LMFDB - Embedded newform 5520.2.a.bh.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5520,2,Mod(1,5520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5520.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:	$44.0774219157$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	5520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +2.56155 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} +2.56155 q^{7} +1.00000 q^{9} +5.12311 q^{11} +2.00000 q^{13} +1.00000 q^{15} +4.56155 q^{17} -1.12311 q^{19} -2.56155 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +8.56155 q^{29} +3.68466 q^{31} -5.12311 q^{33} -2.56155 q^{35} +0.561553 q^{37} -2.00000 q^{39} +3.43845 q^{41} +6.24621 q^{43} -1.00000 q^{45} -8.00000 q^{47} -0.438447 q^{49} -4.56155 q^{51} +0.561553 q^{53} -5.12311 q^{55} +1.12311 q^{57} -6.56155 q^{59} +13.3693 q^{61} +2.56155 q^{63} -2.00000 q^{65} +2.56155 q^{67} -1.00000 q^{69} -11.6847 q^{71} +2.00000 q^{73} -1.00000 q^{75} +13.1231 q^{77} -10.2462 q^{79} +1.00000 q^{81} +0.315342 q^{83} -4.56155 q^{85} -8.56155 q^{87} -3.12311 q^{89} +5.12311 q^{91} -3.68466 q^{93} +1.12311 q^{95} -6.00000 q^{97} +5.12311 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 + q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} + q^{7} + 2 q^{9} + 2 q^{11} + 4 q^{13} + 2 q^{15} + 5 q^{17} + 6 q^{19} - q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} + 13 q^{29} - 5 q^{31} - 2 q^{33} - q^{35} - 3 q^{37} - 4 q^{39} + 11 q^{41} - 4 q^{43} - 2 q^{45} - 16 q^{47} - 5 q^{49} - 5 q^{51} - 3 q^{53} - 2 q^{55} - 6 q^{57} - 9 q^{59} + 2 q^{61} + q^{63} - 4 q^{65} + q^{67} - 2 q^{69} - 11 q^{71} + 4 q^{73} - 2 q^{75} + 18 q^{77} - 4 q^{79} + 2 q^{81} + 13 q^{83} - 5 q^{85} - 13 q^{87} + 2 q^{89} + 2 q^{91} + 5 q^{93} - 6 q^{95} - 12 q^{97} + 2 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + q^7 + 2 * q^9 + 2 * q^11 + 4 * q^13 + 2 * q^15 + 5 * q^17 + 6 * q^19 - q^21 + 2 * q^23 + 2 * q^25 - 2 * q^27 + 13 * q^29 - 5 * q^31 - 2 * q^33 - q^35 - 3 * q^37 - 4 * q^39 + 11 * q^41 - 4 * q^43 - 2 * q^45 - 16 * q^47 - 5 * q^49 - 5 * q^51 - 3 * q^53 - 2 * q^55 - 6 * q^57 - 9 * q^59 + 2 * q^61 + q^63 - 4 * q^65 + q^67 - 2 * q^69 - 11 * q^71 + 4 * q^73 - 2 * q^75 + 18 * q^77 - 4 * q^79 + 2 * q^81 + 13 * q^83 - 5 * q^85 - 13 * q^87 + 2 * q^89 + 2 * q^91 + 5 * q^93 - 6 * q^95 - 12 * q^97 + 2 * 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 $

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.56155	0.968176	0.484088	−	0.875019i	$-0.339151\pi$
$7$	2.56155	0.968176	0.484088	+	0.875019i	$0.339151\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.12311	1.54467	0.772337	−	0.635213i	$-0.219088\pi$
$11$	5.12311	1.54467	0.772337	+	0.635213i	$0.219088\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	4.56155	1.10634	0.553170	−	0.833069i	$-0.313418\pi$
$17$	4.56155	1.10634	0.553170	+	0.833069i	$0.313418\pi$
$18$	0	0
$19$	−1.12311	−0.257658	−0.128829	−	0.991667i	$-0.541122\pi$
$19$	−1.12311	−0.257658	−0.128829	+	0.991667i	$0.541122\pi$
$20$	0	0
$21$	−2.56155	−0.558977
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.56155	1.58984	0.794920	−	0.606714i	$-0.207513\pi$
$29$	8.56155	1.58984	0.794920	+	0.606714i	$0.207513\pi$
$30$	0	0
$31$	3.68466	0.661784	0.330892	−	0.943669i	$-0.392650\pi$
$31$	3.68466	0.661784	0.330892	+	0.943669i	$0.392650\pi$
$32$	0	0
$33$	−5.12311	−0.891818
$34$	0	0
$35$	−2.56155	−0.432981
$36$	0	0
$37$	0.561553	0.0923187	0.0461594	−	0.998934i	$-0.485302\pi$
$37$	0.561553	0.0923187	0.0461594	+	0.998934i	$0.485302\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	0	0
$41$	3.43845	0.536995	0.268498	−	0.963280i	$-0.413473\pi$
$41$	3.43845	0.536995	0.268498	+	0.963280i	$0.413473\pi$
$42$	0	0
$43$	6.24621	0.952538	0.476269	−	0.879300i	$-0.341989\pi$
$43$	6.24621	0.952538	0.476269	+	0.879300i	$0.341989\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	−0.438447	−0.0626353
$50$	0	0
$51$	−4.56155	−0.638745
$52$	0	0
$53$	0.561553	0.0771352	0.0385676	−	0.999256i	$-0.487721\pi$
$53$	0.561553	0.0771352	0.0385676	+	0.999256i	$0.487721\pi$
$54$	0	0
$55$	−5.12311	−0.690799
$56$	0	0
$57$	1.12311	0.148759
$58$	0	0
$59$	−6.56155	−0.854241	−0.427121	−	0.904195i	$-0.640472\pi$
$59$	−6.56155	−0.854241	−0.427121	+	0.904195i	$0.640472\pi$
$60$	0	0
$61$	13.3693	1.71177	0.855883	−	0.517170i	$-0.173014\pi$
$61$	13.3693	1.71177	0.855883	+	0.517170i	$0.173014\pi$
$62$	0	0
$63$	2.56155	0.322725
$64$	0	0
$65$	−2.00000	−0.248069
$66$	0	0
$67$	2.56155	0.312943	0.156472	−	0.987682i	$-0.449988\pi$
$67$	2.56155	0.312943	0.156472	+	0.987682i	$0.449988\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−11.6847	−1.38671	−0.693357	−	0.720594i	$-0.743869\pi$
$71$	−11.6847	−1.38671	−0.693357	+	0.720594i	$0.743869\pi$
$72$	0	0
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	13.1231	1.49552
$78$	0	0
$79$	−10.2462	−1.15279	−0.576394	−	0.817172i	$-0.695541\pi$
$79$	−10.2462	−1.15279	−0.576394	+	0.817172i	$0.695541\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.315342	0.0346132	0.0173066	−	0.999850i	$-0.494491\pi$
$83$	0.315342	0.0346132	0.0173066	+	0.999850i	$0.494491\pi$
$84$	0	0
$85$	−4.56155	−0.494770
$86$	0	0
$87$	−8.56155	−0.917895
$88$	0	0
$89$	−3.12311	−0.331049	−0.165524	−	0.986206i	$-0.552932\pi$
$89$	−3.12311	−0.331049	−0.165524	+	0.986206i	$0.552932\pi$
$90$	0	0
$91$	5.12311	0.537047
$92$	0	0
$93$	−3.68466	−0.382081
$94$	0	0
$95$	1.12311	0.115228
$96$	0	0
$97$	−6.00000	−0.609208	−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000	−0.609208	−0.304604	+	0.952479i	$0.598524\pi$
$98$	0	0
$99$	5.12311	0.514891
$100$	0	0
$101$	6.31534	0.628400	0.314200	−	0.949357i	$-0.398264\pi$
$101$	6.31534	0.628400	0.314200	+	0.949357i	$0.398264\pi$
$102$	0	0
$103$	−14.2462	−1.40372	−0.701860	−	0.712314i	$-0.747647\pi$
$103$	−14.2462	−1.40372	−0.701860	+	0.712314i	$0.747647\pi$
$104$	0	0
$105$	2.56155	0.249982
$106$	0	0
$107$	−7.68466	−0.742904	−0.371452	−	0.928452i	$-0.621140\pi$
$107$	−7.68466	−0.742904	−0.371452	+	0.928452i	$0.621140\pi$
$108$	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
$111$	−0.561553	−0.0533002
$112$	0	0
$113$	2.31534	0.217809	0.108905	−	0.994052i	$-0.465266\pi$
$113$	2.31534	0.217809	0.108905	+	0.994052i	$0.465266\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	2.00000	0.184900
$118$	0	0
$119$	11.6847	1.07113
$120$	0	0
$121$	15.2462	1.38602
$122$	0	0
$123$	−3.43845	−0.310034
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−14.2462	−1.26415	−0.632073	−	0.774909i	$-0.717796\pi$
$127$	−14.2462	−1.26415	−0.632073	+	0.774909i	$0.717796\pi$
$128$	0	0
$129$	−6.24621	−0.549948
$130$	0	0
$131$	10.2462	0.895216	0.447608	−	0.894230i	$-0.352276\pi$
$131$	10.2462	0.895216	0.447608	+	0.894230i	$0.352276\pi$
$132$	0	0
$133$	−2.87689	−0.249458
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	14.0000	1.19610	0.598050	−	0.801459i	$-0.295942\pi$
$137$	14.0000	1.19610	0.598050	+	0.801459i	$0.295942\pi$
$138$	0	0
$139$	12.8078	1.08634	0.543170	−	0.839623i	$-0.317224\pi$
$139$	12.8078	1.08634	0.543170	+	0.839623i	$0.317224\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	10.2462	0.856831
$144$	0	0
$145$	−8.56155	−0.710998
$146$	0	0
$147$	0.438447	0.0361625
$148$	0	0
$149$	−13.3693	−1.09526	−0.547629	−	0.836722i	$-0.684469\pi$
$149$	−13.3693	−1.09526	−0.547629	+	0.836722i	$0.684469\pi$
$150$	0	0
$151$	−10.2462	−0.833825	−0.416912	−	0.908947i	$-0.636888\pi$
$151$	−10.2462	−0.833825	−0.416912	+	0.908947i	$0.636888\pi$
$152$	0	0
$153$	4.56155	0.368780
$154$	0	0
$155$	−3.68466	−0.295959
$156$	0	0
$157$	−17.0540	−1.36106	−0.680528	−	0.732722i	$-0.738249\pi$
$157$	−17.0540	−1.36106	−0.680528	+	0.732722i	$0.738249\pi$
$158$	0	0
$159$	−0.561553	−0.0445340
$160$	0	0
$161$	2.56155	0.201879
$162$	0	0
$163$	−19.3693	−1.51712	−0.758561	−	0.651602i	$-0.774097\pi$
$163$	−19.3693	−1.51712	−0.758561	+	0.651602i	$0.774097\pi$
$164$	0	0
$165$	5.12311	0.398833
$166$	0	0
$167$	12.4924	0.966693	0.483346	−	0.875429i	$-0.339421\pi$
$167$	12.4924	0.966693	0.483346	+	0.875429i	$0.339421\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−1.12311	−0.0858860
$172$	0	0
$173$	15.1231	1.14979	0.574894	−	0.818228i	$-0.305043\pi$
$173$	15.1231	1.14979	0.574894	+	0.818228i	$0.305043\pi$
$174$	0	0
$175$	2.56155	0.193635
$176$	0	0
$177$	6.56155	0.493197
$178$	0	0
$179$	−12.4924	−0.933727	−0.466864	−	0.884329i	$-0.654616\pi$
$179$	−12.4924	−0.933727	−0.466864	+	0.884329i	$0.654616\pi$
$180$	0	0
$181$	0.876894	0.0651790	0.0325895	−	0.999469i	$-0.489625\pi$
$181$	0.876894	0.0651790	0.0325895	+	0.999469i	$0.489625\pi$
$182$	0	0
$183$	−13.3693	−0.988288
$184$	0	0
$185$	−0.561553	−0.0412862
$186$	0	0
$187$	23.3693	1.70893
$188$	0	0
$189$	−2.56155	−0.186326
$190$	0	0
$191$	12.4924	0.903920	0.451960	−	0.892038i	$-0.350725\pi$
$191$	12.4924	0.903920	0.451960	+	0.892038i	$0.350725\pi$
$192$	0	0
$193$	−13.3693	−0.962344	−0.481172	−	0.876626i	$-0.659789\pi$
$193$	−13.3693	−0.962344	−0.481172	+	0.876626i	$0.659789\pi$
$194$	0	0
$195$	2.00000	0.143223
$196$	0	0
$197$	20.2462	1.44248	0.721241	−	0.692684i	$-0.243572\pi$
$197$	20.2462	1.44248	0.721241	+	0.692684i	$0.243572\pi$
$198$	0	0
$199$	2.87689	0.203938	0.101969	−	0.994788i	$-0.467486\pi$
$199$	2.87689	0.203938	0.101969	+	0.994788i	$0.467486\pi$
$200$	0	0
$201$	−2.56155	−0.180678
$202$	0	0
$203$	21.9309	1.53925
$204$	0	0
$205$	−3.43845	−0.240152
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−5.75379	−0.397998
$210$	0	0
$211$	13.4384	0.925141	0.462570	−	0.886583i	$-0.346927\pi$
$211$	13.4384	0.925141	0.462570	+	0.886583i	$0.346927\pi$
$212$	0	0
$213$	11.6847	0.800620
$214$	0	0
$215$	−6.24621	−0.425988
$216$	0	0
$217$	9.43845	0.640724
$218$	0	0
$219$	−2.00000	−0.135147
$220$	0	0
$221$	9.12311	0.613686
$222$	0	0
$223$	19.3693	1.29707	0.648533	−	0.761187i	$-0.275383\pi$
$223$	19.3693	1.29707	0.648533	+	0.761187i	$0.275383\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−8.49242	−0.563662	−0.281831	−	0.959464i	$-0.590942\pi$
$227$	−8.49242	−0.563662	−0.281831	+	0.959464i	$0.590942\pi$
$228$	0	0
$229$	−7.12311	−0.470708	−0.235354	−	0.971910i	$-0.575625\pi$
$229$	−7.12311	−0.470708	−0.235354	+	0.971910i	$0.575625\pi$
$230$	0	0
$231$	−13.1231	−0.863437
$232$	0	0
$233$	13.3693	0.875853	0.437927	−	0.899011i	$-0.355713\pi$
$233$	13.3693	0.875853	0.437927	+	0.899011i	$0.355713\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	0	0
$237$	10.2462	0.665563
$238$	0	0
$239$	19.6847	1.27329	0.636647	−	0.771155i	$-0.280321\pi$
$239$	19.6847	1.27329	0.636647	+	0.771155i	$0.280321\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0.438447	0.0280114
$246$	0	0
$247$	−2.24621	−0.142923
$248$	0	0
$249$	−0.315342	−0.0199840
$250$	0	0
$251$	29.1231	1.83823	0.919117	−	0.393985i	$-0.128904\pi$
$251$	29.1231	1.83823	0.919117	+	0.393985i	$0.128904\pi$
$252$	0	0
$253$	5.12311	0.322087
$254$	0	0
$255$	4.56155	0.285656
$256$	0	0
$257$	23.6155	1.47310	0.736548	−	0.676385i	$-0.236455\pi$
$257$	23.6155	1.47310	0.736548	+	0.676385i	$0.236455\pi$
$258$	0	0
$259$	1.43845	0.0893808
$260$	0	0
$261$	8.56155	0.529947
$262$	0	0
$263$	−30.5616	−1.88451	−0.942253	−	0.334902i	$-0.891297\pi$
$263$	−30.5616	−1.88451	−0.942253	+	0.334902i	$0.891297\pi$
$264$	0	0
$265$	−0.561553	−0.0344959
$266$	0	0
$267$	3.12311	0.191131
$268$	0	0
$269$	11.4384	0.697414	0.348707	−	0.937232i	$-0.386621\pi$
$269$	11.4384	0.697414	0.348707	+	0.937232i	$0.386621\pi$
$270$	0	0
$271$	−11.6847	−0.709792	−0.354896	−	0.934906i	$-0.615484\pi$
$271$	−11.6847	−0.709792	−0.354896	+	0.934906i	$0.615484\pi$
$272$	0	0
$273$	−5.12311	−0.310064
$274$	0	0
$275$	5.12311	0.308935
$276$	0	0
$277$	−8.24621	−0.495467	−0.247733	−	0.968828i	$-0.579686\pi$
$277$	−8.24621	−0.495467	−0.247733	+	0.968828i	$0.579686\pi$
$278$	0	0
$279$	3.68466	0.220595
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	−2.56155	−0.152269	−0.0761343	−	0.997098i	$-0.524258\pi$
$283$	−2.56155	−0.152269	−0.0761343	+	0.997098i	$0.524258\pi$
$284$	0	0
$285$	−1.12311	−0.0665270
$286$	0	0
$287$	8.80776	0.519906
$288$	0	0
$289$	3.80776	0.223986
$290$	0	0
$291$	6.00000	0.351726
$292$	0	0
$293$	7.93087	0.463326	0.231663	−	0.972796i	$-0.425583\pi$
$293$	7.93087	0.463326	0.231663	+	0.972796i	$0.425583\pi$
$294$	0	0
$295$	6.56155	0.382028
$296$	0	0
$297$	−5.12311	−0.297273
$298$	0	0
$299$	2.00000	0.115663
$300$	0	0
$301$	16.0000	0.922225
$302$	0	0
$303$	−6.31534	−0.362807
$304$	0	0
$305$	−13.3693	−0.765525
$306$	0	0
$307$	29.6155	1.69025	0.845124	−	0.534571i	$-0.179527\pi$
$307$	29.6155	1.69025	0.845124	+	0.534571i	$0.179527\pi$
$308$	0	0
$309$	14.2462	0.810439
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	3.43845	0.194353	0.0971763	−	0.995267i	$-0.469019\pi$
$313$	3.43845	0.194353	0.0971763	+	0.995267i	$0.469019\pi$
$314$	0	0
$315$	−2.56155	−0.144327
$316$	0	0
$317$	−11.1231	−0.624736	−0.312368	−	0.949961i	$-0.601122\pi$
$317$	−11.1231	−0.624736	−0.312368	+	0.949961i	$0.601122\pi$
$318$	0	0
$319$	43.8617	2.45579
$320$	0	0
$321$	7.68466	0.428916
$322$	0	0
$323$	−5.12311	−0.285057
$324$	0	0
$325$	2.00000	0.110940
$326$	0	0
$327$	4.24621	0.234816
$328$	0	0
$329$	−20.4924	−1.12978
$330$	0	0
$331$	3.19224	0.175461	0.0877306	−	0.996144i	$-0.472039\pi$
$331$	3.19224	0.175461	0.0877306	+	0.996144i	$0.472039\pi$
$332$	0	0
$333$	0.561553	0.0307729
$334$	0	0
$335$	−2.56155	−0.139953
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	0	0
$339$	−2.31534	−0.125752
$340$	0	0
$341$	18.8769	1.02224
$342$	0	0
$343$	−19.0540	−1.02882
$344$	0	0
$345$	1.00000	0.0538382
$346$	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
$349$	1.05398	0.0564180	0.0282090	−	0.999602i	$-0.491020\pi$
$349$	1.05398	0.0564180	0.0282090	+	0.999602i	$0.491020\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	0	0
$353$	11.1231	0.592023	0.296012	−	0.955184i	$-0.404343\pi$
$353$	11.1231	0.592023	0.296012	+	0.955184i	$0.404343\pi$
$354$	0	0
$355$	11.6847	0.620157
$356$	0	0
$357$	−11.6847	−0.618418
$358$	0	0
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	−17.7386	−0.933612
$362$	0	0
$363$	−15.2462	−0.800219
$364$	0	0
$365$	−2.00000	−0.104685
$366$	0	0
$367$	−17.3002	−0.903062	−0.451531	−	0.892255i	$-0.649122\pi$
$367$	−17.3002	−0.903062	−0.451531	+	0.892255i	$0.649122\pi$
$368$	0	0
$369$	3.43845	0.178998
$370$	0	0
$371$	1.43845	0.0746805
$372$	0	0
$373$	−19.7538	−1.02281	−0.511406	−	0.859339i	$-0.670875\pi$
$373$	−19.7538	−1.02281	−0.511406	+	0.859339i	$0.670875\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	17.1231	0.881885
$378$	0	0
$379$	1.75379	0.0900861	0.0450430	−	0.998985i	$-0.485658\pi$
$379$	1.75379	0.0900861	0.0450430	+	0.998985i	$0.485658\pi$
$380$	0	0
$381$	14.2462	0.729856
$382$	0	0
$383$	21.9309	1.12062	0.560308	−	0.828285i	$-0.310683\pi$
$383$	21.9309	1.12062	0.560308	+	0.828285i	$0.310683\pi$
$384$	0	0
$385$	−13.1231	−0.668815
$386$	0	0
$387$	6.24621	0.317513
$388$	0	0
$389$	30.4924	1.54603	0.773014	−	0.634389i	$-0.218748\pi$
$389$	30.4924	1.54603	0.773014	+	0.634389i	$0.218748\pi$
$390$	0	0
$391$	4.56155	0.230688
$392$	0	0
$393$	−10.2462	−0.516853
$394$	0	0
$395$	10.2462	0.515543
$396$	0	0
$397$	−23.6155	−1.18523	−0.592615	−	0.805486i	$-0.701904\pi$
$397$	−23.6155	−1.18523	−0.592615	+	0.805486i	$0.701904\pi$
$398$	0	0
$399$	2.87689	0.144025
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	7.36932	0.367092
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	2.87689	0.142602
$408$	0	0
$409$	−2.31534	−0.114486	−0.0572431	−	0.998360i	$-0.518231\pi$
$409$	−2.31534	−0.114486	−0.0572431	+	0.998360i	$0.518231\pi$
$410$	0	0
$411$	−14.0000	−0.690569
$412$	0	0
$413$	−16.8078	−0.827056
$414$	0	0
$415$	−0.315342	−0.0154795
$416$	0	0
$417$	−12.8078	−0.627199
$418$	0	0
$419$	26.8769	1.31302	0.656511	−	0.754316i	$-0.272032\pi$
$419$	26.8769	1.31302	0.656511	+	0.754316i	$0.272032\pi$
$420$	0	0
$421$	26.4924	1.29116	0.645581	−	0.763692i	$-0.276615\pi$
$421$	26.4924	1.29116	0.645581	+	0.763692i	$0.276615\pi$
$422$	0	0
$423$	−8.00000	−0.388973
$424$	0	0
$425$	4.56155	0.221268
$426$	0	0
$427$	34.2462	1.65729
$428$	0	0
$429$	−10.2462	−0.494692
$430$	0	0
$431$	−20.4924	−0.987085	−0.493543	−	0.869722i	$-0.664298\pi$
$431$	−20.4924	−0.987085	−0.493543	+	0.869722i	$0.664298\pi$
$432$	0	0
$433$	−7.43845	−0.357469	−0.178734	−	0.983897i	$-0.557200\pi$
$433$	−7.43845	−0.357469	−0.178734	+	0.983897i	$0.557200\pi$
$434$	0	0
$435$	8.56155	0.410495
$436$	0	0
$437$	−1.12311	−0.0537254
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	−0.438447	−0.0208784
$442$	0	0
$443$	−16.4924	−0.783579	−0.391789	−	0.920055i	$-0.628144\pi$
$443$	−16.4924	−0.783579	−0.391789	+	0.920055i	$0.628144\pi$
$444$	0	0
$445$	3.12311	0.148049
$446$	0	0
$447$	13.3693	0.632347
$448$	0	0
$449$	−40.4233	−1.90769	−0.953847	−	0.300294i	$-0.902915\pi$
$449$	−40.4233	−1.90769	−0.953847	+	0.300294i	$0.902915\pi$
$450$	0	0
$451$	17.6155	0.829483
$452$	0	0
$453$	10.2462	0.481409
$454$	0	0
$455$	−5.12311	−0.240175
$456$	0	0
$457$	−10.3153	−0.482531	−0.241266	−	0.970459i	$-0.577563\pi$
$457$	−10.3153	−0.482531	−0.241266	+	0.970459i	$0.577563\pi$
$458$	0	0
$459$	−4.56155	−0.212915
$460$	0	0
$461$	20.2462	0.942960	0.471480	−	0.881877i	$-0.343720\pi$
$461$	20.2462	0.942960	0.471480	+	0.881877i	$0.343720\pi$
$462$	0	0
$463$	−4.63068	−0.215206	−0.107603	−	0.994194i	$-0.534318\pi$
$463$	−4.63068	−0.215206	−0.107603	+	0.994194i	$0.534318\pi$
$464$	0	0
$465$	3.68466	0.170872
$466$	0	0
$467$	−31.6847	−1.46619	−0.733096	−	0.680126i	$-0.761925\pi$
$467$	−31.6847	−1.46619	−0.733096	+	0.680126i	$0.761925\pi$
$468$	0	0
$469$	6.56155	0.302984
$470$	0	0
$471$	17.0540	0.785806
$472$	0	0
$473$	32.0000	1.47136
$474$	0	0
$475$	−1.12311	−0.0515316
$476$	0	0
$477$	0.561553	0.0257117
$478$	0	0
$479$	−22.7386	−1.03895	−0.519477	−	0.854484i	$-0.673873\pi$
$479$	−22.7386	−1.03895	−0.519477	+	0.854484i	$0.673873\pi$
$480$	0	0
$481$	1.12311	0.0512092
$482$	0	0
$483$	−2.56155	−0.116555
$484$	0	0
$485$	6.00000	0.272446
$486$	0	0
$487$	−18.7386	−0.849129	−0.424564	−	0.905398i	$-0.639573\pi$
$487$	−18.7386	−0.849129	−0.424564	+	0.905398i	$0.639573\pi$
$488$	0	0
$489$	19.3693	0.875911
$490$	0	0
$491$	4.31534	0.194749	0.0973743	−	0.995248i	$-0.468956\pi$
$491$	4.31534	0.194749	0.0973743	+	0.995248i	$0.468956\pi$
$492$	0	0
$493$	39.0540	1.75890
$494$	0	0
$495$	−5.12311	−0.230266
$496$	0	0
$497$	−29.9309	−1.34258
$498$	0	0
$499$	−22.4233	−1.00380	−0.501902	−	0.864924i	$-0.667366\pi$
$499$	−22.4233	−1.00380	−0.501902	+	0.864924i	$0.667366\pi$
$500$	0	0
$501$	−12.4924	−0.558120
$502$	0	0
$503$	−12.3153	−0.549114	−0.274557	−	0.961571i	$-0.588531\pi$
$503$	−12.3153	−0.549114	−0.274557	+	0.961571i	$0.588531\pi$
$504$	0	0
$505$	−6.31534	−0.281029
$506$	0	0
$507$	9.00000	0.399704
$508$	0	0
$509$	6.49242	0.287772	0.143886	−	0.989594i	$-0.454040\pi$
$509$	6.49242	0.287772	0.143886	+	0.989594i	$0.454040\pi$
$510$	0	0
$511$	5.12311	0.226633
$512$	0	0
$513$	1.12311	0.0495863
$514$	0	0
$515$	14.2462	0.627763
$516$	0	0
$517$	−40.9848	−1.80251
$518$	0	0
$519$	−15.1231	−0.663831
$520$	0	0
$521$	5.50758	0.241291	0.120646	−	0.992696i	$-0.461504\pi$
$521$	5.50758	0.241291	0.120646	+	0.992696i	$0.461504\pi$
$522$	0	0
$523$	−0.492423	−0.0215321	−0.0107661	−	0.999942i	$-0.503427\pi$
$523$	−0.492423	−0.0215321	−0.0107661	+	0.999942i	$0.503427\pi$
$524$	0	0
$525$	−2.56155	−0.111795
$526$	0	0
$527$	16.8078	0.732158
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−6.56155	−0.284747
$532$	0	0
$533$	6.87689	0.297871
$534$	0	0
$535$	7.68466	0.332237
$536$	0	0
$537$	12.4924	0.539088
$538$	0	0
$539$	−2.24621	−0.0967512
$540$	0	0
$541$	6.00000	0.257960	0.128980	−	0.991647i	$-0.458830\pi$
$541$	6.00000	0.257960	0.128980	+	0.991647i	$0.458830\pi$
$542$	0	0
$543$	−0.876894	−0.0376311
$544$	0	0
$545$	4.24621	0.181888
$546$	0	0
$547$	−9.75379	−0.417042	−0.208521	−	0.978018i	$-0.566865\pi$
$547$	−9.75379	−0.417042	−0.208521	+	0.978018i	$0.566865\pi$
$548$	0	0
$549$	13.3693	0.570589
$550$	0	0
$551$	−9.61553	−0.409635
$552$	0	0
$553$	−26.2462	−1.11610
$554$	0	0
$555$	0.561553	0.0238366
$556$	0	0
$557$	−11.9309	−0.505527	−0.252764	−	0.967528i	$-0.581339\pi$
$557$	−11.9309	−0.505527	−0.252764	+	0.967528i	$0.581339\pi$
$558$	0	0
$559$	12.4924	0.528373
$560$	0	0
$561$	−23.3693	−0.986653
$562$	0	0
$563$	15.6847	0.661030	0.330515	−	0.943801i	$-0.392778\pi$
$563$	15.6847	0.661030	0.330515	+	0.943801i	$0.392778\pi$
$564$	0	0
$565$	−2.31534	−0.0974072
$566$	0	0
$567$	2.56155	0.107575
$568$	0	0
$569$	−22.0000	−0.922288	−0.461144	−	0.887325i	$-0.652561\pi$
$569$	−22.0000	−0.922288	−0.461144	+	0.887325i	$0.652561\pi$
$570$	0	0
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	−12.4924	−0.521878
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	45.8617	1.90925	0.954625	−	0.297812i	$-0.0962568\pi$
$577$	45.8617	1.90925	0.954625	+	0.297812i	$0.0962568\pi$
$578$	0	0
$579$	13.3693	0.555610
$580$	0	0
$581$	0.807764	0.0335117
$582$	0	0
$583$	2.87689	0.119149
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	39.8617	1.64527	0.822635	−	0.568570i	$-0.192503\pi$
$587$	39.8617	1.64527	0.822635	+	0.568570i	$0.192503\pi$
$588$	0	0
$589$	−4.13826	−0.170514
$590$	0	0
$591$	−20.2462	−0.832818
$592$	0	0
$593$	−12.8769	−0.528791	−0.264395	−	0.964414i	$-0.585172\pi$
$593$	−12.8769	−0.528791	−0.264395	+	0.964414i	$0.585172\pi$
$594$	0	0
$595$	−11.6847	−0.479024
$596$	0	0
$597$	−2.87689	−0.117743
$598$	0	0
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	0	0
$601$	0.561553	0.0229062	0.0114531	−	0.999934i	$-0.496354\pi$
$601$	0.561553	0.0229062	0.0114531	+	0.999934i	$0.496354\pi$
$602$	0	0
$603$	2.56155	0.104314
$604$	0	0
$605$	−15.2462	−0.619847
$606$	0	0
$607$	−28.0000	−1.13648	−0.568242	−	0.822861i	$-0.692376\pi$
$607$	−28.0000	−1.13648	−0.568242	+	0.822861i	$0.692376\pi$
$608$	0	0
$609$	−21.9309	−0.888684
$610$	0	0
$611$	−16.0000	−0.647291
$612$	0	0
$613$	20.2462	0.817737	0.408868	−	0.912593i	$-0.365924\pi$
$613$	20.2462	0.817737	0.408868	+	0.912593i	$0.365924\pi$
$614$	0	0
$615$	3.43845	0.138652
$616$	0	0
$617$	25.6847	1.03403	0.517013	−	0.855978i	$-0.327044\pi$
$617$	25.6847	1.03403	0.517013	+	0.855978i	$0.327044\pi$
$618$	0	0
$619$	47.2311	1.89838	0.949188	−	0.314709i	$-0.101907\pi$
$619$	47.2311	1.89838	0.949188	+	0.314709i	$0.101907\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−8.00000	−0.320513
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	5.75379	0.229784
$628$	0	0
$629$	2.56155	0.102136
$630$	0	0
$631$	18.2462	0.726370	0.363185	−	0.931717i	$-0.381689\pi$
$631$	18.2462	0.726370	0.363185	+	0.931717i	$0.381689\pi$
$632$	0	0
$633$	−13.4384	−0.534130
$634$	0	0
$635$	14.2462	0.565344
$636$	0	0
$637$	−0.876894	−0.0347438
$638$	0	0
$639$	−11.6847	−0.462238
$640$	0	0
$641$	−32.2462	−1.27365	−0.636824	−	0.771009i	$-0.719752\pi$
$641$	−32.2462	−1.27365	−0.636824	+	0.771009i	$0.719752\pi$
$642$	0	0
$643$	33.3002	1.31323	0.656616	−	0.754225i	$-0.271987\pi$
$643$	33.3002	1.31323	0.656616	+	0.754225i	$0.271987\pi$
$644$	0	0
$645$	6.24621	0.245944
$646$	0	0
$647$	−34.8769	−1.37115	−0.685576	−	0.728001i	$-0.740450\pi$
$647$	−34.8769	−1.37115	−0.685576	+	0.728001i	$0.740450\pi$
$648$	0	0
$649$	−33.6155	−1.31952
$650$	0	0
$651$	−9.43845	−0.369922
$652$	0	0
$653$	−34.4924	−1.34979	−0.674896	−	0.737912i	$-0.735812\pi$
$653$	−34.4924	−1.34979	−0.674896	+	0.737912i	$0.735812\pi$
$654$	0	0
$655$	−10.2462	−0.400353
$656$	0	0
$657$	2.00000	0.0780274
$658$	0	0
$659$	15.3693	0.598704	0.299352	−	0.954143i	$-0.403230\pi$
$659$	15.3693	0.598704	0.299352	+	0.954143i	$0.403230\pi$
$660$	0	0
$661$	−16.7386	−0.651057	−0.325529	−	0.945532i	$-0.605542\pi$
$661$	−16.7386	−0.651057	−0.325529	+	0.945532i	$0.605542\pi$
$662$	0	0
$663$	−9.12311	−0.354312
$664$	0	0
$665$	2.87689	0.111561
$666$	0	0
$667$	8.56155	0.331505
$668$	0	0
$669$	−19.3693	−0.748861
$670$	0	0
$671$	68.4924	2.64412
$672$	0	0
$673$	18.6307	0.718160	0.359080	−	0.933307i	$-0.383091\pi$
$673$	18.6307	0.718160	0.359080	+	0.933307i	$0.383091\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−11.3002	−0.434301	−0.217151	−	0.976138i	$-0.569676\pi$
$677$	−11.3002	−0.434301	−0.217151	+	0.976138i	$0.569676\pi$
$678$	0	0
$679$	−15.3693	−0.589820
$680$	0	0
$681$	8.49242	0.325430
$682$	0	0
$683$	6.24621	0.239005	0.119502	−	0.992834i	$-0.461870\pi$
$683$	6.24621	0.239005	0.119502	+	0.992834i	$0.461870\pi$
$684$	0	0
$685$	−14.0000	−0.534913
$686$	0	0
$687$	7.12311	0.271763
$688$	0	0
$689$	1.12311	0.0427869
$690$	0	0
$691$	7.50758	0.285602	0.142801	−	0.989751i	$-0.454389\pi$
$691$	7.50758	0.285602	0.142801	+	0.989751i	$0.454389\pi$
$692$	0	0
$693$	13.1231	0.498506
$694$	0	0
$695$	−12.8078	−0.485826
$696$	0	0
$697$	15.6847	0.594099
$698$	0	0
$699$	−13.3693	−0.505674
$700$	0	0
$701$	37.8617	1.43002	0.715009	−	0.699115i	$-0.246423\pi$
$701$	37.8617	1.43002	0.715009	+	0.699115i	$0.246423\pi$
$702$	0	0
$703$	−0.630683	−0.0237867
$704$	0	0
$705$	−8.00000	−0.301297
$706$	0	0
$707$	16.1771	0.608402
$708$	0	0
$709$	−28.8769	−1.08449	−0.542247	−	0.840219i	$-0.682426\pi$
$709$	−28.8769	−1.08449	−0.542247	+	0.840219i	$0.682426\pi$
$710$	0	0
$711$	−10.2462	−0.384263
$712$	0	0
$713$	3.68466	0.137992
$714$	0	0
$715$	−10.2462	−0.383187
$716$	0	0
$717$	−19.6847	−0.735137
$718$	0	0
$719$	−13.3002	−0.496013	−0.248007	−	0.968758i	$-0.579775\pi$
$719$	−13.3002	−0.496013	−0.248007	+	0.968758i	$0.579775\pi$
$720$	0	0
$721$	−36.4924	−1.35905
$722$	0	0
$723$	−2.00000	−0.0743808
$724$	0	0
$725$	8.56155	0.317968
$726$	0	0
$727$	−12.8078	−0.475014	−0.237507	−	0.971386i	$-0.576330\pi$
$727$	−12.8078	−0.475014	−0.237507	+	0.971386i	$0.576330\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	28.4924	1.05383
$732$	0	0
$733$	−27.9309	−1.03165	−0.515825	−	0.856694i	$-0.672515\pi$
$733$	−27.9309	−1.03165	−0.515825	+	0.856694i	$0.672515\pi$
$734$	0	0
$735$	−0.438447	−0.0161724
$736$	0	0
$737$	13.1231	0.483396
$738$	0	0
$739$	25.9309	0.953882	0.476941	−	0.878935i	$-0.341745\pi$
$739$	25.9309	0.953882	0.476941	+	0.878935i	$0.341745\pi$
$740$	0	0
$741$	2.24621	0.0825166
$742$	0	0
$743$	−52.4924	−1.92576	−0.962880	−	0.269929i	$-0.913000\pi$
$743$	−52.4924	−1.92576	−0.962880	+	0.269929i	$0.913000\pi$
$744$	0	0
$745$	13.3693	0.489814
$746$	0	0
$747$	0.315342	0.0115377
$748$	0	0
$749$	−19.6847	−0.719262
$750$	0	0
$751$	−1.61553	−0.0589515	−0.0294757	−	0.999565i	$-0.509384\pi$
$751$	−1.61553	−0.0589515	−0.0294757	+	0.999565i	$0.509384\pi$
$752$	0	0
$753$	−29.1231	−1.06130
$754$	0	0
$755$	10.2462	0.372898
$756$	0	0
$757$	2.17708	0.0791274	0.0395637	−	0.999217i	$-0.487403\pi$
$757$	2.17708	0.0791274	0.0395637	+	0.999217i	$0.487403\pi$
$758$	0	0
$759$	−5.12311	−0.185957
$760$	0	0
$761$	1.19224	0.0432185	0.0216093	−	0.999766i	$-0.493121\pi$
$761$	1.19224	0.0432185	0.0216093	+	0.999766i	$0.493121\pi$
$762$	0	0
$763$	−10.8769	−0.393770
$764$	0	0
$765$	−4.56155	−0.164923
$766$	0	0
$767$	−13.1231	−0.473848
$768$	0	0
$769$	25.3693	0.914841	0.457420	−	0.889251i	$-0.348773\pi$
$769$	25.3693	0.914841	0.457420	+	0.889251i	$0.348773\pi$
$770$	0	0
$771$	−23.6155	−0.850492
$772$	0	0
$773$	−52.7386	−1.89688	−0.948438	−	0.316961i	$-0.897337\pi$
$773$	−52.7386	−1.89688	−0.948438	+	0.316961i	$0.897337\pi$
$774$	0	0
$775$	3.68466	0.132357
$776$	0	0
$777$	−1.43845	−0.0516040
$778$	0	0
$779$	−3.86174	−0.138361
$780$	0	0
$781$	−59.8617	−2.14202
$782$	0	0
$783$	−8.56155	−0.305965
$784$	0	0
$785$	17.0540	0.608682
$786$	0	0
$787$	−32.6695	−1.16454	−0.582271	−	0.812995i	$-0.697836\pi$
$787$	−32.6695	−1.16454	−0.582271	+	0.812995i	$0.697836\pi$
$788$	0	0
$789$	30.5616	1.08802
$790$	0	0
$791$	5.93087	0.210877
$792$	0	0
$793$	26.7386	0.949517
$794$	0	0
$795$	0.561553	0.0199162
$796$	0	0
$797$	−55.7926	−1.97628	−0.988138	−	0.153570i	$-0.950923\pi$
$797$	−55.7926	−1.97628	−0.988138	+	0.153570i	$0.950923\pi$
$798$	0	0
$799$	−36.4924	−1.29101
$800$	0	0
$801$	−3.12311	−0.110350
$802$	0	0
$803$	10.2462	0.361581
$804$	0	0
$805$	−2.56155	−0.0902829
$806$	0	0
$807$	−11.4384	−0.402652
$808$	0	0
$809$	−21.1922	−0.745079	−0.372540	−	0.928016i	$-0.621513\pi$
$809$	−21.1922	−0.745079	−0.372540	+	0.928016i	$0.621513\pi$
$810$	0	0
$811$	−19.5464	−0.686367	−0.343183	−	0.939268i	$-0.611505\pi$
$811$	−19.5464	−0.686367	−0.343183	+	0.939268i	$0.611505\pi$
$812$	0	0
$813$	11.6847	0.409799
$814$	0	0
$815$	19.3693	0.678478
$816$	0	0
$817$	−7.01515	−0.245429
$818$	0	0
$819$	5.12311	0.179016
$820$	0	0
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	0	0
$823$	−52.0000	−1.81261	−0.906303	−	0.422628i	$-0.861108\pi$
$823$	−52.0000	−1.81261	−0.906303	+	0.422628i	$0.861108\pi$
$824$	0	0
$825$	−5.12311	−0.178364
$826$	0	0
$827$	8.94602	0.311084	0.155542	−	0.987829i	$-0.450288\pi$
$827$	8.94602	0.311084	0.155542	+	0.987829i	$0.450288\pi$
$828$	0	0
$829$	5.19224	0.180334	0.0901669	−	0.995927i	$-0.471260\pi$
$829$	5.19224	0.180334	0.0901669	+	0.995927i	$0.471260\pi$
$830$	0	0
$831$	8.24621	0.286058
$832$	0	0
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	−12.4924	−0.432318
$836$	0	0
$837$	−3.68466	−0.127360
$838$	0	0
$839$	5.12311	0.176869	0.0884346	−	0.996082i	$-0.471814\pi$
$839$	5.12311	0.176869	0.0884346	+	0.996082i	$0.471814\pi$
$840$	0	0
$841$	44.3002	1.52759
$842$	0	0
$843$	−10.0000	−0.344418
$844$	0	0
$845$	9.00000	0.309609
$846$	0	0
$847$	39.0540	1.34191
$848$	0	0
$849$	2.56155	0.0879123
$850$	0	0
$851$	0.561553	0.0192498
$852$	0	0
$853$	−49.2311	−1.68564	−0.842820	−	0.538196i	$-0.819106\pi$
$853$	−49.2311	−1.68564	−0.842820	+	0.538196i	$0.819106\pi$
$854$	0	0
$855$	1.12311	0.0384094
$856$	0	0
$857$	42.4924	1.45151	0.725757	−	0.687951i	$-0.241490\pi$
$857$	42.4924	1.45151	0.725757	+	0.687951i	$0.241490\pi$
$858$	0	0
$859$	20.1771	0.688433	0.344217	−	0.938890i	$-0.388145\pi$
$859$	20.1771	0.688433	0.344217	+	0.938890i	$0.388145\pi$
$860$	0	0
$861$	−8.80776	−0.300168
$862$	0	0
$863$	11.8617	0.403778	0.201889	−	0.979408i	$-0.435292\pi$
$863$	11.8617	0.403778	0.201889	+	0.979408i	$0.435292\pi$
$864$	0	0
$865$	−15.1231	−0.514201
$866$	0	0
$867$	−3.80776	−0.129318
$868$	0	0
$869$	−52.4924	−1.78068
$870$	0	0
$871$	5.12311	0.173590
$872$	0	0
$873$	−6.00000	−0.203069
$874$	0	0
$875$	−2.56155	−0.0865963
$876$	0	0
$877$	53.2311	1.79749	0.898743	−	0.438477i	$-0.144482\pi$
$877$	53.2311	1.79749	0.898743	+	0.438477i	$0.144482\pi$
$878$	0	0
$879$	−7.93087	−0.267502
$880$	0	0
$881$	26.6307	0.897211	0.448605	−	0.893730i	$-0.351921\pi$
$881$	26.6307	0.897211	0.448605	+	0.893730i	$0.351921\pi$
$882$	0	0
$883$	−12.0000	−0.403832	−0.201916	−	0.979403i	$-0.564717\pi$
$883$	−12.0000	−0.403832	−0.201916	+	0.979403i	$0.564717\pi$
$884$	0	0
$885$	−6.56155	−0.220564
$886$	0	0
$887$	21.1231	0.709244	0.354622	−	0.935010i	$-0.384609\pi$
$887$	21.1231	0.709244	0.354622	+	0.935010i	$0.384609\pi$
$888$	0	0
$889$	−36.4924	−1.22392
$890$	0	0
$891$	5.12311	0.171630
$892$	0	0
$893$	8.98485	0.300666
$894$	0	0
$895$	12.4924	0.417576
$896$	0	0
$897$	−2.00000	−0.0667781
$898$	0	0
$899$	31.5464	1.05213
$900$	0	0
$901$	2.56155	0.0853377
$902$	0	0
$903$	−16.0000	−0.532447
$904$	0	0
$905$	−0.876894	−0.0291490
$906$	0	0
$907$	14.0691	0.467158	0.233579	−	0.972338i	$-0.424956\pi$
$907$	14.0691	0.467158	0.233579	+	0.972338i	$0.424956\pi$
$908$	0	0
$909$	6.31534	0.209467
$910$	0	0
$911$	−30.7386	−1.01842	−0.509208	−	0.860643i	$-0.670062\pi$
$911$	−30.7386	−1.01842	−0.509208	+	0.860643i	$0.670062\pi$
$912$	0	0
$913$	1.61553	0.0534662
$914$	0	0
$915$	13.3693	0.441976
$916$	0	0
$917$	26.2462	0.866726
$918$	0	0
$919$	48.9848	1.61586	0.807930	−	0.589278i	$-0.200588\pi$
$919$	48.9848	1.61586	0.807930	+	0.589278i	$0.200588\pi$
$920$	0	0
$921$	−29.6155	−0.975865
$922$	0	0
$923$	−23.3693	−0.769210
$924$	0	0
$925$	0.561553	0.0184637
$926$	0	0
$927$	−14.2462	−0.467907
$928$	0	0
$929$	21.0540	0.690759	0.345379	−	0.938463i	$-0.387750\pi$
$929$	21.0540	0.690759	0.345379	+	0.938463i	$0.387750\pi$
$930$	0	0
$931$	0.492423	0.0161385
$932$	0	0
$933$	−8.00000	−0.261908
$934$	0	0
$935$	−23.3693	−0.764258
$936$	0	0
$937$	−34.4924	−1.12682	−0.563409	−	0.826178i	$-0.690511\pi$
$937$	−34.4924	−1.12682	−0.563409	+	0.826178i	$0.690511\pi$
$938$	0	0
$939$	−3.43845	−0.112209
$940$	0	0
$941$	−21.3693	−0.696620	−0.348310	−	0.937379i	$-0.613244\pi$
$941$	−21.3693	−0.696620	−0.348310	+	0.937379i	$0.613244\pi$
$942$	0	0
$943$	3.43845	0.111971
$944$	0	0
$945$	2.56155	0.0833273
$946$	0	0
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	0	0
$949$	4.00000	0.129845
$950$	0	0
$951$	11.1231	0.360691
$952$	0	0
$953$	19.7538	0.639888	0.319944	−	0.947436i	$-0.396336\pi$
$953$	19.7538	0.639888	0.319944	+	0.947436i	$0.396336\pi$
$954$	0	0
$955$	−12.4924	−0.404245
$956$	0	0
$957$	−43.8617	−1.41785
$958$	0	0
$959$	35.8617	1.15804
$960$	0	0
$961$	−17.4233	−0.562042
$962$	0	0
$963$	−7.68466	−0.247635
$964$	0	0
$965$	13.3693	0.430374
$966$	0	0
$967$	12.9848	0.417564	0.208782	−	0.977962i	$-0.433050\pi$
$967$	12.9848	0.417564	0.208782	+	0.977962i	$0.433050\pi$
$968$	0	0
$969$	5.12311	0.164578
$970$	0	0
$971$	−46.7386	−1.49991	−0.749957	−	0.661487i	$-0.769926\pi$
$971$	−46.7386	−1.49991	−0.749957	+	0.661487i	$0.769926\pi$
$972$	0	0
$973$	32.8078	1.05177
$974$	0	0
$975$	−2.00000	−0.0640513
$976$	0	0
$977$	48.4233	1.54920	0.774599	−	0.632452i	$-0.217952\pi$
$977$	48.4233	1.54920	0.774599	+	0.632452i	$0.217952\pi$
$978$	0	0
$979$	−16.0000	−0.511362
$980$	0	0
$981$	−4.24621	−0.135571
$982$	0	0
$983$	−24.1771	−0.771129	−0.385565	−	0.922681i	$-0.625993\pi$
$983$	−24.1771	−0.771129	−0.385565	+	0.922681i	$0.625993\pi$
$984$	0	0
$985$	−20.2462	−0.645098
$986$	0	0
$987$	20.4924	0.652281
$988$	0	0
$989$	6.24621	0.198618
$990$	0	0
$991$	12.9460	0.411244	0.205622	−	0.978631i	$-0.434078\pi$
$991$	12.9460	0.411244	0.205622	+	0.978631i	$0.434078\pi$
$992$	0	0
$993$	−3.19224	−0.101303
$994$	0	0
$995$	−2.87689	−0.0912037
$996$	0	0
$997$	37.8617	1.19909	0.599547	−	0.800340i	$-0.295348\pi$
$997$	37.8617	1.19909	0.599547	+	0.800340i	$0.295348\pi$
$998$	0	0
$999$	−0.561553	−0.0177667

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.a.bh.1.2		2
4.3	odd	2		2760.2.a.p.1.1	✓	2
12.11	even	2		8280.2.a.be.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.p.1.1	✓	2	4.3	odd	2
5520.2.a.bh.1.2		2	1.1	even	1	trivial
8280.2.a.be.1.1		2	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 5520 = 2^{4} \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5520.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +2.56155 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} +2.56155 q^{7} +1.00000 q^{9} +5.12311 q^{11} +2.00000 q^{13} +1.00000 q^{15} +4.56155 q^{17} -1.12311 q^{19} -2.56155 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +8.56155 q^{29} +3.68466 q^{31} -5.12311 q^{33} -2.56155 q^{35} +0.561553 q^{37} -2.00000 q^{39} +3.43845 q^{41} +6.24621 q^{43} -1.00000 q^{45} -8.00000 q^{47} -0.438447 q^{49} -4.56155 q^{51} +0.561553 q^{53} -5.12311 q^{55} +1.12311 q^{57} -6.56155 q^{59} +13.3693 q^{61} +2.56155 q^{63} -2.00000 q^{65} +2.56155 q^{67} -1.00000 q^{69} -11.6847 q^{71} +2.00000 q^{73} -1.00000 q^{75} +13.1231 q^{77} -10.2462 q^{79} +1.00000 q^{81} +0.315342 q^{83} -4.56155 q^{85} -8.56155 q^{87} -3.12311 q^{89} +5.12311 q^{91} -3.68466 q^{93} +1.12311 q^{95} -6.00000 q^{97} +5.12311 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 + q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} + q^{7} + 2 q^{9} + 2 q^{11} + 4 q^{13} + 2 q^{15} + 5 q^{17} + 6 q^{19} - q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} + 13 q^{29} - 5 q^{31} - 2 q^{33} - q^{35} - 3 q^{37} - 4 q^{39} + 11 q^{41} - 4 q^{43} - 2 q^{45} - 16 q^{47} - 5 q^{49} - 5 q^{51} - 3 q^{53} - 2 q^{55} - 6 q^{57} - 9 q^{59} + 2 q^{61} + q^{63} - 4 q^{65} + q^{67} - 2 q^{69} - 11 q^{71} + 4 q^{73} - 2 q^{75} + 18 q^{77} - 4 q^{79} + 2 q^{81} + 13 q^{83} - 5 q^{85} - 13 q^{87} + 2 q^{89} + 2 q^{91} + 5 q^{93} - 6 q^{95} - 12 q^{97} + 2 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + q^7 + 2 * q^9 + 2 * q^11 + 4 * q^13 + 2 * q^15 + 5 * q^17 + 6 * q^19 - q^21 + 2 * q^23 + 2 * q^25 - 2 * q^27 + 13 * q^29 - 5 * q^31 - 2 * q^33 - q^35 - 3 * q^37 - 4 * q^39 + 11 * q^41 - 4 * q^43 - 2 * q^45 - 16 * q^47 - 5 * q^49 - 5 * q^51 - 3 * q^53 - 2 * q^55 - 6 * q^57 - 9 * q^59 + 2 * q^61 + q^63 - 4 * q^65 + q^67 - 2 * q^69 - 11 * q^71 + 4 * q^73 - 2 * q^75 + 18 * q^77 - 4 * q^79 + 2 * q^81 + 13 * q^83 - 5 * q^85 - 13 * q^87 + 2 * q^89 + 2 * q^91 + 5 * q^93 - 6 * q^95 - 12 * q^97 + 2 * q^99

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