LMFDB - Embedded newform 3200.2.d.o.1601.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3200,2,Mod(1601,3200)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3200, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("3200.1601");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3200 = 2^{7} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3200.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:	$25.5521286468$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 3x^{2} + 1 $ x^4 + 3*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1601.2
Root			$0.618034i$ of defining polynomial
Character	$\chi$	$=$	3200.1601
Dual form			3200.2.d.o.1601.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.23607i q^{3} -2.00000 q^{9} +O(q^{10})$	$q-2.23607i q^{3} -2.00000 q^{9} +2.23607i q^{11} +4.00000i q^{13} -3.00000 q^{17} -2.23607i q^{19} +8.94427 q^{23} -2.23607i q^{27} -4.00000i q^{29} -8.94427 q^{31} +5.00000 q^{33} -8.00000i q^{37} +8.94427 q^{39} -5.00000 q^{41} -8.94427i q^{43} +8.94427 q^{47} -7.00000 q^{49} +6.70820i q^{51} -4.00000i q^{53} -5.00000 q^{57} -8.94427i q^{59} -8.00000i q^{61} -6.70820i q^{67} -20.0000i q^{69} +8.94427 q^{71} +9.00000 q^{73} -11.0000 q^{81} -6.70820i q^{83} -8.94427 q^{87} -15.0000 q^{89} +20.0000i q^{93} -2.00000 q^{97} -4.47214i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 8 q^{9}+O(q^{10}) $ 4 * q - 8 * q^9	$ 4 q - 8 q^{9} - 12 q^{17} + 20 q^{33} - 20 q^{41} - 28 q^{49} - 20 q^{57} + 36 q^{73} - 44 q^{81} - 60 q^{89} - 8 q^{97}+O(q^{100}) $ 4 * q - 8 * q^9 - 12 * q^17 + 20 * q^33 - 20 * q^41 - 28 * q^49 - 20 * q^57 + 36 * q^73 - 44 * q^81 - 60 * q^89 - 8 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3200\mathbb{Z}\right)^\times$.

$n$	$901$	$1151$	$2177$
$\chi(n)$	$-1$	$1$	$1$

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
$3$	− 2.23607i	− 1.29099i	−0.763763	−	0.645497i	$-0.776650\pi$
$3$	− 2.23607i	− 1.29099i	0.763763	−	0.645497i	$-0.223350\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	−2.00000	−0.666667
$10$	0	0
$11$	2.23607i	0.674200i	0.941469	+	0.337100i	$0.109446\pi$
$11$	2.23607i	0.674200i	−0.941469	+	0.337100i	$0.890554\pi$
$12$	0	0
$13$	4.00000i	1.10940i	0.832050	+	0.554700i	$0.187167\pi$
$13$	4.00000i	1.10940i	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	− 2.23607i	− 0.512989i	−0.966546	−	0.256495i	$-0.917432\pi$
$19$	− 2.23607i	− 0.512989i	0.966546	−	0.256495i	$-0.0825676\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	8.94427	1.86501	0.932505	−	0.361158i	$-0.117618\pi$
$23$	8.94427	1.86501	0.932505	+	0.361158i	$0.117618\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 2.23607i	− 0.430331i
$28$	0	0
$29$	− 4.00000i	− 0.742781i	−0.928477	−	0.371391i	$-0.878881\pi$
$29$	− 4.00000i	− 0.742781i	0.928477	−	0.371391i	$-0.121119\pi$
$30$	0	0
$31$	−8.94427	−1.60644	−0.803219	−	0.595683i	$-0.796881\pi$
$31$	−8.94427	−1.60644	−0.803219	+	0.595683i	$0.796881\pi$
$32$	0	0
$33$	5.00000	0.870388
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 8.00000i	− 1.31519i	−0.753371	−	0.657596i	$-0.771573\pi$
$37$	− 8.00000i	− 1.31519i	0.753371	−	0.657596i	$-0.228427\pi$
$38$	0	0
$39$	8.94427	1.43223
$40$	0	0
$41$	−5.00000	−0.780869	−0.390434	−	0.920631i	$-0.627675\pi$
$41$	−5.00000	−0.780869	−0.390434	+	0.920631i	$0.627675\pi$
$42$	0	0
$43$	− 8.94427i	− 1.36399i	−0.731357	−	0.681994i	$-0.761113\pi$
$43$	− 8.94427i	− 1.36399i	0.731357	−	0.681994i	$-0.238887\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.94427	1.30466	0.652328	−	0.757937i	$-0.273792\pi$
$47$	8.94427	1.30466	0.652328	+	0.757937i	$0.273792\pi$
$48$	0	0
$49$	−7.00000	−1.00000
$50$	0	0
$51$	6.70820i	0.939336i
$52$	0	0
$53$	− 4.00000i	− 0.549442i	−0.961524	−	0.274721i	$-0.911414\pi$
$53$	− 4.00000i	− 0.549442i	0.961524	−	0.274721i	$-0.0885855\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−5.00000	−0.662266
$58$	0	0
$59$	− 8.94427i	− 1.16445i	−0.813029	−	0.582223i	$-0.802183\pi$
$59$	− 8.94427i	− 1.16445i	0.813029	−	0.582223i	$-0.197817\pi$
$60$	0	0
$61$	− 8.00000i	− 1.02430i	−0.858898	−	0.512148i	$-0.828850\pi$
$61$	− 8.00000i	− 1.02430i	0.858898	−	0.512148i	$-0.171150\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 6.70820i	− 0.819538i	−0.912189	−	0.409769i	$-0.865609\pi$
$67$	− 6.70820i	− 0.819538i	0.912189	−	0.409769i	$-0.134391\pi$
$68$	0	0
$69$	− 20.0000i	− 2.40772i
$70$	0	0
$71$	8.94427	1.06149	0.530745	−	0.847532i	$-0.321912\pi$
$71$	8.94427	1.06149	0.530745	+	0.847532i	$0.321912\pi$
$72$	0	0
$73$	9.00000	1.05337	0.526685	−	0.850060i	$-0.323435\pi$
$73$	9.00000	1.05337	0.526685	+	0.850060i	$0.323435\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	0	0
$83$	− 6.70820i	− 0.736321i	−0.929762	−	0.368161i	$-0.879988\pi$
$83$	− 6.70820i	− 0.736321i	0.929762	−	0.368161i	$-0.120012\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−8.94427	−0.958927
$88$	0	0
$89$	−15.0000	−1.59000	−0.794998	−	0.606612i	$-0.792528\pi$
$89$	−15.0000	−1.59000	−0.794998	+	0.606612i	$0.792528\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	20.0000i	2.07390i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	− 4.47214i	− 0.449467i
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	8.94427	0.881305	0.440653	−	0.897678i	$-0.354747\pi$
$103$	8.94427	0.881305	0.440653	+	0.897678i	$0.354747\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	2.23607i	0.216169i	0.994142	+	0.108084i	$0.0344717\pi$
$107$	2.23607i	0.216169i	−0.994142	+	0.108084i	$0.965528\pi$
$108$	0	0
$109$	0	0	1.00000			$0$
$109$	0	0	−1.00000			$\pi$
$110$	0	0
$111$	−17.8885	−1.69791
$112$	0	0
$113$	−1.00000	−0.0940721	−0.0470360	−	0.998893i	$-0.514978\pi$
$113$	−1.00000	−0.0940721	−0.0470360	+	0.998893i	$0.514978\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 8.00000i	− 0.739600i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	6.00000	0.545455
$122$	0	0
$123$	11.1803i	1.00810i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	8.94427	0.793676	0.396838	−	0.917889i	$-0.370108\pi$
$127$	8.94427	0.793676	0.396838	+	0.917889i	$0.370108\pi$
$128$	0	0
$129$	−20.0000	−1.76090
$130$	0	0
$131$	− 8.94427i	− 0.781465i	−0.920504	−	0.390732i	$-0.872222\pi$
$131$	− 8.94427i	− 0.781465i	0.920504	−	0.390732i	$-0.127778\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−13.0000	−1.11066	−0.555332	−	0.831628i	$-0.687409\pi$
$137$	−13.0000	−1.11066	−0.555332	+	0.831628i	$0.687409\pi$
$138$	0	0
$139$	20.1246i	1.70695i	0.521136	+	0.853474i	$0.325508\pi$
$139$	20.1246i	1.70695i	−0.521136	+	0.853474i	$0.674492\pi$
$140$	0	0
$141$	− 20.0000i	− 1.68430i
$142$	0	0
$143$	−8.94427	−0.747958
$144$	0	0
$145$	0	0
$146$	0	0
$147$	15.6525i	1.29099i
$148$	0	0
$149$	− 20.0000i	− 1.63846i	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	− 20.0000i	− 1.63846i	0.573462	−	0.819232i	$-0.305600\pi$
$150$	0	0
$151$	17.8885	1.45575	0.727875	−	0.685710i	$-0.240508\pi$
$151$	17.8885	1.45575	0.727875	+	0.685710i	$0.240508\pi$
$152$	0	0
$153$	6.00000	0.485071
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 12.0000i	− 0.957704i	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	− 12.0000i	− 0.957704i	0.877896	−	0.478852i	$-0.158947\pi$
$158$	0	0
$159$	−8.94427	−0.709327
$160$	0	0
$161$	0	0
$162$	0	0
$163$	11.1803i	0.875712i	0.899045	+	0.437856i	$0.144262\pi$
$163$	11.1803i	0.875712i	−0.899045	+	0.437856i	$0.855738\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−8.94427	−0.692129	−0.346064	−	0.938211i	$-0.612482\pi$
$167$	−8.94427	−0.692129	−0.346064	+	0.938211i	$0.612482\pi$
$168$	0	0
$169$	−3.00000	−0.230769
$170$	0	0
$171$	4.47214i	0.341993i
$172$	0	0
$173$	24.0000i	1.82469i	0.409426	+	0.912343i	$0.365729\pi$
$173$	24.0000i	1.82469i	−0.409426	+	0.912343i	$0.634271\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−20.0000	−1.50329
$178$	0	0
$179$	15.6525i	1.16992i	0.811062	+	0.584960i	$0.198890\pi$
$179$	15.6525i	1.16992i	−0.811062	+	0.584960i	$0.801110\pi$
$180$	0	0
$181$	20.0000i	1.48659i	0.668965	+	0.743294i	$0.266738\pi$
$181$	20.0000i	1.48659i	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	−17.8885	−1.32236
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 6.70820i	− 0.490552i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.94427	0.647185	0.323592	−	0.946197i	$-0.395109\pi$
$191$	8.94427	0.647185	0.323592	+	0.946197i	$0.395109\pi$
$192$	0	0
$193$	21.0000	1.51161	0.755807	−	0.654795i	$-0.227245\pi$
$193$	21.0000	1.51161	0.755807	+	0.654795i	$0.227245\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 12.0000i	− 0.854965i	−0.904024	−	0.427482i	$-0.859401\pi$
$197$	− 12.0000i	− 0.854965i	0.904024	−	0.427482i	$-0.140599\pi$
$198$	0	0
$199$	−17.8885	−1.26809	−0.634043	−	0.773298i	$-0.718606\pi$
$199$	−17.8885	−1.26809	−0.634043	+	0.773298i	$0.718606\pi$
$200$	0	0
$201$	−15.0000	−1.05802
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−17.8885	−1.24334
$208$	0	0
$209$	5.00000	0.345857
$210$	0	0
$211$	− 6.70820i	− 0.461812i	−0.972976	−	0.230906i	$-0.925831\pi$
$211$	− 6.70820i	− 0.461812i	0.972976	−	0.230906i	$-0.0741690\pi$
$212$	0	0
$213$	− 20.0000i	− 1.37038i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	0	0
$218$	0	0
$219$	− 20.1246i	− 1.35990i
$220$	0	0
$221$	− 12.0000i	− 0.807207i
$222$	0	0
$223$	−17.8885	−1.19791	−0.598953	−	0.800784i	$-0.704416\pi$
$223$	−17.8885	−1.19791	−0.598953	+	0.800784i	$0.704416\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 8.94427i	− 0.593652i	−0.954932	−	0.296826i	$-0.904072\pi$
$227$	− 8.94427i	− 0.593652i	0.954932	−	0.296826i	$-0.0959282\pi$
$228$	0	0
$229$	− 16.0000i	− 1.05731i	−0.848837	−	0.528655i	$-0.822697\pi$
$229$	− 16.0000i	− 1.05731i	0.848837	−	0.528655i	$-0.177303\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−8.94427	−0.578557	−0.289278	−	0.957245i	$-0.593415\pi$
$239$	−8.94427	−0.578557	−0.289278	+	0.957245i	$0.593415\pi$
$240$	0	0
$241$	5.00000	0.322078	0.161039	−	0.986948i	$-0.448515\pi$
$241$	5.00000	0.322078	0.161039	+	0.986948i	$0.448515\pi$
$242$	0	0
$243$	17.8885i	1.14755i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	8.94427	0.569110
$248$	0	0
$249$	−15.0000	−0.950586
$250$	0	0
$251$	− 29.0689i	− 1.83481i	−0.397953	−	0.917406i	$-0.630279\pi$
$251$	− 29.0689i	− 1.83481i	0.397953	−	0.917406i	$-0.369721\pi$
$252$	0	0
$253$	20.0000i	1.25739i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	8.00000i	0.495188i
$262$	0	0
$263$	−8.94427	−0.551527	−0.275764	−	0.961225i	$-0.588931\pi$
$263$	−8.94427	−0.551527	−0.275764	+	0.961225i	$0.588931\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	33.5410i	2.05268i
$268$	0	0
$269$	0	0	1.00000			$0$
$269$	0	0	−1.00000			$\pi$
$270$	0	0
$271$	26.8328	1.62998	0.814989	−	0.579477i	$-0.196743\pi$
$271$	26.8328	1.62998	0.814989	+	0.579477i	$0.196743\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 12.0000i	− 0.721010i	−0.932757	−	0.360505i	$-0.882604\pi$
$277$	− 12.0000i	− 0.721010i	0.932757	−	0.360505i	$-0.117396\pi$
$278$	0	0
$279$	17.8885	1.07096
$280$	0	0
$281$	−10.0000	−0.596550	−0.298275	−	0.954480i	$-0.596411\pi$
$281$	−10.0000	−0.596550	−0.298275	+	0.954480i	$0.596411\pi$
$282$	0	0
$283$	24.5967i	1.46212i	0.682311	+	0.731062i	$0.260975\pi$
$283$	24.5967i	1.46212i	−0.682311	+	0.731062i	$0.739025\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−8.00000	−0.470588
$290$	0	0
$291$	4.47214i	0.262161i
$292$	0	0
$293$	− 24.0000i	− 1.40209i	−0.713115	−	0.701047i	$-0.752716\pi$
$293$	− 24.0000i	− 1.40209i	0.713115	−	0.701047i	$-0.247284\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	5.00000	0.290129
$298$	0	0
$299$	35.7771i	2.06904i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 2.23607i	− 0.127619i	−0.997962	−	0.0638096i	$-0.979675\pi$
$307$	− 2.23607i	− 0.127619i	0.997962	−	0.0638096i	$-0.0203250\pi$
$308$	0	0
$309$	− 20.0000i	− 1.13776i
$310$	0	0
$311$	−17.8885	−1.01437	−0.507183	−	0.861838i	$-0.669313\pi$
$311$	−17.8885	−1.01437	−0.507183	+	0.861838i	$0.669313\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	32.0000i	1.79730i	0.438667	+	0.898650i	$0.355451\pi$
$317$	32.0000i	1.79730i	−0.438667	+	0.898650i	$0.644549\pi$
$318$	0	0
$319$	8.94427	0.500783
$320$	0	0
$321$	5.00000	0.279073
$322$	0	0
$323$	6.70820i	0.373254i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	2.23607i	0.122905i	0.998110	+	0.0614527i	$0.0195733\pi$
$331$	2.23607i	0.122905i	−0.998110	+	0.0614527i	$0.980427\pi$
$332$	0	0
$333$	16.0000i	0.876795i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−33.0000	−1.79762	−0.898812	−	0.438334i	$-0.855569\pi$
$337$	−33.0000	−1.79762	−0.898812	+	0.438334i	$0.855569\pi$
$338$	0	0
$339$	2.23607i	0.121447i
$340$	0	0
$341$	− 20.0000i	− 1.08306i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 33.5410i	− 1.80058i	−0.435294	−	0.900288i	$-0.643356\pi$
$347$	− 33.5410i	− 1.80058i	0.435294	−	0.900288i	$-0.356644\pi$
$348$	0	0
$349$	24.0000i	1.28469i	0.766415	+	0.642345i	$0.222038\pi$
$349$	24.0000i	1.28469i	−0.766415	+	0.642345i	$0.777962\pi$
$350$	0	0
$351$	8.94427	0.477410
$352$	0	0
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−17.8885	−0.944121	−0.472061	−	0.881566i	$-0.656490\pi$
$359$	−17.8885	−0.944121	−0.472061	+	0.881566i	$0.656490\pi$
$360$	0	0
$361$	14.0000	0.736842
$362$	0	0
$363$	− 13.4164i	− 0.704179i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	0	0		−	1.00000i	$-0.5\pi$
$367$	0	0			1.00000i	$0.5\pi$
$368$	0	0
$369$	10.0000	0.520579
$370$	0	0
$371$	0	0
$372$	0	0
$373$	4.00000i	0.207112i	0.994624	+	0.103556i	$0.0330221\pi$
$373$	4.00000i	0.207112i	−0.994624	+	0.103556i	$0.966978\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	16.0000	0.824042
$378$	0	0
$379$	− 29.0689i	− 1.49317i	−0.665291	−	0.746584i	$-0.731693\pi$
$379$	− 29.0689i	− 1.49317i	0.665291	−	0.746584i	$-0.268307\pi$
$380$	0	0
$381$	− 20.0000i	− 1.02463i
$382$	0	0
$383$	17.8885	0.914062	0.457031	−	0.889451i	$-0.348913\pi$
$383$	17.8885	0.914062	0.457031	+	0.889451i	$0.348913\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	17.8885i	0.909326i
$388$	0	0
$389$	20.0000i	1.01404i	0.861934	+	0.507020i	$0.169253\pi$
$389$	20.0000i	1.01404i	−0.861934	+	0.507020i	$0.830747\pi$
$390$	0	0
$391$	−26.8328	−1.35699
$392$	0	0
$393$	−20.0000	−1.00887
$394$	0	0
$395$	0	0
$396$	0	0
$397$	8.00000i	0.401508i	0.979642	+	0.200754i	$0.0643393\pi$
$397$	8.00000i	0.401508i	−0.979642	+	0.200754i	$0.935661\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−27.0000	−1.34832	−0.674158	−	0.738587i	$-0.735493\pi$
$401$	−27.0000	−1.34832	−0.674158	+	0.738587i	$0.735493\pi$
$402$	0	0
$403$	− 35.7771i	− 1.78218i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	17.8885	0.886702
$408$	0	0
$409$	19.0000	0.939490	0.469745	−	0.882802i	$-0.344346\pi$
$409$	19.0000	0.939490	0.469745	+	0.882802i	$0.344346\pi$
$410$	0	0
$411$	29.0689i	1.43386i
$412$	0	0
$413$	0	0
$414$	0	0
$415$	0	0
$416$	0	0
$417$	45.0000	2.20366
$418$	0	0
$419$	− 20.1246i	− 0.983152i	−0.870835	−	0.491576i	$-0.836421\pi$
$419$	− 20.1246i	− 0.983152i	0.870835	−	0.491576i	$-0.163579\pi$
$420$	0	0
$421$	32.0000i	1.55958i	0.626038	+	0.779792i	$0.284675\pi$
$421$	32.0000i	1.55958i	−0.626038	+	0.779792i	$0.715325\pi$
$422$	0	0
$423$	−17.8885	−0.869771
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	0	0
$429$	20.0000i	0.965609i
$430$	0	0
$431$	17.8885	0.861661	0.430830	−	0.902433i	$-0.358221\pi$
$431$	17.8885	0.861661	0.430830	+	0.902433i	$0.358221\pi$
$432$	0	0
$433$	−11.0000	−0.528626	−0.264313	−	0.964437i	$-0.585145\pi$
$433$	−11.0000	−0.528626	−0.264313	+	0.964437i	$0.585145\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 20.0000i	− 0.956730i
$438$	0	0
$439$	−26.8328	−1.28066	−0.640330	−	0.768100i	$-0.721202\pi$
$439$	−26.8328	−1.28066	−0.640330	+	0.768100i	$0.721202\pi$
$440$	0	0
$441$	14.0000	0.666667
$442$	0	0
$443$	− 11.1803i	− 0.531194i	−0.964084	−	0.265597i	$-0.914431\pi$
$443$	− 11.1803i	− 0.531194i	0.964084	−	0.265597i	$-0.0855691\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−44.7214	−2.11525
$448$	0	0
$449$	−11.0000	−0.519122	−0.259561	−	0.965727i	$-0.583578\pi$
$449$	−11.0000	−0.519122	−0.259561	+	0.965727i	$0.583578\pi$
$450$	0	0
$451$	− 11.1803i	− 0.526462i
$452$	0	0
$453$	− 40.0000i	− 1.87936i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−37.0000	−1.73079	−0.865393	−	0.501093i	$-0.832931\pi$
$457$	−37.0000	−1.73079	−0.865393	+	0.501093i	$0.832931\pi$
$458$	0	0
$459$	6.70820i	0.313112i
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	35.7771	1.66270	0.831351	−	0.555748i	$-0.187568\pi$
$463$	35.7771	1.66270	0.831351	+	0.555748i	$0.187568\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 8.94427i	− 0.413892i	−0.978352	−	0.206946i	$-0.933648\pi$
$467$	− 8.94427i	− 0.413892i	0.978352	−	0.206946i	$-0.0663524\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−26.8328	−1.23639
$472$	0	0
$473$	20.0000	0.919601
$474$	0	0
$475$	0	0
$476$	0	0
$477$	8.00000i	0.366295i
$478$	0	0
$479$	26.8328	1.22602	0.613011	−	0.790074i	$-0.289958\pi$
$479$	26.8328	1.22602	0.613011	+	0.790074i	$0.289958\pi$
$480$	0	0
$481$	32.0000	1.45907
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	26.8328	1.21591	0.607955	−	0.793971i	$-0.291990\pi$
$487$	26.8328	1.21591	0.607955	+	0.793971i	$0.291990\pi$
$488$	0	0
$489$	25.0000	1.13054
$490$	0	0
$491$	26.8328i	1.21095i	0.795865	+	0.605474i	$0.207016\pi$
$491$	26.8328i	1.21095i	−0.795865	+	0.605474i	$0.792984\pi$
$492$	0	0
$493$	12.0000i	0.540453i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	26.8328i	1.20120i	0.799549	+	0.600601i	$0.205072\pi$
$499$	26.8328i	1.20120i	−0.799549	+	0.600601i	$0.794928\pi$
$500$	0	0
$501$	20.0000i	0.893534i
$502$	0	0
$503$	17.8885	0.797611	0.398805	−	0.917036i	$-0.369425\pi$
$503$	17.8885	0.797611	0.398805	+	0.917036i	$0.369425\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	6.70820i	0.297922i
$508$	0	0
$509$	4.00000i	0.177297i	0.996063	+	0.0886484i	$0.0282548\pi$
$509$	4.00000i	0.177297i	−0.996063	+	0.0886484i	$0.971745\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−5.00000	−0.220755
$514$	0	0
$515$	0	0
$516$	0	0
$517$	20.0000i	0.879599i
$518$	0	0
$519$	53.6656	2.35566
$520$	0	0
$521$	3.00000	0.131432	0.0657162	−	0.997838i	$-0.479067\pi$
$521$	3.00000	0.131432	0.0657162	+	0.997838i	$0.479067\pi$
$522$	0	0
$523$	− 33.5410i	− 1.46665i	−0.679880	−	0.733323i	$-0.737968\pi$
$523$	− 33.5410i	− 1.46665i	0.679880	−	0.733323i	$-0.262032\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	26.8328	1.16886
$528$	0	0
$529$	57.0000	2.47826
$530$	0	0
$531$	17.8885i	0.776297i
$532$	0	0
$533$	− 20.0000i	− 0.866296i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	35.0000	1.51036
$538$	0	0
$539$	− 15.6525i	− 0.674200i
$540$	0	0
$541$	40.0000i	1.71973i	0.510518	+	0.859867i	$0.329454\pi$
$541$	40.0000i	1.71973i	−0.510518	+	0.859867i	$0.670546\pi$
$542$	0	0
$543$	44.7214	1.91918
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 20.1246i	− 0.860466i	−0.902718	−	0.430233i	$-0.858431\pi$
$547$	− 20.1246i	− 0.860466i	0.902718	−	0.430233i	$-0.141569\pi$
$548$	0	0
$549$	16.0000i	0.682863i
$550$	0	0
$551$	−8.94427	−0.381039
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12.0000i	0.508456i	0.967144	+	0.254228i	$0.0818214\pi$
$557$	12.0000i	0.508456i	−0.967144	+	0.254228i	$0.918179\pi$
$558$	0	0
$559$	35.7771	1.51321
$560$	0	0
$561$	−15.0000	−0.633300
$562$	0	0
$563$	− 8.94427i	− 0.376956i	−0.982077	−	0.188478i	$-0.939645\pi$
$563$	− 8.94427i	− 0.376956i	0.982077	−	0.188478i	$-0.0603554\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	11.0000	0.461144	0.230572	−	0.973055i	$-0.425940\pi$
$569$	11.0000	0.461144	0.230572	+	0.973055i	$0.425940\pi$
$570$	0	0
$571$	− 44.7214i	− 1.87153i	−0.352623	−	0.935765i	$-0.614710\pi$
$571$	− 44.7214i	− 1.87153i	0.352623	−	0.935765i	$-0.385290\pi$
$572$	0	0
$573$	− 20.0000i	− 0.835512i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	23.0000	0.957503	0.478751	−	0.877951i	$-0.341090\pi$
$577$	23.0000	0.957503	0.478751	+	0.877951i	$0.341090\pi$
$578$	0	0
$579$	− 46.9574i	− 1.95148i
$580$	0	0
$581$	0	0
$582$	0	0
$583$	8.94427	0.370434
$584$	0	0
$585$	0	0
$586$	0	0
$587$	6.70820i	0.276877i	0.990371	+	0.138439i	$0.0442084\pi$
$587$	6.70820i	0.276877i	−0.990371	+	0.138439i	$0.955792\pi$
$588$	0	0
$589$	20.0000i	0.824086i
$590$	0	0
$591$	−26.8328	−1.10375
$592$	0	0
$593$	−9.00000	−0.369586	−0.184793	−	0.982777i	$-0.559161\pi$
$593$	−9.00000	−0.369586	−0.184793	+	0.982777i	$0.559161\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	40.0000i	1.63709i
$598$	0	0
$599$	−35.7771	−1.46181	−0.730906	−	0.682478i	$-0.760902\pi$
$599$	−35.7771	−1.46181	−0.730906	+	0.682478i	$0.760902\pi$
$600$	0	0
$601$	25.0000	1.01977	0.509886	−	0.860242i	$-0.329688\pi$
$601$	25.0000	1.01977	0.509886	+	0.860242i	$0.329688\pi$
$602$	0	0
$603$	13.4164i	0.546358i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	17.8885	0.726074	0.363037	−	0.931775i	$-0.381740\pi$
$607$	17.8885	0.726074	0.363037	+	0.931775i	$0.381740\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	35.7771i	1.44739i
$612$	0	0
$613$	4.00000i	0.161558i	0.996732	+	0.0807792i	$0.0257409\pi$
$613$	4.00000i	0.161558i	−0.996732	+	0.0807792i	$0.974259\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	0	0
$619$	− 8.94427i	− 0.359501i	−0.983712	−	0.179750i	$-0.942471\pi$
$619$	− 8.94427i	− 0.359501i	0.983712	−	0.179750i	$-0.0575290\pi$
$620$	0	0
$621$	− 20.0000i	− 0.802572i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 11.1803i	− 0.446500i
$628$	0	0
$629$	24.0000i	0.956943i
$630$	0	0
$631$	−8.94427	−0.356066	−0.178033	−	0.984025i	$-0.556973\pi$
$631$	−8.94427	−0.356066	−0.178033	+	0.984025i	$0.556973\pi$
$632$	0	0
$633$	−15.0000	−0.596196
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 28.0000i	− 1.10940i
$638$	0	0
$639$	−17.8885	−0.707660
$640$	0	0
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	− 8.94427i	− 0.352728i	−0.984325	−	0.176364i	$-0.943566\pi$
$643$	− 8.94427i	− 0.352728i	0.984325	−	0.176364i	$-0.0564335\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17.8885	0.703271	0.351636	−	0.936137i	$-0.385626\pi$
$647$	17.8885	0.703271	0.351636	+	0.936137i	$0.385626\pi$
$648$	0	0
$649$	20.0000	0.785069
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 24.0000i	− 0.939193i	−0.882881	−	0.469596i	$-0.844399\pi$
$653$	− 24.0000i	− 0.939193i	0.882881	−	0.469596i	$-0.155601\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−18.0000	−0.702247
$658$	0	0
$659$	− 6.70820i	− 0.261315i	−0.991428	−	0.130657i	$-0.958291\pi$
$659$	− 6.70820i	− 0.261315i	0.991428	−	0.130657i	$-0.0417087\pi$
$660$	0	0
$661$	28.0000i	1.08907i	0.838737	+	0.544537i	$0.183295\pi$
$661$	28.0000i	1.08907i	−0.838737	+	0.544537i	$0.816705\pi$
$662$	0	0
$663$	−26.8328	−1.04210
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 35.7771i	− 1.38529i
$668$	0	0
$669$	40.0000i	1.54649i
$670$	0	0
$671$	17.8885	0.690580
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8.00000i	0.307465i	0.988113	+	0.153732i	$0.0491294\pi$
$677$	8.00000i	0.307465i	−0.988113	+	0.153732i	$0.950871\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−20.0000	−0.766402
$682$	0	0
$683$	− 15.6525i	− 0.598925i	−0.954108	−	0.299463i	$-0.903193\pi$
$683$	− 15.6525i	− 0.598925i	0.954108	−	0.299463i	$-0.0968074\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−35.7771	−1.36498
$688$	0	0
$689$	16.0000	0.609551
$690$	0	0
$691$	15.6525i	0.595448i	0.954652	+	0.297724i	$0.0962276\pi$
$691$	15.6525i	0.595448i	−0.954652	+	0.297724i	$0.903772\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	15.0000	0.568166
$698$	0	0
$699$	− 13.4164i	− 0.507455i
$700$	0	0
$701$	− 12.0000i	− 0.453234i	−0.973984	−	0.226617i	$-0.927233\pi$
$701$	− 12.0000i	− 0.453234i	0.973984	−	0.226617i	$-0.0727665\pi$
$702$	0	0
$703$	−17.8885	−0.674679
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	16.0000i	0.600893i	0.953799	+	0.300446i	$0.0971356\pi$
$709$	16.0000i	0.600893i	−0.953799	+	0.300446i	$0.902864\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−80.0000	−2.99602
$714$	0	0
$715$	0	0
$716$	0	0
$717$	20.0000i	0.746914i
$718$	0	0
$719$	17.8885	0.667130	0.333565	−	0.942727i	$-0.391748\pi$
$719$	17.8885	0.667130	0.333565	+	0.942727i	$0.391748\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 11.1803i	− 0.415801i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−26.8328	−0.995174	−0.497587	−	0.867414i	$-0.665780\pi$
$727$	−26.8328	−0.995174	−0.497587	+	0.867414i	$0.665780\pi$
$728$	0	0
$729$	7.00000	0.259259
$730$	0	0
$731$	26.8328i	0.992448i
$732$	0	0
$733$	4.00000i	0.147743i	0.997268	+	0.0738717i	$0.0235355\pi$
$733$	4.00000i	0.147743i	−0.997268	+	0.0738717i	$0.976464\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	15.0000	0.552532
$738$	0	0
$739$	26.8328i	0.987061i	0.869728	+	0.493531i	$0.164294\pi$
$739$	26.8328i	0.987061i	−0.869728	+	0.493531i	$0.835706\pi$
$740$	0	0
$741$	− 20.0000i	− 0.734718i
$742$	0	0
$743$	8.94427	0.328134	0.164067	−	0.986449i	$-0.447539\pi$
$743$	8.94427	0.328134	0.164067	+	0.986449i	$0.447539\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	13.4164i	0.490881i
$748$	0	0
$749$	0	0
$750$	0	0
$751$	26.8328	0.979143	0.489572	−	0.871963i	$-0.337153\pi$
$751$	26.8328	0.979143	0.489572	+	0.871963i	$0.337153\pi$
$752$	0	0
$753$	−65.0000	−2.36873
$754$	0	0
$755$	0	0
$756$	0	0
$757$	8.00000i	0.290765i	0.989376	+	0.145382i	$0.0464413\pi$
$757$	8.00000i	0.290765i	−0.989376	+	0.145382i	$0.953559\pi$
$758$	0	0
$759$	44.7214	1.62328
$760$	0	0
$761$	−13.0000	−0.471250	−0.235625	−	0.971844i	$-0.575714\pi$
$761$	−13.0000	−0.471250	−0.235625	+	0.971844i	$0.575714\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	0	0
$766$	0	0
$767$	35.7771	1.29184
$768$	0	0
$769$	5.00000	0.180305	0.0901523	−	0.995928i	$-0.471265\pi$
$769$	5.00000	0.180305	0.0901523	+	0.995928i	$0.471265\pi$
$770$	0	0
$771$	− 40.2492i	− 1.44954i
$772$	0	0
$773$	4.00000i	0.143870i	0.997409	+	0.0719350i	$0.0229174\pi$
$773$	4.00000i	0.143870i	−0.997409	+	0.0719350i	$0.977083\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	11.1803i	0.400577i
$780$	0	0
$781$	20.0000i	0.715656i
$782$	0	0
$783$	−8.94427	−0.319642
$784$	0	0
$785$	0	0
$786$	0	0
$787$	26.8328i	0.956487i	0.878227	+	0.478243i	$0.158726\pi$
$787$	26.8328i	0.956487i	−0.878227	+	0.478243i	$0.841274\pi$
$788$	0	0
$789$	20.0000i	0.712019i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	32.0000	1.13635
$794$	0	0
$795$	0	0
$796$	0	0
$797$	52.0000i	1.84193i	0.389640	+	0.920967i	$0.372599\pi$
$797$	52.0000i	1.84193i	−0.389640	+	0.920967i	$0.627401\pi$
$798$	0	0
$799$	−26.8328	−0.949277
$800$	0	0
$801$	30.0000	1.06000
$802$	0	0
$803$	20.1246i	0.710182i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	10.0000	0.351581	0.175791	−	0.984428i	$-0.443752\pi$
$809$	10.0000	0.351581	0.175791	+	0.984428i	$0.443752\pi$
$810$	0	0
$811$	26.8328i	0.942228i	0.882072	+	0.471114i	$0.156148\pi$
$811$	26.8328i	0.942228i	−0.882072	+	0.471114i	$0.843852\pi$
$812$	0	0
$813$	− 60.0000i	− 2.10429i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−20.0000	−0.699711
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 28.0000i	− 0.977207i	−0.872506	−	0.488603i	$-0.837507\pi$
$821$	− 28.0000i	− 0.977207i	0.872506	−	0.488603i	$-0.162493\pi$
$822$	0	0
$823$	35.7771	1.24711	0.623555	−	0.781779i	$-0.285688\pi$
$823$	35.7771	1.24711	0.623555	+	0.781779i	$0.285688\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	24.5967i	0.855313i	0.903941	+	0.427656i	$0.140661\pi$
$827$	24.5967i	0.855313i	−0.903941	+	0.427656i	$0.859339\pi$
$828$	0	0
$829$	20.0000i	0.694629i	0.937749	+	0.347314i	$0.112906\pi$
$829$	20.0000i	0.694629i	−0.937749	+	0.347314i	$0.887094\pi$
$830$	0	0
$831$	−26.8328	−0.930820
$832$	0	0
$833$	21.0000	0.727607
$834$	0	0
$835$	0	0
$836$	0	0
$837$	20.0000i	0.691301i
$838$	0	0
$839$	−35.7771	−1.23516	−0.617581	−	0.786507i	$-0.711887\pi$
$839$	−35.7771	−1.23516	−0.617581	+	0.786507i	$0.711887\pi$
$840$	0	0
$841$	13.0000	0.448276
$842$	0	0
$843$	22.3607i	0.770143i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	0	0
$848$	0	0
$849$	55.0000	1.88760
$850$	0	0
$851$	− 71.5542i	− 2.45285i
$852$	0	0
$853$	4.00000i	0.136957i	0.997653	+	0.0684787i	$0.0218145\pi$
$853$	4.00000i	0.136957i	−0.997653	+	0.0684787i	$0.978185\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	27.0000	0.922302	0.461151	−	0.887322i	$-0.347437\pi$
$857$	27.0000	0.922302	0.461151	+	0.887322i	$0.347437\pi$
$858$	0	0
$859$	38.0132i	1.29699i	0.761218	+	0.648496i	$0.224602\pi$
$859$	38.0132i	1.29699i	−0.761218	+	0.648496i	$0.775398\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−26.8328	−0.913400	−0.456700	−	0.889621i	$-0.650969\pi$
$863$	−26.8328	−0.913400	−0.456700	+	0.889621i	$0.650969\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	17.8885i	0.607527i
$868$	0	0
$869$	0	0
$870$	0	0
$871$	26.8328	0.909195
$872$	0	0
$873$	4.00000	0.135379
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 32.0000i	− 1.08056i	−0.841484	−	0.540282i	$-0.818318\pi$
$877$	− 32.0000i	− 1.08056i	0.841484	−	0.540282i	$-0.181682\pi$
$878$	0	0
$879$	−53.6656	−1.81010
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	33.5410i	1.12875i	0.825520	+	0.564373i	$0.190882\pi$
$883$	33.5410i	1.12875i	−0.825520	+	0.564373i	$0.809118\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−17.8885	−0.600639	−0.300319	−	0.953839i	$-0.597093\pi$
$887$	−17.8885	−0.600639	−0.300319	+	0.953839i	$0.597093\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	− 24.5967i	− 0.824022i
$892$	0	0
$893$	− 20.0000i	− 0.669274i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	80.0000	2.67112
$898$	0	0
$899$	35.7771i	1.19323i
$900$	0	0
$901$	12.0000i	0.399778i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 8.94427i	− 0.296990i	−0.988913	−	0.148495i	$-0.952557\pi$
$907$	− 8.94427i	− 0.296990i	0.988913	−	0.148495i	$-0.0474428\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−35.7771	−1.18535	−0.592674	−	0.805443i	$-0.701928\pi$
$911$	−35.7771	−1.18535	−0.592674	+	0.805443i	$0.701928\pi$
$912$	0	0
$913$	15.0000	0.496428
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	8.94427	0.295044	0.147522	−	0.989059i	$-0.452870\pi$
$919$	8.94427	0.295044	0.147522	+	0.989059i	$0.452870\pi$
$920$	0	0
$921$	−5.00000	−0.164756
$922$	0	0
$923$	35.7771i	1.17762i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−17.8885	−0.587537
$928$	0	0
$929$	−34.0000	−1.11550	−0.557752	−	0.830008i	$-0.688336\pi$
$929$	−34.0000	−1.11550	−0.557752	+	0.830008i	$0.688336\pi$
$930$	0	0
$931$	15.6525i	0.512989i
$932$	0	0
$933$	40.0000i	1.30954i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−23.0000	−0.751377	−0.375689	−	0.926746i	$-0.622594\pi$
$937$	−23.0000	−0.751377	−0.375689	+	0.926746i	$0.622594\pi$
$938$	0	0
$939$	13.4164i	0.437828i
$940$	0	0
$941$	60.0000i	1.95594i	0.208736	+	0.977972i	$0.433065\pi$
$941$	60.0000i	1.95594i	−0.208736	+	0.977972i	$0.566935\pi$
$942$	0	0
$943$	−44.7214	−1.45633
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 8.94427i	− 0.290650i	−0.989384	−	0.145325i	$-0.953577\pi$
$947$	− 8.94427i	− 0.290650i	0.989384	−	0.145325i	$-0.0464227\pi$
$948$	0	0
$949$	36.0000i	1.16861i
$950$	0	0
$951$	71.5542	2.32030
$952$	0	0
$953$	−21.0000	−0.680257	−0.340128	−	0.940379i	$-0.610471\pi$
$953$	−21.0000	−0.680257	−0.340128	+	0.940379i	$0.610471\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 20.0000i	− 0.646508i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	49.0000	1.58065
$962$	0	0
$963$	− 4.47214i	− 0.144113i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−53.6656	−1.72577	−0.862885	−	0.505400i	$-0.831345\pi$
$967$	−53.6656	−1.72577	−0.862885	+	0.505400i	$0.831345\pi$
$968$	0	0
$969$	15.0000	0.481869
$970$	0	0
$971$	− 51.4296i	− 1.65045i	−0.564802	−	0.825227i	$-0.691047\pi$
$971$	− 51.4296i	− 1.65045i	0.564802	−	0.825227i	$-0.308953\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−27.0000	−0.863807	−0.431903	−	0.901920i	$-0.642158\pi$
$977$	−27.0000	−0.863807	−0.431903	+	0.901920i	$0.642158\pi$
$978$	0	0
$979$	− 33.5410i	− 1.07198i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−17.8885	−0.570556	−0.285278	−	0.958445i	$-0.592086\pi$
$983$	−17.8885	−0.570556	−0.285278	+	0.958445i	$0.592086\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 80.0000i	− 2.54385i
$990$	0	0
$991$	0	0		−	1.00000i	$-0.5\pi$
$991$	0	0			1.00000i	$0.5\pi$
$992$	0	0
$993$	5.00000	0.158670
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 8.00000i	− 0.253363i	−0.991943	−	0.126681i	$-0.959567\pi$
$997$	− 8.00000i	− 0.253363i	0.991943	−	0.126681i	$-0.0404325\pi$
$998$	0	0
$999$	−17.8885	−0.565968

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3200.2.d.o.1601.2	yes	4
4.3	odd	2	inner	3200.2.d.o.1601.4	yes	4
5.2	odd	4		3200.2.f.n.449.2		4
5.3	odd	4		3200.2.f.m.449.4		4
5.4	even	2		3200.2.d.p.1601.3	yes	4
8.3	odd	2	inner	3200.2.d.o.1601.1	✓	4
8.5	even	2	inner	3200.2.d.o.1601.3	yes	4
16.3	odd	4		6400.2.a.bq.1.1		2
16.5	even	4		6400.2.a.bs.1.1		2
16.11	odd	4		6400.2.a.bs.1.2		2
16.13	even	4		6400.2.a.bq.1.2		2
20.3	even	4		3200.2.f.m.449.1		4
20.7	even	4		3200.2.f.n.449.3		4
20.19	odd	2		3200.2.d.p.1601.1	yes	4
40.3	even	4		3200.2.f.n.449.4		4
40.13	odd	4		3200.2.f.n.449.1		4
40.19	odd	2		3200.2.d.p.1601.4	yes	4
40.27	even	4		3200.2.f.m.449.2		4
40.29	even	2		3200.2.d.p.1601.2	yes	4
40.37	odd	4		3200.2.f.m.449.3		4
80.19	odd	4		6400.2.a.bt.1.2		2
80.29	even	4		6400.2.a.bt.1.1		2
80.59	odd	4		6400.2.a.br.1.1		2
80.69	even	4		6400.2.a.br.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3200.2.d.o.1601.1	✓	4	8.3	odd	2	inner
3200.2.d.o.1601.2	yes	4	1.1	even	1	trivial
3200.2.d.o.1601.3	yes	4	8.5	even	2	inner
3200.2.d.o.1601.4	yes	4	4.3	odd	2	inner
3200.2.d.p.1601.1	yes	4	20.19	odd	2
3200.2.d.p.1601.2	yes	4	40.29	even	2
3200.2.d.p.1601.3	yes	4	5.4	even	2
3200.2.d.p.1601.4	yes	4	40.19	odd	2
3200.2.f.m.449.1		4	20.3	even	4
3200.2.f.m.449.2		4	40.27	even	4
3200.2.f.m.449.3		4	40.37	odd	4
3200.2.f.m.449.4		4	5.3	odd	4
3200.2.f.n.449.1		4	40.13	odd	4
3200.2.f.n.449.2		4	5.2	odd	4
3200.2.f.n.449.3		4	20.7	even	4
3200.2.f.n.449.4		4	40.3	even	4
6400.2.a.bq.1.1		2	16.3	odd	4
6400.2.a.bq.1.2		2	16.13	even	4
6400.2.a.br.1.1		2	80.59	odd	4
6400.2.a.br.1.2		2	80.69	even	4
6400.2.a.bs.1.1		2	16.5	even	4
6400.2.a.bs.1.2		2	16.11	odd	4
6400.2.a.bt.1.1		2	80.29	even	4
6400.2.a.bt.1.2		2	80.19	odd	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:	\( N \)	\(=\)	\( 3200 = 2^{7} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3200.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.23607i q^{3} -2.00000 q^{9} +O(q^{10})\)	\(q-2.23607i q^{3} -2.00000 q^{9} +2.23607i q^{11} +4.00000i q^{13} -3.00000 q^{17} -2.23607i q^{19} +8.94427 q^{23} -2.23607i q^{27} -4.00000i q^{29} -8.94427 q^{31} +5.00000 q^{33} -8.00000i q^{37} +8.94427 q^{39} -5.00000 q^{41} -8.94427i q^{43} +8.94427 q^{47} -7.00000 q^{49} +6.70820i q^{51} -4.00000i q^{53} -5.00000 q^{57} -8.94427i q^{59} -8.00000i q^{61} -6.70820i q^{67} -20.0000i q^{69} +8.94427 q^{71} +9.00000 q^{73} -11.0000 q^{81} -6.70820i q^{83} -8.94427 q^{87} -15.0000 q^{89} +20.0000i q^{93} -2.00000 q^{97} -4.47214i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 8 q^{9}+O(q^{10}) \) 4 * q - 8 * q^9	\( 4 q - 8 q^{9} - 12 q^{17} + 20 q^{33} - 20 q^{41} - 28 q^{49} - 20 q^{57} + 36 q^{73} - 44 q^{81} - 60 q^{89} - 8 q^{97}+O(q^{100}) \) 4 * q - 8 * q^9 - 12 * q^17 + 20 * q^33 - 20 * q^41 - 28 * q^49 - 20 * q^57 + 36 * q^73 - 44 * q^81 - 60 * q^89 - 8 * q^97

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