LMFDB - Embedded newform 5775.2.a.bw.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5775,2,Mod(1,5775)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5775, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("5775.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5775.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:	$46.1136071673$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.36147$ of defining polynomial
Character	$\chi$	$=$	5775.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.36147 q^{2} +1.00000 q^{3} +3.57653 q^{4} +2.36147 q^{6} +1.00000 q^{7} +3.72294 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.36147 q^{2} +1.00000 q^{3} +3.57653 q^{4} +2.36147 q^{6} +1.00000 q^{7} +3.72294 q^{8} +1.00000 q^{9} +1.00000 q^{11} +3.57653 q^{12} +3.93800 q^{13} +2.36147 q^{14} +1.63853 q^{16} -4.72294 q^{17} +2.36147 q^{18} +4.78493 q^{19} +1.00000 q^{21} +2.36147 q^{22} -2.72294 q^{23} +3.72294 q^{24} +9.29947 q^{26} +1.00000 q^{27} +3.57653 q^{28} +7.93800 q^{29} +1.15307 q^{31} -3.57653 q^{32} +1.00000 q^{33} -11.1531 q^{34} +3.57653 q^{36} +5.50787 q^{37} +11.2995 q^{38} +3.93800 q^{39} +0.430132 q^{41} +2.36147 q^{42} -6.72294 q^{43} +3.57653 q^{44} -6.43013 q^{46} +8.78493 q^{47} +1.63853 q^{48} +1.00000 q^{49} -4.72294 q^{51} +14.0844 q^{52} +3.15307 q^{53} +2.36147 q^{54} +3.72294 q^{56} +4.78493 q^{57} +18.7453 q^{58} -15.0911 q^{59} +6.00000 q^{61} +2.72294 q^{62} +1.00000 q^{63} -11.7229 q^{64} +2.36147 q^{66} -3.21507 q^{67} -16.8918 q^{68} -2.72294 q^{69} +13.4459 q^{71} +3.72294 q^{72} -11.9380 q^{73} +13.0067 q^{74} +17.1135 q^{76} +1.00000 q^{77} +9.29947 q^{78} -5.44588 q^{79} +1.00000 q^{81} +1.01574 q^{82} +2.84693 q^{83} +3.57653 q^{84} -15.8760 q^{86} +7.93800 q^{87} +3.72294 q^{88} +12.3061 q^{89} +3.93800 q^{91} -9.73868 q^{92} +1.15307 q^{93} +20.7453 q^{94} -3.57653 q^{96} -11.1531 q^{97} +2.36147 q^{98} +1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 6 q^{4} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 6 * q^4 + 3 * q^7 - 3 * q^8 + 3 * q^9	$ 3 q + 3 q^{3} + 6 q^{4} + 3 q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{11} + 6 q^{12} + 12 q^{16} + 12 q^{19} + 3 q^{21} + 6 q^{23} - 3 q^{24} + 9 q^{26} + 3 q^{27} + 6 q^{28} + 12 q^{29} - 6 q^{31} - 6 q^{32} + 3 q^{33} - 24 q^{34} + 6 q^{36} + 15 q^{38} + 6 q^{41} - 6 q^{43} + 6 q^{44} - 24 q^{46} + 24 q^{47} + 12 q^{48} + 3 q^{49} + 21 q^{52} - 3 q^{56} + 12 q^{57} + 9 q^{58} - 24 q^{59} + 18 q^{61} - 6 q^{62} + 3 q^{63} - 21 q^{64} - 12 q^{67} + 6 q^{68} + 6 q^{69} + 12 q^{71} - 3 q^{72} - 24 q^{73} + 39 q^{74} - 3 q^{76} + 3 q^{77} + 9 q^{78} + 12 q^{79} + 3 q^{81} - 30 q^{82} + 18 q^{83} + 6 q^{84} - 24 q^{86} + 12 q^{87} - 3 q^{88} + 18 q^{89} + 18 q^{92} - 6 q^{93} + 15 q^{94} - 6 q^{96} - 24 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 6 * q^4 + 3 * q^7 - 3 * q^8 + 3 * q^9 + 3 * q^11 + 6 * q^12 + 12 * q^16 + 12 * q^19 + 3 * q^21 + 6 * q^23 - 3 * q^24 + 9 * q^26 + 3 * q^27 + 6 * q^28 + 12 * q^29 - 6 * q^31 - 6 * q^32 + 3 * q^33 - 24 * q^34 + 6 * q^36 + 15 * q^38 + 6 * q^41 - 6 * q^43 + 6 * q^44 - 24 * q^46 + 24 * q^47 + 12 * q^48 + 3 * q^49 + 21 * q^52 - 3 * q^56 + 12 * q^57 + 9 * q^58 - 24 * q^59 + 18 * q^61 - 6 * q^62 + 3 * q^63 - 21 * q^64 - 12 * q^67 + 6 * q^68 + 6 * q^69 + 12 * q^71 - 3 * q^72 - 24 * q^73 + 39 * q^74 - 3 * q^76 + 3 * q^77 + 9 * q^78 + 12 * q^79 + 3 * q^81 - 30 * q^82 + 18 * q^83 + 6 * q^84 - 24 * q^86 + 12 * q^87 - 3 * q^88 + 18 * q^89 + 18 * q^92 - 6 * q^93 + 15 * q^94 - 6 * q^96 - 24 * q^97 + 3 * 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 $

$2$	2.36147	1.66981	0.834905	−	0.550394i	$-0.185522\pi$
$2$	2.36147	1.66981	0.834905	+	0.550394i	$0.185522\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	3.57653	1.78827
$5$	0	0
$5$	0	0
$6$	2.36147	0.964066
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	3.72294	1.31626
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	3.57653	1.03246
$13$	3.93800	1.09221	0.546103	−	0.837718i	$-0.316111\pi$
$13$	3.93800	1.09221	0.546103	+	0.837718i	$0.316111\pi$
$14$	2.36147	0.631129
$15$	0	0
$16$	1.63853	0.409633
$17$	−4.72294	−1.14548	−0.572740	−	0.819737i	$-0.694120\pi$
$17$	−4.72294	−1.14548	−0.572740	+	0.819737i	$0.694120\pi$
$18$	2.36147	0.556604
$19$	4.78493	1.09774	0.548870	−	0.835908i	$-0.315058\pi$
$19$	4.78493	1.09774	0.548870	+	0.835908i	$0.315058\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	2.36147	0.503467
$23$	−2.72294	−0.567772	−0.283886	−	0.958858i	$-0.591624\pi$
$23$	−2.72294	−0.567772	−0.283886	+	0.958858i	$0.591624\pi$
$24$	3.72294	0.759941
$25$	0	0
$26$	9.29947	1.82378
$27$	1.00000	0.192450
$28$	3.57653	0.675902
$29$	7.93800	1.47405	0.737025	−	0.675865i	$-0.236230\pi$
$29$	7.93800	1.47405	0.737025	+	0.675865i	$0.236230\pi$
$30$	0	0
$31$	1.15307	0.207097	0.103549	−	0.994624i	$-0.466980\pi$
$31$	1.15307	0.207097	0.103549	+	0.994624i	$0.466980\pi$
$32$	−3.57653	−0.632248
$33$	1.00000	0.174078
$34$	−11.1531	−1.91274
$35$	0	0
$36$	3.57653	0.596089
$37$	5.50787	0.905489	0.452744	−	0.891640i	$-0.350445\pi$
$37$	5.50787	0.905489	0.452744	+	0.891640i	$0.350445\pi$
$38$	11.2995	1.83302
$39$	3.93800	0.630585
$40$	0	0
$41$	0.430132	0.0671753	0.0335877	−	0.999436i	$-0.489307\pi$
$41$	0.430132	0.0671753	0.0335877	+	0.999436i	$0.489307\pi$
$42$	2.36147	0.364383
$43$	−6.72294	−1.02524	−0.512619	−	0.858616i	$-0.671325\pi$
$43$	−6.72294	−1.02524	−0.512619	+	0.858616i	$0.671325\pi$
$44$	3.57653	0.539183
$45$	0	0
$46$	−6.43013	−0.948071
$47$	8.78493	1.28141	0.640707	−	0.767785i	$-0.278641\pi$
$47$	8.78493	1.28141	0.640707	+	0.767785i	$0.278641\pi$
$48$	1.63853	0.236502
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.72294	−0.661344
$52$	14.0844	1.95316
$53$	3.15307	0.433107	0.216554	−	0.976271i	$-0.430518\pi$
$53$	3.15307	0.433107	0.216554	+	0.976271i	$0.430518\pi$
$54$	2.36147	0.321355
$55$	0	0
$56$	3.72294	0.497498
$57$	4.78493	0.633780
$58$	18.7453	2.46138
$59$	−15.0911	−1.96469	−0.982345	−	0.187077i	$-0.940098\pi$
$59$	−15.0911	−1.96469	−0.982345	+	0.187077i	$0.940098\pi$
$60$	0	0
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	2.72294	0.345813
$63$	1.00000	0.125988
$64$	−11.7229	−1.46537
$65$	0	0
$66$	2.36147	0.290677
$67$	−3.21507	−0.392783	−0.196391	−	0.980526i	$-0.562922\pi$
$67$	−3.21507	−0.392783	−0.196391	+	0.980526i	$0.562922\pi$
$68$	−16.8918	−2.04843
$69$	−2.72294	−0.327803
$70$	0	0
$71$	13.4459	1.59573	0.797866	−	0.602835i	$-0.205962\pi$
$71$	13.4459	1.59573	0.797866	+	0.602835i	$0.205962\pi$
$72$	3.72294	0.438752
$73$	−11.9380	−1.39724	−0.698619	−	0.715494i	$-0.746202\pi$
$73$	−11.9380	−1.39724	−0.698619	+	0.715494i	$0.746202\pi$
$74$	13.0067	1.51199
$75$	0	0
$76$	17.1135	1.96305
$77$	1.00000	0.113961
$78$	9.29947	1.05296
$79$	−5.44588	−0.612709	−0.306354	−	0.951918i	$-0.599109\pi$
$79$	−5.44588	−0.612709	−0.306354	+	0.951918i	$0.599109\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	1.01574	0.112170
$83$	2.84693	0.312491	0.156246	−	0.987718i	$-0.450061\pi$
$83$	2.84693	0.312491	0.156246	+	0.987718i	$0.450061\pi$
$84$	3.57653	0.390232
$85$	0	0
$86$	−15.8760	−1.71195
$87$	7.93800	0.851043
$88$	3.72294	0.396866
$89$	12.3061	1.30445	0.652224	−	0.758026i	$-0.273836\pi$
$89$	12.3061	1.30445	0.652224	+	0.758026i	$0.273836\pi$
$90$	0	0
$91$	3.93800	0.412815
$92$	−9.73868	−1.01533
$93$	1.15307	0.119568
$94$	20.7453	2.13972
$95$	0	0
$96$	−3.57653	−0.365029
$97$	−11.1531	−1.13242	−0.566211	−	0.824260i	$-0.691591\pi$
$97$	−11.1531	−1.13242	−0.566211	+	0.824260i	$0.691591\pi$
$98$	2.36147	0.238544
$99$	1.00000	0.100504
$100$	0	0
$101$	16.4750	1.63932	0.819659	−	0.572851i	$-0.194163\pi$
$101$	16.4750	1.63932	0.819659	+	0.572851i	$0.194163\pi$
$102$	−11.1531	−1.10432
$103$	1.56987	0.154684	0.0773418	−	0.997005i	$-0.475357\pi$
$103$	1.56987	0.154684	0.0773418	+	0.997005i	$0.475357\pi$
$104$	14.6609	1.43762
$105$	0	0
$106$	7.44588	0.723207
$107$	4.78493	0.462577	0.231289	−	0.972885i	$-0.425706\pi$
$107$	4.78493	0.462577	0.231289	+	0.972885i	$0.425706\pi$
$108$	3.57653	0.344152
$109$	−15.0291	−1.43952	−0.719762	−	0.694221i	$-0.755749\pi$
$109$	−15.0291	−1.43952	−0.719762	+	0.694221i	$0.755749\pi$
$110$	0	0
$111$	5.50787	0.522784
$112$	1.63853	0.154827
$113$	−7.44588	−0.700449	−0.350225	−	0.936666i	$-0.613895\pi$
$113$	−7.44588	−0.700449	−0.350225	+	0.936666i	$0.613895\pi$
$114$	11.2995	1.05829
$115$	0	0
$116$	28.3905	2.63600
$117$	3.93800	0.364069
$118$	−35.6371	−3.28066
$119$	−4.72294	−0.432951
$120$	0	0
$121$	1.00000	0.0909091
$122$	14.1688	1.28278
$123$	0.430132	0.0387837
$124$	4.12399	0.370346
$125$	0	0
$126$	2.36147	0.210376
$127$	−12.2928	−1.09081	−0.545405	−	0.838173i	$-0.683624\pi$
$127$	−12.2928	−1.09081	−0.545405	+	0.838173i	$0.683624\pi$
$128$	−20.5303	−1.81464
$129$	−6.72294	−0.591922
$130$	0	0
$131$	7.13974	0.623802	0.311901	−	0.950115i	$-0.399034\pi$
$131$	7.13974	0.623802	0.311901	+	0.950115i	$0.399034\pi$
$132$	3.57653	0.311297
$133$	4.78493	0.414906
$134$	−7.59228	−0.655873
$135$	0	0
$136$	−17.5832	−1.50775
$137$	−1.58320	−0.135262	−0.0676310	−	0.997710i	$-0.521544\pi$
$137$	−1.58320	−0.135262	−0.0676310	+	0.997710i	$0.521544\pi$
$138$	−6.43013	−0.547369
$139$	−17.1979	−1.45871	−0.729353	−	0.684138i	$-0.760179\pi$
$139$	−17.1979	−1.45871	−0.729353	+	0.684138i	$0.760179\pi$
$140$	0	0
$141$	8.78493	0.739825
$142$	31.7520	2.66457
$143$	3.93800	0.329312
$144$	1.63853	0.136544
$145$	0	0
$146$	−28.1912	−2.33312
$147$	1.00000	0.0824786
$148$	19.6991	1.61926
$149$	15.9380	1.30569	0.652846	−	0.757491i	$-0.273575\pi$
$149$	15.9380	1.30569	0.652846	+	0.757491i	$0.273575\pi$
$150$	0	0
$151$	6.59894	0.537014	0.268507	−	0.963278i	$-0.413470\pi$
$151$	6.59894	0.537014	0.268507	+	0.963278i	$0.413470\pi$
$152$	17.8140	1.44491
$153$	−4.72294	−0.381827
$154$	2.36147	0.190293
$155$	0	0
$156$	14.0844	1.12765
$157$	−7.32188	−0.584350	−0.292175	−	0.956365i	$-0.594379\pi$
$157$	−7.32188	−0.584350	−0.292175	+	0.956365i	$0.594379\pi$
$158$	−12.8603	−1.02311
$159$	3.15307	0.250055
$160$	0	0
$161$	−2.72294	−0.214598
$162$	2.36147	0.185535
$163$	−14.1068	−1.10493	−0.552466	−	0.833536i	$-0.686313\pi$
$163$	−14.1068	−1.10493	−0.552466	+	0.833536i	$0.686313\pi$
$164$	1.53838	0.120127
$165$	0	0
$166$	6.72294	0.521801
$167$	0.416799	0.0322528	0.0161264	−	0.999870i	$-0.494867\pi$
$167$	0.416799	0.0322528	0.0161264	+	0.999870i	$0.494867\pi$
$168$	3.72294	0.287231
$169$	2.50787	0.192913
$170$	0	0
$171$	4.78493	0.365913
$172$	−24.0448	−1.83340
$173$	8.43013	0.640931	0.320466	−	0.947260i	$-0.396161\pi$
$173$	8.43013	0.640931	0.320466	+	0.947260i	$0.396161\pi$
$174$	18.7453	1.42108
$175$	0	0
$176$	1.63853	0.123509
$177$	−15.0911	−1.13431
$178$	29.0606	2.17818
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	20.5989	1.53111	0.765554	−	0.643372i	$-0.222465\pi$
$181$	20.5989	1.53111	0.765554	+	0.643372i	$0.222465\pi$
$182$	9.29947	0.689323
$183$	6.00000	0.443533
$184$	−10.1373	−0.747334
$185$	0	0
$186$	2.72294	0.199655
$187$	−4.72294	−0.345375
$188$	31.4196	2.29151
$189$	1.00000	0.0727393
$190$	0	0
$191$	−14.5989	−1.05634	−0.528171	−	0.849138i	$-0.677122\pi$
$191$	−14.5989	−1.05634	−0.528171	+	0.849138i	$0.677122\pi$
$192$	−11.7229	−0.846030
$193$	−13.4592	−0.968815	−0.484408	−	0.874842i	$-0.660965\pi$
$193$	−13.4592	−0.968815	−0.484408	+	0.874842i	$0.660965\pi$
$194$	−26.3376	−1.89093
$195$	0	0
$196$	3.57653	0.255467
$197$	−11.4459	−0.815485	−0.407742	−	0.913097i	$-0.633684\pi$
$197$	−11.4459	−0.815485	−0.407742	+	0.913097i	$0.633684\pi$
$198$	2.36147	0.167822
$199$	−13.0291	−0.923607	−0.461803	−	0.886982i	$-0.652797\pi$
$199$	−13.0291	−0.923607	−0.461803	+	0.886982i	$0.652797\pi$
$200$	0	0
$201$	−3.21507	−0.226773
$202$	38.9051	2.73735
$203$	7.93800	0.557139
$204$	−16.8918	−1.18266
$205$	0	0
$206$	3.70719	0.258292
$207$	−2.72294	−0.189257
$208$	6.45254	0.447403
$209$	4.78493	0.330981
$210$	0	0
$211$	2.43013	0.167297	0.0836486	−	0.996495i	$-0.473343\pi$
$211$	2.43013	0.167297	0.0836486	+	0.996495i	$0.473343\pi$
$212$	11.2771	0.774512
$213$	13.4459	0.921296
$214$	11.2995	0.772416
$215$	0	0
$216$	3.72294	0.253314
$217$	1.15307	0.0782755
$218$	−35.4907	−2.40373
$219$	−11.9380	−0.806696
$220$	0	0
$221$	−18.5989	−1.25110
$222$	13.0067	0.872950
$223$	−22.7678	−1.52464	−0.762321	−	0.647199i	$-0.775940\pi$
$223$	−22.7678	−1.52464	−0.762321	+	0.647199i	$0.775940\pi$
$224$	−3.57653	−0.238967
$225$	0	0
$226$	−17.5832	−1.16962
$227$	15.8760	1.05373	0.526864	−	0.849950i	$-0.323368\pi$
$227$	15.8760	1.05373	0.526864	+	0.849950i	$0.323368\pi$
$228$	17.1135	1.13337
$229$	−3.15307	−0.208361	−0.104180	−	0.994558i	$-0.533222\pi$
$229$	−3.15307	−0.208361	−0.104180	+	0.994558i	$0.533222\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	29.5527	1.94023
$233$	−13.1397	−0.860813	−0.430406	−	0.902635i	$-0.641630\pi$
$233$	−13.1397	−0.860813	−0.430406	+	0.902635i	$0.641630\pi$
$234$	9.29947	0.607926
$235$	0	0
$236$	−53.9737	−3.51339
$237$	−5.44588	−0.353748
$238$	−11.1531	−0.722946
$239$	−10.1068	−0.653756	−0.326878	−	0.945067i	$-0.605997\pi$
$239$	−10.1068	−0.653756	−0.326878	+	0.945067i	$0.605997\pi$
$240$	0	0
$241$	2.36814	0.152545	0.0762725	−	0.997087i	$-0.475698\pi$
$241$	2.36814	0.152545	0.0762725	+	0.997087i	$0.475698\pi$
$242$	2.36147	0.151801
$243$	1.00000	0.0641500
$244$	21.4592	1.37379
$245$	0	0
$246$	1.01574	0.0647614
$247$	18.8431	1.19896
$248$	4.29281	0.272593
$249$	2.84693	0.180417
$250$	0	0
$251$	10.2308	0.645763	0.322881	−	0.946439i	$-0.395348\pi$
$251$	10.2308	0.645763	0.322881	+	0.946439i	$0.395348\pi$
$252$	3.57653	0.225301
$253$	−2.72294	−0.171190
$254$	−29.0291	−1.82145
$255$	0	0
$256$	−25.0357	−1.56473
$257$	11.0777	0.691010	0.345505	−	0.938417i	$-0.387708\pi$
$257$	11.0777	0.691010	0.345505	+	0.938417i	$0.387708\pi$
$258$	−15.8760	−0.988397
$259$	5.50787	0.342242
$260$	0	0
$261$	7.93800	0.491350
$262$	16.8603	1.04163
$263$	8.78493	0.541702	0.270851	−	0.962621i	$-0.412695\pi$
$263$	8.78493	0.541702	0.270851	+	0.962621i	$0.412695\pi$
$264$	3.72294	0.229131
$265$	0	0
$266$	11.2995	0.692815
$267$	12.3061	0.753123
$268$	−11.4988	−0.702401
$269$	5.75201	0.350706	0.175353	−	0.984506i	$-0.443893\pi$
$269$	5.75201	0.350706	0.175353	+	0.984506i	$0.443893\pi$
$270$	0	0
$271$	0.784934	0.0476813	0.0238407	−	0.999716i	$-0.492411\pi$
$271$	0.784934	0.0476813	0.0238407	+	0.999716i	$0.492411\pi$
$272$	−7.73868	−0.469226
$273$	3.93800	0.238339
$274$	−3.73868	−0.225862
$275$	0	0
$276$	−9.73868	−0.586200
$277$	11.5699	0.695166	0.347583	−	0.937649i	$-0.387002\pi$
$277$	11.5699	0.695166	0.347583	+	0.937649i	$0.387002\pi$
$278$	−40.6123	−2.43576
$279$	1.15307	0.0690325
$280$	0	0
$281$	−25.2599	−1.50688	−0.753439	−	0.657518i	$-0.771607\pi$
$281$	−25.2599	−1.50688	−0.753439	+	0.657518i	$0.771607\pi$
$282$	20.7453	1.23537
$283$	10.9671	0.651925	0.325963	−	0.945383i	$-0.394312\pi$
$283$	10.9671	0.651925	0.325963	+	0.945383i	$0.394312\pi$
$284$	48.0896	2.85360
$285$	0	0
$286$	9.29947	0.549889
$287$	0.430132	0.0253899
$288$	−3.57653	−0.210749
$289$	5.30614	0.312126
$290$	0	0
$291$	−11.1531	−0.653805
$292$	−42.6967	−2.49863
$293$	−14.7363	−0.860902	−0.430451	−	0.902614i	$-0.641645\pi$
$293$	−14.7363	−0.860902	−0.430451	+	0.902614i	$0.641645\pi$
$294$	2.36147	0.137724
$295$	0	0
$296$	20.5055	1.19186
$297$	1.00000	0.0580259
$298$	37.6371	2.18026
$299$	−10.7229	−0.620123
$300$	0	0
$301$	−6.72294	−0.387504
$302$	15.5832	0.896712
$303$	16.4750	0.946461
$304$	7.84026	0.449670
$305$	0	0
$306$	−11.1531	−0.637579
$307$	−0.860264	−0.0490979	−0.0245489	−	0.999699i	$-0.507815\pi$
$307$	−0.860264	−0.0490979	−0.0245489	+	0.999699i	$0.507815\pi$
$308$	3.57653	0.203792
$309$	1.56987	0.0893067
$310$	0	0
$311$	26.8918	1.52489	0.762446	−	0.647052i	$-0.223998\pi$
$311$	26.8918	1.52489	0.762446	+	0.647052i	$0.223998\pi$
$312$	14.6609	0.830012
$313$	−10.1688	−0.574775	−0.287388	−	0.957814i	$-0.592787\pi$
$313$	−10.1688	−0.574775	−0.287388	+	0.957814i	$0.592787\pi$
$314$	−17.2904	−0.975753
$315$	0	0
$316$	−19.4774	−1.09569
$317$	27.4907	1.54403	0.772016	−	0.635604i	$-0.219249\pi$
$317$	27.4907	1.54403	0.772016	+	0.635604i	$0.219249\pi$
$318$	7.44588	0.417544
$319$	7.93800	0.444443
$320$	0	0
$321$	4.78493	0.267069
$322$	−6.43013	−0.358337
$323$	−22.5989	−1.25744
$324$	3.57653	0.198696
$325$	0	0
$326$	−33.3128	−1.84503
$327$	−15.0291	−0.831110
$328$	1.60135	0.0884200
$329$	8.78493	0.484329
$330$	0	0
$331$	−17.1979	−0.945281	−0.472641	−	0.881255i	$-0.656699\pi$
$331$	−17.1979	−0.945281	−0.472641	+	0.881255i	$0.656699\pi$
$332$	10.1821	0.558818
$333$	5.50787	0.301830
$334$	0.984257	0.0538561
$335$	0	0
$336$	1.63853	0.0893892
$337$	−19.7387	−1.07523	−0.537617	−	0.843189i	$-0.680675\pi$
$337$	−19.7387	−1.07523	−0.537617	+	0.843189i	$0.680675\pi$
$338$	5.92226	0.322128
$339$	−7.44588	−0.404404
$340$	0	0
$341$	1.15307	0.0624422
$342$	11.2995	0.611005
$343$	1.00000	0.0539949
$344$	−25.0291	−1.34948
$345$	0	0
$346$	19.9075	1.07023
$347$	−16.6123	−0.891794	−0.445897	−	0.895084i	$-0.647115\pi$
$347$	−16.6123	−0.891794	−0.445897	+	0.895084i	$0.647115\pi$
$348$	28.3905	1.52189
$349$	6.95375	0.372226	0.186113	−	0.982528i	$-0.440411\pi$
$349$	6.95375	0.372226	0.186113	+	0.982528i	$0.440411\pi$
$350$	0	0
$351$	3.93800	0.210195
$352$	−3.57653	−0.190630
$353$	3.22840	0.171830	0.0859152	−	0.996302i	$-0.472619\pi$
$353$	3.22840	0.171830	0.0859152	+	0.996302i	$0.472619\pi$
$354$	−35.6371	−1.89409
$355$	0	0
$356$	44.0133	2.33270
$357$	−4.72294	−0.249964
$358$	−28.3376	−1.49769
$359$	10.3061	0.543937	0.271969	−	0.962306i	$-0.412325\pi$
$359$	10.3061	0.543937	0.271969	+	0.962306i	$0.412325\pi$
$360$	0	0
$361$	3.89559	0.205031
$362$	48.6438	2.55666
$363$	1.00000	0.0524864
$364$	14.0844	0.738223
$365$	0	0
$366$	14.1688	0.740616
$367$	11.8760	0.619923	0.309961	−	0.950749i	$-0.399684\pi$
$367$	11.8760	0.619923	0.309961	+	0.950749i	$0.399684\pi$
$368$	−4.46162	−0.232578
$369$	0.430132	0.0223918
$370$	0	0
$371$	3.15307	0.163699
$372$	4.12399	0.213819
$373$	−16.7229	−0.865881	−0.432940	−	0.901423i	$-0.642524\pi$
$373$	−16.7229	−0.865881	−0.432940	+	0.901423i	$0.642524\pi$
$374$	−11.1531	−0.576711
$375$	0	0
$376$	32.7058	1.68667
$377$	31.2599	1.60997
$378$	2.36147	0.121461
$379$	−10.9671	−0.563341	−0.281671	−	0.959511i	$-0.590889\pi$
$379$	−10.9671	−0.563341	−0.281671	+	0.959511i	$0.590889\pi$
$380$	0	0
$381$	−12.2928	−0.629780
$382$	−34.4750	−1.76389
$383$	−15.7520	−0.804890	−0.402445	−	0.915444i	$-0.631840\pi$
$383$	−15.7520	−0.804890	−0.402445	+	0.915444i	$0.631840\pi$
$384$	−20.5303	−1.04768
$385$	0	0
$386$	−31.7835	−1.61774
$387$	−6.72294	−0.341746
$388$	−39.8893	−2.02507
$389$	36.7678	1.86420	0.932100	−	0.362202i	$-0.117975\pi$
$389$	36.7678	1.86420	0.932100	+	0.362202i	$0.117975\pi$
$390$	0	0
$391$	12.8603	0.650371
$392$	3.72294	0.188037
$393$	7.13974	0.360152
$394$	−27.0291	−1.36171
$395$	0	0
$396$	3.57653	0.179728
$397$	−11.8627	−0.595371	−0.297685	−	0.954664i	$-0.596215\pi$
$397$	−11.8627	−0.595371	−0.297685	+	0.954664i	$0.596215\pi$
$398$	−30.7678	−1.54225
$399$	4.78493	0.239546
$400$	0	0
$401$	3.40106	0.169841	0.0849203	−	0.996388i	$-0.472936\pi$
$401$	3.40106	0.169841	0.0849203	+	0.996388i	$0.472936\pi$
$402$	−7.59228	−0.378668
$403$	4.54079	0.226193
$404$	58.9232	2.93154
$405$	0	0
$406$	18.7453	0.930316
$407$	5.50787	0.273015
$408$	−17.5832	−0.870498
$409$	−28.0315	−1.38607	−0.693034	−	0.720905i	$-0.743726\pi$
$409$	−28.0315	−1.38607	−0.693034	+	0.720905i	$0.743726\pi$
$410$	0	0
$411$	−1.58320	−0.0780936
$412$	5.61469	0.276616
$413$	−15.0911	−0.742583
$414$	−6.43013	−0.316024
$415$	0	0
$416$	−14.0844	−0.690545
$417$	−17.1979	−0.842184
$418$	11.2995	0.552675
$419$	28.2890	1.38201	0.691003	−	0.722852i	$-0.257169\pi$
$419$	28.2890	1.38201	0.691003	+	0.722852i	$0.257169\pi$
$420$	0	0
$421$	−14.4921	−0.706303	−0.353152	−	0.935566i	$-0.614890\pi$
$421$	−14.4921	−0.706303	−0.353152	+	0.935566i	$0.614890\pi$
$422$	5.73868	0.279355
$423$	8.78493	0.427138
$424$	11.7387	0.570081
$425$	0	0
$426$	31.7520	1.53839
$427$	6.00000	0.290360
$428$	17.1135	0.827211
$429$	3.93800	0.190129
$430$	0	0
$431$	−24.5369	−1.18190	−0.590952	−	0.806707i	$-0.701248\pi$
$431$	−24.5369	−1.18190	−0.590952	+	0.806707i	$0.701248\pi$
$432$	1.63853	0.0788339
$433$	7.32188	0.351867	0.175934	−	0.984402i	$-0.443706\pi$
$433$	7.32188	0.351867	0.175934	+	0.984402i	$0.443706\pi$
$434$	2.72294	0.130705
$435$	0	0
$436$	−53.7520	−2.57425
$437$	−13.0291	−0.623265
$438$	−28.1912	−1.34703
$439$	16.5369	0.789265	0.394633	−	0.918839i	$-0.370872\pi$
$439$	16.5369	0.789265	0.394633	+	0.918839i	$0.370872\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	−43.9208	−2.08910
$443$	−25.1979	−1.19719	−0.598594	−	0.801053i	$-0.704274\pi$
$443$	−25.1979	−1.19719	−0.598594	+	0.801053i	$0.704274\pi$
$444$	19.6991	0.934878
$445$	0	0
$446$	−53.7653	−2.54586
$447$	15.9380	0.753842
$448$	−11.7229	−0.553857
$449$	−36.5198	−1.72347	−0.861737	−	0.507355i	$-0.830623\pi$
$449$	−36.5198	−1.72347	−0.861737	+	0.507355i	$0.830623\pi$
$450$	0	0
$451$	0.430132	0.0202541
$452$	−26.6304	−1.25259
$453$	6.59894	0.310045
$454$	37.4907	1.75953
$455$	0	0
$456$	17.8140	0.834217
$457$	−3.15307	−0.147494	−0.0737472	−	0.997277i	$-0.523496\pi$
$457$	−3.15307	−0.147494	−0.0737472	+	0.997277i	$0.523496\pi$
$458$	−7.44588	−0.347923
$459$	−4.72294	−0.220448
$460$	0	0
$461$	−24.7678	−1.15355	−0.576775	−	0.816903i	$-0.695689\pi$
$461$	−24.7678	−1.15355	−0.576775	+	0.816903i	$0.695689\pi$
$462$	2.36147	0.109865
$463$	35.4287	1.64651	0.823256	−	0.567671i	$-0.192155\pi$
$463$	35.4287	1.64651	0.823256	+	0.567671i	$0.192155\pi$
$464$	13.0067	0.603819
$465$	0	0
$466$	−31.0291	−1.43739
$467$	23.9247	1.10710	0.553551	−	0.832815i	$-0.313272\pi$
$467$	23.9247	1.10710	0.553551	+	0.832815i	$0.313272\pi$
$468$	14.0844	0.651052
$469$	−3.21507	−0.148458
$470$	0	0
$471$	−7.32188	−0.337375
$472$	−56.1831	−2.58604
$473$	−6.72294	−0.309121
$474$	−12.8603	−0.590691
$475$	0	0
$476$	−16.8918	−0.774232
$477$	3.15307	0.144369
$478$	−23.8669	−1.09165
$479$	4.54079	0.207474	0.103737	−	0.994605i	$-0.466920\pi$
$479$	4.54079	0.207474	0.103737	+	0.994605i	$0.466920\pi$
$480$	0	0
$481$	21.6900	0.988980
$482$	5.59228	0.254721
$483$	−2.72294	−0.123898
$484$	3.57653	0.162570
$485$	0	0
$486$	2.36147	0.107118
$487$	−27.1397	−1.22982	−0.614909	−	0.788598i	$-0.710807\pi$
$487$	−27.1397	−1.22982	−0.614909	+	0.788598i	$0.710807\pi$
$488$	22.3376	1.01118
$489$	−14.1068	−0.637932
$490$	0	0
$491$	3.46305	0.156285	0.0781427	−	0.996942i	$-0.475101\pi$
$491$	3.46305	0.156285	0.0781427	+	0.996942i	$0.475101\pi$
$492$	1.53838	0.0693556
$493$	−37.4907	−1.68850
$494$	44.4974	2.00203
$495$	0	0
$496$	1.88934	0.0848339
$497$	13.4459	0.603130
$498$	6.72294	0.301262
$499$	−18.9671	−0.849083	−0.424542	−	0.905408i	$-0.639565\pi$
$499$	−18.9671	−0.849083	−0.424542	+	0.905408i	$0.639565\pi$
$500$	0	0
$501$	0.416799	0.0186212
$502$	24.1597	1.07830
$503$	−5.86267	−0.261404	−0.130702	−	0.991422i	$-0.541723\pi$
$503$	−5.86267	−0.261404	−0.130702	+	0.991422i	$0.541723\pi$
$504$	3.72294	0.165833
$505$	0	0
$506$	−6.43013	−0.285854
$507$	2.50787	0.111378
$508$	−43.9656	−1.95066
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	−11.9380	−0.528106
$512$	−18.0606	−0.798172
$513$	4.78493	0.211260
$514$	26.1597	1.15386
$515$	0	0
$516$	−24.0448	−1.05851
$517$	8.78493	0.386361
$518$	13.0067	0.571480
$519$	8.43013	0.370042
$520$	0	0
$521$	−2.24414	−0.0983177	−0.0491588	−	0.998791i	$-0.515654\pi$
$521$	−2.24414	−0.0983177	−0.0491588	+	0.998791i	$0.515654\pi$
$522$	18.7453	0.820462
$523$	17.9828	0.786334	0.393167	−	0.919467i	$-0.371379\pi$
$523$	17.9828	0.786334	0.393167	+	0.919467i	$0.371379\pi$
$524$	25.5355	1.11552
$525$	0	0
$526$	20.7453	0.904540
$527$	−5.44588	−0.237226
$528$	1.63853	0.0713079
$529$	−15.5856	−0.677635
$530$	0	0
$531$	−15.0911	−0.654897
$532$	17.1135	0.741964
$533$	1.69386	0.0733693
$534$	29.0606	1.25757
$535$	0	0
$536$	−11.9695	−0.517003
$537$	−12.0000	−0.517838
$538$	13.5832	0.585613
$539$	1.00000	0.0430730
$540$	0	0
$541$	28.7678	1.23682	0.618411	−	0.785855i	$-0.287777\pi$
$541$	28.7678	1.23682	0.618411	+	0.785855i	$0.287777\pi$
$542$	1.85360	0.0796188
$543$	20.5989	0.883985
$544$	16.8918	0.724228
$545$	0	0
$546$	9.29947	0.397981
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	−5.66237	−0.241885
$549$	6.00000	0.256074
$550$	0	0
$551$	37.9828	1.61812
$552$	−10.1373	−0.431473
$553$	−5.44588	−0.231582
$554$	27.3219	1.16080
$555$	0	0
$556$	−61.5088	−2.60856
$557$	13.5079	0.572347	0.286173	−	0.958178i	$-0.407617\pi$
$557$	13.5079	0.572347	0.286173	+	0.958178i	$0.407617\pi$
$558$	2.72294	0.115271
$559$	−26.4750	−1.11977
$560$	0	0
$561$	−4.72294	−0.199403
$562$	−59.6504	−2.51620
$563$	−24.7811	−1.04440	−0.522199	−	0.852824i	$-0.674888\pi$
$563$	−24.7811	−1.04440	−0.522199	+	0.852824i	$0.674888\pi$
$564$	31.4196	1.32300
$565$	0	0
$566$	25.8984	1.08859
$567$	1.00000	0.0419961
$568$	50.0582	2.10039
$569$	1.75201	0.0734482	0.0367241	−	0.999325i	$-0.488308\pi$
$569$	1.75201	0.0734482	0.0367241	+	0.999325i	$0.488308\pi$
$570$	0	0
$571$	−3.16640	−0.132510	−0.0662549	−	0.997803i	$-0.521105\pi$
$571$	−3.16640	−0.132510	−0.0662549	+	0.997803i	$0.521105\pi$
$572$	14.0844	0.588899
$573$	−14.5989	−0.609880
$574$	1.01574	0.0423963
$575$	0	0
$576$	−11.7229	−0.488456
$577$	−20.4750	−0.852383	−0.426192	−	0.904633i	$-0.640145\pi$
$577$	−20.4750	−0.852383	−0.426192	+	0.904633i	$0.640145\pi$
$578$	12.5303	0.521191
$579$	−13.4592	−0.559346
$580$	0	0
$581$	2.84693	0.118111
$582$	−26.3376	−1.09173
$583$	3.15307	0.130587
$584$	−44.4444	−1.83912
$585$	0	0
$586$	−34.7992	−1.43754
$587$	16.6609	0.687671	0.343835	−	0.939030i	$-0.388274\pi$
$587$	16.6609	0.687671	0.343835	+	0.939030i	$0.388274\pi$
$588$	3.57653	0.147494
$589$	5.51736	0.227339
$590$	0	0
$591$	−11.4459	−0.470820
$592$	9.02482	0.370918
$593$	−27.9075	−1.14602	−0.573012	−	0.819547i	$-0.694225\pi$
$593$	−27.9075	−1.14602	−0.573012	+	0.819547i	$0.694225\pi$
$594$	2.36147	0.0968922
$595$	0	0
$596$	57.0028	2.33493
$597$	−13.0291	−0.533245
$598$	−25.3219	−1.03549
$599$	−24.3376	−0.994408	−0.497204	−	0.867634i	$-0.665640\pi$
$599$	−24.3376	−0.994408	−0.497204	+	0.867634i	$0.665640\pi$
$600$	0	0
$601$	−9.50787	−0.387834	−0.193917	−	0.981018i	$-0.562119\pi$
$601$	−9.50787	−0.387834	−0.193917	+	0.981018i	$0.562119\pi$
$602$	−15.8760	−0.647058
$603$	−3.21507	−0.130928
$604$	23.6014	0.960325
$605$	0	0
$606$	38.9051	1.58041
$607$	−26.8431	−1.08953	−0.544764	−	0.838590i	$-0.683381\pi$
$607$	−26.8431	−1.08953	−0.544764	+	0.838590i	$0.683381\pi$
$608$	−17.1135	−0.694043
$609$	7.93800	0.321664
$610$	0	0
$611$	34.5951	1.39957
$612$	−16.8918	−0.682809
$613$	23.0291	0.930136	0.465068	−	0.885275i	$-0.346030\pi$
$613$	23.0291	0.930136	0.465068	+	0.885275i	$0.346030\pi$
$614$	−2.03149	−0.0819841
$615$	0	0
$616$	3.72294	0.150001
$617$	−31.4459	−1.26596	−0.632982	−	0.774167i	$-0.718169\pi$
$617$	−31.4459	−1.26596	−0.632982	+	0.774167i	$0.718169\pi$
$618$	3.70719	0.149125
$619$	2.26132	0.0908901	0.0454450	−	0.998967i	$-0.485529\pi$
$619$	2.26132	0.0908901	0.0454450	+	0.998967i	$0.485529\pi$
$620$	0	0
$621$	−2.72294	−0.109268
$622$	63.5040	2.54628
$623$	12.3061	0.493035
$624$	6.45254	0.258308
$625$	0	0
$626$	−24.0133	−0.959766
$627$	4.78493	0.191092
$628$	−26.1870	−1.04497
$629$	−26.0133	−1.03722
$630$	0	0
$631$	10.3061	0.410281	0.205140	−	0.978733i	$-0.434235\pi$
$631$	10.3061	0.410281	0.205140	+	0.978733i	$0.434235\pi$
$632$	−20.2747	−0.806482
$633$	2.43013	0.0965891
$634$	64.9184	2.57824
$635$	0	0
$636$	11.2771	0.447165
$637$	3.93800	0.156029
$638$	18.7453	0.742135
$639$	13.4459	0.531911
$640$	0	0
$641$	4.13733	0.163415	0.0817073	−	0.996656i	$-0.473963\pi$
$641$	4.13733	0.163415	0.0817073	+	0.996656i	$0.473963\pi$
$642$	11.2995	0.445955
$643$	8.44347	0.332978	0.166489	−	0.986043i	$-0.446757\pi$
$643$	8.44347	0.332978	0.166489	+	0.986043i	$0.446757\pi$
$644$	−9.73868	−0.383758
$645$	0	0
$646$	−53.3667	−2.09968
$647$	25.2732	0.993593	0.496796	−	0.867867i	$-0.334510\pi$
$647$	25.2732	0.993593	0.496796	+	0.867867i	$0.334510\pi$
$648$	3.72294	0.146251
$649$	−15.0911	−0.592376
$650$	0	0
$651$	1.15307	0.0451924
$652$	−50.4535	−1.97591
$653$	−47.9075	−1.87477	−0.937383	−	0.348302i	$-0.886759\pi$
$653$	−47.9075	−1.87477	−0.937383	+	0.348302i	$0.886759\pi$
$654$	−35.4907	−1.38780
$655$	0	0
$656$	0.704785	0.0275172
$657$	−11.9380	−0.465746
$658$	20.7453	0.808738
$659$	−28.2890	−1.10198	−0.550991	−	0.834511i	$-0.685750\pi$
$659$	−28.2890	−1.10198	−0.550991	+	0.834511i	$0.685750\pi$
$660$	0	0
$661$	19.7653	0.768783	0.384391	−	0.923170i	$-0.374411\pi$
$661$	19.7653	0.768783	0.384391	+	0.923170i	$0.374411\pi$
$662$	−40.6123	−1.57844
$663$	−18.5989	−0.722323
$664$	10.5989	0.411319
$665$	0	0
$666$	13.0067	0.503998
$667$	−21.6147	−0.836924
$668$	1.49069	0.0576767
$669$	−22.7678	−0.880252
$670$	0	0
$671$	6.00000	0.231627
$672$	−3.57653	−0.137968
$673$	51.5174	1.98585	0.992924	−	0.118750i	$-0.0378887\pi$
$673$	51.5174	1.98585	0.992924	+	0.118750i	$0.0378887\pi$
$674$	−46.6123	−1.79544
$675$	0	0
$676$	8.96949	0.344980
$677$	−46.3376	−1.78090	−0.890450	−	0.455081i	$-0.849610\pi$
$677$	−46.3376	−1.78090	−0.890450	+	0.455081i	$0.849610\pi$
$678$	−17.5832	−0.675279
$679$	−11.1531	−0.428016
$680$	0	0
$681$	15.8760	0.608370
$682$	2.72294	0.104267
$683$	39.2112	1.50038	0.750188	−	0.661225i	$-0.229963\pi$
$683$	39.2112	1.50038	0.750188	+	0.661225i	$0.229963\pi$
$684$	17.1135	0.654350
$685$	0	0
$686$	2.36147	0.0901613
$687$	−3.15307	−0.120297
$688$	−11.0157	−0.419971
$689$	12.4168	0.473042
$690$	0	0
$691$	7.62802	0.290184	0.145092	−	0.989418i	$-0.453652\pi$
$691$	7.62802	0.290184	0.145092	+	0.989418i	$0.453652\pi$
$692$	30.1507	1.14616
$693$	1.00000	0.0379869
$694$	−39.2294	−1.48913
$695$	0	0
$696$	29.5527	1.12019
$697$	−2.03149	−0.0769480
$698$	16.4211	0.621546
$699$	−13.1397	−0.496990
$700$	0	0
$701$	2.61228	0.0986644	0.0493322	−	0.998782i	$-0.484291\pi$
$701$	2.61228	0.0986644	0.0493322	+	0.998782i	$0.484291\pi$
$702$	9.29947	0.350986
$703$	26.3548	0.993990
$704$	−11.7229	−0.441825
$705$	0	0
$706$	7.62376	0.286924
$707$	16.4750	0.619604
$708$	−53.9737	−2.02846
$709$	36.1516	1.35770	0.678852	−	0.734276i	$-0.262478\pi$
$709$	36.1516	1.35770	0.678852	+	0.734276i	$0.262478\pi$
$710$	0	0
$711$	−5.44588	−0.204236
$712$	45.8150	1.71699
$713$	−3.13974	−0.117584
$714$	−11.1531	−0.417393
$715$	0	0
$716$	−42.9184	−1.60394
$717$	−10.1068	−0.377446
$718$	24.3376	0.908272
$719$	19.6767	0.733816	0.366908	−	0.930257i	$-0.380416\pi$
$719$	19.6767	0.733816	0.366908	+	0.930257i	$0.380416\pi$
$720$	0	0
$721$	1.56987	0.0584649
$722$	9.19932	0.342363
$723$	2.36814	0.0880719
$724$	73.6728	2.73803
$725$	0	0
$726$	2.36147	0.0876423
$727$	−2.47495	−0.0917909	−0.0458954	−	0.998946i	$-0.514614\pi$
$727$	−2.47495	−0.0917909	−0.0458954	+	0.998946i	$0.514614\pi$
$728$	14.6609	0.543371
$729$	1.00000	0.0370370
$730$	0	0
$731$	31.7520	1.17439
$732$	21.4592	0.793155
$733$	40.0315	1.47860	0.739298	−	0.673378i	$-0.235157\pi$
$733$	40.0315	1.47860	0.739298	+	0.673378i	$0.235157\pi$
$734$	28.0448	1.03515
$735$	0	0
$736$	9.73868	0.358973
$737$	−3.21507	−0.118428
$738$	1.01574	0.0373900
$739$	25.2771	0.929832	0.464916	−	0.885355i	$-0.346085\pi$
$739$	25.2771	0.929832	0.464916	+	0.885355i	$0.346085\pi$
$740$	0	0
$741$	18.8431	0.692218
$742$	7.44588	0.273347
$743$	−2.10682	−0.0772916	−0.0386458	−	0.999253i	$-0.512304\pi$
$743$	−2.10682	−0.0772916	−0.0386458	+	0.999253i	$0.512304\pi$
$744$	4.29281	0.157382
$745$	0	0
$746$	−39.4907	−1.44586
$747$	2.84693	0.104164
$748$	−16.8918	−0.617624
$749$	4.78493	0.174838
$750$	0	0
$751$	−45.9828	−1.67794	−0.838969	−	0.544180i	$-0.816841\pi$
$751$	−45.9828	−1.67794	−0.838969	+	0.544180i	$0.816841\pi$
$752$	14.3944	0.524909
$753$	10.2308	0.372831
$754$	73.8192	2.68834
$755$	0	0
$756$	3.57653	0.130077
$757$	29.2599	1.06347	0.531734	−	0.846911i	$-0.321540\pi$
$757$	29.2599	1.06347	0.531734	+	0.846911i	$0.321540\pi$
$758$	−25.8984	−0.940673
$759$	−2.72294	−0.0988364
$760$	0	0
$761$	21.8760	0.793005	0.396502	−	0.918034i	$-0.370224\pi$
$761$	21.8760	0.793005	0.396502	+	0.918034i	$0.370224\pi$
$762$	−29.0291	−1.05161
$763$	−15.0291	−0.544089
$764$	−52.2136	−1.88902
$765$	0	0
$766$	−37.1979	−1.34401
$767$	−59.4287	−2.14585
$768$	−25.0357	−0.903400
$769$	−16.2756	−0.586914	−0.293457	−	0.955972i	$-0.594806\pi$
$769$	−16.2756	−0.586914	−0.293457	+	0.955972i	$0.594806\pi$
$770$	0	0
$771$	11.0777	0.398955
$772$	−48.1373	−1.73250
$773$	30.4921	1.09673	0.548363	−	0.836241i	$-0.315251\pi$
$773$	30.4921	1.09673	0.548363	+	0.836241i	$0.315251\pi$
$774$	−15.8760	−0.570651
$775$	0	0
$776$	−41.5222	−1.49056
$777$	5.50787	0.197594
$778$	86.8259	3.11286
$779$	2.05815	0.0737410
$780$	0	0
$781$	13.4459	0.481131
$782$	30.3691	1.08600
$783$	7.93800	0.283681
$784$	1.63853	0.0585190
$785$	0	0
$786$	16.8603	0.601386
$787$	−35.2151	−1.25528	−0.627641	−	0.778503i	$-0.715979\pi$
$787$	−35.2151	−1.25528	−0.627641	+	0.778503i	$0.715979\pi$
$788$	−40.9366	−1.45830
$789$	8.78493	0.312752
$790$	0	0
$791$	−7.44588	−0.264745
$792$	3.72294	0.132289
$793$	23.6280	0.839056
$794$	−28.0133	−0.994156
$795$	0	0
$796$	−46.5989	−1.65166
$797$	14.9804	0.530633	0.265317	−	0.964161i	$-0.414523\pi$
$797$	14.9804	0.530633	0.265317	+	0.964161i	$0.414523\pi$
$798$	11.2995	0.399997
$799$	−41.4907	−1.46784
$800$	0	0
$801$	12.3061	0.434816
$802$	8.03149	0.283602
$803$	−11.9380	−0.421283
$804$	−11.4988	−0.405531
$805$	0	0
$806$	10.7229	0.377699
$807$	5.75201	0.202480
$808$	61.3352	2.15777
$809$	−25.2599	−0.888090	−0.444045	−	0.896004i	$-0.646457\pi$
$809$	−25.2599	−0.888090	−0.444045	+	0.896004i	$0.646457\pi$
$810$	0	0
$811$	29.5212	1.03663	0.518315	−	0.855190i	$-0.326560\pi$
$811$	29.5212	1.03663	0.518315	+	0.855190i	$0.326560\pi$
$812$	28.3905	0.996313
$813$	0.784934	0.0275288
$814$	13.0067	0.455883
$815$	0	0
$816$	−7.73868	−0.270908
$817$	−32.1688	−1.12544
$818$	−66.1955	−2.31447
$819$	3.93800	0.137605
$820$	0	0
$821$	23.7873	0.830184	0.415092	−	0.909779i	$-0.363749\pi$
$821$	23.7873	0.830184	0.415092	+	0.909779i	$0.363749\pi$
$822$	−3.73868	−0.130401
$823$	−17.7692	−0.619395	−0.309698	−	0.950835i	$-0.600228\pi$
$823$	−17.7692	−0.619395	−0.309698	+	0.950835i	$0.600228\pi$
$824$	5.84452	0.203604
$825$	0	0
$826$	−35.6371	−1.23997
$827$	−8.90893	−0.309794	−0.154897	−	0.987931i	$-0.549505\pi$
$827$	−8.90893	−0.309794	−0.154897	+	0.987931i	$0.549505\pi$
$828$	−9.73868	−0.338443
$829$	−32.7678	−1.13807	−0.569036	−	0.822313i	$-0.692683\pi$
$829$	−32.7678	−1.13807	−0.569036	+	0.822313i	$0.692683\pi$
$830$	0	0
$831$	11.5699	0.401354
$832$	−46.1650	−1.60048
$833$	−4.72294	−0.163640
$834$	−40.6123	−1.40629
$835$	0	0
$836$	17.1135	0.591882
$837$	1.15307	0.0398559
$838$	66.8035	2.30769
$839$	6.96708	0.240530	0.120265	−	0.992742i	$-0.461626\pi$
$839$	6.96708	0.240530	0.120265	+	0.992742i	$0.461626\pi$
$840$	0	0
$841$	34.0119	1.17282
$842$	−34.2227	−1.17939
$843$	−25.2599	−0.869997
$844$	8.69145	0.299172
$845$	0	0
$846$	20.7453	0.713240
$847$	1.00000	0.0343604
$848$	5.16640	0.177415
$849$	10.9671	0.376389
$850$	0	0
$851$	−14.9976	−0.514111
$852$	48.0896	1.64752
$853$	−11.1979	−0.383408	−0.191704	−	0.981453i	$-0.561401\pi$
$853$	−11.1979	−0.383408	−0.191704	+	0.981453i	$0.561401\pi$
$854$	14.1688	0.484847
$855$	0	0
$856$	17.8140	0.608870
$857$	44.0315	1.50409	0.752043	−	0.659114i	$-0.229068\pi$
$857$	44.0315	1.50409	0.752043	+	0.659114i	$0.229068\pi$
$858$	9.29947	0.317479
$859$	7.13974	0.243605	0.121802	−	0.992554i	$-0.461133\pi$
$859$	7.13974	0.243605	0.121802	+	0.992554i	$0.461133\pi$
$860$	0	0
$861$	0.430132	0.0146589
$862$	−57.9432	−1.97355
$863$	5.27706	0.179633	0.0898166	−	0.995958i	$-0.471372\pi$
$863$	5.27706	0.179633	0.0898166	+	0.995958i	$0.471372\pi$
$864$	−3.57653	−0.121676
$865$	0	0
$866$	17.2904	0.587552
$867$	5.30614	0.180206
$868$	4.12399	0.139977
$869$	−5.44588	−0.184739
$870$	0	0
$871$	−12.6609	−0.429000
$872$	−55.9523	−1.89478
$873$	−11.1531	−0.377474
$874$	−30.7678	−1.04073
$875$	0	0
$876$	−42.6967	−1.44259
$877$	32.4301	1.09509	0.547544	−	0.836777i	$-0.315563\pi$
$877$	32.4301	1.09509	0.547544	+	0.836777i	$0.315563\pi$
$878$	39.0515	1.31792
$879$	−14.7363	−0.497042
$880$	0	0
$881$	39.5660	1.33301	0.666507	−	0.745499i	$-0.267789\pi$
$881$	39.5660	1.33301	0.666507	+	0.745499i	$0.267789\pi$
$882$	2.36147	0.0795148
$883$	24.4130	0.821561	0.410781	−	0.911734i	$-0.365256\pi$
$883$	24.4130	0.821561	0.410781	+	0.911734i	$0.365256\pi$
$884$	−66.5198	−2.23730
$885$	0	0
$886$	−59.5040	−1.99908
$887$	−36.2928	−1.21859	−0.609297	−	0.792942i	$-0.708548\pi$
$887$	−36.2928	−1.21859	−0.609297	+	0.792942i	$0.708548\pi$
$888$	20.5055	0.688118
$889$	−12.2928	−0.412287
$890$	0	0
$891$	1.00000	0.0335013
$892$	−81.4297	−2.72647
$893$	42.0353	1.40666
$894$	37.6371	1.25877
$895$	0	0
$896$	−20.5303	−0.685869
$897$	−10.7229	−0.358028
$898$	−86.2403	−2.87788
$899$	9.15307	0.305272
$900$	0	0
$901$	−14.8918	−0.496116
$902$	1.01574	0.0338205
$903$	−6.72294	−0.223725
$904$	−27.7205	−0.921971
$905$	0	0
$906$	15.5832	0.517717
$907$	30.3958	1.00928	0.504638	−	0.863331i	$-0.331626\pi$
$907$	30.3958	1.00928	0.504638	+	0.863331i	$0.331626\pi$
$908$	56.7811	1.88435
$909$	16.4750	0.546440
$910$	0	0
$911$	−56.8259	−1.88273	−0.941363	−	0.337395i	$-0.890454\pi$
$911$	−56.8259	−1.88273	−0.941363	+	0.337395i	$0.890454\pi$
$912$	7.84026	0.259617
$913$	2.84693	0.0942196
$914$	−7.44588	−0.246288
$915$	0	0
$916$	−11.2771	−0.372605
$917$	7.13974	0.235775
$918$	−11.1531	−0.368106
$919$	−40.1688	−1.32505	−0.662523	−	0.749041i	$-0.730514\pi$
$919$	−40.1688	−1.32505	−0.662523	+	0.749041i	$0.730514\pi$
$920$	0	0
$921$	−0.860264	−0.0283467
$922$	−58.4883	−1.92621
$923$	52.9499	1.74287
$924$	3.57653	0.117659
$925$	0	0
$926$	83.6638	2.74936
$927$	1.56987	0.0515612
$928$	−28.3905	−0.931965
$929$	−27.3180	−0.896276	−0.448138	−	0.893964i	$-0.647913\pi$
$929$	−27.3180	−0.896276	−0.448138	+	0.893964i	$0.647913\pi$
$930$	0	0
$931$	4.78493	0.156820
$932$	−46.9947	−1.53936
$933$	26.8918	0.880396
$934$	56.4974	1.84865
$935$	0	0
$936$	14.6609	0.479208
$937$	5.75201	0.187910	0.0939551	−	0.995576i	$-0.470049\pi$
$937$	5.75201	0.187910	0.0939551	+	0.995576i	$0.470049\pi$
$938$	−7.59228	−0.247897
$939$	−10.1688	−0.331847
$940$	0	0
$941$	−27.5699	−0.898752	−0.449376	−	0.893343i	$-0.648354\pi$
$941$	−27.5699	−0.898752	−0.449376	+	0.893343i	$0.648354\pi$
$942$	−17.2904	−0.563352
$943$	−1.17122	−0.0381402
$944$	−24.7272	−0.804802
$945$	0	0
$946$	−15.8760	−0.516174
$947$	−37.0739	−1.20474	−0.602370	−	0.798217i	$-0.705777\pi$
$947$	−37.0739	−1.20474	−0.602370	+	0.798217i	$0.705777\pi$
$948$	−19.4774	−0.632595
$949$	−47.0119	−1.52607
$950$	0	0
$951$	27.4907	0.891447
$952$	−17.5832	−0.569875
$953$	12.0620	0.390726	0.195363	−	0.980731i	$-0.437411\pi$
$953$	12.0620	0.390726	0.195363	+	0.980731i	$0.437411\pi$
$954$	7.44588	0.241069
$955$	0	0
$956$	−36.1474	−1.16909
$957$	7.93800	0.256599
$958$	10.7229	0.346442
$959$	−1.58320	−0.0511242
$960$	0	0
$961$	−29.6704	−0.957111
$962$	51.2203	1.65141
$963$	4.78493	0.154192
$964$	8.46972	0.272791
$965$	0	0
$966$	−6.43013	−0.206886
$967$	−6.43013	−0.206779	−0.103390	−	0.994641i	$-0.532969\pi$
$967$	−6.43013	−0.206779	−0.103390	+	0.994641i	$0.532969\pi$
$968$	3.72294	0.119660
$969$	−22.5989	−0.725983
$970$	0	0
$971$	−35.3047	−1.13298	−0.566491	−	0.824068i	$-0.691699\pi$
$971$	−35.3047	−1.13298	−0.566491	+	0.824068i	$0.691699\pi$
$972$	3.57653	0.114717
$973$	−17.1979	−0.551339
$974$	−64.0896	−2.05356
$975$	0	0
$976$	9.83119	0.314689
$977$	−0.261319	−0.00836035	−0.00418017	−	0.999991i	$-0.501331\pi$
$977$	−0.261319	−0.00836035	−0.00418017	+	0.999991i	$0.501331\pi$
$978$	−33.3128	−1.06523
$979$	12.3061	0.393306
$980$	0	0
$981$	−15.0291	−0.479841
$982$	8.17789	0.260967
$983$	16.3376	0.521089	0.260545	−	0.965462i	$-0.416098\pi$
$983$	16.3376	0.521089	0.260545	+	0.965462i	$0.416098\pi$
$984$	1.60135	0.0510493
$985$	0	0
$986$	−88.5331	−2.81947
$987$	8.78493	0.279628
$988$	67.3930	2.14406
$989$	18.3061	0.582101
$990$	0	0
$991$	−36.2623	−1.15191	−0.575955	−	0.817481i	$-0.695370\pi$
$991$	−36.2623	−1.15191	−0.575955	+	0.817481i	$0.695370\pi$
$992$	−4.12399	−0.130937
$993$	−17.1979	−0.545759
$994$	31.7520	1.00711
$995$	0	0
$996$	10.1821	0.322634
$997$	3.47254	0.109977	0.0549883	−	0.998487i	$-0.482488\pi$
$997$	3.47254	0.109977	0.0549883	+	0.998487i	$0.482488\pi$
$998$	−44.7902	−1.41781
$999$	5.50787	0.174261

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5775.2.a.bw.1.3		3
5.4	even	2		231.2.a.d.1.1	✓	3
15.14	odd	2		693.2.a.m.1.3		3
20.19	odd	2		3696.2.a.bp.1.1		3
35.34	odd	2		1617.2.a.s.1.1		3
55.54	odd	2		2541.2.a.bi.1.3		3
105.104	even	2		4851.2.a.bp.1.3		3
165.164	even	2		7623.2.a.cb.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.d.1.1	✓	3	5.4	even	2
693.2.a.m.1.3		3	15.14	odd	2
1617.2.a.s.1.1		3	35.34	odd	2
2541.2.a.bi.1.3		3	55.54	odd	2
3696.2.a.bp.1.1		3	20.19	odd	2
4851.2.a.bp.1.3		3	105.104	even	2
5775.2.a.bw.1.3		3	1.1	even	1	trivial
7623.2.a.cb.1.1		3	165.164	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 5775 = 3 \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5775.a (trivial)

\(f(q)\)	\(=\)	\(q+2.36147 q^{2} +1.00000 q^{3} +3.57653 q^{4} +2.36147 q^{6} +1.00000 q^{7} +3.72294 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.36147 q^{2} +1.00000 q^{3} +3.57653 q^{4} +2.36147 q^{6} +1.00000 q^{7} +3.72294 q^{8} +1.00000 q^{9} +1.00000 q^{11} +3.57653 q^{12} +3.93800 q^{13} +2.36147 q^{14} +1.63853 q^{16} -4.72294 q^{17} +2.36147 q^{18} +4.78493 q^{19} +1.00000 q^{21} +2.36147 q^{22} -2.72294 q^{23} +3.72294 q^{24} +9.29947 q^{26} +1.00000 q^{27} +3.57653 q^{28} +7.93800 q^{29} +1.15307 q^{31} -3.57653 q^{32} +1.00000 q^{33} -11.1531 q^{34} +3.57653 q^{36} +5.50787 q^{37} +11.2995 q^{38} +3.93800 q^{39} +0.430132 q^{41} +2.36147 q^{42} -6.72294 q^{43} +3.57653 q^{44} -6.43013 q^{46} +8.78493 q^{47} +1.63853 q^{48} +1.00000 q^{49} -4.72294 q^{51} +14.0844 q^{52} +3.15307 q^{53} +2.36147 q^{54} +3.72294 q^{56} +4.78493 q^{57} +18.7453 q^{58} -15.0911 q^{59} +6.00000 q^{61} +2.72294 q^{62} +1.00000 q^{63} -11.7229 q^{64} +2.36147 q^{66} -3.21507 q^{67} -16.8918 q^{68} -2.72294 q^{69} +13.4459 q^{71} +3.72294 q^{72} -11.9380 q^{73} +13.0067 q^{74} +17.1135 q^{76} +1.00000 q^{77} +9.29947 q^{78} -5.44588 q^{79} +1.00000 q^{81} +1.01574 q^{82} +2.84693 q^{83} +3.57653 q^{84} -15.8760 q^{86} +7.93800 q^{87} +3.72294 q^{88} +12.3061 q^{89} +3.93800 q^{91} -9.73868 q^{92} +1.15307 q^{93} +20.7453 q^{94} -3.57653 q^{96} -11.1531 q^{97} +2.36147 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 6 q^{4} + 3 q^{7} - 3 q^{8} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 6 * q^4 + 3 * q^7 - 3 * q^8 + 3 * q^9	\( 3 q + 3 q^{3} + 6 q^{4} + 3 q^{7} - 3 q^{8} + 3 q^{9} + 3 q^{11} + 6 q^{12} + 12 q^{16} + 12 q^{19} + 3 q^{21} + 6 q^{23} - 3 q^{24} + 9 q^{26} + 3 q^{27} + 6 q^{28} + 12 q^{29} - 6 q^{31} - 6 q^{32} + 3 q^{33} - 24 q^{34} + 6 q^{36} + 15 q^{38} + 6 q^{41} - 6 q^{43} + 6 q^{44} - 24 q^{46} + 24 q^{47} + 12 q^{48} + 3 q^{49} + 21 q^{52} - 3 q^{56} + 12 q^{57} + 9 q^{58} - 24 q^{59} + 18 q^{61} - 6 q^{62} + 3 q^{63} - 21 q^{64} - 12 q^{67} + 6 q^{68} + 6 q^{69} + 12 q^{71} - 3 q^{72} - 24 q^{73} + 39 q^{74} - 3 q^{76} + 3 q^{77} + 9 q^{78} + 12 q^{79} + 3 q^{81} - 30 q^{82} + 18 q^{83} + 6 q^{84} - 24 q^{86} + 12 q^{87} - 3 q^{88} + 18 q^{89} + 18 q^{92} - 6 q^{93} + 15 q^{94} - 6 q^{96} - 24 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 6 * q^4 + 3 * q^7 - 3 * q^8 + 3 * q^9 + 3 * q^11 + 6 * q^12 + 12 * q^16 + 12 * q^19 + 3 * q^21 + 6 * q^23 - 3 * q^24 + 9 * q^26 + 3 * q^27 + 6 * q^28 + 12 * q^29 - 6 * q^31 - 6 * q^32 + 3 * q^33 - 24 * q^34 + 6 * q^36 + 15 * q^38 + 6 * q^41 - 6 * q^43 + 6 * q^44 - 24 * q^46 + 24 * q^47 + 12 * q^48 + 3 * q^49 + 21 * q^52 - 3 * q^56 + 12 * q^57 + 9 * q^58 - 24 * q^59 + 18 * q^61 - 6 * q^62 + 3 * q^63 - 21 * q^64 - 12 * q^67 + 6 * q^68 + 6 * q^69 + 12 * q^71 - 3 * q^72 - 24 * q^73 + 39 * q^74 - 3 * q^76 + 3 * q^77 + 9 * q^78 + 12 * q^79 + 3 * q^81 - 30 * q^82 + 18 * q^83 + 6 * q^84 - 24 * q^86 + 12 * q^87 - 3 * q^88 + 18 * q^89 + 18 * q^92 - 6 * q^93 + 15 * q^94 - 6 * q^96 - 24 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more