LMFDB - Embedded newform 552.2.a.f.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [552,2,Mod(1,552)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(552, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("552.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 552 = 2^{3} \cdot 3 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	552.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:	$4.40774219157$
Analytic rank:	$0$
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:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	552.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.23607 q^{5} +2.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.23607 q^{5} +2.00000 q^{7} +1.00000 q^{9} +3.23607 q^{11} -4.47214 q^{13} +3.23607 q^{15} -6.47214 q^{17} -1.23607 q^{19} +2.00000 q^{21} +1.00000 q^{23} +5.47214 q^{25} +1.00000 q^{27} -8.47214 q^{29} +1.52786 q^{31} +3.23607 q^{33} +6.47214 q^{35} +7.70820 q^{37} -4.47214 q^{39} -2.00000 q^{41} +5.23607 q^{43} +3.23607 q^{45} -8.94427 q^{47} -3.00000 q^{49} -6.47214 q^{51} +12.1803 q^{53} +10.4721 q^{55} -1.23607 q^{57} -4.00000 q^{59} -11.7082 q^{61} +2.00000 q^{63} -14.4721 q^{65} +7.70820 q^{67} +1.00000 q^{69} +8.94427 q^{71} +4.47214 q^{73} +5.47214 q^{75} +6.47214 q^{77} +14.0000 q^{79} +1.00000 q^{81} -11.2361 q^{83} -20.9443 q^{85} -8.47214 q^{87} -16.9443 q^{89} -8.94427 q^{91} +1.52786 q^{93} -4.00000 q^{95} +4.47214 q^{97} +3.23607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 + 2 * q^5 + 4 * q^7 + 2 * q^9	$ 2 q + 2 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{15} - 4 q^{17} + 2 q^{19} + 4 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} - 8 q^{29} + 12 q^{31} + 2 q^{33} + 4 q^{35} + 2 q^{37} - 4 q^{41} + 6 q^{43} + 2 q^{45} - 6 q^{49} - 4 q^{51} + 2 q^{53} + 12 q^{55} + 2 q^{57} - 8 q^{59} - 10 q^{61} + 4 q^{63} - 20 q^{65} + 2 q^{67} + 2 q^{69} + 2 q^{75} + 4 q^{77} + 28 q^{79} + 2 q^{81} - 18 q^{83} - 24 q^{85} - 8 q^{87} - 16 q^{89} + 12 q^{93} - 8 q^{95} + 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 + 4 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^15 - 4 * q^17 + 2 * q^19 + 4 * q^21 + 2 * q^23 + 2 * q^25 + 2 * q^27 - 8 * q^29 + 12 * q^31 + 2 * q^33 + 4 * q^35 + 2 * q^37 - 4 * q^41 + 6 * q^43 + 2 * q^45 - 6 * q^49 - 4 * q^51 + 2 * q^53 + 12 * q^55 + 2 * q^57 - 8 * q^59 - 10 * q^61 + 4 * q^63 - 20 * q^65 + 2 * q^67 + 2 * q^69 + 2 * q^75 + 4 * q^77 + 28 * q^79 + 2 * q^81 - 18 * q^83 - 24 * q^85 - 8 * q^87 - 16 * q^89 + 12 * q^93 - 8 * q^95 + 2 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.23607	1.44721	0.723607	−	0.690212i	$-0.242483\pi$
$5$	3.23607	1.44721	0.723607	+	0.690212i	$0.242483\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	3.23607	0.975711	0.487856	−	0.872924i	$-0.337779\pi$
$11$	3.23607	0.975711	0.487856	+	0.872924i	$0.337779\pi$
$12$	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
$15$	3.23607	0.835549
$16$	0	0
$17$	−6.47214	−1.56972	−0.784862	−	0.619671i	$-0.787266\pi$
$17$	−6.47214	−1.56972	−0.784862	+	0.619671i	$0.787266\pi$
$18$	0	0
$19$	−1.23607	−0.283573	−0.141787	−	0.989897i	$-0.545285\pi$
$19$	−1.23607	−0.283573	−0.141787	+	0.989897i	$0.545285\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	5.47214	1.09443
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−8.47214	−1.57324	−0.786618	−	0.617440i	$-0.788170\pi$
$29$	−8.47214	−1.57324	−0.786618	+	0.617440i	$0.788170\pi$
$30$	0	0
$31$	1.52786	0.274412	0.137206	−	0.990543i	$-0.456188\pi$
$31$	1.52786	0.274412	0.137206	+	0.990543i	$0.456188\pi$
$32$	0	0
$33$	3.23607	0.563327
$34$	0	0
$35$	6.47214	1.09399
$36$	0	0
$37$	7.70820	1.26722	0.633610	−	0.773652i	$-0.281572\pi$
$37$	7.70820	1.26722	0.633610	+	0.773652i	$0.281572\pi$
$38$	0	0
$39$	−4.47214	−0.716115
$40$	0	0
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	5.23607	0.798493	0.399246	−	0.916844i	$-0.369272\pi$
$43$	5.23607	0.798493	0.399246	+	0.916844i	$0.369272\pi$
$44$	0	0
$45$	3.23607	0.482405
$46$	0	0
$47$	−8.94427	−1.30466	−0.652328	−	0.757937i	$-0.726208\pi$
$47$	−8.94427	−1.30466	−0.652328	+	0.757937i	$0.726208\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	−6.47214	−0.906280
$52$	0	0
$53$	12.1803	1.67310	0.836549	−	0.547892i	$-0.184569\pi$
$53$	12.1803	1.67310	0.836549	+	0.547892i	$0.184569\pi$
$54$	0	0
$55$	10.4721	1.41206
$56$	0	0
$57$	−1.23607	−0.163721
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	−11.7082	−1.49908	−0.749541	−	0.661958i	$-0.769726\pi$
$61$	−11.7082	−1.49908	−0.749541	+	0.661958i	$0.769726\pi$
$62$	0	0
$63$	2.00000	0.251976
$64$	0	0
$65$	−14.4721	−1.79505
$66$	0	0
$67$	7.70820	0.941707	0.470853	−	0.882211i	$-0.343946\pi$
$67$	7.70820	0.941707	0.470853	+	0.882211i	$0.343946\pi$
$68$	0	0
$69$	1.00000	0.120386
$70$	0	0
$71$	8.94427	1.06149	0.530745	−	0.847532i	$-0.321912\pi$
$71$	8.94427	1.06149	0.530745	+	0.847532i	$0.321912\pi$
$72$	0	0
$73$	4.47214	0.523424	0.261712	−	0.965146i	$-0.415713\pi$
$73$	4.47214	0.523424	0.261712	+	0.965146i	$0.415713\pi$
$74$	0	0
$75$	5.47214	0.631868
$76$	0	0
$77$	6.47214	0.737568
$78$	0	0
$79$	14.0000	1.57512	0.787562	−	0.616236i	$-0.211343\pi$
$79$	14.0000	1.57512	0.787562	+	0.616236i	$0.211343\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−11.2361	−1.23332	−0.616659	−	0.787230i	$-0.711514\pi$
$83$	−11.2361	−1.23332	−0.616659	+	0.787230i	$0.711514\pi$
$84$	0	0
$85$	−20.9443	−2.27173
$86$	0	0
$87$	−8.47214	−0.908308
$88$	0	0
$89$	−16.9443	−1.79609	−0.898045	−	0.439904i	$-0.855012\pi$
$89$	−16.9443	−1.79609	−0.898045	+	0.439904i	$0.855012\pi$
$90$	0	0
$91$	−8.94427	−0.937614
$92$	0	0
$93$	1.52786	0.158432
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	4.47214	0.454077	0.227038	−	0.973886i	$-0.427096\pi$
$97$	4.47214	0.454077	0.227038	+	0.973886i	$0.427096\pi$
$98$	0	0
$99$	3.23607	0.325237
$100$	0	0
$101$	−4.47214	−0.444994	−0.222497	−	0.974933i	$-0.571421\pi$
$101$	−4.47214	−0.444994	−0.222497	+	0.974933i	$0.571421\pi$
$102$	0	0
$103$	4.47214	0.440653	0.220326	−	0.975426i	$-0.429288\pi$
$103$	4.47214	0.440653	0.220326	+	0.975426i	$0.429288\pi$
$104$	0	0
$105$	6.47214	0.631616
$106$	0	0
$107$	−11.2361	−1.08623	−0.543116	−	0.839658i	$-0.682756\pi$
$107$	−11.2361	−1.08623	−0.543116	+	0.839658i	$0.682756\pi$
$108$	0	0
$109$	−2.76393	−0.264737	−0.132368	−	0.991201i	$-0.542258\pi$
$109$	−2.76393	−0.264737	−0.132368	+	0.991201i	$0.542258\pi$
$110$	0	0
$111$	7.70820	0.731630
$112$	0	0
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	0	0
$115$	3.23607	0.301765
$116$	0	0
$117$	−4.47214	−0.413449
$118$	0	0
$119$	−12.9443	−1.18660
$120$	0	0
$121$	−0.527864	−0.0479876
$122$	0	0
$123$	−2.00000	−0.180334
$124$	0	0
$125$	1.52786	0.136656
$126$	0	0
$127$	−0.944272	−0.0837906	−0.0418953	−	0.999122i	$-0.513340\pi$
$127$	−0.944272	−0.0837906	−0.0418953	+	0.999122i	$0.513340\pi$
$128$	0	0
$129$	5.23607	0.461010
$130$	0	0
$131$	−10.4721	−0.914955	−0.457477	−	0.889221i	$-0.651247\pi$
$131$	−10.4721	−0.914955	−0.457477	+	0.889221i	$0.651247\pi$
$132$	0	0
$133$	−2.47214	−0.214361
$134$	0	0
$135$	3.23607	0.278516
$136$	0	0
$137$	7.41641	0.633626	0.316813	−	0.948488i	$-0.397387\pi$
$137$	7.41641	0.633626	0.316813	+	0.948488i	$0.397387\pi$
$138$	0	0
$139$	8.94427	0.758643	0.379322	−	0.925265i	$-0.376157\pi$
$139$	8.94427	0.758643	0.379322	+	0.925265i	$0.376157\pi$
$140$	0	0
$141$	−8.94427	−0.753244
$142$	0	0
$143$	−14.4721	−1.21022
$144$	0	0
$145$	−27.4164	−2.27681
$146$	0	0
$147$	−3.00000	−0.247436
$148$	0	0
$149$	8.18034	0.670160	0.335080	−	0.942190i	$-0.391237\pi$
$149$	8.18034	0.670160	0.335080	+	0.942190i	$0.391237\pi$
$150$	0	0
$151$	10.4721	0.852210	0.426105	−	0.904674i	$-0.359885\pi$
$151$	10.4721	0.852210	0.426105	+	0.904674i	$0.359885\pi$
$152$	0	0
$153$	−6.47214	−0.523241
$154$	0	0
$155$	4.94427	0.397133
$156$	0	0
$157$	−0.291796	−0.0232879	−0.0116439	−	0.999932i	$-0.503706\pi$
$157$	−0.291796	−0.0232879	−0.0116439	+	0.999932i	$0.503706\pi$
$158$	0	0
$159$	12.1803	0.965964
$160$	0	0
$161$	2.00000	0.157622
$162$	0	0
$163$	−14.4721	−1.13355	−0.566773	−	0.823874i	$-0.691808\pi$
$163$	−14.4721	−1.13355	−0.566773	+	0.823874i	$0.691808\pi$
$164$	0	0
$165$	10.4721	0.815255
$166$	0	0
$167$	0.944272	0.0730700	0.0365350	−	0.999332i	$-0.488368\pi$
$167$	0.944272	0.0730700	0.0365350	+	0.999332i	$0.488368\pi$
$168$	0	0
$169$	7.00000	0.538462
$170$	0	0
$171$	−1.23607	−0.0945245
$172$	0	0
$173$	12.4721	0.948239	0.474119	−	0.880461i	$-0.342766\pi$
$173$	12.4721	0.948239	0.474119	+	0.880461i	$0.342766\pi$
$174$	0	0
$175$	10.9443	0.827309
$176$	0	0
$177$	−4.00000	−0.300658
$178$	0	0
$179$	−18.4721	−1.38067	−0.690336	−	0.723489i	$-0.742537\pi$
$179$	−18.4721	−1.38067	−0.690336	+	0.723489i	$0.742537\pi$
$180$	0	0
$181$	23.1246	1.71884	0.859419	−	0.511271i	$-0.170825\pi$
$181$	23.1246	1.71884	0.859419	+	0.511271i	$0.170825\pi$
$182$	0	0
$183$	−11.7082	−0.865495
$184$	0	0
$185$	24.9443	1.83394
$186$	0	0
$187$	−20.9443	−1.53160
$188$	0	0
$189$	2.00000	0.145479
$190$	0	0
$191$	26.4721	1.91546	0.957728	−	0.287674i	$-0.0928821\pi$
$191$	26.4721	1.91546	0.957728	+	0.287674i	$0.0928821\pi$
$192$	0	0
$193$	−11.8885	−0.855756	−0.427878	−	0.903836i	$-0.640739\pi$
$193$	−11.8885	−0.855756	−0.427878	+	0.903836i	$0.640739\pi$
$194$	0	0
$195$	−14.4721	−1.03637
$196$	0	0
$197$	−11.8885	−0.847024	−0.423512	−	0.905891i	$-0.639203\pi$
$197$	−11.8885	−0.847024	−0.423512	+	0.905891i	$0.639203\pi$
$198$	0	0
$199$	−22.9443	−1.62648	−0.813238	−	0.581931i	$-0.802297\pi$
$199$	−22.9443	−1.62648	−0.813238	+	0.581931i	$0.802297\pi$
$200$	0	0
$201$	7.70820	0.543695
$202$	0	0
$203$	−16.9443	−1.18925
$204$	0	0
$205$	−6.47214	−0.452034
$206$	0	0
$207$	1.00000	0.0695048
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−19.4164	−1.33668	−0.668340	−	0.743856i	$-0.732995\pi$
$211$	−19.4164	−1.33668	−0.668340	+	0.743856i	$0.732995\pi$
$212$	0	0
$213$	8.94427	0.612851
$214$	0	0
$215$	16.9443	1.15559
$216$	0	0
$217$	3.05573	0.207436
$218$	0	0
$219$	4.47214	0.302199
$220$	0	0
$221$	28.9443	1.94700
$222$	0	0
$223$	18.4721	1.23699	0.618493	−	0.785790i	$-0.287744\pi$
$223$	18.4721	1.23699	0.618493	+	0.785790i	$0.287744\pi$
$224$	0	0
$225$	5.47214	0.364809
$226$	0	0
$227$	−5.70820	−0.378867	−0.189433	−	0.981894i	$-0.560665\pi$
$227$	−5.70820	−0.378867	−0.189433	+	0.981894i	$0.560665\pi$
$228$	0	0
$229$	−23.7082	−1.56668	−0.783341	−	0.621592i	$-0.786486\pi$
$229$	−23.7082	−1.56668	−0.783341	+	0.621592i	$0.786486\pi$
$230$	0	0
$231$	6.47214	0.425835
$232$	0	0
$233$	26.0000	1.70332	0.851658	−	0.524097i	$-0.175597\pi$
$233$	26.0000	1.70332	0.851658	+	0.524097i	$0.175597\pi$
$234$	0	0
$235$	−28.9443	−1.88812
$236$	0	0
$237$	14.0000	0.909398
$238$	0	0
$239$	12.9443	0.837295	0.418648	−	0.908149i	$-0.362504\pi$
$239$	12.9443	0.837295	0.418648	+	0.908149i	$0.362504\pi$
$240$	0	0
$241$	−0.472136	−0.0304130	−0.0152065	−	0.999884i	$-0.504841\pi$
$241$	−0.472136	−0.0304130	−0.0152065	+	0.999884i	$0.504841\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−9.70820	−0.620234
$246$	0	0
$247$	5.52786	0.351730
$248$	0	0
$249$	−11.2361	−0.712057
$250$	0	0
$251$	−25.1246	−1.58585	−0.792926	−	0.609318i	$-0.791443\pi$
$251$	−25.1246	−1.58585	−0.792926	+	0.609318i	$0.791443\pi$
$252$	0	0
$253$	3.23607	0.203450
$254$	0	0
$255$	−20.9443	−1.31158
$256$	0	0
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	15.4164	0.957929
$260$	0	0
$261$	−8.47214	−0.524412
$262$	0	0
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	39.4164	2.42133
$266$	0	0
$267$	−16.9443	−1.03697
$268$	0	0
$269$	23.8885	1.45651	0.728255	−	0.685306i	$-0.240332\pi$
$269$	23.8885	1.45651	0.728255	+	0.685306i	$0.240332\pi$
$270$	0	0
$271$	29.8885	1.81560	0.907800	−	0.419404i	$-0.137761\pi$
$271$	29.8885	1.81560	0.907800	+	0.419404i	$0.137761\pi$
$272$	0	0
$273$	−8.94427	−0.541332
$274$	0	0
$275$	17.7082	1.06784
$276$	0	0
$277$	12.4721	0.749378	0.374689	−	0.927151i	$-0.377749\pi$
$277$	12.4721	0.749378	0.374689	+	0.927151i	$0.377749\pi$
$278$	0	0
$279$	1.52786	0.0914708
$280$	0	0
$281$	4.00000	0.238620	0.119310	−	0.992857i	$-0.461932\pi$
$281$	4.00000	0.238620	0.119310	+	0.992857i	$0.461932\pi$
$282$	0	0
$283$	19.1246	1.13684	0.568420	−	0.822738i	$-0.307555\pi$
$283$	19.1246	1.13684	0.568420	+	0.822738i	$0.307555\pi$
$284$	0	0
$285$	−4.00000	−0.236940
$286$	0	0
$287$	−4.00000	−0.236113
$288$	0	0
$289$	24.8885	1.46403
$290$	0	0
$291$	4.47214	0.262161
$292$	0	0
$293$	−10.6525	−0.622324	−0.311162	−	0.950357i	$-0.600718\pi$
$293$	−10.6525	−0.622324	−0.311162	+	0.950357i	$0.600718\pi$
$294$	0	0
$295$	−12.9443	−0.753645
$296$	0	0
$297$	3.23607	0.187776
$298$	0	0
$299$	−4.47214	−0.258630
$300$	0	0
$301$	10.4721	0.603604
$302$	0	0
$303$	−4.47214	−0.256917
$304$	0	0
$305$	−37.8885	−2.16949
$306$	0	0
$307$	19.4164	1.10815	0.554076	−	0.832466i	$-0.313072\pi$
$307$	19.4164	1.10815	0.554076	+	0.832466i	$0.313072\pi$
$308$	0	0
$309$	4.47214	0.254411
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	6.58359	0.372127	0.186063	−	0.982538i	$-0.440427\pi$
$313$	6.58359	0.372127	0.186063	+	0.982538i	$0.440427\pi$
$314$	0	0
$315$	6.47214	0.364664
$316$	0	0
$317$	−10.3607	−0.581914	−0.290957	−	0.956736i	$-0.593974\pi$
$317$	−10.3607	−0.581914	−0.290957	+	0.956736i	$0.593974\pi$
$318$	0	0
$319$	−27.4164	−1.53502
$320$	0	0
$321$	−11.2361	−0.627136
$322$	0	0
$323$	8.00000	0.445132
$324$	0	0
$325$	−24.4721	−1.35747
$326$	0	0
$327$	−2.76393	−0.152846
$328$	0	0
$329$	−17.8885	−0.986227
$330$	0	0
$331$	17.5279	0.963419	0.481709	−	0.876331i	$-0.340016\pi$
$331$	17.5279	0.963419	0.481709	+	0.876331i	$0.340016\pi$
$332$	0	0
$333$	7.70820	0.422407
$334$	0	0
$335$	24.9443	1.36285
$336$	0	0
$337$	20.8328	1.13484	0.567418	−	0.823430i	$-0.307942\pi$
$337$	20.8328	1.13484	0.567418	+	0.823430i	$0.307942\pi$
$338$	0	0
$339$	−4.00000	−0.217250
$340$	0	0
$341$	4.94427	0.267747
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	3.23607	0.174224
$346$	0	0
$347$	20.3607	1.09302	0.546509	−	0.837453i	$-0.315956\pi$
$347$	20.3607	1.09302	0.546509	+	0.837453i	$0.315956\pi$
$348$	0	0
$349$	−14.0000	−0.749403	−0.374701	−	0.927146i	$-0.622255\pi$
$349$	−14.0000	−0.749403	−0.374701	+	0.927146i	$0.622255\pi$
$350$	0	0
$351$	−4.47214	−0.238705
$352$	0	0
$353$	10.9443	0.582505	0.291252	−	0.956646i	$-0.405928\pi$
$353$	10.9443	0.582505	0.291252	+	0.956646i	$0.405928\pi$
$354$	0	0
$355$	28.9443	1.53620
$356$	0	0
$357$	−12.9443	−0.685084
$358$	0	0
$359$	−20.3607	−1.07460	−0.537298	−	0.843393i	$-0.680555\pi$
$359$	−20.3607	−1.07460	−0.537298	+	0.843393i	$0.680555\pi$
$360$	0	0
$361$	−17.4721	−0.919586
$362$	0	0
$363$	−0.527864	−0.0277057
$364$	0	0
$365$	14.4721	0.757506
$366$	0	0
$367$	26.9443	1.40648	0.703240	−	0.710953i	$-0.251736\pi$
$367$	26.9443	1.40648	0.703240	+	0.710953i	$0.251736\pi$
$368$	0	0
$369$	−2.00000	−0.104116
$370$	0	0
$371$	24.3607	1.26474
$372$	0	0
$373$	−5.23607	−0.271113	−0.135557	−	0.990770i	$-0.543282\pi$
$373$	−5.23607	−0.271113	−0.135557	+	0.990770i	$0.543282\pi$
$374$	0	0
$375$	1.52786	0.0788986
$376$	0	0
$377$	37.8885	1.95136
$378$	0	0
$379$	−25.2361	−1.29629	−0.648145	−	0.761517i	$-0.724455\pi$
$379$	−25.2361	−1.29629	−0.648145	+	0.761517i	$0.724455\pi$
$380$	0	0
$381$	−0.944272	−0.0483765
$382$	0	0
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	0	0
$385$	20.9443	1.06742
$386$	0	0
$387$	5.23607	0.266164
$388$	0	0
$389$	−18.2918	−0.927431	−0.463715	−	0.885984i	$-0.653484\pi$
$389$	−18.2918	−0.927431	−0.463715	+	0.885984i	$0.653484\pi$
$390$	0	0
$391$	−6.47214	−0.327310
$392$	0	0
$393$	−10.4721	−0.528249
$394$	0	0
$395$	45.3050	2.27954
$396$	0	0
$397$	−38.9443	−1.95456	−0.977278	−	0.211959i	$-0.932016\pi$
$397$	−38.9443	−1.95456	−0.977278	+	0.211959i	$0.932016\pi$
$398$	0	0
$399$	−2.47214	−0.123762
$400$	0	0
$401$	−11.4164	−0.570108	−0.285054	−	0.958511i	$-0.592012\pi$
$401$	−11.4164	−0.570108	−0.285054	+	0.958511i	$0.592012\pi$
$402$	0	0
$403$	−6.83282	−0.340367
$404$	0	0
$405$	3.23607	0.160802
$406$	0	0
$407$	24.9443	1.23644
$408$	0	0
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	0	0
$411$	7.41641	0.365824
$412$	0	0
$413$	−8.00000	−0.393654
$414$	0	0
$415$	−36.3607	−1.78488
$416$	0	0
$417$	8.94427	0.438003
$418$	0	0
$419$	−14.2918	−0.698200	−0.349100	−	0.937085i	$-0.613513\pi$
$419$	−14.2918	−0.698200	−0.349100	+	0.937085i	$0.613513\pi$
$420$	0	0
$421$	−9.23607	−0.450138	−0.225069	−	0.974343i	$-0.572261\pi$
$421$	−9.23607	−0.450138	−0.225069	+	0.974343i	$0.572261\pi$
$422$	0	0
$423$	−8.94427	−0.434885
$424$	0	0
$425$	−35.4164	−1.71795
$426$	0	0
$427$	−23.4164	−1.13320
$428$	0	0
$429$	−14.4721	−0.698721
$430$	0	0
$431$	−8.36068	−0.402720	−0.201360	−	0.979517i	$-0.564536\pi$
$431$	−8.36068	−0.402720	−0.201360	+	0.979517i	$0.564536\pi$
$432$	0	0
$433$	−2.58359	−0.124160	−0.0620798	−	0.998071i	$-0.519773\pi$
$433$	−2.58359	−0.124160	−0.0620798	+	0.998071i	$0.519773\pi$
$434$	0	0
$435$	−27.4164	−1.31452
$436$	0	0
$437$	−1.23607	−0.0591292
$438$	0	0
$439$	−28.9443	−1.38143	−0.690717	−	0.723125i	$-0.742705\pi$
$439$	−28.9443	−1.38143	−0.690717	+	0.723125i	$0.742705\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	−26.8328	−1.27487	−0.637433	−	0.770506i	$-0.720004\pi$
$443$	−26.8328	−1.27487	−0.637433	+	0.770506i	$0.720004\pi$
$444$	0	0
$445$	−54.8328	−2.59932
$446$	0	0
$447$	8.18034	0.386917
$448$	0	0
$449$	−10.0000	−0.471929	−0.235965	−	0.971762i	$-0.575825\pi$
$449$	−10.0000	−0.471929	−0.235965	+	0.971762i	$0.575825\pi$
$450$	0	0
$451$	−6.47214	−0.304761
$452$	0	0
$453$	10.4721	0.492024
$454$	0	0
$455$	−28.9443	−1.35693
$456$	0	0
$457$	−2.36068	−0.110428	−0.0552140	−	0.998475i	$-0.517584\pi$
$457$	−2.36068	−0.110428	−0.0552140	+	0.998475i	$0.517584\pi$
$458$	0	0
$459$	−6.47214	−0.302093
$460$	0	0
$461$	9.05573	0.421767	0.210884	−	0.977511i	$-0.432366\pi$
$461$	9.05573	0.421767	0.210884	+	0.977511i	$0.432366\pi$
$462$	0	0
$463$	9.88854	0.459560	0.229780	−	0.973243i	$-0.426199\pi$
$463$	9.88854	0.459560	0.229780	+	0.973243i	$0.426199\pi$
$464$	0	0
$465$	4.94427	0.229285
$466$	0	0
$467$	24.7639	1.14594	0.572969	−	0.819577i	$-0.305792\pi$
$467$	24.7639	1.14594	0.572969	+	0.819577i	$0.305792\pi$
$468$	0	0
$469$	15.4164	0.711864
$470$	0	0
$471$	−0.291796	−0.0134453
$472$	0	0
$473$	16.9443	0.779098
$474$	0	0
$475$	−6.76393	−0.310350
$476$	0	0
$477$	12.1803	0.557699
$478$	0	0
$479$	−20.9443	−0.956968	−0.478484	−	0.878096i	$-0.658814\pi$
$479$	−20.9443	−0.956968	−0.478484	+	0.878096i	$0.658814\pi$
$480$	0	0
$481$	−34.4721	−1.57179
$482$	0	0
$483$	2.00000	0.0910032
$484$	0	0
$485$	14.4721	0.657146
$486$	0	0
$487$	1.88854	0.0855781	0.0427890	−	0.999084i	$-0.486376\pi$
$487$	1.88854	0.0855781	0.0427890	+	0.999084i	$0.486376\pi$
$488$	0	0
$489$	−14.4721	−0.654453
$490$	0	0
$491$	−28.3607	−1.27990	−0.639950	−	0.768417i	$-0.721045\pi$
$491$	−28.3607	−1.27990	−0.639950	+	0.768417i	$0.721045\pi$
$492$	0	0
$493$	54.8328	2.46955
$494$	0	0
$495$	10.4721	0.470688
$496$	0	0
$497$	17.8885	0.802411
$498$	0	0
$499$	−9.52786	−0.426526	−0.213263	−	0.976995i	$-0.568409\pi$
$499$	−9.52786	−0.426526	−0.213263	+	0.976995i	$0.568409\pi$
$500$	0	0
$501$	0.944272	0.0421870
$502$	0	0
$503$	29.3050	1.30664	0.653322	−	0.757080i	$-0.273375\pi$
$503$	29.3050	1.30664	0.653322	+	0.757080i	$0.273375\pi$
$504$	0	0
$505$	−14.4721	−0.644002
$506$	0	0
$507$	7.00000	0.310881
$508$	0	0
$509$	12.4721	0.552818	0.276409	−	0.961040i	$-0.410856\pi$
$509$	12.4721	0.552818	0.276409	+	0.961040i	$0.410856\pi$
$510$	0	0
$511$	8.94427	0.395671
$512$	0	0
$513$	−1.23607	−0.0545737
$514$	0	0
$515$	14.4721	0.637719
$516$	0	0
$517$	−28.9443	−1.27297
$518$	0	0
$519$	12.4721	0.547466
$520$	0	0
$521$	12.3607	0.541531	0.270766	−	0.962645i	$-0.412723\pi$
$521$	12.3607	0.541531	0.270766	+	0.962645i	$0.412723\pi$
$522$	0	0
$523$	7.70820	0.337056	0.168528	−	0.985697i	$-0.446099\pi$
$523$	7.70820	0.337056	0.168528	+	0.985697i	$0.446099\pi$
$524$	0	0
$525$	10.9443	0.477647
$526$	0	0
$527$	−9.88854	−0.430752
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	8.94427	0.387419
$534$	0	0
$535$	−36.3607	−1.57201
$536$	0	0
$537$	−18.4721	−0.797132
$538$	0	0
$539$	−9.70820	−0.418162
$540$	0	0
$541$	19.8885	0.855075	0.427538	−	0.903998i	$-0.359381\pi$
$541$	19.8885	0.855075	0.427538	+	0.903998i	$0.359381\pi$
$542$	0	0
$543$	23.1246	0.992372
$544$	0	0
$545$	−8.94427	−0.383131
$546$	0	0
$547$	27.4164	1.17224	0.586120	−	0.810224i	$-0.300655\pi$
$547$	27.4164	1.17224	0.586120	+	0.810224i	$0.300655\pi$
$548$	0	0
$549$	−11.7082	−0.499694
$550$	0	0
$551$	10.4721	0.446128
$552$	0	0
$553$	28.0000	1.19068
$554$	0	0
$555$	24.9443	1.05883
$556$	0	0
$557$	−7.23607	−0.306602	−0.153301	−	0.988180i	$-0.548990\pi$
$557$	−7.23607	−0.306602	−0.153301	+	0.988180i	$0.548990\pi$
$558$	0	0
$559$	−23.4164	−0.990409
$560$	0	0
$561$	−20.9443	−0.884268
$562$	0	0
$563$	−29.7082	−1.25205	−0.626026	−	0.779802i	$-0.715320\pi$
$563$	−29.7082	−1.25205	−0.626026	+	0.779802i	$0.715320\pi$
$564$	0	0
$565$	−12.9443	−0.544570
$566$	0	0
$567$	2.00000	0.0839921
$568$	0	0
$569$	−20.5836	−0.862909	−0.431455	−	0.902135i	$-0.641999\pi$
$569$	−20.5836	−0.862909	−0.431455	+	0.902135i	$0.641999\pi$
$570$	0	0
$571$	15.7082	0.657368	0.328684	−	0.944440i	$-0.393395\pi$
$571$	15.7082	0.657368	0.328684	+	0.944440i	$0.393395\pi$
$572$	0	0
$573$	26.4721	1.10589
$574$	0	0
$575$	5.47214	0.228204
$576$	0	0
$577$	−35.3050	−1.46976	−0.734882	−	0.678195i	$-0.762763\pi$
$577$	−35.3050	−1.46976	−0.734882	+	0.678195i	$0.762763\pi$
$578$	0	0
$579$	−11.8885	−0.494071
$580$	0	0
$581$	−22.4721	−0.932301
$582$	0	0
$583$	39.4164	1.63246
$584$	0	0
$585$	−14.4721	−0.598349
$586$	0	0
$587$	37.5279	1.54894	0.774470	−	0.632610i	$-0.218016\pi$
$587$	37.5279	1.54894	0.774470	+	0.632610i	$0.218016\pi$
$588$	0	0
$589$	−1.88854	−0.0778161
$590$	0	0
$591$	−11.8885	−0.489029
$592$	0	0
$593$	33.7771	1.38706	0.693529	−	0.720428i	$-0.256055\pi$
$593$	33.7771	1.38706	0.693529	+	0.720428i	$0.256055\pi$
$594$	0	0
$595$	−41.8885	−1.71726
$596$	0	0
$597$	−22.9443	−0.939047
$598$	0	0
$599$	17.8885	0.730906	0.365453	−	0.930830i	$-0.380914\pi$
$599$	17.8885	0.730906	0.365453	+	0.930830i	$0.380914\pi$
$600$	0	0
$601$	8.47214	0.345586	0.172793	−	0.984958i	$-0.444721\pi$
$601$	8.47214	0.345586	0.172793	+	0.984958i	$0.444721\pi$
$602$	0	0
$603$	7.70820	0.313902
$604$	0	0
$605$	−1.70820	−0.0694484
$606$	0	0
$607$	15.4164	0.625733	0.312866	−	0.949797i	$-0.398711\pi$
$607$	15.4164	0.625733	0.312866	+	0.949797i	$0.398711\pi$
$608$	0	0
$609$	−16.9443	−0.686617
$610$	0	0
$611$	40.0000	1.61823
$612$	0	0
$613$	−27.7082	−1.11912	−0.559562	−	0.828789i	$-0.689031\pi$
$613$	−27.7082	−1.11912	−0.559562	+	0.828789i	$0.689031\pi$
$614$	0	0
$615$	−6.47214	−0.260982
$616$	0	0
$617$	−39.7771	−1.60137	−0.800683	−	0.599089i	$-0.795530\pi$
$617$	−39.7771	−1.60137	−0.800683	+	0.599089i	$0.795530\pi$
$618$	0	0
$619$	29.0132	1.16614	0.583069	−	0.812423i	$-0.301852\pi$
$619$	29.0132	1.16614	0.583069	+	0.812423i	$0.301852\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	0	0
$623$	−33.8885	−1.35772
$624$	0	0
$625$	−22.4164	−0.896656
$626$	0	0
$627$	−4.00000	−0.159745
$628$	0	0
$629$	−49.8885	−1.98919
$630$	0	0
$631$	20.4721	0.814983	0.407491	−	0.913209i	$-0.366404\pi$
$631$	20.4721	0.814983	0.407491	+	0.913209i	$0.366404\pi$
$632$	0	0
$633$	−19.4164	−0.771733
$634$	0	0
$635$	−3.05573	−0.121263
$636$	0	0
$637$	13.4164	0.531577
$638$	0	0
$639$	8.94427	0.353830
$640$	0	0
$641$	−4.58359	−0.181041	−0.0905205	−	0.995895i	$-0.528853\pi$
$641$	−4.58359	−0.181041	−0.0905205	+	0.995895i	$0.528853\pi$
$642$	0	0
$643$	29.2361	1.15296	0.576479	−	0.817112i	$-0.304426\pi$
$643$	29.2361	1.15296	0.576479	+	0.817112i	$0.304426\pi$
$644$	0	0
$645$	16.9443	0.667180
$646$	0	0
$647$	−41.8885	−1.64681	−0.823404	−	0.567455i	$-0.807928\pi$
$647$	−41.8885	−1.64681	−0.823404	+	0.567455i	$0.807928\pi$
$648$	0	0
$649$	−12.9443	−0.508107
$650$	0	0
$651$	3.05573	0.119763
$652$	0	0
$653$	40.4721	1.58380	0.791899	−	0.610653i	$-0.209093\pi$
$653$	40.4721	1.58380	0.791899	+	0.610653i	$0.209093\pi$
$654$	0	0
$655$	−33.8885	−1.32413
$656$	0	0
$657$	4.47214	0.174475
$658$	0	0
$659$	41.4853	1.61604	0.808019	−	0.589157i	$-0.200540\pi$
$659$	41.4853	1.61604	0.808019	+	0.589157i	$0.200540\pi$
$660$	0	0
$661$	9.23607	0.359241	0.179621	−	0.983736i	$-0.442513\pi$
$661$	9.23607	0.359241	0.179621	+	0.983736i	$0.442513\pi$
$662$	0	0
$663$	28.9443	1.12410
$664$	0	0
$665$	−8.00000	−0.310227
$666$	0	0
$667$	−8.47214	−0.328042
$668$	0	0
$669$	18.4721	0.714174
$670$	0	0
$671$	−37.8885	−1.46267
$672$	0	0
$673$	21.4164	0.825542	0.412771	−	0.910835i	$-0.364561\pi$
$673$	21.4164	0.825542	0.412771	+	0.910835i	$0.364561\pi$
$674$	0	0
$675$	5.47214	0.210623
$676$	0	0
$677$	21.1246	0.811885	0.405942	−	0.913899i	$-0.366943\pi$
$677$	21.1246	0.811885	0.405942	+	0.913899i	$0.366943\pi$
$678$	0	0
$679$	8.94427	0.343250
$680$	0	0
$681$	−5.70820	−0.218739
$682$	0	0
$683$	−28.3607	−1.08519	−0.542596	−	0.839994i	$-0.682558\pi$
$683$	−28.3607	−1.08519	−0.542596	+	0.839994i	$0.682558\pi$
$684$	0	0
$685$	24.0000	0.916993
$686$	0	0
$687$	−23.7082	−0.904524
$688$	0	0
$689$	−54.4721	−2.07522
$690$	0	0
$691$	46.4721	1.76788	0.883942	−	0.467597i	$-0.154880\pi$
$691$	46.4721	1.76788	0.883942	+	0.467597i	$0.154880\pi$
$692$	0	0
$693$	6.47214	0.245856
$694$	0	0
$695$	28.9443	1.09792
$696$	0	0
$697$	12.9443	0.490299
$698$	0	0
$699$	26.0000	0.983410
$700$	0	0
$701$	−48.1803	−1.81975	−0.909873	−	0.414887i	$-0.863821\pi$
$701$	−48.1803	−1.81975	−0.909873	+	0.414887i	$0.863821\pi$
$702$	0	0
$703$	−9.52786	−0.359350
$704$	0	0
$705$	−28.9443	−1.09010
$706$	0	0
$707$	−8.94427	−0.336384
$708$	0	0
$709$	16.6525	0.625397	0.312698	−	0.949852i	$-0.398767\pi$
$709$	16.6525	0.625397	0.312698	+	0.949852i	$0.398767\pi$
$710$	0	0
$711$	14.0000	0.525041
$712$	0	0
$713$	1.52786	0.0572190
$714$	0	0
$715$	−46.8328	−1.75145
$716$	0	0
$717$	12.9443	0.483413
$718$	0	0
$719$	−6.11146	−0.227919	−0.113959	−	0.993485i	$-0.536353\pi$
$719$	−6.11146	−0.227919	−0.113959	+	0.993485i	$0.536353\pi$
$720$	0	0
$721$	8.94427	0.333102
$722$	0	0
$723$	−0.472136	−0.0175589
$724$	0	0
$725$	−46.3607	−1.72179
$726$	0	0
$727$	−36.8328	−1.36605	−0.683027	−	0.730393i	$-0.739337\pi$
$727$	−36.8328	−1.36605	−0.683027	+	0.730393i	$0.739337\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−33.8885	−1.25341
$732$	0	0
$733$	35.7082	1.31891	0.659456	−	0.751743i	$-0.270787\pi$
$733$	35.7082	1.31891	0.659456	+	0.751743i	$0.270787\pi$
$734$	0	0
$735$	−9.70820	−0.358092
$736$	0	0
$737$	24.9443	0.918834
$738$	0	0
$739$	−40.9443	−1.50616	−0.753080	−	0.657929i	$-0.771433\pi$
$739$	−40.9443	−1.50616	−0.753080	+	0.657929i	$0.771433\pi$
$740$	0	0
$741$	5.52786	0.203071
$742$	0	0
$743$	23.0557	0.845833	0.422916	−	0.906169i	$-0.361006\pi$
$743$	23.0557	0.845833	0.422916	+	0.906169i	$0.361006\pi$
$744$	0	0
$745$	26.4721	0.969864
$746$	0	0
$747$	−11.2361	−0.411106
$748$	0	0
$749$	−22.4721	−0.821114
$750$	0	0
$751$	45.4164	1.65727	0.828634	−	0.559791i	$-0.189118\pi$
$751$	45.4164	1.65727	0.828634	+	0.559791i	$0.189118\pi$
$752$	0	0
$753$	−25.1246	−0.915592
$754$	0	0
$755$	33.8885	1.23333
$756$	0	0
$757$	−18.7639	−0.681987	−0.340993	−	0.940066i	$-0.610763\pi$
$757$	−18.7639	−0.681987	−0.340993	+	0.940066i	$0.610763\pi$
$758$	0	0
$759$	3.23607	0.117462
$760$	0	0
$761$	−27.8885	−1.01096	−0.505479	−	0.862839i	$-0.668684\pi$
$761$	−27.8885	−1.01096	−0.505479	+	0.862839i	$0.668684\pi$
$762$	0	0
$763$	−5.52786	−0.200122
$764$	0	0
$765$	−20.9443	−0.757242
$766$	0	0
$767$	17.8885	0.645918
$768$	0	0
$769$	43.5279	1.56965	0.784827	−	0.619714i	$-0.212752\pi$
$769$	43.5279	1.56965	0.784827	+	0.619714i	$0.212752\pi$
$770$	0	0
$771$	14.0000	0.504198
$772$	0	0
$773$	−31.0132	−1.11547	−0.557733	−	0.830021i	$-0.688329\pi$
$773$	−31.0132	−1.11547	−0.557733	+	0.830021i	$0.688329\pi$
$774$	0	0
$775$	8.36068	0.300324
$776$	0	0
$777$	15.4164	0.553061
$778$	0	0
$779$	2.47214	0.0885735
$780$	0	0
$781$	28.9443	1.03571
$782$	0	0
$783$	−8.47214	−0.302769
$784$	0	0
$785$	−0.944272	−0.0337025
$786$	0	0
$787$	−2.76393	−0.0985235	−0.0492618	−	0.998786i	$-0.515687\pi$
$787$	−2.76393	−0.0985235	−0.0492618	+	0.998786i	$0.515687\pi$
$788$	0	0
$789$	12.0000	0.427211
$790$	0	0
$791$	−8.00000	−0.284447
$792$	0	0
$793$	52.3607	1.85938
$794$	0	0
$795$	39.4164	1.39796
$796$	0	0
$797$	−4.18034	−0.148075	−0.0740376	−	0.997255i	$-0.523588\pi$
$797$	−4.18034	−0.148075	−0.0740376	+	0.997255i	$0.523588\pi$
$798$	0	0
$799$	57.8885	2.04795
$800$	0	0
$801$	−16.9443	−0.598696
$802$	0	0
$803$	14.4721	0.510711
$804$	0	0
$805$	6.47214	0.228113
$806$	0	0
$807$	23.8885	0.840917
$808$	0	0
$809$	−2.00000	−0.0703163	−0.0351581	−	0.999382i	$-0.511193\pi$
$809$	−2.00000	−0.0703163	−0.0351581	+	0.999382i	$0.511193\pi$
$810$	0	0
$811$	53.3050	1.87179	0.935895	−	0.352279i	$-0.114593\pi$
$811$	53.3050	1.87179	0.935895	+	0.352279i	$0.114593\pi$
$812$	0	0
$813$	29.8885	1.04824
$814$	0	0
$815$	−46.8328	−1.64048
$816$	0	0
$817$	−6.47214	−0.226431
$818$	0	0
$819$	−8.94427	−0.312538
$820$	0	0
$821$	42.3607	1.47840	0.739199	−	0.673487i	$-0.235204\pi$
$821$	42.3607	1.47840	0.739199	+	0.673487i	$0.235204\pi$
$822$	0	0
$823$	−32.3607	−1.12802	−0.564011	−	0.825767i	$-0.690743\pi$
$823$	−32.3607	−1.12802	−0.564011	+	0.825767i	$0.690743\pi$
$824$	0	0
$825$	17.7082	0.616521
$826$	0	0
$827$	24.7639	0.861126	0.430563	−	0.902560i	$-0.358315\pi$
$827$	24.7639	0.861126	0.430563	+	0.902560i	$0.358315\pi$
$828$	0	0
$829$	−17.7771	−0.617424	−0.308712	−	0.951156i	$-0.599898\pi$
$829$	−17.7771	−0.617424	−0.308712	+	0.951156i	$0.599898\pi$
$830$	0	0
$831$	12.4721	0.432654
$832$	0	0
$833$	19.4164	0.672739
$834$	0	0
$835$	3.05573	0.105748
$836$	0	0
$837$	1.52786	0.0528107
$838$	0	0
$839$	−24.9443	−0.861172	−0.430586	−	0.902550i	$-0.641693\pi$
$839$	−24.9443	−0.861172	−0.430586	+	0.902550i	$0.641693\pi$
$840$	0	0
$841$	42.7771	1.47507
$842$	0	0
$843$	4.00000	0.137767
$844$	0	0
$845$	22.6525	0.779269
$846$	0	0
$847$	−1.05573	−0.0362752
$848$	0	0
$849$	19.1246	0.656355
$850$	0	0
$851$	7.70820	0.264234
$852$	0	0
$853$	54.0000	1.84892	0.924462	−	0.381273i	$-0.124514\pi$
$853$	54.0000	1.84892	0.924462	+	0.381273i	$0.124514\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	0	0
$857$	29.7771	1.01717	0.508583	−	0.861013i	$-0.330170\pi$
$857$	29.7771	1.01717	0.508583	+	0.861013i	$0.330170\pi$
$858$	0	0
$859$	−55.7771	−1.90309	−0.951545	−	0.307510i	$-0.900504\pi$
$859$	−55.7771	−1.90309	−0.951545	+	0.307510i	$0.900504\pi$
$860$	0	0
$861$	−4.00000	−0.136320
$862$	0	0
$863$	33.8885	1.15358	0.576790	−	0.816893i	$-0.304305\pi$
$863$	33.8885	1.15358	0.576790	+	0.816893i	$0.304305\pi$
$864$	0	0
$865$	40.3607	1.37230
$866$	0	0
$867$	24.8885	0.845259
$868$	0	0
$869$	45.3050	1.53687
$870$	0	0
$871$	−34.4721	−1.16804
$872$	0	0
$873$	4.47214	0.151359
$874$	0	0
$875$	3.05573	0.103302
$876$	0	0
$877$	−1.63932	−0.0553559	−0.0276780	−	0.999617i	$-0.508811\pi$
$877$	−1.63932	−0.0553559	−0.0276780	+	0.999617i	$0.508811\pi$
$878$	0	0
$879$	−10.6525	−0.359299
$880$	0	0
$881$	−15.0557	−0.507240	−0.253620	−	0.967304i	$-0.581621\pi$
$881$	−15.0557	−0.507240	−0.253620	+	0.967304i	$0.581621\pi$
$882$	0	0
$883$	−0.360680	−0.0121378	−0.00606892	−	0.999982i	$-0.501932\pi$
$883$	−0.360680	−0.0121378	−0.00606892	+	0.999982i	$0.501932\pi$
$884$	0	0
$885$	−12.9443	−0.435117
$886$	0	0
$887$	20.0000	0.671534	0.335767	−	0.941945i	$-0.391004\pi$
$887$	20.0000	0.671534	0.335767	+	0.941945i	$0.391004\pi$
$888$	0	0
$889$	−1.88854	−0.0633397
$890$	0	0
$891$	3.23607	0.108412
$892$	0	0
$893$	11.0557	0.369966
$894$	0	0
$895$	−59.7771	−1.99813
$896$	0	0
$897$	−4.47214	−0.149320
$898$	0	0
$899$	−12.9443	−0.431716
$900$	0	0
$901$	−78.8328	−2.62630
$902$	0	0
$903$	10.4721	0.348491
$904$	0	0
$905$	74.8328	2.48753
$906$	0	0
$907$	23.3475	0.775242	0.387621	−	0.921819i	$-0.373297\pi$
$907$	23.3475	0.775242	0.387621	+	0.921819i	$0.373297\pi$
$908$	0	0
$909$	−4.47214	−0.148331
$910$	0	0
$911$	29.3050	0.970916	0.485458	−	0.874260i	$-0.338653\pi$
$911$	29.3050	0.970916	0.485458	+	0.874260i	$0.338653\pi$
$912$	0	0
$913$	−36.3607	−1.20336
$914$	0	0
$915$	−37.8885	−1.25256
$916$	0	0
$917$	−20.9443	−0.691641
$918$	0	0
$919$	26.9443	0.888810	0.444405	−	0.895826i	$-0.353415\pi$
$919$	26.9443	0.888810	0.444405	+	0.895826i	$0.353415\pi$
$920$	0	0
$921$	19.4164	0.639792
$922$	0	0
$923$	−40.0000	−1.31662
$924$	0	0
$925$	42.1803	1.38688
$926$	0	0
$927$	4.47214	0.146884
$928$	0	0
$929$	−22.9443	−0.752777	−0.376389	−	0.926462i	$-0.622834\pi$
$929$	−22.9443	−0.752777	−0.376389	+	0.926462i	$0.622834\pi$
$930$	0	0
$931$	3.70820	0.121531
$932$	0	0
$933$	24.0000	0.785725
$934$	0	0
$935$	−67.7771	−2.21655
$936$	0	0
$937$	−17.7771	−0.580752	−0.290376	−	0.956913i	$-0.593780\pi$
$937$	−17.7771	−0.580752	−0.290376	+	0.956913i	$0.593780\pi$
$938$	0	0
$939$	6.58359	0.214847
$940$	0	0
$941$	−35.5967	−1.16042	−0.580210	−	0.814467i	$-0.697030\pi$
$941$	−35.5967	−1.16042	−0.580210	+	0.814467i	$0.697030\pi$
$942$	0	0
$943$	−2.00000	−0.0651290
$944$	0	0
$945$	6.47214	0.210539
$946$	0	0
$947$	0.583592	0.0189642	0.00948210	−	0.999955i	$-0.496982\pi$
$947$	0.583592	0.0189642	0.00948210	+	0.999955i	$0.496982\pi$
$948$	0	0
$949$	−20.0000	−0.649227
$950$	0	0
$951$	−10.3607	−0.335968
$952$	0	0
$953$	−8.36068	−0.270829	−0.135414	−	0.990789i	$-0.543237\pi$
$953$	−8.36068	−0.270829	−0.135414	+	0.990789i	$0.543237\pi$
$954$	0	0
$955$	85.6656	2.77207
$956$	0	0
$957$	−27.4164	−0.886247
$958$	0	0
$959$	14.8328	0.478977
$960$	0	0
$961$	−28.6656	−0.924698
$962$	0	0
$963$	−11.2361	−0.362077
$964$	0	0
$965$	−38.4721	−1.23846
$966$	0	0
$967$	13.3050	0.427858	0.213929	−	0.976849i	$-0.431374\pi$
$967$	13.3050	0.427858	0.213929	+	0.976849i	$0.431374\pi$
$968$	0	0
$969$	8.00000	0.256997
$970$	0	0
$971$	10.2918	0.330279	0.165140	−	0.986270i	$-0.447192\pi$
$971$	10.2918	0.330279	0.165140	+	0.986270i	$0.447192\pi$
$972$	0	0
$973$	17.8885	0.573480
$974$	0	0
$975$	−24.4721	−0.783736
$976$	0	0
$977$	−0.583592	−0.0186708	−0.00933538	−	0.999956i	$-0.502972\pi$
$977$	−0.583592	−0.0186708	−0.00933538	+	0.999956i	$0.502972\pi$
$978$	0	0
$979$	−54.8328	−1.75246
$980$	0	0
$981$	−2.76393	−0.0882456
$982$	0	0
$983$	29.3050	0.934683	0.467341	−	0.884077i	$-0.345212\pi$
$983$	29.3050	0.934683	0.467341	+	0.884077i	$0.345212\pi$
$984$	0	0
$985$	−38.4721	−1.22582
$986$	0	0
$987$	−17.8885	−0.569399
$988$	0	0
$989$	5.23607	0.166497
$990$	0	0
$991$	29.3050	0.930902	0.465451	−	0.885074i	$-0.345892\pi$
$991$	29.3050	0.930902	0.465451	+	0.885074i	$0.345892\pi$
$992$	0	0
$993$	17.5279	0.556230
$994$	0	0
$995$	−74.2492	−2.35386
$996$	0	0
$997$	8.83282	0.279738	0.139869	−	0.990170i	$-0.455332\pi$
$997$	8.83282	0.279738	0.139869	+	0.990170i	$0.455332\pi$
$998$	0	0
$999$	7.70820	0.243877

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	552.2.a.f.1.2	✓	2
3.2	odd	2		1656.2.a.l.1.1		2
4.3	odd	2		1104.2.a.k.1.2		2
8.3	odd	2		4416.2.a.bj.1.1		2
8.5	even	2		4416.2.a.bd.1.1		2
12.11	even	2		3312.2.a.w.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
552.2.a.f.1.2	✓	2	1.1	even	1	trivial
1104.2.a.k.1.2		2	4.3	odd	2
1656.2.a.l.1.1		2	3.2	odd	2
3312.2.a.w.1.1		2	12.11	even	2
4416.2.a.bd.1.1		2	8.5	even	2
4416.2.a.bj.1.1		2	8.3	odd	2

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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:	\( N \)	\(=\)	\( 552 = 2^{3} \cdot 3 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	552.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.23607 q^{5} +2.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.23607 q^{5} +2.00000 q^{7} +1.00000 q^{9} +3.23607 q^{11} -4.47214 q^{13} +3.23607 q^{15} -6.47214 q^{17} -1.23607 q^{19} +2.00000 q^{21} +1.00000 q^{23} +5.47214 q^{25} +1.00000 q^{27} -8.47214 q^{29} +1.52786 q^{31} +3.23607 q^{33} +6.47214 q^{35} +7.70820 q^{37} -4.47214 q^{39} -2.00000 q^{41} +5.23607 q^{43} +3.23607 q^{45} -8.94427 q^{47} -3.00000 q^{49} -6.47214 q^{51} +12.1803 q^{53} +10.4721 q^{55} -1.23607 q^{57} -4.00000 q^{59} -11.7082 q^{61} +2.00000 q^{63} -14.4721 q^{65} +7.70820 q^{67} +1.00000 q^{69} +8.94427 q^{71} +4.47214 q^{73} +5.47214 q^{75} +6.47214 q^{77} +14.0000 q^{79} +1.00000 q^{81} -11.2361 q^{83} -20.9443 q^{85} -8.47214 q^{87} -16.9443 q^{89} -8.94427 q^{91} +1.52786 q^{93} -4.00000 q^{95} +4.47214 q^{97} +3.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 + 2 * q^5 + 4 * q^7 + 2 * q^9	\( 2 q + 2 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9} + 2 q^{11} + 2 q^{15} - 4 q^{17} + 2 q^{19} + 4 q^{21} + 2 q^{23} + 2 q^{25} + 2 q^{27} - 8 q^{29} + 12 q^{31} + 2 q^{33} + 4 q^{35} + 2 q^{37} - 4 q^{41} + 6 q^{43} + 2 q^{45} - 6 q^{49} - 4 q^{51} + 2 q^{53} + 12 q^{55} + 2 q^{57} - 8 q^{59} - 10 q^{61} + 4 q^{63} - 20 q^{65} + 2 q^{67} + 2 q^{69} + 2 q^{75} + 4 q^{77} + 28 q^{79} + 2 q^{81} - 18 q^{83} - 24 q^{85} - 8 q^{87} - 16 q^{89} + 12 q^{93} - 8 q^{95} + 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 + 4 * q^7 + 2 * q^9 + 2 * q^11 + 2 * q^15 - 4 * q^17 + 2 * q^19 + 4 * q^21 + 2 * q^23 + 2 * q^25 + 2 * q^27 - 8 * q^29 + 12 * q^31 + 2 * q^33 + 4 * q^35 + 2 * q^37 - 4 * q^41 + 6 * q^43 + 2 * q^45 - 6 * q^49 - 4 * q^51 + 2 * q^53 + 12 * q^55 + 2 * q^57 - 8 * q^59 - 10 * q^61 + 4 * q^63 - 20 * q^65 + 2 * q^67 + 2 * q^69 + 2 * q^75 + 4 * q^77 + 28 * q^79 + 2 * q^81 - 18 * q^83 - 24 * q^85 - 8 * q^87 - 16 * q^89 + 12 * q^93 - 8 * q^95 + 2 * q^99

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