LMFDB - Embedded newform 520.2.k.a.441.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [520,2,Mod(441,520)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(520, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("520.441"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 520 = 2^{3} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	520.k (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	no
Analytic conductor:	$4.15222090511$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.350464.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 $ x^6 - 2x^5 + 2x^4 + 2x^3 + 4x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			441.3
Root			$-0.854638 - 0.854638i$ of defining polynomial
Character	$\chi$	$=$	520.441
Dual form			520.2.k.a.441.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.539189 q^{3} -1.00000i q^{5} -1.70928i q^{7} -2.70928 q^{9} -2.09171i q^{11} +(-2.87936 + 2.17009i) q^{13} +0.539189i q^{15} -3.41855 q^{17} -2.24846i q^{19} +0.921622i q^{21} -0.879362 q^{23} -1.00000 q^{25} +3.07838 q^{27} -5.89269 q^{29} -7.85043i q^{31} +1.12783i q^{33} -1.70928 q^{35} -1.36910i q^{37} +(1.55252 - 1.17009i) q^{39} -2.18342i q^{41} -4.38243 q^{43} +2.70928i q^{45} -5.12783i q^{47} +4.07838 q^{49} +1.84324 q^{51} -2.18342 q^{53} -2.09171 q^{55} +1.21235i q^{57} +1.17009i q^{59} -1.02893 q^{61} +4.63090i q^{63} +(2.17009 + 2.87936i) q^{65} +4.63090i q^{67} +0.474142 q^{69} -13.0856i q^{71} +8.20620i q^{73} +0.539189 q^{75} -3.57531 q^{77} +16.0989 q^{79} +6.46800 q^{81} +8.72979i q^{83} +3.41855i q^{85} +3.17727 q^{87} -7.41855i q^{89} +(3.70928 + 4.92162i) q^{91} +4.23287i q^{93} -2.24846 q^{95} +7.57531i q^{97} +5.66701i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 2 q^{9} + 8 q^{13} + 8 q^{17} + 20 q^{23} - 6 q^{25} + 12 q^{27} - 12 q^{29} + 4 q^{35} + 8 q^{39} - 36 q^{43} + 18 q^{49} + 24 q^{51} - 4 q^{53} - 8 q^{55} - 36 q^{61} + 2 q^{65} + 8 q^{69} + 20 q^{77}+ \cdots + 4 q^{95}+O(q^{100}) $ 6 * q - 2 * q^9 + 8 * q^13 + 8 * q^17 + 20 * q^23 - 6 * q^25 + 12 * q^27 - 12 * q^29 + 4 * q^35 + 8 * q^39 - 36 * q^43 + 18 * q^49 + 24 * q^51 - 4 * q^53 - 8 * q^55 - 36 * q^61 + 2 * q^65 + 8 * q^69 + 20 * q^77 + 24 * q^79 - 26 * q^81 - 60 * q^87 + 8 * q^91 + 4 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/520\mathbb{Z}\right)^\times$.

$n$	$41$	$261$	$391$	$417$
$\chi(n)$	$-1$	$1$	$1$	$1$

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
$3$	−0.539189			−0.311301			−0.155650	−	0.987812i	$-0.549747\pi$
$3$	−0.539189			−0.311301			−0.155650	+	0.987812i	$0.549747\pi$
$4$		0			0
$5$		−	1.00000i		−	0.447214i
$5$		−	1.00000i		−	0.447214i
$6$		0			0
$7$		−	1.70928i		−	0.646045i	−0.946391	−	0.323023i	$-0.895301\pi$
$7$		−	1.70928i		−	0.646045i	0.946391	−	0.323023i	$-0.104699\pi$
$8$		0			0
$9$	−2.70928			−0.903092
$10$		0			0
$11$		−	2.09171i		−	0.630674i	−0.948980	−	0.315337i	$-0.897882\pi$
$11$		−	2.09171i		−	0.630674i	0.948980	−	0.315337i	$-0.102118\pi$
$12$		0			0
$13$	−2.87936	+	2.17009i	−0.798591	+	0.601874i
$13$	−2.87936	+	2.17009i	−0.798591	+	0.601874i
$14$		0			0
$15$			0.539189i			0.139218i
$16$		0			0
$17$	−3.41855			−0.829120			−0.414560	−	0.910022i	$-0.636065\pi$
$17$	−3.41855			−0.829120			−0.414560	+	0.910022i	$0.636065\pi$
$18$		0			0
$19$		−	2.24846i		−	0.515833i	−0.966167	−	0.257917i	$-0.916964\pi$
$19$		−	2.24846i		−	0.515833i	0.966167	−	0.257917i	$-0.0830360\pi$
$20$		0			0
$21$			0.921622i			0.201114i
$22$		0			0
$23$	−0.879362			−0.183360			−0.0916798	−	0.995789i	$-0.529224\pi$
$23$	−0.879362			−0.183360			−0.0916798	+	0.995789i	$0.529224\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	3.07838			0.592434
$28$		0			0
$29$	−5.89269			−1.09425			−0.547123	−	0.837052i	$-0.684277\pi$
$29$	−5.89269			−1.09425			−0.547123	+	0.837052i	$0.684277\pi$
$30$		0			0
$31$		−	7.85043i		−	1.40998i	−0.709218	−	0.704990i	$-0.750952\pi$
$31$		−	7.85043i		−	1.40998i	0.709218	−	0.704990i	$-0.249048\pi$
$32$		0			0
$33$			1.12783i			0.196329i
$34$		0			0
$35$	−1.70928			−0.288920
$36$		0			0
$37$		−	1.36910i		−	0.225079i	−0.993647	−	0.112540i	$-0.964102\pi$
$37$		−	1.36910i		−	0.225079i	0.993647	−	0.112540i	$-0.0358985\pi$
$38$		0			0
$39$	1.55252	−	1.17009i	0.248602	−	0.187364i
$40$		0			0
$41$		−	2.18342i		−	0.340993i	−0.985358	−	0.170496i	$-0.945463\pi$
$41$		−	2.18342i		−	0.340993i	0.985358	−	0.170496i	$-0.0545371\pi$
$42$		0			0
$43$	−4.38243			−0.668315			−0.334157	−	0.942517i	$-0.608452\pi$
$43$	−4.38243			−0.668315			−0.334157	+	0.942517i	$0.608452\pi$
$44$		0			0
$45$			2.70928i			0.403875i
$46$		0			0
$47$		−	5.12783i		−	0.747970i	−0.927435	−	0.373985i	$-0.877991\pi$
$47$		−	5.12783i		−	0.747970i	0.927435	−	0.373985i	$-0.122009\pi$
$48$		0			0
$49$	4.07838			0.582625
$50$		0			0
$51$	1.84324			0.258106
$52$		0			0
$53$	−2.18342			−0.299915			−0.149958	−	0.988692i	$-0.547914\pi$
$53$	−2.18342			−0.299915			−0.149958	+	0.988692i	$0.547914\pi$
$54$		0			0
$55$	−2.09171			−0.282046
$56$		0			0
$57$			1.21235i			0.160579i
$58$		0			0
$59$			1.17009i			0.152332i	0.997095	+	0.0761661i	$0.0242679\pi$
$59$			1.17009i			0.152332i	−0.997095	+	0.0761661i	$0.975732\pi$
$60$		0			0
$61$	−1.02893			−0.131741			−0.0658705	−	0.997828i	$-0.520982\pi$
$61$	−1.02893			−0.131741			−0.0658705	+	0.997828i	$0.520982\pi$
$62$		0			0
$63$			4.63090i			0.583438i
$64$		0			0
$65$	2.17009	+	2.87936i	0.269166	+	0.357141i
$66$		0			0
$67$			4.63090i			0.565754i	0.959156	+	0.282877i	$0.0912889\pi$
$67$			4.63090i			0.565754i	−0.959156	+	0.282877i	$0.908711\pi$
$68$		0			0
$69$	0.474142			0.0570800
$70$		0			0
$71$		−	13.0856i		−	1.55297i	−0.630135	−	0.776485i	$-0.717001\pi$
$71$		−	13.0856i		−	1.55297i	0.630135	−	0.776485i	$-0.282999\pi$
$72$		0			0
$73$			8.20620i			0.960464i	0.877142	+	0.480232i	$0.159447\pi$
$73$			8.20620i			0.960464i	−0.877142	+	0.480232i	$0.840553\pi$
$74$		0			0
$75$	0.539189			0.0622602
$76$		0			0
$77$	−3.57531			−0.407444
$78$		0			0
$79$	16.0989			1.81127			0.905634	−	0.424061i	$-0.139396\pi$
$79$	16.0989			1.81127			0.905634	+	0.424061i	$0.139396\pi$
$80$		0			0
$81$	6.46800			0.718667
$82$		0			0
$83$			8.72979i			0.958219i	0.877755	+	0.479110i	$0.159040\pi$
$83$			8.72979i			0.958219i	−0.877755	+	0.479110i	$0.840960\pi$
$84$		0			0
$85$			3.41855i			0.370794i
$86$		0			0
$87$	3.17727			0.340640
$88$		0			0
$89$		−	7.41855i		−	0.786365i	−0.919460	−	0.393182i	$-0.871374\pi$
$89$		−	7.41855i		−	0.786365i	0.919460	−	0.393182i	$-0.128626\pi$
$90$		0			0
$91$	3.70928	+	4.92162i	0.388838	+	0.515926i
$92$		0			0
$93$			4.23287i			0.438928i
$94$		0			0
$95$	−2.24846			−0.230688
$96$		0			0
$97$			7.57531i			0.769156i	0.923093	+	0.384578i	$0.125653\pi$
$97$			7.57531i			0.769156i	−0.923093	+	0.384578i	$0.874347\pi$
$98$		0			0
$99$			5.66701i			0.569556i
$100$		0			0
$101$	15.7587			1.56805			0.784026	−	0.620728i	$-0.213163\pi$
$101$	15.7587			1.56805			0.784026	+	0.620728i	$0.213163\pi$
$102$		0			0
$103$	−0.355771			−0.0350552			−0.0175276	−	0.999846i	$-0.505579\pi$
$103$	−0.355771			−0.0350552			−0.0175276	+	0.999846i	$0.505579\pi$
$104$		0			0
$105$	0.921622			0.0899411
$106$		0			0
$107$	−2.04226			−0.197433			−0.0987164	−	0.995116i	$-0.531474\pi$
$107$	−2.04226			−0.197433			−0.0987164	+	0.995116i	$0.531474\pi$
$108$		0			0
$109$			5.41855i			0.519003i	0.965743	+	0.259502i	$0.0835583\pi$
$109$			5.41855i			0.519003i	−0.965743	+	0.259502i	$0.916442\pi$
$110$		0			0
$111$			0.738205i			0.0700673i
$112$		0			0
$113$	−18.8371			−1.77205			−0.886023	−	0.463642i	$-0.846542\pi$
$113$	−18.8371			−1.77205			−0.886023	+	0.463642i	$0.846542\pi$
$114$		0			0
$115$			0.879362i			0.0820009i
$116$		0			0
$117$	7.80098	−	5.87936i	0.721201	−	0.543547i
$118$		0			0
$119$			5.84324i			0.535649i
$120$		0			0
$121$	6.62475			0.602250
$122$		0			0
$123$			1.17727i			0.106151i
$124$		0			0
$125$			1.00000i			0.0894427i
$126$		0			0
$127$	21.7165			1.92702			0.963512	−	0.267664i	$-0.0862518\pi$
$127$	21.7165			1.92702			0.963512	+	0.267664i	$0.0862518\pi$
$128$		0			0
$129$	2.36296			0.208047
$130$		0			0
$131$	2.73820			0.239238			0.119619	−	0.992820i	$-0.461833\pi$
$131$	2.73820			0.239238			0.119619	+	0.992820i	$0.461833\pi$
$132$		0			0
$133$	−3.84324			−0.333252
$134$		0			0
$135$		−	3.07838i		−	0.264945i
$136$		0			0
$137$		−	18.5958i		−	1.58875i	−0.607429	−	0.794374i	$-0.707799\pi$
$137$		−	18.5958i		−	1.58875i	0.607429	−	0.794374i	$-0.292201\pi$
$138$		0			0
$139$	−13.5753			−1.15144			−0.575721	−	0.817646i	$-0.695279\pi$
$139$	−13.5753			−1.15144			−0.575721	+	0.817646i	$0.695279\pi$
$140$		0			0
$141$			2.76487i			0.232844i
$142$		0			0
$143$	4.53919	+	6.02279i	0.379586	+	0.503651i
$144$		0			0
$145$			5.89269i			0.489362i
$146$		0			0
$147$	−2.19902			−0.181372
$148$		0			0
$149$			8.73820i			0.715862i	0.933748	+	0.357931i	$0.116518\pi$
$149$			8.73820i			0.715862i	−0.933748	+	0.357931i	$0.883482\pi$
$150$		0			0
$151$		−	17.0856i		−	1.39040i	−0.718815	−	0.695202i	$-0.755315\pi$
$151$		−	17.0856i		−	1.39040i	0.718815	−	0.695202i	$-0.244685\pi$
$152$		0			0
$153$	9.26180			0.748772
$154$		0			0
$155$	−7.85043			−0.630562
$156$		0			0
$157$	5.07838			0.405299			0.202649	−	0.979251i	$-0.435045\pi$
$157$	5.07838			0.405299			0.202649	+	0.979251i	$0.435045\pi$
$158$		0			0
$159$	1.17727			0.0933639
$160$		0			0
$161$			1.50307i			0.118459i
$162$		0			0
$163$			12.2329i			0.958152i	0.877773	+	0.479076i	$0.159028\pi$
$163$			12.2329i			0.958152i	−0.877773	+	0.479076i	$0.840972\pi$
$164$		0			0
$165$	1.12783			0.0878011
$166$		0			0
$167$			5.52586i			0.427604i	0.976877	+	0.213802i	$0.0685847\pi$
$167$			5.52586i			0.427604i	−0.976877	+	0.213802i	$0.931415\pi$
$168$		0			0
$169$	3.58145	−	12.4969i	0.275496	−	0.961302i
$170$		0			0
$171$			6.09171i			0.465845i
$172$		0			0
$173$	−5.60197			−0.425910			−0.212955	−	0.977062i	$-0.568309\pi$
$173$	−5.60197			−0.425910			−0.212955	+	0.977062i	$0.568309\pi$
$174$		0			0
$175$			1.70928i			0.129209i
$176$		0			0
$177$		−	0.630898i		−	0.0474212i
$178$		0			0
$179$	−8.28231			−0.619049			−0.309525	−	0.950891i	$-0.600170\pi$
$179$	−8.28231			−0.619049			−0.309525	+	0.950891i	$0.600170\pi$
$180$		0			0
$181$	6.57304			0.488570			0.244285	−	0.969703i	$-0.421447\pi$
$181$	6.57304			0.488570			0.244285	+	0.969703i	$0.421447\pi$
$182$		0			0
$183$	0.554787			0.0410111
$184$		0			0
$185$	−1.36910			−0.100658
$186$		0			0
$187$			7.15061i			0.522905i
$188$		0			0
$189$		−	5.26180i		−	0.382739i
$190$		0			0
$191$	−21.6742			−1.56829			−0.784145	−	0.620577i	$-0.786898\pi$
$191$	−21.6742			−1.56829			−0.784145	+	0.620577i	$0.786898\pi$
$192$		0			0
$193$			7.75872i			0.558485i	0.960221	+	0.279243i	$0.0900834\pi$
$193$			7.75872i			0.558485i	−0.960221	+	0.279243i	$0.909917\pi$
$194$		0			0
$195$	−1.17009	−	1.55252i	−0.0837916	−	0.111178i
$196$		0			0
$197$		−	23.5441i		−	1.67745i	−0.544556	−	0.838724i	$-0.683302\pi$
$197$		−	23.5441i		−	1.67745i	0.544556	−	0.838724i	$-0.316698\pi$
$198$		0			0
$199$	12.5958			0.892894			0.446447	−	0.894810i	$-0.352689\pi$
$199$	12.5958			0.892894			0.446447	+	0.894810i	$0.352689\pi$
$200$		0			0
$201$		−	2.49693i		−	0.176120i
$202$		0			0
$203$			10.0722i			0.706932i
$204$		0			0
$205$	−2.18342			−0.152496
$206$		0			0
$207$	2.38243			0.165591
$208$		0			0
$209$	−4.70313			−0.325322
$210$		0			0
$211$	−16.2823			−1.12092			−0.560460	−	0.828181i	$-0.689376\pi$
$211$	−16.2823			−1.12092			−0.560460	+	0.828181i	$0.689376\pi$
$212$		0			0
$213$			7.05559i			0.483441i
$214$		0			0
$215$			4.38243i			0.298879i
$216$		0			0
$217$	−13.4186			−0.910911
$218$		0			0
$219$		−	4.42469i		−	0.298993i
$220$		0			0
$221$	9.84324	−	7.41855i	0.662128	−	0.499026i
$222$		0			0
$223$		−	19.2846i		−	1.29139i	−0.763595	−	0.645696i	$-0.776568\pi$
$223$		−	19.2846i		−	1.29139i	0.763595	−	0.645696i	$-0.223432\pi$
$224$		0			0
$225$	2.70928			0.180618
$226$		0			0
$227$		−	4.26406i		−	0.283016i	−0.989937	−	0.141508i	$-0.954805\pi$
$227$		−	4.26406i		−	0.283016i	0.989937	−	0.141508i	$-0.0451951\pi$
$228$		0			0
$229$		−	7.91548i		−	0.523070i	−0.965194	−	0.261535i	$-0.915771\pi$
$229$		−	7.91548i		−	0.523070i	0.965194	−	0.261535i	$-0.0842286\pi$
$230$		0			0
$231$	1.92777			0.126838
$232$		0			0
$233$	8.42469			0.551920			0.275960	−	0.961169i	$-0.411004\pi$
$233$	8.42469			0.551920			0.275960	+	0.961169i	$0.411004\pi$
$234$		0			0
$235$	−5.12783			−0.334502
$236$		0			0
$237$	−8.68035			−0.563849
$238$		0			0
$239$		−	18.0917i		−	1.17026i	−0.810941	−	0.585128i	$-0.801044\pi$
$239$		−	18.0917i		−	1.17026i	0.810941	−	0.585128i	$-0.198956\pi$
$240$		0			0
$241$			3.44521i			0.221926i	0.993825	+	0.110963i	$0.0353935\pi$
$241$			3.44521i			0.221926i	−0.993825	+	0.110963i	$0.964607\pi$
$242$		0			0
$243$	−12.7226			−0.816156
$244$		0			0
$245$		−	4.07838i		−	0.260558i
$246$		0			0
$247$	4.87936	+	6.47414i	0.310466	+	0.411940i
$248$		0			0
$249$		−	4.70701i		−	0.298295i
$250$		0			0
$251$	−15.3340			−0.967875			−0.483938	−	0.875102i	$-0.660794\pi$
$251$	−15.3340			−0.967875			−0.483938	+	0.875102i	$0.660794\pi$
$252$		0			0
$253$			1.83937i			0.115640i
$254$		0			0
$255$		−	1.84324i		−	0.115428i
$256$		0			0
$257$	3.16290			0.197296			0.0986481	−	0.995122i	$-0.468548\pi$
$257$	3.16290			0.197296			0.0986481	+	0.995122i	$0.468548\pi$
$258$		0			0
$259$	−2.34017			−0.145411
$260$		0			0
$261$	15.9649			0.988204
$262$		0			0
$263$	−26.9783			−1.66355			−0.831775	−	0.555112i	$-0.812675\pi$
$263$	−26.9783			−1.66355			−0.831775	+	0.555112i	$0.812675\pi$
$264$		0			0
$265$			2.18342i			0.134126i
$266$		0			0
$267$			4.00000i			0.244796i
$268$		0			0
$269$	−11.0784			−0.675461			−0.337730	−	0.941243i	$-0.609659\pi$
$269$	−11.0784			−0.675461			−0.337730	+	0.941243i	$0.609659\pi$
$270$		0			0
$271$			1.96615i			0.119435i	0.998215	+	0.0597176i	$0.0190200\pi$
$271$			1.96615i			0.119435i	−0.998215	+	0.0597176i	$0.980980\pi$
$272$		0			0
$273$	−2.00000	−	2.65368i	−0.121046	−	0.160608i
$274$		0			0
$275$			2.09171i			0.126135i
$276$		0			0
$277$	−15.1773			−0.911914			−0.455957	−	0.890002i	$-0.650703\pi$
$277$	−15.1773			−0.911914			−0.455957	+	0.890002i	$0.650703\pi$
$278$		0			0
$279$			21.2690i			1.27334i
$280$		0			0
$281$		−	5.91548i		−	0.352888i	−0.984311	−	0.176444i	$-0.943541\pi$
$281$		−	5.91548i		−	0.352888i	0.984311	−	0.176444i	$-0.0564594\pi$
$282$		0			0
$283$	19.9844			1.18795			0.593975	−	0.804484i	$-0.297558\pi$
$283$	19.9844			1.18795			0.593975	+	0.804484i	$0.297558\pi$
$284$		0			0
$285$	1.21235			0.0718132
$286$		0			0
$287$	−3.73206			−0.220297
$288$		0			0
$289$	−5.31351			−0.312559
$290$		0			0
$291$		−	4.08452i		−	0.239439i
$292$		0			0
$293$		−	30.5152i		−	1.78272i	−0.453300	−	0.891358i	$-0.649753\pi$
$293$		−	30.5152i		−	1.78272i	0.453300	−	0.891358i	$-0.350247\pi$
$294$		0			0
$295$	1.17009			0.0681251
$296$		0			0
$297$		−	6.43907i		−	0.373633i
$298$		0			0
$299$	2.53200	−	1.90829i	0.146429	−	0.110359i
$300$		0			0
$301$			7.49079i			0.431762i
$302$		0			0
$303$	−8.49693			−0.488136
$304$		0			0
$305$			1.02893i			0.0589163i
$306$		0			0
$307$			15.2534i			0.870557i	0.900296	+	0.435278i	$0.143350\pi$
$307$			15.2534i			0.870557i	−0.900296	+	0.435278i	$0.856650\pi$
$308$		0			0
$309$	0.191828			0.0109127
$310$		0			0
$311$	2.86830			0.162646			0.0813231	−	0.996688i	$-0.474085\pi$
$311$	2.86830			0.162646			0.0813231	+	0.996688i	$0.474085\pi$
$312$		0			0
$313$	14.5236			0.820922			0.410461	−	0.911878i	$-0.365368\pi$
$313$	14.5236			0.820922			0.410461	+	0.911878i	$0.365368\pi$
$314$		0			0
$315$	4.63090			0.260922
$316$		0			0
$317$			8.42082i			0.472960i	0.971636	+	0.236480i	$0.0759938\pi$
$317$			8.42082i			0.472960i	−0.971636	+	0.236480i	$0.924006\pi$
$318$		0			0
$319$			12.3258i			0.690112i
$320$		0			0
$321$	1.10116			0.0614610
$322$		0			0
$323$			7.68649i			0.427688i
$324$		0			0
$325$	2.87936	−	2.17009i	0.159718	−	0.120375i
$326$		0			0
$327$		−	2.92162i		−	0.161566i
$328$		0			0
$329$	−8.76487			−0.483223
$330$		0			0
$331$		−	24.2485i		−	1.33282i	−0.745587	−	0.666408i	$-0.767831\pi$
$331$		−	24.2485i		−	1.33282i	0.745587	−	0.666408i	$-0.232169\pi$
$332$		0			0
$333$			3.70928i			0.203267i
$334$		0			0
$335$	4.63090			0.253013
$336$		0			0
$337$	−27.3028			−1.48728			−0.743640	−	0.668580i	$-0.766902\pi$
$337$	−27.3028			−1.48728			−0.743640	+	0.668580i	$0.766902\pi$
$338$		0			0
$339$	10.1568			0.551639
$340$		0			0
$341$	−16.4208			−0.889237
$342$		0			0
$343$		−	18.9360i		−	1.02245i
$344$		0			0
$345$		−	0.474142i		−	0.0255270i
$346$		0			0
$347$	−4.04226			−0.217000			−0.108500	−	0.994096i	$-0.534605\pi$
$347$	−4.04226			−0.217000			−0.108500	+	0.994096i	$0.534605\pi$
$348$		0			0
$349$		−	29.4908i		−	1.57861i	−0.614004	−	0.789303i	$-0.710442\pi$
$349$		−	29.4908i		−	1.57861i	0.614004	−	0.789303i	$-0.289558\pi$
$350$		0			0
$351$	−8.86376	+	6.68035i	−0.473113	+	0.356570i
$352$		0			0
$353$		−	22.9444i		−	1.22121i	−0.791936	−	0.610604i	$-0.790927\pi$
$353$		−	22.9444i		−	1.22121i	0.791936	−	0.610604i	$-0.209073\pi$
$354$		0			0
$355$	−13.0856			−0.694510
$356$		0			0
$357$		−	3.15061i		−	0.166748i
$358$		0			0
$359$		−	3.22795i		−	0.170364i	−0.996365	−	0.0851822i	$-0.972853\pi$
$359$		−	3.22795i		−	0.170364i	0.996365	−	0.0851822i	$-0.0271472\pi$
$360$		0			0
$361$	13.9444			0.733916
$362$		0			0
$363$	−3.57199			−0.187481
$364$		0			0
$365$	8.20620			0.429532
$366$		0			0
$367$	−1.33525			−0.0696996			−0.0348498	−	0.999393i	$-0.511095\pi$
$367$	−1.33525			−0.0696996			−0.0348498	+	0.999393i	$0.511095\pi$
$368$		0			0
$369$			5.91548i			0.307948i
$370$		0			0
$371$			3.73206i			0.193759i
$372$		0			0
$373$	−10.9216			−0.565500			−0.282750	−	0.959194i	$-0.591247\pi$
$373$	−10.9216			−0.565500			−0.282750	+	0.959194i	$0.591247\pi$
$374$		0			0
$375$		−	0.539189i		−	0.0278436i
$376$		0			0
$377$	16.9672	−	12.7877i	0.873855	−	0.658598i
$378$		0			0
$379$			18.4052i			0.945413i	0.881220	+	0.472706i	$0.156723\pi$
$379$			18.4052i			0.945413i	−0.881220	+	0.472706i	$0.843277\pi$
$380$		0			0
$381$	−11.7093			−0.599884
$382$		0			0
$383$		−	22.8865i		−	1.16945i	−0.811232	−	0.584724i	$-0.801203\pi$
$383$		−	22.8865i		−	1.16945i	0.811232	−	0.584724i	$-0.198797\pi$
$384$		0			0
$385$			3.57531i			0.182214i
$386$		0			0
$387$	11.8732			0.603550
$388$		0			0
$389$	20.5236			1.04059			0.520294	−	0.853987i	$-0.325822\pi$
$389$	20.5236			1.04059			0.520294	+	0.853987i	$0.325822\pi$
$390$		0			0
$391$	3.00614			0.152027
$392$		0			0
$393$	−1.47641			−0.0744750
$394$		0			0
$395$		−	16.0989i		−	0.810023i
$396$		0			0
$397$			17.6658i			0.886621i	0.896368	+	0.443310i	$0.146196\pi$
$397$			17.6658i			0.886621i	−0.896368	+	0.443310i	$0.853804\pi$
$398$		0			0
$399$	2.07223			0.103741
$400$		0			0
$401$			30.6681i			1.53149i	0.643144	+	0.765745i	$0.277629\pi$
$401$			30.6681i			1.53149i	−0.643144	+	0.765745i	$0.722371\pi$
$402$		0			0
$403$	17.0361	+	22.6042i	0.848629	+	1.12600i
$404$		0			0
$405$		−	6.46800i		−	0.321397i
$406$		0			0
$407$	−2.86376			−0.141951
$408$		0			0
$409$			37.0205i			1.83055i	0.402833	+	0.915273i	$0.368025\pi$
$409$			37.0205i			1.83055i	−0.402833	+	0.915273i	$0.631975\pi$
$410$		0			0
$411$			10.0267i			0.494579i
$412$		0			0
$413$	2.00000			0.0984136
$414$		0			0
$415$	8.72979			0.428529
$416$		0			0
$417$	7.31965			0.358445
$418$		0			0
$419$	−28.0144			−1.36859			−0.684296	−	0.729204i	$-0.739890\pi$
$419$	−28.0144			−1.36859			−0.684296	+	0.729204i	$0.739890\pi$
$420$		0			0
$421$			27.8888i			1.35922i	0.733575	+	0.679609i	$0.237850\pi$
$421$			27.8888i			1.35922i	−0.733575	+	0.679609i	$0.762150\pi$
$422$		0			0
$423$			13.8927i			0.675486i
$424$		0			0
$425$	3.41855			0.165824
$426$		0			0
$427$			1.75872i			0.0851106i
$428$		0			0
$429$	−2.44748	−	3.24742i	−0.118165	−	0.156787i
$430$		0			0
$431$		−	7.53692i		−	0.363041i	−0.983387	−	0.181520i	$-0.941898\pi$
$431$		−	7.53692i		−	0.363041i	0.983387	−	0.181520i	$-0.0581018\pi$
$432$		0			0
$433$	13.2039			0.634541			0.317270	−	0.948335i	$-0.397234\pi$
$433$	13.2039			0.634541			0.317270	+	0.948335i	$0.397234\pi$
$434$		0			0
$435$		−	3.17727i		−	0.152339i
$436$		0			0
$437$			1.97721i			0.0945830i
$438$		0			0
$439$	30.5380			1.45750			0.728749	−	0.684781i	$-0.240102\pi$
$439$	30.5380			1.45750			0.728749	+	0.684781i	$0.240102\pi$
$440$		0			0
$441$	−11.0494			−0.526164
$442$		0			0
$443$	9.76979			0.464177			0.232088	−	0.972695i	$-0.425444\pi$
$443$	9.76979			0.464177			0.232088	+	0.972695i	$0.425444\pi$
$444$		0			0
$445$	−7.41855			−0.351673
$446$		0			0
$447$		−	4.71154i		−	0.222848i
$448$		0			0
$449$			10.9627i			0.517360i	0.965963	+	0.258680i	$0.0832875\pi$
$449$			10.9627i			0.517360i	−0.965963	+	0.258680i	$0.916713\pi$
$450$		0			0
$451$	−4.56707			−0.215055
$452$		0			0
$453$			9.21235i			0.432834i
$454$		0			0
$455$	4.92162	−	3.70928i	0.230729	−	0.173894i
$456$		0			0
$457$			19.4596i			0.910281i	0.890420	+	0.455141i	$0.150411\pi$
$457$			19.4596i			0.910281i	−0.890420	+	0.455141i	$0.849589\pi$
$458$		0			0
$459$	−10.5236			−0.491199
$460$		0			0
$461$			12.2823i			0.572044i	0.958223	+	0.286022i	$0.0923331\pi$
$461$			12.2823i			0.572044i	−0.958223	+	0.286022i	$0.907667\pi$
$462$		0			0
$463$		−	35.5357i		−	1.65148i	−0.564048	−	0.825742i	$-0.690757\pi$
$463$		−	35.5357i		−	1.65148i	0.564048	−	0.825742i	$-0.309243\pi$
$464$		0			0
$465$	4.23287			0.196294
$466$		0			0
$467$	35.8420			1.65857			0.829285	−	0.558825i	$-0.188748\pi$
$467$	35.8420			1.65857			0.829285	+	0.558825i	$0.188748\pi$
$468$		0			0
$469$	7.91548			0.365503
$470$		0			0
$471$	−2.73820			−0.126170
$472$		0			0
$473$			9.16677i			0.421489i
$474$		0			0
$475$			2.24846i			0.103167i
$476$		0			0
$477$	5.91548			0.270851
$478$		0			0
$479$			12.6030i			0.575846i	0.957654	+	0.287923i	$0.0929648\pi$
$479$			12.6030i			0.575846i	−0.957654	+	0.287923i	$0.907035\pi$
$480$		0			0
$481$	2.97107	+	3.94214i	0.135469	+	0.179746i
$482$		0			0
$483$		−	0.810439i		−	0.0368763i
$484$		0			0
$485$	7.57531			0.343977
$486$		0			0
$487$		−	11.5525i		−	0.523495i	−0.965136	−	0.261747i	$-0.915701\pi$
$487$		−	11.5525i		−	0.523495i	0.965136	−	0.261747i	$-0.0842987\pi$
$488$		0			0
$489$		−	6.59583i		−	0.298274i
$490$		0			0
$491$	−10.1568			−0.458368			−0.229184	−	0.973383i	$-0.573606\pi$
$491$	−10.1568			−0.458368			−0.229184	+	0.973383i	$0.573606\pi$
$492$		0			0
$493$	20.1445			0.907261
$494$		0			0
$495$	5.66701			0.254713
$496$		0			0
$497$	−22.3668			−1.00329
$498$		0			0
$499$		−	36.9165i		−	1.65261i	−0.563222	−	0.826305i	$-0.690439\pi$
$499$		−	36.9165i		−	1.65261i	0.563222	−	0.826305i	$-0.309561\pi$
$500$		0			0
$501$		−	2.97948i		−	0.133113i
$502$		0			0
$503$	26.3246			1.17375			0.586877	−	0.809676i	$-0.300357\pi$
$503$	26.3246			1.17375			0.586877	+	0.809676i	$0.300357\pi$
$504$		0			0
$505$		−	15.7587i		−	0.701254i
$506$		0			0
$507$	−1.93108	+	6.73820i	−0.0857622	+	0.299254i
$508$		0			0
$509$			32.6491i			1.44715i	0.690247	+	0.723574i	$0.257502\pi$
$509$			32.6491i			1.44715i	−0.690247	+	0.723574i	$0.742498\pi$
$510$		0			0
$511$	14.0267			0.620503
$512$		0			0
$513$		−	6.92162i		−	0.305597i
$514$		0			0
$515$			0.355771i			0.0156772i
$516$		0			0
$517$	−10.7259			−0.471725
$518$		0			0
$519$	3.02052			0.132586
$520$		0			0
$521$	−9.52586			−0.417335			−0.208668	−	0.977987i	$-0.566913\pi$
$521$	−9.52586			−0.417335			−0.208668	+	0.977987i	$0.566913\pi$
$522$		0			0
$523$	37.0193			1.61874			0.809370	−	0.587299i	$-0.199809\pi$
$523$	37.0193			1.61874			0.809370	+	0.587299i	$0.199809\pi$
$524$		0			0
$525$		−	0.921622i		−	0.0402229i
$526$		0			0
$527$			26.8371i			1.16904i
$528$		0			0
$529$	−22.2267			−0.966379
$530$		0			0
$531$		−	3.17009i		−	0.137570i
$532$		0			0
$533$	4.73820	+	6.28685i	0.205234	+	0.272314i
$534$		0			0
$535$			2.04226i			0.0882946i
$536$		0			0
$537$	4.46573			0.192711
$538$		0			0
$539$		−	8.53078i		−	0.367447i
$540$		0			0
$541$			14.5380i			0.625036i	0.949912	+	0.312518i	$0.101172\pi$
$541$			14.5380i			0.625036i	−0.949912	+	0.312518i	$0.898828\pi$
$542$		0			0
$543$	−3.54411			−0.152092
$544$		0			0
$545$	5.41855			0.232105
$546$		0			0
$547$	−7.03612			−0.300843			−0.150421	−	0.988622i	$-0.548063\pi$
$547$	−7.03612			−0.300843			−0.150421	+	0.988622i	$0.548063\pi$
$548$		0			0
$549$	2.78765			0.118974
$550$		0			0
$551$			13.2495i			0.564448i
$552$		0			0
$553$		−	27.5174i		−	1.17016i
$554$		0			0
$555$	0.738205			0.0313350
$556$		0			0
$557$			13.8348i			0.586201i	0.956082	+	0.293100i	$0.0946870\pi$
$557$			13.8348i			0.586201i	−0.956082	+	0.293100i	$0.905313\pi$
$558$		0			0
$559$	12.6186	−	9.51026i	0.533710	−	0.402241i
$560$		0			0
$561$		−	3.85553i		−	0.162781i
$562$		0			0
$563$	27.3328			1.15194			0.575970	−	0.817471i	$-0.304624\pi$
$563$	27.3328			1.15194			0.575970	+	0.817471i	$0.304624\pi$
$564$		0			0
$565$			18.8371i			0.792483i
$566$		0			0
$567$		−	11.0556i		−	0.464291i
$568$		0			0
$569$	16.0183			0.671520			0.335760	−	0.941948i	$-0.391007\pi$
$569$	16.0183			0.671520			0.335760	+	0.941948i	$0.391007\pi$
$570$		0			0
$571$	14.7382			0.616775			0.308387	−	0.951261i	$-0.400211\pi$
$571$	14.7382			0.616775			0.308387	+	0.951261i	$0.400211\pi$
$572$		0			0
$573$	11.6865			0.488210
$574$		0			0
$575$	0.879362			0.0366719
$576$		0			0
$577$			26.5152i			1.10384i	0.833897	+	0.551921i	$0.186105\pi$
$577$			26.5152i			1.10384i	−0.833897	+	0.551921i	$0.813895\pi$
$578$		0			0
$579$		−	4.18342i		−	0.173857i
$580$		0			0
$581$	14.9216			0.619053
$582$		0			0
$583$			4.56707i			0.189149i
$584$		0			0
$585$	−5.87936	−	7.80098i	−0.243082	−	0.322531i
$586$		0			0
$587$			30.9711i			1.27831i	0.769077	+	0.639156i	$0.220716\pi$
$587$			30.9711i			1.27831i	−0.769077	+	0.639156i	$0.779284\pi$
$588$		0			0
$589$	−17.6514			−0.727314
$590$		0			0
$591$			12.6947i			0.522191i
$592$		0			0
$593$			27.8888i			1.14526i	0.819815	+	0.572628i	$0.194076\pi$
$593$			27.8888i			1.14526i	−0.819815	+	0.572628i	$0.805924\pi$
$594$		0			0
$595$	5.84324			0.239550
$596$		0			0
$597$	−6.79153			−0.277959
$598$		0			0
$599$	−21.0784			−0.861239			−0.430620	−	0.902534i	$-0.641705\pi$
$599$	−21.0784			−0.861239			−0.430620	+	0.902534i	$0.641705\pi$
$600$		0			0
$601$	−16.3545			−0.667116			−0.333558	−	0.942730i	$-0.608249\pi$
$601$	−16.3545			−0.667116			−0.333558	+	0.942730i	$0.608249\pi$
$602$		0			0
$603$		−	12.5464i		−	0.510928i
$604$		0			0
$605$		−	6.62475i		−	0.269335i
$606$		0			0
$607$	−33.1617			−1.34599			−0.672995	−	0.739647i	$-0.734993\pi$
$607$	−33.1617			−1.34599			−0.672995	+	0.739647i	$0.734993\pi$
$608$		0			0
$609$		−	5.43084i		−	0.220069i
$610$		0			0
$611$	11.1278	+	14.7649i	0.450184	+	0.597323i
$612$		0			0
$613$		−	40.3012i		−	1.62775i	−0.581039	−	0.813876i	$-0.697354\pi$
$613$		−	40.3012i		−	1.62775i	0.581039	−	0.813876i	$-0.302646\pi$
$614$		0			0
$615$	1.17727			0.0474723
$616$		0			0
$617$		−	14.2823i		−	0.574984i	−0.957783	−	0.287492i	$-0.907178\pi$
$617$		−	14.2823i		−	0.574984i	0.957783	−	0.287492i	$-0.0928215\pi$
$618$		0			0
$619$		−	19.0277i		−	0.764788i	−0.923999	−	0.382394i	$-0.875100\pi$
$619$		−	19.0277i		−	0.764788i	0.923999	−	0.382394i	$-0.124900\pi$
$620$		0			0
$621$	−2.70701			−0.108628
$622$		0			0
$623$	−12.6803			−0.508027
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	2.53588			0.101273
$628$		0			0
$629$			4.68035i			0.186618i
$630$		0			0
$631$			3.25461i			0.129564i	0.997899	+	0.0647819i	$0.0206352\pi$
$631$			3.25461i			0.129564i	−0.997899	+	0.0647819i	$0.979365\pi$
$632$		0			0
$633$	8.77924			0.348944
$634$		0			0
$635$		−	21.7165i		−	0.861792i
$636$		0			0
$637$	−11.7431	+	8.85043i	−0.465280	+	0.350667i
$638$		0			0
$639$			35.4524i			1.40248i
$640$		0			0
$641$	−16.8904			−0.667132			−0.333566	−	0.942727i	$-0.608252\pi$
$641$	−16.8904			−0.667132			−0.333566	+	0.942727i	$0.608252\pi$
$642$		0			0
$643$		−	48.9132i		−	1.92895i	−0.264176	−	0.964474i	$-0.585100\pi$
$643$		−	48.9132i		−	1.92895i	0.264176	−	0.964474i	$-0.414900\pi$
$644$		0			0
$645$		−	2.36296i		−	0.0930414i
$646$		0			0
$647$	0.669282			0.0263122			0.0131561	−	0.999913i	$-0.495812\pi$
$647$	0.669282			0.0263122			0.0131561	+	0.999913i	$0.495812\pi$
$648$		0			0
$649$	2.44748			0.0960720
$650$		0			0
$651$	7.23513			0.283567
$652$		0			0
$653$	−14.6537			−0.573443			−0.286721	−	0.958014i	$-0.592565\pi$
$653$	−14.6537			−0.573443			−0.286721	+	0.958014i	$0.592565\pi$
$654$		0			0
$655$		−	2.73820i		−	0.106991i
$656$		0			0
$657$		−	22.2329i		−	0.867387i
$658$		0			0
$659$	−18.2245			−0.709924			−0.354962	−	0.934881i	$-0.615506\pi$
$659$	−18.2245			−0.709924			−0.354962	+	0.934881i	$0.615506\pi$
$660$		0			0
$661$			0.523590i			0.0203653i	0.999948	+	0.0101827i	$0.00324130\pi$
$661$			0.523590i			0.0203653i	−0.999948	+	0.0101827i	$0.996759\pi$
$662$		0			0
$663$	−5.30737	+	4.00000i	−0.206121	+	0.155347i
$664$		0			0
$665$			3.84324i			0.149035i
$666$		0			0
$667$	5.18181			0.200640
$668$		0			0
$669$			10.3980i			0.402011i
$670$		0			0
$671$			2.15222i			0.0830856i
$672$		0			0
$673$	−23.8843			−0.920671			−0.460336	−	0.887745i	$-0.652271\pi$
$673$	−23.8843			−0.920671			−0.460336	+	0.887745i	$0.652271\pi$
$674$		0			0
$675$	−3.07838			−0.118487
$676$		0			0
$677$	−4.55479			−0.175055			−0.0875273	−	0.996162i	$-0.527897\pi$
$677$	−4.55479			−0.175055			−0.0875273	+	0.996162i	$0.527897\pi$
$678$		0			0
$679$	12.9483			0.496910
$680$		0			0
$681$			2.29914i			0.0881030i
$682$		0			0
$683$		−	17.3112i		−	0.662396i	−0.943561	−	0.331198i	$-0.892547\pi$
$683$		−	17.3112i		−	0.662396i	0.943561	−	0.331198i	$-0.107453\pi$
$684$		0			0
$685$	−18.5958			−0.710510
$686$		0			0
$687$			4.26794i			0.162832i
$688$		0			0
$689$	6.28685	−	4.73820i	0.239510	−	0.180511i
$690$		0			0
$691$			17.6404i			0.671071i	0.942028	+	0.335535i	$0.108917\pi$
$691$			17.6404i			0.671071i	−0.942028	+	0.335535i	$0.891083\pi$
$692$		0			0
$693$	9.68649			0.367959
$694$		0			0
$695$			13.5753i			0.514941i
$696$		0			0
$697$			7.46412i			0.282724i
$698$		0			0
$699$	−4.54250			−0.171813
$700$		0			0
$701$	−32.8059			−1.23906			−0.619531	−	0.784972i	$-0.712677\pi$
$701$	−32.8059			−1.23906			−0.619531	+	0.784972i	$0.712677\pi$
$702$		0			0
$703$	−3.07838			−0.116103
$704$		0			0
$705$	2.76487			0.104131
$706$		0			0
$707$		−	26.9360i		−	1.01303i
$708$		0			0
$709$		−	4.55025i		−	0.170888i	−0.996343	−	0.0854442i	$-0.972769\pi$
$709$		−	4.55025i		−	0.170888i	0.996343	−	0.0854442i	$-0.0272309\pi$
$710$		0			0
$711$	−43.6163			−1.63574
$712$		0			0
$713$			6.90337i			0.258533i
$714$		0			0
$715$	6.02279	−	4.53919i	0.225239	−	0.169756i
$716$		0			0
$717$			9.75485i			0.364301i
$718$		0			0
$719$	14.7382			0.549642			0.274821	−	0.961495i	$-0.411381\pi$
$719$	14.7382			0.549642			0.274821	+	0.961495i	$0.411381\pi$
$720$		0			0
$721$			0.608111i			0.0226472i
$722$		0			0
$723$		−	1.85762i		−	0.0690856i
$724$		0			0
$725$	5.89269			0.218849
$726$		0			0
$727$	−36.6791			−1.36035			−0.680177	−	0.733048i	$-0.738097\pi$
$727$	−36.6791			−1.36035			−0.680177	+	0.733048i	$0.738097\pi$
$728$		0			0
$729$	−12.5441			−0.464597
$730$		0			0
$731$	14.9816			0.554113
$732$		0			0
$733$			12.8227i			0.473618i	0.971556	+	0.236809i	$0.0761016\pi$
$733$			12.8227i			0.473618i	−0.971556	+	0.236809i	$0.923898\pi$
$734$		0			0
$735$			2.19902i			0.0811119i
$736$		0			0
$737$	9.68649			0.356806
$738$		0			0
$739$		−	54.1061i		−	1.99032i	−0.0982451	−	0.995162i	$-0.531323\pi$
$739$		−	54.1061i		−	1.99032i	0.0982451	−	0.995162i	$-0.468677\pi$
$740$		0			0
$741$	−2.63090	−	3.49079i	−0.0966484	−	0.128237i
$742$		0			0
$743$		−	24.0494i		−	0.882289i	−0.897436	−	0.441144i	$-0.854573\pi$
$743$		−	24.0494i		−	0.882289i	0.897436	−	0.441144i	$-0.145427\pi$
$744$		0			0
$745$	8.73820			0.320143
$746$		0			0
$747$		−	23.6514i		−	0.865360i
$748$		0			0
$749$			3.49079i			0.127551i
$750$		0			0
$751$	−31.7009			−1.15678			−0.578390	−	0.815760i	$-0.696319\pi$
$751$	−31.7009			−1.15678			−0.578390	+	0.815760i	$0.696319\pi$
$752$		0			0
$753$	8.26794			0.301300
$754$		0			0
$755$	−17.0856			−0.621807
$756$		0			0
$757$	11.3919			0.414045			0.207023	−	0.978336i	$-0.433623\pi$
$757$	11.3919			0.414045			0.207023	+	0.978336i	$0.433623\pi$
$758$		0			0
$759$		−	0.991767i		−	0.0359989i
$760$		0			0
$761$		−	35.4186i		−	1.28392i	−0.766738	−	0.641961i	$-0.778121\pi$
$761$		−	35.4186i		−	1.28392i	0.766738	−	0.641961i	$-0.221879\pi$
$762$		0			0
$763$	9.26180			0.335300
$764$		0			0
$765$		−	9.26180i		−	0.334861i
$766$		0			0
$767$	−2.53919	−	3.36910i	−0.0916848	−	0.121651i
$768$		0			0
$769$		−	18.1399i		−	0.654143i	−0.945000	−	0.327071i	$-0.893938\pi$
$769$		−	18.1399i		−	0.654143i	0.945000	−	0.327071i	$-0.106062\pi$
$770$		0			0
$771$	−1.70540			−0.0614185
$772$		0			0
$773$		−	29.4680i		−	1.05989i	−0.848032	−	0.529945i	$-0.822213\pi$
$773$		−	29.4680i		−	1.05989i	0.848032	−	0.529945i	$-0.177787\pi$
$774$		0			0
$775$			7.85043i			0.281996i
$776$		0			0
$777$	1.26180			0.0452667
$778$		0			0
$779$	−4.90934			−0.175895
$780$		0			0
$781$	−27.3712			−0.979418
$782$		0			0
$783$	−18.1399			−0.648268
$784$		0			0
$785$		−	5.07838i		−	0.181255i
$786$		0			0
$787$		−	1.79380i		−	0.0639419i	−0.999489	−	0.0319710i	$-0.989822\pi$
$787$		−	1.79380i		−	0.0639419i	0.999489	−	0.0319710i	$-0.0101784\pi$
$788$		0			0
$789$	14.5464			0.517865
$790$		0			0
$791$			32.1978i			1.14482i
$792$		0			0
$793$	2.96266	−	2.23287i	0.105207	−	0.0792914i
$794$		0			0
$795$		−	1.17727i		−	0.0417536i
$796$		0			0
$797$	17.9109			0.634438			0.317219	−	0.948352i	$-0.397251\pi$
$797$	17.9109			0.634438			0.317219	+	0.948352i	$0.397251\pi$
$798$		0			0
$799$			17.5297i			0.620157i
$800$		0			0
$801$			20.0989i			0.710160i
$802$		0			0
$803$	17.1650			0.605739
$804$		0			0
$805$	1.50307			0.0529763
$806$		0			0
$807$	5.97334			0.210271
$808$		0			0
$809$	8.56075			0.300980			0.150490	−	0.988612i	$-0.451915\pi$
$809$	8.56075			0.300980			0.150490	+	0.988612i	$0.451915\pi$
$810$		0			0
$811$			36.8443i			1.29378i	0.762584	+	0.646889i	$0.223930\pi$
$811$			36.8443i			1.29378i	−0.762584	+	0.646889i	$0.776070\pi$
$812$		0			0
$813$		−	1.06013i		−	0.0371803i
$814$		0			0
$815$	12.2329			0.428499
$816$		0			0
$817$			9.85374i			0.344739i
$818$		0			0
$819$	−10.0494	−	13.3340i	−0.351156	−	0.465929i
$820$		0			0
$821$		−	53.0805i		−	1.85252i	−0.376884	−	0.926261i	$-0.623004\pi$
$821$		−	53.0805i		−	1.85252i	0.376884	−	0.926261i	$-0.376996\pi$
$822$		0			0
$823$	−46.0132			−1.60392			−0.801959	−	0.597379i	$-0.796209\pi$
$823$	−46.0132			−1.60392			−0.801959	+	0.597379i	$0.796209\pi$
$824$		0			0
$825$		−	1.12783i		−	0.0392659i
$826$		0			0
$827$		−	27.0843i		−	0.941815i	−0.882183	−	0.470907i	$-0.843927\pi$
$827$		−	27.0843i		−	0.941815i	0.882183	−	0.470907i	$-0.156073\pi$
$828$		0			0
$829$	16.9132			0.587420			0.293710	−	0.955895i	$-0.405110\pi$
$829$	16.9132			0.587420			0.293710	+	0.955895i	$0.405110\pi$
$830$		0			0
$831$	8.18342			0.283880
$832$		0			0
$833$	−13.9421			−0.483067
$834$		0			0
$835$	5.52586			0.191230
$836$		0			0
$837$		−	24.1666i		−	0.835320i
$838$		0			0
$839$			55.3679i			1.91151i	0.294161	+	0.955756i	$0.404960\pi$
$839$			55.3679i			1.91151i	−0.294161	+	0.955756i	$0.595040\pi$
$840$		0			0
$841$	5.72383			0.197373
$842$		0			0
$843$			3.18956i			0.109854i
$844$		0			0
$845$	−12.4969	−	3.58145i	−0.429907	−	0.123206i
$846$		0			0
$847$		−	11.3235i		−	0.389081i
$848$		0			0
$849$	−10.7754			−0.369810
$850$		0			0
$851$			1.20394i			0.0412704i
$852$		0			0
$853$			1.85166i			0.0633995i	0.999497	+	0.0316997i	$0.0100920\pi$
$853$			1.85166i			0.0633995i	−0.999497	+	0.0316997i	$0.989908\pi$
$854$		0			0
$855$	6.09171			0.208332
$856$		0			0
$857$	51.3439			1.75387			0.876937	−	0.480606i	$-0.159583\pi$
$857$	51.3439			1.75387			0.876937	+	0.480606i	$0.159583\pi$
$858$		0			0
$859$	−15.0349			−0.512984			−0.256492	−	0.966546i	$-0.582567\pi$
$859$	−15.0349			−0.512984			−0.256492	+	0.966546i	$0.582567\pi$
$860$		0			0
$861$	2.01229			0.0685785
$862$		0			0
$863$			20.4307i			0.695468i	0.937593	+	0.347734i	$0.113049\pi$
$863$			20.4307i			0.695468i	−0.937593	+	0.347734i	$0.886951\pi$
$864$		0			0
$865$			5.60197i			0.190473i
$866$		0			0
$867$	2.86499			0.0973000
$868$		0			0
$869$		−	33.6742i		−	1.14232i
$870$		0			0
$871$	−10.0494	−	13.3340i	−0.340513	−	0.451806i
$872$		0			0
$873$		−	20.5236i		−	0.694618i
$874$		0			0
$875$	1.70928			0.0577841
$876$		0			0
$877$			36.4079i			1.22941i	0.788758	+	0.614703i	$0.210724\pi$
$877$			36.4079i			1.22941i	−0.788758	+	0.614703i	$0.789276\pi$
$878$		0			0
$879$			16.4534i			0.554961i
$880$		0			0
$881$	−0.814315			−0.0274350			−0.0137175	−	0.999906i	$-0.504367\pi$
$881$	−0.814315			−0.0274350			−0.0137175	+	0.999906i	$0.504367\pi$
$882$		0			0
$883$	−37.4219			−1.25935			−0.629673	−	0.776860i	$-0.716811\pi$
$883$	−37.4219			−1.25935			−0.629673	+	0.776860i	$0.716811\pi$
$884$		0			0
$885$	−0.630898			−0.0212074
$886$		0			0
$887$	−23.4752			−0.788220			−0.394110	−	0.919063i	$-0.628947\pi$
$887$	−23.4752			−0.788220			−0.394110	+	0.919063i	$0.628947\pi$
$888$		0			0
$889$		−	37.1194i		−	1.24495i
$890$		0			0
$891$		−	13.5292i		−	0.453244i
$892$		0			0
$893$	−11.5297			−0.385828
$894$		0			0
$895$			8.28231i			0.276847i
$896$		0			0
$897$	−1.36523	+	1.02893i	−0.0455836	+	0.0343550i
$898$		0			0
$899$			46.2602i			1.54286i
$900$		0			0
$901$	7.46412			0.248666
$902$		0			0
$903$		−	4.03895i		−	0.134408i
$904$		0			0
$905$		−	6.57304i		−	0.218495i
$906$		0			0
$907$	−43.7575			−1.45294			−0.726472	−	0.687196i	$-0.758841\pi$
$907$	−43.7575			−1.45294			−0.726472	+	0.687196i	$0.758841\pi$
$908$		0			0
$909$	−42.6947			−1.41609
$910$		0			0
$911$	−14.2413			−0.471835			−0.235917	−	0.971773i	$-0.575809\pi$
$911$	−14.2413			−0.471835			−0.235917	+	0.971773i	$0.575809\pi$
$912$		0			0
$913$	18.2602			0.604324
$914$		0			0
$915$		−	0.554787i		−	0.0183407i
$916$		0			0
$917$		−	4.68035i		−	0.154559i
$918$		0			0
$919$	53.8408			1.77604			0.888022	−	0.459801i	$-0.152079\pi$
$919$	53.8408			1.77604			0.888022	+	0.459801i	$0.152079\pi$
$920$		0			0
$921$		−	8.22446i		−	0.271005i
$922$		0			0
$923$	28.3968	+	37.6781i	0.934692	+	1.24019i
$924$		0			0
$925$			1.36910i			0.0450158i
$926$		0			0
$927$	0.963883			0.0316581
$928$		0			0
$929$			53.3874i			1.75158i	0.482690	+	0.875791i	$0.339660\pi$
$929$			53.3874i			1.75158i	−0.482690	+	0.875791i	$0.660340\pi$
$930$		0			0
$931$		−	9.17009i		−	0.300537i
$932$		0			0
$933$	−1.54655			−0.0506319
$934$		0			0
$935$	7.15061			0.233850
$936$		0			0
$937$	5.24951			0.171494			0.0857470	−	0.996317i	$-0.472672\pi$
$937$	5.24951			0.171494			0.0857470	+	0.996317i	$0.472672\pi$
$938$		0			0
$939$	−7.83096			−0.255554
$940$		0			0
$941$			47.2639i			1.54076i	0.637586	+	0.770379i	$0.279933\pi$
$941$			47.2639i			1.54076i	−0.637586	+	0.770379i	$0.720067\pi$
$942$		0			0
$943$			1.92001i			0.0625243i
$944$		0			0
$945$	−5.26180			−0.171166
$946$		0			0
$947$		−	22.2595i		−	0.723337i	−0.932307	−	0.361669i	$-0.882207\pi$
$947$		−	22.2595i		−	0.723337i	0.932307	−	0.361669i	$-0.117793\pi$
$948$		0			0
$949$	−17.8082	−	23.6286i	−0.578078	−	0.767018i
$950$		0			0
$951$		−	4.54041i		−	0.147233i
$952$		0			0
$953$	12.0579			0.390592			0.195296	−	0.980744i	$-0.437433\pi$
$953$	12.0579			0.390592			0.195296	+	0.980744i	$0.437433\pi$
$954$		0			0
$955$			21.6742i			0.701361i
$956$		0			0
$957$		−	6.64593i		−	0.214832i
$958$		0			0
$959$	−31.7854			−1.02640
$960$		0			0
$961$	−30.6293			−0.988042
$962$		0			0
$963$	5.53305			0.178300
$964$		0			0
$965$	7.75872			0.249762
$966$		0			0
$967$		−	16.6842i		−	0.536528i	−0.963345	−	0.268264i	$-0.913550\pi$
$967$		−	16.6842i		−	0.536528i	0.963345	−	0.268264i	$-0.0864500\pi$
$968$		0			0
$969$		−	4.14447i		−	0.133140i
$970$		0			0
$971$	43.8531			1.40731			0.703656	−	0.710541i	$-0.251550\pi$
$971$	43.8531			1.40731			0.703656	+	0.710541i	$0.251550\pi$
$972$		0			0
$973$			23.2039i			0.743884i
$974$		0			0
$975$	−1.55252	+	1.17009i	−0.0497204	+	0.0374728i
$976$		0			0
$977$		−	21.5136i		−	0.688280i	−0.938918	−	0.344140i	$-0.888170\pi$
$977$		−	21.5136i		−	0.688280i	0.938918	−	0.344140i	$-0.111830\pi$
$978$		0			0
$979$	−15.5174			−0.495940
$980$		0			0
$981$		−	14.6803i		−	0.468707i
$982$		0			0
$983$			27.9481i			0.891406i	0.895181	+	0.445703i	$0.147046\pi$
$983$			27.9481i			0.891406i	−0.895181	+	0.445703i	$0.852954\pi$
$984$		0			0
$985$	−23.5441			−0.750178
$986$		0			0
$987$	4.72592			0.150428
$988$		0			0
$989$	3.85374			0.122542
$990$		0			0
$991$	20.4436			0.649412			0.324706	−	0.945815i	$-0.394735\pi$
$991$	20.4436			0.649412			0.324706	+	0.945815i	$0.394735\pi$
$992$		0			0
$993$			13.0745i			0.414907i
$994$		0			0
$995$		−	12.5958i		−	0.399315i
$996$		0			0
$997$	14.4514			0.457679			0.228840	−	0.973464i	$-0.426507\pi$
$997$	14.4514			0.457679			0.228840	+	0.973464i	$0.426507\pi$
$998$		0			0
$999$		−	4.21461i		−	0.133344i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	520.2.k.a.441.3	✓	6
3.2	odd	2		4680.2.g.j.2521.4		6
4.3	odd	2		1040.2.k.b.961.3		6
5.2	odd	4		2600.2.f.c.649.4		6
5.3	odd	4		2600.2.f.d.649.3		6
5.4	even	2		2600.2.k.b.2001.4		6
13.5	odd	4		6760.2.a.u.1.2		3
13.8	odd	4		6760.2.a.v.1.2		3
13.12	even	2	inner	520.2.k.a.441.4	yes	6
39.38	odd	2		4680.2.g.j.2521.3		6
52.51	odd	2		1040.2.k.b.961.4		6
65.12	odd	4		2600.2.f.d.649.4		6
65.38	odd	4		2600.2.f.c.649.3		6
65.64	even	2		2600.2.k.b.2001.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.k.a.441.3	✓	6	1.1	even	1	trivial
520.2.k.a.441.4	yes	6	13.12	even	2	inner
1040.2.k.b.961.3		6	4.3	odd	2
1040.2.k.b.961.4		6	52.51	odd	2
2600.2.f.c.649.3		6	65.38	odd	4
2600.2.f.c.649.4		6	5.2	odd	4
2600.2.f.d.649.3		6	5.3	odd	4
2600.2.f.d.649.4		6	65.12	odd	4
2600.2.k.b.2001.3		6	65.64	even	2
2600.2.k.b.2001.4		6	5.4	even	2
4680.2.g.j.2521.3		6	39.38	odd	2
4680.2.g.j.2521.4		6	3.2	odd	2
6760.2.a.u.1.2		3	13.5	odd	4
6760.2.a.v.1.2		3	13.8	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 520 = 2^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	520.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.539189 q^{3} -1.00000i q^{5} -1.70928i q^{7} -2.70928 q^{9} -2.09171i q^{11} +(-2.87936 + 2.17009i) q^{13} +0.539189i q^{15} -3.41855 q^{17} -2.24846i q^{19} +0.921622i q^{21} -0.879362 q^{23} -1.00000 q^{25} +3.07838 q^{27} -5.89269 q^{29} -7.85043i q^{31} +1.12783i q^{33} -1.70928 q^{35} -1.36910i q^{37} +(1.55252 - 1.17009i) q^{39} -2.18342i q^{41} -4.38243 q^{43} +2.70928i q^{45} -5.12783i q^{47} +4.07838 q^{49} +1.84324 q^{51} -2.18342 q^{53} -2.09171 q^{55} +1.21235i q^{57} +1.17009i q^{59} -1.02893 q^{61} +4.63090i q^{63} +(2.17009 + 2.87936i) q^{65} +4.63090i q^{67} +0.474142 q^{69} -13.0856i q^{71} +8.20620i q^{73} +0.539189 q^{75} -3.57531 q^{77} +16.0989 q^{79} +6.46800 q^{81} +8.72979i q^{83} +3.41855i q^{85} +3.17727 q^{87} -7.41855i q^{89} +(3.70928 + 4.92162i) q^{91} +4.23287i q^{93} -2.24846 q^{95} +7.57531i q^{97} +5.66701i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 2 q^{9} + 8 q^{13} + 8 q^{17} + 20 q^{23} - 6 q^{25} + 12 q^{27} - 12 q^{29} + 4 q^{35} + 8 q^{39} - 36 q^{43} + 18 q^{49} + 24 q^{51} - 4 q^{53} - 8 q^{55} - 36 q^{61} + 2 q^{65} + 8 q^{69} + 20 q^{77}+ \cdots + 4 q^{95}+O(q^{100}) \) 6 * q - 2 * q^9 + 8 * q^13 + 8 * q^17 + 20 * q^23 - 6 * q^25 + 12 * q^27 - 12 * q^29 + 4 * q^35 + 8 * q^39 - 36 * q^43 + 18 * q^49 + 24 * q^51 - 4 * q^53 - 8 * q^55 - 36 * q^61 + 2 * q^65 + 8 * q^69 + 20 * q^77 + 24 * q^79 - 26 * q^81 - 60 * q^87 + 8 * q^91 + 4 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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