LMFDB - Embedded newform 425.2.a.i.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [425,2,Mod(1,425)] 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("425.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	425.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$3.39364208590$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1893456.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 10x^{3} + 10x^{2} + 23x - 25 $ x^5 - x^4 - 10x^3 + 10x^2 + 23*x - 25
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.96189$ of defining polynomial
Character	$\chi$	$=$	425.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.150980 q^{2} -1.96189 q^{3} -1.97720 q^{4} -0.296207 q^{6} -1.54475 q^{7} -0.600480 q^{8} +0.849020 q^{9} +4.56006 q^{11} +3.87906 q^{12} -1.09756 q^{13} -0.233227 q^{14} +3.86375 q^{16} +1.00000 q^{17} +0.128185 q^{18} +4.67524 q^{19} +3.03063 q^{21} +0.688480 q^{22} +0.529434 q^{23} +1.17808 q^{24} -0.165710 q^{26} +4.21999 q^{27} +3.05428 q^{28} +8.06670 q^{29} -4.78005 q^{31} +1.78431 q^{32} -8.94634 q^{33} +0.150980 q^{34} -1.67869 q^{36} -5.27917 q^{37} +0.705870 q^{38} +2.15329 q^{39} -0.751460 q^{41} +0.457565 q^{42} +9.49340 q^{43} -9.01617 q^{44} +0.0799341 q^{46} +10.7419 q^{47} -7.58026 q^{48} -4.61376 q^{49} -1.96189 q^{51} +2.17010 q^{52} +0.0227951 q^{53} +0.637136 q^{54} +0.927590 q^{56} -9.17232 q^{57} +1.21791 q^{58} -3.56962 q^{59} +3.92378 q^{61} -0.721694 q^{62} -1.31152 q^{63} -7.45810 q^{64} -1.35072 q^{66} -9.75929 q^{67} -1.97720 q^{68} -1.03869 q^{69} +1.21216 q^{71} -0.509819 q^{72} +10.1135 q^{73} -0.797051 q^{74} -9.24392 q^{76} -7.04414 q^{77} +0.325105 q^{78} +14.4151 q^{79} -10.8262 q^{81} -0.113456 q^{82} -5.08949 q^{83} -5.99217 q^{84} +1.43332 q^{86} -15.8260 q^{87} -2.73822 q^{88} -17.6123 q^{89} +1.69545 q^{91} -1.04680 q^{92} +9.37794 q^{93} +1.62182 q^{94} -3.50062 q^{96} -6.78753 q^{97} -0.696587 q^{98} +3.87158 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - q^{2} + q^{3} + 11 q^{4} + 3 q^{6} + q^{7} - 9 q^{8} + 6 q^{9} + 4 q^{11} + 17 q^{12} - 3 q^{13} - 7 q^{14} + 27 q^{16} + 5 q^{17} - 22 q^{18} + 6 q^{19} - 5 q^{21} + 18 q^{22} + 4 q^{23} - 19 q^{24}+ \cdots - 14 q^{99}+O(q^{100}) $ 5 * q - q^2 + q^3 + 11 * q^4 + 3 * q^6 + q^7 - 9 * q^8 + 6 * q^9 + 4 * q^11 + 17 * q^12 - 3 * q^13 - 7 * q^14 + 27 * q^16 + 5 * q^17 - 22 * q^18 + 6 * q^19 - 5 * q^21 + 18 * q^22 + 4 * q^23 - 19 * q^24 - 5 * q^26 - 5 * q^27 - 15 * q^28 + 2 * q^29 + 21 * q^31 - 9 * q^32 + 12 * q^33 - q^34 + 22 * q^36 - 2 * q^37 - 34 * q^38 + 23 * q^39 - 8 * q^41 + 25 * q^42 - 4 * q^43 - 4 * q^44 - 6 * q^46 - 2 * q^47 + 51 * q^48 + 10 * q^49 + q^51 - 13 * q^52 + 21 * q^53 - 25 * q^54 - 29 * q^56 - 20 * q^57 + 22 * q^58 + 12 * q^59 - 2 * q^61 + 3 * q^62 + 8 * q^63 + 43 * q^64 - 28 * q^66 - 12 * q^67 + 11 * q^68 + 21 * q^71 - 58 * q^72 + 22 * q^73 - 50 * q^74 - 14 * q^76 - 32 * q^77 - 53 * q^78 + 41 * q^79 - 35 * q^81 + 28 * q^82 - 8 * q^83 - 71 * q^84 - 36 * q^86 - 4 * q^87 + 14 * q^88 - 10 * q^89 + 13 * q^91 + 56 * q^92 + 9 * q^93 - 10 * q^94 - 85 * q^96 - 20 * q^97 + 38 * q^98 - 14 * 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 $

$2$	0.150980	0.106759	0.0533796	−	0.998574i	$-0.483001\pi$
$2$	0.150980	0.106759	0.0533796	+	0.998574i	$0.483001\pi$
$3$	−1.96189	−1.13270	−0.566349	−	0.824165i	$-0.691645\pi$
$3$	−1.96189	−1.13270	−0.566349	+	0.824165i	$0.691645\pi$
$4$	−1.97720	−0.988602
$5$	0	0
$5$	0	0
$6$	−0.296207	−0.120926
$7$	−1.54475	−0.583859	−0.291930	−	0.956440i	$-0.594297\pi$
$7$	−1.54475	−0.583859	−0.291930	+	0.956440i	$0.594297\pi$
$8$	−0.600480	−0.212302
$9$	0.849020	0.283007
$10$	0	0
$11$	4.56006	1.37491	0.687455	−	0.726227i	$-0.258728\pi$
$11$	4.56006	1.37491	0.687455	+	0.726227i	$0.258728\pi$
$12$	3.87906	1.11979
$13$	−1.09756	−0.304408	−0.152204	−	0.988349i	$-0.548637\pi$
$13$	−1.09756	−0.304408	−0.152204	+	0.988349i	$0.548637\pi$
$14$	−0.233227	−0.0623324
$15$	0	0
$16$	3.86375	0.965937
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0.128185	0.0302136
$19$	4.67524	1.07257	0.536287	−	0.844036i	$-0.319826\pi$
$19$	4.67524	1.07257	0.536287	+	0.844036i	$0.319826\pi$
$20$	0	0
$21$	3.03063	0.661337
$22$	0.688480	0.146784
$23$	0.529434	0.110395	0.0551973	−	0.998475i	$-0.482421\pi$
$23$	0.529434	0.110395	0.0551973	+	0.998475i	$0.482421\pi$
$24$	1.17808	0.240474
$25$	0	0
$26$	−0.165710	−0.0324984
$27$	4.21999	0.812138
$28$	3.05428	0.577205
$29$	8.06670	1.49795	0.748974	−	0.662599i	$-0.230547\pi$
$29$	8.06670	1.49795	0.748974	+	0.662599i	$0.230547\pi$
$30$	0	0
$31$	−4.78005	−0.858522	−0.429261	−	0.903180i	$-0.641226\pi$
$31$	−4.78005	−0.858522	−0.429261	+	0.903180i	$0.641226\pi$
$32$	1.78431	0.315425
$33$	−8.94634	−1.55736
$34$	0.150980	0.0258929
$35$	0	0
$36$	−1.67869	−0.279781
$37$	−5.27917	−0.867889	−0.433945	−	0.900939i	$-0.642879\pi$
$37$	−5.27917	−0.867889	−0.433945	+	0.900939i	$0.642879\pi$
$38$	0.705870	0.114507
$39$	2.15329	0.344803
$40$	0	0
$41$	−0.751460	−0.117358	−0.0586792	−	0.998277i	$-0.518689\pi$
$41$	−0.751460	−0.117358	−0.0586792	+	0.998277i	$0.518689\pi$
$42$	0.457565	0.0706038
$43$	9.49340	1.44773	0.723865	−	0.689941i	$-0.242364\pi$
$43$	9.49340	1.44773	0.723865	+	0.689941i	$0.242364\pi$
$44$	−9.01617	−1.35924
$45$	0	0
$46$	0.0799341	0.0117856
$47$	10.7419	1.56687	0.783437	−	0.621472i	$-0.213465\pi$
$47$	10.7419	1.56687	0.783437	+	0.621472i	$0.213465\pi$
$48$	−7.58026	−1.09412
$49$	−4.61376	−0.659108
$50$	0	0
$51$	−1.96189	−0.274720
$52$	2.17010	0.300939
$53$	0.0227951	0.00313115	0.00156557	−	0.999999i	$-0.499502\pi$
$53$	0.0227951	0.00313115	0.00156557	+	0.999999i	$0.499502\pi$
$54$	0.637136	0.0867032
$55$	0	0
$56$	0.927590	0.123954
$57$	−9.17232	−1.21490
$58$	1.21791	0.159920
$59$	−3.56962	−0.464725	−0.232362	−	0.972629i	$-0.574646\pi$
$59$	−3.56962	−0.464725	−0.232362	+	0.972629i	$0.574646\pi$
$60$	0	0
$61$	3.92378	0.502389	0.251195	−	0.967937i	$-0.419177\pi$
$61$	3.92378	0.502389	0.251195	+	0.967937i	$0.419177\pi$
$62$	−0.721694	−0.0916552
$63$	−1.31152	−0.165236
$64$	−7.45810	−0.932263
$65$	0	0
$66$	−1.35072	−0.166262
$67$	−9.75929	−1.19229	−0.596144	−	0.802878i	$-0.703301\pi$
$67$	−9.75929	−1.19229	−0.596144	+	0.802878i	$0.703301\pi$
$68$	−1.97720	−0.239771
$69$	−1.03869	−0.125044
$70$	0	0
$71$	1.21216	0.143857	0.0719285	−	0.997410i	$-0.477085\pi$
$71$	1.21216	0.143857	0.0719285	+	0.997410i	$0.477085\pi$
$72$	−0.509819	−0.0600828
$73$	10.1135	1.18369	0.591845	−	0.806052i	$-0.298400\pi$
$73$	10.1135	1.18369	0.591845	+	0.806052i	$0.298400\pi$
$74$	−0.797051	−0.0926552
$75$	0	0
$76$	−9.24392	−1.06035
$77$	−7.04414	−0.802754
$78$	0.325105	0.0368109
$79$	14.4151	1.62183	0.810913	−	0.585166i	$-0.198971\pi$
$79$	14.4151	1.62183	0.810913	+	0.585166i	$0.198971\pi$
$80$	0	0
$81$	−10.8262	−1.20291
$82$	−0.113456	−0.0125291
$83$	−5.08949	−0.558644	−0.279322	−	0.960197i	$-0.590110\pi$
$83$	−5.08949	−0.558644	−0.279322	+	0.960197i	$0.590110\pi$
$84$	−5.99217	−0.653799
$85$	0	0
$86$	1.43332	0.154559
$87$	−15.8260	−1.69672
$88$	−2.73822	−0.291896
$89$	−17.6123	−1.86690	−0.933450	−	0.358707i	$-0.883218\pi$
$89$	−17.6123	−1.86690	−0.933450	+	0.358707i	$0.883218\pi$
$90$	0	0
$91$	1.69545	0.177732
$92$	−1.04680	−0.109136
$93$	9.37794	0.972447
$94$	1.62182	0.167278
$95$	0	0
$96$	−3.50062	−0.357281
$97$	−6.78753	−0.689170	−0.344585	−	0.938755i	$-0.611980\pi$
$97$	−6.78753	−0.689170	−0.344585	+	0.938755i	$0.611980\pi$
$98$	−0.696587	−0.0703659
$99$	3.87158	0.389108
$100$	0	0
$101$	−8.21768	−0.817690	−0.408845	−	0.912604i	$-0.634068\pi$
$101$	−8.21768	−0.817690	−0.408845	+	0.912604i	$0.634068\pi$
$102$	−0.296207	−0.0293289
$103$	17.2696	1.70163	0.850814	−	0.525466i	$-0.176109\pi$
$103$	17.2696	1.70163	0.850814	+	0.525466i	$0.176109\pi$
$104$	0.659062	0.0646264
$105$	0	0
$106$	0.00344161	0.000334279 0
$107$	15.3869	1.48750	0.743752	−	0.668455i	$-0.233044\pi$
$107$	15.3869	1.48750	0.743752	+	0.668455i	$0.233044\pi$
$108$	−8.34379	−0.802881
$109$	17.2221	1.64957	0.824787	−	0.565443i	$-0.191295\pi$
$109$	17.2221	1.64957	0.824787	+	0.565443i	$0.191295\pi$
$110$	0	0
$111$	10.3572	0.983057
$112$	−5.96851	−0.563972
$113$	18.9562	1.78325	0.891624	−	0.452777i	$-0.149567\pi$
$113$	18.9562	1.78325	0.891624	+	0.452777i	$0.149567\pi$
$114$	−1.38484	−0.129702
$115$	0	0
$116$	−15.9495	−1.48088
$117$	−0.931849	−0.0861495
$118$	−0.538943	−0.0496137
$119$	−1.54475	−0.141607
$120$	0	0
$121$	9.79415	0.890377
$122$	0.592414	0.0536347
$123$	1.47428	0.132932
$124$	9.45114	0.848737
$125$	0	0
$126$	−0.198014	−0.0176405
$127$	15.6123	1.38537	0.692684	−	0.721241i	$-0.256428\pi$
$127$	15.6123	1.38537	0.692684	+	0.721241i	$0.256428\pi$
$128$	−4.69465	−0.414952
$129$	−18.6250	−1.63984
$130$	0	0
$131$	−4.79329	−0.418791	−0.209396	−	0.977831i	$-0.567150\pi$
$131$	−4.79329	−0.418791	−0.209396	+	0.977831i	$0.567150\pi$
$132$	17.6888	1.53961
$133$	−7.22207	−0.626233
$134$	−1.47346	−0.127288
$135$	0	0
$136$	−0.600480	−0.0514907
$137$	−15.6750	−1.33921	−0.669603	−	0.742719i	$-0.733536\pi$
$137$	−15.6750	−1.33921	−0.669603	+	0.742719i	$0.733536\pi$
$138$	−0.156822	−0.0133496
$139$	8.33055	0.706588	0.353294	−	0.935512i	$-0.385062\pi$
$139$	8.33055	0.706588	0.353294	+	0.935512i	$0.385062\pi$
$140$	0	0
$141$	−21.0745	−1.77480
$142$	0.183012	0.0153581
$143$	−5.00494	−0.418534
$144$	3.28040	0.273367
$145$	0	0
$146$	1.52693	0.126370
$147$	9.05169	0.746571
$148$	10.4380	0.857998
$149$	−3.10739	−0.254568	−0.127284	−	0.991866i	$-0.540626\pi$
$149$	−3.10739	−0.254568	−0.127284	+	0.991866i	$0.540626\pi$
$150$	0	0
$151$	10.4088	0.847056	0.423528	−	0.905883i	$-0.360791\pi$
$151$	10.4088	0.847056	0.423528	+	0.905883i	$0.360791\pi$
$152$	−2.80739	−0.227709
$153$	0.849020	0.0686392
$154$	−1.06353	−0.0857014
$155$	0	0
$156$	−4.25750	−0.340873
$157$	−0.661967	−0.0528307	−0.0264154	−	0.999651i	$-0.508409\pi$
$157$	−0.661967	−0.0528307	−0.0264154	+	0.999651i	$0.508409\pi$
$158$	2.17640	0.173145
$159$	−0.0447215	−0.00354664
$160$	0	0
$161$	−0.817841	−0.0644549
$162$	−1.63455	−0.128422
$163$	−16.2543	−1.27314	−0.636569	−	0.771220i	$-0.719647\pi$
$163$	−16.2543	−1.27314	−0.636569	+	0.771220i	$0.719647\pi$
$164$	1.48579	0.116021
$165$	0	0
$166$	−0.768414	−0.0596405
$167$	19.9527	1.54398	0.771992	−	0.635632i	$-0.219260\pi$
$167$	19.9527	1.54398	0.771992	+	0.635632i	$0.219260\pi$
$168$	−1.81983	−0.140403
$169$	−11.7954	−0.907336
$170$	0	0
$171$	3.96937	0.303546
$172$	−18.7704	−1.43123
$173$	3.91717	0.297817	0.148908	−	0.988851i	$-0.452424\pi$
$173$	3.91717	0.297817	0.148908	+	0.988851i	$0.452424\pi$
$174$	−2.38941	−0.181141
$175$	0	0
$176$	17.6189	1.32808
$177$	7.00321	0.526393
$178$	−2.65911	−0.199309
$179$	3.14170	0.234821	0.117411	−	0.993083i	$-0.462541\pi$
$179$	3.14170	0.234821	0.117411	+	0.993083i	$0.462541\pi$
$180$	0	0
$181$	0.782087	0.0581320	0.0290660	−	0.999577i	$-0.490747\pi$
$181$	0.782087	0.0581320	0.0290660	+	0.999577i	$0.490747\pi$
$182$	0.255980	0.0189745
$183$	−7.69804	−0.569055
$184$	−0.317914	−0.0234370
$185$	0	0
$186$	1.41589	0.103818
$187$	4.56006	0.333465
$188$	−21.2390	−1.54901
$189$	−6.51882	−0.474174
$190$	0	0
$191$	12.4154	0.898348	0.449174	−	0.893444i	$-0.351718\pi$
$191$	12.4154	0.898348	0.449174	+	0.893444i	$0.351718\pi$
$192$	14.6320	1.05597
$193$	1.70465	0.122704	0.0613518	−	0.998116i	$-0.480459\pi$
$193$	1.70465	0.122704	0.0613518	+	0.998116i	$0.480459\pi$
$194$	−1.02478	−0.0735752
$195$	0	0
$196$	9.12234	0.651596
$197$	−16.2755	−1.15958	−0.579790	−	0.814766i	$-0.696866\pi$
$197$	−16.2755	−1.15958	−0.579790	+	0.814766i	$0.696866\pi$
$198$	0.584533	0.0415409
$199$	17.1946	1.21889	0.609447	−	0.792827i	$-0.291392\pi$
$199$	17.1946	1.21889	0.609447	+	0.792827i	$0.291392\pi$
$200$	0	0
$201$	19.1467	1.35050
$202$	−1.24071	−0.0872959
$203$	−12.4610	−0.874591
$204$	3.87906	0.271589
$205$	0	0
$206$	2.60738	0.181665
$207$	0.449500	0.0312424
$208$	−4.24069	−0.294039
$209$	21.3194	1.47469
$210$	0	0
$211$	2.01038	0.138400	0.0692000	−	0.997603i	$-0.477955\pi$
$211$	2.01038	0.138400	0.0692000	+	0.997603i	$0.477955\pi$
$212$	−0.0450706	−0.00309546
$213$	−2.37813	−0.162947
$214$	2.32312	0.158805
$215$	0	0
$216$	−2.53402	−0.172418
$217$	7.38397	0.501256
$218$	2.60019	0.176107
$219$	−19.8415	−1.34076
$220$	0	0
$221$	−1.09756	−0.0738298
$222$	1.56373	0.104950
$223$	−14.1074	−0.944701	−0.472351	−	0.881411i	$-0.656594\pi$
$223$	−14.1074	−0.944701	−0.472351	+	0.881411i	$0.656594\pi$
$224$	−2.75631	−0.184164
$225$	0	0
$226$	2.86201	0.190378
$227$	−15.7117	−1.04282	−0.521410	−	0.853306i	$-0.674594\pi$
$227$	−15.7117	−1.04282	−0.521410	+	0.853306i	$0.674594\pi$
$228$	18.1356	1.20106
$229$	16.5921	1.09644	0.548219	−	0.836335i	$-0.315306\pi$
$229$	16.5921	1.09644	0.548219	+	0.836335i	$0.315306\pi$
$230$	0	0
$231$	13.8198	0.909279
$232$	−4.84389	−0.318017
$233$	6.56378	0.430007	0.215004	−	0.976613i	$-0.431024\pi$
$233$	6.56378	0.430007	0.215004	+	0.976613i	$0.431024\pi$
$234$	−0.140691	−0.00919726
$235$	0	0
$236$	7.05787	0.459428
$237$	−28.2809	−1.83704
$238$	−0.233227	−0.0151178
$239$	−7.62182	−0.493015	−0.246507	−	0.969141i	$-0.579283\pi$
$239$	−7.62182	−0.493015	−0.246507	+	0.969141i	$0.579283\pi$
$240$	0	0
$241$	6.89554	0.444181	0.222090	−	0.975026i	$-0.428712\pi$
$241$	6.89554	0.444181	0.222090	+	0.975026i	$0.428712\pi$
$242$	1.47872	0.0950560
$243$	8.57991	0.550401
$244$	−7.75812	−0.496663
$245$	0	0
$246$	0.222588	0.0141917
$247$	−5.13136	−0.326500
$248$	2.87032	0.182266
$249$	9.98504	0.632776
$250$	0	0
$251$	−25.8326	−1.63054	−0.815270	−	0.579081i	$-0.803411\pi$
$251$	−25.8326	−1.63054	−0.815270	+	0.579081i	$0.803411\pi$
$252$	2.59314	0.163353
$253$	2.41425	0.151783
$254$	2.35715	0.147901
$255$	0	0
$256$	14.2074	0.887963
$257$	17.8491	1.11340	0.556698	−	0.830715i	$-0.312068\pi$
$257$	17.8491	1.11340	0.556698	+	0.830715i	$0.312068\pi$
$258$	−2.81201	−0.175068
$259$	8.15497	0.506725
$260$	0	0
$261$	6.84879	0.423929
$262$	−0.723692	−0.0447099
$263$	−14.3446	−0.884529	−0.442264	−	0.896885i	$-0.645825\pi$
$263$	−14.3446	−0.884529	−0.442264	+	0.896885i	$0.645825\pi$
$264$	5.37210	0.330630
$265$	0	0
$266$	−1.09039	−0.0668562
$267$	34.5534	2.11464
$268$	19.2961	1.17870
$269$	−3.91172	−0.238502	−0.119251	−	0.992864i	$-0.538049\pi$
$269$	−3.91172	−0.238502	−0.119251	+	0.992864i	$0.538049\pi$
$270$	0	0
$271$	28.4490	1.72816	0.864078	−	0.503358i	$-0.167902\pi$
$271$	28.4490	1.72816	0.864078	+	0.503358i	$0.167902\pi$
$272$	3.86375	0.234274
$273$	−3.32629	−0.201316
$274$	−2.36662	−0.142973
$275$	0	0
$276$	2.05371	0.123619
$277$	25.6839	1.54320	0.771598	−	0.636111i	$-0.219458\pi$
$277$	25.6839	1.54320	0.771598	+	0.636111i	$0.219458\pi$
$278$	1.25775	0.0754348
$279$	−4.05836	−0.242967
$280$	0	0
$281$	−20.7667	−1.23884	−0.619420	−	0.785060i	$-0.712632\pi$
$281$	−20.7667	−1.23884	−0.619420	+	0.785060i	$0.712632\pi$
$282$	−3.18184	−0.189476
$283$	−24.4689	−1.45452	−0.727262	−	0.686360i	$-0.759207\pi$
$283$	−24.4689	−1.45452	−0.727262	+	0.686360i	$0.759207\pi$
$284$	−2.39669	−0.142217
$285$	0	0
$286$	−0.755647	−0.0446824
$287$	1.16082	0.0685208
$288$	1.51491	0.0892672
$289$	1.00000	0.0588235
$290$	0	0
$291$	13.3164	0.780621
$292$	−19.9964	−1.17020
$293$	−28.1952	−1.64718	−0.823591	−	0.567185i	$-0.808033\pi$
$293$	−28.1952	−1.64718	−0.823591	+	0.567185i	$0.808033\pi$
$294$	1.36663	0.0797034
$295$	0	0
$296$	3.17003	0.184254
$297$	19.2434	1.11662
$298$	−0.469156	−0.0271775
$299$	−0.581085	−0.0336050
$300$	0	0
$301$	−14.6649	−0.845271
$302$	1.57153	0.0904311
$303$	16.1222	0.926196
$304$	18.0640	1.03604
$305$	0	0
$306$	0.128185	0.00732787
$307$	−0.0120595	−0.000688275 0	−0.000344137	−	1.00000i	$-0.500110\pi$
$307$	−0.0120595	−0.000688275 0	−0.000344137		1.00000i	$0.500110\pi$
$308$	13.9277	0.793605
$309$	−33.8812	−1.92743
$310$	0	0
$311$	−28.5605	−1.61951	−0.809757	−	0.586765i	$-0.800401\pi$
$311$	−28.5605	−1.61951	−0.809757	+	0.586765i	$0.800401\pi$
$312$	−1.29301	−0.0732022
$313$	−4.11691	−0.232702	−0.116351	−	0.993208i	$-0.537120\pi$
$313$	−4.11691	−0.232702	−0.116351	+	0.993208i	$0.537120\pi$
$314$	−0.0999440	−0.00564017
$315$	0	0
$316$	−28.5016	−1.60334
$317$	20.6297	1.15868	0.579338	−	0.815087i	$-0.303311\pi$
$317$	20.6297	1.15868	0.579338	+	0.815087i	$0.303311\pi$
$318$	−0.00675207	−0.000378637 0
$319$	36.7846	2.05954
$320$	0	0
$321$	−30.1874	−1.68489
$322$	−0.123478	−0.00688116
$323$	4.67524	0.260138
$324$	21.4057	1.18920
$325$	0	0
$326$	−2.45409	−0.135919
$327$	−33.7878	−1.86847
$328$	0.451237	0.0249154
$329$	−16.5936	−0.914834
$330$	0	0
$331$	−0.971759	−0.0534127	−0.0267063	−	0.999643i	$-0.508502\pi$
$331$	−0.971759	−0.0534127	−0.0267063	+	0.999643i	$0.508502\pi$
$332$	10.0630	0.552277
$333$	−4.48212	−0.245618
$334$	3.01246	0.164835
$335$	0	0
$336$	11.7096	0.638810
$337$	−6.01450	−0.327630	−0.163815	−	0.986491i	$-0.552380\pi$
$337$	−6.01450	−0.327630	−0.163815	+	0.986491i	$0.552380\pi$
$338$	−1.78087	−0.0968665
$339$	−37.1900	−2.01988
$340$	0	0
$341$	−21.7973	−1.18039
$342$	0.599298	0.0324063
$343$	17.9403	0.968686
$344$	−5.70060	−0.307356
$345$	0	0
$346$	0.591416	0.0317947
$347$	−5.33466	−0.286380	−0.143190	−	0.989695i	$-0.545736\pi$
$347$	−5.33466	−0.286380	−0.143190	+	0.989695i	$0.545736\pi$
$348$	31.2912	1.67739
$349$	−20.7601	−1.11126	−0.555632	−	0.831429i	$-0.687524\pi$
$349$	−20.7601	−1.11126	−0.555632	+	0.831429i	$0.687524\pi$
$350$	0	0
$351$	−4.63169	−0.247221
$352$	8.13656	0.433680
$353$	15.0618	0.801659	0.400829	−	0.916153i	$-0.368722\pi$
$353$	15.0618	0.801659	0.400829	+	0.916153i	$0.368722\pi$
$354$	1.05735	0.0561974
$355$	0	0
$356$	34.8231	1.84562
$357$	3.03063	0.160398
$358$	0.474335	0.0250694
$359$	−9.72949	−0.513503	−0.256751	−	0.966477i	$-0.582652\pi$
$359$	−9.72949	−0.513503	−0.256751	+	0.966477i	$0.582652\pi$
$360$	0	0
$361$	2.85791	0.150416
$362$	0.118080	0.00620613
$363$	−19.2151	−1.00853
$364$	−3.35225	−0.175706
$365$	0	0
$366$	−1.16225	−0.0607519
$367$	−8.62563	−0.450254	−0.225127	−	0.974329i	$-0.572280\pi$
$367$	−8.62563	−0.450254	−0.225127	+	0.974329i	$0.572280\pi$
$368$	2.04560	0.106634
$369$	−0.638005	−0.0332132
$370$	0	0
$371$	−0.0352126	−0.00182815
$372$	−18.5421	−0.961364
$373$	15.0922	0.781444	0.390722	−	0.920509i	$-0.372225\pi$
$373$	15.0922	0.781444	0.390722	+	0.920509i	$0.372225\pi$
$374$	0.688480	0.0356004
$375$	0	0
$376$	−6.45032	−0.332650
$377$	−8.85368	−0.455988
$378$	−0.984214	−0.0506225
$379$	−10.7124	−0.550261	−0.275131	−	0.961407i	$-0.588721\pi$
$379$	−10.7124	−0.550261	−0.275131	+	0.961407i	$0.588721\pi$
$380$	0	0
$381$	−30.6297	−1.56920
$382$	1.87448	0.0959070
$383$	−15.8670	−0.810764	−0.405382	−	0.914147i	$-0.632861\pi$
$383$	−15.8670	−0.810764	−0.405382	+	0.914147i	$0.632861\pi$
$384$	9.21039	0.470016
$385$	0	0
$386$	0.257369	0.0130997
$387$	8.06009	0.409717
$388$	13.4203	0.681315
$389$	22.3496	1.13317	0.566585	−	0.824003i	$-0.308264\pi$
$389$	22.3496	1.13317	0.566585	+	0.824003i	$0.308264\pi$
$390$	0	0
$391$	0.529434	0.0267746
$392$	2.77047	0.139930
$393$	9.40391	0.474364
$394$	−2.45728	−0.123796
$395$	0	0
$396$	−7.65491	−0.384674
$397$	−31.3003	−1.57092	−0.785458	−	0.618915i	$-0.787572\pi$
$397$	−31.3003	−1.57092	−0.785458	+	0.618915i	$0.787572\pi$
$398$	2.59605	0.130128
$399$	14.1689	0.709333
$400$	0	0
$401$	32.0373	1.59987	0.799934	−	0.600088i	$-0.204868\pi$
$401$	32.0373	1.59987	0.799934	+	0.600088i	$0.204868\pi$
$402$	2.89077	0.144179
$403$	5.24639	0.261341
$404$	16.2480	0.808370
$405$	0	0
$406$	−1.88137	−0.0933707
$407$	−24.0733	−1.19327
$408$	1.17808	0.0583235
$409$	−21.5037	−1.06329	−0.531645	−	0.846968i	$-0.678426\pi$
$409$	−21.5037	−1.06329	−0.531645	+	0.846968i	$0.678426\pi$
$410$	0	0
$411$	30.7527	1.51692
$412$	−34.1456	−1.68223
$413$	5.51416	0.271334
$414$	0.0678656	0.00333541
$415$	0	0
$416$	−1.95839	−0.0960178
$417$	−16.3436	−0.800352
$418$	3.21881	0.157437
$419$	−16.7246	−0.817048	−0.408524	−	0.912748i	$-0.633956\pi$
$419$	−16.7246	−0.817048	−0.408524	+	0.912748i	$0.633956\pi$
$420$	0	0
$421$	−16.9873	−0.827908	−0.413954	−	0.910298i	$-0.635853\pi$
$421$	−16.9873	−0.827908	−0.413954	+	0.910298i	$0.635853\pi$
$422$	0.303528	0.0147755
$423$	9.12012	0.443435
$424$	−0.0136880	−0.000664748 0
$425$	0	0
$426$	−0.359051	−0.0173961
$427$	−6.06125	−0.293325
$428$	−30.4230	−1.47055
$429$	9.81914	0.474073
$430$	0	0
$431$	−17.0941	−0.823392	−0.411696	−	0.911321i	$-0.635064\pi$
$431$	−17.0941	−0.823392	−0.411696	+	0.911321i	$0.635064\pi$
$432$	16.3050	0.784474
$433$	−9.95242	−0.478283	−0.239141	−	0.970985i	$-0.576866\pi$
$433$	−9.95242	−0.478283	−0.239141	+	0.970985i	$0.576866\pi$
$434$	1.11483	0.0535138
$435$	0	0
$436$	−34.0516	−1.63077
$437$	2.47523	0.118406
$438$	−2.99568	−0.143139
$439$	2.92877	0.139782	0.0698912	−	0.997555i	$-0.477735\pi$
$439$	2.92877	0.139782	0.0698912	+	0.997555i	$0.477735\pi$
$440$	0	0
$441$	−3.91717	−0.186532
$442$	−0.165710	−0.00788202
$443$	0.778484	0.0369869	0.0184934	−	0.999829i	$-0.494113\pi$
$443$	0.778484	0.0369869	0.0184934	+	0.999829i	$0.494113\pi$
$444$	−20.4782	−0.971853
$445$	0	0
$446$	−2.12994	−0.100856
$447$	6.09637	0.288349
$448$	11.5209	0.544310
$449$	−31.2804	−1.47621	−0.738107	−	0.674684i	$-0.764280\pi$
$449$	−31.2804	−1.47621	−0.738107	+	0.674684i	$0.764280\pi$
$450$	0	0
$451$	−3.42670	−0.161357
$452$	−37.4803	−1.76292
$453$	−20.4209	−0.959460
$454$	−2.37215	−0.111331
$455$	0	0
$456$	5.50780	0.257926
$457$	3.59265	0.168057	0.0840285	−	0.996463i	$-0.473221\pi$
$457$	3.59265	0.168057	0.0840285	+	0.996463i	$0.473221\pi$
$458$	2.50509	0.117055
$459$	4.21999	0.196972
$460$	0	0
$461$	−6.60392	−0.307575	−0.153788	−	0.988104i	$-0.549147\pi$
$461$	−6.60392	−0.307575	−0.153788	+	0.988104i	$0.549147\pi$
$462$	2.08652	0.0970739
$463$	−10.9502	−0.508898	−0.254449	−	0.967086i	$-0.581894\pi$
$463$	−10.9502	−0.508898	−0.254449	+	0.967086i	$0.581894\pi$
$464$	31.1677	1.44692
$465$	0	0
$466$	0.991002	0.0459073
$467$	−10.1288	−0.468704	−0.234352	−	0.972152i	$-0.575297\pi$
$467$	−10.1288	−0.468704	−0.234352	+	0.972152i	$0.575297\pi$
$468$	1.84246	0.0851676
$469$	15.0756	0.696128
$470$	0	0
$471$	1.29871	0.0598413
$472$	2.14349	0.0986619
$473$	43.2905	1.99050
$474$	−4.26986	−0.196121
$475$	0	0
$476$	3.05428	0.139993
$477$	0.0193535	0.000886135 0
$478$	−1.15075	−0.0526339
$479$	−7.73039	−0.353211	−0.176605	−	0.984282i	$-0.556512\pi$
$479$	−7.73039	−0.353211	−0.176605	+	0.984282i	$0.556512\pi$
$480$	0	0
$481$	5.79420	0.264193
$482$	1.04109	0.0474204
$483$	1.60452	0.0730080
$484$	−19.3650	−0.880229
$485$	0	0
$486$	1.29540	0.0587605
$487$	13.5925	0.615933	0.307966	−	0.951397i	$-0.400352\pi$
$487$	13.5925	0.615933	0.307966	+	0.951397i	$0.400352\pi$
$488$	−2.35615	−0.106658
$489$	31.8892	1.44208
$490$	0	0
$491$	38.8592	1.75369	0.876846	−	0.480772i	$-0.159643\pi$
$491$	38.8592	1.75369	0.876846	+	0.480772i	$0.159643\pi$
$492$	−2.91496	−0.131417
$493$	8.06670	0.363306
$494$	−0.774734	−0.0348569
$495$	0	0
$496$	−18.4689	−0.829279
$497$	−1.87248	−0.0839922
$498$	1.50754	0.0675547
$499$	−26.6894	−1.19478	−0.597391	−	0.801950i	$-0.703796\pi$
$499$	−26.6894	−1.19478	−0.597391	+	0.801950i	$0.703796\pi$
$500$	0	0
$501$	−39.1450	−1.74887
$502$	−3.90022	−0.174075
$503$	−0.591853	−0.0263894	−0.0131947	−	0.999913i	$-0.504200\pi$
$503$	−0.591853	−0.0263894	−0.0131947	+	0.999913i	$0.504200\pi$
$504$	0.787542	0.0350799
$505$	0	0
$506$	0.364504	0.0162042
$507$	23.1412	1.02774
$508$	−30.8687	−1.36958
$509$	11.6397	0.515922	0.257961	−	0.966155i	$-0.416949\pi$
$509$	11.6397	0.515922	0.257961	+	0.966155i	$0.416949\pi$
$510$	0	0
$511$	−15.6227	−0.691109
$512$	11.5343	0.509750
$513$	19.7295	0.871078
$514$	2.69486	0.118865
$515$	0	0
$516$	36.8255	1.62115
$517$	48.9839	2.15431
$518$	1.23124	0.0540976
$519$	−7.68506	−0.337337
$520$	0	0
$521$	−11.5584	−0.506382	−0.253191	−	0.967416i	$-0.581480\pi$
$521$	−11.5584	−0.506382	−0.253191	+	0.967416i	$0.581480\pi$
$522$	1.03403	0.0452584
$523$	14.5087	0.634420	0.317210	−	0.948355i	$-0.397254\pi$
$523$	14.5087	0.634420	0.317210	+	0.948355i	$0.397254\pi$
$524$	9.47731	0.414018
$525$	0	0
$526$	−2.16576	−0.0944316
$527$	−4.78005	−0.208222
$528$	−34.5664	−1.50431
$529$	−22.7197	−0.987813
$530$	0	0
$531$	−3.03068	−0.131520
$532$	14.2795	0.619095
$533$	0.824772	0.0357249
$534$	5.21689	0.225757
$535$	0	0
$536$	5.86026	0.253125
$537$	−6.16367	−0.265982
$538$	−0.590594	−0.0254623
$539$	−21.0390	−0.906214
$540$	0	0
$541$	−3.23241	−0.138972	−0.0694860	−	0.997583i	$-0.522136\pi$
$541$	−3.23241	−0.138972	−0.0694860	+	0.997583i	$0.522136\pi$
$542$	4.29525	0.184497
$543$	−1.53437	−0.0658461
$544$	1.78431	0.0765017
$545$	0	0
$546$	−0.502205	−0.0214924
$547$	38.7154	1.65535	0.827676	−	0.561206i	$-0.189662\pi$
$547$	38.7154	1.65535	0.827676	+	0.561206i	$0.189662\pi$
$548$	30.9927	1.32394
$549$	3.33137	0.142179
$550$	0	0
$551$	37.7138	1.60666
$552$	0.623714	0.0265470
$553$	−22.2677	−0.946919
$554$	3.87777	0.164750
$555$	0	0
$556$	−16.4712	−0.698535
$557$	0.669168	0.0283536	0.0141768	−	0.999900i	$-0.495487\pi$
$557$	0.669168	0.0283536	0.0141768	+	0.999900i	$0.495487\pi$
$558$	−0.612732	−0.0259390
$559$	−10.4196	−0.440701
$560$	0	0
$561$	−8.94634	−0.377715
$562$	−3.13537	−0.132258
$563$	−42.4772	−1.79020	−0.895101	−	0.445864i	$-0.852896\pi$
$563$	−42.4772	−1.79020	−0.895101	+	0.445864i	$0.852896\pi$
$564$	41.6687	1.75457
$565$	0	0
$566$	−3.69432	−0.155284
$567$	16.7238	0.702333
$568$	−0.727878	−0.0305411
$569$	36.3706	1.52474	0.762368	−	0.647143i	$-0.224036\pi$
$569$	36.3706	1.52474	0.762368	+	0.647143i	$0.224036\pi$
$570$	0	0
$571$	26.5970	1.11305	0.556525	−	0.830831i	$-0.312134\pi$
$571$	26.5970	1.11305	0.556525	+	0.830831i	$0.312134\pi$
$572$	9.89578	0.413763
$573$	−24.3577	−1.01756
$574$	0.175260	0.00731523
$575$	0	0
$576$	−6.33207	−0.263836
$577$	−2.30741	−0.0960586	−0.0480293	−	0.998846i	$-0.515294\pi$
$577$	−2.30741	−0.0960586	−0.0480293	+	0.998846i	$0.515294\pi$
$578$	0.150980	0.00627996
$579$	−3.34434	−0.138986
$580$	0	0
$581$	7.86198	0.326170
$582$	2.01052	0.0833386
$583$	0.103947	0.00430504
$584$	−6.07293	−0.251300
$585$	0	0
$586$	−4.25692	−0.175852
$587$	25.8915	1.06866	0.534329	−	0.845277i	$-0.320564\pi$
$587$	25.8915	1.06866	0.534329	+	0.845277i	$0.320564\pi$
$588$	−17.8971	−0.738062
$589$	−22.3479	−0.920829
$590$	0	0
$591$	31.9308	1.31346
$592$	−20.3974	−0.838327
$593$	42.0620	1.72728	0.863640	−	0.504109i	$-0.168179\pi$
$593$	42.0620	1.72728	0.863640	+	0.504109i	$0.168179\pi$
$594$	2.90538	0.119209
$595$	0	0
$596$	6.14396	0.251666
$597$	−33.7340	−1.38064
$598$	−0.0877324	−0.00358765
$599$	23.6399	0.965902	0.482951	−	0.875647i	$-0.339565\pi$
$599$	23.6399	0.965902	0.482951	+	0.875647i	$0.339565\pi$
$600$	0	0
$601$	−9.89731	−0.403720	−0.201860	−	0.979414i	$-0.564699\pi$
$601$	−9.89731	−0.403720	−0.201860	+	0.979414i	$0.564699\pi$
$602$	−2.21411	−0.0902405
$603$	−8.28583	−0.337425
$604$	−20.5803	−0.837402
$605$	0	0
$606$	2.43414	0.0988800
$607$	29.5988	1.20138	0.600688	−	0.799483i	$-0.294893\pi$
$607$	29.5988	1.20138	0.600688	+	0.799483i	$0.294893\pi$
$608$	8.34209	0.338316
$609$	24.4471	0.990648
$610$	0	0
$611$	−11.7899	−0.476969
$612$	−1.67869	−0.0678568
$613$	13.6657	0.551953	0.275977	−	0.961164i	$-0.410999\pi$
$613$	13.6657	0.551953	0.275977	+	0.961164i	$0.410999\pi$
$614$	−0.00182076	−7.34797e−5 0
$615$	0	0
$616$	4.22986	0.170426
$617$	41.8255	1.68383	0.841916	−	0.539609i	$-0.181428\pi$
$617$	41.8255	1.68383	0.841916	+	0.539609i	$0.181428\pi$
$618$	−5.11539	−0.205771
$619$	17.8194	0.716221	0.358110	−	0.933679i	$-0.383421\pi$
$619$	17.8194	0.716221	0.358110	+	0.933679i	$0.383421\pi$
$620$	0	0
$621$	2.23421	0.0896556
$622$	−4.31207	−0.172898
$623$	27.2066	1.09001
$624$	8.31978	0.333058
$625$	0	0
$626$	−0.621573	−0.0248431
$627$	−41.8263	−1.67038
$628$	1.30884	0.0522286
$629$	−5.27917	−0.210494
$630$	0	0
$631$	32.6557	1.30000	0.650001	−	0.759934i	$-0.274769\pi$
$631$	32.6557	1.30000	0.650001	+	0.759934i	$0.274769\pi$
$632$	−8.65599	−0.344317
$633$	−3.94414	−0.156766
$634$	3.11467	0.123699
$635$	0	0
$636$	0.0884235	0.00350622
$637$	5.06387	0.200638
$638$	5.55376	0.219875
$639$	1.02915	0.0407124
$640$	0	0
$641$	25.0356	0.988845	0.494422	−	0.869222i	$-0.335380\pi$
$641$	25.0356	0.988845	0.494422	+	0.869222i	$0.335380\pi$
$642$	−4.55770	−0.179878
$643$	−12.4048	−0.489196	−0.244598	−	0.969625i	$-0.578656\pi$
$643$	−12.4048	−0.489196	−0.244598	+	0.969625i	$0.578656\pi$
$644$	1.61704	0.0637203
$645$	0	0
$646$	0.705870	0.0277721
$647$	−37.9570	−1.49224	−0.746121	−	0.665811i	$-0.768086\pi$
$647$	−37.9570	−1.49224	−0.746121	+	0.665811i	$0.768086\pi$
$648$	6.50093	0.255381
$649$	−16.2777	−0.638955
$650$	0	0
$651$	−14.4865	−0.567773
$652$	32.1382	1.25863
$653$	−36.9847	−1.44732	−0.723662	−	0.690155i	$-0.757543\pi$
$653$	−36.9847	−1.44732	−0.723662	+	0.690155i	$0.757543\pi$
$654$	−5.10130	−0.199477
$655$	0	0
$656$	−2.90345	−0.113361
$657$	8.58652	0.334992
$658$	−2.50531	−0.0976670
$659$	29.0956	1.13341	0.566703	−	0.823922i	$-0.308219\pi$
$659$	29.0956	1.13341	0.566703	+	0.823922i	$0.308219\pi$
$660$	0	0
$661$	33.6207	1.30769	0.653847	−	0.756627i	$-0.273154\pi$
$661$	33.6207	1.30769	0.653847	+	0.756627i	$0.273154\pi$
$662$	−0.146717	−0.00570230
$663$	2.15329	0.0836269
$664$	3.05614	0.118601
$665$	0	0
$666$	−0.676712	−0.0262220
$667$	4.27078	0.165365
$668$	−39.4505	−1.52639
$669$	27.6772	1.07006
$670$	0	0
$671$	17.8927	0.690740
$672$	5.40758	0.208602
$673$	13.1865	0.508304	0.254152	−	0.967164i	$-0.418204\pi$
$673$	13.1865	0.508304	0.254152	+	0.967164i	$0.418204\pi$
$674$	−0.908071	−0.0349776
$675$	0	0
$676$	23.3219	0.896994
$677$	−1.37696	−0.0529208	−0.0264604	−	0.999650i	$-0.508424\pi$
$677$	−1.37696	−0.0529208	−0.0264604	+	0.999650i	$0.508424\pi$
$678$	−5.61496	−0.215641
$679$	10.4850	0.402378
$680$	0	0
$681$	30.8246	1.18120
$682$	−3.29097	−0.126018
$683$	−10.4181	−0.398637	−0.199319	−	0.979935i	$-0.563873\pi$
$683$	−10.4181	−0.398637	−0.199319	+	0.979935i	$0.563873\pi$
$684$	−7.84827	−0.300086
$685$	0	0
$686$	2.70864	0.103416
$687$	−32.5520	−1.24193
$688$	36.6801	1.39842
$689$	−0.0250190	−0.000953146 0
$690$	0	0
$691$	4.39455	0.167177	0.0835883	−	0.996500i	$-0.473362\pi$
$691$	4.39455	0.167177	0.0835883	+	0.996500i	$0.473362\pi$
$692$	−7.74505	−0.294423
$693$	−5.98061	−0.227185
$694$	−0.805430	−0.0305737
$695$	0	0
$696$	9.50319	0.360218
$697$	−0.751460	−0.0284636
$698$	−3.13437	−0.118638
$699$	−12.8774	−0.487069
$700$	0	0
$701$	−1.18383	−0.0447127	−0.0223563	−	0.999750i	$-0.507117\pi$
$701$	−1.18383	−0.0447127	−0.0223563	+	0.999750i	$0.507117\pi$
$702$	−0.699294	−0.0263932
$703$	−24.6814	−0.930876
$704$	−34.0094	−1.28178
$705$	0	0
$706$	2.27404	0.0855845
$707$	12.6942	0.477416
$708$	−13.8468	−0.520394
$709$	−13.3650	−0.501932	−0.250966	−	0.967996i	$-0.580748\pi$
$709$	−13.3650	−0.501932	−0.250966	+	0.967996i	$0.580748\pi$
$710$	0	0
$711$	12.2387	0.458987
$712$	10.5758	0.396346
$713$	−2.53072	−0.0947762
$714$	0.457565	0.0171239
$715$	0	0
$716$	−6.21178	−0.232145
$717$	14.9532	0.558437
$718$	−1.46896	−0.0548212
$719$	14.1925	0.529292	0.264646	−	0.964346i	$-0.414745\pi$
$719$	14.1925	0.529292	0.264646	+	0.964346i	$0.414745\pi$
$720$	0	0
$721$	−26.6772	−0.993512
$722$	0.431488	0.0160583
$723$	−13.5283	−0.503123
$724$	−1.54635	−0.0574695
$725$	0	0
$726$	−2.90110	−0.107670
$727$	−33.0852	−1.22706	−0.613530	−	0.789671i	$-0.710251\pi$
$727$	−33.0852	−1.22706	−0.613530	+	0.789671i	$0.710251\pi$
$728$	−1.01808	−0.0377327
$729$	15.6458	0.579475
$730$	0	0
$731$	9.49340	0.351126
$732$	15.2206	0.562570
$733$	−14.0451	−0.518767	−0.259383	−	0.965774i	$-0.583519\pi$
$733$	−14.0451	−0.518767	−0.259383	+	0.965774i	$0.583519\pi$
$734$	−1.30230	−0.0480688
$735$	0	0
$736$	0.944674	0.0348212
$737$	−44.5030	−1.63929
$738$	−0.0963262	−0.00354582
$739$	−5.14831	−0.189384	−0.0946918	−	0.995507i	$-0.530187\pi$
$739$	−5.14831	−0.189384	−0.0946918	+	0.995507i	$0.530187\pi$
$740$	0	0
$741$	10.0672	0.369827
$742$	−0.00531642	−0.000195172 0
$743$	26.8978	0.986784	0.493392	−	0.869807i	$-0.335757\pi$
$743$	26.8978	0.986784	0.493392	+	0.869807i	$0.335757\pi$
$744$	−5.63127	−0.206452
$745$	0	0
$746$	2.27863	0.0834264
$747$	−4.32108	−0.158100
$748$	−9.01617	−0.329664
$749$	−23.7688	−0.868494
$750$	0	0
$751$	13.5733	0.495295	0.247648	−	0.968850i	$-0.420343\pi$
$751$	13.5733	0.495295	0.247648	+	0.968850i	$0.420343\pi$
$752$	41.5042	1.51350
$753$	50.6808	1.84691
$754$	−1.33673	−0.0486809
$755$	0	0
$756$	12.8890	0.468770
$757$	−9.44665	−0.343344	−0.171672	−	0.985154i	$-0.554917\pi$
$757$	−9.44665	−0.343344	−0.171672	+	0.985154i	$0.554917\pi$
$758$	−1.61737	−0.0587455
$759$	−4.73650	−0.171924
$760$	0	0
$761$	−26.6325	−0.965427	−0.482713	−	0.875778i	$-0.660349\pi$
$761$	−26.6325	−0.965427	−0.482713	+	0.875778i	$0.660349\pi$
$762$	−4.62448	−0.167527
$763$	−26.6037	−0.963120
$764$	−24.5478	−0.888109
$765$	0	0
$766$	−2.39560	−0.0865565
$767$	3.91787	0.141466
$768$	−27.8734	−1.00579
$769$	−17.9587	−0.647609	−0.323805	−	0.946124i	$-0.604962\pi$
$769$	−17.9587	−0.647609	−0.323805	+	0.946124i	$0.604962\pi$
$770$	0	0
$771$	−35.0180	−1.26114
$772$	−3.37045	−0.121305
$773$	−17.0705	−0.613984	−0.306992	−	0.951712i	$-0.599323\pi$
$773$	−17.0705	−0.613984	−0.306992	+	0.951712i	$0.599323\pi$
$774$	1.21691	0.0437411
$775$	0	0
$776$	4.07578	0.146312
$777$	−15.9992	−0.573967
$778$	3.37436	0.120976
$779$	−3.51326	−0.125876
$780$	0	0
$781$	5.52752	0.197790
$782$	0.0799341	0.00285844
$783$	34.0414	1.21654
$784$	−17.8264	−0.636657
$785$	0	0
$786$	1.41981	0.0506428
$787$	−18.3219	−0.653104	−0.326552	−	0.945179i	$-0.605887\pi$
$787$	−18.3219	−0.653104	−0.326552	+	0.945179i	$0.605887\pi$
$788$	32.1800	1.14636
$789$	28.1426	1.00190
$790$	0	0
$791$	−29.2825	−1.04117
$792$	−2.32481	−0.0826084
$793$	−4.30658	−0.152931
$794$	−4.72573	−0.167710
$795$	0	0
$796$	−33.9973	−1.20500
$797$	16.4573	0.582949	0.291474	−	0.956579i	$-0.405854\pi$
$797$	16.4573	0.582949	0.291474	+	0.956579i	$0.405854\pi$
$798$	2.13923	0.0757279
$799$	10.7419	0.380023
$800$	0	0
$801$	−14.9532	−0.528345
$802$	4.83701	0.170801
$803$	46.1180	1.62747
$804$	−37.8569	−1.33511
$805$	0	0
$806$	0.792102	0.0279006
$807$	7.67438	0.270151
$808$	4.93455	0.173597
$809$	19.3869	0.681607	0.340804	−	0.940134i	$-0.389301\pi$
$809$	19.3869	0.681607	0.340804	+	0.940134i	$0.389301\pi$
$810$	0	0
$811$	−27.4015	−0.962196	−0.481098	−	0.876667i	$-0.659762\pi$
$811$	−27.4015	−0.962196	−0.481098	+	0.876667i	$0.659762\pi$
$812$	24.6380	0.864623
$813$	−55.8139	−1.95748
$814$	−3.63460	−0.127393
$815$	0	0
$816$	−7.58026	−0.265362
$817$	44.3840	1.55280
$818$	−3.24664	−0.113516
$819$	1.43947	0.0502992
$820$	0	0
$821$	20.6646	0.721198	0.360599	−	0.932721i	$-0.382572\pi$
$821$	20.6646	0.721198	0.360599	+	0.932721i	$0.382572\pi$
$822$	4.64305	0.161945
$823$	0.983957	0.0342986	0.0171493	−	0.999853i	$-0.494541\pi$
$823$	0.983957	0.0342986	0.0171493	+	0.999853i	$0.494541\pi$
$824$	−10.3701	−0.361259
$825$	0	0
$826$	0.832530	0.0289674
$827$	15.5237	0.539811	0.269906	−	0.962887i	$-0.413008\pi$
$827$	15.5237	0.539811	0.269906	+	0.962887i	$0.413008\pi$
$828$	−0.888753	−0.0308863
$829$	−41.6910	−1.44799	−0.723994	−	0.689806i	$-0.757696\pi$
$829$	−41.6910	−1.44799	−0.723994	+	0.689806i	$0.757696\pi$
$830$	0	0
$831$	−50.3890	−1.74798
$832$	8.18571	0.283788
$833$	−4.61376	−0.159857
$834$	−2.46757	−0.0854449
$835$	0	0
$836$	−42.1528	−1.45789
$837$	−20.1718	−0.697238
$838$	−2.52508	−0.0872274
$839$	11.0633	0.381949	0.190974	−	0.981595i	$-0.438835\pi$
$839$	11.0633	0.381949	0.190974	+	0.981595i	$0.438835\pi$
$840$	0	0
$841$	36.0716	1.24385
$842$	−2.56474	−0.0883869
$843$	40.7421	1.40323
$844$	−3.97493	−0.136823
$845$	0	0
$846$	1.37696	0.0473408
$847$	−15.1295	−0.519855
$848$	0.0880745	0.00302449
$849$	48.0053	1.64754
$850$	0	0
$851$	−2.79497	−0.0958103
$852$	4.70204	0.161089
$853$	16.9575	0.580615	0.290307	−	0.956933i	$-0.406242\pi$
$853$	16.9575	0.580615	0.290307	+	0.956933i	$0.406242\pi$
$854$	−0.915130	−0.0313151
$855$	0	0
$856$	−9.23951	−0.315800
$857$	−21.7394	−0.742605	−0.371302	−	0.928512i	$-0.621089\pi$
$857$	−21.7394	−0.742605	−0.371302	+	0.928512i	$0.621089\pi$
$858$	1.48250	0.0506117
$859$	31.5036	1.07489	0.537444	−	0.843300i	$-0.319390\pi$
$859$	31.5036	1.07489	0.537444	+	0.843300i	$0.319390\pi$
$860$	0	0
$861$	−2.27740	−0.0776134
$862$	−2.58087	−0.0879048
$863$	−12.4798	−0.424817	−0.212408	−	0.977181i	$-0.568131\pi$
$863$	−12.4798	−0.424817	−0.212408	+	0.977181i	$0.568131\pi$
$864$	7.52977	0.256168
$865$	0	0
$866$	−1.50262	−0.0510611
$867$	−1.96189	−0.0666293
$868$	−14.5996	−0.495543
$869$	65.7338	2.22987
$870$	0	0
$871$	10.7114	0.362942
$872$	−10.3415	−0.350208
$873$	−5.76275	−0.195039
$874$	0.373712	0.0126410
$875$	0	0
$876$	39.2307	1.32548
$877$	14.5835	0.492450	0.246225	−	0.969213i	$-0.420810\pi$
$877$	14.5835	0.492450	0.246225	+	0.969213i	$0.420810\pi$
$878$	0.442186	0.0149231
$879$	55.3159	1.86576
$880$	0	0
$881$	−12.4663	−0.420001	−0.210001	−	0.977701i	$-0.567347\pi$
$881$	−12.4663	−0.420001	−0.210001	+	0.977701i	$0.567347\pi$
$882$	−0.591416	−0.0199140
$883$	−52.0737	−1.75242	−0.876209	−	0.481931i	$-0.839936\pi$
$883$	−52.0737	−1.75242	−0.876209	+	0.481931i	$0.839936\pi$
$884$	2.17010	0.0729883
$885$	0	0
$886$	0.117536	0.00394869
$887$	−46.1318	−1.54895	−0.774477	−	0.632602i	$-0.781987\pi$
$887$	−46.1318	−1.54895	−0.774477	+	0.632602i	$0.781987\pi$
$888$	−6.21926	−0.208705
$889$	−24.1171	−0.808860
$890$	0	0
$891$	−49.3682	−1.65390
$892$	27.8932	0.933934
$893$	50.2212	1.68059
$894$	0.920433	0.0307839
$895$	0	0
$896$	7.25204	0.242274
$897$	1.14003	0.0380643
$898$	−4.72273	−0.157600
$899$	−38.5592	−1.28602
$900$	0	0
$901$	0.0227951	0.000759414 0
$902$	−0.517365	−0.0172264
$903$	28.7710	0.957437
$904$	−11.3828	−0.378587
$905$	0	0
$906$	−3.08316	−0.102431
$907$	2.91050	0.0966415	0.0483207	−	0.998832i	$-0.484613\pi$
$907$	2.91050	0.0966415	0.0483207	+	0.998832i	$0.484613\pi$
$908$	31.0652	1.03093
$909$	−6.97697	−0.231412
$910$	0	0
$911$	−41.7029	−1.38168	−0.690839	−	0.723009i	$-0.742759\pi$
$911$	−41.7029	−1.38168	−0.690839	+	0.723009i	$0.742759\pi$
$912$	−35.4396	−1.17352
$913$	−23.2084	−0.768086
$914$	0.542420	0.0179416
$915$	0	0
$916$	−32.8060	−1.08394
$917$	7.40441	0.244515
$918$	0.637136	0.0210286
$919$	−2.60146	−0.0858144	−0.0429072	−	0.999079i	$-0.513662\pi$
$919$	−2.60146	−0.0858144	−0.0429072	+	0.999079i	$0.513662\pi$
$920$	0	0
$921$	0.0236595	0.000779608 0
$922$	−0.997063	−0.0328365
$923$	−1.33042	−0.0437912
$924$	−27.3246	−0.898915
$925$	0	0
$926$	−1.65326	−0.0543296
$927$	14.6623	0.481572
$928$	14.3935	0.472490
$929$	5.89744	0.193489	0.0967443	−	0.995309i	$-0.469157\pi$
$929$	5.89744	0.193489	0.0967443	+	0.995309i	$0.469157\pi$
$930$	0	0
$931$	−21.5704	−0.706943
$932$	−12.9779	−0.425106
$933$	56.0325	1.83442
$934$	−1.52925	−0.0500385
$935$	0	0
$936$	0.559557	0.0182897
$937$	58.2446	1.90277	0.951383	−	0.308010i	$-0.0996629\pi$
$937$	58.2446	1.90277	0.951383	+	0.308010i	$0.0996629\pi$
$938$	2.27613	0.0743181
$939$	8.07694	0.263581
$940$	0	0
$941$	28.2631	0.921350	0.460675	−	0.887569i	$-0.347607\pi$
$941$	28.2631	0.921350	0.460675	+	0.887569i	$0.347607\pi$
$942$	0.196079	0.00638861
$943$	−0.397848	−0.0129557
$944$	−13.7921	−0.448895
$945$	0	0
$946$	6.53602	0.212504
$947$	−31.3759	−1.01958	−0.509789	−	0.860299i	$-0.670277\pi$
$947$	−31.3759	−1.01958	−0.509789	+	0.860299i	$0.670277\pi$
$948$	55.9171	1.81610
$949$	−11.1001	−0.360325
$950$	0	0
$951$	−40.4731	−1.31243
$952$	0.927590	0.0300634
$953$	27.2735	0.883476	0.441738	−	0.897144i	$-0.354362\pi$
$953$	27.2735	0.883476	0.441738	+	0.897144i	$0.354362\pi$
$954$	0.00292200	9.46031e−5 0
$955$	0	0
$956$	15.0699	0.487396
$957$	−72.1675	−2.33284
$958$	−1.16714	−0.0377085
$959$	24.2139	0.781908
$960$	0	0
$961$	−8.15111	−0.262939
$962$	0.874810	0.0282050
$963$	13.0638	0.420974
$964$	−13.6339	−0.439118
$965$	0	0
$966$	0.242250	0.00779428
$967$	−1.30807	−0.0420648	−0.0210324	−	0.999779i	$-0.506695\pi$
$967$	−1.30807	−0.0420648	−0.0210324	+	0.999779i	$0.506695\pi$
$968$	−5.88119	−0.189029
$969$	−9.17232	−0.294657
$970$	0	0
$971$	−41.0902	−1.31865	−0.659324	−	0.751859i	$-0.729157\pi$
$971$	−41.0902	−1.31865	−0.659324	+	0.751859i	$0.729157\pi$
$972$	−16.9642	−0.544128
$973$	−12.8686	−0.412548
$974$	2.05219	0.0657566
$975$	0	0
$976$	15.1605	0.485276
$977$	−60.7556	−1.94375	−0.971873	−	0.235507i	$-0.924325\pi$
$977$	−60.7556	−1.94375	−0.971873	+	0.235507i	$0.924325\pi$
$978$	4.81465	0.153956
$979$	−80.3132	−2.56682
$980$	0	0
$981$	14.6219	0.466840
$982$	5.86698	0.187223
$983$	9.13830	0.291466	0.145733	−	0.989324i	$-0.453446\pi$
$983$	9.13830	0.291466	0.145733	+	0.989324i	$0.453446\pi$
$984$	−0.885278	−0.0282216
$985$	0	0
$986$	1.21791	0.0387863
$987$	32.5548	1.03623
$988$	10.1457	0.322779
$989$	5.02613	0.159822
$990$	0	0
$991$	33.1886	1.05427	0.527135	−	0.849782i	$-0.323266\pi$
$991$	33.1886	1.05427	0.527135	+	0.849782i	$0.323266\pi$
$992$	−8.52909	−0.270799
$993$	1.90649	0.0605005
$994$	−0.282708	−0.00896695
$995$	0	0
$996$	−19.7425	−0.625564
$997$	−30.0746	−0.952471	−0.476235	−	0.879318i	$-0.657999\pi$
$997$	−30.0746	−0.952471	−0.476235	+	0.879318i	$0.657999\pi$
$998$	−4.02958	−0.127554
$999$	−22.2780	−0.704846

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.2.a.i.1.3	✓	5
3.2	odd	2		3825.2.a.bq.1.3		5
4.3	odd	2		6800.2.a.bz.1.4		5
5.2	odd	4		425.2.b.f.324.6		10
5.3	odd	4		425.2.b.f.324.5		10
5.4	even	2		425.2.a.j.1.3	yes	5
15.14	odd	2		3825.2.a.bl.1.3		5
17.16	even	2		7225.2.a.x.1.3		5
20.19	odd	2		6800.2.a.cd.1.2		5
85.84	even	2		7225.2.a.y.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.2.a.i.1.3	✓	5	1.1	even	1	trivial
425.2.a.j.1.3	yes	5	5.4	even	2
425.2.b.f.324.5		10	5.3	odd	4
425.2.b.f.324.6		10	5.2	odd	4
3825.2.a.bl.1.3		5	15.14	odd	2
3825.2.a.bq.1.3		5	3.2	odd	2
6800.2.a.bz.1.4		5	4.3	odd	2
6800.2.a.cd.1.2		5	20.19	odd	2
7225.2.a.x.1.3		5	17.16	even	2
7225.2.a.y.1.3		5	85.84	even	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:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	425.a (trivial)

\(f(q)\)	\(=\)	\(q+0.150980 q^{2} -1.96189 q^{3} -1.97720 q^{4} -0.296207 q^{6} -1.54475 q^{7} -0.600480 q^{8} +0.849020 q^{9} +4.56006 q^{11} +3.87906 q^{12} -1.09756 q^{13} -0.233227 q^{14} +3.86375 q^{16} +1.00000 q^{17} +0.128185 q^{18} +4.67524 q^{19} +3.03063 q^{21} +0.688480 q^{22} +0.529434 q^{23} +1.17808 q^{24} -0.165710 q^{26} +4.21999 q^{27} +3.05428 q^{28} +8.06670 q^{29} -4.78005 q^{31} +1.78431 q^{32} -8.94634 q^{33} +0.150980 q^{34} -1.67869 q^{36} -5.27917 q^{37} +0.705870 q^{38} +2.15329 q^{39} -0.751460 q^{41} +0.457565 q^{42} +9.49340 q^{43} -9.01617 q^{44} +0.0799341 q^{46} +10.7419 q^{47} -7.58026 q^{48} -4.61376 q^{49} -1.96189 q^{51} +2.17010 q^{52} +0.0227951 q^{53} +0.637136 q^{54} +0.927590 q^{56} -9.17232 q^{57} +1.21791 q^{58} -3.56962 q^{59} +3.92378 q^{61} -0.721694 q^{62} -1.31152 q^{63} -7.45810 q^{64} -1.35072 q^{66} -9.75929 q^{67} -1.97720 q^{68} -1.03869 q^{69} +1.21216 q^{71} -0.509819 q^{72} +10.1135 q^{73} -0.797051 q^{74} -9.24392 q^{76} -7.04414 q^{77} +0.325105 q^{78} +14.4151 q^{79} -10.8262 q^{81} -0.113456 q^{82} -5.08949 q^{83} -5.99217 q^{84} +1.43332 q^{86} -15.8260 q^{87} -2.73822 q^{88} -17.6123 q^{89} +1.69545 q^{91} -1.04680 q^{92} +9.37794 q^{93} +1.62182 q^{94} -3.50062 q^{96} -6.78753 q^{97} -0.696587 q^{98} +3.87158 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - q^{2} + q^{3} + 11 q^{4} + 3 q^{6} + q^{7} - 9 q^{8} + 6 q^{9} + 4 q^{11} + 17 q^{12} - 3 q^{13} - 7 q^{14} + 27 q^{16} + 5 q^{17} - 22 q^{18} + 6 q^{19} - 5 q^{21} + 18 q^{22} + 4 q^{23} - 19 q^{24}+ \cdots - 14 q^{99}+O(q^{100}) \) 5 * q - q^2 + q^3 + 11 * q^4 + 3 * q^6 + q^7 - 9 * q^8 + 6 * q^9 + 4 * q^11 + 17 * q^12 - 3 * q^13 - 7 * q^14 + 27 * q^16 + 5 * q^17 - 22 * q^18 + 6 * q^19 - 5 * q^21 + 18 * q^22 + 4 * q^23 - 19 * q^24 - 5 * q^26 - 5 * q^27 - 15 * q^28 + 2 * q^29 + 21 * q^31 - 9 * q^32 + 12 * q^33 - q^34 + 22 * q^36 - 2 * q^37 - 34 * q^38 + 23 * q^39 - 8 * q^41 + 25 * q^42 - 4 * q^43 - 4 * q^44 - 6 * q^46 - 2 * q^47 + 51 * q^48 + 10 * q^49 + q^51 - 13 * q^52 + 21 * q^53 - 25 * q^54 - 29 * q^56 - 20 * q^57 + 22 * q^58 + 12 * q^59 - 2 * q^61 + 3 * q^62 + 8 * q^63 + 43 * q^64 - 28 * q^66 - 12 * q^67 + 11 * q^68 + 21 * q^71 - 58 * q^72 + 22 * q^73 - 50 * q^74 - 14 * q^76 - 32 * q^77 - 53 * q^78 + 41 * q^79 - 35 * q^81 + 28 * q^82 - 8 * q^83 - 71 * q^84 - 36 * q^86 - 4 * q^87 + 14 * q^88 - 10 * q^89 + 13 * q^91 + 56 * q^92 + 9 * q^93 - 10 * q^94 - 85 * q^96 - 20 * q^97 + 38 * q^98 - 14 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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