LMFDB - Embedded newform 9300.2.g.q.3349.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [9300,2,Mod(3349,9300)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("9300.3349"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9300, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9300.g (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,-6,0,4,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$74.2608738798$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.2611456.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{3} + 16x^{2} - 8x + 2 $ x^6 - 2x^3 + 16x^2 - 8*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1860)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3349.5
Root			$-1.52569 + 1.52569i$ of defining polynomial
Character	$\chi$	$=$	9300.3349
Dual form			9300.2.g.q.3349.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000i q^{3} -0.395932i q^{7} -1.00000 q^{9} +2.79186 q^{11} +5.70682i q^{13} -4.25951i q^{17} +6.10275 q^{19} +0.395932 q^{21} -6.36226i q^{23} -1.00000i q^{27} -7.96633 q^{29} +1.00000 q^{31} +2.79186i q^{33} +9.80957i q^{37} -5.70682 q^{39} +3.20814 q^{41} -3.31088i q^{43} -7.05137i q^{47} +6.84324 q^{49} +4.25951 q^{51} -1.05137i q^{53} +6.10275i q^{57} -3.34456 q^{59} +12.6218 q^{61} +0.395932i q^{63} -13.8096i q^{67} +6.36226 q^{69} +13.5501 q^{71} +13.8096i q^{73} -1.10539i q^{77} +0.156762 q^{79} +1.00000 q^{81} +7.74049i q^{83} -7.96633i q^{87} -7.55005 q^{89} +2.25951 q^{91} +1.00000i q^{93} -4.79186i q^{97} -2.79186 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 6 q^{9} + 4 q^{11} - 4 q^{21} - 12 q^{29} + 6 q^{31} - 4 q^{39} + 32 q^{41} + 10 q^{49} + 20 q^{51} - 32 q^{59} + 28 q^{61} - 4 q^{69} + 20 q^{71} + 32 q^{79} + 6 q^{81} + 16 q^{89} + 8 q^{91} - 4 q^{99}+O(q^{100}) $ 6 * q - 6 * q^9 + 4 * q^11 - 4 * q^21 - 12 * q^29 + 6 * q^31 - 4 * q^39 + 32 * q^41 + 10 * q^49 + 20 * q^51 - 32 * q^59 + 28 * q^61 - 4 * q^69 + 20 * q^71 + 32 * q^79 + 6 * q^81 + 16 * q^89 + 8 * q^91 - 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/9300\mathbb{Z}\right)^\times$.

$n$	$1801$	$2977$	$3101$	$4651$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 0.395932i	− 0.149648i	−0.997197	−	0.0748241i	$-0.976160\pi$
$7$	− 0.395932i	− 0.149648i	0.997197	−	0.0748241i	$-0.0238395\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	2.79186	0.841779	0.420889	−	0.907112i	$-0.361718\pi$
$11$	2.79186	0.841779	0.420889	+	0.907112i	$0.361718\pi$
$12$	0	0
$13$	5.70682i	1.58279i	0.611308	+	0.791393i	$0.290644\pi$
$13$	5.70682i	1.58279i	−0.611308	+	0.791393i	$0.709356\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 4.25951i	− 1.03308i	−0.856262	−	0.516542i	$-0.827219\pi$
$17$	− 4.25951i	− 1.03308i	0.856262	−	0.516542i	$-0.172781\pi$
$18$	0	0
$19$	6.10275	1.40007	0.700033	−	0.714110i	$-0.253168\pi$
$19$	6.10275	1.40007	0.700033	+	0.714110i	$0.253168\pi$
$20$	0	0
$21$	0.395932	0.0863994
$22$	0	0
$23$	− 6.36226i	− 1.32662i	−0.748344	−	0.663311i	$-0.769151\pi$
$23$	− 6.36226i	− 1.32662i	0.748344	−	0.663311i	$-0.230849\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.00000i	− 0.192450i
$28$	0	0
$29$	−7.96633	−1.47931	−0.739655	−	0.672986i	$-0.765011\pi$
$29$	−7.96633	−1.47931	−0.739655	+	0.672986i	$0.765011\pi$
$30$	0	0
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	0	0
$33$	2.79186i	0.486001i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.80957i	1.61268i	0.591451	+	0.806341i	$0.298555\pi$
$37$	9.80957i	1.61268i	−0.591451	+	0.806341i	$0.701445\pi$
$38$	0	0
$39$	−5.70682	−0.913822
$40$	0	0
$41$	3.20814	0.501027	0.250513	−	0.968113i	$-0.419401\pi$
$41$	3.20814	0.501027	0.250513	+	0.968113i	$0.419401\pi$
$42$	0	0
$43$	− 3.31088i	− 0.504905i	−0.967609	−	0.252453i	$-0.918763\pi$
$43$	− 3.31088i	− 0.504905i	0.967609	−	0.252453i	$-0.0812372\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 7.05137i	− 1.02855i	−0.857626	−	0.514274i	$-0.828061\pi$
$47$	− 7.05137i	− 1.02855i	0.857626	−	0.514274i	$-0.171939\pi$
$48$	0	0
$49$	6.84324	0.977605
$50$	0	0
$51$	4.25951	0.596451
$52$	0	0
$53$	− 1.05137i	− 0.144417i	−0.997390	−	0.0722087i	$-0.976995\pi$
$53$	− 1.05137i	− 0.144417i	0.997390	−	0.0722087i	$-0.0230048\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	6.10275i	0.808329i
$58$	0	0
$59$	−3.34456	−0.435424	−0.217712	−	0.976013i	$-0.569859\pi$
$59$	−3.34456	−0.435424	−0.217712	+	0.976013i	$0.569859\pi$
$60$	0	0
$61$	12.6218	1.61605	0.808026	−	0.589147i	$-0.200536\pi$
$61$	12.6218	1.61605	0.808026	+	0.589147i	$0.200536\pi$
$62$	0	0
$63$	0.395932i	0.0498827i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 13.8096i	− 1.68711i	−0.537045	−	0.843553i	$-0.680460\pi$
$67$	− 13.8096i	− 1.68711i	0.537045	−	0.843553i	$-0.319540\pi$
$68$	0	0
$69$	6.36226	0.765926
$70$	0	0
$71$	13.5501	1.60810	0.804048	−	0.594565i	$-0.202676\pi$
$71$	13.5501	1.60810	0.804048	+	0.594565i	$0.202676\pi$
$72$	0	0
$73$	13.8096i	1.61629i	0.588985	+	0.808144i	$0.299528\pi$
$73$	13.8096i	1.61629i	−0.588985	+	0.808144i	$0.700472\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1.10539i	− 0.125971i
$78$	0	0
$79$	0.156762	0.0176371	0.00881855	−	0.999961i	$-0.497193\pi$
$79$	0.156762	0.0176371	0.00881855	+	0.999961i	$0.497193\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	7.74049i	0.849629i	0.905280	+	0.424815i	$0.139661\pi$
$83$	7.74049i	0.849629i	−0.905280	+	0.424815i	$0.860339\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 7.96633i	− 0.854080i
$88$	0	0
$89$	−7.55005	−0.800304	−0.400152	−	0.916449i	$-0.631043\pi$
$89$	−7.55005	−0.800304	−0.400152	+	0.916449i	$0.631043\pi$
$90$	0	0
$91$	2.25951	0.236861
$92$	0	0
$93$	1.00000i	0.103695i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 4.79186i	− 0.486540i	−0.969959	−	0.243270i	$-0.921780\pi$
$97$	− 4.79186i	− 0.486540i	0.969959	−	0.243270i	$-0.0782201\pi$
$98$	0	0
$99$	−2.79186	−0.280593
$100$	0	0
$101$	0.272843	0.0271489	0.0135744	−	0.999908i	$-0.495679\pi$
$101$	0.272843	0.0271489	0.0135744	+	0.999908i	$0.495679\pi$
$102$	0	0
$103$	− 2.66877i	− 0.262962i	−0.991319	−	0.131481i	$-0.958027\pi$
$103$	− 2.66877i	− 0.262962i	0.991319	−	0.131481i	$-0.0419733\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 10.2595i	− 0.991824i	−0.868373	−	0.495912i	$-0.834834\pi$
$107$	− 10.2595i	− 0.991824i	0.868373	−	0.495912i	$-0.165166\pi$
$108$	0	0
$109$	13.5704	1.29981	0.649904	−	0.760016i	$-0.274809\pi$
$109$	13.5704	1.29981	0.649904	+	0.760016i	$0.274809\pi$
$110$	0	0
$111$	−9.80957	−0.931083
$112$	0	0
$113$	− 9.93265i	− 0.934386i	−0.884155	−	0.467193i	$-0.845265\pi$
$113$	− 9.93265i	− 0.934386i	0.884155	−	0.467193i	$-0.154735\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 5.70682i	− 0.527595i
$118$	0	0
$119$	−1.68648	−0.154599
$120$	0	0
$121$	−3.20550	−0.291409
$122$	0	0
$123$	3.20814i	0.289268i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	17.8299i	1.58215i	0.611720	+	0.791074i	$0.290478\pi$
$127$	17.8299i	1.58215i	−0.611720	+	0.791074i	$0.709522\pi$
$128$	0	0
$129$	3.31088	0.291507
$130$	0	0
$131$	−14.9637	−1.30738	−0.653692	−	0.756761i	$-0.726781\pi$
$131$	−14.9637	−1.30738	−0.653692	+	0.756761i	$0.726781\pi$
$132$	0	0
$133$	− 2.41627i	− 0.209517i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 14.8813i	− 1.27139i	−0.771939	−	0.635697i	$-0.780713\pi$
$137$	− 14.8813i	− 1.27139i	0.771939	−	0.635697i	$-0.219287\pi$
$138$	0	0
$139$	11.4136	0.968092	0.484046	−	0.875043i	$-0.339167\pi$
$139$	11.4136	0.968092	0.484046	+	0.875043i	$0.339167\pi$
$140$	0	0
$141$	7.05137	0.593833
$142$	0	0
$143$	15.9327i	1.33236i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	6.84324i	0.564421i
$148$	0	0
$149$	8.79186	0.720258	0.360129	−	0.932903i	$-0.382733\pi$
$149$	8.79186	0.720258	0.360129	+	0.932903i	$0.382733\pi$
$150$	0	0
$151$	−23.7759	−1.93485	−0.967427	−	0.253149i	$-0.918534\pi$
$151$	−23.7759	−1.93485	−0.967427	+	0.253149i	$0.918534\pi$
$152$	0	0
$153$	4.25951i	0.344361i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 17.5164i	− 1.39796i	−0.715141	−	0.698980i	$-0.753638\pi$
$157$	− 17.5164i	− 1.39796i	0.715141	−	0.698980i	$-0.246362\pi$
$158$	0	0
$159$	1.05137	0.0833794
$160$	0	0
$161$	−2.51902	−0.198527
$162$	0	0
$163$	− 16.4987i	− 1.29228i	−0.763220	−	0.646138i	$-0.776383\pi$
$163$	− 16.4987i	− 1.29228i	0.763220	−	0.646138i	$-0.223617\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.8946i	0.843050i	0.906817	+	0.421525i	$0.138505\pi$
$167$	10.8946i	0.843050i	−0.906817	+	0.421525i	$0.861495\pi$
$168$	0	0
$169$	−19.5678	−1.50521
$170$	0	0
$171$	−6.10275	−0.466689
$172$	0	0
$173$	− 16.8946i	− 1.28447i	−0.766506	−	0.642237i	$-0.778007\pi$
$173$	− 16.8946i	− 1.28447i	0.766506	−	0.642237i	$-0.221993\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 3.34456i	− 0.251392i
$178$	0	0
$179$	−2.93529	−0.219394	−0.109697	−	0.993965i	$-0.534988\pi$
$179$	−2.93529	−0.219394	−0.109697	+	0.993965i	$0.534988\pi$
$180$	0	0
$181$	7.03804	0.523134	0.261567	−	0.965185i	$-0.415761\pi$
$181$	7.03804	0.523134	0.261567	+	0.965185i	$0.415761\pi$
$182$	0	0
$183$	12.6218i	0.933028i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 11.8920i	− 0.869627i
$188$	0	0
$189$	−0.395932	−0.0287998
$190$	0	0
$191$	12.8609	0.930585	0.465292	−	0.885157i	$-0.345949\pi$
$191$	12.8609	0.930585	0.465292	+	0.885157i	$0.345949\pi$
$192$	0	0
$193$	23.9327i	1.72271i	0.508003	+	0.861355i	$0.330384\pi$
$193$	23.9327i	1.72271i	−0.508003	+	0.861355i	$0.669616\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	19.0868i	1.35988i	0.733269	+	0.679938i	$0.237993\pi$
$197$	19.0868i	1.35988i	−0.733269	+	0.679938i	$0.762007\pi$
$198$	0	0
$199$	14.5324	1.03017	0.515086	−	0.857139i	$-0.327760\pi$
$199$	14.5324	1.03017	0.515086	+	0.857139i	$0.327760\pi$
$200$	0	0
$201$	13.8096	0.974052
$202$	0	0
$203$	3.15412i	0.221376i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	6.36226i	0.442208i
$208$	0	0
$209$	17.0380	1.17855
$210$	0	0
$211$	6.89461	0.474645	0.237322	−	0.971431i	$-0.423730\pi$
$211$	6.89461	0.474645	0.237322	+	0.971431i	$0.423730\pi$
$212$	0	0
$213$	13.5501i	0.928434i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 0.395932i	− 0.0268776i
$218$	0	0
$219$	−13.8096	−0.933164
$220$	0	0
$221$	24.3082	1.63515
$222$	0	0
$223$	− 16.3082i	− 1.09208i	−0.837759	−	0.546040i	$-0.816134\pi$
$223$	− 16.3082i	− 1.09208i	0.837759	−	0.546040i	$-0.183866\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.94599i	0.394649i	0.980338	+	0.197324i	$0.0632253\pi$
$227$	5.94599i	0.394649i	−0.980338	+	0.197324i	$0.936775\pi$
$228$	0	0
$229$	12.3756	0.817802	0.408901	−	0.912579i	$-0.365912\pi$
$229$	12.3756	0.817802	0.408901	+	0.912579i	$0.365912\pi$
$230$	0	0
$231$	1.10539	0.0727292
$232$	0	0
$233$	19.7759i	1.29556i	0.761827	+	0.647781i	$0.224303\pi$
$233$	19.7759i	1.29556i	−0.761827	+	0.647781i	$0.775697\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0.156762i	0.0101828i
$238$	0	0
$239$	29.3463	1.89825	0.949127	−	0.314894i	$-0.101969\pi$
$239$	29.3463	1.89825	0.949127	+	0.314894i	$0.101969\pi$
$240$	0	0
$241$	−13.9327	−0.897481	−0.448741	−	0.893662i	$-0.648127\pi$
$241$	−13.9327	−0.897481	−0.448741	+	0.893662i	$0.648127\pi$
$242$	0	0
$243$	1.00000i	0.0641500i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	34.8273i	2.21601i
$248$	0	0
$249$	−7.74049	−0.490534
$250$	0	0
$251$	−7.68648	−0.485166	−0.242583	−	0.970131i	$-0.577995\pi$
$251$	−7.68648	−0.485166	−0.242583	+	0.970131i	$0.577995\pi$
$252$	0	0
$253$	− 17.7626i	− 1.11672i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	13.3596i	0.833350i	0.909055	+	0.416675i	$0.136805\pi$
$257$	13.3596i	0.833350i	−0.909055	+	0.416675i	$0.863195\pi$
$258$	0	0
$259$	3.88392	0.241335
$260$	0	0
$261$	7.96633	0.493103
$262$	0	0
$263$	13.8299i	0.852789i	0.904537	+	0.426394i	$0.140216\pi$
$263$	13.8299i	0.852789i	−0.904537	+	0.426394i	$0.859784\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 7.55005i	− 0.462056i
$268$	0	0
$269$	−13.0044	−0.792890	−0.396445	−	0.918058i	$-0.629756\pi$
$269$	−13.0044	−0.792890	−0.396445	+	0.918058i	$0.629756\pi$
$270$	0	0
$271$	7.61913	0.462829	0.231415	−	0.972855i	$-0.425665\pi$
$271$	7.61913	0.462829	0.231415	+	0.972855i	$0.425665\pi$
$272$	0	0
$273$	2.25951i	0.136752i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	17.2232i	1.03484i	0.855731	+	0.517421i	$0.173108\pi$
$277$	17.2232i	1.03484i	−0.855731	+	0.517421i	$0.826892\pi$
$278$	0	0
$279$	−1.00000	−0.0598684
$280$	0	0
$281$	−17.6191	−1.05107	−0.525535	−	0.850772i	$-0.676135\pi$
$281$	−17.6191	−1.05107	−0.525535	+	0.850772i	$0.676135\pi$
$282$	0	0
$283$	8.08241i	0.480449i	0.970717	+	0.240225i	$0.0772211\pi$
$283$	8.08241i	0.480449i	−0.970717	+	0.240225i	$0.922779\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 1.27020i	− 0.0749777i
$288$	0	0
$289$	−1.14343	−0.0672606
$290$	0	0
$291$	4.79186	0.280904
$292$	0	0
$293$	29.6731i	1.73352i	0.498722	+	0.866762i	$0.333803\pi$
$293$	29.6731i	1.73352i	−0.498722	+	0.866762i	$0.666197\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 2.79186i	− 0.162000i
$298$	0	0
$299$	36.3082	2.09976
$300$	0	0
$301$	−1.31088	−0.0755581
$302$	0	0
$303$	0.272843i	0.0156744i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	13.9797i	0.797861i	0.916981	+	0.398931i	$0.130619\pi$
$307$	13.9797i	0.797861i	−0.916981	+	0.398931i	$0.869381\pi$
$308$	0	0
$309$	2.66877	0.151821
$310$	0	0
$311$	32.4180	1.83826	0.919128	−	0.393959i	$-0.128895\pi$
$311$	32.4180	1.83826	0.919128	+	0.393959i	$0.128895\pi$
$312$	0	0
$313$	− 22.1178i	− 1.25017i	−0.780556	−	0.625086i	$-0.785064\pi$
$313$	− 22.1178i	− 1.25017i	0.780556	−	0.625086i	$-0.214936\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 25.3596i	− 1.42434i	−0.702008	−	0.712169i	$-0.747713\pi$
$317$	− 25.3596i	− 1.42434i	0.702008	−	0.712169i	$-0.252287\pi$
$318$	0	0
$319$	−22.2409	−1.24525
$320$	0	0
$321$	10.2595	0.572630
$322$	0	0
$323$	− 25.9947i	− 1.44638i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	13.5704i	0.750445i
$328$	0	0
$329$	−2.79186	−0.153920
$330$	0	0
$331$	25.9460	1.42612	0.713060	−	0.701103i	$-0.247309\pi$
$331$	25.9460	1.42612	0.713060	+	0.701103i	$0.247309\pi$
$332$	0	0
$333$	− 9.80957i	− 0.537561i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 0.812204i	− 0.0442436i	−0.999755	−	0.0221218i	$-0.992958\pi$
$337$	− 0.812204i	− 0.0442436i	0.999755	−	0.0221218i	$-0.00704216\pi$
$338$	0	0
$339$	9.93265	0.539468
$340$	0	0
$341$	2.79186	0.151188
$342$	0	0
$343$	− 5.48098i	− 0.295945i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 32.1382i	− 1.72527i	−0.505830	−	0.862633i	$-0.668814\pi$
$347$	− 32.1382i	− 1.72527i	0.505830	−	0.862633i	$-0.331186\pi$
$348$	0	0
$349$	21.2569	1.13785	0.568927	−	0.822388i	$-0.307359\pi$
$349$	21.2569	1.13785	0.568927	+	0.822388i	$0.307359\pi$
$350$	0	0
$351$	5.70682	0.304607
$352$	0	0
$353$	2.88128i	0.153355i	0.997056	+	0.0766775i	$0.0244312\pi$
$353$	2.88128i	0.153355i	−0.997056	+	0.0766775i	$0.975569\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 1.68648i	− 0.0892578i
$358$	0	0
$359$	28.6148	1.51023	0.755115	−	0.655593i	$-0.227581\pi$
$359$	28.6148	1.51023	0.755115	+	0.655593i	$0.227581\pi$
$360$	0	0
$361$	18.2435	0.960186
$362$	0	0
$363$	− 3.20550i	− 0.168245i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	16.0000i	0.835193i	0.908633	+	0.417597i	$0.137127\pi$
$367$	16.0000i	0.835193i	−0.908633	+	0.417597i	$0.862873\pi$
$368$	0	0
$369$	−3.20814	−0.167009
$370$	0	0
$371$	−0.416273	−0.0216118
$372$	0	0
$373$	− 6.13815i	− 0.317821i	−0.987293	−	0.158911i	$-0.949202\pi$
$373$	− 6.13815i	− 0.317821i	0.987293	−	0.158911i	$-0.0507982\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 45.4624i	− 2.34143i
$378$	0	0
$379$	−2.30825	−0.118567	−0.0592833	−	0.998241i	$-0.518882\pi$
$379$	−2.30825	−0.118567	−0.0592833	+	0.998241i	$0.518882\pi$
$380$	0	0
$381$	−17.8299	−0.913454
$382$	0	0
$383$	6.90794i	0.352979i	0.984302	+	0.176490i	$0.0564742\pi$
$383$	6.90794i	0.352979i	−0.984302	+	0.176490i	$0.943526\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	3.31088i	0.168302i
$388$	0	0
$389$	29.1745	1.47920	0.739602	−	0.673044i	$-0.235014\pi$
$389$	29.1745	1.47920	0.739602	+	0.673044i	$0.235014\pi$
$390$	0	0
$391$	−27.1001	−1.37051
$392$	0	0
$393$	− 14.9637i	− 0.754818i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 19.7626i	− 0.991854i	−0.868364	−	0.495927i	$-0.834828\pi$
$397$	− 19.7626i	− 0.991854i	0.868364	−	0.495927i	$-0.165172\pi$
$398$	0	0
$399$	2.41627	0.120965
$400$	0	0
$401$	−19.5501	−0.976283	−0.488142	−	0.872764i	$-0.662325\pi$
$401$	−19.5501	−0.976283	−0.488142	+	0.872764i	$0.662325\pi$
$402$	0	0
$403$	5.70682i	0.284277i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	27.3870i	1.35752i
$408$	0	0
$409$	9.45431	0.467486	0.233743	−	0.972298i	$-0.424903\pi$
$409$	9.45431	0.467486	0.233743	+	0.972298i	$0.424903\pi$
$410$	0	0
$411$	14.8813	0.734039
$412$	0	0
$413$	1.32422i	0.0651605i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	11.4136i	0.558928i
$418$	0	0
$419$	6.40926	0.313113	0.156557	−	0.987669i	$-0.449961\pi$
$419$	6.40926	0.313113	0.156557	+	0.987669i	$0.449961\pi$
$420$	0	0
$421$	−4.05401	−0.197581	−0.0987903	−	0.995108i	$-0.531497\pi$
$421$	−4.05401	−0.197581	−0.0987903	+	0.995108i	$0.531497\pi$
$422$	0	0
$423$	7.05137i	0.342850i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 4.99736i	− 0.241839i
$428$	0	0
$429$	−15.9327	−0.769236
$430$	0	0
$431$	−9.52339	−0.458726	−0.229363	−	0.973341i	$-0.573664\pi$
$431$	−9.52339	−0.458726	−0.229363	+	0.973341i	$0.573664\pi$
$432$	0	0
$433$	30.0151i	1.44243i	0.692710	+	0.721216i	$0.256416\pi$
$433$	30.0151i	1.44243i	−0.692710	+	0.721216i	$0.743584\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 38.8273i	− 1.85736i
$438$	0	0
$439$	31.3730	1.49735	0.748675	−	0.662938i	$-0.230691\pi$
$439$	31.3730	1.49735	0.748675	+	0.662938i	$0.230691\pi$
$440$	0	0
$441$	−6.84324	−0.325868
$442$	0	0
$443$	18.4917i	0.878566i	0.898349	+	0.439283i	$0.144767\pi$
$443$	18.4917i	0.878566i	−0.898349	+	0.439283i	$0.855233\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	8.79186i	0.415841i
$448$	0	0
$449$	−6.44467	−0.304143	−0.152071	−	0.988370i	$-0.548594\pi$
$449$	−6.44467	−0.304143	−0.152071	+	0.988370i	$0.548594\pi$
$450$	0	0
$451$	8.95668	0.421754
$452$	0	0
$453$	− 23.7759i	− 1.11709i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 20.9504i	− 0.980016i	−0.871718	−	0.490008i	$-0.836994\pi$
$457$	− 20.9504i	− 0.980016i	0.871718	−	0.490008i	$-0.163006\pi$
$458$	0	0
$459$	−4.25951	−0.198817
$460$	0	0
$461$	11.8990	0.554191	0.277095	−	0.960842i	$-0.410628\pi$
$461$	11.8990	0.554191	0.277095	+	0.960842i	$0.410628\pi$
$462$	0	0
$463$	− 2.58637i	− 0.120199i	−0.998192	−	0.0600993i	$-0.980858\pi$
$463$	− 2.58637i	− 0.120199i	0.998192	−	0.0600993i	$-0.0191417\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	7.05137i	0.326299i	0.986601	+	0.163149i	$0.0521652\pi$
$467$	7.05137i	0.326299i	−0.986601	+	0.163149i	$0.947835\pi$
$468$	0	0
$469$	−5.46765	−0.252472
$470$	0	0
$471$	17.5164	0.807112
$472$	0	0
$473$	− 9.24354i	− 0.425018i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	1.05137i	0.0481391i
$478$	0	0
$479$	32.9637	1.50615	0.753075	−	0.657935i	$-0.228570\pi$
$479$	32.9637	1.50615	0.753075	+	0.657935i	$0.228570\pi$
$480$	0	0
$481$	−55.9814	−2.55253
$482$	0	0
$483$	− 2.51902i	− 0.114619i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	14.0621i	0.637213i	0.947887	+	0.318607i	$0.103215\pi$
$487$	14.0621i	0.637213i	−0.947887	+	0.318607i	$0.896785\pi$
$488$	0	0
$489$	16.4987	0.746096
$490$	0	0
$491$	15.9327	0.719031	0.359515	−	0.933139i	$-0.382942\pi$
$491$	15.9327	0.719031	0.359515	+	0.933139i	$0.382942\pi$
$492$	0	0
$493$	33.9327i	1.52825i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 5.36490i	− 0.240649i
$498$	0	0
$499$	−21.9947	−0.984619	−0.492309	−	0.870420i	$-0.663847\pi$
$499$	−21.9947	−0.984619	−0.492309	+	0.870420i	$0.663847\pi$
$500$	0	0
$501$	−10.8946	−0.486735
$502$	0	0
$503$	5.67314i	0.252953i	0.991970	+	0.126476i	$0.0403668\pi$
$503$	5.67314i	0.252953i	−0.991970	+	0.126476i	$0.959633\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 19.5678i	− 0.869035i
$508$	0	0
$509$	33.1071	1.46745	0.733724	−	0.679448i	$-0.237781\pi$
$509$	33.1071	1.46745	0.733724	+	0.679448i	$0.237781\pi$
$510$	0	0
$511$	5.46765	0.241874
$512$	0	0
$513$	− 6.10275i	− 0.269443i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 19.6865i	− 0.865810i
$518$	0	0
$519$	16.8946	0.741591
$520$	0	0
$521$	−23.6191	−1.03477	−0.517386	−	0.855752i	$-0.673095\pi$
$521$	−23.6191	−1.03477	−0.517386	+	0.855752i	$0.673095\pi$
$522$	0	0
$523$	− 0.0673457i	− 0.00294482i	−0.999999	−	0.00147241i	$-0.999531\pi$
$523$	− 0.0673457i	− 0.00294482i	0.999999	−	0.00147241i	$-0.000468683\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 4.25951i	− 0.185547i
$528$	0	0
$529$	−17.4783	−0.759928
$530$	0	0
$531$	3.34456	0.145141
$532$	0	0
$533$	18.3082i	0.793018i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 2.93529i	− 0.126667i
$538$	0	0
$539$	19.1054	0.822927
$540$	0	0
$541$	−32.3303	−1.38999	−0.694994	−	0.719015i	$-0.744593\pi$
$541$	−32.3303	−1.38999	−0.694994	+	0.719015i	$0.744593\pi$
$542$	0	0
$543$	7.03804i	0.302031i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	31.8716i	1.36273i	0.731943	+	0.681366i	$0.238614\pi$
$547$	31.8716i	1.36273i	−0.731943	+	0.681366i	$0.761386\pi$
$548$	0	0
$549$	−12.6218	−0.538684
$550$	0	0
$551$	−48.6165	−2.07113
$552$	0	0
$553$	− 0.0620671i	− 0.00263936i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	33.9140i	1.43698i	0.695535	+	0.718492i	$0.255167\pi$
$557$	33.9140i	1.43698i	−0.695535	+	0.718492i	$0.744833\pi$
$558$	0	0
$559$	18.8946	0.799157
$560$	0	0
$561$	11.8920	0.502079
$562$	0	0
$563$	− 3.29755i	− 0.138975i	−0.997583	−	0.0694876i	$-0.977864\pi$
$563$	− 3.29755i	− 0.138975i	0.997583	−	0.0694876i	$-0.0221364\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 0.395932i	− 0.0166276i
$568$	0	0
$569$	17.8636	0.748880	0.374440	−	0.927251i	$-0.377835\pi$
$569$	17.8636	0.748880	0.374440	+	0.927251i	$0.377835\pi$
$570$	0	0
$571$	−10.0354	−0.419969	−0.209984	−	0.977705i	$-0.567341\pi$
$571$	−10.0354	−0.419969	−0.209984	+	0.977705i	$0.567341\pi$
$572$	0	0
$573$	12.8609i	0.537273i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	30.8273i	1.28336i	0.766974	+	0.641678i	$0.221761\pi$
$577$	30.8273i	1.28336i	−0.766974	+	0.641678i	$0.778239\pi$
$578$	0	0
$579$	−23.9327	−0.994607
$580$	0	0
$581$	3.06471	0.127145
$582$	0	0
$583$	− 2.93529i	− 0.121567i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.82991i	− 0.0755283i	−0.999287	−	0.0377641i	$-0.987976\pi$
$587$	− 1.82991i	− 0.0755283i	0.999287	−	0.0377641i	$-0.0120236\pi$
$588$	0	0
$589$	6.10275	0.251459
$590$	0	0
$591$	−19.0868	−0.785125
$592$	0	0
$593$	18.4517i	0.757719i	0.925454	+	0.378860i	$0.123684\pi$
$593$	18.4517i	0.757719i	−0.925454	+	0.378860i	$0.876316\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	14.5324i	0.594770i
$598$	0	0
$599$	−28.6502	−1.17061	−0.585307	−	0.810812i	$-0.699026\pi$
$599$	−28.6502	−1.17061	−0.585307	+	0.810812i	$0.699026\pi$
$600$	0	0
$601$	14.4517	0.589496	0.294748	−	0.955575i	$-0.404764\pi$
$601$	14.4517	0.589496	0.294748	+	0.955575i	$0.404764\pi$
$602$	0	0
$603$	13.8096i	0.562369i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	16.8796i	0.685120i	0.939496	+	0.342560i	$0.111294\pi$
$607$	16.8796i	0.685120i	−0.939496	+	0.342560i	$0.888706\pi$
$608$	0	0
$609$	−3.15412	−0.127812
$610$	0	0
$611$	40.2409	1.62797
$612$	0	0
$613$	36.8336i	1.48769i	0.668350	+	0.743847i	$0.267001\pi$
$613$	36.8336i	1.48769i	−0.668350	+	0.743847i	$0.732999\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 24.9620i	− 1.00493i	−0.864597	−	0.502465i	$-0.832426\pi$
$617$	− 24.9620i	− 1.00493i	0.864597	−	0.502465i	$-0.167574\pi$
$618$	0	0
$619$	47.3950	1.90497	0.952483	−	0.304591i	$-0.0985196\pi$
$619$	47.3950	1.90497	0.952483	+	0.304591i	$0.0985196\pi$
$620$	0	0
$621$	−6.36226	−0.255309
$622$	0	0
$623$	2.98931i	0.119764i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	17.0380i	0.680434i
$628$	0	0
$629$	41.7839	1.66604
$630$	0	0
$631$	−41.9947	−1.67178	−0.835892	−	0.548894i	$-0.815049\pi$
$631$	−41.9947	−1.67178	−0.835892	+	0.548894i	$0.815049\pi$
$632$	0	0
$633$	6.89461i	0.274036i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	39.0531i	1.54734i
$638$	0	0
$639$	−13.5501	−0.536032
$640$	0	0
$641$	−21.2365	−0.838793	−0.419396	−	0.907803i	$-0.637758\pi$
$641$	−21.2365	−0.838793	−0.419396	+	0.907803i	$0.637758\pi$
$642$	0	0
$643$	− 19.9327i	− 0.786067i	−0.919524	−	0.393034i	$-0.871426\pi$
$643$	− 19.9327i	− 0.786067i	0.919524	−	0.393034i	$-0.128574\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	29.4757i	1.15881i	0.815040	+	0.579405i	$0.196715\pi$
$647$	29.4757i	1.15881i	−0.815040	+	0.579405i	$0.803285\pi$
$648$	0	0
$649$	−9.33755	−0.366531
$650$	0	0
$651$	0.395932	0.0155178
$652$	0	0
$653$	12.8139i	0.501448i	0.968059	+	0.250724i	$0.0806687\pi$
$653$	12.8139i	0.501448i	−0.968059	+	0.250724i	$0.919331\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 13.8096i	− 0.538762i
$658$	0	0
$659$	−26.7262	−1.04111	−0.520553	−	0.853829i	$-0.674274\pi$
$659$	−26.7262	−1.04111	−0.520553	+	0.853829i	$0.674274\pi$
$660$	0	0
$661$	17.9327	0.697499	0.348750	−	0.937216i	$-0.386606\pi$
$661$	17.9327	0.697499	0.348750	+	0.937216i	$0.386606\pi$
$662$	0	0
$663$	24.3082i	0.944054i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	50.6838i	1.96249i
$668$	0	0
$669$	16.3082	0.630513
$670$	0	0
$671$	35.2383	1.36036
$672$	0	0
$673$	45.2586i	1.74459i	0.488979	+	0.872295i	$0.337369\pi$
$673$	45.2586i	1.74459i	−0.488979	+	0.872295i	$0.662631\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 36.9433i	− 1.41985i	−0.704278	−	0.709924i	$-0.748729\pi$
$677$	− 36.9433i	− 1.41985i	0.704278	−	0.709924i	$-0.251271\pi$
$678$	0	0
$679$	−1.89725	−0.0728098
$680$	0	0
$681$	−5.94599	−0.227851
$682$	0	0
$683$	− 51.0815i	− 1.95458i	−0.211908	−	0.977290i	$-0.567968\pi$
$683$	− 51.0815i	− 1.95458i	0.211908	−	0.977290i	$-0.432032\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	12.3756i	0.472158i
$688$	0	0
$689$	6.00000	0.228582
$690$	0	0
$691$	43.6191	1.65935	0.829675	−	0.558247i	$-0.188526\pi$
$691$	43.6191	1.65935	0.829675	+	0.558247i	$0.188526\pi$
$692$	0	0
$693$	1.10539i	0.0419902i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 13.6651i	− 0.517602i
$698$	0	0
$699$	−19.7759	−0.747993
$700$	0	0
$701$	45.8600	1.73211	0.866055	−	0.499949i	$-0.166648\pi$
$701$	45.8600	1.73211	0.866055	+	0.499949i	$0.166648\pi$
$702$	0	0
$703$	59.8653i	2.25786i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 0.108027i	− 0.00406278i
$708$	0	0
$709$	15.0380	0.564766	0.282383	−	0.959302i	$-0.408875\pi$
$709$	15.0380	0.564766	0.282383	+	0.959302i	$0.408875\pi$
$710$	0	0
$711$	−0.156762	−0.00587904
$712$	0	0
$713$	− 6.36226i	− 0.238268i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	29.3463i	1.09596i
$718$	0	0
$719$	15.6244	0.582692	0.291346	−	0.956618i	$-0.405897\pi$
$719$	15.6244	0.582692	0.291346	+	0.956618i	$0.405897\pi$
$720$	0	0
$721$	−1.05665	−0.0393518
$722$	0	0
$723$	− 13.9327i	− 0.518161i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 8.77152i	− 0.325318i	−0.986682	−	0.162659i	$-0.947993\pi$
$727$	− 8.77152i	− 0.325318i	0.986682	−	0.162659i	$-0.0520070\pi$
$728$	0	0
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	−14.1027	−0.521609
$732$	0	0
$733$	4.75646i	0.175684i	0.996134	+	0.0878419i	$0.0279970\pi$
$733$	4.75646i	0.175684i	−0.996134	+	0.0878419i	$0.972003\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 38.5544i	− 1.42017i
$738$	0	0
$739$	−17.8078	−0.655072	−0.327536	−	0.944839i	$-0.606218\pi$
$739$	−17.8078	−0.655072	−0.327536	+	0.944839i	$0.606218\pi$
$740$	0	0
$741$	−34.8273	−1.27941
$742$	0	0
$743$	40.3703i	1.48104i	0.672033	+	0.740522i	$0.265421\pi$
$743$	40.3703i	1.48104i	−0.672033	+	0.740522i	$0.734579\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 7.74049i	− 0.283210i
$748$	0	0
$749$	−4.06207	−0.148425
$750$	0	0
$751$	−45.3817	−1.65600	−0.828001	−	0.560727i	$-0.810522\pi$
$751$	−45.3817	−1.65600	−0.828001	+	0.560727i	$0.810522\pi$
$752$	0	0
$753$	− 7.68648i	− 0.280111i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 13.9390i	− 0.506621i	−0.967385	−	0.253310i	$-0.918481\pi$
$757$	− 13.9390i	− 0.506621i	0.967385	−	0.253310i	$-0.0815194\pi$
$758$	0	0
$759$	17.7626	0.644740
$760$	0	0
$761$	−46.5067	−1.68587	−0.842934	−	0.538017i	$-0.819174\pi$
$761$	−46.5067	−1.68587	−0.842934	+	0.538017i	$0.819174\pi$
$762$	0	0
$763$	− 5.37295i	− 0.194514i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 19.0868i	− 0.689184i
$768$	0	0
$769$	7.44098	0.268329	0.134164	−	0.990959i	$-0.457165\pi$
$769$	7.44098	0.268329	0.134164	+	0.990959i	$0.457165\pi$
$770$	0	0
$771$	−13.3596	−0.481135
$772$	0	0
$773$	− 44.9027i	− 1.61504i	−0.589842	−	0.807518i	$-0.700810\pi$
$773$	− 44.9027i	− 1.61504i	0.589842	−	0.807518i	$-0.299190\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	3.88392i	0.139335i
$778$	0	0
$779$	19.5784	0.701471
$780$	0	0
$781$	37.8299	1.35366
$782$	0	0
$783$	7.96633i	0.284693i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	9.45431i	0.337010i	0.985701	+	0.168505i	$0.0538939\pi$
$787$	9.45431i	0.337010i	−0.985701	+	0.168505i	$0.946106\pi$
$788$	0	0
$789$	−13.8299	−0.492358
$790$	0	0
$791$	−3.93265	−0.139829
$792$	0	0
$793$	72.0301i	2.55786i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 44.4650i	− 1.57503i	−0.616295	−	0.787516i	$-0.711367\pi$
$797$	− 44.4650i	− 1.57503i	0.616295	−	0.787516i	$-0.288633\pi$
$798$	0	0
$799$	−30.0354	−1.06258
$800$	0	0
$801$	7.55005	0.266768
$802$	0	0
$803$	38.5544i	1.36056i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 13.0044i	− 0.457775i
$808$	0	0
$809$	−38.1311	−1.34062	−0.670310	−	0.742081i	$-0.733839\pi$
$809$	−38.1311	−1.34062	−0.670310	+	0.742081i	$0.733839\pi$
$810$	0	0
$811$	27.4136	0.962623	0.481311	−	0.876550i	$-0.340161\pi$
$811$	27.4136	0.962623	0.481311	+	0.876550i	$0.340161\pi$
$812$	0	0
$813$	7.61913i	0.267215i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 20.2055i	− 0.706901i
$818$	0	0
$819$	−2.25951	−0.0789537
$820$	0	0
$821$	−2.96369	−0.103433	−0.0517167	−	0.998662i	$-0.516469\pi$
$821$	−2.96369	−0.103433	−0.0517167	+	0.998662i	$0.516469\pi$
$822$	0	0
$823$	− 1.20814i	− 0.0421130i	−0.999778	−	0.0210565i	$-0.993297\pi$
$823$	− 1.20814i	− 0.0421130i	0.999778	−	0.0210565i	$-0.00670298\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 2.70245i	− 0.0939733i	−0.998896	−	0.0469867i	$-0.985038\pi$
$827$	− 2.70245i	− 0.0939733i	0.998896	−	0.0469867i	$-0.0149618\pi$
$828$	0	0
$829$	−28.5544	−0.991736	−0.495868	−	0.868398i	$-0.665150\pi$
$829$	−28.5544	−0.991736	−0.495868	+	0.868398i	$0.665150\pi$
$830$	0	0
$831$	−17.2232	−0.597466
$832$	0	0
$833$	− 29.1488i	− 1.00995i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 1.00000i	− 0.0345651i
$838$	0	0
$839$	43.7289	1.50969	0.754844	−	0.655904i	$-0.227712\pi$
$839$	43.7289	1.50969	0.754844	+	0.655904i	$0.227712\pi$
$840$	0	0
$841$	34.4624	1.18836
$842$	0	0
$843$	− 17.6191i	− 0.606835i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	1.26916i	0.0436088i
$848$	0	0
$849$	−8.08241	−0.277388
$850$	0	0
$851$	62.4110	2.13942
$852$	0	0
$853$	− 44.9300i	− 1.53837i	−0.639024	−	0.769187i	$-0.720661\pi$
$853$	− 44.9300i	− 1.53837i	0.639024	−	0.769187i	$-0.279339\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16.8946i	0.577109i	0.957463	+	0.288554i	$0.0931746\pi$
$857$	16.8946i	0.577109i	−0.957463	+	0.288554i	$0.906825\pi$
$858$	0	0
$859$	−35.9186	−1.22553	−0.612764	−	0.790266i	$-0.709942\pi$
$859$	−35.9186	−1.22553	−0.612764	+	0.790266i	$0.709942\pi$
$860$	0	0
$861$	1.27020	0.0432884
$862$	0	0
$863$	− 48.0708i	− 1.63635i	−0.574970	−	0.818175i	$-0.694986\pi$
$863$	− 48.0708i	− 1.63635i	0.574970	−	0.818175i	$-0.305014\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1.14343i	− 0.0388329i
$868$	0	0
$869$	0.437658	0.0148465
$870$	0	0
$871$	78.8087	2.67033
$872$	0	0
$873$	4.79186i	0.162180i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 28.6979i	− 0.969058i	−0.874775	−	0.484529i	$-0.838991\pi$
$877$	− 28.6979i	− 0.969058i	0.874775	−	0.484529i	$-0.161009\pi$
$878$	0	0
$879$	−29.6731	−1.00085
$880$	0	0
$881$	25.8583	0.871188	0.435594	−	0.900143i	$-0.356538\pi$
$881$	25.8583	0.871188	0.435594	+	0.900143i	$0.356538\pi$
$882$	0	0
$883$	− 17.0328i	− 0.573198i	−0.958051	−	0.286599i	$-0.907475\pi$
$883$	− 17.0328i	− 0.573198i	0.958051	−	0.286599i	$-0.0925248\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	7.23021i	0.242767i	0.992606	+	0.121383i	$0.0387330\pi$
$887$	7.23021i	0.242767i	−0.992606	+	0.121383i	$0.961267\pi$
$888$	0	0
$889$	7.05943	0.236766
$890$	0	0
$891$	2.79186	0.0935310
$892$	0	0
$893$	− 43.0328i	− 1.44004i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	36.3082i	1.21230i
$898$	0	0
$899$	−7.96633	−0.265692
$900$	0	0
$901$	−4.47834	−0.149195
$902$	0	0
$903$	− 1.31088i	− 0.0436235i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	15.2499i	0.506363i	0.967419	+	0.253182i	$0.0814770\pi$
$907$	15.2499i	0.506363i	−0.967419	+	0.253182i	$0.918523\pi$
$908$	0	0
$909$	−0.272843	−0.00904962
$910$	0	0
$911$	−30.4517	−1.00891	−0.504455	−	0.863438i	$-0.668306\pi$
$911$	−30.4517	−1.00891	−0.504455	+	0.863438i	$0.668306\pi$
$912$	0	0
$913$	21.6104i	0.715200i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.92460i	0.195648i
$918$	0	0
$919$	−29.2435	−0.964655	−0.482328	−	0.875991i	$-0.660209\pi$
$919$	−29.2435	−0.964655	−0.482328	+	0.875991i	$0.660209\pi$
$920$	0	0
$921$	−13.9797	−0.460645
$922$	0	0
$923$	77.3277i	2.54527i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	2.66877i	0.0876541i
$928$	0	0
$929$	11.8282	0.388070	0.194035	−	0.980995i	$-0.437842\pi$
$929$	11.8282	0.388070	0.194035	+	0.980995i	$0.437842\pi$
$930$	0	0
$931$	41.7626	1.36871
$932$	0	0
$933$	32.4180i	1.06132i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	20.6572i	0.674840i	0.941354	+	0.337420i	$0.109554\pi$
$937$	20.6572i	0.674840i	−0.941354	+	0.337420i	$0.890446\pi$
$938$	0	0
$939$	22.1178	0.721787
$940$	0	0
$941$	−48.0284	−1.56568	−0.782840	−	0.622222i	$-0.786230\pi$
$941$	−48.0284	−1.56568	−0.782840	+	0.622222i	$0.786230\pi$
$942$	0	0
$943$	− 20.4110i	− 0.664673i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	25.3242i	0.822926i	0.911426	+	0.411463i	$0.134982\pi$
$947$	25.3242i	0.822926i	−0.911426	+	0.411463i	$0.865018\pi$
$948$	0	0
$949$	−78.8087	−2.55824
$950$	0	0
$951$	25.3596	0.822342
$952$	0	0
$953$	− 37.5030i	− 1.21484i	−0.794380	−	0.607421i	$-0.792204\pi$
$953$	− 37.5030i	− 1.21484i	0.794380	−	0.607421i	$-0.207796\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 22.2409i	− 0.718946i
$958$	0	0
$959$	−5.89197	−0.190262
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	10.2595i	0.330608i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 26.1788i	− 0.841855i	−0.907094	−	0.420927i	$-0.861705\pi$
$967$	− 26.1788i	− 0.841855i	0.907094	−	0.420927i	$-0.138295\pi$
$968$	0	0
$969$	25.9947	0.835071
$970$	0	0
$971$	−27.8316	−0.893160	−0.446580	−	0.894744i	$-0.647358\pi$
$971$	−27.8316	−0.893160	−0.446580	+	0.894744i	$0.647358\pi$
$972$	0	0
$973$	− 4.51902i	− 0.144873i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	11.0380i	0.353138i	0.984288	+	0.176569i	$0.0564999\pi$
$977$	11.0380i	0.353138i	−0.984288	+	0.176569i	$0.943500\pi$
$978$	0	0
$979$	−21.0787	−0.673679
$980$	0	0
$981$	−13.5704	−0.433269
$982$	0	0
$983$	− 2.10275i	− 0.0670673i	−0.999438	−	0.0335336i	$-0.989324\pi$
$983$	− 2.10275i	− 0.0670673i	0.999438	−	0.0335336i	$-0.0106761\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 2.79186i	− 0.0888660i
$988$	0	0
$989$	−21.0647	−0.669819
$990$	0	0
$991$	13.1321	0.417153	0.208577	−	0.978006i	$-0.433117\pi$
$991$	13.1321	0.417153	0.208577	+	0.978006i	$0.433117\pi$
$992$	0	0
$993$	25.9460i	0.823371i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 36.3489i	− 1.15118i	−0.817738	−	0.575591i	$-0.804772\pi$
$997$	− 36.3489i	− 1.15118i	0.817738	−	0.575591i	$-0.195228\pi$
$998$	0	0
$999$	9.80957	0.310361

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9300.2.g.q.3349.5		6
5.2	odd	4		1860.2.a.g.1.2	✓	3
5.3	odd	4		9300.2.a.u.1.2		3
5.4	even	2	inner	9300.2.g.q.3349.2		6
15.2	even	4		5580.2.a.j.1.2		3
20.7	even	4		7440.2.a.bn.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1860.2.a.g.1.2	✓	3	5.2	odd	4
5580.2.a.j.1.2		3	15.2	even	4
7440.2.a.bn.1.2		3	20.7	even	4
9300.2.a.u.1.2		3	5.3	odd	4
9300.2.g.q.3349.2		6	5.4	even	2	inner
9300.2.g.q.3349.5		6	1.1	even	1	trivial

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 9300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9300.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000i q^{3} -0.395932i q^{7} -1.00000 q^{9} +2.79186 q^{11} +5.70682i q^{13} -4.25951i q^{17} +6.10275 q^{19} +0.395932 q^{21} -6.36226i q^{23} -1.00000i q^{27} -7.96633 q^{29} +1.00000 q^{31} +2.79186i q^{33} +9.80957i q^{37} -5.70682 q^{39} +3.20814 q^{41} -3.31088i q^{43} -7.05137i q^{47} +6.84324 q^{49} +4.25951 q^{51} -1.05137i q^{53} +6.10275i q^{57} -3.34456 q^{59} +12.6218 q^{61} +0.395932i q^{63} -13.8096i q^{67} +6.36226 q^{69} +13.5501 q^{71} +13.8096i q^{73} -1.10539i q^{77} +0.156762 q^{79} +1.00000 q^{81} +7.74049i q^{83} -7.96633i q^{87} -7.55005 q^{89} +2.25951 q^{91} +1.00000i q^{93} -4.79186i q^{97} -2.79186 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 6 q^{9} + 4 q^{11} - 4 q^{21} - 12 q^{29} + 6 q^{31} - 4 q^{39} + 32 q^{41} + 10 q^{49} + 20 q^{51} - 32 q^{59} + 28 q^{61} - 4 q^{69} + 20 q^{71} + 32 q^{79} + 6 q^{81} + 16 q^{89} + 8 q^{91} - 4 q^{99}+O(q^{100}) \) 6 * q - 6 * q^9 + 4 * q^11 - 4 * q^21 - 12 * q^29 + 6 * q^31 - 4 * q^39 + 32 * q^41 + 10 * q^49 + 20 * q^51 - 32 * q^59 + 28 * q^61 - 4 * q^69 + 20 * q^71 + 32 * q^79 + 6 * q^81 + 16 * q^89 + 8 * q^91 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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