LMFDB - Embedded newform 405.2.a.i.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)

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

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	$-0.564819\pi$
$2$	−0.571993	−0.404460	−0.202230	+	0.979338i	$0.564819\pi$
$3$	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	$-0.632013\pi$
$11$	−2.67282	−0.805887	−0.402943	+	0.915225i	$0.632013\pi$
$12$	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	$-0.605071\pi$
$17$	−2.67282	−0.648255	−0.324127	+	0.946013i	$0.605071\pi$
$18$	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	$-0.288365\pi$
$23$	5.91764	1.23391	0.616957	+	0.786997i	$0.288365\pi$
$24$	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	$-0.156810\pi$
$29$	9.48963	1.76218	0.881090	+	0.472948i	$0.156810\pi$
$30$	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	$-0.463352\pi$
$41$	1.47116	0.229757	0.114879	+	0.993380i	$0.463352\pi$
$42$	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	$-0.669386\pi$
$47$	−6.95684	−1.01476	−0.507380	+	0.861722i	$0.669386\pi$
$48$	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	$-0.474965\pi$
$53$	1.14399	0.157139	0.0785693	+	0.996909i	$0.474965\pi$
$54$	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	$-0.476274\pi$
$59$	1.14399	0.148934	0.0744672	+	0.997223i	$0.476274\pi$
$60$	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	$-0.223252\pi$
$71$	12.8745	1.52792	0.763960	+	0.645263i	$0.223252\pi$
$72$	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	$-0.424453\pi$
$83$	4.28402	0.470232	0.235116	+	0.971967i	$0.424453\pi$
$84$	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	$-0.449173\pi$
$89$	3.00000	0.317999	0.159000	+	0.987279i	$0.449173\pi$
$90$	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	$-0.432966\pi$
$101$	4.20166	0.418081	0.209040	+	0.977907i	$0.432966\pi$
$102$	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	$-0.695411\pi$
$107$	−11.9176	−1.15212	−0.576061	+	0.817407i	$0.695411\pi$
$108$	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	$-0.895426\pi$
$113$	−20.1233	−1.89304	−0.946518	+	0.322650i	$0.895426\pi$
$114$	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	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	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	$-0.356468\pi$
$137$	10.2017	0.871587	0.435793	+	0.900047i	$0.356468\pi$
$138$	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	$-0.192662\pi$
$149$	20.0761	1.64470	0.822351	+	0.568981i	$0.192662\pi$
$150$	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	$-0.683690\pi$
$167$	−14.1008	−1.09116	−0.545578	+	0.838060i	$0.683690\pi$
$168$	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	$-0.553080\pi$
$173$	−4.36638	−0.331970	−0.165985	+	0.986128i	$0.553080\pi$
$174$	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	$-0.308240\pi$
$179$	15.1625	1.13330	0.566648	+	0.823960i	$0.308240\pi$
$180$	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	$-0.467265\pi$
$191$	2.83754	0.205317	0.102659	+	0.994717i	$0.467265\pi$
$192$	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	$-0.566654\pi$
$197$	−5.83528	−0.415747	−0.207873	+	0.978156i	$0.566654\pi$
$198$	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	$-0.471729\pi$
$227$	2.67282	0.177402	0.0887008	+	0.996058i	$0.471729\pi$
$228$	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	$-0.434688\pi$
$233$	6.22013	0.407494	0.203747	+	0.979024i	$0.434688\pi$
$234$	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	$-0.584621\pi$
$239$	−8.12325	−0.525450	−0.262725	+	0.964871i	$0.584621\pi$
$240$	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	$-0.494479\pi$
$251$	0.549569	0.0346885	0.0173443	+	0.999850i	$0.494479\pi$
$252$	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	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	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	$-0.619528\pi$
$263$	−11.8952	−0.733490	−0.366745	+	0.930321i	$0.619528\pi$
$264$	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	$-0.836583\pi$
$269$	−28.5737	−1.74217	−0.871084	+	0.491134i	$0.836583\pi$
$270$	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	$-0.563579\pi$
$281$	−6.65209	−0.396831	−0.198415	+	0.980118i	$0.563579\pi$
$282$	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	$-0.617826\pi$
$293$	−12.3849	−0.723531	−0.361765	+	0.932269i	$0.617826\pi$
$294$	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	$-0.259059\pi$
$311$	24.2201	1.37340	0.686699	+	0.726942i	$0.259059\pi$
$312$	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	$-0.404994\pi$
$317$	10.4712	0.588119	0.294060	+	0.955787i	$0.404994\pi$
$318$	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	$-0.282013\pi$
$347$	23.5658	1.26508	0.632539	+	0.774529i	$0.282013\pi$
$348$	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	$-0.758458\pi$
$353$	−27.2672	−1.45129	−0.725644	+	0.688070i	$0.758458\pi$
$354$	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	$-0.409051\pi$
$359$	10.6807	0.563707	0.281854	+	0.959457i	$0.409051\pi$
$360$	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	$-0.580696\pi$
$383$	−9.81681	−0.501616	−0.250808	+	0.968037i	$0.580696\pi$
$384$	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	$-0.575096\pi$
$389$	−9.22013	−0.467479	−0.233740	+	0.972299i	$0.575096\pi$
$390$	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	$-0.410894\pi$
$401$	11.0656	0.552589	0.276294	+	0.961073i	$0.410894\pi$
$402$	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	$-0.859854\pi$
$419$	−37.0347	−1.80926	−0.904631	+	0.426195i	$0.859854\pi$
$420$	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	$-0.459797\pi$
$431$	5.23030	0.251935	0.125967	+	0.992034i	$0.459797\pi$
$432$	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	$-0.580277\pi$
$443$	−10.5042	−0.499067	−0.249534	+	0.968366i	$0.580277\pi$
$444$	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	$-0.681319\pi$
$449$	−22.8560	−1.07864	−0.539321	+	0.842100i	$0.681319\pi$
$450$	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	$-0.310308\pi$
$461$	24.1025	1.12257	0.561283	+	0.827624i	$0.310308\pi$
$462$	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	$-0.281695\pi$
$467$	27.3720	1.26663	0.633313	+	0.773896i	$0.281695\pi$
$468$	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	$-0.538126\pi$
$479$	−5.23030	−0.238978	−0.119489	+	0.992835i	$0.538126\pi$
$480$	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	$-0.608214\pi$
$491$	−14.7776	−0.666904	−0.333452	+	0.942767i	$0.608214\pi$
$492$	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	$-0.834323\pi$
$503$	−38.9154	−1.73515	−0.867576	+	0.497305i	$0.834323\pi$
$504$	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	$-0.514260\pi$
$509$	−2.02073	−0.0895674	−0.0447837	+	0.998997i	$0.514260\pi$
$510$	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	$-0.331692\pi$
$521$	23.0290	1.00892	0.504460	+	0.863435i	$0.331692\pi$
$522$	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	$-0.401295\pi$
$557$	14.4033	0.610288	0.305144	+	0.952306i	$0.401295\pi$
$558$	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	$-0.712405\pi$
$563$	−29.3681	−1.23772	−0.618858	+	0.785503i	$0.712405\pi$
$564$	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	$-0.938046\pi$
$569$	−46.8066	−1.96224	−0.981118	+	0.193409i	$0.938046\pi$
$570$	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	$-0.621729\pi$
$587$	−18.0824	−0.746339	−0.373169	+	0.927763i	$0.621729\pi$
$588$	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	$-0.449209\pi$
$593$	7.73840	0.317778	0.158889	+	0.987296i	$0.449209\pi$
$594$	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	$-0.306779\pi$
$599$	27.9216	1.14085	0.570423	+	0.821351i	$0.306779\pi$
$600$	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	$-0.639648\pi$
$617$	−21.1025	−0.849556	−0.424778	+	0.905298i	$0.639648\pi$
$618$	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	$-0.673737\pi$
$641$	−26.2857	−1.03822	−0.519112	+	0.854706i	$0.673737\pi$
$642$	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	$-0.651104\pi$
$647$	−23.2527	−0.914159	−0.457079	+	0.889426i	$0.651104\pi$
$648$	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	$-0.619633\pi$
$653$	−18.7591	−0.734102	−0.367051	+	0.930201i	$0.619633\pi$
$654$	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	$-0.498264\pi$
$659$	0.280067	0.0109099	0.00545494	+	0.999985i	$0.498264\pi$
$660$	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	$-0.677138\pi$
$677$	−27.4874	−1.05643	−0.528213	+	0.849112i	$0.677138\pi$
$678$	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	$-0.269832\pi$
$683$	34.5865	1.32342	0.661708	+	0.749762i	$0.269832\pi$
$684$	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	$-0.380143\pi$
$701$	19.4712	0.735416	0.367708	+	0.929941i	$0.380143\pi$
$702$	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	$-0.479616\pi$
$719$	3.43196	0.127990	0.0639952	+	0.997950i	$0.479616\pi$
$720$	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	$-0.345051\pi$
$743$	25.5019	0.935574	0.467787	+	0.883841i	$0.345051\pi$
$744$	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	$-0.453849\pi$
$761$	7.97136	0.288962	0.144481	+	0.989508i	$0.453849\pi$
$762$	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	$-0.794393\pi$
$773$	−44.4033	−1.59708	−0.798538	+	0.601944i	$0.794393\pi$
$774$	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	$-0.298714\pi$
$797$	33.3720	1.18210	0.591049	+	0.806636i	$0.298714\pi$
$798$	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	$-0.671456\pi$
$809$	−29.1809	−1.02595	−0.512973	+	0.858404i	$0.671456\pi$
$810$	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	$-0.364175\pi$
$821$	23.7177	0.827752	0.413876	+	0.910333i	$0.364175\pi$
$822$	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	$-0.862395\pi$
$827$	−52.2241	−1.81601	−0.908005	+	0.418960i	$0.862395\pi$
$828$	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	$-0.357729\pi$
$839$	25.0392	0.864449	0.432225	+	0.901766i	$0.357729\pi$
$840$	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	$-0.397128\pi$
$857$	18.5944	0.635173	0.317587	+	0.948229i	$0.397128\pi$
$858$	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	$-0.341516\pi$
$863$	28.0594	0.955152	0.477576	+	0.878590i	$0.341516\pi$
$864$	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	$-0.471551\pi$
$881$	5.29854	0.178512	0.0892561	+	0.996009i	$0.471551\pi$
$882$	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	$-0.744988\pi$
$887$	−41.4504	−1.39177	−0.695885	+	0.718154i	$0.744988\pi$
$888$	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	$-0.712377\pi$
$911$	−37.3536	−1.23758	−0.618789	+	0.785557i	$0.712377\pi$
$912$	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	$-0.211458\pi$
$929$	47.9955	1.57468	0.787340	+	0.616519i	$0.211458\pi$
$930$	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	$-0.730082\pi$
$941$	−40.5843	−1.32301	−0.661504	+	0.749941i	$0.730082\pi$
$942$	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	$-0.663443\pi$
$947$	−30.2320	−0.982408	−0.491204	+	0.871045i	$0.663443\pi$
$948$	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	$-0.512934\pi$
$953$	−2.50811	−0.0812455	−0.0406227	+	0.999175i	$0.512934\pi$
$954$	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	$-0.618952\pi$
$971$	−22.7512	−0.730122	−0.365061	+	0.930984i	$0.618952\pi$
$972$	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	$-0.716251\pi$
$977$	−39.2778	−1.25661	−0.628304	+	0.777968i	$0.716251\pi$
$978$	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	$-0.681147\pi$
$983$	−33.7899	−1.07773	−0.538865	+	0.842392i	$0.681147\pi$
$984$	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.i.1.2		3
3.2	odd	2		405.2.a.j.1.2		3
4.3	odd	2		6480.2.a.bs.1.2		3
5.2	odd	4		2025.2.b.m.649.3		6
5.3	odd	4		2025.2.b.m.649.4		6
5.4	even	2		2025.2.a.o.1.2		3
9.2	odd	6		45.2.e.b.31.2	yes	6
9.4	even	3		135.2.e.b.46.2		6
9.5	odd	6		45.2.e.b.16.2	✓	6
9.7	even	3		135.2.e.b.91.2		6
12.11	even	2		6480.2.a.bv.1.2		3
15.2	even	4		2025.2.b.l.649.4		6
15.8	even	4		2025.2.b.l.649.3		6
15.14	odd	2		2025.2.a.n.1.2		3
36.7	odd	6		2160.2.q.k.1441.2		6
36.11	even	6		720.2.q.i.481.2		6
36.23	even	6		720.2.q.i.241.2		6
36.31	odd	6		2160.2.q.k.721.2		6
45.2	even	12		225.2.k.b.49.4		12
45.4	even	6		675.2.e.b.451.2		6
45.7	odd	12		675.2.k.b.199.3		12
45.13	odd	12		675.2.k.b.424.3		12
45.14	odd	6		225.2.e.b.151.2		6
45.22	odd	12		675.2.k.b.424.4		12
45.23	even	12		225.2.k.b.124.4		12
45.29	odd	6		225.2.e.b.76.2		6
45.32	even	12		225.2.k.b.124.3		12
45.34	even	6		675.2.e.b.226.2		6
45.38	even	12		225.2.k.b.49.3		12
45.43	odd	12		675.2.k.b.199.4		12

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

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