LMFDB - Embedded newform 88.2.g.a.43.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [88,2,Mod(43,88)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(88, 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("88.43");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 88 = 2^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	88.g (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:	$0.702683537787$
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			43.2
Root			$1.41421i$ of defining polynomial
Character	$\chi$	$=$	88.43
Dual form			88.2.g.a.43.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} +2.00000 q^{3} -2.00000 q^{4} +2.82843i q^{6} -2.82843i q^{8} +1.00000 q^{9} +O(q^{10})$	$q+1.41421i q^{2} +2.00000 q^{3} -2.00000 q^{4} +2.82843i q^{6} -2.82843i q^{8} +1.00000 q^{9} +(-3.00000 - 1.41421i) q^{11} -4.00000 q^{12} +4.00000 q^{16} +5.65685i q^{17} +1.41421i q^{18} -8.48528i q^{19} +(2.00000 - 4.24264i) q^{22} -5.65685i q^{24} +5.00000 q^{25} -4.00000 q^{27} +5.65685i q^{32} +(-6.00000 - 2.82843i) q^{33} -8.00000 q^{34} -2.00000 q^{36} +12.0000 q^{38} +11.3137i q^{41} +8.48528i q^{43} +(6.00000 + 2.82843i) q^{44} +8.00000 q^{48} -7.00000 q^{49} +7.07107i q^{50} +11.3137i q^{51} -5.65685i q^{54} -16.9706i q^{57} -6.00000 q^{59} -8.00000 q^{64} +(4.00000 - 8.48528i) q^{66} +14.0000 q^{67} -11.3137i q^{68} -2.82843i q^{72} -16.9706i q^{73} +10.0000 q^{75} +16.9706i q^{76} -11.0000 q^{81} -16.0000 q^{82} +2.82843i q^{83} -12.0000 q^{86} +(-4.00000 + 8.48528i) q^{88} +18.0000 q^{89} +11.3137i q^{96} +10.0000 q^{97} -9.89949i q^{98} +(-3.00000 - 1.41421i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} - 4 q^{4} + 2 q^{9}+O(q^{10}) $ 2 * q + 4 * q^3 - 4 * q^4 + 2 * q^9	$ 2 q + 4 q^{3} - 4 q^{4} + 2 q^{9} - 6 q^{11} - 8 q^{12} + 8 q^{16} + 4 q^{22} + 10 q^{25} - 8 q^{27} - 12 q^{33} - 16 q^{34} - 4 q^{36} + 24 q^{38} + 12 q^{44} + 16 q^{48} - 14 q^{49} - 12 q^{59} - 16 q^{64} + 8 q^{66} + 28 q^{67} + 20 q^{75} - 22 q^{81} - 32 q^{82} - 24 q^{86} - 8 q^{88} + 36 q^{89} + 20 q^{97} - 6 q^{99}+O(q^{100}) $ 2 * q + 4 * q^3 - 4 * q^4 + 2 * q^9 - 6 * q^11 - 8 * q^12 + 8 * q^16 + 4 * q^22 + 10 * q^25 - 8 * q^27 - 12 * q^33 - 16 * q^34 - 4 * q^36 + 24 * q^38 + 12 * q^44 + 16 * q^48 - 14 * q^49 - 12 * q^59 - 16 * q^64 + 8 * q^66 + 28 * q^67 + 20 * q^75 - 22 * q^81 - 32 * q^82 - 24 * q^86 - 8 * q^88 + 36 * q^89 + 20 * q^97 - 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/88\mathbb{Z}\right)^\times$.

$n$	$23$	$45$	$57$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	2.00000			1.15470			0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000			1.15470			0.577350	+	0.816497i	$0.304087\pi$
$4$	−2.00000			−1.00000
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$			2.82843i			1.15470i
$7$		0			0			−	1.00000i	$-0.5\pi$
$7$		0			0				1.00000i	$0.5\pi$
$8$		−	2.82843i		−	1.00000i
$9$	1.00000			0.333333
$10$		0			0
$11$	−3.00000	−	1.41421i	−0.904534	−	0.426401i
$11$	−3.00000	−	1.41421i	−0.904534	−	0.426401i
$12$	−4.00000			−1.15470
$13$		0			0			−	1.00000i	$-0.5\pi$
$13$		0			0				1.00000i	$0.5\pi$
$14$		0			0
$15$		0			0
$16$	4.00000			1.00000
$17$			5.65685i			1.37199i	0.727607	+	0.685994i	$0.240633\pi$
$17$			5.65685i			1.37199i	−0.727607	+	0.685994i	$0.759367\pi$
$18$			1.41421i			0.333333i
$19$		−	8.48528i		−	1.94666i	−0.229416	−	0.973329i	$-0.573682\pi$
$19$		−	8.48528i		−	1.94666i	0.229416	−	0.973329i	$-0.426318\pi$
$20$		0			0
$21$		0			0
$22$	2.00000	−	4.24264i	0.426401	−	0.904534i
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		−	5.65685i		−	1.15470i
$25$	5.00000			1.00000
$26$		0			0
$27$	−4.00000			−0.769800
$28$		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
$33$	−6.00000	−	2.82843i	−1.04447	−	0.492366i
$34$	−8.00000			−1.37199
$35$		0			0
$36$	−2.00000			−0.333333
$37$		0			0		1.00000			$0$
$37$		0			0		−1.00000			$\pi$
$38$	12.0000			1.94666
$39$		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
$43$			8.48528i			1.29399i	0.762493	+	0.646997i	$0.223975\pi$
$43$			8.48528i			1.29399i	−0.762493	+	0.646997i	$0.776025\pi$
$44$	6.00000	+	2.82843i	0.904534	+	0.426401i
$45$		0			0
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$	8.00000			1.15470
$49$	−7.00000			−1.00000
$50$			7.07107i			1.00000i
$51$			11.3137i			1.58424i
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		−	5.65685i		−	0.769800i
$55$		0			0
$56$		0			0
$57$		−	16.9706i		−	2.24781i
$58$		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
$61$		0			0			−	1.00000i	$-0.5\pi$
$61$		0			0				1.00000i	$0.5\pi$
$62$		0			0
$63$		0			0
$64$	−8.00000			−1.00000
$65$		0			0
$66$	4.00000	−	8.48528i	0.492366	−	1.04447i
$67$	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		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		−	2.82843i		−	0.333333i
$73$		−	16.9706i		−	1.98625i	−0.117041	−	0.993127i	$-0.537341\pi$
$73$		−	16.9706i		−	1.98625i	0.117041	−	0.993127i	$-0.462659\pi$
$74$		0			0
$75$	10.0000			1.15470
$76$			16.9706i			1.94666i
$77$		0			0
$78$		0			0
$79$		0			0			−	1.00000i	$-0.5\pi$
$79$		0			0				1.00000i	$0.5\pi$
$80$		0			0
$81$	−11.0000			−1.22222
$82$	−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
$86$	−12.0000			−1.29399
$87$		0			0
$88$	−4.00000	+	8.48528i	−0.426401	+	0.904534i
$89$	18.0000			1.90800			0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000			1.90800			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
$96$			11.3137i			1.15470i
$97$	10.0000			1.01535			0.507673	−	0.861550i	$-0.330506\pi$
$97$	10.0000			1.01535			0.507673	+	0.861550i	$0.330506\pi$
$98$		−	9.89949i		−	1.00000i
$99$	−3.00000	−	1.41421i	−0.301511	−	0.142134i
$100$	−10.0000			−1.00000
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$	−16.0000			−1.58424
$103$		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	$-0.593662\pi$
$107$		−	19.7990i		−	1.91404i	0.290021	−	0.957020i	$-0.406338\pi$
$108$	8.00000			0.769800
$109$		0			0			−	1.00000i	$-0.5\pi$
$109$		0			0				1.00000i	$0.5\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−18.0000			−1.69330			−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000			−1.69330			−0.846649	+	0.532152i	$0.821383\pi$
$114$	24.0000			2.24781
$115$		0			0
$116$		0			0
$117$		0			0
$118$		−	8.48528i		−	0.781133i
$119$		0			0
$120$		0			0
$121$	7.00000	+	8.48528i	0.636364	+	0.771389i
$122$		0			0
$123$			22.6274i			2.04025i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		−	11.3137i		−	1.00000i
$129$			16.9706i			1.49417i
$130$		0			0
$131$		−	14.1421i		−	1.23560i	−0.786334	−	0.617802i	$-0.788023\pi$
$131$		−	14.1421i		−	1.23560i	0.786334	−	0.617802i	$-0.211977\pi$
$132$	12.0000	+	5.65685i	1.04447	+	0.492366i
$133$		0			0
$134$			19.7990i			1.71037i
$135$		0			0
$136$	16.0000			1.37199
$137$	6.00000			0.512615			0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000			0.512615			0.256307	+	0.966595i	$0.417494\pi$
$138$		0			0
$139$			8.48528i			0.719712i	0.933008	+	0.359856i	$0.117174\pi$
$139$			8.48528i			0.719712i	−0.933008	+	0.359856i	$0.882826\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	4.00000			0.333333
$145$		0			0
$146$	24.0000			1.98625
$147$	−14.0000			−1.15470
$148$		0			0
$149$		0			0			−	1.00000i	$-0.5\pi$
$149$		0			0				1.00000i	$0.5\pi$
$150$			14.1421i			1.15470i
$151$		0			0			−	1.00000i	$-0.5\pi$
$151$		0			0				1.00000i	$0.5\pi$
$152$	−24.0000			−1.94666
$153$			5.65685i			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
$162$		−	15.5563i		−	1.22222i
$163$	−2.00000			−0.156652			−0.0783260	−	0.996928i	$-0.524958\pi$
$163$	−2.00000			−0.156652			−0.0783260	+	0.996928i	$0.524958\pi$
$164$		−	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
$171$		−	8.48528i		−	0.648886i
$172$		−	16.9706i		−	1.29399i
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$		0			0
$176$	−12.0000	−	5.65685i	−0.904534	−	0.426401i
$177$	−12.0000			−0.901975
$178$			25.4558i			1.90800i
$179$	−18.0000			−1.34538			−0.672692	−	0.739923i	$-0.734862\pi$
$179$	−18.0000			−1.34538			−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
$187$	8.00000	−	16.9706i	0.585018	−	1.24101i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$	−16.0000			−1.15470
$193$		−	16.9706i		−	1.22157i	−0.791797	−	0.610784i	$-0.790854\pi$
$193$		−	16.9706i		−	1.22157i	0.791797	−	0.610784i	$-0.209146\pi$
$194$			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	$0.5\pi$
$198$	2.00000	−	4.24264i	0.142134	−	0.301511i
$199$		0			0		1.00000			$0$
$199$		0			0		−1.00000			$\pi$
$200$		−	14.1421i		−	1.00000i
$201$	28.0000			1.97497
$202$		0			0
$203$		0			0
$204$		−	22.6274i		−	1.58424i
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−12.0000	+	25.4558i	−0.830057	+	1.76082i
$210$		0			0
$211$			25.4558i			1.75245i	0.481900	+	0.876226i	$0.339947\pi$
$211$			25.4558i			1.75245i	−0.481900	+	0.876226i	$0.660053\pi$
$212$		0			0
$213$		0			0
$214$	28.0000			1.91404
$215$		0			0
$216$			11.3137i			0.769800i
$217$		0			0
$218$		0			0
$219$		−	33.9411i		−	2.29353i
$220$		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			0.333333
$226$		−	25.4558i		−	1.69330i
$227$		−	2.82843i		−	0.187729i	−0.995585	−	0.0938647i	$-0.970078\pi$
$227$		−	2.82843i		−	0.187729i	0.995585	−	0.0938647i	$-0.0299221\pi$
$228$			33.9411i			2.24781i
$229$		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
$236$	12.0000			0.781133
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$			16.9706i			1.09317i	0.837404	+	0.546585i	$0.184072\pi$
$241$			16.9706i			1.09317i	−0.837404	+	0.546585i	$0.815928\pi$
$242$	−12.0000	+	9.89949i	−0.771389	+	0.636364i
$243$	−10.0000			−0.641500
$244$		0			0
$245$		0			0
$246$	−32.0000			−2.04025
$247$		0			0
$248$		0			0
$249$			5.65685i			0.358489i
$250$		0			0
$251$	6.00000			0.378717			0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000			0.378717			0.189358	+	0.981908i	$0.439359\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	16.0000			1.00000
$257$	30.0000			1.87135			0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000			1.87135			0.935674	+	0.352865i	$0.114792\pi$
$258$	−24.0000			−1.49417
$259$		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	+	16.9706i	−0.492366	+	1.04447i
$265$		0			0
$266$		0			0
$267$	36.0000			2.20316
$268$	−28.0000			−1.71037
$269$		0			0		1.00000			$0$
$269$		0			0		−1.00000			$\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$			22.6274i			1.37199i
$273$		0			0
$274$			8.48528i			0.512615i
$275$	−15.0000	−	7.07107i	−0.904534	−	0.426401i
$276$		0			0
$277$		0			0			−	1.00000i	$-0.5\pi$
$277$		0			0				1.00000i	$0.5\pi$
$278$	−12.0000			−0.719712
$279$		0			0
$280$		0			0
$281$		−	28.2843i		−	1.68730i	−0.536895	−	0.843649i	$-0.680403\pi$
$281$		−	28.2843i		−	1.68730i	0.536895	−	0.843649i	$-0.319597\pi$
$282$		0			0
$283$			25.4558i			1.51319i	0.653882	+	0.756596i	$0.273139\pi$
$283$			25.4558i			1.51319i	−0.653882	+	0.756596i	$0.726861\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$			5.65685i			0.333333i
$289$	−15.0000			−0.882353
$290$		0			0
$291$	20.0000			1.17242
$292$			33.9411i			1.98625i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		−	19.7990i		−	1.15470i
$295$		0			0
$296$		0			0
$297$	12.0000	+	5.65685i	0.696311	+	0.328244i
$298$		0			0
$299$		0			0
$300$	−20.0000			−1.15470
$301$		0			0
$302$		0			0
$303$		0			0
$304$		−	33.9411i		−	1.94666i
$305$		0			0
$306$	−8.00000			−0.457330
$307$			8.48528i			0.484281i	0.970241	+	0.242140i	$0.0778494\pi$
$307$			8.48528i			0.484281i	−0.970241	+	0.242140i	$0.922151\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		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		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		−	39.5980i		−	2.21014i
$322$		0			0
$323$	48.0000			2.67079
$324$	22.0000			1.22222
$325$		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	$-0.753366\pi$
$331$	−26.0000			−1.42909			−0.714545	+	0.699590i	$0.753366\pi$
$332$		−	5.65685i		−	0.310460i
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$		−	33.9411i		−	1.84889i	−0.381314	−	0.924445i	$-0.624528\pi$
$337$		−	33.9411i		−	1.84889i	0.381314	−	0.924445i	$-0.375472\pi$
$338$		−	18.3848i		−	1.00000i
$339$	−36.0000			−1.95525
$340$		0			0
$341$		0			0
$342$	12.0000			0.648886
$343$		0			0
$344$	24.0000			1.29399
$345$		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			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$	8.00000	−	16.9706i	0.426401	−	0.904534i
$353$	−30.0000			−1.59674			−0.798369	−	0.602168i	$-0.794304\pi$
$353$	−30.0000			−1.59674			−0.798369	+	0.602168i	$0.794304\pi$
$354$		−	16.9706i		−	0.901975i
$355$		0			0
$356$	−36.0000			−1.90800
$357$		0			0
$358$		−	25.4558i		−	1.34538i
$359$		0			0			−	1.00000i	$-0.5\pi$
$359$		0			0				1.00000i	$0.5\pi$
$360$		0			0
$361$	−53.0000			−2.78947
$362$		0			0
$363$	14.0000	+	16.9706i	0.734809	+	0.890724i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		1.00000			$0$
$367$		0			0		−1.00000			$\pi$
$368$		0			0
$369$			11.3137i			0.588968i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0			−	1.00000i	$-0.5\pi$
$373$		0			0				1.00000i	$0.5\pi$
$374$	24.0000	+	11.3137i	1.24101	+	0.585018i
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−38.0000			−1.95193			−0.975964	−	0.217930i	$-0.930070\pi$
$379$	−38.0000			−1.95193			−0.975964	+	0.217930i	$0.930070\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		1.00000			$0$
$383$		0			0		−1.00000			$\pi$
$384$		−	22.6274i		−	1.15470i
$385$		0			0
$386$	24.0000			1.22157
$387$			8.48528i			0.431331i
$388$	−20.0000			−1.01535
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$			19.7990i			1.00000i
$393$		−	28.2843i		−	1.42675i
$394$		0			0
$395$		0			0
$396$	6.00000	+	2.82843i	0.301511	+	0.142134i
$397$		0			0		1.00000			$0$
$397$		0			0		−1.00000			$\pi$
$398$		0			0
$399$		0			0
$400$	20.0000			1.00000
$401$	6.00000			0.299626			0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000			0.299626			0.149813	+	0.988714i	$0.452133\pi$
$402$			39.5980i			1.97497i
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$	32.0000			1.58424
$409$			33.9411i			1.67828i	0.543915	+	0.839140i	$0.316941\pi$
$409$			33.9411i			1.67828i	−0.543915	+	0.839140i	$0.683059\pi$
$410$		0			0
$411$	12.0000			0.591916
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$			16.9706i			0.831052i
$418$	−36.0000	−	16.9706i	−1.76082	−	0.830057i
$419$	18.0000			0.879358			0.439679	−	0.898155i	$-0.355092\pi$
$419$	18.0000			0.879358			0.439679	+	0.898155i	$0.355092\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$	−36.0000			−1.75245
$423$		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	$0.5\pi$
$432$	−16.0000			−0.769800
$433$	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
$438$	48.0000			2.29353
$439$		0			0			−	1.00000i	$-0.5\pi$
$439$		0			0				1.00000i	$0.5\pi$
$440$		0			0
$441$	−7.00000			−0.333333
$442$		0			0
$443$	−42.0000			−1.99548			−0.997740	−	0.0671913i	$-0.978596\pi$
$443$	−42.0000			−1.99548			−0.997740	+	0.0671913i	$0.978596\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	42.0000			1.98210			0.991051	−	0.133482i	$-0.0426157\pi$
$449$	42.0000			1.98210			0.991051	+	0.133482i	$0.0426157\pi$
$450$			7.07107i			0.333333i
$451$	16.0000	−	33.9411i	0.753411	−	1.59823i
$452$	36.0000			1.69330
$453$		0			0
$454$	4.00000			0.187729
$455$		0			0
$456$	−48.0000			−2.24781
$457$			33.9411i			1.58770i	0.608114	+	0.793849i	$0.291926\pi$
$457$			33.9411i			1.58770i	−0.608114	+	0.793849i	$0.708074\pi$
$458$		0			0
$459$		−	22.6274i		−	1.05616i
$460$		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
$467$	30.0000			1.38823			0.694117	−	0.719862i	$-0.255795\pi$
$467$	30.0000			1.38823			0.694117	+	0.719862i	$0.255795\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$			16.9706i			0.781133i
$473$	12.0000	−	25.4558i	0.551761	−	1.17046i
$474$		0			0
$475$		−	42.4264i		−	1.94666i
$476$		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
$482$	−24.0000			−1.09317
$483$		0			0
$484$	−14.0000	−	16.9706i	−0.636364	−	0.771389i
$485$		0			0
$486$		−	14.1421i		−	0.641500i
$487$		0			0		1.00000			$0$
$487$		0			0		−1.00000			$\pi$
$488$		0			0
$489$	−4.00000			−0.180886
$490$		0			0
$491$			14.1421i			0.638226i	0.947717	+	0.319113i	$0.103385\pi$
$491$			14.1421i			0.638226i	−0.947717	+	0.319113i	$0.896615\pi$
$492$		−	45.2548i		−	2.04025i
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$	−8.00000			−0.358489
$499$	−14.0000			−0.626726			−0.313363	−	0.949633i	$-0.601456\pi$
$499$	−14.0000			−0.626726			−0.313363	+	0.949633i	$0.601456\pi$
$500$		0			0
$501$		0			0
$502$			8.48528i			0.378717i
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−26.0000			−1.15470
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$			22.6274i			1.00000i
$513$			33.9411i			1.49854i
$514$			42.4264i			1.87135i
$515$		0			0
$516$		−	33.9411i		−	1.49417i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−6.00000			−0.262865			−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000			−0.262865			−0.131432	+	0.991325i	$0.541958\pi$
$522$		0			0
$523$		−	25.4558i		−	1.11311i	−0.830812	−	0.556553i	$-0.812124\pi$
$523$		−	25.4558i		−	1.11311i	0.830812	−	0.556553i	$-0.187876\pi$
$524$			28.2843i			1.23560i
$525$		0			0
$526$		0			0
$527$		0			0
$528$	−24.0000	−	11.3137i	−1.04447	−	0.492366i
$529$	23.0000			1.00000
$530$		0			0
$531$	−6.00000			−0.260378
$532$		0			0
$533$		0			0
$534$			50.9117i			2.20316i
$535$		0			0
$536$		−	39.5980i		−	1.71037i
$537$	−36.0000			−1.55351
$538$		0			0
$539$	21.0000	+	9.89949i	0.904534	+	0.426401i
$540$		0			0
$541$		0			0			−	1.00000i	$-0.5\pi$
$541$		0			0				1.00000i	$0.5\pi$
$542$		0			0
$543$		0			0
$544$	−32.0000			−1.37199
$545$		0			0
$546$		0			0
$547$		−	8.48528i		−	0.362804i	−0.983409	−	0.181402i	$-0.941936\pi$
$547$		−	8.48528i		−	0.362804i	0.983409	−	0.181402i	$-0.0580636\pi$
$548$	−12.0000			−0.512615
$549$		0			0
$550$	10.0000	−	21.2132i	0.426401	−	0.904534i
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		−	16.9706i		−	0.719712i
$557$		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	−	33.9411i	0.675521	−	1.43300i
$562$	40.0000			1.68730
$563$		−	36.7696i		−	1.54965i	−0.632175	−	0.774826i	$-0.717837\pi$
$563$		−	36.7696i		−	1.54965i	0.632175	−	0.774826i	$-0.282163\pi$
$564$		0			0
$565$		0			0
$566$	−36.0000			−1.51319
$567$		0			0
$568$		0			0
$569$			22.6274i			0.948591i	0.880366	+	0.474295i	$0.157297\pi$
$569$			22.6274i			0.948591i	−0.880366	+	0.474295i	$0.842703\pi$
$570$		0			0
$571$			42.4264i			1.77549i	0.460336	+	0.887745i	$0.347729\pi$
$571$			42.4264i			1.77549i	−0.460336	+	0.887745i	$0.652271\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−8.00000			−0.333333
$577$	34.0000			1.41544			0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000			1.41544			0.707719	+	0.706494i	$0.249724\pi$
$578$		−	21.2132i		−	0.882353i
$579$		−	33.9411i		−	1.41055i
$580$		0			0
$581$		0			0
$582$			28.2843i			1.17242i
$583$		0			0
$584$	−48.0000			−1.98625
$585$		0			0
$586$		0			0
$587$	−6.00000			−0.247647			−0.123823	−	0.992304i	$-0.539516\pi$
$587$	−6.00000			−0.247647			−0.123823	+	0.992304i	$0.539516\pi$
$588$	28.0000			1.15470
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	45.2548i		−	1.85839i	−0.369586	−	0.929197i	$-0.620500\pi$
$593$		−	45.2548i		−	1.85839i	0.369586	−	0.929197i	$-0.379500\pi$
$594$	−8.00000	+	16.9706i	−0.328244	+	0.696311i
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		−	28.2843i		−	1.15470i
$601$		−	16.9706i		−	0.692244i	−0.938190	−	0.346122i	$-0.887498\pi$
$601$		−	16.9706i		−	0.692244i	0.938190	−	0.346122i	$-0.112502\pi$
$602$		0			0
$603$	14.0000			0.570124
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0			−	1.00000i	$-0.5\pi$
$607$		0			0				1.00000i	$0.5\pi$
$608$	48.0000			1.94666
$609$		0			0
$610$		0			0
$611$		0			0
$612$		−	11.3137i		−	0.457330i
$613$		0			0			−	1.00000i	$-0.5\pi$
$613$		0			0				1.00000i	$0.5\pi$
$614$	−12.0000			−0.484281
$615$		0			0
$616$		0			0
$617$	−30.0000			−1.20775			−0.603877	−	0.797077i	$-0.706378\pi$
$617$	−30.0000			−1.20775			−0.603877	+	0.797077i	$0.706378\pi$
$618$		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
$627$	−24.0000	+	50.9117i	−0.958468	+	2.03322i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$			50.9117i			2.02356i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−42.0000			−1.65890			−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−42.0000			−1.65890			−0.829450	+	0.558581i	$0.811346\pi$
$642$	56.0000			2.21014
$643$	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
$646$			67.8823i			2.67079i
$647$		0			0		1.00000			$0$
$647$		0			0		−1.00000			$\pi$
$648$			31.1127i			1.22222i
$649$	18.0000	+	8.48528i	0.706562	+	0.333076i
$650$		0			0
$651$		0			0
$652$	4.00000			0.156652
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$			45.2548i			1.76690i
$657$		−	16.9706i		−	0.662085i
$658$		0			0
$659$			48.0833i			1.87306i	0.350590	+	0.936529i	$0.385981\pi$
$659$			48.0833i			1.87306i	−0.350590	+	0.936529i	$0.614019\pi$
$660$		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
$673$			50.9117i			1.96250i	0.192736	+	0.981251i	$0.438264\pi$
$673$			50.9117i			1.96250i	−0.192736	+	0.981251i	$0.561736\pi$
$674$	48.0000			1.84889
$675$	−20.0000			−0.769800
$676$	26.0000			1.00000
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		−	50.9117i		−	1.95525i
$679$		0			0
$680$		0			0
$681$		−	5.65685i		−	0.216771i
$682$		0			0
$683$	−42.0000			−1.60709			−0.803543	−	0.595247i	$-0.797054\pi$
$683$	−42.0000			−1.60709			−0.803543	+	0.595247i	$0.797054\pi$
$684$			16.9706i			0.648886i
$685$		0			0
$686$		0			0
$687$		0			0
$688$			33.9411i			1.29399i
$689$		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
$699$		−	11.3137i		−	0.427924i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	24.0000	+	11.3137i	0.904534	+	0.426401i
$705$		0			0
$706$		−	42.4264i		−	1.59674i
$707$		0			0
$708$	24.0000			0.901975
$709$		0			0		1.00000			$0$
$709$		0			0		−1.00000			$\pi$
$710$		0			0
$711$		0			0
$712$		−	50.9117i		−	1.90800i
$713$		0			0
$714$		0			0
$715$		0			0
$716$	36.0000			1.34538
$717$		0			0
$718$		0			0
$719$		0			0		1.00000			$0$
$719$		0			0		−1.00000			$\pi$
$720$		0			0
$721$		0			0
$722$		−	74.9533i		−	2.78947i
$723$			33.9411i			1.26228i
$724$		0			0
$725$		0			0
$726$	−24.0000	+	19.7990i	−0.890724	+	0.734809i
$727$		0			0		1.00000			$0$
$727$		0			0		−1.00000			$\pi$
$728$		0			0
$729$	13.0000			0.481481
$730$		0			0
$731$	−48.0000			−1.77534
$732$		0			0
$733$		0			0			−	1.00000i	$-0.5\pi$
$733$		0			0				1.00000i	$0.5\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−42.0000	−	19.7990i	−1.54709	−	0.729305i
$738$	−16.0000			−0.588968
$739$		−	42.4264i		−	1.56068i	−0.625355	−	0.780340i	$-0.715046\pi$
$739$		−	42.4264i		−	1.56068i	0.625355	−	0.780340i	$-0.284954\pi$
$740$		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
$747$			2.82843i			0.103487i
$748$	−16.0000	+	33.9411i	−0.585018	+	1.24101i
$749$		0			0
$750$		0			0
$751$		0			0		1.00000			$0$
$751$		0			0		−1.00000			$\pi$
$752$		0			0
$753$	12.0000			0.437304
$754$		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	$-0.934261\pi$
$761$		−	11.3137i		−	0.410122i	0.978749	−	0.205061i	$-0.0657392\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	32.0000			1.15470
$769$		−	50.9117i		−	1.83592i	−0.396670	−	0.917961i	$-0.629834\pi$
$769$		−	50.9117i		−	1.83592i	0.396670	−	0.917961i	$-0.370166\pi$
$770$		0			0
$771$	60.0000			2.16085
$772$			33.9411i			1.22157i
$773$		0			0		1.00000			$0$
$773$		0			0		−1.00000			$\pi$
$774$	−12.0000			−0.431331
$775$		0			0
$776$		−	28.2843i		−	1.01535i
$777$		0			0
$778$		0			0
$779$	96.0000			3.43956
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−28.0000			−1.00000
$785$		0			0
$786$	40.0000			1.42675
$787$		−	25.4558i		−	0.907403i	−0.891154	−	0.453701i	$-0.850103\pi$
$787$		−	25.4558i		−	0.907403i	0.891154	−	0.453701i	$-0.149897\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$	−4.00000	+	8.48528i	−0.142134	+	0.301511i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$			28.2843i			1.00000i
$801$	18.0000			0.635999
$802$			8.48528i			0.299626i
$803$	−24.0000	+	50.9117i	−0.846942	+	1.79663i
$804$	−56.0000			−1.97497
$805$		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
$811$			42.4264i			1.48979i	0.667180	+	0.744896i	$0.267501\pi$
$811$			42.4264i			1.48979i	−0.667180	+	0.744896i	$0.732499\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$			45.2548i			1.58424i
$817$	72.0000			2.51896
$818$	−48.0000			−1.67828
$819$		0			0
$820$		0			0
$821$		0			0			−	1.00000i	$-0.5\pi$
$821$		0			0				1.00000i	$0.5\pi$
$822$			16.9706i			0.591916i
$823$		0			0		1.00000			$0$
$823$		0			0		−1.00000			$\pi$
$824$		0			0
$825$	−30.0000	−	14.1421i	−1.04447	−	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
$834$	−24.0000			−0.831052
$835$		0			0
$836$	24.0000	−	50.9117i	0.830057	−	1.76082i
$837$		0			0
$838$			25.4558i			0.879358i
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	−29.0000			−1.00000
$842$		0			0
$843$		−	56.5685i		−	1.94832i
$844$		−	50.9117i		−	1.75245i
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$			50.9117i			1.74728i
$850$	−40.0000			−1.37199
$851$		0			0
$852$		0			0
$853$		0			0			−	1.00000i	$-0.5\pi$
$853$		0			0				1.00000i	$0.5\pi$
$854$		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	$-0.0462401\pi$
$859$	58.0000			1.97893			0.989467	+	0.144757i	$0.0462401\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		−	22.6274i		−	0.769800i
$865$		0			0
$866$			53.7401i			1.82616i
$867$	−30.0000			−1.01885
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	10.0000			0.338449
$874$		0			0
$875$		0			0
$876$			67.8823i			2.29353i
$877$		0			0			−	1.00000i	$-0.5\pi$
$877$		0			0				1.00000i	$0.5\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	18.0000			0.606435			0.303218	−	0.952921i	$-0.401939\pi$
$881$	18.0000			0.606435			0.303218	+	0.952921i	$0.401939\pi$
$882$		−	9.89949i		−	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
$886$		−	59.3970i		−	1.99548i
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	33.0000	+	15.5563i	1.10554	+	0.521157i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$			59.3970i			1.98210i
$899$		0			0
$900$	−10.0000			−0.333333
$901$		0			0
$902$	48.0000	+	22.6274i	1.59823	+	0.753411i
$903$		0			0
$904$			50.9117i			1.69330i
$905$		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		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$		−	67.8823i		−	2.24781i
$913$	4.00000	−	8.48528i	0.132381	−	0.280822i
$914$	−48.0000			−1.58770
$915$		0			0
$916$		0			0
$917$		0			0
$918$	32.0000			1.05616
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$			16.9706i			0.559199i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−54.0000			−1.77168			−0.885841	−	0.463988i	$-0.846418\pi$
$929$	−54.0000			−1.77168			−0.885841	+	0.463988i	$0.846418\pi$
$930$		0			0
$931$			59.3970i			1.94666i
$932$			11.3137i			0.370593i
$933$		0			0
$934$			42.4264i			1.38823i
$935$		0			0
$936$		0			0
$937$			50.9117i			1.66321i	0.555366	+	0.831606i	$0.312578\pi$
$937$			50.9117i			1.66321i	−0.555366	+	0.831606i	$0.687422\pi$
$938$		0			0
$939$	−20.0000			−0.652675
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$	−24.0000			−0.781133
$945$		0			0
$946$	36.0000	+	16.9706i	1.17046	+	0.551761i
$947$	−30.0000			−0.974869			−0.487435	−	0.873160i	$-0.662067\pi$
$947$	−30.0000			−0.974869			−0.487435	+	0.873160i	$0.662067\pi$
$948$		0			0
$949$		0			0
$950$	60.0000			1.94666
$951$		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
$963$		−	19.7990i		−	0.638014i
$964$		−	33.9411i		−	1.09317i
$965$		0			0
$966$		0			0
$967$		0			0			−	1.00000i	$-0.5\pi$
$967$		0			0				1.00000i	$0.5\pi$
$968$	24.0000	−	19.7990i	0.771389	−	0.636364i
$969$	96.0000			3.08396
$970$		0			0
$971$	54.0000			1.73294			0.866471	−	0.499227i	$-0.166383\pi$
$971$	54.0000			1.73294			0.866471	+	0.499227i	$0.166383\pi$
$972$	20.0000			0.641500
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	6.00000			0.191957			0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000			0.191957			0.0959785	+	0.995383i	$0.469402\pi$
$978$		−	5.65685i		−	0.180886i
$979$	−54.0000	−	25.4558i	−1.72585	−	0.813572i
$980$		0			0
$981$		0			0
$982$	−20.0000			−0.638226
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$	64.0000			2.04025
$985$		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
$993$	−52.0000			−1.65017
$994$		0			0
$995$		0			0
$996$		−	11.3137i		−	0.358489i
$997$		0			0			−	1.00000i	$-0.5\pi$
$997$		0			0				1.00000i	$0.5\pi$
$998$		−	19.7990i		−	0.626726i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	88.2.g.a.43.2	yes	2
3.2	odd	2		792.2.h.b.307.1		2
4.3	odd	2		352.2.g.a.175.2		2
8.3	odd	2	CM	88.2.g.a.43.2	yes	2
8.5	even	2		352.2.g.a.175.2		2
11.2	odd	10		968.2.k.a.403.1		8
11.3	even	5		968.2.k.a.475.2		8
11.4	even	5		968.2.k.a.699.1		8
11.5	even	5		968.2.k.a.723.1		8
11.6	odd	10		968.2.k.a.723.2		8
11.7	odd	10		968.2.k.a.699.2		8
11.8	odd	10		968.2.k.a.475.1		8
11.9	even	5		968.2.k.a.403.2		8
11.10	odd	2	inner	88.2.g.a.43.1	✓	2
12.11	even	2		3168.2.h.b.2287.1		2
16.3	odd	4		2816.2.e.d.2815.2		4
16.5	even	4		2816.2.e.d.2815.2		4
16.11	odd	4		2816.2.e.d.2815.3		4
16.13	even	4		2816.2.e.d.2815.3		4
24.5	odd	2		3168.2.h.b.2287.1		2
24.11	even	2		792.2.h.b.307.1		2
33.32	even	2		792.2.h.b.307.2		2
44.43	even	2		352.2.g.a.175.1		2
88.3	odd	10		968.2.k.a.475.2		8
88.19	even	10		968.2.k.a.475.1		8
88.21	odd	2		352.2.g.a.175.1		2
88.27	odd	10		968.2.k.a.723.1		8
88.35	even	10		968.2.k.a.403.1		8
88.43	even	2	inner	88.2.g.a.43.1	✓	2
88.51	even	10		968.2.k.a.699.2		8
88.59	odd	10		968.2.k.a.699.1		8
88.75	odd	10		968.2.k.a.403.2		8
88.83	even	10		968.2.k.a.723.2		8
132.131	odd	2		3168.2.h.b.2287.2		2
176.21	odd	4		2816.2.e.d.2815.1		4
176.43	even	4		2816.2.e.d.2815.4		4
176.109	odd	4		2816.2.e.d.2815.4		4
176.131	even	4		2816.2.e.d.2815.1		4
264.131	odd	2		792.2.h.b.307.2		2
264.197	even	2		3168.2.h.b.2287.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.g.a.43.1	✓	2	11.10	odd	2	inner
88.2.g.a.43.1	✓	2	88.43	even	2	inner
88.2.g.a.43.2	yes	2	1.1	even	1	trivial
88.2.g.a.43.2	yes	2	8.3	odd	2	CM
352.2.g.a.175.1		2	44.43	even	2
352.2.g.a.175.1		2	88.21	odd	2
352.2.g.a.175.2		2	4.3	odd	2
352.2.g.a.175.2		2	8.5	even	2
792.2.h.b.307.1		2	3.2	odd	2
792.2.h.b.307.1		2	24.11	even	2
792.2.h.b.307.2		2	33.32	even	2
792.2.h.b.307.2		2	264.131	odd	2
968.2.k.a.403.1		8	11.2	odd	10
968.2.k.a.403.1		8	88.35	even	10
968.2.k.a.403.2		8	11.9	even	5
968.2.k.a.403.2		8	88.75	odd	10
968.2.k.a.475.1		8	11.8	odd	10
968.2.k.a.475.1		8	88.19	even	10
968.2.k.a.475.2		8	11.3	even	5
968.2.k.a.475.2		8	88.3	odd	10
968.2.k.a.699.1		8	11.4	even	5
968.2.k.a.699.1		8	88.59	odd	10
968.2.k.a.699.2		8	11.7	odd	10
968.2.k.a.699.2		8	88.51	even	10
968.2.k.a.723.1		8	11.5	even	5
968.2.k.a.723.1		8	88.27	odd	10
968.2.k.a.723.2		8	11.6	odd	10
968.2.k.a.723.2		8	88.83	even	10
2816.2.e.d.2815.1		4	176.21	odd	4
2816.2.e.d.2815.1		4	176.131	even	4
2816.2.e.d.2815.2		4	16.3	odd	4
2816.2.e.d.2815.2		4	16.5	even	4
2816.2.e.d.2815.3		4	16.11	odd	4
2816.2.e.d.2815.3		4	16.13	even	4
2816.2.e.d.2815.4		4	176.43	even	4
2816.2.e.d.2815.4		4	176.109	odd	4
3168.2.h.b.2287.1		2	12.11	even	2
3168.2.h.b.2287.1		2	24.5	odd	2
3168.2.h.b.2287.2		2	132.131	odd	2
3168.2.h.b.2287.2		2	264.197	even	2

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:	\( N \)	\(=\)	\( 88 = 2^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	88.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} +2.00000 q^{3} -2.00000 q^{4} +2.82843i q^{6} -2.82843i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.41421i q^{2} +2.00000 q^{3} -2.00000 q^{4} +2.82843i q^{6} -2.82843i q^{8} +1.00000 q^{9} +(-3.00000 - 1.41421i) q^{11} -4.00000 q^{12} +4.00000 q^{16} +5.65685i q^{17} +1.41421i q^{18} -8.48528i q^{19} +(2.00000 - 4.24264i) q^{22} -5.65685i q^{24} +5.00000 q^{25} -4.00000 q^{27} +5.65685i q^{32} +(-6.00000 - 2.82843i) q^{33} -8.00000 q^{34} -2.00000 q^{36} +12.0000 q^{38} +11.3137i q^{41} +8.48528i q^{43} +(6.00000 + 2.82843i) q^{44} +8.00000 q^{48} -7.00000 q^{49} +7.07107i q^{50} +11.3137i q^{51} -5.65685i q^{54} -16.9706i q^{57} -6.00000 q^{59} -8.00000 q^{64} +(4.00000 - 8.48528i) q^{66} +14.0000 q^{67} -11.3137i q^{68} -2.82843i q^{72} -16.9706i q^{73} +10.0000 q^{75} +16.9706i q^{76} -11.0000 q^{81} -16.0000 q^{82} +2.82843i q^{83} -12.0000 q^{86} +(-4.00000 + 8.48528i) q^{88} +18.0000 q^{89} +11.3137i q^{96} +10.0000 q^{97} -9.89949i q^{98} +(-3.00000 - 1.41421i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} - 4 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q + 4 * q^3 - 4 * q^4 + 2 * q^9	\( 2 q + 4 q^{3} - 4 q^{4} + 2 q^{9} - 6 q^{11} - 8 q^{12} + 8 q^{16} + 4 q^{22} + 10 q^{25} - 8 q^{27} - 12 q^{33} - 16 q^{34} - 4 q^{36} + 24 q^{38} + 12 q^{44} + 16 q^{48} - 14 q^{49} - 12 q^{59} - 16 q^{64} + 8 q^{66} + 28 q^{67} + 20 q^{75} - 22 q^{81} - 32 q^{82} - 24 q^{86} - 8 q^{88} + 36 q^{89} + 20 q^{97} - 6 q^{99}+O(q^{100}) \) 2 * q + 4 * q^3 - 4 * q^4 + 2 * q^9 - 6 * q^11 - 8 * q^12 + 8 * q^16 + 4 * q^22 + 10 * q^25 - 8 * q^27 - 12 * q^33 - 16 * q^34 - 4 * q^36 + 24 * q^38 + 12 * q^44 + 16 * q^48 - 14 * q^49 - 12 * q^59 - 16 * q^64 + 8 * q^66 + 28 * q^67 + 20 * q^75 - 22 * q^81 - 32 * q^82 - 24 * q^86 - 8 * q^88 + 36 * q^89 + 20 * q^97 - 6 * q^99

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