LMFDB - Embedded newform 5520.2.a.bi.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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{6}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 6 $ x^2 - 6
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 345)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.44949$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +2.44949 q^{11} -0.449490 q^{13} +1.00000 q^{15} -0.550510 q^{17} +0.449490 q^{19} -1.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -4.34847 q^{29} -9.89898 q^{31} -2.44949 q^{33} -1.00000 q^{35} -5.89898 q^{37} +0.449490 q^{39} +0.550510 q^{41} -2.00000 q^{43} -1.00000 q^{45} -3.55051 q^{47} -6.00000 q^{49} +0.550510 q^{51} +5.44949 q^{53} -2.44949 q^{55} -0.449490 q^{57} +4.34847 q^{59} +15.3485 q^{61} +1.00000 q^{63} +0.449490 q^{65} +7.00000 q^{67} -1.00000 q^{69} -10.3485 q^{71} +9.34847 q^{73} -1.00000 q^{75} +2.44949 q^{77} +4.00000 q^{79} +1.00000 q^{81} +9.24745 q^{83} +0.550510 q^{85} +4.34847 q^{87} -7.10102 q^{89} -0.449490 q^{91} +9.89898 q^{93} -0.449490 q^{95} +12.8990 q^{97} +2.44949 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + 4 q^{13} + 2 q^{15} - 6 q^{17} - 4 q^{19} - 2 q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} + 6 q^{29} - 10 q^{31} - 2 q^{35} - 2 q^{37} - 4 q^{39} + 6 q^{41} - 4 q^{43} - 2 q^{45} - 12 q^{47} - 12 q^{49} + 6 q^{51} + 6 q^{53} + 4 q^{57} - 6 q^{59} + 16 q^{61} + 2 q^{63} - 4 q^{65} + 14 q^{67} - 2 q^{69} - 6 q^{71} + 4 q^{73} - 2 q^{75} + 8 q^{79} + 2 q^{81} - 6 q^{83} + 6 q^{85} - 6 q^{87} - 24 q^{89} + 4 q^{91} + 10 q^{93} + 4 q^{95} + 16 q^{97}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 + 4 * q^13 + 2 * q^15 - 6 * q^17 - 4 * q^19 - 2 * q^21 + 2 * q^23 + 2 * q^25 - 2 * q^27 + 6 * q^29 - 10 * q^31 - 2 * q^35 - 2 * q^37 - 4 * q^39 + 6 * q^41 - 4 * q^43 - 2 * q^45 - 12 * q^47 - 12 * q^49 + 6 * q^51 + 6 * q^53 + 4 * q^57 - 6 * q^59 + 16 * q^61 + 2 * q^63 - 4 * q^65 + 14 * q^67 - 2 * q^69 - 6 * q^71 + 4 * q^73 - 2 * q^75 + 8 * q^79 + 2 * q^81 - 6 * q^83 + 6 * q^85 - 6 * q^87 - 24 * q^89 + 4 * q^91 + 10 * q^93 + 4 * q^95 + 16 * q^97

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
$7$	1.00000	0.377964	0.188982	−	0.981981i	$-0.439481\pi$
$7$	1.00000	0.377964	0.188982	+	0.981981i	$0.439481\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	2.44949	0.738549	0.369274	−	0.929320i	$-0.379606\pi$
$11$	2.44949	0.738549	0.369274	+	0.929320i	$0.379606\pi$
$12$	0	0
$13$	−0.449490	−0.124666	−0.0623330	−	0.998055i	$-0.519854\pi$
$13$	−0.449490	−0.124666	−0.0623330	+	0.998055i	$0.519854\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−0.550510	−0.133518	−0.0667592	−	0.997769i	$-0.521266\pi$
$17$	−0.550510	−0.133518	−0.0667592	+	0.997769i	$0.521266\pi$
$18$	0	0
$19$	0.449490	0.103120	0.0515600	−	0.998670i	$-0.483581\pi$
$19$	0.449490	0.103120	0.0515600	+	0.998670i	$0.483581\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	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
$29$	−4.34847	−0.807490	−0.403745	−	0.914871i	$-0.632292\pi$
$29$	−4.34847	−0.807490	−0.403745	+	0.914871i	$0.632292\pi$
$30$	0	0
$31$	−9.89898	−1.77791	−0.888955	−	0.457995i	$-0.848568\pi$
$31$	−9.89898	−1.77791	−0.888955	+	0.457995i	$0.848568\pi$
$32$	0	0
$33$	−2.44949	−0.426401
$34$	0	0
$35$	−1.00000	−0.169031
$36$	0	0
$37$	−5.89898	−0.969786	−0.484893	−	0.874573i	$-0.661142\pi$
$37$	−5.89898	−0.969786	−0.484893	+	0.874573i	$0.661142\pi$
$38$	0	0
$39$	0.449490	0.0719760
$40$	0	0
$41$	0.550510	0.0859753	0.0429876	−	0.999076i	$-0.486312\pi$
$41$	0.550510	0.0859753	0.0429876	+	0.999076i	$0.486312\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−3.55051	−0.517895	−0.258948	−	0.965891i	$-0.583376\pi$
$47$	−3.55051	−0.517895	−0.258948	+	0.965891i	$0.583376\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0.550510	0.0770869
$52$	0	0
$53$	5.44949	0.748545	0.374272	−	0.927319i	$-0.377892\pi$
$53$	5.44949	0.748545	0.374272	+	0.927319i	$0.377892\pi$
$54$	0	0
$55$	−2.44949	−0.330289
$56$	0	0
$57$	−0.449490	−0.0595364
$58$	0	0
$59$	4.34847	0.566122	0.283061	−	0.959102i	$-0.408650\pi$
$59$	4.34847	0.566122	0.283061	+	0.959102i	$0.408650\pi$
$60$	0	0
$61$	15.3485	1.96517	0.982585	−	0.185813i	$-0.0594920\pi$
$61$	15.3485	1.96517	0.982585	+	0.185813i	$0.0594920\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	0.449490	0.0557523
$66$	0	0
$67$	7.00000	0.855186	0.427593	−	0.903971i	$-0.359362\pi$
$67$	7.00000	0.855186	0.427593	+	0.903971i	$0.359362\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−10.3485	−1.22814	−0.614069	−	0.789253i	$-0.710468\pi$
$71$	−10.3485	−1.22814	−0.614069	+	0.789253i	$0.710468\pi$
$72$	0	0
$73$	9.34847	1.09416	0.547078	−	0.837082i	$-0.315740\pi$
$73$	9.34847	1.09416	0.547078	+	0.837082i	$0.315740\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	2.44949	0.279145
$78$	0	0
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	9.24745	1.01504	0.507520	−	0.861640i	$-0.330562\pi$
$83$	9.24745	1.01504	0.507520	+	0.861640i	$0.330562\pi$
$84$	0	0
$85$	0.550510	0.0597112
$86$	0	0
$87$	4.34847	0.466205
$88$	0	0
$89$	−7.10102	−0.752707	−0.376353	−	0.926476i	$-0.622822\pi$
$89$	−7.10102	−0.752707	−0.376353	+	0.926476i	$0.622822\pi$
$90$	0	0
$91$	−0.449490	−0.0471193
$92$	0	0
$93$	9.89898	1.02648
$94$	0	0
$95$	−0.449490	−0.0461167
$96$	0	0
$97$	12.8990	1.30969	0.654846	−	0.755762i	$-0.272733\pi$
$97$	12.8990	1.30969	0.654846	+	0.755762i	$0.272733\pi$
$98$	0	0
$99$	2.44949	0.246183
$100$	0	0
$101$	−17.4495	−1.73629	−0.868145	−	0.496311i	$-0.834687\pi$
$101$	−17.4495	−1.73629	−0.868145	+	0.496311i	$0.834687\pi$
$102$	0	0
$103$	−16.6969	−1.64520	−0.822599	−	0.568622i	$-0.807477\pi$
$103$	−16.6969	−1.64520	−0.822599	+	0.568622i	$0.807477\pi$
$104$	0	0
$105$	1.00000	0.0975900
$106$	0	0
$107$	0.550510	0.0532198	0.0266099	−	0.999646i	$-0.491529\pi$
$107$	0.550510	0.0532198	0.0266099	+	0.999646i	$0.491529\pi$
$108$	0	0
$109$	16.4495	1.57558	0.787788	−	0.615947i	$-0.211226\pi$
$109$	16.4495	1.57558	0.787788	+	0.615947i	$0.211226\pi$
$110$	0	0
$111$	5.89898	0.559906
$112$	0	0
$113$	−16.3485	−1.53793	−0.768967	−	0.639288i	$-0.779229\pi$
$113$	−16.3485	−1.53793	−0.768967	+	0.639288i	$0.779229\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	−0.449490	−0.0415553
$118$	0	0
$119$	−0.550510	−0.0504652
$120$	0	0
$121$	−5.00000	−0.454545
$122$	0	0
$123$	−0.550510	−0.0496378
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	6.44949	0.572300	0.286150	−	0.958185i	$-0.407624\pi$
$127$	6.44949	0.572300	0.286150	+	0.958185i	$0.407624\pi$
$128$	0	0
$129$	2.00000	0.176090
$130$	0	0
$131$	−19.5959	−1.71210	−0.856052	−	0.516890i	$-0.827090\pi$
$131$	−19.5959	−1.71210	−0.856052	+	0.516890i	$0.827090\pi$
$132$	0	0
$133$	0.449490	0.0389757
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−7.10102	−0.606681	−0.303341	−	0.952882i	$-0.598102\pi$
$137$	−7.10102	−0.606681	−0.303341	+	0.952882i	$0.598102\pi$
$138$	0	0
$139$	−14.7980	−1.25515	−0.627573	−	0.778558i	$-0.715952\pi$
$139$	−14.7980	−1.25515	−0.627573	+	0.778558i	$0.715952\pi$
$140$	0	0
$141$	3.55051	0.299007
$142$	0	0
$143$	−1.10102	−0.0920720
$144$	0	0
$145$	4.34847	0.361121
$146$	0	0
$147$	6.00000	0.494872
$148$	0	0
$149$	13.3485	1.09355	0.546775	−	0.837280i	$-0.315855\pi$
$149$	13.3485	1.09355	0.546775	+	0.837280i	$0.315855\pi$
$150$	0	0
$151$	−14.0000	−1.13930	−0.569652	−	0.821886i	$-0.692922\pi$
$151$	−14.0000	−1.13930	−0.569652	+	0.821886i	$0.692922\pi$
$152$	0	0
$153$	−0.550510	−0.0445061
$154$	0	0
$155$	9.89898	0.795105
$156$	0	0
$157$	−20.5959	−1.64373	−0.821867	−	0.569680i	$-0.807067\pi$
$157$	−20.5959	−1.64373	−0.821867	+	0.569680i	$0.807067\pi$
$158$	0	0
$159$	−5.44949	−0.432173
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	0	0
$165$	2.44949	0.190693
$166$	0	0
$167$	1.34847	0.104348	0.0521738	−	0.998638i	$-0.483385\pi$
$167$	1.34847	0.104348	0.0521738	+	0.998638i	$0.483385\pi$
$168$	0	0
$169$	−12.7980	−0.984458
$170$	0	0
$171$	0.449490	0.0343733
$172$	0	0
$173$	−19.5959	−1.48985	−0.744925	−	0.667148i	$-0.767515\pi$
$173$	−19.5959	−1.48985	−0.744925	+	0.667148i	$0.767515\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	0	0
$177$	−4.34847	−0.326851
$178$	0	0
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	0.898979	0.0668206	0.0334103	−	0.999442i	$-0.489363\pi$
$181$	0.898979	0.0668206	0.0334103	+	0.999442i	$0.489363\pi$
$182$	0	0
$183$	−15.3485	−1.13459
$184$	0	0
$185$	5.89898	0.433702
$186$	0	0
$187$	−1.34847	−0.0986098
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−6.24745	−0.452050	−0.226025	−	0.974122i	$-0.572573\pi$
$191$	−6.24745	−0.452050	−0.226025	+	0.974122i	$0.572573\pi$
$192$	0	0
$193$	22.6969	1.63376	0.816881	−	0.576807i	$-0.195701\pi$
$193$	22.6969	1.63376	0.816881	+	0.576807i	$0.195701\pi$
$194$	0	0
$195$	−0.449490	−0.0321886
$196$	0	0
$197$	−13.1010	−0.933409	−0.466705	−	0.884413i	$-0.654559\pi$
$197$	−13.1010	−0.933409	−0.466705	+	0.884413i	$0.654559\pi$
$198$	0	0
$199$	−6.89898	−0.489056	−0.244528	−	0.969642i	$-0.578633\pi$
$199$	−6.89898	−0.489056	−0.244528	+	0.969642i	$0.578633\pi$
$200$	0	0
$201$	−7.00000	−0.493742
$202$	0	0
$203$	−4.34847	−0.305203
$204$	0	0
$205$	−0.550510	−0.0384493
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	1.10102	0.0761592
$210$	0	0
$211$	−11.0000	−0.757271	−0.378636	−	0.925546i	$-0.623607\pi$
$211$	−11.0000	−0.757271	−0.378636	+	0.925546i	$0.623607\pi$
$212$	0	0
$213$	10.3485	0.709065
$214$	0	0
$215$	2.00000	0.136399
$216$	0	0
$217$	−9.89898	−0.671987
$218$	0	0
$219$	−9.34847	−0.631711
$220$	0	0
$221$	0.247449	0.0166452
$222$	0	0
$223$	17.5959	1.17831	0.589155	−	0.808020i	$-0.299461\pi$
$223$	17.5959	1.17831	0.589155	+	0.808020i	$0.299461\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−2.20204	−0.146155	−0.0730773	−	0.997326i	$-0.523282\pi$
$227$	−2.20204	−0.146155	−0.0730773	+	0.997326i	$0.523282\pi$
$228$	0	0
$229$	20.4949	1.35434	0.677170	−	0.735826i	$-0.263206\pi$
$229$	20.4949	1.35434	0.677170	+	0.735826i	$0.263206\pi$
$230$	0	0
$231$	−2.44949	−0.161165
$232$	0	0
$233$	2.20204	0.144261	0.0721303	−	0.997395i	$-0.477020\pi$
$233$	2.20204	0.144261	0.0721303	+	0.997395i	$0.477020\pi$
$234$	0	0
$235$	3.55051	0.231610
$236$	0	0
$237$	−4.00000	−0.259828
$238$	0	0
$239$	−11.4495	−0.740606	−0.370303	−	0.928911i	$-0.620746\pi$
$239$	−11.4495	−0.740606	−0.370303	+	0.928911i	$0.620746\pi$
$240$	0	0
$241$	−14.6515	−0.943788	−0.471894	−	0.881655i	$-0.656430\pi$
$241$	−14.6515	−0.943788	−0.471894	+	0.881655i	$0.656430\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	6.00000	0.383326
$246$	0	0
$247$	−0.202041	−0.0128556
$248$	0	0
$249$	−9.24745	−0.586033
$250$	0	0
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	0	0
$253$	2.44949	0.153998
$254$	0	0
$255$	−0.550510	−0.0344743
$256$	0	0
$257$	−10.6515	−0.664424	−0.332212	−	0.943205i	$-0.607795\pi$
$257$	−10.6515	−0.664424	−0.332212	+	0.943205i	$0.607795\pi$
$258$	0	0
$259$	−5.89898	−0.366545
$260$	0	0
$261$	−4.34847	−0.269163
$262$	0	0
$263$	−2.75255	−0.169730	−0.0848648	−	0.996392i	$-0.527046\pi$
$263$	−2.75255	−0.169730	−0.0848648	+	0.996392i	$0.527046\pi$
$264$	0	0
$265$	−5.44949	−0.334759
$266$	0	0
$267$	7.10102	0.434575
$268$	0	0
$269$	−0.550510	−0.0335652	−0.0167826	−	0.999859i	$-0.505342\pi$
$269$	−0.550510	−0.0335652	−0.0167826	+	0.999859i	$0.505342\pi$
$270$	0	0
$271$	−2.30306	−0.139901	−0.0699505	−	0.997550i	$-0.522284\pi$
$271$	−2.30306	−0.139901	−0.0699505	+	0.997550i	$0.522284\pi$
$272$	0	0
$273$	0.449490	0.0272044
$274$	0	0
$275$	2.44949	0.147710
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	0	0
$279$	−9.89898	−0.592636
$280$	0	0
$281$	−4.65153	−0.277487	−0.138744	−	0.990328i	$-0.544306\pi$
$281$	−4.65153	−0.277487	−0.138744	+	0.990328i	$0.544306\pi$
$282$	0	0
$283$	11.8990	0.707321	0.353660	−	0.935374i	$-0.384937\pi$
$283$	11.8990	0.707321	0.353660	+	0.935374i	$0.384937\pi$
$284$	0	0
$285$	0.449490	0.0266255
$286$	0	0
$287$	0.550510	0.0324956
$288$	0	0
$289$	−16.6969	−0.982173
$290$	0	0
$291$	−12.8990	−0.756152
$292$	0	0
$293$	3.24745	0.189718	0.0948590	−	0.995491i	$-0.469760\pi$
$293$	3.24745	0.189718	0.0948590	+	0.995491i	$0.469760\pi$
$294$	0	0
$295$	−4.34847	−0.253178
$296$	0	0
$297$	−2.44949	−0.142134
$298$	0	0
$299$	−0.449490	−0.0259947
$300$	0	0
$301$	−2.00000	−0.115278
$302$	0	0
$303$	17.4495	1.00245
$304$	0	0
$305$	−15.3485	−0.878851
$306$	0	0
$307$	17.3485	0.990129	0.495065	−	0.868856i	$-0.335144\pi$
$307$	17.3485	0.990129	0.495065	+	0.868856i	$0.335144\pi$
$308$	0	0
$309$	16.6969	0.949856
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	0	0
$313$	−9.69694	−0.548103	−0.274052	−	0.961715i	$-0.588364\pi$
$313$	−9.69694	−0.548103	−0.274052	+	0.961715i	$0.588364\pi$
$314$	0	0
$315$	−1.00000	−0.0563436
$316$	0	0
$317$	−18.2474	−1.02488	−0.512439	−	0.858723i	$-0.671258\pi$
$317$	−18.2474	−1.02488	−0.512439	+	0.858723i	$0.671258\pi$
$318$	0	0
$319$	−10.6515	−0.596371
$320$	0	0
$321$	−0.550510	−0.0307265
$322$	0	0
$323$	−0.247449	−0.0137684
$324$	0	0
$325$	−0.449490	−0.0249332
$326$	0	0
$327$	−16.4495	−0.909659
$328$	0	0
$329$	−3.55051	−0.195746
$330$	0	0
$331$	14.5959	0.802264	0.401132	−	0.916020i	$-0.368617\pi$
$331$	14.5959	0.802264	0.401132	+	0.916020i	$0.368617\pi$
$332$	0	0
$333$	−5.89898	−0.323262
$334$	0	0
$335$	−7.00000	−0.382451
$336$	0	0
$337$	−0.202041	−0.0110059	−0.00550294	−	0.999985i	$-0.501752\pi$
$337$	−0.202041	−0.0110059	−0.00550294	+	0.999985i	$0.501752\pi$
$338$	0	0
$339$	16.3485	0.887927
$340$	0	0
$341$	−24.2474	−1.31307
$342$	0	0
$343$	−13.0000	−0.701934
$344$	0	0
$345$	1.00000	0.0538382
$346$	0	0
$347$	19.1010	1.02540	0.512698	−	0.858569i	$-0.328646\pi$
$347$	19.1010	1.02540	0.512698	+	0.858569i	$0.328646\pi$
$348$	0	0
$349$	−25.4949	−1.36471	−0.682355	−	0.731021i	$-0.739044\pi$
$349$	−25.4949	−1.36471	−0.682355	+	0.731021i	$0.739044\pi$
$350$	0	0
$351$	0.449490	0.0239920
$352$	0	0
$353$	−20.4495	−1.08842	−0.544208	−	0.838950i	$-0.683170\pi$
$353$	−20.4495	−1.08842	−0.544208	+	0.838950i	$0.683170\pi$
$354$	0	0
$355$	10.3485	0.549240
$356$	0	0
$357$	0.550510	0.0291361
$358$	0	0
$359$	8.44949	0.445947	0.222974	−	0.974825i	$-0.428424\pi$
$359$	8.44949	0.445947	0.222974	+	0.974825i	$0.428424\pi$
$360$	0	0
$361$	−18.7980	−0.989366
$362$	0	0
$363$	5.00000	0.262432
$364$	0	0
$365$	−9.34847	−0.489321
$366$	0	0
$367$	15.6969	0.819374	0.409687	−	0.912226i	$-0.365638\pi$
$367$	15.6969	0.819374	0.409687	+	0.912226i	$0.365638\pi$
$368$	0	0
$369$	0.550510	0.0286584
$370$	0	0
$371$	5.44949	0.282923
$372$	0	0
$373$	−11.5959	−0.600414	−0.300207	−	0.953874i	$-0.597056\pi$
$373$	−11.5959	−0.600414	−0.300207	+	0.953874i	$0.597056\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	1.95459	0.100667
$378$	0	0
$379$	33.3939	1.71533	0.857664	−	0.514210i	$-0.171915\pi$
$379$	33.3939	1.71533	0.857664	+	0.514210i	$0.171915\pi$
$380$	0	0
$381$	−6.44949	−0.330417
$382$	0	0
$383$	−16.3485	−0.835368	−0.417684	−	0.908592i	$-0.637158\pi$
$383$	−16.3485	−0.835368	−0.417684	+	0.908592i	$0.637158\pi$
$384$	0	0
$385$	−2.44949	−0.124838
$386$	0	0
$387$	−2.00000	−0.101666
$388$	0	0
$389$	−32.6969	−1.65780	−0.828900	−	0.559396i	$-0.811033\pi$
$389$	−32.6969	−1.65780	−0.828900	+	0.559396i	$0.811033\pi$
$390$	0	0
$391$	−0.550510	−0.0278405
$392$	0	0
$393$	19.5959	0.988483
$394$	0	0
$395$	−4.00000	−0.201262
$396$	0	0
$397$	5.79796	0.290991	0.145496	−	0.989359i	$-0.453522\pi$
$397$	5.79796	0.290991	0.145496	+	0.989359i	$0.453522\pi$
$398$	0	0
$399$	−0.449490	−0.0225026
$400$	0	0
$401$	−22.8990	−1.14352	−0.571760	−	0.820421i	$-0.693739\pi$
$401$	−22.8990	−1.14352	−0.571760	+	0.820421i	$0.693739\pi$
$402$	0	0
$403$	4.44949	0.221645
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−14.4495	−0.716235
$408$	0	0
$409$	12.1010	0.598357	0.299178	−	0.954197i	$-0.403287\pi$
$409$	12.1010	0.598357	0.299178	+	0.954197i	$0.403287\pi$
$410$	0	0
$411$	7.10102	0.350268
$412$	0	0
$413$	4.34847	0.213974
$414$	0	0
$415$	−9.24745	−0.453939
$416$	0	0
$417$	14.7980	0.724659
$418$	0	0
$419$	−38.4495	−1.87838	−0.939190	−	0.343397i	$-0.888422\pi$
$419$	−38.4495	−1.87838	−0.939190	+	0.343397i	$0.888422\pi$
$420$	0	0
$421$	2.24745	0.109534	0.0547670	−	0.998499i	$-0.482558\pi$
$421$	2.24745	0.109534	0.0547670	+	0.998499i	$0.482558\pi$
$422$	0	0
$423$	−3.55051	−0.172632
$424$	0	0
$425$	−0.550510	−0.0267037
$426$	0	0
$427$	15.3485	0.742764
$428$	0	0
$429$	1.10102	0.0531578
$430$	0	0
$431$	−24.4949	−1.17988	−0.589939	−	0.807448i	$-0.700848\pi$
$431$	−24.4949	−1.17988	−0.589939	+	0.807448i	$0.700848\pi$
$432$	0	0
$433$	−16.7980	−0.807258	−0.403629	−	0.914923i	$-0.632251\pi$
$433$	−16.7980	−0.807258	−0.403629	+	0.914923i	$0.632251\pi$
$434$	0	0
$435$	−4.34847	−0.208493
$436$	0	0
$437$	0.449490	0.0215020
$438$	0	0
$439$	7.30306	0.348556	0.174278	−	0.984696i	$-0.444241\pi$
$439$	7.30306	0.348556	0.174278	+	0.984696i	$0.444241\pi$
$440$	0	0
$441$	−6.00000	−0.285714
$442$	0	0
$443$	15.5505	0.738827	0.369414	−	0.929265i	$-0.379559\pi$
$443$	15.5505	0.738827	0.369414	+	0.929265i	$0.379559\pi$
$444$	0	0
$445$	7.10102	0.336621
$446$	0	0
$447$	−13.3485	−0.631361
$448$	0	0
$449$	3.24745	0.153257	0.0766283	−	0.997060i	$-0.475585\pi$
$449$	3.24745	0.153257	0.0766283	+	0.997060i	$0.475585\pi$
$450$	0	0
$451$	1.34847	0.0634969
$452$	0	0
$453$	14.0000	0.657777
$454$	0	0
$455$	0.449490	0.0210724
$456$	0	0
$457$	9.89898	0.463055	0.231527	−	0.972828i	$-0.425628\pi$
$457$	9.89898	0.463055	0.231527	+	0.972828i	$0.425628\pi$
$458$	0	0
$459$	0.550510	0.0256956
$460$	0	0
$461$	30.4949	1.42029	0.710144	−	0.704056i	$-0.248630\pi$
$461$	30.4949	1.42029	0.710144	+	0.704056i	$0.248630\pi$
$462$	0	0
$463$	−28.9444	−1.34516	−0.672580	−	0.740025i	$-0.734814\pi$
$463$	−28.9444	−1.34516	−0.672580	+	0.740025i	$0.734814\pi$
$464$	0	0
$465$	−9.89898	−0.459054
$466$	0	0
$467$	−28.8434	−1.33471	−0.667356	−	0.744739i	$-0.732574\pi$
$467$	−28.8434	−1.33471	−0.667356	+	0.744739i	$0.732574\pi$
$468$	0	0
$469$	7.00000	0.323230
$470$	0	0
$471$	20.5959	0.949010
$472$	0	0
$473$	−4.89898	−0.225255
$474$	0	0
$475$	0.449490	0.0206240
$476$	0	0
$477$	5.44949	0.249515
$478$	0	0
$479$	20.9444	0.956973	0.478487	−	0.878095i	$-0.341185\pi$
$479$	20.9444	0.956973	0.478487	+	0.878095i	$0.341185\pi$
$480$	0	0
$481$	2.65153	0.120899
$482$	0	0
$483$	−1.00000	−0.0455016
$484$	0	0
$485$	−12.8990	−0.585712
$486$	0	0
$487$	−34.9444	−1.58348	−0.791741	−	0.610857i	$-0.790825\pi$
$487$	−34.9444	−1.58348	−0.791741	+	0.610857i	$0.790825\pi$
$488$	0	0
$489$	−10.0000	−0.452216
$490$	0	0
$491$	−17.4495	−0.787484	−0.393742	−	0.919221i	$-0.628820\pi$
$491$	−17.4495	−0.787484	−0.393742	+	0.919221i	$0.628820\pi$
$492$	0	0
$493$	2.39388	0.107815
$494$	0	0
$495$	−2.44949	−0.110096
$496$	0	0
$497$	−10.3485	−0.464192
$498$	0	0
$499$	1.00000	0.0447661	0.0223831	−	0.999749i	$-0.492875\pi$
$499$	1.00000	0.0447661	0.0223831	+	0.999749i	$0.492875\pi$
$500$	0	0
$501$	−1.34847	−0.0602452
$502$	0	0
$503$	37.0454	1.65177	0.825887	−	0.563836i	$-0.190675\pi$
$503$	37.0454	1.65177	0.825887	+	0.563836i	$0.190675\pi$
$504$	0	0
$505$	17.4495	0.776492
$506$	0	0
$507$	12.7980	0.568377
$508$	0	0
$509$	19.1010	0.846638	0.423319	−	0.905981i	$-0.360865\pi$
$509$	19.1010	0.846638	0.423319	+	0.905981i	$0.360865\pi$
$510$	0	0
$511$	9.34847	0.413552
$512$	0	0
$513$	−0.449490	−0.0198455
$514$	0	0
$515$	16.6969	0.735755
$516$	0	0
$517$	−8.69694	−0.382491
$518$	0	0
$519$	19.5959	0.860165
$520$	0	0
$521$	19.8434	0.869354	0.434677	−	0.900587i	$-0.356863\pi$
$521$	19.8434	0.869354	0.434677	+	0.900587i	$0.356863\pi$
$522$	0	0
$523$	−25.3939	−1.11040	−0.555198	−	0.831718i	$-0.687358\pi$
$523$	−25.3939	−1.11040	−0.555198	+	0.831718i	$0.687358\pi$
$524$	0	0
$525$	−1.00000	−0.0436436
$526$	0	0
$527$	5.44949	0.237384
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	4.34847	0.188707
$532$	0	0
$533$	−0.247449	−0.0107182
$534$	0	0
$535$	−0.550510	−0.0238006
$536$	0	0
$537$	−6.00000	−0.258919
$538$	0	0
$539$	−14.6969	−0.633042
$540$	0	0
$541$	3.10102	0.133323	0.0666616	−	0.997776i	$-0.478765\pi$
$541$	3.10102	0.133323	0.0666616	+	0.997776i	$0.478765\pi$
$542$	0	0
$543$	−0.898979	−0.0385789
$544$	0	0
$545$	−16.4495	−0.704619
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	0	0
$549$	15.3485	0.655057
$550$	0	0
$551$	−1.95459	−0.0832684
$552$	0	0
$553$	4.00000	0.170097
$554$	0	0
$555$	−5.89898	−0.250398
$556$	0	0
$557$	25.0454	1.06121	0.530604	−	0.847620i	$-0.321965\pi$
$557$	25.0454	1.06121	0.530604	+	0.847620i	$0.321965\pi$
$558$	0	0
$559$	0.898979	0.0380228
$560$	0	0
$561$	1.34847	0.0569324
$562$	0	0
$563$	−0.550510	−0.0232012	−0.0116006	−	0.999933i	$-0.503693\pi$
$563$	−0.550510	−0.0232012	−0.0116006	+	0.999933i	$0.503693\pi$
$564$	0	0
$565$	16.3485	0.687785
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	40.2929	1.68916	0.844582	−	0.535426i	$-0.179849\pi$
$569$	40.2929	1.68916	0.844582	+	0.535426i	$0.179849\pi$
$570$	0	0
$571$	21.1464	0.884950	0.442475	−	0.896781i	$-0.354100\pi$
$571$	21.1464	0.884950	0.442475	+	0.896781i	$0.354100\pi$
$572$	0	0
$573$	6.24745	0.260991
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−28.0000	−1.16566	−0.582828	−	0.812596i	$-0.698054\pi$
$577$	−28.0000	−1.16566	−0.582828	+	0.812596i	$0.698054\pi$
$578$	0	0
$579$	−22.6969	−0.943253
$580$	0	0
$581$	9.24745	0.383649
$582$	0	0
$583$	13.3485	0.552837
$584$	0	0
$585$	0.449490	0.0185841
$586$	0	0
$587$	6.00000	0.247647	0.123823	−	0.992304i	$-0.460484\pi$
$587$	6.00000	0.247647	0.123823	+	0.992304i	$0.460484\pi$
$588$	0	0
$589$	−4.44949	−0.183338
$590$	0	0
$591$	13.1010	0.538904
$592$	0	0
$593$	32.4495	1.33254	0.666270	−	0.745710i	$-0.267890\pi$
$593$	32.4495	1.33254	0.666270	+	0.745710i	$0.267890\pi$
$594$	0	0
$595$	0.550510	0.0225687
$596$	0	0
$597$	6.89898	0.282356
$598$	0	0
$599$	−26.6969	−1.09081	−0.545404	−	0.838174i	$-0.683624\pi$
$599$	−26.6969	−1.09081	−0.545404	+	0.838174i	$0.683624\pi$
$600$	0	0
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	0	0
$603$	7.00000	0.285062
$604$	0	0
$605$	5.00000	0.203279
$606$	0	0
$607$	−8.24745	−0.334754	−0.167377	−	0.985893i	$-0.553530\pi$
$607$	−8.24745	−0.334754	−0.167377	+	0.985893i	$0.553530\pi$
$608$	0	0
$609$	4.34847	0.176209
$610$	0	0
$611$	1.59592	0.0645639
$612$	0	0
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	0	0
$615$	0.550510	0.0221987
$616$	0	0
$617$	17.4495	0.702490	0.351245	−	0.936284i	$-0.385758\pi$
$617$	17.4495	0.702490	0.351245	+	0.936284i	$0.385758\pi$
$618$	0	0
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−7.10102	−0.284496
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−1.10102	−0.0439705
$628$	0	0
$629$	3.24745	0.129484
$630$	0	0
$631$	−34.4495	−1.37141	−0.685706	−	0.727878i	$-0.740507\pi$
$631$	−34.4495	−1.37141	−0.685706	+	0.727878i	$0.740507\pi$
$632$	0	0
$633$	11.0000	0.437211
$634$	0	0
$635$	−6.44949	−0.255940
$636$	0	0
$637$	2.69694	0.106857
$638$	0	0
$639$	−10.3485	−0.409379
$640$	0	0
$641$	−26.9444	−1.06424	−0.532120	−	0.846669i	$-0.678604\pi$
$641$	−26.9444	−1.06424	−0.532120	+	0.846669i	$0.678604\pi$
$642$	0	0
$643$	−19.6969	−0.776771	−0.388386	−	0.921497i	$-0.626967\pi$
$643$	−19.6969	−0.776771	−0.388386	+	0.921497i	$0.626967\pi$
$644$	0	0
$645$	−2.00000	−0.0787499
$646$	0	0
$647$	−46.0454	−1.81023	−0.905116	−	0.425165i	$-0.860216\pi$
$647$	−46.0454	−1.81023	−0.905116	+	0.425165i	$0.860216\pi$
$648$	0	0
$649$	10.6515	0.418109
$650$	0	0
$651$	9.89898	0.387972
$652$	0	0
$653$	−19.8434	−0.776531	−0.388265	−	0.921548i	$-0.626926\pi$
$653$	−19.8434	−0.776531	−0.388265	+	0.921548i	$0.626926\pi$
$654$	0	0
$655$	19.5959	0.765676
$656$	0	0
$657$	9.34847	0.364719
$658$	0	0
$659$	−29.1464	−1.13538	−0.567692	−	0.823241i	$-0.692163\pi$
$659$	−29.1464	−1.13538	−0.567692	+	0.823241i	$0.692163\pi$
$660$	0	0
$661$	−42.0908	−1.63714	−0.818571	−	0.574405i	$-0.805234\pi$
$661$	−42.0908	−1.63714	−0.818571	+	0.574405i	$0.805234\pi$
$662$	0	0
$663$	−0.247449	−0.00961011
$664$	0	0
$665$	−0.449490	−0.0174305
$666$	0	0
$667$	−4.34847	−0.168373
$668$	0	0
$669$	−17.5959	−0.680297
$670$	0	0
$671$	37.5959	1.45137
$672$	0	0
$673$	17.5505	0.676522	0.338261	−	0.941052i	$-0.390161\pi$
$673$	17.5505	0.676522	0.338261	+	0.941052i	$0.390161\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−17.4495	−0.670638	−0.335319	−	0.942105i	$-0.608844\pi$
$677$	−17.4495	−0.670638	−0.335319	+	0.942105i	$0.608844\pi$
$678$	0	0
$679$	12.8990	0.495017
$680$	0	0
$681$	2.20204	0.0843824
$682$	0	0
$683$	6.24745	0.239052	0.119526	−	0.992831i	$-0.461863\pi$
$683$	6.24745	0.239052	0.119526	+	0.992831i	$0.461863\pi$
$684$	0	0
$685$	7.10102	0.271316
$686$	0	0
$687$	−20.4949	−0.781929
$688$	0	0
$689$	−2.44949	−0.0933181
$690$	0	0
$691$	−35.7980	−1.36182	−0.680909	−	0.732368i	$-0.738415\pi$
$691$	−35.7980	−1.36182	−0.680909	+	0.732368i	$0.738415\pi$
$692$	0	0
$693$	2.44949	0.0930484
$694$	0	0
$695$	14.7980	0.561319
$696$	0	0
$697$	−0.303062	−0.0114793
$698$	0	0
$699$	−2.20204	−0.0832888
$700$	0	0
$701$	−7.95459	−0.300441	−0.150220	−	0.988653i	$-0.547998\pi$
$701$	−7.95459	−0.300441	−0.150220	+	0.988653i	$0.547998\pi$
$702$	0	0
$703$	−2.65153	−0.100004
$704$	0	0
$705$	−3.55051	−0.133720
$706$	0	0
$707$	−17.4495	−0.656256
$708$	0	0
$709$	44.7423	1.68033	0.840167	−	0.542328i	$-0.182457\pi$
$709$	44.7423	1.68033	0.840167	+	0.542328i	$0.182457\pi$
$710$	0	0
$711$	4.00000	0.150012
$712$	0	0
$713$	−9.89898	−0.370720
$714$	0	0
$715$	1.10102	0.0411758
$716$	0	0
$717$	11.4495	0.427589
$718$	0	0
$719$	−41.9444	−1.56426	−0.782131	−	0.623114i	$-0.785867\pi$
$719$	−41.9444	−1.56426	−0.782131	+	0.623114i	$0.785867\pi$
$720$	0	0
$721$	−16.6969	−0.621826
$722$	0	0
$723$	14.6515	0.544896
$724$	0	0
$725$	−4.34847	−0.161498
$726$	0	0
$727$	45.6969	1.69481	0.847403	−	0.530951i	$-0.178165\pi$
$727$	45.6969	1.69481	0.847403	+	0.530951i	$0.178165\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.10102	0.0407227
$732$	0	0
$733$	30.5959	1.13009	0.565043	−	0.825061i	$-0.308860\pi$
$733$	30.5959	1.13009	0.565043	+	0.825061i	$0.308860\pi$
$734$	0	0
$735$	−6.00000	−0.221313
$736$	0	0
$737$	17.1464	0.631597
$738$	0	0
$739$	28.7980	1.05935	0.529675	−	0.848201i	$-0.322314\pi$
$739$	28.7980	1.05935	0.529675	+	0.848201i	$0.322314\pi$
$740$	0	0
$741$	0.202041	0.00742216
$742$	0	0
$743$	2.20204	0.0807851	0.0403925	−	0.999184i	$-0.487139\pi$
$743$	2.20204	0.0807851	0.0403925	+	0.999184i	$0.487139\pi$
$744$	0	0
$745$	−13.3485	−0.489050
$746$	0	0
$747$	9.24745	0.338346
$748$	0	0
$749$	0.550510	0.0201152
$750$	0	0
$751$	−45.3485	−1.65479	−0.827395	−	0.561621i	$-0.810178\pi$
$751$	−45.3485	−1.65479	−0.827395	+	0.561621i	$0.810178\pi$
$752$	0	0
$753$	18.0000	0.655956
$754$	0	0
$755$	14.0000	0.509512
$756$	0	0
$757$	25.6969	0.933971	0.466986	−	0.884265i	$-0.345340\pi$
$757$	25.6969	0.933971	0.466986	+	0.884265i	$0.345340\pi$
$758$	0	0
$759$	−2.44949	−0.0889108
$760$	0	0
$761$	−20.1464	−0.730307	−0.365154	−	0.930947i	$-0.618984\pi$
$761$	−20.1464	−0.730307	−0.365154	+	0.930947i	$0.618984\pi$
$762$	0	0
$763$	16.4495	0.595512
$764$	0	0
$765$	0.550510	0.0199037
$766$	0	0
$767$	−1.95459	−0.0705762
$768$	0	0
$769$	5.55051	0.200157	0.100078	−	0.994980i	$-0.468091\pi$
$769$	5.55051	0.200157	0.100078	+	0.994980i	$0.468091\pi$
$770$	0	0
$771$	10.6515	0.383606
$772$	0	0
$773$	52.2929	1.88084	0.940422	−	0.340010i	$-0.110431\pi$
$773$	52.2929	1.88084	0.940422	+	0.340010i	$0.110431\pi$
$774$	0	0
$775$	−9.89898	−0.355582
$776$	0	0
$777$	5.89898	0.211625
$778$	0	0
$779$	0.247449	0.00886577
$780$	0	0
$781$	−25.3485	−0.907040
$782$	0	0
$783$	4.34847	0.155402
$784$	0	0
$785$	20.5959	0.735100
$786$	0	0
$787$	−7.69694	−0.274366	−0.137183	−	0.990546i	$-0.543805\pi$
$787$	−7.69694	−0.274366	−0.137183	+	0.990546i	$0.543805\pi$
$788$	0	0
$789$	2.75255	0.0979934
$790$	0	0
$791$	−16.3485	−0.581285
$792$	0	0
$793$	−6.89898	−0.244990
$794$	0	0
$795$	5.44949	0.193273
$796$	0	0
$797$	−15.2474	−0.540092	−0.270046	−	0.962847i	$-0.587039\pi$
$797$	−15.2474	−0.540092	−0.270046	+	0.962847i	$0.587039\pi$
$798$	0	0
$799$	1.95459	0.0691485
$800$	0	0
$801$	−7.10102	−0.250902
$802$	0	0
$803$	22.8990	0.808087
$804$	0	0
$805$	−1.00000	−0.0352454
$806$	0	0
$807$	0.550510	0.0193789
$808$	0	0
$809$	11.4495	0.402543	0.201271	−	0.979536i	$-0.435493\pi$
$809$	11.4495	0.402543	0.201271	+	0.979536i	$0.435493\pi$
$810$	0	0
$811$	−16.3939	−0.575667	−0.287833	−	0.957680i	$-0.592935\pi$
$811$	−16.3939	−0.575667	−0.287833	+	0.957680i	$0.592935\pi$
$812$	0	0
$813$	2.30306	0.0807719
$814$	0	0
$815$	−10.0000	−0.350285
$816$	0	0
$817$	−0.898979	−0.0314513
$818$	0	0
$819$	−0.449490	−0.0157064
$820$	0	0
$821$	50.6969	1.76934	0.884668	−	0.466222i	$-0.154385\pi$
$821$	50.6969	1.76934	0.884668	+	0.466222i	$0.154385\pi$
$822$	0	0
$823$	35.5959	1.24080	0.620398	−	0.784287i	$-0.286971\pi$
$823$	35.5959	1.24080	0.620398	+	0.784287i	$0.286971\pi$
$824$	0	0
$825$	−2.44949	−0.0852803
$826$	0	0
$827$	2.14643	0.0746386	0.0373193	−	0.999303i	$-0.488118\pi$
$827$	2.14643	0.0746386	0.0373193	+	0.999303i	$0.488118\pi$
$828$	0	0
$829$	1.69694	0.0589371	0.0294686	−	0.999566i	$-0.490619\pi$
$829$	1.69694	0.0589371	0.0294686	+	0.999566i	$0.490619\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	0	0
$833$	3.30306	0.114444
$834$	0	0
$835$	−1.34847	−0.0466657
$836$	0	0
$837$	9.89898	0.342159
$838$	0	0
$839$	−33.1918	−1.14591	−0.572955	−	0.819587i	$-0.694203\pi$
$839$	−33.1918	−1.14591	−0.572955	+	0.819587i	$0.694203\pi$
$840$	0	0
$841$	−10.0908	−0.347959
$842$	0	0
$843$	4.65153	0.160207
$844$	0	0
$845$	12.7980	0.440263
$846$	0	0
$847$	−5.00000	−0.171802
$848$	0	0
$849$	−11.8990	−0.408372
$850$	0	0
$851$	−5.89898	−0.202214
$852$	0	0
$853$	−42.0908	−1.44116	−0.720581	−	0.693371i	$-0.756125\pi$
$853$	−42.0908	−1.44116	−0.720581	+	0.693371i	$0.756125\pi$
$854$	0	0
$855$	−0.449490	−0.0153722
$856$	0	0
$857$	25.5959	0.874340	0.437170	−	0.899379i	$-0.355981\pi$
$857$	25.5959	0.874340	0.437170	+	0.899379i	$0.355981\pi$
$858$	0	0
$859$	40.1918	1.37133	0.685664	−	0.727918i	$-0.259512\pi$
$859$	40.1918	1.37133	0.685664	+	0.727918i	$0.259512\pi$
$860$	0	0
$861$	−0.550510	−0.0187613
$862$	0	0
$863$	42.4949	1.44654	0.723272	−	0.690564i	$-0.242637\pi$
$863$	42.4949	1.44654	0.723272	+	0.690564i	$0.242637\pi$
$864$	0	0
$865$	19.5959	0.666281
$866$	0	0
$867$	16.6969	0.567058
$868$	0	0
$869$	9.79796	0.332373
$870$	0	0
$871$	−3.14643	−0.106613
$872$	0	0
$873$	12.8990	0.436564
$874$	0	0
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	38.4949	1.29988	0.649940	−	0.759985i	$-0.274794\pi$
$877$	38.4949	1.29988	0.649940	+	0.759985i	$0.274794\pi$
$878$	0	0
$879$	−3.24745	−0.109534
$880$	0	0
$881$	3.55051	0.119620	0.0598099	−	0.998210i	$-0.480951\pi$
$881$	3.55051	0.119620	0.0598099	+	0.998210i	$0.480951\pi$
$882$	0	0
$883$	42.4495	1.42854	0.714270	−	0.699871i	$-0.246759\pi$
$883$	42.4495	1.42854	0.714270	+	0.699871i	$0.246759\pi$
$884$	0	0
$885$	4.34847	0.146172
$886$	0	0
$887$	−42.4949	−1.42684	−0.713420	−	0.700737i	$-0.752855\pi$
$887$	−42.4949	−1.42684	−0.713420	+	0.700737i	$0.752855\pi$
$888$	0	0
$889$	6.44949	0.216309
$890$	0	0
$891$	2.44949	0.0820610
$892$	0	0
$893$	−1.59592	−0.0534054
$894$	0	0
$895$	−6.00000	−0.200558
$896$	0	0
$897$	0.449490	0.0150080
$898$	0	0
$899$	43.0454	1.43564
$900$	0	0
$901$	−3.00000	−0.0999445
$902$	0	0
$903$	2.00000	0.0665558
$904$	0	0
$905$	−0.898979	−0.0298831
$906$	0	0
$907$	−19.2020	−0.637593	−0.318797	−	0.947823i	$-0.603279\pi$
$907$	−19.2020	−0.637593	−0.318797	+	0.947823i	$0.603279\pi$
$908$	0	0
$909$	−17.4495	−0.578763
$910$	0	0
$911$	33.7980	1.11978	0.559888	−	0.828568i	$-0.310844\pi$
$911$	33.7980	1.11978	0.559888	+	0.828568i	$0.310844\pi$
$912$	0	0
$913$	22.6515	0.749656
$914$	0	0
$915$	15.3485	0.507405
$916$	0	0
$917$	−19.5959	−0.647114
$918$	0	0
$919$	−13.3939	−0.441823	−0.220912	−	0.975294i	$-0.570903\pi$
$919$	−13.3939	−0.441823	−0.220912	+	0.975294i	$0.570903\pi$
$920$	0	0
$921$	−17.3485	−0.571651
$922$	0	0
$923$	4.65153	0.153107
$924$	0	0
$925$	−5.89898	−0.193957
$926$	0	0
$927$	−16.6969	−0.548399
$928$	0	0
$929$	−40.8434	−1.34003	−0.670014	−	0.742349i	$-0.733712\pi$
$929$	−40.8434	−1.34003	−0.670014	+	0.742349i	$0.733712\pi$
$930$	0	0
$931$	−2.69694	−0.0883886
$932$	0	0
$933$	12.0000	0.392862
$934$	0	0
$935$	1.34847	0.0440997
$936$	0	0
$937$	0.898979	0.0293684	0.0146842	−	0.999892i	$-0.495326\pi$
$937$	0.898979	0.0293684	0.0146842	+	0.999892i	$0.495326\pi$
$938$	0	0
$939$	9.69694	0.316448
$940$	0	0
$941$	−59.1464	−1.92812	−0.964059	−	0.265687i	$-0.914401\pi$
$941$	−59.1464	−1.92812	−0.964059	+	0.265687i	$0.914401\pi$
$942$	0	0
$943$	0.550510	0.0179271
$944$	0	0
$945$	1.00000	0.0325300
$946$	0	0
$947$	−48.4949	−1.57587	−0.787936	−	0.615757i	$-0.788850\pi$
$947$	−48.4949	−1.57587	−0.787936	+	0.615757i	$0.788850\pi$
$948$	0	0
$949$	−4.20204	−0.136404
$950$	0	0
$951$	18.2474	0.591714
$952$	0	0
$953$	49.5959	1.60657	0.803285	−	0.595595i	$-0.203084\pi$
$953$	49.5959	1.60657	0.803285	+	0.595595i	$0.203084\pi$
$954$	0	0
$955$	6.24745	0.202163
$956$	0	0
$957$	10.6515	0.344315
$958$	0	0
$959$	−7.10102	−0.229304
$960$	0	0
$961$	66.9898	2.16096
$962$	0	0
$963$	0.550510	0.0177399
$964$	0	0
$965$	−22.6969	−0.730640
$966$	0	0
$967$	−9.95459	−0.320118	−0.160059	−	0.987107i	$-0.551168\pi$
$967$	−9.95459	−0.320118	−0.160059	+	0.987107i	$0.551168\pi$
$968$	0	0
$969$	0.247449	0.00794920
$970$	0	0
$971$	46.2929	1.48561	0.742804	−	0.669509i	$-0.233495\pi$
$971$	46.2929	1.48561	0.742804	+	0.669509i	$0.233495\pi$
$972$	0	0
$973$	−14.7980	−0.474401
$974$	0	0
$975$	0.449490	0.0143952
$976$	0	0
$977$	34.3485	1.09890	0.549452	−	0.835525i	$-0.314836\pi$
$977$	34.3485	1.09890	0.549452	+	0.835525i	$0.314836\pi$
$978$	0	0
$979$	−17.3939	−0.555911
$980$	0	0
$981$	16.4495	0.525192
$982$	0	0
$983$	−39.7423	−1.26758	−0.633792	−	0.773504i	$-0.718502\pi$
$983$	−39.7423	−1.26758	−0.633792	+	0.773504i	$0.718502\pi$
$984$	0	0
$985$	13.1010	0.417433
$986$	0	0
$987$	3.55051	0.113014
$988$	0	0
$989$	−2.00000	−0.0635963
$990$	0	0
$991$	59.7878	1.89922	0.949610	−	0.313433i	$-0.101479\pi$
$991$	59.7878	1.89922	0.949610	+	0.313433i	$0.101479\pi$
$992$	0	0
$993$	−14.5959	−0.463187
$994$	0	0
$995$	6.89898	0.218712
$996$	0	0
$997$	−48.0908	−1.52305	−0.761526	−	0.648135i	$-0.775549\pi$
$997$	−48.0908	−1.52305	−0.761526	+	0.648135i	$0.775549\pi$
$998$	0	0
$999$	5.89898	0.186635

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.a.bi.1.2		2
4.3	odd	2		345.2.a.i.1.2	✓	2
12.11	even	2		1035.2.a.k.1.1		2
20.3	even	4		1725.2.b.m.1174.2		4
20.7	even	4		1725.2.b.m.1174.3		4
20.19	odd	2		1725.2.a.y.1.1		2
60.59	even	2		5175.2.a.bl.1.2		2
92.91	even	2		7935.2.a.t.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
345.2.a.i.1.2	✓	2	4.3	odd	2
1035.2.a.k.1.1		2	12.11	even	2
1725.2.a.y.1.1		2	20.19	odd	2
1725.2.b.m.1174.2		4	20.3	even	4
1725.2.b.m.1174.3		4	20.7	even	4
5175.2.a.bl.1.2		2	60.59	even	2
5520.2.a.bi.1.2		2	1.1	even	1	trivial
7935.2.a.t.1.2		2	92.91	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +2.44949 q^{11} -0.449490 q^{13} +1.00000 q^{15} -0.550510 q^{17} +0.449490 q^{19} -1.00000 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -4.34847 q^{29} -9.89898 q^{31} -2.44949 q^{33} -1.00000 q^{35} -5.89898 q^{37} +0.449490 q^{39} +0.550510 q^{41} -2.00000 q^{43} -1.00000 q^{45} -3.55051 q^{47} -6.00000 q^{49} +0.550510 q^{51} +5.44949 q^{53} -2.44949 q^{55} -0.449490 q^{57} +4.34847 q^{59} +15.3485 q^{61} +1.00000 q^{63} +0.449490 q^{65} +7.00000 q^{67} -1.00000 q^{69} -10.3485 q^{71} +9.34847 q^{73} -1.00000 q^{75} +2.44949 q^{77} +4.00000 q^{79} +1.00000 q^{81} +9.24745 q^{83} +0.550510 q^{85} +4.34847 q^{87} -7.10102 q^{89} -0.449490 q^{91} +9.89898 q^{93} -0.449490 q^{95} +12.8990 q^{97} +2.44949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 2 q^{9} + 4 q^{13} + 2 q^{15} - 6 q^{17} - 4 q^{19} - 2 q^{21} + 2 q^{23} + 2 q^{25} - 2 q^{27} + 6 q^{29} - 10 q^{31} - 2 q^{35} - 2 q^{37} - 4 q^{39} + 6 q^{41} - 4 q^{43} - 2 q^{45} - 12 q^{47} - 12 q^{49} + 6 q^{51} + 6 q^{53} + 4 q^{57} - 6 q^{59} + 16 q^{61} + 2 q^{63} - 4 q^{65} + 14 q^{67} - 2 q^{69} - 6 q^{71} + 4 q^{73} - 2 q^{75} + 8 q^{79} + 2 q^{81} - 6 q^{83} + 6 q^{85} - 6 q^{87} - 24 q^{89} + 4 q^{91} + 10 q^{93} + 4 q^{95} + 16 q^{97}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 2 * q^9 + 4 * q^13 + 2 * q^15 - 6 * q^17 - 4 * q^19 - 2 * q^21 + 2 * q^23 + 2 * q^25 - 2 * q^27 + 6 * q^29 - 10 * q^31 - 2 * q^35 - 2 * q^37 - 4 * q^39 + 6 * q^41 - 4 * q^43 - 2 * q^45 - 12 * q^47 - 12 * q^49 + 6 * q^51 + 6 * q^53 + 4 * q^57 - 6 * q^59 + 16 * q^61 + 2 * q^63 - 4 * q^65 + 14 * q^67 - 2 * q^69 - 6 * q^71 + 4 * q^73 - 2 * q^75 + 8 * q^79 + 2 * q^81 - 6 * q^83 + 6 * q^85 - 6 * q^87 - 24 * q^89 + 4 * q^91 + 10 * q^93 + 4 * q^95 + 16 * q^97

Introduction

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