LMFDB - Embedded newform 1088.2.m.g.353.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1088,2,Mod(225,1088)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1088, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 2, 3]))

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

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

chi := DirichletCharacter("1088.225");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1088 = 2^{6} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1088.m (of order $4$, degree $2$, 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:	$8.68772373992$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label			353.1
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1088.353
Dual form			1088.2.m.g.225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.414214 + 0.414214i) q^{3} +2.65685i q^{9} +O(q^{10})$	$q+(-0.414214 + 0.414214i) q^{3} +2.65685i q^{9} +(-1.58579 - 1.58579i) q^{11} +(-2.82843 + 3.00000i) q^{17} -2.00000 q^{19} +5.00000i q^{25} +(-2.34315 - 2.34315i) q^{27} +1.31371 q^{33} +(-2.65685 + 2.65685i) q^{41} -8.48528 q^{43} +7.00000i q^{49} +(-0.0710678 - 2.41421i) q^{51} +(0.828427 - 0.828427i) q^{57} -14.1421 q^{59} +8.48528i q^{67} +(7.48528 + 7.48528i) q^{73} +(-2.07107 - 2.07107i) q^{75} -6.02944 q^{81} +2.82843 q^{83} -5.65685 q^{89} +(-3.48528 - 3.48528i) q^{97} +(4.21320 - 4.21320i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3}+O(q^{10}) $ 4 * q + 4 * q^3	$ 4 q + 4 q^{3} - 12 q^{11} - 8 q^{19} - 32 q^{27} - 40 q^{33} + 12 q^{41} + 28 q^{51} - 8 q^{57} - 4 q^{73} + 20 q^{75} - 92 q^{81} + 20 q^{97} - 68 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 12 * q^11 - 8 * q^19 - 32 * q^27 - 40 * q^33 + 12 * q^41 + 28 * q^51 - 8 * q^57 - 4 * q^73 + 20 * q^75 - 92 * q^81 + 20 * q^97 - 68 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1088\mathbb{Z}\right)^\times$.

$n$	$69$	$511$	$513$
$\chi(n)$	$-1$	$1$	$e\left(\frac{1}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−0.414214	+	0.414214i	−0.239146	+	0.239146i	−0.816497	−	0.577350i	$-0.804087\pi$
$3$	−0.414214	+	0.414214i	−0.239146	+	0.239146i	0.577350	+	0.816497i	$0.304087\pi$
$4$		0			0
$5$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$5$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$6$		0			0
$7$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$7$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$8$		0			0
$9$			2.65685i			0.885618i
$10$		0			0
$11$	−1.58579	−	1.58579i	−0.478133	−	0.478133i	0.426401	−	0.904534i	$-0.359781\pi$
$11$	−1.58579	−	1.58579i	−0.478133	−	0.478133i	−0.904534	+	0.426401i	$0.859781\pi$
$12$		0			0
$13$		0			0		1.00000			$0$
$13$		0			0		−1.00000			$\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.82843	+	3.00000i	−0.685994	+	0.727607i
$17$	−2.82843	+	3.00000i	−0.685994	+	0.727607i
$18$		0			0
$19$	−2.00000			−0.458831			−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000			−0.458831			−0.229416	+	0.973329i	$0.573682\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$23$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$24$		0			0
$25$			5.00000i			1.00000i
$26$		0			0
$27$	−2.34315	−	2.34315i	−0.450939	−	0.450939i
$28$		0			0
$29$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$29$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$30$		0			0
$31$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$31$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$32$		0			0
$33$	1.31371			0.228687
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$37$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.65685	+	2.65685i	−0.414931	+	0.414931i	−0.883452	−	0.468521i	$-0.844787\pi$
$41$	−2.65685	+	2.65685i	−0.414931	+	0.414931i	0.468521	+	0.883452i	$0.344787\pi$
$42$		0			0
$43$	−8.48528			−1.29399			−0.646997	−	0.762493i	$-0.723975\pi$
$43$	−8.48528			−1.29399			−0.646997	+	0.762493i	$0.723975\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		0			0
$49$			7.00000i			1.00000i
$50$		0			0
$51$	−0.0710678	−	2.41421i	−0.00995148	−	0.338058i
$52$		0			0
$53$		0			0			−	1.00000i	$-0.5\pi$
$53$		0			0				1.00000i	$0.5\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	0.828427	−	0.828427i	0.109728	−	0.109728i
$58$		0			0
$59$	−14.1421			−1.84115			−0.920575	−	0.390567i	$-0.872279\pi$
$59$	−14.1421			−1.84115			−0.920575	+	0.390567i	$0.872279\pi$
$60$		0			0
$61$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$61$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$			8.48528i			1.03664i	0.855186	+	0.518321i	$0.173443\pi$
$67$			8.48528i			1.03664i	−0.855186	+	0.518321i	$0.826557\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$71$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$72$		0			0
$73$	7.48528	+	7.48528i	0.876086	+	0.876086i	0.993127	−	0.117041i	$-0.0373409\pi$
$73$	7.48528	+	7.48528i	0.876086	+	0.876086i	−0.117041	+	0.993127i	$0.537341\pi$
$74$		0			0
$75$	−2.07107	−	2.07107i	−0.239146	−	0.239146i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$79$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$80$		0			0
$81$	−6.02944			−0.669937
$82$		0			0
$83$	2.82843			0.310460			0.155230	−	0.987878i	$-0.450388\pi$
$83$	2.82843			0.310460			0.155230	+	0.987878i	$0.450388\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−5.65685			−0.599625			−0.299813	−	0.953998i	$-0.596924\pi$
$89$	−5.65685			−0.599625			−0.299813	+	0.953998i	$0.596924\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−3.48528	−	3.48528i	−0.353877	−	0.353877i	0.507673	−	0.861550i	$-0.330506\pi$
$97$	−3.48528	−	3.48528i	−0.353877	−	0.353877i	−0.861550	+	0.507673i	$0.830506\pi$
$98$		0			0
$99$	4.21320	−	4.21320i	0.423443	−	0.423443i
$100$		0			0
$101$		0			0		1.00000			$0$
$101$		0			0		−1.00000			$\pi$
$102$		0			0
$103$		0			0			−	1.00000i	$-0.5\pi$
$103$		0			0				1.00000i	$0.5\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.89949	+	6.89949i	−0.666999	+	0.666999i	−0.957020	−	0.290021i	$-0.906338\pi$
$107$	−6.89949	+	6.89949i	−0.666999	+	0.666999i	0.290021	+	0.957020i	$0.406338\pi$
$108$		0			0
$109$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$109$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−3.34315	+	3.34315i	−0.314497	+	0.314497i	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−3.34315	+	3.34315i	−0.314497	+	0.314497i	0.532152	+	0.846649i	$0.321383\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$		−	5.97056i		−	0.542778i
$122$		0			0
$123$		−	2.20101i		−	0.198458i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		1.00000			$0$
$127$		0			0		−1.00000			$\pi$
$128$		0			0
$129$	3.51472	−	3.51472i	0.309454	−	0.309454i
$130$		0			0
$131$	16.0711	−	16.0711i	1.40414	−	1.40414i	0.617802	−	0.786334i	$-0.288023\pi$
$131$	16.0711	−	16.0711i	1.40414	−	1.40414i	0.786334	−	0.617802i	$-0.211977\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	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
$139$	−6.75736	+	6.75736i	−0.573152	+	0.573152i	−0.933008	−	0.359856i	$-0.882826\pi$
$139$	−6.75736	+	6.75736i	−0.573152	+	0.573152i	0.359856	+	0.933008i	$0.382826\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−2.89949	−	2.89949i	−0.239146	−	0.239146i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$		0			0		1.00000			$0$
$151$		0			0		−1.00000			$\pi$
$152$		0			0
$153$	−7.97056	−	7.51472i	−0.644382	−	0.607529i
$154$		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
$162$		0			0
$163$	13.7279	+	13.7279i	1.07525	+	1.07525i	0.996928	+	0.0783260i	$0.0249575\pi$
$163$	13.7279	+	13.7279i	1.07525	+	1.07525i	0.0783260	+	0.996928i	$0.475042\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$167$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$168$		0			0
$169$	13.0000			1.00000
$170$		0			0
$171$		−	5.31371i		−	0.406349i
$172$		0			0
$173$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$173$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	5.85786	−	5.85786i	0.440304	−	0.440304i
$178$		0			0
$179$	18.0000			1.34538			0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000			1.34538			0.672692	+	0.739923i	$0.265138\pi$
$180$		0			0
$181$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$181$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	9.24264	−	0.272078i	0.675889	−	0.0198963i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0			−	1.00000i	$-0.5\pi$
$191$		0			0				1.00000i	$0.5\pi$
$192$		0			0
$193$	19.4853	−	19.4853i	1.40258	−	1.40258i	0.610784	−	0.791797i	$-0.290854\pi$
$193$	19.4853	−	19.4853i	1.40258	−	1.40258i	0.791797	−	0.610784i	$-0.209146\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$197$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$198$		0			0
$199$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$199$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$200$		0			0
$201$	−3.51472	−	3.51472i	−0.247909	−	0.247909i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3.17157	+	3.17157i	0.219382	+	0.219382i
$210$		0			0
$211$	19.7279	+	19.7279i	1.35813	+	1.35813i	0.876226	+	0.481900i	$0.160053\pi$
$211$	19.7279	+	19.7279i	1.35813	+	1.35813i	0.481900	+	0.876226i	$0.339947\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−6.20101			−0.419025
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		1.00000			$0$
$223$		0			0		−1.00000			$\pi$
$224$		0			0
$225$	−13.2843			−0.885618
$226$		0			0
$227$	13.5858	+	13.5858i	0.901720	+	0.901720i	0.995585	−	0.0938647i	$-0.0299221\pi$
$227$	13.5858	+	13.5858i	0.901720	+	0.901720i	−0.0938647	+	0.995585i	$0.529922\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−17.8284	−	17.8284i	−1.16798	−	1.16798i	−0.982683	−	0.185296i	$-0.940675\pi$
$233$	−17.8284	−	17.8284i	−1.16798	−	1.16798i	−0.185296	−	0.982683i	$-0.559325\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0			−	1.00000i	$-0.5\pi$
$239$		0			0				1.00000i	$0.5\pi$
$240$		0			0
$241$	4.51472	+	4.51472i	0.290819	+	0.290819i	0.837404	−	0.546585i	$-0.184072\pi$
$241$	4.51472	+	4.51472i	0.290819	+	0.290819i	−0.546585	+	0.837404i	$0.684072\pi$
$242$		0			0
$243$	9.52691	−	9.52691i	0.611152	−	0.611152i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	−1.17157	+	1.17157i	−0.0742454	+	0.0742454i
$250$		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
$256$		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
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	2.34315	−	2.34315i	0.143398	−	0.143398i
$268$		0			0
$269$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$269$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	7.92893	−	7.92893i	0.478133	−	0.478133i
$276$		0			0
$277$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$277$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	18.0000i		−	1.07379i	−0.843649	−	0.536895i	$-0.819597\pi$
$281$		−	18.0000i		−	1.07379i	0.843649	−	0.536895i	$-0.180403\pi$
$282$		0			0
$283$	1.72792	+	1.72792i	0.102714	+	0.102714i	0.756596	−	0.653882i	$-0.226861\pi$
$283$	1.72792	+	1.72792i	0.102714	+	0.102714i	−0.653882	+	0.756596i	$0.726861\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.00000	−	16.9706i	−0.0588235	−	0.998268i
$290$		0			0
$291$	2.88730			0.169257
$292$		0			0
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$			7.43146i			0.431217i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$			34.0000i			1.94048i	0.242140	+	0.970241i	$0.422151\pi$
$307$			34.0000i			1.94048i	−0.242140	+	0.970241i	$0.577849\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$311$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$312$		0			0
$313$	21.9706	−	21.9706i	1.24185	−	1.24185i	0.282617	−	0.959233i	$-0.408798\pi$
$313$	21.9706	−	21.9706i	1.24185	−	1.24185i	0.959233	−	0.282617i	$-0.0912024\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$317$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		−	5.71573i		−	0.319021i
$322$		0			0
$323$	5.65685	−	6.00000i	0.314756	−	0.333849i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−25.4558			−1.39918			−0.699590	−	0.714545i	$-0.746634\pi$
$331$	−25.4558			−1.39918			−0.699590	+	0.714545i	$0.746634\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	23.9706	+	23.9706i	1.30576	+	1.30576i	0.924445	+	0.381314i	$0.124528\pi$
$337$	23.9706	+	23.9706i	1.30576	+	1.30576i	0.381314	+	0.924445i	$0.375472\pi$
$338$		0			0
$339$		−	2.76955i		−	0.150421i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	21.3848	+	21.3848i	1.14799	+	1.14799i	0.986947	+	0.161048i	$0.0514875\pi$
$347$	21.3848	+	21.3848i	1.14799	+	1.14799i	0.161048	+	0.986947i	$0.448512\pi$
$348$		0			0
$349$		0			0			−	1.00000i	$-0.5\pi$
$349$		0			0				1.00000i	$0.5\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−22.6274			−1.20434			−0.602168	−	0.798369i	$-0.705696\pi$
$353$	−22.6274			−1.20434			−0.602168	+	0.798369i	$0.705696\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	−15.0000			−0.789474
$362$		0			0
$363$	2.47309	+	2.47309i	0.129803	+	0.129803i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$367$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$368$		0			0
$369$	−7.05887	−	7.05887i	−0.367470	−	0.367470i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		1.00000			$0$
$373$		0			0		−1.00000			$\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−23.2426	+	23.2426i	−1.19389	+	1.19389i	−0.217930	+	0.975964i	$0.569930\pi$
$379$	−23.2426	+	23.2426i	−1.19389	+	1.19389i	−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			$\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		−	22.5442i		−	1.14598i
$388$		0			0
$389$		0			0			−	1.00000i	$-0.5\pi$
$389$		0			0				1.00000i	$0.5\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$			13.3137i			0.671588i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$397$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−22.7990	+	22.7990i	−1.13853	+	1.13853i	−0.149813	+	0.988714i	$0.547867\pi$
$401$	−22.7990	+	22.7990i	−1.13853	+	1.13853i	−0.988714	+	0.149813i	$0.952133\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	33.9411			1.67828			0.839140	−	0.543915i	$-0.183059\pi$
$409$	33.9411			1.67828			0.839140	+	0.543915i	$0.183059\pi$
$410$		0			0
$411$	−2.48528	+	2.48528i	−0.122590	+	0.122590i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$		−	5.59798i		−	0.274134i
$418$		0			0
$419$	−27.3848	−	27.3848i	−1.33783	−	1.33783i	−0.898155	−	0.439679i	$-0.855092\pi$
$419$	−27.3848	−	27.3848i	−1.33783	−	1.33783i	−0.439679	−	0.898155i	$-0.644908\pi$
$420$		0			0
$421$		0			0		1.00000			$0$
$421$		0			0		−1.00000			$\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−15.0000	−	14.1421i	−0.727607	−	0.685994i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$431$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$432$		0			0
$433$			38.0000i			1.82616i	0.407777	+	0.913082i	$0.366304\pi$
$433$			38.0000i			1.82616i	−0.407777	+	0.913082i	$0.633696\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$439$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$440$		0			0
$441$	−18.5980			−0.885618
$442$		0			0
$443$		−	2.82843i		−	0.134383i	−0.997740	−	0.0671913i	$-0.978596\pi$
$443$		−	2.82843i		−	0.134383i	0.997740	−	0.0671913i	$-0.0214038\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	23.8284	−	23.8284i	1.12453	−	1.12453i	0.133482	−	0.991051i	$-0.457384\pi$
$449$	23.8284	−	23.8284i	1.12453	−	1.12453i	0.991051	−	0.133482i	$-0.0426157\pi$
$450$		0			0
$451$	8.42641			0.396784
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$			26.0000i			1.21623i	0.793849	+	0.608114i	$0.208074\pi$
$457$			26.0000i			1.21623i	−0.793849	+	0.608114i	$0.791926\pi$
$458$		0			0
$459$	13.6569	−	0.402020i	0.637447	−	0.0187647i
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−30.0000			−1.38823			−0.694117	−	0.719862i	$-0.744205\pi$
$467$	−30.0000			−1.38823			−0.694117	+	0.719862i	$0.744205\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	13.4558	+	13.4558i	0.618700	+	0.618700i
$474$		0			0
$475$		−	10.0000i		−	0.458831i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$479$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$487$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$488$		0			0
$489$	−11.3726			−0.514286
$490$		0			0
$491$	−42.0000			−1.89543			−0.947717	−	0.319113i	$-0.896615\pi$
$491$	−42.0000			−1.89543			−0.947717	+	0.319113i	$0.896615\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	14.2132	+	14.2132i	0.636270	+	0.636270i	0.949633	−	0.313363i	$-0.101456\pi$
$499$	14.2132	+	14.2132i	0.636270	+	0.636270i	−0.313363	+	0.949633i	$0.601456\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$503$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−5.38478	+	5.38478i	−0.239146	+	0.239146i
$508$		0			0
$509$		0			0		1.00000			$0$
$509$		0			0		−1.00000			$\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	4.68629	+	4.68629i	0.206905	+	0.206905i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−25.6274	+	25.6274i	−1.12276	+	1.12276i	−0.131432	+	0.991325i	$0.541958\pi$
$521$	−25.6274	+	25.6274i	−1.12276	+	1.12276i	−0.991325	+	0.131432i	$0.958042\pi$
$522$		0			0
$523$		−	25.4558i		−	1.11311i	−0.830812	−	0.556553i	$-0.812124\pi$
$523$		−	25.4558i		−	1.11311i	0.830812	−	0.556553i	$-0.187876\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$			23.0000i			1.00000i
$530$		0			0
$531$		−	37.5736i		−	1.63056i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−7.45584	+	7.45584i	−0.321743	+	0.321743i
$538$		0			0
$539$	11.1005	−	11.1005i	0.478133	−	0.478133i
$540$		0			0
$541$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$541$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−27.2426	+	27.2426i	−1.16481	+	1.16481i	−0.181402	+	0.983409i	$0.558064\pi$
$547$	−27.2426	+	27.2426i	−1.16481	+	1.16481i	−0.983409	+	0.181402i	$0.941936\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	−3.71573	+	3.94113i	−0.156878	+	0.166394i
$562$		0			0
$563$	−36.7696			−1.54965			−0.774826	−	0.632175i	$-0.782163\pi$
$563$	−36.7696			−1.54965			−0.774826	+	0.632175i	$0.782163\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		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
$571$	−10.2132	−	10.2132i	−0.427409	−	0.427409i	0.460336	−	0.887745i	$-0.347729\pi$
$571$	−10.2132	−	10.2132i	−0.427409	−	0.427409i	−0.887745	+	0.460336i	$0.847729\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	34.0000			1.41544			0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000			1.41544			0.707719	+	0.706494i	$0.249724\pi$
$578$		0			0
$579$			16.1421i			0.670844i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	6.00000			0.247647			0.123823	−	0.992304i	$-0.460484\pi$
$587$	6.00000			0.247647			0.123823	+	0.992304i	$0.460484\pi$
$588$		0			0
$589$		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	$0.620500\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0			−	1.00000i	$-0.5\pi$
$599$		0			0				1.00000i	$0.5\pi$
$600$		0			0
$601$	14.5147	−	14.5147i	0.592068	−	0.592068i	−0.346122	−	0.938190i	$-0.612502\pi$
$601$	14.5147	−	14.5147i	0.592068	−	0.592068i	0.938190	+	0.346122i	$0.112502\pi$
$602$		0			0
$603$	−22.5442			−0.918069
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$607$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		1.00000			$0$
$613$		0			0		−1.00000			$\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	34.7990	+	34.7990i	1.40095	+	1.40095i	0.797077	+	0.603877i	$0.206378\pi$
$617$	34.7990	+	34.7990i	1.40095	+	1.40095i	0.603877	+	0.797077i	$0.293622\pi$
$618$		0			0
$619$	34.2132	+	34.2132i	1.37514	+	1.37514i	0.852631	+	0.522514i	$0.175006\pi$
$619$	34.2132	+	34.2132i	1.37514	+	1.37514i	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
$626$		0			0
$627$	−2.62742			−0.104929
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		1.00000			$0$
$631$		0			0		−1.00000			$\pi$
$632$		0			0
$633$	−16.3431			−0.649582
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−35.1421	−	35.1421i	−1.38803	−	1.38803i	−0.829450	−	0.558581i	$-0.811346\pi$
$641$	−35.1421	−	35.1421i	−1.38803	−	1.38803i	−0.558581	−	0.829450i	$-0.688654\pi$
$642$		0			0
$643$	−29.2426	+	29.2426i	−1.15322	+	1.15322i	−0.167313	+	0.985904i	$0.553509\pi$
$643$	−29.2426	+	29.2426i	−1.15322	+	1.15322i	−0.985904	+	0.167313i	$0.946491\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$	22.4264	+	22.4264i	0.880313	+	0.880313i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$653$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−19.8873	+	19.8873i	−0.775878	+	0.775878i
$658$		0			0
$659$		−	18.0000i		−	0.701180i	−0.936529	−	0.350590i	$-0.885981\pi$
$659$		−	18.0000i		−	0.701180i	0.936529	−	0.350590i	$-0.114019\pi$
$660$		0			0
$661$		0			0			−	1.00000i	$-0.5\pi$
$661$		0			0				1.00000i	$0.5\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	20.4558	−	20.4558i	0.788515	−	0.788515i	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	20.4558	−	20.4558i	0.788515	−	0.788515i	0.981251	+	0.192736i	$0.0617360\pi$
$674$		0			0
$675$	11.7157	−	11.7157i	0.450939	−	0.450939i
$676$		0			0
$677$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$677$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−11.2548			−0.431286
$682$		0			0
$683$	−5.44365	+	5.44365i	−0.208296	+	0.208296i	−0.803543	−	0.595247i	$-0.797054\pi$
$683$	−5.44365	+	5.44365i	−0.208296	+	0.208296i	0.595247	+	0.803543i	$0.297054\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−35.7279	−	35.7279i	−1.35915	−	1.35915i	−0.874961	−	0.484193i	$-0.839113\pi$
$691$	−35.7279	−	35.7279i	−1.35915	−	1.35915i	−0.484193	−	0.874961i	$-0.660887\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−0.455844	−	15.4853i	−0.0172663	−	0.586547i
$698$		0			0
$699$	14.7696			0.558636
$700$		0			0
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$709$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$719$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−3.74012			−0.139096
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0			−	1.00000i	$-0.5\pi$
$727$		0			0				1.00000i	$0.5\pi$
$728$		0			0
$729$		−	10.1960i		−	0.377628i
$730$		0			0
$731$	24.0000	−	25.4558i	0.887672	−	0.941518i
$732$		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
$737$	13.4558	−	13.4558i	0.495652	−	0.495652i
$738$		0			0
$739$	−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		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$			7.51472i			0.274949i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$751$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$752$		0			0
$753$	12.8873	+	12.8873i	0.469639	+	0.469639i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0			−	1.00000i	$-0.5\pi$
$757$		0			0				1.00000i	$0.5\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−54.0000			−1.95750			−0.978749	−	0.205061i	$-0.934261\pi$
$761$	−54.0000			−1.95750			−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
$768$		0			0
$769$	−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
$771$	4.68629	+	4.68629i	0.168773	+	0.168773i
$772$		0			0
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	5.31371	−	5.31371i	0.190383	−	0.190383i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−37.7279	+	37.7279i	−1.34486	+	1.34486i	−0.453701	+	0.891154i	$0.649897\pi$
$787$	−37.7279	+	37.7279i	−1.34486	+	1.34486i	−0.891154	+	0.453701i	$0.850103\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		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
$800$		0			0
$801$		−	15.0294i		−	0.531039i
$802$		0			0
$803$		−	23.7401i		−	0.837770i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	25.2843	−	25.2843i	0.888948	−	0.888948i	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	25.2843	−	25.2843i	0.888948	−	0.888948i	0.994422	+	0.105474i	$0.0336361\pi$
$810$		0			0
$811$	40.2132	−	40.2132i	1.41208	−	1.41208i	0.667180	−	0.744896i	$-0.267501\pi$
$811$	40.2132	−	40.2132i	1.41208	−	1.41208i	0.744896	−	0.667180i	$-0.232499\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	16.9706			0.593725
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$821$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$822$		0			0
$823$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$823$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$824$		0			0
$825$			6.56854i			0.228687i
$826$		0			0
$827$	36.8995	+	36.8995i	1.28312	+	1.28312i	0.938882	+	0.344239i	$0.111863\pi$
$827$	36.8995	+	36.8995i	1.28312	+	1.28312i	0.344239	+	0.938882i	$0.388137\pi$
$828$		0			0
$829$		0			0		1.00000			$0$
$829$		0			0		−1.00000			$\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−21.0000	−	19.7990i	−0.727607	−	0.685994i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$839$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$840$		0			0
$841$			29.0000i			1.00000i
$842$		0			0
$843$	7.45584	+	7.45584i	0.256793	+	0.256793i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−1.43146			−0.0491275
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$853$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	38.3137	−	38.3137i	1.30877	−	1.30877i	0.386469	−	0.922302i	$-0.373695\pi$
$857$	38.3137	−	38.3137i	1.30877	−	1.30877i	0.922302	−	0.386469i	$-0.126305\pi$
$858$		0			0
$859$	8.48528			0.289514			0.144757	−	0.989467i	$-0.453760\pi$
$859$	8.48528			0.289514			0.144757	+	0.989467i	$0.453760\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	7.44365	+	6.61522i	0.252800	+	0.224665i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	9.25988	−	9.25988i	0.313400	−	0.313400i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$877$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−19.2843	−	19.2843i	−0.649704	−	0.649704i	0.303218	−	0.952921i	$-0.401939\pi$
$881$	−19.2843	−	19.2843i	−0.649704	−	0.649704i	−0.952921	+	0.303218i	$0.901939\pi$
$882$		0			0
$883$			59.3970i			1.99887i	0.0336527	+	0.999434i	$0.489286\pi$
$883$			59.3970i			1.99887i	−0.0336527	+	0.999434i	$0.510714\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$887$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	9.56140	+	9.56140i	0.320319	+	0.320319i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$		0			0
$906$		0			0
$907$	34.6985	+	34.6985i	1.15214	+	1.15214i	0.986122	+	0.166022i	$0.0530924\pi$
$907$	34.6985	+	34.6985i	1.15214	+	1.15214i	0.166022	+	0.986122i	$0.446908\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$911$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$912$		0			0
$913$	−4.48528	−	4.48528i	−0.148441	−	0.148441i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0			−	1.00000i	$-0.5\pi$
$919$		0			0				1.00000i	$0.5\pi$
$920$		0			0
$921$	−14.0833	−	14.0833i	−0.464059	−	0.464059i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	12.8579	−	12.8579i	0.421853	−	0.421853i	−0.463988	−	0.885841i	$-0.653582\pi$
$929$	12.8579	−	12.8579i	0.421853	−	0.421853i	0.885841	+	0.463988i	$0.153582\pi$
$930$		0			0
$931$		−	14.0000i		−	0.458831i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	34.0000i		−	1.11073i	−0.831606	−	0.555366i	$-0.812578\pi$
$937$		−	34.0000i		−	1.11073i	0.831606	−	0.555366i	$-0.187422\pi$
$938$		0			0
$939$			18.2010i			0.593968i
$940$		0			0
$941$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$941$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	41.8701	−	41.8701i	1.36059	−	1.36059i	0.487435	−	0.873160i	$-0.337933\pi$
$947$	41.8701	−	41.8701i	1.36059	−	1.36059i	0.873160	−	0.487435i	$-0.162067\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	45.2548			1.46595			0.732974	−	0.680257i	$-0.238132\pi$
$953$	45.2548			1.46595			0.732974	+	0.680257i	$0.238132\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$		−	31.0000i		−	1.00000i
$962$		0			0
$963$	−18.3310	−	18.3310i	−0.590707	−	0.590707i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		1.00000			$0$
$967$		0			0		−1.00000			$\pi$
$968$		0			0
$969$	0.142136	+	4.82843i	0.00456605	+	0.155111i
$970$		0			0
$971$	31.1127			0.998454			0.499227	−	0.866471i	$-0.333617\pi$
$971$	31.1127			0.998454			0.499227	+	0.866471i	$0.333617\pi$
$972$		0			0
$973$		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	$0.530598\pi$
$978$		0			0
$979$	8.97056	+	8.97056i	0.286700	+	0.286700i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$983$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$991$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$992$		0			0
$993$	10.5442	−	10.5442i	0.334609	−	0.334609i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$997$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1088.2.m.g.353.1	yes	4
4.3	odd	2		1088.2.m.f.353.2	yes	4
8.3	odd	2	CM	1088.2.m.g.353.1	yes	4
8.5	even	2		1088.2.m.f.353.2	yes	4
17.4	even	4		1088.2.m.f.225.2	✓	4
68.55	odd	4	inner	1088.2.m.g.225.1	yes	4
136.21	even	4	inner	1088.2.m.g.225.1	yes	4
136.123	odd	4		1088.2.m.f.225.2	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1088.2.m.f.225.2	✓	4	17.4	even	4
1088.2.m.f.225.2	✓	4	136.123	odd	4
1088.2.m.f.353.2	yes	4	4.3	odd	2
1088.2.m.f.353.2	yes	4	8.5	even	2
1088.2.m.g.225.1	yes	4	68.55	odd	4	inner
1088.2.m.g.225.1	yes	4	136.21	even	4	inner
1088.2.m.g.353.1	yes	4	1.1	even	1	trivial
1088.2.m.g.353.1	yes	4	8.3	odd	2	CM

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:	\( N \)	\(=\)	\( 1088 = 2^{6} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1088.m (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.414214 + 0.414214i) q^{3} +2.65685i q^{9} +O(q^{10})\)	\(q+(-0.414214 + 0.414214i) q^{3} +2.65685i q^{9} +(-1.58579 - 1.58579i) q^{11} +(-2.82843 + 3.00000i) q^{17} -2.00000 q^{19} +5.00000i q^{25} +(-2.34315 - 2.34315i) q^{27} +1.31371 q^{33} +(-2.65685 + 2.65685i) q^{41} -8.48528 q^{43} +7.00000i q^{49} +(-0.0710678 - 2.41421i) q^{51} +(0.828427 - 0.828427i) q^{57} -14.1421 q^{59} +8.48528i q^{67} +(7.48528 + 7.48528i) q^{73} +(-2.07107 - 2.07107i) q^{75} -6.02944 q^{81} +2.82843 q^{83} -5.65685 q^{89} +(-3.48528 - 3.48528i) q^{97} +(4.21320 - 4.21320i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3}+O(q^{10}) \) 4 * q + 4 * q^3	\( 4 q + 4 q^{3} - 12 q^{11} - 8 q^{19} - 32 q^{27} - 40 q^{33} + 12 q^{41} + 28 q^{51} - 8 q^{57} - 4 q^{73} + 20 q^{75} - 92 q^{81} + 20 q^{97} - 68 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 12 * q^11 - 8 * q^19 - 32 * q^27 - 40 * q^33 + 12 * q^41 + 28 * q^51 - 8 * q^57 - 4 * q^73 + 20 * q^75 - 92 * q^81 + 20 * q^97 - 68 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more