LMFDB - Embedded newform 7728.2.a.cg.1.6

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:	$6$
Coefficient field:	$\mathbb{Q}[x]/(x^{6} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 13x^{4} + 7x^{3} + 31x^{2} - 17x + 2 $ x^6 - x^5 - 13x^4 + 7x^3 + 31x^2 - 17x + 2
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			1.6
Root			$0.359645$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} +4.26660 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +4.26660 q^{5} +1.00000 q^{7} +1.00000 q^{9} +0.134803 q^{11} +5.08177 q^{13} -4.26660 q^{15} +1.18483 q^{17} -3.20372 q^{19} -1.00000 q^{21} -1.00000 q^{23} +13.2038 q^{25} -1.00000 q^{27} -3.09550 q^{29} +0.815170 q^{31} -0.134803 q^{33} +4.26660 q^{35} -3.09550 q^{37} -5.08177 q^{39} -0.706943 q^{41} +4.52035 q^{43} +4.26660 q^{45} +8.56883 q^{47} +1.00000 q^{49} -1.18483 q^{51} -0.963705 q^{53} +0.575151 q^{55} +3.20372 q^{57} -2.96370 q^{59} +2.70995 q^{61} +1.00000 q^{63} +21.6818 q^{65} +13.6415 q^{67} +1.00000 q^{69} -3.09851 q^{71} -10.4704 q^{73} -13.2038 q^{75} +0.134803 q^{77} +6.78694 q^{79} +1.00000 q^{81} -9.34836 q^{83} +5.05519 q^{85} +3.09550 q^{87} +12.7006 q^{89} +5.08177 q^{91} -0.815170 q^{93} -13.6690 q^{95} +8.02658 q^{97} +0.134803 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{3} + 6 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q - 6 * q^3 + 6 * q^7 + 6 * q^9	$ 6 q - 6 q^{3} + 6 q^{7} + 6 q^{9} - 5 q^{11} + 2 q^{13} + 10 q^{17} - 3 q^{19} - 6 q^{21} - 6 q^{23} + 18 q^{25} - 6 q^{27} + 3 q^{29} + 2 q^{31} + 5 q^{33} + 3 q^{37} - 2 q^{39} + 4 q^{41} - 6 q^{43} + 2 q^{47} + 6 q^{49} - 10 q^{51} - 4 q^{53} + 15 q^{55} + 3 q^{57} - 16 q^{59} + 22 q^{61} + 6 q^{63} + 35 q^{65} - 9 q^{67} + 6 q^{69} - 11 q^{71} + 24 q^{73} - 18 q^{75} - 5 q^{77} - 18 q^{79} + 6 q^{81} - 2 q^{83} + 13 q^{85} - 3 q^{87} + 7 q^{89} + 2 q^{91} - 2 q^{93} - 5 q^{95} + 37 q^{97} - 5 q^{99}+O(q^{100}) $ 6 * q - 6 * q^3 + 6 * q^7 + 6 * q^9 - 5 * q^11 + 2 * q^13 + 10 * q^17 - 3 * q^19 - 6 * q^21 - 6 * q^23 + 18 * q^25 - 6 * q^27 + 3 * q^29 + 2 * q^31 + 5 * q^33 + 3 * q^37 - 2 * q^39 + 4 * q^41 - 6 * q^43 + 2 * q^47 + 6 * q^49 - 10 * q^51 - 4 * q^53 + 15 * q^55 + 3 * q^57 - 16 * q^59 + 22 * q^61 + 6 * q^63 + 35 * q^65 - 9 * q^67 + 6 * q^69 - 11 * q^71 + 24 * q^73 - 18 * q^75 - 5 * q^77 - 18 * q^79 + 6 * q^81 - 2 * q^83 + 13 * q^85 - 3 * q^87 + 7 * q^89 + 2 * q^91 - 2 * q^93 - 5 * q^95 + 37 * q^97 - 5 * 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
$5$	4.26660	1.90808	0.954040	−	0.299680i	$-0.0968799\pi$
$5$	4.26660	1.90808	0.954040	+	0.299680i	$0.0968799\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	0.134803	0.0406447	0.0203223	−	0.999793i	$-0.493531\pi$
$11$	0.134803	0.0406447	0.0203223	+	0.999793i	$0.493531\pi$
$12$	0	0
$13$	5.08177	1.40943	0.704714	−	0.709491i	$-0.251075\pi$
$13$	5.08177	1.40943	0.704714	+	0.709491i	$0.251075\pi$
$14$	0	0
$15$	−4.26660	−1.10163
$16$	0	0
$17$	1.18483	0.287363	0.143682	−	0.989624i	$-0.454106\pi$
$17$	1.18483	0.287363	0.143682	+	0.989624i	$0.454106\pi$
$18$	0	0
$19$	−3.20372	−0.734985	−0.367492	−	0.930027i	$-0.619784\pi$
$19$	−3.20372	−0.734985	−0.367492	+	0.930027i	$0.619784\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	13.2038	2.64077
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−3.09550	−0.574819	−0.287410	−	0.957808i	$-0.592794\pi$
$29$	−3.09550	−0.574819	−0.287410	+	0.957808i	$0.592794\pi$
$30$	0	0
$31$	0.815170	0.146409	0.0732044	−	0.997317i	$-0.476677\pi$
$31$	0.815170	0.146409	0.0732044	+	0.997317i	$0.476677\pi$
$32$	0	0
$33$	−0.134803	−0.0234662
$34$	0	0
$35$	4.26660	0.721186
$36$	0	0
$37$	−3.09550	−0.508897	−0.254448	−	0.967086i	$-0.581894\pi$
$37$	−3.09550	−0.508897	−0.254448	+	0.967086i	$0.581894\pi$
$38$	0	0
$39$	−5.08177	−0.813734
$40$	0	0
$41$	−0.706943	−0.110406	−0.0552030	−	0.998475i	$-0.517581\pi$
$41$	−0.706943	−0.110406	−0.0552030	+	0.998475i	$0.517581\pi$
$42$	0	0
$43$	4.52035	0.689346	0.344673	−	0.938723i	$-0.387990\pi$
$43$	4.52035	0.689346	0.344673	+	0.938723i	$0.387990\pi$
$44$	0	0
$45$	4.26660	0.636027
$46$	0	0
$47$	8.56883	1.24989	0.624946	−	0.780668i	$-0.285121\pi$
$47$	8.56883	1.24989	0.624946	+	0.780668i	$0.285121\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.18483	−0.165909
$52$	0	0
$53$	−0.963705	−0.132375	−0.0661875	−	0.997807i	$-0.521084\pi$
$53$	−0.963705	−0.132375	−0.0661875	+	0.997807i	$0.521084\pi$
$54$	0	0
$55$	0.575151	0.0775533
$56$	0	0
$57$	3.20372	0.424344
$58$	0	0
$59$	−2.96370	−0.385841	−0.192921	−	0.981214i	$-0.561796\pi$
$59$	−2.96370	−0.385841	−0.192921	+	0.981214i	$0.561796\pi$
$60$	0	0
$61$	2.70995	0.346974	0.173487	−	0.984836i	$-0.444497\pi$
$61$	2.70995	0.346974	0.173487	+	0.984836i	$0.444497\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	21.6818	2.68930
$66$	0	0
$67$	13.6415	1.66658	0.833289	−	0.552837i	$-0.186455\pi$
$67$	13.6415	1.66658	0.833289	+	0.552837i	$0.186455\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−3.09851	−0.367725	−0.183863	−	0.982952i	$-0.558860\pi$
$71$	−3.09851	−0.367725	−0.183863	+	0.982952i	$0.558860\pi$
$72$	0	0
$73$	−10.4704	−1.22547	−0.612736	−	0.790288i	$-0.709931\pi$
$73$	−10.4704	−1.22547	−0.612736	+	0.790288i	$0.709931\pi$
$74$	0	0
$75$	−13.2038	−1.52465
$76$	0	0
$77$	0.134803	0.0153622
$78$	0	0
$79$	6.78694	0.763591	0.381795	−	0.924247i	$-0.375306\pi$
$79$	6.78694	0.763591	0.381795	+	0.924247i	$0.375306\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.34836	−1.02612	−0.513058	−	0.858354i	$-0.671487\pi$
$83$	−9.34836	−1.02612	−0.513058	+	0.858354i	$0.671487\pi$
$84$	0	0
$85$	5.05519	0.548312
$86$	0	0
$87$	3.09550	0.331872
$88$	0	0
$89$	12.7006	1.34626	0.673132	−	0.739523i	$-0.264949\pi$
$89$	12.7006	1.34626	0.673132	+	0.739523i	$0.264949\pi$
$90$	0	0
$91$	5.08177	0.532714
$92$	0	0
$93$	−0.815170	−0.0845292
$94$	0	0
$95$	−13.6690	−1.40241
$96$	0	0
$97$	8.02658	0.814975	0.407488	−	0.913211i	$-0.366405\pi$
$97$	8.02658	0.814975	0.407488	+	0.913211i	$0.366405\pi$
$98$	0	0
$99$	0.134803	0.0135482
$100$	0	0
$101$	10.9355	1.08812	0.544060	−	0.839046i	$-0.316886\pi$
$101$	10.9355	1.08812	0.544060	+	0.839046i	$0.316886\pi$
$102$	0	0
$103$	−0.966715	−0.0952533	−0.0476266	−	0.998865i	$-0.515166\pi$
$103$	−0.966715	−0.0952533	−0.0476266	+	0.998865i	$0.515166\pi$
$104$	0	0
$105$	−4.26660	−0.416377
$106$	0	0
$107$	3.63170	0.351090	0.175545	−	0.984471i	$-0.443831\pi$
$107$	3.63170	0.351090	0.175545	+	0.984471i	$0.443831\pi$
$108$	0	0
$109$	10.4044	0.996564	0.498282	−	0.867015i	$-0.333964\pi$
$109$	10.4044	0.996564	0.498282	+	0.867015i	$0.333964\pi$
$110$	0	0
$111$	3.09550	0.293812
$112$	0	0
$113$	−19.2604	−1.81186	−0.905932	−	0.423422i	$-0.860829\pi$
$113$	−19.2604	−1.81186	−0.905932	+	0.423422i	$0.860829\pi$
$114$	0	0
$115$	−4.26660	−0.397862
$116$	0	0
$117$	5.08177	0.469809
$118$	0	0
$119$	1.18483	0.108613
$120$	0	0
$121$	−10.9818	−0.998348
$122$	0	0
$123$	0.706943	0.0637429
$124$	0	0
$125$	35.0025	3.13072
$126$	0	0
$127$	−6.18495	−0.548825	−0.274413	−	0.961612i	$-0.588483\pi$
$127$	−6.18495	−0.548825	−0.274413	+	0.961612i	$0.588483\pi$
$128$	0	0
$129$	−4.52035	−0.397994
$130$	0	0
$131$	8.96506	0.783281	0.391641	−	0.920118i	$-0.371908\pi$
$131$	8.96506	0.783281	0.391641	+	0.920118i	$0.371908\pi$
$132$	0	0
$133$	−3.20372	−0.277798
$134$	0	0
$135$	−4.26660	−0.367210
$136$	0	0
$137$	−5.48405	−0.468534	−0.234267	−	0.972172i	$-0.575269\pi$
$137$	−5.48405	−0.468534	−0.234267	+	0.972172i	$0.575269\pi$
$138$	0	0
$139$	−10.5362	−0.893669	−0.446835	−	0.894617i	$-0.647449\pi$
$139$	−10.5362	−0.893669	−0.446835	+	0.894617i	$0.647449\pi$
$140$	0	0
$141$	−8.56883	−0.721625
$142$	0	0
$143$	0.685038	0.0572858
$144$	0	0
$145$	−13.2072	−1.09680
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−0.658160	−0.0539186	−0.0269593	−	0.999637i	$-0.508582\pi$
$149$	−0.658160	−0.0539186	−0.0269593	+	0.999637i	$0.508582\pi$
$150$	0	0
$151$	11.8361	0.963207	0.481603	−	0.876389i	$-0.340055\pi$
$151$	11.8361	0.963207	0.481603	+	0.876389i	$0.340055\pi$
$152$	0	0
$153$	1.18483	0.0957878
$154$	0	0
$155$	3.47800	0.279360
$156$	0	0
$157$	7.61145	0.607460	0.303730	−	0.952758i	$-0.401768\pi$
$157$	7.61145	0.607460	0.303730	+	0.952758i	$0.401768\pi$
$158$	0	0
$159$	0.963705	0.0764268
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	1.35905	0.106449	0.0532247	−	0.998583i	$-0.483050\pi$
$163$	1.35905	0.106449	0.0532247	+	0.998583i	$0.483050\pi$
$164$	0	0
$165$	−0.575151	−0.0447754
$166$	0	0
$167$	0.796276	0.0616177	0.0308088	−	0.999525i	$-0.490192\pi$
$167$	0.796276	0.0616177	0.0308088	+	0.999525i	$0.490192\pi$
$168$	0	0
$169$	12.8243	0.986488
$170$	0	0
$171$	−3.20372	−0.244995
$172$	0	0
$173$	−13.9876	−1.06346	−0.531730	−	0.846914i	$-0.678458\pi$
$173$	−13.9876	−1.06346	−0.531730	+	0.846914i	$0.678458\pi$
$174$	0	0
$175$	13.2038	0.998116
$176$	0	0
$177$	2.96370	0.222766
$178$	0	0
$179$	3.62733	0.271119	0.135560	−	0.990769i	$-0.456717\pi$
$179$	3.62733	0.271119	0.135560	+	0.990769i	$0.456717\pi$
$180$	0	0
$181$	14.2069	1.05599	0.527993	−	0.849248i	$-0.322945\pi$
$181$	14.2069	1.05599	0.527993	+	0.849248i	$0.322945\pi$
$182$	0	0
$183$	−2.70995	−0.200326
$184$	0	0
$185$	−13.2072	−0.971015
$186$	0	0
$187$	0.159719	0.0116798
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	14.4614	1.04639	0.523194	−	0.852213i	$-0.324740\pi$
$191$	14.4614	1.04639	0.523194	+	0.852213i	$0.324740\pi$
$192$	0	0
$193$	−5.13025	−0.369283	−0.184642	−	0.982806i	$-0.559112\pi$
$193$	−5.13025	−0.369283	−0.184642	+	0.982806i	$0.559112\pi$
$194$	0	0
$195$	−21.6818	−1.55267
$196$	0	0
$197$	12.7645	0.909433	0.454716	−	0.890636i	$-0.349741\pi$
$197$	12.7645	0.909433	0.454716	+	0.890636i	$0.349741\pi$
$198$	0	0
$199$	−17.8251	−1.26359	−0.631795	−	0.775136i	$-0.717681\pi$
$199$	−17.8251	−1.26359	−0.631795	+	0.775136i	$0.717681\pi$
$200$	0	0
$201$	−13.6415	−0.962200
$202$	0	0
$203$	−3.09550	−0.217261
$204$	0	0
$205$	−3.01624	−0.210663
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−0.431872	−0.0298732
$210$	0	0
$211$	−0.134803	−0.00928023	−0.00464012	−	0.999989i	$-0.501477\pi$
$211$	−0.134803	−0.00928023	−0.00464012	+	0.999989i	$0.501477\pi$
$212$	0	0
$213$	3.09851	0.212306
$214$	0	0
$215$	19.2865	1.31533
$216$	0	0
$217$	0.815170	0.0553374
$218$	0	0
$219$	10.4704	0.707526
$220$	0	0
$221$	6.02103	0.405018
$222$	0	0
$223$	−15.3437	−1.02749	−0.513745	−	0.857943i	$-0.671742\pi$
$223$	−15.3437	−1.02749	−0.513745	+	0.857943i	$0.671742\pi$
$224$	0	0
$225$	13.2038	0.880256
$226$	0	0
$227$	−21.7157	−1.44132	−0.720662	−	0.693287i	$-0.756162\pi$
$227$	−21.7157	−1.44132	−0.720662	+	0.693287i	$0.756162\pi$
$228$	0	0
$229$	−21.5697	−1.42537	−0.712684	−	0.701486i	$-0.752520\pi$
$229$	−21.5697	−1.42537	−0.712684	+	0.701486i	$0.752520\pi$
$230$	0	0
$231$	−0.134803	−0.00886940
$232$	0	0
$233$	28.2185	1.84866	0.924329	−	0.381597i	$-0.124626\pi$
$233$	28.2185	1.84866	0.924329	+	0.381597i	$0.124626\pi$
$234$	0	0
$235$	36.5597	2.38489
$236$	0	0
$237$	−6.78694	−0.440859
$238$	0	0
$239$	19.2719	1.24659	0.623297	−	0.781985i	$-0.285793\pi$
$239$	19.2719	1.24659	0.623297	+	0.781985i	$0.285793\pi$
$240$	0	0
$241$	−10.4786	−0.674987	−0.337493	−	0.941328i	$-0.609579\pi$
$241$	−10.4786	−0.674987	−0.337493	+	0.941328i	$0.609579\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	4.26660	0.272583
$246$	0	0
$247$	−16.2806	−1.03591
$248$	0	0
$249$	9.34836	0.592428
$250$	0	0
$251$	−2.13646	−0.134852	−0.0674262	−	0.997724i	$-0.521479\pi$
$251$	−2.13646	−0.134852	−0.0674262	+	0.997724i	$0.521479\pi$
$252$	0	0
$253$	−0.134803	−0.00847500
$254$	0	0
$255$	−5.05519	−0.316568
$256$	0	0
$257$	−20.1918	−1.25953	−0.629764	−	0.776787i	$-0.716848\pi$
$257$	−20.1918	−1.25953	−0.629764	+	0.776787i	$0.716848\pi$
$258$	0	0
$259$	−3.09550	−0.192345
$260$	0	0
$261$	−3.09550	−0.191606
$262$	0	0
$263$	24.7944	1.52889	0.764443	−	0.644692i	$-0.223014\pi$
$263$	24.7944	1.52889	0.764443	+	0.644692i	$0.223014\pi$
$264$	0	0
$265$	−4.11174	−0.252582
$266$	0	0
$267$	−12.7006	−0.777265
$268$	0	0
$269$	−11.0703	−0.674967	−0.337483	−	0.941331i	$-0.609576\pi$
$269$	−11.0703	−0.674967	−0.337483	+	0.941331i	$0.609576\pi$
$270$	0	0
$271$	−3.98763	−0.242231	−0.121116	−	0.992638i	$-0.538647\pi$
$271$	−3.98763	−0.242231	−0.121116	+	0.992638i	$0.538647\pi$
$272$	0	0
$273$	−5.08177	−0.307562
$274$	0	0
$275$	1.77992	0.107333
$276$	0	0
$277$	−4.87349	−0.292819	−0.146410	−	0.989224i	$-0.546772\pi$
$277$	−4.87349	−0.292819	−0.146410	+	0.989224i	$0.546772\pi$
$278$	0	0
$279$	0.815170	0.0488030
$280$	0	0
$281$	−23.2946	−1.38964	−0.694818	−	0.719185i	$-0.744515\pi$
$281$	−23.2946	−1.38964	−0.694818	+	0.719185i	$0.744515\pi$
$282$	0	0
$283$	−26.9606	−1.60264	−0.801321	−	0.598234i	$-0.795869\pi$
$283$	−26.9606	−1.60264	−0.801321	+	0.598234i	$0.795869\pi$
$284$	0	0
$285$	13.6690	0.809681
$286$	0	0
$287$	−0.706943	−0.0417295
$288$	0	0
$289$	−15.5962	−0.917422
$290$	0	0
$291$	−8.02658	−0.470526
$292$	0	0
$293$	0.131764	0.00769774	0.00384887	−	0.999993i	$-0.498775\pi$
$293$	0.131764	0.00769774	0.00384887	+	0.999993i	$0.498775\pi$
$294$	0	0
$295$	−12.6449	−0.736216
$296$	0	0
$297$	−0.134803	−0.00782207
$298$	0	0
$299$	−5.08177	−0.293886
$300$	0	0
$301$	4.52035	0.260548
$302$	0	0
$303$	−10.9355	−0.628227
$304$	0	0
$305$	11.5623	0.662054
$306$	0	0
$307$	−16.0240	−0.914540	−0.457270	−	0.889328i	$-0.651173\pi$
$307$	−16.0240	−0.914540	−0.457270	+	0.889328i	$0.651173\pi$
$308$	0	0
$309$	0.966715	0.0549945
$310$	0	0
$311$	18.1211	1.02755	0.513776	−	0.857924i	$-0.328246\pi$
$311$	18.1211	1.02755	0.513776	+	0.857924i	$0.328246\pi$
$312$	0	0
$313$	−8.26690	−0.467273	−0.233636	−	0.972324i	$-0.575063\pi$
$313$	−8.26690	−0.467273	−0.233636	+	0.972324i	$0.575063\pi$
$314$	0	0
$315$	4.26660	0.240395
$316$	0	0
$317$	−23.2604	−1.30643	−0.653217	−	0.757171i	$-0.726581\pi$
$317$	−23.2604	−1.30643	−0.653217	+	0.757171i	$0.726581\pi$
$318$	0	0
$319$	−0.417283	−0.0233634
$320$	0	0
$321$	−3.63170	−0.202702
$322$	0	0
$323$	−3.79587	−0.211208
$324$	0	0
$325$	67.0988	3.72197
$326$	0	0
$327$	−10.4044	−0.575367
$328$	0	0
$329$	8.56883	0.472415
$330$	0	0
$331$	−5.40532	−0.297103	−0.148552	−	0.988905i	$-0.547461\pi$
$331$	−5.40532	−0.297103	−0.148552	+	0.988905i	$0.547461\pi$
$332$	0	0
$333$	−3.09550	−0.169632
$334$	0	0
$335$	58.2029	3.17996
$336$	0	0
$337$	−2.16489	−0.117929	−0.0589646	−	0.998260i	$-0.518780\pi$
$337$	−2.16489	−0.117929	−0.0589646	+	0.998260i	$0.518780\pi$
$338$	0	0
$339$	19.2604	1.04608
$340$	0	0
$341$	0.109888	0.00595074
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	4.26660	0.229706
$346$	0	0
$347$	−2.58446	−0.138741	−0.0693706	−	0.997591i	$-0.522099\pi$
$347$	−2.58446	−0.138741	−0.0693706	+	0.997591i	$0.522099\pi$
$348$	0	0
$349$	27.1905	1.45547	0.727737	−	0.685856i	$-0.240572\pi$
$349$	27.1905	1.45547	0.727737	+	0.685856i	$0.240572\pi$
$350$	0	0
$351$	−5.08177	−0.271245
$352$	0	0
$353$	7.62869	0.406034	0.203017	−	0.979175i	$-0.434925\pi$
$353$	7.62869	0.406034	0.203017	+	0.979175i	$0.434925\pi$
$354$	0	0
$355$	−13.2201	−0.701649
$356$	0	0
$357$	−1.18483	−0.0627078
$358$	0	0
$359$	−20.4395	−1.07875	−0.539377	−	0.842064i	$-0.681340\pi$
$359$	−20.4395	−1.07875	−0.539377	+	0.842064i	$0.681340\pi$
$360$	0	0
$361$	−8.73615	−0.459797
$362$	0	0
$363$	10.9818	0.576396
$364$	0	0
$365$	−44.6731	−2.33830
$366$	0	0
$367$	27.9062	1.45669	0.728347	−	0.685209i	$-0.240289\pi$
$367$	27.9062	1.45669	0.728347	+	0.685209i	$0.240289\pi$
$368$	0	0
$369$	−0.706943	−0.0368020
$370$	0	0
$371$	−0.963705	−0.0500331
$372$	0	0
$373$	16.0690	0.832022	0.416011	−	0.909360i	$-0.363428\pi$
$373$	16.0690	0.832022	0.416011	+	0.909360i	$0.363428\pi$
$374$	0	0
$375$	−35.0025	−1.80752
$376$	0	0
$377$	−15.7306	−0.810167
$378$	0	0
$379$	32.2901	1.65863	0.829316	−	0.558780i	$-0.188730\pi$
$379$	32.2901	1.65863	0.829316	+	0.558780i	$0.188730\pi$
$380$	0	0
$381$	6.18495	0.316864
$382$	0	0
$383$	−26.0778	−1.33251	−0.666257	−	0.745722i	$-0.732105\pi$
$383$	−26.0778	−1.33251	−0.666257	+	0.745722i	$0.732105\pi$
$384$	0	0
$385$	0.575151	0.0293124
$386$	0	0
$387$	4.52035	0.229782
$388$	0	0
$389$	21.4111	1.08558	0.542792	−	0.839867i	$-0.317367\pi$
$389$	21.4111	1.08558	0.542792	+	0.839867i	$0.317367\pi$
$390$	0	0
$391$	−1.18483	−0.0599194
$392$	0	0
$393$	−8.96506	−0.452228
$394$	0	0
$395$	28.9571	1.45699
$396$	0	0
$397$	11.5174	0.578043	0.289021	−	0.957323i	$-0.406670\pi$
$397$	11.5174	0.578043	0.289021	+	0.957323i	$0.406670\pi$
$398$	0	0
$399$	3.20372	0.160387
$400$	0	0
$401$	6.34524	0.316866	0.158433	−	0.987370i	$-0.449356\pi$
$401$	6.34524	0.316866	0.158433	+	0.987370i	$0.449356\pi$
$402$	0	0
$403$	4.14250	0.206353
$404$	0	0
$405$	4.26660	0.212009
$406$	0	0
$407$	−0.417283	−0.0206839
$408$	0	0
$409$	38.5554	1.90644	0.953220	−	0.302278i	$-0.0977471\pi$
$409$	38.5554	1.90644	0.953220	+	0.302278i	$0.0977471\pi$
$410$	0	0
$411$	5.48405	0.270508
$412$	0	0
$413$	−2.96370	−0.145834
$414$	0	0
$415$	−39.8857	−1.95791
$416$	0	0
$417$	10.5362	0.515960
$418$	0	0
$419$	−31.2242	−1.52540	−0.762701	−	0.646751i	$-0.776127\pi$
$419$	−31.2242	−1.52540	−0.762701	+	0.646751i	$0.776127\pi$
$420$	0	0
$421$	−2.25074	−0.109694	−0.0548472	−	0.998495i	$-0.517467\pi$
$421$	−2.25074	−0.109694	−0.0548472	+	0.998495i	$0.517467\pi$
$422$	0	0
$423$	8.56883	0.416631
$424$	0	0
$425$	15.6443	0.758860
$426$	0	0
$427$	2.70995	0.131144
$428$	0	0
$429$	−0.685038	−0.0330740
$430$	0	0
$431$	−8.58637	−0.413591	−0.206796	−	0.978384i	$-0.566303\pi$
$431$	−8.58637	−0.413591	−0.206796	+	0.978384i	$0.566303\pi$
$432$	0	0
$433$	37.4639	1.80040	0.900200	−	0.435477i	$-0.143420\pi$
$433$	37.4639	1.80040	0.900200	+	0.435477i	$0.143420\pi$
$434$	0	0
$435$	13.2072	0.633238
$436$	0	0
$437$	3.20372	0.153255
$438$	0	0
$439$	2.71512	0.129585	0.0647927	−	0.997899i	$-0.479361\pi$
$439$	2.71512	0.129585	0.0647927	+	0.997899i	$0.479361\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−15.6189	−0.742074	−0.371037	−	0.928618i	$-0.620998\pi$
$443$	−15.6189	−0.742074	−0.371037	+	0.928618i	$0.620998\pi$
$444$	0	0
$445$	54.1884	2.56878
$446$	0	0
$447$	0.658160	0.0311299
$448$	0	0
$449$	8.54035	0.403044	0.201522	−	0.979484i	$-0.435411\pi$
$449$	8.54035	0.403044	0.201522	+	0.979484i	$0.435411\pi$
$450$	0	0
$451$	−0.0952982	−0.00448742
$452$	0	0
$453$	−11.8361	−0.556108
$454$	0	0
$455$	21.6818	1.01646
$456$	0	0
$457$	−12.2297	−0.572079	−0.286039	−	0.958218i	$-0.592339\pi$
$457$	−12.2297	−0.572079	−0.286039	+	0.958218i	$0.592339\pi$
$458$	0	0
$459$	−1.18483	−0.0553031
$460$	0	0
$461$	−12.5452	−0.584286	−0.292143	−	0.956375i	$-0.594368\pi$
$461$	−12.5452	−0.584286	−0.292143	+	0.956375i	$0.594368\pi$
$462$	0	0
$463$	27.3525	1.27118	0.635590	−	0.772027i	$-0.280757\pi$
$463$	27.3525	1.27118	0.635590	+	0.772027i	$0.280757\pi$
$464$	0	0
$465$	−3.47800	−0.161288
$466$	0	0
$467$	−24.5660	−1.13678	−0.568390	−	0.822759i	$-0.692433\pi$
$467$	−24.5660	−1.13678	−0.568390	+	0.822759i	$0.692433\pi$
$468$	0	0
$469$	13.6415	0.629908
$470$	0	0
$471$	−7.61145	−0.350717
$472$	0	0
$473$	0.609357	0.0280183
$474$	0	0
$475$	−42.3015	−1.94092
$476$	0	0
$477$	−0.963705	−0.0441250
$478$	0	0
$479$	27.0505	1.23597	0.617985	−	0.786190i	$-0.287949\pi$
$479$	27.0505	1.23597	0.617985	+	0.786190i	$0.287949\pi$
$480$	0	0
$481$	−15.7306	−0.717253
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	34.2462	1.55504
$486$	0	0
$487$	−10.0397	−0.454941	−0.227470	−	0.973785i	$-0.573046\pi$
$487$	−10.0397	−0.454941	−0.227470	+	0.973785i	$0.573046\pi$
$488$	0	0
$489$	−1.35905	−0.0614586
$490$	0	0
$491$	16.4088	0.740519	0.370260	−	0.928928i	$-0.379269\pi$
$491$	16.4088	0.740519	0.370260	+	0.928928i	$0.379269\pi$
$492$	0	0
$493$	−3.66764	−0.165182
$494$	0	0
$495$	0.575151	0.0258511
$496$	0	0
$497$	−3.09851	−0.138987
$498$	0	0
$499$	22.7414	1.01804	0.509022	−	0.860753i	$-0.330007\pi$
$499$	22.7414	1.01804	0.509022	+	0.860753i	$0.330007\pi$
$500$	0	0
$501$	−0.796276	−0.0355750
$502$	0	0
$503$	24.6974	1.10120	0.550600	−	0.834769i	$-0.314399\pi$
$503$	24.6974	1.10120	0.550600	+	0.834769i	$0.314399\pi$
$504$	0	0
$505$	46.6573	2.07622
$506$	0	0
$507$	−12.8243	−0.569549
$508$	0	0
$509$	−28.1408	−1.24732	−0.623659	−	0.781696i	$-0.714355\pi$
$509$	−28.1408	−1.24732	−0.623659	+	0.781696i	$0.714355\pi$
$510$	0	0
$511$	−10.4704	−0.463185
$512$	0	0
$513$	3.20372	0.141448
$514$	0	0
$515$	−4.12458	−0.181751
$516$	0	0
$517$	1.15511	0.0508015
$518$	0	0
$519$	13.9876	0.613989
$520$	0	0
$521$	2.89937	0.127024	0.0635118	−	0.997981i	$-0.479770\pi$
$521$	2.89937	0.127024	0.0635118	+	0.997981i	$0.479770\pi$
$522$	0	0
$523$	3.51195	0.153567	0.0767833	−	0.997048i	$-0.475535\pi$
$523$	3.51195	0.153567	0.0767833	+	0.997048i	$0.475535\pi$
$524$	0	0
$525$	−13.2038	−0.576263
$526$	0	0
$527$	0.965838	0.0420726
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−2.96370	−0.128614
$532$	0	0
$533$	−3.59252	−0.155609
$534$	0	0
$535$	15.4950	0.669907
$536$	0	0
$537$	−3.62733	−0.156531
$538$	0	0
$539$	0.134803	0.00580638
$540$	0	0
$541$	−20.9852	−0.902225	−0.451113	−	0.892467i	$-0.648973\pi$
$541$	−20.9852	−0.902225	−0.451113	+	0.892467i	$0.648973\pi$
$542$	0	0
$543$	−14.2069	−0.609674
$544$	0	0
$545$	44.3915	1.90152
$546$	0	0
$547$	−39.6231	−1.69416	−0.847081	−	0.531464i	$-0.821642\pi$
$547$	−39.6231	−1.69416	−0.847081	+	0.531464i	$0.821642\pi$
$548$	0	0
$549$	2.70995	0.115658
$550$	0	0
$551$	9.91712	0.422484
$552$	0	0
$553$	6.78694	0.288610
$554$	0	0
$555$	13.2072	0.560616
$556$	0	0
$557$	−22.3014	−0.944940	−0.472470	−	0.881347i	$-0.656637\pi$
$557$	−22.3014	−0.944940	−0.472470	+	0.881347i	$0.656637\pi$
$558$	0	0
$559$	22.9713	0.971584
$560$	0	0
$561$	−0.159719	−0.00674333
$562$	0	0
$563$	31.7213	1.33689	0.668446	−	0.743761i	$-0.266960\pi$
$563$	31.7213	1.33689	0.668446	+	0.743761i	$0.266960\pi$
$564$	0	0
$565$	−82.1763	−3.45718
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−14.8455	−0.622356	−0.311178	−	0.950352i	$-0.600724\pi$
$569$	−14.8455	−0.622356	−0.311178	+	0.950352i	$0.600724\pi$
$570$	0	0
$571$	12.3795	0.518066	0.259033	−	0.965869i	$-0.416596\pi$
$571$	12.3795	0.518066	0.259033	+	0.965869i	$0.416596\pi$
$572$	0	0
$573$	−14.4614	−0.604133
$574$	0	0
$575$	−13.2038	−0.550638
$576$	0	0
$577$	45.1104	1.87797	0.938986	−	0.343956i	$-0.111767\pi$
$577$	45.1104	1.87797	0.938986	+	0.343956i	$0.111767\pi$
$578$	0	0
$579$	5.13025	0.213206
$580$	0	0
$581$	−9.34836	−0.387835
$582$	0	0
$583$	−0.129910	−0.00538034
$584$	0	0
$585$	21.6818	0.896434
$586$	0	0
$587$	31.9356	1.31812	0.659061	−	0.752090i	$-0.270954\pi$
$587$	31.9356	1.31812	0.659061	+	0.752090i	$0.270954\pi$
$588$	0	0
$589$	−2.61158	−0.107608
$590$	0	0
$591$	−12.7645	−0.525061
$592$	0	0
$593$	−28.0674	−1.15259	−0.576294	−	0.817242i	$-0.695502\pi$
$593$	−28.0674	−1.15259	−0.576294	+	0.817242i	$0.695502\pi$
$594$	0	0
$595$	5.05519	0.207243
$596$	0	0
$597$	17.8251	0.729534
$598$	0	0
$599$	39.8645	1.62882	0.814410	−	0.580290i	$-0.197061\pi$
$599$	39.8645	1.62882	0.814410	+	0.580290i	$0.197061\pi$
$600$	0	0
$601$	6.48421	0.264496	0.132248	−	0.991217i	$-0.457780\pi$
$601$	6.48421	0.264496	0.132248	+	0.991217i	$0.457780\pi$
$602$	0	0
$603$	13.6415	0.555526
$604$	0	0
$605$	−46.8550	−1.90493
$606$	0	0
$607$	−4.04926	−0.164354	−0.0821771	−	0.996618i	$-0.526187\pi$
$607$	−4.04926	−0.164354	−0.0821771	+	0.996618i	$0.526187\pi$
$608$	0	0
$609$	3.09550	0.125436
$610$	0	0
$611$	43.5448	1.76163
$612$	0	0
$613$	−24.6951	−0.997425	−0.498712	−	0.866768i	$-0.666194\pi$
$613$	−24.6951	−0.997425	−0.498712	+	0.866768i	$0.666194\pi$
$614$	0	0
$615$	3.01624	0.121627
$616$	0	0
$617$	−21.9862	−0.885132	−0.442566	−	0.896736i	$-0.645932\pi$
$617$	−21.9862	−0.885132	−0.442566	+	0.896736i	$0.645932\pi$
$618$	0	0
$619$	23.3790	0.939681	0.469841	−	0.882751i	$-0.344311\pi$
$619$	23.3790	0.939681	0.469841	+	0.882751i	$0.344311\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	12.7006	0.508840
$624$	0	0
$625$	83.3222	3.33289
$626$	0	0
$627$	0.431872	0.0172473
$628$	0	0
$629$	−3.66764	−0.146238
$630$	0	0
$631$	6.38303	0.254104	0.127052	−	0.991896i	$-0.459448\pi$
$631$	6.38303	0.254104	0.127052	+	0.991896i	$0.459448\pi$
$632$	0	0
$633$	0.134803	0.00535795
$634$	0	0
$635$	−26.3887	−1.04720
$636$	0	0
$637$	5.08177	0.201347
$638$	0	0
$639$	−3.09851	−0.122575
$640$	0	0
$641$	−22.5743	−0.891630	−0.445815	−	0.895125i	$-0.647086\pi$
$641$	−22.5743	−0.891630	−0.445815	+	0.895125i	$0.647086\pi$
$642$	0	0
$643$	8.00467	0.315673	0.157837	−	0.987465i	$-0.449548\pi$
$643$	8.00467	0.315673	0.157837	+	0.987465i	$0.449548\pi$
$644$	0	0
$645$	−19.2865	−0.759405
$646$	0	0
$647$	33.2180	1.30593	0.652966	−	0.757387i	$-0.273524\pi$
$647$	33.2180	1.30593	0.652966	+	0.757387i	$0.273524\pi$
$648$	0	0
$649$	−0.399517	−0.0156824
$650$	0	0
$651$	−0.815170	−0.0319490
$652$	0	0
$653$	−32.8786	−1.28664	−0.643320	−	0.765597i	$-0.722444\pi$
$653$	−32.8786	−1.28664	−0.643320	+	0.765597i	$0.722444\pi$
$654$	0	0
$655$	38.2503	1.49456
$656$	0	0
$657$	−10.4704	−0.408491
$658$	0	0
$659$	−49.8279	−1.94102	−0.970509	−	0.241065i	$-0.922503\pi$
$659$	−49.8279	−1.94102	−0.970509	+	0.241065i	$0.922503\pi$
$660$	0	0
$661$	11.9378	0.464326	0.232163	−	0.972677i	$-0.425420\pi$
$661$	11.9378	0.464326	0.232163	+	0.972677i	$0.425420\pi$
$662$	0	0
$663$	−6.02103	−0.233837
$664$	0	0
$665$	−13.6690	−0.530061
$666$	0	0
$667$	3.09550	0.119858
$668$	0	0
$669$	15.3437	0.593222
$670$	0	0
$671$	0.365310	0.0141027
$672$	0	0
$673$	28.7293	1.10743	0.553717	−	0.832705i	$-0.313209\pi$
$673$	28.7293	1.10743	0.553717	+	0.832705i	$0.313209\pi$
$674$	0	0
$675$	−13.2038	−0.508216
$676$	0	0
$677$	24.1180	0.926930	0.463465	−	0.886115i	$-0.346606\pi$
$677$	24.1180	0.926930	0.463465	+	0.886115i	$0.346606\pi$
$678$	0	0
$679$	8.02658	0.308032
$680$	0	0
$681$	21.7157	0.832149
$682$	0	0
$683$	0.204132	0.00781090	0.00390545	−	0.999992i	$-0.498757\pi$
$683$	0.204132	0.00781090	0.00390545	+	0.999992i	$0.498757\pi$
$684$	0	0
$685$	−23.3982	−0.894000
$686$	0	0
$687$	21.5697	0.822936
$688$	0	0
$689$	−4.89732	−0.186573
$690$	0	0
$691$	−2.67023	−0.101580	−0.0507902	−	0.998709i	$-0.516174\pi$
$691$	−2.67023	−0.101580	−0.0507902	+	0.998709i	$0.516174\pi$
$692$	0	0
$693$	0.134803	0.00512075
$694$	0	0
$695$	−44.9537	−1.70519
$696$	0	0
$697$	−0.837607	−0.0317266
$698$	0	0
$699$	−28.2185	−1.06732
$700$	0	0
$701$	−28.5921	−1.07991	−0.539954	−	0.841695i	$-0.681558\pi$
$701$	−28.5921	−1.07991	−0.539954	+	0.841695i	$0.681558\pi$
$702$	0	0
$703$	9.91712	0.374031
$704$	0	0
$705$	−36.5597	−1.37692
$706$	0	0
$707$	10.9355	0.411271
$708$	0	0
$709$	12.0014	0.450720	0.225360	−	0.974276i	$-0.427644\pi$
$709$	12.0014	0.450720	0.225360	+	0.974276i	$0.427644\pi$
$710$	0	0
$711$	6.78694	0.254530
$712$	0	0
$713$	−0.815170	−0.0305284
$714$	0	0
$715$	2.92278	0.109306
$716$	0	0
$717$	−19.2719	−0.719721
$718$	0	0
$719$	−34.7077	−1.29438	−0.647189	−	0.762330i	$-0.724055\pi$
$719$	−34.7077	−1.29438	−0.647189	+	0.762330i	$0.724055\pi$
$720$	0	0
$721$	−0.966715	−0.0360024
$722$	0	0
$723$	10.4786	0.389704
$724$	0	0
$725$	−40.8725	−1.51796
$726$	0	0
$727$	−4.76673	−0.176788	−0.0883941	−	0.996086i	$-0.528173\pi$
$727$	−4.76673	−0.176788	−0.0883941	+	0.996086i	$0.528173\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	5.35584	0.198093
$732$	0	0
$733$	−34.9858	−1.29223	−0.646114	−	0.763241i	$-0.723607\pi$
$733$	−34.9858	−1.29223	−0.646114	+	0.763241i	$0.723607\pi$
$734$	0	0
$735$	−4.26660	−0.157376
$736$	0	0
$737$	1.83892	0.0677376
$738$	0	0
$739$	−28.9023	−1.06319	−0.531593	−	0.847000i	$-0.678407\pi$
$739$	−28.9023	−1.06319	−0.531593	+	0.847000i	$0.678407\pi$
$740$	0	0
$741$	16.2806	0.598082
$742$	0	0
$743$	46.8255	1.71786	0.858931	−	0.512091i	$-0.171129\pi$
$743$	46.8255	1.71786	0.858931	+	0.512091i	$0.171129\pi$
$744$	0	0
$745$	−2.80810	−0.102881
$746$	0	0
$747$	−9.34836	−0.342039
$748$	0	0
$749$	3.63170	0.132699
$750$	0	0
$751$	−18.3063	−0.668005	−0.334002	−	0.942572i	$-0.608399\pi$
$751$	−18.3063	−0.668005	−0.334002	+	0.942572i	$0.608399\pi$
$752$	0	0
$753$	2.13646	0.0778571
$754$	0	0
$755$	50.4998	1.83788
$756$	0	0
$757$	41.9564	1.52493	0.762465	−	0.647029i	$-0.223989\pi$
$757$	41.9564	1.52493	0.762465	+	0.647029i	$0.223989\pi$
$758$	0	0
$759$	0.134803	0.00489305
$760$	0	0
$761$	−32.1260	−1.16457	−0.582283	−	0.812986i	$-0.697840\pi$
$761$	−32.1260	−1.16457	−0.582283	+	0.812986i	$0.697840\pi$
$762$	0	0
$763$	10.4044	0.376666
$764$	0	0
$765$	5.05519	0.182771
$766$	0	0
$767$	−15.0609	−0.543816
$768$	0	0
$769$	20.8834	0.753074	0.376537	−	0.926402i	$-0.377115\pi$
$769$	20.8834	0.753074	0.376537	+	0.926402i	$0.377115\pi$
$770$	0	0
$771$	20.1918	0.727189
$772$	0	0
$773$	20.8392	0.749533	0.374766	−	0.927119i	$-0.377723\pi$
$773$	20.8392	0.749533	0.374766	+	0.927119i	$0.377723\pi$
$774$	0	0
$775$	10.7634	0.386632
$776$	0	0
$777$	3.09550	0.111050
$778$	0	0
$779$	2.26485	0.0811467
$780$	0	0
$781$	−0.417689	−0.0149461
$782$	0	0
$783$	3.09550	0.110624
$784$	0	0
$785$	32.4750	1.15908
$786$	0	0
$787$	20.4658	0.729526	0.364763	−	0.931100i	$-0.381150\pi$
$787$	20.4658	0.729526	0.364763	+	0.931100i	$0.381150\pi$
$788$	0	0
$789$	−24.7944	−0.882702
$790$	0	0
$791$	−19.2604	−0.684821
$792$	0	0
$793$	13.7714	0.489035
$794$	0	0
$795$	4.11174	0.145828
$796$	0	0
$797$	−44.7233	−1.58418	−0.792090	−	0.610405i	$-0.791007\pi$
$797$	−44.7233	−1.58418	−0.792090	+	0.610405i	$0.791007\pi$
$798$	0	0
$799$	10.1526	0.359173
$800$	0	0
$801$	12.7006	0.448754
$802$	0	0
$803$	−1.41145	−0.0498089
$804$	0	0
$805$	−4.26660	−0.150378
$806$	0	0
$807$	11.0703	0.389692
$808$	0	0
$809$	−12.5318	−0.440596	−0.220298	−	0.975433i	$-0.570703\pi$
$809$	−12.5318	−0.440596	−0.220298	+	0.975433i	$0.570703\pi$
$810$	0	0
$811$	−17.4654	−0.613293	−0.306646	−	0.951824i	$-0.599207\pi$
$811$	−17.4654	−0.613293	−0.306646	+	0.951824i	$0.599207\pi$
$812$	0	0
$813$	3.98763	0.139852
$814$	0	0
$815$	5.79853	0.203114
$816$	0	0
$817$	−14.4819	−0.506659
$818$	0	0
$819$	5.08177	0.177571
$820$	0	0
$821$	38.4986	1.34361	0.671806	−	0.740728i	$-0.265519\pi$
$821$	38.4986	1.34361	0.671806	+	0.740728i	$0.265519\pi$
$822$	0	0
$823$	46.7622	1.63003	0.815014	−	0.579441i	$-0.196729\pi$
$823$	46.7622	1.63003	0.815014	+	0.579441i	$0.196729\pi$
$824$	0	0
$825$	−1.77992	−0.0619688
$826$	0	0
$827$	−19.8744	−0.691100	−0.345550	−	0.938400i	$-0.612308\pi$
$827$	−19.8744	−0.691100	−0.345550	+	0.938400i	$0.612308\pi$
$828$	0	0
$829$	22.8709	0.794338	0.397169	−	0.917746i	$-0.369993\pi$
$829$	22.8709	0.794338	0.397169	+	0.917746i	$0.369993\pi$
$830$	0	0
$831$	4.87349	0.169059
$832$	0	0
$833$	1.18483	0.0410519
$834$	0	0
$835$	3.39739	0.117571
$836$	0	0
$837$	−0.815170	−0.0281764
$838$	0	0
$839$	52.3977	1.80897	0.904485	−	0.426506i	$-0.140256\pi$
$839$	52.3977	1.80897	0.904485	+	0.426506i	$0.140256\pi$
$840$	0	0
$841$	−19.4179	−0.669583
$842$	0	0
$843$	23.2946	0.802307
$844$	0	0
$845$	54.7163	1.88230
$846$	0	0
$847$	−10.9818	−0.377340
$848$	0	0
$849$	26.9606	0.925286
$850$	0	0
$851$	3.09550	0.106112
$852$	0	0
$853$	−41.5102	−1.42128	−0.710642	−	0.703554i	$-0.751595\pi$
$853$	−41.5102	−1.42128	−0.710642	+	0.703554i	$0.751595\pi$
$854$	0	0
$855$	−13.6690	−0.467470
$856$	0	0
$857$	21.5122	0.734844	0.367422	−	0.930054i	$-0.380240\pi$
$857$	21.5122	0.734844	0.367422	+	0.930054i	$0.380240\pi$
$858$	0	0
$859$	0.149328	0.00509501	0.00254750	−	0.999997i	$-0.499189\pi$
$859$	0.149328	0.00509501	0.00254750	+	0.999997i	$0.499189\pi$
$860$	0	0
$861$	0.706943	0.0240926
$862$	0	0
$863$	−40.8953	−1.39209	−0.696046	−	0.717997i	$-0.745059\pi$
$863$	−40.8953	−1.39209	−0.696046	+	0.717997i	$0.745059\pi$
$864$	0	0
$865$	−59.6796	−2.02917
$866$	0	0
$867$	15.5962	0.529674
$868$	0	0
$869$	0.914901	0.0310359
$870$	0	0
$871$	69.3231	2.34892
$872$	0	0
$873$	8.02658	0.271658
$874$	0	0
$875$	35.0025	1.18330
$876$	0	0
$877$	53.2703	1.79881	0.899404	−	0.437118i	$-0.144001\pi$
$877$	53.2703	1.79881	0.899404	+	0.437118i	$0.144001\pi$
$878$	0	0
$879$	−0.131764	−0.00444429
$880$	0	0
$881$	44.2068	1.48937	0.744683	−	0.667419i	$-0.232601\pi$
$881$	44.2068	1.48937	0.744683	+	0.667419i	$0.232601\pi$
$882$	0	0
$883$	36.9042	1.24193	0.620963	−	0.783840i	$-0.286742\pi$
$883$	36.9042	1.24193	0.620963	+	0.783840i	$0.286742\pi$
$884$	0	0
$885$	12.6449	0.425055
$886$	0	0
$887$	39.6882	1.33260	0.666299	−	0.745685i	$-0.267877\pi$
$887$	39.6882	1.33260	0.666299	+	0.745685i	$0.267877\pi$
$888$	0	0
$889$	−6.18495	−0.207436
$890$	0	0
$891$	0.134803	0.00451608
$892$	0	0
$893$	−27.4522	−0.918652
$894$	0	0
$895$	15.4764	0.517317
$896$	0	0
$897$	5.08177	0.169675
$898$	0	0
$899$	−2.52336	−0.0841587
$900$	0	0
$901$	−1.14183	−0.0380397
$902$	0	0
$903$	−4.52035	−0.150428
$904$	0	0
$905$	60.6149	2.01491
$906$	0	0
$907$	52.6690	1.74884	0.874422	−	0.485166i	$-0.161241\pi$
$907$	52.6690	1.74884	0.874422	+	0.485166i	$0.161241\pi$
$908$	0	0
$909$	10.9355	0.362707
$910$	0	0
$911$	−37.1867	−1.23205	−0.616025	−	0.787726i	$-0.711258\pi$
$911$	−37.1867	−1.23205	−0.616025	+	0.787726i	$0.711258\pi$
$912$	0	0
$913$	−1.26019	−0.0417062
$914$	0	0
$915$	−11.5623	−0.382237
$916$	0	0
$917$	8.96506	0.296052
$918$	0	0
$919$	16.5520	0.546001	0.273001	−	0.962014i	$-0.411984\pi$
$919$	16.5520	0.546001	0.273001	+	0.962014i	$0.411984\pi$
$920$	0	0
$921$	16.0240	0.528010
$922$	0	0
$923$	−15.7459	−0.518282
$924$	0	0
$925$	−40.8725	−1.34388
$926$	0	0
$927$	−0.966715	−0.0317511
$928$	0	0
$929$	−22.0312	−0.722822	−0.361411	−	0.932407i	$-0.617705\pi$
$929$	−22.0312	−0.722822	−0.361411	+	0.932407i	$0.617705\pi$
$930$	0	0
$931$	−3.20372	−0.104998
$932$	0	0
$933$	−18.1211	−0.593257
$934$	0	0
$935$	0.681456	0.0222860
$936$	0	0
$937$	48.2740	1.57704	0.788521	−	0.615008i	$-0.210847\pi$
$937$	48.2740	1.57704	0.788521	+	0.615008i	$0.210847\pi$
$938$	0	0
$939$	8.26690	0.269780
$940$	0	0
$941$	−50.3377	−1.64096	−0.820482	−	0.571673i	$-0.806295\pi$
$941$	−50.3377	−1.64096	−0.820482	+	0.571673i	$0.806295\pi$
$942$	0	0
$943$	0.706943	0.0230212
$944$	0	0
$945$	−4.26660	−0.138792
$946$	0	0
$947$	37.7040	1.22522	0.612608	−	0.790387i	$-0.290121\pi$
$947$	37.7040	1.22522	0.612608	+	0.790387i	$0.290121\pi$
$948$	0	0
$949$	−53.2083	−1.72721
$950$	0	0
$951$	23.2604	0.754270
$952$	0	0
$953$	−5.13282	−0.166268	−0.0831342	−	0.996538i	$-0.526493\pi$
$953$	−5.13282	−0.166268	−0.0831342	+	0.996538i	$0.526493\pi$
$954$	0	0
$955$	61.7008	1.99659
$956$	0	0
$957$	0.417283	0.0134888
$958$	0	0
$959$	−5.48405	−0.177089
$960$	0	0
$961$	−30.3355	−0.978564
$962$	0	0
$963$	3.63170	0.117030
$964$	0	0
$965$	−21.8887	−0.704622
$966$	0	0
$967$	8.94087	0.287519	0.143760	−	0.989613i	$-0.454081\pi$
$967$	8.94087	0.287519	0.143760	+	0.989613i	$0.454081\pi$
$968$	0	0
$969$	3.79587	0.121941
$970$	0	0
$971$	−37.5881	−1.20626	−0.603130	−	0.797643i	$-0.706080\pi$
$971$	−37.5881	−1.20626	−0.603130	+	0.797643i	$0.706080\pi$
$972$	0	0
$973$	−10.5362	−0.337775
$974$	0	0
$975$	−67.0988	−2.14888
$976$	0	0
$977$	−41.4678	−1.32667	−0.663336	−	0.748322i	$-0.730860\pi$
$977$	−41.4678	−1.32667	−0.663336	+	0.748322i	$0.730860\pi$
$978$	0	0
$979$	1.71208	0.0547184
$980$	0	0
$981$	10.4044	0.332188
$982$	0	0
$983$	43.1354	1.37581	0.687904	−	0.725802i	$-0.258531\pi$
$983$	43.1354	1.37581	0.687904	+	0.725802i	$0.258531\pi$
$984$	0	0
$985$	54.4609	1.73527
$986$	0	0
$987$	−8.56883	−0.272749
$988$	0	0
$989$	−4.52035	−0.143739
$990$	0	0
$991$	11.7037	0.371782	0.185891	−	0.982570i	$-0.440483\pi$
$991$	11.7037	0.371782	0.185891	+	0.982570i	$0.440483\pi$
$992$	0	0
$993$	5.40532	0.171533
$994$	0	0
$995$	−76.0526	−2.41103
$996$	0	0
$997$	−42.4545	−1.34455	−0.672274	−	0.740302i	$-0.734682\pi$
$997$	−42.4545	−1.34455	−0.672274	+	0.740302i	$0.734682\pi$
$998$	0	0
$999$	3.09550	0.0979372

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.cg.1.6		6
4.3	odd	2		3864.2.a.x.1.6	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.x.1.6	✓	6	4.3	odd	2
7728.2.a.cg.1.6		6	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +4.26660 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +4.26660 q^{5} +1.00000 q^{7} +1.00000 q^{9} +0.134803 q^{11} +5.08177 q^{13} -4.26660 q^{15} +1.18483 q^{17} -3.20372 q^{19} -1.00000 q^{21} -1.00000 q^{23} +13.2038 q^{25} -1.00000 q^{27} -3.09550 q^{29} +0.815170 q^{31} -0.134803 q^{33} +4.26660 q^{35} -3.09550 q^{37} -5.08177 q^{39} -0.706943 q^{41} +4.52035 q^{43} +4.26660 q^{45} +8.56883 q^{47} +1.00000 q^{49} -1.18483 q^{51} -0.963705 q^{53} +0.575151 q^{55} +3.20372 q^{57} -2.96370 q^{59} +2.70995 q^{61} +1.00000 q^{63} +21.6818 q^{65} +13.6415 q^{67} +1.00000 q^{69} -3.09851 q^{71} -10.4704 q^{73} -13.2038 q^{75} +0.134803 q^{77} +6.78694 q^{79} +1.00000 q^{81} -9.34836 q^{83} +5.05519 q^{85} +3.09550 q^{87} +12.7006 q^{89} +5.08177 q^{91} -0.815170 q^{93} -13.6690 q^{95} +8.02658 q^{97} +0.134803 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 6 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q - 6 * q^3 + 6 * q^7 + 6 * q^9	\( 6 q - 6 q^{3} + 6 q^{7} + 6 q^{9} - 5 q^{11} + 2 q^{13} + 10 q^{17} - 3 q^{19} - 6 q^{21} - 6 q^{23} + 18 q^{25} - 6 q^{27} + 3 q^{29} + 2 q^{31} + 5 q^{33} + 3 q^{37} - 2 q^{39} + 4 q^{41} - 6 q^{43} + 2 q^{47} + 6 q^{49} - 10 q^{51} - 4 q^{53} + 15 q^{55} + 3 q^{57} - 16 q^{59} + 22 q^{61} + 6 q^{63} + 35 q^{65} - 9 q^{67} + 6 q^{69} - 11 q^{71} + 24 q^{73} - 18 q^{75} - 5 q^{77} - 18 q^{79} + 6 q^{81} - 2 q^{83} + 13 q^{85} - 3 q^{87} + 7 q^{89} + 2 q^{91} - 2 q^{93} - 5 q^{95} + 37 q^{97} - 5 q^{99}+O(q^{100}) \) 6 * q - 6 * q^3 + 6 * q^7 + 6 * q^9 - 5 * q^11 + 2 * q^13 + 10 * q^17 - 3 * q^19 - 6 * q^21 - 6 * q^23 + 18 * q^25 - 6 * q^27 + 3 * q^29 + 2 * q^31 + 5 * q^33 + 3 * q^37 - 2 * q^39 + 4 * q^41 - 6 * q^43 + 2 * q^47 + 6 * q^49 - 10 * q^51 - 4 * q^53 + 15 * q^55 + 3 * q^57 - 16 * q^59 + 22 * q^61 + 6 * q^63 + 35 * q^65 - 9 * q^67 + 6 * q^69 - 11 * q^71 + 24 * q^73 - 18 * q^75 - 5 * q^77 - 18 * q^79 + 6 * q^81 - 2 * q^83 + 13 * q^85 - 3 * q^87 + 7 * q^89 + 2 * q^91 - 2 * q^93 - 5 * q^95 + 37 * q^97 - 5 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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