LMFDB - Embedded newform 448.3.d.e.127.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [448,3,Mod(127,448)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(448, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("448.127");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 448 = 2^{6} \cdot 7 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	448.d (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:	$12.2071158433$
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^{14} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			127.3
Root			$-1.27733i$ of defining polynomial
Character	$\chi$	$=$	448.127
Dual form			448.3.d.e.127.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.55467i q^{3} -9.86836 q^{5} +2.64575i q^{7} +2.47367 q^{9} +O(q^{10})$	$q-2.55467i q^{3} -9.86836 q^{5} +2.64575i q^{7} +2.47367 q^{9} +13.1537i q^{11} +5.86836 q^{13} +25.2104i q^{15} -0.570700 q^{17} -15.6640i q^{19} +6.75902 q^{21} -16.4817i q^{23} +72.3844 q^{25} -29.3114i q^{27} +29.7367 q^{29} +54.8014i q^{31} +33.6034 q^{33} -26.1092i q^{35} +42.0853 q^{37} -14.9917i q^{39} +0.773275 q^{41} +41.7931i q^{43} -24.4110 q^{45} -58.4528i q^{47} -7.00000 q^{49} +1.45795i q^{51} -5.65139 q^{53} -129.805i q^{55} -40.0163 q^{57} +42.6434i q^{59} +95.9371 q^{61} +6.54471i q^{63} -57.9110 q^{65} +69.8503i q^{67} -42.1053 q^{69} +92.0882i q^{71} +9.97539 q^{73} -184.918i q^{75} -34.8014 q^{77} +20.1780i q^{79} -52.6180 q^{81} +151.307i q^{83} +5.63187 q^{85} -75.9674i q^{87} +5.79743 q^{89} +15.5262i q^{91} +140.000 q^{93} +154.578i q^{95} +103.696 q^{97} +32.5379i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 40 q^{9}+O(q^{10}) $ 8 * q - 40 * q^9	$ 8 q - 40 q^{9} - 32 q^{13} - 16 q^{17} + 104 q^{25} + 80 q^{29} + 176 q^{37} + 144 q^{41} - 256 q^{45} - 56 q^{49} - 48 q^{53} - 400 q^{57} + 192 q^{61} - 304 q^{65} - 576 q^{69} + 272 q^{73} + 112 q^{77} + 504 q^{81} + 160 q^{85} - 80 q^{89} + 608 q^{93} + 528 q^{97}+O(q^{100}) $ 8 * q - 40 * q^9 - 32 * q^13 - 16 * q^17 + 104 * q^25 + 80 * q^29 + 176 * q^37 + 144 * q^41 - 256 * q^45 - 56 * q^49 - 48 * q^53 - 400 * q^57 + 192 * q^61 - 304 * q^65 - 576 * q^69 + 272 * q^73 + 112 * q^77 + 504 * q^81 + 160 * q^85 - 80 * q^89 + 608 * q^93 + 528 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/448\mathbb{Z}\right)^\times$.

$n$	$127$	$129$	$197$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 2.55467i	− 0.851556i	−0.904828	−	0.425778i	$-0.860000\pi$
$3$	− 2.55467i	− 0.851556i	0.904828	−	0.425778i	$-0.140000\pi$
$4$	0	0
$5$	−9.86836	−1.97367	−0.986836	−	0.161727i	$-0.948294\pi$
$5$	−9.86836	−1.97367	−0.986836	+	0.161727i	$0.948294\pi$
$6$	0	0
$7$	2.64575i	0.377964i
$7$	2.64575i	0.377964i
$8$	0	0
$9$	2.47367	0.274852
$10$	0	0
$11$	13.1537i	1.19579i	0.801574	+	0.597896i	$0.203996\pi$
$11$	13.1537i	1.19579i	−0.801574	+	0.597896i	$0.796004\pi$
$12$	0	0
$13$	5.86836	0.451412	0.225706	−	0.974195i	$-0.427531\pi$
$13$	5.86836	0.451412	0.225706	+	0.974195i	$0.427531\pi$
$14$	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
$19$	− 15.6640i	− 0.824421i	−0.911089	−	0.412211i	$-0.864757\pi$
$19$	− 15.6640i	− 0.824421i	0.911089	−	0.412211i	$-0.135243\pi$
$20$	0	0
$21$	6.75902	0.321858
$22$	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
$27$	− 29.3114i	− 1.08561i
$28$	0	0
$29$	29.7367	1.02540	0.512702	−	0.858567i	$-0.328645\pi$
$29$	29.7367	1.02540	0.512702	+	0.858567i	$0.328645\pi$
$30$	0	0
$31$	54.8014i	1.76779i	0.467687	+	0.883894i	$0.345087\pi$
$31$	54.8014i	1.76779i	−0.467687	+	0.883894i	$0.654913\pi$
$32$	0	0
$33$	33.6034	1.01828
$34$	0	0
$35$	− 26.1092i	− 0.745978i
$36$	0	0
$37$	42.0853	1.13744	0.568721	−	0.822531i	$-0.307439\pi$
$37$	42.0853	1.13744	0.568721	+	0.822531i	$0.307439\pi$
$38$	0	0
$39$	− 14.9917i	− 0.384403i
$40$	0	0
$41$	0.773275	0.0188604	0.00943019	−	0.999956i	$-0.496998\pi$
$41$	0.773275	0.0188604	0.00943019	+	0.999956i	$0.496998\pi$
$42$	0	0
$43$	41.7931i	0.971933i	0.873977	+	0.485967i	$0.161532\pi$
$43$	41.7931i	0.971933i	−0.873977	+	0.485967i	$0.838468\pi$
$44$	0	0
$45$	−24.4110	−0.542467
$46$	0	0
$47$	− 58.4528i	− 1.24368i	−0.783145	−	0.621839i	$-0.786386\pi$
$47$	− 58.4528i	− 1.24368i	0.783145	−	0.621839i	$-0.213614\pi$
$48$	0	0
$49$	−7.00000	−0.142857
$50$	0	0
$51$	1.45795i	0.0285873i
$52$	0	0
$53$	−5.65139	−0.106630	−0.0533150	−	0.998578i	$-0.516979\pi$
$53$	−5.65139	−0.106630	−0.0533150	+	0.998578i	$0.516979\pi$
$54$	0	0
$55$	− 129.805i	− 2.36010i
$56$	0	0
$57$	−40.0163	−0.702041
$58$	0	0
$59$	42.6434i	0.722770i	0.932417	+	0.361385i	$0.117696\pi$
$59$	42.6434i	0.722770i	−0.932417	+	0.361385i	$0.882304\pi$
$60$	0	0
$61$	95.9371	1.57274	0.786370	−	0.617756i	$-0.211958\pi$
$61$	95.9371	1.57274	0.786370	+	0.617756i	$0.211958\pi$
$62$	0	0
$63$	6.54471i	0.103884i
$64$	0	0
$65$	−57.9110	−0.890939
$66$	0	0
$67$	69.8503i	1.04254i	0.853391	+	0.521271i	$0.174542\pi$
$67$	69.8503i	1.04254i	−0.853391	+	0.521271i	$0.825458\pi$
$68$	0	0
$69$	−42.1053	−0.610222
$70$	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	$-0.478235\pi$
$73$	9.97539	0.136649	0.0683246	+	0.997663i	$0.478235\pi$
$74$	0	0
$75$	− 184.918i	− 2.46558i
$76$	0	0
$77$	−34.8014	−0.451967
$78$	0	0
$79$	20.1780i	0.255417i	0.991812	+	0.127709i	$0.0407622\pi$
$79$	20.1780i	0.255417i	−0.991812	+	0.127709i	$0.959238\pi$
$80$	0	0
$81$	−52.6180	−0.649604
$82$	0	0
$83$	151.307i	1.82298i	0.411321	+	0.911491i	$0.365068\pi$
$83$	151.307i	1.82298i	−0.411321	+	0.911491i	$0.634932\pi$
$84$	0	0
$85$	5.63187	0.0662573
$86$	0	0
$87$	− 75.9674i	− 0.873189i
$88$	0	0
$89$	5.79743	0.0651396	0.0325698	−	0.999469i	$-0.489631\pi$
$89$	5.79743	0.0651396	0.0325698	+	0.999469i	$0.489631\pi$
$90$	0	0
$91$	15.5262i	0.170618i
$92$	0	0
$93$	140.000	1.50537
$94$	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
$99$	32.5379i	0.328666i
$100$	0	0
$101$	−41.9176	−0.415025	−0.207513	−	0.978232i	$-0.566537\pi$
$101$	−41.9176	−0.415025	−0.207513	+	0.978232i	$0.566537\pi$
$102$	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
$107$	− 64.8549i	− 0.606121i	−0.952971	−	0.303060i	$-0.901992\pi$
$107$	− 64.8549i	− 0.606121i	0.952971	−	0.303060i	$-0.0980084\pi$
$108$	0	0
$109$	186.432	1.71039	0.855195	−	0.518307i	$-0.173437\pi$
$109$	186.432	1.71039	0.855195	+	0.518307i	$0.173437\pi$
$110$	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
$115$	162.647i	1.41433i
$116$	0	0
$117$	14.5164	0.124071
$118$	0	0
$119$	− 1.50993i	− 0.0126885i
$120$	0	0
$121$	−52.0200	−0.429917
$122$	0	0
$123$	− 1.97546i	− 0.0160607i
$124$	0	0
$125$	−467.606	−3.74085
$126$	0	0
$127$	− 93.3874i	− 0.735334i	−0.929958	−	0.367667i	$-0.880157\pi$
$127$	− 93.3874i	− 0.735334i	0.929958	−	0.367667i	$-0.119843\pi$
$128$	0	0
$129$	106.768	0.827656
$130$	0	0
$131$	− 58.0857i	− 0.443403i	−0.975115	−	0.221701i	$-0.928839\pi$
$131$	− 58.0857i	− 0.443403i	0.975115	−	0.221701i	$-0.0711610\pi$
$132$	0	0
$133$	41.4431	0.311602
$134$	0	0
$135$	289.256i	2.14263i
$136$	0	0
$137$	176.420	1.28774	0.643870	−	0.765135i	$-0.277328\pi$
$137$	176.420	1.28774	0.643870	+	0.765135i	$0.277328\pi$
$138$	0	0
$139$	− 12.3041i	− 0.0885185i	−0.999020	−	0.0442592i	$-0.985907\pi$
$139$	− 12.3041i	− 0.0885185i	0.999020	−	0.0442592i	$-0.0140928\pi$
$140$	0	0
$141$	−149.328	−1.05906
$142$	0	0
$143$	77.1906i	0.539795i
$144$	0	0
$145$	−293.452	−2.02381
$146$	0	0
$147$	17.8827i	0.121651i
$148$	0	0
$149$	−173.782	−1.16632	−0.583160	−	0.812358i	$-0.698184\pi$
$149$	−173.782	−1.16632	−0.583160	+	0.812358i	$0.698184\pi$
$150$	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
$155$	− 540.800i	− 3.48903i
$156$	0	0
$157$	−189.693	−1.20824	−0.604119	−	0.796894i	$-0.706475\pi$
$157$	−189.693	−1.20824	−0.604119	+	0.796894i	$0.706475\pi$
$158$	0	0
$159$	14.4374i	0.0908014i
$160$	0	0
$161$	43.6065	0.270848
$162$	0	0
$163$	− 17.7685i	− 0.109009i	−0.998514	−	0.0545047i	$-0.982642\pi$
$163$	− 17.7685i	− 0.109009i	0.998514	−	0.0545047i	$-0.0173580\pi$
$164$	0	0
$165$	−331.610	−2.00976
$166$	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
$171$	− 38.7475i	− 0.226594i
$172$	0	0
$173$	−43.2891	−0.250226	−0.125113	−	0.992143i	$-0.539929\pi$
$173$	−43.2891	−0.250226	−0.125113	+	0.992143i	$0.539929\pi$
$174$	0	0
$175$	191.511i	1.09435i
$176$	0	0
$177$	108.940	0.615479
$178$	0	0
$179$	190.838i	1.06614i	0.846072	+	0.533068i	$0.178961\pi$
$179$	190.838i	1.06614i	−0.846072	+	0.533068i	$0.821039\pi$
$180$	0	0
$181$	207.153	1.14449	0.572246	−	0.820082i	$-0.306072\pi$
$181$	207.153	1.14449	0.572246	+	0.820082i	$0.306072\pi$
$182$	0	0
$183$	− 245.087i	− 1.33928i
$184$	0	0
$185$	−415.313	−2.24493
$186$	0	0
$187$	− 7.50683i	− 0.0401434i
$188$	0	0
$189$	77.5507	0.410321
$190$	0	0
$191$	315.489i	1.65178i	0.563833	+	0.825889i	$0.309326\pi$
$191$	315.489i	1.65178i	−0.563833	+	0.825889i	$0.690674\pi$
$192$	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
$195$	147.943i	0.758684i
$196$	0	0
$197$	264.248	1.34136	0.670681	−	0.741746i	$-0.266002\pi$
$197$	264.248	1.34136	0.670681	+	0.741746i	$0.266002\pi$
$198$	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
$203$	78.6759i	0.387566i
$204$	0	0
$205$	−7.63095	−0.0372242
$206$	0	0
$207$	− 40.7703i	− 0.196958i
$208$	0	0
$209$	206.040	0.985836
$210$	0	0
$211$	− 247.994i	− 1.17533i	−0.809105	−	0.587664i	$-0.800048\pi$
$211$	− 247.994i	− 1.17533i	0.809105	−	0.587664i	$-0.199952\pi$
$212$	0	0
$213$	235.255	1.10448
$214$	0	0
$215$	− 412.430i	− 1.91828i
$216$	0	0
$217$	−144.991	−0.668161
$218$	0	0
$219$	− 25.4838i	− 0.116364i
$220$	0	0
$221$	−3.34907	−0.0151542
$222$	0	0
$223$	56.7999i	0.254708i	0.991857	+	0.127354i	$0.0406484\pi$
$223$	56.7999i	0.254708i	−0.991857	+	0.127354i	$0.959352\pi$
$224$	0	0
$225$	179.055	0.795800
$226$	0	0
$227$	− 275.095i	− 1.21187i	−0.795513	−	0.605937i	$-0.792798\pi$
$227$	− 275.095i	− 1.21187i	0.795513	−	0.605937i	$-0.207202\pi$
$228$	0	0
$229$	−6.29210	−0.0274764	−0.0137382	−	0.999906i	$-0.504373\pi$
$229$	−6.29210	−0.0274764	−0.0137382	+	0.999906i	$0.504373\pi$
$230$	0	0
$231$	88.9061i	0.384875i
$232$	0	0
$233$	−142.049	−0.609653	−0.304827	−	0.952408i	$-0.598599\pi$
$233$	−142.049	−0.609653	−0.304827	+	0.952408i	$0.598599\pi$
$234$	0	0
$235$	576.833i	2.45461i
$236$	0	0
$237$	51.5480	0.217502
$238$	0	0
$239$	51.3954i	0.215043i	0.994203	+	0.107522i	$0.0342915\pi$
$239$	51.3954i	0.215043i	−0.994203	+	0.107522i	$0.965708\pi$
$240$	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
$243$	− 129.381i	− 0.532433i
$244$	0	0
$245$	69.0785	0.281953
$246$	0	0
$247$	− 91.9220i	− 0.372154i
$248$	0	0
$249$	386.540	1.55237
$250$	0	0
$251$	− 12.4430i	− 0.0495737i	−0.999693	−	0.0247869i	$-0.992109\pi$
$251$	− 12.4430i	− 0.0495737i	0.999693	−	0.0247869i	$-0.00789071\pi$
$252$	0	0
$253$	216.796	0.856900
$254$	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
$259$	111.347i	0.429912i
$260$	0	0
$261$	73.5587	0.281834
$262$	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
$267$	− 14.8105i	− 0.0554700i
$268$	0	0
$269$	9.37496	0.0348512	0.0174256	−	0.999848i	$-0.494453\pi$
$269$	9.37496	0.0348512	0.0174256	+	0.999848i	$0.494453\pi$
$270$	0	0
$271$	203.067i	0.749324i	0.927161	+	0.374662i	$0.122241\pi$
$271$	203.067i	0.749324i	−0.927161	+	0.374662i	$0.877759\pi$
$272$	0	0
$273$	39.6643	0.145291
$274$	0	0
$275$	952.124i	3.46227i
$276$	0	0
$277$	−154.419	−0.557471	−0.278735	−	0.960368i	$-0.589915\pi$
$277$	−154.419	−0.557471	−0.278735	+	0.960368i	$0.589915\pi$
$278$	0	0
$279$	135.561i	0.485880i
$280$	0	0
$281$	−217.495	−0.774005	−0.387002	−	0.922079i	$-0.626489\pi$
$281$	−217.495	−0.774005	−0.387002	+	0.922079i	$0.626489\pi$
$282$	0	0
$283$	431.849i	1.52597i	0.646418	+	0.762983i	$0.276266\pi$
$283$	431.849i	1.52597i	−0.646418	+	0.762983i	$0.723734\pi$
$284$	0	0
$285$	394.896	1.38560
$286$	0	0
$287$	2.04589i	0.00712855i
$288$	0	0
$289$	−288.674	−0.998873
$290$	0	0
$291$	− 264.908i	− 0.910336i
$292$	0	0
$293$	212.019	0.723616	0.361808	−	0.932253i	$-0.382160\pi$
$293$	212.019	0.723616	0.361808	+	0.932253i	$0.382160\pi$
$294$	0	0
$295$	− 420.820i	− 1.42651i
$296$	0	0
$297$	385.554	1.29816
$298$	0	0
$299$	− 96.7206i	− 0.323480i
$300$	0	0
$301$	−110.574	−0.367356
$302$	0	0
$303$	107.086i	0.353418i
$304$	0	0
$305$	−946.741	−3.10407
$306$	0	0
$307$	− 192.868i	− 0.628234i	−0.949384	−	0.314117i	$-0.898292\pi$
$307$	− 192.868i	− 0.628234i	0.949384	−	0.314117i	$-0.101708\pi$
$308$	0	0
$309$	−47.8689	−0.154915
$310$	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	$-0.623312\pi$
$313$	−236.490	−0.755558	−0.377779	+	0.925896i	$0.623312\pi$
$314$	0	0
$315$	− 64.5855i	− 0.205033i
$316$	0	0
$317$	249.012	0.785526	0.392763	−	0.919640i	$-0.371519\pi$
$317$	249.012	0.785526	0.392763	+	0.919640i	$0.371519\pi$
$318$	0	0
$319$	391.148i	1.22617i
$320$	0	0
$321$	−165.683	−0.516146
$322$	0	0
$323$	8.93945i	0.0276763i
$324$	0	0
$325$	424.778	1.30701
$326$	0	0
$327$	− 476.273i	− 1.45649i
$328$	0	0
$329$	154.652	0.470066
$330$	0	0
$331$	− 465.343i	− 1.40587i	−0.711254	−	0.702935i	$-0.751872\pi$
$331$	− 465.343i	− 1.40587i	0.711254	−	0.702935i	$-0.248128\pi$
$332$	0	0
$333$	104.105	0.312628
$334$	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
$339$	216.516i	0.638690i
$340$	0	0
$341$	−720.842	−2.11391
$342$	0	0
$343$	− 18.5203i	− 0.0539949i
$344$	0	0
$345$	415.510	1.20438
$346$	0	0
$347$	− 450.576i	− 1.29849i	−0.760580	−	0.649245i	$-0.775085\pi$
$347$	− 450.576i	− 1.29849i	0.760580	−	0.649245i	$-0.224915\pi$
$348$	0	0
$349$	64.7762	0.185605	0.0928025	−	0.995685i	$-0.470417\pi$
$349$	64.7762	0.185605	0.0928025	+	0.995685i	$0.470417\pi$
$350$	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
$355$	− 908.760i	− 2.55989i
$356$	0	0
$357$	−3.85737	−0.0108050
$358$	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
$363$	132.894i	0.366099i
$364$	0	0
$365$	−98.4407	−0.269701
$366$	0	0
$367$	430.196i	1.17220i	0.810240	+	0.586098i	$0.199337\pi$
$367$	430.196i	1.17220i	−0.810240	+	0.586098i	$0.800663\pi$
$368$	0	0
$369$	1.91283	0.00518381
$370$	0	0
$371$	− 14.9522i	− 0.0403023i
$372$	0	0
$373$	−2.60758	−0.00699083	−0.00349542	−	0.999994i	$-0.501113\pi$
$373$	−2.60758	−0.00699083	−0.00349542	+	0.999994i	$0.501113\pi$
$374$	0	0
$375$	1194.58i	3.18555i
$376$	0	0
$377$	174.506	0.462880
$378$	0	0
$379$	359.118i	0.947541i	0.880648	+	0.473771i	$0.157107\pi$
$379$	359.118i	0.947541i	−0.880648	+	0.473771i	$0.842893\pi$
$380$	0	0
$381$	−238.574	−0.626178
$382$	0	0
$383$	− 470.758i	− 1.22913i	−0.788865	−	0.614567i	$-0.789331\pi$
$383$	− 470.758i	− 1.22913i	0.788865	−	0.614567i	$-0.210669\pi$
$384$	0	0
$385$	343.433	0.892034
$386$	0	0
$387$	103.382i	0.267138i
$388$	0	0
$389$	180.270	0.463418	0.231709	−	0.972785i	$-0.425568\pi$
$389$	180.270	0.463418	0.231709	+	0.972785i	$0.425568\pi$
$390$	0	0
$391$	9.40612i	0.0240566i
$392$	0	0
$393$	−148.390	−0.377582
$394$	0	0
$395$	− 199.123i	− 0.504110i
$396$	0	0
$397$	248.223	0.625248	0.312624	−	0.949877i	$-0.398792\pi$
$397$	248.223	0.625248	0.312624	+	0.949877i	$0.398792\pi$
$398$	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
$403$	321.594i	0.798001i
$404$	0	0
$405$	519.253	1.28211
$406$	0	0
$407$	553.578i	1.36014i
$408$	0	0
$409$	479.494	1.17236	0.586179	−	0.810182i	$-0.300632\pi$
$409$	479.494	1.17236	0.586179	+	0.810182i	$0.300632\pi$
$410$	0	0
$411$	− 450.695i	− 1.09658i
$412$	0	0
$413$	−112.824	−0.273181
$414$	0	0
$415$	− 1493.16i	− 3.59797i
$416$	0	0
$417$	−31.4328	−0.0753785
$418$	0	0
$419$	− 564.644i	− 1.34760i	−0.738914	−	0.673800i	$-0.764661\pi$
$419$	− 564.644i	− 1.34760i	0.738914	−	0.673800i	$-0.235339\pi$
$420$	0	0
$421$	−544.282	−1.29283	−0.646415	−	0.762986i	$-0.723733\pi$
$421$	−544.282	−1.29283	−0.646415	+	0.762986i	$0.723733\pi$
$422$	0	0
$423$	− 144.593i	− 0.341827i
$424$	0	0
$425$	−41.3098	−0.0971996
$426$	0	0
$427$	253.826i	0.594440i
$428$	0	0
$429$	197.196	0.459665
$430$	0	0
$431$	90.3539i	0.209638i	0.994491	+	0.104819i	$0.0334263\pi$
$431$	90.3539i	0.209638i	−0.994491	+	0.104819i	$0.966574\pi$
$432$	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
$435$	749.674i	1.72339i
$436$	0	0
$437$	−258.170	−0.590777
$438$	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
$443$	193.210i	0.436139i	0.975933	+	0.218070i	$0.0699760\pi$
$443$	193.210i	0.436139i	−0.975933	+	0.218070i	$0.930024\pi$
$444$	0	0
$445$	−57.2111	−0.128564
$446$	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
$451$	10.1714i	0.0225531i
$452$	0	0
$453$	−353.651	−0.780686
$454$	0	0
$455$	− 153.218i	− 0.336743i
$456$	0	0
$457$	−426.401	−0.933043	−0.466521	−	0.884510i	$-0.654493\pi$
$457$	−426.401	−0.933043	−0.466521	+	0.884510i	$0.654493\pi$
$458$	0	0
$459$	16.7280i	0.0364445i
$460$	0	0
$461$	−430.182	−0.933150	−0.466575	−	0.884482i	$-0.654512\pi$
$461$	−430.182	−0.933150	−0.466575	+	0.884482i	$0.654512\pi$
$462$	0	0
$463$	183.454i	0.396230i	0.980179	+	0.198115i	$0.0634819\pi$
$463$	183.454i	0.396230i	−0.980179	+	0.198115i	$0.936518\pi$
$464$	0	0
$465$	−1381.56	−2.97111
$466$	0	0
$467$	− 110.381i	− 0.236362i	−0.992992	−	0.118181i	$-0.962294\pi$
$467$	− 110.381i	− 0.236362i	0.992992	−	0.118181i	$-0.0377063\pi$
$468$	0	0
$469$	−184.807	−0.394044
$470$	0	0
$471$	484.604i	1.02888i
$472$	0	0
$473$	−549.735	−1.16223
$474$	0	0
$475$	− 1133.83i	− 2.38701i
$476$	0	0
$477$	−13.9797	−0.0293075
$478$	0	0
$479$	515.593i	1.07639i	0.842819	+	0.538197i	$0.180894\pi$
$479$	515.593i	1.07639i	−0.842819	+	0.538197i	$0.819106\pi$
$480$	0	0
$481$	246.972	0.513455
$482$	0	0
$483$	− 111.400i	− 0.230642i
$484$	0	0
$485$	−1023.30	−2.10991
$486$	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
$491$	556.042i	1.13247i	0.824244	+	0.566234i	$0.191600\pi$
$491$	556.042i	1.13247i	−0.824244	+	0.566234i	$0.808400\pi$
$492$	0	0
$493$	−16.9708	−0.0344234
$494$	0	0
$495$	− 321.096i	− 0.648678i
$496$	0	0
$497$	−243.643	−0.490227
$498$	0	0
$499$	− 150.429i	− 0.301461i	−0.988575	−	0.150730i	$-0.951837\pi$
$499$	− 150.429i	− 0.301461i	0.988575	−	0.150730i	$-0.0481626\pi$
$500$	0	0
$501$	−0.227601	−0.000454293 0
$502$	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
$507$	343.762i	0.678032i
$508$	0	0
$509$	−622.234	−1.22246	−0.611232	−	0.791452i	$-0.709326\pi$
$509$	−622.234	−1.22246	−0.611232	+	0.791452i	$0.709326\pi$
$510$	0	0
$511$	26.3924i	0.0516485i
$512$	0	0
$513$	−459.134	−0.894999
$514$	0	0
$515$	184.911i	0.359051i
$516$	0	0
$517$	768.871	1.48718
$518$	0	0
$519$	110.589i	0.213081i
$520$	0	0
$521$	−644.142	−1.23636	−0.618179	−	0.786037i	$-0.712129\pi$
$521$	−644.142	−1.23636	−0.618179	+	0.786037i	$0.712129\pi$
$522$	0	0
$523$	889.589i	1.70093i	0.526028	+	0.850467i	$0.323681\pi$
$523$	889.589i	1.70093i	−0.526028	+	0.850467i	$0.676319\pi$
$524$	0	0
$525$	489.248	0.931900
$526$	0	0
$527$	− 31.2752i	− 0.0593457i
$528$	0	0
$529$	257.353	0.486490
$530$	0	0
$531$	105.486i	0.198655i
$532$	0	0
$533$	4.53785	0.00851380
$534$	0	0
$535$	640.012i	1.19628i
$536$	0	0
$537$	487.529	0.907875
$538$	0	0
$539$	− 92.0759i	− 0.170827i
$540$	0	0
$541$	−832.015	−1.53792	−0.768961	−	0.639296i	$-0.779226\pi$
$541$	−832.015	−1.53792	−0.768961	+	0.639296i	$0.779226\pi$
$542$	0	0
$543$	− 529.207i	− 0.974599i
$544$	0	0
$545$	−1839.78	−3.37575
$546$	0	0
$547$	− 1014.85i	− 1.85531i	−0.373441	−	0.927654i	$-0.621822\pi$
$547$	− 1014.85i	− 1.85531i	0.373441	−	0.927654i	$-0.378178\pi$
$548$	0	0
$549$	237.316	0.432270
$550$	0	0
$551$	− 465.796i	− 0.845365i
$552$	0	0
$553$	−53.3859	−0.0965387
$554$	0	0
$555$	1060.99i	1.91169i
$556$	0	0
$557$	−473.280	−0.849694	−0.424847	−	0.905265i	$-0.639672\pi$
$557$	−473.280	−0.849694	−0.424847	+	0.905265i	$0.639672\pi$
$558$	0	0
$559$	245.257i	0.438742i
$560$	0	0
$561$	−19.1775	−0.0341844
$562$	0	0
$563$	102.364i	0.181818i	0.995859	+	0.0909092i	$0.0289773\pi$
$563$	102.364i	0.181818i	−0.995859	+	0.0909092i	$0.971023\pi$
$564$	0	0
$565$	836.373	1.48031
$566$	0	0
$567$	− 139.214i	− 0.245527i
$568$	0	0
$569$	−68.4059	−0.120221	−0.0601106	−	0.998192i	$-0.519145\pi$
$569$	−68.4059	−0.120221	−0.0601106	+	0.998192i	$0.519145\pi$
$570$	0	0
$571$	701.419i	1.22841i	0.789148	+	0.614203i	$0.210522\pi$
$571$	701.419i	1.22841i	−0.789148	+	0.614203i	$0.789478\pi$
$572$	0	0
$573$	805.971	1.40658
$574$	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
$579$	79.0421i	0.136515i
$580$	0	0
$581$	−400.322	−0.689022
$582$	0	0
$583$	− 74.3367i	− 0.127507i
$584$	0	0
$585$	−143.253	−0.244876
$586$	0	0
$587$	− 99.0239i	− 0.168695i	−0.996436	−	0.0843475i	$-0.973119\pi$
$587$	− 99.0239i	− 0.168695i	0.996436	−	0.0843475i	$-0.0268806\pi$
$588$	0	0
$589$	858.410	1.45740
$590$	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
$595$	14.9005i	0.0250429i
$596$	0	0
$597$	−335.481	−0.561945
$598$	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	$-0.515371\pi$
$601$	−58.0231	−0.0965443	−0.0482722	+	0.998834i	$0.515371\pi$
$602$	0	0
$603$	172.786i	0.286545i
$604$	0	0
$605$	513.352	0.848515
$606$	0	0
$607$	− 67.3822i	− 0.111009i	−0.998458	−	0.0555043i	$-0.982323\pi$
$607$	− 67.3822i	− 0.111009i	0.998458	−	0.0555043i	$-0.0176767\pi$
$608$	0	0
$609$	200.991	0.330034
$610$	0	0
$611$	− 343.022i	− 0.561411i
$612$	0	0
$613$	521.917	0.851415	0.425707	−	0.904861i	$-0.360025\pi$
$613$	521.917	0.851415	0.425707	+	0.904861i	$0.360025\pi$
$614$	0	0
$615$	19.4946i	0.0316985i
$616$	0	0
$617$	165.257	0.267840	0.133920	−	0.990992i	$-0.457243\pi$
$617$	165.257	0.267840	0.133920	+	0.990992i	$0.457243\pi$
$618$	0	0
$619$	205.392i	0.331812i	0.986142	+	0.165906i	$0.0530549\pi$
$619$	205.392i	0.331812i	−0.986142	+	0.165906i	$0.946945\pi$
$620$	0	0
$621$	−483.103	−0.777943
$622$	0	0
$623$	15.3385i	0.0246205i
$624$	0	0
$625$	2804.90	4.48783
$626$	0	0
$627$	− 526.363i	− 0.839495i
$628$	0	0
$629$	−24.0181	−0.0381846
$630$	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
$635$	921.580i	1.45131i
$636$	0	0
$637$	−41.0785	−0.0644874
$638$	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
$643$	− 607.082i	− 0.944140i	−0.881561	−	0.472070i	$-0.843507\pi$
$643$	− 607.082i	− 0.944140i	0.881561	−	0.472070i	$-0.156493\pi$
$644$	0	0
$645$	−1053.62	−1.63352
$646$	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
$651$	370.404i	0.568977i
$652$	0	0
$653$	850.083	1.30181	0.650906	−	0.759159i	$-0.274389\pi$
$653$	850.083	1.30181	0.650906	+	0.759159i	$0.274389\pi$
$654$	0	0
$655$	573.211i	0.875131i
$656$	0	0
$657$	24.6758	0.0375583
$658$	0	0
$659$	331.172i	0.502537i	0.967917	+	0.251268i	$0.0808477\pi$
$659$	331.172i	0.502537i	−0.967917	+	0.251268i	$0.919152\pi$
$660$	0	0
$661$	−464.323	−0.702456	−0.351228	−	0.936290i	$-0.614236\pi$
$661$	−464.323	−0.702456	−0.351228	+	0.936290i	$0.614236\pi$
$662$	0	0
$663$	8.55577i	0.0129046i
$664$	0	0
$665$	−408.975	−0.615000
$666$	0	0
$667$	− 490.112i	− 0.734801i
$668$	0	0
$669$	145.105	0.216898
$670$	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
$675$	− 2121.69i	− 3.14325i
$676$	0	0
$677$	664.414	0.981410	0.490705	−	0.871326i	$-0.336739\pi$
$677$	664.414	0.981410	0.490705	+	0.871326i	$0.336739\pi$
$678$	0	0
$679$	274.353i	0.404054i
$680$	0	0
$681$	−702.778	−1.03198
$682$	0	0
$683$	− 601.901i	− 0.881261i	−0.897689	−	0.440631i	$-0.854755\pi$
$683$	− 601.901i	− 0.881261i	0.897689	−	0.440631i	$-0.145245\pi$
$684$	0	0
$685$	−1740.98	−2.54157
$686$	0	0
$687$	16.0742i	0.0233977i
$688$	0	0
$689$	−33.1643	−0.0481340
$690$	0	0
$691$	579.981i	0.839336i	0.907678	+	0.419668i	$0.137854\pi$
$691$	579.981i	0.839336i	−0.907678	+	0.419668i	$0.862146\pi$
$692$	0	0
$693$	−86.0872	−0.124224
$694$	0	0
$695$	121.421i	0.174706i
$696$	0	0
$697$	−0.441308	−0.000633154 0
$698$	0	0
$699$	362.889i	0.519154i
$700$	0	0
$701$	−863.561	−1.23190	−0.615949	−	0.787786i	$-0.711227\pi$
$701$	−863.561	−1.23190	−0.615949	+	0.787786i	$0.711227\pi$
$702$	0	0
$703$	− 659.225i	− 0.937731i
$704$	0	0
$705$	1473.62	2.09024
$706$	0	0
$707$	− 110.903i	− 0.156865i
$708$	0	0
$709$	293.022	0.413290	0.206645	−	0.978416i	$-0.433746\pi$
$709$	293.022	0.413290	0.206645	+	0.978416i	$0.433746\pi$
$710$	0	0
$711$	49.9136i	0.0702019i
$712$	0	0
$713$	903.222	1.26679
$714$	0	0
$715$	− 761.744i	− 1.06538i
$716$	0	0
$717$	131.298	0.183122
$718$	0	0
$719$	381.821i	0.531044i	0.964105	+	0.265522i	$0.0855443\pi$
$719$	381.821i	0.531044i	−0.964105	+	0.265522i	$0.914456\pi$
$720$	0	0
$721$	49.5756	0.0687594
$722$	0	0
$723$	− 239.385i	− 0.331099i
$724$	0	0
$725$	2152.48	2.96893
$726$	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
$731$	− 23.8514i	− 0.0326284i
$732$	0	0
$733$	−59.1034	−0.0806322	−0.0403161	−	0.999187i	$-0.512836\pi$
$733$	−59.1034	−0.0806322	−0.0403161	+	0.999187i	$0.512836\pi$
$734$	0	0
$735$	− 176.473i	− 0.240099i
$736$	0	0
$737$	−918.790	−1.24666
$738$	0	0
$739$	− 1285.33i	− 1.73928i	−0.493687	−	0.869640i	$-0.664351\pi$
$739$	− 1285.33i	− 1.73928i	0.493687	−	0.869640i	$-0.335649\pi$
$740$	0	0
$741$	−234.830	−0.316910
$742$	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
$747$	374.284i	0.501050i
$748$	0	0
$749$	171.590	0.229092
$750$	0	0
$751$	− 190.150i	− 0.253195i	−0.991954	−	0.126598i	$-0.959594\pi$
$751$	− 190.150i	− 0.253195i	0.991954	−	0.126598i	$-0.0404057\pi$
$752$	0	0
$753$	−31.7878	−0.0422148
$754$	0	0
$755$	1366.11i	1.80941i
$756$	0	0
$757$	−441.225	−0.582859	−0.291430	−	0.956592i	$-0.594131\pi$
$757$	−441.225	−0.582859	−0.291430	+	0.956592i	$0.594131\pi$
$758$	0	0
$759$	− 553.841i	− 0.729698i
$760$	0	0
$761$	−1044.28	−1.37225	−0.686123	−	0.727486i	$-0.740689\pi$
$761$	−1044.28	−1.37225	−0.686123	+	0.727486i	$0.740689\pi$
$762$	0	0
$763$	493.254i	0.646466i
$764$	0	0
$765$	13.9314	0.0182110
$766$	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
$771$	− 433.998i	− 0.562903i
$772$	0	0
$773$	211.566	0.273694	0.136847	−	0.990592i	$-0.456303\pi$
$773$	211.566	0.273694	0.136847	+	0.990592i	$0.456303\pi$
$774$	0	0
$775$	3966.77i	5.11841i
$776$	0	0
$777$	284.455	0.366095
$778$	0	0
$779$	− 12.1126i	− 0.0155489i
$780$	0	0
$781$	−1211.30	−1.55096
$782$	0	0
$783$	− 871.625i	− 1.11319i
$784$	0	0
$785$	1871.96	2.38466
$786$	0	0
$787$	523.213i	0.664820i	0.943135	+	0.332410i	$0.107862\pi$
$787$	523.213i	0.664820i	−0.943135	+	0.332410i	$0.892138\pi$
$788$	0	0
$789$	−303.396	−0.384533
$790$	0	0
$791$	− 224.235i	− 0.283483i
$792$	0	0
$793$	562.993	0.709953
$794$	0	0
$795$	− 142.474i	− 0.179212i
$796$	0	0
$797$	503.460	0.631693	0.315847	−	0.948810i	$-0.397711\pi$
$797$	503.460	0.631693	0.315847	+	0.948810i	$0.397711\pi$
$798$	0	0
$799$	33.3590i	0.0417510i
$800$	0	0
$801$	14.3409	0.0179038
$802$	0	0
$803$	131.213i	0.163404i
$804$	0	0
$805$	−430.325	−0.534565
$806$	0	0
$807$	− 23.9499i	− 0.0296777i
$808$	0	0
$809$	100.517	0.124248	0.0621242	−	0.998068i	$-0.480212\pi$
$809$	100.517	0.124248	0.0621242	+	0.998068i	$0.480212\pi$
$810$	0	0
$811$	759.830i	0.936905i	0.883489	+	0.468453i	$0.155188\pi$
$811$	759.830i	0.936905i	−0.883489	+	0.468453i	$0.844812\pi$
$812$	0	0
$813$	518.768	0.638091
$814$	0	0
$815$	175.346i	0.215149i
$816$	0	0
$817$	654.648	0.801283
$818$	0	0
$819$	38.4067i	0.0468946i
$820$	0	0
$821$	749.705	0.913160	0.456580	−	0.889682i	$-0.349074\pi$
$821$	749.705	0.913160	0.456580	+	0.889682i	$0.349074\pi$
$822$	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
$827$	− 575.272i	− 0.695613i	−0.937566	−	0.347806i	$-0.886927\pi$
$827$	− 575.272i	− 0.695613i	0.937566	−	0.347806i	$-0.113073\pi$
$828$	0	0
$829$	−157.635	−0.190151	−0.0950756	−	0.995470i	$-0.530309\pi$
$829$	−157.635	−0.190151	−0.0950756	+	0.995470i	$0.530309\pi$
$830$	0	0
$831$	394.490i	0.474718i
$832$	0	0
$833$	3.99490	0.00479580
$834$	0	0
$835$	0.879193i	0.00105293i
$836$	0	0
$837$	1606.31	1.91913
$838$	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
$843$	555.628i	0.659108i
$844$	0	0
$845$	1327.91	1.57149
$846$	0	0
$847$	− 137.632i	− 0.162493i
$848$	0	0
$849$	1103.23	1.29945
$850$	0	0
$851$	− 693.638i	− 0.815086i
$852$	0	0
$853$	−315.373	−0.369722	−0.184861	−	0.982765i	$-0.559183\pi$
$853$	−315.373	−0.369722	−0.184861	+	0.982765i	$0.559183\pi$
$854$	0	0
$855$	382.375i	0.447222i
$856$	0	0
$857$	1222.53	1.42652	0.713260	−	0.700900i	$-0.247218\pi$
$857$	1222.53	1.42652	0.713260	+	0.700900i	$0.247218\pi$
$858$	0	0
$859$	− 1008.23i	− 1.17373i	−0.809687	−	0.586863i	$-0.800363\pi$
$859$	− 1008.23i	− 1.17373i	0.809687	−	0.586863i	$-0.199637\pi$
$860$	0	0
$861$	5.22658	0.00607036
$862$	0	0
$863$	1140.31i	1.32133i	0.750679	+	0.660667i	$0.229726\pi$
$863$	1140.31i	1.32133i	−0.750679	+	0.660667i	$0.770274\pi$
$864$	0	0
$865$	427.192	0.493864
$866$	0	0
$867$	737.467i	0.850597i
$868$	0	0
$869$	−265.415	−0.305426
$870$	0	0
$871$	409.906i	0.470616i
$872$	0	0
$873$	256.508	0.293824
$874$	0	0
$875$	− 1237.17i	− 1.41391i
$876$	0	0
$877$	306.788	0.349816	0.174908	−	0.984585i	$-0.444037\pi$
$877$	306.788	0.349816	0.174908	+	0.984585i	$0.444037\pi$
$878$	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
$883$	458.136i	0.518841i	0.965765	+	0.259420i	$0.0835315\pi$
$883$	458.136i	0.518841i	−0.965765	+	0.259420i	$0.916469\pi$
$884$	0	0
$885$	−1075.06	−1.21475
$886$	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
$891$	− 692.121i	− 0.776791i
$892$	0	0
$893$	−915.605	−1.02531
$894$	0	0
$895$	− 1883.26i	− 2.10420i
$896$	0	0
$897$	−247.089	−0.275462
$898$	0	0
$899$	1629.61i	1.81270i
$900$	0	0
$901$	3.22525	0.00357963
$902$	0	0
$903$	282.481i	0.312825i
$904$	0	0
$905$	−2044.26	−2.25885
$906$	0	0
$907$	375.701i	0.414224i	0.978317	+	0.207112i	$0.0664064\pi$
$907$	375.701i	0.414224i	−0.978317	+	0.207112i	$0.933594\pi$
$908$	0	0
$909$	−103.690	−0.114071
$910$	0	0
$911$	− 41.9614i	− 0.0460608i	−0.999735	−	0.0230304i	$-0.992669\pi$
$911$	− 41.9614i	− 0.0460608i	0.999735	−	0.0230304i	$-0.00733145\pi$
$912$	0	0
$913$	−1990.25	−2.17991
$914$	0	0
$915$	2418.61i	2.64329i
$916$	0	0
$917$	153.680	0.167590
$918$	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
$923$	540.407i	0.585489i
$924$	0	0
$925$	3046.32	3.29332
$926$	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
$931$	109.648i	0.117774i
$932$	0	0
$933$	689.285	0.738784
$934$	0	0
$935$	74.0800i	0.0792300i
$936$	0	0
$937$	−183.028	−0.195334	−0.0976668	−	0.995219i	$-0.531138\pi$
$937$	−183.028	−0.195334	−0.0976668	+	0.995219i	$0.531138\pi$
$938$	0	0
$939$	604.152i	0.643400i
$940$	0	0
$941$	−255.475	−0.271493	−0.135747	−	0.990744i	$-0.543343\pi$
$941$	−255.475	−0.271493	−0.135747	+	0.990744i	$0.543343\pi$
$942$	0	0
$943$	− 12.7449i	− 0.0135153i
$944$	0	0
$945$	−765.298	−0.809839
$946$	0	0
$947$	− 187.173i	− 0.197648i	−0.995105	−	0.0988242i	$-0.968492\pi$
$947$	− 187.173i	− 0.197648i	0.995105	−	0.0988242i	$-0.0315082\pi$
$948$	0	0
$949$	58.5391	0.0616851
$950$	0	0
$951$	− 636.142i	− 0.668920i
$952$	0	0
$953$	362.350	0.380220	0.190110	−	0.981763i	$-0.439115\pi$
$953$	362.350	0.380220	0.190110	+	0.981763i	$0.439115\pi$
$954$	0	0
$955$	− 3113.36i	− 3.26007i
$956$	0	0
$957$	999.253	1.04415
$958$	0	0
$959$	466.764i	0.486720i
$960$	0	0
$961$	−2042.20	−2.12508
$962$	0	0
$963$	− 160.430i	− 0.166594i
$964$	0	0
$965$	305.330	0.316404
$966$	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
$971$	− 223.833i	− 0.230518i	−0.993335	−	0.115259i	$-0.963230\pi$
$971$	− 223.833i	− 0.230518i	0.993335	−	0.115259i	$-0.0367698\pi$
$972$	0	0
$973$	32.5535	0.0334568
$974$	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
$979$	76.2576i	0.0778934i
$980$	0	0
$981$	461.172	0.470104
$982$	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
$987$	− 395.084i	− 0.400287i
$988$	0	0
$989$	688.823	0.696484
$990$	0	0
$991$	− 1607.53i	− 1.62213i	−0.584959	−	0.811063i	$-0.698889\pi$
$991$	− 1607.53i	− 1.62213i	0.584959	−	0.811063i	$-0.301111\pi$
$992$	0	0
$993$	−1188.80	−1.19718
$994$	0	0
$995$	1295.92i	1.30243i
$996$	0	0
$997$	−701.364	−0.703474	−0.351737	−	0.936099i	$-0.614409\pi$
$997$	−701.364	−0.703474	−0.351737	+	0.936099i	$0.614409\pi$
$998$	0	0
$999$	− 1233.58i	− 1.23482i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.d.b.127.3	✓	8	8.3	odd	2
224.3.d.b.127.6	yes	8	8.5	even	2
448.3.d.e.127.3		8	1.1	even	1	trivial
448.3.d.e.127.6		8	4.3	odd	2	inner
1568.3.d.n.1471.3		8	56.13	odd	2
1568.3.d.n.1471.6		8	56.27	even	2
1792.3.g.d.127.3		8	16.5	even	4
1792.3.g.d.127.4		8	16.3	odd	4
1792.3.g.f.127.5		8	16.11	odd	4
1792.3.g.f.127.6		8	16.13	even	4
2016.3.m.c.127.1		8	24.11	even	2
2016.3.m.c.127.2		8	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 448 = 2^{6} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	448.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.55467i q^{3} -9.86836 q^{5} +2.64575i q^{7} +2.47367 q^{9} +O(q^{10})\)	\(q-2.55467i q^{3} -9.86836 q^{5} +2.64575i q^{7} +2.47367 q^{9} +13.1537i q^{11} +5.86836 q^{13} +25.2104i q^{15} -0.570700 q^{17} -15.6640i q^{19} +6.75902 q^{21} -16.4817i q^{23} +72.3844 q^{25} -29.3114i q^{27} +29.7367 q^{29} +54.8014i q^{31} +33.6034 q^{33} -26.1092i q^{35} +42.0853 q^{37} -14.9917i q^{39} +0.773275 q^{41} +41.7931i q^{43} -24.4110 q^{45} -58.4528i q^{47} -7.00000 q^{49} +1.45795i q^{51} -5.65139 q^{53} -129.805i q^{55} -40.0163 q^{57} +42.6434i q^{59} +95.9371 q^{61} +6.54471i q^{63} -57.9110 q^{65} +69.8503i q^{67} -42.1053 q^{69} +92.0882i q^{71} +9.97539 q^{73} -184.918i q^{75} -34.8014 q^{77} +20.1780i q^{79} -52.6180 q^{81} +151.307i q^{83} +5.63187 q^{85} -75.9674i q^{87} +5.79743 q^{89} +15.5262i q^{91} +140.000 q^{93} +154.578i q^{95} +103.696 q^{97} +32.5379i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 40 q^{9}+O(q^{10}) \) 8 * q - 40 * q^9	\( 8 q - 40 q^{9} - 32 q^{13} - 16 q^{17} + 104 q^{25} + 80 q^{29} + 176 q^{37} + 144 q^{41} - 256 q^{45} - 56 q^{49} - 48 q^{53} - 400 q^{57} + 192 q^{61} - 304 q^{65} - 576 q^{69} + 272 q^{73} + 112 q^{77} + 504 q^{81} + 160 q^{85} - 80 q^{89} + 608 q^{93} + 528 q^{97}+O(q^{100}) \) 8 * q - 40 * q^9 - 32 * q^13 - 16 * q^17 + 104 * q^25 + 80 * q^29 + 176 * q^37 + 144 * q^41 - 256 * q^45 - 56 * q^49 - 48 * q^53 - 400 * q^57 + 192 * q^61 - 304 * q^65 - 576 * q^69 + 272 * q^73 + 112 * q^77 + 504 * q^81 + 160 * q^85 - 80 * q^89 + 608 * q^93 + 528 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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