LMFDB - Embedded newform 5520.2.a.bt.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 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			$-1.41421$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} -1.82843 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} -1.82843 q^{7} +1.00000 q^{9} +1.41421 q^{11} +0.585786 q^{13} +1.00000 q^{15} -2.41421 q^{17} +7.41421 q^{19} -1.82843 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.58579 q^{29} +1.00000 q^{31} +1.41421 q^{33} -1.82843 q^{35} -7.48528 q^{37} +0.585786 q^{39} +7.24264 q^{41} +7.65685 q^{43} +1.00000 q^{45} +6.24264 q^{47} -3.65685 q^{49} -2.41421 q^{51} +5.24264 q^{53} +1.41421 q^{55} +7.41421 q^{57} -13.7279 q^{59} +1.41421 q^{61} -1.82843 q^{63} +0.585786 q^{65} +15.4853 q^{67} +1.00000 q^{69} +14.0711 q^{71} +2.24264 q^{73} +1.00000 q^{75} -2.58579 q^{77} -16.9706 q^{79} +1.00000 q^{81} -2.75736 q^{83} -2.41421 q^{85} -3.58579 q^{87} -9.17157 q^{89} -1.07107 q^{91} +1.00000 q^{93} +7.41421 q^{95} +18.1421 q^{97} +1.41421 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + 4 q^{13} + 2 q^{15} - 2 q^{17} + 12 q^{19} + 2 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} - 10 q^{29} + 2 q^{31} + 2 q^{35} + 2 q^{37} + 4 q^{39} + 6 q^{41} + 4 q^{43} + 2 q^{45} + 4 q^{47} + 4 q^{49} - 2 q^{51} + 2 q^{53} + 12 q^{57} - 2 q^{59} + 2 q^{63} + 4 q^{65} + 14 q^{67} + 2 q^{69} + 14 q^{71} - 4 q^{73} + 2 q^{75} - 8 q^{77} + 2 q^{81} - 14 q^{83} - 2 q^{85} - 10 q^{87} - 24 q^{89} + 12 q^{91} + 2 q^{93} + 12 q^{95} + 8 q^{97}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 + 4 * q^13 + 2 * q^15 - 2 * q^17 + 12 * q^19 + 2 * q^21 + 2 * q^23 + 2 * q^25 + 2 * q^27 - 10 * q^29 + 2 * q^31 + 2 * q^35 + 2 * q^37 + 4 * q^39 + 6 * q^41 + 4 * q^43 + 2 * q^45 + 4 * q^47 + 4 * q^49 - 2 * q^51 + 2 * q^53 + 12 * q^57 - 2 * q^59 + 2 * q^63 + 4 * q^65 + 14 * q^67 + 2 * q^69 + 14 * q^71 - 4 * q^73 + 2 * q^75 - 8 * q^77 + 2 * q^81 - 14 * q^83 - 2 * q^85 - 10 * q^87 - 24 * q^89 + 12 * q^91 + 2 * q^93 + 12 * q^95 + 8 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.82843	−0.691080	−0.345540	−	0.938404i	$-0.612304\pi$
$7$	−1.82843	−0.691080	−0.345540	+	0.938404i	$0.612304\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.41421	0.426401	0.213201	−	0.977008i	$-0.431611\pi$
$11$	1.41421	0.426401	0.213201	+	0.977008i	$0.431611\pi$
$12$	0	0
$13$	0.585786	0.162468	0.0812340	−	0.996695i	$-0.474114\pi$
$13$	0.585786	0.162468	0.0812340	+	0.996695i	$0.474114\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−2.41421	−0.585533	−0.292766	−	0.956184i	$-0.594576\pi$
$17$	−2.41421	−0.585533	−0.292766	+	0.956184i	$0.594576\pi$
$18$	0	0
$19$	7.41421	1.70094	0.850469	−	0.526026i	$-0.176318\pi$
$19$	7.41421	1.70094	0.850469	+	0.526026i	$0.176318\pi$
$20$	0	0
$21$	−1.82843	−0.398996
$22$	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
$29$	−3.58579	−0.665864	−0.332932	−	0.942951i	$-0.608038\pi$
$29$	−3.58579	−0.665864	−0.332932	+	0.942951i	$0.608038\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	0	0
$33$	1.41421	0.246183
$34$	0	0
$35$	−1.82843	−0.309061
$36$	0	0
$37$	−7.48528	−1.23057	−0.615286	−	0.788304i	$-0.710960\pi$
$37$	−7.48528	−1.23057	−0.615286	+	0.788304i	$0.710960\pi$
$38$	0	0
$39$	0.585786	0.0938009
$40$	0	0
$41$	7.24264	1.13111	0.565555	−	0.824710i	$-0.308662\pi$
$41$	7.24264	1.13111	0.565555	+	0.824710i	$0.308662\pi$
$42$	0	0
$43$	7.65685	1.16766	0.583830	−	0.811876i	$-0.301554\pi$
$43$	7.65685	1.16766	0.583830	+	0.811876i	$0.301554\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	6.24264	0.910583	0.455291	−	0.890343i	$-0.349535\pi$
$47$	6.24264	0.910583	0.455291	+	0.890343i	$0.349535\pi$
$48$	0	0
$49$	−3.65685	−0.522408
$50$	0	0
$51$	−2.41421	−0.338058
$52$	0	0
$53$	5.24264	0.720132	0.360066	−	0.932927i	$-0.382754\pi$
$53$	5.24264	0.720132	0.360066	+	0.932927i	$0.382754\pi$
$54$	0	0
$55$	1.41421	0.190693
$56$	0	0
$57$	7.41421	0.982037
$58$	0	0
$59$	−13.7279	−1.78722	−0.893612	−	0.448841i	$-0.851837\pi$
$59$	−13.7279	−1.78722	−0.893612	+	0.448841i	$0.851837\pi$
$60$	0	0
$61$	1.41421	0.181071	0.0905357	−	0.995893i	$-0.471142\pi$
$61$	1.41421	0.181071	0.0905357	+	0.995893i	$0.471142\pi$
$62$	0	0
$63$	−1.82843	−0.230360
$64$	0	0
$65$	0.585786	0.0726579
$66$	0	0
$67$	15.4853	1.89183	0.945914	−	0.324417i	$-0.105168\pi$
$67$	15.4853	1.89183	0.945914	+	0.324417i	$0.105168\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	14.0711	1.66993	0.834964	−	0.550304i	$-0.185488\pi$
$71$	14.0711	1.66993	0.834964	+	0.550304i	$0.185488\pi$
$72$	0	0
$73$	2.24264	0.262481	0.131241	−	0.991351i	$-0.458104\pi$
$73$	2.24264	0.262481	0.131241	+	0.991351i	$0.458104\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−2.58579	−0.294678
$78$	0	0
$79$	−16.9706	−1.90934	−0.954669	−	0.297670i	$-0.903790\pi$
$79$	−16.9706	−1.90934	−0.954669	+	0.297670i	$0.903790\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.75736	−0.302660	−0.151330	−	0.988483i	$-0.548356\pi$
$83$	−2.75736	−0.302660	−0.151330	+	0.988483i	$0.548356\pi$
$84$	0	0
$85$	−2.41421	−0.261858
$86$	0	0
$87$	−3.58579	−0.384437
$88$	0	0
$89$	−9.17157	−0.972185	−0.486092	−	0.873907i	$-0.661578\pi$
$89$	−9.17157	−0.972185	−0.486092	+	0.873907i	$0.661578\pi$
$90$	0	0
$91$	−1.07107	−0.112278
$92$	0	0
$93$	1.00000	0.103695
$94$	0	0
$95$	7.41421	0.760682
$96$	0	0
$97$	18.1421	1.84205	0.921027	−	0.389498i	$-0.127351\pi$
$97$	18.1421	1.84205	0.921027	+	0.389498i	$0.127351\pi$
$98$	0	0
$99$	1.41421	0.142134
$100$	0	0
$101$	−10.0711	−1.00211	−0.501054	−	0.865416i	$-0.667054\pi$
$101$	−10.0711	−1.00211	−0.501054	+	0.865416i	$0.667054\pi$
$102$	0	0
$103$	1.51472	0.149250	0.0746248	−	0.997212i	$-0.476224\pi$
$103$	1.51472	0.149250	0.0746248	+	0.997212i	$0.476224\pi$
$104$	0	0
$105$	−1.82843	−0.178436
$106$	0	0
$107$	−3.58579	−0.346651	−0.173326	−	0.984865i	$-0.555451\pi$
$107$	−3.58579	−0.346651	−0.173326	+	0.984865i	$0.555451\pi$
$108$	0	0
$109$	−7.41421	−0.710153	−0.355076	−	0.934837i	$-0.615545\pi$
$109$	−7.41421	−0.710153	−0.355076	+	0.934837i	$0.615545\pi$
$110$	0	0
$111$	−7.48528	−0.710471
$112$	0	0
$113$	18.8995	1.77791	0.888957	−	0.457990i	$-0.151430\pi$
$113$	18.8995	1.77791	0.888957	+	0.457990i	$0.151430\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	0.585786	0.0541560
$118$	0	0
$119$	4.41421	0.404650
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	7.24264	0.653047
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	19.5563	1.73535	0.867673	−	0.497136i	$-0.165615\pi$
$127$	19.5563	1.73535	0.867673	+	0.497136i	$0.165615\pi$
$128$	0	0
$129$	7.65685	0.674148
$130$	0	0
$131$	16.0000	1.39793	0.698963	−	0.715158i	$-0.253645\pi$
$131$	16.0000	1.39793	0.698963	+	0.715158i	$0.253645\pi$
$132$	0	0
$133$	−13.5563	−1.17548
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	2.82843	0.241649	0.120824	−	0.992674i	$-0.461446\pi$
$137$	2.82843	0.241649	0.120824	+	0.992674i	$0.461446\pi$
$138$	0	0
$139$	17.4853	1.48308	0.741541	−	0.670907i	$-0.234095\pi$
$139$	17.4853	1.48308	0.741541	+	0.670907i	$0.234095\pi$
$140$	0	0
$141$	6.24264	0.525725
$142$	0	0
$143$	0.828427	0.0692766
$144$	0	0
$145$	−3.58579	−0.297783
$146$	0	0
$147$	−3.65685	−0.301612
$148$	0	0
$149$	−2.24264	−0.183724	−0.0918621	−	0.995772i	$-0.529282\pi$
$149$	−2.24264	−0.183724	−0.0918621	+	0.995772i	$0.529282\pi$
$150$	0	0
$151$	21.3137	1.73448	0.867242	−	0.497886i	$-0.165890\pi$
$151$	21.3137	1.73448	0.867242	+	0.497886i	$0.165890\pi$
$152$	0	0
$153$	−2.41421	−0.195178
$154$	0	0
$155$	1.00000	0.0803219
$156$	0	0
$157$	−3.00000	−0.239426	−0.119713	−	0.992809i	$-0.538197\pi$
$157$	−3.00000	−0.239426	−0.119713	+	0.992809i	$0.538197\pi$
$158$	0	0
$159$	5.24264	0.415768
$160$	0	0
$161$	−1.82843	−0.144100
$162$	0	0
$163$	−18.9706	−1.48589	−0.742945	−	0.669353i	$-0.766571\pi$
$163$	−18.9706	−1.48589	−0.742945	+	0.669353i	$0.766571\pi$
$164$	0	0
$165$	1.41421	0.110096
$166$	0	0
$167$	1.75736	0.135989	0.0679943	−	0.997686i	$-0.478340\pi$
$167$	1.75736	0.135989	0.0679943	+	0.997686i	$0.478340\pi$
$168$	0	0
$169$	−12.6569	−0.973604
$170$	0	0
$171$	7.41421	0.566979
$172$	0	0
$173$	21.6569	1.64654	0.823270	−	0.567650i	$-0.192147\pi$
$173$	21.6569	1.64654	0.823270	+	0.567650i	$0.192147\pi$
$174$	0	0
$175$	−1.82843	−0.138216
$176$	0	0
$177$	−13.7279	−1.03185
$178$	0	0
$179$	19.6569	1.46922	0.734611	−	0.678488i	$-0.237365\pi$
$179$	19.6569	1.46922	0.734611	+	0.678488i	$0.237365\pi$
$180$	0	0
$181$	−3.51472	−0.261247	−0.130623	−	0.991432i	$-0.541698\pi$
$181$	−3.51472	−0.261247	−0.130623	+	0.991432i	$0.541698\pi$
$182$	0	0
$183$	1.41421	0.104542
$184$	0	0
$185$	−7.48528	−0.550329
$186$	0	0
$187$	−3.41421	−0.249672
$188$	0	0
$189$	−1.82843	−0.132999
$190$	0	0
$191$	−5.75736	−0.416588	−0.208294	−	0.978066i	$-0.566791\pi$
$191$	−5.75736	−0.416588	−0.208294	+	0.978066i	$0.566791\pi$
$192$	0	0
$193$	18.1421	1.30590	0.652950	−	0.757401i	$-0.273531\pi$
$193$	18.1421	1.30590	0.652950	+	0.757401i	$0.273531\pi$
$194$	0	0
$195$	0.585786	0.0419490
$196$	0	0
$197$	−7.17157	−0.510953	−0.255477	−	0.966815i	$-0.582232\pi$
$197$	−7.17157	−0.510953	−0.255477	+	0.966815i	$0.582232\pi$
$198$	0	0
$199$	14.4853	1.02683	0.513417	−	0.858139i	$-0.328379\pi$
$199$	14.4853	1.02683	0.513417	+	0.858139i	$0.328379\pi$
$200$	0	0
$201$	15.4853	1.09225
$202$	0	0
$203$	6.55635	0.460166
$204$	0	0
$205$	7.24264	0.505848
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	10.4853	0.725282
$210$	0	0
$211$	−20.7990	−1.43186	−0.715931	−	0.698171i	$-0.753997\pi$
$211$	−20.7990	−1.43186	−0.715931	+	0.698171i	$0.753997\pi$
$212$	0	0
$213$	14.0711	0.964134
$214$	0	0
$215$	7.65685	0.522193
$216$	0	0
$217$	−1.82843	−0.124122
$218$	0	0
$219$	2.24264	0.151544
$220$	0	0
$221$	−1.41421	−0.0951303
$222$	0	0
$223$	18.9706	1.27036	0.635181	−	0.772363i	$-0.280925\pi$
$223$	18.9706	1.27036	0.635181	+	0.772363i	$0.280925\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	0.686292	0.0455508	0.0227754	−	0.999741i	$-0.492750\pi$
$227$	0.686292	0.0455508	0.0227754	+	0.999741i	$0.492750\pi$
$228$	0	0
$229$	−16.4853	−1.08938	−0.544689	−	0.838638i	$-0.683352\pi$
$229$	−16.4853	−1.08938	−0.544689	+	0.838638i	$0.683352\pi$
$230$	0	0
$231$	−2.58579	−0.170132
$232$	0	0
$233$	−5.65685	−0.370593	−0.185296	−	0.982683i	$-0.559325\pi$
$233$	−5.65685	−0.370593	−0.185296	+	0.982683i	$0.559325\pi$
$234$	0	0
$235$	6.24264	0.407225
$236$	0	0
$237$	−16.9706	−1.10236
$238$	0	0
$239$	−4.41421	−0.285532	−0.142766	−	0.989756i	$-0.545600\pi$
$239$	−4.41421	−0.285532	−0.142766	+	0.989756i	$0.545600\pi$
$240$	0	0
$241$	−15.2132	−0.979969	−0.489984	−	0.871731i	$-0.662997\pi$
$241$	−15.2132	−0.979969	−0.489984	+	0.871731i	$0.662997\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−3.65685	−0.233628
$246$	0	0
$247$	4.34315	0.276348
$248$	0	0
$249$	−2.75736	−0.174741
$250$	0	0
$251$	14.9706	0.944934	0.472467	−	0.881348i	$-0.343364\pi$
$251$	14.9706	0.944934	0.472467	+	0.881348i	$0.343364\pi$
$252$	0	0
$253$	1.41421	0.0889108
$254$	0	0
$255$	−2.41421	−0.151184
$256$	0	0
$257$	−11.4142	−0.711999	−0.356000	−	0.934486i	$-0.615860\pi$
$257$	−11.4142	−0.711999	−0.356000	+	0.934486i	$0.615860\pi$
$258$	0	0
$259$	13.6863	0.850425
$260$	0	0
$261$	−3.58579	−0.221955
$262$	0	0
$263$	−12.4142	−0.765493	−0.382747	−	0.923853i	$-0.625022\pi$
$263$	−12.4142	−0.765493	−0.382747	+	0.923853i	$0.625022\pi$
$264$	0	0
$265$	5.24264	0.322053
$266$	0	0
$267$	−9.17157	−0.561291
$268$	0	0
$269$	12.0711	0.735986	0.367993	−	0.929829i	$-0.380045\pi$
$269$	12.0711	0.735986	0.367993	+	0.929829i	$0.380045\pi$
$270$	0	0
$271$	−18.3137	−1.11248	−0.556239	−	0.831022i	$-0.687756\pi$
$271$	−18.3137	−1.11248	−0.556239	+	0.831022i	$0.687756\pi$
$272$	0	0
$273$	−1.07107	−0.0648240
$274$	0	0
$275$	1.41421	0.0852803
$276$	0	0
$277$	18.9706	1.13983	0.569915	−	0.821703i	$-0.306976\pi$
$277$	18.9706	1.13983	0.569915	+	0.821703i	$0.306976\pi$
$278$	0	0
$279$	1.00000	0.0598684
$280$	0	0
$281$	3.75736	0.224145	0.112073	−	0.993700i	$-0.464251\pi$
$281$	3.75736	0.224145	0.112073	+	0.993700i	$0.464251\pi$
$282$	0	0
$283$	25.6274	1.52339	0.761696	−	0.647935i	$-0.224367\pi$
$283$	25.6274	1.52339	0.761696	+	0.647935i	$0.224367\pi$
$284$	0	0
$285$	7.41421	0.439180
$286$	0	0
$287$	−13.2426	−0.781688
$288$	0	0
$289$	−11.1716	−0.657151
$290$	0	0
$291$	18.1421	1.06351
$292$	0	0
$293$	10.8995	0.636755	0.318378	−	0.947964i	$-0.396862\pi$
$293$	10.8995	0.636755	0.318378	+	0.947964i	$0.396862\pi$
$294$	0	0
$295$	−13.7279	−0.799271
$296$	0	0
$297$	1.41421	0.0820610
$298$	0	0
$299$	0.585786	0.0338769
$300$	0	0
$301$	−14.0000	−0.806947
$302$	0	0
$303$	−10.0711	−0.578568
$304$	0	0
$305$	1.41421	0.0809776
$306$	0	0
$307$	−3.89949	−0.222556	−0.111278	−	0.993789i	$-0.535494\pi$
$307$	−3.89949	−0.222556	−0.111278	+	0.993789i	$0.535494\pi$
$308$	0	0
$309$	1.51472	0.0861693
$310$	0	0
$311$	−31.3137	−1.77564	−0.887819	−	0.460193i	$-0.847780\pi$
$311$	−31.3137	−1.77564	−0.887819	+	0.460193i	$0.847780\pi$
$312$	0	0
$313$	21.1421	1.19502	0.597512	−	0.801860i	$-0.296156\pi$
$313$	21.1421	1.19502	0.597512	+	0.801860i	$0.296156\pi$
$314$	0	0
$315$	−1.82843	−0.103020
$316$	0	0
$317$	−7.41421	−0.416424	−0.208212	−	0.978084i	$-0.566764\pi$
$317$	−7.41421	−0.416424	−0.208212	+	0.978084i	$0.566764\pi$
$318$	0	0
$319$	−5.07107	−0.283925
$320$	0	0
$321$	−3.58579	−0.200139
$322$	0	0
$323$	−17.8995	−0.995955
$324$	0	0
$325$	0.585786	0.0324936
$326$	0	0
$327$	−7.41421	−0.410007
$328$	0	0
$329$	−11.4142	−0.629286
$330$	0	0
$331$	23.8284	1.30973	0.654864	−	0.755746i	$-0.272726\pi$
$331$	23.8284	1.30973	0.654864	+	0.755746i	$0.272726\pi$
$332$	0	0
$333$	−7.48528	−0.410191
$334$	0	0
$335$	15.4853	0.846051
$336$	0	0
$337$	−8.34315	−0.454480	−0.227240	−	0.973839i	$-0.572970\pi$
$337$	−8.34315	−0.454480	−0.227240	+	0.973839i	$0.572970\pi$
$338$	0	0
$339$	18.8995	1.02648
$340$	0	0
$341$	1.41421	0.0765840
$342$	0	0
$343$	19.4853	1.05211
$344$	0	0
$345$	1.00000	0.0538382
$346$	0	0
$347$	−4.48528	−0.240783	−0.120391	−	0.992727i	$-0.538415\pi$
$347$	−4.48528	−0.240783	−0.120391	+	0.992727i	$0.538415\pi$
$348$	0	0
$349$	−15.8284	−0.847276	−0.423638	−	0.905832i	$-0.639247\pi$
$349$	−15.8284	−0.847276	−0.423638	+	0.905832i	$0.639247\pi$
$350$	0	0
$351$	0.585786	0.0312670
$352$	0	0
$353$	−22.0416	−1.17316	−0.586579	−	0.809892i	$-0.699526\pi$
$353$	−22.0416	−1.17316	−0.586579	+	0.809892i	$0.699526\pi$
$354$	0	0
$355$	14.0711	0.746815
$356$	0	0
$357$	4.41421	0.233625
$358$	0	0
$359$	5.07107	0.267641	0.133820	−	0.991006i	$-0.457275\pi$
$359$	5.07107	0.267641	0.133820	+	0.991006i	$0.457275\pi$
$360$	0	0
$361$	35.9706	1.89319
$362$	0	0
$363$	−9.00000	−0.472377
$364$	0	0
$365$	2.24264	0.117385
$366$	0	0
$367$	−3.00000	−0.156599	−0.0782994	−	0.996930i	$-0.524949\pi$
$367$	−3.00000	−0.156599	−0.0782994	+	0.996930i	$0.524949\pi$
$368$	0	0
$369$	7.24264	0.377037
$370$	0	0
$371$	−9.58579	−0.497669
$372$	0	0
$373$	17.6569	0.914237	0.457119	−	0.889406i	$-0.348881\pi$
$373$	17.6569	0.914237	0.457119	+	0.889406i	$0.348881\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−2.10051	−0.108182
$378$	0	0
$379$	14.3431	0.736758	0.368379	−	0.929676i	$-0.379913\pi$
$379$	14.3431	0.736758	0.368379	+	0.929676i	$0.379913\pi$
$380$	0	0
$381$	19.5563	1.00190
$382$	0	0
$383$	−13.7279	−0.701464	−0.350732	−	0.936476i	$-0.614067\pi$
$383$	−13.7279	−0.701464	−0.350732	+	0.936476i	$0.614067\pi$
$384$	0	0
$385$	−2.58579	−0.131784
$386$	0	0
$387$	7.65685	0.389220
$388$	0	0
$389$	−19.4558	−0.986450	−0.493225	−	0.869902i	$-0.664182\pi$
$389$	−19.4558	−0.986450	−0.493225	+	0.869902i	$0.664182\pi$
$390$	0	0
$391$	−2.41421	−0.122092
$392$	0	0
$393$	16.0000	0.807093
$394$	0	0
$395$	−16.9706	−0.853882
$396$	0	0
$397$	−21.6569	−1.08693	−0.543463	−	0.839433i	$-0.682887\pi$
$397$	−21.6569	−1.08693	−0.543463	+	0.839433i	$0.682887\pi$
$398$	0	0
$399$	−13.5563	−0.678666
$400$	0	0
$401$	38.4853	1.92186	0.960932	−	0.276786i	$-0.0892693\pi$
$401$	38.4853	1.92186	0.960932	+	0.276786i	$0.0892693\pi$
$402$	0	0
$403$	0.585786	0.0291801
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−10.5858	−0.524718
$408$	0	0
$409$	19.8284	0.980453	0.490226	−	0.871595i	$-0.336914\pi$
$409$	19.8284	0.980453	0.490226	+	0.871595i	$0.336914\pi$
$410$	0	0
$411$	2.82843	0.139516
$412$	0	0
$413$	25.1005	1.23512
$414$	0	0
$415$	−2.75736	−0.135353
$416$	0	0
$417$	17.4853	0.856258
$418$	0	0
$419$	−35.3553	−1.72722	−0.863611	−	0.504159i	$-0.831802\pi$
$419$	−35.3553	−1.72722	−0.863611	+	0.504159i	$0.831802\pi$
$420$	0	0
$421$	32.8701	1.60199	0.800994	−	0.598672i	$-0.204305\pi$
$421$	32.8701	1.60199	0.800994	+	0.598672i	$0.204305\pi$
$422$	0	0
$423$	6.24264	0.303528
$424$	0	0
$425$	−2.41421	−0.117107
$426$	0	0
$427$	−2.58579	−0.125135
$428$	0	0
$429$	0.828427	0.0399968
$430$	0	0
$431$	−28.4853	−1.37209	−0.686044	−	0.727560i	$-0.740654\pi$
$431$	−28.4853	−1.37209	−0.686044	+	0.727560i	$0.740654\pi$
$432$	0	0
$433$	3.68629	0.177152	0.0885759	−	0.996069i	$-0.471768\pi$
$433$	3.68629	0.177152	0.0885759	+	0.996069i	$0.471768\pi$
$434$	0	0
$435$	−3.58579	−0.171925
$436$	0	0
$437$	7.41421	0.354670
$438$	0	0
$439$	33.7990	1.61314	0.806569	−	0.591140i	$-0.201322\pi$
$439$	33.7990	1.61314	0.806569	+	0.591140i	$0.201322\pi$
$440$	0	0
$441$	−3.65685	−0.174136
$442$	0	0
$443$	−15.8995	−0.755408	−0.377704	−	0.925926i	$-0.623286\pi$
$443$	−15.8995	−0.755408	−0.377704	+	0.925926i	$0.623286\pi$
$444$	0	0
$445$	−9.17157	−0.434774
$446$	0	0
$447$	−2.24264	−0.106073
$448$	0	0
$449$	1.10051	0.0519360	0.0259680	−	0.999663i	$-0.491733\pi$
$449$	1.10051	0.0519360	0.0259680	+	0.999663i	$0.491733\pi$
$450$	0	0
$451$	10.2426	0.482307
$452$	0	0
$453$	21.3137	1.00141
$454$	0	0
$455$	−1.07107	−0.0502124
$456$	0	0
$457$	−10.4558	−0.489104	−0.244552	−	0.969636i	$-0.578641\pi$
$457$	−10.4558	−0.489104	−0.244552	+	0.969636i	$0.578641\pi$
$458$	0	0
$459$	−2.41421	−0.112686
$460$	0	0
$461$	−27.4558	−1.27875	−0.639373	−	0.768897i	$-0.720806\pi$
$461$	−27.4558	−1.27875	−0.639373	+	0.768897i	$0.720806\pi$
$462$	0	0
$463$	−22.0416	−1.02436	−0.512181	−	0.858878i	$-0.671162\pi$
$463$	−22.0416	−1.02436	−0.512181	+	0.858878i	$0.671162\pi$
$464$	0	0
$465$	1.00000	0.0463739
$466$	0	0
$467$	−11.5858	−0.536126	−0.268063	−	0.963401i	$-0.586384\pi$
$467$	−11.5858	−0.536126	−0.268063	+	0.963401i	$0.586384\pi$
$468$	0	0
$469$	−28.3137	−1.30741
$470$	0	0
$471$	−3.00000	−0.138233
$472$	0	0
$473$	10.8284	0.497892
$474$	0	0
$475$	7.41421	0.340187
$476$	0	0
$477$	5.24264	0.240044
$478$	0	0
$479$	−20.1005	−0.918416	−0.459208	−	0.888329i	$-0.651867\pi$
$479$	−20.1005	−0.918416	−0.459208	+	0.888329i	$0.651867\pi$
$480$	0	0
$481$	−4.38478	−0.199929
$482$	0	0
$483$	−1.82843	−0.0831963
$484$	0	0
$485$	18.1421	0.823792
$486$	0	0
$487$	−31.0711	−1.40796	−0.703982	−	0.710218i	$-0.748597\pi$
$487$	−31.0711	−1.40796	−0.703982	+	0.710218i	$0.748597\pi$
$488$	0	0
$489$	−18.9706	−0.857879
$490$	0	0
$491$	−23.1005	−1.04251	−0.521256	−	0.853401i	$-0.674536\pi$
$491$	−23.1005	−1.04251	−0.521256	+	0.853401i	$0.674536\pi$
$492$	0	0
$493$	8.65685	0.389885
$494$	0	0
$495$	1.41421	0.0635642
$496$	0	0
$497$	−25.7279	−1.15406
$498$	0	0
$499$	24.4558	1.09479	0.547397	−	0.836873i	$-0.315619\pi$
$499$	24.4558	1.09479	0.547397	+	0.836873i	$0.315619\pi$
$500$	0	0
$501$	1.75736	0.0785130
$502$	0	0
$503$	−20.7574	−0.925525	−0.462762	−	0.886482i	$-0.653142\pi$
$503$	−20.7574	−0.925525	−0.462762	+	0.886482i	$0.653142\pi$
$504$	0	0
$505$	−10.0711	−0.448157
$506$	0	0
$507$	−12.6569	−0.562111
$508$	0	0
$509$	−20.4853	−0.907994	−0.453997	−	0.891003i	$-0.650002\pi$
$509$	−20.4853	−0.907994	−0.453997	+	0.891003i	$0.650002\pi$
$510$	0	0
$511$	−4.10051	−0.181396
$512$	0	0
$513$	7.41421	0.327346
$514$	0	0
$515$	1.51472	0.0667465
$516$	0	0
$517$	8.82843	0.388274
$518$	0	0
$519$	21.6569	0.950630
$520$	0	0
$521$	−24.7279	−1.08335	−0.541675	−	0.840588i	$-0.682210\pi$
$521$	−24.7279	−1.08335	−0.541675	+	0.840588i	$0.682210\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	−1.82843	−0.0797991
$526$	0	0
$527$	−2.41421	−0.105165
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−13.7279	−0.595741
$532$	0	0
$533$	4.24264	0.183769
$534$	0	0
$535$	−3.58579	−0.155027
$536$	0	0
$537$	19.6569	0.848256
$538$	0	0
$539$	−5.17157	−0.222755
$540$	0	0
$541$	−43.1127	−1.85356	−0.926780	−	0.375605i	$-0.877435\pi$
$541$	−43.1127	−1.85356	−0.926780	+	0.375605i	$0.877435\pi$
$542$	0	0
$543$	−3.51472	−0.150831
$544$	0	0
$545$	−7.41421	−0.317590
$546$	0	0
$547$	−6.34315	−0.271213	−0.135607	−	0.990763i	$-0.543298\pi$
$547$	−6.34315	−0.271213	−0.135607	+	0.990763i	$0.543298\pi$
$548$	0	0
$549$	1.41421	0.0603572
$550$	0	0
$551$	−26.5858	−1.13259
$552$	0	0
$553$	31.0294	1.31951
$554$	0	0
$555$	−7.48528	−0.317732
$556$	0	0
$557$	33.5269	1.42058	0.710290	−	0.703909i	$-0.248564\pi$
$557$	33.5269	1.42058	0.710290	+	0.703909i	$0.248564\pi$
$558$	0	0
$559$	4.48528	0.189707
$560$	0	0
$561$	−3.41421	−0.144148
$562$	0	0
$563$	43.5858	1.83692	0.918461	−	0.395512i	$-0.129433\pi$
$563$	43.5858	1.83692	0.918461	+	0.395512i	$0.129433\pi$
$564$	0	0
$565$	18.8995	0.795108
$566$	0	0
$567$	−1.82843	−0.0767867
$568$	0	0
$569$	18.4853	0.774943	0.387472	−	0.921882i	$-0.373349\pi$
$569$	18.4853	0.774943	0.387472	+	0.921882i	$0.373349\pi$
$570$	0	0
$571$	1.89949	0.0794914	0.0397457	−	0.999210i	$-0.487345\pi$
$571$	1.89949	0.0794914	0.0397457	+	0.999210i	$0.487345\pi$
$572$	0	0
$573$	−5.75736	−0.240517
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	24.2843	1.01097	0.505484	−	0.862836i	$-0.331314\pi$
$577$	24.2843	1.01097	0.505484	+	0.862836i	$0.331314\pi$
$578$	0	0
$579$	18.1421	0.753961
$580$	0	0
$581$	5.04163	0.209162
$582$	0	0
$583$	7.41421	0.307065
$584$	0	0
$585$	0.585786	0.0242193
$586$	0	0
$587$	−36.6274	−1.51178	−0.755888	−	0.654701i	$-0.772794\pi$
$587$	−36.6274	−1.51178	−0.755888	+	0.654701i	$0.772794\pi$
$588$	0	0
$589$	7.41421	0.305497
$590$	0	0
$591$	−7.17157	−0.294999
$592$	0	0
$593$	−6.92893	−0.284537	−0.142269	−	0.989828i	$-0.545440\pi$
$593$	−6.92893	−0.284537	−0.142269	+	0.989828i	$0.545440\pi$
$594$	0	0
$595$	4.41421	0.180965
$596$	0	0
$597$	14.4853	0.592843
$598$	0	0
$599$	18.1421	0.741268	0.370634	−	0.928779i	$-0.379141\pi$
$599$	18.1421	0.741268	0.370634	+	0.928779i	$0.379141\pi$
$600$	0	0
$601$	22.6569	0.924192	0.462096	−	0.886830i	$-0.347097\pi$
$601$	22.6569	0.924192	0.462096	+	0.886830i	$0.347097\pi$
$602$	0	0
$603$	15.4853	0.630609
$604$	0	0
$605$	−9.00000	−0.365902
$606$	0	0
$607$	12.7279	0.516610	0.258305	−	0.966063i	$-0.416836\pi$
$607$	12.7279	0.516610	0.258305	+	0.966063i	$0.416836\pi$
$608$	0	0
$609$	6.55635	0.265677
$610$	0	0
$611$	3.65685	0.147940
$612$	0	0
$613$	−4.62742	−0.186900	−0.0934498	−	0.995624i	$-0.529789\pi$
$613$	−4.62742	−0.186900	−0.0934498	+	0.995624i	$0.529789\pi$
$614$	0	0
$615$	7.24264	0.292051
$616$	0	0
$617$	17.2426	0.694163	0.347081	−	0.937835i	$-0.387173\pi$
$617$	17.2426	0.694163	0.347081	+	0.937835i	$0.387173\pi$
$618$	0	0
$619$	−16.2843	−0.654520	−0.327260	−	0.944934i	$-0.606125\pi$
$619$	−16.2843	−0.654520	−0.327260	+	0.944934i	$0.606125\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	16.7696	0.671858
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	10.4853	0.418742
$628$	0	0
$629$	18.0711	0.720541
$630$	0	0
$631$	−1.41421	−0.0562990	−0.0281495	−	0.999604i	$-0.508961\pi$
$631$	−1.41421	−0.0562990	−0.0281495	+	0.999604i	$0.508961\pi$
$632$	0	0
$633$	−20.7990	−0.826686
$634$	0	0
$635$	19.5563	0.776070
$636$	0	0
$637$	−2.14214	−0.0848745
$638$	0	0
$639$	14.0711	0.556643
$640$	0	0
$641$	−21.4142	−0.845811	−0.422905	−	0.906174i	$-0.638990\pi$
$641$	−21.4142	−0.845811	−0.422905	+	0.906174i	$0.638990\pi$
$642$	0	0
$643$	−31.3431	−1.23605	−0.618027	−	0.786157i	$-0.712068\pi$
$643$	−31.3431	−1.23605	−0.618027	+	0.786157i	$0.712068\pi$
$644$	0	0
$645$	7.65685	0.301488
$646$	0	0
$647$	2.38478	0.0937552	0.0468776	−	0.998901i	$-0.485073\pi$
$647$	2.38478	0.0937552	0.0468776	+	0.998901i	$0.485073\pi$
$648$	0	0
$649$	−19.4142	−0.762075
$650$	0	0
$651$	−1.82843	−0.0716617
$652$	0	0
$653$	−16.7279	−0.654614	−0.327307	−	0.944918i	$-0.606141\pi$
$653$	−16.7279	−0.654614	−0.327307	+	0.944918i	$0.606141\pi$
$654$	0	0
$655$	16.0000	0.625172
$656$	0	0
$657$	2.24264	0.0874937
$658$	0	0
$659$	6.38478	0.248716	0.124358	−	0.992237i	$-0.460313\pi$
$659$	6.38478	0.248716	0.124358	+	0.992237i	$0.460313\pi$
$660$	0	0
$661$	13.1127	0.510025	0.255012	−	0.966938i	$-0.417920\pi$
$661$	13.1127	0.510025	0.255012	+	0.966938i	$0.417920\pi$
$662$	0	0
$663$	−1.41421	−0.0549235
$664$	0	0
$665$	−13.5563	−0.525693
$666$	0	0
$667$	−3.58579	−0.138842
$668$	0	0
$669$	18.9706	0.733444
$670$	0	0
$671$	2.00000	0.0772091
$672$	0	0
$673$	7.27208	0.280318	0.140159	−	0.990129i	$-0.455239\pi$
$673$	7.27208	0.280318	0.140159	+	0.990129i	$0.455239\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−43.1838	−1.65969	−0.829843	−	0.557996i	$-0.811570\pi$
$677$	−43.1838	−1.65969	−0.829843	+	0.557996i	$0.811570\pi$
$678$	0	0
$679$	−33.1716	−1.27301
$680$	0	0
$681$	0.686292	0.0262987
$682$	0	0
$683$	−43.0122	−1.64582	−0.822908	−	0.568175i	$-0.807650\pi$
$683$	−43.0122	−1.64582	−0.822908	+	0.568175i	$0.807650\pi$
$684$	0	0
$685$	2.82843	0.108069
$686$	0	0
$687$	−16.4853	−0.628953
$688$	0	0
$689$	3.07107	0.116998
$690$	0	0
$691$	14.9706	0.569507	0.284754	−	0.958601i	$-0.408088\pi$
$691$	14.9706	0.569507	0.284754	+	0.958601i	$0.408088\pi$
$692$	0	0
$693$	−2.58579	−0.0982259
$694$	0	0
$695$	17.4853	0.663255
$696$	0	0
$697$	−17.4853	−0.662302
$698$	0	0
$699$	−5.65685	−0.213962
$700$	0	0
$701$	49.5563	1.87172	0.935859	−	0.352375i	$-0.114626\pi$
$701$	49.5563	1.87172	0.935859	+	0.352375i	$0.114626\pi$
$702$	0	0
$703$	−55.4975	−2.09313
$704$	0	0
$705$	6.24264	0.235111
$706$	0	0
$707$	18.4142	0.692538
$708$	0	0
$709$	17.6985	0.664681	0.332340	−	0.943160i	$-0.392162\pi$
$709$	17.6985	0.664681	0.332340	+	0.943160i	$0.392162\pi$
$710$	0	0
$711$	−16.9706	−0.636446
$712$	0	0
$713$	1.00000	0.0374503
$714$	0	0
$715$	0.828427	0.0309814
$716$	0	0
$717$	−4.41421	−0.164852
$718$	0	0
$719$	−32.5563	−1.21415	−0.607073	−	0.794646i	$-0.707657\pi$
$719$	−32.5563	−1.21415	−0.607073	+	0.794646i	$0.707657\pi$
$720$	0	0
$721$	−2.76955	−0.103144
$722$	0	0
$723$	−15.2132	−0.565785
$724$	0	0
$725$	−3.58579	−0.133173
$726$	0	0
$727$	−9.00000	−0.333792	−0.166896	−	0.985975i	$-0.553374\pi$
$727$	−9.00000	−0.333792	−0.166896	+	0.985975i	$0.553374\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−18.4853	−0.683703
$732$	0	0
$733$	16.3137	0.602561	0.301280	−	0.953536i	$-0.402586\pi$
$733$	16.3137	0.602561	0.301280	+	0.953536i	$0.402586\pi$
$734$	0	0
$735$	−3.65685	−0.134885
$736$	0	0
$737$	21.8995	0.806678
$738$	0	0
$739$	−17.1421	−0.630584	−0.315292	−	0.948995i	$-0.602102\pi$
$739$	−17.1421	−0.630584	−0.315292	+	0.948995i	$0.602102\pi$
$740$	0	0
$741$	4.34315	0.159549
$742$	0	0
$743$	−4.00000	−0.146746	−0.0733729	−	0.997305i	$-0.523376\pi$
$743$	−4.00000	−0.146746	−0.0733729	+	0.997305i	$0.523376\pi$
$744$	0	0
$745$	−2.24264	−0.0821640
$746$	0	0
$747$	−2.75736	−0.100887
$748$	0	0
$749$	6.55635	0.239564
$750$	0	0
$751$	45.0711	1.64467	0.822333	−	0.569006i	$-0.192672\pi$
$751$	45.0711	1.64467	0.822333	+	0.569006i	$0.192672\pi$
$752$	0	0
$753$	14.9706	0.545558
$754$	0	0
$755$	21.3137	0.775685
$756$	0	0
$757$	−1.82843	−0.0664553	−0.0332277	−	0.999448i	$-0.510579\pi$
$757$	−1.82843	−0.0664553	−0.0332277	+	0.999448i	$0.510579\pi$
$758$	0	0
$759$	1.41421	0.0513327
$760$	0	0
$761$	−23.9289	−0.867423	−0.433712	−	0.901052i	$-0.642796\pi$
$761$	−23.9289	−0.867423	−0.433712	+	0.901052i	$0.642796\pi$
$762$	0	0
$763$	13.5563	0.490773
$764$	0	0
$765$	−2.41421	−0.0872861
$766$	0	0
$767$	−8.04163	−0.290366
$768$	0	0
$769$	−15.2721	−0.550725	−0.275363	−	0.961340i	$-0.588798\pi$
$769$	−15.2721	−0.550725	−0.275363	+	0.961340i	$0.588798\pi$
$770$	0	0
$771$	−11.4142	−0.411073
$772$	0	0
$773$	−13.7990	−0.496315	−0.248158	−	0.968720i	$-0.579825\pi$
$773$	−13.7990	−0.496315	−0.248158	+	0.968720i	$0.579825\pi$
$774$	0	0
$775$	1.00000	0.0359211
$776$	0	0
$777$	13.6863	0.490993
$778$	0	0
$779$	53.6985	1.92395
$780$	0	0
$781$	19.8995	0.712060
$782$	0	0
$783$	−3.58579	−0.128146
$784$	0	0
$785$	−3.00000	−0.107075
$786$	0	0
$787$	19.2843	0.687410	0.343705	−	0.939078i	$-0.388318\pi$
$787$	19.2843	0.687410	0.343705	+	0.939078i	$0.388318\pi$
$788$	0	0
$789$	−12.4142	−0.441958
$790$	0	0
$791$	−34.5563	−1.22868
$792$	0	0
$793$	0.828427	0.0294183
$794$	0	0
$795$	5.24264	0.185937
$796$	0	0
$797$	−34.8995	−1.23620	−0.618102	−	0.786098i	$-0.712098\pi$
$797$	−34.8995	−1.23620	−0.618102	+	0.786098i	$0.712098\pi$
$798$	0	0
$799$	−15.0711	−0.533176
$800$	0	0
$801$	−9.17157	−0.324062
$802$	0	0
$803$	3.17157	0.111922
$804$	0	0
$805$	−1.82843	−0.0644436
$806$	0	0
$807$	12.0711	0.424922
$808$	0	0
$809$	29.7279	1.04518	0.522589	−	0.852585i	$-0.324966\pi$
$809$	29.7279	1.04518	0.522589	+	0.852585i	$0.324966\pi$
$810$	0	0
$811$	5.14214	0.180565	0.0902824	−	0.995916i	$-0.471223\pi$
$811$	5.14214	0.180565	0.0902824	+	0.995916i	$0.471223\pi$
$812$	0	0
$813$	−18.3137	−0.642290
$814$	0	0
$815$	−18.9706	−0.664510
$816$	0	0
$817$	56.7696	1.98612
$818$	0	0
$819$	−1.07107	−0.0374261
$820$	0	0
$821$	33.4558	1.16762	0.583809	−	0.811891i	$-0.301562\pi$
$821$	33.4558	1.16762	0.583809	+	0.811891i	$0.301562\pi$
$822$	0	0
$823$	53.6569	1.87036	0.935180	−	0.354172i	$-0.115237\pi$
$823$	53.6569	1.87036	0.935180	+	0.354172i	$0.115237\pi$
$824$	0	0
$825$	1.41421	0.0492366
$826$	0	0
$827$	−2.27208	−0.0790079	−0.0395039	−	0.999219i	$-0.512578\pi$
$827$	−2.27208	−0.0790079	−0.0395039	+	0.999219i	$0.512578\pi$
$828$	0	0
$829$	−7.82843	−0.271893	−0.135946	−	0.990716i	$-0.543407\pi$
$829$	−7.82843	−0.271893	−0.135946	+	0.990716i	$0.543407\pi$
$830$	0	0
$831$	18.9706	0.658082
$832$	0	0
$833$	8.82843	0.305887
$834$	0	0
$835$	1.75736	0.0608159
$836$	0	0
$837$	1.00000	0.0345651
$838$	0	0
$839$	−28.3431	−0.978514	−0.489257	−	0.872140i	$-0.662732\pi$
$839$	−28.3431	−0.978514	−0.489257	+	0.872140i	$0.662732\pi$
$840$	0	0
$841$	−16.1421	−0.556625
$842$	0	0
$843$	3.75736	0.129410
$844$	0	0
$845$	−12.6569	−0.435409
$846$	0	0
$847$	16.4558	0.565429
$848$	0	0
$849$	25.6274	0.879531
$850$	0	0
$851$	−7.48528	−0.256592
$852$	0	0
$853$	6.48528	0.222052	0.111026	−	0.993818i	$-0.464586\pi$
$853$	6.48528	0.222052	0.111026	+	0.993818i	$0.464586\pi$
$854$	0	0
$855$	7.41421	0.253561
$856$	0	0
$857$	−30.6863	−1.04822	−0.524112	−	0.851649i	$-0.675603\pi$
$857$	−30.6863	−1.04822	−0.524112	+	0.851649i	$0.675603\pi$
$858$	0	0
$859$	53.8284	1.83660	0.918301	−	0.395883i	$-0.129561\pi$
$859$	53.8284	1.83660	0.918301	+	0.395883i	$0.129561\pi$
$860$	0	0
$861$	−13.2426	−0.451308
$862$	0	0
$863$	−46.4853	−1.58238	−0.791189	−	0.611572i	$-0.790537\pi$
$863$	−46.4853	−1.58238	−0.791189	+	0.611572i	$0.790537\pi$
$864$	0	0
$865$	21.6569	0.736355
$866$	0	0
$867$	−11.1716	−0.379407
$868$	0	0
$869$	−24.0000	−0.814144
$870$	0	0
$871$	9.07107	0.307361
$872$	0	0
$873$	18.1421	0.614018
$874$	0	0
$875$	−1.82843	−0.0618121
$876$	0	0
$877$	−7.17157	−0.242167	−0.121083	−	0.992642i	$-0.538637\pi$
$877$	−7.17157	−0.242167	−0.121083	+	0.992642i	$0.538637\pi$
$878$	0	0
$879$	10.8995	0.367631
$880$	0	0
$881$	−10.9289	−0.368205	−0.184103	−	0.982907i	$-0.558938\pi$
$881$	−10.9289	−0.368205	−0.184103	+	0.982907i	$0.558938\pi$
$882$	0	0
$883$	32.9289	1.10815	0.554073	−	0.832468i	$-0.313073\pi$
$883$	32.9289	1.10815	0.554073	+	0.832468i	$0.313073\pi$
$884$	0	0
$885$	−13.7279	−0.461459
$886$	0	0
$887$	4.82843	0.162123	0.0810614	−	0.996709i	$-0.474169\pi$
$887$	4.82843	0.162123	0.0810614	+	0.996709i	$0.474169\pi$
$888$	0	0
$889$	−35.7574	−1.19926
$890$	0	0
$891$	1.41421	0.0473779
$892$	0	0
$893$	46.2843	1.54884
$894$	0	0
$895$	19.6569	0.657056
$896$	0	0
$897$	0.585786	0.0195588
$898$	0	0
$899$	−3.58579	−0.119593
$900$	0	0
$901$	−12.6569	−0.421661
$902$	0	0
$903$	−14.0000	−0.465891
$904$	0	0
$905$	−3.51472	−0.116833
$906$	0	0
$907$	−18.1716	−0.603377	−0.301689	−	0.953407i	$-0.597550\pi$
$907$	−18.1716	−0.603377	−0.301689	+	0.953407i	$0.597550\pi$
$908$	0	0
$909$	−10.0711	−0.334036
$910$	0	0
$911$	−28.9706	−0.959838	−0.479919	−	0.877313i	$-0.659334\pi$
$911$	−28.9706	−0.959838	−0.479919	+	0.877313i	$0.659334\pi$
$912$	0	0
$913$	−3.89949	−0.129054
$914$	0	0
$915$	1.41421	0.0467525
$916$	0	0
$917$	−29.2548	−0.966080
$918$	0	0
$919$	−24.2843	−0.801064	−0.400532	−	0.916283i	$-0.631175\pi$
$919$	−24.2843	−0.801064	−0.400532	+	0.916283i	$0.631175\pi$
$920$	0	0
$921$	−3.89949	−0.128493
$922$	0	0
$923$	8.24264	0.271310
$924$	0	0
$925$	−7.48528	−0.246115
$926$	0	0
$927$	1.51472	0.0497499
$928$	0	0
$929$	−32.7574	−1.07473	−0.537367	−	0.843348i	$-0.680581\pi$
$929$	−32.7574	−1.07473	−0.537367	+	0.843348i	$0.680581\pi$
$930$	0	0
$931$	−27.1127	−0.888583
$932$	0	0
$933$	−31.3137	−1.02516
$934$	0	0
$935$	−3.41421	−0.111657
$936$	0	0
$937$	16.4853	0.538551	0.269275	−	0.963063i	$-0.413216\pi$
$937$	16.4853	0.538551	0.269275	+	0.963063i	$0.413216\pi$
$938$	0	0
$939$	21.1421	0.689948
$940$	0	0
$941$	6.52691	0.212771	0.106386	−	0.994325i	$-0.466072\pi$
$941$	6.52691	0.212771	0.106386	+	0.994325i	$0.466072\pi$
$942$	0	0
$943$	7.24264	0.235853
$944$	0	0
$945$	−1.82843	−0.0594787
$946$	0	0
$947$	28.4853	0.925647	0.462824	−	0.886450i	$-0.346836\pi$
$947$	28.4853	0.925647	0.462824	+	0.886450i	$0.346836\pi$
$948$	0	0
$949$	1.31371	0.0426448
$950$	0	0
$951$	−7.41421	−0.240422
$952$	0	0
$953$	−4.62742	−0.149897	−0.0749484	−	0.997187i	$-0.523879\pi$
$953$	−4.62742	−0.149897	−0.0749484	+	0.997187i	$0.523879\pi$
$954$	0	0
$955$	−5.75736	−0.186304
$956$	0	0
$957$	−5.07107	−0.163924
$958$	0	0
$959$	−5.17157	−0.166999
$960$	0	0
$961$	−30.0000	−0.967742
$962$	0	0
$963$	−3.58579	−0.115550
$964$	0	0
$965$	18.1421	0.584016
$966$	0	0
$967$	−42.3848	−1.36300	−0.681501	−	0.731817i	$-0.738673\pi$
$967$	−42.3848	−1.36300	−0.681501	+	0.731817i	$0.738673\pi$
$968$	0	0
$969$	−17.8995	−0.575015
$970$	0	0
$971$	34.4264	1.10480	0.552398	−	0.833581i	$-0.313713\pi$
$971$	34.4264	1.10480	0.552398	+	0.833581i	$0.313713\pi$
$972$	0	0
$973$	−31.9706	−1.02493
$974$	0	0
$975$	0.585786	0.0187602
$976$	0	0
$977$	−49.8701	−1.59548	−0.797742	−	0.602999i	$-0.793972\pi$
$977$	−49.8701	−1.59548	−0.797742	+	0.602999i	$0.793972\pi$
$978$	0	0
$979$	−12.9706	−0.414541
$980$	0	0
$981$	−7.41421	−0.236718
$982$	0	0
$983$	37.5269	1.19692	0.598461	−	0.801152i	$-0.295779\pi$
$983$	37.5269	1.19692	0.598461	+	0.801152i	$0.295779\pi$
$984$	0	0
$985$	−7.17157	−0.228505
$986$	0	0
$987$	−11.4142	−0.363318
$988$	0	0
$989$	7.65685	0.243474
$990$	0	0
$991$	−50.4558	−1.60278	−0.801391	−	0.598140i	$-0.795907\pi$
$991$	−50.4558	−1.60278	−0.801391	+	0.598140i	$0.795907\pi$
$992$	0	0
$993$	23.8284	0.756172
$994$	0	0
$995$	14.4853	0.459214
$996$	0	0
$997$	19.5147	0.618037	0.309019	−	0.951056i	$-0.399999\pi$
$997$	19.5147	0.618037	0.309019	+	0.951056i	$0.399999\pi$
$998$	0	0
$999$	−7.48528	−0.236824

Twists

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

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} -1.82843 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} -1.82843 q^{7} +1.00000 q^{9} +1.41421 q^{11} +0.585786 q^{13} +1.00000 q^{15} -2.41421 q^{17} +7.41421 q^{19} -1.82843 q^{21} +1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.58579 q^{29} +1.00000 q^{31} +1.41421 q^{33} -1.82843 q^{35} -7.48528 q^{37} +0.585786 q^{39} +7.24264 q^{41} +7.65685 q^{43} +1.00000 q^{45} +6.24264 q^{47} -3.65685 q^{49} -2.41421 q^{51} +5.24264 q^{53} +1.41421 q^{55} +7.41421 q^{57} -13.7279 q^{59} +1.41421 q^{61} -1.82843 q^{63} +0.585786 q^{65} +15.4853 q^{67} +1.00000 q^{69} +14.0711 q^{71} +2.24264 q^{73} +1.00000 q^{75} -2.58579 q^{77} -16.9706 q^{79} +1.00000 q^{81} -2.75736 q^{83} -2.41421 q^{85} -3.58579 q^{87} -9.17157 q^{89} -1.07107 q^{91} +1.00000 q^{93} +7.41421 q^{95} +18.1421 q^{97} +1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} + 4 q^{13} + 2 q^{15} - 2 q^{17} + 12 q^{19} + 2 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} - 10 q^{29} + 2 q^{31} + 2 q^{35} + 2 q^{37} + 4 q^{39} + 6 q^{41} + 4 q^{43} + 2 q^{45} + 4 q^{47} + 4 q^{49} - 2 q^{51} + 2 q^{53} + 12 q^{57} - 2 q^{59} + 2 q^{63} + 4 q^{65} + 14 q^{67} + 2 q^{69} + 14 q^{71} - 4 q^{73} + 2 q^{75} - 8 q^{77} + 2 q^{81} - 14 q^{83} - 2 q^{85} - 10 q^{87} - 24 q^{89} + 12 q^{91} + 2 q^{93} + 12 q^{95} + 8 q^{97}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 + 4 * q^13 + 2 * q^15 - 2 * q^17 + 12 * q^19 + 2 * q^21 + 2 * q^23 + 2 * q^25 + 2 * q^27 - 10 * q^29 + 2 * q^31 + 2 * q^35 + 2 * q^37 + 4 * q^39 + 6 * q^41 + 4 * q^43 + 2 * q^45 + 4 * q^47 + 4 * q^49 - 2 * q^51 + 2 * q^53 + 12 * q^57 - 2 * q^59 + 2 * q^63 + 4 * q^65 + 14 * q^67 + 2 * q^69 + 14 * q^71 - 4 * q^73 + 2 * q^75 - 8 * q^77 + 2 * q^81 - 14 * q^83 - 2 * q^85 - 10 * q^87 - 24 * q^89 + 12 * q^91 + 2 * q^93 + 12 * q^95 + 8 * q^97

Introduction

L-functions

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Database

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