LMFDB - Embedded newform 1088.2.a.o.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1088,2,Mod(1,1088)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1088, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1088.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1088 = 2^{6} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1088.a (trivial)

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:	yes
Analytic conductor:	$8.68772373992$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 136)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	1088.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.23607 q^{3} -2.00000 q^{5} +1.23607 q^{7} -1.47214 q^{9} +O(q^{10})$	$q+1.23607 q^{3} -2.00000 q^{5} +1.23607 q^{7} -1.47214 q^{9} -1.23607 q^{11} -4.47214 q^{13} -2.47214 q^{15} +1.00000 q^{17} -6.47214 q^{19} +1.52786 q^{21} +1.23607 q^{23} -1.00000 q^{25} -5.52786 q^{27} -2.00000 q^{29} -1.23607 q^{31} -1.52786 q^{33} -2.47214 q^{35} +10.9443 q^{37} -5.52786 q^{39} +2.00000 q^{41} -1.52786 q^{43} +2.94427 q^{45} -12.9443 q^{47} -5.47214 q^{49} +1.23607 q^{51} +2.00000 q^{53} +2.47214 q^{55} -8.00000 q^{57} +14.4721 q^{59} -6.94427 q^{61} -1.81966 q^{63} +8.94427 q^{65} -12.0000 q^{67} +1.52786 q^{69} -9.23607 q^{71} +14.9443 q^{73} -1.23607 q^{75} -1.52786 q^{77} -11.7082 q^{79} -2.41641 q^{81} +1.52786 q^{83} -2.00000 q^{85} -2.47214 q^{87} -7.52786 q^{89} -5.52786 q^{91} -1.52786 q^{93} +12.9443 q^{95} +2.00000 q^{97} +1.81966 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 4 q^{5} - 2 q^{7} + 6 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 4 * q^5 - 2 * q^7 + 6 * q^9	$ 2 q - 2 q^{3} - 4 q^{5} - 2 q^{7} + 6 q^{9} + 2 q^{11} + 4 q^{15} + 2 q^{17} - 4 q^{19} + 12 q^{21} - 2 q^{23} - 2 q^{25} - 20 q^{27} - 4 q^{29} + 2 q^{31} - 12 q^{33} + 4 q^{35} + 4 q^{37} - 20 q^{39} + 4 q^{41} - 12 q^{43} - 12 q^{45} - 8 q^{47} - 2 q^{49} - 2 q^{51} + 4 q^{53} - 4 q^{55} - 16 q^{57} + 20 q^{59} + 4 q^{61} - 26 q^{63} - 24 q^{67} + 12 q^{69} - 14 q^{71} + 12 q^{73} + 2 q^{75} - 12 q^{77} - 10 q^{79} + 22 q^{81} + 12 q^{83} - 4 q^{85} + 4 q^{87} - 24 q^{89} - 20 q^{91} - 12 q^{93} + 8 q^{95} + 4 q^{97} + 26 q^{99}+O(q^{100}) $ 2 * q - 2 * q^3 - 4 * q^5 - 2 * q^7 + 6 * q^9 + 2 * q^11 + 4 * q^15 + 2 * q^17 - 4 * q^19 + 12 * q^21 - 2 * q^23 - 2 * q^25 - 20 * q^27 - 4 * q^29 + 2 * q^31 - 12 * q^33 + 4 * q^35 + 4 * q^37 - 20 * q^39 + 4 * q^41 - 12 * q^43 - 12 * q^45 - 8 * q^47 - 2 * q^49 - 2 * q^51 + 4 * q^53 - 4 * q^55 - 16 * q^57 + 20 * q^59 + 4 * q^61 - 26 * q^63 - 24 * q^67 + 12 * q^69 - 14 * q^71 + 12 * q^73 + 2 * q^75 - 12 * q^77 - 10 * q^79 + 22 * q^81 + 12 * q^83 - 4 * q^85 + 4 * q^87 - 24 * q^89 - 20 * q^91 - 12 * q^93 + 8 * q^95 + 4 * q^97 + 26 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.23607	0.713644	0.356822	−	0.934172i	$-0.383860\pi$
$3$	1.23607	0.713644	0.356822	+	0.934172i	$0.383860\pi$
$4$	0	0
$5$	−2.00000	−0.894427	−0.447214	−	0.894427i	$-0.647584\pi$
$5$	−2.00000	−0.894427	−0.447214	+	0.894427i	$0.647584\pi$
$6$	0	0
$7$	1.23607	0.467190	0.233595	−	0.972334i	$-0.424951\pi$
$7$	1.23607	0.467190	0.233595	+	0.972334i	$0.424951\pi$
$8$	0	0
$9$	−1.47214	−0.490712
$10$	0	0
$11$	−1.23607	−0.372689	−0.186344	−	0.982485i	$-0.559664\pi$
$11$	−1.23607	−0.372689	−0.186344	+	0.982485i	$0.559664\pi$
$12$	0	0
$13$	−4.47214	−1.24035	−0.620174	−	0.784465i	$-0.712938\pi$
$13$	−4.47214	−1.24035	−0.620174	+	0.784465i	$0.712938\pi$
$14$	0	0
$15$	−2.47214	−0.638303
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−6.47214	−1.48481	−0.742405	−	0.669951i	$-0.766315\pi$
$19$	−6.47214	−1.48481	−0.742405	+	0.669951i	$0.766315\pi$
$20$	0	0
$21$	1.52786	0.333407
$22$	0	0
$23$	1.23607	0.257738	0.128869	−	0.991662i	$-0.458865\pi$
$23$	1.23607	0.257738	0.128869	+	0.991662i	$0.458865\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	−5.52786	−1.06384
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−1.23607	−0.222004	−0.111002	−	0.993820i	$-0.535406\pi$
$31$	−1.23607	−0.222004	−0.111002	+	0.993820i	$0.535406\pi$
$32$	0	0
$33$	−1.52786	−0.265967
$34$	0	0
$35$	−2.47214	−0.417867
$36$	0	0
$37$	10.9443	1.79923	0.899614	−	0.436687i	$-0.143848\pi$
$37$	10.9443	1.79923	0.899614	+	0.436687i	$0.143848\pi$
$38$	0	0
$39$	−5.52786	−0.885167
$40$	0	0
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−1.52786	−0.232997	−0.116499	−	0.993191i	$-0.537167\pi$
$43$	−1.52786	−0.232997	−0.116499	+	0.993191i	$0.537167\pi$
$44$	0	0
$45$	2.94427	0.438906
$46$	0	0
$47$	−12.9443	−1.88812	−0.944058	−	0.329779i	$-0.893026\pi$
$47$	−12.9443	−1.88812	−0.944058	+	0.329779i	$0.893026\pi$
$48$	0	0
$49$	−5.47214	−0.781734
$50$	0	0
$51$	1.23607	0.173084
$52$	0	0
$53$	2.00000	0.274721	0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000	0.274721	0.137361	+	0.990521i	$0.456138\pi$
$54$	0	0
$55$	2.47214	0.333343
$56$	0	0
$57$	−8.00000	−1.05963
$58$	0	0
$59$	14.4721	1.88411	0.942056	−	0.335456i	$-0.108890\pi$
$59$	14.4721	1.88411	0.942056	+	0.335456i	$0.108890\pi$
$60$	0	0
$61$	−6.94427	−0.889123	−0.444561	−	0.895748i	$-0.646640\pi$
$61$	−6.94427	−0.889123	−0.444561	+	0.895748i	$0.646640\pi$
$62$	0	0
$63$	−1.81966	−0.229256
$64$	0	0
$65$	8.94427	1.10940
$66$	0	0
$67$	−12.0000	−1.46603	−0.733017	−	0.680211i	$-0.761888\pi$
$67$	−12.0000	−1.46603	−0.733017	+	0.680211i	$0.761888\pi$
$68$	0	0
$69$	1.52786	0.183933
$70$	0	0
$71$	−9.23607	−1.09612	−0.548060	−	0.836439i	$-0.684633\pi$
$71$	−9.23607	−1.09612	−0.548060	+	0.836439i	$0.684633\pi$
$72$	0	0
$73$	14.9443	1.74909	0.874547	−	0.484940i	$-0.161159\pi$
$73$	14.9443	1.74909	0.874547	+	0.484940i	$0.161159\pi$
$74$	0	0
$75$	−1.23607	−0.142729
$76$	0	0
$77$	−1.52786	−0.174116
$78$	0	0
$79$	−11.7082	−1.31728	−0.658638	−	0.752460i	$-0.728867\pi$
$79$	−11.7082	−1.31728	−0.658638	+	0.752460i	$0.728867\pi$
$80$	0	0
$81$	−2.41641	−0.268490
$82$	0	0
$83$	1.52786	0.167705	0.0838524	−	0.996478i	$-0.473278\pi$
$83$	1.52786	0.167705	0.0838524	+	0.996478i	$0.473278\pi$
$84$	0	0
$85$	−2.00000	−0.216930
$86$	0	0
$87$	−2.47214	−0.265041
$88$	0	0
$89$	−7.52786	−0.797952	−0.398976	−	0.916961i	$-0.630634\pi$
$89$	−7.52786	−0.797952	−0.398976	+	0.916961i	$0.630634\pi$
$90$	0	0
$91$	−5.52786	−0.579478
$92$	0	0
$93$	−1.52786	−0.158432
$94$	0	0
$95$	12.9443	1.32805
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	0	0
$99$	1.81966	0.182883
$100$	0	0
$101$	13.4164	1.33498	0.667491	−	0.744618i	$-0.267368\pi$
$101$	13.4164	1.33498	0.667491	+	0.744618i	$0.267368\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	0	0
$105$	−3.05573	−0.298209
$106$	0	0
$107$	−3.70820	−0.358486	−0.179243	−	0.983805i	$-0.557365\pi$
$107$	−3.70820	−0.358486	−0.179243	+	0.983805i	$0.557365\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	13.5279	1.28401
$112$	0	0
$113$	14.9443	1.40584	0.702919	−	0.711269i	$-0.251879\pi$
$113$	14.9443	1.40584	0.702919	+	0.711269i	$0.251879\pi$
$114$	0	0
$115$	−2.47214	−0.230528
$116$	0	0
$117$	6.58359	0.608653
$118$	0	0
$119$	1.23607	0.113310
$120$	0	0
$121$	−9.47214	−0.861103
$122$	0	0
$123$	2.47214	0.222905
$124$	0	0
$125$	12.0000	1.07331
$126$	0	0
$127$	−2.47214	−0.219367	−0.109683	−	0.993967i	$-0.534984\pi$
$127$	−2.47214	−0.219367	−0.109683	+	0.993967i	$0.534984\pi$
$128$	0	0
$129$	−1.88854	−0.166277
$130$	0	0
$131$	19.1246	1.67093	0.835463	−	0.549547i	$-0.185200\pi$
$131$	19.1246	1.67093	0.835463	+	0.549547i	$0.185200\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	0	0
$135$	11.0557	0.951526
$136$	0	0
$137$	−7.52786	−0.643149	−0.321574	−	0.946884i	$-0.604212\pi$
$137$	−7.52786	−0.643149	−0.321574	+	0.946884i	$0.604212\pi$
$138$	0	0
$139$	−6.76393	−0.573709	−0.286855	−	0.957974i	$-0.592610\pi$
$139$	−6.76393	−0.573709	−0.286855	+	0.957974i	$0.592610\pi$
$140$	0	0
$141$	−16.0000	−1.34744
$142$	0	0
$143$	5.52786	0.462263
$144$	0	0
$145$	4.00000	0.332182
$146$	0	0
$147$	−6.76393	−0.557880
$148$	0	0
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−5.52786	−0.449851	−0.224926	−	0.974376i	$-0.572214\pi$
$151$	−5.52786	−0.449851	−0.224926	+	0.974376i	$0.572214\pi$
$152$	0	0
$153$	−1.47214	−0.119015
$154$	0	0
$155$	2.47214	0.198567
$156$	0	0
$157$	11.8885	0.948809	0.474405	−	0.880307i	$-0.342663\pi$
$157$	11.8885	0.948809	0.474405	+	0.880307i	$0.342663\pi$
$158$	0	0
$159$	2.47214	0.196053
$160$	0	0
$161$	1.52786	0.120413
$162$	0	0
$163$	−22.1803	−1.73730	−0.868649	−	0.495428i	$-0.835011\pi$
$163$	−22.1803	−1.73730	−0.868649	+	0.495428i	$0.835011\pi$
$164$	0	0
$165$	3.05573	0.237888
$166$	0	0
$167$	19.7082	1.52507	0.762533	−	0.646949i	$-0.223955\pi$
$167$	19.7082	1.52507	0.762533	+	0.646949i	$0.223955\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	9.52786	0.728614
$172$	0	0
$173$	−2.00000	−0.152057	−0.0760286	−	0.997106i	$-0.524224\pi$
$173$	−2.00000	−0.152057	−0.0760286	+	0.997106i	$0.524224\pi$
$174$	0	0
$175$	−1.23607	−0.0934380
$176$	0	0
$177$	17.8885	1.34459
$178$	0	0
$179$	14.4721	1.08170	0.540849	−	0.841120i	$-0.318103\pi$
$179$	14.4721	1.08170	0.540849	+	0.841120i	$0.318103\pi$
$180$	0	0
$181$	−10.0000	−0.743294	−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000	−0.743294	−0.371647	+	0.928374i	$0.621207\pi$
$182$	0	0
$183$	−8.58359	−0.634517
$184$	0	0
$185$	−21.8885	−1.60928
$186$	0	0
$187$	−1.23607	−0.0903902
$188$	0	0
$189$	−6.83282	−0.497014
$190$	0	0
$191$	12.9443	0.936615	0.468307	−	0.883566i	$-0.344864\pi$
$191$	12.9443	0.936615	0.468307	+	0.883566i	$0.344864\pi$
$192$	0	0
$193$	14.9443	1.07571	0.537856	−	0.843037i	$-0.319234\pi$
$193$	14.9443	1.07571	0.537856	+	0.843037i	$0.319234\pi$
$194$	0	0
$195$	11.0557	0.791717
$196$	0	0
$197$	−6.94427	−0.494759	−0.247379	−	0.968919i	$-0.579569\pi$
$197$	−6.94427	−0.494759	−0.247379	+	0.968919i	$0.579569\pi$
$198$	0	0
$199$	−22.1803	−1.57232	−0.786161	−	0.618021i	$-0.787935\pi$
$199$	−22.1803	−1.57232	−0.786161	+	0.618021i	$0.787935\pi$
$200$	0	0
$201$	−14.8328	−1.04623
$202$	0	0
$203$	−2.47214	−0.173510
$204$	0	0
$205$	−4.00000	−0.279372
$206$	0	0
$207$	−1.81966	−0.126475
$208$	0	0
$209$	8.00000	0.553372
$210$	0	0
$211$	−1.23607	−0.0850944	−0.0425472	−	0.999094i	$-0.513547\pi$
$211$	−1.23607	−0.0850944	−0.0425472	+	0.999094i	$0.513547\pi$
$212$	0	0
$213$	−11.4164	−0.782239
$214$	0	0
$215$	3.05573	0.208399
$216$	0	0
$217$	−1.52786	−0.103718
$218$	0	0
$219$	18.4721	1.24823
$220$	0	0
$221$	−4.47214	−0.300828
$222$	0	0
$223$	−5.52786	−0.370173	−0.185087	−	0.982722i	$-0.559257\pi$
$223$	−5.52786	−0.370173	−0.185087	+	0.982722i	$0.559257\pi$
$224$	0	0
$225$	1.47214	0.0981424
$226$	0	0
$227$	−1.23607	−0.0820407	−0.0410204	−	0.999158i	$-0.513061\pi$
$227$	−1.23607	−0.0820407	−0.0410204	+	0.999158i	$0.513061\pi$
$228$	0	0
$229$	24.4721	1.61716	0.808582	−	0.588383i	$-0.200235\pi$
$229$	24.4721	1.61716	0.808582	+	0.588383i	$0.200235\pi$
$230$	0	0
$231$	−1.88854	−0.124257
$232$	0	0
$233$	−23.8885	−1.56499	−0.782495	−	0.622657i	$-0.786053\pi$
$233$	−23.8885	−1.56499	−0.782495	+	0.622657i	$0.786053\pi$
$234$	0	0
$235$	25.8885	1.68878
$236$	0	0
$237$	−14.4721	−0.940066
$238$	0	0
$239$	−17.8885	−1.15711	−0.578557	−	0.815642i	$-0.696384\pi$
$239$	−17.8885	−1.15711	−0.578557	+	0.815642i	$0.696384\pi$
$240$	0	0
$241$	−1.05573	−0.0680054	−0.0340027	−	0.999422i	$-0.510825\pi$
$241$	−1.05573	−0.0680054	−0.0340027	+	0.999422i	$0.510825\pi$
$242$	0	0
$243$	13.5967	0.872232
$244$	0	0
$245$	10.9443	0.699204
$246$	0	0
$247$	28.9443	1.84168
$248$	0	0
$249$	1.88854	0.119682
$250$	0	0
$251$	−8.94427	−0.564557	−0.282279	−	0.959332i	$-0.591090\pi$
$251$	−8.94427	−0.564557	−0.282279	+	0.959332i	$0.591090\pi$
$252$	0	0
$253$	−1.52786	−0.0960560
$254$	0	0
$255$	−2.47214	−0.154811
$256$	0	0
$257$	−22.3607	−1.39482	−0.697410	−	0.716672i	$-0.745665\pi$
$257$	−22.3607	−1.39482	−0.697410	+	0.716672i	$0.745665\pi$
$258$	0	0
$259$	13.5279	0.840581
$260$	0	0
$261$	2.94427	0.182246
$262$	0	0
$263$	15.4164	0.950616	0.475308	−	0.879819i	$-0.342337\pi$
$263$	15.4164	0.950616	0.475308	+	0.879819i	$0.342337\pi$
$264$	0	0
$265$	−4.00000	−0.245718
$266$	0	0
$267$	−9.30495	−0.569454
$268$	0	0
$269$	1.05573	0.0643689	0.0321844	−	0.999482i	$-0.489754\pi$
$269$	1.05573	0.0643689	0.0321844	+	0.999482i	$0.489754\pi$
$270$	0	0
$271$	17.8885	1.08665	0.543326	−	0.839522i	$-0.317165\pi$
$271$	17.8885	1.08665	0.543326	+	0.839522i	$0.317165\pi$
$272$	0	0
$273$	−6.83282	−0.413541
$274$	0	0
$275$	1.23607	0.0745377
$276$	0	0
$277$	20.8328	1.25172	0.625861	−	0.779934i	$-0.284748\pi$
$277$	20.8328	1.25172	0.625861	+	0.779934i	$0.284748\pi$
$278$	0	0
$279$	1.81966	0.108940
$280$	0	0
$281$	19.8885	1.18645	0.593226	−	0.805036i	$-0.297854\pi$
$281$	19.8885	1.18645	0.593226	+	0.805036i	$0.297854\pi$
$282$	0	0
$283$	−1.23607	−0.0734766	−0.0367383	−	0.999325i	$-0.511697\pi$
$283$	−1.23607	−0.0734766	−0.0367383	+	0.999325i	$0.511697\pi$
$284$	0	0
$285$	16.0000	0.947758
$286$	0	0
$287$	2.47214	0.145926
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	2.47214	0.144919
$292$	0	0
$293$	10.0000	0.584206	0.292103	−	0.956387i	$-0.405645\pi$
$293$	10.0000	0.584206	0.292103	+	0.956387i	$0.405645\pi$
$294$	0	0
$295$	−28.9443	−1.68520
$296$	0	0
$297$	6.83282	0.396480
$298$	0	0
$299$	−5.52786	−0.319685
$300$	0	0
$301$	−1.88854	−0.108854
$302$	0	0
$303$	16.5836	0.952702
$304$	0	0
$305$	13.8885	0.795256
$306$	0	0
$307$	5.88854	0.336077	0.168038	−	0.985780i	$-0.446257\pi$
$307$	5.88854	0.336077	0.168038	+	0.985780i	$0.446257\pi$
$308$	0	0
$309$	9.88854	0.562540
$310$	0	0
$311$	14.7639	0.837186	0.418593	−	0.908174i	$-0.362523\pi$
$311$	14.7639	0.837186	0.418593	+	0.908174i	$0.362523\pi$
$312$	0	0
$313$	−7.88854	−0.445887	−0.222943	−	0.974831i	$-0.571567\pi$
$313$	−7.88854	−0.445887	−0.222943	+	0.974831i	$0.571567\pi$
$314$	0	0
$315$	3.63932	0.205052
$316$	0	0
$317$	15.8885	0.892390	0.446195	−	0.894936i	$-0.352779\pi$
$317$	15.8885	0.892390	0.446195	+	0.894936i	$0.352779\pi$
$318$	0	0
$319$	2.47214	0.138413
$320$	0	0
$321$	−4.58359	−0.255831
$322$	0	0
$323$	−6.47214	−0.360119
$324$	0	0
$325$	4.47214	0.248069
$326$	0	0
$327$	−12.3607	−0.683547
$328$	0	0
$329$	−16.0000	−0.882109
$330$	0	0
$331$	−1.52786	−0.0839790	−0.0419895	−	0.999118i	$-0.513370\pi$
$331$	−1.52786	−0.0839790	−0.0419895	+	0.999118i	$0.513370\pi$
$332$	0	0
$333$	−16.1115	−0.882902
$334$	0	0
$335$	24.0000	1.31126
$336$	0	0
$337$	11.8885	0.647610	0.323805	−	0.946124i	$-0.395038\pi$
$337$	11.8885	0.647610	0.323805	+	0.946124i	$0.395038\pi$
$338$	0	0
$339$	18.4721	1.00327
$340$	0	0
$341$	1.52786	0.0827385
$342$	0	0
$343$	−15.4164	−0.832408
$344$	0	0
$345$	−3.05573	−0.164515
$346$	0	0
$347$	−9.23607	−0.495818	−0.247909	−	0.968783i	$-0.579743\pi$
$347$	−9.23607	−0.495818	−0.247909	+	0.968783i	$0.579743\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	0	0
$351$	24.7214	1.31953
$352$	0	0
$353$	11.8885	0.632763	0.316382	−	0.948632i	$-0.397532\pi$
$353$	11.8885	0.632763	0.316382	+	0.948632i	$0.397532\pi$
$354$	0	0
$355$	18.4721	0.980399
$356$	0	0
$357$	1.52786	0.0808631
$358$	0	0
$359$	−13.5279	−0.713973	−0.356987	−	0.934109i	$-0.616196\pi$
$359$	−13.5279	−0.713973	−0.356987	+	0.934109i	$0.616196\pi$
$360$	0	0
$361$	22.8885	1.20466
$362$	0	0
$363$	−11.7082	−0.614521
$364$	0	0
$365$	−29.8885	−1.56444
$366$	0	0
$367$	−14.7639	−0.770671	−0.385335	−	0.922777i	$-0.625914\pi$
$367$	−14.7639	−0.770671	−0.385335	+	0.922777i	$0.625914\pi$
$368$	0	0
$369$	−2.94427	−0.153273
$370$	0	0
$371$	2.47214	0.128347
$372$	0	0
$373$	−12.4721	−0.645783	−0.322891	−	0.946436i	$-0.604655\pi$
$373$	−12.4721	−0.645783	−0.322891	+	0.946436i	$0.604655\pi$
$374$	0	0
$375$	14.8328	0.765963
$376$	0	0
$377$	8.94427	0.460653
$378$	0	0
$379$	−32.6525	−1.67725	−0.838623	−	0.544713i	$-0.816639\pi$
$379$	−32.6525	−1.67725	−0.838623	+	0.544713i	$0.816639\pi$
$380$	0	0
$381$	−3.05573	−0.156550
$382$	0	0
$383$	23.4164	1.19652	0.598261	−	0.801301i	$-0.295858\pi$
$383$	23.4164	1.19652	0.598261	+	0.801301i	$0.295858\pi$
$384$	0	0
$385$	3.05573	0.155734
$386$	0	0
$387$	2.24922	0.114334
$388$	0	0
$389$	−1.41641	−0.0718147	−0.0359074	−	0.999355i	$-0.511432\pi$
$389$	−1.41641	−0.0718147	−0.0359074	+	0.999355i	$0.511432\pi$
$390$	0	0
$391$	1.23607	0.0625106
$392$	0	0
$393$	23.6393	1.19245
$394$	0	0
$395$	23.4164	1.17821
$396$	0	0
$397$	−13.0557	−0.655248	−0.327624	−	0.944808i	$-0.606248\pi$
$397$	−13.0557	−0.655248	−0.327624	+	0.944808i	$0.606248\pi$
$398$	0	0
$399$	−9.88854	−0.495046
$400$	0	0
$401$	30.9443	1.54528	0.772642	−	0.634842i	$-0.218935\pi$
$401$	30.9443	1.54528	0.772642	+	0.634842i	$0.218935\pi$
$402$	0	0
$403$	5.52786	0.275363
$404$	0	0
$405$	4.83282	0.240145
$406$	0	0
$407$	−13.5279	−0.670551
$408$	0	0
$409$	−6.00000	−0.296681	−0.148340	−	0.988936i	$-0.547393\pi$
$409$	−6.00000	−0.296681	−0.148340	+	0.988936i	$0.547393\pi$
$410$	0	0
$411$	−9.30495	−0.458979
$412$	0	0
$413$	17.8885	0.880238
$414$	0	0
$415$	−3.05573	−0.150000
$416$	0	0
$417$	−8.36068	−0.409424
$418$	0	0
$419$	−25.2361	−1.23286	−0.616431	−	0.787409i	$-0.711422\pi$
$419$	−25.2361	−1.23286	−0.616431	+	0.787409i	$0.711422\pi$
$420$	0	0
$421$	−6.36068	−0.310001	−0.155000	−	0.987914i	$-0.549538\pi$
$421$	−6.36068	−0.310001	−0.155000	+	0.987914i	$0.549538\pi$
$422$	0	0
$423$	19.0557	0.926521
$424$	0	0
$425$	−1.00000	−0.0485071
$426$	0	0
$427$	−8.58359	−0.415389
$428$	0	0
$429$	6.83282	0.329891
$430$	0	0
$431$	−37.5967	−1.81097	−0.905486	−	0.424377i	$-0.860493\pi$
$431$	−37.5967	−1.81097	−0.905486	+	0.424377i	$0.860493\pi$
$432$	0	0
$433$	−28.4721	−1.36828	−0.684142	−	0.729349i	$-0.739823\pi$
$433$	−28.4721	−1.36828	−0.684142	+	0.729349i	$0.739823\pi$
$434$	0	0
$435$	4.94427	0.237060
$436$	0	0
$437$	−8.00000	−0.382692
$438$	0	0
$439$	0.652476	0.0311410	0.0155705	−	0.999879i	$-0.495044\pi$
$439$	0.652476	0.0311410	0.0155705	+	0.999879i	$0.495044\pi$
$440$	0	0
$441$	8.05573	0.383606
$442$	0	0
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	0	0
$445$	15.0557	0.713710
$446$	0	0
$447$	−7.41641	−0.350784
$448$	0	0
$449$	−7.88854	−0.372283	−0.186142	−	0.982523i	$-0.559598\pi$
$449$	−7.88854	−0.372283	−0.186142	+	0.982523i	$0.559598\pi$
$450$	0	0
$451$	−2.47214	−0.116408
$452$	0	0
$453$	−6.83282	−0.321034
$454$	0	0
$455$	11.0557	0.518301
$456$	0	0
$457$	37.4164	1.75027	0.875133	−	0.483883i	$-0.160774\pi$
$457$	37.4164	1.75027	0.875133	+	0.483883i	$0.160774\pi$
$458$	0	0
$459$	−5.52786	−0.258019
$460$	0	0
$461$	−31.8885	−1.48520	−0.742599	−	0.669737i	$-0.766407\pi$
$461$	−31.8885	−1.48520	−0.742599	+	0.669737i	$0.766407\pi$
$462$	0	0
$463$	−30.8328	−1.43292	−0.716461	−	0.697627i	$-0.754239\pi$
$463$	−30.8328	−1.43292	−0.716461	+	0.697627i	$0.754239\pi$
$464$	0	0
$465$	3.05573	0.141706
$466$	0	0
$467$	−21.3050	−0.985876	−0.492938	−	0.870065i	$-0.664077\pi$
$467$	−21.3050	−0.985876	−0.492938	+	0.870065i	$0.664077\pi$
$468$	0	0
$469$	−14.8328	−0.684916
$470$	0	0
$471$	14.6950	0.677112
$472$	0	0
$473$	1.88854	0.0868353
$474$	0	0
$475$	6.47214	0.296962
$476$	0	0
$477$	−2.94427	−0.134809
$478$	0	0
$479$	−20.2918	−0.927156	−0.463578	−	0.886056i	$-0.653435\pi$
$479$	−20.2918	−0.927156	−0.463578	+	0.886056i	$0.653435\pi$
$480$	0	0
$481$	−48.9443	−2.23167
$482$	0	0
$483$	1.88854	0.0859317
$484$	0	0
$485$	−4.00000	−0.181631
$486$	0	0
$487$	−38.7639	−1.75656	−0.878281	−	0.478145i	$-0.841309\pi$
$487$	−38.7639	−1.75656	−0.878281	+	0.478145i	$0.841309\pi$
$488$	0	0
$489$	−27.4164	−1.23981
$490$	0	0
$491$	1.52786	0.0689515	0.0344758	−	0.999406i	$-0.489024\pi$
$491$	1.52786	0.0689515	0.0344758	+	0.999406i	$0.489024\pi$
$492$	0	0
$493$	−2.00000	−0.0900755
$494$	0	0
$495$	−3.63932	−0.163575
$496$	0	0
$497$	−11.4164	−0.512096
$498$	0	0
$499$	−38.1803	−1.70919	−0.854593	−	0.519298i	$-0.826194\pi$
$499$	−38.1803	−1.70919	−0.854593	+	0.519298i	$0.826194\pi$
$500$	0	0
$501$	24.3607	1.08835
$502$	0	0
$503$	−27.7082	−1.23545	−0.617724	−	0.786395i	$-0.711945\pi$
$503$	−27.7082	−1.23545	−0.617724	+	0.786395i	$0.711945\pi$
$504$	0	0
$505$	−26.8328	−1.19404
$506$	0	0
$507$	8.65248	0.384270
$508$	0	0
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	18.4721	0.817159
$512$	0	0
$513$	35.7771	1.57960
$514$	0	0
$515$	−16.0000	−0.705044
$516$	0	0
$517$	16.0000	0.703679
$518$	0	0
$519$	−2.47214	−0.108515
$520$	0	0
$521$	−23.8885	−1.04658	−0.523288	−	0.852156i	$-0.675295\pi$
$521$	−23.8885	−1.04658	−0.523288	+	0.852156i	$0.675295\pi$
$522$	0	0
$523$	−24.9443	−1.09074	−0.545368	−	0.838196i	$-0.683610\pi$
$523$	−24.9443	−1.09074	−0.545368	+	0.838196i	$0.683610\pi$
$524$	0	0
$525$	−1.52786	−0.0666815
$526$	0	0
$527$	−1.23607	−0.0538440
$528$	0	0
$529$	−21.4721	−0.933571
$530$	0	0
$531$	−21.3050	−0.924556
$532$	0	0
$533$	−8.94427	−0.387419
$534$	0	0
$535$	7.41641	0.320639
$536$	0	0
$537$	17.8885	0.771948
$538$	0	0
$539$	6.76393	0.291343
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	0	0
$543$	−12.3607	−0.530448
$544$	0	0
$545$	20.0000	0.856706
$546$	0	0
$547$	−11.7082	−0.500607	−0.250303	−	0.968167i	$-0.580530\pi$
$547$	−11.7082	−0.500607	−0.250303	+	0.968167i	$0.580530\pi$
$548$	0	0
$549$	10.2229	0.436303
$550$	0	0
$551$	12.9443	0.551445
$552$	0	0
$553$	−14.4721	−0.615418
$554$	0	0
$555$	−27.0557	−1.14845
$556$	0	0
$557$	5.41641	0.229501	0.114750	−	0.993394i	$-0.463393\pi$
$557$	5.41641	0.229501	0.114750	+	0.993394i	$0.463393\pi$
$558$	0	0
$559$	6.83282	0.288997
$560$	0	0
$561$	−1.52786	−0.0645065
$562$	0	0
$563$	−24.3607	−1.02668	−0.513340	−	0.858185i	$-0.671592\pi$
$563$	−24.3607	−1.02668	−0.513340	+	0.858185i	$0.671592\pi$
$564$	0	0
$565$	−29.8885	−1.25742
$566$	0	0
$567$	−2.98684	−0.125436
$568$	0	0
$569$	0.111456	0.00467249	0.00233624	−	0.999997i	$-0.499256\pi$
$569$	0.111456	0.00467249	0.00233624	+	0.999997i	$0.499256\pi$
$570$	0	0
$571$	−1.23607	−0.0517278	−0.0258639	−	0.999665i	$-0.508234\pi$
$571$	−1.23607	−0.0517278	−0.0258639	+	0.999665i	$0.508234\pi$
$572$	0	0
$573$	16.0000	0.668410
$574$	0	0
$575$	−1.23607	−0.0515476
$576$	0	0
$577$	0.472136	0.0196553	0.00982764	−	0.999952i	$-0.496872\pi$
$577$	0.472136	0.0196553	0.00982764	+	0.999952i	$0.496872\pi$
$578$	0	0
$579$	18.4721	0.767676
$580$	0	0
$581$	1.88854	0.0783500
$582$	0	0
$583$	−2.47214	−0.102385
$584$	0	0
$585$	−13.1672	−0.544396
$586$	0	0
$587$	32.3607	1.33567	0.667834	−	0.744310i	$-0.267222\pi$
$587$	32.3607	1.33567	0.667834	+	0.744310i	$0.267222\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	0	0
$591$	−8.58359	−0.353082
$592$	0	0
$593$	−23.8885	−0.980985	−0.490492	−	0.871445i	$-0.663183\pi$
$593$	−23.8885	−0.980985	−0.490492	+	0.871445i	$0.663183\pi$
$594$	0	0
$595$	−2.47214	−0.101348
$596$	0	0
$597$	−27.4164	−1.12208
$598$	0	0
$599$	6.83282	0.279181	0.139591	−	0.990209i	$-0.455421\pi$
$599$	6.83282	0.279181	0.139591	+	0.990209i	$0.455421\pi$
$600$	0	0
$601$	21.0557	0.858881	0.429441	−	0.903095i	$-0.358711\pi$
$601$	21.0557	0.858881	0.429441	+	0.903095i	$0.358711\pi$
$602$	0	0
$603$	17.6656	0.719400
$604$	0	0
$605$	18.9443	0.770194
$606$	0	0
$607$	19.7082	0.799931	0.399966	−	0.916530i	$-0.369022\pi$
$607$	19.7082	0.799931	0.399966	+	0.916530i	$0.369022\pi$
$608$	0	0
$609$	−3.05573	−0.123824
$610$	0	0
$611$	57.8885	2.34192
$612$	0	0
$613$	−6.00000	−0.242338	−0.121169	−	0.992632i	$-0.538664\pi$
$613$	−6.00000	−0.242338	−0.121169	+	0.992632i	$0.538664\pi$
$614$	0	0
$615$	−4.94427	−0.199372
$616$	0	0
$617$	34.0000	1.36879	0.684394	−	0.729112i	$-0.260067\pi$
$617$	34.0000	1.36879	0.684394	+	0.729112i	$0.260067\pi$
$618$	0	0
$619$	32.6525	1.31241	0.656207	−	0.754581i	$-0.272160\pi$
$619$	32.6525	1.31241	0.656207	+	0.754581i	$0.272160\pi$
$620$	0	0
$621$	−6.83282	−0.274191
$622$	0	0
$623$	−9.30495	−0.372795
$624$	0	0
$625$	−19.0000	−0.760000
$626$	0	0
$627$	9.88854	0.394910
$628$	0	0
$629$	10.9443	0.436377
$630$	0	0
$631$	−33.3050	−1.32585	−0.662925	−	0.748686i	$-0.730685\pi$
$631$	−33.3050	−1.32585	−0.662925	+	0.748686i	$0.730685\pi$
$632$	0	0
$633$	−1.52786	−0.0607271
$634$	0	0
$635$	4.94427	0.196207
$636$	0	0
$637$	24.4721	0.969621
$638$	0	0
$639$	13.5967	0.537879
$640$	0	0
$641$	27.8885	1.10153	0.550766	−	0.834660i	$-0.314336\pi$
$641$	27.8885	1.10153	0.550766	+	0.834660i	$0.314336\pi$
$642$	0	0
$643$	−18.5410	−0.731186	−0.365593	−	0.930775i	$-0.619134\pi$
$643$	−18.5410	−0.731186	−0.365593	+	0.930775i	$0.619134\pi$
$644$	0	0
$645$	3.77709	0.148723
$646$	0	0
$647$	−32.0000	−1.25805	−0.629025	−	0.777385i	$-0.716546\pi$
$647$	−32.0000	−1.25805	−0.629025	+	0.777385i	$0.716546\pi$
$648$	0	0
$649$	−17.8885	−0.702187
$650$	0	0
$651$	−1.88854	−0.0740179
$652$	0	0
$653$	−32.8328	−1.28485	−0.642424	−	0.766350i	$-0.722071\pi$
$653$	−32.8328	−1.28485	−0.642424	+	0.766350i	$0.722071\pi$
$654$	0	0
$655$	−38.2492	−1.49452
$656$	0	0
$657$	−22.0000	−0.858302
$658$	0	0
$659$	−34.8328	−1.35689	−0.678447	−	0.734649i	$-0.737347\pi$
$659$	−34.8328	−1.35689	−0.678447	+	0.734649i	$0.737347\pi$
$660$	0	0
$661$	−4.11146	−0.159917	−0.0799586	−	0.996798i	$-0.525479\pi$
$661$	−4.11146	−0.159917	−0.0799586	+	0.996798i	$0.525479\pi$
$662$	0	0
$663$	−5.52786	−0.214684
$664$	0	0
$665$	16.0000	0.620453
$666$	0	0
$667$	−2.47214	−0.0957215
$668$	0	0
$669$	−6.83282	−0.264172
$670$	0	0
$671$	8.58359	0.331366
$672$	0	0
$673$	−33.7771	−1.30201	−0.651006	−	0.759073i	$-0.725653\pi$
$673$	−33.7771	−1.30201	−0.651006	+	0.759073i	$0.725653\pi$
$674$	0	0
$675$	5.52786	0.212768
$676$	0	0
$677$	−8.11146	−0.311749	−0.155874	−	0.987777i	$-0.549819\pi$
$677$	−8.11146	−0.311749	−0.155874	+	0.987777i	$0.549819\pi$
$678$	0	0
$679$	2.47214	0.0948719
$680$	0	0
$681$	−1.52786	−0.0585479
$682$	0	0
$683$	8.06888	0.308747	0.154374	−	0.988013i	$-0.450664\pi$
$683$	8.06888	0.308747	0.154374	+	0.988013i	$0.450664\pi$
$684$	0	0
$685$	15.0557	0.575250
$686$	0	0
$687$	30.2492	1.15408
$688$	0	0
$689$	−8.94427	−0.340750
$690$	0	0
$691$	37.5967	1.43025	0.715124	−	0.698998i	$-0.246370\pi$
$691$	37.5967	1.43025	0.715124	+	0.698998i	$0.246370\pi$
$692$	0	0
$693$	2.24922	0.0854409
$694$	0	0
$695$	13.5279	0.513141
$696$	0	0
$697$	2.00000	0.0757554
$698$	0	0
$699$	−29.5279	−1.11685
$700$	0	0
$701$	−14.3607	−0.542395	−0.271198	−	0.962524i	$-0.587420\pi$
$701$	−14.3607	−0.542395	−0.271198	+	0.962524i	$0.587420\pi$
$702$	0	0
$703$	−70.8328	−2.67151
$704$	0	0
$705$	32.0000	1.20519
$706$	0	0
$707$	16.5836	0.623690
$708$	0	0
$709$	−14.9443	−0.561244	−0.280622	−	0.959818i	$-0.590541\pi$
$709$	−14.9443	−0.561244	−0.280622	+	0.959818i	$0.590541\pi$
$710$	0	0
$711$	17.2361	0.646403
$712$	0	0
$713$	−1.52786	−0.0572190
$714$	0	0
$715$	−11.0557	−0.413461
$716$	0	0
$717$	−22.1115	−0.825767
$718$	0	0
$719$	8.65248	0.322683	0.161341	−	0.986899i	$-0.448418\pi$
$719$	8.65248	0.322683	0.161341	+	0.986899i	$0.448418\pi$
$720$	0	0
$721$	9.88854	0.368269
$722$	0	0
$723$	−1.30495	−0.0485317
$724$	0	0
$725$	2.00000	0.0742781
$726$	0	0
$727$	−20.9443	−0.776780	−0.388390	−	0.921495i	$-0.626969\pi$
$727$	−20.9443	−0.776780	−0.388390	+	0.921495i	$0.626969\pi$
$728$	0	0
$729$	24.0557	0.890953
$730$	0	0
$731$	−1.52786	−0.0565101
$732$	0	0
$733$	35.8885	1.32557	0.662787	−	0.748808i	$-0.269374\pi$
$733$	35.8885	1.32557	0.662787	+	0.748808i	$0.269374\pi$
$734$	0	0
$735$	13.5279	0.498983
$736$	0	0
$737$	14.8328	0.546374
$738$	0	0
$739$	−43.4164	−1.59710	−0.798549	−	0.601930i	$-0.794399\pi$
$739$	−43.4164	−1.59710	−0.798549	+	0.601930i	$0.794399\pi$
$740$	0	0
$741$	35.7771	1.31430
$742$	0	0
$743$	48.6525	1.78489	0.892443	−	0.451160i	$-0.148990\pi$
$743$	48.6525	1.78489	0.892443	+	0.451160i	$0.148990\pi$
$744$	0	0
$745$	12.0000	0.439646
$746$	0	0
$747$	−2.24922	−0.0822948
$748$	0	0
$749$	−4.58359	−0.167481
$750$	0	0
$751$	35.7082	1.30301	0.651505	−	0.758644i	$-0.274138\pi$
$751$	35.7082	1.30301	0.651505	+	0.758644i	$0.274138\pi$
$752$	0	0
$753$	−11.0557	−0.402893
$754$	0	0
$755$	11.0557	0.402359
$756$	0	0
$757$	−7.52786	−0.273605	−0.136802	−	0.990598i	$-0.543683\pi$
$757$	−7.52786	−0.273605	−0.136802	+	0.990598i	$0.543683\pi$
$758$	0	0
$759$	−1.88854	−0.0685498
$760$	0	0
$761$	34.3607	1.24557	0.622787	−	0.782392i	$-0.286000\pi$
$761$	34.3607	1.24557	0.622787	+	0.782392i	$0.286000\pi$
$762$	0	0
$763$	−12.3607	−0.447487
$764$	0	0
$765$	2.94427	0.106450
$766$	0	0
$767$	−64.7214	−2.33695
$768$	0	0
$769$	16.4721	0.594000	0.297000	−	0.954877i	$-0.404014\pi$
$769$	16.4721	0.594000	0.297000	+	0.954877i	$0.404014\pi$
$770$	0	0
$771$	−27.6393	−0.995406
$772$	0	0
$773$	−27.3050	−0.982091	−0.491045	−	0.871134i	$-0.663385\pi$
$773$	−27.3050	−0.982091	−0.491045	+	0.871134i	$0.663385\pi$
$774$	0	0
$775$	1.23607	0.0444009
$776$	0	0
$777$	16.7214	0.599875
$778$	0	0
$779$	−12.9443	−0.463777
$780$	0	0
$781$	11.4164	0.408511
$782$	0	0
$783$	11.0557	0.395099
$784$	0	0
$785$	−23.7771	−0.848641
$786$	0	0
$787$	−24.6525	−0.878766	−0.439383	−	0.898300i	$-0.644803\pi$
$787$	−24.6525	−0.878766	−0.439383	+	0.898300i	$0.644803\pi$
$788$	0	0
$789$	19.0557	0.678402
$790$	0	0
$791$	18.4721	0.656794
$792$	0	0
$793$	31.0557	1.10282
$794$	0	0
$795$	−4.94427	−0.175355
$796$	0	0
$797$	−15.8885	−0.562801	−0.281401	−	0.959590i	$-0.590799\pi$
$797$	−15.8885	−0.562801	−0.281401	+	0.959590i	$0.590799\pi$
$798$	0	0
$799$	−12.9443	−0.457935
$800$	0	0
$801$	11.0820	0.391565
$802$	0	0
$803$	−18.4721	−0.651868
$804$	0	0
$805$	−3.05573	−0.107700
$806$	0	0
$807$	1.30495	0.0459365
$808$	0	0
$809$	−42.9443	−1.50984	−0.754920	−	0.655817i	$-0.772324\pi$
$809$	−42.9443	−1.50984	−0.754920	+	0.655817i	$0.772324\pi$
$810$	0	0
$811$	27.1246	0.952474	0.476237	−	0.879317i	$-0.342000\pi$
$811$	27.1246	0.952474	0.476237	+	0.879317i	$0.342000\pi$
$812$	0	0
$813$	22.1115	0.775483
$814$	0	0
$815$	44.3607	1.55389
$816$	0	0
$817$	9.88854	0.345956
$818$	0	0
$819$	8.13777	0.284357
$820$	0	0
$821$	−50.0000	−1.74501	−0.872506	−	0.488603i	$-0.837507\pi$
$821$	−50.0000	−1.74501	−0.872506	+	0.488603i	$0.837507\pi$
$822$	0	0
$823$	2.40325	0.0837721	0.0418861	−	0.999122i	$-0.486663\pi$
$823$	2.40325	0.0837721	0.0418861	+	0.999122i	$0.486663\pi$
$824$	0	0
$825$	1.52786	0.0531934
$826$	0	0
$827$	−25.2361	−0.877544	−0.438772	−	0.898598i	$-0.644586\pi$
$827$	−25.2361	−0.877544	−0.438772	+	0.898598i	$0.644586\pi$
$828$	0	0
$829$	43.8885	1.52431	0.762156	−	0.647393i	$-0.224141\pi$
$829$	43.8885	1.52431	0.762156	+	0.647393i	$0.224141\pi$
$830$	0	0
$831$	25.7508	0.893285
$832$	0	0
$833$	−5.47214	−0.189598
$834$	0	0
$835$	−39.4164	−1.36406
$836$	0	0
$837$	6.83282	0.236177
$838$	0	0
$839$	32.0689	1.10714	0.553570	−	0.832802i	$-0.313265\pi$
$839$	32.0689	1.10714	0.553570	+	0.832802i	$0.313265\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	24.5836	0.846704
$844$	0	0
$845$	−14.0000	−0.481615
$846$	0	0
$847$	−11.7082	−0.402299
$848$	0	0
$849$	−1.52786	−0.0524362
$850$	0	0
$851$	13.5279	0.463729
$852$	0	0
$853$	−37.7771	−1.29346	−0.646731	−	0.762718i	$-0.723865\pi$
$853$	−37.7771	−1.29346	−0.646731	+	0.762718i	$0.723865\pi$
$854$	0	0
$855$	−19.0557	−0.651692
$856$	0	0
$857$	−7.88854	−0.269468	−0.134734	−	0.990882i	$-0.543018\pi$
$857$	−7.88854	−0.269468	−0.134734	+	0.990882i	$0.543018\pi$
$858$	0	0
$859$	50.2492	1.71448	0.857241	−	0.514916i	$-0.172177\pi$
$859$	50.2492	1.71448	0.857241	+	0.514916i	$0.172177\pi$
$860$	0	0
$861$	3.05573	0.104139
$862$	0	0
$863$	30.8328	1.04956	0.524781	−	0.851238i	$-0.324147\pi$
$863$	30.8328	1.04956	0.524781	+	0.851238i	$0.324147\pi$
$864$	0	0
$865$	4.00000	0.136004
$866$	0	0
$867$	1.23607	0.0419791
$868$	0	0
$869$	14.4721	0.490934
$870$	0	0
$871$	53.6656	1.81839
$872$	0	0
$873$	−2.94427	−0.0996485
$874$	0	0
$875$	14.8328	0.501441
$876$	0	0
$877$	−3.88854	−0.131307	−0.0656534	−	0.997842i	$-0.520913\pi$
$877$	−3.88854	−0.131307	−0.0656534	+	0.997842i	$0.520913\pi$
$878$	0	0
$879$	12.3607	0.416915
$880$	0	0
$881$	−7.88854	−0.265772	−0.132886	−	0.991131i	$-0.542424\pi$
$881$	−7.88854	−0.265772	−0.132886	+	0.991131i	$0.542424\pi$
$882$	0	0
$883$	−2.11146	−0.0710562	−0.0355281	−	0.999369i	$-0.511311\pi$
$883$	−2.11146	−0.0710562	−0.0355281	+	0.999369i	$0.511311\pi$
$884$	0	0
$885$	−35.7771	−1.20263
$886$	0	0
$887$	−11.1246	−0.373528	−0.186764	−	0.982405i	$-0.559800\pi$
$887$	−11.1246	−0.373528	−0.186764	+	0.982405i	$0.559800\pi$
$888$	0	0
$889$	−3.05573	−0.102486
$890$	0	0
$891$	2.98684	0.100063
$892$	0	0
$893$	83.7771	2.80349
$894$	0	0
$895$	−28.9443	−0.967500
$896$	0	0
$897$	−6.83282	−0.228141
$898$	0	0
$899$	2.47214	0.0824504
$900$	0	0
$901$	2.00000	0.0666297
$902$	0	0
$903$	−2.33437	−0.0776829
$904$	0	0
$905$	20.0000	0.664822
$906$	0	0
$907$	−59.1246	−1.96320	−0.981600	−	0.190947i	$-0.938844\pi$
$907$	−59.1246	−1.96320	−0.981600	+	0.190947i	$0.938844\pi$
$908$	0	0
$909$	−19.7508	−0.655092
$910$	0	0
$911$	−27.7082	−0.918014	−0.459007	−	0.888433i	$-0.651795\pi$
$911$	−27.7082	−0.918014	−0.459007	+	0.888433i	$0.651795\pi$
$912$	0	0
$913$	−1.88854	−0.0625017
$914$	0	0
$915$	17.1672	0.567530
$916$	0	0
$917$	23.6393	0.780639
$918$	0	0
$919$	41.8885	1.38178	0.690888	−	0.722962i	$-0.257220\pi$
$919$	41.8885	1.38178	0.690888	+	0.722962i	$0.257220\pi$
$920$	0	0
$921$	7.27864	0.239839
$922$	0	0
$923$	41.3050	1.35957
$924$	0	0
$925$	−10.9443	−0.359845
$926$	0	0
$927$	−11.7771	−0.386810
$928$	0	0
$929$	−26.9443	−0.884013	−0.442006	−	0.897012i	$-0.645733\pi$
$929$	−26.9443	−0.884013	−0.442006	+	0.897012i	$0.645733\pi$
$930$	0	0
$931$	35.4164	1.16073
$932$	0	0
$933$	18.2492	0.597453
$934$	0	0
$935$	2.47214	0.0808475
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	0	0
$939$	−9.75078	−0.318205
$940$	0	0
$941$	25.7771	0.840309	0.420155	−	0.907453i	$-0.361976\pi$
$941$	25.7771	0.840309	0.420155	+	0.907453i	$0.361976\pi$
$942$	0	0
$943$	2.47214	0.0805038
$944$	0	0
$945$	13.6656	0.444543
$946$	0	0
$947$	37.0132	1.20277	0.601383	−	0.798961i	$-0.294617\pi$
$947$	37.0132	1.20277	0.601383	+	0.798961i	$0.294617\pi$
$948$	0	0
$949$	−66.8328	−2.16949
$950$	0	0
$951$	19.6393	0.636849
$952$	0	0
$953$	−23.5279	−0.762142	−0.381071	−	0.924546i	$-0.624445\pi$
$953$	−23.5279	−0.762142	−0.381071	+	0.924546i	$0.624445\pi$
$954$	0	0
$955$	−25.8885	−0.837734
$956$	0	0
$957$	3.05573	0.0987777
$958$	0	0
$959$	−9.30495	−0.300473
$960$	0	0
$961$	−29.4721	−0.950714
$962$	0	0
$963$	5.45898	0.175913
$964$	0	0
$965$	−29.8885	−0.962146
$966$	0	0
$967$	2.47214	0.0794985	0.0397493	−	0.999210i	$-0.487344\pi$
$967$	2.47214	0.0794985	0.0397493	+	0.999210i	$0.487344\pi$
$968$	0	0
$969$	−8.00000	−0.256997
$970$	0	0
$971$	9.52786	0.305764	0.152882	−	0.988244i	$-0.451145\pi$
$971$	9.52786	0.305764	0.152882	+	0.988244i	$0.451145\pi$
$972$	0	0
$973$	−8.36068	−0.268031
$974$	0	0
$975$	5.52786	0.177033
$976$	0	0
$977$	34.0000	1.08776	0.543878	−	0.839164i	$-0.316955\pi$
$977$	34.0000	1.08776	0.543878	+	0.839164i	$0.316955\pi$
$978$	0	0
$979$	9.30495	0.297388
$980$	0	0
$981$	14.7214	0.470017
$982$	0	0
$983$	6.76393	0.215736	0.107868	−	0.994165i	$-0.465598\pi$
$983$	6.76393	0.215736	0.107868	+	0.994165i	$0.465598\pi$
$984$	0	0
$985$	13.8885	0.442526
$986$	0	0
$987$	−19.7771	−0.629512
$988$	0	0
$989$	−1.88854	−0.0600522
$990$	0	0
$991$	−17.2361	−0.547522	−0.273761	−	0.961798i	$-0.588268\pi$
$991$	−17.2361	−0.547522	−0.273761	+	0.961798i	$0.588268\pi$
$992$	0	0
$993$	−1.88854	−0.0599311
$994$	0	0
$995$	44.3607	1.40633
$996$	0	0
$997$	−29.0557	−0.920204	−0.460102	−	0.887866i	$-0.652187\pi$
$997$	−29.0557	−0.920204	−0.460102	+	0.887866i	$0.652187\pi$
$998$	0	0
$999$	−60.4984	−1.91409

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1088.2.a.o.1.2		2
3.2	odd	2		9792.2.a.da.1.2		2
4.3	odd	2		1088.2.a.s.1.1		2
8.3	odd	2		136.2.a.c.1.2	✓	2
8.5	even	2		272.2.a.f.1.1		2
12.11	even	2		9792.2.a.db.1.1		2
24.5	odd	2		2448.2.a.u.1.2		2
24.11	even	2		1224.2.a.i.1.1		2
40.3	even	4		3400.2.e.f.2449.3		4
40.19	odd	2		3400.2.a.i.1.1		2
40.27	even	4		3400.2.e.f.2449.2		4
40.29	even	2		6800.2.a.bd.1.2		2
56.27	even	2		6664.2.a.i.1.1		2
136.67	odd	2		2312.2.a.m.1.1		2
136.101	even	2		4624.2.a.h.1.2		2
136.115	odd	4		2312.2.b.g.577.3		4
136.123	odd	4		2312.2.b.g.577.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.2.a.c.1.2	✓	2	8.3	odd	2
272.2.a.f.1.1		2	8.5	even	2
1088.2.a.o.1.2		2	1.1	even	1	trivial
1088.2.a.s.1.1		2	4.3	odd	2
1224.2.a.i.1.1		2	24.11	even	2
2312.2.a.m.1.1		2	136.67	odd	2
2312.2.b.g.577.2		4	136.123	odd	4
2312.2.b.g.577.3		4	136.115	odd	4
2448.2.a.u.1.2		2	24.5	odd	2
3400.2.a.i.1.1		2	40.19	odd	2
3400.2.e.f.2449.2		4	40.27	even	4
3400.2.e.f.2449.3		4	40.3	even	4
4624.2.a.h.1.2		2	136.101	even	2
6664.2.a.i.1.1		2	56.27	even	2
6800.2.a.bd.1.2		2	40.29	even	2
9792.2.a.da.1.2		2	3.2	odd	2
9792.2.a.db.1.1		2	12.11	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:	\( N \)	\(=\)	\( 1088 = 2^{6} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1088.a (trivial)

\(f(q)\)	\(=\)	\(q+1.23607 q^{3} -2.00000 q^{5} +1.23607 q^{7} -1.47214 q^{9} +O(q^{10})\)	\(q+1.23607 q^{3} -2.00000 q^{5} +1.23607 q^{7} -1.47214 q^{9} -1.23607 q^{11} -4.47214 q^{13} -2.47214 q^{15} +1.00000 q^{17} -6.47214 q^{19} +1.52786 q^{21} +1.23607 q^{23} -1.00000 q^{25} -5.52786 q^{27} -2.00000 q^{29} -1.23607 q^{31} -1.52786 q^{33} -2.47214 q^{35} +10.9443 q^{37} -5.52786 q^{39} +2.00000 q^{41} -1.52786 q^{43} +2.94427 q^{45} -12.9443 q^{47} -5.47214 q^{49} +1.23607 q^{51} +2.00000 q^{53} +2.47214 q^{55} -8.00000 q^{57} +14.4721 q^{59} -6.94427 q^{61} -1.81966 q^{63} +8.94427 q^{65} -12.0000 q^{67} +1.52786 q^{69} -9.23607 q^{71} +14.9443 q^{73} -1.23607 q^{75} -1.52786 q^{77} -11.7082 q^{79} -2.41641 q^{81} +1.52786 q^{83} -2.00000 q^{85} -2.47214 q^{87} -7.52786 q^{89} -5.52786 q^{91} -1.52786 q^{93} +12.9443 q^{95} +2.00000 q^{97} +1.81966 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 4 q^{5} - 2 q^{7} + 6 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 4 * q^5 - 2 * q^7 + 6 * q^9	\( 2 q - 2 q^{3} - 4 q^{5} - 2 q^{7} + 6 q^{9} + 2 q^{11} + 4 q^{15} + 2 q^{17} - 4 q^{19} + 12 q^{21} - 2 q^{23} - 2 q^{25} - 20 q^{27} - 4 q^{29} + 2 q^{31} - 12 q^{33} + 4 q^{35} + 4 q^{37} - 20 q^{39} + 4 q^{41} - 12 q^{43} - 12 q^{45} - 8 q^{47} - 2 q^{49} - 2 q^{51} + 4 q^{53} - 4 q^{55} - 16 q^{57} + 20 q^{59} + 4 q^{61} - 26 q^{63} - 24 q^{67} + 12 q^{69} - 14 q^{71} + 12 q^{73} + 2 q^{75} - 12 q^{77} - 10 q^{79} + 22 q^{81} + 12 q^{83} - 4 q^{85} + 4 q^{87} - 24 q^{89} - 20 q^{91} - 12 q^{93} + 8 q^{95} + 4 q^{97} + 26 q^{99}+O(q^{100}) \) 2 * q - 2 * q^3 - 4 * q^5 - 2 * q^7 + 6 * q^9 + 2 * q^11 + 4 * q^15 + 2 * q^17 - 4 * q^19 + 12 * q^21 - 2 * q^23 - 2 * q^25 - 20 * q^27 - 4 * q^29 + 2 * q^31 - 12 * q^33 + 4 * q^35 + 4 * q^37 - 20 * q^39 + 4 * q^41 - 12 * q^43 - 12 * q^45 - 8 * q^47 - 2 * q^49 - 2 * q^51 + 4 * q^53 - 4 * q^55 - 16 * q^57 + 20 * q^59 + 4 * q^61 - 26 * q^63 - 24 * q^67 + 12 * q^69 - 14 * q^71 + 12 * q^73 + 2 * q^75 - 12 * q^77 - 10 * q^79 + 22 * q^81 + 12 * q^83 - 4 * q^85 + 4 * q^87 - 24 * q^89 - 20 * q^91 - 12 * q^93 + 8 * q^95 + 4 * q^97 + 26 * q^99

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