LMFDB - Embedded newform 5520.2.a.bw.1.3

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:	$3$
Coefficient field:	3.3.1436.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 11x - 12 $ x^3 - 11*x - 12
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
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			1.3
Root			$-1.28282$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +3.78872 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} +3.78872 q^{7} +1.00000 q^{9} +3.07154 q^{11} -5.07154 q^{13} +1.00000 q^{15} +3.28282 q^{17} -5.07154 q^{19} -3.78872 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +1.28282 q^{29} -3.78872 q^{31} -3.07154 q^{33} -3.78872 q^{35} +3.22307 q^{37} +5.07154 q^{39} -9.42590 q^{41} -12.1431 q^{43} -1.00000 q^{45} -10.0833 q^{47} +7.35436 q^{49} -3.28282 q^{51} +5.84847 q^{53} -3.07154 q^{55} +5.07154 q^{57} +2.29461 q^{59} -3.07154 q^{61} +3.78872 q^{63} +5.07154 q^{65} -3.22307 q^{67} +1.00000 q^{69} -7.28282 q^{71} +9.63719 q^{73} -1.00000 q^{75} +11.6372 q^{77} +8.00000 q^{79} +1.00000 q^{81} -6.71718 q^{83} -3.28282 q^{85} -1.28282 q^{87} +5.01178 q^{89} -19.2146 q^{91} +3.78872 q^{93} +5.07154 q^{95} -1.13130 q^{97} +3.07154 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 4 q^{11} - 2 q^{13} + 3 q^{15} + 6 q^{17} - 2 q^{19} - 2 q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} - 2 q^{31} + 4 q^{33} - 2 q^{35} + 8 q^{37} + 2 q^{39} + 2 q^{41} - 10 q^{43} - 3 q^{45} - 6 q^{47} + 5 q^{49} - 6 q^{51} + 6 q^{53} + 4 q^{55} + 2 q^{57} - 8 q^{59} + 4 q^{61} + 2 q^{63} + 2 q^{65} - 8 q^{67} + 3 q^{69} - 18 q^{71} + 8 q^{73} - 3 q^{75} + 14 q^{77} + 24 q^{79} + 3 q^{81} - 24 q^{83} - 6 q^{85} + 4 q^{89} - 18 q^{91} + 2 q^{93} + 2 q^{95} + 12 q^{97} - 4 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9 - 4 * q^11 - 2 * q^13 + 3 * q^15 + 6 * q^17 - 2 * q^19 - 2 * q^21 - 3 * q^23 + 3 * q^25 - 3 * q^27 - 2 * q^31 + 4 * q^33 - 2 * q^35 + 8 * q^37 + 2 * q^39 + 2 * q^41 - 10 * q^43 - 3 * q^45 - 6 * q^47 + 5 * q^49 - 6 * q^51 + 6 * q^53 + 4 * q^55 + 2 * q^57 - 8 * q^59 + 4 * q^61 + 2 * q^63 + 2 * q^65 - 8 * q^67 + 3 * q^69 - 18 * q^71 + 8 * q^73 - 3 * q^75 + 14 * q^77 + 24 * q^79 + 3 * q^81 - 24 * q^83 - 6 * q^85 + 4 * q^89 - 18 * q^91 + 2 * q^93 + 2 * q^95 + 12 * q^97 - 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	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
$7$	3.78872	1.43200	0.716000	−	0.698100i	$-0.245971\pi$
$7$	3.78872	1.43200	0.716000	+	0.698100i	$0.245971\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.07154	0.926104	0.463052	−	0.886331i	$-0.346754\pi$
$11$	3.07154	0.926104	0.463052	+	0.886331i	$0.346754\pi$
$12$	0	0
$13$	−5.07154	−1.40659	−0.703296	−	0.710897i	$-0.748289\pi$
$13$	−5.07154	−1.40659	−0.703296	+	0.710897i	$0.748289\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	3.28282	0.796202	0.398101	−	0.917342i	$-0.369669\pi$
$17$	3.28282	0.796202	0.398101	+	0.917342i	$0.369669\pi$
$18$	0	0
$19$	−5.07154	−1.16349	−0.581745	−	0.813371i	$-0.697630\pi$
$19$	−5.07154	−1.16349	−0.581745	+	0.813371i	$0.697630\pi$
$20$	0	0
$21$	−3.78872	−0.826765
$22$	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
$29$	1.28282	0.238214	0.119107	−	0.992881i	$-0.461997\pi$
$29$	1.28282	0.238214	0.119107	+	0.992881i	$0.461997\pi$
$30$	0	0
$31$	−3.78872	−0.680473	−0.340237	−	0.940340i	$-0.610507\pi$
$31$	−3.78872	−0.680473	−0.340237	+	0.940340i	$0.610507\pi$
$32$	0	0
$33$	−3.07154	−0.534686
$34$	0	0
$35$	−3.78872	−0.640410
$36$	0	0
$37$	3.22307	0.529869	0.264935	−	0.964266i	$-0.414650\pi$
$37$	3.22307	0.529869	0.264935	+	0.964266i	$0.414650\pi$
$38$	0	0
$39$	5.07154	0.812096
$40$	0	0
$41$	−9.42590	−1.47208	−0.736039	−	0.676939i	$-0.763306\pi$
$41$	−9.42590	−1.47208	−0.736039	+	0.676939i	$0.763306\pi$
$42$	0	0
$43$	−12.1431	−1.85180	−0.925901	−	0.377766i	$-0.876692\pi$
$43$	−12.1431	−1.85180	−0.925901	+	0.377766i	$0.876692\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−10.0833	−1.47080	−0.735402	−	0.677631i	$-0.763007\pi$
$47$	−10.0833	−1.47080	−0.735402	+	0.677631i	$0.763007\pi$
$48$	0	0
$49$	7.35436	1.05062
$50$	0	0
$51$	−3.28282	−0.459687
$52$	0	0
$53$	5.84847	0.803349	0.401675	−	0.915782i	$-0.368428\pi$
$53$	5.84847	0.803349	0.401675	+	0.915782i	$0.368428\pi$
$54$	0	0
$55$	−3.07154	−0.414166
$56$	0	0
$57$	5.07154	0.671742
$58$	0	0
$59$	2.29461	0.298732	0.149366	−	0.988782i	$-0.452277\pi$
$59$	2.29461	0.298732	0.149366	+	0.988782i	$0.452277\pi$
$60$	0	0
$61$	−3.07154	−0.393270	−0.196635	−	0.980477i	$-0.563001\pi$
$61$	−3.07154	−0.393270	−0.196635	+	0.980477i	$0.563001\pi$
$62$	0	0
$63$	3.78872	0.477333
$64$	0	0
$65$	5.07154	0.629047
$66$	0	0
$67$	−3.22307	−0.393760	−0.196880	−	0.980428i	$-0.563081\pi$
$67$	−3.22307	−0.393760	−0.196880	+	0.980428i	$0.563081\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−7.28282	−0.864312	−0.432156	−	0.901799i	$-0.642247\pi$
$71$	−7.28282	−0.864312	−0.432156	+	0.901799i	$0.642247\pi$
$72$	0	0
$73$	9.63719	1.12795	0.563974	−	0.825793i	$-0.309272\pi$
$73$	9.63719	1.12795	0.563974	+	0.825793i	$0.309272\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	11.6372	1.32618
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.71718	−0.737306	−0.368653	−	0.929567i	$-0.620181\pi$
$83$	−6.71718	−0.737306	−0.368653	+	0.929567i	$0.620181\pi$
$84$	0	0
$85$	−3.28282	−0.356072
$86$	0	0
$87$	−1.28282	−0.137533
$88$	0	0
$89$	5.01178	0.531248	0.265624	−	0.964077i	$-0.414422\pi$
$89$	5.01178	0.531248	0.265624	+	0.964077i	$0.414422\pi$
$90$	0	0
$91$	−19.2146	−2.01424
$92$	0	0
$93$	3.78872	0.392871
$94$	0	0
$95$	5.07154	0.520329
$96$	0	0
$97$	−1.13130	−0.114866	−0.0574328	−	0.998349i	$-0.518292\pi$
$97$	−1.13130	−0.114866	−0.0574328	+	0.998349i	$0.518292\pi$
$98$	0	0
$99$	3.07154	0.308701
$100$	0	0
$101$	−6.29461	−0.626337	−0.313168	−	0.949698i	$-0.601390\pi$
$101$	−6.29461	−0.626337	−0.313168	+	0.949698i	$0.601390\pi$
$102$	0	0
$103$	12.7087	1.25223	0.626114	−	0.779732i	$-0.284644\pi$
$103$	12.7087	1.25223	0.626114	+	0.779732i	$0.284644\pi$
$104$	0	0
$105$	3.78872	0.369741
$106$	0	0
$107$	−18.2946	−1.76861	−0.884303	−	0.466913i	$-0.845366\pi$
$107$	−18.2946	−1.76861	−0.884303	+	0.466913i	$0.845366\pi$
$108$	0	0
$109$	−4.50589	−0.431586	−0.215793	−	0.976439i	$-0.569234\pi$
$109$	−4.50589	−0.431586	−0.215793	+	0.976439i	$0.569234\pi$
$110$	0	0
$111$	−3.22307	−0.305920
$112$	0	0
$113$	4.71718	0.443755	0.221877	−	0.975075i	$-0.428782\pi$
$113$	4.71718	0.443755	0.221877	+	0.975075i	$0.428782\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	−5.07154	−0.468864
$118$	0	0
$119$	12.4377	1.14016
$120$	0	0
$121$	−1.56565	−0.142332
$122$	0	0
$123$	9.42590	0.849905
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	12.0833	1.07222	0.536111	−	0.844148i	$-0.319893\pi$
$127$	12.0833	1.07222	0.536111	+	0.844148i	$0.319893\pi$
$128$	0	0
$129$	12.1431	1.06914
$130$	0	0
$131$	−17.1313	−1.49677	−0.748384	−	0.663266i	$-0.769170\pi$
$131$	−17.1313	−1.49677	−0.748384	+	0.663266i	$0.769170\pi$
$132$	0	0
$133$	−19.2146	−1.66612
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−2.70873	−0.231422	−0.115711	−	0.993283i	$-0.536915\pi$
$137$	−2.70873	−0.231422	−0.115711	+	0.993283i	$0.536915\pi$
$138$	0	0
$139$	0.211285	0.0179209	0.00896047	−	0.999960i	$-0.497148\pi$
$139$	0.211285	0.0179209	0.00896047	+	0.999960i	$0.497148\pi$
$140$	0	0
$141$	10.0833	0.849169
$142$	0	0
$143$	−15.5774	−1.30265
$144$	0	0
$145$	−1.28282	−0.106533
$146$	0	0
$147$	−7.35436	−0.606578
$148$	0	0
$149$	−17.6372	−1.44489	−0.722447	−	0.691426i	$-0.756983\pi$
$149$	−17.6372	−1.44489	−0.722447	+	0.691426i	$0.756983\pi$
$150$	0	0
$151$	13.7205	1.11656	0.558280	−	0.829653i	$-0.311462\pi$
$151$	13.7205	1.11656	0.558280	+	0.829653i	$0.311462\pi$
$152$	0	0
$153$	3.28282	0.265401
$154$	0	0
$155$	3.78872	0.304317
$156$	0	0
$157$	13.4857	1.07627	0.538136	−	0.842858i	$-0.319129\pi$
$157$	13.4857	1.07627	0.538136	+	0.842858i	$0.319129\pi$
$158$	0	0
$159$	−5.84847	−0.463814
$160$	0	0
$161$	−3.78872	−0.298593
$162$	0	0
$163$	7.57743	0.593510	0.296755	−	0.954954i	$-0.404096\pi$
$163$	7.57743	0.593510	0.296755	+	0.954954i	$0.404096\pi$
$164$	0	0
$165$	3.07154	0.239119
$166$	0	0
$167$	−14.2028	−1.09905	−0.549524	−	0.835478i	$-0.685191\pi$
$167$	−14.2028	−1.09905	−0.549524	+	0.835478i	$0.685191\pi$
$168$	0	0
$169$	12.7205	0.978501
$170$	0	0
$171$	−5.07154	−0.387830
$172$	0	0
$173$	5.57743	0.424044	0.212022	−	0.977265i	$-0.431995\pi$
$173$	5.57743	0.424044	0.212022	+	0.977265i	$0.431995\pi$
$174$	0	0
$175$	3.78872	0.286400
$176$	0	0
$177$	−2.29461	−0.172473
$178$	0	0
$179$	14.8518	1.11008	0.555038	−	0.831825i	$-0.312704\pi$
$179$	14.8518	1.11008	0.555038	+	0.831825i	$0.312704\pi$
$180$	0	0
$181$	20.8518	1.54990	0.774951	−	0.632021i	$-0.217774\pi$
$181$	20.8518	1.54990	0.774951	+	0.632021i	$0.217774\pi$
$182$	0	0
$183$	3.07154	0.227055
$184$	0	0
$185$	−3.22307	−0.236965
$186$	0	0
$187$	10.0833	0.737366
$188$	0	0
$189$	−3.78872	−0.275588
$190$	0	0
$191$	−19.0951	−1.38167	−0.690837	−	0.723011i	$-0.742758\pi$
$191$	−19.0951	−1.38167	−0.690837	+	0.723011i	$0.742758\pi$
$192$	0	0
$193$	−8.56565	−0.616569	−0.308284	−	0.951294i	$-0.599755\pi$
$193$	−8.56565	−0.616569	−0.308284	+	0.951294i	$0.599755\pi$
$194$	0	0
$195$	−5.07154	−0.363180
$196$	0	0
$197$	3.01178	0.214581	0.107290	−	0.994228i	$-0.465783\pi$
$197$	3.01178	0.214581	0.107290	+	0.994228i	$0.465783\pi$
$198$	0	0
$199$	22.8518	1.61992	0.809961	−	0.586484i	$-0.199488\pi$
$199$	22.8518	1.61992	0.809961	+	0.586484i	$0.199488\pi$
$200$	0	0
$201$	3.22307	0.227338
$202$	0	0
$203$	4.86025	0.341123
$204$	0	0
$205$	9.42590	0.658334
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−15.5774	−1.07751
$210$	0	0
$211$	−8.21128	−0.565288	−0.282644	−	0.959225i	$-0.591212\pi$
$211$	−8.21128	−0.565288	−0.282644	+	0.959225i	$0.591212\pi$
$212$	0	0
$213$	7.28282	0.499011
$214$	0	0
$215$	12.1431	0.828151
$216$	0	0
$217$	−14.3544	−0.974438
$218$	0	0
$219$	−9.63719	−0.651221
$220$	0	0
$221$	−16.6490	−1.11993
$222$	0	0
$223$	−10.2862	−0.688812	−0.344406	−	0.938821i	$-0.611920\pi$
$223$	−10.2862	−0.688812	−0.344406	+	0.938821i	$0.611920\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−23.1549	−1.53684	−0.768421	−	0.639945i	$-0.778957\pi$
$227$	−23.1549	−1.53684	−0.768421	+	0.639945i	$0.778957\pi$
$228$	0	0
$229$	−23.7205	−1.56750	−0.783748	−	0.621079i	$-0.786694\pi$
$229$	−23.7205	−1.56750	−0.783748	+	0.621079i	$0.786694\pi$
$230$	0	0
$231$	−11.6372	−0.765671
$232$	0	0
$233$	2.70873	0.177455	0.0887273	−	0.996056i	$-0.471720\pi$
$233$	2.70873	0.177455	0.0887273	+	0.996056i	$0.471720\pi$
$234$	0	0
$235$	10.0833	0.657763
$236$	0	0
$237$	−8.00000	−0.519656
$238$	0	0
$239$	−17.0033	−1.09985	−0.549927	−	0.835213i	$-0.685344\pi$
$239$	−17.0033	−1.09985	−0.549927	+	0.835213i	$0.685344\pi$
$240$	0	0
$241$	−2.36281	−0.152202	−0.0761011	−	0.997100i	$-0.524247\pi$
$241$	−2.36281	−0.152202	−0.0761011	+	0.997100i	$0.524247\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−7.35436	−0.469853
$246$	0	0
$247$	25.7205	1.63656
$248$	0	0
$249$	6.71718	0.425684
$250$	0	0
$251$	−23.1313	−1.46003	−0.730017	−	0.683429i	$-0.760488\pi$
$251$	−23.1313	−1.46003	−0.730017	+	0.683429i	$0.760488\pi$
$252$	0	0
$253$	−3.07154	−0.193106
$254$	0	0
$255$	3.28282	0.205578
$256$	0	0
$257$	−0.505891	−0.0315566	−0.0157783	−	0.999876i	$-0.505023\pi$
$257$	−0.505891	−0.0315566	−0.0157783	+	0.999876i	$0.505023\pi$
$258$	0	0
$259$	12.2113	0.758772
$260$	0	0
$261$	1.28282	0.0794048
$262$	0	0
$263$	−1.99155	−0.122804	−0.0614021	−	0.998113i	$-0.519557\pi$
$263$	−1.99155	−0.122804	−0.0614021	+	0.998113i	$0.519557\pi$
$264$	0	0
$265$	−5.84847	−0.359269
$266$	0	0
$267$	−5.01178	−0.306716
$268$	0	0
$269$	−3.13975	−0.191434	−0.0957168	−	0.995409i	$-0.530514\pi$
$269$	−3.13975	−0.191434	−0.0957168	+	0.995409i	$0.530514\pi$
$270$	0	0
$271$	−21.9318	−1.33226	−0.666131	−	0.745835i	$-0.732051\pi$
$271$	−21.9318	−1.33226	−0.666131	+	0.745835i	$0.732051\pi$
$272$	0	0
$273$	19.2146	1.16292
$274$	0	0
$275$	3.07154	0.185221
$276$	0	0
$277$	15.8636	0.953151	0.476575	−	0.879134i	$-0.341878\pi$
$277$	15.8636	0.953151	0.476575	+	0.879134i	$0.341878\pi$
$278$	0	0
$279$	−3.78872	−0.226824
$280$	0	0
$281$	−1.37460	−0.0820015	−0.0410008	−	0.999159i	$-0.513055\pi$
$281$	−1.37460	−0.0820015	−0.0410008	+	0.999159i	$0.513055\pi$
$282$	0	0
$283$	−12.3544	−0.734391	−0.367195	−	0.930144i	$-0.619682\pi$
$283$	−12.3544	−0.734391	−0.367195	+	0.930144i	$0.619682\pi$
$284$	0	0
$285$	−5.07154	−0.300412
$286$	0	0
$287$	−35.7121	−2.10802
$288$	0	0
$289$	−6.22307	−0.366063
$290$	0	0
$291$	1.13130	0.0663177
$292$	0	0
$293$	−29.7121	−1.73580	−0.867898	−	0.496742i	$-0.834530\pi$
$293$	−29.7121	−1.73580	−0.867898	+	0.496742i	$0.834530\pi$
$294$	0	0
$295$	−2.29461	−0.133597
$296$	0	0
$297$	−3.07154	−0.178229
$298$	0	0
$299$	5.07154	0.293295
$300$	0	0
$301$	−46.0067	−2.65178
$302$	0	0
$303$	6.29461	0.361616
$304$	0	0
$305$	3.07154	0.175876
$306$	0	0
$307$	30.4890	1.74010	0.870049	−	0.492965i	$-0.164087\pi$
$307$	30.4890	1.74010	0.870049	+	0.492965i	$0.164087\pi$
$308$	0	0
$309$	−12.7087	−0.722974
$310$	0	0
$311$	−4.44613	−0.252117	−0.126059	−	0.992023i	$-0.540233\pi$
$311$	−4.44613	−0.252117	−0.126059	+	0.992023i	$0.540233\pi$
$312$	0	0
$313$	6.35436	0.359170	0.179585	−	0.983742i	$-0.442525\pi$
$313$	6.35436	0.359170	0.179585	+	0.983742i	$0.442525\pi$
$314$	0	0
$315$	−3.78872	−0.213470
$316$	0	0
$317$	2.64897	0.148781	0.0743905	−	0.997229i	$-0.476299\pi$
$317$	2.64897	0.148781	0.0743905	+	0.997229i	$0.476299\pi$
$318$	0	0
$319$	3.94024	0.220611
$320$	0	0
$321$	18.2946	1.02111
$322$	0	0
$323$	−16.6490	−0.926373
$324$	0	0
$325$	−5.07154	−0.281318
$326$	0	0
$327$	4.50589	0.249176
$328$	0	0
$329$	−38.2028	−2.10619
$330$	0	0
$331$	−4.80050	−0.263859	−0.131930	−	0.991259i	$-0.542117\pi$
$331$	−4.80050	−0.263859	−0.131930	+	0.991259i	$0.542117\pi$
$332$	0	0
$333$	3.22307	0.176623
$334$	0	0
$335$	3.22307	0.176095
$336$	0	0
$337$	−14.7323	−0.802519	−0.401260	−	0.915964i	$-0.631427\pi$
$337$	−14.7323	−0.802519	−0.401260	+	0.915964i	$0.631427\pi$
$338$	0	0
$339$	−4.71718	−0.256202
$340$	0	0
$341$	−11.6372	−0.630189
$342$	0	0
$343$	1.34258	0.0724925
$344$	0	0
$345$	−1.00000	−0.0538382
$346$	0	0
$347$	33.5841	1.80289	0.901444	−	0.432895i	$-0.142508\pi$
$347$	33.5841	1.80289	0.901444	+	0.432895i	$0.142508\pi$
$348$	0	0
$349$	−6.92001	−0.370420	−0.185210	−	0.982699i	$-0.559296\pi$
$349$	−6.92001	−0.370420	−0.185210	+	0.982699i	$0.559296\pi$
$350$	0	0
$351$	5.07154	0.270699
$352$	0	0
$353$	8.38638	0.446362	0.223181	−	0.974777i	$-0.428356\pi$
$353$	8.38638	0.446362	0.223181	+	0.974777i	$0.428356\pi$
$354$	0	0
$355$	7.28282	0.386532
$356$	0	0
$357$	−12.4377	−0.658272
$358$	0	0
$359$	−19.2146	−1.01411	−0.507054	−	0.861914i	$-0.669266\pi$
$359$	−19.2146	−1.01411	−0.507054	+	0.861914i	$0.669266\pi$
$360$	0	0
$361$	6.72051	0.353711
$362$	0	0
$363$	1.56565	0.0821752
$364$	0	0
$365$	−9.63719	−0.504433
$366$	0	0
$367$	−16.4974	−0.861159	−0.430580	−	0.902553i	$-0.641691\pi$
$367$	−16.4974	−0.861159	−0.430580	+	0.902553i	$0.641691\pi$
$368$	0	0
$369$	−9.42590	−0.490693
$370$	0	0
$371$	22.1582	1.15040
$372$	0	0
$373$	−5.45792	−0.282600	−0.141300	−	0.989967i	$-0.545128\pi$
$373$	−5.45792	−0.282600	−0.141300	+	0.989967i	$0.545128\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−6.50589	−0.335070
$378$	0	0
$379$	−25.4175	−1.30561	−0.652803	−	0.757527i	$-0.726407\pi$
$379$	−25.4175	−1.30561	−0.652803	+	0.757527i	$0.726407\pi$
$380$	0	0
$381$	−12.0833	−0.619047
$382$	0	0
$383$	−27.7121	−1.41602	−0.708010	−	0.706202i	$-0.750407\pi$
$383$	−27.7121	−1.41602	−0.708010	+	0.706202i	$0.750407\pi$
$384$	0	0
$385$	−11.6372	−0.593086
$386$	0	0
$387$	−12.1431	−0.617267
$388$	0	0
$389$	38.4528	1.94963	0.974817	−	0.223006i	$-0.0715868\pi$
$389$	38.4528	1.94963	0.974817	+	0.223006i	$0.0715868\pi$
$390$	0	0
$391$	−3.28282	−0.166020
$392$	0	0
$393$	17.1313	0.864160
$394$	0	0
$395$	−8.00000	−0.402524
$396$	0	0
$397$	8.28616	0.415870	0.207935	−	0.978143i	$-0.433326\pi$
$397$	8.28616	0.415870	0.207935	+	0.978143i	$0.433326\pi$
$398$	0	0
$399$	19.2146	0.961934
$400$	0	0
$401$	5.72051	0.285669	0.142834	−	0.989747i	$-0.454378\pi$
$401$	5.72051	0.285669	0.142834	+	0.989747i	$0.454378\pi$
$402$	0	0
$403$	19.2146	0.957148
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	9.89978	0.490714
$408$	0	0
$409$	−26.4974	−1.31021	−0.655107	−	0.755536i	$-0.727376\pi$
$409$	−26.4974	−1.31021	−0.655107	+	0.755536i	$0.727376\pi$
$410$	0	0
$411$	2.70873	0.133612
$412$	0	0
$413$	8.69361	0.427785
$414$	0	0
$415$	6.71718	0.329733
$416$	0	0
$417$	−0.211285	−0.0103467
$418$	0	0
$419$	−22.2264	−1.08583	−0.542915	−	0.839787i	$-0.682680\pi$
$419$	−22.2264	−1.08583	−0.542915	+	0.839787i	$0.682680\pi$
$420$	0	0
$421$	18.9351	0.922842	0.461421	−	0.887181i	$-0.347340\pi$
$421$	18.9351	0.922842	0.461421	+	0.887181i	$0.347340\pi$
$422$	0	0
$423$	−10.0833	−0.490268
$424$	0	0
$425$	3.28282	0.159240
$426$	0	0
$427$	−11.6372	−0.563163
$428$	0	0
$429$	15.5774	0.752085
$430$	0	0
$431$	−10.9882	−0.529284	−0.264642	−	0.964347i	$-0.585254\pi$
$431$	−10.9882	−0.529284	−0.264642	+	0.964347i	$0.585254\pi$
$432$	0	0
$433$	18.3544	0.882054	0.441027	−	0.897494i	$-0.354614\pi$
$433$	18.3544	0.882054	0.441027	+	0.897494i	$0.354614\pi$
$434$	0	0
$435$	1.28282	0.0615067
$436$	0	0
$437$	5.07154	0.242605
$438$	0	0
$439$	−31.0184	−1.48043	−0.740215	−	0.672370i	$-0.765276\pi$
$439$	−31.0184	−1.48043	−0.740215	+	0.672370i	$0.765276\pi$
$440$	0	0
$441$	7.35436	0.350208
$442$	0	0
$443$	−7.09510	−0.337099	−0.168549	−	0.985693i	$-0.553908\pi$
$443$	−7.09510	−0.337099	−0.168549	+	0.985693i	$0.553908\pi$
$444$	0	0
$445$	−5.01178	−0.237581
$446$	0	0
$447$	17.6372	0.834210
$448$	0	0
$449$	35.4326	1.67217	0.836083	−	0.548603i	$-0.184840\pi$
$449$	35.4326	1.67217	0.836083	+	0.548603i	$0.184840\pi$
$450$	0	0
$451$	−28.9520	−1.36330
$452$	0	0
$453$	−13.7205	−0.644646
$454$	0	0
$455$	19.2146	0.900795
$456$	0	0
$457$	−0.514342	−0.0240599	−0.0120299	−	0.999928i	$-0.503829\pi$
$457$	−0.514342	−0.0240599	−0.0120299	+	0.999928i	$0.503829\pi$
$458$	0	0
$459$	−3.28282	−0.153229
$460$	0	0
$461$	−16.5892	−0.772637	−0.386318	−	0.922366i	$-0.626253\pi$
$461$	−16.5892	−0.772637	−0.386318	+	0.922366i	$0.626253\pi$
$462$	0	0
$463$	−24.6725	−1.14663	−0.573315	−	0.819335i	$-0.694343\pi$
$463$	−24.6725	−1.14663	−0.573315	+	0.819335i	$0.694343\pi$
$464$	0	0
$465$	−3.78872	−0.175697
$466$	0	0
$467$	−19.4259	−0.898924	−0.449462	−	0.893300i	$-0.648384\pi$
$467$	−19.4259	−0.898924	−0.449462	+	0.893300i	$0.648384\pi$
$468$	0	0
$469$	−12.2113	−0.563865
$470$	0	0
$471$	−13.4857	−0.621386
$472$	0	0
$473$	−37.2979	−1.71496
$474$	0	0
$475$	−5.07154	−0.232698
$476$	0	0
$477$	5.84847	0.267783
$478$	0	0
$479$	19.2146	0.877938	0.438969	−	0.898502i	$-0.355344\pi$
$479$	19.2146	0.877938	0.438969	+	0.898502i	$0.355344\pi$
$480$	0	0
$481$	−16.3459	−0.745309
$482$	0	0
$483$	3.78872	0.172393
$484$	0	0
$485$	1.13130	0.0513695
$486$	0	0
$487$	43.5243	1.97228	0.986138	−	0.165927i	$-0.0530616\pi$
$487$	43.5243	1.97228	0.986138	+	0.165927i	$0.0530616\pi$
$488$	0	0
$489$	−7.57743	−0.342663
$490$	0	0
$491$	−1.82491	−0.0823569	−0.0411784	−	0.999152i	$-0.513111\pi$
$491$	−1.82491	−0.0823569	−0.0411784	+	0.999152i	$0.513111\pi$
$492$	0	0
$493$	4.21128	0.189667
$494$	0	0
$495$	−3.07154	−0.138055
$496$	0	0
$497$	−27.5925	−1.23769
$498$	0	0
$499$	−14.6405	−0.655400	−0.327700	−	0.944782i	$-0.606273\pi$
$499$	−14.6405	−0.655400	−0.327700	+	0.944782i	$0.606273\pi$
$500$	0	0
$501$	14.2028	0.634536
$502$	0	0
$503$	6.88382	0.306934	0.153467	−	0.988154i	$-0.450956\pi$
$503$	6.88382	0.306934	0.153467	+	0.988154i	$0.450956\pi$
$504$	0	0
$505$	6.29461	0.280106
$506$	0	0
$507$	−12.7205	−0.564938
$508$	0	0
$509$	−28.0000	−1.24108	−0.620539	−	0.784176i	$-0.713086\pi$
$509$	−28.0000	−1.24108	−0.620539	+	0.784176i	$0.713086\pi$
$510$	0	0
$511$	36.5126	1.61522
$512$	0	0
$513$	5.07154	0.223914
$514$	0	0
$515$	−12.7087	−0.560013
$516$	0	0
$517$	−30.9713	−1.36212
$518$	0	0
$519$	−5.57743	−0.244822
$520$	0	0
$521$	33.9469	1.48724	0.743621	−	0.668602i	$-0.233107\pi$
$521$	33.9469	1.48724	0.743621	+	0.668602i	$0.233107\pi$
$522$	0	0
$523$	−13.5774	−0.593700	−0.296850	−	0.954924i	$-0.595936\pi$
$523$	−13.5774	−0.593700	−0.296850	+	0.954924i	$0.595936\pi$
$524$	0	0
$525$	−3.78872	−0.165353
$526$	0	0
$527$	−12.4377	−0.541794
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	2.29461	0.0995774
$532$	0	0
$533$	47.8038	2.07061
$534$	0	0
$535$	18.2946	0.790945
$536$	0	0
$537$	−14.8518	−0.640903
$538$	0	0
$539$	22.5892	0.972986
$540$	0	0
$541$	−16.6852	−0.717351	−0.358676	−	0.933462i	$-0.616772\pi$
$541$	−16.6852	−0.717351	−0.358676	+	0.933462i	$0.616772\pi$
$542$	0	0
$543$	−20.8518	−0.894837
$544$	0	0
$545$	4.50589	0.193011
$546$	0	0
$547$	−13.4175	−0.573689	−0.286844	−	0.957977i	$-0.592606\pi$
$547$	−13.4175	−0.573689	−0.286844	+	0.957977i	$0.592606\pi$
$548$	0	0
$549$	−3.07154	−0.131090
$550$	0	0
$551$	−6.50589	−0.277160
$552$	0	0
$553$	30.3097	1.28890
$554$	0	0
$555$	3.22307	0.136812
$556$	0	0
$557$	35.0269	1.48414	0.742069	−	0.670324i	$-0.233845\pi$
$557$	35.0269	1.48414	0.742069	+	0.670324i	$0.233845\pi$
$558$	0	0
$559$	61.5841	2.60473
$560$	0	0
$561$	−10.0833	−0.425718
$562$	0	0
$563$	33.7356	1.42179	0.710894	−	0.703300i	$-0.248291\pi$
$563$	33.7356	1.42179	0.710894	+	0.703300i	$0.248291\pi$
$564$	0	0
$565$	−4.71718	−0.198453
$566$	0	0
$567$	3.78872	0.159111
$568$	0	0
$569$	−19.4579	−0.815718	−0.407859	−	0.913045i	$-0.633725\pi$
$569$	−19.4579	−0.815718	−0.407859	+	0.913045i	$0.633725\pi$
$570$	0	0
$571$	−39.6843	−1.66074	−0.830369	−	0.557215i	$-0.811870\pi$
$571$	−39.6843	−1.66074	−0.830369	+	0.557215i	$0.811870\pi$
$572$	0	0
$573$	19.0951	0.797709
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−37.8636	−1.57628	−0.788141	−	0.615495i	$-0.788956\pi$
$577$	−37.8636	−1.57628	−0.788141	+	0.615495i	$0.788956\pi$
$578$	0	0
$579$	8.56565	0.355976
$580$	0	0
$581$	−25.4495	−1.05582
$582$	0	0
$583$	17.9638	0.743985
$584$	0	0
$585$	5.07154	0.209682
$586$	0	0
$587$	−9.55387	−0.394330	−0.197165	−	0.980370i	$-0.563174\pi$
$587$	−9.55387	−0.394330	−0.197165	+	0.980370i	$0.563174\pi$
$588$	0	0
$589$	19.2146	0.791725
$590$	0	0
$591$	−3.01178	−0.123888
$592$	0	0
$593$	−22.9351	−0.941833	−0.470916	−	0.882178i	$-0.656077\pi$
$593$	−22.9351	−0.941833	−0.470916	+	0.882178i	$0.656077\pi$
$594$	0	0
$595$	−12.4377	−0.509895
$596$	0	0
$597$	−22.8518	−0.935262
$598$	0	0
$599$	25.6969	1.04995	0.524974	−	0.851118i	$-0.324075\pi$
$599$	25.6969	1.04995	0.524974	+	0.851118i	$0.324075\pi$
$600$	0	0
$601$	−17.0800	−0.696707	−0.348354	−	0.937363i	$-0.613259\pi$
$601$	−17.0800	−0.696707	−0.348354	+	0.937363i	$0.613259\pi$
$602$	0	0
$603$	−3.22307	−0.131253
$604$	0	0
$605$	1.56565	0.0636526
$606$	0	0
$607$	24.2028	0.982363	0.491181	−	0.871057i	$-0.336565\pi$
$607$	24.2028	0.982363	0.491181	+	0.871057i	$0.336565\pi$
$608$	0	0
$609$	−4.86025	−0.196947
$610$	0	0
$611$	51.1380	2.06882
$612$	0	0
$613$	3.73741	0.150953	0.0754763	−	0.997148i	$-0.475952\pi$
$613$	3.73741	0.150953	0.0754763	+	0.997148i	$0.475952\pi$
$614$	0	0
$615$	−9.42590	−0.380089
$616$	0	0
$617$	−14.3182	−0.576428	−0.288214	−	0.957566i	$-0.593061\pi$
$617$	−14.3182	−0.576428	−0.288214	+	0.957566i	$0.593061\pi$
$618$	0	0
$619$	37.4175	1.50393	0.751967	−	0.659201i	$-0.229105\pi$
$619$	37.4175	1.50393	0.751967	+	0.659201i	$0.229105\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	18.9882	0.760747
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	15.5774	0.622103
$628$	0	0
$629$	10.5808	0.421883
$630$	0	0
$631$	42.6725	1.69877	0.849383	−	0.527776i	$-0.176974\pi$
$631$	42.6725	1.69877	0.849383	+	0.527776i	$0.176974\pi$
$632$	0	0
$633$	8.21128	0.326369
$634$	0	0
$635$	−12.0833	−0.479512
$636$	0	0
$637$	−37.2979	−1.47780
$638$	0	0
$639$	−7.28282	−0.288104
$640$	0	0
$641$	3.39816	0.134219	0.0671097	−	0.997746i	$-0.478622\pi$
$641$	3.39816	0.134219	0.0671097	+	0.997746i	$0.478622\pi$
$642$	0	0
$643$	−16.5210	−0.651525	−0.325762	−	0.945452i	$-0.605621\pi$
$643$	−16.5210	−0.651525	−0.325762	+	0.945452i	$0.605621\pi$
$644$	0	0
$645$	−12.1431	−0.478133
$646$	0	0
$647$	37.9469	1.49185	0.745923	−	0.666032i	$-0.232008\pi$
$647$	37.9469	1.49185	0.745923	+	0.666032i	$0.232008\pi$
$648$	0	0
$649$	7.04797	0.276657
$650$	0	0
$651$	14.3544	0.562592
$652$	0	0
$653$	19.0715	0.746327	0.373163	−	0.927766i	$-0.378273\pi$
$653$	19.0715	0.746327	0.373163	+	0.927766i	$0.378273\pi$
$654$	0	0
$655$	17.1313	0.669375
$656$	0	0
$657$	9.63719	0.375982
$658$	0	0
$659$	−21.5008	−0.837551	−0.418776	−	0.908090i	$-0.637541\pi$
$659$	−21.5008	−0.837551	−0.418776	+	0.908090i	$0.637541\pi$
$660$	0	0
$661$	2.11951	0.0824395	0.0412197	−	0.999150i	$-0.486876\pi$
$661$	2.11951	0.0824395	0.0412197	+	0.999150i	$0.486876\pi$
$662$	0	0
$663$	16.6490	0.646592
$664$	0	0
$665$	19.2146	0.745111
$666$	0	0
$667$	−1.28282	−0.0496711
$668$	0	0
$669$	10.2862	0.397686
$670$	0	0
$671$	−9.43435	−0.364209
$672$	0	0
$673$	8.03619	0.309772	0.154886	−	0.987932i	$-0.450499\pi$
$673$	8.03619	0.309772	0.154886	+	0.987932i	$0.450499\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	27.5690	1.05956	0.529781	−	0.848134i	$-0.322274\pi$
$677$	27.5690	1.05956	0.529781	+	0.848134i	$0.322274\pi$
$678$	0	0
$679$	−4.28616	−0.164488
$680$	0	0
$681$	23.1549	0.887296
$682$	0	0
$683$	9.35770	0.358062	0.179031	−	0.983843i	$-0.442704\pi$
$683$	9.35770	0.358062	0.179031	+	0.983843i	$0.442704\pi$
$684$	0	0
$685$	2.70873	0.103495
$686$	0	0
$687$	23.7205	0.904994
$688$	0	0
$689$	−29.6608	−1.12998
$690$	0	0
$691$	44.8754	1.70714	0.853570	−	0.520979i	$-0.174433\pi$
$691$	44.8754	1.70714	0.853570	+	0.520979i	$0.174433\pi$
$692$	0	0
$693$	11.6372	0.442060
$694$	0	0
$695$	−0.211285	−0.00801449
$696$	0	0
$697$	−30.9436	−1.17207
$698$	0	0
$699$	−2.70873	−0.102453
$700$	0	0
$701$	−28.9116	−1.09197	−0.545987	−	0.837793i	$-0.683845\pi$
$701$	−28.9116	−1.09197	−0.545987	+	0.837793i	$0.683845\pi$
$702$	0	0
$703$	−16.3459	−0.616498
$704$	0	0
$705$	−10.0833	−0.379760
$706$	0	0
$707$	−23.8485	−0.896914
$708$	0	0
$709$	−52.7751	−1.98201	−0.991006	−	0.133817i	$-0.957277\pi$
$709$	−52.7751	−1.98201	−0.991006	+	0.133817i	$0.957277\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	3.78872	0.141888
$714$	0	0
$715$	15.5774	0.582563
$716$	0	0
$717$	17.0033	0.635001
$718$	0	0
$719$	45.4090	1.69347	0.846735	−	0.532015i	$-0.178565\pi$
$719$	45.4090	1.69347	0.846735	+	0.532015i	$0.178565\pi$
$720$	0	0
$721$	48.1497	1.79319
$722$	0	0
$723$	2.36281	0.0878740
$724$	0	0
$725$	1.28282	0.0476429
$726$	0	0
$727$	−32.2415	−1.19577	−0.597886	−	0.801581i	$-0.703992\pi$
$727$	−32.2415	−1.19577	−0.597886	+	0.801581i	$0.703992\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−39.8636	−1.47441
$732$	0	0
$733$	39.3897	1.45489	0.727446	−	0.686165i	$-0.240707\pi$
$733$	39.3897	1.45489	0.727446	+	0.686165i	$0.240707\pi$
$734$	0	0
$735$	7.35436	0.271270
$736$	0	0
$737$	−9.89978	−0.364663
$738$	0	0
$739$	23.4857	0.863934	0.431967	−	0.901889i	$-0.357820\pi$
$739$	23.4857	0.863934	0.431967	+	0.901889i	$0.357820\pi$
$740$	0	0
$741$	−25.7205	−0.944866
$742$	0	0
$743$	37.1313	1.36222	0.681108	−	0.732183i	$-0.261499\pi$
$743$	37.1313	1.36222	0.681108	+	0.732183i	$0.261499\pi$
$744$	0	0
$745$	17.6372	0.646177
$746$	0	0
$747$	−6.71718	−0.245769
$748$	0	0
$749$	−69.3131	−2.53264
$750$	0	0
$751$	14.0833	0.513908	0.256954	−	0.966424i	$-0.417281\pi$
$751$	14.0833	0.513908	0.256954	+	0.966424i	$0.417281\pi$
$752$	0	0
$753$	23.1313	0.842951
$754$	0	0
$755$	−13.7205	−0.499340
$756$	0	0
$757$	44.8072	1.62854	0.814272	−	0.580483i	$-0.197136\pi$
$757$	44.8072	1.62854	0.814272	+	0.580483i	$0.197136\pi$
$758$	0	0
$759$	3.07154	0.111490
$760$	0	0
$761$	3.56898	0.129375	0.0646877	−	0.997906i	$-0.479395\pi$
$761$	3.56898	0.129375	0.0646877	+	0.997906i	$0.479395\pi$
$762$	0	0
$763$	−17.0715	−0.618031
$764$	0	0
$765$	−3.28282	−0.118691
$766$	0	0
$767$	−11.6372	−0.420194
$768$	0	0
$769$	−43.6439	−1.57384	−0.786919	−	0.617057i	$-0.788325\pi$
$769$	−43.6439	−1.57384	−0.786919	+	0.617057i	$0.788325\pi$
$770$	0	0
$771$	0.505891	0.0182192
$772$	0	0
$773$	13.8805	0.499246	0.249623	−	0.968343i	$-0.419693\pi$
$773$	13.8805	0.499246	0.249623	+	0.968343i	$0.419693\pi$
$774$	0	0
$775$	−3.78872	−0.136095
$776$	0	0
$777$	−12.2113	−0.438077
$778$	0	0
$779$	47.8038	1.71275
$780$	0	0
$781$	−22.3695	−0.800443
$782$	0	0
$783$	−1.28282	−0.0458444
$784$	0	0
$785$	−13.4857	−0.481324
$786$	0	0
$787$	−37.6523	−1.34216	−0.671080	−	0.741385i	$-0.734169\pi$
$787$	−37.6523	−1.34216	−0.671080	+	0.741385i	$0.734169\pi$
$788$	0	0
$789$	1.99155	0.0709010
$790$	0	0
$791$	17.8720	0.635456
$792$	0	0
$793$	15.5774	0.553171
$794$	0	0
$795$	5.84847	0.207424
$796$	0	0
$797$	2.15153	0.0762110	0.0381055	−	0.999274i	$-0.487868\pi$
$797$	2.15153	0.0762110	0.0381055	+	0.999274i	$0.487868\pi$
$798$	0	0
$799$	−33.1018	−1.17106
$800$	0	0
$801$	5.01178	0.177083
$802$	0	0
$803$	29.6010	1.04460
$804$	0	0
$805$	3.78872	0.133535
$806$	0	0
$807$	3.13975	0.110524
$808$	0	0
$809$	14.5572	0.511804	0.255902	−	0.966703i	$-0.417628\pi$
$809$	14.5572	0.511804	0.255902	+	0.966703i	$0.417628\pi$
$810$	0	0
$811$	−2.47388	−0.0868695	−0.0434348	−	0.999056i	$-0.513830\pi$
$811$	−2.47388	−0.0868695	−0.0434348	+	0.999056i	$0.513830\pi$
$812$	0	0
$813$	21.9318	0.769182
$814$	0	0
$815$	−7.57743	−0.265426
$816$	0	0
$817$	61.5841	2.15455
$818$	0	0
$819$	−19.2146	−0.671413
$820$	0	0
$821$	32.3333	1.12844	0.564220	−	0.825625i	$-0.309177\pi$
$821$	32.3333	1.12844	0.564220	+	0.825625i	$0.309177\pi$
$822$	0	0
$823$	−14.7087	−0.512714	−0.256357	−	0.966582i	$-0.582522\pi$
$823$	−14.7087	−0.512714	−0.256357	+	0.966582i	$0.582522\pi$
$824$	0	0
$825$	−3.07154	−0.106937
$826$	0	0
$827$	−19.5623	−0.680248	−0.340124	−	0.940381i	$-0.610469\pi$
$827$	−19.5623	−0.680248	−0.340124	+	0.940381i	$0.610469\pi$
$828$	0	0
$829$	31.5092	1.09436	0.547180	−	0.837015i	$-0.315701\pi$
$829$	31.5092	1.09436	0.547180	+	0.837015i	$0.315701\pi$
$830$	0	0
$831$	−15.8636	−0.550302
$832$	0	0
$833$	24.1431	0.836508
$834$	0	0
$835$	14.2028	0.491509
$836$	0	0
$837$	3.78872	0.130957
$838$	0	0
$839$	−20.9949	−0.724824	−0.362412	−	0.932018i	$-0.618047\pi$
$839$	−20.9949	−0.724824	−0.362412	+	0.932018i	$0.618047\pi$
$840$	0	0
$841$	−27.3544	−0.943254
$842$	0	0
$843$	1.37460	0.0473436
$844$	0	0
$845$	−12.7205	−0.437599
$846$	0	0
$847$	−5.93179	−0.203819
$848$	0	0
$849$	12.3544	0.424001
$850$	0	0
$851$	−3.22307	−0.110485
$852$	0	0
$853$	−50.0067	−1.71220	−0.856098	−	0.516814i	$-0.827118\pi$
$853$	−50.0067	−1.71220	−0.856098	+	0.516814i	$0.827118\pi$
$854$	0	0
$855$	5.07154	0.173443
$856$	0	0
$857$	−13.3939	−0.457526	−0.228763	−	0.973482i	$-0.573468\pi$
$857$	−13.3939	−0.457526	−0.228763	+	0.973482i	$0.573468\pi$
$858$	0	0
$859$	0.633854	0.0216268	0.0108134	−	0.999942i	$-0.496558\pi$
$859$	0.633854	0.0216268	0.0108134	+	0.999942i	$0.496558\pi$
$860$	0	0
$861$	35.7121	1.21706
$862$	0	0
$863$	29.1144	0.991066	0.495533	−	0.868589i	$-0.334973\pi$
$863$	29.1144	0.991066	0.495533	+	0.868589i	$0.334973\pi$
$864$	0	0
$865$	−5.57743	−0.189638
$866$	0	0
$867$	6.22307	0.211346
$868$	0	0
$869$	24.5723	0.833559
$870$	0	0
$871$	16.3459	0.553860
$872$	0	0
$873$	−1.13130	−0.0382886
$874$	0	0
$875$	−3.78872	−0.128082
$876$	0	0
$877$	56.3164	1.90167	0.950835	−	0.309699i	$-0.100228\pi$
$877$	56.3164	1.90167	0.950835	+	0.309699i	$0.100228\pi$
$878$	0	0
$879$	29.7121	1.00216
$880$	0	0
$881$	−1.79717	−0.0605480	−0.0302740	−	0.999542i	$-0.509638\pi$
$881$	−1.79717	−0.0605480	−0.0302740	+	0.999542i	$0.509638\pi$
$882$	0	0
$883$	47.6843	1.60471	0.802353	−	0.596850i	$-0.203581\pi$
$883$	47.6843	1.60471	0.802353	+	0.596850i	$0.203581\pi$
$884$	0	0
$885$	2.29461	0.0771324
$886$	0	0
$887$	−4.87537	−0.163699	−0.0818494	−	0.996645i	$-0.526083\pi$
$887$	−4.87537	−0.163699	−0.0818494	+	0.996645i	$0.526083\pi$
$888$	0	0
$889$	45.7803	1.53542
$890$	0	0
$891$	3.07154	0.102900
$892$	0	0
$893$	51.1380	1.71127
$894$	0	0
$895$	−14.8518	−0.496441
$896$	0	0
$897$	−5.07154	−0.169334
$898$	0	0
$899$	−4.86025	−0.162099
$900$	0	0
$901$	19.1995	0.639628
$902$	0	0
$903$	46.0067	1.53101
$904$	0	0
$905$	−20.8518	−0.693137
$906$	0	0
$907$	25.6692	0.852332	0.426166	−	0.904645i	$-0.359864\pi$
$907$	25.6692	0.852332	0.426166	+	0.904645i	$0.359864\pi$
$908$	0	0
$909$	−6.29461	−0.208779
$910$	0	0
$911$	−17.9764	−0.595586	−0.297793	−	0.954630i	$-0.596250\pi$
$911$	−17.9764	−0.595586	−0.297793	+	0.954630i	$0.596250\pi$
$912$	0	0
$913$	−20.6321	−0.682822
$914$	0	0
$915$	−3.07154	−0.101542
$916$	0	0
$917$	−64.9056	−2.14337
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	0	0
$921$	−30.4890	−1.00465
$922$	0	0
$923$	36.9351	1.21573
$924$	0	0
$925$	3.22307	0.105974
$926$	0	0
$927$	12.7087	0.417409
$928$	0	0
$929$	12.1582	0.398897	0.199449	−	0.979908i	$-0.436085\pi$
$929$	12.1582	0.398897	0.199449	+	0.979908i	$0.436085\pi$
$930$	0	0
$931$	−37.2979	−1.22239
$932$	0	0
$933$	4.44613	0.145560
$934$	0	0
$935$	−10.0833	−0.329760
$936$	0	0
$937$	−9.97643	−0.325916	−0.162958	−	0.986633i	$-0.552103\pi$
$937$	−9.97643	−0.325916	−0.162958	+	0.986633i	$0.552103\pi$
$938$	0	0
$939$	−6.35436	−0.207367
$940$	0	0
$941$	10.7685	0.351042	0.175521	−	0.984476i	$-0.443839\pi$
$941$	10.7685	0.351042	0.175521	+	0.984476i	$0.443839\pi$
$942$	0	0
$943$	9.42590	0.306950
$944$	0	0
$945$	3.78872	0.123247
$946$	0	0
$947$	−17.8569	−0.580272	−0.290136	−	0.956985i	$-0.593701\pi$
$947$	−17.8569	−0.580272	−0.290136	+	0.956985i	$0.593701\pi$
$948$	0	0
$949$	−48.8754	−1.58656
$950$	0	0
$951$	−2.64897	−0.0858987
$952$	0	0
$953$	−23.4175	−0.758566	−0.379283	−	0.925281i	$-0.623829\pi$
$953$	−23.4175	−0.758566	−0.379283	+	0.925281i	$0.623829\pi$
$954$	0	0
$955$	19.0951	0.617903
$956$	0	0
$957$	−3.94024	−0.127370
$958$	0	0
$959$	−10.2626	−0.331396
$960$	0	0
$961$	−16.6456	−0.536956
$962$	0	0
$963$	−18.2946	−0.589535
$964$	0	0
$965$	8.56565	0.275738
$966$	0	0
$967$	32.0833	1.03173	0.515865	−	0.856670i	$-0.327471\pi$
$967$	32.0833	1.03173	0.515865	+	0.856670i	$0.327471\pi$
$968$	0	0
$969$	16.6490	0.534842
$970$	0	0
$971$	−55.1615	−1.77022	−0.885109	−	0.465384i	$-0.845916\pi$
$971$	−55.1615	−1.77022	−0.885109	+	0.465384i	$0.845916\pi$
$972$	0	0
$973$	0.800498	0.0256628
$974$	0	0
$975$	5.07154	0.162419
$976$	0	0
$977$	58.7069	1.87820	0.939101	−	0.343642i	$-0.111661\pi$
$977$	58.7069	1.87820	0.939101	+	0.343642i	$0.111661\pi$
$978$	0	0
$979$	15.3939	0.491991
$980$	0	0
$981$	−4.50589	−0.143862
$982$	0	0
$983$	27.5925	0.880066	0.440033	−	0.897982i	$-0.354967\pi$
$983$	27.5925	0.880066	0.440033	+	0.897982i	$0.354967\pi$
$984$	0	0
$985$	−3.01178	−0.0959634
$986$	0	0
$987$	38.2028	1.21601
$988$	0	0
$989$	12.1431	0.386127
$990$	0	0
$991$	−52.8307	−1.67822	−0.839112	−	0.543959i	$-0.816925\pi$
$991$	−52.8307	−1.67822	−0.839112	+	0.543959i	$0.816925\pi$
$992$	0	0
$993$	4.80050	0.152339
$994$	0	0
$995$	−22.8518	−0.724451
$996$	0	0
$997$	34.4292	1.09038	0.545192	−	0.838311i	$-0.316457\pi$
$997$	34.4292	1.09038	0.545192	+	0.838311i	$0.316457\pi$
$998$	0	0
$999$	−3.22307	−0.101973

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.a.bw.1.3		3
4.3	odd	2		2760.2.a.s.1.1	✓	3
12.11	even	2		8280.2.a.bn.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.s.1.1	✓	3	4.3	odd	2
5520.2.a.bw.1.3		3	1.1	even	1	trivial
8280.2.a.bn.1.1		3	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +3.78872 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} +3.78872 q^{7} +1.00000 q^{9} +3.07154 q^{11} -5.07154 q^{13} +1.00000 q^{15} +3.28282 q^{17} -5.07154 q^{19} -3.78872 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +1.28282 q^{29} -3.78872 q^{31} -3.07154 q^{33} -3.78872 q^{35} +3.22307 q^{37} +5.07154 q^{39} -9.42590 q^{41} -12.1431 q^{43} -1.00000 q^{45} -10.0833 q^{47} +7.35436 q^{49} -3.28282 q^{51} +5.84847 q^{53} -3.07154 q^{55} +5.07154 q^{57} +2.29461 q^{59} -3.07154 q^{61} +3.78872 q^{63} +5.07154 q^{65} -3.22307 q^{67} +1.00000 q^{69} -7.28282 q^{71} +9.63719 q^{73} -1.00000 q^{75} +11.6372 q^{77} +8.00000 q^{79} +1.00000 q^{81} -6.71718 q^{83} -3.28282 q^{85} -1.28282 q^{87} +5.01178 q^{89} -19.2146 q^{91} +3.78872 q^{93} +5.07154 q^{95} -1.13130 q^{97} +3.07154 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 4 q^{11} - 2 q^{13} + 3 q^{15} + 6 q^{17} - 2 q^{19} - 2 q^{21} - 3 q^{23} + 3 q^{25} - 3 q^{27} - 2 q^{31} + 4 q^{33} - 2 q^{35} + 8 q^{37} + 2 q^{39} + 2 q^{41} - 10 q^{43} - 3 q^{45} - 6 q^{47} + 5 q^{49} - 6 q^{51} + 6 q^{53} + 4 q^{55} + 2 q^{57} - 8 q^{59} + 4 q^{61} + 2 q^{63} + 2 q^{65} - 8 q^{67} + 3 q^{69} - 18 q^{71} + 8 q^{73} - 3 q^{75} + 14 q^{77} + 24 q^{79} + 3 q^{81} - 24 q^{83} - 6 q^{85} + 4 q^{89} - 18 q^{91} + 2 q^{93} + 2 q^{95} + 12 q^{97} - 4 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^5 + 2 * q^7 + 3 * q^9 - 4 * q^11 - 2 * q^13 + 3 * q^15 + 6 * q^17 - 2 * q^19 - 2 * q^21 - 3 * q^23 + 3 * q^25 - 3 * q^27 - 2 * q^31 + 4 * q^33 - 2 * q^35 + 8 * q^37 + 2 * q^39 + 2 * q^41 - 10 * q^43 - 3 * q^45 - 6 * q^47 + 5 * q^49 - 6 * q^51 + 6 * q^53 + 4 * q^55 + 2 * q^57 - 8 * q^59 + 4 * q^61 + 2 * q^63 + 2 * q^65 - 8 * q^67 + 3 * q^69 - 18 * q^71 + 8 * q^73 - 3 * q^75 + 14 * q^77 + 24 * q^79 + 3 * q^81 - 24 * q^83 - 6 * q^85 + 4 * q^89 - 18 * q^91 + 2 * q^93 + 2 * q^95 + 12 * q^97 - 4 * q^99

Introduction

L-functions

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