LMFDB - Embedded newform 1520.2.d.j.609.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(609,1520)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1520.609");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$12.1372611072$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.5161984.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 4x^{3} + 25x^{2} - 20x + 8 $ x^6 - 4x^3 + 25x^2 - 20*x + 8
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 190)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			609.1
Root			$0.432320 + 0.432320i$ of defining polynomial
Character	$\chi$	$=$	1520.609
Dual form			1520.2.d.j.609.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.76156i q^{3} +(-2.19388 + 0.432320i) q^{5} -0.761557i q^{7} -4.62620 q^{9} +O(q^{10})$	$q-2.76156i q^{3} +(-2.19388 + 0.432320i) q^{5} -0.761557i q^{7} -4.62620 q^{9} +0.864641 q^{11} +5.62620i q^{13} +(1.19388 + 6.05852i) q^{15} +3.62620i q^{17} +1.00000 q^{19} -2.10308 q^{21} +8.01395i q^{23} +(4.62620 - 1.89692i) q^{25} +4.49084i q^{27} +7.35548 q^{29} -8.11704 q^{31} -2.38776i q^{33} +(0.329237 + 1.67076i) q^{35} -0.476886i q^{37} +15.5371 q^{39} -2.65847 q^{41} -6.86464i q^{43} +(10.1493 - 2.00000i) q^{45} +1.25240i q^{47} +6.42003 q^{49} +10.0140 q^{51} +2.37380i q^{53} +(-1.89692 + 0.373802i) q^{55} -2.76156i q^{57} +4.49084 q^{59} -10.8646 q^{61} +3.52311i q^{63} +(-2.43232 - 12.3432i) q^{65} +1.03228i q^{67} +22.1310 q^{69} +10.1816 q^{71} +16.4017i q^{73} +(-5.23844 - 12.7755i) q^{75} -0.658473i q^{77} -12.5693 q^{79} -1.47689 q^{81} +0.270718i q^{83} +(-1.56768 - 7.95543i) q^{85} -20.3126i q^{87} -0.387755 q^{89} +4.28467 q^{91} +22.4157i q^{93} +(-2.19388 + 0.432320i) q^{95} +8.50479i q^{97} -4.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{5} - 10 q^{9}+O(q^{10}) $ 6 * q + 2 * q^5 - 10 * q^9	$ 6 q + 2 q^{5} - 10 q^{9} - 8 q^{15} + 6 q^{19} - 20 q^{21} + 10 q^{25} + 16 q^{29} - 8 q^{31} - 8 q^{35} + 20 q^{39} + 4 q^{41} + 18 q^{45} + 6 q^{49} + 12 q^{51} - 4 q^{55} + 4 q^{59} - 60 q^{61} - 12 q^{65} + 44 q^{69} + 16 q^{71} - 44 q^{75} - 34 q^{81} - 12 q^{85} + 28 q^{89} - 12 q^{91} + 2 q^{95} - 24 q^{99}+O(q^{100}) $ 6 * q + 2 * q^5 - 10 * q^9 - 8 * q^15 + 6 * q^19 - 20 * q^21 + 10 * q^25 + 16 * q^29 - 8 * q^31 - 8 * q^35 + 20 * q^39 + 4 * q^41 + 18 * q^45 + 6 * q^49 + 12 * q^51 - 4 * q^55 + 4 * q^59 - 60 * q^61 - 12 * q^65 + 44 * q^69 + 16 * q^71 - 44 * q^75 - 34 * q^81 - 12 * q^85 + 28 * q^89 - 12 * q^91 + 2 * q^95 - 24 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times$.

$n$	$191$	$401$	$1141$	$1217$
$\chi(n)$	$1$	$1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		−	2.76156i		−	1.59439i	−0.603725	−	0.797193i	$-0.706317\pi$
$3$		−	2.76156i		−	1.59439i	0.603725	−	0.797193i	$-0.293683\pi$
$4$		0			0
$5$	−2.19388	+	0.432320i	−0.981132	+	0.193340i
$5$	−2.19388	+	0.432320i	−0.981132	+	0.193340i
$6$		0			0
$7$		−	0.761557i		−	0.287842i	−0.989589	−	0.143921i	$-0.954029\pi$
$7$		−	0.761557i		−	0.287842i	0.989589	−	0.143921i	$-0.0459710\pi$
$8$		0			0
$9$	−4.62620			−1.54207
$10$		0			0
$11$	0.864641			0.260699			0.130350	−	0.991468i	$-0.458390\pi$
$11$	0.864641			0.260699			0.130350	+	0.991468i	$0.458390\pi$
$12$		0			0
$13$			5.62620i			1.56043i	0.625514	+	0.780213i	$0.284889\pi$
$13$			5.62620i			1.56043i	−0.625514	+	0.780213i	$0.715111\pi$
$14$		0			0
$15$	1.19388	+	6.05852i	0.308258	+	1.56430i
$16$		0			0
$17$			3.62620i			0.879482i	0.898125	+	0.439741i	$0.144930\pi$
$17$			3.62620i			0.879482i	−0.898125	+	0.439741i	$0.855070\pi$
$18$		0			0
$19$	1.00000			0.229416
$19$	1.00000			0.229416
$20$		0			0
$21$	−2.10308			−0.458930
$22$		0			0
$23$			8.01395i			1.67102i	0.549472	+	0.835512i	$0.314829\pi$
$23$			8.01395i			1.67102i	−0.549472	+	0.835512i	$0.685171\pi$
$24$		0			0
$25$	4.62620	−	1.89692i	0.925240	−	0.379383i
$26$		0			0
$27$			4.49084i			0.864262i
$28$		0			0
$29$	7.35548			1.36588			0.682939	−	0.730475i	$-0.260701\pi$
$29$	7.35548			1.36588			0.682939	+	0.730475i	$0.260701\pi$
$30$		0			0
$31$	−8.11704			−1.45786			−0.728931	−	0.684587i	$-0.759983\pi$
$31$	−8.11704			−1.45786			−0.728931	+	0.684587i	$0.759983\pi$
$32$		0			0
$33$		−	2.38776i		−	0.415655i
$34$		0			0
$35$	0.329237	+	1.67076i	0.0556512	+	0.282411i
$36$		0			0
$37$		−	0.476886i		−	0.0783995i	−0.999231	−	0.0391998i	$-0.987519\pi$
$37$		−	0.476886i		−	0.0783995i	0.999231	−	0.0391998i	$-0.0124809\pi$
$38$		0			0
$39$	15.5371			2.48792
$40$		0			0
$41$	−2.65847			−0.415184			−0.207592	−	0.978216i	$-0.566563\pi$
$41$	−2.65847			−0.415184			−0.207592	+	0.978216i	$0.566563\pi$
$42$		0			0
$43$		−	6.86464i		−	1.04685i	−0.852072	−	0.523424i	$-0.824654\pi$
$43$		−	6.86464i		−	1.04685i	0.852072	−	0.523424i	$-0.175346\pi$
$44$		0			0
$45$	10.1493	−	2.00000i	1.51297	−	0.298142i
$46$		0			0
$47$			1.25240i			0.182681i	0.995820	+	0.0913404i	$0.0291151\pi$
$47$			1.25240i			0.182681i	−0.995820	+	0.0913404i	$0.970885\pi$
$48$		0			0
$49$	6.42003			0.917147
$50$		0			0
$51$	10.0140			1.40223
$52$		0			0
$53$			2.37380i			0.326067i	0.986621	+	0.163033i	$0.0521278\pi$
$53$			2.37380i			0.326067i	−0.986621	+	0.163033i	$0.947872\pi$
$54$		0			0
$55$	−1.89692	+	0.373802i	−0.255780	+	0.0504034i
$56$		0			0
$57$		−	2.76156i		−	0.365777i
$58$		0			0
$59$	4.49084			0.584657			0.292329	−	0.956318i	$-0.405570\pi$
$59$	4.49084			0.584657			0.292329	+	0.956318i	$0.405570\pi$
$60$		0			0
$61$	−10.8646			−1.39107			−0.695537	−	0.718490i	$-0.744834\pi$
$61$	−10.8646			−1.39107			−0.695537	+	0.718490i	$0.744834\pi$
$62$		0			0
$63$			3.52311i			0.443871i
$64$		0			0
$65$	−2.43232	−	12.3432i	−0.301692	−	1.53098i
$66$		0			0
$67$			1.03228i			0.126113i	0.998010	+	0.0630563i	$0.0200848\pi$
$67$			1.03228i			0.126113i	−0.998010	+	0.0630563i	$0.979915\pi$
$68$		0			0
$69$	22.1310			2.66426
$70$		0			0
$71$	10.1816			1.20833			0.604166	−	0.796858i	$-0.293506\pi$
$71$	10.1816			1.20833			0.604166	+	0.796858i	$0.293506\pi$
$72$		0			0
$73$			16.4017i			1.91967i	0.280557	+	0.959837i	$0.409481\pi$
$73$			16.4017i			1.91967i	−0.280557	+	0.959837i	$0.590519\pi$
$74$		0			0
$75$	−5.23844	−	12.7755i	−0.604883	−	1.47519i
$76$		0			0
$77$		−	0.658473i		−	0.0750400i
$78$		0			0
$79$	−12.5693			−1.41416			−0.707081	−	0.707133i	$-0.749988\pi$
$79$	−12.5693			−1.41416			−0.707081	+	0.707133i	$0.749988\pi$
$80$		0			0
$81$	−1.47689			−0.164098
$82$		0			0
$83$			0.270718i			0.0297152i	0.999890	+	0.0148576i	$0.00472949\pi$
$83$			0.270718i			0.0297152i	−0.999890	+	0.0148576i	$0.995271\pi$
$84$		0			0
$85$	−1.56768	−	7.95543i	−0.170039	−	0.862888i
$86$		0			0
$87$		−	20.3126i		−	2.17774i
$88$		0			0
$89$	−0.387755			−0.0411020			−0.0205510	−	0.999789i	$-0.506542\pi$
$89$	−0.387755			−0.0411020			−0.0205510	+	0.999789i	$0.506542\pi$
$90$		0			0
$91$	4.28467			0.449156
$92$		0			0
$93$			22.4157i			2.32440i
$94$		0			0
$95$	−2.19388	+	0.432320i	−0.225087	+	0.0443551i
$96$		0			0
$97$			8.50479i			0.863531i	0.901986	+	0.431765i	$0.142109\pi$
$97$			8.50479i			0.863531i	−0.901986	+	0.431765i	$0.857891\pi$
$98$		0			0
$99$	−4.00000			−0.402015
$100$		0			0
$101$	16.4157			1.63342			0.816710	−	0.577049i	$-0.195796\pi$
$101$	16.4157			1.63342			0.816710	+	0.577049i	$0.195796\pi$
$102$		0			0
$103$			9.64015i			0.949872i	0.880020	+	0.474936i	$0.157529\pi$
$103$			9.64015i			0.949872i	−0.880020	+	0.474936i	$0.842471\pi$
$104$		0			0
$105$	4.61391	−	0.909206i	0.450271	−	0.0887294i
$106$		0			0
$107$			4.28467i			0.414215i	0.978318	+	0.207107i	$0.0664050\pi$
$107$			4.28467i			0.414215i	−0.978318	+	0.207107i	$0.933595\pi$
$108$		0			0
$109$	13.4200			1.28541			0.642703	−	0.766116i	$-0.277813\pi$
$109$	13.4200			1.28541			0.642703	+	0.766116i	$0.277813\pi$
$110$		0			0
$111$	−1.31695			−0.124999
$112$		0			0
$113$		−	10.3232i		−	0.971125i	−0.874202	−	0.485563i	$-0.838615\pi$
$113$		−	10.3232i		−	0.971125i	0.874202	−	0.485563i	$-0.161385\pi$
$114$		0			0
$115$	−3.46460	−	17.5816i	−0.323075	−	1.63950i
$116$		0			0
$117$		−	26.0279i		−	2.40628i
$118$		0			0
$119$	2.76156			0.253152
$120$		0			0
$121$	−10.2524			−0.932036
$122$		0			0
$123$			7.34153i			0.661963i
$124$		0			0
$125$	−9.32924	+	6.16160i	−0.834432	+	0.551110i
$126$		0			0
$127$			16.9817i			1.50688i	0.657517	+	0.753440i	$0.271607\pi$
$127$			16.9817i			1.50688i	−0.657517	+	0.753440i	$0.728393\pi$
$128$		0			0
$129$	−18.9571			−1.66908
$130$		0			0
$131$	−0.541436			−0.0473055			−0.0236528	−	0.999720i	$-0.507530\pi$
$131$	−0.541436			−0.0473055			−0.0236528	+	0.999720i	$0.507530\pi$
$132$		0			0
$133$		−	0.761557i		−	0.0660354i
$134$		0			0
$135$	−1.94148	−	9.85235i	−0.167096	−	0.847955i
$136$		0			0
$137$		−	2.87859i		−	0.245935i	−0.992411	−	0.122967i	$-0.960759\pi$
$137$		−	2.87859i		−	0.245935i	0.992411	−	0.122967i	$-0.0392411\pi$
$138$		0			0
$139$	3.58767			0.304302			0.152151	−	0.988357i	$-0.451380\pi$
$139$	3.58767			0.304302			0.152151	+	0.988357i	$0.451380\pi$
$140$		0			0
$141$	3.45856			0.291264
$142$		0			0
$143$			4.86464i			0.406802i
$144$		0			0
$145$	−16.1370	+	3.17992i	−1.34011	+	0.264078i
$146$		0			0
$147$		−	17.7293i		−	1.46229i
$148$		0			0
$149$	16.8401			1.37959			0.689796	−	0.724004i	$-0.257700\pi$
$149$	16.8401			1.37959			0.689796	+	0.724004i	$0.257700\pi$
$150$		0			0
$151$	16.9817			1.38195			0.690975	−	0.722879i	$-0.257182\pi$
$151$	16.9817			1.38195			0.690975	+	0.722879i	$0.257182\pi$
$152$		0			0
$153$		−	16.7755i		−	1.35622i
$154$		0			0
$155$	17.8078	−	3.50916i	1.43036	−	0.281863i
$156$		0			0
$157$		−	14.8401i		−	1.18437i	−0.805804	−	0.592183i	$-0.798266\pi$
$157$		−	14.8401i		−	1.18437i	0.805804	−	0.592183i	$-0.201734\pi$
$158$		0			0
$159$	6.55539			0.519876
$160$		0			0
$161$	6.10308			0.480990
$162$		0			0
$163$		−	13.3694i		−	1.04717i	−0.851972	−	0.523587i	$-0.824593\pi$
$163$		−	13.3694i		−	1.04717i	0.851972	−	0.523587i	$-0.175407\pi$
$164$		0			0
$165$	1.03228	+	5.23844i	0.0803625	+	0.407812i
$166$		0			0
$167$			9.84632i			0.761931i	0.924589	+	0.380966i	$0.124408\pi$
$167$			9.84632i			0.761931i	−0.924589	+	0.380966i	$0.875592\pi$
$168$		0			0
$169$	−18.6541			−1.43493
$170$		0			0
$171$	−4.62620			−0.353774
$172$		0			0
$173$			2.98168i			0.226693i	0.993556	+	0.113346i	$0.0361570\pi$
$173$			2.98168i			0.226693i	−0.993556	+	0.113346i	$0.963843\pi$
$174$		0			0
$175$	−1.44461	−	3.52311i	−0.109202	−	0.266322i
$176$		0			0
$177$		−	12.4017i		−	0.932169i
$178$		0			0
$179$	11.7938			0.881512			0.440756	−	0.897627i	$-0.354710\pi$
$179$	11.7938			0.881512			0.440756	+	0.897627i	$0.354710\pi$
$180$		0			0
$181$	14.5693			1.08293			0.541465	−	0.840723i	$-0.317870\pi$
$181$	14.5693			1.08293			0.541465	+	0.840723i	$0.317870\pi$
$182$		0			0
$183$			30.0033i			2.21791i
$184$		0			0
$185$	0.206167	+	1.04623i	0.0151577	+	0.0769203i
$186$		0			0
$187$			3.13536i			0.229280i
$188$		0			0
$189$	3.42003			0.248771
$190$		0			0
$191$	−13.2384			−0.957900			−0.478950	−	0.877842i	$-0.658983\pi$
$191$	−13.2384			−0.957900			−0.478950	+	0.877842i	$0.658983\pi$
$192$		0			0
$193$			2.54144i			0.182937i	0.995808	+	0.0914683i	$0.0291560\pi$
$193$			2.54144i			0.182937i	−0.995808	+	0.0914683i	$0.970844\pi$
$194$		0			0
$195$	−34.0864	+	6.71699i	−2.44098	+	0.481014i
$196$		0			0
$197$			19.9109i			1.41859i	0.704911	+	0.709295i	$0.250987\pi$
$197$			19.9109i			1.41859i	−0.704911	+	0.709295i	$0.749013\pi$
$198$		0			0
$199$	−20.3126			−1.43992			−0.719960	−	0.694015i	$-0.755840\pi$
$199$	−20.3126			−1.43992			−0.719960	+	0.694015i	$0.755840\pi$
$200$		0			0
$201$	2.85069			0.201072
$202$		0			0
$203$		−	5.60162i		−	0.393157i
$204$		0			0
$205$	5.83237	−	1.14931i	0.407350	−	0.0802715i
$206$		0			0
$207$		−	37.0741i		−	2.57683i
$208$		0			0
$209$	0.864641			0.0598085
$210$		0			0
$211$	−18.0419			−1.24205			−0.621026	−	0.783790i	$-0.713284\pi$
$211$	−18.0419			−1.24205			−0.621026	+	0.783790i	$0.713284\pi$
$212$		0			0
$213$		−	28.1170i		−	1.92655i
$214$		0			0
$215$	2.96772	+	15.0602i	0.202397	+	1.02710i
$216$		0			0
$217$			6.18159i			0.419634i
$218$		0			0
$219$	45.2943			3.06070
$220$		0			0
$221$	−20.4017			−1.37237
$222$		0			0
$223$			13.5231i			0.905575i	0.891619	+	0.452787i	$0.149570\pi$
$223$			13.5231i			0.905575i	−0.891619	+	0.452787i	$0.850430\pi$
$224$		0			0
$225$	−21.4017	+	8.77551i	−1.42678	+	0.585034i
$226$		0			0
$227$			13.6016i			0.902771i	0.892329	+	0.451386i	$0.149070\pi$
$227$			13.6016i			0.902771i	−0.892329	+	0.451386i	$0.850930\pi$
$228$		0			0
$229$	−13.5877			−0.897898			−0.448949	−	0.893557i	$-0.648202\pi$
$229$	−13.5877			−0.897898			−0.448949	+	0.893557i	$0.648202\pi$
$230$		0			0
$231$	−1.81841			−0.119643
$232$		0			0
$233$			25.5510i			1.67390i	0.547277	+	0.836952i	$0.315664\pi$
$233$			25.5510i			1.67390i	−0.547277	+	0.836952i	$0.684336\pi$
$234$		0			0
$235$	−0.541436	−	2.74760i	−0.0353194	−	0.179234i
$236$		0			0
$237$			34.7110i			2.25472i
$238$		0			0
$239$	−11.3309			−0.732935			−0.366468	−	0.930431i	$-0.619433\pi$
$239$	−11.3309			−0.732935			−0.366468	+	0.930431i	$0.619433\pi$
$240$		0			0
$241$	−1.25240			−0.0806739			−0.0403370	−	0.999186i	$-0.512843\pi$
$241$	−1.25240			−0.0806739			−0.0403370	+	0.999186i	$0.512843\pi$
$242$		0			0
$243$			17.5510i			1.12590i
$244$		0			0
$245$	−14.0848	+	2.77551i	−0.899842	+	0.177321i
$246$		0			0
$247$			5.62620i			0.357986i
$248$		0			0
$249$	0.747604			0.0473775
$250$		0			0
$251$	−10.5939			−0.668682			−0.334341	−	0.942452i	$-0.608514\pi$
$251$	−10.5939			−0.668682			−0.334341	+	0.942452i	$0.608514\pi$
$252$		0			0
$253$			6.92919i			0.435635i
$254$		0			0
$255$	−21.9694	+	4.32924i	−1.37578	+	0.271107i
$256$		0			0
$257$			0.153681i			0.00958637i	0.999989	+	0.00479319i	$0.00152572\pi$
$257$			0.153681i			0.00958637i	−0.999989	+	0.00479319i	$0.998474\pi$
$258$		0			0
$259$	−0.363176			−0.0225666
$260$		0			0
$261$	−34.0279			−2.10627
$262$		0			0
$263$		−	0.504792i		−	0.0311268i	−0.999879	−	0.0155634i	$-0.995046\pi$
$263$		−	0.504792i		−	0.0311268i	0.999879	−	0.0155634i	$-0.00495419\pi$
$264$		0			0
$265$	−1.02624	−	5.20783i	−0.0630416	−	0.319915i
$266$		0			0
$267$			1.07081i			0.0655324i
$268$		0			0
$269$	3.49521			0.213107			0.106553	−	0.994307i	$-0.466019\pi$
$269$	3.49521			0.213107			0.106553	+	0.994307i	$0.466019\pi$
$270$		0			0
$271$	−5.47252			−0.332432			−0.166216	−	0.986089i	$-0.553155\pi$
$271$	−5.47252			−0.332432			−0.166216	+	0.986089i	$0.553155\pi$
$272$		0			0
$273$		−	11.8324i		−	0.716127i
$274$		0			0
$275$	4.00000	−	1.64015i	0.241209	−	0.0989049i
$276$		0			0
$277$		−	12.9538i		−	0.778317i	−0.921171	−	0.389158i	$-0.872766\pi$
$277$		−	12.9538i		−	0.778317i	0.921171	−	0.389158i	$-0.127234\pi$
$278$		0			0
$279$	37.5510			2.24812
$280$		0			0
$281$	0.153681			0.00916785			0.00458393	−	0.999989i	$-0.498541\pi$
$281$	0.153681			0.00916785			0.00458393	+	0.999989i	$0.498541\pi$
$282$		0			0
$283$			18.2341i			1.08390i	0.840410	+	0.541952i	$0.182314\pi$
$283$			18.2341i			1.08390i	−0.840410	+	0.541952i	$0.817686\pi$
$284$		0			0
$285$	1.19388	+	6.05852i	0.0707192	+	0.358876i
$286$		0			0
$287$			2.02458i			0.119507i
$288$		0			0
$289$	3.85069			0.226511
$290$		0			0
$291$	23.4865			1.37680
$292$		0			0
$293$			2.03853i			0.119092i	0.998226	+	0.0595462i	$0.0189654\pi$
$293$			2.03853i			0.119092i	−0.998226	+	0.0595462i	$0.981035\pi$
$294$		0			0
$295$	−9.85235	+	1.94148i	−0.573626	+	0.113037i
$296$		0			0
$297$			3.88296i			0.225312i
$298$		0			0
$299$	−45.0881			−2.60751
$300$		0			0
$301$	−5.22782			−0.301326
$302$		0			0
$303$		−	45.3328i		−	2.60430i
$304$		0			0
$305$	23.8357	−	4.69701i	1.36483	−	0.268950i
$306$		0			0
$307$			16.5414i			0.944070i	0.881580	+	0.472035i	$0.156480\pi$
$307$			16.5414i			0.944070i	−0.881580	+	0.472035i	$0.843520\pi$
$308$		0			0
$309$	26.6218			1.51446
$310$		0			0
$311$	21.4725			1.21759			0.608797	−	0.793326i	$-0.291652\pi$
$311$	21.4725			1.21759			0.608797	+	0.793326i	$0.291652\pi$
$312$		0			0
$313$		−	1.12141i		−	0.0633856i	−0.999498	−	0.0316928i	$-0.989910\pi$
$313$		−	1.12141i		−	0.0633856i	0.999498	−	0.0316928i	$-0.0100898\pi$
$314$		0			0
$315$	−1.52311	−	7.72928i	−0.0858178	−	0.435496i
$316$		0			0
$317$			29.8882i			1.67869i	0.543601	+	0.839344i	$0.317060\pi$
$317$			29.8882i			1.67869i	−0.543601	+	0.839344i	$0.682940\pi$
$318$		0			0
$319$	6.35985			0.356083
$320$		0			0
$321$	11.8324			0.660418
$322$		0			0
$323$			3.62620i			0.201767i
$324$		0			0
$325$	10.6724	+	26.0279i	0.592000	+	1.44377i
$326$		0			0
$327$		−	37.0602i		−	2.04943i
$328$		0			0
$329$	0.953771			0.0525831
$330$		0			0
$331$	−32.3126			−1.77606			−0.888030	−	0.459786i	$-0.847926\pi$
$331$	−32.3126			−1.77606			−0.888030	+	0.459786i	$0.847926\pi$
$332$		0			0
$333$			2.20617i			0.120897i
$334$		0			0
$335$	−0.446274	−	2.26469i	−0.0243825	−	0.123733i
$336$		0			0
$337$		−	26.3511i		−	1.43544i	−0.696334	−	0.717718i	$-0.745187\pi$
$337$		−	26.3511i		−	1.43544i	0.696334	−	0.717718i	$-0.254813\pi$
$338$		0			0
$339$	−28.5081			−1.54835
$340$		0			0
$341$	−7.01832			−0.380063
$342$		0			0
$343$		−	10.2201i		−	0.551835i
$344$		0			0
$345$	−48.5527	+	9.56768i	−2.61399	+	0.515107i
$346$		0			0
$347$		−	2.77551i		−	0.148997i	−0.997221	−	0.0744986i	$-0.976264\pi$
$347$		−	2.77551i		−	0.148997i	0.997221	−	0.0744986i	$-0.0237356\pi$
$348$		0			0
$349$	−11.5510			−0.618312			−0.309156	−	0.951011i	$-0.600047\pi$
$349$	−11.5510			−0.618312			−0.309156	+	0.951011i	$0.600047\pi$
$350$		0			0
$351$	−25.2663			−1.34862
$352$		0			0
$353$		−	8.40171i		−	0.447178i	−0.974684	−	0.223589i	$-0.928223\pi$
$353$		−	8.40171i		−	0.447178i	0.974684	−	0.223589i	$-0.0717773\pi$
$354$		0			0
$355$	−22.3372	+	4.40171i	−1.18553	+	0.233618i
$356$		0			0
$357$		−	7.62620i		−	0.403621i
$358$		0			0
$359$	22.7895			1.20278			0.601391	−	0.798955i	$-0.294613\pi$
$359$	22.7895			1.20278			0.601391	+	0.798955i	$0.294613\pi$
$360$		0			0
$361$	1.00000			0.0526316
$362$		0			0
$363$			28.3126i			1.48602i
$364$		0			0
$365$	−7.09079	−	35.9833i	−0.371149	−	1.88345i
$366$		0			0
$367$			4.06455i			0.212168i	0.994357	+	0.106084i	$0.0338312\pi$
$367$			4.06455i			0.212168i	−0.994357	+	0.106084i	$0.966169\pi$
$368$		0			0
$369$	12.2986			0.640241
$370$		0			0
$371$	1.80779			0.0938556
$372$		0			0
$373$		−	18.4017i		−	0.952804i	−0.879227	−	0.476402i	$-0.841941\pi$
$373$		−	18.4017i		−	0.952804i	0.879227	−	0.476402i	$-0.158059\pi$
$374$		0			0
$375$	17.0156	+	25.7632i	0.878683	+	1.33041i
$376$		0			0
$377$			41.3834i			2.13135i
$378$		0			0
$379$	1.23844			0.0636145			0.0318073	−	0.999494i	$-0.489874\pi$
$379$	1.23844			0.0636145			0.0318073	+	0.999494i	$0.489874\pi$
$380$		0			0
$381$	46.8959			2.40255
$382$		0			0
$383$		−	16.8646i		−	0.861743i	−0.902413	−	0.430871i	$-0.858206\pi$
$383$		−	16.8646i		−	0.861743i	0.902413	−	0.430871i	$-0.141794\pi$
$384$		0			0
$385$	0.284672	+	1.44461i	0.0145082	+	0.0736242i
$386$		0			0
$387$			31.7572i			1.61431i
$388$		0			0
$389$	−8.59392			−0.435729			−0.217865	−	0.975979i	$-0.569909\pi$
$389$	−8.59392			−0.435729			−0.217865	+	0.975979i	$0.569909\pi$
$390$		0			0
$391$	−29.0602			−1.46964
$392$		0			0
$393$			1.49521i			0.0754233i
$394$		0			0
$395$	27.5756	−	5.43398i	1.38748	−	0.273413i
$396$		0			0
$397$		−	16.0558i		−	0.805818i	−0.915240	−	0.402909i	$-0.867999\pi$
$397$		−	16.0558i		−	0.805818i	0.915240	−	0.402909i	$-0.132001\pi$
$398$		0			0
$399$	−2.10308			−0.105286
$400$		0			0
$401$	−14.8925			−0.743698			−0.371849	−	0.928293i	$-0.621276\pi$
$401$	−14.8925			−0.743698			−0.371849	+	0.928293i	$0.621276\pi$
$402$		0			0
$403$		−	45.6681i		−	2.27489i
$404$		0			0
$405$	3.24011	−	0.638488i	0.161002	−	0.0317267i
$406$		0			0
$407$		−	0.412335i		−	0.0204387i
$408$		0			0
$409$	18.3511			0.907404			0.453702	−	0.891153i	$-0.350103\pi$
$409$	18.3511			0.907404			0.453702	+	0.891153i	$0.350103\pi$
$410$		0			0
$411$	−7.94940			−0.392115
$412$		0			0
$413$		−	3.42003i		−	0.168289i
$414$		0			0
$415$	−0.117037	−	0.593923i	−0.00574512	−	0.0291545i
$416$		0			0
$417$		−	9.90754i		−	0.485174i
$418$		0			0
$419$	34.7509			1.69769			0.848847	−	0.528639i	$-0.177297\pi$
$419$	34.7509			1.69769			0.848847	+	0.528639i	$0.177297\pi$
$420$		0			0
$421$	−40.1589			−1.95722			−0.978612	−	0.205713i	$-0.934049\pi$
$421$	−40.1589			−1.95722			−0.978612	+	0.205713i	$0.934049\pi$
$422$		0			0
$423$		−	5.79383i		−	0.281706i
$424$		0			0
$425$	6.87859	+	16.7755i	0.333661	+	0.813732i
$426$		0			0
$427$			8.27405i			0.400409i
$428$		0			0
$429$	13.4340			0.648599
$430$		0			0
$431$	−34.9571			−1.68382			−0.841912	−	0.539615i	$-0.818570\pi$
$431$	−34.9571			−1.68382			−0.841912	+	0.539615i	$0.818570\pi$
$432$		0			0
$433$			1.13536i			0.0545619i	0.999628	+	0.0272809i	$0.00868487\pi$
$433$			1.13536i			0.0545619i	−0.999628	+	0.0272809i	$0.991315\pi$
$434$		0			0
$435$	8.78154	+	44.5633i	0.421043	+	2.13665i
$436$		0			0
$437$			8.01395i			0.383359i
$438$		0			0
$439$	−6.80009			−0.324551			−0.162275	−	0.986746i	$-0.551883\pi$
$439$	−6.80009			−0.324551			−0.162275	+	0.986746i	$0.551883\pi$
$440$		0			0
$441$	−29.7003			−1.41430
$442$		0			0
$443$		−	38.0679i		−	1.80866i	−0.426835	−	0.904330i	$-0.640371\pi$
$443$		−	38.0679i		−	1.80866i	0.426835	−	0.904330i	$-0.359629\pi$
$444$		0			0
$445$	0.850688	−	0.167635i	0.0403265	−	0.00794664i
$446$		0			0
$447$		−	46.5048i		−	2.19960i
$448$		0			0
$449$	18.5414			0.875024			0.437512	−	0.899212i	$-0.355860\pi$
$449$	18.5414			0.875024			0.437512	+	0.899212i	$0.355860\pi$
$450$		0			0
$451$	−2.29862			−0.108238
$452$		0			0
$453$		−	46.8959i		−	2.20336i
$454$		0			0
$455$	−9.40005	+	1.85235i	−0.440681	+	0.0868396i
$456$		0			0
$457$		−	16.3738i		−	0.765934i	−0.923762	−	0.382967i	$-0.874902\pi$
$457$		−	16.3738i		−	0.765934i	0.923762	−	0.382967i	$-0.125098\pi$
$458$		0			0
$459$	−16.2847			−0.760103
$460$		0			0
$461$	1.70470			0.0793959			0.0396979	−	0.999212i	$-0.487360\pi$
$461$	1.70470			0.0793959			0.0396979	+	0.999212i	$0.487360\pi$
$462$		0			0
$463$			10.0279i			0.466036i	0.972472	+	0.233018i	$0.0748602\pi$
$463$			10.0279i			0.466036i	−0.972472	+	0.233018i	$0.925140\pi$
$464$		0			0
$465$	−9.69075	−	49.1772i	−0.449398	−	2.28054i
$466$		0			0
$467$			32.7509i			1.51553i	0.652526	+	0.757766i	$0.273709\pi$
$467$			32.7509i			1.51553i	−0.652526	+	0.757766i	$0.726291\pi$
$468$		0			0
$469$	0.786137			0.0363004
$470$		0			0
$471$	−40.9817			−1.88834
$472$		0			0
$473$		−	5.93545i		−	0.272912i
$474$		0			0
$475$	4.62620	−	1.89692i	0.212265	−	0.0870365i
$476$		0			0
$477$		−	10.9817i		−	0.502816i
$478$		0			0
$479$	27.2803			1.24647			0.623234	−	0.782035i	$-0.285818\pi$
$479$	27.2803			1.24647			0.623234	+	0.782035i	$0.285818\pi$
$480$		0			0
$481$	2.68305			0.122337
$482$		0			0
$483$		−	16.8540i		−	0.766884i
$484$		0			0
$485$	−3.67680	−	18.6585i	−0.166955	−	0.847238i
$486$		0			0
$487$		−	11.0741i		−	0.501817i	−0.968011	−	0.250908i	$-0.919271\pi$
$487$		−	11.0741i		−	0.501817i	0.968011	−	0.250908i	$-0.0807293\pi$
$488$		0			0
$489$	−36.9205			−1.66960
$490$		0			0
$491$	−8.11704			−0.366317			−0.183158	−	0.983083i	$-0.558632\pi$
$491$	−8.11704			−0.366317			−0.183158	+	0.983083i	$0.558632\pi$
$492$		0			0
$493$			26.6724i			1.20127i
$494$		0			0
$495$	8.77551	−	1.72928i	0.394430	−	0.0777254i
$496$		0			0
$497$		−	7.75386i		−	0.347808i
$498$		0			0
$499$	0.295298			0.0132193			0.00660967	−	0.999978i	$-0.497896\pi$
$499$	0.295298			0.0132193			0.00660967	+	0.999978i	$0.497896\pi$
$500$		0			0
$501$	27.1912			1.21481
$502$		0			0
$503$		−	19.6016i		−	0.873993i	−0.899463	−	0.436996i	$-0.856042\pi$
$503$		−	19.6016i		−	0.873993i	0.899463	−	0.436996i	$-0.143958\pi$
$504$		0			0
$505$	−36.0140	+	7.09683i	−1.60260	+	0.315805i
$506$		0			0
$507$			51.5144i			2.28783i
$508$		0			0
$509$	1.79383			0.0795102			0.0397551	−	0.999209i	$-0.487342\pi$
$509$	1.79383			0.0795102			0.0397551	+	0.999209i	$0.487342\pi$
$510$		0			0
$511$	12.4908			0.552562
$512$		0			0
$513$			4.49084i			0.198275i
$514$		0			0
$515$	−4.16763	−	21.1493i	−0.183648	−	0.931950i
$516$		0			0
$517$			1.08287i			0.0476247i
$518$		0			0
$519$	8.23407			0.361436
$520$		0			0
$521$	−2.61850			−0.114719			−0.0573593	−	0.998354i	$-0.518268\pi$
$521$	−2.61850			−0.114719			−0.0573593	+	0.998354i	$0.518268\pi$
$522$		0			0
$523$		−	14.9956i		−	0.655713i	−0.944728	−	0.327857i	$-0.893674\pi$
$523$		−	14.9956i		−	0.655713i	0.944728	−	0.327857i	$-0.106326\pi$
$524$		0			0
$525$	−9.72928	+	3.98937i	−0.424621	+	0.174111i
$526$		0			0
$527$		−	29.4340i		−	1.28216i
$528$		0			0
$529$	−41.2234			−1.79232
$530$		0			0
$531$	−20.7755			−0.901580
$532$		0			0
$533$		−	14.9571i		−	0.647864i
$534$		0			0
$535$	−1.85235	−	9.40005i	−0.0800841	−	0.406399i
$536$		0			0
$537$		−	32.5693i		−	1.40547i
$538$		0			0
$539$	5.55102			0.239099
$540$		0			0
$541$	3.40608			0.146439			0.0732194	−	0.997316i	$-0.476673\pi$
$541$	3.40608			0.146439			0.0732194	+	0.997316i	$0.476673\pi$
$542$		0			0
$543$		−	40.2341i		−	1.72661i
$544$		0			0
$545$	−29.4419	+	5.80175i	−1.26115	+	0.248520i
$546$		0			0
$547$			4.74760i			0.202993i	0.994836	+	0.101496i	$0.0323630\pi$
$547$			4.74760i			0.202993i	−0.994836	+	0.101496i	$0.967637\pi$
$548$		0			0
$549$	50.2620			2.14513
$550$		0			0
$551$	7.35548			0.313354
$552$		0			0
$553$			9.57227i			0.407054i
$554$		0			0
$555$	2.88922	−	0.569343i	0.122641	−	0.0241673i
$556$		0			0
$557$		−	43.0462i		−	1.82393i	−0.410271	−	0.911964i	$-0.634566\pi$
$557$		−	43.0462i		−	1.82393i	0.410271	−	0.911964i	$-0.365434\pi$
$558$		0			0
$559$	38.6218			1.63353
$560$		0			0
$561$	8.65847			0.365561
$562$		0			0
$563$		−	17.0096i		−	0.716869i	−0.933555	−	0.358434i	$-0.883311\pi$
$563$		−	17.0096i		−	0.716869i	0.933555	−	0.358434i	$-0.116689\pi$
$564$		0			0
$565$	4.46293	+	22.6478i	0.187757	+	0.952802i
$566$		0			0
$567$			1.12473i			0.0472343i
$568$		0			0
$569$	19.7572			0.828264			0.414132	−	0.910217i	$-0.364085\pi$
$569$	19.7572			0.828264			0.414132	+	0.910217i	$0.364085\pi$
$570$		0			0
$571$	11.3973			0.476964			0.238482	−	0.971147i	$-0.423350\pi$
$571$	11.3973			0.476964			0.238482	+	0.971147i	$0.423350\pi$
$572$		0			0
$573$			36.5587i			1.52726i
$574$		0			0
$575$	15.2018	+	37.0741i	0.633959	+	1.54610i
$576$		0			0
$577$			18.3372i			0.763386i	0.924289	+	0.381693i	$0.124659\pi$
$577$			18.3372i			0.763386i	−0.924289	+	0.381693i	$0.875341\pi$
$578$		0			0
$579$	7.01832			0.291672
$580$		0			0
$581$	0.206167			0.00855327
$582$		0			0
$583$			2.05249i			0.0850053i
$584$		0			0
$585$	11.2524	+	57.1020i	0.465229	+	2.36088i
$586$		0			0
$587$		−	11.9475i		−	0.493127i	−0.969127	−	0.246563i	$-0.920699\pi$
$587$		−	11.9475i		−	0.493127i	0.969127	−	0.246563i	$-0.0793013\pi$
$588$		0			0
$589$	−8.11704			−0.334457
$590$		0			0
$591$	54.9850			2.26178
$592$		0			0
$593$		−	24.3911i		−	1.00162i	−0.865557	−	0.500811i	$-0.833035\pi$
$593$		−	24.3911i		−	1.00162i	0.865557	−	0.500811i	$-0.166965\pi$
$594$		0			0
$595$	−6.05852	+	1.19388i	−0.248375	+	0.0489442i
$596$		0			0
$597$			56.0943i			2.29579i
$598$		0			0
$599$	−21.0708			−0.860930			−0.430465	−	0.902607i	$-0.641650\pi$
$599$	−21.0708			−0.860930			−0.430465	+	0.902607i	$0.641650\pi$
$600$		0			0
$601$	32.3878			1.32112			0.660562	−	0.750771i	$-0.270318\pi$
$601$	32.3878			1.32112			0.660562	+	0.750771i	$0.270318\pi$
$602$		0			0
$603$		−	4.77551i		−	0.194474i
$604$		0			0
$605$	22.4925	−	4.43232i	0.914450	−	0.180199i
$606$		0			0
$607$			19.0183i			0.771930i	0.922513	+	0.385965i	$0.126131\pi$
$607$			19.0183i			0.771930i	−0.922513	+	0.385965i	$0.873869\pi$
$608$		0			0
$609$	−15.4692			−0.626843
$610$		0			0
$611$	−7.04623			−0.285060
$612$		0			0
$613$		−	23.5756i		−	0.952210i	−0.879389	−	0.476105i	$-0.842048\pi$
$613$		−	23.5756i		−	0.952210i	0.879389	−	0.476105i	$-0.157952\pi$
$614$		0			0
$615$	−3.17389	−	16.1064i	−0.127984	−	0.649473i
$616$		0			0
$617$			26.2707i			1.05762i	0.848740	+	0.528810i	$0.177361\pi$
$617$			26.2707i			1.05762i	−0.848740	+	0.528810i	$0.822639\pi$
$618$		0			0
$619$	11.4985			0.462165			0.231083	−	0.972934i	$-0.425773\pi$
$619$	11.4985			0.462165			0.231083	+	0.972934i	$0.425773\pi$
$620$		0			0
$621$	−35.9894			−1.44420
$622$		0			0
$623$			0.295298i			0.0118309i
$624$		0			0
$625$	17.8034	−	17.5510i	0.712137	−	0.702041i
$626$		0			0
$627$		−	2.38776i		−	0.0953578i
$628$		0			0
$629$	1.72928			0.0689510
$630$		0			0
$631$	45.8130			1.82379			0.911893	−	0.410427i	$-0.134620\pi$
$631$	45.8130			1.82379			0.911893	+	0.410427i	$0.134620\pi$
$632$		0			0
$633$			49.8236i			1.98031i
$634$		0			0
$635$	−7.34153	−	37.2557i	−0.291340	−	1.47845i
$636$		0			0
$637$			36.1204i			1.43114i
$638$		0			0
$639$	−47.1020			−1.86333
$640$		0			0
$641$	1.36943			0.0540894			0.0270447	−	0.999634i	$-0.491390\pi$
$641$	1.36943			0.0540894			0.0270447	+	0.999634i	$0.491390\pi$
$642$		0			0
$643$			24.7389i			0.975606i	0.872954	+	0.487803i	$0.162201\pi$
$643$			24.7389i			0.975606i	−0.872954	+	0.487803i	$0.837799\pi$
$644$		0			0
$645$	41.5896	−	8.19554i	1.63759	−	0.322699i
$646$		0			0
$647$		−	6.82611i		−	0.268362i	−0.990957	−	0.134181i	$-0.957160\pi$
$647$		−	6.82611i		−	0.268362i	0.990957	−	0.134181i	$-0.0428404\pi$
$648$		0			0
$649$	3.88296			0.152420
$650$		0			0
$651$	17.0708			0.669058
$652$		0			0
$653$			6.91713i			0.270688i	0.990799	+	0.135344i	$0.0432140\pi$
$653$			6.91713i			0.270688i	−0.990799	+	0.135344i	$0.956786\pi$
$654$		0			0
$655$	1.18785	−	0.234074i	0.0464130	−	0.00914603i
$656$		0			0
$657$		−	75.8776i		−	2.96027i
$658$		0			0
$659$	9.44461			0.367910			0.183955	−	0.982935i	$-0.441110\pi$
$659$	9.44461			0.367910			0.183955	+	0.982935i	$0.441110\pi$
$660$		0			0
$661$	−22.1955			−0.863306			−0.431653	−	0.902040i	$-0.642070\pi$
$661$	−22.1955			−0.863306			−0.431653	+	0.902040i	$0.642070\pi$
$662$		0			0
$663$			56.3405i			2.18808i
$664$		0			0
$665$	0.329237	+	1.67076i	0.0127673	+	0.0647894i
$666$		0			0
$667$			58.9465i			2.28242i
$668$		0			0
$669$	37.3449			1.44384
$670$		0			0
$671$	−9.39401			−0.362652
$672$		0			0
$673$			12.2986i			0.474077i	0.971500	+	0.237039i	$0.0761768\pi$
$673$			12.2986i			0.474077i	−0.971500	+	0.237039i	$0.923823\pi$
$674$		0			0
$675$	8.51875	+	20.7755i	0.327887	+	0.799650i
$676$		0			0
$677$			35.9527i			1.38178i	0.722962	+	0.690888i	$0.242780\pi$
$677$			35.9527i			1.38178i	−0.722962	+	0.690888i	$0.757220\pi$
$678$		0			0
$679$	6.47689			0.248560
$680$		0			0
$681$	37.5616			1.43937
$682$		0			0
$683$			9.00958i			0.344742i	0.985032	+	0.172371i	$0.0551428\pi$
$683$			9.00958i			0.344742i	−0.985032	+	0.172371i	$0.944857\pi$
$684$		0			0
$685$	1.24448	+	6.31528i	0.0475490	+	0.241295i
$686$		0			0
$687$			37.5231i			1.43160i
$688$		0			0
$689$	−13.3555			−0.508803
$690$		0			0
$691$	9.11078			0.346590			0.173295	−	0.984870i	$-0.444559\pi$
$691$	9.11078			0.346590			0.173295	+	0.984870i	$0.444559\pi$
$692$		0			0
$693$			3.04623i			0.115717i
$694$		0			0
$695$	−7.87090	+	1.55102i	−0.298560	+	0.0588336i
$696$		0			0
$697$		−	9.64015i		−	0.365147i
$698$		0			0
$699$	70.5606			2.66885
$700$		0			0
$701$	−14.7476			−0.557009			−0.278505	−	0.960435i	$-0.589839\pi$
$701$	−14.7476			−0.557009			−0.278505	+	0.960435i	$0.589839\pi$
$702$		0			0
$703$		−	0.476886i		−	0.0179861i
$704$		0			0
$705$	−7.58767	+	1.49521i	−0.285768	+	0.0563128i
$706$		0			0
$707$		−	12.5015i		−	0.470166i
$708$		0			0
$709$	8.63389			0.324253			0.162126	−	0.986770i	$-0.448165\pi$
$709$	8.63389			0.324253			0.162126	+	0.986770i	$0.448165\pi$
$710$		0			0
$711$	58.1483			2.18073
$712$		0			0
$713$		−	65.0496i		−	2.43613i
$714$		0			0
$715$	−2.10308	−	10.6724i	−0.0786509	−	0.399126i
$716$		0			0
$717$			31.2909i			1.16858i
$718$		0			0
$719$	−38.2759			−1.42745			−0.713726	−	0.700425i	$-0.752994\pi$
$719$	−38.2759			−1.42745			−0.713726	+	0.700425i	$0.752994\pi$
$720$		0			0
$721$	7.34153			0.273413
$722$		0			0
$723$			3.45856i			0.128625i
$724$		0			0
$725$	34.0279	−	13.9527i	1.26376	−	0.518191i
$726$		0			0
$727$		−	31.1893i		−	1.15675i	−0.815772	−	0.578373i	$-0.803688\pi$
$727$		−	31.1893i		−	1.15675i	0.815772	−	0.578373i	$-0.196312\pi$
$728$		0			0
$729$	44.0375			1.63102
$730$		0			0
$731$	24.8925			0.920684
$732$		0			0
$733$		−	13.9634i		−	0.515748i	−0.966179	−	0.257874i	$-0.916978\pi$
$733$		−	13.9634i		−	0.515748i	0.966179	−	0.257874i	$-0.0830220\pi$
$734$		0			0
$735$	7.66473	+	38.8959i	0.282718	+	1.43470i
$736$		0			0
$737$			0.892548i			0.0328774i
$738$		0			0
$739$	9.02165			0.331867			0.165933	−	0.986137i	$-0.446936\pi$
$739$	9.02165			0.331867			0.165933	+	0.986137i	$0.446936\pi$
$740$		0			0
$741$	15.5371			0.570768
$742$		0			0
$743$			15.0342i			0.551550i	0.961222	+	0.275775i	$0.0889345\pi$
$743$			15.0342i			0.551550i	−0.961222	+	0.275775i	$0.911066\pi$
$744$		0			0
$745$	−36.9450	+	7.28030i	−1.35356	+	0.266730i
$746$		0			0
$747$		−	1.25240i		−	0.0458228i
$748$		0			0
$749$	3.26302			0.119228
$750$		0			0
$751$	−29.6681			−1.08260			−0.541301	−	0.840829i	$-0.682068\pi$
$751$	−29.6681			−1.08260			−0.541301	+	0.840829i	$0.682068\pi$
$752$		0			0
$753$			29.2557i			1.06614i
$754$		0			0
$755$	−37.2557	+	7.34153i	−1.35587	+	0.267186i
$756$		0			0
$757$		−	10.5819i		−	0.384604i	−0.981336	−	0.192302i	$-0.938405\pi$
$757$		−	10.5819i		−	0.384604i	0.981336	−	0.192302i	$-0.0615953\pi$
$758$		0			0
$759$	19.1354			0.694570
$760$		0			0
$761$	−0.979789			−0.0355173			−0.0177587	−	0.999842i	$-0.505653\pi$
$761$	−0.979789			−0.0355173			−0.0177587	+	0.999842i	$0.505653\pi$
$762$		0			0
$763$		−	10.2201i		−	0.369993i
$764$		0			0
$765$	7.25240	+	36.8034i	0.262211	+	1.33063i
$766$		0			0
$767$			25.2663i			0.912315i
$768$		0			0
$769$	−43.1772			−1.55701			−0.778505	−	0.627638i	$-0.784022\pi$
$769$	−43.1772			−1.55701			−0.778505	+	0.627638i	$0.784022\pi$
$770$		0			0
$771$	0.424399			0.0152844
$772$		0			0
$773$			37.5250i			1.34968i	0.737964	+	0.674840i	$0.235787\pi$
$773$			37.5250i			1.34968i	−0.737964	+	0.674840i	$0.764213\pi$
$774$		0			0
$775$	−37.5510	+	15.3973i	−1.34887	+	0.553089i
$776$		0			0
$777$			1.00293i			0.0359799i
$778$		0			0
$779$	−2.65847			−0.0952497
$780$		0			0
$781$	8.80342			0.315011
$782$		0			0
$783$			33.0323i			1.18048i
$784$		0			0
$785$	6.41566	+	32.5573i	0.228985	+	1.16202i
$786$		0			0
$787$			22.5833i			0.805008i	0.915418	+	0.402504i	$0.131860\pi$
$787$			22.5833i			0.805008i	−0.915418	+	0.402504i	$0.868140\pi$
$788$		0			0
$789$	−1.39401			−0.0496282
$790$		0			0
$791$	−7.86171			−0.279530
$792$		0			0
$793$		−	61.1266i		−	2.17067i
$794$		0			0
$795$	−14.3817	+	2.83403i	−0.510067	+	0.100513i
$796$		0			0
$797$		−	35.9806i		−	1.27450i	−0.770657	−	0.637250i	$-0.780072\pi$
$797$		−	35.9806i		−	1.27450i	0.770657	−	0.637250i	$-0.219928\pi$
$798$		0			0
$799$	−4.54144			−0.160664
$800$		0			0
$801$	1.79383			0.0633820
$802$		0			0
$803$			14.1816i			0.500457i
$804$		0			0
$805$	−13.3894	+	2.63849i	−0.471915	+	0.0929945i
$806$		0			0
$807$		−	9.65222i		−	0.339774i
$808$		0			0
$809$	0.955660			0.0335992			0.0167996	−	0.999859i	$-0.494652\pi$
$809$	0.955660			0.0335992			0.0167996	+	0.999859i	$0.494652\pi$
$810$		0			0
$811$	7.53707			0.264662			0.132331	−	0.991206i	$-0.457754\pi$
$811$	7.53707			0.264662			0.132331	+	0.991206i	$0.457754\pi$
$812$		0			0
$813$			15.1127i			0.530024i
$814$		0			0
$815$	5.77988	+	29.3309i	0.202460	+	1.02742i
$816$		0			0
$817$		−	6.86464i		−	0.240163i
$818$		0			0
$819$	−19.8217			−0.692628
$820$		0			0
$821$	13.3082			0.464460			0.232230	−	0.972661i	$-0.425398\pi$
$821$	13.3082			0.464460			0.232230	+	0.972661i	$0.425398\pi$
$822$		0			0
$823$		−	24.4050i		−	0.850706i	−0.905028	−	0.425353i	$-0.860150\pi$
$823$		−	24.4050i		−	0.850706i	0.905028	−	0.425353i	$-0.139850\pi$
$824$		0			0
$825$	−4.52937	−	11.0462i	−0.157692	−	0.384580i
$826$		0			0
$827$		−	11.6874i		−	0.406411i	−0.979136	−	0.203206i	$-0.934864\pi$
$827$		−	11.6874i		−	0.406411i	0.979136	−	0.203206i	$-0.0651360\pi$
$828$		0			0
$829$	−25.6541			−0.891004			−0.445502	−	0.895281i	$-0.646975\pi$
$829$	−25.6541			−0.891004			−0.445502	+	0.895281i	$0.646975\pi$
$830$		0			0
$831$	−35.7726			−1.24094
$832$		0			0
$833$			23.2803i			0.806615i
$834$		0			0
$835$	−4.25676	−	21.6016i	−0.147311	−	0.747555i
$836$		0			0
$837$		−	36.4523i		−	1.25998i
$838$		0			0
$839$	−2.91713			−0.100710			−0.0503552	−	0.998731i	$-0.516035\pi$
$839$	−2.91713			−0.100710			−0.0503552	+	0.998731i	$0.516035\pi$
$840$		0			0
$841$	25.1031			0.865624
$842$		0			0
$843$		−	0.424399i		−	0.0146171i
$844$		0			0
$845$	40.9248	−	8.06455i	1.40786	−	0.277429i
$846$		0			0
$847$			7.80779i			0.268279i
$848$		0			0
$849$	50.3544			1.72816
$850$		0			0
$851$	3.82174			0.131008
$852$		0			0
$853$		−	6.24281i		−	0.213750i	−0.994272	−	0.106875i	$-0.965916\pi$
$853$		−	6.24281i		−	0.213750i	0.994272	−	0.106875i	$-0.0340844\pi$
$854$		0			0
$855$	10.1493	−	2.00000i	0.347099	−	0.0683986i
$856$		0			0
$857$			23.2158i			0.793035i	0.918027	+	0.396517i	$0.129781\pi$
$857$			23.2158i			0.793035i	−0.918027	+	0.396517i	$0.870219\pi$
$858$		0			0
$859$	−7.13536			−0.243455			−0.121728	−	0.992564i	$-0.538843\pi$
$859$	−7.13536			−0.243455			−0.121728	+	0.992564i	$0.538843\pi$
$860$		0			0
$861$	5.59099			0.190541
$862$		0			0
$863$			7.31362i			0.248959i	0.992222	+	0.124479i	$0.0397260\pi$
$863$			7.31362i			0.248959i	−0.992222	+	0.124479i	$0.960274\pi$
$864$		0			0
$865$	−1.28904	−	6.54144i	−0.0438287	−	0.222416i
$866$		0			0
$867$		−	10.6339i		−	0.361146i
$868$		0			0
$869$	−10.8680			−0.368671
$870$		0			0
$871$	−5.80779			−0.196789
$872$		0			0
$873$		−	39.3449i		−	1.33162i
$874$		0			0
$875$	4.69241	+	7.10475i	0.158632	+	0.240184i
$876$		0			0
$877$			22.0173i			0.743471i	0.928339	+	0.371735i	$0.121237\pi$
$877$			22.0173i			0.743471i	−0.928339	+	0.371735i	$0.878763\pi$
$878$		0			0
$879$	5.62953			0.189879
$880$		0			0
$881$	11.7572			0.396110			0.198055	−	0.980191i	$-0.436538\pi$
$881$	11.7572			0.396110			0.198055	+	0.980191i	$0.436538\pi$
$882$		0			0
$883$		−	55.6560i		−	1.87297i	−0.350703	−	0.936487i	$-0.614057\pi$
$883$		−	55.6560i		−	1.87297i	0.350703	−	0.936487i	$-0.385943\pi$
$884$		0			0
$885$	5.36151	+	27.2078i	0.180225	+	0.914581i
$886$		0			0
$887$		−	6.41566i		−	0.215417i	−0.994183	−	0.107708i	$-0.965649\pi$
$887$		−	6.41566i		−	0.215417i	0.994183	−	0.107708i	$-0.0343513\pi$
$888$		0			0
$889$	12.9325			0.433743
$890$		0			0
$891$	−1.27698			−0.0427803
$892$		0			0
$893$			1.25240i			0.0419098i
$894$		0			0
$895$	−25.8742	+	5.09871i	−0.864880	+	0.170431i
$896$		0			0
$897$			124.513i			4.15738i
$898$		0			0
$899$	−59.7047			−1.99126
$900$		0			0
$901$	−8.60788			−0.286770
$902$		0			0
$903$			14.4369i			0.480430i
$904$		0			0
$905$	−31.9634	+	6.29862i	−1.06250	+	0.209373i
$906$		0			0
$907$		−	57.1160i		−	1.89651i	−0.317516	−	0.948253i	$-0.602849\pi$
$907$		−	57.1160i		−	1.89651i	0.317516	−	0.948253i	$-0.397151\pi$
$908$		0			0
$909$	−75.9421			−2.51884
$910$		0			0
$911$	26.6339			0.882420			0.441210	−	0.897404i	$-0.354549\pi$
$911$	26.6339			0.882420			0.441210	+	0.897404i	$0.354549\pi$
$912$		0			0
$913$			0.234074i			0.00774672i
$914$		0			0
$915$	−12.9711	−	65.8236i	−0.428810	−	2.17606i
$916$		0			0
$917$			0.412335i			0.0136165i
$918$		0			0
$919$	6.63246			0.218785			0.109392	−	0.993999i	$-0.465110\pi$
$919$	6.63246			0.218785			0.109392	+	0.993999i	$0.465110\pi$
$920$		0			0
$921$	45.6801			1.50521
$922$		0			0
$923$			57.2836i			1.88551i
$924$		0			0
$925$	−0.904612	−	2.20617i	−0.0297435	−	0.0725383i
$926$		0			0
$927$		−	44.5972i		−	1.46477i
$928$		0			0
$929$	50.0173			1.64101			0.820507	−	0.571637i	$-0.193691\pi$
$929$	50.0173			1.64101			0.820507	+	0.571637i	$0.193691\pi$
$930$		0			0
$931$	6.42003			0.210408
$932$		0			0
$933$		−	59.2976i		−	1.94132i
$934$		0			0
$935$	−1.35548	−	6.87859i	−0.0443289	−	0.224954i
$936$		0			0
$937$			39.8882i			1.30309i	0.758610	+	0.651545i	$0.225879\pi$
$937$			39.8882i			1.30309i	−0.758610	+	0.651545i	$0.774121\pi$
$938$		0			0
$939$	−3.09683			−0.101061
$940$		0			0
$941$	−5.59829			−0.182499			−0.0912495	−	0.995828i	$-0.529086\pi$
$941$	−5.59829			−0.182499			−0.0912495	+	0.995828i	$0.529086\pi$
$942$		0			0
$943$		−	21.3049i		−	0.693782i
$944$		0			0
$945$	−7.50313	+	1.47855i	−0.244077	+	0.0480972i
$946$		0			0
$947$			12.7110i			0.413051i	0.978441	+	0.206525i	$0.0662156\pi$
$947$			12.7110i			0.413051i	−0.978441	+	0.206525i	$0.933784\pi$
$948$		0			0
$949$	−92.2793			−2.99551
$950$		0			0
$951$	82.5379			2.67648
$952$		0			0
$953$		−	57.0129i		−	1.84683i	−0.383804	−	0.923415i	$-0.625386\pi$
$953$		−	57.0129i		−	1.84683i	0.383804	−	0.923415i	$-0.374614\pi$
$954$		0			0
$955$	29.0435	−	5.72325i	0.939826	−	0.185200i
$956$		0			0
$957$		−	17.5631i		−	0.567734i
$958$		0			0
$959$	−2.19221			−0.0707903
$960$		0			0
$961$	34.8863			1.12536
$962$		0			0
$963$		−	19.8217i		−	0.638747i
$964$		0			0
$965$	−1.09871	−	5.57560i	−0.0353689	−	0.179485i
$966$		0			0
$967$		−	33.0183i		−	1.06180i	−0.847435	−	0.530899i	$-0.821854\pi$
$967$		−	33.0183i		−	1.06180i	0.847435	−	0.530899i	$-0.178146\pi$
$968$		0			0
$969$	10.0140			0.321695
$970$		0			0
$971$	3.04623			0.0977581			0.0488791	−	0.998805i	$-0.484435\pi$
$971$	3.04623			0.0977581			0.0488791	+	0.998805i	$0.484435\pi$
$972$		0			0
$973$		−	2.73221i		−	0.0875907i
$974$		0			0
$975$	71.8776	−	29.4725i	2.30192	−	0.943876i
$976$		0			0
$977$		−	13.4465i		−	0.430192i	−0.976593	−	0.215096i	$-0.930994\pi$
$977$		−	13.4465i		−	0.430192i	0.976593	−	0.215096i	$-0.0690064\pi$
$978$		0			0
$979$	−0.335269			−0.0107152
$980$		0			0
$981$	−62.0837			−1.98218
$982$		0			0
$983$			22.0646i			0.703750i	0.936047	+	0.351875i	$0.114456\pi$
$983$			22.0646i			0.703750i	−0.936047	+	0.351875i	$0.885544\pi$
$984$		0			0
$985$	−8.60788	−	43.6820i	−0.274270	−	1.39182i
$986$		0			0
$987$		−	2.63389i		−	0.0838378i
$988$		0			0
$989$	55.0129			1.74931
$990$		0			0
$991$	−51.9946			−1.65166			−0.825831	−	0.563917i	$-0.809294\pi$
$991$	−51.9946			−1.65166			−0.825831	+	0.563917i	$0.809294\pi$
$992$		0			0
$993$			89.2330i			2.83172i
$994$		0			0
$995$	44.5633	−	8.78154i	1.41275	−	0.278394i
$996$		0			0
$997$			37.7693i			1.19616i	0.801435	+	0.598082i	$0.204070\pi$
$997$			37.7693i			1.19616i	−0.801435	+	0.598082i	$0.795930\pi$
$998$		0			0
$999$	2.14162			0.0677578

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.d.j.609.1		6
4.3	odd	2		190.2.b.b.39.3	✓	6
5.2	odd	4		7600.2.a.bi.1.1		3
5.3	odd	4		7600.2.a.cd.1.3		3
5.4	even	2	inner	1520.2.d.j.609.6		6
12.11	even	2		1710.2.d.d.1369.6		6
20.3	even	4		950.2.a.i.1.1		3
20.7	even	4		950.2.a.n.1.3		3
20.19	odd	2		190.2.b.b.39.4	yes	6
60.23	odd	4		8550.2.a.cl.1.3		3
60.47	odd	4		8550.2.a.ck.1.1		3
60.59	even	2		1710.2.d.d.1369.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.2.b.b.39.3	✓	6	4.3	odd	2
190.2.b.b.39.4	yes	6	20.19	odd	2
950.2.a.i.1.1		3	20.3	even	4
950.2.a.n.1.3		3	20.7	even	4
1520.2.d.j.609.1		6	1.1	even	1	trivial
1520.2.d.j.609.6		6	5.4	even	2	inner
1710.2.d.d.1369.3		6	60.59	even	2
1710.2.d.d.1369.6		6	12.11	even	2
7600.2.a.bi.1.1		3	5.2	odd	4
7600.2.a.cd.1.3		3	5.3	odd	4
8550.2.a.ck.1.1		3	60.47	odd	4
8550.2.a.cl.1.3		3	60.23	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-2.76156i q^{3} +(-2.19388 + 0.432320i) q^{5} -0.761557i q^{7} -4.62620 q^{9} +O(q^{10})\)	\(q-2.76156i q^{3} +(-2.19388 + 0.432320i) q^{5} -0.761557i q^{7} -4.62620 q^{9} +0.864641 q^{11} +5.62620i q^{13} +(1.19388 + 6.05852i) q^{15} +3.62620i q^{17} +1.00000 q^{19} -2.10308 q^{21} +8.01395i q^{23} +(4.62620 - 1.89692i) q^{25} +4.49084i q^{27} +7.35548 q^{29} -8.11704 q^{31} -2.38776i q^{33} +(0.329237 + 1.67076i) q^{35} -0.476886i q^{37} +15.5371 q^{39} -2.65847 q^{41} -6.86464i q^{43} +(10.1493 - 2.00000i) q^{45} +1.25240i q^{47} +6.42003 q^{49} +10.0140 q^{51} +2.37380i q^{53} +(-1.89692 + 0.373802i) q^{55} -2.76156i q^{57} +4.49084 q^{59} -10.8646 q^{61} +3.52311i q^{63} +(-2.43232 - 12.3432i) q^{65} +1.03228i q^{67} +22.1310 q^{69} +10.1816 q^{71} +16.4017i q^{73} +(-5.23844 - 12.7755i) q^{75} -0.658473i q^{77} -12.5693 q^{79} -1.47689 q^{81} +0.270718i q^{83} +(-1.56768 - 7.95543i) q^{85} -20.3126i q^{87} -0.387755 q^{89} +4.28467 q^{91} +22.4157i q^{93} +(-2.19388 + 0.432320i) q^{95} +8.50479i q^{97} -4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{5} - 10 q^{9}+O(q^{10}) \) 6 * q + 2 * q^5 - 10 * q^9	\( 6 q + 2 q^{5} - 10 q^{9} - 8 q^{15} + 6 q^{19} - 20 q^{21} + 10 q^{25} + 16 q^{29} - 8 q^{31} - 8 q^{35} + 20 q^{39} + 4 q^{41} + 18 q^{45} + 6 q^{49} + 12 q^{51} - 4 q^{55} + 4 q^{59} - 60 q^{61} - 12 q^{65} + 44 q^{69} + 16 q^{71} - 44 q^{75} - 34 q^{81} - 12 q^{85} + 28 q^{89} - 12 q^{91} + 2 q^{95} - 24 q^{99}+O(q^{100}) \) 6 * q + 2 * q^5 - 10 * q^9 - 8 * q^15 + 6 * q^19 - 20 * q^21 + 10 * q^25 + 16 * q^29 - 8 * q^31 - 8 * q^35 + 20 * q^39 + 4 * q^41 + 18 * q^45 + 6 * q^49 + 12 * q^51 - 4 * q^55 + 4 * q^59 - 60 * q^61 - 12 * q^65 + 44 * q^69 + 16 * q^71 - 44 * q^75 - 34 * q^81 - 12 * q^85 + 28 * q^89 - 12 * q^91 + 2 * q^95 - 24 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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