LMFDB - Embedded newform 5775.2.a.bz.1.4

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			1.4
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)

$f(q)$	$=$	$q+2.58874 q^{2} -1.00000 q^{3} +4.70156 q^{4} -2.58874 q^{6} +1.00000 q^{7} +6.99364 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.58874 q^{2} -1.00000 q^{3} +4.70156 q^{4} -2.58874 q^{6} +1.00000 q^{7} +6.99364 q^{8} +1.00000 q^{9} -1.00000 q^{11} -4.70156 q^{12} +2.40490 q^{13} +2.58874 q^{14} +8.70156 q^{16} +3.70156 q^{17} +2.58874 q^{18} +4.15641 q^{19} -1.00000 q^{21} -2.58874 q^{22} +0.0692417 q^{23} -6.99364 q^{24} +6.22565 q^{26} -1.00000 q^{27} +4.70156 q^{28} +0.703336 q^{29} +5.22742 q^{31} +8.53879 q^{32} +1.00000 q^{33} +9.58237 q^{34} +4.70156 q^{36} -7.95005 q^{37} +10.7598 q^{38} -2.40490 q^{39} +2.31773 q^{41} -2.58874 q^{42} -4.24849 q^{43} -4.70156 q^{44} +0.179249 q^{46} -13.3532 q^{47} -8.70156 q^{48} +1.00000 q^{49} -3.70156 q^{51} +11.3068 q^{52} +8.51136 q^{53} -2.58874 q^{54} +6.99364 q^{56} -4.15641 q^{57} +1.82075 q^{58} -4.47414 q^{59} +5.02107 q^{61} +13.5324 q^{62} +1.00000 q^{63} +4.70156 q^{64} +2.58874 q^{66} -4.72263 q^{67} +17.4031 q^{68} -0.0692417 q^{69} +4.40490 q^{71} +6.99364 q^{72} +14.2129 q^{73} -20.5806 q^{74} +19.5416 q^{76} -1.00000 q^{77} -6.22565 q^{78} -4.40490 q^{79} +1.00000 q^{81} +6.00000 q^{82} -5.56308 q^{83} -4.70156 q^{84} -10.9982 q^{86} -0.703336 q^{87} -6.99364 q^{88} +15.1968 q^{89} +2.40490 q^{91} +0.325544 q^{92} -5.22742 q^{93} -34.5679 q^{94} -8.53879 q^{96} +8.24494 q^{97} +2.58874 q^{98} -1.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.131988\pi$
$2$	2.58874	1.83051	0.915257	+	0.402871i	$0.131988\pi$
$3$	−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
$13$	2.40490	0.666999	0.333499	−	0.942750i	$-0.391771\pi$
$13$	2.40490	0.666999	0.333499	+	0.942750i	$0.391771\pi$
$14$	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
$19$	4.15641	0.953545	0.476773	−	0.879027i	$-0.341807\pi$
$19$	4.15641	0.953545	0.476773	+	0.879027i	$0.341807\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	−2.58874	−0.551921
$23$	0.0692417	0.0144379	0.00721895	−	0.999974i	$-0.497702\pi$
$23$	0.0692417	0.0144379	0.00721895	+	0.999974i	$0.497702\pi$
$24$	−6.99364	−1.42757
$25$	0	0
$26$	6.22565	1.22095
$27$	−1.00000	−0.192450
$28$	4.70156	0.888512
$29$	0.703336	0.130606	0.0653031	−	0.997865i	$-0.479199\pi$
$29$	0.703336	0.130606	0.0653031	+	0.997865i	$0.479199\pi$
$30$	0	0
$31$	5.22742	0.938873	0.469436	−	0.882966i	$-0.344457\pi$
$31$	5.22742	0.938873	0.469436	+	0.882966i	$0.344457\pi$
$32$	8.53879	1.50946
$33$	1.00000	0.174078
$34$	9.58237	1.64336
$35$	0	0
$36$	4.70156	0.783594
$37$	−7.95005	−1.30698	−0.653490	−	0.756935i	$-0.726696\pi$
$37$	−7.95005	−1.30698	−0.653490	+	0.756935i	$0.726696\pi$
$38$	10.7598	1.74548
$39$	−2.40490	−0.385092
$40$	0	0
$41$	2.31773	0.361969	0.180984	−	0.983486i	$-0.442072\pi$
$41$	2.31773	0.361969	0.180984	+	0.983486i	$0.442072\pi$
$42$	−2.58874	−0.399451
$43$	−4.24849	−0.647889	−0.323944	−	0.946076i	$-0.605009\pi$
$43$	−4.24849	−0.647889	−0.323944	+	0.946076i	$0.605009\pi$
$44$	−4.70156	−0.708787
$45$	0	0
$46$	0.179249	0.0264288
$47$	−13.3532	−1.94776	−0.973881	−	0.227061i	$-0.927088\pi$
$47$	−13.3532	−1.94776	−0.973881	+	0.227061i	$0.927088\pi$
$48$	−8.70156	−1.25596
$49$	1.00000	0.142857
$50$	0	0
$51$	−3.70156	−0.518322
$52$	11.3068	1.56797
$53$	8.51136	1.16912	0.584562	−	0.811349i	$-0.301266\pi$
$53$	8.51136	1.16912	0.584562	+	0.811349i	$0.301266\pi$
$54$	−2.58874	−0.352283
$55$	0	0
$56$	6.99364	0.934564
$57$	−4.15641	−0.550530
$58$	1.82075	0.239076
$59$	−4.47414	−0.582483	−0.291242	−	0.956650i	$-0.594068\pi$
$59$	−4.47414	−0.582483	−0.291242	+	0.956650i	$0.594068\pi$
$60$	0	0
$61$	5.02107	0.642882	0.321441	−	0.946930i	$-0.395833\pi$
$61$	5.02107	0.642882	0.321441	+	0.946930i	$0.395833\pi$
$62$	13.5324	1.71862
$63$	1.00000	0.125988
$64$	4.70156	0.587695
$65$	0	0
$66$	2.58874	0.318652
$67$	−4.72263	−0.576961	−0.288481	−	0.957486i	$-0.593150\pi$
$67$	−4.72263	−0.576961	−0.288481	+	0.957486i	$0.593150\pi$
$68$	17.4031	2.11044
$69$	−0.0692417	−0.00833572
$70$	0	0
$71$	4.40490	0.522765	0.261383	−	0.965235i	$-0.415822\pi$
$71$	4.40490	0.522765	0.261383	+	0.965235i	$0.415822\pi$
$72$	6.99364	0.824208
$73$	14.2129	1.66350	0.831748	−	0.555153i	$-0.187340\pi$
$73$	14.2129	1.66350	0.831748	+	0.555153i	$0.187340\pi$
$74$	−20.5806	−2.39245
$75$	0	0
$76$	19.5416	2.24158
$77$	−1.00000	−0.113961
$78$	−6.22565	−0.704916
$79$	−4.40490	−0.495590	−0.247795	−	0.968813i	$-0.579706\pi$
$79$	−4.40490	−0.495590	−0.247795	+	0.968813i	$0.579706\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	6.00000	0.662589
$83$	−5.56308	−0.610627	−0.305314	−	0.952252i	$-0.598761\pi$
$83$	−5.56308	−0.610627	−0.305314	+	0.952252i	$0.598761\pi$
$84$	−4.70156	−0.512982
$85$	0	0
$86$	−10.9982	−1.18597
$87$	−0.703336	−0.0754055
$88$	−6.99364	−0.745524
$89$	15.1968	1.61085	0.805427	−	0.592695i	$-0.201936\pi$
$89$	15.1968	1.61085	0.805427	+	0.592695i	$0.201936\pi$
$90$	0	0
$91$	2.40490	0.252102
$92$	0.325544	0.0339403
$93$	−5.22742	−0.542058
$94$	−34.5679	−3.56540
$95$	0	0
$96$	−8.53879	−0.871487
$97$	8.24494	0.837147	0.418574	−	0.908183i	$-0.362530\pi$
$97$	8.24494	0.837147	0.418574	+	0.908183i	$0.362530\pi$
$98$	2.58874	0.261502
$99$	−1.00000	−0.100504
$100$	0	0
$101$	11.0354	1.09807	0.549034	−	0.835800i	$-0.314996\pi$
$101$	11.0354	1.09807	0.549034	+	0.835800i	$0.314996\pi$
$102$	−9.58237	−0.948796
$103$	2.84182	0.280013	0.140006	−	0.990151i	$-0.455288\pi$
$103$	2.84182	0.280013	0.140006	+	0.990151i	$0.455288\pi$
$104$	16.8190	1.64924
$105$	0	0
$106$	22.0337	2.14010
$107$	15.3567	1.48459	0.742295	−	0.670073i	$-0.233737\pi$
$107$	15.3567	1.48459	0.742295	+	0.670073i	$0.233737\pi$
$108$	−4.70156	−0.452408
$109$	12.8597	1.23174	0.615870	−	0.787848i	$-0.288805\pi$
$109$	12.8597	1.23174	0.615870	+	0.787848i	$0.288805\pi$
$110$	0	0
$111$	7.95005	0.754586
$112$	8.70156	0.822220
$113$	−16.5114	−1.55326	−0.776629	−	0.629958i	$-0.783072\pi$
$113$	−16.5114	−1.55326	−0.776629	+	0.629958i	$0.783072\pi$
$114$	−10.7598	−1.00775
$115$	0	0
$116$	3.30678	0.307026
$117$	2.40490	0.222333
$118$	−11.5824	−1.06624
$119$	3.70156	0.339322
$120$	0	0
$121$	1.00000	0.0909091
$122$	12.9982	1.17680
$123$	−2.31773	−0.208983
$124$	24.5771	2.20708
$125$	0	0
$126$	2.58874	0.230623
$127$	0.248490	0.0220500	0.0110250	−	0.999939i	$-0.496491\pi$
$127$	0.248490	0.0220500	0.0110250	+	0.999939i	$0.496491\pi$
$128$	−4.90647	−0.433675
$129$	4.24849	0.374059
$130$	0	0
$131$	12.6178	1.10242	0.551212	−	0.834365i	$-0.314166\pi$
$131$	12.6178	1.10242	0.551212	+	0.834365i	$0.314166\pi$
$132$	4.70156	0.409218
$133$	4.15641	0.360406
$134$	−12.2256	−1.05614
$135$	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	$0.631574\pi$
$138$	−0.179249	−0.0152587
$139$	22.4934	1.90787	0.953934	−	0.300016i	$-0.0969921\pi$
$139$	22.4934	1.90787	0.953934	+	0.300016i	$0.0969921\pi$
$140$	0	0
$141$	13.3532	1.12454
$142$	11.4031	0.956929
$143$	−2.40490	−0.201108
$144$	8.70156	0.725130
$145$	0	0
$146$	36.7935	3.04505
$147$	−1.00000	−0.0824786
$148$	−37.3777	−3.07243
$149$	−13.9001	−1.13874	−0.569370	−	0.822081i	$-0.692813\pi$
$149$	−13.9001	−1.13874	−0.569370	+	0.822081i	$0.692813\pi$
$150$	0	0
$151$	2.95183	0.240216	0.120108	−	0.992761i	$-0.461676\pi$
$151$	2.95183	0.240216	0.120108	+	0.992761i	$0.461676\pi$
$152$	29.0684	2.35776
$153$	3.70156	0.299254
$154$	−2.58874	−0.208606
$155$	0	0
$156$	−11.3068	−0.905267
$157$	−5.15463	−0.411385	−0.205692	−	0.978617i	$-0.565945\pi$
$157$	−5.15463	−0.411385	−0.205692	+	0.978617i	$0.565945\pi$
$158$	−11.4031	−0.907184
$159$	−8.51136	−0.674995
$160$	0	0
$161$	0.0692417	0.00545701
$162$	2.58874	0.203390
$163$	−2.32128	−0.181817	−0.0909083	−	0.995859i	$-0.528977\pi$
$163$	−2.32128	−0.181817	−0.0909083	+	0.995859i	$0.528977\pi$
$164$	10.8970	0.850910
$165$	0	0
$166$	−14.4014	−1.11776
$167$	−17.7581	−1.37416	−0.687081	−	0.726581i	$-0.741108\pi$
$167$	−17.7581	−1.37416	−0.687081	+	0.726581i	$0.741108\pi$
$168$	−6.99364	−0.539571
$169$	−7.21647	−0.555113
$170$	0	0
$171$	4.15641	0.317848
$172$	−19.9745	−1.52304
$173$	−20.2094	−1.53649	−0.768245	−	0.640156i	$-0.778870\pi$
$173$	−20.2094	−1.53649	−0.768245	+	0.640156i	$0.778870\pi$
$174$	−1.82075	−0.138031
$175$	0	0
$176$	−8.70156	−0.655905
$177$	4.47414	0.336297
$178$	39.3404	2.94869
$179$	−5.95005	−0.444728	−0.222364	−	0.974964i	$-0.571377\pi$
$179$	−5.95005	−0.444728	−0.222364	+	0.974964i	$0.571377\pi$
$180$	0	0
$181$	−5.54161	−0.411904	−0.205952	−	0.978562i	$-0.566029\pi$
$181$	−5.54161	−0.411904	−0.205952	+	0.978562i	$0.566029\pi$
$182$	6.22565	0.461476
$183$	−5.02107	−0.371168
$184$	0.484251	0.0356995
$185$	0	0
$186$	−13.5324	−0.992246
$187$	−3.70156	−0.270685
$188$	−62.7808	−4.57876
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	25.8966	1.87381	0.936905	−	0.349585i	$-0.113677\pi$
$191$	25.8966	1.87381	0.936905	+	0.349585i	$0.113677\pi$
$192$	−4.70156	−0.339306
$193$	1.31596	0.0947248	0.0473624	−	0.998878i	$-0.484918\pi$
$193$	1.31596	0.0947248	0.0473624	+	0.998878i	$0.484918\pi$
$194$	21.3440	1.53241
$195$	0	0
$196$	4.70156	0.335826
$197$	4.44212	0.316488	0.158244	−	0.987400i	$-0.449417\pi$
$197$	4.44212	0.316488	0.158244	+	0.987400i	$0.449417\pi$
$198$	−2.58874	−0.183974
$199$	−25.7453	−1.82504	−0.912520	−	0.409033i	$-0.865866\pi$
$199$	−25.7453	−1.82504	−0.912520	+	0.409033i	$0.865866\pi$
$200$	0	0
$201$	4.72263	0.333109
$202$	28.5679	2.01003
$203$	0.703336	0.0493645
$204$	−17.4031	−1.21846
$205$	0	0
$206$	7.35672	0.512567
$207$	0.0692417	0.00481263
$208$	20.9264	1.45098
$209$	−4.15641	−0.287505
$210$	0	0
$211$	−24.2129	−1.66689	−0.833443	−	0.552605i	$-0.813634\pi$
$211$	−24.2129	−1.66689	−0.833443	+	0.552605i	$0.813634\pi$
$212$	40.0167	2.74836
$213$	−4.40490	−0.301819
$214$	39.7545	2.71756
$215$	0	0
$216$	−6.99364	−0.475857
$217$	5.22742	0.354861
$218$	33.2905	2.25472
$219$	−14.2129	−0.960420
$220$	0	0
$221$	8.90188	0.598805
$222$	20.5806	1.38128
$223$	−8.38697	−0.561633	−0.280817	−	0.959761i	$-0.590605\pi$
$223$	−8.38697	−0.561633	−0.280817	+	0.959761i	$0.590605\pi$
$224$	8.53879	0.570522
$225$	0	0
$226$	−42.7436	−2.84326
$227$	−2.79542	−0.185538	−0.0927692	−	0.995688i	$-0.529572\pi$
$227$	−2.79542	−0.185538	−0.0927692	+	0.995688i	$0.529572\pi$
$228$	−19.5416	−1.29417
$229$	9.95360	0.657752	0.328876	−	0.944373i	$-0.393330\pi$
$229$	9.95360	0.657752	0.328876	+	0.944373i	$0.393330\pi$
$230$	0	0
$231$	1.00000	0.0657952
$232$	4.91887	0.322940
$233$	2.77612	0.181870	0.0909350	−	0.995857i	$-0.471014\pi$
$233$	2.77612	0.181870	0.0909350	+	0.995857i	$0.471014\pi$
$234$	6.22565	0.406983
$235$	0	0
$236$	−21.0354	−1.36929
$237$	4.40490	0.286129
$238$	9.58237	0.621133
$239$	−14.6034	−0.944618	−0.472309	−	0.881433i	$-0.656579\pi$
$239$	−14.6034	−0.944618	−0.472309	+	0.881433i	$0.656579\pi$
$240$	0	0
$241$	10.5385	0.678842	0.339421	−	0.940635i	$-0.389769\pi$
$241$	10.5385	0.678842	0.339421	+	0.940635i	$0.389769\pi$
$242$	2.58874	0.166410
$243$	−1.00000	−0.0641500
$244$	23.6069	1.51127
$245$	0	0
$246$	−6.00000	−0.382546
$247$	9.99573	0.636013
$248$	36.5587	2.32148
$249$	5.56308	0.352546
$250$	0	0
$251$	−19.3904	−1.22391	−0.611955	−	0.790892i	$-0.709617\pi$
$251$	−19.3904	−1.22391	−0.611955	+	0.790892i	$0.709617\pi$
$252$	4.70156	0.296171
$253$	−0.0692417	−0.00435319
$254$	0.643276	0.0403627
$255$	0	0
$256$	−22.1047	−1.38154
$257$	−2.71771	−0.169526	−0.0847631	−	0.996401i	$-0.527013\pi$
$257$	−2.71771	−0.169526	−0.0847631	+	0.996401i	$0.527013\pi$
$258$	10.9982	0.684720
$259$	−7.95005	−0.493992
$260$	0	0
$261$	0.703336	0.0435354
$262$	32.6642	2.01800
$263$	14.4548	0.891324	0.445662	−	0.895201i	$-0.352968\pi$
$263$	14.4548	0.891324	0.445662	+	0.895201i	$0.352968\pi$
$264$	6.99364	0.430428
$265$	0	0
$266$	10.7598	0.659729
$267$	−15.1968	−0.930027
$268$	−22.2037	−1.35631
$269$	−17.2677	−1.05283	−0.526414	−	0.850228i	$-0.676464\pi$
$269$	−17.2677	−1.05283	−0.526414	+	0.850228i	$0.676464\pi$
$270$	0	0
$271$	15.7051	0.954017	0.477009	−	0.878899i	$-0.341721\pi$
$271$	15.7051	0.954017	0.477009	+	0.878899i	$0.341721\pi$
$272$	32.2094	1.95298
$273$	−2.40490	−0.145551
$274$	−24.3422	−1.47057
$275$	0	0
$276$	−0.325544	−0.0195955
$277$	−15.9837	−0.960369	−0.480184	−	0.877168i	$-0.659430\pi$
$277$	−15.9837	−0.960369	−0.480184	+	0.877168i	$0.659430\pi$
$278$	58.2296	3.49238
$279$	5.22742	0.312958
$280$	0	0
$281$	−5.50302	−0.328283	−0.164141	−	0.986437i	$-0.552485\pi$
$281$	−5.50302	−0.328283	−0.164141	+	0.986437i	$0.552485\pi$
$282$	34.5679	2.05849
$283$	8.09208	0.481024	0.240512	−	0.970646i	$-0.422685\pi$
$283$	8.09208	0.481024	0.240512	+	0.970646i	$0.422685\pi$
$284$	20.7099	1.22891
$285$	0	0
$286$	−6.22565	−0.368130
$287$	2.31773	0.136811
$288$	8.53879	0.503153
$289$	−3.29844	−0.194026
$290$	0	0
$291$	−8.24494	−0.483327
$292$	66.8229	3.91052
$293$	−6.51490	−0.380605	−0.190302	−	0.981726i	$-0.560947\pi$
$293$	−6.51490	−0.380605	−0.190302	+	0.981726i	$0.560947\pi$
$294$	−2.58874	−0.150978
$295$	0	0
$296$	−55.5998	−3.23167
$297$	1.00000	0.0580259
$298$	−35.9837	−2.08448
$299$	0.166519	0.00963006
$300$	0	0
$301$	−4.24849	−0.244879
$302$	7.64150	0.439719
$303$	−11.0354	−0.633970
$304$	36.1672	2.07433
$305$	0	0
$306$	9.58237	0.547788
$307$	22.5679	1.28802	0.644008	−	0.765019i	$-0.277270\pi$
$307$	22.5679	1.28802	0.644008	+	0.765019i	$0.277270\pi$
$308$	−4.70156	−0.267896
$309$	−2.84182	−0.161665
$310$	0	0
$311$	0.0513177	0.00290996	0.00145498	−	0.999999i	$-0.499537\pi$
$311$	0.0513177	0.00290996	0.00145498	+	0.999999i	$0.499537\pi$
$312$	−16.8190	−0.952187
$313$	3.44224	0.194567	0.0972835	−	0.995257i	$-0.468985\pi$
$313$	3.44224	0.194567	0.0972835	+	0.995257i	$0.468985\pi$
$314$	−13.3440	−0.753045
$315$	0	0
$316$	−20.7099	−1.16502
$317$	−31.6582	−1.77810	−0.889050	−	0.457810i	$-0.848634\pi$
$317$	−31.6582	−1.77810	−0.889050	+	0.457810i	$0.848634\pi$
$318$	−22.0337	−1.23559
$319$	−0.703336	−0.0393792
$320$	0	0
$321$	−15.3567	−0.857129
$322$	0.179249	0.00998914
$323$	15.3852	0.856055
$324$	4.70156	0.261198
$325$	0	0
$326$	−6.00918	−0.332818
$327$	−12.8597	−0.711145
$328$	16.2094	0.895013
$329$	−13.3532	−0.736184
$330$	0	0
$331$	−3.55953	−0.195650	−0.0978248	−	0.995204i	$-0.531188\pi$
$331$	−3.55953	−0.195650	−0.0978248	+	0.995204i	$0.531188\pi$
$332$	−26.1552	−1.43545
$333$	−7.95005	−0.435660
$334$	−45.9710	−2.51542
$335$	0	0
$336$	−8.70156	−0.474709
$337$	−4.23221	−0.230543	−0.115272	−	0.993334i	$-0.536774\pi$
$337$	−4.23221	−0.230543	−0.115272	+	0.993334i	$0.536774\pi$
$338$	−18.6815	−1.01614
$339$	16.5114	0.896774
$340$	0	0
$341$	−5.22742	−0.283081
$342$	10.7598	0.581826
$343$	1.00000	0.0539949
$344$	−29.7124	−1.60198
$345$	0	0
$346$	−52.3168	−2.81257
$347$	−15.6160	−0.838313	−0.419157	−	0.907914i	$-0.637674\pi$
$347$	−15.6160	−0.838313	−0.419157	+	0.907914i	$0.637674\pi$
$348$	−3.30678	−0.177262
$349$	2.24357	0.120096	0.0600479	−	0.998195i	$-0.480875\pi$
$349$	2.24357	0.120096	0.0600479	+	0.998195i	$0.480875\pi$
$350$	0	0
$351$	−2.40490	−0.128364
$352$	−8.53879	−0.455119
$353$	−11.9965	−0.638507	−0.319253	−	0.947669i	$-0.603432\pi$
$353$	−11.9965	−0.638507	−0.319253	+	0.947669i	$0.603432\pi$
$354$	11.5824	0.615596
$355$	0	0
$356$	71.4486	3.78677
$357$	−3.70156	−0.195907
$358$	−15.4031	−0.814080
$359$	24.6743	1.30226	0.651131	−	0.758966i	$-0.274295\pi$
$359$	24.6743	1.30226	0.651131	+	0.758966i	$0.274295\pi$
$360$	0	0
$361$	−1.72428	−0.0907514
$362$	−14.3458	−0.753997
$363$	−1.00000	−0.0524864
$364$	11.3068	0.592636
$365$	0	0
$366$	−12.9982	−0.679428
$367$	−22.2808	−1.16305	−0.581524	−	0.813529i	$-0.697543\pi$
$367$	−22.2808	−1.16305	−0.581524	+	0.813529i	$0.697543\pi$
$368$	0.602511	0.0314081
$369$	2.31773	0.120656
$370$	0	0
$371$	8.51136	0.441888
$372$	−24.5771	−1.27426
$373$	−13.6389	−0.706195	−0.353097	−	0.935587i	$-0.614872\pi$
$373$	−13.6389	−0.706195	−0.353097	+	0.935587i	$0.614872\pi$
$374$	−9.58237	−0.495493
$375$	0	0
$376$	−93.3872	−4.81608
$377$	1.69145	0.0871141
$378$	−2.58874	−0.133150
$379$	14.1985	0.729330	0.364665	−	0.931139i	$-0.381183\pi$
$379$	14.1985	0.729330	0.364665	+	0.931139i	$0.381183\pi$
$380$	0	0
$381$	−0.248490	−0.0127305
$382$	67.0394	3.43003
$383$	10.1244	0.517332	0.258666	−	0.965967i	$-0.416717\pi$
$383$	10.1244	0.517332	0.258666	+	0.965967i	$0.416717\pi$
$384$	4.90647	0.250382
$385$	0	0
$386$	3.40667	0.173395
$387$	−4.24849	−0.215963
$388$	38.7641	1.96795
$389$	−20.9342	−1.06141	−0.530703	−	0.847558i	$-0.678072\pi$
$389$	−20.9342	−1.06141	−0.530703	+	0.847558i	$0.678072\pi$
$390$	0	0
$391$	0.256303	0.0129618
$392$	6.99364	0.353232
$393$	−12.6178	−0.636485
$394$	11.4995	0.579335
$395$	0	0
$396$	−4.70156	−0.236262
$397$	24.5679	1.23303	0.616513	−	0.787345i	$-0.288545\pi$
$397$	24.5679	1.23303	0.616513	+	0.787345i	$0.288545\pi$
$398$	−66.6479	−3.34076
$399$	−4.15641	−0.208081
$400$	0	0
$401$	−38.4180	−1.91850	−0.959252	−	0.282551i	$-0.908819\pi$
$401$	−38.4180	−1.91850	−0.959252	+	0.282551i	$0.908819\pi$
$402$	12.2256	0.609760
$403$	12.5714	0.626227
$404$	51.8838	2.58132
$405$	0	0
$406$	1.82075	0.0903624
$407$	7.95005	0.394069
$408$	−25.8874	−1.28162
$409$	−1.17748	−0.0582224	−0.0291112	−	0.999576i	$-0.509268\pi$
$409$	−1.17748	−0.0582224	−0.0291112	+	0.999576i	$0.509268\pi$
$410$	0	0
$411$	9.40312	0.463822
$412$	13.3610	0.658249
$413$	−4.47414	−0.220158
$414$	0.179249	0.00880959
$415$	0	0
$416$	20.5349	1.00681
$417$	−22.4934	−1.10151
$418$	−10.7598	−0.526281
$419$	−37.2585	−1.82020	−0.910098	−	0.414393i	$-0.863994\pi$
$419$	−37.2585	−1.82020	−0.910098	+	0.414393i	$0.863994\pi$
$420$	0	0
$421$	3.10469	0.151313	0.0756566	−	0.997134i	$-0.475895\pi$
$421$	3.10469	0.151313	0.0756566	+	0.997134i	$0.475895\pi$
$422$	−62.6809	−3.05126
$423$	−13.3532	−0.649254
$424$	59.5253	2.89081
$425$	0	0
$426$	−11.4031	−0.552483
$427$	5.02107	0.242986
$428$	72.2006	3.48995
$429$	2.40490	0.116110
$430$	0	0
$431$	−13.5452	−0.652447	−0.326224	−	0.945293i	$-0.605776\pi$
$431$	−13.5452	−0.652447	−0.326224	+	0.945293i	$0.605776\pi$
$432$	−8.70156	−0.418654
$433$	33.7904	1.62386	0.811931	−	0.583754i	$-0.198417\pi$
$433$	33.7904	1.62386	0.811931	+	0.583754i	$0.198417\pi$
$434$	13.5324	0.649577
$435$	0	0
$436$	60.4609	2.89555
$437$	0.287797	0.0137672
$438$	−36.7935	−1.75806
$439$	11.6052	0.553887	0.276943	−	0.960886i	$-0.410679\pi$
$439$	11.6052	0.553887	0.276943	+	0.960886i	$0.410679\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	23.0446	1.09612
$443$	24.7226	1.17461	0.587304	−	0.809367i	$-0.300189\pi$
$443$	24.7226	1.17461	0.587304	+	0.809367i	$0.300189\pi$
$444$	37.3777	1.77387
$445$	0	0
$446$	−21.7117	−1.02808
$447$	13.9001	0.657452
$448$	4.70156	0.222128
$449$	31.5275	1.48788	0.743938	−	0.668249i	$-0.232956\pi$
$449$	31.5275	1.48788	0.743938	+	0.668249i	$0.232956\pi$
$450$	0	0
$451$	−2.31773	−0.109138
$452$	−77.6292	−3.65137
$453$	−2.95183	−0.138689
$454$	−7.23660	−0.339631
$455$	0	0
$456$	−29.0684	−1.36125
$457$	13.3874	0.626235	0.313118	−	0.949714i	$-0.398627\pi$
$457$	13.3874	0.626235	0.313118	+	0.949714i	$0.398627\pi$
$458$	25.7673	1.20402
$459$	−3.70156	−0.172774
$460$	0	0
$461$	3.87070	0.180276	0.0901382	−	0.995929i	$-0.471269\pi$
$461$	3.87070	0.180276	0.0901382	+	0.995929i	$0.471269\pi$
$462$	2.58874	0.120439
$463$	−29.2484	−1.35929	−0.679643	−	0.733543i	$-0.737865\pi$
$463$	−29.2484	−1.35929	−0.679643	+	0.733543i	$0.737865\pi$
$464$	6.12012	0.284119
$465$	0	0
$466$	7.18666	0.332915
$467$	−14.2551	−0.659645	−0.329823	−	0.944043i	$-0.606989\pi$
$467$	−14.2551	−0.659645	−0.329823	+	0.944043i	$0.606989\pi$
$468$	11.3068	0.522656
$469$	−4.72263	−0.218071
$470$	0	0
$471$	5.15463	0.237513
$472$	−31.2905	−1.44026
$473$	4.24849	0.195346
$474$	11.4031	0.523763
$475$	0	0
$476$	17.4031	0.797671
$477$	8.51136	0.389708
$478$	−37.8045	−1.72914
$479$	−32.2339	−1.47280	−0.736401	−	0.676545i	$-0.763476\pi$
$479$	−32.2339	−1.47280	−0.736401	+	0.676545i	$0.763476\pi$
$480$	0	0
$481$	−19.1191	−0.871754
$482$	27.2813	1.24263
$483$	−0.0692417	−0.00315061
$484$	4.70156	0.213707
$485$	0	0
$486$	−2.58874	−0.117428
$487$	−5.27382	−0.238980	−0.119490	−	0.992835i	$-0.538126\pi$
$487$	−5.27382	−0.238980	−0.119490	+	0.992835i	$0.538126\pi$
$488$	35.1155	1.58960
$489$	2.32128	0.104972
$490$	0	0
$491$	35.6647	1.60953	0.804764	−	0.593595i	$-0.202292\pi$
$491$	35.6647	1.60953	0.804764	+	0.593595i	$0.202292\pi$
$492$	−10.8970	−0.491273
$493$	2.60344	0.117253
$494$	25.8763	1.16423
$495$	0	0
$496$	45.4867	2.04242
$497$	4.40490	0.197587
$498$	14.4014	0.645340
$499$	−39.0083	−1.74625	−0.873127	−	0.487494i	$-0.837911\pi$
$499$	−39.0083	−1.74625	−0.873127	+	0.487494i	$0.837911\pi$
$500$	0	0
$501$	17.7581	0.793372
$502$	−50.1966	−2.24039
$503$	32.6821	1.45722	0.728612	−	0.684926i	$-0.240166\pi$
$503$	32.6821	1.45722	0.728612	+	0.684926i	$0.240166\pi$
$504$	6.99364	0.311521
$505$	0	0
$506$	−0.179249	−0.00796857
$507$	7.21647	0.320495
$508$	1.16829	0.0518346
$509$	3.25808	0.144412	0.0722058	−	0.997390i	$-0.476996\pi$
$509$	3.25808	0.144412	0.0722058	+	0.997390i	$0.476996\pi$
$510$	0	0
$511$	14.2129	0.628743
$512$	−47.4103	−2.09526
$513$	−4.15641	−0.183510
$514$	−7.03544	−0.310320
$515$	0	0
$516$	19.9745	0.879330
$517$	13.3532	0.587272
$518$	−20.5806	−0.904260
$519$	20.2094	0.887093
$520$	0	0
$521$	7.75506	0.339755	0.169878	−	0.985465i	$-0.445663\pi$
$521$	7.75506	0.339755	0.169878	+	0.985465i	$0.445663\pi$
$522$	1.82075	0.0796921
$523$	−27.1647	−1.18783	−0.593916	−	0.804527i	$-0.702419\pi$
$523$	−27.1647	−1.18783	−0.593916	+	0.804527i	$0.702419\pi$
$524$	59.3235	2.59156
$525$	0	0
$526$	37.4198	1.63158
$527$	19.3496	0.842883
$528$	8.70156	0.378687
$529$	−22.9952	−0.999792
$530$	0	0
$531$	−4.47414	−0.194161
$532$	19.5416	0.847236
$533$	5.57391	0.241433
$534$	−39.3404	−1.70243
$535$	0	0
$536$	−33.0284	−1.42661
$537$	5.95005	0.256764
$538$	−44.7014	−1.92722
$539$	−1.00000	−0.0430730
$540$	0	0
$541$	36.2087	1.55673	0.778366	−	0.627811i	$-0.216049\pi$
$541$	36.2087	1.55673	0.778366	+	0.627811i	$0.216049\pi$
$542$	40.6564	1.74634
$543$	5.54161	0.237813
$544$	31.6069	1.35513
$545$	0	0
$546$	−6.22565	−0.266433
$547$	11.1160	0.475288	0.237644	−	0.971352i	$-0.423625\pi$
$547$	11.1160	0.475288	0.237644	+	0.971352i	$0.423625\pi$
$548$	−44.2094	−1.88853
$549$	5.02107	0.214294
$550$	0	0
$551$	2.92335	0.124539
$552$	−0.484251	−0.0206111
$553$	−4.40490	−0.187315
$554$	−41.3777	−1.75797
$555$	0	0
$556$	105.754	4.48498
$557$	−34.4772	−1.46084	−0.730422	−	0.682996i	$-0.760677\pi$
$557$	−34.4772	−1.46084	−0.730422	+	0.682996i	$0.760677\pi$
$558$	13.5324	0.572873
$559$	−10.2172	−0.432141
$560$	0	0
$561$	3.70156	0.156280
$562$	−14.2459	−0.600926
$563$	−16.2129	−0.683293	−0.341647	−	0.939829i	$-0.610985\pi$
$563$	−16.2129	−0.683293	−0.341647	+	0.939829i	$0.610985\pi$
$564$	62.7808	2.64355
$565$	0	0
$566$	20.9483	0.880522
$567$	1.00000	0.0419961
$568$	30.8062	1.29260
$569$	3.64807	0.152935	0.0764675	−	0.997072i	$-0.475636\pi$
$569$	3.64807	0.152935	0.0764675	+	0.997072i	$0.475636\pi$
$570$	0	0
$571$	−36.7563	−1.53820	−0.769102	−	0.639126i	$-0.779296\pi$
$571$	−36.7563	−1.53820	−0.769102	+	0.639126i	$0.779296\pi$
$572$	−11.3068	−0.472760
$573$	−25.8966	−1.08184
$574$	6.00000	0.250435
$575$	0	0
$576$	4.70156	0.195898
$577$	2.17433	0.0905186	0.0452593	−	0.998975i	$-0.485589\pi$
$577$	2.17433	0.0905186	0.0452593	+	0.998975i	$0.485589\pi$
$578$	−8.53879	−0.355167
$579$	−1.31596	−0.0546894
$580$	0	0
$581$	−5.56308	−0.230795
$582$	−21.3440	−0.884737
$583$	−8.51136	−0.352504
$584$	99.4000	4.11320
$585$	0	0
$586$	−16.8654	−0.696702
$587$	−13.9501	−0.575780	−0.287890	−	0.957663i	$-0.592954\pi$
$587$	−13.9501	−0.575780	−0.287890	+	0.957663i	$0.592954\pi$
$588$	−4.70156	−0.193889
$589$	21.7273	0.895258
$590$	0	0
$591$	−4.44212	−0.182724
$592$	−69.1779	−2.84319
$593$	10.9483	0.449592	0.224796	−	0.974406i	$-0.427828\pi$
$593$	10.9483	0.449592	0.224796	+	0.974406i	$0.427828\pi$
$594$	2.58874	0.106217
$595$	0	0
$596$	−65.3522	−2.67693
$597$	25.7453	1.05369
$598$	0.431075	0.0176280
$599$	−41.1998	−1.68338	−0.841689	−	0.539963i	$-0.818438\pi$
$599$	−41.1998	−1.68338	−0.841689	+	0.539963i	$0.818438\pi$
$600$	0	0
$601$	21.5110	0.877450	0.438725	−	0.898621i	$-0.355430\pi$
$601$	21.5110	0.877450	0.438725	+	0.898621i	$0.355430\pi$
$602$	−10.9982	−0.448254
$603$	−4.72263	−0.192320
$604$	13.8782	0.564696
$605$	0	0
$606$	−28.5679	−1.16049
$607$	13.8580	0.562478	0.281239	−	0.959638i	$-0.409255\pi$
$607$	13.8580	0.562478	0.281239	+	0.959638i	$0.409255\pi$
$608$	35.4907	1.43934
$609$	−0.703336	−0.0285006
$610$	0	0
$611$	−32.1130	−1.29915
$612$	17.4031	0.703480
$613$	32.0223	1.29337	0.646684	−	0.762758i	$-0.276155\pi$
$613$	32.0223	1.29337	0.646684	+	0.762758i	$0.276155\pi$
$614$	58.4223	2.35773
$615$	0	0
$616$	−6.99364	−0.281782
$617$	3.39958	0.136862	0.0684309	−	0.997656i	$-0.478201\pi$
$617$	3.39958	0.136862	0.0684309	+	0.997656i	$0.478201\pi$
$618$	−7.35672	−0.295931
$619$	−1.91638	−0.0770259	−0.0385129	−	0.999258i	$-0.512262\pi$
$619$	−1.91638	−0.0770259	−0.0385129	+	0.999258i	$0.512262\pi$
$620$	0	0
$621$	−0.0692417	−0.00277857
$622$	0.132848	0.00532672
$623$	15.1968	0.608846
$624$	−20.9264	−0.837725
$625$	0	0
$626$	8.91106	0.356158
$627$	4.15641	0.165991
$628$	−24.2348	−0.967075
$629$	−29.4276	−1.17336
$630$	0	0
$631$	−37.4019	−1.48895	−0.744473	−	0.667653i	$-0.767299\pi$
$631$	−37.4019	−1.48895	−0.744473	+	0.667653i	$0.767299\pi$
$632$	−30.8062	−1.22541
$633$	24.2129	0.962377
$634$	−81.9547	−3.25484
$635$	0	0
$636$	−40.0167	−1.58676
$637$	2.40490	0.0952855
$638$	−1.82075	−0.0720842
$639$	4.40490	0.174255
$640$	0	0
$641$	8.52928	0.336886	0.168443	−	0.985711i	$-0.446126\pi$
$641$	8.52928	0.336886	0.168443	+	0.985711i	$0.446126\pi$
$642$	−39.7545	−1.56899
$643$	−36.3615	−1.43396	−0.716979	−	0.697095i	$-0.754476\pi$
$643$	−36.3615	−1.43396	−0.716979	+	0.697095i	$0.754476\pi$
$644$	0.325544	0.0128282
$645$	0	0
$646$	39.8282	1.56702
$647$	−42.2551	−1.66122	−0.830609	−	0.556856i	$-0.812007\pi$
$647$	−42.2551	−1.66122	−0.830609	+	0.556856i	$0.812007\pi$
$648$	6.99364	0.274736
$649$	4.47414	0.175625
$650$	0	0
$651$	−5.22742	−0.204879
$652$	−10.9136	−0.427411
$653$	21.8110	0.853532	0.426766	−	0.904362i	$-0.359653\pi$
$653$	21.8110	0.853532	0.426766	+	0.904362i	$0.359653\pi$
$654$	−33.2905	−1.30176
$655$	0	0
$656$	20.1679	0.787424
$657$	14.2129	0.554499
$658$	−34.5679	−1.34760
$659$	−46.8164	−1.82371	−0.911853	−	0.410516i	$-0.865348\pi$
$659$	−46.8164	−1.82371	−0.911853	+	0.410516i	$0.865348\pi$
$660$	0	0
$661$	−17.1113	−0.665551	−0.332775	−	0.943006i	$-0.607985\pi$
$661$	−17.1113	−0.665551	−0.332775	+	0.943006i	$0.607985\pi$
$662$	−9.21469	−0.358139
$663$	−8.90188	−0.345720
$664$	−38.9061	−1.50985
$665$	0	0
$666$	−20.5806	−0.797482
$667$	0.0487002	0.00188568
$668$	−83.4907	−3.23035
$669$	8.38697	0.324259
$670$	0	0
$671$	−5.02107	−0.193836
$672$	−8.53879	−0.329391
$673$	48.3707	1.86455	0.932277	−	0.361746i	$-0.117819\pi$
$673$	48.3707	1.86455	0.932277	+	0.361746i	$0.117819\pi$
$674$	−10.9561	−0.422013
$675$	0	0
$676$	−33.9287	−1.30495
$677$	−26.6463	−1.02410	−0.512050	−	0.858956i	$-0.671114\pi$
$677$	−26.6463	−1.02410	−0.512050	+	0.858956i	$0.671114\pi$
$678$	42.7436	1.64156
$679$	8.24494	0.316412
$680$	0	0
$681$	2.79542	0.107121
$682$	−13.5324	−0.518183
$683$	−5.94514	−0.227484	−0.113742	−	0.993510i	$-0.536284\pi$
$683$	−5.94514	−0.227484	−0.113742	+	0.993510i	$0.536284\pi$
$684$	19.5416	0.747192
$685$	0	0
$686$	2.58874	0.0988385
$687$	−9.95360	−0.379754
$688$	−36.9685	−1.40941
$689$	20.4689	0.779805
$690$	0	0
$691$	7.72867	0.294012	0.147006	−	0.989136i	$-0.453036\pi$
$691$	7.72867	0.294012	0.147006	+	0.989136i	$0.453036\pi$
$692$	−95.0156	−3.61195
$693$	−1.00000	−0.0379869
$694$	−40.4258	−1.53454
$695$	0	0
$696$	−4.91887	−0.186449
$697$	8.57923	0.324961
$698$	5.80802	0.219837
$699$	−2.77612	−0.105003
$700$	0	0
$701$	−4.70334	−0.177643	−0.0888213	−	0.996048i	$-0.528310\pi$
$701$	−4.70334	−0.177643	−0.0888213	+	0.996048i	$0.528310\pi$
$702$	−6.22565	−0.234972
$703$	−33.0437	−1.24627
$704$	−4.70156	−0.177197
$705$	0	0
$706$	−31.0557	−1.16880
$707$	11.0354	0.415031
$708$	21.0354	0.790560
$709$	21.8822	0.821803	0.410901	−	0.911680i	$-0.365214\pi$
$709$	21.8822	0.821803	0.410901	+	0.911680i	$0.365214\pi$
$710$	0	0
$711$	−4.40490	−0.165197
$712$	106.281	3.98304
$713$	0.361956	0.0135553
$714$	−9.58237	−0.358611
$715$	0	0
$716$	−27.9745	−1.04546
$717$	14.6034	0.545375
$718$	63.8754	2.38381
$719$	−19.2875	−0.719302	−0.359651	−	0.933087i	$-0.617104\pi$
$719$	−19.2875	−0.719302	−0.359651	+	0.933087i	$0.617104\pi$
$720$	0	0
$721$	2.84182	0.105835
$722$	−4.46370	−0.166122
$723$	−10.5385	−0.391930
$724$	−26.0542	−0.968297
$725$	0	0
$726$	−2.58874	−0.0960771
$727$	−35.4877	−1.31616	−0.658082	−	0.752946i	$-0.728632\pi$
$727$	−35.4877	−1.31616	−0.658082	+	0.752946i	$0.728632\pi$
$728$	16.8190	0.623353
$729$	1.00000	0.0370370
$730$	0	0
$731$	−15.7261	−0.581649
$732$	−23.6069	−0.872535
$733$	−49.4163	−1.82523	−0.912615	−	0.408819i	$-0.865941\pi$
$733$	−49.4163	−1.82523	−0.912615	+	0.408819i	$0.865941\pi$
$734$	−57.6791	−2.12898
$735$	0	0
$736$	0.591240	0.0217934
$737$	4.72263	0.173960
$738$	6.00000	0.220863
$739$	18.1095	0.666168	0.333084	−	0.942897i	$-0.391911\pi$
$739$	18.1095	0.666168	0.333084	+	0.942897i	$0.391911\pi$
$740$	0	0
$741$	−9.99573	−0.367202
$742$	22.0337	0.808882
$743$	45.0972	1.65445	0.827227	−	0.561868i	$-0.189917\pi$
$743$	45.0972	1.65445	0.827227	+	0.561868i	$0.189917\pi$
$744$	−36.5587	−1.34031
$745$	0	0
$746$	−35.3075	−1.29270
$747$	−5.56308	−0.203542
$748$	−17.4031	−0.636321
$749$	15.3567	0.561122
$750$	0	0
$751$	47.9495	1.74970	0.874851	−	0.484391i	$-0.160959\pi$
$751$	47.9495	1.74970	0.874851	+	0.484391i	$0.160959\pi$
$752$	−116.193	−4.23714
$753$	19.3904	0.706625
$754$	4.37872	0.159464
$755$	0	0
$756$	−4.70156	−0.170994
$757$	−13.8158	−0.502145	−0.251073	−	0.967968i	$-0.580783\pi$
$757$	−13.8158	−0.502145	−0.251073	+	0.967968i	$0.580783\pi$
$758$	36.7563	1.33505
$759$	0.0692417	0.00251331
$760$	0	0
$761$	12.2221	0.443051	0.221525	−	0.975155i	$-0.428896\pi$
$761$	12.2221	0.443051	0.221525	+	0.975155i	$0.428896\pi$
$762$	−0.643276	−0.0233034
$763$	12.8597	0.465554
$764$	121.754	4.40492
$765$	0	0
$766$	26.2094	0.946983
$767$	−10.7598	−0.388516
$768$	22.1047	0.797634
$769$	29.3013	1.05663	0.528316	−	0.849048i	$-0.322823\pi$
$769$	29.3013	1.05663	0.528316	+	0.849048i	$0.322823\pi$
$770$	0	0
$771$	2.71771	0.0978760
$772$	6.18706	0.222677
$773$	−43.2356	−1.55508	−0.777539	−	0.628835i	$-0.783532\pi$
$773$	−43.2356	−1.55508	−0.777539	+	0.628835i	$0.783532\pi$
$774$	−10.9982	−0.395323
$775$	0	0
$776$	57.6621	2.06995
$777$	7.95005	0.285207
$778$	−54.1931	−1.94292
$779$	9.63344	0.345154
$780$	0	0
$781$	−4.40490	−0.157620
$782$	0.663500	0.0237267
$783$	−0.703336	−0.0251352
$784$	8.70156	0.310770
$785$	0	0
$786$	−32.6642	−1.16509
$787$	20.1489	0.718230	0.359115	−	0.933293i	$-0.383079\pi$
$787$	20.1489	0.718230	0.359115	+	0.933293i	$0.383079\pi$
$788$	20.8849	0.743993
$789$	−14.4548	−0.514606
$790$	0	0
$791$	−16.5114	−0.587076
$792$	−6.99364	−0.248508
$793$	12.0752	0.428801
$794$	63.5998	2.25707
$795$	0	0
$796$	−121.043	−4.29027
$797$	7.08676	0.251026	0.125513	−	0.992092i	$-0.459942\pi$
$797$	7.08676	0.251026	0.125513	+	0.992092i	$0.459942\pi$
$798$	−10.7598	−0.380894
$799$	−49.4276	−1.74862
$800$	0	0
$801$	15.1968	0.536951
$802$	−99.4542	−3.51185
$803$	−14.2129	−0.501563
$804$	22.2037	0.783065
$805$	0	0
$806$	32.5441	1.14632
$807$	17.2677	0.607850
$808$	77.1779	2.71511
$809$	−12.5391	−0.440852	−0.220426	−	0.975404i	$-0.570745\pi$
$809$	−12.5391	−0.440852	−0.220426	+	0.975404i	$0.570745\pi$
$810$	0	0
$811$	−8.27697	−0.290644	−0.145322	−	0.989384i	$-0.546422\pi$
$811$	−8.27697	−0.290644	−0.145322	+	0.989384i	$0.546422\pi$
$812$	3.30678	0.116045
$813$	−15.7051	−0.550802
$814$	20.5806	0.721350
$815$	0	0
$816$	−32.2094	−1.12755
$817$	−17.6585	−0.617791
$818$	−3.04817	−0.106577
$819$	2.40490	0.0840339
$820$	0	0
$821$	50.1133	1.74897	0.874483	−	0.485056i	$-0.161201\pi$
$821$	50.1133	1.74897	0.874483	+	0.485056i	$0.161201\pi$
$822$	24.3422	0.849032
$823$	52.3851	1.82603	0.913014	−	0.407927i	$-0.133748\pi$
$823$	52.3851	1.82603	0.913014	+	0.407927i	$0.133748\pi$
$824$	19.8746	0.692366
$825$	0	0
$826$	−11.5824	−0.403002
$827$	−27.7003	−0.963234	−0.481617	−	0.876382i	$-0.659950\pi$
$827$	−27.7003	−0.963234	−0.481617	+	0.876382i	$0.659950\pi$
$828$	0.325544	0.0113134
$829$	−7.51970	−0.261170	−0.130585	−	0.991437i	$-0.541686\pi$
$829$	−7.51970	−0.261170	−0.130585	+	0.991437i	$0.541686\pi$
$830$	0	0
$831$	15.9837	0.554469
$832$	11.3068	0.391992
$833$	3.70156	0.128252
$834$	−58.2296	−2.01633
$835$	0	0
$836$	−19.5416	−0.675861
$837$	−5.22742	−0.180686
$838$	−96.4524	−3.33189
$839$	−41.0385	−1.41681	−0.708403	−	0.705809i	$-0.750584\pi$
$839$	−41.0385	−1.41681	−0.708403	+	0.705809i	$0.750584\pi$
$840$	0	0
$841$	−28.5053	−0.982942
$842$	8.03722	0.276981
$843$	5.50302	0.189534
$844$	−113.839	−3.91848
$845$	0	0
$846$	−34.5679	−1.18847
$847$	1.00000	0.0343604
$848$	74.0621	2.54330
$849$	−8.09208	−0.277720
$850$	0	0
$851$	−0.550475	−0.0188700
$852$	−20.7099	−0.709509
$853$	−42.8693	−1.46782	−0.733909	−	0.679248i	$-0.762306\pi$
$853$	−42.8693	−1.46782	−0.733909	+	0.679248i	$0.762306\pi$
$854$	12.9982	0.444790
$855$	0	0
$856$	107.399	3.67083
$857$	−18.7354	−0.639988	−0.319994	−	0.947420i	$-0.603681\pi$
$857$	−18.7354	−0.639988	−0.319994	+	0.947420i	$0.603681\pi$
$858$	6.22565	0.212540
$859$	−47.5569	−1.62262	−0.811310	−	0.584616i	$-0.801245\pi$
$859$	−47.5569	−1.62262	−0.811310	+	0.584616i	$0.801245\pi$
$860$	0	0
$861$	−2.31773	−0.0789881
$862$	−35.0649	−1.19431
$863$	49.6177	1.68901	0.844503	−	0.535551i	$-0.179896\pi$
$863$	49.6177	1.68901	0.844503	+	0.535551i	$0.179896\pi$
$864$	−8.53879	−0.290496
$865$	0	0
$866$	87.4744	2.97250
$867$	3.29844	0.112021
$868$	24.5771	0.834200
$869$	4.40490	0.149426
$870$	0	0
$871$	−11.3574	−0.384832
$872$	89.9364	3.04563
$873$	8.24494	0.279049
$874$	0.745030	0.0252010
$875$	0	0
$876$	−66.8229	−2.25774
$877$	−29.0962	−0.982510	−0.491255	−	0.871016i	$-0.663462\pi$
$877$	−29.0962	−0.982510	−0.491255	+	0.871016i	$0.663462\pi$
$878$	30.0429	1.01390
$879$	6.51490	0.219742
$880$	0	0
$881$	10.8454	0.365390	0.182695	−	0.983170i	$-0.441518\pi$
$881$	10.8454	0.365390	0.182695	+	0.983170i	$0.441518\pi$
$882$	2.58874	0.0871673
$883$	−12.4000	−0.417293	−0.208646	−	0.977991i	$-0.566906\pi$
$883$	−12.4000	−0.417293	−0.208646	+	0.977991i	$0.566906\pi$
$884$	41.8527	1.40766
$885$	0	0
$886$	64.0004	2.15014
$887$	−19.7823	−0.664224	−0.332112	−	0.943240i	$-0.607761\pi$
$887$	−19.7823	−0.664224	−0.332112	+	0.943240i	$0.607761\pi$
$888$	55.5998	1.86581
$889$	0.248490	0.00833410
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	−39.4319	−1.32028
$893$	−55.5012	−1.85728
$894$	35.9837	1.20348
$895$	0	0
$896$	−4.90647	−0.163914
$897$	−0.166519	−0.00555992
$898$	81.6164	2.72358
$899$	3.67663	0.122623
$900$	0	0
$901$	31.5053	1.04959
$902$	−6.00000	−0.199778
$903$	4.24849	0.141381
$904$	−115.474	−3.84062
$905$	0	0
$906$	−7.64150	−0.253872
$907$	9.96278	0.330809	0.165404	−	0.986226i	$-0.447107\pi$
$907$	9.96278	0.330809	0.165404	+	0.986226i	$0.447107\pi$
$908$	−13.1428	−0.436160
$909$	11.0354	0.366023
$910$	0	0
$911$	57.6869	1.91125	0.955627	−	0.294580i	$-0.0951799\pi$
$911$	57.6869	1.91125	0.955627	+	0.294580i	$0.0951799\pi$
$912$	−36.1672	−1.19762
$913$	5.56308	0.184111
$914$	34.6564	1.14633
$915$	0	0
$916$	46.7975	1.54623
$917$	12.6178	0.416677
$918$	−9.58237	−0.316265
$919$	19.2033	0.633460	0.316730	−	0.948516i	$-0.397415\pi$
$919$	19.2033	0.633460	0.316730	+	0.948516i	$0.397415\pi$
$920$	0	0
$921$	−22.5679	−0.743637
$922$	10.0202	0.329998
$923$	10.5933	0.348684
$924$	4.70156	0.154670
$925$	0	0
$926$	−75.7163	−2.48819
$927$	2.84182	0.0933376
$928$	6.00564	0.197145
$929$	39.9422	1.31046	0.655231	−	0.755428i	$-0.272571\pi$
$929$	39.9422	1.31046	0.655231	+	0.755428i	$0.272571\pi$
$930$	0	0
$931$	4.15641	0.136221
$932$	13.0521	0.427536
$933$	−0.0513177	−0.00168007
$934$	−36.9026	−1.20749
$935$	0	0
$936$	16.8190	0.549745
$937$	−29.7826	−0.972954	−0.486477	−	0.873693i	$-0.661718\pi$
$937$	−29.7826	−0.972954	−0.486477	+	0.873693i	$0.661718\pi$
$938$	−12.2256	−0.399182
$939$	−3.44224	−0.112333
$940$	0	0
$941$	26.5208	0.864554	0.432277	−	0.901741i	$-0.357710\pi$
$941$	26.5208	0.864554	0.432277	+	0.901741i	$0.357710\pi$
$942$	13.3440	0.434771
$943$	0.160484	0.00522607
$944$	−38.9320	−1.26713
$945$	0	0
$946$	10.9982	0.357583
$947$	34.6659	1.12649	0.563245	−	0.826290i	$-0.309553\pi$
$947$	34.6659	1.12649	0.563245	+	0.826290i	$0.309553\pi$
$948$	20.7099	0.672626
$949$	34.1806	1.10955
$950$	0	0
$951$	31.6582	1.02659
$952$	25.8874	0.839015
$953$	31.9451	1.03480	0.517402	−	0.855742i	$-0.326899\pi$
$953$	31.9451	1.03480	0.517402	+	0.855742i	$0.326899\pi$
$954$	22.0337	0.713366
$955$	0	0
$956$	−68.6590	−2.22059
$957$	0.703336	0.0227356
$958$	−83.4450	−2.69599
$959$	−9.40312	−0.303643
$960$	0	0
$961$	−3.67405	−0.118518
$962$	−49.4942	−1.59576
$963$	15.3567	0.494864
$964$	49.5472	1.59581
$965$	0	0
$966$	−0.179249	−0.00576723
$967$	−5.97781	−0.192233	−0.0961167	−	0.995370i	$-0.530642\pi$
$967$	−5.97781	−0.192233	−0.0961167	+	0.995370i	$0.530642\pi$
$968$	6.99364	0.224784
$969$	−15.3852	−0.494244
$970$	0	0
$971$	−6.61617	−0.212323	−0.106161	−	0.994349i	$-0.533856\pi$
$971$	−6.61617	−0.212323	−0.106161	+	0.994349i	$0.533856\pi$
$972$	−4.70156	−0.150803
$973$	22.4934	0.721106
$974$	−13.6525	−0.437456
$975$	0	0
$976$	43.6911	1.39852
$977$	−14.9276	−0.477577	−0.238788	−	0.971072i	$-0.576750\pi$
$977$	−14.9276	−0.477577	−0.238788	+	0.971072i	$0.576750\pi$
$978$	6.00918	0.192152
$979$	−15.1968	−0.485691
$980$	0	0
$981$	12.8597	0.410580
$982$	92.3267	2.94626
$983$	15.6774	0.500030	0.250015	−	0.968242i	$-0.419564\pi$
$983$	15.6774	0.500030	0.250015	+	0.968242i	$0.419564\pi$
$984$	−16.2094	−0.516736
$985$	0	0
$986$	6.73962	0.214633
$987$	13.3532	0.425036
$988$	46.9956	1.49513
$989$	−0.294173	−0.00935415
$990$	0	0
$991$	−33.8504	−1.07529	−0.537647	−	0.843170i	$-0.680687\pi$
$991$	−33.8504	−1.07529	−0.537647	+	0.843170i	$0.680687\pi$
$992$	44.6359	1.41719
$993$	3.55953	0.112958
$994$	11.4031	0.361685
$995$	0	0
$996$	26.1552	0.828758
$997$	6.45130	0.204315	0.102157	−	0.994768i	$-0.467425\pi$
$997$	6.45130	0.204315	0.102157	+	0.994768i	$0.467425\pi$
$998$	−100.982	−3.19654
$999$	7.95005	0.251529

Twists

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

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

\(f(q)\)	\(=\)	\(q+2.58874 q^{2} -1.00000 q^{3} +4.70156 q^{4} -2.58874 q^{6} +1.00000 q^{7} +6.99364 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.58874 q^{2} -1.00000 q^{3} +4.70156 q^{4} -2.58874 q^{6} +1.00000 q^{7} +6.99364 q^{8} +1.00000 q^{9} -1.00000 q^{11} -4.70156 q^{12} +2.40490 q^{13} +2.58874 q^{14} +8.70156 q^{16} +3.70156 q^{17} +2.58874 q^{18} +4.15641 q^{19} -1.00000 q^{21} -2.58874 q^{22} +0.0692417 q^{23} -6.99364 q^{24} +6.22565 q^{26} -1.00000 q^{27} +4.70156 q^{28} +0.703336 q^{29} +5.22742 q^{31} +8.53879 q^{32} +1.00000 q^{33} +9.58237 q^{34} +4.70156 q^{36} -7.95005 q^{37} +10.7598 q^{38} -2.40490 q^{39} +2.31773 q^{41} -2.58874 q^{42} -4.24849 q^{43} -4.70156 q^{44} +0.179249 q^{46} -13.3532 q^{47} -8.70156 q^{48} +1.00000 q^{49} -3.70156 q^{51} +11.3068 q^{52} +8.51136 q^{53} -2.58874 q^{54} +6.99364 q^{56} -4.15641 q^{57} +1.82075 q^{58} -4.47414 q^{59} +5.02107 q^{61} +13.5324 q^{62} +1.00000 q^{63} +4.70156 q^{64} +2.58874 q^{66} -4.72263 q^{67} +17.4031 q^{68} -0.0692417 q^{69} +4.40490 q^{71} +6.99364 q^{72} +14.2129 q^{73} -20.5806 q^{74} +19.5416 q^{76} -1.00000 q^{77} -6.22565 q^{78} -4.40490 q^{79} +1.00000 q^{81} +6.00000 q^{82} -5.56308 q^{83} -4.70156 q^{84} -10.9982 q^{86} -0.703336 q^{87} -6.99364 q^{88} +15.1968 q^{89} +2.40490 q^{91} +0.325544 q^{92} -5.22742 q^{93} -34.5679 q^{94} -8.53879 q^{96} +8.24494 q^{97} +2.58874 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} + 6 q^{4} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 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

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