LMFDB - Embedded newform 1568.2.i.n.1537.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1568,2,Mod(961,1568)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1568, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

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

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

chi := DirichletCharacter("1568.961");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1568 = 2^{5} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1568.i (of order $3$, degree $2$, 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:	$12.5205430369$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 2x^{2} + x + 1 $ x^4 - x^3 + 2*x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 224)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1537.2
Root			$-0.309017 + 0.535233i$ of defining polynomial
Character	$\chi$	$=$	1568.1537
Dual form			1568.2.i.n.961.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.618034 - 1.07047i) q^{3} +(1.61803 + 2.80252i) q^{5} +(0.736068 + 1.27491i) q^{9} +O(q^{10})$	\(q+(0.618034 - 1.07047i) q^{3} +(1.61803 + 2.80252i) q^{5} +(0.736068 + 1.27491i) q^{9} +(3.23607 - 5.60503i) q^{11} -0.763932 q^{13} +4.00000 q^{15} +(2.23607 - 3.87298i) q^{17} +(-0.618034 - 1.07047i) q^{19} +(-2.00000 - 3.46410i) q^{23} +(-2.73607 + 4.73901i) q^{25} +5.52786 q^{27} -4.47214 q^{29} +(-1.23607 + 2.14093i) q^{31} +(-4.00000 - 6.92820i) q^{33} +(2.23607 + 3.87298i) q^{37} +(-0.472136 + 0.817763i) q^{39} +8.47214 q^{41} +6.47214 q^{43} +(-2.38197 + 4.12569i) q^{45} +(5.23607 + 9.06914i) q^{47} +(-2.76393 - 4.78727i) q^{51} +(5.00000 - 8.66025i) q^{53} +20.9443 q^{55} -1.52786 q^{57} +(-4.61803 + 7.99867i) q^{59} +(5.61803 + 9.73072i) q^{61} +(-1.23607 - 2.14093i) q^{65} +(-2.00000 + 3.46410i) q^{67} -4.94427 q^{69} -4.94427 q^{71} +(-1.47214 + 2.54981i) q^{73} +(3.38197 + 5.85774i) q^{75} +(-6.47214 - 11.2101i) q^{79} +(1.20820 - 2.09267i) q^{81} +9.23607 q^{83} +14.4721 q^{85} +(-2.76393 + 4.78727i) q^{87} +(-3.00000 - 5.19615i) q^{89} +(1.52786 + 2.64634i) q^{93} +(2.00000 - 3.46410i) q^{95} -12.4721 q^{97} +9.52786 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{3} + 2 q^{5} - 6 q^{9}+O(q^{10}) $ 4 * q - 2 * q^3 + 2 * q^5 - 6 * q^9	$ 4 q - 2 q^{3} + 2 q^{5} - 6 q^{9} + 4 q^{11} - 12 q^{13} + 16 q^{15} + 2 q^{19} - 8 q^{23} - 2 q^{25} + 40 q^{27} + 4 q^{31} - 16 q^{33} + 16 q^{39} + 16 q^{41} + 8 q^{43} - 14 q^{45} + 12 q^{47} - 20 q^{51} + 20 q^{53} + 48 q^{55} - 24 q^{57} - 14 q^{59} + 18 q^{61} + 4 q^{65} - 8 q^{67} + 16 q^{69} + 16 q^{71} + 12 q^{73} + 18 q^{75} - 8 q^{79} - 22 q^{81} + 28 q^{83} + 40 q^{85} - 20 q^{87} - 12 q^{89} + 24 q^{93} + 8 q^{95} - 32 q^{97} + 56 q^{99}+O(q^{100}) $ 4 * q - 2 * q^3 + 2 * q^5 - 6 * q^9 + 4 * q^11 - 12 * q^13 + 16 * q^15 + 2 * q^19 - 8 * q^23 - 2 * q^25 + 40 * q^27 + 4 * q^31 - 16 * q^33 + 16 * q^39 + 16 * q^41 + 8 * q^43 - 14 * q^45 + 12 * q^47 - 20 * q^51 + 20 * q^53 + 48 * q^55 - 24 * q^57 - 14 * q^59 + 18 * q^61 + 4 * q^65 - 8 * q^67 + 16 * q^69 + 16 * q^71 + 12 * q^73 + 18 * q^75 - 8 * q^79 - 22 * q^81 + 28 * q^83 + 40 * q^85 - 20 * q^87 - 12 * q^89 + 24 * q^93 + 8 * q^95 - 32 * q^97 + 56 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1568\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$3$	0.618034	−	1.07047i	0.356822	−	0.618034i	−0.630606	−	0.776103i	$-0.717194\pi$
$3$	0.618034	−	1.07047i	0.356822	−	0.618034i	0.987428	+	0.158069i	$0.0505269\pi$
$4$		0			0
$5$	1.61803	+	2.80252i	0.723607	+	1.25332i	0.959545	+	0.281556i	$0.0908504\pi$
$5$	1.61803	+	2.80252i	0.723607	+	1.25332i	−0.235938	+	0.971768i	$0.575816\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	0.736068	+	1.27491i	0.245356	+	0.424969i
$10$		0			0
$11$	3.23607	−	5.60503i	0.975711	−	1.68998i	0.298143	−	0.954521i	$-0.403633\pi$
$11$	3.23607	−	5.60503i	0.975711	−	1.68998i	0.677568	−	0.735460i	$-0.263034\pi$
$12$		0			0
$13$	−0.763932			−0.211877			−0.105938	−	0.994373i	$-0.533785\pi$
$13$	−0.763932			−0.211877			−0.105938	+	0.994373i	$0.533785\pi$
$14$		0			0
$15$	4.00000			1.03280
$16$		0			0
$17$	2.23607	−	3.87298i	0.542326	−	0.939336i	−0.456444	−	0.889752i	$-0.650877\pi$
$17$	2.23607	−	3.87298i	0.542326	−	0.939336i	0.998770	−	0.0495842i	$-0.0157896\pi$
$18$		0			0
$19$	−0.618034	−	1.07047i	−0.141787	−	0.245582i	0.786383	−	0.617740i	$-0.211951\pi$
$19$	−0.618034	−	1.07047i	−0.141787	−	0.245582i	−0.928170	+	0.372158i	$0.878618\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.00000	−	3.46410i	−0.417029	−	0.722315i	0.578610	−	0.815604i	$-0.303595\pi$
$23$	−2.00000	−	3.46410i	−0.417029	−	0.722315i	−0.995639	+	0.0932891i	$0.970262\pi$
$24$		0			0
$25$	−2.73607	+	4.73901i	−0.547214	+	0.947802i
$26$		0			0
$27$	5.52786			1.06384
$28$		0			0
$29$	−4.47214			−0.830455			−0.415227	−	0.909718i	$-0.636298\pi$
$29$	−4.47214			−0.830455			−0.415227	+	0.909718i	$0.636298\pi$
$30$		0			0
$31$	−1.23607	+	2.14093i	−0.222004	+	0.384523i	−0.955417	−	0.295261i	$-0.904593\pi$
$31$	−1.23607	+	2.14093i	−0.222004	+	0.384523i	0.733412	+	0.679784i	$0.237927\pi$
$32$		0			0
$33$	−4.00000	−	6.92820i	−0.696311	−	1.20605i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	2.23607	+	3.87298i	0.367607	+	0.636715i	0.989191	−	0.146633i	$-0.0468437\pi$
$37$	2.23607	+	3.87298i	0.367607	+	0.636715i	−0.621584	+	0.783348i	$0.713510\pi$
$38$		0			0
$39$	−0.472136	+	0.817763i	−0.0756023	+	0.130947i
$40$		0			0
$41$	8.47214			1.32313			0.661563	−	0.749890i	$-0.269894\pi$
$41$	8.47214			1.32313			0.661563	+	0.749890i	$0.269894\pi$
$42$		0			0
$43$	6.47214			0.986991			0.493496	−	0.869748i	$-0.335719\pi$
$43$	6.47214			0.986991			0.493496	+	0.869748i	$0.335719\pi$
$44$		0			0
$45$	−2.38197	+	4.12569i	−0.355083	+	0.615021i
$46$		0			0
$47$	5.23607	+	9.06914i	0.763759	+	1.32287i	0.940900	+	0.338684i	$0.109982\pi$
$47$	5.23607	+	9.06914i	0.763759	+	1.32287i	−0.177141	+	0.984185i	$0.556685\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	−2.76393	−	4.78727i	−0.387028	−	0.670352i
$52$		0			0
$53$	5.00000	−	8.66025i	0.686803	−	1.18958i	−0.286064	−	0.958211i	$-0.592347\pi$
$53$	5.00000	−	8.66025i	0.686803	−	1.18958i	0.972867	−	0.231367i	$-0.0743197\pi$
$54$		0			0
$55$	20.9443			2.82413
$56$		0			0
$57$	−1.52786			−0.202371
$58$		0			0
$59$	−4.61803	+	7.99867i	−0.601217	+	1.04134i	0.391420	+	0.920212i	$0.371984\pi$
$59$	−4.61803	+	7.99867i	−0.601217	+	1.04134i	−0.992637	+	0.121126i	$0.961349\pi$
$60$		0			0
$61$	5.61803	+	9.73072i	0.719316	+	1.24589i	0.961271	+	0.275604i	$0.0888778\pi$
$61$	5.61803	+	9.73072i	0.719316	+	1.24589i	−0.241956	+	0.970287i	$0.577789\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.23607	−	2.14093i	−0.153315	−	0.265550i
$66$		0			0
$67$	−2.00000	+	3.46410i	−0.244339	+	0.423207i	−0.961946	−	0.273241i	$-0.911904\pi$
$67$	−2.00000	+	3.46410i	−0.244339	+	0.423207i	0.717607	+	0.696449i	$0.245238\pi$
$68$		0			0
$69$	−4.94427			−0.595220
$70$		0			0
$71$	−4.94427			−0.586777			−0.293389	−	0.955993i	$-0.594783\pi$
$71$	−4.94427			−0.586777			−0.293389	+	0.955993i	$0.594783\pi$
$72$		0			0
$73$	−1.47214	+	2.54981i	−0.172300	+	0.298433i	−0.939224	−	0.343306i	$-0.888453\pi$
$73$	−1.47214	+	2.54981i	−0.172300	+	0.298433i	0.766923	+	0.641739i	$0.221787\pi$
$74$		0			0
$75$	3.38197	+	5.85774i	0.390516	+	0.676393i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−6.47214	−	11.2101i	−0.728172	−	1.26123i	−0.957655	−	0.287918i	$-0.907037\pi$
$79$	−6.47214	−	11.2101i	−0.728172	−	1.26123i	0.229483	−	0.973313i	$-0.426297\pi$
$80$		0			0
$81$	1.20820	−	2.09267i	0.134245	−	0.232519i
$82$		0			0
$83$	9.23607			1.01379			0.506895	−	0.862008i	$-0.330793\pi$
$83$	9.23607			1.01379			0.506895	+	0.862008i	$0.330793\pi$
$84$		0			0
$85$	14.4721			1.56972
$86$		0			0
$87$	−2.76393	+	4.78727i	−0.296325	+	0.513249i
$88$		0			0
$89$	−3.00000	−	5.19615i	−0.317999	−	0.550791i	0.662071	−	0.749441i	$-0.269678\pi$
$89$	−3.00000	−	5.19615i	−0.317999	−	0.550791i	−0.980071	+	0.198650i	$0.936344\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	1.52786	+	2.64634i	0.158432	+	0.274412i
$94$		0			0
$95$	2.00000	−	3.46410i	0.205196	−	0.355409i
$96$		0			0
$97$	−12.4721			−1.26635			−0.633177	−	0.774007i	$-0.718249\pi$
$97$	−12.4721			−1.26635			−0.633177	+	0.774007i	$0.718249\pi$
$98$		0			0
$99$	9.52786			0.957586
$100$		0			0
$101$	−0.854102	+	1.47935i	−0.0849863	+	0.147201i	−0.905385	−	0.424591i	$-0.860418\pi$
$101$	−0.854102	+	1.47935i	−0.0849863	+	0.147201i	0.820399	+	0.571791i	$0.193751\pi$
$102$		0			0
$103$	−2.76393	−	4.78727i	−0.272338	−	0.471704i	0.697122	−	0.716953i	$-0.254464\pi$
$103$	−2.76393	−	4.78727i	−0.272338	−	0.471704i	−0.969460	+	0.245249i	$0.921130\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−4.47214	−	7.74597i	−0.432338	−	0.748831i	0.564736	−	0.825271i	$-0.308978\pi$
$107$	−4.47214	−	7.74597i	−0.432338	−	0.748831i	−0.997074	+	0.0764405i	$0.975644\pi$
$108$		0			0
$109$	−4.23607	+	7.33708i	−0.405742	+	0.702765i	−0.994407	−	0.105611i	$-0.966320\pi$
$109$	−4.23607	+	7.33708i	−0.405742	+	0.702765i	0.588666	+	0.808377i	$0.299653\pi$
$110$		0			0
$111$	5.52786			0.524682
$112$		0			0
$113$	−12.4721			−1.17328			−0.586640	−	0.809848i	$-0.699550\pi$
$113$	−12.4721			−1.17328			−0.586640	+	0.809848i	$0.699550\pi$
$114$		0			0
$115$	6.47214	−	11.2101i	0.603530	−	1.04534i
$116$		0			0
$117$	−0.562306	−	0.973942i	−0.0519852	−	0.0900410i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−15.4443	−	26.7503i	−1.40402	−	2.43184i
$122$		0			0
$123$	5.23607	−	9.06914i	0.472120	−	0.817736i
$124$		0			0
$125$	−1.52786			−0.136656
$126$		0			0
$127$	8.94427			0.793676			0.396838	−	0.917889i	$-0.370108\pi$
$127$	8.94427			0.793676			0.396838	+	0.917889i	$0.370108\pi$
$128$		0			0
$129$	4.00000	−	6.92820i	0.352180	−	0.609994i
$130$		0			0
$131$	5.85410	+	10.1396i	0.511475	+	0.885901i	0.999912	+	0.0133016i	$0.00423415\pi$
$131$	5.85410	+	10.1396i	0.511475	+	0.885901i	−0.488436	+	0.872600i	$0.662433\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	8.94427	+	15.4919i	0.769800	+	1.33333i
$136$		0			0
$137$	−7.47214	+	12.9421i	−0.638388	+	1.10572i	0.347399	+	0.937717i	$0.387065\pi$
$137$	−7.47214	+	12.9421i	−0.638388	+	1.10572i	−0.985787	+	0.168002i	$0.946268\pi$
$138$		0			0
$139$	−1.23607			−0.104842			−0.0524210	−	0.998625i	$-0.516694\pi$
$139$	−1.23607			−0.104842			−0.0524210	+	0.998625i	$0.516694\pi$
$140$		0			0
$141$	12.9443			1.09010
$142$		0			0
$143$	−2.47214	+	4.28187i	−0.206730	+	0.358068i
$144$		0			0
$145$	−7.23607	−	12.5332i	−0.600923	−	1.04083i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.47214	−	2.54981i	−0.120602	−	0.208889i	0.799403	−	0.600795i	$-0.205149\pi$
$149$	−1.47214	−	2.54981i	−0.120602	−	0.208889i	−0.920005	+	0.391906i	$0.871816\pi$
$150$		0			0
$151$	4.47214	−	7.74597i	0.363937	−	0.630358i	−0.624668	−	0.780891i	$-0.714766\pi$
$151$	4.47214	−	7.74597i	0.363937	−	0.630358i	0.988605	+	0.150533i	$0.0480989\pi$
$152$		0			0
$153$	6.58359			0.532252
$154$		0			0
$155$	−8.00000			−0.642575
$156$		0			0
$157$	0.381966	−	0.661585i	0.0304842	−	0.0528002i	−0.850381	−	0.526168i	$-0.823628\pi$
$157$	0.381966	−	0.661585i	0.0304842	−	0.0528002i	0.880865	+	0.473368i	$0.156962\pi$
$158$		0			0
$159$	−6.18034	−	10.7047i	−0.490133	−	0.848935i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1.70820	+	2.95870i	0.133797	+	0.231743i	0.925137	−	0.379633i	$-0.123950\pi$
$163$	1.70820	+	2.95870i	0.133797	+	0.231743i	−0.791340	+	0.611376i	$0.790616\pi$
$164$		0			0
$165$	12.9443	−	22.4201i	1.00771	−	1.74541i
$166$		0			0
$167$	−23.4164			−1.81202			−0.906008	−	0.423261i	$-0.860886\pi$
$167$	−23.4164			−1.81202			−0.906008	+	0.423261i	$0.860886\pi$
$168$		0			0
$169$	−12.4164			−0.955108
$170$		0			0
$171$	0.909830	−	1.57587i	0.0695764	−	0.120510i
$172$		0			0
$173$	2.85410	+	4.94345i	0.216993	+	0.375844i	0.953887	−	0.300165i	$-0.0970417\pi$
$173$	2.85410	+	4.94345i	0.216993	+	0.375844i	−0.736894	+	0.676008i	$0.763708\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	5.70820	+	9.88690i	0.429055	+	0.743145i
$178$		0			0
$179$	3.52786	−	6.11044i	0.263685	−	0.456716i	−0.703533	−	0.710662i	$-0.748395\pi$
$179$	3.52786	−	6.11044i	0.263685	−	0.456716i	0.967218	+	0.253947i	$0.0817287\pi$
$180$		0			0
$181$	12.1803			0.905358			0.452679	−	0.891674i	$-0.350468\pi$
$181$	12.1803			0.905358			0.452679	+	0.891674i	$0.350468\pi$
$182$		0			0
$183$	13.8885			1.02667
$184$		0			0
$185$	−7.23607	+	12.5332i	−0.532006	+	0.921462i
$186$		0			0
$187$	−14.4721	−	25.0665i	−1.05831	−	1.83304i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.47214	−	4.28187i	−0.178877	−	0.309825i	0.762619	−	0.646848i	$-0.223913\pi$
$191$	−2.47214	−	4.28187i	−0.178877	−	0.309825i	−0.941496	+	0.337023i	$0.890580\pi$
$192$		0			0
$193$	−0.236068	+	0.408882i	−0.0169925	+	0.0294320i	−0.874397	−	0.485212i	$-0.838742\pi$
$193$	−0.236068	+	0.408882i	−0.0169925	+	0.0294320i	0.857404	+	0.514644i	$0.172076\pi$
$194$		0			0
$195$	−3.05573			−0.218825
$196$		0			0
$197$	10.9443			0.779747			0.389874	−	0.920868i	$-0.372519\pi$
$197$	10.9443			0.779747			0.389874	+	0.920868i	$0.372519\pi$
$198$		0			0
$199$	7.70820	−	13.3510i	0.546420	−	0.946427i	−0.452096	−	0.891969i	$-0.649324\pi$
$199$	7.70820	−	13.3510i	0.546420	−	0.946427i	0.998516	−	0.0544581i	$-0.0173431\pi$
$200$		0			0
$201$	2.47214	+	4.28187i	0.174371	+	0.302019i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	13.7082	+	23.7433i	0.957422	+	1.65830i
$206$		0			0
$207$	2.94427	−	5.09963i	0.204641	−	0.354449i
$208$		0			0
$209$	−8.00000			−0.553372
$210$		0			0
$211$	−12.0000			−0.826114			−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000			−0.826114			−0.413057	+	0.910705i	$0.635539\pi$
$212$		0			0
$213$	−3.05573	+	5.29268i	−0.209375	+	0.362648i
$214$		0			0
$215$	10.4721	+	18.1383i	0.714194	+	1.23702i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	1.81966	+	3.15174i	0.122961	+	0.212975i
$220$		0			0
$221$	−1.70820	+	2.95870i	−0.114906	+	0.199023i
$222$		0			0
$223$	−12.9443			−0.866813			−0.433406	−	0.901199i	$-0.642688\pi$
$223$	−12.9443			−0.866813			−0.433406	+	0.901199i	$0.642688\pi$
$224$		0			0
$225$	−8.05573			−0.537049
$226$		0			0
$227$	−8.61803	+	14.9269i	−0.571999	+	0.990731i	0.424362	+	0.905493i	$0.360499\pi$
$227$	−8.61803	+	14.9269i	−0.571999	+	0.990731i	−0.996361	+	0.0852385i	$0.972835\pi$
$228$		0			0
$229$	11.7984	+	20.4354i	0.779658	+	1.35041i	0.932139	+	0.362102i	$0.117941\pi$
$229$	11.7984	+	20.4354i	0.779658	+	1.35041i	−0.152480	+	0.988307i	$0.548726\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	7.94427	+	13.7599i	0.520447	+	0.901440i	0.999717	+	0.0237728i	$0.00756784\pi$
$233$	7.94427	+	13.7599i	0.520447	+	0.901440i	−0.479271	+	0.877667i	$0.659099\pi$
$234$		0			0
$235$	−16.9443	+	29.3483i	−1.10532	+	1.91447i
$236$		0			0
$237$	−16.0000			−1.03931
$238$		0			0
$239$	−13.8885			−0.898375			−0.449188	−	0.893437i	$-0.648287\pi$
$239$	−13.8885			−0.898375			−0.449188	+	0.893437i	$0.648287\pi$
$240$		0			0
$241$	6.23607	−	10.8012i	0.401700	−	0.695766i	−0.592231	−	0.805768i	$-0.701753\pi$
$241$	6.23607	−	10.8012i	0.401700	−	0.695766i	0.993931	+	0.110003i	$0.0350860\pi$
$242$		0			0
$243$	6.79837	+	11.7751i	0.436116	+	0.755375i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	0.472136	+	0.817763i	0.0300413	+	0.0520330i
$248$		0			0
$249$	5.70820	−	9.88690i	0.361743	−	0.626557i
$250$		0			0
$251$	−4.29180			−0.270896			−0.135448	−	0.990784i	$-0.543247\pi$
$251$	−4.29180			−0.270896			−0.135448	+	0.990784i	$0.543247\pi$
$252$		0			0
$253$	−25.8885			−1.62760
$254$		0			0
$255$	8.94427	−	15.4919i	0.560112	−	0.970143i
$256$		0			0
$257$	−7.00000	−	12.1244i	−0.436648	−	0.756297i	0.560781	−	0.827964i	$-0.310501\pi$
$257$	−7.00000	−	12.1244i	−0.436648	−	0.756297i	−0.997429	+	0.0716680i	$0.977168\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−3.29180	−	5.70156i	−0.203757	−	0.352918i
$262$		0			0
$263$	−5.52786	+	9.57454i	−0.340863	+	0.590392i	−0.984593	−	0.174861i	$-0.944052\pi$
$263$	−5.52786	+	9.57454i	−0.340863	+	0.590392i	0.643730	+	0.765252i	$0.277386\pi$
$264$		0			0
$265$	32.3607			1.98790
$266$		0			0
$267$	−7.41641			−0.453877
$268$		0			0
$269$	−2.09017	+	3.62028i	−0.127440	+	0.220732i	−0.922684	−	0.385557i	$-0.874009\pi$
$269$	−2.09017	+	3.62028i	−0.127440	+	0.220732i	0.795244	+	0.606289i	$0.207343\pi$
$270$		0			0
$271$	−12.0000	−	20.7846i	−0.728948	−	1.26258i	−0.957328	−	0.289003i	$-0.906676\pi$
$271$	−12.0000	−	20.7846i	−0.728948	−	1.26258i	0.228380	−	0.973572i	$-0.426657\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	17.7082	+	30.6715i	1.06784	+	1.84956i
$276$		0			0
$277$	−3.94427	+	6.83168i	−0.236988	+	0.410476i	−0.959849	−	0.280518i	$-0.909494\pi$
$277$	−3.94427	+	6.83168i	−0.236988	+	0.410476i	0.722860	+	0.690994i	$0.242827\pi$
$278$		0			0
$279$	−3.63932			−0.217880
$280$		0			0
$281$	26.0000			1.55103			0.775515	−	0.631329i	$-0.217490\pi$
$281$	26.0000			1.55103			0.775515	+	0.631329i	$0.217490\pi$
$282$		0			0
$283$	−3.09017	+	5.35233i	−0.183692	+	0.318163i	−0.943135	−	0.332410i	$-0.892138\pi$
$283$	−3.09017	+	5.35233i	−0.183692	+	0.318163i	0.759443	+	0.650574i	$0.225471\pi$
$284$		0			0
$285$	−2.47214	−	4.28187i	−0.146437	−	0.253636i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.50000	−	2.59808i	−0.0882353	−	0.152828i
$290$		0			0
$291$	−7.70820	+	13.3510i	−0.451863	+	0.782650i
$292$		0			0
$293$	12.7639			0.745677			0.372838	−	0.927896i	$-0.378385\pi$
$293$	12.7639			0.745677			0.372838	+	0.927896i	$0.378385\pi$
$294$		0			0
$295$	−29.8885			−1.74018
$296$		0			0
$297$	17.8885	−	30.9839i	1.03800	−	1.79787i
$298$		0			0
$299$	1.52786	+	2.64634i	0.0883587	+	0.153042i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	1.05573	+	1.82857i	0.0606500	+	0.105049i
$304$		0			0
$305$	−18.1803	+	31.4893i	−1.04100	+	1.80307i
$306$		0			0
$307$	−1.81966			−0.103853			−0.0519267	−	0.998651i	$-0.516536\pi$
$307$	−1.81966			−0.103853			−0.0519267	+	0.998651i	$0.516536\pi$
$308$		0			0
$309$	−6.83282			−0.388705
$310$		0			0
$311$	4.00000	−	6.92820i	0.226819	−	0.392862i	−0.730044	−	0.683400i	$-0.760501\pi$
$311$	4.00000	−	6.92820i	0.226819	−	0.392862i	0.956864	+	0.290537i	$0.0938340\pi$
$312$		0			0
$313$	−4.23607	−	7.33708i	−0.239437	−	0.414717i	0.721116	−	0.692814i	$-0.243629\pi$
$313$	−4.23607	−	7.33708i	−0.239437	−	0.414717i	−0.960553	+	0.278098i	$0.910296\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−4.52786	−	7.84249i	−0.254310	−	0.440478i	0.710398	−	0.703800i	$-0.248515\pi$
$317$	−4.52786	−	7.84249i	−0.254310	−	0.440478i	−0.964708	+	0.263322i	$0.915182\pi$
$318$		0			0
$319$	−14.4721	+	25.0665i	−0.810284	+	1.40345i
$320$		0			0
$321$	−11.0557			−0.617071
$322$		0			0
$323$	−5.52786			−0.307579
$324$		0			0
$325$	2.09017	−	3.62028i	0.115942	−	0.200817i
$326$		0			0
$327$	5.23607	+	9.06914i	0.289555	+	0.501524i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−11.2361	−	19.4614i	−0.617590	−	1.06970i	−0.989924	−	0.141599i	$-0.954776\pi$
$331$	−11.2361	−	19.4614i	−0.617590	−	1.06970i	0.372334	−	0.928099i	$-0.378558\pi$
$332$		0			0
$333$	−3.29180	+	5.70156i	−0.180389	+	0.312443i
$334$		0			0
$335$	−12.9443			−0.707221
$336$		0			0
$337$	10.3607			0.564382			0.282191	−	0.959358i	$-0.408939\pi$
$337$	10.3607			0.564382			0.282191	+	0.959358i	$0.408939\pi$
$338$		0			0
$339$	−7.70820	+	13.3510i	−0.418652	+	0.725127i
$340$		0			0
$341$	8.00000	+	13.8564i	0.433224	+	0.750366i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−8.00000	−	13.8564i	−0.430706	−	0.746004i
$346$		0			0
$347$	−3.23607	+	5.60503i	−0.173721	+	0.300894i	−0.939718	−	0.341950i	$-0.888913\pi$
$347$	−3.23607	+	5.60503i	−0.173721	+	0.300894i	0.765997	+	0.642844i	$0.222246\pi$
$348$		0			0
$349$	−26.6525			−1.42667			−0.713337	−	0.700821i	$-0.752817\pi$
$349$	−26.6525			−1.42667			−0.713337	+	0.700821i	$0.752817\pi$
$350$		0			0
$351$	−4.22291			−0.225402
$352$		0			0
$353$	−7.94427	+	13.7599i	−0.422831	+	0.732365i	−0.996215	−	0.0869220i	$-0.972297\pi$
$353$	−7.94427	+	13.7599i	−0.422831	+	0.732365i	0.573384	+	0.819287i	$0.305630\pi$
$354$		0			0
$355$	−8.00000	−	13.8564i	−0.424596	−	0.735422i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	8.47214	+	14.6742i	0.447142	+	0.774473i	0.998199	−	0.0599949i	$-0.0191084\pi$
$359$	8.47214	+	14.6742i	0.447142	+	0.774473i	−0.551056	+	0.834468i	$0.685775\pi$
$360$		0			0
$361$	8.73607	−	15.1313i	0.459793	−	0.796385i
$362$		0			0
$363$	−38.1803			−2.00395
$364$		0			0
$365$	−9.52786			−0.498711
$366$		0			0
$367$	−11.4164	+	19.7738i	−0.595932	+	1.03218i	0.397483	+	0.917610i	$0.369884\pi$
$367$	−11.4164	+	19.7738i	−0.595932	+	1.03218i	−0.993415	+	0.114574i	$0.963450\pi$
$368$		0			0
$369$	6.23607	+	10.8012i	0.324637	+	0.562287i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	−1.47214	−	2.54981i	−0.0762243	−	0.132024i	0.825394	−	0.564558i	$-0.190953\pi$
$373$	−1.47214	−	2.54981i	−0.0762243	−	0.132024i	−0.901618	+	0.432533i	$0.857620\pi$
$374$		0			0
$375$	−0.944272	+	1.63553i	−0.0487620	+	0.0844582i
$376$		0			0
$377$	3.41641			0.175954
$378$		0			0
$379$	−4.58359			−0.235443			−0.117722	−	0.993047i	$-0.537559\pi$
$379$	−4.58359			−0.235443			−0.117722	+	0.993047i	$0.537559\pi$
$380$		0			0
$381$	5.52786	−	9.57454i	0.283201	−	0.490519i
$382$		0			0
$383$	−7.70820	−	13.3510i	−0.393871	−	0.682204i	0.599086	−	0.800685i	$-0.295531\pi$
$383$	−7.70820	−	13.3510i	−0.393871	−	0.682204i	−0.992956	+	0.118481i	$0.962198\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	4.76393	+	8.25137i	0.242164	+	0.419441i
$388$		0			0
$389$	2.23607	−	3.87298i	0.113373	−	0.196368i	−0.803755	−	0.594960i	$-0.797168\pi$
$389$	2.23607	−	3.87298i	0.113373	−	0.196368i	0.917128	+	0.398592i	$0.130501\pi$
$390$		0			0
$391$	−17.8885			−0.904663
$392$		0			0
$393$	14.4721			0.730023
$394$		0			0
$395$	20.9443	−	36.2765i	1.05382	−	1.82527i
$396$		0			0
$397$	−7.61803	−	13.1948i	−0.382338	−	0.662229i	0.609058	−	0.793126i	$-0.291548\pi$
$397$	−7.61803	−	13.1948i	−0.382338	−	0.662229i	−0.991396	+	0.130897i	$0.958214\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	11.7639	+	20.3757i	0.587463	+	1.01752i	0.994563	+	0.104132i	$0.0332065\pi$
$401$	11.7639	+	20.3757i	0.587463	+	1.01752i	−0.407101	+	0.913383i	$0.633460\pi$
$402$		0			0
$403$	0.944272	−	1.63553i	0.0470375	−	0.0814714i
$404$		0			0
$405$	7.81966			0.388562
$406$		0			0
$407$	28.9443			1.43471
$408$		0			0
$409$	−10.7082	+	18.5472i	−0.529487	+	0.917098i	0.469922	+	0.882708i	$0.344282\pi$
$409$	−10.7082	+	18.5472i	−0.529487	+	0.917098i	−0.999408	+	0.0343897i	$0.989051\pi$
$410$		0			0
$411$	9.23607	+	15.9973i	0.455582	+	0.789091i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	14.9443	+	25.8842i	0.733585	+	1.27061i
$416$		0			0
$417$	−0.763932	+	1.32317i	−0.0374099	+	0.0647959i
$418$		0			0
$419$	−22.1803			−1.08358			−0.541790	−	0.840514i	$-0.682253\pi$
$419$	−22.1803			−1.08358			−0.541790	+	0.840514i	$0.682253\pi$
$420$		0			0
$421$	−2.00000			−0.0974740			−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000			−0.0974740			−0.0487370	+	0.998812i	$0.515520\pi$
$422$		0			0
$423$	−7.70820	+	13.3510i	−0.374786	+	0.649148i
$424$		0			0
$425$	12.2361	+	21.1935i	0.593536	+	1.02804i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	3.05573	+	5.29268i	0.147532	+	0.255533i
$430$		0			0
$431$	14.0000	−	24.2487i	0.674356	−	1.16802i	−0.302300	−	0.953213i	$-0.597755\pi$
$431$	14.0000	−	24.2487i	0.674356	−	1.16802i	0.976657	−	0.214807i	$-0.0689121\pi$
$432$		0			0
$433$	−9.41641			−0.452524			−0.226262	−	0.974067i	$-0.572651\pi$
$433$	−9.41641			−0.452524			−0.226262	+	0.974067i	$0.572651\pi$
$434$		0			0
$435$	−17.8885			−0.857690
$436$		0			0
$437$	−2.47214	+	4.28187i	−0.118258	+	0.204829i
$438$		0			0
$439$	−16.0000	−	27.7128i	−0.763638	−	1.32266i	−0.940963	−	0.338508i	$-0.890078\pi$
$439$	−16.0000	−	27.7128i	−0.763638	−	1.32266i	0.177325	−	0.984152i	$-0.443256\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	6.94427	+	12.0278i	0.329932	+	0.571460i	0.982498	−	0.186273i	$-0.0596407\pi$
$443$	6.94427	+	12.0278i	0.329932	+	0.571460i	−0.652566	+	0.757732i	$0.726307\pi$
$444$		0			0
$445$	9.70820	−	16.8151i	0.460213	−	0.797112i
$446$		0			0
$447$	−3.63932			−0.172134
$448$		0			0
$449$	−7.88854			−0.372283			−0.186142	−	0.982523i	$-0.559598\pi$
$449$	−7.88854			−0.372283			−0.186142	+	0.982523i	$0.559598\pi$
$450$		0			0
$451$	27.4164	−	47.4866i	1.29099	−	2.23606i
$452$		0			0
$453$	−5.52786	−	9.57454i	−0.259722	−	0.449851i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	3.76393	+	6.51932i	0.176069	+	0.304961i	0.940531	−	0.339708i	$-0.110328\pi$
$457$	3.76393	+	6.51932i	0.176069	+	0.304961i	−0.764462	+	0.644669i	$0.776995\pi$
$458$		0			0
$459$	12.3607	−	21.4093i	0.576947	−	0.999302i
$460$		0			0
$461$	−21.7082			−1.01105			−0.505526	−	0.862811i	$-0.668702\pi$
$461$	−21.7082			−1.01105			−0.505526	+	0.862811i	$0.668702\pi$
$462$		0			0
$463$	−35.7771			−1.66270			−0.831351	−	0.555748i	$-0.812432\pi$
$463$	−35.7771			−1.66270			−0.831351	+	0.555748i	$0.812432\pi$
$464$		0			0
$465$	−4.94427	+	8.56373i	−0.229285	+	0.397133i
$466$		0			0
$467$	−16.0344	−	27.7725i	−0.741985	−	1.28516i	−0.951590	−	0.307370i	$-0.900551\pi$
$467$	−16.0344	−	27.7725i	−0.741985	−	1.28516i	0.209605	−	0.977786i	$-0.432782\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−0.472136	−	0.817763i	−0.0217549	−	0.0376806i
$472$		0			0
$473$	20.9443	−	36.2765i	0.963019	−	1.66800i
$474$		0			0
$475$	6.76393			0.310350
$476$		0			0
$477$	14.7214			0.674045
$478$		0			0
$479$	−4.29180	+	7.43361i	−0.196097	+	0.339650i	−0.947260	−	0.320467i	$-0.896160\pi$
$479$	−4.29180	+	7.43361i	−0.196097	+	0.339650i	0.751162	+	0.660117i	$0.229493\pi$
$480$		0			0
$481$	−1.70820	−	2.95870i	−0.0778874	−	0.134905i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−20.1803	−	34.9534i	−0.916342	−	1.58715i
$486$		0			0
$487$	−10.0000	+	17.3205i	−0.453143	+	0.784867i	−0.998579	−	0.0532853i	$-0.983031\pi$
$487$	−10.0000	+	17.3205i	−0.453143	+	0.784867i	0.545436	+	0.838152i	$0.316364\pi$
$488$		0			0
$489$	4.22291			0.190967
$490$		0			0
$491$	37.8885			1.70989			0.854943	−	0.518722i	$-0.173592\pi$
$491$	37.8885			1.70989			0.854943	+	0.518722i	$0.173592\pi$
$492$		0			0
$493$	−10.0000	+	17.3205i	−0.450377	+	0.780076i
$494$		0			0
$495$	15.4164	+	26.7020i	0.692916	+	1.20017i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−10.9443	−	18.9560i	−0.489933	−	0.848589i	0.510000	−	0.860174i	$-0.329645\pi$
$499$	−10.9443	−	18.9560i	−0.489933	−	0.848589i	−0.999933	+	0.0115857i	$0.996312\pi$
$500$		0			0
$501$	−14.4721	+	25.0665i	−0.646567	+	1.11989i
$502$		0			0
$503$	4.94427			0.220454			0.110227	−	0.993906i	$-0.464842\pi$
$503$	4.94427			0.220454			0.110227	+	0.993906i	$0.464842\pi$
$504$		0			0
$505$	−5.52786			−0.245987
$506$		0			0
$507$	−7.67376	+	13.2913i	−0.340804	+	0.590289i
$508$		0			0
$509$	−20.5623	−	35.6150i	−0.911408	−	1.57861i	−0.812077	−	0.583551i	$-0.801663\pi$
$509$	−20.5623	−	35.6150i	−0.911408	−	1.57861i	−0.0993316	−	0.995054i	$-0.531670\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−3.41641	−	5.91739i	−0.150838	−	0.261259i
$514$		0			0
$515$	8.94427	−	15.4919i	0.394132	−	0.682656i
$516$		0			0
$517$	67.7771			2.98083
$518$		0			0
$519$	7.05573			0.309712
$520$		0			0
$521$	−3.29180	+	5.70156i	−0.144216	+	0.249790i	−0.929080	−	0.369878i	$-0.879399\pi$
$521$	−3.29180	+	5.70156i	−0.144216	+	0.249790i	0.784864	+	0.619668i	$0.212733\pi$
$522$		0			0
$523$	−2.14590	−	3.71680i	−0.0938336	−	0.162525i	0.815288	−	0.579056i	$-0.196579\pi$
$523$	−2.14590	−	3.71680i	−0.0938336	−	0.162525i	−0.909121	+	0.416532i	$0.863245\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	5.52786	+	9.57454i	0.240798	+	0.417074i
$528$		0			0
$529$	3.50000	−	6.06218i	0.152174	−	0.263573i
$530$		0			0
$531$	−13.5967			−0.590049
$532$		0			0
$533$	−6.47214			−0.280339
$534$		0			0
$535$	14.4721	−	25.0665i	0.625685	−	1.08372i
$536$		0			0
$537$	−4.36068	−	7.55292i	−0.188177	−	0.325933i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	2.52786	+	4.37839i	0.108681	+	0.188242i	0.915236	−	0.402917i	$-0.132004\pi$
$541$	2.52786	+	4.37839i	0.108681	+	0.188242i	−0.806555	+	0.591159i	$0.798671\pi$
$542$		0			0
$543$	7.52786	−	13.0386i	0.323052	−	0.559542i
$544$		0			0
$545$	−27.4164			−1.17439
$546$		0			0
$547$	−4.58359			−0.195980			−0.0979901	−	0.995187i	$-0.531241\pi$
$547$	−4.58359			−0.195980			−0.0979901	+	0.995187i	$0.531241\pi$
$548$		0			0
$549$	−8.27051	+	14.3249i	−0.352977	+	0.611374i
$550$		0			0
$551$	2.76393	+	4.78727i	0.117747	+	0.203945i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	8.94427	+	15.4919i	0.379663	+	0.657596i
$556$		0			0
$557$	−4.52786	+	7.84249i	−0.191852	+	0.332297i	−0.945864	−	0.324563i	$-0.894783\pi$
$557$	−4.52786	+	7.84249i	−0.191852	+	0.332297i	0.754012	+	0.656860i	$0.228116\pi$
$558$		0			0
$559$	−4.94427			−0.209120
$560$		0			0
$561$	−35.7771			−1.51051
$562$		0			0
$563$	−8.90983	+	15.4323i	−0.375505	+	0.650393i	−0.990402	−	0.138214i	$-0.955864\pi$
$563$	−8.90983	+	15.4323i	−0.375505	+	0.650393i	0.614898	+	0.788607i	$0.289197\pi$
$564$		0			0
$565$	−20.1803	−	34.9534i	−0.848993	−	1.47050i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−13.1803	−	22.8290i	−0.552549	−	0.957042i	−0.998090	−	0.0617808i	$-0.980322\pi$
$569$	−13.1803	−	22.8290i	−0.552549	−	0.957042i	0.445541	−	0.895261i	$-0.353011\pi$
$570$		0			0
$571$	7.23607	−	12.5332i	0.302820	−	0.524500i	−0.673954	−	0.738774i	$-0.735405\pi$
$571$	7.23607	−	12.5332i	0.302820	−	0.524500i	0.976774	+	0.214274i	$0.0687386\pi$
$572$		0			0
$573$	−6.11146			−0.255310
$574$		0			0
$575$	21.8885			0.912815
$576$		0			0
$577$	−3.00000	+	5.19615i	−0.124892	+	0.216319i	−0.921691	−	0.387926i	$-0.873192\pi$
$577$	−3.00000	+	5.19615i	−0.124892	+	0.216319i	0.796799	+	0.604245i	$0.206525\pi$
$578$		0			0
$579$	0.291796	+	0.505406i	0.0121266	+	0.0210039i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−32.3607	−	56.0503i	−1.34024	−	2.32137i
$584$		0			0
$585$	1.81966	−	3.15174i	0.0752337	−	0.130309i
$586$		0			0
$587$	3.70820			0.153054			0.0765270	−	0.997068i	$-0.475617\pi$
$587$	3.70820			0.153054			0.0765270	+	0.997068i	$0.475617\pi$
$588$		0			0
$589$	3.05573			0.125909
$590$		0			0
$591$	6.76393	−	11.7155i	0.278231	−	0.481910i
$592$		0			0
$593$	16.4164	+	28.4341i	0.674141	+	1.16765i	0.976719	+	0.214522i	$0.0688194\pi$
$593$	16.4164	+	28.4341i	0.674141	+	1.16765i	−0.302578	+	0.953125i	$0.597847\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−9.52786	−	16.5027i	−0.389950	−	0.675412i
$598$		0			0
$599$	−8.94427	+	15.4919i	−0.365453	+	0.632983i	−0.988849	−	0.148923i	$-0.952419\pi$
$599$	−8.94427	+	15.4919i	−0.365453	+	0.632983i	0.623396	+	0.781907i	$0.285753\pi$
$600$		0			0
$601$	−29.7771			−1.21463			−0.607316	−	0.794460i	$-0.707754\pi$
$601$	−29.7771			−1.21463			−0.607316	+	0.794460i	$0.707754\pi$
$602$		0			0
$603$	−5.88854			−0.239800
$604$		0			0
$605$	49.9787	−	86.5657i	2.03192	−	3.51940i
$606$		0			0
$607$	4.94427	+	8.56373i	0.200682	+	0.347591i	0.948748	−	0.316033i	$-0.102351\pi$
$607$	4.94427	+	8.56373i	0.200682	+	0.347591i	−0.748066	+	0.663624i	$0.769018\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−4.00000	−	6.92820i	−0.161823	−	0.280285i
$612$		0			0
$613$	−14.7082	+	25.4754i	−0.594059	+	1.02894i	0.399620	+	0.916681i	$0.369142\pi$
$613$	−14.7082	+	25.4754i	−0.594059	+	1.02894i	−0.993679	+	0.112259i	$0.964191\pi$
$614$		0			0
$615$	33.8885			1.36652
$616$		0			0
$617$	34.3607			1.38331			0.691654	−	0.722229i	$-0.256882\pi$
$617$	34.3607			1.38331			0.691654	+	0.722229i	$0.256882\pi$
$618$		0			0
$619$	−24.0344	+	41.6289i	−0.966026	+	1.67321i	−0.259192	+	0.965826i	$0.583456\pi$
$619$	−24.0344	+	41.6289i	−0.966026	+	1.67321i	−0.706833	+	0.707380i	$0.749877\pi$
$620$		0			0
$621$	−11.0557	−	19.1491i	−0.443651	−	0.768426i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	11.2082	+	19.4132i	0.448328	+	0.776527i
$626$		0			0
$627$	−4.94427	+	8.56373i	−0.197455	+	0.342002i
$628$		0			0
$629$	20.0000			0.797452
$630$		0			0
$631$	−44.9443			−1.78920			−0.894602	−	0.446865i	$-0.852541\pi$
$631$	−44.9443			−1.78920			−0.894602	+	0.446865i	$0.852541\pi$
$632$		0			0
$633$	−7.41641	+	12.8456i	−0.294776	+	0.510567i
$634$		0			0
$635$	14.4721	+	25.0665i	0.574309	+	0.994733i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−3.63932	−	6.30349i	−0.143969	−	0.249362i
$640$		0			0
$641$	−7.29180	+	12.6298i	−0.288009	+	0.498846i	−0.973334	−	0.229391i	$-0.926327\pi$
$641$	−7.29180	+	12.6298i	−0.288009	+	0.498846i	0.685326	+	0.728237i	$0.259660\pi$
$642$		0			0
$643$	43.7082			1.72368			0.861842	−	0.507177i	$-0.169311\pi$
$643$	43.7082			1.72368			0.861842	+	0.507177i	$0.169311\pi$
$644$		0			0
$645$	25.8885			1.01936
$646$		0			0
$647$	−10.1803	+	17.6329i	−0.400230	+	0.693219i	−0.993753	−	0.111598i	$-0.964403\pi$
$647$	−10.1803	+	17.6329i	−0.400230	+	0.693219i	0.593523	+	0.804817i	$0.297737\pi$
$648$		0			0
$649$	29.8885	+	51.7685i	1.17323	+	2.03209i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	4.70820	+	8.15485i	0.184246	+	0.319124i	0.943322	−	0.331878i	$-0.107682\pi$
$653$	4.70820	+	8.15485i	0.184246	+	0.319124i	−0.759076	+	0.651002i	$0.774349\pi$
$654$		0			0
$655$	−18.9443	+	32.8124i	−0.740214	+	1.28209i
$656$		0			0
$657$	−4.33437			−0.169100
$658$		0			0
$659$	37.3050			1.45319			0.726597	−	0.687064i	$-0.241101\pi$
$659$	37.3050			1.45319			0.726597	+	0.687064i	$0.241101\pi$
$660$		0			0
$661$	−11.3262	+	19.6176i	−0.440540	+	0.763037i	−0.997730	−	0.0673481i	$-0.978546\pi$
$661$	−11.3262	+	19.6176i	−0.440540	+	0.763037i	0.557190	+	0.830385i	$0.311880\pi$
$662$		0			0
$663$	2.11146	+	3.65715i	0.0820022	+	0.142032i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	8.94427	+	15.4919i	0.346324	+	0.599850i
$668$		0			0
$669$	−8.00000	+	13.8564i	−0.309298	+	0.535720i
$670$		0			0
$671$	72.7214			2.80738
$672$		0			0
$673$	−2.94427			−0.113493			−0.0567467	−	0.998389i	$-0.518073\pi$
$673$	−2.94427			−0.113493			−0.0567467	+	0.998389i	$0.518073\pi$
$674$		0			0
$675$	−15.1246	+	26.1966i	−0.582147	+	1.00831i
$676$		0			0
$677$	9.90983	+	17.1643i	0.380866	+	0.659679i	0.991186	−	0.132476i	$-0.0422927\pi$
$677$	9.90983	+	17.1643i	0.380866	+	0.659679i	−0.610321	+	0.792155i	$0.708959\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	10.6525	+	18.4506i	0.408204	+	0.707030i
$682$		0			0
$683$	−11.8885	+	20.5916i	−0.454902	+	0.787914i	−0.998683	−	0.0513135i	$-0.983659\pi$
$683$	−11.8885	+	20.5916i	−0.454902	+	0.787914i	0.543780	+	0.839228i	$0.316993\pi$
$684$		0			0
$685$	−48.3607			−1.84777
$686$		0			0
$687$	29.1672			1.11280
$688$		0			0
$689$	−3.81966	+	6.61585i	−0.145517	+	0.252044i
$690$		0			0
$691$	−7.09017	−	12.2805i	−0.269723	−	0.467174i	0.699067	−	0.715056i	$-0.253599\pi$
$691$	−7.09017	−	12.2805i	−0.269723	−	0.467174i	−0.968790	+	0.247882i	$0.920265\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−2.00000	−	3.46410i	−0.0758643	−	0.131401i
$696$		0			0
$697$	18.9443	−	32.8124i	0.717565	−	1.24286i
$698$		0			0
$699$	19.6393			0.742827
$700$		0			0
$701$	−41.4164			−1.56428			−0.782138	−	0.623105i	$-0.785871\pi$
$701$	−41.4164			−1.56428			−0.782138	+	0.623105i	$0.785871\pi$
$702$		0			0
$703$	2.76393	−	4.78727i	0.104244	−	0.180555i
$704$		0			0
$705$	20.9443	+	36.2765i	0.788807	+	1.36625i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.819660	−	1.41969i	−0.0307830	−	0.0533177i	0.850223	−	0.526422i	$-0.176467\pi$
$709$	−0.819660	−	1.41969i	−0.0307830	−	0.0533177i	−0.881006	+	0.473104i	$0.843133\pi$
$710$		0			0
$711$	9.52786	−	16.5027i	0.357323	−	0.618901i
$712$		0			0
$713$	9.88854			0.370329
$714$		0			0
$715$	−16.0000			−0.598366
$716$		0			0
$717$	−8.58359	+	14.8672i	−0.320560	+	0.555226i
$718$		0			0
$719$	25.5967	+	44.3349i	0.954598	+	1.65341i	0.735286	+	0.677757i	$0.237048\pi$
$719$	25.5967	+	44.3349i	0.954598	+	1.65341i	0.219311	+	0.975655i	$0.429619\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−7.70820	−	13.3510i	−0.286671	−	0.496529i
$724$		0			0
$725$	12.2361	−	21.1935i	0.454436	−	0.787107i
$726$		0			0
$727$	−12.3607			−0.458432			−0.229216	−	0.973376i	$-0.573616\pi$
$727$	−12.3607			−0.458432			−0.229216	+	0.973376i	$0.573616\pi$
$728$		0			0
$729$	24.0557			0.890953
$730$		0			0
$731$	14.4721	−	25.0665i	0.535271	−	0.927117i
$732$		0			0
$733$	−2.38197	−	4.12569i	−0.0879799	−	0.152386i	0.818677	−	0.574254i	$-0.194708\pi$
$733$	−2.38197	−	4.12569i	−0.0879799	−	0.152386i	−0.906657	+	0.421868i	$0.861374\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	12.9443	+	22.4201i	0.476808	+	0.825856i
$738$		0			0
$739$	1.70820	−	2.95870i	0.0628373	−	0.108837i	−0.832895	−	0.553431i	$-0.813318\pi$
$739$	1.70820	−	2.95870i	0.0628373	−	0.108837i	0.895733	+	0.444593i	$0.146652\pi$
$740$		0			0
$741$	1.16718			0.0428776
$742$		0			0
$743$	24.9443			0.915117			0.457558	−	0.889180i	$-0.348724\pi$
$743$	24.9443			0.915117			0.457558	+	0.889180i	$0.348724\pi$
$744$		0			0
$745$	4.76393	−	8.25137i	0.174537	−	0.302307i
$746$		0			0
$747$	6.79837	+	11.7751i	0.248739	+	0.430829i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−18.0000	−	31.1769i	−0.656829	−	1.13766i	−0.981432	−	0.191811i	$-0.938564\pi$
$751$	−18.0000	−	31.1769i	−0.656829	−	1.13766i	0.324603	−	0.945851i	$-0.394769\pi$
$752$		0			0
$753$	−2.65248	+	4.59422i	−0.0966616	+	0.167423i
$754$		0			0
$755$	28.9443			1.05339
$756$		0			0
$757$	39.3050			1.42856			0.714281	−	0.699859i	$-0.246754\pi$
$757$	39.3050			1.42856			0.714281	+	0.699859i	$0.246754\pi$
$758$		0			0
$759$	−16.0000	+	27.7128i	−0.580763	+	1.00591i
$760$		0			0
$761$	−1.76393	−	3.05522i	−0.0639425	−	0.110752i	0.832282	−	0.554353i	$-0.187034\pi$
$761$	−1.76393	−	3.05522i	−0.0639425	−	0.110752i	−0.896224	+	0.443601i	$0.853701\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	10.6525	+	18.4506i	0.385141	+	0.667084i
$766$		0			0
$767$	3.52786	−	6.11044i	0.127384	−	0.220635i
$768$		0			0
$769$	18.3607			0.662103			0.331052	−	0.943613i	$-0.392597\pi$
$769$	18.3607			0.662103			0.331052	+	0.943613i	$0.392597\pi$
$770$		0			0
$771$	−17.3050			−0.623223
$772$		0			0
$773$	20.0902	−	34.7972i	0.722593	−	1.25157i	−0.237364	−	0.971421i	$-0.576283\pi$
$773$	20.0902	−	34.7972i	0.722593	−	1.25157i	0.959957	−	0.280147i	$-0.0903833\pi$
$774$		0			0
$775$	−6.76393	−	11.7155i	−0.242968	−	0.420832i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−5.23607	−	9.06914i	−0.187602	−	0.324936i
$780$		0			0
$781$	−16.0000	+	27.7128i	−0.572525	+	0.991642i
$782$		0			0
$783$	−24.7214			−0.883469
$784$		0			0
$785$	2.47214			0.0882343
$786$		0			0
$787$	−6.14590	+	10.6450i	−0.219078	+	0.379454i	−0.954526	−	0.298127i	$-0.903638\pi$
$787$	−6.14590	+	10.6450i	−0.219078	+	0.379454i	0.735449	+	0.677580i	$0.236971\pi$
$788$		0			0
$789$	6.83282	+	11.8348i	0.243255	+	0.421329i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−4.29180	−	7.43361i	−0.152406	−	0.263975i
$794$

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:	\( N \)	\(=\)	\( 1568 = 2^{5} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1568.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.618034 - 1.07047i) q^{3} +(1.61803 + 2.80252i) q^{5} +(0.736068 + 1.27491i) q^{9} +O(q^{10})\)	\(q+(0.618034 - 1.07047i) q^{3} +(1.61803 + 2.80252i) q^{5} +(0.736068 + 1.27491i) q^{9} +(3.23607 - 5.60503i) q^{11} -0.763932 q^{13} +4.00000 q^{15} +(2.23607 - 3.87298i) q^{17} +(-0.618034 - 1.07047i) q^{19} +(-2.00000 - 3.46410i) q^{23} +(-2.73607 + 4.73901i) q^{25} +5.52786 q^{27} -4.47214 q^{29} +(-1.23607 + 2.14093i) q^{31} +(-4.00000 - 6.92820i) q^{33} +(2.23607 + 3.87298i) q^{37} +(-0.472136 + 0.817763i) q^{39} +8.47214 q^{41} +6.47214 q^{43} +(-2.38197 + 4.12569i) q^{45} +(5.23607 + 9.06914i) q^{47} +(-2.76393 - 4.78727i) q^{51} +(5.00000 - 8.66025i) q^{53} +20.9443 q^{55} -1.52786 q^{57} +(-4.61803 + 7.99867i) q^{59} +(5.61803 + 9.73072i) q^{61} +(-1.23607 - 2.14093i) q^{65} +(-2.00000 + 3.46410i) q^{67} -4.94427 q^{69} -4.94427 q^{71} +(-1.47214 + 2.54981i) q^{73} +(3.38197 + 5.85774i) q^{75} +(-6.47214 - 11.2101i) q^{79} +(1.20820 - 2.09267i) q^{81} +9.23607 q^{83} +14.4721 q^{85} +(-2.76393 + 4.78727i) q^{87} +(-3.00000 - 5.19615i) q^{89} +(1.52786 + 2.64634i) q^{93} +(2.00000 - 3.46410i) q^{95} -12.4721 q^{97} +9.52786 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{3} + 2 q^{5} - 6 q^{9}+O(q^{10}) \) 4 * q - 2 * q^3 + 2 * q^5 - 6 * q^9	\( 4 q - 2 q^{3} + 2 q^{5} - 6 q^{9} + 4 q^{11} - 12 q^{13} + 16 q^{15} + 2 q^{19} - 8 q^{23} - 2 q^{25} + 40 q^{27} + 4 q^{31} - 16 q^{33} + 16 q^{39} + 16 q^{41} + 8 q^{43} - 14 q^{45} + 12 q^{47} - 20 q^{51} + 20 q^{53} + 48 q^{55} - 24 q^{57} - 14 q^{59} + 18 q^{61} + 4 q^{65} - 8 q^{67} + 16 q^{69} + 16 q^{71} + 12 q^{73} + 18 q^{75} - 8 q^{79} - 22 q^{81} + 28 q^{83} + 40 q^{85} - 20 q^{87} - 12 q^{89} + 24 q^{93} + 8 q^{95} - 32 q^{97} + 56 q^{99}+O(q^{100}) \) 4 * q - 2 * q^3 + 2 * q^5 - 6 * q^9 + 4 * q^11 - 12 * q^13 + 16 * q^15 + 2 * q^19 - 8 * q^23 - 2 * q^25 + 40 * q^27 + 4 * q^31 - 16 * q^33 + 16 * q^39 + 16 * q^41 + 8 * q^43 - 14 * q^45 + 12 * q^47 - 20 * q^51 + 20 * q^53 + 48 * q^55 - 24 * q^57 - 14 * q^59 + 18 * q^61 + 4 * q^65 - 8 * q^67 + 16 * q^69 + 16 * q^71 + 12 * q^73 + 18 * q^75 - 8 * q^79 - 22 * q^81 + 28 * q^83 + 40 * q^85 - 20 * q^87 - 12 * q^89 + 24 * q^93 + 8 * q^95 - 32 * q^97 + 56 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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