LMFDB - Embedded newform 9025.2.a.bw.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

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")

//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);

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.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:	$72.0649878242$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.71593280.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{4} + 13x^{2} - 5 $ x^6 - 8x^4 + 13x^2 - 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.755530$ 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)

$f(q)$	$=$	$q+0.755530 q^{2} +2.51596 q^{3} -1.42917 q^{4} +1.90089 q^{6} -3.75923 q^{7} -2.59085 q^{8} +3.33006 q^{9} +O(q^{10})$	\(q+0.755530 q^{2} +2.51596 q^{3} -1.42917 q^{4} +1.90089 q^{6} -3.75923 q^{7} -2.59085 q^{8} +3.33006 q^{9} -1.90089 q^{11} -3.59575 q^{12} -5.10681 q^{13} -2.84021 q^{14} +0.900885 q^{16} -3.75923 q^{17} +2.51596 q^{18} -9.45808 q^{21} -1.43618 q^{22} +4.13183 q^{23} -6.51846 q^{24} -3.85835 q^{26} +0.830415 q^{27} +5.37260 q^{28} +2.84021 q^{29} +6.61787 q^{31} +5.86234 q^{32} -4.78255 q^{33} -2.84021 q^{34} -4.75923 q^{36} -7.87214 q^{37} -12.8485 q^{39} +5.96194 q^{41} -7.14587 q^{42} +12.5185 q^{43} +2.71669 q^{44} +3.12172 q^{46} +12.8911 q^{47} +2.26659 q^{48} +7.13183 q^{49} -9.45808 q^{51} +7.29851 q^{52} +5.35618 q^{53} +0.627404 q^{54} +9.73959 q^{56} +2.14587 q^{58} -13.2357 q^{59} -9.74941 q^{61} +5.00000 q^{62} -12.5185 q^{63} +2.62740 q^{64} -3.61336 q^{66} -5.86234 q^{67} +5.37260 q^{68} +10.3955 q^{69} -15.4200 q^{71} -8.62767 q^{72} +0.868171 q^{73} -5.94764 q^{74} +7.14587 q^{77} -9.70745 q^{78} -0.655930 q^{79} -7.90089 q^{81} +4.50443 q^{82} +0.868171 q^{83} +13.5172 q^{84} +9.45808 q^{86} +7.14587 q^{87} +4.92490 q^{88} +3.77765 q^{89} +19.1977 q^{91} -5.90510 q^{92} +16.6503 q^{93} +9.73959 q^{94} +14.7494 q^{96} +4.09448 q^{97} +5.38831 q^{98} -6.33006 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 4 q^{4} + 10 q^{6} + 4 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q + 4 * q^4 + 10 * q^6 + 4 * q^7 + 6 * q^9	$ 6 q + 4 q^{4} + 10 q^{6} + 4 q^{7} + 6 q^{9} - 10 q^{11} + 4 q^{16} + 4 q^{17} + 8 q^{23} + 14 q^{24} + 2 q^{26} + 42 q^{28} - 2 q^{36} - 10 q^{39} + 20 q^{42} + 22 q^{43} - 34 q^{44} + 34 q^{47} + 26 q^{49} - 6 q^{54} - 50 q^{58} + 10 q^{61} + 30 q^{62} - 22 q^{63} + 6 q^{64} - 58 q^{66} + 42 q^{68} + 22 q^{73} + 30 q^{74} - 20 q^{77} - 46 q^{81} + 20 q^{82} + 22 q^{83} - 20 q^{87} + 54 q^{92} + 30 q^{93} + 20 q^{96} - 24 q^{99}+O(q^{100}) $ 6 * q + 4 * q^4 + 10 * q^6 + 4 * q^7 + 6 * q^9 - 10 * q^11 + 4 * q^16 + 4 * q^17 + 8 * q^23 + 14 * q^24 + 2 * q^26 + 42 * q^28 - 2 * q^36 - 10 * q^39 + 20 * q^42 + 22 * q^43 - 34 * q^44 + 34 * q^47 + 26 * q^49 - 6 * q^54 - 50 * q^58 + 10 * q^61 + 30 * q^62 - 22 * q^63 + 6 * q^64 - 58 * q^66 + 42 * q^68 + 22 * q^73 + 30 * q^74 - 20 * q^77 - 46 * q^81 + 20 * q^82 + 22 * q^83 - 20 * q^87 + 54 * q^92 + 30 * q^93 + 20 * q^96 - 24 * 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 $

$2$	0.755530	0.534241	0.267120	−	0.963663i	$-0.413928\pi$
$2$	0.755530	0.534241	0.267120	+	0.963663i	$0.413928\pi$
$3$	2.51596	1.45259	0.726295	−	0.687383i	$-0.241240\pi$
$3$	2.51596	1.45259	0.726295	+	0.687383i	$0.241240\pi$
$4$	−1.42917	−0.714587
$5$	0	0
$5$	0	0
$6$	1.90089	0.776033
$7$	−3.75923	−1.42086	−0.710428	−	0.703770i	$-0.751499\pi$
$7$	−3.75923	−1.42086	−0.710428	+	0.703770i	$0.751499\pi$
$8$	−2.59085	−0.916002
$9$	3.33006	1.11002
$10$	0	0
$11$	−1.90089	−0.573138	−0.286569	−	0.958060i	$-0.592515\pi$
$11$	−1.90089	−0.573138	−0.286569	+	0.958060i	$0.592515\pi$
$12$	−3.59575	−1.03800
$13$	−5.10681	−1.41637	−0.708187	−	0.706025i	$-0.750486\pi$
$13$	−5.10681	−1.41637	−0.708187	+	0.706025i	$0.750486\pi$
$14$	−2.84021	−0.759079
$15$	0	0
$16$	0.900885	0.225221
$17$	−3.75923	−0.911748	−0.455874	−	0.890044i	$-0.650673\pi$
$17$	−3.75923	−0.911748	−0.455874	+	0.890044i	$0.650673\pi$
$18$	2.51596	0.593018
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−9.45808	−2.06392
$22$	−1.43618	−0.306194
$23$	4.13183	0.861546	0.430773	−	0.902460i	$-0.358241\pi$
$23$	4.13183	0.861546	0.430773	+	0.902460i	$0.358241\pi$
$24$	−6.51846	−1.33058
$25$	0	0
$26$	−3.85835	−0.756684
$27$	0.830415	0.159813
$28$	5.37260	1.01533
$29$	2.84021	0.527415	0.263707	−	0.964603i	$-0.415055\pi$
$29$	2.84021	0.527415	0.263707	+	0.964603i	$0.415055\pi$
$30$	0	0
$31$	6.61787	1.18860	0.594302	−	0.804242i	$-0.297428\pi$
$31$	6.61787	1.18860	0.594302	+	0.804242i	$0.297428\pi$
$32$	5.86234	1.03632
$33$	−4.78255	−0.832535
$34$	−2.84021	−0.487093
$35$	0	0
$36$	−4.75923	−0.793205
$37$	−7.87214	−1.29417	−0.647086	−	0.762417i	$-0.724012\pi$
$37$	−7.87214	−1.29417	−0.647086	+	0.762417i	$0.724012\pi$
$38$	0	0
$39$	−12.8485	−2.05741
$40$	0	0
$41$	5.96194	0.931098	0.465549	−	0.885022i	$-0.345857\pi$
$41$	5.96194	0.931098	0.465549	+	0.885022i	$0.345857\pi$
$42$	−7.14587	−1.10263
$43$	12.5185	1.90905	0.954524	−	0.298134i	$-0.0963643\pi$
$43$	12.5185	1.90905	0.954524	+	0.298134i	$0.0963643\pi$
$44$	2.71669	0.409557
$45$	0	0
$46$	3.12172	0.460273
$47$	12.8911	1.88035	0.940177	−	0.340686i	$-0.110659\pi$
$47$	12.8911	1.88035	0.940177	+	0.340686i	$0.110659\pi$
$48$	2.26659	0.327154
$49$	7.13183	1.01883
$50$	0	0
$51$	−9.45808	−1.32440
$52$	7.29851	1.01212
$53$	5.35618	0.735727	0.367864	−	0.929880i	$-0.380089\pi$
$53$	5.35618	0.735727	0.367864	+	0.929880i	$0.380089\pi$
$54$	0.627404	0.0853788
$55$	0	0
$56$	9.73959	1.30151
$57$	0	0
$58$	2.14587	0.281766
$59$	−13.2357	−1.72315	−0.861573	−	0.507634i	$-0.830520\pi$
$59$	−13.2357	−1.72315	−0.861573	+	0.507634i	$0.830520\pi$
$60$	0	0
$61$	−9.74941	−1.24828	−0.624142	−	0.781311i	$-0.714551\pi$
$61$	−9.74941	−1.24828	−0.624142	+	0.781311i	$0.714551\pi$
$62$	5.00000	0.635001
$63$	−12.5185	−1.57718
$64$	2.62740	0.328425
$65$	0	0
$66$	−3.61336	−0.444774
$67$	−5.86234	−0.716198	−0.358099	−	0.933684i	$-0.616575\pi$
$67$	−5.86234	−0.716198	−0.358099	+	0.933684i	$0.616575\pi$
$68$	5.37260	0.651523
$69$	10.3955	1.25147
$70$	0	0
$71$	−15.4200	−1.83002	−0.915010	−	0.403432i	$-0.867817\pi$
$71$	−15.4200	−1.83002	−0.915010	+	0.403432i	$0.867817\pi$
$72$	−8.62767	−1.01678
$73$	0.868171	0.101612	0.0508059	−	0.998709i	$-0.483821\pi$
$73$	0.868171	0.101612	0.0508059	+	0.998709i	$0.483821\pi$
$74$	−5.94764	−0.691399
$75$	0	0
$76$	0	0
$77$	7.14587	0.814347
$78$	−9.70745	−1.09915
$79$	−0.655930	−0.0737979	−0.0368989	−	0.999319i	$-0.511748\pi$
$79$	−0.655930	−0.0737979	−0.0368989	+	0.999319i	$0.511748\pi$
$80$	0	0
$81$	−7.90089	−0.877876
$82$	4.50443	0.497431
$83$	0.868171	0.0952942	0.0476471	−	0.998864i	$-0.484828\pi$
$83$	0.868171	0.0952942	0.0476471	+	0.998864i	$0.484828\pi$
$84$	13.5172	1.47485
$85$	0	0
$86$	9.45808	1.01989
$87$	7.14587	0.766118
$88$	4.92490	0.524996
$89$	3.77765	0.400430	0.200215	−	0.979752i	$-0.435836\pi$
$89$	3.77765	0.400430	0.200215	+	0.979752i	$0.435836\pi$
$90$	0	0
$91$	19.1977	2.01246
$92$	−5.90510	−0.615649
$93$	16.6503	1.72656
$94$	9.73959	1.00456
$95$	0	0
$96$	14.7494	1.50536
$97$	4.09448	0.415732	0.207866	−	0.978157i	$-0.433348\pi$
$97$	4.09448	0.415732	0.207866	+	0.978157i	$0.433348\pi$
$98$	5.38831	0.544302
$99$	−6.33006	−0.636195
$100$	0	0
$101$	3.09911	0.308373	0.154187	−	0.988042i	$-0.450724\pi$
$101$	3.09911	0.308373	0.154187	+	0.988042i	$0.450724\pi$
$102$	−7.14587	−0.707546
$103$	16.7492	1.65034	0.825172	−	0.564881i	$-0.191078\pi$
$103$	16.7492	1.65034	0.825172	+	0.564881i	$0.191078\pi$
$104$	13.2309	1.29740
$105$	0	0
$106$	4.04675	0.393055
$107$	12.4802	1.20651	0.603253	−	0.797550i	$-0.293871\pi$
$107$	12.4802	1.20651	0.603253	+	0.797550i	$0.293871\pi$
$108$	−1.18681	−0.114201
$109$	7.27380	0.696703	0.348352	−	0.937364i	$-0.386742\pi$
$109$	7.27380	0.696703	0.348352	+	0.937364i	$0.386742\pi$
$110$	0	0
$111$	−19.8060	−1.87990
$112$	−3.38664	−0.320007
$113$	10.3132	0.970185	0.485093	−	0.874463i	$-0.338786\pi$
$113$	10.3132	0.970185	0.485093	+	0.874463i	$0.338786\pi$
$114$	0	0
$115$	0	0
$116$	−4.05916	−0.376884
$117$	−17.0060	−1.57220
$118$	−10.0000	−0.920575
$119$	14.1318	1.29546
$120$	0	0
$121$	−7.38664	−0.671512
$122$	−7.36598	−0.666884
$123$	15.0000	1.35250
$124$	−9.45808	−0.849361
$125$	0	0
$126$	−9.45808	−0.842593
$127$	8.80957	0.781723	0.390862	−	0.920449i	$-0.372177\pi$
$127$	8.80957	0.781723	0.390862	+	0.920449i	$0.372177\pi$
$128$	−9.73959	−0.860866
$129$	31.4960	2.77307
$130$	0	0
$131$	−7.13183	−0.623111	−0.311555	−	0.950228i	$-0.600850\pi$
$131$	−7.13183	−0.623111	−0.311555	+	0.950228i	$0.600850\pi$
$132$	6.83510	0.594919
$133$	0	0
$134$	−4.42917	−0.382622
$135$	0	0
$136$	9.73959	0.835163
$137$	−8.38664	−0.716519	−0.358259	−	0.933622i	$-0.616630\pi$
$137$	−8.38664	−0.716519	−0.358259	+	0.933622i	$0.616630\pi$
$138$	7.85413	0.668588
$139$	6.32023	0.536075	0.268038	−	0.963408i	$-0.413625\pi$
$139$	6.32023	0.536075	0.268038	+	0.963408i	$0.413625\pi$
$140$	0	0
$141$	32.4334	2.73139
$142$	−11.6503	−0.977671
$143$	9.70745	0.811778
$144$	3.00000	0.250000
$145$	0	0
$146$	0.655930	0.0542851
$147$	17.9434	1.47995
$148$	11.2507	0.924798
$149$	15.3301	1.25589	0.627944	−	0.778259i	$-0.283897\pi$
$149$	15.3301	1.25589	0.627944	+	0.778259i	$0.283897\pi$
$150$	0	0
$151$	14.4826	1.17858	0.589288	−	0.807923i	$-0.299408\pi$
$151$	14.4826	1.17858	0.589288	+	0.807923i	$0.299408\pi$
$152$	0	0
$153$	−12.5185	−1.01206
$154$	5.39892	0.435057
$155$	0	0
$156$	18.3628	1.47020
$157$	3.38664	0.270283	0.135141	−	0.990826i	$-0.456851\pi$
$157$	3.38664	0.270283	0.135141	+	0.990826i	$0.456851\pi$
$158$	−0.495575	−0.0394258
$159$	13.4759	1.06871
$160$	0	0
$161$	−15.5325	−1.22413
$162$	−5.96936	−0.468997
$163$	4.62740	0.362446	0.181223	−	0.983442i	$-0.441994\pi$
$163$	4.62740	0.362446	0.181223	+	0.983442i	$0.441994\pi$
$164$	−8.52064	−0.665351
$165$	0	0
$166$	0.655930	0.0509100
$167$	−0.930015	−0.0719668	−0.0359834	−	0.999352i	$-0.511456\pi$
$167$	−0.930015	−0.0719668	−0.0359834	+	0.999352i	$0.511456\pi$
$168$	24.5044	1.89056
$169$	13.0795	1.00611
$170$	0	0
$171$	0	0
$172$	−17.8911	−1.36418
$173$	12.3806	0.941280	0.470640	−	0.882325i	$-0.344023\pi$
$173$	12.3806	0.941280	0.470640	+	0.882325i	$0.344023\pi$
$174$	5.39892	0.409291
$175$	0	0
$176$	−1.71248	−0.129083
$177$	−33.3006	−2.50303
$178$	2.85413	0.213926
$179$	11.0514	0.826024	0.413012	−	0.910726i	$-0.364477\pi$
$179$	11.0514	0.826024	0.413012	+	0.910726i	$0.364477\pi$
$180$	0	0
$181$	2.18428	0.162357	0.0811783	−	0.996700i	$-0.474132\pi$
$181$	2.18428	0.162357	0.0811783	+	0.996700i	$0.474132\pi$
$182$	14.5044	1.07514
$183$	−24.5291	−1.81325
$184$	−10.7049	−0.789178
$185$	0	0
$186$	12.5798	0.922396
$187$	7.14587	0.522558
$188$	−18.4236	−1.34368
$189$	−3.12172	−0.227072
$190$	0	0
$191$	−14.5414	−1.05218	−0.526088	−	0.850430i	$-0.676342\pi$
$191$	−14.5414	−1.05218	−0.526088	+	0.850430i	$0.676342\pi$
$192$	6.61044	0.477068
$193$	17.3376	1.24799	0.623995	−	0.781428i	$-0.285508\pi$
$193$	17.3376	1.24799	0.623995	+	0.781428i	$0.285508\pi$
$194$	3.09351	0.222101
$195$	0	0
$196$	−10.1926	−0.728044
$197$	18.7592	1.33654	0.668270	−	0.743919i	$-0.267035\pi$
$197$	18.7592	1.33654	0.668270	+	0.743919i	$0.267035\pi$
$198$	−4.78255	−0.339881
$199$	0.495575	0.0351304	0.0175652	−	0.999846i	$-0.494409\pi$
$199$	0.495575	0.0351304	0.0175652	+	0.999846i	$0.494409\pi$
$200$	0	0
$201$	−14.7494	−1.04034
$202$	2.34148	0.164746
$203$	−10.6770	−0.749380
$204$	13.5172	0.946396
$205$	0	0
$206$	12.6545	0.881681
$207$	13.7592	0.956333
$208$	−4.60064	−0.318997
$209$	0	0
$210$	0	0
$211$	3.49614	0.240685	0.120342	−	0.992732i	$-0.461601\pi$
$211$	3.49614	0.240685	0.120342	+	0.992732i	$0.461601\pi$
$212$	−7.65491	−0.525741
$213$	−38.7962	−2.65827
$214$	9.42917	0.644565
$215$	0	0
$216$	−2.15148	−0.146389
$217$	−24.8781	−1.68884
$218$	5.49557	0.372207
$219$	2.18428	0.147600
$220$	0	0
$221$	19.1977	1.29138
$222$	−14.9640	−1.00432
$223$	−26.7957	−1.79437	−0.897187	−	0.441651i	$-0.854393\pi$
$223$	−26.7957	−1.79437	−0.897187	+	0.441651i	$0.854393\pi$
$224$	−22.0379	−1.47247
$225$	0	0
$226$	7.79195	0.518313
$227$	−14.6645	−0.973316	−0.486658	−	0.873593i	$-0.661784\pi$
$227$	−14.6645	−0.973316	−0.486658	+	0.873593i	$0.661784\pi$
$228$	0	0
$229$	19.8714	1.31314	0.656570	−	0.754265i	$-0.272007\pi$
$229$	19.8714	1.31314	0.656570	+	0.754265i	$0.272007\pi$
$230$	0	0
$231$	17.9787	1.18291
$232$	−7.35856	−0.483113
$233$	18.0140	1.18014	0.590069	−	0.807352i	$-0.299100\pi$
$233$	18.0140	1.18014	0.590069	+	0.807352i	$0.299100\pi$
$234$	−12.8485	−0.839934
$235$	0	0
$236$	18.9162	1.23134
$237$	−1.65029	−0.107198
$238$	10.6770	0.692089
$239$	−23.9280	−1.54777	−0.773887	−	0.633324i	$-0.781690\pi$
$239$	−23.9280	−1.54777	−0.773887	+	0.633324i	$0.781690\pi$
$240$	0	0
$241$	12.9542	0.834455	0.417228	−	0.908802i	$-0.363002\pi$
$241$	12.9542	0.834455	0.417228	+	0.908802i	$0.363002\pi$
$242$	−5.58083	−0.358749
$243$	−22.3696	−1.43501
$244$	13.9336	0.892007
$245$	0	0
$246$	11.3330	0.722563
$247$	0	0
$248$	−17.1459	−1.08876
$249$	2.18428	0.138423
$250$	0	0
$251$	17.8911	1.12927	0.564637	−	0.825339i	$-0.309016\pi$
$251$	17.8911	1.12927	0.564637	+	0.825339i	$0.309016\pi$
$252$	17.8911	1.12703
$253$	−7.85413	−0.493785
$254$	6.65590	0.417628
$255$	0	0
$256$	−12.6134	−0.788335
$257$	5.68785	0.354798	0.177399	−	0.984139i	$-0.443232\pi$
$257$	5.68785	0.354798	0.177399	+	0.984139i	$0.443232\pi$
$258$	23.7962	1.48148
$259$	29.5932	1.83883
$260$	0	0
$261$	9.45808	0.585441
$262$	−5.38831	−0.332891
$263$	15.3726	0.947915	0.473957	−	0.880548i	$-0.342825\pi$
$263$	15.3726	0.947915	0.473957	+	0.880548i	$0.342825\pi$
$264$	12.3909	0.762604
$265$	0	0
$266$	0	0
$267$	9.50443	0.581661
$268$	8.37830	0.511786
$269$	8.80215	0.536677	0.268338	−	0.963325i	$-0.413525\pi$
$269$	8.80215	0.536677	0.268338	+	0.963325i	$0.413525\pi$
$270$	0	0
$271$	−18.0991	−1.09944	−0.549721	−	0.835348i	$-0.685266\pi$
$271$	−18.0991	−1.09944	−0.549721	+	0.835348i	$0.685266\pi$
$272$	−3.38664	−0.205345
$273$	48.3006	2.92328
$274$	−6.33636	−0.382793
$275$	0	0
$276$	−14.8570	−0.894286
$277$	−13.2637	−0.796936	−0.398468	−	0.917182i	$-0.630458\pi$
$277$	−13.2637	−0.796936	−0.398468	+	0.917182i	$0.630458\pi$
$278$	4.77513	0.286393
$279$	22.0379	1.31937
$280$	0	0
$281$	−13.2357	−0.789578	−0.394789	−	0.918772i	$-0.629182\pi$
$281$	−13.2357	−0.789578	−0.394789	+	0.918772i	$0.629182\pi$
$282$	24.5044	1.45922
$283$	0.122979	0.00731032	0.00365516	−	0.999993i	$-0.498837\pi$
$283$	0.122979	0.00731032	0.00365516	+	0.999993i	$0.498837\pi$
$284$	22.0379	1.30771
$285$	0	0
$286$	7.33427	0.433685
$287$	−22.4123	−1.32296
$288$	19.5219	1.15034
$289$	−2.86817	−0.168716
$290$	0	0
$291$	10.3016	0.603888
$292$	−1.24077	−0.0726104
$293$	0.605762	0.0353890	0.0176945	−	0.999843i	$-0.494367\pi$
$293$	0.605762	0.0353890	0.0176945	+	0.999843i	$0.494367\pi$
$294$	13.5568	0.790648
$295$	0	0
$296$	20.3955	1.18546
$297$	−1.57852	−0.0915952
$298$	11.5823	0.670946
$299$	−21.1004	−1.22027
$300$	0	0
$301$	−47.0598	−2.71248
$302$	10.9420	0.629643
$303$	7.79725	0.447940
$304$	0	0
$305$	0	0
$306$	−9.45808	−0.540683
$307$	−0.556329	−0.0317514	−0.0158757	−	0.999874i	$-0.505054\pi$
$307$	−0.556329	−0.0317514	−0.0158757	+	0.999874i	$0.505054\pi$
$308$	−10.2127	−0.581922
$309$	42.1403	2.39728
$310$	0	0
$311$	−13.5512	−0.768417	−0.384208	−	0.923246i	$-0.625526\pi$
$311$	−13.5512	−0.768417	−0.384208	+	0.923246i	$0.625526\pi$
$312$	33.2885	1.88459
$313$	18.0140	1.01821	0.509107	−	0.860703i	$-0.329976\pi$
$313$	18.0140	1.01821	0.509107	+	0.860703i	$0.329976\pi$
$314$	2.55871	0.144396
$315$	0	0
$316$	0.937438	0.0527350
$317$	23.7309	1.33286	0.666429	−	0.745569i	$-0.267822\pi$
$317$	23.7309	1.33286	0.666429	+	0.745569i	$0.267822\pi$
$318$	10.1815	0.570949
$319$	−5.39892	−0.302282
$320$	0	0
$321$	31.3997	1.75256
$322$	−11.7353	−0.653982
$323$	0	0
$324$	11.2917	0.627319
$325$	0	0
$326$	3.49614	0.193634
$327$	18.3006	1.01202
$328$	−15.4465	−0.852888
$329$	−48.4605	−2.67171
$330$	0	0
$331$	−27.7183	−1.52354	−0.761768	−	0.647850i	$-0.775668\pi$
$331$	−27.7183	−1.52354	−0.761768	+	0.647850i	$0.775668\pi$
$332$	−1.24077	−0.0680960
$333$	−26.2147	−1.43656
$334$	−0.702655	−0.0384476
$335$	0	0
$336$	−8.52064	−0.464839
$337$	4.90760	0.267334	0.133667	−	0.991026i	$-0.457325\pi$
$337$	4.90760	0.267334	0.133667	+	0.991026i	$0.457325\pi$
$338$	9.88194	0.537506
$339$	25.9476	1.40928
$340$	0	0
$341$	−12.5798	−0.681235
$342$	0	0
$343$	−0.495575	−0.0267585
$344$	−32.4334	−1.74869
$345$	0	0
$346$	9.35392	0.502870
$347$	−14.5414	−0.780621	−0.390310	−	0.920683i	$-0.627632\pi$
$347$	−14.5414	−0.780621	−0.390310	+	0.920683i	$0.627632\pi$
$348$	−10.2127	−0.547458
$349$	2.85413	0.152778	0.0763891	−	0.997078i	$-0.475661\pi$
$349$	2.85413	0.152778	0.0763891	+	0.997078i	$0.475661\pi$
$350$	0	0
$351$	−4.24077	−0.226355
$352$	−11.1436	−0.593957
$353$	−22.9280	−1.22033	−0.610167	−	0.792273i	$-0.708898\pi$
$353$	−22.9280	−1.22033	−0.610167	+	0.792273i	$0.708898\pi$
$354$	−25.1596	−1.33722
$355$	0	0
$356$	−5.39892	−0.286142
$357$	35.5551	1.88178
$358$	8.34971	0.441296
$359$	1.57083	0.0829050	0.0414525	−	0.999140i	$-0.486801\pi$
$359$	1.57083	0.0829050	0.0414525	+	0.999140i	$0.486801\pi$
$360$	0	0
$361$	0	0
$362$	1.65029	0.0867375
$363$	−18.5845	−0.975433
$364$	−27.4368	−1.43808
$365$	0	0
$366$	−18.5325	−0.968710
$367$	5.90510	0.308244	0.154122	−	0.988052i	$-0.450745\pi$
$367$	5.90510	0.308244	0.154122	+	0.988052i	$0.450745\pi$
$368$	3.72230	0.194038
$369$	19.8536	1.03354
$370$	0	0
$371$	−20.1351	−1.04536
$372$	−23.7962	−1.23377
$373$	10.4383	0.540473	0.270236	−	0.962794i	$-0.412898\pi$
$373$	10.4383	0.540473	0.270236	+	0.962794i	$0.412898\pi$
$374$	5.39892	0.279172
$375$	0	0
$376$	−33.3987	−1.72241
$377$	−14.5044	−0.747016
$378$	−2.35856	−0.121311
$379$	−25.5340	−1.31160	−0.655798	−	0.754937i	$-0.727667\pi$
$379$	−25.5340	−1.31160	−0.655798	+	0.754937i	$0.727667\pi$
$380$	0	0
$381$	22.1645	1.13552
$382$	−10.9864	−0.562115
$383$	−24.5860	−1.25629	−0.628143	−	0.778098i	$-0.716185\pi$
$383$	−24.5860	−1.25629	−0.628143	+	0.778098i	$0.716185\pi$
$384$	−24.5044	−1.25049
$385$	0	0
$386$	13.0991	0.666727
$387$	41.6872	2.11908
$388$	−5.85173	−0.297077
$389$	−15.5273	−0.787266	−0.393633	−	0.919268i	$-0.628782\pi$
$389$	−15.5273	−0.787266	−0.393633	+	0.919268i	$0.628782\pi$
$390$	0	0
$391$	−15.5325	−0.785513
$392$	−18.4775	−0.933253
$393$	−17.9434	−0.905125
$394$	14.1732	0.714034
$395$	0	0
$396$	9.04675	0.454616
$397$	8.88221	0.445785	0.222893	−	0.974843i	$-0.428450\pi$
$397$	8.88221	0.445785	0.222893	+	0.974843i	$0.428450\pi$
$398$	0.374422	0.0187681
$399$	0	0
$400$	0	0
$401$	−6.33636	−0.316423	−0.158211	−	0.987405i	$-0.550573\pi$
$401$	−6.33636	−0.316423	−0.158211	+	0.987405i	$0.550573\pi$
$402$	−11.1436	−0.555794
$403$	−33.7962	−1.68351
$404$	−4.42917	−0.220360
$405$	0	0
$406$	−8.06682	−0.400350
$407$	14.9640	0.741739
$408$	24.5044	1.21315
$409$	−22.6938	−1.12214	−0.561068	−	0.827769i	$-0.689610\pi$
$409$	−22.6938	−1.12214	−0.561068	+	0.827769i	$0.689610\pi$
$410$	0	0
$411$	−21.1004	−1.04081
$412$	−23.9375	−1.17931
$413$	49.7562	2.44834
$414$	10.3955	0.510912
$415$	0	0
$416$	−29.9378	−1.46782
$417$	15.9015	0.778698
$418$	0	0
$419$	−28.6643	−1.40034	−0.700172	−	0.713974i	$-0.746893\pi$
$419$	−28.6643	−1.40034	−0.700172	+	0.713974i	$0.746893\pi$
$420$	0	0
$421$	−12.2983	−0.599382	−0.299691	−	0.954036i	$-0.596884\pi$
$421$	−12.2983	−0.599382	−0.299691	+	0.954036i	$0.596884\pi$
$422$	2.64144	0.128583
$423$	42.9280	2.08723
$424$	−13.8770	−0.673928
$425$	0	0
$426$	−29.3117	−1.42016
$427$	36.6503	1.77363
$428$	−17.8364	−0.862154
$429$	24.4236	1.17918
$430$	0	0
$431$	7.20878	0.347235	0.173617	−	0.984813i	$-0.444454\pi$
$431$	7.20878	0.347235	0.173617	+	0.984813i	$0.444454\pi$
$432$	0.748108	0.0359934
$433$	−0.954732	−0.0458815	−0.0229407	−	0.999737i	$-0.507303\pi$
$433$	−0.954732	−0.0458815	−0.0229407	+	0.999737i	$0.507303\pi$
$434$	−18.7962	−0.902245
$435$	0	0
$436$	−10.3955	−0.497855
$437$	0	0
$438$	1.65029	0.0788541
$439$	10.3955	0.496151	0.248076	−	0.968741i	$-0.420202\pi$
$439$	10.3955	0.496151	0.248076	+	0.968741i	$0.420202\pi$
$440$	0	0
$441$	23.7494	1.13092
$442$	14.5044	0.689905
$443$	−23.4236	−1.11289	−0.556444	−	0.830885i	$-0.687834\pi$
$443$	−23.4236	−1.11289	−0.556444	+	0.830885i	$0.687834\pi$
$444$	28.3062	1.34335
$445$	0	0
$446$	−20.2450	−0.958628
$447$	38.5698	1.82429
$448$	−9.87702	−0.466645
$449$	37.4579	1.76775	0.883874	−	0.467725i	$-0.154926\pi$
$449$	37.4579	1.76775	0.883874	+	0.467725i	$0.154926\pi$
$450$	0	0
$451$	−11.3330	−0.533648
$452$	−14.7394	−0.693282
$453$	36.4376	1.71199
$454$	−11.0795	−0.519985
$455$	0	0
$456$	0	0
$457$	−10.4095	−0.486937	−0.243469	−	0.969909i	$-0.578285\pi$
$457$	−10.4095	−0.486937	−0.243469	+	0.969909i	$0.578285\pi$
$458$	15.0135	0.701533
$459$	−3.12172	−0.145710
$460$	0	0
$461$	20.9280	0.974714	0.487357	−	0.873203i	$-0.337961\pi$
$461$	20.9280	0.974714	0.487357	+	0.873203i	$0.337961\pi$
$462$	13.5835	0.631960
$463$	9.62740	0.447423	0.223712	−	0.974655i	$-0.428183\pi$
$463$	9.62740	0.447423	0.223712	+	0.974655i	$0.428183\pi$
$464$	2.55871	0.118785
$465$	0	0
$466$	13.6102	0.630478
$467$	11.6134	0.537402	0.268701	−	0.963224i	$-0.413406\pi$
$467$	11.6134	0.537402	0.268701	+	0.963224i	$0.413406\pi$
$468$	24.3045	1.12347
$469$	22.0379	1.01762
$470$	0	0
$471$	8.52064	0.392611
$472$	34.2917	1.57841
$473$	−23.7962	−1.09415
$474$	−1.24685	−0.0572696
$475$	0	0
$476$	−20.1968	−0.925721
$477$	17.8364	0.816672
$478$	−18.0783	−0.826883
$479$	−10.5750	−0.483186	−0.241593	−	0.970378i	$-0.577670\pi$
$479$	−10.5750	−0.483186	−0.241593	+	0.970378i	$0.577670\pi$
$480$	0	0
$481$	40.2015	1.83303
$482$	9.78731	0.445800
$483$	−39.0792	−1.77816
$484$	10.5568	0.479854
$485$	0	0
$486$	−16.9009	−0.766640
$487$	20.5441	0.930943	0.465471	−	0.885063i	$-0.345885\pi$
$487$	20.5441	0.930943	0.465471	+	0.885063i	$0.345885\pi$
$488$	25.2592	1.14343
$489$	11.6424	0.526486
$490$	0	0
$491$	19.2057	0.866741	0.433370	−	0.901216i	$-0.357324\pi$
$491$	19.2057	0.866741	0.433370	+	0.901216i	$0.357324\pi$
$492$	−21.4376	−0.966482
$493$	−10.6770	−0.480869
$494$	0	0
$495$	0	0
$496$	5.96194	0.267699
$497$	57.9674	2.60019
$498$	1.65029	0.0739514
$499$	4.92053	0.220273	0.110137	−	0.993916i	$-0.464871\pi$
$499$	4.92053	0.220273	0.110137	+	0.993916i	$0.464871\pi$
$500$	0	0
$501$	−2.33988	−0.104538
$502$	13.5172	0.603304
$503$	41.2777	1.84048	0.920241	−	0.391353i	$-0.127993\pi$
$503$	41.2777	1.84048	0.920241	+	0.391353i	$0.127993\pi$
$504$	32.4334	1.44470
$505$	0	0
$506$	−5.93404	−0.263800
$507$	32.9074	1.46147
$508$	−12.5904	−0.558609
$509$	−19.1977	−0.850922	−0.425461	−	0.904977i	$-0.639888\pi$
$509$	−19.1977	−0.850922	−0.425461	+	0.904977i	$0.639888\pi$
$510$	0	0
$511$	−3.26366	−0.144376
$512$	9.94940	0.439705
$513$	0	0
$514$	4.29734	0.189548
$515$	0	0
$516$	−45.0132	−1.98160
$517$	−24.5044	−1.07770
$518$	22.3586	0.982379
$519$	31.1491	1.36729
$520$	0	0
$521$	−17.6043	−0.771259	−0.385629	−	0.922654i	$-0.626016\pi$
$521$	−17.6043	−0.771259	−0.385629	+	0.922654i	$0.626016\pi$
$522$	7.14587	0.312766
$523$	9.50825	0.415767	0.207883	−	0.978154i	$-0.433343\pi$
$523$	9.50825	0.415767	0.207883	+	0.978154i	$0.433343\pi$
$524$	10.1926	0.445267
$525$	0	0
$526$	11.6145	0.506415
$527$	−24.8781	−1.08371
$528$	−4.30853	−0.187505
$529$	−5.92799	−0.257739
$530$	0	0
$531$	−44.0758	−1.91273
$532$	0	0
$533$	−30.4465	−1.31878
$534$	7.18088	0.310747
$535$	0	0
$536$	15.1884	0.656039
$537$	27.8050	1.19987
$538$	6.65029	0.286715
$539$	−13.5568	−0.583932
$540$	0	0
$541$	31.3997	1.34998	0.674989	−	0.737827i	$-0.264148\pi$
$541$	31.3997	1.34998	0.674989	+	0.737827i	$0.264148\pi$
$542$	−13.6744	−0.587367
$543$	5.49557	0.235838
$544$	−22.0379	−0.944867
$545$	0	0
$546$	36.4926	1.56174
$547$	−18.3178	−0.783214	−0.391607	−	0.920133i	$-0.628081\pi$
$547$	−18.3178	−0.783214	−0.391607	+	0.920133i	$0.628081\pi$
$548$	11.9860	0.512015
$549$	−32.4661	−1.38562
$550$	0	0
$551$	0	0
$552$	−26.9332	−1.14635
$553$	2.46579	0.104856
$554$	−10.0211	−0.425756
$555$	0	0
$556$	−9.03271	−0.383072
$557$	39.5414	1.67542	0.837710	−	0.546115i	$-0.183894\pi$
$557$	39.5414	1.67542	0.837710	+	0.546115i	$0.183894\pi$
$558$	16.6503	0.704863
$559$	−63.9294	−2.70392
$560$	0	0
$561$	17.9787	0.759062
$562$	−10.0000	−0.421825
$563$	−32.3832	−1.36479	−0.682395	−	0.730983i	$-0.739062\pi$
$563$	−32.3832	−1.36479	−0.682395	+	0.730983i	$0.739062\pi$
$564$	−46.3530	−1.95181
$565$	0	0
$566$	0.0929141	0.00390547
$567$	29.7013	1.24734
$568$	39.9509	1.67630
$569$	−19.4792	−0.816610	−0.408305	−	0.912846i	$-0.633880\pi$
$569$	−19.4792	−0.816610	−0.408305	+	0.912846i	$0.633880\pi$
$570$	0	0
$571$	25.2450	1.05647	0.528235	−	0.849098i	$-0.322854\pi$
$571$	25.2450	1.05647	0.528235	+	0.849098i	$0.322854\pi$
$572$	−13.8736	−0.580086
$573$	−36.5855	−1.52838
$574$	−16.9332	−0.706777
$575$	0	0
$576$	8.74941	0.364559
$577$	−17.0598	−0.710210	−0.355105	−	0.934826i	$-0.615555\pi$
$577$	−17.0598	−0.710210	−0.355105	+	0.934826i	$0.615555\pi$
$578$	−2.16699	−0.0901349
$579$	43.6208	1.81282
$580$	0	0
$581$	−3.26366	−0.135399
$582$	7.78314	0.322622
$583$	−10.1815	−0.421674
$584$	−2.24930	−0.0930766
$585$	0	0
$586$	0.457671	0.0189062
$587$	−32.0598	−1.32325	−0.661625	−	0.749835i	$-0.730133\pi$
$587$	−32.0598	−1.32325	−0.661625	+	0.749835i	$0.730133\pi$
$588$	−25.6442	−1.05755
$589$	0	0
$590$	0	0
$591$	47.1975	1.94145
$592$	−7.09189	−0.291475
$593$	22.9649	0.943056	0.471528	−	0.881851i	$-0.343703\pi$
$593$	22.9649	0.943056	0.471528	+	0.881851i	$0.343703\pi$
$594$	−1.19262	−0.0489339
$595$	0	0
$596$	−21.9093	−0.897441
$597$	1.24685	0.0510301
$598$	−15.9420	−0.651918
$599$	−14.7641	−0.603244	−0.301622	−	0.953428i	$-0.597528\pi$
$599$	−14.7641	−0.603244	−0.301622	+	0.953428i	$0.597528\pi$
$600$	0	0
$601$	39.9887	1.63117	0.815587	−	0.578635i	$-0.196414\pi$
$601$	39.9887	1.63117	0.815587	+	0.578635i	$0.196414\pi$
$602$	−35.5551	−1.44912
$603$	−19.5219	−0.794994
$604$	−20.6981	−0.842195
$605$	0	0
$606$	5.89106	0.239308
$607$	46.8164	1.90022	0.950109	−	0.311917i	$-0.100971\pi$
$607$	46.8164	1.90022	0.950109	+	0.311917i	$0.100971\pi$
$608$	0	0
$609$	−26.8630	−1.08854
$610$	0	0
$611$	−65.8321	−2.66328
$612$	17.8911	0.723203
$613$	−0.372596	−0.0150490	−0.00752451	−	0.999972i	$-0.502395\pi$
$613$	−0.372596	−0.0150490	−0.00752451	+	0.999972i	$0.502395\pi$
$614$	−0.420324	−0.0169629
$615$	0	0
$616$	−18.5138	−0.745944
$617$	25.0369	1.00795	0.503974	−	0.863719i	$-0.331871\pi$
$617$	25.0369	1.00795	0.503974	+	0.863719i	$0.331871\pi$
$618$	31.8383	1.28072
$619$	30.3670	1.22055	0.610276	−	0.792189i	$-0.291058\pi$
$619$	30.3670	1.22055	0.610276	+	0.792189i	$0.291058\pi$
$620$	0	0
$621$	3.43113	0.137687
$622$	−10.2383	−0.410520
$623$	−14.2011	−0.568954
$624$	−11.5750	−0.463373
$625$	0	0
$626$	13.6102	0.543971
$627$	0	0
$628$	−4.84009	−0.193141
$629$	29.5932	1.17996
$630$	0	0
$631$	−4.75502	−0.189294	−0.0946471	−	0.995511i	$-0.530172\pi$
$631$	−4.75502	−0.189294	−0.0946471	+	0.995511i	$0.530172\pi$
$632$	1.69941	0.0675990
$633$	8.79616	0.349616
$634$	17.9294	0.712067
$635$	0	0
$636$	−19.2594	−0.763687
$637$	−36.4209	−1.44305
$638$	−4.07905	−0.161491
$639$	−51.3496	−2.03136
$640$	0	0
$641$	46.2601	1.82716	0.913581	−	0.406656i	$-0.133305\pi$
$641$	46.2601	1.82716	0.913581	+	0.406656i	$0.133305\pi$
$642$	23.7234	0.936289
$643$	6.77327	0.267112	0.133556	−	0.991041i	$-0.457360\pi$
$643$	6.77327	0.267112	0.133556	+	0.991041i	$0.457360\pi$
$644$	22.1986	0.874749
$645$	0	0
$646$	0	0
$647$	9.09490	0.357557	0.178779	−	0.983889i	$-0.442785\pi$
$647$	9.09490	0.357557	0.178779	+	0.983889i	$0.442785\pi$
$648$	20.4700	0.804136
$649$	25.1596	0.987601
$650$	0	0
$651$	−62.5923	−2.45319
$652$	−6.61336	−0.258999
$653$	15.1230	0.591808	0.295904	−	0.955218i	$-0.404379\pi$
$653$	15.1230	0.591808	0.295904	+	0.955218i	$0.404379\pi$
$654$	13.8267	0.540665
$655$	0	0
$656$	5.37102	0.209703
$657$	2.89106	0.112791
$658$	−36.6134	−1.42734
$659$	1.80986	0.0705022	0.0352511	−	0.999378i	$-0.488777\pi$
$659$	1.80986	0.0705022	0.0352511	+	0.999378i	$0.488777\pi$
$660$	0	0
$661$	−7.49029	−0.291339	−0.145669	−	0.989333i	$-0.546534\pi$
$661$	−7.49029	−0.291339	−0.145669	+	0.989333i	$0.546534\pi$
$662$	−20.9420	−0.813935
$663$	48.3006	1.87584
$664$	−2.24930	−0.0872897
$665$	0	0
$666$	−19.8060	−0.767466
$667$	11.7353	0.454392
$668$	1.32915	0.0514265
$669$	−67.4170	−2.60649
$670$	0	0
$671$	18.5325	0.715439
$672$	−55.4465	−2.13889
$673$	23.0576	0.888806	0.444403	−	0.895827i	$-0.353416\pi$
$673$	23.0576	0.888806	0.444403	+	0.895827i	$0.353416\pi$
$674$	3.70784	0.142821
$675$	0	0
$676$	−18.6928	−0.718955
$677$	2.17686	0.0836636	0.0418318	−	0.999125i	$-0.486681\pi$
$677$	2.17686	0.0836636	0.0418318	+	0.999125i	$0.486681\pi$
$678$	19.6042	0.752896
$679$	−15.3921	−0.590695
$680$	0	0
$681$	−36.8953	−1.41383
$682$	−9.50443	−0.363943
$683$	−34.6679	−1.32653	−0.663264	−	0.748385i	$-0.730829\pi$
$683$	−34.6679	−1.32653	−0.663264	+	0.748385i	$0.730829\pi$
$684$	0	0
$685$	0	0
$686$	−0.374422	−0.0142955
$687$	49.9957	1.90745
$688$	11.2777	0.429958
$689$	−27.3529	−1.04206
$690$	0	0
$691$	44.7059	1.70069	0.850346	−	0.526224i	$-0.176393\pi$
$691$	44.7059	1.70069	0.850346	+	0.526224i	$0.176393\pi$
$692$	−17.6940	−0.672626
$693$	23.7962	0.903941
$694$	−10.9864	−0.417039
$695$	0	0
$696$	−18.5138	−0.701765
$697$	−22.4123	−0.848927
$698$	2.15638	0.0816203
$699$	45.3226	1.71426
$700$	0	0
$701$	−38.0500	−1.43713	−0.718564	−	0.695461i	$-0.755200\pi$
$701$	−38.0500	−1.43713	−0.718564	+	0.695461i	$0.755200\pi$
$702$	−3.20403	−0.120928
$703$	0	0
$704$	−4.99439	−0.188233
$705$	0	0
$706$	−17.3228	−0.651952
$707$	−11.6503	−0.438154
$708$	47.5923	1.78863
$709$	−21.1547	−0.794482	−0.397241	−	0.917714i	$-0.630032\pi$
$709$	−21.1547	−0.794482	−0.397241	+	0.917714i	$0.630032\pi$
$710$	0	0
$711$	−2.18428	−0.0819171
$712$	−9.78731	−0.366795
$713$	27.3439	1.02404
$714$	26.8630	1.00532
$715$	0	0
$716$	−15.7944	−0.590266
$717$	−60.2019	−2.24828
$718$	1.18681	0.0442912
$719$	34.8714	1.30048	0.650242	−	0.759727i	$-0.274667\pi$
$719$	34.8714	1.30048	0.650242	+	0.759727i	$0.274667\pi$
$720$	0	0
$721$	−62.9640	−2.34490
$722$	0	0
$723$	32.5923	1.21212
$724$	−3.12172	−0.116018
$725$	0	0
$726$	−14.0411	−0.521116
$727$	−53.8331	−1.99656	−0.998279	−	0.0586359i	$-0.981325\pi$
$727$	−53.8331	−1.99656	−0.998279	+	0.0586359i	$0.981325\pi$
$728$	−49.7382	−1.84342
$729$	−32.5783	−1.20660
$730$	0	0
$731$	−47.0598	−1.74057
$732$	35.0564	1.29572
$733$	44.8279	1.65576	0.827878	−	0.560908i	$-0.189548\pi$
$733$	44.8279	1.65576	0.827878	+	0.560908i	$0.189548\pi$
$734$	4.46148	0.164676
$735$	0	0
$736$	24.2222	0.892841
$737$	11.1436	0.410481
$738$	15.0000	0.552158
$739$	−2.85413	−0.104991	−0.0524955	−	0.998621i	$-0.516718\pi$
$739$	−2.85413	−0.104991	−0.0524955	+	0.998621i	$0.516718\pi$
$740$	0	0
$741$	0	0
$742$	−15.2127	−0.558475
$743$	6.28619	0.230618	0.115309	−	0.993330i	$-0.463214\pi$
$743$	6.28619	0.230618	0.115309	+	0.993330i	$0.463214\pi$
$744$	−43.1383	−1.58153
$745$	0	0
$746$	7.88643	0.288743
$747$	2.89106	0.105778
$748$	−10.2127	−0.373413
$749$	−46.9160	−1.71427
$750$	0	0
$751$	−28.9373	−1.05594	−0.527968	−	0.849264i	$-0.677046\pi$
$751$	−28.9373	−1.05594	−0.527968	+	0.849264i	$0.677046\pi$
$752$	11.6134	0.423496
$753$	45.0132	1.64037
$754$	−10.9585	−0.399086
$755$	0	0
$756$	4.46148	0.162263
$757$	−36.6872	−1.33342	−0.666710	−	0.745317i	$-0.732298\pi$
$757$	−36.6872	−1.33342	−0.666710	+	0.745317i	$0.732298\pi$
$758$	−19.2917	−0.700707
$759$	−19.7607	−0.717267
$760$	0	0
$761$	5.24498	0.190131	0.0950653	−	0.995471i	$-0.469694\pi$
$761$	5.24498	0.190131	0.0950653	+	0.995471i	$0.469694\pi$
$762$	16.7460	0.606643
$763$	−27.3439	−0.989915
$764$	20.7821	0.751871
$765$	0	0
$766$	−18.5755	−0.671159
$767$	67.5923	2.44062
$768$	−31.7347	−1.14513
$769$	10.3291	0.372476	0.186238	−	0.982505i	$-0.440370\pi$
$769$	10.3291	0.372476	0.186238	+	0.982505i	$0.440370\pi$
$770$	0	0
$771$	14.3104	0.515377
$772$	−24.7785	−0.891798
$773$	−17.2126	−0.619094	−0.309547	−	0.950884i	$-0.600177\pi$
$773$	−17.2126	−0.619094	−0.309547	+	0.950884i	$0.600177\pi$
$774$	31.4960	1.13210
$775$	0	0
$776$	−10.6082	−0.380811
$777$	74.4553	2.67107
$778$	−11.7314	−0.420590
$779$	0	0
$780$	0	0
$781$	29.3117	1.04885
$782$	−11.7353	−0.419653
$783$	2.35856	0.0842879
$784$	6.42496	0.229463
$785$	0	0
$786$	−13.5568	−0.483555
$787$	10.3379	0.368507	0.184254	−	0.982879i	$-0.441013\pi$
$787$	10.3379	0.368507	0.184254	+	0.982879i	$0.441013\pi$
$788$	−26.8102	−0.955074
$789$	38.6768	1.37693
$790$	0	0
$791$	−38.7698	−1.37849
$792$	16.4002	0.582756
$793$	49.7883	1.76804
$794$	6.71078	0.238157
$795$	0	0
$796$	−0.708263	−0.0251037
$797$	20.4092	0.722931	0.361465	−	0.932386i	$-0.382277\pi$
$797$	20.4092	0.722931	0.361465	+	0.932386i	$0.382277\pi$
$798$	0	0
$799$	−48.4605	−1.71441
$800$	0	0
$801$	12.5798	0.444486
$802$	−4.78731	−0.169046
$803$	−1.65029	−0.0582376
$804$	21.0795	0.743415
$805$	0	0
$806$	−25.5340	−0.899398
$807$	22.1459	0.779571
$808$	−8.02933	−0.282471
$809$	−23.1370	−0.813454	−0.406727	−	0.913550i	$-0.633330\pi$
$809$	−23.1370	−0.813454	−0.406727	+	0.913550i	$0.633330\pi$
$810$	0	0
$811$	10.6770	0.374921	0.187461	−	0.982272i	$-0.439974\pi$
$811$	10.6770	0.374921	0.187461	+	0.982272i	$0.439974\pi$
$812$	15.2593	0.535497
$813$	−45.5367	−1.59704
$814$	11.3058	0.396267
$815$	0	0
$816$	−8.52064	−0.298282
$817$	0	0
$818$	−17.1459	−0.599491
$819$	63.9294	2.23387
$820$	0	0
$821$	−17.6591	−0.616308	−0.308154	−	0.951336i	$-0.599711\pi$
$821$	−17.6591	−0.616308	−0.308154	+	0.951336i	$0.599711\pi$
$822$	−15.9420	−0.556042
$823$	18.6363	0.649619	0.324809	−	0.945779i	$-0.394700\pi$
$823$	18.6363	0.649619	0.324809	+	0.945779i	$0.394700\pi$
$824$	−43.3945	−1.51172
$825$	0	0
$826$	37.5923	1.30800
$827$	−11.9592	−0.415862	−0.207931	−	0.978143i	$-0.566673\pi$
$827$	−11.9592	−0.415862	−0.207931	+	0.978143i	$0.566673\pi$
$828$	−19.6643	−0.683383
$829$	29.5282	1.02556	0.512778	−	0.858521i	$-0.328616\pi$
$829$	29.5282	1.02556	0.512778	+	0.858521i	$0.328616\pi$
$830$	0	0
$831$	−33.3708	−1.15762
$832$	−13.4176	−0.465173
$833$	−26.8102	−0.928918
$834$	12.0140	0.416012
$835$	0	0
$836$	0	0
$837$	5.49557	0.189955
$838$	−21.6568	−0.748121
$839$	27.4089	0.946260	0.473130	−	0.880992i	$-0.343124\pi$
$839$	27.4089	0.946260	0.473130	+	0.880992i	$0.343124\pi$
$840$	0	0
$841$	−20.9332	−0.721834
$842$	−9.29174	−0.320214
$843$	−33.3006	−1.14693
$844$	−4.99660	−0.171990
$845$	0	0
$846$	32.4334	1.11508
$847$	27.7681	0.954123
$848$	4.82530	0.165701
$849$	0.309409	0.0106189
$850$	0	0
$851$	−32.5263	−1.11499
$852$	55.4465	1.89956
$853$	31.3146	1.07219	0.536096	−	0.844157i	$-0.319898\pi$
$853$	31.3146	1.07219	0.536096	+	0.844157i	$0.319898\pi$
$854$	27.6904	0.947546
$855$	0	0
$856$	−32.3343	−1.10516
$857$	−13.8192	−0.472056	−0.236028	−	0.971746i	$-0.575846\pi$
$857$	−13.8192	−0.472056	−0.236028	+	0.971746i	$0.575846\pi$
$858$	18.4527	0.629966
$859$	42.4605	1.44873	0.724367	−	0.689415i	$-0.242132\pi$
$859$	42.4605	1.44873	0.724367	+	0.689415i	$0.242132\pi$
$860$	0	0
$861$	−56.3885	−1.92171
$862$	5.44646	0.185507
$863$	14.2233	0.484168	0.242084	−	0.970255i	$-0.422169\pi$
$863$	14.2233	0.484168	0.242084	+	0.970255i	$0.422169\pi$
$864$	4.86817	0.165619
$865$	0	0
$866$	−0.721329	−0.0245118
$867$	−7.21621	−0.245075
$868$	35.5551	1.20682
$869$	1.24685	0.0422964
$870$	0	0
$871$	29.9378	1.01440
$872$	−18.8453	−0.638182
$873$	13.6349	0.461471
$874$	0	0
$875$	0	0
$876$	−3.12172	−0.105473
$877$	−15.7591	−0.532148	−0.266074	−	0.963953i	$-0.585726\pi$
$877$	−15.7591	−0.532148	−0.266074	+	0.963953i	$0.585726\pi$
$878$	7.85413	0.265064
$879$	1.52407	0.0514057
$880$	0	0
$881$	−17.3473	−0.584447	−0.292223	−	0.956350i	$-0.594395\pi$
$881$	−17.3473	−0.584447	−0.292223	+	0.956350i	$0.594395\pi$
$882$	17.9434	0.604186
$883$	16.9860	0.571623	0.285812	−	0.958286i	$-0.407737\pi$
$883$	16.9860	0.571623	0.285812	+	0.958286i	$0.407737\pi$
$884$	−27.4368	−0.922800
$885$	0	0
$886$	−17.6972	−0.594550
$887$	−41.7096	−1.40047	−0.700235	−	0.713912i	$-0.746922\pi$
$887$	−41.7096	−1.40047	−0.700235	+	0.713912i	$0.746922\pi$
$888$	51.3142	1.72199
$889$	−33.1172	−1.11072
$890$	0	0
$891$	15.0187	0.503145
$892$	38.2957	1.28224
$893$	0	0
$894$	29.1407	0.974610
$895$	0	0
$896$	36.6134	1.22317
$897$	−53.0879	−1.77255
$898$	28.3006	0.944403
$899$	18.7962	0.626887
$900$	0	0
$901$	−20.1351	−0.670798
$902$	−8.56239	−0.285097
$903$	−118.401	−3.94013
$904$	−26.7199	−0.888692
$905$	0	0
$906$	27.5297	0.914614
$907$	37.2760	1.23773	0.618865	−	0.785498i	$-0.287593\pi$
$907$	37.2760	1.23773	0.618865	+	0.785498i	$0.287593\pi$
$908$	20.9581	0.695519
$909$	10.3202	0.342301
$910$	0	0
$911$	6.61787	0.219260	0.109630	−	0.993972i	$-0.465033\pi$
$911$	6.61787	0.219260	0.109630	+	0.993972i	$0.465033\pi$
$912$	0	0
$913$	−1.65029	−0.0546167
$914$	−7.86471	−0.260142
$915$	0	0
$916$	−28.3997	−0.938353
$917$	26.8102	0.885351
$918$	−2.35856	−0.0778440
$919$	8.92799	0.294507	0.147254	−	0.989099i	$-0.452957\pi$
$919$	8.92799	0.294507	0.147254	+	0.989099i	$0.452957\pi$
$920$	0	0
$921$	−1.39970	−0.0461218
$922$	15.8117	0.520732
$923$	78.7470	2.59199
$924$	−25.6947	−0.845294
$925$	0	0
$926$	7.27380	0.239032
$927$	55.7757	1.83192
$928$	16.6503	0.546573
$929$	−40.9420	−1.34326	−0.671632	−	0.740885i	$-0.734406\pi$
$929$	−40.9420	−1.34326	−0.671632	+	0.740885i	$0.734406\pi$
$930$	0	0
$931$	0	0
$932$	−25.7452	−0.843312
$933$	−34.0942	−1.11620
$934$	8.77425	0.287102
$935$	0	0
$936$	44.0598	1.44014
$937$	−18.0510	−0.589699	−0.294850	−	0.955544i	$-0.595270\pi$
$937$	−18.0510	−0.589699	−0.294850	+	0.955544i	$0.595270\pi$
$938$	16.6503	0.543651
$939$	45.3226	1.47905
$940$	0	0
$941$	−19.9465	−0.650238	−0.325119	−	0.945673i	$-0.605404\pi$
$941$	−19.9465	−0.650238	−0.325119	+	0.945673i	$0.605404\pi$
$942$	6.43761	0.209749
$943$	24.6337	0.802184
$944$	−11.9239	−0.388089
$945$	0	0
$946$	−17.9787	−0.584539
$947$	−31.8963	−1.03649	−0.518244	−	0.855233i	$-0.673414\pi$
$947$	−31.8963	−1.03649	−0.518244	+	0.855233i	$0.673414\pi$
$948$	2.35856	0.0766024
$949$	−4.43358	−0.143920
$950$	0	0
$951$	59.7059	1.93610
$952$	−36.6134	−1.18665
$953$	−35.6053	−1.15337	−0.576684	−	0.816967i	$-0.695654\pi$
$953$	−35.6053	−1.15337	−0.576684	+	0.816967i	$0.695654\pi$
$954$	13.4759	0.436299
$955$	0	0
$956$	34.1973	1.10602
$957$	−13.5835	−0.439091
$958$	−7.98977	−0.258138
$959$	31.5273	1.01807
$960$	0	0
$961$	12.7962	0.412779
$962$	30.3734	0.979279
$963$	41.5598	1.33925
$964$	−18.5138	−0.596291
$965$	0	0
$966$	−29.5255	−0.949968
$967$	22.1089	0.710976	0.355488	−	0.934681i	$-0.384315\pi$
$967$	22.1089	0.710976	0.355488	+	0.934681i	$0.384315\pi$
$968$	19.1376	0.615107
$969$	0	0
$970$	0	0
$971$	8.86716	0.284561	0.142280	−	0.989826i	$-0.454557\pi$
$971$	8.86716	0.284561	0.142280	+	0.989826i	$0.454557\pi$
$972$	31.9700	1.02544
$973$	−23.7592	−0.761686
$974$	15.5217	0.497348
$975$	0	0
$976$	−8.78310	−0.281140
$977$	−5.47062	−0.175021	−0.0875103	−	0.996164i	$-0.527891\pi$
$977$	−5.47062	−0.175021	−0.0875103	+	0.996164i	$0.527891\pi$
$978$	8.79616	0.281270
$979$	−7.18088	−0.229502
$980$	0	0
$981$	24.2222	0.773354
$982$	14.5105	0.463048
$983$	−6.91667	−0.220607	−0.110304	−	0.993898i	$-0.535182\pi$
$983$	−6.91667	−0.220607	−0.110304	+	0.993898i	$0.535182\pi$
$984$	−38.8627	−1.23890
$985$	0	0
$986$	−8.06682	−0.256900
$987$	−121.925	−3.88091
$988$	0	0
$989$	51.7242	1.64473
$990$	0	0
$991$	27.5018	0.873624	0.436812	−	0.899553i	$-0.356107\pi$
$991$	27.5018	0.873624	0.436812	+	0.899553i	$0.356107\pi$
$992$	38.7962	1.23178
$993$	−69.7382	−2.21307
$994$	43.7962	1.38913
$995$	0	0
$996$	−3.12172	−0.0989156
$997$	−26.6872	−0.845193	−0.422596	−	0.906318i	$-0.638881\pi$
$997$	−26.6872	−0.845193	−0.422596	+	0.906318i	$0.638881\pi$
$998$	3.71761	0.117679
$999$	−6.53714	−0.206826

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.bw.1.4	yes	6
5.4	even	2		9025.2.a.bv.1.3	✓	6
19.18	odd	2	inner	9025.2.a.bw.1.3	yes	6
95.94	odd	2		9025.2.a.bv.1.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9025.2.a.bv.1.3	✓	6	5.4	even	2
9025.2.a.bv.1.4	yes	6	95.94	odd	2
9025.2.a.bw.1.3	yes	6	19.18	odd	2	inner
9025.2.a.bw.1.4	yes	6	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+0.755530 q^{2} +2.51596 q^{3} -1.42917 q^{4} +1.90089 q^{6} -3.75923 q^{7} -2.59085 q^{8} +3.33006 q^{9} +O(q^{10})\)	\(q+0.755530 q^{2} +2.51596 q^{3} -1.42917 q^{4} +1.90089 q^{6} -3.75923 q^{7} -2.59085 q^{8} +3.33006 q^{9} -1.90089 q^{11} -3.59575 q^{12} -5.10681 q^{13} -2.84021 q^{14} +0.900885 q^{16} -3.75923 q^{17} +2.51596 q^{18} -9.45808 q^{21} -1.43618 q^{22} +4.13183 q^{23} -6.51846 q^{24} -3.85835 q^{26} +0.830415 q^{27} +5.37260 q^{28} +2.84021 q^{29} +6.61787 q^{31} +5.86234 q^{32} -4.78255 q^{33} -2.84021 q^{34} -4.75923 q^{36} -7.87214 q^{37} -12.8485 q^{39} +5.96194 q^{41} -7.14587 q^{42} +12.5185 q^{43} +2.71669 q^{44} +3.12172 q^{46} +12.8911 q^{47} +2.26659 q^{48} +7.13183 q^{49} -9.45808 q^{51} +7.29851 q^{52} +5.35618 q^{53} +0.627404 q^{54} +9.73959 q^{56} +2.14587 q^{58} -13.2357 q^{59} -9.74941 q^{61} +5.00000 q^{62} -12.5185 q^{63} +2.62740 q^{64} -3.61336 q^{66} -5.86234 q^{67} +5.37260 q^{68} +10.3955 q^{69} -15.4200 q^{71} -8.62767 q^{72} +0.868171 q^{73} -5.94764 q^{74} +7.14587 q^{77} -9.70745 q^{78} -0.655930 q^{79} -7.90089 q^{81} +4.50443 q^{82} +0.868171 q^{83} +13.5172 q^{84} +9.45808 q^{86} +7.14587 q^{87} +4.92490 q^{88} +3.77765 q^{89} +19.1977 q^{91} -5.90510 q^{92} +16.6503 q^{93} +9.73959 q^{94} +14.7494 q^{96} +4.09448 q^{97} +5.38831 q^{98} -6.33006 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 4 q^{4} + 10 q^{6} + 4 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q + 4 * q^4 + 10 * q^6 + 4 * q^7 + 6 * q^9	\( 6 q + 4 q^{4} + 10 q^{6} + 4 q^{7} + 6 q^{9} - 10 q^{11} + 4 q^{16} + 4 q^{17} + 8 q^{23} + 14 q^{24} + 2 q^{26} + 42 q^{28} - 2 q^{36} - 10 q^{39} + 20 q^{42} + 22 q^{43} - 34 q^{44} + 34 q^{47} + 26 q^{49} - 6 q^{54} - 50 q^{58} + 10 q^{61} + 30 q^{62} - 22 q^{63} + 6 q^{64} - 58 q^{66} + 42 q^{68} + 22 q^{73} + 30 q^{74} - 20 q^{77} - 46 q^{81} + 20 q^{82} + 22 q^{83} - 20 q^{87} + 54 q^{92} + 30 q^{93} + 20 q^{96} - 24 q^{99}+O(q^{100}) \) 6 * q + 4 * q^4 + 10 * q^6 + 4 * q^7 + 6 * q^9 - 10 * q^11 + 4 * q^16 + 4 * q^17 + 8 * q^23 + 14 * q^24 + 2 * q^26 + 42 * q^28 - 2 * q^36 - 10 * q^39 + 20 * q^42 + 22 * q^43 - 34 * q^44 + 34 * q^47 + 26 * q^49 - 6 * q^54 - 50 * q^58 + 10 * q^61 + 30 * q^62 - 22 * q^63 + 6 * q^64 - 58 * q^66 + 42 * q^68 + 22 * q^73 + 30 * q^74 - 20 * q^77 - 46 * q^81 + 20 * q^82 + 22 * q^83 - 20 * q^87 + 54 * q^92 + 30 * q^93 + 20 * q^96 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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