LMFDB - Embedded newform 1792.3.g.e.127.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1792,3,Mod(127,1792)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1792, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1792.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1792 = 2^{8} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1792.g (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$48.8284633734$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.49787136.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 3x^{6} + 5x^{4} + 12x^{2} + 16 $ x^8 + 3x^6 + 5x^4 + 12*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 112)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.7
Root			$-1.09445 - 0.895644i$ of defining polynomial
Character	$\chi$	$=$	1792.127
Dual form			1792.3.g.e.127.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.37780 q^{3} -5.58258i q^{5} -2.64575i q^{7} +10.1652 q^{9} +O(q^{10})$	$q+4.37780 q^{3} -5.58258i q^{5} -2.64575i q^{7} +10.1652 q^{9} -1.82740 q^{11} +20.7477i q^{13} -24.4394i q^{15} +25.1652 q^{17} +30.6446 q^{19} -11.5826i q^{21} -3.27340i q^{23} -6.16515 q^{25} +5.10080 q^{27} -53.8258i q^{29} +29.1588i q^{31} -8.00000 q^{33} -14.7701 q^{35} +17.1652i q^{37} +90.8294i q^{39} -7.49545 q^{41} +57.2530 q^{43} -56.7477i q^{45} -37.2312i q^{47} -7.00000 q^{49} +110.168 q^{51} -46.0000i q^{53} +10.2016i q^{55} +134.156 q^{57} -75.1058 q^{59} -28.0871i q^{61} -26.8945i q^{63} +115.826 q^{65} +81.3906 q^{67} -14.3303i q^{69} -32.1304i q^{71} +45.6515 q^{73} -26.9898 q^{75} +4.83485i q^{77} -10.9644i q^{79} -69.1561 q^{81} -143.705 q^{83} -140.486i q^{85} -235.639i q^{87} -59.3212 q^{89} +54.8933 q^{91} +127.652i q^{93} -171.076i q^{95} +17.1652 q^{97} -18.5758 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^9	$ 8 q + 8 q^{9} + 128 q^{17} + 24 q^{25} - 64 q^{33} + 160 q^{41} - 56 q^{49} + 560 q^{57} + 560 q^{65} - 368 q^{73} - 40 q^{81} + 112 q^{89} + 64 q^{97}+O(q^{100}) $ 8 * q + 8 * q^9 + 128 * q^17 + 24 * q^25 - 64 * q^33 + 160 * q^41 - 56 * q^49 + 560 * q^57 + 560 * q^65 - 368 * q^73 - 40 * q^81 + 112 * q^89 + 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1792\mathbb{Z}\right)^\times$.

$n$	$1023$	$1025$	$1541$
$\chi(n)$	$-1$	$1$	$-1$

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
$3$	4.37780	1.45927	0.729634	−	0.683838i	$-0.239691\pi$
$3$	4.37780	1.45927	0.729634	+	0.683838i	$0.239691\pi$
$4$	0	0
$5$	− 5.58258i	− 1.11652i	−0.829668	−	0.558258i	$-0.811470\pi$
$5$	− 5.58258i	− 1.11652i	0.829668	−	0.558258i	$-0.188530\pi$
$6$	0	0
$7$	− 2.64575i	− 0.377964i
$7$	− 2.64575i	− 0.377964i
$8$	0	0
$9$	10.1652	1.12946
$10$	0	0
$11$	−1.82740	−0.166127	−0.0830637	−	0.996544i	$-0.526470\pi$
$11$	−1.82740	−0.166127	−0.0830637	+	0.996544i	$0.526470\pi$
$12$	0	0
$13$	20.7477i	1.59598i	0.602671	+	0.797990i	$0.294103\pi$
$13$	20.7477i	1.59598i	−0.602671	+	0.797990i	$0.705897\pi$
$14$	0	0
$15$	− 24.4394i	− 1.62929i
$16$	0	0
$17$	25.1652	1.48030	0.740152	−	0.672440i	$-0.234754\pi$
$17$	25.1652	1.48030	0.740152	+	0.672440i	$0.234754\pi$
$18$	0	0
$19$	30.6446	1.61287	0.806437	−	0.591320i	$-0.201393\pi$
$19$	30.6446	1.61287	0.806437	+	0.591320i	$0.201393\pi$
$20$	0	0
$21$	− 11.5826i	− 0.551551i
$22$	0	0
$23$	− 3.27340i	− 0.142322i	−0.997465	−	0.0711609i	$-0.977330\pi$
$23$	− 3.27340i	− 0.142322i	0.997465	−	0.0711609i	$-0.0226704\pi$
$24$	0	0
$25$	−6.16515	−0.246606
$26$	0	0
$27$	5.10080	0.188919
$28$	0	0
$29$	− 53.8258i	− 1.85606i	−0.372505	−	0.928030i	$-0.621501\pi$
$29$	− 53.8258i	− 1.85606i	0.372505	−	0.928030i	$-0.378499\pi$
$30$	0	0
$31$	29.1588i	0.940607i	0.882505	+	0.470303i	$0.155856\pi$
$31$	29.1588i	0.940607i	−0.882505	+	0.470303i	$0.844144\pi$
$32$	0	0
$33$	−8.00000	−0.242424
$34$	0	0
$35$	−14.7701	−0.422003
$36$	0	0
$37$	17.1652i	0.463923i	0.972725	+	0.231962i	$0.0745143\pi$
$37$	17.1652i	0.463923i	−0.972725	+	0.231962i	$0.925486\pi$
$38$	0	0
$39$	90.8294i	2.32896i
$40$	0	0
$41$	−7.49545	−0.182816	−0.0914080	−	0.995814i	$-0.529137\pi$
$41$	−7.49545	−0.182816	−0.0914080	+	0.995814i	$0.529137\pi$
$42$	0	0
$43$	57.2530	1.33147	0.665733	−	0.746190i	$-0.268119\pi$
$43$	57.2530	1.33147	0.665733	+	0.746190i	$0.268119\pi$
$44$	0	0
$45$	− 56.7477i	− 1.26106i
$46$	0	0
$47$	− 37.2312i	− 0.792154i	−0.918217	−	0.396077i	$-0.870371\pi$
$47$	− 37.2312i	− 0.792154i	0.918217	−	0.396077i	$-0.129629\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	110.168	2.16016
$52$	0	0
$53$	− 46.0000i	− 0.867925i	−0.900931	−	0.433962i	$-0.857115\pi$
$53$	− 46.0000i	− 0.867925i	0.900931	−	0.433962i	$-0.142885\pi$
$54$	0	0
$55$	10.2016i	0.185484i
$56$	0	0
$57$	134.156	2.35362
$58$	0	0
$59$	−75.1058	−1.27298	−0.636490	−	0.771285i	$-0.719614\pi$
$59$	−75.1058	−1.27298	−0.636490	+	0.771285i	$0.719614\pi$
$60$	0	0
$61$	− 28.0871i	− 0.460445i	−0.973138	−	0.230222i	$-0.926055\pi$
$61$	− 28.0871i	− 0.460445i	0.973138	−	0.230222i	$-0.0739453\pi$
$62$	0	0
$63$	− 26.8945i	− 0.426896i
$64$	0	0
$65$	115.826	1.78193
$66$	0	0
$67$	81.3906	1.21479	0.607393	−	0.794402i	$-0.292215\pi$
$67$	81.3906	1.21479	0.607393	+	0.794402i	$0.292215\pi$
$68$	0	0
$69$	− 14.3303i	− 0.207686i
$70$	0	0
$71$	− 32.1304i	− 0.452541i	−0.974064	−	0.226271i	$-0.927347\pi$
$71$	− 32.1304i	− 0.452541i	0.974064	−	0.226271i	$-0.0726533\pi$
$72$	0	0
$73$	45.6515	0.625363	0.312682	−	0.949858i	$-0.398773\pi$
$73$	45.6515	0.625363	0.312682	+	0.949858i	$0.398773\pi$
$74$	0	0
$75$	−26.9898	−0.359864
$76$	0	0
$77$	4.83485i	0.0627902i
$78$	0	0
$79$	− 10.9644i	− 0.138790i	−0.997589	−	0.0693950i	$-0.977893\pi$
$79$	− 10.9644i	− 0.138790i	0.997589	−	0.0693950i	$-0.0221069\pi$
$80$	0	0
$81$	−69.1561	−0.853779
$82$	0	0
$83$	−143.705	−1.73138	−0.865691	−	0.500579i	$-0.833120\pi$
$83$	−143.705	−1.73138	−0.865691	+	0.500579i	$0.833120\pi$
$84$	0	0
$85$	− 140.486i	− 1.65278i
$86$	0	0
$87$	− 235.639i	− 2.70849i
$88$	0	0
$89$	−59.3212	−0.666530	−0.333265	−	0.942833i	$-0.608150\pi$
$89$	−59.3212	−0.666530	−0.333265	+	0.942833i	$0.608150\pi$
$90$	0	0
$91$	54.8933	0.603223
$92$	0	0
$93$	127.652i	1.37260i
$94$	0	0
$95$	− 171.076i	− 1.80080i
$96$	0	0
$97$	17.1652	0.176960	0.0884802	−	0.996078i	$-0.471799\pi$
$97$	17.1652	0.176960	0.0884802	+	0.996078i	$0.471799\pi$
$98$	0	0
$99$	−18.5758	−0.187634
$100$	0	0
$101$	8.74773i	0.0866112i	0.999062	+	0.0433056i	$0.0137889\pi$
$101$	8.74773i	0.0866112i	−0.999062	+	0.0433056i	$0.986211\pi$
$102$	0	0
$103$	− 54.7424i	− 0.531480i	−0.964045	−	0.265740i	$-0.914384\pi$
$103$	− 54.7424i	− 0.531480i	0.964045	−	0.265740i	$-0.0856162\pi$
$104$	0	0
$105$	−64.6606	−0.615815
$106$	0	0
$107$	127.377	1.19044	0.595222	−	0.803561i	$-0.297064\pi$
$107$	127.377	1.19044	0.595222	+	0.803561i	$0.297064\pi$
$108$	0	0
$109$	48.5045i	0.444996i	0.974933	+	0.222498i	$0.0714211\pi$
$109$	48.5045i	0.444996i	−0.974933	+	0.222498i	$0.928579\pi$
$110$	0	0
$111$	75.1456i	0.676988i
$112$	0	0
$113$	−36.5045	−0.323049	−0.161525	−	0.986869i	$-0.551641\pi$
$113$	−36.5045	−0.323049	−0.161525	+	0.986869i	$0.551641\pi$
$114$	0	0
$115$	−18.2740	−0.158904
$116$	0	0
$117$	210.904i	1.80260i
$118$	0	0
$119$	− 66.5807i	− 0.559502i
$120$	0	0
$121$	−117.661	−0.972402
$122$	0	0
$123$	−32.8136	−0.266777
$124$	0	0
$125$	− 105.147i	− 0.841176i
$126$	0	0
$127$	− 41.9506i	− 0.330320i	−0.986267	−	0.165160i	$-0.947186\pi$
$127$	− 41.9506i	− 0.330320i	0.986267	−	0.165160i	$-0.0528140\pi$
$128$	0	0
$129$	250.642	1.94296
$130$	0	0
$131$	47.3930	0.361779	0.180889	−	0.983503i	$-0.442102\pi$
$131$	47.3930	0.361779	0.180889	+	0.983503i	$0.442102\pi$
$132$	0	0
$133$	− 81.0780i	− 0.609609i
$134$	0	0
$135$	− 28.4756i	− 0.210930i
$136$	0	0
$137$	−153.652	−1.12154	−0.560772	−	0.827970i	$-0.689496\pi$
$137$	−153.652	−1.12154	−0.560772	+	0.827970i	$0.689496\pi$
$138$	0	0
$139$	147.439	1.06071	0.530356	−	0.847775i	$-0.322058\pi$
$139$	147.439	1.06071	0.530356	+	0.847775i	$0.322058\pi$
$140$	0	0
$141$	− 162.991i	− 1.15596i
$142$	0	0
$143$	− 37.9144i	− 0.265136i
$144$	0	0
$145$	−300.486	−2.07232
$146$	0	0
$147$	−30.6446	−0.208467
$148$	0	0
$149$	78.3121i	0.525585i	0.964852	+	0.262792i	$0.0846434\pi$
$149$	78.3121i	0.525585i	−0.964852	+	0.262792i	$0.915357\pi$
$150$	0	0
$151$	− 154.327i	− 1.02204i	−0.859570	−	0.511018i	$-0.829268\pi$
$151$	− 154.327i	− 1.02204i	0.859570	−	0.511018i	$-0.170732\pi$
$152$	0	0
$153$	255.808	1.67194
$154$	0	0
$155$	162.781	1.05020
$156$	0	0
$157$	161.408i	1.02808i	0.857767	+	0.514039i	$0.171851\pi$
$157$	161.408i	1.02808i	−0.857767	+	0.514039i	$0.828149\pi$
$158$	0	0
$159$	− 201.379i	− 1.26653i
$160$	0	0
$161$	−8.66061	−0.0537926
$162$	0	0
$163$	139.788	0.857594	0.428797	−	0.903401i	$-0.358938\pi$
$163$	139.788	0.857594	0.428797	+	0.903401i	$0.358938\pi$
$164$	0	0
$165$	44.6606i	0.270670i
$166$	0	0
$167$	− 115.348i	− 0.690709i	−0.938472	−	0.345355i	$-0.887759\pi$
$167$	− 115.348i	− 0.690709i	0.938472	−	0.345355i	$-0.112241\pi$
$168$	0	0
$169$	−261.468	−1.54715
$170$	0	0
$171$	311.507	1.82168
$172$	0	0
$173$	85.7205i	0.495494i	0.968825	+	0.247747i	$0.0796902\pi$
$173$	85.7205i	0.495494i	−0.968825	+	0.247747i	$0.920310\pi$
$174$	0	0
$175$	16.3115i	0.0932083i
$176$	0	0
$177$	−328.798	−1.85762
$178$	0	0
$179$	−112.758	−0.629934	−0.314967	−	0.949103i	$-0.601994\pi$
$179$	−112.758	−0.629934	−0.314967	+	0.949103i	$0.601994\pi$
$180$	0	0
$181$	65.9311i	0.364260i	0.983274	+	0.182130i	$0.0582992\pi$
$181$	65.9311i	0.364260i	−0.983274	+	0.182130i	$0.941701\pi$
$182$	0	0
$183$	− 122.960i	− 0.671912i
$184$	0	0
$185$	95.8258	0.517977
$186$	0	0
$187$	−45.9868	−0.245919
$188$	0	0
$189$	− 13.4955i	− 0.0714045i
$190$	0	0
$191$	207.243i	1.08504i	0.840043	+	0.542520i	$0.182530\pi$
$191$	207.243i	1.08504i	−0.840043	+	0.542520i	$0.817470\pi$
$192$	0	0
$193$	127.183	0.658981	0.329491	−	0.944159i	$-0.393123\pi$
$193$	127.183	0.658981	0.329491	+	0.944159i	$0.393123\pi$
$194$	0	0
$195$	507.062	2.60032
$196$	0	0
$197$	227.670i	1.15568i	0.816149	+	0.577842i	$0.196105\pi$
$197$	227.670i	1.15568i	−0.816149	+	0.577842i	$0.803895\pi$
$198$	0	0
$199$	171.378i	0.861194i	0.902544	+	0.430597i	$0.141697\pi$
$199$	171.378i	0.861194i	−0.902544	+	0.430597i	$0.858303\pi$
$200$	0	0
$201$	356.312	1.77270
$202$	0	0
$203$	−142.410	−0.701525
$204$	0	0
$205$	41.8439i	0.204117i
$206$	0	0
$207$	− 33.2746i	− 0.160747i
$208$	0	0
$209$	−56.0000	−0.267943
$210$	0	0
$211$	149.910	0.710473	0.355237	−	0.934776i	$-0.384400\pi$
$211$	149.910	0.710473	0.355237	+	0.934776i	$0.384400\pi$
$212$	0	0
$213$	− 140.661i	− 0.660378i
$214$	0	0
$215$	− 319.619i	− 1.48660i
$216$	0	0
$217$	77.1470	0.355516
$218$	0	0
$219$	199.853	0.912572
$220$	0	0
$221$	522.120i	2.36253i
$222$	0	0
$223$	85.2676i	0.382366i	0.981554	+	0.191183i	$0.0612324\pi$
$223$	85.2676i	0.382366i	−0.981554	+	0.191183i	$0.938768\pi$
$224$	0	0
$225$	−62.6697	−0.278532
$226$	0	0
$227$	−154.510	−0.680660	−0.340330	−	0.940306i	$-0.610539\pi$
$227$	−154.510	−0.680660	−0.340330	+	0.940306i	$0.610539\pi$
$228$	0	0
$229$	330.555i	1.44347i	0.692168	+	0.721736i	$0.256656\pi$
$229$	330.555i	1.44347i	−0.692168	+	0.721736i	$0.743344\pi$
$230$	0	0
$231$	21.1660i	0.0916278i
$232$	0	0
$233$	−213.339	−0.915620	−0.457810	−	0.889050i	$-0.651366\pi$
$233$	−213.339	−0.915620	−0.457810	+	0.889050i	$0.651366\pi$
$234$	0	0
$235$	−207.846	−0.884451
$236$	0	0
$237$	− 48.0000i	− 0.202532i
$238$	0	0
$239$	266.545i	1.11525i	0.830092	+	0.557626i	$0.188288\pi$
$239$	266.545i	1.11525i	−0.830092	+	0.557626i	$0.811712\pi$
$240$	0	0
$241$	26.4864	0.109902	0.0549510	−	0.998489i	$-0.482500\pi$
$241$	26.4864	0.109902	0.0549510	+	0.998489i	$0.482500\pi$
$242$	0	0
$243$	−348.659	−1.43481
$244$	0	0
$245$	39.0780i	0.159502i
$246$	0	0
$247$	635.806i	2.57411i
$248$	0	0
$249$	−629.111	−2.52655
$250$	0	0
$251$	372.796	1.48524	0.742622	−	0.669711i	$-0.233582\pi$
$251$	372.796	1.48524	0.742622	+	0.669711i	$0.233582\pi$
$252$	0	0
$253$	5.98182i	0.0236435i
$254$	0	0
$255$	− 615.021i	− 2.41185i
$256$	0	0
$257$	−206.000	−0.801556	−0.400778	−	0.916175i	$-0.631260\pi$
$257$	−206.000	−0.801556	−0.400778	+	0.916175i	$0.631260\pi$
$258$	0	0
$259$	45.4147	0.175346
$260$	0	0
$261$	− 547.147i	− 2.09635i
$262$	0	0
$263$	− 465.493i	− 1.76994i	−0.465652	−	0.884968i	$-0.654180\pi$
$263$	− 465.493i	− 1.76994i	0.465652	−	0.884968i	$-0.345820\pi$
$264$	0	0
$265$	−256.798	−0.969051
$266$	0	0
$267$	−259.697	−0.972646
$268$	0	0
$269$	390.417i	1.45137i	0.688029	+	0.725683i	$0.258476\pi$
$269$	390.417i	1.45137i	−0.688029	+	0.725683i	$0.741524\pi$
$270$	0	0
$271$	− 133.543i	− 0.492778i	−0.969171	−	0.246389i	$-0.920756\pi$
$271$	− 133.543i	− 0.492778i	0.969171	−	0.246389i	$-0.0792441\pi$
$272$	0	0
$273$	240.312	0.880264
$274$	0	0
$275$	11.2662	0.0409680
$276$	0	0
$277$	306.624i	1.10695i	0.832867	+	0.553473i	$0.186698\pi$
$277$	306.624i	1.10695i	−0.832867	+	0.553473i	$0.813302\pi$
$278$	0	0
$279$	296.404i	1.06238i
$280$	0	0
$281$	112.642	0.400863	0.200431	−	0.979708i	$-0.435766\pi$
$281$	112.642	0.400863	0.200431	+	0.979708i	$0.435766\pi$
$282$	0	0
$283$	−474.368	−1.67621	−0.838106	−	0.545507i	$-0.816337\pi$
$283$	−474.368	−1.67621	−0.838106	+	0.545507i	$0.816337\pi$
$284$	0	0
$285$	− 748.936i	− 2.62785i
$286$	0	0
$287$	19.8311i	0.0690979i
$288$	0	0
$289$	344.285	1.19130
$290$	0	0
$291$	75.1456	0.258232
$292$	0	0
$293$	17.0962i	0.0583489i	0.999574	+	0.0291744i	$0.00928783\pi$
$293$	17.0962i	0.0583489i	−0.999574	+	0.0291744i	$0.990712\pi$
$294$	0	0
$295$	419.284i	1.42130i
$296$	0	0
$297$	−9.32121	−0.0313845
$298$	0	0
$299$	67.9156	0.227143
$300$	0	0
$301$	− 151.477i	− 0.503247i
$302$	0	0
$303$	38.2958i	0.126389i
$304$	0	0
$305$	−156.798	−0.514093
$306$	0	0
$307$	508.468	1.65625	0.828124	−	0.560544i	$-0.189408\pi$
$307$	508.468	1.65625	0.828124	+	0.560544i	$0.189408\pi$
$308$	0	0
$309$	− 239.652i	− 0.775571i
$310$	0	0
$311$	31.2084i	0.100349i	0.998740	+	0.0501743i	$0.0159777\pi$
$311$	31.2084i	0.100349i	−0.998740	+	0.0501743i	$0.984022\pi$
$312$	0	0
$313$	175.183	0.559691	0.279846	−	0.960045i	$-0.409717\pi$
$313$	175.183	0.559691	0.279846	+	0.960045i	$0.409717\pi$
$314$	0	0
$315$	−150.140	−0.476636
$316$	0	0
$317$	− 217.652i	− 0.686598i	−0.939226	−	0.343299i	$-0.888456\pi$
$317$	− 217.652i	− 0.686598i	0.939226	−	0.343299i	$-0.111544\pi$
$318$	0	0
$319$	98.3612i	0.308342i
$320$	0	0
$321$	557.633	1.73718
$322$	0	0
$323$	771.176	2.38754
$324$	0	0
$325$	− 127.913i	− 0.393578i
$326$	0	0
$327$	212.343i	0.649368i
$328$	0	0
$329$	−98.5045	−0.299406
$330$	0	0
$331$	601.467	1.81712	0.908560	−	0.417754i	$-0.137183\pi$
$331$	601.467	1.81712	0.908560	+	0.417754i	$0.137183\pi$
$332$	0	0
$333$	174.486i	0.523983i
$334$	0	0
$335$	− 454.369i	− 1.35633i
$336$	0	0
$337$	−445.477	−1.32189	−0.660946	−	0.750434i	$-0.729845\pi$
$337$	−445.477	−1.32189	−0.660946	+	0.750434i	$0.729845\pi$
$338$	0	0
$339$	−159.810	−0.471415
$340$	0	0
$341$	− 53.2848i	− 0.156261i
$342$	0	0
$343$	18.5203i	0.0539949i
$344$	0	0
$345$	−80.0000	−0.231884
$346$	0	0
$347$	404.426	1.16549	0.582746	−	0.812654i	$-0.301978\pi$
$347$	404.426	1.16549	0.582746	+	0.812654i	$0.301978\pi$
$348$	0	0
$349$	− 157.895i	− 0.452420i	−0.974079	−	0.226210i	$-0.927366\pi$
$349$	− 157.895i	− 0.452420i	0.974079	−	0.226210i	$-0.0726336\pi$
$350$	0	0
$351$	105.830i	0.301510i
$352$	0	0
$353$	−321.267	−0.910104	−0.455052	−	0.890465i	$-0.650379\pi$
$353$	−321.267	−0.910104	−0.455052	+	0.890465i	$0.650379\pi$
$354$	0	0
$355$	−179.370	−0.505269
$356$	0	0
$357$	− 291.477i	− 0.816463i
$358$	0	0
$359$	− 173.968i	− 0.484590i	−0.970203	−	0.242295i	$-0.922100\pi$
$359$	− 173.968i	− 0.484590i	0.970203	−	0.242295i	$-0.0779002\pi$
$360$	0	0
$361$	578.092	1.60136
$362$	0	0
$363$	−515.095	−1.41899
$364$	0	0
$365$	− 254.853i	− 0.698227i
$366$	0	0
$367$	− 278.050i	− 0.757630i	−0.925472	−	0.378815i	$-0.876332\pi$
$367$	− 278.050i	− 0.757630i	0.925472	−	0.378815i	$-0.123668\pi$
$368$	0	0
$369$	−76.1924	−0.206484
$370$	0	0
$371$	−121.705	−0.328045
$372$	0	0
$373$	− 342.973i	− 0.919498i	−0.888049	−	0.459749i	$-0.847939\pi$
$373$	− 342.973i	− 0.919498i	0.888049	−	0.459749i	$-0.152061\pi$
$374$	0	0
$375$	− 460.313i	− 1.22750i
$376$	0	0
$377$	1116.76	2.96223
$378$	0	0
$379$	−145.731	−0.384515	−0.192257	−	0.981345i	$-0.561581\pi$
$379$	−145.731	−0.384515	−0.192257	+	0.981345i	$0.561581\pi$
$380$	0	0
$381$	− 183.652i	− 0.482025i
$382$	0	0
$383$	224.071i	0.585040i	0.956259	+	0.292520i	$0.0944939\pi$
$383$	224.071i	0.585040i	−0.956259	+	0.292520i	$0.905506\pi$
$384$	0	0
$385$	26.9909	0.0701063
$386$	0	0
$387$	581.986	1.50384
$388$	0	0
$389$	99.8076i	0.256575i	0.991737	+	0.128287i	$0.0409480\pi$
$389$	99.8076i	0.256575i	−0.991737	+	0.128287i	$0.959052\pi$
$390$	0	0
$391$	− 82.3756i	− 0.210679i
$392$	0	0
$393$	207.477	0.527932
$394$	0	0
$395$	−61.2096	−0.154961
$396$	0	0
$397$	163.354i	0.411470i	0.978608	+	0.205735i	$0.0659586\pi$
$397$	163.354i	0.411470i	−0.978608	+	0.205735i	$0.934041\pi$
$398$	0	0
$399$	− 354.944i	− 0.889583i
$400$	0	0
$401$	−123.459	−0.307878	−0.153939	−	0.988080i	$-0.549196\pi$
$401$	−123.459	−0.307878	−0.153939	+	0.988080i	$0.549196\pi$
$402$	0	0
$403$	−604.979	−1.50119
$404$	0	0
$405$	386.069i	0.953257i
$406$	0	0
$407$	− 31.3676i	− 0.0770703i
$408$	0	0
$409$	−7.84394	−0.0191783	−0.00958917	−	0.999954i	$-0.503052\pi$
$409$	−7.84394	−0.0191783	−0.00958917	+	0.999954i	$0.503052\pi$
$410$	0	0
$411$	−672.656	−1.63663
$412$	0	0
$413$	198.711i	0.481141i
$414$	0	0
$415$	802.242i	1.93311i
$416$	0	0
$417$	645.459	1.54786
$418$	0	0
$419$	−364.565	−0.870083	−0.435041	−	0.900410i	$-0.643266\pi$
$419$	−364.565	−0.870083	−0.435041	+	0.900410i	$0.643266\pi$
$420$	0	0
$421$	708.570i	1.68306i	0.540208	+	0.841532i	$0.318346\pi$
$421$	708.570i	1.68306i	−0.540208	+	0.841532i	$0.681654\pi$
$422$	0	0
$423$	− 378.461i	− 0.894707i
$424$	0	0
$425$	−155.147	−0.365052
$426$	0	0
$427$	−74.3115	−0.174032
$428$	0	0
$429$	− 165.982i	− 0.386904i
$430$	0	0
$431$	80.9463i	0.187810i	0.995581	+	0.0939052i	$0.0299350\pi$
$431$	80.9463i	0.187810i	−0.995581	+	0.0939052i	$0.970065\pi$
$432$	0	0
$433$	295.495	0.682438	0.341219	−	0.939984i	$-0.389160\pi$
$433$	295.495	0.682438	0.341219	+	0.939984i	$0.389160\pi$
$434$	0	0
$435$	−1315.47	−3.02407
$436$	0	0
$437$	− 100.312i	− 0.229547i
$438$	0	0
$439$	− 11.5680i	− 0.0263508i	−0.999913	−	0.0131754i	$-0.995806\pi$
$439$	− 11.5680i	− 0.0263508i	0.999913	−	0.0131754i	$-0.00419398\pi$
$440$	0	0
$441$	−71.1561	−0.161352
$442$	0	0
$443$	−284.660	−0.642573	−0.321287	−	0.946982i	$-0.604115\pi$
$443$	−284.660	−0.642573	−0.321287	+	0.946982i	$0.604115\pi$
$444$	0	0
$445$	331.165i	0.744191i
$446$	0	0
$447$	342.835i	0.766969i
$448$	0	0
$449$	154.624	0.344375	0.172187	−	0.985064i	$-0.444917\pi$
$449$	154.624	0.344375	0.172187	+	0.985064i	$0.444917\pi$
$450$	0	0
$451$	13.6972	0.0303707
$452$	0	0
$453$	− 675.615i	− 1.49142i
$454$	0	0
$455$	− 306.446i	− 0.673508i
$456$	0	0
$457$	−410.450	−0.898140	−0.449070	−	0.893497i	$-0.648245\pi$
$457$	−410.450	−0.898140	−0.449070	+	0.893497i	$0.648245\pi$
$458$	0	0
$459$	128.362	0.279657
$460$	0	0
$461$	595.078i	1.29084i	0.763827	+	0.645421i	$0.223318\pi$
$461$	595.078i	1.29084i	−0.763827	+	0.645421i	$0.776682\pi$
$462$	0	0
$463$	− 511.924i	− 1.10567i	−0.833291	−	0.552834i	$-0.813546\pi$
$463$	− 511.924i	− 1.10567i	0.833291	−	0.552834i	$-0.186454\pi$
$464$	0	0
$465$	712.624	1.53253
$466$	0	0
$467$	−365.122	−0.781846	−0.390923	−	0.920423i	$-0.627844\pi$
$467$	−365.122	−0.781846	−0.390923	+	0.920423i	$0.627844\pi$
$468$	0	0
$469$	− 215.339i	− 0.459146i
$470$	0	0
$471$	706.614i	1.50024i
$472$	0	0
$473$	−104.624	−0.221193
$474$	0	0
$475$	−188.929	−0.397745
$476$	0	0
$477$	− 467.597i	− 0.980287i
$478$	0	0
$479$	202.745i	0.423268i	0.977349	+	0.211634i	$0.0678785\pi$
$479$	202.745i	0.423268i	−0.977349	+	0.211634i	$0.932122\pi$
$480$	0	0
$481$	−356.138	−0.740411
$482$	0	0
$483$	−37.9144	−0.0784978
$484$	0	0
$485$	− 95.8258i	− 0.197579i
$486$	0	0
$487$	821.120i	1.68608i	0.537853	+	0.843039i	$0.319236\pi$
$487$	821.120i	1.68608i	−0.537853	+	0.843039i	$0.680764\pi$
$488$	0	0
$489$	611.964	1.25146
$490$	0	0
$491$	316.505	0.644613	0.322307	−	0.946635i	$-0.395542\pi$
$491$	316.505	0.644613	0.322307	+	0.946635i	$0.395542\pi$
$492$	0	0
$493$	− 1354.53i	− 2.74753i
$494$	0	0
$495$	103.701i	0.209497i
$496$	0	0
$497$	−85.0091	−0.171044
$498$	0	0
$499$	−176.415	−0.353538	−0.176769	−	0.984252i	$-0.556565\pi$
$499$	−176.415	−0.353538	−0.176769	+	0.984252i	$0.556565\pi$
$500$	0	0
$501$	− 504.973i	− 1.00793i
$502$	0	0
$503$	883.918i	1.75729i	0.477474	+	0.878646i	$0.341553\pi$
$503$	883.918i	1.75729i	−0.477474	+	0.878646i	$0.658447\pi$
$504$	0	0
$505$	48.8348	0.0967027
$506$	0	0
$507$	−1144.66	−2.25770
$508$	0	0
$509$	− 519.216i	− 1.02007i	−0.860153	−	0.510035i	$-0.829632\pi$
$509$	− 519.216i	− 1.02007i	0.860153	−	0.510035i	$-0.170368\pi$
$510$	0	0
$511$	− 120.783i	− 0.236365i
$512$	0	0
$513$	156.312	0.304702
$514$	0	0
$515$	−305.604	−0.593405
$516$	0	0
$517$	68.0364i	0.131598i
$518$	0	0
$519$	375.267i	0.723058i
$520$	0	0
$521$	−1037.40	−1.99118	−0.995590	−	0.0938137i	$-0.970094\pi$
$521$	−1037.40	−1.99118	−0.995590	+	0.0938137i	$0.970094\pi$
$522$	0	0
$523$	525.900	1.00555	0.502773	−	0.864419i	$-0.332313\pi$
$523$	525.900	1.00555	0.502773	+	0.864419i	$0.332313\pi$
$524$	0	0
$525$	71.4083i	0.136016i
$526$	0	0
$527$	733.786i	1.39238i
$528$	0	0
$529$	518.285	0.979745
$530$	0	0
$531$	−763.462	−1.43778
$532$	0	0
$533$	− 155.514i	− 0.291770i
$534$	0	0
$535$	− 711.094i	− 1.32915i
$536$	0	0
$537$	−493.633	−0.919243
$538$	0	0
$539$	12.7918	0.0237325
$540$	0	0
$541$	776.221i	1.43479i	0.696667	+	0.717395i	$0.254666\pi$
$541$	776.221i	1.43479i	−0.696667	+	0.717395i	$0.745334\pi$
$542$	0	0
$543$	288.633i	0.531553i
$544$	0	0
$545$	270.780	0.496845
$546$	0	0
$547$	153.041	0.279782	0.139891	−	0.990167i	$-0.455325\pi$
$547$	153.041	0.279782	0.139891	+	0.990167i	$0.455325\pi$
$548$	0	0
$549$	− 285.510i	− 0.520054i
$550$	0	0
$551$	− 1649.47i	− 2.99359i
$552$	0	0
$553$	−29.0091	−0.0524577
$554$	0	0
$555$	419.506	0.755867
$556$	0	0
$557$	− 481.579i	− 0.864594i	−0.901731	−	0.432297i	$-0.857703\pi$
$557$	− 481.579i	− 0.864594i	0.901731	−	0.432297i	$-0.142297\pi$
$558$	0	0
$559$	1187.87i	2.12499i
$560$	0	0
$561$	−201.321	−0.358861
$562$	0	0
$563$	−489.909	−0.870176	−0.435088	−	0.900388i	$-0.643283\pi$
$563$	−489.909	−0.870176	−0.435088	+	0.900388i	$0.643283\pi$
$564$	0	0
$565$	203.789i	0.360689i
$566$	0	0
$567$	182.970i	0.322698i
$568$	0	0
$569$	−97.8985	−0.172054	−0.0860268	−	0.996293i	$-0.527417\pi$
$569$	−97.8985	−0.172054	−0.0860268	+	0.996293i	$0.527417\pi$
$570$	0	0
$571$	−752.680	−1.31818	−0.659089	−	0.752065i	$-0.729058\pi$
$571$	−752.680	−1.31818	−0.659089	+	0.752065i	$0.729058\pi$
$572$	0	0
$573$	907.267i	1.58336i
$574$	0	0
$575$	20.1810i	0.0350974i
$576$	0	0
$577$	−924.533	−1.60231	−0.801155	−	0.598456i	$-0.795781\pi$
$577$	−924.533	−1.60231	−0.801155	+	0.598456i	$0.795781\pi$
$578$	0	0
$579$	556.783	0.961629
$580$	0	0
$581$	380.207i	0.654401i
$582$	0	0
$583$	84.0604i	0.144186i
$584$	0	0
$585$	1177.39	2.01263
$586$	0	0
$587$	1110.24	1.89138	0.945688	−	0.325077i	$-0.105390\pi$
$587$	1110.24	1.89138	0.945688	+	0.325077i	$0.105390\pi$
$588$	0	0
$589$	893.561i	1.51708i
$590$	0	0
$591$	996.693i	1.68645i
$592$	0	0
$593$	44.9909	0.0758700	0.0379350	−	0.999280i	$-0.487922\pi$
$593$	44.9909	0.0758700	0.0379350	+	0.999280i	$0.487922\pi$
$594$	0	0
$595$	−371.692	−0.624692
$596$	0	0
$597$	750.258i	1.25671i
$598$	0	0
$599$	988.382i	1.65005i	0.565094	+	0.825026i	$0.308840\pi$
$599$	988.382i	1.65005i	−0.565094	+	0.825026i	$0.691160\pi$
$600$	0	0
$601$	−732.570	−1.21892	−0.609459	−	0.792818i	$-0.708613\pi$
$601$	−732.570	−1.21892	−0.609459	+	0.792818i	$0.708613\pi$
$602$	0	0
$603$	827.348	1.37205
$604$	0	0
$605$	656.849i	1.08570i
$606$	0	0
$607$	− 145.874i	− 0.240319i	−0.992755	−	0.120160i	$-0.961659\pi$
$607$	− 145.874i	− 0.240319i	0.992755	−	0.120160i	$-0.0383406\pi$
$608$	0	0
$609$	−623.441	−1.02371
$610$	0	0
$611$	772.463	1.26426
$612$	0	0
$613$	100.577i	0.164074i	0.996629	+	0.0820369i	$0.0261425\pi$
$613$	100.577i	0.164074i	−0.996629	+	0.0820369i	$0.973857\pi$
$614$	0	0
$615$	183.184i	0.297861i
$616$	0	0
$617$	342.102	0.554459	0.277230	−	0.960804i	$-0.410584\pi$
$617$	342.102	0.554459	0.277230	+	0.960804i	$0.410584\pi$
$618$	0	0
$619$	−512.123	−0.827340	−0.413670	−	0.910427i	$-0.635753\pi$
$619$	−512.123	−0.827340	−0.413670	+	0.910427i	$0.635753\pi$
$620$	0	0
$621$	− 16.6970i	− 0.0268872i
$622$	0	0
$623$	156.949i	0.251925i
$624$	0	0
$625$	−741.120	−1.18579
$626$	0	0
$627$	−245.157	−0.391000
$628$	0	0
$629$	431.964i	0.686747i
$630$	0	0
$631$	− 530.517i	− 0.840755i	−0.907349	−	0.420378i	$-0.861898\pi$
$631$	− 530.517i	− 0.840755i	0.907349	−	0.420378i	$-0.138102\pi$
$632$	0	0
$633$	656.276	1.03677
$634$	0	0
$635$	−234.193	−0.368807
$636$	0	0
$637$	− 145.234i	− 0.227997i
$638$	0	0
$639$	− 326.611i	− 0.511128i
$640$	0	0
$641$	−1054.80	−1.64555	−0.822776	−	0.568366i	$-0.807576\pi$
$641$	−1054.80	−1.64555	−0.822776	+	0.568366i	$0.807576\pi$
$642$	0	0
$643$	107.316	0.166899	0.0834493	−	0.996512i	$-0.473406\pi$
$643$	107.316	0.166899	0.0834493	+	0.996512i	$0.473406\pi$
$644$	0	0
$645$	− 1399.23i	− 2.16935i
$646$	0	0
$647$	202.142i	0.312429i	0.987723	+	0.156215i	$0.0499291\pi$
$647$	202.142i	0.312429i	−0.987723	+	0.156215i	$0.950071\pi$
$648$	0	0
$649$	137.248	0.211477
$650$	0	0
$651$	337.734	0.518793
$652$	0	0
$653$	− 1278.73i	− 1.95823i	−0.203301	−	0.979116i	$-0.565167\pi$
$653$	− 1278.73i	− 1.95823i	0.203301	−	0.979116i	$-0.434833\pi$
$654$	0	0
$655$	− 264.575i	− 0.403931i
$656$	0	0
$657$	464.055	0.706324
$658$	0	0
$659$	835.500	1.26783	0.633915	−	0.773403i	$-0.281447\pi$
$659$	835.500	1.26783	0.633915	+	0.773403i	$0.281447\pi$
$660$	0	0
$661$	998.381i	1.51041i	0.655489	+	0.755205i	$0.272463\pi$
$661$	998.381i	1.51041i	−0.655489	+	0.755205i	$0.727537\pi$
$662$	0	0
$663$	2285.74i	3.44757i
$664$	0	0
$665$	−452.624	−0.680638
$666$	0	0
$667$	−176.193	−0.264158
$668$	0	0
$669$	373.285i	0.557974i
$670$	0	0
$671$	51.3264i	0.0764925i
$672$	0	0
$673$	−572.330	−0.850416	−0.425208	−	0.905096i	$-0.639799\pi$
$673$	−572.330	−0.850416	−0.425208	+	0.905096i	$0.639799\pi$
$674$	0	0
$675$	−31.4472	−0.0465885
$676$	0	0
$677$	1079.77i	1.59493i	0.603365	+	0.797465i	$0.293826\pi$
$677$	1079.77i	1.59493i	−0.603365	+	0.797465i	$0.706174\pi$
$678$	0	0
$679$	− 45.4147i	− 0.0668847i
$680$	0	0
$681$	−676.414	−0.993265
$682$	0	0
$683$	−444.708	−0.651110	−0.325555	−	0.945523i	$-0.605551\pi$
$683$	−444.708	−0.651110	−0.325555	+	0.945523i	$0.605551\pi$
$684$	0	0
$685$	857.771i	1.25222i
$686$	0	0
$687$	1447.11i	2.10641i
$688$	0	0
$689$	954.395	1.38519
$690$	0	0
$691$	−214.227	−0.310025	−0.155012	−	0.987913i	$-0.549542\pi$
$691$	−214.227	−0.310025	−0.155012	+	0.987913i	$0.549542\pi$
$692$	0	0
$693$	49.1470i	0.0709191i
$694$	0	0
$695$	− 823.090i	− 1.18430i
$696$	0	0
$697$	−188.624	−0.270623
$698$	0	0
$699$	−933.958	−1.33613
$700$	0	0
$701$	− 573.477i	− 0.818085i	−0.912515	−	0.409042i	$-0.865863\pi$
$701$	− 573.477i	− 0.818085i	0.912515	−	0.409042i	$-0.134137\pi$
$702$	0	0
$703$	526.019i	0.748250i
$704$	0	0
$705$	−909.909	−1.29065
$706$	0	0
$707$	23.1443	0.0327359
$708$	0	0
$709$	− 436.817i	− 0.616102i	−0.951370	−	0.308051i	$-0.900323\pi$
$709$	− 436.817i	− 0.616102i	0.951370	−	0.308051i	$-0.0996768\pi$
$710$	0	0
$711$	− 111.455i	− 0.156758i
$712$	0	0
$713$	95.4485	0.133869
$714$	0	0
$715$	−211.660	−0.296028
$716$	0	0
$717$	1166.88i	1.62745i
$718$	0	0
$719$	− 1279.32i	− 1.77930i	−0.456639	−	0.889652i	$-0.650947\pi$
$719$	− 1279.32i	− 1.77930i	0.456639	−	0.889652i	$-0.349053\pi$
$720$	0	0
$721$	−144.835	−0.200881
$722$	0	0
$723$	115.952	0.160376
$724$	0	0
$725$	331.844i	0.457716i
$726$	0	0
$727$	166.834i	0.229483i	0.993395	+	0.114741i	$0.0366040\pi$
$727$	166.834i	0.229483i	−0.993395	+	0.114741i	$0.963396\pi$
$728$	0	0
$729$	−903.955	−1.23999
$730$	0	0
$731$	1440.78	1.97097
$732$	0	0
$733$	− 105.299i	− 0.143655i	−0.997417	−	0.0718276i	$-0.977117\pi$
$733$	− 105.299i	− 0.143655i	0.997417	−	0.0718276i	$-0.0228831\pi$
$734$	0	0
$735$	171.076i	0.232756i
$736$	0	0
$737$	−148.733	−0.201809
$738$	0	0
$739$	898.713	1.21612	0.608060	−	0.793891i	$-0.291948\pi$
$739$	898.713	1.21612	0.608060	+	0.793891i	$0.291948\pi$
$740$	0	0
$741$	2783.43i	3.75632i
$742$	0	0
$743$	1154.15i	1.55337i	0.629891	+	0.776683i	$0.283099\pi$
$743$	1154.15i	1.55337i	−0.629891	+	0.776683i	$0.716901\pi$
$744$	0	0
$745$	437.183	0.586823
$746$	0	0
$747$	−1460.78	−1.95553
$748$	0	0
$749$	− 337.009i	− 0.449945i
$750$	0	0
$751$	− 561.025i	− 0.747038i	−0.927623	−	0.373519i	$-0.878151\pi$
$751$	− 561.025i	− 0.747038i	0.927623	−	0.373519i	$-0.121849\pi$
$752$	0	0
$753$	1632.03	2.16737
$754$	0	0
$755$	−861.545	−1.14112
$756$	0	0
$757$	− 860.120i	− 1.13622i	−0.822952	−	0.568111i	$-0.807675\pi$
$757$	− 860.120i	− 1.13622i	0.822952	−	0.568111i	$-0.192325\pi$
$758$	0	0
$759$	26.1872i	0.0345023i
$760$	0	0
$761$	−1336.47	−1.75620	−0.878100	−	0.478477i	$-0.841189\pi$
$761$	−1336.47	−1.75620	−0.878100	+	0.478477i	$0.841189\pi$
$762$	0	0
$763$	128.331	0.168193
$764$	0	0
$765$	− 1428.07i	− 1.86675i
$766$	0	0
$767$	− 1558.28i	− 2.03165i
$768$	0	0
$769$	−716.323	−0.931499	−0.465749	−	0.884917i	$-0.654215\pi$
$769$	−716.323	−0.931499	−0.465749	+	0.884917i	$0.654215\pi$
$770$	0	0
$771$	−901.827	−1.16969
$772$	0	0
$773$	− 98.6644i	− 0.127638i	−0.997961	−	0.0638191i	$-0.979672\pi$
$773$	− 98.6644i	− 0.127638i	0.997961	−	0.0638191i	$-0.0203281\pi$
$774$	0	0
$775$	− 179.768i	− 0.231959i
$776$	0	0
$777$	198.817	0.255877
$778$	0	0
$779$	−229.695	−0.294859
$780$	0	0
$781$	58.7152i	0.0751795i
$782$	0	0
$783$	− 274.555i	− 0.350644i
$784$	0	0
$785$	901.074	1.14787
$786$	0	0
$787$	−673.903	−0.856293	−0.428147	−	0.903709i	$-0.640833\pi$
$787$	−673.903	−0.856293	−0.428147	+	0.903709i	$0.640833\pi$
$788$	0	0
$789$	− 2037.84i	− 2.58281i
$790$	0	0
$791$	96.5819i	0.122101i
$792$	0	0
$793$	582.744	0.734860
$794$	0	0
$795$	−1124.21	−1.41410
$796$	0	0
$797$	1097.76i	1.37736i	0.725065	+	0.688681i	$0.241810\pi$
$797$	1097.76i	1.37736i	−0.725065	+	0.688681i	$0.758190\pi$
$798$	0	0
$799$	− 936.929i	− 1.17263i
$800$	0	0
$801$	−603.009	−0.752820
$802$	0	0
$803$	−83.4236	−0.103890
$804$	0	0
$805$	48.3485i	0.0600602i
$806$	0	0
$807$	1709.17i	2.11793i
$808$	0	0
$809$	340.853	0.421326	0.210663	−	0.977559i	$-0.432438\pi$
$809$	340.853	0.421326	0.210663	+	0.977559i	$0.432438\pi$
$810$	0	0
$811$	112.655	0.138909	0.0694547	−	0.997585i	$-0.477874\pi$
$811$	112.655	0.138909	0.0694547	+	0.997585i	$0.477874\pi$
$812$	0	0
$813$	− 584.624i	− 0.719095i
$814$	0	0
$815$	− 780.376i	− 0.957517i
$816$	0	0
$817$	1754.50	2.14749
$818$	0	0
$819$	557.999	0.681317
$820$	0	0
$821$	− 1267.91i	− 1.54435i	−0.635412	−	0.772174i	$-0.719170\pi$
$821$	− 1267.91i	− 1.54435i	0.635412	−	0.772174i	$-0.280830\pi$
$822$	0	0
$823$	− 743.669i	− 0.903608i	−0.892117	−	0.451804i	$-0.850781\pi$
$823$	− 743.669i	− 0.903608i	0.892117	−	0.451804i	$-0.149219\pi$
$824$	0	0
$825$	49.3212	0.0597833
$826$	0	0
$827$	−1473.29	−1.78149	−0.890745	−	0.454503i	$-0.849817\pi$
$827$	−1473.29	−1.78149	−0.890745	+	0.454503i	$0.849817\pi$
$828$	0	0
$829$	379.840i	0.458191i	0.973404	+	0.229095i	$0.0735768\pi$
$829$	379.840i	0.458191i	−0.973404	+	0.229095i	$0.926423\pi$
$830$	0	0
$831$	1342.34i	1.61533i
$832$	0	0
$833$	−176.156	−0.211472
$834$	0	0
$835$	−643.941	−0.771187
$836$	0	0
$837$	148.733i	0.177698i
$838$	0	0
$839$	− 1080.75i	− 1.28814i	−0.764965	−	0.644072i	$-0.777244\pi$
$839$	− 1080.75i	− 1.28814i	0.764965	−	0.644072i	$-0.222756\pi$
$840$	0	0
$841$	−2056.21	−2.44496
$842$	0	0
$843$	493.126	0.584966
$844$	0	0
$845$	1459.67i	1.72742i
$846$	0	0
$847$	311.301i	0.367533i
$848$	0	0
$849$	−2076.69	−2.44604
$850$	0	0
$851$	56.1884	0.0660264
$852$	0	0
$853$	− 125.372i	− 0.146978i	−0.997296	−	0.0734888i	$-0.976587\pi$
$853$	− 125.372i	− 0.146978i	0.997296	−	0.0734888i	$-0.0234133\pi$
$854$	0	0
$855$	− 1739.01i	− 2.03393i
$856$	0	0
$857$	−721.717	−0.842143	−0.421072	−	0.907027i	$-0.638346\pi$
$857$	−721.717	−0.842143	−0.421072	+	0.907027i	$0.638346\pi$
$858$	0	0
$859$	−104.742	−0.121935	−0.0609676	−	0.998140i	$-0.519419\pi$
$859$	−104.742	−0.121935	−0.0609676	+	0.998140i	$0.519419\pi$
$860$	0	0
$861$	86.8167i	0.100832i
$862$	0	0
$863$	− 425.005i	− 0.492474i	−0.969210	−	0.246237i	$-0.920806\pi$
$863$	− 425.005i	− 0.492474i	0.969210	−	0.246237i	$-0.0791941\pi$
$864$	0	0
$865$	478.541	0.553226
$866$	0	0
$867$	1507.21	1.73842
$868$	0	0
$869$	20.0364i	0.0230568i
$870$	0	0
$871$	1688.67i	1.93877i
$872$	0	0
$873$	174.486	0.199870
$874$	0	0
$875$	−278.193	−0.317935
$876$	0	0
$877$	469.368i	0.535197i	0.963530	+	0.267599i	$0.0862302\pi$
$877$	469.368i	0.535197i	−0.963530	+	0.267599i	$0.913770\pi$
$878$	0	0
$879$	74.8438i	0.0851466i
$880$	0	0
$881$	−1116.11	−1.26687	−0.633435	−	0.773796i	$-0.718356\pi$
$881$	−1116.11	−1.26687	−0.633435	+	0.773796i	$0.718356\pi$
$882$	0	0
$883$	−271.281	−0.307227	−0.153613	−	0.988131i	$-0.549091\pi$
$883$	−271.281	−0.307227	−0.153613	+	0.988131i	$0.549091\pi$
$884$	0	0
$885$	1835.54i	2.07406i
$886$	0	0
$887$	− 211.421i	− 0.238355i	−0.992873	−	0.119178i	$-0.961974\pi$
$887$	− 211.421i	− 0.238355i	0.992873	−	0.119178i	$-0.0380258\pi$
$888$	0	0
$889$	−110.991	−0.124849
$890$	0	0
$891$	126.376	0.141836
$892$	0	0
$893$	− 1140.94i	− 1.27764i
$894$	0	0
$895$	629.481i	0.703331i
$896$	0	0
$897$	297.321	0.331462
$898$	0	0
$899$	1569.50	1.74582
$900$	0	0
$901$	− 1157.60i	− 1.28479i
$902$	0	0
$903$	− 663.138i	− 0.734372i
$904$	0	0
$905$	368.065	0.406702
$906$	0	0
$907$	−410.290	−0.452359	−0.226179	−	0.974086i	$-0.572624\pi$
$907$	−410.290	−0.452359	−0.226179	+	0.974086i	$0.572624\pi$
$908$	0	0
$909$	88.9220i	0.0978239i
$910$	0	0
$911$	608.982i	0.668476i	0.942489	+	0.334238i	$0.108479\pi$
$911$	608.982i	0.668476i	−0.942489	+	0.334238i	$0.891521\pi$
$912$	0	0
$913$	262.606	0.287630
$914$	0	0
$915$	−686.433	−0.750200
$916$	0	0
$917$	− 125.390i	− 0.136740i
$918$	0	0
$919$	− 497.942i	− 0.541830i	−0.962603	−	0.270915i	$-0.912674\pi$
$919$	− 497.942i	− 0.541830i	0.962603	−	0.270915i	$-0.0873262\pi$
$920$	0	0
$921$	2225.97	2.41691
$922$	0	0
$923$	666.633	0.722246
$924$	0	0
$925$	− 105.826i	− 0.114406i
$926$	0	0
$927$	− 556.465i	− 0.600286i
$928$	0	0
$929$	−65.4045	−0.0704032	−0.0352016	−	0.999380i	$-0.511207\pi$
$929$	−65.4045	−0.0704032	−0.0352016	+	0.999380i	$0.511207\pi$
$930$	0	0
$931$	−214.512	−0.230411
$932$	0	0
$933$	136.624i	0.146435i
$934$	0	0
$935$	256.725i	0.274572i
$936$	0	0
$937$	600.918	0.641321	0.320661	−	0.947194i	$-0.396095\pi$
$937$	600.918	0.641321	0.320661	+	0.947194i	$0.396095\pi$
$938$	0	0
$939$	766.918	0.816739
$940$	0	0
$941$	1482.87i	1.57584i	0.615776	+	0.787921i	$0.288843\pi$
$941$	1482.87i	1.57584i	−0.615776	+	0.787921i	$0.711157\pi$
$942$	0	0
$943$	24.5356i	0.0260187i
$944$	0	0
$945$	−75.3394	−0.0797242
$946$	0	0
$947$	−806.865	−0.852023	−0.426011	−	0.904718i	$-0.640082\pi$
$947$	−806.865	−0.852023	−0.426011	+	0.904718i	$0.640082\pi$
$948$	0	0
$949$	947.165i	0.998067i
$950$	0	0
$951$	− 952.835i	− 1.00193i
$952$	0	0
$953$	−899.945	−0.944329	−0.472164	−	0.881510i	$-0.656527\pi$
$953$	−899.945	−0.944329	−0.472164	+	0.881510i	$0.656527\pi$
$954$	0	0
$955$	1156.95	1.21146
$956$	0	0
$957$	430.606i	0.449954i
$958$	0	0
$959$	406.524i	0.423904i
$960$	0	0
$961$	110.764	0.115259
$962$	0	0
$963$	1294.81	1.34456
$964$	0	0
$965$	− 710.011i	− 0.735762i
$966$	0	0
$967$	1725.92i	1.78482i	0.451228	+	0.892409i	$0.350986\pi$
$967$	1725.92i	1.78482i	−0.451228	+	0.892409i	$0.649014\pi$
$968$	0	0
$969$	3376.06	3.48406
$970$	0	0
$971$	657.951	0.677601	0.338800	−	0.940858i	$-0.389979\pi$
$971$	657.951	0.677601	0.338800	+	0.940858i	$0.389979\pi$
$972$	0	0
$973$	− 390.087i	− 0.400912i
$974$	0	0
$975$	− 559.977i	− 0.574336i
$976$	0	0
$977$	46.9364	0.0480413	0.0240207	−	0.999711i	$-0.492353\pi$
$977$	46.9364	0.0480413	0.0240207	+	0.999711i	$0.492353\pi$
$978$	0	0
$979$	108.404	0.110729
$980$	0	0
$981$	493.056i	0.502606i
$982$	0	0
$983$	377.095i	0.383616i	0.981432	+	0.191808i	$0.0614351\pi$
$983$	377.095i	0.383616i	−0.981432	+	0.191808i	$0.938565\pi$
$984$	0	0
$985$	1270.98	1.29034
$986$	0	0
$987$	−431.233	−0.436913
$988$	0	0
$989$	− 187.412i	− 0.189497i
$990$	0	0
$991$	926.283i	0.934696i	0.884074	+	0.467348i	$0.154790\pi$
$991$	926.283i	0.934696i	−0.884074	+	0.467348i	$0.845210\pi$
$992$	0	0
$993$	2633.10	2.65166
$994$	0	0
$995$	956.729	0.961537
$996$	0	0
$997$	− 56.6462i	− 0.0568167i	−0.999596	−	0.0284083i	$-0.990956\pi$
$997$	− 56.6462i	− 0.0568167i	0.999596	−	0.0284083i	$-0.00904387\pi$
$998$	0	0
$999$	87.5560i	0.0876437i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.3.g.e.127.7		8
4.3	odd	2	inner	1792.3.g.e.127.1		8
8.3	odd	2	inner	1792.3.g.e.127.8		8
8.5	even	2	inner	1792.3.g.e.127.2		8
16.3	odd	4		112.3.d.b.15.1	✓	4
16.5	even	4		448.3.d.b.127.1		4
16.11	odd	4		448.3.d.b.127.4		4
16.13	even	4		112.3.d.b.15.4	yes	4
48.29	odd	4		1008.3.m.d.127.4		4
48.35	even	4		1008.3.m.d.127.3		4
112.3	even	12		784.3.r.n.79.2		4
112.13	odd	4		784.3.d.j.687.1		4
112.19	even	12		784.3.r.i.655.1		4
112.45	odd	12		784.3.r.i.79.1		4
112.51	odd	12		784.3.r.o.655.2		4
112.61	odd	12		784.3.r.n.655.2		4
112.67	odd	12		784.3.r.j.79.1		4
112.83	even	4		784.3.d.j.687.4		4
112.93	even	12		784.3.r.j.655.1		4
112.109	even	12		784.3.r.o.79.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
112.3.d.b.15.1	✓	4	16.3	odd	4
112.3.d.b.15.4	yes	4	16.13	even	4
448.3.d.b.127.1		4	16.5	even	4
448.3.d.b.127.4		4	16.11	odd	4
784.3.d.j.687.1		4	112.13	odd	4
784.3.d.j.687.4		4	112.83	even	4
784.3.r.i.79.1		4	112.45	odd	12
784.3.r.i.655.1		4	112.19	even	12
784.3.r.j.79.1		4	112.67	odd	12
784.3.r.j.655.1		4	112.93	even	12
784.3.r.n.79.2		4	112.3	even	12
784.3.r.n.655.2		4	112.61	odd	12
784.3.r.o.79.2		4	112.109	even	12
784.3.r.o.655.2		4	112.51	odd	12
1008.3.m.d.127.3		4	48.35	even	4
1008.3.m.d.127.4		4	48.29	odd	4
1792.3.g.e.127.1		8	4.3	odd	2	inner
1792.3.g.e.127.2		8	8.5	even	2	inner
1792.3.g.e.127.7		8	1.1	even	1	trivial
1792.3.g.e.127.8		8	8.3	odd	2	inner

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:	\( N \)	\(=\)	\( 1792 = 2^{8} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1792.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.37780 q^{3} -5.58258i q^{5} -2.64575i q^{7} +10.1652 q^{9} +O(q^{10})\)	\(q+4.37780 q^{3} -5.58258i q^{5} -2.64575i q^{7} +10.1652 q^{9} -1.82740 q^{11} +20.7477i q^{13} -24.4394i q^{15} +25.1652 q^{17} +30.6446 q^{19} -11.5826i q^{21} -3.27340i q^{23} -6.16515 q^{25} +5.10080 q^{27} -53.8258i q^{29} +29.1588i q^{31} -8.00000 q^{33} -14.7701 q^{35} +17.1652i q^{37} +90.8294i q^{39} -7.49545 q^{41} +57.2530 q^{43} -56.7477i q^{45} -37.2312i q^{47} -7.00000 q^{49} +110.168 q^{51} -46.0000i q^{53} +10.2016i q^{55} +134.156 q^{57} -75.1058 q^{59} -28.0871i q^{61} -26.8945i q^{63} +115.826 q^{65} +81.3906 q^{67} -14.3303i q^{69} -32.1304i q^{71} +45.6515 q^{73} -26.9898 q^{75} +4.83485i q^{77} -10.9644i q^{79} -69.1561 q^{81} -143.705 q^{83} -140.486i q^{85} -235.639i q^{87} -59.3212 q^{89} +54.8933 q^{91} +127.652i q^{93} -171.076i q^{95} +17.1652 q^{97} -18.5758 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^9	\( 8 q + 8 q^{9} + 128 q^{17} + 24 q^{25} - 64 q^{33} + 160 q^{41} - 56 q^{49} + 560 q^{57} + 560 q^{65} - 368 q^{73} - 40 q^{81} + 112 q^{89} + 64 q^{97}+O(q^{100}) \) 8 * q + 8 * q^9 + 128 * q^17 + 24 * q^25 - 64 * q^33 + 160 * q^41 - 56 * q^49 + 560 * q^57 + 560 * q^65 - 368 * q^73 - 40 * q^81 + 112 * q^89 + 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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