LMFDB - Embedded newform 9405.2.a.z.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9405,2,Mod(1,9405)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9405, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9405.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9405.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$75.0993031010$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.7281497.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 5x^{4} + 7x^{3} + 6x^{2} - 2x - 1 $ x^6 - 2x^5 - 5x^4 + 7x^3 + 6x^2 - 2*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1045)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.59744$ of defining polynomial
Character	$\chi$	$=$	9405.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.21244 q^{2} -0.529980 q^{4} +1.00000 q^{5} -3.37010 q^{7} +3.06746 q^{8} +O(q^{10})$	\(q-1.21244 q^{2} -0.529980 q^{4} +1.00000 q^{5} -3.37010 q^{7} +3.06746 q^{8} -1.21244 q^{10} +1.00000 q^{11} -0.439786 q^{13} +4.08605 q^{14} -2.65916 q^{16} +3.08872 q^{17} -1.00000 q^{19} -0.529980 q^{20} -1.21244 q^{22} -1.45571 q^{23} +1.00000 q^{25} +0.533216 q^{26} +1.78608 q^{28} -5.85137 q^{29} -6.80721 q^{31} -2.91083 q^{32} -3.74490 q^{34} -3.37010 q^{35} -2.73402 q^{37} +1.21244 q^{38} +3.06746 q^{40} +3.67980 q^{41} +4.59330 q^{43} -0.529980 q^{44} +1.76497 q^{46} -0.210971 q^{47} +4.35755 q^{49} -1.21244 q^{50} +0.233078 q^{52} -5.17998 q^{53} +1.00000 q^{55} -10.3376 q^{56} +7.09446 q^{58} +10.8732 q^{59} -5.66286 q^{61} +8.25336 q^{62} +8.84754 q^{64} -0.439786 q^{65} +6.14764 q^{67} -1.63696 q^{68} +4.08605 q^{70} +7.15175 q^{71} +4.15351 q^{73} +3.31484 q^{74} +0.529980 q^{76} -3.37010 q^{77} +15.4881 q^{79} -2.65916 q^{80} -4.46155 q^{82} -3.35537 q^{83} +3.08872 q^{85} -5.56911 q^{86} +3.06746 q^{88} +6.26557 q^{89} +1.48212 q^{91} +0.771498 q^{92} +0.255791 q^{94} -1.00000 q^{95} -6.70633 q^{97} -5.28329 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 5 q^{7} + 12 q^{8}+O(q^{10}) $ 6 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 5 * q^7 + 12 * q^8	$ 6 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 5 q^{7} + 12 q^{8} + 2 q^{10} + 6 q^{11} - 5 q^{13} + 8 q^{14} + 4 q^{16} - q^{17} - 6 q^{19} + 4 q^{20} + 2 q^{22} - 4 q^{23} + 6 q^{25} + 14 q^{26} + 10 q^{28} + 9 q^{29} - 21 q^{31} + q^{32} + 5 q^{35} - 3 q^{37} - 2 q^{38} + 12 q^{40} + 23 q^{41} + 7 q^{43} + 4 q^{44} - 12 q^{46} + 18 q^{47} - 3 q^{49} + 2 q^{50} + 13 q^{52} + 17 q^{53} + 6 q^{55} + 2 q^{56} + 23 q^{58} + 29 q^{59} + 17 q^{61} - 2 q^{62} - 18 q^{64} - 5 q^{65} + 8 q^{67} + q^{68} + 8 q^{70} + 12 q^{71} + 2 q^{73} + 37 q^{74} - 4 q^{76} + 5 q^{77} + 3 q^{79} + 4 q^{80} + 24 q^{82} + 11 q^{83} - q^{85} + 12 q^{86} + 12 q^{88} + 22 q^{89} - 18 q^{91} + 15 q^{92} + 22 q^{94} - 6 q^{95} - 2 q^{97} + q^{98}+O(q^{100}) $ 6 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 5 * q^7 + 12 * q^8 + 2 * q^10 + 6 * q^11 - 5 * q^13 + 8 * q^14 + 4 * q^16 - q^17 - 6 * q^19 + 4 * q^20 + 2 * q^22 - 4 * q^23 + 6 * q^25 + 14 * q^26 + 10 * q^28 + 9 * q^29 - 21 * q^31 + q^32 + 5 * q^35 - 3 * q^37 - 2 * q^38 + 12 * q^40 + 23 * q^41 + 7 * q^43 + 4 * q^44 - 12 * q^46 + 18 * q^47 - 3 * q^49 + 2 * q^50 + 13 * q^52 + 17 * q^53 + 6 * q^55 + 2 * q^56 + 23 * q^58 + 29 * q^59 + 17 * q^61 - 2 * q^62 - 18 * q^64 - 5 * q^65 + 8 * q^67 + q^68 + 8 * q^70 + 12 * q^71 + 2 * q^73 + 37 * q^74 - 4 * q^76 + 5 * q^77 + 3 * q^79 + 4 * q^80 + 24 * q^82 + 11 * q^83 - q^85 + 12 * q^86 + 12 * q^88 + 22 * q^89 - 18 * q^91 + 15 * q^92 + 22 * q^94 - 6 * q^95 - 2 * q^97 + q^98

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.21244	−0.857327	−0.428664	−	0.903464i	$-0.641015\pi$
$2$	−1.21244	−0.857327	−0.428664	+	0.903464i	$0.641015\pi$
$3$	0	0
$3$	0	0
$4$	−0.529980	−0.264990
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.37010	−1.27378	−0.636888	−	0.770956i	$-0.719779\pi$
$7$	−3.37010	−1.27378	−0.636888	+	0.770956i	$0.719779\pi$
$8$	3.06746	1.08451
$9$	0	0
$10$	−1.21244	−0.383408
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−0.439786	−0.121975	−0.0609873	−	0.998139i	$-0.519425\pi$
$13$	−0.439786	−0.121975	−0.0609873	+	0.998139i	$0.519425\pi$
$14$	4.08605	1.09204
$15$	0	0
$16$	−2.65916	−0.664790
$17$	3.08872	0.749125	0.374563	−	0.927202i	$-0.377793\pi$
$17$	3.08872	0.749125	0.374563	+	0.927202i	$0.377793\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−0.529980	−0.118507
$21$	0	0
$22$	−1.21244	−0.258494
$23$	−1.45571	−0.303537	−0.151768	−	0.988416i	$-0.548497\pi$
$23$	−1.45571	−0.303537	−0.151768	+	0.988416i	$0.548497\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.533216	0.104572
$27$	0	0
$28$	1.78608	0.337538
$29$	−5.85137	−1.08657	−0.543286	−	0.839548i	$-0.682820\pi$
$29$	−5.85137	−1.08657	−0.543286	+	0.839548i	$0.682820\pi$
$30$	0	0
$31$	−6.80721	−1.22261	−0.611306	−	0.791394i	$-0.709355\pi$
$31$	−6.80721	−1.22261	−0.611306	+	0.791394i	$0.709355\pi$
$32$	−2.91083	−0.514567
$33$	0	0
$34$	−3.74490	−0.642245
$35$	−3.37010	−0.569650
$36$	0	0
$37$	−2.73402	−0.449470	−0.224735	−	0.974420i	$-0.572152\pi$
$37$	−2.73402	−0.449470	−0.224735	+	0.974420i	$0.572152\pi$
$38$	1.21244	0.196684
$39$	0	0
$40$	3.06746	0.485008
$41$	3.67980	0.574687	0.287344	−	0.957828i	$-0.407228\pi$
$41$	3.67980	0.574687	0.287344	+	0.957828i	$0.407228\pi$
$42$	0	0
$43$	4.59330	0.700471	0.350236	−	0.936662i	$-0.386102\pi$
$43$	4.59330	0.700471	0.350236	+	0.936662i	$0.386102\pi$
$44$	−0.529980	−0.0798975
$45$	0	0
$46$	1.76497	0.260230
$47$	−0.210971	−0.0307733	−0.0153867	−	0.999882i	$-0.504898\pi$
$47$	−0.210971	−0.0307733	−0.0153867	+	0.999882i	$0.504898\pi$
$48$	0	0
$49$	4.35755	0.622507
$50$	−1.21244	−0.171465
$51$	0	0
$52$	0.233078	0.0323221
$53$	−5.17998	−0.711525	−0.355762	−	0.934576i	$-0.615779\pi$
$53$	−5.17998	−0.711525	−0.355762	+	0.934576i	$0.615779\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	−10.3376	−1.38142
$57$	0	0
$58$	7.09446	0.931548
$59$	10.8732	1.41557	0.707785	−	0.706428i	$-0.249695\pi$
$59$	10.8732	1.41557	0.707785	+	0.706428i	$0.249695\pi$
$60$	0	0
$61$	−5.66286	−0.725055	−0.362527	−	0.931973i	$-0.618086\pi$
$61$	−5.66286	−0.725055	−0.362527	+	0.931973i	$0.618086\pi$
$62$	8.25336	1.04818
$63$	0	0
$64$	8.84754	1.10594
$65$	−0.439786	−0.0545487
$66$	0	0
$67$	6.14764	0.751054	0.375527	−	0.926811i	$-0.377462\pi$
$67$	6.14764	0.751054	0.375527	+	0.926811i	$0.377462\pi$
$68$	−1.63696	−0.198511
$69$	0	0
$70$	4.08605	0.488377
$71$	7.15175	0.848756	0.424378	−	0.905485i	$-0.360493\pi$
$71$	7.15175	0.848756	0.424378	+	0.905485i	$0.360493\pi$
$72$	0	0
$73$	4.15351	0.486132	0.243066	−	0.970010i	$-0.421847\pi$
$73$	4.15351	0.486132	0.243066	+	0.970010i	$0.421847\pi$
$74$	3.31484	0.385343
$75$	0	0
$76$	0.529980	0.0607929
$77$	−3.37010	−0.384058
$78$	0	0
$79$	15.4881	1.74254	0.871272	−	0.490800i	$-0.163295\pi$
$79$	15.4881	1.74254	0.871272	+	0.490800i	$0.163295\pi$
$80$	−2.65916	−0.297303
$81$	0	0
$82$	−4.46155	−0.492695
$83$	−3.35537	−0.368299	−0.184150	−	0.982898i	$-0.558953\pi$
$83$	−3.35537	−0.368299	−0.184150	+	0.982898i	$0.558953\pi$
$84$	0	0
$85$	3.08872	0.335019
$86$	−5.56911	−0.600533
$87$	0	0
$88$	3.06746	0.326992
$89$	6.26557	0.664149	0.332075	−	0.943253i	$-0.392251\pi$
$89$	6.26557	0.664149	0.332075	+	0.943253i	$0.392251\pi$
$90$	0	0
$91$	1.48212	0.155368
$92$	0.771498	0.0804342
$93$	0	0
$94$	0.255791	0.0263828
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−6.70633	−0.680924	−0.340462	−	0.940258i	$-0.610584\pi$
$97$	−6.70633	−0.680924	−0.340462	+	0.940258i	$0.610584\pi$
$98$	−5.28329	−0.533692
$99$	0	0
$100$	−0.529980	−0.0529980
$101$	−19.2123	−1.91169	−0.955846	−	0.293868i	$-0.905057\pi$
$101$	−19.2123	−1.91169	−0.955846	+	0.293868i	$0.905057\pi$
$102$	0	0
$103$	−15.0086	−1.47884	−0.739418	−	0.673246i	$-0.764899\pi$
$103$	−15.0086	−1.47884	−0.739418	+	0.673246i	$0.764899\pi$
$104$	−1.34902	−0.132283
$105$	0	0
$106$	6.28043	0.610010
$107$	7.63909	0.738499	0.369250	−	0.929330i	$-0.379615\pi$
$107$	7.63909	0.738499	0.369250	+	0.929330i	$0.379615\pi$
$108$	0	0
$109$	0.0458556	0.00439217	0.00219609	−	0.999998i	$-0.499301\pi$
$109$	0.0458556	0.00439217	0.00219609	+	0.999998i	$0.499301\pi$
$110$	−1.21244	−0.115602
$111$	0	0
$112$	8.96163	0.846795
$113$	−8.43432	−0.793434	−0.396717	−	0.917941i	$-0.629851\pi$
$113$	−8.43432	−0.793434	−0.396717	+	0.917941i	$0.629851\pi$
$114$	0	0
$115$	−1.45571	−0.135746
$116$	3.10111	0.287931
$117$	0	0
$118$	−13.1831	−1.21361
$119$	−10.4093	−0.954218
$120$	0	0
$121$	1.00000	0.0909091
$122$	6.86590	0.621609
$123$	0	0
$124$	3.60769	0.323980
$125$	1.00000	0.0894427
$126$	0	0
$127$	1.50239	0.133316	0.0666579	−	0.997776i	$-0.478766\pi$
$127$	1.50239	0.133316	0.0666579	+	0.997776i	$0.478766\pi$
$128$	−4.90548	−0.433588
$129$	0	0
$130$	0.533216	0.0467661
$131$	0.608483	0.0531634	0.0265817	−	0.999647i	$-0.491538\pi$
$131$	0.608483	0.0531634	0.0265817	+	0.999647i	$0.491538\pi$
$132$	0	0
$133$	3.37010	0.292224
$134$	−7.45367	−0.643899
$135$	0	0
$136$	9.47453	0.812434
$137$	21.4534	1.83289	0.916443	−	0.400166i	$-0.131047\pi$
$137$	21.4534	1.83289	0.916443	+	0.400166i	$0.131047\pi$
$138$	0	0
$139$	−8.85639	−0.751189	−0.375595	−	0.926784i	$-0.622561\pi$
$139$	−8.85639	−0.751189	−0.375595	+	0.926784i	$0.622561\pi$
$140$	1.78608	0.150952
$141$	0	0
$142$	−8.67109	−0.727662
$143$	−0.439786	−0.0367767
$144$	0	0
$145$	−5.85137	−0.485930
$146$	−5.03590	−0.416774
$147$	0	0
$148$	1.44898	0.119105
$149$	2.25215	0.184503	0.0922515	−	0.995736i	$-0.470594\pi$
$149$	2.25215	0.184503	0.0922515	+	0.995736i	$0.470594\pi$
$150$	0	0
$151$	−17.3616	−1.41286	−0.706432	−	0.707781i	$-0.749696\pi$
$151$	−17.3616	−1.41286	−0.706432	+	0.707781i	$0.749696\pi$
$152$	−3.06746	−0.248804
$153$	0	0
$154$	4.08605	0.329264
$155$	−6.80721	−0.546769
$156$	0	0
$157$	−3.75886	−0.299990	−0.149995	−	0.988687i	$-0.547926\pi$
$157$	−3.75886	−0.299990	−0.149995	+	0.988687i	$0.547926\pi$
$158$	−18.7784	−1.49393
$159$	0	0
$160$	−2.91083	−0.230122
$161$	4.90589	0.386638
$162$	0	0
$163$	−11.8435	−0.927655	−0.463828	−	0.885925i	$-0.653524\pi$
$163$	−11.8435	−0.927655	−0.463828	+	0.885925i	$0.653524\pi$
$164$	−1.95022	−0.152286
$165$	0	0
$166$	4.06819	0.315753
$167$	−6.33806	−0.490454	−0.245227	−	0.969466i	$-0.578863\pi$
$167$	−6.33806	−0.490454	−0.245227	+	0.969466i	$0.578863\pi$
$168$	0	0
$169$	−12.8066	−0.985122
$170$	−3.74490	−0.287221
$171$	0	0
$172$	−2.43436	−0.185618
$173$	8.21796	0.624800	0.312400	−	0.949951i	$-0.398867\pi$
$173$	8.21796	0.624800	0.312400	+	0.949951i	$0.398867\pi$
$174$	0	0
$175$	−3.37010	−0.254755
$176$	−2.65916	−0.200442
$177$	0	0
$178$	−7.59665	−0.569393
$179$	−11.2448	−0.840476	−0.420238	−	0.907414i	$-0.638053\pi$
$179$	−11.2448	−0.840476	−0.420238	+	0.907414i	$0.638053\pi$
$180$	0	0
$181$	8.27670	0.615202	0.307601	−	0.951515i	$-0.400474\pi$
$181$	8.27670	0.615202	0.307601	+	0.951515i	$0.400474\pi$
$182$	−1.79699	−0.133202
$183$	0	0
$184$	−4.46533	−0.329189
$185$	−2.73402	−0.201009
$186$	0	0
$187$	3.08872	0.225870
$188$	0.111811	0.00815462
$189$	0	0
$190$	1.21244	0.0879599
$191$	−3.21777	−0.232830	−0.116415	−	0.993201i	$-0.537140\pi$
$191$	−3.21777	−0.232830	−0.116415	+	0.993201i	$0.537140\pi$
$192$	0	0
$193$	19.0447	1.37087	0.685433	−	0.728136i	$-0.259613\pi$
$193$	19.0447	1.37087	0.685433	+	0.728136i	$0.259613\pi$
$194$	8.13104	0.583775
$195$	0	0
$196$	−2.30941	−0.164958
$197$	−2.04266	−0.145534	−0.0727668	−	0.997349i	$-0.523183\pi$
$197$	−2.04266	−0.145534	−0.0727668	+	0.997349i	$0.523183\pi$
$198$	0	0
$199$	4.61202	0.326937	0.163469	−	0.986549i	$-0.447732\pi$
$199$	4.61202	0.326937	0.163469	+	0.986549i	$0.447732\pi$
$200$	3.06746	0.216902
$201$	0	0
$202$	23.2938	1.63895
$203$	19.7197	1.38405
$204$	0	0
$205$	3.67980	0.257008
$206$	18.1970	1.26785
$207$	0	0
$208$	1.16946	0.0810876
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	−16.5102	−1.13661	−0.568303	−	0.822819i	$-0.692400\pi$
$211$	−16.5102	−1.13661	−0.568303	+	0.822819i	$0.692400\pi$
$212$	2.74528	0.188547
$213$	0	0
$214$	−9.26197	−0.633135
$215$	4.59330	0.313260
$216$	0	0
$217$	22.9410	1.55733
$218$	−0.0555973	−0.00376553
$219$	0	0
$220$	−0.529980	−0.0357312
$221$	−1.35838	−0.0913742
$222$	0	0
$223$	3.43650	0.230125	0.115062	−	0.993358i	$-0.463293\pi$
$223$	3.43650	0.230125	0.115062	+	0.993358i	$0.463293\pi$
$224$	9.80979	0.655444
$225$	0	0
$226$	10.2261	0.680233
$227$	9.66762	0.641663	0.320831	−	0.947136i	$-0.396038\pi$
$227$	9.66762	0.641663	0.320831	+	0.947136i	$0.396038\pi$
$228$	0	0
$229$	11.7897	0.779084	0.389542	−	0.921009i	$-0.372633\pi$
$229$	11.7897	0.779084	0.389542	+	0.921009i	$0.372633\pi$
$230$	1.76497	0.116379
$231$	0	0
$232$	−17.9488	−1.17840
$233$	−2.00908	−0.131619	−0.0658097	−	0.997832i	$-0.520963\pi$
$233$	−2.00908	−0.131619	−0.0658097	+	0.997832i	$0.520963\pi$
$234$	0	0
$235$	−0.210971	−0.0137622
$236$	−5.76258	−0.375112
$237$	0	0
$238$	12.6207	0.818077
$239$	6.36104	0.411461	0.205731	−	0.978609i	$-0.434043\pi$
$239$	6.36104	0.411461	0.205731	+	0.978609i	$0.434043\pi$
$240$	0	0
$241$	−28.3775	−1.82796	−0.913978	−	0.405763i	$-0.867006\pi$
$241$	−28.3775	−1.82796	−0.913978	+	0.405763i	$0.867006\pi$
$242$	−1.21244	−0.0779388
$243$	0	0
$244$	3.00120	0.192132
$245$	4.35755	0.278394
$246$	0	0
$247$	0.439786	0.0279829
$248$	−20.8808	−1.32594
$249$	0	0
$250$	−1.21244	−0.0766817
$251$	17.7932	1.12310	0.561550	−	0.827443i	$-0.310205\pi$
$251$	17.7932	1.12310	0.561550	+	0.827443i	$0.310205\pi$
$252$	0	0
$253$	−1.45571	−0.0915198
$254$	−1.82157	−0.114295
$255$	0	0
$256$	−11.7475	−0.734217
$257$	22.5168	1.40456	0.702279	−	0.711902i	$-0.252166\pi$
$257$	22.5168	1.40456	0.702279	+	0.711902i	$0.252166\pi$
$258$	0	0
$259$	9.21391	0.572524
$260$	0.233078	0.0144549
$261$	0	0
$262$	−0.737751	−0.0455784
$263$	5.74826	0.354453	0.177226	−	0.984170i	$-0.443288\pi$
$263$	5.74826	0.354453	0.177226	+	0.984170i	$0.443288\pi$
$264$	0	0
$265$	−5.17998	−0.318204
$266$	−4.08605	−0.250532
$267$	0	0
$268$	−3.25813	−0.199022
$269$	5.69265	0.347087	0.173544	−	0.984826i	$-0.444478\pi$
$269$	5.69265	0.347087	0.173544	+	0.984826i	$0.444478\pi$
$270$	0	0
$271$	−16.0881	−0.977282	−0.488641	−	0.872485i	$-0.662507\pi$
$271$	−16.0881	−0.977282	−0.488641	+	0.872485i	$0.662507\pi$
$272$	−8.21341	−0.498011
$273$	0	0
$274$	−26.0110	−1.57138
$275$	1.00000	0.0603023
$276$	0	0
$277$	−3.44717	−0.207120	−0.103560	−	0.994623i	$-0.533023\pi$
$277$	−3.44717	−0.207120	−0.103560	+	0.994623i	$0.533023\pi$
$278$	10.7379	0.644015
$279$	0	0
$280$	−10.3376	−0.617792
$281$	8.74423	0.521637	0.260819	−	0.965388i	$-0.416008\pi$
$281$	8.74423	0.521637	0.260819	+	0.965388i	$0.416008\pi$
$282$	0	0
$283$	30.1987	1.79513	0.897564	−	0.440885i	$-0.145335\pi$
$283$	30.1987	1.79513	0.897564	+	0.440885i	$0.145335\pi$
$284$	−3.79028	−0.224912
$285$	0	0
$286$	0.533216	0.0315297
$287$	−12.4013	−0.732024
$288$	0	0
$289$	−7.45980	−0.438812
$290$	7.09446	0.416601
$291$	0	0
$292$	−2.20128	−0.128820
$293$	11.1213	0.649714	0.324857	−	0.945763i	$-0.394684\pi$
$293$	11.1213	0.649714	0.324857	+	0.945763i	$0.394684\pi$
$294$	0	0
$295$	10.8732	0.633062
$296$	−8.38649	−0.487455
$297$	0	0
$298$	−2.73060	−0.158179
$299$	0.640201	0.0370238
$300$	0	0
$301$	−15.4799	−0.892244
$302$	21.0499	1.21129
$303$	0	0
$304$	2.65916	0.152513
$305$	−5.66286	−0.324254
$306$	0	0
$307$	24.8969	1.42094	0.710469	−	0.703728i	$-0.248483\pi$
$307$	24.8969	1.42094	0.710469	+	0.703728i	$0.248483\pi$
$308$	1.78608	0.101772
$309$	0	0
$310$	8.25336	0.468760
$311$	−9.24020	−0.523963	−0.261982	−	0.965073i	$-0.584376\pi$
$311$	−9.24020	−0.523963	−0.261982	+	0.965073i	$0.584376\pi$
$312$	0	0
$313$	−14.8438	−0.839021	−0.419511	−	0.907750i	$-0.637798\pi$
$313$	−14.8438	−0.839021	−0.419511	+	0.907750i	$0.637798\pi$
$314$	4.55740	0.257189
$315$	0	0
$316$	−8.20837	−0.461757
$317$	−33.8110	−1.89902	−0.949509	−	0.313740i	$-0.898418\pi$
$317$	−33.8110	−1.89902	−0.949509	+	0.313740i	$0.898418\pi$
$318$	0	0
$319$	−5.85137	−0.327614
$320$	8.84754	0.494593
$321$	0	0
$322$	−5.94811	−0.331475
$323$	−3.08872	−0.171861
$324$	0	0
$325$	−0.439786	−0.0243949
$326$	14.3596	0.795304
$327$	0	0
$328$	11.2876	0.623255
$329$	0.710993	0.0391983
$330$	0	0
$331$	11.5116	0.632737	0.316369	−	0.948636i	$-0.397536\pi$
$331$	11.5116	0.632737	0.316369	+	0.948636i	$0.397536\pi$
$332$	1.77828	0.0975956
$333$	0	0
$334$	7.68455	0.420480
$335$	6.14764	0.335882
$336$	0	0
$337$	25.3591	1.38140	0.690698	−	0.723143i	$-0.257303\pi$
$337$	25.3591	1.38140	0.690698	+	0.723143i	$0.257303\pi$
$338$	15.5273	0.844572
$339$	0	0
$340$	−1.63696	−0.0887766
$341$	−6.80721	−0.368631
$342$	0	0
$343$	8.90531	0.480841
$344$	14.0897	0.759668
$345$	0	0
$346$	−9.96382	−0.535658
$347$	9.66755	0.518982	0.259491	−	0.965746i	$-0.416445\pi$
$347$	9.66755	0.518982	0.259491	+	0.965746i	$0.416445\pi$
$348$	0	0
$349$	30.9349	1.65591	0.827953	−	0.560797i	$-0.189505\pi$
$349$	30.9349	1.65591	0.827953	+	0.560797i	$0.189505\pi$
$350$	4.08605	0.218409
$351$	0	0
$352$	−2.91083	−0.155148
$353$	−10.9128	−0.580830	−0.290415	−	0.956901i	$-0.593793\pi$
$353$	−10.9128	−0.580830	−0.290415	+	0.956901i	$0.593793\pi$
$354$	0	0
$355$	7.15175	0.379575
$356$	−3.32063	−0.175993
$357$	0	0
$358$	13.6337	0.720563
$359$	29.0146	1.53133	0.765667	−	0.643237i	$-0.222409\pi$
$359$	29.0146	1.53133	0.765667	+	0.643237i	$0.222409\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−10.0350	−0.527429
$363$	0	0
$364$	−0.785494	−0.0411711
$365$	4.15351	0.217405
$366$	0	0
$367$	21.1172	1.10231	0.551154	−	0.834404i	$-0.314188\pi$
$367$	21.1172	1.10231	0.551154	+	0.834404i	$0.314188\pi$
$368$	3.87097	0.201788
$369$	0	0
$370$	3.31484	0.172331
$371$	17.4570	0.906324
$372$	0	0
$373$	24.3430	1.26043	0.630215	−	0.776421i	$-0.282967\pi$
$373$	24.3430	1.26043	0.630215	+	0.776421i	$0.282967\pi$
$374$	−3.74490	−0.193644
$375$	0	0
$376$	−0.647146	−0.0333740
$377$	2.57335	0.132534
$378$	0	0
$379$	12.5458	0.644433	0.322216	−	0.946666i	$-0.395572\pi$
$379$	12.5458	0.644433	0.322216	+	0.946666i	$0.395572\pi$
$380$	0.529980	0.0271874
$381$	0	0
$382$	3.90137	0.199611
$383$	−8.27908	−0.423041	−0.211521	−	0.977374i	$-0.567842\pi$
$383$	−8.27908	−0.423041	−0.211521	+	0.977374i	$0.567842\pi$
$384$	0	0
$385$	−3.37010	−0.171756
$386$	−23.0906	−1.17528
$387$	0	0
$388$	3.55422	0.180438
$389$	7.91740	0.401428	0.200714	−	0.979650i	$-0.435674\pi$
$389$	7.91740	0.401428	0.200714	+	0.979650i	$0.435674\pi$
$390$	0	0
$391$	−4.49629	−0.227387
$392$	13.3666	0.675116
$393$	0	0
$394$	2.47661	0.124770
$395$	15.4881	0.779290
$396$	0	0
$397$	−24.3775	−1.22347	−0.611737	−	0.791062i	$-0.709529\pi$
$397$	−24.3775	−1.22347	−0.611737	+	0.791062i	$0.709529\pi$
$398$	−5.59181	−0.280292
$399$	0	0
$400$	−2.65916	−0.132958
$401$	−32.6120	−1.62857	−0.814283	−	0.580469i	$-0.802869\pi$
$401$	−32.6120	−1.62857	−0.814283	+	0.580469i	$0.802869\pi$
$402$	0	0
$403$	2.99372	0.149128
$404$	10.1821	0.506579
$405$	0	0
$406$	−23.9090	−1.18658
$407$	−2.73402	−0.135520
$408$	0	0
$409$	−36.7429	−1.81682	−0.908409	−	0.418083i	$-0.862702\pi$
$409$	−36.7429	−1.81682	−0.908409	+	0.418083i	$0.862702\pi$
$410$	−4.46155	−0.220340
$411$	0	0
$412$	7.95423	0.391877
$413$	−36.6437	−1.80312
$414$	0	0
$415$	−3.35537	−0.164708
$416$	1.28014	0.0627642
$417$	0	0
$418$	1.21244	0.0593026
$419$	20.7229	1.01238	0.506191	−	0.862422i	$-0.331053\pi$
$419$	20.7229	1.01238	0.506191	+	0.862422i	$0.331053\pi$
$420$	0	0
$421$	−23.3391	−1.13748	−0.568738	−	0.822518i	$-0.692568\pi$
$421$	−23.3391	−1.13748	−0.568738	+	0.822518i	$0.692568\pi$
$422$	20.0176	0.974443
$423$	0	0
$424$	−15.8894	−0.771656
$425$	3.08872	0.149825
$426$	0	0
$427$	19.0844	0.923558
$428$	−4.04857	−0.195695
$429$	0	0
$430$	−5.56911	−0.268567
$431$	−2.35724	−0.113544	−0.0567721	−	0.998387i	$-0.518081\pi$
$431$	−2.35724	−0.113544	−0.0567721	+	0.998387i	$0.518081\pi$
$432$	0	0
$433$	37.3199	1.79348	0.896741	−	0.442555i	$-0.145928\pi$
$433$	37.3199	1.79348	0.896741	+	0.442555i	$0.145928\pi$
$434$	−27.8146	−1.33515
$435$	0	0
$436$	−0.0243026	−0.00116388
$437$	1.45571	0.0696361
$438$	0	0
$439$	26.3625	1.25821	0.629106	−	0.777320i	$-0.283421\pi$
$439$	26.3625	1.25821	0.629106	+	0.777320i	$0.283421\pi$
$440$	3.06746	0.146235
$441$	0	0
$442$	1.64695	0.0783376
$443$	7.31633	0.347609	0.173805	−	0.984780i	$-0.444394\pi$
$443$	7.31633	0.347609	0.173805	+	0.984780i	$0.444394\pi$
$444$	0	0
$445$	6.26557	0.297017
$446$	−4.16656	−0.197292
$447$	0	0
$448$	−29.8171	−1.40872
$449$	0.872582	0.0411797	0.0205898	−	0.999788i	$-0.493446\pi$
$449$	0.872582	0.0411797	0.0205898	+	0.999788i	$0.493446\pi$
$450$	0	0
$451$	3.67980	0.173275
$452$	4.47002	0.210252
$453$	0	0
$454$	−11.7215	−0.550115
$455$	1.48212	0.0694829
$456$	0	0
$457$	−2.04749	−0.0957774	−0.0478887	−	0.998853i	$-0.515249\pi$
$457$	−2.04749	−0.0957774	−0.0478887	+	0.998853i	$0.515249\pi$
$458$	−14.2943	−0.667930
$459$	0	0
$460$	0.771498	0.0359713
$461$	−26.6538	−1.24139	−0.620695	−	0.784052i	$-0.713149\pi$
$461$	−26.6538	−1.24139	−0.620695	+	0.784052i	$0.713149\pi$
$462$	0	0
$463$	−19.5335	−0.907798	−0.453899	−	0.891053i	$-0.649967\pi$
$463$	−19.5335	−0.907798	−0.453899	+	0.891053i	$0.649967\pi$
$464$	15.5597	0.722343
$465$	0	0
$466$	2.43590	0.112841
$467$	29.3029	1.35598	0.677988	−	0.735073i	$-0.262852\pi$
$467$	29.3029	1.35598	0.677988	+	0.735073i	$0.262852\pi$
$468$	0	0
$469$	−20.7182	−0.956675
$470$	0.255791	0.0117987
$471$	0	0
$472$	33.3531	1.53520
$473$	4.59330	0.211200
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	5.51671	0.252858
$477$	0	0
$478$	−7.71240	−0.352757
$479$	0.965639	0.0441212	0.0220606	−	0.999757i	$-0.492977\pi$
$479$	0.965639	0.0441212	0.0220606	+	0.999757i	$0.492977\pi$
$480$	0	0
$481$	1.20238	0.0548239
$482$	34.4061	1.56716
$483$	0	0
$484$	−0.529980	−0.0240900
$485$	−6.70633	−0.304519
$486$	0	0
$487$	−3.36125	−0.152313	−0.0761564	−	0.997096i	$-0.524265\pi$
$487$	−3.36125	−0.152313	−0.0761564	+	0.997096i	$0.524265\pi$
$488$	−17.3706	−0.786329
$489$	0	0
$490$	−5.28329	−0.238675
$491$	32.5832	1.47046	0.735230	−	0.677818i	$-0.237074\pi$
$491$	32.5832	1.47046	0.735230	+	0.677818i	$0.237074\pi$
$492$	0	0
$493$	−18.0733	−0.813978
$494$	−0.533216	−0.0239905
$495$	0	0
$496$	18.1015	0.812780
$497$	−24.1021	−1.08113
$498$	0	0
$499$	7.53868	0.337477	0.168739	−	0.985661i	$-0.446031\pi$
$499$	7.53868	0.337477	0.168739	+	0.985661i	$0.446031\pi$
$500$	−0.529980	−0.0237014
$501$	0	0
$502$	−21.5733	−0.962864
$503$	9.02729	0.402507	0.201254	−	0.979539i	$-0.435498\pi$
$503$	9.02729	0.402507	0.201254	+	0.979539i	$0.435498\pi$
$504$	0	0
$505$	−19.2123	−0.854935
$506$	1.76497	0.0784624
$507$	0	0
$508$	−0.796238	−0.0353273
$509$	20.8922	0.926031	0.463015	−	0.886350i	$-0.346767\pi$
$509$	20.8922	0.926031	0.463015	+	0.886350i	$0.346767\pi$
$510$	0	0
$511$	−13.9977	−0.619223
$512$	24.0541	1.06305
$513$	0	0
$514$	−27.3003	−1.20417
$515$	−15.0086	−0.661356
$516$	0	0
$517$	−0.210971	−0.00927850
$518$	−11.1713	−0.490841
$519$	0	0
$520$	−1.34902	−0.0591586
$521$	43.6640	1.91296	0.956478	−	0.291803i	$-0.0942553\pi$
$521$	43.6640	1.91296	0.956478	+	0.291803i	$0.0942553\pi$
$522$	0	0
$523$	26.4677	1.15735	0.578675	−	0.815558i	$-0.303570\pi$
$523$	26.4677	1.15735	0.578675	+	0.815558i	$0.303570\pi$
$524$	−0.322484	−0.0140878
$525$	0	0
$526$	−6.96944	−0.303882
$527$	−21.0256	−0.915889
$528$	0	0
$529$	−20.8809	−0.907865
$530$	6.28043	0.272805
$531$	0	0
$532$	−1.78608	−0.0774365
$533$	−1.61832	−0.0700973
$534$	0	0
$535$	7.63909	0.330267
$536$	18.8576	0.814526
$537$	0	0
$538$	−6.90202	−0.297567
$539$	4.35755	0.187693
$540$	0	0
$541$	−1.04162	−0.0447829	−0.0223915	−	0.999749i	$-0.507128\pi$
$541$	−1.04162	−0.0447829	−0.0223915	+	0.999749i	$0.507128\pi$
$542$	19.5059	0.837851
$543$	0	0
$544$	−8.99075	−0.385475
$545$	0.0458556	0.00196424
$546$	0	0
$547$	−5.27609	−0.225589	−0.112795	−	0.993618i	$-0.535980\pi$
$547$	−5.27609	−0.225589	−0.112795	+	0.993618i	$0.535980\pi$
$548$	−11.3699	−0.485696
$549$	0	0
$550$	−1.21244	−0.0516988
$551$	5.85137	0.249277
$552$	0	0
$553$	−52.1963	−2.21961
$554$	4.17949	0.177570
$555$	0	0
$556$	4.69371	0.199058
$557$	12.9219	0.547517	0.273759	−	0.961798i	$-0.411733\pi$
$557$	12.9219	0.547517	0.273759	+	0.961798i	$0.411733\pi$
$558$	0	0
$559$	−2.02007	−0.0854397
$560$	8.96163	0.378698
$561$	0	0
$562$	−10.6019	−0.447214
$563$	8.76657	0.369467	0.184733	−	0.982789i	$-0.440858\pi$
$563$	8.76657	0.369467	0.184733	+	0.982789i	$0.440858\pi$
$564$	0	0
$565$	−8.43432	−0.354834
$566$	−36.6142	−1.53901
$567$	0	0
$568$	21.9377	0.920485
$569$	14.7828	0.619726	0.309863	−	0.950781i	$-0.399717\pi$
$569$	14.7828	0.619726	0.309863	+	0.950781i	$0.399717\pi$
$570$	0	0
$571$	22.6278	0.946943	0.473472	−	0.880809i	$-0.343001\pi$
$571$	22.6278	0.946943	0.473472	+	0.880809i	$0.343001\pi$
$572$	0.233078	0.00974547
$573$	0	0
$574$	15.0358	0.627584
$575$	−1.45571	−0.0607074
$576$	0	0
$577$	3.73371	0.155436	0.0777182	−	0.996975i	$-0.475237\pi$
$577$	3.73371	0.155436	0.0777182	+	0.996975i	$0.475237\pi$
$578$	9.04459	0.376205
$579$	0	0
$580$	3.10111	0.128767
$581$	11.3079	0.469131
$582$	0	0
$583$	−5.17998	−0.214533
$584$	12.7407	0.527215
$585$	0	0
$586$	−13.4840	−0.557018
$587$	19.6389	0.810584	0.405292	−	0.914187i	$-0.367170\pi$
$587$	19.6389	0.810584	0.405292	+	0.914187i	$0.367170\pi$
$588$	0	0
$589$	6.80721	0.280486
$590$	−13.1831	−0.542741
$591$	0	0
$592$	7.27020	0.298803
$593$	32.8110	1.34739	0.673693	−	0.739011i	$-0.264707\pi$
$593$	32.8110	1.34739	0.673693	+	0.739011i	$0.264707\pi$
$594$	0	0
$595$	−10.4093	−0.426739
$596$	−1.19359	−0.0488914
$597$	0	0
$598$	−0.776208	−0.0317415
$599$	41.5943	1.69950	0.849750	−	0.527187i	$-0.176753\pi$
$599$	41.5943	1.69950	0.849750	+	0.527187i	$0.176753\pi$
$600$	0	0
$601$	−16.1727	−0.659698	−0.329849	−	0.944034i	$-0.606998\pi$
$601$	−16.1727	−0.659698	−0.329849	+	0.944034i	$0.606998\pi$
$602$	18.7685	0.764945
$603$	0	0
$604$	9.20127	0.374395
$605$	1.00000	0.0406558
$606$	0	0
$607$	−14.9877	−0.608330	−0.304165	−	0.952619i	$-0.598377\pi$
$607$	−14.9877	−0.608330	−0.304165	+	0.952619i	$0.598377\pi$
$608$	2.91083	0.118050
$609$	0	0
$610$	6.86590	0.277992
$611$	0.0927821	0.00375356
$612$	0	0
$613$	19.1145	0.772027	0.386014	−	0.922493i	$-0.373852\pi$
$613$	19.1145	0.772027	0.386014	+	0.922493i	$0.373852\pi$
$614$	−30.1860	−1.21821
$615$	0	0
$616$	−10.3376	−0.416515
$617$	1.20030	0.0483223	0.0241612	−	0.999708i	$-0.492309\pi$
$617$	1.20030	0.0483223	0.0241612	+	0.999708i	$0.492309\pi$
$618$	0	0
$619$	−44.2783	−1.77969	−0.889847	−	0.456258i	$-0.849189\pi$
$619$	−44.2783	−1.77969	−0.889847	+	0.456258i	$0.849189\pi$
$620$	3.60769	0.144888
$621$	0	0
$622$	11.2032	0.449208
$623$	−21.1156	−0.845978
$624$	0	0
$625$	1.00000	0.0400000
$626$	17.9973	0.719316
$627$	0	0
$628$	1.99212	0.0794942
$629$	−8.44462	−0.336709
$630$	0	0
$631$	7.76811	0.309244	0.154622	−	0.987974i	$-0.450584\pi$
$631$	7.76811	0.309244	0.154622	+	0.987974i	$0.450584\pi$
$632$	47.5090	1.88981
$633$	0	0
$634$	40.9940	1.62808
$635$	1.50239	0.0596206
$636$	0	0
$637$	−1.91639	−0.0759301
$638$	7.09446	0.280872
$639$	0	0
$640$	−4.90548	−0.193906
$641$	−27.0513	−1.06846	−0.534231	−	0.845339i	$-0.679399\pi$
$641$	−27.0513	−1.06846	−0.534231	+	0.845339i	$0.679399\pi$
$642$	0	0
$643$	−15.1507	−0.597485	−0.298742	−	0.954334i	$-0.596567\pi$
$643$	−15.1507	−0.597485	−0.298742	+	0.954334i	$0.596567\pi$
$644$	−2.60002	−0.102455
$645$	0	0
$646$	3.74490	0.147341
$647$	22.7874	0.895866	0.447933	−	0.894067i	$-0.352160\pi$
$647$	22.7874	0.895866	0.447933	+	0.894067i	$0.352160\pi$
$648$	0	0
$649$	10.8732	0.426810
$650$	0.533216	0.0209144
$651$	0	0
$652$	6.27682	0.245819
$653$	−36.1144	−1.41327	−0.706633	−	0.707580i	$-0.749787\pi$
$653$	−36.1144	−1.41327	−0.706633	+	0.707580i	$0.749787\pi$
$654$	0	0
$655$	0.608483	0.0237754
$656$	−9.78517	−0.382047
$657$	0	0
$658$	−0.862040	−0.0336058
$659$	5.96009	0.232172	0.116086	−	0.993239i	$-0.462965\pi$
$659$	5.96009	0.232172	0.116086	+	0.993239i	$0.462965\pi$
$660$	0	0
$661$	47.5667	1.85013	0.925065	−	0.379808i	$-0.124010\pi$
$661$	47.5667	1.85013	0.925065	+	0.379808i	$0.124010\pi$
$662$	−13.9572	−0.542463
$663$	0	0
$664$	−10.2924	−0.399425
$665$	3.37010	0.130687
$666$	0	0
$667$	8.51791	0.329815
$668$	3.35905	0.129965
$669$	0	0
$670$	−7.45367	−0.287960
$671$	−5.66286	−0.218612
$672$	0	0
$673$	23.3646	0.900639	0.450319	−	0.892868i	$-0.351310\pi$
$673$	23.3646	0.900639	0.450319	+	0.892868i	$0.351310\pi$
$674$	−30.7465	−1.18431
$675$	0	0
$676$	6.78723	0.261047
$677$	22.7258	0.873423	0.436712	−	0.899602i	$-0.356143\pi$
$677$	22.7258	0.873423	0.436712	+	0.899602i	$0.356143\pi$
$678$	0	0
$679$	22.6010	0.867346
$680$	9.47453	0.363331
$681$	0	0
$682$	8.25336	0.316038
$683$	6.08405	0.232800	0.116400	−	0.993202i	$-0.462865\pi$
$683$	6.08405	0.232800	0.116400	+	0.993202i	$0.462865\pi$
$684$	0	0
$685$	21.4534	0.819691
$686$	−10.7972	−0.412239
$687$	0	0
$688$	−12.2143	−0.465667
$689$	2.27808	0.0867880
$690$	0	0
$691$	10.2313	0.389215	0.194608	−	0.980881i	$-0.437657\pi$
$691$	10.2313	0.389215	0.194608	+	0.980881i	$0.437657\pi$
$692$	−4.35535	−0.165566
$693$	0	0
$694$	−11.7214	−0.444937
$695$	−8.85639	−0.335942
$696$	0	0
$697$	11.3659	0.430513
$698$	−37.5068	−1.41965
$699$	0	0
$700$	1.78608	0.0675076
$701$	−20.1154	−0.759749	−0.379875	−	0.925038i	$-0.624033\pi$
$701$	−20.1154	−0.759749	−0.379875	+	0.925038i	$0.624033\pi$
$702$	0	0
$703$	2.73402	0.103115
$704$	8.84754	0.333454
$705$	0	0
$706$	13.2312	0.497962
$707$	64.7472	2.43507
$708$	0	0
$709$	−18.6469	−0.700301	−0.350150	−	0.936694i	$-0.613870\pi$
$709$	−18.6469	−0.700301	−0.350150	+	0.936694i	$0.613870\pi$
$710$	−8.67109	−0.325420
$711$	0	0
$712$	19.2194	0.720277
$713$	9.90934	0.371108
$714$	0	0
$715$	−0.439786	−0.0164471
$716$	5.95952	0.222718
$717$	0	0
$718$	−35.1786	−1.31285
$719$	25.8776	0.965073	0.482537	−	0.875876i	$-0.339716\pi$
$719$	25.8776	0.965073	0.482537	+	0.875876i	$0.339716\pi$
$720$	0	0
$721$	50.5803	1.88371
$722$	−1.21244	−0.0451225
$723$	0	0
$724$	−4.38648	−0.163022
$725$	−5.85137	−0.217314
$726$	0	0
$727$	48.1800	1.78690	0.893450	−	0.449164i	$-0.148278\pi$
$727$	48.1800	1.78690	0.893450	+	0.449164i	$0.148278\pi$
$728$	4.54634	0.168499
$729$	0	0
$730$	−5.03590	−0.186387
$731$	14.1874	0.524741
$732$	0	0
$733$	5.47157	0.202097	0.101049	−	0.994881i	$-0.467780\pi$
$733$	5.47157	0.202097	0.101049	+	0.994881i	$0.467780\pi$
$734$	−25.6034	−0.945038
$735$	0	0
$736$	4.23733	0.156190
$737$	6.14764	0.226451
$738$	0	0
$739$	11.0454	0.406311	0.203156	−	0.979146i	$-0.434880\pi$
$739$	11.0454	0.406311	0.203156	+	0.979146i	$0.434880\pi$
$740$	1.44898	0.0532654
$741$	0	0
$742$	−21.1657	−0.777016
$743$	16.8303	0.617443	0.308721	−	0.951153i	$-0.400099\pi$
$743$	16.8303	0.617443	0.308721	+	0.951153i	$0.400099\pi$
$744$	0	0
$745$	2.25215	0.0825122
$746$	−29.5145	−1.08060
$747$	0	0
$748$	−1.63696	−0.0598532
$749$	−25.7445	−0.940683
$750$	0	0
$751$	−8.34710	−0.304590	−0.152295	−	0.988335i	$-0.548666\pi$
$751$	−8.34710	−0.304590	−0.152295	+	0.988335i	$0.548666\pi$
$752$	0.561007	0.0204578
$753$	0	0
$754$	−3.12004	−0.113625
$755$	−17.3616	−0.631852
$756$	0	0
$757$	2.18852	0.0795431	0.0397716	−	0.999209i	$-0.487337\pi$
$757$	2.18852	0.0795431	0.0397716	+	0.999209i	$0.487337\pi$
$758$	−15.2110	−0.552490
$759$	0	0
$760$	−3.06746	−0.111268
$761$	−33.4322	−1.21192	−0.605958	−	0.795496i	$-0.707210\pi$
$761$	−33.4322	−1.21192	−0.605958	+	0.795496i	$0.707210\pi$
$762$	0	0
$763$	−0.154538	−0.00559464
$764$	1.70535	0.0616975
$765$	0	0
$766$	10.0379	0.362685
$767$	−4.78188	−0.172664
$768$	0	0
$769$	35.7102	1.28774	0.643871	−	0.765134i	$-0.277327\pi$
$769$	35.7102	1.28774	0.643871	+	0.765134i	$0.277327\pi$
$770$	4.08605	0.147251
$771$	0	0
$772$	−10.0933	−0.363266
$773$	−12.1707	−0.437749	−0.218875	−	0.975753i	$-0.570239\pi$
$773$	−12.1707	−0.437749	−0.218875	+	0.975753i	$0.570239\pi$
$774$	0	0
$775$	−6.80721	−0.244522
$776$	−20.5714	−0.738469
$777$	0	0
$778$	−9.59940	−0.344155
$779$	−3.67980	−0.131842
$780$	0	0
$781$	7.15175	0.255910
$782$	5.45150	0.194945
$783$	0	0
$784$	−11.5874	−0.413837
$785$	−3.75886	−0.134159
$786$	0	0
$787$	47.9347	1.70869	0.854344	−	0.519707i	$-0.173959\pi$
$787$	47.9347	1.70869	0.854344	+	0.519707i	$0.173959\pi$
$788$	1.08257	0.0385649
$789$	0	0
$790$	−18.7784	−0.668106
$791$	28.4245	1.01066
$792$	0	0
$793$	2.49044	0.0884383
$794$	29.5564	1.04892
$795$	0	0
$796$	−2.44428	−0.0866351
$797$	−50.1321	−1.77577	−0.887885	−	0.460066i	$-0.847826\pi$
$797$	−50.1321	−1.77577	−0.887885	+	0.460066i	$0.847826\pi$
$798$	0	0
$799$	−0.651631	−0.0230531
$800$	−2.91083	−0.102913
$801$	0	0
$802$	39.5402	1.39621
$803$	4.15351	0.146574
$804$	0	0
$805$	4.90589	0.172910
$806$	−3.62971	−0.127851
$807$	0	0
$808$	−58.9328	−2.07325
$809$	39.8704	1.40177	0.700884	−	0.713275i	$-0.252789\pi$
$809$	39.8704	1.40177	0.700884	+	0.713275i	$0.252789\pi$
$810$	0	0
$811$	−54.0597	−1.89829	−0.949147	−	0.314835i	$-0.898051\pi$
$811$	−54.0597	−1.89829	−0.949147	+	0.314835i	$0.898051\pi$
$812$	−10.4510	−0.366759
$813$	0	0
$814$	3.31484	0.116185
$815$	−11.8435	−0.414860
$816$	0	0
$817$	−4.59330	−0.160699
$818$	44.5487	1.55761
$819$	0	0
$820$	−1.95022	−0.0681046
$821$	−28.2714	−0.986677	−0.493339	−	0.869837i	$-0.664224\pi$
$821$	−28.2714	−0.986677	−0.493339	+	0.869837i	$0.664224\pi$
$822$	0	0
$823$	−28.9439	−1.00892	−0.504461	−	0.863434i	$-0.668309\pi$
$823$	−28.9439	−1.00892	−0.504461	+	0.863434i	$0.668309\pi$
$824$	−46.0381	−1.60381
$825$	0	0
$826$	44.4285	1.54586
$827$	6.79361	0.236237	0.118118	−	0.993000i	$-0.462314\pi$
$827$	6.79361	0.236237	0.118118	+	0.993000i	$0.462314\pi$
$828$	0	0
$829$	4.79089	0.166394	0.0831972	−	0.996533i	$-0.473487\pi$
$829$	4.79089	0.166394	0.0831972	+	0.996533i	$0.473487\pi$
$830$	4.06819	0.141209
$831$	0	0
$832$	−3.89102	−0.134897
$833$	13.4593	0.466336
$834$	0	0
$835$	−6.33806	−0.219338
$836$	0.529980	0.0183297
$837$	0	0
$838$	−25.1254	−0.867942
$839$	46.8222	1.61648	0.808241	−	0.588852i	$-0.200420\pi$
$839$	46.8222	1.61648	0.808241	+	0.588852i	$0.200420\pi$
$840$	0	0
$841$	5.23853	0.180639
$842$	28.2973	0.975190
$843$	0	0
$844$	8.75005	0.301189
$845$	−12.8066	−0.440560
$846$	0	0
$847$	−3.37010	−0.115798
$848$	13.7744	0.473015
$849$	0	0
$850$	−3.74490	−0.128449
$851$	3.97994	0.136431
$852$	0	0
$853$	4.99555	0.171044	0.0855222	−	0.996336i	$-0.472744\pi$
$853$	4.99555	0.171044	0.0855222	+	0.996336i	$0.472744\pi$
$854$	−23.1387	−0.791791
$855$	0	0
$856$	23.4326	0.800910
$857$	−3.32386	−0.113541	−0.0567705	−	0.998387i	$-0.518080\pi$
$857$	−3.32386	−0.113541	−0.0567705	+	0.998387i	$0.518080\pi$
$858$	0	0
$859$	−32.5496	−1.11058	−0.555288	−	0.831658i	$-0.687392\pi$
$859$	−32.5496	−1.11058	−0.555288	+	0.831658i	$0.687392\pi$
$860$	−2.43436	−0.0830108
$861$	0	0
$862$	2.85802	0.0973446
$863$	−7.39803	−0.251832	−0.125916	−	0.992041i	$-0.540187\pi$
$863$	−7.39803	−0.251832	−0.125916	+	0.992041i	$0.540187\pi$
$864$	0	0
$865$	8.21796	0.279419
$866$	−45.2483	−1.53760
$867$	0	0
$868$	−12.1583	−0.412678
$869$	15.4881	0.525397
$870$	0	0
$871$	−2.70365	−0.0916096
$872$	0.140660	0.00476335
$873$	0	0
$874$	−1.76497	−0.0597009
$875$	−3.37010	−0.113930
$876$	0	0
$877$	40.5912	1.37067	0.685333	−	0.728230i	$-0.259657\pi$
$877$	40.5912	1.37067	0.685333	+	0.728230i	$0.259657\pi$
$878$	−31.9630	−1.07870
$879$	0	0
$880$	−2.65916	−0.0896403
$881$	−16.4133	−0.552978	−0.276489	−	0.961017i	$-0.589171\pi$
$881$	−16.4133	−0.552978	−0.276489	+	0.961017i	$0.589171\pi$
$882$	0	0
$883$	41.3719	1.39227	0.696137	−	0.717909i	$-0.254900\pi$
$883$	41.3719	1.39227	0.696137	+	0.717909i	$0.254900\pi$
$884$	0.719912	0.0242133
$885$	0	0
$886$	−8.87064	−0.298015
$887$	−11.9068	−0.399791	−0.199896	−	0.979817i	$-0.564060\pi$
$887$	−11.9068	−0.399791	−0.199896	+	0.979817i	$0.564060\pi$
$888$	0	0
$889$	−5.06321	−0.169814
$890$	−7.59665	−0.254640
$891$	0	0
$892$	−1.82128	−0.0609808
$893$	0.210971	0.00705988
$894$	0	0
$895$	−11.2448	−0.375872
$896$	16.5320	0.552294
$897$	0	0
$898$	−1.05796	−0.0353045
$899$	39.8315	1.32846
$900$	0	0
$901$	−15.9995	−0.533021
$902$	−4.46155	−0.148553
$903$	0	0
$904$	−25.8719	−0.860487
$905$	8.27670	0.275127
$906$	0	0
$907$	−18.1787	−0.603614	−0.301807	−	0.953369i	$-0.597590\pi$
$907$	−18.1787	−0.603614	−0.301807	+	0.953369i	$0.597590\pi$
$908$	−5.12365	−0.170034
$909$	0	0
$910$	−1.79699	−0.0595696
$911$	−32.5463	−1.07831	−0.539154	−	0.842207i	$-0.681256\pi$
$911$	−32.5463	−1.07831	−0.539154	+	0.842207i	$0.681256\pi$
$912$	0	0
$913$	−3.35537	−0.111046
$914$	2.48246	0.0821125
$915$	0	0
$916$	−6.24830	−0.206450
$917$	−2.05065	−0.0677183
$918$	0	0
$919$	−22.8963	−0.755280	−0.377640	−	0.925952i	$-0.623264\pi$
$919$	−22.8963	−0.755280	−0.377640	+	0.925952i	$0.623264\pi$
$920$	−4.46533	−0.147218
$921$	0	0
$922$	32.3162	1.06428
$923$	−3.14524	−0.103527
$924$	0	0
$925$	−2.73402	−0.0898940
$926$	23.6832	0.778280
$927$	0	0
$928$	17.0324	0.559115
$929$	39.9363	1.31027	0.655135	−	0.755512i	$-0.272612\pi$
$929$	39.9363	1.31027	0.655135	+	0.755512i	$0.272612\pi$
$930$	0	0
$931$	−4.35755	−0.142813
$932$	1.06477	0.0348778
$933$	0	0
$934$	−35.5281	−1.16252
$935$	3.08872	0.101012
$936$	0	0
$937$	51.4680	1.68139	0.840693	−	0.541511i	$-0.182148\pi$
$937$	51.4680	1.68139	0.840693	+	0.541511i	$0.182148\pi$
$938$	25.1196	0.820184
$939$	0	0
$940$	0.111811	0.00364686
$941$	−56.3979	−1.83852	−0.919259	−	0.393652i	$-0.871211\pi$
$941$	−56.3979	−1.83852	−0.919259	+	0.393652i	$0.871211\pi$
$942$	0	0
$943$	−5.35672	−0.174439
$944$	−28.9136	−0.941057
$945$	0	0
$946$	−5.56911	−0.181068
$947$	−26.6837	−0.867103	−0.433552	−	0.901129i	$-0.642740\pi$
$947$	−26.6837	−0.867103	−0.433552	+	0.901129i	$0.642740\pi$
$948$	0	0
$949$	−1.82666	−0.0592957
$950$	1.21244	0.0393369
$951$	0	0
$952$	−31.9301	−1.03486
$953$	14.9629	0.484695	0.242348	−	0.970190i	$-0.422083\pi$
$953$	14.9629	0.484695	0.242348	+	0.970190i	$0.422083\pi$
$954$	0	0
$955$	−3.21777	−0.104125
$956$	−3.37122	−0.109033
$957$	0	0
$958$	−1.17078	−0.0378263
$959$	−72.2999	−2.33469
$960$	0	0
$961$	15.3382	0.494779
$962$	−1.45782	−0.0470021
$963$	0	0
$964$	15.0395	0.484390
$965$	19.0447	0.613070
$966$	0	0
$967$	19.7493	0.635094	0.317547	−	0.948242i	$-0.397141\pi$
$967$	19.7493	0.635094	0.317547	+	0.948242i	$0.397141\pi$
$968$	3.06746	0.0985919
$969$	0	0
$970$	8.13104	0.261072
$971$	53.2159	1.70778	0.853889	−	0.520455i	$-0.174238\pi$
$971$	53.2159	1.70778	0.853889	+	0.520455i	$0.174238\pi$
$972$	0	0
$973$	29.8469	0.956847
$974$	4.07533	0.130582
$975$	0	0
$976$	15.0585	0.482009
$977$	49.9728	1.59877	0.799385	−	0.600819i	$-0.205159\pi$
$977$	49.9728	1.59877	0.799385	+	0.600819i	$0.205159\pi$
$978$	0	0
$979$	6.26557	0.200249
$980$	−2.30941	−0.0737715
$981$	0	0
$982$	−39.5053	−1.26067
$983$	−29.2283	−0.932239	−0.466119	−	0.884722i	$-0.654348\pi$
$983$	−29.2283	−0.932239	−0.466119	+	0.884722i	$0.654348\pi$
$984$	0	0
$985$	−2.04266	−0.0650846
$986$	21.9128	0.697846
$987$	0	0
$988$	−0.233078	−0.00741519
$989$	−6.68651	−0.212619
$990$	0	0
$991$	−7.92503	−0.251747	−0.125873	−	0.992046i	$-0.540173\pi$
$991$	−7.92503	−0.251747	−0.125873	+	0.992046i	$0.540173\pi$
$992$	19.8147	0.629116
$993$	0	0
$994$	29.2224	0.926879
$995$	4.61202	0.146211
$996$	0	0
$997$	11.9229	0.377601	0.188800	−	0.982015i	$-0.439540\pi$
$997$	11.9229	0.377601	0.188800	+	0.982015i	$0.439540\pi$
$998$	−9.14022	−0.289329
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9405.2.a.z.1.2		6
3.2	odd	2		1045.2.a.f.1.5	✓	6
15.14	odd	2		5225.2.a.l.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.f.1.5	✓	6	3.2	odd	2
5225.2.a.l.1.2		6	15.14	odd	2
9405.2.a.z.1.2		6	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9405.a (trivial)

\(f(q)\)	\(=\)	\(q-1.21244 q^{2} -0.529980 q^{4} +1.00000 q^{5} -3.37010 q^{7} +3.06746 q^{8} +O(q^{10})\)	\(q-1.21244 q^{2} -0.529980 q^{4} +1.00000 q^{5} -3.37010 q^{7} +3.06746 q^{8} -1.21244 q^{10} +1.00000 q^{11} -0.439786 q^{13} +4.08605 q^{14} -2.65916 q^{16} +3.08872 q^{17} -1.00000 q^{19} -0.529980 q^{20} -1.21244 q^{22} -1.45571 q^{23} +1.00000 q^{25} +0.533216 q^{26} +1.78608 q^{28} -5.85137 q^{29} -6.80721 q^{31} -2.91083 q^{32} -3.74490 q^{34} -3.37010 q^{35} -2.73402 q^{37} +1.21244 q^{38} +3.06746 q^{40} +3.67980 q^{41} +4.59330 q^{43} -0.529980 q^{44} +1.76497 q^{46} -0.210971 q^{47} +4.35755 q^{49} -1.21244 q^{50} +0.233078 q^{52} -5.17998 q^{53} +1.00000 q^{55} -10.3376 q^{56} +7.09446 q^{58} +10.8732 q^{59} -5.66286 q^{61} +8.25336 q^{62} +8.84754 q^{64} -0.439786 q^{65} +6.14764 q^{67} -1.63696 q^{68} +4.08605 q^{70} +7.15175 q^{71} +4.15351 q^{73} +3.31484 q^{74} +0.529980 q^{76} -3.37010 q^{77} +15.4881 q^{79} -2.65916 q^{80} -4.46155 q^{82} -3.35537 q^{83} +3.08872 q^{85} -5.56911 q^{86} +3.06746 q^{88} +6.26557 q^{89} +1.48212 q^{91} +0.771498 q^{92} +0.255791 q^{94} -1.00000 q^{95} -6.70633 q^{97} -5.28329 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 5 q^{7} + 12 q^{8}+O(q^{10}) \) 6 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 5 * q^7 + 12 * q^8	\( 6 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 5 q^{7} + 12 q^{8} + 2 q^{10} + 6 q^{11} - 5 q^{13} + 8 q^{14} + 4 q^{16} - q^{17} - 6 q^{19} + 4 q^{20} + 2 q^{22} - 4 q^{23} + 6 q^{25} + 14 q^{26} + 10 q^{28} + 9 q^{29} - 21 q^{31} + q^{32} + 5 q^{35} - 3 q^{37} - 2 q^{38} + 12 q^{40} + 23 q^{41} + 7 q^{43} + 4 q^{44} - 12 q^{46} + 18 q^{47} - 3 q^{49} + 2 q^{50} + 13 q^{52} + 17 q^{53} + 6 q^{55} + 2 q^{56} + 23 q^{58} + 29 q^{59} + 17 q^{61} - 2 q^{62} - 18 q^{64} - 5 q^{65} + 8 q^{67} + q^{68} + 8 q^{70} + 12 q^{71} + 2 q^{73} + 37 q^{74} - 4 q^{76} + 5 q^{77} + 3 q^{79} + 4 q^{80} + 24 q^{82} + 11 q^{83} - q^{85} + 12 q^{86} + 12 q^{88} + 22 q^{89} - 18 q^{91} + 15 q^{92} + 22 q^{94} - 6 q^{95} - 2 q^{97} + q^{98}+O(q^{100}) \) 6 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 5 * q^7 + 12 * q^8 + 2 * q^10 + 6 * q^11 - 5 * q^13 + 8 * q^14 + 4 * q^16 - q^17 - 6 * q^19 + 4 * q^20 + 2 * q^22 - 4 * q^23 + 6 * q^25 + 14 * q^26 + 10 * q^28 + 9 * q^29 - 21 * q^31 + q^32 + 5 * q^35 - 3 * q^37 - 2 * q^38 + 12 * q^40 + 23 * q^41 + 7 * q^43 + 4 * q^44 - 12 * q^46 + 18 * q^47 - 3 * q^49 + 2 * q^50 + 13 * q^52 + 17 * q^53 + 6 * q^55 + 2 * q^56 + 23 * q^58 + 29 * q^59 + 17 * q^61 - 2 * q^62 - 18 * q^64 - 5 * q^65 + 8 * q^67 + q^68 + 8 * q^70 + 12 * q^71 + 2 * q^73 + 37 * q^74 - 4 * q^76 + 5 * q^77 + 3 * q^79 + 4 * q^80 + 24 * q^82 + 11 * q^83 - q^85 + 12 * q^86 + 12 * q^88 + 22 * q^89 - 18 * q^91 + 15 * q^92 + 22 * q^94 - 6 * q^95 - 2 * q^97 + q^98

Introduction

L-functions

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