LMFDB - Embedded newform 7728.2.a.cg.1.3

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			1.3
Root			$1.67441$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -0.344522 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -0.344522 q^{5} +1.00000 q^{7} +1.00000 q^{9} +5.25319 q^{11} -4.27021 q^{13} +0.344522 q^{15} +5.92569 q^{17} -4.09944 q^{19} -1.00000 q^{21} -1.00000 q^{23} -4.88130 q^{25} -1.00000 q^{27} +3.38242 q^{29} -3.92569 q^{31} -5.25319 q^{33} -0.344522 q^{35} +3.38242 q^{37} +4.27021 q^{39} +11.4075 q^{41} +0.427422 q^{43} -0.344522 q^{45} +13.2234 q^{47} +1.00000 q^{49} -5.92569 q^{51} -4.21529 q^{53} -1.80984 q^{55} +4.09944 q^{57} -6.21529 q^{59} +5.44335 q^{61} +1.00000 q^{63} +1.47118 q^{65} -7.18793 q^{67} +1.00000 q^{69} -11.4685 q^{71} +12.2258 q^{73} +4.88130 q^{75} +5.25319 q^{77} -1.91710 q^{79} +1.00000 q^{81} +4.61473 q^{83} -2.04153 q^{85} -3.38242 q^{87} +7.62569 q^{89} -4.27021 q^{91} +3.92569 q^{93} +1.41235 q^{95} +5.77132 q^{97} +5.25319 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−0.344522	−0.154075	−0.0770376	−	0.997028i	$-0.524546\pi$
$5$	−0.344522	−0.154075	−0.0770376	+	0.997028i	$0.524546\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.25319	1.58390	0.791948	−	0.610589i	$-0.209067\pi$
$11$	5.25319	1.58390	0.791948	+	0.610589i	$0.209067\pi$
$12$	0	0
$13$	−4.27021	−1.18434	−0.592171	−	0.805812i	$-0.701729\pi$
$13$	−4.27021	−1.18434	−0.592171	+	0.805812i	$0.701729\pi$
$14$	0	0
$15$	0.344522	0.0889553
$16$	0	0
$17$	5.92569	1.43719	0.718595	−	0.695429i	$-0.244786\pi$
$17$	5.92569	1.43719	0.718595	+	0.695429i	$0.244786\pi$
$18$	0	0
$19$	−4.09944	−0.940477	−0.470238	−	0.882539i	$-0.655832\pi$
$19$	−4.09944	−0.940477	−0.470238	+	0.882539i	$0.655832\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−4.88130	−0.976261
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	3.38242	0.628099	0.314050	−	0.949407i	$-0.398314\pi$
$29$	3.38242	0.628099	0.314050	+	0.949407i	$0.398314\pi$
$30$	0	0
$31$	−3.92569	−0.705074	−0.352537	−	0.935798i	$-0.614681\pi$
$31$	−3.92569	−0.705074	−0.352537	+	0.935798i	$0.614681\pi$
$32$	0	0
$33$	−5.25319	−0.914462
$34$	0	0
$35$	−0.344522	−0.0582349
$36$	0	0
$37$	3.38242	0.556066	0.278033	−	0.960571i	$-0.410318\pi$
$37$	3.38242	0.556066	0.278033	+	0.960571i	$0.410318\pi$
$38$	0	0
$39$	4.27021	0.683781
$40$	0	0
$41$	11.4075	1.78156	0.890780	−	0.454435i	$-0.150159\pi$
$41$	11.4075	1.78156	0.890780	+	0.454435i	$0.150159\pi$
$42$	0	0
$43$	0.427422	0.0651812	0.0325906	−	0.999469i	$-0.489624\pi$
$43$	0.427422	0.0651812	0.0325906	+	0.999469i	$0.489624\pi$
$44$	0	0
$45$	−0.344522	−0.0513584
$46$	0	0
$47$	13.2234	1.92883	0.964415	−	0.264392i	$-0.0851713\pi$
$47$	13.2234	1.92883	0.964415	+	0.264392i	$0.0851713\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−5.92569	−0.829762
$52$	0	0
$53$	−4.21529	−0.579015	−0.289507	−	0.957176i	$-0.593491\pi$
$53$	−4.21529	−0.579015	−0.289507	+	0.957176i	$0.593491\pi$
$54$	0	0
$55$	−1.80984	−0.244039
$56$	0	0
$57$	4.09944	0.542985
$58$	0	0
$59$	−6.21529	−0.809162	−0.404581	−	0.914502i	$-0.632583\pi$
$59$	−6.21529	−0.809162	−0.404581	+	0.914502i	$0.632583\pi$
$60$	0	0
$61$	5.44335	0.696949	0.348474	−	0.937318i	$-0.386700\pi$
$61$	5.44335	0.696949	0.348474	+	0.937318i	$0.386700\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	1.47118	0.182478
$66$	0	0
$67$	−7.18793	−0.878145	−0.439073	−	0.898452i	$-0.644693\pi$
$67$	−7.18793	−0.878145	−0.439073	+	0.898452i	$0.644693\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	−11.4685	−1.36106	−0.680529	−	0.732721i	$-0.738250\pi$
$71$	−11.4685	−1.36106	−0.680529	+	0.732721i	$0.738250\pi$
$72$	0	0
$73$	12.2258	1.43092	0.715462	−	0.698651i	$-0.246216\pi$
$73$	12.2258	1.43092	0.715462	+	0.698651i	$0.246216\pi$
$74$	0	0
$75$	4.88130	0.563644
$76$	0	0
$77$	5.25319	0.598656
$78$	0	0
$79$	−1.91710	−0.215691	−0.107845	−	0.994168i	$-0.534395\pi$
$79$	−1.91710	−0.215691	−0.107845	+	0.994168i	$0.534395\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.61473	0.506533	0.253266	−	0.967397i	$-0.418495\pi$
$83$	4.61473	0.506533	0.253266	+	0.967397i	$0.418495\pi$
$84$	0	0
$85$	−2.04153	−0.221435
$86$	0	0
$87$	−3.38242	−0.362633
$88$	0	0
$89$	7.62569	0.808321	0.404161	−	0.914688i	$-0.367564\pi$
$89$	7.62569	0.808321	0.404161	+	0.914688i	$0.367564\pi$
$90$	0	0
$91$	−4.27021	−0.447640
$92$	0	0
$93$	3.92569	0.407075
$94$	0	0
$95$	1.41235	0.144904
$96$	0	0
$97$	5.77132	0.585989	0.292995	−	0.956114i	$-0.405348\pi$
$97$	5.77132	0.585989	0.292995	+	0.956114i	$0.405348\pi$
$98$	0	0
$99$	5.25319	0.527965
$100$	0	0
$101$	10.2239	1.01731	0.508657	−	0.860969i	$-0.330142\pi$
$101$	10.2239	1.01731	0.508657	+	0.860969i	$0.330142\pi$
$102$	0	0
$103$	−19.0662	−1.87865	−0.939323	−	0.343033i	$-0.888546\pi$
$103$	−19.0662	−1.87865	−0.939323	+	0.343033i	$0.888546\pi$
$104$	0	0
$105$	0.344522	0.0336220
$106$	0	0
$107$	2.77943	0.268698	0.134349	−	0.990934i	$-0.457106\pi$
$107$	2.77943	0.268698	0.134349	+	0.990934i	$0.457106\pi$
$108$	0	0
$109$	−2.65201	−0.254016	−0.127008	−	0.991902i	$-0.540537\pi$
$109$	−2.65201	−0.254016	−0.127008	+	0.991902i	$0.540537\pi$
$110$	0	0
$111$	−3.38242	−0.321045
$112$	0	0
$113$	−2.70797	−0.254744	−0.127372	−	0.991855i	$-0.540654\pi$
$113$	−2.70797	−0.254744	−0.127372	+	0.991855i	$0.540654\pi$
$114$	0	0
$115$	0.344522	0.0321269
$116$	0	0
$117$	−4.27021	−0.394781
$118$	0	0
$119$	5.92569	0.543207
$120$	0	0
$121$	16.5960	1.50872
$122$	0	0
$123$	−11.4075	−1.02858
$124$	0	0
$125$	3.40433	0.304493
$126$	0	0
$127$	8.05506	0.714771	0.357386	−	0.933957i	$-0.383668\pi$
$127$	8.05506	0.714771	0.357386	+	0.933957i	$0.383668\pi$
$128$	0	0
$129$	−0.427422	−0.0376324
$130$	0	0
$131$	20.8461	1.82133	0.910666	−	0.413144i	$-0.135569\pi$
$131$	20.8461	1.82133	0.910666	+	0.413144i	$0.135569\pi$
$132$	0	0
$133$	−4.09944	−0.355467
$134$	0	0
$135$	0.344522	0.0296518
$136$	0	0
$137$	−4.64271	−0.396654	−0.198327	−	0.980136i	$-0.563551\pi$
$137$	−4.64271	−0.396654	−0.198327	+	0.980136i	$0.563551\pi$
$138$	0	0
$139$	−16.1618	−1.37083	−0.685415	−	0.728153i	$-0.740379\pi$
$139$	−16.1618	−1.37083	−0.685415	+	0.728153i	$0.740379\pi$
$140$	0	0
$141$	−13.2234	−1.11361
$142$	0	0
$143$	−22.4322	−1.87587
$144$	0	0
$145$	−1.16532	−0.0967745
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	−16.5315	−1.35431	−0.677157	−	0.735839i	$-0.736788\pi$
$149$	−16.5315	−1.35431	−0.677157	+	0.735839i	$0.736788\pi$
$150$	0	0
$151$	−5.24886	−0.427146	−0.213573	−	0.976927i	$-0.568510\pi$
$151$	−5.24886	−0.427146	−0.213573	+	0.976927i	$0.568510\pi$
$152$	0	0
$153$	5.92569	0.479063
$154$	0	0
$155$	1.35249	0.108634
$156$	0	0
$157$	1.97487	0.157612	0.0788059	−	0.996890i	$-0.474889\pi$
$157$	1.97487	0.157612	0.0788059	+	0.996890i	$0.474889\pi$
$158$	0	0
$159$	4.21529	0.334294
$160$	0	0
$161$	−1.00000	−0.0788110
$162$	0	0
$163$	3.83373	0.300281	0.150140	−	0.988665i	$-0.452027\pi$
$163$	3.83373	0.300281	0.150140	+	0.988665i	$0.452027\pi$
$164$	0	0
$165$	1.80984	0.140896
$166$	0	0
$167$	−0.0994434	−0.00769516	−0.00384758	−	0.999993i	$-0.501225\pi$
$167$	−0.0994434	−0.00769516	−0.00384758	+	0.999993i	$0.501225\pi$
$168$	0	0
$169$	5.23468	0.402668
$170$	0	0
$171$	−4.09944	−0.313492
$172$	0	0
$173$	−19.7430	−1.50103	−0.750517	−	0.660851i	$-0.770195\pi$
$173$	−19.7430	−1.50103	−0.750517	+	0.660851i	$0.770195\pi$
$174$	0	0
$175$	−4.88130	−0.368992
$176$	0	0
$177$	6.21529	0.467170
$178$	0	0
$179$	−20.7023	−1.54736	−0.773680	−	0.633577i	$-0.781586\pi$
$179$	−20.7023	−1.54736	−0.773680	+	0.633577i	$0.781586\pi$
$180$	0	0
$181$	10.9696	0.815363	0.407682	−	0.913124i	$-0.366337\pi$
$181$	10.9696	0.815363	0.407682	+	0.913124i	$0.366337\pi$
$182$	0	0
$183$	−5.44335	−0.402384
$184$	0	0
$185$	−1.16532	−0.0856760
$186$	0	0
$187$	31.1287	2.27636
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−24.3669	−1.76313	−0.881565	−	0.472062i	$-0.843510\pi$
$191$	−24.3669	−1.76313	−0.881565	+	0.472062i	$0.843510\pi$
$192$	0	0
$193$	−4.52577	−0.325772	−0.162886	−	0.986645i	$-0.552080\pi$
$193$	−4.52577	−0.325772	−0.162886	+	0.986645i	$0.552080\pi$
$194$	0	0
$195$	−1.47118	−0.105354
$196$	0	0
$197$	−8.79477	−0.626601	−0.313301	−	0.949654i	$-0.601435\pi$
$197$	−8.79477	−0.626601	−0.313301	+	0.949654i	$0.601435\pi$
$198$	0	0
$199$	25.8738	1.83414	0.917072	−	0.398721i	$-0.130546\pi$
$199$	25.8738	1.83414	0.917072	+	0.398721i	$0.130546\pi$
$200$	0	0
$201$	7.18793	0.506998
$202$	0	0
$203$	3.38242	0.237399
$204$	0	0
$205$	−3.93016	−0.274494
$206$	0	0
$207$	−1.00000	−0.0695048
$208$	0	0
$209$	−21.5351	−1.48962
$210$	0	0
$211$	−5.25319	−0.361644	−0.180822	−	0.983516i	$-0.557876\pi$
$211$	−5.25319	−0.361644	−0.180822	+	0.983516i	$0.557876\pi$
$212$	0	0
$213$	11.4685	0.785807
$214$	0	0
$215$	−0.147256	−0.0100428
$216$	0	0
$217$	−3.92569	−0.266493
$218$	0	0
$219$	−12.2258	−0.826145
$220$	0	0
$221$	−25.3039	−1.70213
$222$	0	0
$223$	19.6857	1.31825	0.659127	−	0.752032i	$-0.270926\pi$
$223$	19.6857	1.31825	0.659127	+	0.752032i	$0.270926\pi$
$224$	0	0
$225$	−4.88130	−0.325420
$226$	0	0
$227$	29.0365	1.92722	0.963609	−	0.267317i	$-0.0861372\pi$
$227$	29.0365	1.92722	0.963609	+	0.267317i	$0.0861372\pi$
$228$	0	0
$229$	28.8658	1.90751	0.953754	−	0.300589i	$-0.0971832\pi$
$229$	28.8658	1.90751	0.953754	+	0.300589i	$0.0971832\pi$
$230$	0	0
$231$	−5.25319	−0.345634
$232$	0	0
$233$	20.1723	1.32153	0.660766	−	0.750592i	$-0.270231\pi$
$233$	20.1723	1.32153	0.660766	+	0.750592i	$0.270231\pi$
$234$	0	0
$235$	−4.55576	−0.297185
$236$	0	0
$237$	1.91710	0.124529
$238$	0	0
$239$	−11.0393	−0.714073	−0.357037	−	0.934090i	$-0.616213\pi$
$239$	−11.0393	−0.714073	−0.357037	+	0.934090i	$0.616213\pi$
$240$	0	0
$241$	4.08897	0.263393	0.131697	−	0.991290i	$-0.457957\pi$
$241$	4.08897	0.263393	0.131697	+	0.991290i	$0.457957\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−0.344522	−0.0220107
$246$	0	0
$247$	17.5055	1.11385
$248$	0	0
$249$	−4.61473	−0.292447
$250$	0	0
$251$	20.8510	1.31611	0.658053	−	0.752972i	$-0.271380\pi$
$251$	20.8510	1.31611	0.658053	+	0.752972i	$0.271380\pi$
$252$	0	0
$253$	−5.25319	−0.330265
$254$	0	0
$255$	2.04153	0.127846
$256$	0	0
$257$	28.8733	1.80107	0.900534	−	0.434786i	$-0.143176\pi$
$257$	28.8733	1.80107	0.900534	+	0.434786i	$0.143176\pi$
$258$	0	0
$259$	3.38242	0.210173
$260$	0	0
$261$	3.38242	0.209366
$262$	0	0
$263$	26.0039	1.60347	0.801735	−	0.597679i	$-0.203910\pi$
$263$	26.0039	1.60347	0.801735	+	0.597679i	$0.203910\pi$
$264$	0	0
$265$	1.45226	0.0892117
$266$	0	0
$267$	−7.62569	−0.466685
$268$	0	0
$269$	−15.4771	−0.943653	−0.471827	−	0.881691i	$-0.656405\pi$
$269$	−15.4771	−0.943653	−0.471827	+	0.881691i	$0.656405\pi$
$270$	0	0
$271$	−9.74301	−0.591846	−0.295923	−	0.955212i	$-0.595627\pi$
$271$	−9.74301	−0.591846	−0.295923	+	0.955212i	$0.595627\pi$
$272$	0	0
$273$	4.27021	0.258445
$274$	0	0
$275$	−25.6424	−1.54629
$276$	0	0
$277$	11.0971	0.666758	0.333379	−	0.942793i	$-0.391811\pi$
$277$	11.0971	0.666758	0.333379	+	0.942793i	$0.391811\pi$
$278$	0	0
$279$	−3.92569	−0.235025
$280$	0	0
$281$	32.1231	1.91631	0.958153	−	0.286258i	$-0.0924114\pi$
$281$	32.1231	1.91631	0.958153	+	0.286258i	$0.0924114\pi$
$282$	0	0
$283$	−14.1379	−0.840412	−0.420206	−	0.907429i	$-0.638042\pi$
$283$	−14.1379	−0.840412	−0.420206	+	0.907429i	$0.638042\pi$
$284$	0	0
$285$	−1.41235	−0.0836604
$286$	0	0
$287$	11.4075	0.673366
$288$	0	0
$289$	18.1138	1.06552
$290$	0	0
$291$	−5.77132	−0.338321
$292$	0	0
$293$	18.8139	1.09912	0.549559	−	0.835455i	$-0.314796\pi$
$293$	18.8139	1.09912	0.549559	+	0.835455i	$0.314796\pi$
$294$	0	0
$295$	2.14131	0.124672
$296$	0	0
$297$	−5.25319	−0.304821
$298$	0	0
$299$	4.27021	0.246953
$300$	0	0
$301$	0.427422	0.0246362
$302$	0	0
$303$	−10.2239	−0.587346
$304$	0	0
$305$	−1.87535	−0.107382
$306$	0	0
$307$	0.453024	0.0258555	0.0129277	−	0.999916i	$-0.495885\pi$
$307$	0.453024	0.0258555	0.0129277	+	0.999916i	$0.495885\pi$
$308$	0	0
$309$	19.0662	1.08464
$310$	0	0
$311$	20.3654	1.15482	0.577408	−	0.816456i	$-0.304064\pi$
$311$	20.3654	1.15482	0.577408	+	0.816456i	$0.304064\pi$
$312$	0	0
$313$	33.0795	1.86977	0.934883	−	0.354955i	$-0.115504\pi$
$313$	33.0795	1.86977	0.934883	+	0.354955i	$0.115504\pi$
$314$	0	0
$315$	−0.344522	−0.0194116
$316$	0	0
$317$	−6.70797	−0.376757	−0.188378	−	0.982097i	$-0.560323\pi$
$317$	−6.70797	−0.376757	−0.188378	+	0.982097i	$0.560323\pi$
$318$	0	0
$319$	17.7685	0.994844
$320$	0	0
$321$	−2.77943	−0.155133
$322$	0	0
$323$	−24.2920	−1.35164
$324$	0	0
$325$	20.8442	1.15623
$326$	0	0
$327$	2.65201	0.146656
$328$	0	0
$329$	13.2234	0.729029
$330$	0	0
$331$	28.4169	1.56194	0.780968	−	0.624572i	$-0.214726\pi$
$331$	28.4169	1.56194	0.780968	+	0.624572i	$0.214726\pi$
$332$	0	0
$333$	3.38242	0.185355
$334$	0	0
$335$	2.47640	0.135300
$336$	0	0
$337$	7.90962	0.430864	0.215432	−	0.976519i	$-0.430884\pi$
$337$	7.90962	0.430864	0.215432	+	0.976519i	$0.430884\pi$
$338$	0	0
$339$	2.70797	0.147077
$340$	0	0
$341$	−20.6224	−1.11676
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	−0.344522	−0.0185485
$346$	0	0
$347$	−20.5950	−1.10560	−0.552799	−	0.833315i	$-0.686440\pi$
$347$	−20.5950	−1.10560	−0.552799	+	0.833315i	$0.686440\pi$
$348$	0	0
$349$	−2.57336	−0.137749	−0.0688745	−	0.997625i	$-0.521941\pi$
$349$	−2.57336	−0.137749	−0.0688745	+	0.997625i	$0.521941\pi$
$350$	0	0
$351$	4.27021	0.227927
$352$	0	0
$353$	−8.07146	−0.429601	−0.214800	−	0.976658i	$-0.568910\pi$
$353$	−8.07146	−0.429601	−0.214800	+	0.976658i	$0.568910\pi$
$354$	0	0
$355$	3.95115	0.209705
$356$	0	0
$357$	−5.92569	−0.313621
$358$	0	0
$359$	29.3916	1.55123	0.775615	−	0.631207i	$-0.217440\pi$
$359$	29.3916	1.55123	0.775615	+	0.631207i	$0.217440\pi$
$360$	0	0
$361$	−2.19456	−0.115503
$362$	0	0
$363$	−16.5960	−0.871062
$364$	0	0
$365$	−4.21207	−0.220470
$366$	0	0
$367$	−8.01627	−0.418446	−0.209223	−	0.977868i	$-0.567093\pi$
$367$	−8.01627	−0.418446	−0.209223	+	0.977868i	$0.567093\pi$
$368$	0	0
$369$	11.4075	0.593853
$370$	0	0
$371$	−4.21529	−0.218847
$372$	0	0
$373$	21.2771	1.10169	0.550843	−	0.834609i	$-0.314306\pi$
$373$	21.2771	1.10169	0.550843	+	0.834609i	$0.314306\pi$
$374$	0	0
$375$	−3.40433	−0.175799
$376$	0	0
$377$	−14.4436	−0.743885
$378$	0	0
$379$	−19.0621	−0.979156	−0.489578	−	0.871960i	$-0.662849\pi$
$379$	−19.0621	−0.979156	−0.489578	+	0.871960i	$0.662849\pi$
$380$	0	0
$381$	−8.05506	−0.412673
$382$	0	0
$383$	−21.4173	−1.09437	−0.547187	−	0.837010i	$-0.684302\pi$
$383$	−21.4173	−1.09437	−0.547187	+	0.837010i	$0.684302\pi$
$384$	0	0
$385$	−1.80984	−0.0922380
$386$	0	0
$387$	0.427422	0.0217271
$388$	0	0
$389$	−8.71598	−0.441918	−0.220959	−	0.975283i	$-0.570919\pi$
$389$	−8.71598	−0.441918	−0.220959	+	0.975283i	$0.570919\pi$
$390$	0	0
$391$	−5.92569	−0.299675
$392$	0	0
$393$	−20.8461	−1.05155
$394$	0	0
$395$	0.660484	0.0332326
$396$	0	0
$397$	2.00734	0.100746	0.0503729	−	0.998730i	$-0.483959\pi$
$397$	2.00734	0.100746	0.0503729	+	0.998730i	$0.483959\pi$
$398$	0	0
$399$	4.09944	0.205229
$400$	0	0
$401$	−3.48488	−0.174026	−0.0870132	−	0.996207i	$-0.527732\pi$
$401$	−3.48488	−0.174026	−0.0870132	+	0.996207i	$0.527732\pi$
$402$	0	0
$403$	16.7635	0.835049
$404$	0	0
$405$	−0.344522	−0.0171195
$406$	0	0
$407$	17.7685	0.880751
$408$	0	0
$409$	−26.0375	−1.28747	−0.643735	−	0.765249i	$-0.722616\pi$
$409$	−26.0375	−1.28747	−0.643735	+	0.765249i	$0.722616\pi$
$410$	0	0
$411$	4.64271	0.229008
$412$	0	0
$413$	−6.21529	−0.305834
$414$	0	0
$415$	−1.58988	−0.0780441
$416$	0	0
$417$	16.1618	0.791449
$418$	0	0
$419$	1.05749	0.0516619	0.0258309	−	0.999666i	$-0.491777\pi$
$419$	1.05749	0.0516619	0.0258309	+	0.999666i	$0.491777\pi$
$420$	0	0
$421$	12.0789	0.588692	0.294346	−	0.955699i	$-0.404898\pi$
$421$	12.0789	0.588692	0.294346	+	0.955699i	$0.404898\pi$
$422$	0	0
$423$	13.2234	0.642943
$424$	0	0
$425$	−28.9251	−1.40307
$426$	0	0
$427$	5.44335	0.263422
$428$	0	0
$429$	22.4322	1.08304
$430$	0	0
$431$	33.5580	1.61643	0.808215	−	0.588887i	$-0.200434\pi$
$431$	33.5580	1.61643	0.808215	+	0.588887i	$0.200434\pi$
$432$	0	0
$433$	−3.32458	−0.159769	−0.0798845	−	0.996804i	$-0.525455\pi$
$433$	−3.32458	−0.159769	−0.0798845	+	0.996804i	$0.525455\pi$
$434$	0	0
$435$	1.16532	0.0558728
$436$	0	0
$437$	4.09944	0.196103
$438$	0	0
$439$	−1.27069	−0.0606466	−0.0303233	−	0.999540i	$-0.509654\pi$
$439$	−1.27069	−0.0606466	−0.0303233	+	0.999540i	$0.509654\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−19.8959	−0.945283	−0.472641	−	0.881255i	$-0.656699\pi$
$443$	−19.8959	−0.945283	−0.472641	+	0.881255i	$0.656699\pi$
$444$	0	0
$445$	−2.62722	−0.124542
$446$	0	0
$447$	16.5315	0.781913
$448$	0	0
$449$	4.96825	0.234466	0.117233	−	0.993104i	$-0.462598\pi$
$449$	4.96825	0.234466	0.117233	+	0.993104i	$0.462598\pi$
$450$	0	0
$451$	59.9260	2.82180
$452$	0	0
$453$	5.24886	0.246613
$454$	0	0
$455$	1.47118	0.0689701
$456$	0	0
$457$	6.32582	0.295909	0.147955	−	0.988994i	$-0.452731\pi$
$457$	6.32582	0.295909	0.147955	+	0.988994i	$0.452731\pi$
$458$	0	0
$459$	−5.92569	−0.276587
$460$	0	0
$461$	9.80977	0.456886	0.228443	−	0.973557i	$-0.426636\pi$
$461$	9.80977	0.456886	0.228443	+	0.973557i	$0.426636\pi$
$462$	0	0
$463$	−30.1326	−1.40038	−0.700191	−	0.713955i	$-0.746902\pi$
$463$	−30.1326	−1.40038	−0.700191	+	0.713955i	$0.746902\pi$
$464$	0	0
$465$	−1.35249	−0.0627201
$466$	0	0
$467$	−4.82257	−0.223162	−0.111581	−	0.993755i	$-0.535591\pi$
$467$	−4.82257	−0.223162	−0.111581	+	0.993755i	$0.535591\pi$
$468$	0	0
$469$	−7.18793	−0.331908
$470$	0	0
$471$	−1.97487	−0.0909972
$472$	0	0
$473$	2.24533	0.103240
$474$	0	0
$475$	20.0106	0.918151
$476$	0	0
$477$	−4.21529	−0.193005
$478$	0	0
$479$	−1.11252	−0.0508322	−0.0254161	−	0.999677i	$-0.508091\pi$
$479$	−1.11252	−0.0508322	−0.0254161	+	0.999677i	$0.508091\pi$
$480$	0	0
$481$	−14.4436	−0.658573
$482$	0	0
$483$	1.00000	0.0455016
$484$	0	0
$485$	−1.98835	−0.0902864
$486$	0	0
$487$	29.3939	1.33196	0.665982	−	0.745968i	$-0.268013\pi$
$487$	29.3939	1.33196	0.665982	+	0.745968i	$0.268013\pi$
$488$	0	0
$489$	−3.83373	−0.173367
$490$	0	0
$491$	26.8297	1.21081	0.605403	−	0.795919i	$-0.293012\pi$
$491$	26.8297	1.21081	0.605403	+	0.795919i	$0.293012\pi$
$492$	0	0
$493$	20.0432	0.902698
$494$	0	0
$495$	−1.80984	−0.0813463
$496$	0	0
$497$	−11.4685	−0.514431
$498$	0	0
$499$	18.9113	0.846588	0.423294	−	0.905992i	$-0.360874\pi$
$499$	18.9113	0.846588	0.423294	+	0.905992i	$0.360874\pi$
$500$	0	0
$501$	0.0994434	0.00444280
$502$	0	0
$503$	36.8235	1.64188	0.820940	−	0.571014i	$-0.193450\pi$
$503$	36.8235	1.64188	0.820940	+	0.571014i	$0.193450\pi$
$504$	0	0
$505$	−3.52235	−0.156743
$506$	0	0
$507$	−5.23468	−0.232480
$508$	0	0
$509$	−33.3169	−1.47675	−0.738374	−	0.674392i	$-0.764406\pi$
$509$	−33.3169	−1.47675	−0.738374	+	0.674392i	$0.764406\pi$
$510$	0	0
$511$	12.2258	0.540839
$512$	0	0
$513$	4.09944	0.180995
$514$	0	0
$515$	6.56873	0.289453
$516$	0	0
$517$	69.4650	3.05507
$518$	0	0
$519$	19.7430	0.866622
$520$	0	0
$521$	−12.3151	−0.539535	−0.269767	−	0.962926i	$-0.586947\pi$
$521$	−12.3151	−0.539535	−0.269767	+	0.962926i	$0.586947\pi$
$522$	0	0
$523$	−33.8414	−1.47978	−0.739891	−	0.672727i	$-0.765123\pi$
$523$	−33.8414	−1.47978	−0.739891	+	0.672727i	$0.765123\pi$
$524$	0	0
$525$	4.88130	0.213038
$526$	0	0
$527$	−23.2624	−1.01333
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−6.21529	−0.269721
$532$	0	0
$533$	−48.7126	−2.10998
$534$	0	0
$535$	−0.957576	−0.0413996
$536$	0	0
$537$	20.7023	0.893369
$538$	0	0
$539$	5.25319	0.226271
$540$	0	0
$541$	0.549334	0.0236177	0.0118089	−	0.999930i	$-0.496241\pi$
$541$	0.549334	0.0236177	0.0118089	+	0.999930i	$0.496241\pi$
$542$	0	0
$543$	−10.9696	−0.470750
$544$	0	0
$545$	0.913676	0.0391376
$546$	0	0
$547$	34.7847	1.48729	0.743644	−	0.668576i	$-0.233096\pi$
$547$	34.7847	1.48729	0.743644	+	0.668576i	$0.233096\pi$
$548$	0	0
$549$	5.44335	0.232316
$550$	0	0
$551$	−13.8660	−0.590713
$552$	0	0
$553$	−1.91710	−0.0815234
$554$	0	0
$555$	1.16532	0.0494651
$556$	0	0
$557$	4.84790	0.205412	0.102706	−	0.994712i	$-0.467250\pi$
$557$	4.84790	0.205412	0.102706	+	0.994712i	$0.467250\pi$
$558$	0	0
$559$	−1.82518	−0.0771969
$560$	0	0
$561$	−31.1287	−1.31426
$562$	0	0
$563$	4.12604	0.173892	0.0869459	−	0.996213i	$-0.472289\pi$
$563$	4.12604	0.173892	0.0869459	+	0.996213i	$0.472289\pi$
$564$	0	0
$565$	0.932955	0.0392497
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	37.1372	1.55687	0.778437	−	0.627723i	$-0.216013\pi$
$569$	37.1372	1.55687	0.778437	+	0.627723i	$0.216013\pi$
$570$	0	0
$571$	1.88401	0.0788435	0.0394217	−	0.999223i	$-0.487448\pi$
$571$	1.88401	0.0788435	0.0394217	+	0.999223i	$0.487448\pi$
$572$	0	0
$573$	24.3669	1.01794
$574$	0	0
$575$	4.88130	0.203564
$576$	0	0
$577$	10.7097	0.445851	0.222926	−	0.974835i	$-0.428439\pi$
$577$	10.7097	0.445851	0.222926	+	0.974835i	$0.428439\pi$
$578$	0	0
$579$	4.52577	0.188084
$580$	0	0
$581$	4.61473	0.191451
$582$	0	0
$583$	−22.1437	−0.917098
$584$	0	0
$585$	1.47118	0.0608259
$586$	0	0
$587$	40.6547	1.67800	0.838999	−	0.544132i	$-0.183141\pi$
$587$	40.6547	1.67800	0.838999	+	0.544132i	$0.183141\pi$
$588$	0	0
$589$	16.0931	0.663106
$590$	0	0
$591$	8.79477	0.361768
$592$	0	0
$593$	−27.0570	−1.11110	−0.555549	−	0.831484i	$-0.687492\pi$
$593$	−27.0570	−1.11110	−0.555549	+	0.831484i	$0.687492\pi$
$594$	0	0
$595$	−2.04153	−0.0836947
$596$	0	0
$597$	−25.8738	−1.05894
$598$	0	0
$599$	1.36825	0.0559052	0.0279526	−	0.999609i	$-0.491101\pi$
$599$	1.36825	0.0559052	0.0279526	+	0.999609i	$0.491101\pi$
$600$	0	0
$601$	16.3000	0.664891	0.332445	−	0.943123i	$-0.392126\pi$
$601$	16.3000	0.664891	0.332445	+	0.943123i	$0.392126\pi$
$602$	0	0
$603$	−7.18793	−0.292715
$604$	0	0
$605$	−5.71768	−0.232457
$606$	0	0
$607$	23.3125	0.946225	0.473113	−	0.881002i	$-0.343130\pi$
$607$	23.3125	0.946225	0.473113	+	0.881002i	$0.343130\pi$
$608$	0	0
$609$	−3.38242	−0.137063
$610$	0	0
$611$	−56.4667	−2.28440
$612$	0	0
$613$	9.44955	0.381664	0.190832	−	0.981623i	$-0.438881\pi$
$613$	9.44955	0.381664	0.190832	+	0.981623i	$0.438881\pi$
$614$	0	0
$615$	3.93016	0.158479
$616$	0	0
$617$	10.5258	0.423754	0.211877	−	0.977296i	$-0.432042\pi$
$617$	10.5258	0.423754	0.211877	+	0.977296i	$0.432042\pi$
$618$	0	0
$619$	−3.92451	−0.157740	−0.0788698	−	0.996885i	$-0.525131\pi$
$619$	−3.92451	−0.157740	−0.0788698	+	0.996885i	$0.525131\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	7.62569	0.305517
$624$	0	0
$625$	23.2337	0.929346
$626$	0	0
$627$	21.5351	0.860031
$628$	0	0
$629$	20.0432	0.799173
$630$	0	0
$631$	−29.6373	−1.17984	−0.589921	−	0.807461i	$-0.700841\pi$
$631$	−29.6373	−1.17984	−0.589921	+	0.807461i	$0.700841\pi$
$632$	0	0
$633$	5.25319	0.208795
$634$	0	0
$635$	−2.77515	−0.110128
$636$	0	0
$637$	−4.27021	−0.169192
$638$	0	0
$639$	−11.4685	−0.453686
$640$	0	0
$641$	22.1384	0.874415	0.437207	−	0.899361i	$-0.355968\pi$
$641$	22.1384	0.874415	0.437207	+	0.899361i	$0.355968\pi$
$642$	0	0
$643$	−5.25333	−0.207171	−0.103586	−	0.994621i	$-0.533032\pi$
$643$	−5.25333	−0.207171	−0.103586	+	0.994621i	$0.533032\pi$
$644$	0	0
$645$	0.147256	0.00579821
$646$	0	0
$647$	9.20222	0.361776	0.180888	−	0.983504i	$-0.442103\pi$
$647$	9.20222	0.361776	0.180888	+	0.983504i	$0.442103\pi$
$648$	0	0
$649$	−32.6501	−1.28163
$650$	0	0
$651$	3.92569	0.153860
$652$	0	0
$653$	−4.27615	−0.167339	−0.0836693	−	0.996494i	$-0.526664\pi$
$653$	−4.27615	−0.167339	−0.0836693	+	0.996494i	$0.526664\pi$
$654$	0	0
$655$	−7.18195	−0.280622
$656$	0	0
$657$	12.2258	0.476975
$658$	0	0
$659$	5.02375	0.195698	0.0978488	−	0.995201i	$-0.468804\pi$
$659$	5.02375	0.195698	0.0978488	+	0.995201i	$0.468804\pi$
$660$	0	0
$661$	0.316236	0.0123002	0.00615008	−	0.999981i	$-0.498042\pi$
$661$	0.316236	0.0123002	0.00615008	+	0.999981i	$0.498042\pi$
$662$	0	0
$663$	25.3039	0.982723
$664$	0	0
$665$	1.41235	0.0547686
$666$	0	0
$667$	−3.38242	−0.130968
$668$	0	0
$669$	−19.6857	−0.761094
$670$	0	0
$671$	28.5949	1.10389
$672$	0	0
$673$	−6.08066	−0.234392	−0.117196	−	0.993109i	$-0.537391\pi$
$673$	−6.08066	−0.234392	−0.117196	+	0.993109i	$0.537391\pi$
$674$	0	0
$675$	4.88130	0.187881
$676$	0	0
$677$	10.2880	0.395401	0.197701	−	0.980262i	$-0.436653\pi$
$677$	10.2880	0.395401	0.197701	+	0.980262i	$0.436653\pi$
$678$	0	0
$679$	5.77132	0.221483
$680$	0	0
$681$	−29.0365	−1.11268
$682$	0	0
$683$	−20.2920	−0.776452	−0.388226	−	0.921564i	$-0.626912\pi$
$683$	−20.2920	−0.776452	−0.388226	+	0.921564i	$0.626912\pi$
$684$	0	0
$685$	1.59952	0.0611145
$686$	0	0
$687$	−28.8658	−1.10130
$688$	0	0
$689$	18.0002	0.685752
$690$	0	0
$691$	−49.5235	−1.88396	−0.941982	−	0.335665i	$-0.891039\pi$
$691$	−49.5235	−1.88396	−0.941982	+	0.335665i	$0.891039\pi$
$692$	0	0
$693$	5.25319	0.199552
$694$	0	0
$695$	5.56812	0.211211
$696$	0	0
$697$	67.5976	2.56044
$698$	0	0
$699$	−20.1723	−0.762987
$700$	0	0
$701$	24.3317	0.918997	0.459498	−	0.888179i	$-0.348029\pi$
$701$	24.3317	0.918997	0.459498	+	0.888179i	$0.348029\pi$
$702$	0	0
$703$	−13.8660	−0.522967
$704$	0	0
$705$	4.55576	0.171580
$706$	0	0
$707$	10.2239	0.384508
$708$	0	0
$709$	20.6308	0.774806	0.387403	−	0.921910i	$-0.373372\pi$
$709$	20.6308	0.774806	0.387403	+	0.921910i	$0.373372\pi$
$710$	0	0
$711$	−1.91710	−0.0718969
$712$	0	0
$713$	3.92569	0.147018
$714$	0	0
$715$	7.72840	0.289026
$716$	0	0
$717$	11.0393	0.412270
$718$	0	0
$719$	−27.4827	−1.02493	−0.512466	−	0.858707i	$-0.671268\pi$
$719$	−27.4827	−1.02493	−0.512466	+	0.858707i	$0.671268\pi$
$720$	0	0
$721$	−19.0662	−0.710062
$722$	0	0
$723$	−4.08897	−0.152070
$724$	0	0
$725$	−16.5106	−0.613189
$726$	0	0
$727$	−34.1646	−1.26709	−0.633547	−	0.773704i	$-0.718402\pi$
$727$	−34.1646	−1.26709	−0.633547	+	0.773704i	$0.718402\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.53277	0.0936778
$732$	0	0
$733$	−15.3037	−0.565254	−0.282627	−	0.959230i	$-0.591206\pi$
$733$	−15.3037	−0.565254	−0.282627	+	0.959230i	$0.591206\pi$
$734$	0	0
$735$	0.344522	0.0127079
$736$	0	0
$737$	−37.7595	−1.39089
$738$	0	0
$739$	−31.9956	−1.17698	−0.588488	−	0.808506i	$-0.700277\pi$
$739$	−31.9956	−1.17698	−0.588488	+	0.808506i	$0.700277\pi$
$740$	0	0
$741$	−17.5055	−0.643080
$742$	0	0
$743$	37.2082	1.36504	0.682518	−	0.730869i	$-0.260885\pi$
$743$	37.2082	1.36504	0.682518	+	0.730869i	$0.260885\pi$
$744$	0	0
$745$	5.69547	0.208666
$746$	0	0
$747$	4.61473	0.168844
$748$	0	0
$749$	2.77943	0.101558
$750$	0	0
$751$	−44.8984	−1.63836	−0.819182	−	0.573533i	$-0.805572\pi$
$751$	−44.8984	−1.63836	−0.819182	+	0.573533i	$0.805572\pi$
$752$	0	0
$753$	−20.8510	−0.759854
$754$	0	0
$755$	1.80835	0.0658126
$756$	0	0
$757$	5.17542	0.188104	0.0940519	−	0.995567i	$-0.470018\pi$
$757$	5.17542	0.188104	0.0940519	+	0.995567i	$0.470018\pi$
$758$	0	0
$759$	5.25319	0.190679
$760$	0	0
$761$	−1.65051	−0.0598310	−0.0299155	−	0.999552i	$-0.509524\pi$
$761$	−1.65051	−0.0598310	−0.0299155	+	0.999552i	$0.509524\pi$
$762$	0	0
$763$	−2.65201	−0.0960091
$764$	0	0
$765$	−2.04153	−0.0738118
$766$	0	0
$767$	26.5406	0.958325
$768$	0	0
$769$	−13.8380	−0.499010	−0.249505	−	0.968374i	$-0.580268\pi$
$769$	−13.8380	−0.499010	−0.249505	+	0.968374i	$0.580268\pi$
$770$	0	0
$771$	−28.8733	−1.03985
$772$	0	0
$773$	−30.0167	−1.07963	−0.539814	−	0.841785i	$-0.681505\pi$
$773$	−30.0167	−1.07963	−0.539814	+	0.841785i	$0.681505\pi$
$774$	0	0
$775$	19.1625	0.688336
$776$	0	0
$777$	−3.38242	−0.121344
$778$	0	0
$779$	−46.7646	−1.67552
$780$	0	0
$781$	−60.2460	−2.15577
$782$	0	0
$783$	−3.38242	−0.120878
$784$	0	0
$785$	−0.680387	−0.0242841
$786$	0	0
$787$	11.0275	0.393088	0.196544	−	0.980495i	$-0.437028\pi$
$787$	11.0275	0.393088	0.196544	+	0.980495i	$0.437028\pi$
$788$	0	0
$789$	−26.0039	−0.925764
$790$	0	0
$791$	−2.70797	−0.0962842
$792$	0	0
$793$	−23.2442	−0.825426
$794$	0	0
$795$	−1.45226	−0.0515064
$796$	0	0
$797$	15.0962	0.534734	0.267367	−	0.963595i	$-0.413846\pi$
$797$	15.0962	0.534734	0.267367	+	0.963595i	$0.413846\pi$
$798$	0	0
$799$	78.3577	2.77210
$800$	0	0
$801$	7.62569	0.269440
$802$	0	0
$803$	64.2245	2.26643
$804$	0	0
$805$	0.344522	0.0121428
$806$	0	0
$807$	15.4771	0.544818
$808$	0	0
$809$	5.31985	0.187036	0.0935179	−	0.995618i	$-0.470189\pi$
$809$	5.31985	0.187036	0.0935179	+	0.995618i	$0.470189\pi$
$810$	0	0
$811$	17.4925	0.614246	0.307123	−	0.951670i	$-0.400634\pi$
$811$	17.4925	0.614246	0.307123	+	0.951670i	$0.400634\pi$
$812$	0	0
$813$	9.74301	0.341702
$814$	0	0
$815$	−1.32081	−0.0462658
$816$	0	0
$817$	−1.75219	−0.0613014
$818$	0	0
$819$	−4.27021	−0.149213
$820$	0	0
$821$	−9.87245	−0.344551	−0.172275	−	0.985049i	$-0.555112\pi$
$821$	−9.87245	−0.344551	−0.172275	+	0.985049i	$0.555112\pi$
$822$	0	0
$823$	21.1574	0.737499	0.368750	−	0.929529i	$-0.379786\pi$
$823$	21.1574	0.737499	0.368750	+	0.929529i	$0.379786\pi$
$824$	0	0
$825$	25.6424	0.892754
$826$	0	0
$827$	45.2735	1.57432	0.787158	−	0.616752i	$-0.211552\pi$
$827$	45.2735	1.57432	0.787158	+	0.616752i	$0.211552\pi$
$828$	0	0
$829$	−45.5214	−1.58102	−0.790512	−	0.612447i	$-0.790185\pi$
$829$	−45.5214	−1.58102	−0.790512	+	0.612447i	$0.790185\pi$
$830$	0	0
$831$	−11.0971	−0.384953
$832$	0	0
$833$	5.92569	0.205313
$834$	0	0
$835$	0.0342605	0.00118563
$836$	0	0
$837$	3.92569	0.135692
$838$	0	0
$839$	4.67921	0.161544	0.0807721	−	0.996733i	$-0.474261\pi$
$839$	4.67921	0.161544	0.0807721	+	0.996733i	$0.474261\pi$
$840$	0	0
$841$	−17.5592	−0.605491
$842$	0	0
$843$	−32.1231	−1.10638
$844$	0	0
$845$	−1.80347	−0.0620411
$846$	0	0
$847$	16.5960	0.570244
$848$	0	0
$849$	14.1379	0.485212
$850$	0	0
$851$	−3.38242	−0.115948
$852$	0	0
$853$	−2.62476	−0.0898701	−0.0449350	−	0.998990i	$-0.514308\pi$
$853$	−2.62476	−0.0898701	−0.0449350	+	0.998990i	$0.514308\pi$
$854$	0	0
$855$	1.41235	0.0483014
$856$	0	0
$857$	−5.00391	−0.170930	−0.0854652	−	0.996341i	$-0.527238\pi$
$857$	−5.00391	−0.170930	−0.0854652	+	0.996341i	$0.527238\pi$
$858$	0	0
$859$	−22.0548	−0.752498	−0.376249	−	0.926518i	$-0.622786\pi$
$859$	−22.0548	−0.752498	−0.376249	+	0.926518i	$0.622786\pi$
$860$	0	0
$861$	−11.4075	−0.388768
$862$	0	0
$863$	42.5095	1.44704	0.723520	−	0.690304i	$-0.242523\pi$
$863$	42.5095	1.44704	0.723520	+	0.690304i	$0.242523\pi$
$864$	0	0
$865$	6.80191	0.231272
$866$	0	0
$867$	−18.1138	−0.615175
$868$	0	0
$869$	−10.0709	−0.341631
$870$	0	0
$871$	30.6940	1.04003
$872$	0	0
$873$	5.77132	0.195330
$874$	0	0
$875$	3.40433	0.115087
$876$	0	0
$877$	−11.6710	−0.394100	−0.197050	−	0.980393i	$-0.563136\pi$
$877$	−11.6710	−0.394100	−0.197050	+	0.980393i	$0.563136\pi$
$878$	0	0
$879$	−18.8139	−0.634576
$880$	0	0
$881$	11.3316	0.381770	0.190885	−	0.981612i	$-0.438864\pi$
$881$	11.3316	0.381770	0.190885	+	0.981612i	$0.438864\pi$
$882$	0	0
$883$	45.7931	1.54106	0.770531	−	0.637403i	$-0.219991\pi$
$883$	45.7931	1.54106	0.770531	+	0.637403i	$0.219991\pi$
$884$	0	0
$885$	−2.14131	−0.0719792
$886$	0	0
$887$	30.9379	1.03879	0.519396	−	0.854534i	$-0.326157\pi$
$887$	30.9379	1.03879	0.519396	+	0.854534i	$0.326157\pi$
$888$	0	0
$889$	8.05506	0.270158
$890$	0	0
$891$	5.25319	0.175988
$892$	0	0
$893$	−54.2086	−1.81402
$894$	0	0
$895$	7.13240	0.238410
$896$	0	0
$897$	−4.27021	−0.142578
$898$	0	0
$899$	−13.2783	−0.442857
$900$	0	0
$901$	−24.9785	−0.832154
$902$	0	0
$903$	−0.427422	−0.0142237
$904$	0	0
$905$	−3.77927	−0.125627
$906$	0	0
$907$	21.4057	0.710764	0.355382	−	0.934721i	$-0.384351\pi$
$907$	21.4057	0.710764	0.355382	+	0.934721i	$0.384351\pi$
$908$	0	0
$909$	10.2239	0.339105
$910$	0	0
$911$	5.64009	0.186865	0.0934323	−	0.995626i	$-0.470216\pi$
$911$	5.64009	0.186865	0.0934323	+	0.995626i	$0.470216\pi$
$912$	0	0
$913$	24.2420	0.802294
$914$	0	0
$915$	1.87535	0.0619973
$916$	0	0
$917$	20.8461	0.688399
$918$	0	0
$919$	8.17099	0.269536	0.134768	−	0.990877i	$-0.456971\pi$
$919$	8.17099	0.269536	0.134768	+	0.990877i	$0.456971\pi$
$920$	0	0
$921$	−0.453024	−0.0149277
$922$	0	0
$923$	48.9728	1.61196
$924$	0	0
$925$	−16.5106	−0.542866
$926$	0	0
$927$	−19.0662	−0.626216
$928$	0	0
$929$	−25.0179	−0.820812	−0.410406	−	0.911903i	$-0.634613\pi$
$929$	−25.0179	−0.820812	−0.410406	+	0.911903i	$0.634613\pi$
$930$	0	0
$931$	−4.09944	−0.134354
$932$	0	0
$933$	−20.3654	−0.666733
$934$	0	0
$935$	−10.7245	−0.350730
$936$	0	0
$937$	47.6638	1.55711	0.778554	−	0.627578i	$-0.215954\pi$
$937$	47.6638	1.55711	0.778554	+	0.627578i	$0.215954\pi$
$938$	0	0
$939$	−33.0795	−1.07951
$940$	0	0
$941$	−41.9548	−1.36769	−0.683844	−	0.729628i	$-0.739693\pi$
$941$	−41.9548	−1.36769	−0.683844	+	0.729628i	$0.739693\pi$
$942$	0	0
$943$	−11.4075	−0.371481
$944$	0	0
$945$	0.344522	0.0112073
$946$	0	0
$947$	−37.8772	−1.23084	−0.615422	−	0.788198i	$-0.711014\pi$
$947$	−37.8772	−1.23084	−0.615422	+	0.788198i	$0.711014\pi$
$948$	0	0
$949$	−52.2068	−1.69471
$950$	0	0
$951$	6.70797	0.217521
$952$	0	0
$953$	43.9858	1.42484	0.712419	−	0.701754i	$-0.247599\pi$
$953$	43.9858	1.42484	0.712419	+	0.701754i	$0.247599\pi$
$954$	0	0
$955$	8.39496	0.271655
$956$	0	0
$957$	−17.7685	−0.574373
$958$	0	0
$959$	−4.64271	−0.149921
$960$	0	0
$961$	−15.5890	−0.502871
$962$	0	0
$963$	2.77943	0.0895659
$964$	0	0
$965$	1.55923	0.0501933
$966$	0	0
$967$	−36.4517	−1.17221	−0.586103	−	0.810236i	$-0.699339\pi$
$967$	−36.4517	−1.17221	−0.586103	+	0.810236i	$0.699339\pi$
$968$	0	0
$969$	24.2920	0.780372
$970$	0	0
$971$	−29.5929	−0.949681	−0.474840	−	0.880072i	$-0.657494\pi$
$971$	−29.5929	−0.949681	−0.474840	+	0.880072i	$0.657494\pi$
$972$	0	0
$973$	−16.1618	−0.518125
$974$	0	0
$975$	−20.8442	−0.667548
$976$	0	0
$977$	−23.5306	−0.752810	−0.376405	−	0.926455i	$-0.622840\pi$
$977$	−23.5306	−0.752810	−0.376405	+	0.926455i	$0.622840\pi$
$978$	0	0
$979$	40.0592	1.28030
$980$	0	0
$981$	−2.65201	−0.0846721
$982$	0	0
$983$	−26.9241	−0.858747	−0.429373	−	0.903127i	$-0.641266\pi$
$983$	−26.9241	−0.858747	−0.429373	+	0.903127i	$0.641266\pi$
$984$	0	0
$985$	3.03000	0.0965437
$986$	0	0
$987$	−13.2234	−0.420905
$988$	0	0
$989$	−0.427422	−0.0135912
$990$	0	0
$991$	2.49583	0.0792828	0.0396414	−	0.999214i	$-0.487378\pi$
$991$	2.49583	0.0792828	0.0396414	+	0.999214i	$0.487378\pi$
$992$	0	0
$993$	−28.4169	−0.901784
$994$	0	0
$995$	−8.91410	−0.282596
$996$	0	0
$997$	19.5983	0.620684	0.310342	−	0.950625i	$-0.399556\pi$
$997$	19.5983	0.620684	0.310342	+	0.950625i	$0.399556\pi$
$998$	0	0
$999$	−3.38242	−0.107015

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7728.2.a.cg.1.3		6
4.3	odd	2		3864.2.a.x.1.3	✓	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3864.2.a.x.1.3	✓	6	4.3	odd	2
7728.2.a.cg.1.3		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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -0.344522 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -0.344522 q^{5} +1.00000 q^{7} +1.00000 q^{9} +5.25319 q^{11} -4.27021 q^{13} +0.344522 q^{15} +5.92569 q^{17} -4.09944 q^{19} -1.00000 q^{21} -1.00000 q^{23} -4.88130 q^{25} -1.00000 q^{27} +3.38242 q^{29} -3.92569 q^{31} -5.25319 q^{33} -0.344522 q^{35} +3.38242 q^{37} +4.27021 q^{39} +11.4075 q^{41} +0.427422 q^{43} -0.344522 q^{45} +13.2234 q^{47} +1.00000 q^{49} -5.92569 q^{51} -4.21529 q^{53} -1.80984 q^{55} +4.09944 q^{57} -6.21529 q^{59} +5.44335 q^{61} +1.00000 q^{63} +1.47118 q^{65} -7.18793 q^{67} +1.00000 q^{69} -11.4685 q^{71} +12.2258 q^{73} +4.88130 q^{75} +5.25319 q^{77} -1.91710 q^{79} +1.00000 q^{81} +4.61473 q^{83} -2.04153 q^{85} -3.38242 q^{87} +7.62569 q^{89} -4.27021 q^{91} +3.92569 q^{93} +1.41235 q^{95} +5.77132 q^{97} +5.25319 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{3} + 6 q^{7} + 6 q^{9}+O(q^{10}) \) 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

Related objects

Downloads

Learn more