LMFDB - Embedded newform 3920.2.k.d.2351.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3920,2,Mod(2351,3920)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3920, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("3920.2351");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3920 = 2^{4} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3920.k (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:	$31.3013575923$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 6 x^{11} + 43 x^{10} - 160 x^{9} + 572 x^{8} - 1394 x^{7} + 3039 x^{6} - 4844 x^{5} + \cdots + 657 $ x^12 - 6x^11 + 43x^10 - 160x^9 + 572x^8 - 1394x^7 + 3039x^6 - 4844x^5 + 6526x^4 - 6318x^3 + 4737x^2 - 2196*x + 657
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	no (minimal twist has level 560)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2351.10
Root			$0.500000 + 2.22465i$ of defining polynomial
Character	$\chi$	$=$	3920.2351
Dual form			3920.2.k.d.2351.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.35862 q^{3} +1.00000i q^{5} -1.15414 q^{9} +O(q^{10})$	$q+1.35862 q^{3} +1.00000i q^{5} -1.15414 q^{9} -1.40452i q^{11} -4.85382i q^{13} +1.35862i q^{15} +5.88522i q^{17} -3.15414 q^{19} +5.68074i q^{23} -1.00000 q^{25} -5.64392 q^{27} -6.01637 q^{29} -2.46952 q^{31} -1.90821i q^{33} -2.12177 q^{37} -6.59451i q^{39} -8.16113i q^{41} -11.8173i q^{43} -1.15414i q^{45} -2.57249 q^{47} +7.99580i q^{51} +0.236544 q^{53} +1.40452 q^{55} -4.28529 q^{57} -10.3318 q^{59} -9.09613i q^{61} +4.85382 q^{65} +2.64739i q^{67} +7.71799i q^{69} +10.1663i q^{71} +14.1307i q^{73} -1.35862 q^{75} -9.89318i q^{79} -4.20554 q^{81} -2.05288 q^{83} -5.88522 q^{85} -8.17399 q^{87} +10.7608i q^{89} -3.35515 q^{93} -3.15414i q^{95} +9.86223i q^{97} +1.62101i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 4 q^{3} + 20 q^{9}+O(q^{10}) $ 12 * q - 4 * q^3 + 20 * q^9	$ 12 q - 4 q^{3} + 20 q^{9} - 4 q^{19} - 12 q^{25} + 20 q^{27} + 16 q^{29} - 8 q^{31} + 8 q^{37} - 24 q^{47} + 24 q^{53} + 24 q^{55} + 16 q^{57} - 48 q^{59} + 4 q^{65} + 4 q^{75} + 12 q^{81} + 36 q^{83} - 16 q^{85} + 60 q^{87} - 56 q^{93}+O(q^{100}) $ 12 * q - 4 * q^3 + 20 * q^9 - 4 * q^19 - 12 * q^25 + 20 * q^27 + 16 * q^29 - 8 * q^31 + 8 * q^37 - 24 * q^47 + 24 * q^53 + 24 * q^55 + 16 * q^57 - 48 * q^59 + 4 * q^65 + 4 * q^75 + 12 * q^81 + 36 * q^83 - 16 * q^85 + 60 * q^87 - 56 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3920\mathbb{Z}\right)^\times$.

$n$	$981$	$1471$	$3041$	$3137$
$\chi(n)$	$1$	$-1$	$-1$	$1$

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
$3$	1.35862	0.784402	0.392201	−	0.919880i	$-0.371714\pi$
$3$	1.35862	0.784402	0.392201	+	0.919880i	$0.371714\pi$
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−1.15414	−0.384714
$10$	0	0
$11$	− 1.40452i	− 0.423478i	−0.977326	−	0.211739i	$-0.932087\pi$
$11$	− 1.40452i	− 0.423478i	0.977326	−	0.211739i	$-0.0679127\pi$
$12$	0	0
$13$	− 4.85382i	− 1.34621i	−0.739548	−	0.673103i	$-0.764961\pi$
$13$	− 4.85382i	− 1.34621i	0.739548	−	0.673103i	$-0.235039\pi$
$14$	0	0
$15$	1.35862i	0.350795i
$16$	0	0
$17$	5.88522i	1.42738i	0.700464	+	0.713688i	$0.252977\pi$
$17$	5.88522i	1.42738i	−0.700464	+	0.713688i	$0.747023\pi$
$18$	0	0
$19$	−3.15414	−0.723610	−0.361805	−	0.932254i	$-0.617839\pi$
$19$	−3.15414	−0.723610	−0.361805	+	0.932254i	$0.617839\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.68074i	1.18452i	0.805748	+	0.592258i	$0.201763\pi$
$23$	5.68074i	1.18452i	−0.805748	+	0.592258i	$0.798237\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−5.64392	−1.08617
$28$	0	0
$29$	−6.01637	−1.11721	−0.558606	−	0.829433i	$-0.688664\pi$
$29$	−6.01637	−1.11721	−0.558606	+	0.829433i	$0.688664\pi$
$30$	0	0
$31$	−2.46952	−0.443539	−0.221769	−	0.975099i	$-0.571183\pi$
$31$	−2.46952	−0.443539	−0.221769	+	0.975099i	$0.571183\pi$
$32$	0	0
$33$	− 1.90821i	− 0.332177i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.12177	−0.348816	−0.174408	−	0.984673i	$-0.555801\pi$
$37$	−2.12177	−0.348816	−0.174408	+	0.984673i	$0.555801\pi$
$38$	0	0
$39$	− 6.59451i	− 1.05597i
$40$	0	0
$41$	− 8.16113i	− 1.27455i	−0.770635	−	0.637277i	$-0.780061\pi$
$41$	− 8.16113i	− 1.27455i	0.770635	−	0.637277i	$-0.219939\pi$
$42$	0	0
$43$	− 11.8173i	− 1.80212i	−0.433692	−	0.901061i	$-0.642789\pi$
$43$	− 11.8173i	− 1.80212i	0.433692	−	0.901061i	$-0.357211\pi$
$44$	0	0
$45$	− 1.15414i	− 0.172049i
$46$	0	0
$47$	−2.57249	−0.375237	−0.187618	−	0.982242i	$-0.560077\pi$
$47$	−2.57249	−0.375237	−0.187618	+	0.982242i	$0.560077\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	7.99580i	1.11964i
$52$	0	0
$53$	0.236544	0.0324918	0.0162459	−	0.999868i	$-0.494829\pi$
$53$	0.236544	0.0324918	0.0162459	+	0.999868i	$0.494829\pi$
$54$	0	0
$55$	1.40452	0.189385
$56$	0	0
$57$	−4.28529	−0.567601
$58$	0	0
$59$	−10.3318	−1.34508	−0.672540	−	0.740061i	$-0.734797\pi$
$59$	−10.3318	−1.34508	−0.672540	+	0.740061i	$0.734797\pi$
$60$	0	0
$61$	− 9.09613i	− 1.16464i	−0.812960	−	0.582320i	$-0.802145\pi$
$61$	− 9.09613i	− 1.16464i	0.812960	−	0.582320i	$-0.197855\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.85382	0.602042
$66$	0	0
$67$	2.64739i	0.323431i	0.986837	+	0.161715i	$0.0517026\pi$
$67$	2.64739i	0.323431i	−0.986837	+	0.161715i	$0.948297\pi$
$68$	0	0
$69$	7.71799i	0.929137i
$70$	0	0
$71$	10.1663i	1.20652i	0.797545	+	0.603260i	$0.206132\pi$
$71$	10.1663i	1.20652i	−0.797545	+	0.603260i	$0.793868\pi$
$72$	0	0
$73$	14.1307i	1.65387i	0.562296	+	0.826936i	$0.309918\pi$
$73$	14.1307i	1.65387i	−0.562296	+	0.826936i	$0.690082\pi$
$74$	0	0
$75$	−1.35862	−0.156880
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 9.89318i	− 1.11307i	−0.830824	−	0.556535i	$-0.812131\pi$
$79$	− 9.89318i	− 1.11307i	0.830824	−	0.556535i	$-0.187869\pi$
$80$	0	0
$81$	−4.20554	−0.467282
$82$	0	0
$83$	−2.05288	−0.225333	−0.112667	−	0.993633i	$-0.535939\pi$
$83$	−2.05288	−0.225333	−0.112667	+	0.993633i	$0.535939\pi$
$84$	0	0
$85$	−5.88522	−0.638342
$86$	0	0
$87$	−8.17399	−0.876344
$88$	0	0
$89$	10.7608i	1.14064i	0.821421	+	0.570322i	$0.193181\pi$
$89$	10.7608i	1.14064i	−0.821421	+	0.570322i	$0.806819\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−3.35515	−0.347913
$94$	0	0
$95$	− 3.15414i	− 0.323608i
$96$	0	0
$97$	9.86223i	1.00136i	0.865633	+	0.500679i	$0.166916\pi$
$97$	9.86223i	1.00136i	−0.865633	+	0.500679i	$0.833084\pi$
$98$	0	0
$99$	1.62101i	0.162918i
$100$	0	0
$101$	4.17713i	0.415640i	0.978167	+	0.207820i	$0.0666368\pi$
$101$	4.17713i	0.415640i	−0.978167	+	0.207820i	$0.933363\pi$
$102$	0	0
$103$	2.30543	0.227161	0.113580	−	0.993529i	$-0.463768\pi$
$103$	2.30543	0.227161	0.113580	+	0.993529i	$0.463768\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	6.05708i	0.585560i	0.956180	+	0.292780i	$0.0945803\pi$
$107$	6.05708i	0.585560i	−0.956180	+	0.292780i	$0.905420\pi$
$108$	0	0
$109$	−13.9845	−1.33947	−0.669737	−	0.742599i	$-0.733593\pi$
$109$	−13.9845	−1.33947	−0.669737	+	0.742599i	$0.733593\pi$
$110$	0	0
$111$	−2.88268	−0.273612
$112$	0	0
$113$	−4.42306	−0.416086	−0.208043	−	0.978120i	$-0.566709\pi$
$113$	−4.42306	−0.416086	−0.208043	+	0.978120i	$0.566709\pi$
$114$	0	0
$115$	−5.68074	−0.529732
$116$	0	0
$117$	5.60199i	0.517904i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	9.02733	0.820666
$122$	0	0
$123$	− 11.0879i	− 0.999763i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	− 14.5070i	− 1.28729i	−0.765324	−	0.643646i	$-0.777421\pi$
$127$	− 14.5070i	− 1.28729i	0.765324	−	0.643646i	$-0.222579\pi$
$128$	0	0
$129$	− 16.0553i	− 1.41359i
$130$	0	0
$131$	−18.9204	−1.65308	−0.826541	−	0.562877i	$-0.809695\pi$
$131$	−18.9204	−1.65308	−0.826541	+	0.562877i	$0.809695\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 5.64392i	− 0.485751i
$136$	0	0
$137$	−14.2358	−1.21624	−0.608122	−	0.793844i	$-0.708077\pi$
$137$	−14.2358	−1.21624	−0.608122	+	0.793844i	$0.708077\pi$
$138$	0	0
$139$	0.381300	0.0323415	0.0161707	−	0.999869i	$-0.494852\pi$
$139$	0.381300	0.0323415	0.0161707	+	0.999869i	$0.494852\pi$
$140$	0	0
$141$	−3.49505	−0.294336
$142$	0	0
$143$	−6.81727	−0.570089
$144$	0	0
$145$	− 6.01637i	− 0.499633i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	23.4654	1.92236	0.961182	−	0.275915i	$-0.0889809\pi$
$149$	23.4654	1.92236	0.961182	+	0.275915i	$0.0889809\pi$
$150$	0	0
$151$	− 20.3265i	− 1.65414i	−0.562096	−	0.827072i	$-0.690005\pi$
$151$	− 20.3265i	− 1.65414i	0.562096	−	0.827072i	$-0.309995\pi$
$152$	0	0
$153$	− 6.79237i	− 0.549131i
$154$	0	0
$155$	− 2.46952i	− 0.198357i
$156$	0	0
$157$	− 1.95961i	− 0.156394i	−0.996938	−	0.0781969i	$-0.975084\pi$
$157$	− 1.95961i	− 0.156394i	0.996938	−	0.0781969i	$-0.0249163\pi$
$158$	0	0
$159$	0.321375	0.0254867
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 3.50392i	− 0.274448i	−0.990540	−	0.137224i	$-0.956182\pi$
$163$	− 3.50392i	− 0.274448i	0.990540	−	0.137224i	$-0.0438181\pi$
$164$	0	0
$165$	1.90821	0.148554
$166$	0	0
$167$	−1.37463	−0.106372	−0.0531859	−	0.998585i	$-0.516938\pi$
$167$	−1.37463	−0.106372	−0.0531859	+	0.998585i	$0.516938\pi$
$168$	0	0
$169$	−10.5595	−0.812272
$170$	0	0
$171$	3.64032	0.278382
$172$	0	0
$173$	− 6.49745i	− 0.493992i	−0.969017	−	0.246996i	$-0.920557\pi$
$173$	− 6.49745i	− 0.493992i	0.969017	−	0.246996i	$-0.0794434\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−14.0370	−1.05508
$178$	0	0
$179$	2.76180i	0.206427i	0.994659	+	0.103213i	$0.0329124\pi$
$179$	2.76180i	0.206427i	−0.994659	+	0.103213i	$0.967088\pi$
$180$	0	0
$181$	18.5675i	1.38011i	0.723756	+	0.690056i	$0.242414\pi$
$181$	18.5675i	1.38011i	−0.723756	+	0.690056i	$0.757586\pi$
$182$	0	0
$183$	− 12.3582i	− 0.913546i
$184$	0	0
$185$	− 2.12177i	− 0.155995i
$186$	0	0
$187$	8.26590	0.604463
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 4.10576i	− 0.297083i	−0.988906	−	0.148541i	$-0.952542\pi$
$191$	− 4.10576i	− 0.297083i	0.988906	−	0.148541i	$-0.0474578\pi$
$192$	0	0
$193$	−16.7006	−1.20213	−0.601067	−	0.799198i	$-0.705258\pi$
$193$	−16.7006	−1.20213	−0.601067	+	0.799198i	$0.705258\pi$
$194$	0	0
$195$	6.59451	0.472243
$196$	0	0
$197$	8.20826	0.584814	0.292407	−	0.956294i	$-0.405544\pi$
$197$	8.20826	0.584814	0.292407	+	0.956294i	$0.405544\pi$
$198$	0	0
$199$	−4.08402	−0.289509	−0.144754	−	0.989468i	$-0.546239\pi$
$199$	−4.08402	−0.289509	−0.144754	+	0.989468i	$0.546239\pi$
$200$	0	0
$201$	3.59681i	0.253700i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	8.16113	0.569998
$206$	0	0
$207$	− 6.55637i	− 0.455699i
$208$	0	0
$209$	4.43005i	0.306433i
$210$	0	0
$211$	− 22.2041i	− 1.52859i	−0.644866	−	0.764295i	$-0.723087\pi$
$211$	− 22.2041i	− 1.52859i	0.644866	−	0.764295i	$-0.276913\pi$
$212$	0	0
$213$	13.8122i	0.946397i
$214$	0	0
$215$	11.8173	0.805934
$216$	0	0
$217$	0	0
$218$	0	0
$219$	19.1983i	1.29730i
$220$	0	0
$221$	28.5658	1.92154
$222$	0	0
$223$	19.5716	1.31061	0.655307	−	0.755363i	$-0.272539\pi$
$223$	19.5716	1.31061	0.655307	+	0.755363i	$0.272539\pi$
$224$	0	0
$225$	1.15414	0.0769427
$226$	0	0
$227$	−9.33849	−0.619817	−0.309909	−	0.950766i	$-0.600298\pi$
$227$	−9.33849	−0.619817	−0.309909	+	0.950766i	$0.600298\pi$
$228$	0	0
$229$	1.08771i	0.0718780i	0.999354	+	0.0359390i	$0.0114422\pi$
$229$	1.08771i	0.0718780i	−0.999354	+	0.0359390i	$0.988558\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−22.9671	−1.50463	−0.752313	−	0.658806i	$-0.771062\pi$
$233$	−22.9671	−1.50463	−0.752313	+	0.658806i	$0.771062\pi$
$234$	0	0
$235$	− 2.57249i	− 0.167811i
$236$	0	0
$237$	− 13.4411i	− 0.873094i
$238$	0	0
$239$	− 14.3265i	− 0.926701i	−0.886175	−	0.463351i	$-0.846647\pi$
$239$	− 14.3265i	− 0.926701i	0.886175	−	0.463351i	$-0.153353\pi$
$240$	0	0
$241$	7.81348i	0.503311i	0.967817	+	0.251655i	$0.0809750\pi$
$241$	7.81348i	0.503311i	−0.967817	+	0.251655i	$0.919025\pi$
$242$	0	0
$243$	11.2180	0.719635
$244$	0	0
$245$	0	0
$246$	0	0
$247$	15.3096i	0.974128i
$248$	0	0
$249$	−2.78909	−0.176752
$250$	0	0
$251$	−29.2322	−1.84512	−0.922561	−	0.385851i	$-0.873908\pi$
$251$	−29.2322	−1.84512	−0.922561	+	0.385851i	$0.873908\pi$
$252$	0	0
$253$	7.97870	0.501617
$254$	0	0
$255$	−7.99580	−0.500717
$256$	0	0
$257$	21.7922i	1.35936i	0.733509	+	0.679680i	$0.237881\pi$
$257$	21.7922i	1.35936i	−0.733509	+	0.679680i	$0.762119\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	6.94374	0.429807
$262$	0	0
$263$	− 26.2422i	− 1.61816i	−0.587697	−	0.809081i	$-0.699965\pi$
$263$	− 26.2422i	− 1.61816i	0.587697	−	0.809081i	$-0.300035\pi$
$264$	0	0
$265$	0.236544i	0.0145308i
$266$	0	0
$267$	14.6199i	0.894723i
$268$	0	0
$269$	15.1802i	0.925550i	0.886476	+	0.462775i	$0.153146\pi$
$269$	15.1802i	0.925550i	−0.886476	+	0.462775i	$0.846854\pi$
$270$	0	0
$271$	−18.3501	−1.11469	−0.557344	−	0.830282i	$-0.688179\pi$
$271$	−18.3501	−1.11469	−0.557344	+	0.830282i	$0.688179\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.40452i	0.0846956i
$276$	0	0
$277$	0.598871	0.0359827	0.0179913	−	0.999838i	$-0.494273\pi$
$277$	0.598871	0.0359827	0.0179913	+	0.999838i	$0.494273\pi$
$278$	0	0
$279$	2.85017	0.170635
$280$	0	0
$281$	9.80257	0.584772	0.292386	−	0.956300i	$-0.405551\pi$
$281$	9.80257	0.584772	0.292386	+	0.956300i	$0.405551\pi$
$282$	0	0
$283$	−26.1121	−1.55220	−0.776101	−	0.630608i	$-0.782806\pi$
$283$	−26.1121	−1.55220	−0.776101	+	0.630608i	$0.782806\pi$
$284$	0	0
$285$	− 4.28529i	− 0.253839i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−17.6358	−1.03740
$290$	0	0
$291$	13.3991i	0.785467i
$292$	0	0
$293$	− 7.95961i	− 0.465005i	−0.972596	−	0.232503i	$-0.925309\pi$
$293$	− 7.95961i	− 0.465005i	0.972596	−	0.232503i	$-0.0746914\pi$
$294$	0	0
$295$	− 10.3318i	− 0.601538i
$296$	0	0
$297$	7.92698i	0.459970i
$298$	0	0
$299$	27.5733	1.59460
$300$	0	0
$301$	0	0
$302$	0	0
$303$	5.67515i	0.326029i
$304$	0	0
$305$	9.09613	0.520843
$306$	0	0
$307$	−9.49330	−0.541811	−0.270906	−	0.962606i	$-0.587323\pi$
$307$	−9.49330	−0.541811	−0.270906	+	0.962606i	$0.587323\pi$
$308$	0	0
$309$	3.13221	0.178185
$310$	0	0
$311$	−4.80484	−0.272457	−0.136229	−	0.990677i	$-0.543498\pi$
$311$	−4.80484	−0.272457	−0.136229	+	0.990677i	$0.543498\pi$
$312$	0	0
$313$	9.59657i	0.542430i	0.962519	+	0.271215i	$0.0874255\pi$
$313$	9.59657i	0.542430i	−0.962519	+	0.271215i	$0.912575\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	11.9880	0.673314	0.336657	−	0.941627i	$-0.390704\pi$
$317$	11.9880	0.673314	0.336657	+	0.941627i	$0.390704\pi$
$318$	0	0
$319$	8.45010i	0.473115i
$320$	0	0
$321$	8.22929i	0.459314i
$322$	0	0
$323$	− 18.5628i	− 1.03286i
$324$	0	0
$325$	4.85382i	0.269241i
$326$	0	0
$327$	−18.9997	−1.05069
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 20.6373i	− 1.13433i	−0.823604	−	0.567165i	$-0.808040\pi$
$331$	− 20.6373i	− 1.13433i	0.823604	−	0.567165i	$-0.191960\pi$
$332$	0	0
$333$	2.44882	0.134194
$334$	0	0
$335$	−2.64739	−0.144643
$336$	0	0
$337$	26.6075	1.44941	0.724703	−	0.689062i	$-0.241977\pi$
$337$	26.6075	1.44941	0.724703	+	0.689062i	$0.241977\pi$
$338$	0	0
$339$	−6.00927	−0.326379
$340$	0	0
$341$	3.46849i	0.187829i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−7.71799	−0.415523
$346$	0	0
$347$	− 11.5652i	− 0.620854i	−0.950597	−	0.310427i	$-0.899528\pi$
$347$	− 11.5652i	− 0.620854i	0.950597	−	0.310427i	$-0.100472\pi$
$348$	0	0
$349$	1.33101i	0.0712474i	0.999365	+	0.0356237i	$0.0113418\pi$
$349$	1.33101i	0.0712474i	−0.999365	+	0.0356237i	$0.988658\pi$
$350$	0	0
$351$	27.3945i	1.46221i
$352$	0	0
$353$	2.10701i	0.112145i	0.998427	+	0.0560725i	$0.0178578\pi$
$353$	2.10701i	0.112145i	−0.998427	+	0.0560725i	$0.982142\pi$
$354$	0	0
$355$	−10.1663	−0.539572
$356$	0	0
$357$	0	0
$358$	0	0
$359$	20.3234i	1.07263i	0.844019	+	0.536313i	$0.180183\pi$
$359$	20.3234i	1.07263i	−0.844019	+	0.536313i	$0.819817\pi$
$360$	0	0
$361$	−9.05140	−0.476389
$362$	0	0
$363$	12.2647	0.643732
$364$	0	0
$365$	−14.1307	−0.739634
$366$	0	0
$367$	32.7963	1.71195	0.855976	−	0.517015i	$-0.172957\pi$
$367$	32.7963	1.71195	0.855976	+	0.517015i	$0.172957\pi$
$368$	0	0
$369$	9.41909i	0.490338i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	17.7839	0.920816	0.460408	−	0.887707i	$-0.347703\pi$
$373$	17.7839	0.920816	0.460408	+	0.887707i	$0.347703\pi$
$374$	0	0
$375$	− 1.35862i	− 0.0701590i
$376$	0	0
$377$	29.2024i	1.50400i
$378$	0	0
$379$	12.6132i	0.647896i	0.946075	+	0.323948i	$0.105010\pi$
$379$	12.6132i	0.647896i	−0.946075	+	0.323948i	$0.894990\pi$
$380$	0	0
$381$	− 19.7096i	− 1.00975i
$382$	0	0
$383$	5.84772	0.298805	0.149402	−	0.988776i	$-0.452265\pi$
$383$	5.84772	0.298805	0.149402	+	0.988776i	$0.452265\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	13.6388i	0.693301i
$388$	0	0
$389$	14.9301	0.756988	0.378494	−	0.925604i	$-0.376442\pi$
$389$	14.9301	0.756988	0.378494	+	0.925604i	$0.376442\pi$
$390$	0	0
$391$	−33.4324	−1.69075
$392$	0	0
$393$	−25.7057	−1.29668
$394$	0	0
$395$	9.89318	0.497780
$396$	0	0
$397$	13.6505i	0.685100i	0.939500	+	0.342550i	$0.111291\pi$
$397$	13.6505i	0.685100i	−0.939500	+	0.342550i	$0.888709\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	10.6415	0.531409	0.265705	−	0.964054i	$-0.414395\pi$
$401$	10.6415	0.531409	0.265705	+	0.964054i	$0.414395\pi$
$402$	0	0
$403$	11.9866i	0.597095i
$404$	0	0
$405$	− 4.20554i	− 0.208975i
$406$	0	0
$407$	2.98006i	0.147716i
$408$	0	0
$409$	14.5700i	0.720438i	0.932868	+	0.360219i	$0.117298\pi$
$409$	14.5700i	0.720438i	−0.932868	+	0.360219i	$0.882702\pi$
$410$	0	0
$411$	−19.3411	−0.954024
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 2.05288i	− 0.100772i
$416$	0	0
$417$	0.518044	0.0253687
$418$	0	0
$419$	22.9120	1.11932	0.559662	−	0.828721i	$-0.310931\pi$
$419$	22.9120	1.11932	0.559662	+	0.828721i	$0.310931\pi$
$420$	0	0
$421$	−11.0383	−0.537975	−0.268988	−	0.963144i	$-0.586689\pi$
$421$	−11.0383	−0.537975	−0.268988	+	0.963144i	$0.586689\pi$
$422$	0	0
$423$	2.96902	0.144359
$424$	0	0
$425$	− 5.88522i	− 0.285475i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−9.26211	−0.447179
$430$	0	0
$431$	− 10.3024i	− 0.496247i	−0.968728	−	0.248124i	$-0.920186\pi$
$431$	− 10.3024i	− 0.496247i	0.968728	−	0.248124i	$-0.0798139\pi$
$432$	0	0
$433$	2.59940i	0.124919i	0.998047	+	0.0624596i	$0.0198944\pi$
$433$	2.59940i	0.124919i	−0.998047	+	0.0624596i	$0.980106\pi$
$434$	0	0
$435$	− 8.17399i	− 0.391913i
$436$	0	0
$437$	− 17.9178i	− 0.857127i
$438$	0	0
$439$	−24.3499	−1.16216	−0.581079	−	0.813847i	$-0.697369\pi$
$439$	−24.3499	−1.16216	−0.581079	+	0.813847i	$0.697369\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 17.7408i	− 0.842891i	−0.906854	−	0.421446i	$-0.861523\pi$
$443$	− 17.7408i	− 0.842891i	0.906854	−	0.421446i	$-0.138477\pi$
$444$	0	0
$445$	−10.7608	−0.510111
$446$	0	0
$447$	31.8807	1.50791
$448$	0	0
$449$	36.6118	1.72782	0.863909	−	0.503647i	$-0.168009\pi$
$449$	36.6118	1.72782	0.863909	+	0.503647i	$0.168009\pi$
$450$	0	0
$451$	−11.4625	−0.539746
$452$	0	0
$453$	− 27.6160i	− 1.29751i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−5.51915	−0.258175	−0.129087	−	0.991633i	$-0.541205\pi$
$457$	−5.51915	−0.258175	−0.129087	+	0.991633i	$0.541205\pi$
$458$	0	0
$459$	− 33.2157i	− 1.55038i
$460$	0	0
$461$	− 20.3030i	− 0.945604i	−0.881169	−	0.472802i	$-0.843243\pi$
$461$	− 20.3030i	− 0.945604i	0.881169	−	0.472802i	$-0.156757\pi$
$462$	0	0
$463$	− 36.6622i	− 1.70384i	−0.523674	−	0.851919i	$-0.675439\pi$
$463$	− 36.6622i	− 1.70384i	0.523674	−	0.851919i	$-0.324561\pi$
$464$	0	0
$465$	− 3.35515i	− 0.155591i
$466$	0	0
$467$	12.7060	0.587964	0.293982	−	0.955811i	$-0.405019\pi$
$467$	12.7060	0.587964	0.293982	+	0.955811i	$0.405019\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	− 2.66237i	− 0.122676i
$472$	0	0
$473$	−16.5976	−0.763159
$474$	0	0
$475$	3.15414	0.144722
$476$	0	0
$477$	−0.273005	−0.0125001
$478$	0	0
$479$	16.3915	0.748948	0.374474	−	0.927237i	$-0.377823\pi$
$479$	16.3915	0.748948	0.374474	+	0.927237i	$0.377823\pi$
$480$	0	0
$481$	10.2987i	0.469579i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−9.86223	−0.447821
$486$	0	0
$487$	21.1677i	0.959199i	0.877487	+	0.479600i	$0.159218\pi$
$487$	21.1677i	0.959199i	−0.877487	+	0.479600i	$0.840782\pi$
$488$	0	0
$489$	− 4.76051i	− 0.215278i
$490$	0	0
$491$	35.0600i	1.58224i	0.611664	+	0.791118i	$0.290501\pi$
$491$	35.0600i	1.58224i	−0.611664	+	0.791118i	$0.709499\pi$
$492$	0	0
$493$	− 35.4077i	− 1.59468i
$494$	0	0
$495$	−1.62101	−0.0728590
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 18.2199i	− 0.815633i	−0.913064	−	0.407816i	$-0.866290\pi$
$499$	− 18.2199i	− 0.815633i	0.913064	−	0.407816i	$-0.133710\pi$
$500$	0	0
$501$	−1.86760	−0.0834382
$502$	0	0
$503$	43.1348	1.92328	0.961642	−	0.274307i	$-0.0884485\pi$
$503$	43.1348	1.92328	0.961642	+	0.274307i	$0.0884485\pi$
$504$	0	0
$505$	−4.17713	−0.185880
$506$	0	0
$507$	−14.3464	−0.637148
$508$	0	0
$509$	− 16.6502i	− 0.738008i	−0.929428	−	0.369004i	$-0.879699\pi$
$509$	− 16.6502i	− 0.738008i	0.929428	−	0.369004i	$-0.120301\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	17.8017	0.785964
$514$	0	0
$515$	2.30543i	0.101589i
$516$	0	0
$517$	3.61311i	0.158904i
$518$	0	0
$519$	− 8.82759i	− 0.387488i
$520$	0	0
$521$	− 10.5328i	− 0.461452i	−0.973019	−	0.230726i	$-0.925890\pi$
$521$	− 10.5328i	− 0.461452i	0.973019	−	0.230726i	$-0.0741101\pi$
$522$	0	0
$523$	29.0897	1.27200	0.636001	−	0.771688i	$-0.280587\pi$
$523$	29.0897	1.27200	0.636001	+	0.771688i	$0.280587\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 14.5337i	− 0.633097i
$528$	0	0
$529$	−9.27079	−0.403078
$530$	0	0
$531$	11.9243	0.517470
$532$	0	0
$533$	−39.6126	−1.71581
$534$	0	0
$535$	−6.05708	−0.261870
$536$	0	0
$537$	3.75225i	0.161921i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	34.2927	1.47436	0.737178	−	0.675698i	$-0.236158\pi$
$541$	34.2927	1.47436	0.737178	+	0.675698i	$0.236158\pi$
$542$	0	0
$543$	25.2262i	1.08256i
$544$	0	0
$545$	− 13.9845i	− 0.599031i
$546$	0	0
$547$	37.1639i	1.58902i	0.607254	+	0.794508i	$0.292271\pi$
$547$	37.1639i	1.58902i	−0.607254	+	0.794508i	$0.707729\pi$
$548$	0	0
$549$	10.4982i	0.448053i
$550$	0	0
$551$	18.9765	0.808425
$552$	0	0
$553$	0	0
$554$	0	0
$555$	− 2.88268i	− 0.122363i
$556$	0	0
$557$	−35.6255	−1.50950	−0.754750	−	0.656013i	$-0.772242\pi$
$557$	−35.6255	−1.50950	−0.754750	+	0.656013i	$0.772242\pi$
$558$	0	0
$559$	−57.3590	−2.42603
$560$	0	0
$561$	11.2303	0.474142
$562$	0	0
$563$	−21.3297	−0.898940	−0.449470	−	0.893295i	$-0.648387\pi$
$563$	−21.3297	−0.898940	−0.449470	+	0.893295i	$0.648387\pi$
$564$	0	0
$565$	− 4.42306i	− 0.186080i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−18.1167	−0.759489	−0.379745	−	0.925091i	$-0.623988\pi$
$569$	−18.1167	−0.759489	−0.379745	+	0.925091i	$0.623988\pi$
$570$	0	0
$571$	40.4361i	1.69220i	0.533026	+	0.846099i	$0.321055\pi$
$571$	40.4361i	1.69220i	−0.533026	+	0.846099i	$0.678945\pi$
$572$	0	0
$573$	− 5.57819i	− 0.233032i
$574$	0	0
$575$	− 5.68074i	− 0.236903i
$576$	0	0
$577$	20.5730i	0.856466i	0.903668	+	0.428233i	$0.140864\pi$
$577$	20.5730i	0.856466i	−0.903668	+	0.428233i	$0.859136\pi$
$578$	0	0
$579$	−22.6898	−0.942957
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 0.332231i	− 0.0137596i
$584$	0	0
$585$	−5.60199	−0.231614
$586$	0	0
$587$	−16.6666	−0.687904	−0.343952	−	0.938987i	$-0.611766\pi$
$587$	−16.6666	−0.687904	−0.343952	+	0.938987i	$0.611766\pi$
$588$	0	0
$589$	7.78921	0.320949
$590$	0	0
$591$	11.1519	0.458729
$592$	0	0
$593$	2.94213i	0.120819i	0.998174	+	0.0604094i	$0.0192406\pi$
$593$	2.94213i	0.120819i	−0.998174	+	0.0604094i	$0.980759\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−5.54865	−0.227091
$598$	0	0
$599$	43.1824i	1.76438i	0.470891	+	0.882192i	$0.343933\pi$
$599$	43.1824i	1.76438i	−0.470891	+	0.882192i	$0.656067\pi$
$600$	0	0
$601$	42.4270i	1.73063i	0.501226	+	0.865316i	$0.332883\pi$
$601$	42.4270i	1.73063i	−0.501226	+	0.865316i	$0.667117\pi$
$602$	0	0
$603$	− 3.05547i	− 0.124428i
$604$	0	0
$605$	9.02733i	0.367013i
$606$	0	0
$607$	−4.87792	−0.197989	−0.0989943	−	0.995088i	$-0.531563\pi$
$607$	−4.87792	−0.197989	−0.0989943	+	0.995088i	$0.531563\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12.4864i	0.505146i
$612$	0	0
$613$	44.3972	1.79319	0.896594	−	0.442854i	$-0.146034\pi$
$613$	44.3972	1.79319	0.896594	+	0.442854i	$0.146034\pi$
$614$	0	0
$615$	11.0879	0.447108
$616$	0	0
$617$	−10.0274	−0.403690	−0.201845	−	0.979417i	$-0.564694\pi$
$617$	−10.0274	−0.403690	−0.201845	+	0.979417i	$0.564694\pi$
$618$	0	0
$619$	28.9063	1.16184	0.580921	−	0.813960i	$-0.302692\pi$
$619$	28.9063	1.16184	0.580921	+	0.813960i	$0.302692\pi$
$620$	0	0
$621$	− 32.0616i	− 1.28659i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	6.01877i	0.240366i
$628$	0	0
$629$	− 12.4871i	− 0.497892i
$630$	0	0
$631$	40.2236i	1.60127i	0.599149	+	0.800637i	$0.295506\pi$
$631$	40.2236i	1.60127i	−0.599149	+	0.800637i	$0.704494\pi$
$632$	0	0
$633$	− 30.1670i	− 1.19903i
$634$	0	0
$635$	14.5070	0.575694
$636$	0	0
$637$	0	0
$638$	0	0
$639$	− 11.7334i	− 0.464165i
$640$	0	0
$641$	27.5977	1.09004	0.545021	−	0.838422i	$-0.316522\pi$
$641$	27.5977	1.09004	0.545021	+	0.838422i	$0.316522\pi$
$642$	0	0
$643$	10.5963	0.417876	0.208938	−	0.977929i	$-0.432999\pi$
$643$	10.5963	0.417876	0.208938	+	0.977929i	$0.432999\pi$
$644$	0	0
$645$	16.0553	0.632176
$646$	0	0
$647$	27.5785	1.08422	0.542111	−	0.840307i	$-0.317625\pi$
$647$	27.5785	1.08422	0.542111	+	0.840307i	$0.317625\pi$
$648$	0	0
$649$	14.5111i	0.569612i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	24.3915	0.954514	0.477257	−	0.878764i	$-0.341631\pi$
$653$	24.3915	0.954514	0.477257	+	0.878764i	$0.341631\pi$
$654$	0	0
$655$	− 18.9204i	− 0.739281i
$656$	0	0
$657$	− 16.3088i	− 0.636267i
$658$	0	0
$659$	27.3144i	1.06402i	0.846739	+	0.532009i	$0.178563\pi$
$659$	27.3144i	1.06402i	−0.846739	+	0.532009i	$0.821437\pi$
$660$	0	0
$661$	− 22.2539i	− 0.865576i	−0.901496	−	0.432788i	$-0.857530\pi$
$661$	− 22.2539i	− 0.865576i	0.901496	−	0.432788i	$-0.142470\pi$
$662$	0	0
$663$	38.8102	1.50726
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 34.1774i	− 1.32336i
$668$	0	0
$669$	26.5905	1.02805
$670$	0	0
$671$	−12.7757	−0.493199
$672$	0	0
$673$	−24.5193	−0.945150	−0.472575	−	0.881291i	$-0.656675\pi$
$673$	−24.5193	−0.945150	−0.472575	+	0.881291i	$0.656675\pi$
$674$	0	0
$675$	5.64392	0.217234
$676$	0	0
$677$	− 39.7486i	− 1.52766i	−0.645416	−	0.763831i	$-0.723316\pi$
$677$	− 39.7486i	− 1.52766i	0.645416	−	0.763831i	$-0.276684\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−12.6875	−0.486186
$682$	0	0
$683$	0.144817i	0.00554128i	0.999996	+	0.00277064i	$0.000881924\pi$
$683$	0.144817i	0.00554128i	−0.999996	+	0.00277064i	$0.999118\pi$
$684$	0	0
$685$	− 14.2358i	− 0.543921i
$686$	0	0
$687$	1.47779i	0.0563813i
$688$	0	0
$689$	− 1.14814i	− 0.0437407i
$690$	0	0
$691$	−28.4140	−1.08092	−0.540461	−	0.841369i	$-0.681750\pi$
$691$	−28.4140	−1.08092	−0.540461	+	0.841369i	$0.681750\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0.381300i	0.0144635i
$696$	0	0
$697$	48.0301	1.81927
$698$	0	0
$699$	−31.2037	−1.18023
$700$	0	0
$701$	29.7062	1.12199	0.560993	−	0.827820i	$-0.310419\pi$
$701$	29.7062	1.12199	0.560993	+	0.827820i	$0.310419\pi$
$702$	0	0
$703$	6.69235	0.252407
$704$	0	0
$705$	− 3.49505i	− 0.131631i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−36.9605	−1.38808	−0.694040	−	0.719936i	$-0.744171\pi$
$709$	−36.9605	−1.38808	−0.694040	+	0.719936i	$0.744171\pi$
$710$	0	0
$711$	11.4181i	0.428213i
$712$	0	0
$713$	− 14.0287i	− 0.525379i
$714$	0	0
$715$	− 6.81727i	− 0.254952i
$716$	0	0
$717$	− 19.4643i	− 0.726906i
$718$	0	0
$719$	−20.0606	−0.748134	−0.374067	−	0.927402i	$-0.622037\pi$
$719$	−20.0606	−0.748134	−0.374067	+	0.927402i	$0.622037\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	10.6156i	0.394798i
$724$	0	0
$725$	6.01637	0.223442
$726$	0	0
$727$	−49.1552	−1.82307	−0.911533	−	0.411226i	$-0.865101\pi$
$727$	−49.1552	−1.82307	−0.911533	+	0.411226i	$0.865101\pi$
$728$	0	0
$729$	27.8577	1.03177
$730$	0	0
$731$	69.5475	2.57231
$732$	0	0
$733$	− 33.1883i	− 1.22584i	−0.790145	−	0.612919i	$-0.789995\pi$
$733$	− 33.1883i	− 1.22584i	0.790145	−	0.612919i	$-0.210005\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.71831	0.136966
$738$	0	0
$739$	16.8142i	0.618520i	0.950977	+	0.309260i	$0.100081\pi$
$739$	16.8142i	0.618520i	−0.950977	+	0.309260i	$0.899919\pi$
$740$	0	0
$741$	20.8000i	0.764108i
$742$	0	0
$743$	34.8645i	1.27906i	0.768768	+	0.639528i	$0.220870\pi$
$743$	34.8645i	1.27906i	−0.768768	+	0.639528i	$0.779130\pi$
$744$	0	0
$745$	23.4654i	0.859707i
$746$	0	0
$747$	2.36931	0.0866887
$748$	0	0
$749$	0	0
$750$	0	0
$751$	11.9552i	0.436252i	0.975921	+	0.218126i	$0.0699944\pi$
$751$	11.9552i	0.436252i	−0.975921	+	0.218126i	$0.930006\pi$
$752$	0	0
$753$	−39.7156	−1.44732
$754$	0	0
$755$	20.3265	0.739755
$756$	0	0
$757$	−37.3810	−1.35863	−0.679317	−	0.733845i	$-0.737724\pi$
$757$	−37.3810	−1.35863	−0.679317	+	0.733845i	$0.737724\pi$
$758$	0	0
$759$	10.8401	0.393469
$760$	0	0
$761$	49.9640i	1.81119i	0.424139	+	0.905597i	$0.360577\pi$
$761$	49.9640i	1.81119i	−0.424139	+	0.905597i	$0.639423\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	6.79237	0.245579
$766$	0	0
$767$	50.1484i	1.81075i
$768$	0	0
$769$	− 31.6786i	− 1.14236i	−0.820825	−	0.571179i	$-0.806486\pi$
$769$	− 31.6786i	− 1.14236i	0.820825	−	0.571179i	$-0.193514\pi$
$770$	0	0
$771$	29.6074i	1.06628i
$772$	0	0
$773$	42.8353i	1.54068i	0.637635	+	0.770339i	$0.279913\pi$
$773$	42.8353i	1.54068i	−0.637635	+	0.770339i	$0.720087\pi$
$774$	0	0
$775$	2.46952	0.0887078
$776$	0	0
$777$	0	0
$778$	0	0
$779$	25.7413i	0.922280i
$780$	0	0
$781$	14.2788	0.510935
$782$	0	0
$783$	33.9559	1.21348
$784$	0	0
$785$	1.95961	0.0699414
$786$	0	0
$787$	20.5752	0.733426	0.366713	−	0.930334i	$-0.380483\pi$
$787$	20.5752	0.733426	0.366713	+	0.930334i	$0.380483\pi$
$788$	0	0
$789$	− 35.6533i	− 1.26929i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−44.1509	−1.56785
$794$	0	0
$795$	0.321375i	0.0113980i
$796$	0	0
$797$	− 0.739835i	− 0.0262063i	−0.999914	−	0.0131032i	$-0.995829\pi$
$797$	− 0.739835i	− 0.0262063i	0.999914	−	0.0131032i	$-0.00417098\pi$
$798$	0	0
$799$	− 15.1397i	− 0.535604i
$800$	0	0
$801$	− 12.4195i	− 0.438821i
$802$	0	0
$803$	19.8468	0.700379
$804$	0	0
$805$	0	0
$806$	0	0
$807$	20.6241i	0.726003i
$808$	0	0
$809$	−40.8951	−1.43780	−0.718898	−	0.695115i	$-0.755353\pi$
$809$	−40.8951	−1.43780	−0.718898	+	0.695115i	$0.755353\pi$
$810$	0	0
$811$	−8.03340	−0.282091	−0.141045	−	0.990003i	$-0.545046\pi$
$811$	−8.03340	−0.282091	−0.141045	+	0.990003i	$0.545046\pi$
$812$	0	0
$813$	−24.9308	−0.874363
$814$	0	0
$815$	3.50392	0.122737
$816$	0	0
$817$	37.2735i	1.30403i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−14.2115	−0.495986	−0.247993	−	0.968762i	$-0.579771\pi$
$821$	−14.2115	−0.495986	−0.247993	+	0.968762i	$0.579771\pi$
$822$	0	0
$823$	− 16.3437i	− 0.569707i	−0.958571	−	0.284853i	$-0.908055\pi$
$823$	− 16.3437i	− 0.569707i	0.958571	−	0.284853i	$-0.0919449\pi$
$824$	0	0
$825$	1.90821i	0.0664354i
$826$	0	0
$827$	− 14.6839i	− 0.510609i	−0.966861	−	0.255305i	$-0.917824\pi$
$827$	− 14.6839i	− 0.510609i	0.966861	−	0.255305i	$-0.0821758\pi$
$828$	0	0
$829$	15.0857i	0.523946i	0.965075	+	0.261973i	$0.0843732\pi$
$829$	15.0857i	0.523946i	−0.965075	+	0.261973i	$0.915627\pi$
$830$	0	0
$831$	0.813640	0.0282249
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 1.37463i	− 0.0475709i
$836$	0	0
$837$	13.9378	0.481759
$838$	0	0
$839$	20.9688	0.723924	0.361962	−	0.932193i	$-0.382107\pi$
$839$	20.9688	0.723924	0.361962	+	0.932193i	$0.382107\pi$
$840$	0	0
$841$	7.19674	0.248163
$842$	0	0
$843$	13.3180	0.458696
$844$	0	0
$845$	− 10.5595i	− 0.363259i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−35.4765	−1.21755
$850$	0	0
$851$	− 12.0532i	− 0.413178i
$852$	0	0
$853$	− 24.6110i	− 0.842665i	−0.906906	−	0.421333i	$-0.861562\pi$
$853$	− 24.6110i	− 0.842665i	0.906906	−	0.421333i	$-0.138438\pi$
$854$	0	0
$855$	3.64032i	0.124496i
$856$	0	0
$857$	− 28.7218i	− 0.981119i	−0.871408	−	0.490559i	$-0.836792\pi$
$857$	− 28.7218i	− 0.981119i	0.871408	−	0.490559i	$-0.163208\pi$
$858$	0	0
$859$	38.9017	1.32731	0.663655	−	0.748039i	$-0.269004\pi$
$859$	38.9017	1.32731	0.663655	+	0.748039i	$0.269004\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 53.7150i	− 1.82848i	−0.405172	−	0.914240i	$-0.632789\pi$
$863$	− 53.7150i	− 1.82848i	0.405172	−	0.914240i	$-0.367211\pi$
$864$	0	0
$865$	6.49745	0.220920
$866$	0	0
$867$	−23.9605	−0.813740
$868$	0	0
$869$	−13.8951	−0.471361
$870$	0	0
$871$	12.8500	0.435405
$872$	0	0
$873$	− 11.3824i	− 0.385236i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−26.0205	−0.878648	−0.439324	−	0.898329i	$-0.644782\pi$
$877$	−26.0205	−0.878648	−0.439324	+	0.898329i	$0.644782\pi$
$878$	0	0
$879$	− 10.8141i	− 0.364751i
$880$	0	0
$881$	− 29.9674i	− 1.00963i	−0.863228	−	0.504814i	$-0.831561\pi$
$881$	− 29.9674i	− 1.00963i	0.863228	−	0.504814i	$-0.168439\pi$
$882$	0	0
$883$	− 47.1260i	− 1.58592i	−0.609277	−	0.792958i	$-0.708540\pi$
$883$	− 47.1260i	− 1.58592i	0.609277	−	0.792958i	$-0.291460\pi$
$884$	0	0
$885$	− 14.0370i	− 0.471847i
$886$	0	0
$887$	−23.1194	−0.776274	−0.388137	−	0.921602i	$-0.626881\pi$
$887$	−23.1194	−0.776274	−0.388137	+	0.921602i	$0.626881\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	5.90675i	0.197884i
$892$	0	0
$893$	8.11400	0.271525
$894$	0	0
$895$	−2.76180	−0.0923168
$896$	0	0
$897$	37.4617	1.25081
$898$	0	0
$899$	14.8576	0.495527
$900$	0	0
$901$	1.39212i	0.0463781i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−18.5675	−0.617204
$906$	0	0
$907$	− 8.07340i	− 0.268073i	−0.990976	−	0.134036i	$-0.957206\pi$
$907$	− 8.07340i	− 0.268073i	0.990976	−	0.134036i	$-0.0427939\pi$
$908$	0	0
$909$	− 4.82100i	− 0.159902i
$910$	0	0
$911$	48.9927i	1.62320i	0.584213	+	0.811600i	$0.301403\pi$
$911$	48.9927i	1.62320i	−0.584213	+	0.811600i	$0.698597\pi$
$912$	0	0
$913$	2.88331i	0.0954236i
$914$	0	0
$915$	12.3582	0.408550
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 31.4639i	− 1.03790i	−0.854806	−	0.518948i	$-0.826324\pi$
$919$	− 31.4639i	− 1.03790i	0.854806	−	0.518948i	$-0.173676\pi$
$920$	0	0
$921$	−12.8978	−0.424998
$922$	0	0
$923$	49.3454	1.62423
$924$	0	0
$925$	2.12177	0.0697632
$926$	0	0
$927$	−2.66079	−0.0873918
$928$	0	0
$929$	− 10.2934i	− 0.337716i	−0.985640	−	0.168858i	$-0.945992\pi$
$929$	− 10.2934i	− 0.337716i	0.985640	−	0.168858i	$-0.0540080\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−6.52797	−0.213716
$934$	0	0
$935$	8.26590i	0.270324i
$936$	0	0
$937$	9.74494i	0.318353i	0.987250	+	0.159177i	$0.0508839\pi$
$937$	9.74494i	0.318353i	−0.987250	+	0.159177i	$0.949116\pi$
$938$	0	0
$939$	13.0381i	0.425483i
$940$	0	0
$941$	0.662543i	0.0215983i	0.999942	+	0.0107991i	$0.00343754\pi$
$941$	0.662543i	0.0215983i	−0.999942	+	0.0107991i	$0.996562\pi$
$942$	0	0
$943$	46.3612	1.50973
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 10.8781i	− 0.353492i	−0.984256	−	0.176746i	$-0.943443\pi$
$947$	− 10.8781i	− 0.353492i	0.984256	−	0.176746i	$-0.0565571\pi$
$948$	0	0
$949$	68.5878	2.22645
$950$	0	0
$951$	16.2872	0.528149
$952$	0	0
$953$	58.0080	1.87906	0.939531	−	0.342463i	$-0.111261\pi$
$953$	58.0080	1.87906	0.939531	+	0.342463i	$0.111261\pi$
$954$	0	0
$955$	4.10576	0.132859
$956$	0	0
$957$	11.4805i	0.371112i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−24.9015	−0.803273
$962$	0	0
$963$	− 6.99072i	− 0.225273i
$964$	0	0
$965$	− 16.7006i	− 0.537611i
$966$	0	0
$967$	13.3474i	0.429222i	0.976700	+	0.214611i	$0.0688484\pi$
$967$	13.3474i	0.429222i	−0.976700	+	0.214611i	$0.931152\pi$
$968$	0	0
$969$	− 25.2199i	− 0.810180i
$970$	0	0
$971$	−41.9613	−1.34660	−0.673300	−	0.739369i	$-0.735124\pi$
$971$	−41.9613	−1.34660	−0.673300	+	0.739369i	$0.735124\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	6.59451i	0.211193i
$976$	0	0
$977$	−3.23243	−0.103415	−0.0517074	−	0.998662i	$-0.516466\pi$
$977$	−3.23243	−0.103415	−0.0517074	+	0.998662i	$0.516466\pi$
$978$	0	0
$979$	15.1137	0.483037
$980$	0	0
$981$	16.1401	0.515314
$982$	0	0
$983$	−20.0060	−0.638092	−0.319046	−	0.947739i	$-0.603362\pi$
$983$	−20.0060	−0.638092	−0.319046	+	0.947739i	$0.603362\pi$
$984$	0	0
$985$	8.20826i	0.261537i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	67.1310	2.13464
$990$	0	0
$991$	26.3713i	0.837711i	0.908053	+	0.418856i	$0.137569\pi$
$991$	26.3713i	0.837711i	−0.908053	+	0.418856i	$0.862431\pi$
$992$	0	0
$993$	− 28.0384i	− 0.889771i
$994$	0	0
$995$	− 4.08402i	− 0.129472i
$996$	0	0
$997$	− 32.0692i	− 1.01564i	−0.861463	−	0.507821i	$-0.830451\pi$
$997$	− 32.0692i	− 1.01564i	0.861463	−	0.507821i	$-0.169549\pi$
$998$	0	0
$999$	11.9751	0.378874

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3920.2.k.d.2351.10		12
4.3	odd	2		3920.2.k.e.2351.4		12
7.4	even	3		560.2.bs.c.271.2	yes	12
7.5	odd	6		560.2.bs.b.31.5	✓	12
7.6	odd	2		3920.2.k.e.2351.3		12
28.11	odd	6		560.2.bs.b.271.5	yes	12
28.19	even	6		560.2.bs.c.31.2	yes	12
28.27	even	2	inner	3920.2.k.d.2351.9		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
560.2.bs.b.31.5	✓	12	7.5	odd	6
560.2.bs.b.271.5	yes	12	28.11	odd	6
560.2.bs.c.31.2	yes	12	28.19	even	6
560.2.bs.c.271.2	yes	12	7.4	even	3
3920.2.k.d.2351.9		12	28.27	even	2	inner
3920.2.k.d.2351.10		12	1.1	even	1	trivial
3920.2.k.e.2351.3		12	7.6	odd	2
3920.2.k.e.2351.4		12	4.3	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:	\( N \)	\(=\)	\( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3920.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.35862 q^{3} +1.00000i q^{5} -1.15414 q^{9} +O(q^{10})\)	\(q+1.35862 q^{3} +1.00000i q^{5} -1.15414 q^{9} -1.40452i q^{11} -4.85382i q^{13} +1.35862i q^{15} +5.88522i q^{17} -3.15414 q^{19} +5.68074i q^{23} -1.00000 q^{25} -5.64392 q^{27} -6.01637 q^{29} -2.46952 q^{31} -1.90821i q^{33} -2.12177 q^{37} -6.59451i q^{39} -8.16113i q^{41} -11.8173i q^{43} -1.15414i q^{45} -2.57249 q^{47} +7.99580i q^{51} +0.236544 q^{53} +1.40452 q^{55} -4.28529 q^{57} -10.3318 q^{59} -9.09613i q^{61} +4.85382 q^{65} +2.64739i q^{67} +7.71799i q^{69} +10.1663i q^{71} +14.1307i q^{73} -1.35862 q^{75} -9.89318i q^{79} -4.20554 q^{81} -2.05288 q^{83} -5.88522 q^{85} -8.17399 q^{87} +10.7608i q^{89} -3.35515 q^{93} -3.15414i q^{95} +9.86223i q^{97} +1.62101i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 4 q^{3} + 20 q^{9}+O(q^{10}) \) 12 * q - 4 * q^3 + 20 * q^9	\( 12 q - 4 q^{3} + 20 q^{9} - 4 q^{19} - 12 q^{25} + 20 q^{27} + 16 q^{29} - 8 q^{31} + 8 q^{37} - 24 q^{47} + 24 q^{53} + 24 q^{55} + 16 q^{57} - 48 q^{59} + 4 q^{65} + 4 q^{75} + 12 q^{81} + 36 q^{83} - 16 q^{85} + 60 q^{87} - 56 q^{93}+O(q^{100}) \) 12 * q - 4 * q^3 + 20 * q^9 - 4 * q^19 - 12 * q^25 + 20 * q^27 + 16 * q^29 - 8 * q^31 + 8 * q^37 - 24 * q^47 + 24 * q^53 + 24 * q^55 + 16 * q^57 - 48 * q^59 + 4 * q^65 + 4 * q^75 + 12 * q^81 + 36 * q^83 - 16 * q^85 + 60 * q^87 - 56 * q^93

Introduction

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