LMFDB - Embedded newform 1536.2.f.k.767.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1536,2,Mod(767,1536)]

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, 1, 1]))

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

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

chi := DirichletCharacter("1536.767");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1536 = 2^{9} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1536.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$12.2650217505$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.157351936.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + x^{4} + 16 $ x^8 + x^4 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{10} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			767.3
Root			$-0.581861 + 1.28897i$ of defining polynomial
Character	$\chi$	$=$	1536.767
Dual form			1536.2.f.k.767.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.41421 + 1.00000i) q^{3} -3.74166 q^{5} -3.74166i q^{7} +(1.00000 - 2.82843i) q^{9} +O(q^{10})$	$q+(-1.41421 + 1.00000i) q^{3} -3.74166 q^{5} -3.74166i q^{7} +(1.00000 - 2.82843i) q^{9} -5.29150i q^{13} +(5.29150 - 3.74166i) q^{15} +2.82843i q^{17} -5.65685 q^{19} +(3.74166 + 5.29150i) q^{21} +9.00000 q^{25} +(1.41421 + 5.00000i) q^{27} +3.74166 q^{29} +3.74166i q^{31} +14.0000i q^{35} -5.29150i q^{37} +(5.29150 + 7.48331i) q^{39} -8.48528i q^{41} +(-3.74166 + 10.5830i) q^{45} -10.5830 q^{47} -7.00000 q^{49} +(-2.82843 - 4.00000i) q^{51} +3.74166 q^{53} +(8.00000 - 5.65685i) q^{57} +6.00000i q^{59} +5.29150i q^{61} +(-10.5830 - 3.74166i) q^{63} +19.7990i q^{65} -2.82843 q^{67} -10.5830 q^{71} -8.00000 q^{73} +(-12.7279 + 9.00000i) q^{75} +11.2250i q^{79} +(-7.00000 - 5.65685i) q^{81} +8.00000i q^{83} -10.5830i q^{85} +(-5.29150 + 3.74166i) q^{87} +11.3137i q^{89} -19.7990 q^{91} +(-3.74166 - 5.29150i) q^{93} +21.1660 q^{95} -6.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9}+O(q^{10}) $ 8 * q + 8 * q^9	$ 8 q + 8 q^{9} + 72 q^{25} - 56 q^{49} + 64 q^{57} - 64 q^{73} - 56 q^{81} - 48 q^{97}+O(q^{100}) $ 8 * q + 8 * q^9 + 72 * q^25 - 56 * q^49 + 64 * q^57 - 64 * q^73 - 56 * q^81 - 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
$3$	−1.41421	+	1.00000i	−0.816497	+	0.577350i
$3$	−1.41421	+	1.00000i	−0.816497	+	0.577350i
$4$		0			0
$5$	−3.74166			−1.67332			−0.836660	−	0.547723i	$-0.815495\pi$
$5$	−3.74166			−1.67332			−0.836660	+	0.547723i	$0.815495\pi$
$6$		0			0
$7$		−	3.74166i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$7$		−	3.74166i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$8$		0			0
$9$	1.00000	−	2.82843i	0.333333	−	0.942809i
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		0			0
$13$		−	5.29150i		−	1.46760i	−0.679366	−	0.733799i	$-0.737745\pi$
$13$		−	5.29150i		−	1.46760i	0.679366	−	0.733799i	$-0.262255\pi$
$14$		0			0
$15$	5.29150	−	3.74166i	1.36626	−	0.966092i
$16$		0			0
$17$			2.82843i			0.685994i	0.939336	+	0.342997i	$0.111442\pi$
$17$			2.82843i			0.685994i	−0.939336	+	0.342997i	$0.888558\pi$
$18$		0			0
$19$	−5.65685			−1.29777			−0.648886	−	0.760886i	$-0.724765\pi$
$19$	−5.65685			−1.29777			−0.648886	+	0.760886i	$0.724765\pi$
$20$		0			0
$21$	3.74166	+	5.29150i	0.816497	+	1.15470i
$22$		0			0
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$		0			0
$25$	9.00000			1.80000
$26$		0			0
$27$	1.41421	+	5.00000i	0.272166	+	0.962250i
$28$		0			0
$29$	3.74166			0.694808			0.347404	−	0.937715i	$-0.387063\pi$
$29$	3.74166			0.694808			0.347404	+	0.937715i	$0.387063\pi$
$30$		0			0
$31$			3.74166i			0.672022i	0.941858	+	0.336011i	$0.109078\pi$
$31$			3.74166i			0.672022i	−0.941858	+	0.336011i	$0.890922\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			14.0000i			2.36643i
$36$		0			0
$37$		−	5.29150i		−	0.869918i	−0.900450	−	0.434959i	$-0.856763\pi$
$37$		−	5.29150i		−	0.869918i	0.900450	−	0.434959i	$-0.143237\pi$
$38$		0			0
$39$	5.29150	+	7.48331i	0.847319	+	1.19829i
$40$		0			0
$41$		−	8.48528i		−	1.32518i	−0.748983	−	0.662589i	$-0.769458\pi$
$41$		−	8.48528i		−	1.32518i	0.748983	−	0.662589i	$-0.230542\pi$
$42$		0			0
$43$		0			0			−	1.00000i	$-0.5\pi$
$43$		0			0				1.00000i	$0.5\pi$
$44$		0			0
$45$	−3.74166	+	10.5830i	−0.557773	+	1.57762i
$46$		0			0
$47$	−10.5830			−1.54369			−0.771845	−	0.635811i	$-0.780666\pi$
$47$	−10.5830			−1.54369			−0.771845	+	0.635811i	$0.780666\pi$
$48$		0			0
$49$	−7.00000			−1.00000
$50$		0			0
$51$	−2.82843	−	4.00000i	−0.396059	−	0.560112i
$52$		0			0
$53$	3.74166			0.513956			0.256978	−	0.966417i	$-0.417273\pi$
$53$	3.74166			0.513956			0.256978	+	0.966417i	$0.417273\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	8.00000	−	5.65685i	1.05963	−	0.749269i
$58$		0			0
$59$			6.00000i			0.781133i	0.920575	+	0.390567i	$0.127721\pi$
$59$			6.00000i			0.781133i	−0.920575	+	0.390567i	$0.872279\pi$
$60$		0			0
$61$			5.29150i			0.677507i	0.940875	+	0.338754i	$0.110005\pi$
$61$			5.29150i			0.677507i	−0.940875	+	0.338754i	$0.889995\pi$
$62$		0			0
$63$	−10.5830	−	3.74166i	−1.33333	−	0.471405i
$64$		0			0
$65$			19.7990i			2.45576i
$66$		0			0
$67$	−2.82843			−0.345547			−0.172774	−	0.984962i	$-0.555273\pi$
$67$	−2.82843			−0.345547			−0.172774	+	0.984962i	$0.555273\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−10.5830			−1.25597			−0.627986	−	0.778225i	$-0.716120\pi$
$71$	−10.5830			−1.25597			−0.627986	+	0.778225i	$0.716120\pi$
$72$		0			0
$73$	−8.00000			−0.936329			−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000			−0.936329			−0.468165	+	0.883641i	$0.655085\pi$
$74$		0			0
$75$	−12.7279	+	9.00000i	−1.46969	+	1.03923i
$76$		0			0
$77$		0			0
$78$		0			0
$79$			11.2250i			1.26291i	0.775413	+	0.631454i	$0.217542\pi$
$79$			11.2250i			1.26291i	−0.775413	+	0.631454i	$0.782458\pi$
$80$		0			0
$81$	−7.00000	−	5.65685i	−0.777778	−	0.628539i
$82$		0			0
$83$			8.00000i			0.878114i	0.898459	+	0.439057i	$0.144687\pi$
$83$			8.00000i			0.878114i	−0.898459	+	0.439057i	$0.855313\pi$
$84$		0			0
$85$		−	10.5830i		−	1.14789i
$86$		0			0
$87$	−5.29150	+	3.74166i	−0.567309	+	0.401148i
$88$		0			0
$89$			11.3137i			1.19925i	0.800281	+	0.599625i	$0.204684\pi$
$89$			11.3137i			1.19925i	−0.800281	+	0.599625i	$0.795316\pi$
$90$		0			0
$91$	−19.7990			−2.07550
$92$		0			0
$93$	−3.74166	−	5.29150i	−0.387992	−	0.548703i
$94$		0			0
$95$	21.1660			2.17159
$96$		0			0
$97$	−6.00000			−0.609208			−0.304604	−	0.952479i	$-0.598524\pi$
$97$	−6.00000			−0.609208			−0.304604	+	0.952479i	$0.598524\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	11.2250			1.11693			0.558463	−	0.829529i	$-0.311391\pi$
$101$	11.2250			1.11693			0.558463	+	0.829529i	$0.311391\pi$
$102$		0			0
$103$			3.74166i			0.368676i	0.982863	+	0.184338i	$0.0590142\pi$
$103$			3.74166i			0.368676i	−0.982863	+	0.184338i	$0.940986\pi$
$104$		0			0
$105$	−14.0000	−	19.7990i	−1.36626	−	1.93218i
$106$		0			0
$107$		−	2.00000i		−	0.193347i	−0.995316	−	0.0966736i	$-0.969180\pi$
$107$		−	2.00000i		−	0.193347i	0.995316	−	0.0966736i	$-0.0308203\pi$
$108$		0			0
$109$		−	5.29150i		−	0.506834i	−0.967357	−	0.253417i	$-0.918446\pi$
$109$		−	5.29150i		−	0.506834i	0.967357	−	0.253417i	$-0.0815545\pi$
$110$		0			0
$111$	5.29150	+	7.48331i	0.502247	+	0.710285i
$112$		0			0
$113$			11.3137i			1.06430i	0.846649	+	0.532152i	$0.178617\pi$
$113$			11.3137i			1.06430i	−0.846649	+	0.532152i	$0.821383\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−14.9666	−	5.29150i	−1.38367	−	0.489200i
$118$		0			0
$119$	10.5830			0.970143
$120$		0			0
$121$	11.0000			1.00000
$122$		0			0
$123$	8.48528	+	12.0000i	0.765092	+	1.08200i
$124$		0			0
$125$	−14.9666			−1.33866
$126$		0			0
$127$		−	11.2250i		−	0.996055i	−0.867161	−	0.498028i	$-0.834058\pi$
$127$		−	11.2250i		−	0.996055i	0.867161	−	0.498028i	$-0.165942\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			2.00000i			0.174741i	0.996176	+	0.0873704i	$0.0278464\pi$
$131$			2.00000i			0.174741i	−0.996176	+	0.0873704i	$0.972154\pi$
$132$		0			0
$133$			21.1660i			1.83533i
$134$		0			0
$135$	−5.29150	−	18.7083i	−0.455420	−	1.61015i
$136$		0			0
$137$			2.82843i			0.241649i	0.992674	+	0.120824i	$0.0385538\pi$
$137$			2.82843i			0.241649i	−0.992674	+	0.120824i	$0.961446\pi$
$138$		0			0
$139$	2.82843			0.239904			0.119952	−	0.992780i	$-0.461726\pi$
$139$	2.82843			0.239904			0.119952	+	0.992780i	$0.461726\pi$
$140$		0			0
$141$	14.9666	−	10.5830i	1.26042	−	0.891250i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−14.0000			−1.16264
$146$		0			0
$147$	9.89949	−	7.00000i	0.816497	−	0.577350i
$148$		0			0
$149$	−18.7083			−1.53264			−0.766321	−	0.642458i	$-0.777915\pi$
$149$	−18.7083			−1.53264			−0.766321	+	0.642458i	$0.777915\pi$
$150$		0			0
$151$		−	11.2250i		−	0.913475i	−0.889601	−	0.456738i	$-0.849018\pi$
$151$		−	11.2250i		−	0.913475i	0.889601	−	0.456738i	$-0.150982\pi$
$152$		0			0
$153$	8.00000	+	2.82843i	0.646762	+	0.228665i
$154$		0			0
$155$		−	14.0000i		−	1.12451i
$156$		0			0
$157$		−	5.29150i		−	0.422308i	−0.977453	−	0.211154i	$-0.932278\pi$
$157$		−	5.29150i		−	0.422308i	0.977453	−	0.211154i	$-0.0677221\pi$
$158$		0			0
$159$	−5.29150	+	3.74166i	−0.419643	+	0.296733i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	16.9706			1.32924			0.664619	−	0.747183i	$-0.268594\pi$
$163$	16.9706			1.32924			0.664619	+	0.747183i	$0.268594\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	21.1660			1.63788			0.818938	−	0.573883i	$-0.194563\pi$
$167$	21.1660			1.63788			0.818938	+	0.573883i	$0.194563\pi$
$168$		0			0
$169$	−15.0000			−1.15385
$170$		0			0
$171$	−5.65685	+	16.0000i	−0.432590	+	1.22355i
$172$		0			0
$173$	3.74166			0.284473			0.142236	−	0.989833i	$-0.454571\pi$
$173$	3.74166			0.284473			0.142236	+	0.989833i	$0.454571\pi$
$174$		0			0
$175$		−	33.6749i		−	2.54558i
$176$		0			0
$177$	−6.00000	−	8.48528i	−0.450988	−	0.637793i
$178$		0			0
$179$			22.0000i			1.64436i	0.569230	+	0.822179i	$0.307242\pi$
$179$			22.0000i			1.64436i	−0.569230	+	0.822179i	$0.692758\pi$
$180$		0			0
$181$			15.8745i			1.17994i	0.807424	+	0.589971i	$0.200861\pi$
$181$			15.8745i			1.17994i	−0.807424	+	0.589971i	$0.799139\pi$
$182$		0			0
$183$	−5.29150	−	7.48331i	−0.391159	−	0.553183i
$184$		0			0
$185$			19.7990i			1.45565i
$186$		0			0
$187$		0			0
$188$		0			0
$189$	18.7083	−	5.29150i	1.36083	−	0.384900i
$190$		0			0
$191$	−10.5830			−0.765759			−0.382880	−	0.923798i	$-0.625068\pi$
$191$	−10.5830			−0.765759			−0.382880	+	0.923798i	$0.625068\pi$
$192$		0			0
$193$	20.0000			1.43963			0.719816	−	0.694165i	$-0.244226\pi$
$193$	20.0000			1.43963			0.719816	+	0.694165i	$0.244226\pi$
$194$		0			0
$195$	−19.7990	−	28.0000i	−1.41784	−	2.00512i
$196$		0			0
$197$	3.74166			0.266582			0.133291	−	0.991077i	$-0.457446\pi$
$197$	3.74166			0.266582			0.133291	+	0.991077i	$0.457446\pi$
$198$		0			0
$199$			18.7083i			1.32620i	0.748533	+	0.663098i	$0.230759\pi$
$199$			18.7083i			1.32620i	−0.748533	+	0.663098i	$0.769241\pi$
$200$		0			0
$201$	4.00000	−	2.82843i	0.282138	−	0.199502i
$202$		0			0
$203$		−	14.0000i		−	0.982607i
$204$		0			0
$205$			31.7490i			2.21745i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	−8.48528			−0.584151			−0.292075	−	0.956395i	$-0.594346\pi$
$211$	−8.48528			−0.584151			−0.292075	+	0.956395i	$0.594346\pi$
$212$		0			0
$213$	14.9666	−	10.5830i	1.02550	−	0.725136i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	14.0000			0.950382
$218$		0			0
$219$	11.3137	−	8.00000i	0.764510	−	0.540590i
$220$		0			0
$221$	14.9666			1.00676
$222$		0			0
$223$			11.2250i			0.751680i	0.926685	+	0.375840i	$0.122646\pi$
$223$			11.2250i			0.751680i	−0.926685	+	0.375840i	$0.877354\pi$
$224$		0			0
$225$	9.00000	−	25.4558i	0.600000	−	1.69706i
$226$		0			0
$227$		−	8.00000i		−	0.530979i	−0.964114	−	0.265489i	$-0.914466\pi$
$227$		−	8.00000i		−	0.530979i	0.964114	−	0.265489i	$-0.0855335\pi$
$228$		0			0
$229$		−	5.29150i		−	0.349672i	−0.984598	−	0.174836i	$-0.944060\pi$
$229$		−	5.29150i		−	0.349672i	0.984598	−	0.174836i	$-0.0559396\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	5.65685i		−	0.370593i	−0.982683	−	0.185296i	$-0.940675\pi$
$233$		−	5.65685i		−	0.370593i	0.982683	−	0.185296i	$-0.0593245\pi$
$234$		0			0
$235$	39.5980			2.58309
$236$		0			0
$237$	−11.2250	−	15.8745i	−0.729140	−	1.03116i
$238$		0			0
$239$	10.5830			0.684558			0.342279	−	0.939598i	$-0.388801\pi$
$239$	10.5830			0.684558			0.342279	+	0.939598i	$0.388801\pi$
$240$		0			0
$241$	20.0000			1.28831			0.644157	−	0.764894i	$-0.277208\pi$
$241$	20.0000			1.28831			0.644157	+	0.764894i	$0.277208\pi$
$242$		0			0
$243$	15.5563	+	1.00000i	0.997940	+	0.0641500i
$244$		0			0
$245$	26.1916			1.67332
$246$		0			0
$247$			29.9333i			1.90461i
$248$		0			0
$249$	−8.00000	−	11.3137i	−0.506979	−	0.716977i
$250$		0			0
$251$		−	24.0000i		−	1.51487i	−0.652913	−	0.757433i	$-0.726453\pi$
$251$		−	24.0000i		−	1.51487i	0.652913	−	0.757433i	$-0.273547\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$	10.5830	+	14.9666i	0.662733	+	0.937247i
$256$		0			0
$257$			11.3137i			0.705730i	0.935674	+	0.352865i	$0.114792\pi$
$257$			11.3137i			0.705730i	−0.935674	+	0.352865i	$0.885208\pi$
$258$		0			0
$259$	−19.7990			−1.23025
$260$		0			0
$261$	3.74166	−	10.5830i	0.231603	−	0.655072i
$262$		0			0
$263$	−10.5830			−0.652576			−0.326288	−	0.945270i	$-0.605798\pi$
$263$	−10.5830			−0.652576			−0.326288	+	0.945270i	$0.605798\pi$
$264$		0			0
$265$	−14.0000			−0.860013
$266$		0			0
$267$	−11.3137	−	16.0000i	−0.692388	−	0.979184i
$268$		0			0
$269$	−18.7083			−1.14066			−0.570332	−	0.821414i	$-0.693186\pi$
$269$	−18.7083			−1.14066			−0.570332	+	0.821414i	$0.693186\pi$
$270$		0			0
$271$			11.2250i			0.681868i	0.940087	+	0.340934i	$0.110743\pi$
$271$			11.2250i			0.681868i	−0.940087	+	0.340934i	$0.889257\pi$
$272$		0			0
$273$	28.0000	−	19.7990i	1.69464	−	1.19829i
$274$		0			0
$275$		0			0
$276$		0			0
$277$			15.8745i			0.953807i	0.878956	+	0.476903i	$0.158241\pi$
$277$			15.8745i			0.953807i	−0.878956	+	0.476903i	$0.841759\pi$
$278$		0			0
$279$	10.5830	+	3.74166i	0.633588	+	0.224007i
$280$		0			0
$281$			28.2843i			1.68730i	0.536895	+	0.843649i	$0.319597\pi$
$281$			28.2843i			1.68730i	−0.536895	+	0.843649i	$0.680403\pi$
$282$		0			0
$283$	−14.1421			−0.840663			−0.420331	−	0.907371i	$-0.638086\pi$
$283$	−14.1421			−0.840663			−0.420331	+	0.907371i	$0.638086\pi$
$284$		0			0
$285$	−29.9333	+	21.1660i	−1.77309	+	1.25377i
$286$		0			0
$287$	−31.7490			−1.87409
$288$		0			0
$289$	9.00000			0.529412
$290$		0			0
$291$	8.48528	−	6.00000i	0.497416	−	0.351726i
$292$		0			0
$293$	−26.1916			−1.53013			−0.765065	−	0.643953i	$-0.777293\pi$
$293$	−26.1916			−1.53013			−0.765065	+	0.643953i	$0.777293\pi$
$294$		0			0
$295$		−	22.4499i		−	1.30709i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−15.8745	+	11.2250i	−0.911967	+	0.644858i
$304$		0			0
$305$		−	19.7990i		−	1.13369i
$306$		0			0
$307$	2.82843			0.161427			0.0807134	−	0.996737i	$-0.474280\pi$
$307$	2.82843			0.161427			0.0807134	+	0.996737i	$0.474280\pi$
$308$		0			0
$309$	−3.74166	−	5.29150i	−0.212855	−	0.301023i
$310$		0			0
$311$	−31.7490			−1.80032			−0.900161	−	0.435558i	$-0.856551\pi$
$311$	−31.7490			−1.80032			−0.900161	+	0.435558i	$0.856551\pi$
$312$		0			0
$313$	−4.00000			−0.226093			−0.113047	−	0.993590i	$-0.536061\pi$
$313$	−4.00000			−0.226093			−0.113047	+	0.993590i	$0.536061\pi$
$314$		0			0
$315$	39.5980	+	14.0000i	2.23109	+	0.788811i
$316$		0			0
$317$	−3.74166			−0.210152			−0.105076	−	0.994464i	$-0.533509\pi$
$317$	−3.74166			−0.210152			−0.105076	+	0.994464i	$0.533509\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	2.00000	+	2.82843i	0.111629	+	0.157867i
$322$		0			0
$323$		−	16.0000i		−	0.890264i
$324$		0			0
$325$		−	47.6235i		−	2.64168i
$326$		0			0
$327$	5.29150	+	7.48331i	0.292621	+	0.413828i
$328$		0			0
$329$			39.5980i			2.18311i
$330$		0			0
$331$	19.7990			1.08825			0.544125	−	0.839004i	$-0.316862\pi$
$331$	19.7990			1.08825			0.544125	+	0.839004i	$0.316862\pi$
$332$		0			0
$333$	−14.9666	−	5.29150i	−0.820166	−	0.289973i
$334$		0			0
$335$	10.5830			0.578211
$336$		0			0
$337$		0			0			−	1.00000i	$-0.5\pi$
$337$		0			0				1.00000i	$0.5\pi$
$338$		0			0
$339$	−11.3137	−	16.0000i	−0.614476	−	0.869001i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$		−	32.0000i		−	1.71785i	−0.512101	−	0.858925i	$-0.671133\pi$
$347$		−	32.0000i		−	1.71785i	0.512101	−	0.858925i	$-0.328867\pi$
$348$		0			0
$349$			5.29150i			0.283248i	0.989921	+	0.141624i	$0.0452323\pi$
$349$			5.29150i			0.283248i	−0.989921	+	0.141624i	$0.954768\pi$
$350$		0			0
$351$	26.4575	−	7.48331i	1.41220	−	0.399430i
$352$		0			0
$353$		−	16.9706i		−	0.903252i	−0.892207	−	0.451626i	$-0.850844\pi$
$353$		−	16.9706i		−	0.903252i	0.892207	−	0.451626i	$-0.149156\pi$
$354$		0			0
$355$	39.5980			2.10164
$356$		0			0
$357$	−14.9666	+	10.5830i	−0.792118	+	0.560112i
$358$		0			0
$359$	−31.7490			−1.67565			−0.837824	−	0.545940i	$-0.816173\pi$
$359$	−31.7490			−1.67565			−0.837824	+	0.545940i	$0.816173\pi$
$360$		0			0
$361$	13.0000			0.684211
$362$		0			0
$363$	−15.5563	+	11.0000i	−0.816497	+	0.577350i
$364$		0			0
$365$	29.9333			1.56678
$366$		0			0
$367$			33.6749i			1.75782i	0.476991	+	0.878908i	$0.341727\pi$
$367$			33.6749i			1.75782i	−0.476991	+	0.878908i	$0.658273\pi$
$368$		0			0
$369$	−24.0000	−	8.48528i	−1.24939	−	0.441726i
$370$		0			0
$371$		−	14.0000i		−	0.726844i
$372$		0			0
$373$		−	15.8745i		−	0.821951i	−0.911646	−	0.410975i	$-0.865188\pi$
$373$		−	15.8745i		−	0.821951i	0.911646	−	0.410975i	$-0.134812\pi$
$374$		0			0
$375$	21.1660	−	14.9666i	1.09301	−	0.772873i
$376$		0			0
$377$		−	19.7990i		−	1.01970i
$378$		0			0
$379$	5.65685			0.290573			0.145287	−	0.989390i	$-0.453590\pi$
$379$	5.65685			0.290573			0.145287	+	0.989390i	$0.453590\pi$
$380$		0			0
$381$	11.2250	+	15.8745i	0.575073	+	0.813276i
$382$		0			0
$383$	10.5830			0.540766			0.270383	−	0.962753i	$-0.412850\pi$
$383$	10.5830			0.540766			0.270383	+	0.962753i	$0.412850\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−11.2250			−0.569129			−0.284564	−	0.958657i	$-0.591849\pi$
$389$	−11.2250			−0.569129			−0.284564	+	0.958657i	$0.591849\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	−2.00000	−	2.82843i	−0.100887	−	0.142675i
$394$		0			0
$395$		−	42.0000i		−	2.11325i
$396$		0			0
$397$		−	26.4575i		−	1.32786i	−0.747793	−	0.663932i	$-0.768886\pi$
$397$		−	26.4575i		−	1.32786i	0.747793	−	0.663932i	$-0.231114\pi$
$398$		0			0
$399$	−21.1660	−	29.9333i	−1.05963	−	1.49854i
$400$		0			0
$401$		−	2.82843i		−	0.141245i	−0.997503	−	0.0706225i	$-0.977501\pi$
$401$		−	2.82843i		−	0.141245i	0.997503	−	0.0706225i	$-0.0224986\pi$
$402$		0			0
$403$	19.7990			0.986258
$404$		0			0
$405$	26.1916	+	21.1660i	1.30147	+	1.05175i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−34.0000			−1.68119			−0.840596	−	0.541663i	$-0.817795\pi$
$409$	−34.0000			−1.68119			−0.840596	+	0.541663i	$0.817795\pi$
$410$		0			0
$411$	−2.82843	−	4.00000i	−0.139516	−	0.197305i
$412$		0			0
$413$	22.4499			1.10469
$414$		0			0
$415$		−	29.9333i		−	1.46937i
$416$		0			0
$417$	−4.00000	+	2.82843i	−0.195881	+	0.138509i
$418$		0			0
$419$			24.0000i			1.17248i	0.810139	+	0.586238i	$0.199392\pi$
$419$			24.0000i			1.17248i	−0.810139	+	0.586238i	$0.800608\pi$
$420$		0			0
$421$			37.0405i			1.80524i	0.430434	+	0.902622i	$0.358361\pi$
$421$			37.0405i			1.80524i	−0.430434	+	0.902622i	$0.641639\pi$
$422$		0			0
$423$	−10.5830	+	29.9333i	−0.514563	+	1.45540i
$424$		0			0
$425$			25.4558i			1.23479i
$426$		0			0
$427$	19.7990			0.958140
$428$		0			0
$429$		0			0
$430$		0			0
$431$	10.5830			0.509765			0.254883	−	0.966972i	$-0.417963\pi$
$431$	10.5830			0.509765			0.254883	+	0.966972i	$0.417963\pi$
$432$		0			0
$433$	22.0000			1.05725			0.528626	−	0.848855i	$-0.322707\pi$
$433$	22.0000			1.05725			0.528626	+	0.848855i	$0.322707\pi$
$434$		0			0
$435$	19.7990	−	14.0000i	0.949289	−	0.671249i
$436$		0			0
$437$		0			0
$438$		0			0
$439$		−	41.1582i		−	1.96438i	−0.187903	−	0.982188i	$-0.560169\pi$
$439$		−	41.1582i		−	1.96438i	0.187903	−	0.982188i	$-0.439831\pi$
$440$		0			0
$441$	−7.00000	+	19.7990i	−0.333333	+	0.942809i
$442$		0			0
$443$			40.0000i			1.90046i	0.311553	+	0.950229i	$0.399151\pi$
$443$			40.0000i			1.90046i	−0.311553	+	0.950229i	$0.600849\pi$
$444$		0			0
$445$		−	42.3320i		−	2.00673i
$446$		0			0
$447$	26.4575	−	18.7083i	1.25140	−	0.884872i
$448$		0			0
$449$		−	19.7990i		−	0.934372i	−0.884159	−	0.467186i	$-0.845268\pi$
$449$		−	19.7990i		−	0.934372i	0.884159	−	0.467186i	$-0.154732\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	11.2250	+	15.8745i	0.527395	+	0.745849i
$454$		0			0
$455$	74.0810			3.47297
$456$		0			0
$457$	2.00000			0.0935561			0.0467780	−	0.998905i	$-0.485105\pi$
$457$	2.00000			0.0935561			0.0467780	+	0.998905i	$0.485105\pi$
$458$		0			0
$459$	−14.1421	+	4.00000i	−0.660098	+	0.186704i
$460$		0			0
$461$	−11.2250			−0.522799			−0.261400	−	0.965231i	$-0.584184\pi$
$461$	−11.2250			−0.522799			−0.261400	+	0.965231i	$0.584184\pi$
$462$		0			0
$463$		−	11.2250i		−	0.521669i	−0.965384	−	0.260834i	$-0.916002\pi$
$463$		−	11.2250i		−	0.521669i	0.965384	−	0.260834i	$-0.0839976\pi$
$464$		0			0
$465$	14.0000	+	19.7990i	0.649234	+	0.918156i
$466$		0			0
$467$		−	24.0000i		−	1.11059i	−0.831654	−	0.555294i	$-0.812606\pi$
$467$		−	24.0000i		−	1.11059i	0.831654	−	0.555294i	$-0.187394\pi$
$468$		0			0
$469$			10.5830i			0.488678i
$470$		0			0
$471$	5.29150	+	7.48331i	0.243820	+	0.344813i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−50.9117			−2.33599
$476$		0			0
$477$	3.74166	−	10.5830i	0.171319	−	0.484563i
$478$		0			0
$479$	−21.1660			−0.967100			−0.483550	−	0.875317i	$-0.660653\pi$
$479$	−21.1660			−0.967100			−0.483550	+	0.875317i	$0.660653\pi$
$480$		0			0
$481$	−28.0000			−1.27669
$482$		0			0
$483$		0			0
$484$		0			0
$485$	22.4499			1.01940
$486$		0			0
$487$		−	18.7083i		−	0.847753i	−0.905720	−	0.423877i	$-0.860669\pi$
$487$		−	18.7083i		−	0.847753i	0.905720	−	0.423877i	$-0.139331\pi$
$488$		0			0
$489$	−24.0000	+	16.9706i	−1.08532	+	0.767435i
$490$		0			0
$491$			18.0000i			0.812329i	0.913800	+	0.406164i	$0.133134\pi$
$491$			18.0000i			0.812329i	−0.913800	+	0.406164i	$0.866866\pi$
$492$		0			0
$493$			10.5830i			0.476635i
$494$		0			0
$495$		0			0
$496$		0			0
$497$			39.5980i			1.77621i
$498$		0			0
$499$	−25.4558			−1.13956			−0.569780	−	0.821797i	$-0.692972\pi$
$499$	−25.4558			−1.13956			−0.569780	+	0.821797i	$0.692972\pi$
$500$		0			0
$501$	−29.9333	+	21.1660i	−1.33732	+	0.945628i
$502$		0			0
$503$	31.7490			1.41562			0.707809	−	0.706404i	$-0.249684\pi$
$503$	31.7490			1.41562			0.707809	+	0.706404i	$0.249684\pi$
$504$		0			0
$505$	−42.0000			−1.86898
$506$		0			0
$507$	21.2132	−	15.0000i	0.942111	−	0.666173i
$508$		0			0
$509$	−11.2250			−0.497538			−0.248769	−	0.968563i	$-0.580026\pi$
$509$	−11.2250			−0.497538			−0.248769	+	0.968563i	$0.580026\pi$
$510$		0			0
$511$			29.9333i			1.32417i
$512$		0			0
$513$	−8.00000	−	28.2843i	−0.353209	−	1.24878i
$514$		0			0
$515$		−	14.0000i		−	0.616914i
$516$		0			0
$517$		0			0
$518$		0			0
$519$	−5.29150	+	3.74166i	−0.232271	+	0.164241i
$520$		0			0
$521$		−	25.4558i		−	1.11524i	−0.830096	−	0.557620i	$-0.811714\pi$
$521$		−	25.4558i		−	1.11524i	0.830096	−	0.557620i	$-0.188286\pi$
$522$		0			0
$523$	−33.9411			−1.48414			−0.742071	−	0.670321i	$-0.766156\pi$
$523$	−33.9411			−1.48414			−0.742071	+	0.670321i	$0.766156\pi$
$524$		0			0
$525$	33.6749	+	47.6235i	1.46969	+	2.07846i
$526$		0			0
$527$	−10.5830			−0.461003
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$	16.9706	+	6.00000i	0.736460	+	0.260378i
$532$		0			0
$533$	−44.8999			−1.94483
$534$		0			0
$535$			7.48331i			0.323532i
$536$		0			0
$537$	−22.0000	−	31.1127i	−0.949370	−	1.34261i
$538$		0			0
$539$		0			0
$540$		0			0
$541$			5.29150i			0.227499i	0.993509	+	0.113750i	$0.0362862\pi$
$541$			5.29150i			0.227499i	−0.993509	+	0.113750i	$0.963714\pi$
$542$		0			0
$543$	−15.8745	−	22.4499i	−0.681240	−	0.963419i
$544$		0			0
$545$			19.7990i			0.848096i
$546$		0			0
$547$	39.5980			1.69309			0.846544	−	0.532319i	$-0.178679\pi$
$547$	39.5980			1.69309			0.846544	+	0.532319i	$0.178679\pi$
$548$		0			0
$549$	14.9666	+	5.29150i	0.638760	+	0.225836i
$550$		0			0
$551$	−21.1660			−0.901702
$552$		0			0
$553$	42.0000			1.78602
$554$		0			0
$555$	−19.7990	−	28.0000i	−0.840420	−	1.18853i
$556$		0			0
$557$	−3.74166			−0.158539			−0.0792696	−	0.996853i	$-0.525259\pi$
$557$	−3.74166			−0.158539			−0.0792696	+	0.996853i	$0.525259\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	16.0000i		−	0.674320i	−0.941447	−	0.337160i	$-0.890534\pi$
$563$		−	16.0000i		−	0.674320i	0.941447	−	0.337160i	$-0.109466\pi$
$564$		0			0
$565$		−	42.3320i		−	1.78092i
$566$		0			0
$567$	−21.1660	+	26.1916i	−0.888889	+	1.09994i
$568$		0			0
$569$			2.82843i			0.118574i	0.998241	+	0.0592869i	$0.0188827\pi$
$569$			2.82843i			0.118574i	−0.998241	+	0.0592869i	$0.981117\pi$
$570$		0			0
$571$	8.48528			0.355098			0.177549	−	0.984112i	$-0.443183\pi$
$571$	8.48528			0.355098			0.177549	+	0.984112i	$0.443183\pi$
$572$		0			0
$573$	14.9666	−	10.5830i	0.625240	−	0.442111i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−8.00000			−0.333044			−0.166522	−	0.986038i	$-0.553254\pi$
$577$	−8.00000			−0.333044			−0.166522	+	0.986038i	$0.553254\pi$
$578$		0			0
$579$	−28.2843	+	20.0000i	−1.17545	+	0.831172i
$580$		0			0
$581$	29.9333			1.24184
$582$		0			0
$583$		0			0
$584$		0			0
$585$	56.0000	+	19.7990i	2.31532	+	0.818587i
$586$		0			0
$587$			18.0000i			0.742940i	0.928445	+	0.371470i	$0.121146\pi$
$587$			18.0000i			0.742940i	−0.928445	+	0.371470i	$0.878854\pi$
$588$		0			0
$589$		−	21.1660i		−	0.872130i
$590$		0			0
$591$	−5.29150	+	3.74166i	−0.217663	+	0.153911i
$592$		0			0
$593$		−	5.65685i		−	0.232299i	−0.993232	−	0.116150i	$-0.962945\pi$
$593$		−	5.65685i		−	0.232299i	0.993232	−	0.116150i	$-0.0370552\pi$
$594$		0			0
$595$	−39.5980			−1.62336
$596$		0			0
$597$	−18.7083	−	26.4575i	−0.765679	−	1.08283i
$598$		0			0
$599$	−21.1660			−0.864820			−0.432410	−	0.901677i	$-0.642337\pi$
$599$	−21.1660			−0.864820			−0.432410	+	0.901677i	$0.642337\pi$
$600$		0			0
$601$	−8.00000			−0.326327			−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000			−0.326327			−0.163163	+	0.986599i	$0.552170\pi$
$602$		0			0
$603$	−2.82843	+	8.00000i	−0.115182	+	0.325785i
$604$		0			0
$605$	−41.1582			−1.67332
$606$		0			0
$607$			18.7083i			0.759346i	0.925121	+	0.379673i	$0.123963\pi$
$607$			18.7083i			0.759346i	−0.925121	+	0.379673i	$0.876037\pi$
$608$		0			0
$609$	14.0000	+	19.7990i	0.567309	+	0.802296i
$610$		0			0
$611$			56.0000i			2.26552i
$612$		0			0
$613$		−	15.8745i		−	0.641165i	−0.947221	−	0.320583i	$-0.896121\pi$
$613$		−	15.8745i		−	0.641165i	0.947221	−	0.320583i	$-0.103879\pi$
$614$		0			0
$615$	−31.7490	−	44.8999i	−1.28024	−	1.81054i
$616$		0			0
$617$			5.65685i			0.227736i	0.993496	+	0.113868i	$0.0363242\pi$
$617$			5.65685i			0.227736i	−0.993496	+	0.113868i	$0.963676\pi$
$618$		0			0
$619$	42.4264			1.70526			0.852631	−	0.522514i	$-0.175006\pi$
$619$	42.4264			1.70526			0.852631	+	0.522514i	$0.175006\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	42.3320			1.69600
$624$		0			0
$625$	11.0000			0.440000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	14.9666			0.596759
$630$		0			0
$631$			11.2250i			0.446859i	0.974720	+	0.223430i	$0.0717253\pi$
$631$			11.2250i			0.446859i	−0.974720	+	0.223430i	$0.928275\pi$
$632$		0			0
$633$	12.0000	−	8.48528i	0.476957	−	0.337260i
$634$		0			0
$635$			42.0000i			1.66672i
$636$		0			0
$637$			37.0405i			1.46760i
$638$		0			0
$639$	−10.5830	+	29.9333i	−0.418657	+	1.18414i
$640$		0			0
$641$		−	19.7990i		−	0.782013i	−0.920388	−	0.391007i	$-0.872127\pi$
$641$		−	19.7990i		−	0.782013i	0.920388	−	0.391007i	$-0.127873\pi$
$642$		0			0
$643$	−11.3137			−0.446169			−0.223085	−	0.974799i	$-0.571613\pi$
$643$	−11.3137			−0.446169			−0.223085	+	0.974799i	$0.571613\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−21.1660			−0.832122			−0.416061	−	0.909337i	$-0.636590\pi$
$647$	−21.1660			−0.832122			−0.416061	+	0.909337i	$0.636590\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−19.7990	+	14.0000i	−0.775984	+	0.548703i
$652$		0			0
$653$	−11.2250			−0.439267			−0.219634	−	0.975582i	$-0.570486\pi$
$653$	−11.2250			−0.439267			−0.219634	+	0.975582i	$0.570486\pi$
$654$		0			0
$655$		−	7.48331i		−	0.292397i
$656$		0			0
$657$	−8.00000	+	22.6274i	−0.312110	+	0.882780i
$658$		0			0
$659$			2.00000i			0.0779089i	0.999241	+	0.0389545i	$0.0124027\pi$
$659$			2.00000i			0.0779089i	−0.999241	+	0.0389545i	$0.987597\pi$
$660$		0			0
$661$		−	37.0405i		−	1.44071i	−0.693606	−	0.720355i	$-0.743979\pi$
$661$		−	37.0405i		−	1.44071i	0.693606	−	0.720355i	$-0.256021\pi$
$662$		0			0
$663$	−21.1660	+	14.9666i	−0.822020	+	0.581256i
$664$		0			0
$665$		−	79.1960i		−	3.07109i
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−11.2250	−	15.8745i	−0.433982	−	0.613744i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−10.0000			−0.385472			−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000			−0.385472			−0.192736	+	0.981251i	$0.561736\pi$
$674$		0			0
$675$	12.7279	+	45.0000i	0.489898	+	1.73205i
$676$		0			0
$677$	11.2250			0.431411			0.215705	−	0.976458i	$-0.430795\pi$
$677$	11.2250			0.431411			0.215705	+	0.976458i	$0.430795\pi$
$678$		0			0
$679$			22.4499i			0.861550i
$680$		0			0
$681$	8.00000	+	11.3137i	0.306561	+	0.433542i
$682$		0			0
$683$		−	8.00000i		−	0.306111i	−0.988218	−	0.153056i	$-0.951089\pi$
$683$		−	8.00000i		−	0.306111i	0.988218	−	0.153056i	$-0.0489114\pi$
$684$		0			0
$685$		−	10.5830i		−	0.404356i
$686$		0			0
$687$	5.29150	+	7.48331i	0.201883	+	0.285506i
$688$		0			0
$689$		−	19.7990i		−	0.754281i
$690$		0			0
$691$	11.3137			0.430394			0.215197	−	0.976571i	$-0.430961\pi$
$691$	11.3137			0.430394			0.215197	+	0.976571i	$0.430961\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−10.5830			−0.401436
$696$		0			0
$697$	24.0000			0.909065
$698$		0			0
$699$	5.65685	+	8.00000i	0.213962	+	0.302588i
$700$		0			0
$701$	−18.7083			−0.706602			−0.353301	−	0.935510i	$-0.614941\pi$
$701$	−18.7083			−0.706602			−0.353301	+	0.935510i	$0.614941\pi$
$702$		0			0
$703$			29.9333i			1.12895i
$704$		0			0
$705$	−56.0000	+	39.5980i	−2.10908	+	1.49135i
$706$		0			0
$707$		−	42.0000i		−	1.57957i
$708$		0			0
$709$			37.0405i			1.39109i	0.718485	+	0.695543i	$0.244836\pi$
$709$			37.0405i			1.39109i	−0.718485	+	0.695543i	$0.755164\pi$
$710$		0			0
$711$	31.7490	+	11.2250i	1.19068	+	0.420969i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−14.9666	+	10.5830i	−0.558939	+	0.395230i
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	14.0000			0.521387
$722$		0			0
$723$	−28.2843	+	20.0000i	−1.05190	+	0.743808i
$724$		0			0
$725$	33.6749			1.25066
$726$		0			0
$727$			3.74166i			0.138770i	0.997590	+	0.0693852i	$0.0221038\pi$
$727$			3.74166i			0.138770i	−0.997590	+	0.0693852i	$0.977896\pi$
$728$		0			0
$729$	−23.0000	+	14.1421i	−0.851852	+	0.523783i
$730$		0			0
$731$		0			0
$732$		0			0
$733$			26.4575i			0.977231i	0.872499	+	0.488615i	$0.162498\pi$
$733$			26.4575i			0.977231i	−0.872499	+	0.488615i	$0.837502\pi$
$734$		0			0
$735$	−37.0405	+	26.1916i	−1.36626	+	0.966092i
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−31.1127			−1.14450			−0.572250	−	0.820080i	$-0.693929\pi$
$739$	−31.1127			−1.14450			−0.572250	+	0.820080i	$0.693929\pi$
$740$		0			0
$741$	−29.9333	−	42.3320i	−1.09963	−	1.55511i
$742$		0			0
$743$	21.1660			0.776506			0.388253	−	0.921553i	$-0.373079\pi$
$743$	21.1660			0.776506			0.388253	+	0.921553i	$0.373079\pi$
$744$		0			0
$745$	70.0000			2.56460
$746$		0			0
$747$	22.6274	+	8.00000i	0.827894	+	0.292705i
$748$		0			0
$749$	−7.48331			−0.273434
$750$		0			0
$751$		−	3.74166i		−	0.136535i	−0.997667	−	0.0682675i	$-0.978253\pi$
$751$		−	3.74166i		−	0.136535i	0.997667	−	0.0682675i	$-0.0217471\pi$
$752$		0			0
$753$	24.0000	+	33.9411i	0.874609	+	1.23688i
$754$		0			0
$755$			42.0000i			1.52854i
$756$		0			0
$757$			37.0405i			1.34626i	0.739524	+	0.673130i	$0.235051\pi$
$757$			37.0405i			1.34626i	−0.739524	+	0.673130i	$0.764949\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	48.0833i		−	1.74302i	−0.490381	−	0.871508i	$-0.663142\pi$
$761$		−	48.0833i		−	1.74302i	0.490381	−	0.871508i	$-0.336858\pi$
$762$		0			0
$763$	−19.7990			−0.716772
$764$		0			0
$765$	−29.9333	−	10.5830i	−1.08224	−	0.382629i
$766$		0			0
$767$	31.7490			1.14639
$768$		0			0
$769$	−4.00000			−0.144244			−0.0721218	−	0.997396i	$-0.522977\pi$
$769$	−4.00000			−0.144244			−0.0721218	+	0.997396i	$0.522977\pi$
$770$		0			0
$771$	−11.3137	−	16.0000i	−0.407453	−	0.576226i
$772$		0			0
$773$	−18.7083			−0.672890			−0.336445	−	0.941703i	$-0.609225\pi$
$773$	−18.7083			−0.672890			−0.336445	+	0.941703i	$0.609225\pi$
$774$		0			0
$775$			33.6749i			1.20964i
$776$		0			0
$777$	28.0000	−	19.7990i	1.00449	−	0.710285i
$778$		0			0
$779$			48.0000i			1.71978i
$780$		0			0
$781$		0			0
$782$		0			0
$783$	5.29150	+	18.7083i	0.189103	+	0.668580i
$784$		0			0
$785$			19.7990i			0.706656i
$786$		0			0
$787$	−16.9706			−0.604935			−0.302468	−	0.953160i	$-0.597810\pi$
$787$	−16.9706			−0.604935			−0.302468	+	0.953160i	$0.597810\pi$
$788$		0			0
$789$	14.9666	−	10.5830i	0.532826	−	0.376765i
$790$		0			0
$791$	42.3320			1.50515
$792$		0			0
$793$	28.0000			0.994309
$794$		0			0
$795$	19.7990	−	14.0000i	0.702198	−	0.496529i
$796$		0			0
$797$	41.1582			1.45790			0.728950	−	0.684567i	$-0.240009\pi$
$797$	41.1582			1.45790			0.728950	+	0.684567i	$0.240009\pi$
$798$		0			0
$799$		−	29.9333i		−	1.05896i
$800$		0			0
$801$	32.0000	+	11.3137i	1.13066	+	0.399750i
$802$		0			0
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$	26.4575	−	18.7083i	0.931349	−	0.658563i
$808$		0			0
$809$			2.82843i			0.0994422i	0.998763	+	0.0497211i	$0.0158332\pi$
$809$			2.82843i			0.0994422i	−0.998763	+	0.0497211i	$0.984167\pi$
$810$		0			0
$811$	−16.9706			−0.595917			−0.297959	−	0.954579i	$-0.596306\pi$
$811$	−16.9706			−0.595917			−0.297959	+	0.954579i	$0.596306\pi$
$812$		0			0
$813$	−11.2250	−	15.8745i	−0.393677	−	0.556743i
$814$		0			0
$815$	−63.4980			−2.22424
$816$		0			0
$817$		0			0
$818$		0			0
$819$	−19.7990	+	56.0000i	−0.691833	+	1.95680i
$820$		0			0
$821$	−3.74166			−0.130585			−0.0652924	−	0.997866i	$-0.520798\pi$
$821$	−3.74166			−0.130585			−0.0652924	+	0.997866i	$0.520798\pi$
$822$		0			0
$823$		−	18.7083i		−	0.652130i	−0.945347	−	0.326065i	$-0.894277\pi$
$823$		−	18.7083i		−	0.652130i	0.945347	−	0.326065i	$-0.105723\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	18.0000i		−	0.625921i	−0.949766	−	0.312961i	$-0.898679\pi$
$827$		−	18.0000i		−	0.625921i	0.949766	−	0.312961i	$-0.101321\pi$
$828$		0			0
$829$			37.0405i			1.28647i	0.765669	+	0.643235i	$0.222408\pi$
$829$			37.0405i			1.28647i	−0.765669	+	0.643235i	$0.777592\pi$
$830$		0			0
$831$	−15.8745	−	22.4499i	−0.550681	−	0.778780i
$832$		0			0
$833$		−	19.7990i		−	0.685994i
$834$		0			0
$835$	−79.1960			−2.74069
$836$		0			0
$837$	−18.7083	+	5.29150i	−0.646653	+	0.182901i
$838$		0			0
$839$	−31.7490			−1.09610			−0.548049	−	0.836446i	$-0.684629\pi$
$839$	−31.7490			−1.09610			−0.548049	+	0.836446i	$0.684629\pi$
$840$		0			0
$841$	−15.0000			−0.517241
$842$		0			0
$843$	−28.2843	−	40.0000i	−0.974162	−	1.37767i
$844$		0			0
$845$	56.1249			1.93075
$846$		0			0
$847$		−	41.1582i		−	1.41421i
$848$		0			0
$849$	20.0000	−	14.1421i	0.686398	−	0.485357i
$850$		0			0
$851$		0			0
$852$		0			0
$853$			47.6235i			1.63060i	0.579040	+	0.815299i	$0.303428\pi$
$853$			47.6235i			1.63060i	−0.579040	+	0.815299i	$0.696572\pi$
$854$		0			0
$855$	21.1660	−	59.8665i	0.723862	−	2.04739i
$856$		0			0
$857$		−	36.7696i		−	1.25602i	−0.778204	−	0.628012i	$-0.783869\pi$
$857$		−	36.7696i		−	1.25602i	0.778204	−	0.628012i	$-0.216131\pi$
$858$		0			0
$859$	−45.2548			−1.54408			−0.772038	−	0.635577i	$-0.780762\pi$
$859$	−45.2548			−1.54408			−0.772038	+	0.635577i	$0.780762\pi$
$860$		0			0
$861$	44.8999	−	31.7490i	1.53018	−	1.08200i
$862$		0			0
$863$	31.7490			1.08075			0.540375	−	0.841425i	$-0.318283\pi$
$863$	31.7490			1.08075			0.540375	+	0.841425i	$0.318283\pi$
$864$		0			0
$865$	−14.0000			−0.476014
$866$		0			0
$867$	−12.7279	+	9.00000i	−0.432263	+	0.305656i
$868$		0			0
$869$		0			0
$870$		0			0
$871$			14.9666i			0.507125i
$872$		0			0
$873$	−6.00000	+	16.9706i	−0.203069	+	0.574367i
$874$		0			0
$875$			56.0000i			1.89315i
$876$		0			0
$877$			26.4575i			0.893407i	0.894682	+	0.446703i	$0.147402\pi$
$877$			26.4575i			0.893407i	−0.894682	+	0.446703i	$0.852598\pi$
$878$		0			0
$879$	37.0405	−	26.1916i	1.24935	−	0.883421i
$880$		0			0
$881$		−	16.9706i		−	0.571753i	−0.958267	−	0.285876i	$-0.907715\pi$
$881$		−	16.9706i		−	0.571753i	0.958267	−	0.285876i	$-0.0922847\pi$
$882$		0			0
$883$	11.3137			0.380737			0.190368	−	0.981713i	$-0.439032\pi$
$883$	11.3137			0.380737			0.190368	+	0.981713i	$0.439032\pi$
$884$		0			0
$885$	22.4499	+	31.7490i	0.754647	+	1.06723i
$886$		0			0
$887$	−31.7490			−1.06603			−0.533014	−	0.846107i	$-0.678941\pi$
$887$	−31.7490			−1.06603			−0.533014	+	0.846107i	$0.678941\pi$
$888$		0			0
$889$	−42.0000			−1.40863
$890$		0			0
$891$		0			0
$892$		0			0
$893$	59.8665			2.00336
$894$		0			0
$895$		−	82.3165i		−	2.75154i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			14.0000i			0.466926i
$900$		0			0
$901$			10.5830i			0.352571i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	59.3970i		−	1.97442i
$906$		0			0
$907$	16.9706			0.563498			0.281749	−	0.959488i	$-0.409085\pi$
$907$	16.9706			0.563498			0.281749	+	0.959488i	$0.409085\pi$
$908$		0			0
$909$	11.2250	−	31.7490i	0.372309	−	1.05305i
$910$		0			0
$911$	−31.7490			−1.05189			−0.525946	−	0.850518i	$-0.676289\pi$
$911$	−31.7490			−1.05189			−0.525946	+	0.850518i	$0.676289\pi$
$912$		0			0
$913$		0			0
$914$		0			0
$915$	19.7990	+	28.0000i	0.654534	+	0.925651i
$916$		0			0
$917$	7.48331			0.247121
$918$		0			0
$919$			11.2250i			0.370278i	0.982712	+	0.185139i	$0.0592735\pi$
$919$			11.2250i			0.370278i	−0.982712	+	0.185139i	$0.940727\pi$
$920$		0			0
$921$	−4.00000	+	2.82843i	−0.131804	+	0.0931998i
$922$		0			0
$923$			56.0000i			1.84326i
$924$		0			0
$925$		−	47.6235i		−	1.56585i
$926$		0			0
$927$	10.5830	+	3.74166i	0.347591	+	0.122892i
$928$		0			0
$929$			53.7401i			1.76316i	0.472038	+	0.881578i	$0.343518\pi$
$929$			53.7401i			1.76316i	−0.472038	+	0.881578i	$0.656482\pi$
$930$		0			0
$931$	39.5980			1.29777
$932$		0			0
$933$	44.8999	−	31.7490i	1.46996	−	1.03942i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	46.0000			1.50275			0.751377	−	0.659873i	$-0.229390\pi$
$937$	46.0000			1.50275			0.751377	+	0.659873i	$0.229390\pi$
$938$		0			0
$939$	5.65685	−	4.00000i	0.184604	−	0.130535i
$940$		0			0
$941$	−41.1582			−1.34172			−0.670860	−	0.741584i	$-0.734075\pi$
$941$	−41.1582			−1.34172			−0.670860	+	0.741584i	$0.734075\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	−70.0000	+	19.7990i	−2.27710	+	0.644061i
$946$		0			0
$947$		−	34.0000i		−	1.10485i	−0.833562	−	0.552426i	$-0.813702\pi$
$947$		−	34.0000i		−	1.10485i	0.833562	−	0.552426i	$-0.186298\pi$
$948$		0			0
$949$			42.3320i			1.37416i
$950$		0			0
$951$	5.29150	−	3.74166i	0.171589	−	0.121332i
$952$		0			0
$953$			14.1421i			0.458109i	0.973414	+	0.229054i	$0.0735634\pi$
$953$			14.1421i			0.458109i	−0.973414	+	0.229054i	$0.926437\pi$
$954$		0			0
$955$	39.5980			1.28136
$956$		0			0
$957$		0			0
$958$		0			0
$959$	10.5830			0.341743
$960$		0			0
$961$	17.0000			0.548387
$962$		0			0
$963$	−5.65685	−	2.00000i	−0.182290	−	0.0644491i
$964$		0			0
$965$	−74.8331			−2.40896
$966$		0			0
$967$		−	48.6415i		−	1.56421i	−0.623149	−	0.782103i	$-0.714147\pi$
$967$		−	48.6415i		−	1.56421i	0.623149	−	0.782103i	$-0.285853\pi$
$968$		0			0
$969$	16.0000	+	22.6274i	0.513994	+	0.726897i
$970$		0			0
$971$			16.0000i			0.513464i	0.966483	+	0.256732i	$0.0826458\pi$
$971$			16.0000i			0.513464i	−0.966483	+	0.256732i	$0.917354\pi$
$972$		0			0
$973$		−	10.5830i		−	0.339276i
$974$		0			0
$975$	47.6235	+	67.3498i	1.52517	+	2.15692i
$976$		0			0
$977$			42.4264i			1.35734i	0.734443	+	0.678671i	$0.237444\pi$
$977$			42.4264i			1.35734i	−0.734443	+	0.678671i	$0.762556\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−14.9666	−	5.29150i	−0.477848	−	0.168945i
$982$		0			0
$983$	31.7490			1.01264			0.506318	−	0.862347i	$-0.331006\pi$
$983$	31.7490			1.01264			0.506318	+	0.862347i	$0.331006\pi$
$984$		0			0
$985$	−14.0000			−0.446077
$986$		0			0
$987$	−39.5980	−	56.0000i	−1.26042	−	1.78250i
$988$		0			0
$989$		0			0
$990$		0			0
$991$		−	3.74166i		−	0.118858i	−0.998233	−	0.0594288i	$-0.981072\pi$
$991$		−	3.74166i		−	0.118858i	0.998233	−	0.0594288i	$-0.0189279\pi$
$992$		0			0
$993$	−28.0000	+	19.7990i	−0.888553	+	0.628302i
$994$		0			0
$995$		−	70.0000i		−	2.21915i
$996$		0			0
$997$		−	26.4575i		−	0.837918i	−0.908005	−	0.418959i	$-0.862395\pi$
$997$		−	26.4575i		−	0.837918i	0.908005	−	0.418959i	$-0.137605\pi$
$998$		0			0
$999$	26.4575	−	7.48331i	0.837079	−	0.236762i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1536.2.f.k.767.3		8
3.2	odd	2	inner	1536.2.f.k.767.2		8
4.3	odd	2	inner	1536.2.f.k.767.5		8
8.3	odd	2	inner	1536.2.f.k.767.4		8
8.5	even	2	inner	1536.2.f.k.767.6		8
12.11	even	2	inner	1536.2.f.k.767.8		8
16.3	odd	4		1536.2.c.i.1535.4	yes	4
16.5	even	4		1536.2.c.i.1535.3	yes	4
16.11	odd	4		1536.2.c.e.1535.1	✓	4
16.13	even	4		1536.2.c.e.1535.2	yes	4
24.5	odd	2	inner	1536.2.f.k.767.7		8
24.11	even	2	inner	1536.2.f.k.767.1		8
48.5	odd	4		1536.2.c.e.1535.4	yes	4
48.11	even	4		1536.2.c.i.1535.2	yes	4
48.29	odd	4		1536.2.c.i.1535.1	yes	4
48.35	even	4		1536.2.c.e.1535.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1536.2.c.e.1535.1	✓	4	16.11	odd	4
1536.2.c.e.1535.2	yes	4	16.13	even	4
1536.2.c.e.1535.3	yes	4	48.35	even	4
1536.2.c.e.1535.4	yes	4	48.5	odd	4
1536.2.c.i.1535.1	yes	4	48.29	odd	4
1536.2.c.i.1535.2	yes	4	48.11	even	4
1536.2.c.i.1535.3	yes	4	16.5	even	4
1536.2.c.i.1535.4	yes	4	16.3	odd	4
1536.2.f.k.767.1		8	24.11	even	2	inner
1536.2.f.k.767.2		8	3.2	odd	2	inner
1536.2.f.k.767.3		8	1.1	even	1	trivial
1536.2.f.k.767.4		8	8.3	odd	2	inner
1536.2.f.k.767.5		8	4.3	odd	2	inner
1536.2.f.k.767.6		8	8.5	even	2	inner
1536.2.f.k.767.7		8	24.5	odd	2	inner
1536.2.f.k.767.8		8	12.11	even	2	inner

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.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.41421 + 1.00000i) q^{3} -3.74166 q^{5} -3.74166i q^{7} +(1.00000 - 2.82843i) q^{9} +O(q^{10})\)	\(q+(-1.41421 + 1.00000i) q^{3} -3.74166 q^{5} -3.74166i q^{7} +(1.00000 - 2.82843i) q^{9} -5.29150i q^{13} +(5.29150 - 3.74166i) q^{15} +2.82843i q^{17} -5.65685 q^{19} +(3.74166 + 5.29150i) q^{21} +9.00000 q^{25} +(1.41421 + 5.00000i) q^{27} +3.74166 q^{29} +3.74166i q^{31} +14.0000i q^{35} -5.29150i q^{37} +(5.29150 + 7.48331i) q^{39} -8.48528i q^{41} +(-3.74166 + 10.5830i) q^{45} -10.5830 q^{47} -7.00000 q^{49} +(-2.82843 - 4.00000i) q^{51} +3.74166 q^{53} +(8.00000 - 5.65685i) q^{57} +6.00000i q^{59} +5.29150i q^{61} +(-10.5830 - 3.74166i) q^{63} +19.7990i q^{65} -2.82843 q^{67} -10.5830 q^{71} -8.00000 q^{73} +(-12.7279 + 9.00000i) q^{75} +11.2250i q^{79} +(-7.00000 - 5.65685i) q^{81} +8.00000i q^{83} -10.5830i q^{85} +(-5.29150 + 3.74166i) q^{87} +11.3137i q^{89} -19.7990 q^{91} +(-3.74166 - 5.29150i) q^{93} +21.1660 q^{95} -6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9}+O(q^{10}) \) 8 * q + 8 * q^9	\( 8 q + 8 q^{9} + 72 q^{25} - 56 q^{49} + 64 q^{57} - 64 q^{73} - 56 q^{81} - 48 q^{97}+O(q^{100}) \) 8 * q + 8 * q^9 + 72 * q^25 - 56 * q^49 + 64 * q^57 - 64 * q^73 - 56 * q^81 - 48 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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