LMFDB - Embedded newform 3420.2.f.e.1369.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3420,2,Mod(1369,3420)] 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("3420.1369"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3420, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3420.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$27.3088374913$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 2x^{8} + 18x^{7} - 7x^{6} - 48x^{5} - 35x^{4} + 450x^{3} - 250x^{2} - 1250x + 3125 $ x^10 - 2x^9 - 2x^8 + 18x^7 - 7x^6 - 48x^5 - 35x^4 + 450x^3 - 250x^2 - 1250*x + 3125
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	no (minimal twist has level 1140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1369.10
Root			$1.16684 - 1.90748i$ of defining polynomial
Character	$\chi$	$=$	3420.1369
Dual form			3420.2.f.e.1369.9

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.90748 + 1.16684i) q^{5} -1.36950i q^{7} +0.469639 q^{11} +1.03583i q^{13} -5.65411i q^{17} +1.00000 q^{19} -2.30950i q^{23} +(2.27698 + 4.45144i) q^{25} -6.21295 q^{29} +9.98778 q^{31} +(1.59798 - 2.61230i) q^{35} -8.64245i q^{37} -0.654673 q^{41} +8.26043i q^{43} -8.39311i q^{47} +5.12447 q^{49} -9.24878i q^{53} +(0.895829 + 0.547992i) q^{55} +5.62993 q^{59} +9.06374 q^{61} +(-1.20864 + 1.97583i) q^{65} +5.62993i q^{67} -10.3456 q^{71} -3.89195i q^{73} -0.643172i q^{77} -12.5692 q^{79} -3.61884i q^{83} +(6.59742 - 10.7851i) q^{85} +11.8429 q^{89} +1.41857 q^{91} +(1.90748 + 1.16684i) q^{95} -5.65483i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 2 q^{5} - 12 q^{11} + 10 q^{19} - 8 q^{25} + 16 q^{31} + 22 q^{35} - 24 q^{41} - 6 q^{49} - 10 q^{55} - 12 q^{59} + 32 q^{65} - 4 q^{71} - 24 q^{79} - 10 q^{85} - 12 q^{89} + 32 q^{91} + 2 q^{95}+O(q^{100}) $ 10 * q + 2 * q^5 - 12 * q^11 + 10 * q^19 - 8 * q^25 + 16 * q^31 + 22 * q^35 - 24 * q^41 - 6 * q^49 - 10 * q^55 - 12 * q^59 + 32 * q^65 - 4 * q^71 - 24 * q^79 - 10 * q^85 - 12 * q^89 + 32 * q^91 + 2 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times$.

$n$	$1711$	$1901$	$2737$	$3061$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$3$		0			0
$3$		0			0
$4$		0			0
$5$	1.90748	+	1.16684i	0.853052	+	0.521825i
$5$	1.90748	+	1.16684i	0.853052	+	0.521825i
$6$		0			0
$7$		−	1.36950i		−	0.517623i	−0.965928	−	0.258811i	$-0.916669\pi$
$7$		−	1.36950i		−	0.517623i	0.965928	−	0.258811i	$-0.0833308\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.469639			0.141602			0.0708008	−	0.997490i	$-0.477445\pi$
$11$	0.469639			0.141602			0.0708008	+	0.997490i	$0.477445\pi$
$12$		0			0
$13$			1.03583i			0.287287i	0.989629	+	0.143644i	$0.0458819\pi$
$13$			1.03583i			0.287287i	−0.989629	+	0.143644i	$0.954118\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	5.65411i		−	1.37132i	−0.727921	−	0.685661i	$-0.759513\pi$
$17$		−	5.65411i		−	1.37132i	0.727921	−	0.685661i	$-0.240487\pi$
$18$		0			0
$19$	1.00000			0.229416
$19$	1.00000			0.229416
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	2.30950i		−	0.481564i	−0.970579	−	0.240782i	$-0.922596\pi$
$23$		−	2.30950i		−	0.481564i	0.970579	−	0.240782i	$-0.0774038\pi$
$24$		0			0
$25$	2.27698	+	4.45144i	0.455397	+	0.890288i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6.21295			−1.15372			−0.576858	−	0.816845i	$-0.695721\pi$
$29$	−6.21295			−1.15372			−0.576858	+	0.816845i	$0.695721\pi$
$30$		0			0
$31$	9.98778			1.79386			0.896929	−	0.442174i	$-0.145793\pi$
$31$	9.98778			1.79386			0.896929	+	0.442174i	$0.145793\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.59798	−	2.61230i	0.270109	−	0.441560i
$36$		0			0
$37$		−	8.64245i		−	1.42081i	−0.703793	−	0.710405i	$-0.748512\pi$
$37$		−	8.64245i		−	1.42081i	0.703793	−	0.710405i	$-0.251488\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.654673			−0.102243			−0.0511214	−	0.998692i	$-0.516280\pi$
$41$	−0.654673			−0.102243			−0.0511214	+	0.998692i	$0.516280\pi$
$42$		0			0
$43$			8.26043i			1.25970i	0.776715	+	0.629852i	$0.216884\pi$
$43$			8.26043i			1.25970i	−0.776715	+	0.629852i	$0.783116\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	8.39311i		−	1.22426i	−0.790757	−	0.612131i	$-0.790313\pi$
$47$		−	8.39311i		−	1.22426i	0.790757	−	0.612131i	$-0.209687\pi$
$48$		0			0
$49$	5.12447			0.732066
$50$		0			0
$51$		0			0
$52$		0			0
$53$		−	9.24878i		−	1.27042i	−0.772341	−	0.635209i	$-0.780914\pi$
$53$		−	9.24878i		−	1.27042i	0.772341	−	0.635209i	$-0.219086\pi$
$54$		0			0
$55$	0.895829	+	0.547992i	0.120794	+	0.0738913i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	5.62993			0.732955			0.366477	−	0.930427i	$-0.380564\pi$
$59$	5.62993			0.732955			0.366477	+	0.930427i	$0.380564\pi$
$60$		0			0
$61$	9.06374			1.16049			0.580247	−	0.814441i	$-0.302956\pi$
$61$	9.06374			1.16049			0.580247	+	0.814441i	$0.302956\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.20864	+	1.97583i	−0.149914	+	0.245071i
$66$		0			0
$67$			5.62993i			0.687806i	0.939005	+	0.343903i	$0.111749\pi$
$67$			5.62993i			0.687806i	−0.939005	+	0.343903i	$0.888251\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−10.3456			−1.22780			−0.613900	−	0.789384i	$-0.710400\pi$
$71$	−10.3456			−1.22780			−0.613900	+	0.789384i	$0.710400\pi$
$72$		0			0
$73$		−	3.89195i		−	0.455518i	−0.973718	−	0.227759i	$-0.926860\pi$
$73$		−	3.89195i		−	0.455518i	0.973718	−	0.227759i	$-0.0731399\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	0.643172i		−	0.0732962i
$78$		0			0
$79$	−12.5692			−1.41415			−0.707073	−	0.707140i	$-0.749985\pi$
$79$	−12.5692			−1.41415			−0.707073	+	0.707140i	$0.749985\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	3.61884i		−	0.397220i	−0.980079	−	0.198610i	$-0.936357\pi$
$83$		−	3.61884i		−	0.397220i	0.980079	−	0.198610i	$-0.0636427\pi$
$84$		0			0
$85$	6.59742	−	10.7851i	0.715591	−	1.16981i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	11.8429			1.25534			0.627671	−	0.778478i	$-0.284008\pi$
$89$	11.8429			1.25534			0.627671	+	0.778478i	$0.284008\pi$
$90$		0			0
$91$	1.41857			0.148706
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.90748	+	1.16684i	0.195704	+	0.119715i
$96$		0			0
$97$		−	5.65483i		−	0.574161i	−0.957907	−	0.287080i	$-0.907315\pi$
$97$		−	5.65483i		−	0.574161i	0.957907	−	0.287080i	$-0.0926847\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	18.9865			1.88923			0.944614	−	0.328185i	$-0.106437\pi$
$101$	18.9865			1.88923			0.944614	+	0.328185i	$0.106437\pi$
$102$		0			0
$103$			15.3082i			1.50836i	0.656666	+	0.754182i	$0.271966\pi$
$103$			15.3082i			1.50836i	−0.656666	+	0.754182i	$0.728034\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			9.23656i			0.892932i	0.894801	+	0.446466i	$0.147318\pi$
$107$			9.23656i			0.892932i	−0.894801	+	0.446466i	$0.852682\pi$
$108$		0			0
$109$	10.2489			0.981670			0.490835	−	0.871253i	$-0.336692\pi$
$109$	10.2489			0.981670			0.490835	+	0.871253i	$0.336692\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$		−	11.1771i		−	1.05146i	−0.850653	−	0.525728i	$-0.823793\pi$
$113$		−	11.1771i		−	1.05146i	0.850653	−	0.525728i	$-0.176207\pi$
$114$		0			0
$115$	2.69481	−	4.40533i	0.251292	−	0.410799i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−7.74331			−0.709828
$120$		0			0
$121$	−10.7794			−0.979949
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−0.850795	+	11.1479i	−0.0760974	+	0.997100i
$126$		0			0
$127$			4.20028i			0.372714i	0.982482	+	0.186357i	$0.0596681\pi$
$127$			4.20028i			0.372714i	−0.982482	+	0.186357i	$0.940332\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−5.87599			−0.513388			−0.256694	−	0.966493i	$-0.582633\pi$
$131$	−5.87599			−0.513388			−0.256694	+	0.966493i	$0.582633\pi$
$132$		0			0
$133$		−	1.36950i		−	0.118751i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			10.2355i			0.874481i	0.899345	+	0.437241i	$0.144044\pi$
$137$			10.2355i			0.874481i	−0.899345	+	0.437241i	$0.855956\pi$
$138$		0			0
$139$	9.56491			0.811285			0.405642	−	0.914032i	$-0.367048\pi$
$139$	9.56491			0.811285			0.405642	+	0.914032i	$0.367048\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			0.486466i			0.0406803i
$144$		0			0
$145$	−11.8511	−	7.24950i	−0.984180	−	0.602038i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−3.26763			−0.267694			−0.133847	−	0.991002i	$-0.542733\pi$
$149$	−3.26763			−0.267694			−0.133847	+	0.991002i	$0.542733\pi$
$150$		0			0
$151$	3.57050			0.290563			0.145281	−	0.989390i	$-0.453591\pi$
$151$	3.57050			0.290563			0.145281	+	0.989390i	$0.453591\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	19.0515	+	11.6541i	1.53026	+	0.936080i
$156$		0			0
$157$		−	16.2740i		−	1.29880i	−0.760445	−	0.649402i	$-0.775019\pi$
$157$		−	16.2740i		−	1.29880i	0.760445	−	0.649402i	$-0.224981\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−3.16286			−0.249268
$162$		0			0
$163$		−	13.0114i		−	1.01913i	−0.860432	−	0.509565i	$-0.829806\pi$
$163$		−	13.0114i		−	1.01913i	0.860432	−	0.509565i	$-0.170194\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	17.4174i		−	1.34780i	−0.738822	−	0.673901i	$-0.764617\pi$
$167$		−	17.4174i		−	1.34780i	0.738822	−	0.673901i	$-0.235383\pi$
$168$		0			0
$169$	11.9271			0.917466
$170$		0			0
$171$		0			0
$172$		0			0
$173$			3.93943i			0.299509i	0.988723	+	0.149755i	$0.0478484\pi$
$173$			3.93943i			0.299509i	−0.988723	+	0.149755i	$0.952152\pi$
$174$		0			0
$175$	6.09626	−	3.11833i	0.460834	−	0.235724i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−18.7379			−1.40053			−0.700267	−	0.713881i	$-0.746936\pi$
$179$	−18.7379			−1.40053			−0.700267	+	0.713881i	$0.746936\pi$
$180$		0			0
$181$	19.1384			1.42255			0.711274	−	0.702915i	$-0.248119\pi$
$181$	19.1384			1.42255			0.711274	+	0.702915i	$0.248119\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	10.0843	−	16.4853i	0.741415	−	1.21203i
$186$		0			0
$187$		−	2.65539i		−	0.194181i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.23195			−0.0891408			−0.0445704	−	0.999006i	$-0.514192\pi$
$191$	−1.23195			−0.0891408			−0.0445704	+	0.999006i	$0.514192\pi$
$192$		0			0
$193$		−	4.34404i		−	0.312691i	−0.987702	−	0.156346i	$-0.950029\pi$
$193$		−	4.34404i		−	0.312691i	0.987702	−	0.156346i	$-0.0499713\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			22.8304i			1.62660i	0.581847	+	0.813298i	$0.302330\pi$
$197$			22.8304i			1.62660i	−0.581847	+	0.813298i	$0.697670\pi$
$198$		0			0
$199$	5.05152			0.358093			0.179047	−	0.983841i	$-0.442699\pi$
$199$	5.05152			0.358093			0.179047	+	0.983841i	$0.442699\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$			8.50864i			0.597190i
$204$		0			0
$205$	−1.24878	−	0.763896i	−0.0872184	−	0.0533528i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.469639			0.0324856
$210$		0			0
$211$	−9.90390			−0.681813			−0.340906	−	0.940097i	$-0.610734\pi$
$211$	−9.90390			−0.681813			−0.340906	+	0.940097i	$0.610734\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−9.63857	+	15.7566i	−0.657345	+	1.07459i
$216$		0			0
$217$		−	13.6783i		−	0.928542i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.85669			0.393963
$222$		0			0
$223$			16.1189i			1.07940i	0.841858	+	0.539700i	$0.181462\pi$
$223$			16.1189i			1.07940i	−0.841858	+	0.539700i	$0.818538\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		−	2.55812i		−	0.169789i	−0.996390	−	0.0848943i	$-0.972945\pi$
$227$		−	2.55812i		−	0.169789i	0.996390	−	0.0848943i	$-0.0270553\pi$
$228$		0			0
$229$	−16.3268			−1.07890			−0.539452	−	0.842016i	$-0.681369\pi$
$229$	−16.3268			−1.07890			−0.539452	+	0.842016i	$0.681369\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	9.71468i		−	0.636430i	−0.948019	−	0.318215i	$-0.896917\pi$
$233$		−	9.71468i		−	0.636430i	0.948019	−	0.318215i	$-0.103083\pi$
$234$		0			0
$235$	9.79339	−	16.0097i	0.638850	−	1.04436i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−5.53036			−0.357729			−0.178865	−	0.983874i	$-0.557242\pi$
$239$	−5.53036			−0.357729			−0.178865	+	0.983874i	$0.557242\pi$
$240$		0			0
$241$	−3.60806			−0.232416			−0.116208	−	0.993225i	$-0.537074\pi$
$241$	−3.60806			−0.232416			−0.116208	+	0.993225i	$0.537074\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	9.77483	+	5.97941i	0.624491	+	0.382011i
$246$		0			0
$247$			1.03583i			0.0659082i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	15.0084			0.947320			0.473660	−	0.880708i	$-0.342933\pi$
$251$	15.0084			0.947320			0.473660	+	0.880708i	$0.342933\pi$
$252$		0			0
$253$		−	1.08463i		−	0.0681902i
$254$		0			0
$255$		0			0
$256$		0			0
$257$		−	1.91612i		−	0.119525i	−0.998213	−	0.0597623i	$-0.980966\pi$
$257$		−	1.91612i		−	0.119525i	0.998213	−	0.0597623i	$-0.0190343\pi$
$258$		0			0
$259$	−11.8359			−0.735444
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	12.3448i		−	0.761211i	−0.924738	−	0.380605i	$-0.875716\pi$
$263$		−	12.3448i		−	0.761211i	0.924738	−	0.380605i	$-0.124284\pi$
$264$		0			0
$265$	10.7918	−	17.6419i	0.662936	−	1.08373i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.701891			0.0427951			0.0213975	−	0.999771i	$-0.493188\pi$
$269$	0.701891			0.0427951			0.0213975	+	0.999771i	$0.493188\pi$
$270$		0			0
$271$	26.2159			1.59250			0.796251	−	0.604967i	$-0.206814\pi$
$271$	26.2159			1.59250			0.796251	+	0.604967i	$0.206814\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.06936	+	2.09057i	0.0644849	+	0.126066i
$276$		0			0
$277$			6.11410i			0.367360i	0.982986	+	0.183680i	$0.0588011\pi$
$277$			6.11410i			0.367360i	−0.982986	+	0.183680i	$0.941199\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	22.4838			1.34127			0.670634	−	0.741788i	$-0.266022\pi$
$281$	22.4838			1.34127			0.670634	+	0.741788i	$0.266022\pi$
$282$		0			0
$283$			28.4861i			1.69332i	0.532134	+	0.846660i	$0.321390\pi$
$283$			28.4861i			1.69332i	−0.532134	+	0.846660i	$0.678610\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$			0.896575i			0.0529232i
$288$		0			0
$289$	−14.9689			−0.880525
$290$		0			0
$291$		0			0
$292$		0			0
$293$			8.55699i			0.499905i	0.968258	+	0.249952i	$0.0804150\pi$
$293$			8.55699i			0.499905i	−0.968258	+	0.249952i	$0.919585\pi$
$294$		0			0
$295$	10.7390	+	6.56921i	0.625249	+	0.382474i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.39224			0.138347
$300$		0			0
$301$	11.3127			0.652052
$302$		0			0
$303$		0			0
$304$		0			0
$305$	17.2889	+	10.5759i	0.989962	+	0.605575i
$306$		0			0
$307$		−	7.85352i		−	0.448224i	−0.974563	−	0.224112i	$-0.928052\pi$
$307$		−	7.85352i		−	0.448224i	0.974563	−	0.224112i	$-0.0719481\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	31.8265			1.80472			0.902358	−	0.430988i	$-0.141835\pi$
$311$	31.8265			1.80472			0.902358	+	0.430988i	$0.141835\pi$
$312$		0			0
$313$		−	10.1991i		−	0.576490i	−0.957557	−	0.288245i	$-0.906928\pi$
$313$		−	10.1991i		−	0.576490i	0.957557	−	0.288245i	$-0.0930717\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			17.9130i			1.00609i	0.864260	+	0.503046i	$0.167787\pi$
$317$			17.9130i			1.00609i	−0.864260	+	0.503046i	$0.832213\pi$
$318$		0			0
$319$	−2.91785			−0.163368
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	5.65411i		−	0.314603i
$324$		0			0
$325$	−4.61093	+	2.35857i	−0.255768	+	0.130830i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−11.4944			−0.633706
$330$		0			0
$331$	−17.1674			−0.943604			−0.471802	−	0.881704i	$-0.656396\pi$
$331$	−17.1674			−0.943604			−0.471802	+	0.881704i	$0.656396\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−6.56921	+	10.7390i	−0.358914	+	0.586734i
$336$		0			0
$337$		−	16.4010i		−	0.893420i	−0.894679	−	0.446710i	$-0.852596\pi$
$337$		−	16.4010i		−	0.893420i	0.894679	−	0.446710i	$-0.147404\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	4.69065			0.254013
$342$		0			0
$343$		−	16.6045i		−	0.896557i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			18.4648i			0.991241i	0.868539	+	0.495620i	$0.165059\pi$
$347$			18.4648i			0.991241i	−0.868539	+	0.495620i	$0.834941\pi$
$348$		0			0
$349$	14.6561			0.784524			0.392262	−	0.919854i	$-0.371693\pi$
$349$	14.6561			0.784524			0.392262	+	0.919854i	$0.371693\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		−	21.7986i		−	1.16022i	−0.814537	−	0.580111i	$-0.803009\pi$
$353$		−	21.7986i		−	1.16022i	0.814537	−	0.580111i	$-0.196991\pi$
$354$		0			0
$355$	−19.7341	−	12.0717i	−1.04738	−	0.640697i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−16.1494			−0.852331			−0.426165	−	0.904645i	$-0.640136\pi$
$359$	−16.1494			−0.852331			−0.426165	+	0.904645i	$0.640136\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4.54127	−	7.42383i	0.237701	−	0.388581i
$366$		0			0
$367$			7.33454i			0.382860i	0.981506	+	0.191430i	$0.0613125\pi$
$367$			7.33454i			0.382860i	−0.981506	+	0.191430i	$0.938687\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−12.6662			−0.657597
$372$		0			0
$373$		−	28.2727i		−	1.46390i	−0.681356	−	0.731952i	$-0.738609\pi$
$373$		−	28.2727i		−	1.46390i	0.681356	−	0.731952i	$-0.261391\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	6.43555i		−	0.331448i
$378$		0			0
$379$	−24.9357			−1.28086			−0.640430	−	0.768017i	$-0.721244\pi$
$379$	−24.9357			−1.28086			−0.640430	+	0.768017i	$0.721244\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	33.0435i		−	1.68844i	−0.535995	−	0.844221i	$-0.680064\pi$
$383$		−	33.0435i		−	1.68844i	0.535995	−	0.844221i	$-0.319936\pi$
$384$		0			0
$385$	0.750476	−	1.22684i	0.0382478	−	0.0625255i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	10.0087			0.507460			0.253730	−	0.967275i	$-0.418343\pi$
$389$	10.0087			0.507460			0.253730	+	0.967275i	$0.418343\pi$
$390$		0			0
$391$	−13.0582			−0.660379
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−23.9756	−	14.6662i	−1.20634	−	0.737937i
$396$		0			0
$397$			20.8784i			1.04786i	0.851762	+	0.523930i	$0.175535\pi$
$397$			20.8784i			1.04786i	−0.851762	+	0.523930i	$0.824465\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	14.1327			0.705752			0.352876	−	0.935670i	$-0.385204\pi$
$401$	14.1327			0.705752			0.352876	+	0.935670i	$0.385204\pi$
$402$		0			0
$403$			10.3456i			0.515352i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	4.05884i		−	0.201189i
$408$		0			0
$409$	−5.23452			−0.258830			−0.129415	−	0.991590i	$-0.541310\pi$
$409$	−5.23452			−0.258830			−0.129415	+	0.991590i	$0.541310\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	7.71020i		−	0.379394i
$414$		0			0
$415$	4.22260	−	6.90288i	0.207279	−	0.338849i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	7.46737			0.364805			0.182402	−	0.983224i	$-0.441613\pi$
$419$	7.46737			0.364805			0.182402	+	0.983224i	$0.441613\pi$
$420$		0			0
$421$	−17.9502			−0.874840			−0.437420	−	0.899257i	$-0.644108\pi$
$421$	−17.9502			−0.874840			−0.437420	+	0.899257i	$0.644108\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	25.1689	−	12.8743i	1.22087	−	0.624496i
$426$		0			0
$427$		−	12.4128i		−	0.600698i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−7.79859			−0.375645			−0.187823	−	0.982203i	$-0.560143\pi$
$431$	−7.79859			−0.375645			−0.187823	+	0.982203i	$0.560143\pi$
$432$		0			0
$433$			27.7020i			1.33128i	0.746275	+	0.665638i	$0.231840\pi$
$433$			27.7020i			1.33128i	−0.746275	+	0.665638i	$0.768160\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		−	2.30950i		−	0.110478i
$438$		0			0
$439$	−13.2513			−0.632448			−0.316224	−	0.948685i	$-0.602415\pi$
$439$	−13.2513			−0.632448			−0.316224	+	0.948685i	$0.602415\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	21.0107i		−	0.998247i	−0.866531	−	0.499124i	$-0.833655\pi$
$443$		−	21.0107i		−	0.998247i	0.866531	−	0.499124i	$-0.166345\pi$
$444$		0			0
$445$	22.5901	+	13.8187i	1.07087	+	0.655070i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−22.1295			−1.04436			−0.522178	−	0.852836i	$-0.674880\pi$
$449$	−22.1295			−1.04436			−0.522178	+	0.852836i	$0.674880\pi$
$450$		0			0
$451$	−0.307460			−0.0144777
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.70590	+	1.65524i	0.126854	+	0.0775988i
$456$		0			0
$457$		−	6.05857i		−	0.283408i	−0.989909	−	0.141704i	$-0.954742\pi$
$457$		−	6.05857i		−	0.283408i	0.989909	−	0.141704i	$-0.0452581\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−25.6858			−1.19631			−0.598154	−	0.801381i	$-0.704099\pi$
$461$	−25.6858			−1.19631			−0.598154	+	0.801381i	$0.704099\pi$
$462$		0			0
$463$		−	16.2858i		−	0.756864i	−0.925629	−	0.378432i	$-0.876463\pi$
$463$		−	16.2858i		−	0.756864i	0.925629	−	0.378432i	$-0.123537\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			22.1537i			1.02515i	0.858642	+	0.512576i	$0.171309\pi$
$467$			22.1537i			1.02515i	−0.858642	+	0.512576i	$0.828691\pi$
$468$		0			0
$469$	7.71020			0.356024
$470$		0			0
$471$		0			0
$472$		0			0
$473$			3.87942i			0.178376i
$474$		0			0
$475$	2.27698	+	4.45144i	0.104475	+	0.204246i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−41.0475			−1.87551			−0.937755	−	0.347299i	$-0.887099\pi$
$479$	−41.0475			−1.87551			−0.937755	+	0.347299i	$0.887099\pi$
$480$		0			0
$481$	8.95210			0.408181
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6.59826	−	10.7865i	0.299611	−	0.489789i
$486$		0			0
$487$			8.51686i			0.385936i	0.981205	+	0.192968i	$0.0618113\pi$
$487$			8.51686i			0.385936i	−0.981205	+	0.192968i	$0.938189\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	18.8706			0.851616			0.425808	−	0.904814i	$-0.359990\pi$
$491$	18.8706			0.851616			0.425808	+	0.904814i	$0.359990\pi$
$492$		0			0
$493$			35.1287i			1.58212i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			14.1684i			0.635538i
$498$		0			0
$499$	−0.426482			−0.0190920			−0.00954598	−	0.999954i	$-0.503039\pi$
$499$	−0.426482			−0.0190920			−0.00954598	+	0.999954i	$0.503039\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	0.950368i		−	0.0423748i	−0.999776	−	0.0211874i	$-0.993255\pi$
$503$		−	0.950368i		−	0.0423748i	0.999776	−	0.0211874i	$-0.00674466\pi$
$504$		0			0
$505$	36.2164	+	22.1541i	1.61161	+	0.985846i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7.69412			−0.341036			−0.170518	−	0.985355i	$-0.554544\pi$
$509$	−7.69412			−0.341036			−0.170518	+	0.985355i	$0.554544\pi$
$510$		0			0
$511$	−5.33003			−0.235787
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−17.8622	+	29.2002i	−0.787102	+	1.28671i
$516$		0			0
$517$		−	3.94173i		−	0.173357i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−29.2578			−1.28181			−0.640905	−	0.767620i	$-0.721441\pi$
$521$	−29.2578			−1.28181			−0.640905	+	0.767620i	$0.721441\pi$
$522$		0			0
$523$			21.3885i			0.935253i	0.883926	+	0.467627i	$0.154891\pi$
$523$			21.3885i			0.935253i	−0.883926	+	0.467627i	$0.845109\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	56.4720i		−	2.45996i
$528$		0			0
$529$	17.6662			0.768096
$530$		0			0
$531$		0			0
$532$		0			0
$533$		−	0.678129i		−	0.0293730i
$534$		0			0
$535$	−10.7776	+	17.6186i	−0.465954	+	0.761718i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.40665			0.103662
$540$		0			0
$541$	−13.7453			−0.590959			−0.295479	−	0.955349i	$-0.595479\pi$
$541$	−13.7453			−0.590959			−0.295479	+	0.955349i	$0.595479\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	19.5497	+	11.9588i	0.837416	+	0.512260i
$546$		0			0
$547$			11.2613i			0.481499i	0.970587	+	0.240749i	$0.0773932\pi$
$547$			11.2613i			0.481499i	−0.970587	+	0.240749i	$0.922607\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6.21295			−0.264681
$552$		0			0
$553$			17.2136i			0.731995i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	15.1412i		−	0.641552i	−0.947155	−	0.320776i	$-0.896056\pi$
$557$		−	15.1412i		−	0.641552i	0.947155	−	0.320776i	$-0.103944\pi$
$558$		0			0
$559$	−8.55639			−0.361897
$560$		0			0
$561$		0			0
$562$		0			0
$563$			32.2672i			1.35990i	0.733258	+	0.679950i	$0.237999\pi$
$563$			32.2672i			1.35990i	−0.733258	+	0.679950i	$0.762001\pi$
$564$		0			0
$565$	13.0419	−	21.3202i	0.548676	−	0.896946i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−19.5476			−0.819480			−0.409740	−	0.912202i	$-0.634381\pi$
$569$	−19.5476			−0.819480			−0.409740	+	0.912202i	$0.634381\pi$
$570$		0			0
$571$	4.19023			0.175356			0.0876778	−	0.996149i	$-0.472055\pi$
$571$	4.19023			0.175356			0.0876778	+	0.996149i	$0.472055\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	10.2806	−	5.25869i	0.428731	−	0.219303i
$576$		0			0
$577$		−	26.1553i		−	1.08886i	−0.838807	−	0.544429i	$-0.816746\pi$
$577$		−	26.1553i		−	1.08886i	0.838807	−	0.544429i	$-0.183254\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−4.95601			−0.205610
$582$		0			0
$583$		−	4.34359i		−	0.179893i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	25.3145i		−	1.04484i	−0.852688	−	0.522421i	$-0.825029\pi$
$587$		−	25.3145i		−	1.04484i	0.852688	−	0.522421i	$-0.174971\pi$
$588$		0			0
$589$	9.98778			0.411539
$590$		0			0
$591$		0			0
$592$		0			0
$593$			38.2001i			1.56869i	0.620326	+	0.784344i	$0.287000\pi$
$593$			38.2001i			1.56869i	−0.620326	+	0.784344i	$0.713000\pi$
$594$		0			0
$595$	−14.7702	−	9.03518i	−0.605521	−	0.370406i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−9.23255			−0.377232			−0.188616	−	0.982051i	$-0.560400\pi$
$599$	−9.23255			−0.377232			−0.188616	+	0.982051i	$0.560400\pi$
$600$		0			0
$601$	−40.1997			−1.63978			−0.819891	−	0.572520i	$-0.805966\pi$
$601$	−40.1997			−1.63978			−0.819891	+	0.572520i	$0.805966\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−20.5616	−	12.5778i	−0.835948	−	0.511362i
$606$		0			0
$607$			46.1042i			1.87131i	0.352915	+	0.935655i	$0.385191\pi$
$607$			46.1042i			1.87131i	−0.352915	+	0.935655i	$0.614809\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	8.69382			0.351714
$612$		0			0
$613$			26.4347i			1.06769i	0.845583	+	0.533843i	$0.179253\pi$
$613$			26.4347i			1.06769i	−0.845583	+	0.533843i	$0.820747\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$			15.8944i			0.639885i	0.947437	+	0.319943i	$0.103664\pi$
$617$			15.8944i			0.639885i	−0.947437	+	0.319943i	$0.896336\pi$
$618$		0			0
$619$	−15.7102			−0.631446			−0.315723	−	0.948851i	$-0.602247\pi$
$619$	−15.7102			−0.631446			−0.315723	+	0.948851i	$0.602247\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	16.2188i		−	0.649794i
$624$		0			0
$625$	−14.6307	+	20.2717i	−0.585227	+	0.810869i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−48.8654			−1.94839
$630$		0			0
$631$	−15.9822			−0.636241			−0.318120	−	0.948050i	$-0.603052\pi$
$631$	−15.9822			−0.636241			−0.318120	+	0.948050i	$0.603052\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−4.90103	+	8.01196i	−0.194492	+	0.317945i
$636$		0			0
$637$			5.30807i			0.210313i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−12.8015			−0.505627			−0.252814	−	0.967515i	$-0.581356\pi$
$641$	−12.8015			−0.505627			−0.252814	+	0.967515i	$0.581356\pi$
$642$		0			0
$643$		−	11.1413i		−	0.439370i	−0.975571	−	0.219685i	$-0.929497\pi$
$643$		−	11.1413i		−	0.439370i	0.975571	−	0.219685i	$-0.0705030\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			32.1287i			1.26311i	0.775332	+	0.631554i	$0.217583\pi$
$647$			32.1287i			1.26311i	−0.775332	+	0.631554i	$0.782417\pi$
$648$		0			0
$649$	2.64404			0.103788
$650$		0			0
$651$		0			0
$652$		0			0
$653$			26.2685i			1.02797i	0.857800	+	0.513983i	$0.171831\pi$
$653$			26.2685i			1.02797i	−0.857800	+	0.513983i	$0.828169\pi$
$654$		0			0
$655$	−11.2084	−	6.85632i	−0.437946	−	0.267899i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−5.37783			−0.209491			−0.104745	−	0.994499i	$-0.533403\pi$
$659$	−5.37783			−0.209491			−0.104745	+	0.994499i	$0.533403\pi$
$660$		0			0
$661$	42.3710			1.64804			0.824020	−	0.566561i	$-0.191726\pi$
$661$	42.3710			1.64804			0.824020	+	0.566561i	$0.191726\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	1.59798	−	2.61230i	0.0619672	−	0.101301i
$666$		0			0
$667$			14.3488i			0.555588i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	4.25669			0.164328
$672$		0			0
$673$			46.4252i			1.78956i	0.446507	+	0.894780i	$0.352668\pi$
$673$			46.4252i			1.78956i	−0.446507	+	0.894780i	$0.647332\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			37.9124i			1.45709i	0.684996	+	0.728546i	$0.259804\pi$
$677$			37.9124i			1.45709i	−0.684996	+	0.728546i	$0.740196\pi$
$678$		0			0
$679$	−7.74429			−0.297199
$680$		0			0
$681$		0			0
$682$		0			0
$683$			17.0932i			0.654055i	0.945015	+	0.327027i	$0.106047\pi$
$683$			17.0932i			0.654055i	−0.945015	+	0.327027i	$0.893953\pi$
$684$		0			0
$685$	−11.9432	+	19.5241i	−0.456326	+	0.745978i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	9.58015			0.364975
$690$		0			0
$691$	9.07279			0.345145			0.172573	−	0.984997i	$-0.444792\pi$
$691$	9.07279			0.345145			0.172573	+	0.984997i	$0.444792\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	18.2449	+	11.1607i	0.692068	+	0.423349i
$696$		0			0
$697$			3.70159i			0.140208i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−29.9511			−1.13124			−0.565619	−	0.824667i	$-0.691363\pi$
$701$	−29.9511			−1.13124			−0.565619	+	0.824667i	$0.691363\pi$
$702$		0			0
$703$		−	8.64245i		−	0.325956i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	26.0020i		−	0.977907i
$708$		0			0
$709$	−26.6347			−1.00029			−0.500143	−	0.865943i	$-0.666719\pi$
$709$	−26.6347			−1.00029			−0.500143	+	0.865943i	$0.666719\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	23.0668i		−	0.863857i
$714$		0			0
$715$	−0.567626	+	0.927925i	−0.0212280	+	0.0347024i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−10.5194			−0.392308			−0.196154	−	0.980573i	$-0.562845\pi$
$719$	−10.5194			−0.392308			−0.196154	+	0.980573i	$0.562845\pi$
$720$		0			0
$721$	20.9646			0.780763
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−14.1468	−	27.6566i	−0.525399	−	1.02714i
$726$		0			0
$727$			24.0343i			0.891382i	0.895187	+	0.445691i	$0.147042\pi$
$727$			24.0343i			0.891382i	−0.895187	+	0.445691i	$0.852958\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	46.7054			1.72746
$732$		0			0
$733$		−	17.5589i		−	0.648552i	−0.945963	−	0.324276i	$-0.894879\pi$
$733$		−	17.5589i		−	0.648552i	0.945963	−	0.324276i	$-0.105121\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			2.64404i			0.0973944i
$738$		0			0
$739$	−30.7753			−1.13209			−0.566044	−	0.824375i	$-0.691527\pi$
$739$	−30.7753			−1.13209			−0.566044	+	0.824375i	$0.691527\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			30.6409i			1.12410i	0.827102	+	0.562052i	$0.189988\pi$
$743$			30.6409i			1.12410i	−0.827102	+	0.562052i	$0.810012\pi$
$744$		0			0
$745$	−6.23294	−	3.81279i	−0.228357	−	0.139690i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	12.6495			0.462202
$750$		0			0
$751$	−5.83299			−0.212849			−0.106424	−	0.994321i	$-0.533940\pi$
$751$	−5.83299			−0.212849			−0.106424	+	0.994321i	$0.533940\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	6.81066	+	4.16618i	0.247865	+	0.151623i
$756$		0			0
$757$		−	46.8068i		−	1.70122i	−0.525796	−	0.850611i	$-0.676232\pi$
$757$		−	46.8068i		−	1.70122i	0.525796	−	0.850611i	$-0.323768\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−2.40317			−0.0871150			−0.0435575	−	0.999051i	$-0.513869\pi$
$761$	−2.40317			−0.0871150			−0.0435575	+	0.999051i	$0.513869\pi$
$762$		0			0
$763$		−	14.0359i		−	0.508135i
$764$		0			0
$765$		0			0
$766$		0			0
$767$			5.83165i			0.210569i
$768$		0			0
$769$	−52.3618			−1.88821			−0.944107	−	0.329639i	$-0.893073\pi$
$769$	−52.3618			−1.88821			−0.944107	+	0.329639i	$0.893073\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			0.420892i			0.0151384i	0.999971	+	0.00756922i	$0.00240938\pi$
$773$			0.420892i			0.0151384i	−0.999971	+	0.00756922i	$0.997591\pi$
$774$		0			0
$775$	22.7420	+	44.4600i	0.816918	+	1.59705i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.654673			−0.0234561
$780$		0			0
$781$	−4.85871			−0.173858
$782$		0			0
$783$		0			0
$784$		0			0
$785$	18.9891	−	31.0423i	0.677749	−	1.10795i
$786$		0			0
$787$			49.5598i			1.76662i	0.468794	+	0.883308i	$0.344689\pi$
$787$			49.5598i			1.76662i	−0.468794	+	0.883308i	$0.655311\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−15.3071			−0.544257
$792$		0			0
$793$			9.38848i			0.333395i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	39.1979i		−	1.38846i	−0.719753	−	0.694230i	$-0.755745\pi$
$797$		−	39.1979i		−	1.38846i	0.719753	−	0.694230i	$-0.244255\pi$
$798$		0			0
$799$	−47.4556			−1.67886
$800$		0			0
$801$		0			0
$802$		0			0
$803$		−	1.82781i		−	0.0645021i
$804$		0			0
$805$	−6.03311	−	3.69054i	−0.212639	−	0.130075i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−54.1920			−1.90529			−0.952644	−	0.304087i	$-0.901649\pi$
$809$	−54.1920			−1.90529			−0.952644	+	0.304087i	$0.901649\pi$
$810$		0			0
$811$	−1.44752			−0.0508294			−0.0254147	−	0.999677i	$-0.508091\pi$
$811$	−1.44752			−0.0508294			−0.0254147	+	0.999677i	$0.508091\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	15.1822	−	24.8190i	0.531808	−	0.869372i
$816$		0			0
$817$			8.26043i			0.288996i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	51.6786			1.80360			0.901798	−	0.432158i	$-0.142248\pi$
$821$	51.6786			1.80360			0.901798	+	0.432158i	$0.142248\pi$
$822$		0			0
$823$			19.4492i			0.677958i	0.940794	+	0.338979i	$0.110082\pi$
$823$			19.4492i			0.677958i	−0.940794	+	0.338979i	$0.889918\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			19.5628i			0.680265i	0.940378	+	0.340132i	$0.110472\pi$
$827$			19.5628i			0.680265i	−0.940378	+	0.340132i	$0.889528\pi$
$828$		0			0
$829$	7.55828			0.262510			0.131255	−	0.991349i	$-0.458099\pi$
$829$	7.55828			0.262510			0.131255	+	0.991349i	$0.458099\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	28.9743i		−	1.00390i
$834$		0			0
$835$	20.3233	−	33.2235i	0.703317	−	1.14975i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−43.3540			−1.49675			−0.748373	−	0.663278i	$-0.769165\pi$
$839$	−43.3540			−1.49675			−0.748373	+	0.663278i	$0.769165\pi$
$840$		0			0
$841$	9.60073			0.331060
$842$		0			0
$843$		0			0
$844$		0			0
$845$	22.7507	+	13.9169i	0.782647	+	0.478757i
$846$		0			0
$847$			14.7625i			0.507244i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−19.9597			−0.684211
$852$		0			0
$853$		−	20.8731i		−	0.714680i	−0.933974	−	0.357340i	$-0.883684\pi$
$853$		−	20.8731i		−	0.714680i	0.933974	−	0.357340i	$-0.116316\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	38.1539i		−	1.30331i	−0.758515	−	0.651656i	$-0.774075\pi$
$857$		−	38.1539i		−	1.30331i	0.758515	−	0.651656i	$-0.225925\pi$
$858$		0			0
$859$	3.36607			0.114849			0.0574244	−	0.998350i	$-0.481711\pi$
$859$	3.36607			0.114849			0.0574244	+	0.998350i	$0.481711\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			37.7606i			1.28539i	0.766124	+	0.642693i	$0.222183\pi$
$863$			37.7606i			1.28539i	−0.766124	+	0.642693i	$0.777817\pi$
$864$		0			0
$865$	−4.59667	+	7.51440i	−0.156292	+	0.255497i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−5.90300			−0.200245
$870$		0			0
$871$	−5.83165			−0.197598
$872$		0			0
$873$		0			0
$874$		0			0
$875$	15.2671	+	1.16517i	0.516122	+	0.0393898i
$876$		0			0
$877$			10.1910i			0.344125i	0.985086	+	0.172063i	$0.0550431\pi$
$877$			10.1910i			0.344125i	−0.985086	+	0.172063i	$0.944957\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−32.8786			−1.10771			−0.553853	−	0.832614i	$-0.686843\pi$
$881$	−32.8786			−1.10771			−0.553853	+	0.832614i	$0.686843\pi$
$882$		0			0
$883$		−	53.1963i		−	1.79020i	−0.445867	−	0.895099i	$-0.647104\pi$
$883$		−	53.1963i		−	1.79020i	0.445867	−	0.895099i	$-0.352896\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			22.2768i			0.747983i	0.927432	+	0.373992i	$0.122011\pi$
$887$			22.2768i			0.747983i	−0.927432	+	0.373992i	$0.877989\pi$
$888$		0			0
$889$	5.75228			0.192925
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	8.39311i		−	0.280865i
$894$		0			0
$895$	−35.7422	−	21.8640i	−1.19473	−	0.730834i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−62.0536			−2.06960
$900$		0			0
$901$	−52.2936			−1.74215
$902$		0			0
$903$		0			0
$904$		0			0
$905$	36.5062	+	22.3314i	1.21351	+	0.742321i
$906$		0			0
$907$			25.4129i			0.843822i	0.906637	+	0.421911i	$0.138641\pi$
$907$			25.4129i			0.843822i	−0.906637	+	0.421911i	$0.861359\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	15.4450			0.511714			0.255857	−	0.966715i	$-0.417642\pi$
$911$	15.4450			0.511714			0.255857	+	0.966715i	$0.417642\pi$
$912$		0			0
$913$		−	1.69955i		−	0.0562469i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			8.04718i			0.265741i
$918$		0			0
$919$	−15.8354			−0.522362			−0.261181	−	0.965290i	$-0.584112\pi$
$919$	−15.8354			−0.522362			−0.261181	+	0.965290i	$0.584112\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		−	10.7163i		−	0.352731i
$924$		0			0
$925$	38.4714	−	19.6787i	1.26493	−	0.647033i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−33.1079			−1.08624			−0.543118	−	0.839656i	$-0.682756\pi$
$929$	−33.1079			−1.08624			−0.543118	+	0.839656i	$0.682756\pi$
$930$		0			0
$931$	5.12447			0.167948
$932$		0			0
$933$		0			0
$934$		0			0
$935$	3.09841	−	5.06511i	0.101329	−	0.165647i
$936$		0			0
$937$			8.90605i			0.290948i	0.989362	+	0.145474i	$0.0464707\pi$
$937$			8.90605i			0.290948i	−0.989362	+	0.145474i	$0.953529\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	30.0334			0.979061			0.489531	−	0.871986i	$-0.337168\pi$
$941$	30.0334			0.979061			0.489531	+	0.871986i	$0.337168\pi$
$942$		0			0
$943$			1.51197i			0.0492364i
$944$		0			0
$945$		0			0
$946$		0			0
$947$			47.2376i			1.53502i	0.641040	+	0.767508i	$0.278503\pi$
$947$			47.2376i			1.53502i	−0.641040	+	0.767508i	$0.721497\pi$
$948$		0			0
$949$	4.03139			0.130865
$950$		0			0
$951$		0			0
$952$		0			0
$953$			24.0163i			0.777964i	0.921245	+	0.388982i	$0.127173\pi$
$953$			24.0163i			0.777964i	−0.921245	+	0.388982i	$0.872827\pi$
$954$		0			0
$955$	−2.34993	−	1.43749i	−0.0760418	−	0.0465159i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	14.0176			0.452651
$960$		0			0
$961$	68.7558			2.21793
$962$		0			0
$963$		0			0
$964$		0			0
$965$	5.06879	−	8.28619i	0.163170	−	0.266742i
$966$		0			0
$967$			22.7242i			0.730760i	0.930859	+	0.365380i	$0.119061\pi$
$967$			22.7242i			0.730760i	−0.930859	+	0.365380i	$0.880939\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	12.8708			0.413043			0.206522	−	0.978442i	$-0.433786\pi$
$971$	12.8708			0.413043			0.206522	+	0.978442i	$0.433786\pi$
$972$		0			0
$973$		−	13.0992i		−	0.419940i
$974$		0			0
$975$		0			0
$976$		0			0
$977$			48.7912i			1.56097i	0.625175	+	0.780485i	$0.285028\pi$
$977$			48.7912i			1.56097i	−0.625175	+	0.780485i	$0.714972\pi$
$978$		0			0
$979$	5.56188			0.177759
$980$		0			0
$981$		0			0
$982$		0			0
$983$			4.87123i			0.155368i	0.996978	+	0.0776840i	$0.0247525\pi$
$983$			4.87123i			0.155368i	−0.996978	+	0.0776840i	$0.975247\pi$
$984$		0			0
$985$	−26.6393	+	43.5485i	−0.848799	+	1.38757i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	19.0775			0.606628
$990$		0			0
$991$	−42.8104			−1.35992			−0.679959	−	0.733250i	$-0.738003\pi$
$991$	−42.8104			−1.35992			−0.679959	+	0.733250i	$0.738003\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	9.63570	+	5.89430i	0.305472	+	0.186862i
$996$		0			0
$997$		−	38.0563i		−	1.20526i	−0.798022	−	0.602628i	$-0.794120\pi$
$997$		−	38.0563i		−	1.20526i	0.798022	−	0.602628i	$-0.205880\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3420.2.f.e.1369.10		10
3.2	odd	2		1140.2.f.b.229.6	yes	10
5.4	even	2	inner	3420.2.f.e.1369.9		10
15.2	even	4		5700.2.a.bd.1.4		5
15.8	even	4		5700.2.a.bc.1.2		5
15.14	odd	2		1140.2.f.b.229.1	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1140.2.f.b.229.1	✓	10	15.14	odd	2
1140.2.f.b.229.6	yes	10	3.2	odd	2
3420.2.f.e.1369.9		10	5.4	even	2	inner
3420.2.f.e.1369.10		10	1.1	even	1	trivial
5700.2.a.bc.1.2		5	15.8	even	4
5700.2.a.bd.1.4		5	15.2	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3420.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.90748 + 1.16684i) q^{5} -1.36950i q^{7} +0.469639 q^{11} +1.03583i q^{13} -5.65411i q^{17} +1.00000 q^{19} -2.30950i q^{23} +(2.27698 + 4.45144i) q^{25} -6.21295 q^{29} +9.98778 q^{31} +(1.59798 - 2.61230i) q^{35} -8.64245i q^{37} -0.654673 q^{41} +8.26043i q^{43} -8.39311i q^{47} +5.12447 q^{49} -9.24878i q^{53} +(0.895829 + 0.547992i) q^{55} +5.62993 q^{59} +9.06374 q^{61} +(-1.20864 + 1.97583i) q^{65} +5.62993i q^{67} -10.3456 q^{71} -3.89195i q^{73} -0.643172i q^{77} -12.5692 q^{79} -3.61884i q^{83} +(6.59742 - 10.7851i) q^{85} +11.8429 q^{89} +1.41857 q^{91} +(1.90748 + 1.16684i) q^{95} -5.65483i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 2 q^{5} - 12 q^{11} + 10 q^{19} - 8 q^{25} + 16 q^{31} + 22 q^{35} - 24 q^{41} - 6 q^{49} - 10 q^{55} - 12 q^{59} + 32 q^{65} - 4 q^{71} - 24 q^{79} - 10 q^{85} - 12 q^{89} + 32 q^{91} + 2 q^{95}+O(q^{100}) \) 10 * q + 2 * q^5 - 12 * q^11 + 10 * q^19 - 8 * q^25 + 16 * q^31 + 22 * q^35 - 24 * q^41 - 6 * q^49 - 10 * q^55 - 12 * q^59 + 32 * q^65 - 4 * q^71 - 24 * q^79 - 10 * q^85 - 12 * q^89 + 32 * q^91 + 2 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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