LMFDB - Embedded newform 704.2.a.m.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [704,2,Mod(1,704)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(704, 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("704.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 704 = 2^{6} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	704.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:	$5.62146830230$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 88)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	704.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.56155 q^{3} -3.56155 q^{5} +3.12311 q^{7} -0.561553 q^{9} +O(q^{10})$	$q+1.56155 q^{3} -3.56155 q^{5} +3.12311 q^{7} -0.561553 q^{9} +1.00000 q^{11} +5.12311 q^{13} -5.56155 q^{15} +2.00000 q^{17} +4.00000 q^{19} +4.87689 q^{21} +2.43845 q^{23} +7.68466 q^{25} -5.56155 q^{27} +5.12311 q^{29} -5.56155 q^{31} +1.56155 q^{33} -11.1231 q^{35} +7.56155 q^{37} +8.00000 q^{39} -1.12311 q^{41} +7.12311 q^{43} +2.00000 q^{45} +8.00000 q^{47} +2.75379 q^{49} +3.12311 q^{51} -12.2462 q^{53} -3.56155 q^{55} +6.24621 q^{57} -7.80776 q^{59} -1.12311 q^{61} -1.75379 q^{63} -18.2462 q^{65} -9.56155 q^{67} +3.80776 q^{69} -8.68466 q^{71} +5.12311 q^{73} +12.0000 q^{75} +3.12311 q^{77} -11.1231 q^{79} -7.00000 q^{81} -0.876894 q^{83} -7.12311 q^{85} +8.00000 q^{87} +2.68466 q^{89} +16.0000 q^{91} -8.68466 q^{93} -14.2462 q^{95} +15.5616 q^{97} -0.561553 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q - q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9	$ 2 q - q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} + 2 q^{11} + 2 q^{13} - 7 q^{15} + 4 q^{17} + 8 q^{19} + 18 q^{21} + 9 q^{23} + 3 q^{25} - 7 q^{27} + 2 q^{29} - 7 q^{31} - q^{33} - 14 q^{35} + 11 q^{37} + 16 q^{39} + 6 q^{41} + 6 q^{43} + 4 q^{45} + 16 q^{47} + 22 q^{49} - 2 q^{51} - 8 q^{53} - 3 q^{55} - 4 q^{57} + 5 q^{59} + 6 q^{61} - 20 q^{63} - 20 q^{65} - 15 q^{67} - 13 q^{69} - 5 q^{71} + 2 q^{73} + 24 q^{75} - 2 q^{77} - 14 q^{79} - 14 q^{81} - 10 q^{83} - 6 q^{85} + 16 q^{87} - 7 q^{89} + 32 q^{91} - 5 q^{93} - 12 q^{95} + 27 q^{97} + 3 q^{99}+O(q^{100}) $ 2 * q - q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9 + 2 * q^11 + 2 * q^13 - 7 * q^15 + 4 * q^17 + 8 * q^19 + 18 * q^21 + 9 * q^23 + 3 * q^25 - 7 * q^27 + 2 * q^29 - 7 * q^31 - q^33 - 14 * q^35 + 11 * q^37 + 16 * q^39 + 6 * q^41 + 6 * q^43 + 4 * q^45 + 16 * q^47 + 22 * q^49 - 2 * q^51 - 8 * q^53 - 3 * q^55 - 4 * q^57 + 5 * q^59 + 6 * q^61 - 20 * q^63 - 20 * q^65 - 15 * q^67 - 13 * q^69 - 5 * q^71 + 2 * q^73 + 24 * q^75 - 2 * q^77 - 14 * q^79 - 14 * q^81 - 10 * q^83 - 6 * q^85 + 16 * q^87 - 7 * q^89 + 32 * q^91 - 5 * q^93 - 12 * q^95 + 27 * q^97 + 3 * 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
$3$	1.56155	0.901563	0.450781	−	0.892634i	$-0.351145\pi$
$3$	1.56155	0.901563	0.450781	+	0.892634i	$0.351145\pi$
$4$	0	0
$5$	−3.56155	−1.59277	−0.796387	−	0.604787i	$-0.793258\pi$
$5$	−3.56155	−1.59277	−0.796387	+	0.604787i	$0.793258\pi$
$6$	0	0
$7$	3.12311	1.18042	0.590211	−	0.807249i	$-0.299044\pi$
$7$	3.12311	1.18042	0.590211	+	0.807249i	$0.299044\pi$
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.12311	1.42089	0.710447	−	0.703751i	$-0.248493\pi$
$13$	5.12311	1.42089	0.710447	+	0.703751i	$0.248493\pi$
$14$	0	0
$15$	−5.56155	−1.43599
$16$	0	0
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	4.87689	1.06423
$22$	0	0
$23$	2.43845	0.508451	0.254226	−	0.967145i	$-0.418179\pi$
$23$	2.43845	0.508451	0.254226	+	0.967145i	$0.418179\pi$
$24$	0	0
$25$	7.68466	1.53693
$26$	0	0
$27$	−5.56155	−1.07032
$28$	0	0
$29$	5.12311	0.951337	0.475668	−	0.879625i	$-0.342206\pi$
$29$	5.12311	0.951337	0.475668	+	0.879625i	$0.342206\pi$
$30$	0	0
$31$	−5.56155	−0.998884	−0.499442	−	0.866347i	$-0.666462\pi$
$31$	−5.56155	−0.998884	−0.499442	+	0.866347i	$0.666462\pi$
$32$	0	0
$33$	1.56155	0.271831
$34$	0	0
$35$	−11.1231	−1.88015
$36$	0	0
$37$	7.56155	1.24311	0.621556	−	0.783370i	$-0.286501\pi$
$37$	7.56155	1.24311	0.621556	+	0.783370i	$0.286501\pi$
$38$	0	0
$39$	8.00000	1.28103
$40$	0	0
$41$	−1.12311	−0.175400	−0.0876998	−	0.996147i	$-0.527952\pi$
$41$	−1.12311	−0.175400	−0.0876998	+	0.996147i	$0.527952\pi$
$42$	0	0
$43$	7.12311	1.08626	0.543132	−	0.839648i	$-0.317238\pi$
$43$	7.12311	1.08626	0.543132	+	0.839648i	$0.317238\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	2.75379	0.393398
$50$	0	0
$51$	3.12311	0.437322
$52$	0	0
$53$	−12.2462	−1.68215	−0.841073	−	0.540921i	$-0.818076\pi$
$53$	−12.2462	−1.68215	−0.841073	+	0.540921i	$0.818076\pi$
$54$	0	0
$55$	−3.56155	−0.480240
$56$	0	0
$57$	6.24621	0.827331
$58$	0	0
$59$	−7.80776	−1.01648	−0.508242	−	0.861214i	$-0.669705\pi$
$59$	−7.80776	−1.01648	−0.508242	+	0.861214i	$0.669705\pi$
$60$	0	0
$61$	−1.12311	−0.143799	−0.0718995	−	0.997412i	$-0.522906\pi$
$61$	−1.12311	−0.143799	−0.0718995	+	0.997412i	$0.522906\pi$
$62$	0	0
$63$	−1.75379	−0.220957
$64$	0	0
$65$	−18.2462	−2.26316
$66$	0	0
$67$	−9.56155	−1.16813	−0.584065	−	0.811707i	$-0.698539\pi$
$67$	−9.56155	−1.16813	−0.584065	+	0.811707i	$0.698539\pi$
$68$	0	0
$69$	3.80776	0.458401
$70$	0	0
$71$	−8.68466	−1.03068	−0.515340	−	0.856986i	$-0.672334\pi$
$71$	−8.68466	−1.03068	−0.515340	+	0.856986i	$0.672334\pi$
$72$	0	0
$73$	5.12311	0.599614	0.299807	−	0.954000i	$-0.403078\pi$
$73$	5.12311	0.599614	0.299807	+	0.954000i	$0.403078\pi$
$74$	0	0
$75$	12.0000	1.38564
$76$	0	0
$77$	3.12311	0.355911
$78$	0	0
$79$	−11.1231	−1.25145	−0.625724	−	0.780045i	$-0.715196\pi$
$79$	−11.1231	−1.25145	−0.625724	+	0.780045i	$0.715196\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−0.876894	−0.0962517	−0.0481258	−	0.998841i	$-0.515325\pi$
$83$	−0.876894	−0.0962517	−0.0481258	+	0.998841i	$0.515325\pi$
$84$	0	0
$85$	−7.12311	−0.772609
$86$	0	0
$87$	8.00000	0.857690
$88$	0	0
$89$	2.68466	0.284573	0.142287	−	0.989825i	$-0.454555\pi$
$89$	2.68466	0.284573	0.142287	+	0.989825i	$0.454555\pi$
$90$	0	0
$91$	16.0000	1.67726
$92$	0	0
$93$	−8.68466	−0.900557
$94$	0	0
$95$	−14.2462	−1.46163
$96$	0	0
$97$	15.5616	1.58004	0.790018	−	0.613083i	$-0.210071\pi$
$97$	15.5616	1.58004	0.790018	+	0.613083i	$0.210071\pi$
$98$	0	0
$99$	−0.561553	−0.0564382
$100$	0	0
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	−17.3693	−1.69507
$106$	0	0
$107$	13.3693	1.29246	0.646230	−	0.763142i	$-0.276345\pi$
$107$	13.3693	1.29246	0.646230	+	0.763142i	$0.276345\pi$
$108$	0	0
$109$	−12.2462	−1.17297	−0.586487	−	0.809959i	$-0.699490\pi$
$109$	−12.2462	−1.17297	−0.586487	+	0.809959i	$0.699490\pi$
$110$	0	0
$111$	11.8078	1.12074
$112$	0	0
$113$	−0.438447	−0.0412456	−0.0206228	−	0.999787i	$-0.506565\pi$
$113$	−0.438447	−0.0412456	−0.0206228	+	0.999787i	$0.506565\pi$
$114$	0	0
$115$	−8.68466	−0.809849
$116$	0	0
$117$	−2.87689	−0.265969
$118$	0	0
$119$	6.24621	0.572589
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−1.75379	−0.158134
$124$	0	0
$125$	−9.56155	−0.855211
$126$	0	0
$127$	6.24621	0.554262	0.277131	−	0.960832i	$-0.410616\pi$
$127$	6.24621	0.554262	0.277131	+	0.960832i	$0.410616\pi$
$128$	0	0
$129$	11.1231	0.979335
$130$	0	0
$131$	−13.3693	−1.16808	−0.584041	−	0.811724i	$-0.698529\pi$
$131$	−13.3693	−1.16808	−0.584041	+	0.811724i	$0.698529\pi$
$132$	0	0
$133$	12.4924	1.08323
$134$	0	0
$135$	19.8078	1.70478
$136$	0	0
$137$	−8.43845	−0.720945	−0.360473	−	0.932770i	$-0.617385\pi$
$137$	−8.43845	−0.720945	−0.360473	+	0.932770i	$0.617385\pi$
$138$	0	0
$139$	−15.1231	−1.28273	−0.641363	−	0.767238i	$-0.721631\pi$
$139$	−15.1231	−1.28273	−0.641363	+	0.767238i	$0.721631\pi$
$140$	0	0
$141$	12.4924	1.05205
$142$	0	0
$143$	5.12311	0.428416
$144$	0	0
$145$	−18.2462	−1.51527
$146$	0	0
$147$	4.30019	0.354673
$148$	0	0
$149$	−4.24621	−0.347863	−0.173932	−	0.984758i	$-0.555647\pi$
$149$	−4.24621	−0.347863	−0.173932	+	0.984758i	$0.555647\pi$
$150$	0	0
$151$	−9.36932	−0.762464	−0.381232	−	0.924479i	$-0.624500\pi$
$151$	−9.36932	−0.762464	−0.381232	+	0.924479i	$0.624500\pi$
$152$	0	0
$153$	−1.12311	−0.0907977
$154$	0	0
$155$	19.8078	1.59100
$156$	0	0
$157$	4.43845	0.354227	0.177113	−	0.984190i	$-0.443324\pi$
$157$	4.43845	0.354227	0.177113	+	0.984190i	$0.443324\pi$
$158$	0	0
$159$	−19.1231	−1.51656
$160$	0	0
$161$	7.61553	0.600188
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	0	0
$165$	−5.56155	−0.432966
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	13.2462	1.01894
$170$	0	0
$171$	−2.24621	−0.171772
$172$	0	0
$173$	−12.2462	−0.931062	−0.465531	−	0.885032i	$-0.654137\pi$
$173$	−12.2462	−0.931062	−0.465531	+	0.885032i	$0.654137\pi$
$174$	0	0
$175$	24.0000	1.81423
$176$	0	0
$177$	−12.1922	−0.916425
$178$	0	0
$179$	6.43845	0.481232	0.240616	−	0.970620i	$-0.422651\pi$
$179$	6.43845	0.481232	0.240616	+	0.970620i	$0.422651\pi$
$180$	0	0
$181$	1.31534	0.0977686	0.0488843	−	0.998804i	$-0.484433\pi$
$181$	1.31534	0.0977686	0.0488843	+	0.998804i	$0.484433\pi$
$182$	0	0
$183$	−1.75379	−0.129644
$184$	0	0
$185$	−26.9309	−1.98000
$186$	0	0
$187$	2.00000	0.146254
$188$	0	0
$189$	−17.3693	−1.26343
$190$	0	0
$191$	−10.4384	−0.755300	−0.377650	−	0.925949i	$-0.623268\pi$
$191$	−10.4384	−0.755300	−0.377650	+	0.925949i	$0.623268\pi$
$192$	0	0
$193$	−9.12311	−0.656696	−0.328348	−	0.944557i	$-0.606492\pi$
$193$	−9.12311	−0.656696	−0.328348	+	0.944557i	$0.606492\pi$
$194$	0	0
$195$	−28.4924	−2.04038
$196$	0	0
$197$	14.4924	1.03254	0.516271	−	0.856425i	$-0.327320\pi$
$197$	14.4924	1.03254	0.516271	+	0.856425i	$0.327320\pi$
$198$	0	0
$199$	−12.4924	−0.885564	−0.442782	−	0.896629i	$-0.646009\pi$
$199$	−12.4924	−0.885564	−0.442782	+	0.896629i	$0.646009\pi$
$200$	0	0
$201$	−14.9309	−1.05314
$202$	0	0
$203$	16.0000	1.12298
$204$	0	0
$205$	4.00000	0.279372
$206$	0	0
$207$	−1.36932	−0.0951741
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	−8.49242	−0.584642	−0.292321	−	0.956320i	$-0.594428\pi$
$211$	−8.49242	−0.584642	−0.292321	+	0.956320i	$0.594428\pi$
$212$	0	0
$213$	−13.5616	−0.929222
$214$	0	0
$215$	−25.3693	−1.73017
$216$	0	0
$217$	−17.3693	−1.17911
$218$	0	0
$219$	8.00000	0.540590
$220$	0	0
$221$	10.2462	0.689235
$222$	0	0
$223$	−11.8078	−0.790706	−0.395353	−	0.918529i	$-0.629378\pi$
$223$	−11.8078	−0.790706	−0.395353	+	0.918529i	$0.629378\pi$
$224$	0	0
$225$	−4.31534	−0.287689
$226$	0	0
$227$	−23.1231	−1.53473	−0.767367	−	0.641208i	$-0.778434\pi$
$227$	−23.1231	−1.53473	−0.767367	+	0.641208i	$0.778434\pi$
$228$	0	0
$229$	−14.6847	−0.970390	−0.485195	−	0.874406i	$-0.661251\pi$
$229$	−14.6847	−0.970390	−0.485195	+	0.874406i	$0.661251\pi$
$230$	0	0
$231$	4.87689	0.320876
$232$	0	0
$233$	−7.36932	−0.482780	−0.241390	−	0.970428i	$-0.577603\pi$
$233$	−7.36932	−0.482780	−0.241390	+	0.970428i	$0.577603\pi$
$234$	0	0
$235$	−28.4924	−1.85864
$236$	0	0
$237$	−17.3693	−1.12826
$238$	0	0
$239$	−4.87689	−0.315460	−0.157730	−	0.987482i	$-0.550418\pi$
$239$	−4.87689	−0.315460	−0.157730	+	0.987482i	$0.550418\pi$
$240$	0	0
$241$	29.1231	1.87598	0.937992	−	0.346657i	$-0.112683\pi$
$241$	29.1231	1.87598	0.937992	+	0.346657i	$0.112683\pi$
$242$	0	0
$243$	5.75379	0.369106
$244$	0	0
$245$	−9.80776	−0.626595
$246$	0	0
$247$	20.4924	1.30390
$248$	0	0
$249$	−1.36932	−0.0867769
$250$	0	0
$251$	−1.56155	−0.0985643	−0.0492822	−	0.998785i	$-0.515693\pi$
$251$	−1.56155	−0.0985643	−0.0492822	+	0.998785i	$0.515693\pi$
$252$	0	0
$253$	2.43845	0.153304
$254$	0	0
$255$	−11.1231	−0.696556
$256$	0	0
$257$	11.7538	0.733181	0.366591	−	0.930382i	$-0.380525\pi$
$257$	11.7538	0.733181	0.366591	+	0.930382i	$0.380525\pi$
$258$	0	0
$259$	23.6155	1.46740
$260$	0	0
$261$	−2.87689	−0.178075
$262$	0	0
$263$	19.1231	1.17918	0.589591	−	0.807702i	$-0.299289\pi$
$263$	19.1231	1.17918	0.589591	+	0.807702i	$0.299289\pi$
$264$	0	0
$265$	43.6155	2.67928
$266$	0	0
$267$	4.19224	0.256561
$268$	0	0
$269$	20.7386	1.26446	0.632228	−	0.774782i	$-0.282140\pi$
$269$	20.7386	1.26446	0.632228	+	0.774782i	$0.282140\pi$
$270$	0	0
$271$	−28.4924	−1.73079	−0.865396	−	0.501089i	$-0.832933\pi$
$271$	−28.4924	−1.73079	−0.865396	+	0.501089i	$0.832933\pi$
$272$	0	0
$273$	24.9848	1.51215
$274$	0	0
$275$	7.68466	0.463402
$276$	0	0
$277$	18.0000	1.08152	0.540758	−	0.841178i	$-0.318138\pi$
$277$	18.0000	1.08152	0.540758	+	0.841178i	$0.318138\pi$
$278$	0	0
$279$	3.12311	0.186975
$280$	0	0
$281$	16.2462	0.969168	0.484584	−	0.874745i	$-0.338971\pi$
$281$	16.2462	0.969168	0.484584	+	0.874745i	$0.338971\pi$
$282$	0	0
$283$	−20.0000	−1.18888	−0.594438	−	0.804141i	$-0.702626\pi$
$283$	−20.0000	−1.18888	−0.594438	+	0.804141i	$0.702626\pi$
$284$	0	0
$285$	−22.2462	−1.31775
$286$	0	0
$287$	−3.50758	−0.207046
$288$	0	0
$289$	−13.0000	−0.764706
$290$	0	0
$291$	24.3002	1.42450
$292$	0	0
$293$	3.36932	0.196838	0.0984188	−	0.995145i	$-0.468622\pi$
$293$	3.36932	0.196838	0.0984188	+	0.995145i	$0.468622\pi$
$294$	0	0
$295$	27.8078	1.61903
$296$	0	0
$297$	−5.56155	−0.322714
$298$	0	0
$299$	12.4924	0.722455
$300$	0	0
$301$	22.2462	1.28225
$302$	0	0
$303$	3.12311	0.179418
$304$	0	0
$305$	4.00000	0.229039
$306$	0	0
$307$	32.4924	1.85444	0.927220	−	0.374516i	$-0.122191\pi$
$307$	32.4924	1.85444	0.927220	+	0.374516i	$0.122191\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.75379	0.553087	0.276543	−	0.961001i	$-0.410811\pi$
$311$	9.75379	0.553087	0.276543	+	0.961001i	$0.410811\pi$
$312$	0	0
$313$	−9.80776	−0.554368	−0.277184	−	0.960817i	$-0.589401\pi$
$313$	−9.80776	−0.554368	−0.277184	+	0.960817i	$0.589401\pi$
$314$	0	0
$315$	6.24621	0.351934
$316$	0	0
$317$	14.1922	0.797115	0.398558	−	0.917143i	$-0.369511\pi$
$317$	14.1922	0.797115	0.398558	+	0.917143i	$0.369511\pi$
$318$	0	0
$319$	5.12311	0.286839
$320$	0	0
$321$	20.8769	1.16523
$322$	0	0
$323$	8.00000	0.445132
$324$	0	0
$325$	39.3693	2.18382
$326$	0	0
$327$	−19.1231	−1.05751
$328$	0	0
$329$	24.9848	1.37746
$330$	0	0
$331$	−34.9309	−1.91997	−0.959987	−	0.280044i	$-0.909651\pi$
$331$	−34.9309	−1.91997	−0.959987	+	0.280044i	$0.909651\pi$
$332$	0	0
$333$	−4.24621	−0.232691
$334$	0	0
$335$	34.0540	1.86057
$336$	0	0
$337$	−16.7386	−0.911811	−0.455906	−	0.890028i	$-0.650685\pi$
$337$	−16.7386	−0.911811	−0.455906	+	0.890028i	$0.650685\pi$
$338$	0	0
$339$	−0.684658	−0.0371855
$340$	0	0
$341$	−5.56155	−0.301175
$342$	0	0
$343$	−13.2614	−0.716046
$344$	0	0
$345$	−13.5616	−0.730129
$346$	0	0
$347$	−22.7386	−1.22067	−0.610337	−	0.792142i	$-0.708966\pi$
$347$	−22.7386	−1.22067	−0.610337	+	0.792142i	$0.708966\pi$
$348$	0	0
$349$	32.2462	1.72610	0.863050	−	0.505118i	$-0.168551\pi$
$349$	32.2462	1.72610	0.863050	+	0.505118i	$0.168551\pi$
$350$	0	0
$351$	−28.4924	−1.52081
$352$	0	0
$353$	−24.0540	−1.28026	−0.640132	−	0.768265i	$-0.721120\pi$
$353$	−24.0540	−1.28026	−0.640132	+	0.768265i	$0.721120\pi$
$354$	0	0
$355$	30.9309	1.64164
$356$	0	0
$357$	9.75379	0.516225
$358$	0	0
$359$	4.49242	0.237101	0.118550	−	0.992948i	$-0.462175\pi$
$359$	4.49242	0.237101	0.118550	+	0.992948i	$0.462175\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	1.56155	0.0819603
$364$	0	0
$365$	−18.2462	−0.955050
$366$	0	0
$367$	22.9309	1.19698	0.598491	−	0.801130i	$-0.295767\pi$
$367$	22.9309	1.19698	0.598491	+	0.801130i	$0.295767\pi$
$368$	0	0
$369$	0.630683	0.0328321
$370$	0	0
$371$	−38.2462	−1.98564
$372$	0	0
$373$	8.24621	0.426973	0.213486	−	0.976946i	$-0.431518\pi$
$373$	8.24621	0.426973	0.213486	+	0.976946i	$0.431518\pi$
$374$	0	0
$375$	−14.9309	−0.771027
$376$	0	0
$377$	26.2462	1.35175
$378$	0	0
$379$	−0.192236	−0.00987450	−0.00493725	−	0.999988i	$-0.501572\pi$
$379$	−0.192236	−0.00987450	−0.00493725	+	0.999988i	$0.501572\pi$
$380$	0	0
$381$	9.75379	0.499702
$382$	0	0
$383$	−2.05398	−0.104953	−0.0524766	−	0.998622i	$-0.516712\pi$
$383$	−2.05398	−0.104953	−0.0524766	+	0.998622i	$0.516712\pi$
$384$	0	0
$385$	−11.1231	−0.566886
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	−3.56155	−0.180578	−0.0902889	−	0.995916i	$-0.528779\pi$
$389$	−3.56155	−0.180578	−0.0902889	+	0.995916i	$0.528779\pi$
$390$	0	0
$391$	4.87689	0.246635
$392$	0	0
$393$	−20.8769	−1.05310
$394$	0	0
$395$	39.6155	1.99327
$396$	0	0
$397$	−10.4924	−0.526600	−0.263300	−	0.964714i	$-0.584811\pi$
$397$	−10.4924	−0.526600	−0.263300	+	0.964714i	$0.584811\pi$
$398$	0	0
$399$	19.5076	0.976600
$400$	0	0
$401$	30.4924	1.52272	0.761359	−	0.648330i	$-0.224532\pi$
$401$	30.4924	1.52272	0.761359	+	0.648330i	$0.224532\pi$
$402$	0	0
$403$	−28.4924	−1.41931
$404$	0	0
$405$	24.9309	1.23882
$406$	0	0
$407$	7.56155	0.374812
$408$	0	0
$409$	22.4924	1.11218	0.556089	−	0.831123i	$-0.312301\pi$
$409$	22.4924	1.11218	0.556089	+	0.831123i	$0.312301\pi$
$410$	0	0
$411$	−13.1771	−0.649977
$412$	0	0
$413$	−24.3845	−1.19988
$414$	0	0
$415$	3.12311	0.153307
$416$	0	0
$417$	−23.6155	−1.15646
$418$	0	0
$419$	32.4924	1.58736	0.793679	−	0.608336i	$-0.208163\pi$
$419$	32.4924	1.58736	0.793679	+	0.608336i	$0.208163\pi$
$420$	0	0
$421$	−2.49242	−0.121473	−0.0607366	−	0.998154i	$-0.519345\pi$
$421$	−2.49242	−0.121473	−0.0607366	+	0.998154i	$0.519345\pi$
$422$	0	0
$423$	−4.49242	−0.218429
$424$	0	0
$425$	15.3693	0.745521
$426$	0	0
$427$	−3.50758	−0.169744
$428$	0	0
$429$	8.00000	0.386244
$430$	0	0
$431$	27.1231	1.30647	0.653237	−	0.757153i	$-0.273411\pi$
$431$	27.1231	1.30647	0.653237	+	0.757153i	$0.273411\pi$
$432$	0	0
$433$	−22.6847	−1.09016	−0.545078	−	0.838386i	$-0.683500\pi$
$433$	−22.6847	−1.09016	−0.545078	+	0.838386i	$0.683500\pi$
$434$	0	0
$435$	−28.4924	−1.36611
$436$	0	0
$437$	9.75379	0.466587
$438$	0	0
$439$	4.49242	0.214412	0.107206	−	0.994237i	$-0.465810\pi$
$439$	4.49242	0.214412	0.107206	+	0.994237i	$0.465810\pi$
$440$	0	0
$441$	−1.54640	−0.0736380
$442$	0	0
$443$	−11.3153	−0.537608	−0.268804	−	0.963195i	$-0.586628\pi$
$443$	−11.3153	−0.537608	−0.268804	+	0.963195i	$0.586628\pi$
$444$	0	0
$445$	−9.56155	−0.453261
$446$	0	0
$447$	−6.63068	−0.313621
$448$	0	0
$449$	−36.5464	−1.72473	−0.862366	−	0.506286i	$-0.831018\pi$
$449$	−36.5464	−1.72473	−0.862366	+	0.506286i	$0.831018\pi$
$450$	0	0
$451$	−1.12311	−0.0528850
$452$	0	0
$453$	−14.6307	−0.687409
$454$	0	0
$455$	−56.9848	−2.67149
$456$	0	0
$457$	23.8617	1.11621	0.558103	−	0.829772i	$-0.311530\pi$
$457$	23.8617	1.11621	0.558103	+	0.829772i	$0.311530\pi$
$458$	0	0
$459$	−11.1231	−0.519182
$460$	0	0
$461$	−1.12311	−0.0523082	−0.0261541	−	0.999658i	$-0.508326\pi$
$461$	−1.12311	−0.0523082	−0.0261541	+	0.999658i	$0.508326\pi$
$462$	0	0
$463$	15.3153	0.711764	0.355882	−	0.934531i	$-0.384180\pi$
$463$	15.3153	0.711764	0.355882	+	0.934531i	$0.384180\pi$
$464$	0	0
$465$	30.9309	1.43438
$466$	0	0
$467$	−28.3002	−1.30958	−0.654788	−	0.755812i	$-0.727242\pi$
$467$	−28.3002	−1.30958	−0.654788	+	0.755812i	$0.727242\pi$
$468$	0	0
$469$	−29.8617	−1.37889
$470$	0	0
$471$	6.93087	0.319358
$472$	0	0
$473$	7.12311	0.327521
$474$	0	0
$475$	30.7386	1.41039
$476$	0	0
$477$	6.87689	0.314871
$478$	0	0
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	38.7386	1.76633
$482$	0	0
$483$	11.8920	0.541107
$484$	0	0
$485$	−55.4233	−2.51664
$486$	0	0
$487$	14.9309	0.676582	0.338291	−	0.941041i	$-0.390151\pi$
$487$	14.9309	0.676582	0.338291	+	0.941041i	$0.390151\pi$
$488$	0	0
$489$	6.24621	0.282463
$490$	0	0
$491$	−13.7538	−0.620700	−0.310350	−	0.950622i	$-0.600446\pi$
$491$	−13.7538	−0.620700	−0.310350	+	0.950622i	$0.600446\pi$
$492$	0	0
$493$	10.2462	0.461466
$494$	0	0
$495$	2.00000	0.0898933
$496$	0	0
$497$	−27.1231	−1.21664
$498$	0	0
$499$	28.9848	1.29754	0.648770	−	0.760985i	$-0.275284\pi$
$499$	28.9848	1.29754	0.648770	+	0.760985i	$0.275284\pi$
$500$	0	0
$501$	12.4924	0.558120
$502$	0	0
$503$	−31.6155	−1.40967	−0.704833	−	0.709373i	$-0.748978\pi$
$503$	−31.6155	−1.40967	−0.704833	+	0.709373i	$0.748978\pi$
$504$	0	0
$505$	−7.12311	−0.316974
$506$	0	0
$507$	20.6847	0.918638
$508$	0	0
$509$	18.3002	0.811142	0.405571	−	0.914064i	$-0.367073\pi$
$509$	18.3002	0.811142	0.405571	+	0.914064i	$0.367073\pi$
$510$	0	0
$511$	16.0000	0.707798
$512$	0	0
$513$	−22.2462	−0.982194
$514$	0	0
$515$	0	0
$516$	0	0
$517$	8.00000	0.351840
$518$	0	0
$519$	−19.1231	−0.839411
$520$	0	0
$521$	1.31534	0.0576262	0.0288131	−	0.999585i	$-0.490827\pi$
$521$	1.31534	0.0576262	0.0288131	+	0.999585i	$0.490827\pi$
$522$	0	0
$523$	12.0000	0.524723	0.262362	−	0.964970i	$-0.415499\pi$
$523$	12.0000	0.524723	0.262362	+	0.964970i	$0.415499\pi$
$524$	0	0
$525$	37.4773	1.63564
$526$	0	0
$527$	−11.1231	−0.484530
$528$	0	0
$529$	−17.0540	−0.741477
$530$	0	0
$531$	4.38447	0.190270
$532$	0	0
$533$	−5.75379	−0.249224
$534$	0	0
$535$	−47.6155	−2.05860
$536$	0	0
$537$	10.0540	0.433861
$538$	0	0
$539$	2.75379	0.118614
$540$	0	0
$541$	23.8617	1.02590	0.512948	−	0.858420i	$-0.328553\pi$
$541$	23.8617	1.02590	0.512948	+	0.858420i	$0.328553\pi$
$542$	0	0
$543$	2.05398	0.0881445
$544$	0	0
$545$	43.6155	1.86828
$546$	0	0
$547$	42.2462	1.80632	0.903159	−	0.429307i	$-0.141242\pi$
$547$	42.2462	1.80632	0.903159	+	0.429307i	$0.141242\pi$
$548$	0	0
$549$	0.630683	0.0269169
$550$	0	0
$551$	20.4924	0.873007
$552$	0	0
$553$	−34.7386	−1.47724
$554$	0	0
$555$	−42.0540	−1.78509
$556$	0	0
$557$	3.75379	0.159053	0.0795266	−	0.996833i	$-0.474659\pi$
$557$	3.75379	0.159053	0.0795266	+	0.996833i	$0.474659\pi$
$558$	0	0
$559$	36.4924	1.54347
$560$	0	0
$561$	3.12311	0.131858
$562$	0	0
$563$	−24.4924	−1.03223	−0.516116	−	0.856519i	$-0.672623\pi$
$563$	−24.4924	−1.03223	−0.516116	+	0.856519i	$0.672623\pi$
$564$	0	0
$565$	1.56155	0.0656950
$566$	0	0
$567$	−21.8617	−0.918107
$568$	0	0
$569$	−26.8769	−1.12674	−0.563369	−	0.826205i	$-0.690495\pi$
$569$	−26.8769	−1.12674	−0.563369	+	0.826205i	$0.690495\pi$
$570$	0	0
$571$	−16.4924	−0.690186	−0.345093	−	0.938568i	$-0.612153\pi$
$571$	−16.4924	−0.690186	−0.345093	+	0.938568i	$0.612153\pi$
$572$	0	0
$573$	−16.3002	−0.680950
$574$	0	0
$575$	18.7386	0.781455
$576$	0	0
$577$	15.5616	0.647836	0.323918	−	0.946085i	$-0.395000\pi$
$577$	15.5616	0.647836	0.323918	+	0.946085i	$0.395000\pi$
$578$	0	0
$579$	−14.2462	−0.592052
$580$	0	0
$581$	−2.73863	−0.113618
$582$	0	0
$583$	−12.2462	−0.507186
$584$	0	0
$585$	10.2462	0.423629
$586$	0	0
$587$	24.4924	1.01091	0.505455	−	0.862853i	$-0.331325\pi$
$587$	24.4924	1.01091	0.505455	+	0.862853i	$0.331325\pi$
$588$	0	0
$589$	−22.2462	−0.916639
$590$	0	0
$591$	22.6307	0.930902
$592$	0	0
$593$	3.36932	0.138361	0.0691806	−	0.997604i	$-0.477962\pi$
$593$	3.36932	0.138361	0.0691806	+	0.997604i	$0.477962\pi$
$594$	0	0
$595$	−22.2462	−0.912006
$596$	0	0
$597$	−19.5076	−0.798392
$598$	0	0
$599$	−16.0000	−0.653742	−0.326871	−	0.945069i	$-0.605994\pi$
$599$	−16.0000	−0.653742	−0.326871	+	0.945069i	$0.605994\pi$
$600$	0	0
$601$	3.75379	0.153120	0.0765601	−	0.997065i	$-0.475606\pi$
$601$	3.75379	0.153120	0.0765601	+	0.997065i	$0.475606\pi$
$602$	0	0
$603$	5.36932	0.218655
$604$	0	0
$605$	−3.56155	−0.144798
$606$	0	0
$607$	−45.8617	−1.86147	−0.930735	−	0.365694i	$-0.880832\pi$
$607$	−45.8617	−1.86147	−0.930735	+	0.365694i	$0.880832\pi$
$608$	0	0
$609$	24.9848	1.01244
$610$	0	0
$611$	40.9848	1.65807
$612$	0	0
$613$	−11.8617	−0.479091	−0.239546	−	0.970885i	$-0.576998\pi$
$613$	−11.8617	−0.479091	−0.239546	+	0.970885i	$0.576998\pi$
$614$	0	0
$615$	6.24621	0.251872
$616$	0	0
$617$	−2.49242	−0.100341	−0.0501706	−	0.998741i	$-0.515976\pi$
$617$	−2.49242	−0.100341	−0.0501706	+	0.998741i	$0.515976\pi$
$618$	0	0
$619$	−18.9309	−0.760896	−0.380448	−	0.924802i	$-0.624230\pi$
$619$	−18.9309	−0.760896	−0.380448	+	0.924802i	$0.624230\pi$
$620$	0	0
$621$	−13.5616	−0.544206
$622$	0	0
$623$	8.38447	0.335917
$624$	0	0
$625$	−4.36932	−0.174773
$626$	0	0
$627$	6.24621	0.249450
$628$	0	0
$629$	15.1231	0.602998
$630$	0	0
$631$	−42.0540	−1.67414	−0.837071	−	0.547094i	$-0.815734\pi$
$631$	−42.0540	−1.67414	−0.837071	+	0.547094i	$0.815734\pi$
$632$	0	0
$633$	−13.2614	−0.527092
$634$	0	0
$635$	−22.2462	−0.882814
$636$	0	0
$637$	14.1080	0.558977
$638$	0	0
$639$	4.87689	0.192927
$640$	0	0
$641$	−46.3002	−1.82875	−0.914374	−	0.404871i	$-0.867316\pi$
$641$	−46.3002	−1.82875	−0.914374	+	0.404871i	$0.867316\pi$
$642$	0	0
$643$	9.17708	0.361909	0.180954	−	0.983491i	$-0.442081\pi$
$643$	9.17708	0.361909	0.180954	+	0.983491i	$0.442081\pi$
$644$	0	0
$645$	−39.6155	−1.55986
$646$	0	0
$647$	13.5616	0.533160	0.266580	−	0.963813i	$-0.414106\pi$
$647$	13.5616	0.533160	0.266580	+	0.963813i	$0.414106\pi$
$648$	0	0
$649$	−7.80776	−0.306482
$650$	0	0
$651$	−27.1231	−1.06304
$652$	0	0
$653$	−35.1771	−1.37659	−0.688293	−	0.725433i	$-0.741640\pi$
$653$	−35.1771	−1.37659	−0.688293	+	0.725433i	$0.741640\pi$
$654$	0	0
$655$	47.6155	1.86049
$656$	0	0
$657$	−2.87689	−0.112238
$658$	0	0
$659$	11.6155	0.452477	0.226238	−	0.974072i	$-0.427357\pi$
$659$	11.6155	0.452477	0.226238	+	0.974072i	$0.427357\pi$
$660$	0	0
$661$	−41.8078	−1.62613	−0.813067	−	0.582170i	$-0.802204\pi$
$661$	−41.8078	−1.62613	−0.813067	+	0.582170i	$0.802204\pi$
$662$	0	0
$663$	16.0000	0.621389
$664$	0	0
$665$	−44.4924	−1.72534
$666$	0	0
$667$	12.4924	0.483709
$668$	0	0
$669$	−18.4384	−0.712872
$670$	0	0
$671$	−1.12311	−0.0433570
$672$	0	0
$673$	33.2311	1.28096	0.640482	−	0.767974i	$-0.278735\pi$
$673$	33.2311	1.28096	0.640482	+	0.767974i	$0.278735\pi$
$674$	0	0
$675$	−42.7386	−1.64501
$676$	0	0
$677$	20.7386	0.797050	0.398525	−	0.917157i	$-0.369522\pi$
$677$	20.7386	0.797050	0.398525	+	0.917157i	$0.369522\pi$
$678$	0	0
$679$	48.6004	1.86511
$680$	0	0
$681$	−36.1080	−1.38366
$682$	0	0
$683$	−6.73863	−0.257847	−0.128923	−	0.991655i	$-0.541152\pi$
$683$	−6.73863	−0.257847	−0.128923	+	0.991655i	$0.541152\pi$
$684$	0	0
$685$	30.0540	1.14830
$686$	0	0
$687$	−22.9309	−0.874867
$688$	0	0
$689$	−62.7386	−2.39015
$690$	0	0
$691$	9.94602	0.378365	0.189182	−	0.981942i	$-0.439416\pi$
$691$	9.94602	0.378365	0.189182	+	0.981942i	$0.439416\pi$
$692$	0	0
$693$	−1.75379	−0.0666209
$694$	0	0
$695$	53.8617	2.04309
$696$	0	0
$697$	−2.24621	−0.0850813
$698$	0	0
$699$	−11.5076	−0.435257
$700$	0	0
$701$	−50.4924	−1.90707	−0.953536	−	0.301278i	$-0.902587\pi$
$701$	−50.4924	−1.90707	−0.953536	+	0.301278i	$0.902587\pi$
$702$	0	0
$703$	30.2462	1.14076
$704$	0	0
$705$	−44.4924	−1.67568
$706$	0	0
$707$	6.24621	0.234913
$708$	0	0
$709$	−2.19224	−0.0823311	−0.0411656	−	0.999152i	$-0.513107\pi$
$709$	−2.19224	−0.0823311	−0.0411656	+	0.999152i	$0.513107\pi$
$710$	0	0
$711$	6.24621	0.234251
$712$	0	0
$713$	−13.5616	−0.507884
$714$	0	0
$715$	−18.2462	−0.682370
$716$	0	0
$717$	−7.61553	−0.284407
$718$	0	0
$719$	35.4233	1.32107	0.660533	−	0.750797i	$-0.270330\pi$
$719$	35.4233	1.32107	0.660533	+	0.750797i	$0.270330\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	45.4773	1.69132
$724$	0	0
$725$	39.3693	1.46214
$726$	0	0
$727$	−23.3153	−0.864718	−0.432359	−	0.901702i	$-0.642319\pi$
$727$	−23.3153	−0.864718	−0.432359	+	0.901702i	$0.642319\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	14.2462	0.526915
$732$	0	0
$733$	−1.12311	−0.0414829	−0.0207414	−	0.999785i	$-0.506603\pi$
$733$	−1.12311	−0.0414829	−0.0207414	+	0.999785i	$0.506603\pi$
$734$	0	0
$735$	−15.3153	−0.564915
$736$	0	0
$737$	−9.56155	−0.352204
$738$	0	0
$739$	2.63068	0.0967712	0.0483856	−	0.998829i	$-0.484592\pi$
$739$	2.63068	0.0967712	0.0483856	+	0.998829i	$0.484592\pi$
$740$	0	0
$741$	32.0000	1.17555
$742$	0	0
$743$	10.7386	0.393962	0.196981	−	0.980407i	$-0.436886\pi$
$743$	10.7386	0.393962	0.196981	+	0.980407i	$0.436886\pi$
$744$	0	0
$745$	15.1231	0.554068
$746$	0	0
$747$	0.492423	0.0180168
$748$	0	0
$749$	41.7538	1.52565
$750$	0	0
$751$	5.56155	0.202944	0.101472	−	0.994838i	$-0.467645\pi$
$751$	5.56155	0.202944	0.101472	+	0.994838i	$0.467645\pi$
$752$	0	0
$753$	−2.43845	−0.0888620
$754$	0	0
$755$	33.3693	1.21443
$756$	0	0
$757$	−15.7538	−0.572581	−0.286291	−	0.958143i	$-0.592422\pi$
$757$	−15.7538	−0.572581	−0.286291	+	0.958143i	$0.592422\pi$
$758$	0	0
$759$	3.80776	0.138213
$760$	0	0
$761$	5.12311	0.185712	0.0928562	−	0.995680i	$-0.470400\pi$
$761$	5.12311	0.185712	0.0928562	+	0.995680i	$0.470400\pi$
$762$	0	0
$763$	−38.2462	−1.38461
$764$	0	0
$765$	4.00000	0.144620
$766$	0	0
$767$	−40.0000	−1.44432
$768$	0	0
$769$	25.6155	0.923720	0.461860	−	0.886953i	$-0.347182\pi$
$769$	25.6155	0.923720	0.461860	+	0.886953i	$0.347182\pi$
$770$	0	0
$771$	18.3542	0.661009
$772$	0	0
$773$	−40.7386	−1.46527	−0.732633	−	0.680623i	$-0.761709\pi$
$773$	−40.7386	−1.46527	−0.732633	+	0.680623i	$0.761709\pi$
$774$	0	0
$775$	−42.7386	−1.53522
$776$	0	0
$777$	36.8769	1.32295
$778$	0	0
$779$	−4.49242	−0.160958
$780$	0	0
$781$	−8.68466	−0.310762
$782$	0	0
$783$	−28.4924	−1.01824
$784$	0	0
$785$	−15.8078	−0.564203
$786$	0	0
$787$	29.7538	1.06061	0.530304	−	0.847808i	$-0.322078\pi$
$787$	29.7538	1.06061	0.530304	+	0.847808i	$0.322078\pi$
$788$	0	0
$789$	29.8617	1.06311
$790$	0	0
$791$	−1.36932	−0.0486873
$792$	0	0
$793$	−5.75379	−0.204323
$794$	0	0
$795$	68.1080	2.41554
$796$	0	0
$797$	14.1922	0.502715	0.251357	−	0.967894i	$-0.419123\pi$
$797$	14.1922	0.502715	0.251357	+	0.967894i	$0.419123\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	0	0
$801$	−1.50758	−0.0532676
$802$	0	0
$803$	5.12311	0.180790
$804$	0	0
$805$	−27.1231	−0.955964
$806$	0	0
$807$	32.3845	1.13999
$808$	0	0
$809$	−45.6155	−1.60376	−0.801878	−	0.597487i	$-0.796166\pi$
$809$	−45.6155	−1.60376	−0.801878	+	0.597487i	$0.796166\pi$
$810$	0	0
$811$	7.12311	0.250126	0.125063	−	0.992149i	$-0.460087\pi$
$811$	7.12311	0.250126	0.125063	+	0.992149i	$0.460087\pi$
$812$	0	0
$813$	−44.4924	−1.56042
$814$	0	0
$815$	−14.2462	−0.499023
$816$	0	0
$817$	28.4924	0.996824
$818$	0	0
$819$	−8.98485	−0.313956
$820$	0	0
$821$	42.9848	1.50018	0.750091	−	0.661335i	$-0.230010\pi$
$821$	42.9848	1.50018	0.750091	+	0.661335i	$0.230010\pi$
$822$	0	0
$823$	−54.5464	−1.90137	−0.950684	−	0.310161i	$-0.899617\pi$
$823$	−54.5464	−1.90137	−0.950684	+	0.310161i	$0.899617\pi$
$824$	0	0
$825$	12.0000	0.417786
$826$	0	0
$827$	−38.7386	−1.34707	−0.673537	−	0.739153i	$-0.735226\pi$
$827$	−38.7386	−1.34707	−0.673537	+	0.739153i	$0.735226\pi$
$828$	0	0
$829$	−15.0691	−0.523373	−0.261686	−	0.965153i	$-0.584279\pi$
$829$	−15.0691	−0.523373	−0.261686	+	0.965153i	$0.584279\pi$
$830$	0	0
$831$	28.1080	0.975054
$832$	0	0
$833$	5.50758	0.190826
$834$	0	0
$835$	−28.4924	−0.986021
$836$	0	0
$837$	30.9309	1.06913
$838$	0	0
$839$	−19.8078	−0.683840	−0.341920	−	0.939729i	$-0.611077\pi$
$839$	−19.8078	−0.683840	−0.341920	+	0.939729i	$0.611077\pi$
$840$	0	0
$841$	−2.75379	−0.0949582
$842$	0	0
$843$	25.3693	0.873766
$844$	0	0
$845$	−47.1771	−1.62294
$846$	0	0
$847$	3.12311	0.107311
$848$	0	0
$849$	−31.2311	−1.07185
$850$	0	0
$851$	18.4384	0.632062
$852$	0	0
$853$	46.4924	1.59187	0.795935	−	0.605382i	$-0.206980\pi$
$853$	46.4924	1.59187	0.795935	+	0.605382i	$0.206980\pi$
$854$	0	0
$855$	8.00000	0.273594
$856$	0	0
$857$	30.1080	1.02847	0.514234	−	0.857650i	$-0.328076\pi$
$857$	30.1080	1.02847	0.514234	+	0.857650i	$0.328076\pi$
$858$	0	0
$859$	−30.0540	−1.02543	−0.512714	−	0.858559i	$-0.671360\pi$
$859$	−30.0540	−1.02543	−0.512714	+	0.858559i	$0.671360\pi$
$860$	0	0
$861$	−5.47727	−0.186665
$862$	0	0
$863$	−36.4924	−1.24222	−0.621108	−	0.783725i	$-0.713317\pi$
$863$	−36.4924	−1.24222	−0.621108	+	0.783725i	$0.713317\pi$
$864$	0	0
$865$	43.6155	1.48297
$866$	0	0
$867$	−20.3002	−0.689430
$868$	0	0
$869$	−11.1231	−0.377326
$870$	0	0
$871$	−48.9848	−1.65979
$872$	0	0
$873$	−8.73863	−0.295758
$874$	0	0
$875$	−29.8617	−1.00951
$876$	0	0
$877$	−55.3693	−1.86969	−0.934844	−	0.355057i	$-0.884461\pi$
$877$	−55.3693	−1.86969	−0.934844	+	0.355057i	$0.884461\pi$
$878$	0	0
$879$	5.26137	0.177461
$880$	0	0
$881$	34.3002	1.15560	0.577801	−	0.816177i	$-0.303911\pi$
$881$	34.3002	1.15560	0.577801	+	0.816177i	$0.303911\pi$
$882$	0	0
$883$	−8.49242	−0.285793	−0.142896	−	0.989738i	$-0.545642\pi$
$883$	−8.49242	−0.285793	−0.142896	+	0.989738i	$0.545642\pi$
$884$	0	0
$885$	43.4233	1.45966
$886$	0	0
$887$	31.6155	1.06155	0.530773	−	0.847514i	$-0.321902\pi$
$887$	31.6155	1.06155	0.530773	+	0.847514i	$0.321902\pi$
$888$	0	0
$889$	19.5076	0.654263
$890$	0	0
$891$	−7.00000	−0.234509
$892$	0	0
$893$	32.0000	1.07084
$894$	0	0
$895$	−22.9309	−0.766494
$896$	0	0
$897$	19.5076	0.651339
$898$	0	0
$899$	−28.4924	−0.950275
$900$	0	0
$901$	−24.4924	−0.815961
$902$	0	0
$903$	34.7386	1.15603
$904$	0	0
$905$	−4.68466	−0.155723
$906$	0	0
$907$	−16.4924	−0.547622	−0.273811	−	0.961784i	$-0.588284\pi$
$907$	−16.4924	−0.547622	−0.273811	+	0.961784i	$0.588284\pi$
$908$	0	0
$909$	−1.12311	−0.0372511
$910$	0	0
$911$	−26.7386	−0.885890	−0.442945	−	0.896549i	$-0.646066\pi$
$911$	−26.7386	−0.885890	−0.442945	+	0.896549i	$0.646066\pi$
$912$	0	0
$913$	−0.876894	−0.0290210
$914$	0	0
$915$	6.24621	0.206493
$916$	0	0
$917$	−41.7538	−1.37883
$918$	0	0
$919$	6.63068	0.218726	0.109363	−	0.994002i	$-0.465119\pi$
$919$	6.63068	0.218726	0.109363	+	0.994002i	$0.465119\pi$
$920$	0	0
$921$	50.7386	1.67189
$922$	0	0
$923$	−44.4924	−1.46449
$924$	0	0
$925$	58.1080	1.91058
$926$	0	0
$927$	0	0
$928$	0	0
$929$	46.4924	1.52537	0.762683	−	0.646772i	$-0.223881\pi$
$929$	46.4924	1.52537	0.762683	+	0.646772i	$0.223881\pi$
$930$	0	0
$931$	11.0152	0.361007
$932$	0	0
$933$	15.2311	0.498642
$934$	0	0
$935$	−7.12311	−0.232950
$936$	0	0
$937$	−42.1080	−1.37561	−0.687803	−	0.725897i	$-0.741425\pi$
$937$	−42.1080	−1.37561	−0.687803	+	0.725897i	$0.741425\pi$
$938$	0	0
$939$	−15.3153	−0.499797
$940$	0	0
$941$	32.2462	1.05120	0.525598	−	0.850733i	$-0.323842\pi$
$941$	32.2462	1.05120	0.525598	+	0.850733i	$0.323842\pi$
$942$	0	0
$943$	−2.73863	−0.0891822
$944$	0	0
$945$	61.8617	2.01236
$946$	0	0
$947$	12.6847	0.412196	0.206098	−	0.978531i	$-0.433923\pi$
$947$	12.6847	0.412196	0.206098	+	0.978531i	$0.433923\pi$
$948$	0	0
$949$	26.2462	0.851988
$950$	0	0
$951$	22.1619	0.718650
$952$	0	0
$953$	0.246211	0.00797556	0.00398778	−	0.999992i	$-0.498731\pi$
$953$	0.246211	0.00797556	0.00398778	+	0.999992i	$0.498731\pi$
$954$	0	0
$955$	37.1771	1.20302
$956$	0	0
$957$	8.00000	0.258603
$958$	0	0
$959$	−26.3542	−0.851020
$960$	0	0
$961$	−0.0691303	−0.00223001
$962$	0	0
$963$	−7.50758	−0.241928
$964$	0	0
$965$	32.4924	1.04597
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	0	0
$969$	12.4924	0.401314
$970$	0	0
$971$	34.5464	1.10865	0.554323	−	0.832301i	$-0.312977\pi$
$971$	34.5464	1.10865	0.554323	+	0.832301i	$0.312977\pi$
$972$	0	0
$973$	−47.2311	−1.51416
$974$	0	0
$975$	61.4773	1.96885
$976$	0	0
$977$	53.8078	1.72146	0.860731	−	0.509059i	$-0.170007\pi$
$977$	53.8078	1.72146	0.860731	+	0.509059i	$0.170007\pi$
$978$	0	0
$979$	2.68466	0.0858021
$980$	0	0
$981$	6.87689	0.219562
$982$	0	0
$983$	30.9309	0.986542	0.493271	−	0.869876i	$-0.335801\pi$
$983$	30.9309	0.986542	0.493271	+	0.869876i	$0.335801\pi$
$984$	0	0
$985$	−51.6155	−1.64461
$986$	0	0
$987$	39.0152	1.24187
$988$	0	0
$989$	17.3693	0.552312
$990$	0	0
$991$	−4.49242	−0.142707	−0.0713533	−	0.997451i	$-0.522732\pi$
$991$	−4.49242	−0.142707	−0.0713533	+	0.997451i	$0.522732\pi$
$992$	0	0
$993$	−54.5464	−1.73098
$994$	0	0
$995$	44.4924	1.41050
$996$	0	0
$997$	−52.2462	−1.65465	−0.827327	−	0.561721i	$-0.810140\pi$
$997$	−52.2462	−1.65465	−0.827327	+	0.561721i	$0.810140\pi$
$998$	0	0
$999$	−42.0540	−1.33053

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	704.2.a.m.1.2		2
3.2	odd	2		6336.2.a.cu.1.2		2
4.3	odd	2		704.2.a.p.1.1		2
8.3	odd	2		176.2.a.d.1.2		2
8.5	even	2		88.2.a.b.1.1	✓	2
11.10	odd	2		7744.2.a.by.1.2		2
12.11	even	2		6336.2.a.cx.1.2		2
16.3	odd	4		2816.2.c.p.1409.2		4
16.5	even	4		2816.2.c.w.1409.2		4
16.11	odd	4		2816.2.c.p.1409.3		4
16.13	even	4		2816.2.c.w.1409.3		4
24.5	odd	2		792.2.a.h.1.1		2
24.11	even	2		1584.2.a.t.1.1		2
40.3	even	4		4400.2.b.v.4049.3		4
40.13	odd	4		2200.2.b.g.1849.2		4
40.19	odd	2		4400.2.a.bp.1.1		2
40.27	even	4		4400.2.b.v.4049.2		4
40.29	even	2		2200.2.a.o.1.2		2
40.37	odd	4		2200.2.b.g.1849.3		4
44.43	even	2		7744.2.a.cl.1.1		2
56.13	odd	2		4312.2.a.n.1.2		2
56.27	even	2		8624.2.a.cb.1.1		2
88.5	even	10		968.2.i.r.729.1		8
88.13	odd	10		968.2.i.q.81.1		8
88.21	odd	2		968.2.a.j.1.1		2
88.29	odd	10		968.2.i.q.753.2		8
88.37	even	10		968.2.i.r.753.2		8
88.43	even	2		1936.2.a.r.1.2		2
88.53	even	10		968.2.i.r.81.1		8
88.61	odd	10		968.2.i.q.729.1		8
88.69	even	10		968.2.i.r.9.2		8
88.85	odd	10		968.2.i.q.9.2		8
264.197	even	2		8712.2.a.bb.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
88.2.a.b.1.1	✓	2	8.5	even	2
176.2.a.d.1.2		2	8.3	odd	2
704.2.a.m.1.2		2	1.1	even	1	trivial
704.2.a.p.1.1		2	4.3	odd	2
792.2.a.h.1.1		2	24.5	odd	2
968.2.a.j.1.1		2	88.21	odd	2
968.2.i.q.9.2		8	88.85	odd	10
968.2.i.q.81.1		8	88.13	odd	10
968.2.i.q.729.1		8	88.61	odd	10
968.2.i.q.753.2		8	88.29	odd	10
968.2.i.r.9.2		8	88.69	even	10
968.2.i.r.81.1		8	88.53	even	10
968.2.i.r.729.1		8	88.5	even	10
968.2.i.r.753.2		8	88.37	even	10
1584.2.a.t.1.1		2	24.11	even	2
1936.2.a.r.1.2		2	88.43	even	2
2200.2.a.o.1.2		2	40.29	even	2
2200.2.b.g.1849.2		4	40.13	odd	4
2200.2.b.g.1849.3		4	40.37	odd	4
2816.2.c.p.1409.2		4	16.3	odd	4
2816.2.c.p.1409.3		4	16.11	odd	4
2816.2.c.w.1409.2		4	16.5	even	4
2816.2.c.w.1409.3		4	16.13	even	4
4312.2.a.n.1.2		2	56.13	odd	2
4400.2.a.bp.1.1		2	40.19	odd	2
4400.2.b.v.4049.2		4	40.27	even	4
4400.2.b.v.4049.3		4	40.3	even	4
6336.2.a.cu.1.2		2	3.2	odd	2
6336.2.a.cx.1.2		2	12.11	even	2
7744.2.a.by.1.2		2	11.10	odd	2
7744.2.a.cl.1.1		2	44.43	even	2
8624.2.a.cb.1.1		2	56.27	even	2
8712.2.a.bb.1.1		2	264.197	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 704 = 2^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	704.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56155 q^{3} -3.56155 q^{5} +3.12311 q^{7} -0.561553 q^{9} +O(q^{10})\)	\(q+1.56155 q^{3} -3.56155 q^{5} +3.12311 q^{7} -0.561553 q^{9} +1.00000 q^{11} +5.12311 q^{13} -5.56155 q^{15} +2.00000 q^{17} +4.00000 q^{19} +4.87689 q^{21} +2.43845 q^{23} +7.68466 q^{25} -5.56155 q^{27} +5.12311 q^{29} -5.56155 q^{31} +1.56155 q^{33} -11.1231 q^{35} +7.56155 q^{37} +8.00000 q^{39} -1.12311 q^{41} +7.12311 q^{43} +2.00000 q^{45} +8.00000 q^{47} +2.75379 q^{49} +3.12311 q^{51} -12.2462 q^{53} -3.56155 q^{55} +6.24621 q^{57} -7.80776 q^{59} -1.12311 q^{61} -1.75379 q^{63} -18.2462 q^{65} -9.56155 q^{67} +3.80776 q^{69} -8.68466 q^{71} +5.12311 q^{73} +12.0000 q^{75} +3.12311 q^{77} -11.1231 q^{79} -7.00000 q^{81} -0.876894 q^{83} -7.12311 q^{85} +8.00000 q^{87} +2.68466 q^{89} +16.0000 q^{91} -8.68466 q^{93} -14.2462 q^{95} +15.5616 q^{97} -0.561553 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q - q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9	\( 2 q - q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} + 2 q^{11} + 2 q^{13} - 7 q^{15} + 4 q^{17} + 8 q^{19} + 18 q^{21} + 9 q^{23} + 3 q^{25} - 7 q^{27} + 2 q^{29} - 7 q^{31} - q^{33} - 14 q^{35} + 11 q^{37} + 16 q^{39} + 6 q^{41} + 6 q^{43} + 4 q^{45} + 16 q^{47} + 22 q^{49} - 2 q^{51} - 8 q^{53} - 3 q^{55} - 4 q^{57} + 5 q^{59} + 6 q^{61} - 20 q^{63} - 20 q^{65} - 15 q^{67} - 13 q^{69} - 5 q^{71} + 2 q^{73} + 24 q^{75} - 2 q^{77} - 14 q^{79} - 14 q^{81} - 10 q^{83} - 6 q^{85} + 16 q^{87} - 7 q^{89} + 32 q^{91} - 5 q^{93} - 12 q^{95} + 27 q^{97} + 3 q^{99}+O(q^{100}) \) 2 * q - q^3 - 3 * q^5 - 2 * q^7 + 3 * q^9 + 2 * q^11 + 2 * q^13 - 7 * q^15 + 4 * q^17 + 8 * q^19 + 18 * q^21 + 9 * q^23 + 3 * q^25 - 7 * q^27 + 2 * q^29 - 7 * q^31 - q^33 - 14 * q^35 + 11 * q^37 + 16 * q^39 + 6 * q^41 + 6 * q^43 + 4 * q^45 + 16 * q^47 + 22 * q^49 - 2 * q^51 - 8 * q^53 - 3 * q^55 - 4 * q^57 + 5 * q^59 + 6 * q^61 - 20 * q^63 - 20 * q^65 - 15 * q^67 - 13 * q^69 - 5 * q^71 + 2 * q^73 + 24 * q^75 - 2 * q^77 - 14 * q^79 - 14 * q^81 - 10 * q^83 - 6 * q^85 + 16 * q^87 - 7 * q^89 + 32 * q^91 - 5 * q^93 - 12 * q^95 + 27 * q^97 + 3 * q^99

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