LMFDB - Embedded newform 1792.3.g.f.127.5

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:	8.0.1997017344.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 14x^{6} + 53x^{4} + 56x^{2} + 4 $ x^8 + 14x^6 + 53x^4 + 56*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{12} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.5
Root			$1.27733i$ of defining polynomial
Character	$\chi$	$=$	1792.127
Dual form			1792.3.g.f.127.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.55467 q^{3} -9.86836i q^{5} +2.64575i q^{7} -2.47367 q^{9} +O(q^{10})$	$q+2.55467 q^{3} -9.86836i q^{5} +2.64575i q^{7} -2.47367 q^{9} +13.1537 q^{11} -5.86836i q^{13} -25.2104i q^{15} -0.570700 q^{17} +15.6640 q^{19} +6.75902i q^{21} -16.4817i q^{23} -72.3844 q^{25} -29.3114 q^{27} -29.7367i q^{29} -54.8014i q^{31} +33.6034 q^{33} +26.1092 q^{35} +42.0853i q^{37} -14.9917i q^{39} -0.773275 q^{41} +41.7931 q^{43} +24.4110i q^{45} +58.4528i q^{47} -7.00000 q^{49} -1.45795 q^{51} -5.65139i q^{53} -129.805i q^{55} +40.0163 q^{57} +42.6434 q^{59} -95.9371i q^{61} -6.54471i q^{63} -57.9110 q^{65} -69.8503 q^{67} -42.1053i q^{69} +92.0882i q^{71} -9.97539 q^{73} -184.918 q^{75} +34.8014i q^{77} -20.1780i q^{79} -52.6180 q^{81} -151.307 q^{83} +5.63187i q^{85} -75.9674i q^{87} -5.79743 q^{89} +15.5262 q^{91} -140.000i q^{93} -154.578i q^{95} +103.696 q^{97} -32.5379 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{3} + 40 q^{9}+O(q^{10}) $ 8 * q + 8 * q^3 + 40 * q^9	$ 8 q + 8 q^{3} + 40 q^{9} + 32 q^{11} - 16 q^{17} + 88 q^{19} - 104 q^{25} + 176 q^{27} + 56 q^{35} - 144 q^{41} - 224 q^{43} - 56 q^{49} + 16 q^{51} + 400 q^{57} + 232 q^{59} - 304 q^{65} - 368 q^{67} - 272 q^{73} - 664 q^{75} + 504 q^{81} - 424 q^{83} + 80 q^{89} + 56 q^{91} + 528 q^{97} + 544 q^{99}+O(q^{100}) $ 8 * q + 8 * q^3 + 40 * q^9 + 32 * q^11 - 16 * q^17 + 88 * q^19 - 104 * q^25 + 176 * q^27 + 56 * q^35 - 144 * q^41 - 224 * q^43 - 56 * q^49 + 16 * q^51 + 400 * q^57 + 232 * q^59 - 304 * q^65 - 368 * q^67 - 272 * q^73 - 664 * q^75 + 504 * q^81 - 424 * q^83 + 80 * q^89 + 56 * q^91 + 528 * q^97 + 544 * q^99

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
$3$	2.55467	0.851556	0.425778	−	0.904828i	$-0.360000\pi$
$3$	2.55467	0.851556	0.425778	+	0.904828i	$0.360000\pi$
$4$	0	0
$5$	− 9.86836i	− 1.97367i	−0.161727	−	0.986836i	$-0.551706\pi$
$5$	− 9.86836i	− 1.97367i	0.161727	−	0.986836i	$-0.448294\pi$
$6$	0	0
$7$	2.64575i	0.377964i
$7$	2.64575i	0.377964i
$8$	0	0
$9$	−2.47367	−0.274852
$10$	0	0
$11$	13.1537	1.19579	0.597896	−	0.801574i	$-0.296004\pi$
$11$	13.1537	1.19579	0.597896	+	0.801574i	$0.296004\pi$
$12$	0	0
$13$	− 5.86836i	− 0.451412i	−0.974195	−	0.225706i	$-0.927531\pi$
$13$	− 5.86836i	− 0.451412i	0.974195	−	0.225706i	$-0.0724689\pi$
$14$	0	0
$15$	− 25.2104i	− 1.68069i
$16$	0	0
$17$	−0.570700	−0.0335706	−0.0167853	−	0.999859i	$-0.505343\pi$
$17$	−0.570700	−0.0335706	−0.0167853	+	0.999859i	$0.505343\pi$
$18$	0	0
$19$	15.6640	0.824421	0.412211	−	0.911089i	$-0.364757\pi$
$19$	15.6640	0.824421	0.412211	+	0.911089i	$0.364757\pi$
$20$	0	0
$21$	6.75902i	0.321858i
$22$	0	0
$23$	− 16.4817i	− 0.716596i	−0.933607	−	0.358298i	$-0.883357\pi$
$23$	− 16.4817i	− 0.716596i	0.933607	−	0.358298i	$-0.116643\pi$
$24$	0	0
$25$	−72.3844	−2.89538
$26$	0	0
$27$	−29.3114	−1.08561
$28$	0	0
$29$	− 29.7367i	− 1.02540i	−0.858567	−	0.512702i	$-0.828645\pi$
$29$	− 29.7367i	− 1.02540i	0.858567	−	0.512702i	$-0.171355\pi$
$30$	0	0
$31$	− 54.8014i	− 1.76779i	−0.467687	−	0.883894i	$-0.654913\pi$
$31$	− 54.8014i	− 1.76779i	0.467687	−	0.883894i	$-0.345087\pi$
$32$	0	0
$33$	33.6034	1.01828
$34$	0	0
$35$	26.1092	0.745978
$36$	0	0
$37$	42.0853i	1.13744i	0.822531	+	0.568721i	$0.192561\pi$
$37$	42.0853i	1.13744i	−0.822531	+	0.568721i	$0.807439\pi$
$38$	0	0
$39$	− 14.9917i	− 0.384403i
$40$	0	0
$41$	−0.773275	−0.0188604	−0.00943019	−	0.999956i	$-0.503002\pi$
$41$	−0.773275	−0.0188604	−0.00943019	+	0.999956i	$0.503002\pi$
$42$	0	0
$43$	41.7931	0.971933	0.485967	−	0.873977i	$-0.338468\pi$
$43$	41.7931	0.971933	0.485967	+	0.873977i	$0.338468\pi$
$44$	0	0
$45$	24.4110i	0.542467i
$46$	0	0
$47$	58.4528i	1.24368i	0.783145	+	0.621839i	$0.213614\pi$
$47$	58.4528i	1.24368i	−0.783145	+	0.621839i	$0.786386\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	−1.45795	−0.0285873
$52$	0	0
$53$	− 5.65139i	− 0.106630i	−0.998578	−	0.0533150i	$-0.983021\pi$
$53$	− 5.65139i	− 0.106630i	0.998578	−	0.0533150i	$-0.0169787\pi$
$54$	0	0
$55$	− 129.805i	− 2.36010i
$56$	0	0
$57$	40.0163	0.702041
$58$	0	0
$59$	42.6434	0.722770	0.361385	−	0.932417i	$-0.382304\pi$
$59$	42.6434	0.722770	0.361385	+	0.932417i	$0.382304\pi$
$60$	0	0
$61$	− 95.9371i	− 1.57274i	−0.617756	−	0.786370i	$-0.711958\pi$
$61$	− 95.9371i	− 1.57274i	0.617756	−	0.786370i	$-0.288042\pi$
$62$	0	0
$63$	− 6.54471i	− 0.103884i
$64$	0	0
$65$	−57.9110	−0.890939
$66$	0	0
$67$	−69.8503	−1.04254	−0.521271	−	0.853391i	$-0.674542\pi$
$67$	−69.8503	−1.04254	−0.521271	+	0.853391i	$0.674542\pi$
$68$	0	0
$69$	− 42.1053i	− 0.610222i
$70$	0	0
$71$	92.0882i	1.29702i	0.761207	+	0.648509i	$0.224607\pi$
$71$	92.0882i	1.29702i	−0.761207	+	0.648509i	$0.775393\pi$
$72$	0	0
$73$	−9.97539	−0.136649	−0.0683246	−	0.997663i	$-0.521765\pi$
$73$	−9.97539	−0.136649	−0.0683246	+	0.997663i	$0.521765\pi$
$74$	0	0
$75$	−184.918	−2.46558
$76$	0	0
$77$	34.8014i	0.451967i
$78$	0	0
$79$	− 20.1780i	− 0.255417i	−0.991812	−	0.127709i	$-0.959238\pi$
$79$	− 20.1780i	− 0.255417i	0.991812	−	0.127709i	$-0.0407622\pi$
$80$	0	0
$81$	−52.6180	−0.649604
$82$	0	0
$83$	−151.307	−1.82298	−0.911491	−	0.411321i	$-0.865068\pi$
$83$	−151.307	−1.82298	−0.911491	+	0.411321i	$0.865068\pi$
$84$	0	0
$85$	5.63187i	0.0662573i
$86$	0	0
$87$	− 75.9674i	− 0.873189i
$88$	0	0
$89$	−5.79743	−0.0651396	−0.0325698	−	0.999469i	$-0.510369\pi$
$89$	−5.79743	−0.0651396	−0.0325698	+	0.999469i	$0.510369\pi$
$90$	0	0
$91$	15.5262	0.170618
$92$	0	0
$93$	− 140.000i	− 1.50537i
$94$	0	0
$95$	− 154.578i	− 1.62714i
$96$	0	0
$97$	103.696	1.06903	0.534513	−	0.845160i	$-0.320495\pi$
$97$	103.696	1.06903	0.534513	+	0.845160i	$0.320495\pi$
$98$	0	0
$99$	−32.5379	−0.328666
$100$	0	0
$101$	− 41.9176i	− 0.415025i	−0.978232	−	0.207513i	$-0.933463\pi$
$101$	− 41.9176i	− 0.415025i	0.978232	−	0.207513i	$-0.0665368\pi$
$102$	0	0
$103$	− 18.7378i	− 0.181920i	−0.995855	−	0.0909602i	$-0.971006\pi$
$103$	− 18.7378i	− 0.181920i	0.995855	−	0.0909602i	$-0.0289936\pi$
$104$	0	0
$105$	66.7004	0.635242
$106$	0	0
$107$	−64.8549	−0.606121	−0.303060	−	0.952971i	$-0.598008\pi$
$107$	−64.8549	−0.606121	−0.303060	+	0.952971i	$0.598008\pi$
$108$	0	0
$109$	− 186.432i	− 1.71039i	−0.518307	−	0.855195i	$-0.673437\pi$
$109$	− 186.432i	− 1.71039i	0.518307	−	0.855195i	$-0.326563\pi$
$110$	0	0
$111$	107.514i	0.968595i
$112$	0	0
$113$	−84.7530	−0.750027	−0.375013	−	0.927019i	$-0.622362\pi$
$113$	−84.7530	−0.750027	−0.375013	+	0.927019i	$0.622362\pi$
$114$	0	0
$115$	−162.647	−1.41433
$116$	0	0
$117$	14.5164i	0.124071i
$118$	0	0
$119$	− 1.50993i	− 0.0126885i
$120$	0	0
$121$	52.0200	0.429917
$122$	0	0
$123$	−1.97546	−0.0160607
$124$	0	0
$125$	467.606i	3.74085i
$126$	0	0
$127$	93.3874i	0.735334i	0.929958	+	0.367667i	$0.119843\pi$
$127$	93.3874i	0.735334i	−0.929958	+	0.367667i	$0.880157\pi$
$128$	0	0
$129$	106.768	0.827656
$130$	0	0
$131$	58.0857	0.443403	0.221701	−	0.975115i	$-0.428839\pi$
$131$	58.0857	0.443403	0.221701	+	0.975115i	$0.428839\pi$
$132$	0	0
$133$	41.4431i	0.311602i
$134$	0	0
$135$	289.256i	2.14263i
$136$	0	0
$137$	−176.420	−1.28774	−0.643870	−	0.765135i	$-0.722672\pi$
$137$	−176.420	−1.28774	−0.643870	+	0.765135i	$0.722672\pi$
$138$	0	0
$139$	−12.3041	−0.0885185	−0.0442592	−	0.999020i	$-0.514093\pi$
$139$	−12.3041	−0.0885185	−0.0442592	+	0.999020i	$0.514093\pi$
$140$	0	0
$141$	149.328i	1.05906i
$142$	0	0
$143$	− 77.1906i	− 0.539795i
$144$	0	0
$145$	−293.452	−2.02381
$146$	0	0
$147$	−17.8827	−0.121651
$148$	0	0
$149$	− 173.782i	− 1.16632i	−0.812358	−	0.583160i	$-0.801816\pi$
$149$	− 173.782i	− 1.16632i	0.812358	−	0.583160i	$-0.198184\pi$
$150$	0	0
$151$	− 138.433i	− 0.916776i	−0.888752	−	0.458388i	$-0.848427\pi$
$151$	− 138.433i	− 0.916776i	0.888752	−	0.458388i	$-0.151573\pi$
$152$	0	0
$153$	1.41172	0.00922695
$154$	0	0
$155$	−540.800	−3.48903
$156$	0	0
$157$	189.693i	1.20824i	0.796894	+	0.604119i	$0.206475\pi$
$157$	189.693i	1.20824i	−0.796894	+	0.604119i	$0.793525\pi$
$158$	0	0
$159$	− 14.4374i	− 0.0908014i
$160$	0	0
$161$	43.6065	0.270848
$162$	0	0
$163$	17.7685	0.109009	0.0545047	−	0.998514i	$-0.482642\pi$
$163$	17.7685	0.109009	0.0545047	+	0.998514i	$0.482642\pi$
$164$	0	0
$165$	− 331.610i	− 2.00976i
$166$	0	0
$167$	− 0.0890922i	0 0.000533486i	−1.00000		0.000266743i	$-0.999915\pi$
$167$	− 0.0890922i	0 0.000533486i	1.00000		0.000266743i	$-8.49069e-5\pi$
$168$	0	0
$169$	134.562	0.796227
$170$	0	0
$171$	−38.7475	−0.226594
$172$	0	0
$173$	43.2891i	0.250226i	0.992143	+	0.125113i	$0.0399293\pi$
$173$	43.2891i	0.250226i	−0.992143	+	0.125113i	$0.960071\pi$
$174$	0	0
$175$	− 191.511i	− 1.09435i
$176$	0	0
$177$	108.940	0.615479
$178$	0	0
$179$	−190.838	−1.06614	−0.533068	−	0.846072i	$-0.678961\pi$
$179$	−190.838	−1.06614	−0.533068	+	0.846072i	$0.678961\pi$
$180$	0	0
$181$	207.153i	1.14449i	0.820082	+	0.572246i	$0.193928\pi$
$181$	207.153i	1.14449i	−0.820082	+	0.572246i	$0.806072\pi$
$182$	0	0
$183$	− 245.087i	− 1.33928i
$184$	0	0
$185$	415.313	2.24493
$186$	0	0
$187$	−7.50683	−0.0401434
$188$	0	0
$189$	− 77.5507i	− 0.410321i
$190$	0	0
$191$	− 315.489i	− 1.65178i	−0.563833	−	0.825889i	$-0.690674\pi$
$191$	− 315.489i	− 1.65178i	0.563833	−	0.825889i	$-0.309326\pi$
$192$	0	0
$193$	−30.9403	−0.160312	−0.0801561	−	0.996782i	$-0.525542\pi$
$193$	−30.9403	−0.160312	−0.0801561	+	0.996782i	$0.525542\pi$
$194$	0	0
$195$	−147.943	−0.758684
$196$	0	0
$197$	264.248i	1.34136i	0.741746	+	0.670681i	$0.233998\pi$
$197$	264.248i	1.34136i	−0.741746	+	0.670681i	$0.766002\pi$
$198$	0	0
$199$	− 131.321i	− 0.659903i	−0.943998	−	0.329951i	$-0.892968\pi$
$199$	− 131.321i	− 0.659903i	0.943998	−	0.329951i	$-0.107032\pi$
$200$	0	0
$201$	−178.444	−0.887783
$202$	0	0
$203$	78.6759	0.387566
$204$	0	0
$205$	7.63095i	0.0372242i
$206$	0	0
$207$	40.7703i	0.196958i
$208$	0	0
$209$	206.040	0.985836
$210$	0	0
$211$	247.994	1.17533	0.587664	−	0.809105i	$-0.300048\pi$
$211$	247.994	1.17533	0.587664	+	0.809105i	$0.300048\pi$
$212$	0	0
$213$	235.255i	1.10448i
$214$	0	0
$215$	− 412.430i	− 1.91828i
$216$	0	0
$217$	144.991	0.668161
$218$	0	0
$219$	−25.4838	−0.116364
$220$	0	0
$221$	3.34907i	0.0151542i
$222$	0	0
$223$	− 56.7999i	− 0.254708i	−0.991857	−	0.127354i	$-0.959352\pi$
$223$	− 56.7999i	− 0.254708i	0.991857	−	0.127354i	$-0.0406484\pi$
$224$	0	0
$225$	179.055	0.795800
$226$	0	0
$227$	275.095	1.21187	0.605937	−	0.795513i	$-0.292798\pi$
$227$	275.095	1.21187	0.605937	+	0.795513i	$0.292798\pi$
$228$	0	0
$229$	− 6.29210i	− 0.0274764i	−0.999906	−	0.0137382i	$-0.995627\pi$
$229$	− 6.29210i	− 0.0274764i	0.999906	−	0.0137382i	$-0.00437314\pi$
$230$	0	0
$231$	88.9061i	0.384875i
$232$	0	0
$233$	142.049	0.609653	0.304827	−	0.952408i	$-0.401401\pi$
$233$	142.049	0.609653	0.304827	+	0.952408i	$0.401401\pi$
$234$	0	0
$235$	576.833	2.45461
$236$	0	0
$237$	− 51.5480i	− 0.217502i
$238$	0	0
$239$	− 51.3954i	− 0.215043i	−0.994203	−	0.107522i	$-0.965708\pi$
$239$	− 51.3954i	− 0.215043i	0.994203	−	0.107522i	$-0.0342915\pi$
$240$	0	0
$241$	93.7048	0.388817	0.194408	−	0.980921i	$-0.437721\pi$
$241$	93.7048	0.388817	0.194408	+	0.980921i	$0.437721\pi$
$242$	0	0
$243$	129.381	0.532433
$244$	0	0
$245$	69.0785i	0.281953i
$246$	0	0
$247$	− 91.9220i	− 0.372154i
$248$	0	0
$249$	−386.540	−1.55237
$250$	0	0
$251$	−12.4430	−0.0495737	−0.0247869	−	0.999693i	$-0.507891\pi$
$251$	−12.4430	−0.0495737	−0.0247869	+	0.999693i	$0.507891\pi$
$252$	0	0
$253$	− 216.796i	− 0.856900i
$254$	0	0
$255$	14.3876i	0.0564219i
$256$	0	0
$257$	169.884	0.661029	0.330514	−	0.943801i	$-0.392778\pi$
$257$	169.884	0.661029	0.330514	+	0.943801i	$0.392778\pi$
$258$	0	0
$259$	−111.347	−0.429912
$260$	0	0
$261$	73.5587i	0.281834i
$262$	0	0
$263$	− 118.762i	− 0.451565i	−0.974178	−	0.225782i	$-0.927506\pi$
$263$	− 118.762i	− 0.451565i	0.974178	−	0.225782i	$-0.0724938\pi$
$264$	0	0
$265$	−55.7699	−0.210452
$266$	0	0
$267$	−14.8105	−0.0554700
$268$	0	0
$269$	− 9.37496i	− 0.0348512i	−0.999848	−	0.0174256i	$-0.994453\pi$
$269$	− 9.37496i	− 0.0348512i	0.999848	−	0.0174256i	$-0.00554701\pi$
$270$	0	0
$271$	− 203.067i	− 0.749324i	−0.927161	−	0.374662i	$-0.877759\pi$
$271$	− 203.067i	− 0.749324i	0.927161	−	0.374662i	$-0.122241\pi$
$272$	0	0
$273$	39.6643	0.145291
$274$	0	0
$275$	−952.124	−3.46227
$276$	0	0
$277$	− 154.419i	− 0.557471i	−0.960368	−	0.278735i	$-0.910085\pi$
$277$	− 154.419i	− 0.557471i	0.960368	−	0.278735i	$-0.0899152\pi$
$278$	0	0
$279$	135.561i	0.485880i
$280$	0	0
$281$	217.495	0.774005	0.387002	−	0.922079i	$-0.373511\pi$
$281$	217.495	0.774005	0.387002	+	0.922079i	$0.373511\pi$
$282$	0	0
$283$	431.849	1.52597	0.762983	−	0.646418i	$-0.223734\pi$
$283$	431.849	1.52597	0.762983	+	0.646418i	$0.223734\pi$
$284$	0	0
$285$	− 394.896i	− 1.38560i
$286$	0	0
$287$	− 2.04589i	− 0.00712855i
$288$	0	0
$289$	−288.674	−0.998873
$290$	0	0
$291$	264.908	0.910336
$292$	0	0
$293$	212.019i	0.723616i	0.932253	+	0.361808i	$0.117840\pi$
$293$	212.019i	0.723616i	−0.932253	+	0.361808i	$0.882160\pi$
$294$	0	0
$295$	− 420.820i	− 1.42651i
$296$	0	0
$297$	−385.554	−1.29816
$298$	0	0
$299$	−96.7206	−0.323480
$300$	0	0
$301$	110.574i	0.367356i
$302$	0	0
$303$	− 107.086i	− 0.353418i
$304$	0	0
$305$	−946.741	−3.10407
$306$	0	0
$307$	192.868	0.628234	0.314117	−	0.949384i	$-0.398292\pi$
$307$	192.868	0.628234	0.314117	+	0.949384i	$0.398292\pi$
$308$	0	0
$309$	− 47.8689i	− 0.154915i
$310$	0	0
$311$	269.814i	0.867569i	0.901017	+	0.433785i	$0.142822\pi$
$311$	269.814i	0.867569i	−0.901017	+	0.433785i	$0.857178\pi$
$312$	0	0
$313$	236.490	0.755558	0.377779	−	0.925896i	$-0.376688\pi$
$313$	236.490	0.755558	0.377779	+	0.925896i	$0.376688\pi$
$314$	0	0
$315$	−64.5855	−0.205033
$316$	0	0
$317$	− 249.012i	− 0.785526i	−0.919640	−	0.392763i	$-0.871519\pi$
$317$	− 249.012i	− 0.785526i	0.919640	−	0.392763i	$-0.128481\pi$
$318$	0	0
$319$	− 391.148i	− 1.22617i
$320$	0	0
$321$	−165.683	−0.516146
$322$	0	0
$323$	−8.93945	−0.0276763
$324$	0	0
$325$	424.778i	1.30701i
$326$	0	0
$327$	− 476.273i	− 1.45649i
$328$	0	0
$329$	−154.652	−0.470066
$330$	0	0
$331$	−465.343	−1.40587	−0.702935	−	0.711254i	$-0.748128\pi$
$331$	−465.343	−1.40587	−0.702935	+	0.711254i	$0.748128\pi$
$332$	0	0
$333$	− 104.105i	− 0.312628i
$334$	0	0
$335$	689.308i	2.05763i
$336$	0	0
$337$	626.243	1.85829	0.929144	−	0.369718i	$-0.120546\pi$
$337$	626.243	1.85829	0.929144	+	0.369718i	$0.120546\pi$
$338$	0	0
$339$	−216.516	−0.638690
$340$	0	0
$341$	− 720.842i	− 2.11391i
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	0	0
$345$	−415.510	−1.20438
$346$	0	0
$347$	−450.576	−1.29849	−0.649245	−	0.760580i	$-0.724915\pi$
$347$	−450.576	−1.29849	−0.649245	+	0.760580i	$0.724915\pi$
$348$	0	0
$349$	− 64.7762i	− 0.185605i	−0.995685	−	0.0928025i	$-0.970417\pi$
$349$	− 64.7762i	− 0.185605i	0.995685	−	0.0928025i	$-0.0295825\pi$
$350$	0	0
$351$	172.010i	0.490057i
$352$	0	0
$353$	589.179	1.66906	0.834532	−	0.550960i	$-0.185738\pi$
$353$	589.179	1.66906	0.834532	+	0.550960i	$0.185738\pi$
$354$	0	0
$355$	908.760	2.55989
$356$	0	0
$357$	− 3.85737i	− 0.0108050i
$358$	0	0
$359$	406.643i	1.13271i	0.824161	+	0.566355i	$0.191647\pi$
$359$	406.643i	1.13271i	−0.824161	+	0.566355i	$0.808353\pi$
$360$	0	0
$361$	−115.639	−0.320329
$362$	0	0
$363$	132.894	0.366099
$364$	0	0
$365$	98.4407i	0.269701i
$366$	0	0
$367$	− 430.196i	− 1.17220i	−0.810240	−	0.586098i	$-0.800663\pi$
$367$	− 430.196i	− 1.17220i	0.810240	−	0.586098i	$-0.199337\pi$
$368$	0	0
$369$	1.91283	0.00518381
$370$	0	0
$371$	14.9522	0.0403023
$372$	0	0
$373$	− 2.60758i	− 0.00699083i	−0.999994	−	0.00349542i	$-0.998887\pi$
$373$	− 2.60758i	− 0.00699083i	0.999994	−	0.00349542i	$-0.00111263\pi$
$374$	0	0
$375$	1194.58i	3.18555i
$376$	0	0
$377$	−174.506	−0.462880
$378$	0	0
$379$	359.118	0.947541	0.473771	−	0.880648i	$-0.342893\pi$
$379$	359.118	0.947541	0.473771	+	0.880648i	$0.342893\pi$
$380$	0	0
$381$	238.574i	0.626178i
$382$	0	0
$383$	470.758i	1.22913i	0.788865	+	0.614567i	$0.210669\pi$
$383$	470.758i	1.22913i	−0.788865	+	0.614567i	$0.789331\pi$
$384$	0	0
$385$	343.433	0.892034
$386$	0	0
$387$	−103.382	−0.267138
$388$	0	0
$389$	180.270i	0.463418i	0.972785	+	0.231709i	$0.0744318\pi$
$389$	180.270i	0.463418i	−0.972785	+	0.231709i	$0.925568\pi$
$390$	0	0
$391$	9.40612i	0.0240566i
$392$	0	0
$393$	148.390	0.377582
$394$	0	0
$395$	−199.123	−0.504110
$396$	0	0
$397$	− 248.223i	− 0.625248i	−0.949877	−	0.312624i	$-0.898792\pi$
$397$	− 248.223i	− 0.625248i	0.949877	−	0.312624i	$-0.101208\pi$
$398$	0	0
$399$	105.873i	0.265347i
$400$	0	0
$401$	369.502	0.921452	0.460726	−	0.887542i	$-0.347589\pi$
$401$	369.502	0.921452	0.460726	+	0.887542i	$0.347589\pi$
$402$	0	0
$403$	−321.594	−0.798001
$404$	0	0
$405$	519.253i	1.28211i
$406$	0	0
$407$	553.578i	1.36014i
$408$	0	0
$409$	−479.494	−1.17236	−0.586179	−	0.810182i	$-0.699368\pi$
$409$	−479.494	−1.17236	−0.586179	+	0.810182i	$0.699368\pi$
$410$	0	0
$411$	−450.695	−1.09658
$412$	0	0
$413$	112.824i	0.273181i
$414$	0	0
$415$	1493.16i	3.59797i
$416$	0	0
$417$	−31.4328	−0.0753785
$418$	0	0
$419$	564.644	1.34760	0.673800	−	0.738914i	$-0.264661\pi$
$419$	564.644	1.34760	0.673800	+	0.738914i	$0.264661\pi$
$420$	0	0
$421$	− 544.282i	− 1.29283i	−0.762986	−	0.646415i	$-0.776267\pi$
$421$	− 544.282i	− 1.29283i	0.762986	−	0.646415i	$-0.223733\pi$
$422$	0	0
$423$	− 144.593i	− 0.341827i
$424$	0	0
$425$	41.3098	0.0971996
$426$	0	0
$427$	253.826	0.594440
$428$	0	0
$429$	− 197.196i	− 0.459665i
$430$	0	0
$431$	− 90.3539i	− 0.209638i	−0.994491	−	0.104819i	$-0.966574\pi$
$431$	− 90.3539i	− 0.209638i	0.994491	−	0.104819i	$-0.0334263\pi$
$432$	0	0
$433$	−131.383	−0.303425	−0.151713	−	0.988425i	$-0.548479\pi$
$433$	−131.383	−0.303425	−0.151713	+	0.988425i	$0.548479\pi$
$434$	0	0
$435$	−749.674	−1.72339
$436$	0	0
$437$	− 258.170i	− 0.590777i
$438$	0	0
$439$	− 128.655i	− 0.293064i	−0.989206	−	0.146532i	$-0.953189\pi$
$439$	− 128.655i	− 0.293064i	0.989206	−	0.146532i	$-0.0468111\pi$
$440$	0	0
$441$	17.3157	0.0392646
$442$	0	0
$443$	193.210	0.436139	0.218070	−	0.975933i	$-0.430024\pi$
$443$	193.210	0.436139	0.218070	+	0.975933i	$0.430024\pi$
$444$	0	0
$445$	57.2111i	0.128564i
$446$	0	0
$447$	− 443.954i	− 0.993186i
$448$	0	0
$449$	−31.5046	−0.0701661	−0.0350830	−	0.999384i	$-0.511170\pi$
$449$	−31.5046	−0.0701661	−0.0350830	+	0.999384i	$0.511170\pi$
$450$	0	0
$451$	−10.1714	−0.0225531
$452$	0	0
$453$	− 353.651i	− 0.780686i
$454$	0	0
$455$	− 153.218i	− 0.336743i
$456$	0	0
$457$	426.401	0.933043	0.466521	−	0.884510i	$-0.345507\pi$
$457$	426.401	0.933043	0.466521	+	0.884510i	$0.345507\pi$
$458$	0	0
$459$	16.7280	0.0364445
$460$	0	0
$461$	430.182i	0.933150i	0.884482	+	0.466575i	$0.154512\pi$
$461$	430.182i	0.933150i	−0.884482	+	0.466575i	$0.845488\pi$
$462$	0	0
$463$	− 183.454i	− 0.396230i	−0.980179	−	0.198115i	$-0.936518\pi$
$463$	− 183.454i	− 0.396230i	0.980179	−	0.198115i	$-0.0634819\pi$
$464$	0	0
$465$	−1381.56	−2.97111
$466$	0	0
$467$	110.381	0.236362	0.118181	−	0.992992i	$-0.462294\pi$
$467$	110.381	0.236362	0.118181	+	0.992992i	$0.462294\pi$
$468$	0	0
$469$	− 184.807i	− 0.394044i
$470$	0	0
$471$	484.604i	1.02888i
$472$	0	0
$473$	549.735	1.16223
$474$	0	0
$475$	−1133.83	−2.38701
$476$	0	0
$477$	13.9797i	0.0293075i
$478$	0	0
$479$	− 515.593i	− 1.07639i	−0.842819	−	0.538197i	$-0.819106\pi$
$479$	− 515.593i	− 1.07639i	0.842819	−	0.538197i	$-0.180894\pi$
$480$	0	0
$481$	246.972	0.513455
$482$	0	0
$483$	111.400	0.230642
$484$	0	0
$485$	− 1023.30i	− 2.10991i
$486$	0	0
$487$	− 59.2995i	− 0.121765i	−0.998145	−	0.0608824i	$-0.980609\pi$
$487$	− 59.2995i	− 0.121765i	0.998145	−	0.0608824i	$-0.0193915\pi$
$488$	0	0
$489$	45.3927	0.0928276
$490$	0	0
$491$	556.042	1.13247	0.566234	−	0.824244i	$-0.308400\pi$
$491$	556.042	1.13247	0.566234	+	0.824244i	$0.308400\pi$
$492$	0	0
$493$	16.9708i	0.0344234i
$494$	0	0
$495$	321.096i	0.648678i
$496$	0	0
$497$	−243.643	−0.490227
$498$	0	0
$499$	150.429	0.301461	0.150730	−	0.988575i	$-0.451837\pi$
$499$	150.429	0.301461	0.150730	+	0.988575i	$0.451837\pi$
$500$	0	0
$501$	− 0.227601i	0 0.000454293i
$502$	0	0
$503$	415.893i	0.826825i	0.910544	+	0.413412i	$0.135663\pi$
$503$	415.893i	0.826825i	−0.910544	+	0.413412i	$0.864337\pi$
$504$	0	0
$505$	−413.658	−0.819124
$506$	0	0
$507$	343.762	0.678032
$508$	0	0
$509$	622.234i	1.22246i	0.791452	+	0.611232i	$0.209326\pi$
$509$	622.234i	1.22246i	−0.791452	+	0.611232i	$0.790674\pi$
$510$	0	0
$511$	− 26.3924i	− 0.0516485i
$512$	0	0
$513$	−459.134	−0.894999
$514$	0	0
$515$	−184.911	−0.359051
$516$	0	0
$517$	768.871i	1.48718i
$518$	0	0
$519$	110.589i	0.213081i
$520$	0	0
$521$	644.142	1.23636	0.618179	−	0.786037i	$-0.287871\pi$
$521$	644.142	1.23636	0.618179	+	0.786037i	$0.287871\pi$
$522$	0	0
$523$	889.589	1.70093	0.850467	−	0.526028i	$-0.176319\pi$
$523$	889.589	1.70093	0.850467	+	0.526028i	$0.176319\pi$
$524$	0	0
$525$	− 489.248i	− 0.931900i
$526$	0	0
$527$	31.2752i	0.0593457i
$528$	0	0
$529$	257.353	0.486490
$530$	0	0
$531$	−105.486	−0.198655
$532$	0	0
$533$	4.53785i	0.00851380i
$534$	0	0
$535$	640.012i	1.19628i
$536$	0	0
$537$	−487.529	−0.907875
$538$	0	0
$539$	−92.0759	−0.170827
$540$	0	0
$541$	832.015i	1.53792i	0.639296	+	0.768961i	$0.279226\pi$
$541$	832.015i	1.53792i	−0.639296	+	0.768961i	$0.720774\pi$
$542$	0	0
$543$	529.207i	0.974599i
$544$	0	0
$545$	−1839.78	−3.37575
$546$	0	0
$547$	1014.85	1.85531	0.927654	−	0.373441i	$-0.121822\pi$
$547$	1014.85	1.85531	0.927654	+	0.373441i	$0.121822\pi$
$548$	0	0
$549$	237.316i	0.432270i
$550$	0	0
$551$	− 465.796i	− 0.845365i
$552$	0	0
$553$	53.3859	0.0965387
$554$	0	0
$555$	1060.99	1.91169
$556$	0	0
$557$	473.280i	0.849694i	0.905265	+	0.424847i	$0.139672\pi$
$557$	473.280i	0.849694i	−0.905265	+	0.424847i	$0.860328\pi$
$558$	0	0
$559$	− 245.257i	− 0.438742i
$560$	0	0
$561$	−19.1775	−0.0341844
$562$	0	0
$563$	−102.364	−0.181818	−0.0909092	−	0.995859i	$-0.528977\pi$
$563$	−102.364	−0.181818	−0.0909092	+	0.995859i	$0.528977\pi$
$564$	0	0
$565$	836.373i	1.48031i
$566$	0	0
$567$	− 139.214i	− 0.245527i
$568$	0	0
$569$	68.4059	0.120221	0.0601106	−	0.998192i	$-0.480855\pi$
$569$	68.4059	0.120221	0.0601106	+	0.998192i	$0.480855\pi$
$570$	0	0
$571$	701.419	1.22841	0.614203	−	0.789148i	$-0.289478\pi$
$571$	701.419	1.22841	0.614203	+	0.789148i	$0.289478\pi$
$572$	0	0
$573$	− 805.971i	− 1.40658i
$574$	0	0
$575$	1193.02i	2.07482i
$576$	0	0
$577$	44.4710	0.0770727	0.0385364	−	0.999257i	$-0.487730\pi$
$577$	44.4710	0.0770727	0.0385364	+	0.999257i	$0.487730\pi$
$578$	0	0
$579$	−79.0421	−0.136515
$580$	0	0
$581$	− 400.322i	− 0.689022i
$582$	0	0
$583$	− 74.3367i	− 0.127507i
$584$	0	0
$585$	143.253	0.244876
$586$	0	0
$587$	−99.0239	−0.168695	−0.0843475	−	0.996436i	$-0.526881\pi$
$587$	−99.0239	−0.168695	−0.0843475	+	0.996436i	$0.526881\pi$
$588$	0	0
$589$	− 858.410i	− 1.45740i
$590$	0	0
$591$	675.066i	1.14224i
$592$	0	0
$593$	−443.870	−0.748516	−0.374258	−	0.927325i	$-0.622103\pi$
$593$	−443.870	−0.748516	−0.374258	+	0.927325i	$0.622103\pi$
$594$	0	0
$595$	−14.9005	−0.0250429
$596$	0	0
$597$	− 335.481i	− 0.561945i
$598$	0	0
$599$	− 969.972i	− 1.61932i	−0.586900	−	0.809659i	$-0.699652\pi$
$599$	− 969.972i	− 1.61932i	0.586900	−	0.809659i	$-0.300348\pi$
$600$	0	0
$601$	58.0231	0.0965443	0.0482722	−	0.998834i	$-0.484629\pi$
$601$	58.0231	0.0965443	0.0482722	+	0.998834i	$0.484629\pi$
$602$	0	0
$603$	172.786	0.286545
$604$	0	0
$605$	− 513.352i	− 0.848515i
$606$	0	0
$607$	67.3822i	0.111009i	0.998458	+	0.0555043i	$0.0176767\pi$
$607$	67.3822i	0.111009i	−0.998458	+	0.0555043i	$0.982323\pi$
$608$	0	0
$609$	200.991	0.330034
$610$	0	0
$611$	343.022	0.561411
$612$	0	0
$613$	521.917i	0.851415i	0.904861	+	0.425707i	$0.139975\pi$
$613$	521.917i	0.851415i	−0.904861	+	0.425707i	$0.860025\pi$
$614$	0	0
$615$	19.4946i	0.0316985i
$616$	0	0
$617$	−165.257	−0.267840	−0.133920	−	0.990992i	$-0.542757\pi$
$617$	−165.257	−0.267840	−0.133920	+	0.990992i	$0.542757\pi$
$618$	0	0
$619$	205.392	0.331812	0.165906	−	0.986142i	$-0.446945\pi$
$619$	205.392	0.331812	0.165906	+	0.986142i	$0.446945\pi$
$620$	0	0
$621$	483.103i	0.777943i
$622$	0	0
$623$	− 15.3385i	− 0.0246205i
$624$	0	0
$625$	2804.90	4.48783
$626$	0	0
$627$	526.363	0.839495
$628$	0	0
$629$	− 24.0181i	− 0.0381846i
$630$	0	0
$631$	631.682i	1.00108i	0.865713	+	0.500540i	$0.166865\pi$
$631$	631.682i	1.00108i	−0.865713	+	0.500540i	$0.833135\pi$
$632$	0	0
$633$	633.543	1.00086
$634$	0	0
$635$	921.580	1.45131
$636$	0	0
$637$	41.0785i	0.0644874i
$638$	0	0
$639$	− 227.796i	− 0.356488i
$640$	0	0
$641$	817.936	1.27603	0.638016	−	0.770023i	$-0.279755\pi$
$641$	817.936	1.27603	0.638016	+	0.770023i	$0.279755\pi$
$642$	0	0
$643$	607.082	0.944140	0.472070	−	0.881561i	$-0.343507\pi$
$643$	607.082	0.944140	0.472070	+	0.881561i	$0.343507\pi$
$644$	0	0
$645$	− 1053.62i	− 1.63352i
$646$	0	0
$647$	− 621.063i	− 0.959911i	−0.877293	−	0.479956i	$-0.840653\pi$
$647$	− 621.063i	− 0.959911i	0.877293	−	0.479956i	$-0.159347\pi$
$648$	0	0
$649$	560.919	0.864282
$650$	0	0
$651$	370.404	0.568977
$652$	0	0
$653$	− 850.083i	− 1.30181i	−0.759159	−	0.650906i	$-0.774389\pi$
$653$	− 850.083i	− 1.30181i	0.759159	−	0.650906i	$-0.225611\pi$
$654$	0	0
$655$	− 573.211i	− 0.875131i
$656$	0	0
$657$	24.6758	0.0375583
$658$	0	0
$659$	−331.172	−0.502537	−0.251268	−	0.967917i	$-0.580848\pi$
$659$	−331.172	−0.502537	−0.251268	+	0.967917i	$0.580848\pi$
$660$	0	0
$661$	− 464.323i	− 0.702456i	−0.936290	−	0.351228i	$-0.885764\pi$
$661$	− 464.323i	− 0.702456i	0.936290	−	0.351228i	$-0.114236\pi$
$662$	0	0
$663$	8.55577i	0.0129046i
$664$	0	0
$665$	408.975	0.615000
$666$	0	0
$667$	−490.112	−0.734801
$668$	0	0
$669$	− 145.105i	− 0.216898i
$670$	0	0
$671$	− 1261.93i	− 1.88067i
$672$	0	0
$673$	−844.655	−1.25506	−0.627529	−	0.778593i	$-0.715934\pi$
$673$	−844.655	−1.25506	−0.627529	+	0.778593i	$0.715934\pi$
$674$	0	0
$675$	2121.69	3.14325
$676$	0	0
$677$	664.414i	0.981410i	0.871326	+	0.490705i	$0.163261\pi$
$677$	664.414i	0.981410i	−0.871326	+	0.490705i	$0.836739\pi$
$678$	0	0
$679$	274.353i	0.404054i
$680$	0	0
$681$	702.778	1.03198
$682$	0	0
$683$	−601.901	−0.881261	−0.440631	−	0.897689i	$-0.645245\pi$
$683$	−601.901	−0.881261	−0.440631	+	0.897689i	$0.645245\pi$
$684$	0	0
$685$	1740.98i	2.54157i
$686$	0	0
$687$	− 16.0742i	− 0.0233977i
$688$	0	0
$689$	−33.1643	−0.0481340
$690$	0	0
$691$	−579.981	−0.839336	−0.419668	−	0.907678i	$-0.637854\pi$
$691$	−579.981	−0.839336	−0.419668	+	0.907678i	$0.637854\pi$
$692$	0	0
$693$	− 86.0872i	− 0.124224i
$694$	0	0
$695$	121.421i	0.174706i
$696$	0	0
$697$	0.441308	0.000633154 0
$698$	0	0
$699$	362.889	0.519154
$700$	0	0
$701$	863.561i	1.23190i	0.787786	+	0.615949i	$0.211227\pi$
$701$	863.561i	1.23190i	−0.787786	+	0.615949i	$0.788773\pi$
$702$	0	0
$703$	659.225i	0.937731i
$704$	0	0
$705$	1473.62	2.09024
$706$	0	0
$707$	110.903	0.156865
$708$	0	0
$709$	293.022i	0.413290i	0.978416	+	0.206645i	$0.0662545\pi$
$709$	293.022i	0.413290i	−0.978416	+	0.206645i	$0.933746\pi$
$710$	0	0
$711$	49.9136i	0.0702019i
$712$	0	0
$713$	−903.222	−1.26679
$714$	0	0
$715$	−761.744	−1.06538
$716$	0	0
$717$	− 131.298i	− 0.183122i
$718$	0	0
$719$	− 381.821i	− 0.531044i	−0.964105	−	0.265522i	$-0.914456\pi$
$719$	− 381.821i	− 0.531044i	0.964105	−	0.265522i	$-0.0855443\pi$
$720$	0	0
$721$	49.5756	0.0687594
$722$	0	0
$723$	239.385	0.331099
$724$	0	0
$725$	2152.48i	2.96893i
$726$	0	0
$727$	387.165i	0.532552i	0.963897	+	0.266276i	$0.0857932\pi$
$727$	387.165i	0.532552i	−0.963897	+	0.266276i	$0.914207\pi$
$728$	0	0
$729$	804.088	1.10300
$730$	0	0
$731$	−23.8514	−0.0326284
$732$	0	0
$733$	59.1034i	0.0806322i	0.999187	+	0.0403161i	$0.0128365\pi$
$733$	59.1034i	0.0806322i	−0.999187	+	0.0403161i	$0.987164\pi$
$734$	0	0
$735$	176.473i	0.240099i
$736$	0	0
$737$	−918.790	−1.24666
$738$	0	0
$739$	1285.33	1.73928	0.869640	−	0.493687i	$-0.164351\pi$
$739$	1285.33	1.73928	0.869640	+	0.493687i	$0.164351\pi$
$740$	0	0
$741$	− 234.830i	− 0.316910i
$742$	0	0
$743$	− 1450.21i	− 1.95183i	−0.218141	−	0.975917i	$-0.569999\pi$
$743$	− 1450.21i	− 1.95183i	0.218141	−	0.975917i	$-0.430001\pi$
$744$	0	0
$745$	−1714.94	−2.30193
$746$	0	0
$747$	374.284	0.501050
$748$	0	0
$749$	− 171.590i	− 0.229092i
$750$	0	0
$751$	190.150i	0.253195i	0.991954	+	0.126598i	$0.0404057\pi$
$751$	190.150i	0.253195i	−0.991954	+	0.126598i	$0.959594\pi$
$752$	0	0
$753$	−31.7878	−0.0422148
$754$	0	0
$755$	−1366.11	−1.80941
$756$	0	0
$757$	− 441.225i	− 0.582859i	−0.956592	−	0.291430i	$-0.905869\pi$
$757$	− 441.225i	− 0.582859i	0.956592	−	0.291430i	$-0.0941310\pi$
$758$	0	0
$759$	− 553.841i	− 0.729698i
$760$	0	0
$761$	1044.28	1.37225	0.686123	−	0.727486i	$-0.259311\pi$
$761$	1044.28	1.37225	0.686123	+	0.727486i	$0.259311\pi$
$762$	0	0
$763$	493.254	0.646466
$764$	0	0
$765$	− 13.9314i	− 0.0182110i
$766$	0	0
$767$	− 250.247i	− 0.326267i
$768$	0	0
$769$	−1141.60	−1.48452	−0.742262	−	0.670110i	$-0.766247\pi$
$769$	−1141.60	−1.48452	−0.742262	+	0.670110i	$0.766247\pi$
$770$	0	0
$771$	433.998	0.562903
$772$	0	0
$773$	211.566i	0.273694i	0.990592	+	0.136847i	$0.0436969\pi$
$773$	211.566i	0.273694i	−0.990592	+	0.136847i	$0.956303\pi$
$774$	0	0
$775$	3966.77i	5.11841i
$776$	0	0
$777$	−284.455	−0.366095
$778$	0	0
$779$	−12.1126	−0.0155489
$780$	0	0
$781$	1211.30i	1.55096i
$782$	0	0
$783$	871.625i	1.11319i
$784$	0	0
$785$	1871.96	2.38466
$786$	0	0
$787$	−523.213	−0.664820	−0.332410	−	0.943135i	$-0.607862\pi$
$787$	−523.213	−0.664820	−0.332410	+	0.943135i	$0.607862\pi$
$788$	0	0
$789$	− 303.396i	− 0.384533i
$790$	0	0
$791$	− 224.235i	− 0.283483i
$792$	0	0
$793$	−562.993	−0.709953
$794$	0	0
$795$	−142.474	−0.179212
$796$	0	0
$797$	− 503.460i	− 0.631693i	−0.948810	−	0.315847i	$-0.897711\pi$
$797$	− 503.460i	− 0.631693i	0.948810	−	0.315847i	$-0.102289\pi$
$798$	0	0
$799$	− 33.3590i	− 0.0417510i
$800$	0	0
$801$	14.3409	0.0179038
$802$	0	0
$803$	−131.213	−0.163404
$804$	0	0
$805$	− 430.325i	− 0.534565i
$806$	0	0
$807$	− 23.9499i	− 0.0296777i
$808$	0	0
$809$	−100.517	−0.124248	−0.0621242	−	0.998068i	$-0.519788\pi$
$809$	−100.517	−0.124248	−0.0621242	+	0.998068i	$0.519788\pi$
$810$	0	0
$811$	759.830	0.936905	0.468453	−	0.883489i	$-0.344812\pi$
$811$	759.830	0.936905	0.468453	+	0.883489i	$0.344812\pi$
$812$	0	0
$813$	− 518.768i	− 0.638091i
$814$	0	0
$815$	− 175.346i	− 0.215149i
$816$	0	0
$817$	654.648	0.801283
$818$	0	0
$819$	−38.4067	−0.0468946
$820$	0	0
$821$	749.705i	0.913160i	0.889682	+	0.456580i	$0.150926\pi$
$821$	749.705i	0.913160i	−0.889682	+	0.456580i	$0.849074\pi$
$822$	0	0
$823$	− 424.800i	− 0.516160i	−0.966124	−	0.258080i	$-0.916910\pi$
$823$	− 424.800i	− 0.516160i	0.966124	−	0.258080i	$-0.0830899\pi$
$824$	0	0
$825$	−2432.36	−2.94832
$826$	0	0
$827$	−575.272	−0.695613	−0.347806	−	0.937566i	$-0.613073\pi$
$827$	−575.272	−0.695613	−0.347806	+	0.937566i	$0.613073\pi$
$828$	0	0
$829$	157.635i	0.190151i	0.995470	+	0.0950756i	$0.0303093\pi$
$829$	157.635i	0.190151i	−0.995470	+	0.0950756i	$0.969691\pi$
$830$	0	0
$831$	− 394.490i	− 0.474718i
$832$	0	0
$833$	3.99490	0.00479580
$834$	0	0
$835$	−0.879193	−0.00105293
$836$	0	0
$837$	1606.31i	1.91913i
$838$	0	0
$839$	759.895i	0.905716i	0.891583	+	0.452858i	$0.149595\pi$
$839$	759.895i	0.905716i	−0.891583	+	0.452858i	$0.850405\pi$
$840$	0	0
$841$	−43.2720	−0.0514530
$842$	0	0
$843$	555.628	0.659108
$844$	0	0
$845$	− 1327.91i	− 1.57149i
$846$	0	0
$847$	137.632i	0.162493i
$848$	0	0
$849$	1103.23	1.29945
$850$	0	0
$851$	693.638	0.815086
$852$	0	0
$853$	− 315.373i	− 0.369722i	−0.982765	−	0.184861i	$-0.940817\pi$
$853$	− 315.373i	− 0.369722i	0.982765	−	0.184861i	$-0.0591834\pi$
$854$	0	0
$855$	382.375i	0.447222i
$856$	0	0
$857$	−1222.53	−1.42652	−0.713260	−	0.700900i	$-0.752782\pi$
$857$	−1222.53	−1.42652	−0.713260	+	0.700900i	$0.752782\pi$
$858$	0	0
$859$	−1008.23	−1.17373	−0.586863	−	0.809687i	$-0.699637\pi$
$859$	−1008.23	−1.17373	−0.586863	+	0.809687i	$0.699637\pi$
$860$	0	0
$861$	− 5.22658i	− 0.00607036i
$862$	0	0
$863$	− 1140.31i	− 1.32133i	−0.750679	−	0.660667i	$-0.770274\pi$
$863$	− 1140.31i	− 1.32133i	0.750679	−	0.660667i	$-0.229726\pi$
$864$	0	0
$865$	427.192	0.493864
$866$	0	0
$867$	−737.467	−0.850597
$868$	0	0
$869$	− 265.415i	− 0.305426i
$870$	0	0
$871$	409.906i	0.470616i
$872$	0	0
$873$	−256.508	−0.293824
$874$	0	0
$875$	−1237.17	−1.41391
$876$	0	0
$877$	− 306.788i	− 0.349816i	−0.984585	−	0.174908i	$-0.944037\pi$
$877$	− 306.788i	− 0.349816i	0.984585	−	0.174908i	$-0.0559627\pi$
$878$	0	0
$879$	541.640i	0.616200i
$880$	0	0
$881$	−248.968	−0.282597	−0.141299	−	0.989967i	$-0.545128\pi$
$881$	−248.968	−0.282597	−0.141299	+	0.989967i	$0.545128\pi$
$882$	0	0
$883$	−458.136	−0.518841	−0.259420	−	0.965765i	$-0.583531\pi$
$883$	−458.136	−0.518841	−0.259420	+	0.965765i	$0.583531\pi$
$884$	0	0
$885$	− 1075.06i	− 1.21475i
$886$	0	0
$887$	− 312.628i	− 0.352456i	−0.984349	−	0.176228i	$-0.943610\pi$
$887$	− 312.628i	− 0.352456i	0.984349	−	0.176228i	$-0.0563895\pi$
$888$	0	0
$889$	−247.080	−0.277930
$890$	0	0
$891$	−692.121	−0.776791
$892$	0	0
$893$	915.605i	1.02531i
$894$	0	0
$895$	1883.26i	2.10420i
$896$	0	0
$897$	−247.089	−0.275462
$898$	0	0
$899$	−1629.61	−1.81270
$900$	0	0
$901$	3.22525i	0.00357963i
$902$	0	0
$903$	282.481i	0.312825i
$904$	0	0
$905$	2044.26	2.25885
$906$	0	0
$907$	375.701	0.414224	0.207112	−	0.978317i	$-0.433594\pi$
$907$	375.701	0.414224	0.207112	+	0.978317i	$0.433594\pi$
$908$	0	0
$909$	103.690i	0.114071i
$910$	0	0
$911$	41.9614i	0.0460608i	0.999735	+	0.0230304i	$0.00733145\pi$
$911$	41.9614i	0.0460608i	−0.999735	+	0.0230304i	$0.992669\pi$
$912$	0	0
$913$	−1990.25	−2.17991
$914$	0	0
$915$	−2418.61	−2.64329
$916$	0	0
$917$	153.680i	0.167590i
$918$	0	0
$919$	− 1770.71i	− 1.92678i	−0.268101	−	0.963391i	$-0.586396\pi$
$919$	− 1770.71i	− 1.92678i	0.268101	−	0.963391i	$-0.413604\pi$
$920$	0	0
$921$	492.713	0.534976
$922$	0	0
$923$	540.407	0.585489
$924$	0	0
$925$	− 3046.32i	− 3.29332i
$926$	0	0
$927$	46.3511i	0.0500012i
$928$	0	0
$929$	1668.30	1.79581	0.897903	−	0.440193i	$-0.145090\pi$
$929$	1668.30	1.79581	0.897903	+	0.440193i	$0.145090\pi$
$930$	0	0
$931$	−109.648	−0.117774
$932$	0	0
$933$	689.285i	0.738784i
$934$	0	0
$935$	74.0800i	0.0792300i
$936$	0	0
$937$	183.028	0.195334	0.0976668	−	0.995219i	$-0.468862\pi$
$937$	183.028	0.195334	0.0976668	+	0.995219i	$0.468862\pi$
$938$	0	0
$939$	604.152	0.643400
$940$	0	0
$941$	255.475i	0.271493i	0.990744	+	0.135747i	$0.0433433\pi$
$941$	255.475i	0.271493i	−0.990744	+	0.135747i	$0.956657\pi$
$942$	0	0
$943$	12.7449i	0.0135153i
$944$	0	0
$945$	−765.298	−0.809839
$946$	0	0
$947$	187.173	0.197648	0.0988242	−	0.995105i	$-0.468492\pi$
$947$	187.173	0.197648	0.0988242	+	0.995105i	$0.468492\pi$
$948$	0	0
$949$	58.5391i	0.0616851i
$950$	0	0
$951$	− 636.142i	− 0.668920i
$952$	0	0
$953$	−362.350	−0.380220	−0.190110	−	0.981763i	$-0.560885\pi$
$953$	−362.350	−0.380220	−0.190110	+	0.981763i	$0.560885\pi$
$954$	0	0
$955$	−3113.36	−3.26007
$956$	0	0
$957$	− 999.253i	− 1.04415i
$958$	0	0
$959$	− 466.764i	− 0.486720i
$960$	0	0
$961$	−2042.20	−2.12508
$962$	0	0
$963$	160.430	0.166594
$964$	0	0
$965$	305.330i	0.316404i
$966$	0	0
$967$	1473.76i	1.52406i	0.647545	+	0.762028i	$0.275796\pi$
$967$	1473.76i	1.52406i	−0.647545	+	0.762028i	$0.724204\pi$
$968$	0	0
$969$	−22.8373	−0.0235680
$970$	0	0
$971$	−223.833	−0.230518	−0.115259	−	0.993335i	$-0.536770\pi$
$971$	−223.833	−0.230518	−0.115259	+	0.993335i	$0.536770\pi$
$972$	0	0
$973$	− 32.5535i	− 0.0334568i
$974$	0	0
$975$	1085.17i	1.11299i
$976$	0	0
$977$	825.305	0.844734	0.422367	−	0.906425i	$-0.361199\pi$
$977$	825.305	0.844734	0.422367	+	0.906425i	$0.361199\pi$
$978$	0	0
$979$	−76.2576	−0.0778934
$980$	0	0
$981$	461.172i	0.470104i
$982$	0	0
$983$	− 411.945i	− 0.419069i	−0.977801	−	0.209535i	$-0.932805\pi$
$983$	− 411.945i	− 0.419069i	0.977801	−	0.209535i	$-0.0671949\pi$
$984$	0	0
$985$	2607.69	2.64741
$986$	0	0
$987$	−395.084	−0.400287
$988$	0	0
$989$	− 688.823i	− 0.696484i
$990$	0	0
$991$	1607.53i	1.62213i	0.584959	+	0.811063i	$0.301111\pi$
$991$	1607.53i	1.62213i	−0.584959	+	0.811063i	$0.698889\pi$
$992$	0	0
$993$	−1188.80	−1.19718
$994$	0	0
$995$	−1295.92	−1.30243
$996$	0	0
$997$	− 701.364i	− 0.703474i	−0.936099	−	0.351737i	$-0.885591\pi$
$997$	− 701.364i	− 0.703474i	0.936099	−	0.351737i	$-0.114409\pi$
$998$	0	0
$999$	− 1233.58i	− 1.23482i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1792.3.g.f.127.5		8
4.3	odd	2		1792.3.g.d.127.3		8
8.3	odd	2	inner	1792.3.g.f.127.6		8
8.5	even	2		1792.3.g.d.127.4		8
16.3	odd	4		448.3.d.e.127.3		8
16.5	even	4		224.3.d.b.127.3	✓	8
16.11	odd	4		224.3.d.b.127.6	yes	8
16.13	even	4		448.3.d.e.127.6		8
48.5	odd	4		2016.3.m.c.127.1		8
48.11	even	4		2016.3.m.c.127.2		8
112.27	even	4		1568.3.d.n.1471.3		8
112.69	odd	4		1568.3.d.n.1471.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.d.b.127.3	✓	8	16.5	even	4
224.3.d.b.127.6	yes	8	16.11	odd	4
448.3.d.e.127.3		8	16.3	odd	4
448.3.d.e.127.6		8	16.13	even	4
1568.3.d.n.1471.3		8	112.27	even	4
1568.3.d.n.1471.6		8	112.69	odd	4
1792.3.g.d.127.3		8	4.3	odd	2
1792.3.g.d.127.4		8	8.5	even	2
1792.3.g.f.127.5		8	1.1	even	1	trivial
1792.3.g.f.127.6		8	8.3	odd	2	inner
2016.3.m.c.127.1		8	48.5	odd	4
2016.3.m.c.127.2		8	48.11	even	4

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)

\(f(q)\)	\(=\)	\(q+2.55467 q^{3} -9.86836i q^{5} +2.64575i q^{7} -2.47367 q^{9} +O(q^{10})\)	\(q+2.55467 q^{3} -9.86836i q^{5} +2.64575i q^{7} -2.47367 q^{9} +13.1537 q^{11} -5.86836i q^{13} -25.2104i q^{15} -0.570700 q^{17} +15.6640 q^{19} +6.75902i q^{21} -16.4817i q^{23} -72.3844 q^{25} -29.3114 q^{27} -29.7367i q^{29} -54.8014i q^{31} +33.6034 q^{33} +26.1092 q^{35} +42.0853i q^{37} -14.9917i q^{39} -0.773275 q^{41} +41.7931 q^{43} +24.4110i q^{45} +58.4528i q^{47} -7.00000 q^{49} -1.45795 q^{51} -5.65139i q^{53} -129.805i q^{55} +40.0163 q^{57} +42.6434 q^{59} -95.9371i q^{61} -6.54471i q^{63} -57.9110 q^{65} -69.8503 q^{67} -42.1053i q^{69} +92.0882i q^{71} -9.97539 q^{73} -184.918 q^{75} +34.8014i q^{77} -20.1780i q^{79} -52.6180 q^{81} -151.307 q^{83} +5.63187i q^{85} -75.9674i q^{87} -5.79743 q^{89} +15.5262 q^{91} -140.000i q^{93} -154.578i q^{95} +103.696 q^{97} -32.5379 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{3} + 40 q^{9}+O(q^{10}) \) 8 * q + 8 * q^3 + 40 * q^9	\( 8 q + 8 q^{3} + 40 q^{9} + 32 q^{11} - 16 q^{17} + 88 q^{19} - 104 q^{25} + 176 q^{27} + 56 q^{35} - 144 q^{41} - 224 q^{43} - 56 q^{49} + 16 q^{51} + 400 q^{57} + 232 q^{59} - 304 q^{65} - 368 q^{67} - 272 q^{73} - 664 q^{75} + 504 q^{81} - 424 q^{83} + 80 q^{89} + 56 q^{91} + 528 q^{97} + 544 q^{99}+O(q^{100}) \) 8 * q + 8 * q^3 + 40 * q^9 + 32 * q^11 - 16 * q^17 + 88 * q^19 - 104 * q^25 + 176 * q^27 + 56 * q^35 - 144 * q^41 - 224 * q^43 - 56 * q^49 + 16 * q^51 + 400 * q^57 + 232 * q^59 - 304 * q^65 - 368 * q^67 - 272 * q^73 - 664 * q^75 + 504 * q^81 - 424 * q^83 + 80 * q^89 + 56 * q^91 + 528 * q^97 + 544 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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