LMFDB - Embedded newform 4560.2.a.bh.1.2

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:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 285)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +1.00000 q^{5} +2.73205 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +1.00000 q^{5} +2.73205 q^{7} +1.00000 q^{9} -4.73205 q^{11} +0.732051 q^{13} -1.00000 q^{15} -1.00000 q^{19} -2.73205 q^{21} -3.46410 q^{23} +1.00000 q^{25} -1.00000 q^{27} +8.19615 q^{29} -8.92820 q^{31} +4.73205 q^{33} +2.73205 q^{35} -6.19615 q^{37} -0.732051 q^{39} +1.26795 q^{41} -4.19615 q^{43} +1.00000 q^{45} +3.46410 q^{47} +0.464102 q^{49} -9.46410 q^{53} -4.73205 q^{55} +1.00000 q^{57} -2.53590 q^{59} -6.53590 q^{61} +2.73205 q^{63} +0.732051 q^{65} -8.00000 q^{67} +3.46410 q^{69} +4.39230 q^{71} -16.9282 q^{73} -1.00000 q^{75} -12.9282 q^{77} +10.9282 q^{79} +1.00000 q^{81} +12.9282 q^{83} -8.19615 q^{87} +10.7321 q^{89} +2.00000 q^{91} +8.92820 q^{93} -1.00000 q^{95} -6.19615 q^{97} -4.73205 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 6 q^{11} - 2 q^{13} - 2 q^{15} - 2 q^{19} - 2 q^{21} + 2 q^{25} - 2 q^{27} + 6 q^{29} - 4 q^{31} + 6 q^{33} + 2 q^{35} - 2 q^{37} + 2 q^{39} + 6 q^{41} + 2 q^{43} + 2 q^{45} - 6 q^{49} - 12 q^{53} - 6 q^{55} + 2 q^{57} - 12 q^{59} - 20 q^{61} + 2 q^{63} - 2 q^{65} - 16 q^{67} - 12 q^{71} - 20 q^{73} - 2 q^{75} - 12 q^{77} + 8 q^{79} + 2 q^{81} + 12 q^{83} - 6 q^{87} + 18 q^{89} + 4 q^{91} + 4 q^{93} - 2 q^{95} - 2 q^{97} - 6 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 6 * q^11 - 2 * q^13 - 2 * q^15 - 2 * q^19 - 2 * q^21 + 2 * q^25 - 2 * q^27 + 6 * q^29 - 4 * q^31 + 6 * q^33 + 2 * q^35 - 2 * q^37 + 2 * q^39 + 6 * q^41 + 2 * q^43 + 2 * q^45 - 6 * q^49 - 12 * q^53 - 6 * q^55 + 2 * q^57 - 12 * q^59 - 20 * q^61 + 2 * q^63 - 2 * q^65 - 16 * q^67 - 12 * q^71 - 20 * q^73 - 2 * q^75 - 12 * q^77 + 8 * q^79 + 2 * q^81 + 12 * q^83 - 6 * q^87 + 18 * q^89 + 4 * q^91 + 4 * q^93 - 2 * q^95 - 2 * q^97 - 6 * 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
$7$	2.73205	1.03262	0.516309	−	0.856402i	$-0.327306\pi$
$7$	2.73205	1.03262	0.516309	+	0.856402i	$0.327306\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.73205	−1.42677	−0.713384	−	0.700774i	$-0.752838\pi$
$11$	−4.73205	−1.42677	−0.713384	+	0.700774i	$0.752838\pi$
$12$	0	0
$13$	0.732051	0.203034	0.101517	−	0.994834i	$-0.467630\pi$
$13$	0.732051	0.203034	0.101517	+	0.994834i	$0.467630\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−2.73205	−0.596182
$22$	0	0
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	8.19615	1.52199	0.760994	−	0.648759i	$-0.224712\pi$
$29$	8.19615	1.52199	0.760994	+	0.648759i	$0.224712\pi$
$30$	0	0
$31$	−8.92820	−1.60355	−0.801776	−	0.597624i	$-0.796111\pi$
$31$	−8.92820	−1.60355	−0.801776	+	0.597624i	$0.796111\pi$
$32$	0	0
$33$	4.73205	0.823744
$34$	0	0
$35$	2.73205	0.461801
$36$	0	0
$37$	−6.19615	−1.01864	−0.509321	−	0.860577i	$-0.670103\pi$
$37$	−6.19615	−1.01864	−0.509321	+	0.860577i	$0.670103\pi$
$38$	0	0
$39$	−0.732051	−0.117222
$40$	0	0
$41$	1.26795	0.198020	0.0990102	−	0.995086i	$-0.468432\pi$
$41$	1.26795	0.198020	0.0990102	+	0.995086i	$0.468432\pi$
$42$	0	0
$43$	−4.19615	−0.639907	−0.319954	−	0.947433i	$-0.603667\pi$
$43$	−4.19615	−0.639907	−0.319954	+	0.947433i	$0.603667\pi$
$44$	0	0
$45$	1.00000	0.149071
$46$	0	0
$47$	3.46410	0.505291	0.252646	−	0.967559i	$-0.418699\pi$
$47$	3.46410	0.505291	0.252646	+	0.967559i	$0.418699\pi$
$48$	0	0
$49$	0.464102	0.0663002
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−9.46410	−1.29999	−0.649997	−	0.759937i	$-0.725230\pi$
$53$	−9.46410	−1.29999	−0.649997	+	0.759937i	$0.725230\pi$
$54$	0	0
$55$	−4.73205	−0.638070
$56$	0	0
$57$	1.00000	0.132453
$58$	0	0
$59$	−2.53590	−0.330146	−0.165073	−	0.986281i	$-0.552786\pi$
$59$	−2.53590	−0.330146	−0.165073	+	0.986281i	$0.552786\pi$
$60$	0	0
$61$	−6.53590	−0.836836	−0.418418	−	0.908255i	$-0.637415\pi$
$61$	−6.53590	−0.836836	−0.418418	+	0.908255i	$0.637415\pi$
$62$	0	0
$63$	2.73205	0.344206
$64$	0	0
$65$	0.732051	0.0907997
$66$	0	0
$67$	−8.00000	−0.977356	−0.488678	−	0.872464i	$-0.662521\pi$
$67$	−8.00000	−0.977356	−0.488678	+	0.872464i	$0.662521\pi$
$68$	0	0
$69$	3.46410	0.417029
$70$	0	0
$71$	4.39230	0.521271	0.260635	−	0.965437i	$-0.416068\pi$
$71$	4.39230	0.521271	0.260635	+	0.965437i	$0.416068\pi$
$72$	0	0
$73$	−16.9282	−1.98130	−0.990648	−	0.136441i	$-0.956434\pi$
$73$	−16.9282	−1.98130	−0.990648	+	0.136441i	$0.956434\pi$
$74$	0	0
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−12.9282	−1.47331
$78$	0	0
$79$	10.9282	1.22952	0.614759	−	0.788715i	$-0.289253\pi$
$79$	10.9282	1.22952	0.614759	+	0.788715i	$0.289253\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.9282	1.41905	0.709527	−	0.704678i	$-0.248908\pi$
$83$	12.9282	1.41905	0.709527	+	0.704678i	$0.248908\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−8.19615	−0.878720
$88$	0	0
$89$	10.7321	1.13760	0.568798	−	0.822478i	$-0.307409\pi$
$89$	10.7321	1.13760	0.568798	+	0.822478i	$0.307409\pi$
$90$	0	0
$91$	2.00000	0.209657
$92$	0	0
$93$	8.92820	0.925812
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−6.19615	−0.629124	−0.314562	−	0.949237i	$-0.601858\pi$
$97$	−6.19615	−0.629124	−0.314562	+	0.949237i	$0.601858\pi$
$98$	0	0
$99$	−4.73205	−0.475589
$100$	0	0
$101$	−10.3923	−1.03407	−0.517036	−	0.855963i	$-0.672965\pi$
$101$	−10.3923	−1.03407	−0.517036	+	0.855963i	$0.672965\pi$
$102$	0	0
$103$	−9.85641	−0.971181	−0.485590	−	0.874187i	$-0.661395\pi$
$103$	−9.85641	−0.971181	−0.485590	+	0.874187i	$0.661395\pi$
$104$	0	0
$105$	−2.73205	−0.266621
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−14.3923	−1.37853	−0.689266	−	0.724508i	$-0.742067\pi$
$109$	−14.3923	−1.37853	−0.689266	+	0.724508i	$0.742067\pi$
$110$	0	0
$111$	6.19615	0.588113
$112$	0	0
$113$	18.9282	1.78062	0.890308	−	0.455359i	$-0.150489\pi$
$113$	18.9282	1.78062	0.890308	+	0.455359i	$0.150489\pi$
$114$	0	0
$115$	−3.46410	−0.323029
$116$	0	0
$117$	0.732051	0.0676781
$118$	0	0
$119$	0	0
$120$	0	0
$121$	11.3923	1.03566
$122$	0	0
$123$	−1.26795	−0.114327
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	4.00000	0.354943	0.177471	−	0.984126i	$-0.443208\pi$
$127$	4.00000	0.354943	0.177471	+	0.984126i	$0.443208\pi$
$128$	0	0
$129$	4.19615	0.369451
$130$	0	0
$131$	9.12436	0.797199	0.398599	−	0.917125i	$-0.369496\pi$
$131$	9.12436	0.797199	0.398599	+	0.917125i	$0.369496\pi$
$132$	0	0
$133$	−2.73205	−0.236899
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	19.8564	1.69645	0.848224	−	0.529638i	$-0.177672\pi$
$137$	19.8564	1.69645	0.848224	+	0.529638i	$0.177672\pi$
$138$	0	0
$139$	8.39230	0.711826	0.355913	−	0.934519i	$-0.384170\pi$
$139$	8.39230	0.711826	0.355913	+	0.934519i	$0.384170\pi$
$140$	0	0
$141$	−3.46410	−0.291730
$142$	0	0
$143$	−3.46410	−0.289683
$144$	0	0
$145$	8.19615	0.680653
$146$	0	0
$147$	−0.464102	−0.0382785
$148$	0	0
$149$	−19.8564	−1.62670	−0.813350	−	0.581775i	$-0.802359\pi$
$149$	−19.8564	−1.62670	−0.813350	+	0.581775i	$0.802359\pi$
$150$	0	0
$151$	−14.0000	−1.13930	−0.569652	−	0.821886i	$-0.692922\pi$
$151$	−14.0000	−1.13930	−0.569652	+	0.821886i	$0.692922\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−8.92820	−0.717131
$156$	0	0
$157$	6.39230	0.510161	0.255081	−	0.966920i	$-0.417898\pi$
$157$	6.39230	0.510161	0.255081	+	0.966920i	$0.417898\pi$
$158$	0	0
$159$	9.46410	0.750552
$160$	0	0
$161$	−9.46410	−0.745876
$162$	0	0
$163$	−9.26795	−0.725922	−0.362961	−	0.931804i	$-0.618234\pi$
$163$	−9.26795	−0.725922	−0.362961	+	0.931804i	$0.618234\pi$
$164$	0	0
$165$	4.73205	0.368390
$166$	0	0
$167$	−3.46410	−0.268060	−0.134030	−	0.990977i	$-0.542792\pi$
$167$	−3.46410	−0.268060	−0.134030	+	0.990977i	$0.542792\pi$
$168$	0	0
$169$	−12.4641	−0.958777
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	0	0
$173$	6.92820	0.526742	0.263371	−	0.964695i	$-0.415166\pi$
$173$	6.92820	0.526742	0.263371	+	0.964695i	$0.415166\pi$
$174$	0	0
$175$	2.73205	0.206524
$176$	0	0
$177$	2.53590	0.190610
$178$	0	0
$179$	−23.3205	−1.74306	−0.871528	−	0.490345i	$-0.836871\pi$
$179$	−23.3205	−1.74306	−0.871528	+	0.490345i	$0.836871\pi$
$180$	0	0
$181$	−2.39230	−0.177819	−0.0889093	−	0.996040i	$-0.528338\pi$
$181$	−2.39230	−0.177819	−0.0889093	+	0.996040i	$0.528338\pi$
$182$	0	0
$183$	6.53590	0.483148
$184$	0	0
$185$	−6.19615	−0.455550
$186$	0	0
$187$	0	0
$188$	0	0
$189$	−2.73205	−0.198727
$190$	0	0
$191$	−0.339746	−0.0245832	−0.0122916	−	0.999924i	$-0.503913\pi$
$191$	−0.339746	−0.0245832	−0.0122916	+	0.999924i	$0.503913\pi$
$192$	0	0
$193$	17.1244	1.23264	0.616319	−	0.787497i	$-0.288623\pi$
$193$	17.1244	1.23264	0.616319	+	0.787497i	$0.288623\pi$
$194$	0	0
$195$	−0.732051	−0.0524232
$196$	0	0
$197$	−24.0000	−1.70993	−0.854965	−	0.518686i	$-0.826421\pi$
$197$	−24.0000	−1.70993	−0.854965	+	0.518686i	$0.826421\pi$
$198$	0	0
$199$	15.3205	1.08604	0.543021	−	0.839719i	$-0.317280\pi$
$199$	15.3205	1.08604	0.543021	+	0.839719i	$0.317280\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	0	0
$203$	22.3923	1.57163
$204$	0	0
$205$	1.26795	0.0885574
$206$	0	0
$207$	−3.46410	−0.240772
$208$	0	0
$209$	4.73205	0.327323
$210$	0	0
$211$	−1.07180	−0.0737855	−0.0368928	−	0.999319i	$-0.511746\pi$
$211$	−1.07180	−0.0737855	−0.0368928	+	0.999319i	$0.511746\pi$
$212$	0	0
$213$	−4.39230	−0.300956
$214$	0	0
$215$	−4.19615	−0.286175
$216$	0	0
$217$	−24.3923	−1.65586
$218$	0	0
$219$	16.9282	1.14390
$220$	0	0
$221$	0	0
$222$	0	0
$223$	17.8564	1.19575	0.597877	−	0.801588i	$-0.296011\pi$
$223$	17.8564	1.19575	0.597877	+	0.801588i	$0.296011\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−10.3923	−0.689761	−0.344881	−	0.938647i	$-0.612081\pi$
$227$	−10.3923	−0.689761	−0.344881	+	0.938647i	$0.612081\pi$
$228$	0	0
$229$	−18.5359	−1.22489	−0.612443	−	0.790515i	$-0.709813\pi$
$229$	−18.5359	−1.22489	−0.612443	+	0.790515i	$0.709813\pi$
$230$	0	0
$231$	12.9282	0.850613
$232$	0	0
$233$	−7.85641	−0.514690	−0.257345	−	0.966320i	$-0.582848\pi$
$233$	−7.85641	−0.514690	−0.257345	+	0.966320i	$0.582848\pi$
$234$	0	0
$235$	3.46410	0.225973
$236$	0	0
$237$	−10.9282	−0.709863
$238$	0	0
$239$	9.80385	0.634158	0.317079	−	0.948399i	$-0.397298\pi$
$239$	9.80385	0.634158	0.317079	+	0.948399i	$0.397298\pi$
$240$	0	0
$241$	−3.07180	−0.197872	−0.0989359	−	0.995094i	$-0.531544\pi$
$241$	−3.07180	−0.197872	−0.0989359	+	0.995094i	$0.531544\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	0.464102	0.0296504
$246$	0	0
$247$	−0.732051	−0.0465793
$248$	0	0
$249$	−12.9282	−0.819292
$250$	0	0
$251$	−28.0526	−1.77066	−0.885331	−	0.464961i	$-0.846068\pi$
$251$	−28.0526	−1.77066	−0.885331	+	0.464961i	$0.846068\pi$
$252$	0	0
$253$	16.3923	1.03058
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−24.0000	−1.49708	−0.748539	−	0.663090i	$-0.769245\pi$
$257$	−24.0000	−1.49708	−0.748539	+	0.663090i	$0.769245\pi$
$258$	0	0
$259$	−16.9282	−1.05187
$260$	0	0
$261$	8.19615	0.507329
$262$	0	0
$263$	−6.00000	−0.369976	−0.184988	−	0.982741i	$-0.559225\pi$
$263$	−6.00000	−0.369976	−0.184988	+	0.982741i	$0.559225\pi$
$264$	0	0
$265$	−9.46410	−0.581375
$266$	0	0
$267$	−10.7321	−0.656791
$268$	0	0
$269$	0.588457	0.0358789	0.0179394	−	0.999839i	$-0.494289\pi$
$269$	0.588457	0.0358789	0.0179394	+	0.999839i	$0.494289\pi$
$270$	0	0
$271$	−0.392305	−0.0238308	−0.0119154	−	0.999929i	$-0.503793\pi$
$271$	−0.392305	−0.0238308	−0.0119154	+	0.999929i	$0.503793\pi$
$272$	0	0
$273$	−2.00000	−0.121046
$274$	0	0
$275$	−4.73205	−0.285353
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	0	0
$279$	−8.92820	−0.534518
$280$	0	0
$281$	1.26795	0.0756395	0.0378198	−	0.999285i	$-0.487959\pi$
$281$	1.26795	0.0756395	0.0378198	+	0.999285i	$0.487959\pi$
$282$	0	0
$283$	−24.9808	−1.48495	−0.742476	−	0.669873i	$-0.766349\pi$
$283$	−24.9808	−1.48495	−0.742476	+	0.669873i	$0.766349\pi$
$284$	0	0
$285$	1.00000	0.0592349
$286$	0	0
$287$	3.46410	0.204479
$288$	0	0
$289$	−17.0000	−1.00000
$290$	0	0
$291$	6.19615	0.363225
$292$	0	0
$293$	27.7128	1.61900	0.809500	−	0.587120i	$-0.199738\pi$
$293$	27.7128	1.61900	0.809500	+	0.587120i	$0.199738\pi$
$294$	0	0
$295$	−2.53590	−0.147646
$296$	0	0
$297$	4.73205	0.274581
$298$	0	0
$299$	−2.53590	−0.146655
$300$	0	0
$301$	−11.4641	−0.660780
$302$	0	0
$303$	10.3923	0.597022
$304$	0	0
$305$	−6.53590	−0.374244
$306$	0	0
$307$	32.3923	1.84873	0.924363	−	0.381514i	$-0.124597\pi$
$307$	32.3923	1.84873	0.924363	+	0.381514i	$0.124597\pi$
$308$	0	0
$309$	9.85641	0.560711
$310$	0	0
$311$	−32.4449	−1.83978	−0.919890	−	0.392177i	$-0.871722\pi$
$311$	−32.4449	−1.83978	−0.919890	+	0.392177i	$0.871722\pi$
$312$	0	0
$313$	6.39230	0.361314	0.180657	−	0.983546i	$-0.442178\pi$
$313$	6.39230	0.361314	0.180657	+	0.983546i	$0.442178\pi$
$314$	0	0
$315$	2.73205	0.153934
$316$	0	0
$317$	−11.3205	−0.635823	−0.317912	−	0.948120i	$-0.602982\pi$
$317$	−11.3205	−0.635823	−0.317912	+	0.948120i	$0.602982\pi$
$318$	0	0
$319$	−38.7846	−2.17152
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	0.732051	0.0406069
$326$	0	0
$327$	14.3923	0.795896
$328$	0	0
$329$	9.46410	0.521773
$330$	0	0
$331$	25.7128	1.41330	0.706652	−	0.707561i	$-0.250205\pi$
$331$	25.7128	1.41330	0.706652	+	0.707561i	$0.250205\pi$
$332$	0	0
$333$	−6.19615	−0.339547
$334$	0	0
$335$	−8.00000	−0.437087
$336$	0	0
$337$	5.12436	0.279141	0.139571	−	0.990212i	$-0.455428\pi$
$337$	5.12436	0.279141	0.139571	+	0.990212i	$0.455428\pi$
$338$	0	0
$339$	−18.9282	−1.02804
$340$	0	0
$341$	42.2487	2.28790
$342$	0	0
$343$	−17.8564	−0.964155
$344$	0	0
$345$	3.46410	0.186501
$346$	0	0
$347$	0.928203	0.0498286	0.0249143	−	0.999690i	$-0.492069\pi$
$347$	0.928203	0.0498286	0.0249143	+	0.999690i	$0.492069\pi$
$348$	0	0
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	0	0
$351$	−0.732051	−0.0390740
$352$	0	0
$353$	−14.7846	−0.786905	−0.393453	−	0.919345i	$-0.628719\pi$
$353$	−14.7846	−0.786905	−0.393453	+	0.919345i	$0.628719\pi$
$354$	0	0
$355$	4.39230	0.233119
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−0.339746	−0.0179311	−0.00896555	−	0.999960i	$-0.502854\pi$
$359$	−0.339746	−0.0179311	−0.00896555	+	0.999960i	$0.502854\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−11.3923	−0.597941
$364$	0	0
$365$	−16.9282	−0.886063
$366$	0	0
$367$	−16.1962	−0.845432	−0.422716	−	0.906262i	$-0.638923\pi$
$367$	−16.1962	−0.845432	−0.422716	+	0.906262i	$0.638923\pi$
$368$	0	0
$369$	1.26795	0.0660068
$370$	0	0
$371$	−25.8564	−1.34240
$372$	0	0
$373$	−6.19615	−0.320825	−0.160412	−	0.987050i	$-0.551282\pi$
$373$	−6.19615	−0.320825	−0.160412	+	0.987050i	$0.551282\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	6.00000	0.309016
$378$	0	0
$379$	−20.9282	−1.07501	−0.537505	−	0.843261i	$-0.680633\pi$
$379$	−20.9282	−1.07501	−0.537505	+	0.843261i	$0.680633\pi$
$380$	0	0
$381$	−4.00000	−0.204926
$382$	0	0
$383$	17.0718	0.872328	0.436164	−	0.899867i	$-0.356337\pi$
$383$	17.0718	0.872328	0.436164	+	0.899867i	$0.356337\pi$
$384$	0	0
$385$	−12.9282	−0.658882
$386$	0	0
$387$	−4.19615	−0.213302
$388$	0	0
$389$	7.85641	0.398336	0.199168	−	0.979965i	$-0.436176\pi$
$389$	7.85641	0.398336	0.199168	+	0.979965i	$0.436176\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−9.12436	−0.460263
$394$	0	0
$395$	10.9282	0.549858
$396$	0	0
$397$	8.92820	0.448094	0.224047	−	0.974578i	$-0.428073\pi$
$397$	8.92820	0.448094	0.224047	+	0.974578i	$0.428073\pi$
$398$	0	0
$399$	2.73205	0.136774
$400$	0	0
$401$	−34.0526	−1.70050	−0.850252	−	0.526376i	$-0.823550\pi$
$401$	−34.0526	−1.70050	−0.850252	+	0.526376i	$0.823550\pi$
$402$	0	0
$403$	−6.53590	−0.325576
$404$	0	0
$405$	1.00000	0.0496904
$406$	0	0
$407$	29.3205	1.45336
$408$	0	0
$409$	−26.3923	−1.30502	−0.652508	−	0.757782i	$-0.726283\pi$
$409$	−26.3923	−1.30502	−0.652508	+	0.757782i	$0.726283\pi$
$410$	0	0
$411$	−19.8564	−0.979444
$412$	0	0
$413$	−6.92820	−0.340915
$414$	0	0
$415$	12.9282	0.634621
$416$	0	0
$417$	−8.39230	−0.410973
$418$	0	0
$419$	28.0526	1.37046	0.685229	−	0.728328i	$-0.259702\pi$
$419$	28.0526	1.37046	0.685229	+	0.728328i	$0.259702\pi$
$420$	0	0
$421$	−18.7846	−0.915506	−0.457753	−	0.889079i	$-0.651346\pi$
$421$	−18.7846	−0.915506	−0.457753	+	0.889079i	$0.651346\pi$
$422$	0	0
$423$	3.46410	0.168430
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−17.8564	−0.864132
$428$	0	0
$429$	3.46410	0.167248
$430$	0	0
$431$	11.3205	0.545290	0.272645	−	0.962115i	$-0.412102\pi$
$431$	11.3205	0.545290	0.272645	+	0.962115i	$0.412102\pi$
$432$	0	0
$433$	−10.5885	−0.508849	−0.254424	−	0.967093i	$-0.581886\pi$
$433$	−10.5885	−0.508849	−0.254424	+	0.967093i	$0.581886\pi$
$434$	0	0
$435$	−8.19615	−0.392975
$436$	0	0
$437$	3.46410	0.165710
$438$	0	0
$439$	−26.9282	−1.28521	−0.642607	−	0.766196i	$-0.722147\pi$
$439$	−26.9282	−1.28521	−0.642607	+	0.766196i	$0.722147\pi$
$440$	0	0
$441$	0.464102	0.0221001
$442$	0	0
$443$	5.32051	0.252785	0.126392	−	0.991980i	$-0.459660\pi$
$443$	5.32051	0.252785	0.126392	+	0.991980i	$0.459660\pi$
$444$	0	0
$445$	10.7321	0.508748
$446$	0	0
$447$	19.8564	0.939176
$448$	0	0
$449$	−5.66025	−0.267124	−0.133562	−	0.991040i	$-0.542642\pi$
$449$	−5.66025	−0.267124	−0.133562	+	0.991040i	$0.542642\pi$
$450$	0	0
$451$	−6.00000	−0.282529
$452$	0	0
$453$	14.0000	0.657777
$454$	0	0
$455$	2.00000	0.0937614
$456$	0	0
$457$	4.53590	0.212180	0.106090	−	0.994357i	$-0.466167\pi$
$457$	4.53590	0.212180	0.106090	+	0.994357i	$0.466167\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.00000	0.279448	0.139724	−	0.990190i	$-0.455378\pi$
$461$	6.00000	0.279448	0.139724	+	0.990190i	$0.455378\pi$
$462$	0	0
$463$	35.5167	1.65060	0.825300	−	0.564695i	$-0.191006\pi$
$463$	35.5167	1.65060	0.825300	+	0.564695i	$0.191006\pi$
$464$	0	0
$465$	8.92820	0.414036
$466$	0	0
$467$	−20.5359	−0.950288	−0.475144	−	0.879908i	$-0.657604\pi$
$467$	−20.5359	−0.950288	−0.475144	+	0.879908i	$0.657604\pi$
$468$	0	0
$469$	−21.8564	−1.00924
$470$	0	0
$471$	−6.39230	−0.294542
$472$	0	0
$473$	19.8564	0.912999
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	−9.46410	−0.433331
$478$	0	0
$479$	−25.5167	−1.16589	−0.582943	−	0.812513i	$-0.698099\pi$
$479$	−25.5167	−1.16589	−0.582943	+	0.812513i	$0.698099\pi$
$480$	0	0
$481$	−4.53590	−0.206819
$482$	0	0
$483$	9.46410	0.430632
$484$	0	0
$485$	−6.19615	−0.281353
$486$	0	0
$487$	32.3923	1.46784	0.733918	−	0.679238i	$-0.237690\pi$
$487$	32.3923	1.46784	0.733918	+	0.679238i	$0.237690\pi$
$488$	0	0
$489$	9.26795	0.419111
$490$	0	0
$491$	−16.0526	−0.724442	−0.362221	−	0.932092i	$-0.617981\pi$
$491$	−16.0526	−0.724442	−0.362221	+	0.932092i	$0.617981\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−4.73205	−0.212690
$496$	0	0
$497$	12.0000	0.538274
$498$	0	0
$499$	−10.5359	−0.471652	−0.235826	−	0.971795i	$-0.575779\pi$
$499$	−10.5359	−0.471652	−0.235826	+	0.971795i	$0.575779\pi$
$500$	0	0
$501$	3.46410	0.154765
$502$	0	0
$503$	23.0718	1.02872	0.514360	−	0.857574i	$-0.328029\pi$
$503$	23.0718	1.02872	0.514360	+	0.857574i	$0.328029\pi$
$504$	0	0
$505$	−10.3923	−0.462451
$506$	0	0
$507$	12.4641	0.553550
$508$	0	0
$509$	−10.0526	−0.445572	−0.222786	−	0.974867i	$-0.571515\pi$
$509$	−10.0526	−0.445572	−0.222786	+	0.974867i	$0.571515\pi$
$510$	0	0
$511$	−46.2487	−2.04592
$512$	0	0
$513$	1.00000	0.0441511
$514$	0	0
$515$	−9.85641	−0.434325
$516$	0	0
$517$	−16.3923	−0.720933
$518$	0	0
$519$	−6.92820	−0.304114
$520$	0	0
$521$	37.2679	1.63274	0.816369	−	0.577530i	$-0.195983\pi$
$521$	37.2679	1.63274	0.816369	+	0.577530i	$0.195983\pi$
$522$	0	0
$523$	−8.67949	−0.379528	−0.189764	−	0.981830i	$-0.560772\pi$
$523$	−8.67949	−0.379528	−0.189764	+	0.981830i	$0.560772\pi$
$524$	0	0
$525$	−2.73205	−0.119236
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	−2.53590	−0.110049
$532$	0	0
$533$	0.928203	0.0402049
$534$	0	0
$535$	0	0
$536$	0	0
$537$	23.3205	1.00635
$538$	0	0
$539$	−2.19615	−0.0945950
$540$	0	0
$541$	41.7128	1.79337	0.896687	−	0.442665i	$-0.145967\pi$
$541$	41.7128	1.79337	0.896687	+	0.442665i	$0.145967\pi$
$542$	0	0
$543$	2.39230	0.102664
$544$	0	0
$545$	−14.3923	−0.616499
$546$	0	0
$547$	−43.3205	−1.85225	−0.926126	−	0.377215i	$-0.876882\pi$
$547$	−43.3205	−1.85225	−0.926126	+	0.377215i	$0.876882\pi$
$548$	0	0
$549$	−6.53590	−0.278945
$550$	0	0
$551$	−8.19615	−0.349168
$552$	0	0
$553$	29.8564	1.26962
$554$	0	0
$555$	6.19615	0.263012
$556$	0	0
$557$	0.928203	0.0393292	0.0196646	−	0.999807i	$-0.493740\pi$
$557$	0.928203	0.0393292	0.0196646	+	0.999807i	$0.493740\pi$
$558$	0	0
$559$	−3.07180	−0.129923
$560$	0	0
$561$	0	0
$562$	0	0
$563$	27.4641	1.15747	0.578737	−	0.815514i	$-0.303546\pi$
$563$	27.4641	1.15747	0.578737	+	0.815514i	$0.303546\pi$
$564$	0	0
$565$	18.9282	0.796315
$566$	0	0
$567$	2.73205	0.114735
$568$	0	0
$569$	22.0526	0.924491	0.462246	−	0.886752i	$-0.347044\pi$
$569$	22.0526	0.924491	0.462246	+	0.886752i	$0.347044\pi$
$570$	0	0
$571$	34.2487	1.43326	0.716632	−	0.697452i	$-0.245683\pi$
$571$	34.2487	1.43326	0.716632	+	0.697452i	$0.245683\pi$
$572$	0	0
$573$	0.339746	0.0141931
$574$	0	0
$575$	−3.46410	−0.144463
$576$	0	0
$577$	15.1769	0.631823	0.315912	−	0.948789i	$-0.397690\pi$
$577$	15.1769	0.631823	0.315912	+	0.948789i	$0.397690\pi$
$578$	0	0
$579$	−17.1244	−0.711664
$580$	0	0
$581$	35.3205	1.46534
$582$	0	0
$583$	44.7846	1.85479
$584$	0	0
$585$	0.732051	0.0302666
$586$	0	0
$587$	3.46410	0.142979	0.0714894	−	0.997441i	$-0.477225\pi$
$587$	3.46410	0.142979	0.0714894	+	0.997441i	$0.477225\pi$
$588$	0	0
$589$	8.92820	0.367880
$590$	0	0
$591$	24.0000	0.987228
$592$	0	0
$593$	−38.7846	−1.59269	−0.796347	−	0.604841i	$-0.793237\pi$
$593$	−38.7846	−1.59269	−0.796347	+	0.604841i	$0.793237\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−15.3205	−0.627027
$598$	0	0
$599$	13.8564	0.566157	0.283079	−	0.959097i	$-0.408644\pi$
$599$	13.8564	0.566157	0.283079	+	0.959097i	$0.408644\pi$
$600$	0	0
$601$	−47.1769	−1.92439	−0.962193	−	0.272368i	$-0.912193\pi$
$601$	−47.1769	−1.92439	−0.962193	+	0.272368i	$0.912193\pi$
$602$	0	0
$603$	−8.00000	−0.325785
$604$	0	0
$605$	11.3923	0.463163
$606$	0	0
$607$	11.6077	0.471142	0.235571	−	0.971857i	$-0.424304\pi$
$607$	11.6077	0.471142	0.235571	+	0.971857i	$0.424304\pi$
$608$	0	0
$609$	−22.3923	−0.907382
$610$	0	0
$611$	2.53590	0.102591
$612$	0	0
$613$	42.3923	1.71221	0.856105	−	0.516803i	$-0.172878\pi$
$613$	42.3923	1.71221	0.856105	+	0.516803i	$0.172878\pi$
$614$	0	0
$615$	−1.26795	−0.0511286
$616$	0	0
$617$	27.7128	1.11568	0.557838	−	0.829950i	$-0.311631\pi$
$617$	27.7128	1.11568	0.557838	+	0.829950i	$0.311631\pi$
$618$	0	0
$619$	15.3205	0.615783	0.307892	−	0.951421i	$-0.400377\pi$
$619$	15.3205	0.615783	0.307892	+	0.951421i	$0.400377\pi$
$620$	0	0
$621$	3.46410	0.139010
$622$	0	0
$623$	29.3205	1.17470
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−4.73205	−0.188980
$628$	0	0
$629$	0	0
$630$	0	0
$631$	34.9282	1.39047	0.695235	−	0.718783i	$-0.255300\pi$
$631$	34.9282	1.39047	0.695235	+	0.718783i	$0.255300\pi$
$632$	0	0
$633$	1.07180	0.0426001
$634$	0	0
$635$	4.00000	0.158735
$636$	0	0
$637$	0.339746	0.0134612
$638$	0	0
$639$	4.39230	0.173757
$640$	0	0
$641$	48.5885	1.91913	0.959564	−	0.281489i	$-0.0908284\pi$
$641$	48.5885	1.91913	0.959564	+	0.281489i	$0.0908284\pi$
$642$	0	0
$643$	12.1962	0.480969	0.240485	−	0.970653i	$-0.422694\pi$
$643$	12.1962	0.480969	0.240485	+	0.970653i	$0.422694\pi$
$644$	0	0
$645$	4.19615	0.165223
$646$	0	0
$647$	4.14359	0.162901	0.0814507	−	0.996677i	$-0.474045\pi$
$647$	4.14359	0.162901	0.0814507	+	0.996677i	$0.474045\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	24.3923	0.956010
$652$	0	0
$653$	17.0718	0.668071	0.334036	−	0.942560i	$-0.391589\pi$
$653$	17.0718	0.668071	0.334036	+	0.942560i	$0.391589\pi$
$654$	0	0
$655$	9.12436	0.356518
$656$	0	0
$657$	−16.9282	−0.660432
$658$	0	0
$659$	−5.07180	−0.197569	−0.0987846	−	0.995109i	$-0.531495\pi$
$659$	−5.07180	−0.197569	−0.0987846	+	0.995109i	$0.531495\pi$
$660$	0	0
$661$	39.1769	1.52381	0.761903	−	0.647692i	$-0.224265\pi$
$661$	39.1769	1.52381	0.761903	+	0.647692i	$0.224265\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−2.73205	−0.105944
$666$	0	0
$667$	−28.3923	−1.09935
$668$	0	0
$669$	−17.8564	−0.690369
$670$	0	0
$671$	30.9282	1.19397
$672$	0	0
$673$	17.1244	0.660095	0.330048	−	0.943964i	$-0.392935\pi$
$673$	17.1244	0.660095	0.330048	+	0.943964i	$0.392935\pi$
$674$	0	0
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	0.679492	0.0261150	0.0130575	−	0.999915i	$-0.495844\pi$
$677$	0.679492	0.0261150	0.0130575	+	0.999915i	$0.495844\pi$
$678$	0	0
$679$	−16.9282	−0.649645
$680$	0	0
$681$	10.3923	0.398234
$682$	0	0
$683$	5.07180	0.194067	0.0970335	−	0.995281i	$-0.469065\pi$
$683$	5.07180	0.194067	0.0970335	+	0.995281i	$0.469065\pi$
$684$	0	0
$685$	19.8564	0.758674
$686$	0	0
$687$	18.5359	0.707189
$688$	0	0
$689$	−6.92820	−0.263944
$690$	0	0
$691$	−12.3923	−0.471425	−0.235713	−	0.971823i	$-0.575742\pi$
$691$	−12.3923	−0.471425	−0.235713	+	0.971823i	$0.575742\pi$
$692$	0	0
$693$	−12.9282	−0.491102
$694$	0	0
$695$	8.39230	0.318338
$696$	0	0
$697$	0	0
$698$	0	0
$699$	7.85641	0.297157
$700$	0	0
$701$	−33.7128	−1.27332	−0.636658	−	0.771147i	$-0.719684\pi$
$701$	−33.7128	−1.27332	−0.636658	+	0.771147i	$0.719684\pi$
$702$	0	0
$703$	6.19615	0.233692
$704$	0	0
$705$	−3.46410	−0.130466
$706$	0	0
$707$	−28.3923	−1.06780
$708$	0	0
$709$	−29.1769	−1.09576	−0.547881	−	0.836556i	$-0.684565\pi$
$709$	−29.1769	−1.09576	−0.547881	+	0.836556i	$0.684565\pi$
$710$	0	0
$711$	10.9282	0.409840
$712$	0	0
$713$	30.9282	1.15827
$714$	0	0
$715$	−3.46410	−0.129550
$716$	0	0
$717$	−9.80385	−0.366131
$718$	0	0
$719$	11.6603	0.434854	0.217427	−	0.976077i	$-0.430234\pi$
$719$	11.6603	0.434854	0.217427	+	0.976077i	$0.430234\pi$
$720$	0	0
$721$	−26.9282	−1.00286
$722$	0	0
$723$	3.07180	0.114241
$724$	0	0
$725$	8.19615	0.304397
$726$	0	0
$727$	−25.6603	−0.951686	−0.475843	−	0.879530i	$-0.657857\pi$
$727$	−25.6603	−0.951686	−0.475843	+	0.879530i	$0.657857\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−18.7846	−0.693825	−0.346913	−	0.937897i	$-0.612770\pi$
$733$	−18.7846	−0.693825	−0.346913	+	0.937897i	$0.612770\pi$
$734$	0	0
$735$	−0.464102	−0.0171186
$736$	0	0
$737$	37.8564	1.39446
$738$	0	0
$739$	−6.14359	−0.225996	−0.112998	−	0.993595i	$-0.536045\pi$
$739$	−6.14359	−0.225996	−0.112998	+	0.993595i	$0.536045\pi$
$740$	0	0
$741$	0.732051	0.0268926
$742$	0	0
$743$	3.21539	0.117961	0.0589806	−	0.998259i	$-0.481215\pi$
$743$	3.21539	0.117961	0.0589806	+	0.998259i	$0.481215\pi$
$744$	0	0
$745$	−19.8564	−0.727482
$746$	0	0
$747$	12.9282	0.473018
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−26.0000	−0.948753	−0.474377	−	0.880322i	$-0.657327\pi$
$751$	−26.0000	−0.948753	−0.474377	+	0.880322i	$0.657327\pi$
$752$	0	0
$753$	28.0526	1.02229
$754$	0	0
$755$	−14.0000	−0.509512
$756$	0	0
$757$	32.2487	1.17210	0.586050	−	0.810275i	$-0.300682\pi$
$757$	32.2487	1.17210	0.586050	+	0.810275i	$0.300682\pi$
$758$	0	0
$759$	−16.3923	−0.595003
$760$	0	0
$761$	6.00000	0.217500	0.108750	−	0.994069i	$-0.465315\pi$
$761$	6.00000	0.217500	0.108750	+	0.994069i	$0.465315\pi$
$762$	0	0
$763$	−39.3205	−1.42350
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−1.85641	−0.0670310
$768$	0	0
$769$	−20.6410	−0.744334	−0.372167	−	0.928166i	$-0.621385\pi$
$769$	−20.6410	−0.744334	−0.372167	+	0.928166i	$0.621385\pi$
$770$	0	0
$771$	24.0000	0.864339
$772$	0	0
$773$	−25.1769	−0.905551	−0.452775	−	0.891625i	$-0.649566\pi$
$773$	−25.1769	−0.905551	−0.452775	+	0.891625i	$0.649566\pi$
$774$	0	0
$775$	−8.92820	−0.320711
$776$	0	0
$777$	16.9282	0.607296
$778$	0	0
$779$	−1.26795	−0.0454290
$780$	0	0
$781$	−20.7846	−0.743732
$782$	0	0
$783$	−8.19615	−0.292907
$784$	0	0
$785$	6.39230	0.228151
$786$	0	0
$787$	−8.67949	−0.309390	−0.154695	−	0.987962i	$-0.549440\pi$
$787$	−8.67949	−0.309390	−0.154695	+	0.987962i	$0.549440\pi$
$788$	0	0
$789$	6.00000	0.213606
$790$	0	0
$791$	51.7128	1.83870
$792$	0	0
$793$	−4.78461	−0.169906
$794$	0	0
$795$	9.46410	0.335657
$796$	0	0
$797$	44.7846	1.58635	0.793176	−	0.608992i	$-0.208426\pi$
$797$	44.7846	1.58635	0.793176	+	0.608992i	$0.208426\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	10.7321	0.379198
$802$	0	0
$803$	80.1051	2.82685
$804$	0	0
$805$	−9.46410	−0.333566
$806$	0	0
$807$	−0.588457	−0.0207147
$808$	0	0
$809$	14.7846	0.519799	0.259900	−	0.965636i	$-0.416311\pi$
$809$	14.7846	0.519799	0.259900	+	0.965636i	$0.416311\pi$
$810$	0	0
$811$	−37.5692	−1.31923	−0.659617	−	0.751602i	$-0.729281\pi$
$811$	−37.5692	−1.31923	−0.659617	+	0.751602i	$0.729281\pi$
$812$	0	0
$813$	0.392305	0.0137587
$814$	0	0
$815$	−9.26795	−0.324642
$816$	0	0
$817$	4.19615	0.146805
$818$	0	0
$819$	2.00000	0.0698857
$820$	0	0
$821$	−32.5359	−1.13551	−0.567755	−	0.823197i	$-0.692188\pi$
$821$	−32.5359	−1.13551	−0.567755	+	0.823197i	$0.692188\pi$
$822$	0	0
$823$	−12.9808	−0.452481	−0.226240	−	0.974071i	$-0.572644\pi$
$823$	−12.9808	−0.452481	−0.226240	+	0.974071i	$0.572644\pi$
$824$	0	0
$825$	4.73205	0.164749
$826$	0	0
$827$	5.32051	0.185012	0.0925061	−	0.995712i	$-0.470512\pi$
$827$	5.32051	0.185012	0.0925061	+	0.995712i	$0.470512\pi$
$828$	0	0
$829$	34.1051	1.18452	0.592260	−	0.805747i	$-0.298236\pi$
$829$	34.1051	1.18452	0.592260	+	0.805747i	$0.298236\pi$
$830$	0	0
$831$	−2.00000	−0.0693792
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−3.46410	−0.119880
$836$	0	0
$837$	8.92820	0.308604
$838$	0	0
$839$	19.6077	0.676933	0.338466	−	0.940978i	$-0.390092\pi$
$839$	19.6077	0.676933	0.338466	+	0.940978i	$0.390092\pi$
$840$	0	0
$841$	38.1769	1.31645
$842$	0	0
$843$	−1.26795	−0.0436705
$844$	0	0
$845$	−12.4641	−0.428778
$846$	0	0
$847$	31.1244	1.06945
$848$	0	0
$849$	24.9808	0.857338
$850$	0	0
$851$	21.4641	0.735780
$852$	0	0
$853$	27.1769	0.930520	0.465260	−	0.885174i	$-0.345961\pi$
$853$	27.1769	0.930520	0.465260	+	0.885174i	$0.345961\pi$
$854$	0	0
$855$	−1.00000	−0.0341993
$856$	0	0
$857$	−42.2487	−1.44319	−0.721594	−	0.692316i	$-0.756590\pi$
$857$	−42.2487	−1.44319	−0.721594	+	0.692316i	$0.756590\pi$
$858$	0	0
$859$	−32.0000	−1.09183	−0.545913	−	0.837842i	$-0.683817\pi$
$859$	−32.0000	−1.09183	−0.545913	+	0.837842i	$0.683817\pi$
$860$	0	0
$861$	−3.46410	−0.118056
$862$	0	0
$863$	12.0000	0.408485	0.204242	−	0.978920i	$-0.434527\pi$
$863$	12.0000	0.408485	0.204242	+	0.978920i	$0.434527\pi$
$864$	0	0
$865$	6.92820	0.235566
$866$	0	0
$867$	17.0000	0.577350
$868$	0	0
$869$	−51.7128	−1.75424
$870$	0	0
$871$	−5.85641	−0.198437
$872$	0	0
$873$	−6.19615	−0.209708
$874$	0	0
$875$	2.73205	0.0923602
$876$	0	0
$877$	53.1244	1.79388	0.896941	−	0.442150i	$-0.145784\pi$
$877$	53.1244	1.79388	0.896941	+	0.442150i	$0.145784\pi$
$878$	0	0
$879$	−27.7128	−0.934730
$880$	0	0
$881$	8.53590	0.287582	0.143791	−	0.989608i	$-0.454071\pi$
$881$	8.53590	0.287582	0.143791	+	0.989608i	$0.454071\pi$
$882$	0	0
$883$	−36.9808	−1.24450	−0.622251	−	0.782818i	$-0.713782\pi$
$883$	−36.9808	−1.24450	−0.622251	+	0.782818i	$0.713782\pi$
$884$	0	0
$885$	2.53590	0.0852433
$886$	0	0
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	0	0
$889$	10.9282	0.366520
$890$	0	0
$891$	−4.73205	−0.158530
$892$	0	0
$893$	−3.46410	−0.115922
$894$	0	0
$895$	−23.3205	−0.779519
$896$	0	0
$897$	2.53590	0.0846712
$898$	0	0
$899$	−73.1769	−2.44059
$900$	0	0
$901$	0	0
$902$	0	0
$903$	11.4641	0.381501
$904$	0	0
$905$	−2.39230	−0.0795229
$906$	0	0
$907$	32.3923	1.07557	0.537784	−	0.843082i	$-0.319261\pi$
$907$	32.3923	1.07557	0.537784	+	0.843082i	$0.319261\pi$
$908$	0	0
$909$	−10.3923	−0.344691
$910$	0	0
$911$	54.9282	1.81985	0.909926	−	0.414770i	$-0.136138\pi$
$911$	54.9282	1.81985	0.909926	+	0.414770i	$0.136138\pi$
$912$	0	0
$913$	−61.1769	−2.02466
$914$	0	0
$915$	6.53590	0.216070
$916$	0	0
$917$	24.9282	0.823202
$918$	0	0
$919$	−51.4256	−1.69637	−0.848187	−	0.529696i	$-0.822306\pi$
$919$	−51.4256	−1.69637	−0.848187	+	0.529696i	$0.822306\pi$
$920$	0	0
$921$	−32.3923	−1.06736
$922$	0	0
$923$	3.21539	0.105836
$924$	0	0
$925$	−6.19615	−0.203728
$926$	0	0
$927$	−9.85641	−0.323727
$928$	0	0
$929$	−1.60770	−0.0527468	−0.0263734	−	0.999652i	$-0.508396\pi$
$929$	−1.60770	−0.0527468	−0.0263734	+	0.999652i	$0.508396\pi$
$930$	0	0
$931$	−0.464102	−0.0152103
$932$	0	0
$933$	32.4449	1.06220
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−16.2487	−0.530822	−0.265411	−	0.964135i	$-0.585508\pi$
$937$	−16.2487	−0.530822	−0.265411	+	0.964135i	$0.585508\pi$
$938$	0	0
$939$	−6.39230	−0.208605
$940$	0	0
$941$	0.588457	0.0191832	0.00959158	−	0.999954i	$-0.496947\pi$
$941$	0.588457	0.0191832	0.00959158	+	0.999954i	$0.496947\pi$
$942$	0	0
$943$	−4.39230	−0.143033
$944$	0	0
$945$	−2.73205	−0.0888736
$946$	0	0
$947$	−28.1436	−0.914544	−0.457272	−	0.889327i	$-0.651173\pi$
$947$	−28.1436	−0.914544	−0.457272	+	0.889327i	$0.651173\pi$
$948$	0	0
$949$	−12.3923	−0.402271
$950$	0	0
$951$	11.3205	0.367093
$952$	0	0
$953$	37.8564	1.22629	0.613145	−	0.789971i	$-0.289904\pi$
$953$	37.8564	1.22629	0.613145	+	0.789971i	$0.289904\pi$
$954$	0	0
$955$	−0.339746	−0.0109939
$956$	0	0
$957$	38.7846	1.25373
$958$	0	0
$959$	54.2487	1.75178
$960$	0	0
$961$	48.7128	1.57138
$962$	0	0
$963$	0	0
$964$	0	0
$965$	17.1244	0.551253
$966$	0	0
$967$	−4.87564	−0.156790	−0.0783951	−	0.996922i	$-0.524980\pi$
$967$	−4.87564	−0.156790	−0.0783951	+	0.996922i	$0.524980\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	27.7128	0.889346	0.444673	−	0.895693i	$-0.353320\pi$
$971$	27.7128	0.889346	0.444673	+	0.895693i	$0.353320\pi$
$972$	0	0
$973$	22.9282	0.735044
$974$	0	0
$975$	−0.732051	−0.0234444
$976$	0	0
$977$	−39.0333	−1.24879	−0.624393	−	0.781110i	$-0.714654\pi$
$977$	−39.0333	−1.24879	−0.624393	+	0.781110i	$0.714654\pi$
$978$	0	0
$979$	−50.7846	−1.62308
$980$	0	0
$981$	−14.3923	−0.459511
$982$	0	0
$983$	41.3205	1.31792	0.658960	−	0.752178i	$-0.270997\pi$
$983$	41.3205	1.31792	0.658960	+	0.752178i	$0.270997\pi$
$984$	0	0
$985$	−24.0000	−0.764704
$986$	0	0
$987$	−9.46410	−0.301246
$988$	0	0
$989$	14.5359	0.462215
$990$	0	0
$991$	−13.0718	−0.415239	−0.207620	−	0.978210i	$-0.566572\pi$
$991$	−13.0718	−0.415239	−0.207620	+	0.978210i	$0.566572\pi$
$992$	0	0
$993$	−25.7128	−0.815971
$994$	0	0
$995$	15.3205	0.485693
$996$	0	0
$997$	−17.6077	−0.557641	−0.278821	−	0.960343i	$-0.589944\pi$
$997$	−17.6077	−0.557641	−0.278821	+	0.960343i	$0.589944\pi$
$998$	0	0
$999$	6.19615	0.196038

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4560.2.a.bh.1.2		2
4.3	odd	2		285.2.a.e.1.1	✓	2
12.11	even	2		855.2.a.f.1.2		2
20.3	even	4		1425.2.c.k.799.4		4
20.7	even	4		1425.2.c.k.799.1		4
20.19	odd	2		1425.2.a.o.1.2		2
60.59	even	2		4275.2.a.t.1.1		2
76.75	even	2		5415.2.a.r.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
285.2.a.e.1.1	✓	2	4.3	odd	2
855.2.a.f.1.2		2	12.11	even	2
1425.2.a.o.1.2		2	20.19	odd	2
1425.2.c.k.799.1		4	20.7	even	4
1425.2.c.k.799.4		4	20.3	even	4
4275.2.a.t.1.1		2	60.59	even	2
4560.2.a.bh.1.2		2	1.1	even	1	trivial
5415.2.a.r.1.2		2	76.75	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +1.00000 q^{5} +2.73205 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +1.00000 q^{5} +2.73205 q^{7} +1.00000 q^{9} -4.73205 q^{11} +0.732051 q^{13} -1.00000 q^{15} -1.00000 q^{19} -2.73205 q^{21} -3.46410 q^{23} +1.00000 q^{25} -1.00000 q^{27} +8.19615 q^{29} -8.92820 q^{31} +4.73205 q^{33} +2.73205 q^{35} -6.19615 q^{37} -0.732051 q^{39} +1.26795 q^{41} -4.19615 q^{43} +1.00000 q^{45} +3.46410 q^{47} +0.464102 q^{49} -9.46410 q^{53} -4.73205 q^{55} +1.00000 q^{57} -2.53590 q^{59} -6.53590 q^{61} +2.73205 q^{63} +0.732051 q^{65} -8.00000 q^{67} +3.46410 q^{69} +4.39230 q^{71} -16.9282 q^{73} -1.00000 q^{75} -12.9282 q^{77} +10.9282 q^{79} +1.00000 q^{81} +12.9282 q^{83} -8.19615 q^{87} +10.7321 q^{89} +2.00000 q^{91} +8.92820 q^{93} -1.00000 q^{95} -6.19615 q^{97} -4.73205 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9	\( 2 q - 2 q^{3} + 2 q^{5} + 2 q^{7} + 2 q^{9} - 6 q^{11} - 2 q^{13} - 2 q^{15} - 2 q^{19} - 2 q^{21} + 2 q^{25} - 2 q^{27} + 6 q^{29} - 4 q^{31} + 6 q^{33} + 2 q^{35} - 2 q^{37} + 2 q^{39} + 6 q^{41} + 2 q^{43} + 2 q^{45} - 6 q^{49} - 12 q^{53} - 6 q^{55} + 2 q^{57} - 12 q^{59} - 20 q^{61} + 2 q^{63} - 2 q^{65} - 16 q^{67} - 12 q^{71} - 20 q^{73} - 2 q^{75} - 12 q^{77} + 8 q^{79} + 2 q^{81} + 12 q^{83} - 6 q^{87} + 18 q^{89} + 4 q^{91} + 4 q^{93} - 2 q^{95} - 2 q^{97} - 6 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 + 2 * q^5 + 2 * q^7 + 2 * q^9 - 6 * q^11 - 2 * q^13 - 2 * q^15 - 2 * q^19 - 2 * q^21 + 2 * q^25 - 2 * q^27 + 6 * q^29 - 4 * q^31 + 6 * q^33 + 2 * q^35 - 2 * q^37 + 2 * q^39 + 6 * q^41 + 2 * q^43 + 2 * q^45 - 6 * q^49 - 12 * q^53 - 6 * q^55 + 2 * q^57 - 12 * q^59 - 20 * q^61 + 2 * q^63 - 2 * q^65 - 16 * q^67 - 12 * q^71 - 20 * q^73 - 2 * q^75 - 12 * q^77 + 8 * q^79 + 2 * q^81 + 12 * q^83 - 6 * q^87 + 18 * q^89 + 4 * q^91 + 4 * q^93 - 2 * q^95 - 2 * q^97 - 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more