LMFDB - Embedded newform 3200.2.f.p.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3200,2,Mod(449,3200)]

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, 1]))

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

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

chi := DirichletCharacter("3200.449");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3200 = 2^{7} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3200.f (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:	$25.5521286468$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 25 $ x^4 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 640)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			449.2
Root			$-1.58114 - 1.58114i$ of defining polynomial
Character	$\chi$	$=$	3200.449
Dual form			3200.2.f.p.449.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.16228 q^{3} +3.16228i q^{7} +7.00000 q^{9} +O(q^{10})$	$q-3.16228 q^{3} +3.16228i q^{7} +7.00000 q^{9} -6.00000 q^{13} -2.00000i q^{17} +6.32456i q^{19} -10.0000i q^{21} -3.16228i q^{23} -12.6491 q^{27} -4.00000i q^{29} -6.32456 q^{31} -2.00000 q^{37} +18.9737 q^{39} -3.16228 q^{43} +9.48683i q^{47} -3.00000 q^{49} +6.32456i q^{51} +6.00000 q^{53} -20.0000i q^{57} +6.32456i q^{59} -2.00000i q^{61} +22.1359i q^{63} -9.48683 q^{67} +10.0000i q^{69} +6.32456 q^{71} +14.0000i q^{73} -12.6491 q^{79} +19.0000 q^{81} +3.16228 q^{83} +12.6491i q^{87} -10.0000 q^{89} -18.9737i q^{91} +20.0000 q^{93} +2.00000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 28 q^{9}+O(q^{10}) $ 4 * q + 28 * q^9	$ 4 q + 28 q^{9} - 24 q^{13} - 8 q^{37} - 12 q^{49} + 24 q^{53} + 76 q^{81} - 40 q^{89} + 80 q^{93}+O(q^{100}) $ 4 * q + 28 * q^9 - 24 * q^13 - 8 * q^37 - 12 * q^49 + 24 * q^53 + 76 * q^81 - 40 * q^89 + 80 * q^93

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
$3$	−3.16228	−1.82574	−0.912871	−	0.408248i	$-0.866140\pi$
$3$	−3.16228	−1.82574	−0.912871	+	0.408248i	$0.866140\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	3.16228i	1.19523i	0.801784	+	0.597614i	$0.203885\pi$
$7$	3.16228i	1.19523i	−0.801784	+	0.597614i	$0.796115\pi$
$8$	0	0
$9$	7.00000	2.33333
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	0	0
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.00000i	− 0.485071i	−0.970143	−	0.242536i	$-0.922021\pi$
$17$	− 2.00000i	− 0.485071i	0.970143	−	0.242536i	$-0.0779791\pi$
$18$	0	0
$19$	6.32456i	1.45095i	0.688247	+	0.725476i	$0.258380\pi$
$19$	6.32456i	1.45095i	−0.688247	+	0.725476i	$0.741620\pi$
$20$	0	0
$21$	− 10.0000i	− 2.18218i
$22$	0	0
$23$	− 3.16228i	− 0.659380i	−0.944089	−	0.329690i	$-0.893056\pi$
$23$	− 3.16228i	− 0.659380i	0.944089	−	0.329690i	$-0.106944\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−12.6491	−2.43432
$28$	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
$31$	−6.32456	−1.13592	−0.567962	−	0.823055i	$-0.692268\pi$
$31$	−6.32456	−1.13592	−0.567962	+	0.823055i	$0.692268\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.00000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	0	0
$39$	18.9737	3.03822
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−3.16228	−0.482243	−0.241121	−	0.970495i	$-0.577515\pi$
$43$	−3.16228	−0.482243	−0.241121	+	0.970495i	$0.577515\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.48683i	1.38380i	0.721995	+	0.691898i	$0.243225\pi$
$47$	9.48683i	1.38380i	−0.721995	+	0.691898i	$0.756775\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	6.32456i	0.885615i
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 20.0000i	− 2.64906i
$58$	0	0
$59$	6.32456i	0.823387i	0.911322	+	0.411693i	$0.135063\pi$
$59$	6.32456i	0.823387i	−0.911322	+	0.411693i	$0.864937\pi$
$60$	0	0
$61$	− 2.00000i	− 0.256074i	−0.991769	−	0.128037i	$-0.959132\pi$
$61$	− 2.00000i	− 0.256074i	0.991769	−	0.128037i	$-0.0408676\pi$
$62$	0	0
$63$	22.1359i	2.78887i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−9.48683	−1.15900	−0.579501	−	0.814972i	$-0.696752\pi$
$67$	−9.48683	−1.15900	−0.579501	+	0.814972i	$0.696752\pi$
$68$	0	0
$69$	10.0000i	1.20386i
$70$	0	0
$71$	6.32456	0.750587	0.375293	−	0.926906i	$-0.377542\pi$
$71$	6.32456	0.750587	0.375293	+	0.926906i	$0.377542\pi$
$72$	0	0
$73$	14.0000i	1.63858i	0.573382	+	0.819288i	$0.305631\pi$
$73$	14.0000i	1.63858i	−0.573382	+	0.819288i	$0.694369\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−12.6491	−1.42314	−0.711568	−	0.702617i	$-0.752015\pi$
$79$	−12.6491	−1.42314	−0.711568	+	0.702617i	$0.752015\pi$
$80$	0	0
$81$	19.0000	2.11111
$82$	0	0
$83$	3.16228	0.347105	0.173553	−	0.984825i	$-0.444475\pi$
$83$	3.16228	0.347105	0.173553	+	0.984825i	$0.444475\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	12.6491i	1.35613i
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	− 18.9737i	− 1.98898i
$92$	0	0
$93$	20.0000	2.07390
$94$	0	0
$95$	0	0
$96$	0	0
$97$	2.00000i	0.203069i	0.994832	+	0.101535i	$0.0323753\pi$
$97$	2.00000i	0.203069i	−0.994832	+	0.101535i	$0.967625\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	0	0	1.00000			$0$
$101$	0	0	−1.00000			$\pi$
$102$	0	0
$103$	− 15.8114i	− 1.55794i	−0.627060	−	0.778971i	$-0.715742\pi$
$103$	− 15.8114i	− 1.55794i	0.627060	−	0.778971i	$-0.284258\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.8114	1.52854	0.764272	−	0.644894i	$-0.223098\pi$
$107$	15.8114	1.52854	0.764272	+	0.644894i	$0.223098\pi$
$108$	0	0
$109$	− 10.0000i	− 0.957826i	−0.877862	−	0.478913i	$-0.841031\pi$
$109$	− 10.0000i	− 0.957826i	0.877862	−	0.478913i	$-0.158969\pi$
$110$	0	0
$111$	6.32456	0.600300
$112$	0	0
$113$	− 6.00000i	− 0.564433i	−0.959351	−	0.282216i	$-0.908930\pi$
$113$	− 6.00000i	− 0.564433i	0.959351	−	0.282216i	$-0.0910696\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−42.0000	−3.88290
$118$	0	0
$119$	6.32456	0.579771
$120$	0	0
$121$	11.0000	1.00000
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 9.48683i	− 0.841820i	−0.907102	−	0.420910i	$-0.861711\pi$
$127$	− 9.48683i	− 0.841820i	0.907102	−	0.420910i	$-0.138289\pi$
$128$	0	0
$129$	10.0000	0.880451
$130$	0	0
$131$	− 12.6491i	− 1.10516i	−0.833461	−	0.552579i	$-0.813644\pi$
$131$	− 12.6491i	− 1.10516i	0.833461	−	0.552579i	$-0.186356\pi$
$132$	0	0
$133$	−20.0000	−1.73422
$134$	0	0
$135$	0	0
$136$	0	0
$137$	18.0000i	1.53784i	0.639343	+	0.768922i	$0.279207\pi$
$137$	18.0000i	1.53784i	−0.639343	+	0.768922i	$0.720793\pi$
$138$	0	0
$139$	6.32456i	0.536442i	0.963357	+	0.268221i	$0.0864357\pi$
$139$	6.32456i	0.536442i	−0.963357	+	0.268221i	$0.913564\pi$
$140$	0	0
$141$	− 30.0000i	− 2.52646i
$142$	0	0
$143$	0	0
$144$	0	0
$145$	0	0
$146$	0	0
$147$	9.48683	0.782461
$148$	0	0
$149$	− 10.0000i	− 0.819232i	−0.912258	−	0.409616i	$-0.865663\pi$
$149$	− 10.0000i	− 0.819232i	0.912258	−	0.409616i	$-0.134337\pi$
$150$	0	0
$151$	6.32456	0.514685	0.257343	−	0.966320i	$-0.417153\pi$
$151$	6.32456	0.514685	0.257343	+	0.966320i	$0.417153\pi$
$152$	0	0
$153$	− 14.0000i	− 1.13183i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	0	0
$159$	−18.9737	−1.50471
$160$	0	0
$161$	10.0000	0.788110
$162$	0	0
$163$	9.48683	0.743066	0.371533	−	0.928420i	$-0.378832\pi$
$163$	9.48683	0.743066	0.371533	+	0.928420i	$0.378832\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 15.8114i	− 1.22352i	−0.791043	−	0.611761i	$-0.790461\pi$
$167$	− 15.8114i	− 1.22352i	0.791043	−	0.611761i	$-0.209539\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	44.2719i	3.38556i
$172$	0	0
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 20.0000i	− 1.50329i
$178$	0	0
$179$	− 18.9737i	− 1.41816i	−0.705129	−	0.709079i	$-0.749111\pi$
$179$	− 18.9737i	− 1.41816i	0.705129	−	0.709079i	$-0.250889\pi$
$180$	0	0
$181$	0	0	1.00000			$0$
$181$	0	0	−1.00000			$\pi$
$182$	0	0
$183$	6.32456i	0.467525i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	0	0
$188$	0	0
$189$	− 40.0000i	− 2.90957i
$190$	0	0
$191$	−18.9737	−1.37289	−0.686443	−	0.727183i	$-0.740829\pi$
$191$	−18.9737	−1.37289	−0.686443	+	0.727183i	$0.740829\pi$
$192$	0	0
$193$	− 14.0000i	− 1.00774i	−0.863779	−	0.503871i	$-0.831909\pi$
$193$	− 14.0000i	− 1.00774i	0.863779	−	0.503871i	$-0.168091\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−18.0000	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$197$	−18.0000	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$198$	0	0
$199$	0	0		−	1.00000i	$-0.5\pi$
$199$	0	0			1.00000i	$0.5\pi$
$200$	0	0
$201$	30.0000	2.11604
$202$	0	0
$203$	12.6491	0.887794
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 22.1359i	− 1.53855i
$208$	0	0
$209$	0	0
$210$	0	0
$211$	0	0	1.00000			$0$
$211$	0	0	−1.00000			$\pi$
$212$	0	0
$213$	−20.0000	−1.37038
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 20.0000i	− 1.35769i
$218$	0	0
$219$	− 44.2719i	− 2.99162i
$220$	0	0
$221$	12.0000i	0.807207i
$222$	0	0
$223$	− 22.1359i	− 1.48233i	−0.671322	−	0.741166i	$-0.734273\pi$
$223$	− 22.1359i	− 1.48233i	0.671322	−	0.741166i	$-0.265727\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.48683	0.629663	0.314832	−	0.949148i	$-0.398052\pi$
$227$	9.48683	0.629663	0.314832	+	0.949148i	$0.398052\pi$
$228$	0	0
$229$	4.00000i	0.264327i	0.991228	+	0.132164i	$0.0421925\pi$
$229$	4.00000i	0.264327i	−0.991228	+	0.132164i	$0.957808\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 14.0000i	− 0.917170i	−0.888650	−	0.458585i	$-0.848356\pi$
$233$	− 14.0000i	− 0.917170i	0.888650	−	0.458585i	$-0.151644\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	40.0000	2.59828
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	0	0		−	1.00000i	$-0.5\pi$
$241$	0	0			1.00000i	$0.5\pi$
$242$	0	0
$243$	−22.1359	−1.42002
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 37.9473i	− 2.41453i
$248$	0	0
$249$	−10.0000	−0.633724
$250$	0	0
$251$	− 12.6491i	− 0.798405i	−0.916863	−	0.399202i	$-0.869287\pi$
$251$	− 12.6491i	− 0.798405i	0.916863	−	0.399202i	$-0.130713\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 18.0000i	− 1.12281i	−0.827541	−	0.561405i	$-0.810261\pi$
$257$	− 18.0000i	− 1.12281i	0.827541	−	0.561405i	$-0.189739\pi$
$258$	0	0
$259$	− 6.32456i	− 0.392989i
$260$	0	0
$261$	− 28.0000i	− 1.73316i
$262$	0	0
$263$	− 22.1359i	− 1.36496i	−0.730904	−	0.682480i	$-0.760901\pi$
$263$	− 22.1359i	− 1.36496i	0.730904	−	0.682480i	$-0.239099\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	31.6228	1.93528
$268$	0	0
$269$	10.0000i	0.609711i	0.952399	+	0.304855i	$0.0986081\pi$
$269$	10.0000i	0.609711i	−0.952399	+	0.304855i	$0.901392\pi$
$270$	0	0
$271$	−6.32456	−0.384189	−0.192095	−	0.981376i	$-0.561528\pi$
$271$	−6.32456	−0.384189	−0.192095	+	0.981376i	$0.561528\pi$
$272$	0	0
$273$	60.0000i	3.63137i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	2.00000	0.120168	0.0600842	−	0.998193i	$-0.480863\pi$
$277$	2.00000	0.120168	0.0600842	+	0.998193i	$0.480863\pi$
$278$	0	0
$279$	−44.2719	−2.65049
$280$	0	0
$281$	0	0		−	1.00000i	$-0.5\pi$
$281$	0	0			1.00000i	$0.5\pi$
$282$	0	0
$283$	3.16228	0.187978	0.0939889	−	0.995573i	$-0.470038\pi$
$283$	3.16228	0.187978	0.0939889	+	0.995573i	$0.470038\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	13.0000	0.764706
$290$	0	0
$291$	− 6.32456i	− 0.370752i
$292$	0	0
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	18.9737i	1.09728i
$300$	0	0
$301$	− 10.0000i	− 0.576390i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	15.8114	0.902404	0.451202	−	0.892422i	$-0.350995\pi$
$307$	15.8114	0.902404	0.451202	+	0.892422i	$0.350995\pi$
$308$	0	0
$309$	50.0000i	2.84440i
$310$	0	0
$311$	−18.9737	−1.07590	−0.537949	−	0.842977i	$-0.680801\pi$
$311$	−18.9737	−1.07590	−0.537949	+	0.842977i	$0.680801\pi$
$312$	0	0
$313$	− 6.00000i	− 0.339140i	−0.985518	−	0.169570i	$-0.945762\pi$
$313$	− 6.00000i	− 0.339140i	0.985518	−	0.169570i	$-0.0542379\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−22.0000	−1.23564	−0.617822	−	0.786318i	$-0.711985\pi$
$317$	−22.0000	−1.23564	−0.617822	+	0.786318i	$0.711985\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	−50.0000	−2.79073
$322$	0	0
$323$	12.6491	0.703815
$324$	0	0
$325$	0	0
$326$	0	0
$327$	31.6228i	1.74874i
$328$	0	0
$329$	−30.0000	−1.65395
$330$	0	0
$331$	12.6491i	0.695258i	0.937632	+	0.347629i	$0.113013\pi$
$331$	12.6491i	0.695258i	−0.937632	+	0.347629i	$0.886987\pi$
$332$	0	0
$333$	−14.0000	−0.767195
$334$	0	0
$335$	0	0
$336$	0	0
$337$	18.0000i	0.980522i	0.871576	+	0.490261i	$0.163099\pi$
$337$	18.0000i	0.980522i	−0.871576	+	0.490261i	$0.836901\pi$
$338$	0	0
$339$	18.9737i	1.03051i
$340$	0	0
$341$	0	0
$342$	0	0
$343$	12.6491i	0.682988i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−15.8114	−0.848800	−0.424400	−	0.905475i	$-0.639515\pi$
$347$	−15.8114	−0.848800	−0.424400	+	0.905475i	$0.639515\pi$
$348$	0	0
$349$	4.00000i	0.214115i	0.994253	+	0.107058i	$0.0341429\pi$
$349$	4.00000i	0.214115i	−0.994253	+	0.107058i	$0.965857\pi$
$350$	0	0
$351$	75.8947	4.05096
$352$	0	0
$353$	6.00000i	0.319348i	0.987170	+	0.159674i	$0.0510443\pi$
$353$	6.00000i	0.319348i	−0.987170	+	0.159674i	$0.948956\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	−20.0000	−1.05851
$358$	0	0
$359$	−12.6491	−0.667595	−0.333797	−	0.942645i	$-0.608330\pi$
$359$	−12.6491	−0.667595	−0.333797	+	0.942645i	$0.608330\pi$
$360$	0	0
$361$	−21.0000	−1.10526
$362$	0	0
$363$	−34.7851	−1.82574
$364$	0	0
$365$	0	0
$366$	0	0
$367$	3.16228i	0.165070i	0.996588	+	0.0825348i	$0.0263016\pi$
$367$	3.16228i	0.165070i	−0.996588	+	0.0825348i	$0.973698\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	18.9737i	0.985064i
$372$	0	0
$373$	14.0000	0.724893	0.362446	−	0.932005i	$-0.381942\pi$
$373$	14.0000	0.724893	0.362446	+	0.932005i	$0.381942\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	24.0000i	1.23606i
$378$	0	0
$379$	6.32456i	0.324871i	0.986719	+	0.162435i	$0.0519349\pi$
$379$	6.32456i	0.324871i	−0.986719	+	0.162435i	$0.948065\pi$
$380$	0	0
$381$	30.0000i	1.53695i
$382$	0	0
$383$	− 22.1359i	− 1.13109i	−0.824716	−	0.565547i	$-0.808665\pi$
$383$	− 22.1359i	− 1.13109i	0.824716	−	0.565547i	$-0.191335\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−22.1359	−1.12523
$388$	0	0
$389$	30.0000i	1.52106i	0.649303	+	0.760530i	$0.275061\pi$
$389$	30.0000i	1.52106i	−0.649303	+	0.760530i	$0.724939\pi$
$390$	0	0
$391$	−6.32456	−0.319847
$392$	0	0
$393$	40.0000i	2.01773i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	22.0000	1.10415	0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000	1.10415	0.552074	+	0.833795i	$0.313837\pi$
$398$	0	0
$399$	63.2456	3.16624
$400$	0	0
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	0	0
$403$	37.9473	1.89029
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	0	0
$409$	36.0000	1.78009	0.890043	−	0.455877i	$-0.150674\pi$
$409$	36.0000	1.78009	0.890043	+	0.455877i	$0.150674\pi$
$410$	0	0
$411$	− 56.9210i	− 2.80771i
$412$	0	0
$413$	−20.0000	−0.984136
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 20.0000i	− 0.979404i
$418$	0	0
$419$	− 31.6228i	− 1.54487i	−0.635092	−	0.772437i	$-0.719038\pi$
$419$	− 31.6228i	− 1.54487i	0.635092	−	0.772437i	$-0.280962\pi$
$420$	0	0
$421$	− 22.0000i	− 1.07221i	−0.844150	−	0.536107i	$-0.819894\pi$
$421$	− 22.0000i	− 1.07221i	0.844150	−	0.536107i	$-0.180106\pi$
$422$	0	0
$423$	66.4078i	3.22886i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	6.32456	0.306067
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−31.6228	−1.52322	−0.761608	−	0.648038i	$-0.775590\pi$
$431$	−31.6228	−1.52322	−0.761608	+	0.648038i	$0.775590\pi$
$432$	0	0
$433$	− 26.0000i	− 1.24948i	−0.780833	−	0.624740i	$-0.785205\pi$
$433$	− 26.0000i	− 1.24948i	0.780833	−	0.624740i	$-0.214795\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	20.0000	0.956730
$438$	0	0
$439$	12.6491	0.603709	0.301855	−	0.953354i	$-0.402394\pi$
$439$	12.6491	0.603709	0.301855	+	0.953354i	$0.402394\pi$
$440$	0	0
$441$	−21.0000	−1.00000
$442$	0	0
$443$	41.1096	1.95318	0.976588	−	0.215117i	$-0.0690133\pi$
$443$	41.1096	1.95318	0.976588	+	0.215117i	$0.0690133\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	31.6228i	1.49571i
$448$	0	0
$449$	−4.00000	−0.188772	−0.0943858	−	0.995536i	$-0.530089\pi$
$449$	−4.00000	−0.188772	−0.0943858	+	0.995536i	$0.530089\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	−20.0000	−0.939682
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 18.0000i	− 0.842004i	−0.907060	−	0.421002i	$-0.861678\pi$
$457$	− 18.0000i	− 0.842004i	0.907060	−	0.421002i	$-0.138322\pi$
$458$	0	0
$459$	25.2982i	1.18082i
$460$	0	0
$461$	40.0000i	1.86299i	0.363760	+	0.931493i	$0.381493\pi$
$461$	40.0000i	1.86299i	−0.363760	+	0.931493i	$0.618507\pi$
$462$	0	0
$463$	− 3.16228i	− 0.146964i	−0.997297	−	0.0734818i	$-0.976589\pi$
$463$	− 3.16228i	− 0.146964i	0.997297	−	0.0734818i	$-0.0234111\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.16228	−0.146333	−0.0731664	−	0.997320i	$-0.523310\pi$
$467$	−3.16228	−0.146333	−0.0731664	+	0.997320i	$0.523310\pi$
$468$	0	0
$469$	− 30.0000i	− 1.38527i
$470$	0	0
$471$	−6.32456	−0.291420
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	0	0
$477$	42.0000	1.92305
$478$	0	0
$479$	25.2982	1.15591	0.577953	−	0.816070i	$-0.303852\pi$
$479$	25.2982	1.15591	0.577953	+	0.816070i	$0.303852\pi$
$480$	0	0
$481$	12.0000	0.547153
$482$	0	0
$483$	−31.6228	−1.43889
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 9.48683i	− 0.429889i	−0.976626	−	0.214945i	$-0.931043\pi$
$487$	− 9.48683i	− 0.429889i	0.976626	−	0.214945i	$-0.0689571\pi$
$488$	0	0
$489$	−30.0000	−1.35665
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0	0
$493$	−8.00000	−0.360302
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20.0000i	0.897123i
$498$	0	0
$499$	31.6228i	1.41563i	0.706398	+	0.707815i	$0.250319\pi$
$499$	31.6228i	1.41563i	−0.706398	+	0.707815i	$0.749681\pi$
$500$	0	0
$501$	50.0000i	2.23384i
$502$	0	0
$503$	34.7851i	1.55099i	0.631354	+	0.775494i	$0.282499\pi$
$503$	34.7851i	1.55099i	−0.631354	+	0.775494i	$0.717501\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−72.7324	−3.23016
$508$	0	0
$509$	− 36.0000i	− 1.59567i	−0.602875	−	0.797836i	$-0.705978\pi$
$509$	− 36.0000i	− 1.59567i	0.602875	−	0.797836i	$-0.294022\pi$
$510$	0	0
$511$	−44.2719	−1.95847
$512$	0	0
$513$	− 80.0000i	− 3.53209i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−44.2719	−1.94332
$520$	0	0
$521$	18.0000	0.788594	0.394297	−	0.918983i	$-0.370988\pi$
$521$	18.0000	0.788594	0.394297	+	0.918983i	$0.370988\pi$
$522$	0	0
$523$	3.16228	0.138277	0.0691384	−	0.997607i	$-0.477975\pi$
$523$	3.16228	0.138277	0.0691384	+	0.997607i	$0.477975\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12.6491i	0.551004i
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	44.2719i	1.92124i
$532$	0	0
$533$	0	0
$534$	0	0
$535$	0	0
$536$	0	0
$537$	60.0000i	2.58919i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	− 40.0000i	− 1.71973i	−0.510518	−	0.859867i	$-0.670546\pi$
$541$	− 40.0000i	− 1.71973i	0.510518	−	0.859867i	$-0.329454\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−15.8114	−0.676046	−0.338023	−	0.941138i	$-0.609758\pi$
$547$	−15.8114	−0.676046	−0.338023	+	0.941138i	$0.609758\pi$
$548$	0	0
$549$	− 14.0000i	− 0.597505i
$550$	0	0
$551$	25.2982	1.07774
$552$	0	0
$553$	− 40.0000i	− 1.70097i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.00000	−0.0847427	−0.0423714	−	0.999102i	$-0.513491\pi$
$557$	−2.00000	−0.0847427	−0.0423714	+	0.999102i	$0.513491\pi$
$558$	0	0
$559$	18.9737	0.802501
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−28.4605	−1.19947	−0.599734	−	0.800200i	$-0.704727\pi$
$563$	−28.4605	−1.19947	−0.599734	+	0.800200i	$0.704727\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	60.0833i	2.52326i
$568$	0	0
$569$	4.00000	0.167689	0.0838444	−	0.996479i	$-0.473280\pi$
$569$	4.00000	0.167689	0.0838444	+	0.996479i	$0.473280\pi$
$570$	0	0
$571$	− 12.6491i	− 0.529349i	−0.964338	−	0.264674i	$-0.914736\pi$
$571$	− 12.6491i	− 0.529349i	0.964338	−	0.264674i	$-0.0852645\pi$
$572$	0	0
$573$	60.0000	2.50654
$574$	0	0
$575$	0	0
$576$	0	0
$577$	22.0000i	0.915872i	0.888985	+	0.457936i	$0.151411\pi$
$577$	22.0000i	0.915872i	−0.888985	+	0.457936i	$0.848589\pi$
$578$	0	0
$579$	44.2719i	1.83988i
$580$	0	0
$581$	10.0000i	0.414870i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.16228	−0.130521	−0.0652606	−	0.997868i	$-0.520788\pi$
$587$	−3.16228	−0.130521	−0.0652606	+	0.997868i	$0.520788\pi$
$588$	0	0
$589$	− 40.0000i	− 1.64817i
$590$	0	0
$591$	56.9210	2.34142
$592$	0	0
$593$	46.0000i	1.88899i	0.328521	+	0.944497i	$0.393450\pi$
$593$	46.0000i	1.88899i	−0.328521	+	0.944497i	$0.606550\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	37.9473	1.55049	0.775243	−	0.631663i	$-0.217627\pi$
$599$	37.9473	1.55049	0.775243	+	0.631663i	$0.217627\pi$
$600$	0	0
$601$	−40.0000	−1.63163	−0.815817	−	0.578310i	$-0.803712\pi$
$601$	−40.0000	−1.63163	−0.815817	+	0.578310i	$0.803712\pi$
$602$	0	0
$603$	−66.4078	−2.70434
$604$	0	0
$605$	0	0
$606$	0	0
$607$	34.7851i	1.41188i	0.708271	+	0.705941i	$0.249476\pi$
$607$	34.7851i	1.41188i	−0.708271	+	0.705941i	$0.750524\pi$
$608$	0	0
$609$	−40.0000	−1.62088
$610$	0	0
$611$	− 56.9210i	− 2.30278i
$612$	0	0
$613$	−46.0000	−1.85792	−0.928961	−	0.370177i	$-0.879297\pi$
$613$	−46.0000	−1.85792	−0.928961	+	0.370177i	$0.879297\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 42.0000i	− 1.69086i	−0.534089	−	0.845428i	$-0.679345\pi$
$617$	− 42.0000i	− 1.69086i	0.534089	−	0.845428i	$-0.320655\pi$
$618$	0	0
$619$	− 31.6228i	− 1.27103i	−0.772090	−	0.635513i	$-0.780789\pi$
$619$	− 31.6228i	− 1.27103i	0.772090	−	0.635513i	$-0.219211\pi$
$620$	0	0
$621$	40.0000i	1.60514i
$622$	0	0
$623$	− 31.6228i	− 1.26694i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	4.00000i	0.159490i
$630$	0	0
$631$	−6.32456	−0.251777	−0.125888	−	0.992044i	$-0.540178\pi$
$631$	−6.32456	−0.251777	−0.125888	+	0.992044i	$0.540178\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	18.0000	0.713186
$638$	0	0
$639$	44.2719	1.75137
$640$	0	0
$641$	−40.0000	−1.57991	−0.789953	−	0.613168i	$-0.789895\pi$
$641$	−40.0000	−1.57991	−0.789953	+	0.613168i	$0.789895\pi$
$642$	0	0
$643$	−9.48683	−0.374124	−0.187062	−	0.982348i	$-0.559897\pi$
$643$	−9.48683	−0.374124	−0.187062	+	0.982348i	$0.559897\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 9.48683i	− 0.372966i	−0.982458	−	0.186483i	$-0.940291\pi$
$647$	− 9.48683i	− 0.372966i	0.982458	−	0.186483i	$-0.0597089\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	63.2456i	2.47879i
$652$	0	0
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	98.0000i	3.82334i
$658$	0	0
$659$	44.2719i	1.72459i	0.506408	+	0.862294i	$0.330973\pi$
$659$	44.2719i	1.72459i	−0.506408	+	0.862294i	$0.669027\pi$
$660$	0	0
$661$	22.0000i	0.855701i	0.903850	+	0.427850i	$0.140729\pi$
$661$	22.0000i	0.855701i	−0.903850	+	0.427850i	$0.859271\pi$
$662$	0	0
$663$	− 37.9473i	− 1.47375i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−12.6491	−0.489776
$668$	0	0
$669$	70.0000i	2.70636i
$670$	0	0
$671$	0	0
$672$	0	0
$673$	− 26.0000i	− 1.00223i	−0.865382	−	0.501113i	$-0.832924\pi$
$673$	− 26.0000i	− 1.00223i	0.865382	−	0.501113i	$-0.167076\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	−6.32456	−0.242714
$680$	0	0
$681$	−30.0000	−1.14960
$682$	0	0
$683$	22.1359	0.847008	0.423504	−	0.905894i	$-0.360800\pi$
$683$	22.1359	0.847008	0.423504	+	0.905894i	$0.360800\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 12.6491i	− 0.482594i
$688$	0	0
$689$	−36.0000	−1.37149
$690$	0	0
$691$	− 37.9473i	− 1.44358i	−0.692110	−	0.721792i	$-0.743319\pi$
$691$	− 37.9473i	− 1.44358i	0.692110	−	0.721792i	$-0.256681\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	0	0
$698$	0	0
$699$	44.2719i	1.67452i
$700$	0	0
$701$	2.00000i	0.0755390i	0.999286	+	0.0377695i	$0.0120253\pi$
$701$	2.00000i	0.0755390i	−0.999286	+	0.0377695i	$0.987975\pi$
$702$	0	0
$703$	− 12.6491i	− 0.477070i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	− 4.00000i	− 0.150223i	−0.997175	−	0.0751116i	$-0.976069\pi$
$709$	− 4.00000i	− 0.150223i	0.997175	−	0.0751116i	$-0.0239313\pi$
$710$	0	0
$711$	−88.5438	−3.32065
$712$	0	0
$713$	20.0000i	0.749006i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−50.5964	−1.88693	−0.943464	−	0.331474i	$-0.892454\pi$
$719$	−50.5964	−1.88693	−0.943464	+	0.331474i	$0.892454\pi$
$720$	0	0
$721$	50.0000	1.86210
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	3.16228i	0.117282i	0.998279	+	0.0586412i	$0.0186768\pi$
$727$	3.16228i	0.117282i	−0.998279	+	0.0586412i	$0.981323\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	0	0
$731$	6.32456i	0.233922i
$732$	0	0
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 18.9737i	− 0.697958i	−0.937131	−	0.348979i	$-0.886529\pi$
$739$	− 18.9737i	− 0.697958i	0.937131	−	0.348979i	$-0.113471\pi$
$740$	0	0
$741$	120.000i	4.40831i
$742$	0	0
$743$	− 9.48683i	− 0.348038i	−0.984742	−	0.174019i	$-0.944325\pi$
$743$	− 9.48683i	− 0.348038i	0.984742	−	0.174019i	$-0.0556754\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	22.1359	0.809912
$748$	0	0
$749$	50.0000i	1.82696i
$750$	0	0
$751$	−6.32456	−0.230786	−0.115393	−	0.993320i	$-0.536813\pi$
$751$	−6.32456	−0.230786	−0.115393	+	0.993320i	$0.536813\pi$
$752$	0	0
$753$	40.0000i	1.45768i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−18.0000	−0.652499	−0.326250	−	0.945284i	$-0.605785\pi$
$761$	−18.0000	−0.652499	−0.326250	+	0.945284i	$0.605785\pi$
$762$	0	0
$763$	31.6228	1.14482
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 37.9473i	− 1.37020i
$768$	0	0
$769$	−30.0000	−1.08183	−0.540914	−	0.841078i	$-0.681921\pi$
$769$	−30.0000	−1.08183	−0.540914	+	0.841078i	$0.681921\pi$
$770$	0	0
$771$	56.9210i	2.04996i
$772$	0	0
$773$	34.0000	1.22290	0.611448	−	0.791285i	$-0.290588\pi$
$773$	34.0000	1.22290	0.611448	+	0.791285i	$0.290588\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	20.0000i	0.717496i
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	50.5964i	1.80817i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	3.16228	0.112723	0.0563615	−	0.998410i	$-0.482050\pi$
$787$	3.16228	0.112723	0.0563615	+	0.998410i	$0.482050\pi$
$788$	0	0
$789$	70.0000i	2.49207i
$790$	0	0
$791$	18.9737	0.674626
$792$	0	0
$793$	12.0000i	0.426132i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	0	0
$799$	18.9737	0.671240
$800$	0	0
$801$	−70.0000	−2.47333
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 31.6228i	− 1.11317i
$808$	0	0
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	− 12.6491i	− 0.444170i	−0.975027	−	0.222085i	$-0.928714\pi$
$811$	− 12.6491i	− 0.444170i	0.975027	−	0.222085i	$-0.0712863\pi$
$812$	0	0
$813$	20.0000	0.701431
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 20.0000i	− 0.699711i
$818$	0	0
$819$	− 132.816i	− 4.64095i
$820$	0	0
$821$	− 42.0000i	− 1.46581i	−0.680331	−	0.732905i	$-0.738164\pi$
$821$	− 42.0000i	− 1.46581i	0.680331	−	0.732905i	$-0.261836\pi$
$822$	0	0
$823$	− 28.4605i	− 0.992071i	−0.868302	−	0.496035i	$-0.834789\pi$
$823$	− 28.4605i	− 0.992071i	0.868302	−	0.496035i	$-0.165211\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9.48683	0.329890	0.164945	−	0.986303i	$-0.447255\pi$
$827$	9.48683	0.329890	0.164945	+	0.986303i	$0.447255\pi$
$828$	0	0
$829$	10.0000i	0.347314i	0.984806	+	0.173657i	$0.0555585\pi$
$829$	10.0000i	0.347314i	−0.984806	+	0.173657i	$0.944442\pi$
$830$	0	0
$831$	−6.32456	−0.219396
$832$	0	0
$833$	6.00000i	0.207888i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	80.0000	2.76520
$838$	0	0
$839$	12.6491	0.436696	0.218348	−	0.975871i	$-0.429933\pi$
$839$	12.6491	0.436696	0.218348	+	0.975871i	$0.429933\pi$
$840$	0	0
$841$	13.0000	0.448276
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	34.7851i	1.19523i
$848$	0	0
$849$	−10.0000	−0.343199
$850$	0	0
$851$	6.32456i	0.216803i
$852$	0	0
$853$	−46.0000	−1.57501	−0.787505	−	0.616308i	$-0.788628\pi$
$853$	−46.0000	−1.57501	−0.787505	+	0.616308i	$0.788628\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	18.0000i	0.614868i	0.951569	+	0.307434i	$0.0994704\pi$
$857$	18.0000i	0.614868i	−0.951569	+	0.307434i	$0.900530\pi$
$858$	0	0
$859$	− 44.2719i	− 1.51054i	−0.655415	−	0.755269i	$-0.727506\pi$
$859$	− 44.2719i	− 1.51054i	0.655415	−	0.755269i	$-0.272494\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	34.7851i	1.18410i	0.805902	+	0.592049i	$0.201681\pi$
$863$	34.7851i	1.18410i	−0.805902	+	0.592049i	$0.798319\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−41.1096	−1.39616
$868$	0	0
$869$	0	0
$870$	0	0
$871$	56.9210	1.92869
$872$	0	0
$873$	14.0000i	0.473828i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−38.0000	−1.28317	−0.641584	−	0.767052i	$-0.721723\pi$
$877$	−38.0000	−1.28317	−0.641584	+	0.767052i	$0.721723\pi$
$878$	0	0
$879$	−82.2192	−2.77319
$880$	0	0
$881$	−40.0000	−1.34763	−0.673817	−	0.738898i	$-0.735346\pi$
$881$	−40.0000	−1.34763	−0.673817	+	0.738898i	$0.735346\pi$
$882$	0	0
$883$	−3.16228	−0.106419	−0.0532096	−	0.998583i	$-0.516945\pi$
$883$	−3.16228	−0.106419	−0.0532096	+	0.998583i	$0.516945\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 47.4342i	− 1.59268i	−0.604847	−	0.796342i	$-0.706766\pi$
$887$	− 47.4342i	− 1.59268i	0.604847	−	0.796342i	$-0.293234\pi$
$888$	0	0
$889$	30.0000	1.00617
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−60.0000	−2.00782
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 60.0000i	− 2.00334i
$898$	0	0
$899$	25.2982i	0.843743i
$900$	0	0
$901$	− 12.0000i	− 0.399778i
$902$	0	0
$903$	31.6228i	1.05234i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	53.7587	1.78503	0.892515	−	0.451019i	$-0.148939\pi$
$907$	53.7587	1.78503	0.892515	+	0.451019i	$0.148939\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−31.6228	−1.04771	−0.523855	−	0.851808i	$-0.675507\pi$
$911$	−31.6228	−1.04771	−0.523855	+	0.851808i	$0.675507\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	0	0
$916$	0	0
$917$	40.0000	1.32092
$918$	0	0
$919$	12.6491	0.417256	0.208628	−	0.977995i	$-0.433100\pi$
$919$	12.6491	0.417256	0.208628	+	0.977995i	$0.433100\pi$
$920$	0	0
$921$	−50.0000	−1.64756
$922$	0	0
$923$	−37.9473	−1.24905
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 110.680i	− 3.63520i
$928$	0	0
$929$	4.00000	0.131236	0.0656179	−	0.997845i	$-0.479098\pi$
$929$	4.00000	0.131236	0.0656179	+	0.997845i	$0.479098\pi$
$930$	0	0
$931$	− 18.9737i	− 0.621837i
$932$	0	0
$933$	60.0000	1.96431
$934$	0	0
$935$	0	0
$936$	0	0
$937$	38.0000i	1.24141i	0.784046	+	0.620703i	$0.213153\pi$
$937$	38.0000i	1.24141i	−0.784046	+	0.620703i	$0.786847\pi$
$938$	0	0
$939$	18.9737i	0.619182i
$940$	0	0
$941$	0	0	1.00000			$0$
$941$	0	0	−1.00000			$\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	15.8114	0.513801	0.256901	−	0.966438i	$-0.417299\pi$
$947$	15.8114	0.513801	0.256901	+	0.966438i	$0.417299\pi$
$948$	0	0
$949$	− 84.0000i	− 2.72676i
$950$	0	0
$951$	69.5701	2.25597
$952$	0	0
$953$	− 6.00000i	− 0.194359i	−0.995267	−	0.0971795i	$-0.969018\pi$
$953$	− 6.00000i	− 0.194359i	0.995267	−	0.0971795i	$-0.0309821\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−56.9210	−1.83807
$960$	0	0
$961$	9.00000	0.290323
$962$	0	0
$963$	110.680	3.56660
$964$	0	0
$965$	0	0
$966$	0	0
$967$	22.1359i	0.711844i	0.934516	+	0.355922i	$0.115833\pi$
$967$	22.1359i	0.711844i	−0.934516	+	0.355922i	$0.884167\pi$
$968$	0	0
$969$	−40.0000	−1.28499
$970$	0	0
$971$	0	0	1.00000			$0$
$971$	0	0	−1.00000			$\pi$
$972$	0	0
$973$	−20.0000	−0.641171
$974$	0	0
$975$	0	0
$976$	0	0
$977$	2.00000i	0.0639857i	0.999488	+	0.0319928i	$0.0101854\pi$
$977$	2.00000i	0.0639857i	−0.999488	+	0.0319928i	$0.989815\pi$
$978$	0	0
$979$	0	0
$980$	0	0
$981$	− 70.0000i	− 2.23493i
$982$	0	0
$983$	− 47.4342i	− 1.51291i	−0.654043	−	0.756457i	$-0.726928\pi$
$983$	− 47.4342i	− 1.51291i	0.654043	−	0.756457i	$-0.273072\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	94.8683	3.01969
$988$	0	0
$989$	10.0000i	0.317982i
$990$	0	0
$991$	−44.2719	−1.40634	−0.703171	−	0.711020i	$-0.748233\pi$
$991$	−44.2719	−1.40634	−0.703171	+	0.711020i	$0.748233\pi$
$992$	0	0
$993$	− 40.0000i	− 1.26936i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	18.0000	0.570066	0.285033	−	0.958518i	$-0.407995\pi$
$997$	18.0000	0.570066	0.285033	+	0.958518i	$0.407995\pi$
$998$	0	0
$999$	25.2982	0.800400

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3200.2.f.p.449.2		4
4.3	odd	2	inner	3200.2.f.p.449.3		4
5.2	odd	4		3200.2.d.j.1601.3		4
5.3	odd	4		640.2.d.a.321.2	yes	4
5.4	even	2		3200.2.f.q.449.3		4
8.3	odd	2		3200.2.f.q.449.1		4
8.5	even	2		3200.2.f.q.449.4		4
15.8	even	4		5760.2.k.t.2881.2		4
20.3	even	4		640.2.d.a.321.4	yes	4
20.7	even	4		3200.2.d.j.1601.2		4
20.19	odd	2		3200.2.f.q.449.2		4
40.3	even	4		640.2.d.a.321.1	✓	4
40.13	odd	4		640.2.d.a.321.3	yes	4
40.19	odd	2	inner	3200.2.f.p.449.4		4
40.27	even	4		3200.2.d.j.1601.4		4
40.29	even	2	inner	3200.2.f.p.449.1		4
40.37	odd	4		3200.2.d.j.1601.1		4
60.23	odd	4		5760.2.k.t.2881.1		4
80.3	even	4		1280.2.a.l.1.1		2
80.13	odd	4		1280.2.a.l.1.2		2
80.27	even	4		6400.2.a.ca.1.1		2
80.37	odd	4		6400.2.a.ca.1.2		2
80.43	even	4		1280.2.a.h.1.2		2
80.53	odd	4		1280.2.a.h.1.1		2
80.67	even	4		6400.2.a.bz.1.2		2
80.77	odd	4		6400.2.a.bz.1.1		2
120.53	even	4		5760.2.k.t.2881.4		4
120.83	odd	4		5760.2.k.t.2881.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
640.2.d.a.321.1	✓	4	40.3	even	4
640.2.d.a.321.2	yes	4	5.3	odd	4
640.2.d.a.321.3	yes	4	40.13	odd	4
640.2.d.a.321.4	yes	4	20.3	even	4
1280.2.a.h.1.1		2	80.53	odd	4
1280.2.a.h.1.2		2	80.43	even	4
1280.2.a.l.1.1		2	80.3	even	4
1280.2.a.l.1.2		2	80.13	odd	4
3200.2.d.j.1601.1		4	40.37	odd	4
3200.2.d.j.1601.2		4	20.7	even	4
3200.2.d.j.1601.3		4	5.2	odd	4
3200.2.d.j.1601.4		4	40.27	even	4
3200.2.f.p.449.1		4	40.29	even	2	inner
3200.2.f.p.449.2		4	1.1	even	1	trivial
3200.2.f.p.449.3		4	4.3	odd	2	inner
3200.2.f.p.449.4		4	40.19	odd	2	inner
3200.2.f.q.449.1		4	8.3	odd	2
3200.2.f.q.449.2		4	20.19	odd	2
3200.2.f.q.449.3		4	5.4	even	2
3200.2.f.q.449.4		4	8.5	even	2
5760.2.k.t.2881.1		4	60.23	odd	4
5760.2.k.t.2881.2		4	15.8	even	4
5760.2.k.t.2881.3		4	120.83	odd	4
5760.2.k.t.2881.4		4	120.53	even	4
6400.2.a.bz.1.1		2	80.77	odd	4
6400.2.a.bz.1.2		2	80.67	even	4
6400.2.a.ca.1.1		2	80.27	even	4
6400.2.a.ca.1.2		2	80.37	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.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.16228 q^{3} +3.16228i q^{7} +7.00000 q^{9} +O(q^{10})\)	\(q-3.16228 q^{3} +3.16228i q^{7} +7.00000 q^{9} -6.00000 q^{13} -2.00000i q^{17} +6.32456i q^{19} -10.0000i q^{21} -3.16228i q^{23} -12.6491 q^{27} -4.00000i q^{29} -6.32456 q^{31} -2.00000 q^{37} +18.9737 q^{39} -3.16228 q^{43} +9.48683i q^{47} -3.00000 q^{49} +6.32456i q^{51} +6.00000 q^{53} -20.0000i q^{57} +6.32456i q^{59} -2.00000i q^{61} +22.1359i q^{63} -9.48683 q^{67} +10.0000i q^{69} +6.32456 q^{71} +14.0000i q^{73} -12.6491 q^{79} +19.0000 q^{81} +3.16228 q^{83} +12.6491i q^{87} -10.0000 q^{89} -18.9737i q^{91} +20.0000 q^{93} +2.00000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 28 q^{9}+O(q^{10}) \) 4 * q + 28 * q^9	\( 4 q + 28 q^{9} - 24 q^{13} - 8 q^{37} - 12 q^{49} + 24 q^{53} + 76 q^{81} - 40 q^{89} + 80 q^{93}+O(q^{100}) \) 4 * q + 28 * q^9 - 24 * q^13 - 8 * q^37 - 12 * q^49 + 24 * q^53 + 76 * q^81 - 40 * q^89 + 80 * q^93

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