LMFDB - Embedded newform 3920.2.k.a.2351.8

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:	$8$
Coefficient field:	$\Q(\zeta_{16})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 1 $ x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2351.8
Root			$0.923880 + 0.382683i$ of defining polynomial
Character	$\chi$	$=$	3920.2351
Dual form			3920.2.k.a.2351.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.26197 q^{3} +1.00000i q^{5} +2.11652 q^{9} +O(q^{10})$	$q+2.26197 q^{3} +1.00000i q^{5} +2.11652 q^{9} -1.28130i q^{11} -2.26197i q^{13} +2.26197i q^{15} +0.0698487i q^{17} +2.44834 q^{19} -2.21824i q^{23} -1.00000 q^{25} -1.99841 q^{27} +1.98361 q^{29} +6.31703 q^{31} -2.89828i q^{33} +8.48822 q^{37} -5.11652i q^{39} -4.20692i q^{41} +5.01095i q^{43} +2.11652i q^{45} +10.6211 q^{47} +0.157996i q^{51} +4.01480 q^{53} +1.28130 q^{55} +5.53808 q^{57} +6.89149 q^{59} -6.73925i q^{61} +2.26197 q^{65} -11.1519i q^{67} -5.01761i q^{69} +14.5335i q^{71} +11.8490i q^{73} -2.26197 q^{75} -5.86709i q^{79} -10.8699 q^{81} -5.79655 q^{83} -0.0698487 q^{85} +4.48687 q^{87} -17.5854i q^{89} +14.2889 q^{93} +2.44834i q^{95} +7.06081i q^{97} -2.71191i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} + 16 q^{9}+O(q^{10}) $ 8 * q - 8 * q^3 + 16 * q^9	$ 8 q - 8 q^{3} + 16 q^{9} + 16 q^{19} - 8 q^{25} - 8 q^{27} + 8 q^{29} + 48 q^{31} + 24 q^{47} + 32 q^{53} - 8 q^{55} - 32 q^{57} + 16 q^{59} - 8 q^{65} + 8 q^{75} - 8 q^{81} - 48 q^{83} - 24 q^{85} - 24 q^{87} - 32 q^{93}+O(q^{100}) $ 8 * q - 8 * q^3 + 16 * q^9 + 16 * q^19 - 8 * q^25 - 8 * q^27 + 8 * q^29 + 48 * q^31 + 24 * q^47 + 32 * q^53 - 8 * q^55 - 32 * q^57 + 16 * q^59 - 8 * q^65 + 8 * q^75 - 8 * q^81 - 48 * q^83 - 24 * q^85 - 24 * q^87 - 32 * 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
$3$	2.26197	1.30595	0.652975	−	0.757379i	$-0.273521\pi$
$3$	2.26197	1.30595	0.652975	+	0.757379i	$0.273521\pi$
$4$	0	0
$5$	1.00000i	0.447214i
$5$	1.00000i	0.447214i
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	2.11652	0.705507
$10$	0	0
$11$	− 1.28130i	− 0.386328i	−0.981166	−	0.193164i	$-0.938125\pi$
$11$	− 1.28130i	− 0.386328i	0.981166	−	0.193164i	$-0.0618749\pi$
$12$	0	0
$13$	− 2.26197i	− 0.627358i	−0.949529	−	0.313679i	$-0.898438\pi$
$13$	− 2.26197i	− 0.627358i	0.949529	−	0.313679i	$-0.101562\pi$
$14$	0	0
$15$	2.26197i	0.584039i
$16$	0	0
$17$	0.0698487i	0.0169408i	0.999964	+	0.00847040i	$0.00269625\pi$
$17$	0.0698487i	0.0169408i	−0.999964	+	0.00847040i	$0.997304\pi$
$18$	0	0
$19$	2.44834	0.561688	0.280844	−	0.959753i	$-0.409386\pi$
$19$	2.44834	0.561688	0.280844	+	0.959753i	$0.409386\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 2.21824i	− 0.462536i	−0.972890	−	0.231268i	$-0.925713\pi$
$23$	− 2.21824i	− 0.462536i	0.972890	−	0.231268i	$-0.0742874\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−1.99841	−0.384594
$28$	0	0
$29$	1.98361	0.368347	0.184174	−	0.982894i	$-0.441039\pi$
$29$	1.98361	0.368347	0.184174	+	0.982894i	$0.441039\pi$
$30$	0	0
$31$	6.31703	1.13457	0.567286	−	0.823521i	$-0.307994\pi$
$31$	6.31703	1.13457	0.567286	+	0.823521i	$0.307994\pi$
$32$	0	0
$33$	− 2.89828i	− 0.504525i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	8.48822	1.39546	0.697728	−	0.716363i	$-0.254195\pi$
$37$	8.48822	1.39546	0.697728	+	0.716363i	$0.254195\pi$
$38$	0	0
$39$	− 5.11652i	− 0.819299i
$40$	0	0
$41$	− 4.20692i	− 0.657011i	−0.944502	−	0.328505i	$-0.893455\pi$
$41$	− 4.20692i	− 0.657011i	0.944502	−	0.328505i	$-0.106545\pi$
$42$	0	0
$43$	5.01095i	0.764163i	0.924129	+	0.382081i	$0.124793\pi$
$43$	5.01095i	0.764163i	−0.924129	+	0.382081i	$0.875207\pi$
$44$	0	0
$45$	2.11652i	0.315512i
$46$	0	0
$47$	10.6211	1.54925	0.774626	−	0.632420i	$-0.217938\pi$
$47$	10.6211	1.54925	0.774626	+	0.632420i	$0.217938\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0.157996i	0.0221239i
$52$	0	0
$53$	4.01480	0.551475	0.275737	−	0.961233i	$-0.411078\pi$
$53$	4.01480	0.551475	0.275737	+	0.961233i	$0.411078\pi$
$54$	0	0
$55$	1.28130	0.172771
$56$	0	0
$57$	5.53808	0.733537
$58$	0	0
$59$	6.89149	0.897195	0.448598	−	0.893734i	$-0.351924\pi$
$59$	6.89149	0.897195	0.448598	+	0.893734i	$0.351924\pi$
$60$	0	0
$61$	− 6.73925i	− 0.862872i	−0.902144	−	0.431436i	$-0.858007\pi$
$61$	− 6.73925i	− 0.862872i	0.902144	−	0.431436i	$-0.141993\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2.26197	0.280563
$66$	0	0
$67$	− 11.1519i	− 1.36242i	−0.732089	−	0.681209i	$-0.761455\pi$
$67$	− 11.1519i	− 1.36242i	0.732089	−	0.681209i	$-0.238545\pi$
$68$	0	0
$69$	− 5.01761i	− 0.604049i
$70$	0	0
$71$	14.5335i	1.72481i	0.506215	+	0.862407i	$0.331044\pi$
$71$	14.5335i	1.72481i	−0.506215	+	0.862407i	$0.668956\pi$
$72$	0	0
$73$	11.8490i	1.38682i	0.720544	+	0.693409i	$0.243892\pi$
$73$	11.8490i	1.38682i	−0.720544	+	0.693409i	$0.756108\pi$
$74$	0	0
$75$	−2.26197	−0.261190
$76$	0	0
$77$	0	0
$78$	0	0
$79$	− 5.86709i	− 0.660099i	−0.943964	−	0.330050i	$-0.892935\pi$
$79$	− 5.86709i	− 0.660099i	0.943964	−	0.330050i	$-0.107065\pi$
$80$	0	0
$81$	−10.8699	−1.20777
$82$	0	0
$83$	−5.79655	−0.636254	−0.318127	−	0.948048i	$-0.603054\pi$
$83$	−5.79655	−0.636254	−0.318127	+	0.948048i	$0.603054\pi$
$84$	0	0
$85$	−0.0698487	−0.00757616
$86$	0	0
$87$	4.48687	0.481043
$88$	0	0
$89$	− 17.5854i	− 1.86405i	−0.362394	−	0.932025i	$-0.618041\pi$
$89$	− 17.5854i	− 1.86405i	0.362394	−	0.932025i	$-0.381959\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	14.2889	1.48169
$94$	0	0
$95$	2.44834i	0.251195i
$96$	0	0
$97$	7.06081i	0.716916i	0.933546	+	0.358458i	$0.116697\pi$
$97$	7.06081i	0.716916i	−0.933546	+	0.358458i	$0.883303\pi$
$98$	0	0
$99$	− 2.71191i	− 0.272557i
$100$	0	0
$101$	5.98226i	0.595257i	0.954682	+	0.297629i	$0.0961957\pi$
$101$	5.98226i	0.595257i	−0.954682	+	0.297629i	$0.903804\pi$
$102$	0	0
$103$	9.29610	0.915972	0.457986	−	0.888959i	$-0.348571\pi$
$103$	9.29610	0.915972	0.457986	+	0.888959i	$0.348571\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.50981i	0.339306i	0.985504	+	0.169653i	$0.0542647\pi$
$107$	3.50981i	0.339306i	−0.985504	+	0.169653i	$0.945735\pi$
$108$	0	0
$109$	7.28772	0.698037	0.349018	−	0.937116i	$-0.386515\pi$
$109$	7.28772	0.698037	0.349018	+	0.937116i	$0.386515\pi$
$110$	0	0
$111$	19.2001	1.82240
$112$	0	0
$113$	−1.14771	−0.107967	−0.0539835	−	0.998542i	$-0.517192\pi$
$113$	−1.14771	−0.107967	−0.0539835	+	0.998542i	$0.517192\pi$
$114$	0	0
$115$	2.21824	0.206852
$116$	0	0
$117$	− 4.78751i	− 0.442606i
$118$	0	0
$119$	0	0
$120$	0	0
$121$	9.35826	0.850751
$122$	0	0
$123$	− 9.51594i	− 0.858023i
$124$	0	0
$125$	− 1.00000i	− 0.0894427i
$126$	0	0
$127$	− 14.9398i	− 1.32569i	−0.748757	−	0.662844i	$-0.769349\pi$
$127$	− 14.9398i	− 1.32569i	0.748757	−	0.662844i	$-0.230651\pi$
$128$	0	0
$129$	11.3346i	0.997959i
$130$	0	0
$131$	−17.9275	−1.56634	−0.783168	−	0.621811i	$-0.786397\pi$
$131$	−17.9275	−1.56634	−0.783168	+	0.621811i	$0.786397\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	− 1.99841i	− 0.171996i
$136$	0	0
$137$	−12.6382	−1.07976	−0.539878	−	0.841743i	$-0.681530\pi$
$137$	−12.6382	−1.07976	−0.539878	+	0.841743i	$0.681530\pi$
$138$	0	0
$139$	−12.3894	−1.05086	−0.525429	−	0.850837i	$-0.676095\pi$
$139$	−12.3894	−1.05086	−0.525429	+	0.850837i	$0.676095\pi$
$140$	0	0
$141$	24.0247	2.02325
$142$	0	0
$143$	−2.89828	−0.242366
$144$	0	0
$145$	1.98361i	0.164730i
$146$	0	0
$147$	0	0
$148$	0	0
$149$	6.40424	0.524656	0.262328	−	0.964979i	$-0.415510\pi$
$149$	6.40424	0.524656	0.262328	+	0.964979i	$0.415510\pi$
$150$	0	0
$151$	23.6469i	1.92436i	0.272422	+	0.962178i	$0.412175\pi$
$151$	23.6469i	1.92436i	−0.272422	+	0.962178i	$0.587825\pi$
$152$	0	0
$153$	0.147836i	0.0119519i
$154$	0	0
$155$	6.31703i	0.507396i
$156$	0	0
$157$	2.34315i	0.187003i	0.995619	+	0.0935017i	$0.0298061\pi$
$157$	2.34315i	0.187003i	−0.995619	+	0.0935017i	$0.970194\pi$
$158$	0	0
$159$	9.08136	0.720199
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 0.918827i	− 0.0719681i	−0.999352	−	0.0359840i	$-0.988543\pi$
$163$	− 0.918827i	− 0.0719681i	0.999352	−	0.0359840i	$-0.0114565\pi$
$164$	0	0
$165$	2.89828	0.225630
$166$	0	0
$167$	−13.6234	−1.05421	−0.527106	−	0.849800i	$-0.676723\pi$
$167$	−13.6234	−1.05421	−0.527106	+	0.849800i	$0.676723\pi$
$168$	0	0
$169$	7.88348	0.606422
$170$	0	0
$171$	5.18196	0.396275
$172$	0	0
$173$	3.92790i	0.298633i	0.988789	+	0.149316i	$0.0477073\pi$
$173$	3.92790i	0.298633i	−0.988789	+	0.149316i	$0.952293\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	15.5884	1.17169
$178$	0	0
$179$	− 0.999626i	− 0.0747156i	−0.999302	−	0.0373578i	$-0.988106\pi$
$179$	− 0.999626i	− 0.0747156i	0.999302	−	0.0373578i	$-0.0118941\pi$
$180$	0	0
$181$	1.09666i	0.0815141i	0.999169	+	0.0407570i	$0.0129770\pi$
$181$	1.09666i	0.0815141i	−0.999169	+	0.0407570i	$0.987023\pi$
$182$	0	0
$183$	− 15.2440i	− 1.12687i
$184$	0	0
$185$	8.48822i	0.624067i
$186$	0	0
$187$	0.0894975	0.00654471
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 5.36558i	− 0.388239i	−0.980978	−	0.194120i	$-0.937815\pi$
$191$	− 5.36558i	− 0.388239i	0.980978	−	0.194120i	$-0.0621850\pi$
$192$	0	0
$193$	13.2036	0.950416	0.475208	−	0.879873i	$-0.342373\pi$
$193$	13.2036	0.950416	0.475208	+	0.879873i	$0.342373\pi$
$194$	0	0
$195$	5.11652	0.366402
$196$	0	0
$197$	23.1264	1.64769	0.823845	−	0.566815i	$-0.191825\pi$
$197$	23.1264	1.64769	0.823845	+	0.566815i	$0.191825\pi$
$198$	0	0
$199$	−20.9597	−1.48580	−0.742898	−	0.669404i	$-0.766549\pi$
$199$	−20.9597	−1.48580	−0.742898	+	0.669404i	$0.766549\pi$
$200$	0	0
$201$	− 25.2252i	− 1.77925i
$202$	0	0
$203$	0	0
$204$	0	0
$205$	4.20692	0.293824
$206$	0	0
$207$	− 4.69496i	− 0.326322i
$208$	0	0
$209$	− 3.13707i	− 0.216996i
$210$	0	0
$211$	24.9864i	1.72014i	0.510179	+	0.860068i	$0.329579\pi$
$211$	24.9864i	1.72014i	−0.510179	+	0.860068i	$0.670421\pi$
$212$	0	0
$213$	32.8745i	2.25252i
$214$	0	0
$215$	−5.01095	−0.341744
$216$	0	0
$217$	0	0
$218$	0	0
$219$	26.8021i	1.81112i
$220$	0	0
$221$	0.157996	0.0106280
$222$	0	0
$223$	0.0721013	0.00482826	0.00241413	−	0.999997i	$-0.499232\pi$
$223$	0.0721013	0.00482826	0.00241413	+	0.999997i	$0.499232\pi$
$224$	0	0
$225$	−2.11652	−0.141101
$226$	0	0
$227$	−21.9302	−1.45556	−0.727778	−	0.685813i	$-0.759447\pi$
$227$	−21.9302	−1.45556	−0.727778	+	0.685813i	$0.759447\pi$
$228$	0	0
$229$	4.73259i	0.312738i	0.987699	+	0.156369i	$0.0499790\pi$
$229$	4.73259i	0.312738i	−0.987699	+	0.156369i	$0.950021\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5.33145	0.349275	0.174637	−	0.984633i	$-0.444125\pi$
$233$	5.33145	0.349275	0.174637	+	0.984633i	$0.444125\pi$
$234$	0	0
$235$	10.6211i	0.692846i
$236$	0	0
$237$	− 13.2712i	− 0.862057i
$238$	0	0
$239$	− 18.4393i	− 1.19274i	−0.802710	−	0.596370i	$-0.796609\pi$
$239$	− 18.4393i	− 1.19274i	0.802710	−	0.596370i	$-0.203391\pi$
$240$	0	0
$241$	− 19.6021i	− 1.26268i	−0.775505	−	0.631342i	$-0.782505\pi$
$241$	− 19.6021i	− 1.26268i	0.775505	−	0.631342i	$-0.217495\pi$
$242$	0	0
$243$	−18.5922	−1.19269
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 5.53808i	− 0.352380i
$248$	0	0
$249$	−13.1116	−0.830917
$250$	0	0
$251$	15.4391	0.974505	0.487252	−	0.873261i	$-0.337999\pi$
$251$	15.4391	0.974505	0.487252	+	0.873261i	$0.337999\pi$
$252$	0	0
$253$	−2.84225	−0.178691
$254$	0	0
$255$	−0.157996	−0.00989409
$256$	0	0
$257$	− 15.8216i	− 0.986927i	−0.869767	−	0.493463i	$-0.835731\pi$
$257$	− 15.8216i	− 0.986927i	0.869767	−	0.493463i	$-0.164269\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	4.19835	0.259872
$262$	0	0
$263$	21.5945i	1.33157i	0.746142	+	0.665786i	$0.231904\pi$
$263$	21.5945i	1.33157i	−0.746142	+	0.665786i	$0.768096\pi$
$264$	0	0
$265$	4.01480i	0.246627i
$266$	0	0
$267$	− 39.7777i	− 2.43436i
$268$	0	0
$269$	0.710688i	0.0433314i	0.999765	+	0.0216657i	$0.00689695\pi$
$269$	0.710688i	0.0433314i	−0.999765	+	0.0216657i	$0.993103\pi$
$270$	0	0
$271$	−17.5687	−1.06722	−0.533610	−	0.845731i	$-0.679165\pi$
$271$	−17.5687	−1.06722	−0.533610	+	0.845731i	$0.679165\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.28130i	0.0772656i
$276$	0	0
$277$	4.92471	0.295897	0.147949	−	0.988995i	$-0.452733\pi$
$277$	4.92471	0.295897	0.147949	+	0.988995i	$0.452733\pi$
$278$	0	0
$279$	13.3701	0.800448
$280$	0	0
$281$	4.87707	0.290941	0.145471	−	0.989363i	$-0.453530\pi$
$281$	4.87707	0.290941	0.145471	+	0.989363i	$0.453530\pi$
$282$	0	0
$283$	−17.8838	−1.06308	−0.531541	−	0.847033i	$-0.678387\pi$
$283$	−17.8838	−1.06308	−0.531541	+	0.847033i	$0.678387\pi$
$284$	0	0
$285$	5.53808i	0.328048i
$286$	0	0
$287$	0	0
$288$	0	0
$289$	16.9951	0.999713
$290$	0	0
$291$	15.9714i	0.936257i
$292$	0	0
$293$	− 24.0836i	− 1.40698i	−0.710705	−	0.703490i	$-0.751624\pi$
$293$	− 24.0836i	− 1.40698i	0.710705	−	0.703490i	$-0.248376\pi$
$294$	0	0
$295$	6.89149i	0.401238i
$296$	0	0
$297$	2.56057i	0.148579i
$298$	0	0
$299$	−5.01761	−0.290176
$300$	0	0
$301$	0	0
$302$	0	0
$303$	13.5317i	0.777376i
$304$	0	0
$305$	6.73925	0.385888
$306$	0	0
$307$	−29.1042	−1.66106	−0.830531	−	0.556972i	$-0.811963\pi$
$307$	−29.1042	−1.66106	−0.830531	+	0.556972i	$0.811963\pi$
$308$	0	0
$309$	21.0275	1.19621
$310$	0	0
$311$	−12.3693	−0.701398	−0.350699	−	0.936488i	$-0.614056\pi$
$311$	−12.3693	−0.701398	−0.350699	+	0.936488i	$0.614056\pi$
$312$	0	0
$313$	3.08027i	0.174107i	0.996204	+	0.0870536i	$0.0277451\pi$
$313$	3.08027i	0.174107i	−0.996204	+	0.0870536i	$0.972255\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−17.9983	−1.01088	−0.505442	−	0.862861i	$-0.668671\pi$
$317$	−17.9983	−1.01088	−0.505442	+	0.862861i	$0.668671\pi$
$318$	0	0
$319$	− 2.54161i	− 0.142303i
$320$	0	0
$321$	7.93909i	0.443117i
$322$	0	0
$323$	0.171014i	0.00951545i
$324$	0	0
$325$	2.26197i	0.125472i
$326$	0	0
$327$	16.4846	0.911602
$328$	0	0
$329$	0	0
$330$	0	0
$331$	2.09011i	0.114883i	0.998349	+	0.0574415i	$0.0182943\pi$
$331$	2.09011i	0.114883i	−0.998349	+	0.0574415i	$0.981706\pi$
$332$	0	0
$333$	17.9655	0.984503
$334$	0	0
$335$	11.1519	0.609292
$336$	0	0
$337$	−8.22975	−0.448303	−0.224152	−	0.974554i	$-0.571961\pi$
$337$	−8.22975	−0.448303	−0.224152	+	0.974554i	$0.571961\pi$
$338$	0	0
$339$	−2.59608	−0.141000
$340$	0	0
$341$	− 8.09403i	− 0.438316i
$342$	0	0
$343$	0	0
$344$	0	0
$345$	5.01761	0.270139
$346$	0	0
$347$	− 10.8760i	− 0.583854i	−0.956441	−	0.291927i	$-0.905704\pi$
$347$	− 10.8760i	− 0.583854i	0.956441	−	0.291927i	$-0.0942965\pi$
$348$	0	0
$349$	7.39422i	0.395804i	0.980222	+	0.197902i	$0.0634127\pi$
$349$	7.39422i	0.395804i	−0.980222	+	0.197902i	$0.936587\pi$
$350$	0	0
$351$	4.52034i	0.241278i
$352$	0	0
$353$	29.6796i	1.57969i	0.613309	+	0.789843i	$0.289838\pi$
$353$	29.6796i	1.57969i	−0.613309	+	0.789843i	$0.710162\pi$
$354$	0	0
$355$	−14.5335	−0.771361
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 27.3548i	− 1.44373i	−0.692033	−	0.721866i	$-0.743285\pi$
$359$	− 27.3548i	− 1.44373i	0.692033	−	0.721866i	$-0.256715\pi$
$360$	0	0
$361$	−13.0056	−0.684507
$362$	0	0
$363$	21.1681	1.11104
$364$	0	0
$365$	−11.8490	−0.620204
$366$	0	0
$367$	−13.7326	−0.716835	−0.358418	−	0.933561i	$-0.616684\pi$
$367$	−13.7326	−0.716835	−0.358418	+	0.933561i	$0.616684\pi$
$368$	0	0
$369$	− 8.90403i	− 0.463525i
$370$	0	0
$371$	0	0
$372$	0	0
$373$	26.3020	1.36187	0.680933	−	0.732345i	$-0.261574\pi$
$373$	26.3020	1.36187	0.680933	+	0.732345i	$0.261574\pi$
$374$	0	0
$375$	− 2.26197i	− 0.116808i
$376$	0	0
$377$	− 4.48687i	− 0.231086i
$378$	0	0
$379$	− 17.0292i	− 0.874732i	−0.899284	−	0.437366i	$-0.855911\pi$
$379$	− 17.0292i	− 0.874732i	0.899284	−	0.437366i	$-0.144089\pi$
$380$	0	0
$381$	− 33.7933i	− 1.73128i
$382$	0	0
$383$	8.58770	0.438811	0.219405	−	0.975634i	$-0.429588\pi$
$383$	8.58770	0.438811	0.219405	+	0.975634i	$0.429588\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10.6058i	0.539122i
$388$	0	0
$389$	−20.3201	−1.03027	−0.515136	−	0.857109i	$-0.672258\pi$
$389$	−20.3201	−1.03027	−0.515136	+	0.857109i	$0.672258\pi$
$390$	0	0
$391$	0.154942	0.00783573
$392$	0	0
$393$	−40.5516	−2.04556
$394$	0	0
$395$	5.86709	0.295205
$396$	0	0
$397$	− 18.2011i	− 0.913487i	−0.889598	−	0.456744i	$-0.849016\pi$
$397$	− 18.2011i	− 0.913487i	0.889598	−	0.456744i	$-0.150984\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−1.91270	−0.0955156	−0.0477578	−	0.998859i	$-0.515208\pi$
$401$	−1.91270	−0.0955156	−0.0477578	+	0.998859i	$0.515208\pi$
$402$	0	0
$403$	− 14.2889i	− 0.711783i
$404$	0	0
$405$	− 10.8699i	− 0.540130i
$406$	0	0
$407$	− 10.8760i	− 0.539103i
$408$	0	0
$409$	− 7.51299i	− 0.371494i	−0.982598	−	0.185747i	$-0.940530\pi$
$409$	− 7.51299i	− 0.371494i	0.982598	−	0.185747i	$-0.0594704\pi$
$410$	0	0
$411$	−28.5873	−1.41011
$412$	0	0
$413$	0	0
$414$	0	0
$415$	− 5.79655i	− 0.284542i
$416$	0	0
$417$	−28.0246	−1.37237
$418$	0	0
$419$	−33.5844	−1.64070	−0.820352	−	0.571859i	$-0.806223\pi$
$419$	−33.5844	−1.64070	−0.820352	+	0.571859i	$0.806223\pi$
$420$	0	0
$421$	25.7594	1.25544	0.627719	−	0.778440i	$-0.283989\pi$
$421$	25.7594	1.25544	0.627719	+	0.778440i	$0.283989\pi$
$422$	0	0
$423$	22.4798	1.09301
$424$	0	0
$425$	− 0.0698487i	− 0.00338816i
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−6.55582	−0.316518
$430$	0	0
$431$	23.7343i	1.14324i	0.820518	+	0.571621i	$0.193685\pi$
$431$	23.7343i	1.14324i	−0.820518	+	0.571621i	$0.806315\pi$
$432$	0	0
$433$	33.1945i	1.59523i	0.603169	+	0.797614i	$0.293905\pi$
$433$	33.1945i	1.59523i	−0.603169	+	0.797614i	$0.706095\pi$
$434$	0	0
$435$	4.48687i	0.215129i
$436$	0	0
$437$	− 5.43102i	− 0.259801i
$438$	0	0
$439$	0.628450	0.0299943	0.0149972	−	0.999888i	$-0.495226\pi$
$439$	0.628450	0.0299943	0.0149972	+	0.999888i	$0.495226\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0.315992i	0.0150132i	0.999972	+	0.00750661i	$0.00238945\pi$
$443$	0.315992i	0.0150132i	−0.999972	+	0.00750661i	$0.997611\pi$
$444$	0	0
$445$	17.5854	0.833629
$446$	0	0
$447$	14.4862	0.685174
$448$	0	0
$449$	−28.7179	−1.35528	−0.677642	−	0.735392i	$-0.736998\pi$
$449$	−28.7179	−1.35528	−0.677642	+	0.735392i	$0.736998\pi$
$450$	0	0
$451$	−5.39035	−0.253821
$452$	0	0
$453$	53.4886i	2.51311i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	1.67975	0.0785752	0.0392876	−	0.999228i	$-0.487491\pi$
$457$	1.67975	0.0785752	0.0392876	+	0.999228i	$0.487491\pi$
$458$	0	0
$459$	− 0.139586i	− 0.00651533i
$460$	0	0
$461$	− 5.34502i	− 0.248943i	−0.992223	−	0.124471i	$-0.960277\pi$
$461$	− 5.34502i	− 0.248943i	0.992223	−	0.124471i	$-0.0397235\pi$
$462$	0	0
$463$	− 0.161599i	− 0.00751013i	−0.999993	−	0.00375507i	$-0.998805\pi$
$463$	− 0.161599i	− 0.00751013i	0.999993	−	0.00375507i	$-0.00119528\pi$
$464$	0	0
$465$	14.2889i	0.662634i
$466$	0	0
$467$	18.5406	0.857958	0.428979	−	0.903315i	$-0.358874\pi$
$467$	18.5406	0.857958	0.428979	+	0.903315i	$0.358874\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	5.30013i	0.244217i
$472$	0	0
$473$	6.42055	0.295217
$474$	0	0
$475$	−2.44834	−0.112338
$476$	0	0
$477$	8.49740	0.389069
$478$	0	0
$479$	−5.41716	−0.247516	−0.123758	−	0.992312i	$-0.539495\pi$
$479$	−5.41716	−0.247516	−0.123758	+	0.992312i	$0.539495\pi$
$480$	0	0
$481$	− 19.2001i	− 0.875451i
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7.06081	−0.320615
$486$	0	0
$487$	− 7.19297i	− 0.325945i	−0.986631	−	0.162972i	$-0.947892\pi$
$487$	− 7.19297i	− 0.325945i	0.986631	−	0.162972i	$-0.0521081\pi$
$488$	0	0
$489$	− 2.07836i	− 0.0939867i
$490$	0	0
$491$	32.6329i	1.47270i	0.676599	+	0.736352i	$0.263453\pi$
$491$	32.6329i	1.47270i	−0.676599	+	0.736352i	$0.736547\pi$
$492$	0	0
$493$	0.138553i	0.00624010i
$494$	0	0
$495$	2.71191	0.121891
$496$	0	0
$497$	0	0
$498$	0	0
$499$	− 7.96951i	− 0.356764i	−0.983961	−	0.178382i	$-0.942914\pi$
$499$	− 7.96951i	− 0.356764i	0.983961	−	0.178382i	$-0.0570863\pi$
$500$	0	0
$501$	−30.8158	−1.37675
$502$	0	0
$503$	−26.7556	−1.19297	−0.596486	−	0.802623i	$-0.703437\pi$
$503$	−26.7556	−1.19297	−0.596486	+	0.802623i	$0.703437\pi$
$504$	0	0
$505$	−5.98226	−0.266207
$506$	0	0
$507$	17.8322	0.791956
$508$	0	0
$509$	26.4518i	1.17246i	0.810146	+	0.586228i	$0.199388\pi$
$509$	26.4518i	1.17246i	−0.810146	+	0.586228i	$0.800612\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−4.89278	−0.216022
$514$	0	0
$515$	9.29610i	0.409635i
$516$	0	0
$517$	− 13.6089i	− 0.598519i
$518$	0	0
$519$	8.88480i	0.389999i
$520$	0	0
$521$	− 19.6460i	− 0.860706i	−0.902661	−	0.430353i	$-0.858389\pi$
$521$	− 19.6460i	− 0.860706i	0.902661	−	0.430353i	$-0.141611\pi$
$522$	0	0
$523$	10.7807	0.471407	0.235703	−	0.971825i	$-0.424261\pi$
$523$	10.7807	0.471407	0.235703	+	0.971825i	$0.424261\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0.441236i	0.0192206i
$528$	0	0
$529$	18.0794	0.786061
$530$	0	0
$531$	14.5860	0.632977
$532$	0	0
$533$	−9.51594	−0.412181
$534$	0	0
$535$	−3.50981	−0.151742
$536$	0	0
$537$	− 2.26113i	− 0.0975748i
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−9.74203	−0.418842	−0.209421	−	0.977826i	$-0.567158\pi$
$541$	−9.74203	−0.418842	−0.209421	+	0.977826i	$0.567158\pi$
$542$	0	0
$543$	2.48061i	0.106453i
$544$	0	0
$545$	7.28772i	0.312172i
$546$	0	0
$547$	20.0170i	0.855865i	0.903811	+	0.427932i	$0.140758\pi$
$547$	20.0170i	0.855865i	−0.903811	+	0.427932i	$0.859242\pi$
$548$	0	0
$549$	− 14.2638i	− 0.608762i
$550$	0	0
$551$	4.85656	0.206896
$552$	0	0
$553$	0	0
$554$	0	0
$555$	19.2001i	0.815000i
$556$	0	0
$557$	24.6069	1.04263	0.521314	−	0.853365i	$-0.325442\pi$
$557$	24.6069	1.04263	0.521314	+	0.853365i	$0.325442\pi$
$558$	0	0
$559$	11.3346	0.479404
$560$	0	0
$561$	0.202441	0.00854706
$562$	0	0
$563$	−5.48345	−0.231100	−0.115550	−	0.993302i	$-0.536863\pi$
$563$	−5.48345	−0.231100	−0.115550	+	0.993302i	$0.536863\pi$
$564$	0	0
$565$	− 1.14771i	− 0.0482843i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−20.6101	−0.864021	−0.432011	−	0.901868i	$-0.642196\pi$
$569$	−20.6101	−0.864021	−0.432011	+	0.901868i	$0.642196\pi$
$570$	0	0
$571$	− 19.2191i	− 0.804294i	−0.915575	−	0.402147i	$-0.868264\pi$
$571$	− 19.2191i	− 0.804294i	0.915575	−	0.402147i	$-0.131736\pi$
$572$	0	0
$573$	− 12.1368i	− 0.507021i
$574$	0	0
$575$	2.21824i	0.0925072i
$576$	0	0
$577$	− 47.9735i	− 1.99716i	−0.0532429	−	0.998582i	$-0.516956\pi$
$577$	− 47.9735i	− 1.99716i	0.0532429	−	0.998582i	$-0.483044\pi$
$578$	0	0
$579$	29.8662	1.24120
$580$	0	0
$581$	0	0
$582$	0	0
$583$	− 5.14418i	− 0.213050i
$584$	0	0
$585$	4.78751	0.197939
$586$	0	0
$587$	−0.534337	−0.0220544	−0.0110272	−	0.999939i	$-0.503510\pi$
$587$	−0.534337	−0.0220544	−0.0110272	+	0.999939i	$0.503510\pi$
$588$	0	0
$589$	15.4662	0.637275
$590$	0	0
$591$	52.3114	2.15180
$592$	0	0
$593$	29.7714i	1.22257i	0.791412	+	0.611283i	$0.209346\pi$
$593$	29.7714i	1.22257i	−0.791412	+	0.611283i	$0.790654\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−47.4104	−1.94038
$598$	0	0
$599$	− 29.0758i	− 1.18800i	−0.804463	−	0.594002i	$-0.797547\pi$
$599$	− 29.0758i	− 1.18800i	0.804463	−	0.594002i	$-0.202453\pi$
$600$	0	0
$601$	23.6224i	0.963576i	0.876288	+	0.481788i	$0.160012\pi$
$601$	23.6224i	0.963576i	−0.876288	+	0.481788i	$0.839988\pi$
$602$	0	0
$603$	− 23.6032i	− 0.961195i
$604$	0	0
$605$	9.35826i	0.380467i
$606$	0	0
$607$	12.8327	0.520864	0.260432	−	0.965492i	$-0.416135\pi$
$607$	12.8327	0.520864	0.260432	+	0.965492i	$0.416135\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 24.0247i	− 0.971936i
$612$	0	0
$613$	36.4044	1.47036	0.735180	−	0.677871i	$-0.237097\pi$
$613$	36.4044	1.47036	0.735180	+	0.677871i	$0.237097\pi$
$614$	0	0
$615$	9.51594	0.383720
$616$	0	0
$617$	−8.80275	−0.354386	−0.177193	−	0.984176i	$-0.556702\pi$
$617$	−8.80275	−0.354386	−0.177193	+	0.984176i	$0.556702\pi$
$618$	0	0
$619$	41.8467	1.68196	0.840980	−	0.541066i	$-0.181979\pi$
$619$	41.8467	1.68196	0.840980	+	0.541066i	$0.181979\pi$
$620$	0	0
$621$	4.43296i	0.177888i
$622$	0	0
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	− 7.09597i	− 0.283386i
$628$	0	0
$629$	0.592892i	0.0236401i
$630$	0	0
$631$	28.7721i	1.14540i	0.819765	+	0.572700i	$0.194104\pi$
$631$	28.7721i	1.14540i	−0.819765	+	0.572700i	$0.805896\pi$
$632$	0	0
$633$	56.5186i	2.24641i
$634$	0	0
$635$	14.9398	0.592866
$636$	0	0
$637$	0	0
$638$	0	0
$639$	30.7605i	1.21687i
$640$	0	0
$641$	−28.3384	−1.11930	−0.559650	−	0.828729i	$-0.689064\pi$
$641$	−28.3384	−1.11930	−0.559650	+	0.828729i	$0.689064\pi$
$642$	0	0
$643$	1.69981	0.0670340	0.0335170	−	0.999438i	$-0.489329\pi$
$643$	1.69981	0.0670340	0.0335170	+	0.999438i	$0.489329\pi$
$644$	0	0
$645$	−11.3346	−0.446301
$646$	0	0
$647$	4.36235	0.171502	0.0857508	−	0.996317i	$-0.472671\pi$
$647$	4.36235	0.171502	0.0857508	+	0.996317i	$0.472671\pi$
$648$	0	0
$649$	− 8.83009i	− 0.346612i
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−10.2849	−0.402481	−0.201240	−	0.979542i	$-0.564497\pi$
$653$	−10.2849	−0.402481	−0.201240	+	0.979542i	$0.564497\pi$
$654$	0	0
$655$	− 17.9275i	− 0.700486i
$656$	0	0
$657$	25.0786i	0.978409i
$658$	0	0
$659$	19.5800i	0.762729i	0.924425	+	0.381364i	$0.124546\pi$
$659$	19.5800i	0.762729i	−0.924425	+	0.381364i	$0.875454\pi$
$660$	0	0
$661$	− 41.3927i	− 1.60999i	−0.593281	−	0.804995i	$-0.702168\pi$
$661$	− 41.3927i	− 1.60999i	0.593281	−	0.804995i	$-0.297832\pi$
$662$	0	0
$663$	0.357382	0.0138796
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 4.40013i	− 0.170374i
$668$	0	0
$669$	0.163091	0.00630547
$670$	0	0
$671$	−8.63503	−0.333352
$672$	0	0
$673$	−47.6784	−1.83787	−0.918934	−	0.394412i	$-0.870948\pi$
$673$	−47.6784	−1.83787	−0.918934	+	0.394412i	$0.870948\pi$
$674$	0	0
$675$	1.99841	0.0769187
$676$	0	0
$677$	− 4.60650i	− 0.177042i	−0.996074	−	0.0885211i	$-0.971786\pi$
$677$	− 4.60650i	− 0.177042i	0.996074	−	0.0885211i	$-0.0282141\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−49.6054	−1.90088
$682$	0	0
$683$	− 41.3757i	− 1.58320i	−0.611041	−	0.791599i	$-0.709249\pi$
$683$	− 41.3757i	− 1.58320i	0.611041	−	0.791599i	$-0.290751\pi$
$684$	0	0
$685$	− 12.6382i	− 0.482881i
$686$	0	0
$687$	10.7050i	0.408421i
$688$	0	0
$689$	− 9.08136i	− 0.345972i
$690$	0	0
$691$	26.1121	0.993353	0.496676	−	0.867936i	$-0.334554\pi$
$691$	26.1121	0.993353	0.496676	+	0.867936i	$0.334554\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 12.3894i	− 0.469958i
$696$	0	0
$697$	0.293848	0.0111303
$698$	0	0
$699$	12.0596	0.456136
$700$	0	0
$701$	36.8190	1.39063	0.695317	−	0.718703i	$-0.255264\pi$
$701$	36.8190	1.39063	0.695317	+	0.718703i	$0.255264\pi$
$702$	0	0
$703$	20.7821	0.783811
$704$	0	0
$705$	24.0247i	0.904823i
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−0.0518671	−0.00194791	−0.000973955	−	1.00000i	$-0.500310\pi$
$709$	−0.0518671	−0.00194791	−0.000973955		1.00000i	$0.500310\pi$
$710$	0	0
$711$	− 12.4178i	− 0.465705i
$712$	0	0
$713$	− 14.0127i	− 0.524780i
$714$	0	0
$715$	− 2.89828i	− 0.108389i
$716$	0	0
$717$	− 41.7092i	− 1.55766i
$718$	0	0
$719$	−15.2869	−0.570104	−0.285052	−	0.958512i	$-0.592011\pi$
$719$	−15.2869	−0.570104	−0.285052	+	0.958512i	$0.592011\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	− 44.3395i	− 1.64900i
$724$	0	0
$725$	−1.98361	−0.0736695
$726$	0	0
$727$	−27.4261	−1.01718	−0.508588	−	0.861010i	$-0.669832\pi$
$727$	−27.4261	−1.01718	−0.508588	+	0.861010i	$0.669832\pi$
$728$	0	0
$729$	−9.44534	−0.349827
$730$	0	0
$731$	−0.350009	−0.0129455
$732$	0	0
$733$	29.6566i	1.09539i	0.836677	+	0.547697i	$0.184495\pi$
$733$	29.6566i	1.09539i	−0.836677	+	0.547697i	$0.815505\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−14.2889	−0.526340
$738$	0	0
$739$	38.0930i	1.40127i	0.713519	+	0.700636i	$0.247100\pi$
$739$	38.0930i	1.40127i	−0.713519	+	0.700636i	$0.752900\pi$
$740$	0	0
$741$	− 12.5270i	− 0.460190i
$742$	0	0
$743$	35.6723i	1.30869i	0.756197	+	0.654345i	$0.227055\pi$
$743$	35.6723i	1.30869i	−0.756197	+	0.654345i	$0.772945\pi$
$744$	0	0
$745$	6.40424i	0.234633i
$746$	0	0
$747$	−12.2685	−0.448882
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 0.970408i	− 0.0354107i	−0.999843	−	0.0177053i	$-0.994364\pi$
$751$	− 0.970408i	− 0.0354107i	0.999843	−	0.0177053i	$-0.00563608\pi$
$752$	0	0
$753$	34.9227	1.27265
$754$	0	0
$755$	−23.6469	−0.860598
$756$	0	0
$757$	15.0405	0.546658	0.273329	−	0.961921i	$-0.411875\pi$
$757$	15.0405	0.546658	0.273329	+	0.961921i	$0.411875\pi$
$758$	0	0
$759$	−6.42908	−0.233361
$760$	0	0
$761$	− 10.2355i	− 0.371035i	−0.982641	−	0.185517i	$-0.940604\pi$
$761$	− 10.2355i	− 0.371035i	0.982641	−	0.185517i	$-0.0593961\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−0.147836	−0.00534503
$766$	0	0
$767$	− 15.5884i	− 0.562863i
$768$	0	0
$769$	40.4890i	1.46007i	0.683409	+	0.730036i	$0.260497\pi$
$769$	40.4890i	1.46007i	−0.683409	+	0.730036i	$0.739503\pi$
$770$	0	0
$771$	− 35.7881i	− 1.28888i
$772$	0	0
$773$	16.3932i	0.589624i	0.955555	+	0.294812i	$0.0952570\pi$
$773$	16.3932i	0.589624i	−0.955555	+	0.294812i	$0.904743\pi$
$774$	0	0
$775$	−6.31703	−0.226914
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 10.3000i	− 0.369035i
$780$	0	0
$781$	18.6219	0.666344
$782$	0	0
$783$	−3.96406	−0.141664
$784$	0	0
$785$	−2.34315	−0.0836305
$786$	0	0
$787$	7.18768	0.256213	0.128106	−	0.991760i	$-0.459110\pi$
$787$	7.18768	0.256213	0.128106	+	0.991760i	$0.459110\pi$
$788$	0	0
$789$	48.8461i	1.73897i
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−15.2440	−0.541330
$794$	0	0
$795$	9.08136i	0.322083i
$796$	0	0
$797$	16.4615i	0.583096i	0.956556	+	0.291548i	$0.0941703\pi$
$797$	16.4615i	0.583096i	−0.956556	+	0.291548i	$0.905830\pi$
$798$	0	0
$799$	0.741873i	0.0262456i
$800$	0	0
$801$	− 37.2199i	− 1.31510i
$802$	0	0
$803$	15.1821	0.535766
$804$	0	0
$805$	0	0
$806$	0	0
$807$	1.60756i	0.0565887i
$808$	0	0
$809$	−25.2638	−0.888228	−0.444114	−	0.895970i	$-0.646481\pi$
$809$	−25.2638	−0.888228	−0.444114	+	0.895970i	$0.646481\pi$
$810$	0	0
$811$	15.4145	0.541275	0.270637	−	0.962681i	$-0.412766\pi$
$811$	15.4145	0.541275	0.270637	+	0.962681i	$0.412766\pi$
$812$	0	0
$813$	−39.7398	−1.39374
$814$	0	0
$815$	0.918827	0.0321851
$816$	0	0
$817$	12.2685i	0.429221i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−51.3340	−1.79157	−0.895784	−	0.444490i	$-0.853385\pi$
$821$	−51.3340	−1.79157	−0.895784	+	0.444490i	$0.853385\pi$
$822$	0	0
$823$	23.2020i	0.808772i	0.914588	+	0.404386i	$0.132515\pi$
$823$	23.2020i	0.808772i	−0.914588	+	0.404386i	$0.867485\pi$
$824$	0	0
$825$	2.89828i	0.100905i
$826$	0	0
$827$	13.0659i	0.454345i	0.973855	+	0.227172i	$0.0729481\pi$
$827$	13.0659i	0.454345i	−0.973855	+	0.227172i	$0.927052\pi$
$828$	0	0
$829$	− 40.5361i	− 1.40788i	−0.710261	−	0.703938i	$-0.751423\pi$
$829$	− 40.5361i	− 1.40788i	0.710261	−	0.703938i	$-0.248577\pi$
$830$	0	0
$831$	11.1396	0.386427
$832$	0	0
$833$	0	0
$834$	0	0
$835$	− 13.6234i	− 0.471458i
$836$	0	0
$837$	−12.6240	−0.436349
$838$	0	0
$839$	−1.80249	−0.0622290	−0.0311145	−	0.999516i	$-0.509906\pi$
$839$	−1.80249	−0.0622290	−0.0311145	+	0.999516i	$0.509906\pi$
$840$	0	0
$841$	−25.0653	−0.864320
$842$	0	0
$843$	11.0318	0.379955
$844$	0	0
$845$	7.88348i	0.271200i
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−40.4527	−1.38833
$850$	0	0
$851$	− 18.8290i	− 0.645448i
$852$	0	0
$853$	20.3256i	0.695937i	0.937506	+	0.347968i	$0.113128\pi$
$853$	20.3256i	0.695937i	−0.937506	+	0.347968i	$0.886872\pi$
$854$	0	0
$855$	5.18196i	0.177219i
$856$	0	0
$857$	21.3433i	0.729073i	0.931189	+	0.364537i	$0.118773\pi$
$857$	21.3433i	0.729073i	−0.931189	+	0.364537i	$0.881227\pi$
$858$	0	0
$859$	−0.152184	−0.00519245	−0.00259623	−	0.999997i	$-0.500826\pi$
$859$	−0.152184	−0.00519245	−0.00259623	+	0.999997i	$0.500826\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	52.9477i	1.80236i	0.433445	+	0.901180i	$0.357298\pi$
$863$	52.9477i	1.80236i	−0.433445	+	0.901180i	$0.642702\pi$
$864$	0	0
$865$	−3.92790	−0.133553
$866$	0	0
$867$	38.4425	1.30558
$868$	0	0
$869$	−7.51753	−0.255015
$870$	0	0
$871$	−25.2252	−0.854724
$872$	0	0
$873$	14.9443i	0.505789i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	13.7875	0.465572	0.232786	−	0.972528i	$-0.425216\pi$
$877$	13.7875	0.465572	0.232786	+	0.972528i	$0.425216\pi$
$878$	0	0
$879$	− 54.4765i	− 1.83745i
$880$	0	0
$881$	− 12.6991i	− 0.427844i	−0.976851	−	0.213922i	$-0.931376\pi$
$881$	− 12.6991i	− 0.427844i	0.976851	−	0.213922i	$-0.0686239\pi$
$882$	0	0
$883$	41.9953i	1.41326i	0.707586	+	0.706628i	$0.249784\pi$
$883$	41.9953i	1.41326i	−0.707586	+	0.706628i	$0.750216\pi$
$884$	0	0
$885$	15.5884i	0.523997i
$886$	0	0
$887$	−27.5309	−0.924398	−0.462199	−	0.886776i	$-0.652939\pi$
$887$	−27.5309	−0.924398	−0.462199	+	0.886776i	$0.652939\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	13.9277i	0.466594i
$892$	0	0
$893$	26.0042	0.870196
$894$	0	0
$895$	0.999626	0.0334138
$896$	0	0
$897$	−11.3497	−0.378955
$898$	0	0
$899$	12.5305	0.417916
$900$	0	0
$901$	0.280428i	0.00934243i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−1.09666	−0.0364542
$906$	0	0
$907$	28.2562i	0.938233i	0.883136	+	0.469117i	$0.155428\pi$
$907$	28.2562i	0.938233i	−0.883136	+	0.469117i	$0.844572\pi$
$908$	0	0
$909$	12.6616i	0.419958i
$910$	0	0
$911$	− 47.4861i	− 1.57328i	−0.617410	−	0.786642i	$-0.711818\pi$
$911$	− 47.4861i	− 1.57328i	0.617410	−	0.786642i	$-0.288182\pi$
$912$	0	0
$913$	7.42715i	0.245803i
$914$	0	0
$915$	15.2440	0.503951
$916$	0	0
$917$	0	0
$918$	0	0
$919$	− 43.9408i	− 1.44947i	−0.689027	−	0.724736i	$-0.741962\pi$
$919$	− 43.9408i	− 1.44947i	0.689027	−	0.724736i	$-0.258038\pi$
$920$	0	0
$921$	−65.8328	−2.16927
$922$	0	0
$923$	32.8745	1.08208
$924$	0	0
$925$	−8.48822	−0.279091
$926$	0	0
$927$	19.6754	0.646224
$928$	0	0
$929$	− 30.0003i	− 0.984277i	−0.870517	−	0.492139i	$-0.836215\pi$
$929$	− 30.0003i	− 0.984277i	0.870517	−	0.492139i	$-0.163785\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−27.9790	−0.915991
$934$	0	0
$935$	0.0894975i	0.00292688i
$936$	0	0
$937$	42.7802i	1.39757i	0.715332	+	0.698784i	$0.246275\pi$
$937$	42.7802i	1.39757i	−0.715332	+	0.698784i	$0.753725\pi$
$938$	0	0
$939$	6.96749i	0.227375i
$940$	0	0
$941$	− 39.7822i	− 1.29686i	−0.761273	−	0.648432i	$-0.775425\pi$
$941$	− 39.7822i	− 1.29686i	0.761273	−	0.648432i	$-0.224575\pi$
$942$	0	0
$943$	−9.33198	−0.303891
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 33.7788i	− 1.09766i	−0.835933	−	0.548832i	$-0.815073\pi$
$947$	− 33.7788i	− 1.09766i	0.835933	−	0.548832i	$-0.184927\pi$
$948$	0	0
$949$	26.8021	0.870032
$950$	0	0
$951$	−40.7116	−1.32016
$952$	0	0
$953$	−40.3496	−1.30705	−0.653526	−	0.756904i	$-0.726711\pi$
$953$	−40.3496	−1.30705	−0.653526	+	0.756904i	$0.726711\pi$
$954$	0	0
$955$	5.36558	0.173626
$956$	0	0
$957$	− 5.74905i	− 0.185840i
$958$	0	0
$959$	0	0
$960$	0	0
$961$	8.90481	0.287252
$962$	0	0
$963$	7.42858i	0.239383i
$964$	0	0
$965$	13.2036i	0.425039i
$966$	0	0
$967$	14.3062i	0.460057i	0.973184	+	0.230028i	$0.0738819\pi$
$967$	14.3062i	0.460057i	−0.973184	+	0.230028i	$0.926118\pi$
$968$	0	0
$969$	0.386828i	0.0124267i
$970$	0	0
$971$	44.9837	1.44360	0.721798	−	0.692103i	$-0.243316\pi$
$971$	44.9837	1.44360	0.721798	+	0.692103i	$0.243316\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	5.11652i	0.163860i
$976$	0	0
$977$	−44.8058	−1.43346	−0.716732	−	0.697349i	$-0.754363\pi$
$977$	−44.8058	−1.43346	−0.716732	+	0.697349i	$0.754363\pi$
$978$	0	0
$979$	−22.5323	−0.720134
$980$	0	0
$981$	15.4246	0.492470
$982$	0	0
$983$	−27.9903	−0.892751	−0.446375	−	0.894846i	$-0.647285\pi$
$983$	−27.9903	−0.892751	−0.446375	+	0.894846i	$0.647285\pi$
$984$	0	0
$985$	23.1264i	0.736870i
$986$	0	0
$987$	0	0
$988$	0	0
$989$	11.1155	0.353453
$990$	0	0
$991$	− 14.2555i	− 0.452840i	−0.974030	−	0.226420i	$-0.927298\pi$
$991$	− 14.2555i	− 0.452840i	0.974030	−	0.226420i	$-0.0727022\pi$
$992$	0	0
$993$	4.72778i	0.150032i
$994$	0	0
$995$	− 20.9597i	− 0.664469i
$996$	0	0
$997$	− 53.5810i	− 1.69693i	−0.529254	−	0.848464i	$-0.677528\pi$
$997$	− 53.5810i	− 1.69693i	0.529254	−	0.848464i	$-0.322472\pi$
$998$	0	0
$999$	−16.9629	−0.536683

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3920.2.k.a.2351.8	yes	8
4.3	odd	2		3920.2.k.c.2351.2	yes	8
7.6	odd	2		3920.2.k.c.2351.1	yes	8
28.27	even	2	inner	3920.2.k.a.2351.7	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3920.2.k.a.2351.7	✓	8	28.27	even	2	inner
3920.2.k.a.2351.8	yes	8	1.1	even	1	trivial
3920.2.k.c.2351.1	yes	8	7.6	odd	2
3920.2.k.c.2351.2	yes	8	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)

\(f(q)\)	\(=\)	\(q+2.26197 q^{3} +1.00000i q^{5} +2.11652 q^{9} +O(q^{10})\)	\(q+2.26197 q^{3} +1.00000i q^{5} +2.11652 q^{9} -1.28130i q^{11} -2.26197i q^{13} +2.26197i q^{15} +0.0698487i q^{17} +2.44834 q^{19} -2.21824i q^{23} -1.00000 q^{25} -1.99841 q^{27} +1.98361 q^{29} +6.31703 q^{31} -2.89828i q^{33} +8.48822 q^{37} -5.11652i q^{39} -4.20692i q^{41} +5.01095i q^{43} +2.11652i q^{45} +10.6211 q^{47} +0.157996i q^{51} +4.01480 q^{53} +1.28130 q^{55} +5.53808 q^{57} +6.89149 q^{59} -6.73925i q^{61} +2.26197 q^{65} -11.1519i q^{67} -5.01761i q^{69} +14.5335i q^{71} +11.8490i q^{73} -2.26197 q^{75} -5.86709i q^{79} -10.8699 q^{81} -5.79655 q^{83} -0.0698487 q^{85} +4.48687 q^{87} -17.5854i q^{89} +14.2889 q^{93} +2.44834i q^{95} +7.06081i q^{97} -2.71191i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} + 16 q^{9}+O(q^{10}) \) 8 * q - 8 * q^3 + 16 * q^9	\( 8 q - 8 q^{3} + 16 q^{9} + 16 q^{19} - 8 q^{25} - 8 q^{27} + 8 q^{29} + 48 q^{31} + 24 q^{47} + 32 q^{53} - 8 q^{55} - 32 q^{57} + 16 q^{59} - 8 q^{65} + 8 q^{75} - 8 q^{81} - 48 q^{83} - 24 q^{85} - 24 q^{87} - 32 q^{93}+O(q^{100}) \) 8 * q - 8 * q^3 + 16 * q^9 + 16 * q^19 - 8 * q^25 - 8 * q^27 + 8 * q^29 + 48 * q^31 + 24 * q^47 + 32 * q^53 - 8 * q^55 - 32 * q^57 + 16 * q^59 - 8 * q^65 + 8 * q^75 - 8 * q^81 - 48 * q^83 - 24 * q^85 - 24 * q^87 - 32 * q^93

Introduction

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Database

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