LMFDB - Embedded newform 1568.3.d.n.1471.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,3,Mod(1471,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, 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("1568.1471");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1568.d (of order $2$, degree $1$, 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:	$42.7249054517$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1471.6
Root			$1.27733i$ of defining polynomial
Character	$\chi$	$=$	1568.1471
Dual form			1568.3.d.n.1471.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.55467i q^{3} -9.86836 q^{5} +2.47367 q^{9} +O(q^{10})$	$q+2.55467i q^{3} -9.86836 q^{5} +2.47367 q^{9} +13.1537i q^{11} +5.86836 q^{13} -25.2104i q^{15} +0.570700 q^{17} +15.6640i q^{19} +16.4817i q^{23} +72.3844 q^{25} +29.3114i q^{27} -29.7367 q^{29} +54.8014i q^{31} -33.6034 q^{33} -42.0853 q^{37} +14.9917i q^{39} -0.773275 q^{41} +41.7931i q^{43} -24.4110 q^{45} -58.4528i q^{47} +1.45795i q^{51} +5.65139 q^{53} -129.805i q^{55} -40.0163 q^{57} -42.6434i q^{59} +95.9371 q^{61} -57.9110 q^{65} +69.8503i q^{67} -42.1053 q^{69} -92.0882i q^{71} -9.97539 q^{73} +184.918i q^{75} -20.1780i q^{79} -52.6180 q^{81} -151.307i q^{83} -5.63187 q^{85} -75.9674i q^{87} -5.79743 q^{89} -140.000 q^{93} -154.578i q^{95} -103.696 q^{97} +32.5379i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 48 q^{53} - 400 q^{57} + 192 q^{61} - 304 q^{65} - 576 q^{69} - 272 q^{73} + 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 + 48 * q^53 - 400 * q^57 + 192 * q^61 - 304 * q^65 - 576 * q^69 - 272 * q^73 + 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 $\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$1$	$-1$	$1$

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	$0.140000\pi$
$3$	2.55467i	0.851556i	−0.904828	+	0.425778i	$0.860000\pi$
$4$	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
$7$	0	0
$7$	0	0
$8$	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	$-0.494657\pi$
$17$	0.570700	0.0335706	0.0167853	+	0.999859i	$0.494657\pi$
$18$	0	0
$19$	15.6640i	0.824421i	0.911089	+	0.412211i	$0.135243\pi$
$19$	15.6640i	0.824421i	−0.911089	+	0.412211i	$0.864757\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	16.4817i	0.716596i	0.933607	+	0.358298i	$0.116643\pi$
$23$	16.4817i	0.716596i	−0.933607	+	0.358298i	$0.883357\pi$
$24$	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	$-0.671355\pi$
$29$	−29.7367	−1.02540	−0.512702	+	0.858567i	$0.671355\pi$
$30$	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
$35$	0	0
$36$	0	0
$37$	−42.0853	−1.13744	−0.568721	−	0.822531i	$-0.692561\pi$
$37$	−42.0853	−1.13744	−0.568721	+	0.822531i	$0.692561\pi$
$38$	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
$43$	41.7931i	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
$49$	0	0
$50$	0	0
$51$	1.45795i	0.0285873i
$52$	0	0
$53$	5.65139	0.106630	0.0533150	−	0.998578i	$-0.483021\pi$
$53$	5.65139	0.106630	0.0533150	+	0.998578i	$0.483021\pi$
$54$	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	$-0.882304\pi$
$59$	− 42.6434i	− 0.722770i	0.932417	−	0.361385i	$-0.117696\pi$
$60$	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
$63$	0	0
$64$	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	$-0.775393\pi$
$71$	− 92.0882i	− 1.29702i	0.761207	−	0.648509i	$-0.224607\pi$
$72$	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
$75$	184.918i	2.46558i
$76$	0	0
$77$	0	0
$78$	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
$83$	− 151.307i	− 1.82298i	−0.411321	−	0.911491i	$-0.634932\pi$
$83$	− 151.307i	− 1.82298i	0.411321	−	0.911491i	$-0.365068\pi$
$84$	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	$-0.510369\pi$
$89$	−5.79743	−0.0651396	−0.0325698	+	0.999469i	$0.510369\pi$
$90$	0	0
$91$	0	0
$92$	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	$-0.679505\pi$
$97$	−103.696	−1.06903	−0.534513	+	0.845160i	$0.679505\pi$
$98$	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
$105$	0	0
$106$	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	$-0.826563\pi$
$109$	−186.432	−1.71039	−0.855195	+	0.518307i	$0.826563\pi$
$110$	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
$119$	0	0
$120$	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	$0.119843\pi$
$127$	93.3874i	0.735334i	−0.929958	+	0.367667i	$0.880157\pi$
$128$	0	0
$129$	−106.768	−0.827656
$130$	0	0
$131$	58.0857i	0.443403i	0.975115	+	0.221701i	$0.0711610\pi$
$131$	58.0857i	0.443403i	−0.975115	+	0.221701i	$0.928839\pi$
$132$	0	0
$133$	0	0
$134$	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	$0.0140928\pi$
$139$	12.3041i	0.0885185i	−0.999020	+	0.0442592i	$0.985907\pi$
$140$	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
$147$	0	0
$148$	0	0
$149$	173.782	1.16632	0.583160	−	0.812358i	$-0.301816\pi$
$149$	173.782	1.16632	0.583160	+	0.812358i	$0.301816\pi$
$150$	0	0
$151$	138.433i	0.916776i	0.888752	+	0.458388i	$0.151573\pi$
$151$	138.433i	0.916776i	−0.888752	+	0.458388i	$0.848427\pi$
$152$	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
$161$	0	0
$162$	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
$175$	0	0
$176$	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
$189$	0	0
$190$	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
$195$	− 147.943i	− 0.758684i
$196$	0	0
$197$	−264.248	−1.34136	−0.670681	−	0.741746i	$-0.733998\pi$
$197$	−264.248	−1.34136	−0.670681	+	0.741746i	$0.733998\pi$
$198$	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
$203$	0	0
$204$	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
$217$	0	0
$218$	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	$0.207202\pi$
$227$	275.095i	1.21187i	−0.795513	+	0.605937i	$0.792798\pi$
$228$	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
$231$	0	0
$232$	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	$-0.965708\pi$
$239$	− 51.3954i	− 0.215043i	0.994203	−	0.107522i	$-0.0342915\pi$
$240$	0	0
$241$	−93.7048	−0.388817	−0.194408	−	0.980921i	$-0.562279\pi$
$241$	−93.7048	−0.388817	−0.194408	+	0.980921i	$0.562279\pi$
$242$	0	0
$243$	129.381i	0.532433i
$244$	0	0
$245$	0	0
$246$	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	$0.00789071\pi$
$251$	12.4430i	0.0495737i	−0.999693	+	0.0247869i	$0.992109\pi$
$252$	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	$-0.607222\pi$
$257$	−169.884	−0.661029	−0.330514	+	0.943801i	$0.607222\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−73.5587	−0.281834
$262$	0	0
$263$	118.762i	0.451565i	0.974178	+	0.225782i	$0.0724938\pi$
$263$	118.762i	0.451565i	−0.974178	+	0.225782i	$0.927506\pi$
$264$	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
$273$	0	0
$274$	0	0
$275$	952.124i	3.46227i
$276$	0	0
$277$	154.419	0.557471	0.278735	−	0.960368i	$-0.410085\pi$
$277$	154.419	0.557471	0.278735	+	0.960368i	$0.410085\pi$
$278$	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	$-0.723734\pi$
$283$	− 431.849i	− 1.52597i	0.646418	−	0.762983i	$-0.276266\pi$
$284$	0	0
$285$	394.896	1.38560
$286$	0	0
$287$	0	0
$288$	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
$301$	0	0
$302$	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	$0.101708\pi$
$307$	192.868i	0.628234i	−0.949384	+	0.314117i	$0.898292\pi$
$308$	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	$-0.376688\pi$
$313$	236.490	0.755558	0.377779	+	0.925896i	$0.376688\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−249.012	−0.785526	−0.392763	−	0.919640i	$-0.628481\pi$
$317$	−249.012	−0.785526	−0.392763	+	0.919640i	$0.628481\pi$
$318$	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
$329$	0	0
$330$	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
$343$	0	0
$344$	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	$-0.814262\pi$
$353$	−589.179	−1.66906	−0.834532	+	0.550960i	$0.814262\pi$
$354$	0	0
$355$	908.760i	2.55989i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 406.643i	− 1.13271i	−0.824161	−	0.566355i	$-0.808353\pi$
$359$	− 406.643i	− 1.13271i	0.824161	−	0.566355i	$-0.191647\pi$
$360$	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
$371$	0	0
$372$	0	0
$373$	2.60758	0.00699083	0.00349542	−	0.999994i	$-0.498887\pi$
$373$	2.60758	0.00699083	0.00349542	+	0.999994i	$0.498887\pi$
$374$	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
$385$	0	0
$386$	0	0
$387$	103.382i	0.267138i
$388$	0	0
$389$	−180.270	−0.463418	−0.231709	−	0.972785i	$-0.574432\pi$
$389$	−180.270	−0.463418	−0.231709	+	0.972785i	$0.574432\pi$
$390$	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
$399$	0	0
$400$	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	$-0.699368\pi$
$409$	−479.494	−1.17236	−0.586179	+	0.810182i	$0.699368\pi$
$410$	0	0
$411$	450.695i	1.09658i
$412$	0	0
$413$	0	0
$414$	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	$0.235339\pi$
$419$	564.644i	1.34760i	−0.738914	+	0.673800i	$0.764661\pi$
$420$	0	0
$421$	544.282	1.29283	0.646415	−	0.762986i	$-0.276267\pi$
$421$	544.282	1.29283	0.646415	+	0.762986i	$0.276267\pi$
$422$	0	0
$423$	− 144.593i	− 0.341827i
$424$	0	0
$425$	41.3098	0.0971996
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−197.196	−0.459665
$430$	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	$-0.451521\pi$
$433$	131.383	0.303425	0.151713	+	0.988425i	$0.451521\pi$
$434$	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
$441$	0	0
$442$	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
$455$	0	0
$456$	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	$-0.936518\pi$
$463$	− 183.454i	− 0.396230i	0.980179	−	0.198115i	$-0.0634819\pi$
$464$	0	0
$465$	1381.56	2.97111
$466$	0	0
$467$	110.381i	0.236362i	0.992992	+	0.118181i	$0.0377063\pi$
$467$	110.381i	0.236362i	−0.992992	+	0.118181i	$0.962294\pi$
$468$	0	0
$469$	0	0
$470$	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
$483$	0	0
$484$	0	0
$485$	1023.30	2.10991
$486$	0	0
$487$	59.2995i	0.121765i	0.998145	+	0.0608824i	$0.0193915\pi$
$487$	59.2995i	0.121765i	−0.998145	+	0.0608824i	$0.980609\pi$
$488$	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
$497$	0	0
$498$	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
$511$	0	0
$512$	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	$-0.287871\pi$
$521$	644.142	1.23636	0.618179	+	0.786037i	$0.287871\pi$
$522$	0	0
$523$	− 889.589i	− 1.70093i	−0.526028	−	0.850467i	$-0.676319\pi$
$523$	− 889.589i	− 1.70093i	0.526028	−	0.850467i	$-0.323681\pi$
$524$	0	0
$525$	0	0
$526$	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
$539$	0	0
$540$	0	0
$541$	832.015	1.53792	0.768961	−	0.639296i	$-0.220774\pi$
$541$	832.015	1.53792	0.768961	+	0.639296i	$0.220774\pi$
$542$	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
$553$	0	0
$554$	0	0
$555$	1060.99i	1.91169i
$556$	0	0
$557$	473.280	0.849694	0.424847	−	0.905265i	$-0.360328\pi$
$557$	473.280	0.849694	0.424847	+	0.905265i	$0.360328\pi$
$558$	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	$-0.971023\pi$
$563$	− 102.364i	− 0.181818i	0.995859	−	0.0909092i	$-0.0289773\pi$
$564$	0	0
$565$	836.373	1.48031
$566$	0	0
$567$	0	0
$568$	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	$-0.512270\pi$
$577$	−44.4710	−0.0770727	−0.0385364	+	0.999257i	$0.512270\pi$
$578$	0	0
$579$	− 79.0421i	− 0.136515i
$580$	0	0
$581$	0	0
$582$	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	$0.0268806\pi$
$587$	99.0239i	0.168695i	−0.996436	+	0.0843475i	$0.973119\pi$
$588$	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	$-0.377897\pi$
$593$	443.870	0.748516	0.374258	+	0.927325i	$0.377897\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	335.481	0.561945
$598$	0	0
$599$	969.972i	1.61932i	0.586900	+	0.809659i	$0.300348\pi$
$599$	969.972i	1.61932i	−0.586900	+	0.809659i	$0.699652\pi$
$600$	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
$603$	172.786i	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
$609$	0	0
$610$	0	0
$611$	− 343.022i	− 0.561411i
$612$	0	0
$613$	−521.917	−0.851415	−0.425707	−	0.904861i	$-0.639975\pi$
$613$	−521.917	−0.851415	−0.425707	+	0.904861i	$0.639975\pi$
$614$	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	$-0.946945\pi$
$619$	− 205.392i	− 0.331812i	0.986142	−	0.165906i	$-0.0530549\pi$
$620$	0	0
$621$	−483.103	−0.777943
$622$	0	0
$623$	0	0
$624$	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	$-0.833135\pi$
$631$	− 631.682i	− 1.00108i	0.865713	−	0.500540i	$-0.166865\pi$
$632$	0	0
$633$	633.543	1.00086
$634$	0	0
$635$	− 921.580i	− 1.45131i
$636$	0	0
$637$	0	0
$638$	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	$0.156493\pi$
$643$	607.082i	0.944140i	−0.881561	+	0.472070i	$0.843507\pi$
$644$	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
$651$	0	0
$652$	0	0
$653$	−850.083	−1.30181	−0.650906	−	0.759159i	$-0.725611\pi$
$653$	−850.083	−1.30181	−0.650906	+	0.759159i	$0.725611\pi$
$654$	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
$665$	0	0
$666$	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
$679$	0	0
$680$	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	$-0.862146\pi$
$691$	− 579.981i	− 0.839336i	0.907678	−	0.419668i	$-0.137854\pi$
$692$	0	0
$693$	0	0
$694$	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	$-0.288773\pi$
$701$	863.561	1.23190	0.615949	+	0.787786i	$0.288773\pi$
$702$	0	0
$703$	− 659.225i	− 0.937731i
$704$	0	0
$705$	−1473.62	−2.09024
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−293.022	−0.413290	−0.206645	−	0.978416i	$-0.566254\pi$
$709$	−293.022	−0.413290	−0.206645	+	0.978416i	$0.566254\pi$
$710$	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
$721$	0	0
$722$	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
$735$	0	0
$736$	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	$0.430001\pi$
$743$	1450.21i	1.95183i	−0.218141	+	0.975917i	$0.569999\pi$
$744$	0	0
$745$	−1714.94	−2.30193
$746$	0	0
$747$	− 374.284i	− 0.501050i
$748$	0	0
$749$	0	0
$750$	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
$755$	− 1366.11i	− 1.80941i
$756$	0	0
$757$	441.225	0.582859	0.291430	−	0.956592i	$-0.405869\pi$
$757$	441.225	0.582859	0.291430	+	0.956592i	$0.405869\pi$
$758$	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
$763$	0	0
$764$	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	$-0.233753\pi$
$769$	1141.60	1.48452	0.742262	+	0.670110i	$0.233753\pi$
$770$	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
$777$	0	0
$778$	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	$-0.892138\pi$
$787$	− 523.213i	− 0.664820i	0.943135	−	0.332410i	$-0.107862\pi$
$788$	0	0
$789$	−303.396	−0.384533
$790$	0	0
$791$	0	0
$792$	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
$805$	0	0
$806$	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	$-0.844812\pi$
$811$	− 759.830i	− 0.936905i	0.883489	−	0.468453i	$-0.155188\pi$
$812$	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
$819$	0	0
$820$	0	0
$821$	−749.705	−0.913160	−0.456580	−	0.889682i	$-0.650926\pi$
$821$	−749.705	−0.913160	−0.456580	+	0.889682i	$0.650926\pi$
$822$	0	0
$823$	424.800i	0.516160i	0.966124	+	0.258080i	$0.0830899\pi$
$823$	424.800i	0.516160i	−0.966124	+	0.258080i	$0.916910\pi$
$824$	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
$833$	0	0
$834$	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
$847$	0	0
$848$	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	$-0.752782\pi$
$857$	−1222.53	−1.42652	−0.713260	+	0.700900i	$0.752782\pi$
$858$	0	0
$859$	1008.23i	1.17373i	0.809687	+	0.586863i	$0.199637\pi$
$859$	1008.23i	1.17373i	−0.809687	+	0.586863i	$0.800363\pi$
$860$	0	0
$861$	0	0
$862$	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
$867$	− 737.467i	− 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
$875$	0	0
$876$	0	0
$877$	−306.788	−0.349816	−0.174908	−	0.984585i	$-0.555963\pi$
$877$	−306.788	−0.349816	−0.174908	+	0.984585i	$0.555963\pi$
$878$	0	0
$879$	541.640i	0.616200i
$880$	0	0
$881$	248.968	0.282597	0.141299	−	0.989967i	$-0.454872\pi$
$881$	248.968	0.282597	0.141299	+	0.989967i	$0.454872\pi$
$882$	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
$889$	0	0
$890$	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
$903$	0	0
$904$	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	$0.00733145\pi$
$911$	41.9614i	0.0460608i	−0.999735	+	0.0230304i	$0.992669\pi$
$912$	0	0
$913$	1990.25	2.17991
$914$	0	0
$915$	− 2418.61i	− 2.64329i
$916$	0	0
$917$	0	0
$918$	0	0
$919$	1770.71i	1.92678i	0.268101	+	0.963391i	$0.413604\pi$
$919$	1770.71i	1.92678i	−0.268101	+	0.963391i	$0.586396\pi$
$920$	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	$-0.854910\pi$
$929$	−1668.30	−1.79581	−0.897903	+	0.440193i	$0.854910\pi$
$930$	0	0
$931$	0	0
$932$	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	$-0.468862\pi$
$937$	183.028	0.195334	0.0976668	+	0.995219i	$0.468862\pi$
$938$	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
$945$	0	0
$946$	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
$959$	0	0
$960$	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	$-0.724204\pi$
$967$	− 1473.76i	− 1.52406i	0.647545	−	0.762028i	$-0.275796\pi$
$968$	0	0
$969$	−22.8373	−0.0235680
$970$	0	0
$971$	223.833i	0.230518i	0.993335	+	0.115259i	$0.0367698\pi$
$971$	223.833i	0.230518i	−0.993335	+	0.115259i	$0.963230\pi$
$972$	0	0
$973$	0	0
$974$	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
$987$	0	0
$988$	0	0
$989$	−688.823	−0.696484
$990$	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
$995$	1295.92i	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	1568.3.d.n.1471.6		8
4.3	odd	2	inner	1568.3.d.n.1471.3		8
7.6	odd	2		224.3.d.b.127.3	✓	8
21.20	even	2		2016.3.m.c.127.1		8
28.27	even	2		224.3.d.b.127.6	yes	8
56.13	odd	2		448.3.d.e.127.6		8
56.27	even	2		448.3.d.e.127.3		8
84.83	odd	2		2016.3.m.c.127.2		8
112.13	odd	4		1792.3.g.f.127.5		8
112.27	even	4		1792.3.g.f.127.6		8
112.69	odd	4		1792.3.g.d.127.4		8
112.83	even	4		1792.3.g.d.127.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
224.3.d.b.127.3	✓	8	7.6	odd	2
224.3.d.b.127.6	yes	8	28.27	even	2
448.3.d.e.127.3		8	56.27	even	2
448.3.d.e.127.6		8	56.13	odd	2
1568.3.d.n.1471.3		8	4.3	odd	2	inner
1568.3.d.n.1471.6		8	1.1	even	1	trivial
1792.3.g.d.127.3		8	112.83	even	4
1792.3.g.d.127.4		8	112.69	odd	4
1792.3.g.f.127.5		8	112.13	odd	4
1792.3.g.f.127.6		8	112.27	even	4
2016.3.m.c.127.1		8	21.20	even	2
2016.3.m.c.127.2		8	84.83	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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1568.d (of order \(2\), degree \(1\), minimal)

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

Introduction

L-functions

Modular forms

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Database

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