LMFDB - Embedded newform 2752.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2752,2,Mod(1,2752)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2752, 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("2752.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2752 = 2^{6} \cdot 43 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2752.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:	$21.9748306363$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 43)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	2752.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.41421 q^{3} -3.41421 q^{5} -3.41421 q^{7} -1.00000 q^{9} +O(q^{10})$	$q-1.41421 q^{3} -3.41421 q^{5} -3.41421 q^{7} -1.00000 q^{9} +3.82843 q^{11} +1.82843 q^{13} +4.82843 q^{15} +2.17157 q^{17} -0.828427 q^{19} +4.82843 q^{21} +6.65685 q^{23} +6.65685 q^{25} +5.65685 q^{27} +4.24264 q^{29} -3.00000 q^{31} -5.41421 q^{33} +11.6569 q^{35} -8.48528 q^{37} -2.58579 q^{39} +1.82843 q^{41} -1.00000 q^{43} +3.41421 q^{45} +6.00000 q^{47} +4.65685 q^{49} -3.07107 q^{51} -13.8284 q^{53} -13.0711 q^{55} +1.17157 q^{57} +4.82843 q^{59} +0.242641 q^{61} +3.41421 q^{63} -6.24264 q^{65} +7.48528 q^{67} -9.41421 q^{69} -3.17157 q^{71} -16.2426 q^{73} -9.41421 q^{75} -13.0711 q^{77} +4.82843 q^{79} -5.00000 q^{81} -3.34315 q^{83} -7.41421 q^{85} -6.00000 q^{87} -1.75736 q^{89} -6.24264 q^{91} +4.24264 q^{93} +2.82843 q^{95} +1.82843 q^{97} -3.82843 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 4 q^{5} - 4 q^{7} - 2 q^{9}+O(q^{10}) $ 2 * q - 4 * q^5 - 4 * q^7 - 2 * q^9	$ 2 q - 4 q^{5} - 4 q^{7} - 2 q^{9} + 2 q^{11} - 2 q^{13} + 4 q^{15} + 10 q^{17} + 4 q^{19} + 4 q^{21} + 2 q^{23} + 2 q^{25} - 6 q^{31} - 8 q^{33} + 12 q^{35} - 8 q^{39} - 2 q^{41} - 2 q^{43} + 4 q^{45} + 12 q^{47} - 2 q^{49} + 8 q^{51} - 22 q^{53} - 12 q^{55} + 8 q^{57} + 4 q^{59} - 8 q^{61} + 4 q^{63} - 4 q^{65} - 2 q^{67} - 16 q^{69} - 12 q^{71} - 24 q^{73} - 16 q^{75} - 12 q^{77} + 4 q^{79} - 10 q^{81} - 18 q^{83} - 12 q^{85} - 12 q^{87} - 12 q^{89} - 4 q^{91} - 2 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q - 4 * q^5 - 4 * q^7 - 2 * q^9 + 2 * q^11 - 2 * q^13 + 4 * q^15 + 10 * q^17 + 4 * q^19 + 4 * q^21 + 2 * q^23 + 2 * q^25 - 6 * q^31 - 8 * q^33 + 12 * q^35 - 8 * q^39 - 2 * q^41 - 2 * q^43 + 4 * q^45 + 12 * q^47 - 2 * q^49 + 8 * q^51 - 22 * q^53 - 12 * q^55 + 8 * q^57 + 4 * q^59 - 8 * q^61 + 4 * q^63 - 4 * q^65 - 2 * q^67 - 16 * q^69 - 12 * q^71 - 24 * q^73 - 16 * q^75 - 12 * q^77 + 4 * q^79 - 10 * q^81 - 18 * q^83 - 12 * q^85 - 12 * q^87 - 12 * q^89 - 4 * q^91 - 2 * q^97 - 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
$3$	−1.41421	−0.816497	−0.408248	−	0.912871i	$-0.633860\pi$
$3$	−1.41421	−0.816497	−0.408248	+	0.912871i	$0.633860\pi$
$4$	0	0
$5$	−3.41421	−1.52688	−0.763441	−	0.645877i	$-0.776492\pi$
$5$	−3.41421	−1.52688	−0.763441	+	0.645877i	$0.776492\pi$
$6$	0	0
$7$	−3.41421	−1.29045	−0.645226	−	0.763992i	$-0.723237\pi$
$7$	−3.41421	−1.29045	−0.645226	+	0.763992i	$0.723237\pi$
$8$	0	0
$9$	−1.00000	−0.333333
$10$	0	0
$11$	3.82843	1.15431	0.577157	−	0.816633i	$-0.304162\pi$
$11$	3.82843	1.15431	0.577157	+	0.816633i	$0.304162\pi$
$12$	0	0
$13$	1.82843	0.507114	0.253557	−	0.967320i	$-0.418399\pi$
$13$	1.82843	0.507114	0.253557	+	0.967320i	$0.418399\pi$
$14$	0	0
$15$	4.82843	1.24669
$16$	0	0
$17$	2.17157	0.526684	0.263342	−	0.964703i	$-0.415175\pi$
$17$	2.17157	0.526684	0.263342	+	0.964703i	$0.415175\pi$
$18$	0	0
$19$	−0.828427	−0.190054	−0.0950271	−	0.995475i	$-0.530294\pi$
$19$	−0.828427	−0.190054	−0.0950271	+	0.995475i	$0.530294\pi$
$20$	0	0
$21$	4.82843	1.05365
$22$	0	0
$23$	6.65685	1.38805	0.694025	−	0.719951i	$-0.255836\pi$
$23$	6.65685	1.38805	0.694025	+	0.719951i	$0.255836\pi$
$24$	0	0
$25$	6.65685	1.33137
$26$	0	0
$27$	5.65685	1.08866
$28$	0	0
$29$	4.24264	0.787839	0.393919	−	0.919145i	$-0.371119\pi$
$29$	4.24264	0.787839	0.393919	+	0.919145i	$0.371119\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	0	0
$33$	−5.41421	−0.942494
$34$	0	0
$35$	11.6569	1.97037
$36$	0	0
$37$	−8.48528	−1.39497	−0.697486	−	0.716599i	$-0.745698\pi$
$37$	−8.48528	−1.39497	−0.697486	+	0.716599i	$0.745698\pi$
$38$	0	0
$39$	−2.58579	−0.414057
$40$	0	0
$41$	1.82843	0.285552	0.142776	−	0.989755i	$-0.454397\pi$
$41$	1.82843	0.285552	0.142776	+	0.989755i	$0.454397\pi$
$42$	0	0
$43$	−1.00000	−0.152499
$43$	−1.00000	−0.152499
$44$	0	0
$45$	3.41421	0.508961
$46$	0	0
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	4.65685	0.665265
$50$	0	0
$51$	−3.07107	−0.430036
$52$	0	0
$53$	−13.8284	−1.89948	−0.949740	−	0.313039i	$-0.898653\pi$
$53$	−13.8284	−1.89948	−0.949740	+	0.313039i	$0.898653\pi$
$54$	0	0
$55$	−13.0711	−1.76250
$56$	0	0
$57$	1.17157	0.155179
$58$	0	0
$59$	4.82843	0.628608	0.314304	−	0.949322i	$-0.398229\pi$
$59$	4.82843	0.628608	0.314304	+	0.949322i	$0.398229\pi$
$60$	0	0
$61$	0.242641	0.0310670	0.0155335	−	0.999879i	$-0.495055\pi$
$61$	0.242641	0.0310670	0.0155335	+	0.999879i	$0.495055\pi$
$62$	0	0
$63$	3.41421	0.430150
$64$	0	0
$65$	−6.24264	−0.774304
$66$	0	0
$67$	7.48528	0.914473	0.457236	−	0.889345i	$-0.348839\pi$
$67$	7.48528	0.914473	0.457236	+	0.889345i	$0.348839\pi$
$68$	0	0
$69$	−9.41421	−1.13334
$70$	0	0
$71$	−3.17157	−0.376396	−0.188198	−	0.982131i	$-0.560265\pi$
$71$	−3.17157	−0.376396	−0.188198	+	0.982131i	$0.560265\pi$
$72$	0	0
$73$	−16.2426	−1.90106	−0.950529	−	0.310637i	$-0.899458\pi$
$73$	−16.2426	−1.90106	−0.950529	+	0.310637i	$0.899458\pi$
$74$	0	0
$75$	−9.41421	−1.08706
$76$	0	0
$77$	−13.0711	−1.48959
$78$	0	0
$79$	4.82843	0.543240	0.271620	−	0.962405i	$-0.412441\pi$
$79$	4.82843	0.543240	0.271620	+	0.962405i	$0.412441\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	0	0
$83$	−3.34315	−0.366958	−0.183479	−	0.983024i	$-0.558736\pi$
$83$	−3.34315	−0.366958	−0.183479	+	0.983024i	$0.558736\pi$
$84$	0	0
$85$	−7.41421	−0.804184
$86$	0	0
$87$	−6.00000	−0.643268
$88$	0	0
$89$	−1.75736	−0.186280	−0.0931399	−	0.995653i	$-0.529690\pi$
$89$	−1.75736	−0.186280	−0.0931399	+	0.995653i	$0.529690\pi$
$90$	0	0
$91$	−6.24264	−0.654407
$92$	0	0
$93$	4.24264	0.439941
$94$	0	0
$95$	2.82843	0.290191
$96$	0	0
$97$	1.82843	0.185649	0.0928243	−	0.995683i	$-0.470411\pi$
$97$	1.82843	0.185649	0.0928243	+	0.995683i	$0.470411\pi$
$98$	0	0
$99$	−3.82843	−0.384771
$100$	0	0
$101$	−5.82843	−0.579950	−0.289975	−	0.957034i	$-0.593647\pi$
$101$	−5.82843	−0.579950	−0.289975	+	0.957034i	$0.593647\pi$
$102$	0	0
$103$	0.514719	0.0507167	0.0253584	−	0.999678i	$-0.491927\pi$
$103$	0.514719	0.0507167	0.0253584	+	0.999678i	$0.491927\pi$
$104$	0	0
$105$	−16.4853	−1.60880
$106$	0	0
$107$	0.343146	0.0331732	0.0165866	−	0.999862i	$-0.494720\pi$
$107$	0.343146	0.0331732	0.0165866	+	0.999862i	$0.494720\pi$
$108$	0	0
$109$	19.9706	1.91283	0.956416	−	0.292006i	$-0.0943227\pi$
$109$	19.9706	1.91283	0.956416	+	0.292006i	$0.0943227\pi$
$110$	0	0
$111$	12.0000	1.13899
$112$	0	0
$113$	−6.82843	−0.642364	−0.321182	−	0.947017i	$-0.604080\pi$
$113$	−6.82843	−0.642364	−0.321182	+	0.947017i	$0.604080\pi$
$114$	0	0
$115$	−22.7279	−2.11939
$116$	0	0
$117$	−1.82843	−0.169038
$118$	0	0
$119$	−7.41421	−0.679660
$120$	0	0
$121$	3.65685	0.332441
$122$	0	0
$123$	−2.58579	−0.233153
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	3.82843	0.339718	0.169859	−	0.985468i	$-0.445669\pi$
$127$	3.82843	0.339718	0.169859	+	0.985468i	$0.445669\pi$
$128$	0	0
$129$	1.41421	0.124515
$130$	0	0
$131$	−9.65685	−0.843723	−0.421862	−	0.906660i	$-0.638623\pi$
$131$	−9.65685	−0.843723	−0.421862	+	0.906660i	$0.638623\pi$
$132$	0	0
$133$	2.82843	0.245256
$134$	0	0
$135$	−19.3137	−1.66226
$136$	0	0
$137$	−14.4853	−1.23756	−0.618781	−	0.785564i	$-0.712373\pi$
$137$	−14.4853	−1.23756	−0.618781	+	0.785564i	$0.712373\pi$
$138$	0	0
$139$	−5.48528	−0.465255	−0.232628	−	0.972566i	$-0.574732\pi$
$139$	−5.48528	−0.465255	−0.232628	+	0.972566i	$0.574732\pi$
$140$	0	0
$141$	−8.48528	−0.714590
$142$	0	0
$143$	7.00000	0.585369
$144$	0	0
$145$	−14.4853	−1.20294
$146$	0	0
$147$	−6.58579	−0.543187
$148$	0	0
$149$	−8.48528	−0.695141	−0.347571	−	0.937654i	$-0.612993\pi$
$149$	−8.48528	−0.695141	−0.347571	+	0.937654i	$0.612993\pi$
$150$	0	0
$151$	18.2426	1.48457	0.742283	−	0.670087i	$-0.233743\pi$
$151$	18.2426	1.48457	0.742283	+	0.670087i	$0.233743\pi$
$152$	0	0
$153$	−2.17157	−0.175561
$154$	0	0
$155$	10.2426	0.822709
$156$	0	0
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	0	0
$159$	19.5563	1.55092
$160$	0	0
$161$	−22.7279	−1.79121
$162$	0	0
$163$	11.7574	0.920907	0.460454	−	0.887684i	$-0.347687\pi$
$163$	11.7574	0.920907	0.460454	+	0.887684i	$0.347687\pi$
$164$	0	0
$165$	18.4853	1.43908
$166$	0	0
$167$	−14.3137	−1.10763	−0.553814	−	0.832640i	$-0.686828\pi$
$167$	−14.3137	−1.10763	−0.553814	+	0.832640i	$0.686828\pi$
$168$	0	0
$169$	−9.65685	−0.742835
$170$	0	0
$171$	0.828427	0.0633514
$172$	0	0
$173$	−23.6569	−1.79860	−0.899299	−	0.437335i	$-0.855922\pi$
$173$	−23.6569	−1.79860	−0.899299	+	0.437335i	$0.855922\pi$
$174$	0	0
$175$	−22.7279	−1.71807
$176$	0	0
$177$	−6.82843	−0.513256
$178$	0	0
$179$	4.58579	0.342758	0.171379	−	0.985205i	$-0.445178\pi$
$179$	4.58579	0.342758	0.171379	+	0.985205i	$0.445178\pi$
$180$	0	0
$181$	−7.31371	−0.543624	−0.271812	−	0.962350i	$-0.587623\pi$
$181$	−7.31371	−0.543624	−0.271812	+	0.962350i	$0.587623\pi$
$182$	0	0
$183$	−0.343146	−0.0253661
$184$	0	0
$185$	28.9706	2.12996
$186$	0	0
$187$	8.31371	0.607959
$188$	0	0
$189$	−19.3137	−1.40487
$190$	0	0
$191$	−6.14214	−0.444429	−0.222215	−	0.974998i	$-0.571329\pi$
$191$	−6.14214	−0.444429	−0.222215	+	0.974998i	$0.571329\pi$
$192$	0	0
$193$	15.9706	1.14959	0.574793	−	0.818299i	$-0.305083\pi$
$193$	15.9706	1.14959	0.574793	+	0.818299i	$0.305083\pi$
$194$	0	0
$195$	8.82843	0.632217
$196$	0	0
$197$	14.1421	1.00759	0.503793	−	0.863825i	$-0.331938\pi$
$197$	14.1421	1.00759	0.503793	+	0.863825i	$0.331938\pi$
$198$	0	0
$199$	7.65685	0.542780	0.271390	−	0.962469i	$-0.412517\pi$
$199$	7.65685	0.542780	0.271390	+	0.962469i	$0.412517\pi$
$200$	0	0
$201$	−10.5858	−0.746664
$202$	0	0
$203$	−14.4853	−1.01667
$204$	0	0
$205$	−6.24264	−0.436005
$206$	0	0
$207$	−6.65685	−0.462683
$208$	0	0
$209$	−3.17157	−0.219382
$210$	0	0
$211$	0.142136	0.00978502	0.00489251	−	0.999988i	$-0.498443\pi$
$211$	0.142136	0.00978502	0.00489251	+	0.999988i	$0.498443\pi$
$212$	0	0
$213$	4.48528	0.307326
$214$	0	0
$215$	3.41421	0.232847
$216$	0	0
$217$	10.2426	0.695316
$218$	0	0
$219$	22.9706	1.55221
$220$	0	0
$221$	3.97056	0.267089
$222$	0	0
$223$	4.10051	0.274590	0.137295	−	0.990530i	$-0.456159\pi$
$223$	4.10051	0.274590	0.137295	+	0.990530i	$0.456159\pi$
$224$	0	0
$225$	−6.65685	−0.443790
$226$	0	0
$227$	6.34315	0.421009	0.210505	−	0.977593i	$-0.432489\pi$
$227$	6.34315	0.421009	0.210505	+	0.977593i	$0.432489\pi$
$228$	0	0
$229$	25.9706	1.71618	0.858092	−	0.513497i	$-0.171650\pi$
$229$	25.9706	1.71618	0.858092	+	0.513497i	$0.171650\pi$
$230$	0	0
$231$	18.4853	1.21624
$232$	0	0
$233$	4.82843	0.316321	0.158160	−	0.987413i	$-0.449444\pi$
$233$	4.82843	0.316321	0.158160	+	0.987413i	$0.449444\pi$
$234$	0	0
$235$	−20.4853	−1.33631
$236$	0	0
$237$	−6.82843	−0.443554
$238$	0	0
$239$	14.4853	0.936975	0.468487	−	0.883470i	$-0.344799\pi$
$239$	14.4853	0.936975	0.468487	+	0.883470i	$0.344799\pi$
$240$	0	0
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	0	0
$243$	−9.89949	−0.635053
$244$	0	0
$245$	−15.8995	−1.01578
$246$	0	0
$247$	−1.51472	−0.0963792
$248$	0	0
$249$	4.72792	0.299620
$250$	0	0
$251$	−23.1421	−1.46072	−0.730359	−	0.683063i	$-0.760647\pi$
$251$	−23.1421	−1.46072	−0.730359	+	0.683063i	$0.760647\pi$
$252$	0	0
$253$	25.4853	1.60225
$254$	0	0
$255$	10.4853	0.656614
$256$	0	0
$257$	−1.75736	−0.109621	−0.0548105	−	0.998497i	$-0.517455\pi$
$257$	−1.75736	−0.109621	−0.0548105	+	0.998497i	$0.517455\pi$
$258$	0	0
$259$	28.9706	1.80014
$260$	0	0
$261$	−4.24264	−0.262613
$262$	0	0
$263$	0.142136	0.00876446	0.00438223	−	0.999990i	$-0.498605\pi$
$263$	0.142136	0.00876446	0.00438223	+	0.999990i	$0.498605\pi$
$264$	0	0
$265$	47.2132	2.90028
$266$	0	0
$267$	2.48528	0.152097
$268$	0	0
$269$	3.00000	0.182913	0.0914566	−	0.995809i	$-0.470848\pi$
$269$	3.00000	0.182913	0.0914566	+	0.995809i	$0.470848\pi$
$270$	0	0
$271$	−25.9706	−1.57760	−0.788800	−	0.614650i	$-0.789297\pi$
$271$	−25.9706	−1.57760	−0.788800	+	0.614650i	$0.789297\pi$
$272$	0	0
$273$	8.82843	0.534321
$274$	0	0
$275$	25.4853	1.53682
$276$	0	0
$277$	−7.89949	−0.474635	−0.237317	−	0.971432i	$-0.576268\pi$
$277$	−7.89949	−0.474635	−0.237317	+	0.971432i	$0.576268\pi$
$278$	0	0
$279$	3.00000	0.179605
$280$	0	0
$281$	−8.65685	−0.516425	−0.258212	−	0.966088i	$-0.583133\pi$
$281$	−8.65685	−0.516425	−0.258212	+	0.966088i	$0.583133\pi$
$282$	0	0
$283$	−18.3137	−1.08864	−0.544318	−	0.838879i	$-0.683212\pi$
$283$	−18.3137	−1.08864	−0.544318	+	0.838879i	$0.683212\pi$
$284$	0	0
$285$	−4.00000	−0.236940
$286$	0	0
$287$	−6.24264	−0.368491
$288$	0	0
$289$	−12.2843	−0.722604
$290$	0	0
$291$	−2.58579	−0.151581
$292$	0	0
$293$	−10.3431	−0.604253	−0.302127	−	0.953268i	$-0.597697\pi$
$293$	−10.3431	−0.604253	−0.302127	+	0.953268i	$0.597697\pi$
$294$	0	0
$295$	−16.4853	−0.959810
$296$	0	0
$297$	21.6569	1.25666
$298$	0	0
$299$	12.1716	0.703900
$300$	0	0
$301$	3.41421	0.196792
$302$	0	0
$303$	8.24264	0.473527
$304$	0	0
$305$	−0.828427	−0.0474356
$306$	0	0
$307$	−26.7990	−1.52950	−0.764750	−	0.644328i	$-0.777137\pi$
$307$	−26.7990	−1.52950	−0.764750	+	0.644328i	$0.777137\pi$
$308$	0	0
$309$	−0.727922	−0.0414100
$310$	0	0
$311$	−13.9706	−0.792198	−0.396099	−	0.918208i	$-0.629636\pi$
$311$	−13.9706	−0.792198	−0.396099	+	0.918208i	$0.629636\pi$
$312$	0	0
$313$	−25.2132	−1.42513	−0.712567	−	0.701604i	$-0.752468\pi$
$313$	−25.2132	−1.42513	−0.712567	+	0.701604i	$0.752468\pi$
$314$	0	0
$315$	−11.6569	−0.656789
$316$	0	0
$317$	−25.8284	−1.45067	−0.725334	−	0.688397i	$-0.758315\pi$
$317$	−25.8284	−1.45067	−0.725334	+	0.688397i	$0.758315\pi$
$318$	0	0
$319$	16.2426	0.909413
$320$	0	0
$321$	−0.485281	−0.0270858
$322$	0	0
$323$	−1.79899	−0.100098
$324$	0	0
$325$	12.1716	0.675157
$326$	0	0
$327$	−28.2426	−1.56182
$328$	0	0
$329$	−20.4853	−1.12939
$330$	0	0
$331$	−21.4142	−1.17703	−0.588516	−	0.808486i	$-0.700288\pi$
$331$	−21.4142	−1.17703	−0.588516	+	0.808486i	$0.700288\pi$
$332$	0	0
$333$	8.48528	0.464991
$334$	0	0
$335$	−25.5563	−1.39629
$336$	0	0
$337$	−5.00000	−0.272367	−0.136184	−	0.990684i	$-0.543484\pi$
$337$	−5.00000	−0.272367	−0.136184	+	0.990684i	$0.543484\pi$
$338$	0	0
$339$	9.65685	0.524488
$340$	0	0
$341$	−11.4853	−0.621963
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	32.1421	1.73047
$346$	0	0
$347$	4.58579	0.246178	0.123089	−	0.992396i	$-0.460720\pi$
$347$	4.58579	0.246178	0.123089	+	0.992396i	$0.460720\pi$
$348$	0	0
$349$	20.2426	1.08356	0.541782	−	0.840519i	$-0.317750\pi$
$349$	20.2426	1.08356	0.541782	+	0.840519i	$0.317750\pi$
$350$	0	0
$351$	10.3431	0.552076
$352$	0	0
$353$	5.82843	0.310216	0.155108	−	0.987898i	$-0.450427\pi$
$353$	5.82843	0.310216	0.155108	+	0.987898i	$0.450427\pi$
$354$	0	0
$355$	10.8284	0.574713
$356$	0	0
$357$	10.4853	0.554940
$358$	0	0
$359$	−21.3431	−1.12645	−0.563224	−	0.826304i	$-0.690439\pi$
$359$	−21.3431	−1.12645	−0.563224	+	0.826304i	$0.690439\pi$
$360$	0	0
$361$	−18.3137	−0.963879
$362$	0	0
$363$	−5.17157	−0.271437
$364$	0	0
$365$	55.4558	2.90269
$366$	0	0
$367$	9.79899	0.511503	0.255752	−	0.966743i	$-0.417677\pi$
$367$	9.79899	0.511503	0.255752	+	0.966743i	$0.417677\pi$
$368$	0	0
$369$	−1.82843	−0.0951841
$370$	0	0
$371$	47.2132	2.45119
$372$	0	0
$373$	−8.48528	−0.439351	−0.219676	−	0.975573i	$-0.570500\pi$
$373$	−8.48528	−0.439351	−0.219676	+	0.975573i	$0.570500\pi$
$374$	0	0
$375$	8.00000	0.413118
$376$	0	0
$377$	7.75736	0.399524
$378$	0	0
$379$	−30.3137	−1.55711	−0.778555	−	0.627576i	$-0.784047\pi$
$379$	−30.3137	−1.55711	−0.778555	+	0.627576i	$0.784047\pi$
$380$	0	0
$381$	−5.41421	−0.277379
$382$	0	0
$383$	−20.4853	−1.04675	−0.523374	−	0.852103i	$-0.675327\pi$
$383$	−20.4853	−1.04675	−0.523374	+	0.852103i	$0.675327\pi$
$384$	0	0
$385$	44.6274	2.27442
$386$	0	0
$387$	1.00000	0.0508329
$388$	0	0
$389$	16.6274	0.843044	0.421522	−	0.906818i	$-0.361496\pi$
$389$	16.6274	0.843044	0.421522	+	0.906818i	$0.361496\pi$
$390$	0	0
$391$	14.4558	0.731063
$392$	0	0
$393$	13.6569	0.688897
$394$	0	0
$395$	−16.4853	−0.829465
$396$	0	0
$397$	−29.4558	−1.47835	−0.739173	−	0.673515i	$-0.764784\pi$
$397$	−29.4558	−1.47835	−0.739173	+	0.673515i	$0.764784\pi$
$398$	0	0
$399$	−4.00000	−0.200250
$400$	0	0
$401$	−12.5147	−0.624955	−0.312478	−	0.949925i	$-0.601159\pi$
$401$	−12.5147	−0.624955	−0.312478	+	0.949925i	$0.601159\pi$
$402$	0	0
$403$	−5.48528	−0.273241
$404$	0	0
$405$	17.0711	0.848268
$406$	0	0
$407$	−32.4853	−1.61024
$408$	0	0
$409$	−16.8701	−0.834171	−0.417085	−	0.908867i	$-0.636948\pi$
$409$	−16.8701	−0.834171	−0.417085	+	0.908867i	$0.636948\pi$
$410$	0	0
$411$	20.4853	1.01046
$412$	0	0
$413$	−16.4853	−0.811188
$414$	0	0
$415$	11.4142	0.560302
$416$	0	0
$417$	7.75736	0.379880
$418$	0	0
$419$	23.8995	1.16757	0.583783	−	0.811909i	$-0.301572\pi$
$419$	23.8995	1.16757	0.583783	+	0.811909i	$0.301572\pi$
$420$	0	0
$421$	15.6569	0.763068	0.381534	−	0.924355i	$-0.375396\pi$
$421$	15.6569	0.763068	0.381534	+	0.924355i	$0.375396\pi$
$422$	0	0
$423$	−6.00000	−0.291730
$424$	0	0
$425$	14.4558	0.701211
$426$	0	0
$427$	−0.828427	−0.0400904
$428$	0	0
$429$	−9.89949	−0.477952
$430$	0	0
$431$	23.2843	1.12156	0.560782	−	0.827964i	$-0.310501\pi$
$431$	23.2843	1.12156	0.560782	+	0.827964i	$0.310501\pi$
$432$	0	0
$433$	21.7574	1.04559	0.522796	−	0.852458i	$-0.324889\pi$
$433$	21.7574	1.04559	0.522796	+	0.852458i	$0.324889\pi$
$434$	0	0
$435$	20.4853	0.982194
$436$	0	0
$437$	−5.51472	−0.263805
$438$	0	0
$439$	11.4853	0.548163	0.274081	−	0.961707i	$-0.411626\pi$
$439$	11.4853	0.548163	0.274081	+	0.961707i	$0.411626\pi$
$440$	0	0
$441$	−4.65685	−0.221755
$442$	0	0
$443$	−7.85786	−0.373338	−0.186669	−	0.982423i	$-0.559769\pi$
$443$	−7.85786	−0.373338	−0.186669	+	0.982423i	$0.559769\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	0	0
$447$	12.0000	0.567581
$448$	0	0
$449$	33.2132	1.56743	0.783714	−	0.621122i	$-0.213323\pi$
$449$	33.2132	1.56743	0.783714	+	0.621122i	$0.213323\pi$
$450$	0	0
$451$	7.00000	0.329617
$452$	0	0
$453$	−25.7990	−1.21214
$454$	0	0
$455$	21.3137	0.999202
$456$	0	0
$457$	0.727922	0.0340508	0.0170254	−	0.999855i	$-0.494580\pi$
$457$	0.727922	0.0340508	0.0170254	+	0.999855i	$0.494580\pi$
$458$	0	0
$459$	12.2843	0.573381
$460$	0	0
$461$	−42.6274	−1.98536	−0.992678	−	0.120788i	$-0.961458\pi$
$461$	−42.6274	−1.98536	−0.992678	+	0.120788i	$0.961458\pi$
$462$	0	0
$463$	−30.7279	−1.42805	−0.714024	−	0.700121i	$-0.753129\pi$
$463$	−30.7279	−1.42805	−0.714024	+	0.700121i	$0.753129\pi$
$464$	0	0
$465$	−14.4853	−0.671739
$466$	0	0
$467$	−23.6569	−1.09471	−0.547354	−	0.836901i	$-0.684365\pi$
$467$	−23.6569	−1.09471	−0.547354	+	0.836901i	$0.684365\pi$
$468$	0	0
$469$	−25.5563	−1.18008
$470$	0	0
$471$	−14.1421	−0.651635
$472$	0	0
$473$	−3.82843	−0.176031
$474$	0	0
$475$	−5.51472	−0.253033
$476$	0	0
$477$	13.8284	0.633160
$478$	0	0
$479$	−6.65685	−0.304159	−0.152080	−	0.988368i	$-0.548597\pi$
$479$	−6.65685	−0.304159	−0.152080	+	0.988368i	$0.548597\pi$
$480$	0	0
$481$	−15.5147	−0.707410
$482$	0	0
$483$	32.1421	1.46252
$484$	0	0
$485$	−6.24264	−0.283464
$486$	0	0
$487$	22.3431	1.01246	0.506232	−	0.862397i	$-0.331038\pi$
$487$	22.3431	1.01246	0.506232	+	0.862397i	$0.331038\pi$
$488$	0	0
$489$	−16.6274	−0.751918
$490$	0	0
$491$	−13.4142	−0.605375	−0.302687	−	0.953090i	$-0.597884\pi$
$491$	−13.4142	−0.605375	−0.302687	+	0.953090i	$0.597884\pi$
$492$	0	0
$493$	9.21320	0.414942
$494$	0	0
$495$	13.0711	0.587501
$496$	0	0
$497$	10.8284	0.485721
$498$	0	0
$499$	25.7574	1.15306	0.576529	−	0.817077i	$-0.304407\pi$
$499$	25.7574	1.15306	0.576529	+	0.817077i	$0.304407\pi$
$500$	0	0
$501$	20.2426	0.904374
$502$	0	0
$503$	−30.7696	−1.37195	−0.685973	−	0.727627i	$-0.740623\pi$
$503$	−30.7696	−1.37195	−0.685973	+	0.727627i	$0.740623\pi$
$504$	0	0
$505$	19.8995	0.885516
$506$	0	0
$507$	13.6569	0.606522
$508$	0	0
$509$	−0.514719	−0.0228145	−0.0114073	−	0.999935i	$-0.503631\pi$
$509$	−0.514719	−0.0228145	−0.0114073	+	0.999935i	$0.503631\pi$
$510$	0	0
$511$	55.4558	2.45322
$512$	0	0
$513$	−4.68629	−0.206905
$514$	0	0
$515$	−1.75736	−0.0774385
$516$	0	0
$517$	22.9706	1.01024
$518$	0	0
$519$	33.4558	1.46855
$520$	0	0
$521$	16.9289	0.741670	0.370835	−	0.928699i	$-0.379072\pi$
$521$	16.9289	0.741670	0.370835	+	0.928699i	$0.379072\pi$
$522$	0	0
$523$	39.2132	1.71467	0.857337	−	0.514756i	$-0.172117\pi$
$523$	39.2132	1.71467	0.857337	+	0.514756i	$0.172117\pi$
$524$	0	0
$525$	32.1421	1.40280
$526$	0	0
$527$	−6.51472	−0.283786
$528$	0	0
$529$	21.3137	0.926683
$530$	0	0
$531$	−4.82843	−0.209536
$532$	0	0
$533$	3.34315	0.144808
$534$	0	0
$535$	−1.17157	−0.0506515
$536$	0	0
$537$	−6.48528	−0.279861
$538$	0	0
$539$	17.8284	0.767925
$540$	0	0
$541$	−8.79899	−0.378298	−0.189149	−	0.981948i	$-0.560573\pi$
$541$	−8.79899	−0.378298	−0.189149	+	0.981948i	$0.560573\pi$
$542$	0	0
$543$	10.3431	0.443867
$544$	0	0
$545$	−68.1838	−2.92067
$546$	0	0
$547$	−9.00000	−0.384812	−0.192406	−	0.981315i	$-0.561629\pi$
$547$	−9.00000	−0.384812	−0.192406	+	0.981315i	$0.561629\pi$
$548$	0	0
$549$	−0.242641	−0.0103557
$550$	0	0
$551$	−3.51472	−0.149732
$552$	0	0
$553$	−16.4853	−0.701025
$554$	0	0
$555$	−40.9706	−1.73910
$556$	0	0
$557$	9.00000	0.381342	0.190671	−	0.981654i	$-0.438934\pi$
$557$	9.00000	0.381342	0.190671	+	0.981654i	$0.438934\pi$
$558$	0	0
$559$	−1.82843	−0.0773342
$560$	0	0
$561$	−11.7574	−0.496396
$562$	0	0
$563$	−39.6274	−1.67010	−0.835048	−	0.550177i	$-0.814560\pi$
$563$	−39.6274	−1.67010	−0.835048	+	0.550177i	$0.814560\pi$
$564$	0	0
$565$	23.3137	0.980815
$566$	0	0
$567$	17.0711	0.716917
$568$	0	0
$569$	13.3431	0.559374	0.279687	−	0.960091i	$-0.409769\pi$
$569$	13.3431	0.559374	0.279687	+	0.960091i	$0.409769\pi$
$570$	0	0
$571$	3.07107	0.128520	0.0642601	−	0.997933i	$-0.479531\pi$
$571$	3.07107	0.128520	0.0642601	+	0.997933i	$0.479531\pi$
$572$	0	0
$573$	8.68629	0.362875
$574$	0	0
$575$	44.3137	1.84801
$576$	0	0
$577$	33.0711	1.37677	0.688383	−	0.725347i	$-0.258321\pi$
$577$	33.0711	1.37677	0.688383	+	0.725347i	$0.258321\pi$
$578$	0	0
$579$	−22.5858	−0.938633
$580$	0	0
$581$	11.4142	0.473541
$582$	0	0
$583$	−52.9411	−2.19260
$584$	0	0
$585$	6.24264	0.258101
$586$	0	0
$587$	−33.7990	−1.39503	−0.697517	−	0.716568i	$-0.745712\pi$
$587$	−33.7990	−1.39503	−0.697517	+	0.716568i	$0.745712\pi$
$588$	0	0
$589$	2.48528	0.102404
$590$	0	0
$591$	−20.0000	−0.822690
$592$	0	0
$593$	−9.07107	−0.372504	−0.186252	−	0.982502i	$-0.559634\pi$
$593$	−9.07107	−0.372504	−0.186252	+	0.982502i	$0.559634\pi$
$594$	0	0
$595$	25.3137	1.03776
$596$	0	0
$597$	−10.8284	−0.443178
$598$	0	0
$599$	36.6569	1.49776	0.748879	−	0.662706i	$-0.230592\pi$
$599$	36.6569	1.49776	0.748879	+	0.662706i	$0.230592\pi$
$600$	0	0
$601$	−24.9706	−1.01857	−0.509285	−	0.860598i	$-0.670090\pi$
$601$	−24.9706	−1.01857	−0.509285	+	0.860598i	$0.670090\pi$
$602$	0	0
$603$	−7.48528	−0.304824
$604$	0	0
$605$	−12.4853	−0.507599
$606$	0	0
$607$	−32.9706	−1.33823	−0.669117	−	0.743157i	$-0.733327\pi$
$607$	−32.9706	−1.33823	−0.669117	+	0.743157i	$0.733327\pi$
$608$	0	0
$609$	20.4853	0.830105
$610$	0	0
$611$	10.9706	0.443821
$612$	0	0
$613$	−6.82843	−0.275798	−0.137899	−	0.990446i	$-0.544035\pi$
$613$	−6.82843	−0.275798	−0.137899	+	0.990446i	$0.544035\pi$
$614$	0	0
$615$	8.82843	0.355997
$616$	0	0
$617$	19.9706	0.803985	0.401992	−	0.915643i	$-0.368318\pi$
$617$	19.9706	0.803985	0.401992	+	0.915643i	$0.368318\pi$
$618$	0	0
$619$	−28.9706	−1.16443	−0.582213	−	0.813037i	$-0.697813\pi$
$619$	−28.9706	−1.16443	−0.582213	+	0.813037i	$0.697813\pi$
$620$	0	0
$621$	37.6569	1.51112
$622$	0	0
$623$	6.00000	0.240385
$624$	0	0
$625$	−13.9706	−0.558823
$626$	0	0
$627$	4.48528	0.179125
$628$	0	0
$629$	−18.4264	−0.734709
$630$	0	0
$631$	46.7696	1.86187	0.930933	−	0.365189i	$-0.118996\pi$
$631$	46.7696	1.86187	0.930933	+	0.365189i	$0.118996\pi$
$632$	0	0
$633$	−0.201010	−0.00798944
$634$	0	0
$635$	−13.0711	−0.518710
$636$	0	0
$637$	8.51472	0.337365
$638$	0	0
$639$	3.17157	0.125465
$640$	0	0
$641$	−29.5563	−1.16741	−0.583703	−	0.811967i	$-0.698397\pi$
$641$	−29.5563	−1.16741	−0.583703	+	0.811967i	$0.698397\pi$
$642$	0	0
$643$	26.4853	1.04448	0.522239	−	0.852799i	$-0.325097\pi$
$643$	26.4853	1.04448	0.522239	+	0.852799i	$0.325097\pi$
$644$	0	0
$645$	−4.82843	−0.190119
$646$	0	0
$647$	−1.17157	−0.0460593	−0.0230296	−	0.999735i	$-0.507331\pi$
$647$	−1.17157	−0.0460593	−0.0230296	+	0.999735i	$0.507331\pi$
$648$	0	0
$649$	18.4853	0.725611
$650$	0	0
$651$	−14.4853	−0.567723
$652$	0	0
$653$	−15.1716	−0.593710	−0.296855	−	0.954923i	$-0.595938\pi$
$653$	−15.1716	−0.593710	−0.296855	+	0.954923i	$0.595938\pi$
$654$	0	0
$655$	32.9706	1.28827
$656$	0	0
$657$	16.2426	0.633686
$658$	0	0
$659$	−16.3137	−0.635492	−0.317746	−	0.948176i	$-0.602926\pi$
$659$	−16.3137	−0.635492	−0.317746	+	0.948176i	$0.602926\pi$
$660$	0	0
$661$	17.0000	0.661223	0.330612	−	0.943767i	$-0.392745\pi$
$661$	17.0000	0.661223	0.330612	+	0.943767i	$0.392745\pi$
$662$	0	0
$663$	−5.61522	−0.218077
$664$	0	0
$665$	−9.65685	−0.374477
$666$	0	0
$667$	28.2426	1.09356
$668$	0	0
$669$	−5.79899	−0.224202
$670$	0	0
$671$	0.928932	0.0358610
$672$	0	0
$673$	−16.7279	−0.644814	−0.322407	−	0.946601i	$-0.604492\pi$
$673$	−16.7279	−0.644814	−0.322407	+	0.946601i	$0.604492\pi$
$674$	0	0
$675$	37.6569	1.44941
$676$	0	0
$677$	−11.1716	−0.429358	−0.214679	−	0.976685i	$-0.568871\pi$
$677$	−11.1716	−0.429358	−0.214679	+	0.976685i	$0.568871\pi$
$678$	0	0
$679$	−6.24264	−0.239571
$680$	0	0
$681$	−8.97056	−0.343753
$682$	0	0
$683$	46.4558	1.77758	0.888792	−	0.458311i	$-0.151546\pi$
$683$	46.4558	1.77758	0.888792	+	0.458311i	$0.151546\pi$
$684$	0	0
$685$	49.4558	1.88961
$686$	0	0
$687$	−36.7279	−1.40126
$688$	0	0
$689$	−25.2843	−0.963254
$690$	0	0
$691$	26.7279	1.01678	0.508389	−	0.861128i	$-0.330241\pi$
$691$	26.7279	1.01678	0.508389	+	0.861128i	$0.330241\pi$
$692$	0	0
$693$	13.0711	0.496529
$694$	0	0
$695$	18.7279	0.710391
$696$	0	0
$697$	3.97056	0.150396
$698$	0	0
$699$	−6.82843	−0.258275
$700$	0	0
$701$	29.6569	1.12012	0.560062	−	0.828451i	$-0.310777\pi$
$701$	29.6569	1.12012	0.560062	+	0.828451i	$0.310777\pi$
$702$	0	0
$703$	7.02944	0.265120
$704$	0	0
$705$	28.9706	1.09109
$706$	0	0
$707$	19.8995	0.748398
$708$	0	0
$709$	38.1127	1.43135	0.715676	−	0.698432i	$-0.246119\pi$
$709$	38.1127	1.43135	0.715676	+	0.698432i	$0.246119\pi$
$710$	0	0
$711$	−4.82843	−0.181080
$712$	0	0
$713$	−19.9706	−0.747903
$714$	0	0
$715$	−23.8995	−0.893790
$716$	0	0
$717$	−20.4853	−0.765037
$718$	0	0
$719$	−22.6274	−0.843860	−0.421930	−	0.906628i	$-0.638647\pi$
$719$	−22.6274	−0.843860	−0.421930	+	0.906628i	$0.638647\pi$
$720$	0	0
$721$	−1.75736	−0.0654475
$722$	0	0
$723$	−5.65685	−0.210381
$724$	0	0
$725$	28.2426	1.04891
$726$	0	0
$727$	−24.9706	−0.926107	−0.463053	−	0.886330i	$-0.653246\pi$
$727$	−24.9706	−0.926107	−0.463053	+	0.886330i	$0.653246\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	0	0
$731$	−2.17157	−0.0803185
$732$	0	0
$733$	−16.9706	−0.626822	−0.313411	−	0.949618i	$-0.601472\pi$
$733$	−16.9706	−0.626822	−0.313411	+	0.949618i	$0.601472\pi$
$734$	0	0
$735$	22.4853	0.829382
$736$	0	0
$737$	28.6569	1.05559
$738$	0	0
$739$	39.4558	1.45141	0.725703	−	0.688008i	$-0.241515\pi$
$739$	39.4558	1.45141	0.725703	+	0.688008i	$0.241515\pi$
$740$	0	0
$741$	2.14214	0.0786933
$742$	0	0
$743$	15.1127	0.554431	0.277216	−	0.960808i	$-0.410588\pi$
$743$	15.1127	0.554431	0.277216	+	0.960808i	$0.410588\pi$
$744$	0	0
$745$	28.9706	1.06140
$746$	0	0
$747$	3.34315	0.122319
$748$	0	0
$749$	−1.17157	−0.0428083
$750$	0	0
$751$	−11.7574	−0.429032	−0.214516	−	0.976720i	$-0.568817\pi$
$751$	−11.7574	−0.429032	−0.214516	+	0.976720i	$0.568817\pi$
$752$	0	0
$753$	32.7279	1.19267
$754$	0	0
$755$	−62.2843	−2.26676
$756$	0	0
$757$	−3.51472	−0.127745	−0.0638723	−	0.997958i	$-0.520345\pi$
$757$	−3.51472	−0.127745	−0.0638723	+	0.997958i	$0.520345\pi$
$758$	0	0
$759$	−36.0416	−1.30823
$760$	0	0
$761$	19.1127	0.692835	0.346417	−	0.938080i	$-0.387398\pi$
$761$	19.1127	0.692835	0.346417	+	0.938080i	$0.387398\pi$
$762$	0	0
$763$	−68.1838	−2.46842
$764$	0	0
$765$	7.41421	0.268061
$766$	0	0
$767$	8.82843	0.318776
$768$	0	0
$769$	44.7696	1.61443	0.807216	−	0.590257i	$-0.200973\pi$
$769$	44.7696	1.61443	0.807216	+	0.590257i	$0.200973\pi$
$770$	0	0
$771$	2.48528	0.0895052
$772$	0	0
$773$	−8.10051	−0.291355	−0.145677	−	0.989332i	$-0.546536\pi$
$773$	−8.10051	−0.291355	−0.145677	+	0.989332i	$0.546536\pi$
$774$	0	0
$775$	−19.9706	−0.717364
$776$	0	0
$777$	−40.9706	−1.46981
$778$	0	0
$779$	−1.51472	−0.0542704
$780$	0	0
$781$	−12.1421	−0.434480
$782$	0	0
$783$	24.0000	0.857690
$784$	0	0
$785$	−34.1421	−1.21859
$786$	0	0
$787$	−9.79899	−0.349296	−0.174648	−	0.984631i	$-0.555879\pi$
$787$	−9.79899	−0.349296	−0.174648	+	0.984631i	$0.555879\pi$
$788$	0	0
$789$	−0.201010	−0.00715615
$790$	0	0
$791$	23.3137	0.828940
$792$	0	0
$793$	0.443651	0.0157545
$794$	0	0
$795$	−66.7696	−2.36807
$796$	0	0
$797$	−35.3137	−1.25088	−0.625438	−	0.780274i	$-0.715080\pi$
$797$	−35.3137	−1.25088	−0.625438	+	0.780274i	$0.715080\pi$
$798$	0	0
$799$	13.0294	0.460948
$800$	0	0
$801$	1.75736	0.0620932
$802$	0	0
$803$	−62.1838	−2.19442
$804$	0	0
$805$	77.5980	2.73497
$806$	0	0
$807$	−4.24264	−0.149348
$808$	0	0
$809$	−5.65685	−0.198884	−0.0994422	−	0.995043i	$-0.531706\pi$
$809$	−5.65685	−0.198884	−0.0994422	+	0.995043i	$0.531706\pi$
$810$	0	0
$811$	−23.2721	−0.817193	−0.408597	−	0.912715i	$-0.633982\pi$
$811$	−23.2721	−0.817193	−0.408597	+	0.912715i	$0.633982\pi$
$812$	0	0
$813$	36.7279	1.28810
$814$	0	0
$815$	−40.1421	−1.40612
$816$	0	0
$817$	0.828427	0.0289830
$818$	0	0
$819$	6.24264	0.218136
$820$	0	0
$821$	52.1127	1.81875	0.909373	−	0.415982i	$-0.136562\pi$
$821$	52.1127	1.81875	0.909373	+	0.415982i	$0.136562\pi$
$822$	0	0
$823$	−54.6569	−1.90522	−0.952609	−	0.304197i	$-0.901612\pi$
$823$	−54.6569	−1.90522	−0.952609	+	0.304197i	$0.901612\pi$
$824$	0	0
$825$	−36.0416	−1.25481
$826$	0	0
$827$	1.65685	0.0576145	0.0288072	−	0.999585i	$-0.490829\pi$
$827$	1.65685	0.0576145	0.0288072	+	0.999585i	$0.490829\pi$
$828$	0	0
$829$	−23.7990	−0.826573	−0.413287	−	0.910601i	$-0.635619\pi$
$829$	−23.7990	−0.826573	−0.413287	+	0.910601i	$0.635619\pi$
$830$	0	0
$831$	11.1716	0.387538
$832$	0	0
$833$	10.1127	0.350384
$834$	0	0
$835$	48.8701	1.69122
$836$	0	0
$837$	−16.9706	−0.586588
$838$	0	0
$839$	−48.8701	−1.68718	−0.843591	−	0.536986i	$-0.819563\pi$
$839$	−48.8701	−1.68718	−0.843591	+	0.536986i	$0.819563\pi$
$840$	0	0
$841$	−11.0000	−0.379310
$842$	0	0
$843$	12.2426	0.421659
$844$	0	0
$845$	32.9706	1.13422
$846$	0	0
$847$	−12.4853	−0.428999
$848$	0	0
$849$	25.8995	0.888868
$850$	0	0
$851$	−56.4853	−1.93629
$852$	0	0
$853$	−46.5980	−1.59548	−0.797742	−	0.602999i	$-0.793972\pi$
$853$	−46.5980	−1.59548	−0.797742	+	0.602999i	$0.793972\pi$
$854$	0	0
$855$	−2.82843	−0.0967302
$856$	0	0
$857$	7.65685	0.261553	0.130777	−	0.991412i	$-0.458253\pi$
$857$	7.65685	0.261553	0.130777	+	0.991412i	$0.458253\pi$
$858$	0	0
$859$	−16.9706	−0.579028	−0.289514	−	0.957174i	$-0.593494\pi$
$859$	−16.9706	−0.579028	−0.289514	+	0.957174i	$0.593494\pi$
$860$	0	0
$861$	8.82843	0.300872
$862$	0	0
$863$	−47.2548	−1.60857	−0.804287	−	0.594242i	$-0.797452\pi$
$863$	−47.2548	−1.60857	−0.804287	+	0.594242i	$0.797452\pi$
$864$	0	0
$865$	80.7696	2.74625
$866$	0	0
$867$	17.3726	0.590004
$868$	0	0
$869$	18.4853	0.627070
$870$	0	0
$871$	13.6863	0.463742
$872$	0	0
$873$	−1.82843	−0.0618829
$874$	0	0
$875$	19.3137	0.652923
$876$	0	0
$877$	−36.7990	−1.24261	−0.621307	−	0.783567i	$-0.713398\pi$
$877$	−36.7990	−1.24261	−0.621307	+	0.783567i	$0.713398\pi$
$878$	0	0
$879$	14.6274	0.493371
$880$	0	0
$881$	36.2548	1.22146	0.610728	−	0.791840i	$-0.290877\pi$
$881$	36.2548	1.22146	0.610728	+	0.791840i	$0.290877\pi$
$882$	0	0
$883$	8.02944	0.270212	0.135106	−	0.990831i	$-0.456862\pi$
$883$	8.02944	0.270212	0.135106	+	0.990831i	$0.456862\pi$
$884$	0	0
$885$	23.3137	0.783682
$886$	0	0
$887$	58.9706	1.98004	0.990019	−	0.140935i	$-0.0450108\pi$
$887$	58.9706	1.98004	0.990019	+	0.140935i	$0.0450108\pi$
$888$	0	0
$889$	−13.0711	−0.438390
$890$	0	0
$891$	−19.1421	−0.641286
$892$	0	0
$893$	−4.97056	−0.166334
$894$	0	0
$895$	−15.6569	−0.523351
$896$	0	0
$897$	−17.2132	−0.574732
$898$	0	0
$899$	−12.7279	−0.424500
$900$	0	0
$901$	−30.0294	−1.00043
$902$	0	0
$903$	−4.82843	−0.160680
$904$	0	0
$905$	24.9706	0.830050
$906$	0	0
$907$	19.9706	0.663112	0.331556	−	0.943436i	$-0.392426\pi$
$907$	19.9706	0.663112	0.331556	+	0.943436i	$0.392426\pi$
$908$	0	0
$909$	5.82843	0.193317
$910$	0	0
$911$	−4.24264	−0.140565	−0.0702825	−	0.997527i	$-0.522390\pi$
$911$	−4.24264	−0.140565	−0.0702825	+	0.997527i	$0.522390\pi$
$912$	0	0
$913$	−12.7990	−0.423585
$914$	0	0
$915$	1.17157	0.0387310
$916$	0	0
$917$	32.9706	1.08878
$918$	0	0
$919$	−12.5147	−0.412822	−0.206411	−	0.978465i	$-0.566178\pi$
$919$	−12.5147	−0.412822	−0.206411	+	0.978465i	$0.566178\pi$
$920$	0	0
$921$	37.8995	1.24883
$922$	0	0
$923$	−5.79899	−0.190876
$924$	0	0
$925$	−56.4853	−1.85722
$926$	0	0
$927$	−0.514719	−0.0169056
$928$	0	0
$929$	39.1716	1.28518	0.642589	−	0.766211i	$-0.277860\pi$
$929$	39.1716	1.28518	0.642589	+	0.766211i	$0.277860\pi$
$930$	0	0
$931$	−3.85786	−0.126436
$932$	0	0
$933$	19.7574	0.646827
$934$	0	0
$935$	−28.3848	−0.928281
$936$	0	0
$937$	−51.0122	−1.66650	−0.833248	−	0.552900i	$-0.813521\pi$
$937$	−51.0122	−1.66650	−0.833248	+	0.552900i	$0.813521\pi$
$938$	0	0
$939$	35.6569	1.16362
$940$	0	0
$941$	31.6274	1.03102	0.515512	−	0.856882i	$-0.327602\pi$
$941$	31.6274	1.03102	0.515512	+	0.856882i	$0.327602\pi$
$942$	0	0
$943$	12.1716	0.396361
$944$	0	0
$945$	65.9411	2.14506
$946$	0	0
$947$	5.82843	0.189398	0.0946992	−	0.995506i	$-0.469811\pi$
$947$	5.82843	0.189398	0.0946992	+	0.995506i	$0.469811\pi$
$948$	0	0
$949$	−29.6985	−0.964054
$950$	0	0
$951$	36.5269	1.18447
$952$	0	0
$953$	−24.0416	−0.778785	−0.389392	−	0.921072i	$-0.627315\pi$
$953$	−24.0416	−0.778785	−0.389392	+	0.921072i	$0.627315\pi$
$954$	0	0
$955$	20.9706	0.678591
$956$	0	0
$957$	−22.9706	−0.742533
$958$	0	0
$959$	49.4558	1.59701
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	−0.343146	−0.0110577
$964$	0	0
$965$	−54.5269	−1.75528
$966$	0	0
$967$	−40.9411	−1.31658	−0.658289	−	0.752765i	$-0.728719\pi$
$967$	−40.9411	−1.31658	−0.658289	+	0.752765i	$0.728719\pi$
$968$	0	0
$969$	2.54416	0.0817301
$970$	0	0
$971$	3.14214	0.100836	0.0504180	−	0.998728i	$-0.483945\pi$
$971$	3.14214	0.100836	0.0504180	+	0.998728i	$0.483945\pi$
$972$	0	0
$973$	18.7279	0.600390
$974$	0	0
$975$	−17.2132	−0.551264
$976$	0	0
$977$	−23.3137	−0.745872	−0.372936	−	0.927857i	$-0.621649\pi$
$977$	−23.3137	−0.745872	−0.372936	+	0.927857i	$0.621649\pi$
$978$	0	0
$979$	−6.72792	−0.215025
$980$	0	0
$981$	−19.9706	−0.637611
$982$	0	0
$983$	−62.5269	−1.99430	−0.997149	−	0.0754527i	$-0.975960\pi$
$983$	−62.5269	−1.99430	−0.997149	+	0.0754527i	$0.975960\pi$
$984$	0	0
$985$	−48.2843	−1.53846
$986$	0	0
$987$	28.9706	0.922143
$988$	0	0
$989$	−6.65685	−0.211676
$990$	0	0
$991$	11.5563	0.367100	0.183550	−	0.983010i	$-0.441241\pi$
$991$	11.5563	0.367100	0.183550	+	0.983010i	$0.441241\pi$
$992$	0	0
$993$	30.2843	0.961042
$994$	0	0
$995$	−26.1421	−0.828761
$996$	0	0
$997$	−2.92893	−0.0927602	−0.0463801	−	0.998924i	$-0.514769\pi$
$997$	−2.92893	−0.0927602	−0.0463801	+	0.998924i	$0.514769\pi$
$998$	0	0
$999$	−48.0000	−1.51865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2752.2.a.l.1.1		2
4.3	odd	2		2752.2.a.m.1.2		2
8.3	odd	2		688.2.a.f.1.1		2
8.5	even	2		43.2.a.b.1.1	✓	2
24.5	odd	2		387.2.a.h.1.2		2
24.11	even	2		6192.2.a.bd.1.1		2
40.13	odd	4		1075.2.b.f.474.4		4
40.29	even	2		1075.2.a.i.1.2		2
40.37	odd	4		1075.2.b.f.474.1		4
56.13	odd	2		2107.2.a.b.1.1		2
88.21	odd	2		5203.2.a.f.1.2		2
104.77	even	2		7267.2.a.b.1.2		2
120.29	odd	2		9675.2.a.bf.1.1		2
344.85	odd	2		1849.2.a.f.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.2.a.b.1.1	✓	2	8.5	even	2
387.2.a.h.1.2		2	24.5	odd	2
688.2.a.f.1.1		2	8.3	odd	2
1075.2.a.i.1.2		2	40.29	even	2
1075.2.b.f.474.1		4	40.37	odd	4
1075.2.b.f.474.4		4	40.13	odd	4
1849.2.a.f.1.2		2	344.85	odd	2
2107.2.a.b.1.1		2	56.13	odd	2
2752.2.a.l.1.1		2	1.1	even	1	trivial
2752.2.a.m.1.2		2	4.3	odd	2
5203.2.a.f.1.2		2	88.21	odd	2
6192.2.a.bd.1.1		2	24.11	even	2
7267.2.a.b.1.2		2	104.77	even	2
9675.2.a.bf.1.1		2	120.29	odd	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:	\( N \)	\(=\)	\( 2752 = 2^{6} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2752.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{3} -3.41421 q^{5} -3.41421 q^{7} -1.00000 q^{9} +O(q^{10})\)	\(q-1.41421 q^{3} -3.41421 q^{5} -3.41421 q^{7} -1.00000 q^{9} +3.82843 q^{11} +1.82843 q^{13} +4.82843 q^{15} +2.17157 q^{17} -0.828427 q^{19} +4.82843 q^{21} +6.65685 q^{23} +6.65685 q^{25} +5.65685 q^{27} +4.24264 q^{29} -3.00000 q^{31} -5.41421 q^{33} +11.6569 q^{35} -8.48528 q^{37} -2.58579 q^{39} +1.82843 q^{41} -1.00000 q^{43} +3.41421 q^{45} +6.00000 q^{47} +4.65685 q^{49} -3.07107 q^{51} -13.8284 q^{53} -13.0711 q^{55} +1.17157 q^{57} +4.82843 q^{59} +0.242641 q^{61} +3.41421 q^{63} -6.24264 q^{65} +7.48528 q^{67} -9.41421 q^{69} -3.17157 q^{71} -16.2426 q^{73} -9.41421 q^{75} -13.0711 q^{77} +4.82843 q^{79} -5.00000 q^{81} -3.34315 q^{83} -7.41421 q^{85} -6.00000 q^{87} -1.75736 q^{89} -6.24264 q^{91} +4.24264 q^{93} +2.82843 q^{95} +1.82843 q^{97} -3.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 4 q^{5} - 4 q^{7} - 2 q^{9}+O(q^{10}) \) 2 * q - 4 * q^5 - 4 * q^7 - 2 * q^9	\( 2 q - 4 q^{5} - 4 q^{7} - 2 q^{9} + 2 q^{11} - 2 q^{13} + 4 q^{15} + 10 q^{17} + 4 q^{19} + 4 q^{21} + 2 q^{23} + 2 q^{25} - 6 q^{31} - 8 q^{33} + 12 q^{35} - 8 q^{39} - 2 q^{41} - 2 q^{43} + 4 q^{45} + 12 q^{47} - 2 q^{49} + 8 q^{51} - 22 q^{53} - 12 q^{55} + 8 q^{57} + 4 q^{59} - 8 q^{61} + 4 q^{63} - 4 q^{65} - 2 q^{67} - 16 q^{69} - 12 q^{71} - 24 q^{73} - 16 q^{75} - 12 q^{77} + 4 q^{79} - 10 q^{81} - 18 q^{83} - 12 q^{85} - 12 q^{87} - 12 q^{89} - 4 q^{91} - 2 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q - 4 * q^5 - 4 * q^7 - 2 * q^9 + 2 * q^11 - 2 * q^13 + 4 * q^15 + 10 * q^17 + 4 * q^19 + 4 * q^21 + 2 * q^23 + 2 * q^25 - 6 * q^31 - 8 * q^33 + 12 * q^35 - 8 * q^39 - 2 * q^41 - 2 * q^43 + 4 * q^45 + 12 * q^47 - 2 * q^49 + 8 * q^51 - 22 * q^53 - 12 * q^55 + 8 * q^57 + 4 * q^59 - 8 * q^61 + 4 * q^63 - 4 * q^65 - 2 * q^67 - 16 * q^69 - 12 * q^71 - 24 * q^73 - 16 * q^75 - 12 * q^77 + 4 * q^79 - 10 * q^81 - 18 * q^83 - 12 * q^85 - 12 * q^87 - 12 * q^89 - 4 * q^91 - 2 * q^97 - 2 * q^99

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