LMFDB - Embedded newform 4400.2.b.t.4049.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4400,2,Mod(4049,4400)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4400, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

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

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

chi := DirichletCharacter("4400.4049");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.b (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$35.1341768894$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 440)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			4049.3
Root			$1.56155i$ of defining polynomial
Character	$\chi$	$=$	4400.4049
Dual form			4400.2.b.t.4049.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155i q^{3} -1.56155i q^{7} +0.561553 q^{9} +O(q^{10})$	$q+1.56155i q^{3} -1.56155i q^{7} +0.561553 q^{9} -1.00000 q^{11} -2.00000i q^{13} +3.56155i q^{17} -1.56155 q^{19} +2.43845 q^{21} -3.12311i q^{23} +5.56155i q^{27} +2.68466 q^{29} -2.43845 q^{31} -1.56155i q^{33} +6.68466i q^{37} +3.12311 q^{39} +2.00000 q^{41} -6.24621i q^{43} +4.87689i q^{47} +4.56155 q^{49} -5.56155 q^{51} -0.438447i q^{53} -2.43845i q^{57} -7.12311 q^{59} +14.6847 q^{61} -0.876894i q^{63} +10.2462i q^{67} +4.87689 q^{69} +8.68466 q^{71} +2.00000i q^{73} +1.56155i q^{77} +9.36932 q^{79} -7.00000 q^{81} -3.12311i q^{83} +4.19224i q^{87} +8.43845 q^{89} -3.12311 q^{91} -3.80776i q^{93} -1.12311i q^{97} -0.561553 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{9}+O(q^{10}) $ 4 * q - 6 * q^9	$ 4 q - 6 q^{9} - 4 q^{11} + 2 q^{19} + 18 q^{21} - 14 q^{29} - 18 q^{31} - 4 q^{39} + 8 q^{41} + 10 q^{49} - 14 q^{51} - 12 q^{59} + 34 q^{61} + 36 q^{69} + 10 q^{71} - 12 q^{79} - 28 q^{81} + 42 q^{89} + 4 q^{91} + 6 q^{99}+O(q^{100}) $ 4 * q - 6 * q^9 - 4 * q^11 + 2 * q^19 + 18 * q^21 - 14 * q^29 - 18 * q^31 - 4 * q^39 + 8 * q^41 + 10 * q^49 - 14 * q^51 - 12 * q^59 + 34 * q^61 + 36 * q^69 + 10 * q^71 - 12 * q^79 - 28 * q^81 + 42 * q^89 + 4 * q^91 + 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times$.

$n$	$177$	$1201$	$2751$	$3301$
$\chi(n)$	$-1$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.56155i	0.901563i	0.892634	+	0.450781i	$0.148855\pi$
$3$	1.56155i	0.901563i	−0.892634	+	0.450781i	$0.851145\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.56155i	− 0.590211i	−0.955465	−	0.295106i	$-0.904645\pi$
$7$	− 1.56155i	− 0.590211i	0.955465	−	0.295106i	$-0.0953549\pi$
$8$	0	0
$9$	0.561553	0.187184
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	− 2.00000i	− 0.554700i	−0.960769	−	0.277350i	$-0.910544\pi$
$13$	− 2.00000i	− 0.554700i	0.960769	−	0.277350i	$-0.0894562\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.56155i	0.863803i	0.901921	+	0.431902i	$0.142157\pi$
$17$	3.56155i	0.863803i	−0.901921	+	0.431902i	$0.857843\pi$
$18$	0	0
$19$	−1.56155	−0.358245	−0.179122	−	0.983827i	$-0.557326\pi$
$19$	−1.56155	−0.358245	−0.179122	+	0.983827i	$0.557326\pi$
$20$	0	0
$21$	2.43845	0.532113
$22$	0	0
$23$	− 3.12311i	− 0.651213i	−0.945505	−	0.325606i	$-0.894432\pi$
$23$	− 3.12311i	− 0.651213i	0.945505	−	0.325606i	$-0.105568\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	5.56155i	1.07032i
$28$	0	0
$29$	2.68466	0.498529	0.249264	−	0.968436i	$-0.419811\pi$
$29$	2.68466	0.498529	0.249264	+	0.968436i	$0.419811\pi$
$30$	0	0
$31$	−2.43845	−0.437958	−0.218979	−	0.975730i	$-0.570273\pi$
$31$	−2.43845	−0.437958	−0.218979	+	0.975730i	$0.570273\pi$
$32$	0	0
$33$	− 1.56155i	− 0.271831i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	6.68466i	1.09895i	0.835510	+	0.549476i	$0.185172\pi$
$37$	6.68466i	1.09895i	−0.835510	+	0.549476i	$0.814828\pi$
$38$	0	0
$39$	3.12311	0.500097
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	− 6.24621i	− 0.952538i	−0.879300	−	0.476269i	$-0.841989\pi$
$43$	− 6.24621i	− 0.952538i	0.879300	−	0.476269i	$-0.158011\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.87689i	0.711368i	0.934606	+	0.355684i	$0.115752\pi$
$47$	4.87689i	0.711368i	−0.934606	+	0.355684i	$0.884248\pi$
$48$	0	0
$49$	4.56155	0.651650
$50$	0	0
$51$	−5.56155	−0.778773
$52$	0	0
$53$	− 0.438447i	− 0.0602254i	−0.999547	−	0.0301127i	$-0.990413\pi$
$53$	− 0.438447i	− 0.0602254i	0.999547	−	0.0301127i	$-0.00958661\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 2.43845i	− 0.322980i
$58$	0	0
$59$	−7.12311	−0.927349	−0.463675	−	0.886006i	$-0.653469\pi$
$59$	−7.12311	−0.927349	−0.463675	+	0.886006i	$0.653469\pi$
$60$	0	0
$61$	14.6847	1.88018	0.940089	−	0.340929i	$-0.110742\pi$
$61$	14.6847	1.88018	0.940089	+	0.340929i	$0.110742\pi$
$62$	0	0
$63$	− 0.876894i	− 0.110478i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	10.2462i	1.25177i	0.779914	+	0.625887i	$0.215263\pi$
$67$	10.2462i	1.25177i	−0.779914	+	0.625887i	$0.784737\pi$
$68$	0	0
$69$	4.87689	0.587109
$70$	0	0
$71$	8.68466	1.03068	0.515340	−	0.856986i	$-0.327666\pi$
$71$	8.68466	1.03068	0.515340	+	0.856986i	$0.327666\pi$
$72$	0	0
$73$	2.00000i	0.234082i	0.993127	+	0.117041i	$0.0373409\pi$
$73$	2.00000i	0.234082i	−0.993127	+	0.117041i	$0.962659\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	1.56155i	0.177955i
$78$	0	0
$79$	9.36932	1.05413	0.527065	−	0.849825i	$-0.323292\pi$
$79$	9.36932	1.05413	0.527065	+	0.849825i	$0.323292\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	− 3.12311i	− 0.342805i	−0.985201	−	0.171403i	$-0.945170\pi$
$83$	− 3.12311i	− 0.342805i	0.985201	−	0.171403i	$-0.0548299\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	4.19224i	0.449455i
$88$	0	0
$89$	8.43845	0.894474	0.447237	−	0.894416i	$-0.352408\pi$
$89$	8.43845	0.894474	0.447237	+	0.894416i	$0.352408\pi$
$90$	0	0
$91$	−3.12311	−0.327390
$92$	0	0
$93$	− 3.80776i	− 0.394847i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 1.12311i	− 0.114034i	−0.998373	−	0.0570170i	$-0.981841\pi$
$97$	− 1.12311i	− 0.114034i	0.998373	−	0.0570170i	$-0.0181589\pi$
$98$	0	0
$99$	−0.561553	−0.0564382
$100$	0	0
$101$	12.2462	1.21854	0.609272	−	0.792961i	$-0.291462\pi$
$101$	12.2462	1.21854	0.609272	+	0.792961i	$0.291462\pi$
$102$	0	0
$103$	− 6.24621i	− 0.615457i	−0.951474	−	0.307729i	$-0.900431\pi$
$103$	− 6.24621i	− 0.615457i	0.951474	−	0.307729i	$-0.0995689\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16.0000i	1.54678i	0.633932	+	0.773389i	$0.281440\pi$
$107$	16.0000i	1.54678i	−0.633932	+	0.773389i	$0.718560\pi$
$108$	0	0
$109$	14.4924	1.38812	0.694061	−	0.719916i	$-0.255820\pi$
$109$	14.4924	1.38812	0.694061	+	0.719916i	$0.255820\pi$
$110$	0	0
$111$	−10.4384	−0.990774
$112$	0	0
$113$	10.8769i	1.02321i	0.859220	+	0.511606i	$0.170949\pi$
$113$	10.8769i	1.02321i	−0.859220	+	0.511606i	$0.829051\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 1.12311i	− 0.103831i
$118$	0	0
$119$	5.56155	0.509827
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	3.12311i	0.281601i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	2.24621i	0.199319i	0.995022	+	0.0996595i	$0.0317754\pi$
$127$	2.24621i	0.199319i	−0.995022	+	0.0996595i	$0.968225\pi$
$128$	0	0
$129$	9.75379	0.858773
$130$	0	0
$131$	−4.68466	−0.409301	−0.204650	−	0.978835i	$-0.565606\pi$
$131$	−4.68466	−0.409301	−0.204650	+	0.978835i	$0.565606\pi$
$132$	0	0
$133$	2.43845i	0.211440i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 21.6155i	− 1.84674i	−0.383912	−	0.923370i	$-0.625423\pi$
$137$	− 21.6155i	− 1.84674i	0.383912	−	0.923370i	$-0.374577\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	−7.61553	−0.641343
$142$	0	0
$143$	2.00000i	0.167248i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	7.12311i	0.587504i
$148$	0	0
$149$	−14.6847	−1.20301	−0.601507	−	0.798867i	$-0.705433\pi$
$149$	−14.6847	−1.20301	−0.601507	+	0.798867i	$0.705433\pi$
$150$	0	0
$151$	−4.87689	−0.396876	−0.198438	−	0.980113i	$-0.563587\pi$
$151$	−4.87689	−0.396876	−0.198438	+	0.980113i	$0.563587\pi$
$152$	0	0
$153$	2.00000i	0.161690i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 5.80776i	− 0.463510i	−0.972774	−	0.231755i	$-0.925553\pi$
$157$	− 5.80776i	− 0.463510i	0.972774	−	0.231755i	$-0.0744468\pi$
$158$	0	0
$159$	0.684658	0.0542969
$160$	0	0
$161$	−4.87689	−0.384353
$162$	0	0
$163$	23.8078i	1.86477i	0.361469	+	0.932384i	$0.382275\pi$
$163$	23.8078i	1.86477i	−0.361469	+	0.932384i	$0.617725\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	6.43845i	0.498222i	0.968475	+	0.249111i	$0.0801384\pi$
$167$	6.43845i	0.498222i	−0.968475	+	0.249111i	$0.919862\pi$
$168$	0	0
$169$	9.00000	0.692308
$170$	0	0
$171$	−0.876894	−0.0670578
$172$	0	0
$173$	12.2462i	0.931062i	0.885032	+	0.465531i	$0.154137\pi$
$173$	12.2462i	0.931062i	−0.885032	+	0.465531i	$0.845863\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 11.1231i	− 0.836064i
$178$	0	0
$179$	10.2462	0.765838	0.382919	−	0.923782i	$-0.374919\pi$
$179$	10.2462	0.765838	0.382919	+	0.923782i	$0.374919\pi$
$180$	0	0
$181$	12.2462	0.910254	0.455127	−	0.890427i	$-0.349594\pi$
$181$	12.2462	0.910254	0.455127	+	0.890427i	$0.349594\pi$
$182$	0	0
$183$	22.9309i	1.69510i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 3.56155i	− 0.260447i
$188$	0	0
$189$	8.68466	0.631716
$190$	0	0
$191$	6.24621	0.451960	0.225980	−	0.974132i	$-0.427442\pi$
$191$	6.24621	0.451960	0.225980	+	0.974132i	$0.427442\pi$
$192$	0	0
$193$	− 25.8078i	− 1.85768i	−0.370477	−	0.928842i	$-0.620806\pi$
$193$	− 25.8078i	− 1.85768i	0.370477	−	0.928842i	$-0.379194\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.3693i	1.38001i	0.723806	+	0.690003i	$0.242391\pi$
$197$	19.3693i	1.38001i	−0.723806	+	0.690003i	$0.757609\pi$
$198$	0	0
$199$	19.8078	1.40414	0.702068	−	0.712110i	$-0.252260\pi$
$199$	19.8078	1.40414	0.702068	+	0.712110i	$0.252260\pi$
$200$	0	0
$201$	−16.0000	−1.12855
$202$	0	0
$203$	− 4.19224i	− 0.294237i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 1.75379i	− 0.121897i
$208$	0	0
$209$	1.56155	0.108015
$210$	0	0
$211$	−20.6847	−1.42399	−0.711995	−	0.702184i	$-0.752208\pi$
$211$	−20.6847	−1.42399	−0.711995	+	0.702184i	$0.752208\pi$
$212$	0	0
$213$	13.5616i	0.929222i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.80776i	0.258488i
$218$	0	0
$219$	−3.12311	−0.211040
$220$	0	0
$221$	7.12311	0.479152
$222$	0	0
$223$	4.87689i	0.326581i	0.986578	+	0.163291i	$0.0522108\pi$
$223$	4.87689i	0.326581i	−0.986578	+	0.163291i	$0.947789\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 11.1231i	− 0.738266i	−0.929376	−	0.369133i	$-0.879655\pi$
$227$	− 11.1231i	− 0.738266i	0.929376	−	0.369133i	$-0.120345\pi$
$228$	0	0
$229$	−24.7386	−1.63477	−0.817387	−	0.576088i	$-0.804578\pi$
$229$	−24.7386	−1.63477	−0.817387	+	0.576088i	$0.804578\pi$
$230$	0	0
$231$	−2.43845	−0.160438
$232$	0	0
$233$	8.93087i	0.585081i	0.956253	+	0.292540i	$0.0945006\pi$
$233$	8.93087i	0.585081i	−0.956253	+	0.292540i	$0.905499\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	14.6307i	0.950365i
$238$	0	0
$239$	−15.6155	−1.01008	−0.505042	−	0.863095i	$-0.668523\pi$
$239$	−15.6155	−1.01008	−0.505042	+	0.863095i	$0.668523\pi$
$240$	0	0
$241$	0.246211	0.0158599	0.00792993	−	0.999969i	$-0.497476\pi$
$241$	0.246211	0.0158599	0.00792993	+	0.999969i	$0.497476\pi$
$242$	0	0
$243$	5.75379i	0.369106i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	3.12311i	0.198718i
$248$	0	0
$249$	4.87689	0.309061
$250$	0	0
$251$	−11.6155	−0.733166	−0.366583	−	0.930385i	$-0.619472\pi$
$251$	−11.6155	−0.733166	−0.366583	+	0.930385i	$0.619472\pi$
$252$	0	0
$253$	3.12311i	0.196348i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	20.7386i	1.29364i	0.762643	+	0.646820i	$0.223902\pi$
$257$	20.7386i	1.29364i	−0.762643	+	0.646820i	$0.776098\pi$
$258$	0	0
$259$	10.4384	0.648614
$260$	0	0
$261$	1.50758	0.0933167
$262$	0	0
$263$	− 15.8078i	− 0.974748i	−0.873193	−	0.487374i	$-0.837955\pi$
$263$	− 15.8078i	− 0.974748i	0.873193	−	0.487374i	$-0.162045\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	13.1771i	0.806424i
$268$	0	0
$269$	−21.6155	−1.31792	−0.658961	−	0.752177i	$-0.729004\pi$
$269$	−21.6155	−1.31792	−0.658961	+	0.752177i	$0.729004\pi$
$270$	0	0
$271$	4.49242	0.272895	0.136448	−	0.990647i	$-0.456431\pi$
$271$	4.49242	0.272895	0.136448	+	0.990647i	$0.456431\pi$
$272$	0	0
$273$	− 4.87689i	− 0.295163i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1.12311i	− 0.0674809i	−0.999431	−	0.0337404i	$-0.989258\pi$
$277$	− 1.12311i	− 0.0674809i	0.999431	−	0.0337404i	$-0.0107420\pi$
$278$	0	0
$279$	−1.36932	−0.0819789
$280$	0	0
$281$	8.24621	0.491928	0.245964	−	0.969279i	$-0.420896\pi$
$281$	8.24621	0.491928	0.245964	+	0.969279i	$0.420896\pi$
$282$	0	0
$283$	1.36932i	0.0813974i	0.999171	+	0.0406987i	$0.0129584\pi$
$283$	1.36932i	0.0813974i	−0.999171	+	0.0406987i	$0.987042\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 3.12311i	− 0.184351i
$288$	0	0
$289$	4.31534	0.253844
$290$	0	0
$291$	1.75379	0.102809
$292$	0	0
$293$	18.4924i	1.08034i	0.841556	+	0.540169i	$0.181640\pi$
$293$	18.4924i	1.08034i	−0.841556	+	0.540169i	$0.818360\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 5.56155i	− 0.322714i
$298$	0	0
$299$	−6.24621	−0.361228
$300$	0	0
$301$	−9.75379	−0.562199
$302$	0	0
$303$	19.1231i	1.09859i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 15.6155i	− 0.891225i	−0.895226	−	0.445613i	$-0.852986\pi$
$307$	− 15.6155i	− 0.891225i	0.895226	−	0.445613i	$-0.147014\pi$
$308$	0	0
$309$	9.75379	0.554874
$310$	0	0
$311$	29.5616	1.67628	0.838141	−	0.545454i	$-0.183643\pi$
$311$	29.5616	1.67628	0.838141	+	0.545454i	$0.183643\pi$
$312$	0	0
$313$	2.49242i	0.140880i	0.997516	+	0.0704400i	$0.0224403\pi$
$313$	2.49242i	0.140880i	−0.997516	+	0.0704400i	$0.977560\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	33.8078i	1.89883i	0.314019	+	0.949417i	$0.398324\pi$
$317$	33.8078i	1.89883i	−0.314019	+	0.949417i	$0.601676\pi$
$318$	0	0
$319$	−2.68466	−0.150312
$320$	0	0
$321$	−24.9848	−1.39452
$322$	0	0
$323$	− 5.56155i	− 0.309453i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	22.6307i	1.25148i
$328$	0	0
$329$	7.61553	0.419858
$330$	0	0
$331$	−24.4924	−1.34623	−0.673113	−	0.739540i	$-0.735043\pi$
$331$	−24.4924	−1.34623	−0.673113	+	0.739540i	$0.735043\pi$
$332$	0	0
$333$	3.75379i	0.205706i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	14.6847i	0.799924i	0.916532	+	0.399962i	$0.130977\pi$
$337$	14.6847i	0.799924i	−0.916532	+	0.399962i	$0.869023\pi$
$338$	0	0
$339$	−16.9848	−0.922490
$340$	0	0
$341$	2.43845	0.132049
$342$	0	0
$343$	− 18.0540i	− 0.974823i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	27.1231i	1.45604i	0.685553	+	0.728022i	$0.259560\pi$
$347$	27.1231i	1.45604i	−0.685553	+	0.728022i	$0.740440\pi$
$348$	0	0
$349$	22.4924	1.20399	0.601996	−	0.798499i	$-0.294372\pi$
$349$	22.4924	1.20399	0.601996	+	0.798499i	$0.294372\pi$
$350$	0	0
$351$	11.1231	0.593707
$352$	0	0
$353$	1.50758i	0.0802403i	0.999195	+	0.0401201i	$0.0127741\pi$
$353$	1.50758i	0.0802403i	−0.999195	+	0.0401201i	$0.987226\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	8.68466i	0.459641i
$358$	0	0
$359$	9.75379	0.514785	0.257393	−	0.966307i	$-0.417137\pi$
$359$	9.75379	0.514785	0.257393	+	0.966307i	$0.417137\pi$
$360$	0	0
$361$	−16.5616	−0.871661
$362$	0	0
$363$	1.56155i	0.0819603i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 34.7386i	− 1.81334i	−0.421839	−	0.906671i	$-0.638615\pi$
$367$	− 34.7386i	− 1.81334i	0.421839	−	0.906671i	$-0.361385\pi$
$368$	0	0
$369$	1.12311	0.0584665
$370$	0	0
$371$	−0.684658	−0.0355457
$372$	0	0
$373$	− 2.00000i	− 0.103556i	−0.998659	−	0.0517780i	$-0.983511\pi$
$373$	− 2.00000i	− 0.103556i	0.998659	−	0.0517780i	$-0.0164888\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 5.36932i	− 0.276534i
$378$	0	0
$379$	−16.8769	−0.866908	−0.433454	−	0.901176i	$-0.642705\pi$
$379$	−16.8769	−0.866908	−0.433454	+	0.901176i	$0.642705\pi$
$380$	0	0
$381$	−3.50758	−0.179699
$382$	0	0
$383$	− 6.63068i	− 0.338812i	−0.985546	−	0.169406i	$-0.945815\pi$
$383$	− 6.63068i	− 0.338812i	0.985546	−	0.169406i	$-0.0541849\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 3.50758i	− 0.178300i
$388$	0	0
$389$	−15.7538	−0.798749	−0.399374	−	0.916788i	$-0.630773\pi$
$389$	−15.7538	−0.798749	−0.399374	+	0.916788i	$0.630773\pi$
$390$	0	0
$391$	11.1231	0.562520
$392$	0	0
$393$	− 7.31534i	− 0.369010i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 18.0000i	− 0.903394i	−0.892171	−	0.451697i	$-0.850819\pi$
$397$	− 18.0000i	− 0.903394i	0.892171	−	0.451697i	$-0.149181\pi$
$398$	0	0
$399$	−3.80776	−0.190627
$400$	0	0
$401$	−11.5616	−0.577356	−0.288678	−	0.957426i	$-0.593216\pi$
$401$	−11.5616	−0.577356	−0.288678	+	0.957426i	$0.593216\pi$
$402$	0	0
$403$	4.87689i	0.242935i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 6.68466i	− 0.331346i
$408$	0	0
$409$	1.12311	0.0555340	0.0277670	−	0.999614i	$-0.491160\pi$
$409$	1.12311	0.0555340	0.0277670	+	0.999614i	$0.491160\pi$
$410$	0	0
$411$	33.7538	1.66495
$412$	0	0
$413$	11.1231i	0.547332i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	18.7386i	0.917635i
$418$	0	0
$419$	22.7386	1.11085	0.555427	−	0.831565i	$-0.312555\pi$
$419$	22.7386	1.11085	0.555427	+	0.831565i	$0.312555\pi$
$420$	0	0
$421$	−33.6155	−1.63832	−0.819160	−	0.573565i	$-0.805560\pi$
$421$	−33.6155	−1.63832	−0.819160	+	0.573565i	$0.805560\pi$
$422$	0	0
$423$	2.73863i	0.133157i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 22.9309i	− 1.10970i
$428$	0	0
$429$	−3.12311	−0.150785
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	− 8.63068i	− 0.414764i	−0.978260	−	0.207382i	$-0.933506\pi$
$433$	− 8.63068i	− 0.414764i	0.978260	−	0.207382i	$-0.0664943\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.87689i	0.233293i
$438$	0	0
$439$	−17.7538	−0.847342	−0.423671	−	0.905816i	$-0.639259\pi$
$439$	−17.7538	−0.847342	−0.423671	+	0.905816i	$0.639259\pi$
$440$	0	0
$441$	2.56155	0.121979
$442$	0	0
$443$	− 4.00000i	− 0.190046i	−0.995475	−	0.0950229i	$-0.969708\pi$
$443$	− 4.00000i	− 0.190046i	0.995475	−	0.0950229i	$-0.0302924\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 22.9309i	− 1.08459i
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	−2.00000	−0.0941763
$452$	0	0
$453$	− 7.61553i	− 0.357809i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	24.4384i	1.14318i	0.820539	+	0.571591i	$0.193674\pi$
$457$	24.4384i	1.14318i	−0.820539	+	0.571591i	$0.806326\pi$
$458$	0	0
$459$	−19.8078	−0.924547
$460$	0	0
$461$	16.4384	0.765615	0.382807	−	0.923828i	$-0.374957\pi$
$461$	16.4384	0.765615	0.382807	+	0.923828i	$0.374957\pi$
$462$	0	0
$463$	17.3693i	0.807221i	0.914931	+	0.403610i	$0.132245\pi$
$463$	17.3693i	0.807221i	−0.914931	+	0.403610i	$0.867755\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	18.9309i	0.876016i	0.898971	+	0.438008i	$0.144316\pi$
$467$	18.9309i	0.876016i	−0.898971	+	0.438008i	$0.855684\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	0	0
$471$	9.06913	0.417883
$472$	0	0
$473$	6.24621i	0.287201i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 0.246211i	− 0.0112732i
$478$	0	0
$479$	32.9848	1.50712	0.753558	−	0.657381i	$-0.228336\pi$
$479$	32.9848	1.50712	0.753558	+	0.657381i	$0.228336\pi$
$480$	0	0
$481$	13.3693	0.609588
$482$	0	0
$483$	− 7.61553i	− 0.346519i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 14.2462i	− 0.645557i	−0.946474	−	0.322779i	$-0.895383\pi$
$487$	− 14.2462i	− 0.645557i	0.946474	−	0.322779i	$-0.104617\pi$
$488$	0	0
$489$	−37.1771	−1.68121
$490$	0	0
$491$	30.0540	1.35632	0.678158	−	0.734916i	$-0.262778\pi$
$491$	30.0540	1.35632	0.678158	+	0.734916i	$0.262778\pi$
$492$	0	0
$493$	9.56155i	0.430631i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 13.5616i	− 0.608319i
$498$	0	0
$499$	−16.4924	−0.738302	−0.369151	−	0.929369i	$-0.620352\pi$
$499$	−16.4924	−0.738302	−0.369151	+	0.929369i	$0.620352\pi$
$500$	0	0
$501$	−10.0540	−0.449178
$502$	0	0
$503$	2.24621i	0.100154i	0.998745	+	0.0500768i	$0.0159466\pi$
$503$	2.24621i	0.100154i	−0.998745	+	0.0500768i	$0.984053\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	14.0540i	0.624159i
$508$	0	0
$509$	3.36932	0.149342	0.0746712	−	0.997208i	$-0.476209\pi$
$509$	3.36932	0.149342	0.0746712	+	0.997208i	$0.476209\pi$
$510$	0	0
$511$	3.12311	0.138158
$512$	0	0
$513$	− 8.68466i	− 0.383437i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 4.87689i	− 0.214486i
$518$	0	0
$519$	−19.1231	−0.839411
$520$	0	0
$521$	−40.7386	−1.78479	−0.892396	−	0.451253i	$-0.850977\pi$
$521$	−40.7386	−1.78479	−0.892396	+	0.451253i	$0.850977\pi$
$522$	0	0
$523$	− 36.4924i	− 1.59570i	−0.602855	−	0.797851i	$-0.705970\pi$
$523$	− 36.4924i	− 1.59570i	0.602855	−	0.797851i	$-0.294030\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 8.68466i	− 0.378310i
$528$	0	0
$529$	13.2462	0.575922
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	− 4.00000i	− 0.173259i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	16.0000i	0.690451i
$538$	0	0
$539$	−4.56155	−0.196480
$540$	0	0
$541$	36.5464	1.57125	0.785626	−	0.618701i	$-0.212341\pi$
$541$	36.5464	1.57125	0.785626	+	0.618701i	$0.212341\pi$
$542$	0	0
$543$	19.1231i	0.820651i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	16.0000i	0.684111i	0.939680	+	0.342055i	$0.111123\pi$
$547$	16.0000i	0.684111i	−0.939680	+	0.342055i	$0.888877\pi$
$548$	0	0
$549$	8.24621	0.351940
$550$	0	0
$551$	−4.19224	−0.178595
$552$	0	0
$553$	− 14.6307i	− 0.622160i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	13.5076i	0.572334i	0.958180	+	0.286167i	$0.0923813\pi$
$557$	13.5076i	0.572334i	−0.958180	+	0.286167i	$0.907619\pi$
$558$	0	0
$559$	−12.4924	−0.528373
$560$	0	0
$561$	5.56155	0.234809
$562$	0	0
$563$	− 11.1231i	− 0.468783i	−0.972142	−	0.234392i	$-0.924690\pi$
$563$	− 11.1231i	− 0.468783i	0.972142	−	0.234392i	$-0.0753098\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	10.9309i	0.459053i
$568$	0	0
$569$	−46.1080	−1.93295	−0.966473	−	0.256769i	$-0.917342\pi$
$569$	−46.1080	−1.93295	−0.966473	+	0.256769i	$0.917342\pi$
$570$	0	0
$571$	31.4233	1.31502	0.657512	−	0.753444i	$-0.271609\pi$
$571$	31.4233	1.31502	0.657512	+	0.753444i	$0.271609\pi$
$572$	0	0
$573$	9.75379i	0.407470i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 24.7386i	− 1.02988i	−0.857225	−	0.514941i	$-0.827814\pi$
$577$	− 24.7386i	− 1.02988i	0.857225	−	0.514941i	$-0.172186\pi$
$578$	0	0
$579$	40.3002	1.67482
$580$	0	0
$581$	−4.87689	−0.202328
$582$	0	0
$583$	0.438447i	0.0181586i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 44.6847i	− 1.84433i	−0.386793	−	0.922167i	$-0.626417\pi$
$587$	− 44.6847i	− 1.84433i	0.386793	−	0.922167i	$-0.373583\pi$
$588$	0	0
$589$	3.80776	0.156896
$590$	0	0
$591$	−30.2462	−1.24416
$592$	0	0
$593$	− 34.4924i	− 1.41643i	−0.705995	−	0.708217i	$-0.749500\pi$
$593$	− 34.4924i	− 1.41643i	0.705995	−	0.708217i	$-0.250500\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	30.9309i	1.26592i
$598$	0	0
$599$	−21.5616	−0.880981	−0.440491	−	0.897757i	$-0.645195\pi$
$599$	−21.5616	−0.880981	−0.440491	+	0.897757i	$0.645195\pi$
$600$	0	0
$601$	−30.0000	−1.22373	−0.611863	−	0.790964i	$-0.709580\pi$
$601$	−30.0000	−1.22373	−0.611863	+	0.790964i	$0.709580\pi$
$602$	0	0
$603$	5.75379i	0.234312i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 0.192236i	− 0.00780262i	−0.999992	−	0.00390131i	$-0.998758\pi$
$607$	− 0.192236i	− 0.00780262i	0.999992	−	0.00390131i	$-0.00124183\pi$
$608$	0	0
$609$	6.54640	0.265273
$610$	0	0
$611$	9.75379	0.394596
$612$	0	0
$613$	32.7386i	1.32230i	0.750253	+	0.661150i	$0.229932\pi$
$613$	32.7386i	1.32230i	−0.750253	+	0.661150i	$0.770068\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 31.3693i	− 1.26288i	−0.775424	−	0.631441i	$-0.782464\pi$
$617$	− 31.3693i	− 1.26288i	0.775424	−	0.631441i	$-0.217536\pi$
$618$	0	0
$619$	24.4924	0.984434	0.492217	−	0.870473i	$-0.336187\pi$
$619$	24.4924	0.984434	0.492217	+	0.870473i	$0.336187\pi$
$620$	0	0
$621$	17.3693	0.697007
$622$	0	0
$623$	− 13.1771i	− 0.527929i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	2.43845i	0.0973822i
$628$	0	0
$629$	−23.8078	−0.949278
$630$	0	0
$631$	−3.80776	−0.151585	−0.0757923	−	0.997124i	$-0.524149\pi$
$631$	−3.80776	−0.151585	−0.0757923	+	0.997124i	$0.524149\pi$
$632$	0	0
$633$	− 32.3002i	− 1.28382i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 9.12311i	− 0.361471i
$638$	0	0
$639$	4.87689	0.192927
$640$	0	0
$641$	7.17708	0.283478	0.141739	−	0.989904i	$-0.454731\pi$
$641$	7.17708	0.283478	0.141739	+	0.989904i	$0.454731\pi$
$642$	0	0
$643$	11.3153i	0.446234i	0.974792	+	0.223117i	$0.0716231\pi$
$643$	11.3153i	0.446234i	−0.974792	+	0.223117i	$0.928377\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	4.87689i	0.191731i	0.995394	+	0.0958653i	$0.0305618\pi$
$647$	4.87689i	0.191731i	−0.995394	+	0.0958653i	$0.969438\pi$
$648$	0	0
$649$	7.12311	0.279606
$650$	0	0
$651$	−5.94602	−0.233043
$652$	0	0
$653$	− 9.80776i	− 0.383807i	−0.981414	−	0.191904i	$-0.938534\pi$
$653$	− 9.80776i	− 0.383807i	0.981414	−	0.191904i	$-0.0614661\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.12311i	0.0438165i
$658$	0	0
$659$	39.8078	1.55069	0.775345	−	0.631538i	$-0.217576\pi$
$659$	39.8078	1.55069	0.775345	+	0.631538i	$0.217576\pi$
$660$	0	0
$661$	−0.246211	−0.00957651	−0.00478825	−	0.999989i	$-0.501524\pi$
$661$	−0.246211	−0.00957651	−0.00478825	+	0.999989i	$0.501524\pi$
$662$	0	0
$663$	11.1231i	0.431986i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 8.38447i	− 0.324648i
$668$	0	0
$669$	−7.61553	−0.294433
$670$	0	0
$671$	−14.6847	−0.566895
$672$	0	0
$673$	− 50.7926i	− 1.95791i	−0.204073	−	0.978956i	$-0.565418\pi$
$673$	− 50.7926i	− 1.95791i	0.204073	−	0.978956i	$-0.434582\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 13.6155i	− 0.523287i	−0.965165	−	0.261644i	$-0.915735\pi$
$677$	− 13.6155i	− 0.523287i	0.965165	−	0.261644i	$-0.0842645\pi$
$678$	0	0
$679$	−1.75379	−0.0673042
$680$	0	0
$681$	17.3693	0.665594
$682$	0	0
$683$	2.93087i	0.112147i	0.998427	+	0.0560733i	$0.0178581\pi$
$683$	2.93087i	0.112147i	−0.998427	+	0.0560733i	$0.982142\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 38.6307i	− 1.47385i
$688$	0	0
$689$	−0.876894	−0.0334070
$690$	0	0
$691$	21.3693	0.812927	0.406464	−	0.913667i	$-0.366762\pi$
$691$	21.3693	0.812927	0.406464	+	0.913667i	$0.366762\pi$
$692$	0	0
$693$	0.876894i	0.0333105i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	7.12311i	0.269807i
$698$	0	0
$699$	−13.9460	−0.527487
$700$	0	0
$701$	46.6847	1.76326	0.881628	−	0.471946i	$-0.156448\pi$
$701$	46.6847	1.76326	0.881628	+	0.471946i	$0.156448\pi$
$702$	0	0
$703$	− 10.4384i	− 0.393693i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 19.1231i	− 0.719198i
$708$	0	0
$709$	−34.4924	−1.29539	−0.647695	−	0.761900i	$-0.724267\pi$
$709$	−34.4924	−1.29539	−0.647695	+	0.761900i	$0.724267\pi$
$710$	0	0
$711$	5.26137	0.197317
$712$	0	0
$713$	7.61553i	0.285204i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	− 24.3845i	− 0.910655i
$718$	0	0
$719$	53.1771	1.98317	0.991585	−	0.129455i	$-0.0413229\pi$
$719$	53.1771	1.98317	0.991585	+	0.129455i	$0.0413229\pi$
$720$	0	0
$721$	−9.75379	−0.363250
$722$	0	0
$723$	0.384472i	0.0142987i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	23.6155i	0.875851i	0.899011	+	0.437926i	$0.144287\pi$
$727$	23.6155i	0.875851i	−0.899011	+	0.437926i	$0.855713\pi$
$728$	0	0
$729$	−29.9848	−1.11055
$730$	0	0
$731$	22.2462	0.822806
$732$	0	0
$733$	10.8769i	0.401747i	0.979617	+	0.200874i	$0.0643781\pi$
$733$	10.8769i	0.401747i	−0.979617	+	0.200874i	$0.935622\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 10.2462i	− 0.377424i
$738$	0	0
$739$	6.73863	0.247885	0.123942	−	0.992289i	$-0.460446\pi$
$739$	6.73863	0.247885	0.123942	+	0.992289i	$0.460446\pi$
$740$	0	0
$741$	−4.87689	−0.179157
$742$	0	0
$743$	1.56155i	0.0572878i	0.999590	+	0.0286439i	$0.00911889\pi$
$743$	1.56155i	0.0572878i	−0.999590	+	0.0286439i	$0.990881\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 1.75379i	− 0.0641678i
$748$	0	0
$749$	24.9848	0.912926
$750$	0	0
$751$	−29.1771	−1.06469	−0.532343	−	0.846529i	$-0.678688\pi$
$751$	−29.1771	−1.06469	−0.532343	+	0.846529i	$0.678688\pi$
$752$	0	0
$753$	− 18.1383i	− 0.660995i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 26.0000i	− 0.944986i	−0.881334	−	0.472493i	$-0.843354\pi$
$757$	− 26.0000i	− 0.944986i	0.881334	−	0.472493i	$-0.156646\pi$
$758$	0	0
$759$	−4.87689	−0.177020
$760$	0	0
$761$	12.7386	0.461775	0.230888	−	0.972980i	$-0.425837\pi$
$761$	12.7386	0.461775	0.230888	+	0.972980i	$0.425837\pi$
$762$	0	0
$763$	− 22.6307i	− 0.819286i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	14.2462i	0.514401i
$768$	0	0
$769$	24.7386	0.892098	0.446049	−	0.895009i	$-0.352831\pi$
$769$	24.7386	0.892098	0.446049	+	0.895009i	$0.352831\pi$
$770$	0	0
$771$	−32.3845	−1.16630
$772$	0	0
$773$	25.3153i	0.910530i	0.890356	+	0.455265i	$0.150455\pi$
$773$	25.3153i	0.910530i	−0.890356	+	0.455265i	$0.849545\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	16.3002i	0.584766i
$778$	0	0
$779$	−3.12311	−0.111897
$780$	0	0
$781$	−8.68466	−0.310762
$782$	0	0
$783$	14.9309i	0.533586i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	5.26137i	0.187547i	0.995594	+	0.0937737i	$0.0298930\pi$
$787$	5.26137i	0.187547i	−0.995594	+	0.0937737i	$0.970107\pi$
$788$	0	0
$789$	24.6847	0.878797
$790$	0	0
$791$	16.9848	0.603912
$792$	0	0
$793$	− 29.3693i	− 1.04294i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 40.2462i	− 1.42559i	−0.701370	−	0.712797i	$-0.747428\pi$
$797$	− 40.2462i	− 1.42559i	0.701370	−	0.712797i	$-0.252572\pi$
$798$	0	0
$799$	−17.3693	−0.614482
$800$	0	0
$801$	4.73863	0.167431
$802$	0	0
$803$	− 2.00000i	− 0.0705785i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 33.7538i	− 1.18819i
$808$	0	0
$809$	26.4924	0.931424	0.465712	−	0.884936i	$-0.345798\pi$
$809$	26.4924	0.931424	0.465712	+	0.884936i	$0.345798\pi$
$810$	0	0
$811$	−47.8078	−1.67876	−0.839379	−	0.543547i	$-0.817081\pi$
$811$	−47.8078	−1.67876	−0.839379	+	0.543547i	$0.817081\pi$
$812$	0	0
$813$	7.01515i	0.246032i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	9.75379i	0.341242i
$818$	0	0
$819$	−1.75379	−0.0612823
$820$	0	0
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	0	0
$823$	− 19.5076i	− 0.679991i	−0.940427	−	0.339996i	$-0.889574\pi$
$823$	− 19.5076i	− 0.679991i	0.940427	−	0.339996i	$-0.110426\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 28.4924i	− 0.990779i	−0.868671	−	0.495389i	$-0.835025\pi$
$827$	− 28.4924i	− 0.990779i	0.868671	−	0.495389i	$-0.164975\pi$
$828$	0	0
$829$	19.3693	0.672724	0.336362	−	0.941733i	$-0.390803\pi$
$829$	19.3693	0.672724	0.336362	+	0.941733i	$0.390803\pi$
$830$	0	0
$831$	1.75379	0.0608383
$832$	0	0
$833$	16.2462i	0.562898i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 13.5616i	− 0.468756i
$838$	0	0
$839$	44.4924	1.53605	0.768025	−	0.640420i	$-0.221240\pi$
$839$	44.4924	1.53605	0.768025	+	0.640420i	$0.221240\pi$
$840$	0	0
$841$	−21.7926	−0.751469
$842$	0	0
$843$	12.8769i	0.443504i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 1.56155i	− 0.0536556i
$848$	0	0
$849$	−2.13826	−0.0733849
$850$	0	0
$851$	20.8769	0.715651
$852$	0	0
$853$	20.2462i	0.693217i	0.938010	+	0.346609i	$0.112667\pi$
$853$	20.2462i	0.693217i	−0.938010	+	0.346609i	$0.887333\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 29.4233i	− 1.00508i	−0.864554	−	0.502540i	$-0.832399\pi$
$857$	− 29.4233i	− 1.00508i	0.864554	−	0.502540i	$-0.167601\pi$
$858$	0	0
$859$	−47.1231	−1.60782	−0.803910	−	0.594751i	$-0.797251\pi$
$859$	−47.1231	−1.60782	−0.803910	+	0.594751i	$0.797251\pi$
$860$	0	0
$861$	4.87689	0.166204
$862$	0	0
$863$	− 31.6155i	− 1.07621i	−0.842879	−	0.538103i	$-0.819141\pi$
$863$	− 31.6155i	− 1.07621i	0.842879	−	0.538103i	$-0.180859\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	6.73863i	0.228856i
$868$	0	0
$869$	−9.36932	−0.317832
$870$	0	0
$871$	20.4924	0.694359
$872$	0	0
$873$	− 0.630683i	− 0.0213454i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 15.7538i	− 0.531968i	−0.963977	−	0.265984i	$-0.914303\pi$
$877$	− 15.7538i	− 0.531968i	0.963977	−	0.265984i	$-0.0856968\pi$
$878$	0	0
$879$	−28.8769	−0.973993
$880$	0	0
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	0	0
$883$	34.9309i	1.17552i	0.809036	+	0.587759i	$0.199990\pi$
$883$	34.9309i	1.17552i	−0.809036	+	0.587759i	$0.800010\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	13.7538i	0.461807i	0.972977	+	0.230904i	$0.0741682\pi$
$887$	13.7538i	0.461807i	−0.972977	+	0.230904i	$0.925832\pi$
$888$	0	0
$889$	3.50758	0.117640
$890$	0	0
$891$	7.00000	0.234509
$892$	0	0
$893$	− 7.61553i	− 0.254844i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 9.75379i	− 0.325670i
$898$	0	0
$899$	−6.54640	−0.218335
$900$	0	0
$901$	1.56155	0.0520229
$902$	0	0
$903$	− 15.2311i	− 0.506858i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 46.0540i	− 1.52920i	−0.644507	−	0.764599i	$-0.722937\pi$
$907$	− 46.0540i	− 1.52920i	0.644507	−	0.764599i	$-0.277063\pi$
$908$	0	0
$909$	6.87689	0.228092
$910$	0	0
$911$	30.9309	1.02479	0.512393	−	0.858751i	$-0.328759\pi$
$911$	30.9309	1.02479	0.512393	+	0.858751i	$0.328759\pi$
$912$	0	0
$913$	3.12311i	0.103360i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.31534i	0.241574i
$918$	0	0
$919$	8.00000	0.263896	0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000	0.263896	0.131948	+	0.991257i	$0.457877\pi$
$920$	0	0
$921$	24.3845	0.803496
$922$	0	0
$923$	− 17.3693i	− 0.571718i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 3.50758i	− 0.115204i
$928$	0	0
$929$	−7.94602	−0.260701	−0.130350	−	0.991468i	$-0.541610\pi$
$929$	−7.94602	−0.260701	−0.130350	+	0.991468i	$0.541610\pi$
$930$	0	0
$931$	−7.12311	−0.233450
$932$	0	0
$933$	46.1619i	1.51127i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 42.0000i	− 1.37208i	−0.727564	−	0.686040i	$-0.759347\pi$
$937$	− 42.0000i	− 1.37208i	0.727564	−	0.686040i	$-0.240653\pi$
$938$	0	0
$939$	−3.89205	−0.127012
$940$	0	0
$941$	−32.9309	−1.07352	−0.536758	−	0.843736i	$-0.680351\pi$
$941$	−32.9309	−1.07352	−0.536758	+	0.843736i	$0.680351\pi$
$942$	0	0
$943$	− 6.24621i	− 0.203405i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	27.3153i	0.887629i	0.896119	+	0.443815i	$0.146375\pi$
$947$	27.3153i	0.887629i	−0.896119	+	0.443815i	$0.853625\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	0	0
$951$	−52.7926	−1.71192
$952$	0	0
$953$	− 11.1771i	− 0.362061i	−0.983477	−	0.181031i	$-0.942057\pi$
$953$	− 11.1771i	− 0.362061i	0.983477	−	0.181031i	$-0.0579433\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 4.19224i	− 0.135516i
$958$	0	0
$959$	−33.7538	−1.08997
$960$	0	0
$961$	−25.0540	−0.808193
$962$	0	0
$963$	8.98485i	0.289533i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	58.5464i	1.88273i	0.337397	+	0.941363i	$0.390454\pi$
$967$	58.5464i	1.88273i	−0.337397	+	0.941363i	$0.609546\pi$
$968$	0	0
$969$	8.68466	0.278991
$970$	0	0
$971$	10.2462	0.328817	0.164408	−	0.986392i	$-0.447429\pi$
$971$	10.2462	0.328817	0.164408	+	0.986392i	$0.447429\pi$
$972$	0	0
$973$	− 18.7386i	− 0.600733i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 33.1231i	− 1.05970i	−0.848091	−	0.529851i	$-0.822248\pi$
$977$	− 33.1231i	− 1.05970i	0.848091	−	0.529851i	$-0.177752\pi$
$978$	0	0
$979$	−8.43845	−0.269694
$980$	0	0
$981$	8.13826	0.259835
$982$	0	0
$983$	27.1231i	0.865093i	0.901612	+	0.432546i	$0.142385\pi$
$983$	27.1231i	0.865093i	−0.901612	+	0.432546i	$0.857615\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	11.8920i	0.378528i
$988$	0	0
$989$	−19.5076	−0.620305
$990$	0	0
$991$	−32.0000	−1.01651	−0.508257	−	0.861206i	$-0.669710\pi$
$991$	−32.0000	−1.01651	−0.508257	+	0.861206i	$0.669710\pi$
$992$	0	0
$993$	− 38.2462i	− 1.21371i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 43.8617i	− 1.38912i	−0.719437	−	0.694558i	$-0.755600\pi$
$997$	− 43.8617i	− 1.38912i	0.719437	−	0.694558i	$-0.244400\pi$
$998$	0	0
$999$	−37.1771	−1.17623

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.b.t.4049.3		4
4.3	odd	2		2200.2.b.i.1849.2		4
5.2	odd	4		4400.2.a.bj.1.2		2
5.3	odd	4		880.2.a.o.1.1		2
5.4	even	2	inner	4400.2.b.t.4049.2		4
15.8	even	4		7920.2.a.bu.1.1		2
20.3	even	4		440.2.a.e.1.2	✓	2
20.7	even	4		2200.2.a.s.1.1		2
20.19	odd	2		2200.2.b.i.1849.3		4
40.3	even	4		3520.2.a.bp.1.1		2
40.13	odd	4		3520.2.a.bk.1.2		2
55.43	even	4		9680.2.a.bs.1.1		2
60.23	odd	4		3960.2.a.w.1.2		2
220.43	odd	4		4840.2.a.j.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
440.2.a.e.1.2	✓	2	20.3	even	4
880.2.a.o.1.1		2	5.3	odd	4
2200.2.a.s.1.1		2	20.7	even	4
2200.2.b.i.1849.2		4	4.3	odd	2
2200.2.b.i.1849.3		4	20.19	odd	2
3520.2.a.bk.1.2		2	40.13	odd	4
3520.2.a.bp.1.1		2	40.3	even	4
3960.2.a.w.1.2		2	60.23	odd	4
4400.2.a.bj.1.2		2	5.2	odd	4
4400.2.b.t.4049.2		4	5.4	even	2	inner
4400.2.b.t.4049.3		4	1.1	even	1	trivial
4840.2.a.j.1.2		2	220.43	odd	4
7920.2.a.bu.1.1		2	15.8	even	4
9680.2.a.bs.1.1		2	55.43	even	4

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:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.56155i q^{3} -1.56155i q^{7} +0.561553 q^{9} +O(q^{10})\)	\(q+1.56155i q^{3} -1.56155i q^{7} +0.561553 q^{9} -1.00000 q^{11} -2.00000i q^{13} +3.56155i q^{17} -1.56155 q^{19} +2.43845 q^{21} -3.12311i q^{23} +5.56155i q^{27} +2.68466 q^{29} -2.43845 q^{31} -1.56155i q^{33} +6.68466i q^{37} +3.12311 q^{39} +2.00000 q^{41} -6.24621i q^{43} +4.87689i q^{47} +4.56155 q^{49} -5.56155 q^{51} -0.438447i q^{53} -2.43845i q^{57} -7.12311 q^{59} +14.6847 q^{61} -0.876894i q^{63} +10.2462i q^{67} +4.87689 q^{69} +8.68466 q^{71} +2.00000i q^{73} +1.56155i q^{77} +9.36932 q^{79} -7.00000 q^{81} -3.12311i q^{83} +4.19224i q^{87} +8.43845 q^{89} -3.12311 q^{91} -3.80776i q^{93} -1.12311i q^{97} -0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{9}+O(q^{10}) \) 4 * q - 6 * q^9	\( 4 q - 6 q^{9} - 4 q^{11} + 2 q^{19} + 18 q^{21} - 14 q^{29} - 18 q^{31} - 4 q^{39} + 8 q^{41} + 10 q^{49} - 14 q^{51} - 12 q^{59} + 34 q^{61} + 36 q^{69} + 10 q^{71} - 12 q^{79} - 28 q^{81} + 42 q^{89} + 4 q^{91} + 6 q^{99}+O(q^{100}) \) 4 * q - 6 * q^9 - 4 * q^11 + 2 * q^19 + 18 * q^21 - 14 * q^29 - 18 * q^31 - 4 * q^39 + 8 * q^41 + 10 * q^49 - 14 * q^51 - 12 * q^59 + 34 * q^61 + 36 * q^69 + 10 * q^71 - 12 * q^79 - 28 * q^81 + 42 * q^89 + 4 * q^91 + 6 * q^99

Introduction

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