LMFDB - Embedded newform 1536.2.c.i.1535.2

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:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 8x^{2} + 9 $ x^4 + 8*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1535.2
Root			$1.16372i$ of defining polynomial
Character	$\chi$	$=$	1536.1535
Dual form			1536.2.c.i.1535.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.00000 - 1.41421i) q^{3} +3.74166i q^{5} -3.74166i q^{7} +(-1.00000 - 2.82843i) q^{9} +O(q^{10})$	$q+(1.00000 - 1.41421i) q^{3} +3.74166i q^{5} -3.74166i q^{7} +(-1.00000 - 2.82843i) q^{9} -5.29150 q^{13} +(5.29150 + 3.74166i) q^{15} -2.82843i q^{17} -5.65685i q^{19} +(-5.29150 - 3.74166i) q^{21} -9.00000 q^{25} +(-5.00000 - 1.41421i) q^{27} +3.74166i q^{29} -3.74166i q^{31} +14.0000 q^{35} +5.29150 q^{37} +(-5.29150 + 7.48331i) q^{39} -8.48528i q^{41} +(10.5830 - 3.74166i) q^{45} -10.5830 q^{47} -7.00000 q^{49} +(-4.00000 - 2.82843i) q^{51} -3.74166i q^{53} +(-8.00000 - 5.65685i) q^{57} -6.00000 q^{59} +5.29150 q^{61} +(-10.5830 + 3.74166i) q^{63} -19.7990i q^{65} -2.82843i q^{67} +10.5830 q^{71} +8.00000 q^{73} +(-9.00000 + 12.7279i) q^{75} -11.2250i q^{79} +(-7.00000 + 5.65685i) q^{81} +8.00000 q^{83} +10.5830 q^{85} +(5.29150 + 3.74166i) q^{87} +11.3137i q^{89} +19.7990i q^{91} +(-5.29150 - 3.74166i) q^{93} +21.1660 q^{95} -6.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 4 * q^9	$ 4 q + 4 q^{3} - 4 q^{9} - 36 q^{25} - 20 q^{27} + 56 q^{35} - 28 q^{49} - 16 q^{51} - 32 q^{57} - 24 q^{59} + 32 q^{73} - 36 q^{75} - 28 q^{81} + 32 q^{83} - 24 q^{97}+O(q^{100}) $ 4 * q + 4 * q^3 - 4 * q^9 - 36 * q^25 - 20 * q^27 + 56 * q^35 - 28 * q^49 - 16 * q^51 - 32 * q^57 - 24 * q^59 + 32 * q^73 - 36 * q^75 - 28 * q^81 + 32 * q^83 - 24 * 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
$3$	1.00000	−	1.41421i	0.577350	−	0.816497i
$3$	1.00000	−	1.41421i	0.577350	−	0.816497i
$4$		0			0
$5$			3.74166i			1.67332i	0.547723	+	0.836660i	$0.315495\pi$
$5$			3.74166i			1.67332i	−0.547723	+	0.836660i	$0.684505\pi$
$6$		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			−	1.00000i	$-0.5\pi$
$11$		0			0				1.00000i	$0.5\pi$
$12$		0			0
$13$	−5.29150			−1.46760			−0.733799	−	0.679366i	$-0.762255\pi$
$13$	−5.29150			−1.46760			−0.733799	+	0.679366i	$0.762255\pi$
$14$		0			0
$15$	5.29150	+	3.74166i	1.36626	+	0.966092i
$16$		0			0
$17$		−	2.82843i		−	0.685994i	−0.939336	−	0.342997i	$-0.888558\pi$
$17$		−	2.82843i		−	0.685994i	0.939336	−	0.342997i	$-0.111442\pi$
$18$		0			0
$19$		−	5.65685i		−	1.29777i	−0.760886	−	0.648886i	$-0.775235\pi$
$19$		−	5.65685i		−	1.29777i	0.760886	−	0.648886i	$-0.224765\pi$
$20$		0			0
$21$	−5.29150	−	3.74166i	−1.15470	−	0.816497i
$22$		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
$27$	−5.00000	−	1.41421i	−0.962250	−	0.272166i
$28$		0			0
$29$			3.74166i			0.694808i	0.937715	+	0.347404i	$0.112937\pi$
$29$			3.74166i			0.694808i	−0.937715	+	0.347404i	$0.887063\pi$
$30$		0			0
$31$		−	3.74166i		−	0.672022i	−0.941858	−	0.336011i	$-0.890922\pi$
$31$		−	3.74166i		−	0.672022i	0.941858	−	0.336011i	$-0.109078\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	14.0000			2.36643
$36$		0			0
$37$	5.29150			0.869918			0.434959	−	0.900450i	$-0.356763\pi$
$37$	5.29150			0.869918			0.434959	+	0.900450i	$0.356763\pi$
$38$		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		1.00000			$0$
$43$		0			0		−1.00000			$\pi$
$44$		0			0
$45$	10.5830	−	3.74166i	1.57762	−	0.557773i
$46$		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
$51$	−4.00000	−	2.82843i	−0.560112	−	0.396059i
$52$		0			0
$53$		−	3.74166i		−	0.513956i	−0.966417	−	0.256978i	$-0.917273\pi$
$53$		−	3.74166i		−	0.513956i	0.966417	−	0.256978i	$-0.0827268\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−8.00000	−	5.65685i	−1.05963	−	0.749269i
$58$		0			0
$59$	−6.00000			−0.781133			−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000			−0.781133			−0.390567	+	0.920575i	$0.627721\pi$
$60$		0			0
$61$	5.29150			0.677507			0.338754	−	0.940875i	$-0.389995\pi$
$61$	5.29150			0.677507			0.338754	+	0.940875i	$0.389995\pi$
$62$		0			0
$63$	−10.5830	+	3.74166i	−1.33333	+	0.471405i
$64$		0			0
$65$		−	19.7990i		−	2.45576i
$66$		0			0
$67$		−	2.82843i		−	0.345547i	−0.984962	−	0.172774i	$-0.944727\pi$
$67$		−	2.82843i		−	0.345547i	0.984962	−	0.172774i	$-0.0552729\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	10.5830			1.25597			0.627986	−	0.778225i	$-0.283880\pi$
$71$	10.5830			1.25597			0.627986	+	0.778225i	$0.283880\pi$
$72$		0			0
$73$	8.00000			0.936329			0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000			0.936329			0.468165	+	0.883641i	$0.344915\pi$
$74$		0			0
$75$	−9.00000	+	12.7279i	−1.03923	+	1.46969i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		−	11.2250i		−	1.26291i	−0.775413	−	0.631454i	$-0.782458\pi$
$79$		−	11.2250i		−	1.26291i	0.775413	−	0.631454i	$-0.217542\pi$
$80$		0			0
$81$	−7.00000	+	5.65685i	−0.777778	+	0.628539i
$82$		0			0
$83$	8.00000			0.878114			0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000			0.878114			0.439057	+	0.898459i	$0.355313\pi$
$84$		0			0
$85$	10.5830			1.14789
$86$		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
$91$			19.7990i			2.07550i
$92$		0			0
$93$	−5.29150	−	3.74166i	−0.548703	−	0.387992i
$94$		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
$101$		−	11.2250i		−	1.11693i	−0.829529	−	0.558463i	$-0.811391\pi$
$101$		−	11.2250i		−	1.11693i	0.829529	−	0.558463i	$-0.188609\pi$
$102$		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
$107$	2.00000			0.193347			0.0966736	−	0.995316i	$-0.469180\pi$
$107$	2.00000			0.193347			0.0966736	+	0.995316i	$0.469180\pi$
$108$		0			0
$109$	−5.29150			−0.506834			−0.253417	−	0.967357i	$-0.581554\pi$
$109$	−5.29150			−0.506834			−0.253417	+	0.967357i	$0.581554\pi$
$110$		0			0
$111$	5.29150	−	7.48331i	0.502247	−	0.710285i
$112$		0			0
$113$		−	11.3137i		−	1.06430i	−0.846649	−	0.532152i	$-0.821383\pi$
$113$		−	11.3137i		−	1.06430i	0.846649	−	0.532152i	$-0.178617\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	5.29150	+	14.9666i	0.489200	+	1.38367i
$118$		0			0
$119$	−10.5830			−0.970143
$120$		0			0
$121$	−11.0000			−1.00000
$122$		0			0
$123$	−12.0000	−	8.48528i	−1.08200	−	0.765092i
$124$		0			0
$125$		−	14.9666i		−	1.33866i
$126$		0			0
$127$			11.2250i			0.996055i	0.867161	+	0.498028i	$0.165942\pi$
$127$			11.2250i			0.996055i	−0.867161	+	0.498028i	$0.834058\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	2.00000			0.174741			0.0873704	−	0.996176i	$-0.472154\pi$
$131$	2.00000			0.174741			0.0873704	+	0.996176i	$0.472154\pi$
$132$		0			0
$133$	−21.1660			−1.83533
$134$		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
$139$		−	2.82843i		−	0.239904i	−0.992780	−	0.119952i	$-0.961726\pi$
$139$		−	2.82843i		−	0.239904i	0.992780	−	0.119952i	$-0.0382741\pi$
$140$		0			0
$141$	−10.5830	+	14.9666i	−0.891250	+	1.26042i
$142$		0			0
$143$		0			0
$144$		0			0
$145$	−14.0000			−1.16264
$146$		0			0
$147$	−7.00000	+	9.89949i	−0.577350	+	0.816497i
$148$		0			0
$149$			18.7083i			1.53264i	0.642458	+	0.766321i	$0.277915\pi$
$149$			18.7083i			1.53264i	−0.642458	+	0.766321i	$0.722085\pi$
$150$		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
$155$	14.0000			1.12451
$156$		0			0
$157$	−5.29150			−0.422308			−0.211154	−	0.977453i	$-0.567722\pi$
$157$	−5.29150			−0.422308			−0.211154	+	0.977453i	$0.567722\pi$
$158$		0			0
$159$	−5.29150	−	3.74166i	−0.419643	−	0.296733i
$160$		0			0
$161$		0			0
$162$		0			0
$163$			16.9706i			1.32924i	0.747183	+	0.664619i	$0.231406\pi$
$163$			16.9706i			1.32924i	−0.747183	+	0.664619i	$0.768594\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−21.1660			−1.63788			−0.818938	−	0.573883i	$-0.805437\pi$
$167$	−21.1660			−1.63788			−0.818938	+	0.573883i	$0.805437\pi$
$168$		0			0
$169$	15.0000			1.15385
$170$		0			0
$171$	−16.0000	+	5.65685i	−1.22355	+	0.432590i
$172$		0			0
$173$			3.74166i			0.284473i	0.989833	+	0.142236i	$0.0454293\pi$
$173$			3.74166i			0.284473i	−0.989833	+	0.142236i	$0.954571\pi$
$174$		0			0
$175$			33.6749i			2.54558i
$176$		0			0
$177$	−6.00000	+	8.48528i	−0.450988	+	0.637793i
$178$		0			0
$179$	22.0000			1.64436			0.822179	−	0.569230i	$-0.192758\pi$
$179$	22.0000			1.64436			0.822179	+	0.569230i	$0.192758\pi$
$180$		0			0
$181$	−15.8745			−1.17994			−0.589971	−	0.807424i	$-0.700861\pi$
$181$	−15.8745			−1.17994			−0.589971	+	0.807424i	$0.700861\pi$
$182$		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
$189$	−5.29150	+	18.7083i	−0.384900	+	1.36083i
$190$		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
$195$	−28.0000	−	19.7990i	−2.00512	−	1.41784i
$196$		0			0
$197$		−	3.74166i		−	0.266582i	−0.991077	−	0.133291i	$-0.957446\pi$
$197$		−	3.74166i		−	0.266582i	0.991077	−	0.133291i	$-0.0425545\pi$
$198$		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
$203$	14.0000			0.982607
$204$		0			0
$205$	31.7490			2.21745
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$		−	8.48528i		−	0.584151i	−0.956395	−	0.292075i	$-0.905654\pi$
$211$		−	8.48528i		−	0.584151i	0.956395	−	0.292075i	$-0.0943458\pi$
$212$		0			0
$213$	10.5830	−	14.9666i	0.725136	−	1.02550i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−14.0000			−0.950382
$218$		0			0
$219$	8.00000	−	11.3137i	0.540590	−	0.764510i
$220$		0			0
$221$			14.9666i			1.00676i
$222$		0			0
$223$		−	11.2250i		−	0.751680i	−0.926685	−	0.375840i	$-0.877354\pi$
$223$		−	11.2250i		−	0.751680i	0.926685	−	0.375840i	$-0.122646\pi$
$224$		0			0
$225$	9.00000	+	25.4558i	0.600000	+	1.69706i
$226$		0			0
$227$	−8.00000			−0.530979			−0.265489	−	0.964114i	$-0.585534\pi$
$227$	−8.00000			−0.530979			−0.265489	+	0.964114i	$0.585534\pi$
$228$		0			0
$229$	5.29150			0.349672			0.174836	−	0.984598i	$-0.444060\pi$
$229$	5.29150			0.349672			0.174836	+	0.984598i	$0.444060\pi$
$230$		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
$235$		−	39.5980i		−	2.58309i
$236$		0			0
$237$	−15.8745	−	11.2250i	−1.03116	−	0.729140i
$238$		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
$243$	1.00000	+	15.5563i	0.0641500	+	0.997940i
$244$		0			0
$245$		−	26.1916i		−	1.67332i
$246$		0			0
$247$			29.9333i			1.90461i
$248$		0			0
$249$	8.00000	−	11.3137i	0.506979	−	0.716977i
$250$		0			0
$251$	24.0000			1.51487			0.757433	−	0.652913i	$-0.226453\pi$
$251$	24.0000			1.51487			0.757433	+	0.652913i	$0.226453\pi$
$252$		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	$-0.885208\pi$
$257$		−	11.3137i		−	0.705730i	0.935674	−	0.352865i	$-0.114792\pi$
$258$		0			0
$259$		−	19.7990i		−	1.23025i
$260$		0			0
$261$	10.5830	−	3.74166i	0.655072	−	0.231603i
$262$		0			0
$263$	10.5830			0.652576			0.326288	−	0.945270i	$-0.394202\pi$
$263$	10.5830			0.652576			0.326288	+	0.945270i	$0.394202\pi$
$264$		0			0
$265$	14.0000			0.860013
$266$		0			0
$267$	16.0000	+	11.3137i	0.979184	+	0.692388i
$268$		0			0
$269$		−	18.7083i		−	1.14066i	−0.821414	−	0.570332i	$-0.806814\pi$
$269$		−	18.7083i		−	1.14066i	0.821414	−	0.570332i	$-0.193186\pi$
$270$		0			0
$271$		−	11.2250i		−	0.681868i	−0.940087	−	0.340934i	$-0.889257\pi$
$271$		−	11.2250i		−	0.681868i	0.940087	−	0.340934i	$-0.110743\pi$
$272$		0			0
$273$	28.0000	+	19.7990i	1.69464	+	1.19829i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−15.8745			−0.953807			−0.476903	−	0.878956i	$-0.658241\pi$
$277$	−15.8745			−0.953807			−0.476903	+	0.878956i	$0.658241\pi$
$278$		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
$283$			14.1421i			0.840663i	0.907371	+	0.420331i	$0.138086\pi$
$283$			14.1421i			0.840663i	−0.907371	+	0.420331i	$0.861914\pi$
$284$		0			0
$285$	21.1660	−	29.9333i	1.25377	−	1.77309i
$286$		0			0
$287$	−31.7490			−1.87409
$288$		0			0
$289$	9.00000			0.529412
$290$		0			0
$291$	−6.00000	+	8.48528i	−0.351726	+	0.497416i
$292$		0			0
$293$			26.1916i			1.53013i	0.643953	+	0.765065i	$0.277293\pi$
$293$			26.1916i			1.53013i	−0.643953	+	0.765065i	$0.722707\pi$
$294$		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
$307$			2.82843i			0.161427i	0.996737	+	0.0807134i	$0.0257199\pi$
$307$			2.82843i			0.161427i	−0.996737	+	0.0807134i	$0.974280\pi$
$308$		0			0
$309$	5.29150	+	3.74166i	0.301023	+	0.212855i
$310$		0			0
$311$	31.7490			1.80032			0.900161	−	0.435558i	$-0.143449\pi$
$311$	31.7490			1.80032			0.900161	+	0.435558i	$0.143449\pi$
$312$		0			0
$313$	4.00000			0.226093			0.113047	−	0.993590i	$-0.463939\pi$
$313$	4.00000			0.226093			0.113047	+	0.993590i	$0.463939\pi$
$314$		0			0
$315$	−14.0000	−	39.5980i	−0.788811	−	2.23109i
$316$		0			0
$317$		−	3.74166i		−	0.210152i	−0.994464	−	0.105076i	$-0.966491\pi$
$317$		−	3.74166i		−	0.210152i	0.994464	−	0.105076i	$-0.0335087\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	2.00000	−	2.82843i	0.111629	−	0.157867i
$322$		0			0
$323$	−16.0000			−0.890264
$324$		0			0
$325$	47.6235			2.64168
$326$		0			0
$327$	−5.29150	+	7.48331i	−0.292621	+	0.413828i
$328$		0			0
$329$			39.5980i			2.18311i
$330$		0			0
$331$		−	19.7990i		−	1.08825i	−0.839004	−	0.544125i	$-0.816862\pi$
$331$		−	19.7990i		−	1.08825i	0.839004	−	0.544125i	$-0.183138\pi$
$332$		0			0
$333$	−5.29150	−	14.9666i	−0.289973	−	0.820166i
$334$		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
$339$	−16.0000	−	11.3137i	−0.869001	−	0.614476i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	32.0000			1.71785			0.858925	−	0.512101i	$-0.171133\pi$
$347$	32.0000			1.71785			0.858925	+	0.512101i	$0.171133\pi$
$348$		0			0
$349$	5.29150			0.283248			0.141624	−	0.989921i	$-0.454768\pi$
$349$	5.29150			0.283248			0.141624	+	0.989921i	$0.454768\pi$
$350$		0			0
$351$	26.4575	+	7.48331i	1.41220	+	0.399430i
$352$		0			0
$353$			16.9706i			0.903252i	0.892207	+	0.451626i	$0.149156\pi$
$353$			16.9706i			0.903252i	−0.892207	+	0.451626i	$0.850844\pi$
$354$		0			0
$355$			39.5980i			2.10164i
$356$		0			0
$357$	−10.5830	+	14.9666i	−0.560112	+	0.792118i
$358$		0			0
$359$	31.7490			1.67565			0.837824	−	0.545940i	$-0.183827\pi$
$359$	31.7490			1.67565			0.837824	+	0.545940i	$0.183827\pi$
$360$		0			0
$361$	−13.0000			−0.684211
$362$		0			0
$363$	−11.0000	+	15.5563i	−0.577350	+	0.816497i
$364$		0			0
$365$			29.9333i			1.56678i
$366$		0			0
$367$		−	33.6749i		−	1.75782i	−0.476991	−	0.878908i	$-0.658273\pi$
$367$		−	33.6749i		−	1.75782i	0.476991	−	0.878908i	$-0.341727\pi$
$368$		0			0
$369$	−24.0000	+	8.48528i	−1.24939	+	0.441726i
$370$		0			0
$371$	−14.0000			−0.726844
$372$		0			0
$373$	15.8745			0.821951			0.410975	−	0.911646i	$-0.365188\pi$
$373$	15.8745			0.821951			0.410975	+	0.911646i	$0.365188\pi$
$374$		0			0
$375$	−21.1660	−	14.9666i	−1.09301	−	0.772873i
$376$		0			0
$377$		−	19.7990i		−	1.01970i
$378$		0			0
$379$		−	5.65685i		−	0.290573i	−0.989390	−	0.145287i	$-0.953590\pi$
$379$		−	5.65685i		−	0.290573i	0.989390	−	0.145287i	$-0.0464104\pi$
$380$		0			0
$381$	15.8745	+	11.2250i	0.813276	+	0.575073i
$382$		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
$389$			11.2250i			0.569129i	0.958657	+	0.284564i	$0.0918489\pi$
$389$			11.2250i			0.569129i	−0.958657	+	0.284564i	$0.908151\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	2.00000	−	2.82843i	0.100887	−	0.142675i
$394$		0			0
$395$	42.0000			2.11325
$396$		0			0
$397$	−26.4575			−1.32786			−0.663932	−	0.747793i	$-0.731114\pi$
$397$	−26.4575			−1.32786			−0.663932	+	0.747793i	$0.731114\pi$
$398$		0			0
$399$	−21.1660	+	29.9333i	−1.05963	+	1.49854i
$400$		0			0
$401$			2.82843i			0.141245i	0.997503	+	0.0706225i	$0.0224986\pi$
$401$			2.82843i			0.141245i	−0.997503	+	0.0706225i	$0.977501\pi$
$402$		0			0
$403$			19.7990i			0.986258i
$404$		0			0
$405$	−21.1660	−	26.1916i	−1.05175	−	1.30147i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	34.0000			1.68119			0.840596	−	0.541663i	$-0.182205\pi$
$409$	34.0000			1.68119			0.840596	+	0.541663i	$0.182205\pi$
$410$		0			0
$411$	4.00000	+	2.82843i	0.197305	+	0.139516i
$412$		0			0
$413$			22.4499i			1.10469i
$414$		0			0
$415$			29.9333i			1.46937i
$416$		0			0
$417$	−4.00000	−	2.82843i	−0.195881	−	0.138509i
$418$		0			0
$419$	24.0000			1.17248			0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000			1.17248			0.586238	+	0.810139i	$0.300608\pi$
$420$		0			0
$421$	−37.0405			−1.80524			−0.902622	−	0.430434i	$-0.858361\pi$
$421$	−37.0405			−1.80524			−0.902622	+	0.430434i	$0.858361\pi$
$422$		0			0
$423$	10.5830	+	29.9333i	0.514563	+	1.45540i
$424$		0			0
$425$			25.4558i			1.23479i
$426$		0			0
$427$		−	19.7990i		−	0.958140i
$428$		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
$435$	−14.0000	+	19.7990i	−0.671249	+	0.949289i
$436$		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
$443$	−40.0000			−1.90046			−0.950229	−	0.311553i	$-0.899151\pi$
$443$	−40.0000			−1.90046			−0.950229	+	0.311553i	$0.899151\pi$
$444$		0			0
$445$	−42.3320			−2.00673
$446$		0			0
$447$	26.4575	+	18.7083i	1.25140	+	0.884872i
$448$		0			0
$449$			19.7990i			0.934372i	0.884159	+	0.467186i	$0.154732\pi$
$449$			19.7990i			0.934372i	−0.884159	+	0.467186i	$0.845268\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−15.8745	−	11.2250i	−0.745849	−	0.527395i
$454$		0			0
$455$	−74.0810			−3.47297
$456$		0			0
$457$	−2.00000			−0.0935561			−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000			−0.0935561			−0.0467780	+	0.998905i	$0.514895\pi$
$458$		0			0
$459$	−4.00000	+	14.1421i	−0.186704	+	0.660098i
$460$		0			0
$461$		−	11.2250i		−	0.522799i	−0.965231	−	0.261400i	$-0.915816\pi$
$461$		−	11.2250i		−	0.522799i	0.965231	−	0.261400i	$-0.0841840\pi$
$462$		0			0
$463$			11.2250i			0.521669i	0.965384	+	0.260834i	$0.0839976\pi$
$463$			11.2250i			0.521669i	−0.965384	+	0.260834i	$0.916002\pi$
$464$		0			0
$465$	14.0000	−	19.7990i	0.649234	−	0.918156i
$466$		0			0
$467$	−24.0000			−1.11059			−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000			−1.11059			−0.555294	+	0.831654i	$0.687394\pi$
$468$		0			0
$469$	−10.5830			−0.488678
$470$		0			0
$471$	−5.29150	+	7.48331i	−0.243820	+	0.344813i
$472$		0			0
$473$		0			0
$474$		0			0
$475$			50.9117i			2.33599i
$476$		0			0
$477$	−10.5830	+	3.74166i	−0.484563	+	0.171319i
$478$		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
$485$		−	22.4499i		−	1.01940i
$486$		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
$491$	−18.0000			−0.812329			−0.406164	−	0.913800i	$-0.633134\pi$
$491$	−18.0000			−0.812329			−0.406164	+	0.913800i	$0.633134\pi$
$492$		0			0
$493$	10.5830			0.476635
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	39.5980i		−	1.77621i
$498$		0			0
$499$		−	25.4558i		−	1.13956i	−0.821797	−	0.569780i	$-0.807028\pi$
$499$		−	25.4558i		−	1.13956i	0.821797	−	0.569780i	$-0.192972\pi$
$500$		0			0
$501$	−21.1660	+	29.9333i	−0.945628	+	1.33732i
$502$		0			0
$503$	−31.7490			−1.41562			−0.707809	−	0.706404i	$-0.750316\pi$
$503$	−31.7490			−1.41562			−0.707809	+	0.706404i	$0.750316\pi$
$504$		0			0
$505$	42.0000			1.86898
$506$		0			0
$507$	15.0000	−	21.2132i	0.666173	−	0.942111i
$508$		0			0
$509$		−	11.2250i		−	0.497538i	−0.968563	−	0.248769i	$-0.919974\pi$
$509$		−	11.2250i		−	0.497538i	0.968563	−	0.248769i	$-0.0800260\pi$
$510$		0			0
$511$		−	29.9333i		−	1.32417i
$512$		0			0
$513$	−8.00000	+	28.2843i	−0.353209	+	1.24878i
$514$		0			0
$515$	−14.0000			−0.616914
$516$		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
$523$			33.9411i			1.48414i	0.670321	+	0.742071i	$0.266156\pi$
$523$			33.9411i			1.48414i	−0.670321	+	0.742071i	$0.733844\pi$
$524$		0			0
$525$	47.6235	+	33.6749i	2.07846	+	1.46969i
$526$		0			0
$527$	−10.5830			−0.461003
$528$		0			0
$529$	−23.0000			−1.00000
$530$		0			0
$531$	6.00000	+	16.9706i	0.260378	+	0.736460i
$532$		0			0
$533$			44.8999i			1.94483i
$534$		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
$541$	5.29150			0.227499			0.113750	−	0.993509i	$-0.463714\pi$
$541$	5.29150			0.227499			0.113750	+	0.993509i	$0.463714\pi$
$542$		0			0
$543$	−15.8745	+	22.4499i	−0.681240	+	0.963419i
$544$		0			0
$545$		−	19.7990i		−	0.848096i
$546$		0			0
$547$			39.5980i			1.69309i	0.532319	+	0.846544i	$0.321321\pi$
$547$			39.5980i			1.69309i	−0.532319	+	0.846544i	$0.678679\pi$
$548$		0			0
$549$	−5.29150	−	14.9666i	−0.225836	−	0.638760i
$550$		0			0
$551$	21.1660			0.901702
$552$		0			0
$553$	−42.0000			−1.78602
$554$		0			0
$555$	28.0000	+	19.7990i	1.18853	+	0.840420i
$556$		0			0
$557$		−	3.74166i		−	0.158539i	−0.996853	−	0.0792696i	$-0.974741\pi$
$557$		−	3.74166i		−	0.158539i	0.996853	−	0.0792696i	$-0.0252588\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−16.0000			−0.674320			−0.337160	−	0.941447i	$-0.609466\pi$
$563$	−16.0000			−0.674320			−0.337160	+	0.941447i	$0.609466\pi$
$564$		0			0
$565$	42.3320			1.78092
$566$		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
$571$		−	8.48528i		−	0.355098i	−0.984112	−	0.177549i	$-0.943183\pi$
$571$		−	8.48528i		−	0.355098i	0.984112	−	0.177549i	$-0.0568168\pi$
$572$		0			0
$573$	−10.5830	+	14.9666i	−0.442111	+	0.625240i
$574$		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
$579$	20.0000	−	28.2843i	0.831172	−	1.17545i
$580$		0			0
$581$		−	29.9333i		−	1.24184i
$582$		0			0
$583$		0			0
$584$		0			0
$585$	−56.0000	+	19.7990i	−2.31532	+	0.818587i
$586$		0			0
$587$	−18.0000			−0.742940			−0.371470	−	0.928445i	$-0.621146\pi$
$587$	−18.0000			−0.742940			−0.371470	+	0.928445i	$0.621146\pi$
$588$		0			0
$589$	−21.1660			−0.872130
$590$		0			0
$591$	−5.29150	−	3.74166i	−0.217663	−	0.153911i
$592$		0			0
$593$			5.65685i			0.232299i	0.993232	+	0.116150i	$0.0370552\pi$
$593$			5.65685i			0.232299i	−0.993232	+	0.116150i	$0.962945\pi$
$594$		0			0
$595$		−	39.5980i		−	1.62336i
$596$		0			0
$597$	26.4575	+	18.7083i	1.08283	+	0.765679i
$598$		0			0
$599$	21.1660			0.864820			0.432410	−	0.901677i	$-0.357663\pi$
$599$	21.1660			0.864820			0.432410	+	0.901677i	$0.357663\pi$
$600$		0			0
$601$	8.00000			0.326327			0.163163	−	0.986599i	$-0.447830\pi$
$601$	8.00000			0.326327			0.163163	+	0.986599i	$0.447830\pi$
$602$		0			0
$603$	−8.00000	+	2.82843i	−0.325785	+	0.115182i
$604$		0			0
$605$		−	41.1582i		−	1.67332i
$606$		0			0
$607$		−	18.7083i		−	0.759346i	−0.925121	−	0.379673i	$-0.876037\pi$
$607$		−	18.7083i		−	0.759346i	0.925121	−	0.379673i	$-0.123963\pi$
$608$		0			0
$609$	14.0000	−	19.7990i	0.567309	−	0.802296i
$610$		0			0
$611$	56.0000			2.26552
$612$		0			0
$613$	15.8745			0.641165			0.320583	−	0.947221i	$-0.396121\pi$
$613$	15.8745			0.641165			0.320583	+	0.947221i	$0.396121\pi$
$614$		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
$619$		−	42.4264i		−	1.70526i	−0.522514	−	0.852631i	$-0.675006\pi$
$619$		−	42.4264i		−	1.70526i	0.522514	−	0.852631i	$-0.324994\pi$
$620$		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
$629$		−	14.9666i		−	0.596759i
$630$		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
$635$	−42.0000			−1.66672
$636$		0			0
$637$	37.0405			1.46760
$638$		0			0
$639$	−10.5830	−	29.9333i	−0.418657	−	1.18414i
$640$		0			0
$641$			19.7990i			0.782013i	0.920388	+	0.391007i	$0.127873\pi$
$641$			19.7990i			0.782013i	−0.920388	+	0.391007i	$0.872127\pi$
$642$		0			0
$643$		−	11.3137i		−	0.446169i	−0.974799	−	0.223085i	$-0.928387\pi$
$643$		−	11.3137i		−	0.446169i	0.974799	−	0.223085i	$-0.0716126\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	21.1660			0.832122			0.416061	−	0.909337i	$-0.363410\pi$
$647$	21.1660			0.832122			0.416061	+	0.909337i	$0.363410\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−14.0000	+	19.7990i	−0.548703	+	0.775984i
$652$		0			0
$653$		−	11.2250i		−	0.439267i	−0.975582	−	0.219634i	$-0.929514\pi$
$653$		−	11.2250i		−	0.439267i	0.975582	−	0.219634i	$-0.0704862\pi$
$654$		0			0
$655$			7.48331i			0.292397i
$656$		0			0
$657$	−8.00000	−	22.6274i	−0.312110	−	0.882780i
$658$		0			0
$659$	2.00000			0.0779089			0.0389545	−	0.999241i	$-0.487597\pi$
$659$	2.00000			0.0779089			0.0389545	+	0.999241i	$0.487597\pi$
$660$		0			0
$661$	37.0405			1.44071			0.720355	−	0.693606i	$-0.243979\pi$
$661$	37.0405			1.44071			0.720355	+	0.693606i	$0.243979\pi$
$662$		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
$669$	−15.8745	−	11.2250i	−0.613744	−	0.433982i
$670$		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
$675$	45.0000	+	12.7279i	1.73205	+	0.489898i
$676$		0			0
$677$		−	11.2250i		−	0.431411i	−0.976458	−	0.215705i	$-0.930795\pi$
$677$		−	11.2250i		−	0.431411i	0.976458	−	0.215705i	$-0.0692051\pi$
$678$		0			0
$679$			22.4499i			0.861550i
$680$		0			0
$681$	−8.00000	+	11.3137i	−0.306561	+	0.433542i
$682$		0			0
$683$	8.00000			0.306111			0.153056	−	0.988218i	$-0.451089\pi$
$683$	8.00000			0.306111			0.153056	+	0.988218i	$0.451089\pi$
$684$		0			0
$685$	−10.5830			−0.404356
$686$		0			0
$687$	5.29150	−	7.48331i	0.201883	−	0.285506i
$688$		0			0
$689$			19.7990i			0.754281i
$690$		0			0
$691$			11.3137i			0.430394i	0.976571	+	0.215197i	$0.0690393\pi$
$691$			11.3137i			0.430394i	−0.976571	+	0.215197i	$0.930961\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	10.5830			0.401436
$696$		0			0
$697$	−24.0000			−0.909065
$698$		0			0
$699$	−8.00000	−	5.65685i	−0.302588	−	0.213962i
$700$		0			0
$701$		−	18.7083i		−	0.706602i	−0.935510	−	0.353301i	$-0.885059\pi$
$701$		−	18.7083i		−	0.706602i	0.935510	−	0.353301i	$-0.114941\pi$
$702$		0			0
$703$		−	29.9333i		−	1.12895i
$704$		0			0
$705$	−56.0000	−	39.5980i	−2.10908	−	1.49135i
$706$		0			0
$707$	−42.0000			−1.57957
$708$		0			0
$709$	−37.0405			−1.39109			−0.695543	−	0.718485i	$-0.744836\pi$
$709$	−37.0405			−1.39109			−0.695543	+	0.718485i	$0.744836\pi$
$710$		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
$717$	10.5830	−	14.9666i	0.395230	−	0.558939i
$718$		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
$723$	20.0000	−	28.2843i	0.743808	−	1.05190i
$724$		0			0
$725$		−	33.6749i		−	1.25066i
$726$		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
$733$	26.4575			0.977231			0.488615	−	0.872499i	$-0.337502\pi$
$733$	26.4575			0.977231			0.488615	+	0.872499i	$0.337502\pi$
$734$		0			0
$735$	−37.0405	−	26.1916i	−1.36626	−	0.966092i
$736$		0			0
$737$		0			0
$738$		0			0
$739$		−	31.1127i		−	1.14450i	−0.820080	−	0.572250i	$-0.806071\pi$
$739$		−	31.1127i		−	1.14450i	0.820080	−	0.572250i	$-0.193929\pi$
$740$		0			0
$741$	42.3320	+	29.9333i	1.55511	+	1.09963i
$742$		0			0
$743$	−21.1660			−0.776506			−0.388253	−	0.921553i	$-0.626921\pi$
$743$	−21.1660			−0.776506			−0.388253	+	0.921553i	$0.626921\pi$
$744$		0			0
$745$	−70.0000			−2.56460
$746$		0			0
$747$	−8.00000	−	22.6274i	−0.292705	−	0.827894i
$748$		0			0
$749$		−	7.48331i		−	0.273434i
$750$		0			0
$751$			3.74166i			0.136535i	0.997667	+	0.0682675i	$0.0217471\pi$
$751$			3.74166i			0.136535i	−0.997667	+	0.0682675i	$0.978253\pi$
$752$		0			0
$753$	24.0000	−	33.9411i	0.874609	−	1.23688i
$754$		0			0
$755$	42.0000			1.52854
$756$		0			0
$757$	−37.0405			−1.34626			−0.673130	−	0.739524i	$-0.735051\pi$
$757$	−37.0405			−1.34626			−0.673130	+	0.739524i	$0.735051\pi$
$758$		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
$763$			19.7990i			0.716772i
$764$		0			0
$765$	−10.5830	−	29.9333i	−0.382629	−	1.08224i
$766$		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
$771$	−16.0000	−	11.3137i	−0.576226	−	0.407453i
$772$		0			0
$773$			18.7083i			0.672890i	0.941703	+	0.336445i	$0.109225\pi$
$773$			18.7083i			0.672890i	−0.941703	+	0.336445i	$0.890775\pi$
$774$		0			0
$775$			33.6749i			1.20964i
$776$		0			0
$777$	−28.0000	−	19.7990i	−1.00449	−	0.710285i
$778$		0			0
$779$	−48.0000			−1.71978
$780$		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
$787$		−	16.9706i		−	0.604935i	−0.953160	−	0.302468i	$-0.902190\pi$
$787$		−	16.9706i		−	0.604935i	0.953160	−	0.302468i	$-0.0978104\pi$
$788$		0			0
$789$	10.5830	−	14.9666i	0.376765	−	0.532826i
$790$		0			0
$791$	−42.3320			−1.50515
$792$		0			0
$793$	−28.0000			−0.994309
$794$		0			0
$795$	14.0000	−	19.7990i	0.496529	−	0.702198i
$796$		0			0
$797$			41.1582i			1.45790i	0.684567	+	0.728950i	$0.259991\pi$
$797$			41.1582i			1.45790i	−0.684567	+	0.728950i	$0.740009\pi$
$798$		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
$811$			16.9706i			0.595917i	0.954579	+	0.297959i	$0.0963057\pi$
$811$			16.9706i			0.595917i	−0.954579	+	0.297959i	$0.903694\pi$
$812$		0			0
$813$	−15.8745	−	11.2250i	−0.556743	−	0.393677i
$814$		0			0
$815$	−63.4980			−2.22424
$816$		0			0
$817$		0			0
$818$		0			0
$819$	56.0000	−	19.7990i	1.95680	−	0.691833i
$820$		0			0
$821$			3.74166i			0.130585i	0.997866	+	0.0652924i	$0.0207980\pi$
$821$			3.74166i			0.130585i	−0.997866	+	0.0652924i	$0.979202\pi$
$822$		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
$827$	18.0000			0.625921			0.312961	−	0.949766i	$-0.398679\pi$
$827$	18.0000			0.625921			0.312961	+	0.949766i	$0.398679\pi$
$828$		0			0
$829$	37.0405			1.28647			0.643235	−	0.765669i	$-0.277592\pi$
$829$	37.0405			1.28647			0.643235	+	0.765669i	$0.277592\pi$
$830$		0			0
$831$	−15.8745	+	22.4499i	−0.550681	+	0.778780i
$832$		0			0
$833$			19.7990i			0.685994i
$834$		0			0
$835$		−	79.1960i		−	2.74069i
$836$		0			0
$837$	−5.29150	+	18.7083i	−0.182901	+	0.646653i
$838$		0			0
$839$	31.7490			1.09610			0.548049	−	0.836446i	$-0.315371\pi$
$839$	31.7490			1.09610			0.548049	+	0.836446i	$0.315371\pi$
$840$		0			0
$841$	15.0000			0.517241
$842$		0			0
$843$	40.0000	+	28.2843i	1.37767	+	0.974162i
$844$		0			0
$845$			56.1249i			1.93075i
$846$		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
$853$	−47.6235			−1.63060			−0.815299	−	0.579040i	$-0.803428\pi$
$853$	−47.6235			−1.63060			−0.815299	+	0.579040i	$0.803428\pi$
$854$		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
$859$			45.2548i			1.54408i	0.635577	+	0.772038i	$0.280762\pi$
$859$			45.2548i			1.54408i	−0.635577	+	0.772038i	$0.719238\pi$
$860$		0			0
$861$	−31.7490	+	44.8999i	−1.08200	+	1.53018i
$862$		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
$867$	9.00000	−	12.7279i	0.305656	−	0.432263i
$868$		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
$875$	−56.0000			−1.89315
$876$		0			0
$877$	26.4575			0.893407			0.446703	−	0.894682i	$-0.352598\pi$
$877$	26.4575			0.893407			0.446703	+	0.894682i	$0.352598\pi$
$878$		0			0
$879$	37.0405	+	26.1916i	1.24935	+	0.883421i
$880$		0			0
$881$			16.9706i			0.571753i	0.958267	+	0.285876i	$0.0922847\pi$
$881$			16.9706i			0.571753i	−0.958267	+	0.285876i	$0.907715\pi$
$882$		0			0
$883$			11.3137i			0.380737i	0.981713	+	0.190368i	$0.0609682\pi$
$883$			11.3137i			0.380737i	−0.981713	+	0.190368i	$0.939032\pi$
$884$		0			0
$885$	−31.7490	−	22.4499i	−1.06723	−	0.754647i
$886$		0			0
$887$	31.7490			1.06603			0.533014	−	0.846107i	$-0.321059\pi$
$887$	31.7490			1.06603			0.533014	+	0.846107i	$0.321059\pi$
$888$		0			0
$889$	42.0000			1.40863
$890$		0			0
$891$		0			0
$892$		0			0
$893$			59.8665i			2.00336i
$894$		0			0
$895$			82.3165i			2.75154i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	14.0000			0.466926
$900$		0			0
$901$	−10.5830			−0.352571
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	59.3970i		−	1.97442i
$906$		0			0
$907$		−	16.9706i		−	0.563498i	−0.959488	−	0.281749i	$-0.909085\pi$
$907$		−	16.9706i		−	0.563498i	0.959488	−	0.281749i	$-0.0909146\pi$
$908$		0			0
$909$	−31.7490	+	11.2250i	−1.05305	+	0.372309i
$910$		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
$915$	28.0000	+	19.7990i	0.925651	+	0.654534i
$916$		0			0
$917$		−	7.48331i		−	0.247121i
$918$		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
$923$	−56.0000			−1.84326
$924$		0			0
$925$	−47.6235			−1.56585
$926$		0			0
$927$	10.5830	−	3.74166i	0.347591	−	0.122892i
$928$		0			0
$929$		−	53.7401i		−	1.76316i	−0.472038	−	0.881578i	$-0.656482\pi$
$929$		−	53.7401i		−	1.76316i	0.472038	−	0.881578i	$-0.343518\pi$
$930$		0			0
$931$			39.5980i			1.29777i
$932$		0			0
$933$	31.7490	−	44.8999i	1.03942	−	1.46996i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−46.0000			−1.50275			−0.751377	−	0.659873i	$-0.770610\pi$
$937$	−46.0000			−1.50275			−0.751377	+	0.659873i	$0.770610\pi$
$938$		0			0
$939$	4.00000	−	5.65685i	0.130535	−	0.184604i
$940$		0			0
$941$		−	41.1582i		−	1.34172i	−0.741584	−	0.670860i	$-0.765925\pi$
$941$		−	41.1582i		−	1.34172i	0.741584	−	0.670860i	$-0.234075\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$	−70.0000	−	19.7990i	−2.27710	−	0.644061i
$946$		0			0
$947$	−34.0000			−1.10485			−0.552426	−	0.833562i	$-0.686298\pi$
$947$	−34.0000			−1.10485			−0.552426	+	0.833562i	$0.686298\pi$
$948$		0			0
$949$	−42.3320			−1.37416
$950$		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
$955$		−	39.5980i		−	1.28136i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	10.5830			0.341743
$960$		0			0
$961$	17.0000			0.548387
$962$		0			0
$963$	−2.00000	−	5.65685i	−0.0644491	−	0.182290i
$964$		0			0
$965$			74.8331i			2.40896i
$966$		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
$971$	−16.0000			−0.513464			−0.256732	−	0.966483i	$-0.582646\pi$
$971$	−16.0000			−0.513464			−0.256732	+	0.966483i	$0.582646\pi$
$972$		0			0
$973$	−10.5830			−0.339276
$974$		0			0
$975$	47.6235	−	67.3498i	1.52517	−	2.15692i
$976$		0			0
$977$		−	42.4264i		−	1.35734i	−0.734443	−	0.678671i	$-0.762556\pi$
$977$		−	42.4264i		−	1.35734i	0.734443	−	0.678671i	$-0.237444\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	5.29150	+	14.9666i	0.168945	+	0.477848i
$982$		0			0
$983$	−31.7490			−1.01264			−0.506318	−	0.862347i	$-0.668994\pi$
$983$	−31.7490			−1.01264			−0.506318	+	0.862347i	$0.668994\pi$
$984$		0			0
$985$	14.0000			0.446077
$986$		0			0
$987$	56.0000	+	39.5980i	1.78250	+	1.26042i
$988$		0			0
$989$		0			0
$990$		0			0
$991$			3.74166i			0.118858i	0.998233	+	0.0594288i	$0.0189279\pi$
$991$			3.74166i			0.118858i	−0.998233	+	0.0594288i	$0.981072\pi$
$992$		0			0
$993$	−28.0000	−	19.7990i	−0.888553	−	0.628302i
$994$		0			0
$995$	−70.0000			−2.21915
$996$		0			0
$997$	26.4575			0.837918			0.418959	−	0.908005i	$-0.362395\pi$
$997$	26.4575			0.837918			0.418959	+	0.908005i	$0.362395\pi$
$998$		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.c.i.1535.2	yes	4
3.2	odd	2		1536.2.c.e.1535.1	✓	4
4.3	odd	2		1536.2.c.e.1535.4	yes	4
8.3	odd	2	inner	1536.2.c.i.1535.1	yes	4
8.5	even	2		1536.2.c.e.1535.3	yes	4
12.11	even	2	inner	1536.2.c.i.1535.3	yes	4
16.3	odd	4		1536.2.f.k.767.2		8
16.5	even	4		1536.2.f.k.767.1		8
16.11	odd	4		1536.2.f.k.767.7		8
16.13	even	4		1536.2.f.k.767.8		8
24.5	odd	2	inner	1536.2.c.i.1535.4	yes	4
24.11	even	2		1536.2.c.e.1535.2	yes	4
48.5	odd	4		1536.2.f.k.767.4		8
48.11	even	4		1536.2.f.k.767.6		8
48.29	odd	4		1536.2.f.k.767.5		8
48.35	even	4		1536.2.f.k.767.3		8

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

\(f(q)\)	\(=\)	\(q+(1.00000 - 1.41421i) q^{3} +3.74166i q^{5} -3.74166i q^{7} +(-1.00000 - 2.82843i) q^{9} +O(q^{10})\)	\(q+(1.00000 - 1.41421i) q^{3} +3.74166i q^{5} -3.74166i q^{7} +(-1.00000 - 2.82843i) q^{9} -5.29150 q^{13} +(5.29150 + 3.74166i) q^{15} -2.82843i q^{17} -5.65685i q^{19} +(-5.29150 - 3.74166i) q^{21} -9.00000 q^{25} +(-5.00000 - 1.41421i) q^{27} +3.74166i q^{29} -3.74166i q^{31} +14.0000 q^{35} +5.29150 q^{37} +(-5.29150 + 7.48331i) q^{39} -8.48528i q^{41} +(10.5830 - 3.74166i) q^{45} -10.5830 q^{47} -7.00000 q^{49} +(-4.00000 - 2.82843i) q^{51} -3.74166i q^{53} +(-8.00000 - 5.65685i) q^{57} -6.00000 q^{59} +5.29150 q^{61} +(-10.5830 + 3.74166i) q^{63} -19.7990i q^{65} -2.82843i q^{67} +10.5830 q^{71} +8.00000 q^{73} +(-9.00000 + 12.7279i) q^{75} -11.2250i q^{79} +(-7.00000 + 5.65685i) q^{81} +8.00000 q^{83} +10.5830 q^{85} +(5.29150 + 3.74166i) q^{87} +11.3137i q^{89} +19.7990i q^{91} +(-5.29150 - 3.74166i) q^{93} +21.1660 q^{95} -6.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 4 * q^9	\( 4 q + 4 q^{3} - 4 q^{9} - 36 q^{25} - 20 q^{27} + 56 q^{35} - 28 q^{49} - 16 q^{51} - 32 q^{57} - 24 q^{59} + 32 q^{73} - 36 q^{75} - 28 q^{81} + 32 q^{83} - 24 q^{97}+O(q^{100}) \) 4 * q + 4 * q^3 - 4 * q^9 - 36 * q^25 - 20 * q^27 + 56 * q^35 - 28 * q^49 - 16 * q^51 - 32 * q^57 - 24 * q^59 + 32 * q^73 - 36 * q^75 - 28 * q^81 + 32 * q^83 - 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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