LMFDB - Embedded newform 4560.2.a.bv.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4560,2,Mod(1,4560)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4560, 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("4560.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4560.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:	$36.4117833217$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2280)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.34292$ of defining polynomial
Character	$\chi$	$=$	4560.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} -3.83221 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} -3.83221 q^{7} +1.00000 q^{9} +1.14637 q^{11} -3.53948 q^{13} +1.00000 q^{15} -6.97858 q^{17} +1.00000 q^{19} -3.83221 q^{21} +8.97858 q^{23} +1.00000 q^{25} +1.00000 q^{27} +0.853635 q^{29} +4.39312 q^{31} +1.14637 q^{33} -3.83221 q^{35} +1.83221 q^{37} -3.53948 q^{39} -8.51806 q^{41} +10.1249 q^{43} +1.00000 q^{45} -0.978577 q^{47} +7.68585 q^{49} -6.97858 q^{51} +6.97858 q^{53} +1.14637 q^{55} +1.00000 q^{57} +6.29273 q^{59} +10.0575 q^{61} -3.83221 q^{63} -3.53948 q^{65} -0.585462 q^{67} +8.97858 q^{69} +11.0790 q^{71} +7.37169 q^{73} +1.00000 q^{75} -4.39312 q^{77} +1.00000 q^{81} -12.6430 q^{83} -6.97858 q^{85} +0.853635 q^{87} +7.93260 q^{89} +13.5640 q^{91} +4.39312 q^{93} +1.00000 q^{95} +1.24675 q^{97} +1.14637 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 2 q^{11} + 3 q^{15} - 6 q^{17} + 3 q^{19} + 2 q^{21} + 12 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} + 4 q^{31} + 2 q^{33} + 2 q^{35} - 8 q^{37} + 14 q^{43} + 3 q^{45} + 12 q^{47} + 11 q^{49} - 6 q^{51} + 6 q^{53} + 2 q^{55} + 3 q^{57} + 16 q^{59} - 6 q^{61} + 2 q^{63} + 4 q^{67} + 12 q^{69} + 12 q^{71} - 2 q^{73} + 3 q^{75} - 4 q^{77} + 3 q^{81} + 4 q^{83} - 6 q^{85} + 4 q^{87} + 4 q^{89} + 20 q^{91} + 4 q^{93} + 3 q^{95} - 4 q^{97} + 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^5 + 2 * q^7 + 3 * q^9 + 2 * q^11 + 3 * q^15 - 6 * q^17 + 3 * q^19 + 2 * q^21 + 12 * q^23 + 3 * q^25 + 3 * q^27 + 4 * q^29 + 4 * q^31 + 2 * q^33 + 2 * q^35 - 8 * q^37 + 14 * q^43 + 3 * q^45 + 12 * q^47 + 11 * q^49 - 6 * q^51 + 6 * q^53 + 2 * q^55 + 3 * q^57 + 16 * q^59 - 6 * q^61 + 2 * q^63 + 4 * q^67 + 12 * q^69 + 12 * q^71 - 2 * q^73 + 3 * q^75 - 4 * q^77 + 3 * q^81 + 4 * q^83 - 6 * q^85 + 4 * q^87 + 4 * q^89 + 20 * q^91 + 4 * q^93 + 3 * q^95 - 4 * q^97 + 2 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−3.83221	−1.44844	−0.724220	−	0.689569i	$-0.757800\pi$
$7$	−3.83221	−1.44844	−0.724220	+	0.689569i	$0.757800\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.14637	0.345642	0.172821	−	0.984953i	$-0.444712\pi$
$11$	1.14637	0.345642	0.172821	+	0.984953i	$0.444712\pi$
$12$	0	0
$13$	−3.53948	−0.981675	−0.490838	−	0.871251i	$-0.663309\pi$
$13$	−3.53948	−0.981675	−0.490838	+	0.871251i	$0.663309\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	−6.97858	−1.69255	−0.846277	−	0.532743i	$-0.821161\pi$
$17$	−6.97858	−1.69255	−0.846277	+	0.532743i	$0.821161\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−3.83221	−0.836257
$22$	0	0
$23$	8.97858	1.87216	0.936081	−	0.351784i	$-0.114425\pi$
$23$	8.97858	1.87216	0.936081	+	0.351784i	$0.114425\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	0.853635	0.158516	0.0792580	−	0.996854i	$-0.474745\pi$
$29$	0.853635	0.158516	0.0792580	+	0.996854i	$0.474745\pi$
$30$	0	0
$31$	4.39312	0.789027	0.394513	−	0.918890i	$-0.370913\pi$
$31$	4.39312	0.789027	0.394513	+	0.918890i	$0.370913\pi$
$32$	0	0
$33$	1.14637	0.199557
$34$	0	0
$35$	−3.83221	−0.647762
$36$	0	0
$37$	1.83221	0.301214	0.150607	−	0.988594i	$-0.451877\pi$
$37$	1.83221	0.301214	0.150607	+	0.988594i	$0.451877\pi$
$38$	0	0
$39$	−3.53948	−0.566771
$40$	0	0
$41$	−8.51806	−1.33030	−0.665149	−	0.746711i	$-0.731632\pi$
$41$	−8.51806	−1.33030	−0.665149	+	0.746711i	$0.731632\pi$
$42$	0	0
$43$	10.1249	1.54404	0.772020	−	0.635599i	$-0.219247\pi$
$43$	10.1249	1.54404	0.772020	+	0.635599i	$0.219247\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	−0.978577	−0.142740	−0.0713701	−	0.997450i	$-0.522737\pi$
$47$	−0.978577	−0.142740	−0.0713701	+	0.997450i	$0.522737\pi$
$48$	0	0
$49$	7.68585	1.09798
$50$	0	0
$51$	−6.97858	−0.977196
$52$	0	0
$53$	6.97858	0.958581	0.479291	−	0.877656i	$-0.340894\pi$
$53$	6.97858	0.958581	0.479291	+	0.877656i	$0.340894\pi$
$54$	0	0
$55$	1.14637	0.154576
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	6.29273	0.819244	0.409622	−	0.912255i	$-0.365661\pi$
$59$	6.29273	0.819244	0.409622	+	0.912255i	$0.365661\pi$
$60$	0	0
$61$	10.0575	1.28774	0.643868	−	0.765137i	$-0.277329\pi$
$61$	10.0575	1.28774	0.643868	+	0.765137i	$0.277329\pi$
$62$	0	0
$63$	−3.83221	−0.482813
$64$	0	0
$65$	−3.53948	−0.439019
$66$	0	0
$67$	−0.585462	−0.0715256	−0.0357628	−	0.999360i	$-0.511386\pi$
$67$	−0.585462	−0.0715256	−0.0357628	+	0.999360i	$0.511386\pi$
$68$	0	0
$69$	8.97858	1.08089
$70$	0	0
$71$	11.0790	1.31483	0.657415	−	0.753528i	$-0.271650\pi$
$71$	11.0790	1.31483	0.657415	+	0.753528i	$0.271650\pi$
$72$	0	0
$73$	7.37169	0.862791	0.431396	−	0.902163i	$-0.358021\pi$
$73$	7.37169	0.862791	0.431396	+	0.902163i	$0.358021\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−4.39312	−0.500642
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−12.6430	−1.38775	−0.693875	−	0.720096i	$-0.744098\pi$
$83$	−12.6430	−1.38775	−0.693875	+	0.720096i	$0.744098\pi$
$84$	0	0
$85$	−6.97858	−0.756933
$86$	0	0
$87$	0.853635	0.0915192
$88$	0	0
$89$	7.93260	0.840853	0.420427	−	0.907326i	$-0.361880\pi$
$89$	7.93260	0.840853	0.420427	+	0.907326i	$0.361880\pi$
$90$	0	0
$91$	13.5640	1.42190
$92$	0	0
$93$	4.39312	0.455545
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	1.24675	0.126588	0.0632941	−	0.997995i	$-0.479839\pi$
$97$	1.24675	0.126588	0.0632941	+	0.997995i	$0.479839\pi$
$98$	0	0
$99$	1.14637	0.115214
$100$	0	0
$101$	9.07896	0.903390	0.451695	−	0.892172i	$-0.350819\pi$
$101$	9.07896	0.903390	0.451695	+	0.892172i	$0.350819\pi$
$102$	0	0
$103$	18.7434	1.84684	0.923420	−	0.383790i	$-0.125381\pi$
$103$	18.7434	1.84684	0.923420	+	0.383790i	$0.125381\pi$
$104$	0	0
$105$	−3.83221	−0.373986
$106$	0	0
$107$	−1.37169	−0.132607	−0.0663033	−	0.997800i	$-0.521120\pi$
$107$	−1.37169	−0.132607	−0.0663033	+	0.997800i	$0.521120\pi$
$108$	0	0
$109$	4.87819	0.467246	0.233623	−	0.972327i	$-0.424942\pi$
$109$	4.87819	0.467246	0.233623	+	0.972327i	$0.424942\pi$
$110$	0	0
$111$	1.83221	0.173906
$112$	0	0
$113$	−12.6858	−1.19338	−0.596692	−	0.802470i	$-0.703519\pi$
$113$	−12.6858	−1.19338	−0.596692	+	0.802470i	$0.703519\pi$
$114$	0	0
$115$	8.97858	0.837257
$116$	0	0
$117$	−3.53948	−0.327225
$118$	0	0
$119$	26.7434	2.45156
$120$	0	0
$121$	−9.68585	−0.880531
$122$	0	0
$123$	−8.51806	−0.768047
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	2.62831	0.233225	0.116612	−	0.993178i	$-0.462797\pi$
$127$	2.62831	0.233225	0.116612	+	0.993178i	$0.462797\pi$
$128$	0	0
$129$	10.1249	0.891451
$130$	0	0
$131$	19.8898	1.73778	0.868888	−	0.495009i	$-0.164835\pi$
$131$	19.8898	1.73778	0.868888	+	0.495009i	$0.164835\pi$
$132$	0	0
$133$	−3.83221	−0.332295
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−19.3717	−1.65504	−0.827518	−	0.561440i	$-0.810248\pi$
$137$	−19.3717	−1.65504	−0.827518	+	0.561440i	$0.810248\pi$
$138$	0	0
$139$	−9.70727	−0.823360	−0.411680	−	0.911329i	$-0.635058\pi$
$139$	−9.70727	−0.823360	−0.411680	+	0.911329i	$0.635058\pi$
$140$	0	0
$141$	−0.978577	−0.0824111
$142$	0	0
$143$	−4.05754	−0.339308
$144$	0	0
$145$	0.853635	0.0708905
$146$	0	0
$147$	7.68585	0.633918
$148$	0	0
$149$	21.3288	1.74733	0.873664	−	0.486530i	$-0.161738\pi$
$149$	21.3288	1.74733	0.873664	+	0.486530i	$0.161738\pi$
$150$	0	0
$151$	−14.3503	−1.16781	−0.583904	−	0.811823i	$-0.698476\pi$
$151$	−14.3503	−1.16781	−0.583904	+	0.811823i	$0.698476\pi$
$152$	0	0
$153$	−6.97858	−0.564185
$154$	0	0
$155$	4.39312	0.352864
$156$	0	0
$157$	1.07896	0.0861105	0.0430552	−	0.999073i	$-0.486291\pi$
$157$	1.07896	0.0861105	0.0430552	+	0.999073i	$0.486291\pi$
$158$	0	0
$159$	6.97858	0.553437
$160$	0	0
$161$	−34.4078	−2.71172
$162$	0	0
$163$	−2.46052	−0.192723	−0.0963614	−	0.995346i	$-0.530720\pi$
$163$	−2.46052	−0.192723	−0.0963614	+	0.995346i	$0.530720\pi$
$164$	0	0
$165$	1.14637	0.0892444
$166$	0	0
$167$	−4.05754	−0.313982	−0.156991	−	0.987600i	$-0.550179\pi$
$167$	−4.05754	−0.313982	−0.156991	+	0.987600i	$0.550179\pi$
$168$	0	0
$169$	−0.472077	−0.0363136
$170$	0	0
$171$	1.00000	0.0764719
$172$	0	0
$173$	−8.68585	−0.660373	−0.330186	−	0.943916i	$-0.607112\pi$
$173$	−8.68585	−0.660373	−0.330186	+	0.943916i	$0.607112\pi$
$174$	0	0
$175$	−3.83221	−0.289688
$176$	0	0
$177$	6.29273	0.472991
$178$	0	0
$179$	11.6644	0.871840	0.435920	−	0.899985i	$-0.356423\pi$
$179$	11.6644	0.871840	0.435920	+	0.899985i	$0.356423\pi$
$180$	0	0
$181$	10.2499	0.761868	0.380934	−	0.924602i	$-0.375603\pi$
$181$	10.2499	0.761868	0.380934	+	0.924602i	$0.375603\pi$
$182$	0	0
$183$	10.0575	0.743475
$184$	0	0
$185$	1.83221	0.134707
$186$	0	0
$187$	−8.00000	−0.585018
$188$	0	0
$189$	−3.83221	−0.278752
$190$	0	0
$191$	12.0246	0.870067	0.435033	−	0.900414i	$-0.356737\pi$
$191$	12.0246	0.870067	0.435033	+	0.900414i	$0.356737\pi$
$192$	0	0
$193$	−15.2039	−1.09440	−0.547200	−	0.837002i	$-0.684306\pi$
$193$	−15.2039	−1.09440	−0.547200	+	0.837002i	$0.684306\pi$
$194$	0	0
$195$	−3.53948	−0.253467
$196$	0	0
$197$	−1.02142	−0.0727734	−0.0363867	−	0.999338i	$-0.511585\pi$
$197$	−1.02142	−0.0727734	−0.0363867	+	0.999338i	$0.511585\pi$
$198$	0	0
$199$	−10.2927	−0.729632	−0.364816	−	0.931080i	$-0.618868\pi$
$199$	−10.2927	−0.729632	−0.364816	+	0.931080i	$0.618868\pi$
$200$	0	0
$201$	−0.585462	−0.0412953
$202$	0	0
$203$	−3.27131	−0.229601
$204$	0	0
$205$	−8.51806	−0.594927
$206$	0	0
$207$	8.97858	0.624054
$208$	0	0
$209$	1.14637	0.0792958
$210$	0	0
$211$	19.3288	1.33065	0.665326	−	0.746553i	$-0.268292\pi$
$211$	19.3288	1.33065	0.665326	+	0.746553i	$0.268292\pi$
$212$	0	0
$213$	11.0790	0.759118
$214$	0	0
$215$	10.1249	0.690515
$216$	0	0
$217$	−16.8353	−1.14286
$218$	0	0
$219$	7.37169	0.498133
$220$	0	0
$221$	24.7005	1.66154
$222$	0	0
$223$	9.17092	0.614130	0.307065	−	0.951688i	$-0.400653\pi$
$223$	9.17092	0.614130	0.307065	+	0.951688i	$0.400653\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	23.2713	1.54457	0.772285	−	0.635276i	$-0.219114\pi$
$227$	23.2713	1.54457	0.772285	+	0.635276i	$0.219114\pi$
$228$	0	0
$229$	−10.6430	−0.703309	−0.351655	−	0.936130i	$-0.614381\pi$
$229$	−10.6430	−0.703309	−0.351655	+	0.936130i	$0.614381\pi$
$230$	0	0
$231$	−4.39312	−0.289046
$232$	0	0
$233$	22.7005	1.48716	0.743581	−	0.668646i	$-0.233126\pi$
$233$	22.7005	1.48716	0.743581	+	0.668646i	$0.233126\pi$
$234$	0	0
$235$	−0.978577	−0.0638353
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−6.31729	−0.408631	−0.204316	−	0.978905i	$-0.565497\pi$
$239$	−6.31729	−0.408631	−0.204316	+	0.978905i	$0.565497\pi$
$240$	0	0
$241$	17.3288	1.11625	0.558125	−	0.829757i	$-0.311521\pi$
$241$	17.3288	1.11625	0.558125	+	0.829757i	$0.311521\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	7.68585	0.491031
$246$	0	0
$247$	−3.53948	−0.225212
$248$	0	0
$249$	−12.6430	−0.801218
$250$	0	0
$251$	−24.4752	−1.54486	−0.772431	−	0.635099i	$-0.780959\pi$
$251$	−24.4752	−1.54486	−0.772431	+	0.635099i	$0.780959\pi$
$252$	0	0
$253$	10.2927	0.647098
$254$	0	0
$255$	−6.97858	−0.437015
$256$	0	0
$257$	−17.2713	−1.07735	−0.538677	−	0.842512i	$-0.681076\pi$
$257$	−17.2713	−1.07735	−0.538677	+	0.842512i	$0.681076\pi$
$258$	0	0
$259$	−7.02142	−0.436290
$260$	0	0
$261$	0.853635	0.0528386
$262$	0	0
$263$	5.89962	0.363786	0.181893	−	0.983318i	$-0.441778\pi$
$263$	5.89962	0.363786	0.181893	+	0.983318i	$0.441778\pi$
$264$	0	0
$265$	6.97858	0.428691
$266$	0	0
$267$	7.93260	0.485467
$268$	0	0
$269$	13.7747	0.839857	0.419928	−	0.907557i	$-0.362055\pi$
$269$	13.7747	0.839857	0.419928	+	0.907557i	$0.362055\pi$
$270$	0	0
$271$	−14.9933	−0.910776	−0.455388	−	0.890293i	$-0.650500\pi$
$271$	−14.9933	−0.910776	−0.455388	+	0.890293i	$0.650500\pi$
$272$	0	0
$273$	13.5640	0.820933
$274$	0	0
$275$	1.14637	0.0691284
$276$	0	0
$277$	15.1709	0.911532	0.455766	−	0.890100i	$-0.349365\pi$
$277$	15.1709	0.911532	0.455766	+	0.890100i	$0.349365\pi$
$278$	0	0
$279$	4.39312	0.263009
$280$	0	0
$281$	6.81079	0.406298	0.203149	−	0.979148i	$-0.434882\pi$
$281$	6.81079	0.406298	0.203149	+	0.979148i	$0.434882\pi$
$282$	0	0
$283$	−23.8322	−1.41668	−0.708339	−	0.705872i	$-0.750555\pi$
$283$	−23.8322	−1.41668	−0.708339	+	0.705872i	$0.750555\pi$
$284$	0	0
$285$	1.00000	0.0592349
$286$	0	0
$287$	32.6430	1.92686
$288$	0	0
$289$	31.7005	1.86474
$290$	0	0
$291$	1.24675	0.0730858
$292$	0	0
$293$	−7.22846	−0.422291	−0.211146	−	0.977455i	$-0.567719\pi$
$293$	−7.22846	−0.422291	−0.211146	+	0.977455i	$0.567719\pi$
$294$	0	0
$295$	6.29273	0.366377
$296$	0	0
$297$	1.14637	0.0665189
$298$	0	0
$299$	−31.7795	−1.83786
$300$	0	0
$301$	−38.8009	−2.23645
$302$	0	0
$303$	9.07896	0.521573
$304$	0	0
$305$	10.0575	0.575893
$306$	0	0
$307$	−6.29273	−0.359145	−0.179573	−	0.983745i	$-0.557471\pi$
$307$	−6.29273	−0.359145	−0.179573	+	0.983745i	$0.557471\pi$
$308$	0	0
$309$	18.7434	1.06627
$310$	0	0
$311$	−26.1825	−1.48467	−0.742336	−	0.670028i	$-0.766282\pi$
$311$	−26.1825	−1.48467	−0.742336	+	0.670028i	$0.766282\pi$
$312$	0	0
$313$	13.0790	0.739267	0.369633	−	0.929178i	$-0.379483\pi$
$313$	13.0790	0.739267	0.369633	+	0.929178i	$0.379483\pi$
$314$	0	0
$315$	−3.83221	−0.215921
$316$	0	0
$317$	−34.3074	−1.92690	−0.963448	−	0.267894i	$-0.913672\pi$
$317$	−34.3074	−1.92690	−0.963448	+	0.267894i	$0.913672\pi$
$318$	0	0
$319$	0.978577	0.0547898
$320$	0	0
$321$	−1.37169	−0.0765604
$322$	0	0
$323$	−6.97858	−0.388298
$324$	0	0
$325$	−3.53948	−0.196335
$326$	0	0
$327$	4.87819	0.269765
$328$	0	0
$329$	3.75011	0.206751
$330$	0	0
$331$	−4.19235	−0.230432	−0.115216	−	0.993340i	$-0.536756\pi$
$331$	−4.19235	−0.230432	−0.115216	+	0.993340i	$0.536756\pi$
$332$	0	0
$333$	1.83221	0.100405
$334$	0	0
$335$	−0.585462	−0.0319872
$336$	0	0
$337$	−23.9901	−1.30683	−0.653413	−	0.757002i	$-0.726663\pi$
$337$	−23.9901	−1.30683	−0.653413	+	0.757002i	$0.726663\pi$
$338$	0	0
$339$	−12.6858	−0.689001
$340$	0	0
$341$	5.03612	0.272721
$342$	0	0
$343$	−2.62831	−0.141915
$344$	0	0
$345$	8.97858	0.483390
$346$	0	0
$347$	24.0575	1.29148	0.645738	−	0.763559i	$-0.276550\pi$
$347$	24.0575	1.29148	0.645738	+	0.763559i	$0.276550\pi$
$348$	0	0
$349$	22.7862	1.21972	0.609859	−	0.792510i	$-0.291226\pi$
$349$	22.7862	1.21972	0.609859	+	0.792510i	$0.291226\pi$
$350$	0	0
$351$	−3.53948	−0.188924
$352$	0	0
$353$	2.00000	0.106449	0.0532246	−	0.998583i	$-0.483050\pi$
$353$	2.00000	0.106449	0.0532246	+	0.998583i	$0.483050\pi$
$354$	0	0
$355$	11.0790	0.588010
$356$	0	0
$357$	26.7434	1.41541
$358$	0	0
$359$	4.02456	0.212408	0.106204	−	0.994344i	$-0.466130\pi$
$359$	4.02456	0.212408	0.106204	+	0.994344i	$0.466130\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−9.68585	−0.508375
$364$	0	0
$365$	7.37169	0.385852
$366$	0	0
$367$	−7.24675	−0.378277	−0.189139	−	0.981950i	$-0.560570\pi$
$367$	−7.24675	−0.378277	−0.189139	+	0.981950i	$0.560570\pi$
$368$	0	0
$369$	−8.51806	−0.443432
$370$	0	0
$371$	−26.7434	−1.38845
$372$	0	0
$373$	−33.4109	−1.72995	−0.864977	−	0.501812i	$-0.832667\pi$
$373$	−33.4109	−1.72995	−0.864977	+	0.501812i	$0.832667\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−3.02142	−0.155611
$378$	0	0
$379$	−19.5212	−1.00274	−0.501368	−	0.865234i	$-0.667170\pi$
$379$	−19.5212	−1.00274	−0.501368	+	0.865234i	$0.667170\pi$
$380$	0	0
$381$	2.62831	0.134652
$382$	0	0
$383$	4.20077	0.214649	0.107325	−	0.994224i	$-0.465772\pi$
$383$	4.20077	0.214649	0.107325	+	0.994224i	$0.465772\pi$
$384$	0	0
$385$	−4.39312	−0.223894
$386$	0	0
$387$	10.1249	0.514680
$388$	0	0
$389$	−6.70054	−0.339731	−0.169865	−	0.985467i	$-0.554333\pi$
$389$	−6.70054	−0.339731	−0.169865	+	0.985467i	$0.554333\pi$
$390$	0	0
$391$	−62.6577	−3.16874
$392$	0	0
$393$	19.8898	1.00331
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−4.62831	−0.232288	−0.116144	−	0.993232i	$-0.537053\pi$
$397$	−4.62831	−0.232288	−0.116144	+	0.993232i	$0.537053\pi$
$398$	0	0
$399$	−3.83221	−0.191851
$400$	0	0
$401$	31.6461	1.58033	0.790166	−	0.612892i	$-0.209994\pi$
$401$	31.6461	1.58033	0.790166	+	0.612892i	$0.209994\pi$
$402$	0	0
$403$	−15.5493	−0.774568
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	2.10038	0.104112
$408$	0	0
$409$	−30.9504	−1.53040	−0.765200	−	0.643793i	$-0.777360\pi$
$409$	−30.9504	−1.53040	−0.765200	+	0.643793i	$0.777360\pi$
$410$	0	0
$411$	−19.3717	−0.955535
$412$	0	0
$413$	−24.1151	−1.18663
$414$	0	0
$415$	−12.6430	−0.620620
$416$	0	0
$417$	−9.70727	−0.475367
$418$	0	0
$419$	4.56090	0.222815	0.111407	−	0.993775i	$-0.464464\pi$
$419$	4.56090	0.222815	0.111407	+	0.993775i	$0.464464\pi$
$420$	0	0
$421$	−15.7564	−0.767919	−0.383960	−	0.923350i	$-0.625440\pi$
$421$	−15.7564	−0.767919	−0.383960	+	0.923350i	$0.625440\pi$
$422$	0	0
$423$	−0.978577	−0.0475800
$424$	0	0
$425$	−6.97858	−0.338511
$426$	0	0
$427$	−38.5426	−1.86521
$428$	0	0
$429$	−4.05754	−0.195900
$430$	0	0
$431$	18.4078	0.886673	0.443336	−	0.896355i	$-0.353795\pi$
$431$	18.4078	0.886673	0.443336	+	0.896355i	$0.353795\pi$
$432$	0	0
$433$	31.4538	1.51157	0.755786	−	0.654818i	$-0.227255\pi$
$433$	31.4538	1.51157	0.755786	+	0.654818i	$0.227255\pi$
$434$	0	0
$435$	0.853635	0.0409286
$436$	0	0
$437$	8.97858	0.429504
$438$	0	0
$439$	11.9143	0.568639	0.284319	−	0.958730i	$-0.408232\pi$
$439$	11.9143	0.568639	0.284319	+	0.958730i	$0.408232\pi$
$440$	0	0
$441$	7.68585	0.365993
$442$	0	0
$443$	38.2646	1.81800	0.909002	−	0.416791i	$-0.136845\pi$
$443$	38.2646	1.81800	0.909002	+	0.416791i	$0.136845\pi$
$444$	0	0
$445$	7.93260	0.376041
$446$	0	0
$447$	21.3288	1.00882
$448$	0	0
$449$	28.1825	1.33001	0.665007	−	0.746837i	$-0.268429\pi$
$449$	28.1825	1.33001	0.665007	+	0.746837i	$0.268429\pi$
$450$	0	0
$451$	−9.76481	−0.459807
$452$	0	0
$453$	−14.3503	−0.674234
$454$	0	0
$455$	13.5640	0.635892
$456$	0	0
$457$	20.2927	0.949254	0.474627	−	0.880187i	$-0.342583\pi$
$457$	20.2927	0.949254	0.474627	+	0.880187i	$0.342583\pi$
$458$	0	0
$459$	−6.97858	−0.325732
$460$	0	0
$461$	−26.1151	−1.21630	−0.608150	−	0.793822i	$-0.708088\pi$
$461$	−26.1151	−1.21630	−0.608150	+	0.793822i	$0.708088\pi$
$462$	0	0
$463$	−27.1611	−1.26228	−0.631141	−	0.775668i	$-0.717413\pi$
$463$	−27.1611	−1.26228	−0.631141	+	0.775668i	$0.717413\pi$
$464$	0	0
$465$	4.39312	0.203726
$466$	0	0
$467$	1.56404	0.0723751	0.0361875	−	0.999345i	$-0.488479\pi$
$467$	1.56404	0.0723751	0.0361875	+	0.999345i	$0.488479\pi$
$468$	0	0
$469$	2.24361	0.103600
$470$	0	0
$471$	1.07896	0.0497159
$472$	0	0
$473$	11.6069	0.533685
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	6.97858	0.319527
$478$	0	0
$479$	17.3471	0.792611	0.396305	−	0.918119i	$-0.370292\pi$
$479$	17.3471	0.792611	0.396305	+	0.918119i	$0.370292\pi$
$480$	0	0
$481$	−6.48508	−0.295694
$482$	0	0
$483$	−34.4078	−1.56561
$484$	0	0
$485$	1.24675	0.0566120
$486$	0	0
$487$	30.9933	1.40444	0.702220	−	0.711960i	$-0.252192\pi$
$487$	30.9933	1.40444	0.702220	+	0.711960i	$0.252192\pi$
$488$	0	0
$489$	−2.46052	−0.111269
$490$	0	0
$491$	39.4685	1.78119	0.890594	−	0.454800i	$-0.150289\pi$
$491$	39.4685	1.78119	0.890594	+	0.454800i	$0.150289\pi$
$492$	0	0
$493$	−5.95715	−0.268297
$494$	0	0
$495$	1.14637	0.0515253
$496$	0	0
$497$	−42.4569	−1.90445
$498$	0	0
$499$	18.2070	0.815059	0.407530	−	0.913192i	$-0.366390\pi$
$499$	18.2070	0.815059	0.407530	+	0.913192i	$0.366390\pi$
$500$	0	0
$501$	−4.05754	−0.181277
$502$	0	0
$503$	9.81392	0.437581	0.218791	−	0.975772i	$-0.429789\pi$
$503$	9.81392	0.437581	0.218791	+	0.975772i	$0.429789\pi$
$504$	0	0
$505$	9.07896	0.404008
$506$	0	0
$507$	−0.472077	−0.0209657
$508$	0	0
$509$	17.9754	0.796747	0.398374	−	0.917223i	$-0.369575\pi$
$509$	17.9754	0.796747	0.398374	+	0.917223i	$0.369575\pi$
$510$	0	0
$511$	−28.2499	−1.24970
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	18.7434	0.825932
$516$	0	0
$517$	−1.12181	−0.0493370
$518$	0	0
$519$	−8.68585	−0.381266
$520$	0	0
$521$	18.6100	0.815320	0.407660	−	0.913134i	$-0.366345\pi$
$521$	18.6100	0.815320	0.407660	+	0.913134i	$0.366345\pi$
$522$	0	0
$523$	−34.2070	−1.49577	−0.747885	−	0.663829i	$-0.768930\pi$
$523$	−34.2070	−1.49577	−0.747885	+	0.663829i	$0.768930\pi$
$524$	0	0
$525$	−3.83221	−0.167251
$526$	0	0
$527$	−30.6577	−1.33547
$528$	0	0
$529$	57.6148	2.50499
$530$	0	0
$531$	6.29273	0.273081
$532$	0	0
$533$	30.1495	1.30592
$534$	0	0
$535$	−1.37169	−0.0593034
$536$	0	0
$537$	11.6644	0.503357
$538$	0	0
$539$	8.81079	0.379508
$540$	0	0
$541$	1.91431	0.0823026	0.0411513	−	0.999153i	$-0.486897\pi$
$541$	1.91431	0.0823026	0.0411513	+	0.999153i	$0.486897\pi$
$542$	0	0
$543$	10.2499	0.439865
$544$	0	0
$545$	4.87819	0.208959
$546$	0	0
$547$	−2.37842	−0.101694	−0.0508470	−	0.998706i	$-0.516192\pi$
$547$	−2.37842	−0.101694	−0.0508470	+	0.998706i	$0.516192\pi$
$548$	0	0
$549$	10.0575	0.429245
$550$	0	0
$551$	0.853635	0.0363661
$552$	0	0
$553$	0	0
$554$	0	0
$555$	1.83221	0.0777731
$556$	0	0
$557$	7.17092	0.303842	0.151921	−	0.988393i	$-0.451454\pi$
$557$	7.17092	0.303842	0.151921	+	0.988393i	$0.451454\pi$
$558$	0	0
$559$	−35.8370	−1.51575
$560$	0	0
$561$	−8.00000	−0.337760
$562$	0	0
$563$	0.442232	0.0186379	0.00931893	−	0.999957i	$-0.497034\pi$
$563$	0.442232	0.0186379	0.00931893	+	0.999957i	$0.497034\pi$
$564$	0	0
$565$	−12.6858	−0.533698
$566$	0	0
$567$	−3.83221	−0.160938
$568$	0	0
$569$	−6.56090	−0.275047	−0.137524	−	0.990498i	$-0.543914\pi$
$569$	−6.56090	−0.275047	−0.137524	+	0.990498i	$0.543914\pi$
$570$	0	0
$571$	−24.6514	−1.03163	−0.515815	−	0.856700i	$-0.672511\pi$
$571$	−24.6514	−1.03163	−0.515815	+	0.856700i	$0.672511\pi$
$572$	0	0
$573$	12.0246	0.502333
$574$	0	0
$575$	8.97858	0.374433
$576$	0	0
$577$	−17.0790	−0.711006	−0.355503	−	0.934675i	$-0.615690\pi$
$577$	−17.0790	−0.711006	−0.355503	+	0.934675i	$0.615690\pi$
$578$	0	0
$579$	−15.2039	−0.631853
$580$	0	0
$581$	48.4507	2.01007
$582$	0	0
$583$	8.00000	0.331326
$584$	0	0
$585$	−3.53948	−0.146340
$586$	0	0
$587$	−15.6069	−0.644165	−0.322083	−	0.946712i	$-0.604383\pi$
$587$	−15.6069	−0.644165	−0.322083	+	0.946712i	$0.604383\pi$
$588$	0	0
$589$	4.39312	0.181015
$590$	0	0
$591$	−1.02142	−0.0420157
$592$	0	0
$593$	−25.4145	−1.04365	−0.521825	−	0.853053i	$-0.674749\pi$
$593$	−25.4145	−1.04365	−0.521825	+	0.853053i	$0.674749\pi$
$594$	0	0
$595$	26.7434	1.09637
$596$	0	0
$597$	−10.2927	−0.421253
$598$	0	0
$599$	45.8715	1.87426	0.937129	−	0.348984i	$-0.113473\pi$
$599$	45.8715	1.87426	0.937129	+	0.348984i	$0.113473\pi$
$600$	0	0
$601$	−46.5657	−1.89946	−0.949728	−	0.313077i	$-0.898640\pi$
$601$	−46.5657	−1.89946	−0.949728	+	0.313077i	$0.898640\pi$
$602$	0	0
$603$	−0.585462	−0.0238419
$604$	0	0
$605$	−9.68585	−0.393786
$606$	0	0
$607$	32.4507	1.31713	0.658566	−	0.752523i	$-0.271163\pi$
$607$	32.4507	1.31713	0.658566	+	0.752523i	$0.271163\pi$
$608$	0	0
$609$	−3.27131	−0.132560
$610$	0	0
$611$	3.46365	0.140124
$612$	0	0
$613$	4.20704	0.169921	0.0849604	−	0.996384i	$-0.472924\pi$
$613$	4.20704	0.169921	0.0849604	+	0.996384i	$0.472924\pi$
$614$	0	0
$615$	−8.51806	−0.343481
$616$	0	0
$617$	20.9357	0.842841	0.421420	−	0.906865i	$-0.361532\pi$
$617$	20.9357	0.842841	0.421420	+	0.906865i	$0.361532\pi$
$618$	0	0
$619$	−34.4935	−1.38641	−0.693205	−	0.720740i	$-0.743802\pi$
$619$	−34.4935	−1.38641	−0.693205	+	0.720740i	$0.743802\pi$
$620$	0	0
$621$	8.97858	0.360298
$622$	0	0
$623$	−30.3994	−1.21793
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	1.14637	0.0457814
$628$	0	0
$629$	−12.7862	−0.509820
$630$	0	0
$631$	46.6577	1.85741	0.928707	−	0.370815i	$-0.120922\pi$
$631$	46.6577	1.85741	0.928707	+	0.370815i	$0.120922\pi$
$632$	0	0
$633$	19.3288	0.768253
$634$	0	0
$635$	2.62831	0.104301
$636$	0	0
$637$	−27.2039	−1.07786
$638$	0	0
$639$	11.0790	0.438277
$640$	0	0
$641$	−9.23833	−0.364892	−0.182446	−	0.983216i	$-0.558401\pi$
$641$	−9.23833	−0.364892	−0.182446	+	0.983216i	$0.558401\pi$
$642$	0	0
$643$	−6.21063	−0.244923	−0.122462	−	0.992473i	$-0.539079\pi$
$643$	−6.21063	−0.244923	−0.122462	+	0.992473i	$0.539079\pi$
$644$	0	0
$645$	10.1249	0.398669
$646$	0	0
$647$	23.4721	0.922783	0.461391	−	0.887197i	$-0.347350\pi$
$647$	23.4721	0.922783	0.461391	+	0.887197i	$0.347350\pi$
$648$	0	0
$649$	7.21377	0.283165
$650$	0	0
$651$	−16.8353	−0.659829
$652$	0	0
$653$	22.5939	0.884167	0.442083	−	0.896974i	$-0.354239\pi$
$653$	22.5939	0.884167	0.442083	+	0.896974i	$0.354239\pi$
$654$	0	0
$655$	19.8898	0.777157
$656$	0	0
$657$	7.37169	0.287597
$658$	0	0
$659$	−17.8715	−0.696173	−0.348087	−	0.937462i	$-0.613169\pi$
$659$	−17.8715	−0.696173	−0.348087	+	0.937462i	$0.613169\pi$
$660$	0	0
$661$	10.2499	0.398674	0.199337	−	0.979931i	$-0.436121\pi$
$661$	10.2499	0.398674	0.199337	+	0.979931i	$0.436121\pi$
$662$	0	0
$663$	24.7005	0.959289
$664$	0	0
$665$	−3.83221	−0.148607
$666$	0	0
$667$	7.66442	0.296768
$668$	0	0
$669$	9.17092	0.354568
$670$	0	0
$671$	11.5296	0.445096
$672$	0	0
$673$	0.124943	0.00481618	0.00240809	−	0.999997i	$-0.499233\pi$
$673$	0.124943	0.00481618	0.00240809	+	0.999997i	$0.499233\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	1.72196	0.0661804	0.0330902	−	0.999452i	$-0.489465\pi$
$677$	1.72196	0.0661804	0.0330902	+	0.999452i	$0.489465\pi$
$678$	0	0
$679$	−4.77781	−0.183355
$680$	0	0
$681$	23.2713	0.891758
$682$	0	0
$683$	13.9572	0.534056	0.267028	−	0.963689i	$-0.413958\pi$
$683$	13.9572	0.534056	0.267028	+	0.963689i	$0.413958\pi$
$684$	0	0
$685$	−19.3717	−0.740154
$686$	0	0
$687$	−10.6430	−0.406056
$688$	0	0
$689$	−24.7005	−0.941016
$690$	0	0
$691$	33.8223	1.28666	0.643331	−	0.765588i	$-0.277552\pi$
$691$	33.8223	1.28666	0.643331	+	0.765588i	$0.277552\pi$
$692$	0	0
$693$	−4.39312	−0.166881
$694$	0	0
$695$	−9.70727	−0.368218
$696$	0	0
$697$	59.4439	2.25160
$698$	0	0
$699$	22.7005	0.858613
$700$	0	0
$701$	−15.7564	−0.595110	−0.297555	−	0.954705i	$-0.596171\pi$
$701$	−15.7564	−0.595110	−0.297555	+	0.954705i	$0.596171\pi$
$702$	0	0
$703$	1.83221	0.0691032
$704$	0	0
$705$	−0.978577	−0.0368553
$706$	0	0
$707$	−34.7925	−1.30851
$708$	0	0
$709$	5.07054	0.190428	0.0952141	−	0.995457i	$-0.469646\pi$
$709$	5.07054	0.190428	0.0952141	+	0.995457i	$0.469646\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	39.4439	1.47719
$714$	0	0
$715$	−4.05754	−0.151743
$716$	0	0
$717$	−6.31729	−0.235923
$718$	0	0
$719$	−14.4324	−0.538236	−0.269118	−	0.963107i	$-0.586732\pi$
$719$	−14.4324	−0.538236	−0.269118	+	0.963107i	$0.586732\pi$
$720$	0	0
$721$	−71.8286	−2.67504
$722$	0	0
$723$	17.3288	0.644467
$724$	0	0
$725$	0.853635	0.0317032
$726$	0	0
$727$	−18.0393	−0.669039	−0.334519	−	0.942389i	$-0.608574\pi$
$727$	−18.0393	−0.669039	−0.334519	+	0.942389i	$0.608574\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−70.6577	−2.61337
$732$	0	0
$733$	30.9013	1.14137	0.570683	−	0.821171i	$-0.306679\pi$
$733$	30.9013	1.14137	0.570683	+	0.821171i	$0.306679\pi$
$734$	0	0
$735$	7.68585	0.283497
$736$	0	0
$737$	−0.671153	−0.0247223
$738$	0	0
$739$	−38.0722	−1.40051	−0.700255	−	0.713893i	$-0.746930\pi$
$739$	−38.0722	−1.40051	−0.700255	+	0.713893i	$0.746930\pi$
$740$	0	0
$741$	−3.53948	−0.130026
$742$	0	0
$743$	−20.0294	−0.734807	−0.367403	−	0.930062i	$-0.619753\pi$
$743$	−20.0294	−0.734807	−0.367403	+	0.930062i	$0.619753\pi$
$744$	0	0
$745$	21.3288	0.781428
$746$	0	0
$747$	−12.6430	−0.462583
$748$	0	0
$749$	5.25662	0.192073
$750$	0	0
$751$	−4.39312	−0.160307	−0.0801535	−	0.996783i	$-0.525541\pi$
$751$	−4.39312	−0.160307	−0.0801535	+	0.996783i	$0.525541\pi$
$752$	0	0
$753$	−24.4752	−0.891926
$754$	0	0
$755$	−14.3503	−0.522260
$756$	0	0
$757$	−22.9210	−0.833079	−0.416540	−	0.909118i	$-0.636757\pi$
$757$	−22.9210	−0.833079	−0.416540	+	0.909118i	$0.636757\pi$
$758$	0	0
$759$	10.2927	0.373602
$760$	0	0
$761$	5.41454	0.196277	0.0981384	−	0.995173i	$-0.468711\pi$
$761$	5.41454	0.196277	0.0981384	+	0.995173i	$0.468711\pi$
$762$	0	0
$763$	−18.6943	−0.676778
$764$	0	0
$765$	−6.97858	−0.252311
$766$	0	0
$767$	−22.2730	−0.804231
$768$	0	0
$769$	17.3288	0.624894	0.312447	−	0.949935i	$-0.398851\pi$
$769$	17.3288	0.624894	0.312447	+	0.949935i	$0.398851\pi$
$770$	0	0
$771$	−17.2713	−0.622011
$772$	0	0
$773$	−41.6363	−1.49755	−0.748776	−	0.662823i	$-0.769358\pi$
$773$	−41.6363	−1.49755	−0.748776	+	0.662823i	$0.769358\pi$
$774$	0	0
$775$	4.39312	0.157805
$776$	0	0
$777$	−7.02142	−0.251892
$778$	0	0
$779$	−8.51806	−0.305191
$780$	0	0
$781$	12.7005	0.454461
$782$	0	0
$783$	0.853635	0.0305064
$784$	0	0
$785$	1.07896	0.0385098
$786$	0	0
$787$	51.1083	1.82182	0.910908	−	0.412610i	$-0.135383\pi$
$787$	51.1083	1.82182	0.910908	+	0.412610i	$0.135383\pi$
$788$	0	0
$789$	5.89962	0.210032
$790$	0	0
$791$	48.6148	1.72854
$792$	0	0
$793$	−35.5985	−1.26414
$794$	0	0
$795$	6.97858	0.247505
$796$	0	0
$797$	−9.35700	−0.331442	−0.165721	−	0.986173i	$-0.552995\pi$
$797$	−9.35700	−0.331442	−0.165721	+	0.986173i	$0.552995\pi$
$798$	0	0
$799$	6.82908	0.241595
$800$	0	0
$801$	7.93260	0.280284
$802$	0	0
$803$	8.45065	0.298217
$804$	0	0
$805$	−34.4078	−1.21272
$806$	0	0
$807$	13.7747	0.484891
$808$	0	0
$809$	12.6283	0.443988	0.221994	−	0.975048i	$-0.428744\pi$
$809$	12.6283	0.443988	0.221994	+	0.975048i	$0.428744\pi$
$810$	0	0
$811$	2.15792	0.0757749	0.0378875	−	0.999282i	$-0.487937\pi$
$811$	2.15792	0.0757749	0.0378875	+	0.999282i	$0.487937\pi$
$812$	0	0
$813$	−14.9933	−0.525837
$814$	0	0
$815$	−2.46052	−0.0861882
$816$	0	0
$817$	10.1249	0.354227
$818$	0	0
$819$	13.5640	0.473966
$820$	0	0
$821$	−24.8782	−0.868255	−0.434127	−	0.900851i	$-0.642943\pi$
$821$	−24.8782	−0.868255	−0.434127	+	0.900851i	$0.642943\pi$
$822$	0	0
$823$	−15.6314	−0.544878	−0.272439	−	0.962173i	$-0.587830\pi$
$823$	−15.6314	−0.544878	−0.272439	+	0.962173i	$0.587830\pi$
$824$	0	0
$825$	1.14637	0.0399113
$826$	0	0
$827$	−43.1856	−1.50171	−0.750856	−	0.660466i	$-0.770359\pi$
$827$	−43.1856	−1.50171	−0.750856	+	0.660466i	$0.770359\pi$
$828$	0	0
$829$	29.6644	1.03029	0.515144	−	0.857104i	$-0.327738\pi$
$829$	29.6644	1.03029	0.515144	+	0.857104i	$0.327738\pi$
$830$	0	0
$831$	15.1709	0.526274
$832$	0	0
$833$	−53.6363	−1.85839
$834$	0	0
$835$	−4.05754	−0.140417
$836$	0	0
$837$	4.39312	0.151848
$838$	0	0
$839$	−41.7367	−1.44091	−0.720455	−	0.693502i	$-0.756067\pi$
$839$	−41.7367	−1.44091	−0.720455	+	0.693502i	$0.756067\pi$
$840$	0	0
$841$	−28.2713	−0.974873
$842$	0	0
$843$	6.81079	0.234576
$844$	0	0
$845$	−0.472077	−0.0162399
$846$	0	0
$847$	37.1182	1.27540
$848$	0	0
$849$	−23.8322	−0.817919
$850$	0	0
$851$	16.4507	0.563921
$852$	0	0
$853$	−50.5657	−1.73134	−0.865669	−	0.500617i	$-0.833106\pi$
$853$	−50.5657	−1.73134	−0.865669	+	0.500617i	$0.833106\pi$
$854$	0	0
$855$	1.00000	0.0341993
$856$	0	0
$857$	47.0080	1.60576	0.802881	−	0.596140i	$-0.203300\pi$
$857$	47.0080	1.60576	0.802881	+	0.596140i	$0.203300\pi$
$858$	0	0
$859$	44.6148	1.52224	0.761119	−	0.648612i	$-0.224650\pi$
$859$	44.6148	1.52224	0.761119	+	0.648612i	$0.224650\pi$
$860$	0	0
$861$	32.6430	1.11247
$862$	0	0
$863$	0.384694	0.0130951	0.00654756	−	0.999979i	$-0.497916\pi$
$863$	0.384694	0.0130951	0.00654756	+	0.999979i	$0.497916\pi$
$864$	0	0
$865$	−8.68585	−0.295328
$866$	0	0
$867$	31.7005	1.07661
$868$	0	0
$869$	0	0
$870$	0	0
$871$	2.07223	0.0702149
$872$	0	0
$873$	1.24675	0.0421961
$874$	0	0
$875$	−3.83221	−0.129552
$876$	0	0
$877$	12.2400	0.413316	0.206658	−	0.978413i	$-0.433741\pi$
$877$	12.2400	0.413316	0.206658	+	0.978413i	$0.433741\pi$
$878$	0	0
$879$	−7.22846	−0.243810
$880$	0	0
$881$	−1.46365	−0.0493118	−0.0246559	−	0.999696i	$-0.507849\pi$
$881$	−1.46365	−0.0493118	−0.0246559	+	0.999696i	$0.507849\pi$
$882$	0	0
$883$	−55.9473	−1.88278	−0.941388	−	0.337325i	$-0.890478\pi$
$883$	−55.9473	−1.88278	−0.941388	+	0.337325i	$0.890478\pi$
$884$	0	0
$885$	6.29273	0.211528
$886$	0	0
$887$	12.5855	0.422578	0.211289	−	0.977424i	$-0.432234\pi$
$887$	12.5855	0.422578	0.211289	+	0.977424i	$0.432234\pi$
$888$	0	0
$889$	−10.0722	−0.337812
$890$	0	0
$891$	1.14637	0.0384047
$892$	0	0
$893$	−0.978577	−0.0327468
$894$	0	0
$895$	11.6644	0.389899
$896$	0	0
$897$	−31.7795	−1.06109
$898$	0	0
$899$	3.75011	0.125073
$900$	0	0
$901$	−48.7005	−1.62245
$902$	0	0
$903$	−38.8009	−1.29121
$904$	0	0
$905$	10.2499	0.340718
$906$	0	0
$907$	−45.2369	−1.50207	−0.751033	−	0.660265i	$-0.770444\pi$
$907$	−45.2369	−1.50207	−0.751033	+	0.660265i	$0.770444\pi$
$908$	0	0
$909$	9.07896	0.301130
$910$	0	0
$911$	−14.9442	−0.495122	−0.247561	−	0.968872i	$-0.579629\pi$
$911$	−14.9442	−0.495122	−0.247561	+	0.968872i	$0.579629\pi$
$912$	0	0
$913$	−14.4935	−0.479665
$914$	0	0
$915$	10.0575	0.332492
$916$	0	0
$917$	−76.2217	−2.51706
$918$	0	0
$919$	45.3717	1.49667	0.748337	−	0.663319i	$-0.230853\pi$
$919$	45.3717	1.49667	0.748337	+	0.663319i	$0.230853\pi$
$920$	0	0
$921$	−6.29273	−0.207353
$922$	0	0
$923$	−39.2138	−1.29074
$924$	0	0
$925$	1.83221	0.0602427
$926$	0	0
$927$	18.7434	0.615614
$928$	0	0
$929$	−46.8353	−1.53662	−0.768309	−	0.640079i	$-0.778902\pi$
$929$	−46.8353	−1.53662	−0.768309	+	0.640079i	$0.778902\pi$
$930$	0	0
$931$	7.68585	0.251893
$932$	0	0
$933$	−26.1825	−0.857176
$934$	0	0
$935$	−8.00000	−0.261628
$936$	0	0
$937$	4.96388	0.162163	0.0810815	−	0.996707i	$-0.474163\pi$
$937$	4.96388	0.162163	0.0810815	+	0.996707i	$0.474163\pi$
$938$	0	0
$939$	13.0790	0.426816
$940$	0	0
$941$	33.6890	1.09823	0.549115	−	0.835747i	$-0.314965\pi$
$941$	33.6890	1.09823	0.549115	+	0.835747i	$0.314965\pi$
$942$	0	0
$943$	−76.4800	−2.49053
$944$	0	0
$945$	−3.83221	−0.124662
$946$	0	0
$947$	−22.1004	−0.718166	−0.359083	−	0.933306i	$-0.616911\pi$
$947$	−22.1004	−0.718166	−0.359083	+	0.933306i	$0.616911\pi$
$948$	0	0
$949$	−26.0920	−0.846981
$950$	0	0
$951$	−34.3074	−1.11249
$952$	0	0
$953$	26.6430	0.863051	0.431526	−	0.902101i	$-0.357975\pi$
$953$	26.6430	0.863051	0.431526	+	0.902101i	$0.357975\pi$
$954$	0	0
$955$	12.0246	0.389106
$956$	0	0
$957$	0.978577	0.0316329
$958$	0	0
$959$	74.2364	2.39722
$960$	0	0
$961$	−11.7005	−0.377437
$962$	0	0
$963$	−1.37169	−0.0442022
$964$	0	0
$965$	−15.2039	−0.489431
$966$	0	0
$967$	−6.91117	−0.222248	−0.111124	−	0.993807i	$-0.535445\pi$
$967$	−6.91117	−0.222248	−0.111124	+	0.993807i	$0.535445\pi$
$968$	0	0
$969$	−6.97858	−0.224184
$970$	0	0
$971$	−3.71354	−0.119173	−0.0595866	−	0.998223i	$-0.518978\pi$
$971$	−3.71354	−0.119173	−0.0595866	+	0.998223i	$0.518978\pi$
$972$	0	0
$973$	37.2003	1.19259
$974$	0	0
$975$	−3.53948	−0.113354
$976$	0	0
$977$	32.4653	1.03866	0.519329	−	0.854574i	$-0.326182\pi$
$977$	32.4653	1.03866	0.519329	+	0.854574i	$0.326182\pi$
$978$	0	0
$979$	9.09365	0.290634
$980$	0	0
$981$	4.87819	0.155749
$982$	0	0
$983$	1.71569	0.0547220	0.0273610	−	0.999626i	$-0.491290\pi$
$983$	1.71569	0.0547220	0.0273610	+	0.999626i	$0.491290\pi$
$984$	0	0
$985$	−1.02142	−0.0325452
$986$	0	0
$987$	3.75011	0.119367
$988$	0	0
$989$	90.9076	2.89069
$990$	0	0
$991$	38.1579	1.21213	0.606063	−	0.795417i	$-0.292748\pi$
$991$	38.1579	1.21213	0.606063	+	0.795417i	$0.292748\pi$
$992$	0	0
$993$	−4.19235	−0.133040
$994$	0	0
$995$	−10.2927	−0.326302
$996$	0	0
$997$	−0.0919626	−0.00291248	−0.00145624	−	0.999999i	$-0.500464\pi$
$997$	−0.0919626	−0.00291248	−0.00145624	+	0.999999i	$0.500464\pi$
$998$	0	0
$999$	1.83221	0.0579686

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.a.bv.1.1		3
4.3	odd	2		2280.2.a.s.1.3	✓	3
12.11	even	2		6840.2.a.bf.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.2.a.s.1.3	✓	3	4.3	odd	2
4560.2.a.bv.1.1		3	1.1	even	1	trivial
6840.2.a.bf.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:	\( N \)	\(=\)	\( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4560.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} -3.83221 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} -3.83221 q^{7} +1.00000 q^{9} +1.14637 q^{11} -3.53948 q^{13} +1.00000 q^{15} -6.97858 q^{17} +1.00000 q^{19} -3.83221 q^{21} +8.97858 q^{23} +1.00000 q^{25} +1.00000 q^{27} +0.853635 q^{29} +4.39312 q^{31} +1.14637 q^{33} -3.83221 q^{35} +1.83221 q^{37} -3.53948 q^{39} -8.51806 q^{41} +10.1249 q^{43} +1.00000 q^{45} -0.978577 q^{47} +7.68585 q^{49} -6.97858 q^{51} +6.97858 q^{53} +1.14637 q^{55} +1.00000 q^{57} +6.29273 q^{59} +10.0575 q^{61} -3.83221 q^{63} -3.53948 q^{65} -0.585462 q^{67} +8.97858 q^{69} +11.0790 q^{71} +7.37169 q^{73} +1.00000 q^{75} -4.39312 q^{77} +1.00000 q^{81} -12.6430 q^{83} -6.97858 q^{85} +0.853635 q^{87} +7.93260 q^{89} +13.5640 q^{91} +4.39312 q^{93} +1.00000 q^{95} +1.24675 q^{97} +1.14637 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{5} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^5 + 2 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{5} + 2 q^{7} + 3 q^{9} + 2 q^{11} + 3 q^{15} - 6 q^{17} + 3 q^{19} + 2 q^{21} + 12 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} + 4 q^{31} + 2 q^{33} + 2 q^{35} - 8 q^{37} + 14 q^{43} + 3 q^{45} + 12 q^{47} + 11 q^{49} - 6 q^{51} + 6 q^{53} + 2 q^{55} + 3 q^{57} + 16 q^{59} - 6 q^{61} + 2 q^{63} + 4 q^{67} + 12 q^{69} + 12 q^{71} - 2 q^{73} + 3 q^{75} - 4 q^{77} + 3 q^{81} + 4 q^{83} - 6 q^{85} + 4 q^{87} + 4 q^{89} + 20 q^{91} + 4 q^{93} + 3 q^{95} - 4 q^{97} + 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^5 + 2 * q^7 + 3 * q^9 + 2 * q^11 + 3 * q^15 - 6 * q^17 + 3 * q^19 + 2 * q^21 + 12 * q^23 + 3 * q^25 + 3 * q^27 + 4 * q^29 + 4 * q^31 + 2 * q^33 + 2 * q^35 - 8 * q^37 + 14 * q^43 + 3 * q^45 + 12 * q^47 + 11 * q^49 - 6 * q^51 + 6 * q^53 + 2 * q^55 + 3 * q^57 + 16 * q^59 - 6 * q^61 + 2 * q^63 + 4 * q^67 + 12 * q^69 + 12 * q^71 - 2 * q^73 + 3 * q^75 - 4 * q^77 + 3 * q^81 + 4 * q^83 - 6 * q^85 + 4 * q^87 + 4 * q^89 + 20 * q^91 + 4 * q^93 + 3 * q^95 - 4 * q^97 + 2 * q^99

Introduction

L-functions

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Varieties

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Database

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