LMFDB - Embedded newform 4560.2.a.bu.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:	3.3.1524.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 7x + 1 $ x^3 - x^2 - 7*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1140)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$0.140435$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.00000 q^{5} -4.98028 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.00000 q^{5} -4.98028 q^{7} +1.00000 q^{9} +5.26115 q^{11} +6.98028 q^{13} +1.00000 q^{15} +0.280871 q^{17} -1.00000 q^{19} -4.98028 q^{21} -0.280871 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.26115 q^{29} -2.28087 q^{31} +5.26115 q^{33} -4.98028 q^{35} -2.98028 q^{37} +6.98028 q^{39} +0.738851 q^{41} +5.54202 q^{43} +1.00000 q^{45} +4.28087 q^{47} +17.8032 q^{49} +0.280871 q^{51} +5.71913 q^{53} +5.26115 q^{55} -1.00000 q^{57} -10.5223 q^{59} -2.24143 q^{61} -4.98028 q^{63} +6.98028 q^{65} -0.280871 q^{69} -14.5223 q^{71} +5.43826 q^{73} +1.00000 q^{75} -26.2020 q^{77} +16.4829 q^{79} +1.00000 q^{81} -10.8032 q^{83} +0.280871 q^{85} -3.26115 q^{87} +16.6600 q^{89} -34.7637 q^{91} -2.28087 q^{93} -1.00000 q^{95} +2.98028 q^{97} +5.26115 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} + 3 q^{5} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 + 3 * q^5 + 3 * q^9	$ 3 q + 3 q^{3} + 3 q^{5} + 3 q^{9} + 2 q^{11} + 6 q^{13} + 3 q^{15} + 2 q^{17} - 3 q^{19} - 2 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} - 8 q^{31} + 2 q^{33} + 6 q^{37} + 6 q^{39} + 16 q^{41} + 4 q^{43} + 3 q^{45} + 14 q^{47} + 27 q^{49} + 2 q^{51} + 16 q^{53} + 2 q^{55} - 3 q^{57} - 4 q^{59} + 22 q^{61} + 6 q^{65} - 2 q^{69} - 16 q^{71} + 14 q^{73} + 3 q^{75} - 20 q^{77} - 8 q^{79} + 3 q^{81} - 6 q^{83} + 2 q^{85} + 4 q^{87} + 4 q^{89} - 48 q^{91} - 8 q^{93} - 3 q^{95} - 6 q^{97} + 2 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 + 3 * q^5 + 3 * q^9 + 2 * q^11 + 6 * q^13 + 3 * q^15 + 2 * q^17 - 3 * q^19 - 2 * q^23 + 3 * q^25 + 3 * q^27 + 4 * q^29 - 8 * q^31 + 2 * q^33 + 6 * q^37 + 6 * q^39 + 16 * q^41 + 4 * q^43 + 3 * q^45 + 14 * q^47 + 27 * q^49 + 2 * q^51 + 16 * q^53 + 2 * q^55 - 3 * q^57 - 4 * q^59 + 22 * q^61 + 6 * q^65 - 2 * q^69 - 16 * q^71 + 14 * q^73 + 3 * q^75 - 20 * q^77 - 8 * q^79 + 3 * q^81 - 6 * q^83 + 2 * q^85 + 4 * q^87 + 4 * q^89 - 48 * q^91 - 8 * q^93 - 3 * q^95 - 6 * 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
$7$	−4.98028	−1.88237	−0.941184	−	0.337894i	$-0.890285\pi$
$7$	−4.98028	−1.88237	−0.941184	+	0.337894i	$0.890285\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.26115	1.58630	0.793148	−	0.609029i	$-0.208441\pi$
$11$	5.26115	1.58630	0.793148	+	0.609029i	$0.208441\pi$
$12$	0	0
$13$	6.98028	1.93598	0.967990	−	0.250987i	$-0.0807552\pi$
$13$	6.98028	1.93598	0.967990	+	0.250987i	$0.0807552\pi$
$14$	0	0
$15$	1.00000	0.258199
$16$	0	0
$17$	0.280871	0.0681212	0.0340606	−	0.999420i	$-0.489156\pi$
$17$	0.280871	0.0681212	0.0340606	+	0.999420i	$0.489156\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−4.98028	−1.08679
$22$	0	0
$23$	−0.280871	−0.0585656	−0.0292828	−	0.999571i	$-0.509322\pi$
$23$	−0.280871	−0.0585656	−0.0292828	+	0.999571i	$0.509322\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−3.26115	−0.605580	−0.302790	−	0.953057i	$-0.597918\pi$
$29$	−3.26115	−0.605580	−0.302790	+	0.953057i	$0.597918\pi$
$30$	0	0
$31$	−2.28087	−0.409656	−0.204828	−	0.978798i	$-0.565664\pi$
$31$	−2.28087	−0.409656	−0.204828	+	0.978798i	$0.565664\pi$
$32$	0	0
$33$	5.26115	0.915848
$34$	0	0
$35$	−4.98028	−0.841821
$36$	0	0
$37$	−2.98028	−0.489955	−0.244977	−	0.969529i	$-0.578781\pi$
$37$	−2.98028	−0.489955	−0.244977	+	0.969529i	$0.578781\pi$
$38$	0	0
$39$	6.98028	1.11774
$40$	0	0
$41$	0.738851	0.115389	0.0576946	−	0.998334i	$-0.481625\pi$
$41$	0.738851	0.115389	0.0576946	+	0.998334i	$0.481625\pi$
$42$	0	0
$43$	5.54202	0.845150	0.422575	−	0.906328i	$-0.361126\pi$
$43$	5.54202	0.845150	0.422575	+	0.906328i	$0.361126\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	4.28087	0.624429	0.312215	−	0.950012i	$-0.398929\pi$
$47$	4.28087	0.624429	0.312215	+	0.950012i	$0.398929\pi$
$48$	0	0
$49$	17.8032	2.54331
$50$	0	0
$51$	0.280871	0.0393298
$52$	0	0
$53$	5.71913	0.785583	0.392791	−	0.919628i	$-0.371509\pi$
$53$	5.71913	0.785583	0.392791	+	0.919628i	$0.371509\pi$
$54$	0	0
$55$	5.26115	0.709413
$56$	0	0
$57$	−1.00000	−0.132453
$58$	0	0
$59$	−10.5223	−1.36989	−0.684943	−	0.728596i	$-0.740173\pi$
$59$	−10.5223	−1.36989	−0.684943	+	0.728596i	$0.740173\pi$
$60$	0	0
$61$	−2.24143	−0.286985	−0.143493	−	0.989651i	$-0.545833\pi$
$61$	−2.24143	−0.286985	−0.143493	+	0.989651i	$0.545833\pi$
$62$	0	0
$63$	−4.98028	−0.627456
$64$	0	0
$65$	6.98028	0.865797
$66$	0	0
$67$	0	0		−	1.00000i	$-0.5\pi$
$67$	0	0			1.00000i	$0.5\pi$
$68$	0	0
$69$	−0.280871	−0.0338129
$70$	0	0
$71$	−14.5223	−1.72348	−0.861740	−	0.507350i	$-0.830625\pi$
$71$	−14.5223	−1.72348	−0.861740	+	0.507350i	$0.830625\pi$
$72$	0	0
$73$	5.43826	0.636500	0.318250	−	0.948007i	$-0.396905\pi$
$73$	5.43826	0.636500	0.318250	+	0.948007i	$0.396905\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	−26.2020	−2.98599
$78$	0	0
$79$	16.4829	1.85447	0.927233	−	0.374485i	$-0.122181\pi$
$79$	16.4829	1.85447	0.927233	+	0.374485i	$0.122181\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−10.8032	−1.18580	−0.592901	−	0.805275i	$-0.702017\pi$
$83$	−10.8032	−1.18580	−0.592901	+	0.805275i	$0.702017\pi$
$84$	0	0
$85$	0.280871	0.0304647
$86$	0	0
$87$	−3.26115	−0.349632
$88$	0	0
$89$	16.6600	1.76595	0.882976	−	0.469418i	$-0.155536\pi$
$89$	16.6600	1.76595	0.882976	+	0.469418i	$0.155536\pi$
$90$	0	0
$91$	−34.7637	−3.64423
$92$	0	0
$93$	−2.28087	−0.236515
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	2.98028	0.302601	0.151301	−	0.988488i	$-0.451654\pi$
$97$	2.98028	0.302601	0.151301	+	0.988488i	$0.451654\pi$
$98$	0	0
$99$	5.26115	0.528765
$100$	0	0
$101$	13.0840	1.30191	0.650955	−	0.759116i	$-0.274369\pi$
$101$	13.0840	1.30191	0.650955	+	0.759116i	$0.274369\pi$
$102$	0	0
$103$	−4.56174	−0.449482	−0.224741	−	0.974419i	$-0.572154\pi$
$103$	−4.56174	−0.449482	−0.224741	+	0.974419i	$0.572154\pi$
$104$	0	0
$105$	−4.98028	−0.486025
$106$	0	0
$107$	16.0000	1.54678	0.773389	−	0.633932i	$-0.218560\pi$
$107$	16.0000	1.54678	0.773389	+	0.633932i	$0.218560\pi$
$108$	0	0
$109$	−4.52230	−0.433158	−0.216579	−	0.976265i	$-0.569490\pi$
$109$	−4.52230	−0.433158	−0.216579	+	0.976265i	$0.569490\pi$
$110$	0	0
$111$	−2.98028	−0.282875
$112$	0	0
$113$	−0.241427	−0.0227115	−0.0113557	−	0.999936i	$-0.503615\pi$
$113$	−0.241427	−0.0227115	−0.0113557	+	0.999936i	$0.503615\pi$
$114$	0	0
$115$	−0.280871	−0.0261913
$116$	0	0
$117$	6.98028	0.645327
$118$	0	0
$119$	−1.39881	−0.128229
$120$	0	0
$121$	16.6797	1.51634
$122$	0	0
$123$	0.738851	0.0666200
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	0	0
$129$	5.54202	0.487948
$130$	0	0
$131$	15.7834	1.37901	0.689503	−	0.724283i	$-0.257829\pi$
$131$	15.7834	1.37901	0.689503	+	0.724283i	$0.257829\pi$
$132$	0	0
$133$	4.98028	0.431845
$134$	0	0
$135$	1.00000	0.0860663
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	10.5223	0.892490	0.446245	−	0.894911i	$-0.352761\pi$
$139$	10.5223	0.892490	0.446245	+	0.894911i	$0.352761\pi$
$140$	0	0
$141$	4.28087	0.360514
$142$	0	0
$143$	36.7243	3.07104
$144$	0	0
$145$	−3.26115	−0.270824
$146$	0	0
$147$	17.8032	1.46838
$148$	0	0
$149$	−5.43826	−0.445519	−0.222760	−	0.974873i	$-0.571507\pi$
$149$	−5.43826	−0.445519	−0.222760	+	0.974873i	$0.571507\pi$
$150$	0	0
$151$	14.2020	1.15574	0.577870	−	0.816128i	$-0.303884\pi$
$151$	14.2020	1.15574	0.577870	+	0.816128i	$0.303884\pi$
$152$	0	0
$153$	0.280871	0.0227071
$154$	0	0
$155$	−2.28087	−0.183204
$156$	0	0
$157$	−15.3988	−1.22896	−0.614480	−	0.788933i	$-0.710634\pi$
$157$	−15.3988	−1.22896	−0.614480	+	0.788933i	$0.710634\pi$
$158$	0	0
$159$	5.71913	0.453556
$160$	0	0
$161$	1.39881	0.110242
$162$	0	0
$163$	18.9408	1.48356	0.741780	−	0.670643i	$-0.233982\pi$
$163$	18.9408	1.48356	0.741780	+	0.670643i	$0.233982\pi$
$164$	0	0
$165$	5.26115	0.409580
$166$	0	0
$167$	−11.6797	−0.903801	−0.451901	−	0.892068i	$-0.649254\pi$
$167$	−11.6797	−0.903801	−0.451901	+	0.892068i	$0.649254\pi$
$168$	0	0
$169$	35.7243	2.74802
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	−16.2414	−1.23481	−0.617406	−	0.786644i	$-0.711817\pi$
$173$	−16.2414	−1.23481	−0.617406	+	0.786644i	$0.711817\pi$
$174$	0	0
$175$	−4.98028	−0.376474
$176$	0	0
$177$	−10.5223	−0.790904
$178$	0	0
$179$	21.9606	1.64141	0.820705	−	0.571353i	$-0.193581\pi$
$179$	21.9606	1.64141	0.820705	+	0.571353i	$0.193581\pi$
$180$	0	0
$181$	19.9606	1.48366	0.741828	−	0.670590i	$-0.233959\pi$
$181$	19.9606	1.48366	0.741828	+	0.670590i	$0.233959\pi$
$182$	0	0
$183$	−2.24143	−0.165691
$184$	0	0
$185$	−2.98028	−0.219114
$186$	0	0
$187$	1.47770	0.108060
$188$	0	0
$189$	−4.98028	−0.362262
$190$	0	0
$191$	19.2217	1.39083	0.695417	−	0.718607i	$-0.255220\pi$
$191$	19.2217	1.39083	0.695417	+	0.718607i	$0.255220\pi$
$192$	0	0
$193$	−6.41854	−0.462016	−0.231008	−	0.972952i	$-0.574202\pi$
$193$	−6.41854	−0.462016	−0.231008	+	0.972952i	$0.574202\pi$
$194$	0	0
$195$	6.98028	0.499868
$196$	0	0
$197$	−16.2809	−1.15996	−0.579982	−	0.814629i	$-0.696940\pi$
$197$	−16.2809	−1.15996	−0.579982	+	0.814629i	$0.696940\pi$
$198$	0	0
$199$	25.3988	1.80047	0.900237	−	0.435400i	$-0.143393\pi$
$199$	25.3988	1.80047	0.900237	+	0.435400i	$0.143393\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	16.2414	1.13992
$204$	0	0
$205$	0.738851	0.0516036
$206$	0	0
$207$	−0.280871	−0.0195219
$208$	0	0
$209$	−5.26115	−0.363921
$210$	0	0
$211$	−27.3594	−1.88350	−0.941748	−	0.336318i	$-0.890818\pi$
$211$	−27.3594	−1.88350	−0.941748	+	0.336318i	$0.890818\pi$
$212$	0	0
$213$	−14.5223	−0.995051
$214$	0	0
$215$	5.54202	0.377963
$216$	0	0
$217$	11.3594	0.771124
$218$	0	0
$219$	5.43826	0.367483
$220$	0	0
$221$	1.96056	0.131881
$222$	0	0
$223$	−4.56174	−0.305477	−0.152738	−	0.988267i	$-0.548809\pi$
$223$	−4.56174	−0.305477	−0.152738	+	0.988267i	$0.548809\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−20.7243	−1.37552	−0.687759	−	0.725939i	$-0.741406\pi$
$227$	−20.7243	−1.37552	−0.687759	+	0.725939i	$0.741406\pi$
$228$	0	0
$229$	−2.32031	−0.153331	−0.0766654	−	0.997057i	$-0.524427\pi$
$229$	−2.32031	−0.153331	−0.0766654	+	0.997057i	$0.524427\pi$
$230$	0	0
$231$	−26.2020	−1.72396
$232$	0	0
$233$	23.6063	1.54650	0.773251	−	0.634100i	$-0.218629\pi$
$233$	23.6063	1.54650	0.773251	+	0.634100i	$0.218629\pi$
$234$	0	0
$235$	4.28087	0.279253
$236$	0	0
$237$	16.4829	1.07068
$238$	0	0
$239$	16.6994	1.08019	0.540097	−	0.841603i	$-0.318387\pi$
$239$	16.6994	1.08019	0.540097	+	0.841603i	$0.318387\pi$
$240$	0	0
$241$	−2.00000	−0.128831	−0.0644157	−	0.997923i	$-0.520518\pi$
$241$	−2.00000	−0.128831	−0.0644157	+	0.997923i	$0.520518\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	17.8032	1.13740
$246$	0	0
$247$	−6.98028	−0.444144
$248$	0	0
$249$	−10.8032	−0.684623
$250$	0	0
$251$	12.1377	0.766123	0.383061	−	0.923723i	$-0.374870\pi$
$251$	12.1377	0.766123	0.383061	+	0.923723i	$0.374870\pi$
$252$	0	0
$253$	−1.47770	−0.0929024
$254$	0	0
$255$	0.280871	0.0175888
$256$	0	0
$257$	8.80317	0.549127	0.274563	−	0.961569i	$-0.411467\pi$
$257$	8.80317	0.549127	0.274563	+	0.961569i	$0.411467\pi$
$258$	0	0
$259$	14.8426	0.922275
$260$	0	0
$261$	−3.26115	−0.201860
$262$	0	0
$263$	−13.1179	−0.808887	−0.404444	−	0.914563i	$-0.632535\pi$
$263$	−13.1179	−0.808887	−0.404444	+	0.914563i	$0.632535\pi$
$264$	0	0
$265$	5.71913	0.351323
$266$	0	0
$267$	16.6600	1.01957
$268$	0	0
$269$	13.2217	0.806142	0.403071	−	0.915169i	$-0.367943\pi$
$269$	13.2217	0.806142	0.403071	+	0.915169i	$0.367943\pi$
$270$	0	0
$271$	−21.9606	−1.33401	−0.667004	−	0.745054i	$-0.732424\pi$
$271$	−21.9606	−1.33401	−0.667004	+	0.745054i	$0.732424\pi$
$272$	0	0
$273$	−34.7637	−2.10400
$274$	0	0
$275$	5.26115	0.317259
$276$	0	0
$277$	−19.6063	−1.17803	−0.589015	−	0.808122i	$-0.700484\pi$
$277$	−19.6063	−1.17803	−0.589015	+	0.808122i	$0.700484\pi$
$278$	0	0
$279$	−2.28087	−0.136552
$280$	0	0
$281$	−5.78345	−0.345011	−0.172506	−	0.985009i	$-0.555186\pi$
$281$	−5.78345	−0.345011	−0.172506	+	0.985009i	$0.555186\pi$
$282$	0	0
$283$	−0.418536	−0.0248794	−0.0124397	−	0.999923i	$-0.503960\pi$
$283$	−0.418536	−0.0248794	−0.0124397	+	0.999923i	$0.503960\pi$
$284$	0	0
$285$	−1.00000	−0.0592349
$286$	0	0
$287$	−3.67969	−0.217205
$288$	0	0
$289$	−16.9211	−0.995360
$290$	0	0
$291$	2.98028	0.174707
$292$	0	0
$293$	24.2414	1.41620	0.708100	−	0.706113i	$-0.249553\pi$
$293$	24.2414	1.41620	0.708100	+	0.706113i	$0.249553\pi$
$294$	0	0
$295$	−10.5223	−0.612632
$296$	0	0
$297$	5.26115	0.305283
$298$	0	0
$299$	−1.96056	−0.113382
$300$	0	0
$301$	−27.6008	−1.59088
$302$	0	0
$303$	13.0840	0.751658
$304$	0	0
$305$	−2.24143	−0.128344
$306$	0	0
$307$	9.39881	0.536419	0.268209	−	0.963361i	$-0.413568\pi$
$307$	9.39881	0.536419	0.268209	+	0.963361i	$0.413568\pi$
$308$	0	0
$309$	−4.56174	−0.259508
$310$	0	0
$311$	19.2217	1.08996	0.544981	−	0.838448i	$-0.316537\pi$
$311$	19.2217	1.08996	0.544981	+	0.838448i	$0.316537\pi$
$312$	0	0
$313$	−31.9606	−1.80652	−0.903259	−	0.429096i	$-0.858832\pi$
$313$	−31.9606	−1.80652	−0.903259	+	0.429096i	$0.858832\pi$
$314$	0	0
$315$	−4.98028	−0.280607
$316$	0	0
$317$	−9.71913	−0.545881	−0.272940	−	0.962031i	$-0.587996\pi$
$317$	−9.71913	−0.545881	−0.272940	+	0.962031i	$0.587996\pi$
$318$	0	0
$319$	−17.1574	−0.960629
$320$	0	0
$321$	16.0000	0.893033
$322$	0	0
$323$	−0.280871	−0.0156281
$324$	0	0
$325$	6.98028	0.387196
$326$	0	0
$327$	−4.52230	−0.250084
$328$	0	0
$329$	−21.3199	−1.17541
$330$	0	0
$331$	18.2020	1.00047	0.500236	−	0.865889i	$-0.333247\pi$
$331$	18.2020	1.00047	0.500236	+	0.865889i	$0.333247\pi$
$332$	0	0
$333$	−2.98028	−0.163318
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−8.37909	−0.456438	−0.228219	−	0.973610i	$-0.573290\pi$
$337$	−8.37909	−0.456438	−0.228219	+	0.973610i	$0.573290\pi$
$338$	0	0
$339$	−0.241427	−0.0131125
$340$	0	0
$341$	−12.0000	−0.649836
$342$	0	0
$343$	−53.8028	−2.90508
$344$	0	0
$345$	−0.280871	−0.0151216
$346$	0	0
$347$	29.6008	1.58905	0.794527	−	0.607229i	$-0.207719\pi$
$347$	29.6008	1.58905	0.794527	+	0.607229i	$0.207719\pi$
$348$	0	0
$349$	11.0446	0.591204	0.295602	−	0.955311i	$-0.404480\pi$
$349$	11.0446	0.591204	0.295602	+	0.955311i	$0.404480\pi$
$350$	0	0
$351$	6.98028	0.372580
$352$	0	0
$353$	−19.0446	−1.01364	−0.506821	−	0.862051i	$-0.669179\pi$
$353$	−19.0446	−1.01364	−0.506821	+	0.862051i	$0.669179\pi$
$354$	0	0
$355$	−14.5223	−0.770764
$356$	0	0
$357$	−1.39881	−0.0740331
$358$	0	0
$359$	−29.7440	−1.56983	−0.784914	−	0.619604i	$-0.787293\pi$
$359$	−29.7440	−1.56983	−0.784914	+	0.619604i	$0.787293\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	16.6797	0.875456
$364$	0	0
$365$	5.43826	0.284651
$366$	0	0
$367$	−27.5026	−1.43562	−0.717811	−	0.696238i	$-0.754856\pi$
$367$	−27.5026	−1.43562	−0.717811	+	0.696238i	$0.754856\pi$
$368$	0	0
$369$	0.738851	0.0384631
$370$	0	0
$371$	−28.4829	−1.47876
$372$	0	0
$373$	−16.4580	−0.852162	−0.426081	−	0.904685i	$-0.640106\pi$
$373$	−16.4580	−0.852162	−0.426081	+	0.904685i	$0.640106\pi$
$374$	0	0
$375$	1.00000	0.0516398
$376$	0	0
$377$	−22.7637	−1.17239
$378$	0	0
$379$	−2.84261	−0.146015	−0.0730076	−	0.997331i	$-0.523260\pi$
$379$	−2.84261	−0.146015	−0.0730076	+	0.997331i	$0.523260\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	−15.4383	−0.788858	−0.394429	−	0.918926i	$-0.629058\pi$
$383$	−15.4383	−0.788858	−0.394429	+	0.918926i	$0.629058\pi$
$384$	0	0
$385$	−26.2020	−1.33538
$386$	0	0
$387$	5.54202	0.281717
$388$	0	0
$389$	−18.4829	−0.937118	−0.468559	−	0.883432i	$-0.655227\pi$
$389$	−18.4829	−0.937118	−0.468559	+	0.883432i	$0.655227\pi$
$390$	0	0
$391$	−0.0788884	−0.00398956
$392$	0	0
$393$	15.7834	0.796170
$394$	0	0
$395$	16.4829	0.829342
$396$	0	0
$397$	−34.9657	−1.75488	−0.877439	−	0.479688i	$-0.840750\pi$
$397$	−34.9657	−1.75488	−0.877439	+	0.479688i	$0.840750\pi$
$398$	0	0
$399$	4.98028	0.249326
$400$	0	0
$401$	19.8229	0.989908	0.494954	−	0.868919i	$-0.335185\pi$
$401$	19.8229	0.989908	0.494954	+	0.868919i	$0.335185\pi$
$402$	0	0
$403$	−15.9211	−0.793087
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	−15.6797	−0.777213
$408$	0	0
$409$	−9.64578	−0.476953	−0.238477	−	0.971148i	$-0.576648\pi$
$409$	−9.64578	−0.476953	−0.238477	+	0.971148i	$0.576648\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	0	0
$413$	52.4040	2.57863
$414$	0	0
$415$	−10.8032	−0.530307
$416$	0	0
$417$	10.5223	0.515279
$418$	0	0
$419$	−6.65996	−0.325360	−0.162680	−	0.986679i	$-0.552014\pi$
$419$	−6.65996	−0.325360	−0.162680	+	0.986679i	$0.552014\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	4.28087	0.208143
$424$	0	0
$425$	0.280871	0.0136242
$426$	0	0
$427$	11.1629	0.540212
$428$	0	0
$429$	36.7243	1.77306
$430$	0	0
$431$	20.8371	1.00369	0.501843	−	0.864959i	$-0.332655\pi$
$431$	20.8371	1.00369	0.501843	+	0.864959i	$0.332655\pi$
$432$	0	0
$433$	9.58146	0.460456	0.230228	−	0.973137i	$-0.426053\pi$
$433$	9.58146	0.460456	0.230228	+	0.973137i	$0.426053\pi$
$434$	0	0
$435$	−3.26115	−0.156360
$436$	0	0
$437$	0.280871	0.0134359
$438$	0	0
$439$	−5.04459	−0.240765	−0.120383	−	0.992728i	$-0.538412\pi$
$439$	−5.04459	−0.240765	−0.120383	+	0.992728i	$0.538412\pi$
$440$	0	0
$441$	17.8032	0.847770
$442$	0	0
$443$	−31.6402	−1.50327	−0.751637	−	0.659577i	$-0.770735\pi$
$443$	−31.6402	−1.50327	−0.751637	+	0.659577i	$0.770735\pi$
$444$	0	0
$445$	16.6600	0.789758
$446$	0	0
$447$	−5.43826	−0.257221
$448$	0	0
$449$	−7.82289	−0.369185	−0.184593	−	0.982815i	$-0.559097\pi$
$449$	−7.82289	−0.369185	−0.184593	+	0.982815i	$0.559097\pi$
$450$	0	0
$451$	3.88721	0.183041
$452$	0	0
$453$	14.2020	0.667267
$454$	0	0
$455$	−34.7637	−1.62975
$456$	0	0
$457$	−15.4777	−0.724016	−0.362008	−	0.932175i	$-0.617909\pi$
$457$	−15.4777	−0.724016	−0.362008	+	0.932175i	$0.617909\pi$
$458$	0	0
$459$	0.280871	0.0131099
$460$	0	0
$461$	9.92111	0.462072	0.231036	−	0.972945i	$-0.425788\pi$
$461$	9.92111	0.462072	0.231036	+	0.972945i	$0.425788\pi$
$462$	0	0
$463$	21.4631	0.997476	0.498738	−	0.866753i	$-0.333797\pi$
$463$	21.4631	0.997476	0.498738	+	0.866753i	$0.333797\pi$
$464$	0	0
$465$	−2.28087	−0.105773
$466$	0	0
$467$	32.7637	1.51612	0.758062	−	0.652182i	$-0.226146\pi$
$467$	32.7637	1.51612	0.758062	+	0.652182i	$0.226146\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−15.3988	−0.709540
$472$	0	0
$473$	29.1574	1.34066
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	5.71913	0.261861
$478$	0	0
$479$	27.7834	1.26946	0.634729	−	0.772735i	$-0.281112\pi$
$479$	27.7834	1.26946	0.634729	+	0.772735i	$0.281112\pi$
$480$	0	0
$481$	−20.8032	−0.948543
$482$	0	0
$483$	1.39881	0.0636483
$484$	0	0
$485$	2.98028	0.135327
$486$	0	0
$487$	29.3199	1.32861	0.664306	−	0.747460i	$-0.268727\pi$
$487$	29.3199	1.32861	0.664306	+	0.747460i	$0.268727\pi$
$488$	0	0
$489$	18.9408	0.856534
$490$	0	0
$491$	−4.69941	−0.212081	−0.106041	−	0.994362i	$-0.533817\pi$
$491$	−4.69941	−0.212081	−0.106041	+	0.994362i	$0.533817\pi$
$492$	0	0
$493$	−0.915961	−0.0412528
$494$	0	0
$495$	5.26115	0.236471
$496$	0	0
$497$	72.3251	3.24422
$498$	0	0
$499$	−30.4434	−1.36283	−0.681417	−	0.731895i	$-0.738636\pi$
$499$	−30.4434	−1.36283	−0.681417	+	0.731895i	$0.738636\pi$
$500$	0	0
$501$	−11.6797	−0.521810
$502$	0	0
$503$	13.6797	0.609947	0.304974	−	0.952361i	$-0.401352\pi$
$503$	13.6797	0.609947	0.304974	+	0.952361i	$0.401352\pi$
$504$	0	0
$505$	13.0840	0.582232
$506$	0	0
$507$	35.7243	1.58657
$508$	0	0
$509$	−17.7046	−0.784741	−0.392370	−	0.919807i	$-0.628345\pi$
$509$	−17.7046	−0.784741	−0.392370	+	0.919807i	$0.628345\pi$
$510$	0	0
$511$	−27.0840	−1.19813
$512$	0	0
$513$	−1.00000	−0.0441511
$514$	0	0
$515$	−4.56174	−0.201014
$516$	0	0
$517$	22.5223	0.990530
$518$	0	0
$519$	−16.2414	−0.712919
$520$	0	0
$521$	−9.30059	−0.407466	−0.203733	−	0.979026i	$-0.565308\pi$
$521$	−9.30059	−0.407466	−0.203733	+	0.979026i	$0.565308\pi$
$522$	0	0
$523$	−7.64578	−0.334327	−0.167163	−	0.985929i	$-0.553461\pi$
$523$	−7.64578	−0.334327	−0.167163	+	0.985929i	$0.553461\pi$
$524$	0	0
$525$	−4.98028	−0.217357
$526$	0	0
$527$	−0.640630	−0.0279063
$528$	0	0
$529$	−22.9211	−0.996570
$530$	0	0
$531$	−10.5223	−0.456629
$532$	0	0
$533$	5.15739	0.223391
$534$	0	0
$535$	16.0000	0.691740
$536$	0	0
$537$	21.9606	0.947668
$538$	0	0
$539$	93.6651	4.03444
$540$	0	0
$541$	−12.8765	−0.553605	−0.276802	−	0.960927i	$-0.589275\pi$
$541$	−12.8765	−0.553605	−0.276802	+	0.960927i	$0.589275\pi$
$542$	0	0
$543$	19.9606	0.856589
$544$	0	0
$545$	−4.52230	−0.193714
$546$	0	0
$547$	−13.4777	−0.576265	−0.288132	−	0.957591i	$-0.593034\pi$
$547$	−13.4777	−0.576265	−0.288132	+	0.957591i	$0.593034\pi$
$548$	0	0
$549$	−2.24143	−0.0956618
$550$	0	0
$551$	3.26115	0.138930
$552$	0	0
$553$	−82.0892	−3.49079
$554$	0	0
$555$	−2.98028	−0.126506
$556$	0	0
$557$	−3.68522	−0.156148	−0.0780740	−	0.996948i	$-0.524877\pi$
$557$	−3.68522	−0.156148	−0.0780740	+	0.996948i	$0.524877\pi$
$558$	0	0
$559$	38.6848	1.63619
$560$	0	0
$561$	1.47770	0.0623887
$562$	0	0
$563$	−12.7243	−0.536264	−0.268132	−	0.963382i	$-0.586406\pi$
$563$	−12.7243	−0.536264	−0.268132	+	0.963382i	$0.586406\pi$
$564$	0	0
$565$	−0.241427	−0.0101569
$566$	0	0
$567$	−4.98028	−0.209152
$568$	0	0
$569$	25.3006	1.06066	0.530328	−	0.847793i	$-0.322069\pi$
$569$	25.3006	1.06066	0.530328	+	0.847793i	$0.322069\pi$
$570$	0	0
$571$	−23.0052	−0.962736	−0.481368	−	0.876519i	$-0.659860\pi$
$571$	−23.0052	−0.962736	−0.481368	+	0.876519i	$0.659860\pi$
$572$	0	0
$573$	19.2217	0.802998
$574$	0	0
$575$	−0.280871	−0.0117131
$576$	0	0
$577$	−25.0840	−1.04426	−0.522131	−	0.852865i	$-0.674863\pi$
$577$	−25.0840	−1.04426	−0.522131	+	0.852865i	$0.674863\pi$
$578$	0	0
$579$	−6.41854	−0.266745
$580$	0	0
$581$	53.8028	2.23212
$582$	0	0
$583$	30.0892	1.24617
$584$	0	0
$585$	6.98028	0.288599
$586$	0	0
$587$	28.6848	1.18395	0.591975	−	0.805956i	$-0.298348\pi$
$587$	28.6848	1.18395	0.591975	+	0.805956i	$0.298348\pi$
$588$	0	0
$589$	2.28087	0.0939816
$590$	0	0
$591$	−16.2809	−0.669706
$592$	0	0
$593$	2.56174	0.105198	0.0525991	−	0.998616i	$-0.483249\pi$
$593$	2.56174	0.105198	0.0525991	+	0.998616i	$0.483249\pi$
$594$	0	0
$595$	−1.39881	−0.0573458
$596$	0	0
$597$	25.3988	1.03950
$598$	0	0
$599$	−33.0446	−1.35017	−0.675083	−	0.737742i	$-0.735892\pi$
$599$	−33.0446	−1.35017	−0.675083	+	0.737742i	$0.735892\pi$
$600$	0	0
$601$	−22.8371	−0.931544	−0.465772	−	0.884905i	$-0.654223\pi$
$601$	−22.8371	−0.931544	−0.465772	+	0.884905i	$0.654223\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	16.6797	0.678126
$606$	0	0
$607$	28.1286	1.14171	0.570853	−	0.821052i	$-0.306613\pi$
$607$	28.1286	1.14171	0.570853	+	0.821052i	$0.306613\pi$
$608$	0	0
$609$	16.2414	0.658136
$610$	0	0
$611$	29.8817	1.20888
$612$	0	0
$613$	15.9606	0.644641	0.322320	−	0.946631i	$-0.395537\pi$
$613$	15.9606	0.644641	0.322320	+	0.946631i	$0.395537\pi$
$614$	0	0
$615$	0.738851	0.0297934
$616$	0	0
$617$	−6.59565	−0.265531	−0.132765	−	0.991147i	$-0.542386\pi$
$617$	−6.59565	−0.265531	−0.132765	+	0.991147i	$0.542386\pi$
$618$	0	0
$619$	−33.8817	−1.36182	−0.680910	−	0.732367i	$-0.738415\pi$
$619$	−33.8817	−1.36182	−0.680910	+	0.732367i	$0.738415\pi$
$620$	0	0
$621$	−0.280871	−0.0112710
$622$	0	0
$623$	−82.9712	−3.32417
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−5.26115	−0.210110
$628$	0	0
$629$	−0.837073	−0.0333763
$630$	0	0
$631$	−26.7976	−1.06680	−0.533398	−	0.845864i	$-0.679085\pi$
$631$	−26.7976	−1.06680	−0.533398	+	0.845864i	$0.679085\pi$
$632$	0	0
$633$	−27.3594	−1.08744
$634$	0	0
$635$	−4.00000	−0.158735
$636$	0	0
$637$	124.271	4.92380
$638$	0	0
$639$	−14.5223	−0.574493
$640$	0	0
$641$	25.7834	1.01838	0.509192	−	0.860653i	$-0.329944\pi$
$641$	25.7834	1.01838	0.509192	+	0.860653i	$0.329944\pi$
$642$	0	0
$643$	19.4237	0.765995	0.382998	−	0.923749i	$-0.374892\pi$
$643$	19.4237	0.765995	0.382998	+	0.923749i	$0.374892\pi$
$644$	0	0
$645$	5.54202	0.218217
$646$	0	0
$647$	8.55620	0.336379	0.168190	−	0.985755i	$-0.446208\pi$
$647$	8.55620	0.336379	0.168190	+	0.985755i	$0.446208\pi$
$648$	0	0
$649$	−55.3594	−2.17305
$650$	0	0
$651$	11.3594	0.445209
$652$	0	0
$653$	−0.842612	−0.0329740	−0.0164870	−	0.999864i	$-0.505248\pi$
$653$	−0.842612	−0.0329740	−0.0164870	+	0.999864i	$0.505248\pi$
$654$	0	0
$655$	15.7834	0.616710
$656$	0	0
$657$	5.43826	0.212167
$658$	0	0
$659$	3.43826	0.133936	0.0669678	−	0.997755i	$-0.478668\pi$
$659$	3.43826	0.133936	0.0669678	+	0.997755i	$0.478668\pi$
$660$	0	0
$661$	9.08404	0.353328	0.176664	−	0.984271i	$-0.443469\pi$
$661$	9.08404	0.353328	0.176664	+	0.984271i	$0.443469\pi$
$662$	0	0
$663$	1.96056	0.0761417
$664$	0	0
$665$	4.98028	0.193127
$666$	0	0
$667$	0.915961	0.0354662
$668$	0	0
$669$	−4.56174	−0.176367
$670$	0	0
$671$	−11.7925	−0.455244
$672$	0	0
$673$	2.98028	0.114881	0.0574406	−	0.998349i	$-0.481706\pi$
$673$	2.98028	0.114881	0.0574406	+	0.998349i	$0.481706\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−48.9996	−1.88321	−0.941604	−	0.336722i	$-0.890682\pi$
$677$	−48.9996	−1.88321	−0.941604	+	0.336722i	$0.890682\pi$
$678$	0	0
$679$	−14.8426	−0.569607
$680$	0	0
$681$	−20.7243	−0.794156
$682$	0	0
$683$	−27.9211	−1.06837	−0.534186	−	0.845367i	$-0.679382\pi$
$683$	−27.9211	−1.06837	−0.534186	+	0.845367i	$0.679382\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	0	0
$687$	−2.32031	−0.0885255
$688$	0	0
$689$	39.9211	1.52087
$690$	0	0
$691$	9.96056	0.378917	0.189459	−	0.981889i	$-0.439327\pi$
$691$	9.96056	0.378917	0.189459	+	0.981889i	$0.439327\pi$
$692$	0	0
$693$	−26.2020	−0.995331
$694$	0	0
$695$	10.5223	0.399133
$696$	0	0
$697$	0.207522	0.00786045
$698$	0	0
$699$	23.6063	0.892874
$700$	0	0
$701$	−10.4829	−0.395932	−0.197966	−	0.980209i	$-0.563434\pi$
$701$	−10.4829	−0.395932	−0.197966	+	0.980209i	$0.563434\pi$
$702$	0	0
$703$	2.98028	0.112403
$704$	0	0
$705$	4.28087	0.161227
$706$	0	0
$707$	−65.1621	−2.45067
$708$	0	0
$709$	16.5562	0.621781	0.310891	−	0.950446i	$-0.399373\pi$
$709$	16.5562	0.621781	0.310891	+	0.950446i	$0.399373\pi$
$710$	0	0
$711$	16.4829	0.618155
$712$	0	0
$713$	0.640630	0.0239918
$714$	0	0
$715$	36.7243	1.37341
$716$	0	0
$717$	16.6994	0.623651
$718$	0	0
$719$	4.21655	0.157251	0.0786255	−	0.996904i	$-0.474947\pi$
$719$	4.21655	0.157251	0.0786255	+	0.996904i	$0.474947\pi$
$720$	0	0
$721$	22.7187	0.846090
$722$	0	0
$723$	−2.00000	−0.0743808
$724$	0	0
$725$	−3.26115	−0.121116
$726$	0	0
$727$	−26.4580	−0.981272	−0.490636	−	0.871365i	$-0.663236\pi$
$727$	−26.4580	−0.981272	−0.490636	+	0.871365i	$0.663236\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	1.55659	0.0575726
$732$	0	0
$733$	−6.56174	−0.242363	−0.121182	−	0.992630i	$-0.538668\pi$
$733$	−6.56174	−0.242363	−0.121182	+	0.992630i	$0.538668\pi$
$734$	0	0
$735$	17.8032	0.656680
$736$	0	0
$737$	0	0
$738$	0	0
$739$	−44.9657	−1.65409	−0.827045	−	0.562136i	$-0.809980\pi$
$739$	−44.9657	−1.65409	−0.827045	+	0.562136i	$0.809980\pi$
$740$	0	0
$741$	−6.98028	−0.256427
$742$	0	0
$743$	24.4040	0.895295	0.447647	−	0.894210i	$-0.352262\pi$
$743$	24.4040	0.895295	0.447647	+	0.894210i	$0.352262\pi$
$744$	0	0
$745$	−5.43826	−0.199242
$746$	0	0
$747$	−10.8032	−0.395267
$748$	0	0
$749$	−79.6844	−2.91161
$750$	0	0
$751$	−5.23628	−0.191074	−0.0955372	−	0.995426i	$-0.530457\pi$
$751$	−5.23628	−0.191074	−0.0955372	+	0.995426i	$0.530457\pi$
$752$	0	0
$753$	12.1377	0.442321
$754$	0	0
$755$	14.2020	0.516863
$756$	0	0
$757$	20.5223	0.745896	0.372948	−	0.927852i	$-0.378347\pi$
$757$	20.5223	0.745896	0.372948	+	0.927852i	$0.378347\pi$
$758$	0	0
$759$	−1.47770	−0.0536372
$760$	0	0
$761$	−13.5171	−0.489996	−0.244998	−	0.969524i	$-0.578787\pi$
$761$	−13.5171	−0.489996	−0.244998	+	0.969524i	$0.578787\pi$
$762$	0	0
$763$	22.5223	0.815362
$764$	0	0
$765$	0.280871	0.0101549
$766$	0	0
$767$	−73.4486	−2.65207
$768$	0	0
$769$	33.9211	1.22323	0.611613	−	0.791157i	$-0.290521\pi$
$769$	33.9211	1.22323	0.611613	+	0.791157i	$0.290521\pi$
$770$	0	0
$771$	8.80317	0.317038
$772$	0	0
$773$	−5.79802	−0.208540	−0.104270	−	0.994549i	$-0.533251\pi$
$773$	−5.79802	−0.208540	−0.104270	+	0.994549i	$0.533251\pi$
$774$	0	0
$775$	−2.28087	−0.0819313
$776$	0	0
$777$	14.8426	0.532476
$778$	0	0
$779$	−0.738851	−0.0264721
$780$	0	0
$781$	−76.4040	−2.73395
$782$	0	0
$783$	−3.26115	−0.116544
$784$	0	0
$785$	−15.3988	−0.549607
$786$	0	0
$787$	−11.5669	−0.412315	−0.206158	−	0.978519i	$-0.566096\pi$
$787$	−11.5669	−0.412315	−0.206158	+	0.978519i	$0.566096\pi$
$788$	0	0
$789$	−13.1179	−0.467011
$790$	0	0
$791$	1.20237	0.0427514
$792$	0	0
$793$	−15.6458	−0.555598
$794$	0	0
$795$	5.71913	0.202837
$796$	0	0
$797$	−17.3649	−0.615097	−0.307548	−	0.951532i	$-0.599509\pi$
$797$	−17.3649	−0.615097	−0.307548	+	0.951532i	$0.599509\pi$
$798$	0	0
$799$	1.20237	0.0425368
$800$	0	0
$801$	16.6600	0.588651
$802$	0	0
$803$	28.6115	1.00968
$804$	0	0
$805$	1.39881	0.0493017
$806$	0	0
$807$	13.2217	0.465426
$808$	0	0
$809$	26.4829	0.931088	0.465544	−	0.885025i	$-0.345859\pi$
$809$	26.4829	0.931088	0.465544	+	0.885025i	$0.345859\pi$
$810$	0	0
$811$	−35.8422	−1.25859	−0.629295	−	0.777166i	$-0.716656\pi$
$811$	−35.8422	−1.25859	−0.629295	+	0.777166i	$0.716656\pi$
$812$	0	0
$813$	−21.9606	−0.770190
$814$	0	0
$815$	18.9408	0.663468
$816$	0	0
$817$	−5.54202	−0.193891
$818$	0	0
$819$	−34.7637	−1.21474
$820$	0	0
$821$	−12.5223	−0.437031	−0.218516	−	0.975833i	$-0.570121\pi$
$821$	−12.5223	−0.437031	−0.218516	+	0.975833i	$0.570121\pi$
$822$	0	0
$823$	−4.06432	−0.141673	−0.0708366	−	0.997488i	$-0.522567\pi$
$823$	−4.06432	−0.141673	−0.0708366	+	0.997488i	$0.522567\pi$
$824$	0	0
$825$	5.26115	0.183170
$826$	0	0
$827$	33.8478	1.17700	0.588501	−	0.808496i	$-0.299718\pi$
$827$	33.8478	1.17700	0.588501	+	0.808496i	$0.299718\pi$
$828$	0	0
$829$	−9.08404	−0.315502	−0.157751	−	0.987479i	$-0.550424\pi$
$829$	−9.08404	−0.315502	−0.157751	+	0.987479i	$0.550424\pi$
$830$	0	0
$831$	−19.6063	−0.680136
$832$	0	0
$833$	5.00039	0.173253
$834$	0	0
$835$	−11.6797	−0.404192
$836$	0	0
$837$	−2.28087	−0.0788384
$838$	0	0
$839$	−12.2753	−0.423792	−0.211896	−	0.977292i	$-0.567964\pi$
$839$	−12.2753	−0.423792	−0.211896	+	0.977292i	$0.567964\pi$
$840$	0	0
$841$	−18.3649	−0.633273
$842$	0	0
$843$	−5.78345	−0.199192
$844$	0	0
$845$	35.7243	1.22895
$846$	0	0
$847$	−83.0695	−2.85430
$848$	0	0
$849$	−0.418536	−0.0143641
$850$	0	0
$851$	0.837073	0.0286945
$852$	0	0
$853$	−11.4777	−0.392989	−0.196495	−	0.980505i	$-0.562956\pi$
$853$	−11.4777	−0.392989	−0.196495	+	0.980505i	$0.562956\pi$
$854$	0	0
$855$	−1.00000	−0.0341993
$856$	0	0
$857$	−46.7637	−1.59742	−0.798709	−	0.601717i	$-0.794483\pi$
$857$	−46.7637	−1.59742	−0.798709	+	0.601717i	$0.794483\pi$
$858$	0	0
$859$	−30.7298	−1.04849	−0.524244	−	0.851568i	$-0.675652\pi$
$859$	−30.7298	−1.04849	−0.524244	+	0.851568i	$0.675652\pi$
$860$	0	0
$861$	−3.67969	−0.125403
$862$	0	0
$863$	−44.0000	−1.49778	−0.748889	−	0.662696i	$-0.769412\pi$
$863$	−44.0000	−1.49778	−0.748889	+	0.662696i	$0.769412\pi$
$864$	0	0
$865$	−16.2414	−0.552225
$866$	0	0
$867$	−16.9211	−0.574671
$868$	0	0
$869$	86.7187	2.94173
$870$	0	0
$871$	0	0
$872$	0	0
$873$	2.98028	0.100867
$874$	0	0
$875$	−4.98028	−0.168364
$876$	0	0
$877$	−31.1878	−1.05314	−0.526569	−	0.850133i	$-0.676522\pi$
$877$	−31.1878	−1.05314	−0.526569	+	0.850133i	$0.676522\pi$
$878$	0	0
$879$	24.2414	0.817643
$880$	0	0
$881$	1.56689	0.0527899	0.0263950	−	0.999652i	$-0.491597\pi$
$881$	1.56689	0.0527899	0.0263950	+	0.999652i	$0.491597\pi$
$882$	0	0
$883$	36.0643	1.21366	0.606830	−	0.794831i	$-0.292441\pi$
$883$	36.0643	1.21366	0.606830	+	0.794831i	$0.292441\pi$
$884$	0	0
$885$	−10.5223	−0.353703
$886$	0	0
$887$	9.12348	0.306337	0.153168	−	0.988200i	$-0.451052\pi$
$887$	9.12348	0.306337	0.153168	+	0.988200i	$0.451052\pi$
$888$	0	0
$889$	19.9211	0.668133
$890$	0	0
$891$	5.26115	0.176255
$892$	0	0
$893$	−4.28087	−0.143254
$894$	0	0
$895$	21.9606	0.734060
$896$	0	0
$897$	−1.96056	−0.0654611
$898$	0	0
$899$	7.43826	0.248080
$900$	0	0
$901$	1.60634	0.0535148
$902$	0	0
$903$	−27.6008	−0.918497
$904$	0	0
$905$	19.9606	0.663511
$906$	0	0
$907$	15.7136	0.521761	0.260881	−	0.965371i	$-0.415987\pi$
$907$	15.7136	0.521761	0.260881	+	0.965371i	$0.415987\pi$
$908$	0	0
$909$	13.0840	0.433970
$910$	0	0
$911$	47.3594	1.56909	0.784543	−	0.620074i	$-0.212898\pi$
$911$	47.3594	1.56909	0.784543	+	0.620074i	$0.212898\pi$
$912$	0	0
$913$	−56.8371	−1.88103
$914$	0	0
$915$	−2.24143	−0.0740993
$916$	0	0
$917$	−78.6059	−2.59580
$918$	0	0
$919$	−24.4829	−0.807615	−0.403807	−	0.914844i	$-0.632313\pi$
$919$	−24.4829	−0.807615	−0.403807	+	0.914844i	$0.632313\pi$
$920$	0	0
$921$	9.39881	0.309701
$922$	0	0
$923$	−101.370	−3.33662
$924$	0	0
$925$	−2.98028	−0.0979909
$926$	0	0
$927$	−4.56174	−0.149827
$928$	0	0
$929$	51.3988	1.68634	0.843170	−	0.537647i	$-0.180687\pi$
$929$	51.3988	1.68634	0.843170	+	0.537647i	$0.180687\pi$
$930$	0	0
$931$	−17.8032	−0.583475
$932$	0	0
$933$	19.2217	0.629290
$934$	0	0
$935$	1.47770	0.0483260
$936$	0	0
$937$	21.5669	0.704560	0.352280	−	0.935895i	$-0.385407\pi$
$937$	21.5669	0.704560	0.352280	+	0.935895i	$0.385407\pi$
$938$	0	0
$939$	−31.9606	−1.04299
$940$	0	0
$941$	−28.8675	−0.941053	−0.470527	−	0.882386i	$-0.655936\pi$
$941$	−28.8675	−0.941053	−0.470527	+	0.882386i	$0.655936\pi$
$942$	0	0
$943$	−0.207522	−0.00675784
$944$	0	0
$945$	−4.98028	−0.162008
$946$	0	0
$947$	−26.2414	−0.852732	−0.426366	−	0.904551i	$-0.640206\pi$
$947$	−26.2414	−0.852732	−0.426366	+	0.904551i	$0.640206\pi$
$948$	0	0
$949$	37.9606	1.23225
$950$	0	0
$951$	−9.71913	−0.315164
$952$	0	0
$953$	−47.0391	−1.52374	−0.761872	−	0.647727i	$-0.775720\pi$
$953$	−47.0391	−1.52374	−0.761872	+	0.647727i	$0.775720\pi$
$954$	0	0
$955$	19.2217	0.622000
$956$	0	0
$957$	−17.1574	−0.554620
$958$	0	0
$959$	29.8817	0.964929
$960$	0	0
$961$	−25.7976	−0.832182
$962$	0	0
$963$	16.0000	0.515593
$964$	0	0
$965$	−6.41854	−0.206620
$966$	0	0
$967$	53.9460	1.73479	0.867393	−	0.497624i	$-0.165794\pi$
$967$	53.9460	1.73479	0.867393	+	0.497624i	$0.165794\pi$
$968$	0	0
$969$	−0.280871	−0.00902287
$970$	0	0
$971$	13.6852	0.439180	0.219590	−	0.975592i	$-0.429528\pi$
$971$	13.6852	0.439180	0.219590	+	0.975592i	$0.429528\pi$
$972$	0	0
$973$	−52.4040	−1.67999
$974$	0	0
$975$	6.98028	0.223548
$976$	0	0
$977$	50.6848	1.62155	0.810776	−	0.585357i	$-0.199046\pi$
$977$	50.6848	1.62155	0.810776	+	0.585357i	$0.199046\pi$
$978$	0	0
$979$	87.6505	2.80132
$980$	0	0
$981$	−4.52230	−0.144386
$982$	0	0
$983$	−3.19683	−0.101963	−0.0509816	−	0.998700i	$-0.516235\pi$
$983$	−3.19683	−0.101963	−0.0509816	+	0.998700i	$0.516235\pi$
$984$	0	0
$985$	−16.2809	−0.518752
$986$	0	0
$987$	−21.3199	−0.678621
$988$	0	0
$989$	−1.55659	−0.0494967
$990$	0	0
$991$	−5.04459	−0.160247	−0.0801234	−	0.996785i	$-0.525531\pi$
$991$	−5.04459	−0.160247	−0.0801234	+	0.996785i	$0.525531\pi$
$992$	0	0
$993$	18.2020	0.577622
$994$	0	0
$995$	25.3988	0.805197
$996$	0	0
$997$	−1.64578	−0.0521224	−0.0260612	−	0.999660i	$-0.508296\pi$
$997$	−1.64578	−0.0521224	−0.0260612	+	0.999660i	$0.508296\pi$
$998$	0	0
$999$	−2.98028	−0.0942918

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.a.bu.1.1		3
4.3	odd	2		1140.2.a.f.1.3	✓	3
12.11	even	2		3420.2.a.k.1.3		3
20.3	even	4		5700.2.f.p.3649.1		6
20.7	even	4		5700.2.f.p.3649.6		6
20.19	odd	2		5700.2.a.z.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1140.2.a.f.1.3	✓	3	4.3	odd	2
3420.2.a.k.1.3		3	12.11	even	2
4560.2.a.bu.1.1		3	1.1	even	1	trivial
5700.2.a.z.1.1		3	20.19	odd	2
5700.2.f.p.3649.1		6	20.3	even	4
5700.2.f.p.3649.6		6	20.7	even	4

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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.00000 q^{5} -4.98028 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.00000 q^{5} -4.98028 q^{7} +1.00000 q^{9} +5.26115 q^{11} +6.98028 q^{13} +1.00000 q^{15} +0.280871 q^{17} -1.00000 q^{19} -4.98028 q^{21} -0.280871 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.26115 q^{29} -2.28087 q^{31} +5.26115 q^{33} -4.98028 q^{35} -2.98028 q^{37} +6.98028 q^{39} +0.738851 q^{41} +5.54202 q^{43} +1.00000 q^{45} +4.28087 q^{47} +17.8032 q^{49} +0.280871 q^{51} +5.71913 q^{53} +5.26115 q^{55} -1.00000 q^{57} -10.5223 q^{59} -2.24143 q^{61} -4.98028 q^{63} +6.98028 q^{65} -0.280871 q^{69} -14.5223 q^{71} +5.43826 q^{73} +1.00000 q^{75} -26.2020 q^{77} +16.4829 q^{79} +1.00000 q^{81} -10.8032 q^{83} +0.280871 q^{85} -3.26115 q^{87} +16.6600 q^{89} -34.7637 q^{91} -2.28087 q^{93} -1.00000 q^{95} +2.98028 q^{97} +5.26115 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} + 3 q^{5} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 + 3 * q^5 + 3 * q^9	\( 3 q + 3 q^{3} + 3 q^{5} + 3 q^{9} + 2 q^{11} + 6 q^{13} + 3 q^{15} + 2 q^{17} - 3 q^{19} - 2 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} - 8 q^{31} + 2 q^{33} + 6 q^{37} + 6 q^{39} + 16 q^{41} + 4 q^{43} + 3 q^{45} + 14 q^{47} + 27 q^{49} + 2 q^{51} + 16 q^{53} + 2 q^{55} - 3 q^{57} - 4 q^{59} + 22 q^{61} + 6 q^{65} - 2 q^{69} - 16 q^{71} + 14 q^{73} + 3 q^{75} - 20 q^{77} - 8 q^{79} + 3 q^{81} - 6 q^{83} + 2 q^{85} + 4 q^{87} + 4 q^{89} - 48 q^{91} - 8 q^{93} - 3 q^{95} - 6 q^{97} + 2 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 + 3 * q^5 + 3 * q^9 + 2 * q^11 + 6 * q^13 + 3 * q^15 + 2 * q^17 - 3 * q^19 - 2 * q^23 + 3 * q^25 + 3 * q^27 + 4 * q^29 - 8 * q^31 + 2 * q^33 + 6 * q^37 + 6 * q^39 + 16 * q^41 + 4 * q^43 + 3 * q^45 + 14 * q^47 + 27 * q^49 + 2 * q^51 + 16 * q^53 + 2 * q^55 - 3 * q^57 - 4 * q^59 + 22 * q^61 + 6 * q^65 - 2 * q^69 - 16 * q^71 + 14 * q^73 + 3 * q^75 - 20 * q^77 - 8 * q^79 + 3 * q^81 - 6 * q^83 + 2 * q^85 + 4 * q^87 + 4 * q^89 - 48 * q^91 - 8 * q^93 - 3 * q^95 - 6 * q^97 + 2 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more