LMFDB - Embedded newform 24.2.f.a.11.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [24,2,Mod(11,24)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("24.11"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(24, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 24 = 2^{3} \cdot 3 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	24.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$0.191640964851$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-2}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 2 $ x^2 + 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			11.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	24.11
Dual form			24.2.f.a.11.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.41421i q^{2} +(-1.00000 - 1.41421i) q^{3} -2.00000 q^{4} +(2.00000 - 1.41421i) q^{6} -2.82843i q^{8} +(-1.00000 + 2.82843i) q^{9} +2.82843i q^{11} +(2.00000 + 2.82843i) q^{12} +4.00000 q^{16} -5.65685i q^{17} +(-4.00000 - 1.41421i) q^{18} +2.00000 q^{19} -4.00000 q^{22} +(-4.00000 + 2.82843i) q^{24} -5.00000 q^{25} +(5.00000 - 1.41421i) q^{27} +5.65685i q^{32} +(4.00000 - 2.82843i) q^{33} +8.00000 q^{34} +(2.00000 - 5.65685i) q^{36} +2.82843i q^{38} +11.3137i q^{41} -10.0000 q^{43} -5.65685i q^{44} +(-4.00000 - 5.65685i) q^{48} +7.00000 q^{49} -7.07107i q^{50} +(-8.00000 + 5.65685i) q^{51} +(2.00000 + 7.07107i) q^{54} +(-2.00000 - 2.82843i) q^{57} -14.1421i q^{59} -8.00000 q^{64} +(4.00000 + 5.65685i) q^{66} +14.0000 q^{67} +11.3137i q^{68} +(8.00000 + 2.82843i) q^{72} +2.00000 q^{73} +(5.00000 + 7.07107i) q^{75} -4.00000 q^{76} +(-7.00000 - 5.65685i) q^{81} -16.0000 q^{82} +2.82843i q^{83} -14.1421i q^{86} +8.00000 q^{88} -5.65685i q^{89} +(8.00000 - 5.65685i) q^{96} -10.0000 q^{97} +9.89949i q^{98} +(-8.00000 - 2.82843i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 4 q^{4} + 4 q^{6} - 2 q^{9} + 4 q^{12} + 8 q^{16} - 8 q^{18} + 4 q^{19} - 8 q^{22} - 8 q^{24} - 10 q^{25} + 10 q^{27} + 8 q^{33} + 16 q^{34} + 4 q^{36} - 20 q^{43} - 8 q^{48} + 14 q^{49}+ \cdots - 16 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 4 * q^4 + 4 * q^6 - 2 * q^9 + 4 * q^12 + 8 * q^16 - 8 * q^18 + 4 * q^19 - 8 * q^22 - 8 * q^24 - 10 * q^25 + 10 * q^27 + 8 * q^33 + 16 * q^34 + 4 * q^36 - 20 * q^43 - 8 * q^48 + 14 * q^49 - 16 * q^51 + 4 * q^54 - 4 * q^57 - 16 * q^64 + 8 * q^66 + 28 * q^67 + 16 * q^72 + 4 * q^73 + 10 * q^75 - 8 * q^76 - 14 * q^81 - 32 * q^82 + 16 * q^88 + 16 * q^96 - 20 * q^97 - 16 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/24\mathbb{Z}\right)^\times$.

$n$	$7$	$13$	$17$
$\chi(n)$	$-1$	$-1$	$-1$

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 $

$2$			1.41421i			1.00000i
$2$			1.41421i			1.00000i
$3$	−1.00000	−	1.41421i	−0.577350	−	0.816497i
$3$	−1.00000	−	1.41421i	−0.577350	−	0.816497i
$4$	−2.00000			−1.00000
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$	2.00000	−	1.41421i	0.816497	−	0.577350i
$7$		0			0		1.00000			$0$
$7$		0			0		−1.00000			$\pi$
$8$		−	2.82843i		−	1.00000i
$9$	−1.00000	+	2.82843i	−0.333333	+	0.942809i
$10$		0			0
$11$			2.82843i			0.852803i	0.904534	+	0.426401i	$0.140219\pi$
$11$			2.82843i			0.852803i	−0.904534	+	0.426401i	$0.859781\pi$
$12$	2.00000	+	2.82843i	0.577350	+	0.816497i
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$	4.00000			1.00000
$17$		−	5.65685i		−	1.37199i	−0.727607	−	0.685994i	$-0.759367\pi$
$17$		−	5.65685i		−	1.37199i	0.727607	−	0.685994i	$-0.240633\pi$
$18$	−4.00000	−	1.41421i	−0.942809	−	0.333333i
$19$	2.00000			0.458831			0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000			0.458831			0.229416	+	0.973329i	$0.426318\pi$
$20$		0			0
$21$		0			0
$22$	−4.00000			−0.852803
$23$		0			0			−	1.00000i	$-0.5\pi$
$23$		0			0				1.00000i	$0.5\pi$
$24$	−4.00000	+	2.82843i	−0.816497	+	0.577350i
$25$	−5.00000			−1.00000
$26$		0			0
$27$	5.00000	−	1.41421i	0.962250	−	0.272166i
$28$		0			0
$29$		0			0			−	1.00000i	$-0.5\pi$
$29$		0			0				1.00000i	$0.5\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$			5.65685i			1.00000i
$33$	4.00000	−	2.82843i	0.696311	−	0.492366i
$34$	8.00000			1.37199
$35$		0			0
$36$	2.00000	−	5.65685i	0.333333	−	0.942809i
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$			2.82843i			0.458831i
$39$		0			0
$40$		0			0
$41$			11.3137i			1.76690i	0.468521	+	0.883452i	$0.344787\pi$
$41$			11.3137i			1.76690i	−0.468521	+	0.883452i	$0.655213\pi$
$42$		0			0
$43$	−10.0000			−1.52499			−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000			−1.52499			−0.762493	+	0.646997i	$0.776025\pi$
$44$		−	5.65685i		−	0.852803i
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$	−4.00000	−	5.65685i	−0.577350	−	0.816497i
$49$	7.00000			1.00000
$50$		−	7.07107i		−	1.00000i
$51$	−8.00000	+	5.65685i	−1.12022	+	0.792118i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$	2.00000	+	7.07107i	0.272166	+	0.962250i
$55$		0			0
$56$		0			0
$57$	−2.00000	−	2.82843i	−0.264906	−	0.374634i
$58$		0			0
$59$		−	14.1421i		−	1.84115i	−0.390567	−	0.920575i	$-0.627721\pi$
$59$		−	14.1421i		−	1.84115i	0.390567	−	0.920575i	$-0.372279\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$		0			0
$64$	−8.00000			−1.00000
$65$		0			0
$66$	4.00000	+	5.65685i	0.492366	+	0.696311i
$67$	14.0000			1.71037			0.855186	−	0.518321i	$-0.173443\pi$
$67$	14.0000			1.71037			0.855186	+	0.518321i	$0.173443\pi$
$68$			11.3137i			1.37199i
$69$		0			0
$70$		0			0
$71$		0			0			−	1.00000i	$-0.5\pi$
$71$		0			0				1.00000i	$0.5\pi$
$72$	8.00000	+	2.82843i	0.942809	+	0.333333i
$73$	2.00000			0.234082			0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000			0.234082			0.117041	+	0.993127i	$0.462659\pi$
$74$		0			0
$75$	5.00000	+	7.07107i	0.577350	+	0.816497i
$76$	−4.00000			−0.458831
$77$		0			0
$78$		0			0
$79$		0			0		1.00000			$0$
$79$		0			0		−1.00000			$\pi$
$80$		0			0
$81$	−7.00000	−	5.65685i	−0.777778	−	0.628539i
$82$	−16.0000			−1.76690
$83$			2.82843i			0.310460i	0.987878	+	0.155230i	$0.0496119\pi$
$83$			2.82843i			0.310460i	−0.987878	+	0.155230i	$0.950388\pi$
$84$		0			0
$85$		0			0
$86$		−	14.1421i		−	1.52499i
$87$		0			0
$88$	8.00000			0.852803
$89$		−	5.65685i		−	0.599625i	−0.953998	−	0.299813i	$-0.903076\pi$
$89$		−	5.65685i		−	0.599625i	0.953998	−	0.299813i	$-0.0969242\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$	8.00000	−	5.65685i	0.816497	−	0.577350i
$97$	−10.0000			−1.01535			−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000			−1.01535			−0.507673	+	0.861550i	$0.669494\pi$
$98$			9.89949i			1.00000i
$99$	−8.00000	−	2.82843i	−0.804030	−	0.284268i
$100$	10.0000			1.00000
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$	−8.00000	−	11.3137i	−0.792118	−	1.12022i
$103$		0			0		1.00000			$0$
$103$		0			0		−1.00000			$\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			19.7990i			1.91404i	0.290021	+	0.957020i	$0.406338\pi$
$107$			19.7990i			1.91404i	−0.290021	+	0.957020i	$0.593662\pi$
$108$	−10.0000	+	2.82843i	−0.962250	+	0.272166i
$109$		0			0		1.00000			$0$
$109$		0			0		−1.00000			$\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$			11.3137i			1.06430i	0.846649	+	0.532152i	$0.178617\pi$
$113$			11.3137i			1.06430i	−0.846649	+	0.532152i	$0.821383\pi$
$114$	4.00000	−	2.82843i	0.374634	−	0.264906i
$115$		0			0
$116$		0			0
$117$		0			0
$118$	20.0000			1.84115
$119$		0			0
$120$		0			0
$121$	3.00000			0.272727
$122$		0			0
$123$	16.0000	−	11.3137i	1.44267	−	1.02012i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		−	11.3137i		−	1.00000i
$129$	10.0000	+	14.1421i	0.880451	+	1.24515i
$130$		0			0
$131$		−	14.1421i		−	1.23560i	−0.786334	−	0.617802i	$-0.788023\pi$
$131$		−	14.1421i		−	1.23560i	0.786334	−	0.617802i	$-0.211977\pi$
$132$	−8.00000	+	5.65685i	−0.696311	+	0.492366i
$133$		0			0
$134$			19.7990i			1.71037i
$135$		0			0
$136$	−16.0000			−1.37199
$137$		−	22.6274i		−	1.93319i	−0.256307	−	0.966595i	$-0.582506\pi$
$137$		−	22.6274i		−	1.93319i	0.256307	−	0.966595i	$-0.417494\pi$
$138$		0			0
$139$	−22.0000			−1.86602			−0.933008	−	0.359856i	$-0.882826\pi$
$139$	−22.0000			−1.86602			−0.933008	+	0.359856i	$0.882826\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−4.00000	+	11.3137i	−0.333333	+	0.942809i
$145$		0			0
$146$			2.82843i			0.234082i
$147$	−7.00000	−	9.89949i	−0.577350	−	0.816497i
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$	−10.0000	+	7.07107i	−0.816497	+	0.577350i
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		−	5.65685i		−	0.458831i
$153$	16.0000	+	5.65685i	1.29352	+	0.457330i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		1.00000			$0$
$157$		0			0		−1.00000			$\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$	8.00000	−	9.89949i	0.628539	−	0.777778i
$163$	2.00000			0.156652			0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000			0.156652			0.0783260	+	0.996928i	$0.475042\pi$
$164$		−	22.6274i		−	1.76690i
$165$		0			0
$166$	−4.00000			−0.310460
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$	−2.00000	+	5.65685i	−0.152944	+	0.432590i
$172$	20.0000			1.52499
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		0			0
$176$			11.3137i			0.852803i
$177$	−20.0000	+	14.1421i	−1.50329	+	1.06299i
$178$	8.00000			0.599625
$179$			19.7990i			1.47985i	0.672692	+	0.739923i	$0.265138\pi$
$179$			19.7990i			1.47985i	−0.672692	+	0.739923i	$0.734862\pi$
$180$		0			0
$181$		0			0		1.00000			$0$
$181$		0			0		−1.00000			$\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	16.0000			1.17004
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$	8.00000	+	11.3137i	0.577350	+	0.816497i
$193$	−22.0000			−1.58359			−0.791797	−	0.610784i	$-0.790854\pi$
$193$	−22.0000			−1.58359			−0.791797	+	0.610784i	$0.790854\pi$
$194$		−	14.1421i		−	1.01535i
$195$		0			0
$196$	−14.0000			−1.00000
$197$		0			0			−	1.00000i	$-0.5\pi$
$197$		0			0				1.00000i	$0.5\pi$
$198$	4.00000	−	11.3137i	0.284268	−	0.804030i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$			14.1421i			1.00000i
$201$	−14.0000	−	19.7990i	−0.987484	−	1.39651i
$202$		0			0
$203$		0			0
$204$	16.0000	−	11.3137i	1.12022	−	0.792118i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$			5.65685i			0.391293i
$210$		0			0
$211$	14.0000			0.963800			0.481900	−	0.876226i	$-0.339947\pi$
$211$	14.0000			0.963800			0.481900	+	0.876226i	$0.339947\pi$
$212$		0			0
$213$		0			0
$214$	−28.0000			−1.91404
$215$		0			0
$216$	−4.00000	−	14.1421i	−0.272166	−	0.962250i
$217$		0			0
$218$		0			0
$219$	−2.00000	−	2.82843i	−0.135147	−	0.191127i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	5.00000	−	14.1421i	0.333333	−	0.942809i
$226$	−16.0000			−1.06430
$227$			2.82843i			0.187729i	0.995585	+	0.0938647i	$0.0299221\pi$
$227$			2.82843i			0.187729i	−0.995585	+	0.0938647i	$0.970078\pi$
$228$	4.00000	+	5.65685i	0.264906	+	0.374634i
$229$		0			0		1.00000			$0$
$229$		0			0		−1.00000			$\pi$
$230$		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
$235$		0			0
$236$			28.2843i			1.84115i
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	26.0000			1.67481			0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000			1.67481			0.837404	+	0.546585i	$0.184072\pi$
$242$			4.24264i			0.272727i
$243$	−1.00000	+	15.5563i	−0.0641500	+	0.997940i
$244$		0			0
$245$		0			0
$246$	16.0000	+	22.6274i	1.02012	+	1.44267i
$247$		0			0
$248$		0			0
$249$	4.00000	−	2.82843i	0.253490	−	0.179244i
$250$		0			0
$251$		−	31.1127i		−	1.96382i	−0.189358	−	0.981908i	$-0.560641\pi$
$251$		−	31.1127i		−	1.96382i	0.189358	−	0.981908i	$-0.439359\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$			11.3137i			0.705730i	0.935674	+	0.352865i	$0.114792\pi$
$257$			11.3137i			0.705730i	−0.935674	+	0.352865i	$0.885208\pi$
$258$	−20.0000	+	14.1421i	−1.24515	+	0.880451i
$259$		0			0
$260$		0			0
$261$		0			0
$262$	20.0000			1.23560
$263$		0			0			−	1.00000i	$-0.5\pi$
$263$		0			0				1.00000i	$0.5\pi$
$264$	−8.00000	−	11.3137i	−0.492366	−	0.696311i
$265$		0			0
$266$		0			0
$267$	−8.00000	+	5.65685i	−0.489592	+	0.346194i
$268$	−28.0000			−1.71037
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		1.00000			$0$
$271$		0			0		−1.00000			$\pi$
$272$		−	22.6274i		−	1.37199i
$273$		0			0
$274$	32.0000			1.93319
$275$		−	14.1421i		−	0.852803i
$276$		0			0
$277$		0			0		1.00000			$0$
$277$		0			0		−1.00000			$\pi$
$278$		−	31.1127i		−	1.86602i
$279$		0			0
$280$		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
$283$	−22.0000			−1.30776			−0.653882	−	0.756596i	$-0.726861\pi$
$283$	−22.0000			−1.30776			−0.653882	+	0.756596i	$0.726861\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$	−16.0000	−	5.65685i	−0.942809	−	0.333333i
$289$	−15.0000			−0.882353
$290$		0			0
$291$	10.0000	+	14.1421i	0.586210	+	0.829027i
$292$	−4.00000			−0.234082
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$	14.0000	−	9.89949i	0.816497	−	0.577350i
$295$		0			0
$296$		0			0
$297$	4.00000	+	14.1421i	0.232104	+	0.820610i
$298$		0			0
$299$		0			0
$300$	−10.0000	−	14.1421i	−0.577350	−	0.816497i
$301$		0			0
$302$		0			0
$303$		0			0
$304$	8.00000			0.458831
$305$		0			0
$306$	−8.00000	+	22.6274i	−0.457330	+	1.29352i
$307$	−34.0000			−1.94048			−0.970241	−	0.242140i	$-0.922151\pi$
$307$	−34.0000			−1.94048			−0.970241	+	0.242140i	$0.922151\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$	−10.0000			−0.565233			−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000			−0.565233			−0.282617	+	0.959233i	$0.591202\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0			−	1.00000i	$-0.5\pi$
$317$		0			0				1.00000i	$0.5\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	28.0000	−	19.7990i	1.56281	−	1.10507i
$322$		0			0
$323$		−	11.3137i		−	0.629512i
$324$	14.0000	+	11.3137i	0.777778	+	0.628539i
$325$		0			0
$326$			2.82843i			0.156652i
$327$		0			0
$328$	32.0000			1.76690
$329$		0			0
$330$		0			0
$331$	26.0000			1.42909			0.714545	−	0.699590i	$-0.246634\pi$
$331$	26.0000			1.42909			0.714545	+	0.699590i	$0.246634\pi$
$332$		−	5.65685i		−	0.310460i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	14.0000			0.762629			0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000			0.762629			0.381314	+	0.924445i	$0.375472\pi$
$338$			18.3848i			1.00000i
$339$	16.0000	−	11.3137i	0.869001	−	0.614476i
$340$		0			0
$341$		0			0
$342$	−8.00000	−	2.82843i	−0.432590	−	0.152944i
$343$		0			0
$344$			28.2843i			1.52499i
$345$		0			0
$346$		0			0
$347$			36.7696i			1.97389i	0.161048	+	0.986947i	$0.448512\pi$
$347$			36.7696i			1.97389i	−0.161048	+	0.986947i	$0.551488\pi$
$348$		0			0
$349$		0			0		1.00000			$0$
$349$		0			0		−1.00000			$\pi$
$350$		0			0
$351$		0			0
$352$	−16.0000			−0.852803
$353$		−	22.6274i		−	1.20434i	−0.798369	−	0.602168i	$-0.794304\pi$
$353$		−	22.6274i		−	1.20434i	0.798369	−	0.602168i	$-0.205696\pi$
$354$	−20.0000	−	28.2843i	−1.06299	−	1.50329i
$355$		0			0
$356$			11.3137i			0.599625i
$357$		0			0
$358$	−28.0000			−1.47985
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$	−3.00000	−	4.24264i	−0.157459	−	0.222681i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$	−32.0000	−	11.3137i	−1.66585	−	0.588968i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$			22.6274i			1.17004i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	38.0000			1.95193			0.975964	−	0.217930i	$-0.0699304\pi$
$379$	38.0000			1.95193			0.975964	+	0.217930i	$0.0699304\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$	−16.0000	+	11.3137i	−0.816497	+	0.577350i
$385$		0			0
$386$		−	31.1127i		−	1.58359i
$387$	10.0000	−	28.2843i	0.508329	−	1.43777i
$388$	20.0000			1.01535
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		−	19.7990i		−	1.00000i
$393$	−20.0000	+	14.1421i	−1.00887	+	0.713376i
$394$		0			0
$395$		0			0
$396$	16.0000	+	5.65685i	0.804030	+	0.284268i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	−20.0000			−1.00000
$401$		−	39.5980i		−	1.97743i	−0.149813	−	0.988714i	$-0.547867\pi$
$401$		−	39.5980i		−	1.97743i	0.149813	−	0.988714i	$-0.452133\pi$
$402$	28.0000	−	19.7990i	1.39651	−	0.987484i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	16.0000	+	22.6274i	0.792118	+	1.12022i
$409$	−22.0000			−1.08783			−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000			−1.08783			−0.543915	+	0.839140i	$0.683059\pi$
$410$		0			0
$411$	−32.0000	+	22.6274i	−1.57844	+	1.11613i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	22.0000	+	31.1127i	1.07734	+	1.52360i
$418$	−8.00000			−0.391293
$419$			36.7696i			1.79631i	0.439679	+	0.898155i	$0.355092\pi$
$419$			36.7696i			1.79631i	−0.439679	+	0.898155i	$0.644908\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$			19.7990i			0.963800i
$423$		0			0
$424$		0			0
$425$			28.2843i			1.37199i
$426$		0			0
$427$		0			0
$428$		−	39.5980i		−	1.91404i
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$	20.0000	−	5.65685i	0.962250	−	0.272166i
$433$	38.0000			1.82616			0.913082	−	0.407777i	$-0.133696\pi$
$433$	38.0000			1.82616			0.913082	+	0.407777i	$0.133696\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$	4.00000	−	2.82843i	0.191127	−	0.135147i
$439$		0			0		1.00000			$0$
$439$		0			0		−1.00000			$\pi$
$440$		0			0
$441$	−7.00000	+	19.7990i	−0.333333	+	0.942809i
$442$		0			0
$443$			2.82843i			0.134383i	0.997740	+	0.0671913i	$0.0214038\pi$
$443$			2.82843i			0.134383i	−0.997740	+	0.0671913i	$0.978596\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	5.65685i		−	0.266963i	−0.991051	−	0.133482i	$-0.957384\pi$
$449$		−	5.65685i		−	0.266963i	0.991051	−	0.133482i	$-0.0426157\pi$
$450$	20.0000	+	7.07107i	0.942809	+	0.333333i
$451$	−32.0000			−1.50682
$452$		−	22.6274i		−	1.06430i
$453$		0			0
$454$	−4.00000			−0.187729
$455$		0			0
$456$	−8.00000	+	5.65685i	−0.374634	+	0.264906i
$457$	26.0000			1.21623			0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000			1.21623			0.608114	+	0.793849i	$0.291926\pi$
$458$		0			0
$459$	−8.00000	−	28.2843i	−0.373408	−	1.32020i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0		1.00000			$0$
$463$		0			0		−1.00000			$\pi$
$464$		0			0
$465$		0			0
$466$	8.00000			0.370593
$467$		−	31.1127i		−	1.43972i	−0.694117	−	0.719862i	$-0.744205\pi$
$467$		−	31.1127i		−	1.43972i	0.694117	−	0.719862i	$-0.255795\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$	−40.0000			−1.84115
$473$		−	28.2843i		−	1.30051i
$474$		0			0
$475$	−10.0000			−0.458831
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$		0			0
$482$			36.7696i			1.67481i
$483$		0			0
$484$	−6.00000			−0.272727
$485$		0			0
$486$	−22.0000	−	1.41421i	−0.997940	−	0.0641500i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	−2.00000	−	2.82843i	−0.0904431	−	0.127906i
$490$		0			0
$491$		−	14.1421i		−	0.638226i	−0.947717	−	0.319113i	$-0.896615\pi$
$491$		−	14.1421i		−	0.638226i	0.947717	−	0.319113i	$-0.103385\pi$
$492$	−32.0000	+	22.6274i	−1.44267	+	1.02012i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	4.00000	+	5.65685i	0.179244	+	0.253490i
$499$	14.0000			0.626726			0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000			0.626726			0.313363	+	0.949633i	$0.398544\pi$
$500$		0			0
$501$		0			0
$502$	44.0000			1.96382
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−13.0000	−	18.3848i	−0.577350	−	0.816497i
$508$		0			0
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$		0			0
$512$			22.6274i			1.00000i
$513$	10.0000	−	2.82843i	0.441511	−	0.124878i
$514$	−16.0000			−0.705730
$515$		0			0
$516$	−20.0000	−	28.2843i	−0.880451	−	1.24515i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$			45.2548i			1.98265i	0.131432	+	0.991325i	$0.458042\pi$
$521$			45.2548i			1.98265i	−0.131432	+	0.991325i	$0.541958\pi$
$522$		0			0
$523$	38.0000			1.66162			0.830812	−	0.556553i	$-0.187876\pi$
$523$	38.0000			1.66162			0.830812	+	0.556553i	$0.187876\pi$
$524$			28.2843i			1.23560i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	16.0000	−	11.3137i	0.696311	−	0.492366i
$529$	−23.0000			−1.00000
$530$		0			0
$531$	40.0000	+	14.1421i	1.73585	+	0.613716i
$532$		0			0
$533$		0			0
$534$	−8.00000	−	11.3137i	−0.346194	−	0.489592i
$535$		0			0
$536$		−	39.5980i		−	1.71037i
$537$	28.0000	−	19.7990i	1.20829	−	0.854389i
$538$		0			0
$539$			19.7990i			0.852803i
$540$		0			0
$541$		0			0		1.00000			$0$
$541$		0			0		−1.00000			$\pi$
$542$		0			0
$543$		0			0
$544$	32.0000			1.37199
$545$		0			0
$546$		0			0
$547$	−46.0000			−1.96682			−0.983409	−	0.181402i	$-0.941936\pi$
$547$	−46.0000			−1.96682			−0.983409	+	0.181402i	$0.941936\pi$
$548$			45.2548i			1.93319i
$549$		0			0
$550$	20.0000			0.852803
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$	44.0000			1.86602
$557$		0			0			−	1.00000i	$-0.5\pi$
$557$		0			0				1.00000i	$0.5\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−16.0000	−	22.6274i	−0.675521	−	0.955330i
$562$	−40.0000			−1.68730
$563$			36.7696i			1.54965i	0.632175	+	0.774826i	$0.282163\pi$
$563$			36.7696i			1.54965i	−0.632175	+	0.774826i	$0.717837\pi$
$564$		0			0
$565$		0			0
$566$		−	31.1127i		−	1.30776i
$567$		0			0
$568$		0			0
$569$		−	22.6274i		−	0.948591i	−0.880366	−	0.474295i	$-0.842703\pi$
$569$		−	22.6274i		−	0.948591i	0.880366	−	0.474295i	$-0.157297\pi$
$570$		0			0
$571$	−22.0000			−0.920671			−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000			−0.920671			−0.460336	+	0.887745i	$0.652271\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	8.00000	−	22.6274i	0.333333	−	0.942809i
$577$	−34.0000			−1.41544			−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000			−1.41544			−0.707719	+	0.706494i	$0.750276\pi$
$578$		−	21.2132i		−	0.882353i
$579$	22.0000	+	31.1127i	0.914289	+	1.29300i
$580$		0			0
$581$		0			0
$582$	−20.0000	+	14.1421i	−0.829027	+	0.586210i
$583$		0			0
$584$		−	5.65685i		−	0.234082i
$585$		0			0
$586$		0			0
$587$		−	48.0833i		−	1.98461i	−0.123823	−	0.992304i	$-0.539516\pi$
$587$		−	48.0833i		−	1.98461i	0.123823	−	0.992304i	$-0.460484\pi$
$588$	14.0000	+	19.7990i	0.577350	+	0.816497i
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			45.2548i			1.85839i	0.369586	+	0.929197i	$0.379500\pi$
$593$			45.2548i			1.85839i	−0.369586	+	0.929197i	$0.620500\pi$
$594$	−20.0000	+	5.65685i	−0.820610	+	0.232104i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$	20.0000	−	14.1421i	0.816497	−	0.577350i
$601$	−46.0000			−1.87638			−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000			−1.87638			−0.938190	+	0.346122i	$0.887498\pi$
$602$		0			0
$603$	−14.0000	+	39.5980i	−0.570124	+	1.61255i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		1.00000			$0$
$607$		0			0		−1.00000			$\pi$
$608$			11.3137i			0.458831i
$609$		0			0
$610$		0			0
$611$		0			0
$612$	−32.0000	−	11.3137i	−1.29352	−	0.457330i
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		−	48.0833i		−	1.94048i
$615$		0			0
$616$		0			0
$617$		−	39.5980i		−	1.59415i	−0.603877	−	0.797077i	$-0.706378\pi$
$617$		−	39.5980i		−	1.59415i	0.603877	−	0.797077i	$-0.293622\pi$
$618$		0			0
$619$	26.0000			1.04503			0.522514	−	0.852631i	$-0.324994\pi$
$619$	26.0000			1.04503			0.522514	+	0.852631i	$0.324994\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	25.0000			1.00000
$626$		−	14.1421i		−	0.565233i
$627$	8.00000	−	5.65685i	0.319489	−	0.225913i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	−14.0000	−	19.7990i	−0.556450	−	0.786939i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$			28.2843i			1.11716i	0.829450	+	0.558581i	$0.188654\pi$
$641$			28.2843i			1.11716i	−0.829450	+	0.558581i	$0.811346\pi$
$642$	28.0000	+	39.5980i	1.10507	+	1.56281i
$643$	50.0000			1.97181			0.985904	−	0.167313i	$-0.0535092\pi$
$643$	50.0000			1.97181			0.985904	+	0.167313i	$0.0535092\pi$
$644$		0			0
$645$		0			0
$646$	16.0000			0.629512
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$	−16.0000	+	19.7990i	−0.628539	+	0.777778i
$649$	40.0000			1.57014
$650$		0			0
$651$		0			0
$652$	−4.00000			−0.156652
$653$		0			0			−	1.00000i	$-0.5\pi$
$653$		0			0				1.00000i	$0.5\pi$
$654$		0			0
$655$		0			0
$656$			45.2548i			1.76690i
$657$	−2.00000	+	5.65685i	−0.0780274	+	0.220695i
$658$		0			0
$659$		−	48.0833i		−	1.87306i	−0.350590	−	0.936529i	$-0.614019\pi$
$659$		−	48.0833i		−	1.87306i	0.350590	−	0.936529i	$-0.385981\pi$
$660$		0			0
$661$		0			0		1.00000			$0$
$661$		0			0		−1.00000			$\pi$
$662$			36.7696i			1.42909i
$663$		0			0
$664$	8.00000			0.310460
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		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	$0.561736\pi$
$674$			19.7990i			0.762629i
$675$	−25.0000	+	7.07107i	−0.962250	+	0.272166i
$676$	−26.0000			−1.00000
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$	16.0000	+	22.6274i	0.614476	+	0.869001i
$679$		0			0
$680$		0			0
$681$	4.00000	−	2.82843i	0.153280	−	0.108386i
$682$		0			0
$683$		−	31.1127i		−	1.19049i	−0.803543	−	0.595247i	$-0.797054\pi$
$683$		−	31.1127i		−	1.19049i	0.803543	−	0.595247i	$-0.202946\pi$
$684$	4.00000	−	11.3137i	0.152944	−	0.432590i
$685$		0			0
$686$		0			0
$687$		0			0
$688$	−40.0000			−1.52499
$689$		0			0
$690$		0			0
$691$	−46.0000			−1.74992			−0.874961	−	0.484193i	$-0.839113\pi$
$691$	−46.0000			−1.74992			−0.874961	+	0.484193i	$0.839113\pi$
$692$		0			0
$693$		0			0
$694$	−52.0000			−1.97389
$695$		0			0
$696$		0			0
$697$	64.0000			2.42417
$698$		0			0
$699$	−8.00000	+	5.65685i	−0.302588	+	0.213962i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		−	22.6274i		−	0.852803i
$705$		0			0
$706$	32.0000			1.20434
$707$		0			0
$708$	40.0000	−	28.2843i	1.50329	−	1.06299i
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$	−16.0000			−0.599625
$713$		0			0
$714$		0			0
$715$		0			0
$716$		−	39.5980i		−	1.47985i
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$		0			0
$722$		−	21.2132i		−	0.789474i
$723$	−26.0000	−	36.7696i	−0.966950	−	1.36747i
$724$		0			0
$725$		0			0
$726$	6.00000	−	4.24264i	0.222681	−	0.157459i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	23.0000	−	14.1421i	0.851852	−	0.523783i
$730$		0			0
$731$			56.5685i			2.09226i
$732$		0			0
$733$		0			0		1.00000			$0$
$733$		0			0		−1.00000			$\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			39.5980i			1.45861i
$738$	16.0000	−	45.2548i	0.588968	−	1.66585i
$739$	−34.0000			−1.25071			−0.625355	−	0.780340i	$-0.715046\pi$
$739$	−34.0000			−1.25071			−0.625355	+	0.780340i	$0.715046\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0			−	1.00000i	$-0.5\pi$
$743$		0			0				1.00000i	$0.5\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−8.00000	−	2.82843i	−0.292705	−	0.103487i
$748$	−32.0000			−1.17004
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	−44.0000	+	31.1127i	−1.60345	+	1.13381i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		1.00000			$0$
$757$		0			0		−1.00000			$\pi$
$758$			53.7401i			1.95193i
$759$		0			0
$760$		0			0
$761$			11.3137i			0.410122i	0.978749	+	0.205061i	$0.0657392\pi$
$761$			11.3137i			0.410122i	−0.978749	+	0.205061i	$0.934261\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−16.0000	−	22.6274i	−0.577350	−	0.816497i
$769$	−22.0000			−0.793340			−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000			−0.793340			−0.396670	+	0.917961i	$0.629834\pi$
$770$		0			0
$771$	16.0000	−	11.3137i	0.576226	−	0.407453i
$772$	44.0000			1.58359
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$	40.0000	+	14.1421i	1.43777	+	0.508329i
$775$		0			0
$776$			28.2843i			1.01535i
$777$		0			0
$778$		0			0
$779$			22.6274i			0.810711i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	28.0000			1.00000
$785$		0			0
$786$	−20.0000	−	28.2843i	−0.713376	−	1.00887i
$787$	50.0000			1.78231			0.891154	−	0.453701i	$-0.149897\pi$
$787$	50.0000			1.78231			0.891154	+	0.453701i	$0.149897\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−8.00000	+	22.6274i	−0.284268	+	0.804030i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		−	28.2843i		−	1.00000i
$801$	16.0000	+	5.65685i	0.565332	+	0.199875i
$802$	56.0000			1.97743
$803$			5.65685i			0.199626i
$804$	28.0000	+	39.5980i	0.987484	+	1.39651i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	56.5685i		−	1.98884i	−0.105474	−	0.994422i	$-0.533636\pi$
$809$		−	56.5685i		−	1.98884i	0.105474	−	0.994422i	$-0.466364\pi$
$810$		0			0
$811$	38.0000			1.33436			0.667180	−	0.744896i	$-0.267501\pi$
$811$	38.0000			1.33436			0.667180	+	0.744896i	$0.267501\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$	−32.0000	+	22.6274i	−1.12022	+	0.792118i
$817$	−20.0000			−0.699711
$818$		−	31.1127i		−	1.08783i
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$	−32.0000	−	45.2548i	−1.11613	−	1.57844i
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$	−20.0000	+	14.1421i	−0.696311	+	0.492366i
$826$		0			0
$827$			19.7990i			0.688478i	0.938882	+	0.344239i	$0.111863\pi$
$827$			19.7990i			0.688478i	−0.938882	+	0.344239i	$0.888137\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	39.5980i		−	1.37199i
$834$	−44.0000	+	31.1127i	−1.52360	+	1.07734i
$835$		0			0
$836$		−	11.3137i		−	0.391293i
$837$		0			0
$838$	−52.0000			−1.79631
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$	40.0000	−	28.2843i	1.37767	−	0.974162i
$844$	−28.0000			−0.963800
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	22.0000	+	31.1127i	0.755038	+	1.06779i
$850$	−40.0000			−1.37199
$851$		0			0
$852$		0			0
$853$		0			0		1.00000			$0$
$853$		0			0		−1.00000			$\pi$
$854$		0			0
$855$		0			0
$856$	56.0000			1.91404
$857$		−	22.6274i		−	0.772938i	−0.922302	−	0.386469i	$-0.873695\pi$
$857$		−	22.6274i		−	0.772938i	0.922302	−	0.386469i	$-0.126305\pi$
$858$		0			0
$859$	−58.0000			−1.97893			−0.989467	−	0.144757i	$-0.953760\pi$
$859$	−58.0000			−1.97893			−0.989467	+	0.144757i	$0.953760\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$	8.00000	+	28.2843i	0.272166	+	0.962250i
$865$		0			0
$866$			53.7401i			1.82616i
$867$	15.0000	+	21.2132i	0.509427	+	0.720438i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	10.0000	−	28.2843i	0.338449	−	0.957278i
$874$		0			0
$875$		0			0
$876$	4.00000	+	5.65685i	0.135147	+	0.191127i
$877$		0			0		1.00000			$0$
$877$		0			0		−1.00000			$\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	56.5685i		−	1.90584i	−0.303218	−	0.952921i	$-0.598061\pi$
$881$		−	56.5685i		−	1.90584i	0.303218	−	0.952921i	$-0.401939\pi$
$882$	−28.0000	−	9.89949i	−0.942809	−	0.333333i
$883$	2.00000			0.0673054			0.0336527	−	0.999434i	$-0.489286\pi$
$883$	2.00000			0.0673054			0.0336527	+	0.999434i	$0.489286\pi$
$884$		0			0
$885$		0			0
$886$	−4.00000			−0.134383
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	16.0000	−	19.7990i	0.536020	−	0.663291i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	8.00000			0.266963
$899$		0			0
$900$	−10.0000	+	28.2843i	−0.333333	+	0.942809i
$901$		0			0
$902$		−	45.2548i		−	1.50682i
$903$		0			0
$904$	32.0000			1.06430
$905$		0			0
$906$		0			0
$907$	−10.0000			−0.332045			−0.166022	−	0.986122i	$-0.553092\pi$
$907$	−10.0000			−0.332045			−0.166022	+	0.986122i	$0.553092\pi$
$908$		−	5.65685i		−	0.187729i
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$	−8.00000	−	11.3137i	−0.264906	−	0.374634i
$913$	−8.00000			−0.264761
$914$			36.7696i			1.21623i
$915$		0			0
$916$		0			0
$917$		0			0
$918$	40.0000	−	11.3137i	1.32020	−	0.373408i
$919$		0			0		1.00000			$0$
$919$		0			0		−1.00000			$\pi$
$920$		0			0
$921$	34.0000	+	48.0833i	1.12034	+	1.58440i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$			28.2843i			0.927977i	0.885841	+	0.463988i	$0.153582\pi$
$929$			28.2843i			0.927977i	−0.885841	+	0.463988i	$0.846418\pi$
$930$		0			0
$931$	14.0000			0.458831
$932$			11.3137i			0.370593i
$933$		0			0
$934$	44.0000			1.43972
$935$		0			0
$936$		0			0
$937$	−34.0000			−1.11073			−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000			−1.11073			−0.555366	+	0.831606i	$0.687422\pi$
$938$		0			0
$939$	10.0000	+	14.1421i	0.326338	+	0.461511i
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		−	56.5685i		−	1.84115i
$945$		0			0
$946$	40.0000			1.30051
$947$			53.7401i			1.74632i	0.487435	+	0.873160i	$0.337933\pi$
$947$			53.7401i			1.74632i	−0.487435	+	0.873160i	$0.662067\pi$
$948$		0			0
$949$		0			0
$950$		−	14.1421i		−	0.458831i
$951$		0			0
$952$		0			0
$953$			45.2548i			1.46595i	0.680257	+	0.732974i	$0.261868\pi$
$953$			45.2548i			1.46595i	−0.680257	+	0.732974i	$0.738132\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	31.0000			1.00000
$962$		0			0
$963$	−56.0000	−	19.7990i	−1.80457	−	0.638014i
$964$	−52.0000			−1.67481
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		−	8.48528i		−	0.272727i
$969$	−16.0000	+	11.3137i	−0.513994	+	0.363449i
$970$		0			0
$971$		−	31.1127i		−	0.998454i	−0.866471	−	0.499227i	$-0.833617\pi$
$971$		−	31.1127i		−	0.998454i	0.866471	−	0.499227i	$-0.166383\pi$
$972$	2.00000	−	31.1127i	0.0641500	−	0.997940i
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$			62.2254i			1.99077i	0.0959785	+	0.995383i	$0.469402\pi$
$977$			62.2254i			1.99077i	−0.0959785	+	0.995383i	$0.530598\pi$
$978$	4.00000	−	2.82843i	0.127906	−	0.0904431i
$979$	16.0000			0.511362
$980$		0			0
$981$		0			0
$982$	20.0000			0.638226
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$	−32.0000	−	45.2548i	−1.02012	−	1.44267i
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$		0			0
$993$	−26.0000	−	36.7696i	−0.825085	−	1.16685i
$994$		0			0
$995$		0			0
$996$	−8.00000	+	5.65685i	−0.253490	+	0.179244i
$997$		0			0		1.00000			$0$
$997$		0			0		−1.00000			$\pi$
$998$			19.7990i			0.626726i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	24.2.f.a.11.2	yes	2
3.2	odd	2	inner	24.2.f.a.11.1	✓	2
4.3	odd	2		96.2.f.a.47.2		2
5.2	odd	4		600.2.m.a.299.2		4
5.3	odd	4		600.2.m.a.299.3		4
5.4	even	2		600.2.b.a.251.1		2
8.3	odd	2	CM	24.2.f.a.11.2	yes	2
8.5	even	2		96.2.f.a.47.2		2
9.2	odd	6		648.2.l.b.539.1		4
9.4	even	3		648.2.l.b.107.1		4
9.5	odd	6		648.2.l.b.107.2		4
9.7	even	3		648.2.l.b.539.2		4
12.11	even	2		96.2.f.a.47.1		2
15.2	even	4		600.2.m.a.299.4		4
15.8	even	4		600.2.m.a.299.1		4
15.14	odd	2		600.2.b.a.251.2		2
16.3	odd	4		768.2.c.h.767.2		4
16.5	even	4		768.2.c.h.767.2		4
16.11	odd	4		768.2.c.h.767.3		4
16.13	even	4		768.2.c.h.767.3		4
20.3	even	4		2400.2.m.a.1199.2		4
20.7	even	4		2400.2.m.a.1199.3		4
20.19	odd	2		2400.2.b.a.2351.1		2
24.5	odd	2		96.2.f.a.47.1		2
24.11	even	2	inner	24.2.f.a.11.1	✓	2
36.7	odd	6		2592.2.p.b.2159.1		4
36.11	even	6		2592.2.p.b.2159.2		4
36.23	even	6		2592.2.p.b.431.1		4
36.31	odd	6		2592.2.p.b.431.2		4
40.3	even	4		600.2.m.a.299.3		4
40.13	odd	4		2400.2.m.a.1199.2		4
40.19	odd	2		600.2.b.a.251.1		2
40.27	even	4		600.2.m.a.299.2		4
40.29	even	2		2400.2.b.a.2351.1		2
40.37	odd	4		2400.2.m.a.1199.3		4
48.5	odd	4		768.2.c.h.767.4		4
48.11	even	4		768.2.c.h.767.1		4
48.29	odd	4		768.2.c.h.767.1		4
48.35	even	4		768.2.c.h.767.4		4
60.23	odd	4		2400.2.m.a.1199.4		4
60.47	odd	4		2400.2.m.a.1199.1		4
60.59	even	2		2400.2.b.a.2351.2		2
72.5	odd	6		2592.2.p.b.431.1		4
72.11	even	6		648.2.l.b.539.1		4
72.13	even	6		2592.2.p.b.431.2		4
72.29	odd	6		2592.2.p.b.2159.2		4
72.43	odd	6		648.2.l.b.539.2		4
72.59	even	6		648.2.l.b.107.2		4
72.61	even	6		2592.2.p.b.2159.1		4
72.67	odd	6		648.2.l.b.107.1		4
120.29	odd	2		2400.2.b.a.2351.2		2
120.53	even	4		2400.2.m.a.1199.4		4
120.59	even	2		600.2.b.a.251.2		2
120.77	even	4		2400.2.m.a.1199.1		4
120.83	odd	4		600.2.m.a.299.1		4
120.107	odd	4		600.2.m.a.299.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
24.2.f.a.11.1	✓	2	3.2	odd	2	inner
24.2.f.a.11.1	✓	2	24.11	even	2	inner
24.2.f.a.11.2	yes	2	1.1	even	1	trivial
24.2.f.a.11.2	yes	2	8.3	odd	2	CM
96.2.f.a.47.1		2	12.11	even	2
96.2.f.a.47.1		2	24.5	odd	2
96.2.f.a.47.2		2	4.3	odd	2
96.2.f.a.47.2		2	8.5	even	2
600.2.b.a.251.1		2	5.4	even	2
600.2.b.a.251.1		2	40.19	odd	2
600.2.b.a.251.2		2	15.14	odd	2
600.2.b.a.251.2		2	120.59	even	2
600.2.m.a.299.1		4	15.8	even	4
600.2.m.a.299.1		4	120.83	odd	4
600.2.m.a.299.2		4	5.2	odd	4
600.2.m.a.299.2		4	40.27	even	4
600.2.m.a.299.3		4	5.3	odd	4
600.2.m.a.299.3		4	40.3	even	4
600.2.m.a.299.4		4	15.2	even	4
600.2.m.a.299.4		4	120.107	odd	4
648.2.l.b.107.1		4	9.4	even	3
648.2.l.b.107.1		4	72.67	odd	6
648.2.l.b.107.2		4	9.5	odd	6
648.2.l.b.107.2		4	72.59	even	6
648.2.l.b.539.1		4	9.2	odd	6
648.2.l.b.539.1		4	72.11	even	6
648.2.l.b.539.2		4	9.7	even	3
648.2.l.b.539.2		4	72.43	odd	6
768.2.c.h.767.1		4	48.11	even	4
768.2.c.h.767.1		4	48.29	odd	4
768.2.c.h.767.2		4	16.3	odd	4
768.2.c.h.767.2		4	16.5	even	4
768.2.c.h.767.3		4	16.11	odd	4
768.2.c.h.767.3		4	16.13	even	4
768.2.c.h.767.4		4	48.5	odd	4
768.2.c.h.767.4		4	48.35	even	4
2400.2.b.a.2351.1		2	20.19	odd	2
2400.2.b.a.2351.1		2	40.29	even	2
2400.2.b.a.2351.2		2	60.59	even	2
2400.2.b.a.2351.2		2	120.29	odd	2
2400.2.m.a.1199.1		4	60.47	odd	4
2400.2.m.a.1199.1		4	120.77	even	4
2400.2.m.a.1199.2		4	20.3	even	4
2400.2.m.a.1199.2		4	40.13	odd	4
2400.2.m.a.1199.3		4	20.7	even	4
2400.2.m.a.1199.3		4	40.37	odd	4
2400.2.m.a.1199.4		4	60.23	odd	4
2400.2.m.a.1199.4		4	120.53	even	4
2592.2.p.b.431.1		4	36.23	even	6
2592.2.p.b.431.1		4	72.5	odd	6
2592.2.p.b.431.2		4	36.31	odd	6
2592.2.p.b.431.2		4	72.13	even	6
2592.2.p.b.2159.1		4	36.7	odd	6
2592.2.p.b.2159.1		4	72.61	even	6
2592.2.p.b.2159.2		4	36.11	even	6
2592.2.p.b.2159.2		4	72.29	odd	6

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 24 = 2^{3} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	24.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +(-1.00000 - 1.41421i) q^{3} -2.00000 q^{4} +(2.00000 - 1.41421i) q^{6} -2.82843i q^{8} +(-1.00000 + 2.82843i) q^{9} +2.82843i q^{11} +(2.00000 + 2.82843i) q^{12} +4.00000 q^{16} -5.65685i q^{17} +(-4.00000 - 1.41421i) q^{18} +2.00000 q^{19} -4.00000 q^{22} +(-4.00000 + 2.82843i) q^{24} -5.00000 q^{25} +(5.00000 - 1.41421i) q^{27} +5.65685i q^{32} +(4.00000 - 2.82843i) q^{33} +8.00000 q^{34} +(2.00000 - 5.65685i) q^{36} +2.82843i q^{38} +11.3137i q^{41} -10.0000 q^{43} -5.65685i q^{44} +(-4.00000 - 5.65685i) q^{48} +7.00000 q^{49} -7.07107i q^{50} +(-8.00000 + 5.65685i) q^{51} +(2.00000 + 7.07107i) q^{54} +(-2.00000 - 2.82843i) q^{57} -14.1421i q^{59} -8.00000 q^{64} +(4.00000 + 5.65685i) q^{66} +14.0000 q^{67} +11.3137i q^{68} +(8.00000 + 2.82843i) q^{72} +2.00000 q^{73} +(5.00000 + 7.07107i) q^{75} -4.00000 q^{76} +(-7.00000 - 5.65685i) q^{81} -16.0000 q^{82} +2.82843i q^{83} -14.1421i q^{86} +8.00000 q^{88} -5.65685i q^{89} +(8.00000 - 5.65685i) q^{96} -10.0000 q^{97} +9.89949i q^{98} +(-8.00000 - 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 4 q^{4} + 4 q^{6} - 2 q^{9} + 4 q^{12} + 8 q^{16} - 8 q^{18} + 4 q^{19} - 8 q^{22} - 8 q^{24} - 10 q^{25} + 10 q^{27} + 8 q^{33} + 16 q^{34} + 4 q^{36} - 20 q^{43} - 8 q^{48} + 14 q^{49}+ \cdots - 16 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 4 * q^4 + 4 * q^6 - 2 * q^9 + 4 * q^12 + 8 * q^16 - 8 * q^18 + 4 * q^19 - 8 * q^22 - 8 * q^24 - 10 * q^25 + 10 * q^27 + 8 * q^33 + 16 * q^34 + 4 * q^36 - 20 * q^43 - 8 * q^48 + 14 * q^49 - 16 * q^51 + 4 * q^54 - 4 * q^57 - 16 * q^64 + 8 * q^66 + 28 * q^67 + 16 * q^72 + 4 * q^73 + 10 * q^75 - 8 * q^76 - 14 * q^81 - 32 * q^82 + 16 * q^88 + 16 * q^96 - 20 * q^97 - 16 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more