LMFDB - Embedded newform 9405.2.a.bq.1.4

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			1.4
Root			$1.15628$ 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)

$f(q)$	$=$	$q-1.15628 q^{2} -0.663011 q^{4} +1.00000 q^{5} +0.0469121 q^{7} +3.07919 q^{8} +O(q^{10})$	\(q-1.15628 q^{2} -0.663011 q^{4} +1.00000 q^{5} +0.0469121 q^{7} +3.07919 q^{8} -1.15628 q^{10} +1.00000 q^{11} -1.87376 q^{13} -0.0542437 q^{14} -2.23440 q^{16} -5.00426 q^{17} -1.00000 q^{19} -0.663011 q^{20} -1.15628 q^{22} +2.91713 q^{23} +1.00000 q^{25} +2.16660 q^{26} -0.0311032 q^{28} -3.22377 q^{29} +10.2267 q^{31} -3.57479 q^{32} +5.78633 q^{34} +0.0469121 q^{35} -3.30522 q^{37} +1.15628 q^{38} +3.07919 q^{40} -5.42564 q^{41} +7.61593 q^{43} -0.663011 q^{44} -3.37302 q^{46} +6.15429 q^{47} -6.99780 q^{49} -1.15628 q^{50} +1.24232 q^{52} -8.81703 q^{53} +1.00000 q^{55} +0.144452 q^{56} +3.72759 q^{58} +9.17264 q^{59} +9.52082 q^{61} -11.8250 q^{62} +8.60226 q^{64} -1.87376 q^{65} -15.2971 q^{67} +3.31787 q^{68} -0.0542437 q^{70} +11.9556 q^{71} -4.30517 q^{73} +3.82177 q^{74} +0.663011 q^{76} +0.0469121 q^{77} -12.8200 q^{79} -2.23440 q^{80} +6.27357 q^{82} -7.52064 q^{83} -5.00426 q^{85} -8.80617 q^{86} +3.07919 q^{88} -7.16636 q^{89} -0.0879022 q^{91} -1.93409 q^{92} -7.11610 q^{94} -1.00000 q^{95} +7.86692 q^{97} +8.09143 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.15628	−0.817615	−0.408808	−	0.912621i	$-0.634055\pi$
$2$	−1.15628	−0.817615	−0.408808	+	0.912621i	$0.634055\pi$
$3$	0	0
$3$	0	0
$4$	−0.663011	−0.331505
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.0469121	0.0177311	0.00886556	−	0.999961i	$-0.497178\pi$
$7$	0.0469121	0.0177311	0.00886556	+	0.999961i	$0.497178\pi$
$8$	3.07919	1.08866
$9$	0	0
$10$	−1.15628	−0.365649
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−1.87376	−0.519688	−0.259844	−	0.965651i	$-0.583671\pi$
$13$	−1.87376	−0.519688	−0.259844	+	0.965651i	$0.583671\pi$
$14$	−0.0542437	−0.0144972
$15$	0	0
$16$	−2.23440	−0.558599
$17$	−5.00426	−1.21371	−0.606855	−	0.794812i	$-0.707569\pi$
$17$	−5.00426	−1.21371	−0.606855	+	0.794812i	$0.707569\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−0.663011	−0.148254
$21$	0	0
$22$	−1.15628	−0.246520
$23$	2.91713	0.608263	0.304132	−	0.952630i	$-0.401634\pi$
$23$	2.91713	0.608263	0.304132	+	0.952630i	$0.401634\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.16660	0.424905
$27$	0	0
$28$	−0.0311032	−0.00587796
$29$	−3.22377	−0.598639	−0.299319	−	0.954153i	$-0.596760\pi$
$29$	−3.22377	−0.598639	−0.299319	+	0.954153i	$0.596760\pi$
$30$	0	0
$31$	10.2267	1.83678	0.918389	−	0.395679i	$-0.129491\pi$
$31$	10.2267	1.83678	0.918389	+	0.395679i	$0.129491\pi$
$32$	−3.57479	−0.631940
$33$	0	0
$34$	5.78633	0.992348
$35$	0.0469121	0.00792960
$36$	0	0
$37$	−3.30522	−0.543375	−0.271687	−	0.962386i	$-0.587582\pi$
$37$	−3.30522	−0.543375	−0.271687	+	0.962386i	$0.587582\pi$
$38$	1.15628	0.187574
$39$	0	0
$40$	3.07919	0.486863
$41$	−5.42564	−0.847342	−0.423671	−	0.905816i	$-0.639259\pi$
$41$	−5.42564	−0.847342	−0.423671	+	0.905816i	$0.639259\pi$
$42$	0	0
$43$	7.61593	1.16142	0.580709	−	0.814111i	$-0.302775\pi$
$43$	7.61593	1.16142	0.580709	+	0.814111i	$0.302775\pi$
$44$	−0.663011	−0.0999526
$45$	0	0
$46$	−3.37302	−0.497325
$47$	6.15429	0.897696	0.448848	−	0.893608i	$-0.351835\pi$
$47$	6.15429	0.897696	0.448848	+	0.893608i	$0.351835\pi$
$48$	0	0
$49$	−6.99780	−0.999686
$50$	−1.15628	−0.163523
$51$	0	0
$52$	1.24232	0.172279
$53$	−8.81703	−1.21111	−0.605556	−	0.795803i	$-0.707049\pi$
$53$	−8.81703	−1.21111	−0.605556	+	0.795803i	$0.707049\pi$
$54$	0	0
$55$	1.00000	0.134840
$56$	0.144452	0.0193031
$57$	0	0
$58$	3.72759	0.489456
$59$	9.17264	1.19418	0.597088	−	0.802176i	$-0.296324\pi$
$59$	9.17264	1.19418	0.597088	+	0.802176i	$0.296324\pi$
$60$	0	0
$61$	9.52082	1.21902	0.609508	−	0.792780i	$-0.291367\pi$
$61$	9.52082	1.21902	0.609508	+	0.792780i	$0.291367\pi$
$62$	−11.8250	−1.50178
$63$	0	0
$64$	8.60226	1.07528
$65$	−1.87376	−0.232412
$66$	0	0
$67$	−15.2971	−1.86883	−0.934417	−	0.356182i	$-0.884078\pi$
$67$	−15.2971	−1.86883	−0.934417	+	0.356182i	$0.884078\pi$
$68$	3.31787	0.402351
$69$	0	0
$70$	−0.0542437	−0.00648336
$71$	11.9556	1.41887	0.709435	−	0.704771i	$-0.248950\pi$
$71$	11.9556	1.41887	0.709435	+	0.704771i	$0.248950\pi$
$72$	0	0
$73$	−4.30517	−0.503882	−0.251941	−	0.967743i	$-0.581069\pi$
$73$	−4.30517	−0.503882	−0.251941	+	0.967743i	$0.581069\pi$
$74$	3.82177	0.444271
$75$	0	0
$76$	0.663011	0.0760525
$77$	0.0469121	0.00534613
$78$	0	0
$79$	−12.8200	−1.44236	−0.721180	−	0.692748i	$-0.756400\pi$
$79$	−12.8200	−1.44236	−0.721180	+	0.692748i	$0.756400\pi$
$80$	−2.23440	−0.249813
$81$	0	0
$82$	6.27357	0.692800
$83$	−7.52064	−0.825498	−0.412749	−	0.910845i	$-0.635431\pi$
$83$	−7.52064	−0.825498	−0.412749	+	0.910845i	$0.635431\pi$
$84$	0	0
$85$	−5.00426	−0.542788
$86$	−8.80617	−0.949594
$87$	0	0
$88$	3.07919	0.328243
$89$	−7.16636	−0.759633	−0.379816	−	0.925062i	$-0.624013\pi$
$89$	−7.16636	−0.759633	−0.379816	+	0.925062i	$0.624013\pi$
$90$	0	0
$91$	−0.0879022	−0.00921465
$92$	−1.93409	−0.201643
$93$	0	0
$94$	−7.11610	−0.733970
$95$	−1.00000	−0.102598
$96$	0	0
$97$	7.86692	0.798764	0.399382	−	0.916785i	$-0.369225\pi$
$97$	7.86692	0.798764	0.399382	+	0.916785i	$0.369225\pi$
$98$	8.09143	0.817358
$99$	0	0
$100$	−0.663011	−0.0663011
$101$	5.07442	0.504924	0.252462	−	0.967607i	$-0.418760\pi$
$101$	5.07442	0.504924	0.252462	+	0.967607i	$0.418760\pi$
$102$	0	0
$103$	6.79814	0.669841	0.334920	−	0.942246i	$-0.391291\pi$
$103$	6.79814	0.669841	0.334920	+	0.942246i	$0.391291\pi$
$104$	−5.76967	−0.565763
$105$	0	0
$106$	10.1950	0.990224
$107$	11.7036	1.13143	0.565715	−	0.824601i	$-0.308600\pi$
$107$	11.7036	1.13143	0.565715	+	0.824601i	$0.308600\pi$
$108$	0	0
$109$	−11.7108	−1.12169	−0.560846	−	0.827920i	$-0.689524\pi$
$109$	−11.7108	−1.12169	−0.560846	+	0.827920i	$0.689524\pi$
$110$	−1.15628	−0.110247
$111$	0	0
$112$	−0.104820	−0.00990458
$113$	−4.25241	−0.400033	−0.200016	−	0.979793i	$-0.564100\pi$
$113$	−4.25241	−0.400033	−0.200016	+	0.979793i	$0.564100\pi$
$114$	0	0
$115$	2.91713	0.272024
$116$	2.13739	0.198452
$117$	0	0
$118$	−10.6062	−0.976376
$119$	−0.234760	−0.0215204
$120$	0	0
$121$	1.00000	0.0909091
$122$	−11.0088	−0.996686
$123$	0	0
$124$	−6.78044	−0.608902
$125$	1.00000	0.0894427
$126$	0	0
$127$	−16.9048	−1.50006	−0.750030	−	0.661404i	$-0.769961\pi$
$127$	−16.9048	−1.50006	−0.750030	+	0.661404i	$0.769961\pi$
$128$	−2.79706	−0.247227
$129$	0	0
$130$	2.16660	0.190023
$131$	1.19404	0.104324	0.0521621	−	0.998639i	$-0.483389\pi$
$131$	1.19404	0.104324	0.0521621	+	0.998639i	$0.483389\pi$
$132$	0	0
$133$	−0.0469121	−0.00406780
$134$	17.6877	1.52799
$135$	0	0
$136$	−15.4091	−1.32132
$137$	9.12665	0.779742	0.389871	−	0.920869i	$-0.372520\pi$
$137$	9.12665	0.779742	0.389871	+	0.920869i	$0.372520\pi$
$138$	0	0
$139$	−15.2894	−1.29683	−0.648416	−	0.761286i	$-0.724568\pi$
$139$	−15.2894	−1.29683	−0.648416	+	0.761286i	$0.724568\pi$
$140$	−0.0311032	−0.00262870
$141$	0	0
$142$	−13.8241	−1.16009
$143$	−1.87376	−0.156692
$144$	0	0
$145$	−3.22377	−0.267719
$146$	4.97800	0.411982
$147$	0	0
$148$	2.19140	0.180132
$149$	−0.555862	−0.0455380	−0.0227690	−	0.999741i	$-0.507248\pi$
$149$	−0.555862	−0.0455380	−0.0227690	+	0.999741i	$0.507248\pi$
$150$	0	0
$151$	−19.7766	−1.60940	−0.804699	−	0.593682i	$-0.797674\pi$
$151$	−19.7766	−1.60940	−0.804699	+	0.593682i	$0.797674\pi$
$152$	−3.07919	−0.249756
$153$	0	0
$154$	−0.0542437	−0.00437108
$155$	10.2267	0.821432
$156$	0	0
$157$	−1.50792	−0.120345	−0.0601724	−	0.998188i	$-0.519165\pi$
$157$	−1.50792	−0.120345	−0.0601724	+	0.998188i	$0.519165\pi$
$158$	14.8235	1.17930
$159$	0	0
$160$	−3.57479	−0.282612
$161$	0.136849	0.0107852
$162$	0	0
$163$	1.96539	0.153941	0.0769707	−	0.997033i	$-0.475475\pi$
$163$	1.96539	0.153941	0.0769707	+	0.997033i	$0.475475\pi$
$164$	3.59726	0.280898
$165$	0	0
$166$	8.69599	0.674940
$167$	10.3275	0.799163	0.399581	−	0.916698i	$-0.369156\pi$
$167$	10.3275	0.799163	0.399581	+	0.916698i	$0.369156\pi$
$168$	0	0
$169$	−9.48902	−0.729924
$170$	5.78633	0.443791
$171$	0	0
$172$	−5.04945	−0.385017
$173$	−13.8971	−1.05657	−0.528287	−	0.849066i	$-0.677165\pi$
$173$	−13.8971	−1.05657	−0.528287	+	0.849066i	$0.677165\pi$
$174$	0	0
$175$	0.0469121	0.00354622
$176$	−2.23440	−0.168424
$177$	0	0
$178$	8.28634	0.621087
$179$	12.3082	0.919955	0.459978	−	0.887931i	$-0.347857\pi$
$179$	12.3082	0.919955	0.459978	+	0.887931i	$0.347857\pi$
$180$	0	0
$181$	9.30181	0.691398	0.345699	−	0.938345i	$-0.387642\pi$
$181$	9.30181	0.691398	0.345699	+	0.938345i	$0.387642\pi$
$182$	0.101640	0.00753404
$183$	0	0
$184$	8.98240	0.662191
$185$	−3.30522	−0.243005
$186$	0	0
$187$	−5.00426	−0.365947
$188$	−4.08036	−0.297591
$189$	0	0
$190$	1.15628	0.0838856
$191$	−15.3785	−1.11275	−0.556376	−	0.830931i	$-0.687809\pi$
$191$	−15.3785	−1.11275	−0.556376	+	0.830931i	$0.687809\pi$
$192$	0	0
$193$	−16.0645	−1.15635	−0.578174	−	0.815913i	$-0.696235\pi$
$193$	−16.0645	−1.15635	−0.578174	+	0.815913i	$0.696235\pi$
$194$	−9.09638	−0.653082
$195$	0	0
$196$	4.63962	0.331401
$197$	−5.11396	−0.364354	−0.182177	−	0.983266i	$-0.558314\pi$
$197$	−5.11396	−0.364354	−0.182177	+	0.983266i	$0.558314\pi$
$198$	0	0
$199$	12.7142	0.901289	0.450644	−	0.892704i	$-0.351194\pi$
$199$	12.7142	0.901289	0.450644	+	0.892704i	$0.351194\pi$
$200$	3.07919	0.217732
$201$	0	0
$202$	−5.86746	−0.412833
$203$	−0.151234	−0.0106145
$204$	0	0
$205$	−5.42564	−0.378943
$206$	−7.86057	−0.547672
$207$	0	0
$208$	4.18673	0.290297
$209$	−1.00000	−0.0691714
$210$	0	0
$211$	24.5572	1.69059	0.845293	−	0.534303i	$-0.179426\pi$
$211$	24.5572	1.69059	0.845293	+	0.534303i	$0.179426\pi$
$212$	5.84579	0.401490
$213$	0	0
$214$	−13.5327	−0.925075
$215$	7.61593	0.519402
$216$	0	0
$217$	0.479759	0.0325681
$218$	13.5410	0.917112
$219$	0	0
$220$	−0.663011	−0.0447002
$221$	9.37678	0.630751
$222$	0	0
$223$	9.27268	0.620944	0.310472	−	0.950582i	$-0.399513\pi$
$223$	9.27268	0.620944	0.310472	+	0.950582i	$0.399513\pi$
$224$	−0.167701	−0.0112050
$225$	0	0
$226$	4.91698	0.327073
$227$	19.4301	1.28962	0.644812	−	0.764341i	$-0.276936\pi$
$227$	19.4301	1.28962	0.644812	+	0.764341i	$0.276936\pi$
$228$	0	0
$229$	−14.6972	−0.971219	−0.485609	−	0.874176i	$-0.661402\pi$
$229$	−14.6972	−0.971219	−0.485609	+	0.874176i	$0.661402\pi$
$230$	−3.37302	−0.222411
$231$	0	0
$232$	−9.92660	−0.651713
$233$	12.1613	0.796713	0.398357	−	0.917231i	$-0.369581\pi$
$233$	12.1613	0.796713	0.398357	+	0.917231i	$0.369581\pi$
$234$	0	0
$235$	6.15429	0.401462
$236$	−6.08156	−0.395876
$237$	0	0
$238$	0.271449	0.0175954
$239$	15.7989	1.02195	0.510975	−	0.859596i	$-0.329285\pi$
$239$	15.7989	1.02195	0.510975	+	0.859596i	$0.329285\pi$
$240$	0	0
$241$	−26.6468	−1.71647	−0.858237	−	0.513254i	$-0.828440\pi$
$241$	−26.6468	−1.71647	−0.858237	+	0.513254i	$0.828440\pi$
$242$	−1.15628	−0.0743287
$243$	0	0
$244$	−6.31240	−0.404110
$245$	−6.99780	−0.447073
$246$	0	0
$247$	1.87376	0.119225
$248$	31.4901	1.99963
$249$	0	0
$250$	−1.15628	−0.0731297
$251$	0.278608	0.0175856	0.00879278	−	0.999961i	$-0.497201\pi$
$251$	0.278608	0.0175856	0.00879278	+	0.999961i	$0.497201\pi$
$252$	0	0
$253$	2.91713	0.183398
$254$	19.5467	1.22647
$255$	0	0
$256$	−13.9703	−0.873146
$257$	−19.5416	−1.21897	−0.609487	−	0.792796i	$-0.708625\pi$
$257$	−19.5416	−1.21897	−0.609487	+	0.792796i	$0.708625\pi$
$258$	0	0
$259$	−0.155055	−0.00963464
$260$	1.24232	0.0770457
$261$	0	0
$262$	−1.38065	−0.0852970
$263$	−14.1771	−0.874197	−0.437098	−	0.899414i	$-0.643994\pi$
$263$	−14.1771	−0.874197	−0.437098	+	0.899414i	$0.643994\pi$
$264$	0	0
$265$	−8.81703	−0.541626
$266$	0.0542437	0.00332589
$267$	0	0
$268$	10.1421	0.619528
$269$	9.08760	0.554081	0.277040	−	0.960858i	$-0.410646\pi$
$269$	9.08760	0.554081	0.277040	+	0.960858i	$0.410646\pi$
$270$	0	0
$271$	14.1825	0.861528	0.430764	−	0.902465i	$-0.358244\pi$
$271$	14.1825	0.861528	0.430764	+	0.902465i	$0.358244\pi$
$272$	11.1815	0.677977
$273$	0	0
$274$	−10.5530	−0.637529
$275$	1.00000	0.0603023
$276$	0	0
$277$	−18.5324	−1.11350	−0.556752	−	0.830678i	$-0.687953\pi$
$277$	−18.5324	−1.11350	−0.556752	+	0.830678i	$0.687953\pi$
$278$	17.6789	1.06031
$279$	0	0
$280$	0.144452	0.00863263
$281$	−4.51996	−0.269638	−0.134819	−	0.990870i	$-0.543045\pi$
$281$	−4.51996	−0.269638	−0.134819	+	0.990870i	$0.543045\pi$
$282$	0	0
$283$	−5.96830	−0.354779	−0.177389	−	0.984141i	$-0.556765\pi$
$283$	−5.96830	−0.354779	−0.177389	+	0.984141i	$0.556765\pi$
$284$	−7.92669	−0.470363
$285$	0	0
$286$	2.16660	0.128114
$287$	−0.254528	−0.0150243
$288$	0	0
$289$	8.04257	0.473092
$290$	3.72759	0.218891
$291$	0	0
$292$	2.85438	0.167040
$293$	−6.72428	−0.392836	−0.196418	−	0.980520i	$-0.562931\pi$
$293$	−6.72428	−0.392836	−0.196418	+	0.980520i	$0.562931\pi$
$294$	0	0
$295$	9.17264	0.534052
$296$	−10.1774	−0.591550
$297$	0	0
$298$	0.642734	0.0372326
$299$	−5.46601	−0.316107
$300$	0	0
$301$	0.357280	0.0205933
$302$	22.8674	1.31587
$303$	0	0
$304$	2.23440	0.128151
$305$	9.52082	0.545161
$306$	0	0
$307$	15.2996	0.873193	0.436596	−	0.899658i	$-0.356184\pi$
$307$	15.2996	0.873193	0.436596	+	0.899658i	$0.356184\pi$
$308$	−0.0311032	−0.00177227
$309$	0	0
$310$	−11.8250	−0.671615
$311$	6.38725	0.362187	0.181094	−	0.983466i	$-0.442036\pi$
$311$	6.38725	0.362187	0.181094	+	0.983466i	$0.442036\pi$
$312$	0	0
$313$	21.3557	1.20709	0.603547	−	0.797327i	$-0.293754\pi$
$313$	21.3557	1.20709	0.603547	+	0.797327i	$0.293754\pi$
$314$	1.74358	0.0983957
$315$	0	0
$316$	8.49978	0.478150
$317$	−1.71697	−0.0964348	−0.0482174	−	0.998837i	$-0.515354\pi$
$317$	−1.71697	−0.0964348	−0.0482174	+	0.998837i	$0.515354\pi$
$318$	0	0
$319$	−3.22377	−0.180496
$320$	8.60226	0.480881
$321$	0	0
$322$	−0.158236	−0.00881814
$323$	5.00426	0.278444
$324$	0	0
$325$	−1.87376	−0.103938
$326$	−2.27255	−0.125865
$327$	0	0
$328$	−16.7066	−0.922467
$329$	0.288711	0.0159171
$330$	0	0
$331$	8.26215	0.454129	0.227064	−	0.973880i	$-0.427087\pi$
$331$	8.26215	0.454129	0.227064	+	0.973880i	$0.427087\pi$
$332$	4.98627	0.273657
$333$	0	0
$334$	−11.9415	−0.653408
$335$	−15.2971	−0.835768
$336$	0	0
$337$	−9.59850	−0.522864	−0.261432	−	0.965222i	$-0.584195\pi$
$337$	−9.59850	−0.522864	−0.261432	+	0.965222i	$0.584195\pi$
$338$	10.9720	0.596797
$339$	0	0
$340$	3.31787	0.179937
$341$	10.2267	0.553809
$342$	0	0
$343$	−0.656667	−0.0354567
$344$	23.4509	1.26439
$345$	0	0
$346$	16.0689	0.863871
$347$	−3.19185	−0.171347	−0.0856737	−	0.996323i	$-0.527304\pi$
$347$	−3.19185	−0.171347	−0.0856737	+	0.996323i	$0.527304\pi$
$348$	0	0
$349$	0.304843	0.0163179	0.00815893	−	0.999967i	$-0.497403\pi$
$349$	0.304843	0.0163179	0.00815893	+	0.999967i	$0.497403\pi$
$350$	−0.0542437	−0.00289945
$351$	0	0
$352$	−3.57479	−0.190537
$353$	−3.80537	−0.202539	−0.101270	−	0.994859i	$-0.532290\pi$
$353$	−3.80537	−0.202539	−0.101270	+	0.994859i	$0.532290\pi$
$354$	0	0
$355$	11.9556	0.634538
$356$	4.75137	0.251822
$357$	0	0
$358$	−14.2317	−0.752169
$359$	34.6549	1.82902	0.914508	−	0.404568i	$-0.132578\pi$
$359$	34.6549	1.82902	0.914508	+	0.404568i	$0.132578\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	−10.7555	−0.565298
$363$	0	0
$364$	0.0582801	0.00305471
$365$	−4.30517	−0.225343
$366$	0	0
$367$	−6.70983	−0.350250	−0.175125	−	0.984546i	$-0.556033\pi$
$367$	−6.70983	−0.350250	−0.175125	+	0.984546i	$0.556033\pi$
$368$	−6.51802	−0.339775
$369$	0	0
$370$	3.82177	0.198684
$371$	−0.413626	−0.0214744
$372$	0	0
$373$	−13.3029	−0.688797	−0.344398	−	0.938824i	$-0.611917\pi$
$373$	−13.3029	−0.688797	−0.344398	+	0.938824i	$0.611917\pi$
$374$	5.78633	0.299204
$375$	0	0
$376$	18.9502	0.977284
$377$	6.04057	0.311105
$378$	0	0
$379$	1.49141	0.0766088	0.0383044	−	0.999266i	$-0.487804\pi$
$379$	1.49141	0.0766088	0.0383044	+	0.999266i	$0.487804\pi$
$380$	0.663011	0.0340117
$381$	0	0
$382$	17.7819	0.909803
$383$	33.5154	1.71256	0.856280	−	0.516513i	$-0.172770\pi$
$383$	33.5154	1.71256	0.856280	+	0.516513i	$0.172770\pi$
$384$	0	0
$385$	0.0469121	0.00239086
$386$	18.5751	0.945448
$387$	0	0
$388$	−5.21585	−0.264795
$389$	−31.8932	−1.61705	−0.808524	−	0.588464i	$-0.799733\pi$
$389$	−31.8932	−1.61705	−0.808524	+	0.588464i	$0.799733\pi$
$390$	0	0
$391$	−14.5981	−0.738255
$392$	−21.5476	−1.08832
$393$	0	0
$394$	5.91318	0.297902
$395$	−12.8200	−0.645043
$396$	0	0
$397$	−29.9150	−1.50139	−0.750694	−	0.660650i	$-0.770281\pi$
$397$	−29.9150	−1.50139	−0.750694	+	0.660650i	$0.770281\pi$
$398$	−14.7013	−0.736907
$399$	0	0
$400$	−2.23440	−0.111720
$401$	31.6134	1.57870	0.789349	−	0.613945i	$-0.210418\pi$
$401$	31.6134	1.57870	0.789349	+	0.613945i	$0.210418\pi$
$402$	0	0
$403$	−19.1625	−0.954552
$404$	−3.36440	−0.167385
$405$	0	0
$406$	0.174869	0.00867860
$407$	−3.30522	−0.163834
$408$	0	0
$409$	25.1565	1.24391	0.621954	−	0.783054i	$-0.286339\pi$
$409$	25.1565	1.24391	0.621954	+	0.783054i	$0.286339\pi$
$410$	6.27357	0.309830
$411$	0	0
$412$	−4.50724	−0.222056
$413$	0.430308	0.0211741
$414$	0	0
$415$	−7.52064	−0.369174
$416$	6.69831	0.328412
$417$	0	0
$418$	1.15628	0.0565556
$419$	23.6870	1.15719	0.578594	−	0.815616i	$-0.303602\pi$
$419$	23.6870	1.15719	0.578594	+	0.815616i	$0.303602\pi$
$420$	0	0
$421$	−34.3877	−1.67595	−0.837976	−	0.545707i	$-0.816261\pi$
$421$	−34.3877	−1.67595	−0.837976	+	0.545707i	$0.816261\pi$
$422$	−28.3950	−1.38225
$423$	0	0
$424$	−27.1493	−1.31849
$425$	−5.00426	−0.242742
$426$	0	0
$427$	0.446642	0.0216145
$428$	−7.75962	−0.375075
$429$	0	0
$430$	−8.80617	−0.424671
$431$	−40.4927	−1.95046	−0.975231	−	0.221187i	$-0.929007\pi$
$431$	−40.4927	−1.95046	−0.975231	+	0.221187i	$0.929007\pi$
$432$	0	0
$433$	−27.6554	−1.32903	−0.664517	−	0.747273i	$-0.731363\pi$
$433$	−27.6554	−1.32903	−0.664517	+	0.747273i	$0.731363\pi$
$434$	−0.554736	−0.0266282
$435$	0	0
$436$	7.76438	0.371847
$437$	−2.91713	−0.139545
$438$	0	0
$439$	−7.74381	−0.369592	−0.184796	−	0.982777i	$-0.559162\pi$
$439$	−7.74381	−0.369592	−0.184796	+	0.982777i	$0.559162\pi$
$440$	3.07919	0.146795
$441$	0	0
$442$	−10.8422	−0.515711
$443$	−33.3827	−1.58606	−0.793030	−	0.609183i	$-0.791497\pi$
$443$	−33.3827	−1.58606	−0.793030	+	0.609183i	$0.791497\pi$
$444$	0	0
$445$	−7.16636	−0.339718
$446$	−10.7218	−0.507694
$447$	0	0
$448$	0.403550	0.0190660
$449$	−36.3627	−1.71606	−0.858032	−	0.513596i	$-0.828313\pi$
$449$	−36.3627	−1.71606	−0.858032	+	0.513596i	$0.828313\pi$
$450$	0	0
$451$	−5.42564	−0.255483
$452$	2.81939	0.132613
$453$	0	0
$454$	−22.4667	−1.05442
$455$	−0.0879022	−0.00412092
$456$	0	0
$457$	−16.4867	−0.771215	−0.385607	−	0.922663i	$-0.626008\pi$
$457$	−16.4867	−0.771215	−0.385607	+	0.922663i	$0.626008\pi$
$458$	16.9941	0.794083
$459$	0	0
$460$	−1.93409	−0.0901773
$461$	14.3984	0.670601	0.335301	−	0.942111i	$-0.391162\pi$
$461$	14.3984	0.670601	0.335301	+	0.942111i	$0.391162\pi$
$462$	0	0
$463$	−24.6294	−1.14462	−0.572312	−	0.820036i	$-0.693953\pi$
$463$	−24.6294	−1.14462	−0.572312	+	0.820036i	$0.693953\pi$
$464$	7.20317	0.334399
$465$	0	0
$466$	−14.0619	−0.651405
$467$	−2.98666	−0.138206	−0.0691030	−	0.997610i	$-0.522014\pi$
$467$	−2.98666	−0.138206	−0.0691030	+	0.997610i	$0.522014\pi$
$468$	0	0
$469$	−0.717618	−0.0331365
$470$	−7.11610	−0.328241
$471$	0	0
$472$	28.2443	1.30005
$473$	7.61593	0.350181
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0.155649	0.00713414
$477$	0	0
$478$	−18.2680	−0.835561
$479$	−15.9921	−0.730698	−0.365349	−	0.930871i	$-0.619050\pi$
$479$	−15.9921	−0.730698	−0.365349	+	0.930871i	$0.619050\pi$
$480$	0	0
$481$	6.19319	0.282385
$482$	30.8113	1.40342
$483$	0	0
$484$	−0.663011	−0.0301368
$485$	7.86692	0.357218
$486$	0	0
$487$	6.63089	0.300474	0.150237	−	0.988650i	$-0.451996\pi$
$487$	6.63089	0.300474	0.150237	+	0.988650i	$0.451996\pi$
$488$	29.3164	1.32709
$489$	0	0
$490$	8.09143	0.365534
$491$	9.19461	0.414947	0.207474	−	0.978241i	$-0.433476\pi$
$491$	9.19461	0.414947	0.207474	+	0.978241i	$0.433476\pi$
$492$	0	0
$493$	16.1326	0.726574
$494$	−2.16660	−0.0974799
$495$	0	0
$496$	−22.8506	−1.02602
$497$	0.560863	0.0251581
$498$	0	0
$499$	−15.4987	−0.693818	−0.346909	−	0.937899i	$-0.612769\pi$
$499$	−15.4987	−0.693818	−0.346909	+	0.937899i	$0.612769\pi$
$500$	−0.663011	−0.0296507
$501$	0	0
$502$	−0.322149	−0.0143782
$503$	−18.2296	−0.812817	−0.406409	−	0.913691i	$-0.633219\pi$
$503$	−18.2296	−0.812817	−0.406409	+	0.913691i	$0.633219\pi$
$504$	0	0
$505$	5.07442	0.225809
$506$	−3.37302	−0.149949
$507$	0	0
$508$	11.2081	0.497278
$509$	−26.1709	−1.16000	−0.580002	−	0.814615i	$-0.696948\pi$
$509$	−26.1709	−1.16000	−0.580002	+	0.814615i	$0.696948\pi$
$510$	0	0
$511$	−0.201965	−0.00893440
$512$	21.7478	0.961125
$513$	0	0
$514$	22.5957	0.996652
$515$	6.79814	0.299562
$516$	0	0
$517$	6.15429	0.270665
$518$	0.179287	0.00787743
$519$	0	0
$520$	−5.76967	−0.253017
$521$	37.1007	1.62541	0.812705	−	0.582675i	$-0.197994\pi$
$521$	37.1007	1.62541	0.812705	+	0.582675i	$0.197994\pi$
$522$	0	0
$523$	−40.3366	−1.76380	−0.881899	−	0.471438i	$-0.843735\pi$
$523$	−40.3366	−1.76380	−0.881899	+	0.471438i	$0.843735\pi$
$524$	−0.791665	−0.0345840
$525$	0	0
$526$	16.3927	0.714757
$527$	−51.1773	−2.22932
$528$	0	0
$529$	−14.4904	−0.630016
$530$	10.1950	0.442842
$531$	0	0
$532$	0.0311032	0.00134850
$533$	10.1664	0.440354
$534$	0	0
$535$	11.7036	0.505991
$536$	−47.1026	−2.03452
$537$	0	0
$538$	−10.5078	−0.453025
$539$	−6.99780	−0.301417
$540$	0	0
$541$	−16.0072	−0.688205	−0.344102	−	0.938932i	$-0.611817\pi$
$541$	−16.0072	−0.688205	−0.344102	+	0.938932i	$0.611817\pi$
$542$	−16.3990	−0.704398
$543$	0	0
$544$	17.8892	0.766992
$545$	−11.7108	−0.501635
$546$	0	0
$547$	−38.2938	−1.63733	−0.818663	−	0.574274i	$-0.805284\pi$
$547$	−38.2938	−1.63733	−0.818663	+	0.574274i	$0.805284\pi$
$548$	−6.05107	−0.258489
$549$	0	0
$550$	−1.15628	−0.0493041
$551$	3.22377	0.137337
$552$	0	0
$553$	−0.601412	−0.0255747
$554$	21.4287	0.910419
$555$	0	0
$556$	10.1370	0.429907
$557$	6.72990	0.285155	0.142578	−	0.989784i	$-0.454461\pi$
$557$	6.72990	0.285155	0.142578	+	0.989784i	$0.454461\pi$
$558$	0	0
$559$	−14.2704	−0.603576
$560$	−0.104820	−0.00442946
$561$	0	0
$562$	5.22635	0.220460
$563$	7.84003	0.330418	0.165209	−	0.986259i	$-0.447170\pi$
$563$	7.84003	0.330418	0.165209	+	0.986259i	$0.447170\pi$
$564$	0	0
$565$	−4.25241	−0.178900
$566$	6.90104	0.290073
$567$	0	0
$568$	36.8136	1.54467
$569$	−7.74955	−0.324878	−0.162439	−	0.986719i	$-0.551936\pi$
$569$	−7.74955	−0.324878	−0.162439	+	0.986719i	$0.551936\pi$
$570$	0	0
$571$	21.9093	0.916875	0.458437	−	0.888727i	$-0.348409\pi$
$571$	21.9093	0.916875	0.458437	+	0.888727i	$0.348409\pi$
$572$	1.24232	0.0519442
$573$	0	0
$574$	0.294307	0.0122841
$575$	2.91713	0.121653
$576$	0	0
$577$	0.986738	0.0410784	0.0205392	−	0.999789i	$-0.493462\pi$
$577$	0.986738	0.0410784	0.0205392	+	0.999789i	$0.493462\pi$
$578$	−9.29948	−0.386808
$579$	0	0
$580$	2.13739	0.0887504
$581$	−0.352810	−0.0146370
$582$	0	0
$583$	−8.81703	−0.365164
$584$	−13.2565	−0.548556
$585$	0	0
$586$	7.77516	0.321189
$587$	−9.62391	−0.397221	−0.198611	−	0.980078i	$-0.563643\pi$
$587$	−9.62391	−0.397221	−0.198611	+	0.980078i	$0.563643\pi$
$588$	0	0
$589$	−10.2267	−0.421386
$590$	−10.6062	−0.436649
$591$	0	0
$592$	7.38517	0.303529
$593$	17.3125	0.710937	0.355469	−	0.934688i	$-0.384321\pi$
$593$	17.3125	0.710937	0.355469	+	0.934688i	$0.384321\pi$
$594$	0	0
$595$	−0.234760	−0.00962423
$596$	0.368543	0.0150961
$597$	0	0
$598$	6.32025	0.258454
$599$	−18.8090	−0.768517	−0.384258	−	0.923226i	$-0.625543\pi$
$599$	−18.8090	−0.768517	−0.384258	+	0.923226i	$0.625543\pi$
$600$	0	0
$601$	10.5076	0.428615	0.214308	−	0.976766i	$-0.431251\pi$
$601$	10.5076	0.428615	0.214308	+	0.976766i	$0.431251\pi$
$602$	−0.413116	−0.0168374
$603$	0	0
$604$	13.1121	0.533524
$605$	1.00000	0.0406558
$606$	0	0
$607$	26.2377	1.06496	0.532478	−	0.846444i	$-0.321261\pi$
$607$	26.2377	1.06496	0.532478	+	0.846444i	$0.321261\pi$
$608$	3.57479	0.144977
$609$	0	0
$610$	−11.0088	−0.445732
$611$	−11.5317	−0.466522
$612$	0	0
$613$	−6.31407	−0.255023	−0.127511	−	0.991837i	$-0.540699\pi$
$613$	−6.31407	−0.255023	−0.127511	+	0.991837i	$0.540699\pi$
$614$	−17.6906	−0.713936
$615$	0	0
$616$	0.144452	0.00582012
$617$	−29.6280	−1.19278	−0.596390	−	0.802695i	$-0.703399\pi$
$617$	−29.6280	−1.19278	−0.596390	+	0.802695i	$0.703399\pi$
$618$	0	0
$619$	−28.4137	−1.14204	−0.571022	−	0.820935i	$-0.693453\pi$
$619$	−28.4137	−1.14204	−0.571022	+	0.820935i	$0.693453\pi$
$620$	−6.78044	−0.272309
$621$	0	0
$622$	−7.38546	−0.296130
$623$	−0.336189	−0.0134691
$624$	0	0
$625$	1.00000	0.0400000
$626$	−24.6932	−0.986938
$627$	0	0
$628$	0.999764	0.0398949
$629$	16.5402	0.659499
$630$	0	0
$631$	−48.4902	−1.93036	−0.965182	−	0.261578i	$-0.915757\pi$
$631$	−48.4902	−1.93036	−0.965182	+	0.261578i	$0.915757\pi$
$632$	−39.4752	−1.57024
$633$	0	0
$634$	1.98531	0.0788466
$635$	−16.9048	−0.670847
$636$	0	0
$637$	13.1122	0.519525
$638$	3.72759	0.147577
$639$	0	0
$640$	−2.79706	−0.110563
$641$	−35.9842	−1.42129	−0.710646	−	0.703550i	$-0.751597\pi$
$641$	−35.9842	−1.42129	−0.710646	+	0.703550i	$0.751597\pi$
$642$	0	0
$643$	28.5333	1.12524	0.562621	−	0.826715i	$-0.309793\pi$
$643$	28.5333	1.12524	0.562621	+	0.826715i	$0.309793\pi$
$644$	−0.0907322	−0.00357535
$645$	0	0
$646$	−5.78633	−0.227660
$647$	−12.1377	−0.477184	−0.238592	−	0.971120i	$-0.576686\pi$
$647$	−12.1377	−0.477184	−0.238592	+	0.971120i	$0.576686\pi$
$648$	0	0
$649$	9.17264	0.360057
$650$	2.16660	0.0849810
$651$	0	0
$652$	−1.30308	−0.0510324
$653$	−15.0963	−0.590764	−0.295382	−	0.955379i	$-0.595447\pi$
$653$	−15.0963	−0.590764	−0.295382	+	0.955379i	$0.595447\pi$
$654$	0	0
$655$	1.19404	0.0466552
$656$	12.1230	0.473324
$657$	0	0
$658$	−0.333831	−0.0130141
$659$	51.0712	1.98945	0.994726	−	0.102570i	$-0.0327065\pi$
$659$	51.0712	1.98945	0.994726	+	0.102570i	$0.0327065\pi$
$660$	0	0
$661$	12.1093	0.470997	0.235499	−	0.971875i	$-0.424328\pi$
$661$	12.1093	0.470997	0.235499	+	0.971875i	$0.424328\pi$
$662$	−9.55338	−0.371303
$663$	0	0
$664$	−23.1575	−0.898686
$665$	−0.0469121	−0.00181917
$666$	0	0
$667$	−9.40414	−0.364130
$668$	−6.84721	−0.264927
$669$	0	0
$670$	17.6877	0.683336
$671$	9.52082	0.367547
$672$	0	0
$673$	42.7915	1.64949	0.824746	−	0.565504i	$-0.191318\pi$
$673$	42.7915	1.64949	0.824746	+	0.565504i	$0.191318\pi$
$674$	11.0986	0.427501
$675$	0	0
$676$	6.29132	0.241974
$677$	−43.8946	−1.68701	−0.843503	−	0.537125i	$-0.819510\pi$
$677$	−43.8946	−1.68701	−0.843503	+	0.537125i	$0.819510\pi$
$678$	0	0
$679$	0.369054	0.0141630
$680$	−15.4091	−0.590911
$681$	0	0
$682$	−11.8250	−0.452803
$683$	−10.4301	−0.399095	−0.199547	−	0.979888i	$-0.563947\pi$
$683$	−10.4301	−0.399095	−0.199547	+	0.979888i	$0.563947\pi$
$684$	0	0
$685$	9.12665	0.348711
$686$	0.759292	0.0289899
$687$	0	0
$688$	−17.0170	−0.648767
$689$	16.5210	0.629401
$690$	0	0
$691$	−42.6195	−1.62132	−0.810661	−	0.585516i	$-0.800892\pi$
$691$	−42.6195	−1.62132	−0.810661	+	0.585516i	$0.800892\pi$
$692$	9.21390	0.350260
$693$	0	0
$694$	3.69068	0.140096
$695$	−15.2894	−0.579961
$696$	0	0
$697$	27.1513	1.02843
$698$	−0.352484	−0.0133417
$699$	0	0
$700$	−0.0311032	−0.00117559
$701$	3.45984	0.130676	0.0653381	−	0.997863i	$-0.479187\pi$
$701$	3.45984	0.130676	0.0653381	+	0.997863i	$0.479187\pi$
$702$	0	0
$703$	3.30522	0.124659
$704$	8.60226	0.324210
$705$	0	0
$706$	4.40008	0.165599
$707$	0.238052	0.00895286
$708$	0	0
$709$	15.8360	0.594733	0.297366	−	0.954763i	$-0.403892\pi$
$709$	15.8360	0.594733	0.297366	+	0.954763i	$0.403892\pi$
$710$	−13.8241	−0.518808
$711$	0	0
$712$	−22.0666	−0.826981
$713$	29.8327	1.11724
$714$	0	0
$715$	−1.87376	−0.0700747
$716$	−8.16044	−0.304970
$717$	0	0
$718$	−40.0709	−1.49543
$719$	−11.0933	−0.413709	−0.206855	−	0.978372i	$-0.566323\pi$
$719$	−11.0933	−0.413709	−0.206855	+	0.978372i	$0.566323\pi$
$720$	0	0
$721$	0.318915	0.0118770
$722$	−1.15628	−0.0430324
$723$	0	0
$724$	−6.16720	−0.229202
$725$	−3.22377	−0.119728
$726$	0	0
$727$	−6.48006	−0.240332	−0.120166	−	0.992754i	$-0.538343\pi$
$727$	−6.48006	−0.240332	−0.120166	+	0.992754i	$0.538343\pi$
$728$	−0.270668	−0.0100316
$729$	0	0
$730$	4.97800	0.184244
$731$	−38.1121	−1.40963
$732$	0	0
$733$	−6.22067	−0.229766	−0.114883	−	0.993379i	$-0.536649\pi$
$733$	−6.22067	−0.229766	−0.114883	+	0.993379i	$0.536649\pi$
$734$	7.75846	0.286370
$735$	0	0
$736$	−10.4281	−0.384386
$737$	−15.2971	−0.563475
$738$	0	0
$739$	−32.3499	−1.19001	−0.595006	−	0.803721i	$-0.702850\pi$
$739$	−32.3499	−1.19001	−0.595006	+	0.803721i	$0.702850\pi$
$740$	2.19140	0.0805573
$741$	0	0
$742$	0.478268	0.0175578
$743$	25.4904	0.935151	0.467575	−	0.883953i	$-0.345128\pi$
$743$	25.4904	0.935151	0.467575	+	0.883953i	$0.345128\pi$
$744$	0	0
$745$	−0.555862	−0.0203652
$746$	15.3819	0.563171
$747$	0	0
$748$	3.31787	0.121314
$749$	0.549041	0.0200615
$750$	0	0
$751$	−35.8291	−1.30742	−0.653710	−	0.756745i	$-0.726789\pi$
$751$	−35.8291	−1.30742	−0.653710	+	0.756745i	$0.726789\pi$
$752$	−13.7511	−0.501452
$753$	0	0
$754$	−6.98461	−0.254364
$755$	−19.7766	−0.719745
$756$	0	0
$757$	21.9243	0.796852	0.398426	−	0.917200i	$-0.369556\pi$
$757$	21.9243	0.796852	0.398426	+	0.917200i	$0.369556\pi$
$758$	−1.72450	−0.0626365
$759$	0	0
$760$	−3.07919	−0.111694
$761$	−19.7395	−0.715556	−0.357778	−	0.933807i	$-0.616465\pi$
$761$	−19.7395	−0.715556	−0.357778	+	0.933807i	$0.616465\pi$
$762$	0	0
$763$	−0.549379	−0.0198888
$764$	10.1961	0.368883
$765$	0	0
$766$	−38.7533	−1.40021
$767$	−17.1873	−0.620599
$768$	0	0
$769$	7.89802	0.284810	0.142405	−	0.989808i	$-0.454516\pi$
$769$	7.89802	0.284810	0.142405	+	0.989808i	$0.454516\pi$
$770$	−0.0542437	−0.00195481
$771$	0	0
$772$	10.6509	0.383336
$773$	30.8245	1.10868	0.554340	−	0.832290i	$-0.312971\pi$
$773$	30.8245	1.10868	0.554340	+	0.832290i	$0.312971\pi$
$774$	0	0
$775$	10.2267	0.367356
$776$	24.2237	0.869582
$777$	0	0
$778$	36.8775	1.32212
$779$	5.42564	0.194394
$780$	0	0
$781$	11.9556	0.427805
$782$	16.8795	0.603609
$783$	0	0
$784$	15.6359	0.558423
$785$	−1.50792	−0.0538198
$786$	0	0
$787$	−22.0705	−0.786727	−0.393364	−	0.919383i	$-0.628689\pi$
$787$	−22.0705	−0.786727	−0.393364	+	0.919383i	$0.628689\pi$
$788$	3.39061	0.120785
$789$	0	0
$790$	14.8235	0.527397
$791$	−0.199489	−0.00709303
$792$	0	0
$793$	−17.8398	−0.633508
$794$	34.5901	1.22756
$795$	0	0
$796$	−8.42968	−0.298782
$797$	−11.0344	−0.390857	−0.195429	−	0.980718i	$-0.562610\pi$
$797$	−11.0344	−0.390857	−0.195429	+	0.980718i	$0.562610\pi$
$798$	0	0
$799$	−30.7976	−1.08954
$800$	−3.57479	−0.126388
$801$	0	0
$802$	−36.5540	−1.29077
$803$	−4.30517	−0.151926
$804$	0	0
$805$	0.136849	0.00482328
$806$	22.1573	0.780456
$807$	0	0
$808$	15.6251	0.549690
$809$	53.6537	1.88636	0.943182	−	0.332278i	$-0.107817\pi$
$809$	53.6537	1.88636	0.943182	+	0.332278i	$0.107817\pi$
$810$	0	0
$811$	−8.79329	−0.308774	−0.154387	−	0.988010i	$-0.549340\pi$
$811$	−8.79329	−0.308774	−0.154387	+	0.988010i	$0.549340\pi$
$812$	0.100270	0.00351877
$813$	0	0
$814$	3.82177	0.133953
$815$	1.96539	0.0688447
$816$	0	0
$817$	−7.61593	−0.266448
$818$	−29.0880	−1.01704
$819$	0	0
$820$	3.59726	0.125622
$821$	8.73800	0.304958	0.152479	−	0.988307i	$-0.451274\pi$
$821$	8.73800	0.304958	0.152479	+	0.988307i	$0.451274\pi$
$822$	0	0
$823$	32.1801	1.12173	0.560864	−	0.827908i	$-0.310469\pi$
$823$	32.1801	1.12173	0.560864	+	0.827908i	$0.310469\pi$
$824$	20.9328	0.729228
$825$	0	0
$826$	−0.497558	−0.0173122
$827$	28.5060	0.991251	0.495626	−	0.868536i	$-0.334939\pi$
$827$	28.5060	0.991251	0.495626	+	0.868536i	$0.334939\pi$
$828$	0	0
$829$	−46.6692	−1.62089	−0.810443	−	0.585817i	$-0.800774\pi$
$829$	−46.6692	−1.62089	−0.810443	+	0.585817i	$0.800774\pi$
$830$	8.69599	0.301842
$831$	0	0
$832$	−16.1186	−0.558812
$833$	35.0188	1.21333
$834$	0	0
$835$	10.3275	0.357396
$836$	0.663011	0.0229307
$837$	0	0
$838$	−27.3889	−0.946134
$839$	−38.1471	−1.31698	−0.658492	−	0.752587i	$-0.728805\pi$
$839$	−38.1471	−1.31698	−0.658492	+	0.752587i	$0.728805\pi$
$840$	0	0
$841$	−18.6073	−0.641632
$842$	39.7618	1.37028
$843$	0	0
$844$	−16.2817	−0.560438
$845$	−9.48902	−0.326432
$846$	0	0
$847$	0.0469121	0.00161192
$848$	19.7007	0.676526
$849$	0	0
$850$	5.78633	0.198470
$851$	−9.64175	−0.330515
$852$	0	0
$853$	−44.2618	−1.51549	−0.757747	−	0.652549i	$-0.773700\pi$
$853$	−44.2618	−1.51549	−0.757747	+	0.652549i	$0.773700\pi$
$854$	−0.516444	−0.0176724
$855$	0	0
$856$	36.0377	1.23174
$857$	10.8305	0.369964	0.184982	−	0.982742i	$-0.440777\pi$
$857$	10.8305	0.369964	0.184982	+	0.982742i	$0.440777\pi$
$858$	0	0
$859$	43.3660	1.47963	0.739814	−	0.672811i	$-0.234913\pi$
$859$	43.3660	1.47963	0.739814	+	0.672811i	$0.234913\pi$
$860$	−5.04945	−0.172185
$861$	0	0
$862$	46.8210	1.59473
$863$	−9.03229	−0.307463	−0.153731	−	0.988113i	$-0.549129\pi$
$863$	−9.03229	−0.307463	−0.153731	+	0.988113i	$0.549129\pi$
$864$	0	0
$865$	−13.8971	−0.472514
$866$	31.9775	1.08664
$867$	0	0
$868$	−0.318085	−0.0107965
$869$	−12.8200	−0.434888
$870$	0	0
$871$	28.6631	0.971211
$872$	−36.0598	−1.22114
$873$	0	0
$874$	3.37302	0.114094
$875$	0.0469121	0.00158592
$876$	0	0
$877$	−38.6696	−1.30578	−0.652890	−	0.757453i	$-0.726444\pi$
$877$	−38.6696	−1.30578	−0.652890	+	0.757453i	$0.726444\pi$
$878$	8.95403	0.302184
$879$	0	0
$880$	−2.23440	−0.0753215
$881$	18.7191	0.630662	0.315331	−	0.948982i	$-0.397885\pi$
$881$	18.7191	0.630662	0.315331	+	0.948982i	$0.397885\pi$
$882$	0	0
$883$	−10.5433	−0.354812	−0.177406	−	0.984138i	$-0.556770\pi$
$883$	−10.5433	−0.354812	−0.177406	+	0.984138i	$0.556770\pi$
$884$	−6.21691	−0.209097
$885$	0	0
$886$	38.5998	1.29679
$887$	−44.5870	−1.49708	−0.748542	−	0.663087i	$-0.769246\pi$
$887$	−44.5870	−1.49708	−0.748542	+	0.663087i	$0.769246\pi$
$888$	0	0
$889$	−0.793041	−0.0265977
$890$	8.28634	0.277759
$891$	0	0
$892$	−6.14788	−0.205846
$893$	−6.15429	−0.205945
$894$	0	0
$895$	12.3082	0.411416
$896$	−0.131216	−0.00438362
$897$	0	0
$898$	42.0456	1.40308
$899$	−32.9687	−1.09957
$900$	0	0
$901$	44.1227	1.46994
$902$	6.27357	0.208887
$903$	0	0
$904$	−13.0940	−0.435499
$905$	9.30181	0.309203
$906$	0	0
$907$	−44.9964	−1.49408	−0.747041	−	0.664778i	$-0.768526\pi$
$907$	−44.9964	−1.49408	−0.747041	+	0.664778i	$0.768526\pi$
$908$	−12.8824	−0.427517
$909$	0	0
$910$	0.101640	0.00336933
$911$	15.2697	0.505909	0.252955	−	0.967478i	$-0.418598\pi$
$911$	15.2697	0.505909	0.252955	+	0.967478i	$0.418598\pi$
$912$	0	0
$913$	−7.52064	−0.248897
$914$	19.0633	0.630557
$915$	0	0
$916$	9.74441	0.321964
$917$	0.0560152	0.00184978
$918$	0	0
$919$	24.5389	0.809465	0.404732	−	0.914435i	$-0.367365\pi$
$919$	24.5389	0.809465	0.404732	+	0.914435i	$0.367365\pi$
$920$	8.98240	0.296141
$921$	0	0
$922$	−16.6486	−0.548294
$923$	−22.4020	−0.737370
$924$	0	0
$925$	−3.30522	−0.108675
$926$	28.4785	0.935862
$927$	0	0
$928$	11.5243	0.378304
$929$	36.2820	1.19037	0.595187	−	0.803587i	$-0.297078\pi$
$929$	36.2820	1.19037	0.595187	+	0.803587i	$0.297078\pi$
$930$	0	0
$931$	6.99780	0.229344
$932$	−8.06307	−0.264115
$933$	0	0
$934$	3.45342	0.112999
$935$	−5.00426	−0.163657
$936$	0	0
$937$	−3.61732	−0.118173	−0.0590863	−	0.998253i	$-0.518819\pi$
$937$	−3.61732	−0.118173	−0.0590863	+	0.998253i	$0.518819\pi$
$938$	0.829769	0.0270929
$939$	0	0
$940$	−4.08036	−0.133087
$941$	−55.8558	−1.82085	−0.910424	−	0.413677i	$-0.864244\pi$
$941$	−55.8558	−1.82085	−0.910424	+	0.413677i	$0.864244\pi$
$942$	0	0
$943$	−15.8273	−0.515407
$944$	−20.4953	−0.667065
$945$	0	0
$946$	−8.80617	−0.286313
$947$	−42.5423	−1.38244	−0.691219	−	0.722646i	$-0.742926\pi$
$947$	−42.5423	−1.38244	−0.691219	+	0.722646i	$0.742926\pi$
$948$	0	0
$949$	8.06687	0.261862
$950$	1.15628	0.0375148
$951$	0	0
$952$	−0.722872	−0.0234284
$953$	−41.4625	−1.34310	−0.671551	−	0.740958i	$-0.734372\pi$
$953$	−41.4625	−1.34310	−0.671551	+	0.740958i	$0.734372\pi$
$954$	0	0
$955$	−15.3785	−0.497638
$956$	−10.4749	−0.338782
$957$	0	0
$958$	18.4914	0.597430
$959$	0.428151	0.0138257
$960$	0	0
$961$	73.5864	2.37375
$962$	−7.16108	−0.230883
$963$	0	0
$964$	17.6671	0.569020
$965$	−16.0645	−0.517135
$966$	0	0
$967$	−18.6194	−0.598761	−0.299380	−	0.954134i	$-0.596780\pi$
$967$	−18.6194	−0.598761	−0.299380	+	0.954134i	$0.596780\pi$
$968$	3.07919	0.0989690
$969$	0	0
$970$	−9.09638	−0.292067
$971$	27.8375	0.893347	0.446674	−	0.894697i	$-0.352609\pi$
$971$	27.8375	0.893347	0.446674	+	0.894697i	$0.352609\pi$
$972$	0	0
$973$	−0.717259	−0.0229943
$974$	−7.66718	−0.245672
$975$	0	0
$976$	−21.2733	−0.680941
$977$	−6.02554	−0.192774	−0.0963870	−	0.995344i	$-0.530729\pi$
$977$	−6.02554	−0.192774	−0.0963870	+	0.995344i	$0.530729\pi$
$978$	0	0
$979$	−7.16636	−0.229038
$980$	4.63962	0.148207
$981$	0	0
$982$	−10.6316	−0.339267
$983$	−25.2611	−0.805704	−0.402852	−	0.915265i	$-0.631981\pi$
$983$	−25.2611	−0.805704	−0.402852	+	0.915265i	$0.631981\pi$
$984$	0	0
$985$	−5.11396	−0.162944
$986$	−18.6538	−0.594058
$987$	0	0
$988$	−1.24232	−0.0395236
$989$	22.2167	0.706449
$990$	0	0
$991$	18.3215	0.582001	0.291000	−	0.956723i	$-0.406012\pi$
$991$	18.3215	0.582001	0.291000	+	0.956723i	$0.406012\pi$
$992$	−36.5585	−1.16073
$993$	0	0
$994$	−0.648516	−0.0205697
$995$	12.7142	0.403069
$996$	0	0
$997$	3.03304	0.0960574	0.0480287	−	0.998846i	$-0.484706\pi$
$997$	3.03304	0.0960574	0.0480287	+	0.998846i	$0.484706\pi$
$998$	17.9209	0.567276
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9405.2.a.bq.1.4	✓	14
3.2	odd	2		9405.2.a.br.1.11	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9405.2.a.bq.1.4	✓	14	1.1	even	1	trivial
9405.2.a.br.1.11	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)

\(f(q)\)	\(=\)	\(q-1.15628 q^{2} -0.663011 q^{4} +1.00000 q^{5} +0.0469121 q^{7} +3.07919 q^{8} +O(q^{10})\)	\(q-1.15628 q^{2} -0.663011 q^{4} +1.00000 q^{5} +0.0469121 q^{7} +3.07919 q^{8} -1.15628 q^{10} +1.00000 q^{11} -1.87376 q^{13} -0.0542437 q^{14} -2.23440 q^{16} -5.00426 q^{17} -1.00000 q^{19} -0.663011 q^{20} -1.15628 q^{22} +2.91713 q^{23} +1.00000 q^{25} +2.16660 q^{26} -0.0311032 q^{28} -3.22377 q^{29} +10.2267 q^{31} -3.57479 q^{32} +5.78633 q^{34} +0.0469121 q^{35} -3.30522 q^{37} +1.15628 q^{38} +3.07919 q^{40} -5.42564 q^{41} +7.61593 q^{43} -0.663011 q^{44} -3.37302 q^{46} +6.15429 q^{47} -6.99780 q^{49} -1.15628 q^{50} +1.24232 q^{52} -8.81703 q^{53} +1.00000 q^{55} +0.144452 q^{56} +3.72759 q^{58} +9.17264 q^{59} +9.52082 q^{61} -11.8250 q^{62} +8.60226 q^{64} -1.87376 q^{65} -15.2971 q^{67} +3.31787 q^{68} -0.0542437 q^{70} +11.9556 q^{71} -4.30517 q^{73} +3.82177 q^{74} +0.663011 q^{76} +0.0469121 q^{77} -12.8200 q^{79} -2.23440 q^{80} +6.27357 q^{82} -7.52064 q^{83} -5.00426 q^{85} -8.80617 q^{86} +3.07919 q^{88} -7.16636 q^{89} -0.0879022 q^{91} -1.93409 q^{92} -7.11610 q^{94} -1.00000 q^{95} +7.86692 q^{97} +8.09143 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - q^{2} + 11 q^{4} + 14 q^{5} - 12 q^{7} - 9 q^{8}+O(q^{10}) \) 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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Database

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