LMFDB - Embedded newform 4025.2.a.be.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4025,2,Mod(1,4025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4025 = 5^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4025.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:	$32.1397868136$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	no (minimal twist has level 805)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Character	$\chi$	$=$	4025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.556064 q^{2} -2.94180 q^{3} -1.69079 q^{4} +1.63583 q^{6} +1.00000 q^{7} +2.05232 q^{8} +5.65418 q^{9} +O(q^{10})$	\(q-0.556064 q^{2} -2.94180 q^{3} -1.69079 q^{4} +1.63583 q^{6} +1.00000 q^{7} +2.05232 q^{8} +5.65418 q^{9} -5.83673 q^{11} +4.97397 q^{12} -4.53368 q^{13} -0.556064 q^{14} +2.24037 q^{16} -3.00055 q^{17} -3.14408 q^{18} +3.78067 q^{19} -2.94180 q^{21} +3.24559 q^{22} +1.00000 q^{23} -6.03750 q^{24} +2.52102 q^{26} -7.80805 q^{27} -1.69079 q^{28} -6.57253 q^{29} +6.03143 q^{31} -5.35042 q^{32} +17.1705 q^{33} +1.66850 q^{34} -9.56004 q^{36} -2.03637 q^{37} -2.10230 q^{38} +13.3372 q^{39} +8.22829 q^{41} +1.63583 q^{42} -8.07411 q^{43} +9.86869 q^{44} -0.556064 q^{46} -8.34866 q^{47} -6.59071 q^{48} +1.00000 q^{49} +8.82701 q^{51} +7.66552 q^{52} -5.57566 q^{53} +4.34178 q^{54} +2.05232 q^{56} -11.1220 q^{57} +3.65475 q^{58} -11.3874 q^{59} -1.32833 q^{61} -3.35386 q^{62} +5.65418 q^{63} -1.50556 q^{64} -9.54788 q^{66} +0.934565 q^{67} +5.07331 q^{68} -2.94180 q^{69} -0.463853 q^{71} +11.6042 q^{72} -13.8633 q^{73} +1.13235 q^{74} -6.39233 q^{76} -5.83673 q^{77} -7.41632 q^{78} +5.34650 q^{79} +6.00719 q^{81} -4.57546 q^{82} -9.37885 q^{83} +4.97397 q^{84} +4.48972 q^{86} +19.3351 q^{87} -11.9788 q^{88} -15.5150 q^{89} -4.53368 q^{91} -1.69079 q^{92} -17.7432 q^{93} +4.64239 q^{94} +15.7399 q^{96} -3.55398 q^{97} -0.556064 q^{98} -33.0019 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 2 q^{2} - q^{3} + 30 q^{4} + 6 q^{6} + 21 q^{7} + 6 q^{8} + 30 q^{9}+O(q^{10}) $ 21 * q + 2 * q^2 - q^3 + 30 * q^4 + 6 * q^6 + 21 * q^7 + 6 * q^8 + 30 * q^9	$ 21 q + 2 q^{2} - q^{3} + 30 q^{4} + 6 q^{6} + 21 q^{7} + 6 q^{8} + 30 q^{9} + 7 q^{11} - 22 q^{12} - 3 q^{13} + 2 q^{14} + 56 q^{16} + 7 q^{17} + 24 q^{19} - q^{21} + 4 q^{22} + 21 q^{23} + 24 q^{24} - 2 q^{26} - 19 q^{27} + 30 q^{28} + 11 q^{29} + 46 q^{31} - 6 q^{32} - 3 q^{33} + 28 q^{34} + 58 q^{36} + 24 q^{37} - 4 q^{38} + 31 q^{39} + 14 q^{41} + 6 q^{42} + 18 q^{43} + 12 q^{44} + 2 q^{46} - 25 q^{47} - 36 q^{48} + 21 q^{49} + 17 q^{51} - 8 q^{52} + 22 q^{53} - 6 q^{54} + 6 q^{56} + 40 q^{57} + 6 q^{58} + 10 q^{59} + 38 q^{61} - 54 q^{62} + 30 q^{63} + 100 q^{64} + 38 q^{66} + 12 q^{67} + 18 q^{68} - q^{69} + 56 q^{71} + 42 q^{72} - 40 q^{73} - 20 q^{74} + 60 q^{76} + 7 q^{77} + 38 q^{78} + 49 q^{79} + 57 q^{81} + 16 q^{82} - 2 q^{83} - 22 q^{84} + 16 q^{86} - 23 q^{87} + 12 q^{88} + 28 q^{89} - 3 q^{91} + 30 q^{92} - 30 q^{93} + 66 q^{94} + 46 q^{96} - q^{97} + 2 q^{98} - 2 q^{99}+O(q^{100}) $ 21 * q + 2 * q^2 - q^3 + 30 * q^4 + 6 * q^6 + 21 * q^7 + 6 * q^8 + 30 * q^9 + 7 * q^11 - 22 * q^12 - 3 * q^13 + 2 * q^14 + 56 * q^16 + 7 * q^17 + 24 * q^19 - q^21 + 4 * q^22 + 21 * q^23 + 24 * q^24 - 2 * q^26 - 19 * q^27 + 30 * q^28 + 11 * q^29 + 46 * q^31 - 6 * q^32 - 3 * q^33 + 28 * q^34 + 58 * q^36 + 24 * q^37 - 4 * q^38 + 31 * q^39 + 14 * q^41 + 6 * q^42 + 18 * q^43 + 12 * q^44 + 2 * q^46 - 25 * q^47 - 36 * q^48 + 21 * q^49 + 17 * q^51 - 8 * q^52 + 22 * q^53 - 6 * q^54 + 6 * q^56 + 40 * q^57 + 6 * q^58 + 10 * q^59 + 38 * q^61 - 54 * q^62 + 30 * q^63 + 100 * q^64 + 38 * q^66 + 12 * q^67 + 18 * q^68 - q^69 + 56 * q^71 + 42 * q^72 - 40 * q^73 - 20 * q^74 + 60 * q^76 + 7 * q^77 + 38 * q^78 + 49 * q^79 + 57 * q^81 + 16 * q^82 - 2 * q^83 - 22 * q^84 + 16 * q^86 - 23 * q^87 + 12 * q^88 + 28 * q^89 - 3 * q^91 + 30 * q^92 - 30 * q^93 + 66 * q^94 + 46 * q^96 - q^97 + 2 * q^98 - 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.556064	−0.393197	−0.196598	−	0.980484i	$-0.562989\pi$
$2$	−0.556064	−0.393197	−0.196598	+	0.980484i	$0.562989\pi$
$3$	−2.94180	−1.69845	−0.849224	−	0.528033i	$-0.822930\pi$
$3$	−2.94180	−1.69845	−0.849224	+	0.528033i	$0.822930\pi$
$4$	−1.69079	−0.845396
$5$	0	0
$5$	0	0
$6$	1.63583	0.667824
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.05232	0.725604
$9$	5.65418	1.88473
$10$	0	0
$11$	−5.83673	−1.75984	−0.879920	−	0.475123i	$-0.842404\pi$
$11$	−5.83673	−1.75984	−0.879920	+	0.475123i	$0.842404\pi$
$12$	4.97397	1.43586
$13$	−4.53368	−1.25742	−0.628709	−	0.777641i	$-0.716416\pi$
$13$	−4.53368	−1.25742	−0.628709	+	0.777641i	$0.716416\pi$
$14$	−0.556064	−0.148614
$15$	0	0
$16$	2.24037	0.560092
$17$	−3.00055	−0.727740	−0.363870	−	0.931450i	$-0.618545\pi$
$17$	−3.00055	−0.727740	−0.363870	+	0.931450i	$0.618545\pi$
$18$	−3.14408	−0.741068
$19$	3.78067	0.867346	0.433673	−	0.901070i	$-0.357217\pi$
$19$	3.78067	0.867346	0.433673	+	0.901070i	$0.357217\pi$
$20$	0	0
$21$	−2.94180	−0.641953
$22$	3.24559	0.691963
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−6.03750	−1.23240
$25$	0	0
$26$	2.52102	0.494412
$27$	−7.80805	−1.50266
$28$	−1.69079	−0.319530
$29$	−6.57253	−1.22049	−0.610244	−	0.792213i	$-0.708929\pi$
$29$	−6.57253	−1.22049	−0.610244	+	0.792213i	$0.708929\pi$
$30$	0	0
$31$	6.03143	1.08328	0.541638	−	0.840612i	$-0.317804\pi$
$31$	6.03143	1.08328	0.541638	+	0.840612i	$0.317804\pi$
$32$	−5.35042	−0.945830
$33$	17.1705	2.98900
$34$	1.66850	0.286145
$35$	0	0
$36$	−9.56004	−1.59334
$37$	−2.03637	−0.334777	−0.167388	−	0.985891i	$-0.553533\pi$
$37$	−2.03637	−0.334777	−0.167388	+	0.985891i	$0.553533\pi$
$38$	−2.10230	−0.341037
$39$	13.3372	2.13566
$40$	0	0
$41$	8.22829	1.28504	0.642521	−	0.766268i	$-0.277888\pi$
$41$	8.22829	1.28504	0.642521	+	0.766268i	$0.277888\pi$
$42$	1.63583	0.252414
$43$	−8.07411	−1.23129	−0.615645	−	0.788024i	$-0.711104\pi$
$43$	−8.07411	−1.23129	−0.615645	+	0.788024i	$0.711104\pi$
$44$	9.86869	1.48776
$45$	0	0
$46$	−0.556064	−0.0819872
$47$	−8.34866	−1.21778	−0.608889	−	0.793256i	$-0.708384\pi$
$47$	−8.34866	−1.21778	−0.608889	+	0.793256i	$0.708384\pi$
$48$	−6.59071	−0.951287
$49$	1.00000	0.142857
$50$	0	0
$51$	8.82701	1.23603
$52$	7.66552	1.06302
$53$	−5.57566	−0.765876	−0.382938	−	0.923774i	$-0.625088\pi$
$53$	−5.57566	−0.765876	−0.382938	+	0.923774i	$0.625088\pi$
$54$	4.34178	0.590841
$55$	0	0
$56$	2.05232	0.274252
$57$	−11.1220	−1.47314
$58$	3.65475	0.479892
$59$	−11.3874	−1.48252	−0.741259	−	0.671219i	$-0.765771\pi$
$59$	−11.3874	−1.48252	−0.741259	+	0.671219i	$0.765771\pi$
$60$	0	0
$61$	−1.32833	−0.170076	−0.0850378	−	0.996378i	$-0.527101\pi$
$61$	−1.32833	−0.170076	−0.0850378	+	0.996378i	$0.527101\pi$
$62$	−3.35386	−0.425941
$63$	5.65418	0.712359
$64$	−1.50556	−0.188195
$65$	0	0
$66$	−9.54788	−1.17526
$67$	0.934565	0.114175	0.0570876	−	0.998369i	$-0.481819\pi$
$67$	0.934565	0.114175	0.0570876	+	0.998369i	$0.481819\pi$
$68$	5.07331	0.615229
$69$	−2.94180	−0.354151
$70$	0	0
$71$	−0.463853	−0.0550493	−0.0275246	−	0.999621i	$-0.508762\pi$
$71$	−0.463853	−0.0550493	−0.0275246	+	0.999621i	$0.508762\pi$
$72$	11.6042	1.36756
$73$	−13.8633	−1.62258	−0.811288	−	0.584646i	$-0.801233\pi$
$73$	−13.8633	−1.62258	−0.811288	+	0.584646i	$0.801233\pi$
$74$	1.13235	0.131633
$75$	0	0
$76$	−6.39233	−0.733251
$77$	−5.83673	−0.665157
$78$	−7.41632	−0.839733
$79$	5.34650	0.601528	0.300764	−	0.953699i	$-0.402758\pi$
$79$	5.34650	0.601528	0.300764	+	0.953699i	$0.402758\pi$
$80$	0	0
$81$	6.00719	0.667465
$82$	−4.57546	−0.505274
$83$	−9.37885	−1.02946	−0.514731	−	0.857352i	$-0.672108\pi$
$83$	−9.37885	−1.02946	−0.514731	+	0.857352i	$0.672108\pi$
$84$	4.97397	0.542705
$85$	0	0
$86$	4.48972	0.484139
$87$	19.3351	2.07294
$88$	−11.9788	−1.27695
$89$	−15.5150	−1.64459	−0.822293	−	0.569065i	$-0.807305\pi$
$89$	−15.5150	−1.64459	−0.822293	+	0.569065i	$0.807305\pi$
$90$	0	0
$91$	−4.53368	−0.475259
$92$	−1.69079	−0.176277
$93$	−17.7432	−1.83989
$94$	4.64239	0.478826
$95$	0	0
$96$	15.7399	1.60644
$97$	−3.55398	−0.360852	−0.180426	−	0.983589i	$-0.557748\pi$
$97$	−3.55398	−0.360852	−0.180426	+	0.983589i	$0.557748\pi$
$98$	−0.556064	−0.0561709
$99$	−33.0019	−3.31681
$100$	0	0
$101$	−3.38799	−0.337118	−0.168559	−	0.985692i	$-0.553911\pi$
$101$	−3.38799	−0.337118	−0.168559	+	0.985692i	$0.553911\pi$
$102$	−4.90838	−0.486002
$103$	2.78534	0.274448	0.137224	−	0.990540i	$-0.456182\pi$
$103$	2.78534	0.274448	0.137224	+	0.990540i	$0.456182\pi$
$104$	−9.30455	−0.912386
$105$	0	0
$106$	3.10043	0.301140
$107$	−1.73675	−0.167898	−0.0839488	−	0.996470i	$-0.526753\pi$
$107$	−1.73675	−0.167898	−0.0839488	+	0.996470i	$0.526753\pi$
$108$	13.2018	1.27034
$109$	3.56604	0.341565	0.170783	−	0.985309i	$-0.445370\pi$
$109$	3.56604	0.341565	0.170783	+	0.985309i	$0.445370\pi$
$110$	0	0
$111$	5.99059	0.568601
$112$	2.24037	0.211695
$113$	0.659452	0.0620360	0.0310180	−	0.999519i	$-0.490125\pi$
$113$	0.659452	0.0620360	0.0310180	+	0.999519i	$0.490125\pi$
$114$	6.18453	0.579234
$115$	0	0
$116$	11.1128	1.03180
$117$	−25.6342	−2.36989
$118$	6.33214	0.582921
$119$	−3.00055	−0.275060
$120$	0	0
$121$	23.0674	2.09703
$122$	0.738638	0.0668731
$123$	−24.2060	−2.18258
$124$	−10.1979	−0.915798
$125$	0	0
$126$	−3.14408	−0.280097
$127$	−17.8961	−1.58802	−0.794010	−	0.607904i	$-0.792010\pi$
$127$	−17.8961	−1.58802	−0.794010	+	0.607904i	$0.792010\pi$
$128$	11.5380	1.01983
$129$	23.7524	2.09128
$130$	0	0
$131$	−15.0918	−1.31857	−0.659287	−	0.751891i	$-0.729142\pi$
$131$	−15.0918	−1.31857	−0.659287	+	0.751891i	$0.729142\pi$
$132$	−29.0317	−2.52689
$133$	3.78067	0.327826
$134$	−0.519678	−0.0448933
$135$	0	0
$136$	−6.15808	−0.528051
$137$	6.30784	0.538915	0.269458	−	0.963012i	$-0.413156\pi$
$137$	6.30784	0.538915	0.269458	+	0.963012i	$0.413156\pi$
$138$	1.63583	0.139251
$139$	17.7445	1.50507	0.752534	−	0.658553i	$-0.228831\pi$
$139$	17.7445	1.50507	0.752534	+	0.658553i	$0.228831\pi$
$140$	0	0
$141$	24.5601	2.06833
$142$	0.257932	0.0216452
$143$	26.4619	2.21285
$144$	12.6674	1.05562
$145$	0	0
$146$	7.70888	0.637991
$147$	−2.94180	−0.242635
$148$	3.44308	0.283019
$149$	−13.9107	−1.13961	−0.569805	−	0.821780i	$-0.692981\pi$
$149$	−13.9107	−1.13961	−0.569805	+	0.821780i	$0.692981\pi$
$150$	0	0
$151$	11.1839	0.910136	0.455068	−	0.890457i	$-0.349615\pi$
$151$	11.1839	0.910136	0.455068	+	0.890457i	$0.349615\pi$
$152$	7.75914	0.629349
$153$	−16.9656	−1.37159
$154$	3.24559	0.261537
$155$	0	0
$156$	−22.5504	−1.80548
$157$	−8.49448	−0.677933	−0.338967	−	0.940798i	$-0.610077\pi$
$157$	−8.49448	−0.677933	−0.338967	+	0.940798i	$0.610077\pi$
$158$	−2.97299	−0.236519
$159$	16.4025	1.30080
$160$	0	0
$161$	1.00000	0.0788110
$162$	−3.34038	−0.262445
$163$	14.1094	1.10514	0.552569	−	0.833467i	$-0.313648\pi$
$163$	14.1094	1.10514	0.552569	+	0.833467i	$0.313648\pi$
$164$	−13.9123	−1.08637
$165$	0	0
$166$	5.21524	0.404781
$167$	−7.76284	−0.600706	−0.300353	−	0.953828i	$-0.597105\pi$
$167$	−7.76284	−0.600706	−0.300353	+	0.953828i	$0.597105\pi$
$168$	−6.03750	−0.465803
$169$	7.55427	0.581098
$170$	0	0
$171$	21.3766	1.63471
$172$	13.6516	1.04093
$173$	−1.75008	−0.133056	−0.0665279	−	0.997785i	$-0.521192\pi$
$173$	−1.75008	−0.133056	−0.0665279	+	0.997785i	$0.521192\pi$
$174$	−10.7515	−0.815072
$175$	0	0
$176$	−13.0764	−0.985671
$177$	33.4995	2.51798
$178$	8.62732	0.646645
$179$	10.9293	0.816895	0.408447	−	0.912782i	$-0.366070\pi$
$179$	10.9293	0.816895	0.408447	+	0.912782i	$0.366070\pi$
$180$	0	0
$181$	−4.67911	−0.347795	−0.173898	−	0.984764i	$-0.555636\pi$
$181$	−4.67911	−0.347795	−0.173898	+	0.984764i	$0.555636\pi$
$182$	2.52102	0.186870
$183$	3.90769	0.288865
$184$	2.05232	0.151299
$185$	0	0
$186$	9.86638	0.723438
$187$	17.5134	1.28070
$188$	14.1159	1.02950
$189$	−7.80805	−0.567952
$190$	0	0
$191$	−4.40537	−0.318761	−0.159381	−	0.987217i	$-0.550950\pi$
$191$	−4.40537	−0.318761	−0.159381	+	0.987217i	$0.550950\pi$
$192$	4.42905	0.319639
$193$	−3.64903	−0.262663	−0.131332	−	0.991338i	$-0.541925\pi$
$193$	−3.64903	−0.262663	−0.131332	+	0.991338i	$0.541925\pi$
$194$	1.97624	0.141886
$195$	0	0
$196$	−1.69079	−0.120771
$197$	2.55924	0.182338	0.0911690	−	0.995835i	$-0.470940\pi$
$197$	2.55924	0.182338	0.0911690	+	0.995835i	$0.470940\pi$
$198$	18.3512	1.30416
$199$	−22.5810	−1.60072	−0.800362	−	0.599517i	$-0.795360\pi$
$199$	−22.5810	−1.60072	−0.800362	+	0.599517i	$0.795360\pi$
$200$	0	0
$201$	−2.74930	−0.193921
$202$	1.88394	0.132553
$203$	−6.57253	−0.461301
$204$	−14.9246	−1.04493
$205$	0	0
$206$	−1.54883	−0.107912
$207$	5.65418	0.392992
$208$	−10.1571	−0.704269
$209$	−22.0667	−1.52639
$210$	0	0
$211$	0.522552	0.0359739	0.0179870	−	0.999838i	$-0.494274\pi$
$211$	0.522552	0.0359739	0.0179870	+	0.999838i	$0.494274\pi$
$212$	9.42729	0.647469
$213$	1.36456	0.0934983
$214$	0.965742	0.0660167
$215$	0	0
$216$	−16.0246	−1.09034
$217$	6.03143	0.409440
$218$	−1.98295	−0.134302
$219$	40.7830	2.75586
$220$	0	0
$221$	13.6035	0.915073
$222$	−3.33115	−0.223572
$223$	−8.32105	−0.557219	−0.278609	−	0.960405i	$-0.589873\pi$
$223$	−8.32105	−0.557219	−0.278609	+	0.960405i	$0.589873\pi$
$224$	−5.35042	−0.357490
$225$	0	0
$226$	−0.366697	−0.0243924
$227$	17.2081	1.14214	0.571071	−	0.820901i	$-0.306528\pi$
$227$	17.2081	1.14214	0.571071	+	0.820901i	$0.306528\pi$
$228$	18.8050	1.24539
$229$	−0.742660	−0.0490763	−0.0245382	−	0.999699i	$-0.507812\pi$
$229$	−0.742660	−0.0490763	−0.0245382	+	0.999699i	$0.507812\pi$
$230$	0	0
$231$	17.1705	1.12973
$232$	−13.4889	−0.885591
$233$	26.4704	1.73414	0.867068	−	0.498190i	$-0.166002\pi$
$233$	26.4704	1.73414	0.867068	+	0.498190i	$0.166002\pi$
$234$	14.2543	0.931831
$235$	0	0
$236$	19.2538	1.25332
$237$	−15.7283	−1.02166
$238$	1.66850	0.108153
$239$	30.4200	1.96771	0.983854	−	0.178975i	$-0.0572782\pi$
$239$	30.4200	1.96771	0.983854	+	0.178975i	$0.0572782\pi$
$240$	0	0
$241$	−9.00232	−0.579891	−0.289945	−	0.957043i	$-0.593637\pi$
$241$	−9.00232	−0.579891	−0.289945	+	0.957043i	$0.593637\pi$
$242$	−12.8269	−0.824546
$243$	5.75222	0.369005
$244$	2.24594	0.143781
$245$	0	0
$246$	13.4601	0.858182
$247$	−17.1404	−1.09062
$248$	12.3784	0.786029
$249$	27.5907	1.74849
$250$	0	0
$251$	7.47145	0.471594	0.235797	−	0.971802i	$-0.424230\pi$
$251$	7.47145	0.471594	0.235797	+	0.971802i	$0.424230\pi$
$252$	−9.56004	−0.602226
$253$	−5.83673	−0.366952
$254$	9.95137	0.624404
$255$	0	0
$256$	−3.40477	−0.212798
$257$	−20.9292	−1.30553	−0.652765	−	0.757561i	$-0.726391\pi$
$257$	−20.9292	−1.30553	−0.652765	+	0.757561i	$0.726391\pi$
$258$	−13.2079	−0.822285
$259$	−2.03637	−0.126534
$260$	0	0
$261$	−37.1623	−2.30029
$262$	8.39199	0.518459
$263$	22.5996	1.39355	0.696776	−	0.717289i	$-0.254617\pi$
$263$	22.5996	1.39355	0.696776	+	0.717289i	$0.254617\pi$
$264$	35.2392	2.16883
$265$	0	0
$266$	−2.10230	−0.128900
$267$	45.6420	2.79324
$268$	−1.58016	−0.0965234
$269$	−29.3453	−1.78921	−0.894607	−	0.446853i	$-0.852545\pi$
$269$	−29.3453	−1.78921	−0.894607	+	0.446853i	$0.852545\pi$
$270$	0	0
$271$	9.19954	0.558832	0.279416	−	0.960170i	$-0.409859\pi$
$271$	9.19954	0.558832	0.279416	+	0.960170i	$0.409859\pi$
$272$	−6.72233	−0.407601
$273$	13.3372	0.807203
$274$	−3.50756	−0.211900
$275$	0	0
$276$	4.97397	0.299398
$277$	22.6276	1.35956	0.679779	−	0.733417i	$-0.262076\pi$
$277$	22.6276	1.35956	0.679779	+	0.733417i	$0.262076\pi$
$278$	−9.86707	−0.591788
$279$	34.1028	2.04168
$280$	0	0
$281$	30.5333	1.82146	0.910731	−	0.413001i	$-0.135519\pi$
$281$	30.5333	1.82146	0.910731	+	0.413001i	$0.135519\pi$
$282$	−13.6570	−0.813261
$283$	2.35942	0.140253	0.0701263	−	0.997538i	$-0.477660\pi$
$283$	2.35942	0.140253	0.0701263	+	0.997538i	$0.477660\pi$
$284$	0.784280	0.0465385
$285$	0	0
$286$	−14.7145	−0.870086
$287$	8.22829	0.485701
$288$	−30.2522	−1.78263
$289$	−7.99671	−0.470395
$290$	0	0
$291$	10.4551	0.612888
$292$	23.4400	1.37172
$293$	24.7967	1.44864	0.724320	−	0.689464i	$-0.242154\pi$
$293$	24.7967	1.44864	0.724320	+	0.689464i	$0.242154\pi$
$294$	1.63583	0.0954034
$295$	0	0
$296$	−4.17927	−0.242915
$297$	45.5735	2.64444
$298$	7.73525	0.448091
$299$	−4.53368	−0.262190
$300$	0	0
$301$	−8.07411	−0.465384
$302$	−6.21899	−0.357862
$303$	9.96678	0.572577
$304$	8.47009	0.485793
$305$	0	0
$306$	9.43398	0.539304
$307$	11.4272	0.652186	0.326093	−	0.945338i	$-0.394268\pi$
$307$	11.4272	0.652186	0.326093	+	0.945338i	$0.394268\pi$
$308$	9.86869	0.562321
$309$	−8.19392	−0.466136
$310$	0	0
$311$	22.5386	1.27805	0.639024	−	0.769186i	$-0.279338\pi$
$311$	22.5386	1.27805	0.639024	+	0.769186i	$0.279338\pi$
$312$	27.3721	1.54964
$313$	10.3999	0.587838	0.293919	−	0.955830i	$-0.405040\pi$
$313$	10.3999	0.587838	0.293919	+	0.955830i	$0.405040\pi$
$314$	4.72347	0.266561
$315$	0	0
$316$	−9.03982	−0.508529
$317$	−33.7523	−1.89572	−0.947859	−	0.318691i	$-0.896757\pi$
$317$	−33.7523	−1.89572	−0.947859	+	0.318691i	$0.896757\pi$
$318$	−9.12083	−0.511471
$319$	38.3621	2.14786
$320$	0	0
$321$	5.10915	0.285165
$322$	−0.556064	−0.0309882
$323$	−11.3441	−0.631202
$324$	−10.1569	−0.564273
$325$	0	0
$326$	−7.84575	−0.434536
$327$	−10.4906	−0.580130
$328$	16.8871	0.932432
$329$	−8.34866	−0.460277
$330$	0	0
$331$	14.1287	0.776584	0.388292	−	0.921536i	$-0.373065\pi$
$331$	14.1287	0.776584	0.388292	+	0.921536i	$0.373065\pi$
$332$	15.8577	0.870304
$333$	−11.5140	−0.630963
$334$	4.31663	0.236196
$335$	0	0
$336$	−6.59071	−0.359553
$337$	31.3327	1.70680	0.853400	−	0.521257i	$-0.174537\pi$
$337$	31.3327	1.70680	0.853400	+	0.521257i	$0.174537\pi$
$338$	−4.20066	−0.228486
$339$	−1.93997	−0.105365
$340$	0	0
$341$	−35.2038	−1.90639
$342$	−11.8868	−0.642762
$343$	1.00000	0.0539949
$344$	−16.5706	−0.893429
$345$	0	0
$346$	0.973154	0.0523171
$347$	−4.54684	−0.244087	−0.122044	−	0.992525i	$-0.538945\pi$
$347$	−4.54684	−0.244087	−0.122044	+	0.992525i	$0.538945\pi$
$348$	−32.6916	−1.75245
$349$	−7.94309	−0.425184	−0.212592	−	0.977141i	$-0.568190\pi$
$349$	−7.94309	−0.425184	−0.212592	+	0.977141i	$0.568190\pi$
$350$	0	0
$351$	35.3992	1.88947
$352$	31.2289	1.66451
$353$	−1.27705	−0.0679703	−0.0339851	−	0.999422i	$-0.510820\pi$
$353$	−1.27705	−0.0679703	−0.0339851	+	0.999422i	$0.510820\pi$
$354$	−18.6279	−0.990061
$355$	0	0
$356$	26.2326	1.39033
$357$	8.82701	0.467175
$358$	−6.07739	−0.321200
$359$	26.1035	1.37769	0.688846	−	0.724908i	$-0.258118\pi$
$359$	26.1035	1.37769	0.688846	+	0.724908i	$0.258118\pi$
$360$	0	0
$361$	−4.70652	−0.247712
$362$	2.60188	0.136752
$363$	−67.8595	−3.56170
$364$	7.66552	0.401782
$365$	0	0
$366$	−2.17292	−0.113581
$367$	−14.4492	−0.754244	−0.377122	−	0.926164i	$-0.623086\pi$
$367$	−14.4492	−0.754244	−0.377122	+	0.926164i	$0.623086\pi$
$368$	2.24037	0.116787
$369$	46.5242	2.42195
$370$	0	0
$371$	−5.57566	−0.289474
$372$	30.0002	1.55544
$373$	3.35638	0.173787	0.0868934	−	0.996218i	$-0.472306\pi$
$373$	3.35638	0.173787	0.0868934	+	0.996218i	$0.472306\pi$
$374$	−9.73856	−0.503569
$375$	0	0
$376$	−17.1341	−0.883624
$377$	29.7978	1.53466
$378$	4.34178	0.223317
$379$	−19.1322	−0.982756	−0.491378	−	0.870947i	$-0.663507\pi$
$379$	−19.1322	−0.982756	−0.491378	+	0.870947i	$0.663507\pi$
$380$	0	0
$381$	52.6467	2.69717
$382$	2.44967	0.125336
$383$	17.0711	0.872294	0.436147	−	0.899876i	$-0.356343\pi$
$383$	17.0711	0.872294	0.436147	+	0.899876i	$0.356343\pi$
$384$	−33.9425	−1.73212
$385$	0	0
$386$	2.02910	0.103278
$387$	−45.6524	−2.32064
$388$	6.00904	0.305063
$389$	−1.49125	−0.0756096	−0.0378048	−	0.999285i	$-0.512037\pi$
$389$	−1.49125	−0.0756096	−0.0378048	+	0.999285i	$0.512037\pi$
$390$	0	0
$391$	−3.00055	−0.151744
$392$	2.05232	0.103658
$393$	44.3969	2.23953
$394$	−1.42310	−0.0716947
$395$	0	0
$396$	55.7993	2.80402
$397$	28.8111	1.44599	0.722995	−	0.690854i	$-0.242765\pi$
$397$	28.8111	1.44599	0.722995	+	0.690854i	$0.242765\pi$
$398$	12.5565	0.629399
$399$	−11.1220	−0.556795
$400$	0	0
$401$	−20.2553	−1.01150	−0.505750	−	0.862680i	$-0.668784\pi$
$401$	−20.2553	−1.01150	−0.505750	+	0.862680i	$0.668784\pi$
$402$	1.52879	0.0762490
$403$	−27.3446	−1.36213
$404$	5.72839	0.284998
$405$	0	0
$406$	3.65475	0.181382
$407$	11.8857	0.589154
$408$	18.1158	0.896866
$409$	−31.8965	−1.57718	−0.788589	−	0.614921i	$-0.789188\pi$
$409$	−31.8965	−1.57718	−0.788589	+	0.614921i	$0.789188\pi$
$410$	0	0
$411$	−18.5564	−0.915320
$412$	−4.70944	−0.232018
$413$	−11.3874	−0.560339
$414$	−3.14408	−0.154523
$415$	0	0
$416$	24.2571	1.18930
$417$	−52.2007	−2.55628
$418$	12.2705	0.600171
$419$	17.1127	0.836008	0.418004	−	0.908445i	$-0.362730\pi$
$419$	17.1127	0.836008	0.418004	+	0.908445i	$0.362730\pi$
$420$	0	0
$421$	−28.6851	−1.39802	−0.699012	−	0.715110i	$-0.746377\pi$
$421$	−28.6851	−1.39802	−0.699012	+	0.715110i	$0.746377\pi$
$422$	−0.290572	−0.0141448
$423$	−47.2048	−2.29518
$424$	−11.4430	−0.555723
$425$	0	0
$426$	−0.758784	−0.0367632
$427$	−1.32833	−0.0642825
$428$	2.93648	0.141940
$429$	−77.8455	−3.75841
$430$	0	0
$431$	−8.90443	−0.428911	−0.214456	−	0.976734i	$-0.568798\pi$
$431$	−8.90443	−0.428911	−0.214456	+	0.976734i	$0.568798\pi$
$432$	−17.4929	−0.841628
$433$	−1.16674	−0.0560701	−0.0280350	−	0.999607i	$-0.508925\pi$
$433$	−1.16674	−0.0560701	−0.0280350	+	0.999607i	$0.508925\pi$
$434$	−3.35386	−0.160990
$435$	0	0
$436$	−6.02944	−0.288758
$437$	3.78067	0.180854
$438$	−22.6780	−1.08360
$439$	11.1182	0.530642	0.265321	−	0.964160i	$-0.414522\pi$
$439$	11.1182	0.530642	0.265321	+	0.964160i	$0.414522\pi$
$440$	0	0
$441$	5.65418	0.269247
$442$	−7.56443	−0.359803
$443$	−35.4840	−1.68589	−0.842947	−	0.537996i	$-0.819182\pi$
$443$	−35.4840	−1.68589	−0.842947	+	0.537996i	$0.819182\pi$
$444$	−10.1288	−0.480693
$445$	0	0
$446$	4.62704	0.219097
$447$	40.9225	1.93557
$448$	−1.50556	−0.0711309
$449$	−10.8685	−0.512917	−0.256458	−	0.966555i	$-0.582556\pi$
$449$	−10.8685	−0.512917	−0.256458	+	0.966555i	$0.582556\pi$
$450$	0	0
$451$	−48.0263	−2.26147
$452$	−1.11500	−0.0524450
$453$	−32.9009	−1.54582
$454$	−9.56881	−0.449086
$455$	0	0
$456$	−22.8258	−1.06892
$457$	−33.9574	−1.58846	−0.794230	−	0.607617i	$-0.792126\pi$
$457$	−33.9574	−1.58846	−0.794230	+	0.607617i	$0.792126\pi$
$458$	0.412966	0.0192966
$459$	23.4284	1.09355
$460$	0	0
$461$	1.09199	0.0508592	0.0254296	−	0.999677i	$-0.491905\pi$
$461$	1.09199	0.0508592	0.0254296	+	0.999677i	$0.491905\pi$
$462$	−9.54788	−0.444208
$463$	−12.3391	−0.573446	−0.286723	−	0.958014i	$-0.592566\pi$
$463$	−12.3391	−0.573446	−0.286723	+	0.958014i	$0.592566\pi$
$464$	−14.7249	−0.683586
$465$	0	0
$466$	−14.7193	−0.681856
$467$	5.83611	0.270063	0.135031	−	0.990841i	$-0.456886\pi$
$467$	5.83611	0.270063	0.135031	+	0.990841i	$0.456886\pi$
$468$	43.3422	2.00349
$469$	0.934565	0.0431542
$470$	0	0
$471$	24.9890	1.15143
$472$	−23.3706	−1.07572
$473$	47.1264	2.16687
$474$	8.74595	0.401715
$475$	0	0
$476$	5.07331	0.232535
$477$	−31.5258	−1.44347
$478$	−16.9155	−0.773696
$479$	13.6388	0.623175	0.311587	−	0.950218i	$-0.399139\pi$
$479$	13.6388	0.623175	0.311587	+	0.950218i	$0.399139\pi$
$480$	0	0
$481$	9.23225	0.420954
$482$	5.00587	0.228011
$483$	−2.94180	−0.133856
$484$	−39.0021	−1.77282
$485$	0	0
$486$	−3.19860	−0.145092
$487$	28.9247	1.31070	0.655351	−	0.755324i	$-0.272521\pi$
$487$	28.9247	1.31070	0.655351	+	0.755324i	$0.272521\pi$
$488$	−2.72616	−0.123407
$489$	−41.5071	−1.87702
$490$	0	0
$491$	−19.5800	−0.883635	−0.441817	−	0.897105i	$-0.645666\pi$
$491$	−19.5800	−0.883635	−0.441817	+	0.897105i	$0.645666\pi$
$492$	40.9273	1.84514
$493$	19.7212	0.888198
$494$	9.53114	0.428826
$495$	0	0
$496$	13.5126	0.606734
$497$	−0.463853	−0.0208067
$498$	−15.3422	−0.687499
$499$	24.8556	1.11269	0.556345	−	0.830951i	$-0.312203\pi$
$499$	24.8556	1.11269	0.556345	+	0.830951i	$0.312203\pi$
$500$	0	0
$501$	22.8367	1.02027
$502$	−4.15460	−0.185429
$503$	18.8310	0.839632	0.419816	−	0.907609i	$-0.362095\pi$
$503$	18.8310	0.839632	0.419816	+	0.907609i	$0.362095\pi$
$504$	11.6042	0.516891
$505$	0	0
$506$	3.24559	0.144284
$507$	−22.2232	−0.986965
$508$	30.2586	1.34251
$509$	−16.7777	−0.743659	−0.371830	−	0.928301i	$-0.621269\pi$
$509$	−16.7777	−0.743659	−0.371830	+	0.928301i	$0.621269\pi$
$510$	0	0
$511$	−13.8633	−0.613276
$512$	−21.1828	−0.936156
$513$	−29.5197	−1.30333
$514$	11.6380	0.513330
$515$	0	0
$516$	−40.1604	−1.76796
$517$	48.7288	2.14309
$518$	1.13235	0.0497527
$519$	5.14837	0.225988
$520$	0	0
$521$	−41.5077	−1.81849	−0.909244	−	0.416265i	$-0.863339\pi$
$521$	−41.5077	−1.81849	−0.909244	+	0.416265i	$0.863339\pi$
$522$	20.6646	0.904465
$523$	18.9013	0.826497	0.413248	−	0.910618i	$-0.364394\pi$
$523$	18.9013	0.826497	0.413248	+	0.910618i	$0.364394\pi$
$524$	25.5171	1.11472
$525$	0	0
$526$	−12.5668	−0.547940
$527$	−18.0976	−0.788344
$528$	38.4681	1.67411
$529$	1.00000	0.0434783
$530$	0	0
$531$	−64.3866	−2.79414
$532$	−6.39233	−0.277143
$533$	−37.3045	−1.61584
$534$	−25.3798	−1.09829
$535$	0	0
$536$	1.91802	0.0828460
$537$	−32.1518	−1.38745
$538$	16.3179	0.703513
$539$	−5.83673	−0.251406
$540$	0	0
$541$	18.1259	0.779292	0.389646	−	0.920965i	$-0.372597\pi$
$541$	18.1259	0.779292	0.389646	+	0.920965i	$0.372597\pi$
$542$	−5.11553	−0.219731
$543$	13.7650	0.590712
$544$	16.0542	0.688318
$545$	0	0
$546$	−7.41632	−0.317389
$547$	−3.93211	−0.168125	−0.0840625	−	0.996460i	$-0.526790\pi$
$547$	−3.93211	−0.168125	−0.0840625	+	0.996460i	$0.526790\pi$
$548$	−10.6653	−0.455597
$549$	−7.51063	−0.320546
$550$	0	0
$551$	−24.8486	−1.05859
$552$	−6.03750	−0.256973
$553$	5.34650	0.227356
$554$	−12.5824	−0.534574
$555$	0	0
$556$	−30.0023	−1.27238
$557$	0.373738	0.0158358	0.00791790	−	0.999969i	$-0.497480\pi$
$557$	0.373738	0.0158358	0.00791790	+	0.999969i	$0.497480\pi$
$558$	−18.9633	−0.802781
$559$	36.6054	1.54825
$560$	0	0
$561$	−51.5208	−2.17521
$562$	−16.9784	−0.716192
$563$	24.9236	1.05041	0.525203	−	0.850977i	$-0.323989\pi$
$563$	24.9236	1.05041	0.525203	+	0.850977i	$0.323989\pi$
$564$	−41.5260	−1.74856
$565$	0	0
$566$	−1.31199	−0.0551469
$567$	6.00719	0.252278
$568$	−0.951974	−0.0399439
$569$	36.0336	1.51061	0.755304	−	0.655374i	$-0.227489\pi$
$569$	36.0336	1.51061	0.755304	+	0.655374i	$0.227489\pi$
$570$	0	0
$571$	19.2769	0.806711	0.403356	−	0.915043i	$-0.367844\pi$
$571$	19.2769	0.806711	0.403356	+	0.915043i	$0.367844\pi$
$572$	−44.7415	−1.87074
$573$	12.9597	0.541400
$574$	−4.57546	−0.190976
$575$	0	0
$576$	−8.51269	−0.354695
$577$	−30.5824	−1.27316	−0.636581	−	0.771210i	$-0.719652\pi$
$577$	−30.5824	−1.27316	−0.636581	+	0.771210i	$0.719652\pi$
$578$	4.44668	0.184958
$579$	10.7347	0.446120
$580$	0	0
$581$	−9.37885	−0.389100
$582$	−5.81370	−0.240986
$583$	32.5436	1.34782
$584$	−28.4519	−1.17735
$585$	0	0
$586$	−13.7886	−0.569600
$587$	14.0638	0.580476	0.290238	−	0.956954i	$-0.406266\pi$
$587$	14.0638	0.580476	0.290238	+	0.956954i	$0.406266\pi$
$588$	4.97397	0.205123
$589$	22.8029	0.939575
$590$	0	0
$591$	−7.52875	−0.309692
$592$	−4.56221	−0.187506
$593$	−20.1065	−0.825677	−0.412838	−	0.910804i	$-0.635463\pi$
$593$	−20.1065	−0.825677	−0.412838	+	0.910804i	$0.635463\pi$
$594$	−25.3418	−1.03979
$595$	0	0
$596$	23.5201	0.963422
$597$	66.4287	2.71875
$598$	2.52102	0.103092
$599$	30.6532	1.25246	0.626228	−	0.779640i	$-0.284598\pi$
$599$	30.6532	1.25246	0.626228	+	0.779640i	$0.284598\pi$
$600$	0	0
$601$	−30.3579	−1.23832	−0.619162	−	0.785263i	$-0.712528\pi$
$601$	−30.3579	−1.23832	−0.619162	+	0.785263i	$0.712528\pi$
$602$	4.48972	0.182987
$603$	5.28420	0.215189
$604$	−18.9097	−0.769426
$605$	0	0
$606$	−5.54217	−0.225135
$607$	−42.9081	−1.74159	−0.870794	−	0.491648i	$-0.836395\pi$
$607$	−42.9081	−1.74159	−0.870794	+	0.491648i	$0.836395\pi$
$608$	−20.2282	−0.820361
$609$	19.3351	0.783497
$610$	0	0
$611$	37.8502	1.53125
$612$	28.6854	1.15954
$613$	26.0709	1.05299	0.526497	−	0.850177i	$-0.323505\pi$
$613$	26.0709	1.05299	0.526497	+	0.850177i	$0.323505\pi$
$614$	−6.35427	−0.256437
$615$	0	0
$616$	−11.9788	−0.482640
$617$	−11.6484	−0.468947	−0.234474	−	0.972122i	$-0.575337\pi$
$617$	−11.6484	−0.468947	−0.234474	+	0.972122i	$0.575337\pi$
$618$	4.55635	0.183283
$619$	−33.7130	−1.35504	−0.677519	−	0.735505i	$-0.736945\pi$
$619$	−33.7130	−1.35504	−0.677519	+	0.735505i	$0.736945\pi$
$620$	0	0
$621$	−7.80805	−0.313326
$622$	−12.5329	−0.502524
$623$	−15.5150	−0.621595
$624$	29.8802	1.19616
$625$	0	0
$626$	−5.78302	−0.231136
$627$	64.9159	2.59249
$628$	14.3624	0.573122
$629$	6.11022	0.243630
$630$	0	0
$631$	15.1337	0.602463	0.301232	−	0.953551i	$-0.402602\pi$
$631$	15.1337	0.602463	0.301232	+	0.953551i	$0.402602\pi$
$632$	10.9727	0.436471
$633$	−1.53724	−0.0610999
$634$	18.7684	0.745390
$635$	0	0
$636$	−27.7332	−1.09969
$637$	−4.53368	−0.179631
$638$	−21.3318	−0.844533
$639$	−2.62271	−0.103753
$640$	0	0
$641$	−27.8573	−1.10030	−0.550148	−	0.835067i	$-0.685429\pi$
$641$	−27.8573	−1.10030	−0.550148	+	0.835067i	$0.685429\pi$
$642$	−2.84102	−0.112126
$643$	−6.31762	−0.249142	−0.124571	−	0.992211i	$-0.539756\pi$
$643$	−6.31762	−0.249142	−0.124571	+	0.992211i	$0.539756\pi$
$644$	−1.69079	−0.0666266
$645$	0	0
$646$	6.30804	0.248186
$647$	−41.2329	−1.62103	−0.810516	−	0.585717i	$-0.800813\pi$
$647$	−41.2329	−1.62103	−0.810516	+	0.585717i	$0.800813\pi$
$648$	12.3287	0.484315
$649$	66.4654	2.60899
$650$	0	0
$651$	−17.7432	−0.695413
$652$	−23.8561	−0.934279
$653$	6.44419	0.252181	0.126090	−	0.992019i	$-0.459757\pi$
$653$	6.44419	0.252181	0.126090	+	0.992019i	$0.459757\pi$
$654$	5.83343	0.228105
$655$	0	0
$656$	18.4344	0.719742
$657$	−78.3855	−3.05811
$658$	4.64239	0.180979
$659$	38.5261	1.50076	0.750382	−	0.661004i	$-0.229869\pi$
$659$	38.5261	1.50076	0.750382	+	0.661004i	$0.229869\pi$
$660$	0	0
$661$	36.0103	1.40064	0.700319	−	0.713830i	$-0.253041\pi$
$661$	36.0103	1.40064	0.700319	+	0.713830i	$0.253041\pi$
$662$	−7.85646	−0.305350
$663$	−40.0188	−1.55420
$664$	−19.2484	−0.746981
$665$	0	0
$666$	6.40251	0.248092
$667$	−6.57253	−0.254490
$668$	13.1253	0.507835
$669$	24.4789	0.946407
$670$	0	0
$671$	7.75311	0.299306
$672$	15.7399	0.607178
$673$	22.1405	0.853452	0.426726	−	0.904381i	$-0.359667\pi$
$673$	22.1405	0.853452	0.426726	+	0.904381i	$0.359667\pi$
$674$	−17.4230	−0.671108
$675$	0	0
$676$	−12.7727	−0.491258
$677$	16.0708	0.617652	0.308826	−	0.951119i	$-0.400064\pi$
$677$	16.0708	0.617652	0.308826	+	0.951119i	$0.400064\pi$
$678$	1.07875	0.0414291
$679$	−3.55398	−0.136389
$680$	0	0
$681$	−50.6228	−1.93987
$682$	19.5756	0.749587
$683$	−7.83259	−0.299706	−0.149853	−	0.988708i	$-0.547880\pi$
$683$	−7.83259	−0.299706	−0.149853	+	0.988708i	$0.547880\pi$
$684$	−36.1434	−1.38198
$685$	0	0
$686$	−0.556064	−0.0212306
$687$	2.18475	0.0833536
$688$	−18.0890	−0.689635
$689$	25.2783	0.963026
$690$	0	0
$691$	32.3186	1.22946	0.614729	−	0.788738i	$-0.289265\pi$
$691$	32.3186	1.22946	0.614729	+	0.788738i	$0.289265\pi$
$692$	2.95902	0.112485
$693$	−33.0019	−1.25364
$694$	2.52833	0.0959742
$695$	0	0
$696$	39.6817	1.50413
$697$	−24.6894	−0.935177
$698$	4.41686	0.167181
$699$	−77.8707	−2.94534
$700$	0	0
$701$	−28.3506	−1.07079	−0.535395	−	0.844602i	$-0.679837\pi$
$701$	−28.3506	−1.07079	−0.535395	+	0.844602i	$0.679837\pi$
$702$	−19.6842	−0.742934
$703$	−7.69884	−0.290367
$704$	8.78752	0.331192
$705$	0	0
$706$	0.710119	0.0267257
$707$	−3.38799	−0.127418
$708$	−56.6408	−2.12869
$709$	21.7165	0.815580	0.407790	−	0.913076i	$-0.366299\pi$
$709$	21.7165	0.815580	0.407790	+	0.913076i	$0.366299\pi$
$710$	0	0
$711$	30.2300	1.13371
$712$	−31.8417	−1.19332
$713$	6.03143	0.225879
$714$	−4.90838	−0.183692
$715$	0	0
$716$	−18.4792	−0.690600
$717$	−89.4895	−3.34205
$718$	−14.5152	−0.541704
$719$	−22.0087	−0.820786	−0.410393	−	0.911909i	$-0.634608\pi$
$719$	−22.0087	−0.820786	−0.410393	+	0.911909i	$0.634608\pi$
$720$	0	0
$721$	2.78534	0.103732
$722$	2.61713	0.0973994
$723$	26.4830	0.984914
$724$	7.91140	0.294025
$725$	0	0
$726$	37.7342	1.40045
$727$	42.1067	1.56165	0.780825	−	0.624749i	$-0.214799\pi$
$727$	42.1067	1.56165	0.780825	+	0.624749i	$0.214799\pi$
$728$	−9.30455	−0.344850
$729$	−34.9435	−1.29420
$730$	0	0
$731$	24.2268	0.896059
$732$	−6.60709	−0.244205
$733$	−28.8687	−1.06629	−0.533144	−	0.846024i	$-0.678990\pi$
$733$	−28.8687	−1.06629	−0.533144	+	0.846024i	$0.678990\pi$
$734$	8.03470	0.296566
$735$	0	0
$736$	−5.35042	−0.197219
$737$	−5.45480	−0.200930
$738$	−25.8704	−0.952304
$739$	11.8863	0.437244	0.218622	−	0.975810i	$-0.429844\pi$
$739$	11.8863	0.437244	0.218622	+	0.975810i	$0.429844\pi$
$740$	0	0
$741$	50.4235	1.85235
$742$	3.10043	0.113820
$743$	−27.7349	−1.01750	−0.508748	−	0.860916i	$-0.669891\pi$
$743$	−27.7349	−1.01750	−0.508748	+	0.860916i	$0.669891\pi$
$744$	−36.4148	−1.33503
$745$	0	0
$746$	−1.86636	−0.0683323
$747$	−53.0297	−1.94025
$748$	−29.6115	−1.08270
$749$	−1.73675	−0.0634593
$750$	0	0
$751$	47.5057	1.73351	0.866754	−	0.498735i	$-0.166202\pi$
$751$	47.5057	1.73351	0.866754	+	0.498735i	$0.166202\pi$
$752$	−18.7041	−0.682067
$753$	−21.9795	−0.800978
$754$	−16.5695	−0.603425
$755$	0	0
$756$	13.2018	0.480145
$757$	−11.7959	−0.428731	−0.214365	−	0.976754i	$-0.568768\pi$
$757$	−11.7959	−0.428731	−0.214365	+	0.976754i	$0.568768\pi$
$758$	10.6387	0.386416
$759$	17.1705	0.623249
$760$	0	0
$761$	16.7265	0.606335	0.303168	−	0.952937i	$-0.401956\pi$
$761$	16.7265	0.606335	0.303168	+	0.952937i	$0.401956\pi$
$762$	−29.2749	−1.06052
$763$	3.56604	0.129099
$764$	7.44857	0.269480
$765$	0	0
$766$	−9.49264	−0.342983
$767$	51.6270	1.86414
$768$	10.0161	0.361426
$769$	8.92416	0.321814	0.160907	−	0.986970i	$-0.448558\pi$
$769$	8.92416	0.321814	0.160907	+	0.986970i	$0.448558\pi$
$770$	0	0
$771$	61.5696	2.21737
$772$	6.16976	0.222055
$773$	29.0376	1.04441	0.522205	−	0.852820i	$-0.325110\pi$
$773$	29.0376	1.04441	0.522205	+	0.852820i	$0.325110\pi$
$774$	25.3857	0.912469
$775$	0	0
$776$	−7.29389	−0.261835
$777$	5.99059	0.214911
$778$	0.829232	0.0297294
$779$	31.1085	1.11458
$780$	0	0
$781$	2.70738	0.0968779
$782$	1.66850	0.0596653
$783$	51.3187	1.83398
$784$	2.24037	0.0800131
$785$	0	0
$786$	−24.6875	−0.880575
$787$	38.9409	1.38809	0.694047	−	0.719929i	$-0.255826\pi$
$787$	38.9409	1.38809	0.694047	+	0.719929i	$0.255826\pi$
$788$	−4.32714	−0.154148
$789$	−66.4835	−2.36687
$790$	0	0
$791$	0.659452	0.0234474
$792$	−67.7303	−2.40669
$793$	6.02224	0.213856
$794$	−16.0208	−0.568558
$795$	0	0
$796$	38.1798	1.35325
$797$	−49.2406	−1.74419	−0.872097	−	0.489334i	$-0.837240\pi$
$797$	−49.2406	−1.74419	−0.872097	+	0.489334i	$0.837240\pi$
$798$	6.18453	0.218930
$799$	25.0506	0.886225
$800$	0	0
$801$	−87.7245	−3.09959
$802$	11.2632	0.397718
$803$	80.9163	2.85547
$804$	4.64850	0.163940
$805$	0	0
$806$	15.2053	0.535585
$807$	86.3280	3.03889
$808$	−6.95323	−0.244614
$809$	−35.3578	−1.24311	−0.621557	−	0.783369i	$-0.713500\pi$
$809$	−35.3578	−1.24311	−0.621557	+	0.783369i	$0.713500\pi$
$810$	0	0
$811$	−34.5859	−1.21447	−0.607237	−	0.794521i	$-0.707722\pi$
$811$	−34.5859	−1.21447	−0.607237	+	0.794521i	$0.707722\pi$
$812$	11.1128	0.389983
$813$	−27.0632	−0.949148
$814$	−6.60922	−0.231653
$815$	0	0
$816$	19.7757	0.692289
$817$	−30.5256	−1.06795
$818$	17.7365	0.620141
$819$	−25.6342	−0.895733
$820$	0	0
$821$	−27.7500	−0.968482	−0.484241	−	0.874935i	$-0.660904\pi$
$821$	−27.7500	−0.968482	−0.484241	+	0.874935i	$0.660904\pi$
$822$	10.3185	0.359901
$823$	47.5114	1.65614	0.828072	−	0.560621i	$-0.189438\pi$
$823$	47.5114	1.65614	0.828072	+	0.560621i	$0.189438\pi$
$824$	5.71641	0.199141
$825$	0	0
$826$	6.33214	0.220323
$827$	−0.848974	−0.0295217	−0.0147609	−	0.999891i	$-0.504699\pi$
$827$	−0.848974	−0.0295217	−0.0147609	+	0.999891i	$0.504699\pi$
$828$	−9.56004	−0.332234
$829$	27.4250	0.952511	0.476255	−	0.879307i	$-0.341994\pi$
$829$	27.4250	0.952511	0.476255	+	0.879307i	$0.341994\pi$
$830$	0	0
$831$	−66.5657	−2.30914
$832$	6.82572	0.236639
$833$	−3.00055	−0.103963
$834$	29.0269	1.00512
$835$	0	0
$836$	37.3103	1.29040
$837$	−47.0937	−1.62780
$838$	−9.51573	−0.328715
$839$	−18.3548	−0.633677	−0.316838	−	0.948480i	$-0.602621\pi$
$839$	−18.3548	−0.633677	−0.316838	+	0.948480i	$0.602621\pi$
$840$	0	0
$841$	14.1982	0.489593
$842$	15.9507	0.549699
$843$	−89.8227	−3.09366
$844$	−0.883527	−0.0304122
$845$	0	0
$846$	26.2489	0.902455
$847$	23.0674	0.792604
$848$	−12.4915	−0.428961
$849$	−6.94092	−0.238212
$850$	0	0
$851$	−2.03637	−0.0698058
$852$	−2.30719	−0.0790431
$853$	−1.87925	−0.0643444	−0.0321722	−	0.999482i	$-0.510242\pi$
$853$	−1.87925	−0.0643444	−0.0321722	+	0.999482i	$0.510242\pi$
$854$	0.738638	0.0252757
$855$	0	0
$856$	−3.56435	−0.121827
$857$	44.2840	1.51271	0.756357	−	0.654160i	$-0.226978\pi$
$857$	44.2840	1.51271	0.756357	+	0.654160i	$0.226978\pi$
$858$	43.2870	1.47780
$859$	−39.2482	−1.33913	−0.669566	−	0.742753i	$-0.733520\pi$
$859$	−39.2482	−1.33913	−0.669566	+	0.742753i	$0.733520\pi$
$860$	0	0
$861$	−24.2060	−0.824937
$862$	4.95143	0.168646
$863$	−26.3069	−0.895496	−0.447748	−	0.894160i	$-0.647774\pi$
$863$	−26.3069	−0.895496	−0.447748	+	0.894160i	$0.647774\pi$
$864$	41.7764	1.42126
$865$	0	0
$866$	0.648783	0.0220466
$867$	23.5247	0.798941
$868$	−10.1979	−0.346139
$869$	−31.2060	−1.05859
$870$	0	0
$871$	−4.23702	−0.143566
$872$	7.31865	0.247841
$873$	−20.0948	−0.680107
$874$	−2.10230	−0.0711112
$875$	0	0
$876$	−68.9557	−2.32980
$877$	31.8646	1.07599	0.537995	−	0.842948i	$-0.319182\pi$
$877$	31.8646	1.07599	0.537995	+	0.842948i	$0.319182\pi$
$878$	−6.18242	−0.208646
$879$	−72.9470	−2.46044
$880$	0	0
$881$	18.1105	0.610159	0.305079	−	0.952327i	$-0.401317\pi$
$881$	18.1105	0.610159	0.305079	+	0.952327i	$0.401317\pi$
$882$	−3.14408	−0.105867
$883$	35.3555	1.18981	0.594904	−	0.803797i	$-0.297190\pi$
$883$	35.3555	1.18981	0.594904	+	0.803797i	$0.297190\pi$
$884$	−23.0008	−0.773599
$885$	0	0
$886$	19.7314	0.662888
$887$	33.1313	1.11244	0.556220	−	0.831035i	$-0.312251\pi$
$887$	33.1313	1.11244	0.556220	+	0.831035i	$0.312251\pi$
$888$	12.2946	0.412579
$889$	−17.8961	−0.600215
$890$	0	0
$891$	−35.0623	−1.17463
$892$	14.0692	0.471071
$893$	−31.5635	−1.05623
$894$	−22.7555	−0.761059
$895$	0	0
$896$	11.5380	0.385458
$897$	13.3372	0.445315
$898$	6.04359	0.201677
$899$	−39.6418	−1.32213
$900$	0	0
$901$	16.7300	0.557359
$902$	26.7057	0.889202
$903$	23.7524	0.790430
$904$	1.35340	0.0450136
$905$	0	0
$906$	18.2950	0.607811
$907$	20.1891	0.670369	0.335185	−	0.942152i	$-0.391201\pi$
$907$	20.1891	0.670369	0.335185	+	0.942152i	$0.391201\pi$
$908$	−29.0953	−0.965563
$909$	−19.1563	−0.635374
$910$	0	0
$911$	46.2226	1.53142	0.765711	−	0.643185i	$-0.222387\pi$
$911$	46.2226	1.53142	0.765711	+	0.643185i	$0.222387\pi$
$912$	−24.9173	−0.825094
$913$	54.7418	1.81169
$914$	18.8825	0.624577
$915$	0	0
$916$	1.25568	0.0414890
$917$	−15.0918	−0.498374
$918$	−13.0277	−0.429979
$919$	0.945128	0.0311769	0.0155885	−	0.999878i	$-0.495038\pi$
$919$	0.945128	0.0311769	0.0155885	+	0.999878i	$0.495038\pi$
$920$	0	0
$921$	−33.6166	−1.10770
$922$	−0.607219	−0.0199977
$923$	2.10296	0.0692199
$924$	−29.0317	−0.955073
$925$	0	0
$926$	6.86132	0.225477
$927$	15.7488	0.517260
$928$	35.1658	1.15437
$929$	52.3596	1.71786	0.858931	−	0.512091i	$-0.171129\pi$
$929$	52.3596	1.71786	0.858931	+	0.512091i	$0.171129\pi$
$930$	0	0
$931$	3.78067	0.123907
$932$	−44.7560	−1.46603
$933$	−66.3041	−2.17070
$934$	−3.24525	−0.106188
$935$	0	0
$936$	−52.6096	−1.71960
$937$	3.44931	0.112684	0.0563420	−	0.998412i	$-0.482056\pi$
$937$	3.44931	0.112684	0.0563420	+	0.998412i	$0.482056\pi$
$938$	−0.519678	−0.0169681
$939$	−30.5945	−0.998412
$940$	0	0
$941$	−21.9612	−0.715914	−0.357957	−	0.933738i	$-0.616527\pi$
$941$	−21.9612	−0.715914	−0.357957	+	0.933738i	$0.616527\pi$
$942$	−13.8955	−0.452740
$943$	8.22829	0.267950
$944$	−25.5120	−0.830346
$945$	0	0
$946$	−26.2053	−0.852007
$947$	3.89069	0.126430	0.0632152	−	0.998000i	$-0.479865\pi$
$947$	3.89069	0.126430	0.0632152	+	0.998000i	$0.479865\pi$
$948$	26.5933	0.863711
$949$	62.8518	2.04026
$950$	0	0
$951$	99.2924	3.21978
$952$	−6.15808	−0.199584
$953$	21.1223	0.684219	0.342110	−	0.939660i	$-0.388859\pi$
$953$	21.1223	0.684219	0.342110	+	0.939660i	$0.388859\pi$
$954$	17.5304	0.567566
$955$	0	0
$956$	−51.4339	−1.66349
$957$	−112.853	−3.64804
$958$	−7.58407	−0.245030
$959$	6.30784	0.203691
$960$	0	0
$961$	5.37814	0.173488
$962$	−5.13372	−0.165518
$963$	−9.81987	−0.316441
$964$	15.2211	0.490238
$965$	0	0
$966$	1.63583	0.0526319
$967$	37.2720	1.19859	0.599293	−	0.800530i	$-0.295449\pi$
$967$	37.2720	1.19859	0.599293	+	0.800530i	$0.295449\pi$
$968$	47.3415	1.52162
$969$	33.3720	1.07206
$970$	0	0
$971$	41.1224	1.31968	0.659840	−	0.751406i	$-0.270624\pi$
$971$	41.1224	1.31968	0.659840	+	0.751406i	$0.270624\pi$
$972$	−9.72582	−0.311956
$973$	17.7445	0.568862
$974$	−16.0840	−0.515364
$975$	0	0
$976$	−2.97595	−0.0952579
$977$	−8.63075	−0.276122	−0.138061	−	0.990424i	$-0.544087\pi$
$977$	−8.63075	−0.276122	−0.138061	+	0.990424i	$0.544087\pi$
$978$	23.0806	0.738037
$979$	90.5567	2.89421
$980$	0	0
$981$	20.1630	0.643756
$982$	10.8878	0.347442
$983$	−36.0054	−1.14839	−0.574197	−	0.818717i	$-0.694686\pi$
$983$	−36.0054	−1.14839	−0.574197	+	0.818717i	$0.694686\pi$
$984$	−49.6783	−1.58369
$985$	0	0
$986$	−10.9662	−0.349237
$987$	24.5601	0.781756
$988$	28.9808	0.922002
$989$	−8.07411	−0.256742
$990$	0	0
$991$	−59.7253	−1.89724	−0.948619	−	0.316420i	$-0.897519\pi$
$991$	−59.7253	−1.89724	−0.948619	+	0.316420i	$0.897519\pi$
$992$	−32.2707	−1.02460
$993$	−41.5638	−1.31899
$994$	0.257932	0.00818111
$995$	0	0
$996$	−46.6501	−1.47817
$997$	−33.0350	−1.04623	−0.523114	−	0.852263i	$-0.675230\pi$
$997$	−33.0350	−1.04623	−0.523114	+	0.852263i	$0.675230\pi$
$998$	−13.8213	−0.437506
$999$	15.9001	0.503056

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4025.2.a.be.1.9		21
5.2	odd	4		805.2.c.c.484.18	✓	42
5.3	odd	4		805.2.c.c.484.25	yes	42
5.4	even	2		4025.2.a.bd.1.13		21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
805.2.c.c.484.18	✓	42	5.2	odd	4
805.2.c.c.484.25	yes	42	5.3	odd	4
4025.2.a.bd.1.13		21	5.4	even	2
4025.2.a.be.1.9		21	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4025.a (trivial)

\(f(q)\)	\(=\)	\(q-0.556064 q^{2} -2.94180 q^{3} -1.69079 q^{4} +1.63583 q^{6} +1.00000 q^{7} +2.05232 q^{8} +5.65418 q^{9} +O(q^{10})\)	\(q-0.556064 q^{2} -2.94180 q^{3} -1.69079 q^{4} +1.63583 q^{6} +1.00000 q^{7} +2.05232 q^{8} +5.65418 q^{9} -5.83673 q^{11} +4.97397 q^{12} -4.53368 q^{13} -0.556064 q^{14} +2.24037 q^{16} -3.00055 q^{17} -3.14408 q^{18} +3.78067 q^{19} -2.94180 q^{21} +3.24559 q^{22} +1.00000 q^{23} -6.03750 q^{24} +2.52102 q^{26} -7.80805 q^{27} -1.69079 q^{28} -6.57253 q^{29} +6.03143 q^{31} -5.35042 q^{32} +17.1705 q^{33} +1.66850 q^{34} -9.56004 q^{36} -2.03637 q^{37} -2.10230 q^{38} +13.3372 q^{39} +8.22829 q^{41} +1.63583 q^{42} -8.07411 q^{43} +9.86869 q^{44} -0.556064 q^{46} -8.34866 q^{47} -6.59071 q^{48} +1.00000 q^{49} +8.82701 q^{51} +7.66552 q^{52} -5.57566 q^{53} +4.34178 q^{54} +2.05232 q^{56} -11.1220 q^{57} +3.65475 q^{58} -11.3874 q^{59} -1.32833 q^{61} -3.35386 q^{62} +5.65418 q^{63} -1.50556 q^{64} -9.54788 q^{66} +0.934565 q^{67} +5.07331 q^{68} -2.94180 q^{69} -0.463853 q^{71} +11.6042 q^{72} -13.8633 q^{73} +1.13235 q^{74} -6.39233 q^{76} -5.83673 q^{77} -7.41632 q^{78} +5.34650 q^{79} +6.00719 q^{81} -4.57546 q^{82} -9.37885 q^{83} +4.97397 q^{84} +4.48972 q^{86} +19.3351 q^{87} -11.9788 q^{88} -15.5150 q^{89} -4.53368 q^{91} -1.69079 q^{92} -17.7432 q^{93} +4.64239 q^{94} +15.7399 q^{96} -3.55398 q^{97} -0.556064 q^{98} -33.0019 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 2 q^{2} - q^{3} + 30 q^{4} + 6 q^{6} + 21 q^{7} + 6 q^{8} + 30 q^{9}+O(q^{10}) \) 21 * q + 2 * q^2 - q^3 + 30 * q^4 + 6 * q^6 + 21 * q^7 + 6 * q^8 + 30 * q^9	\( 21 q + 2 q^{2} - q^{3} + 30 q^{4} + 6 q^{6} + 21 q^{7} + 6 q^{8} + 30 q^{9} + 7 q^{11} - 22 q^{12} - 3 q^{13} + 2 q^{14} + 56 q^{16} + 7 q^{17} + 24 q^{19} - q^{21} + 4 q^{22} + 21 q^{23} + 24 q^{24} - 2 q^{26} - 19 q^{27} + 30 q^{28} + 11 q^{29} + 46 q^{31} - 6 q^{32} - 3 q^{33} + 28 q^{34} + 58 q^{36} + 24 q^{37} - 4 q^{38} + 31 q^{39} + 14 q^{41} + 6 q^{42} + 18 q^{43} + 12 q^{44} + 2 q^{46} - 25 q^{47} - 36 q^{48} + 21 q^{49} + 17 q^{51} - 8 q^{52} + 22 q^{53} - 6 q^{54} + 6 q^{56} + 40 q^{57} + 6 q^{58} + 10 q^{59} + 38 q^{61} - 54 q^{62} + 30 q^{63} + 100 q^{64} + 38 q^{66} + 12 q^{67} + 18 q^{68} - q^{69} + 56 q^{71} + 42 q^{72} - 40 q^{73} - 20 q^{74} + 60 q^{76} + 7 q^{77} + 38 q^{78} + 49 q^{79} + 57 q^{81} + 16 q^{82} - 2 q^{83} - 22 q^{84} + 16 q^{86} - 23 q^{87} + 12 q^{88} + 28 q^{89} - 3 q^{91} + 30 q^{92} - 30 q^{93} + 66 q^{94} + 46 q^{96} - q^{97} + 2 q^{98} - 2 q^{99}+O(q^{100}) \) 21 * q + 2 * q^2 - q^3 + 30 * q^4 + 6 * q^6 + 21 * q^7 + 6 * q^8 + 30 * q^9 + 7 * q^11 - 22 * q^12 - 3 * q^13 + 2 * q^14 + 56 * q^16 + 7 * q^17 + 24 * q^19 - q^21 + 4 * q^22 + 21 * q^23 + 24 * q^24 - 2 * q^26 - 19 * q^27 + 30 * q^28 + 11 * q^29 + 46 * q^31 - 6 * q^32 - 3 * q^33 + 28 * q^34 + 58 * q^36 + 24 * q^37 - 4 * q^38 + 31 * q^39 + 14 * q^41 + 6 * q^42 + 18 * q^43 + 12 * q^44 + 2 * q^46 - 25 * q^47 - 36 * q^48 + 21 * q^49 + 17 * q^51 - 8 * q^52 + 22 * q^53 - 6 * q^54 + 6 * q^56 + 40 * q^57 + 6 * q^58 + 10 * q^59 + 38 * q^61 - 54 * q^62 + 30 * q^63 + 100 * q^64 + 38 * q^66 + 12 * q^67 + 18 * q^68 - q^69 + 56 * q^71 + 42 * q^72 - 40 * q^73 - 20 * q^74 + 60 * q^76 + 7 * q^77 + 38 * q^78 + 49 * q^79 + 57 * q^81 + 16 * q^82 - 2 * q^83 - 22 * q^84 + 16 * q^86 - 23 * q^87 + 12 * q^88 + 28 * q^89 - 3 * q^91 + 30 * q^92 - 30 * q^93 + 66 * q^94 + 46 * q^96 - q^97 + 2 * q^98 - 2 * q^99

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