LMFDB - Embedded newform 1536.2.c.m.1535.5

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:	$16$
Coefficient field:	16.0.29960650073923649536.7
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 7x^{12} + 40x^{8} - 112x^{4} + 256 $ x^16 - 7x^12 + 40x^8 - 112*x^4 + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{24} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1535.5
Root			$0.481610 + 1.32968i$ of defining polynomial
Character	$\chi$	$=$	1536.1535
Dual form			1536.2.c.m.1535.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.468213 - 1.66757i) q^{3} -3.02045i q^{5} +3.62258i q^{7} +(-2.56155 + 1.56155i) q^{9} +O(q^{10})$	\(q+(-0.468213 - 1.66757i) q^{3} -3.02045i q^{5} +3.62258i q^{7} +(-2.56155 + 1.56155i) q^{9} +3.33513 q^{11} -1.32431 q^{13} +(-5.03680 + 1.41421i) q^{15} +7.12311i q^{17} +5.20798i q^{19} +(6.04090 - 1.69614i) q^{21} +4.41674 q^{23} -4.12311 q^{25} +(3.80335 + 3.54042i) q^{27} +6.41273i q^{29} -0.794156i q^{31} +(-1.56155 - 5.56155i) q^{33} +10.9418 q^{35} -8.10887 q^{37} +(0.620058 + 2.20837i) q^{39} +4.00000i q^{41} -1.46228i q^{43} +(4.71659 + 7.73704i) q^{45} +5.65685 q^{47} -6.12311 q^{49} +(11.8782 - 3.33513i) q^{51} -3.76412i q^{53} -10.0736i q^{55} +(8.68466 - 2.43845i) q^{57} +5.73384 q^{59} -13.4061 q^{61} +(-5.65685 - 9.27944i) q^{63} +4.00000i q^{65} +7.08084i q^{67} +(-2.06798 - 7.36520i) q^{69} -10.0736 q^{71} +10.2462 q^{73} +(1.93049 + 6.87555i) q^{75} +12.0818i q^{77} -4.86270i q^{79} +(4.12311 - 8.00000i) q^{81} -0.410574 q^{83} +21.5150 q^{85} +(10.6937 - 3.00252i) q^{87} +4.87689i q^{89} -4.79741i q^{91} +(-1.32431 + 0.371834i) q^{93} +15.7304 q^{95} +1.12311 q^{97} +(-8.54312 + 5.20798i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 8 q^{9}+O(q^{10}) $ 16 * q - 8 * q^9	$ 16 q - 8 q^{9} + 8 q^{33} - 32 q^{49} + 40 q^{57} + 32 q^{73} - 48 q^{97}+O(q^{100}) $ 16 * q - 8 * q^9 + 8 * q^33 - 32 * q^49 + 40 * q^57 + 32 * q^73 - 48 * 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
$3$	−0.468213	−	1.66757i	−0.270323	−	0.962770i
$3$	−0.468213	−	1.66757i	−0.270323	−	0.962770i
$4$		0			0
$5$		−	3.02045i		−	1.35079i	−0.737458	−	0.675393i	$-0.763974\pi$
$5$		−	3.02045i		−	1.35079i	0.737458	−	0.675393i	$-0.236026\pi$
$6$		0			0
$7$			3.62258i			1.36921i	0.728915	+	0.684604i	$0.240025\pi$
$7$			3.62258i			1.36921i	−0.728915	+	0.684604i	$0.759975\pi$
$8$		0			0
$9$	−2.56155	+	1.56155i	−0.853851	+	0.520518i
$10$		0			0
$11$	3.33513			1.00558			0.502790	−	0.864409i	$-0.332307\pi$
$11$	3.33513			1.00558			0.502790	+	0.864409i	$0.332307\pi$
$12$		0			0
$13$	−1.32431			−0.367297			−0.183648	−	0.982992i	$-0.558791\pi$
$13$	−1.32431			−0.367297			−0.183648	+	0.982992i	$0.558791\pi$
$14$		0			0
$15$	−5.03680	+	1.41421i	−1.30050	+	0.365148i
$16$		0			0
$17$			7.12311i			1.72761i	0.503829	+	0.863803i	$0.331924\pi$
$17$			7.12311i			1.72761i	−0.503829	+	0.863803i	$0.668076\pi$
$18$		0			0
$19$			5.20798i			1.19479i	0.801946	+	0.597397i	$0.203798\pi$
$19$			5.20798i			1.19479i	−0.801946	+	0.597397i	$0.796202\pi$
$20$		0			0
$21$	6.04090	−	1.69614i	1.31823	−	0.370128i
$22$		0			0
$23$	4.41674			0.920954			0.460477	−	0.887672i	$-0.347678\pi$
$23$	4.41674			0.920954			0.460477	+	0.887672i	$0.347678\pi$
$24$		0			0
$25$	−4.12311			−0.824621
$26$		0			0
$27$	3.80335	+	3.54042i	0.731954	+	0.681354i
$28$		0			0
$29$			6.41273i			1.19081i	0.803424	+	0.595407i	$0.203009\pi$
$29$			6.41273i			1.19081i	−0.803424	+	0.595407i	$0.796991\pi$
$30$		0			0
$31$		−	0.794156i		−	0.142635i	−0.997454	−	0.0713173i	$-0.977280\pi$
$31$		−	0.794156i		−	0.142635i	0.997454	−	0.0713173i	$-0.0227203\pi$
$32$		0			0
$33$	−1.56155	−	5.56155i	−0.271831	−	0.968142i
$34$		0			0
$35$	10.9418			1.84951
$36$		0			0
$37$	−8.10887			−1.33309			−0.666545	−	0.745465i	$-0.732228\pi$
$37$	−8.10887			−1.33309			−0.666545	+	0.745465i	$0.732228\pi$
$38$		0			0
$39$	0.620058	+	2.20837i	0.0992887	+	0.353622i
$40$		0			0
$41$			4.00000i			0.624695i	0.949968	+	0.312348i	$0.101115\pi$
$41$			4.00000i			0.624695i	−0.949968	+	0.312348i	$0.898885\pi$
$42$		0			0
$43$		−	1.46228i		−	0.222995i	−0.993765	−	0.111498i	$-0.964435\pi$
$43$		−	1.46228i		−	0.222995i	0.993765	−	0.111498i	$-0.0355648\pi$
$44$		0			0
$45$	4.71659	+	7.73704i	0.703108	+	1.15337i
$46$		0			0
$47$	5.65685			0.825137			0.412568	−	0.910927i	$-0.364632\pi$
$47$	5.65685			0.825137			0.412568	+	0.910927i	$0.364632\pi$
$48$		0			0
$49$	−6.12311			−0.874729
$50$		0			0
$51$	11.8782	−	3.33513i	1.66329	−	0.467012i
$52$		0			0
$53$		−	3.76412i		−	0.517041i	−0.966006	−	0.258521i	$-0.916765\pi$
$53$		−	3.76412i		−	0.517041i	0.966006	−	0.258521i	$-0.0832350\pi$
$54$		0			0
$55$		−	10.0736i		−	1.35832i
$56$		0			0
$57$	8.68466	−	2.43845i	1.15031	−	0.322980i
$58$		0			0
$59$	5.73384			0.746482			0.373241	−	0.927734i	$-0.378246\pi$
$59$	5.73384			0.746482			0.373241	+	0.927734i	$0.378246\pi$
$60$		0			0
$61$	−13.4061			−1.71648			−0.858238	−	0.513253i	$-0.828440\pi$
$61$	−13.4061			−1.71648			−0.858238	+	0.513253i	$0.828440\pi$
$62$		0			0
$63$	−5.65685	−	9.27944i	−0.712697	−	1.16910i
$64$		0			0
$65$			4.00000i			0.496139i
$66$		0			0
$67$			7.08084i			0.865062i	0.901619	+	0.432531i	$0.142379\pi$
$67$			7.08084i			0.865062i	−0.901619	+	0.432531i	$0.857621\pi$
$68$		0			0
$69$	−2.06798	−	7.36520i	−0.248955	−	0.886666i
$70$		0			0
$71$	−10.0736			−1.19552			−0.597758	−	0.801677i	$-0.703942\pi$
$71$	−10.0736			−1.19552			−0.597758	+	0.801677i	$0.703942\pi$
$72$		0			0
$73$	10.2462			1.19923			0.599614	−	0.800289i	$-0.295321\pi$
$73$	10.2462			1.19923			0.599614	+	0.800289i	$0.295321\pi$
$74$		0			0
$75$	1.93049	+	6.87555i	0.222914	+	0.793920i
$76$		0			0
$77$			12.0818i			1.37685i
$78$		0			0
$79$		−	4.86270i		−	0.547096i	−0.961858	−	0.273548i	$-0.911803\pi$
$79$		−	4.86270i		−	0.547096i	0.961858	−	0.273548i	$-0.0881973\pi$
$80$		0			0
$81$	4.12311	−	8.00000i	0.458123	−	0.888889i
$82$		0			0
$83$	−0.410574			−0.0450663			−0.0225331	−	0.999746i	$-0.507173\pi$
$83$	−0.410574			−0.0450663			−0.0225331	+	0.999746i	$0.507173\pi$
$84$		0			0
$85$	21.5150			2.33363
$86$		0			0
$87$	10.6937	−	3.00252i	1.14648	−	0.321904i
$88$		0			0
$89$			4.87689i			0.516950i	0.966018	+	0.258475i	$0.0832199\pi$
$89$			4.87689i			0.516950i	−0.966018	+	0.258475i	$0.916780\pi$
$90$		0			0
$91$		−	4.79741i		−	0.502905i
$92$		0			0
$93$	−1.32431	+	0.371834i	−0.137324	+	0.0385574i
$94$		0			0
$95$	15.7304			1.61391
$96$		0			0
$97$	1.12311			0.114034			0.0570170	−	0.998373i	$-0.481841\pi$
$97$	1.12311			0.114034			0.0570170	+	0.998373i	$0.481841\pi$
$98$		0			0
$99$	−8.54312	+	5.20798i	−0.858616	+	0.523422i
$100$		0			0
$101$		−	17.7509i		−	1.76628i	−0.469113	−	0.883138i	$-0.655426\pi$
$101$		−	17.7509i		−	1.76628i	0.469113	−	0.883138i	$-0.344574\pi$
$102$		0			0
$103$			2.03427i			0.200443i	0.994965	+	0.100221i	$0.0319551\pi$
$103$			2.03427i			0.200443i	−0.994965	+	0.100221i	$0.968045\pi$
$104$		0			0
$105$	−5.12311	−	18.2462i	−0.499964	−	1.78065i
$106$		0			0
$107$	16.1498			1.56126			0.780630	−	0.624993i	$-0.214898\pi$
$107$	16.1498			1.56126			0.780630	+	0.624993i	$0.214898\pi$
$108$		0			0
$109$	8.10887			0.776689			0.388344	−	0.921514i	$-0.373047\pi$
$109$	8.10887			0.776689			0.388344	+	0.921514i	$0.373047\pi$
$110$		0			0
$111$	3.79668	+	13.5221i	0.360365	+	1.28346i
$112$		0			0
$113$		−	12.4924i		−	1.17519i	−0.809156	−	0.587594i	$-0.800075\pi$
$113$		−	12.4924i		−	1.17519i	0.809156	−	0.587594i	$-0.199925\pi$
$114$		0			0
$115$		−	13.3405i		−	1.24401i
$116$		0			0
$117$	3.39228	−	2.06798i	0.313617	−	0.191184i
$118$		0			0
$119$	−25.8040			−2.36545
$120$		0			0
$121$	0.123106			0.0111914
$122$		0			0
$123$	6.67026	−	1.87285i	0.601437	−	0.168869i
$124$		0			0
$125$		−	2.64861i		−	0.236899i
$126$		0			0
$127$			19.3530i			1.71730i	0.512559	+	0.858652i	$0.328697\pi$
$127$			19.3530i			1.71730i	−0.512559	+	0.858652i	$0.671303\pi$
$128$		0			0
$129$	−2.43845	+	0.684658i	−0.214693	+	0.0602808i
$130$		0			0
$131$	9.47954			0.828232			0.414116	−	0.910224i	$-0.364091\pi$
$131$	9.47954			0.828232			0.414116	+	0.910224i	$0.364091\pi$
$132$		0			0
$133$	−18.8664			−1.63592
$134$		0			0
$135$	10.6937	−	11.4878i	0.920363	−	0.988713i
$136$		0			0
$137$			4.00000i			0.341743i	0.985293	+	0.170872i	$0.0546583\pi$
$137$			4.00000i			0.341743i	−0.985293	+	0.170872i	$0.945342\pi$
$138$		0			0
$139$			10.0054i			0.848647i	0.905511	+	0.424323i	$0.139488\pi$
$139$			10.0054i			0.848647i	−0.905511	+	0.424323i	$0.860512\pi$
$140$		0			0
$141$	−2.64861	−	9.43318i	−0.223054	−	0.794417i
$142$		0			0
$143$	−4.41674			−0.369346
$144$		0			0
$145$	19.3693			1.60853
$146$		0			0
$147$	2.86692	+	10.2107i	0.236459	+	0.842163i
$148$		0			0
$149$			2.27678i			0.186521i	0.995642	+	0.0932605i	$0.0297289\pi$
$149$			2.27678i			0.186521i	−0.995642	+	0.0932605i	$0.970271\pi$
$150$		0			0
$151$			10.8677i			0.884405i	0.896915	+	0.442202i	$0.145803\pi$
$151$			10.8677i			0.884405i	−0.896915	+	0.442202i	$0.854197\pi$
$152$		0			0
$153$	−11.1231	−	18.2462i	−0.899250	−	1.47512i
$154$		0			0
$155$	−2.39871			−0.192669
$156$		0			0
$157$	3.97292			0.317074			0.158537	−	0.987353i	$-0.449322\pi$
$157$	3.97292			0.317074			0.158537	+	0.987353i	$0.449322\pi$
$158$		0			0
$159$	−6.27691	+	1.76241i	−0.497792	+	0.139768i
$160$		0			0
$161$			16.0000i			1.26098i
$162$		0			0
$163$		−	5.20798i		−	0.407921i	−0.978979	−	0.203960i	$-0.934619\pi$
$163$		−	5.20798i		−	0.407921i	0.978979	−	0.203960i	$-0.0653814\pi$
$164$		0			0
$165$	−16.7984	+	4.71659i	−1.30775	+	0.367186i
$166$		0			0
$167$	−6.89697			−0.533704			−0.266852	−	0.963738i	$-0.585983\pi$
$167$	−6.89697			−0.533704			−0.266852	+	0.963738i	$0.585983\pi$
$168$		0			0
$169$	−11.2462			−0.865093
$170$		0			0
$171$	−8.13254	−	13.3405i	−0.621911	−	1.02018i
$172$		0			0
$173$		−	3.02045i		−	0.229640i	−0.993386	−	0.114820i	$-0.963371\pi$
$173$		−	3.02045i		−	0.229640i	0.993386	−	0.114820i	$-0.0366292\pi$
$174$		0			0
$175$		−	14.9363i		−	1.12908i
$176$		0			0
$177$	−2.68466	−	9.56155i	−0.201791	−	0.718690i
$178$		0			0
$179$	−1.98813			−0.148600			−0.0743000	−	0.997236i	$-0.523672\pi$
$179$	−1.98813			−0.148600			−0.0743000	+	0.997236i	$0.523672\pi$
$180$		0			0
$181$	20.1907			1.50076			0.750380	−	0.661007i	$-0.229870\pi$
$181$	20.1907			1.50076			0.750380	+	0.661007i	$0.229870\pi$
$182$		0			0
$183$	6.27691	+	22.3556i	0.464003	+	1.65257i
$184$		0			0
$185$			24.4924i			1.80072i
$186$		0			0
$187$			23.7565i			1.73725i
$188$		0			0
$189$	−12.8255	+	13.7779i	−0.932915	+	1.00220i
$190$		0			0
$191$	−3.17662			−0.229852			−0.114926	−	0.993374i	$-0.536663\pi$
$191$	−3.17662			−0.229852			−0.114926	+	0.993374i	$0.536663\pi$
$192$		0			0
$193$	4.00000			0.287926			0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000			0.287926			0.143963	+	0.989583i	$0.454015\pi$
$194$		0			0
$195$	6.67026	−	1.87285i	0.477668	−	0.134118i
$196$		0			0
$197$			19.2382i			1.37066i	0.728231	+	0.685332i	$0.240343\pi$
$197$			19.2382i			1.37066i	−0.728231	+	0.685332i	$0.759657\pi$
$198$		0			0
$199$			2.03427i			0.144206i	0.997397	+	0.0721028i	$0.0229710\pi$
$199$			2.03427i			0.144206i	−0.997397	+	0.0721028i	$0.977029\pi$
$200$		0			0
$201$	11.8078	−	3.31534i	0.832855	−	0.233846i
$202$		0			0
$203$	−23.2306			−1.63047
$204$		0			0
$205$	12.0818			0.843829
$206$		0			0
$207$	−11.3137	+	6.89697i	−0.786357	+	0.479373i
$208$		0			0
$209$			17.3693i			1.20146i
$210$		0			0
$211$			19.6002i			1.34933i	0.738122	+	0.674667i	$0.235713\pi$
$211$			19.6002i			1.34933i	−0.738122	+	0.674667i	$0.764287\pi$
$212$		0			0
$213$	4.71659	+	16.7984i	0.323175	+	1.15101i
$214$		0			0
$215$	−4.41674			−0.301219
$216$		0			0
$217$	2.87689			0.195296
$218$		0			0
$219$	−4.79741	−	17.0862i	−0.324179	−	1.15458i
$220$		0			0
$221$		−	9.43318i		−	0.634544i
$222$		0			0
$223$			8.93124i			0.598080i	0.954241	+	0.299040i	$0.0966664\pi$
$223$			8.93124i			0.598080i	−0.954241	+	0.299040i	$0.903334\pi$
$224$		0			0
$225$	10.5616	−	6.43845i	0.704104	−	0.429230i
$226$		0			0
$227$	−13.7511			−0.912693			−0.456346	−	0.889802i	$-0.650842\pi$
$227$	−13.7511			−0.912693			−0.456346	+	0.889802i	$0.650842\pi$
$228$		0			0
$229$	1.32431			0.0875127			0.0437563	−	0.999042i	$-0.486067\pi$
$229$	1.32431			0.0875127			0.0437563	+	0.999042i	$0.486067\pi$
$230$		0			0
$231$	20.1472	−	5.65685i	1.32559	−	0.372194i
$232$		0			0
$233$			12.8769i			0.843593i	0.906690	+	0.421797i	$0.138600\pi$
$233$			12.8769i			0.843593i	−0.906690	+	0.421797i	$0.861400\pi$
$234$		0			0
$235$		−	17.0862i		−	1.11458i
$236$		0			0
$237$	−8.10887	+	2.27678i	−0.526728	+	0.147893i
$238$		0			0
$239$	25.8040			1.66912			0.834562	−	0.550914i	$-0.185721\pi$
$239$	25.8040			1.66912			0.834562	+	0.550914i	$0.185721\pi$
$240$		0			0
$241$	−24.4924			−1.57770			−0.788848	−	0.614589i	$-0.789322\pi$
$241$	−24.4924			−1.57770			−0.788848	+	0.614589i	$0.789322\pi$
$242$		0			0
$243$	−15.2710	−	3.12985i	−0.979636	−	0.200780i
$244$		0			0
$245$			18.4945i			1.18157i
$246$		0			0
$247$		−	6.89697i		−	0.438844i
$248$		0			0
$249$	0.192236	+	0.684658i	0.0121825	+	0.0433885i
$250$		0			0
$251$	−20.4214			−1.28899			−0.644493	−	0.764611i	$-0.722931\pi$
$251$	−20.4214			−1.28899			−0.644493	+	0.764611i	$0.722931\pi$
$252$		0			0
$253$	14.7304			0.926093
$254$		0			0
$255$	−10.0736	−	35.8776i	−0.630833	−	2.24674i
$256$		0			0
$257$		0			0		1.00000			$0$
$257$		0			0		−1.00000			$\pi$
$258$		0			0
$259$		−	29.3751i		−	1.82528i
$260$		0			0
$261$	−10.0138	−	16.4265i	−0.619840	−	1.01678i
$262$		0			0
$263$	7.59336			0.468227			0.234113	−	0.972209i	$-0.424781\pi$
$263$	7.59336			0.468227			0.234113	+	0.972209i	$0.424781\pi$
$264$		0			0
$265$	−11.3693			−0.698412
$266$		0			0
$267$	8.13254	−	2.28343i	0.497704	−	0.139743i
$268$		0			0
$269$		−	3.76412i		−	0.229502i	−0.993394	−	0.114751i	$-0.963393\pi$
$269$		−	3.76412i		−	0.229502i	0.993394	−	0.114751i	$-0.0366071\pi$
$270$		0			0
$271$		−	25.0099i		−	1.51924i	−0.650366	−	0.759621i	$-0.725384\pi$
$271$		−	25.0099i		−	1.51924i	0.650366	−	0.759621i	$-0.274616\pi$
$272$		0			0
$273$	−8.00000	+	2.24621i	−0.484182	+	0.135947i
$274$		0			0
$275$	−13.7511			−0.829223
$276$		0			0
$277$	−18.7033			−1.12377			−0.561887	−	0.827214i	$-0.689924\pi$
$277$	−18.7033			−1.12377			−0.561887	+	0.827214i	$0.689924\pi$
$278$		0			0
$279$	1.24012	+	2.03427i	0.0742438	+	0.121789i
$280$		0			0
$281$		−	9.36932i		−	0.558927i	−0.960156	−	0.279463i	$-0.909843\pi$
$281$		−	9.36932i		−	0.558927i	0.960156	−	0.279463i	$-0.0901565\pi$
$282$		0			0
$283$		−	17.4968i		−	1.04008i	−0.854143	−	0.520039i	$-0.825918\pi$
$283$		−	17.4968i		−	1.04008i	0.854143	−	0.520039i	$-0.174082\pi$
$284$		0			0
$285$	−7.36520	−	26.2316i	−0.436277	−	1.55382i
$286$		0			0
$287$	−14.4903			−0.855337
$288$		0			0
$289$	−33.7386			−1.98463
$290$		0			0
$291$	−0.525853	−	1.87285i	−0.0308260	−	0.109789i
$292$		0			0
$293$			0.371834i			0.0217228i	0.999941	+	0.0108614i	$0.00345736\pi$
$293$			0.371834i			0.0217228i	−0.999941	+	0.0108614i	$0.996543\pi$
$294$		0			0
$295$		−	17.3188i		−	1.00834i
$296$		0			0
$297$	12.6847	+	11.8078i	0.736039	+	0.685156i
$298$		0			0
$299$	−5.84912			−0.338263
$300$		0			0
$301$	5.29723			0.305327
$302$		0			0
$303$	−29.6007	+	8.31118i	−1.70052	+	0.477465i
$304$		0			0
$305$			40.4924i			2.31859i
$306$		0			0
$307$		−	23.3459i		−	1.33242i	−0.745763	−	0.666211i	$-0.767915\pi$
$307$		−	23.3459i		−	1.33242i	0.745763	−	0.666211i	$-0.232085\pi$
$308$		0			0
$309$	3.39228	−	0.952473i	0.192980	−	0.0541843i
$310$		0			0
$311$	18.9071			1.07212			0.536061	−	0.844179i	$-0.319912\pi$
$311$	18.9071			1.07212			0.536061	+	0.844179i	$0.319912\pi$
$312$		0			0
$313$	19.6155			1.10874			0.554368	−	0.832272i	$-0.312960\pi$
$313$	19.6155			1.10874			0.554368	+	0.832272i	$0.312960\pi$
$314$		0			0
$315$	−28.0281	+	17.0862i	−1.57920	+	0.962700i
$316$		0			0
$317$		−	27.9277i		−	1.56858i	−0.620397	−	0.784288i	$-0.713029\pi$
$317$		−	27.9277i		−	1.56858i	0.620397	−	0.784288i	$-0.286971\pi$
$318$		0			0
$319$			21.3873i			1.19746i
$320$		0			0
$321$	−7.56155	−	26.9309i	−0.422045	−	1.50313i
$322$		0			0
$323$	−37.0970			−2.06413
$324$		0			0
$325$	5.46026			0.302881
$326$		0			0
$327$	−3.79668	−	13.5221i	−0.209957	−	0.747773i
$328$		0			0
$329$			20.4924i			1.12978i
$330$		0			0
$331$			30.0162i			1.64984i	0.565250	+	0.824919i	$0.308780\pi$
$331$			30.0162i			1.64984i	−0.565250	+	0.824919i	$0.691220\pi$
$332$		0			0
$333$	20.7713	−	12.6624i	1.13826	−	0.693897i
$334$		0			0
$335$	21.3873			1.16851
$336$		0			0
$337$	7.12311			0.388020			0.194010	−	0.981000i	$-0.437851\pi$
$337$	7.12311			0.388020			0.194010	+	0.981000i	$0.437851\pi$
$338$		0			0
$339$	−20.8319	+	5.84912i	−1.13144	+	0.317680i
$340$		0			0
$341$		−	2.64861i		−	0.143430i
$342$		0			0
$343$			3.17662i			0.171521i
$344$		0			0
$345$	−22.2462	+	6.24621i	−1.19770	+	0.336285i
$346$		0			0
$347$	21.2425			1.14036			0.570179	−	0.821521i	$-0.306874\pi$
$347$	21.2425			1.14036			0.570179	+	0.821521i	$0.306874\pi$
$348$		0			0
$349$	−22.8393			−1.22256			−0.611279	−	0.791415i	$-0.709345\pi$
$349$	−22.8393			−1.22256			−0.611279	+	0.791415i	$0.709345\pi$
$350$		0			0
$351$	−5.03680	−	4.68860i	−0.268844	−	0.250259i
$352$		0			0
$353$		−	8.00000i		−	0.425797i	−0.977074	−	0.212899i	$-0.931710\pi$
$353$		−	8.00000i		−	0.425797i	0.977074	−	0.212899i	$-0.0682904\pi$
$354$		0			0
$355$			30.4268i			1.61488i
$356$		0			0
$357$	12.0818	+	43.0299i	0.639436	+	2.27739i
$358$		0			0
$359$	1.24012			0.0654508			0.0327254	−	0.999464i	$-0.489581\pi$
$359$	1.24012			0.0654508			0.0327254	+	0.999464i	$0.489581\pi$
$360$		0			0
$361$	−8.12311			−0.427532
$362$		0			0
$363$	−0.0576397	−	0.205287i	−0.00302530	−	0.0107748i
$364$		0			0
$365$		−	30.9481i		−	1.61990i
$366$		0			0
$367$		−	23.4216i		−	1.22260i	−0.791400	−	0.611298i	$-0.790648\pi$
$367$		−	23.4216i		−	1.22260i	0.791400	−	0.611298i	$-0.209352\pi$
$368$		0			0
$369$	−6.24621	−	10.2462i	−0.325165	−	0.533396i
$370$		0			0
$371$	13.6358			0.707937
$372$		0			0
$373$	−1.32431			−0.0685700			−0.0342850	−	0.999412i	$-0.510915\pi$
$373$	−1.32431			−0.0685700			−0.0342850	+	0.999412i	$0.510915\pi$
$374$		0			0
$375$	−4.41674	+	1.24012i	−0.228079	+	0.0640393i
$376$		0			0
$377$		−	8.49242i		−	0.437382i
$378$		0			0
$379$		−	12.6994i		−	0.652324i	−0.945314	−	0.326162i	$-0.894244\pi$
$379$		−	12.6994i		−	0.652324i	0.945314	−	0.326162i	$-0.105756\pi$
$380$		0			0
$381$	32.2725	−	9.06134i	1.65337	−	0.464227i
$382$		0			0
$383$	34.6375			1.76989			0.884947	−	0.465691i	$-0.154194\pi$
$383$	34.6375			1.76989			0.884947	+	0.465691i	$0.154194\pi$
$384$		0			0
$385$	36.4924			1.85983
$386$		0			0
$387$	2.28343	+	3.74571i	0.116073	+	0.190405i
$388$		0			0
$389$			20.3995i			1.03429i	0.855897	+	0.517147i	$0.173006\pi$
$389$			20.3995i			1.03429i	−0.855897	+	0.517147i	$0.826994\pi$
$390$		0			0
$391$			31.4609i			1.59105i
$392$		0			0
$393$	−4.43845	−	15.8078i	−0.223890	−	0.797396i
$394$		0			0
$395$	−14.6875			−0.739010
$396$		0			0
$397$	−5.46026			−0.274042			−0.137021	−	0.990568i	$-0.543753\pi$
$397$	−5.46026			−0.274042			−0.137021	+	0.990568i	$0.543753\pi$
$398$		0			0
$399$	8.83348	+	31.4609i	0.442227	+	1.57501i
$400$		0			0
$401$		−	7.12311i		−	0.355711i	−0.984057	−	0.177855i	$-0.943084\pi$
$401$		−	7.12311i		−	0.355711i	0.984057	−	0.177855i	$-0.0569160\pi$
$402$		0			0
$403$			1.05171i			0.0523892i
$404$		0			0
$405$	−24.1636	−	12.4536i	−1.20070	−	0.618826i
$406$		0			0
$407$	−27.0442			−1.34053
$408$		0			0
$409$	14.0000			0.692255			0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000			0.692255			0.346128	+	0.938187i	$0.387496\pi$
$410$		0			0
$411$	6.67026	−	1.87285i	0.329020	−	0.0923810i
$412$		0			0
$413$			20.7713i			1.02209i
$414$		0			0
$415$			1.24012i			0.0608749i
$416$		0			0
$417$	16.6847	−	4.68466i	0.817051	−	0.229409i
$418$		0			0
$419$	−16.6757			−0.814659			−0.407330	−	0.913281i	$-0.633540\pi$
$419$	−16.6757			−0.814659			−0.407330	+	0.913281i	$0.633540\pi$
$420$		0			0
$421$	−9.27015			−0.451799			−0.225900	−	0.974151i	$-0.572532\pi$
$421$	−9.27015			−0.451799			−0.225900	+	0.974151i	$0.572532\pi$
$422$		0			0
$423$	−14.4903	+	8.83348i	−0.704544	+	0.429498i
$424$		0			0
$425$		−	29.3693i		−	1.42462i
$426$		0			0
$427$		−	48.5647i		−	2.35021i
$428$		0			0
$429$	2.06798	+	7.36520i	0.0998428	+	0.355595i
$430$		0			0
$431$	−37.1177			−1.78790			−0.893950	−	0.448168i	$-0.852077\pi$
$431$	−37.1177			−1.78790			−0.893950	+	0.448168i	$0.852077\pi$
$432$		0			0
$433$	23.8617			1.14672			0.573361	−	0.819303i	$-0.305639\pi$
$433$	23.8617			1.14672			0.573361	+	0.819303i	$0.305639\pi$
$434$		0			0
$435$	−9.06897	−	32.2996i	−0.434824	−	1.54865i
$436$		0			0
$437$			23.0023i			1.10035i
$438$		0			0
$439$		−	15.6327i		−	0.746107i	−0.927810	−	0.373054i	$-0.878311\pi$
$439$		−	15.6327i		−	0.746107i	0.927810	−	0.373054i	$-0.121689\pi$
$440$		0			0
$441$	15.6847	−	9.56155i	0.746888	−	0.455312i
$442$		0			0
$443$	−17.4968			−0.831298			−0.415649	−	0.909525i	$-0.636446\pi$
$443$	−17.4968			−0.831298			−0.415649	+	0.909525i	$0.636446\pi$
$444$		0			0
$445$	14.7304			0.698288
$446$		0			0
$447$	3.79668	−	1.06602i	0.179577	−	0.0504209i
$448$		0			0
$449$			23.1231i			1.09125i	0.838030	+	0.545623i	$0.183707\pi$
$449$			23.1231i			1.09125i	−0.838030	+	0.545623i	$0.816293\pi$
$450$		0			0
$451$			13.3405i			0.628181i
$452$		0			0
$453$	18.1227	−	5.08842i	0.851478	−	0.239075i
$454$		0			0
$455$	−14.4903			−0.679317
$456$		0			0
$457$	8.24621			0.385741			0.192871	−	0.981224i	$-0.438220\pi$
$457$	8.24621			0.385741			0.192871	+	0.981224i	$0.438220\pi$
$458$		0			0
$459$	−25.2188	+	27.0916i	−1.17711	+	1.26453i
$460$		0			0
$461$			6.41273i			0.298671i	0.988787	+	0.149335i	$0.0477134\pi$
$461$			6.41273i			0.298671i	−0.988787	+	0.149335i	$0.952287\pi$
$462$		0			0
$463$			16.8728i			0.784145i	0.919934	+	0.392073i	$0.128242\pi$
$463$			16.8728i			0.784145i	−0.919934	+	0.392073i	$0.871758\pi$
$464$		0			0
$465$	1.12311	+	4.00000i	0.0520828	+	0.185496i
$466$		0			0
$467$	15.8545			0.733659			0.366830	−	0.930288i	$-0.380443\pi$
$467$	15.8545			0.733659			0.366830	+	0.930288i	$0.380443\pi$
$468$		0			0
$469$	−25.6509			−1.18445
$470$		0			0
$471$	−1.86017	−	6.62511i	−0.0857123	−	0.305269i
$472$		0			0
$473$		−	4.87689i		−	0.224240i
$474$		0			0
$475$		−	21.4731i		−	0.985252i
$476$		0			0
$477$	5.87787	+	9.64198i	0.269129	+	0.441476i
$478$		0			0
$479$	20.1472			0.920548			0.460274	−	0.887777i	$-0.347751\pi$
$479$	20.1472			0.920548			0.460274	+	0.887777i	$0.347751\pi$
$480$		0			0
$481$	10.7386			0.489640
$482$		0			0
$483$	26.6811	−	7.49141i	1.21403	−	0.340871i
$484$		0			0
$485$		−	3.39228i		−	0.154036i
$486$		0			0
$487$			35.0835i			1.58978i	0.606751	+	0.794892i	$0.292473\pi$
$487$			35.0835i			1.58978i	−0.606751	+	0.794892i	$0.707527\pi$
$488$		0			0
$489$	−8.68466	+	2.43845i	−0.392734	+	0.110270i
$490$		0			0
$491$	−29.4903			−1.33088			−0.665440	−	0.746451i	$-0.731756\pi$
$491$	−29.4903			−1.33088			−0.665440	+	0.746451i	$0.731756\pi$
$492$		0			0
$493$	−45.6786			−2.05726
$494$		0			0
$495$	15.7304	+	25.8040i	0.707031	+	1.15981i
$496$		0			0
$497$		−	36.4924i		−	1.63691i
$498$		0			0
$499$		−	16.6757i		−	0.746505i	−0.927730	−	0.373253i	$-0.878243\pi$
$499$		−	16.6757i		−	0.746505i	0.927730	−	0.373253i	$-0.121757\pi$
$500$		0			0
$501$	3.22925	+	11.5012i	0.144272	+	0.513834i
$502$		0			0
$503$	21.3873			0.953613			0.476806	−	0.879008i	$-0.341794\pi$
$503$	21.3873			0.953613			0.476806	+	0.879008i	$0.341794\pi$
$504$		0			0
$505$	−53.6155			−2.38586
$506$		0			0
$507$	5.26562	+	18.7538i	0.233855	+	0.832885i
$508$		0			0
$509$			15.8459i			0.702358i	0.936308	+	0.351179i	$0.114219\pi$
$509$			15.8459i			0.702358i	−0.936308	+	0.351179i	$0.885781\pi$
$510$		0			0
$511$			37.1177i			1.64199i
$512$		0			0
$513$	−18.4384	+	19.8078i	−0.814077	+	0.874534i
$514$		0			0
$515$	6.14441			0.270755
$516$		0			0
$517$	18.8664			0.829741
$518$		0			0
$519$	−5.03680	+	1.41421i	−0.221091	+	0.0620771i
$520$		0			0
$521$		−	0.492423i		−	0.0215734i	−0.999942	−	0.0107867i	$-0.996566\pi$
$521$		−	0.492423i		−	0.0215734i	0.999942	−	0.0107867i	$-0.00343358\pi$
$522$		0			0
$523$			14.8028i			0.647282i	0.946180	+	0.323641i	$0.104907\pi$
$523$			14.8028i			0.647282i	−0.946180	+	0.323641i	$0.895093\pi$
$524$		0			0
$525$	−24.9073	+	6.99337i	−1.08704	+	0.305216i
$526$		0			0
$527$	5.65685			0.246416
$528$		0			0
$529$	−3.49242			−0.151844
$530$		0			0
$531$	−14.6875	+	8.95369i	−0.637384	+	0.388557i
$532$		0			0
$533$		−	5.29723i		−	0.229448i
$534$		0			0
$535$		−	48.7797i		−	2.10893i
$536$		0			0
$537$	0.930870	+	3.31534i	0.0401700	+	0.143068i
$538$		0			0
$539$	−20.4214			−0.879610
$540$		0			0
$541$	5.46026			0.234755			0.117377	−	0.993087i	$-0.462551\pi$
$541$	5.46026			0.234755			0.117377	+	0.993087i	$0.462551\pi$
$542$		0			0
$543$	−9.45353	−	33.6693i	−0.405690	−	1.44489i
$544$		0			0
$545$		−	24.4924i		−	1.04914i
$546$		0			0
$547$		−	8.13254i		−	0.347722i	−0.984770	−	0.173861i	$-0.944376\pi$
$547$		−	8.13254i		−	0.347722i	0.984770	−	0.173861i	$-0.0556244\pi$
$548$		0			0
$549$	34.3404	−	20.9343i	1.46561	−	0.893455i
$550$		0			0
$551$	−33.3974			−1.42278
$552$		0			0
$553$	17.6155			0.749088
$554$		0			0
$555$	40.8427	−	11.4677i	1.73368	−	0.486776i
$556$		0			0
$557$		−	23.7917i		−	1.00809i	−0.863678	−	0.504044i	$-0.831845\pi$
$557$		−	23.7917i		−	1.00809i	0.863678	−	0.504044i	$-0.168155\pi$
$558$		0			0
$559$			1.93651i			0.0819055i
$560$		0			0
$561$	39.6155	−	11.1231i	1.67257	−	0.469618i
$562$		0			0
$563$	4.15628			0.175166			0.0875831	−	0.996157i	$-0.472086\pi$
$563$	4.15628			0.175166			0.0875831	+	0.996157i	$0.472086\pi$
$564$		0			0
$565$	−37.7327			−1.58743
$566$		0			0
$567$	28.9807	+	14.9363i	1.21707	+	0.627265i
$568$		0			0
$569$			0.492423i			0.0206434i	0.999947	+	0.0103217i	$0.00328556\pi$
$569$			0.492423i			0.0206434i	−0.999947	+	0.0103217i	$0.996714\pi$
$570$		0			0
$571$			16.6757i			0.697855i	0.937150	+	0.348927i	$0.113454\pi$
$571$			16.6757i			0.697855i	−0.937150	+	0.348927i	$0.886546\pi$
$572$		0			0
$573$	1.48734	+	5.29723i	0.0621344	+	0.221295i
$574$		0			0
$575$	−18.2107			−0.759438
$576$		0			0
$577$	−33.8617			−1.40968			−0.704841	−	0.709365i	$-0.748982\pi$
$577$	−33.8617			−1.40968			−0.704841	+	0.709365i	$0.748982\pi$
$578$		0			0
$579$	−1.87285	−	6.67026i	−0.0778331	−	0.277207i
$580$		0			0
$581$		−	1.48734i		−	0.0617051i
$582$		0			0
$583$		−	12.5538i		−	0.519926i
$584$		0			0
$585$	−6.24621	−	10.2462i	−0.258249	−	0.423629i
$586$		0			0
$587$	40.1369			1.65663			0.828313	−	0.560266i	$-0.189301\pi$
$587$	40.1369			1.65663			0.828313	+	0.560266i	$0.189301\pi$
$588$		0			0
$589$	4.13595			0.170419
$590$		0			0
$591$	32.0810	−	9.00757i	1.31963	−	0.370522i
$592$		0			0
$593$		−	32.9848i		−	1.35453i	−0.735741	−	0.677263i	$-0.763166\pi$
$593$		−	32.9848i		−	1.35453i	0.735741	−	0.677263i	$-0.236834\pi$
$594$		0			0
$595$			77.9398i			3.19522i
$596$		0			0
$597$	3.39228	−	0.952473i	0.138837	−	0.0389821i
$598$		0			0
$599$	−15.7304			−0.642729			−0.321364	−	0.946956i	$-0.604141\pi$
$599$	−15.7304			−0.642729			−0.321364	+	0.946956i	$0.604141\pi$
$600$		0			0
$601$	26.2462			1.07061			0.535303	−	0.844660i	$-0.320198\pi$
$601$	26.2462			1.07061			0.535303	+	0.844660i	$0.320198\pi$
$602$		0			0
$603$	−11.0571	−	18.1379i	−0.450280	−	0.738634i
$604$		0			0
$605$		−	0.371834i		−	0.0151172i
$606$		0			0
$607$		−	0.794156i		−	0.0322338i	−0.999870	−	0.0161169i	$-0.994870\pi$
$607$		−	0.794156i		−	0.0322338i	0.999870	−	0.0161169i	$-0.00513039\pi$
$608$		0			0
$609$	10.8769	+	38.7386i	0.440754	+	1.56977i
$610$		0			0
$611$	−7.49141			−0.303070
$612$		0			0
$613$	32.2725			1.30347			0.651736	−	0.758446i	$-0.274041\pi$
$613$	32.2725			1.30347			0.651736	+	0.758446i	$0.274041\pi$
$614$		0			0
$615$	−5.65685	−	20.1472i	−0.228106	−	0.812413i
$616$		0			0
$617$		−	12.8769i		−	0.518404i	−0.965823	−	0.259202i	$-0.916540\pi$
$617$		−	12.8769i		−	0.518404i	0.965823	−	0.259202i	$-0.0834596\pi$
$618$		0			0
$619$			3.33513i			0.134050i	0.997751	+	0.0670251i	$0.0213508\pi$
$619$			3.33513i			0.134050i	−0.997751	+	0.0670251i	$0.978649\pi$
$620$		0			0
$621$	16.7984	+	15.6371i	0.674096	+	0.627495i
$622$		0			0
$623$	−17.6670			−0.707812
$624$		0			0
$625$	−28.6155			−1.14462
$626$		0			0
$627$	28.9645	−	8.13254i	1.15673	−	0.324782i
$628$		0			0
$629$		−	57.7603i		−	2.30306i
$630$		0			0
$631$		−	7.69113i		−	0.306179i	−0.988212	−	0.153089i	$-0.951078\pi$
$631$		−	7.69113i		−	0.306179i	0.988212	−	0.153089i	$-0.0489223\pi$
$632$		0			0
$633$	32.6847	−	9.17708i	1.29910	−	0.364756i
$634$		0			0
$635$	58.4548			2.31971
$636$		0			0
$637$	8.10887			0.321285
$638$		0			0
$639$	25.8040	−	15.7304i	1.02079	−	0.622287i
$640$		0			0
$641$		−	2.63068i		−	0.103906i	−0.998650	−	0.0519529i	$-0.983455\pi$
$641$		−	2.63068i		−	0.103906i	0.998650	−	0.0519529i	$-0.0165446\pi$
$642$		0			0
$643$		−	17.7274i		−	0.699099i	−0.936918	−	0.349550i	$-0.886335\pi$
$643$		−	17.7274i		−	0.699099i	0.936918	−	0.349550i	$-0.113665\pi$
$644$		0			0
$645$	2.06798	+	7.36520i	0.0814264	+	0.290005i
$646$		0			0
$647$	29.5244			1.16072			0.580362	−	0.814359i	$-0.302911\pi$
$647$	29.5244			1.16072			0.580362	+	0.814359i	$0.302911\pi$
$648$		0			0
$649$	19.1231			0.750648
$650$		0			0
$651$	−1.34700	−	4.79741i	−0.0527931	−	0.188025i
$652$		0			0
$653$			19.9819i			0.781951i	0.920401	+	0.390975i	$0.127862\pi$
$653$			19.9819i			0.781951i	−0.920401	+	0.390975i	$0.872138\pi$
$654$		0			0
$655$		−	28.6325i		−	1.11876i
$656$		0			0
$657$	−26.2462	+	16.0000i	−1.02396	+	0.624219i
$658$		0			0
$659$	−14.2770			−0.556151			−0.278076	−	0.960559i	$-0.589697\pi$
$659$	−14.2770			−0.556151			−0.278076	+	0.960559i	$0.589697\pi$
$660$		0			0
$661$	−24.3266			−0.946196			−0.473098	−	0.881010i	$-0.656864\pi$
$661$	−24.3266			−0.946196			−0.473098	+	0.881010i	$0.656864\pi$
$662$		0			0
$663$	−15.7304	+	4.41674i	−0.610920	+	0.171532i
$664$		0			0
$665$			56.9848i			2.20978i
$666$		0			0
$667$			28.3234i			1.09668i
$668$		0			0
$669$	14.8934	−	4.18173i	0.575813	−	0.161675i
$670$		0			0
$671$	−44.7111			−1.72605
$672$		0			0
$673$	−4.63068			−0.178500			−0.0892499	−	0.996009i	$-0.528447\pi$
$673$	−4.63068			−0.178500			−0.0892499	+	0.996009i	$0.528447\pi$
$674$		0			0
$675$	−15.6816	−	14.5975i	−0.603585	−	0.561859i
$676$		0			0
$677$		−	1.85917i		−	0.0714537i	−0.999362	−	0.0357269i	$-0.988625\pi$
$677$		−	1.85917i		−	0.0714537i	0.999362	−	0.0357269i	$-0.0113746\pi$
$678$		0			0
$679$			4.06854i			0.156136i
$680$		0			0
$681$	6.43845	+	22.9309i	0.246722	+	0.878713i
$682$		0			0
$683$	42.0745			1.60993			0.804967	−	0.593319i	$-0.202183\pi$
$683$	42.0745			1.60993			0.804967	+	0.593319i	$0.202183\pi$
$684$		0			0
$685$	12.0818			0.461622
$686$		0			0
$687$	−0.620058	−	2.20837i	−0.0236567	−	0.0842545i
$688$		0			0
$689$			4.98485i			0.189907i
$690$		0			0
$691$		−	28.1433i		−	1.07062i	−0.844655	−	0.535311i	$-0.820194\pi$
$691$		−	28.1433i		−	1.07062i	0.844655	−	0.535311i	$-0.179806\pi$
$692$		0			0
$693$	−18.8664	−	30.9481i	−0.716674	−	1.17562i
$694$		0			0
$695$	30.2208			1.14634
$696$		0			0
$697$	−28.4924			−1.07923
$698$		0			0
$699$	21.4731	−	6.02913i	0.812186	−	0.228043i
$700$		0			0
$701$			5.66906i			0.214118i	0.994253	+	0.107059i	$0.0341433\pi$
$701$			5.66906i			0.214118i	−0.994253	+	0.107059i	$0.965857\pi$
$702$		0			0
$703$		−	42.2309i		−	1.59277i
$704$		0			0
$705$	−28.4924	+	8.00000i	−1.07309	+	0.301297i
$706$		0			0
$707$	64.3039			2.41840
$708$		0			0
$709$	40.2183			1.51043			0.755215	−	0.655477i	$-0.227532\pi$
$709$	40.2183			1.51043			0.755215	+	0.655477i	$0.227532\pi$
$710$		0			0
$711$	7.59336	+	12.4561i	0.284773	+	0.467139i
$712$		0			0
$713$		−	3.50758i		−	0.131360i
$714$		0			0
$715$			13.3405i			0.498907i
$716$		0			0
$717$	−12.0818	−	43.0299i	−0.451203	−	1.60698i
$718$		0			0
$719$	−13.7939			−0.514427			−0.257214	−	0.966355i	$-0.582804\pi$
$719$	−13.7939			−0.514427			−0.257214	+	0.966355i	$0.582804\pi$
$720$		0			0
$721$	−7.36932			−0.274448
$722$		0			0
$723$	11.4677	+	40.8427i	0.426487	+	1.51896i
$724$		0			0
$725$		−	26.4404i		−	0.981970i
$726$		0			0
$727$			22.1815i			0.822665i	0.911485	+	0.411332i	$0.134936\pi$
$727$			22.1815i			0.822665i	−0.911485	+	0.411332i	$0.865064\pi$
$728$		0			0
$729$	1.93087	+	26.9309i	0.0715137	+	0.997440i
$730$		0			0
$731$	10.4160			0.385249
$732$		0			0
$733$	28.4626			1.05129			0.525644	−	0.850704i	$-0.323824\pi$
$733$	28.4626			1.05129			0.525644	+	0.850704i	$0.323824\pi$
$734$		0			0
$735$	30.8408	−	8.65938i	1.13758	−	0.319406i
$736$		0			0
$737$			23.6155i			0.869889i
$738$		0			0
$739$			6.25969i			0.230266i	0.993350	+	0.115133i	$0.0367295\pi$
$739$			6.25969i			0.230266i	−0.993350	+	0.115133i	$0.963271\pi$
$740$		0			0
$741$	−11.5012	+	3.22925i	−0.422505	+	0.118630i
$742$		0			0
$743$	33.3974			1.22523			0.612616	−	0.790381i	$-0.290117\pi$
$743$	33.3974			1.22523			0.612616	+	0.790381i	$0.290117\pi$
$744$		0			0
$745$	6.87689			0.251950
$746$		0			0
$747$	1.05171	−	0.641132i	0.0384799	−	0.0234578i
$748$		0			0
$749$			58.5040i			2.13769i
$750$		0			0
$751$		−	31.3631i		−	1.14446i	−0.820094	−	0.572228i	$-0.806079\pi$
$751$		−	31.3631i		−	1.14446i	0.820094	−	0.572228i	$-0.193921\pi$
$752$		0			0
$753$	9.56155	+	34.0540i	0.348442	+	1.24100i
$754$		0			0
$755$	32.8255			1.19464
$756$		0			0
$757$	−22.8393			−0.830108			−0.415054	−	0.909797i	$-0.636237\pi$
$757$	−22.8393			−0.830108			−0.415054	+	0.909797i	$0.636237\pi$
$758$		0			0
$759$	−6.89697	−	24.5639i	−0.250344	−	0.891614i
$760$		0			0
$761$		−	44.9848i		−	1.63070i	−0.578969	−	0.815350i	$-0.696545\pi$
$761$		−	44.9848i		−	1.63070i	0.578969	−	0.815350i	$-0.303455\pi$
$762$		0			0
$763$			29.3751i			1.06345i
$764$		0			0
$765$	−55.1117	+	33.5968i	−1.99257	+	1.21469i
$766$		0			0
$767$	−7.59336			−0.274180
$768$		0			0
$769$	36.0000			1.29819			0.649097	−	0.760706i	$-0.275147\pi$
$769$	36.0000			1.29819			0.649097	+	0.760706i	$0.275147\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	7.15640i		−	0.257398i	−0.991684	−	0.128699i	$-0.958920\pi$
$773$		−	7.15640i		−	0.257398i	0.991684	−	0.128699i	$-0.0410801\pi$
$774$		0			0
$775$			3.27439i			0.117619i
$776$		0			0
$777$	−48.9848	+	13.7538i	−1.75732	+	0.493414i
$778$		0			0
$779$	−20.8319			−0.746382
$780$		0			0
$781$	−33.5968			−1.20219
$782$		0			0
$783$	−22.7037	+	24.3898i	−0.811366	+	0.871621i
$784$		0			0
$785$		−	12.0000i		−	0.428298i
$786$		0			0
$787$		−	11.0571i		−	0.394143i	−0.980389	−	0.197072i	$-0.936857\pi$
$787$		−	11.0571i		−	0.394143i	0.980389	−	0.197072i	$-0.0631431\pi$
$788$		0			0
$789$	−3.55531	−	12.6624i	−0.126572	−	0.450794i
$790$		0			0
$791$	45.2548			1.60908
$792$		0			0
$793$	17.7538			0.630456
$794$		0			0
$795$	5.32326	+	18.9591i	0.188797	+	0.672410i
$796$		0			0
$797$			5.66906i			0.200808i	0.994947	+	0.100404i	$0.0320136\pi$
$797$			5.66906i			0.200808i	−0.994947	+	0.100404i	$0.967986\pi$
$798$		0			0
$799$			40.2944i			1.42551i
$800$		0			0
$801$	−7.61553	−	12.4924i	−0.269081	−	0.441398i
$802$		0			0
$803$	34.1725			1.20592
$804$		0			0
$805$	48.3272			1.70331
$806$		0			0
$807$	−6.27691	+	1.76241i	−0.220958	+	0.0620397i
$808$		0			0
$809$			44.9848i			1.58158i	0.612086	+	0.790791i	$0.290331\pi$
$809$			44.9848i			1.58158i	−0.612086	+	0.790791i	$0.709669\pi$
$810$		0			0
$811$		−	13.9817i		−	0.490962i	−0.969401	−	0.245481i	$-0.921054\pi$
$811$		−	13.9817i		−	0.490962i	0.969401	−	0.245481i	$-0.0789460\pi$
$812$		0			0
$813$	−41.7056	+	11.7100i	−1.46268	+	0.410686i
$814$		0			0
$815$	−15.7304			−0.551014
$816$		0			0
$817$	7.61553			0.266434
$818$		0			0
$819$	7.49141	+	12.2888i	0.261771	+	0.429406i
$820$		0			0
$821$		−	7.15640i		−	0.249760i	−0.992172	−	0.124880i	$-0.960145\pi$
$821$		−	7.15640i		−	0.249760i	0.992172	−	0.124880i	$-0.0398546\pi$
$822$		0			0
$823$			30.1230i			1.05002i	0.851095	+	0.525011i	$0.175939\pi$
$823$			30.1230i			1.05002i	−0.851095	+	0.525011i	$0.824061\pi$
$824$		0			0
$825$	6.43845	+	22.9309i	0.224158	+	0.798350i
$826$		0			0
$827$	−12.1735			−0.423316			−0.211658	−	0.977344i	$-0.567886\pi$
$827$	−12.1735			−0.423316			−0.211658	+	0.977344i	$0.567886\pi$
$828$		0			0
$829$	−25.4879			−0.885231			−0.442616	−	0.896711i	$-0.645949\pi$
$829$	−25.4879			−0.885231			−0.442616	+	0.896711i	$0.645949\pi$
$830$		0			0
$831$	8.75714	+	31.1890i	0.303782	+	1.08194i
$832$		0			0
$833$		−	43.6155i		−	1.51119i
$834$		0			0
$835$			20.8319i			0.720919i
$836$		0			0
$837$	2.81164	−	3.02045i	0.0971846	−	0.104402i
$838$		0			0
$839$	−12.5538			−0.433406			−0.216703	−	0.976238i	$-0.569530\pi$
$839$	−12.5538			−0.433406			−0.216703	+	0.976238i	$0.569530\pi$
$840$		0			0
$841$	−12.1231			−0.418038
$842$		0			0
$843$	−15.6240	+	4.38684i	−0.538118	+	0.151091i
$844$		0			0
$845$			33.9686i			1.16856i
$846$		0			0
$847$			0.445960i			0.0153234i
$848$		0			0
$849$	−29.1771	+	8.19224i	−1.00135	+	0.281157i
$850$		0			0
$851$	−35.8148			−1.22771
$852$		0			0
$853$	33.4337			1.14475			0.572375	−	0.819992i	$-0.306022\pi$
$853$	33.4337			1.14475			0.572375	+	0.819992i	$0.306022\pi$
$854$		0			0
$855$	−40.2944	+	24.5639i	−1.37804	+	0.840068i
$856$		0			0
$857$		−	52.0000i		−	1.77629i	−0.459567	−	0.888143i	$-0.651995\pi$
$857$		−	52.0000i		−	1.77629i	0.459567	−	0.888143i	$-0.348005\pi$
$858$		0			0
$859$		−	41.4839i		−	1.41541i	−0.706507	−	0.707706i	$-0.749730\pi$
$859$		−	41.4839i		−	1.41541i	0.706507	−	0.707706i	$-0.250270\pi$
$860$		0			0
$861$	6.78456	+	24.1636i	0.231217	+	0.823493i
$862$		0			0
$863$	−30.7645			−1.04724			−0.523618	−	0.851953i	$-0.675418\pi$
$863$	−30.7645			−1.04724			−0.523618	+	0.851953i	$0.675418\pi$
$864$		0			0
$865$	−9.12311			−0.310195
$866$		0			0
$867$	15.7969	+	56.2614i	0.536490	+	1.91074i
$868$		0			0
$869$		−	16.2177i		−	0.550149i
$870$		0			0
$871$		−	9.37720i		−	0.317734i
$872$		0			0
$873$	−2.87689	+	1.75379i	−0.0973681	+	0.0593568i
$874$		0			0
$875$	9.59482			0.324364
$876$		0			0
$877$	25.4879			0.860665			0.430332	−	0.902670i	$-0.358396\pi$
$877$	25.4879			0.860665			0.430332	+	0.902670i	$0.358396\pi$
$878$		0			0
$879$	0.620058	−	0.174098i	0.0209140	−	0.00587217i
$880$		0			0
$881$			52.4924i			1.76851i	0.467000	+	0.884257i	$0.345335\pi$
$881$			52.4924i			1.76851i	−0.467000	+	0.884257i	$0.654665\pi$
$882$		0			0
$883$		−	46.0507i		−	1.54973i	−0.632127	−	0.774865i	$-0.717818\pi$
$883$		−	46.0507i		−	1.54973i	0.632127	−	0.774865i	$-0.282182\pi$
$884$		0			0
$885$	−28.8802	+	8.10887i	−0.970796	+	0.272577i
$886$		0			0
$887$	−39.0543			−1.31131			−0.655657	−	0.755059i	$-0.727608\pi$
$887$	−39.0543			−1.31131			−0.655657	+	0.755059i	$0.727608\pi$
$888$		0			0
$889$	−70.1080			−2.35135
$890$		0			0
$891$	13.7511	−	26.6811i	0.460679	−	0.893849i
$892$		0			0
$893$			29.4608i			0.985868i
$894$		0			0
$895$			6.00505i			0.200727i
$896$		0			0
$897$	2.73863	+	9.75379i	0.0914403	+	0.325670i
$898$		0			0
$899$	5.09271			0.169851
$900$		0			0
$901$	26.8122			0.893244
$902$		0			0
$903$	−2.48023	−	8.83348i	−0.0825369	−	0.293960i
$904$		0			0
$905$		−	60.9848i		−	2.02720i
$906$		0			0
$907$			23.1154i			0.767533i	0.923430	+	0.383767i	$0.125373\pi$
$907$			23.1154i			0.767533i	−0.923430	+	0.383767i	$0.874627\pi$
$908$		0			0
$909$	27.7189	+	45.4697i	0.919378	+	1.50814i
$910$		0			0
$911$	12.0101			0.397912			0.198956	−	0.980008i	$-0.436245\pi$
$911$	12.0101			0.397912			0.198956	+	0.980008i	$0.436245\pi$
$912$		0			0
$913$	−1.36932			−0.0453178
$914$		0			0
$915$	67.5238	−	18.9591i	2.23227	−	0.626768i
$916$		0			0
$917$			34.3404i			1.13402i
$918$		0			0
$919$			3.62258i			0.119498i	0.998213	+	0.0597490i	$0.0190300\pi$
$919$			3.62258i			0.119498i	−0.998213	+	0.0597490i	$0.980970\pi$
$920$		0			0
$921$	−38.9309	+	10.9309i	−1.28282	+	0.360184i
$922$		0			0
$923$	13.3405			0.439109
$924$		0			0
$925$	33.4337			1.09929
$926$		0			0
$927$	−3.17662	−	5.21089i	−0.104334	−	0.171148i
$928$		0			0
$929$		−	39.1231i		−	1.28359i	−0.766877	−	0.641794i	$-0.778191\pi$
$929$		−	39.1231i		−	1.28359i	0.766877	−	0.641794i	$-0.221809\pi$
$930$		0			0
$931$		−	31.8890i		−	1.04512i
$932$		0			0
$933$	−8.85254	−	31.5288i	−0.289819	−	1.03221i
$934$		0			0
$935$	71.7553			2.34665
$936$		0			0
$937$	1.61553			0.0527770			0.0263885	−	0.999652i	$-0.491599\pi$
$937$	1.61553			0.0527770			0.0263885	+	0.999652i	$0.491599\pi$
$938$		0			0
$939$	−9.18425	−	32.7102i	−0.299717	−	1.06746i
$940$		0			0
$941$			1.11550i			0.0363643i	0.999835	+	0.0181822i	$0.00578788\pi$
$941$			1.11550i			0.0363643i	−0.999835	+	0.0181822i	$0.994212\pi$
$942$		0			0
$943$			17.6670i			0.575315i
$944$		0			0
$945$	41.6155	+	38.7386i	1.35375	+	1.26017i
$946$		0			0
$947$	27.6175			0.897448			0.448724	−	0.893670i	$-0.351879\pi$
$947$	27.6175			0.897448			0.448724	+	0.893670i	$0.351879\pi$
$948$		0			0
$949$	−13.5691			−0.440473
$950$		0			0
$951$	−46.5713	+	13.0761i	−1.51018	+	0.424022i
$952$		0			0
$953$		−	8.49242i		−	0.275097i	−0.990495	−	0.137548i	$-0.956078\pi$
$953$		−	8.49242i		−	0.275097i	0.990495	−	0.137548i	$-0.0439222\pi$
$954$		0			0
$955$			9.59482i			0.310481i
$956$		0			0
$957$	35.6647	−	10.0138i	1.15288	−	0.323701i
$958$		0			0
$959$	−14.4903			−0.467917
$960$		0			0
$961$	30.3693			0.979655
$962$		0			0
$963$	−41.3686	+	25.2188i	−1.33308	+	0.812664i
$964$		0			0
$965$		−	12.0818i		−	0.388927i
$966$		0			0
$967$		−	11.5641i		−	0.371878i	−0.982561	−	0.185939i	$-0.940467\pi$
$967$		−	11.5641i		−	0.371878i	0.982561	−	0.185939i	$-0.0595326\pi$
$968$		0			0
$969$	17.3693	+	61.8617i	0.557983	+	1.98729i
$970$		0			0
$971$	53.7727			1.72565			0.862824	−	0.505505i	$-0.168694\pi$
$971$	53.7727			1.72565			0.862824	+	0.505505i	$0.168694\pi$
$972$		0			0
$973$	−36.2454			−1.16197
$974$		0			0
$975$	−2.55656	−	9.10534i	−0.0818756	−	0.291604i
$976$		0			0
$977$			11.6155i			0.371614i	0.982586	+	0.185807i	$0.0594899\pi$
$977$			11.6155i			0.371614i	−0.982586	+	0.185807i	$0.940510\pi$
$978$		0			0
$979$			16.2651i			0.519834i
$980$		0			0
$981$	−20.7713	+	12.6624i	−0.663177	+	0.404280i
$982$		0			0
$983$	30.2208			0.963893			0.481947	−	0.876201i	$-0.339930\pi$
$983$	30.2208			0.963893			0.481947	+	0.876201i	$0.339930\pi$
$984$		0			0
$985$	58.1080			1.85147
$986$		0			0
$987$	34.1725	−	9.59482i	1.08772	−	0.305407i
$988$		0			0
$989$		−	6.45850i		−	0.205369i
$990$		0			0
$991$			44.2651i			1.40613i	0.711126	+	0.703064i	$0.248185\pi$
$991$			44.2651i			1.40613i	−0.711126	+	0.703064i	$0.751815\pi$
$992$		0			0
$993$	50.0540	−	14.0540i	1.58841	−	0.445989i
$994$		0			0
$995$	6.14441			0.194791
$996$		0			0
$997$	−24.0006			−0.760105			−0.380053	−	0.924965i	$-0.624094\pi$
$997$	−24.0006			−0.760105			−0.380053	+	0.924965i	$0.624094\pi$
$998$		0			0
$999$	−30.8408	−	28.7088i	−0.975761	−	0.908306i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1536.2.c.m.1535.5	✓	16
3.2	odd	2	inner	1536.2.c.m.1535.10	yes	16
4.3	odd	2	inner	1536.2.c.m.1535.11	yes	16
8.3	odd	2	inner	1536.2.c.m.1535.6	yes	16
8.5	even	2	inner	1536.2.c.m.1535.12	yes	16
12.11	even	2	inner	1536.2.c.m.1535.8	yes	16
16.3	odd	4		1536.2.f.l.767.3		16
16.5	even	4		1536.2.f.l.767.4		16
16.11	odd	4		1536.2.f.l.767.14		16
16.13	even	4		1536.2.f.l.767.13		16
24.5	odd	2	inner	1536.2.c.m.1535.7	yes	16
24.11	even	2	inner	1536.2.c.m.1535.9	yes	16
48.5	odd	4		1536.2.f.l.767.1		16
48.11	even	4		1536.2.f.l.767.15		16
48.29	odd	4		1536.2.f.l.767.16		16
48.35	even	4		1536.2.f.l.767.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.c.m.1535.5	✓	16	1.1	even	1	trivial
1536.2.c.m.1535.6	yes	16	8.3	odd	2	inner
1536.2.c.m.1535.7	yes	16	24.5	odd	2	inner
1536.2.c.m.1535.8	yes	16	12.11	even	2	inner
1536.2.c.m.1535.9	yes	16	24.11	even	2	inner
1536.2.c.m.1535.10	yes	16	3.2	odd	2	inner
1536.2.c.m.1535.11	yes	16	4.3	odd	2	inner
1536.2.c.m.1535.12	yes	16	8.5	even	2	inner
1536.2.f.l.767.1		16	48.5	odd	4
1536.2.f.l.767.2		16	48.35	even	4
1536.2.f.l.767.3		16	16.3	odd	4
1536.2.f.l.767.4		16	16.5	even	4
1536.2.f.l.767.13		16	16.13	even	4
1536.2.f.l.767.14		16	16.11	odd	4
1536.2.f.l.767.15		16	48.11	even	4
1536.2.f.l.767.16		16	48.29	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)

\(f(q)\)	\(=\)	\(q+(-0.468213 - 1.66757i) q^{3} -3.02045i q^{5} +3.62258i q^{7} +(-2.56155 + 1.56155i) q^{9} +O(q^{10})\)	\(q+(-0.468213 - 1.66757i) q^{3} -3.02045i q^{5} +3.62258i q^{7} +(-2.56155 + 1.56155i) q^{9} +3.33513 q^{11} -1.32431 q^{13} +(-5.03680 + 1.41421i) q^{15} +7.12311i q^{17} +5.20798i q^{19} +(6.04090 - 1.69614i) q^{21} +4.41674 q^{23} -4.12311 q^{25} +(3.80335 + 3.54042i) q^{27} +6.41273i q^{29} -0.794156i q^{31} +(-1.56155 - 5.56155i) q^{33} +10.9418 q^{35} -8.10887 q^{37} +(0.620058 + 2.20837i) q^{39} +4.00000i q^{41} -1.46228i q^{43} +(4.71659 + 7.73704i) q^{45} +5.65685 q^{47} -6.12311 q^{49} +(11.8782 - 3.33513i) q^{51} -3.76412i q^{53} -10.0736i q^{55} +(8.68466 - 2.43845i) q^{57} +5.73384 q^{59} -13.4061 q^{61} +(-5.65685 - 9.27944i) q^{63} +4.00000i q^{65} +7.08084i q^{67} +(-2.06798 - 7.36520i) q^{69} -10.0736 q^{71} +10.2462 q^{73} +(1.93049 + 6.87555i) q^{75} +12.0818i q^{77} -4.86270i q^{79} +(4.12311 - 8.00000i) q^{81} -0.410574 q^{83} +21.5150 q^{85} +(10.6937 - 3.00252i) q^{87} +4.87689i q^{89} -4.79741i q^{91} +(-1.32431 + 0.371834i) q^{93} +15.7304 q^{95} +1.12311 q^{97} +(-8.54312 + 5.20798i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 8 q^{9}+O(q^{10}) \) 16 * q - 8 * q^9	\( 16 q - 8 q^{9} + 8 q^{33} - 32 q^{49} + 40 q^{57} + 32 q^{73} - 48 q^{97}+O(q^{100}) \) 16 * q - 8 * q^9 + 8 * q^33 - 32 * q^49 + 40 * q^57 + 32 * q^73 - 48 * q^97

Introduction

L-functions

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