LMFDB - Embedded newform 7728.2.a.cf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7728,2,Mod(1,7728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7728, 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("7728.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7728.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:	$61.7083906820$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.17679757.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 14x^{3} + 17x^{2} + 23x - 10 $ x^5 - x^4 - 14x^3 + 17x^2 + 23*x - 10
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 3864)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.26247$ of defining polynomial
Character	$\chi$	$=$	7728.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} -3.26247 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.26247 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.38124 q^{11} -3.69307 q^{13} +3.26247 q^{15} -4.95554 q^{17} -4.43060 q^{19} +1.00000 q^{21} +1.00000 q^{23} +5.64371 q^{25} -1.00000 q^{27} -6.66239 q^{29} -6.95554 q^{31} -1.38124 q^{33} +3.26247 q^{35} +6.66239 q^{37} +3.69307 q^{39} +6.66239 q^{41} -1.35629 q^{43} -3.26247 q^{45} -11.6179 q^{47} +1.00000 q^{49} +4.95554 q^{51} -9.06856 q^{53} -4.50626 q^{55} +4.43060 q^{57} +4.78116 q^{59} -7.92486 q^{61} -1.00000 q^{63} +12.0485 q^{65} -11.3061 q^{67} -1.00000 q^{69} -12.4498 q^{71} +5.90618 q^{73} -5.64371 q^{75} -1.38124 q^{77} -1.38124 q^{79} +1.00000 q^{81} -17.4805 q^{83} +16.1673 q^{85} +6.66239 q^{87} -2.11302 q^{89} +3.69307 q^{91} +6.95554 q^{93} +14.4547 q^{95} -0.474237 q^{97} +1.38124 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} - 6 q^{5} - 5 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 - 6 * q^5 - 5 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} - 6 q^{5} - 5 q^{7} + 5 q^{9} + 4 q^{13} + 6 q^{15} + 8 q^{17} - 10 q^{19} + 5 q^{21} + 5 q^{23} + 11 q^{25} - 5 q^{27} - 3 q^{29} - 2 q^{31} + 6 q^{35} + 3 q^{37} - 4 q^{39} + 3 q^{41} - 24 q^{43} - 6 q^{45} + 5 q^{47} + 5 q^{49} - 8 q^{51} + 9 q^{53} - 15 q^{55} + 10 q^{57} - 3 q^{59} + q^{61} - 5 q^{63} - 15 q^{65} - 9 q^{67} - 5 q^{69} - q^{71} + 2 q^{73} - 11 q^{75} + 5 q^{81} - 34 q^{83} + 9 q^{85} + 3 q^{87} + 11 q^{89} - 4 q^{91} + 2 q^{93} - 27 q^{95} + 47 q^{97}+O(q^{100}) $ 5 * q - 5 * q^3 - 6 * q^5 - 5 * q^7 + 5 * q^9 + 4 * q^13 + 6 * q^15 + 8 * q^17 - 10 * q^19 + 5 * q^21 + 5 * q^23 + 11 * q^25 - 5 * q^27 - 3 * q^29 - 2 * q^31 + 6 * q^35 + 3 * q^37 - 4 * q^39 + 3 * q^41 - 24 * q^43 - 6 * q^45 + 5 * q^47 + 5 * q^49 - 8 * q^51 + 9 * q^53 - 15 * q^55 + 10 * q^57 - 3 * q^59 + q^61 - 5 * q^63 - 15 * q^65 - 9 * q^67 - 5 * q^69 - q^71 + 2 * q^73 - 11 * q^75 + 5 * q^81 - 34 * q^83 + 9 * q^85 + 3 * q^87 + 11 * q^89 - 4 * q^91 + 2 * q^93 - 27 * q^95 + 47 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	−3.26247	−1.45902	−0.729510	−	0.683970i	$-0.760252\pi$
$5$	−3.26247	−1.45902	−0.729510	+	0.683970i	$0.760252\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.38124	0.416460	0.208230	−	0.978080i	$-0.433230\pi$
$11$	1.38124	0.416460	0.208230	+	0.978080i	$0.433230\pi$
$12$	0	0
$13$	−3.69307	−1.02427	−0.512137	−	0.858904i	$-0.671146\pi$
$13$	−3.69307	−1.02427	−0.512137	+	0.858904i	$0.671146\pi$
$14$	0	0
$15$	3.26247	0.842366
$16$	0	0
$17$	−4.95554	−1.20190	−0.600948	−	0.799288i	$-0.705210\pi$
$17$	−4.95554	−1.20190	−0.600948	+	0.799288i	$0.705210\pi$
$18$	0	0
$19$	−4.43060	−1.01645	−0.508225	−	0.861224i	$-0.669698\pi$
$19$	−4.43060	−1.01645	−0.508225	+	0.861224i	$0.669698\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	5.64371	1.12874
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.66239	−1.23718	−0.618588	−	0.785716i	$-0.712295\pi$
$29$	−6.66239	−1.23718	−0.618588	+	0.785716i	$0.712295\pi$
$30$	0	0
$31$	−6.95554	−1.24925	−0.624626	−	0.780924i	$-0.714749\pi$
$31$	−6.95554	−1.24925	−0.624626	+	0.780924i	$0.714749\pi$
$32$	0	0
$33$	−1.38124	−0.240443
$34$	0	0
$35$	3.26247	0.551458
$36$	0	0
$37$	6.66239	1.09529	0.547645	−	0.836711i	$-0.315524\pi$
$37$	6.66239	1.09529	0.547645	+	0.836711i	$0.315524\pi$
$38$	0	0
$39$	3.69307	0.591365
$40$	0	0
$41$	6.66239	1.04049	0.520246	−	0.854017i	$-0.325840\pi$
$41$	6.66239	1.04049	0.520246	+	0.854017i	$0.325840\pi$
$42$	0	0
$43$	−1.35629	−0.206832	−0.103416	−	0.994638i	$-0.532977\pi$
$43$	−1.35629	−0.206832	−0.103416	+	0.994638i	$0.532977\pi$
$44$	0	0
$45$	−3.26247	−0.486340
$46$	0	0
$47$	−11.6179	−1.69465	−0.847325	−	0.531075i	$-0.821788\pi$
$47$	−11.6179	−1.69465	−0.847325	+	0.531075i	$0.821788\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	4.95554	0.693915
$52$	0	0
$53$	−9.06856	−1.24566	−0.622831	−	0.782356i	$-0.714018\pi$
$53$	−9.06856	−1.24566	−0.622831	+	0.782356i	$0.714018\pi$
$54$	0	0
$55$	−4.50626	−0.607624
$56$	0	0
$57$	4.43060	0.586848
$58$	0	0
$59$	4.78116	0.622455	0.311227	−	0.950336i	$-0.399260\pi$
$59$	4.78116	0.622455	0.311227	+	0.950336i	$0.399260\pi$
$60$	0	0
$61$	−7.92486	−1.01467	−0.507337	−	0.861748i	$-0.669370\pi$
$61$	−7.92486	−1.01467	−0.507337	+	0.861748i	$0.669370\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	12.0485	1.49444
$66$	0	0
$67$	−11.3061	−1.38126	−0.690630	−	0.723208i	$-0.742667\pi$
$67$	−11.3061	−1.38126	−0.690630	+	0.723208i	$0.742667\pi$
$68$	0	0
$69$	−1.00000	−0.120386
$70$	0	0
$71$	−12.4498	−1.47752	−0.738760	−	0.673969i	$-0.764588\pi$
$71$	−12.4498	−1.47752	−0.738760	+	0.673969i	$0.764588\pi$
$72$	0	0
$73$	5.90618	0.691266	0.345633	−	0.938370i	$-0.387664\pi$
$73$	5.90618	0.691266	0.345633	+	0.938370i	$0.387664\pi$
$74$	0	0
$75$	−5.64371	−0.651680
$76$	0	0
$77$	−1.38124	−0.157407
$78$	0	0
$79$	−1.38124	−0.155402	−0.0777009	−	0.996977i	$-0.524758\pi$
$79$	−1.38124	−0.155402	−0.0777009	+	0.996977i	$0.524758\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−17.4805	−1.91873	−0.959366	−	0.282164i	$-0.908948\pi$
$83$	−17.4805	−1.91873	−0.959366	+	0.282164i	$0.908948\pi$
$84$	0	0
$85$	16.1673	1.75359
$86$	0	0
$87$	6.66239	0.714284
$88$	0	0
$89$	−2.11302	−0.223980	−0.111990	−	0.993709i	$-0.535722\pi$
$89$	−2.11302	−0.223980	−0.111990	+	0.993709i	$0.535722\pi$
$90$	0	0
$91$	3.69307	0.387139
$92$	0	0
$93$	6.95554	0.721256
$94$	0	0
$95$	14.4547	1.48302
$96$	0	0
$97$	−0.474237	−0.0481515	−0.0240757	−	0.999710i	$-0.507664\pi$
$97$	−0.474237	−0.0481515	−0.0240757	+	0.999710i	$0.507664\pi$
$98$	0	0
$99$	1.38124	0.138820
$100$	0	0
$101$	4.35547	0.433385	0.216693	−	0.976240i	$-0.430473\pi$
$101$	4.35547	0.433385	0.216693	+	0.976240i	$0.430473\pi$
$102$	0	0
$103$	5.80609	0.572091	0.286046	−	0.958216i	$-0.407659\pi$
$103$	5.80609	0.572091	0.286046	+	0.958216i	$0.407659\pi$
$104$	0	0
$105$	−3.26247	−0.318384
$106$	0	0
$107$	9.92486	0.959473	0.479736	−	0.877413i	$-0.340732\pi$
$107$	9.92486	0.959473	0.479736	+	0.877413i	$0.340732\pi$
$108$	0	0
$109$	−4.12367	−0.394976	−0.197488	−	0.980305i	$-0.563278\pi$
$109$	−4.12367	−0.394976	−0.197488	+	0.980305i	$0.563278\pi$
$110$	0	0
$111$	−6.66239	−0.632366
$112$	0	0
$113$	−3.02493	−0.284561	−0.142281	−	0.989826i	$-0.545444\pi$
$113$	−3.02493	−0.284561	−0.142281	+	0.989826i	$0.545444\pi$
$114$	0	0
$115$	−3.26247	−0.304227
$116$	0	0
$117$	−3.69307	−0.341425
$118$	0	0
$119$	4.95554	0.454274
$120$	0	0
$121$	−9.09217	−0.826561
$122$	0	0
$123$	−6.66239	−0.600728
$124$	0	0
$125$	−2.10009	−0.187838
$126$	0	0
$127$	12.5548	1.11406	0.557029	−	0.830493i	$-0.311941\pi$
$127$	12.5548	1.11406	0.557029	+	0.830493i	$0.311941\pi$
$128$	0	0
$129$	1.35629	0.119415
$130$	0	0
$131$	−19.5299	−1.70633	−0.853166	−	0.521639i	$-0.825321\pi$
$131$	−19.5299	−1.70633	−0.853166	+	0.521639i	$0.825321\pi$
$132$	0	0
$133$	4.43060	0.384182
$134$	0	0
$135$	3.26247	0.280789
$136$	0	0
$137$	17.1873	1.46841	0.734207	−	0.678926i	$-0.237554\pi$
$137$	17.1873	1.46841	0.734207	+	0.678926i	$0.237554\pi$
$138$	0	0
$139$	−11.5499	−0.979647	−0.489824	−	0.871822i	$-0.662939\pi$
$139$	−11.5499	−0.979647	−0.489824	+	0.871822i	$0.662939\pi$
$140$	0	0
$141$	11.6179	0.978407
$142$	0	0
$143$	−5.10102	−0.426569
$144$	0	0
$145$	21.7359	1.80506
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−8.52494	−0.698390	−0.349195	−	0.937050i	$-0.613545\pi$
$149$	−8.52494	−0.698390	−0.349195	+	0.937050i	$0.613545\pi$
$150$	0	0
$151$	−6.04363	−0.491824	−0.245912	−	0.969292i	$-0.579087\pi$
$151$	−6.04363	−0.491824	−0.245912	+	0.969292i	$0.579087\pi$
$152$	0	0
$153$	−4.95554	−0.400632
$154$	0	0
$155$	22.6922	1.82269
$156$	0	0
$157$	9.76246	0.779129	0.389564	−	0.920999i	$-0.372626\pi$
$157$	9.76246	0.779129	0.389564	+	0.920999i	$0.372626\pi$
$158$	0	0
$159$	9.06856	0.719184
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	6.60498	0.517342	0.258671	−	0.965965i	$-0.416715\pi$
$163$	6.60498	0.517342	0.258671	+	0.965965i	$0.416715\pi$
$164$	0	0
$165$	4.50626	0.350812
$166$	0	0
$167$	−8.61928	−0.666980	−0.333490	−	0.942754i	$-0.608226\pi$
$167$	−8.61928	−0.666980	−0.333490	+	0.942754i	$0.608226\pi$
$168$	0	0
$169$	0.638783	0.0491372
$170$	0	0
$171$	−4.43060	−0.338817
$172$	0	0
$173$	−6.43060	−0.488910	−0.244455	−	0.969661i	$-0.578609\pi$
$173$	−6.43060	−0.488910	−0.244455	+	0.969661i	$0.578609\pi$
$174$	0	0
$175$	−5.64371	−0.426624
$176$	0	0
$177$	−4.78116	−0.359374
$178$	0	0
$179$	−17.6985	−1.32285	−0.661424	−	0.750012i	$-0.730048\pi$
$179$	−17.6985	−1.32285	−0.661424	+	0.750012i	$0.730048\pi$
$180$	0	0
$181$	−2.14372	−0.159342	−0.0796709	−	0.996821i	$-0.525387\pi$
$181$	−2.14372	−0.159342	−0.0796709	+	0.996821i	$0.525387\pi$
$182$	0	0
$183$	7.92486	0.585823
$184$	0	0
$185$	−21.7359	−1.59805
$186$	0	0
$187$	−6.84480	−0.500541
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	16.7011	1.20845	0.604225	−	0.796813i	$-0.293483\pi$
$191$	16.7011	1.20845	0.604225	+	0.796813i	$0.293483\pi$
$192$	0	0
$193$	20.6299	1.48498	0.742488	−	0.669860i	$-0.233646\pi$
$193$	20.6299	1.48498	0.742488	+	0.669860i	$0.233646\pi$
$194$	0	0
$195$	−12.0485	−0.862814
$196$	0	0
$197$	−19.7923	−1.41014	−0.705072	−	0.709136i	$-0.749085\pi$
$197$	−19.7923	−1.41014	−0.705072	+	0.709136i	$0.749085\pi$
$198$	0	0
$199$	−20.0348	−1.42023	−0.710113	−	0.704088i	$-0.751356\pi$
$199$	−20.0348	−1.42023	−0.710113	+	0.704088i	$0.751356\pi$
$200$	0	0
$201$	11.3061	0.797471
$202$	0	0
$203$	6.66239	0.467608
$204$	0	0
$205$	−21.7359	−1.51810
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−6.11973	−0.423311
$210$	0	0
$211$	−17.3910	−1.19725	−0.598625	−	0.801030i	$-0.704286\pi$
$211$	−17.3910	−1.19725	−0.598625	+	0.801030i	$0.704286\pi$
$212$	0	0
$213$	12.4498	0.853047
$214$	0	0
$215$	4.42485	0.301772
$216$	0	0
$217$	6.95554	0.472173
$218$	0	0
$219$	−5.90618	−0.399103
$220$	0	0
$221$	18.3012	1.23107
$222$	0	0
$223$	−7.79316	−0.521869	−0.260934	−	0.965357i	$-0.584031\pi$
$223$	−7.79316	−0.521869	−0.260934	+	0.965357i	$0.584031\pi$
$224$	0	0
$225$	5.64371	0.376247
$226$	0	0
$227$	3.31186	0.219816	0.109908	−	0.993942i	$-0.464944\pi$
$227$	3.31186	0.219816	0.109908	+	0.993942i	$0.464944\pi$
$228$	0	0
$229$	16.8799	1.11545	0.557727	−	0.830024i	$-0.311674\pi$
$229$	16.8799	1.11545	0.557727	+	0.830024i	$0.311674\pi$
$230$	0	0
$231$	1.38124	0.0908790
$232$	0	0
$233$	18.3560	1.20254	0.601270	−	0.799046i	$-0.294662\pi$
$233$	18.3560	1.20254	0.601270	+	0.799046i	$0.294662\pi$
$234$	0	0
$235$	37.9032	2.47253
$236$	0	0
$237$	1.38124	0.0897213
$238$	0	0
$239$	12.1673	0.787038	0.393519	−	0.919317i	$-0.371258\pi$
$239$	12.1673	0.787038	0.393519	+	0.919317i	$0.371258\pi$
$240$	0	0
$241$	30.5740	1.96944	0.984722	−	0.174133i	$-0.0557122\pi$
$241$	30.5740	1.96944	0.984722	+	0.174133i	$0.0557122\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−3.26247	−0.208432
$246$	0	0
$247$	16.3625	1.04112
$248$	0	0
$249$	17.4805	1.10778
$250$	0	0
$251$	−9.33054	−0.588938	−0.294469	−	0.955661i	$-0.595143\pi$
$251$	−9.33054	−0.588938	−0.294469	+	0.955661i	$0.595143\pi$
$252$	0	0
$253$	1.38124	0.0868379
$254$	0	0
$255$	−16.1673	−1.01244
$256$	0	0
$257$	7.93864	0.495199	0.247599	−	0.968862i	$-0.420358\pi$
$257$	7.93864	0.495199	0.247599	+	0.968862i	$0.420358\pi$
$258$	0	0
$259$	−6.66239	−0.413981
$260$	0	0
$261$	−6.66239	−0.412392
$262$	0	0
$263$	−4.66239	−0.287495	−0.143748	−	0.989614i	$-0.545915\pi$
$263$	−4.66239	−0.287495	−0.143748	+	0.989614i	$0.545915\pi$
$264$	0	0
$265$	29.5859	1.81745
$266$	0	0
$267$	2.11302	0.129315
$268$	0	0
$269$	−8.17438	−0.498401	−0.249200	−	0.968452i	$-0.580168\pi$
$269$	−8.17438	−0.498401	−0.249200	+	0.968452i	$0.580168\pi$
$270$	0	0
$271$	−6.71800	−0.408089	−0.204045	−	0.978962i	$-0.565409\pi$
$271$	−6.71800	−0.408089	−0.204045	+	0.978962i	$0.565409\pi$
$272$	0	0
$273$	−3.69307	−0.223515
$274$	0	0
$275$	7.79533	0.470076
$276$	0	0
$277$	22.0735	1.32627	0.663134	−	0.748501i	$-0.269226\pi$
$277$	22.0735	1.32627	0.663134	+	0.748501i	$0.269226\pi$
$278$	0	0
$279$	−6.95554	−0.416417
$280$	0	0
$281$	11.1922	0.667673	0.333836	−	0.942631i	$-0.391657\pi$
$281$	11.1922	0.667673	0.333836	+	0.942631i	$0.391657\pi$
$282$	0	0
$283$	21.9303	1.30362	0.651810	−	0.758382i	$-0.274010\pi$
$283$	21.9303	1.30362	0.651810	+	0.758382i	$0.274010\pi$
$284$	0	0
$285$	−14.4547	−0.856223
$286$	0	0
$287$	−6.66239	−0.393269
$288$	0	0
$289$	7.55740	0.444553
$290$	0	0
$291$	0.474237	0.0278003
$292$	0	0
$293$	−6.86120	−0.400836	−0.200418	−	0.979710i	$-0.564230\pi$
$293$	−6.86120	−0.400836	−0.200418	+	0.979710i	$0.564230\pi$
$294$	0	0
$295$	−15.5984	−0.908174
$296$	0	0
$297$	−1.38124	−0.0801477
$298$	0	0
$299$	−3.69307	−0.213576
$300$	0	0
$301$	1.35629	0.0781752
$302$	0	0
$303$	−4.35547	−0.250215
$304$	0	0
$305$	25.8546	1.48043
$306$	0	0
$307$	14.8110	0.845310	0.422655	−	0.906291i	$-0.361098\pi$
$307$	14.8110	0.845310	0.422655	+	0.906291i	$0.361098\pi$
$308$	0	0
$309$	−5.80609	−0.330297
$310$	0	0
$311$	−6.64946	−0.377056	−0.188528	−	0.982068i	$-0.560372\pi$
$311$	−6.64946	−0.377056	−0.188528	+	0.982068i	$0.560372\pi$
$312$	0	0
$313$	−13.5727	−0.767172	−0.383586	−	0.923505i	$-0.625311\pi$
$313$	−13.5727	−0.767172	−0.383586	+	0.923505i	$0.625311\pi$
$314$	0	0
$315$	3.26247	0.183819
$316$	0	0
$317$	−19.1122	−1.07345	−0.536724	−	0.843758i	$-0.680338\pi$
$317$	−19.1122	−1.07345	−0.536724	+	0.843758i	$0.680338\pi$
$318$	0	0
$319$	−9.20237	−0.515234
$320$	0	0
$321$	−9.92486	−0.553952
$322$	0	0
$323$	21.9560	1.22167
$324$	0	0
$325$	−20.8426	−1.15614
$326$	0	0
$327$	4.12367	0.228040
$328$	0	0
$329$	11.6179	0.640518
$330$	0	0
$331$	14.2874	0.785306	0.392653	−	0.919687i	$-0.371557\pi$
$331$	14.2874	0.785306	0.392653	+	0.919687i	$0.371557\pi$
$332$	0	0
$333$	6.66239	0.365097
$334$	0	0
$335$	36.8858	2.01529
$336$	0	0
$337$	9.39992	0.512046	0.256023	−	0.966671i	$-0.417588\pi$
$337$	9.39992	0.512046	0.256023	+	0.966671i	$0.417588\pi$
$338$	0	0
$339$	3.02493	0.164292
$340$	0	0
$341$	−9.60728	−0.520263
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	3.26247	0.175645
$346$	0	0
$347$	−7.77363	−0.417310	−0.208655	−	0.977989i	$-0.566909\pi$
$347$	−7.77363	−0.417310	−0.208655	+	0.977989i	$0.566909\pi$
$348$	0	0
$349$	12.1678	0.651329	0.325664	−	0.945485i	$-0.394412\pi$
$349$	12.1678	0.651329	0.325664	+	0.945485i	$0.394412\pi$
$350$	0	0
$351$	3.69307	0.197122
$352$	0	0
$353$	−15.1473	−0.806209	−0.403104	−	0.915154i	$-0.632069\pi$
$353$	−15.1473	−0.806209	−0.403104	+	0.915154i	$0.632069\pi$
$354$	0	0
$355$	40.6171	2.15573
$356$	0	0
$357$	−4.95554	−0.262275
$358$	0	0
$359$	7.97505	0.420907	0.210453	−	0.977604i	$-0.432506\pi$
$359$	7.97505	0.420907	0.210453	+	0.977604i	$0.432506\pi$
$360$	0	0
$361$	0.630236	0.0331703
$362$	0	0
$363$	9.09217	0.477215
$364$	0	0
$365$	−19.2687	−1.00857
$366$	0	0
$367$	31.0797	1.62235	0.811175	−	0.584804i	$-0.198829\pi$
$367$	31.0797	1.62235	0.811175	+	0.584804i	$0.198829\pi$
$368$	0	0
$369$	6.66239	0.346830
$370$	0	0
$371$	9.06856	0.470816
$372$	0	0
$373$	8.09217	0.418997	0.209498	−	0.977809i	$-0.432817\pi$
$373$	8.09217	0.418997	0.209498	+	0.977809i	$0.432817\pi$
$374$	0	0
$375$	2.10009	0.108448
$376$	0	0
$377$	24.6047	1.26721
$378$	0	0
$379$	10.1939	0.523626	0.261813	−	0.965119i	$-0.415680\pi$
$379$	10.1939	0.523626	0.261813	+	0.965119i	$0.415680\pi$
$380$	0	0
$381$	−12.5548	−0.643202
$382$	0	0
$383$	−14.6572	−0.748946	−0.374473	−	0.927238i	$-0.622176\pi$
$383$	−14.6572	−0.748946	−0.374473	+	0.927238i	$0.622176\pi$
$384$	0	0
$385$	4.50626	0.229660
$386$	0	0
$387$	−1.35629	−0.0689440
$388$	0	0
$389$	26.8547	1.36159	0.680793	−	0.732476i	$-0.261636\pi$
$389$	26.8547	1.36159	0.680793	+	0.732476i	$0.261636\pi$
$390$	0	0
$391$	−4.95554	−0.250613
$392$	0	0
$393$	19.5299	0.985152
$394$	0	0
$395$	4.50626	0.226734
$396$	0	0
$397$	−2.09924	−0.105358	−0.0526790	−	0.998611i	$-0.516776\pi$
$397$	−2.09924	−0.105358	−0.0526790	+	0.998611i	$0.516776\pi$
$398$	0	0
$399$	−4.43060	−0.221808
$400$	0	0
$401$	−30.8547	−1.54081	−0.770404	−	0.637556i	$-0.779945\pi$
$401$	−30.8547	−1.54081	−0.770404	+	0.637556i	$0.779945\pi$
$402$	0	0
$403$	25.6873	1.27958
$404$	0	0
$405$	−3.26247	−0.162113
$406$	0	0
$407$	9.20237	0.456145
$408$	0	0
$409$	29.5672	1.46201	0.731003	−	0.682374i	$-0.239053\pi$
$409$	29.5672	1.46201	0.731003	+	0.682374i	$0.239053\pi$
$410$	0	0
$411$	−17.1873	−0.847789
$412$	0	0
$413$	−4.78116	−0.235266
$414$	0	0
$415$	57.0296	2.79947
$416$	0	0
$417$	11.5499	0.565599
$418$	0	0
$419$	−5.49426	−0.268412	−0.134206	−	0.990953i	$-0.542848\pi$
$419$	−5.49426	−0.268412	−0.134206	+	0.990953i	$0.542848\pi$
$420$	0	0
$421$	−33.5807	−1.63662	−0.818311	−	0.574775i	$-0.805089\pi$
$421$	−33.5807	−1.63662	−0.818311	+	0.574775i	$0.805089\pi$
$422$	0	0
$423$	−11.6179	−0.564883
$424$	0	0
$425$	−27.9677	−1.35663
$426$	0	0
$427$	7.92486	0.383511
$428$	0	0
$429$	5.10102	0.246280
$430$	0	0
$431$	−10.8911	−0.524604	−0.262302	−	0.964986i	$-0.584482\pi$
$431$	−10.8911	−0.524604	−0.262302	+	0.964986i	$0.584482\pi$
$432$	0	0
$433$	7.05563	0.339072	0.169536	−	0.985524i	$-0.445773\pi$
$433$	7.05563	0.339072	0.169536	+	0.985524i	$0.445773\pi$
$434$	0	0
$435$	−21.7359	−1.04215
$436$	0	0
$437$	−4.43060	−0.211944
$438$	0	0
$439$	32.4903	1.55068	0.775339	−	0.631545i	$-0.217579\pi$
$439$	32.4903	1.55068	0.775339	+	0.631545i	$0.217579\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	26.0161	1.23606	0.618031	−	0.786154i	$-0.287931\pi$
$443$	26.0161	1.23606	0.618031	+	0.786154i	$0.287931\pi$
$444$	0	0
$445$	6.89367	0.326791
$446$	0	0
$447$	8.52494	0.403216
$448$	0	0
$449$	−16.8310	−0.794305	−0.397152	−	0.917753i	$-0.630002\pi$
$449$	−16.8310	−0.794305	−0.397152	+	0.917753i	$0.630002\pi$
$450$	0	0
$451$	9.20237	0.433323
$452$	0	0
$453$	6.04363	0.283955
$454$	0	0
$455$	−12.0485	−0.564844
$456$	0	0
$457$	10.7812	0.504322	0.252161	−	0.967685i	$-0.418859\pi$
$457$	10.7812	0.504322	0.252161	+	0.967685i	$0.418859\pi$
$458$	0	0
$459$	4.95554	0.231305
$460$	0	0
$461$	16.1776	0.753468	0.376734	−	0.926322i	$-0.377047\pi$
$461$	16.1776	0.753468	0.376734	+	0.926322i	$0.377047\pi$
$462$	0	0
$463$	6.94979	0.322984	0.161492	−	0.986874i	$-0.448369\pi$
$463$	6.94979	0.322984	0.161492	+	0.986874i	$0.448369\pi$
$464$	0	0
$465$	−22.6922	−1.05233
$466$	0	0
$467$	23.0881	1.06839	0.534195	−	0.845361i	$-0.320615\pi$
$467$	23.0881	1.06839	0.534195	+	0.845361i	$0.320615\pi$
$468$	0	0
$469$	11.3061	0.522067
$470$	0	0
$471$	−9.76246	−0.449830
$472$	0	0
$473$	−1.87336	−0.0861373
$474$	0	0
$475$	−25.0050	−1.14731
$476$	0	0
$477$	−9.06856	−0.415221
$478$	0	0
$479$	−17.1161	−0.782056	−0.391028	−	0.920379i	$-0.627881\pi$
$479$	−17.1161	−0.782056	−0.391028	+	0.920379i	$0.627881\pi$
$480$	0	0
$481$	−24.6047	−1.12188
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	1.54718	0.0702540
$486$	0	0
$487$	23.2372	1.05298	0.526490	−	0.850181i	$-0.323508\pi$
$487$	23.2372	1.05298	0.526490	+	0.850181i	$0.323508\pi$
$488$	0	0
$489$	−6.60498	−0.298688
$490$	0	0
$491$	−38.8245	−1.75212	−0.876062	−	0.482199i	$-0.839838\pi$
$491$	−38.8245	−1.75212	−0.876062	+	0.482199i	$0.839838\pi$
$492$	0	0
$493$	33.0158	1.48696
$494$	0	0
$495$	−4.50626	−0.202541
$496$	0	0
$497$	12.4498	0.558450
$498$	0	0
$499$	14.2074	0.636009	0.318004	−	0.948089i	$-0.396987\pi$
$499$	14.2074	0.636009	0.318004	+	0.948089i	$0.396987\pi$
$500$	0	0
$501$	8.61928	0.385081
$502$	0	0
$503$	5.54360	0.247177	0.123588	−	0.992334i	$-0.460560\pi$
$503$	5.54360	0.247177	0.123588	+	0.992334i	$0.460560\pi$
$504$	0	0
$505$	−14.2096	−0.632318
$506$	0	0
$507$	−0.638783	−0.0283694
$508$	0	0
$509$	38.7613	1.71806	0.859032	−	0.511921i	$-0.171066\pi$
$509$	38.7613	1.71806	0.859032	+	0.511921i	$0.171066\pi$
$510$	0	0
$511$	−5.90618	−0.261274
$512$	0	0
$513$	4.43060	0.195616
$514$	0	0
$515$	−18.9422	−0.834693
$516$	0	0
$517$	−16.0472	−0.705754
$518$	0	0
$519$	6.43060	0.282272
$520$	0	0
$521$	−23.1865	−1.01582	−0.507909	−	0.861411i	$-0.669581\pi$
$521$	−23.1865	−1.01582	−0.507909	+	0.861411i	$0.669581\pi$
$522$	0	0
$523$	23.8617	1.04340	0.521700	−	0.853129i	$-0.325298\pi$
$523$	23.8617	1.04340	0.521700	+	0.853129i	$0.325298\pi$
$524$	0	0
$525$	5.64371	0.246312
$526$	0	0
$527$	34.4685	1.50147
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	4.78116	0.207485
$532$	0	0
$533$	−24.6047	−1.06575
$534$	0	0
$535$	−32.3796	−1.39989
$536$	0	0
$537$	17.6985	0.763746
$538$	0	0
$539$	1.38124	0.0594943
$540$	0	0
$541$	−30.6247	−1.31666	−0.658329	−	0.752730i	$-0.728737\pi$
$541$	−30.6247	−1.31666	−0.658329	+	0.752730i	$0.728737\pi$
$542$	0	0
$543$	2.14372	0.0919960
$544$	0	0
$545$	13.4534	0.576279
$546$	0	0
$547$	−25.9911	−1.11130	−0.555650	−	0.831416i	$-0.687531\pi$
$547$	−25.9911	−1.11130	−0.555650	+	0.831416i	$0.687531\pi$
$548$	0	0
$549$	−7.92486	−0.338225
$550$	0	0
$551$	29.5184	1.25753
$552$	0	0
$553$	1.38124	0.0587363
$554$	0	0
$555$	21.7359	0.922636
$556$	0	0
$557$	−24.8871	−1.05450	−0.527250	−	0.849710i	$-0.676777\pi$
$557$	−24.8871	−1.05450	−0.527250	+	0.849710i	$0.676777\pi$
$558$	0	0
$559$	5.00887	0.211853
$560$	0	0
$561$	6.84480	0.288988
$562$	0	0
$563$	38.3925	1.61805	0.809026	−	0.587774i	$-0.199995\pi$
$563$	38.3925	1.61805	0.809026	+	0.587774i	$0.199995\pi$
$564$	0	0
$565$	9.86874	0.415181
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	46.8543	1.96424	0.982118	−	0.188267i	$-0.0602872\pi$
$569$	46.8543	1.96424	0.982118	+	0.188267i	$0.0602872\pi$
$570$	0	0
$571$	14.2523	0.596438	0.298219	−	0.954497i	$-0.403607\pi$
$571$	14.2523	0.596438	0.298219	+	0.954497i	$0.403607\pi$
$572$	0	0
$573$	−16.7011	−0.697699
$574$	0	0
$575$	5.64371	0.235359
$576$	0	0
$577$	25.5010	1.06162	0.530810	−	0.847491i	$-0.321888\pi$
$577$	25.5010	1.06162	0.530810	+	0.847491i	$0.321888\pi$
$578$	0	0
$579$	−20.6299	−0.857351
$580$	0	0
$581$	17.4805	0.725213
$582$	0	0
$583$	−12.5259	−0.518769
$584$	0	0
$585$	12.0485	0.498146
$586$	0	0
$587$	−26.9747	−1.11337	−0.556683	−	0.830725i	$-0.687926\pi$
$587$	−26.9747	−1.11337	−0.556683	+	0.830725i	$0.687926\pi$
$588$	0	0
$589$	30.8172	1.26980
$590$	0	0
$591$	19.7923	0.814147
$592$	0	0
$593$	−32.8284	−1.34810	−0.674051	−	0.738685i	$-0.735447\pi$
$593$	−32.8284	−1.34810	−0.674051	+	0.738685i	$0.735447\pi$
$594$	0	0
$595$	−16.1673	−0.662795
$596$	0	0
$597$	20.0348	0.819968
$598$	0	0
$599$	16.4987	0.674117	0.337059	−	0.941484i	$-0.390568\pi$
$599$	16.4987	0.674117	0.337059	+	0.941484i	$0.390568\pi$
$600$	0	0
$601$	41.1168	1.67719	0.838594	−	0.544757i	$-0.183378\pi$
$601$	41.1168	1.67719	0.838594	+	0.544757i	$0.183378\pi$
$602$	0	0
$603$	−11.3061	−0.460420
$604$	0	0
$605$	29.6629	1.20597
$606$	0	0
$607$	−39.1239	−1.58799	−0.793995	−	0.607924i	$-0.792002\pi$
$607$	−39.1239	−1.58799	−0.793995	+	0.607924i	$0.792002\pi$
$608$	0	0
$609$	−6.66239	−0.269974
$610$	0	0
$611$	42.9059	1.73579
$612$	0	0
$613$	15.9596	0.644603	0.322302	−	0.946637i	$-0.395543\pi$
$613$	15.9596	0.644603	0.322302	+	0.946637i	$0.395543\pi$
$614$	0	0
$615$	21.7359	0.876474
$616$	0	0
$617$	12.7861	0.514748	0.257374	−	0.966312i	$-0.417143\pi$
$617$	12.7861	0.514748	0.257374	+	0.966312i	$0.417143\pi$
$618$	0	0
$619$	−10.0356	−0.403365	−0.201683	−	0.979451i	$-0.564641\pi$
$619$	−10.0356	−0.403365	−0.201683	+	0.979451i	$0.564641\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	0	0
$623$	2.11302	0.0846564
$624$	0	0
$625$	−21.3671	−0.854683
$626$	0	0
$627$	6.11973	0.244398
$628$	0	0
$629$	−33.0158	−1.31643
$630$	0	0
$631$	−2.94357	−0.117182	−0.0585908	−	0.998282i	$-0.518661\pi$
$631$	−2.94357	−0.117182	−0.0585908	+	0.998282i	$0.518661\pi$
$632$	0	0
$633$	17.3910	0.691232
$634$	0	0
$635$	−40.9596	−1.62543
$636$	0	0
$637$	−3.69307	−0.146325
$638$	0	0
$639$	−12.4498	−0.492507
$640$	0	0
$641$	−7.40619	−0.292527	−0.146264	−	0.989246i	$-0.546725\pi$
$641$	−7.40619	−0.292527	−0.146264	+	0.989246i	$0.546725\pi$
$642$	0	0
$643$	−12.1354	−0.478572	−0.239286	−	0.970949i	$-0.576913\pi$
$643$	−12.1354	−0.478572	−0.239286	+	0.970949i	$0.576913\pi$
$644$	0	0
$645$	−4.42485	−0.174228
$646$	0	0
$647$	−17.6554	−0.694105	−0.347052	−	0.937846i	$-0.612817\pi$
$647$	−17.6554	−0.694105	−0.347052	+	0.937846i	$0.612817\pi$
$648$	0	0
$649$	6.60394	0.259227
$650$	0	0
$651$	−6.95554	−0.272609
$652$	0	0
$653$	−22.6860	−0.887771	−0.443885	−	0.896084i	$-0.646400\pi$
$653$	−22.6860	−0.887771	−0.443885	+	0.896084i	$0.646400\pi$
$654$	0	0
$655$	63.7156	2.48957
$656$	0	0
$657$	5.90618	0.230422
$658$	0	0
$659$	−25.8672	−1.00764	−0.503821	−	0.863808i	$-0.668073\pi$
$659$	−25.8672	−1.00764	−0.503821	+	0.863808i	$0.668073\pi$
$660$	0	0
$661$	−2.61881	−0.101860	−0.0509299	−	0.998702i	$-0.516219\pi$
$661$	−2.61881	−0.101860	−0.0509299	+	0.998702i	$0.516219\pi$
$662$	0	0
$663$	−18.3012	−0.710759
$664$	0	0
$665$	−14.4547	−0.560530
$666$	0	0
$667$	−6.66239	−0.257969
$668$	0	0
$669$	7.79316	0.301301
$670$	0	0
$671$	−10.9461	−0.422571
$672$	0	0
$673$	19.9774	0.770071	0.385036	−	0.922902i	$-0.374189\pi$
$673$	19.9774	0.770071	0.385036	+	0.922902i	$0.374189\pi$
$674$	0	0
$675$	−5.64371	−0.217227
$676$	0	0
$677$	2.36253	0.0907996	0.0453998	−	0.998969i	$-0.485544\pi$
$677$	2.36253	0.0907996	0.0453998	+	0.998969i	$0.485544\pi$
$678$	0	0
$679$	0.474237	0.0181995
$680$	0	0
$681$	−3.31186	−0.126911
$682$	0	0
$683$	−28.7185	−1.09888	−0.549442	−	0.835532i	$-0.685160\pi$
$683$	−28.7185	−1.09888	−0.549442	+	0.835532i	$0.685160\pi$
$684$	0	0
$685$	−56.0732	−2.14245
$686$	0	0
$687$	−16.8799	−0.644008
$688$	0	0
$689$	33.4909	1.27590
$690$	0	0
$691$	39.3546	1.49712	0.748561	−	0.663066i	$-0.230745\pi$
$691$	39.3546	1.49712	0.748561	+	0.663066i	$0.230745\pi$
$692$	0	0
$693$	−1.38124	−0.0524690
$694$	0	0
$695$	37.6811	1.42933
$696$	0	0
$697$	−33.0158	−1.25056
$698$	0	0
$699$	−18.3560	−0.694287
$700$	0	0
$701$	−18.0811	−0.682913	−0.341456	−	0.939898i	$-0.610920\pi$
$701$	−18.0811	−0.682913	−0.341456	+	0.939898i	$0.610920\pi$
$702$	0	0
$703$	−29.5184	−1.11331
$704$	0	0
$705$	−37.9032	−1.42752
$706$	0	0
$707$	−4.35547	−0.163804
$708$	0	0
$709$	4.26381	0.160131	0.0800654	−	0.996790i	$-0.474487\pi$
$709$	4.26381	0.160131	0.0800654	+	0.996790i	$0.474487\pi$
$710$	0	0
$711$	−1.38124	−0.0518006
$712$	0	0
$713$	−6.95554	−0.260487
$714$	0	0
$715$	16.6419	0.622373
$716$	0	0
$717$	−12.1673	−0.454396
$718$	0	0
$719$	22.0979	0.824113	0.412057	−	0.911158i	$-0.364811\pi$
$719$	22.0979	0.824113	0.412057	+	0.911158i	$0.364811\pi$
$720$	0	0
$721$	−5.80609	−0.216230
$722$	0	0
$723$	−30.5740	−1.13706
$724$	0	0
$725$	−37.6006	−1.39645
$726$	0	0
$727$	33.2408	1.23283	0.616416	−	0.787421i	$-0.288584\pi$
$727$	33.2408	1.23283	0.616416	+	0.787421i	$0.288584\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	6.72115	0.248591
$732$	0	0
$733$	−5.76246	−0.212841	−0.106421	−	0.994321i	$-0.533939\pi$
$733$	−5.76246	−0.212841	−0.106421	+	0.994321i	$0.533939\pi$
$734$	0	0
$735$	3.26247	0.120338
$736$	0	0
$737$	−15.6165	−0.575240
$738$	0	0
$739$	−42.5282	−1.56443	−0.782213	−	0.623012i	$-0.785909\pi$
$739$	−42.5282	−1.56443	−0.782213	+	0.623012i	$0.785909\pi$
$740$	0	0
$741$	−16.3625	−0.601093
$742$	0	0
$743$	−39.1667	−1.43689	−0.718443	−	0.695586i	$-0.755145\pi$
$743$	−39.1667	−1.43689	−0.718443	+	0.695586i	$0.755145\pi$
$744$	0	0
$745$	27.8124	1.01897
$746$	0	0
$747$	−17.4805	−0.639577
$748$	0	0
$749$	−9.92486	−0.362647
$750$	0	0
$751$	−24.3284	−0.887757	−0.443878	−	0.896087i	$-0.646398\pi$
$751$	−24.3284	−0.887757	−0.443878	+	0.896087i	$0.646398\pi$
$752$	0	0
$753$	9.33054	0.340024
$754$	0	0
$755$	19.7172	0.717582
$756$	0	0
$757$	6.55740	0.238333	0.119166	−	0.992874i	$-0.461978\pi$
$757$	6.55740	0.238333	0.119166	+	0.992874i	$0.461978\pi$
$758$	0	0
$759$	−1.38124	−0.0501359
$760$	0	0
$761$	47.6105	1.72588	0.862940	−	0.505306i	$-0.168620\pi$
$761$	47.6105	1.72588	0.862940	+	0.505306i	$0.168620\pi$
$762$	0	0
$763$	4.12367	0.149287
$764$	0	0
$765$	16.1673	0.584530
$766$	0	0
$767$	−17.6572	−0.637564
$768$	0	0
$769$	−37.2301	−1.34255	−0.671276	−	0.741207i	$-0.734254\pi$
$769$	−37.2301	−1.34255	−0.671276	+	0.741207i	$0.734254\pi$
$770$	0	0
$771$	−7.93864	−0.285903
$772$	0	0
$773$	−1.88742	−0.0678858	−0.0339429	−	0.999424i	$-0.510806\pi$
$773$	−1.88742	−0.0678858	−0.0339429	+	0.999424i	$0.510806\pi$
$774$	0	0
$775$	−39.2551	−1.41008
$776$	0	0
$777$	6.66239	0.239012
$778$	0	0
$779$	−29.5184	−1.05761
$780$	0	0
$781$	−17.1962	−0.615328
$782$	0	0
$783$	6.66239	0.238095
$784$	0	0
$785$	−31.8497	−1.13677
$786$	0	0
$787$	−7.01819	−0.250171	−0.125086	−	0.992146i	$-0.539921\pi$
$787$	−7.01819	−0.250171	−0.125086	+	0.992146i	$0.539921\pi$
$788$	0	0
$789$	4.66239	0.165986
$790$	0	0
$791$	3.02493	0.107554
$792$	0	0
$793$	29.2671	1.03931
$794$	0	0
$795$	−29.5859	−1.04930
$796$	0	0
$797$	−50.2394	−1.77957	−0.889786	−	0.456378i	$-0.849146\pi$
$797$	−50.2394	−1.77957	−0.889786	+	0.456378i	$0.849146\pi$
$798$	0	0
$799$	57.5732	2.03679
$800$	0	0
$801$	−2.11302	−0.0746599
$802$	0	0
$803$	8.15786	0.287885
$804$	0	0
$805$	3.26247	0.114987
$806$	0	0
$807$	8.17438	0.287752
$808$	0	0
$809$	−40.8084	−1.43475	−0.717374	−	0.696689i	$-0.754656\pi$
$809$	−40.8084	−1.43475	−0.717374	+	0.696689i	$0.754656\pi$
$810$	0	0
$811$	−48.0220	−1.68628	−0.843141	−	0.537693i	$-0.819296\pi$
$811$	−48.0220	−1.68628	−0.843141	+	0.537693i	$0.819296\pi$
$812$	0	0
$813$	6.71800	0.235611
$814$	0	0
$815$	−21.5486	−0.754813
$816$	0	0
$817$	6.00918	0.210234
$818$	0	0
$819$	3.69307	0.129046
$820$	0	0
$821$	−33.6611	−1.17478	−0.587389	−	0.809304i	$-0.699844\pi$
$821$	−33.6611	−1.17478	−0.587389	+	0.809304i	$0.699844\pi$
$822$	0	0
$823$	−45.3595	−1.58113	−0.790567	−	0.612375i	$-0.790214\pi$
$823$	−45.3595	−1.58113	−0.790567	+	0.612375i	$0.790214\pi$
$824$	0	0
$825$	−7.79533	−0.271398
$826$	0	0
$827$	35.1070	1.22079	0.610394	−	0.792098i	$-0.291011\pi$
$827$	35.1070	1.22079	0.610394	+	0.792098i	$0.291011\pi$
$828$	0	0
$829$	−56.6665	−1.96811	−0.984054	−	0.177868i	$-0.943080\pi$
$829$	−56.6665	−1.96811	−0.984054	+	0.177868i	$0.943080\pi$
$830$	0	0
$831$	−22.0735	−0.765721
$832$	0	0
$833$	−4.95554	−0.171699
$834$	0	0
$835$	28.1201	0.973137
$836$	0	0
$837$	6.95554	0.240419
$838$	0	0
$839$	22.4836	0.776220	0.388110	−	0.921613i	$-0.373128\pi$
$839$	22.4836	0.776220	0.388110	+	0.921613i	$0.373128\pi$
$840$	0	0
$841$	15.3875	0.530603
$842$	0	0
$843$	−11.1922	−0.385481
$844$	0	0
$845$	−2.08401	−0.0716922
$846$	0	0
$847$	9.09217	0.312411
$848$	0	0
$849$	−21.9303	−0.752645
$850$	0	0
$851$	6.66239	0.228384
$852$	0	0
$853$	−6.85715	−0.234784	−0.117392	−	0.993086i	$-0.537453\pi$
$853$	−6.85715	−0.234784	−0.117392	+	0.993086i	$0.537453\pi$
$854$	0	0
$855$	14.4547	0.494341
$856$	0	0
$857$	57.9645	1.98003	0.990015	−	0.140960i	$-0.0450190\pi$
$857$	57.9645	1.98003	0.990015	+	0.140960i	$0.0450190\pi$
$858$	0	0
$859$	−20.0935	−0.685581	−0.342791	−	0.939412i	$-0.611372\pi$
$859$	−20.0935	−0.685581	−0.342791	+	0.939412i	$0.611372\pi$
$860$	0	0
$861$	6.66239	0.227054
$862$	0	0
$863$	−27.8606	−0.948385	−0.474192	−	0.880421i	$-0.657260\pi$
$863$	−27.8606	−0.948385	−0.474192	+	0.880421i	$0.657260\pi$
$864$	0	0
$865$	20.9796	0.713329
$866$	0	0
$867$	−7.55740	−0.256663
$868$	0	0
$869$	−1.90783	−0.0647186
$870$	0	0
$871$	41.7543	1.41479
$872$	0	0
$873$	−0.474237	−0.0160505
$874$	0	0
$875$	2.10009	0.0709960
$876$	0	0
$877$	18.3987	0.621279	0.310639	−	0.950528i	$-0.399457\pi$
$877$	18.3987	0.621279	0.310639	+	0.950528i	$0.399457\pi$
$878$	0	0
$879$	6.86120	0.231423
$880$	0	0
$881$	−58.7826	−1.98044	−0.990218	−	0.139527i	$-0.955442\pi$
$881$	−58.7826	−1.98044	−0.990218	+	0.139527i	$0.955442\pi$
$882$	0	0
$883$	2.60013	0.0875012	0.0437506	−	0.999042i	$-0.486069\pi$
$883$	2.60013	0.0875012	0.0437506	+	0.999042i	$0.486069\pi$
$884$	0	0
$885$	15.5984	0.524335
$886$	0	0
$887$	−0.150385	−0.00504942	−0.00252471	−	0.999997i	$-0.500804\pi$
$887$	−0.150385	−0.00504942	−0.00252471	+	0.999997i	$0.500804\pi$
$888$	0	0
$889$	−12.5548	−0.421074
$890$	0	0
$891$	1.38124	0.0462733
$892$	0	0
$893$	51.4745	1.72253
$894$	0	0
$895$	57.7408	1.93006
$896$	0	0
$897$	3.69307	0.123308
$898$	0	0
$899$	46.3406	1.54554
$900$	0	0
$901$	44.9396	1.49716
$902$	0	0
$903$	−1.35629	−0.0451345
$904$	0	0
$905$	6.99383	0.232483
$906$	0	0
$907$	36.0847	1.19817	0.599086	−	0.800685i	$-0.295531\pi$
$907$	36.0847	1.19817	0.599086	+	0.800685i	$0.295531\pi$
$908$	0	0
$909$	4.35547	0.144462
$910$	0	0
$911$	−54.6506	−1.81065	−0.905327	−	0.424715i	$-0.860374\pi$
$911$	−54.6506	−1.81065	−0.905327	+	0.424715i	$0.860374\pi$
$912$	0	0
$913$	−24.1448	−0.799075
$914$	0	0
$915$	−25.8546	−0.854728
$916$	0	0
$917$	19.5299	0.644933
$918$	0	0
$919$	3.13121	0.103289	0.0516445	−	0.998666i	$-0.483554\pi$
$919$	3.13121	0.103289	0.0516445	+	0.998666i	$0.483554\pi$
$920$	0	0
$921$	−14.8110	−0.488040
$922$	0	0
$923$	45.9780	1.51339
$924$	0	0
$925$	37.6006	1.23630
$926$	0	0
$927$	5.80609	0.190697
$928$	0	0
$929$	−16.1335	−0.529322	−0.264661	−	0.964341i	$-0.585260\pi$
$929$	−16.1335	−0.529322	−0.264661	+	0.964341i	$0.585260\pi$
$930$	0	0
$931$	−4.43060	−0.145207
$932$	0	0
$933$	6.64946	0.217694
$934$	0	0
$935$	22.3310	0.730300
$936$	0	0
$937$	−49.7600	−1.62559	−0.812794	−	0.582552i	$-0.802054\pi$
$937$	−49.7600	−1.62559	−0.812794	+	0.582552i	$0.802054\pi$
$938$	0	0
$939$	13.5727	0.442927
$940$	0	0
$941$	6.10603	0.199051	0.0995255	−	0.995035i	$-0.468268\pi$
$941$	6.10603	0.199051	0.0995255	+	0.995035i	$0.468268\pi$
$942$	0	0
$943$	6.66239	0.216957
$944$	0	0
$945$	−3.26247	−0.106128
$946$	0	0
$947$	54.9530	1.78573	0.892867	−	0.450320i	$-0.148690\pi$
$947$	54.9530	1.78573	0.892867	+	0.450320i	$0.148690\pi$
$948$	0	0
$949$	−21.8120	−0.708046
$950$	0	0
$951$	19.1122	0.619755
$952$	0	0
$953$	−17.3127	−0.560812	−0.280406	−	0.959882i	$-0.590469\pi$
$953$	−17.3127	−0.560812	−0.280406	+	0.959882i	$0.590469\pi$
$954$	0	0
$955$	−54.4869	−1.76316
$956$	0	0
$957$	9.20237	0.297470
$958$	0	0
$959$	−17.1873	−0.555008
$960$	0	0
$961$	17.3796	0.560631
$962$	0	0
$963$	9.92486	0.319824
$964$	0	0
$965$	−67.3045	−2.16661
$966$	0	0
$967$	−13.4121	−0.431303	−0.215651	−	0.976470i	$-0.569187\pi$
$967$	−13.4121	−0.431303	−0.215651	+	0.976470i	$0.569187\pi$
$968$	0	0
$969$	−21.9560	−0.705330
$970$	0	0
$971$	−32.2466	−1.03484	−0.517421	−	0.855731i	$-0.673108\pi$
$971$	−32.2466	−1.03484	−0.517421	+	0.855731i	$0.673108\pi$
$972$	0	0
$973$	11.5499	0.370272
$974$	0	0
$975$	20.8426	0.667499
$976$	0	0
$977$	17.3536	0.555191	0.277595	−	0.960698i	$-0.410463\pi$
$977$	17.3536	0.555191	0.277595	+	0.960698i	$0.410463\pi$
$978$	0	0
$979$	−2.91859	−0.0932786
$980$	0	0
$981$	−4.12367	−0.131659
$982$	0	0
$983$	5.80746	0.185229	0.0926146	−	0.995702i	$-0.470478\pi$
$983$	5.80746	0.185229	0.0926146	+	0.995702i	$0.470478\pi$
$984$	0	0
$985$	64.5718	2.05743
$986$	0	0
$987$	−11.6179	−0.369803
$988$	0	0
$989$	−1.35629	−0.0431275
$990$	0	0
$991$	24.9000	0.790974	0.395487	−	0.918472i	$-0.370576\pi$
$991$	24.9000	0.790974	0.395487	+	0.918472i	$0.370576\pi$
$992$	0	0
$993$	−14.2874	−0.453397
$994$	0	0
$995$	65.3628	2.07214
$996$	0	0
$997$	7.50804	0.237782	0.118891	−	0.992907i	$-0.462066\pi$
$997$	7.50804	0.237782	0.118891	+	0.992907i	$0.462066\pi$
$998$	0	0
$999$	−6.66239	−0.210789

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.cf.1.2		5
4.3	odd	2		3864.2.a.v.1.2	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.v.1.2	✓	5	4.3	odd	2
7728.2.a.cf.1.2		5	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7728.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.26247 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.26247 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.38124 q^{11} -3.69307 q^{13} +3.26247 q^{15} -4.95554 q^{17} -4.43060 q^{19} +1.00000 q^{21} +1.00000 q^{23} +5.64371 q^{25} -1.00000 q^{27} -6.66239 q^{29} -6.95554 q^{31} -1.38124 q^{33} +3.26247 q^{35} +6.66239 q^{37} +3.69307 q^{39} +6.66239 q^{41} -1.35629 q^{43} -3.26247 q^{45} -11.6179 q^{47} +1.00000 q^{49} +4.95554 q^{51} -9.06856 q^{53} -4.50626 q^{55} +4.43060 q^{57} +4.78116 q^{59} -7.92486 q^{61} -1.00000 q^{63} +12.0485 q^{65} -11.3061 q^{67} -1.00000 q^{69} -12.4498 q^{71} +5.90618 q^{73} -5.64371 q^{75} -1.38124 q^{77} -1.38124 q^{79} +1.00000 q^{81} -17.4805 q^{83} +16.1673 q^{85} +6.66239 q^{87} -2.11302 q^{89} +3.69307 q^{91} +6.95554 q^{93} +14.4547 q^{95} -0.474237 q^{97} +1.38124 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} - 6 q^{5} - 5 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 - 6 * q^5 - 5 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} - 6 q^{5} - 5 q^{7} + 5 q^{9} + 4 q^{13} + 6 q^{15} + 8 q^{17} - 10 q^{19} + 5 q^{21} + 5 q^{23} + 11 q^{25} - 5 q^{27} - 3 q^{29} - 2 q^{31} + 6 q^{35} + 3 q^{37} - 4 q^{39} + 3 q^{41} - 24 q^{43} - 6 q^{45} + 5 q^{47} + 5 q^{49} - 8 q^{51} + 9 q^{53} - 15 q^{55} + 10 q^{57} - 3 q^{59} + q^{61} - 5 q^{63} - 15 q^{65} - 9 q^{67} - 5 q^{69} - q^{71} + 2 q^{73} - 11 q^{75} + 5 q^{81} - 34 q^{83} + 9 q^{85} + 3 q^{87} + 11 q^{89} - 4 q^{91} + 2 q^{93} - 27 q^{95} + 47 q^{97}+O(q^{100}) \) 5 * q - 5 * q^3 - 6 * q^5 - 5 * q^7 + 5 * q^9 + 4 * q^13 + 6 * q^15 + 8 * q^17 - 10 * q^19 + 5 * q^21 + 5 * q^23 + 11 * q^25 - 5 * q^27 - 3 * q^29 - 2 * q^31 + 6 * q^35 + 3 * q^37 - 4 * q^39 + 3 * q^41 - 24 * q^43 - 6 * q^45 + 5 * q^47 + 5 * q^49 - 8 * q^51 + 9 * q^53 - 15 * q^55 + 10 * q^57 - 3 * q^59 + q^61 - 5 * q^63 - 15 * q^65 - 9 * q^67 - 5 * q^69 - q^71 + 2 * q^73 - 11 * q^75 + 5 * q^81 - 34 * q^83 + 9 * q^85 + 3 * q^87 + 11 * q^89 - 4 * q^91 + 2 * q^93 - 27 * q^95 + 47 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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