LMFDB - Embedded newform 2744.2.a.f.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2744,2,Mod(1,2744)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2744, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("2744.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2744 = 2^{3} \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2744.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:	$21.9109503146$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 10x^{7} + 32x^{6} + 27x^{5} - 103x^{4} - 8x^{3} + 105x^{2} - 35x - 7 $ x^9 - 3x^8 - 10x^7 + 32x^6 + 27x^5 - 103x^4 - 8x^3 + 105x^2 - 35x - 7
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$-2.38979$ of defining polynomial
Character	$\chi$	$=$	2744.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.38979 q^{3} -1.92162 q^{5} +8.49066 q^{9} +O(q^{10})$	$q+3.38979 q^{3} -1.92162 q^{5} +8.49066 q^{9} -2.11130 q^{11} -4.15414 q^{13} -6.51390 q^{15} +6.65131 q^{17} +4.19834 q^{19} +3.79015 q^{23} -1.30736 q^{25} +18.6122 q^{27} +0.00635776 q^{29} +1.20718 q^{31} -7.15687 q^{33} -0.0945169 q^{37} -14.0816 q^{39} +3.28616 q^{41} +11.6366 q^{43} -16.3159 q^{45} +4.93649 q^{47} +22.5465 q^{51} +12.9414 q^{53} +4.05713 q^{55} +14.2315 q^{57} -12.5956 q^{59} +7.79892 q^{61} +7.98269 q^{65} -10.9774 q^{67} +12.8478 q^{69} -9.76835 q^{71} +3.98823 q^{73} -4.43167 q^{75} -7.11633 q^{79} +37.6193 q^{81} -2.61219 q^{83} -12.7813 q^{85} +0.0215514 q^{87} +3.84429 q^{89} +4.09210 q^{93} -8.06764 q^{95} +8.91447 q^{97} -17.9264 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 6 q^{3} - 5 q^{5} + 5 q^{9}+O(q^{10}) $ 9 * q + 6 * q^3 - 5 * q^5 + 5 * q^9	$ 9 q + 6 q^{3} - 5 q^{5} + 5 q^{9} - 3 q^{11} - 11 q^{15} + 8 q^{17} + 22 q^{19} - 4 q^{23} + 2 q^{25} + 30 q^{27} - 3 q^{29} + 26 q^{31} + 7 q^{33} - 6 q^{39} + 15 q^{41} + 14 q^{43} - 12 q^{45} + 30 q^{47} + 11 q^{51} + 20 q^{53} + 35 q^{55} + 31 q^{57} + 25 q^{59} + 26 q^{61} + 8 q^{65} + 15 q^{67} - 8 q^{69} - 21 q^{71} + 35 q^{73} + 31 q^{75} - 4 q^{79} + 29 q^{81} + 32 q^{83} + 5 q^{85} + 33 q^{87} + 9 q^{89} + 19 q^{93} - 45 q^{95} + 22 q^{97} + 20 q^{99}+O(q^{100}) $ 9 * q + 6 * q^3 - 5 * q^5 + 5 * q^9 - 3 * q^11 - 11 * q^15 + 8 * q^17 + 22 * q^19 - 4 * q^23 + 2 * q^25 + 30 * q^27 - 3 * q^29 + 26 * q^31 + 7 * q^33 - 6 * q^39 + 15 * q^41 + 14 * q^43 - 12 * q^45 + 30 * q^47 + 11 * q^51 + 20 * q^53 + 35 * q^55 + 31 * q^57 + 25 * q^59 + 26 * q^61 + 8 * q^65 + 15 * q^67 - 8 * q^69 - 21 * q^71 + 35 * q^73 + 31 * q^75 - 4 * q^79 + 29 * q^81 + 32 * q^83 + 5 * q^85 + 33 * q^87 + 9 * q^89 + 19 * q^93 - 45 * q^95 + 22 * q^97 + 20 * 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
$3$	3.38979	1.95709	0.978547	−	0.206022i	$-0.0660520\pi$
$3$	3.38979	1.95709	0.978547	+	0.206022i	$0.0660520\pi$
$4$	0	0
$5$	−1.92162	−0.859376	−0.429688	−	0.902977i	$-0.641377\pi$
$5$	−1.92162	−0.859376	−0.429688	+	0.902977i	$0.641377\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	8.49066	2.83022
$10$	0	0
$11$	−2.11130	−0.636582	−0.318291	−	0.947993i	$-0.603109\pi$
$11$	−2.11130	−0.636582	−0.318291	+	0.947993i	$0.603109\pi$
$12$	0	0
$13$	−4.15414	−1.15215	−0.576075	−	0.817397i	$-0.695416\pi$
$13$	−4.15414	−1.15215	−0.576075	+	0.817397i	$0.695416\pi$
$14$	0	0
$15$	−6.51390	−1.68188
$16$	0	0
$17$	6.65131	1.61318	0.806590	−	0.591111i	$-0.201311\pi$
$17$	6.65131	1.61318	0.806590	+	0.591111i	$0.201311\pi$
$18$	0	0
$19$	4.19834	0.963166	0.481583	−	0.876400i	$-0.340062\pi$
$19$	4.19834	0.963166	0.481583	+	0.876400i	$0.340062\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.79015	0.790302	0.395151	−	0.918616i	$-0.370692\pi$
$23$	3.79015	0.790302	0.395151	+	0.918616i	$0.370692\pi$
$24$	0	0
$25$	−1.30736	−0.261472
$26$	0	0
$27$	18.6122	3.58191
$28$	0	0
$29$	0.00635776	0.00118061	0.000590303	−	1.00000i	$-0.499812\pi$
$29$	0.00635776	0.00118061	0.000590303		1.00000i	$0.499812\pi$
$30$	0	0
$31$	1.20718	0.216817	0.108408	−	0.994106i	$-0.465425\pi$
$31$	1.20718	0.216817	0.108408	+	0.994106i	$0.465425\pi$
$32$	0	0
$33$	−7.15687	−1.24585
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.0945169	−0.0155385	−0.00776924	−	0.999970i	$-0.502473\pi$
$37$	−0.0945169	−0.0155385	−0.00776924	+	0.999970i	$0.502473\pi$
$38$	0	0
$39$	−14.0816	−2.25487
$40$	0	0
$41$	3.28616	0.513213	0.256606	−	0.966516i	$-0.417396\pi$
$41$	3.28616	0.513213	0.256606	+	0.966516i	$0.417396\pi$
$42$	0	0
$43$	11.6366	1.77457	0.887283	−	0.461226i	$-0.152590\pi$
$43$	11.6366	1.77457	0.887283	+	0.461226i	$0.152590\pi$
$44$	0	0
$45$	−16.3159	−2.43222
$46$	0	0
$47$	4.93649	0.720061	0.360031	−	0.932940i	$-0.382766\pi$
$47$	4.93649	0.720061	0.360031	+	0.932940i	$0.382766\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	22.5465	3.15715
$52$	0	0
$53$	12.9414	1.77763	0.888816	−	0.458265i	$-0.151529\pi$
$53$	12.9414	1.77763	0.888816	+	0.458265i	$0.151529\pi$
$54$	0	0
$55$	4.05713	0.547064
$56$	0	0
$57$	14.2315	1.88501
$58$	0	0
$59$	−12.5956	−1.63981	−0.819905	−	0.572499i	$-0.805974\pi$
$59$	−12.5956	−1.63981	−0.819905	+	0.572499i	$0.805974\pi$
$60$	0	0
$61$	7.79892	0.998549	0.499274	−	0.866444i	$-0.333600\pi$
$61$	7.79892	0.998549	0.499274	+	0.866444i	$0.333600\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	7.98269	0.990131
$66$	0	0
$67$	−10.9774	−1.34110	−0.670549	−	0.741865i	$-0.733941\pi$
$67$	−10.9774	−1.34110	−0.670549	+	0.741865i	$0.733941\pi$
$68$	0	0
$69$	12.8478	1.54670
$70$	0	0
$71$	−9.76835	−1.15929	−0.579645	−	0.814869i	$-0.696809\pi$
$71$	−9.76835	−1.15929	−0.579645	+	0.814869i	$0.696809\pi$
$72$	0	0
$73$	3.98823	0.466787	0.233393	−	0.972382i	$-0.425017\pi$
$73$	3.98823	0.466787	0.233393	+	0.972382i	$0.425017\pi$
$74$	0	0
$75$	−4.43167	−0.511726
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−7.11633	−0.800649	−0.400325	−	0.916373i	$-0.631103\pi$
$79$	−7.11633	−0.800649	−0.400325	+	0.916373i	$0.631103\pi$
$80$	0	0
$81$	37.6193	4.17992
$82$	0	0
$83$	−2.61219	−0.286725	−0.143363	−	0.989670i	$-0.545791\pi$
$83$	−2.61219	−0.286725	−0.143363	+	0.989670i	$0.545791\pi$
$84$	0	0
$85$	−12.7813	−1.38633
$86$	0	0
$87$	0.0215514	0.00231056
$88$	0	0
$89$	3.84429	0.407494	0.203747	−	0.979024i	$-0.434688\pi$
$89$	3.84429	0.407494	0.203747	+	0.979024i	$0.434688\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	4.09210	0.424331
$94$	0	0
$95$	−8.06764	−0.827722
$96$	0	0
$97$	8.91447	0.905127	0.452563	−	0.891732i	$-0.350510\pi$
$97$	8.91447	0.905127	0.452563	+	0.891732i	$0.350510\pi$
$98$	0	0
$99$	−17.9264	−1.80167
$100$	0	0
$101$	−15.9713	−1.58920	−0.794600	−	0.607134i	$-0.792319\pi$
$101$	−15.9713	−1.58920	−0.794600	+	0.607134i	$0.792319\pi$
$102$	0	0
$103$	5.20713	0.513074	0.256537	−	0.966534i	$-0.417419\pi$
$103$	5.20713	0.513074	0.256537	+	0.966534i	$0.417419\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.79781	0.367148	0.183574	−	0.983006i	$-0.441233\pi$
$107$	3.79781	0.367148	0.183574	+	0.983006i	$0.441233\pi$
$108$	0	0
$109$	−15.7105	−1.50479	−0.752394	−	0.658713i	$-0.771101\pi$
$109$	−15.7105	−1.50479	−0.752394	+	0.658713i	$0.771101\pi$
$110$	0	0
$111$	−0.320392	−0.0304103
$112$	0	0
$113$	−9.81967	−0.923757	−0.461878	−	0.886943i	$-0.652824\pi$
$113$	−9.81967	−0.923757	−0.461878	+	0.886943i	$0.652824\pi$
$114$	0	0
$115$	−7.28325	−0.679167
$116$	0	0
$117$	−35.2713	−3.26084
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−6.54239	−0.594763
$122$	0	0
$123$	11.1394	1.00441
$124$	0	0
$125$	12.1204	1.08408
$126$	0	0
$127$	18.2855	1.62258	0.811289	−	0.584645i	$-0.198766\pi$
$127$	18.2855	1.62258	0.811289	+	0.584645i	$0.198766\pi$
$128$	0	0
$129$	39.4456	3.47299
$130$	0	0
$131$	4.32421	0.377808	0.188904	−	0.981996i	$-0.439506\pi$
$131$	4.32421	0.377808	0.188904	+	0.981996i	$0.439506\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−35.7656	−3.07821
$136$	0	0
$137$	−2.68371	−0.229285	−0.114642	−	0.993407i	$-0.536572\pi$
$137$	−2.68371	−0.229285	−0.114642	+	0.993407i	$0.536572\pi$
$138$	0	0
$139$	−9.01876	−0.764961	−0.382481	−	0.923963i	$-0.624930\pi$
$139$	−9.01876	−0.764961	−0.382481	+	0.923963i	$0.624930\pi$
$140$	0	0
$141$	16.7337	1.40923
$142$	0	0
$143$	8.77065	0.733438
$144$	0	0
$145$	−0.0122172	−0.00101459
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−14.1694	−1.16080	−0.580401	−	0.814331i	$-0.697104\pi$
$149$	−14.1694	−1.16080	−0.580401	+	0.814331i	$0.697104\pi$
$150$	0	0
$151$	1.85542	0.150992	0.0754960	−	0.997146i	$-0.475946\pi$
$151$	1.85542	0.150992	0.0754960	+	0.997146i	$0.475946\pi$
$152$	0	0
$153$	56.4740	4.56565
$154$	0	0
$155$	−2.31975	−0.186327
$156$	0	0
$157$	12.2815	0.980169	0.490085	−	0.871675i	$-0.336966\pi$
$157$	12.2815	0.980169	0.490085	+	0.871675i	$0.336966\pi$
$158$	0	0
$159$	43.8684	3.47899
$160$	0	0
$161$	0	0
$162$	0	0
$163$	1.01374	0.0794023	0.0397012	−	0.999212i	$-0.487359\pi$
$163$	1.01374	0.0794023	0.0397012	+	0.999212i	$0.487359\pi$
$164$	0	0
$165$	13.7528	1.07066
$166$	0	0
$167$	−3.96217	−0.306602	−0.153301	−	0.988180i	$-0.548990\pi$
$167$	−3.96217	−0.306602	−0.153301	+	0.988180i	$0.548990\pi$
$168$	0	0
$169$	4.25684	0.327449
$170$	0	0
$171$	35.6467	2.72597
$172$	0	0
$173$	−3.14544	−0.239143	−0.119572	−	0.992826i	$-0.538152\pi$
$173$	−3.14544	−0.239143	−0.119572	+	0.992826i	$0.538152\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−42.6965	−3.20926
$178$	0	0
$179$	−15.2870	−1.14261	−0.571303	−	0.820739i	$-0.693562\pi$
$179$	−15.2870	−1.14261	−0.571303	+	0.820739i	$0.693562\pi$
$180$	0	0
$181$	−3.35310	−0.249234	−0.124617	−	0.992205i	$-0.539770\pi$
$181$	−3.35310	−0.249234	−0.124617	+	0.992205i	$0.539770\pi$
$182$	0	0
$183$	26.4367	1.95425
$184$	0	0
$185$	0.181626	0.0133534
$186$	0	0
$187$	−14.0429	−1.02692
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−20.4865	−1.48235	−0.741177	−	0.671309i	$-0.765732\pi$
$191$	−20.4865	−1.48235	−0.741177	+	0.671309i	$0.765732\pi$
$192$	0	0
$193$	6.03120	0.434135	0.217068	−	0.976157i	$-0.430351\pi$
$193$	6.03120	0.434135	0.217068	+	0.976157i	$0.430351\pi$
$194$	0	0
$195$	27.0596	1.93778
$196$	0	0
$197$	−11.9483	−0.851280	−0.425640	−	0.904893i	$-0.639951\pi$
$197$	−11.9483	−0.851280	−0.425640	+	0.904893i	$0.639951\pi$
$198$	0	0
$199$	−5.15843	−0.365671	−0.182836	−	0.983144i	$-0.558528\pi$
$199$	−5.15843	−0.365671	−0.182836	+	0.983144i	$0.558528\pi$
$200$	0	0
$201$	−37.2109	−2.62466
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−6.31477	−0.441043
$206$	0	0
$207$	32.1809	2.23673
$208$	0	0
$209$	−8.86399	−0.613135
$210$	0	0
$211$	−19.7648	−1.36066	−0.680331	−	0.732905i	$-0.738164\pi$
$211$	−19.7648	−1.36066	−0.680331	+	0.732905i	$0.738164\pi$
$212$	0	0
$213$	−33.1126	−2.26884
$214$	0	0
$215$	−22.3612	−1.52502
$216$	0	0
$217$	0	0
$218$	0	0
$219$	13.5192	0.913546
$220$	0	0
$221$	−27.6305	−1.85863
$222$	0	0
$223$	21.7883	1.45905	0.729527	−	0.683953i	$-0.239740\pi$
$223$	21.7883	1.45905	0.729527	+	0.683953i	$0.239740\pi$
$224$	0	0
$225$	−11.1004	−0.740023
$226$	0	0
$227$	7.28136	0.483281	0.241640	−	0.970366i	$-0.422315\pi$
$227$	7.28136	0.483281	0.241640	+	0.970366i	$0.422315\pi$
$228$	0	0
$229$	25.0876	1.65783	0.828917	−	0.559371i	$-0.188957\pi$
$229$	25.0876	1.65783	0.828917	+	0.559371i	$0.188957\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.9212	0.715475	0.357737	−	0.933822i	$-0.383548\pi$
$233$	10.9212	0.715475	0.357737	+	0.933822i	$0.383548\pi$
$234$	0	0
$235$	−9.48608	−0.618804
$236$	0	0
$237$	−24.1228	−1.56695
$238$	0	0
$239$	−1.10377	−0.0713971	−0.0356985	−	0.999363i	$-0.511366\pi$
$239$	−1.10377	−0.0713971	−0.0356985	+	0.999363i	$0.511366\pi$
$240$	0	0
$241$	−5.04637	−0.325065	−0.162533	−	0.986703i	$-0.551966\pi$
$241$	−5.04637	−0.325065	−0.162533	+	0.986703i	$0.551966\pi$
$242$	0	0
$243$	71.6849	4.59859
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−17.4405	−1.10971
$248$	0	0
$249$	−8.85477	−0.561148
$250$	0	0
$251$	−7.56595	−0.477559	−0.238779	−	0.971074i	$-0.576747\pi$
$251$	−7.56595	−0.477559	−0.238779	+	0.971074i	$0.576747\pi$
$252$	0	0
$253$	−8.00217	−0.503092
$254$	0	0
$255$	−43.3260	−2.71318
$256$	0	0
$257$	−15.6037	−0.973332	−0.486666	−	0.873588i	$-0.661787\pi$
$257$	−15.6037	−0.973332	−0.486666	+	0.873588i	$0.661787\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0.0539815	0.00334137
$262$	0	0
$263$	−17.2355	−1.06279	−0.531394	−	0.847125i	$-0.678331\pi$
$263$	−17.2355	−1.06279	−0.531394	+	0.847125i	$0.678331\pi$
$264$	0	0
$265$	−24.8684	−1.52765
$266$	0	0
$267$	13.0313	0.797504
$268$	0	0
$269$	6.19949	0.377990	0.188995	−	0.981978i	$-0.439477\pi$
$269$	6.19949	0.377990	0.188995	+	0.981978i	$0.439477\pi$
$270$	0	0
$271$	20.6393	1.25375	0.626874	−	0.779121i	$-0.284334\pi$
$271$	20.6393	1.25375	0.626874	+	0.779121i	$0.284334\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.76024	0.166449
$276$	0	0
$277$	−24.3296	−1.46182	−0.730912	−	0.682472i	$-0.760905\pi$
$277$	−24.3296	−1.46182	−0.730912	+	0.682472i	$0.760905\pi$
$278$	0	0
$279$	10.2498	0.613639
$280$	0	0
$281$	−9.77604	−0.583190	−0.291595	−	0.956542i	$-0.594186\pi$
$281$	−9.77604	−0.583190	−0.291595	+	0.956542i	$0.594186\pi$
$282$	0	0
$283$	−14.4028	−0.856157	−0.428078	−	0.903742i	$-0.640809\pi$
$283$	−14.4028	−0.856157	−0.428078	+	0.903742i	$0.640809\pi$
$284$	0	0
$285$	−27.3476	−1.61993
$286$	0	0
$287$	0	0
$288$	0	0
$289$	27.2400	1.60235
$290$	0	0
$291$	30.2181	1.77142
$292$	0	0
$293$	−29.2654	−1.70970	−0.854852	−	0.518871i	$-0.826352\pi$
$293$	−29.2654	−1.70970	−0.854852	+	0.518871i	$0.826352\pi$
$294$	0	0
$295$	24.2041	1.40921
$296$	0	0
$297$	−39.2959	−2.28018
$298$	0	0
$299$	−15.7448	−0.910546
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−54.1392	−3.11021
$304$	0	0
$305$	−14.9866	−0.858130
$306$	0	0
$307$	−9.10718	−0.519774	−0.259887	−	0.965639i	$-0.583685\pi$
$307$	−9.10718	−0.519774	−0.259887	+	0.965639i	$0.583685\pi$
$308$	0	0
$309$	17.6511	1.00413
$310$	0	0
$311$	12.9695	0.735433	0.367717	−	0.929938i	$-0.380140\pi$
$311$	12.9695	0.735433	0.367717	+	0.929938i	$0.380140\pi$
$312$	0	0
$313$	10.7635	0.608390	0.304195	−	0.952610i	$-0.401613\pi$
$313$	10.7635	0.608390	0.304195	+	0.952610i	$0.401613\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.96178	0.391013	0.195506	−	0.980702i	$-0.437365\pi$
$317$	6.96178	0.391013	0.195506	+	0.980702i	$0.437365\pi$
$318$	0	0
$319$	−0.0134232	−0.000751553 0
$320$	0	0
$321$	12.8738	0.718544
$322$	0	0
$323$	27.9245	1.55376
$324$	0	0
$325$	5.43095	0.301255
$326$	0	0
$327$	−53.2551	−2.94501
$328$	0	0
$329$	0	0
$330$	0	0
$331$	7.19098	0.395252	0.197626	−	0.980277i	$-0.436677\pi$
$331$	7.19098	0.395252	0.197626	+	0.980277i	$0.436677\pi$
$332$	0	0
$333$	−0.802510	−0.0439773
$334$	0	0
$335$	21.0944	1.15251
$336$	0	0
$337$	−15.9379	−0.868194	−0.434097	−	0.900866i	$-0.642932\pi$
$337$	−15.9379	−0.868194	−0.434097	+	0.900866i	$0.642932\pi$
$338$	0	0
$339$	−33.2866	−1.80788
$340$	0	0
$341$	−2.54873	−0.138022
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−24.6887	−1.32919
$346$	0	0
$347$	−25.0656	−1.34559	−0.672795	−	0.739829i	$-0.734906\pi$
$347$	−25.0656	−1.34559	−0.672795	+	0.739829i	$0.734906\pi$
$348$	0	0
$349$	8.30976	0.444812	0.222406	−	0.974954i	$-0.428609\pi$
$349$	8.30976	0.444812	0.222406	+	0.974954i	$0.428609\pi$
$350$	0	0
$351$	−77.3174	−4.12690
$352$	0	0
$353$	−23.9515	−1.27481	−0.637406	−	0.770528i	$-0.719993\pi$
$353$	−23.9515	−1.27481	−0.637406	+	0.770528i	$0.719993\pi$
$354$	0	0
$355$	18.7711	0.996266
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7.47867	0.394709	0.197355	−	0.980332i	$-0.436765\pi$
$359$	7.47867	0.394709	0.197355	+	0.980332i	$0.436765\pi$
$360$	0	0
$361$	−1.37390	−0.0723106
$362$	0	0
$363$	−22.1773	−1.16401
$364$	0	0
$365$	−7.66388	−0.401146
$366$	0	0
$367$	24.7527	1.29208	0.646040	−	0.763303i	$-0.276424\pi$
$367$	24.7527	1.29208	0.646040	+	0.763303i	$0.276424\pi$
$368$	0	0
$369$	27.9017	1.45250
$370$	0	0
$371$	0	0
$372$	0	0
$373$	26.3468	1.36419	0.682093	−	0.731266i	$-0.261070\pi$
$373$	26.3468	1.36419	0.682093	+	0.731266i	$0.261070\pi$
$374$	0	0
$375$	41.0855	2.12165
$376$	0	0
$377$	−0.0264110	−0.00136024
$378$	0	0
$379$	15.9846	0.821072	0.410536	−	0.911844i	$-0.365342\pi$
$379$	15.9846	0.821072	0.410536	+	0.911844i	$0.365342\pi$
$380$	0	0
$381$	61.9841	3.17554
$382$	0	0
$383$	3.00287	0.153440	0.0767198	−	0.997053i	$-0.475555\pi$
$383$	3.00287	0.153440	0.0767198	+	0.997053i	$0.475555\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	98.8024	5.02241
$388$	0	0
$389$	−9.01878	−0.457270	−0.228635	−	0.973512i	$-0.573426\pi$
$389$	−9.01878	−0.457270	−0.228635	+	0.973512i	$0.573426\pi$
$390$	0	0
$391$	25.2095	1.27490
$392$	0	0
$393$	14.6582	0.739406
$394$	0	0
$395$	13.6749	0.688059
$396$	0	0
$397$	−28.8950	−1.45020	−0.725099	−	0.688645i	$-0.758206\pi$
$397$	−28.8950	−1.45020	−0.725099	+	0.688645i	$0.758206\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−34.6945	−1.73256	−0.866280	−	0.499558i	$-0.833496\pi$
$401$	−34.6945	−1.73256	−0.866280	+	0.499558i	$0.833496\pi$
$402$	0	0
$403$	−5.01481	−0.249805
$404$	0	0
$405$	−72.2901	−3.59213
$406$	0	0
$407$	0.199554	0.00989152
$408$	0	0
$409$	12.5406	0.620095	0.310047	−	0.950721i	$-0.399655\pi$
$409$	12.5406	0.620095	0.310047	+	0.950721i	$0.399655\pi$
$410$	0	0
$411$	−9.09720	−0.448732
$412$	0	0
$413$	0	0
$414$	0	0
$415$	5.01965	0.246405
$416$	0	0
$417$	−30.5717	−1.49710
$418$	0	0
$419$	13.8664	0.677420	0.338710	−	0.940891i	$-0.390009\pi$
$419$	13.8664	0.677420	0.338710	+	0.940891i	$0.390009\pi$
$420$	0	0
$421$	−23.1687	−1.12917	−0.564586	−	0.825374i	$-0.690964\pi$
$421$	−23.1687	−1.12917	−0.564586	+	0.825374i	$0.690964\pi$
$422$	0	0
$423$	41.9141	2.03793
$424$	0	0
$425$	−8.69566	−0.421802
$426$	0	0
$427$	0	0
$428$	0	0
$429$	29.7306	1.43541
$430$	0	0
$431$	16.0752	0.774317	0.387158	−	0.922013i	$-0.373457\pi$
$431$	16.0752	0.774317	0.387158	+	0.922013i	$0.373457\pi$
$432$	0	0
$433$	9.35430	0.449539	0.224769	−	0.974412i	$-0.427837\pi$
$433$	9.35430	0.449539	0.224769	+	0.974412i	$0.427837\pi$
$434$	0	0
$435$	−0.0414138	−0.00198564
$436$	0	0
$437$	15.9124	0.761192
$438$	0	0
$439$	30.7719	1.46866	0.734332	−	0.678790i	$-0.237495\pi$
$439$	30.7719	1.46866	0.734332	+	0.678790i	$0.237495\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	32.8925	1.56277	0.781385	−	0.624049i	$-0.214513\pi$
$443$	32.8925	1.56277	0.781385	+	0.624049i	$0.214513\pi$
$444$	0	0
$445$	−7.38728	−0.350191
$446$	0	0
$447$	−48.0312	−2.27180
$448$	0	0
$449$	−14.5414	−0.686249	−0.343125	−	0.939290i	$-0.611485\pi$
$449$	−14.5414	−0.686249	−0.343125	+	0.939290i	$0.611485\pi$
$450$	0	0
$451$	−6.93809	−0.326702
$452$	0	0
$453$	6.28948	0.295506
$454$	0	0
$455$	0	0
$456$	0	0
$457$	10.3987	0.486430	0.243215	−	0.969972i	$-0.421798\pi$
$457$	10.3987	0.486430	0.243215	+	0.969972i	$0.421798\pi$
$458$	0	0
$459$	123.795	5.77827
$460$	0	0
$461$	21.7240	1.01179	0.505895	−	0.862595i	$-0.331162\pi$
$461$	21.7240	1.01179	0.505895	+	0.862595i	$0.331162\pi$
$462$	0	0
$463$	−6.47521	−0.300929	−0.150464	−	0.988615i	$-0.548077\pi$
$463$	−6.47521	−0.300929	−0.150464	+	0.988615i	$0.548077\pi$
$464$	0	0
$465$	−7.86347	−0.364660
$466$	0	0
$467$	1.05813	0.0489642	0.0244821	−	0.999700i	$-0.492206\pi$
$467$	1.05813	0.0489642	0.0244821	+	0.999700i	$0.492206\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	41.6316	1.91828
$472$	0	0
$473$	−24.5684	−1.12966
$474$	0	0
$475$	−5.48875	−0.251841
$476$	0	0
$477$	109.881	5.03109
$478$	0	0
$479$	−3.13835	−0.143395	−0.0716974	−	0.997426i	$-0.522842\pi$
$479$	−3.13835	−0.143395	−0.0716974	+	0.997426i	$0.522842\pi$
$480$	0	0
$481$	0.392636	0.0179027
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−17.1303	−0.777845
$486$	0	0
$487$	3.56976	0.161761	0.0808806	−	0.996724i	$-0.474227\pi$
$487$	3.56976	0.161761	0.0808806	+	0.996724i	$0.474227\pi$
$488$	0	0
$489$	3.43637	0.155398
$490$	0	0
$491$	10.0843	0.455099	0.227550	−	0.973766i	$-0.426929\pi$
$491$	10.0843	0.455099	0.227550	+	0.973766i	$0.426929\pi$
$492$	0	0
$493$	0.0422874	0.00190453
$494$	0	0
$495$	34.4477	1.54831
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−5.06146	−0.226582	−0.113291	−	0.993562i	$-0.536139\pi$
$499$	−5.06146	−0.226582	−0.113291	+	0.993562i	$0.536139\pi$
$500$	0	0
$501$	−13.4309	−0.600049
$502$	0	0
$503$	−16.5923	−0.739815	−0.369908	−	0.929069i	$-0.620611\pi$
$503$	−16.5923	−0.739815	−0.369908	+	0.929069i	$0.620611\pi$
$504$	0	0
$505$	30.6908	1.36572
$506$	0	0
$507$	14.4298	0.640849
$508$	0	0
$509$	−16.2866	−0.721891	−0.360945	−	0.932587i	$-0.617546\pi$
$509$	−16.2866	−0.721891	−0.360945	+	0.932587i	$0.617546\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	78.1403	3.44998
$514$	0	0
$515$	−10.0061	−0.440923
$516$	0	0
$517$	−10.4224	−0.458378
$518$	0	0
$519$	−10.6624	−0.468026
$520$	0	0
$521$	−9.30998	−0.407878	−0.203939	−	0.978984i	$-0.565374\pi$
$521$	−9.30998	−0.407878	−0.203939	+	0.978984i	$0.565374\pi$
$522$	0	0
$523$	−11.0679	−0.483966	−0.241983	−	0.970280i	$-0.577798\pi$
$523$	−11.0679	−0.483966	−0.241983	+	0.970280i	$0.577798\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	8.02936	0.349764
$528$	0	0
$529$	−8.63473	−0.375423
$530$	0	0
$531$	−106.945	−4.64102
$532$	0	0
$533$	−13.6512	−0.591298
$534$	0	0
$535$	−7.29797	−0.315519
$536$	0	0
$537$	−51.8198	−2.23619
$538$	0	0
$539$	0	0
$540$	0	0
$541$	39.9900	1.71930	0.859652	−	0.510880i	$-0.170680\pi$
$541$	39.9900	1.71930	0.859652	+	0.510880i	$0.170680\pi$
$542$	0	0
$543$	−11.3663	−0.487775
$544$	0	0
$545$	30.1896	1.29318
$546$	0	0
$547$	28.7307	1.22844	0.614218	−	0.789136i	$-0.289472\pi$
$547$	28.7307	1.22844	0.614218	+	0.789136i	$0.289472\pi$
$548$	0	0
$549$	66.2179	2.82611
$550$	0	0
$551$	0.0266921	0.00113712
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0.615673	0.0261339
$556$	0	0
$557$	20.4221	0.865312	0.432656	−	0.901559i	$-0.357576\pi$
$557$	20.4221	0.865312	0.432656	+	0.901559i	$0.357576\pi$
$558$	0	0
$559$	−48.3400	−2.04456
$560$	0	0
$561$	−47.6026	−2.00978
$562$	0	0
$563$	14.5113	0.611580	0.305790	−	0.952099i	$-0.401079\pi$
$563$	14.5113	0.611580	0.305790	+	0.952099i	$0.401079\pi$
$564$	0	0
$565$	18.8697	0.793855
$566$	0	0
$567$	0	0
$568$	0	0
$569$	20.7309	0.869087	0.434543	−	0.900651i	$-0.356910\pi$
$569$	20.7309	0.869087	0.434543	+	0.900651i	$0.356910\pi$
$570$	0	0
$571$	−17.2698	−0.722718	−0.361359	−	0.932427i	$-0.617687\pi$
$571$	−17.2698	−0.722718	−0.361359	+	0.932427i	$0.617687\pi$
$572$	0	0
$573$	−69.4450	−2.90111
$574$	0	0
$575$	−4.95510	−0.206642
$576$	0	0
$577$	−24.7422	−1.03003	−0.515015	−	0.857181i	$-0.672214\pi$
$577$	−24.7422	−1.03003	−0.515015	+	0.857181i	$0.672214\pi$
$578$	0	0
$579$	20.4445	0.849643
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−27.3231	−1.13161
$584$	0	0
$585$	67.7783	2.80229
$586$	0	0
$587$	−4.75509	−0.196264	−0.0981318	−	0.995173i	$-0.531287\pi$
$587$	−4.75509	−0.196264	−0.0981318	+	0.995173i	$0.531287\pi$
$588$	0	0
$589$	5.06818	0.208831
$590$	0	0
$591$	−40.5021	−1.66603
$592$	0	0
$593$	−6.75782	−0.277510	−0.138755	−	0.990327i	$-0.544310\pi$
$593$	−6.75782	−0.277510	−0.138755	+	0.990327i	$0.544310\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−17.4860	−0.715653
$598$	0	0
$599$	0.601674	0.0245837	0.0122919	−	0.999924i	$-0.496087\pi$
$599$	0.601674	0.0245837	0.0122919	+	0.999924i	$0.496087\pi$
$600$	0	0
$601$	12.9396	0.527817	0.263909	−	0.964548i	$-0.414988\pi$
$601$	12.9396	0.527817	0.263909	+	0.964548i	$0.414988\pi$
$602$	0	0
$603$	−93.2050	−3.79560
$604$	0	0
$605$	12.5720	0.511125
$606$	0	0
$607$	35.6040	1.44512	0.722561	−	0.691308i	$-0.242965\pi$
$607$	35.6040	1.44512	0.722561	+	0.691308i	$0.242965\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−20.5069	−0.829619
$612$	0	0
$613$	−41.0156	−1.65660	−0.828302	−	0.560282i	$-0.810693\pi$
$613$	−41.0156	−1.65660	−0.828302	+	0.560282i	$0.810693\pi$
$614$	0	0
$615$	−21.4057	−0.863163
$616$	0	0
$617$	−1.64285	−0.0661387	−0.0330693	−	0.999453i	$-0.510528\pi$
$617$	−1.64285	−0.0661387	−0.0330693	+	0.999453i	$0.510528\pi$
$618$	0	0
$619$	−45.5684	−1.83155	−0.915774	−	0.401695i	$-0.868421\pi$
$619$	−45.5684	−1.83155	−0.915774	+	0.401695i	$0.868421\pi$
$620$	0	0
$621$	70.5430	2.83079
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−16.7540	−0.670160
$626$	0	0
$627$	−30.0470	−1.19996
$628$	0	0
$629$	−0.628661	−0.0250664
$630$	0	0
$631$	−24.7025	−0.983390	−0.491695	−	0.870768i	$-0.663622\pi$
$631$	−24.7025	−0.983390	−0.491695	+	0.870768i	$0.663622\pi$
$632$	0	0
$633$	−66.9983	−2.66294
$634$	0	0
$635$	−35.1379	−1.39441
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−82.9397	−3.28104
$640$	0	0
$641$	−31.4052	−1.24043	−0.620216	−	0.784431i	$-0.712955\pi$
$641$	−31.4052	−1.24043	−0.620216	+	0.784431i	$0.712955\pi$
$642$	0	0
$643$	27.4517	1.08259	0.541294	−	0.840833i	$-0.317935\pi$
$643$	27.4517	1.08259	0.541294	+	0.840833i	$0.317935\pi$
$644$	0	0
$645$	−75.7996	−2.98461
$646$	0	0
$647$	−8.22281	−0.323272	−0.161636	−	0.986850i	$-0.551677\pi$
$647$	−8.22281	−0.323272	−0.161636	+	0.986850i	$0.551677\pi$
$648$	0	0
$649$	26.5932	1.04387
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−36.3074	−1.42082	−0.710409	−	0.703789i	$-0.751490\pi$
$653$	−36.3074	−1.42082	−0.710409	+	0.703789i	$0.751490\pi$
$654$	0	0
$655$	−8.30951	−0.324679
$656$	0	0
$657$	33.8627	1.32111
$658$	0	0
$659$	18.0033	0.701308	0.350654	−	0.936505i	$-0.385959\pi$
$659$	18.0033	0.701308	0.350654	+	0.936505i	$0.385959\pi$
$660$	0	0
$661$	−34.8553	−1.35571	−0.677857	−	0.735194i	$-0.737091\pi$
$661$	−34.8553	−1.35571	−0.677857	+	0.735194i	$0.737091\pi$
$662$	0	0
$663$	−93.6614	−3.63751
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.0240969	0.000933035 0
$668$	0	0
$669$	73.8577	2.85550
$670$	0	0
$671$	−16.4659	−0.635659
$672$	0	0
$673$	11.8164	0.455487	0.227744	−	0.973721i	$-0.426865\pi$
$673$	11.8164	0.455487	0.227744	+	0.973721i	$0.426865\pi$
$674$	0	0
$675$	−24.3328	−0.936570
$676$	0	0
$677$	−21.9820	−0.844837	−0.422418	−	0.906401i	$-0.638819\pi$
$677$	−21.9820	−0.844837	−0.422418	+	0.906401i	$0.638819\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	24.6823	0.945826
$682$	0	0
$683$	10.9224	0.417934	0.208967	−	0.977923i	$-0.432990\pi$
$683$	10.9224	0.417934	0.208967	+	0.977923i	$0.432990\pi$
$684$	0	0
$685$	5.15708	0.197042
$686$	0	0
$687$	85.0416	3.24454
$688$	0	0
$689$	−53.7601	−2.04810
$690$	0	0
$691$	27.7373	1.05518	0.527589	−	0.849500i	$-0.323096\pi$
$691$	27.7373	1.05518	0.527589	+	0.849500i	$0.323096\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	17.3307	0.657390
$696$	0	0
$697$	21.8573	0.827904
$698$	0	0
$699$	37.0207	1.40025
$700$	0	0
$701$	−33.4793	−1.26450	−0.632248	−	0.774766i	$-0.717868\pi$
$701$	−33.4793	−1.26450	−0.632248	+	0.774766i	$0.717868\pi$
$702$	0	0
$703$	−0.396814	−0.0149661
$704$	0	0
$705$	−32.1558	−1.21106
$706$	0	0
$707$	0	0
$708$	0	0
$709$	12.0644	0.453087	0.226543	−	0.974001i	$-0.427257\pi$
$709$	12.0644	0.453087	0.226543	+	0.974001i	$0.427257\pi$
$710$	0	0
$711$	−60.4223	−2.26601
$712$	0	0
$713$	4.57542	0.171351
$714$	0	0
$715$	−16.8539	−0.630300
$716$	0	0
$717$	−3.74155	−0.139731
$718$	0	0
$719$	8.83175	0.329369	0.164684	−	0.986346i	$-0.447339\pi$
$719$	8.83175	0.329369	0.164684	+	0.986346i	$0.447339\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−17.1061	−0.636184
$724$	0	0
$725$	−0.00831188	−0.000308696 0
$726$	0	0
$727$	−37.5394	−1.39226	−0.696129	−	0.717917i	$-0.745096\pi$
$727$	−37.5394	−1.39226	−0.696129	+	0.717917i	$0.745096\pi$
$728$	0	0
$729$	130.139	4.81995
$730$	0	0
$731$	77.3987	2.86269
$732$	0	0
$733$	9.35237	0.345438	0.172719	−	0.984971i	$-0.444745\pi$
$733$	9.35237	0.345438	0.172719	+	0.984971i	$0.444745\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	23.1766	0.853720
$738$	0	0
$739$	6.25481	0.230087	0.115043	−	0.993360i	$-0.463299\pi$
$739$	6.25481	0.230087	0.115043	+	0.993360i	$0.463299\pi$
$740$	0	0
$741$	−59.1196	−2.17181
$742$	0	0
$743$	36.0845	1.32381	0.661906	−	0.749587i	$-0.269748\pi$
$743$	36.0845	1.32381	0.661906	+	0.749587i	$0.269748\pi$
$744$	0	0
$745$	27.2282	0.997565
$746$	0	0
$747$	−22.1792	−0.811495
$748$	0	0
$749$	0	0
$750$	0	0
$751$	14.4229	0.526300	0.263150	−	0.964755i	$-0.415239\pi$
$751$	14.4229	0.526300	0.263150	+	0.964755i	$0.415239\pi$
$752$	0	0
$753$	−25.6470	−0.934627
$754$	0	0
$755$	−3.56542	−0.129759
$756$	0	0
$757$	−12.6369	−0.459296	−0.229648	−	0.973274i	$-0.573758\pi$
$757$	−12.6369	−0.459296	−0.229648	+	0.973274i	$0.573758\pi$
$758$	0	0
$759$	−27.1257	−0.984599
$760$	0	0
$761$	51.6136	1.87099	0.935496	−	0.353337i	$-0.114953\pi$
$761$	51.6136	1.87099	0.935496	+	0.353337i	$0.114953\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−108.522	−3.92362
$766$	0	0
$767$	52.3239	1.88931
$768$	0	0
$769$	−0.0964427	−0.00347781	−0.00173891	−	0.999998i	$-0.500554\pi$
$769$	−0.0964427	−0.00347781	−0.00173891	+	0.999998i	$0.500554\pi$
$770$	0	0
$771$	−52.8932	−1.90490
$772$	0	0
$773$	−35.2895	−1.26927	−0.634637	−	0.772811i	$-0.718850\pi$
$773$	−35.2895	−1.26927	−0.634637	+	0.772811i	$0.718850\pi$
$774$	0	0
$775$	−1.57823	−0.0566915
$776$	0	0
$777$	0	0
$778$	0	0
$779$	13.7965	0.494309
$780$	0	0
$781$	20.6240	0.737983
$782$	0	0
$783$	0.118332	0.00422883
$784$	0	0
$785$	−23.6004	−0.842334
$786$	0	0
$787$	25.6393	0.913942	0.456971	−	0.889482i	$-0.348934\pi$
$787$	25.6393	0.913942	0.456971	+	0.889482i	$0.348934\pi$
$788$	0	0
$789$	−58.4247	−2.07997
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−32.3978	−1.15048
$794$	0	0
$795$	−84.2986	−2.98976
$796$	0	0
$797$	−7.21469	−0.255557	−0.127779	−	0.991803i	$-0.540785\pi$
$797$	−7.21469	−0.255557	−0.127779	+	0.991803i	$0.540785\pi$
$798$	0	0
$799$	32.8342	1.16159
$800$	0	0
$801$	32.6406	1.15330
$802$	0	0
$803$	−8.42037	−0.297148
$804$	0	0
$805$	0	0
$806$	0	0
$807$	21.0150	0.739761
$808$	0	0
$809$	16.8902	0.593828	0.296914	−	0.954904i	$-0.404042\pi$
$809$	16.8902	0.593828	0.296914	+	0.954904i	$0.404042\pi$
$810$	0	0
$811$	−4.04701	−0.142110	−0.0710548	−	0.997472i	$-0.522637\pi$
$811$	−4.04701	−0.142110	−0.0710548	+	0.997472i	$0.522637\pi$
$812$	0	0
$813$	69.9628	2.45370
$814$	0	0
$815$	−1.94803	−0.0682365
$816$	0	0
$817$	48.8545	1.70920
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−28.4421	−0.992637	−0.496319	−	0.868140i	$-0.665315\pi$
$821$	−28.4421	−0.992637	−0.496319	+	0.868140i	$0.665315\pi$
$822$	0	0
$823$	−46.3581	−1.61594	−0.807972	−	0.589222i	$-0.799435\pi$
$823$	−46.3581	−1.61594	−0.807972	+	0.589222i	$0.799435\pi$
$824$	0	0
$825$	9.35662	0.325756
$826$	0	0
$827$	30.6281	1.06504	0.532521	−	0.846417i	$-0.321245\pi$
$827$	30.6281	1.06504	0.532521	+	0.846417i	$0.321245\pi$
$828$	0	0
$829$	29.5514	1.02636	0.513181	−	0.858281i	$-0.328467\pi$
$829$	29.5514	1.02636	0.513181	+	0.858281i	$0.328467\pi$
$830$	0	0
$831$	−82.4721	−2.86093
$832$	0	0
$833$	0	0
$834$	0	0
$835$	7.61381	0.263487
$836$	0	0
$837$	22.4683	0.776618
$838$	0	0
$839$	−39.0760	−1.34905	−0.674527	−	0.738250i	$-0.735652\pi$
$839$	−39.0760	−1.34905	−0.674527	+	0.738250i	$0.735652\pi$
$840$	0	0
$841$	−29.0000	−0.999999
$842$	0	0
$843$	−33.1387	−1.14136
$844$	0	0
$845$	−8.18005	−0.281402
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−48.8224	−1.67558
$850$	0	0
$851$	−0.358234	−0.0122801
$852$	0	0
$853$	6.76369	0.231584	0.115792	−	0.993273i	$-0.463059\pi$
$853$	6.76369	0.231584	0.115792	+	0.993273i	$0.463059\pi$
$854$	0	0
$855$	−68.4996	−2.34264
$856$	0	0
$857$	20.6326	0.704796	0.352398	−	0.935850i	$-0.385366\pi$
$857$	20.6326	0.704796	0.352398	+	0.935850i	$0.385366\pi$
$858$	0	0
$859$	15.8264	0.539989	0.269994	−	0.962862i	$-0.412978\pi$
$859$	15.8264	0.539989	0.269994	+	0.962862i	$0.412978\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−30.2039	−1.02815	−0.514077	−	0.857744i	$-0.671866\pi$
$863$	−30.2039	−1.02815	−0.514077	+	0.857744i	$0.671866\pi$
$864$	0	0
$865$	6.04435	0.205514
$866$	0	0
$867$	92.3377	3.13595
$868$	0	0
$869$	15.0247	0.509679
$870$	0	0
$871$	45.6015	1.54515
$872$	0	0
$873$	75.6897	2.56171
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−47.5728	−1.60642	−0.803209	−	0.595697i	$-0.796876\pi$
$877$	−47.5728	−1.60642	−0.803209	+	0.595697i	$0.796876\pi$
$878$	0	0
$879$	−99.2036	−3.34605
$880$	0	0
$881$	−54.1899	−1.82570	−0.912852	−	0.408291i	$-0.866125\pi$
$881$	−54.1899	−1.82570	−0.912852	+	0.408291i	$0.866125\pi$
$882$	0	0
$883$	−47.7160	−1.60577	−0.802885	−	0.596134i	$-0.796703\pi$
$883$	−47.7160	−1.60577	−0.802885	+	0.596134i	$0.796703\pi$
$884$	0	0
$885$	82.0466	2.75797
$886$	0	0
$887$	17.1012	0.574201	0.287101	−	0.957900i	$-0.407309\pi$
$887$	17.1012	0.574201	0.287101	+	0.957900i	$0.407309\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−79.4258	−2.66086
$892$	0	0
$893$	20.7251	0.693539
$894$	0	0
$895$	29.3759	0.981929
$896$	0	0
$897$	−53.3716	−1.78202
$898$	0	0
$899$	0.00767499	0.000255975 0
$900$	0	0
$901$	86.0770	2.86764
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.44340	0.214186
$906$	0	0
$907$	−41.9220	−1.39200	−0.695999	−	0.718042i	$-0.745038\pi$
$907$	−41.9220	−1.39200	−0.695999	+	0.718042i	$0.745038\pi$
$908$	0	0
$909$	−135.606	−4.49778
$910$	0	0
$911$	29.7965	0.987203	0.493602	−	0.869688i	$-0.335680\pi$
$911$	29.7965	0.987203	0.493602	+	0.869688i	$0.335680\pi$
$912$	0	0
$913$	5.51513	0.182524
$914$	0	0
$915$	−50.8013	−1.67944
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−25.4747	−0.840332	−0.420166	−	0.907447i	$-0.638028\pi$
$919$	−25.4747	−0.840332	−0.420166	+	0.907447i	$0.638028\pi$
$920$	0	0
$921$	−30.8714	−1.01725
$922$	0	0
$923$	40.5790	1.33568
$924$	0	0
$925$	0.123568	0.00406288
$926$	0	0
$927$	44.2119	1.45211
$928$	0	0
$929$	−26.0730	−0.855427	−0.427714	−	0.903914i	$-0.640681\pi$
$929$	−26.0730	−0.855427	−0.427714	+	0.903914i	$0.640681\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	43.9639	1.43931
$934$	0	0
$935$	26.9853	0.882513
$936$	0	0
$937$	−16.8602	−0.550798	−0.275399	−	0.961330i	$-0.588810\pi$
$937$	−16.8602	−0.550798	−0.275399	+	0.961330i	$0.588810\pi$
$938$	0	0
$939$	36.4861	1.19068
$940$	0	0
$941$	−42.9732	−1.40089	−0.700443	−	0.713708i	$-0.747014\pi$
$941$	−42.9732	−1.40089	−0.700443	+	0.713708i	$0.747014\pi$
$942$	0	0
$943$	12.4551	0.405593
$944$	0	0
$945$	0	0
$946$	0	0
$947$	30.5119	0.991503	0.495752	−	0.868464i	$-0.334893\pi$
$947$	30.5119	0.991503	0.495752	+	0.868464i	$0.334893\pi$
$948$	0	0
$949$	−16.5676	−0.537808
$950$	0	0
$951$	23.5990	0.765249
$952$	0	0
$953$	22.5111	0.729207	0.364603	−	0.931163i	$-0.381205\pi$
$953$	22.5111	0.729207	0.364603	+	0.931163i	$0.381205\pi$
$954$	0	0
$955$	39.3674	1.27390
$956$	0	0
$957$	−0.0455017	−0.00147086
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.5427	−0.952991
$962$	0	0
$963$	32.2459	1.03911
$964$	0	0
$965$	−11.5897	−0.373085
$966$	0	0
$967$	9.64077	0.310026	0.155013	−	0.987912i	$-0.450458\pi$
$967$	9.64077	0.310026	0.155013	+	0.987912i	$0.450458\pi$
$968$	0	0
$969$	94.6581	3.04086
$970$	0	0
$971$	−37.4069	−1.20044	−0.600222	−	0.799834i	$-0.704921\pi$
$971$	−37.4069	−1.20044	−0.600222	+	0.799834i	$0.704921\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	18.4098	0.589585
$976$	0	0
$977$	10.8402	0.346808	0.173404	−	0.984851i	$-0.444523\pi$
$977$	10.8402	0.346808	0.173404	+	0.984851i	$0.444523\pi$
$978$	0	0
$979$	−8.11647	−0.259403
$980$	0	0
$981$	−133.392	−4.25888
$982$	0	0
$983$	48.9114	1.56003	0.780015	−	0.625760i	$-0.215211\pi$
$983$	48.9114	1.56003	0.780015	+	0.625760i	$0.215211\pi$
$984$	0	0
$985$	22.9601	0.731570
$986$	0	0
$987$	0	0
$988$	0	0
$989$	44.1045	1.40244
$990$	0	0
$991$	38.0096	1.20741	0.603707	−	0.797207i	$-0.293690\pi$
$991$	38.0096	1.20741	0.603707	+	0.797207i	$0.293690\pi$
$992$	0	0
$993$	24.3759	0.773545
$994$	0	0
$995$	9.91256	0.314249
$996$	0	0
$997$	9.50776	0.301114	0.150557	−	0.988601i	$-0.451893\pi$
$997$	9.50776	0.301114	0.150557	+	0.988601i	$0.451893\pi$
$998$	0	0
$999$	−1.75916	−0.0556575

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2744.2.a.f.1.9	yes	9
4.3	odd	2		5488.2.a.r.1.1		9
7.6	odd	2		2744.2.a.c.1.1	✓	9
28.27	even	2		5488.2.a.u.1.9		9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2744.2.a.c.1.1	✓	9	7.6	odd	2
2744.2.a.f.1.9	yes	9	1.1	even	1	trivial
5488.2.a.r.1.1		9	4.3	odd	2
5488.2.a.u.1.9		9	28.27	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:	\( N \)	\(=\)	\( 2744 = 2^{3} \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2744.a (trivial)

\(f(q)\)	\(=\)	\(q+3.38979 q^{3} -1.92162 q^{5} +8.49066 q^{9} +O(q^{10})\)	\(q+3.38979 q^{3} -1.92162 q^{5} +8.49066 q^{9} -2.11130 q^{11} -4.15414 q^{13} -6.51390 q^{15} +6.65131 q^{17} +4.19834 q^{19} +3.79015 q^{23} -1.30736 q^{25} +18.6122 q^{27} +0.00635776 q^{29} +1.20718 q^{31} -7.15687 q^{33} -0.0945169 q^{37} -14.0816 q^{39} +3.28616 q^{41} +11.6366 q^{43} -16.3159 q^{45} +4.93649 q^{47} +22.5465 q^{51} +12.9414 q^{53} +4.05713 q^{55} +14.2315 q^{57} -12.5956 q^{59} +7.79892 q^{61} +7.98269 q^{65} -10.9774 q^{67} +12.8478 q^{69} -9.76835 q^{71} +3.98823 q^{73} -4.43167 q^{75} -7.11633 q^{79} +37.6193 q^{81} -2.61219 q^{83} -12.7813 q^{85} +0.0215514 q^{87} +3.84429 q^{89} +4.09210 q^{93} -8.06764 q^{95} +8.91447 q^{97} -17.9264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 6 q^{3} - 5 q^{5} + 5 q^{9}+O(q^{10}) \) 9 * q + 6 * q^3 - 5 * q^5 + 5 * q^9	\( 9 q + 6 q^{3} - 5 q^{5} + 5 q^{9} - 3 q^{11} - 11 q^{15} + 8 q^{17} + 22 q^{19} - 4 q^{23} + 2 q^{25} + 30 q^{27} - 3 q^{29} + 26 q^{31} + 7 q^{33} - 6 q^{39} + 15 q^{41} + 14 q^{43} - 12 q^{45} + 30 q^{47} + 11 q^{51} + 20 q^{53} + 35 q^{55} + 31 q^{57} + 25 q^{59} + 26 q^{61} + 8 q^{65} + 15 q^{67} - 8 q^{69} - 21 q^{71} + 35 q^{73} + 31 q^{75} - 4 q^{79} + 29 q^{81} + 32 q^{83} + 5 q^{85} + 33 q^{87} + 9 q^{89} + 19 q^{93} - 45 q^{95} + 22 q^{97} + 20 q^{99}+O(q^{100}) \) 9 * q + 6 * q^3 - 5 * q^5 + 5 * q^9 - 3 * q^11 - 11 * q^15 + 8 * q^17 + 22 * q^19 - 4 * q^23 + 2 * q^25 + 30 * q^27 - 3 * q^29 + 26 * q^31 + 7 * q^33 - 6 * q^39 + 15 * q^41 + 14 * q^43 - 12 * q^45 + 30 * q^47 + 11 * q^51 + 20 * q^53 + 35 * q^55 + 31 * q^57 + 25 * q^59 + 26 * q^61 + 8 * q^65 + 15 * q^67 - 8 * q^69 - 21 * q^71 + 35 * q^73 + 31 * q^75 - 4 * q^79 + 29 * q^81 + 32 * q^83 + 5 * q^85 + 33 * q^87 + 9 * q^89 + 19 * q^93 - 45 * q^95 + 22 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more