LMFDB - Embedded newform 5775.2.a.bz.1.1

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:	$4$
Coefficient field:	4.4.13448.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 2 $ x^4 - 7*x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1155)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.58874$ 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)

$f(q)$	$=$	$q-2.58874 q^{2} -1.00000 q^{3} +4.70156 q^{4} +2.58874 q^{6} +1.00000 q^{7} -6.99364 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.58874 q^{2} -1.00000 q^{3} +4.70156 q^{4} +2.58874 q^{6} +1.00000 q^{7} -6.99364 q^{8} +1.00000 q^{9} -1.00000 q^{11} -4.70156 q^{12} -6.40490 q^{13} -2.58874 q^{14} +8.70156 q^{16} +3.70156 q^{17} -2.58874 q^{18} +7.24672 q^{19} -1.00000 q^{21} +2.58874 q^{22} +7.33388 q^{23} +6.99364 q^{24} +16.5806 q^{26} -1.00000 q^{27} +4.70156 q^{28} -8.10646 q^{29} +6.77258 q^{31} -8.53879 q^{32} +1.00000 q^{33} -9.58237 q^{34} +4.70156 q^{36} +3.95005 q^{37} -18.7598 q^{38} +6.40490 q^{39} -2.31773 q^{41} +2.58874 q^{42} +7.65161 q^{43} -4.70156 q^{44} -18.9855 q^{46} -1.45307 q^{47} -8.70156 q^{48} +1.00000 q^{49} -3.70156 q^{51} -30.1130 q^{52} -9.10823 q^{53} +2.58874 q^{54} -6.99364 q^{56} -7.24672 q^{57} +20.9855 q^{58} -2.92898 q^{59} -8.42419 q^{61} -17.5324 q^{62} +1.00000 q^{63} +4.70156 q^{64} -2.58874 q^{66} +8.72263 q^{67} +17.4031 q^{68} -7.33388 q^{69} -4.40490 q^{71} -6.99364 q^{72} -3.40667 q^{73} -10.2256 q^{74} +34.0709 q^{76} -1.00000 q^{77} -16.5806 q^{78} +4.40490 q^{79} +1.00000 q^{81} +6.00000 q^{82} +8.96620 q^{83} -4.70156 q^{84} -19.8080 q^{86} +8.10646 q^{87} +6.99364 q^{88} +0.206355 q^{89} -6.40490 q^{91} +34.4807 q^{92} -6.77258 q^{93} +3.76162 q^{94} +8.53879 q^{96} +13.9644 q^{97} -2.58874 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} + 6 q^{4} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 + 6 * q^4 + 4 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} + 6 q^{4} + 4 q^{7} + 4 q^{9} - 4 q^{11} - 6 q^{12} - 8 q^{13} + 22 q^{16} + 2 q^{17} + 10 q^{19} - 4 q^{21} + 2 q^{23} + 20 q^{26} - 4 q^{27} + 6 q^{28} - 2 q^{29} + 24 q^{31} + 4 q^{33} + 6 q^{36} - 8 q^{37} - 16 q^{38} + 8 q^{39} - 6 q^{43} - 6 q^{44} - 12 q^{46} - 4 q^{47} - 22 q^{48} + 4 q^{49} - 2 q^{51} - 12 q^{52} - 14 q^{53} - 10 q^{57} + 20 q^{58} - 2 q^{59} + 6 q^{61} - 8 q^{62} + 4 q^{63} + 6 q^{64} + 8 q^{67} + 44 q^{68} - 2 q^{69} - 4 q^{73} - 36 q^{74} + 56 q^{76} - 4 q^{77} - 20 q^{78} + 4 q^{81} + 24 q^{82} - 6 q^{83} - 6 q^{84} - 36 q^{86} + 2 q^{87} + 18 q^{89} - 8 q^{91} + 44 q^{92} - 24 q^{93} - 36 q^{94} + 6 q^{97} - 4 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 + 6 * q^4 + 4 * q^7 + 4 * q^9 - 4 * q^11 - 6 * q^12 - 8 * q^13 + 22 * q^16 + 2 * q^17 + 10 * q^19 - 4 * q^21 + 2 * q^23 + 20 * q^26 - 4 * q^27 + 6 * q^28 - 2 * q^29 + 24 * q^31 + 4 * q^33 + 6 * q^36 - 8 * q^37 - 16 * q^38 + 8 * q^39 - 6 * q^43 - 6 * q^44 - 12 * q^46 - 4 * q^47 - 22 * q^48 + 4 * q^49 - 2 * q^51 - 12 * q^52 - 14 * q^53 - 10 * q^57 + 20 * q^58 - 2 * q^59 + 6 * q^61 - 8 * q^62 + 4 * q^63 + 6 * q^64 + 8 * q^67 + 44 * q^68 - 2 * q^69 - 4 * q^73 - 36 * q^74 + 56 * q^76 - 4 * q^77 - 20 * q^78 + 4 * q^81 + 24 * q^82 - 6 * q^83 - 6 * q^84 - 36 * q^86 + 2 * q^87 + 18 * q^89 - 8 * q^91 + 44 * q^92 - 24 * q^93 - 36 * q^94 + 6 * q^97 - 4 * 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 $

$2$	−2.58874	−1.83051	−0.915257	−	0.402871i	$-0.868012\pi$
$2$	−2.58874	−1.83051	−0.915257	+	0.402871i	$0.868012\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	4.70156	2.35078
$5$	0	0
$5$	0	0
$6$	2.58874	1.05685
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−6.99364	−2.47262
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−4.70156	−1.35722
$13$	−6.40490	−1.77640	−0.888200	−	0.459458i	$-0.848044\pi$
$13$	−6.40490	−1.77640	−0.888200	+	0.459458i	$0.848044\pi$
$14$	−2.58874	−0.691869
$15$	0	0
$16$	8.70156	2.17539
$17$	3.70156	0.897761	0.448880	−	0.893592i	$-0.351823\pi$
$17$	3.70156	0.897761	0.448880	+	0.893592i	$0.351823\pi$
$18$	−2.58874	−0.610171
$19$	7.24672	1.66251	0.831255	−	0.555891i	$-0.187623\pi$
$19$	7.24672	1.66251	0.831255	+	0.555891i	$0.187623\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	2.58874	0.551921
$23$	7.33388	1.52922	0.764610	−	0.644493i	$-0.222932\pi$
$23$	7.33388	1.52922	0.764610	+	0.644493i	$0.222932\pi$
$24$	6.99364	1.42757
$25$	0	0
$26$	16.5806	3.25172
$27$	−1.00000	−0.192450
$28$	4.70156	0.888512
$29$	−8.10646	−1.50533	−0.752666	−	0.658403i	$-0.771232\pi$
$29$	−8.10646	−1.50533	−0.752666	+	0.658403i	$0.771232\pi$
$30$	0	0
$31$	6.77258	1.21639	0.608195	−	0.793787i	$-0.291894\pi$
$31$	6.77258	1.21639	0.608195	+	0.793787i	$0.291894\pi$
$32$	−8.53879	−1.50946
$33$	1.00000	0.174078
$34$	−9.58237	−1.64336
$35$	0	0
$36$	4.70156	0.783594
$37$	3.95005	0.649385	0.324692	−	0.945820i	$-0.394739\pi$
$37$	3.95005	0.649385	0.324692	+	0.945820i	$0.394739\pi$
$38$	−18.7598	−3.04325
$39$	6.40490	1.02560
$40$	0	0
$41$	−2.31773	−0.361969	−0.180984	−	0.983486i	$-0.557928\pi$
$41$	−2.31773	−0.361969	−0.180984	+	0.983486i	$0.557928\pi$
$42$	2.58874	0.399451
$43$	7.65161	1.16686	0.583430	−	0.812163i	$-0.301710\pi$
$43$	7.65161	1.16686	0.583430	+	0.812163i	$0.301710\pi$
$44$	−4.70156	−0.708787
$45$	0	0
$46$	−18.9855	−2.79926
$47$	−1.45307	−0.211952	−0.105976	−	0.994369i	$-0.533797\pi$
$47$	−1.45307	−0.211952	−0.105976	+	0.994369i	$0.533797\pi$
$48$	−8.70156	−1.25596
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.70156	−0.518322
$52$	−30.1130	−4.17593
$53$	−9.10823	−1.25111	−0.625556	−	0.780179i	$-0.715128\pi$
$53$	−9.10823	−1.25111	−0.625556	+	0.780179i	$0.715128\pi$
$54$	2.58874	0.352283
$55$	0	0
$56$	−6.99364	−0.934564
$57$	−7.24672	−0.959851
$58$	20.9855	2.75553
$59$	−2.92898	−0.381321	−0.190661	−	0.981656i	$-0.561063\pi$
$59$	−2.92898	−0.381321	−0.190661	+	0.981656i	$0.561063\pi$
$60$	0	0
$61$	−8.42419	−1.07861	−0.539304	−	0.842111i	$-0.681312\pi$
$61$	−8.42419	−1.07861	−0.539304	+	0.842111i	$0.681312\pi$
$62$	−17.5324	−2.22662
$63$	1.00000	0.125988
$64$	4.70156	0.587695
$65$	0	0
$66$	−2.58874	−0.318652
$67$	8.72263	1.06564	0.532819	−	0.846229i	$-0.321132\pi$
$67$	8.72263	1.06564	0.532819	+	0.846229i	$0.321132\pi$
$68$	17.4031	2.11044
$69$	−7.33388	−0.882896
$70$	0	0
$71$	−4.40490	−0.522765	−0.261383	−	0.965235i	$-0.584178\pi$
$71$	−4.40490	−0.522765	−0.261383	+	0.965235i	$0.584178\pi$
$72$	−6.99364	−0.824208
$73$	−3.40667	−0.398721	−0.199360	−	0.979926i	$-0.563886\pi$
$73$	−3.40667	−0.398721	−0.199360	+	0.979926i	$0.563886\pi$
$74$	−10.2256	−1.18871
$75$	0	0
$76$	34.0709	3.90820
$77$	−1.00000	−0.113961
$78$	−16.5806	−1.87738
$79$	4.40490	0.495590	0.247795	−	0.968813i	$-0.420294\pi$
$79$	4.40490	0.495590	0.247795	+	0.968813i	$0.420294\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	8.96620	0.984169	0.492084	−	0.870548i	$-0.336235\pi$
$83$	8.96620	0.984169	0.492084	+	0.870548i	$0.336235\pi$
$84$	−4.70156	−0.512982
$85$	0	0
$86$	−19.8080	−2.13595
$87$	8.10646	0.869104
$88$	6.99364	0.745524
$89$	0.206355	0.0218736	0.0109368	−	0.999940i	$-0.496519\pi$
$89$	0.206355	0.0218736	0.0109368	+	0.999940i	$0.496519\pi$
$90$	0	0
$91$	−6.40490	−0.671416
$92$	34.4807	3.59486
$93$	−6.77258	−0.702284
$94$	3.76162	0.387982
$95$	0	0
$96$	8.53879	0.871487
$97$	13.9644	1.41787	0.708936	−	0.705272i	$-0.249175\pi$
$97$	13.9644	1.41787	0.708936	+	0.705272i	$0.249175\pi$
$98$	−2.58874	−0.261502
$99$	−1.00000	−0.100504
$100$	0	0
$101$	3.77080	0.375209	0.187605	−	0.982245i	$-0.439928\pi$
$101$	3.77080	0.375209	0.187605	+	0.982245i	$0.439928\pi$
$102$	9.58237	0.948796
$103$	8.56131	0.843570	0.421785	−	0.906696i	$-0.361404\pi$
$103$	8.56131	0.843570	0.421785	+	0.906696i	$0.361404\pi$
$104$	44.7935	4.39237
$105$	0	0
$106$	23.5788	2.29018
$107$	−14.1630	−1.36919	−0.684593	−	0.728925i	$-0.740020\pi$
$107$	−14.1630	−1.36919	−0.684593	+	0.728925i	$0.740020\pi$
$108$	−4.70156	−0.452408
$109$	7.14026	0.683913	0.341956	−	0.939716i	$-0.388911\pi$
$109$	7.14026	0.683913	0.341956	+	0.939716i	$0.388911\pi$
$110$	0	0
$111$	−3.95005	−0.374922
$112$	8.70156	0.822220
$113$	1.10823	0.104254	0.0521269	−	0.998640i	$-0.483400\pi$
$113$	1.10823	0.104254	0.0521269	+	0.998640i	$0.483400\pi$
$114$	18.7598	1.75702
$115$	0	0
$116$	−38.1130	−3.53871
$117$	−6.40490	−0.592133
$118$	7.58237	0.698014
$119$	3.70156	0.339322
$120$	0	0
$121$	1.00000	0.0909091
$122$	21.8080	1.97441
$123$	2.31773	0.208983
$124$	31.8417	2.85947
$125$	0	0
$126$	−2.58874	−0.230623
$127$	−11.6516	−1.03391	−0.516957	−	0.856011i	$-0.672935\pi$
$127$	−11.6516	−1.03391	−0.516957	+	0.856011i	$0.672935\pi$
$128$	4.90647	0.433675
$129$	−7.65161	−0.673687
$130$	0	0
$131$	−13.8116	−1.20672	−0.603361	−	0.797468i	$-0.706172\pi$
$131$	−13.8116	−1.20672	−0.603361	+	0.797468i	$0.706172\pi$
$132$	4.70156	0.409218
$133$	7.24672	0.628370
$134$	−22.5806	−1.95067
$135$	0	0
$136$	−25.8874	−2.21982
$137$	−9.40312	−0.803363	−0.401682	−	0.915779i	$-0.631574\pi$
$137$	−9.40312	−0.803363	−0.401682	+	0.915779i	$0.631574\pi$
$138$	18.9855	1.61615
$139$	16.3128	1.38363	0.691817	−	0.722072i	$-0.256810\pi$
$139$	16.3128	1.38363	0.691817	+	0.722072i	$0.256810\pi$
$140$	0	0
$141$	1.45307	0.122371
$142$	11.4031	0.956929
$143$	6.40490	0.535604
$144$	8.70156	0.725130
$145$	0	0
$146$	8.81898	0.729864
$147$	−1.00000	−0.0824786
$148$	18.5714	1.52656
$149$	9.90010	0.811048	0.405524	−	0.914084i	$-0.367089\pi$
$149$	9.90010	0.811048	0.405524	+	0.914084i	$0.367089\pi$
$150$	0	0
$151$	−17.7581	−1.44513	−0.722566	−	0.691302i	$-0.757037\pi$
$151$	−17.7581	−1.44513	−0.722566	+	0.691302i	$0.757037\pi$
$152$	−50.6809	−4.11076
$153$	3.70156	0.299254
$154$	2.58874	0.208606
$155$	0	0
$156$	30.1130	2.41097
$157$	−17.0547	−1.36112	−0.680558	−	0.732694i	$-0.738263\pi$
$157$	−17.0547	−1.36112	−0.680558	+	0.732694i	$0.738263\pi$
$158$	−11.4031	−0.907184
$159$	9.10823	0.722330
$160$	0	0
$161$	7.33388	0.577991
$162$	−2.58874	−0.203390
$163$	19.9338	1.56133	0.780667	−	0.624947i	$-0.214880\pi$
$163$	19.9338	1.56133	0.780667	+	0.624947i	$0.214880\pi$
$164$	−10.8970	−0.850910
$165$	0	0
$166$	−23.2111	−1.80153
$167$	2.95183	0.228419	0.114210	−	0.993457i	$-0.463566\pi$
$167$	2.95183	0.228419	0.114210	+	0.993457i	$0.463566\pi$
$168$	6.99364	0.539571
$169$	28.0227	2.15559
$170$	0	0
$171$	7.24672	0.554170
$172$	35.9745	2.74303
$173$	−20.2094	−1.53649	−0.768245	−	0.640156i	$-0.778870\pi$
$173$	−20.2094	−1.53649	−0.768245	+	0.640156i	$0.778870\pi$
$174$	−20.9855	−1.59091
$175$	0	0
$176$	−8.70156	−0.655905
$177$	2.92898	0.220156
$178$	−0.534199	−0.0400399
$179$	5.95005	0.444728	0.222364	−	0.974964i	$-0.428623\pi$
$179$	5.95005	0.444728	0.222364	+	0.974964i	$0.428623\pi$
$180$	0	0
$181$	−20.0709	−1.49186	−0.745929	−	0.666026i	$-0.767994\pi$
$181$	−20.0709	−1.49186	−0.745929	+	0.666026i	$0.767994\pi$
$182$	16.5806	1.22904
$183$	8.42419	0.622734
$184$	−51.2905	−3.78119
$185$	0	0
$186$	17.5324	1.28554
$187$	−3.70156	−0.270685
$188$	−6.83171	−0.498253
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	19.7159	1.42660	0.713298	−	0.700861i	$-0.247201\pi$
$191$	19.7159	1.42660	0.713298	+	0.700861i	$0.247201\pi$
$192$	−4.70156	−0.339306
$193$	5.49029	0.395200	0.197600	−	0.980283i	$-0.436685\pi$
$193$	5.49029	0.395200	0.197600	+	0.980283i	$0.436685\pi$
$194$	−36.1502	−2.59544
$195$	0	0
$196$	4.70156	0.335826
$197$	−20.4421	−1.45644	−0.728220	−	0.685343i	$-0.759652\pi$
$197$	−20.4421	−1.45644	−0.728220	+	0.685343i	$0.759652\pi$
$198$	2.58874	0.183974
$199$	22.9391	1.62611	0.813055	−	0.582187i	$-0.197803\pi$
$199$	22.9391	1.62611	0.813055	+	0.582187i	$0.197803\pi$
$200$	0	0
$201$	−8.72263	−0.615247
$202$	−9.76162	−0.686825
$203$	−8.10646	−0.568962
$204$	−17.4031	−1.21846
$205$	0	0
$206$	−22.1630	−1.54417
$207$	7.33388	0.509740
$208$	−55.7326	−3.86436
$209$	−7.24672	−0.501266
$210$	0	0
$211$	−6.59333	−0.453903	−0.226952	−	0.973906i	$-0.572876\pi$
$211$	−6.59333	−0.453903	−0.226952	+	0.973906i	$0.572876\pi$
$212$	−42.8229	−2.94109
$213$	4.40490	0.301819
$214$	36.6642	2.50631
$215$	0	0
$216$	6.99364	0.475857
$217$	6.77258	0.459753
$218$	−18.4843	−1.25191
$219$	3.40667	0.230202
$220$	0	0
$221$	−23.7081	−1.59478
$222$	10.2256	0.686301
$223$	−11.0162	−0.737696	−0.368848	−	0.929490i	$-0.620248\pi$
$223$	−11.0162	−0.737696	−0.368848	+	0.929490i	$0.620248\pi$
$224$	−8.53879	−0.570522
$225$	0	0
$226$	−2.86893	−0.190838
$227$	21.0048	1.39414	0.697068	−	0.717005i	$-0.254487\pi$
$227$	21.0048	1.39414	0.697068	+	0.717005i	$0.254487\pi$
$228$	−34.0709	−2.25640
$229$	−19.5661	−1.29296	−0.646482	−	0.762929i	$-0.723760\pi$
$229$	−19.5661	−1.29296	−0.646482	+	0.762929i	$0.723760\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	56.6936	3.72212
$233$	−16.3886	−1.07365	−0.536827	−	0.843692i	$-0.680377\pi$
$233$	−16.3886	−1.07365	−0.536827	+	0.843692i	$0.680377\pi$
$234$	16.5806	1.08391
$235$	0	0
$236$	−13.7708	−0.896403
$237$	−4.40490	−0.286129
$238$	−9.58237	−0.621133
$239$	18.0066	1.16475	0.582374	−	0.812921i	$-0.302124\pi$
$239$	18.0066	1.16475	0.582374	+	0.812921i	$0.302124\pi$
$240$	0	0
$241$	27.0740	1.74399	0.871996	−	0.489513i	$-0.162826\pi$
$241$	27.0740	1.74399	0.871996	+	0.489513i	$0.162826\pi$
$242$	−2.58874	−0.166410
$243$	−1.00000	−0.0641500
$244$	−39.6069	−2.53557
$245$	0	0
$246$	−6.00000	−0.382546
$247$	−46.4145	−2.95328
$248$	−47.3649	−3.00768
$249$	−8.96620	−0.568210
$250$	0	0
$251$	8.58415	0.541827	0.270913	−	0.962604i	$-0.412674\pi$
$251$	8.58415	0.541827	0.270913	+	0.962604i	$0.412674\pi$
$252$	4.70156	0.296171
$253$	−7.33388	−0.461077
$254$	30.1630	1.89259
$255$	0	0
$256$	−22.1047	−1.38154
$257$	−0.0885359	−0.00552272	−0.00276136	−	0.999996i	$-0.500879\pi$
$257$	−0.0885359	−0.00552272	−0.00276136	+	0.999996i	$0.500879\pi$
$258$	19.8080	1.23319
$259$	3.95005	0.245444
$260$	0	0
$261$	−8.10646	−0.501777
$262$	35.7545	2.20892
$263$	17.5452	1.08188	0.540940	−	0.841061i	$-0.318068\pi$
$263$	17.5452	1.08188	0.540940	+	0.841061i	$0.318068\pi$
$264$	−6.99364	−0.430428
$265$	0	0
$266$	−18.7598	−1.15024
$267$	−0.206355	−0.0126287
$268$	41.0100	2.50508
$269$	12.2520	0.747020	0.373510	−	0.927626i	$-0.378154\pi$
$269$	12.2520	0.747020	0.373510	+	0.927626i	$0.378154\pi$
$270$	0	0
$271$	−1.91448	−0.116296	−0.0581482	−	0.998308i	$-0.518520\pi$
$271$	−1.91448	−0.116296	−0.0581482	+	0.998308i	$0.518520\pi$
$272$	32.2094	1.95298
$273$	6.40490	0.387642
$274$	24.3422	1.47057
$275$	0	0
$276$	−34.4807	−2.07549
$277$	−5.62877	−0.338200	−0.169100	−	0.985599i	$-0.554086\pi$
$277$	−5.62877	−0.338200	−0.169100	+	0.985599i	$0.554086\pi$
$278$	−42.2296	−2.53276
$279$	6.77258	0.405464
$280$	0	0
$281$	−29.3032	−1.74808	−0.874042	−	0.485850i	$-0.838510\pi$
$281$	−29.3032	−1.74808	−0.874042	+	0.485850i	$0.838510\pi$
$282$	−3.76162	−0.224001
$283$	−6.89833	−0.410063	−0.205032	−	0.978755i	$-0.565730\pi$
$283$	−6.89833	−0.410063	−0.205032	+	0.978755i	$0.565730\pi$
$284$	−20.7099	−1.22891
$285$	0	0
$286$	−16.5806	−0.980431
$287$	−2.31773	−0.136811
$288$	−8.53879	−0.503153
$289$	−3.29844	−0.194026
$290$	0	0
$291$	−13.9644	−0.818609
$292$	−16.0167	−0.937305
$293$	28.7243	1.67809	0.839045	−	0.544062i	$-0.183114\pi$
$293$	28.7243	1.67809	0.839045	+	0.544062i	$0.183114\pi$
$294$	2.58874	0.150978
$295$	0	0
$296$	−27.6252	−1.60568
$297$	1.00000	0.0580259
$298$	−25.6288	−1.48463
$299$	−46.9728	−2.71651
$300$	0	0
$301$	7.65161	0.441032
$302$	45.9710	2.64533
$303$	−3.77080	−0.216627
$304$	63.0578	3.61661
$305$	0	0
$306$	−9.58237	−0.547788
$307$	−15.7616	−0.899563	−0.449782	−	0.893139i	$-0.648498\pi$
$307$	−15.7616	−0.899563	−0.449782	+	0.893139i	$0.648498\pi$
$308$	−4.70156	−0.267896
$309$	−8.56131	−0.487036
$310$	0	0
$311$	18.7549	1.06349	0.531747	−	0.846903i	$-0.321536\pi$
$311$	18.7549	1.06349	0.531747	+	0.846903i	$0.321536\pi$
$312$	−44.7935	−2.53593
$313$	−8.45786	−0.478067	−0.239033	−	0.971011i	$-0.576831\pi$
$313$	−8.45786	−0.478067	−0.239033	+	0.971011i	$0.576831\pi$
$314$	44.1502	2.49154
$315$	0	0
$316$	20.7099	1.16502
$317$	12.8519	0.721836	0.360918	−	0.932597i	$-0.382463\pi$
$317$	12.8519	0.721836	0.360918	+	0.932597i	$0.382463\pi$
$318$	−23.5788	−1.32223
$319$	8.10646	0.453875
$320$	0	0
$321$	14.1630	0.790500
$322$	−18.9855	−1.05802
$323$	26.8242	1.49254
$324$	4.70156	0.261198
$325$	0	0
$326$	−51.6033	−2.85804
$327$	−7.14026	−0.394857
$328$	16.2094	0.895013
$329$	−1.45307	−0.0801104
$330$	0	0
$331$	−6.64984	−0.365508	−0.182754	−	0.983159i	$-0.558501\pi$
$331$	−6.64984	−0.365508	−0.182754	+	0.983159i	$0.558501\pi$
$332$	42.1552	2.31356
$333$	3.95005	0.216462
$334$	−7.64150	−0.418124
$335$	0	0
$336$	−8.70156	−0.474709
$337$	18.0228	0.981767	0.490883	−	0.871225i	$-0.336674\pi$
$337$	18.0228	0.981767	0.490883	+	0.871225i	$0.336674\pi$
$338$	−72.5435	−3.94584
$339$	−1.10823	−0.0601910
$340$	0	0
$341$	−6.77258	−0.366756
$342$	−18.7598	−1.01442
$343$	1.00000	0.0539949
$344$	−53.5126	−2.88521
$345$	0	0
$346$	52.3168	2.81257
$347$	2.00355	0.107556	0.0537780	−	0.998553i	$-0.482874\pi$
$347$	2.00355	0.107556	0.0537780	+	0.998553i	$0.482874\pi$
$348$	38.1130	2.04307
$349$	1.15955	0.0620693	0.0310347	−	0.999518i	$-0.490120\pi$
$349$	1.15955	0.0620693	0.0310347	+	0.999518i	$0.490120\pi$
$350$	0	0
$351$	6.40490	0.341868
$352$	8.53879	0.455119
$353$	−29.6160	−1.57630	−0.788151	−	0.615481i	$-0.788962\pi$
$353$	−29.6160	−1.57630	−0.788151	+	0.615481i	$0.788962\pi$
$354$	−7.58237	−0.402999
$355$	0	0
$356$	0.970191	0.0514200
$357$	−3.70156	−0.195907
$358$	−15.4031	−0.814080
$359$	−22.4650	−1.18566	−0.592828	−	0.805329i	$-0.701988\pi$
$359$	−22.4650	−1.18566	−0.592828	+	0.805329i	$0.701988\pi$
$360$	0	0
$361$	33.5149	1.76394
$362$	51.9583	2.73087
$363$	−1.00000	−0.0524864
$364$	−30.1130	−1.57835
$365$	0	0
$366$	−21.8080	−1.13992
$367$	−5.12233	−0.267384	−0.133692	−	0.991023i	$-0.542683\pi$
$367$	−5.12233	−0.267384	−0.133692	+	0.991023i	$0.542683\pi$
$368$	63.8162	3.32665
$369$	−2.31773	−0.120656
$370$	0	0
$371$	−9.10823	−0.472876
$372$	−31.8417	−1.65091
$373$	26.2358	1.35844	0.679218	−	0.733936i	$-0.262319\pi$
$373$	26.2358	1.35844	0.679218	+	0.733936i	$0.262319\pi$
$374$	9.58237	0.495493
$375$	0	0
$376$	10.1623	0.524078
$377$	51.9210	2.67407
$378$	2.58874	0.133150
$379$	−9.60167	−0.493205	−0.246602	−	0.969117i	$-0.579314\pi$
$379$	−9.60167	−0.493205	−0.246602	+	0.969117i	$0.579314\pi$
$380$	0	0
$381$	11.6516	0.596930
$382$	−51.0394	−2.61140
$383$	−10.1244	−0.517332	−0.258666	−	0.965967i	$-0.583283\pi$
$383$	−10.1244	−0.517332	−0.258666	+	0.965967i	$0.583283\pi$
$384$	−4.90647	−0.250382
$385$	0	0
$386$	−14.2129	−0.723419
$387$	7.65161	0.388953
$388$	65.6546	3.33311
$389$	16.9342	0.858597	0.429298	−	0.903163i	$-0.358761\pi$
$389$	16.9342	0.858597	0.429298	+	0.903163i	$0.358761\pi$
$390$	0	0
$391$	27.1468	1.37287
$392$	−6.99364	−0.353232
$393$	13.8116	0.696701
$394$	52.9193	2.66603
$395$	0	0
$396$	−4.70156	−0.236262
$397$	−13.7616	−0.690676	−0.345338	−	0.938478i	$-0.612236\pi$
$397$	−13.7616	−0.690676	−0.345338	+	0.938478i	$0.612236\pi$
$398$	−59.3833	−2.97662
$399$	−7.24672	−0.362790
$400$	0	0
$401$	35.6118	1.77837	0.889184	−	0.457550i	$-0.151273\pi$
$401$	35.6118	1.77837	0.889184	+	0.457550i	$0.151273\pi$
$402$	22.5806	1.12622
$403$	−43.3777	−2.16080
$404$	17.7287	0.882034
$405$	0	0
$406$	20.9855	1.04149
$407$	−3.95005	−0.195797
$408$	25.8874	1.28162
$409$	9.17748	0.453797	0.226898	−	0.973918i	$-0.427141\pi$
$409$	9.17748	0.453797	0.226898	+	0.973918i	$0.427141\pi$
$410$	0	0
$411$	9.40312	0.463822
$412$	40.2515	1.98305
$413$	−2.92898	−0.144126
$414$	−18.9855	−0.933086
$415$	0	0
$416$	54.6901	2.68140
$417$	−16.3128	−0.798842
$418$	18.7598	0.917574
$419$	37.8554	1.84935	0.924677	−	0.380751i	$-0.124335\pi$
$419$	37.8554	1.84935	0.924677	+	0.380751i	$0.124335\pi$
$420$	0	0
$421$	3.10469	0.151313	0.0756566	−	0.997134i	$-0.475895\pi$
$421$	3.10469	0.151313	0.0756566	+	0.997134i	$0.475895\pi$
$422$	17.0684	0.830877
$423$	−1.45307	−0.0706508
$424$	63.6997	3.09353
$425$	0	0
$426$	−11.4031	−0.552483
$427$	−8.42419	−0.407675
$428$	−66.5881	−3.21866
$429$	−6.40490	−0.309231
$430$	0	0
$431$	−10.4548	−0.503592	−0.251796	−	0.967780i	$-0.581021\pi$
$431$	−10.4548	−0.503592	−0.251796	+	0.967780i	$0.581021\pi$
$432$	−8.70156	−0.418654
$433$	7.82212	0.375907	0.187954	−	0.982178i	$-0.439815\pi$
$433$	7.82212	0.375907	0.187954	+	0.982178i	$0.439815\pi$
$434$	−17.5324	−0.841583
$435$	0	0
$436$	33.5704	1.60773
$437$	53.1466	2.54235
$438$	−8.81898	−0.421387
$439$	−29.8146	−1.42297	−0.711486	−	0.702700i	$-0.751978\pi$
$439$	−29.8146	−1.42297	−0.711486	+	0.702700i	$0.751978\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	61.3741	2.91927
$443$	11.2774	0.535804	0.267902	−	0.963446i	$-0.413670\pi$
$443$	11.2774	0.535804	0.267902	+	0.963446i	$0.413670\pi$
$444$	−18.5714	−0.881360
$445$	0	0
$446$	28.5179	1.35036
$447$	−9.90010	−0.468259
$448$	4.70156	0.222128
$449$	11.2787	0.532277	0.266138	−	0.963935i	$-0.414252\pi$
$449$	11.2787	0.532277	0.266138	+	0.963935i	$0.414252\pi$
$450$	0	0
$451$	2.31773	0.109138
$452$	5.21043	0.245078
$453$	17.7581	0.834347
$454$	−54.3759	−2.55199
$455$	0	0
$456$	50.6809	2.37335
$457$	0.403250	0.0188632	0.00943162	−	0.999956i	$-0.496998\pi$
$457$	0.403250	0.0188632	0.00943162	+	0.999956i	$0.496998\pi$
$458$	50.6515	2.36679
$459$	−3.70156	−0.172774
$460$	0	0
$461$	34.9355	1.62711	0.813555	−	0.581487i	$-0.197529\pi$
$461$	34.9355	1.62711	0.813555	+	0.581487i	$0.197529\pi$
$462$	−2.58874	−0.120439
$463$	−4.36413	−0.202818	−0.101409	−	0.994845i	$-0.532335\pi$
$463$	−4.36413	−0.202818	−0.101409	+	0.994845i	$0.532335\pi$
$464$	−70.5389	−3.27468
$465$	0	0
$466$	42.4258	1.96534
$467$	30.2551	1.40004	0.700018	−	0.714125i	$-0.253175\pi$
$467$	30.2551	1.40004	0.700018	+	0.714125i	$0.253175\pi$
$468$	−30.1130	−1.39198
$469$	8.72263	0.402774
$470$	0	0
$471$	17.0547	0.785841
$472$	20.4843	0.942864
$473$	−7.65161	−0.351822
$474$	11.4031	0.523763
$475$	0	0
$476$	17.4031	0.797671
$477$	−9.10823	−0.417037
$478$	−46.6143	−2.13209
$479$	11.8151	0.539846	0.269923	−	0.962882i	$-0.413002\pi$
$479$	11.8151	0.539846	0.269923	+	0.962882i	$0.413002\pi$
$480$	0	0
$481$	−25.2997	−1.15357
$482$	−70.0876	−3.19240
$483$	−7.33388	−0.333703
$484$	4.70156	0.213707
$485$	0	0
$486$	2.58874	0.117428
$487$	−36.3387	−1.64666	−0.823331	−	0.567561i	$-0.807887\pi$
$487$	−36.3387	−1.64666	−0.823331	+	0.567561i	$0.807887\pi$
$488$	58.9157	2.66699
$489$	−19.9338	−0.901437
$490$	0	0
$491$	−41.4554	−1.87085	−0.935427	−	0.353519i	$-0.884985\pi$
$491$	−41.4554	−1.87085	−0.935427	+	0.353519i	$0.884985\pi$
$492$	10.8970	0.491273
$493$	−30.0066	−1.35143
$494$	120.155	5.40602
$495$	0	0
$496$	58.9320	2.64613
$497$	−4.40490	−0.197587
$498$	23.2111	1.04012
$499$	2.41146	0.107952	0.0539759	−	0.998542i	$-0.482811\pi$
$499$	2.41146	0.107952	0.0539759	+	0.998542i	$0.482811\pi$
$500$	0	0
$501$	−2.95183	−0.131878
$502$	−22.2221	−0.991821
$503$	24.3335	1.08498	0.542488	−	0.840063i	$-0.317482\pi$
$503$	24.3335	1.08498	0.542488	+	0.840063i	$0.317482\pi$
$504$	−6.99364	−0.311521
$505$	0	0
$506$	18.9855	0.844008
$507$	−28.0227	−1.24453
$508$	−54.7808	−2.43050
$509$	21.3388	0.945826	0.472913	−	0.881109i	$-0.343203\pi$
$509$	21.3388	0.945826	0.472913	+	0.881109i	$0.343203\pi$
$510$	0	0
$511$	−3.40667	−0.150702
$512$	47.4103	2.09526
$513$	−7.24672	−0.319950
$514$	0.229196	0.0101094
$515$	0	0
$516$	−35.9745	−1.58369
$517$	1.45307	0.0639060
$518$	−10.2256	−0.449289
$519$	20.2094	0.887093
$520$	0	0
$521$	2.03557	0.0891800	0.0445900	−	0.999005i	$-0.485802\pi$
$521$	2.03557	0.0891800	0.0445900	+	0.999005i	$0.485802\pi$
$522$	20.9855	0.918510
$523$	11.1647	0.488200	0.244100	−	0.969750i	$-0.421507\pi$
$523$	11.1647	0.488200	0.244100	+	0.969750i	$0.421507\pi$
$524$	−64.9359	−2.83674
$525$	0	0
$526$	−45.4198	−1.98040
$527$	25.0691	1.09203
$528$	8.70156	0.378687
$529$	30.7858	1.33851
$530$	0	0
$531$	−2.92898	−0.127107
$532$	34.0709	1.47716
$533$	14.8448	0.643001
$534$	0.534199	0.0231170
$535$	0	0
$536$	−61.0029	−2.63492
$537$	−5.95005	−0.256764
$538$	−31.7173	−1.36743
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	−37.8212	−1.62606	−0.813029	−	0.582223i	$-0.802183\pi$
$541$	−37.8212	−1.62606	−0.813029	+	0.582223i	$0.802183\pi$
$542$	4.95609	0.212882
$543$	20.0709	0.861324
$544$	−31.6069	−1.35513
$545$	0	0
$546$	−16.5806	−0.709584
$547$	32.2871	1.38050	0.690248	−	0.723573i	$-0.257501\pi$
$547$	32.2871	1.38050	0.690248	+	0.723573i	$0.257501\pi$
$548$	−44.2094	−1.88853
$549$	−8.42419	−0.359536
$550$	0	0
$551$	−58.7452	−2.50263
$552$	51.2905	2.18307
$553$	4.40490	0.187315
$554$	14.5714	0.619080
$555$	0	0
$556$	76.6957	3.25262
$557$	−17.9416	−0.760209	−0.380105	−	0.924943i	$-0.624112\pi$
$557$	−17.9416	−0.760209	−0.380105	+	0.924943i	$0.624112\pi$
$558$	−17.5324	−0.742207
$559$	−49.0078	−2.07281
$560$	0	0
$561$	3.70156	0.156280
$562$	75.8584	3.19989
$563$	1.40667	0.0592841	0.0296421	−	0.999561i	$-0.490563\pi$
$563$	1.40667	0.0592841	0.0296421	+	0.999561i	$0.490563\pi$
$564$	6.83171	0.287667
$565$	0	0
$566$	17.8580	0.750626
$567$	1.00000	0.0419961
$568$	30.8062	1.29260
$569$	9.36755	0.392708	0.196354	−	0.980533i	$-0.437090\pi$
$569$	9.36755	0.392708	0.196354	+	0.980533i	$0.437090\pi$
$570$	0	0
$571$	−24.8562	−1.04020	−0.520100	−	0.854106i	$-0.674105\pi$
$571$	−24.8562	−1.04020	−0.520100	+	0.854106i	$0.674105\pi$
$572$	30.1130	1.25909
$573$	−19.7159	−0.823645
$574$	6.00000	0.250435
$575$	0	0
$576$	4.70156	0.195898
$577$	−6.17433	−0.257041	−0.128520	−	0.991707i	$-0.541023\pi$
$577$	−6.17433	−0.257041	−0.128520	+	0.991707i	$0.541023\pi$
$578$	8.53879	0.355167
$579$	−5.49029	−0.228169
$580$	0	0
$581$	8.96620	0.371981
$582$	36.1502	1.49848
$583$	9.10823	0.377224
$584$	23.8250	0.985886
$585$	0	0
$586$	−74.3596	−3.07177
$587$	−2.04995	−0.0846104	−0.0423052	−	0.999105i	$-0.513470\pi$
$587$	−2.04995	−0.0846104	−0.0423052	+	0.999105i	$0.513470\pi$
$588$	−4.70156	−0.193889
$589$	49.0790	2.02226
$590$	0	0
$591$	20.4421	0.840876
$592$	34.3716	1.41267
$593$	7.85797	0.322688	0.161344	−	0.986898i	$-0.448417\pi$
$593$	7.85797	0.322688	0.161344	+	0.986898i	$0.448417\pi$
$594$	−2.58874	−0.106217
$595$	0	0
$596$	46.5460	1.90660
$597$	−22.9391	−0.938835
$598$	121.600	4.97260
$599$	−11.2190	−0.458394	−0.229197	−	0.973380i	$-0.573610\pi$
$599$	−11.2190	−0.458394	−0.229197	+	0.973380i	$0.573610\pi$
$600$	0	0
$601$	19.5047	0.795612	0.397806	−	0.917470i	$-0.369772\pi$
$601$	19.5047	0.795612	0.397806	+	0.917470i	$0.369772\pi$
$602$	−19.8080	−0.807315
$603$	8.72263	0.355213
$604$	−83.4907	−3.39719
$605$	0	0
$606$	9.76162	0.396539
$607$	16.9483	0.687909	0.343955	−	0.938986i	$-0.388233\pi$
$607$	16.9483	0.687909	0.343955	+	0.938986i	$0.388233\pi$
$608$	−61.8782	−2.50949
$609$	8.10646	0.328490
$610$	0	0
$611$	9.30678	0.376512
$612$	17.4031	0.703480
$613$	12.3964	0.500687	0.250344	−	0.968157i	$-0.419456\pi$
$613$	12.3964	0.500687	0.250344	+	0.968157i	$0.419456\pi$
$614$	40.8027	1.64666
$615$	0	0
$616$	6.99364	0.281782
$617$	21.0192	0.846200	0.423100	−	0.906083i	$-0.360942\pi$
$617$	21.0192	0.846200	0.423100	+	0.906083i	$0.360942\pi$
$618$	22.1630	0.891525
$619$	11.5289	0.463385	0.231692	−	0.972789i	$-0.425574\pi$
$619$	11.5289	0.463385	0.231692	+	0.972789i	$0.425574\pi$
$620$	0	0
$621$	−7.33388	−0.294299
$622$	−48.5516	−1.94674
$623$	0.206355	0.00826744
$624$	55.7326	2.23109
$625$	0	0
$626$	21.8952	0.875108
$627$	7.24672	0.289406
$628$	−80.1839	−3.19969
$629$	14.6214	0.582992
$630$	0	0
$631$	33.9988	1.35347	0.676734	−	0.736227i	$-0.263395\pi$
$631$	33.9988	1.35347	0.676734	+	0.736227i	$0.263395\pi$
$632$	−30.8062	−1.22541
$633$	6.59333	0.262061
$634$	−33.2703	−1.32133
$635$	0	0
$636$	42.8229	1.69804
$637$	−6.40490	−0.253771
$638$	−20.9855	−0.830824
$639$	−4.40490	−0.174255
$640$	0	0
$641$	−20.5293	−0.810858	−0.405429	−	0.914127i	$-0.632878\pi$
$641$	−20.5293	−0.810858	−0.405429	+	0.914127i	$0.632878\pi$
$642$	−36.6642	−1.44702
$643$	16.9584	0.668774	0.334387	−	0.942436i	$-0.391471\pi$
$643$	16.9584	0.668774	0.334387	+	0.942436i	$0.391471\pi$
$644$	34.4807	1.35873
$645$	0	0
$646$	−69.4407	−2.73211
$647$	2.25506	0.0886554	0.0443277	−	0.999017i	$-0.485885\pi$
$647$	2.25506	0.0886554	0.0443277	+	0.999017i	$0.485885\pi$
$648$	−6.99364	−0.274736
$649$	2.92898	0.114973
$650$	0	0
$651$	−6.77258	−0.265438
$652$	93.7199	3.67035
$653$	−1.98917	−0.0778422	−0.0389211	−	0.999242i	$-0.512392\pi$
$653$	−1.98917	−0.0778422	−0.0389211	+	0.999242i	$0.512392\pi$
$654$	18.4843	0.722791
$655$	0	0
$656$	−20.1679	−0.787424
$657$	−3.40667	−0.132907
$658$	3.76162	0.146643
$659$	3.41324	0.132961	0.0664804	−	0.997788i	$-0.478823\pi$
$659$	3.41324	0.132961	0.0664804	+	0.997788i	$0.478823\pi$
$660$	0	0
$661$	15.4988	0.602832	0.301416	−	0.953493i	$-0.402541\pi$
$661$	15.4988	0.602832	0.301416	+	0.953493i	$0.402541\pi$
$662$	17.2147	0.669068
$663$	23.7081	0.920747
$664$	−62.7064	−2.43348
$665$	0	0
$666$	−10.2256	−0.396236
$667$	−59.4518	−2.30198
$668$	13.8782	0.536963
$669$	11.0162	0.425909
$670$	0	0
$671$	8.42419	0.325212
$672$	8.53879	0.329391
$673$	40.6449	1.56675	0.783373	−	0.621551i	$-0.213497\pi$
$673$	40.6449	1.56675	0.783373	+	0.621551i	$0.213497\pi$
$674$	−46.6564	−1.79714
$675$	0	0
$676$	131.751	5.06733
$677$	−41.1756	−1.58251	−0.791253	−	0.611489i	$-0.790571\pi$
$677$	−41.1756	−1.58251	−0.791253	+	0.611489i	$0.790571\pi$
$678$	2.86893	0.110180
$679$	13.9644	0.535906
$680$	0	0
$681$	−21.0048	−0.804905
$682$	17.5324	0.671351
$683$	−4.86111	−0.186005	−0.0930027	−	0.995666i	$-0.529647\pi$
$683$	−4.86111	−0.186005	−0.0930027	+	0.995666i	$0.529647\pi$
$684$	34.0709	1.30273
$685$	0	0
$686$	−2.58874	−0.0988385
$687$	19.5661	0.746493
$688$	66.5810	2.53838
$689$	58.3373	2.22247
$690$	0	0
$691$	41.8838	1.59334	0.796668	−	0.604417i	$-0.206594\pi$
$691$	41.8838	1.59334	0.796668	+	0.604417i	$0.206594\pi$
$692$	−95.0156	−3.61195
$693$	−1.00000	−0.0379869
$694$	−5.18666	−0.196883
$695$	0	0
$696$	−56.6936	−2.14897
$697$	−8.57923	−0.324961
$698$	−3.00177	−0.113619
$699$	16.3886	0.619875
$700$	0	0
$701$	4.10646	0.155099	0.0775494	−	0.996989i	$-0.475290\pi$
$701$	4.10646	0.155099	0.0775494	+	0.996989i	$0.475290\pi$
$702$	−16.5806	−0.625794
$703$	28.6249	1.07961
$704$	−4.70156	−0.177197
$705$	0	0
$706$	76.6682	2.88544
$707$	3.77080	0.141816
$708$	13.7708	0.517539
$709$	9.52094	0.357567	0.178783	−	0.983888i	$-0.442784\pi$
$709$	9.52094	0.357567	0.178783	+	0.983888i	$0.442784\pi$
$710$	0	0
$711$	4.40490	0.165197
$712$	−1.44317	−0.0540851
$713$	49.6693	1.86013
$714$	9.58237	0.358611
$715$	0	0
$716$	27.9745	1.04546
$717$	−18.0066	−0.672467
$718$	58.1559	2.17036
$719$	17.4969	0.652523	0.326261	−	0.945280i	$-0.394211\pi$
$719$	17.4969	0.652523	0.326261	+	0.945280i	$0.394211\pi$
$720$	0	0
$721$	8.56131	0.318840
$722$	−86.7613	−3.22892
$723$	−27.0740	−1.00689
$724$	−94.3645	−3.50703
$725$	0	0
$726$	2.58874	0.0960771
$727$	46.8908	1.73908	0.869542	−	0.493860i	$-0.164414\pi$
$727$	46.8908	1.73908	0.869542	+	0.493860i	$0.164414\pi$
$728$	44.7935	1.66016
$729$	1.00000	0.0370370
$730$	0	0
$731$	28.3229	1.04756
$732$	39.6069	1.46391
$733$	15.8038	0.583725	0.291863	−	0.956460i	$-0.405725\pi$
$733$	15.8038	0.583725	0.291863	+	0.956460i	$0.405725\pi$
$734$	13.2604	0.489449
$735$	0	0
$736$	−62.6225	−2.30830
$737$	−8.72263	−0.321302
$738$	6.00000	0.220863
$739$	−5.69073	−0.209337	−0.104668	−	0.994507i	$-0.533378\pi$
$739$	−5.69073	−0.209337	−0.104668	+	0.994507i	$0.533378\pi$
$740$	0	0
$741$	46.4145	1.70508
$742$	23.5788	0.865606
$743$	−22.2909	−0.817774	−0.408887	−	0.912585i	$-0.634083\pi$
$743$	−22.2909	−0.817774	−0.408887	+	0.912585i	$0.634083\pi$
$744$	47.3649	1.73648
$745$	0	0
$746$	−67.9175	−2.48664
$747$	8.96620	0.328056
$748$	−17.4031	−0.636321
$749$	−14.1630	−0.517504
$750$	0	0
$751$	38.6786	1.41140	0.705701	−	0.708510i	$-0.250632\pi$
$751$	38.6786	1.41140	0.705701	+	0.708510i	$0.250632\pi$
$752$	−12.6440	−0.461079
$753$	−8.58415	−0.312824
$754$	−134.410	−4.89492
$755$	0	0
$756$	−4.70156	−0.170994
$757$	−43.7967	−1.59182	−0.795908	−	0.605417i	$-0.793006\pi$
$757$	−43.7967	−1.59182	−0.795908	+	0.605417i	$0.793006\pi$
$758$	24.8562	0.902818
$759$	7.33388	0.266203
$760$	0	0
$761$	40.1966	1.45713	0.728564	−	0.684978i	$-0.240188\pi$
$761$	40.1966	1.45713	0.728564	+	0.684978i	$0.240188\pi$
$762$	−30.1630	−1.09269
$763$	7.14026	0.258495
$764$	92.6957	3.35361
$765$	0	0
$766$	26.2094	0.946983
$767$	18.7598	0.677379
$768$	22.1047	0.797634
$769$	1.32679	0.0478452	0.0239226	−	0.999714i	$-0.492384\pi$
$769$	1.32679	0.0478452	0.0239226	+	0.999714i	$0.492384\pi$
$770$	0	0
$771$	0.0885359	0.00318854
$772$	25.8129	0.929028
$773$	9.62314	0.346120	0.173060	−	0.984911i	$-0.444635\pi$
$773$	9.62314	0.346120	0.173060	+	0.984911i	$0.444635\pi$
$774$	−19.8080	−0.711985
$775$	0	0
$776$	−97.6621	−3.50587
$777$	−3.95005	−0.141707
$778$	−43.8381	−1.57167
$779$	−16.7959	−0.601777
$780$	0	0
$781$	4.40490	0.157620
$782$	−70.2760	−2.51306
$783$	8.10646	0.289701
$784$	8.70156	0.310770
$785$	0	0
$786$	−35.7545	−1.27532
$787$	−44.1489	−1.57374	−0.786869	−	0.617120i	$-0.788299\pi$
$787$	−44.1489	−1.57374	−0.786869	+	0.617120i	$0.788299\pi$
$788$	−96.1099	−3.42377
$789$	−17.5452	−0.624624
$790$	0	0
$791$	1.10823	0.0394042
$792$	6.99364	0.248508
$793$	53.9561	1.91604
$794$	35.6252	1.26429
$795$	0	0
$796$	107.850	3.82263
$797$	18.5257	0.656215	0.328108	−	0.944640i	$-0.393589\pi$
$797$	18.5257	0.656215	0.328108	+	0.944640i	$0.393589\pi$
$798$	18.7598	0.664091
$799$	−5.37864	−0.190282
$800$	0	0
$801$	0.206355	0.00729119
$802$	−92.1895	−3.25533
$803$	3.40667	0.120219
$804$	−41.0100	−1.44631
$805$	0	0
$806$	112.293	3.95537
$807$	−12.2520	−0.431292
$808$	−26.3716	−0.927751
$809$	38.1516	1.34134	0.670670	−	0.741756i	$-0.266007\pi$
$809$	38.1516	1.34134	0.670670	+	0.741756i	$0.266007\pi$
$810$	0	0
$811$	−37.3355	−1.31103	−0.655514	−	0.755183i	$-0.727548\pi$
$811$	−37.3355	−1.31103	−0.655514	+	0.755183i	$0.727548\pi$
$812$	−38.1130	−1.33750
$813$	1.91448	0.0671438
$814$	10.2256	0.358409
$815$	0	0
$816$	−32.2094	−1.12755
$817$	55.4491	1.93992
$818$	−23.7581	−0.830682
$819$	−6.40490	−0.223805
$820$	0	0
$821$	−19.9039	−0.694652	−0.347326	−	0.937744i	$-0.612910\pi$
$821$	−19.9039	−0.694652	−0.347326	+	0.937744i	$0.612910\pi$
$822$	−24.3422	−0.849032
$823$	50.8399	1.77217	0.886084	−	0.463525i	$-0.153415\pi$
$823$	50.8399	1.77217	0.886084	+	0.463525i	$0.153415\pi$
$824$	−59.8746	−2.08583
$825$	0	0
$826$	7.58237	0.263824
$827$	43.7003	1.51961	0.759804	−	0.650152i	$-0.225295\pi$
$827$	43.7003	1.51961	0.759804	+	0.650152i	$0.225295\pi$
$828$	34.4807	1.19829
$829$	51.5197	1.78935	0.894677	−	0.446715i	$-0.147406\pi$
$829$	51.5197	1.78935	0.894677	+	0.446715i	$0.147406\pi$
$830$	0	0
$831$	5.62877	0.195260
$832$	−30.1130	−1.04398
$833$	3.70156	0.128252
$834$	42.2296	1.46229
$835$	0	0
$836$	−34.0709	−1.17837
$837$	−6.77258	−0.234095
$838$	−97.9976	−3.38527
$839$	−18.7834	−0.648475	−0.324238	−	0.945976i	$-0.605108\pi$
$839$	−18.7834	−0.648475	−0.324238	+	0.945976i	$0.605108\pi$
$840$	0	0
$841$	36.7147	1.26602
$842$	−8.03722	−0.276981
$843$	29.3032	1.00926
$844$	−30.9989	−1.06703
$845$	0	0
$846$	3.76162	0.129327
$847$	1.00000	0.0343604
$848$	−79.2559	−2.72166
$849$	6.89833	0.236750
$850$	0	0
$851$	28.9692	0.993052
$852$	20.7099	0.709509
$853$	10.4506	0.357821	0.178911	−	0.983865i	$-0.442743\pi$
$853$	10.4506	0.357821	0.178911	+	0.983865i	$0.442743\pi$
$854$	21.8080	0.746255
$855$	0	0
$856$	99.0507	3.38548
$857$	−33.2646	−1.13630	−0.568149	−	0.822926i	$-0.692340\pi$
$857$	−33.2646	−1.13630	−0.568149	+	0.822926i	$0.692340\pi$
$858$	16.5806	0.566052
$859$	27.5569	0.940230	0.470115	−	0.882605i	$-0.344212\pi$
$859$	27.5569	0.940230	0.470115	+	0.882605i	$0.344212\pi$
$860$	0	0
$861$	2.31773	0.0789881
$862$	27.0649	0.921832
$863$	10.2042	0.347354	0.173677	−	0.984803i	$-0.444435\pi$
$863$	10.2042	0.347354	0.173677	+	0.984803i	$0.444435\pi$
$864$	8.53879	0.290496
$865$	0	0
$866$	−20.2494	−0.688103
$867$	3.29844	0.112021
$868$	31.8417	1.08078
$869$	−4.40490	−0.149426
$870$	0	0
$871$	−55.8676	−1.89300
$872$	−49.9364	−1.69106
$873$	13.9644	0.472624
$874$	−137.583	−4.65380
$875$	0	0
$876$	16.0167	0.541153
$877$	−57.5319	−1.94271	−0.971357	−	0.237625i	$-0.923631\pi$
$877$	−57.5319	−1.94271	−0.971357	+	0.237625i	$0.923631\pi$
$878$	77.1821	2.60477
$879$	−28.7243	−0.968846
$880$	0	0
$881$	−1.05474	−0.0355351	−0.0177675	−	0.999842i	$-0.505656\pi$
$881$	−1.05474	−0.0355351	−0.0177675	+	0.999842i	$0.505656\pi$
$882$	−2.58874	−0.0871673
$883$	−14.4063	−0.484810	−0.242405	−	0.970175i	$-0.577936\pi$
$883$	−14.4063	−0.484810	−0.242405	+	0.970175i	$0.577936\pi$
$884$	−111.465	−3.74898
$885$	0	0
$886$	−29.1942	−0.980797
$887$	16.3792	0.549958	0.274979	−	0.961450i	$-0.411329\pi$
$887$	16.3792	0.549958	0.274979	+	0.961450i	$0.411329\pi$
$888$	27.6252	0.927042
$889$	−11.6516	−0.390783
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	−51.7931	−1.73416
$893$	−10.5300	−0.352373
$894$	25.6288	0.857154
$895$	0	0
$896$	4.90647	0.163914
$897$	46.9728	1.56838
$898$	−29.1977	−0.974340
$899$	−54.9016	−1.83107
$900$	0	0
$901$	−33.7147	−1.12320
$902$	−6.00000	−0.199778
$903$	−7.65161	−0.254630
$904$	−7.75058	−0.257780
$905$	0	0
$906$	−45.9710	−1.52728
$907$	26.0372	0.864552	0.432276	−	0.901741i	$-0.357711\pi$
$907$	26.0372	0.864552	0.432276	+	0.901741i	$0.357711\pi$
$908$	98.7553	3.27731
$909$	3.77080	0.125070
$910$	0	0
$911$	25.5381	0.846114	0.423057	−	0.906103i	$-0.360957\pi$
$911$	25.5381	0.846114	0.423057	+	0.906103i	$0.360957\pi$
$912$	−63.0578	−2.08805
$913$	−8.96620	−0.296738
$914$	−1.04391	−0.0345294
$915$	0	0
$916$	−91.9912	−3.03948
$917$	−13.8116	−0.456098
$918$	9.58237	0.316265
$919$	−28.3971	−0.936733	−0.468367	−	0.883534i	$-0.655157\pi$
$919$	−28.3971	−0.936733	−0.468367	+	0.883534i	$0.655157\pi$
$920$	0	0
$921$	15.7616	0.519363
$922$	−90.4390	−2.97845
$923$	28.2129	0.928640
$924$	4.70156	0.154670
$925$	0	0
$926$	11.2976	0.371262
$927$	8.56131	0.281190
$928$	69.2194	2.27224
$929$	−10.7485	−0.352646	−0.176323	−	0.984332i	$-0.556420\pi$
$929$	−10.7485	−0.352646	−0.176323	+	0.984332i	$0.556420\pi$
$930$	0	0
$931$	7.24672	0.237502
$932$	−77.0521	−2.52393
$933$	−18.7549	−0.614009
$934$	−78.3224	−2.56279
$935$	0	0
$936$	44.7935	1.46412
$937$	34.9763	1.14263	0.571313	−	0.820732i	$-0.306434\pi$
$937$	34.9763	1.14263	0.571313	+	0.820732i	$0.306434\pi$
$938$	−22.5806	−0.737283
$939$	8.45786	0.276012
$940$	0	0
$941$	25.8979	0.844248	0.422124	−	0.906538i	$-0.361285\pi$
$941$	25.8979	0.844248	0.422124	+	0.906538i	$0.361285\pi$
$942$	−44.1502	−1.43849
$943$	−16.9980	−0.553530
$944$	−25.4867	−0.829523
$945$	0	0
$946$	19.8080	0.644014
$947$	15.9623	0.518704	0.259352	−	0.965783i	$-0.416491\pi$
$947$	15.9623	0.518704	0.259352	+	0.965783i	$0.416491\pi$
$948$	−20.7099	−0.672626
$949$	21.8194	0.708287
$950$	0	0
$951$	−12.8519	−0.416752
$952$	−25.8874	−0.839015
$953$	30.8611	0.999690	0.499845	−	0.866115i	$-0.333390\pi$
$953$	30.8611	0.999690	0.499845	+	0.866115i	$0.333390\pi$
$954$	23.5788	0.763393
$955$	0	0
$956$	84.6590	2.73807
$957$	−8.10646	−0.262045
$958$	−30.5862	−0.988196
$959$	−9.40312	−0.303643
$960$	0	0
$961$	14.8678	0.479607
$962$	65.4942	2.11162
$963$	−14.1630	−0.456395
$964$	127.290	4.09974
$965$	0	0
$966$	18.9855	0.610848
$967$	38.9934	1.25394	0.626972	−	0.779042i	$-0.284294\pi$
$967$	38.9934	1.25394	0.626972	+	0.779042i	$0.284294\pi$
$968$	−6.99364	−0.224784
$969$	−26.8242	−0.861717
$970$	0	0
$971$	−1.98071	−0.0635639	−0.0317819	−	0.999495i	$-0.510118\pi$
$971$	−1.98071	−0.0635639	−0.0317819	+	0.999495i	$0.510118\pi$
$972$	−4.70156	−0.150803
$973$	16.3128	0.522965
$974$	94.0713	3.01424
$975$	0	0
$976$	−73.3036	−2.34639
$977$	−9.66926	−0.309347	−0.154674	−	0.987966i	$-0.549433\pi$
$977$	−9.66926	−0.309347	−0.154674	+	0.987966i	$0.549433\pi$
$978$	51.6033	1.65009
$979$	−0.206355	−0.00659513
$980$	0	0
$981$	7.14026	0.227971
$982$	107.317	3.42463
$983$	31.1289	0.992858	0.496429	−	0.868077i	$-0.334644\pi$
$983$	31.1289	0.992858	0.496429	+	0.868077i	$0.334644\pi$
$984$	−16.2094	−0.516736
$985$	0	0
$986$	77.6791	2.47381
$987$	1.45307	0.0462518
$988$	−218.221	−6.94252
$989$	56.1160	1.78439
$990$	0	0
$991$	30.4473	0.967191	0.483595	−	0.875292i	$-0.339331\pi$
$991$	30.4473	0.967191	0.483595	+	0.875292i	$0.339331\pi$
$992$	−57.8296	−1.83609
$993$	6.64984	0.211026
$994$	11.4031	0.361685
$995$	0	0
$996$	−42.1552	−1.33574
$997$	27.1612	0.860204	0.430102	−	0.902780i	$-0.358478\pi$
$997$	27.1612	0.860204	0.430102	+	0.902780i	$0.358478\pi$
$998$	−6.24264	−0.197607
$999$	−3.95005	−0.124974

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5775.2.a.bz.1.1		4
5.4	even	2		1155.2.a.u.1.4	✓	4
15.14	odd	2		3465.2.a.bl.1.1		4
35.34	odd	2		8085.2.a.bn.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1155.2.a.u.1.4	✓	4	5.4	even	2
3465.2.a.bl.1.1		4	15.14	odd	2
5775.2.a.bz.1.1		4	1.1	even	1	trivial
8085.2.a.bn.1.4		4	35.34	odd	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)

\(f(q)\)	\(=\)	\(q-2.58874 q^{2} -1.00000 q^{3} +4.70156 q^{4} +2.58874 q^{6} +1.00000 q^{7} -6.99364 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.58874 q^{2} -1.00000 q^{3} +4.70156 q^{4} +2.58874 q^{6} +1.00000 q^{7} -6.99364 q^{8} +1.00000 q^{9} -1.00000 q^{11} -4.70156 q^{12} -6.40490 q^{13} -2.58874 q^{14} +8.70156 q^{16} +3.70156 q^{17} -2.58874 q^{18} +7.24672 q^{19} -1.00000 q^{21} +2.58874 q^{22} +7.33388 q^{23} +6.99364 q^{24} +16.5806 q^{26} -1.00000 q^{27} +4.70156 q^{28} -8.10646 q^{29} +6.77258 q^{31} -8.53879 q^{32} +1.00000 q^{33} -9.58237 q^{34} +4.70156 q^{36} +3.95005 q^{37} -18.7598 q^{38} +6.40490 q^{39} -2.31773 q^{41} +2.58874 q^{42} +7.65161 q^{43} -4.70156 q^{44} -18.9855 q^{46} -1.45307 q^{47} -8.70156 q^{48} +1.00000 q^{49} -3.70156 q^{51} -30.1130 q^{52} -9.10823 q^{53} +2.58874 q^{54} -6.99364 q^{56} -7.24672 q^{57} +20.9855 q^{58} -2.92898 q^{59} -8.42419 q^{61} -17.5324 q^{62} +1.00000 q^{63} +4.70156 q^{64} -2.58874 q^{66} +8.72263 q^{67} +17.4031 q^{68} -7.33388 q^{69} -4.40490 q^{71} -6.99364 q^{72} -3.40667 q^{73} -10.2256 q^{74} +34.0709 q^{76} -1.00000 q^{77} -16.5806 q^{78} +4.40490 q^{79} +1.00000 q^{81} +6.00000 q^{82} +8.96620 q^{83} -4.70156 q^{84} -19.8080 q^{86} +8.10646 q^{87} +6.99364 q^{88} +0.206355 q^{89} -6.40490 q^{91} +34.4807 q^{92} -6.77258 q^{93} +3.76162 q^{94} +8.53879 q^{96} +13.9644 q^{97} -2.58874 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 6 q^{4} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 + 6 * q^4 + 4 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} + 6 q^{4} + 4 q^{7} + 4 q^{9} - 4 q^{11} - 6 q^{12} - 8 q^{13} + 22 q^{16} + 2 q^{17} + 10 q^{19} - 4 q^{21} + 2 q^{23} + 20 q^{26} - 4 q^{27} + 6 q^{28} - 2 q^{29} + 24 q^{31} + 4 q^{33} + 6 q^{36} - 8 q^{37} - 16 q^{38} + 8 q^{39} - 6 q^{43} - 6 q^{44} - 12 q^{46} - 4 q^{47} - 22 q^{48} + 4 q^{49} - 2 q^{51} - 12 q^{52} - 14 q^{53} - 10 q^{57} + 20 q^{58} - 2 q^{59} + 6 q^{61} - 8 q^{62} + 4 q^{63} + 6 q^{64} + 8 q^{67} + 44 q^{68} - 2 q^{69} - 4 q^{73} - 36 q^{74} + 56 q^{76} - 4 q^{77} - 20 q^{78} + 4 q^{81} + 24 q^{82} - 6 q^{83} - 6 q^{84} - 36 q^{86} + 2 q^{87} + 18 q^{89} - 8 q^{91} + 44 q^{92} - 24 q^{93} - 36 q^{94} + 6 q^{97} - 4 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 + 6 * q^4 + 4 * q^7 + 4 * q^9 - 4 * q^11 - 6 * q^12 - 8 * q^13 + 22 * q^16 + 2 * q^17 + 10 * q^19 - 4 * q^21 + 2 * q^23 + 20 * q^26 - 4 * q^27 + 6 * q^28 - 2 * q^29 + 24 * q^31 + 4 * q^33 + 6 * q^36 - 8 * q^37 - 16 * q^38 + 8 * q^39 - 6 * q^43 - 6 * q^44 - 12 * q^46 - 4 * q^47 - 22 * q^48 + 4 * q^49 - 2 * q^51 - 12 * q^52 - 14 * q^53 - 10 * q^57 + 20 * q^58 - 2 * q^59 + 6 * q^61 - 8 * q^62 + 4 * q^63 + 6 * q^64 + 8 * q^67 + 44 * q^68 - 2 * q^69 - 4 * q^73 - 36 * q^74 + 56 * q^76 - 4 * q^77 - 20 * q^78 + 4 * q^81 + 24 * q^82 - 6 * q^83 - 6 * q^84 - 36 * q^86 + 2 * q^87 + 18 * q^89 - 8 * q^91 + 44 * q^92 - 24 * q^93 - 36 * q^94 + 6 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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