LMFDB - Embedded newform 9405.2.a.bq.1.9

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:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - x^{13} - 19 x^{12} + 15 x^{11} + 137 x^{10} - 80 x^{9} - 467 x^{8} + 193 x^{7} + 766 x^{6} + \cdots + 1 $ x^14 - x^13 - 19x^12 + 15x^11 + 137x^10 - 80x^9 - 467x^8 + 193x^7 + 766x^6 - 240x^5 - 567x^4 + 142x^3 + 142x^2 - 25x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-0.668880$ 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)

$f(q)$	$=$	$q+0.668880 q^{2} -1.55260 q^{4} +1.00000 q^{5} +3.89603 q^{7} -2.37626 q^{8} +O(q^{10})$	\(q+0.668880 q^{2} -1.55260 q^{4} +1.00000 q^{5} +3.89603 q^{7} -2.37626 q^{8} +0.668880 q^{10} +1.00000 q^{11} -6.80088 q^{13} +2.60598 q^{14} +1.51576 q^{16} +0.0568207 q^{17} -1.00000 q^{19} -1.55260 q^{20} +0.668880 q^{22} -4.46264 q^{23} +1.00000 q^{25} -4.54897 q^{26} -6.04897 q^{28} +6.58592 q^{29} -1.78033 q^{31} +5.76639 q^{32} +0.0380062 q^{34} +3.89603 q^{35} +6.97736 q^{37} -0.668880 q^{38} -2.37626 q^{40} -11.5363 q^{41} +1.20936 q^{43} -1.55260 q^{44} -2.98497 q^{46} -10.6133 q^{47} +8.17905 q^{49} +0.668880 q^{50} +10.5590 q^{52} +11.8636 q^{53} +1.00000 q^{55} -9.25799 q^{56} +4.40519 q^{58} +2.36872 q^{59} +8.76860 q^{61} -1.19083 q^{62} +0.825498 q^{64} -6.80088 q^{65} -5.77629 q^{67} -0.0882197 q^{68} +2.60598 q^{70} -2.17320 q^{71} -12.9448 q^{73} +4.66701 q^{74} +1.55260 q^{76} +3.89603 q^{77} -16.3860 q^{79} +1.51576 q^{80} -7.71640 q^{82} -10.8040 q^{83} +0.0568207 q^{85} +0.808914 q^{86} -2.37626 q^{88} +4.33886 q^{89} -26.4964 q^{91} +6.92869 q^{92} -7.09903 q^{94} -1.00000 q^{95} -17.7624 q^{97} +5.47080 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - q^{2} + 11 q^{4} + 14 q^{5} - 12 q^{7} - 9 q^{8}+O(q^{10}) $ 14 * q - q^2 + 11 * q^4 + 14 * q^5 - 12 * q^7 - 9 * q^8	$ 14 q - q^{2} + 11 q^{4} + 14 q^{5} - 12 q^{7} - 9 q^{8} - q^{10} + 14 q^{11} - 10 q^{13} + 8 q^{14} + 13 q^{16} - 16 q^{17} - 14 q^{19} + 11 q^{20} - q^{22} - 12 q^{23} + 14 q^{25} - 12 q^{26} - 33 q^{28} - 4 q^{31} - 24 q^{32} - 2 q^{34} - 12 q^{35} - 14 q^{37} + q^{38} - 9 q^{40} - 18 q^{41} - 20 q^{43} + 11 q^{44} - 17 q^{46} - 8 q^{47} + 10 q^{49} - q^{50} - 26 q^{52} - 20 q^{53} + 14 q^{55} + 11 q^{56} - 36 q^{58} + 2 q^{59} + 4 q^{61} - 38 q^{62} + 3 q^{64} - 10 q^{65} - 22 q^{67} - 48 q^{68} + 8 q^{70} - 28 q^{73} + 19 q^{74} - 11 q^{76} - 12 q^{77} - 14 q^{79} + 13 q^{80} - 24 q^{82} - 10 q^{83} - 16 q^{85} + 23 q^{86} - 9 q^{88} + 26 q^{89} - 42 q^{91} - 12 q^{92} - 56 q^{94} - 14 q^{95} - 32 q^{97} + 4 q^{98}+O(q^{100}) $ 14 * q - q^2 + 11 * q^4 + 14 * q^5 - 12 * q^7 - 9 * q^8 - q^10 + 14 * q^11 - 10 * q^13 + 8 * q^14 + 13 * q^16 - 16 * q^17 - 14 * q^19 + 11 * q^20 - q^22 - 12 * q^23 + 14 * q^25 - 12 * q^26 - 33 * q^28 - 4 * q^31 - 24 * q^32 - 2 * q^34 - 12 * q^35 - 14 * q^37 + q^38 - 9 * q^40 - 18 * q^41 - 20 * q^43 + 11 * q^44 - 17 * q^46 - 8 * q^47 + 10 * q^49 - q^50 - 26 * q^52 - 20 * q^53 + 14 * q^55 + 11 * q^56 - 36 * q^58 + 2 * q^59 + 4 * q^61 - 38 * q^62 + 3 * q^64 - 10 * q^65 - 22 * q^67 - 48 * q^68 + 8 * q^70 - 28 * q^73 + 19 * q^74 - 11 * q^76 - 12 * q^77 - 14 * q^79 + 13 * q^80 - 24 * q^82 - 10 * q^83 - 16 * q^85 + 23 * q^86 - 9 * q^88 + 26 * q^89 - 42 * q^91 - 12 * q^92 - 56 * q^94 - 14 * q^95 - 32 * q^97 + 4 * 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 $

$2$	0.668880	0.472970	0.236485	−	0.971635i	$-0.424005\pi$
$2$	0.668880	0.472970	0.236485	+	0.971635i	$0.424005\pi$
$3$	0	0
$3$	0	0
$4$	−1.55260	−0.776300
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	3.89603	1.47256	0.736280	−	0.676677i	$-0.236580\pi$
$7$	3.89603	1.47256	0.736280	+	0.676677i	$0.236580\pi$
$8$	−2.37626	−0.840136
$9$	0	0
$10$	0.668880	0.211518
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−6.80088	−1.88623	−0.943113	−	0.332473i	$-0.892117\pi$
$13$	−6.80088	−1.88623	−0.943113	+	0.332473i	$0.892117\pi$
$14$	2.60598	0.696477
$15$	0	0
$16$	1.51576	0.378941
$17$	0.0568207	0.0137810	0.00689052	−	0.999976i	$-0.497807\pi$
$17$	0.0568207	0.0137810	0.00689052	+	0.999976i	$0.497807\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−1.55260	−0.347172
$21$	0	0
$22$	0.668880	0.142606
$23$	−4.46264	−0.930524	−0.465262	−	0.885173i	$-0.654040\pi$
$23$	−4.46264	−0.930524	−0.465262	+	0.885173i	$0.654040\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−4.54897	−0.892127
$27$	0	0
$28$	−6.04897	−1.14315
$29$	6.58592	1.22298	0.611488	−	0.791254i	$-0.290571\pi$
$29$	6.58592	1.22298	0.611488	+	0.791254i	$0.290571\pi$
$30$	0	0
$31$	−1.78033	−0.319757	−0.159879	−	0.987137i	$-0.551110\pi$
$31$	−1.78033	−0.319757	−0.159879	+	0.987137i	$0.551110\pi$
$32$	5.76639	1.01936
$33$	0	0
$34$	0.0380062	0.00651801
$35$	3.89603	0.658549
$36$	0	0
$37$	6.97736	1.14707	0.573535	−	0.819181i	$-0.305572\pi$
$37$	6.97736	1.14707	0.573535	+	0.819181i	$0.305572\pi$
$38$	−0.668880	−0.108507
$39$	0	0
$40$	−2.37626	−0.375720
$41$	−11.5363	−1.80167	−0.900833	−	0.434166i	$-0.857043\pi$
$41$	−11.5363	−1.80167	−0.900833	+	0.434166i	$0.857043\pi$
$42$	0	0
$43$	1.20936	0.184425	0.0922125	−	0.995739i	$-0.470606\pi$
$43$	1.20936	0.184425	0.0922125	+	0.995739i	$0.470606\pi$
$44$	−1.55260	−0.234063
$45$	0	0
$46$	−2.98497	−0.440110
$47$	−10.6133	−1.54811	−0.774055	−	0.633119i	$-0.781775\pi$
$47$	−10.6133	−1.54811	−0.774055	+	0.633119i	$0.781775\pi$
$48$	0	0
$49$	8.17905	1.16844
$50$	0.668880	0.0945939
$51$	0	0
$52$	10.5590	1.46428
$53$	11.8636	1.62960	0.814799	−	0.579744i	$-0.196848\pi$
$53$	11.8636	1.62960	0.814799	+	0.579744i	$0.196848\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	−9.25799	−1.23715
$57$	0	0
$58$	4.40519	0.578430
$59$	2.36872	0.308381	0.154190	−	0.988041i	$-0.450723\pi$
$59$	2.36872	0.308381	0.154190	+	0.988041i	$0.450723\pi$
$60$	0	0
$61$	8.76860	1.12270	0.561352	−	0.827577i	$-0.310281\pi$
$61$	8.76860	1.12270	0.561352	+	0.827577i	$0.310281\pi$
$62$	−1.19083	−0.151235
$63$	0	0
$64$	0.825498	0.103187
$65$	−6.80088	−0.843546
$66$	0	0
$67$	−5.77629	−0.705686	−0.352843	−	0.935683i	$-0.614785\pi$
$67$	−5.77629	−0.705686	−0.352843	+	0.935683i	$0.614785\pi$
$68$	−0.0882197	−0.0106982
$69$	0	0
$70$	2.60598	0.311474
$71$	−2.17320	−0.257911	−0.128956	−	0.991650i	$-0.541162\pi$
$71$	−2.17320	−0.257911	−0.128956	+	0.991650i	$0.541162\pi$
$72$	0	0
$73$	−12.9448	−1.51507	−0.757537	−	0.652793i	$-0.773597\pi$
$73$	−12.9448	−1.51507	−0.757537	+	0.652793i	$0.773597\pi$
$74$	4.66701	0.542529
$75$	0	0
$76$	1.55260	0.178095
$77$	3.89603	0.443994
$78$	0	0
$79$	−16.3860	−1.84357	−0.921784	−	0.387705i	$-0.873268\pi$
$79$	−16.3860	−1.84357	−0.921784	+	0.387705i	$0.873268\pi$
$80$	1.51576	0.169467
$81$	0	0
$82$	−7.71640	−0.852133
$83$	−10.8040	−1.18589	−0.592944	−	0.805244i	$-0.702035\pi$
$83$	−10.8040	−1.18589	−0.592944	+	0.805244i	$0.702035\pi$
$84$	0	0
$85$	0.0568207	0.00616307
$86$	0.808914	0.0872275
$87$	0	0
$88$	−2.37626	−0.253311
$89$	4.33886	0.459918	0.229959	−	0.973200i	$-0.426141\pi$
$89$	4.33886	0.459918	0.229959	+	0.973200i	$0.426141\pi$
$90$	0	0
$91$	−26.4964	−2.77758
$92$	6.92869	0.722365
$93$	0	0
$94$	−7.09903	−0.732209
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−17.7624	−1.80350	−0.901751	−	0.432256i	$-0.857718\pi$
$97$	−17.7624	−1.80350	−0.901751	+	0.432256i	$0.857718\pi$
$98$	5.47080	0.552635
$99$	0	0
$100$	−1.55260	−0.155260
$101$	8.10361	0.806339	0.403170	−	0.915125i	$-0.367908\pi$
$101$	8.10361	0.806339	0.403170	+	0.915125i	$0.367908\pi$
$102$	0	0
$103$	−4.81204	−0.474144	−0.237072	−	0.971492i	$-0.576188\pi$
$103$	−4.81204	−0.474144	−0.237072	+	0.971492i	$0.576188\pi$
$104$	16.1607	1.58469
$105$	0	0
$106$	7.93536	0.770750
$107$	−2.11970	−0.204919	−0.102460	−	0.994737i	$-0.532671\pi$
$107$	−2.11970	−0.204919	−0.102460	+	0.994737i	$0.532671\pi$
$108$	0	0
$109$	−7.06913	−0.677100	−0.338550	−	0.940948i	$-0.609936\pi$
$109$	−7.06913	−0.677100	−0.338550	+	0.940948i	$0.609936\pi$
$110$	0.668880	0.0637752
$111$	0	0
$112$	5.90546	0.558013
$113$	−16.8601	−1.58607	−0.793033	−	0.609178i	$-0.791499\pi$
$113$	−16.8601	−1.58607	−0.793033	+	0.609178i	$0.791499\pi$
$114$	0	0
$115$	−4.46264	−0.416143
$116$	−10.2253	−0.949395
$117$	0	0
$118$	1.58439	0.145855
$119$	0.221375	0.0202934
$120$	0	0
$121$	1.00000	0.0909091
$122$	5.86515	0.531005
$123$	0	0
$124$	2.76414	0.248227
$125$	1.00000	0.0894427
$126$	0	0
$127$	5.59069	0.496094	0.248047	−	0.968748i	$-0.420211\pi$
$127$	5.59069	0.496094	0.248047	+	0.968748i	$0.420211\pi$
$128$	−10.9806	−0.970559
$129$	0	0
$130$	−4.54897	−0.398971
$131$	−13.3756	−1.16863	−0.584314	−	0.811528i	$-0.698636\pi$
$131$	−13.3756	−1.16863	−0.584314	+	0.811528i	$0.698636\pi$
$132$	0	0
$133$	−3.89603	−0.337829
$134$	−3.86364	−0.333768
$135$	0	0
$136$	−0.135021	−0.0115779
$137$	16.0030	1.36723	0.683614	−	0.729844i	$-0.260407\pi$
$137$	16.0030	1.36723	0.683614	+	0.729844i	$0.260407\pi$
$138$	0	0
$139$	8.07448	0.684868	0.342434	−	0.939542i	$-0.388749\pi$
$139$	8.07448	0.684868	0.342434	+	0.939542i	$0.388749\pi$
$140$	−6.04897	−0.511232
$141$	0	0
$142$	−1.45361	−0.121984
$143$	−6.80088	−0.568718
$144$	0	0
$145$	6.58592	0.546931
$146$	−8.65851	−0.716584
$147$	0	0
$148$	−10.8330	−0.890470
$149$	7.70363	0.631106	0.315553	−	0.948908i	$-0.397810\pi$
$149$	7.70363	0.631106	0.315553	+	0.948908i	$0.397810\pi$
$150$	0	0
$151$	18.6836	1.52045	0.760224	−	0.649661i	$-0.225089\pi$
$151$	18.6836	1.52045	0.760224	+	0.649661i	$0.225089\pi$
$152$	2.37626	0.192740
$153$	0	0
$154$	2.60598	0.209996
$155$	−1.78033	−0.143000
$156$	0	0
$157$	4.93622	0.393953	0.196977	−	0.980408i	$-0.436888\pi$
$157$	4.93622	0.393953	0.196977	+	0.980408i	$0.436888\pi$
$158$	−10.9603	−0.871951
$159$	0	0
$160$	5.76639	0.455873
$161$	−17.3866	−1.37025
$162$	0	0
$163$	−4.54495	−0.355988	−0.177994	−	0.984032i	$-0.556961\pi$
$163$	−4.54495	−0.355988	−0.177994	+	0.984032i	$0.556961\pi$
$164$	17.9112	1.39863
$165$	0	0
$166$	−7.22655	−0.560889
$167$	−14.0033	−1.08361	−0.541804	−	0.840505i	$-0.682259\pi$
$167$	−14.0033	−1.08361	−0.541804	+	0.840505i	$0.682259\pi$
$168$	0	0
$169$	33.2520	2.55785
$170$	0.0380062	0.00291494
$171$	0	0
$172$	−1.87765	−0.143169
$173$	7.18534	0.546291	0.273146	−	0.961973i	$-0.411936\pi$
$173$	7.18534	0.546291	0.273146	+	0.961973i	$0.411936\pi$
$174$	0	0
$175$	3.89603	0.294512
$176$	1.51576	0.114255
$177$	0	0
$178$	2.90218	0.217527
$179$	4.52870	0.338491	0.169245	−	0.985574i	$-0.445867\pi$
$179$	4.52870	0.338491	0.169245	+	0.985574i	$0.445867\pi$
$180$	0	0
$181$	−26.2652	−1.95228	−0.976139	−	0.217145i	$-0.930325\pi$
$181$	−26.2652	−1.95228	−0.976139	+	0.217145i	$0.930325\pi$
$182$	−17.7229	−1.31371
$183$	0	0
$184$	10.6044	0.781767
$185$	6.97736	0.512985
$186$	0	0
$187$	0.0568207	0.00415514
$188$	16.4782	1.20180
$189$	0	0
$190$	−0.668880	−0.0485257
$191$	2.34524	0.169695	0.0848477	−	0.996394i	$-0.472960\pi$
$191$	2.34524	0.169695	0.0848477	+	0.996394i	$0.472960\pi$
$192$	0	0
$193$	−0.0862928	−0.00621149	−0.00310575	−	0.999995i	$-0.500989\pi$
$193$	−0.0862928	−0.00621149	−0.00310575	+	0.999995i	$0.500989\pi$
$194$	−11.8809	−0.853002
$195$	0	0
$196$	−12.6988	−0.907056
$197$	−23.8804	−1.70141	−0.850704	−	0.525646i	$-0.823824\pi$
$197$	−23.8804	−1.70141	−0.850704	+	0.525646i	$0.823824\pi$
$198$	0	0
$199$	17.0473	1.20845	0.604225	−	0.796813i	$-0.293483\pi$
$199$	17.0473	1.20845	0.604225	+	0.796813i	$0.293483\pi$
$200$	−2.37626	−0.168027
$201$	0	0
$202$	5.42034	0.381374
$203$	25.6590	1.80091
$204$	0	0
$205$	−11.5363	−0.805730
$206$	−3.21868	−0.224256
$207$	0	0
$208$	−10.3085	−0.714768
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	10.0018	0.688551	0.344276	−	0.938869i	$-0.388124\pi$
$211$	10.0018	0.688551	0.344276	+	0.938869i	$0.388124\pi$
$212$	−18.4195	−1.26506
$213$	0	0
$214$	−1.41783	−0.0969206
$215$	1.20936	0.0824774
$216$	0	0
$217$	−6.93623	−0.470862
$218$	−4.72840	−0.320248
$219$	0	0
$220$	−1.55260	−0.104676
$221$	−0.386431	−0.0259941
$222$	0	0
$223$	−21.9090	−1.46713	−0.733567	−	0.679617i	$-0.762146\pi$
$223$	−21.9090	−1.46713	−0.733567	+	0.679617i	$0.762146\pi$
$224$	22.4660	1.50107
$225$	0	0
$226$	−11.2774	−0.750162
$227$	20.8284	1.38243	0.691215	−	0.722649i	$-0.257076\pi$
$227$	20.8284	1.38243	0.691215	+	0.722649i	$0.257076\pi$
$228$	0	0
$229$	11.9137	0.787282	0.393641	−	0.919264i	$-0.371215\pi$
$229$	11.9137	0.787282	0.393641	+	0.919264i	$0.371215\pi$
$230$	−2.98497	−0.196823
$231$	0	0
$232$	−15.6499	−1.02747
$233$	11.4272	0.748623	0.374311	−	0.927303i	$-0.377879\pi$
$233$	11.4272	0.748623	0.374311	+	0.927303i	$0.377879\pi$
$234$	0	0
$235$	−10.6133	−0.692336
$236$	−3.67767	−0.239396
$237$	0	0
$238$	0.148073	0.00959817
$239$	−17.4200	−1.12681	−0.563403	−	0.826182i	$-0.690508\pi$
$239$	−17.4200	−1.12681	−0.563403	+	0.826182i	$0.690508\pi$
$240$	0	0
$241$	−0.309762	−0.0199535	−0.00997676	−	0.999950i	$-0.503176\pi$
$241$	−0.309762	−0.0199535	−0.00997676	+	0.999950i	$0.503176\pi$
$242$	0.668880	0.0429972
$243$	0	0
$244$	−13.6141	−0.871555
$245$	8.17905	0.522540
$246$	0	0
$247$	6.80088	0.432730
$248$	4.23054	0.268639
$249$	0	0
$250$	0.668880	0.0423037
$251$	−8.04266	−0.507648	−0.253824	−	0.967250i	$-0.581688\pi$
$251$	−8.04266	−0.507648	−0.253824	+	0.967250i	$0.581688\pi$
$252$	0	0
$253$	−4.46264	−0.280564
$254$	3.73950	0.234637
$255$	0	0
$256$	−8.99572	−0.562232
$257$	2.12457	0.132527	0.0662636	−	0.997802i	$-0.478892\pi$
$257$	2.12457	0.132527	0.0662636	+	0.997802i	$0.478892\pi$
$258$	0	0
$259$	27.1840	1.68913
$260$	10.5590	0.654844
$261$	0	0
$262$	−8.94665	−0.552726
$263$	3.52317	0.217248	0.108624	−	0.994083i	$-0.465356\pi$
$263$	3.52317	0.217248	0.108624	+	0.994083i	$0.465356\pi$
$264$	0	0
$265$	11.8636	0.728778
$266$	−2.60598	−0.159783
$267$	0	0
$268$	8.96826	0.547824
$269$	−12.2486	−0.746807	−0.373404	−	0.927669i	$-0.621809\pi$
$269$	−12.2486	−0.746807	−0.373404	+	0.927669i	$0.621809\pi$
$270$	0	0
$271$	−29.8817	−1.81518	−0.907591	−	0.419856i	$-0.862081\pi$
$271$	−29.8817	−1.81518	−0.907591	+	0.419856i	$0.862081\pi$
$272$	0.0861267	0.00522220
$273$	0	0
$274$	10.7041	0.646658
$275$	1.00000	0.0603023
$276$	0	0
$277$	−16.0830	−0.966331	−0.483166	−	0.875529i	$-0.660513\pi$
$277$	−16.0830	−0.966331	−0.483166	+	0.875529i	$0.660513\pi$
$278$	5.40086	0.323922
$279$	0	0
$280$	−9.25799	−0.553271
$281$	−4.26544	−0.254455	−0.127227	−	0.991874i	$-0.540608\pi$
$281$	−4.26544	−0.254455	−0.127227	+	0.991874i	$0.540608\pi$
$282$	0	0
$283$	−19.0032	−1.12963	−0.564813	−	0.825219i	$-0.691052\pi$
$283$	−19.0032	−1.12963	−0.564813	+	0.825219i	$0.691052\pi$
$284$	3.37411	0.200216
$285$	0	0
$286$	−4.54897	−0.268987
$287$	−44.9457	−2.65306
$288$	0	0
$289$	−16.9968	−0.999810
$290$	4.40519	0.258682
$291$	0	0
$292$	20.0981	1.17615
$293$	−16.7602	−0.979140	−0.489570	−	0.871964i	$-0.662846\pi$
$293$	−16.7602	−0.979140	−0.489570	+	0.871964i	$0.662846\pi$
$294$	0	0
$295$	2.36872	0.137912
$296$	−16.5800	−0.963695
$297$	0	0
$298$	5.15281	0.298494
$299$	30.3499	1.75518
$300$	0	0
$301$	4.71169	0.271577
$302$	12.4971	0.719126
$303$	0	0
$304$	−1.51576	−0.0869350
$305$	8.76860	0.502089
$306$	0	0
$307$	−6.87699	−0.392490	−0.196245	−	0.980555i	$-0.562875\pi$
$307$	−6.87699	−0.392490	−0.196245	+	0.980555i	$0.562875\pi$
$308$	−6.04897	−0.344672
$309$	0	0
$310$	−1.19083	−0.0676345
$311$	−16.3883	−0.929296	−0.464648	−	0.885495i	$-0.653819\pi$
$311$	−16.3883	−0.929296	−0.464648	+	0.885495i	$0.653819\pi$
$312$	0	0
$313$	34.0276	1.92336	0.961678	−	0.274182i	$-0.0884069\pi$
$313$	34.0276	1.92336	0.961678	+	0.274182i	$0.0884069\pi$
$314$	3.30174	0.186328
$315$	0	0
$316$	25.4409	1.43116
$317$	20.9595	1.17720	0.588601	−	0.808424i	$-0.299679\pi$
$317$	20.9595	1.17720	0.588601	+	0.808424i	$0.299679\pi$
$318$	0	0
$319$	6.58592	0.368741
$320$	0.825498	0.0461467
$321$	0	0
$322$	−11.6295	−0.648088
$323$	−0.0568207	−0.00316159
$324$	0	0
$325$	−6.80088	−0.377245
$326$	−3.04003	−0.168372
$327$	0	0
$328$	27.4133	1.51364
$329$	−41.3498	−2.27969
$330$	0	0
$331$	14.2518	0.783351	0.391676	−	0.920103i	$-0.371896\pi$
$331$	14.2518	0.783351	0.391676	+	0.920103i	$0.371896\pi$
$332$	16.7742	0.920605
$333$	0	0
$334$	−9.36654	−0.512514
$335$	−5.77629	−0.315592
$336$	0	0
$337$	−19.4996	−1.06221	−0.531106	−	0.847305i	$-0.678224\pi$
$337$	−19.4996	−1.06221	−0.531106	+	0.847305i	$0.678224\pi$
$338$	22.2416	1.20978
$339$	0	0
$340$	−0.0882197	−0.00478439
$341$	−1.78033	−0.0964104
$342$	0	0
$343$	4.59361	0.248032
$344$	−2.87375	−0.154942
$345$	0	0
$346$	4.80613	0.258379
$347$	5.67690	0.304752	0.152376	−	0.988323i	$-0.451308\pi$
$347$	5.67690	0.304752	0.152376	+	0.988323i	$0.451308\pi$
$348$	0	0
$349$	−17.3074	−0.926443	−0.463222	−	0.886242i	$-0.653307\pi$
$349$	−17.3074	−0.926443	−0.463222	+	0.886242i	$0.653307\pi$
$350$	2.60598	0.139295
$351$	0	0
$352$	5.76639	0.307350
$353$	27.8575	1.48271	0.741354	−	0.671114i	$-0.234184\pi$
$353$	27.8575	1.48271	0.741354	+	0.671114i	$0.234184\pi$
$354$	0	0
$355$	−2.17320	−0.115341
$356$	−6.73651	−0.357034
$357$	0	0
$358$	3.02916	0.160096
$359$	29.1436	1.53814	0.769069	−	0.639166i	$-0.220720\pi$
$359$	29.1436	1.53814	0.769069	+	0.639166i	$0.220720\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−17.5683	−0.923369
$363$	0	0
$364$	41.1384	2.15624
$365$	−12.9448	−0.677561
$366$	0	0
$367$	−24.4348	−1.27548	−0.637742	−	0.770250i	$-0.720131\pi$
$367$	−24.4348	−1.27548	−0.637742	+	0.770250i	$0.720131\pi$
$368$	−6.76430	−0.352613
$369$	0	0
$370$	4.66701	0.242627
$371$	46.2211	2.39968
$372$	0	0
$373$	−21.3183	−1.10382	−0.551910	−	0.833903i	$-0.686101\pi$
$373$	−21.3183	−1.10382	−0.551910	+	0.833903i	$0.686101\pi$
$374$	0.0380062	0.00196525
$375$	0	0
$376$	25.2200	1.30062
$377$	−44.7901	−2.30681
$378$	0	0
$379$	−25.4652	−1.30806	−0.654030	−	0.756469i	$-0.726923\pi$
$379$	−25.4652	−1.30806	−0.654030	+	0.756469i	$0.726923\pi$
$380$	1.55260	0.0796467
$381$	0	0
$382$	1.56868	0.0802608
$383$	−30.9469	−1.58131	−0.790656	−	0.612260i	$-0.790260\pi$
$383$	−30.9469	−1.58131	−0.790656	+	0.612260i	$0.790260\pi$
$384$	0	0
$385$	3.89603	0.198560
$386$	−0.0577195	−0.00293785
$387$	0	0
$388$	27.5779	1.40006
$389$	9.84272	0.499045	0.249523	−	0.968369i	$-0.419726\pi$
$389$	9.84272	0.499045	0.249523	+	0.968369i	$0.419726\pi$
$390$	0	0
$391$	−0.253570	−0.0128236
$392$	−19.4356	−0.981645
$393$	0	0
$394$	−15.9731	−0.804714
$395$	−16.3860	−0.824468
$396$	0	0
$397$	−30.8372	−1.54768	−0.773838	−	0.633383i	$-0.781666\pi$
$397$	−30.8372	−1.54768	−0.773838	+	0.633383i	$0.781666\pi$
$398$	11.4026	0.571561
$399$	0	0
$400$	1.51576	0.0757882
$401$	9.62765	0.480782	0.240391	−	0.970676i	$-0.422724\pi$
$401$	9.62765	0.480782	0.240391	+	0.970676i	$0.422724\pi$
$402$	0	0
$403$	12.1078	0.603134
$404$	−12.5817	−0.625961
$405$	0	0
$406$	17.1628	0.851774
$407$	6.97736	0.345855
$408$	0	0
$409$	−37.7394	−1.86610	−0.933048	−	0.359753i	$-0.882861\pi$
$409$	−37.7394	−1.86610	−0.933048	+	0.359753i	$0.882861\pi$
$410$	−7.71640	−0.381086
$411$	0	0
$412$	7.47117	0.368078
$413$	9.22859	0.454109
$414$	0	0
$415$	−10.8040	−0.530345
$416$	−39.2165	−1.92275
$417$	0	0
$418$	−0.668880	−0.0327160
$419$	−7.18768	−0.351141	−0.175571	−	0.984467i	$-0.556177\pi$
$419$	−7.18768	−0.351141	−0.175571	+	0.984467i	$0.556177\pi$
$420$	0	0
$421$	−33.3224	−1.62404	−0.812018	−	0.583633i	$-0.801631\pi$
$421$	−33.3224	−1.62404	−0.812018	+	0.583633i	$0.801631\pi$
$422$	6.69000	0.325664
$423$	0	0
$424$	−28.1911	−1.36908
$425$	0.0568207	0.00275621
$426$	0	0
$427$	34.1627	1.65325
$428$	3.29105	0.159079
$429$	0	0
$430$	0.808914	0.0390093
$431$	6.40724	0.308626	0.154313	−	0.988022i	$-0.450684\pi$
$431$	6.40724	0.308626	0.154313	+	0.988022i	$0.450684\pi$
$432$	0	0
$433$	−35.3701	−1.69978	−0.849890	−	0.526961i	$-0.823332\pi$
$433$	−35.3701	−1.69978	−0.849890	+	0.526961i	$0.823332\pi$
$434$	−4.63950	−0.222703
$435$	0	0
$436$	10.9755	0.525632
$437$	4.46264	0.213477
$438$	0	0
$439$	4.20908	0.200888	0.100444	−	0.994943i	$-0.467974\pi$
$439$	4.20908	0.200888	0.100444	+	0.994943i	$0.467974\pi$
$440$	−2.37626	−0.113284
$441$	0	0
$442$	−0.258476	−0.0122944
$443$	−6.57863	−0.312560	−0.156280	−	0.987713i	$-0.549950\pi$
$443$	−6.57863	−0.312560	−0.156280	+	0.987713i	$0.549950\pi$
$444$	0	0
$445$	4.33886	0.205682
$446$	−14.6545	−0.693910
$447$	0	0
$448$	3.21616	0.151949
$449$	11.0802	0.522908	0.261454	−	0.965216i	$-0.415798\pi$
$449$	11.0802	0.522908	0.261454	+	0.965216i	$0.415798\pi$
$450$	0	0
$451$	−11.5363	−0.543223
$452$	26.1770	1.23126
$453$	0	0
$454$	13.9317	0.653847
$455$	−26.4964	−1.24217
$456$	0	0
$457$	−2.43597	−0.113950	−0.0569749	−	0.998376i	$-0.518146\pi$
$457$	−2.43597	−0.113950	−0.0569749	+	0.998376i	$0.518146\pi$
$458$	7.96886	0.372360
$459$	0	0
$460$	6.92869	0.323052
$461$	−31.2959	−1.45759	−0.728797	−	0.684730i	$-0.759920\pi$
$461$	−31.2959	−1.45759	−0.728797	+	0.684730i	$0.759920\pi$
$462$	0	0
$463$	15.5863	0.724358	0.362179	−	0.932109i	$-0.382033\pi$
$463$	15.5863	0.724358	0.362179	+	0.932109i	$0.382033\pi$
$464$	9.98270	0.463435
$465$	0	0
$466$	7.64345	0.354076
$467$	9.26451	0.428710	0.214355	−	0.976756i	$-0.431235\pi$
$467$	9.26451	0.428710	0.214355	+	0.976756i	$0.431235\pi$
$468$	0	0
$469$	−22.5046	−1.03917
$470$	−7.09903	−0.327454
$471$	0	0
$472$	−5.62870	−0.259082
$473$	1.20936	0.0556062
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	−0.343707	−0.0157538
$477$	0	0
$478$	−11.6519	−0.532945
$479$	−11.8239	−0.540250	−0.270125	−	0.962825i	$-0.587065\pi$
$479$	−11.8239	−0.540250	−0.270125	+	0.962825i	$0.587065\pi$
$480$	0	0
$481$	−47.4522	−2.16363
$482$	−0.207194	−0.00943741
$483$	0	0
$484$	−1.55260	−0.0705727
$485$	−17.7624	−0.806551
$486$	0	0
$487$	26.1763	1.18616	0.593082	−	0.805142i	$-0.297911\pi$
$487$	26.1763	1.18616	0.593082	+	0.805142i	$0.297911\pi$
$488$	−20.8365	−0.943225
$489$	0	0
$490$	5.47080	0.247146
$491$	−12.7626	−0.575967	−0.287983	−	0.957635i	$-0.592985\pi$
$491$	−12.7626	−0.575967	−0.287983	+	0.957635i	$0.592985\pi$
$492$	0	0
$493$	0.374217	0.0168539
$494$	4.54897	0.204668
$495$	0	0
$496$	−2.69856	−0.121169
$497$	−8.46685	−0.379790
$498$	0	0
$499$	−11.9443	−0.534701	−0.267351	−	0.963599i	$-0.586148\pi$
$499$	−11.9443	−0.534701	−0.267351	+	0.963599i	$0.586148\pi$
$500$	−1.55260	−0.0694344
$501$	0	0
$502$	−5.37958	−0.240102
$503$	−25.5076	−1.13733	−0.568665	−	0.822569i	$-0.692540\pi$
$503$	−25.5076	−1.13733	−0.568665	+	0.822569i	$0.692540\pi$
$504$	0	0
$505$	8.10361	0.360606
$506$	−2.98497	−0.132698
$507$	0	0
$508$	−8.68011	−0.385117
$509$	−25.7342	−1.14065	−0.570325	−	0.821419i	$-0.693183\pi$
$509$	−25.7342	−1.14065	−0.570325	+	0.821419i	$0.693183\pi$
$510$	0	0
$511$	−50.4333	−2.23104
$512$	15.9442	0.704640
$513$	0	0
$514$	1.42108	0.0626814
$515$	−4.81204	−0.212044
$516$	0	0
$517$	−10.6133	−0.466773
$518$	18.1828	0.798908
$519$	0	0
$520$	16.1607	0.708693
$521$	38.7255	1.69660	0.848298	−	0.529519i	$-0.177628\pi$
$521$	38.7255	1.69660	0.848298	+	0.529519i	$0.177628\pi$
$522$	0	0
$523$	8.02095	0.350732	0.175366	−	0.984503i	$-0.443889\pi$
$523$	8.02095	0.350732	0.175366	+	0.984503i	$0.443889\pi$
$524$	20.7669	0.907206
$525$	0	0
$526$	2.35658	0.102752
$527$	−0.101160	−0.00440658
$528$	0	0
$529$	−3.08488	−0.134125
$530$	7.93536	0.344690
$531$	0	0
$532$	6.04897	0.262256
$533$	78.4570	3.39835
$534$	0	0
$535$	−2.11970	−0.0916427
$536$	13.7260	0.592872
$537$	0	0
$538$	−8.19281	−0.353217
$539$	8.17905	0.352297
$540$	0	0
$541$	−2.24006	−0.0963076	−0.0481538	−	0.998840i	$-0.515334\pi$
$541$	−2.24006	−0.0963076	−0.0481538	+	0.998840i	$0.515334\pi$
$542$	−19.9873	−0.858526
$543$	0	0
$544$	0.327650	0.0140479
$545$	−7.06913	−0.302808
$546$	0	0
$547$	39.4365	1.68618	0.843092	−	0.537769i	$-0.180733\pi$
$547$	39.4365	1.68618	0.843092	+	0.537769i	$0.180733\pi$
$548$	−24.8462	−1.06138
$549$	0	0
$550$	0.668880	0.0285211
$551$	−6.58592	−0.280570
$552$	0	0
$553$	−63.8403	−2.71476
$554$	−10.7576	−0.457045
$555$	0	0
$556$	−12.5364	−0.531663
$557$	32.7415	1.38730	0.693650	−	0.720312i	$-0.256001\pi$
$557$	32.7415	1.38730	0.693650	+	0.720312i	$0.256001\pi$
$558$	0	0
$559$	−8.22469	−0.347867
$560$	5.90546	0.249551
$561$	0	0
$562$	−2.85307	−0.120349
$563$	46.4415	1.95728	0.978638	−	0.205589i	$-0.0659110\pi$
$563$	46.4415	1.95728	0.978638	+	0.205589i	$0.0659110\pi$
$564$	0	0
$565$	−16.8601	−0.709311
$566$	−12.7109	−0.534278
$567$	0	0
$568$	5.16409	0.216681
$569$	9.63033	0.403724	0.201862	−	0.979414i	$-0.435301\pi$
$569$	9.63033	0.403724	0.201862	+	0.979414i	$0.435301\pi$
$570$	0	0
$571$	−4.93999	−0.206732	−0.103366	−	0.994643i	$-0.532961\pi$
$571$	−4.93999	−0.206732	−0.103366	+	0.994643i	$0.532961\pi$
$572$	10.5590	0.441496
$573$	0	0
$574$	−30.0633	−1.25482
$575$	−4.46264	−0.186105
$576$	0	0
$577$	−5.57704	−0.232175	−0.116088	−	0.993239i	$-0.537035\pi$
$577$	−5.57704	−0.232175	−0.116088	+	0.993239i	$0.537035\pi$
$578$	−11.3688	−0.472880
$579$	0	0
$580$	−10.2253	−0.424583
$581$	−42.0925	−1.74629
$582$	0	0
$583$	11.8636	0.491342
$584$	30.7602	1.27287
$585$	0	0
$586$	−11.2105	−0.463103
$587$	−39.0684	−1.61253	−0.806264	−	0.591556i	$-0.798514\pi$
$587$	−39.0684	−1.61253	−0.806264	+	0.591556i	$0.798514\pi$
$588$	0	0
$589$	1.78033	0.0733573
$590$	1.58439	0.0652282
$591$	0	0
$592$	10.5760	0.434672
$593$	36.2793	1.48981	0.744905	−	0.667170i	$-0.232495\pi$
$593$	36.2793	1.48981	0.744905	+	0.667170i	$0.232495\pi$
$594$	0	0
$595$	0.221375	0.00907549
$596$	−11.9607	−0.489927
$597$	0	0
$598$	20.3004	0.830146
$599$	40.5451	1.65663	0.828313	−	0.560265i	$-0.189301\pi$
$599$	40.5451	1.65663	0.828313	+	0.560265i	$0.189301\pi$
$600$	0	0
$601$	−5.75685	−0.234827	−0.117413	−	0.993083i	$-0.537460\pi$
$601$	−5.75685	−0.234827	−0.117413	+	0.993083i	$0.537460\pi$
$602$	3.15155	0.128448
$603$	0	0
$604$	−29.0081	−1.18032
$605$	1.00000	0.0406558
$606$	0	0
$607$	−25.9320	−1.05255	−0.526274	−	0.850315i	$-0.676411\pi$
$607$	−25.9320	−1.05255	−0.526274	+	0.850315i	$0.676411\pi$
$608$	−5.76639	−0.233858
$609$	0	0
$610$	5.86515	0.237473
$611$	72.1798	2.92008
$612$	0	0
$613$	3.82959	0.154676	0.0773379	−	0.997005i	$-0.475358\pi$
$613$	3.82959	0.154676	0.0773379	+	0.997005i	$0.475358\pi$
$614$	−4.59988	−0.185636
$615$	0	0
$616$	−9.25799	−0.373015
$617$	39.1870	1.57761	0.788804	−	0.614645i	$-0.210701\pi$
$617$	39.1870	1.57761	0.788804	+	0.614645i	$0.210701\pi$
$618$	0	0
$619$	−5.08107	−0.204226	−0.102113	−	0.994773i	$-0.532560\pi$
$619$	−5.08107	−0.204226	−0.102113	+	0.994773i	$0.532560\pi$
$620$	2.76414	0.111011
$621$	0	0
$622$	−10.9618	−0.439529
$623$	16.9043	0.677257
$624$	0	0
$625$	1.00000	0.0400000
$626$	22.7604	0.909689
$627$	0	0
$628$	−7.66397	−0.305826
$629$	0.396458	0.0158078
$630$	0	0
$631$	15.2354	0.606513	0.303256	−	0.952909i	$-0.401926\pi$
$631$	15.2354	0.606513	0.303256	+	0.952909i	$0.401926\pi$
$632$	38.9374	1.54885
$633$	0	0
$634$	14.0194	0.556780
$635$	5.59069	0.221860
$636$	0	0
$637$	−55.6247	−2.20393
$638$	4.40519	0.174403
$639$	0	0
$640$	−10.9806	−0.434047
$641$	−3.86675	−0.152728	−0.0763638	−	0.997080i	$-0.524331\pi$
$641$	−3.86675	−0.152728	−0.0763638	+	0.997080i	$0.524331\pi$
$642$	0	0
$643$	7.19042	0.283563	0.141781	−	0.989898i	$-0.454717\pi$
$643$	7.19042	0.283563	0.141781	+	0.989898i	$0.454717\pi$
$644$	26.9944	1.06373
$645$	0	0
$646$	−0.0380062	−0.00149533
$647$	2.92924	0.115160	0.0575802	−	0.998341i	$-0.481662\pi$
$647$	2.92924	0.115160	0.0575802	+	0.998341i	$0.481662\pi$
$648$	0	0
$649$	2.36872	0.0929803
$650$	−4.54897	−0.178425
$651$	0	0
$652$	7.05649	0.276354
$653$	32.9802	1.29062	0.645308	−	0.763923i	$-0.276729\pi$
$653$	32.9802	1.29062	0.645308	+	0.763923i	$0.276729\pi$
$654$	0	0
$655$	−13.3756	−0.522626
$656$	−17.4863	−0.682725
$657$	0	0
$658$	−27.6580	−1.07822
$659$	2.61697	0.101943	0.0509714	−	0.998700i	$-0.483768\pi$
$659$	2.61697	0.101943	0.0509714	+	0.998700i	$0.483768\pi$
$660$	0	0
$661$	−32.9463	−1.28146	−0.640732	−	0.767764i	$-0.721369\pi$
$661$	−32.9463	−1.28146	−0.640732	+	0.767764i	$0.721369\pi$
$662$	9.53276	0.370501
$663$	0	0
$664$	25.6730	0.996307
$665$	−3.89603	−0.151082
$666$	0	0
$667$	−29.3906	−1.13801
$668$	21.7415	0.841205
$669$	0	0
$670$	−3.86364	−0.149266
$671$	8.76860	0.338508
$672$	0	0
$673$	−13.8813	−0.535084	−0.267542	−	0.963546i	$-0.586211\pi$
$673$	−13.8813	−0.535084	−0.267542	+	0.963546i	$0.586211\pi$
$674$	−13.0429	−0.502394
$675$	0	0
$676$	−51.6270	−1.98565
$677$	−24.8929	−0.956710	−0.478355	−	0.878166i	$-0.658767\pi$
$677$	−24.8929	−0.956710	−0.478355	+	0.878166i	$0.658767\pi$
$678$	0	0
$679$	−69.2030	−2.65577
$680$	−0.135021	−0.00517781
$681$	0	0
$682$	−1.19083	−0.0455992
$683$	−1.09099	−0.0417454	−0.0208727	−	0.999782i	$-0.506644\pi$
$683$	−1.09099	−0.0417454	−0.0208727	+	0.999782i	$0.506644\pi$
$684$	0	0
$685$	16.0030	0.611443
$686$	3.07258	0.117312
$687$	0	0
$688$	1.83310	0.0698862
$689$	−80.6833	−3.07379
$690$	0	0
$691$	33.9615	1.29196	0.645979	−	0.763355i	$-0.276449\pi$
$691$	33.9615	1.29196	0.645979	+	0.763355i	$0.276449\pi$
$692$	−11.1560	−0.424086
$693$	0	0
$694$	3.79717	0.144139
$695$	8.07448	0.306282
$696$	0	0
$697$	−0.655500	−0.0248288
$698$	−11.5766	−0.438180
$699$	0	0
$700$	−6.04897	−0.228630
$701$	31.3898	1.18558	0.592789	−	0.805358i	$-0.298027\pi$
$701$	31.3898	1.18558	0.592789	+	0.805358i	$0.298027\pi$
$702$	0	0
$703$	−6.97736	−0.263156
$704$	0.825498	0.0311121
$705$	0	0
$706$	18.6334	0.701276
$707$	31.5719	1.18738
$708$	0	0
$709$	−16.3408	−0.613690	−0.306845	−	0.951760i	$-0.599273\pi$
$709$	−16.3408	−0.613690	−0.306845	+	0.951760i	$0.599273\pi$
$710$	−1.45361	−0.0545530
$711$	0	0
$712$	−10.3103	−0.386394
$713$	7.94497	0.297542
$714$	0	0
$715$	−6.80088	−0.254339
$716$	−7.03126	−0.262770
$717$	0	0
$718$	19.4935	0.727493
$719$	18.9364	0.706208	0.353104	−	0.935584i	$-0.385126\pi$
$719$	18.9364	0.706208	0.353104	+	0.935584i	$0.385126\pi$
$720$	0	0
$721$	−18.7479	−0.698206
$722$	0.668880	0.0248931
$723$	0	0
$724$	40.7794	1.51555
$725$	6.58592	0.244595
$726$	0	0
$727$	35.9501	1.33332	0.666658	−	0.745364i	$-0.267724\pi$
$727$	35.9501	1.33332	0.666658	+	0.745364i	$0.267724\pi$
$728$	62.9625	2.33355
$729$	0	0
$730$	−8.65851	−0.320466
$731$	0.0687164	0.00254157
$732$	0	0
$733$	19.7627	0.729953	0.364976	−	0.931017i	$-0.381077\pi$
$733$	19.7627	0.729953	0.364976	+	0.931017i	$0.381077\pi$
$734$	−16.3439	−0.603265
$735$	0	0
$736$	−25.7333	−0.948542
$737$	−5.77629	−0.212772
$738$	0	0
$739$	−19.9335	−0.733267	−0.366634	−	0.930365i	$-0.619490\pi$
$739$	−19.9335	−0.733267	−0.366634	+	0.930365i	$0.619490\pi$
$740$	−10.8330	−0.398230
$741$	0	0
$742$	30.9164	1.13498
$743$	−33.4842	−1.22842	−0.614208	−	0.789144i	$-0.710525\pi$
$743$	−33.4842	−1.22842	−0.614208	+	0.789144i	$0.710525\pi$
$744$	0	0
$745$	7.70363	0.282239
$746$	−14.2594	−0.522074
$747$	0	0
$748$	−0.0882197	−0.00322563
$749$	−8.25842	−0.301756
$750$	0	0
$751$	25.8425	0.943005	0.471502	−	0.881865i	$-0.343712\pi$
$751$	25.8425	0.943005	0.471502	+	0.881865i	$0.343712\pi$
$752$	−16.0873	−0.586642
$753$	0	0
$754$	−29.9592	−1.09105
$755$	18.6836	0.679965
$756$	0	0
$757$	−21.1284	−0.767923	−0.383962	−	0.923349i	$-0.625440\pi$
$757$	−21.1284	−0.767923	−0.383962	+	0.923349i	$0.625440\pi$
$758$	−17.0332	−0.618673
$759$	0	0
$760$	2.37626	0.0861961
$761$	14.9022	0.540205	0.270102	−	0.962832i	$-0.412942\pi$
$761$	14.9022	0.540205	0.270102	+	0.962832i	$0.412942\pi$
$762$	0	0
$763$	−27.5415	−0.997071
$764$	−3.64121	−0.131735
$765$	0	0
$766$	−20.6998	−0.747913
$767$	−16.1094	−0.581675
$768$	0	0
$769$	34.8080	1.25521	0.627604	−	0.778533i	$-0.284036\pi$
$769$	34.8080	1.25521	0.627604	+	0.778533i	$0.284036\pi$
$770$	2.60598	0.0939129
$771$	0	0
$772$	0.133978	0.00482198
$773$	−4.74138	−0.170536	−0.0852678	−	0.996358i	$-0.527175\pi$
$773$	−4.74138	−0.170536	−0.0852678	+	0.996358i	$0.527175\pi$
$774$	0	0
$775$	−1.78033	−0.0639514
$776$	42.2082	1.51519
$777$	0	0
$778$	6.58360	0.236033
$779$	11.5363	0.413331
$780$	0	0
$781$	−2.17320	−0.0777632
$782$	−0.169608	−0.00606517
$783$	0	0
$784$	12.3975	0.442768
$785$	4.93622	0.176181
$786$	0	0
$787$	−2.15635	−0.0768654	−0.0384327	−	0.999261i	$-0.512237\pi$
$787$	−2.15635	−0.0768654	−0.0384327	+	0.999261i	$0.512237\pi$
$788$	37.0767	1.32080
$789$	0	0
$790$	−10.9603	−0.389949
$791$	−65.6875	−2.33558
$792$	0	0
$793$	−59.6342	−2.11767
$794$	−20.6264	−0.732004
$795$	0	0
$796$	−26.4676	−0.938120
$797$	44.4960	1.57613	0.788064	−	0.615593i	$-0.211084\pi$
$797$	44.4960	1.57613	0.788064	+	0.615593i	$0.211084\pi$
$798$	0	0
$799$	−0.603055	−0.0213346
$800$	5.76639	0.203873
$801$	0	0
$802$	6.43974	0.227395
$803$	−12.9448	−0.456812
$804$	0	0
$805$	−17.3866	−0.612796
$806$	8.09869	0.285264
$807$	0	0
$808$	−19.2563	−0.677435
$809$	−38.7402	−1.36203	−0.681017	−	0.732268i	$-0.738462\pi$
$809$	−38.7402	−1.36203	−0.681017	+	0.732268i	$0.738462\pi$
$810$	0	0
$811$	−43.8012	−1.53807	−0.769035	−	0.639207i	$-0.779263\pi$
$811$	−43.8012	−1.53807	−0.769035	+	0.639207i	$0.779263\pi$
$812$	−39.8381	−1.39804
$813$	0	0
$814$	4.66701	0.163579
$815$	−4.54495	−0.159203
$816$	0	0
$817$	−1.20936	−0.0423100
$818$	−25.2432	−0.882606
$819$	0	0
$820$	17.9112	0.625488
$821$	20.9165	0.729991	0.364995	−	0.931009i	$-0.381071\pi$
$821$	20.9165	0.729991	0.364995	+	0.931009i	$0.381071\pi$
$822$	0	0
$823$	−48.5646	−1.69286	−0.846428	−	0.532503i	$-0.821252\pi$
$823$	−48.5646	−1.69286	−0.846428	+	0.532503i	$0.821252\pi$
$824$	11.4347	0.398346
$825$	0	0
$826$	6.17282	0.214780
$827$	−20.9369	−0.728049	−0.364024	−	0.931389i	$-0.618598\pi$
$827$	−20.9369	−0.728049	−0.364024	+	0.931389i	$0.618598\pi$
$828$	0	0
$829$	−27.0846	−0.940688	−0.470344	−	0.882483i	$-0.655870\pi$
$829$	−27.0846	−0.940688	−0.470344	+	0.882483i	$0.655870\pi$
$830$	−7.22655	−0.250837
$831$	0	0
$832$	−5.61411	−0.194634
$833$	0.464739	0.0161023
$834$	0	0
$835$	−14.0033	−0.484605
$836$	1.55260	0.0536978
$837$	0	0
$838$	−4.80770	−0.166079
$839$	−11.8505	−0.409123	−0.204562	−	0.978854i	$-0.565577\pi$
$839$	−11.8505	−0.409123	−0.204562	+	0.978854i	$0.565577\pi$
$840$	0	0
$841$	14.3744	0.495669
$842$	−22.2887	−0.768120
$843$	0	0
$844$	−15.5288	−0.534522
$845$	33.2520	1.14390
$846$	0	0
$847$	3.89603	0.133869
$848$	17.9825	0.617521
$849$	0	0
$850$	0.0380062	0.00130360
$851$	−31.1374	−1.06738
$852$	0	0
$853$	3.81601	0.130658	0.0653288	−	0.997864i	$-0.479190\pi$
$853$	3.81601	0.130658	0.0653288	+	0.997864i	$0.479190\pi$
$854$	22.8508	0.781938
$855$	0	0
$856$	5.03697	0.172160
$857$	17.1685	0.586465	0.293233	−	0.956041i	$-0.405269\pi$
$857$	17.1685	0.586465	0.293233	+	0.956041i	$0.405269\pi$
$858$	0	0
$859$	18.3593	0.626410	0.313205	−	0.949685i	$-0.398597\pi$
$859$	18.3593	0.626410	0.313205	+	0.949685i	$0.398597\pi$
$860$	−1.87765	−0.0640272
$861$	0	0
$862$	4.28567	0.145971
$863$	−12.8422	−0.437154	−0.218577	−	0.975820i	$-0.570141\pi$
$863$	−12.8422	−0.437154	−0.218577	+	0.975820i	$0.570141\pi$
$864$	0	0
$865$	7.18534	0.244309
$866$	−23.6584	−0.803944
$867$	0	0
$868$	10.7692	0.365530
$869$	−16.3860	−0.555856
$870$	0	0
$871$	39.2839	1.33108
$872$	16.7981	0.568856
$873$	0	0
$874$	2.98497	0.100968
$875$	3.89603	0.131710
$876$	0	0
$877$	−13.2698	−0.448091	−0.224045	−	0.974579i	$-0.571926\pi$
$877$	−13.2698	−0.448091	−0.224045	+	0.974579i	$0.571926\pi$
$878$	2.81537	0.0950141
$879$	0	0
$880$	1.51576	0.0510964
$881$	26.1922	0.882438	0.441219	−	0.897400i	$-0.354546\pi$
$881$	26.1922	0.882438	0.441219	+	0.897400i	$0.354546\pi$
$882$	0	0
$883$	−15.6704	−0.527350	−0.263675	−	0.964612i	$-0.584935\pi$
$883$	−15.6704	−0.527350	−0.263675	+	0.964612i	$0.584935\pi$
$884$	0.599972	0.0201792
$885$	0	0
$886$	−4.40032	−0.147831
$887$	−58.0544	−1.94928	−0.974638	−	0.223789i	$-0.928157\pi$
$887$	−58.0544	−1.94928	−0.974638	+	0.223789i	$0.928157\pi$
$888$	0	0
$889$	21.7815	0.730528
$890$	2.90218	0.0972811
$891$	0	0
$892$	34.0159	1.13894
$893$	10.6133	0.355161
$894$	0	0
$895$	4.52870	0.151378
$896$	−42.7808	−1.42921
$897$	0	0
$898$	7.41133	0.247319
$899$	−11.7251	−0.391055
$900$	0	0
$901$	0.674100	0.0224575
$902$	−7.71640	−0.256928
$903$	0	0
$904$	40.0641	1.33251
$905$	−26.2652	−0.873086
$906$	0	0
$907$	48.7833	1.61982	0.809912	−	0.586552i	$-0.199515\pi$
$907$	48.7833	1.61982	0.809912	+	0.586552i	$0.199515\pi$
$908$	−32.3382	−1.07318
$909$	0	0
$910$	−17.7229	−0.587510
$911$	−37.2878	−1.23540	−0.617700	−	0.786414i	$-0.711935\pi$
$911$	−37.2878	−1.23540	−0.617700	+	0.786414i	$0.711935\pi$
$912$	0	0
$913$	−10.8040	−0.357559
$914$	−1.62937	−0.0538948
$915$	0	0
$916$	−18.4973	−0.611166
$917$	−52.1116	−1.72088
$918$	0	0
$919$	−26.5993	−0.877430	−0.438715	−	0.898626i	$-0.644566\pi$
$919$	−26.5993	−0.877430	−0.438715	+	0.898626i	$0.644566\pi$
$920$	10.6044	0.349617
$921$	0	0
$922$	−20.9332	−0.689398
$923$	14.7797	0.486479
$924$	0	0
$925$	6.97736	0.229414
$926$	10.4254	0.342599
$927$	0	0
$928$	37.9770	1.24666
$929$	−29.6174	−0.971714	−0.485857	−	0.874038i	$-0.661492\pi$
$929$	−29.6174	−0.971714	−0.485857	+	0.874038i	$0.661492\pi$
$930$	0	0
$931$	−8.17905	−0.268058
$932$	−17.7419	−0.581155
$933$	0	0
$934$	6.19684	0.202767
$935$	0.0568207	0.00185823
$936$	0	0
$937$	−41.8250	−1.36636	−0.683181	−	0.730249i	$-0.739404\pi$
$937$	−41.8250	−1.36636	−0.683181	+	0.730249i	$0.739404\pi$
$938$	−15.0529	−0.491494
$939$	0	0
$940$	16.4782	0.537460
$941$	31.8359	1.03782	0.518910	−	0.854829i	$-0.326338\pi$
$941$	31.8359	1.03782	0.518910	+	0.854829i	$0.326338\pi$
$942$	0	0
$943$	51.4823	1.67649
$944$	3.59041	0.116858
$945$	0	0
$946$	0.808914	0.0263001
$947$	−50.6277	−1.64518	−0.822590	−	0.568635i	$-0.807472\pi$
$947$	−50.6277	−1.64518	−0.822590	+	0.568635i	$0.807472\pi$
$948$	0	0
$949$	88.0360	2.85777
$950$	−0.668880	−0.0217013
$951$	0	0
$952$	−0.526045	−0.0170492
$953$	−34.8709	−1.12958	−0.564789	−	0.825235i	$-0.691042\pi$
$953$	−34.8709	−1.12958	−0.564789	+	0.825235i	$0.691042\pi$
$954$	0	0
$955$	2.34524	0.0758901
$956$	27.0463	0.874739
$957$	0	0
$958$	−7.90881	−0.255522
$959$	62.3482	2.01333
$960$	0	0
$961$	−27.8304	−0.897755
$962$	−31.7398	−1.02333
$963$	0	0
$964$	0.480936	0.0154899
$965$	−0.0862928	−0.00277786
$966$	0	0
$967$	47.9785	1.54289	0.771443	−	0.636299i	$-0.219535\pi$
$967$	47.9785	1.54289	0.771443	+	0.636299i	$0.219535\pi$
$968$	−2.37626	−0.0763760
$969$	0	0
$970$	−11.8809	−0.381474
$971$	32.5155	1.04347	0.521737	−	0.853107i	$-0.325284\pi$
$971$	32.5155	1.04347	0.521737	+	0.853107i	$0.325284\pi$
$972$	0	0
$973$	31.4584	1.00851
$974$	17.5088	0.561019
$975$	0	0
$976$	13.2911	0.425439
$977$	−44.1129	−1.41130	−0.705649	−	0.708562i	$-0.749344\pi$
$977$	−44.1129	−1.41130	−0.705649	+	0.708562i	$0.749344\pi$
$978$	0	0
$979$	4.33886	0.138670
$980$	−12.6988	−0.405648
$981$	0	0
$982$	−8.53663	−0.272415
$983$	9.60938	0.306492	0.153246	−	0.988188i	$-0.451027\pi$
$983$	9.60938	0.306492	0.153246	+	0.988188i	$0.451027\pi$
$984$	0	0
$985$	−23.8804	−0.760892
$986$	0.250306	0.00797137
$987$	0	0
$988$	−10.5590	−0.335928
$989$	−5.39692	−0.171612
$990$	0	0
$991$	8.65223	0.274847	0.137423	−	0.990512i	$-0.456118\pi$
$991$	8.65223	0.274847	0.137423	+	0.990512i	$0.456118\pi$
$992$	−10.2661	−0.325949
$993$	0	0
$994$	−5.66331	−0.179629
$995$	17.0473	0.540436
$996$	0	0
$997$	36.8661	1.16756	0.583780	−	0.811912i	$-0.301573\pi$
$997$	36.8661	1.16756	0.583780	+	0.811912i	$0.301573\pi$
$998$	−7.98932	−0.252898
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9405.2.a.bq.1.9	✓	14
3.2	odd	2		9405.2.a.br.1.6	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9405.2.a.bq.1.9	✓	14	1.1	even	1	trivial
9405.2.a.br.1.6	yes	14	3.2	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:	\( N \)	\(=\)	\( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9405.a (trivial)

\(f(q)\)	\(=\)	\(q+0.668880 q^{2} -1.55260 q^{4} +1.00000 q^{5} +3.89603 q^{7} -2.37626 q^{8} +O(q^{10})\)	\(q+0.668880 q^{2} -1.55260 q^{4} +1.00000 q^{5} +3.89603 q^{7} -2.37626 q^{8} +0.668880 q^{10} +1.00000 q^{11} -6.80088 q^{13} +2.60598 q^{14} +1.51576 q^{16} +0.0568207 q^{17} -1.00000 q^{19} -1.55260 q^{20} +0.668880 q^{22} -4.46264 q^{23} +1.00000 q^{25} -4.54897 q^{26} -6.04897 q^{28} +6.58592 q^{29} -1.78033 q^{31} +5.76639 q^{32} +0.0380062 q^{34} +3.89603 q^{35} +6.97736 q^{37} -0.668880 q^{38} -2.37626 q^{40} -11.5363 q^{41} +1.20936 q^{43} -1.55260 q^{44} -2.98497 q^{46} -10.6133 q^{47} +8.17905 q^{49} +0.668880 q^{50} +10.5590 q^{52} +11.8636 q^{53} +1.00000 q^{55} -9.25799 q^{56} +4.40519 q^{58} +2.36872 q^{59} +8.76860 q^{61} -1.19083 q^{62} +0.825498 q^{64} -6.80088 q^{65} -5.77629 q^{67} -0.0882197 q^{68} +2.60598 q^{70} -2.17320 q^{71} -12.9448 q^{73} +4.66701 q^{74} +1.55260 q^{76} +3.89603 q^{77} -16.3860 q^{79} +1.51576 q^{80} -7.71640 q^{82} -10.8040 q^{83} +0.0568207 q^{85} +0.808914 q^{86} -2.37626 q^{88} +4.33886 q^{89} -26.4964 q^{91} +6.92869 q^{92} -7.09903 q^{94} -1.00000 q^{95} -17.7624 q^{97} +5.47080 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - q^{2} + 11 q^{4} + 14 q^{5} - 12 q^{7} - 9 q^{8}+O(q^{10}) \) 14 * q - q^2 + 11 * q^4 + 14 * q^5 - 12 * q^7 - 9 * q^8	\( 14 q - q^{2} + 11 q^{4} + 14 q^{5} - 12 q^{7} - 9 q^{8} - q^{10} + 14 q^{11} - 10 q^{13} + 8 q^{14} + 13 q^{16} - 16 q^{17} - 14 q^{19} + 11 q^{20} - q^{22} - 12 q^{23} + 14 q^{25} - 12 q^{26} - 33 q^{28} - 4 q^{31} - 24 q^{32} - 2 q^{34} - 12 q^{35} - 14 q^{37} + q^{38} - 9 q^{40} - 18 q^{41} - 20 q^{43} + 11 q^{44} - 17 q^{46} - 8 q^{47} + 10 q^{49} - q^{50} - 26 q^{52} - 20 q^{53} + 14 q^{55} + 11 q^{56} - 36 q^{58} + 2 q^{59} + 4 q^{61} - 38 q^{62} + 3 q^{64} - 10 q^{65} - 22 q^{67} - 48 q^{68} + 8 q^{70} - 28 q^{73} + 19 q^{74} - 11 q^{76} - 12 q^{77} - 14 q^{79} + 13 q^{80} - 24 q^{82} - 10 q^{83} - 16 q^{85} + 23 q^{86} - 9 q^{88} + 26 q^{89} - 42 q^{91} - 12 q^{92} - 56 q^{94} - 14 q^{95} - 32 q^{97} + 4 q^{98}+O(q^{100}) \) 14 * q - q^2 + 11 * q^4 + 14 * q^5 - 12 * q^7 - 9 * q^8 - q^10 + 14 * q^11 - 10 * q^13 + 8 * q^14 + 13 * q^16 - 16 * q^17 - 14 * q^19 + 11 * q^20 - q^22 - 12 * q^23 + 14 * q^25 - 12 * q^26 - 33 * q^28 - 4 * q^31 - 24 * q^32 - 2 * q^34 - 12 * q^35 - 14 * q^37 + q^38 - 9 * q^40 - 18 * q^41 - 20 * q^43 + 11 * q^44 - 17 * q^46 - 8 * q^47 + 10 * q^49 - q^50 - 26 * q^52 - 20 * q^53 + 14 * q^55 + 11 * q^56 - 36 * q^58 + 2 * q^59 + 4 * q^61 - 38 * q^62 + 3 * q^64 - 10 * q^65 - 22 * q^67 - 48 * q^68 + 8 * q^70 - 28 * q^73 + 19 * q^74 - 11 * q^76 - 12 * q^77 - 14 * q^79 + 13 * q^80 - 24 * q^82 - 10 * q^83 - 16 * q^85 + 23 * q^86 - 9 * q^88 + 26 * q^89 - 42 * q^91 - 12 * q^92 - 56 * q^94 - 14 * q^95 - 32 * q^97 + 4 * q^98

Introduction

L-functions

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