LMFDB - Embedded newform 7728.2.a.ch.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} - 3x^{5} - 11x^{4} + 23x^{3} + 9x^{2} - 23x + 2 $ x^6 - 3x^5 - 11x^4 + 23x^3 + 9x^2 - 23*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
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			$4.26681$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +3.93128 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.93128 q^{5} +1.00000 q^{7} +1.00000 q^{9} -2.69846 q^{11} -1.59242 q^{13} +3.93128 q^{15} -1.68098 q^{17} -2.69846 q^{19} +1.00000 q^{21} +1.00000 q^{23} +10.4550 q^{25} +1.00000 q^{27} +4.39770 q^{29} +8.29927 q^{31} -2.69846 q^{33} +3.93128 q^{35} +6.24043 q^{37} -1.59242 q^{39} +2.60466 q^{41} -10.2306 q^{43} +3.93128 q^{45} +3.07632 q^{47} +1.00000 q^{49} -1.68098 q^{51} +2.38932 q^{53} -10.6084 q^{55} -2.69846 q^{57} -1.63053 q^{59} -1.29167 q^{61} +1.00000 q^{63} -6.26027 q^{65} -11.0771 q^{67} +1.00000 q^{69} +7.70607 q^{71} +9.36643 q^{73} +10.4550 q^{75} -2.69846 q^{77} +7.54355 q^{79} +1.00000 q^{81} -2.16174 q^{83} -6.60842 q^{85} +4.39770 q^{87} +5.84350 q^{89} -1.59242 q^{91} +8.29927 q^{93} -10.6084 q^{95} +7.64198 q^{97} -2.69846 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{3} + 2 q^{5} + 6 q^{7} + 6 q^{9}+O(q^{10}) $ 6 * q + 6 * q^3 + 2 * q^5 + 6 * q^7 + 6 * q^9	$ 6 q + 6 q^{3} + 2 q^{5} + 6 q^{7} + 6 q^{9} + 3 q^{11} + 2 q^{15} + 6 q^{17} + 3 q^{19} + 6 q^{21} + 6 q^{23} + 10 q^{25} + 6 q^{27} + 5 q^{29} + 4 q^{31} + 3 q^{33} + 2 q^{35} + q^{37} + 12 q^{41} + 6 q^{43} + 2 q^{45} + 6 q^{47} + 6 q^{49} + 6 q^{51} + 10 q^{53} - 3 q^{55} + 3 q^{57} + 14 q^{59} + 4 q^{61} + 6 q^{63} + 27 q^{65} - 7 q^{67} + 6 q^{69} - 7 q^{71} + 10 q^{73} + 10 q^{75} + 3 q^{77} - 14 q^{79} + 6 q^{81} - 14 q^{83} + 21 q^{85} + 5 q^{87} + 25 q^{89} + 4 q^{93} - 3 q^{95} + 11 q^{97} + 3 q^{99}+O(q^{100}) $ 6 * q + 6 * q^3 + 2 * q^5 + 6 * q^7 + 6 * q^9 + 3 * q^11 + 2 * q^15 + 6 * q^17 + 3 * q^19 + 6 * q^21 + 6 * q^23 + 10 * q^25 + 6 * q^27 + 5 * q^29 + 4 * q^31 + 3 * q^33 + 2 * q^35 + q^37 + 12 * q^41 + 6 * q^43 + 2 * q^45 + 6 * q^47 + 6 * q^49 + 6 * q^51 + 10 * q^53 - 3 * q^55 + 3 * q^57 + 14 * q^59 + 4 * q^61 + 6 * q^63 + 27 * q^65 - 7 * q^67 + 6 * q^69 - 7 * q^71 + 10 * q^73 + 10 * q^75 + 3 * q^77 - 14 * q^79 + 6 * q^81 - 14 * q^83 + 21 * q^85 + 5 * q^87 + 25 * q^89 + 4 * q^93 - 3 * q^95 + 11 * q^97 + 3 * 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
$5$	3.93128	1.75812	0.879062	−	0.476708i	$-0.158170\pi$
$5$	3.93128	1.75812	0.879062	+	0.476708i	$0.158170\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.69846	−0.813617	−0.406808	−	0.913513i	$-0.633358\pi$
$11$	−2.69846	−0.813617	−0.406808	+	0.913513i	$0.633358\pi$
$12$	0	0
$13$	−1.59242	−0.441659	−0.220830	−	0.975312i	$-0.570876\pi$
$13$	−1.59242	−0.441659	−0.220830	+	0.975312i	$0.570876\pi$
$14$	0	0
$15$	3.93128	1.01505
$16$	0	0
$17$	−1.68098	−0.407698	−0.203849	−	0.979002i	$-0.565345\pi$
$17$	−1.68098	−0.407698	−0.203849	+	0.979002i	$0.565345\pi$
$18$	0	0
$19$	−2.69846	−0.619070	−0.309535	−	0.950888i	$-0.600173\pi$
$19$	−2.69846	−0.619070	−0.309535	+	0.950888i	$0.600173\pi$
$20$	0	0
$21$	1.00000	0.218218
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	10.4550	2.09100
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	4.39770	0.816633	0.408317	−	0.912840i	$-0.366116\pi$
$29$	4.39770	0.816633	0.408317	+	0.912840i	$0.366116\pi$
$30$	0	0
$31$	8.29927	1.49059	0.745297	−	0.666733i	$-0.232308\pi$
$31$	8.29927	1.49059	0.745297	+	0.666733i	$0.232308\pi$
$32$	0	0
$33$	−2.69846	−0.469742
$34$	0	0
$35$	3.93128	0.664508
$36$	0	0
$37$	6.24043	1.02592	0.512960	−	0.858412i	$-0.328549\pi$
$37$	6.24043	1.02592	0.512960	+	0.858412i	$0.328549\pi$
$38$	0	0
$39$	−1.59242	−0.254992
$40$	0	0
$41$	2.60466	0.406779	0.203390	−	0.979098i	$-0.434804\pi$
$41$	2.60466	0.406779	0.203390	+	0.979098i	$0.434804\pi$
$42$	0	0
$43$	−10.2306	−1.56015	−0.780073	−	0.625689i	$-0.784818\pi$
$43$	−10.2306	−1.56015	−0.780073	+	0.625689i	$0.784818\pi$
$44$	0	0
$45$	3.93128	0.586041
$46$	0	0
$47$	3.07632	0.448728	0.224364	−	0.974505i	$-0.427970\pi$
$47$	3.07632	0.448728	0.224364	+	0.974505i	$0.427970\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.68098	−0.235385
$52$	0	0
$53$	2.38932	0.328198	0.164099	−	0.986444i	$-0.447528\pi$
$53$	2.38932	0.328198	0.164099	+	0.986444i	$0.447528\pi$
$54$	0	0
$55$	−10.6084	−1.43044
$56$	0	0
$57$	−2.69846	−0.357420
$58$	0	0
$59$	−1.63053	−0.212276	−0.106138	−	0.994351i	$-0.533849\pi$
$59$	−1.63053	−0.212276	−0.106138	+	0.994351i	$0.533849\pi$
$60$	0	0
$61$	−1.29167	−0.165381	−0.0826905	−	0.996575i	$-0.526351\pi$
$61$	−1.29167	−0.165381	−0.0826905	+	0.996575i	$0.526351\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	−6.26027	−0.776491
$66$	0	0
$67$	−11.0771	−1.35329	−0.676643	−	0.736311i	$-0.736566\pi$
$67$	−11.0771	−1.35329	−0.676643	+	0.736311i	$0.736566\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	7.70607	0.914542	0.457271	−	0.889327i	$-0.348827\pi$
$71$	7.70607	0.914542	0.457271	+	0.889327i	$0.348827\pi$
$72$	0	0
$73$	9.36643	1.09626	0.548129	−	0.836394i	$-0.315340\pi$
$73$	9.36643	1.09626	0.548129	+	0.836394i	$0.315340\pi$
$74$	0	0
$75$	10.4550	1.20724
$76$	0	0
$77$	−2.69846	−0.307518
$78$	0	0
$79$	7.54355	0.848716	0.424358	−	0.905495i	$-0.360500\pi$
$79$	7.54355	0.848716	0.424358	+	0.905495i	$0.360500\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−2.16174	−0.237282	−0.118641	−	0.992937i	$-0.537854\pi$
$83$	−2.16174	−0.237282	−0.118641	+	0.992937i	$0.537854\pi$
$84$	0	0
$85$	−6.60842	−0.716784
$86$	0	0
$87$	4.39770	0.471483
$88$	0	0
$89$	5.84350	0.619410	0.309705	−	0.950833i	$-0.399770\pi$
$89$	5.84350	0.619410	0.309705	+	0.950833i	$0.399770\pi$
$90$	0	0
$91$	−1.59242	−0.166931
$92$	0	0
$93$	8.29927	0.860595
$94$	0	0
$95$	−10.6084	−1.08840
$96$	0	0
$97$	7.64198	0.775926	0.387963	−	0.921675i	$-0.373179\pi$
$97$	7.64198	0.775926	0.387963	+	0.921675i	$0.373179\pi$
$98$	0	0
$99$	−2.69846	−0.271206
$100$	0	0
$101$	−0.232042	−0.0230890	−0.0115445	−	0.999933i	$-0.503675\pi$
$101$	−0.232042	−0.0230890	−0.0115445	+	0.999933i	$0.503675\pi$
$102$	0	0
$103$	10.2612	1.01106	0.505532	−	0.862808i	$-0.331296\pi$
$103$	10.2612	1.01106	0.505532	+	0.862808i	$0.331296\pi$
$104$	0	0
$105$	3.93128	0.383654
$106$	0	0
$107$	17.8420	1.72486	0.862428	−	0.506180i	$-0.168943\pi$
$107$	17.8420	1.72486	0.862428	+	0.506180i	$0.168943\pi$
$108$	0	0
$109$	19.7741	1.89402	0.947008	−	0.321210i	$-0.104090\pi$
$109$	19.7741	1.89402	0.947008	+	0.321210i	$0.104090\pi$
$110$	0	0
$111$	6.24043	0.592315
$112$	0	0
$113$	12.2412	1.15156	0.575778	−	0.817606i	$-0.304699\pi$
$113$	12.2412	1.15156	0.575778	+	0.817606i	$0.304699\pi$
$114$	0	0
$115$	3.93128	0.366594
$116$	0	0
$117$	−1.59242	−0.147220
$118$	0	0
$119$	−1.68098	−0.154095
$120$	0	0
$121$	−3.71830	−0.338028
$122$	0	0
$123$	2.60466	0.234854
$124$	0	0
$125$	21.4451	1.91811
$126$	0	0
$127$	9.24804	0.820631	0.410315	−	0.911944i	$-0.365419\pi$
$127$	9.24804	0.820631	0.410315	+	0.911944i	$0.365419\pi$
$128$	0	0
$129$	−10.2306	−0.900750
$130$	0	0
$131$	0.855736	0.0747660	0.0373830	−	0.999301i	$-0.488098\pi$
$131$	0.855736	0.0747660	0.0373830	+	0.999301i	$0.488098\pi$
$132$	0	0
$133$	−2.69846	−0.233986
$134$	0	0
$135$	3.93128	0.338351
$136$	0	0
$137$	−2.46723	−0.210789	−0.105395	−	0.994430i	$-0.533611\pi$
$137$	−2.46723	−0.210789	−0.105395	+	0.994430i	$0.533611\pi$
$138$	0	0
$139$	−1.50523	−0.127672	−0.0638358	−	0.997960i	$-0.520333\pi$
$139$	−1.50523	−0.127672	−0.0638358	+	0.997960i	$0.520333\pi$
$140$	0	0
$141$	3.07632	0.259073
$142$	0	0
$143$	4.29710	0.359341
$144$	0	0
$145$	17.2886	1.43574
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−1.63577	−0.134008	−0.0670038	−	0.997753i	$-0.521344\pi$
$149$	−1.63577	−0.134008	−0.0670038	+	0.997753i	$0.521344\pi$
$150$	0	0
$151$	0.726816	0.0591474	0.0295737	−	0.999563i	$-0.490585\pi$
$151$	0.726816	0.0591474	0.0295737	+	0.999563i	$0.490585\pi$
$152$	0	0
$153$	−1.68098	−0.135899
$154$	0	0
$155$	32.6268	2.62065
$156$	0	0
$157$	6.96278	0.555690	0.277845	−	0.960626i	$-0.410380\pi$
$157$	6.96278	0.555690	0.277845	+	0.960626i	$0.410380\pi$
$158$	0	0
$159$	2.38932	0.189485
$160$	0	0
$161$	1.00000	0.0788110
$162$	0	0
$163$	−4.19708	−0.328741	−0.164370	−	0.986399i	$-0.552559\pi$
$163$	−4.19708	−0.328741	−0.164370	+	0.986399i	$0.552559\pi$
$164$	0	0
$165$	−10.6084	−0.825864
$166$	0	0
$167$	1.29691	0.100358	0.0501790	−	0.998740i	$-0.484021\pi$
$167$	1.29691	0.100358	0.0501790	+	0.998740i	$0.484021\pi$
$168$	0	0
$169$	−10.4642	−0.804937
$170$	0	0
$171$	−2.69846	−0.206357
$172$	0	0
$173$	−14.9405	−1.13590	−0.567952	−	0.823062i	$-0.692264\pi$
$173$	−14.9405	−1.13590	−0.567952	+	0.823062i	$0.692264\pi$
$174$	0	0
$175$	10.4550	0.790323
$176$	0	0
$177$	−1.63053	−0.122558
$178$	0	0
$179$	−12.8474	−0.960263	−0.480132	−	0.877196i	$-0.659411\pi$
$179$	−12.8474	−0.960263	−0.480132	+	0.877196i	$0.659411\pi$
$180$	0	0
$181$	9.55867	0.710490	0.355245	−	0.934773i	$-0.384397\pi$
$181$	9.55867	0.710490	0.355245	+	0.934773i	$0.384397\pi$
$182$	0	0
$183$	−1.29167	−0.0954827
$184$	0	0
$185$	24.5329	1.80369
$186$	0	0
$187$	4.53607	0.331710
$188$	0	0
$189$	1.00000	0.0727393
$190$	0	0
$191$	−12.2222	−0.884365	−0.442183	−	0.896925i	$-0.645796\pi$
$191$	−12.2222	−0.884365	−0.442183	+	0.896925i	$0.645796\pi$
$192$	0	0
$193$	−1.67784	−0.120774	−0.0603868	−	0.998175i	$-0.519233\pi$
$193$	−1.67784	−0.120774	−0.0603868	+	0.998175i	$0.519233\pi$
$194$	0	0
$195$	−6.26027	−0.448307
$196$	0	0
$197$	−9.43525	−0.672234	−0.336117	−	0.941820i	$-0.609114\pi$
$197$	−9.43525	−0.672234	−0.336117	+	0.941820i	$0.609114\pi$
$198$	0	0
$199$	9.55028	0.677001	0.338501	−	0.940966i	$-0.390080\pi$
$199$	9.55028	0.677001	0.338501	+	0.940966i	$0.390080\pi$
$200$	0	0
$201$	−11.0771	−0.781320
$202$	0	0
$203$	4.39770	0.308658
$204$	0	0
$205$	10.2397	0.715168
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	7.28170	0.503685
$210$	0	0
$211$	−23.7063	−1.63201	−0.816004	−	0.578046i	$-0.803815\pi$
$211$	−23.7063	−1.63201	−0.816004	+	0.578046i	$0.803815\pi$
$212$	0	0
$213$	7.70607	0.528011
$214$	0	0
$215$	−40.2192	−2.74293
$216$	0	0
$217$	8.29927	0.563391
$218$	0	0
$219$	9.36643	0.632925
$220$	0	0
$221$	2.67684	0.180064
$222$	0	0
$223$	10.6313	0.711926	0.355963	−	0.934500i	$-0.384153\pi$
$223$	10.6313	0.711926	0.355963	+	0.934500i	$0.384153\pi$
$224$	0	0
$225$	10.4550	0.696999
$226$	0	0
$227$	−12.2230	−0.811267	−0.405634	−	0.914036i	$-0.632949\pi$
$227$	−12.2230	−0.811267	−0.405634	+	0.914036i	$0.632949\pi$
$228$	0	0
$229$	−15.6259	−1.03259	−0.516294	−	0.856411i	$-0.672689\pi$
$229$	−15.6259	−1.03259	−0.516294	+	0.856411i	$0.672689\pi$
$230$	0	0
$231$	−2.69846	−0.177546
$232$	0	0
$233$	−22.3685	−1.46541	−0.732705	−	0.680546i	$-0.761742\pi$
$233$	−22.3685	−1.46541	−0.732705	+	0.680546i	$0.761742\pi$
$234$	0	0
$235$	12.0939	0.788919
$236$	0	0
$237$	7.54355	0.490006
$238$	0	0
$239$	−20.4193	−1.32081	−0.660406	−	0.750909i	$-0.729616\pi$
$239$	−20.4193	−1.32081	−0.660406	+	0.750909i	$0.729616\pi$
$240$	0	0
$241$	8.00158	0.515427	0.257714	−	0.966221i	$-0.417031\pi$
$241$	8.00158	0.515427	0.257714	+	0.966221i	$0.417031\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.93128	0.251161
$246$	0	0
$247$	4.29710	0.273418
$248$	0	0
$249$	−2.16174	−0.136995
$250$	0	0
$251$	−31.4694	−1.98633	−0.993166	−	0.116707i	$-0.962766\pi$
$251$	−31.4694	−1.98633	−0.993166	+	0.116707i	$0.962766\pi$
$252$	0	0
$253$	−2.69846	−0.169651
$254$	0	0
$255$	−6.60842	−0.413835
$256$	0	0
$257$	−6.89989	−0.430403	−0.215202	−	0.976570i	$-0.569041\pi$
$257$	−6.89989	−0.430403	−0.215202	+	0.976570i	$0.569041\pi$
$258$	0	0
$259$	6.24043	0.387761
$260$	0	0
$261$	4.39770	0.272211
$262$	0	0
$263$	−3.80925	−0.234888	−0.117444	−	0.993079i	$-0.537470\pi$
$263$	−3.80925	−0.234888	−0.117444	+	0.993079i	$0.537470\pi$
$264$	0	0
$265$	9.39308	0.577012
$266$	0	0
$267$	5.84350	0.357616
$268$	0	0
$269$	17.8222	1.08664	0.543320	−	0.839526i	$-0.317167\pi$
$269$	17.8222	1.08664	0.543320	+	0.839526i	$0.317167\pi$
$270$	0	0
$271$	−28.6062	−1.73770	−0.868851	−	0.495074i	$-0.835141\pi$
$271$	−28.6062	−1.73770	−0.868851	+	0.495074i	$0.835141\pi$
$272$	0	0
$273$	−1.59242	−0.0963779
$274$	0	0
$275$	−28.2124	−1.70127
$276$	0	0
$277$	−0.806180	−0.0484387	−0.0242193	−	0.999707i	$-0.507710\pi$
$277$	−0.806180	−0.0484387	−0.0242193	+	0.999707i	$0.507710\pi$
$278$	0	0
$279$	8.29927	0.496865
$280$	0	0
$281$	18.1136	1.08057	0.540285	−	0.841482i	$-0.318317\pi$
$281$	18.1136	1.08057	0.540285	+	0.841482i	$0.318317\pi$
$282$	0	0
$283$	3.23431	0.192260	0.0961298	−	0.995369i	$-0.469354\pi$
$283$	3.23431	0.192260	0.0961298	+	0.995369i	$0.469354\pi$
$284$	0	0
$285$	−10.6084	−0.628389
$286$	0	0
$287$	2.60466	0.153748
$288$	0	0
$289$	−14.1743	−0.833782
$290$	0	0
$291$	7.64198	0.447981
$292$	0	0
$293$	13.6703	0.798629	0.399315	−	0.916814i	$-0.369248\pi$
$293$	13.6703	0.798629	0.399315	+	0.916814i	$0.369248\pi$
$294$	0	0
$295$	−6.41006	−0.373208
$296$	0	0
$297$	−2.69846	−0.156581
$298$	0	0
$299$	−1.59242	−0.0920923
$300$	0	0
$301$	−10.2306	−0.589680
$302$	0	0
$303$	−0.232042	−0.0133304
$304$	0	0
$305$	−5.07791	−0.290760
$306$	0	0
$307$	−15.8940	−0.907118	−0.453559	−	0.891226i	$-0.649846\pi$
$307$	−15.8940	−0.907118	−0.453559	+	0.891226i	$0.649846\pi$
$308$	0	0
$309$	10.2612	0.583738
$310$	0	0
$311$	12.9375	0.733620	0.366810	−	0.930296i	$-0.380450\pi$
$311$	12.9375	0.733620	0.366810	+	0.930296i	$0.380450\pi$
$312$	0	0
$313$	−13.9670	−0.789464	−0.394732	−	0.918796i	$-0.629163\pi$
$313$	−13.9670	−0.789464	−0.394732	+	0.918796i	$0.629163\pi$
$314$	0	0
$315$	3.93128	0.221503
$316$	0	0
$317$	33.0258	1.85491	0.927457	−	0.373930i	$-0.121990\pi$
$317$	33.0258	1.85491	0.927457	+	0.373930i	$0.121990\pi$
$318$	0	0
$319$	−11.8670	−0.664426
$320$	0	0
$321$	17.8420	0.995846
$322$	0	0
$323$	4.53607	0.252394
$324$	0	0
$325$	−16.6488	−0.923508
$326$	0	0
$327$	19.7741	1.09351
$328$	0	0
$329$	3.07632	0.169603
$330$	0	0
$331$	9.51818	0.523166	0.261583	−	0.965181i	$-0.415755\pi$
$331$	9.51818	0.523166	0.261583	+	0.965181i	$0.415755\pi$
$332$	0	0
$333$	6.24043	0.341973
$334$	0	0
$335$	−43.5473	−2.37925
$336$	0	0
$337$	−23.2420	−1.26607	−0.633037	−	0.774121i	$-0.718192\pi$
$337$	−23.2420	−1.26607	−0.633037	+	0.774121i	$0.718192\pi$
$338$	0	0
$339$	12.2412	0.664851
$340$	0	0
$341$	−22.3953	−1.21277
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	3.93128	0.211653
$346$	0	0
$347$	25.2712	1.35663	0.678314	−	0.734772i	$-0.262711\pi$
$347$	25.2712	1.35663	0.678314	+	0.734772i	$0.262711\pi$
$348$	0	0
$349$	−3.31241	−0.177309	−0.0886546	−	0.996062i	$-0.528257\pi$
$349$	−3.31241	−0.177309	−0.0886546	+	0.996062i	$0.528257\pi$
$350$	0	0
$351$	−1.59242	−0.0849973
$352$	0	0
$353$	−27.7291	−1.47587	−0.737936	−	0.674871i	$-0.764199\pi$
$353$	−27.7291	−1.47587	−0.737936	+	0.674871i	$0.764199\pi$
$354$	0	0
$355$	30.2947	1.60788
$356$	0	0
$357$	−1.68098	−0.0889670
$358$	0	0
$359$	−14.2809	−0.753717	−0.376858	−	0.926271i	$-0.622996\pi$
$359$	−14.2809	−0.753717	−0.376858	+	0.926271i	$0.622996\pi$
$360$	0	0
$361$	−11.7183	−0.616753
$362$	0	0
$363$	−3.71830	−0.195160
$364$	0	0
$365$	36.8221	1.92736
$366$	0	0
$367$	35.9110	1.87454	0.937269	−	0.348607i	$-0.113345\pi$
$367$	35.9110	1.87454	0.937269	+	0.348607i	$0.113345\pi$
$368$	0	0
$369$	2.60466	0.135593
$370$	0	0
$371$	2.38932	0.124047
$372$	0	0
$373$	7.16923	0.371208	0.185604	−	0.982625i	$-0.440576\pi$
$373$	7.16923	0.371208	0.185604	+	0.982625i	$0.440576\pi$
$374$	0	0
$375$	21.4451	1.10742
$376$	0	0
$377$	−7.00301	−0.360673
$378$	0	0
$379$	−20.8434	−1.07066	−0.535328	−	0.844644i	$-0.679812\pi$
$379$	−20.8434	−1.07066	−0.535328	+	0.844644i	$0.679812\pi$
$380$	0	0
$381$	9.24804	0.473791
$382$	0	0
$383$	−13.0398	−0.666304	−0.333152	−	0.942873i	$-0.608112\pi$
$383$	−13.0398	−0.666304	−0.333152	+	0.942873i	$0.608112\pi$
$384$	0	0
$385$	−10.6084	−0.540655
$386$	0	0
$387$	−10.2306	−0.520048
$388$	0	0
$389$	8.35563	0.423647	0.211824	−	0.977308i	$-0.432060\pi$
$389$	8.35563	0.423647	0.211824	+	0.977308i	$0.432060\pi$
$390$	0	0
$391$	−1.68098	−0.0850109
$392$	0	0
$393$	0.855736	0.0431662
$394$	0	0
$395$	29.6558	1.49215
$396$	0	0
$397$	−19.2960	−0.968439	−0.484219	−	0.874947i	$-0.660896\pi$
$397$	−19.2960	−0.968439	−0.484219	+	0.874947i	$0.660896\pi$
$398$	0	0
$399$	−2.69846	−0.135092
$400$	0	0
$401$	26.6589	1.33128	0.665640	−	0.746273i	$-0.268158\pi$
$401$	26.6589	1.33128	0.665640	+	0.746273i	$0.268158\pi$
$402$	0	0
$403$	−13.2160	−0.658334
$404$	0	0
$405$	3.93128	0.195347
$406$	0	0
$407$	−16.8396	−0.834706
$408$	0	0
$409$	−4.60910	−0.227905	−0.113953	−	0.993486i	$-0.536351\pi$
$409$	−4.60910	−0.227905	−0.113953	+	0.993486i	$0.536351\pi$
$410$	0	0
$411$	−2.46723	−0.121699
$412$	0	0
$413$	−1.63053	−0.0802329
$414$	0	0
$415$	−8.49842	−0.417171
$416$	0	0
$417$	−1.50523	−0.0737112
$418$	0	0
$419$	−22.7226	−1.11007	−0.555037	−	0.831826i	$-0.687296\pi$
$419$	−22.7226	−1.11007	−0.555037	+	0.831826i	$0.687296\pi$
$420$	0	0
$421$	8.03667	0.391683	0.195842	−	0.980636i	$-0.437256\pi$
$421$	8.03667	0.391683	0.195842	+	0.980636i	$0.437256\pi$
$422$	0	0
$423$	3.07632	0.149576
$424$	0	0
$425$	−17.5747	−0.852496
$426$	0	0
$427$	−1.29167	−0.0625081
$428$	0	0
$429$	4.29710	0.207466
$430$	0	0
$431$	−16.3215	−0.786178	−0.393089	−	0.919500i	$-0.628594\pi$
$431$	−16.3215	−0.786178	−0.393089	+	0.919500i	$0.628594\pi$
$432$	0	0
$433$	−36.3297	−1.74590	−0.872948	−	0.487814i	$-0.837795\pi$
$433$	−36.3297	−1.74590	−0.872948	+	0.487814i	$0.837795\pi$
$434$	0	0
$435$	17.2886	0.828926
$436$	0	0
$437$	−2.69846	−0.129085
$438$	0	0
$439$	−32.7223	−1.56175	−0.780875	−	0.624687i	$-0.785227\pi$
$439$	−32.7223	−1.56175	−0.780875	+	0.624687i	$0.785227\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	22.8360	1.08497	0.542486	−	0.840065i	$-0.317483\pi$
$443$	22.8360	1.08497	0.542486	+	0.840065i	$0.317483\pi$
$444$	0	0
$445$	22.9725	1.08900
$446$	0	0
$447$	−1.63577	−0.0773693
$448$	0	0
$449$	25.8511	1.21999	0.609995	−	0.792405i	$-0.291171\pi$
$449$	25.8511	1.21999	0.609995	+	0.792405i	$0.291171\pi$
$450$	0	0
$451$	−7.02857	−0.330963
$452$	0	0
$453$	0.726816	0.0341488
$454$	0	0
$455$	−6.26027	−0.293486
$456$	0	0
$457$	26.6361	1.24598	0.622991	−	0.782229i	$-0.285917\pi$
$457$	26.6361	1.24598	0.622991	+	0.782229i	$0.285917\pi$
$458$	0	0
$459$	−1.68098	−0.0784615
$460$	0	0
$461$	0.647907	0.0301760	0.0150880	−	0.999886i	$-0.495197\pi$
$461$	0.647907	0.0301760	0.0150880	+	0.999886i	$0.495197\pi$
$462$	0	0
$463$	7.94371	0.369176	0.184588	−	0.982816i	$-0.440905\pi$
$463$	7.94371	0.369176	0.184588	+	0.982816i	$0.440905\pi$
$464$	0	0
$465$	32.6268	1.51303
$466$	0	0
$467$	5.54750	0.256707	0.128354	−	0.991728i	$-0.459031\pi$
$467$	5.54750	0.256707	0.128354	+	0.991728i	$0.459031\pi$
$468$	0	0
$469$	−11.0771	−0.511494
$470$	0	0
$471$	6.96278	0.320828
$472$	0	0
$473$	27.6068	1.26936
$474$	0	0
$475$	−28.2124	−1.29447
$476$	0	0
$477$	2.38932	0.109399
$478$	0	0
$479$	−1.71594	−0.0784034	−0.0392017	−	0.999231i	$-0.512481\pi$
$479$	−1.71594	−0.0784034	−0.0392017	+	0.999231i	$0.512481\pi$
$480$	0	0
$481$	−9.93741	−0.453107
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	30.0428	1.36417
$486$	0	0
$487$	−17.7399	−0.803873	−0.401936	−	0.915668i	$-0.631663\pi$
$487$	−17.7399	−0.803873	−0.401936	+	0.915668i	$0.631663\pi$
$488$	0	0
$489$	−4.19708	−0.189799
$490$	0	0
$491$	11.0497	0.498666	0.249333	−	0.968418i	$-0.419789\pi$
$491$	11.0497	0.498666	0.249333	+	0.968418i	$0.419789\pi$
$492$	0	0
$493$	−7.39246	−0.332940
$494$	0	0
$495$	−10.6084	−0.476813
$496$	0	0
$497$	7.70607	0.345664
$498$	0	0
$499$	19.0954	0.854828	0.427414	−	0.904056i	$-0.359425\pi$
$499$	19.0954	0.854828	0.427414	+	0.904056i	$0.359425\pi$
$500$	0	0
$501$	1.29691	0.0579417
$502$	0	0
$503$	8.59024	0.383020	0.191510	−	0.981491i	$-0.438662\pi$
$503$	8.59024	0.383020	0.191510	+	0.981491i	$0.438662\pi$
$504$	0	0
$505$	−0.912222	−0.0405933
$506$	0	0
$507$	−10.4642	−0.464731
$508$	0	0
$509$	−42.7427	−1.89454	−0.947269	−	0.320440i	$-0.896169\pi$
$509$	−42.7427	−1.89454	−0.947269	+	0.320440i	$0.896169\pi$
$510$	0	0
$511$	9.36643	0.414347
$512$	0	0
$513$	−2.69846	−0.119140
$514$	0	0
$515$	40.3396	1.77757
$516$	0	0
$517$	−8.30134	−0.365093
$518$	0	0
$519$	−14.9405	−0.655814
$520$	0	0
$521$	15.2763	0.669265	0.334633	−	0.942349i	$-0.391388\pi$
$521$	15.2763	0.669265	0.334633	+	0.942349i	$0.391388\pi$
$522$	0	0
$523$	−3.77637	−0.165129	−0.0825645	−	0.996586i	$-0.526311\pi$
$523$	−3.77637	−0.165129	−0.0825645	+	0.996586i	$0.526311\pi$
$524$	0	0
$525$	10.4550	0.456293
$526$	0	0
$527$	−13.9509	−0.607712
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−1.63053	−0.0707588
$532$	0	0
$533$	−4.14772	−0.179658
$534$	0	0
$535$	70.1421	3.03251
$536$	0	0
$537$	−12.8474	−0.554408
$538$	0	0
$539$	−2.69846	−0.116231
$540$	0	0
$541$	−22.2494	−0.956578	−0.478289	−	0.878203i	$-0.658743\pi$
$541$	−22.2494	−0.956578	−0.478289	+	0.878203i	$0.658743\pi$
$542$	0	0
$543$	9.55867	0.410202
$544$	0	0
$545$	77.7376	3.32991
$546$	0	0
$547$	28.2224	1.20670	0.603351	−	0.797476i	$-0.293832\pi$
$547$	28.2224	1.20670	0.603351	+	0.797476i	$0.293832\pi$
$548$	0	0
$549$	−1.29167	−0.0551270
$550$	0	0
$551$	−11.8670	−0.505553
$552$	0	0
$553$	7.54355	0.320784
$554$	0	0
$555$	24.5329	1.04136
$556$	0	0
$557$	−18.6304	−0.789397	−0.394699	−	0.918811i	$-0.629151\pi$
$557$	−18.6304	−0.789397	−0.394699	+	0.918811i	$0.629151\pi$
$558$	0	0
$559$	16.2914	0.689052
$560$	0	0
$561$	4.53607	0.191513
$562$	0	0
$563$	−11.4535	−0.482709	−0.241354	−	0.970437i	$-0.577592\pi$
$563$	−11.4535	−0.482709	−0.241354	+	0.970437i	$0.577592\pi$
$564$	0	0
$565$	48.1237	2.02458
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−2.98194	−0.125009	−0.0625046	−	0.998045i	$-0.519909\pi$
$569$	−2.98194	−0.125009	−0.0625046	+	0.998045i	$0.519909\pi$
$570$	0	0
$571$	−30.6458	−1.28249	−0.641243	−	0.767338i	$-0.721581\pi$
$571$	−30.6458	−1.28249	−0.641243	+	0.767338i	$0.721581\pi$
$572$	0	0
$573$	−12.2222	−0.510588
$574$	0	0
$575$	10.4550	0.436003
$576$	0	0
$577$	−19.0107	−0.791424	−0.395712	−	0.918375i	$-0.629502\pi$
$577$	−19.0107	−0.791424	−0.395712	+	0.918375i	$0.629502\pi$
$578$	0	0
$579$	−1.67784	−0.0697286
$580$	0	0
$581$	−2.16174	−0.0896842
$582$	0	0
$583$	−6.44748	−0.267027
$584$	0	0
$585$	−6.26027	−0.258830
$586$	0	0
$587$	−18.8626	−0.778544	−0.389272	−	0.921123i	$-0.627273\pi$
$587$	−18.8626	−0.778544	−0.389272	+	0.921123i	$0.627273\pi$
$588$	0	0
$589$	−22.3953	−0.922781
$590$	0	0
$591$	−9.43525	−0.388114
$592$	0	0
$593$	−45.7993	−1.88075	−0.940376	−	0.340137i	$-0.889526\pi$
$593$	−45.7993	−1.88075	−0.940376	+	0.340137i	$0.889526\pi$
$594$	0	0
$595$	−6.60842	−0.270919
$596$	0	0
$597$	9.55028	0.390867
$598$	0	0
$599$	−7.42959	−0.303565	−0.151782	−	0.988414i	$-0.548501\pi$
$599$	−7.42959	−0.303565	−0.151782	+	0.988414i	$0.548501\pi$
$600$	0	0
$601$	−26.7887	−1.09273	−0.546367	−	0.837546i	$-0.683990\pi$
$601$	−26.7887	−1.09273	−0.546367	+	0.837546i	$0.683990\pi$
$602$	0	0
$603$	−11.0771	−0.451096
$604$	0	0
$605$	−14.6177	−0.594294
$606$	0	0
$607$	−23.1182	−0.938339	−0.469170	−	0.883108i	$-0.655447\pi$
$607$	−23.1182	−0.938339	−0.469170	+	0.883108i	$0.655447\pi$
$608$	0	0
$609$	4.39770	0.178204
$610$	0	0
$611$	−4.89881	−0.198185
$612$	0	0
$613$	−45.5827	−1.84107	−0.920533	−	0.390665i	$-0.872245\pi$
$613$	−45.5827	−1.84107	−0.920533	+	0.390665i	$0.872245\pi$
$614$	0	0
$615$	10.2397	0.412903
$616$	0	0
$617$	−20.3459	−0.819093	−0.409547	−	0.912289i	$-0.634313\pi$
$617$	−20.3459	−0.819093	−0.409547	+	0.912289i	$0.634313\pi$
$618$	0	0
$619$	33.7563	1.35678	0.678391	−	0.734701i	$-0.262678\pi$
$619$	33.7563	1.35678	0.678391	+	0.734701i	$0.262678\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	5.84350	0.234115
$624$	0	0
$625$	32.0319	1.28128
$626$	0	0
$627$	7.28170	0.290803
$628$	0	0
$629$	−10.4901	−0.418266
$630$	0	0
$631$	−3.54510	−0.141128	−0.0705642	−	0.997507i	$-0.522480\pi$
$631$	−3.54510	−0.141128	−0.0705642	+	0.997507i	$0.522480\pi$
$632$	0	0
$633$	−23.7063	−0.942240
$634$	0	0
$635$	36.3567	1.44277
$636$	0	0
$637$	−1.59242	−0.0630941
$638$	0	0
$639$	7.70607	0.304847
$640$	0	0
$641$	22.3914	0.884407	0.442204	−	0.896915i	$-0.354197\pi$
$641$	22.3914	0.884407	0.442204	+	0.896915i	$0.354197\pi$
$642$	0	0
$643$	2.35721	0.0929594	0.0464797	−	0.998919i	$-0.485200\pi$
$643$	2.35721	0.0929594	0.0464797	+	0.998919i	$0.485200\pi$
$644$	0	0
$645$	−40.2192	−1.58363
$646$	0	0
$647$	12.4923	0.491123	0.245561	−	0.969381i	$-0.421028\pi$
$647$	12.4923	0.491123	0.245561	+	0.969381i	$0.421028\pi$
$648$	0	0
$649$	4.39991	0.172712
$650$	0	0
$651$	8.29927	0.325274
$652$	0	0
$653$	−40.2732	−1.57601	−0.788007	−	0.615666i	$-0.788887\pi$
$653$	−40.2732	−1.57601	−0.788007	+	0.615666i	$0.788887\pi$
$654$	0	0
$655$	3.36414	0.131448
$656$	0	0
$657$	9.36643	0.365419
$658$	0	0
$659$	32.5709	1.26878	0.634391	−	0.773013i	$-0.281251\pi$
$659$	32.5709	1.26878	0.634391	+	0.773013i	$0.281251\pi$
$660$	0	0
$661$	−33.7307	−1.31197	−0.655985	−	0.754774i	$-0.727747\pi$
$661$	−33.7307	−1.31197	−0.655985	+	0.754774i	$0.727747\pi$
$662$	0	0
$663$	2.67684	0.103960
$664$	0	0
$665$	−10.6084	−0.411377
$666$	0	0
$667$	4.39770	0.170280
$668$	0	0
$669$	10.6313	0.411031
$670$	0	0
$671$	3.48551	0.134557
$672$	0	0
$673$	13.9469	0.537613	0.268806	−	0.963194i	$-0.413371\pi$
$673$	13.9469	0.537613	0.268806	+	0.963194i	$0.413371\pi$
$674$	0	0
$675$	10.4550	0.402413
$676$	0	0
$677$	−1.23877	−0.0476100	−0.0238050	−	0.999717i	$-0.507578\pi$
$677$	−1.23877	−0.0476100	−0.0238050	+	0.999717i	$0.507578\pi$
$678$	0	0
$679$	7.64198	0.293272
$680$	0	0
$681$	−12.2230	−0.468385
$682$	0	0
$683$	41.9616	1.60562	0.802808	−	0.596237i	$-0.203338\pi$
$683$	41.9616	1.60562	0.802808	+	0.596237i	$0.203338\pi$
$684$	0	0
$685$	−9.69937	−0.370594
$686$	0	0
$687$	−15.6259	−0.596165
$688$	0	0
$689$	−3.80481	−0.144952
$690$	0	0
$691$	−18.8108	−0.715597	−0.357799	−	0.933799i	$-0.616473\pi$
$691$	−18.8108	−0.715597	−0.357799	+	0.933799i	$0.616473\pi$
$692$	0	0
$693$	−2.69846	−0.102506
$694$	0	0
$695$	−5.91747	−0.224462
$696$	0	0
$697$	−4.37839	−0.165843
$698$	0	0
$699$	−22.3685	−0.846055
$700$	0	0
$701$	26.0550	0.984085	0.492042	−	0.870571i	$-0.336251\pi$
$701$	26.0550	0.984085	0.492042	+	0.870571i	$0.336251\pi$
$702$	0	0
$703$	−16.8396	−0.635116
$704$	0	0
$705$	12.0939	0.455483
$706$	0	0
$707$	−0.232042	−0.00872683
$708$	0	0
$709$	44.1870	1.65948	0.829739	−	0.558151i	$-0.188489\pi$
$709$	44.1870	1.65948	0.829739	+	0.558151i	$0.188489\pi$
$710$	0	0
$711$	7.54355	0.282905
$712$	0	0
$713$	8.29927	0.310810
$714$	0	0
$715$	16.8931	0.631766
$716$	0	0
$717$	−20.4193	−0.762571
$718$	0	0
$719$	33.8677	1.26305	0.631526	−	0.775354i	$-0.282429\pi$
$719$	33.8677	1.26305	0.631526	+	0.775354i	$0.282429\pi$
$720$	0	0
$721$	10.2612	0.382146
$722$	0	0
$723$	8.00158	0.297582
$724$	0	0
$725$	45.9780	1.70758
$726$	0	0
$727$	40.8066	1.51343	0.756717	−	0.653743i	$-0.226802\pi$
$727$	40.8066	1.51343	0.756717	+	0.653743i	$0.226802\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	17.1974	0.636068
$732$	0	0
$733$	13.0460	0.481863	0.240932	−	0.970542i	$-0.422547\pi$
$733$	13.0460	0.481863	0.240932	+	0.970542i	$0.422547\pi$
$734$	0	0
$735$	3.93128	0.145008
$736$	0	0
$737$	29.8912	1.10106
$738$	0	0
$739$	46.5452	1.71219	0.856096	−	0.516816i	$-0.172883\pi$
$739$	46.5452	1.71219	0.856096	+	0.516816i	$0.172883\pi$
$740$	0	0
$741$	4.29710	0.157858
$742$	0	0
$743$	1.36944	0.0502398	0.0251199	−	0.999684i	$-0.492003\pi$
$743$	1.36944	0.0502398	0.0251199	+	0.999684i	$0.492003\pi$
$744$	0	0
$745$	−6.43068	−0.235602
$746$	0	0
$747$	−2.16174	−0.0790940
$748$	0	0
$749$	17.8420	0.651934
$750$	0	0
$751$	−10.5582	−0.385276	−0.192638	−	0.981270i	$-0.561704\pi$
$751$	−10.5582	−0.385276	−0.192638	+	0.981270i	$0.561704\pi$
$752$	0	0
$753$	−31.4694	−1.14681
$754$	0	0
$755$	2.85732	0.103988
$756$	0	0
$757$	−34.1021	−1.23946	−0.619731	−	0.784814i	$-0.712758\pi$
$757$	−34.1021	−1.23946	−0.619731	+	0.784814i	$0.712758\pi$
$758$	0	0
$759$	−2.69846	−0.0979480
$760$	0	0
$761$	20.4495	0.741293	0.370646	−	0.928774i	$-0.379136\pi$
$761$	20.4495	0.741293	0.370646	+	0.928774i	$0.379136\pi$
$762$	0	0
$763$	19.7741	0.715871
$764$	0	0
$765$	−6.60842	−0.238928
$766$	0	0
$767$	2.59649	0.0937538
$768$	0	0
$769$	−53.6421	−1.93438	−0.967192	−	0.254046i	$-0.918238\pi$
$769$	−53.6421	−1.93438	−0.967192	+	0.254046i	$0.918238\pi$
$770$	0	0
$771$	−6.89989	−0.248493
$772$	0	0
$773$	−0.609681	−0.0219287	−0.0109644	−	0.999940i	$-0.503490\pi$
$773$	−0.609681	−0.0219287	−0.0109644	+	0.999940i	$0.503490\pi$
$774$	0	0
$775$	86.7688	3.11683
$776$	0	0
$777$	6.24043	0.223874
$778$	0	0
$779$	−7.02857	−0.251825
$780$	0	0
$781$	−20.7945	−0.744087
$782$	0	0
$783$	4.39770	0.157161
$784$	0	0
$785$	27.3727	0.976972
$786$	0	0
$787$	−6.68256	−0.238208	−0.119104	−	0.992882i	$-0.538002\pi$
$787$	−6.68256	−0.238208	−0.119104	+	0.992882i	$0.538002\pi$
$788$	0	0
$789$	−3.80925	−0.135613
$790$	0	0
$791$	12.2412	0.435247
$792$	0	0
$793$	2.05688	0.0730420
$794$	0	0
$795$	9.39308	0.333138
$796$	0	0
$797$	6.15012	0.217848	0.108924	−	0.994050i	$-0.465259\pi$
$797$	6.15012	0.217848	0.108924	+	0.994050i	$0.465259\pi$
$798$	0	0
$799$	−5.17125	−0.182946
$800$	0	0
$801$	5.84350	0.206470
$802$	0	0
$803$	−25.2750	−0.891934
$804$	0	0
$805$	3.93128	0.138560
$806$	0	0
$807$	17.8222	0.627371
$808$	0	0
$809$	1.64664	0.0578928	0.0289464	−	0.999581i	$-0.490785\pi$
$809$	1.64664	0.0578928	0.0289464	+	0.999581i	$0.490785\pi$
$810$	0	0
$811$	53.6438	1.88369	0.941844	−	0.336050i	$-0.109091\pi$
$811$	53.6438	1.88369	0.941844	+	0.336050i	$0.109091\pi$
$812$	0	0
$813$	−28.6062	−1.00326
$814$	0	0
$815$	−16.4999	−0.577967
$816$	0	0
$817$	27.6068	0.965839
$818$	0	0
$819$	−1.59242	−0.0556438
$820$	0	0
$821$	−17.1819	−0.599652	−0.299826	−	0.953994i	$-0.596929\pi$
$821$	−17.1819	−0.599652	−0.299826	+	0.953994i	$0.596929\pi$
$822$	0	0
$823$	40.0941	1.39759	0.698797	−	0.715320i	$-0.253719\pi$
$823$	40.0941	1.39759	0.698797	+	0.715320i	$0.253719\pi$
$824$	0	0
$825$	−28.2124	−0.982230
$826$	0	0
$827$	9.90076	0.344283	0.172142	−	0.985072i	$-0.444931\pi$
$827$	9.90076	0.344283	0.172142	+	0.985072i	$0.444931\pi$
$828$	0	0
$829$	−43.3433	−1.50538	−0.752688	−	0.658377i	$-0.771243\pi$
$829$	−43.3433	−1.50538	−0.752688	+	0.658377i	$0.771243\pi$
$830$	0	0
$831$	−0.806180	−0.0279661
$832$	0	0
$833$	−1.68098	−0.0582426
$834$	0	0
$835$	5.09852	0.176442
$836$	0	0
$837$	8.29927	0.286865
$838$	0	0
$839$	−14.0056	−0.483527	−0.241763	−	0.970335i	$-0.577726\pi$
$839$	−14.0056	−0.483527	−0.241763	+	0.970335i	$0.577726\pi$
$840$	0	0
$841$	−9.66020	−0.333110
$842$	0	0
$843$	18.1136	0.623867
$844$	0	0
$845$	−41.1377	−1.41518
$846$	0	0
$847$	−3.71830	−0.127762
$848$	0	0
$849$	3.23431	0.111001
$850$	0	0
$851$	6.24043	0.213919
$852$	0	0
$853$	−5.58205	−0.191126	−0.0955628	−	0.995423i	$-0.530465\pi$
$853$	−5.58205	−0.191126	−0.0955628	+	0.995423i	$0.530465\pi$
$854$	0	0
$855$	−10.6084	−0.362800
$856$	0	0
$857$	32.5663	1.11244	0.556221	−	0.831034i	$-0.312251\pi$
$857$	32.5663	1.11244	0.556221	+	0.831034i	$0.312251\pi$
$858$	0	0
$859$	13.4543	0.459055	0.229527	−	0.973302i	$-0.426282\pi$
$859$	13.4543	0.459055	0.229527	+	0.973302i	$0.426282\pi$
$860$	0	0
$861$	2.60466	0.0887665
$862$	0	0
$863$	9.98498	0.339893	0.169946	−	0.985453i	$-0.445641\pi$
$863$	9.98498	0.339893	0.169946	+	0.985453i	$0.445641\pi$
$864$	0	0
$865$	−58.7352	−1.99706
$866$	0	0
$867$	−14.1743	−0.481384
$868$	0	0
$869$	−20.3560	−0.690529
$870$	0	0
$871$	17.6395	0.597691
$872$	0	0
$873$	7.64198	0.258642
$874$	0	0
$875$	21.4451	0.724977
$876$	0	0
$877$	−55.8448	−1.88575	−0.942873	−	0.333152i	$-0.891888\pi$
$877$	−55.8448	−1.88575	−0.942873	+	0.333152i	$0.891888\pi$
$878$	0	0
$879$	13.6703	0.461089
$880$	0	0
$881$	−13.6444	−0.459692	−0.229846	−	0.973227i	$-0.573822\pi$
$881$	−13.6444	−0.459692	−0.229846	+	0.973227i	$0.573822\pi$
$882$	0	0
$883$	35.6072	1.19828	0.599138	−	0.800646i	$-0.295510\pi$
$883$	35.6072	1.19828	0.599138	+	0.800646i	$0.295510\pi$
$884$	0	0
$885$	−6.41006	−0.215472
$886$	0	0
$887$	20.4118	0.685362	0.342681	−	0.939452i	$-0.388665\pi$
$887$	20.4118	0.685362	0.342681	+	0.939452i	$0.388665\pi$
$888$	0	0
$889$	9.24804	0.310169
$890$	0	0
$891$	−2.69846	−0.0904019
$892$	0	0
$893$	−8.30134	−0.277794
$894$	0	0
$895$	−50.5070	−1.68826
$896$	0	0
$897$	−1.59242	−0.0531695
$898$	0	0
$899$	36.4977	1.21727
$900$	0	0
$901$	−4.01640	−0.133806
$902$	0	0
$903$	−10.2306	−0.340452
$904$	0	0
$905$	37.5778	1.24913
$906$	0	0
$907$	−46.4107	−1.54104	−0.770521	−	0.637414i	$-0.780004\pi$
$907$	−46.4107	−1.54104	−0.770521	+	0.637414i	$0.780004\pi$
$908$	0	0
$909$	−0.232042	−0.00769634
$910$	0	0
$911$	21.2375	0.703630	0.351815	−	0.936070i	$-0.385565\pi$
$911$	21.2375	0.703630	0.351815	+	0.936070i	$0.385565\pi$
$912$	0	0
$913$	5.83338	0.193057
$914$	0	0
$915$	−5.07791	−0.167870
$916$	0	0
$917$	0.855736	0.0282589
$918$	0	0
$919$	−18.0974	−0.596979	−0.298490	−	0.954413i	$-0.596483\pi$
$919$	−18.0974	−0.596979	−0.298490	+	0.954413i	$0.596483\pi$
$920$	0	0
$921$	−15.8940	−0.523725
$922$	0	0
$923$	−12.2713	−0.403916
$924$	0	0
$925$	65.2436	2.14520
$926$	0	0
$927$	10.2612	0.337021
$928$	0	0
$929$	−23.6583	−0.776204	−0.388102	−	0.921616i	$-0.626869\pi$
$929$	−23.6583	−0.776204	−0.388102	+	0.921616i	$0.626869\pi$
$930$	0	0
$931$	−2.69846	−0.0884385
$932$	0	0
$933$	12.9375	0.423556
$934$	0	0
$935$	17.8326	0.583187
$936$	0	0
$937$	4.66452	0.152383	0.0761917	−	0.997093i	$-0.475724\pi$
$937$	4.66452	0.152383	0.0761917	+	0.997093i	$0.475724\pi$
$938$	0	0
$939$	−13.9670	−0.455797
$940$	0	0
$941$	21.0877	0.687439	0.343719	−	0.939072i	$-0.388313\pi$
$941$	21.0877	0.687439	0.343719	+	0.939072i	$0.388313\pi$
$942$	0	0
$943$	2.60466	0.0848194
$944$	0	0
$945$	3.93128	0.127885
$946$	0	0
$947$	4.95508	0.161018	0.0805092	−	0.996754i	$-0.474345\pi$
$947$	4.95508	0.161018	0.0805092	+	0.996754i	$0.474345\pi$
$948$	0	0
$949$	−14.9153	−0.484172
$950$	0	0
$951$	33.0258	1.07094
$952$	0	0
$953$	39.8355	1.29040	0.645199	−	0.764015i	$-0.276775\pi$
$953$	39.8355	1.29040	0.645199	+	0.764015i	$0.276775\pi$
$954$	0	0
$955$	−48.0488	−1.55482
$956$	0	0
$957$	−11.8670	−0.383607
$958$	0	0
$959$	−2.46723	−0.0796709
$960$	0	0
$961$	37.8779	1.22187
$962$	0	0
$963$	17.8420	0.574952
$964$	0	0
$965$	−6.59606	−0.212335
$966$	0	0
$967$	−61.3251	−1.97208	−0.986041	−	0.166504i	$-0.946752\pi$
$967$	−61.3251	−1.97208	−0.986041	+	0.166504i	$0.946752\pi$
$968$	0	0
$969$	4.53607	0.145719
$970$	0	0
$971$	29.7183	0.953705	0.476852	−	0.878983i	$-0.341778\pi$
$971$	29.7183	0.953705	0.476852	+	0.878983i	$0.341778\pi$
$972$	0	0
$973$	−1.50523	−0.0482553
$974$	0	0
$975$	−16.6488	−0.533188
$976$	0	0
$977$	16.4741	0.527054	0.263527	−	0.964652i	$-0.415114\pi$
$977$	16.4741	0.527054	0.263527	+	0.964652i	$0.415114\pi$
$978$	0	0
$979$	−15.7685	−0.503962
$980$	0	0
$981$	19.7741	0.631339
$982$	0	0
$983$	−32.8673	−1.04830	−0.524152	−	0.851625i	$-0.675617\pi$
$983$	−32.8673	−1.04830	−0.524152	+	0.851625i	$0.675617\pi$
$984$	0	0
$985$	−37.0926	−1.18187
$986$	0	0
$987$	3.07632	0.0979205
$988$	0	0
$989$	−10.2306	−0.325313
$990$	0	0
$991$	−58.4640	−1.85717	−0.928585	−	0.371120i	$-0.878974\pi$
$991$	−58.4640	−1.85717	−0.928585	+	0.371120i	$0.878974\pi$
$992$	0	0
$993$	9.51818	0.302050
$994$	0	0
$995$	37.5449	1.19025
$996$	0	0
$997$	−34.5053	−1.09279	−0.546396	−	0.837527i	$-0.684001\pi$
$997$	−34.5053	−1.09279	−0.546396	+	0.837527i	$0.684001\pi$
$998$	0	0
$999$	6.24043	0.197438

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.93128 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.93128 q^{5} +1.00000 q^{7} +1.00000 q^{9} -2.69846 q^{11} -1.59242 q^{13} +3.93128 q^{15} -1.68098 q^{17} -2.69846 q^{19} +1.00000 q^{21} +1.00000 q^{23} +10.4550 q^{25} +1.00000 q^{27} +4.39770 q^{29} +8.29927 q^{31} -2.69846 q^{33} +3.93128 q^{35} +6.24043 q^{37} -1.59242 q^{39} +2.60466 q^{41} -10.2306 q^{43} +3.93128 q^{45} +3.07632 q^{47} +1.00000 q^{49} -1.68098 q^{51} +2.38932 q^{53} -10.6084 q^{55} -2.69846 q^{57} -1.63053 q^{59} -1.29167 q^{61} +1.00000 q^{63} -6.26027 q^{65} -11.0771 q^{67} +1.00000 q^{69} +7.70607 q^{71} +9.36643 q^{73} +10.4550 q^{75} -2.69846 q^{77} +7.54355 q^{79} +1.00000 q^{81} -2.16174 q^{83} -6.60842 q^{85} +4.39770 q^{87} +5.84350 q^{89} -1.59242 q^{91} +8.29927 q^{93} -10.6084 q^{95} +7.64198 q^{97} -2.69846 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{3} + 2 q^{5} + 6 q^{7} + 6 q^{9}+O(q^{10}) \) 6 * q + 6 * q^3 + 2 * q^5 + 6 * q^7 + 6 * q^9	\( 6 q + 6 q^{3} + 2 q^{5} + 6 q^{7} + 6 q^{9} + 3 q^{11} + 2 q^{15} + 6 q^{17} + 3 q^{19} + 6 q^{21} + 6 q^{23} + 10 q^{25} + 6 q^{27} + 5 q^{29} + 4 q^{31} + 3 q^{33} + 2 q^{35} + q^{37} + 12 q^{41} + 6 q^{43} + 2 q^{45} + 6 q^{47} + 6 q^{49} + 6 q^{51} + 10 q^{53} - 3 q^{55} + 3 q^{57} + 14 q^{59} + 4 q^{61} + 6 q^{63} + 27 q^{65} - 7 q^{67} + 6 q^{69} - 7 q^{71} + 10 q^{73} + 10 q^{75} + 3 q^{77} - 14 q^{79} + 6 q^{81} - 14 q^{83} + 21 q^{85} + 5 q^{87} + 25 q^{89} + 4 q^{93} - 3 q^{95} + 11 q^{97} + 3 q^{99}+O(q^{100}) \) 6 * q + 6 * q^3 + 2 * q^5 + 6 * q^7 + 6 * q^9 + 3 * q^11 + 2 * q^15 + 6 * q^17 + 3 * q^19 + 6 * q^21 + 6 * q^23 + 10 * q^25 + 6 * q^27 + 5 * q^29 + 4 * q^31 + 3 * q^33 + 2 * q^35 + q^37 + 12 * q^41 + 6 * q^43 + 2 * q^45 + 6 * q^47 + 6 * q^49 + 6 * q^51 + 10 * q^53 - 3 * q^55 + 3 * q^57 + 14 * q^59 + 4 * q^61 + 6 * q^63 + 27 * q^65 - 7 * q^67 + 6 * q^69 - 7 * q^71 + 10 * q^73 + 10 * q^75 + 3 * q^77 - 14 * q^79 + 6 * q^81 - 14 * q^83 + 21 * q^85 + 5 * q^87 + 25 * q^89 + 4 * q^93 - 3 * q^95 + 11 * q^97 + 3 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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