LMFDB - Embedded newform 5520.2.a.bz.1.1

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:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
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			1.1
Root			$-0.363328$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} -4.86799 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} -4.86799 q^{7} +1.00000 q^{9} -4.77801 q^{11} +2.77801 q^{13} -1.00000 q^{15} -0.636672 q^{17} +3.50466 q^{19} +4.86799 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +7.36333 q^{29} +5.15066 q^{31} +4.77801 q^{33} -4.86799 q^{35} +2.86799 q^{37} -2.77801 q^{39} -6.19269 q^{41} +8.28267 q^{43} +1.00000 q^{45} -7.50466 q^{47} +16.6974 q^{49} +0.636672 q^{51} -4.91934 q^{53} -4.77801 q^{55} -3.50466 q^{57} +8.65533 q^{59} -15.0607 q^{61} -4.86799 q^{63} +2.77801 q^{65} +13.4333 q^{67} -1.00000 q^{69} +11.0993 q^{71} -15.2406 q^{73} -1.00000 q^{75} +23.2593 q^{77} +1.45331 q^{79} +1.00000 q^{81} -6.63667 q^{83} -0.636672 q^{85} -7.36333 q^{87} +9.29200 q^{89} -13.5233 q^{91} -5.15066 q^{93} +3.50466 q^{95} -14.7453 q^{97} -4.77801 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 8 q^{11} + 2 q^{13} - 3 q^{15} - 4 q^{17} + 2 q^{21} + 3 q^{23} + 3 q^{25} - 3 q^{27} + 20 q^{29} - 14 q^{31} + 8 q^{33} - 2 q^{35} - 4 q^{37} - 2 q^{39} - 8 q^{41} + 8 q^{43} + 3 q^{45} - 12 q^{47} + 29 q^{49} + 4 q^{51} - 8 q^{55} - 14 q^{59} - 22 q^{61} - 2 q^{63} + 2 q^{65} - 6 q^{67} - 3 q^{69} + 6 q^{71} - 10 q^{73} - 3 q^{75} - 8 q^{77} - 4 q^{79} + 3 q^{81} - 22 q^{83} - 4 q^{85} - 20 q^{87} - 10 q^{89} + 12 q^{91} + 14 q^{93} + 2 q^{97} - 8 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9 - 8 * q^11 + 2 * q^13 - 3 * q^15 - 4 * q^17 + 2 * q^21 + 3 * q^23 + 3 * q^25 - 3 * q^27 + 20 * q^29 - 14 * q^31 + 8 * q^33 - 2 * q^35 - 4 * q^37 - 2 * q^39 - 8 * q^41 + 8 * q^43 + 3 * q^45 - 12 * q^47 + 29 * q^49 + 4 * q^51 - 8 * q^55 - 14 * q^59 - 22 * q^61 - 2 * q^63 + 2 * q^65 - 6 * q^67 - 3 * q^69 + 6 * q^71 - 10 * q^73 - 3 * q^75 - 8 * q^77 - 4 * q^79 + 3 * q^81 - 22 * q^83 - 4 * q^85 - 20 * q^87 - 10 * q^89 + 12 * q^91 + 14 * q^93 + 2 * q^97 - 8 * 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
$7$	−4.86799	−1.83993	−0.919964	−	0.392003i	$-0.871782\pi$
$7$	−4.86799	−1.83993	−0.919964	+	0.392003i	$0.871782\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.77801	−1.44062	−0.720312	−	0.693650i	$-0.756001\pi$
$11$	−4.77801	−1.44062	−0.720312	+	0.693650i	$0.756001\pi$
$12$	0	0
$13$	2.77801	0.770481	0.385240	−	0.922816i	$-0.374119\pi$
$13$	2.77801	0.770481	0.385240	+	0.922816i	$0.374119\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	−0.636672	−0.154416	−0.0772078	−	0.997015i	$-0.524600\pi$
$17$	−0.636672	−0.154416	−0.0772078	+	0.997015i	$0.524600\pi$
$18$	0	0
$19$	3.50466	0.804025	0.402013	−	0.915634i	$-0.368311\pi$
$19$	3.50466	0.804025	0.402013	+	0.915634i	$0.368311\pi$
$20$	0	0
$21$	4.86799	1.06228
$22$	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
$29$	7.36333	1.36734	0.683668	−	0.729793i	$-0.260384\pi$
$29$	7.36333	1.36734	0.683668	+	0.729793i	$0.260384\pi$
$30$	0	0
$31$	5.15066	0.925087	0.462543	−	0.886597i	$-0.346937\pi$
$31$	5.15066	0.925087	0.462543	+	0.886597i	$0.346937\pi$
$32$	0	0
$33$	4.77801	0.831744
$34$	0	0
$35$	−4.86799	−0.822841
$36$	0	0
$37$	2.86799	0.471495	0.235748	−	0.971814i	$-0.424246\pi$
$37$	2.86799	0.471495	0.235748	+	0.971814i	$0.424246\pi$
$38$	0	0
$39$	−2.77801	−0.444837
$40$	0	0
$41$	−6.19269	−0.967135	−0.483568	−	0.875307i	$-0.660659\pi$
$41$	−6.19269	−0.967135	−0.483568	+	0.875307i	$0.660659\pi$
$42$	0	0
$43$	8.28267	1.26310	0.631548	−	0.775337i	$-0.282420\pi$
$43$	8.28267	1.26310	0.631548	+	0.775337i	$0.282420\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−7.50466	−1.09467	−0.547334	−	0.836914i	$-0.684357\pi$
$47$	−7.50466	−1.09467	−0.547334	+	0.836914i	$0.684357\pi$
$48$	0	0
$49$	16.6974	2.38534
$50$	0	0
$51$	0.636672	0.0891519
$52$	0	0
$53$	−4.91934	−0.675724	−0.337862	−	0.941196i	$-0.609704\pi$
$53$	−4.91934	−0.675724	−0.337862	+	0.941196i	$0.609704\pi$
$54$	0	0
$55$	−4.77801	−0.644266
$56$	0	0
$57$	−3.50466	−0.464204
$58$	0	0
$59$	8.65533	1.12683	0.563414	−	0.826175i	$-0.309488\pi$
$59$	8.65533	1.12683	0.563414	+	0.826175i	$0.309488\pi$
$60$	0	0
$61$	−15.0607	−1.92832	−0.964161	−	0.265317i	$-0.914523\pi$
$61$	−15.0607	−1.92832	−0.964161	+	0.265317i	$0.914523\pi$
$62$	0	0
$63$	−4.86799	−0.613309
$64$	0	0
$65$	2.77801	0.344569
$66$	0	0
$67$	13.4333	1.64114	0.820572	−	0.571544i	$-0.193655\pi$
$67$	13.4333	1.64114	0.820572	+	0.571544i	$0.193655\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	11.0993	1.31725	0.658623	−	0.752473i	$-0.271139\pi$
$71$	11.0993	1.31725	0.658623	+	0.752473i	$0.271139\pi$
$72$	0	0
$73$	−15.2406	−1.78378	−0.891892	−	0.452249i	$-0.850622\pi$
$73$	−15.2406	−1.78378	−0.891892	+	0.452249i	$0.850622\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	23.2593	2.65064
$78$	0	0
$79$	1.45331	0.163510	0.0817552	−	0.996652i	$-0.473947\pi$
$79$	1.45331	0.163510	0.0817552	+	0.996652i	$0.473947\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.63667	−0.728469	−0.364235	−	0.931307i	$-0.618669\pi$
$83$	−6.63667	−0.728469	−0.364235	+	0.931307i	$0.618669\pi$
$84$	0	0
$85$	−0.636672	−0.0690567
$86$	0	0
$87$	−7.36333	−0.789432
$88$	0	0
$89$	9.29200	0.984950	0.492475	−	0.870327i	$-0.336092\pi$
$89$	9.29200	0.984950	0.492475	+	0.870327i	$0.336092\pi$
$90$	0	0
$91$	−13.5233	−1.41763
$92$	0	0
$93$	−5.15066	−0.534099
$94$	0	0
$95$	3.50466	0.359571
$96$	0	0
$97$	−14.7453	−1.49716	−0.748580	−	0.663045i	$-0.769264\pi$
$97$	−14.7453	−1.49716	−0.748580	+	0.663045i	$0.769264\pi$
$98$	0	0
$99$	−4.77801	−0.480208
$100$	0	0
$101$	−17.3820	−1.72957	−0.864786	−	0.502140i	$-0.832546\pi$
$101$	−17.3820	−1.72957	−0.864786	+	0.502140i	$0.832546\pi$
$102$	0	0
$103$	−15.2920	−1.50677	−0.753383	−	0.657582i	$-0.771579\pi$
$103$	−15.2920	−1.50677	−0.753383	+	0.657582i	$0.771579\pi$
$104$	0	0
$105$	4.86799	0.475067
$106$	0	0
$107$	−15.3820	−1.48703	−0.743516	−	0.668718i	$-0.766843\pi$
$107$	−15.3820	−1.48703	−0.743516	+	0.668718i	$0.766843\pi$
$108$	0	0
$109$	−15.2406	−1.45979	−0.729895	−	0.683560i	$-0.760431\pi$
$109$	−15.2406	−1.45979	−0.729895	+	0.683560i	$0.760431\pi$
$110$	0	0
$111$	−2.86799	−0.272218
$112$	0	0
$113$	6.19269	0.582559	0.291280	−	0.956638i	$-0.405919\pi$
$113$	6.19269	0.582559	0.291280	+	0.956638i	$0.405919\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	2.77801	0.256827
$118$	0	0
$119$	3.09931	0.284114
$120$	0	0
$121$	11.8294	1.07540
$122$	0	0
$123$	6.19269	0.558376
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	4.77801	0.423980	0.211990	−	0.977272i	$-0.432006\pi$
$127$	4.77801	0.423980	0.211990	+	0.977272i	$0.432006\pi$
$128$	0	0
$129$	−8.28267	−0.729249
$130$	0	0
$131$	−6.54669	−0.571987	−0.285993	−	0.958232i	$-0.592324\pi$
$131$	−6.54669	−0.571987	−0.285993	+	0.958232i	$0.592324\pi$
$132$	0	0
$133$	−17.0607	−1.47935
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	−10.7453	−0.918034	−0.459017	−	0.888427i	$-0.651798\pi$
$137$	−10.7453	−0.918034	−0.459017	+	0.888427i	$0.651798\pi$
$138$	0	0
$139$	−14.4240	−1.22343	−0.611714	−	0.791079i	$-0.709520\pi$
$139$	−14.4240	−1.22343	−0.611714	+	0.791079i	$0.709520\pi$
$140$	0	0
$141$	7.50466	0.632007
$142$	0	0
$143$	−13.2733	−1.10997
$144$	0	0
$145$	7.36333	0.611491
$146$	0	0
$147$	−16.6974	−1.37717
$148$	0	0
$149$	6.67531	0.546862	0.273431	−	0.961892i	$-0.411841\pi$
$149$	6.67531	0.546862	0.273431	+	0.961892i	$0.411841\pi$
$150$	0	0
$151$	−18.3013	−1.48934	−0.744671	−	0.667432i	$-0.767393\pi$
$151$	−18.3013	−1.48934	−0.744671	+	0.667432i	$0.767393\pi$
$152$	0	0
$153$	−0.636672	−0.0514719
$154$	0	0
$155$	5.15066	0.413711
$156$	0	0
$157$	18.4427	1.47188	0.735942	−	0.677044i	$-0.236739\pi$
$157$	18.4427	1.47188	0.735942	+	0.677044i	$0.236739\pi$
$158$	0	0
$159$	4.91934	0.390129
$160$	0	0
$161$	−4.86799	−0.383652
$162$	0	0
$163$	0.282672	0.0221406	0.0110703	−	0.999939i	$-0.496476\pi$
$163$	0.282672	0.0221406	0.0110703	+	0.999939i	$0.496476\pi$
$164$	0	0
$165$	4.77801	0.371967
$166$	0	0
$167$	−6.51399	−0.504068	−0.252034	−	0.967718i	$-0.581099\pi$
$167$	−6.51399	−0.504068	−0.252034	+	0.967718i	$0.581099\pi$
$168$	0	0
$169$	−5.28267	−0.406359
$170$	0	0
$171$	3.50466	0.268008
$172$	0	0
$173$	8.30133	0.631138	0.315569	−	0.948903i	$-0.397805\pi$
$173$	8.30133	0.631138	0.315569	+	0.948903i	$0.397805\pi$
$174$	0	0
$175$	−4.86799	−0.367986
$176$	0	0
$177$	−8.65533	−0.650574
$178$	0	0
$179$	−5.17064	−0.386472	−0.193236	−	0.981152i	$-0.561898\pi$
$179$	−5.17064	−0.386472	−0.193236	+	0.981152i	$0.561898\pi$
$180$	0	0
$181$	21.8573	1.62464	0.812322	−	0.583209i	$-0.198203\pi$
$181$	21.8573	1.62464	0.812322	+	0.583209i	$0.198203\pi$
$182$	0	0
$183$	15.0607	1.11332
$184$	0	0
$185$	2.86799	0.210859
$186$	0	0
$187$	3.04202	0.222455
$188$	0	0
$189$	4.86799	0.354094
$190$	0	0
$191$	8.49534	0.614701	0.307350	−	0.951596i	$-0.400558\pi$
$191$	8.49534	0.614701	0.307350	+	0.951596i	$0.400558\pi$
$192$	0	0
$193$	7.83869	0.564241	0.282121	−	0.959379i	$-0.408962\pi$
$193$	7.83869	0.564241	0.282121	+	0.959379i	$0.408962\pi$
$194$	0	0
$195$	−2.77801	−0.198937
$196$	0	0
$197$	−17.5747	−1.25214	−0.626072	−	0.779765i	$-0.715338\pi$
$197$	−17.5747	−1.25214	−0.626072	+	0.779765i	$0.715338\pi$
$198$	0	0
$199$	−4.17997	−0.296310	−0.148155	−	0.988964i	$-0.547333\pi$
$199$	−4.17997	−0.296310	−0.148155	+	0.988964i	$0.547333\pi$
$200$	0	0
$201$	−13.4333	−0.947515
$202$	0	0
$203$	−35.8446	−2.51580
$204$	0	0
$205$	−6.19269	−0.432516
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−16.7453	−1.15830
$210$	0	0
$211$	22.4240	1.54373	0.771866	−	0.635785i	$-0.219324\pi$
$211$	22.4240	1.54373	0.771866	+	0.635785i	$0.219324\pi$
$212$	0	0
$213$	−11.0993	−0.760512
$214$	0	0
$215$	8.28267	0.564874
$216$	0	0
$217$	−25.0734	−1.70209
$218$	0	0
$219$	15.2406	1.02987
$220$	0	0
$221$	−1.76868	−0.118974
$222$	0	0
$223$	12.2827	0.822509	0.411254	−	0.911521i	$-0.365091\pi$
$223$	12.2827	0.822509	0.411254	+	0.911521i	$0.365091\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−24.5653	−1.63046	−0.815230	−	0.579138i	$-0.803389\pi$
$227$	−24.5653	−1.63046	−0.815230	+	0.579138i	$0.803389\pi$
$228$	0	0
$229$	−5.82003	−0.384598	−0.192299	−	0.981336i	$-0.561594\pi$
$229$	−5.82003	−0.384598	−0.192299	+	0.981336i	$0.561594\pi$
$230$	0	0
$231$	−23.2593	−1.53035
$232$	0	0
$233$	−2.28267	−0.149543	−0.0747714	−	0.997201i	$-0.523823\pi$
$233$	−2.28267	−0.149543	−0.0747714	+	0.997201i	$0.523823\pi$
$234$	0	0
$235$	−7.50466	−0.489550
$236$	0	0
$237$	−1.45331	−0.0944028
$238$	0	0
$239$	−2.53397	−0.163909	−0.0819544	−	0.996636i	$-0.526116\pi$
$239$	−2.53397	−0.163909	−0.0819544	+	0.996636i	$0.526116\pi$
$240$	0	0
$241$	2.67531	0.172332	0.0861658	−	0.996281i	$-0.472539\pi$
$241$	2.67531	0.172332	0.0861658	+	0.996281i	$0.472539\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	16.6974	1.06675
$246$	0	0
$247$	9.73599	0.619486
$248$	0	0
$249$	6.63667	0.420582
$250$	0	0
$251$	−12.2827	−0.775275	−0.387638	−	0.921812i	$-0.626709\pi$
$251$	−12.2827	−0.775275	−0.387638	+	0.921812i	$0.626709\pi$
$252$	0	0
$253$	−4.77801	−0.300391
$254$	0	0
$255$	0.636672	0.0398699
$256$	0	0
$257$	9.89004	0.616924	0.308462	−	0.951237i	$-0.400186\pi$
$257$	9.89004	0.616924	0.308462	+	0.951237i	$0.400186\pi$
$258$	0	0
$259$	−13.9614	−0.867517
$260$	0	0
$261$	7.36333	0.455779
$262$	0	0
$263$	4.26995	0.263297	0.131648	−	0.991296i	$-0.457973\pi$
$263$	4.26995	0.263297	0.131648	+	0.991296i	$0.457973\pi$
$264$	0	0
$265$	−4.91934	−0.302193
$266$	0	0
$267$	−9.29200	−0.568661
$268$	0	0
$269$	30.5526	1.86283	0.931413	−	0.363963i	$-0.118577\pi$
$269$	30.5526	1.86283	0.931413	+	0.363963i	$0.118577\pi$
$270$	0	0
$271$	23.6974	1.43951	0.719756	−	0.694227i	$-0.244254\pi$
$271$	23.6974	1.43951	0.719756	+	0.694227i	$0.244254\pi$
$272$	0	0
$273$	13.5233	0.818469
$274$	0	0
$275$	−4.77801	−0.288125
$276$	0	0
$277$	20.5840	1.23677	0.618386	−	0.785874i	$-0.287787\pi$
$277$	20.5840	1.23677	0.618386	+	0.785874i	$0.287787\pi$
$278$	0	0
$279$	5.15066	0.308362
$280$	0	0
$281$	8.61670	0.514029	0.257014	−	0.966408i	$-0.417261\pi$
$281$	8.61670	0.514029	0.257014	+	0.966408i	$0.417261\pi$
$282$	0	0
$283$	−7.02930	−0.417849	−0.208924	−	0.977932i	$-0.566996\pi$
$283$	−7.02930	−0.417849	−0.208924	+	0.977932i	$0.566996\pi$
$284$	0	0
$285$	−3.50466	−0.207598
$286$	0	0
$287$	30.1460	1.77946
$288$	0	0
$289$	−16.5946	−0.976156
$290$	0	0
$291$	14.7453	0.864386
$292$	0	0
$293$	−21.2793	−1.24315	−0.621574	−	0.783355i	$-0.713507\pi$
$293$	−21.2793	−1.24315	−0.621574	+	0.783355i	$0.713507\pi$
$294$	0	0
$295$	8.65533	0.503933
$296$	0	0
$297$	4.77801	0.277248
$298$	0	0
$299$	2.77801	0.160656
$300$	0	0
$301$	−40.3200	−2.32401
$302$	0	0
$303$	17.3820	0.998569
$304$	0	0
$305$	−15.0607	−0.862372
$306$	0	0
$307$	−13.0607	−0.745412	−0.372706	−	0.927949i	$-0.621570\pi$
$307$	−13.0607	−0.745412	−0.372706	+	0.927949i	$0.621570\pi$
$308$	0	0
$309$	15.2920	0.869932
$310$	0	0
$311$	−5.45331	−0.309229	−0.154615	−	0.987975i	$-0.549414\pi$
$311$	−5.45331	−0.309229	−0.154615	+	0.987975i	$0.549414\pi$
$312$	0	0
$313$	4.60398	0.260232	0.130116	−	0.991499i	$-0.458465\pi$
$313$	4.60398	0.260232	0.130116	+	0.991499i	$0.458465\pi$
$314$	0	0
$315$	−4.86799	−0.274280
$316$	0	0
$317$	−30.3527	−1.70478	−0.852388	−	0.522910i	$-0.824847\pi$
$317$	−30.3527	−1.70478	−0.852388	+	0.522910i	$0.824847\pi$
$318$	0	0
$319$	−35.1820	−1.96982
$320$	0	0
$321$	15.3820	0.858539
$322$	0	0
$323$	−2.23132	−0.124154
$324$	0	0
$325$	2.77801	0.154096
$326$	0	0
$327$	15.2406	0.842810
$328$	0	0
$329$	36.5327	2.01411
$330$	0	0
$331$	−6.42401	−0.353095	−0.176548	−	0.984292i	$-0.556493\pi$
$331$	−6.42401	−0.353095	−0.176548	+	0.984292i	$0.556493\pi$
$332$	0	0
$333$	2.86799	0.157165
$334$	0	0
$335$	13.4333	0.733942
$336$	0	0
$337$	−25.1120	−1.36794	−0.683970	−	0.729510i	$-0.739748\pi$
$337$	−25.1120	−1.36794	−0.683970	+	0.729510i	$0.739748\pi$
$338$	0	0
$339$	−6.19269	−0.336341
$340$	0	0
$341$	−24.6099	−1.33270
$342$	0	0
$343$	−47.2066	−2.54892
$344$	0	0
$345$	−1.00000	−0.0538382
$346$	0	0
$347$	−8.46264	−0.454298	−0.227149	−	0.973860i	$-0.572941\pi$
$347$	−8.46264	−0.454298	−0.227149	+	0.973860i	$0.572941\pi$
$348$	0	0
$349$	−6.40535	−0.342871	−0.171435	−	0.985195i	$-0.554840\pi$
$349$	−6.40535	−0.342871	−0.171435	+	0.985195i	$0.554840\pi$
$350$	0	0
$351$	−2.77801	−0.148279
$352$	0	0
$353$	−25.7873	−1.37252	−0.686261	−	0.727356i	$-0.740749\pi$
$353$	−25.7873	−1.37252	−0.686261	+	0.727356i	$0.740749\pi$
$354$	0	0
$355$	11.0993	0.589090
$356$	0	0
$357$	−3.09931	−0.164033
$358$	0	0
$359$	−5.94865	−0.313958	−0.156979	−	0.987602i	$-0.550175\pi$
$359$	−5.94865	−0.313958	−0.156979	+	0.987602i	$0.550175\pi$
$360$	0	0
$361$	−6.71733	−0.353544
$362$	0	0
$363$	−11.8294	−0.620880
$364$	0	0
$365$	−15.2406	−0.797732
$366$	0	0
$367$	10.5013	0.548162	0.274081	−	0.961707i	$-0.411626\pi$
$367$	10.5013	0.548162	0.274081	+	0.961707i	$0.411626\pi$
$368$	0	0
$369$	−6.19269	−0.322378
$370$	0	0
$371$	23.9473	1.24328
$372$	0	0
$373$	−7.17064	−0.371282	−0.185641	−	0.982618i	$-0.559436\pi$
$373$	−7.17064	−0.371282	−0.185641	+	0.982618i	$0.559436\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	20.4554	1.05351
$378$	0	0
$379$	7.11203	0.365321	0.182660	−	0.983176i	$-0.441529\pi$
$379$	7.11203	0.365321	0.182660	+	0.983176i	$0.441529\pi$
$380$	0	0
$381$	−4.77801	−0.244785
$382$	0	0
$383$	−7.80731	−0.398935	−0.199468	−	0.979904i	$-0.563921\pi$
$383$	−7.80731	−0.398935	−0.199468	+	0.979904i	$0.563921\pi$
$384$	0	0
$385$	23.2593	1.18540
$386$	0	0
$387$	8.28267	0.421032
$388$	0	0
$389$	27.5933	1.39904	0.699519	−	0.714614i	$-0.253398\pi$
$389$	27.5933	1.39904	0.699519	+	0.714614i	$0.253398\pi$
$390$	0	0
$391$	−0.636672	−0.0321979
$392$	0	0
$393$	6.54669	0.330237
$394$	0	0
$395$	1.45331	0.0731241
$396$	0	0
$397$	−14.6426	−0.734892	−0.367446	−	0.930045i	$-0.619768\pi$
$397$	−14.6426	−0.734892	−0.367446	+	0.930045i	$0.619768\pi$
$398$	0	0
$399$	17.0607	0.854102
$400$	0	0
$401$	−3.55602	−0.177579	−0.0887895	−	0.996050i	$-0.528300\pi$
$401$	−3.55602	−0.177579	−0.0887895	+	0.996050i	$0.528300\pi$
$402$	0	0
$403$	14.3086	0.712762
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−13.7033	−0.679247
$408$	0	0
$409$	28.9894	1.43343	0.716716	−	0.697366i	$-0.245645\pi$
$409$	28.9894	1.43343	0.716716	+	0.697366i	$0.245645\pi$
$410$	0	0
$411$	10.7453	0.530027
$412$	0	0
$413$	−42.1341	−2.07328
$414$	0	0
$415$	−6.63667	−0.325781
$416$	0	0
$417$	14.4240	0.706347
$418$	0	0
$419$	−28.3527	−1.38512	−0.692560	−	0.721361i	$-0.743517\pi$
$419$	−28.3527	−1.38512	−0.692560	+	0.721361i	$0.743517\pi$
$420$	0	0
$421$	9.32469	0.454458	0.227229	−	0.973841i	$-0.427033\pi$
$421$	9.32469	0.454458	0.227229	+	0.973841i	$0.427033\pi$
$422$	0	0
$423$	−7.50466	−0.364889
$424$	0	0
$425$	−0.636672	−0.0308831
$426$	0	0
$427$	73.3153	3.54798
$428$	0	0
$429$	13.2733	0.640843
$430$	0	0
$431$	3.89730	0.187726	0.0938631	−	0.995585i	$-0.470078\pi$
$431$	3.89730	0.187726	0.0938631	+	0.995585i	$0.470078\pi$
$432$	0	0
$433$	−28.4613	−1.36776	−0.683882	−	0.729593i	$-0.739710\pi$
$433$	−28.4613	−1.36776	−0.683882	+	0.729593i	$0.739710\pi$
$434$	0	0
$435$	−7.36333	−0.353045
$436$	0	0
$437$	3.50466	0.167651
$438$	0	0
$439$	−19.0866	−0.910953	−0.455477	−	0.890248i	$-0.650531\pi$
$439$	−19.0866	−0.910953	−0.455477	+	0.890248i	$0.650531\pi$
$440$	0	0
$441$	16.6974	0.795112
$442$	0	0
$443$	−6.47668	−0.307716	−0.153858	−	0.988093i	$-0.549170\pi$
$443$	−6.47668	−0.307716	−0.153858	+	0.988093i	$0.549170\pi$
$444$	0	0
$445$	9.29200	0.440483
$446$	0	0
$447$	−6.67531	−0.315731
$448$	0	0
$449$	16.8166	0.793626	0.396813	−	0.917899i	$-0.370116\pi$
$449$	16.8166	0.793626	0.396813	+	0.917899i	$0.370116\pi$
$450$	0	0
$451$	29.5887	1.39328
$452$	0	0
$453$	18.3013	0.859871
$454$	0	0
$455$	−13.5233	−0.633983
$456$	0	0
$457$	−4.52671	−0.211751	−0.105875	−	0.994379i	$-0.533764\pi$
$457$	−4.52671	−0.211751	−0.105875	+	0.994379i	$0.533764\pi$
$458$	0	0
$459$	0.636672	0.0297173
$460$	0	0
$461$	−8.44398	−0.393276	−0.196638	−	0.980476i	$-0.563002\pi$
$461$	−8.44398	−0.393276	−0.196638	+	0.980476i	$0.563002\pi$
$462$	0	0
$463$	30.5140	1.41811	0.709053	−	0.705155i	$-0.249123\pi$
$463$	30.5140	1.41811	0.709053	+	0.705155i	$0.249123\pi$
$464$	0	0
$465$	−5.15066	−0.238856
$466$	0	0
$467$	−12.2700	−0.567786	−0.283893	−	0.958856i	$-0.591626\pi$
$467$	−12.2700	−0.567786	−0.283893	+	0.958856i	$0.591626\pi$
$468$	0	0
$469$	−65.3934	−3.01959
$470$	0	0
$471$	−18.4427	−0.849793
$472$	0	0
$473$	−39.5747	−1.81965
$474$	0	0
$475$	3.50466	0.160805
$476$	0	0
$477$	−4.91934	−0.225241
$478$	0	0
$479$	1.06068	0.0484637	0.0242319	−	0.999706i	$-0.492286\pi$
$479$	1.06068	0.0484637	0.0242319	+	0.999706i	$0.492286\pi$
$480$	0	0
$481$	7.96731	0.363278
$482$	0	0
$483$	4.86799	0.221501
$484$	0	0
$485$	−14.7453	−0.669550
$486$	0	0
$487$	5.13795	0.232823	0.116411	−	0.993201i	$-0.462861\pi$
$487$	5.13795	0.232823	0.116411	+	0.993201i	$0.462861\pi$
$488$	0	0
$489$	−0.282672	−0.0127829
$490$	0	0
$491$	−2.81664	−0.127113	−0.0635566	−	0.997978i	$-0.520244\pi$
$491$	−2.81664	−0.127113	−0.0635566	+	0.997978i	$0.520244\pi$
$492$	0	0
$493$	−4.68802	−0.211138
$494$	0	0
$495$	−4.77801	−0.214755
$496$	0	0
$497$	−54.0314	−2.42364
$498$	0	0
$499$	−13.2534	−0.593302	−0.296651	−	0.954986i	$-0.595870\pi$
$499$	−13.2534	−0.593302	−0.296651	+	0.954986i	$0.595870\pi$
$500$	0	0
$501$	6.51399	0.291024
$502$	0	0
$503$	28.2300	1.25871	0.629357	−	0.777116i	$-0.283318\pi$
$503$	28.2300	1.25871	0.629357	+	0.777116i	$0.283318\pi$
$504$	0	0
$505$	−17.3820	−0.773488
$506$	0	0
$507$	5.28267	0.234612
$508$	0	0
$509$	2.70800	0.120030	0.0600150	−	0.998197i	$-0.480885\pi$
$509$	2.70800	0.120030	0.0600150	+	0.998197i	$0.480885\pi$
$510$	0	0
$511$	74.1914	3.28203
$512$	0	0
$513$	−3.50466	−0.154735
$514$	0	0
$515$	−15.2920	−0.673846
$516$	0	0
$517$	35.8573	1.57700
$518$	0	0
$519$	−8.30133	−0.364388
$520$	0	0
$521$	23.4206	1.02608	0.513038	−	0.858366i	$-0.328520\pi$
$521$	23.4206	1.02608	0.513038	+	0.858366i	$0.328520\pi$
$522$	0	0
$523$	−2.01866	−0.0882697	−0.0441349	−	0.999026i	$-0.514053\pi$
$523$	−2.01866	−0.0882697	−0.0441349	+	0.999026i	$0.514053\pi$
$524$	0	0
$525$	4.86799	0.212457
$526$	0	0
$527$	−3.27928	−0.142848
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	8.65533	0.375609
$532$	0	0
$533$	−17.2033	−0.745159
$534$	0	0
$535$	−15.3820	−0.665021
$536$	0	0
$537$	5.17064	0.223130
$538$	0	0
$539$	−79.7801	−3.43637
$540$	0	0
$541$	11.8387	0.508985	0.254492	−	0.967075i	$-0.418092\pi$
$541$	11.8387	0.508985	0.254492	+	0.967075i	$0.418092\pi$
$542$	0	0
$543$	−21.8573	−0.937988
$544$	0	0
$545$	−15.2406	−0.652838
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	0	0
$549$	−15.0607	−0.642774
$550$	0	0
$551$	25.8060	1.09937
$552$	0	0
$553$	−7.07472	−0.300848
$554$	0	0
$555$	−2.86799	−0.121740
$556$	0	0
$557$	−36.9193	−1.56432	−0.782161	−	0.623076i	$-0.785883\pi$
$557$	−36.9193	−1.56432	−0.782161	+	0.623076i	$0.785883\pi$
$558$	0	0
$559$	23.0093	0.973191
$560$	0	0
$561$	−3.04202	−0.128434
$562$	0	0
$563$	−37.7860	−1.59249	−0.796245	−	0.604974i	$-0.793184\pi$
$563$	−37.7860	−1.59249	−0.796245	+	0.604974i	$0.793184\pi$
$564$	0	0
$565$	6.19269	0.260528
$566$	0	0
$567$	−4.86799	−0.204436
$568$	0	0
$569$	−25.9346	−1.08724	−0.543618	−	0.839333i	$-0.682946\pi$
$569$	−25.9346	−1.08724	−0.543618	+	0.839333i	$0.682946\pi$
$570$	0	0
$571$	−2.69396	−0.112739	−0.0563694	−	0.998410i	$-0.517952\pi$
$571$	−2.69396	−0.112739	−0.0563694	+	0.998410i	$0.517952\pi$
$572$	0	0
$573$	−8.49534	−0.354898
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	17.3947	0.724151	0.362075	−	0.932149i	$-0.382068\pi$
$577$	17.3947	0.724151	0.362075	+	0.932149i	$0.382068\pi$
$578$	0	0
$579$	−7.83869	−0.325765
$580$	0	0
$581$	32.3073	1.34033
$582$	0	0
$583$	23.5047	0.973464
$584$	0	0
$585$	2.77801	0.114856
$586$	0	0
$587$	−25.2080	−1.04044	−0.520222	−	0.854031i	$-0.674151\pi$
$587$	−25.2080	−1.04044	−0.520222	+	0.854031i	$0.674151\pi$
$588$	0	0
$589$	18.0514	0.743793
$590$	0	0
$591$	17.5747	0.722925
$592$	0	0
$593$	44.7313	1.83689	0.918447	−	0.395545i	$-0.129444\pi$
$593$	44.7313	1.83689	0.918447	+	0.395545i	$0.129444\pi$
$594$	0	0
$595$	3.09931	0.127059
$596$	0	0
$597$	4.17997	0.171075
$598$	0	0
$599$	−23.0093	−0.940136	−0.470068	−	0.882630i	$-0.655771\pi$
$599$	−23.0093	−0.940136	−0.470068	+	0.882630i	$0.655771\pi$
$600$	0	0
$601$	40.8867	1.66780	0.833901	−	0.551915i	$-0.186103\pi$
$601$	40.8867	1.66780	0.833901	+	0.551915i	$0.186103\pi$
$602$	0	0
$603$	13.4333	0.547048
$604$	0	0
$605$	11.8294	0.480932
$606$	0	0
$607$	2.65665	0.107830	0.0539150	−	0.998546i	$-0.482830\pi$
$607$	2.65665	0.107830	0.0539150	+	0.998546i	$0.482830\pi$
$608$	0	0
$609$	35.8446	1.45250
$610$	0	0
$611$	−20.8480	−0.843420
$612$	0	0
$613$	−38.0000	−1.53481	−0.767403	−	0.641165i	$-0.778451\pi$
$613$	−38.0000	−1.53481	−0.767403	+	0.641165i	$0.778451\pi$
$614$	0	0
$615$	6.19269	0.249713
$616$	0	0
$617$	−40.0314	−1.61160	−0.805801	−	0.592186i	$-0.798265\pi$
$617$	−40.0314	−1.61160	−0.805801	+	0.592186i	$0.798265\pi$
$618$	0	0
$619$	−22.9066	−0.920695	−0.460348	−	0.887739i	$-0.652275\pi$
$619$	−22.9066	−0.920695	−0.460348	+	0.887739i	$0.652275\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	−45.2334	−1.81224
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	16.7453	0.668743
$628$	0	0
$629$	−1.82597	−0.0728062
$630$	0	0
$631$	31.1566	1.24032	0.620162	−	0.784473i	$-0.287067\pi$
$631$	31.1566	1.24032	0.620162	+	0.784473i	$0.287067\pi$
$632$	0	0
$633$	−22.4240	−0.891274
$634$	0	0
$635$	4.77801	0.189609
$636$	0	0
$637$	46.3854	1.83786
$638$	0	0
$639$	11.0993	0.439082
$640$	0	0
$641$	−37.7474	−1.49093	−0.745466	−	0.666544i	$-0.767773\pi$
$641$	−37.7474	−1.49093	−0.745466	+	0.666544i	$0.767773\pi$
$642$	0	0
$643$	−21.8187	−0.860446	−0.430223	−	0.902723i	$-0.641565\pi$
$643$	−21.8187	−0.860446	−0.430223	+	0.902723i	$0.641565\pi$
$644$	0	0
$645$	−8.28267	−0.326130
$646$	0	0
$647$	−32.4554	−1.27595	−0.637976	−	0.770056i	$-0.720228\pi$
$647$	−32.4554	−1.27595	−0.637976	+	0.770056i	$0.720228\pi$
$648$	0	0
$649$	−41.3552	−1.62333
$650$	0	0
$651$	25.0734	0.982704
$652$	0	0
$653$	15.0607	0.589370	0.294685	−	0.955594i	$-0.404785\pi$
$653$	15.0607	0.589370	0.294685	+	0.955594i	$0.404785\pi$
$654$	0	0
$655$	−6.54669	−0.255800
$656$	0	0
$657$	−15.2406	−0.594594
$658$	0	0
$659$	−32.2500	−1.25628	−0.628140	−	0.778100i	$-0.716184\pi$
$659$	−32.2500	−1.25628	−0.628140	+	0.778100i	$0.716184\pi$
$660$	0	0
$661$	−37.0093	−1.43950	−0.719748	−	0.694235i	$-0.755743\pi$
$661$	−37.0093	−1.43950	−0.719748	+	0.694235i	$0.755743\pi$
$662$	0	0
$663$	1.76868	0.0686898
$664$	0	0
$665$	−17.0607	−0.661585
$666$	0	0
$667$	7.36333	0.285109
$668$	0	0
$669$	−12.2827	−0.474876
$670$	0	0
$671$	71.9600	2.77799
$672$	0	0
$673$	−0.411290	−0.0158541	−0.00792704	−	0.999969i	$-0.502523\pi$
$673$	−0.411290	−0.0158541	−0.00792704	+	0.999969i	$0.502523\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	−1.98728	−0.0763774	−0.0381887	−	0.999271i	$-0.512159\pi$
$677$	−1.98728	−0.0763774	−0.0381887	+	0.999271i	$0.512159\pi$
$678$	0	0
$679$	71.7801	2.75467
$680$	0	0
$681$	24.5653	0.941346
$682$	0	0
$683$	−17.3179	−0.662652	−0.331326	−	0.943516i	$-0.607496\pi$
$683$	−17.3179	−0.662652	−0.331326	+	0.943516i	$0.607496\pi$
$684$	0	0
$685$	−10.7453	−0.410557
$686$	0	0
$687$	5.82003	0.222048
$688$	0	0
$689$	−13.6660	−0.520632
$690$	0	0
$691$	5.94139	0.226021	0.113011	−	0.993594i	$-0.463951\pi$
$691$	5.94139	0.226021	0.113011	+	0.993594i	$0.463951\pi$
$692$	0	0
$693$	23.2593	0.883548
$694$	0	0
$695$	−14.4240	−0.547134
$696$	0	0
$697$	3.94271	0.149341
$698$	0	0
$699$	2.28267	0.0863386
$700$	0	0
$701$	34.2127	1.29219	0.646097	−	0.763255i	$-0.276400\pi$
$701$	34.2127	1.29219	0.646097	+	0.763255i	$0.276400\pi$
$702$	0	0
$703$	10.0514	0.379094
$704$	0	0
$705$	7.50466	0.282642
$706$	0	0
$707$	84.6154	3.18229
$708$	0	0
$709$	18.6354	0.699865	0.349933	−	0.936775i	$-0.386204\pi$
$709$	18.6354	0.699865	0.349933	+	0.936775i	$0.386204\pi$
$710$	0	0
$711$	1.45331	0.0545035
$712$	0	0
$713$	5.15066	0.192894
$714$	0	0
$715$	−13.2733	−0.496395
$716$	0	0
$717$	2.53397	0.0946328
$718$	0	0
$719$	−10.5594	−0.393799	−0.196900	−	0.980424i	$-0.563087\pi$
$719$	−10.5594	−0.393799	−0.196900	+	0.980424i	$0.563087\pi$
$720$	0	0
$721$	74.4413	2.77234
$722$	0	0
$723$	−2.67531	−0.0994957
$724$	0	0
$725$	7.36333	0.273467
$726$	0	0
$727$	47.3493	1.75609	0.878044	−	0.478580i	$-0.158848\pi$
$727$	47.3493	1.75609	0.878044	+	0.478580i	$0.158848\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−5.27334	−0.195042
$732$	0	0
$733$	−42.1600	−1.55721	−0.778607	−	0.627511i	$-0.784074\pi$
$733$	−42.1600	−1.55721	−0.778607	+	0.627511i	$0.784074\pi$
$734$	0	0
$735$	−16.6974	−0.615891
$736$	0	0
$737$	−64.1846	−2.36427
$738$	0	0
$739$	−13.2933	−0.489003	−0.244501	−	0.969649i	$-0.578624\pi$
$739$	−13.2933	−0.489003	−0.244501	+	0.969649i	$0.578624\pi$
$740$	0	0
$741$	−9.73599	−0.357660
$742$	0	0
$743$	−3.47197	−0.127374	−0.0636871	−	0.997970i	$-0.520286\pi$
$743$	−3.47197	−0.127374	−0.0636871	+	0.997970i	$0.520286\pi$
$744$	0	0
$745$	6.67531	0.244564
$746$	0	0
$747$	−6.63667	−0.242823
$748$	0	0
$749$	74.8794	2.73603
$750$	0	0
$751$	−22.5513	−0.822909	−0.411454	−	0.911430i	$-0.634979\pi$
$751$	−22.5513	−0.822909	−0.411454	+	0.911430i	$0.634979\pi$
$752$	0	0
$753$	12.2827	0.447605
$754$	0	0
$755$	−18.3013	−0.666054
$756$	0	0
$757$	−13.9028	−0.505304	−0.252652	−	0.967557i	$-0.581303\pi$
$757$	−13.9028	−0.505304	−0.252652	+	0.967557i	$0.581303\pi$
$758$	0	0
$759$	4.77801	0.173431
$760$	0	0
$761$	29.8701	1.08279	0.541394	−	0.840769i	$-0.317897\pi$
$761$	29.8701	1.08279	0.541394	+	0.840769i	$0.317897\pi$
$762$	0	0
$763$	74.1914	2.68591
$764$	0	0
$765$	−0.636672	−0.0230189
$766$	0	0
$767$	24.0446	0.868199
$768$	0	0
$769$	10.4299	0.376114	0.188057	−	0.982158i	$-0.439781\pi$
$769$	10.4299	0.376114	0.188057	+	0.982158i	$0.439781\pi$
$770$	0	0
$771$	−9.89004	−0.356181
$772$	0	0
$773$	−50.5000	−1.81636	−0.908179	−	0.418583i	$-0.862527\pi$
$773$	−50.5000	−1.81636	−0.908179	+	0.418583i	$0.862527\pi$
$774$	0	0
$775$	5.15066	0.185017
$776$	0	0
$777$	13.9614	0.500861
$778$	0	0
$779$	−21.7033	−0.777601
$780$	0	0
$781$	−53.0326	−1.89766
$782$	0	0
$783$	−7.36333	−0.263144
$784$	0	0
$785$	18.4427	0.658247
$786$	0	0
$787$	−41.2907	−1.47185	−0.735927	−	0.677061i	$-0.763253\pi$
$787$	−41.2907	−1.47185	−0.735927	+	0.677061i	$0.763253\pi$
$788$	0	0
$789$	−4.26995	−0.152014
$790$	0	0
$791$	−30.1460	−1.07187
$792$	0	0
$793$	−41.8387	−1.48574
$794$	0	0
$795$	4.91934	0.174471
$796$	0	0
$797$	25.4847	0.902714	0.451357	−	0.892343i	$-0.350940\pi$
$797$	25.4847	0.902714	0.451357	+	0.892343i	$0.350940\pi$
$798$	0	0
$799$	4.77801	0.169034
$800$	0	0
$801$	9.29200	0.328317
$802$	0	0
$803$	72.8199	2.56976
$804$	0	0
$805$	−4.86799	−0.171574
$806$	0	0
$807$	−30.5526	−1.07550
$808$	0	0
$809$	−30.9380	−1.08772	−0.543861	−	0.839175i	$-0.683038\pi$
$809$	−30.9380	−1.08772	−0.543861	+	0.839175i	$0.683038\pi$
$810$	0	0
$811$	−31.5946	−1.10944	−0.554719	−	0.832038i	$-0.687174\pi$
$811$	−31.5946	−1.10944	−0.554719	+	0.832038i	$0.687174\pi$
$812$	0	0
$813$	−23.6974	−0.831102
$814$	0	0
$815$	0.282672	0.00990158
$816$	0	0
$817$	29.0280	1.01556
$818$	0	0
$819$	−13.5233	−0.472543
$820$	0	0
$821$	−56.7640	−1.98108	−0.990538	−	0.137238i	$-0.956178\pi$
$821$	−56.7640	−1.98108	−0.990538	+	0.137238i	$0.956178\pi$
$822$	0	0
$823$	10.9066	0.380181	0.190091	−	0.981767i	$-0.439122\pi$
$823$	10.9066	0.380181	0.190091	+	0.981767i	$0.439122\pi$
$824$	0	0
$825$	4.77801	0.166349
$826$	0	0
$827$	−8.75803	−0.304547	−0.152273	−	0.988338i	$-0.548659\pi$
$827$	−8.75803	−0.304547	−0.152273	+	0.988338i	$0.548659\pi$
$828$	0	0
$829$	21.2720	0.738808	0.369404	−	0.929269i	$-0.379562\pi$
$829$	21.2720	0.738808	0.369404	+	0.929269i	$0.379562\pi$
$830$	0	0
$831$	−20.5840	−0.714051
$832$	0	0
$833$	−10.6307	−0.368333
$834$	0	0
$835$	−6.51399	−0.225426
$836$	0	0
$837$	−5.15066	−0.178033
$838$	0	0
$839$	14.6240	0.504875	0.252437	−	0.967613i	$-0.418768\pi$
$839$	14.6240	0.504875	0.252437	+	0.967613i	$0.418768\pi$
$840$	0	0
$841$	25.2186	0.869607
$842$	0	0
$843$	−8.61670	−0.296775
$844$	0	0
$845$	−5.28267	−0.181729
$846$	0	0
$847$	−57.5852	−1.97865
$848$	0	0
$849$	7.02930	0.241245
$850$	0	0
$851$	2.86799	0.0983135
$852$	0	0
$853$	−52.6867	−1.80396	−0.901979	−	0.431779i	$-0.857886\pi$
$853$	−52.6867	−1.80396	−0.901979	+	0.431779i	$0.857886\pi$
$854$	0	0
$855$	3.50466	0.119857
$856$	0	0
$857$	−6.77075	−0.231284	−0.115642	−	0.993291i	$-0.536893\pi$
$857$	−6.77075	−0.231284	−0.115642	+	0.993291i	$0.536893\pi$
$858$	0	0
$859$	47.9146	1.63483	0.817413	−	0.576052i	$-0.195407\pi$
$859$	47.9146	1.63483	0.817413	+	0.576052i	$0.195407\pi$
$860$	0	0
$861$	−30.1460	−1.02737
$862$	0	0
$863$	27.8573	0.948275	0.474138	−	0.880451i	$-0.342760\pi$
$863$	27.8573	0.948275	0.474138	+	0.880451i	$0.342760\pi$
$864$	0	0
$865$	8.30133	0.282254
$866$	0	0
$867$	16.5946	0.563584
$868$	0	0
$869$	−6.94394	−0.235557
$870$	0	0
$871$	37.3179	1.26447
$872$	0	0
$873$	−14.7453	−0.499053
$874$	0	0
$875$	−4.86799	−0.164568
$876$	0	0
$877$	12.4440	0.420203	0.210102	−	0.977680i	$-0.432620\pi$
$877$	12.4440	0.420203	0.210102	+	0.977680i	$0.432620\pi$
$878$	0	0
$879$	21.2793	0.717732
$880$	0	0
$881$	−7.70329	−0.259530	−0.129765	−	0.991545i	$-0.541422\pi$
$881$	−7.70329	−0.259530	−0.129765	+	0.991545i	$0.541422\pi$
$882$	0	0
$883$	47.9274	1.61288	0.806442	−	0.591313i	$-0.201390\pi$
$883$	47.9274	1.61288	0.806442	+	0.591313i	$0.201390\pi$
$884$	0	0
$885$	−8.65533	−0.290946
$886$	0	0
$887$	7.61462	0.255674	0.127837	−	0.991795i	$-0.459197\pi$
$887$	7.61462	0.255674	0.127837	+	0.991795i	$0.459197\pi$
$888$	0	0
$889$	−23.2593	−0.780092
$890$	0	0
$891$	−4.77801	−0.160069
$892$	0	0
$893$	−26.3013	−0.880140
$894$	0	0
$895$	−5.17064	−0.172835
$896$	0	0
$897$	−2.77801	−0.0927550
$898$	0	0
$899$	37.9260	1.26490
$900$	0	0
$901$	3.13201	0.104342
$902$	0	0
$903$	40.3200	1.34177
$904$	0	0
$905$	21.8573	0.726563
$906$	0	0
$907$	−50.0759	−1.66274	−0.831372	−	0.555716i	$-0.812444\pi$
$907$	−50.0759	−1.66274	−0.831372	+	0.555716i	$0.812444\pi$
$908$	0	0
$909$	−17.3820	−0.576524
$910$	0	0
$911$	−14.5840	−0.483190	−0.241595	−	0.970377i	$-0.577670\pi$
$911$	−14.5840	−0.483190	−0.241595	+	0.970377i	$0.577670\pi$
$912$	0	0
$913$	31.7101	1.04945
$914$	0	0
$915$	15.0607	0.497891
$916$	0	0
$917$	31.8692	1.05241
$918$	0	0
$919$	−8.03731	−0.265127	−0.132563	−	0.991175i	$-0.542321\pi$
$919$	−8.03731	−0.265127	−0.132563	+	0.991175i	$0.542321\pi$
$920$	0	0
$921$	13.0607	0.430364
$922$	0	0
$923$	30.8340	1.01491
$924$	0	0
$925$	2.86799	0.0942990
$926$	0	0
$927$	−15.2920	−0.502255
$928$	0	0
$929$	14.3353	0.470327	0.235164	−	0.971956i	$-0.424437\pi$
$929$	14.3353	0.470327	0.235164	+	0.971956i	$0.424437\pi$
$930$	0	0
$931$	58.5186	1.91787
$932$	0	0
$933$	5.45331	0.178533
$934$	0	0
$935$	3.04202	0.0994848
$936$	0	0
$937$	14.7080	0.480489	0.240245	−	0.970712i	$-0.422772\pi$
$937$	14.7080	0.480489	0.240245	+	0.970712i	$0.422772\pi$
$938$	0	0
$939$	−4.60398	−0.150245
$940$	0	0
$941$	2.77801	0.0905605	0.0452802	−	0.998974i	$-0.485582\pi$
$941$	2.77801	0.0905605	0.0452802	+	0.998974i	$0.485582\pi$
$942$	0	0
$943$	−6.19269	−0.201662
$944$	0	0
$945$	4.86799	0.158356
$946$	0	0
$947$	32.6680	1.06157	0.530784	−	0.847507i	$-0.321897\pi$
$947$	32.6680	1.06157	0.530784	+	0.847507i	$0.321897\pi$
$948$	0	0
$949$	−42.3386	−1.37437
$950$	0	0
$951$	30.3527	0.984253
$952$	0	0
$953$	−25.4347	−0.823909	−0.411955	−	0.911204i	$-0.635154\pi$
$953$	−25.4347	−0.823909	−0.411955	+	0.911204i	$0.635154\pi$
$954$	0	0
$955$	8.49534	0.274903
$956$	0	0
$957$	35.1820	1.13727
$958$	0	0
$959$	52.3081	1.68912
$960$	0	0
$961$	−4.47065	−0.144215
$962$	0	0
$963$	−15.3820	−0.495678
$964$	0	0
$965$	7.83869	0.252336
$966$	0	0
$967$	46.4740	1.49450	0.747252	−	0.664541i	$-0.231373\pi$
$967$	46.4740	1.49450	0.747252	+	0.664541i	$0.231373\pi$
$968$	0	0
$969$	2.23132	0.0716804
$970$	0	0
$971$	−59.6120	−1.91304	−0.956520	−	0.291667i	$-0.905790\pi$
$971$	−59.6120	−1.91304	−0.956520	+	0.291667i	$0.905790\pi$
$972$	0	0
$973$	70.2160	2.25102
$974$	0	0
$975$	−2.77801	−0.0889675
$976$	0	0
$977$	11.2207	0.358981	0.179491	−	0.983760i	$-0.442555\pi$
$977$	11.2207	0.358981	0.179491	+	0.983760i	$0.442555\pi$
$978$	0	0
$979$	−44.3973	−1.41894
$980$	0	0
$981$	−15.2406	−0.486596
$982$	0	0
$983$	−4.29539	−0.137002	−0.0685008	−	0.997651i	$-0.521822\pi$
$983$	−4.29539	−0.137002	−0.0685008	+	0.997651i	$0.521822\pi$
$984$	0	0
$985$	−17.5747	−0.559976
$986$	0	0
$987$	−36.5327	−1.16285
$988$	0	0
$989$	8.28267	0.263374
$990$	0	0
$991$	−26.3841	−0.838117	−0.419059	−	0.907959i	$-0.637640\pi$
$991$	−26.3841	−0.838117	−0.419059	+	0.907959i	$0.637640\pi$
$992$	0	0
$993$	6.42401	0.203860
$994$	0	0
$995$	−4.17997	−0.132514
$996$	0	0
$997$	13.2547	0.419780	0.209890	−	0.977725i	$-0.432689\pi$
$997$	13.2547	0.419780	0.209890	+	0.977725i	$0.432689\pi$
$998$	0	0
$999$	−2.86799	−0.0907393

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} -4.86799 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} -4.86799 q^{7} +1.00000 q^{9} -4.77801 q^{11} +2.77801 q^{13} -1.00000 q^{15} -0.636672 q^{17} +3.50466 q^{19} +4.86799 q^{21} +1.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} +7.36333 q^{29} +5.15066 q^{31} +4.77801 q^{33} -4.86799 q^{35} +2.86799 q^{37} -2.77801 q^{39} -6.19269 q^{41} +8.28267 q^{43} +1.00000 q^{45} -7.50466 q^{47} +16.6974 q^{49} +0.636672 q^{51} -4.91934 q^{53} -4.77801 q^{55} -3.50466 q^{57} +8.65533 q^{59} -15.0607 q^{61} -4.86799 q^{63} +2.77801 q^{65} +13.4333 q^{67} -1.00000 q^{69} +11.0993 q^{71} -15.2406 q^{73} -1.00000 q^{75} +23.2593 q^{77} +1.45331 q^{79} +1.00000 q^{81} -6.63667 q^{83} -0.636672 q^{85} -7.36333 q^{87} +9.29200 q^{89} -13.5233 q^{91} -5.15066 q^{93} +3.50466 q^{95} -14.7453 q^{97} -4.77801 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 3 q^{5} - 2 q^{7} + 3 q^{9} - 8 q^{11} + 2 q^{13} - 3 q^{15} - 4 q^{17} + 2 q^{21} + 3 q^{23} + 3 q^{25} - 3 q^{27} + 20 q^{29} - 14 q^{31} + 8 q^{33} - 2 q^{35} - 4 q^{37} - 2 q^{39} - 8 q^{41} + 8 q^{43} + 3 q^{45} - 12 q^{47} + 29 q^{49} + 4 q^{51} - 8 q^{55} - 14 q^{59} - 22 q^{61} - 2 q^{63} + 2 q^{65} - 6 q^{67} - 3 q^{69} + 6 q^{71} - 10 q^{73} - 3 q^{75} - 8 q^{77} - 4 q^{79} + 3 q^{81} - 22 q^{83} - 4 q^{85} - 20 q^{87} - 10 q^{89} + 12 q^{91} + 14 q^{93} + 2 q^{97} - 8 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 3 * q^5 - 2 * q^7 + 3 * q^9 - 8 * q^11 + 2 * q^13 - 3 * q^15 - 4 * q^17 + 2 * q^21 + 3 * q^23 + 3 * q^25 - 3 * q^27 + 20 * q^29 - 14 * q^31 + 8 * q^33 - 2 * q^35 - 4 * q^37 - 2 * q^39 - 8 * q^41 + 8 * q^43 + 3 * q^45 - 12 * q^47 + 29 * q^49 + 4 * q^51 - 8 * q^55 - 14 * q^59 - 22 * q^61 - 2 * q^63 + 2 * q^65 - 6 * q^67 - 3 * q^69 + 6 * q^71 - 10 * q^73 - 3 * q^75 - 8 * q^77 - 4 * q^79 + 3 * q^81 - 22 * q^83 - 4 * q^85 - 20 * q^87 - 10 * q^89 + 12 * q^91 + 14 * q^93 + 2 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more