LMFDB - Embedded newform 2325.2.a.q.1.2

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)

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")

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);

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,-1,-3,1,0,1,-2,-3,3,0,6] 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:	$0$
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			1.2
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)

$f(q)$	$=$	\(q-0.311108 q^{2} -1.00000 q^{3} -1.90321 q^{4} +0.311108 q^{6} -0.688892 q^{7} +1.21432 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.90321 q^{12} -3.73975 q^{13} +0.214320 q^{14} +3.42864 q^{16} -2.28100 q^{17} -0.311108 q^{18} +0.688892 q^{21} -0.622216 q^{22} -2.90321 q^{23} -1.21432 q^{24} +1.16346 q^{26} -1.00000 q^{27} +1.31111 q^{28} -4.02074 q^{29} +1.00000 q^{31} -3.49532 q^{32} -2.00000 q^{33} +0.709636 q^{34} -1.90321 q^{36} +8.79060 q^{37} +3.73975 q^{39} +2.75557 q^{41} -0.214320 q^{42} -1.05086 q^{43} -3.80642 q^{44} +0.903212 q^{46} -4.70964 q^{47} -3.42864 q^{48} -6.52543 q^{49} +2.28100 q^{51} +7.11753 q^{52} -2.14764 q^{53} +0.311108 q^{54} -0.836535 q^{56} +1.25088 q^{58} -6.44938 q^{59} -5.05086 q^{61} -0.311108 q^{62} -0.688892 q^{63} -5.76986 q^{64} +0.622216 q^{66} +5.44446 q^{67} +4.34122 q^{68} +2.90321 q^{69} +8.96989 q^{71} +1.21432 q^{72} +8.92396 q^{73} -2.73483 q^{74} -1.37778 q^{77} -1.16346 q^{78} -8.38271 q^{79} +1.00000 q^{81} -0.857279 q^{82} +10.5161 q^{83} -1.31111 q^{84} +0.326929 q^{86} +4.02074 q^{87} +2.42864 q^{88} +13.5002 q^{89} +2.57628 q^{91} +5.52543 q^{92} -1.00000 q^{93} +1.46520 q^{94} +3.49532 q^{96} +12.0000 q^{97} +2.03011 q^{98} +2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} - 3 q^{3} + q^{4} + q^{6} - 2 q^{7} - 3 q^{8} + 3 q^{9} + 6 q^{11} - q^{12} + 2 q^{13} - 6 q^{14} - 3 q^{16} - q^{18} + 2 q^{21} - 2 q^{22} - 2 q^{23} + 3 q^{24} + 10 q^{26} - 3 q^{27}+ \cdots + 6 q^{99}+O(q^{100}) $ 3 * q - q^2 - 3 * q^3 + q^4 + q^6 - 2 * q^7 - 3 * q^8 + 3 * q^9 + 6 * q^11 - q^12 + 2 * q^13 - 6 * q^14 - 3 * q^16 - q^18 + 2 * q^21 - 2 * q^22 - 2 * q^23 + 3 * q^24 + 10 * q^26 - 3 * q^27 + 4 * q^28 + 8 * q^29 + 3 * q^31 + 3 * q^32 - 6 * q^33 - 18 * q^34 + q^36 - 2 * q^39 + 8 * q^41 + 6 * q^42 + 10 * q^43 + 2 * q^44 - 4 * q^46 + 6 * q^47 + 3 * q^48 - 13 * q^49 + 8 * q^52 + q^54 + 4 * q^56 - 10 * q^58 + 14 * q^59 - 2 * q^61 - q^62 - 2 * q^63 - 11 * q^64 + 2 * q^66 + 16 * q^67 + 20 * q^68 + 2 * q^69 + 20 * q^71 - 3 * q^72 - 28 * q^74 - 4 * q^77 - 10 * q^78 + 8 * q^79 + 3 * q^81 + 24 * q^82 - 2 * q^83 - 4 * q^84 + 14 * q^86 - 8 * q^87 - 6 * q^88 - 6 * q^89 - 12 * q^91 + 10 * q^92 - 3 * q^93 - 16 * q^94 - 3 * q^96 + 36 * q^97 + 13 * q^98 + 6 * 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 $

$2$	−0.311108	−0.219986	−0.109993	−	0.993932i	$-0.535083\pi$
$2$	−0.311108	−0.219986	−0.109993	+	0.993932i	$0.535083\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.90321	−0.951606
$5$	0	0
$5$	0	0
$6$	0.311108	0.127009
$7$	−0.688892	−0.260377	−0.130188	−	0.991489i	$-0.541558\pi$
$7$	−0.688892	−0.260377	−0.130188	+	0.991489i	$0.541558\pi$
$8$	1.21432	0.429327
$9$	1.00000	0.333333
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	1.90321	0.549410
$13$	−3.73975	−1.03722	−0.518610	−	0.855011i	$-0.673550\pi$
$13$	−3.73975	−1.03722	−0.518610	+	0.855011i	$0.673550\pi$
$14$	0.214320	0.0572794
$15$	0	0
$16$	3.42864	0.857160
$17$	−2.28100	−0.553223	−0.276611	−	0.960982i	$-0.589211\pi$
$17$	−2.28100	−0.553223	−0.276611	+	0.960982i	$0.589211\pi$
$18$	−0.311108	−0.0733288
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	0	0
$21$	0.688892	0.150329
$22$	−0.622216	−0.132657
$23$	−2.90321	−0.605362	−0.302681	−	0.953092i	$-0.597882\pi$
$23$	−2.90321	−0.605362	−0.302681	+	0.953092i	$0.597882\pi$
$24$	−1.21432	−0.247872
$25$	0	0
$26$	1.16346	0.228174
$27$	−1.00000	−0.192450
$28$	1.31111	0.247776
$29$	−4.02074	−0.746633	−0.373317	−	0.927704i	$-0.621780\pi$
$29$	−4.02074	−0.746633	−0.373317	+	0.927704i	$0.621780\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	−3.49532	−0.617890
$33$	−2.00000	−0.348155
$34$	0.709636	0.121702
$35$	0	0
$36$	−1.90321	−0.317202
$37$	8.79060	1.44517	0.722583	−	0.691284i	$-0.242955\pi$
$37$	8.79060	1.44517	0.722583	+	0.691284i	$0.242955\pi$
$38$	0	0
$39$	3.73975	0.598839
$40$	0	0
$41$	2.75557	0.430348	0.215174	−	0.976576i	$-0.430968\pi$
$41$	2.75557	0.430348	0.215174	+	0.976576i	$0.430968\pi$
$42$	−0.214320	−0.0330703
$43$	−1.05086	−0.160254	−0.0801270	−	0.996785i	$-0.525533\pi$
$43$	−1.05086	−0.160254	−0.0801270	+	0.996785i	$0.525533\pi$
$44$	−3.80642	−0.573840
$45$	0	0
$46$	0.903212	0.133171
$47$	−4.70964	−0.686971	−0.343485	−	0.939158i	$-0.611608\pi$
$47$	−4.70964	−0.686971	−0.343485	+	0.939158i	$0.611608\pi$
$48$	−3.42864	−0.494881
$49$	−6.52543	−0.932204
$50$	0	0
$51$	2.28100	0.319403
$52$	7.11753	0.987024
$53$	−2.14764	−0.295001	−0.147501	−	0.989062i	$-0.547123\pi$
$53$	−2.14764	−0.295001	−0.147501	+	0.989062i	$0.547123\pi$
$54$	0.311108	0.0423364
$55$	0	0
$56$	−0.836535	−0.111787
$57$	0	0
$58$	1.25088	0.164249
$59$	−6.44938	−0.839638	−0.419819	−	0.907608i	$-0.637906\pi$
$59$	−6.44938	−0.839638	−0.419819	+	0.907608i	$0.637906\pi$
$60$	0	0
$61$	−5.05086	−0.646696	−0.323348	−	0.946280i	$-0.604808\pi$
$61$	−5.05086	−0.646696	−0.323348	+	0.946280i	$0.604808\pi$
$62$	−0.311108	−0.0395107
$63$	−0.688892	−0.0867923
$64$	−5.76986	−0.721232
$65$	0	0
$66$	0.622216	0.0765895
$67$	5.44446	0.665147	0.332573	−	0.943077i	$-0.392083\pi$
$67$	5.44446	0.665147	0.332573	+	0.943077i	$0.392083\pi$
$68$	4.34122	0.526450
$69$	2.90321	0.349506
$70$	0	0
$71$	8.96989	1.06453	0.532265	−	0.846578i	$-0.321341\pi$
$71$	8.96989	1.06453	0.532265	+	0.846578i	$0.321341\pi$
$72$	1.21432	0.143109
$73$	8.92396	1.04447	0.522235	−	0.852802i	$-0.325098\pi$
$73$	8.92396	1.04447	0.522235	+	0.852802i	$0.325098\pi$
$74$	−2.73483	−0.317917
$75$	0	0
$76$	0	0
$77$	−1.37778	−0.157013
$78$	−1.16346	−0.131736
$79$	−8.38271	−0.943128	−0.471564	−	0.881832i	$-0.656310\pi$
$79$	−8.38271	−0.943128	−0.471564	+	0.881832i	$0.656310\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−0.857279	−0.0946706
$83$	10.5161	1.15429	0.577144	−	0.816643i	$-0.304167\pi$
$83$	10.5161	1.15429	0.577144	+	0.816643i	$0.304167\pi$
$84$	−1.31111	−0.143054
$85$	0	0
$86$	0.326929	0.0352537
$87$	4.02074	0.431069
$88$	2.42864	0.258894
$89$	13.5002	1.43102	0.715511	−	0.698601i	$-0.246194\pi$
$89$	13.5002	1.43102	0.715511	+	0.698601i	$0.246194\pi$
$90$	0	0
$91$	2.57628	0.270068
$92$	5.52543	0.576066
$93$	−1.00000	−0.103695
$94$	1.46520	0.151124
$95$	0	0
$96$	3.49532	0.356739
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	2.03011	0.205072
$99$	2.00000	0.201008
$100$	0	0
$101$	8.29529	0.825412	0.412706	−	0.910864i	$-0.364584\pi$
$101$	8.29529	0.825412	0.412706	+	0.910864i	$0.364584\pi$
$102$	−0.709636	−0.0702644
$103$	−2.98418	−0.294040	−0.147020	−	0.989134i	$-0.546968\pi$
$103$	−2.98418	−0.294040	−0.147020	+	0.989134i	$0.546968\pi$
$104$	−4.54125	−0.445306
$105$	0	0
$106$	0.668149	0.0648963
$107$	−1.52543	−0.147469	−0.0737343	−	0.997278i	$-0.523492\pi$
$107$	−1.52543	−0.147469	−0.0737343	+	0.997278i	$0.523492\pi$
$108$	1.90321	0.183137
$109$	3.76049	0.360190	0.180095	−	0.983649i	$-0.442360\pi$
$109$	3.76049	0.360190	0.180095	+	0.983649i	$0.442360\pi$
$110$	0	0
$111$	−8.79060	−0.834367
$112$	−2.36196	−0.223185
$113$	9.18421	0.863978	0.431989	−	0.901879i	$-0.357812\pi$
$113$	9.18421	0.863978	0.431989	+	0.901879i	$0.357812\pi$
$114$	0	0
$115$	0	0
$116$	7.65233	0.710501
$117$	−3.73975	−0.345740
$118$	2.00645	0.184709
$119$	1.57136	0.144046
$120$	0	0
$121$	−7.00000	−0.636364
$122$	1.57136	0.142264
$123$	−2.75557	−0.248461
$124$	−1.90321	−0.170913
$125$	0	0
$126$	0.214320	0.0190931
$127$	11.4795	1.01864	0.509320	−	0.860577i	$-0.329897\pi$
$127$	11.4795	1.01864	0.509320	+	0.860577i	$0.329897\pi$
$128$	8.78568	0.776552
$129$	1.05086	0.0925226
$130$	0	0
$131$	13.0716	1.14207	0.571035	−	0.820925i	$-0.306542\pi$
$131$	13.0716	1.14207	0.571035	+	0.820925i	$0.306542\pi$
$132$	3.80642	0.331307
$133$	0	0
$134$	−1.69381	−0.146323
$135$	0	0
$136$	−2.76986	−0.237513
$137$	11.1383	0.951607	0.475804	−	0.879552i	$-0.342157\pi$
$137$	11.1383	0.951607	0.475804	+	0.879552i	$0.342157\pi$
$138$	−0.903212	−0.0768865
$139$	10.2351	0.868127	0.434063	−	0.900882i	$-0.357079\pi$
$139$	10.2351	0.868127	0.434063	+	0.900882i	$0.357079\pi$
$140$	0	0
$141$	4.70964	0.396623
$142$	−2.79060	−0.234182
$143$	−7.47949	−0.625467
$144$	3.42864	0.285720
$145$	0	0
$146$	−2.77631	−0.229769
$147$	6.52543	0.538208
$148$	−16.7304	−1.37523
$149$	−1.51114	−0.123797	−0.0618986	−	0.998082i	$-0.519716\pi$
$149$	−1.51114	−0.123797	−0.0618986	+	0.998082i	$0.519716\pi$
$150$	0	0
$151$	−1.76049	−0.143267	−0.0716334	−	0.997431i	$-0.522821\pi$
$151$	−1.76049	−0.143267	−0.0716334	+	0.997431i	$0.522821\pi$
$152$	0	0
$153$	−2.28100	−0.184408
$154$	0.428639	0.0345408
$155$	0	0
$156$	−7.11753	−0.569859
$157$	3.43801	0.274383	0.137191	−	0.990545i	$-0.456192\pi$
$157$	3.43801	0.274383	0.137191	+	0.990545i	$0.456192\pi$
$158$	2.60793	0.207475
$159$	2.14764	0.170319
$160$	0	0
$161$	2.00000	0.157622
$162$	−0.311108	−0.0244429
$163$	10.5970	0.830023	0.415012	−	0.909816i	$-0.363778\pi$
$163$	10.5970	0.830023	0.415012	+	0.909816i	$0.363778\pi$
$164$	−5.24443	−0.409521
$165$	0	0
$166$	−3.27163	−0.253928
$167$	20.0415	1.55086	0.775428	−	0.631435i	$-0.217534\pi$
$167$	20.0415	1.55086	0.775428	+	0.631435i	$0.217534\pi$
$168$	0.836535	0.0645401
$169$	0.985710	0.0758238
$170$	0	0
$171$	0	0
$172$	2.00000	0.152499
$173$	18.8988	1.43685	0.718423	−	0.695606i	$-0.244864\pi$
$173$	18.8988	1.43685	0.718423	+	0.695606i	$0.244864\pi$
$174$	−1.25088	−0.0948293
$175$	0	0
$176$	6.85728	0.516887
$177$	6.44938	0.484765
$178$	−4.20003	−0.314806
$179$	16.2953	1.21797	0.608983	−	0.793183i	$-0.291578\pi$
$179$	16.2953	1.21797	0.608983	+	0.793183i	$0.291578\pi$
$180$	0	0
$181$	−1.70471	−0.126710	−0.0633552	−	0.997991i	$-0.520180\pi$
$181$	−1.70471	−0.126710	−0.0633552	+	0.997991i	$0.520180\pi$
$182$	−0.801502	−0.0594113
$183$	5.05086	0.373370
$184$	−3.52543	−0.259898
$185$	0	0
$186$	0.311108	0.0228115
$187$	−4.56199	−0.333606
$188$	8.96343	0.653726
$189$	0.688892	0.0501095
$190$	0	0
$191$	−3.53188	−0.255558	−0.127779	−	0.991803i	$-0.540785\pi$
$191$	−3.53188	−0.255558	−0.127779	+	0.991803i	$0.540785\pi$
$192$	5.76986	0.416404
$193$	−17.4193	−1.25387	−0.626933	−	0.779073i	$-0.715690\pi$
$193$	−17.4193	−1.25387	−0.626933	+	0.779073i	$0.715690\pi$
$194$	−3.73329	−0.268035
$195$	0	0
$196$	12.4193	0.887091
$197$	−10.0143	−0.713489	−0.356744	−	0.934202i	$-0.616113\pi$
$197$	−10.0143	−0.713489	−0.356744	+	0.934202i	$0.616113\pi$
$198$	−0.622216	−0.0442189
$199$	27.1798	1.92672	0.963361	−	0.268208i	$-0.0864313\pi$
$199$	27.1798	1.92672	0.963361	+	0.268208i	$0.0864313\pi$
$200$	0	0
$201$	−5.44446	−0.384023
$202$	−2.58073	−0.181579
$203$	2.76986	0.194406
$204$	−4.34122	−0.303946
$205$	0	0
$206$	0.928401	0.0646848
$207$	−2.90321	−0.201787
$208$	−12.8222	−0.889063
$209$	0	0
$210$	0	0
$211$	−1.93978	−0.133540	−0.0667699	−	0.997768i	$-0.521269\pi$
$211$	−1.93978	−0.133540	−0.0667699	+	0.997768i	$0.521269\pi$
$212$	4.08742	0.280725
$213$	−8.96989	−0.614607
$214$	0.474572	0.0324411
$215$	0	0
$216$	−1.21432	−0.0826240
$217$	−0.688892	−0.0467650
$218$	−1.16992	−0.0792369
$219$	−8.92396	−0.603025
$220$	0	0
$221$	8.53035	0.573813
$222$	2.73483	0.183549
$223$	−23.5526	−1.57720	−0.788600	−	0.614906i	$-0.789194\pi$
$223$	−23.5526	−1.57720	−0.788600	+	0.614906i	$0.789194\pi$
$224$	2.40790	0.160884
$225$	0	0
$226$	−2.85728	−0.190063
$227$	−19.8020	−1.31430	−0.657152	−	0.753758i	$-0.728239\pi$
$227$	−19.8020	−1.31430	−0.657152	+	0.753758i	$0.728239\pi$
$228$	0	0
$229$	−24.4099	−1.61305	−0.806526	−	0.591199i	$-0.798655\pi$
$229$	−24.4099	−1.61305	−0.806526	+	0.591199i	$0.798655\pi$
$230$	0	0
$231$	1.37778	0.0906516
$232$	−4.88247	−0.320550
$233$	−3.13828	−0.205595	−0.102798	−	0.994702i	$-0.532779\pi$
$233$	−3.13828	−0.205595	−0.102798	+	0.994702i	$0.532779\pi$
$234$	1.16346	0.0760581
$235$	0	0
$236$	12.2745	0.799005
$237$	8.38271	0.544515
$238$	−0.488863	−0.0316883
$239$	6.81579	0.440877	0.220438	−	0.975401i	$-0.429251\pi$
$239$	6.81579	0.440877	0.220438	+	0.975401i	$0.429251\pi$
$240$	0	0
$241$	17.0923	1.10101	0.550507	−	0.834830i	$-0.314434\pi$
$241$	17.0923	1.10101	0.550507	+	0.834830i	$0.314434\pi$
$242$	2.17775	0.139991
$243$	−1.00000	−0.0641500
$244$	9.61285	0.615400
$245$	0	0
$246$	0.857279	0.0546581
$247$	0	0
$248$	1.21432	0.0771094
$249$	−10.5161	−0.666428
$250$	0	0
$251$	16.1748	1.02095	0.510473	−	0.859894i	$-0.329470\pi$
$251$	16.1748	1.02095	0.510473	+	0.859894i	$0.329470\pi$
$252$	1.31111	0.0825920
$253$	−5.80642	−0.365047
$254$	−3.57136	−0.224087
$255$	0	0
$256$	8.80642	0.550401
$257$	−19.0464	−1.18808	−0.594041	−	0.804435i	$-0.702468\pi$
$257$	−19.0464	−1.18808	−0.594041	+	0.804435i	$0.702468\pi$
$258$	−0.326929	−0.0203537
$259$	−6.05578	−0.376288
$260$	0	0
$261$	−4.02074	−0.248878
$262$	−4.06668	−0.251240
$263$	6.23506	0.384470	0.192235	−	0.981349i	$-0.438426\pi$
$263$	6.23506	0.384470	0.192235	+	0.981349i	$0.438426\pi$
$264$	−2.42864	−0.149472
$265$	0	0
$266$	0	0
$267$	−13.5002	−0.826201
$268$	−10.3620	−0.632958
$269$	20.8365	1.27043	0.635213	−	0.772337i	$-0.280912\pi$
$269$	20.8365	1.27043	0.635213	+	0.772337i	$0.280912\pi$
$270$	0	0
$271$	6.62222	0.402271	0.201135	−	0.979563i	$-0.435537\pi$
$271$	6.62222	0.402271	0.201135	+	0.979563i	$0.435537\pi$
$272$	−7.82071	−0.474200
$273$	−2.57628	−0.155924
$274$	−3.46520	−0.209341
$275$	0	0
$276$	−5.52543	−0.332592
$277$	−4.79060	−0.287839	−0.143920	−	0.989589i	$-0.545971\pi$
$277$	−4.79060	−0.287839	−0.143920	+	0.989589i	$0.545971\pi$
$278$	−3.18421	−0.190976
$279$	1.00000	0.0598684
$280$	0	0
$281$	−20.8256	−1.24235	−0.621177	−	0.783671i	$-0.713345\pi$
$281$	−20.8256	−1.24235	−0.621177	+	0.783671i	$0.713345\pi$
$282$	−1.46520	−0.0872517
$283$	11.8731	0.705783	0.352891	−	0.935664i	$-0.385199\pi$
$283$	11.8731	0.705783	0.352891	+	0.935664i	$0.385199\pi$
$284$	−17.0716	−1.01301
$285$	0	0
$286$	2.32693	0.137594
$287$	−1.89829	−0.112053
$288$	−3.49532	−0.205963
$289$	−11.7971	−0.693944
$290$	0	0
$291$	−12.0000	−0.703452
$292$	−16.9842	−0.993924
$293$	18.0143	1.05241	0.526203	−	0.850359i	$-0.323615\pi$
$293$	18.0143	1.05241	0.526203	+	0.850359i	$0.323615\pi$
$294$	−2.03011	−0.118399
$295$	0	0
$296$	10.6746	0.620449
$297$	−2.00000	−0.116052
$298$	0.470127	0.0272337
$299$	10.8573	0.627893
$300$	0	0
$301$	0.723926	0.0417264
$302$	0.547702	0.0315167
$303$	−8.29529	−0.476552
$304$	0	0
$305$	0	0
$306$	0.709636	0.0405672
$307$	−10.9240	−0.623463	−0.311732	−	0.950170i	$-0.600909\pi$
$307$	−10.9240	−0.623463	−0.311732	+	0.950170i	$0.600909\pi$
$308$	2.62222	0.149415
$309$	2.98418	0.169764
$310$	0	0
$311$	5.10324	0.289378	0.144689	−	0.989477i	$-0.453782\pi$
$311$	5.10324	0.289378	0.144689	+	0.989477i	$0.453782\pi$
$312$	4.54125	0.257098
$313$	22.1082	1.24963	0.624814	−	0.780774i	$-0.285175\pi$
$313$	22.1082	1.24963	0.624814	+	0.780774i	$0.285175\pi$
$314$	−1.06959	−0.0603605
$315$	0	0
$316$	15.9541	0.897486
$317$	1.06515	0.0598245	0.0299123	−	0.999553i	$-0.490477\pi$
$317$	1.06515	0.0598245	0.0299123	+	0.999553i	$0.490477\pi$
$318$	−0.668149	−0.0374679
$319$	−8.04149	−0.450237
$320$	0	0
$321$	1.52543	0.0851411
$322$	−0.622216	−0.0346747
$323$	0	0
$324$	−1.90321	−0.105734
$325$	0	0
$326$	−3.29682	−0.182594
$327$	−3.76049	−0.207956
$328$	3.34614	0.184760
$329$	3.24443	0.178871
$330$	0	0
$331$	−24.9447	−1.37108	−0.685542	−	0.728033i	$-0.740435\pi$
$331$	−24.9447	−1.37108	−0.685542	+	0.728033i	$0.740435\pi$
$332$	−20.0143	−1.09843
$333$	8.79060	0.481722
$334$	−6.23506	−0.341167
$335$	0	0
$336$	2.36196	0.128856
$337$	−0.453829	−0.0247216	−0.0123608	−	0.999924i	$-0.503935\pi$
$337$	−0.453829	−0.0247216	−0.0123608	+	0.999924i	$0.503935\pi$
$338$	−0.306662	−0.0166802
$339$	−9.18421	−0.498818
$340$	0	0
$341$	2.00000	0.108306
$342$	0	0
$343$	9.31756	0.503101
$344$	−1.27607	−0.0688013
$345$	0	0
$346$	−5.87955	−0.316087
$347$	29.7560	1.59739	0.798694	−	0.601737i	$-0.205525\pi$
$347$	29.7560	1.59739	0.798694	+	0.601737i	$0.205525\pi$
$348$	−7.65233	−0.410208
$349$	−9.65878	−0.517023	−0.258511	−	0.966008i	$-0.583232\pi$
$349$	−9.65878	−0.517023	−0.258511	+	0.966008i	$0.583232\pi$
$350$	0	0
$351$	3.73975	0.199613
$352$	−6.99063	−0.372602
$353$	16.8617	0.897459	0.448730	−	0.893668i	$-0.351877\pi$
$353$	16.8617	0.897459	0.448730	+	0.893668i	$0.351877\pi$
$354$	−2.00645	−0.106642
$355$	0	0
$356$	−25.6938	−1.36177
$357$	−1.57136	−0.0831652
$358$	−5.06959	−0.267936
$359$	8.64296	0.456158	0.228079	−	0.973643i	$-0.426756\pi$
$359$	8.64296	0.456158	0.228079	+	0.973643i	$0.426756\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	0.530350	0.0278746
$363$	7.00000	0.367405
$364$	−4.90321	−0.256998
$365$	0	0
$366$	−1.57136	−0.0821363
$367$	−6.95899	−0.363256	−0.181628	−	0.983367i	$-0.558137\pi$
$367$	−6.95899	−0.363256	−0.181628	+	0.983367i	$0.558137\pi$
$368$	−9.95407	−0.518892
$369$	2.75557	0.143449
$370$	0	0
$371$	1.47949	0.0768115
$372$	1.90321	0.0986769
$373$	−21.0005	−1.08736	−0.543682	−	0.839291i	$-0.682970\pi$
$373$	−21.0005	−1.08736	−0.543682	+	0.839291i	$0.682970\pi$
$374$	1.41927	0.0733888
$375$	0	0
$376$	−5.71900	−0.294935
$377$	15.0366	0.774422
$378$	−0.214320	−0.0110234
$379$	29.7146	1.52633	0.763167	−	0.646201i	$-0.223643\pi$
$379$	29.7146	1.52633	0.763167	+	0.646201i	$0.223643\pi$
$380$	0	0
$381$	−11.4795	−0.588112
$382$	1.09880	0.0562193
$383$	−4.05578	−0.207241	−0.103620	−	0.994617i	$-0.533043\pi$
$383$	−4.05578	−0.207241	−0.103620	+	0.994617i	$0.533043\pi$
$384$	−8.78568	−0.448342
$385$	0	0
$386$	5.41927	0.275834
$387$	−1.05086	−0.0534180
$388$	−22.8385	−1.15945
$389$	−10.9985	−0.557644	−0.278822	−	0.960343i	$-0.589944\pi$
$389$	−10.9985	−0.557644	−0.278822	+	0.960343i	$0.589944\pi$
$390$	0	0
$391$	6.62222	0.334900
$392$	−7.92396	−0.400220
$393$	−13.0716	−0.659375
$394$	3.11552	0.156958
$395$	0	0
$396$	−3.80642	−0.191280
$397$	−0.193576	−0.00971531	−0.00485765	−	0.999988i	$-0.501546\pi$
$397$	−0.193576	−0.00971531	−0.00485765	+	0.999988i	$0.501546\pi$
$398$	−8.45584	−0.423853
$399$	0	0
$400$	0	0
$401$	−38.6242	−1.92880	−0.964401	−	0.264445i	$-0.914811\pi$
$401$	−38.6242	−1.92880	−0.964401	+	0.264445i	$0.914811\pi$
$402$	1.69381	0.0844798
$403$	−3.73975	−0.186290
$404$	−15.7877	−0.785467
$405$	0	0
$406$	−0.861725	−0.0427667
$407$	17.5812	0.871468
$408$	2.76986	0.137128
$409$	2.94914	0.145826	0.0729129	−	0.997338i	$-0.476770\pi$
$409$	2.94914	0.145826	0.0729129	+	0.997338i	$0.476770\pi$
$410$	0	0
$411$	−11.1383	−0.549411
$412$	5.67952	0.279810
$413$	4.44293	0.218622
$414$	0.903212	0.0443904
$415$	0	0
$416$	13.0716	0.640888
$417$	−10.2351	−0.501213
$418$	0	0
$419$	−26.0020	−1.27028	−0.635141	−	0.772397i	$-0.719058\pi$
$419$	−26.0020	−1.27028	−0.635141	+	0.772397i	$0.719058\pi$
$420$	0	0
$421$	−20.0143	−0.975437	−0.487718	−	0.873001i	$-0.662171\pi$
$421$	−20.0143	−0.975437	−0.487718	+	0.873001i	$0.662171\pi$
$422$	0.603480	0.0293769
$423$	−4.70964	−0.228990
$424$	−2.60793	−0.126652
$425$	0	0
$426$	2.79060	0.135205
$427$	3.47949	0.168385
$428$	2.90321	0.140332
$429$	7.47949	0.361113
$430$	0	0
$431$	−2.77631	−0.133730	−0.0668651	−	0.997762i	$-0.521300\pi$
$431$	−2.77631	−0.133730	−0.0668651	+	0.997762i	$0.521300\pi$
$432$	−3.42864	−0.164960
$433$	30.7590	1.47818	0.739091	−	0.673606i	$-0.235256\pi$
$433$	30.7590	1.47818	0.739091	+	0.673606i	$0.235256\pi$
$434$	0.214320	0.0102877
$435$	0	0
$436$	−7.15701	−0.342759
$437$	0	0
$438$	2.77631	0.132657
$439$	26.8385	1.28093	0.640467	−	0.767986i	$-0.278741\pi$
$439$	26.8385	1.28093	0.640467	+	0.767986i	$0.278741\pi$
$440$	0	0
$441$	−6.52543	−0.310735
$442$	−2.65386	−0.126231
$443$	11.7190	0.556787	0.278393	−	0.960467i	$-0.410198\pi$
$443$	11.7190	0.556787	0.278393	+	0.960467i	$0.410198\pi$
$444$	16.7304	0.793989
$445$	0	0
$446$	7.32741	0.346963
$447$	1.51114	0.0714743
$448$	3.97481	0.187792
$449$	33.8084	1.59552	0.797759	−	0.602976i	$-0.206019\pi$
$449$	33.8084	1.59552	0.797759	+	0.602976i	$0.206019\pi$
$450$	0	0
$451$	5.51114	0.259509
$452$	−17.4795	−0.822166
$453$	1.76049	0.0827151
$454$	6.16055	0.289129
$455$	0	0
$456$	0	0
$457$	−5.61930	−0.262860	−0.131430	−	0.991325i	$-0.541957\pi$
$457$	−5.61930	−0.262860	−0.131430	+	0.991325i	$0.541957\pi$
$458$	7.59411	0.354850
$459$	2.28100	0.106468
$460$	0	0
$461$	−2.93825	−0.136848	−0.0684239	−	0.997656i	$-0.521797\pi$
$461$	−2.93825	−0.136848	−0.0684239	+	0.997656i	$0.521797\pi$
$462$	−0.428639	−0.0199421
$463$	1.61285	0.0749554	0.0374777	−	0.999297i	$-0.488068\pi$
$463$	1.61285	0.0749554	0.0374777	+	0.999297i	$0.488068\pi$
$464$	−13.7857	−0.639984
$465$	0	0
$466$	0.976342	0.0452282
$467$	11.5857	0.536120	0.268060	−	0.963402i	$-0.413617\pi$
$467$	11.5857	0.536120	0.268060	+	0.963402i	$0.413617\pi$
$468$	7.11753	0.329008
$469$	−3.75065	−0.173189
$470$	0	0
$471$	−3.43801	−0.158415
$472$	−7.83161	−0.360479
$473$	−2.10171	−0.0966367
$474$	−2.60793	−0.119786
$475$	0	0
$476$	−2.99063	−0.137075
$477$	−2.14764	−0.0983338
$478$	−2.12045	−0.0969869
$479$	26.0020	1.18806	0.594031	−	0.804442i	$-0.297536\pi$
$479$	26.0020	1.18806	0.594031	+	0.804442i	$0.297536\pi$
$480$	0	0
$481$	−32.8746	−1.49895
$482$	−5.31756	−0.242208
$483$	−2.00000	−0.0910032
$484$	13.3225	0.605567
$485$	0	0
$486$	0.311108	0.0141121
$487$	−40.8988	−1.85330	−0.926650	−	0.375925i	$-0.877325\pi$
$487$	−40.8988	−1.85330	−0.926650	+	0.375925i	$0.877325\pi$
$488$	−6.13335	−0.277644
$489$	−10.5970	−0.479214
$490$	0	0
$491$	16.6953	0.753450	0.376725	−	0.926325i	$-0.377050\pi$
$491$	16.6953	0.753450	0.376725	+	0.926325i	$0.377050\pi$
$492$	5.24443	0.236437
$493$	9.17130	0.413055
$494$	0	0
$495$	0	0
$496$	3.42864	0.153950
$497$	−6.17929	−0.277179
$498$	3.27163	0.146605
$499$	−26.3684	−1.18041	−0.590206	−	0.807253i	$-0.700954\pi$
$499$	−26.3684	−1.18041	−0.590206	+	0.807253i	$0.700954\pi$
$500$	0	0
$501$	−20.0415	−0.895388
$502$	−5.03212	−0.224594
$503$	9.47505	0.422472	0.211236	−	0.977435i	$-0.432251\pi$
$503$	9.47505	0.422472	0.211236	+	0.977435i	$0.432251\pi$
$504$	−0.836535	−0.0372622
$505$	0	0
$506$	1.80642	0.0803053
$507$	−0.985710	−0.0437769
$508$	−21.8479	−0.969344
$509$	21.9003	0.970714	0.485357	−	0.874316i	$-0.338690\pi$
$509$	21.9003	0.970714	0.485357	+	0.874316i	$0.338690\pi$
$510$	0	0
$511$	−6.14764	−0.271956
$512$	−20.3111	−0.897633
$513$	0	0
$514$	5.92549	0.261362
$515$	0	0
$516$	−2.00000	−0.0880451
$517$	−9.41927	−0.414259
$518$	1.88400	0.0827782
$519$	−18.8988	−0.829564
$520$	0	0
$521$	−36.2034	−1.58610	−0.793050	−	0.609156i	$-0.791508\pi$
$521$	−36.2034	−1.58610	−0.793050	+	0.609156i	$0.791508\pi$
$522$	1.25088	0.0547497
$523$	14.7971	0.647030	0.323515	−	0.946223i	$-0.395135\pi$
$523$	14.7971	0.647030	0.323515	+	0.946223i	$0.395135\pi$
$524$	−24.8780	−1.08680
$525$	0	0
$526$	−1.93978	−0.0845783
$527$	−2.28100	−0.0993618
$528$	−6.85728	−0.298425
$529$	−14.5714	−0.633537
$530$	0	0
$531$	−6.44938	−0.279879
$532$	0	0
$533$	−10.3051	−0.446365
$534$	4.20003	0.181753
$535$	0	0
$536$	6.61132	0.285565
$537$	−16.2953	−0.703194
$538$	−6.48241	−0.279476
$539$	−13.0509	−0.562140
$540$	0	0
$541$	−41.9956	−1.80553	−0.902765	−	0.430134i	$-0.858466\pi$
$541$	−41.9956	−1.80553	−0.902765	+	0.430134i	$0.858466\pi$
$542$	−2.06022	−0.0884942
$543$	1.70471	0.0731563
$544$	7.97280	0.341831
$545$	0	0
$546$	0.801502	0.0343011
$547$	34.8004	1.48796	0.743980	−	0.668202i	$-0.232936\pi$
$547$	34.8004	1.48796	0.743980	+	0.668202i	$0.232936\pi$
$548$	−21.1985	−0.905555
$549$	−5.05086	−0.215565
$550$	0	0
$551$	0	0
$552$	3.52543	0.150052
$553$	5.77478	0.245569
$554$	1.49039	0.0633208
$555$	0	0
$556$	−19.4795	−0.826115
$557$	−26.3096	−1.11477	−0.557386	−	0.830253i	$-0.688196\pi$
$557$	−26.3096	−1.11477	−0.557386	+	0.830253i	$0.688196\pi$
$558$	−0.311108	−0.0131702
$559$	3.92993	0.166218
$560$	0	0
$561$	4.56199	0.192607
$562$	6.47902	0.273301
$563$	31.8306	1.34150	0.670749	−	0.741684i	$-0.265973\pi$
$563$	31.8306	1.34150	0.670749	+	0.741684i	$0.265973\pi$
$564$	−8.96343	−0.377429
$565$	0	0
$566$	−3.69381	−0.155263
$567$	−0.688892	−0.0289308
$568$	10.8923	0.457031
$569$	23.6622	0.991970	0.495985	−	0.868331i	$-0.334807\pi$
$569$	23.6622	0.991970	0.495985	+	0.868331i	$0.334807\pi$
$570$	0	0
$571$	18.8889	0.790477	0.395238	−	0.918579i	$-0.370662\pi$
$571$	18.8889	0.790477	0.395238	+	0.918579i	$0.370662\pi$
$572$	14.2351	0.595198
$573$	3.53188	0.147546
$574$	0.590573	0.0246500
$575$	0	0
$576$	−5.76986	−0.240411
$577$	−1.11108	−0.0462548	−0.0231274	−	0.999733i	$-0.507362\pi$
$577$	−1.11108	−0.0462548	−0.0231274	+	0.999733i	$0.507362\pi$
$578$	3.67016	0.152658
$579$	17.4193	0.723920
$580$	0	0
$581$	−7.24443	−0.300550
$582$	3.73329	0.154750
$583$	−4.29529	−0.177893
$584$	10.8365	0.448419
$585$	0	0
$586$	−5.60439	−0.231515
$587$	17.3778	0.717258	0.358629	−	0.933480i	$-0.383244\pi$
$587$	17.3778	0.717258	0.358629	+	0.933480i	$0.383244\pi$
$588$	−12.4193	−0.512162
$589$	0	0
$590$	0	0
$591$	10.0143	0.411933
$592$	30.1398	1.23874
$593$	37.1753	1.52661	0.763304	−	0.646040i	$-0.223576\pi$
$593$	37.1753	1.52661	0.763304	+	0.646040i	$0.223576\pi$
$594$	0.622216	0.0255298
$595$	0	0
$596$	2.87601	0.117806
$597$	−27.1798	−1.11239
$598$	−3.37778	−0.138128
$599$	15.6207	0.638244	0.319122	−	0.947714i	$-0.396612\pi$
$599$	15.6207	0.638244	0.319122	+	0.947714i	$0.396612\pi$
$600$	0	0
$601$	−26.7239	−1.09009	−0.545046	−	0.838406i	$-0.683488\pi$
$601$	−26.7239	−1.09009	−0.545046	+	0.838406i	$0.683488\pi$
$602$	−0.225219	−0.00917924
$603$	5.44446	0.221716
$604$	3.35059	0.136333
$605$	0	0
$606$	2.58073	0.104835
$607$	−14.3936	−0.584218	−0.292109	−	0.956385i	$-0.594357\pi$
$607$	−14.3936	−0.584218	−0.292109	+	0.956385i	$0.594357\pi$
$608$	0	0
$609$	−2.76986	−0.112240
$610$	0	0
$611$	17.6128	0.712540
$612$	4.34122	0.175483
$613$	−6.09526	−0.246185	−0.123093	−	0.992395i	$-0.539281\pi$
$613$	−6.09526	−0.246185	−0.123093	+	0.992395i	$0.539281\pi$
$614$	3.39853	0.137153
$615$	0	0
$616$	−1.67307	−0.0674099
$617$	−1.34614	−0.0541936	−0.0270968	−	0.999633i	$-0.508626\pi$
$617$	−1.34614	−0.0541936	−0.0270968	+	0.999633i	$0.508626\pi$
$618$	−0.928401	−0.0373458
$619$	5.71900	0.229866	0.114933	−	0.993373i	$-0.463335\pi$
$619$	5.71900	0.229866	0.114933	+	0.993373i	$0.463335\pi$
$620$	0	0
$621$	2.90321	0.116502
$622$	−1.58766	−0.0636593
$623$	−9.30021	−0.372605
$624$	12.8222	0.513301
$625$	0	0
$626$	−6.87802	−0.274901
$627$	0	0
$628$	−6.54326	−0.261104
$629$	−20.0513	−0.799499
$630$	0	0
$631$	37.3274	1.48598	0.742990	−	0.669302i	$-0.233407\pi$
$631$	37.3274	1.48598	0.742990	+	0.669302i	$0.233407\pi$
$632$	−10.1793	−0.404910
$633$	1.93978	0.0770992
$634$	−0.331375	−0.0131606
$635$	0	0
$636$	−4.08742	−0.162077
$637$	24.4035	0.966900
$638$	2.50177	0.0990460
$639$	8.96989	0.354843
$640$	0	0
$641$	36.1323	1.42714	0.713570	−	0.700584i	$-0.247077\pi$
$641$	36.1323	1.42714	0.713570	+	0.700584i	$0.247077\pi$
$642$	−0.474572	−0.0187299
$643$	3.10123	0.122301	0.0611504	−	0.998129i	$-0.480523\pi$
$643$	3.10123	0.122301	0.0611504	+	0.998129i	$0.480523\pi$
$644$	−3.80642	−0.149994
$645$	0	0
$646$	0	0
$647$	−29.6227	−1.16459	−0.582294	−	0.812978i	$-0.697845\pi$
$647$	−29.6227	−1.16459	−0.582294	+	0.812978i	$0.697845\pi$
$648$	1.21432	0.0477030
$649$	−12.8988	−0.506321
$650$	0	0
$651$	0.688892	0.0269998
$652$	−20.1684	−0.789855
$653$	25.2114	0.986599	0.493299	−	0.869860i	$-0.335791\pi$
$653$	25.2114	0.986599	0.493299	+	0.869860i	$0.335791\pi$
$654$	1.16992	0.0457474
$655$	0	0
$656$	9.44785	0.368877
$657$	8.92396	0.348157
$658$	−1.00937	−0.0393493
$659$	40.6242	1.58250	0.791248	−	0.611496i	$-0.209432\pi$
$659$	40.6242	1.58250	0.791248	+	0.611496i	$0.209432\pi$
$660$	0	0
$661$	−38.7783	−1.50830	−0.754151	−	0.656701i	$-0.771951\pi$
$661$	−38.7783	−1.50830	−0.754151	+	0.656701i	$0.771951\pi$
$662$	7.76049	0.301620
$663$	−8.53035	−0.331291
$664$	12.7699	0.495567
$665$	0	0
$666$	−2.73483	−0.105972
$667$	11.6731	0.451983
$668$	−38.1432	−1.47580
$669$	23.5526	0.910597
$670$	0	0
$671$	−10.1017	−0.389972
$672$	−2.40790	−0.0928866
$673$	−40.1813	−1.54888	−0.774438	−	0.632650i	$-0.781967\pi$
$673$	−40.1813	−1.54888	−0.774438	+	0.632650i	$0.781967\pi$
$674$	0.141190	0.00543842
$675$	0	0
$676$	−1.87601	−0.0721544
$677$	−10.6811	−0.410506	−0.205253	−	0.978709i	$-0.565802\pi$
$677$	−10.6811	−0.410506	−0.205253	+	0.978709i	$0.565802\pi$
$678$	2.85728	0.109733
$679$	−8.26671	−0.317247
$680$	0	0
$681$	19.8020	0.758813
$682$	−0.622216	−0.0238259
$683$	−31.6271	−1.21018	−0.605089	−	0.796158i	$-0.706863\pi$
$683$	−31.6271	−1.21018	−0.605089	+	0.796158i	$0.706863\pi$
$684$	0	0
$685$	0	0
$686$	−2.89877	−0.110675
$687$	24.4099	0.931296
$688$	−3.60300	−0.137363
$689$	8.03164	0.305981
$690$	0	0
$691$	10.6508	0.405175	0.202588	−	0.979264i	$-0.435065\pi$
$691$	10.6508	0.405175	0.202588	+	0.979264i	$0.435065\pi$
$692$	−35.9684	−1.36731
$693$	−1.37778	−0.0523377
$694$	−9.25734	−0.351404
$695$	0	0
$696$	4.88247	0.185069
$697$	−6.28544	−0.238078
$698$	3.00492	0.113738
$699$	3.13828	0.118700
$700$	0	0
$701$	39.3590	1.48657	0.743285	−	0.668974i	$-0.233266\pi$
$701$	39.3590	1.48657	0.743285	+	0.668974i	$0.233266\pi$
$702$	−1.16346	−0.0439121
$703$	0	0
$704$	−11.5397	−0.434919
$705$	0	0
$706$	−5.24581	−0.197429
$707$	−5.71456	−0.214918
$708$	−12.2745	−0.461306
$709$	20.1847	0.758052	0.379026	−	0.925386i	$-0.376259\pi$
$709$	20.1847	0.758052	0.379026	+	0.925386i	$0.376259\pi$
$710$	0	0
$711$	−8.38271	−0.314376
$712$	16.3936	0.614376
$713$	−2.90321	−0.108726
$714$	0.488863	0.0182952
$715$	0	0
$716$	−31.0134	−1.15902
$717$	−6.81579	−0.254540
$718$	−2.68889	−0.100349
$719$	24.1017	0.898842	0.449421	−	0.893320i	$-0.351630\pi$
$719$	24.1017	0.898842	0.449421	+	0.893320i	$0.351630\pi$
$720$	0	0
$721$	2.05578	0.0765611
$722$	5.91105	0.219986
$723$	−17.0923	−0.635671
$724$	3.24443	0.120578
$725$	0	0
$726$	−2.17775	−0.0808241
$727$	14.1180	0.523608	0.261804	−	0.965121i	$-0.415683\pi$
$727$	14.1180	0.523608	0.261804	+	0.965121i	$0.415683\pi$
$728$	3.12843	0.115947
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.39700	0.0886561
$732$	−9.61285	−0.355301
$733$	23.1240	0.854104	0.427052	−	0.904227i	$-0.359552\pi$
$733$	23.1240	0.854104	0.427052	+	0.904227i	$0.359552\pi$
$734$	2.16500	0.0799115
$735$	0	0
$736$	10.1476	0.374047
$737$	10.8889	0.401099
$738$	−0.857279	−0.0315569
$739$	−0.128907	−0.00474193	−0.00237097	−	0.999997i	$-0.500755\pi$
$739$	−0.128907	−0.00474193	−0.00237097	+	0.999997i	$0.500755\pi$
$740$	0	0
$741$	0	0
$742$	−0.460282	−0.0168975
$743$	−28.0129	−1.02769	−0.513847	−	0.857882i	$-0.671780\pi$
$743$	−28.0129	−1.02769	−0.513847	+	0.857882i	$0.671780\pi$
$744$	−1.21432	−0.0445191
$745$	0	0
$746$	6.53341	0.239205
$747$	10.5161	0.384763
$748$	8.68244	0.317461
$749$	1.05086	0.0383974
$750$	0	0
$751$	−3.56247	−0.129996	−0.0649982	−	0.997885i	$-0.520704\pi$
$751$	−3.56247	−0.129996	−0.0649982	+	0.997885i	$0.520704\pi$
$752$	−16.1476	−0.588844
$753$	−16.1748	−0.589444
$754$	−4.67799	−0.170362
$755$	0	0
$756$	−1.31111	−0.0476845
$757$	−27.0573	−0.983415	−0.491707	−	0.870761i	$-0.663627\pi$
$757$	−27.0573	−0.983415	−0.491707	+	0.870761i	$0.663627\pi$
$758$	−9.24443	−0.335773
$759$	5.80642	0.210760
$760$	0	0
$761$	13.1447	0.476496	0.238248	−	0.971204i	$-0.423427\pi$
$761$	13.1447	0.476496	0.238248	+	0.971204i	$0.423427\pi$
$762$	3.57136	0.129377
$763$	−2.59057	−0.0937850
$764$	6.72192	0.243190
$765$	0	0
$766$	1.26178	0.0455901
$767$	24.1191	0.870889
$768$	−8.80642	−0.317774
$769$	25.7418	0.928271	0.464136	−	0.885764i	$-0.346365\pi$
$769$	25.7418	0.928271	0.464136	+	0.885764i	$0.346365\pi$
$770$	0	0
$771$	19.0464	0.685940
$772$	33.1526	1.19319
$773$	−50.3007	−1.80919	−0.904595	−	0.426272i	$-0.859827\pi$
$773$	−50.3007	−1.80919	−0.904595	+	0.426272i	$0.859827\pi$
$774$	0.326929	0.0117512
$775$	0	0
$776$	14.5718	0.523098
$777$	6.05578	0.217250
$778$	3.42171	0.122674
$779$	0	0
$780$	0	0
$781$	17.9398	0.641936
$782$	−2.06022	−0.0736734
$783$	4.02074	0.143690
$784$	−22.3733	−0.799048
$785$	0	0
$786$	4.06668	0.145054
$787$	−8.70519	−0.310307	−0.155153	−	0.987890i	$-0.549587\pi$
$787$	−8.70519	−0.310307	−0.155153	+	0.987890i	$0.549587\pi$
$788$	19.0593	0.678960
$789$	−6.23506	−0.221974
$790$	0	0
$791$	−6.32693	−0.224960
$792$	2.42864	0.0862979
$793$	18.8889	0.670765
$794$	0.0602231	0.00213724
$795$	0	0
$796$	−51.7288	−1.83348
$797$	−39.0464	−1.38309	−0.691547	−	0.722331i	$-0.743071\pi$
$797$	−39.0464	−1.38309	−0.691547	+	0.722331i	$0.743071\pi$
$798$	0	0
$799$	10.7427	0.380048
$800$	0	0
$801$	13.5002	0.477007
$802$	12.0163	0.424310
$803$	17.8479	0.629839
$804$	10.3620	0.365438
$805$	0	0
$806$	1.16346	0.0409813
$807$	−20.8365	−0.733481
$808$	10.0731	0.354371
$809$	−24.0435	−0.845324	−0.422662	−	0.906287i	$-0.638904\pi$
$809$	−24.0435	−0.845324	−0.422662	+	0.906287i	$0.638904\pi$
$810$	0	0
$811$	45.5812	1.60057	0.800286	−	0.599618i	$-0.204681\pi$
$811$	45.5812	1.60057	0.800286	+	0.599618i	$0.204681\pi$
$812$	−5.27163	−0.184998
$813$	−6.62222	−0.232251
$814$	−5.46965	−0.191711
$815$	0	0
$816$	7.82071	0.273780
$817$	0	0
$818$	−0.917502	−0.0320797
$819$	2.57628	0.0900226
$820$	0	0
$821$	−32.2054	−1.12398	−0.561989	−	0.827145i	$-0.689964\pi$
$821$	−32.2054	−1.12398	−0.561989	+	0.827145i	$0.689964\pi$
$822$	3.46520	0.120863
$823$	−33.1052	−1.15398	−0.576988	−	0.816752i	$-0.695772\pi$
$823$	−33.1052	−1.15398	−0.576988	+	0.816752i	$0.695772\pi$
$824$	−3.62375	−0.126239
$825$	0	0
$826$	−1.38223	−0.0480939
$827$	0.147643	0.00513406	0.00256703	−	0.999997i	$-0.499183\pi$
$827$	0.147643	0.00513406	0.00256703	+	0.999997i	$0.499183\pi$
$828$	5.52543	0.192022
$829$	−20.9175	−0.726495	−0.363247	−	0.931693i	$-0.618332\pi$
$829$	−20.9175	−0.726495	−0.363247	+	0.931693i	$0.618332\pi$
$830$	0	0
$831$	4.79060	0.166184
$832$	21.5778	0.748076
$833$	14.8845	0.515717
$834$	3.18421	0.110260
$835$	0	0
$836$	0	0
$837$	−1.00000	−0.0345651
$838$	8.08943	0.279445
$839$	−42.9066	−1.48130	−0.740650	−	0.671891i	$-0.765482\pi$
$839$	−42.9066	−1.48130	−0.740650	+	0.671891i	$0.765482\pi$
$840$	0	0
$841$	−12.8336	−0.442539
$842$	6.22660	0.214583
$843$	20.8256	0.717273
$844$	3.69181	0.127077
$845$	0	0
$846$	1.46520	0.0503748
$847$	4.82225	0.165694
$848$	−7.36349	−0.252863
$849$	−11.8731	−0.407484
$850$	0	0
$851$	−25.5210	−0.874848
$852$	17.0716	0.584863
$853$	49.2226	1.68535	0.842675	−	0.538422i	$-0.180979\pi$
$853$	49.2226	1.68535	0.842675	+	0.538422i	$0.180979\pi$
$854$	−1.08250	−0.0370423
$855$	0	0
$856$	−1.85236	−0.0633123
$857$	−34.2449	−1.16978	−0.584892	−	0.811111i	$-0.698863\pi$
$857$	−34.2449	−1.16978	−0.584892	+	0.811111i	$0.698863\pi$
$858$	−2.32693	−0.0794401
$859$	14.5718	0.497185	0.248592	−	0.968608i	$-0.420032\pi$
$859$	14.5718	0.497185	0.248592	+	0.968608i	$0.420032\pi$
$860$	0	0
$861$	1.89829	0.0646935
$862$	0.863732	0.0294188
$863$	−44.6164	−1.51876	−0.759380	−	0.650648i	$-0.774497\pi$
$863$	−44.6164	−1.51876	−0.759380	+	0.650648i	$0.774497\pi$
$864$	3.49532	0.118913
$865$	0	0
$866$	−9.56935	−0.325180
$867$	11.7971	0.400649
$868$	1.31111	0.0445019
$869$	−16.7654	−0.568728
$870$	0	0
$871$	−20.3609	−0.689903
$872$	4.56644	0.154639
$873$	12.0000	0.406138
$874$	0	0
$875$	0	0
$876$	16.9842	0.573842
$877$	−8.05038	−0.271842	−0.135921	−	0.990720i	$-0.543399\pi$
$877$	−8.05038	−0.271842	−0.135921	+	0.990720i	$0.543399\pi$
$878$	−8.34968	−0.281788
$879$	−18.0143	−0.607607
$880$	0	0
$881$	−28.7130	−0.967366	−0.483683	−	0.875243i	$-0.660701\pi$
$881$	−28.7130	−0.967366	−0.483683	+	0.875243i	$0.660701\pi$
$882$	2.03011	0.0683574
$883$	26.4197	0.889095	0.444548	−	0.895755i	$-0.353364\pi$
$883$	26.4197	0.889095	0.444548	+	0.895755i	$0.353364\pi$
$884$	−16.2351	−0.546044
$885$	0	0
$886$	−3.64587	−0.122486
$887$	44.7926	1.50399	0.751994	−	0.659170i	$-0.229092\pi$
$887$	44.7926	1.50399	0.751994	+	0.659170i	$0.229092\pi$
$888$	−10.6746	−0.358216
$889$	−7.90813	−0.265230
$890$	0	0
$891$	2.00000	0.0670025
$892$	44.8256	1.50087
$893$	0	0
$894$	−0.470127	−0.0157234
$895$	0	0
$896$	−6.05239	−0.202196
$897$	−10.8573	−0.362514
$898$	−10.5181	−0.350992
$899$	−4.02074	−0.134099
$900$	0	0
$901$	4.89877	0.163202
$902$	−1.71456	−0.0570885
$903$	−0.723926	−0.0240907
$904$	11.1526	0.370929
$905$	0	0
$906$	−0.547702	−0.0181962
$907$	−7.21924	−0.239711	−0.119855	−	0.992791i	$-0.538243\pi$
$907$	−7.21924	−0.239711	−0.119855	+	0.992791i	$0.538243\pi$
$908$	37.6874	1.25070
$909$	8.29529	0.275137
$910$	0	0
$911$	21.1052	0.699248	0.349624	−	0.936890i	$-0.386309\pi$
$911$	21.1052	0.699248	0.349624	+	0.936890i	$0.386309\pi$
$912$	0	0
$913$	21.0321	0.696062
$914$	1.74821	0.0578256
$915$	0	0
$916$	46.4572	1.53499
$917$	−9.00492	−0.297369
$918$	−0.709636	−0.0234215
$919$	16.4415	0.542357	0.271178	−	0.962529i	$-0.412587\pi$
$919$	16.4415	0.542357	0.271178	+	0.962529i	$0.412587\pi$
$920$	0	0
$921$	10.9240	0.359957
$922$	0.914111	0.0301046
$923$	−33.5451	−1.10415
$924$	−2.62222	−0.0862646
$925$	0	0
$926$	−0.501770	−0.0164892
$927$	−2.98418	−0.0980133
$928$	14.0538	0.461338
$929$	6.26470	0.205538	0.102769	−	0.994705i	$-0.467230\pi$
$929$	6.26470	0.205538	0.102769	+	0.994705i	$0.467230\pi$
$930$	0	0
$931$	0	0
$932$	5.97280	0.195646
$933$	−5.10324	−0.167073
$934$	−3.60439	−0.117939
$935$	0	0
$936$	−4.54125	−0.148435
$937$	−37.1437	−1.21343	−0.606715	−	0.794919i	$-0.707513\pi$
$937$	−37.1437	−1.21343	−0.606715	+	0.794919i	$0.707513\pi$
$938$	1.16686	0.0380992
$939$	−22.1082	−0.721473
$940$	0	0
$941$	39.1318	1.27566	0.637830	−	0.770177i	$-0.279832\pi$
$941$	39.1318	1.27566	0.637830	+	0.770177i	$0.279832\pi$
$942$	1.06959	0.0348492
$943$	−8.00000	−0.260516
$944$	−22.1126	−0.719704
$945$	0	0
$946$	0.653858	0.0212588
$947$	53.0968	1.72541	0.862707	−	0.505704i	$-0.168767\pi$
$947$	53.0968	1.72541	0.862707	+	0.505704i	$0.168767\pi$
$948$	−15.9541	−0.518164
$949$	−33.3733	−1.08334
$950$	0	0
$951$	−1.06515	−0.0345397
$952$	1.90813	0.0618430
$953$	6.57628	0.213027	0.106513	−	0.994311i	$-0.466031\pi$
$953$	6.57628	0.213027	0.106513	+	0.994311i	$0.466031\pi$
$954$	0.668149	0.0216321
$955$	0	0
$956$	−12.9719	−0.419541
$957$	8.04149	0.259944
$958$	−8.08943	−0.261358
$959$	−7.67307	−0.247776
$960$	0	0
$961$	1.00000	0.0322581
$962$	10.2276	0.329750
$963$	−1.52543	−0.0491562
$964$	−32.5303	−1.04773
$965$	0	0
$966$	0.622216	0.0200195
$967$	−54.2449	−1.74440	−0.872199	−	0.489151i	$-0.837307\pi$
$967$	−54.2449	−1.74440	−0.872199	+	0.489151i	$0.837307\pi$
$968$	−8.50024	−0.273208
$969$	0	0
$970$	0	0
$971$	−42.8879	−1.37634	−0.688169	−	0.725551i	$-0.741585\pi$
$971$	−42.8879	−1.37634	−0.688169	+	0.725551i	$0.741585\pi$
$972$	1.90321	0.0610456
$973$	−7.05086	−0.226040
$974$	12.7239	0.407701
$975$	0	0
$976$	−17.3176	−0.554322
$977$	−11.6316	−0.372127	−0.186064	−	0.982538i	$-0.559573\pi$
$977$	−11.6316	−0.372127	−0.186064	+	0.982538i	$0.559573\pi$
$978$	3.29682	0.105421
$979$	27.0005	0.862939
$980$	0	0
$981$	3.76049	0.120063
$982$	−5.19405	−0.165749
$983$	30.3595	0.968318	0.484159	−	0.874980i	$-0.339126\pi$
$983$	30.3595	0.968318	0.484159	+	0.874980i	$0.339126\pi$
$984$	−3.34614	−0.106671
$985$	0	0
$986$	−2.85326	−0.0908664
$987$	−3.24443	−0.103271
$988$	0	0
$989$	3.05086	0.0970115
$990$	0	0
$991$	42.5589	1.35193	0.675964	−	0.736934i	$-0.263727\pi$
$991$	42.5589	1.35193	0.675964	+	0.736934i	$0.263727\pi$
$992$	−3.49532	−0.110976
$993$	24.9447	0.791596
$994$	1.92242	0.0609756
$995$	0	0
$996$	20.0143	0.634177
$997$	33.5339	1.06203	0.531014	−	0.847363i	$-0.321811\pi$
$997$	33.5339	1.06203	0.531014	+	0.847363i	$0.321811\pi$
$998$	8.20342	0.259675
$999$	−8.79060	−0.278122

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2325.2.a.q.1.2		3
3.2	odd	2		6975.2.a.be.1.2		3
5.2	odd	4		2325.2.c.o.1024.3		6
5.3	odd	4		2325.2.c.o.1024.4		6
5.4	even	2		465.2.a.f.1.2	✓	3
15.14	odd	2		1395.2.a.i.1.2		3
20.19	odd	2		7440.2.a.bp.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
465.2.a.f.1.2	✓	3	5.4	even	2
1395.2.a.i.1.2		3	15.14	odd	2
2325.2.a.q.1.2		3	1.1	even	1	trivial
2325.2.c.o.1024.3		6	5.2	odd	4
2325.2.c.o.1024.4		6	5.3	odd	4
6975.2.a.be.1.2		3	3.2	odd	2
7440.2.a.bp.1.2		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)

\(f(q)\)	\(=\)	\(q-0.311108 q^{2} -1.00000 q^{3} -1.90321 q^{4} +0.311108 q^{6} -0.688892 q^{7} +1.21432 q^{8} +1.00000 q^{9} +2.00000 q^{11} +1.90321 q^{12} -3.73975 q^{13} +0.214320 q^{14} +3.42864 q^{16} -2.28100 q^{17} -0.311108 q^{18} +0.688892 q^{21} -0.622216 q^{22} -2.90321 q^{23} -1.21432 q^{24} +1.16346 q^{26} -1.00000 q^{27} +1.31111 q^{28} -4.02074 q^{29} +1.00000 q^{31} -3.49532 q^{32} -2.00000 q^{33} +0.709636 q^{34} -1.90321 q^{36} +8.79060 q^{37} +3.73975 q^{39} +2.75557 q^{41} -0.214320 q^{42} -1.05086 q^{43} -3.80642 q^{44} +0.903212 q^{46} -4.70964 q^{47} -3.42864 q^{48} -6.52543 q^{49} +2.28100 q^{51} +7.11753 q^{52} -2.14764 q^{53} +0.311108 q^{54} -0.836535 q^{56} +1.25088 q^{58} -6.44938 q^{59} -5.05086 q^{61} -0.311108 q^{62} -0.688892 q^{63} -5.76986 q^{64} +0.622216 q^{66} +5.44446 q^{67} +4.34122 q^{68} +2.90321 q^{69} +8.96989 q^{71} +1.21432 q^{72} +8.92396 q^{73} -2.73483 q^{74} -1.37778 q^{77} -1.16346 q^{78} -8.38271 q^{79} +1.00000 q^{81} -0.857279 q^{82} +10.5161 q^{83} -1.31111 q^{84} +0.326929 q^{86} +4.02074 q^{87} +2.42864 q^{88} +13.5002 q^{89} +2.57628 q^{91} +5.52543 q^{92} -1.00000 q^{93} +1.46520 q^{94} +3.49532 q^{96} +12.0000 q^{97} +2.03011 q^{98} +2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} - 3 q^{3} + q^{4} + q^{6} - 2 q^{7} - 3 q^{8} + 3 q^{9} + 6 q^{11} - q^{12} + 2 q^{13} - 6 q^{14} - 3 q^{16} - q^{18} + 2 q^{21} - 2 q^{22} - 2 q^{23} + 3 q^{24} + 10 q^{26} - 3 q^{27}+ \cdots + 6 q^{99}+O(q^{100}) \) 3 * q - q^2 - 3 * q^3 + q^4 + q^6 - 2 * q^7 - 3 * q^8 + 3 * q^9 + 6 * q^11 - q^12 + 2 * q^13 - 6 * q^14 - 3 * q^16 - q^18 + 2 * q^21 - 2 * q^22 - 2 * q^23 + 3 * q^24 + 10 * q^26 - 3 * q^27 + 4 * q^28 + 8 * q^29 + 3 * q^31 + 3 * q^32 - 6 * q^33 - 18 * q^34 + q^36 - 2 * q^39 + 8 * q^41 + 6 * q^42 + 10 * q^43 + 2 * q^44 - 4 * q^46 + 6 * q^47 + 3 * q^48 - 13 * q^49 + 8 * q^52 + q^54 + 4 * q^56 - 10 * q^58 + 14 * q^59 - 2 * q^61 - q^62 - 2 * q^63 - 11 * q^64 + 2 * q^66 + 16 * q^67 + 20 * q^68 + 2 * q^69 + 20 * q^71 - 3 * q^72 - 28 * q^74 - 4 * q^77 - 10 * q^78 + 8 * q^79 + 3 * q^81 + 24 * q^82 - 2 * q^83 - 4 * q^84 + 14 * q^86 - 8 * q^87 - 6 * q^88 - 6 * q^89 - 12 * q^91 + 10 * q^92 - 3 * q^93 - 16 * q^94 - 3 * q^96 + 36 * q^97 + 13 * q^98 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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