LMFDB - Embedded newform 2325.2.a.p.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2325,2,Mod(1,2325)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2325.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2325 = 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2325.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-3,-3,5,0,3,-2,-9,3,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$18.5652184699$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 465)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.311108$ of defining polynomial
Character	$\chi$	$=$	2325.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.21432 q^{2} -1.00000 q^{3} -0.525428 q^{4} -1.21432 q^{6} +1.59210 q^{7} -3.06668 q^{8} +1.00000 q^{9} +0.622216 q^{11} +0.525428 q^{12} -0.214320 q^{13} +1.93332 q^{14} -2.67307 q^{16} -3.52543 q^{17} +1.21432 q^{18} +1.80642 q^{19} -1.59210 q^{21} +0.755569 q^{22} -6.90321 q^{23} +3.06668 q^{24} -0.260253 q^{26} -1.00000 q^{27} -0.836535 q^{28} +9.73975 q^{29} -1.00000 q^{31} +2.88739 q^{32} -0.622216 q^{33} -4.28100 q^{34} -0.525428 q^{36} +4.83654 q^{37} +2.19358 q^{38} +0.214320 q^{39} -7.47949 q^{41} -1.93332 q^{42} -8.23506 q^{43} -0.326929 q^{44} -8.38271 q^{46} -11.4652 q^{47} +2.67307 q^{48} -4.46520 q^{49} +3.52543 q^{51} +0.112610 q^{52} -13.7605 q^{53} -1.21432 q^{54} -4.88247 q^{56} -1.80642 q^{57} +11.8272 q^{58} -4.26025 q^{59} +2.85728 q^{61} -1.21432 q^{62} +1.59210 q^{63} +8.85236 q^{64} -0.755569 q^{66} +2.08097 q^{67} +1.85236 q^{68} +6.90321 q^{69} +1.31111 q^{71} -3.06668 q^{72} +1.65233 q^{73} +5.87310 q^{74} -0.949145 q^{76} +0.990632 q^{77} +0.260253 q^{78} -5.19850 q^{79} +1.00000 q^{81} -9.08250 q^{82} +5.65878 q^{83} +0.836535 q^{84} -10.0000 q^{86} -9.73975 q^{87} -1.90813 q^{88} -1.93332 q^{89} -0.341219 q^{91} +3.62714 q^{92} +1.00000 q^{93} -13.9224 q^{94} -2.88739 q^{96} -6.91750 q^{97} -5.42219 q^{98} +0.622216 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{2} - 3 q^{3} + 5 q^{4} + 3 q^{6} - 2 q^{7} - 9 q^{8} + 3 q^{9} + 2 q^{11} - 5 q^{12} + 6 q^{13} + 6 q^{14} + 5 q^{16} - 4 q^{17} - 3 q^{18} - 8 q^{19} + 2 q^{21} + 2 q^{22} - 14 q^{23} + 9 q^{24}+ \cdots + 2 q^{99}+O(q^{100}) $ 3 * q - 3 * q^2 - 3 * q^3 + 5 * q^4 + 3 * q^6 - 2 * q^7 - 9 * q^8 + 3 * q^9 + 2 * q^11 - 5 * q^12 + 6 * q^13 + 6 * q^14 + 5 * q^16 - 4 * q^17 - 3 * q^18 - 8 * q^19 + 2 * q^21 + 2 * q^22 - 14 * q^23 + 9 * q^24 - 14 * q^26 - 3 * q^27 + 4 * q^28 + 16 * q^29 - 3 * q^31 - 11 * q^32 - 2 * q^33 - 6 * q^34 + 5 * q^36 + 8 * q^37 + 20 * q^38 - 6 * q^39 + 4 * q^41 - 6 * q^42 + 2 * q^43 - 14 * q^44 + 8 * q^46 - 14 * q^47 - 5 * q^48 + 7 * q^49 + 4 * q^51 + 20 * q^52 - 8 * q^53 + 3 * q^54 - 28 * q^56 + 8 * q^57 + 2 * q^58 - 26 * q^59 - 18 * q^61 + 3 * q^62 - 2 * q^63 + 33 * q^64 - 2 * q^66 + 12 * q^68 + 14 * q^69 + 4 * q^71 - 9 * q^72 + 12 * q^73 + 4 * q^74 - 16 * q^76 - 24 * q^77 + 14 * q^78 + 4 * q^79 + 3 * q^81 - 40 * q^82 + 10 * q^83 - 4 * q^84 - 30 * q^86 - 16 * q^87 + 34 * q^88 - 6 * q^89 - 8 * q^91 - 22 * q^92 + 3 * q^93 + 4 * q^94 + 11 * q^96 - 8 * q^97 - 17 * q^98 + 2 * q^99

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 $

$2$	1.21432	0.858654	0.429327	−	0.903149i	$-0.358751\pi$
$2$	1.21432	0.858654	0.429327	+	0.903149i	$0.358751\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−0.525428	−0.262714
$5$	0	0
$5$	0	0
$6$	−1.21432	−0.495744
$7$	1.59210	0.601759	0.300879	−	0.953662i	$-0.402720\pi$
$7$	1.59210	0.601759	0.300879	+	0.953662i	$0.402720\pi$
$8$	−3.06668	−1.08423
$9$	1.00000	0.333333
$10$	0	0
$11$	0.622216	0.187605	0.0938025	−	0.995591i	$-0.470098\pi$
$11$	0.622216	0.187605	0.0938025	+	0.995591i	$0.470098\pi$
$12$	0.525428	0.151678
$13$	−0.214320	−0.0594416	−0.0297208	−	0.999558i	$-0.509462\pi$
$13$	−0.214320	−0.0594416	−0.0297208	+	0.999558i	$0.509462\pi$
$14$	1.93332	0.516702
$15$	0	0
$16$	−2.67307	−0.668268
$17$	−3.52543	−0.855042	−0.427521	−	0.904005i	$-0.640613\pi$
$17$	−3.52543	−0.855042	−0.427521	+	0.904005i	$0.640613\pi$
$18$	1.21432	0.286218
$19$	1.80642	0.414422	0.207211	−	0.978296i	$-0.433561\pi$
$19$	1.80642	0.414422	0.207211	+	0.978296i	$0.433561\pi$
$20$	0	0
$21$	−1.59210	−0.347426
$22$	0.755569	0.161088
$23$	−6.90321	−1.43942	−0.719710	−	0.694275i	$-0.755725\pi$
$23$	−6.90321	−1.43942	−0.719710	+	0.694275i	$0.755725\pi$
$24$	3.06668	0.625983
$25$	0	0
$26$	−0.260253	−0.0510398
$27$	−1.00000	−0.192450
$28$	−0.836535	−0.158090
$29$	9.73975	1.80863	0.904313	−	0.426870i	$-0.140384\pi$
$29$	9.73975	1.80863	0.904313	+	0.426870i	$0.140384\pi$
$30$	0	0
$31$	−1.00000	−0.179605
$31$	−1.00000	−0.179605
$32$	2.88739	0.510423
$33$	−0.622216	−0.108314
$34$	−4.28100	−0.734185
$35$	0	0
$36$	−0.525428	−0.0875713
$37$	4.83654	0.795122	0.397561	−	0.917576i	$-0.369857\pi$
$37$	4.83654	0.795122	0.397561	+	0.917576i	$0.369857\pi$
$38$	2.19358	0.355845
$39$	0.214320	0.0343186
$40$	0	0
$41$	−7.47949	−1.16810	−0.584050	−	0.811717i	$-0.698533\pi$
$41$	−7.47949	−1.16810	−0.584050	+	0.811717i	$0.698533\pi$
$42$	−1.93332	−0.298318
$43$	−8.23506	−1.25584	−0.627918	−	0.778280i	$-0.716093\pi$
$43$	−8.23506	−1.25584	−0.627918	+	0.778280i	$0.716093\pi$
$44$	−0.326929	−0.0492864
$45$	0	0
$46$	−8.38271	−1.23596
$47$	−11.4652	−1.67237	−0.836186	−	0.548446i	$-0.815220\pi$
$47$	−11.4652	−1.67237	−0.836186	+	0.548446i	$0.815220\pi$
$48$	2.67307	0.385825
$49$	−4.46520	−0.637886
$50$	0	0
$51$	3.52543	0.493659
$52$	0.112610	0.0156161
$53$	−13.7605	−1.89015	−0.945074	−	0.326855i	$-0.894011\pi$
$53$	−13.7605	−1.89015	−0.945074	+	0.326855i	$0.894011\pi$
$54$	−1.21432	−0.165248
$55$	0	0
$56$	−4.88247	−0.652447
$57$	−1.80642	−0.239267
$58$	11.8272	1.55298
$59$	−4.26025	−0.554638	−0.277319	−	0.960778i	$-0.589446\pi$
$59$	−4.26025	−0.554638	−0.277319	+	0.960778i	$0.589446\pi$
$60$	0	0
$61$	2.85728	0.365837	0.182919	−	0.983128i	$-0.441446\pi$
$61$	2.85728	0.365837	0.182919	+	0.983128i	$0.441446\pi$
$62$	−1.21432	−0.154219
$63$	1.59210	0.200586
$64$	8.85236	1.10654
$65$	0	0
$66$	−0.755569	−0.0930041
$67$	2.08097	0.254231	0.127115	−	0.991888i	$-0.459428\pi$
$67$	2.08097	0.254231	0.127115	+	0.991888i	$0.459428\pi$
$68$	1.85236	0.224631
$69$	6.90321	0.831049
$70$	0	0
$71$	1.31111	0.155600	0.0777999	−	0.996969i	$-0.475210\pi$
$71$	1.31111	0.155600	0.0777999	+	0.996969i	$0.475210\pi$
$72$	−3.06668	−0.361411
$73$	1.65233	0.193390	0.0966951	−	0.995314i	$-0.469173\pi$
$73$	1.65233	0.193390	0.0966951	+	0.995314i	$0.469173\pi$
$74$	5.87310	0.682734
$75$	0	0
$76$	−0.949145	−0.108874
$77$	0.990632	0.112893
$78$	0.260253	0.0294678
$79$	−5.19850	−0.584877	−0.292438	−	0.956284i	$-0.594467\pi$
$79$	−5.19850	−0.584877	−0.292438	+	0.956284i	$0.594467\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−9.08250	−1.00299
$83$	5.65878	0.621132	0.310566	−	0.950552i	$-0.399481\pi$
$83$	5.65878	0.621132	0.310566	+	0.950552i	$0.399481\pi$
$84$	0.836535	0.0912735
$85$	0	0
$86$	−10.0000	−1.07833
$87$	−9.73975	−1.04421
$88$	−1.90813	−0.203408
$89$	−1.93332	−0.204932	−0.102466	−	0.994737i	$-0.532673\pi$
$89$	−1.93332	−0.204932	−0.102466	+	0.994737i	$0.532673\pi$
$90$	0	0
$91$	−0.341219	−0.0357695
$92$	3.62714	0.378155
$93$	1.00000	0.103695
$94$	−13.9224	−1.43599
$95$	0	0
$96$	−2.88739	−0.294693
$97$	−6.91750	−0.702366	−0.351183	−	0.936307i	$-0.614220\pi$
$97$	−6.91750	−0.702366	−0.351183	+	0.936307i	$0.614220\pi$
$98$	−5.42219	−0.547723
$99$	0.622216	0.0625350
$100$	0	0
$101$	−9.28592	−0.923983	−0.461992	−	0.886884i	$-0.652865\pi$
$101$	−9.28592	−0.923983	−0.461992	+	0.886884i	$0.652865\pi$
$102$	4.28100	0.423882
$103$	−0.274543	−0.0270515	−0.0135258	−	0.999909i	$-0.504306\pi$
$103$	−0.274543	−0.0270515	−0.0135258	+	0.999909i	$0.504306\pi$
$104$	0.657249	0.0644486
$105$	0	0
$106$	−16.7096	−1.62298
$107$	8.18913	0.791673	0.395837	−	0.918321i	$-0.370455\pi$
$107$	8.18913	0.791673	0.395837	+	0.918321i	$0.370455\pi$
$108$	0.525428	0.0505593
$109$	−9.13828	−0.875288	−0.437644	−	0.899148i	$-0.644187\pi$
$109$	−9.13828	−0.875288	−0.437644	+	0.899148i	$0.644187\pi$
$110$	0	0
$111$	−4.83654	−0.459064
$112$	−4.25581	−0.402136
$113$	−7.67307	−0.721822	−0.360911	−	0.932600i	$-0.617534\pi$
$113$	−7.67307	−0.721822	−0.360911	+	0.932600i	$0.617534\pi$
$114$	−2.19358	−0.205447
$115$	0	0
$116$	−5.11753	−0.475151
$117$	−0.214320	−0.0198139
$118$	−5.17331	−0.476242
$119$	−5.61285	−0.514529
$120$	0	0
$121$	−10.6128	−0.964804
$122$	3.46965	0.314127
$123$	7.47949	0.674403
$124$	0.525428	0.0471848
$125$	0	0
$126$	1.93332	0.172234
$127$	−13.6731	−1.21329	−0.606644	−	0.794973i	$-0.707485\pi$
$127$	−13.6731	−1.21329	−0.606644	+	0.794973i	$0.707485\pi$
$128$	4.97481	0.439715
$129$	8.23506	0.725057
$130$	0	0
$131$	−22.2701	−1.94575	−0.972874	−	0.231337i	$-0.925690\pi$
$131$	−22.2701	−1.94575	−0.972874	+	0.231337i	$0.925690\pi$
$132$	0.326929	0.0284555
$133$	2.87601	0.249382
$134$	2.52696	0.218296
$135$	0	0
$136$	10.8113	0.927065
$137$	18.1891	1.55400	0.777001	−	0.629499i	$-0.216740\pi$
$137$	18.1891	1.55400	0.777001	+	0.629499i	$0.216740\pi$
$138$	8.38271	0.713583
$139$	−4.13335	−0.350586	−0.175293	−	0.984516i	$-0.556087\pi$
$139$	−4.13335	−0.350586	−0.175293	+	0.984516i	$0.556087\pi$
$140$	0	0
$141$	11.4652	0.965544
$142$	1.59210	0.133606
$143$	−0.133353	−0.0111515
$144$	−2.67307	−0.222756
$145$	0	0
$146$	2.00645	0.166055
$147$	4.46520	0.368284
$148$	−2.54125	−0.208889
$149$	6.23506	0.510796	0.255398	−	0.966836i	$-0.417793\pi$
$149$	6.23506	0.510796	0.255398	+	0.966836i	$0.417793\pi$
$150$	0	0
$151$	8.38271	0.682175	0.341087	−	0.940032i	$-0.389205\pi$
$151$	8.38271	0.682175	0.341087	+	0.940032i	$0.389205\pi$
$152$	−5.53972	−0.449330
$153$	−3.52543	−0.285014
$154$	1.20294	0.0969360
$155$	0	0
$156$	−0.112610	−0.00901598
$157$	10.2351	0.816847	0.408423	−	0.912793i	$-0.366079\pi$
$157$	10.2351	0.816847	0.408423	+	0.912793i	$0.366079\pi$
$158$	−6.31264	−0.502207
$159$	13.7605	1.09128
$160$	0	0
$161$	−10.9906	−0.866183
$162$	1.21432	0.0954060
$163$	−5.82717	−0.456419	−0.228209	−	0.973612i	$-0.573287\pi$
$163$	−5.82717	−0.456419	−0.228209	+	0.973612i	$0.573287\pi$
$164$	3.92993	0.306876
$165$	0	0
$166$	6.87157	0.533337
$167$	−23.1842	−1.79405	−0.897024	−	0.441982i	$-0.854276\pi$
$167$	−23.1842	−1.79405	−0.897024	+	0.441982i	$0.854276\pi$
$168$	4.88247	0.376691
$169$	−12.9541	−0.996467
$170$	0	0
$171$	1.80642	0.138141
$172$	4.32693	0.329925
$173$	−1.57136	−0.119468	−0.0597342	−	0.998214i	$-0.519025\pi$
$173$	−1.57136	−0.119468	−0.0597342	+	0.998214i	$0.519025\pi$
$174$	−11.8272	−0.896615
$175$	0	0
$176$	−1.66323	−0.125370
$177$	4.26025	0.320220
$178$	−2.34767	−0.175966
$179$	1.53972	0.115084	0.0575420	−	0.998343i	$-0.481674\pi$
$179$	1.53972	0.115084	0.0575420	+	0.998343i	$0.481674\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−0.414349	−0.0307136
$183$	−2.85728	−0.211216
$184$	21.1699	1.56067
$185$	0	0
$186$	1.21432	0.0890382
$187$	−2.19358	−0.160410
$188$	6.02413	0.439355
$189$	−1.59210	−0.115809
$190$	0	0
$191$	9.11753	0.659721	0.329861	−	0.944030i	$-0.392998\pi$
$191$	9.11753	0.659721	0.329861	+	0.944030i	$0.392998\pi$
$192$	−8.85236	−0.638864
$193$	4.72393	0.340036	0.170018	−	0.985441i	$-0.445617\pi$
$193$	4.72393	0.340036	0.170018	+	0.985441i	$0.445617\pi$
$194$	−8.40006	−0.603089
$195$	0	0
$196$	2.34614	0.167582
$197$	−15.7003	−1.11860	−0.559299	−	0.828966i	$-0.688930\pi$
$197$	−15.7003	−1.11860	−0.559299	+	0.828966i	$0.688930\pi$
$198$	0.755569	0.0536959
$199$	6.28100	0.445248	0.222624	−	0.974904i	$-0.428538\pi$
$199$	6.28100	0.445248	0.222624	+	0.974904i	$0.428538\pi$
$200$	0	0
$201$	−2.08097	−0.146780
$202$	−11.2761	−0.793382
$203$	15.5067	1.08836
$204$	−1.85236	−0.129691
$205$	0	0
$206$	−0.333383	−0.0232279
$207$	−6.90321	−0.479806
$208$	0.572892	0.0397229
$209$	1.12399	0.0777477
$210$	0	0
$211$	19.4795	1.34102	0.670512	−	0.741899i	$-0.266075\pi$
$211$	19.4795	1.34102	0.670512	+	0.741899i	$0.266075\pi$
$212$	7.23014	0.496568
$213$	−1.31111	−0.0898356
$214$	9.94422	0.679773
$215$	0	0
$216$	3.06668	0.208661
$217$	−1.59210	−0.108079
$218$	−11.0968	−0.751569
$219$	−1.65233	−0.111654
$220$	0	0
$221$	0.755569	0.0508251
$222$	−5.87310	−0.394177
$223$	15.4193	1.03255	0.516275	−	0.856423i	$-0.327318\pi$
$223$	15.4193	1.03255	0.516275	+	0.856423i	$0.327318\pi$
$224$	4.59703	0.307152
$225$	0	0
$226$	−9.31756	−0.619795
$227$	12.8430	0.852419	0.426210	−	0.904624i	$-0.359849\pi$
$227$	12.8430	0.852419	0.426210	+	0.904624i	$0.359849\pi$
$228$	0.949145	0.0628587
$229$	−3.08250	−0.203697	−0.101849	−	0.994800i	$-0.532476\pi$
$229$	−3.08250	−0.203697	−0.101849	+	0.994800i	$0.532476\pi$
$230$	0	0
$231$	−0.990632	−0.0651788
$232$	−29.8687	−1.96097
$233$	−8.76986	−0.574533	−0.287266	−	0.957851i	$-0.592746\pi$
$233$	−8.76986	−0.574533	−0.287266	+	0.957851i	$0.592746\pi$
$234$	−0.260253	−0.0170133
$235$	0	0
$236$	2.23845	0.145711
$237$	5.19850	0.337679
$238$	−6.81579	−0.441802
$239$	3.12399	0.202074	0.101037	−	0.994883i	$-0.467784\pi$
$239$	3.12399	0.202074	0.101037	+	0.994883i	$0.467784\pi$
$240$	0	0
$241$	29.5625	1.90429	0.952143	−	0.305653i	$-0.0988747\pi$
$241$	29.5625	1.90429	0.952143	+	0.305653i	$0.0988747\pi$
$242$	−12.8874	−0.828433
$243$	−1.00000	−0.0641500
$244$	−1.50129	−0.0961104
$245$	0	0
$246$	9.08250	0.579079
$247$	−0.387152	−0.0246339
$248$	3.06668	0.194734
$249$	−5.65878	−0.358611
$250$	0	0
$251$	21.4193	1.35197	0.675986	−	0.736914i	$-0.263718\pi$
$251$	21.4193	1.35197	0.675986	+	0.736914i	$0.263718\pi$
$252$	−0.836535	−0.0526968
$253$	−4.29529	−0.270042
$254$	−16.6035	−1.04179
$255$	0	0
$256$	−11.6637	−0.728981
$257$	17.6271	1.09955	0.549775	−	0.835313i	$-0.314713\pi$
$257$	17.6271	1.09955	0.549775	+	0.835313i	$0.314713\pi$
$258$	10.0000	0.622573
$259$	7.70027	0.478471
$260$	0	0
$261$	9.73975	0.602875
$262$	−27.0430	−1.67072
$263$	14.7052	0.906761	0.453380	−	0.891317i	$-0.350218\pi$
$263$	14.7052	0.906761	0.453380	+	0.891317i	$0.350218\pi$
$264$	1.90813	0.117438
$265$	0	0
$266$	3.49240	0.214133
$267$	1.93332	0.118317
$268$	−1.09340	−0.0667899
$269$	1.35260	0.0824692	0.0412346	−	0.999149i	$-0.486871\pi$
$269$	1.35260	0.0824692	0.0412346	+	0.999149i	$0.486871\pi$
$270$	0	0
$271$	12.1334	0.737049	0.368524	−	0.929618i	$-0.379863\pi$
$271$	12.1334	0.737049	0.368524	+	0.929618i	$0.379863\pi$
$272$	9.42372	0.571397
$273$	0.341219	0.0206515
$274$	22.0874	1.33435
$275$	0	0
$276$	−3.62714	−0.218328
$277$	−18.9382	−1.13789	−0.568944	−	0.822376i	$-0.692648\pi$
$277$	−18.9382	−1.13789	−0.568944	+	0.822376i	$0.692648\pi$
$278$	−5.01921	−0.301032
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	12.9175	0.770594	0.385297	−	0.922793i	$-0.374099\pi$
$281$	12.9175	0.770594	0.385297	+	0.922793i	$0.374099\pi$
$282$	13.9224	0.829068
$283$	−26.5827	−1.58018	−0.790090	−	0.612991i	$-0.789966\pi$
$283$	−26.5827	−1.58018	−0.790090	+	0.612991i	$0.789966\pi$
$284$	−0.688892	−0.0408782
$285$	0	0
$286$	−0.161933	−0.00957532
$287$	−11.9081	−0.702915
$288$	2.88739	0.170141
$289$	−4.57136	−0.268904
$290$	0	0
$291$	6.91750	0.405511
$292$	−0.868178	−0.0508063
$293$	7.99555	0.467105	0.233553	−	0.972344i	$-0.424965\pi$
$293$	7.99555	0.467105	0.233553	+	0.972344i	$0.424965\pi$
$294$	5.42219	0.316228
$295$	0	0
$296$	−14.8321	−0.862098
$297$	−0.622216	−0.0361046
$298$	7.57136	0.438597
$299$	1.47949	0.0855614
$300$	0	0
$301$	−13.1111	−0.755710
$302$	10.1793	0.585752
$303$	9.28592	0.533462
$304$	−4.82870	−0.276945
$305$	0	0
$306$	−4.28100	−0.244728
$307$	17.3669	0.991180	0.495590	−	0.868556i	$-0.334952\pi$
$307$	17.3669	0.991180	0.495590	+	0.868556i	$0.334952\pi$
$308$	−0.520505	−0.0296585
$309$	0.274543	0.0156182
$310$	0	0
$311$	−27.5877	−1.56435	−0.782176	−	0.623057i	$-0.785890\pi$
$311$	−27.5877	−1.56435	−0.782176	+	0.623057i	$0.785890\pi$
$312$	−0.657249	−0.0372094
$313$	13.7255	0.775809	0.387904	−	0.921700i	$-0.373199\pi$
$313$	13.7255	0.775809	0.387904	+	0.921700i	$0.373199\pi$
$314$	12.4286	0.701389
$315$	0	0
$316$	2.73143	0.153655
$317$	24.9447	1.40103	0.700517	−	0.713636i	$-0.252953\pi$
$317$	24.9447	1.40103	0.700517	+	0.713636i	$0.252953\pi$
$318$	16.7096	0.937030
$319$	6.06022	0.339307
$320$	0	0
$321$	−8.18913	−0.457073
$322$	−13.3461	−0.743751
$323$	−6.36842	−0.354348
$324$	−0.525428	−0.0291904
$325$	0	0
$326$	−7.07604	−0.391906
$327$	9.13828	0.505348
$328$	22.9372	1.26649
$329$	−18.2538	−1.00636
$330$	0	0
$331$	19.1798	1.05422	0.527108	−	0.849799i	$-0.323276\pi$
$331$	19.1798	1.05422	0.527108	+	0.849799i	$0.323276\pi$
$332$	−2.97328	−0.163180
$333$	4.83654	0.265041
$334$	−28.1530	−1.54047
$335$	0	0
$336$	4.25581	0.232173
$337$	25.3067	1.37854	0.689271	−	0.724504i	$-0.257931\pi$
$337$	25.3067	1.37854	0.689271	+	0.724504i	$0.257931\pi$
$338$	−15.7304	−0.855620
$339$	7.67307	0.416744
$340$	0	0
$341$	−0.622216	−0.0336949
$342$	2.19358	0.118615
$343$	−18.2538	−0.985613
$344$	25.2543	1.36162
$345$	0	0
$346$	−1.90813	−0.102582
$347$	−19.0638	−1.02340	−0.511698	−	0.859165i	$-0.670983\pi$
$347$	−19.0638	−1.02340	−0.511698	+	0.859165i	$0.670983\pi$
$348$	5.11753	0.274328
$349$	−1.23014	−0.0658479	−0.0329240	−	0.999458i	$-0.510482\pi$
$349$	−1.23014	−0.0658479	−0.0329240	+	0.999458i	$0.510482\pi$
$350$	0	0
$351$	0.214320	0.0114395
$352$	1.79658	0.0957580
$353$	−4.50315	−0.239679	−0.119839	−	0.992793i	$-0.538238\pi$
$353$	−4.50315	−0.239679	−0.119839	+	0.992793i	$0.538238\pi$
$354$	5.17331	0.274958
$355$	0	0
$356$	1.01582	0.0538384
$357$	5.61285	0.297063
$358$	1.86971	0.0988172
$359$	18.9240	0.998768	0.499384	−	0.866381i	$-0.333560\pi$
$359$	18.9240	0.998768	0.499384	+	0.866381i	$0.333560\pi$
$360$	0	0
$361$	−15.7368	−0.828254
$362$	2.42864	0.127646
$363$	10.6128	0.557030
$364$	0.179286	0.00939714
$365$	0	0
$366$	−3.46965	−0.181362
$367$	9.71456	0.507096	0.253548	−	0.967323i	$-0.418402\pi$
$367$	9.71456	0.507096	0.253548	+	0.967323i	$0.418402\pi$
$368$	18.4528	0.961917
$369$	−7.47949	−0.389367
$370$	0	0
$371$	−21.9081	−1.13741
$372$	−0.525428	−0.0272421
$373$	35.1941	1.82228	0.911139	−	0.412098i	$-0.135204\pi$
$373$	35.1941	1.82228	0.911139	+	0.412098i	$0.135204\pi$
$374$	−2.66370	−0.137737
$375$	0	0
$376$	35.1601	1.81324
$377$	−2.08742	−0.107508
$378$	−1.93332	−0.0994394
$379$	−22.6637	−1.16416	−0.582078	−	0.813133i	$-0.697760\pi$
$379$	−22.6637	−1.16416	−0.582078	+	0.813133i	$0.697760\pi$
$380$	0	0
$381$	13.6731	0.700493
$382$	11.0716	0.566472
$383$	−28.7926	−1.47123	−0.735617	−	0.677398i	$-0.763108\pi$
$383$	−28.7926	−1.47123	−0.735617	+	0.677398i	$0.763108\pi$
$384$	−4.97481	−0.253870
$385$	0	0
$386$	5.73636	0.291973
$387$	−8.23506	−0.418612
$388$	3.63465	0.184521
$389$	−25.0672	−1.27096	−0.635478	−	0.772119i	$-0.719197\pi$
$389$	−25.0672	−1.27096	−0.635478	+	0.772119i	$0.719197\pi$
$390$	0	0
$391$	24.3368	1.23076
$392$	13.6933	0.691618
$393$	22.2701	1.12338
$394$	−19.0651	−0.960488
$395$	0	0
$396$	−0.326929	−0.0164288
$397$	29.0509	1.45802	0.729010	−	0.684503i	$-0.239981\pi$
$397$	29.0509	1.45802	0.729010	+	0.684503i	$0.239981\pi$
$398$	7.62714	0.382314
$399$	−2.87601	−0.143981
$400$	0	0
$401$	11.2607	0.562334	0.281167	−	0.959659i	$-0.409279\pi$
$401$	11.2607	0.562334	0.281167	+	0.959659i	$0.409279\pi$
$402$	−2.52696	−0.126033
$403$	0.214320	0.0106760
$404$	4.87908	0.242743
$405$	0	0
$406$	18.8301	0.934521
$407$	3.00937	0.149169
$408$	−10.8113	−0.535241
$409$	−25.6128	−1.26647	−0.633237	−	0.773958i	$-0.718274\pi$
$409$	−25.6128	−1.26647	−0.633237	+	0.773958i	$0.718274\pi$
$410$	0	0
$411$	−18.1891	−0.897204
$412$	0.144252	0.00710680
$413$	−6.78277	−0.333758
$414$	−8.38271	−0.411988
$415$	0	0
$416$	−0.618825	−0.0303404
$417$	4.13335	0.202411
$418$	1.36488	0.0667583
$419$	−4.36196	−0.213096	−0.106548	−	0.994308i	$-0.533980\pi$
$419$	−4.36196	−0.213096	−0.106548	+	0.994308i	$0.533980\pi$
$420$	0	0
$421$	5.77923	0.281662	0.140831	−	0.990034i	$-0.455023\pi$
$421$	5.77923	0.281662	0.140831	+	0.990034i	$0.455023\pi$
$422$	23.6543	1.15148
$423$	−11.4652	−0.557457
$424$	42.1990	2.04936
$425$	0	0
$426$	−1.59210	−0.0771377
$427$	4.54909	0.220146
$428$	−4.30279	−0.207983
$429$	0.133353	0.00643835
$430$	0	0
$431$	−4.62867	−0.222955	−0.111478	−	0.993767i	$-0.535558\pi$
$431$	−4.62867	−0.222955	−0.111478	+	0.993767i	$0.535558\pi$
$432$	2.67307	0.128608
$433$	−6.81780	−0.327643	−0.163821	−	0.986490i	$-0.552382\pi$
$433$	−6.81780	−0.327643	−0.163821	+	0.986490i	$0.552382\pi$
$434$	−1.93332	−0.0928025
$435$	0	0
$436$	4.80150	0.229950
$437$	−12.4701	−0.596527
$438$	−2.00645	−0.0958721
$439$	9.71456	0.463651	0.231825	−	0.972757i	$-0.425530\pi$
$439$	9.71456	0.463651	0.231825	+	0.972757i	$0.425530\pi$
$440$	0	0
$441$	−4.46520	−0.212629
$442$	0.917502	0.0436411
$443$	−31.1481	−1.47989	−0.739946	−	0.672666i	$-0.765149\pi$
$443$	−31.1481	−1.47989	−0.739946	+	0.672666i	$0.765149\pi$
$444$	2.54125	0.120602
$445$	0	0
$446$	18.7239	0.886604
$447$	−6.23506	−0.294908
$448$	14.0939	0.665873
$449$	28.1180	1.32697	0.663485	−	0.748189i	$-0.269076\pi$
$449$	28.1180	1.32697	0.663485	+	0.748189i	$0.269076\pi$
$450$	0	0
$451$	−4.65386	−0.219142
$452$	4.03164	0.189633
$453$	−8.38271	−0.393854
$454$	15.5955	0.731933
$455$	0	0
$456$	5.53972	0.259421
$457$	41.4400	1.93848	0.969241	−	0.246113i	$-0.0791535\pi$
$457$	41.4400	1.93848	0.969241	+	0.246113i	$0.0791535\pi$
$458$	−3.74314	−0.174905
$459$	3.52543	0.164553
$460$	0	0
$461$	10.3432	0.481732	0.240866	−	0.970558i	$-0.422569\pi$
$461$	10.3432	0.481732	0.240866	+	0.970558i	$0.422569\pi$
$462$	−1.20294	−0.0559660
$463$	−13.3047	−0.618320	−0.309160	−	0.951010i	$-0.600048\pi$
$463$	−13.3047	−0.618320	−0.309160	+	0.951010i	$0.600048\pi$
$464$	−26.0350	−1.20865
$465$	0	0
$466$	−10.6494	−0.493325
$467$	6.22077	0.287863	0.143932	−	0.989588i	$-0.454025\pi$
$467$	6.22077	0.287863	0.143932	+	0.989588i	$0.454025\pi$
$468$	0.112610	0.00520538
$469$	3.31312	0.152985
$470$	0	0
$471$	−10.2351	−0.471607
$472$	13.0648	0.601357
$473$	−5.12399	−0.235601
$474$	6.31264	0.289949
$475$	0	0
$476$	2.94914	0.135174
$477$	−13.7605	−0.630050
$478$	3.79352	0.173511
$479$	−24.8825	−1.13691	−0.568454	−	0.822715i	$-0.692458\pi$
$479$	−24.8825	−1.13691	−0.568454	+	0.822715i	$0.692458\pi$
$480$	0	0
$481$	−1.03657	−0.0472633
$482$	35.8983	1.63512
$483$	10.9906	0.500091
$484$	5.57628	0.253467
$485$	0	0
$486$	−1.21432	−0.0550827
$487$	9.84791	0.446251	0.223126	−	0.974790i	$-0.428374\pi$
$487$	9.84791	0.446251	0.223126	+	0.974790i	$0.428374\pi$
$488$	−8.76235	−0.396653
$489$	5.82717	0.263514
$490$	0	0
$491$	−9.84791	−0.444430	−0.222215	−	0.974998i	$-0.571329\pi$
$491$	−9.84791	−0.444430	−0.222215	+	0.974998i	$0.571329\pi$
$492$	−3.92993	−0.177175
$493$	−34.3368	−1.54645
$494$	−0.470127	−0.0211520
$495$	0	0
$496$	2.67307	0.120024
$497$	2.08742	0.0936336
$498$	−6.87157	−0.307922
$499$	0.653858	0.0292707	0.0146354	−	0.999893i	$-0.495341\pi$
$499$	0.653858	0.0292707	0.0146354	+	0.999893i	$0.495341\pi$
$500$	0	0
$501$	23.1842	1.03579
$502$	26.0098	1.16088
$503$	−13.9541	−0.622181	−0.311091	−	0.950380i	$-0.600694\pi$
$503$	−13.9541	−0.622181	−0.311091	+	0.950380i	$0.600694\pi$
$504$	−4.88247	−0.217482
$505$	0	0
$506$	−5.21585	−0.231873
$507$	12.9541	0.575310
$508$	7.18421	0.318748
$509$	−42.7116	−1.89316	−0.946580	−	0.322469i	$-0.895487\pi$
$509$	−42.7116	−1.89316	−0.946580	+	0.322469i	$0.895487\pi$
$510$	0	0
$511$	2.63068	0.116374
$512$	−24.1131	−1.06566
$513$	−1.80642	−0.0797556
$514$	21.4050	0.944133
$515$	0	0
$516$	−4.32693	−0.190482
$517$	−7.13383	−0.313745
$518$	9.35059	0.410841
$519$	1.57136	0.0689751
$520$	0	0
$521$	−12.6035	−0.552168	−0.276084	−	0.961133i	$-0.589037\pi$
$521$	−12.6035	−0.552168	−0.276084	+	0.961133i	$0.589037\pi$
$522$	11.8272	0.517661
$523$	29.8479	1.30516	0.652579	−	0.757721i	$-0.273687\pi$
$523$	29.8479	1.30516	0.652579	+	0.757721i	$0.273687\pi$
$524$	11.7013	0.511175
$525$	0	0
$526$	17.8568	0.778594
$527$	3.52543	0.153570
$528$	1.66323	0.0723826
$529$	24.6543	1.07193
$530$	0	0
$531$	−4.26025	−0.184879
$532$	−1.51114	−0.0655161
$533$	1.60300	0.0694338
$534$	2.34767	0.101594
$535$	0	0
$536$	−6.38165	−0.275645
$537$	−1.53972	−0.0664437
$538$	1.64248	0.0708125
$539$	−2.77832	−0.119671
$540$	0	0
$541$	−13.9541	−0.599932	−0.299966	−	0.953950i	$-0.596975\pi$
$541$	−13.9541	−0.599932	−0.299966	+	0.953950i	$0.596975\pi$
$542$	14.7338	0.632870
$543$	−2.00000	−0.0858282
$544$	−10.1793	−0.436433
$545$	0	0
$546$	0.414349	0.0177325
$547$	−5.74419	−0.245604	−0.122802	−	0.992431i	$-0.539188\pi$
$547$	−5.74419	−0.245604	−0.122802	+	0.992431i	$0.539188\pi$
$548$	−9.55707	−0.408258
$549$	2.85728	0.121946
$550$	0	0
$551$	17.5941	0.749534
$552$	−21.1699	−0.901052
$553$	−8.27655	−0.351955
$554$	−22.9971	−0.977053
$555$	0	0
$556$	2.17178	0.0921039
$557$	14.7511	0.625025	0.312513	−	0.949914i	$-0.398829\pi$
$557$	14.7511	0.625025	0.312513	+	0.949914i	$0.398829\pi$
$558$	−1.21432	−0.0514063
$559$	1.76494	0.0746489
$560$	0	0
$561$	2.19358	0.0926129
$562$	15.6860	0.661673
$563$	−32.9260	−1.38766	−0.693832	−	0.720137i	$-0.744079\pi$
$563$	−32.9260	−1.38766	−0.693832	+	0.720137i	$0.744079\pi$
$564$	−6.02413	−0.253662
$565$	0	0
$566$	−32.2799	−1.35683
$567$	1.59210	0.0668621
$568$	−4.02074	−0.168707
$569$	−3.07604	−0.128954	−0.0644772	−	0.997919i	$-0.520538\pi$
$569$	−3.07604	−0.128954	−0.0644772	+	0.997919i	$0.520538\pi$
$570$	0	0
$571$	4.52051	0.189177	0.0945886	−	0.995516i	$-0.469846\pi$
$571$	4.52051	0.189177	0.0945886	+	0.995516i	$0.469846\pi$
$572$	0.0700674	0.00292966
$573$	−9.11753	−0.380890
$574$	−14.4603	−0.603561
$575$	0	0
$576$	8.85236	0.368848
$577$	30.6923	1.27774	0.638868	−	0.769316i	$-0.279403\pi$
$577$	30.6923	1.27774	0.638868	+	0.769316i	$0.279403\pi$
$578$	−5.55109	−0.230895
$579$	−4.72393	−0.196320
$580$	0	0
$581$	9.00937	0.373772
$582$	8.40006	0.348194
$583$	−8.56199	−0.354602
$584$	−5.06715	−0.209680
$585$	0	0
$586$	9.70916	0.401082
$587$	−14.8256	−0.611919	−0.305960	−	0.952044i	$-0.598977\pi$
$587$	−14.8256	−0.611919	−0.305960	+	0.952044i	$0.598977\pi$
$588$	−2.34614	−0.0967532
$589$	−1.80642	−0.0744324
$590$	0	0
$591$	15.7003	0.645823
$592$	−12.9284	−0.531354
$593$	−23.9684	−0.984262	−0.492131	−	0.870521i	$-0.663782\pi$
$593$	−23.9684	−0.984262	−0.492131	+	0.870521i	$0.663782\pi$
$594$	−0.755569	−0.0310014
$595$	0	0
$596$	−3.27607	−0.134193
$597$	−6.28100	−0.257064
$598$	1.79658	0.0734676
$599$	−39.1086	−1.59794	−0.798968	−	0.601374i	$-0.794620\pi$
$599$	−39.1086	−1.59794	−0.798968	+	0.601374i	$0.794620\pi$
$600$	0	0
$601$	22.3082	0.909970	0.454985	−	0.890499i	$-0.349645\pi$
$601$	22.3082	0.909970	0.454985	+	0.890499i	$0.349645\pi$
$602$	−15.9210	−0.648893
$603$	2.08097	0.0847435
$604$	−4.40451	−0.179217
$605$	0	0
$606$	11.2761	0.458059
$607$	−2.82669	−0.114732	−0.0573659	−	0.998353i	$-0.518270\pi$
$607$	−2.82669	−0.114732	−0.0573659	+	0.998353i	$0.518270\pi$
$608$	5.21585	0.211531
$609$	−15.5067	−0.628363
$610$	0	0
$611$	2.45722	0.0994085
$612$	1.85236	0.0748771
$613$	36.6844	1.48167	0.740835	−	0.671687i	$-0.234430\pi$
$613$	36.6844	1.48167	0.740835	+	0.671687i	$0.234430\pi$
$614$	21.0890	0.851081
$615$	0	0
$616$	−3.03795	−0.122402
$617$	−1.46659	−0.0590426	−0.0295213	−	0.999564i	$-0.509398\pi$
$617$	−1.46659	−0.0590426	−0.0295213	+	0.999564i	$0.509398\pi$
$618$	0.333383	0.0134106
$619$	−15.9541	−0.641248	−0.320624	−	0.947207i	$-0.603893\pi$
$619$	−15.9541	−0.641248	−0.320624	+	0.947207i	$0.603893\pi$
$620$	0	0
$621$	6.90321	0.277016
$622$	−33.5002	−1.34324
$623$	−3.07805	−0.123320
$624$	−0.572892	−0.0229340
$625$	0	0
$626$	16.6671	0.666151
$627$	−1.12399	−0.0448876
$628$	−5.37778	−0.214597
$629$	−17.0509	−0.679862
$630$	0	0
$631$	−32.0830	−1.27720	−0.638602	−	0.769538i	$-0.720487\pi$
$631$	−32.0830	−1.27720	−0.638602	+	0.769538i	$0.720487\pi$
$632$	15.9421	0.634143
$633$	−19.4795	−0.774240
$634$	30.2908	1.20300
$635$	0	0
$636$	−7.23014	−0.286694
$637$	0.956981	0.0379170
$638$	7.35905	0.291348
$639$	1.31111	0.0518666
$640$	0	0
$641$	−14.8923	−0.588211	−0.294105	−	0.955773i	$-0.595022\pi$
$641$	−14.8923	−0.588211	−0.294105	+	0.955773i	$0.595022\pi$
$642$	−9.94422	−0.392467
$643$	−12.5205	−0.493761	−0.246880	−	0.969046i	$-0.579405\pi$
$643$	−12.5205	−0.493761	−0.246880	+	0.969046i	$0.579405\pi$
$644$	5.77478	0.227558
$645$	0	0
$646$	−7.73329	−0.304262
$647$	−45.6227	−1.79361	−0.896807	−	0.442423i	$-0.854119\pi$
$647$	−45.6227	−1.79361	−0.896807	+	0.442423i	$0.854119\pi$
$648$	−3.06668	−0.120470
$649$	−2.65080	−0.104053
$650$	0	0
$651$	1.59210	0.0623995
$652$	3.06175	0.119908
$653$	18.7797	0.734907	0.367453	−	0.930042i	$-0.380230\pi$
$653$	18.7797	0.734907	0.367453	+	0.930042i	$0.380230\pi$
$654$	11.0968	0.433919
$655$	0	0
$656$	19.9932	0.780604
$657$	1.65233	0.0644634
$658$	−22.1659	−0.864119
$659$	−28.9240	−1.12672	−0.563359	−	0.826212i	$-0.690491\pi$
$659$	−28.9240	−1.12672	−0.563359	+	0.826212i	$0.690491\pi$
$660$	0	0
$661$	−24.3269	−0.946208	−0.473104	−	0.881007i	$-0.656866\pi$
$661$	−24.3269	−0.946208	−0.473104	+	0.881007i	$0.656866\pi$
$662$	23.2904	0.905206
$663$	−0.755569	−0.0293439
$664$	−17.3536	−0.673452
$665$	0	0
$666$	5.87310	0.227578
$667$	−67.2355	−2.60337
$668$	12.1816	0.471321
$669$	−15.4193	−0.596143
$670$	0	0
$671$	1.77784	0.0686329
$672$	−4.59703	−0.177334
$673$	36.5511	1.40894	0.704471	−	0.709733i	$-0.251185\pi$
$673$	36.5511	1.40894	0.704471	+	0.709733i	$0.251185\pi$
$674$	30.7304	1.18369
$675$	0	0
$676$	6.80642	0.261786
$677$	32.1575	1.23591	0.617956	−	0.786212i	$-0.287961\pi$
$677$	32.1575	1.23591	0.617956	+	0.786212i	$0.287961\pi$
$678$	9.31756	0.357839
$679$	−11.0134	−0.422655
$680$	0	0
$681$	−12.8430	−0.492144
$682$	−0.755569	−0.0289322
$683$	38.2623	1.46406	0.732032	−	0.681270i	$-0.238572\pi$
$683$	38.2623	1.46406	0.732032	+	0.681270i	$0.238572\pi$
$684$	−0.949145	−0.0362915
$685$	0	0
$686$	−22.1659	−0.846300
$687$	3.08250	0.117605
$688$	22.0129	0.839234
$689$	2.94914	0.112353
$690$	0	0
$691$	−12.3082	−0.468226	−0.234113	−	0.972209i	$-0.575219\pi$
$691$	−12.3082	−0.468226	−0.234113	+	0.972209i	$0.575219\pi$
$692$	0.825636	0.0313860
$693$	0.990632	0.0376310
$694$	−23.1495	−0.878743
$695$	0	0
$696$	29.8687	1.13217
$697$	26.3684	0.998775
$698$	−1.49378	−0.0565406
$699$	8.76986	0.331707
$700$	0	0
$701$	31.5210	1.19053	0.595266	−	0.803529i	$-0.297047\pi$
$701$	31.5210	1.19053	0.595266	+	0.803529i	$0.297047\pi$
$702$	0.260253	0.00982261
$703$	8.73683	0.329516
$704$	5.50807	0.207593
$705$	0	0
$706$	−5.46827	−0.205801
$707$	−14.7841	−0.556015
$708$	−2.23845	−0.0841263
$709$	−7.51114	−0.282087	−0.141043	−	0.990003i	$-0.545046\pi$
$709$	−7.51114	−0.282087	−0.141043	+	0.990003i	$0.545046\pi$
$710$	0	0
$711$	−5.19850	−0.194959
$712$	5.92888	0.222194
$713$	6.90321	0.258527
$714$	6.81579	0.255075
$715$	0	0
$716$	−0.809010	−0.0302341
$717$	−3.12399	−0.116667
$718$	22.9797	0.857596
$719$	32.3180	1.20526	0.602630	−	0.798021i	$-0.294120\pi$
$719$	32.3180	1.20526	0.602630	+	0.798021i	$0.294120\pi$
$720$	0	0
$721$	−0.437101	−0.0162785
$722$	−19.1095	−0.711184
$723$	−29.5625	−1.09944
$724$	−1.05086	−0.0390547
$725$	0	0
$726$	12.8874	0.478296
$727$	−6.12245	−0.227069	−0.113535	−	0.993534i	$-0.536217\pi$
$727$	−6.12245	−0.227069	−0.113535	+	0.993534i	$0.536217\pi$
$728$	1.04641	0.0387825
$729$	1.00000	0.0370370
$730$	0	0
$731$	29.0321	1.07379
$732$	1.50129	0.0554894
$733$	30.9403	1.14280	0.571402	−	0.820670i	$-0.306400\pi$
$733$	30.9403	1.14280	0.571402	+	0.820670i	$0.306400\pi$
$734$	11.7966	0.435420
$735$	0	0
$736$	−19.9323	−0.734713
$737$	1.29481	0.0476949
$738$	−9.08250	−0.334331
$739$	−12.6780	−0.466368	−0.233184	−	0.972433i	$-0.574914\pi$
$739$	−12.6780	−0.466368	−0.233184	+	0.972433i	$0.574914\pi$
$740$	0	0
$741$	0.387152	0.0142224
$742$	−26.6035	−0.976644
$743$	−38.4385	−1.41017	−0.705086	−	0.709122i	$-0.749091\pi$
$743$	−38.4385	−1.41017	−0.705086	+	0.709122i	$0.749091\pi$
$744$	−3.06668	−0.112430
$745$	0	0
$746$	42.7368	1.56471
$747$	5.65878	0.207044
$748$	1.15257	0.0421420
$749$	13.0379	0.476396
$750$	0	0
$751$	−13.4608	−0.491190	−0.245595	−	0.969373i	$-0.578983\pi$
$751$	−13.4608	−0.491190	−0.245595	+	0.969373i	$0.578983\pi$
$752$	30.6473	1.11759
$753$	−21.4193	−0.780562
$754$	−2.53480	−0.0923118
$755$	0	0
$756$	0.836535	0.0304245
$757$	20.5511	0.746942	0.373471	−	0.927642i	$-0.378168\pi$
$757$	20.5511	0.746942	0.373471	+	0.927642i	$0.378168\pi$
$758$	−27.5210	−0.999607
$759$	4.29529	0.155909
$760$	0	0
$761$	42.3432	1.53494	0.767470	−	0.641084i	$-0.221515\pi$
$761$	42.3432	1.53494	0.767470	+	0.641084i	$0.221515\pi$
$762$	16.6035	0.601481
$763$	−14.5491	−0.526712
$764$	−4.79060	−0.173318
$765$	0	0
$766$	−34.9634	−1.26328
$767$	0.913056	0.0329686
$768$	11.6637	0.420878
$769$	40.5388	1.46187	0.730933	−	0.682449i	$-0.239085\pi$
$769$	40.5388	1.46187	0.730933	+	0.682449i	$0.239085\pi$
$770$	0	0
$771$	−17.6271	−0.634826
$772$	−2.48208	−0.0893320
$773$	−26.2810	−0.945262	−0.472631	−	0.881260i	$-0.656696\pi$
$773$	−26.2810	−0.945262	−0.472631	+	0.881260i	$0.656696\pi$
$774$	−10.0000	−0.359443
$775$	0	0
$776$	21.2137	0.761529
$777$	−7.70027	−0.276246
$778$	−30.4395	−1.09131
$779$	−13.5111	−0.484087
$780$	0	0
$781$	0.815792	0.0291913
$782$	29.5526	1.05680
$783$	−9.73975	−0.348070
$784$	11.9358	0.426279
$785$	0	0
$786$	27.0430	0.964593
$787$	0.755569	0.0269331	0.0134666	−	0.999909i	$-0.495713\pi$
$787$	0.755569	0.0269331	0.0134666	+	0.999909i	$0.495713\pi$
$788$	8.24935	0.293871
$789$	−14.7052	−0.523519
$790$	0	0
$791$	−12.2163	−0.434363
$792$	−1.90813	−0.0678026
$793$	−0.612371	−0.0217459
$794$	35.2770	1.25193
$795$	0	0
$796$	−3.30021	−0.116973
$797$	−1.89384	−0.0670834	−0.0335417	−	0.999437i	$-0.510679\pi$
$797$	−1.89384	−0.0670834	−0.0335417	+	0.999437i	$0.510679\pi$
$798$	−3.49240	−0.123630
$799$	40.4197	1.42995
$800$	0	0
$801$	−1.93332	−0.0683106
$802$	13.6741	0.482850
$803$	1.02810	0.0362810
$804$	1.09340	0.0385611
$805$	0	0
$806$	0.260253	0.00916701
$807$	−1.35260	−0.0476136
$808$	28.4769	1.00181
$809$	−28.6671	−1.00788	−0.503941	−	0.863738i	$-0.668117\pi$
$809$	−28.6671	−1.00788	−0.503941	+	0.863738i	$0.668117\pi$
$810$	0	0
$811$	−38.0228	−1.33516	−0.667580	−	0.744538i	$-0.732670\pi$
$811$	−38.0228	−1.33516	−0.667580	+	0.744538i	$0.732670\pi$
$812$	−8.14764	−0.285926
$813$	−12.1334	−0.425535
$814$	3.65433	0.128084
$815$	0	0
$816$	−9.42372	−0.329896
$817$	−14.8760	−0.520446
$818$	−31.1022	−1.08746
$819$	−0.341219	−0.0119232
$820$	0	0
$821$	−27.2321	−0.950409	−0.475204	−	0.879876i	$-0.657626\pi$
$821$	−27.2321	−0.950409	−0.475204	+	0.879876i	$0.657626\pi$
$822$	−22.0874	−0.770387
$823$	−14.5749	−0.508049	−0.254025	−	0.967198i	$-0.581754\pi$
$823$	−14.5749	−0.508049	−0.254025	+	0.967198i	$0.581754\pi$
$824$	0.841934	0.0293302
$825$	0	0
$826$	−8.23645	−0.286583
$827$	−35.0781	−1.21978	−0.609892	−	0.792485i	$-0.708787\pi$
$827$	−35.0781	−1.21978	−0.609892	+	0.792485i	$0.708787\pi$
$828$	3.62714	0.126052
$829$	18.9077	0.656690	0.328345	−	0.944558i	$-0.393509\pi$
$829$	18.9077	0.656690	0.328345	+	0.944558i	$0.393509\pi$
$830$	0	0
$831$	18.9382	0.656960
$832$	−1.89723	−0.0657748
$833$	15.7418	0.545419
$834$	5.01921	0.173801
$835$	0	0
$836$	−0.590573	−0.0204254
$837$	1.00000	0.0345651
$838$	−5.29682	−0.182976
$839$	−39.5274	−1.36464	−0.682319	−	0.731054i	$-0.739029\pi$
$839$	−39.5274	−1.36464	−0.682319	+	0.731054i	$0.739029\pi$
$840$	0	0
$841$	65.8627	2.27113
$842$	7.01783	0.241850
$843$	−12.9175	−0.444902
$844$	−10.2351	−0.352305
$845$	0	0
$846$	−13.9224	−0.478663
$847$	−16.8968	−0.580579
$848$	36.7828	1.26313
$849$	26.5827	0.912317
$850$	0	0
$851$	−33.3876	−1.14451
$852$	0.688892	0.0236011
$853$	−41.8894	−1.43427	−0.717133	−	0.696937i	$-0.754546\pi$
$853$	−41.8894	−1.43427	−0.717133	+	0.696937i	$0.754546\pi$
$854$	5.52404	0.189029
$855$	0	0
$856$	−25.1134	−0.858359
$857$	−43.2859	−1.47862	−0.739309	−	0.673366i	$-0.764848\pi$
$857$	−43.2859	−1.47862	−0.739309	+	0.673366i	$0.764848\pi$
$858$	0.161933	0.00552831
$859$	−49.6513	−1.69408	−0.847040	−	0.531530i	$-0.821617\pi$
$859$	−49.6513	−1.69408	−0.847040	+	0.531530i	$0.821617\pi$
$860$	0	0
$861$	11.9081	0.405828
$862$	−5.62068	−0.191441
$863$	31.6958	1.07894	0.539469	−	0.842005i	$-0.318625\pi$
$863$	31.6958	1.07894	0.539469	+	0.842005i	$0.318625\pi$
$864$	−2.88739	−0.0982310
$865$	0	0
$866$	−8.27899	−0.281331
$867$	4.57136	0.155252
$868$	0.836535	0.0283939
$869$	−3.23459	−0.109726
$870$	0	0
$871$	−0.445992	−0.0151119
$872$	28.0241	0.949017
$873$	−6.91750	−0.234122
$874$	−15.1427	−0.512210
$875$	0	0
$876$	0.868178	0.0293330
$877$	24.0000	0.810422	0.405211	−	0.914223i	$-0.367198\pi$
$877$	24.0000	0.810422	0.405211	+	0.914223i	$0.367198\pi$
$878$	11.7966	0.398115
$879$	−7.99555	−0.269683
$880$	0	0
$881$	−45.6577	−1.53825	−0.769124	−	0.639100i	$-0.779307\pi$
$881$	−45.6577	−1.53825	−0.769124	+	0.639100i	$0.779307\pi$
$882$	−5.42219	−0.182574
$883$	−7.63158	−0.256823	−0.128412	−	0.991721i	$-0.540988\pi$
$883$	−7.63158	−0.256823	−0.128412	+	0.991721i	$0.540988\pi$
$884$	−0.396997	−0.0133524
$885$	0	0
$886$	−37.8238	−1.27071
$887$	−13.9541	−0.468532	−0.234266	−	0.972173i	$-0.575269\pi$
$887$	−13.9541	−0.468532	−0.234266	+	0.972173i	$0.575269\pi$
$888$	14.8321	0.497732
$889$	−21.7690	−0.730107
$890$	0	0
$891$	0.622216	0.0208450
$892$	−8.10171	−0.271265
$893$	−20.7110	−0.693068
$894$	−7.57136	−0.253224
$895$	0	0
$896$	7.92042	0.264603
$897$	−1.47949	−0.0493989
$898$	34.1443	1.13941
$899$	−9.73975	−0.324839
$900$	0	0
$901$	48.5116	1.61616
$902$	−5.65127	−0.188167
$903$	13.1111	0.436309
$904$	23.5308	0.782624
$905$	0	0
$906$	−10.1793	−0.338184
$907$	−50.6242	−1.68095	−0.840475	−	0.541851i	$-0.817724\pi$
$907$	−50.6242	−1.68095	−0.840475	+	0.541851i	$0.817724\pi$
$908$	−6.74806	−0.223942
$909$	−9.28592	−0.307994
$910$	0	0
$911$	40.5847	1.34463	0.672316	−	0.740264i	$-0.265300\pi$
$911$	40.5847	1.34463	0.672316	+	0.740264i	$0.265300\pi$
$912$	4.82870	0.159894
$913$	3.52098	0.116527
$914$	50.3214	1.66448
$915$	0	0
$916$	1.61963	0.0535141
$917$	−35.4563	−1.17087
$918$	4.28100	0.141294
$919$	53.3274	1.75911	0.879554	−	0.475798i	$-0.157841\pi$
$919$	53.3274	1.75911	0.879554	+	0.475798i	$0.157841\pi$
$920$	0	0
$921$	−17.3669	−0.572258
$922$	12.5600	0.413641
$923$	−0.280996	−0.00924911
$924$	0.520505	0.0171234
$925$	0	0
$926$	−16.1561	−0.530923
$927$	−0.274543	−0.00901717
$928$	28.1225	0.923165
$929$	4.67905	0.153515	0.0767573	−	0.997050i	$-0.475543\pi$
$929$	4.67905	0.153515	0.0767573	+	0.997050i	$0.475543\pi$
$930$	0	0
$931$	−8.06605	−0.264354
$932$	4.60793	0.150938
$933$	27.5877	0.903179
$934$	7.55401	0.247175
$935$	0	0
$936$	0.657249	0.0214829
$937$	19.8952	0.649949	0.324974	−	0.945723i	$-0.394644\pi$
$937$	19.8952	0.649949	0.324974	+	0.945723i	$0.394644\pi$
$938$	4.02318	0.131362
$939$	−13.7255	−0.447913
$940$	0	0
$941$	−1.27946	−0.0417094	−0.0208547	−	0.999783i	$-0.506639\pi$
$941$	−1.27946	−0.0417094	−0.0208547	+	0.999783i	$0.506639\pi$
$942$	−12.4286	−0.404947
$943$	51.6325	1.68139
$944$	11.3880	0.370646
$945$	0	0
$946$	−6.22216	−0.202300
$947$	35.6686	1.15907	0.579537	−	0.814946i	$-0.303233\pi$
$947$	35.6686	1.15907	0.579537	+	0.814946i	$0.303233\pi$
$948$	−2.73143	−0.0887129
$949$	−0.354126	−0.0114954
$950$	0	0
$951$	−24.9447	−0.808887
$952$	17.2128	0.557870
$953$	−39.7832	−1.28871	−0.644353	−	0.764728i	$-0.722873\pi$
$953$	−39.7832	−1.28871	−0.644353	+	0.764728i	$0.722873\pi$
$954$	−16.7096	−0.540994
$955$	0	0
$956$	−1.64143	−0.0530876
$957$	−6.06022	−0.195899
$958$	−30.2153	−0.976211
$959$	28.9590	0.935135
$960$	0	0
$961$	1.00000	0.0322581
$962$	−1.25872	−0.0405828
$963$	8.18913	0.263891
$964$	−15.5329	−0.500282
$965$	0	0
$966$	13.3461	0.429405
$967$	52.9304	1.70213	0.851064	−	0.525063i	$-0.175958\pi$
$967$	52.9304	1.70213	0.851064	+	0.525063i	$0.175958\pi$
$968$	32.5462	1.04607
$969$	6.36842	0.204583
$970$	0	0
$971$	−54.9624	−1.76383	−0.881913	−	0.471412i	$-0.843745\pi$
$971$	−54.9624	−1.76383	−0.881913	+	0.471412i	$0.843745\pi$
$972$	0.525428	0.0168531
$973$	−6.58073	−0.210968
$974$	11.9585	0.383175
$975$	0	0
$976$	−7.63771	−0.244477
$977$	−2.97773	−0.0952659	−0.0476329	−	0.998865i	$-0.515168\pi$
$977$	−2.97773	−0.0952659	−0.0476329	+	0.998865i	$0.515168\pi$
$978$	7.07604	0.226267
$979$	−1.20294	−0.0384463
$980$	0	0
$981$	−9.13828	−0.291763
$982$	−11.9585	−0.381611
$983$	26.9876	0.860770	0.430385	−	0.902645i	$-0.358378\pi$
$983$	26.9876	0.860770	0.430385	+	0.902645i	$0.358378\pi$
$984$	−22.9372	−0.731211
$985$	0	0
$986$	−41.6958	−1.32787
$987$	18.2538	0.581025
$988$	0.203420	0.00647167
$989$	56.8484	1.80767
$990$	0	0
$991$	0.520505	0.0165344	0.00826720	−	0.999966i	$-0.497368\pi$
$991$	0.520505	0.0165344	0.00826720	+	0.999966i	$0.497368\pi$
$992$	−2.88739	−0.0916747
$993$	−19.1798	−0.608651
$994$	2.53480	0.0803988
$995$	0	0
$996$	2.97328	0.0942120
$997$	25.2128	0.798497	0.399249	−	0.916843i	$-0.369271\pi$
$997$	25.2128	0.798497	0.399249	+	0.916843i	$0.369271\pi$
$998$	0.793993	0.0251334
$999$	−4.83654	−0.153021

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2325.2.a.p.1.3		3
3.2	odd	2		6975.2.a.bi.1.1		3
5.2	odd	4		2325.2.c.l.1024.4		6
5.3	odd	4		2325.2.c.l.1024.3		6
5.4	even	2		465.2.a.g.1.1	✓	3
15.14	odd	2		1395.2.a.h.1.3		3
20.19	odd	2		7440.2.a.bm.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.g.1.1	✓	3	5.4	even	2
1395.2.a.h.1.3		3	15.14	odd	2
2325.2.a.p.1.3		3	1.1	even	1	trivial
2325.2.c.l.1024.3		6	5.3	odd	4
2325.2.c.l.1024.4		6	5.2	odd	4
6975.2.a.bi.1.1		3	3.2	odd	2
7440.2.a.bm.1.3		3	20.19	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2325 = 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2325.a (trivial)

\(f(q)\)	\(=\)	\(q+1.21432 q^{2} -1.00000 q^{3} -0.525428 q^{4} -1.21432 q^{6} +1.59210 q^{7} -3.06668 q^{8} +1.00000 q^{9} +0.622216 q^{11} +0.525428 q^{12} -0.214320 q^{13} +1.93332 q^{14} -2.67307 q^{16} -3.52543 q^{17} +1.21432 q^{18} +1.80642 q^{19} -1.59210 q^{21} +0.755569 q^{22} -6.90321 q^{23} +3.06668 q^{24} -0.260253 q^{26} -1.00000 q^{27} -0.836535 q^{28} +9.73975 q^{29} -1.00000 q^{31} +2.88739 q^{32} -0.622216 q^{33} -4.28100 q^{34} -0.525428 q^{36} +4.83654 q^{37} +2.19358 q^{38} +0.214320 q^{39} -7.47949 q^{41} -1.93332 q^{42} -8.23506 q^{43} -0.326929 q^{44} -8.38271 q^{46} -11.4652 q^{47} +2.67307 q^{48} -4.46520 q^{49} +3.52543 q^{51} +0.112610 q^{52} -13.7605 q^{53} -1.21432 q^{54} -4.88247 q^{56} -1.80642 q^{57} +11.8272 q^{58} -4.26025 q^{59} +2.85728 q^{61} -1.21432 q^{62} +1.59210 q^{63} +8.85236 q^{64} -0.755569 q^{66} +2.08097 q^{67} +1.85236 q^{68} +6.90321 q^{69} +1.31111 q^{71} -3.06668 q^{72} +1.65233 q^{73} +5.87310 q^{74} -0.949145 q^{76} +0.990632 q^{77} +0.260253 q^{78} -5.19850 q^{79} +1.00000 q^{81} -9.08250 q^{82} +5.65878 q^{83} +0.836535 q^{84} -10.0000 q^{86} -9.73975 q^{87} -1.90813 q^{88} -1.93332 q^{89} -0.341219 q^{91} +3.62714 q^{92} +1.00000 q^{93} -13.9224 q^{94} -2.88739 q^{96} -6.91750 q^{97} -5.42219 q^{98} +0.622216 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{2} - 3 q^{3} + 5 q^{4} + 3 q^{6} - 2 q^{7} - 9 q^{8} + 3 q^{9} + 2 q^{11} - 5 q^{12} + 6 q^{13} + 6 q^{14} + 5 q^{16} - 4 q^{17} - 3 q^{18} - 8 q^{19} + 2 q^{21} + 2 q^{22} - 14 q^{23} + 9 q^{24}+ \cdots + 2 q^{99}+O(q^{100}) \) 3 * q - 3 * q^2 - 3 * q^3 + 5 * q^4 + 3 * q^6 - 2 * q^7 - 9 * q^8 + 3 * q^9 + 2 * q^11 - 5 * q^12 + 6 * q^13 + 6 * q^14 + 5 * q^16 - 4 * q^17 - 3 * q^18 - 8 * q^19 + 2 * q^21 + 2 * q^22 - 14 * q^23 + 9 * q^24 - 14 * q^26 - 3 * q^27 + 4 * q^28 + 16 * q^29 - 3 * q^31 - 11 * q^32 - 2 * q^33 - 6 * q^34 + 5 * q^36 + 8 * q^37 + 20 * q^38 - 6 * q^39 + 4 * q^41 - 6 * q^42 + 2 * q^43 - 14 * q^44 + 8 * q^46 - 14 * q^47 - 5 * q^48 + 7 * q^49 + 4 * q^51 + 20 * q^52 - 8 * q^53 + 3 * q^54 - 28 * q^56 + 8 * q^57 + 2 * q^58 - 26 * q^59 - 18 * q^61 + 3 * q^62 - 2 * q^63 + 33 * q^64 - 2 * q^66 + 12 * q^68 + 14 * q^69 + 4 * q^71 - 9 * q^72 + 12 * q^73 + 4 * q^74 - 16 * q^76 - 24 * q^77 + 14 * q^78 + 4 * q^79 + 3 * q^81 - 40 * q^82 + 10 * q^83 - 4 * q^84 - 30 * q^86 - 16 * q^87 + 34 * q^88 - 6 * q^89 - 8 * q^91 - 22 * q^92 + 3 * q^93 + 4 * q^94 + 11 * q^96 - 8 * q^97 - 17 * q^98 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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