142317	−0.0711583	−	0.997465i	$-0.522670\pi$
$307$	−2.49359	−0.142317	−0.0711583	+	0.997465i	$0.522670\pi$
$308$	−6.38485	−0.363811
$309$	0	0
$310$	3.98153	0.226136
$311$	−24.2201	−1.37340	−0.686699	−	0.726942i	$-0.740941\pi$
$311$	−24.2201	−1.37340	−0.686699	+	0.726942i	$0.740941\pi$
$312$	0	0
$313$	−35.0841	−1.98307	−0.991534	−	0.129848i	$-0.958551\pi$
$313$	−35.0841	−1.98307	−0.991534	+	0.129848i	$0.958551\pi$
$314$	0.115349	0.00650950
$315$	0	0
$316$	0.481728	0.0270993
$317$	−10.4712	−0.588119	−0.294060	−	0.955787i	$-0.595006\pi$
$317$	−10.4712	−0.588119	−0.294060	+	0.955787i	$0.595006\pi$
$318$	0	0
$319$	−25.3641	−1.42012
$320$	−1.18319	−0.0661423
$321$	0	0
$322$	−4.83359	−0.269365
$323$	12.4896	0.694942
$324$	0	0
$325$	4.67282	0.259202
$326$	10.1911	0.564433
$327$	0	0
$328$	3.09066	0.170653
$329$	9.93442	0.547702
$330$	0	0
$331$	16.7776	0.922181	0.461090	−	0.887353i	$-0.347458\pi$
$331$	16.7776	0.922181	0.461090	+	0.887353i	$0.347458\pi$
$332$	7.16641	0.393308
$333$	0	0
$334$	8.06558	0.441329
$335$	−6.59046	−0.360076
$336$	0	0
$337$	−27.1809	−1.48064	−0.740320	−	0.672255i	$-0.765326\pi$
$337$	−27.1809	−1.48064	−0.740320	+	0.672255i	$0.765326\pi$
$338$	5.05372	0.274886
$339$	0	0
$340$	−4.47116	−0.242483
$341$	18.6050	1.00752
$342$	0	0
$343$	−17.0801	−0.922239
$344$	−0.989833	−0.0533682
$345$	0	0
$346$	2.49754	0.134269
$347$	−23.5658	−1.26508	−0.632539	−	0.774529i	$-0.717987\pi$
$347$	−23.5658	−1.26508	−0.632539	+	0.774529i	$0.717987\pi$
$348$	0	0
$349$	−10.7120	−0.573402	−0.286701	−	0.958020i	$-0.592559\pi$
$349$	−10.7120	−0.573402	−0.286701	+	0.958020i	$0.592559\pi$
$350$	0.816810	0.0436603
$351$	0	0
$352$	14.5081	0.773285
$353$	27.2672	1.45129	0.725644	−	0.688070i	$-0.241542\pi$
$353$	27.2672	1.45129	0.725644	+	0.688070i	$0.241542\pi$
$354$	0	0
$355$	−12.8745	−0.683307
$356$	5.01847	0.265978
$357$	0	0
$358$	−8.67282	−0.458373
$359$	−10.6807	−0.563707	−0.281854	−	0.959457i	$-0.590949\pi$
$359$	−10.6807	−0.563707	−0.281854	+	0.959457i	$0.590949\pi$
$360$	0	0
$361$	2.83528	0.149225
$362$	1.83133	0.0962525
$363$	0	0
$364$	−11.1625	−0.585072
$365$	−1.71203	−0.0896116
$366$	0	0
$367$	−8.47116	−0.442191	−0.221096	−	0.975252i	$-0.570963\pi$
$367$	−8.47116	−0.442191	−0.221096	+	0.975252i	$0.570963\pi$
$368$	−12.6873	−0.661373
$369$	0	0
$370$	−1.03920	−0.0540256
$371$	−1.63362	−0.0848133
$372$	0	0
$373$	10.1233	0.524162	0.262081	−	0.965046i	$-0.415591\pi$
$373$	10.1233	0.524162	0.262081	+	0.965046i	$0.415591\pi$
$374$	4.08631	0.211298
$375$	0	0
$376$	−14.6151	−0.753719
$377$	−44.3434	−2.28380
$378$	0	0
$379$	11.9216	0.612371	0.306186	−	0.951972i	$-0.400947\pi$
$379$	11.9216	0.612371	0.306186	+	0.951972i	$0.400947\pi$
$380$	−7.81681	−0.400994
$381$	0	0
$382$	−1.62306	−0.0830427
$383$	9.81681	0.501616	0.250808	−	0.968037i	$-0.419304\pi$
$383$	9.81681	0.501616	0.250808	+	0.968037i	$0.419304\pi$
$384$	0	0
$385$	3.81681	0.194523
$386$	−10.7467	−0.546993
$387$	0	0
$388$	−13.1070	−0.665409
$389$	9.22013	0.467479	0.233740	−	0.972299i	$-0.424904\pi$
$389$	9.22013	0.467479	0.233740	+	0.972299i	$0.424904\pi$
$390$	0	0
$391$	−15.8168	−0.799890
$392$	10.4218	0.526380
$393$	0	0
$394$	3.33774	0.168153
$395$	−0.287973	−0.0144895
$396$	0	0
$397$	−22.9793	−1.15330	−0.576648	−	0.816993i	$-0.695640\pi$
$397$	−22.9793	−1.15330	−0.576648	+	0.816993i	$0.695640\pi$
$398$	7.47947	0.374912
$399$	0	0
$400$	2.14399	0.107199
$401$	−11.0656	−0.552589	−0.276294	−	0.961073i	$-0.589106\pi$
$401$	−11.0656	−0.552589	−0.276294	+	0.961073i	$0.589106\pi$
$402$	0	0
$403$	32.5266	1.62026
$404$	7.02864	0.349688
$405$	0	0
$406$	−7.75123	−0.384687
$407$	−4.85601	−0.240704
$408$	0	0
$409$	−17.6336	−0.871926	−0.435963	−	0.899964i	$-0.643592\pi$
$409$	−17.6336	−0.871926	−0.435963	+	0.899964i	$0.643592\pi$
$410$	−0.841495	−0.0415585
$411$	0	0
$412$	3.03920	0.149731
$413$	−1.63362	−0.0803852
$414$	0	0
$415$	−4.28402	−0.210294
$416$	25.3641	1.24358
$417$	0	0
$418$	7.14399	0.349424
$419$	37.0347	1.80926	0.904631	−	0.426195i	$-0.140146\pi$
$419$	37.0347	1.80926	0.904631	+	0.426195i	$0.140146\pi$
$420$	0	0
$421$	5.05767	0.246496	0.123248	−	0.992376i	$-0.460669\pi$
$421$	5.05767	0.246496	0.123248	+	0.992376i	$0.460669\pi$
$422$	4.79608	0.233469
$423$	0	0
$424$	2.40332	0.116716
$425$	2.67282	0.129651
$426$	0	0
$427$	−3.61120	−0.174758
$428$	−19.9361	−0.963648
$429$	0	0
$430$	0.269502	0.0129965
$431$	−5.23030	−0.251935	−0.125967	−	0.992034i	$-0.540203\pi$
$431$	−5.23030	−0.251935	−0.125967	+	0.992034i	$0.540203\pi$
$432$	0	0
$433$	34.3434	1.65044	0.825219	−	0.564813i	$-0.191052\pi$
$433$	34.3434	1.65044	0.825219	+	0.564813i	$0.191052\pi$
$434$	5.68565	0.272920
$435$	0	0
$436$	27.8560	1.33406
$437$	−27.6521	−1.32278
$438$	0	0
$439$	−19.5473	−0.932942	−0.466471	−	0.884536i	$-0.654475\pi$
$439$	−19.5473	−0.932942	−0.466471	+	0.884536i	$0.654475\pi$
$440$	−5.61515	−0.267692
$441$	0	0
$442$	7.14399	0.339805
$443$	10.5042	0.499067	0.249534	−	0.968366i	$-0.419723\pi$
$443$	10.5042	0.499067	0.249534	+	0.968366i	$0.419723\pi$
$444$	0	0
$445$	−3.00000	−0.142214
$446$	5.24313	0.248269
$447$	0	0
$448$	−1.68960	−0.0798262
$449$	22.8560	1.07864	0.539321	−	0.842100i	$-0.318681\pi$
$449$	22.8560	1.07864	0.539321	+	0.842100i	$0.318681\pi$
$450$	0	0
$451$	−3.93216	−0.185158
$452$	−33.6627	−1.58336
$453$	0	0
$454$	−1.52884	−0.0717519
$455$	6.67282	0.312827
$456$	0	0
$457$	−14.0264	−0.656126	−0.328063	−	0.944656i	$-0.606396\pi$
$457$	−14.0264	−0.656126	−0.328063	+	0.944656i	$0.606396\pi$
$458$	1.45704	0.0680832
$459$	0	0
$460$	9.89917	0.461551
$461$	−24.1025	−1.12257	−0.561283	−	0.827624i	$-0.689692\pi$
$461$	−24.1025	−1.12257	−0.561283	+	0.827624i	$0.689692\pi$
$462$	0	0
$463$	−33.9401	−1.57733	−0.788664	−	0.614824i	$-0.789227\pi$
$463$	−33.9401	−1.57733	−0.788664	+	0.614824i	$0.789227\pi$
$464$	−20.3456	−0.944523
$465$	0	0
$466$	−3.55787	−0.164815
$467$	−27.3720	−1.26663	−0.633313	−	0.773896i	$-0.718305\pi$
$467$	−27.3720	−1.26663	−0.633313	+	0.773896i	$0.718305\pi$
$468$	0	0
$469$	−9.41123	−0.434570
$470$	3.97927	0.183550
$471$	0	0
$472$	2.40332	0.110622
$473$	1.25934	0.0579043
$474$	0	0
$475$	4.67282	0.214404
$476$	−6.38485	−0.292649
$477$	0	0
$478$	4.64645	0.212524
$479$	5.23030	0.238978	0.119489	−	0.992835i	$-0.461874\pi$
$479$	5.23030	0.238978	0.119489	+	0.992835i	$0.461874\pi$
$480$	0	0
$481$	−8.48963	−0.387094
$482$	−15.0801	−0.686880
$483$	0	0
$484$	6.45043	0.293201
$485$	7.83528	0.355782
$486$	0	0
$487$	−24.0185	−1.08838	−0.544190	−	0.838962i	$-0.683163\pi$
$487$	−24.0185	−1.08838	−0.544190	+	0.838962i	$0.683163\pi$
$488$	5.31266	0.240493
$489$	0	0
$490$	−2.83754	−0.128187
$491$	14.7776	0.666904	0.333452	−	0.942767i	$-0.391786\pi$
$491$	14.7776	0.666904	0.333452	+	0.942767i	$0.391786\pi$
$492$	0	0
$493$	−25.3641	−1.14234
$494$	12.4896	0.561935
$495$	0	0
$496$	14.9239	0.670101
$497$	−18.3849	−0.824673
$498$	0	0
$499$	24.8560	1.11271	0.556354	−	0.830945i	$-0.312200\pi$
$499$	24.8560	1.11271	0.556354	+	0.830945i	$0.312200\pi$
$500$	−1.67282	−0.0748110
$501$	0	0
$502$	−0.314350	−0.0140301
$503$	38.9154	1.73515	0.867576	−	0.497305i	$-0.165677\pi$
$503$	38.9154	1.73515	0.867576	+	0.497305i	$0.165677\pi$
$504$	0	0
$505$	−4.20166	−0.186971
$506$	−9.04711	−0.402193
$507$	0	0
$508$	−3.65870	−0.162329
$509$	2.02073	0.0895674	0.0447837	−	0.998997i	$-0.485740\pi$
$509$	2.02073	0.0895674	0.0447837	+	0.998997i	$0.485740\pi$
$510$	0	0
$511$	−2.44479	−0.108151
$512$	20.6459	0.912428
$513$	0	0
$514$	−10.2959	−0.454132
$515$	−1.81681	−0.0800582
$516$	0	0
$517$	18.5944	0.817782
$518$	−1.48399	−0.0652027
$519$	0	0
$520$	−9.81681	−0.430496
$521$	−23.0290	−1.00892	−0.504460	−	0.863435i	$-0.668308\pi$
$521$	−23.0290	−1.00892	−0.504460	+	0.863435i	$0.668308\pi$
$522$	0	0
$523$	−41.1170	−1.79792	−0.898961	−	0.438028i	$-0.855677\pi$
$523$	−41.1170	−1.79792	−0.898961	+	0.438028i	$0.855677\pi$
$524$	10.0369	0.438466
$525$	0	0
$526$	6.80398	0.296668
$527$	18.6050	0.810446
$528$	0	0
$529$	12.0185	0.522542
$530$	−0.654353	−0.0284233
$531$	0	0
$532$	−11.1625	−0.483954
$533$	−6.87448	−0.297767
$534$	0	0
$535$	11.9176	0.515245
$536$	13.8454	0.598032
$537$	0	0
$538$	16.3440	0.704638
$539$	−13.2593	−0.571120
$540$	0	0
$541$	5.20957	0.223977	0.111988	−	0.993710i	$-0.464278\pi$
$541$	5.20957	0.223977	0.111988	+	0.993710i	$0.464278\pi$
$542$	−13.3641	−0.574038
$543$	0	0
$544$	14.5081	0.622030
$545$	−16.6521	−0.713297
$546$	0	0
$547$	40.0409	1.71203	0.856013	−	0.516955i	$-0.172935\pi$
$547$	40.0409	1.71203	0.856013	+	0.516955i	$0.172935\pi$
$548$	17.0656	0.729005
$549$	0	0
$550$	1.52884	0.0651898
$551$	−44.3434	−1.88909
$552$	0	0
$553$	−0.411227	−0.0174872
$554$	−8.62345	−0.366375
$555$	0	0
$556$	−13.3826	−0.567548
$557$	−14.4033	−0.610288	−0.305144	−	0.952306i	$-0.598705\pi$
$557$	−14.4033	−0.610288	−0.305144	+	0.952306i	$0.598705\pi$
$558$	0	0
$559$	2.20166	0.0931203
$560$	3.06163	0.129377
$561$	0	0
$562$	3.80495	0.160502
$563$	29.3681	1.23772	0.618858	−	0.785503i	$-0.287595\pi$
$563$	29.3681	1.23772	0.618858	+	0.785503i	$0.287595\pi$
$564$	0	0
$565$	20.1233	0.846592
$566$	15.3849	0.646674
$567$	0	0
$568$	27.0471	1.13487
$569$	46.8066	1.96224	0.981118	−	0.193409i	$-0.0619543\pi$
$569$	46.8066	1.96224	0.981118	+	0.193409i	$0.0619543\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	−20.8930	−0.873578
$573$	0	0
$574$	−1.20166	−0.0501564
$575$	−5.91764	−0.246783
$576$	0	0
$577$	28.2386	1.17559	0.587794	−	0.809010i	$-0.299996\pi$
$577$	28.2386	1.17559	0.587794	+	0.809010i	$0.299996\pi$
$578$	−5.63757	−0.234492
$579$	0	0
$580$	15.8745	0.659152
$581$	−6.11761	−0.253801
$582$	0	0
$583$	−3.05767	−0.126636
$584$	3.59668	0.148832
$585$	0	0
$586$	7.08405	0.292639
$587$	18.0824	0.746339	0.373169	−	0.927763i	$-0.378271\pi$
$587$	18.0824	0.746339	0.373169	+	0.927763i	$0.378271\pi$
$588$	0	0
$589$	32.5266	1.34023
$590$	−0.654353	−0.0269393
$591$	0	0
$592$	−3.89522	−0.160092
$593$	−7.73840	−0.317778	−0.158889	−	0.987296i	$-0.550791\pi$
$593$	−7.73840	−0.317778	−0.158889	+	0.987296i	$0.550791\pi$
$594$	0	0
$595$	3.81681	0.156474
$596$	33.5839	1.37565
$597$	0	0
$598$	−15.8168	−0.646797
$599$	−27.9216	−1.14085	−0.570423	−	0.821351i	$-0.693221\pi$
$599$	−27.9216	−1.14085	−0.570423	+	0.821351i	$0.693221\pi$
$600$	0	0
$601$	38.4403	1.56801	0.784006	−	0.620754i	$-0.213173\pi$
$601$	38.4403	1.56801	0.784006	+	0.620754i	$0.213173\pi$
$602$	0.384851	0.0156853
$603$	0	0
$604$	−5.08405	−0.206867
$605$	−3.85601	−0.156769
$606$	0	0
$607$	−0.639834	−0.0259701	−0.0129850	−	0.999916i	$-0.504133\pi$
$607$	−0.639834	−0.0259701	−0.0129850	+	0.999916i	$0.504133\pi$
$608$	25.3641	1.02865
$609$	0	0
$610$	−1.44648	−0.0585662
$611$	32.5081	1.31514
$612$	0	0
$613$	42.7467	1.72652	0.863262	−	0.504757i	$-0.168418\pi$
$613$	42.7467	1.72652	0.863262	+	0.504757i	$0.168418\pi$
$614$	−1.42631	−0.0575614
$615$	0	0
$616$	−8.01847	−0.323073
$617$	21.1025	0.849556	0.424778	−	0.905298i	$-0.360352\pi$
$617$	21.1025	0.849556	0.424778	+	0.905298i	$0.360352\pi$
$618$	0	0
$619$	−13.6521	−0.548724	−0.274362	−	0.961626i	$-0.588467\pi$
$619$	−13.6521	−0.548724	−0.274362	+	0.961626i	$0.588467\pi$
$620$	−11.6442	−0.467642
$621$	0	0
$622$	−13.8538	−0.555485
$623$	−4.28402	−0.171636
$624$	0	0
$625$	1.00000	0.0400000
$626$	−20.0678	−0.802072
$627$	0	0
$628$	−0.337343	−0.0134615
$629$	−4.85601	−0.193622
$630$	0	0
$631$	33.2593	1.32403	0.662017	−	0.749489i	$-0.269701\pi$
$631$	33.2593	1.32403	0.662017	+	0.749489i	$0.269701\pi$
$632$	0.604983	0.0240649
$633$	0	0
$634$	−5.98943	−0.237871
$635$	2.18714	0.0867941
$636$	0	0
$637$	−23.1809	−0.918462
$638$	−14.5081	−0.574381
$639$	0	0
$640$	−11.5328	−0.455874
$641$	26.2857	1.03822	0.519112	−	0.854706i	$-0.326263\pi$
$641$	26.2857	1.03822	0.519112	+	0.854706i	$0.326263\pi$
$642$	0	0
$643$	20.5826	0.811697	0.405848	−	0.913940i	$-0.366976\pi$
$643$	20.5826	0.811697	0.405848	+	0.913940i	$0.366976\pi$
$644$	14.1361	0.557040
$645$	0	0
$646$	7.14399	0.281076
$647$	23.2527	0.914159	0.457079	−	0.889426i	$-0.348896\pi$
$647$	23.2527	0.914159	0.457079	+	0.889426i	$0.348896\pi$
$648$	0	0
$649$	−3.05767	−0.120024
$650$	2.67282	0.104837
$651$	0	0
$652$	−29.8044	−1.16723
$653$	18.7591	0.734102	0.367051	−	0.930201i	$-0.380367\pi$
$653$	18.7591	0.734102	0.367051	+	0.930201i	$0.380367\pi$
$654$	0	0
$655$	−6.00000	−0.234439
$656$	−3.15415	−0.123149
$657$	0	0
$658$	5.68242	0.221524
$659$	−0.280067	−0.0109099	−0.00545494	−	0.999985i	$-0.501736\pi$
$659$	−0.280067	−0.0109099	−0.00545494	+	0.999985i	$0.501736\pi$
$660$	0	0
$661$	−39.7859	−1.54749	−0.773746	−	0.633496i	$-0.781619\pi$
$661$	−39.7859	−1.54749	−0.773746	+	0.633496i	$0.781619\pi$
$662$	9.59668	0.372985
$663$	0	0
$664$	9.00000	0.349268
$665$	6.67282	0.258761
$666$	0	0
$667$	56.1562	2.17438
$668$	−23.5882	−0.912655
$669$	0	0
$670$	−3.76970	−0.145636
$671$	−6.75914	−0.260934
$672$	0	0
$673$	33.5288	1.29244	0.646221	−	0.763150i	$-0.276348\pi$
$673$	33.5288	1.29244	0.646221	+	0.763150i	$0.276348\pi$
$674$	−15.5473	−0.598860
$675$	0	0
$676$	−14.7799	−0.568456
$677$	27.4874	1.05643	0.528213	−	0.849112i	$-0.322862\pi$
$677$	27.4874	1.05643	0.528213	+	0.849112i	$0.322862\pi$
$678$	0	0
$679$	11.1888	0.429388
$680$	−5.61515	−0.215331
$681$	0	0
$682$	10.6419	0.407500
$683$	−34.5865	−1.32342	−0.661708	−	0.749762i	$-0.730168\pi$
$683$	−34.5865	−1.32342	−0.661708	+	0.749762i	$0.730168\pi$
$684$	0	0
$685$	−10.2017	−0.389785
$686$	−9.76970	−0.373009
$687$	0	0
$688$	1.01017	0.0385122
$689$	−5.34565	−0.203653
$690$	0	0
$691$	40.7282	1.54938	0.774688	−	0.632344i	$-0.217907\pi$
$691$	40.7282	1.54938	0.774688	+	0.632344i	$0.217907\pi$
$692$	−7.30418	−0.277663
$693$	0	0
$694$	−13.4795	−0.511674
$695$	8.00000	0.303457
$696$	0	0
$697$	−3.93216	−0.148941
$698$	−6.12721	−0.231918
$699$	0	0
$700$	−2.38880	−0.0902883
$701$	−19.4712	−0.735416	−0.367708	−	0.929941i	$-0.619857\pi$
$701$	−19.4712	−0.735416	−0.367708	+	0.929941i	$0.619857\pi$
$702$	0	0
$703$	−8.48963	−0.320193
$704$	−3.16246	−0.119190
$705$	0	0
$706$	15.5967	0.586989
$707$	−6.00000	−0.225653
$708$	0	0
$709$	15.0863	0.566578	0.283289	−	0.959035i	$-0.408574\pi$
$709$	15.0863	0.566578	0.283289	+	0.959035i	$0.408574\pi$
$710$	−7.36412	−0.276370
$711$	0	0
$712$	6.30249	0.236196
$713$	−41.1915	−1.54263
$714$	0	0
$715$	12.4896	0.467086
$716$	25.3641	0.947902
$717$	0	0
$718$	−6.10931	−0.227997
$719$	−3.43196	−0.127990	−0.0639952	−	0.997950i	$-0.520384\pi$
$719$	−3.43196	−0.127990	−0.0639952	+	0.997950i	$0.520384\pi$
$720$	0	0
$721$	−2.59442	−0.0966211
$722$	1.62176	0.0603557
$723$	0	0
$724$	−5.35581	−0.199047
$725$	−9.48963	−0.352436
$726$	0	0
$727$	−35.7714	−1.32669	−0.663344	−	0.748315i	$-0.730863\pi$
$727$	−35.7714	−1.32669	−0.663344	+	0.748315i	$0.730863\pi$
$728$	−14.0185	−0.519559
$729$	0	0
$730$	−0.979268	−0.0362443
$731$	1.25934	0.0465782
$732$	0	0
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	−4.84545	−0.178849
$735$	0	0
$736$	−32.1210	−1.18400
$737$	−17.6151	−0.648862
$738$	0	0
$739$	6.08631	0.223889	0.111944	−	0.993714i	$-0.464292\pi$
$739$	6.08631	0.223889	0.111944	+	0.993714i	$0.464292\pi$
$740$	3.03920	0.111723
$741$	0	0
$742$	−0.934420	−0.0343036
$743$	−25.5019	−0.935574	−0.467787	−	0.883841i	$-0.654949\pi$
$743$	−25.5019	−0.935574	−0.467787	+	0.883841i	$0.654949\pi$
$744$	0	0
$745$	−20.0761	−0.735533
$746$	5.79043	0.212003
$747$	0	0
$748$	−11.9506	−0.436958
$749$	17.0185	0.621841
$750$	0	0
$751$	−18.3928	−0.671161	−0.335581	−	0.942011i	$-0.608932\pi$
$751$	−18.3928	−0.671161	−0.335581	+	0.942011i	$0.608932\pi$
$752$	14.9154	0.543908
$753$	0	0
$754$	−25.3641	−0.923707
$755$	3.03920	0.110608
$756$	0	0
$757$	−41.8986	−1.52283	−0.761415	−	0.648264i	$-0.775495\pi$
$757$	−41.8986	−1.52283	−0.761415	+	0.648264i	$0.775495\pi$
$758$	6.81907	0.247680
$759$	0	0
$760$	−9.81681	−0.356093
$761$	−7.97136	−0.288962	−0.144481	−	0.989508i	$-0.546151\pi$
$761$	−7.97136	−0.288962	−0.144481	+	0.989508i	$0.546151\pi$
$762$	0	0
$763$	−23.7793	−0.860868
$764$	4.74671	0.171730
$765$	0	0
$766$	5.61515	0.202884
$767$	−5.34565	−0.193020
$768$	0	0
$769$	6.02864	0.217398	0.108699	−	0.994075i	$-0.465331\pi$
$769$	6.02864	0.217398	0.108699	+	0.994075i	$0.465331\pi$
$770$	2.18319	0.0786767
$771$	0	0
$772$	31.4293	1.13117
$773$	44.4033	1.59708	0.798538	−	0.601944i	$-0.205607\pi$
$773$	44.4033	1.59708	0.798538	+	0.601944i	$0.205607\pi$
$774$	0	0
$775$	6.96080	0.250039
$776$	−16.4606	−0.590901
$777$	0	0
$778$	5.27385	0.189077
$779$	−6.87448	−0.246304
$780$	0	0
$781$	−34.4112	−1.23133
$782$	−9.04711	−0.323524
$783$	0	0
$784$	−10.6359	−0.379853
$785$	0.201661	0.00719758
$786$	0	0
$787$	16.8375	0.600194	0.300097	−	0.953909i	$-0.402981\pi$
$787$	16.8375	0.600194	0.300097	+	0.953909i	$0.402981\pi$
$788$	−9.76140	−0.347735
$789$	0	0
$790$	−0.164719	−0.00586043
$791$	28.7361	1.02174
$792$	0	0
$793$	−11.8168	−0.419627
$794$	−13.1440	−0.466463
$795$	0	0
$796$	−21.8741	−0.775306
$797$	−33.3720	−1.18210	−0.591049	−	0.806636i	$-0.701286\pi$
$797$	−33.3720	−1.18210	−0.591049	+	0.806636i	$0.701286\pi$
$798$	0	0
$799$	18.5944	0.657823
$800$	5.42801	0.191909
$801$	0	0
$802$	−6.32944	−0.223500
$803$	−4.57595	−0.161482
$804$	0	0
$805$	−8.45043	−0.297839
$806$	18.6050	0.655333
$807$	0	0
$808$	8.82698	0.310532
$809$	29.1809	1.02595	0.512973	−	0.858404i	$-0.328544\pi$
$809$	29.1809	1.02595	0.512973	+	0.858404i	$0.328544\pi$
$810$	0	0
$811$	−15.5552	−0.546217	−0.273109	−	0.961983i	$-0.588052\pi$
$811$	−15.5552	−0.546217	−0.273109	+	0.961983i	$0.588052\pi$
$812$	22.6689	0.795521
$813$	0	0
$814$	−2.77761	−0.0973551
$815$	17.8168	0.624096
$816$	0	0
$817$	2.20166	0.0770264
$818$	−10.0863	−0.352660
$819$	0	0
$820$	2.46100	0.0859417
$821$	−23.7177	−0.827752	−0.413876	−	0.910333i	$-0.635825\pi$
$821$	−23.7177	−0.827752	−0.413876	+	0.910333i	$0.635825\pi$
$822$	0	0
$823$	19.3681	0.675129	0.337564	−	0.941302i	$-0.390397\pi$
$823$	19.3681	0.675129	0.337564	+	0.941302i	$0.390397\pi$
$824$	3.81681	0.132965
$825$	0	0
$826$	−0.934420	−0.0325126
$827$	52.2241	1.81601	0.908005	−	0.418960i	$-0.137605\pi$
$827$	52.2241	1.81601	0.908005	+	0.418960i	$0.137605\pi$
$828$	0	0
$829$	−26.6442	−0.925391	−0.462695	−	0.886517i	$-0.653118\pi$
$829$	−26.6442	−0.925391	−0.462695	+	0.886517i	$0.653118\pi$
$830$	−2.45043	−0.0850557
$831$	0	0
$832$	−5.52884	−0.191678
$833$	−13.2593	−0.459409
$834$	0	0
$835$	14.1008	0.487979
$836$	−20.8930	−0.722598
$837$	0	0
$838$	21.1836	0.731775
$839$	−25.0392	−0.864449	−0.432225	−	0.901766i	$-0.642271\pi$
$839$	−25.0392	−0.864449	−0.432225	+	0.901766i	$0.642271\pi$
$840$	0	0
$841$	61.0532	2.10528
$842$	2.89296	0.0996978
$843$	0	0
$844$	−14.0264	−0.482808
$845$	8.83528	0.303943
$846$	0	0
$847$	−5.50641	−0.189203
$848$	−2.45269	−0.0842258
$849$	0	0
$850$	1.52884	0.0524387
$851$	10.7512	0.368547
$852$	0	0
$853$	−10.8745	−0.372335	−0.186168	−	0.982518i	$-0.559607\pi$
$853$	−10.8745	−0.372335	−0.186168	+	0.982518i	$0.559607\pi$
$854$	−2.06558	−0.0706827
$855$	0	0
$856$	−25.0369	−0.855745
$857$	−18.5944	−0.635173	−0.317587	−	0.948229i	$-0.602872\pi$
$857$	−18.5944	−0.635173	−0.317587	+	0.948229i	$0.602872\pi$
$858$	0	0
$859$	−4.66492	−0.159165	−0.0795825	−	0.996828i	$-0.525359\pi$
$859$	−4.66492	−0.159165	−0.0795825	+	0.996828i	$0.525359\pi$
$860$	−0.788172	−0.0268765
$861$	0	0
$862$	−2.99170	−0.101898
$863$	−28.0594	−0.955152	−0.477576	−	0.878590i	$-0.658484\pi$
$863$	−28.0594	−0.955152	−0.477576	+	0.878590i	$0.658484\pi$
$864$	0	0
$865$	4.36638	0.148461
$866$	19.6442	0.667537
$867$	0	0
$868$	−16.6280	−0.564390
$869$	−0.769701	−0.0261103
$870$	0	0
$871$	−30.7961	−1.04349
$872$	34.9832	1.18468
$873$	0	0
$874$	−15.8168	−0.535012
$875$	1.42801	0.0482754
$876$	0	0
$877$	34.6683	1.17067	0.585333	−	0.810793i	$-0.300964\pi$
$877$	34.6683	1.17067	0.585333	+	0.810793i	$0.300964\pi$
$878$	−11.1809	−0.377338
$879$	0	0
$880$	5.73050	0.193175
$881$	−5.29854	−0.178512	−0.0892561	−	0.996009i	$-0.528449\pi$
$881$	−5.29854	−0.178512	−0.0892561	+	0.996009i	$0.528449\pi$
$882$	0	0
$883$	−10.3025	−0.346706	−0.173353	−	0.984860i	$-0.555460\pi$
$883$	−10.3025	−0.346706	−0.173353	+	0.984860i	$0.555460\pi$
$884$	−20.8930	−0.702706
$885$	0	0
$886$	6.00830	0.201853
$887$	41.4504	1.39177	0.695885	−	0.718154i	$-0.255012\pi$
$887$	41.4504	1.39177	0.695885	+	0.718154i	$0.255012\pi$
$888$	0	0
$889$	3.12325	0.104751
$890$	−1.71598	−0.0575198
$891$	0	0
$892$	−15.3338	−0.513413
$893$	32.5081	1.08784
$894$	0	0
$895$	−15.1625	−0.506825
$896$	−16.4689	−0.550187
$897$	0	0
$898$	13.0735	0.436268
$899$	−66.0554	−2.20307
$900$	0	0
$901$	−3.05767	−0.101866
$902$	−2.24917	−0.0748891
$903$	0	0
$904$	−42.2755	−1.40606
$905$	3.20166	0.106427
$906$	0	0
$907$	−16.7921	−0.557573	−0.278787	−	0.960353i	$-0.589932\pi$
$907$	−16.7921	−0.557573	−0.278787	+	0.960353i	$0.589932\pi$
$908$	4.47116	0.148381
$909$	0	0
$910$	3.81681	0.126526
$911$	37.3536	1.23758	0.618789	−	0.785557i	$-0.287623\pi$
$911$	37.3536	1.23758	0.618789	+	0.785557i	$0.287623\pi$
$912$	0	0
$913$	−11.4504	−0.378954
$914$	−8.02299	−0.265377
$915$	0	0
$916$	−4.26120	−0.140794
$917$	−8.56804	−0.282942
$918$	0	0
$919$	37.1316	1.22486	0.612429	−	0.790526i	$-0.290193\pi$
$919$	37.1316	1.22486	0.612429	+	0.790526i	$0.290193\pi$
$920$	12.4320	0.409870
$921$	0	0
$922$	−13.7865	−0.454034
$923$	−60.1602	−1.98020
$924$	0	0
$925$	−1.81681	−0.0597364
$926$	−19.4135	−0.637967
$927$	0	0
$928$	−51.5098	−1.69089
$929$	−47.9955	−1.57468	−0.787340	−	0.616519i	$-0.788542\pi$
$929$	−47.9955	−1.57468	−0.787340	+	0.616519i	$0.788542\pi$
$930$	0	0
$931$	−23.1809	−0.759724
$932$	10.4052	0.340833
$933$	0	0
$934$	−15.6566	−0.512300
$935$	7.14399	0.233633
$936$	0	0
$937$	22.0079	0.718967	0.359483	−	0.933151i	$-0.382953\pi$
$937$	22.0079	0.718967	0.359483	+	0.933151i	$0.382953\pi$
$938$	−5.38316	−0.175766
$939$	0	0
$940$	−11.6376	−0.379576
$941$	40.5843	1.32301	0.661504	−	0.749941i	$-0.269918\pi$
$941$	40.5843	1.32301	0.661504	+	0.749941i	$0.269918\pi$
$942$	0	0
$943$	8.70581	0.283500
$944$	−2.45269	−0.0798283
$945$	0	0
$946$	0.720331	0.0234200
$947$	30.2320	0.982408	0.491204	−	0.871045i	$-0.336557\pi$
$947$	30.2320	0.982408	0.491204	+	0.871045i	$0.336557\pi$
$948$	0	0
$949$	−8.00000	−0.259691
$950$	2.67282	0.0867179
$951$	0	0
$952$	−8.01847	−0.259880
$953$	2.50811	0.0812455	0.0406227	−	0.999175i	$-0.487066\pi$
$953$	2.50811	0.0812455	0.0406227	+	0.999175i	$0.487066\pi$
$954$	0	0
$955$	−2.83754	−0.0918207
$956$	−13.5888	−0.439492
$957$	0	0
$958$	2.99170	0.0966573
$959$	−14.5680	−0.470427
$960$	0	0
$961$	17.4527	0.562990
$962$	−4.85601	−0.156564
$963$	0	0
$964$	44.1025	1.42045
$965$	−18.7882	−0.604813
$966$	0	0
$967$	−21.7529	−0.699527	−0.349763	−	0.936838i	$-0.613738\pi$
$967$	−21.7529	−0.699527	−0.349763	+	0.936838i	$0.613738\pi$
$968$	8.10083	0.260371
$969$	0	0
$970$	4.48173	0.143900
$971$	22.7512	0.730122	0.365061	−	0.930984i	$-0.381048\pi$
$971$	22.7512	0.730122	0.365061	+	0.930984i	$0.381048\pi$
$972$	0	0
$973$	11.4241	0.366238
$974$	−13.7384	−0.440207
$975$	0	0
$976$	−5.42179	−0.173547
$977$	39.2778	1.25661	0.628304	−	0.777968i	$-0.283749\pi$
$977$	39.2778	1.25661	0.628304	+	0.777968i	$0.283749\pi$
$978$	0	0
$979$	−8.01847	−0.256271
$980$	8.29854	0.265087
$981$	0	0
$982$	8.45269	0.269736
$983$	33.7899	1.07773	0.538865	−	0.842392i	$-0.318853\pi$
$983$	33.7899	1.07773	0.538865	+	0.842392i	$0.318853\pi$
$984$	0	0
$985$	5.83528	0.185928
$986$	−14.5081	−0.462032
$987$	0	0
$988$	−36.5266	−1.16207
$989$	−2.78817	−0.0886587
$990$	0	0
$991$	−23.7983	−0.755979	−0.377990	−	0.925810i	$-0.623384\pi$
$991$	−23.7983	−0.755979	−0.377990	+	0.925810i	$0.623384\pi$
$992$	37.7833	1.19962
$993$	0	0
$994$	−10.5160	−0.333548
$995$	13.0761	0.414542
$996$	0	0
$997$	51.6785	1.63667	0.818337	−	0.574739i	$-0.194896\pi$
$997$	51.6785	1.63667	0.818337	+	0.574739i	$0.194896\pi$
$998$	14.2175	0.450046
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	405.2.a.j.1.2		3
3.2	odd	2		405.2.a.i.1.2		3
4.3	odd	2		6480.2.a.bv.1.2		3
5.2	odd	4		2025.2.b.l.649.4		6
5.3	odd	4		2025.2.b.l.649.3		6
5.4	even	2		2025.2.a.n.1.2		3
9.2	odd	6		135.2.e.b.91.2		6
9.4	even	3		45.2.e.b.16.2	✓	6
9.5	odd	6		135.2.e.b.46.2		6
9.7	even	3		45.2.e.b.31.2	yes	6
12.11	even	2		6480.2.a.bs.1.2		3
15.2	even	4		2025.2.b.m.649.3		6
15.8	even	4		2025.2.b.m.649.4		6
15.14	odd	2		2025.2.a.o.1.2		3
36.7	odd	6		720.2.q.i.481.2		6
36.11	even	6		2160.2.q.k.1441.2		6
36.23	even	6		2160.2.q.k.721.2		6
36.31	odd	6		720.2.q.i.241.2		6
45.2	even	12		675.2.k.b.199.3		12
45.4	even	6		225.2.e.b.151.2		6
45.7	odd	12		225.2.k.b.49.4		12
45.13	odd	12		225.2.k.b.124.4		12
45.14	odd	6		675.2.e.b.451.2		6
45.22	odd	12		225.2.k.b.124.3		12
45.23	even	12		675.2.k.b.424.3		12
45.29	odd	6		675.2.e.b.226.2		6
45.32	even	12		675.2.k.b.424.4		12
45.34	even	6		225.2.e.b.76.2		6
45.38	even	12		675.2.k.b.199.4		12
45.43	odd	12		225.2.k.b.49.3		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.2.e.b.16.2	✓	6	9.4	even	3
45.2.e.b.31.2	yes	6	9.7	even	3
135.2.e.b.46.2		6	9.5	odd	6
135.2.e.b.91.2		6	9.2	odd	6
225.2.e.b.76.2		6	45.34	even	6
225.2.e.b.151.2		6	45.4	even	6
225.2.k.b.49.3		12	45.43	odd	12
225.2.k.b.49.4		12	45.7	odd	12
225.2.k.b.124.3		12	45.22	odd	12
225.2.k.b.124.4		12	45.13	odd	12
405.2.a.i.1.2		3	3.2	odd	2
405.2.a.j.1.2		3	1.1	even	1	trivial
675.2.e.b.226.2		6	45.29	odd	6
675.2.e.b.451.2		6	45.14	odd	6
675.2.k.b.199.3		12	45.2	even	12
675.2.k.b.199.4		12	45.38	even	12
675.2.k.b.424.3		12	45.23	even	12
675.2.k.b.424.4		12	45.32	even	12
720.2.q.i.241.2		6	36.31	odd	6
720.2.q.i.481.2		6	36.7	odd	6
2025.2.a.n.1.2		3	5.4	even	2
2025.2.a.o.1.2		3	15.14	odd	2
2025.2.b.l.649.3		6	5.3	odd	4
2025.2.b.l.649.4		6	5.2	odd	4
2025.2.b.m.649.3		6	15.2	even	4
2025.2.b.m.649.4		6	15.8	even	4
2160.2.q.k.721.2		6	36.23	even	6
2160.2.q.k.1441.2		6	36.11	even	6
6480.2.a.bs.1.2		3	12.11	even	2
6480.2.a.bv.1.2		3	4.3	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:	\( N \)	\(=\)	\( 405 = 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	405.a (trivial)

\(f(q)\)	\(=\)	\(q+0.571993 q^{2} -1.67282 q^{4} +1.00000 q^{5} +1.42801 q^{7} -2.10083 q^{8} +O(q^{10})\)	\(q+0.571993 q^{2} -1.67282 q^{4} +1.00000 q^{5} +1.42801 q^{7} -2.10083 q^{8} +0.571993 q^{10} +2.67282 q^{11} +4.67282 q^{13} +0.816810 q^{14} +2.14399 q^{16} +2.67282 q^{17} +4.67282 q^{19} -1.67282 q^{20} +1.52884 q^{22} -5.91764 q^{23} +1.00000 q^{25} +2.67282 q^{26} -2.38880 q^{28} -9.48963 q^{29} +6.96080 q^{31} +5.42801 q^{32} +1.52884 q^{34} +1.42801 q^{35} -1.81681 q^{37} +2.67282 q^{38} -2.10083 q^{40} -1.47116 q^{41} +0.471163 q^{43} -4.47116 q^{44} -3.38485 q^{46} +6.95684 q^{47} -4.96080 q^{49} +0.571993 q^{50} -7.81681 q^{52} -1.14399 q^{53} +2.67282 q^{55} -3.00000 q^{56} -5.42801 q^{58} -1.14399 q^{59} -2.52884 q^{61} +3.98153 q^{62} -1.18319 q^{64} +4.67282 q^{65} -6.59046 q^{67} -4.47116 q^{68} +0.816810 q^{70} -12.8745 q^{71} -1.71203 q^{73} -1.03920 q^{74} -7.81681 q^{76} +3.81681 q^{77} -0.287973 q^{79} +2.14399 q^{80} -0.841495 q^{82} -4.28402 q^{83} +2.67282 q^{85} +0.269502 q^{86} -5.61515 q^{88} -3.00000 q^{89} +6.67282 q^{91} +9.89917 q^{92} +3.97927 q^{94} +4.67282 q^{95} +7.83528 q^{97} -2.83754 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 5 q^{4} + 3 q^{5} + 5 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + q^2 + 5 * q^4 + 3 * q^5 + 5 * q^7 + 3 * q^8	\( 3 q + q^{2} + 5 q^{4} + 3 q^{5} + 5 q^{7} + 3 q^{8} + q^{10} - 2 q^{11} + 4 q^{13} - 9 q^{14} + 5 q^{16} - 2 q^{17} + 4 q^{19} + 5 q^{20} - 4 q^{22} + 3 q^{23} + 3 q^{25} - 2 q^{26} + 5 q^{28} - 7 q^{29} + 8 q^{31} + 17 q^{32} - 4 q^{34} + 5 q^{35} + 6 q^{37} - 2 q^{38} + 3 q^{40} - 13 q^{41} + 10 q^{43} - 22 q^{44} - 3 q^{46} + 13 q^{47} - 2 q^{49} + q^{50} - 12 q^{52} - 2 q^{53} - 2 q^{55} - 9 q^{56} - 17 q^{58} - 2 q^{59} + q^{61} + 42 q^{62} - 15 q^{64} + 4 q^{65} + 11 q^{67} - 22 q^{68} - 9 q^{70} - 10 q^{71} - 8 q^{73} - 16 q^{74} - 12 q^{76} + 2 q^{79} + 5 q^{80} - 29 q^{82} - 15 q^{83} - 2 q^{85} + 28 q^{86} - 24 q^{88} - 9 q^{89} + 10 q^{91} + 39 q^{92} - 31 q^{94} + 4 q^{95} - 18 q^{97} - 40 q^{98}+O(q^{100}) \) 3 * q + q^2 + 5 * q^4 + 3 * q^5 + 5 * q^7 + 3 * q^8 + q^10 - 2 * q^11 + 4 * q^13 - 9 * q^14 + 5 * q^16 - 2 * q^17 + 4 * q^19 + 5 * q^20 - 4 * q^22 + 3 * q^23 + 3 * q^25 - 2 * q^26 + 5 * q^28 - 7 * q^29 + 8 * q^31 + 17 * q^32 - 4 * q^34 + 5 * q^35 + 6 * q^37 - 2 * q^38 + 3 * q^40 - 13 * q^41 + 10 * q^43 - 22 * q^44 - 3 * q^46 + 13 * q^47 - 2 * q^49 + q^50 - 12 * q^52 - 2 * q^53 - 2 * q^55 - 9 * q^56 - 17 * q^58 - 2 * q^59 + q^61 + 42 * q^62 - 15 * q^64 + 4 * q^65 + 11 * q^67 - 22 * q^68 - 9 * q^70 - 10 * q^71 - 8 * q^73 - 16 * q^74 - 12 * q^76 + 2 * q^79 + 5 * q^80 - 29 * q^82 - 15 * q^83 - 2 * q^85 + 28 * q^86 - 24 * q^88 - 9 * q^89 + 10 * q^91 + 39 * q^92 - 31 * q^94 + 4 * q^95 - 18 * q^97 - 40 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more