LMFDB - Embedded newform 9405.2.a.bh.1.5

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:	$1$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 9x^{7} + 29x^{6} + 23x^{5} - 84x^{4} - 23x^{3} + 89x^{2} + 8x - 27 $ x^9 - 3x^8 - 9x^7 + 29x^6 + 23x^5 - 84x^4 - 23x^3 + 89x^2 + 8x - 27
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1045)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.682920$ 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)

$f(q)$	$=$	$q-0.682920 q^{2} -1.53362 q^{4} -1.00000 q^{5} +0.856948 q^{7} +2.41318 q^{8} +O(q^{10})$	\(q-0.682920 q^{2} -1.53362 q^{4} -1.00000 q^{5} +0.856948 q^{7} +2.41318 q^{8} +0.682920 q^{10} -1.00000 q^{11} +5.50630 q^{13} -0.585228 q^{14} +1.41923 q^{16} +3.50630 q^{17} +1.00000 q^{19} +1.53362 q^{20} +0.682920 q^{22} -6.75237 q^{23} +1.00000 q^{25} -3.76036 q^{26} -1.31423 q^{28} +0.474680 q^{29} -5.19745 q^{31} -5.79558 q^{32} -2.39452 q^{34} -0.856948 q^{35} +0.420904 q^{37} -0.682920 q^{38} -2.41318 q^{40} -8.25141 q^{41} -1.53320 q^{43} +1.53362 q^{44} +4.61133 q^{46} -2.48849 q^{47} -6.26564 q^{49} -0.682920 q^{50} -8.44457 q^{52} +4.79753 q^{53} +1.00000 q^{55} +2.06797 q^{56} -0.324169 q^{58} -1.75917 q^{59} +3.42433 q^{61} +3.54944 q^{62} +1.11946 q^{64} -5.50630 q^{65} -5.12424 q^{67} -5.37733 q^{68} +0.585228 q^{70} +12.1937 q^{71} -12.7638 q^{73} -0.287444 q^{74} -1.53362 q^{76} -0.856948 q^{77} +10.8691 q^{79} -1.41923 q^{80} +5.63505 q^{82} -2.94669 q^{83} -3.50630 q^{85} +1.04705 q^{86} -2.41318 q^{88} -3.08736 q^{89} +4.71861 q^{91} +10.3556 q^{92} +1.69944 q^{94} -1.00000 q^{95} -8.88210 q^{97} +4.27893 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - 3 q^{2} + 9 q^{4} - 9 q^{5} + 13 q^{7} - 9 q^{8}+O(q^{10}) $ 9 * q - 3 * q^2 + 9 * q^4 - 9 * q^5 + 13 * q^7 - 9 * q^8	$ 9 q - 3 q^{2} + 9 q^{4} - 9 q^{5} + 13 q^{7} - 9 q^{8} + 3 q^{10} - 9 q^{11} + 5 q^{13} + 2 q^{14} + q^{16} - 13 q^{17} + 9 q^{19} - 9 q^{20} + 3 q^{22} - 8 q^{23} + 9 q^{25} - 8 q^{26} + 10 q^{28} + 3 q^{29} - 9 q^{31} - 6 q^{32} - 2 q^{34} - 13 q^{35} - 7 q^{37} - 3 q^{38} + 9 q^{40} - 9 q^{41} + 23 q^{43} - 9 q^{44} - 32 q^{46} - 20 q^{47} - 4 q^{49} - 3 q^{50} + 9 q^{52} + 5 q^{53} + 9 q^{55} - 4 q^{56} + 3 q^{58} - 19 q^{59} + q^{61} - 18 q^{62} + 23 q^{64} - 5 q^{65} - 10 q^{67} - 9 q^{68} - 2 q^{70} + 12 q^{73} - 5 q^{74} + 9 q^{76} - 13 q^{77} - 21 q^{79} - q^{80} - 14 q^{82} - 47 q^{83} + 13 q^{85} - 12 q^{86} + 9 q^{88} + 2 q^{89} + 2 q^{91} + 19 q^{92} + 26 q^{94} - 9 q^{95} - 32 q^{97}+O(q^{100}) $ 9 * q - 3 * q^2 + 9 * q^4 - 9 * q^5 + 13 * q^7 - 9 * q^8 + 3 * q^10 - 9 * q^11 + 5 * q^13 + 2 * q^14 + q^16 - 13 * q^17 + 9 * q^19 - 9 * q^20 + 3 * q^22 - 8 * q^23 + 9 * q^25 - 8 * q^26 + 10 * q^28 + 3 * q^29 - 9 * q^31 - 6 * q^32 - 2 * q^34 - 13 * q^35 - 7 * q^37 - 3 * q^38 + 9 * q^40 - 9 * q^41 + 23 * q^43 - 9 * q^44 - 32 * q^46 - 20 * q^47 - 4 * q^49 - 3 * q^50 + 9 * q^52 + 5 * q^53 + 9 * q^55 - 4 * q^56 + 3 * q^58 - 19 * q^59 + q^61 - 18 * q^62 + 23 * q^64 - 5 * q^65 - 10 * q^67 - 9 * q^68 - 2 * q^70 + 12 * q^73 - 5 * q^74 + 9 * q^76 - 13 * q^77 - 21 * q^79 - q^80 - 14 * q^82 - 47 * q^83 + 13 * q^85 - 12 * q^86 + 9 * q^88 + 2 * q^89 + 2 * q^91 + 19 * q^92 + 26 * q^94 - 9 * q^95 - 32 * q^97

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 $

$2$	−0.682920	−0.482898	−0.241449	−	0.970414i	$-0.577623\pi$
$2$	−0.682920	−0.482898	−0.241449	+	0.970414i	$0.577623\pi$
$3$	0	0
$3$	0	0
$4$	−1.53362	−0.766810
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.856948	0.323896	0.161948	−	0.986799i	$-0.448222\pi$
$7$	0.856948	0.323896	0.161948	+	0.986799i	$0.448222\pi$
$8$	2.41318	0.853188
$9$	0	0
$10$	0.682920	0.215958
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.50630	1.52717	0.763586	−	0.645706i	$-0.223437\pi$
$13$	5.50630	1.52717	0.763586	+	0.645706i	$0.223437\pi$
$14$	−0.585228	−0.156409
$15$	0	0
$16$	1.41923	0.354807
$17$	3.50630	0.850402	0.425201	−	0.905099i	$-0.360203\pi$
$17$	3.50630	0.850402	0.425201	+	0.905099i	$0.360203\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	1.53362	0.342928
$21$	0	0
$22$	0.682920	0.145599
$23$	−6.75237	−1.40797	−0.703984	−	0.710216i	$-0.748597\pi$
$23$	−6.75237	−1.40797	−0.703984	+	0.710216i	$0.748597\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−3.76036	−0.737468
$27$	0	0
$28$	−1.31423	−0.248367
$29$	0.474680	0.0881458	0.0440729	−	0.999028i	$-0.485967\pi$
$29$	0.474680	0.0881458	0.0440729	+	0.999028i	$0.485967\pi$
$30$	0	0
$31$	−5.19745	−0.933489	−0.466745	−	0.884392i	$-0.654573\pi$
$31$	−5.19745	−0.933489	−0.466745	+	0.884392i	$0.654573\pi$
$32$	−5.79558	−1.02452
$33$	0	0
$34$	−2.39452	−0.410657
$35$	−0.856948	−0.144851
$36$	0	0
$37$	0.420904	0.0691961	0.0345981	−	0.999401i	$-0.488985\pi$
$37$	0.420904	0.0691961	0.0345981	+	0.999401i	$0.488985\pi$
$38$	−0.682920	−0.110784
$39$	0	0
$40$	−2.41318	−0.381557
$41$	−8.25141	−1.28865	−0.644327	−	0.764750i	$-0.722862\pi$
$41$	−8.25141	−1.28865	−0.644327	+	0.764750i	$0.722862\pi$
$42$	0	0
$43$	−1.53320	−0.233811	−0.116905	−	0.993143i	$-0.537297\pi$
$43$	−1.53320	−0.233811	−0.116905	+	0.993143i	$0.537297\pi$
$44$	1.53362	0.231202
$45$	0	0
$46$	4.61133	0.679904
$47$	−2.48849	−0.362984	−0.181492	−	0.983392i	$-0.558093\pi$
$47$	−2.48849	−0.362984	−0.181492	+	0.983392i	$0.558093\pi$
$48$	0	0
$49$	−6.26564	−0.895091
$50$	−0.682920	−0.0965795
$51$	0	0
$52$	−8.44457	−1.17105
$53$	4.79753	0.658991	0.329495	−	0.944157i	$-0.393121\pi$
$53$	4.79753	0.658991	0.329495	+	0.944157i	$0.393121\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	2.06797	0.276344
$57$	0	0
$58$	−0.324169	−0.0425654
$59$	−1.75917	−0.229025	−0.114513	−	0.993422i	$-0.536531\pi$
$59$	−1.75917	−0.229025	−0.114513	+	0.993422i	$0.536531\pi$
$60$	0	0
$61$	3.42433	0.438440	0.219220	−	0.975675i	$-0.429649\pi$
$61$	3.42433	0.438440	0.219220	+	0.975675i	$0.429649\pi$
$62$	3.54944	0.450780
$63$	0	0
$64$	1.11946	0.139933
$65$	−5.50630	−0.682972
$66$	0	0
$67$	−5.12424	−0.626026	−0.313013	−	0.949749i	$-0.601338\pi$
$67$	−5.12424	−0.626026	−0.313013	+	0.949749i	$0.601338\pi$
$68$	−5.37733	−0.652097
$69$	0	0
$70$	0.585228	0.0699481
$71$	12.1937	1.44713	0.723563	−	0.690258i	$-0.242503\pi$
$71$	12.1937	1.44713	0.723563	+	0.690258i	$0.242503\pi$
$72$	0	0
$73$	−12.7638	−1.49389	−0.746945	−	0.664886i	$-0.768480\pi$
$73$	−12.7638	−1.49389	−0.746945	+	0.664886i	$0.768480\pi$
$74$	−0.287444	−0.0334146
$75$	0	0
$76$	−1.53362	−0.175918
$77$	−0.856948	−0.0976583
$78$	0	0
$79$	10.8691	1.22287	0.611437	−	0.791293i	$-0.290592\pi$
$79$	10.8691	1.22287	0.611437	+	0.791293i	$0.290592\pi$
$80$	−1.41923	−0.158675
$81$	0	0
$82$	5.63505	0.622288
$83$	−2.94669	−0.323441	−0.161721	−	0.986837i	$-0.551704\pi$
$83$	−2.94669	−0.323441	−0.161721	+	0.986837i	$0.551704\pi$
$84$	0	0
$85$	−3.50630	−0.380311
$86$	1.04705	0.112907
$87$	0	0
$88$	−2.41318	−0.257246
$89$	−3.08736	−0.327259	−0.163629	−	0.986522i	$-0.552320\pi$
$89$	−3.08736	−0.327259	−0.163629	+	0.986522i	$0.552320\pi$
$90$	0	0
$91$	4.71861	0.494645
$92$	10.3556	1.07964
$93$	0	0
$94$	1.69944	0.175284
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−8.88210	−0.901840	−0.450920	−	0.892564i	$-0.648904\pi$
$97$	−8.88210	−0.901840	−0.450920	+	0.892564i	$0.648904\pi$
$98$	4.27893	0.432238
$99$	0	0
$100$	−1.53362	−0.153362
$101$	2.09965	0.208923	0.104462	−	0.994529i	$-0.466688\pi$
$101$	2.09965	0.208923	0.104462	+	0.994529i	$0.466688\pi$
$102$	0	0
$103$	−3.91556	−0.385812	−0.192906	−	0.981217i	$-0.561791\pi$
$103$	−3.91556	−0.385812	−0.192906	+	0.981217i	$0.561791\pi$
$104$	13.2877	1.30297
$105$	0	0
$106$	−3.27633	−0.318225
$107$	18.0439	1.74437	0.872184	−	0.489178i	$-0.162703\pi$
$107$	18.0439	1.74437	0.872184	+	0.489178i	$0.162703\pi$
$108$	0	0
$109$	3.34558	0.320448	0.160224	−	0.987081i	$-0.448778\pi$
$109$	3.34558	0.320448	0.160224	+	0.987081i	$0.448778\pi$
$110$	−0.682920	−0.0651139
$111$	0	0
$112$	1.21621	0.114921
$113$	−7.26819	−0.683733	−0.341867	−	0.939748i	$-0.611059\pi$
$113$	−7.26819	−0.683733	−0.341867	+	0.939748i	$0.611059\pi$
$114$	0	0
$115$	6.75237	0.629662
$116$	−0.727978	−0.0675911
$117$	0	0
$118$	1.20138	0.110596
$119$	3.00472	0.275442
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.33854	−0.211722
$123$	0	0
$124$	7.97091	0.715809
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−16.9303	−1.50232	−0.751162	−	0.660118i	$-0.770506\pi$
$127$	−16.9303	−1.50232	−0.751162	+	0.660118i	$0.770506\pi$
$128$	10.8267	0.956951
$129$	0	0
$130$	3.76036	0.329806
$131$	−5.26619	−0.460109	−0.230055	−	0.973178i	$-0.573891\pi$
$131$	−5.26619	−0.460109	−0.230055	+	0.973178i	$0.573891\pi$
$132$	0	0
$133$	0.856948	0.0743069
$134$	3.49945	0.302307
$135$	0	0
$136$	8.46133	0.725553
$137$	20.6078	1.76064	0.880321	−	0.474378i	$-0.157327\pi$
$137$	20.6078	1.76064	0.880321	+	0.474378i	$0.157327\pi$
$138$	0	0
$139$	1.26192	0.107034	0.0535171	−	0.998567i	$-0.482957\pi$
$139$	1.26192	0.107034	0.0535171	+	0.998567i	$0.482957\pi$
$140$	1.31423	0.111073
$141$	0	0
$142$	−8.32733	−0.698814
$143$	−5.50630	−0.460460
$144$	0	0
$145$	−0.474680	−0.0394200
$146$	8.71666	0.721396
$147$	0	0
$148$	−0.645506	−0.0530603
$149$	11.3117	0.926692	0.463346	−	0.886177i	$-0.346649\pi$
$149$	11.3117	0.926692	0.463346	+	0.886177i	$0.346649\pi$
$150$	0	0
$151$	−3.70074	−0.301162	−0.150581	−	0.988598i	$-0.548114\pi$
$151$	−3.70074	−0.301162	−0.150581	+	0.988598i	$0.548114\pi$
$152$	2.41318	0.195735
$153$	0	0
$154$	0.585228	0.0471590
$155$	5.19745	0.417469
$156$	0	0
$157$	2.91493	0.232637	0.116318	−	0.993212i	$-0.462891\pi$
$157$	2.91493	0.232637	0.116318	+	0.993212i	$0.462891\pi$
$158$	−7.42276	−0.590523
$159$	0	0
$160$	5.79558	0.458181
$161$	−5.78644	−0.456035
$162$	0	0
$163$	18.4278	1.44338	0.721690	−	0.692216i	$-0.243366\pi$
$163$	18.4278	1.44338	0.721690	+	0.692216i	$0.243366\pi$
$164$	12.6545	0.988152
$165$	0	0
$166$	2.01235	0.156189
$167$	−15.1433	−1.17182	−0.585912	−	0.810374i	$-0.699264\pi$
$167$	−15.1433	−1.17182	−0.585912	+	0.810374i	$0.699264\pi$
$168$	0	0
$169$	17.3193	1.33226
$170$	2.39452	0.183652
$171$	0	0
$172$	2.35135	0.179288
$173$	0.110615	0.00840989	0.00420495	−	0.999991i	$-0.498662\pi$
$173$	0.110615	0.00840989	0.00420495	+	0.999991i	$0.498662\pi$
$174$	0	0
$175$	0.856948	0.0647792
$176$	−1.41923	−0.106978
$177$	0	0
$178$	2.10842	0.158033
$179$	9.42389	0.704375	0.352187	−	0.935930i	$-0.385438\pi$
$179$	9.42389	0.704375	0.352187	+	0.935930i	$0.385438\pi$
$180$	0	0
$181$	−12.2876	−0.913334	−0.456667	−	0.889638i	$-0.650957\pi$
$181$	−12.2876	−0.913334	−0.456667	+	0.889638i	$0.650957\pi$
$182$	−3.22244	−0.238863
$183$	0	0
$184$	−16.2947	−1.20126
$185$	−0.420904	−0.0309454
$186$	0	0
$187$	−3.50630	−0.256406
$188$	3.81640	0.278340
$189$	0	0
$190$	0.682920	0.0495443
$191$	−16.4558	−1.19070	−0.595351	−	0.803466i	$-0.702987\pi$
$191$	−16.4558	−1.19070	−0.595351	+	0.803466i	$0.702987\pi$
$192$	0	0
$193$	2.79958	0.201518	0.100759	−	0.994911i	$-0.467873\pi$
$193$	2.79958	0.201518	0.100759	+	0.994911i	$0.467873\pi$
$194$	6.06577	0.435497
$195$	0	0
$196$	9.60911	0.686365
$197$	7.19454	0.512589	0.256295	−	0.966599i	$-0.417498\pi$
$197$	7.19454	0.512589	0.256295	+	0.966599i	$0.417498\pi$
$198$	0	0
$199$	−11.9287	−0.845607	−0.422803	−	0.906221i	$-0.638954\pi$
$199$	−11.9287	−0.845607	−0.422803	+	0.906221i	$0.638954\pi$
$200$	2.41318	0.170638
$201$	0	0
$202$	−1.43390	−0.100889
$203$	0.406776	0.0285501
$204$	0	0
$205$	8.25141	0.576303
$206$	2.67402	0.186308
$207$	0	0
$208$	7.81470	0.541852
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	18.0430	1.24213	0.621065	−	0.783759i	$-0.286700\pi$
$211$	18.0430	1.24213	0.621065	+	0.783759i	$0.286700\pi$
$212$	−7.35758	−0.505321
$213$	0	0
$214$	−12.3225	−0.842351
$215$	1.53320	0.104563
$216$	0	0
$217$	−4.45395	−0.302354
$218$	−2.28476	−0.154744
$219$	0	0
$220$	−1.53362	−0.103397
$221$	19.3067	1.29871
$222$	0	0
$223$	−24.5001	−1.64065	−0.820323	−	0.571901i	$-0.806206\pi$
$223$	−24.5001	−1.64065	−0.820323	+	0.571901i	$0.806206\pi$
$224$	−4.96652	−0.331839
$225$	0	0
$226$	4.96359	0.330173
$227$	−16.6675	−1.10626	−0.553131	−	0.833094i	$-0.686567\pi$
$227$	−16.6675	−1.10626	−0.553131	+	0.833094i	$0.686567\pi$
$228$	0	0
$229$	14.8344	0.980282	0.490141	−	0.871643i	$-0.336945\pi$
$229$	14.8344	0.980282	0.490141	+	0.871643i	$0.336945\pi$
$230$	−4.61133	−0.304062
$231$	0	0
$232$	1.14549	0.0752050
$233$	−28.8509	−1.89009	−0.945043	−	0.326947i	$-0.893980\pi$
$233$	−28.8509	−1.89009	−0.945043	+	0.326947i	$0.893980\pi$
$234$	0	0
$235$	2.48849	0.162331
$236$	2.69790	0.175619
$237$	0	0
$238$	−2.05198	−0.133010
$239$	−8.29173	−0.536347	−0.268174	−	0.963371i	$-0.586420\pi$
$239$	−8.29173	−0.536347	−0.268174	+	0.963371i	$0.586420\pi$
$240$	0	0
$241$	3.48268	0.224339	0.112170	−	0.993689i	$-0.464220\pi$
$241$	3.48268	0.224339	0.112170	+	0.993689i	$0.464220\pi$
$242$	−0.682920	−0.0438998
$243$	0	0
$244$	−5.25162	−0.336200
$245$	6.26564	0.400297
$246$	0	0
$247$	5.50630	0.350357
$248$	−12.5424	−0.796442
$249$	0	0
$250$	0.682920	0.0431917
$251$	28.8156	1.81883	0.909413	−	0.415895i	$-0.136532\pi$
$251$	28.8156	1.81883	0.909413	+	0.415895i	$0.136532\pi$
$252$	0	0
$253$	6.75237	0.424518
$254$	11.5621	0.725469
$255$	0	0
$256$	−9.63267	−0.602042
$257$	8.44177	0.526583	0.263292	−	0.964716i	$-0.415192\pi$
$257$	8.44177	0.526583	0.263292	+	0.964716i	$0.415192\pi$
$258$	0	0
$259$	0.360693	0.0224124
$260$	8.44457	0.523710
$261$	0	0
$262$	3.59639	0.222186
$263$	−21.4210	−1.32088	−0.660438	−	0.750881i	$-0.729629\pi$
$263$	−21.4210	−1.32088	−0.660438	+	0.750881i	$0.729629\pi$
$264$	0	0
$265$	−4.79753	−0.294710
$266$	−0.585228	−0.0358826
$267$	0	0
$268$	7.85864	0.480043
$269$	−10.3642	−0.631916	−0.315958	−	0.948773i	$-0.602326\pi$
$269$	−10.3642	−0.631916	−0.315958	+	0.948773i	$0.602326\pi$
$270$	0	0
$271$	−26.1079	−1.58594	−0.792972	−	0.609258i	$-0.791467\pi$
$271$	−26.1079	−1.58594	−0.792972	+	0.609258i	$0.791467\pi$
$272$	4.97624	0.301729
$273$	0	0
$274$	−14.0735	−0.850210
$275$	−1.00000	−0.0603023
$276$	0	0
$277$	−18.7663	−1.12756	−0.563779	−	0.825925i	$-0.690653\pi$
$277$	−18.7663	−1.12756	−0.563779	+	0.825925i	$0.690653\pi$
$278$	−0.861788	−0.0516866
$279$	0	0
$280$	−2.06797	−0.123585
$281$	−24.6322	−1.46944	−0.734718	−	0.678373i	$-0.762685\pi$
$281$	−24.6322	−1.46944	−0.734718	+	0.678373i	$0.762685\pi$
$282$	0	0
$283$	−13.7870	−0.819550	−0.409775	−	0.912187i	$-0.634393\pi$
$283$	−13.7870	−0.819550	−0.409775	+	0.912187i	$0.634393\pi$
$284$	−18.7005	−1.10967
$285$	0	0
$286$	3.76036	0.222355
$287$	−7.07103	−0.417390
$288$	0	0
$289$	−4.70587	−0.276816
$290$	0.324169	0.0190358
$291$	0	0
$292$	19.5748	1.14553
$293$	3.53033	0.206244	0.103122	−	0.994669i	$-0.467117\pi$
$293$	3.53033	0.206244	0.103122	+	0.994669i	$0.467117\pi$
$294$	0	0
$295$	1.75917	0.102423
$296$	1.01572	0.0590373
$297$	0	0
$298$	−7.72501	−0.447498
$299$	−37.1806	−2.15021
$300$	0	0
$301$	−1.31387	−0.0757304
$302$	2.52731	0.145430
$303$	0	0
$304$	1.41923	0.0813984
$305$	−3.42433	−0.196076
$306$	0	0
$307$	5.59192	0.319148	0.159574	−	0.987186i	$-0.448988\pi$
$307$	5.59192	0.319148	0.159574	+	0.987186i	$0.448988\pi$
$308$	1.31423	0.0748854
$309$	0	0
$310$	−3.54944	−0.201595
$311$	3.38109	0.191724	0.0958621	−	0.995395i	$-0.469439\pi$
$311$	3.38109	0.191724	0.0958621	+	0.995395i	$0.469439\pi$
$312$	0	0
$313$	13.5676	0.766886	0.383443	−	0.923565i	$-0.374738\pi$
$313$	13.5676	0.766886	0.383443	+	0.923565i	$0.374738\pi$
$314$	−1.99066	−0.112340
$315$	0	0
$316$	−16.6691	−0.937712
$317$	9.02947	0.507145	0.253573	−	0.967316i	$-0.418394\pi$
$317$	9.02947	0.507145	0.253573	+	0.967316i	$0.418394\pi$
$318$	0	0
$319$	−0.474680	−0.0265770
$320$	−1.11946	−0.0625799
$321$	0	0
$322$	3.95167	0.220218
$323$	3.50630	0.195096
$324$	0	0
$325$	5.50630	0.305434
$326$	−12.5847	−0.697005
$327$	0	0
$328$	−19.9121	−1.09946
$329$	−2.13251	−0.117569
$330$	0	0
$331$	−18.6966	−1.02766	−0.513829	−	0.857892i	$-0.671774\pi$
$331$	−18.6966	−1.02766	−0.513829	+	0.857892i	$0.671774\pi$
$332$	4.51910	0.248018
$333$	0	0
$334$	10.3417	0.565871
$335$	5.12424	0.279967
$336$	0	0
$337$	29.3264	1.59751	0.798755	−	0.601657i	$-0.205492\pi$
$337$	29.3264	1.59751	0.798755	+	0.601657i	$0.205492\pi$
$338$	−11.8277	−0.643343
$339$	0	0
$340$	5.37733	0.291627
$341$	5.19745	0.281458
$342$	0	0
$343$	−11.3680	−0.613813
$344$	−3.69989	−0.199485
$345$	0	0
$346$	−0.0755411	−0.00406112
$347$	−11.0048	−0.590768	−0.295384	−	0.955379i	$-0.595448\pi$
$347$	−11.0048	−0.590768	−0.295384	+	0.955379i	$0.595448\pi$
$348$	0	0
$349$	−24.1430	−1.29235	−0.646174	−	0.763190i	$-0.723632\pi$
$349$	−24.1430	−1.29235	−0.646174	+	0.763190i	$0.723632\pi$
$350$	−0.585228	−0.0312817
$351$	0	0
$352$	5.79558	0.308906
$353$	−13.1068	−0.697602	−0.348801	−	0.937197i	$-0.613411\pi$
$353$	−13.1068	−0.697602	−0.348801	+	0.937197i	$0.613411\pi$
$354$	0	0
$355$	−12.1937	−0.647175
$356$	4.73483	0.250945
$357$	0	0
$358$	−6.43577	−0.340141
$359$	11.2347	0.592942	0.296471	−	0.955042i	$-0.404190\pi$
$359$	11.2347	0.592942	0.296471	+	0.955042i	$0.404190\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	8.39149	0.441047
$363$	0	0
$364$	−7.23656	−0.379299
$365$	12.7638	0.668088
$366$	0	0
$367$	0.796321	0.0415676	0.0207838	−	0.999784i	$-0.493384\pi$
$367$	0.796321	0.0415676	0.0207838	+	0.999784i	$0.493384\pi$
$368$	−9.58316	−0.499557
$369$	0	0
$370$	0.287444	0.0149435
$371$	4.11123	0.213445
$372$	0	0
$373$	4.22370	0.218695	0.109347	−	0.994004i	$-0.465124\pi$
$373$	4.22370	0.218695	0.109347	+	0.994004i	$0.465124\pi$
$374$	2.39452	0.123818
$375$	0	0
$376$	−6.00518	−0.309694
$377$	2.61373	0.134614
$378$	0	0
$379$	−22.5751	−1.15961	−0.579803	−	0.814757i	$-0.696870\pi$
$379$	−22.5751	−1.15961	−0.579803	+	0.814757i	$0.696870\pi$
$380$	1.53362	0.0786730
$381$	0	0
$382$	11.2380	0.574987
$383$	−32.5159	−1.66148	−0.830742	−	0.556658i	$-0.812083\pi$
$383$	−32.5159	−1.66148	−0.830742	+	0.556658i	$0.812083\pi$
$384$	0	0
$385$	0.856948	0.0436741
$386$	−1.91189	−0.0973128
$387$	0	0
$388$	13.6218	0.691540
$389$	35.4867	1.79925	0.899624	−	0.436665i	$-0.143840\pi$
$389$	35.4867	1.79925	0.899624	+	0.436665i	$0.143840\pi$
$390$	0	0
$391$	−23.6758	−1.19734
$392$	−15.1201	−0.763681
$393$	0	0
$394$	−4.91329	−0.247528
$395$	−10.8691	−0.546886
$396$	0	0
$397$	11.7223	0.588325	0.294162	−	0.955755i	$-0.404959\pi$
$397$	11.7223	0.588325	0.294162	+	0.955755i	$0.404959\pi$
$398$	8.14639	0.408341
$399$	0	0
$400$	1.41923	0.0709614
$401$	33.5093	1.67337	0.836687	−	0.547681i	$-0.184489\pi$
$401$	33.5093	1.67337	0.836687	+	0.547681i	$0.184489\pi$
$402$	0	0
$403$	−28.6187	−1.42560
$404$	−3.22007	−0.160205
$405$	0	0
$406$	−0.277796	−0.0137868
$407$	−0.420904	−0.0208634
$408$	0	0
$409$	−6.50358	−0.321581	−0.160791	−	0.986989i	$-0.551404\pi$
$409$	−6.50358	−0.321581	−0.160791	+	0.986989i	$0.551404\pi$
$410$	−5.63505	−0.278295
$411$	0	0
$412$	6.00499	0.295844
$413$	−1.50752	−0.0741803
$414$	0	0
$415$	2.94669	0.144647
$416$	−31.9122	−1.56462
$417$	0	0
$418$	0.682920	0.0334027
$419$	7.49790	0.366296	0.183148	−	0.983085i	$-0.441371\pi$
$419$	7.49790	0.366296	0.183148	+	0.983085i	$0.441371\pi$
$420$	0	0
$421$	11.5280	0.561840	0.280920	−	0.959731i	$-0.409360\pi$
$421$	11.5280	0.561840	0.280920	+	0.959731i	$0.409360\pi$
$422$	−12.3219	−0.599822
$423$	0	0
$424$	11.5773	0.562243
$425$	3.50630	0.170080
$426$	0	0
$427$	2.93447	0.142009
$428$	−27.6724	−1.33760
$429$	0	0
$430$	−1.04705	−0.0504934
$431$	−22.4459	−1.08118	−0.540590	−	0.841286i	$-0.681799\pi$
$431$	−22.4459	−1.08118	−0.540590	+	0.841286i	$0.681799\pi$
$432$	0	0
$433$	−27.4999	−1.32156	−0.660781	−	0.750579i	$-0.729775\pi$
$433$	−27.4999	−1.32156	−0.660781	+	0.750579i	$0.729775\pi$
$434$	3.04169	0.146006
$435$	0	0
$436$	−5.13084	−0.245723
$437$	−6.75237	−0.323010
$438$	0	0
$439$	8.18489	0.390644	0.195322	−	0.980739i	$-0.437425\pi$
$439$	8.18489	0.390644	0.195322	+	0.980739i	$0.437425\pi$
$440$	2.41318	0.115044
$441$	0	0
$442$	−13.1850	−0.627144
$443$	11.1175	0.528209	0.264104	−	0.964494i	$-0.414924\pi$
$443$	11.1175	0.528209	0.264104	+	0.964494i	$0.414924\pi$
$444$	0	0
$445$	3.08736	0.146355
$446$	16.7316	0.792264
$447$	0	0
$448$	0.959323	0.0453237
$449$	−11.1426	−0.525854	−0.262927	−	0.964816i	$-0.584688\pi$
$449$	−11.1426	−0.525854	−0.262927	+	0.964816i	$0.584688\pi$
$450$	0	0
$451$	8.25141	0.388544
$452$	11.1466	0.524294
$453$	0	0
$454$	11.3826	0.534212
$455$	−4.71861	−0.221212
$456$	0	0
$457$	18.3241	0.857164	0.428582	−	0.903503i	$-0.359013\pi$
$457$	18.3241	0.857164	0.428582	+	0.903503i	$0.359013\pi$
$458$	−10.1307	−0.473376
$459$	0	0
$460$	−10.3556	−0.482831
$461$	−31.6216	−1.47276	−0.736382	−	0.676566i	$-0.763467\pi$
$461$	−31.6216	−1.47276	−0.736382	+	0.676566i	$0.763467\pi$
$462$	0	0
$463$	−7.76077	−0.360674	−0.180337	−	0.983605i	$-0.557719\pi$
$463$	−7.76077	−0.360674	−0.180337	+	0.983605i	$0.557719\pi$
$464$	0.673679	0.0312748
$465$	0	0
$466$	19.7029	0.912718
$467$	−31.2308	−1.44519	−0.722594	−	0.691273i	$-0.757050\pi$
$467$	−31.2308	−1.44519	−0.722594	+	0.691273i	$0.757050\pi$
$468$	0	0
$469$	−4.39121	−0.202767
$470$	−1.69944	−0.0783894
$471$	0	0
$472$	−4.24521	−0.195401
$473$	1.53320	0.0704966
$474$	0	0
$475$	1.00000	0.0458831
$476$	−4.60809	−0.211212
$477$	0	0
$478$	5.66259	0.259001
$479$	18.2169	0.832353	0.416177	−	0.909284i	$-0.363370\pi$
$479$	18.2169	0.832353	0.416177	+	0.909284i	$0.363370\pi$
$480$	0	0
$481$	2.31762	0.105674
$482$	−2.37840	−0.108333
$483$	0	0
$484$	−1.53362	−0.0697100
$485$	8.88210	0.403315
$486$	0	0
$487$	39.2194	1.77720	0.888600	−	0.458683i	$-0.151679\pi$
$487$	39.2194	1.77720	0.888600	+	0.458683i	$0.151679\pi$
$488$	8.26352	0.374072
$489$	0	0
$490$	−4.27893	−0.193302
$491$	−36.5478	−1.64938	−0.824688	−	0.565587i	$-0.808650\pi$
$491$	−36.5478	−1.64938	−0.824688	+	0.565587i	$0.808650\pi$
$492$	0	0
$493$	1.66437	0.0749594
$494$	−3.76036	−0.169187
$495$	0	0
$496$	−7.37637	−0.331209
$497$	10.4494	0.468719
$498$	0	0
$499$	31.4475	1.40778	0.703892	−	0.710307i	$-0.251444\pi$
$499$	31.4475	1.40778	0.703892	+	0.710307i	$0.251444\pi$
$500$	1.53362	0.0685856
$501$	0	0
$502$	−19.6788	−0.878307
$503$	−20.7020	−0.923057	−0.461528	−	0.887125i	$-0.652699\pi$
$503$	−20.7020	−0.923057	−0.461528	+	0.887125i	$0.652699\pi$
$504$	0	0
$505$	−2.09965	−0.0934334
$506$	−4.61133	−0.204999
$507$	0	0
$508$	25.9647	1.15200
$509$	−23.5196	−1.04249	−0.521245	−	0.853407i	$-0.674532\pi$
$509$	−23.5196	−1.04249	−0.521245	+	0.853407i	$0.674532\pi$
$510$	0	0
$511$	−10.9379	−0.483865
$512$	−15.0750	−0.666226
$513$	0	0
$514$	−5.76506	−0.254286
$515$	3.91556	0.172540
$516$	0	0
$517$	2.48849	0.109444
$518$	−0.246324	−0.0108229
$519$	0	0
$520$	−13.2877	−0.582704
$521$	−28.6887	−1.25687	−0.628437	−	0.777860i	$-0.716305\pi$
$521$	−28.6887	−1.25687	−0.628437	+	0.777860i	$0.716305\pi$
$522$	0	0
$523$	5.70672	0.249537	0.124769	−	0.992186i	$-0.460181\pi$
$523$	5.70672	0.249537	0.124769	+	0.992186i	$0.460181\pi$
$524$	8.07634	0.352816
$525$	0	0
$526$	14.6288	0.637848
$527$	−18.2238	−0.793842
$528$	0	0
$529$	22.5945	0.982371
$530$	3.27633	0.142315
$531$	0	0
$532$	−1.31423	−0.0569792
$533$	−45.4347	−1.96800
$534$	0	0
$535$	−18.0439	−0.780105
$536$	−12.3657	−0.534118
$537$	0	0
$538$	7.07792	0.305151
$539$	6.26564	0.269880
$540$	0	0
$541$	3.50382	0.150641	0.0753205	−	0.997159i	$-0.476002\pi$
$541$	3.50382	0.150641	0.0753205	+	0.997159i	$0.476002\pi$
$542$	17.8296	0.765849
$543$	0	0
$544$	−20.3210	−0.871257
$545$	−3.34558	−0.143309
$546$	0	0
$547$	28.3788	1.21339	0.606696	−	0.794934i	$-0.292495\pi$
$547$	28.3788	1.21339	0.606696	+	0.794934i	$0.292495\pi$
$548$	−31.6045	−1.35008
$549$	0	0
$550$	0.682920	0.0291198
$551$	0.474680	0.0202220
$552$	0	0
$553$	9.31430	0.396084
$554$	12.8159	0.544495
$555$	0	0
$556$	−1.93530	−0.0820749
$557$	0.440370	0.0186591	0.00932953	−	0.999956i	$-0.497030\pi$
$557$	0.440370	0.0186591	0.00932953	+	0.999956i	$0.497030\pi$
$558$	0	0
$559$	−8.44226	−0.357069
$560$	−1.21621	−0.0513941
$561$	0	0
$562$	16.8219	0.709587
$563$	−44.5147	−1.87607	−0.938036	−	0.346537i	$-0.887358\pi$
$563$	−44.5147	−1.87607	−0.938036	+	0.346537i	$0.887358\pi$
$564$	0	0
$565$	7.26819	0.305775
$566$	9.41540	0.395759
$567$	0	0
$568$	29.4256	1.23467
$569$	−5.32611	−0.223282	−0.111641	−	0.993749i	$-0.535611\pi$
$569$	−5.32611	−0.223282	−0.111641	+	0.993749i	$0.535611\pi$
$570$	0	0
$571$	11.2850	0.472264	0.236132	−	0.971721i	$-0.424120\pi$
$571$	11.2850	0.472264	0.236132	+	0.971721i	$0.424120\pi$
$572$	8.44457	0.353085
$573$	0	0
$574$	4.82895	0.201557
$575$	−6.75237	−0.281593
$576$	0	0
$577$	44.9805	1.87256	0.936281	−	0.351251i	$-0.114243\pi$
$577$	44.9805	1.87256	0.936281	+	0.351251i	$0.114243\pi$
$578$	3.21374	0.133674
$579$	0	0
$580$	0.727978	0.0302277
$581$	−2.52516	−0.104761
$582$	0	0
$583$	−4.79753	−0.198693
$584$	−30.8014	−1.27457
$585$	0	0
$586$	−2.41094	−0.0995948
$587$	12.8765	0.531470	0.265735	−	0.964046i	$-0.414385\pi$
$587$	12.8765	0.531470	0.265735	+	0.964046i	$0.414385\pi$
$588$	0	0
$589$	−5.19745	−0.214157
$590$	−1.20138	−0.0494599
$591$	0	0
$592$	0.597359	0.0245513
$593$	25.1716	1.03367	0.516837	−	0.856084i	$-0.327109\pi$
$593$	25.1716	1.03367	0.516837	+	0.856084i	$0.327109\pi$
$594$	0	0
$595$	−3.00472	−0.123181
$596$	−17.3479	−0.710597
$597$	0	0
$598$	25.3914	1.03833
$599$	−30.2614	−1.23645	−0.618224	−	0.786002i	$-0.712147\pi$
$599$	−30.2614	−1.23645	−0.618224	+	0.786002i	$0.712147\pi$
$600$	0	0
$601$	33.1672	1.35292	0.676458	−	0.736481i	$-0.263514\pi$
$601$	33.1672	1.35292	0.676458	+	0.736481i	$0.263514\pi$
$602$	0.897271	0.0365700
$603$	0	0
$604$	5.67552	0.230934
$605$	−1.00000	−0.0406558
$606$	0	0
$607$	−10.4625	−0.424660	−0.212330	−	0.977198i	$-0.568105\pi$
$607$	−10.4625	−0.424660	−0.212330	+	0.977198i	$0.568105\pi$
$608$	−5.79558	−0.235042
$609$	0	0
$610$	2.33854	0.0946848
$611$	−13.7024	−0.554339
$612$	0	0
$613$	−9.24317	−0.373328	−0.186664	−	0.982424i	$-0.559768\pi$
$613$	−9.24317	−0.373328	−0.186664	+	0.982424i	$0.559768\pi$
$614$	−3.81884	−0.154116
$615$	0	0
$616$	−2.06797	−0.0833210
$617$	1.95689	0.0787814	0.0393907	−	0.999224i	$-0.487458\pi$
$617$	1.95689	0.0787814	0.0393907	+	0.999224i	$0.487458\pi$
$618$	0	0
$619$	−48.3987	−1.94531	−0.972653	−	0.232263i	$-0.925387\pi$
$619$	−48.3987	−1.94531	−0.972653	+	0.232263i	$0.925387\pi$
$620$	−7.97091	−0.320119
$621$	0	0
$622$	−2.30902	−0.0925832
$623$	−2.64570	−0.105998
$624$	0	0
$625$	1.00000	0.0400000
$626$	−9.26559	−0.370327
$627$	0	0
$628$	−4.47039	−0.178388
$629$	1.47581	0.0588445
$630$	0	0
$631$	30.8039	1.22628	0.613141	−	0.789973i	$-0.289906\pi$
$631$	30.8039	1.22628	0.613141	+	0.789973i	$0.289906\pi$
$632$	26.2292	1.04334
$633$	0	0
$634$	−6.16641	−0.244899
$635$	16.9303	0.671860
$636$	0	0
$637$	−34.5005	−1.36696
$638$	0.324169	0.0128340
$639$	0	0
$640$	−10.8267	−0.427961
$641$	7.12059	0.281246	0.140623	−	0.990063i	$-0.455089\pi$
$641$	7.12059	0.281246	0.140623	+	0.990063i	$0.455089\pi$
$642$	0	0
$643$	−43.8135	−1.72783	−0.863917	−	0.503634i	$-0.831996\pi$
$643$	−43.8135	−1.72783	−0.863917	+	0.503634i	$0.831996\pi$
$644$	8.87419	0.349692
$645$	0	0
$646$	−2.39452	−0.0942112
$647$	−19.4584	−0.764988	−0.382494	−	0.923958i	$-0.624935\pi$
$647$	−19.4584	−0.764988	−0.382494	+	0.923958i	$0.624935\pi$
$648$	0	0
$649$	1.75917	0.0690536
$650$	−3.76036	−0.147494
$651$	0	0
$652$	−28.2613	−1.10680
$653$	35.4429	1.38699	0.693493	−	0.720463i	$-0.256071\pi$
$653$	35.4429	1.38699	0.693493	+	0.720463i	$0.256071\pi$
$654$	0	0
$655$	5.26619	0.205767
$656$	−11.7106	−0.457224
$657$	0	0
$658$	1.45633	0.0567738
$659$	7.91185	0.308202	0.154101	−	0.988055i	$-0.450752\pi$
$659$	7.91185	0.308202	0.154101	+	0.988055i	$0.450752\pi$
$660$	0	0
$661$	−37.9285	−1.47525	−0.737625	−	0.675211i	$-0.764053\pi$
$661$	−37.9285	−1.47525	−0.737625	+	0.675211i	$0.764053\pi$
$662$	12.7683	0.496254
$663$	0	0
$664$	−7.11090	−0.275956
$665$	−0.856948	−0.0332310
$666$	0	0
$667$	−3.20522	−0.124106
$668$	23.2241	0.898567
$669$	0	0
$670$	−3.49945	−0.135196
$671$	−3.42433	−0.132195
$672$	0	0
$673$	−5.19834	−0.200381	−0.100191	−	0.994968i	$-0.531945\pi$
$673$	−5.19834	−0.200381	−0.100191	+	0.994968i	$0.531945\pi$
$674$	−20.0276	−0.771434
$675$	0	0
$676$	−26.5613	−1.02159
$677$	12.7010	0.488137	0.244069	−	0.969758i	$-0.421518\pi$
$677$	12.7010	0.488137	0.244069	+	0.969758i	$0.421518\pi$
$678$	0	0
$679$	−7.61150	−0.292103
$680$	−8.46133	−0.324477
$681$	0	0
$682$	−3.54944	−0.135915
$683$	−45.3282	−1.73444	−0.867219	−	0.497928i	$-0.834095\pi$
$683$	−45.3282	−1.73444	−0.867219	+	0.497928i	$0.834095\pi$
$684$	0	0
$685$	−20.6078	−0.787383
$686$	7.76342	0.296409
$687$	0	0
$688$	−2.17596	−0.0829578
$689$	26.4166	1.00639
$690$	0	0
$691$	30.5149	1.16084	0.580421	−	0.814317i	$-0.302888\pi$
$691$	30.5149	1.16084	0.580421	+	0.814317i	$0.302888\pi$
$692$	−0.169641	−0.00644879
$693$	0	0
$694$	7.51539	0.285280
$695$	−1.26192	−0.0478672
$696$	0	0
$697$	−28.9319	−1.09587
$698$	16.4878	0.624072
$699$	0	0
$700$	−1.31423	−0.0496733
$701$	−0.621590	−0.0234771	−0.0117386	−	0.999931i	$-0.503737\pi$
$701$	−0.621590	−0.0234771	−0.0117386	+	0.999931i	$0.503737\pi$
$702$	0	0
$703$	0.420904	0.0158747
$704$	−1.11946	−0.0421914
$705$	0	0
$706$	8.95087	0.336870
$707$	1.79930	0.0676695
$708$	0	0
$709$	−21.9142	−0.823004	−0.411502	−	0.911409i	$-0.634996\pi$
$709$	−21.9142	−0.823004	−0.411502	+	0.911409i	$0.634996\pi$
$710$	8.32733	0.312519
$711$	0	0
$712$	−7.45035	−0.279214
$713$	35.0951	1.31432
$714$	0	0
$715$	5.50630	0.205924
$716$	−14.4527	−0.540121
$717$	0	0
$718$	−7.67238	−0.286331
$719$	43.0880	1.60691	0.803455	−	0.595365i	$-0.202993\pi$
$719$	43.0880	1.60691	0.803455	+	0.595365i	$0.202993\pi$
$720$	0	0
$721$	−3.35544	−0.124963
$722$	−0.682920	−0.0254157
$723$	0	0
$724$	18.8446	0.700353
$725$	0.474680	0.0176292
$726$	0	0
$727$	30.8306	1.14344	0.571721	−	0.820448i	$-0.306276\pi$
$727$	30.8306	1.14344	0.571721	+	0.820448i	$0.306276\pi$
$728$	11.3869	0.422025
$729$	0	0
$730$	−8.71666	−0.322618
$731$	−5.37586	−0.198833
$732$	0	0
$733$	−17.6688	−0.652610	−0.326305	−	0.945264i	$-0.605804\pi$
$733$	−17.6688	−0.652610	−0.326305	+	0.945264i	$0.605804\pi$
$734$	−0.543824	−0.0200729
$735$	0	0
$736$	39.1339	1.44250
$737$	5.12424	0.188754
$738$	0	0
$739$	−6.72966	−0.247555	−0.123777	−	0.992310i	$-0.539501\pi$
$739$	−6.72966	−0.247555	−0.123777	+	0.992310i	$0.539501\pi$
$740$	0.645506	0.0237293
$741$	0	0
$742$	−2.80764	−0.103072
$743$	−48.4780	−1.77848	−0.889242	−	0.457437i	$-0.848767\pi$
$743$	−48.4780	−1.77848	−0.889242	+	0.457437i	$0.848767\pi$
$744$	0	0
$745$	−11.3117	−0.414429
$746$	−2.88445	−0.105607
$747$	0	0
$748$	5.37733	0.196615
$749$	15.4627	0.564994
$750$	0	0
$751$	−36.2684	−1.32345	−0.661726	−	0.749746i	$-0.730176\pi$
$751$	−36.2684	−1.32345	−0.661726	+	0.749746i	$0.730176\pi$
$752$	−3.53174	−0.128789
$753$	0	0
$754$	−1.78497	−0.0650047
$755$	3.70074	0.134684
$756$	0	0
$757$	−5.13939	−0.186794	−0.0933972	−	0.995629i	$-0.529773\pi$
$757$	−5.13939	−0.186794	−0.0933972	+	0.995629i	$0.529773\pi$
$758$	15.4170	0.559971
$759$	0	0
$760$	−2.41318	−0.0875353
$761$	−10.2119	−0.370179	−0.185090	−	0.982722i	$-0.559258\pi$
$761$	−10.2119	−0.370179	−0.185090	+	0.982722i	$0.559258\pi$
$762$	0	0
$763$	2.86699	0.103792
$764$	25.2370	0.913042
$765$	0	0
$766$	22.2057	0.802326
$767$	−9.68654	−0.349761
$768$	0	0
$769$	−49.2015	−1.77425	−0.887126	−	0.461527i	$-0.847302\pi$
$769$	−49.2015	−1.77425	−0.887126	+	0.461527i	$0.847302\pi$
$770$	−0.585228	−0.0210901
$771$	0	0
$772$	−4.29350	−0.154526
$773$	40.7240	1.46474	0.732371	−	0.680906i	$-0.238414\pi$
$773$	40.7240	1.46474	0.732371	+	0.680906i	$0.238414\pi$
$774$	0	0
$775$	−5.19745	−0.186698
$776$	−21.4341	−0.769440
$777$	0	0
$778$	−24.2346	−0.868853
$779$	−8.25141	−0.295637
$780$	0	0
$781$	−12.1937	−0.436325
$782$	16.1687	0.578192
$783$	0	0
$784$	−8.89238	−0.317585
$785$	−2.91493	−0.104038
$786$	0	0
$787$	7.23417	0.257870	0.128935	−	0.991653i	$-0.458844\pi$
$787$	7.23417	0.257870	0.128935	+	0.991653i	$0.458844\pi$
$788$	−11.0337	−0.393059
$789$	0	0
$790$	7.42276	0.264090
$791$	−6.22846	−0.221459
$792$	0	0
$793$	18.8554	0.669574
$794$	−8.00539	−0.284101
$795$	0	0
$796$	18.2942	0.648420
$797$	−43.9784	−1.55779	−0.778897	−	0.627152i	$-0.784220\pi$
$797$	−43.9784	−1.55779	−0.778897	+	0.627152i	$0.784220\pi$
$798$	0	0
$799$	−8.72540	−0.308682
$800$	−5.79558	−0.204905
$801$	0	0
$802$	−22.8842	−0.808068
$803$	12.7638	0.450425
$804$	0	0
$805$	5.78644	0.203945
$806$	19.5443	0.688419
$807$	0	0
$808$	5.06685	0.178251
$809$	−11.9798	−0.421187	−0.210593	−	0.977574i	$-0.567540\pi$
$809$	−11.9798	−0.421187	−0.210593	+	0.977574i	$0.567540\pi$
$810$	0	0
$811$	30.0059	1.05365	0.526825	−	0.849974i	$-0.323382\pi$
$811$	30.0059	1.05365	0.526825	+	0.849974i	$0.323382\pi$
$812$	−0.623840	−0.0218925
$813$	0	0
$814$	0.287444	0.0100749
$815$	−18.4278	−0.645499
$816$	0	0
$817$	−1.53320	−0.0536399
$818$	4.44143	0.155291
$819$	0	0
$820$	−12.6545	−0.441915
$821$	−26.5123	−0.925284	−0.462642	−	0.886545i	$-0.653099\pi$
$821$	−26.5123	−0.925284	−0.462642	+	0.886545i	$0.653099\pi$
$822$	0	0
$823$	13.2244	0.460972	0.230486	−	0.973076i	$-0.425968\pi$
$823$	13.2244	0.460972	0.230486	+	0.973076i	$0.425968\pi$
$824$	−9.44896	−0.329170
$825$	0	0
$826$	1.02952	0.0358215
$827$	−25.4465	−0.884861	−0.442431	−	0.896803i	$-0.645884\pi$
$827$	−25.4465	−0.884861	−0.442431	+	0.896803i	$0.645884\pi$
$828$	0	0
$829$	−28.6999	−0.996789	−0.498395	−	0.866950i	$-0.666077\pi$
$829$	−28.6999	−0.996789	−0.498395	+	0.866950i	$0.666077\pi$
$830$	−2.01235	−0.0698498
$831$	0	0
$832$	6.16410	0.213702
$833$	−21.9692	−0.761188
$834$	0	0
$835$	15.1433	0.524056
$836$	1.53362	0.0530413
$837$	0	0
$838$	−5.12047	−0.176884
$839$	4.10582	0.141749	0.0708743	−	0.997485i	$-0.477421\pi$
$839$	4.10582	0.141749	0.0708743	+	0.997485i	$0.477421\pi$
$840$	0	0
$841$	−28.7747	−0.992230
$842$	−7.87271	−0.271311
$843$	0	0
$844$	−27.6711	−0.952478
$845$	−17.3193	−0.595803
$846$	0	0
$847$	0.856948	0.0294451
$848$	6.80879	0.233815
$849$	0	0
$850$	−2.39452	−0.0821315
$851$	−2.84210	−0.0974259
$852$	0	0
$853$	−51.2689	−1.75541	−0.877707	−	0.479198i	$-0.840928\pi$
$853$	−51.2689	−1.75541	−0.877707	+	0.479198i	$0.840928\pi$
$854$	−2.00401	−0.0685758
$855$	0	0
$856$	43.5431	1.48827
$857$	34.2836	1.17110	0.585552	−	0.810635i	$-0.300878\pi$
$857$	34.2836	1.17110	0.585552	+	0.810635i	$0.300878\pi$
$858$	0	0
$859$	39.5431	1.34919	0.674596	−	0.738187i	$-0.264318\pi$
$859$	39.5431	1.34919	0.674596	+	0.738187i	$0.264318\pi$
$860$	−2.35135	−0.0801802
$861$	0	0
$862$	15.3288	0.522100
$863$	13.1157	0.446462	0.223231	−	0.974766i	$-0.428340\pi$
$863$	13.1157	0.446462	0.223231	+	0.974766i	$0.428340\pi$
$864$	0	0
$865$	−0.110615	−0.00376102
$866$	18.7802	0.638179
$867$	0	0
$868$	6.83066	0.231848
$869$	−10.8691	−0.368711
$870$	0	0
$871$	−28.2156	−0.956050
$872$	8.07349	0.273403
$873$	0	0
$874$	4.61133	0.155981
$875$	−0.856948	−0.0289701
$876$	0	0
$877$	−34.1036	−1.15160	−0.575798	−	0.817592i	$-0.695309\pi$
$877$	−34.1036	−1.15160	−0.575798	+	0.817592i	$0.695309\pi$
$878$	−5.58963	−0.188641
$879$	0	0
$880$	1.41923	0.0478422
$881$	−39.1343	−1.31847	−0.659234	−	0.751938i	$-0.729119\pi$
$881$	−39.1343	−1.31847	−0.659234	+	0.751938i	$0.729119\pi$
$882$	0	0
$883$	44.1281	1.48503	0.742515	−	0.669829i	$-0.233633\pi$
$883$	44.1281	1.48503	0.742515	+	0.669829i	$0.233633\pi$
$884$	−29.6092	−0.995864
$885$	0	0
$886$	−7.59238	−0.255071
$887$	48.8255	1.63940	0.819700	−	0.572793i	$-0.194140\pi$
$887$	48.8255	1.63940	0.819700	+	0.572793i	$0.194140\pi$
$888$	0	0
$889$	−14.5084	−0.486597
$890$	−2.10842	−0.0706743
$891$	0	0
$892$	37.5738	1.25806
$893$	−2.48849	−0.0832742
$894$	0	0
$895$	−9.42389	−0.315006
$896$	9.27789	0.309953
$897$	0	0
$898$	7.60954	0.253934
$899$	−2.46712	−0.0822832
$900$	0	0
$901$	16.8216	0.560407
$902$	−5.63505	−0.187627
$903$	0	0
$904$	−17.5394	−0.583353
$905$	12.2876	0.408455
$906$	0	0
$907$	−25.4676	−0.845638	−0.422819	−	0.906214i	$-0.638959\pi$
$907$	−25.4676	−0.845638	−0.422819	+	0.906214i	$0.638959\pi$
$908$	25.5617	0.848293
$909$	0	0
$910$	3.22244	0.106823
$911$	23.4140	0.775740	0.387870	−	0.921714i	$-0.373211\pi$
$911$	23.4140	0.775740	0.387870	+	0.921714i	$0.373211\pi$
$912$	0	0
$913$	2.94669	0.0975212
$914$	−12.5139	−0.413923
$915$	0	0
$916$	−22.7503	−0.751690
$917$	−4.51286	−0.149028
$918$	0	0
$919$	−18.7511	−0.618542	−0.309271	−	0.950974i	$-0.600085\pi$
$919$	−18.7511	−0.618542	−0.309271	+	0.950974i	$0.600085\pi$
$920$	16.2947	0.537220
$921$	0	0
$922$	21.5950	0.711194
$923$	67.1422	2.21001
$924$	0	0
$925$	0.420904	0.0138392
$926$	5.29999	0.174169
$927$	0	0
$928$	−2.75105	−0.0903075
$929$	−40.7701	−1.33762	−0.668812	−	0.743432i	$-0.733197\pi$
$929$	−40.7701	−1.33762	−0.668812	+	0.743432i	$0.733197\pi$
$930$	0	0
$931$	−6.26564	−0.205348
$932$	44.2463	1.44934
$933$	0	0
$934$	21.3281	0.697878
$935$	3.50630	0.114668
$936$	0	0
$937$	−6.54322	−0.213758	−0.106879	−	0.994272i	$-0.534086\pi$
$937$	−6.54322	−0.213758	−0.106879	+	0.994272i	$0.534086\pi$
$938$	2.99885	0.0979159
$939$	0	0
$940$	−3.81640	−0.124477
$941$	34.5714	1.12700	0.563498	−	0.826117i	$-0.309455\pi$
$941$	34.5714	1.12700	0.563498	+	0.826117i	$0.309455\pi$
$942$	0	0
$943$	55.7166	1.81438
$944$	−2.49667	−0.0812597
$945$	0	0
$946$	−1.04705	−0.0340427
$947$	−17.6584	−0.573822	−0.286911	−	0.957957i	$-0.592628\pi$
$947$	−17.6584	−0.573822	−0.286911	+	0.957957i	$0.592628\pi$
$948$	0	0
$949$	−70.2813	−2.28143
$950$	−0.682920	−0.0221569
$951$	0	0
$952$	7.25093	0.235004
$953$	42.7340	1.38429	0.692145	−	0.721758i	$-0.256666\pi$
$953$	42.7340	1.38429	0.692145	+	0.721758i	$0.256666\pi$
$954$	0	0
$955$	16.4558	0.532498
$956$	12.7164	0.411276
$957$	0	0
$958$	−12.4407	−0.401941
$959$	17.6598	0.570265
$960$	0	0
$961$	−3.98652	−0.128597
$962$	−1.58275	−0.0510299
$963$	0	0
$964$	−5.34111	−0.172026
$965$	−2.79958	−0.0901218
$966$	0	0
$967$	−3.62541	−0.116585	−0.0582927	−	0.998300i	$-0.518566\pi$
$967$	−3.62541	−0.116585	−0.0582927	+	0.998300i	$0.518566\pi$
$968$	2.41318	0.0775626
$969$	0	0
$970$	−6.06577	−0.194760
$971$	−2.86101	−0.0918142	−0.0459071	−	0.998946i	$-0.514618\pi$
$971$	−2.86101	−0.0918142	−0.0459071	+	0.998946i	$0.514618\pi$
$972$	0	0
$973$	1.08140	0.0346680
$974$	−26.7837	−0.858206
$975$	0	0
$976$	4.85990	0.155562
$977$	−29.4990	−0.943756	−0.471878	−	0.881664i	$-0.656424\pi$
$977$	−29.4990	−0.943756	−0.471878	+	0.881664i	$0.656424\pi$
$978$	0	0
$979$	3.08736	0.0986723
$980$	−9.60911	−0.306952
$981$	0	0
$982$	24.9592	0.796480
$983$	−31.1737	−0.994287	−0.497144	−	0.867668i	$-0.665618\pi$
$983$	−31.1737	−0.994287	−0.497144	+	0.867668i	$0.665618\pi$
$984$	0	0
$985$	−7.19454	−0.229237
$986$	−1.13663	−0.0361977
$987$	0	0
$988$	−8.44457	−0.268657
$989$	10.3527	0.329198
$990$	0	0
$991$	−55.9145	−1.77618	−0.888092	−	0.459666i	$-0.847969\pi$
$991$	−55.9145	−1.77618	−0.888092	+	0.459666i	$0.847969\pi$
$992$	30.1222	0.956382
$993$	0	0
$994$	−7.13609	−0.226343
$995$	11.9287	0.378167
$996$	0	0
$997$	42.1768	1.33575	0.667877	−	0.744272i	$-0.267203\pi$
$997$	42.1768	1.33575	0.667877	+	0.744272i	$0.267203\pi$
$998$	−21.4762	−0.679816
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9405.2.a.bh.1.5		9
3.2	odd	2		1045.2.a.k.1.5	✓	9
15.14	odd	2		5225.2.a.p.1.5		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.k.1.5	✓	9	3.2	odd	2
5225.2.a.p.1.5		9	15.14	odd	2
9405.2.a.bh.1.5		9	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)

\(f(q)\)	\(=\)	\(q-0.682920 q^{2} -1.53362 q^{4} -1.00000 q^{5} +0.856948 q^{7} +2.41318 q^{8} +O(q^{10})\)	\(q-0.682920 q^{2} -1.53362 q^{4} -1.00000 q^{5} +0.856948 q^{7} +2.41318 q^{8} +0.682920 q^{10} -1.00000 q^{11} +5.50630 q^{13} -0.585228 q^{14} +1.41923 q^{16} +3.50630 q^{17} +1.00000 q^{19} +1.53362 q^{20} +0.682920 q^{22} -6.75237 q^{23} +1.00000 q^{25} -3.76036 q^{26} -1.31423 q^{28} +0.474680 q^{29} -5.19745 q^{31} -5.79558 q^{32} -2.39452 q^{34} -0.856948 q^{35} +0.420904 q^{37} -0.682920 q^{38} -2.41318 q^{40} -8.25141 q^{41} -1.53320 q^{43} +1.53362 q^{44} +4.61133 q^{46} -2.48849 q^{47} -6.26564 q^{49} -0.682920 q^{50} -8.44457 q^{52} +4.79753 q^{53} +1.00000 q^{55} +2.06797 q^{56} -0.324169 q^{58} -1.75917 q^{59} +3.42433 q^{61} +3.54944 q^{62} +1.11946 q^{64} -5.50630 q^{65} -5.12424 q^{67} -5.37733 q^{68} +0.585228 q^{70} +12.1937 q^{71} -12.7638 q^{73} -0.287444 q^{74} -1.53362 q^{76} -0.856948 q^{77} +10.8691 q^{79} -1.41923 q^{80} +5.63505 q^{82} -2.94669 q^{83} -3.50630 q^{85} +1.04705 q^{86} -2.41318 q^{88} -3.08736 q^{89} +4.71861 q^{91} +10.3556 q^{92} +1.69944 q^{94} -1.00000 q^{95} -8.88210 q^{97} +4.27893 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - 3 q^{2} + 9 q^{4} - 9 q^{5} + 13 q^{7} - 9 q^{8}+O(q^{10}) \) 9 * q - 3 * q^2 + 9 * q^4 - 9 * q^5 + 13 * q^7 - 9 * q^8	\( 9 q - 3 q^{2} + 9 q^{4} - 9 q^{5} + 13 q^{7} - 9 q^{8} + 3 q^{10} - 9 q^{11} + 5 q^{13} + 2 q^{14} + q^{16} - 13 q^{17} + 9 q^{19} - 9 q^{20} + 3 q^{22} - 8 q^{23} + 9 q^{25} - 8 q^{26} + 10 q^{28} + 3 q^{29} - 9 q^{31} - 6 q^{32} - 2 q^{34} - 13 q^{35} - 7 q^{37} - 3 q^{38} + 9 q^{40} - 9 q^{41} + 23 q^{43} - 9 q^{44} - 32 q^{46} - 20 q^{47} - 4 q^{49} - 3 q^{50} + 9 q^{52} + 5 q^{53} + 9 q^{55} - 4 q^{56} + 3 q^{58} - 19 q^{59} + q^{61} - 18 q^{62} + 23 q^{64} - 5 q^{65} - 10 q^{67} - 9 q^{68} - 2 q^{70} + 12 q^{73} - 5 q^{74} + 9 q^{76} - 13 q^{77} - 21 q^{79} - q^{80} - 14 q^{82} - 47 q^{83} + 13 q^{85} - 12 q^{86} + 9 q^{88} + 2 q^{89} + 2 q^{91} + 19 q^{92} + 26 q^{94} - 9 q^{95} - 32 q^{97}+O(q^{100}) \) 9 * q - 3 * q^2 + 9 * q^4 - 9 * q^5 + 13 * q^7 - 9 * q^8 + 3 * q^10 - 9 * q^11 + 5 * q^13 + 2 * q^14 + q^16 - 13 * q^17 + 9 * q^19 - 9 * q^20 + 3 * q^22 - 8 * q^23 + 9 * q^25 - 8 * q^26 + 10 * q^28 + 3 * q^29 - 9 * q^31 - 6 * q^32 - 2 * q^34 - 13 * q^35 - 7 * q^37 - 3 * q^38 + 9 * q^40 - 9 * q^41 + 23 * q^43 - 9 * q^44 - 32 * q^46 - 20 * q^47 - 4 * q^49 - 3 * q^50 + 9 * q^52 + 5 * q^53 + 9 * q^55 - 4 * q^56 + 3 * q^58 - 19 * q^59 + q^61 - 18 * q^62 + 23 * q^64 - 5 * q^65 - 10 * q^67 - 9 * q^68 - 2 * q^70 + 12 * q^73 - 5 * q^74 + 9 * q^76 - 13 * q^77 - 21 * q^79 - q^80 - 14 * q^82 - 47 * q^83 + 13 * q^85 - 12 * q^86 + 9 * q^88 + 2 * q^89 + 2 * q^91 + 19 * q^92 + 26 * q^94 - 9 * q^95 - 32 * q^97

Introduction

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