LMFDB - Embedded newform 9025.2.a.bp.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9025,2,Mod(1,9025)] 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("9025.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9025, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,1,3,5,0,2,4,12,1,0,-2,6,7,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	yes
Analytic conductor:	$72.0649878242$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.7537.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 4x + 3 $ x^4 - x^3 - 5x^2 + 4x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$1.37933$ of defining polynomial
Character	$\chi$	$=$	9025.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.09744 q^{2} -0.379334 q^{3} -0.795629 q^{4} -0.416295 q^{6} -1.89307 q^{7} -3.06803 q^{8} -2.85611 q^{9} +0.134400 q^{11} +0.301809 q^{12} +3.51373 q^{13} -2.07752 q^{14} -1.77572 q^{16} +1.66123 q^{17} -3.13440 q^{18} +0.718104 q^{21} +0.147496 q^{22} -5.36984 q^{23} +1.16381 q^{24} +3.85611 q^{26} +2.22142 q^{27} +1.50618 q^{28} -4.97059 q^{29} -6.56472 q^{31} +4.18732 q^{32} -0.0509824 q^{33} +1.82310 q^{34} +2.27240 q^{36} -1.69819 q^{37} -1.33288 q^{39} -10.6327 q^{41} +0.788075 q^{42} -8.50784 q^{43} -0.106932 q^{44} -5.89307 q^{46} +11.1154 q^{47} +0.673589 q^{48} -3.41630 q^{49} -0.630160 q^{51} -2.79563 q^{52} -0.264847 q^{53} +2.43787 q^{54} +5.80799 q^{56} -5.45492 q^{58} +6.89667 q^{59} +9.17589 q^{61} -7.20437 q^{62} +5.40680 q^{63} +8.14676 q^{64} -0.0559501 q^{66} -2.95354 q^{67} -1.32172 q^{68} +2.03696 q^{69} -1.32835 q^{71} +8.76262 q^{72} +6.34237 q^{73} -1.86366 q^{74} -0.254428 q^{77} -1.46275 q^{78} +1.46728 q^{79} +7.72566 q^{81} -11.6688 q^{82} -7.44736 q^{83} -0.571345 q^{84} -9.33683 q^{86} +1.88551 q^{87} -0.412343 q^{88} -9.73608 q^{89} -6.65174 q^{91} +4.27240 q^{92} +2.49022 q^{93} +12.1985 q^{94} -1.58839 q^{96} +17.4689 q^{97} -3.74917 q^{98} -0.383860 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + 3 q^{3} + 5 q^{4} + 2 q^{6} + 4 q^{7} + 12 q^{8} + q^{9} - 2 q^{11} + 6 q^{12} + 7 q^{13} + q^{14} + 7 q^{16} + q^{17} - 10 q^{18} + 4 q^{21} + 2 q^{22} - 2 q^{23} + 23 q^{24} + 3 q^{26}+ \cdots + 38 q^{99}+O(q^{100}) $ 4 * q + q^2 + 3 * q^3 + 5 * q^4 + 2 * q^6 + 4 * q^7 + 12 * q^8 + q^9 - 2 * q^11 + 6 * q^12 + 7 * q^13 + q^14 + 7 * q^16 + q^17 - 10 * q^18 + 4 * q^21 + 2 * q^22 - 2 * q^23 + 23 * q^24 + 3 * q^26 + 12 * q^27 + 19 * q^28 + q^29 + 30 * q^32 + 19 * q^33 - 15 * q^34 - 7 * q^36 - 2 * q^37 + 15 * q^39 + 8 * q^41 + 15 * q^42 - q^43 - 12 * q^44 - 12 * q^46 + 12 * q^47 + 23 * q^48 - 10 * q^49 - 22 * q^51 - 3 * q^52 - 5 * q^53 - 34 * q^54 + 41 * q^56 + 27 * q^58 + 5 * q^59 - 37 * q^62 + 3 * q^63 + 56 * q^64 - 31 * q^66 + 4 * q^67 - 16 * q^68 + 9 * q^69 - 20 * q^71 + 17 * q^72 + 20 * q^73 + 25 * q^74 - 14 * q^77 - 18 * q^78 - 17 * q^79 + 12 * q^81 - 21 * q^82 - q^83 + 20 * q^84 - 8 * q^86 + 16 * q^87 - 7 * q^88 - 11 * q^89 - 6 * q^91 + q^92 + 8 * q^93 + 31 * q^94 + 21 * q^96 + q^97 + 9 * q^98 + 38 * q^99

Coefficient data

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

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

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

$2$	1.09744	0.776006	0.388003	−	0.921658i	$-0.373165\pi$
$2$	1.09744	0.776006	0.388003	+	0.921658i	$0.373165\pi$
$3$	−0.379334	−0.219008	−0.109504	−	0.993986i	$-0.534926\pi$
$3$	−0.379334	−0.219008	−0.109504	+	0.993986i	$0.534926\pi$
$4$	−0.795629	−0.397815
$5$	0	0
$5$	0	0
$6$	−0.416295	−0.169952
$7$	−1.89307	−0.715512	−0.357756	−	0.933815i	$-0.616458\pi$
$7$	−1.89307	−0.715512	−0.357756	+	0.933815i	$0.616458\pi$
$8$	−3.06803	−1.08471
$9$	−2.85611	−0.952035
$10$	0	0
$11$	0.134400	0.0405231	0.0202615	−	0.999795i	$-0.493550\pi$
$11$	0.134400	0.0405231	0.0202615	+	0.999795i	$0.493550\pi$
$12$	0.301809	0.0871248
$13$	3.51373	0.974534	0.487267	−	0.873253i	$-0.337994\pi$
$13$	3.51373	0.974534	0.487267	+	0.873253i	$0.337994\pi$
$14$	−2.07752	−0.555242
$15$	0	0
$16$	−1.77572	−0.443929
$17$	1.66123	0.402907	0.201454	−	0.979498i	$-0.435433\pi$
$17$	1.66123	0.402907	0.201454	+	0.979498i	$0.435433\pi$
$18$	−3.13440	−0.738785
$19$	0	0
$19$	0	0
$20$	0	0
$21$	0.718104	0.156703
$22$	0.147496	0.0314462
$23$	−5.36984	−1.11969	−0.559844	−	0.828598i	$-0.689139\pi$
$23$	−5.36984	−1.11969	−0.559844	+	0.828598i	$0.689139\pi$
$24$	1.16381	0.237561
$25$	0	0
$26$	3.85611	0.756245
$27$	2.22142	0.427512
$28$	1.50618	0.284641
$29$	−4.97059	−0.923016	−0.461508	−	0.887136i	$-0.652691\pi$
$29$	−4.97059	−0.923016	−0.461508	+	0.887136i	$0.652691\pi$
$30$	0	0
$31$	−6.56472	−1.17906	−0.589529	−	0.807747i	$-0.700687\pi$
$31$	−6.56472	−1.17906	−0.589529	+	0.807747i	$0.700687\pi$
$32$	4.18732	0.740221
$33$	−0.0509824	−0.00887490
$34$	1.82310	0.312658
$35$	0	0
$36$	2.27240	0.378734
$37$	−1.69819	−0.279181	−0.139590	−	0.990209i	$-0.544579\pi$
$37$	−1.69819	−0.279181	−0.139590	+	0.990209i	$0.544579\pi$
$38$	0	0
$39$	−1.33288	−0.213431
$40$	0	0
$41$	−10.6327	−1.66056	−0.830278	−	0.557349i	$-0.811818\pi$
$41$	−10.6327	−1.66056	−0.830278	+	0.557349i	$0.811818\pi$
$42$	0.788075	0.121603
$43$	−8.50784	−1.29743	−0.648717	−	0.761030i	$-0.724694\pi$
$43$	−8.50784	−1.29743	−0.648717	+	0.761030i	$0.724694\pi$
$44$	−0.106932	−0.0161207
$45$	0	0
$46$	−5.89307	−0.868885
$47$	11.1154	1.62135	0.810675	−	0.585497i	$-0.199101\pi$
$47$	11.1154	1.62135	0.810675	+	0.585497i	$0.199101\pi$
$48$	0.673589	0.0972242
$49$	−3.41630	−0.488042
$50$	0	0
$51$	−0.630160	−0.0882401
$52$	−2.79563	−0.387684
$53$	−0.264847	−0.0363796	−0.0181898	−	0.999835i	$-0.505790\pi$
$53$	−0.264847	−0.0363796	−0.0181898	+	0.999835i	$0.505790\pi$
$54$	2.43787	0.331752
$55$	0	0
$56$	5.80799	0.776125
$57$	0	0
$58$	−5.45492	−0.716266
$59$	6.89667	0.897870	0.448935	−	0.893564i	$-0.351804\pi$
$59$	6.89667	0.897870	0.448935	+	0.893564i	$0.351804\pi$
$60$	0	0
$61$	9.17589	1.17485	0.587426	−	0.809278i	$-0.300141\pi$
$61$	9.17589	1.17485	0.587426	+	0.809278i	$0.300141\pi$
$62$	−7.20437	−0.914956
$63$	5.40680	0.681193
$64$	8.14676	1.01834
$65$	0	0
$66$	−0.0559501	−0.00688698
$67$	−2.95354	−0.360833	−0.180416	−	0.983590i	$-0.557745\pi$
$67$	−2.95354	−0.360833	−0.180416	+	0.983590i	$0.557745\pi$
$68$	−1.32172	−0.160282
$69$	2.03696	0.245221
$70$	0	0
$71$	−1.32835	−0.157646	−0.0788232	−	0.996889i	$-0.525116\pi$
$71$	−1.32835	−0.157646	−0.0788232	+	0.996889i	$0.525116\pi$
$72$	8.76262	1.03268
$73$	6.34237	0.742319	0.371159	−	0.928569i	$-0.378960\pi$
$73$	6.34237	0.742319	0.371159	+	0.928569i	$0.378960\pi$
$74$	−1.86366	−0.216646
$75$	0	0
$76$	0	0
$77$	−0.254428	−0.0289948
$78$	−1.46275	−0.165624
$79$	1.46728	0.165082	0.0825408	−	0.996588i	$-0.473697\pi$
$79$	1.46728	0.165082	0.0825408	+	0.996588i	$0.473697\pi$
$80$	0	0
$81$	7.72566	0.858406
$82$	−11.6688	−1.28860
$83$	−7.44736	−0.817454	−0.408727	−	0.912657i	$-0.634027\pi$
$83$	−7.44736	−0.817454	−0.408727	+	0.912657i	$0.634027\pi$
$84$	−0.571345	−0.0623388
$85$	0	0
$86$	−9.33683	−1.00682
$87$	1.88551	0.202148
$88$	−0.412343	−0.0439559
$89$	−9.73608	−1.03202	−0.516011	−	0.856582i	$-0.672584\pi$
$89$	−9.73608	−1.03202	−0.516011	+	0.856582i	$0.672584\pi$
$90$	0	0
$91$	−6.65174	−0.697291
$92$	4.27240	0.445429
$93$	2.49022	0.258224
$94$	12.1985	1.25818
$95$	0	0
$96$	−1.58839	−0.162115
$97$	17.4689	1.77370	0.886851	−	0.462055i	$-0.152888\pi$
$97$	17.4689	1.77370	0.886851	+	0.462055i	$0.152888\pi$
$98$	−3.74917	−0.378724
$99$	−0.383860	−0.0385794
$100$	0	0
$101$	5.39731	0.537052	0.268526	−	0.963272i	$-0.413463\pi$
$101$	5.39731	0.537052	0.268526	+	0.963272i	$0.413463\pi$
$102$	−0.691562	−0.0684749
$103$	2.14750	0.211599	0.105800	−	0.994387i	$-0.466260\pi$
$103$	2.14750	0.211599	0.105800	+	0.994387i	$0.466260\pi$
$104$	−10.7802	−1.05709
$105$	0	0
$106$	−0.290654	−0.0282308
$107$	−1.00093	−0.0967631	−0.0483815	−	0.998829i	$-0.515406\pi$
$107$	−1.00093	−0.0967631	−0.0483815	+	0.998829i	$0.515406\pi$
$108$	−1.76743	−0.170071
$109$	−16.2629	−1.55770	−0.778852	−	0.627208i	$-0.784198\pi$
$109$	−16.2629	−1.55770	−0.778852	+	0.627208i	$0.784198\pi$
$110$	0	0
$111$	0.644181	0.0611430
$112$	3.36155	0.317637
$113$	0.843010	0.0793037	0.0396519	−	0.999214i	$-0.487375\pi$
$113$	0.843010	0.0793037	0.0396519	+	0.999214i	$0.487375\pi$
$114$	0	0
$115$	0	0
$116$	3.95475	0.367189
$117$	−10.0356	−0.927791
$118$	7.56867	0.696752
$119$	−3.14482	−0.288285
$120$	0	0
$121$	−10.9819	−0.998358
$122$	10.0700	0.911692
$123$	4.03336	0.363676
$124$	5.22308	0.469046
$125$	0	0
$126$	5.93363	0.528610
$127$	18.7397	1.66288	0.831439	−	0.555616i	$-0.187518\pi$
$127$	18.7397	1.66288	0.831439	+	0.555616i	$0.187518\pi$
$128$	0.565920	0.0500208
$129$	3.22731	0.284149
$130$	0	0
$131$	2.88644	0.252189	0.126095	−	0.992018i	$-0.459756\pi$
$131$	2.88644	0.252189	0.126095	+	0.992018i	$0.459756\pi$
$132$	0.0405631	0.00353057
$133$	0	0
$134$	−3.24133	−0.280008
$135$	0	0
$136$	−5.09670	−0.437039
$137$	18.8316	1.60889	0.804445	−	0.594027i	$-0.202463\pi$
$137$	18.8316	1.60889	0.804445	+	0.594027i	$0.202463\pi$
$138$	2.23544	0.190293
$139$	−18.1795	−1.54196	−0.770982	−	0.636857i	$-0.780234\pi$
$139$	−18.1795	−1.54196	−0.770982	+	0.636857i	$0.780234\pi$
$140$	0	0
$141$	−4.21645	−0.355089
$142$	−1.45778	−0.122334
$143$	0.472245	0.0394912
$144$	5.07163	0.422636
$145$	0	0
$146$	6.96036	0.576044
$147$	1.29592	0.106885
$148$	1.35113	0.111062
$149$	−22.2543	−1.82315	−0.911573	−	0.411138i	$-0.865131\pi$
$149$	−22.2543	−1.82315	−0.911573	+	0.411138i	$0.865131\pi$
$150$	0	0
$151$	−3.33482	−0.271384	−0.135692	−	0.990751i	$-0.543326\pi$
$151$	−3.33482	−0.271384	−0.135692	+	0.990751i	$0.543326\pi$
$152$	0	0
$153$	−4.74465	−0.383582
$154$	−0.279219	−0.0225001
$155$	0	0
$156$	1.06048	0.0849061
$157$	−7.26291	−0.579643	−0.289822	−	0.957081i	$-0.593596\pi$
$157$	−7.26291	−0.579643	−0.289822	+	0.957081i	$0.593596\pi$
$158$	1.61025	0.128104
$159$	0.100466	0.00796744
$160$	0	0
$161$	10.1655	0.801151
$162$	8.47843	0.666129
$163$	19.7783	1.54916	0.774578	−	0.632478i	$-0.217962\pi$
$163$	19.7783	1.54916	0.774578	+	0.632478i	$0.217962\pi$
$164$	8.45972	0.660593
$165$	0	0
$166$	−8.17302	−0.634350
$167$	−3.40320	−0.263348	−0.131674	−	0.991293i	$-0.542035\pi$
$167$	−3.40320	−0.263348	−0.131674	+	0.991293i	$0.542035\pi$
$168$	−2.20317	−0.169978
$169$	−0.653675	−0.0502827
$170$	0	0
$171$	0	0
$172$	6.76909	0.516138
$173$	−10.5857	−0.804817	−0.402409	−	0.915460i	$-0.631827\pi$
$173$	−10.5857	−0.804817	−0.402409	+	0.915460i	$0.631827\pi$
$174$	2.06923	0.156868
$175$	0	0
$176$	−0.238656	−0.0179894
$177$	−2.61614	−0.196641
$178$	−10.6847	−0.800855
$179$	14.6024	1.09144	0.545718	−	0.837969i	$-0.316257\pi$
$179$	14.6024	1.09144	0.545718	+	0.837969i	$0.316257\pi$
$180$	0	0
$181$	5.43261	0.403803	0.201901	−	0.979406i	$-0.435288\pi$
$181$	5.43261	0.403803	0.201901	+	0.979406i	$0.435288\pi$
$182$	−7.29987	−0.541102
$183$	−3.48072	−0.257303
$184$	16.4748	1.21454
$185$	0	0
$186$	2.73286	0.200383
$187$	0.223269	0.0163271
$188$	−8.84375	−0.644996
$189$	−4.20530	−0.305890
$190$	0	0
$191$	20.4758	1.48157	0.740787	−	0.671740i	$-0.234453\pi$
$191$	20.4758	1.48157	0.740787	+	0.671740i	$0.234453\pi$
$192$	−3.09034	−0.223026
$193$	11.0235	0.793490	0.396745	−	0.917929i	$-0.370140\pi$
$193$	11.0235	0.793490	0.396745	+	0.917929i	$0.370140\pi$
$194$	19.1711	1.37640
$195$	0	0
$196$	2.71810	0.194150
$197$	19.8532	1.41448	0.707242	−	0.706971i	$-0.249939\pi$
$197$	19.8532	1.41448	0.707242	+	0.706971i	$0.249939\pi$
$198$	−0.421263	−0.0299379
$199$	21.0026	1.48883	0.744417	−	0.667715i	$-0.232728\pi$
$199$	21.0026	1.48883	0.744417	+	0.667715i	$0.232728\pi$
$200$	0	0
$201$	1.12038	0.0790254
$202$	5.92321	0.416756
$203$	9.40967	0.660429
$204$	0.501374	0.0351032
$205$	0	0
$206$	2.35674	0.164202
$207$	15.3368	1.06598
$208$	−6.23939	−0.432624
$209$	0	0
$210$	0	0
$211$	12.8257	0.882956	0.441478	−	0.897272i	$-0.354454\pi$
$211$	12.8257	0.882956	0.441478	+	0.897272i	$0.354454\pi$
$212$	0.210720	0.0144723
$213$	0.503889	0.0345259
$214$	−1.09845	−0.0750887
$215$	0	0
$216$	−6.81538	−0.463728
$217$	12.4275	0.843630
$218$	−17.8475	−1.20879
$219$	−2.40588	−0.162574
$220$	0	0
$221$	5.83712	0.392647
$222$	0.706949	0.0474473
$223$	20.3944	1.36571	0.682856	−	0.730553i	$-0.260737\pi$
$223$	20.3944	1.36571	0.682856	+	0.730553i	$0.260737\pi$
$224$	−7.92688	−0.529637
$225$	0	0
$226$	0.925152	0.0615402
$227$	25.4172	1.68700	0.843500	−	0.537129i	$-0.180491\pi$
$227$	25.4172	1.68700	0.843500	+	0.537129i	$0.180491\pi$
$228$	0	0
$229$	2.21553	0.146406	0.0732030	−	0.997317i	$-0.476678\pi$
$229$	2.21553	0.146406	0.0732030	+	0.997317i	$0.476678\pi$
$230$	0	0
$231$	0.0965132	0.00635010
$232$	15.2499	1.00121
$233$	14.1576	0.927498	0.463749	−	0.885967i	$-0.346504\pi$
$233$	14.1576	0.927498	0.463749	+	0.885967i	$0.346504\pi$
$234$	−11.0134	−0.719972
$235$	0	0
$236$	−5.48719	−0.357186
$237$	−0.556588	−0.0361543
$238$	−3.45125	−0.223711
$239$	3.01476	0.195008	0.0975042	−	0.995235i	$-0.468914\pi$
$239$	3.01476	0.195008	0.0975042	+	0.995235i	$0.468914\pi$
$240$	0	0
$241$	23.7792	1.53175	0.765877	−	0.642987i	$-0.222305\pi$
$241$	23.7792	1.53175	0.765877	+	0.642987i	$0.222305\pi$
$242$	−12.0520	−0.774732
$243$	−9.59486	−0.615511
$244$	−7.30060	−0.467373
$245$	0	0
$246$	4.42636	0.282215
$247$	0	0
$248$	20.1407	1.27894
$249$	2.82504	0.179029
$250$	0	0
$251$	17.1899	1.08502	0.542509	−	0.840050i	$-0.317475\pi$
$251$	17.1899	1.08502	0.542509	+	0.840050i	$0.317475\pi$
$252$	−4.30181	−0.270988
$253$	−0.721706	−0.0453733
$254$	20.5656	1.29040
$255$	0	0
$256$	−15.6725	−0.979529
$257$	−19.5429	−1.21905	−0.609525	−	0.792767i	$-0.708640\pi$
$257$	−19.5429	−1.21905	−0.609525	+	0.792767i	$0.708640\pi$
$258$	3.54178	0.220501
$259$	3.21479	0.199757
$260$	0	0
$261$	14.1965	0.878744
$262$	3.16769	0.195700
$263$	−8.81360	−0.543470	−0.271735	−	0.962372i	$-0.587597\pi$
$263$	−8.81360	−0.543470	−0.271735	+	0.962372i	$0.587597\pi$
$264$	0.156416	0.00962672
$265$	0	0
$266$	0	0
$267$	3.69322	0.226022
$268$	2.34993	0.143545
$269$	−0.288362	−0.0175818	−0.00879088	−	0.999961i	$-0.502798\pi$
$269$	−0.288362	−0.0175818	−0.00879088	+	0.999961i	$0.502798\pi$
$270$	0	0
$271$	−24.8712	−1.51082	−0.755409	−	0.655253i	$-0.772562\pi$
$271$	−24.8712	−1.51082	−0.755409	+	0.655253i	$0.772562\pi$
$272$	−2.94987	−0.178862
$273$	2.52323	0.152713
$274$	20.6665	1.24851
$275$	0	0
$276$	−1.62067	−0.0975526
$277$	4.40486	0.264662	0.132331	−	0.991206i	$-0.457754\pi$
$277$	4.40486	0.264662	0.132331	+	0.991206i	$0.457754\pi$
$278$	−19.9509	−1.19657
$279$	18.7495	1.12250
$280$	0	0
$281$	32.7214	1.95200	0.975998	−	0.217778i	$-0.0698809\pi$
$281$	32.7214	1.95200	0.975998	+	0.217778i	$0.0698809\pi$
$282$	−4.62729	−0.275551
$283$	1.32893	0.0789964	0.0394982	−	0.999220i	$-0.487424\pi$
$283$	1.32893	0.0789964	0.0394982	+	0.999220i	$0.487424\pi$
$284$	1.05688	0.0627140
$285$	0	0
$286$	0.518260	0.0306454
$287$	20.1285	1.18815
$288$	−11.9594	−0.704717
$289$	−14.2403	−0.837666
$290$	0	0
$291$	−6.62656	−0.388456
$292$	−5.04618	−0.295305
$293$	−7.72365	−0.451220	−0.225610	−	0.974218i	$-0.572438\pi$
$293$	−7.72365	−0.451220	−0.225610	+	0.974218i	$0.572438\pi$
$294$	1.42219	0.0829437
$295$	0	0
$296$	5.21010	0.302831
$297$	0.298558	0.0173241
$298$	−24.4228	−1.41477
$299$	−18.8682	−1.09118
$300$	0	0
$301$	16.1059	0.928330
$302$	−3.65976	−0.210595
$303$	−2.04738	−0.117619
$304$	0	0
$305$	0	0
$306$	−5.20696	−0.297662
$307$	−9.10002	−0.519366	−0.259683	−	0.965694i	$-0.583618\pi$
$307$	−9.10002	−0.519366	−0.259683	+	0.965694i	$0.583618\pi$
$308$	0.202430	0.0115345
$309$	−0.814618	−0.0463420
$310$	0	0
$311$	−12.4569	−0.706364	−0.353182	−	0.935555i	$-0.614900\pi$
$311$	−12.4569	−0.706364	−0.353182	+	0.935555i	$0.614900\pi$
$312$	4.08931	0.231512
$313$	−2.04553	−0.115620	−0.0578101	−	0.998328i	$-0.518412\pi$
$313$	−2.04553	−0.115620	−0.0578101	+	0.998328i	$0.518412\pi$
$314$	−7.97059	−0.449807
$315$	0	0
$316$	−1.16741	−0.0656719
$317$	23.5713	1.32389	0.661947	−	0.749551i	$-0.269730\pi$
$317$	23.5713	1.32389	0.661947	+	0.749551i	$0.269730\pi$
$318$	0.110255	0.00618278
$319$	−0.668047	−0.0374035
$320$	0	0
$321$	0.379685	0.0211919
$322$	11.1560	0.621698
$323$	0	0
$324$	−6.14676	−0.341487
$325$	0	0
$326$	21.7055	1.20215
$327$	6.16907	0.341150
$328$	32.6216	1.80123
$329$	−21.0422	−1.16010
$330$	0	0
$331$	18.7175	1.02881	0.514403	−	0.857549i	$-0.328014\pi$
$331$	18.7175	1.02881	0.514403	+	0.857549i	$0.328014\pi$
$332$	5.92534	0.325195
$333$	4.85021	0.265790
$334$	−3.73480	−0.204359
$335$	0	0
$336$	−1.27515	−0.0695651
$337$	−32.6881	−1.78063	−0.890316	−	0.455343i	$-0.849517\pi$
$337$	−32.6881	−1.78063	−0.890316	+	0.455343i	$0.849517\pi$
$338$	−0.717368	−0.0390197
$339$	−0.319782	−0.0173682
$340$	0	0
$341$	−0.882297	−0.0477791
$342$	0	0
$343$	19.7188	1.06471
$344$	26.1023	1.40734
$345$	0	0
$346$	−11.6172	−0.624543
$347$	2.56666	0.137785	0.0688927	−	0.997624i	$-0.478053\pi$
$347$	2.56666	0.137785	0.0688927	+	0.997624i	$0.478053\pi$
$348$	−1.50017	−0.0804175
$349$	16.6195	0.889619	0.444810	−	0.895625i	$-0.353271\pi$
$349$	16.6195	0.889619	0.444810	+	0.895625i	$0.353271\pi$
$350$	0	0
$351$	7.80547	0.416625
$352$	0.562776	0.0299961
$353$	−28.3629	−1.50961	−0.754803	−	0.655951i	$-0.772268\pi$
$353$	−28.3629	−1.50961	−0.754803	+	0.655951i	$0.772268\pi$
$354$	−2.87105	−0.152595
$355$	0	0
$356$	7.74631	0.410553
$357$	1.19294	0.0631369
$358$	16.0252	0.846961
$359$	17.3885	0.917733	0.458866	−	0.888505i	$-0.348256\pi$
$359$	17.3885	0.917733	0.458866	+	0.888505i	$0.348256\pi$
$360$	0	0
$361$	0	0
$362$	5.96195	0.313353
$363$	4.16582	0.218649
$364$	5.29231	0.277393
$365$	0	0
$366$	−3.81988	−0.199668
$367$	−25.8048	−1.34700	−0.673501	−	0.739186i	$-0.735210\pi$
$367$	−25.8048	−1.34700	−0.673501	+	0.739186i	$0.735210\pi$
$368$	9.53531	0.497062
$369$	30.3683	1.58091
$370$	0	0
$371$	0.501374	0.0260300
$372$	−1.98129	−0.102725
$373$	27.0663	1.40144	0.700719	−	0.713437i	$-0.252862\pi$
$373$	27.0663	1.40144	0.700719	+	0.713437i	$0.252862\pi$
$374$	0.245024	0.0126699
$375$	0	0
$376$	−34.1024	−1.75870
$377$	−17.4653	−0.899511
$378$	−4.61505	−0.237373
$379$	−12.4028	−0.637092	−0.318546	−	0.947907i	$-0.603194\pi$
$379$	−12.4028	−0.637092	−0.318546	+	0.947907i	$0.603194\pi$
$380$	0	0
$381$	−7.10859	−0.364184
$382$	22.4709	1.14971
$383$	−5.35942	−0.273854	−0.136927	−	0.990581i	$-0.543723\pi$
$383$	−5.35942	−0.273854	−0.136927	+	0.990581i	$0.543723\pi$
$384$	−0.214673	−0.0109550
$385$	0	0
$386$	12.0976	0.615753
$387$	24.2993	1.23520
$388$	−13.8988	−0.705605
$389$	8.56933	0.434482	0.217241	−	0.976118i	$-0.430294\pi$
$389$	8.56933	0.434482	0.217241	+	0.976118i	$0.430294\pi$
$390$	0	0
$391$	−8.92053	−0.451131
$392$	10.4813	0.529386
$393$	−1.09492	−0.0552316
$394$	21.7877	1.09765
$395$	0	0
$396$	0.305411	0.0153475
$397$	10.6445	0.534234	0.267117	−	0.963664i	$-0.413929\pi$
$397$	10.6445	0.534234	0.267117	+	0.963664i	$0.413929\pi$
$398$	23.0490	1.15534
$399$	0	0
$400$	0	0
$401$	−7.65209	−0.382127	−0.191063	−	0.981578i	$-0.561194\pi$
$401$	−7.65209	−0.382127	−0.191063	+	0.981578i	$0.561194\pi$
$402$	1.22955	0.0613242
$403$	−23.0667	−1.14903
$404$	−4.29426	−0.213647
$405$	0	0
$406$	10.3265	0.512497
$407$	−0.228237	−0.0113133
$408$	1.93335	0.0957152
$409$	17.6887	0.874650	0.437325	−	0.899304i	$-0.355926\pi$
$409$	17.6887	0.874650	0.437325	+	0.899304i	$0.355926\pi$
$410$	0	0
$411$	−7.14345	−0.352361
$412$	−1.70861	−0.0841772
$413$	−13.0559	−0.642437
$414$	16.8312	0.827210
$415$	0	0
$416$	14.7131	0.721371
$417$	6.89609	0.337703
$418$	0	0
$419$	1.18732	0.0580045	0.0290023	−	0.999579i	$-0.490767\pi$
$419$	1.18732	0.0580045	0.0290023	+	0.999579i	$0.490767\pi$
$420$	0	0
$421$	−33.3673	−1.62622	−0.813111	−	0.582109i	$-0.802228\pi$
$421$	−33.3673	−1.62622	−0.813111	+	0.582109i	$0.802228\pi$
$422$	14.0754	0.685180
$423$	−31.7468	−1.54358
$424$	0.812560	0.0394614
$425$	0	0
$426$	0.552987	0.0267923
$427$	−17.3706	−0.840621
$428$	0.796365	0.0384938
$429$	−0.179139	−0.00864890
$430$	0	0
$431$	6.17598	0.297486	0.148743	−	0.988876i	$-0.452477\pi$
$431$	6.17598	0.297486	0.148743	+	0.988876i	$0.452477\pi$
$432$	−3.94461	−0.189785
$433$	−18.5552	−0.891707	−0.445854	−	0.895106i	$-0.647100\pi$
$433$	−18.5552	−0.891707	−0.445854	+	0.895106i	$0.647100\pi$
$434$	13.6384	0.654662
$435$	0	0
$436$	12.9392	0.619677
$437$	0	0
$438$	−2.64030	−0.126158
$439$	0.227312	0.0108490	0.00542450	−	0.999985i	$-0.498273\pi$
$439$	0.227312	0.0108490	0.00542450	+	0.999985i	$0.498273\pi$
$440$	0	0
$441$	9.75730	0.464633
$442$	6.40588	0.304696
$443$	34.9827	1.66208	0.831038	−	0.556215i	$-0.187747\pi$
$443$	34.9827	1.66208	0.831038	+	0.556215i	$0.187747\pi$
$444$	−0.512529	−0.0243236
$445$	0	0
$446$	22.3816	1.05980
$447$	8.44182	0.399285
$448$	−15.4224	−0.728638
$449$	16.9509	0.799961	0.399980	−	0.916524i	$-0.369017\pi$
$449$	16.9509	0.799961	0.399980	+	0.916524i	$0.369017\pi$
$450$	0	0
$451$	−1.42904	−0.0672909
$452$	−0.670724	−0.0315482
$453$	1.26501	0.0594353
$454$	27.8938	1.30912
$455$	0	0
$456$	0	0
$457$	−1.60241	−0.0749578	−0.0374789	−	0.999297i	$-0.511933\pi$
$457$	−1.60241	−0.0749578	−0.0374789	+	0.999297i	$0.511933\pi$
$458$	2.43140	0.113612
$459$	3.69029	0.172248
$460$	0	0
$461$	−8.74162	−0.407138	−0.203569	−	0.979061i	$-0.565254\pi$
$461$	−8.74162	−0.407138	−0.203569	+	0.979061i	$0.565254\pi$
$462$	0.105917	0.00492772
$463$	−21.1886	−0.984718	−0.492359	−	0.870392i	$-0.663865\pi$
$463$	−21.1886	−0.984718	−0.492359	+	0.870392i	$0.663865\pi$
$464$	8.82636	0.409753
$465$	0	0
$466$	15.5371	0.719744
$467$	−20.4516	−0.946388	−0.473194	−	0.880958i	$-0.656899\pi$
$467$	−20.4516	−0.946388	−0.473194	+	0.880958i	$0.656899\pi$
$468$	7.98461	0.369089
$469$	5.59126	0.258180
$470$	0	0
$471$	2.75507	0.126947
$472$	−21.1592	−0.973931
$473$	−1.14345	−0.0525760
$474$	−0.610821	−0.0280559
$475$	0	0
$476$	2.50211	0.114684
$477$	0.756432	0.0346347
$478$	3.30851	0.151328
$479$	23.5491	1.07599	0.537994	−	0.842949i	$-0.319182\pi$
$479$	23.5491	1.07599	0.537994	+	0.842949i	$0.319182\pi$
$480$	0	0
$481$	−5.96699	−0.272071
$482$	26.0962	1.18865
$483$	−3.85611	−0.175459
$484$	8.73755	0.397161
$485$	0	0
$486$	−10.5298	−0.477640
$487$	36.0392	1.63309	0.816546	−	0.577280i	$-0.195886\pi$
$487$	36.0392	1.63309	0.816546	+	0.577280i	$0.195886\pi$
$488$	−28.1519	−1.27438
$489$	−7.50258	−0.339278
$490$	0	0
$491$	20.0595	0.905271	0.452635	−	0.891696i	$-0.350484\pi$
$491$	20.0595	0.905271	0.452635	+	0.891696i	$0.350484\pi$
$492$	−3.20906	−0.144676
$493$	−8.25729	−0.371890
$494$	0	0
$495$	0	0
$496$	11.6571	0.523418
$497$	2.51466	0.112798
$498$	3.10030	0.138928
$499$	−36.8729	−1.65066	−0.825328	−	0.564653i	$-0.809010\pi$
$499$	−36.8729	−1.65066	−0.825328	+	0.564653i	$0.809010\pi$
$500$	0	0
$501$	1.29095	0.0576753
$502$	18.8649	0.841980
$503$	21.6487	0.965268	0.482634	−	0.875822i	$-0.339680\pi$
$503$	21.6487	0.965268	0.482634	+	0.875822i	$0.339680\pi$
$504$	−16.5882	−0.738899
$505$	0	0
$506$	−0.792028	−0.0352099
$507$	0.247961	0.0110123
$508$	−14.9098	−0.661517
$509$	36.4558	1.61588	0.807938	−	0.589267i	$-0.200583\pi$
$509$	36.4558	1.61588	0.807938	+	0.589267i	$0.200583\pi$
$510$	0	0
$511$	−12.0065	−0.531138
$512$	−18.3314	−0.810141
$513$	0	0
$514$	−21.4471	−0.945990
$515$	0	0
$516$	−2.56774	−0.113039
$517$	1.49391	0.0657021
$518$	3.52803	0.155013
$519$	4.01552	0.176262
$520$	0	0
$521$	22.6092	0.990528	0.495264	−	0.868742i	$-0.335071\pi$
$521$	22.6092	0.990528	0.495264	+	0.868742i	$0.335071\pi$
$522$	15.5798	0.681910
$523$	0.532911	0.0233026	0.0116513	−	0.999932i	$-0.496291\pi$
$523$	0.532911	0.0233026	0.0116513	+	0.999932i	$0.496291\pi$
$524$	−2.29654	−0.100325
$525$	0	0
$526$	−9.67238	−0.421736
$527$	−10.9055	−0.475051
$528$	0.0905303	0.00393983
$529$	5.83518	0.253703
$530$	0	0
$531$	−19.6976	−0.854804
$532$	0	0
$533$	−37.3606	−1.61827
$534$	4.05308	0.175394
$535$	0	0
$536$	9.06156	0.391400
$537$	−5.53919	−0.239034
$538$	−0.316460	−0.0136436
$539$	−0.459150	−0.0197770
$540$	0	0
$541$	5.01640	0.215672	0.107836	−	0.994169i	$-0.465608\pi$
$541$	5.01640	0.215672	0.107836	+	0.994169i	$0.465608\pi$
$542$	−27.2946	−1.17240
$543$	−2.06077	−0.0884362
$544$	6.95610	0.298240
$545$	0	0
$546$	2.76909	0.118506
$547$	22.6299	0.967584	0.483792	−	0.875183i	$-0.339259\pi$
$547$	22.6299	0.967584	0.483792	+	0.875183i	$0.339259\pi$
$548$	−14.9830	−0.640040
$549$	−26.2073	−1.11850
$550$	0	0
$551$	0	0
$552$	−6.24946	−0.265995
$553$	−2.77766	−0.118118
$554$	4.83406	0.205380
$555$	0	0
$556$	14.4641	0.613416
$557$	−35.2554	−1.49382	−0.746910	−	0.664925i	$-0.768463\pi$
$557$	−35.2554	−1.49382	−0.746910	+	0.664925i	$0.768463\pi$
$558$	20.5764	0.871070
$559$	−29.8943	−1.26439
$560$	0	0
$561$	−0.0846935	−0.00357576
$562$	35.9097	1.51476
$563$	−24.6295	−1.03801	−0.519005	−	0.854771i	$-0.673698\pi$
$563$	−24.6295	−1.03801	−0.519005	+	0.854771i	$0.673698\pi$
$564$	3.35473	0.141260
$565$	0	0
$566$	1.45841	0.0613017
$567$	−14.6252	−0.614200
$568$	4.07542	0.171001
$569$	20.0193	0.839252	0.419626	−	0.907697i	$-0.362161\pi$
$569$	20.0193	0.839252	0.419626	+	0.907697i	$0.362161\pi$
$570$	0	0
$571$	−16.6121	−0.695195	−0.347597	−	0.937644i	$-0.613002\pi$
$571$	−16.6121	−0.695195	−0.347597	+	0.937644i	$0.613002\pi$
$572$	−0.375732	−0.0157102
$573$	−7.76715	−0.324477
$574$	22.0898	0.922010
$575$	0	0
$576$	−23.2680	−0.969500
$577$	−12.4486	−0.518244	−0.259122	−	0.965845i	$-0.583433\pi$
$577$	−12.4486	−0.518244	−0.259122	+	0.965845i	$0.583433\pi$
$578$	−15.6279	−0.650034
$579$	−4.18159	−0.173781
$580$	0	0
$581$	14.0984	0.584899
$582$	−7.27224	−0.301444
$583$	−0.0355955	−0.00147421
$584$	−19.4586	−0.805202
$585$	0	0
$586$	−8.47623	−0.350150
$587$	−4.51144	−0.186207	−0.0931036	−	0.995656i	$-0.529679\pi$
$587$	−4.51144	−0.186207	−0.0931036	+	0.995656i	$0.529679\pi$
$588$	−1.03107	−0.0425206
$589$	0	0
$590$	0	0
$591$	−7.53101	−0.309784
$592$	3.01550	0.123936
$593$	−19.6082	−0.805213	−0.402606	−	0.915373i	$-0.631896\pi$
$593$	−19.6082	−0.805213	−0.402606	+	0.915373i	$0.631896\pi$
$594$	0.327650	0.0134436
$595$	0	0
$596$	17.7062	0.725274
$597$	−7.96699	−0.326067
$598$	−20.7067	−0.846759
$599$	−10.7759	−0.440292	−0.220146	−	0.975467i	$-0.570653\pi$
$599$	−10.7759	−0.440292	−0.220146	+	0.975467i	$0.570653\pi$
$600$	0	0
$601$	−15.0244	−0.612860	−0.306430	−	0.951893i	$-0.599134\pi$
$601$	−15.0244	−0.612860	−0.306430	+	0.951893i	$0.599134\pi$
$602$	17.6753	0.720389
$603$	8.43563	0.343526
$604$	2.65328	0.107960
$605$	0	0
$606$	−2.24687	−0.0912730
$607$	25.1901	1.02243	0.511217	−	0.859452i	$-0.329195\pi$
$607$	25.1901	1.02243	0.511217	+	0.859452i	$0.329195\pi$
$608$	0	0
$609$	−3.56940	−0.144640
$610$	0	0
$611$	39.0566	1.58006
$612$	3.77498	0.152595
$613$	−16.2351	−0.655728	−0.327864	−	0.944725i	$-0.606329\pi$
$613$	−16.2351	−0.655728	−0.327864	+	0.944725i	$0.606329\pi$
$614$	−9.98672	−0.403031
$615$	0	0
$616$	0.780593	0.0314510
$617$	−6.51826	−0.262415	−0.131208	−	0.991355i	$-0.541885\pi$
$617$	−6.51826	−0.262415	−0.131208	+	0.991355i	$0.541885\pi$
$618$	−0.893993	−0.0359617
$619$	−4.39112	−0.176494	−0.0882470	−	0.996099i	$-0.528126\pi$
$619$	−4.39112	−0.176494	−0.0882470	+	0.996099i	$0.528126\pi$
$620$	0	0
$621$	−11.9287	−0.478681
$622$	−13.6706	−0.548142
$623$	18.4311	0.738425
$624$	2.36681	0.0947483
$625$	0	0
$626$	−2.24484	−0.0897220
$627$	0	0
$628$	5.77858	0.230590
$629$	−2.82108	−0.112484
$630$	0	0
$631$	−34.6209	−1.37824	−0.689118	−	0.724650i	$-0.742002\pi$
$631$	−34.6209	−1.37824	−0.689118	+	0.724650i	$0.742002\pi$
$632$	−4.50165	−0.179066
$633$	−4.86521	−0.193375
$634$	25.8680	1.02735
$635$	0	0
$636$	−0.0799333	−0.00316956
$637$	−12.0040	−0.475614
$638$	−0.733141	−0.0290253
$639$	3.79391	0.150085
$640$	0	0
$641$	−7.41241	−0.292773	−0.146386	−	0.989227i	$-0.546764\pi$
$641$	−7.41241	−0.292773	−0.146386	+	0.989227i	$0.546764\pi$
$642$	0.416681	0.0164451
$643$	16.5459	0.652506	0.326253	−	0.945282i	$-0.394214\pi$
$643$	16.5459	0.652506	0.326253	+	0.945282i	$0.394214\pi$
$644$	−8.08794	−0.318710
$645$	0	0
$646$	0	0
$647$	−29.4822	−1.15907	−0.579533	−	0.814949i	$-0.696765\pi$
$647$	−29.4822	−1.15907	−0.579533	+	0.814949i	$0.696765\pi$
$648$	−23.7026	−0.931124
$649$	0.926912	0.0363845
$650$	0	0
$651$	−4.71415	−0.184762
$652$	−15.7362	−0.616277
$653$	−6.57421	−0.257269	−0.128634	−	0.991692i	$-0.541059\pi$
$653$	−6.57421	−0.257269	−0.128634	+	0.991692i	$0.541059\pi$
$654$	6.77017	0.264735
$655$	0	0
$656$	18.8807	0.737169
$657$	−18.1145	−0.706713
$658$	−23.0925	−0.900241
$659$	26.3370	1.02594	0.512972	−	0.858405i	$-0.328544\pi$
$659$	26.3370	1.02594	0.512972	+	0.858405i	$0.328544\pi$
$660$	0	0
$661$	−3.78419	−0.147188	−0.0735941	−	0.997288i	$-0.523447\pi$
$661$	−3.78419	−0.147188	−0.0735941	+	0.997288i	$0.523447\pi$
$662$	20.5413	0.798359
$663$	−2.21422	−0.0859930
$664$	22.8487	0.886703
$665$	0	0
$666$	5.32281	0.206255
$667$	26.6913	1.03349
$668$	2.70769	0.104763
$669$	−7.73630	−0.299103
$670$	0	0
$671$	1.23324	0.0476086
$672$	3.00694	0.115995
$673$	−21.0431	−0.811150	−0.405575	−	0.914062i	$-0.632929\pi$
$673$	−21.0431	−0.811150	−0.405575	+	0.914062i	$0.632929\pi$
$674$	−35.8731	−1.38178
$675$	0	0
$676$	0.520083	0.0200032
$677$	−15.2744	−0.587043	−0.293522	−	0.955952i	$-0.594827\pi$
$677$	−15.2744	−0.587043	−0.293522	+	0.955952i	$0.594827\pi$
$678$	−0.350941	−0.0134778
$679$	−33.0699	−1.26911
$680$	0	0
$681$	−9.64161	−0.369467
$682$	−0.968267	−0.0370769
$683$	−8.60225	−0.329156	−0.164578	−	0.986364i	$-0.552626\pi$
$683$	−8.60225	−0.329156	−0.164578	+	0.986364i	$0.552626\pi$
$684$	0	0
$685$	0	0
$686$	21.6401	0.826223
$687$	−0.840424	−0.0320642
$688$	15.1075	0.575968
$689$	−0.930603	−0.0354532
$690$	0	0
$691$	34.1079	1.29753	0.648763	−	0.760990i	$-0.275286\pi$
$691$	34.1079	1.29753	0.648763	+	0.760990i	$0.275286\pi$
$692$	8.42231	0.320168
$693$	0.726674	0.0276040
$694$	2.81675	0.106922
$695$	0	0
$696$	−5.78481	−0.219273
$697$	−17.6634	−0.669050
$698$	18.2388	0.690350
$699$	−5.37047	−0.203130
$700$	0	0
$701$	10.7293	0.405239	0.202619	−	0.979258i	$-0.435055\pi$
$701$	10.7293	0.405239	0.202619	+	0.979258i	$0.435055\pi$
$702$	8.56603	0.323304
$703$	0	0
$704$	1.09492	0.0412665
$705$	0	0
$706$	−31.1266	−1.17146
$707$	−10.2175	−0.384267
$708$	2.08148	0.0782267
$709$	−28.8476	−1.08339	−0.541697	−	0.840574i	$-0.682218\pi$
$709$	−28.8476	−1.08339	−0.541697	+	0.840574i	$0.682218\pi$
$710$	0	0
$711$	−4.19070	−0.157164
$712$	29.8706	1.11945
$713$	35.2515	1.32018
$714$	1.30917	0.0489946
$715$	0	0
$716$	−11.6181	−0.434189
$717$	−1.14360	−0.0427085
$718$	19.0829	0.712166
$719$	−20.0555	−0.747944	−0.373972	−	0.927440i	$-0.622004\pi$
$719$	−20.0555	−0.747944	−0.373972	+	0.927440i	$0.622004\pi$
$720$	0	0
$721$	−4.06535	−0.151402
$722$	0	0
$723$	−9.02026	−0.335467
$724$	−4.32234	−0.160639
$725$	0	0
$726$	4.57173	0.169673
$727$	0.780521	0.0289479	0.0144740	−	0.999895i	$-0.495393\pi$
$727$	0.780521	0.0289479	0.0144740	+	0.999895i	$0.495393\pi$
$728$	20.4077	0.756361
$729$	−19.5373	−0.723604
$730$	0	0
$731$	−14.1335	−0.522745
$732$	2.76937	0.102359
$733$	26.8391	0.991326	0.495663	−	0.868515i	$-0.334925\pi$
$733$	26.8391	0.991326	0.495663	+	0.868515i	$0.334925\pi$
$734$	−28.3192	−1.04528
$735$	0	0
$736$	−22.4853	−0.828817
$737$	−0.396956	−0.0146221
$738$	33.3273	1.22679
$739$	20.6530	0.759735	0.379867	−	0.925041i	$-0.375970\pi$
$739$	20.6530	0.759735	0.379867	+	0.925041i	$0.375970\pi$
$740$	0	0
$741$	0	0
$742$	0.550227	0.0201995
$743$	1.47317	0.0540454	0.0270227	−	0.999635i	$-0.491397\pi$
$743$	1.47317	0.0540454	0.0270227	+	0.999635i	$0.491397\pi$
$744$	−7.64007	−0.280098
$745$	0	0
$746$	29.7036	1.08752
$747$	21.2705	0.778245
$748$	−0.177639	−0.00649514
$749$	1.89482	0.0692352
$750$	0	0
$751$	−15.5624	−0.567881	−0.283940	−	0.958842i	$-0.591642\pi$
$751$	−15.5624	−0.567881	−0.283940	+	0.958842i	$0.591642\pi$
$752$	−19.7378	−0.719764
$753$	−6.52071	−0.237628
$754$	−19.1671	−0.698026
$755$	0	0
$756$	3.34586	0.121688
$757$	−20.1756	−0.733295	−0.366647	−	0.930360i	$-0.619494\pi$
$757$	−20.1756	−0.733295	−0.366647	+	0.930360i	$0.619494\pi$
$758$	−13.6114	−0.494387
$759$	0.273767	0.00993713
$760$	0	0
$761$	−15.1076	−0.547649	−0.273825	−	0.961780i	$-0.588289\pi$
$761$	−15.1076	−0.547649	−0.273825	+	0.961780i	$0.588289\pi$
$762$	−7.80124	−0.282609
$763$	30.7868	1.11456
$764$	−16.2911	−0.589392
$765$	0	0
$766$	−5.88163	−0.212512
$767$	24.2331	0.875005
$768$	5.94509	0.214525
$769$	51.6580	1.86284	0.931418	−	0.363951i	$-0.118573\pi$
$769$	51.6580	1.86284	0.931418	+	0.363951i	$0.118573\pi$
$770$	0	0
$771$	7.41327	0.266982
$772$	−8.77063	−0.315662
$773$	30.1558	1.08463	0.542314	−	0.840176i	$-0.317548\pi$
$773$	30.1558	1.08463	0.542314	+	0.840176i	$0.317548\pi$
$774$	26.6670	0.958525
$775$	0	0
$776$	−53.5952	−1.92396
$777$	−1.21948	−0.0437485
$778$	9.40431	0.337161
$779$	0	0
$780$	0	0
$781$	−0.178530	−0.00638832
$782$	−9.78974	−0.350080
$783$	−11.0418	−0.394601
$784$	6.06637	0.216656
$785$	0	0
$786$	−1.20161	−0.0428601
$787$	43.0969	1.53624	0.768119	−	0.640307i	$-0.221193\pi$
$787$	43.0969	1.53624	0.768119	+	0.640307i	$0.221193\pi$
$788$	−15.7958	−0.562703
$789$	3.34330	0.119025
$790$	0	0
$791$	−1.59588	−0.0567428
$792$	1.17770	0.0418476
$793$	32.2416	1.14493
$794$	11.6817	0.414569
$795$	0	0
$796$	−16.7103	−0.592280
$797$	−50.5062	−1.78902	−0.894510	−	0.447048i	$-0.852475\pi$
$797$	−50.5062	−1.78902	−0.894510	+	0.447048i	$0.852475\pi$
$798$	0	0
$799$	18.4652	0.653253
$800$	0	0
$801$	27.8073	0.982522
$802$	−8.39769	−0.296533
$803$	0.852414	0.0300810
$804$	−0.891406	−0.0314375
$805$	0	0
$806$	−25.3142	−0.891656
$807$	0.109386	0.00385056
$808$	−16.5591	−0.582547
$809$	−30.0872	−1.05781	−0.528905	−	0.848681i	$-0.677397\pi$
$809$	−30.0872	−1.05781	−0.528905	+	0.848681i	$0.677397\pi$
$810$	0	0
$811$	−22.9509	−0.805916	−0.402958	−	0.915218i	$-0.632018\pi$
$811$	−22.9509	−0.805916	−0.402958	+	0.915218i	$0.632018\pi$
$812$	−7.48661	−0.262728
$813$	9.43449	0.330882
$814$	−0.250476	−0.00877917
$815$	0	0
$816$	1.11899	0.0391723
$817$	0	0
$818$	19.4123	0.678734
$819$	18.9981	0.663846
$820$	0	0
$821$	−38.0807	−1.32903	−0.664513	−	0.747277i	$-0.731361\pi$
$821$	−38.0807	−1.32903	−0.664513	+	0.747277i	$0.731361\pi$
$822$	−7.83950	−0.273434
$823$	−53.7932	−1.87511	−0.937556	−	0.347835i	$-0.886917\pi$
$823$	−53.7932	−1.87511	−0.937556	+	0.347835i	$0.886917\pi$
$824$	−6.58858	−0.229524
$825$	0	0
$826$	−14.3280	−0.498535
$827$	32.4092	1.12698	0.563490	−	0.826123i	$-0.309458\pi$
$827$	32.4092	1.12698	0.563490	+	0.826123i	$0.309458\pi$
$828$	−12.2024	−0.424064
$829$	18.5305	0.643592	0.321796	−	0.946809i	$-0.395713\pi$
$829$	18.5305	0.643592	0.321796	+	0.946809i	$0.395713\pi$
$830$	0	0
$831$	−1.67091	−0.0579633
$832$	28.6255	0.992412
$833$	−5.67525	−0.196636
$834$	7.56804	0.262060
$835$	0	0
$836$	0	0
$837$	−14.5830	−0.504062
$838$	1.30301	0.0450118
$839$	−0.826608	−0.0285377	−0.0142688	−	0.999898i	$-0.504542\pi$
$839$	−0.826608	−0.0285377	−0.0142688	+	0.999898i	$0.504542\pi$
$840$	0	0
$841$	−4.29321	−0.148042
$842$	−36.6185	−1.26196
$843$	−12.4123	−0.427504
$844$	−10.2045	−0.351253
$845$	0	0
$846$	−34.8401	−1.19783
$847$	20.7895	0.714337
$848$	0.470294	0.0161500
$849$	−0.504106	−0.0173009
$850$	0	0
$851$	9.11901	0.312596
$852$	−0.400908	−0.0137349
$853$	7.34938	0.251638	0.125819	−	0.992053i	$-0.459844\pi$
$853$	7.34938	0.251638	0.125819	+	0.992053i	$0.459844\pi$
$854$	−19.0631	−0.652327
$855$	0	0
$856$	3.07087	0.104960
$857$	−45.3496	−1.54911	−0.774556	−	0.632506i	$-0.782026\pi$
$857$	−45.3496	−1.54911	−0.774556	+	0.632506i	$0.782026\pi$
$858$	−0.196594	−0.00671160
$859$	29.6285	1.01091	0.505456	−	0.862852i	$-0.331324\pi$
$859$	29.6285	1.01091	0.505456	+	0.862852i	$0.331324\pi$
$860$	0	0
$861$	−7.63542	−0.260215
$862$	6.77775	0.230851
$863$	41.0807	1.39840	0.699201	−	0.714925i	$-0.253539\pi$
$863$	41.0807	1.39840	0.699201	+	0.714925i	$0.253539\pi$
$864$	9.30180	0.316454
$865$	0	0
$866$	−20.3632	−0.691970
$867$	5.40183	0.183456
$868$	−9.88764	−0.335608
$869$	0.197202	0.00668962
$870$	0	0
$871$	−10.3780	−0.351644
$872$	49.8951	1.68966
$873$	−49.8931	−1.68863
$874$	0	0
$875$	0	0
$876$	1.91419	0.0646743
$877$	54.6307	1.84475	0.922374	−	0.386298i	$-0.126246\pi$
$877$	54.6307	1.84475	0.922374	+	0.386298i	$0.126246\pi$
$878$	0.249460	0.00841888
$879$	2.92984	0.0988211
$880$	0	0
$881$	−15.4805	−0.521552	−0.260776	−	0.965399i	$-0.583978\pi$
$881$	−15.4805	−0.521552	−0.260776	+	0.965399i	$0.583978\pi$
$882$	10.7080	0.360558
$883$	45.3495	1.52613	0.763066	−	0.646321i	$-0.223693\pi$
$883$	45.3495	1.52613	0.763066	+	0.646321i	$0.223693\pi$
$884$	−4.64418	−0.156201
$885$	0	0
$886$	38.3913	1.28978
$887$	−31.1399	−1.04558	−0.522788	−	0.852463i	$-0.675108\pi$
$887$	−31.1399	−1.04558	−0.522788	+	0.852463i	$0.675108\pi$
$888$	−1.97637	−0.0663226
$889$	−35.4755	−1.18981
$890$	0	0
$891$	1.03833	0.0347853
$892$	−16.2264	−0.543301
$893$	0	0
$894$	9.26438	0.309847
$895$	0	0
$896$	−1.07133	−0.0357905
$897$	7.15734	0.238977
$898$	18.6025	0.620775
$899$	32.6305	1.08829
$900$	0	0
$901$	−0.439972	−0.0146576
$902$	−1.56828	−0.0522181
$903$	−6.10952	−0.203312
$904$	−2.58638	−0.0860218
$905$	0	0
$906$	1.38827	0.0461222
$907$	43.5415	1.44577	0.722886	−	0.690967i	$-0.242815\pi$
$907$	43.5415	1.44577	0.722886	+	0.690967i	$0.242815\pi$
$908$	−20.2227	−0.671113
$909$	−15.4153	−0.511293
$910$	0	0
$911$	−5.72789	−0.189774	−0.0948868	−	0.995488i	$-0.530249\pi$
$911$	−5.72789	−0.189774	−0.0948868	+	0.995488i	$0.530249\pi$
$912$	0	0
$913$	−1.00093	−0.0331258
$914$	−1.75855	−0.0581677
$915$	0	0
$916$	−1.76274	−0.0582425
$917$	−5.46422	−0.180445
$918$	4.04986	0.133665
$919$	7.93860	0.261870	0.130935	−	0.991391i	$-0.458202\pi$
$919$	7.93860	0.261870	0.130935	+	0.991391i	$0.458202\pi$
$920$	0	0
$921$	3.45195	0.113746
$922$	−9.59339	−0.315941
$923$	−4.66747	−0.153632
$924$	−0.0767887	−0.00252616
$925$	0	0
$926$	−23.2532	−0.764147
$927$	−6.13347	−0.201450
$928$	−20.8135	−0.683236
$929$	−28.9567	−0.950039	−0.475019	−	0.879975i	$-0.657559\pi$
$929$	−28.9567	−0.950039	−0.475019	+	0.879975i	$0.657559\pi$
$930$	0	0
$931$	0	0
$932$	−11.2642	−0.368972
$933$	4.72531	0.154700
$934$	−22.4444	−0.734403
$935$	0	0
$936$	30.7895	1.00639
$937$	14.6816	0.479627	0.239813	−	0.970819i	$-0.422914\pi$
$937$	14.6816	0.479627	0.239813	+	0.970819i	$0.422914\pi$
$938$	6.13606	0.200349
$939$	0.775939	0.0253218
$940$	0	0
$941$	0.154755	0.00504485	0.00252243	−	0.999997i	$-0.499197\pi$
$941$	0.154755	0.00504485	0.00252243	+	0.999997i	$0.499197\pi$
$942$	3.02352	0.0985114
$943$	57.0961	1.85931
$944$	−12.2465	−0.398590
$945$	0	0
$946$	−1.25487	−0.0407993
$947$	7.51807	0.244304	0.122152	−	0.992511i	$-0.461020\pi$
$947$	7.51807	0.244304	0.122152	+	0.992511i	$0.461020\pi$
$948$	0.442838	0.0143827
$949$	22.2854	0.723415
$950$	0	0
$951$	−8.94137	−0.289944
$952$	9.64840	0.312706
$953$	22.6743	0.734493	0.367247	−	0.930124i	$-0.380300\pi$
$953$	22.6743	0.734493	0.367247	+	0.930124i	$0.380300\pi$
$954$	0.830138	0.0268767
$955$	0	0
$956$	−2.39863	−0.0775772
$957$	0.253413	0.00819167
$958$	25.8437	0.834973
$959$	−35.6494	−1.15118
$960$	0	0
$961$	12.0955	0.390177
$962$	−6.54840	−0.211129
$963$	2.85875	0.0921219
$964$	−18.9194	−0.609354
$965$	0	0
$966$	−4.23184	−0.136157
$967$	−28.8344	−0.927253	−0.463627	−	0.886031i	$-0.653452\pi$
$967$	−28.8344	−0.927253	−0.463627	+	0.886031i	$0.653452\pi$
$968$	33.6929	1.08293
$969$	0	0
$970$	0	0
$971$	−26.3661	−0.846130	−0.423065	−	0.906099i	$-0.639046\pi$
$971$	−26.3661	−0.846130	−0.423065	+	0.906099i	$0.639046\pi$
$972$	7.63395	0.244859
$973$	34.4150	1.10329
$974$	39.5508	1.26729
$975$	0	0
$976$	−16.2938	−0.521551
$977$	4.59218	0.146917	0.0734585	−	0.997298i	$-0.476596\pi$
$977$	4.59218	0.146917	0.0734585	+	0.997298i	$0.476596\pi$
$978$	−8.23362	−0.263282
$979$	−1.30853	−0.0418207
$980$	0	0
$981$	46.4486	1.48299
$982$	22.0140	0.702496
$983$	6.91473	0.220546	0.110273	−	0.993901i	$-0.464828\pi$
$983$	6.91473	0.220546	0.110273	+	0.993901i	$0.464828\pi$
$984$	−12.3745	−0.394484
$985$	0	0
$986$	−9.06187	−0.288589
$987$	7.98203	0.254071
$988$	0	0
$989$	45.6857	1.45272
$990$	0	0
$991$	38.0070	1.20733	0.603666	−	0.797237i	$-0.293706\pi$
$991$	38.0070	1.20733	0.603666	+	0.797237i	$0.293706\pi$
$992$	−27.4886	−0.872763
$993$	−7.10017	−0.225317
$994$	2.75968	0.0875318
$995$	0	0
$996$	−2.24768	−0.0712205
$997$	−16.2265	−0.513899	−0.256950	−	0.966425i	$-0.582717\pi$
$997$	−16.2265	−0.513899	−0.256950	+	0.966425i	$0.582717\pi$
$998$	−40.4657	−1.28092
$999$	−3.77239	−0.119353

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.bp.1.3		4
5.4	even	2		1805.2.a.i.1.2		4
19.8	odd	6		475.2.e.e.26.3		8
19.12	odd	6		475.2.e.e.201.3		8
19.18	odd	2		9025.2.a.bg.1.2		4
95.8	even	12		475.2.j.c.349.5		16
95.12	even	12		475.2.j.c.49.5		16
95.27	even	12		475.2.j.c.349.4		16
95.69	odd	6		95.2.e.c.11.2	✓	8
95.84	odd	6		95.2.e.c.26.2	yes	8
95.88	even	12		475.2.j.c.49.4		16
95.94	odd	2		1805.2.a.o.1.3		4
285.164	even	6		855.2.k.h.676.3		8
285.179	even	6		855.2.k.h.406.3		8
380.179	even	6		1520.2.q.o.881.2		8
380.259	even	6		1520.2.q.o.961.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.e.c.11.2	✓	8	95.69	odd	6
95.2.e.c.26.2	yes	8	95.84	odd	6
475.2.e.e.26.3		8	19.8	odd	6
475.2.e.e.201.3		8	19.12	odd	6
475.2.j.c.49.4		16	95.88	even	12
475.2.j.c.49.5		16	95.12	even	12
475.2.j.c.349.4		16	95.27	even	12
475.2.j.c.349.5		16	95.8	even	12
855.2.k.h.406.3		8	285.179	even	6
855.2.k.h.676.3		8	285.164	even	6
1520.2.q.o.881.2		8	380.179	even	6
1520.2.q.o.961.2		8	380.259	even	6
1805.2.a.i.1.2		4	5.4	even	2
1805.2.a.o.1.3		4	95.94	odd	2
9025.2.a.bg.1.2		4	19.18	odd	2
9025.2.a.bp.1.3		4	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q+1.09744 q^{2} -0.379334 q^{3} -0.795629 q^{4} -0.416295 q^{6} -1.89307 q^{7} -3.06803 q^{8} -2.85611 q^{9} +0.134400 q^{11} +0.301809 q^{12} +3.51373 q^{13} -2.07752 q^{14} -1.77572 q^{16} +1.66123 q^{17} -3.13440 q^{18} +0.718104 q^{21} +0.147496 q^{22} -5.36984 q^{23} +1.16381 q^{24} +3.85611 q^{26} +2.22142 q^{27} +1.50618 q^{28} -4.97059 q^{29} -6.56472 q^{31} +4.18732 q^{32} -0.0509824 q^{33} +1.82310 q^{34} +2.27240 q^{36} -1.69819 q^{37} -1.33288 q^{39} -10.6327 q^{41} +0.788075 q^{42} -8.50784 q^{43} -0.106932 q^{44} -5.89307 q^{46} +11.1154 q^{47} +0.673589 q^{48} -3.41630 q^{49} -0.630160 q^{51} -2.79563 q^{52} -0.264847 q^{53} +2.43787 q^{54} +5.80799 q^{56} -5.45492 q^{58} +6.89667 q^{59} +9.17589 q^{61} -7.20437 q^{62} +5.40680 q^{63} +8.14676 q^{64} -0.0559501 q^{66} -2.95354 q^{67} -1.32172 q^{68} +2.03696 q^{69} -1.32835 q^{71} +8.76262 q^{72} +6.34237 q^{73} -1.86366 q^{74} -0.254428 q^{77} -1.46275 q^{78} +1.46728 q^{79} +7.72566 q^{81} -11.6688 q^{82} -7.44736 q^{83} -0.571345 q^{84} -9.33683 q^{86} +1.88551 q^{87} -0.412343 q^{88} -9.73608 q^{89} -6.65174 q^{91} +4.27240 q^{92} +2.49022 q^{93} +12.1985 q^{94} -1.58839 q^{96} +17.4689 q^{97} -3.74917 q^{98} -0.383860 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + 3 q^{3} + 5 q^{4} + 2 q^{6} + 4 q^{7} + 12 q^{8} + q^{9} - 2 q^{11} + 6 q^{12} + 7 q^{13} + q^{14} + 7 q^{16} + q^{17} - 10 q^{18} + 4 q^{21} + 2 q^{22} - 2 q^{23} + 23 q^{24} + 3 q^{26}+ \cdots + 38 q^{99}+O(q^{100}) \) 4 * q + q^2 + 3 * q^3 + 5 * q^4 + 2 * q^6 + 4 * q^7 + 12 * q^8 + q^9 - 2 * q^11 + 6 * q^12 + 7 * q^13 + q^14 + 7 * q^16 + q^17 - 10 * q^18 + 4 * q^21 + 2 * q^22 - 2 * q^23 + 23 * q^24 + 3 * q^26 + 12 * q^27 + 19 * q^28 + q^29 + 30 * q^32 + 19 * q^33 - 15 * q^34 - 7 * q^36 - 2 * q^37 + 15 * q^39 + 8 * q^41 + 15 * q^42 - q^43 - 12 * q^44 - 12 * q^46 + 12 * q^47 + 23 * q^48 - 10 * q^49 - 22 * q^51 - 3 * q^52 - 5 * q^53 - 34 * q^54 + 41 * q^56 + 27 * q^58 + 5 * q^59 - 37 * q^62 + 3 * q^63 + 56 * q^64 - 31 * q^66 + 4 * q^67 - 16 * q^68 + 9 * q^69 - 20 * q^71 + 17 * q^72 + 20 * q^73 + 25 * q^74 - 14 * q^77 - 18 * q^78 - 17 * q^79 + 12 * q^81 - 21 * q^82 - q^83 + 20 * q^84 - 8 * q^86 + 16 * q^87 - 7 * q^88 - 11 * q^89 - 6 * q^91 + q^92 + 8 * q^93 + 31 * q^94 + 21 * q^96 + q^97 + 9 * q^98 + 38 * q^99

Introduction

L-functions

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