LMFDB - Embedded newform 2600.2.f.c.649.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2600,2,Mod(649,2600)] 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("2600.649"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2600 = 2^{3} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2600.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$20.7611045255$
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:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 520)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.5
Root			$0.403032 + 0.403032i$ of defining polynomial
Character	$\chi$	$=$	2600.649
Dual form			2600.2.f.c.649.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.67513i q^{3} -0.806063 q^{7} +0.193937 q^{9} +3.83146i q^{11} +(-1.48119 - 3.28726i) q^{13} +1.61213i q^{17} +0.869067i q^{19} -1.35026i q^{21} -5.28726i q^{23} +5.35026i q^{27} -8.46898 q^{29} +10.4060i q^{31} -6.41819 q^{33} +6.15633 q^{37} +(5.50659 - 2.48119i) q^{39} +9.66291i q^{41} +0.974607i q^{43} -2.41819 q^{47} -6.35026 q^{49} -2.70052 q^{51} -9.66291i q^{53} -1.45580 q^{57} +2.48119i q^{59} -13.1187 q^{61} -0.156325 q^{63} +0.156325 q^{67} +8.85685 q^{69} -1.64481i q^{71} +2.93207 q^{73} -3.08840i q^{77} +3.53690 q^{79} -8.38058 q^{81} -15.6932 q^{83} -14.1866i q^{87} +2.38787i q^{89} +(1.19394 + 2.64974i) q^{91} -17.4314 q^{93} -7.08840 q^{97} +0.743059i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{7} + 2 q^{9} + 2 q^{13} + 12 q^{29} - 36 q^{33} + 16 q^{37} - 8 q^{39} - 12 q^{47} - 18 q^{49} + 24 q^{51} - 28 q^{57} - 36 q^{61} + 20 q^{63} - 20 q^{67} - 8 q^{69} - 24 q^{79} - 26 q^{81} - 28 q^{83}+ \cdots - 4 q^{97}+O(q^{100}) $ 6 * q - 4 * q^7 + 2 * q^9 + 2 * q^13 + 12 * q^29 - 36 * q^33 + 16 * q^37 - 8 * q^39 - 12 * q^47 - 18 * q^49 + 24 * q^51 - 28 * q^57 - 36 * q^61 + 20 * q^63 - 20 * q^67 - 8 * q^69 - 24 * q^79 - 26 * q^81 - 28 * q^83 + 8 * q^91 - 20 * q^93 - 4 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2600\mathbb{Z}\right)^\times$.

$n$	$1301$	$1601$	$1951$	$1977$
$\chi(n)$	$1$	$-1$	$1$	$-1$

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
$3$			1.67513i			0.967137i	0.875306	+	0.483569i	$0.160660\pi$
$3$			1.67513i			0.967137i	−0.875306	+	0.483569i	$0.839340\pi$
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	−0.806063			−0.304663			−0.152332	−	0.988329i	$-0.548678\pi$
$7$	−0.806063			−0.304663			−0.152332	+	0.988329i	$0.548678\pi$
$8$		0			0
$9$	0.193937			0.0646455
$10$		0			0
$11$			3.83146i			1.15523i	0.816310	+	0.577614i	$0.196016\pi$
$11$			3.83146i			1.15523i	−0.816310	+	0.577614i	$0.803984\pi$
$12$		0			0
$13$	−1.48119	−	3.28726i	−0.410809	−	0.911721i
$13$	−1.48119	−	3.28726i	−0.410809	−	0.911721i
$14$		0			0
$15$		0			0
$16$		0			0
$17$			1.61213i			0.390998i	0.980704	+	0.195499i	$0.0626327\pi$
$17$			1.61213i			0.390998i	−0.980704	+	0.195499i	$0.937367\pi$
$18$		0			0
$19$			0.869067i			0.199378i	0.995019	+	0.0996889i	$0.0317847\pi$
$19$			0.869067i			0.199378i	−0.995019	+	0.0996889i	$0.968215\pi$
$20$		0			0
$21$		−	1.35026i		−	0.294651i
$22$		0			0
$23$		−	5.28726i		−	1.10247i	−0.834350	−	0.551235i	$-0.814157\pi$
$23$		−	5.28726i		−	1.10247i	0.834350	−	0.551235i	$-0.185843\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$			5.35026i			1.02966i
$28$		0			0
$29$	−8.46898			−1.57265			−0.786325	−	0.617814i	$-0.788019\pi$
$29$	−8.46898			−1.57265			−0.786325	+	0.617814i	$0.788019\pi$
$30$		0			0
$31$			10.4060i			1.86897i	0.356006	+	0.934484i	$0.384138\pi$
$31$			10.4060i			1.86897i	−0.356006	+	0.934484i	$0.615862\pi$
$32$		0			0
$33$	−6.41819			−1.11726
$34$		0			0
$35$		0			0
$36$		0			0
$37$	6.15633			1.01209			0.506047	−	0.862506i	$-0.331106\pi$
$37$	6.15633			1.01209			0.506047	+	0.862506i	$0.331106\pi$
$38$		0			0
$39$	5.50659	−	2.48119i	0.881760	−	0.397309i
$40$		0			0
$41$			9.66291i			1.50909i	0.656246	+	0.754547i	$0.272143\pi$
$41$			9.66291i			1.50909i	−0.656246	+	0.754547i	$0.727857\pi$
$42$		0			0
$43$			0.974607i			0.148626i	0.997235	+	0.0743131i	$0.0236764\pi$
$43$			0.974607i			0.148626i	−0.997235	+	0.0743131i	$0.976324\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.41819			−0.352729			−0.176365	−	0.984325i	$-0.556434\pi$
$47$	−2.41819			−0.352729			−0.176365	+	0.984325i	$0.556434\pi$
$48$		0			0
$49$	−6.35026			−0.907180
$50$		0			0
$51$	−2.70052			−0.378149
$52$		0			0
$53$		−	9.66291i		−	1.32730i	−0.748042	−	0.663652i	$-0.769006\pi$
$53$		−	9.66291i		−	1.32730i	0.748042	−	0.663652i	$-0.230994\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−1.45580			−0.192826
$58$		0			0
$59$			2.48119i			0.323024i	0.986871	+	0.161512i	$0.0516370\pi$
$59$			2.48119i			0.323024i	−0.986871	+	0.161512i	$0.948363\pi$
$60$		0			0
$61$	−13.1187			−1.67968			−0.839840	−	0.542835i	$-0.817351\pi$
$61$	−13.1187			−1.67968			−0.839840	+	0.542835i	$0.817351\pi$
$62$		0			0
$63$	−0.156325			−0.0196951
$64$		0			0
$65$		0			0
$66$		0			0
$67$	0.156325			0.0190982			0.00954908	−	0.999954i	$-0.496960\pi$
$67$	0.156325			0.0190982			0.00954908	+	0.999954i	$0.496960\pi$
$68$		0			0
$69$	8.85685			1.06624
$70$		0			0
$71$		−	1.64481i		−	0.195203i	−0.995226	−	0.0976017i	$-0.968883\pi$
$71$		−	1.64481i		−	0.195203i	0.995226	−	0.0976017i	$-0.0311171\pi$
$72$		0			0
$73$	2.93207			0.343173			0.171587	−	0.985169i	$-0.445111\pi$
$73$	2.93207			0.343173			0.171587	+	0.985169i	$0.445111\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		−	3.08840i		−	0.351955i
$78$		0			0
$79$	3.53690			0.397933			0.198966	−	0.980006i	$-0.436242\pi$
$79$	3.53690			0.397933			0.198966	+	0.980006i	$0.436242\pi$
$80$		0			0
$81$	−8.38058			−0.931175
$82$		0			0
$83$	−15.6932			−1.72256			−0.861278	−	0.508134i	$-0.830335\pi$
$83$	−15.6932			−1.72256			−0.861278	+	0.508134i	$0.830335\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		−	14.1866i		−	1.52097i
$88$		0			0
$89$			2.38787i			0.253114i	0.991959	+	0.126557i	$0.0403927\pi$
$89$			2.38787i			0.253114i	−0.991959	+	0.126557i	$0.959607\pi$
$90$		0			0
$91$	1.19394	+	2.64974i	0.125159	+	0.277768i
$92$		0			0
$93$	−17.4314			−1.80755
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−7.08840			−0.719718			−0.359859	−	0.933007i	$-0.617175\pi$
$97$	−7.08840			−0.719718			−0.359859	+	0.933007i	$0.617175\pi$
$98$		0			0
$99$			0.743059i			0.0746803i
$100$		0			0
$101$	3.42548			0.340848			0.170424	−	0.985371i	$-0.445486\pi$
$101$	3.42548			0.340848			0.170424	+	0.985371i	$0.445486\pi$
$102$		0			0
$103$			13.3380i			1.31424i	0.753788	+	0.657118i	$0.228225\pi$
$103$			13.3380i			1.31424i	−0.753788	+	0.657118i	$0.771775\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		−	5.93700i		−	0.573951i	−0.957938	−	0.286976i	$-0.907350\pi$
$107$		−	5.93700i		−	0.573951i	0.957938	−	0.286976i	$-0.0926498\pi$
$108$		0			0
$109$		−	0.387873i		−	0.0371515i	−0.999827	−	0.0185758i	$-0.994087\pi$
$109$		−	0.387873i		−	0.0371515i	0.999827	−	0.0185758i	$-0.00591319\pi$
$110$		0			0
$111$			10.3127i			0.978833i
$112$		0			0
$113$			8.77575i			0.825553i	0.910832	+	0.412776i	$0.135441\pi$
$113$			8.77575i			0.825553i	−0.910832	+	0.412776i	$0.864559\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−0.287258	−	0.637519i	−0.0265570	−	0.0589387i
$118$		0			0
$119$		−	1.29948i		−	0.119123i
$120$		0			0
$121$	−3.68006			−0.334550
$122$		0			0
$123$	−16.1866			−1.45950
$124$		0			0
$125$		0			0
$126$		0			0
$127$			5.48849i			0.487025i	0.969898	+	0.243512i	$0.0782996\pi$
$127$			5.48849i			0.487025i	−0.969898	+	0.243512i	$0.921700\pi$
$128$		0			0
$129$	−1.63259			−0.143742
$130$		0			0
$131$	12.3127			1.07576			0.537881	−	0.843021i	$-0.319225\pi$
$131$	12.3127			1.07576			0.537881	+	0.843021i	$0.319225\pi$
$132$		0			0
$133$		−	0.700523i		−	0.0607431i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−3.79877			−0.324551			−0.162275	−	0.986746i	$-0.551883\pi$
$137$	−3.79877			−0.324551			−0.162275	+	0.986746i	$0.551883\pi$
$138$		0			0
$139$	13.0884			1.11014			0.555072	−	0.831803i	$-0.312691\pi$
$139$	13.0884			1.11014			0.555072	+	0.831803i	$0.312691\pi$
$140$		0			0
$141$		−	4.05079i		−	0.341138i
$142$		0			0
$143$	12.5950	−	5.67513i	1.05325	−	0.474578i
$144$		0			0
$145$		0			0
$146$		0			0
$147$		−	10.6375i		−	0.877368i
$148$		0			0
$149$		−	18.3127i		−	1.50023i	−0.661307	−	0.750115i	$-0.729998\pi$
$149$		−	18.3127i		−	1.50023i	0.661307	−	0.750115i	$-0.270002\pi$
$150$		0			0
$151$		−	5.64481i		−	0.459368i	−0.973265	−	0.229684i	$-0.926231\pi$
$151$		−	5.64481i		−	0.459368i	0.973265	−	0.229684i	$-0.0737693\pi$
$152$		0			0
$153$			0.312650i			0.0252763i
$154$		0			0
$155$		0			0
$156$		0			0
$157$			7.35026i			0.586615i	0.956018	+	0.293307i	$0.0947559\pi$
$157$			7.35026i			0.586615i	−0.956018	+	0.293307i	$0.905244\pi$
$158$		0			0
$159$	16.1866			1.28368
$160$		0			0
$161$			4.26187i			0.335882i
$162$		0			0
$163$	−9.43136			−0.738721			−0.369361	−	0.929286i	$-0.620423\pi$
$163$	−9.43136			−0.738721			−0.369361	+	0.929286i	$0.620423\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−14.8568			−1.14966			−0.574829	−	0.818274i	$-0.694931\pi$
$167$	−14.8568			−1.14966			−0.574829	+	0.818274i	$0.694931\pi$
$168$		0			0
$169$	−8.61213	+	9.73813i	−0.662471	+	0.749087i
$170$		0			0
$171$			0.168544i			0.0128889i
$172$		0			0
$173$		−	11.2750i		−	0.857225i	−0.903488	−	0.428613i	$-0.859003\pi$
$173$		−	11.2750i		−	0.857225i	0.903488	−	0.428613i	$-0.140997\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−4.15633			−0.312409
$178$		0			0
$179$	−23.1998			−1.73404			−0.867018	−	0.498277i	$-0.833966\pi$
$179$	−23.1998			−1.73404			−0.867018	+	0.498277i	$0.833966\pi$
$180$		0			0
$181$	−22.3938			−1.66451			−0.832257	−	0.554390i	$-0.812952\pi$
$181$	−22.3938			−1.66451			−0.832257	+	0.554390i	$0.812952\pi$
$182$		0			0
$183$		−	21.9756i		−	1.62448i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	−6.17679			−0.451692
$188$		0			0
$189$		−	4.31265i		−	0.313699i
$190$		0			0
$191$	−1.55149			−0.112262			−0.0561310	−	0.998423i	$-0.517876\pi$
$191$	−1.55149			−0.112262			−0.0561310	+	0.998423i	$0.517876\pi$
$192$		0			0
$193$	−4.57452			−0.329281			−0.164640	−	0.986354i	$-0.552646\pi$
$193$	−4.57452			−0.329281			−0.164640	+	0.986354i	$0.552646\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−17.5125			−1.24771			−0.623856	−	0.781539i	$-0.714435\pi$
$197$	−17.5125			−1.24771			−0.623856	+	0.781539i	$0.714435\pi$
$198$		0			0
$199$	9.79877			0.694616			0.347308	−	0.937751i	$-0.387096\pi$
$199$	9.79877			0.694616			0.347308	+	0.937751i	$0.387096\pi$
$200$		0			0
$201$			0.261865i			0.0184705i
$202$		0			0
$203$	6.82653			0.479129
$204$		0			0
$205$		0			0
$206$		0			0
$207$		−	1.02539i		−	0.0712697i
$208$		0			0
$209$	−3.32979			−0.230327
$210$		0			0
$211$	15.1998			1.04640			0.523199	−	0.852210i	$-0.324738\pi$
$211$	15.1998			1.04640			0.523199	+	0.852210i	$0.324738\pi$
$212$		0			0
$213$	2.75528			0.188789
$214$		0			0
$215$		0			0
$216$		0			0
$217$		−	8.38787i		−	0.569406i
$218$		0			0
$219$			4.91160i			0.331895i
$220$		0			0
$221$	5.29948	−	2.38787i	0.356481	−	0.160626i
$222$		0			0
$223$	−16.2823			−1.09035			−0.545173	−	0.838324i	$-0.683536\pi$
$223$	−16.2823			−1.09035			−0.545173	+	0.838324i	$0.683536\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	23.1695			1.53781			0.768907	−	0.639361i	$-0.220801\pi$
$227$	23.1695			1.53781			0.768907	+	0.639361i	$0.220801\pi$
$228$		0			0
$229$			0.126008i			0.00832684i	0.999991	+	0.00416342i	$0.00132526\pi$
$229$			0.126008i			0.00832684i	−0.999991	+	0.00416342i	$0.998675\pi$
$230$		0			0
$231$	5.17347			0.340389
$232$		0			0
$233$		−	8.91160i		−	0.583819i	−0.956446	−	0.291909i	$-0.905709\pi$
$233$		−	8.91160i		−	0.583819i	0.956446	−	0.291909i	$-0.0942905\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			5.92478i			0.384856i
$238$		0			0
$239$			12.1685i			0.787118i	0.919299	+	0.393559i	$0.128756\pi$
$239$			12.1685i			0.787118i	−0.919299	+	0.393559i	$0.871244\pi$
$240$		0			0
$241$		−	17.9756i		−	1.15791i	−0.815360	−	0.578954i	$-0.803461\pi$
$241$		−	17.9756i		−	1.15791i	0.815360	−	0.578954i	$-0.196539\pi$
$242$		0			0
$243$			2.01222i			0.129084i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	2.85685	−	1.28726i	0.181777	−	0.0819062i
$248$		0			0
$249$		−	26.2882i		−	1.66595i
$250$		0			0
$251$	−2.51388			−0.158675			−0.0793374	−	0.996848i	$-0.525280\pi$
$251$	−2.51388			−0.158675			−0.0793374	+	0.996848i	$0.525280\pi$
$252$		0			0
$253$	20.2579			1.27360
$254$		0			0
$255$		0			0
$256$		0			0
$257$			13.2243i			0.824906i	0.910979	+	0.412453i	$0.135328\pi$
$257$			13.2243i			0.824906i	−0.910979	+	0.412453i	$0.864672\pi$
$258$		0			0
$259$	−4.96239			−0.308348
$260$		0			0
$261$	−1.64244			−0.101665
$262$		0			0
$263$			1.17584i			0.0725053i	0.999343	+	0.0362526i	$0.0115421\pi$
$263$			1.17584i			0.0725053i	−0.999343	+	0.0362526i	$0.988458\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−4.00000			−0.244796
$268$		0			0
$269$	13.3503			0.813980			0.406990	−	0.913433i	$-0.366578\pi$
$269$	13.3503			0.813980			0.406990	+	0.913433i	$0.366578\pi$
$270$		0			0
$271$			32.0689i			1.94805i	0.226449	+	0.974023i	$0.427288\pi$
$271$			32.0689i			1.94805i	−0.226449	+	0.974023i	$0.572712\pi$
$272$		0			0
$273$	−4.43866	+	2.00000i	−0.268640	+	0.121046i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			2.18664i			0.131383i	0.997840	+	0.0656913i	$0.0209253\pi$
$277$			2.18664i			0.131383i	−0.997840	+	0.0656913i	$0.979075\pi$
$278$		0			0
$279$			2.01810i			0.120820i
$280$		0			0
$281$			1.87399i			0.111793i	0.998437	+	0.0558965i	$0.0178017\pi$
$281$			1.87399i			0.111793i	−0.998437	+	0.0558965i	$0.982198\pi$
$282$		0			0
$283$			0.300432i			0.0178588i	0.999960	+	0.00892940i	$0.00284235\pi$
$283$			0.300432i			0.0178588i	−0.999960	+	0.00892940i	$0.997158\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	7.78892i		−	0.459765i
$288$		0			0
$289$	14.4010			0.847120
$290$		0			0
$291$		−	11.8740i		−	0.696066i
$292$		0			0
$293$	22.6312			1.32213			0.661064	−	0.750330i	$-0.270105\pi$
$293$	22.6312			1.32213			0.661064	+	0.750330i	$0.270105\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−20.4993			−1.18949
$298$		0			0
$299$	−17.3806	+	7.83146i	−1.00514	+	0.452905i
$300$		0			0
$301$		−	0.785595i		−	0.0452810i
$302$		0			0
$303$			5.73813i			0.329647i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	28.3185			1.61622			0.808112	−	0.589029i	$-0.200490\pi$
$307$	28.3185			1.61622			0.808112	+	0.589029i	$0.200490\pi$
$308$		0			0
$309$	−22.3430			−1.27105
$310$		0			0
$311$	33.3766			1.89261			0.946307	−	0.323270i	$-0.104782\pi$
$311$	33.3766			1.89261			0.946307	+	0.323270i	$0.104782\pi$
$312$		0			0
$313$			4.62530i			0.261437i	0.991419	+	0.130719i	$0.0417285\pi$
$313$			4.62530i			0.261437i	−0.991419	+	0.130719i	$0.958272\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−31.8700			−1.79000			−0.894999	−	0.446067i	$-0.852824\pi$
$317$	−31.8700			−1.79000			−0.894999	+	0.446067i	$0.852824\pi$
$318$		0			0
$319$		−	32.4485i		−	1.81677i
$320$		0			0
$321$	9.94525			0.555089
$322$		0			0
$323$	−1.40105			−0.0779563
$324$		0			0
$325$		0			0
$326$		0			0
$327$	0.649738			0.0359306
$328$		0			0
$329$	1.94921			0.107464
$330$		0			0
$331$		−	22.8691i		−	1.25700i	−0.777811	−	0.628499i	$-0.783670\pi$
$331$		−	22.8691i		−	1.25700i	0.777811	−	0.628499i	$-0.216330\pi$
$332$		0			0
$333$	1.19394			0.0654273
$334$		0			0
$335$		0			0
$336$		0			0
$337$			26.0870i			1.42105i	0.703673	+	0.710524i	$0.251542\pi$
$337$			26.0870i			1.42105i	−0.703673	+	0.710524i	$0.748458\pi$
$338$		0			0
$339$	−14.7005			−0.798423
$340$		0			0
$341$	−39.8700			−2.15908
$342$		0			0
$343$	10.7612			0.581048
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	7.93700i		−	0.426080i	−0.977043	−	0.213040i	$-0.931663\pi$
$347$		−	7.93700i		−	0.426080i	0.977043	−	0.213040i	$-0.0683365\pi$
$348$		0			0
$349$			21.2144i			1.13558i	0.823173	+	0.567791i	$0.192202\pi$
$349$			21.2144i			1.13558i	−0.823173	+	0.567791i	$0.807798\pi$
$350$		0			0
$351$	17.5877	−	7.92478i	0.938761	−	0.422993i
$352$		0			0
$353$	−27.2447			−1.45009			−0.725045	−	0.688701i	$-0.758181\pi$
$353$	−27.2447			−1.45009			−0.725045	+	0.688701i	$0.758181\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	2.17679			0.115208
$358$		0			0
$359$			23.7562i			1.25381i	0.779097	+	0.626903i	$0.215678\pi$
$359$			23.7562i			1.25381i	−0.779097	+	0.626903i	$0.784322\pi$
$360$		0			0
$361$	18.2447			0.960249
$362$		0			0
$363$		−	6.16457i		−	0.323556i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		−	36.2252i		−	1.89094i	−0.325708	−	0.945470i	$-0.605603\pi$
$367$		−	36.2252i		−	1.89094i	0.325708	−	0.945470i	$-0.394397\pi$
$368$		0			0
$369$			1.87399i			0.0975561i
$370$		0			0
$371$			7.78892i			0.404381i
$372$		0			0
$373$			8.64974i			0.447866i	0.974604	+	0.223933i	$0.0718898\pi$
$373$			8.64974i			0.447866i	−0.974604	+	0.223933i	$0.928110\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	12.5442	+	27.8397i	0.646059	+	1.43382i
$378$		0			0
$379$		−	21.5696i		−	1.10796i	−0.832531	−	0.553978i	$-0.813109\pi$
$379$		−	21.5696i		−	1.10796i	0.832531	−	0.553978i	$-0.186891\pi$
$380$		0			0
$381$	−9.19394			−0.471020
$382$		0			0
$383$	−3.00729			−0.153666			−0.0768328	−	0.997044i	$-0.524481\pi$
$383$	−3.00729			−0.153666			−0.0768328	+	0.997044i	$0.524481\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$			0.189012i			0.00960802i
$388$		0			0
$389$	−1.37470			−0.0697000			−0.0348500	−	0.999393i	$-0.511095\pi$
$389$	−1.37470			−0.0697000			−0.0348500	+	0.999393i	$0.511095\pi$
$390$		0			0
$391$	8.52373			0.431064
$392$		0			0
$393$			20.6253i			1.04041i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	36.4544			1.82959			0.914797	−	0.403915i	$-0.132351\pi$
$397$	36.4544			1.82959			0.914797	+	0.403915i	$0.132351\pi$
$398$		0			0
$399$	1.17347			0.0587469
$400$		0			0
$401$			5.02776i			0.251074i	0.992089	+	0.125537i	$0.0400655\pi$
$401$			5.02776i			0.251074i	−0.992089	+	0.125537i	$0.959935\pi$
$402$		0			0
$403$	34.2071	−	15.4133i	1.70398	−	0.767789i
$404$		0			0
$405$		0			0
$406$		0			0
$407$			23.5877i			1.16920i
$408$		0			0
$409$		−	15.1128i		−	0.747282i	−0.927573	−	0.373641i	$-0.878109\pi$
$409$		−	15.1128i		−	0.747282i	0.927573	−	0.373641i	$-0.121891\pi$
$410$		0			0
$411$		−	6.36344i		−	0.313885i
$412$		0			0
$413$		−	2.00000i		−	0.0984136i
$414$		0			0
$415$		0			0
$416$		0			0
$417$			21.9248i			1.07366i
$418$		0			0
$419$	0.589104			0.0287796			0.0143898	−	0.999896i	$-0.495419\pi$
$419$	0.589104			0.0287796			0.0143898	+	0.999896i	$0.495419\pi$
$420$		0			0
$421$			36.4894i			1.77839i	0.457532	+	0.889193i	$0.348734\pi$
$421$			36.4894i			1.77839i	−0.457532	+	0.889193i	$0.651266\pi$
$422$		0			0
$423$	−0.468976			−0.0228024
$424$		0			0
$425$		0			0
$426$		0			0
$427$	10.5745			0.511737
$428$		0			0
$429$	9.50659	+	21.0982i	0.458982	+	1.01863i
$430$		0			0
$431$			19.8070i			0.954071i	0.878884	+	0.477035i	$0.158289\pi$
$431$			19.8070i			0.954071i	−0.878884	+	0.477035i	$0.841711\pi$
$432$		0			0
$433$			20.5501i			0.987574i	0.869583	+	0.493787i	$0.164388\pi$
$433$			20.5501i			0.987574i	−0.869583	+	0.493787i	$0.835612\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4.59498			0.219808
$438$		0			0
$439$	16.0362			0.765366			0.382683	−	0.923880i	$-0.375000\pi$
$439$	16.0362			0.765366			0.382683	+	0.923880i	$0.375000\pi$
$440$		0			0
$441$	−1.23155			−0.0586451
$442$		0			0
$443$			39.2384i			1.86427i	0.362109	+	0.932136i	$0.382057\pi$
$443$			39.2384i			1.86427i	−0.362109	+	0.932136i	$0.617943\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	30.6761			1.45093
$448$		0			0
$449$			35.1246i			1.65763i	0.559521	+	0.828816i	$0.310985\pi$
$449$			35.1246i			1.65763i	−0.559521	+	0.828816i	$0.689015\pi$
$450$		0			0
$451$	−37.0230			−1.74335
$452$		0			0
$453$	9.45580			0.444272
$454$		0			0
$455$		0			0
$456$		0			0
$457$	29.3865			1.37464			0.687320	−	0.726354i	$-0.258787\pi$
$457$	29.3865			1.37464			0.687320	+	0.726354i	$0.258787\pi$
$458$		0			0
$459$	−8.62530			−0.402595
$460$		0			0
$461$		−	19.1998i		−	0.894225i	−0.894478	−	0.447112i	$-0.852452\pi$
$461$		−	19.1998i		−	0.894225i	0.894478	−	0.447112i	$-0.147548\pi$
$462$		0			0
$463$	39.5183			1.83657			0.918287	−	0.395916i	$-0.129573\pi$
$463$	39.5183			1.83657			0.918287	+	0.395916i	$0.129573\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	16.4119i		−	0.759450i	−0.925099	−	0.379725i	$-0.876019\pi$
$467$		−	16.4119i		−	0.759450i	0.925099	−	0.379725i	$-0.123981\pi$
$468$		0			0
$469$	−0.126008			−0.00581851
$470$		0			0
$471$	−12.3127			−0.567337
$472$		0			0
$473$	−3.73417			−0.171697
$474$		0			0
$475$		0			0
$476$		0			0
$477$		−	1.87399i		−	0.0858042i
$478$		0			0
$479$			23.5042i			1.07394i	0.843603	+	0.536968i	$0.180430\pi$
$479$			23.5042i			1.07394i	−0.843603	+	0.536968i	$0.819570\pi$
$480$		0			0
$481$	−9.11871	−	20.2374i	−0.415778	−	0.922747i
$482$		0			0
$483$	−7.13918			−0.324844
$484$		0			0
$485$		0			0
$486$		0			0
$487$	4.49341			0.203616			0.101808	−	0.994804i	$-0.467537\pi$
$487$	4.49341			0.203616			0.101808	+	0.994804i	$0.467537\pi$
$488$		0			0
$489$		−	15.7988i		−	0.714445i
$490$		0			0
$491$	−14.7005			−0.663425			−0.331713	−	0.943380i	$-0.607626\pi$
$491$	−14.7005			−0.663425			−0.331713	+	0.943380i	$0.607626\pi$
$492$		0			0
$493$		−	13.6531i		−	0.614903i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			1.32582i			0.0594713i
$498$		0			0
$499$			9.89683i			0.443043i	0.975155	+	0.221521i	$0.0711023\pi$
$499$			9.89683i			0.443043i	−0.975155	+	0.221521i	$0.928898\pi$
$500$		0			0
$501$		−	24.8872i		−	1.11188i
$502$		0			0
$503$			1.26282i			0.0563064i	0.999604	+	0.0281532i	$0.00896262\pi$
$503$			1.26282i			0.0563064i	−0.999604	+	0.0281532i	$0.991037\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−16.3127	−	14.4264i	−0.724470	−	0.640701i
$508$		0			0
$509$			22.5256i			0.998431i	0.866478	+	0.499216i	$0.166378\pi$
$509$			22.5256i			0.998431i	−0.866478	+	0.499216i	$0.833622\pi$
$510$		0			0
$511$	−2.36344			−0.104552
$512$		0			0
$513$	−4.64974			−0.205291
$514$		0			0
$515$		0			0
$516$		0			0
$517$		−	9.26519i		−	0.407483i
$518$		0			0
$519$	18.8872			0.829055
$520$		0			0
$521$	−18.8568			−0.826134			−0.413067	−	0.910701i	$-0.635542\pi$
$521$	−18.8568			−0.826134			−0.413067	+	0.910701i	$0.635542\pi$
$522$		0			0
$523$			32.5985i			1.42543i	0.701452	+	0.712716i	$0.252535\pi$
$523$			32.5985i			1.42543i	−0.701452	+	0.712716i	$0.747465\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−16.7757			−0.730763
$528$		0			0
$529$	−4.95509			−0.215439
$530$		0			0
$531$			0.481194i			0.0208821i
$532$		0			0
$533$	31.7645	−	14.3127i	1.37587	−	0.619950i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		−	38.8627i		−	1.67705i
$538$		0			0
$539$		−	24.3307i		−	1.04800i
$540$		0			0
$541$		−	32.0362i		−	1.37734i	−0.725073	−	0.688672i	$-0.758194\pi$
$541$		−	32.0362i		−	1.37734i	0.725073	−	0.688672i	$-0.241806\pi$
$542$		0			0
$543$		−	37.5125i		−	1.60981i
$544$		0			0
$545$		0			0
$546$		0			0
$547$		−	5.41327i		−	0.231455i	−0.993281	−	0.115727i	$-0.963080\pi$
$547$		−	5.41327i		−	0.231455i	0.993281	−	0.115727i	$-0.0369199\pi$
$548$		0			0
$549$	−2.54420			−0.108584
$550$		0			0
$551$		−	7.36011i		−	0.313551i
$552$		0			0
$553$	−2.85097			−0.121236
$554$		0			0
$555$		0			0
$556$		0			0
$557$	24.7064			1.04684			0.523422	−	0.852074i	$-0.324655\pi$
$557$	24.7064			1.04684			0.523422	+	0.852074i	$0.324655\pi$
$558$		0			0
$559$	3.20379	−	1.44358i	0.135506	−	0.0610571i
$560$		0			0
$561$		−	10.3469i		−	0.436848i
$562$		0			0
$563$			33.1974i			1.39911i	0.714581	+	0.699553i	$0.246617\pi$
$563$			33.1974i			1.39911i	−0.714581	+	0.699553i	$0.753383\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	6.75528			0.283695
$568$		0			0
$569$	34.3693			1.44084			0.720418	−	0.693540i	$-0.243950\pi$
$569$	34.3693			1.44084			0.720418	+	0.693540i	$0.243950\pi$
$570$		0			0
$571$	24.3127			1.01745			0.508726	−	0.860928i	$-0.330116\pi$
$571$	24.3127			1.01745			0.508726	+	0.860928i	$0.330116\pi$
$572$		0			0
$573$		−	2.59895i		−	0.108573i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	26.6312			1.10867			0.554335	−	0.832293i	$-0.312973\pi$
$577$	26.6312			1.10867			0.554335	+	0.832293i	$0.312973\pi$
$578$		0			0
$579$		−	7.66291i		−	0.318460i
$580$		0			0
$581$	12.6497			0.524800
$582$		0			0
$583$	37.0230			1.53334
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−18.8813			−0.779314			−0.389657	−	0.920960i	$-0.627406\pi$
$587$	−18.8813			−0.779314			−0.389657	+	0.920960i	$0.627406\pi$
$588$		0			0
$589$	−9.04349			−0.372631
$590$		0			0
$591$		−	29.3357i		−	1.20671i
$592$		0			0
$593$	36.4894			1.49844			0.749221	−	0.662320i	$-0.230428\pi$
$593$	36.4894			1.49844			0.749221	+	0.662320i	$0.230428\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			16.4142i			0.671789i
$598$		0			0
$599$	23.3503			0.954066			0.477033	−	0.878885i	$-0.341712\pi$
$599$	23.3503			0.954066			0.477033	+	0.878885i	$0.341712\pi$
$600$		0			0
$601$	18.3733			0.749462			0.374731	−	0.927134i	$-0.377735\pi$
$601$	18.3733			0.749462			0.374731	+	0.927134i	$0.377735\pi$
$602$		0			0
$603$	0.0303172			0.00123461
$604$		0			0
$605$		0			0
$606$		0			0
$607$			4.48707i			0.182125i	0.995845	+	0.0910624i	$0.0290263\pi$
$607$			4.48707i			0.182125i	−0.995845	+	0.0910624i	$0.970974\pi$
$608$		0			0
$609$			11.4353i			0.463383i
$610$		0			0
$611$	3.58181	+	7.94921i	0.144905	+	0.321591i
$612$		0			0
$613$	−38.3536			−1.54909			−0.774543	−	0.632521i	$-0.782020\pi$
$613$	−38.3536			−1.54909			−0.774543	+	0.632521i	$0.782020\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−17.1998			−0.692439			−0.346219	−	0.938154i	$-0.612535\pi$
$617$	−17.1998			−0.692439			−0.346219	+	0.938154i	$0.612535\pi$
$618$		0			0
$619$		−	16.5926i		−	0.666913i	−0.942765	−	0.333457i	$-0.891785\pi$
$619$		−	16.5926i		−	0.666913i	0.942765	−	0.333457i	$-0.108215\pi$
$620$		0			0
$621$	28.2882			1.13517
$622$		0			0
$623$		−	1.92478i		−	0.0771146i
$624$		0			0
$625$		0			0
$626$		0			0
$627$		−	5.57784i		−	0.222757i
$628$		0			0
$629$			9.92478i			0.395727i
$630$		0			0
$631$			7.39280i			0.294303i	0.989114	+	0.147151i	$0.0470104\pi$
$631$			7.39280i			0.294303i	−0.989114	+	0.147151i	$0.952990\pi$
$632$		0			0
$633$			25.4617i			1.01201i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	9.40597	+	20.8749i	0.372678	+	0.827096i
$638$		0			0
$639$		−	0.318990i		−	0.0126190i
$640$		0			0
$641$	25.9511			1.02501			0.512504	−	0.858685i	$-0.328718\pi$
$641$	25.9511			1.02501			0.512504	+	0.858685i	$0.328718\pi$
$642$		0			0
$643$	−12.6439			−0.498625			−0.249313	−	0.968423i	$-0.580205\pi$
$643$	−12.6439			−0.498625			−0.249313	+	0.968423i	$0.580205\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			22.7391i			0.893966i	0.894542	+	0.446983i	$0.147502\pi$
$647$			22.7391i			0.893966i	−0.894542	+	0.446983i	$0.852498\pi$
$648$		0			0
$649$	−9.50659			−0.373166
$650$		0			0
$651$	14.0508			0.550694
$652$		0			0
$653$			16.4387i			0.643294i	0.946860	+	0.321647i	$0.104237\pi$
$653$			16.4387i			0.643294i	−0.946860	+	0.321647i	$0.895763\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	0.568636			0.0221846
$658$		0			0
$659$	−37.4372			−1.45835			−0.729174	−	0.684328i	$-0.760096\pi$
$659$	−37.4372			−1.45835			−0.729174	+	0.684328i	$0.760096\pi$
$660$		0			0
$661$		−	18.6253i		−	0.724440i	−0.932093	−	0.362220i	$-0.882019\pi$
$661$		−	18.6253i		−	0.724440i	0.932093	−	0.362220i	$-0.117981\pi$
$662$		0			0
$663$	4.00000	+	8.87732i	0.155347	+	0.344766i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			44.7777i			1.73380i
$668$		0			0
$669$		−	27.2750i		−	1.05451i
$670$		0			0
$671$		−	50.2638i		−	1.94041i
$672$		0			0
$673$		−	24.4749i		−	0.943436i	−0.881749	−	0.471718i	$-0.843634\pi$
$673$		−	24.4749i		−	0.943436i	0.881749	−	0.471718i	$-0.156366\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	25.9756i		−	0.998322i	−0.866509	−	0.499161i	$-0.833642\pi$
$677$		−	25.9756i		−	0.998322i	0.866509	−	0.499161i	$-0.166358\pi$
$678$		0			0
$679$	5.71370			0.219272
$680$		0			0
$681$			38.8119i			1.48728i
$682$		0			0
$683$	2.08110			0.0796312			0.0398156	−	0.999207i	$-0.487323\pi$
$683$	2.08110			0.0796312			0.0398156	+	0.999207i	$0.487323\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−0.211080			−0.00805320
$688$		0			0
$689$	−31.7645	+	14.3127i	−1.21013	+	0.545269i
$690$		0			0
$691$			27.6204i			1.05073i	0.850877	+	0.525364i	$0.176071\pi$
$691$			27.6204i			1.05073i	−0.850877	+	0.525364i	$0.823929\pi$
$692$		0			0
$693$		−	0.598953i		−	0.0227523i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−15.5778			−0.590053
$698$		0			0
$699$	14.9281			0.564633
$700$		0			0
$701$	17.8251			0.673245			0.336623	−	0.941640i	$-0.390715\pi$
$701$	17.8251			0.673245			0.336623	+	0.941640i	$0.390715\pi$
$702$		0			0
$703$			5.35026i			0.201789i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−2.76116			−0.103844
$708$		0			0
$709$		−	30.9887i		−	1.16381i	−0.813258	−	0.581903i	$-0.802308\pi$
$709$		−	30.9887i		−	1.16381i	0.813258	−	0.581903i	$-0.197692\pi$
$710$		0			0
$711$	0.685935			0.0257246
$712$		0			0
$713$	55.0191			2.06048
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−20.3839			−0.761251
$718$		0			0
$719$	−24.3127			−0.906709			−0.453354	−	0.891330i	$-0.649773\pi$
$719$	−24.3127			−0.906709			−0.453354	+	0.891330i	$0.649773\pi$
$720$		0			0
$721$		−	10.7513i		−	0.400400i
$722$		0			0
$723$	30.1114			1.11986
$724$		0			0
$725$		0			0
$726$		0			0
$727$			25.6361i			0.950791i	0.879772	+	0.475395i	$0.157695\pi$
$727$			25.6361i			0.950791i	−0.879772	+	0.475395i	$0.842305\pi$
$728$		0			0
$729$	−28.5125			−1.05602
$730$		0			0
$731$	−1.57119			−0.0581126
$732$		0			0
$733$	30.1866			1.11497			0.557485	−	0.830187i	$-0.311767\pi$
$733$	30.1866			1.11497			0.557485	+	0.830187i	$0.311767\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			0.598953i			0.0220627i
$738$		0			0
$739$			20.7576i			0.763582i	0.924249	+	0.381791i	$0.124693\pi$
$739$			20.7576i			0.763582i	−0.924249	+	0.381791i	$0.875307\pi$
$740$		0			0
$741$	2.15633	+	4.78560i	0.0792146	+	0.175803i
$742$		0			0
$743$	−14.2315			−0.522105			−0.261052	−	0.965325i	$-0.584070\pi$
$743$	−14.2315			−0.522105			−0.261052	+	0.965325i	$0.584070\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−3.04349			−0.111356
$748$		0			0
$749$			4.78560i			0.174862i
$750$		0			0
$751$	4.81194			0.175590			0.0877951	−	0.996139i	$-0.472018\pi$
$751$	4.81194			0.175590			0.0877951	+	0.996139i	$0.472018\pi$
$752$		0			0
$753$		−	4.21108i		−	0.153460i
$754$		0			0
$755$		0			0
$756$		0			0
$757$			22.7513i			0.826910i	0.910524	+	0.413455i	$0.135678\pi$
$757$			22.7513i			0.826910i	−0.910524	+	0.413455i	$0.864322\pi$
$758$		0			0
$759$			33.9346i			1.23175i
$760$		0			0
$761$		−	30.3879i		−	1.10156i	−0.834651	−	0.550780i	$-0.814330\pi$
$761$		−	30.3879i		−	1.10156i	0.834651	−	0.550780i	$-0.185670\pi$
$762$		0			0
$763$			0.312650i			0.0113187i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	8.15633	−	3.67513i	0.294508	−	0.132701i
$768$		0			0
$769$		−	45.3112i		−	1.63396i	−0.576662	−	0.816982i	$-0.695645\pi$
$769$		−	45.3112i		−	1.63396i	0.576662	−	0.816982i	$-0.304355\pi$
$770$		0			0
$771$	−22.1524			−0.797798
$772$		0			0
$773$	−14.6194			−0.525824			−0.262912	−	0.964820i	$-0.584683\pi$
$773$	−14.6194			−0.525824			−0.262912	+	0.964820i	$0.584683\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		−	8.31265i		−	0.298215i
$778$		0			0
$779$	−8.39772			−0.300880
$780$		0			0
$781$	6.30203			0.225504
$782$		0			0
$783$		−	45.3112i		−	1.61929i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	7.06793			0.251944			0.125972	−	0.992034i	$-0.459795\pi$
$787$	7.06793			0.251944			0.125972	+	0.992034i	$0.459795\pi$
$788$		0			0
$789$	−1.96968			−0.0701226
$790$		0			0
$791$		−	7.07381i		−	0.251516i
$792$		0			0
$793$	19.4314	+	43.1246i	0.690028	+	1.53140i
$794$		0			0
$795$		0			0
$796$		0			0
$797$		−	46.8383i		−	1.65910i	−0.558434	−	0.829549i	$-0.688598\pi$
$797$		−	46.8383i		−	1.65910i	0.558434	−	0.829549i	$-0.311402\pi$
$798$		0			0
$799$		−	3.89843i		−	0.137917i
$800$		0			0
$801$			0.463096i			0.0163627i
$802$		0			0
$803$			11.2341i			0.396443i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			22.3634i			0.787230i
$808$		0			0
$809$	31.4412			1.10541			0.552707	−	0.833376i	$-0.313595\pi$
$809$	31.4412			1.10541			0.552707	+	0.833376i	$0.313595\pi$
$810$		0			0
$811$			13.0703i			0.458960i	0.973313	+	0.229480i	$0.0737026\pi$
$811$			13.0703i			0.458960i	−0.973313	+	0.229480i	$0.926297\pi$
$812$		0			0
$813$	−53.7196			−1.88403
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.847000			−0.0296328
$818$		0			0
$819$	0.231548	+	0.513881i	0.00809094	+	0.0179565i
$820$		0			0
$821$		−	16.8919i		−	0.589532i	−0.955570	−	0.294766i	$-0.904758\pi$
$821$		−	16.8919i		−	0.589532i	0.955570	−	0.294766i	$-0.0952416\pi$
$822$		0			0
$823$		−	29.1222i		−	1.01514i	−0.861611	−	0.507568i	$-0.830544\pi$
$823$		−	29.1222i		−	1.01514i	0.861611	−	0.507568i	$-0.169456\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−32.0665			−1.11506			−0.557531	−	0.830156i	$-0.688251\pi$
$827$	−32.0665			−1.11506			−0.557531	+	0.830156i	$0.688251\pi$
$828$		0			0
$829$	19.3561			0.672267			0.336133	−	0.941814i	$-0.390881\pi$
$829$	19.3561			0.672267			0.336133	+	0.941814i	$0.390881\pi$
$830$		0			0
$831$	−3.66291			−0.127065
$832$		0			0
$833$		−	10.2374i		−	0.354706i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−55.6747			−1.92440
$838$		0			0
$839$		−	12.4450i		−	0.429649i	−0.976653	−	0.214825i	$-0.931082\pi$
$839$		−	12.4450i		−	0.429649i	0.976653	−	0.214825i	$-0.0689180\pi$
$840$		0			0
$841$	42.7235			1.47323
$842$		0			0
$843$	−3.13918			−0.108119
$844$		0			0
$845$		0			0
$846$		0			0
$847$	2.96636			0.101925
$848$		0			0
$849$	−0.503262			−0.0172719
$850$		0			0
$851$		−	32.5501i		−	1.11580i
$852$		0			0
$853$	31.3054			1.07187			0.535937	−	0.844258i	$-0.319958\pi$
$853$	31.3054			1.07187			0.535937	+	0.844258i	$0.319958\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	45.8613i		−	1.56659i	−0.621649	−	0.783296i	$-0.713537\pi$
$857$		−	45.8613i		−	1.56659i	0.621649	−	0.783296i	$-0.286463\pi$
$858$		0			0
$859$	−34.2981			−1.17023			−0.585117	−	0.810949i	$-0.698952\pi$
$859$	−34.2981			−1.17023			−0.585117	+	0.810949i	$0.698952\pi$
$860$		0			0
$861$	13.0475			0.444656
$862$		0			0
$863$	−40.5052			−1.37881			−0.689406	−	0.724375i	$-0.742128\pi$
$863$	−40.5052			−1.37881			−0.689406	+	0.724375i	$0.742128\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			24.1236i			0.819282i
$868$		0			0
$869$			13.5515i			0.459703i
$870$		0			0
$871$	−0.231548	−	0.513881i	−0.00784570	−	0.0174122i
$872$		0			0
$873$	−1.37470			−0.0465265
$874$		0			0
$875$		0			0
$876$		0			0
$877$	31.1002			1.05018			0.525089	−	0.851047i	$-0.324032\pi$
$877$	31.1002			1.05018			0.525089	+	0.851047i	$0.324032\pi$
$878$		0			0
$879$			37.9102i			1.27868i
$880$		0			0
$881$	15.8192			0.532964			0.266482	−	0.963840i	$-0.414139\pi$
$881$	15.8192			0.532964			0.266482	+	0.963840i	$0.414139\pi$
$882$		0			0
$883$			41.6408i			1.40133i	0.713492	+	0.700663i	$0.247112\pi$
$883$			41.6408i			1.40133i	−0.713492	+	0.700663i	$0.752888\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			5.08603i			0.170772i	0.996348	+	0.0853860i	$0.0272124\pi$
$887$			5.08603i			0.170772i	−0.996348	+	0.0853860i	$0.972788\pi$
$888$		0			0
$889$		−	4.42407i		−	0.148379i
$890$		0			0
$891$		−	32.1098i		−	1.07572i
$892$		0			0
$893$		−	2.10157i		−	0.0703264i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	−13.1187	−	29.1147i	−0.438021	−	0.972113i
$898$		0			0
$899$		−	88.1279i		−	2.93923i
$900$		0			0
$901$	15.5778			0.518973
$902$		0			0
$903$	1.31598			0.0437929
$904$		0			0
$905$		0			0
$906$		0			0
$907$			16.2858i			0.540763i	0.962753	+	0.270381i	$0.0871498\pi$
$907$			16.2858i			0.540763i	−0.962753	+	0.270381i	$0.912850\pi$
$908$		0			0
$909$	0.664327			0.0220343
$910$		0			0
$911$	−26.5745			−0.880453			−0.440226	−	0.897887i	$-0.645102\pi$
$911$	−26.5745			−0.880453			−0.440226	+	0.897887i	$0.645102\pi$
$912$		0			0
$913$		−	60.1279i		−	1.98994i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−9.92478			−0.327745
$918$		0			0
$919$	46.1232			1.52146			0.760732	−	0.649067i	$-0.224840\pi$
$919$	46.1232			1.52146			0.760732	+	0.649067i	$0.224840\pi$
$920$		0			0
$921$			47.4372i			1.56311i
$922$		0			0
$923$	−5.40693	+	2.43629i	−0.177971	+	0.0801914i
$924$		0			0
$925$		0			0
$926$		0			0
$927$			2.58673i			0.0849595i
$928$		0			0
$929$		−	7.78701i		−	0.255484i	−0.991807	−	0.127742i	$-0.959227\pi$
$929$		−	7.78701i		−	0.255484i	0.991807	−	0.127742i	$-0.0407729\pi$
$930$		0			0
$931$		−	5.51881i		−	0.180872i
$932$		0			0
$933$			55.9102i			1.83042i
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	15.3601i		−	0.501793i	−0.968014	−	0.250896i	$-0.919275\pi$
$937$		−	15.3601i		−	0.501793i	0.968014	−	0.250896i	$-0.0807254\pi$
$938$		0			0
$939$	−7.74798			−0.252846
$940$		0			0
$941$		−	0.771007i		−	0.0251341i	−0.999921	−	0.0125671i	$-0.996000\pi$
$941$		−	0.771007i		−	0.0251341i	0.999921	−	0.0125671i	$-0.00400032\pi$
$942$		0			0
$943$	51.0903			1.66373
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−15.7948			−0.513262			−0.256631	−	0.966509i	$-0.582613\pi$
$947$	−15.7948			−0.513262			−0.256631	+	0.966509i	$0.582613\pi$
$948$		0			0
$949$	−4.34297	−	9.63847i	−0.140979	−	0.312878i
$950$		0			0
$951$		−	53.3865i		−	1.73117i
$952$		0			0
$953$		−	36.2374i		−	1.17385i	−0.809643	−	0.586923i	$-0.800339\pi$
$953$		−	36.2374i		−	1.17385i	0.809643	−	0.586923i	$-0.199661\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	54.3555			1.75706
$958$		0			0
$959$	3.06205			0.0988787
$960$		0			0
$961$	−77.2842			−2.49304
$962$		0			0
$963$		−	1.15140i		−	0.0371034i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−20.8832			−0.671558			−0.335779	−	0.941941i	$-0.609000\pi$
$967$	−20.8832			−0.671558			−0.335779	+	0.941941i	$0.609000\pi$
$968$		0			0
$969$		−	2.34694i		−	0.0753945i
$970$		0			0
$971$	−45.0757			−1.44655			−0.723274	−	0.690561i	$-0.757364\pi$
$971$	−45.0757			−1.44655			−0.723274	+	0.690561i	$0.757364\pi$
$972$		0			0
$973$	−10.5501			−0.338220
$974$		0			0
$975$		0			0
$976$		0			0
$977$	19.8094			0.633758			0.316879	−	0.948466i	$-0.397365\pi$
$977$	19.8094			0.633758			0.316879	+	0.948466i	$0.397365\pi$
$978$		0			0
$979$	−9.14903			−0.292404
$980$		0			0
$981$		−	0.0752228i		−	0.00240168i
$982$		0			0
$983$	−57.6542			−1.83888			−0.919442	−	0.393226i	$-0.871359\pi$
$983$	−57.6542			−1.83888			−0.919442	+	0.393226i	$0.871359\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$			3.26519i			0.103932i
$988$		0			0
$989$	5.15300			0.163856
$990$		0			0
$991$	50.4650			1.60307			0.801537	−	0.597945i	$-0.204016\pi$
$991$	50.4650			1.60307			0.801537	+	0.597945i	$0.204016\pi$
$992$		0			0
$993$	38.3087			1.21569
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	1.45183i		−	0.0459800i	−0.999736	−	0.0229900i	$-0.992681\pi$
$997$		−	1.45183i		−	0.0459800i	0.999736	−	0.0229900i	$-0.00731859\pi$
$998$		0			0
$999$			32.9380i			1.04211i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2600.2.f.c.649.5		6
5.2	odd	4		2600.2.k.b.2001.5		6
5.3	odd	4		520.2.k.a.441.1	✓	6
5.4	even	2		2600.2.f.d.649.2		6
13.12	even	2		2600.2.f.d.649.5		6
15.8	even	4		4680.2.g.j.2521.5		6
20.3	even	4		1040.2.k.b.961.5		6
65.8	even	4		6760.2.a.v.1.1		3
65.12	odd	4		2600.2.k.b.2001.6		6
65.18	even	4		6760.2.a.u.1.1		3
65.38	odd	4		520.2.k.a.441.2	yes	6
65.64	even	2	inner	2600.2.f.c.649.2		6
195.38	even	4		4680.2.g.j.2521.2		6
260.103	even	4		1040.2.k.b.961.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
520.2.k.a.441.1	✓	6	5.3	odd	4
520.2.k.a.441.2	yes	6	65.38	odd	4
1040.2.k.b.961.5		6	20.3	even	4
1040.2.k.b.961.6		6	260.103	even	4
2600.2.f.c.649.2		6	65.64	even	2	inner
2600.2.f.c.649.5		6	1.1	even	1	trivial
2600.2.f.d.649.2		6	5.4	even	2
2600.2.f.d.649.5		6	13.12	even	2
2600.2.k.b.2001.5		6	5.2	odd	4
2600.2.k.b.2001.6		6	65.12	odd	4
4680.2.g.j.2521.2		6	195.38	even	4
4680.2.g.j.2521.5		6	15.8	even	4
6760.2.a.u.1.1		3	65.18	even	4
6760.2.a.v.1.1		3	65.8	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:	\( N \)	\(=\)	\( 2600 = 2^{3} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2600.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.67513i q^{3} -0.806063 q^{7} +0.193937 q^{9} +3.83146i q^{11} +(-1.48119 - 3.28726i) q^{13} +1.61213i q^{17} +0.869067i q^{19} -1.35026i q^{21} -5.28726i q^{23} +5.35026i q^{27} -8.46898 q^{29} +10.4060i q^{31} -6.41819 q^{33} +6.15633 q^{37} +(5.50659 - 2.48119i) q^{39} +9.66291i q^{41} +0.974607i q^{43} -2.41819 q^{47} -6.35026 q^{49} -2.70052 q^{51} -9.66291i q^{53} -1.45580 q^{57} +2.48119i q^{59} -13.1187 q^{61} -0.156325 q^{63} +0.156325 q^{67} +8.85685 q^{69} -1.64481i q^{71} +2.93207 q^{73} -3.08840i q^{77} +3.53690 q^{79} -8.38058 q^{81} -15.6932 q^{83} -14.1866i q^{87} +2.38787i q^{89} +(1.19394 + 2.64974i) q^{91} -17.4314 q^{93} -7.08840 q^{97} +0.743059i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{7} + 2 q^{9} + 2 q^{13} + 12 q^{29} - 36 q^{33} + 16 q^{37} - 8 q^{39} - 12 q^{47} - 18 q^{49} + 24 q^{51} - 28 q^{57} - 36 q^{61} + 20 q^{63} - 20 q^{67} - 8 q^{69} - 24 q^{79} - 26 q^{81} - 28 q^{83}+ \cdots - 4 q^{97}+O(q^{100}) \) 6 * q - 4 * q^7 + 2 * q^9 + 2 * q^13 + 12 * q^29 - 36 * q^33 + 16 * q^37 - 8 * q^39 - 12 * q^47 - 18 * q^49 + 24 * q^51 - 28 * q^57 - 36 * q^61 + 20 * q^63 - 20 * q^67 - 8 * q^69 - 24 * q^79 - 26 * q^81 - 28 * q^83 + 8 * q^91 - 20 * q^93 - 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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