LMFDB - Embedded newform 4400.2.a.bp.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4400,2,Mod(1,4400)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4400.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4400, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4400 = 2^{4} \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4400.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,1,0,0,0,-2,0,3,0,2,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$35.1341768894$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 88)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	4400.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{3} -5.12311 q^{7} +3.56155 q^{9} +1.00000 q^{11} -3.12311 q^{13} -2.00000 q^{17} +4.00000 q^{19} -13.1231 q^{21} +6.56155 q^{23} +1.43845 q^{27} +3.12311 q^{29} +1.43845 q^{31} +2.56155 q^{33} +3.43845 q^{37} -8.00000 q^{39} +7.12311 q^{41} +1.12311 q^{43} +8.00000 q^{47} +19.2462 q^{49} -5.12311 q^{51} +4.24621 q^{53} +10.2462 q^{57} +12.8078 q^{59} -7.12311 q^{61} -18.2462 q^{63} +5.43845 q^{67} +16.8078 q^{69} -3.68466 q^{71} +3.12311 q^{73} -5.12311 q^{77} +2.87689 q^{79} -7.00000 q^{81} +9.12311 q^{83} +8.00000 q^{87} -9.68466 q^{89} +16.0000 q^{91} +3.68466 q^{93} -11.4384 q^{97} +3.56155 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} - 2 q^{7} + 3 q^{9} + 2 q^{11} + 2 q^{13} - 4 q^{17} + 8 q^{19} - 18 q^{21} + 9 q^{23} + 7 q^{27} - 2 q^{29} + 7 q^{31} + q^{33} + 11 q^{37} - 16 q^{39} + 6 q^{41} - 6 q^{43} + 16 q^{47} + 22 q^{49}+ \cdots + 3 q^{99}+O(q^{100}) $ 2 * q + q^3 - 2 * q^7 + 3 * q^9 + 2 * q^11 + 2 * q^13 - 4 * q^17 + 8 * q^19 - 18 * q^21 + 9 * q^23 + 7 * q^27 - 2 * q^29 + 7 * q^31 + q^33 + 11 * q^37 - 16 * q^39 + 6 * q^41 - 6 * q^43 + 16 * q^47 + 22 * q^49 - 2 * q^51 - 8 * q^53 + 4 * q^57 + 5 * q^59 - 6 * q^61 - 20 * q^63 + 15 * q^67 + 13 * q^69 + 5 * q^71 - 2 * q^73 - 2 * q^77 + 14 * q^79 - 14 * q^81 + 10 * q^83 + 16 * q^87 - 7 * q^89 + 32 * q^91 - 5 * q^93 - 27 * q^97 + 3 * q^99

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
$3$	2.56155	1.47891	0.739457	−	0.673204i	$-0.235083\pi$
$3$	2.56155	1.47891	0.739457	+	0.673204i	$0.235083\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−5.12311	−1.93635	−0.968176	−	0.250270i	$-0.919480\pi$
$7$	−5.12311	−1.93635	−0.968176	+	0.250270i	$0.919480\pi$
$8$	0	0
$9$	3.56155	1.18718
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−3.12311	−0.866194	−0.433097	−	0.901347i	$-0.642579\pi$
$13$	−3.12311	−0.866194	−0.433097	+	0.901347i	$0.642579\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	−13.1231	−2.86370
$22$	0	0
$23$	6.56155	1.36818	0.684089	−	0.729398i	$-0.260200\pi$
$23$	6.56155	1.36818	0.684089	+	0.729398i	$0.260200\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	1.43845	0.276829
$28$	0	0
$29$	3.12311	0.579946	0.289973	−	0.957035i	$-0.406354\pi$
$29$	3.12311	0.579946	0.289973	+	0.957035i	$0.406354\pi$
$30$	0	0
$31$	1.43845	0.258353	0.129176	−	0.991622i	$-0.458767\pi$
$31$	1.43845	0.258353	0.129176	+	0.991622i	$0.458767\pi$
$32$	0	0
$33$	2.56155	0.445909
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.43845	0.565277	0.282639	−	0.959226i	$-0.408790\pi$
$37$	3.43845	0.565277	0.282639	+	0.959226i	$0.408790\pi$
$38$	0	0
$39$	−8.00000	−1.28103
$40$	0	0
$41$	7.12311	1.11244	0.556221	−	0.831034i	$-0.312251\pi$
$41$	7.12311	1.11244	0.556221	+	0.831034i	$0.312251\pi$
$42$	0	0
$43$	1.12311	0.171272	0.0856360	−	0.996326i	$-0.472708\pi$
$43$	1.12311	0.171272	0.0856360	+	0.996326i	$0.472708\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	19.2462	2.74946
$50$	0	0
$51$	−5.12311	−0.717378
$52$	0	0
$53$	4.24621	0.583262	0.291631	−	0.956531i	$-0.405802\pi$
$53$	4.24621	0.583262	0.291631	+	0.956531i	$0.405802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	10.2462	1.35714
$58$	0	0
$59$	12.8078	1.66743	0.833714	−	0.552196i	$-0.186210\pi$
$59$	12.8078	1.66743	0.833714	+	0.552196i	$0.186210\pi$
$60$	0	0
$61$	−7.12311	−0.912020	−0.456010	−	0.889975i	$-0.650722\pi$
$61$	−7.12311	−0.912020	−0.456010	+	0.889975i	$0.650722\pi$
$62$	0	0
$63$	−18.2462	−2.29881
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.43845	0.664412	0.332206	−	0.943207i	$-0.392207\pi$
$67$	5.43845	0.664412	0.332206	+	0.943207i	$0.392207\pi$
$68$	0	0
$69$	16.8078	2.02342
$70$	0	0
$71$	−3.68466	−0.437289	−0.218644	−	0.975805i	$-0.570163\pi$
$71$	−3.68466	−0.437289	−0.218644	+	0.975805i	$0.570163\pi$
$72$	0	0
$73$	3.12311	0.365532	0.182766	−	0.983156i	$-0.441495\pi$
$73$	3.12311	0.365532	0.182766	+	0.983156i	$0.441495\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.12311	−0.583832
$78$	0	0
$79$	2.87689	0.323676	0.161838	−	0.986817i	$-0.448258\pi$
$79$	2.87689	0.323676	0.161838	+	0.986817i	$0.448258\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	9.12311	1.00139	0.500695	−	0.865624i	$-0.333078\pi$
$83$	9.12311	1.00139	0.500695	+	0.865624i	$0.333078\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	8.00000	0.857690
$88$	0	0
$89$	−9.68466	−1.02657	−0.513286	−	0.858218i	$-0.671572\pi$
$89$	−9.68466	−1.02657	−0.513286	+	0.858218i	$0.671572\pi$
$90$	0	0
$91$	16.0000	1.67726
$92$	0	0
$93$	3.68466	0.382081
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−11.4384	−1.16140	−0.580699	−	0.814118i	$-0.697221\pi$
$97$	−11.4384	−1.16140	−0.580699	+	0.814118i	$0.697221\pi$
$98$	0	0
$99$	3.56155	0.357950
$100$	0	0
$101$	−2.00000	−0.199007	−0.0995037	−	0.995037i	$-0.531726\pi$
$101$	−2.00000	−0.199007	−0.0995037	+	0.995037i	$0.531726\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	11.3693	1.09911	0.549557	−	0.835456i	$-0.314797\pi$
$107$	11.3693	1.09911	0.549557	+	0.835456i	$0.314797\pi$
$108$	0	0
$109$	−4.24621	−0.406713	−0.203357	−	0.979105i	$-0.565185\pi$
$109$	−4.24621	−0.406713	−0.203357	+	0.979105i	$0.565185\pi$
$110$	0	0
$111$	8.80776	0.835996
$112$	0	0
$113$	4.56155	0.429115	0.214557	−	0.976711i	$-0.431169\pi$
$113$	4.56155	0.429115	0.214557	+	0.976711i	$0.431169\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−11.1231	−1.02833
$118$	0	0
$119$	10.2462	0.939269
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	18.2462	1.64521
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−10.2462	−0.909204	−0.454602	−	0.890695i	$-0.650219\pi$
$127$	−10.2462	−0.909204	−0.454602	+	0.890695i	$0.650219\pi$
$128$	0	0
$129$	2.87689	0.253296
$130$	0	0
$131$	11.3693	0.993342	0.496671	−	0.867939i	$-0.334556\pi$
$131$	11.3693	0.993342	0.496671	+	0.867939i	$0.334556\pi$
$132$	0	0
$133$	−20.4924	−1.77692
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12.5616	1.07321	0.536603	−	0.843835i	$-0.319707\pi$
$137$	12.5616	1.07321	0.536603	+	0.843835i	$0.319707\pi$
$138$	0	0
$139$	−6.87689	−0.583291	−0.291645	−	0.956527i	$-0.594203\pi$
$139$	−6.87689	−0.583291	−0.291645	+	0.956527i	$0.594203\pi$
$140$	0	0
$141$	20.4924	1.72577
$142$	0	0
$143$	−3.12311	−0.261167
$144$	0	0
$145$	0	0
$146$	0	0
$147$	49.3002	4.06621
$148$	0	0
$149$	−12.2462	−1.00325	−0.501624	−	0.865086i	$-0.667264\pi$
$149$	−12.2462	−1.00325	−0.501624	+	0.865086i	$0.667264\pi$
$150$	0	0
$151$	−15.3693	−1.25074	−0.625369	−	0.780329i	$-0.715051\pi$
$151$	−15.3693	−1.25074	−0.625369	+	0.780329i	$0.715051\pi$
$152$	0	0
$153$	−7.12311	−0.575869
$154$	0	0
$155$	0	0
$156$	0	0
$157$	8.56155	0.683286	0.341643	−	0.939830i	$-0.389017\pi$
$157$	8.56155	0.683286	0.341643	+	0.939830i	$0.389017\pi$
$158$	0	0
$159$	10.8769	0.862594
$160$	0	0
$161$	−33.6155	−2.64927
$162$	0	0
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	−3.24621	−0.249709
$170$	0	0
$171$	14.2462	1.08944
$172$	0	0
$173$	4.24621	0.322833	0.161417	−	0.986886i	$-0.448394\pi$
$173$	4.24621	0.322833	0.161417	+	0.986886i	$0.448394\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	32.8078	2.46598
$178$	0	0
$179$	10.5616	0.789408	0.394704	−	0.918808i	$-0.370847\pi$
$179$	10.5616	0.789408	0.394704	+	0.918808i	$0.370847\pi$
$180$	0	0
$181$	−13.6847	−1.01717	−0.508586	−	0.861011i	$-0.669832\pi$
$181$	−13.6847	−1.01717	−0.508586	+	0.861011i	$0.669832\pi$
$182$	0	0
$183$	−18.2462	−1.34880
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−2.00000	−0.146254
$188$	0	0
$189$	−7.36932	−0.536039
$190$	0	0
$191$	14.5616	1.05364	0.526818	−	0.849978i	$-0.323385\pi$
$191$	14.5616	1.05364	0.526818	+	0.849978i	$0.323385\pi$
$192$	0	0
$193$	0.876894	0.0631202	0.0315601	−	0.999502i	$-0.489952\pi$
$193$	0.876894	0.0631202	0.0315601	+	0.999502i	$0.489952\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−18.4924	−1.31753	−0.658765	−	0.752349i	$-0.728921\pi$
$197$	−18.4924	−1.31753	−0.658765	+	0.752349i	$0.728921\pi$
$198$	0	0
$199$	−20.4924	−1.45267	−0.726335	−	0.687341i	$-0.758778\pi$
$199$	−20.4924	−1.45267	−0.726335	+	0.687341i	$0.758778\pi$
$200$	0	0
$201$	13.9309	0.982608
$202$	0	0
$203$	−16.0000	−1.12298
$204$	0	0
$205$	0	0
$206$	0	0
$207$	23.3693	1.62428
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	24.4924	1.68613	0.843064	−	0.537813i	$-0.180749\pi$
$211$	24.4924	1.68613	0.843064	+	0.537813i	$0.180749\pi$
$212$	0	0
$213$	−9.43845	−0.646712
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−7.36932	−0.500262
$218$	0	0
$219$	8.00000	0.540590
$220$	0	0
$221$	6.24621	0.420166
$222$	0	0
$223$	8.80776	0.589812	0.294906	−	0.955526i	$-0.404712\pi$
$223$	8.80776	0.589812	0.294906	+	0.955526i	$0.404712\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	14.8769	0.987414	0.493707	−	0.869628i	$-0.335642\pi$
$227$	14.8769	0.987414	0.493707	+	0.869628i	$0.335642\pi$
$228$	0	0
$229$	2.31534	0.153002	0.0765010	−	0.997070i	$-0.475625\pi$
$229$	2.31534	0.153002	0.0765010	+	0.997070i	$0.475625\pi$
$230$	0	0
$231$	−13.1231	−0.863437
$232$	0	0
$233$	−17.3693	−1.13790	−0.568951	−	0.822371i	$-0.692651\pi$
$233$	−17.3693	−1.13790	−0.568951	+	0.822371i	$0.692651\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	7.36932	0.478689
$238$	0	0
$239$	13.1231	0.848863	0.424432	−	0.905460i	$-0.360474\pi$
$239$	13.1231	0.848863	0.424432	+	0.905460i	$0.360474\pi$
$240$	0	0
$241$	20.8769	1.34480	0.672399	−	0.740188i	$-0.265264\pi$
$241$	20.8769	1.34480	0.672399	+	0.740188i	$0.265264\pi$
$242$	0	0
$243$	−22.2462	−1.42710
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−12.4924	−0.794874
$248$	0	0
$249$	23.3693	1.48097
$250$	0	0
$251$	2.56155	0.161684	0.0808419	−	0.996727i	$-0.474239\pi$
$251$	2.56155	0.161684	0.0808419	+	0.996727i	$0.474239\pi$
$252$	0	0
$253$	6.56155	0.412521
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−28.2462	−1.76195	−0.880975	−	0.473163i	$-0.843112\pi$
$257$	−28.2462	−1.76195	−0.880975	+	0.473163i	$0.843112\pi$
$258$	0	0
$259$	−17.6155	−1.09458
$260$	0	0
$261$	11.1231	0.688503
$262$	0	0
$263$	10.8769	0.670698	0.335349	−	0.942094i	$-0.391146\pi$
$263$	10.8769	0.670698	0.335349	+	0.942094i	$0.391146\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−24.8078	−1.51821
$268$	0	0
$269$	28.7386	1.75223	0.876113	−	0.482106i	$-0.160128\pi$
$269$	28.7386	1.75223	0.876113	+	0.482106i	$0.160128\pi$
$270$	0	0
$271$	−4.49242	−0.272895	−0.136448	−	0.990647i	$-0.543569\pi$
$271$	−4.49242	−0.272895	−0.136448	+	0.990647i	$0.543569\pi$
$272$	0	0
$273$	40.9848	2.48052
$274$	0	0
$275$	0	0
$276$	0	0
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	0	0
$279$	5.12311	0.306712
$280$	0	0
$281$	−0.246211	−0.0146877	−0.00734387	−	0.999973i	$-0.502338\pi$
$281$	−0.246211	−0.0146877	−0.00734387	+	0.999973i	$0.502338\pi$
$282$	0	0
$283$	20.0000	1.18888	0.594438	−	0.804141i	$-0.297374\pi$
$283$	20.0000	1.18888	0.594438	+	0.804141i	$0.297374\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−36.4924	−2.15408
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	−29.3002	−1.71761
$292$	0	0
$293$	−21.3693	−1.24841	−0.624204	−	0.781261i	$-0.714577\pi$
$293$	−21.3693	−1.24841	−0.624204	+	0.781261i	$0.714577\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	1.43845	0.0834672
$298$	0	0
$299$	−20.4924	−1.18511
$300$	0	0
$301$	−5.75379	−0.331643
$302$	0	0
$303$	−5.12311	−0.294315
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0.492423	0.0281040	0.0140520	−	0.999901i	$-0.495527\pi$
$307$	0.492423	0.0281040	0.0140520	+	0.999901i	$0.495527\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−26.2462	−1.48829	−0.744143	−	0.668020i	$-0.767142\pi$
$311$	−26.2462	−1.48829	−0.744143	+	0.668020i	$0.767142\pi$
$312$	0	0
$313$	−10.8078	−0.610891	−0.305445	−	0.952210i	$-0.598805\pi$
$313$	−10.8078	−0.610891	−0.305445	+	0.952210i	$0.598805\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	34.8078	1.95500	0.977499	−	0.210938i	$-0.0676519\pi$
$317$	34.8078	1.95500	0.977499	+	0.210938i	$0.0676519\pi$
$318$	0	0
$319$	3.12311	0.174860
$320$	0	0
$321$	29.1231	1.62549
$322$	0	0
$323$	−8.00000	−0.445132
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−10.8769	−0.601494
$328$	0	0
$329$	−40.9848	−2.25957
$330$	0	0
$331$	−6.06913	−0.333590	−0.166795	−	0.985992i	$-0.553342\pi$
$331$	−6.06913	−0.333590	−0.166795	+	0.985992i	$0.553342\pi$
$332$	0	0
$333$	12.2462	0.671088
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−32.7386	−1.78339	−0.891694	−	0.452640i	$-0.850482\pi$
$337$	−32.7386	−1.78339	−0.891694	+	0.452640i	$0.850482\pi$
$338$	0	0
$339$	11.6847	0.634624
$340$	0	0
$341$	1.43845	0.0778963
$342$	0	0
$343$	−62.7386	−3.38757
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−26.7386	−1.43541	−0.717703	−	0.696350i	$-0.754806\pi$
$347$	−26.7386	−1.43541	−0.717703	+	0.696350i	$0.754806\pi$
$348$	0	0
$349$	−15.7538	−0.843281	−0.421640	−	0.906763i	$-0.638546\pi$
$349$	−15.7538	−0.843281	−0.421640	+	0.906763i	$0.638546\pi$
$350$	0	0
$351$	−4.49242	−0.239788
$352$	0	0
$353$	−13.0540	−0.694793	−0.347397	−	0.937718i	$-0.612934\pi$
$353$	−13.0540	−0.694793	−0.347397	+	0.937718i	$0.612934\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	26.2462	1.38910
$358$	0	0
$359$	28.4924	1.50377	0.751886	−	0.659293i	$-0.229144\pi$
$359$	28.4924	1.50377	0.751886	+	0.659293i	$0.229144\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	2.56155	0.134447
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−5.93087	−0.309589	−0.154794	−	0.987947i	$-0.549472\pi$
$367$	−5.93087	−0.309589	−0.154794	+	0.987947i	$0.549472\pi$
$368$	0	0
$369$	25.3693	1.32067
$370$	0	0
$371$	−21.7538	−1.12940
$372$	0	0
$373$	−8.24621	−0.426973	−0.213486	−	0.976946i	$-0.568482\pi$
$373$	−8.24621	−0.426973	−0.213486	+	0.976946i	$0.568482\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9.75379	−0.502346
$378$	0	0
$379$	−20.8078	−1.06882	−0.534412	−	0.845224i	$-0.679467\pi$
$379$	−20.8078	−1.06882	−0.534412	+	0.845224i	$0.679467\pi$
$380$	0	0
$381$	−26.2462	−1.34463
$382$	0	0
$383$	35.0540	1.79117	0.895587	−	0.444886i	$-0.146756\pi$
$383$	35.0540	1.79117	0.895587	+	0.444886i	$0.146756\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	4.00000	0.203331
$388$	0	0
$389$	−0.561553	−0.0284719	−0.0142359	−	0.999899i	$-0.504532\pi$
$389$	−0.561553	−0.0284719	−0.0142359	+	0.999899i	$0.504532\pi$
$390$	0	0
$391$	−13.1231	−0.663664
$392$	0	0
$393$	29.1231	1.46907
$394$	0	0
$395$	0	0
$396$	0	0
$397$	22.4924	1.12886	0.564431	−	0.825480i	$-0.309096\pi$
$397$	22.4924	1.12886	0.564431	+	0.825480i	$0.309096\pi$
$398$	0	0
$399$	−52.4924	−2.62791
$400$	0	0
$401$	−2.49242	−0.124466	−0.0622328	−	0.998062i	$-0.519822\pi$
$401$	−2.49242	−0.124466	−0.0622328	+	0.998062i	$0.519822\pi$
$402$	0	0
$403$	−4.49242	−0.223784
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.43845	0.170437
$408$	0	0
$409$	−10.4924	−0.518817	−0.259408	−	0.965768i	$-0.583528\pi$
$409$	−10.4924	−0.518817	−0.259408	+	0.965768i	$0.583528\pi$
$410$	0	0
$411$	32.1771	1.58718
$412$	0	0
$413$	−65.6155	−3.22873
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−17.6155	−0.862636
$418$	0	0
$419$	−0.492423	−0.0240564	−0.0120282	−	0.999928i	$-0.503829\pi$
$419$	−0.492423	−0.0240564	−0.0120282	+	0.999928i	$0.503829\pi$
$420$	0	0
$421$	−30.4924	−1.48611	−0.743055	−	0.669230i	$-0.766624\pi$
$421$	−30.4924	−1.48611	−0.743055	+	0.669230i	$0.766624\pi$
$422$	0	0
$423$	28.4924	1.38535
$424$	0	0
$425$	0	0
$426$	0	0
$427$	36.4924	1.76599
$428$	0	0
$429$	−8.00000	−0.386244
$430$	0	0
$431$	−18.8769	−0.909268	−0.454634	−	0.890678i	$-0.650230\pi$
$431$	−18.8769	−0.909268	−0.454634	+	0.890678i	$0.650230\pi$
$432$	0	0
$433$	10.3153	0.495724	0.247862	−	0.968795i	$-0.420272\pi$
$433$	10.3153	0.495724	0.247862	+	0.968795i	$0.420272\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	26.2462	1.25553
$438$	0	0
$439$	28.4924	1.35987	0.679935	−	0.733273i	$-0.262008\pi$
$439$	28.4924	1.35987	0.679935	+	0.733273i	$0.262008\pi$
$440$	0	0
$441$	68.5464	3.26411
$442$	0	0
$443$	23.6847	1.12529	0.562646	−	0.826698i	$-0.309783\pi$
$443$	23.6847	1.12529	0.562646	+	0.826698i	$0.309783\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−31.3693	−1.48372
$448$	0	0
$449$	33.5464	1.58315	0.791576	−	0.611070i	$-0.209261\pi$
$449$	33.5464	1.58315	0.791576	+	0.611070i	$0.209261\pi$
$450$	0	0
$451$	7.12311	0.335414
$452$	0	0
$453$	−39.3693	−1.84973
$454$	0	0
$455$	0	0
$456$	0	0
$457$	33.8617	1.58399	0.791993	−	0.610531i	$-0.209044\pi$
$457$	33.8617	1.58399	0.791993	+	0.610531i	$0.209044\pi$
$458$	0	0
$459$	−2.87689	−0.134282
$460$	0	0
$461$	−7.12311	−0.331756	−0.165878	−	0.986146i	$-0.553046\pi$
$461$	−7.12311	−0.331756	−0.165878	+	0.986146i	$0.553046\pi$
$462$	0	0
$463$	27.6847	1.28662	0.643308	−	0.765608i	$-0.277562\pi$
$463$	27.6847	1.28662	0.643308	+	0.765608i	$0.277562\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−25.3002	−1.17075	−0.585377	−	0.810762i	$-0.699053\pi$
$467$	−25.3002	−1.17075	−0.585377	+	0.810762i	$0.699053\pi$
$468$	0	0
$469$	−27.8617	−1.28654
$470$	0	0
$471$	21.9309	1.01052
$472$	0	0
$473$	1.12311	0.0516405
$474$	0	0
$475$	0	0
$476$	0	0
$477$	15.1231	0.692439
$478$	0	0
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	−10.7386	−0.489640
$482$	0	0
$483$	−86.1080	−3.91805
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−13.9309	−0.631268	−0.315634	−	0.948881i	$-0.602217\pi$
$487$	−13.9309	−0.631268	−0.315634	+	0.948881i	$0.602217\pi$
$488$	0	0
$489$	−10.2462	−0.463350
$490$	0	0
$491$	−30.2462	−1.36499	−0.682496	−	0.730889i	$-0.739106\pi$
$491$	−30.2462	−1.36499	−0.682496	+	0.730889i	$0.739106\pi$
$492$	0	0
$493$	−6.24621	−0.281315
$494$	0	0
$495$	0	0
$496$	0	0
$497$	18.8769	0.846744
$498$	0	0
$499$	−36.9848	−1.65567	−0.827835	−	0.560972i	$-0.810427\pi$
$499$	−36.9848	−1.65567	−0.827835	+	0.560972i	$0.810427\pi$
$500$	0	0
$501$	20.4924	0.915534
$502$	0	0
$503$	9.61553	0.428735	0.214368	−	0.976753i	$-0.431231\pi$
$503$	9.61553	0.428735	0.214368	+	0.976753i	$0.431231\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−8.31534	−0.369297
$508$	0	0
$509$	35.3002	1.56465	0.782327	−	0.622868i	$-0.214033\pi$
$509$	35.3002	1.56465	0.782327	+	0.622868i	$0.214033\pi$
$510$	0	0
$511$	−16.0000	−0.707798
$512$	0	0
$513$	5.75379	0.254036
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.00000	0.351840
$518$	0	0
$519$	10.8769	0.477443
$520$	0	0
$521$	13.6847	0.599536	0.299768	−	0.954012i	$-0.403091\pi$
$521$	13.6847	0.599536	0.299768	+	0.954012i	$0.403091\pi$
$522$	0	0
$523$	−12.0000	−0.524723	−0.262362	−	0.964970i	$-0.584501\pi$
$523$	−12.0000	−0.524723	−0.262362	+	0.964970i	$0.584501\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.87689	−0.125319
$528$	0	0
$529$	20.0540	0.871912
$530$	0	0
$531$	45.6155	1.97955
$532$	0	0
$533$	−22.2462	−0.963590
$534$	0	0
$535$	0	0
$536$	0	0
$537$	27.0540	1.16747
$538$	0	0
$539$	19.2462	0.828993
$540$	0	0
$541$	33.8617	1.45583	0.727915	−	0.685667i	$-0.240490\pi$
$541$	33.8617	1.45583	0.727915	+	0.685667i	$0.240490\pi$
$542$	0	0
$543$	−35.0540	−1.50431
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−25.7538	−1.10115	−0.550576	−	0.834785i	$-0.685592\pi$
$547$	−25.7538	−1.10115	−0.550576	+	0.834785i	$0.685592\pi$
$548$	0	0
$549$	−25.3693	−1.08274
$550$	0	0
$551$	12.4924	0.532195
$552$	0	0
$553$	−14.7386	−0.626750
$554$	0	0
$555$	0	0
$556$	0	0
$557$	20.2462	0.857860	0.428930	−	0.903338i	$-0.358891\pi$
$557$	20.2462	0.857860	0.428930	+	0.903338i	$0.358891\pi$
$558$	0	0
$559$	−3.50758	−0.148355
$560$	0	0
$561$	−5.12311	−0.216298
$562$	0	0
$563$	−8.49242	−0.357913	−0.178956	−	0.983857i	$-0.557272\pi$
$563$	−8.49242	−0.357913	−0.178956	+	0.983857i	$0.557272\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	35.8617	1.50605
$568$	0	0
$569$	−35.1231	−1.47244	−0.736219	−	0.676744i	$-0.763390\pi$
$569$	−35.1231	−1.47244	−0.736219	+	0.676744i	$0.763390\pi$
$570$	0	0
$571$	16.4924	0.690186	0.345093	−	0.938568i	$-0.387847\pi$
$571$	16.4924	0.690186	0.345093	+	0.938568i	$0.387847\pi$
$572$	0	0
$573$	37.3002	1.55824
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−11.4384	−0.476189	−0.238094	−	0.971242i	$-0.576523\pi$
$577$	−11.4384	−0.476189	−0.238094	+	0.971242i	$0.576523\pi$
$578$	0	0
$579$	2.24621	0.0933494
$580$	0	0
$581$	−46.7386	−1.93905
$582$	0	0
$583$	4.24621	0.175860
$584$	0	0
$585$	0	0
$586$	0	0
$587$	8.49242	0.350520	0.175260	−	0.984522i	$-0.443923\pi$
$587$	8.49242	0.350520	0.175260	+	0.984522i	$0.443923\pi$
$588$	0	0
$589$	5.75379	0.237081
$590$	0	0
$591$	−47.3693	−1.94851
$592$	0	0
$593$	21.3693	0.877533	0.438766	−	0.898601i	$-0.355416\pi$
$593$	21.3693	0.877533	0.438766	+	0.898601i	$0.355416\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−52.4924	−2.14837
$598$	0	0
$599$	16.0000	0.653742	0.326871	−	0.945069i	$-0.394006\pi$
$599$	16.0000	0.653742	0.326871	+	0.945069i	$0.394006\pi$
$600$	0	0
$601$	20.2462	0.825860	0.412930	−	0.910763i	$-0.364505\pi$
$601$	20.2462	0.825860	0.412930	+	0.910763i	$0.364505\pi$
$602$	0	0
$603$	19.3693	0.788780
$604$	0	0
$605$	0	0
$606$	0	0
$607$	11.8617	0.481453	0.240727	−	0.970593i	$-0.422614\pi$
$607$	11.8617	0.481453	0.240727	+	0.970593i	$0.422614\pi$
$608$	0	0
$609$	−40.9848	−1.66079
$610$	0	0
$611$	−24.9848	−1.01078
$612$	0	0
$613$	45.8617	1.85234	0.926169	−	0.377108i	$-0.123082\pi$
$613$	45.8617	1.85234	0.926169	+	0.377108i	$0.123082\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−30.4924	−1.22758	−0.613789	−	0.789470i	$-0.710356\pi$
$617$	−30.4924	−1.22758	−0.613789	+	0.789470i	$0.710356\pi$
$618$	0	0
$619$	9.93087	0.399155	0.199578	−	0.979882i	$-0.436043\pi$
$619$	9.93087	0.399155	0.199578	+	0.979882i	$0.436043\pi$
$620$	0	0
$621$	9.43845	0.378752
$622$	0	0
$623$	49.6155	1.98780
$624$	0	0
$625$	0	0
$626$	0	0
$627$	10.2462	0.409194
$628$	0	0
$629$	−6.87689	−0.274200
$630$	0	0
$631$	4.94602	0.196898	0.0984491	−	0.995142i	$-0.468612\pi$
$631$	4.94602	0.196898	0.0984491	+	0.995142i	$0.468612\pi$
$632$	0	0
$633$	62.7386	2.49364
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−60.1080	−2.38156
$638$	0	0
$639$	−13.1231	−0.519142
$640$	0	0
$641$	7.30019	0.288340	0.144170	−	0.989553i	$-0.453949\pi$
$641$	7.30019	0.288340	0.144170	+	0.989553i	$0.453949\pi$
$642$	0	0
$643$	36.1771	1.42668	0.713342	−	0.700816i	$-0.247180\pi$
$643$	36.1771	1.42668	0.713342	+	0.700816i	$0.247180\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	9.43845	0.371064	0.185532	−	0.982638i	$-0.440599\pi$
$647$	9.43845	0.371064	0.185532	+	0.982638i	$0.440599\pi$
$648$	0	0
$649$	12.8078	0.502749
$650$	0	0
$651$	−18.8769	−0.739844
$652$	0	0
$653$	10.1771	0.398260	0.199130	−	0.979973i	$-0.436188\pi$
$653$	10.1771	0.398260	0.199130	+	0.979973i	$0.436188\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	11.1231	0.433954
$658$	0	0
$659$	−29.6155	−1.15366	−0.576829	−	0.816865i	$-0.695710\pi$
$659$	−29.6155	−1.15366	−0.576829	+	0.816865i	$0.695710\pi$
$660$	0	0
$661$	21.1922	0.824282	0.412141	−	0.911120i	$-0.364781\pi$
$661$	21.1922	0.824282	0.412141	+	0.911120i	$0.364781\pi$
$662$	0	0
$663$	16.0000	0.621389
$664$	0	0
$665$	0	0
$666$	0	0
$667$	20.4924	0.793470
$668$	0	0
$669$	22.5616	0.872280
$670$	0	0
$671$	−7.12311	−0.274984
$672$	0	0
$673$	49.2311	1.89772	0.948859	−	0.315701i	$-0.102239\pi$
$673$	49.2311	1.89772	0.948859	+	0.315701i	$0.102239\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−28.7386	−1.10452	−0.552258	−	0.833673i	$-0.686234\pi$
$677$	−28.7386	−1.10452	−0.552258	+	0.833673i	$0.686234\pi$
$678$	0	0
$679$	58.6004	2.24888
$680$	0	0
$681$	38.1080	1.46030
$682$	0	0
$683$	−42.7386	−1.63535	−0.817674	−	0.575681i	$-0.804737\pi$
$683$	−42.7386	−1.63535	−0.817674	+	0.575681i	$0.804737\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	5.93087	0.226277
$688$	0	0
$689$	−13.2614	−0.505218
$690$	0	0
$691$	47.0540	1.79002	0.895009	−	0.446049i	$-0.147169\pi$
$691$	47.0540	1.79002	0.895009	+	0.446049i	$0.147169\pi$
$692$	0	0
$693$	−18.2462	−0.693116
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−14.2462	−0.539614
$698$	0	0
$699$	−44.4924	−1.68286
$700$	0	0
$701$	17.5076	0.661252	0.330626	−	0.943762i	$-0.392740\pi$
$701$	17.5076	0.661252	0.330626	+	0.943762i	$0.392740\pi$
$702$	0	0
$703$	13.7538	0.518734
$704$	0	0
$705$	0	0
$706$	0	0
$707$	10.2462	0.385348
$708$	0	0
$709$	22.8078	0.856564	0.428282	−	0.903645i	$-0.359119\pi$
$709$	22.8078	0.856564	0.428282	+	0.903645i	$0.359119\pi$
$710$	0	0
$711$	10.2462	0.384263
$712$	0	0
$713$	9.43845	0.353473
$714$	0	0
$715$	0	0
$716$	0	0
$717$	33.6155	1.25539
$718$	0	0
$719$	26.4233	0.985423	0.492711	−	0.870193i	$-0.336006\pi$
$719$	26.4233	0.985423	0.492711	+	0.870193i	$0.336006\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	53.4773	1.98884
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−35.6847	−1.32347	−0.661735	−	0.749737i	$-0.730180\pi$
$727$	−35.6847	−1.32347	−0.661735	+	0.749737i	$0.730180\pi$
$728$	0	0
$729$	−35.9848	−1.33277
$730$	0	0
$731$	−2.24621	−0.0830791
$732$	0	0
$733$	7.12311	0.263098	0.131549	−	0.991310i	$-0.458005\pi$
$733$	7.12311	0.263098	0.131549	+	0.991310i	$0.458005\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.43845	0.200328
$738$	0	0
$739$	27.3693	1.00680	0.503398	−	0.864054i	$-0.332083\pi$
$739$	27.3693	1.00680	0.503398	+	0.864054i	$0.332083\pi$
$740$	0	0
$741$	−32.0000	−1.17555
$742$	0	0
$743$	−38.7386	−1.42118	−0.710591	−	0.703605i	$-0.751572\pi$
$743$	−38.7386	−1.42118	−0.710591	+	0.703605i	$0.751572\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	32.4924	1.18884
$748$	0	0
$749$	−58.2462	−2.12827
$750$	0	0
$751$	−1.43845	−0.0524897	−0.0262448	−	0.999656i	$-0.508355\pi$
$751$	−1.43845	−0.0524897	−0.0262448	+	0.999656i	$0.508355\pi$
$752$	0	0
$753$	6.56155	0.239116
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−32.2462	−1.17201	−0.586004	−	0.810308i	$-0.699300\pi$
$757$	−32.2462	−1.17201	−0.586004	+	0.810308i	$0.699300\pi$
$758$	0	0
$759$	16.8078	0.610083
$760$	0	0
$761$	−3.12311	−0.113212	−0.0566062	−	0.998397i	$-0.518028\pi$
$761$	−3.12311	−0.113212	−0.0566062	+	0.998397i	$0.518028\pi$
$762$	0	0
$763$	21.7538	0.787540
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−40.0000	−1.44432
$768$	0	0
$769$	−15.6155	−0.563110	−0.281555	−	0.959545i	$-0.590850\pi$
$769$	−15.6155	−0.563110	−0.281555	+	0.959545i	$0.590850\pi$
$770$	0	0
$771$	−72.3542	−2.60577
$772$	0	0
$773$	8.73863	0.314307	0.157153	−	0.987574i	$-0.449768\pi$
$773$	8.73863	0.314307	0.157153	+	0.987574i	$0.449768\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−45.1231	−1.61878
$778$	0	0
$779$	28.4924	1.02085
$780$	0	0
$781$	−3.68466	−0.131847
$782$	0	0
$783$	4.49242	0.160546
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−46.2462	−1.64850	−0.824250	−	0.566226i	$-0.808403\pi$
$787$	−46.2462	−1.64850	−0.824250	+	0.566226i	$0.808403\pi$
$788$	0	0
$789$	27.8617	0.991904
$790$	0	0
$791$	−23.3693	−0.830917
$792$	0	0
$793$	22.2462	0.789986
$794$	0	0
$795$	0	0
$796$	0	0
$797$	34.8078	1.23295	0.616477	−	0.787373i	$-0.288559\pi$
$797$	34.8078	1.23295	0.616477	+	0.787373i	$0.288559\pi$
$798$	0	0
$799$	−16.0000	−0.566039
$800$	0	0
$801$	−34.4924	−1.21873
$802$	0	0
$803$	3.12311	0.110212
$804$	0	0
$805$	0	0
$806$	0	0
$807$	73.6155	2.59139
$808$	0	0
$809$	−4.38447	−0.154150	−0.0770749	−	0.997025i	$-0.524558\pi$
$809$	−4.38447	−0.154150	−0.0770749	+	0.997025i	$0.524558\pi$
$810$	0	0
$811$	−1.12311	−0.0394376	−0.0197188	−	0.999806i	$-0.506277\pi$
$811$	−1.12311	−0.0394376	−0.0197188	+	0.999806i	$0.506277\pi$
$812$	0	0
$813$	−11.5076	−0.403588
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.49242	0.157170
$818$	0	0
$819$	56.9848	1.99121
$820$	0	0
$821$	22.9848	0.802177	0.401088	−	0.916039i	$-0.368632\pi$
$821$	22.9848	0.802177	0.401088	+	0.916039i	$0.368632\pi$
$822$	0	0
$823$	15.5464	0.541913	0.270957	−	0.962592i	$-0.412660\pi$
$823$	15.5464	0.541913	0.270957	+	0.962592i	$0.412660\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−10.7386	−0.373419	−0.186709	−	0.982415i	$-0.559782\pi$
$827$	−10.7386	−0.373419	−0.186709	+	0.982415i	$0.559782\pi$
$828$	0	0
$829$	43.9309	1.52578	0.762891	−	0.646527i	$-0.223779\pi$
$829$	43.9309	1.52578	0.762891	+	0.646527i	$0.223779\pi$
$830$	0	0
$831$	46.1080	1.59947
$832$	0	0
$833$	−38.4924	−1.33368
$834$	0	0
$835$	0	0
$836$	0	0
$837$	2.06913	0.0715196
$838$	0	0
$839$	−0.807764	−0.0278871	−0.0139436	−	0.999903i	$-0.504439\pi$
$839$	−0.807764	−0.0278871	−0.0139436	+	0.999903i	$0.504439\pi$
$840$	0	0
$841$	−19.2462	−0.663662
$842$	0	0
$843$	−0.630683	−0.0217219
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−5.12311	−0.176032
$848$	0	0
$849$	51.2311	1.75825
$850$	0	0
$851$	22.5616	0.773400
$852$	0	0
$853$	13.5076	0.462491	0.231245	−	0.972895i	$-0.425720\pi$
$853$	13.5076	0.462491	0.231245	+	0.972895i	$0.425720\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	44.1080	1.50670	0.753349	−	0.657620i	$-0.228437\pi$
$857$	44.1080	1.50670	0.753349	+	0.657620i	$0.228437\pi$
$858$	0	0
$859$	7.05398	0.240679	0.120339	−	0.992733i	$-0.461602\pi$
$859$	7.05398	0.240679	0.120339	+	0.992733i	$0.461602\pi$
$860$	0	0
$861$	−93.4773	−3.18570
$862$	0	0
$863$	−3.50758	−0.119399	−0.0596997	−	0.998216i	$-0.519014\pi$
$863$	−3.50758	−0.119399	−0.0596997	+	0.998216i	$0.519014\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−33.3002	−1.13093
$868$	0	0
$869$	2.87689	0.0975920
$870$	0	0
$871$	−16.9848	−0.575510
$872$	0	0
$873$	−40.7386	−1.37879
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−30.6307	−1.03432	−0.517162	−	0.855887i	$-0.673012\pi$
$877$	−30.6307	−1.03432	−0.517162	+	0.855887i	$0.673012\pi$
$878$	0	0
$879$	−54.7386	−1.84629
$880$	0	0
$881$	−19.3002	−0.650240	−0.325120	−	0.945673i	$-0.605405\pi$
$881$	−19.3002	−0.650240	−0.325120	+	0.945673i	$0.605405\pi$
$882$	0	0
$883$	−24.4924	−0.824236	−0.412118	−	0.911131i	$-0.635211\pi$
$883$	−24.4924	−0.824236	−0.412118	+	0.911131i	$0.635211\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−9.61553	−0.322858	−0.161429	−	0.986884i	$-0.551610\pi$
$887$	−9.61553	−0.322858	−0.161429	+	0.986884i	$0.551610\pi$
$888$	0	0
$889$	52.4924	1.76054
$890$	0	0
$891$	−7.00000	−0.234509
$892$	0	0
$893$	32.0000	1.07084
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−52.4924	−1.75267
$898$	0	0
$899$	4.49242	0.149831
$900$	0	0
$901$	−8.49242	−0.282924
$902$	0	0
$903$	−14.7386	−0.490471
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−16.4924	−0.547622	−0.273811	−	0.961784i	$-0.588284\pi$
$907$	−16.4924	−0.547622	−0.273811	+	0.961784i	$0.588284\pi$
$908$	0	0
$909$	−7.12311	−0.236259
$910$	0	0
$911$	−22.7386	−0.753365	−0.376682	−	0.926343i	$-0.622935\pi$
$911$	−22.7386	−0.753365	−0.376682	+	0.926343i	$0.622935\pi$
$912$	0	0
$913$	9.12311	0.301931
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−58.2462	−1.92346
$918$	0	0
$919$	−31.3693	−1.03478	−0.517389	−	0.855750i	$-0.673096\pi$
$919$	−31.3693	−1.03478	−0.517389	+	0.855750i	$0.673096\pi$
$920$	0	0
$921$	1.26137	0.0415634
$922$	0	0
$923$	11.5076	0.378777
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	13.5076	0.443169	0.221585	−	0.975141i	$-0.428877\pi$
$929$	13.5076	0.443169	0.221585	+	0.975141i	$0.428877\pi$
$930$	0	0
$931$	76.9848	2.52308
$932$	0	0
$933$	−67.2311	−2.20105
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−32.1080	−1.04892	−0.524461	−	0.851435i	$-0.675733\pi$
$937$	−32.1080	−1.04892	−0.524461	+	0.851435i	$0.675733\pi$
$938$	0	0
$939$	−27.6847	−0.903455
$940$	0	0
$941$	−15.7538	−0.513559	−0.256779	−	0.966470i	$-0.582661\pi$
$941$	−15.7538	−0.513559	−0.256779	+	0.966470i	$0.582661\pi$
$942$	0	0
$943$	46.7386	1.52202
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−0.315342	−0.0102472	−0.00512361	−	0.999987i	$-0.501631\pi$
$947$	−0.315342	−0.0102472	−0.00512361	+	0.999987i	$0.501631\pi$
$948$	0	0
$949$	−9.75379	−0.316621
$950$	0	0
$951$	89.1619	2.89127
$952$	0	0
$953$	16.2462	0.526266	0.263133	−	0.964760i	$-0.415244\pi$
$953$	16.2462	0.526266	0.263133	+	0.964760i	$0.415244\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	8.00000	0.258603
$958$	0	0
$959$	−64.3542	−2.07810
$960$	0	0
$961$	−28.9309	−0.933254
$962$	0	0
$963$	40.4924	1.30485
$964$	0	0
$965$	0	0
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	0	0
$969$	−20.4924	−0.658311
$970$	0	0
$971$	−35.5464	−1.14074	−0.570369	−	0.821389i	$-0.693200\pi$
$971$	−35.5464	−1.14074	−0.570369	+	0.821389i	$0.693200\pi$
$972$	0	0
$973$	35.2311	1.12946
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−33.1922	−1.06191	−0.530957	−	0.847399i	$-0.678167\pi$
$977$	−33.1922	−1.06191	−0.530957	+	0.847399i	$0.678167\pi$
$978$	0	0
$979$	−9.68466	−0.309523
$980$	0	0
$981$	−15.1231	−0.482844
$982$	0	0
$983$	2.06913	0.0659950	0.0329975	−	0.999455i	$-0.489495\pi$
$983$	2.06913	0.0659950	0.0329975	+	0.999455i	$0.489495\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−104.985	−3.34170
$988$	0	0
$989$	7.36932	0.234331
$990$	0	0
$991$	−28.4924	−0.905092	−0.452546	−	0.891741i	$-0.649484\pi$
$991$	−28.4924	−0.905092	−0.452546	+	0.891741i	$0.649484\pi$
$992$	0	0
$993$	−15.5464	−0.493350
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−35.7538	−1.13233	−0.566167	−	0.824291i	$-0.691574\pi$
$997$	−35.7538	−1.13233	−0.566167	+	0.824291i	$0.691574\pi$
$998$	0	0
$999$	4.94602	0.156485

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4400.2.a.bp.1.2		2
4.3	odd	2		2200.2.a.o.1.1		2
5.2	odd	4		4400.2.b.v.4049.1		4
5.3	odd	4		4400.2.b.v.4049.4		4
5.4	even	2		176.2.a.d.1.1		2
15.14	odd	2		1584.2.a.t.1.2		2
20.3	even	4		2200.2.b.g.1849.1		4
20.7	even	4		2200.2.b.g.1849.4		4
20.19	odd	2		88.2.a.b.1.2	✓	2
35.34	odd	2		8624.2.a.cb.1.2		2
40.19	odd	2		704.2.a.m.1.1		2
40.29	even	2		704.2.a.p.1.2		2
55.54	odd	2		1936.2.a.r.1.1		2
60.59	even	2		792.2.a.h.1.2		2
80.19	odd	4		2816.2.c.w.1409.4		4
80.29	even	4		2816.2.c.p.1409.1		4
80.59	odd	4		2816.2.c.w.1409.1		4
80.69	even	4		2816.2.c.p.1409.4		4
120.29	odd	2		6336.2.a.cx.1.1		2
120.59	even	2		6336.2.a.cu.1.1		2
140.139	even	2		4312.2.a.n.1.1		2
220.19	even	10		968.2.i.q.9.1		8
220.39	even	10		968.2.i.q.729.2		8
220.59	odd	10		968.2.i.r.753.1		8
220.79	even	10		968.2.i.q.81.2		8
220.119	odd	10		968.2.i.r.81.2		8
220.139	even	10		968.2.i.q.753.1		8
220.159	odd	10		968.2.i.r.729.2		8
220.179	odd	10		968.2.i.r.9.1		8
220.219	even	2		968.2.a.j.1.2		2
440.109	odd	2		7744.2.a.cl.1.2		2
440.219	even	2		7744.2.a.by.1.1		2
660.659	odd	2		8712.2.a.bb.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.a.b.1.2	✓	2	20.19	odd	2
176.2.a.d.1.1		2	5.4	even	2
704.2.a.m.1.1		2	40.19	odd	2
704.2.a.p.1.2		2	40.29	even	2
792.2.a.h.1.2		2	60.59	even	2
968.2.a.j.1.2		2	220.219	even	2
968.2.i.q.9.1		8	220.19	even	10
968.2.i.q.81.2		8	220.79	even	10
968.2.i.q.729.2		8	220.39	even	10
968.2.i.q.753.1		8	220.139	even	10
968.2.i.r.9.1		8	220.179	odd	10
968.2.i.r.81.2		8	220.119	odd	10
968.2.i.r.729.2		8	220.159	odd	10
968.2.i.r.753.1		8	220.59	odd	10
1584.2.a.t.1.2		2	15.14	odd	2
1936.2.a.r.1.1		2	55.54	odd	2
2200.2.a.o.1.1		2	4.3	odd	2
2200.2.b.g.1849.1		4	20.3	even	4
2200.2.b.g.1849.4		4	20.7	even	4
2816.2.c.p.1409.1		4	80.29	even	4
2816.2.c.p.1409.4		4	80.69	even	4
2816.2.c.w.1409.1		4	80.59	odd	4
2816.2.c.w.1409.4		4	80.19	odd	4
4312.2.a.n.1.1		2	140.139	even	2
4400.2.a.bp.1.2		2	1.1	even	1	trivial
4400.2.b.v.4049.1		4	5.2	odd	4
4400.2.b.v.4049.4		4	5.3	odd	4
6336.2.a.cu.1.1		2	120.59	even	2
6336.2.a.cx.1.1		2	120.29	odd	2
7744.2.a.by.1.1		2	440.219	even	2
7744.2.a.cl.1.2		2	440.109	odd	2
8624.2.a.cb.1.2		2	35.34	odd	2
8712.2.a.bb.1.2		2	660.659	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4400.a (trivial)

\(f(q)\)	\(=\)	\(q+2.56155 q^{3} -5.12311 q^{7} +3.56155 q^{9} +1.00000 q^{11} -3.12311 q^{13} -2.00000 q^{17} +4.00000 q^{19} -13.1231 q^{21} +6.56155 q^{23} +1.43845 q^{27} +3.12311 q^{29} +1.43845 q^{31} +2.56155 q^{33} +3.43845 q^{37} -8.00000 q^{39} +7.12311 q^{41} +1.12311 q^{43} +8.00000 q^{47} +19.2462 q^{49} -5.12311 q^{51} +4.24621 q^{53} +10.2462 q^{57} +12.8078 q^{59} -7.12311 q^{61} -18.2462 q^{63} +5.43845 q^{67} +16.8078 q^{69} -3.68466 q^{71} +3.12311 q^{73} -5.12311 q^{77} +2.87689 q^{79} -7.00000 q^{81} +9.12311 q^{83} +8.00000 q^{87} -9.68466 q^{89} +16.0000 q^{91} +3.68466 q^{93} -11.4384 q^{97} +3.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} - 2 q^{7} + 3 q^{9} + 2 q^{11} + 2 q^{13} - 4 q^{17} + 8 q^{19} - 18 q^{21} + 9 q^{23} + 7 q^{27} - 2 q^{29} + 7 q^{31} + q^{33} + 11 q^{37} - 16 q^{39} + 6 q^{41} - 6 q^{43} + 16 q^{47} + 22 q^{49}+ \cdots + 3 q^{99}+O(q^{100}) \) 2 * q + q^3 - 2 * q^7 + 3 * q^9 + 2 * q^11 + 2 * q^13 - 4 * q^17 + 8 * q^19 - 18 * q^21 + 9 * q^23 + 7 * q^27 - 2 * q^29 + 7 * q^31 + q^33 + 11 * q^37 - 16 * q^39 + 6 * q^41 - 6 * q^43 + 16 * q^47 + 22 * q^49 - 2 * q^51 - 8 * q^53 + 4 * q^57 + 5 * q^59 - 6 * q^61 - 20 * q^63 + 15 * q^67 + 13 * q^69 + 5 * q^71 - 2 * q^73 - 2 * q^77 + 14 * q^79 - 14 * q^81 + 10 * q^83 + 16 * q^87 - 7 * q^89 + 32 * q^91 - 5 * q^93 - 27 * q^97 + 3 * q^99

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