LMFDB - Embedded newform 5520.2.a.bw.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:	$3$
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			1.2
Root			$3.76644$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.00000 q^{5} +1.34683 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.00000 q^{5} +1.34683 q^{7} +1.00000 q^{9} -4.41960 q^{11} +2.41960 q^{13} +1.00000 q^{15} -1.76644 q^{17} +2.41960 q^{19} -1.34683 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -3.76644 q^{29} -1.34683 q^{31} +4.41960 q^{33} -1.34683 q^{35} +10.8797 q^{37} -2.41960 q^{39} +10.6056 q^{41} +2.83920 q^{43} -1.00000 q^{45} -7.80694 q^{47} -5.18604 q^{49} +1.76644 q^{51} -9.29931 q^{53} +4.41960 q^{55} -2.41960 q^{57} +2.46010 q^{59} +4.41960 q^{61} +1.34683 q^{63} -2.41960 q^{65} -10.8797 q^{67} +1.00000 q^{69} -2.23356 q^{71} -7.95247 q^{73} -1.00000 q^{75} -5.95247 q^{77} +8.00000 q^{79} +1.00000 q^{81} -11.7664 q^{83} +1.76644 q^{85} +3.76644 q^{87} +10.2265 q^{89} +3.25880 q^{91} +1.34683 q^{93} -2.41960 q^{95} +19.0657 q^{97} -4.41960 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	1.34683	0.509056	0.254528	−	0.967065i	$-0.418080\pi$
$7$	1.34683	0.509056	0.254528	+	0.967065i	$0.418080\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.41960	−1.33256	−0.666280	−	0.745702i	$-0.732114\pi$
$11$	−4.41960	−1.33256	−0.666280	+	0.745702i	$0.732114\pi$
$12$	0	0
$13$	2.41960	0.671077	0.335538	−	0.942027i	$-0.391082\pi$
$13$	2.41960	0.671077	0.335538	+	0.942027i	$0.391082\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−1.76644	−0.428424	−0.214212	−	0.976787i	$-0.568718\pi$
$17$	−1.76644	−0.428424	−0.214212	+	0.976787i	$0.568718\pi$
$18$	0	0
$19$	2.41960	0.555094	0.277547	−	0.960712i	$-0.410478\pi$
$19$	2.41960	0.555094	0.277547	+	0.960712i	$0.410478\pi$
$20$	0	0
$21$	−1.34683	−0.293903
$22$	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
$29$	−3.76644	−0.699410	−0.349705	−	0.936860i	$-0.613718\pi$
$29$	−3.76644	−0.699410	−0.349705	+	0.936860i	$0.613718\pi$
$30$	0	0
$31$	−1.34683	−0.241899	−0.120949	−	0.992659i	$-0.538594\pi$
$31$	−1.34683	−0.241899	−0.120949	+	0.992659i	$0.538594\pi$
$32$	0	0
$33$	4.41960	0.769354
$34$	0	0
$35$	−1.34683	−0.227657
$36$	0	0
$37$	10.8797	1.78861	0.894306	−	0.447455i	$-0.147670\pi$
$37$	10.8797	1.78861	0.894306	+	0.447455i	$0.147670\pi$
$38$	0	0
$39$	−2.41960	−0.387446
$40$	0	0
$41$	10.6056	1.65632	0.828161	−	0.560490i	$-0.189387\pi$
$41$	10.6056	1.65632	0.828161	+	0.560490i	$0.189387\pi$
$42$	0	0
$43$	2.83920	0.432974	0.216487	−	0.976285i	$-0.430540\pi$
$43$	2.83920	0.432974	0.216487	+	0.976285i	$0.430540\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−7.80694	−1.13876	−0.569380	−	0.822075i	$-0.692817\pi$
$47$	−7.80694	−1.13876	−0.569380	+	0.822075i	$0.692817\pi$
$48$	0	0
$49$	−5.18604	−0.740862
$50$	0	0
$51$	1.76644	0.247350
$52$	0	0
$53$	−9.29931	−1.27736	−0.638679	−	0.769473i	$-0.720519\pi$
$53$	−9.29931	−1.27736	−0.638679	+	0.769473i	$0.720519\pi$
$54$	0	0
$55$	4.41960	0.595939
$56$	0	0
$57$	−2.41960	−0.320484
$58$	0	0
$59$	2.46010	0.320278	0.160139	−	0.987094i	$-0.448806\pi$
$59$	2.46010	0.320278	0.160139	+	0.987094i	$0.448806\pi$
$60$	0	0
$61$	4.41960	0.565872	0.282936	−	0.959139i	$-0.408692\pi$
$61$	4.41960	0.565872	0.282936	+	0.959139i	$0.408692\pi$
$62$	0	0
$63$	1.34683	0.169685
$64$	0	0
$65$	−2.41960	−0.300115
$66$	0	0
$67$	−10.8797	−1.32917	−0.664584	−	0.747214i	$-0.731391\pi$
$67$	−10.8797	−1.32917	−0.664584	+	0.747214i	$0.731391\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−2.23356	−0.265075	−0.132538	−	0.991178i	$-0.542313\pi$
$71$	−2.23356	−0.265075	−0.132538	+	0.991178i	$0.542313\pi$
$72$	0	0
$73$	−7.95247	−0.930766	−0.465383	−	0.885109i	$-0.654083\pi$
$73$	−7.95247	−0.930766	−0.465383	+	0.885109i	$0.654083\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−5.95247	−0.678347
$78$	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
$83$	−11.7664	−1.29153	−0.645767	−	0.763534i	$-0.723462\pi$
$83$	−11.7664	−1.29153	−0.645767	+	0.763534i	$0.723462\pi$
$84$	0	0
$85$	1.76644	0.191597
$86$	0	0
$87$	3.76644	0.403804
$88$	0	0
$89$	10.2265	1.08401	0.542006	−	0.840375i	$-0.317665\pi$
$89$	10.2265	1.08401	0.542006	+	0.840375i	$0.317665\pi$
$90$	0	0
$91$	3.25880	0.341615
$92$	0	0
$93$	1.34683	0.139660
$94$	0	0
$95$	−2.41960	−0.248246
$96$	0	0
$97$	19.0657	1.93583	0.967916	−	0.251272i	$-0.0808490\pi$
$97$	19.0657	1.93583	0.967916	+	0.251272i	$0.0808490\pi$
$98$	0	0
$99$	−4.41960	−0.444187
$100$	0	0
$101$	−6.46010	−0.642804	−0.321402	−	0.946943i	$-0.604154\pi$
$101$	−6.46010	−0.642804	−0.321402	+	0.946943i	$0.604154\pi$
$102$	0	0
$103$	−12.3721	−1.21906	−0.609528	−	0.792764i	$-0.708641\pi$
$103$	−12.3721	−1.21906	−0.609528	+	0.792764i	$0.708641\pi$
$104$	0	0
$105$	1.34683	0.131438
$106$	0	0
$107$	−18.4601	−1.78461	−0.892303	−	0.451437i	$-0.850911\pi$
$107$	−18.4601	−1.78461	−0.892303	+	0.451437i	$0.850911\pi$
$108$	0	0
$109$	−7.11327	−0.681328	−0.340664	−	0.940185i	$-0.610652\pi$
$109$	−7.11327	−0.681328	−0.340664	+	0.940185i	$0.610652\pi$
$110$	0	0
$111$	−10.8797	−1.03266
$112$	0	0
$113$	9.76644	0.918749	0.459374	−	0.888243i	$-0.348074\pi$
$113$	9.76644	0.918749	0.459374	+	0.888243i	$0.348074\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	2.41960	0.223692
$118$	0	0
$119$	−2.37910	−0.218091
$120$	0	0
$121$	8.53287	0.775716
$122$	0	0
$123$	−10.6056	−0.956278
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	9.80694	0.870225	0.435113	−	0.900376i	$-0.356709\pi$
$127$	9.80694	0.870225	0.435113	+	0.900376i	$0.356709\pi$
$128$	0	0
$129$	−2.83920	−0.249978
$130$	0	0
$131$	3.06574	0.267855	0.133928	−	0.990991i	$-0.457241\pi$
$131$	3.06574	0.267855	0.133928	+	0.990991i	$0.457241\pi$
$132$	0	0
$133$	3.25880	0.282574
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	22.3721	1.91138	0.955688	−	0.294383i	$-0.0951141\pi$
$137$	22.3721	1.91138	0.955688	+	0.294383i	$0.0951141\pi$
$138$	0	0
$139$	2.65317	0.225039	0.112519	−	0.993650i	$-0.464108\pi$
$139$	2.65317	0.225039	0.112519	+	0.993650i	$0.464108\pi$
$140$	0	0
$141$	7.80694	0.657463
$142$	0	0
$143$	−10.6937	−0.894250
$144$	0	0
$145$	3.76644	0.312785
$146$	0	0
$147$	5.18604	0.427737
$148$	0	0
$149$	−0.0475283	−0.00389367	−0.00194683	−	0.999998i	$-0.500620\pi$
$149$	−0.0475283	−0.00389367	−0.00194683	+	0.999998i	$0.500620\pi$
$150$	0	0
$151$	−6.14553	−0.500116	−0.250058	−	0.968231i	$-0.580450\pi$
$151$	−6.14553	−0.500116	−0.250058	+	0.968231i	$0.580450\pi$
$152$	0	0
$153$	−1.76644	−0.142808
$154$	0	0
$155$	1.34683	0.108180
$156$	0	0
$157$	−19.2518	−1.53646	−0.768230	−	0.640174i	$-0.778862\pi$
$157$	−19.2518	−1.53646	−0.768230	+	0.640174i	$0.778862\pi$
$158$	0	0
$159$	9.29931	0.737483
$160$	0	0
$161$	−1.34683	−0.106145
$162$	0	0
$163$	2.69367	0.210984	0.105492	−	0.994420i	$-0.466358\pi$
$163$	2.69367	0.210984	0.105492	+	0.994420i	$0.466358\pi$
$164$	0	0
$165$	−4.41960	−0.344065
$166$	0	0
$167$	13.4853	1.04353	0.521764	−	0.853090i	$-0.325274\pi$
$167$	13.4853	1.04353	0.521764	+	0.853090i	$0.325274\pi$
$168$	0	0
$169$	−7.14553	−0.549656
$170$	0	0
$171$	2.41960	0.185031
$172$	0	0
$173$	0.693669	0.0527387	0.0263694	−	0.999652i	$-0.491605\pi$
$173$	0.693669	0.0527387	0.0263694	+	0.999652i	$0.491605\pi$
$174$	0	0
$175$	1.34683	0.101811
$176$	0	0
$177$	−2.46010	−0.184913
$178$	0	0
$179$	−25.2113	−1.88438	−0.942190	−	0.335080i	$-0.891237\pi$
$179$	−25.2113	−1.88438	−0.942190	+	0.335080i	$0.891237\pi$
$180$	0	0
$181$	−19.2113	−1.42796	−0.713981	−	0.700165i	$-0.753110\pi$
$181$	−19.2113	−1.42796	−0.713981	+	0.700165i	$0.753110\pi$
$182$	0	0
$183$	−4.41960	−0.326706
$184$	0	0
$185$	−10.8797	−0.799892
$186$	0	0
$187$	7.80694	0.570900
$188$	0	0
$189$	−1.34683	−0.0979678
$190$	0	0
$191$	−22.0335	−1.59429	−0.797143	−	0.603790i	$-0.793656\pi$
$191$	−22.0335	−1.59429	−0.797143	+	0.603790i	$0.793656\pi$
$192$	0	0
$193$	1.53287	0.110338	0.0551692	−	0.998477i	$-0.482430\pi$
$193$	1.53287	0.110338	0.0551692	+	0.998477i	$0.482430\pi$
$194$	0	0
$195$	2.41960	0.173271
$196$	0	0
$197$	8.22654	0.586117	0.293058	−	0.956095i	$-0.405327\pi$
$197$	8.22654	0.586117	0.293058	+	0.956095i	$0.405327\pi$
$198$	0	0
$199$	−17.2113	−1.22007	−0.610037	−	0.792373i	$-0.708846\pi$
$199$	−17.2113	−1.22007	−0.610037	+	0.792373i	$0.708846\pi$
$200$	0	0
$201$	10.8797	0.767395
$202$	0	0
$203$	−5.07277	−0.356038
$204$	0	0
$205$	−10.6056	−0.740730
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−10.6937	−0.739697
$210$	0	0
$211$	−10.6532	−0.733394	−0.366697	−	0.930340i	$-0.619511\pi$
$211$	−10.6532	−0.733394	−0.366697	+	0.930340i	$0.619511\pi$
$212$	0	0
$213$	2.23356	0.153041
$214$	0	0
$215$	−2.83920	−0.193632
$216$	0	0
$217$	−1.81396	−0.123140
$218$	0	0
$219$	7.95247	0.537378
$220$	0	0
$221$	−4.27407	−0.287505
$222$	0	0
$223$	19.6784	1.31776	0.658882	−	0.752247i	$-0.271030\pi$
$223$	19.6784	1.31776	0.658882	+	0.752247i	$0.271030\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−13.3873	−0.888549	−0.444274	−	0.895891i	$-0.646538\pi$
$227$	−13.3873	−0.888549	−0.444274	+	0.895891i	$0.646538\pi$
$228$	0	0
$229$	−3.85447	−0.254710	−0.127355	−	0.991857i	$-0.540649\pi$
$229$	−3.85447	−0.254710	−0.127355	+	0.991857i	$0.540649\pi$
$230$	0	0
$231$	5.95247	0.391644
$232$	0	0
$233$	−22.3721	−1.46564	−0.732822	−	0.680421i	$-0.761797\pi$
$233$	−22.3721	−1.46564	−0.732822	+	0.680421i	$0.761797\pi$
$234$	0	0
$235$	7.80694	0.509269
$236$	0	0
$237$	−8.00000	−0.519656
$238$	0	0
$239$	7.91197	0.511783	0.255891	−	0.966706i	$-0.417631\pi$
$239$	7.91197	0.511783	0.255891	+	0.966706i	$0.417631\pi$
$240$	0	0
$241$	−19.9525	−1.28525	−0.642626	−	0.766180i	$-0.722155\pi$
$241$	−19.9525	−1.28525	−0.642626	+	0.766180i	$0.722155\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	5.18604	0.331324
$246$	0	0
$247$	5.85447	0.372511
$248$	0	0
$249$	11.7664	0.745668
$250$	0	0
$251$	−2.93426	−0.185209	−0.0926044	−	0.995703i	$-0.529519\pi$
$251$	−2.93426	−0.185209	−0.0926044	+	0.995703i	$0.529519\pi$
$252$	0	0
$253$	4.41960	0.277858
$254$	0	0
$255$	−1.76644	−0.110618
$256$	0	0
$257$	−3.11327	−0.194200	−0.0971002	−	0.995275i	$-0.530957\pi$
$257$	−3.11327	−0.194200	−0.0971002	+	0.995275i	$0.530957\pi$
$258$	0	0
$259$	14.6532	0.910503
$260$	0	0
$261$	−3.76644	−0.233137
$262$	0	0
$263$	28.1385	1.73509	0.867547	−	0.497355i	$-0.165695\pi$
$263$	28.1385	1.73509	0.867547	+	0.497355i	$0.165695\pi$
$264$	0	0
$265$	9.29931	0.571252
$266$	0	0
$267$	−10.2265	−0.625854
$268$	0	0
$269$	−13.0728	−0.797061	−0.398530	−	0.917155i	$-0.630480\pi$
$269$	−13.0728	−0.797061	−0.398530	+	0.917155i	$0.630480\pi$
$270$	0	0
$271$	−4.50763	−0.273819	−0.136910	−	0.990584i	$-0.543717\pi$
$271$	−4.50763	−0.273819	−0.136910	+	0.990584i	$0.543717\pi$
$272$	0	0
$273$	−3.25880	−0.197232
$274$	0	0
$275$	−4.41960	−0.266512
$276$	0	0
$277$	−18.9847	−1.14068	−0.570341	−	0.821408i	$-0.693189\pi$
$277$	−18.9847	−1.14068	−0.570341	+	0.821408i	$0.693189\pi$
$278$	0	0
$279$	−1.34683	−0.0806329
$280$	0	0
$281$	−24.1790	−1.44240	−0.721199	−	0.692727i	$-0.756409\pi$
$281$	−24.1790	−1.44240	−0.721199	+	0.692727i	$0.756409\pi$
$282$	0	0
$283$	0.186036	0.0110587	0.00552935	−	0.999985i	$-0.498240\pi$
$283$	0.186036	0.0110587	0.00552935	+	0.999985i	$0.498240\pi$
$284$	0	0
$285$	2.41960	0.143325
$286$	0	0
$287$	14.2840	0.843160
$288$	0	0
$289$	−13.8797	−0.816453
$290$	0	0
$291$	−19.0657	−1.11765
$292$	0	0
$293$	20.2840	1.18501	0.592503	−	0.805568i	$-0.298140\pi$
$293$	20.2840	1.18501	0.592503	+	0.805568i	$0.298140\pi$
$294$	0	0
$295$	−2.46010	−0.143233
$296$	0	0
$297$	4.41960	0.256451
$298$	0	0
$299$	−2.41960	−0.139929
$300$	0	0
$301$	3.82394	0.220408
$302$	0	0
$303$	6.46010	0.371123
$304$	0	0
$305$	−4.41960	−0.253066
$306$	0	0
$307$	−27.1637	−1.55032	−0.775158	−	0.631767i	$-0.782330\pi$
$307$	−27.1637	−1.55032	−0.775158	+	0.631767i	$0.782330\pi$
$308$	0	0
$309$	12.3721	0.703823
$310$	0	0
$311$	−19.7594	−1.12045	−0.560227	−	0.828339i	$-0.689286\pi$
$311$	−19.7594	−1.12045	−0.560227	+	0.828339i	$0.689286\pi$
$312$	0	0
$313$	−6.18604	−0.349655	−0.174828	−	0.984599i	$-0.555937\pi$
$313$	−6.18604	−0.349655	−0.174828	+	0.984599i	$0.555937\pi$
$314$	0	0
$315$	−1.34683	−0.0758855
$316$	0	0
$317$	−9.72593	−0.546263	−0.273131	−	0.961977i	$-0.588059\pi$
$317$	−9.72593	−0.546263	−0.273131	+	0.961977i	$0.588059\pi$
$318$	0	0
$319$	16.6461	0.932005
$320$	0	0
$321$	18.4601	1.03034
$322$	0	0
$323$	−4.27407	−0.237816
$324$	0	0
$325$	2.41960	0.134215
$326$	0	0
$327$	7.11327	0.393365
$328$	0	0
$329$	−10.5147	−0.579692
$330$	0	0
$331$	−7.57338	−0.416270	−0.208135	−	0.978100i	$-0.566739\pi$
$331$	−7.57338	−0.416270	−0.208135	+	0.978100i	$0.566739\pi$
$332$	0	0
$333$	10.8797	0.596204
$334$	0	0
$335$	10.8797	0.594422
$336$	0	0
$337$	−0.0810083	−0.00441280	−0.00220640	−	0.999998i	$-0.500702\pi$
$337$	−0.0810083	−0.00441280	−0.00220640	+	0.999998i	$0.500702\pi$
$338$	0	0
$339$	−9.76644	−0.530440
$340$	0	0
$341$	5.95247	0.322344
$342$	0	0
$343$	−16.4126	−0.886196
$344$	0	0
$345$	−1.00000	−0.0538382
$346$	0	0
$347$	−21.1303	−1.13433	−0.567166	−	0.823603i	$-0.691960\pi$
$347$	−21.1303	−1.13433	−0.567166	+	0.823603i	$0.691960\pi$
$348$	0	0
$349$	15.7189	0.841414	0.420707	−	0.907197i	$-0.361782\pi$
$349$	15.7189	0.841414	0.420707	+	0.907197i	$0.361782\pi$
$350$	0	0
$351$	−2.41960	−0.129149
$352$	0	0
$353$	36.4056	1.93767	0.968836	−	0.247703i	$-0.0796757\pi$
$353$	36.4056	1.93767	0.968836	+	0.247703i	$0.0796757\pi$
$354$	0	0
$355$	2.23356	0.118545
$356$	0	0
$357$	2.37910	0.125915
$358$	0	0
$359$	3.25880	0.171993	0.0859965	−	0.996295i	$-0.472593\pi$
$359$	3.25880	0.171993	0.0859965	+	0.996295i	$0.472593\pi$
$360$	0	0
$361$	−13.1455	−0.691870
$362$	0	0
$363$	−8.53287	−0.447860
$364$	0	0
$365$	7.95247	0.416251
$366$	0	0
$367$	11.0252	0.575513	0.287756	−	0.957704i	$-0.407091\pi$
$367$	11.0252	0.575513	0.287756	+	0.957704i	$0.407091\pi$
$368$	0	0
$369$	10.6056	0.552107
$370$	0	0
$371$	−12.5246	−0.650246
$372$	0	0
$373$	−25.9860	−1.34550	−0.672751	−	0.739869i	$-0.734887\pi$
$373$	−25.9860	−1.34550	−0.672751	+	0.739869i	$0.734887\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−9.11327	−0.469357
$378$	0	0
$379$	24.7441	1.27102	0.635511	−	0.772092i	$-0.280790\pi$
$379$	24.7441	1.27102	0.635511	+	0.772092i	$0.280790\pi$
$380$	0	0
$381$	−9.80694	−0.502425
$382$	0	0
$383$	22.2840	1.13866	0.569331	−	0.822109i	$-0.307202\pi$
$383$	22.2840	1.13866	0.569331	+	0.822109i	$0.307202\pi$
$384$	0	0
$385$	5.95247	0.303366
$386$	0	0
$387$	2.83920	0.144325
$388$	0	0
$389$	3.93548	0.199537	0.0997683	−	0.995011i	$-0.468190\pi$
$389$	3.93548	0.199537	0.0997683	+	0.995011i	$0.468190\pi$
$390$	0	0
$391$	1.76644	0.0893325
$392$	0	0
$393$	−3.06574	−0.154646
$394$	0	0
$395$	−8.00000	−0.402524
$396$	0	0
$397$	−21.6784	−1.08801	−0.544004	−	0.839083i	$-0.683092\pi$
$397$	−21.6784	−1.08801	−0.544004	+	0.839083i	$0.683092\pi$
$398$	0	0
$399$	−3.25880	−0.163144
$400$	0	0
$401$	−14.1455	−0.706394	−0.353197	−	0.935549i	$-0.614905\pi$
$401$	−14.1455	−0.706394	−0.353197	+	0.935549i	$0.614905\pi$
$402$	0	0
$403$	−3.25880	−0.162333
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−48.0840	−2.38343
$408$	0	0
$409$	1.02524	0.0506947	0.0253474	−	0.999679i	$-0.491931\pi$
$409$	1.02524	0.0506947	0.0253474	+	0.999679i	$0.491931\pi$
$410$	0	0
$411$	−22.3721	−1.10353
$412$	0	0
$413$	3.31335	0.163040
$414$	0	0
$415$	11.7664	0.577592
$416$	0	0
$417$	−2.65317	−0.129926
$418$	0	0
$419$	−4.96774	−0.242690	−0.121345	−	0.992610i	$-0.538721\pi$
$419$	−4.96774	−0.242690	−0.121345	+	0.992610i	$0.538721\pi$
$420$	0	0
$421$	−23.4043	−1.14066	−0.570329	−	0.821417i	$-0.693184\pi$
$421$	−23.4043	−1.14066	−0.570329	+	0.821417i	$0.693184\pi$
$422$	0	0
$423$	−7.80694	−0.379586
$424$	0	0
$425$	−1.76644	−0.0856847
$426$	0	0
$427$	5.95247	0.288060
$428$	0	0
$429$	10.6937	0.516295
$430$	0	0
$431$	−5.77346	−0.278098	−0.139049	−	0.990286i	$-0.544405\pi$
$431$	−5.77346	−0.278098	−0.139049	+	0.990286i	$0.544405\pi$
$432$	0	0
$433$	5.81396	0.279401	0.139701	−	0.990194i	$-0.455386\pi$
$433$	5.81396	0.279401	0.139701	+	0.990194i	$0.455386\pi$
$434$	0	0
$435$	−3.76644	−0.180587
$436$	0	0
$437$	−2.41960	−0.115745
$438$	0	0
$439$	13.5974	0.648968	0.324484	−	0.945891i	$-0.394809\pi$
$439$	13.5974	0.648968	0.324484	+	0.945891i	$0.394809\pi$
$440$	0	0
$441$	−5.18604	−0.246954
$442$	0	0
$443$	−10.0335	−0.476705	−0.238353	−	0.971179i	$-0.576607\pi$
$443$	−10.0335	−0.476705	−0.238353	+	0.971179i	$0.576607\pi$
$444$	0	0
$445$	−10.2265	−0.484785
$446$	0	0
$447$	0.0475283	0.00224801
$448$	0	0
$449$	−34.4296	−1.62483	−0.812416	−	0.583078i	$-0.801848\pi$
$449$	−34.4296	−1.62483	−0.812416	+	0.583078i	$0.801848\pi$
$450$	0	0
$451$	−46.8727	−2.20715
$452$	0	0
$453$	6.14553	0.288742
$454$	0	0
$455$	−3.25880	−0.152775
$456$	0	0
$457$	−33.2518	−1.55545	−0.777726	−	0.628603i	$-0.783627\pi$
$457$	−33.2518	−1.55545	−0.777726	+	0.628603i	$0.783627\pi$
$458$	0	0
$459$	1.76644	0.0824501
$460$	0	0
$461$	−16.9202	−0.788053	−0.394026	−	0.919099i	$-0.628918\pi$
$461$	−16.9202	−0.788053	−0.394026	+	0.919099i	$0.628918\pi$
$462$	0	0
$463$	−22.7271	−1.05622	−0.528110	−	0.849176i	$-0.677099\pi$
$463$	−22.7271	−1.05622	−0.528110	+	0.849176i	$0.677099\pi$
$464$	0	0
$465$	−1.34683	−0.0624580
$466$	0	0
$467$	0.605637	0.0280255	0.0140128	−	0.999902i	$-0.495539\pi$
$467$	0.605637	0.0280255	0.0140128	+	0.999902i	$0.495539\pi$
$468$	0	0
$469$	−14.6532	−0.676620
$470$	0	0
$471$	19.2518	0.887075
$472$	0	0
$473$	−12.5481	−0.576964
$474$	0	0
$475$	2.41960	0.111019
$476$	0	0
$477$	−9.29931	−0.425786
$478$	0	0
$479$	−3.25880	−0.148898	−0.0744492	−	0.997225i	$-0.523720\pi$
$479$	−3.25880	−0.148898	−0.0744492	+	0.997225i	$0.523720\pi$
$480$	0	0
$481$	26.3245	1.20030
$482$	0	0
$483$	1.34683	0.0612831
$484$	0	0
$485$	−19.0657	−0.865731
$486$	0	0
$487$	1.51588	0.0686909	0.0343454	−	0.999410i	$-0.489065\pi$
$487$	1.51588	0.0686909	0.0343454	+	0.999410i	$0.489065\pi$
$488$	0	0
$489$	−2.69367	−0.121812
$490$	0	0
$491$	23.7524	1.07193	0.535965	−	0.844240i	$-0.319948\pi$
$491$	23.7524	1.07193	0.535965	+	0.844240i	$0.319948\pi$
$492$	0	0
$493$	6.65317	0.299643
$494$	0	0
$495$	4.41960	0.198646
$496$	0	0
$497$	−3.00824	−0.134938
$498$	0	0
$499$	27.8644	1.24738	0.623692	−	0.781670i	$-0.285632\pi$
$499$	27.8644	1.24738	0.623692	+	0.781670i	$0.285632\pi$
$500$	0	0
$501$	−13.4853	−0.602481
$502$	0	0
$503$	7.38031	0.329072	0.164536	−	0.986371i	$-0.447387\pi$
$503$	7.38031	0.329072	0.164536	+	0.986371i	$0.447387\pi$
$504$	0	0
$505$	6.46010	0.287471
$506$	0	0
$507$	7.14553	0.317344
$508$	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
$511$	−10.7107	−0.473812
$512$	0	0
$513$	−2.41960	−0.106828
$514$	0	0
$515$	12.3721	0.545179
$516$	0	0
$517$	34.5036	1.51746
$518$	0	0
$519$	−0.693669	−0.0304487
$520$	0	0
$521$	−3.17779	−0.139222	−0.0696108	−	0.997574i	$-0.522176\pi$
$521$	−3.17779	−0.139222	−0.0696108	+	0.997574i	$0.522176\pi$
$522$	0	0
$523$	−8.69367	−0.380148	−0.190074	−	0.981770i	$-0.560873\pi$
$523$	−8.69367	−0.380148	−0.190074	+	0.981770i	$0.560873\pi$
$524$	0	0
$525$	−1.34683	−0.0587807
$526$	0	0
$527$	2.37910	0.103635
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	2.46010	0.106759
$532$	0	0
$533$	25.6614	1.11152
$534$	0	0
$535$	18.4601	0.798100
$536$	0	0
$537$	25.2113	1.08795
$538$	0	0
$539$	22.9202	0.987243
$540$	0	0
$541$	18.8252	0.809357	0.404678	−	0.914459i	$-0.367384\pi$
$541$	18.8252	0.809357	0.404678	+	0.914459i	$0.367384\pi$
$542$	0	0
$543$	19.2113	0.824435
$544$	0	0
$545$	7.11327	0.304699
$546$	0	0
$547$	36.7441	1.57107	0.785533	−	0.618820i	$-0.212389\pi$
$547$	36.7441	1.57107	0.785533	+	0.618820i	$0.212389\pi$
$548$	0	0
$549$	4.41960	0.188624
$550$	0	0
$551$	−9.11327	−0.388238
$552$	0	0
$553$	10.7747	0.458186
$554$	0	0
$555$	10.8797	0.461818
$556$	0	0
$557$	20.5411	0.870355	0.435177	−	0.900345i	$-0.356686\pi$
$557$	20.5411	0.870355	0.435177	+	0.900345i	$0.356686\pi$
$558$	0	0
$559$	6.86973	0.290559
$560$	0	0
$561$	−7.80694	−0.329609
$562$	0	0
$563$	−5.83096	−0.245746	−0.122873	−	0.992422i	$-0.539211\pi$
$563$	−5.83096	−0.245746	−0.122873	+	0.992422i	$0.539211\pi$
$564$	0	0
$565$	−9.76644	−0.410877
$566$	0	0
$567$	1.34683	0.0565617
$568$	0	0
$569$	−39.9860	−1.67630	−0.838149	−	0.545441i	$-0.816362\pi$
$569$	−39.9860	−1.67630	−0.838149	+	0.545441i	$0.816362\pi$
$570$	0	0
$571$	−42.9537	−1.79756	−0.898778	−	0.438404i	$-0.855544\pi$
$571$	−42.9537	−1.79756	−0.898778	+	0.438404i	$0.855544\pi$
$572$	0	0
$573$	22.0335	0.920462
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	−3.01527	−0.125527	−0.0627636	−	0.998028i	$-0.519991\pi$
$577$	−3.01527	−0.125527	−0.0627636	+	0.998028i	$0.519991\pi$
$578$	0	0
$579$	−1.53287	−0.0637039
$580$	0	0
$581$	−15.8474	−0.657463
$582$	0	0
$583$	41.0992	1.70216
$584$	0	0
$585$	−2.41960	−0.100038
$586$	0	0
$587$	5.75941	0.237716	0.118858	−	0.992911i	$-0.462077\pi$
$587$	5.75941	0.237716	0.118858	+	0.992911i	$0.462077\pi$
$588$	0	0
$589$	−3.25880	−0.134277
$590$	0	0
$591$	−8.22654	−0.338395
$592$	0	0
$593$	19.4043	0.796841	0.398420	−	0.917203i	$-0.369559\pi$
$593$	19.4043	0.796841	0.398420	+	0.917203i	$0.369559\pi$
$594$	0	0
$595$	2.37910	0.0975335
$596$	0	0
$597$	17.2113	0.704411
$598$	0	0
$599$	−4.59861	−0.187894	−0.0939471	−	0.995577i	$-0.529948\pi$
$599$	−4.59861	−0.187894	−0.0939471	+	0.995577i	$0.529948\pi$
$600$	0	0
$601$	−39.7189	−1.62017	−0.810084	−	0.586314i	$-0.800579\pi$
$601$	−39.7189	−1.62017	−0.810084	+	0.586314i	$0.800579\pi$
$602$	0	0
$603$	−10.8797	−0.443056
$604$	0	0
$605$	−8.53287	−0.346911
$606$	0	0
$607$	−3.48534	−0.141466	−0.0707328	−	0.997495i	$-0.522534\pi$
$607$	−3.48534	−0.141466	−0.0707328	+	0.997495i	$0.522534\pi$
$608$	0	0
$609$	5.07277	0.205559
$610$	0	0
$611$	−18.8897	−0.764195
$612$	0	0
$613$	44.1315	1.78245	0.891227	−	0.453558i	$-0.149845\pi$
$613$	44.1315	1.78245	0.891227	+	0.453558i	$0.149845\pi$
$614$	0	0
$615$	10.6056	0.427661
$616$	0	0
$617$	−24.9132	−1.00297	−0.501484	−	0.865167i	$-0.667212\pi$
$617$	−24.9132	−1.00297	−0.501484	+	0.865167i	$0.667212\pi$
$618$	0	0
$619$	−12.7441	−0.512230	−0.256115	−	0.966646i	$-0.582443\pi$
$619$	−12.7441	−0.512230	−0.256115	+	0.966646i	$0.582443\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	13.7735	0.551822
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	10.6937	0.427064
$628$	0	0
$629$	−19.2183	−0.766284
$630$	0	0
$631$	40.7271	1.62132	0.810661	−	0.585516i	$-0.199108\pi$
$631$	40.7271	1.62132	0.810661	+	0.585516i	$0.199108\pi$
$632$	0	0
$633$	10.6532	0.423425
$634$	0	0
$635$	−9.80694	−0.389177
$636$	0	0
$637$	−12.5481	−0.497175
$638$	0	0
$639$	−2.23356	−0.0883584
$640$	0	0
$641$	36.6321	1.44688	0.723440	−	0.690387i	$-0.242560\pi$
$641$	36.6321	1.44688	0.723440	+	0.690387i	$0.242560\pi$
$642$	0	0
$643$	0.572157	0.0225637	0.0112818	−	0.999936i	$-0.496409\pi$
$643$	0.572157	0.0225637	0.0112818	+	0.999936i	$0.496409\pi$
$644$	0	0
$645$	2.83920	0.111793
$646$	0	0
$647$	0.822206	0.0323243	0.0161621	−	0.999869i	$-0.494855\pi$
$647$	0.822206	0.0323243	0.0161621	+	0.999869i	$0.494855\pi$
$648$	0	0
$649$	−10.8727	−0.426790
$650$	0	0
$651$	1.81396	0.0710948
$652$	0	0
$653$	11.5804	0.453176	0.226588	−	0.973991i	$-0.427243\pi$
$653$	11.5804	0.453176	0.226588	+	0.973991i	$0.427243\pi$
$654$	0	0
$655$	−3.06574	−0.119788
$656$	0	0
$657$	−7.95247	−0.310255
$658$	0	0
$659$	30.9372	1.20514	0.602571	−	0.798065i	$-0.294143\pi$
$659$	30.9372	1.20514	0.602571	+	0.798065i	$0.294143\pi$
$660$	0	0
$661$	−23.2923	−0.905965	−0.452982	−	0.891519i	$-0.649640\pi$
$661$	−23.2923	−0.905965	−0.452982	+	0.891519i	$0.649640\pi$
$662$	0	0
$663$	4.27407	0.165991
$664$	0	0
$665$	−3.25880	−0.126371
$666$	0	0
$667$	3.76644	0.145837
$668$	0	0
$669$	−19.6784	−0.760811
$670$	0	0
$671$	−19.5329	−0.754058
$672$	0	0
$673$	−15.0992	−0.582032	−0.291016	−	0.956718i	$-0.593993\pi$
$673$	−15.0992	−0.582032	−0.291016	+	0.956718i	$0.593993\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−7.44484	−0.286128	−0.143064	−	0.989713i	$-0.545696\pi$
$677$	−7.44484	−0.286128	−0.143064	+	0.989713i	$0.545696\pi$
$678$	0	0
$679$	25.6784	0.985447
$680$	0	0
$681$	13.3873	0.513004
$682$	0	0
$683$	−28.0980	−1.07514	−0.537570	−	0.843219i	$-0.680658\pi$
$683$	−28.0980	−1.07514	−0.537570	+	0.843219i	$0.680658\pi$
$684$	0	0
$685$	−22.3721	−0.854793
$686$	0	0
$687$	3.85447	0.147057
$688$	0	0
$689$	−22.5006	−0.857205
$690$	0	0
$691$	15.2418	0.579826	0.289913	−	0.957053i	$-0.406374\pi$
$691$	15.2418	0.579826	0.289913	+	0.957053i	$0.406374\pi$
$692$	0	0
$693$	−5.95247	−0.226116
$694$	0	0
$695$	−2.65317	−0.100640
$696$	0	0
$697$	−18.7342	−0.709607
$698$	0	0
$699$	22.3721	0.846189
$700$	0	0
$701$	23.8574	0.901082	0.450541	−	0.892756i	$-0.351231\pi$
$701$	23.8574	0.901082	0.450541	+	0.892756i	$0.351231\pi$
$702$	0	0
$703$	26.3245	0.992849
$704$	0	0
$705$	−7.80694	−0.294026
$706$	0	0
$707$	−8.70069	−0.327223
$708$	0	0
$709$	34.8421	1.30852	0.654262	−	0.756268i	$-0.272979\pi$
$709$	34.8421	1.30852	0.654262	+	0.756268i	$0.272979\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	0	0
$713$	1.34683	0.0504394
$714$	0	0
$715$	10.6937	0.399921
$716$	0	0
$717$	−7.91197	−0.295478
$718$	0	0
$719$	−34.8827	−1.30090	−0.650452	−	0.759548i	$-0.725420\pi$
$719$	−34.8827	−1.30090	−0.650452	+	0.759548i	$0.725420\pi$
$720$	0	0
$721$	−16.6631	−0.620568
$722$	0	0
$723$	19.9525	0.742040
$724$	0	0
$725$	−3.76644	−0.139882
$726$	0	0
$727$	4.71769	0.174969	0.0874847	−	0.996166i	$-0.472117\pi$
$727$	4.71769	0.174969	0.0874847	+	0.996166i	$0.472117\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.01527	−0.185496
$732$	0	0
$733$	42.4936	1.56954	0.784768	−	0.619789i	$-0.212782\pi$
$733$	42.4936	1.56954	0.784768	+	0.619789i	$0.212782\pi$
$734$	0	0
$735$	−5.18604	−0.191290
$736$	0	0
$737$	48.0840	1.77120
$738$	0	0
$739$	−9.25178	−0.340332	−0.170166	−	0.985415i	$-0.554430\pi$
$739$	−9.25178	−0.340332	−0.170166	+	0.985415i	$0.554430\pi$
$740$	0	0
$741$	−5.85447	−0.215069
$742$	0	0
$743$	16.9343	0.621258	0.310629	−	0.950531i	$-0.399460\pi$
$743$	16.9343	0.621258	0.310629	+	0.950531i	$0.399460\pi$
$744$	0	0
$745$	0.0475283	0.00174130
$746$	0	0
$747$	−11.7664	−0.430511
$748$	0	0
$749$	−24.8627	−0.908464
$750$	0	0
$751$	11.8069	0.430841	0.215421	−	0.976521i	$-0.430888\pi$
$751$	11.8069	0.430841	0.215421	+	0.976521i	$0.430888\pi$
$752$	0	0
$753$	2.93426	0.106930
$754$	0	0
$755$	6.14553	0.223659
$756$	0	0
$757$	−2.25056	−0.0817980	−0.0408990	−	0.999163i	$-0.513022\pi$
$757$	−2.25056	−0.0817980	−0.0408990	+	0.999163i	$0.513022\pi$
$758$	0	0
$759$	−4.41960	−0.160421
$760$	0	0
$761$	−31.4448	−1.13987	−0.569937	−	0.821688i	$-0.693033\pi$
$761$	−31.4448	−1.13987	−0.569937	+	0.821688i	$0.693033\pi$
$762$	0	0
$763$	−9.58040	−0.346834
$764$	0	0
$765$	1.76644	0.0638656
$766$	0	0
$767$	5.95247	0.214931
$768$	0	0
$769$	23.7764	0.857399	0.428700	−	0.903447i	$-0.358972\pi$
$769$	23.7764	0.857399	0.428700	+	0.903447i	$0.358972\pi$
$770$	0	0
$771$	3.11327	0.112122
$772$	0	0
$773$	39.2923	1.41325	0.706623	−	0.707591i	$-0.250218\pi$
$773$	39.2923	1.41325	0.706623	+	0.707591i	$0.250218\pi$
$774$	0	0
$775$	−1.34683	−0.0483797
$776$	0	0
$777$	−14.6532	−0.525679
$778$	0	0
$779$	25.6614	0.919415
$780$	0	0
$781$	9.87146	0.353229
$782$	0	0
$783$	3.76644	0.134601
$784$	0	0
$785$	19.2518	0.687125
$786$	0	0
$787$	−0.362101	−0.0129075	−0.00645375	−	0.999979i	$-0.502054\pi$
$787$	−0.362101	−0.0129075	−0.00645375	+	0.999979i	$0.502054\pi$
$788$	0	0
$789$	−28.1385	−1.00176
$790$	0	0
$791$	13.1538	0.467694
$792$	0	0
$793$	10.6937	0.379743
$794$	0	0
$795$	−9.29931	−0.329812
$796$	0	0
$797$	17.2993	0.612773	0.306386	−	0.951907i	$-0.400880\pi$
$797$	17.2993	0.612773	0.306386	+	0.951907i	$0.400880\pi$
$798$	0	0
$799$	13.7905	0.487871
$800$	0	0
$801$	10.2265	0.361337
$802$	0	0
$803$	35.1468	1.24030
$804$	0	0
$805$	1.34683	0.0474697
$806$	0	0
$807$	13.0728	0.460183
$808$	0	0
$809$	−25.6714	−0.902558	−0.451279	−	0.892383i	$-0.649032\pi$
$809$	−25.6714	−0.902558	−0.451279	+	0.892383i	$0.649032\pi$
$810$	0	0
$811$	35.4783	1.24581	0.622906	−	0.782297i	$-0.285952\pi$
$811$	35.4783	1.24581	0.622906	+	0.782297i	$0.285952\pi$
$812$	0	0
$813$	4.50763	0.158090
$814$	0	0
$815$	−2.69367	−0.0943551
$816$	0	0
$817$	6.86973	0.240342
$818$	0	0
$819$	3.25880	0.113872
$820$	0	0
$821$	23.2278	0.810654	0.405327	−	0.914172i	$-0.367158\pi$
$821$	23.2278	0.810654	0.405327	+	0.914172i	$0.367158\pi$
$822$	0	0
$823$	10.3721	0.361548	0.180774	−	0.983525i	$-0.442140\pi$
$823$	10.3721	0.361548	0.180774	+	0.983525i	$0.442140\pi$
$824$	0	0
$825$	4.41960	0.153871
$826$	0	0
$827$	−34.3791	−1.19548	−0.597739	−	0.801691i	$-0.703934\pi$
$827$	−34.3791	−1.19548	−0.597739	+	0.801691i	$0.703934\pi$
$828$	0	0
$829$	9.20130	0.319574	0.159787	−	0.987151i	$-0.448919\pi$
$829$	9.20130	0.319574	0.159787	+	0.987151i	$0.448919\pi$
$830$	0	0
$831$	18.9847	0.658573
$832$	0	0
$833$	9.16080	0.317403
$834$	0	0
$835$	−13.4853	−0.466680
$836$	0	0
$837$	1.34683	0.0465534
$838$	0	0
$839$	34.0505	1.17555	0.587776	−	0.809023i	$-0.300003\pi$
$839$	34.0505	1.17555	0.587776	+	0.809023i	$0.300003\pi$
$840$	0	0
$841$	−14.8140	−0.510826
$842$	0	0
$843$	24.1790	0.832769
$844$	0	0
$845$	7.14553	0.245814
$846$	0	0
$847$	11.4924	0.394882
$848$	0	0
$849$	−0.186036	−0.00638475
$850$	0	0
$851$	−10.8797	−0.372952
$852$	0	0
$853$	−0.176065	−0.00602834	−0.00301417	−	0.999995i	$-0.500959\pi$
$853$	−0.176065	−0.00602834	−0.00301417	+	0.999995i	$0.500959\pi$
$854$	0	0
$855$	−2.41960	−0.0827486
$856$	0	0
$857$	47.1972	1.61223	0.806113	−	0.591761i	$-0.201567\pi$
$857$	47.1972	1.61223	0.806113	+	0.591761i	$0.201567\pi$
$858$	0	0
$859$	7.95950	0.271574	0.135787	−	0.990738i	$-0.456644\pi$
$859$	7.95950	0.271574	0.135787	+	0.990738i	$0.456644\pi$
$860$	0	0
$861$	−14.2840	−0.486799
$862$	0	0
$863$	−51.3428	−1.74773	−0.873864	−	0.486171i	$-0.838393\pi$
$863$	−51.3428	−1.74773	−0.873864	+	0.486171i	$0.838393\pi$
$864$	0	0
$865$	−0.693669	−0.0235855
$866$	0	0
$867$	13.8797	0.471380
$868$	0	0
$869$	−35.3568	−1.19940
$870$	0	0
$871$	−26.3245	−0.891973
$872$	0	0
$873$	19.0657	0.645278
$874$	0	0
$875$	−1.34683	−0.0455313
$876$	0	0
$877$	−13.0493	−0.440642	−0.220321	−	0.975427i	$-0.570711\pi$
$877$	−13.0493	−0.440642	−0.220321	+	0.975427i	$0.570711\pi$
$878$	0	0
$879$	−20.2840	−0.684164
$880$	0	0
$881$	−29.4853	−0.993386	−0.496693	−	0.867926i	$-0.665453\pi$
$881$	−29.4853	−0.993386	−0.496693	+	0.867926i	$0.665453\pi$
$882$	0	0
$883$	50.9537	1.71473	0.857364	−	0.514710i	$-0.172101\pi$
$883$	50.9537	1.71473	0.857364	+	0.514710i	$0.172101\pi$
$884$	0	0
$885$	2.46010	0.0826955
$886$	0	0
$887$	24.7582	0.831299	0.415649	−	0.909525i	$-0.363554\pi$
$887$	24.7582	0.831299	0.415649	+	0.909525i	$0.363554\pi$
$888$	0	0
$889$	13.2083	0.442993
$890$	0	0
$891$	−4.41960	−0.148062
$892$	0	0
$893$	−18.8897	−0.632119
$894$	0	0
$895$	25.2113	0.842720
$896$	0	0
$897$	2.41960	0.0807881
$898$	0	0
$899$	5.07277	0.169186
$900$	0	0
$901$	16.4266	0.547250
$902$	0	0
$903$	−3.82394	−0.127253
$904$	0	0
$905$	19.2113	0.638604
$906$	0	0
$907$	48.6391	1.61504	0.807518	−	0.589843i	$-0.200811\pi$
$907$	48.6391	1.61504	0.807518	+	0.589843i	$0.200811\pi$
$908$	0	0
$909$	−6.46010	−0.214268
$910$	0	0
$911$	−7.54692	−0.250041	−0.125020	−	0.992154i	$-0.539900\pi$
$911$	−7.54692	−0.250041	−0.125020	+	0.992154i	$0.539900\pi$
$912$	0	0
$913$	52.0029	1.72105
$914$	0	0
$915$	4.41960	0.146107
$916$	0	0
$917$	4.12905	0.136353
$918$	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
$921$	27.1637	0.895076
$922$	0	0
$923$	−5.40433	−0.177886
$924$	0	0
$925$	10.8797	0.357723
$926$	0	0
$927$	−12.3721	−0.406352
$928$	0	0
$929$	−22.5246	−0.739009	−0.369505	−	0.929229i	$-0.620473\pi$
$929$	−22.5246	−0.739009	−0.369505	+	0.929229i	$0.620473\pi$
$930$	0	0
$931$	−12.5481	−0.411249
$932$	0	0
$933$	19.7594	0.646894
$934$	0	0
$935$	−7.80694	−0.255314
$936$	0	0
$937$	0.453081	0.0148015	0.00740075	−	0.999973i	$-0.497644\pi$
$937$	0.453081	0.0148015	0.00740075	+	0.999973i	$0.497644\pi$
$938$	0	0
$939$	6.18604	0.201874
$940$	0	0
$941$	−27.0182	−0.880769	−0.440384	−	0.897809i	$-0.645158\pi$
$941$	−27.0182	−0.880769	−0.440384	+	0.897809i	$0.645158\pi$
$942$	0	0
$943$	−10.6056	−0.345367
$944$	0	0
$945$	1.34683	0.0438125
$946$	0	0
$947$	−32.8392	−1.06713	−0.533565	−	0.845759i	$-0.679148\pi$
$947$	−32.8392	−1.06713	−0.533565	+	0.845759i	$0.679148\pi$
$948$	0	0
$949$	−19.2418	−0.624615
$950$	0	0
$951$	9.72593	0.315385
$952$	0	0
$953$	26.7441	0.866328	0.433164	−	0.901315i	$-0.357397\pi$
$953$	26.7441	0.866328	0.433164	+	0.901315i	$0.357397\pi$
$954$	0	0
$955$	22.0335	0.712987
$956$	0	0
$957$	−16.6461	−0.538093
$958$	0	0
$959$	30.1315	0.972996
$960$	0	0
$961$	−29.1860	−0.941485
$962$	0	0
$963$	−18.4601	−0.594869
$964$	0	0
$965$	−1.53287	−0.0493449
$966$	0	0
$967$	29.8069	0.958527	0.479263	−	0.877671i	$-0.340904\pi$
$967$	29.8069	0.958527	0.479263	+	0.877671i	$0.340904\pi$
$968$	0	0
$969$	4.27407	0.137303
$970$	0	0
$971$	4.43660	0.142377	0.0711886	−	0.997463i	$-0.477321\pi$
$971$	4.43660	0.142377	0.0711886	+	0.997463i	$0.477321\pi$
$972$	0	0
$973$	3.57338	0.114557
$974$	0	0
$975$	−2.41960	−0.0774892
$976$	0	0
$977$	−46.3345	−1.48237	−0.741186	−	0.671299i	$-0.765736\pi$
$977$	−46.3345	−1.48237	−0.741186	+	0.671299i	$0.765736\pi$
$978$	0	0
$979$	−45.1972	−1.44451
$980$	0	0
$981$	−7.11327	−0.227109
$982$	0	0
$983$	3.00824	0.0959480	0.0479740	−	0.998849i	$-0.484724\pi$
$983$	3.00824	0.0959480	0.0479740	+	0.998849i	$0.484724\pi$
$984$	0	0
$985$	−8.22654	−0.262119
$986$	0	0
$987$	10.5147	0.334685
$988$	0	0
$989$	−2.83920	−0.0902814
$990$	0	0
$991$	−16.2025	−0.514690	−0.257345	−	0.966320i	$-0.582848\pi$
$991$	−16.2025	−0.514690	−0.257345	+	0.966320i	$0.582848\pi$
$992$	0	0
$993$	7.57338	0.240334
$994$	0	0
$995$	17.2113	0.545634
$996$	0	0
$997$	−10.5176	−0.333096	−0.166548	−	0.986033i	$-0.553262\pi$
$997$	−10.5176	−0.333096	−0.166548	+	0.986033i	$0.553262\pi$
$998$	0	0
$999$	−10.8797	−0.344219

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5520.2.a.bw.1.2		3
4.3	odd	2		2760.2.a.s.1.2	✓	3
12.11	even	2		8280.2.a.bn.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.s.1.2	✓	3	4.3	odd	2
5520.2.a.bw.1.2		3	1.1	even	1	trivial
8280.2.a.bn.1.2		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.00000 q^{5} +1.34683 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.00000 q^{5} +1.34683 q^{7} +1.00000 q^{9} -4.41960 q^{11} +2.41960 q^{13} +1.00000 q^{15} -1.76644 q^{17} +2.41960 q^{19} -1.34683 q^{21} -1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -3.76644 q^{29} -1.34683 q^{31} +4.41960 q^{33} -1.34683 q^{35} +10.8797 q^{37} -2.41960 q^{39} +10.6056 q^{41} +2.83920 q^{43} -1.00000 q^{45} -7.80694 q^{47} -5.18604 q^{49} +1.76644 q^{51} -9.29931 q^{53} +4.41960 q^{55} -2.41960 q^{57} +2.46010 q^{59} +4.41960 q^{61} +1.34683 q^{63} -2.41960 q^{65} -10.8797 q^{67} +1.00000 q^{69} -2.23356 q^{71} -7.95247 q^{73} -1.00000 q^{75} -5.95247 q^{77} +8.00000 q^{79} +1.00000 q^{81} -11.7664 q^{83} +1.76644 q^{85} +3.76644 q^{87} +10.2265 q^{89} +3.25880 q^{91} +1.34683 q^{93} -2.41960 q^{95} +19.0657 q^{97} -4.41960 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) 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

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