LMFDB - Embedded newform 1536.2.c.k.1535.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1536,2,Mod(1535,1536)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1536, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1536.1535");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1536 = 2^{9} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1536.c (of order $2$, degree $1$, 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:	$12.2650217505$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.40960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1535.4
Root			$-0.437016 + 0.437016i$ of defining polynomial
Character	$\chi$	$=$	1536.1535
Dual form			1536.2.c.k.1535.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 + 1.58114i) q^{3} +2.00000i q^{5} +(-2.00000 - 2.23607i) q^{9} +O(q^{10})$	$q+(-0.707107 + 1.58114i) q^{3} +2.00000i q^{5} +(-2.00000 - 2.23607i) q^{9} +1.41421 q^{11} +4.47214 q^{13} +(-3.16228 - 1.41421i) q^{15} +4.47214i q^{17} +3.16228i q^{19} -6.32456 q^{23} +1.00000 q^{25} +(4.94975 - 1.58114i) q^{27} +6.00000i q^{29} +8.48528i q^{31} +(-1.00000 + 2.23607i) q^{33} +4.47214 q^{37} +(-3.16228 + 7.07107i) q^{39} -8.94427i q^{41} -3.16228i q^{43} +(4.47214 - 4.00000i) q^{45} -12.6491 q^{47} +7.00000 q^{49} +(-7.07107 - 3.16228i) q^{51} +6.00000i q^{53} +2.82843i q^{55} +(-5.00000 - 2.23607i) q^{57} -9.89949 q^{59} -13.4164 q^{61} +8.94427i q^{65} -3.16228i q^{67} +(4.47214 - 10.0000i) q^{69} -6.32456 q^{71} -4.00000 q^{73} +(-0.707107 + 1.58114i) q^{75} -2.82843i q^{79} +(-1.00000 + 8.94427i) q^{81} +12.7279 q^{83} -8.94427 q^{85} +(-9.48683 - 4.24264i) q^{87} -13.4164i q^{89} +(-13.4164 - 6.00000i) q^{93} -6.32456 q^{95} +8.00000 q^{97} +(-2.82843 - 3.16228i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{9}+O(q^{10}) $ 8 * q - 16 * q^9	$ 8 q - 16 q^{9} + 8 q^{25} - 8 q^{33} + 56 q^{49} - 40 q^{57} - 32 q^{73} - 8 q^{81} + 64 q^{97}+O(q^{100}) $ 8 * q - 16 * q^9 + 8 * q^25 - 8 * q^33 + 56 * q^49 - 40 * q^57 - 32 * q^73 - 8 * q^81 + 64 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1536\mathbb{Z}\right)^\times$.

$n$	$511$	$517$	$1025$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.707107	+	1.58114i	−0.408248	+	0.912871i
$3$	−0.707107	+	1.58114i	−0.408248	+	0.912871i
$4$		0			0
$5$			2.00000i			0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$			2.00000i			0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$		0			0
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		0			0
$9$	−2.00000	−	2.23607i	−0.666667	−	0.745356i
$10$		0			0
$11$	1.41421			0.426401			0.213201	−	0.977008i	$-0.431611\pi$
$11$	1.41421			0.426401			0.213201	+	0.977008i	$0.431611\pi$
$12$		0			0
$13$	4.47214			1.24035			0.620174	−	0.784465i	$-0.287062\pi$
$13$	4.47214			1.24035			0.620174	+	0.784465i	$0.287062\pi$
$14$		0			0
$15$	−3.16228	−	1.41421i	−0.816497	−	0.365148i
$16$		0			0
$17$			4.47214i			1.08465i	0.840168	+	0.542326i	$0.182456\pi$
$17$			4.47214i			1.08465i	−0.840168	+	0.542326i	$0.817544\pi$
$18$		0			0
$19$			3.16228i			0.725476i	0.931891	+	0.362738i	$0.118158\pi$
$19$			3.16228i			0.725476i	−0.931891	+	0.362738i	$0.881842\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.32456			−1.31876			−0.659380	−	0.751809i	$-0.729181\pi$
$23$	−6.32456			−1.31876			−0.659380	+	0.751809i	$0.729181\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	4.94975	−	1.58114i	0.952579	−	0.304290i
$28$		0			0
$29$			6.00000i			1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$			6.00000i			1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$		0			0
$31$			8.48528i			1.52400i	0.647576	+	0.762001i	$0.275783\pi$
$31$			8.48528i			1.52400i	−0.647576	+	0.762001i	$0.724217\pi$
$32$		0			0
$33$	−1.00000	+	2.23607i	−0.174078	+	0.389249i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	4.47214			0.735215			0.367607	−	0.929981i	$-0.380177\pi$
$37$	4.47214			0.735215			0.367607	+	0.929981i	$0.380177\pi$
$38$		0			0
$39$	−3.16228	+	7.07107i	−0.506370	+	1.13228i
$40$		0			0
$41$		−	8.94427i		−	1.39686i	−0.715678	−	0.698430i	$-0.753882\pi$
$41$		−	8.94427i		−	1.39686i	0.715678	−	0.698430i	$-0.246118\pi$
$42$		0			0
$43$		−	3.16228i		−	0.482243i	−0.970495	−	0.241121i	$-0.922485\pi$
$43$		−	3.16228i		−	0.482243i	0.970495	−	0.241121i	$-0.0775152\pi$
$44$		0			0
$45$	4.47214	−	4.00000i	0.666667	−	0.596285i
$46$		0			0
$47$	−12.6491			−1.84506			−0.922531	−	0.385922i	$-0.873883\pi$
$47$	−12.6491			−1.84506			−0.922531	+	0.385922i	$0.873883\pi$
$48$		0			0
$49$	7.00000			1.00000
$50$		0			0
$51$	−7.07107	−	3.16228i	−0.990148	−	0.442807i
$52$		0			0
$53$			6.00000i			0.824163i	0.911147	+	0.412082i	$0.135198\pi$
$53$			6.00000i			0.824163i	−0.911147	+	0.412082i	$0.864802\pi$
$54$		0			0
$55$			2.82843i			0.381385i
$56$		0			0
$57$	−5.00000	−	2.23607i	−0.662266	−	0.296174i
$58$		0			0
$59$	−9.89949			−1.28880			−0.644402	−	0.764687i	$-0.722894\pi$
$59$	−9.89949			−1.28880			−0.644402	+	0.764687i	$0.722894\pi$
$60$		0			0
$61$	−13.4164			−1.71780			−0.858898	−	0.512148i	$-0.828850\pi$
$61$	−13.4164			−1.71780			−0.858898	+	0.512148i	$0.828850\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$			8.94427i			1.10940i
$66$		0			0
$67$		−	3.16228i		−	0.386334i	−0.981166	−	0.193167i	$-0.938124\pi$
$67$		−	3.16228i		−	0.386334i	0.981166	−	0.193167i	$-0.0618759\pi$
$68$		0			0
$69$	4.47214	−	10.0000i	0.538382	−	1.20386i
$70$		0			0
$71$	−6.32456			−0.750587			−0.375293	−	0.926906i	$-0.622458\pi$
$71$	−6.32456			−0.750587			−0.375293	+	0.926906i	$0.622458\pi$
$72$		0			0
$73$	−4.00000			−0.468165			−0.234082	−	0.972217i	$-0.575209\pi$
$73$	−4.00000			−0.468165			−0.234082	+	0.972217i	$0.575209\pi$
$74$		0			0
$75$	−0.707107	+	1.58114i	−0.0816497	+	0.182574i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		−	2.82843i		−	0.318223i	−0.987261	−	0.159111i	$-0.949137\pi$
$79$		−	2.82843i		−	0.318223i	0.987261	−	0.159111i	$-0.0508629\pi$
$80$		0			0
$81$	−1.00000	+	8.94427i	−0.111111	+	0.993808i
$82$		0			0
$83$	12.7279			1.39707			0.698535	−	0.715575i	$-0.253835\pi$
$83$	12.7279			1.39707			0.698535	+	0.715575i	$0.253835\pi$
$84$		0			0
$85$	−8.94427			−0.970143
$86$		0			0
$87$	−9.48683	−	4.24264i	−1.01710	−	0.454859i
$88$		0			0
$89$		−	13.4164i		−	1.42214i	−0.703123	−	0.711068i	$-0.748212\pi$
$89$		−	13.4164i		−	1.42214i	0.703123	−	0.711068i	$-0.251788\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−13.4164	−	6.00000i	−1.39122	−	0.622171i
$94$		0			0
$95$	−6.32456			−0.648886
$96$		0			0
$97$	8.00000			0.812277			0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000			0.812277			0.406138	+	0.913812i	$0.366875\pi$
$98$		0			0
$99$	−2.82843	−	3.16228i	−0.284268	−	0.317821i
$100$		0			0
$101$		−	2.00000i		−	0.199007i	−0.995037	−	0.0995037i	$-0.968274\pi$
$101$		−	2.00000i		−	0.199007i	0.995037	−	0.0995037i	$-0.0317255\pi$
$102$		0			0
$103$		−	5.65685i		−	0.557386i	−0.960380	−	0.278693i	$-0.910099\pi$
$103$		−	5.65685i		−	0.557386i	0.960380	−	0.278693i	$-0.0899013\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−7.07107			−0.683586			−0.341793	−	0.939775i	$-0.611034\pi$
$107$	−7.07107			−0.683586			−0.341793	+	0.939775i	$0.611034\pi$
$108$		0			0
$109$	13.4164			1.28506			0.642529	−	0.766261i	$-0.277885\pi$
$109$	13.4164			1.28506			0.642529	+	0.766261i	$0.277885\pi$
$110$		0			0
$111$	−3.16228	+	7.07107i	−0.300150	+	0.671156i
$112$		0			0
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		−	12.6491i		−	1.17954i
$116$		0			0
$117$	−8.94427	−	10.0000i	−0.826898	−	0.924500i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−9.00000			−0.818182
$122$		0			0
$123$	14.1421	+	6.32456i	1.27515	+	0.570266i
$124$		0			0
$125$			12.0000i			1.07331i
$126$		0			0
$127$			2.82843i			0.250982i	0.992095	+	0.125491i	$0.0400507\pi$
$127$			2.82843i			0.250982i	−0.992095	+	0.125491i	$0.959949\pi$
$128$		0			0
$129$	5.00000	+	2.23607i	0.440225	+	0.196875i
$130$		0			0
$131$	12.7279			1.11204			0.556022	−	0.831168i	$-0.312327\pi$
$131$	12.7279			1.11204			0.556022	+	0.831168i	$0.312327\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	3.16228	+	9.89949i	0.272166	+	0.852013i
$136$		0			0
$137$			8.94427i			0.764161i	0.924129	+	0.382080i	$0.124792\pi$
$137$			8.94427i			0.764161i	−0.924129	+	0.382080i	$0.875208\pi$
$138$		0			0
$139$			15.8114i			1.34110i	0.741862	+	0.670552i	$0.233943\pi$
$139$			15.8114i			1.34110i	−0.741862	+	0.670552i	$0.766057\pi$
$140$		0			0
$141$	8.94427	−	20.0000i	0.753244	−	1.68430i
$142$		0			0
$143$	6.32456			0.528886
$144$		0			0
$145$	−12.0000			−0.996546
$146$		0			0
$147$	−4.94975	+	11.0680i	−0.408248	+	0.912871i
$148$		0			0
$149$			10.0000i			0.819232i	0.912258	+	0.409616i	$0.134337\pi$
$149$			10.0000i			0.819232i	−0.912258	+	0.409616i	$0.865663\pi$
$150$		0			0
$151$			22.6274i			1.84139i	0.390279	+	0.920697i	$0.372378\pi$
$151$			22.6274i			1.84139i	−0.390279	+	0.920697i	$0.627622\pi$
$152$		0			0
$153$	10.0000	−	8.94427i	0.808452	−	0.723102i
$154$		0			0
$155$	−16.9706			−1.36311
$156$		0			0
$157$	−13.4164			−1.07075			−0.535373	−	0.844616i	$-0.679829\pi$
$157$	−13.4164			−1.07075			−0.535373	+	0.844616i	$0.679829\pi$
$158$		0			0
$159$	−9.48683	−	4.24264i	−0.752355	−	0.336463i
$160$		0			0
$161$		0			0
$162$		0			0
$163$			9.48683i			0.743066i	0.928420	+	0.371533i	$0.121168\pi$
$163$			9.48683i			0.743066i	−0.928420	+	0.371533i	$0.878832\pi$
$164$		0			0
$165$	−4.47214	−	2.00000i	−0.348155	−	0.155700i
$166$		0			0
$167$	−6.32456			−0.489409			−0.244704	−	0.969598i	$-0.578691\pi$
$167$	−6.32456			−0.489409			−0.244704	+	0.969598i	$0.578691\pi$
$168$		0			0
$169$	7.00000			0.538462
$170$		0			0
$171$	7.07107	−	6.32456i	0.540738	−	0.483651i
$172$		0			0
$173$		−	6.00000i		−	0.456172i	−0.973641	−	0.228086i	$-0.926753\pi$
$173$		−	6.00000i		−	0.456172i	0.973641	−	0.228086i	$-0.0732467\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	7.00000	−	15.6525i	0.526152	−	1.17651i
$178$		0			0
$179$	4.24264			0.317110			0.158555	−	0.987350i	$-0.449317\pi$
$179$	4.24264			0.317110			0.158555	+	0.987350i	$0.449317\pi$
$180$		0			0
$181$	−4.47214			−0.332411			−0.166206	−	0.986091i	$-0.553152\pi$
$181$	−4.47214			−0.332411			−0.166206	+	0.986091i	$0.553152\pi$
$182$		0			0
$183$	9.48683	−	21.2132i	0.701287	−	1.56813i
$184$		0			0
$185$			8.94427i			0.657596i
$186$		0			0
$187$			6.32456i			0.462497i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	−16.0000			−1.15171			−0.575853	−	0.817554i	$-0.695330\pi$
$193$	−16.0000			−1.15171			−0.575853	+	0.817554i	$0.695330\pi$
$194$		0			0
$195$	−14.1421	−	6.32456i	−1.01274	−	0.452911i
$196$		0			0
$197$			18.0000i			1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$			18.0000i			1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$		0			0
$199$		−	16.9706i		−	1.20301i	−0.798869	−	0.601506i	$-0.794568\pi$
$199$		−	16.9706i		−	1.20301i	0.798869	−	0.601506i	$-0.205432\pi$
$200$		0			0
$201$	5.00000	+	2.23607i	0.352673	+	0.157720i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	17.8885			1.24939
$206$		0			0
$207$	12.6491	+	14.1421i	0.879174	+	0.982946i
$208$		0			0
$209$			4.47214i			0.309344i
$210$		0			0
$211$			15.8114i			1.08850i	0.838923	+	0.544250i	$0.183186\pi$
$211$			15.8114i			1.08850i	−0.838923	+	0.544250i	$0.816814\pi$
$212$		0			0
$213$	4.47214	−	10.0000i	0.306426	−	0.685189i
$214$		0			0
$215$	6.32456			0.431331
$216$		0			0
$217$		0			0
$218$		0			0
$219$	2.82843	−	6.32456i	0.191127	−	0.427374i
$220$		0			0
$221$			20.0000i			1.34535i
$222$		0			0
$223$		−	19.7990i		−	1.32584i	−0.748691	−	0.662919i	$-0.769317\pi$
$223$		−	19.7990i		−	1.32584i	0.748691	−	0.662919i	$-0.230683\pi$
$224$		0			0
$225$	−2.00000	−	2.23607i	−0.133333	−	0.149071i
$226$		0			0
$227$	7.07107			0.469323			0.234662	−	0.972077i	$-0.424602\pi$
$227$	7.07107			0.469323			0.234662	+	0.972077i	$0.424602\pi$
$228$		0			0
$229$	13.4164			0.886581			0.443291	−	0.896378i	$-0.353811\pi$
$229$	13.4164			0.886581			0.443291	+	0.896378i	$0.353811\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			4.47214i			0.292979i	0.989212	+	0.146490i	$0.0467975\pi$
$233$			4.47214i			0.292979i	−0.989212	+	0.146490i	$0.953202\pi$
$234$		0			0
$235$		−	25.2982i		−	1.65027i
$236$		0			0
$237$	4.47214	+	2.00000i	0.290496	+	0.129914i
$238$		0			0
$239$	25.2982			1.63641			0.818203	−	0.574930i	$-0.194971\pi$
$239$	25.2982			1.63641			0.818203	+	0.574930i	$0.194971\pi$
$240$		0			0
$241$	8.00000			0.515325			0.257663	−	0.966235i	$-0.417048\pi$
$241$	8.00000			0.515325			0.257663	+	0.966235i	$0.417048\pi$
$242$		0			0
$243$	−13.4350	−	7.90569i	−0.861858	−	0.507151i
$244$		0			0
$245$			14.0000i			0.894427i
$246$		0			0
$247$			14.1421i			0.899843i
$248$		0			0
$249$	−9.00000	+	20.1246i	−0.570352	+	1.27535i
$250$		0			0
$251$	15.5563			0.981908			0.490954	−	0.871185i	$-0.336648\pi$
$251$	15.5563			0.981908			0.490954	+	0.871185i	$0.336648\pi$
$252$		0			0
$253$	−8.94427			−0.562322
$254$		0			0
$255$	6.32456	−	14.1421i	0.396059	−	0.885615i
$256$		0			0
$257$		−	17.8885i		−	1.11586i	−0.829889	−	0.557928i	$-0.811596\pi$
$257$		−	17.8885i		−	1.11586i	0.829889	−	0.557928i	$-0.188404\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	13.4164	−	12.0000i	0.830455	−	0.742781i
$262$		0			0
$263$	18.9737			1.16997			0.584983	−	0.811045i	$-0.301101\pi$
$263$	18.9737			1.16997			0.584983	+	0.811045i	$0.301101\pi$
$264$		0			0
$265$	−12.0000			−0.737154
$266$		0			0
$267$	21.2132	+	9.48683i	1.29823	+	0.580585i
$268$		0			0
$269$			30.0000i			1.82913i	0.404436	+	0.914566i	$0.367468\pi$
$269$			30.0000i			1.82913i	−0.404436	+	0.914566i	$0.632532\pi$
$270$		0			0
$271$		−	8.48528i		−	0.515444i	−0.966219	−	0.257722i	$-0.917028\pi$
$271$		−	8.48528i		−	0.515444i	0.966219	−	0.257722i	$-0.0829719\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.41421			0.0852803
$276$		0			0
$277$	4.47214			0.268705			0.134352	−	0.990934i	$-0.457105\pi$
$277$	4.47214			0.268705			0.134352	+	0.990934i	$0.457105\pi$
$278$		0			0
$279$	18.9737	−	16.9706i	1.13592	−	1.01600i
$280$		0			0
$281$			13.4164i			0.800356i	0.916437	+	0.400178i	$0.131052\pi$
$281$			13.4164i			0.800356i	−0.916437	+	0.400178i	$0.868948\pi$
$282$		0			0
$283$		−	28.4605i		−	1.69180i	−0.533341	−	0.845901i	$-0.679064\pi$
$283$		−	28.4605i		−	1.69180i	0.533341	−	0.845901i	$-0.320936\pi$
$284$		0			0
$285$	4.47214	−	10.0000i	0.264906	−	0.592349i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−3.00000			−0.176471
$290$		0			0
$291$	−5.65685	+	12.6491i	−0.331611	+	0.741504i
$292$		0			0
$293$		−	6.00000i		−	0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$		−	6.00000i		−	0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$		0			0
$295$		−	19.7990i		−	1.15274i
$296$		0			0
$297$	7.00000	−	2.23607i	0.406181	−	0.129750i
$298$		0			0
$299$	−28.2843			−1.63572
$300$		0			0
$301$		0			0
$302$		0			0
$303$	3.16228	+	1.41421i	0.181668	+	0.0812444i
$304$		0			0
$305$		−	26.8328i		−	1.53644i
$306$		0			0
$307$		−	15.8114i		−	0.902404i	−0.892422	−	0.451202i	$-0.850995\pi$
$307$		−	15.8114i		−	0.902404i	0.892422	−	0.451202i	$-0.149005\pi$
$308$		0			0
$309$	8.94427	+	4.00000i	0.508822	+	0.227552i
$310$		0			0
$311$	18.9737			1.07590			0.537949	−	0.842977i	$-0.319199\pi$
$311$	18.9737			1.07590			0.537949	+	0.842977i	$0.319199\pi$
$312$		0			0
$313$	26.0000			1.46961			0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000			1.46961			0.734803	+	0.678280i	$0.237274\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	2.00000i		−	0.112331i	−0.998421	−	0.0561656i	$-0.982113\pi$
$317$		−	2.00000i		−	0.112331i	0.998421	−	0.0561656i	$-0.0178875\pi$
$318$		0			0
$319$			8.48528i			0.475085i
$320$		0			0
$321$	5.00000	−	11.1803i	0.279073	−	0.624026i
$322$		0			0
$323$	−14.1421			−0.786889
$324$		0			0
$325$	4.47214			0.248069
$326$		0			0
$327$	−9.48683	+	21.2132i	−0.524623	+	1.17309i
$328$		0			0
$329$		0			0
$330$		0			0
$331$			9.48683i			0.521443i	0.965414	+	0.260722i	$0.0839605\pi$
$331$			9.48683i			0.521443i	−0.965414	+	0.260722i	$0.916039\pi$
$332$		0			0
$333$	−8.94427	−	10.0000i	−0.490143	−	0.547997i
$334$		0			0
$335$	6.32456			0.345547
$336$		0			0
$337$	−22.0000			−1.19842			−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000			−1.19842			−0.599208	+	0.800593i	$0.704518\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			12.0000i			0.649836i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	20.0000	+	8.94427i	1.07676	+	0.481543i
$346$		0			0
$347$	−7.07107			−0.379595			−0.189797	−	0.981823i	$-0.560783\pi$
$347$	−7.07107			−0.379595			−0.189797	+	0.981823i	$0.560783\pi$
$348$		0			0
$349$	13.4164			0.718164			0.359082	−	0.933306i	$-0.383090\pi$
$349$	13.4164			0.718164			0.359082	+	0.933306i	$0.383090\pi$
$350$		0			0
$351$	22.1359	−	7.07107i	1.18153	−	0.377426i
$352$		0			0
$353$		−	17.8885i		−	0.952111i	−0.879415	−	0.476056i	$-0.842066\pi$
$353$		−	17.8885i		−	0.952111i	0.879415	−	0.476056i	$-0.157934\pi$
$354$		0			0
$355$		−	12.6491i		−	0.671345i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	18.9737			1.00139			0.500696	−	0.865623i	$-0.333077\pi$
$359$	18.9737			1.00139			0.500696	+	0.865623i	$0.333077\pi$
$360$		0			0
$361$	9.00000			0.473684
$362$		0			0
$363$	6.36396	−	14.2302i	0.334021	−	0.746894i
$364$		0			0
$365$		−	8.00000i		−	0.418739i
$366$		0			0
$367$		−	14.1421i		−	0.738213i	−0.929387	−	0.369107i	$-0.879664\pi$
$367$		−	14.1421i		−	0.738213i	0.929387	−	0.369107i	$-0.120336\pi$
$368$		0			0
$369$	−20.0000	+	17.8885i	−1.04116	+	0.931240i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	4.47214			0.231558			0.115779	−	0.993275i	$-0.463063\pi$
$373$	4.47214			0.231558			0.115779	+	0.993275i	$0.463063\pi$
$374$		0			0
$375$	−18.9737	−	8.48528i	−0.979796	−	0.438178i
$376$		0			0
$377$			26.8328i			1.38196i
$378$		0			0
$379$			34.7851i			1.78679i	0.449274	+	0.893394i	$0.351683\pi$
$379$			34.7851i			1.78679i	−0.449274	+	0.893394i	$0.648317\pi$
$380$		0			0
$381$	−4.47214	−	2.00000i	−0.229114	−	0.102463i
$382$		0			0
$383$	12.6491			0.646339			0.323170	−	0.946341i	$-0.395252\pi$
$383$	12.6491			0.646339			0.323170	+	0.946341i	$0.395252\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−7.07107	+	6.32456i	−0.359443	+	0.321495i
$388$		0			0
$389$			6.00000i			0.304212i	0.988364	+	0.152106i	$0.0486055\pi$
$389$			6.00000i			0.304212i	−0.988364	+	0.152106i	$0.951394\pi$
$390$		0			0
$391$		−	28.2843i		−	1.43040i
$392$		0			0
$393$	−9.00000	+	20.1246i	−0.453990	+	1.01515i
$394$		0			0
$395$	5.65685			0.284627
$396$		0			0
$397$	−22.3607			−1.12225			−0.561125	−	0.827731i	$-0.689631\pi$
$397$	−22.3607			−1.12225			−0.561125	+	0.827731i	$0.689631\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	4.47214i		−	0.223328i	−0.993746	−	0.111664i	$-0.964382\pi$
$401$		−	4.47214i		−	0.223328i	0.993746	−	0.111664i	$-0.0356180\pi$
$402$		0			0
$403$			37.9473i			1.89029i
$404$		0			0
$405$	−17.8885	−	2.00000i	−0.888889	−	0.0993808i
$406$		0			0
$407$	6.32456			0.313497
$408$		0			0
$409$	−10.0000			−0.494468			−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000			−0.494468			−0.247234	+	0.968956i	$0.579522\pi$
$410$		0			0
$411$	−14.1421	−	6.32456i	−0.697580	−	0.311967i
$412$		0			0
$413$		0			0
$414$		0			0
$415$			25.4558i			1.24958i
$416$		0			0
$417$	−25.0000	−	11.1803i	−1.22426	−	0.547504i
$418$		0			0
$419$	38.1838			1.86540			0.932700	−	0.360654i	$-0.117447\pi$
$419$	38.1838			1.86540			0.932700	+	0.360654i	$0.117447\pi$
$420$		0			0
$421$	−22.3607			−1.08979			−0.544896	−	0.838503i	$-0.683431\pi$
$421$	−22.3607			−1.08979			−0.544896	+	0.838503i	$0.683431\pi$
$422$		0			0
$423$	25.2982	+	28.2843i	1.23004	+	1.37523i
$424$		0			0
$425$			4.47214i			0.216930i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−4.47214	+	10.0000i	−0.215917	+	0.482805i
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	16.0000			0.768911			0.384455	−	0.923144i	$-0.374389\pi$
$433$	16.0000			0.768911			0.384455	+	0.923144i	$0.374389\pi$
$434$		0			0
$435$	8.48528	−	18.9737i	0.406838	−	0.909718i
$436$		0			0
$437$		−	20.0000i		−	0.956730i
$438$		0			0
$439$			16.9706i			0.809961i	0.914325	+	0.404980i	$0.132722\pi$
$439$			16.9706i			0.809961i	−0.914325	+	0.404980i	$0.867278\pi$
$440$		0			0
$441$	−14.0000	−	15.6525i	−0.666667	−	0.745356i
$442$		0			0
$443$	7.07107			0.335957			0.167978	−	0.985791i	$-0.446276\pi$
$443$	7.07107			0.335957			0.167978	+	0.985791i	$0.446276\pi$
$444$		0			0
$445$	26.8328			1.27200
$446$		0			0
$447$	−15.8114	−	7.07107i	−0.747853	−	0.334450i
$448$		0			0
$449$			4.47214i			0.211053i	0.994416	+	0.105527i	$0.0336528\pi$
$449$			4.47214i			0.211053i	−0.994416	+	0.105527i	$0.966347\pi$
$450$		0			0
$451$		−	12.6491i		−	0.595623i
$452$		0			0
$453$	−35.7771	−	16.0000i	−1.68095	−	0.751746i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	22.0000			1.02912			0.514558	−	0.857455i	$-0.327956\pi$
$457$	22.0000			1.02912			0.514558	+	0.857455i	$0.327956\pi$
$458$		0			0
$459$	7.07107	+	22.1359i	0.330049	+	1.03322i
$460$		0			0
$461$		−	10.0000i		−	0.465746i	−0.972507	−	0.232873i	$-0.925187\pi$
$461$		−	10.0000i		−	0.465746i	0.972507	−	0.232873i	$-0.0748127\pi$
$462$		0			0
$463$		−	14.1421i		−	0.657241i	−0.944462	−	0.328620i	$-0.893416\pi$
$463$		−	14.1421i		−	0.657241i	0.944462	−	0.328620i	$-0.106584\pi$
$464$		0			0
$465$	12.0000	−	26.8328i	0.556487	−	1.24434i
$466$		0			0
$467$	9.89949			0.458094			0.229047	−	0.973415i	$-0.426439\pi$
$467$	9.89949			0.458094			0.229047	+	0.973415i	$0.426439\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	9.48683	−	21.2132i	0.437130	−	0.977453i
$472$		0			0
$473$		−	4.47214i		−	0.205629i
$474$		0			0
$475$			3.16228i			0.145095i
$476$		0			0
$477$	13.4164	−	12.0000i	0.614295	−	0.549442i
$478$		0			0
$479$	37.9473			1.73386			0.866929	−	0.498432i	$-0.166091\pi$
$479$	37.9473			1.73386			0.866929	+	0.498432i	$0.166091\pi$
$480$		0			0
$481$	20.0000			0.911922
$482$		0			0
$483$		0			0
$484$		0			0
$485$			16.0000i			0.726523i
$486$		0			0
$487$		−	16.9706i		−	0.769010i	−0.923123	−	0.384505i	$-0.874372\pi$
$487$		−	16.9706i		−	0.769010i	0.923123	−	0.384505i	$-0.125628\pi$
$488$		0			0
$489$	−15.0000	−	6.70820i	−0.678323	−	0.303355i
$490$		0			0
$491$	−15.5563			−0.702048			−0.351024	−	0.936366i	$-0.614166\pi$
$491$	−15.5563			−0.702048			−0.351024	+	0.936366i	$0.614166\pi$
$492$		0			0
$493$	−26.8328			−1.20849
$494$		0			0
$495$	6.32456	−	5.65685i	0.284268	−	0.254257i
$496$		0			0
$497$		0			0
$498$		0			0
$499$		−	34.7851i		−	1.55719i	−0.627525	−	0.778596i	$-0.715932\pi$
$499$		−	34.7851i		−	1.55719i	0.627525	−	0.778596i	$-0.284068\pi$
$500$		0			0
$501$	4.47214	−	10.0000i	0.199800	−	0.446767i
$502$		0			0
$503$	−18.9737			−0.845994			−0.422997	−	0.906131i	$-0.639022\pi$
$503$	−18.9737			−0.845994			−0.422997	+	0.906131i	$0.639022\pi$
$504$		0			0
$505$	4.00000			0.177998
$506$		0			0
$507$	−4.94975	+	11.0680i	−0.219826	+	0.491546i
$508$		0			0
$509$		−	30.0000i		−	1.32973i	−0.746965	−	0.664863i	$-0.768490\pi$
$509$		−	30.0000i		−	1.32973i	0.746965	−	0.664863i	$-0.231510\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	5.00000	+	15.6525i	0.220755	+	0.691074i
$514$		0			0
$515$	11.3137			0.498542
$516$		0			0
$517$	−17.8885			−0.786737
$518$		0			0
$519$	9.48683	+	4.24264i	0.416426	+	0.186231i
$520$		0			0
$521$			8.94427i			0.391856i	0.980618	+	0.195928i	$0.0627718\pi$
$521$			8.94427i			0.391856i	−0.980618	+	0.195928i	$0.937228\pi$
$522$		0			0
$523$			28.4605i			1.24449i	0.782822	+	0.622245i	$0.213779\pi$
$523$			28.4605i			1.24449i	−0.782822	+	0.622245i	$0.786221\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−37.9473			−1.65301
$528$		0			0
$529$	17.0000			0.739130
$530$		0			0
$531$	19.7990	+	22.1359i	0.859203	+	0.960618i
$532$		0			0
$533$		−	40.0000i		−	1.73259i
$534$		0			0
$535$		−	14.1421i		−	0.611418i
$536$		0			0
$537$	−3.00000	+	6.70820i	−0.129460	+	0.289480i
$538$		0			0
$539$	9.89949			0.426401
$540$		0			0
$541$	4.47214			0.192272			0.0961361	−	0.995368i	$-0.469352\pi$
$541$	4.47214			0.192272			0.0961361	+	0.995368i	$0.469352\pi$
$542$		0			0
$543$	3.16228	−	7.07107i	0.135706	−	0.303449i
$544$		0			0
$545$			26.8328i			1.14939i
$546$		0			0
$547$			15.8114i			0.676046i	0.941138	+	0.338023i	$0.109758\pi$
$547$			15.8114i			0.676046i	−0.941138	+	0.338023i	$0.890242\pi$
$548$		0			0
$549$	26.8328	+	30.0000i	1.14520	+	1.28037i
$550$		0			0
$551$	−18.9737			−0.808305
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−14.1421	−	6.32456i	−0.600300	−	0.268462i
$556$		0			0
$557$			18.0000i			0.762684i	0.924434	+	0.381342i	$0.124538\pi$
$557$			18.0000i			0.762684i	−0.924434	+	0.381342i	$0.875462\pi$
$558$		0			0
$559$		−	14.1421i		−	0.598149i
$560$		0			0
$561$	−10.0000	−	4.47214i	−0.422200	−	0.188814i
$562$		0			0
$563$	−35.3553			−1.49005			−0.745025	−	0.667037i	$-0.767562\pi$
$563$	−35.3553			−1.49005			−0.745025	+	0.667037i	$0.767562\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	44.7214i		−	1.87482i	−0.348232	−	0.937408i	$-0.613218\pi$
$569$		−	44.7214i		−	1.87482i	0.348232	−	0.937408i	$-0.386782\pi$
$570$		0			0
$571$		−	3.16228i		−	0.132337i	−0.997808	−	0.0661686i	$-0.978922\pi$
$571$		−	3.16228i		−	0.132337i	0.997808	−	0.0661686i	$-0.0210775\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−6.32456			−0.263752
$576$		0			0
$577$	−22.0000			−0.915872			−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000			−0.915872			−0.457936	+	0.888985i	$0.651411\pi$
$578$		0			0
$579$	11.3137	−	25.2982i	0.470182	−	1.05136i
$580$		0			0
$581$		0			0
$582$		0			0
$583$			8.48528i			0.351424i
$584$		0			0
$585$	20.0000	−	17.8885i	0.826898	−	0.739600i
$586$		0			0
$587$	−7.07107			−0.291854			−0.145927	−	0.989295i	$-0.546616\pi$
$587$	−7.07107			−0.291854			−0.145927	+	0.989295i	$0.546616\pi$
$588$		0			0
$589$	−26.8328			−1.10563
$590$		0			0
$591$	−28.4605	−	12.7279i	−1.17071	−	0.523557i
$592$		0			0
$593$			17.8885i			0.734594i	0.930104	+	0.367297i	$0.119717\pi$
$593$			17.8885i			0.734594i	−0.930104	+	0.367297i	$0.880283\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	26.8328	+	12.0000i	1.09819	+	0.491127i
$598$		0			0
$599$	6.32456			0.258414			0.129207	−	0.991618i	$-0.458757\pi$
$599$	6.32456			0.258414			0.129207	+	0.991618i	$0.458757\pi$
$600$		0			0
$601$	−20.0000			−0.815817			−0.407909	−	0.913023i	$-0.633742\pi$
$601$	−20.0000			−0.815817			−0.407909	+	0.913023i	$0.633742\pi$
$602$		0			0
$603$	−7.07107	+	6.32456i	−0.287956	+	0.257556i
$604$		0			0
$605$		−	18.0000i		−	0.731804i
$606$		0			0
$607$			42.4264i			1.72203i	0.508576	+	0.861017i	$0.330172\pi$
$607$			42.4264i			1.72203i	−0.508576	+	0.861017i	$0.669828\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−56.5685			−2.28852
$612$		0			0
$613$	−22.3607			−0.903139			−0.451570	−	0.892236i	$-0.649136\pi$
$613$	−22.3607			−0.903139			−0.451570	+	0.892236i	$0.649136\pi$
$614$		0			0
$615$	−12.6491	+	28.2843i	−0.510061	+	1.14053i
$616$		0			0
$617$			31.3050i			1.26029i	0.776478	+	0.630145i	$0.217005\pi$
$617$			31.3050i			1.26029i	−0.776478	+	0.630145i	$0.782995\pi$
$618$		0			0
$619$		−	41.1096i		−	1.65233i	−0.563425	−	0.826167i	$-0.690517\pi$
$619$		−	41.1096i		−	1.65233i	0.563425	−	0.826167i	$-0.309483\pi$
$620$		0			0
$621$	−31.3050	+	10.0000i	−1.25622	+	0.401286i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−19.0000			−0.760000
$626$		0			0
$627$	−7.07107	−	3.16228i	−0.282391	−	0.126289i
$628$		0			0
$629$			20.0000i			0.797452i
$630$		0			0
$631$			22.6274i			0.900783i	0.892831	+	0.450392i	$0.148716\pi$
$631$			22.6274i			0.900783i	−0.892831	+	0.450392i	$0.851284\pi$
$632$		0			0
$633$	−25.0000	−	11.1803i	−0.993661	−	0.444379i
$634$		0			0
$635$	−5.65685			−0.224485
$636$		0			0
$637$	31.3050			1.24035
$638$		0			0
$639$	12.6491	+	14.1421i	0.500391	+	0.559454i
$640$		0			0
$641$		−	4.47214i		−	0.176639i	−0.996092	−	0.0883194i	$-0.971850\pi$
$641$		−	4.47214i		−	0.176639i	0.996092	−	0.0883194i	$-0.0281496\pi$
$642$		0			0
$643$			15.8114i			0.623540i	0.950158	+	0.311770i	$0.100922\pi$
$643$			15.8114i			0.623540i	−0.950158	+	0.311770i	$0.899078\pi$
$644$		0			0
$645$	−4.47214	+	10.0000i	−0.176090	+	0.393750i
$646$		0			0
$647$	−18.9737			−0.745932			−0.372966	−	0.927845i	$-0.621659\pi$
$647$	−18.9737			−0.745932			−0.372966	+	0.927845i	$0.621659\pi$
$648$		0			0
$649$	−14.0000			−0.549548
$650$		0			0
$651$		0			0
$652$		0			0
$653$			6.00000i			0.234798i	0.993085	+	0.117399i	$0.0374557\pi$
$653$			6.00000i			0.234798i	−0.993085	+	0.117399i	$0.962544\pi$
$654$		0			0
$655$			25.4558i			0.994642i
$656$		0			0
$657$	8.00000	+	8.94427i	0.312110	+	0.348949i
$658$		0			0
$659$	38.1838			1.48743			0.743714	−	0.668498i	$-0.233062\pi$
$659$	38.1838			1.48743			0.743714	+	0.668498i	$0.233062\pi$
$660$		0			0
$661$	31.3050			1.21762			0.608811	−	0.793315i	$-0.291647\pi$
$661$	31.3050			1.21762			0.608811	+	0.793315i	$0.291647\pi$
$662$		0			0
$663$	−31.6228	−	14.1421i	−1.22813	−	0.549235i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	37.9473i		−	1.46933i
$668$		0			0
$669$	31.3050	+	14.0000i	1.21032	+	0.541271i
$670$		0			0
$671$	−18.9737			−0.732470
$672$		0			0
$673$	24.0000			0.925132			0.462566	−	0.886585i	$-0.346929\pi$
$673$	24.0000			0.925132			0.462566	+	0.886585i	$0.346929\pi$
$674$		0			0
$675$	4.94975	−	1.58114i	0.190516	−	0.0608581i
$676$		0			0
$677$		−	42.0000i		−	1.61419i	−0.590421	−	0.807096i	$-0.701038\pi$
$677$		−	42.0000i		−	1.61419i	0.590421	−	0.807096i	$-0.298962\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−5.00000	+	11.1803i	−0.191600	+	0.428432i
$682$		0			0
$683$	43.8406			1.67751			0.838757	−	0.544505i	$-0.183283\pi$
$683$	43.8406			1.67751			0.838757	+	0.544505i	$0.183283\pi$
$684$		0			0
$685$	−17.8885			−0.683486
$686$		0			0
$687$	−9.48683	+	21.2132i	−0.361945	+	0.809334i
$688$		0			0
$689$			26.8328i			1.02225i
$690$		0			0
$691$		−	28.4605i		−	1.08269i	−0.840801	−	0.541344i	$-0.817916\pi$
$691$		−	28.4605i		−	1.08269i	0.840801	−	0.541344i	$-0.182084\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−31.6228			−1.19952
$696$		0			0
$697$	40.0000			1.51511
$698$		0			0
$699$	−7.07107	−	3.16228i	−0.267452	−	0.119608i
$700$		0			0
$701$			10.0000i			0.377695i	0.982006	+	0.188847i	$0.0604752\pi$
$701$			10.0000i			0.377695i	−0.982006	+	0.188847i	$0.939525\pi$
$702$		0			0
$703$			14.1421i			0.533381i
$704$		0			0
$705$	40.0000	+	17.8885i	1.50649	+	0.673722i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	40.2492			1.51159			0.755796	−	0.654808i	$-0.227250\pi$
$709$	40.2492			1.51159			0.755796	+	0.654808i	$0.227250\pi$
$710$		0			0
$711$	−6.32456	+	5.65685i	−0.237189	+	0.212149i
$712$		0			0
$713$		−	53.6656i		−	2.00979i
$714$		0			0
$715$			12.6491i			0.473050i
$716$		0			0
$717$	−17.8885	+	40.0000i	−0.668060	+	1.49383i
$718$		0			0
$719$	−12.6491			−0.471732			−0.235866	−	0.971786i	$-0.575793\pi$
$719$	−12.6491			−0.471732			−0.235866	+	0.971786i	$0.575793\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−5.65685	+	12.6491i	−0.210381	+	0.470425i
$724$		0			0
$725$			6.00000i			0.222834i
$726$		0			0
$727$			45.2548i			1.67841i	0.543816	+	0.839204i	$0.316979\pi$
$727$			45.2548i			1.67841i	−0.543816	+	0.839204i	$0.683021\pi$
$728$		0			0
$729$	22.0000	−	15.6525i	0.814815	−	0.579721i
$730$		0			0
$731$	14.1421			0.523066
$732$		0			0
$733$	13.4164			0.495546			0.247773	−	0.968818i	$-0.420301\pi$
$733$	13.4164			0.495546			0.247773	+	0.968818i	$0.420301\pi$
$734$		0			0
$735$	−22.1359	−	9.89949i	−0.816497	−	0.365148i
$736$		0			0
$737$		−	4.47214i		−	0.164733i
$738$		0			0
$739$			3.16228i			0.116326i	0.998307	+	0.0581631i	$0.0185244\pi$
$739$			3.16228i			0.116326i	−0.998307	+	0.0581631i	$0.981476\pi$
$740$		0			0
$741$	−22.3607	−	10.0000i	−0.821440	−	0.367359i
$742$		0			0
$743$	18.9737			0.696076			0.348038	−	0.937480i	$-0.386848\pi$
$743$	18.9737			0.696076			0.348038	+	0.937480i	$0.386848\pi$
$744$		0			0
$745$	−20.0000			−0.732743
$746$		0			0
$747$	−25.4558	−	28.4605i	−0.931381	−	1.04132i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		−	19.7990i		−	0.722475i	−0.932474	−	0.361238i	$-0.882354\pi$
$751$		−	19.7990i		−	0.722475i	0.932474	−	0.361238i	$-0.117646\pi$
$752$		0			0
$753$	−11.0000	+	24.5967i	−0.400862	+	0.896355i
$754$		0			0
$755$	−45.2548			−1.64699
$756$		0			0
$757$	13.4164			0.487628			0.243814	−	0.969822i	$-0.421601\pi$
$757$	13.4164			0.487628			0.243814	+	0.969822i	$0.421601\pi$
$758$		0			0
$759$	6.32456	−	14.1421i	0.229567	−	0.513327i
$760$		0			0
$761$		−	26.8328i		−	0.972689i	−0.873767	−	0.486344i	$-0.838330\pi$
$761$		−	26.8328i		−	0.972689i	0.873767	−	0.486344i	$-0.161670\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	17.8885	+	20.0000i	0.646762	+	0.723102i
$766$		0			0
$767$	−44.2719			−1.59857
$768$		0			0
$769$		0			0			−	1.00000i	$-0.5\pi$
$769$		0			0				1.00000i	$0.5\pi$
$770$		0			0
$771$	28.2843	+	12.6491i	1.01863	+	0.455547i
$772$		0			0
$773$			14.0000i			0.503545i	0.967786	+	0.251773i	$0.0810135\pi$
$773$			14.0000i			0.503545i	−0.967786	+	0.251773i	$0.918987\pi$
$774$		0			0
$775$			8.48528i			0.304800i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	28.2843			1.01339
$780$		0			0
$781$	−8.94427			−0.320051
$782$		0			0
$783$	9.48683	+	29.6985i	0.339032	+	1.06134i
$784$		0			0
$785$		−	26.8328i		−	0.957704i
$786$		0			0
$787$			47.4342i			1.69085i	0.534098	+	0.845423i	$0.320651\pi$
$787$			47.4342i			1.69085i	−0.534098	+	0.845423i	$0.679349\pi$
$788$		0			0
$789$	−13.4164	+	30.0000i	−0.477637	+	1.06803i
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−60.0000			−2.13066
$794$		0			0
$795$	8.48528	−	18.9737i	0.300942	−	0.672927i
$796$		0			0
$797$		−	38.0000i		−	1.34603i	−0.739629	−	0.673015i	$-0.764999\pi$
$797$		−	38.0000i		−	1.34603i	0.739629	−	0.673015i	$-0.235001\pi$
$798$		0			0
$799$		−	56.5685i		−	2.00125i
$800$		0			0
$801$	−30.0000	+	26.8328i	−1.06000	+	0.948091i
$802$		0			0
$803$	−5.65685			−0.199626
$804$		0			0
$805$		0			0
$806$		0			0
$807$	−47.4342	−	21.2132i	−1.66976	−	0.746740i
$808$		0			0
$809$		−	44.7214i		−	1.57232i	−0.618023	−	0.786160i	$-0.712066\pi$
$809$		−	44.7214i		−	1.57232i	0.618023	−	0.786160i	$-0.287934\pi$
$810$		0			0
$811$			3.16228i			0.111043i	0.998458	+	0.0555213i	$0.0176821\pi$
$811$			3.16228i			0.111043i	−0.998458	+	0.0555213i	$0.982318\pi$
$812$		0			0
$813$	13.4164	+	6.00000i	0.470534	+	0.210429i
$814$		0			0
$815$	−18.9737			−0.664619
$816$		0			0
$817$	10.0000			0.349856
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	18.0000i		−	0.628204i	−0.949389	−	0.314102i	$-0.898297\pi$
$821$		−	18.0000i		−	0.628204i	0.949389	−	0.314102i	$-0.101703\pi$
$822$		0			0
$823$		−	5.65685i		−	0.197186i	−0.995128	−	0.0985928i	$-0.968566\pi$
$823$		−	5.65685i		−	0.197186i	0.995128	−	0.0985928i	$-0.0314341\pi$
$824$		0			0
$825$	−1.00000	+	2.23607i	−0.0348155	+	0.0778499i
$826$		0			0
$827$	32.5269			1.13107			0.565536	−	0.824724i	$-0.308669\pi$
$827$	32.5269			1.13107			0.565536	+	0.824724i	$0.308669\pi$
$828$		0			0
$829$	40.2492			1.39791			0.698957	−	0.715164i	$-0.253648\pi$
$829$	40.2492			1.39791			0.698957	+	0.715164i	$0.253648\pi$
$830$		0			0
$831$	−3.16228	+	7.07107i	−0.109698	+	0.245293i
$832$		0			0
$833$			31.3050i			1.08465i
$834$		0			0
$835$		−	12.6491i		−	0.437741i
$836$		0			0
$837$	13.4164	+	42.0000i	0.463739	+	1.45173i
$838$		0			0
$839$	−18.9737			−0.655044			−0.327522	−	0.944844i	$-0.606214\pi$
$839$	−18.9737			−0.655044			−0.327522	+	0.944844i	$0.606214\pi$
$840$		0			0
$841$	−7.00000			−0.241379
$842$		0			0
$843$	−21.2132	−	9.48683i	−0.730622	−	0.326744i
$844$		0			0
$845$			14.0000i			0.481615i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	45.0000	+	20.1246i	1.54440	+	0.690675i
$850$		0			0
$851$	−28.2843			−0.969572
$852$		0			0
$853$	4.47214			0.153123			0.0765615	−	0.997065i	$-0.475606\pi$
$853$	4.47214			0.153123			0.0765615	+	0.997065i	$0.475606\pi$
$854$		0			0
$855$	12.6491	+	14.1421i	0.432590	+	0.483651i
$856$		0			0
$857$			26.8328i			0.916592i	0.888800	+	0.458296i	$0.151540\pi$
$857$			26.8328i			0.916592i	−0.888800	+	0.458296i	$0.848460\pi$
$858$		0			0
$859$			34.7851i			1.18685i	0.804889	+	0.593425i	$0.202225\pi$
$859$			34.7851i			1.18685i	−0.804889	+	0.593425i	$0.797775\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−25.2982			−0.861161			−0.430581	−	0.902552i	$-0.641691\pi$
$863$	−25.2982			−0.861161			−0.430581	+	0.902552i	$0.641691\pi$
$864$		0			0
$865$	12.0000			0.408012
$866$		0			0
$867$	2.12132	−	4.74342i	0.0720438	−	0.161095i
$868$		0			0
$869$		−	4.00000i		−	0.135691i
$870$		0			0
$871$		−	14.1421i		−	0.479188i
$872$		0			0
$873$	−16.0000	−	17.8885i	−0.541518	−	0.605435i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	13.4164			0.453040			0.226520	−	0.974007i	$-0.427265\pi$
$877$	13.4164			0.453040			0.226520	+	0.974007i	$0.427265\pi$
$878$		0			0
$879$	9.48683	+	4.24264i	0.319983	+	0.143101i
$880$		0			0
$881$			53.6656i			1.80804i	0.427489	+	0.904021i	$0.359398\pi$
$881$			53.6656i			1.80804i	−0.427489	+	0.904021i	$0.640602\pi$
$882$		0			0
$883$			3.16228i			0.106419i	0.998583	+	0.0532096i	$0.0169451\pi$
$883$			3.16228i			0.106419i	−0.998583	+	0.0532096i	$0.983055\pi$
$884$		0			0
$885$	31.3050	+	14.0000i	1.05230	+	0.470605i
$886$		0			0
$887$	−31.6228			−1.06179			−0.530894	−	0.847438i	$-0.678144\pi$
$887$	−31.6228			−1.06179			−0.530894	+	0.847438i	$0.678144\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−1.41421	+	12.6491i	−0.0473779	+	0.423761i
$892$		0			0
$893$		−	40.0000i		−	1.33855i
$894$		0			0
$895$			8.48528i			0.283632i
$896$		0			0
$897$	20.0000	−	44.7214i	0.667781	−	1.49320i
$898$		0			0
$899$	−50.9117			−1.69800
$900$		0			0
$901$	−26.8328			−0.893931
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	8.94427i		−	0.297318i
$906$		0			0
$907$			22.1359i			0.735012i	0.930021	+	0.367506i	$0.119788\pi$
$907$			22.1359i			0.735012i	−0.930021	+	0.367506i	$0.880212\pi$
$908$		0			0
$909$	−4.47214	+	4.00000i	−0.148331	+	0.132672i
$910$		0			0
$911$	37.9473			1.25725			0.628626	−	0.777708i	$-0.283618\pi$
$911$	37.9473			1.25725			0.628626	+	0.777708i	$0.283618\pi$
$912$		0			0
$913$	18.0000			0.595713
$914$		0			0
$915$	42.4264	+	18.9737i	1.40257	+	0.627250i
$916$		0			0
$917$		0			0
$918$		0			0
$919$			11.3137i			0.373205i	0.982436	+	0.186602i	$0.0597476\pi$
$919$			11.3137i			0.373205i	−0.982436	+	0.186602i	$0.940252\pi$
$920$		0			0
$921$	25.0000	+	11.1803i	0.823778	+	0.368405i
$922$		0			0
$923$	−28.2843			−0.930988
$924$		0			0
$925$	4.47214			0.147043
$926$		0			0
$927$	−12.6491	+	11.3137i	−0.415451	+	0.371591i
$928$		0			0
$929$			31.3050i			1.02708i	0.858065	+	0.513541i	$0.171667\pi$
$929$			31.3050i			1.02708i	−0.858065	+	0.513541i	$0.828333\pi$
$930$		0			0
$931$			22.1359i			0.725476i
$932$		0			0
$933$	−13.4164	+	30.0000i	−0.439233	+	0.982156i
$934$		0			0
$935$	−12.6491			−0.413670
$936$		0			0
$937$	−12.0000			−0.392023			−0.196011	−	0.980602i	$-0.562799\pi$
$937$	−12.0000			−0.392023			−0.196011	+	0.980602i	$0.562799\pi$
$938$		0			0
$939$	−18.3848	+	41.1096i	−0.599965	+	1.34156i
$940$		0			0
$941$		−	18.0000i		−	0.586783i	−0.955992	−	0.293392i	$-0.905216\pi$
$941$		−	18.0000i		−	0.586783i	0.955992	−	0.293392i	$-0.0947840\pi$
$942$		0			0
$943$			56.5685i			1.84213i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	18.3848			0.597425			0.298712	−	0.954343i	$-0.403443\pi$
$947$	18.3848			0.597425			0.298712	+	0.954343i	$0.403443\pi$
$948$		0			0
$949$	−17.8885			−0.580687
$950$		0			0
$951$	3.16228	+	1.41421i	0.102544	+	0.0458590i
$952$		0			0
$953$		−	8.94427i		−	0.289733i	−0.989451	−	0.144867i	$-0.953725\pi$
$953$		−	8.94427i		−	0.289733i	0.989451	−	0.144867i	$-0.0462753\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−13.4164	−	6.00000i	−0.433691	−	0.193952i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−41.0000			−1.32258
$962$		0			0
$963$	14.1421	+	15.8114i	0.455724	+	0.509515i
$964$		0			0
$965$		−	32.0000i		−	1.03012i
$966$		0			0
$967$			16.9706i			0.545737i	0.962051	+	0.272868i	$0.0879723\pi$
$967$			16.9706i			0.545737i	−0.962051	+	0.272868i	$0.912028\pi$
$968$		0			0
$969$	10.0000	−	22.3607i	0.321246	−	0.718329i
$970$		0			0
$971$	55.1543			1.76999			0.884993	−	0.465604i	$-0.154163\pi$
$971$	55.1543			1.76999			0.884993	+	0.465604i	$0.154163\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−3.16228	+	7.07107i	−0.101274	+	0.226455i
$976$		0			0
$977$		−	31.3050i		−	1.00153i	−0.865582	−	0.500767i	$-0.833051\pi$
$977$		−	31.3050i		−	1.00153i	0.865582	−	0.500767i	$-0.166949\pi$
$978$		0			0
$979$		−	18.9737i		−	0.606401i
$980$		0			0
$981$	−26.8328	−	30.0000i	−0.856706	−	0.957826i
$982$		0			0
$983$	−6.32456			−0.201722			−0.100861	−	0.994901i	$-0.532160\pi$
$983$	−6.32456			−0.201722			−0.100861	+	0.994901i	$0.532160\pi$
$984$		0			0
$985$	−36.0000			−1.14706
$986$		0			0
$987$		0			0
$988$		0			0
$989$			20.0000i			0.635963i
$990$		0			0
$991$		−	8.48528i		−	0.269544i	−0.990877	−	0.134772i	$-0.956970\pi$
$991$		−	8.48528i		−	0.269544i	0.990877	−	0.134772i	$-0.0430302\pi$
$992$		0			0
$993$	−15.0000	−	6.70820i	−0.476011	−	0.212878i
$994$		0			0
$995$	33.9411			1.07601
$996$		0			0
$997$	−13.4164			−0.424902			−0.212451	−	0.977172i	$-0.568145\pi$
$997$	−13.4164			−0.424902			−0.212451	+	0.977172i	$0.568145\pi$
$998$		0			0
$999$	22.1359	−	7.07107i	0.700350	−	0.223719i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1536.2.c.k.1535.4	yes	8
3.2	odd	2	inner	1536.2.c.k.1535.7	yes	8
4.3	odd	2	inner	1536.2.c.k.1535.6	yes	8
8.3	odd	2	inner	1536.2.c.k.1535.3	yes	8
8.5	even	2	inner	1536.2.c.k.1535.5	yes	8
12.11	even	2	inner	1536.2.c.k.1535.1	✓	8
16.3	odd	4		1536.2.f.i.767.4		4
16.5	even	4		1536.2.f.c.767.4		4
16.11	odd	4		1536.2.f.c.767.1		4
16.13	even	4		1536.2.f.i.767.1		4
24.5	odd	2	inner	1536.2.c.k.1535.2	yes	8
24.11	even	2	inner	1536.2.c.k.1535.8	yes	8
48.5	odd	4		1536.2.f.i.767.3		4
48.11	even	4		1536.2.f.i.767.2		4
48.29	odd	4		1536.2.f.c.767.2		4
48.35	even	4		1536.2.f.c.767.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.c.k.1535.1	✓	8	12.11	even	2	inner
1536.2.c.k.1535.2	yes	8	24.5	odd	2	inner
1536.2.c.k.1535.3	yes	8	8.3	odd	2	inner
1536.2.c.k.1535.4	yes	8	1.1	even	1	trivial
1536.2.c.k.1535.5	yes	8	8.5	even	2	inner
1536.2.c.k.1535.6	yes	8	4.3	odd	2	inner
1536.2.c.k.1535.7	yes	8	3.2	odd	2	inner
1536.2.c.k.1535.8	yes	8	24.11	even	2	inner
1536.2.f.c.767.1		4	16.11	odd	4
1536.2.f.c.767.2		4	48.29	odd	4
1536.2.f.c.767.3		4	48.35	even	4
1536.2.f.c.767.4		4	16.5	even	4
1536.2.f.i.767.1		4	16.13	even	4
1536.2.f.i.767.2		4	48.11	even	4
1536.2.f.i.767.3		4	48.5	odd	4
1536.2.f.i.767.4		4	16.3	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:	\( N \)	\(=\)	\( 1536 = 2^{9} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1536.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 + 1.58114i) q^{3} +2.00000i q^{5} +(-2.00000 - 2.23607i) q^{9} +O(q^{10})\)	\(q+(-0.707107 + 1.58114i) q^{3} +2.00000i q^{5} +(-2.00000 - 2.23607i) q^{9} +1.41421 q^{11} +4.47214 q^{13} +(-3.16228 - 1.41421i) q^{15} +4.47214i q^{17} +3.16228i q^{19} -6.32456 q^{23} +1.00000 q^{25} +(4.94975 - 1.58114i) q^{27} +6.00000i q^{29} +8.48528i q^{31} +(-1.00000 + 2.23607i) q^{33} +4.47214 q^{37} +(-3.16228 + 7.07107i) q^{39} -8.94427i q^{41} -3.16228i q^{43} +(4.47214 - 4.00000i) q^{45} -12.6491 q^{47} +7.00000 q^{49} +(-7.07107 - 3.16228i) q^{51} +6.00000i q^{53} +2.82843i q^{55} +(-5.00000 - 2.23607i) q^{57} -9.89949 q^{59} -13.4164 q^{61} +8.94427i q^{65} -3.16228i q^{67} +(4.47214 - 10.0000i) q^{69} -6.32456 q^{71} -4.00000 q^{73} +(-0.707107 + 1.58114i) q^{75} -2.82843i q^{79} +(-1.00000 + 8.94427i) q^{81} +12.7279 q^{83} -8.94427 q^{85} +(-9.48683 - 4.24264i) q^{87} -13.4164i q^{89} +(-13.4164 - 6.00000i) q^{93} -6.32456 q^{95} +8.00000 q^{97} +(-2.82843 - 3.16228i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{9}+O(q^{10}) \) 8 * q - 16 * q^9	\( 8 q - 16 q^{9} + 8 q^{25} - 8 q^{33} + 56 q^{49} - 40 q^{57} - 32 q^{73} - 8 q^{81} + 64 q^{97}+O(q^{100}) \) 8 * q - 16 * q^9 + 8 * q^25 - 8 * q^33 + 56 * q^49 - 40 * q^57 - 32 * q^73 - 8 * q^81 + 64 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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