LMFDB - Embedded newform 5408.2.a.bf.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5408,2,Mod(1,5408)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5408, 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("5408.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5408 = 2^{5} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5408.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:	$43.1830974131$
Analytic rank:	$0$
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 416)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	5408.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421 q^{3} -2.82843 q^{5} +4.41421 q^{7} +2.82843 q^{9} +O(q^{10})$	$q+2.41421 q^{3} -2.82843 q^{5} +4.41421 q^{7} +2.82843 q^{9} +3.24264 q^{11} -6.82843 q^{15} -5.82843 q^{17} -1.24264 q^{19} +10.6569 q^{21} -1.24264 q^{23} +3.00000 q^{25} -0.414214 q^{27} +8.65685 q^{29} +5.65685 q^{31} +7.82843 q^{33} -12.4853 q^{35} +7.48528 q^{37} +5.82843 q^{41} -4.07107 q^{43} -8.00000 q^{45} +6.00000 q^{47} +12.4853 q^{49} -14.0711 q^{51} -2.82843 q^{53} -9.17157 q^{55} -3.00000 q^{57} -1.24264 q^{59} +7.00000 q^{61} +12.4853 q^{63} +13.2426 q^{67} -3.00000 q^{69} -7.24264 q^{71} -12.4853 q^{73} +7.24264 q^{75} +14.3137 q^{77} +6.00000 q^{79} -9.48528 q^{81} +4.00000 q^{83} +16.4853 q^{85} +20.8995 q^{87} +3.34315 q^{89} +13.6569 q^{93} +3.51472 q^{95} +9.00000 q^{97} +9.17157 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 6 q^{7}+O(q^{10}) $ 2 * q + 2 * q^3 + 6 * q^7	$ 2 q + 2 q^{3} + 6 q^{7} - 2 q^{11} - 8 q^{15} - 6 q^{17} + 6 q^{19} + 10 q^{21} + 6 q^{23} + 6 q^{25} + 2 q^{27} + 6 q^{29} + 10 q^{33} - 8 q^{35} - 2 q^{37} + 6 q^{41} + 6 q^{43} - 16 q^{45} + 12 q^{47} + 8 q^{49} - 14 q^{51} - 24 q^{55} - 6 q^{57} + 6 q^{59} + 14 q^{61} + 8 q^{63} + 18 q^{67} - 6 q^{69} - 6 q^{71} - 8 q^{73} + 6 q^{75} + 6 q^{77} + 12 q^{79} - 2 q^{81} + 8 q^{83} + 16 q^{85} + 22 q^{87} + 18 q^{89} + 16 q^{93} + 24 q^{95} + 18 q^{97} + 24 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 6 * q^7 - 2 * q^11 - 8 * q^15 - 6 * q^17 + 6 * q^19 + 10 * q^21 + 6 * q^23 + 6 * q^25 + 2 * q^27 + 6 * q^29 + 10 * q^33 - 8 * q^35 - 2 * q^37 + 6 * q^41 + 6 * q^43 - 16 * q^45 + 12 * q^47 + 8 * q^49 - 14 * q^51 - 24 * q^55 - 6 * q^57 + 6 * q^59 + 14 * q^61 + 8 * q^63 + 18 * q^67 - 6 * q^69 - 6 * q^71 - 8 * q^73 + 6 * q^75 + 6 * q^77 + 12 * q^79 - 2 * q^81 + 8 * q^83 + 16 * q^85 + 22 * q^87 + 18 * q^89 + 16 * q^93 + 24 * q^95 + 18 * q^97 + 24 * 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
$3$	2.41421	1.39385	0.696923	−	0.717146i	$-0.254552\pi$
$3$	2.41421	1.39385	0.696923	+	0.717146i	$0.254552\pi$
$4$	0	0
$5$	−2.82843	−1.26491	−0.632456	−	0.774597i	$-0.717953\pi$
$5$	−2.82843	−1.26491	−0.632456	+	0.774597i	$0.717953\pi$
$6$	0	0
$7$	4.41421	1.66842	0.834208	−	0.551450i	$-0.185925\pi$
$7$	4.41421	1.66842	0.834208	+	0.551450i	$0.185925\pi$
$8$	0	0
$9$	2.82843	0.942809
$10$	0	0
$11$	3.24264	0.977693	0.488846	−	0.872370i	$-0.337418\pi$
$11$	3.24264	0.977693	0.488846	+	0.872370i	$0.337418\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−6.82843	−1.76309
$16$	0	0
$17$	−5.82843	−1.41360	−0.706801	−	0.707413i	$-0.749862\pi$
$17$	−5.82843	−1.41360	−0.706801	+	0.707413i	$0.749862\pi$
$18$	0	0
$19$	−1.24264	−0.285081	−0.142541	−	0.989789i	$-0.545527\pi$
$19$	−1.24264	−0.285081	−0.142541	+	0.989789i	$0.545527\pi$
$20$	0	0
$21$	10.6569	2.32552
$22$	0	0
$23$	−1.24264	−0.259108	−0.129554	−	0.991572i	$-0.541355\pi$
$23$	−1.24264	−0.259108	−0.129554	+	0.991572i	$0.541355\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	−0.414214	−0.0797154
$28$	0	0
$29$	8.65685	1.60754	0.803769	−	0.594942i	$-0.202825\pi$
$29$	8.65685	1.60754	0.803769	+	0.594942i	$0.202825\pi$
$30$	0	0
$31$	5.65685	1.01600	0.508001	−	0.861357i	$-0.330385\pi$
$31$	5.65685	1.01600	0.508001	+	0.861357i	$0.330385\pi$
$32$	0	0
$33$	7.82843	1.36275
$34$	0	0
$35$	−12.4853	−2.11040
$36$	0	0
$37$	7.48528	1.23057	0.615286	−	0.788304i	$-0.289040\pi$
$37$	7.48528	1.23057	0.615286	+	0.788304i	$0.289040\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.82843	0.910247	0.455124	−	0.890428i	$-0.349595\pi$
$41$	5.82843	0.910247	0.455124	+	0.890428i	$0.349595\pi$
$42$	0	0
$43$	−4.07107	−0.620832	−0.310416	−	0.950601i	$-0.600468\pi$
$43$	−4.07107	−0.620832	−0.310416	+	0.950601i	$0.600468\pi$
$44$	0	0
$45$	−8.00000	−1.19257
$46$	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
$49$	12.4853	1.78361
$50$	0	0
$51$	−14.0711	−1.97034
$52$	0	0
$53$	−2.82843	−0.388514	−0.194257	−	0.980951i	$-0.562230\pi$
$53$	−2.82843	−0.388514	−0.194257	+	0.980951i	$0.562230\pi$
$54$	0	0
$55$	−9.17157	−1.23669
$56$	0	0
$57$	−3.00000	−0.397360
$58$	0	0
$59$	−1.24264	−0.161778	−0.0808890	−	0.996723i	$-0.525776\pi$
$59$	−1.24264	−0.161778	−0.0808890	+	0.996723i	$0.525776\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	0	0
$63$	12.4853	1.57300
$64$	0	0
$65$	0	0
$66$	0	0
$67$	13.2426	1.61785	0.808923	−	0.587915i	$-0.200051\pi$
$67$	13.2426	1.61785	0.808923	+	0.587915i	$0.200051\pi$
$68$	0	0
$69$	−3.00000	−0.361158
$70$	0	0
$71$	−7.24264	−0.859543	−0.429772	−	0.902938i	$-0.641406\pi$
$71$	−7.24264	−0.859543	−0.429772	+	0.902938i	$0.641406\pi$
$72$	0	0
$73$	−12.4853	−1.46129	−0.730646	−	0.682757i	$-0.760781\pi$
$73$	−12.4853	−1.46129	−0.730646	+	0.682757i	$0.760781\pi$
$74$	0	0
$75$	7.24264	0.836308
$76$	0	0
$77$	14.3137	1.63120
$78$	0	0
$79$	6.00000	0.675053	0.337526	−	0.941316i	$-0.390410\pi$
$79$	6.00000	0.675053	0.337526	+	0.941316i	$0.390410\pi$
$80$	0	0
$81$	−9.48528	−1.05392
$82$	0	0
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	16.4853	1.78808
$86$	0	0
$87$	20.8995	2.24066
$88$	0	0
$89$	3.34315	0.354373	0.177186	−	0.984177i	$-0.443300\pi$
$89$	3.34315	0.354373	0.177186	+	0.984177i	$0.443300\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	13.6569	1.41615
$94$	0	0
$95$	3.51472	0.360603
$96$	0	0
$97$	9.00000	0.913812	0.456906	−	0.889515i	$-0.348958\pi$
$97$	9.00000	0.913812	0.456906	+	0.889515i	$0.348958\pi$
$98$	0	0
$99$	9.17157	0.921778
$100$	0	0
$101$	11.8284	1.17697	0.588486	−	0.808507i	$-0.299724\pi$
$101$	11.8284	1.17697	0.588486	+	0.808507i	$0.299724\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	−30.1421	−2.94157
$106$	0	0
$107$	−19.2426	−1.86026	−0.930128	−	0.367235i	$-0.880304\pi$
$107$	−19.2426	−1.86026	−0.930128	+	0.367235i	$0.880304\pi$
$108$	0	0
$109$	8.48528	0.812743	0.406371	−	0.913708i	$-0.366794\pi$
$109$	8.48528	0.812743	0.406371	+	0.913708i	$0.366794\pi$
$110$	0	0
$111$	18.0711	1.71523
$112$	0	0
$113$	−0.171573	−0.0161402	−0.00807011	−	0.999967i	$-0.502569\pi$
$113$	−0.171573	−0.0161402	−0.00807011	+	0.999967i	$0.502569\pi$
$114$	0	0
$115$	3.51472	0.327749
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−25.7279	−2.35847
$120$	0	0
$121$	−0.485281	−0.0441165
$122$	0	0
$123$	14.0711	1.26875
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	−1.58579	−0.140716	−0.0703579	−	0.997522i	$-0.522414\pi$
$127$	−1.58579	−0.140716	−0.0703579	+	0.997522i	$0.522414\pi$
$128$	0	0
$129$	−9.82843	−0.865345
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	−5.48528	−0.475634
$134$	0	0
$135$	1.17157	0.100833
$136$	0	0
$137$	−9.34315	−0.798239	−0.399119	−	0.916899i	$-0.630684\pi$
$137$	−9.34315	−0.798239	−0.399119	+	0.916899i	$0.630684\pi$
$138$	0	0
$139$	1.24264	0.105399	0.0526997	−	0.998610i	$-0.483217\pi$
$139$	1.24264	0.105399	0.0526997	+	0.998610i	$0.483217\pi$
$140$	0	0
$141$	14.4853	1.21988
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−24.4853	−2.03339
$146$	0	0
$147$	30.1421	2.48608
$148$	0	0
$149$	3.34315	0.273881	0.136941	−	0.990579i	$-0.456273\pi$
$149$	3.34315	0.273881	0.136941	+	0.990579i	$0.456273\pi$
$150$	0	0
$151$	−16.9706	−1.38104	−0.690522	−	0.723311i	$-0.742619\pi$
$151$	−16.9706	−1.38104	−0.690522	+	0.723311i	$0.742619\pi$
$152$	0	0
$153$	−16.4853	−1.33276
$154$	0	0
$155$	−16.0000	−1.28515
$156$	0	0
$157$	−0.485281	−0.0387297	−0.0193648	−	0.999812i	$-0.506164\pi$
$157$	−0.485281	−0.0387297	−0.0193648	+	0.999812i	$0.506164\pi$
$158$	0	0
$159$	−6.82843	−0.541529
$160$	0	0
$161$	−5.48528	−0.432301
$162$	0	0
$163$	16.7574	1.31254	0.656269	−	0.754527i	$-0.272134\pi$
$163$	16.7574	1.31254	0.656269	+	0.754527i	$0.272134\pi$
$164$	0	0
$165$	−22.1421	−1.72376
$166$	0	0
$167$	−15.7279	−1.21706	−0.608532	−	0.793530i	$-0.708241\pi$
$167$	−15.7279	−1.21706	−0.608532	+	0.793530i	$0.708241\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−3.51472	−0.268777
$172$	0	0
$173$	−9.00000	−0.684257	−0.342129	−	0.939653i	$-0.611148\pi$
$173$	−9.00000	−0.684257	−0.342129	+	0.939653i	$0.611148\pi$
$174$	0	0
$175$	13.2426	1.00105
$176$	0	0
$177$	−3.00000	−0.225494
$178$	0	0
$179$	12.7574	0.953530	0.476765	−	0.879031i	$-0.341809\pi$
$179$	12.7574	0.953530	0.476765	+	0.879031i	$0.341809\pi$
$180$	0	0
$181$	−0.485281	−0.0360707	−0.0180353	−	0.999837i	$-0.505741\pi$
$181$	−0.485281	−0.0360707	−0.0180353	+	0.999837i	$0.505741\pi$
$182$	0	0
$183$	16.8995	1.24925
$184$	0	0
$185$	−21.1716	−1.55656
$186$	0	0
$187$	−18.8995	−1.38207
$188$	0	0
$189$	−1.82843	−0.132999
$190$	0	0
$191$	−6.75736	−0.488945	−0.244473	−	0.969656i	$-0.578615\pi$
$191$	−6.75736	−0.488945	−0.244473	+	0.969656i	$0.578615\pi$
$192$	0	0
$193$	19.4853	1.40258	0.701291	−	0.712875i	$-0.252608\pi$
$193$	19.4853	1.40258	0.701291	+	0.712875i	$0.252608\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−3.00000	−0.213741	−0.106871	−	0.994273i	$-0.534083\pi$
$197$	−3.00000	−0.213741	−0.106871	+	0.994273i	$0.534083\pi$
$198$	0	0
$199$	24.2132	1.71643	0.858214	−	0.513292i	$-0.171574\pi$
$199$	24.2132	1.71643	0.858214	+	0.513292i	$0.171574\pi$
$200$	0	0
$201$	31.9706	2.25503
$202$	0	0
$203$	38.2132	2.68204
$204$	0	0
$205$	−16.4853	−1.15138
$206$	0	0
$207$	−3.51472	−0.244290
$208$	0	0
$209$	−4.02944	−0.278722
$210$	0	0
$211$	−13.5858	−0.935284	−0.467642	−	0.883918i	$-0.654896\pi$
$211$	−13.5858	−0.935284	−0.467642	+	0.883918i	$0.654896\pi$
$212$	0	0
$213$	−17.4853	−1.19807
$214$	0	0
$215$	11.5147	0.785297
$216$	0	0
$217$	24.9706	1.69511
$218$	0	0
$219$	−30.1421	−2.03682
$220$	0	0
$221$	0	0
$222$	0	0
$223$	6.89949	0.462024	0.231012	−	0.972951i	$-0.425796\pi$
$223$	6.89949	0.462024	0.231012	+	0.972951i	$0.425796\pi$
$224$	0	0
$225$	8.48528	0.565685
$226$	0	0
$227$	15.7279	1.04390	0.521949	−	0.852976i	$-0.325205\pi$
$227$	15.7279	1.04390	0.521949	+	0.852976i	$0.325205\pi$
$228$	0	0
$229$	−8.48528	−0.560723	−0.280362	−	0.959894i	$-0.590454\pi$
$229$	−8.48528	−0.560723	−0.280362	+	0.959894i	$0.590454\pi$
$230$	0	0
$231$	34.5563	2.27364
$232$	0	0
$233$	−20.4853	−1.34204	−0.671018	−	0.741441i	$-0.734143\pi$
$233$	−20.4853	−1.34204	−0.671018	+	0.741441i	$0.734143\pi$
$234$	0	0
$235$	−16.9706	−1.10704
$236$	0	0
$237$	14.4853	0.940920
$238$	0	0
$239$	−8.97056	−0.580257	−0.290129	−	0.956988i	$-0.593698\pi$
$239$	−8.97056	−0.580257	−0.290129	+	0.956988i	$0.593698\pi$
$240$	0	0
$241$	−30.4558	−1.96183	−0.980917	−	0.194429i	$-0.937715\pi$
$241$	−30.4558	−1.96183	−0.980917	+	0.194429i	$0.937715\pi$
$242$	0	0
$243$	−21.6569	−1.38929
$244$	0	0
$245$	−35.3137	−2.25611
$246$	0	0
$247$	0	0
$248$	0	0
$249$	9.65685	0.611978
$250$	0	0
$251$	−17.7279	−1.11898	−0.559488	−	0.828838i	$-0.689002\pi$
$251$	−17.7279	−1.11898	−0.559488	+	0.828838i	$0.689002\pi$
$252$	0	0
$253$	−4.02944	−0.253329
$254$	0	0
$255$	39.7990	2.49231
$256$	0	0
$257$	16.7990	1.04789	0.523946	−	0.851751i	$-0.324459\pi$
$257$	16.7990	1.04789	0.523946	+	0.851751i	$0.324459\pi$
$258$	0	0
$259$	33.0416	2.05311
$260$	0	0
$261$	24.4853	1.51560
$262$	0	0
$263$	−9.24264	−0.569926	−0.284963	−	0.958539i	$-0.591981\pi$
$263$	−9.24264	−0.569926	−0.284963	+	0.958539i	$0.591981\pi$
$264$	0	0
$265$	8.00000	0.491436
$266$	0	0
$267$	8.07107	0.493941
$268$	0	0
$269$	5.48528	0.334444	0.167222	−	0.985919i	$-0.446520\pi$
$269$	5.48528	0.334444	0.167222	+	0.985919i	$0.446520\pi$
$270$	0	0
$271$	25.2426	1.53338	0.766691	−	0.642017i	$-0.221902\pi$
$271$	25.2426	1.53338	0.766691	+	0.642017i	$0.221902\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	9.72792	0.586616
$276$	0	0
$277$	13.4853	0.810252	0.405126	−	0.914261i	$-0.367228\pi$
$277$	13.4853	0.810252	0.405126	+	0.914261i	$0.367228\pi$
$278$	0	0
$279$	16.0000	0.957895
$280$	0	0
$281$	2.14214	0.127789	0.0638945	−	0.997957i	$-0.479648\pi$
$281$	2.14214	0.127789	0.0638945	+	0.997957i	$0.479648\pi$
$282$	0	0
$283$	10.4142	0.619061	0.309530	−	0.950890i	$-0.399828\pi$
$283$	10.4142	0.619061	0.309530	+	0.950890i	$0.399828\pi$
$284$	0	0
$285$	8.48528	0.502625
$286$	0	0
$287$	25.7279	1.51867
$288$	0	0
$289$	16.9706	0.998268
$290$	0	0
$291$	21.7279	1.27371
$292$	0	0
$293$	−2.31371	−0.135168	−0.0675841	−	0.997714i	$-0.521529\pi$
$293$	−2.31371	−0.135168	−0.0675841	+	0.997714i	$0.521529\pi$
$294$	0	0
$295$	3.51472	0.204635
$296$	0	0
$297$	−1.34315	−0.0779372
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−17.9706	−1.03581
$302$	0	0
$303$	28.5563	1.64052
$304$	0	0
$305$	−19.7990	−1.13369
$306$	0	0
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−24.0000	−1.36092	−0.680458	−	0.732787i	$-0.738219\pi$
$311$	−24.0000	−1.36092	−0.680458	+	0.732787i	$0.738219\pi$
$312$	0	0
$313$	20.4853	1.15790	0.578948	−	0.815364i	$-0.303463\pi$
$313$	20.4853	1.15790	0.578948	+	0.815364i	$0.303463\pi$
$314$	0	0
$315$	−35.3137	−1.98970
$316$	0	0
$317$	17.3137	0.972435	0.486217	−	0.873838i	$-0.338376\pi$
$317$	17.3137	0.972435	0.486217	+	0.873838i	$0.338376\pi$
$318$	0	0
$319$	28.0711	1.57168
$320$	0	0
$321$	−46.4558	−2.59291
$322$	0	0
$323$	7.24264	0.402991
$324$	0	0
$325$	0	0
$326$	0	0
$327$	20.4853	1.13284
$328$	0	0
$329$	26.4853	1.46018
$330$	0	0
$331$	−7.24264	−0.398092	−0.199046	−	0.979990i	$-0.563784\pi$
$331$	−7.24264	−0.398092	−0.199046	+	0.979990i	$0.563784\pi$
$332$	0	0
$333$	21.1716	1.16020
$334$	0	0
$335$	−37.4558	−2.04643
$336$	0	0
$337$	4.48528	0.244329	0.122164	−	0.992510i	$-0.461016\pi$
$337$	4.48528	0.244329	0.122164	+	0.992510i	$0.461016\pi$
$338$	0	0
$339$	−0.414214	−0.0224970
$340$	0	0
$341$	18.3431	0.993337
$342$	0	0
$343$	24.2132	1.30739
$344$	0	0
$345$	8.48528	0.456832
$346$	0	0
$347$	31.2426	1.67719	0.838596	−	0.544753i	$-0.183377\pi$
$347$	31.2426	1.67719	0.838596	+	0.544753i	$0.183377\pi$
$348$	0	0
$349$	−34.4558	−1.84438	−0.922190	−	0.386738i	$-0.873602\pi$
$349$	−34.4558	−1.84438	−0.922190	+	0.386738i	$0.873602\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.6569	0.780106	0.390053	−	0.920792i	$-0.372457\pi$
$353$	14.6569	0.780106	0.390053	+	0.920792i	$0.372457\pi$
$354$	0	0
$355$	20.4853	1.08725
$356$	0	0
$357$	−62.1127	−3.28735
$358$	0	0
$359$	−8.00000	−0.422224	−0.211112	−	0.977462i	$-0.567708\pi$
$359$	−8.00000	−0.422224	−0.211112	+	0.977462i	$0.567708\pi$
$360$	0	0
$361$	−17.4558	−0.918729
$362$	0	0
$363$	−1.17157	−0.0614916
$364$	0	0
$365$	35.3137	1.84840
$366$	0	0
$367$	−8.27208	−0.431799	−0.215899	−	0.976416i	$-0.569268\pi$
$367$	−8.27208	−0.431799	−0.215899	+	0.976416i	$0.569268\pi$
$368$	0	0
$369$	16.4853	0.858189
$370$	0	0
$371$	−12.4853	−0.648204
$372$	0	0
$373$	37.4853	1.94091	0.970457	−	0.241274i	$-0.0775651\pi$
$373$	37.4853	1.94091	0.970457	+	0.241274i	$0.0775651\pi$
$374$	0	0
$375$	13.6569	0.705237
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−27.3848	−1.40666	−0.703331	−	0.710863i	$-0.748305\pi$
$379$	−27.3848	−1.40666	−0.703331	+	0.710863i	$0.748305\pi$
$380$	0	0
$381$	−3.82843	−0.196136
$382$	0	0
$383$	15.7279	0.803659	0.401830	−	0.915714i	$-0.368374\pi$
$383$	15.7279	0.803659	0.401830	+	0.915714i	$0.368374\pi$
$384$	0	0
$385$	−40.4853	−2.06332
$386$	0	0
$387$	−11.5147	−0.585326
$388$	0	0
$389$	−16.6274	−0.843044	−0.421522	−	0.906818i	$-0.638504\pi$
$389$	−16.6274	−0.843044	−0.421522	+	0.906818i	$0.638504\pi$
$390$	0	0
$391$	7.24264	0.366276
$392$	0	0
$393$	9.65685	0.487124
$394$	0	0
$395$	−16.9706	−0.853882
$396$	0	0
$397$	11.4853	0.576430	0.288215	−	0.957566i	$-0.406938\pi$
$397$	11.4853	0.576430	0.288215	+	0.957566i	$0.406938\pi$
$398$	0	0
$399$	−13.2426	−0.662961
$400$	0	0
$401$	−11.1421	−0.556412	−0.278206	−	0.960521i	$-0.589740\pi$
$401$	−11.1421	−0.556412	−0.278206	+	0.960521i	$0.589740\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	26.8284	1.33312
$406$	0	0
$407$	24.2721	1.20312
$408$	0	0
$409$	10.5147	0.519919	0.259960	−	0.965619i	$-0.416291\pi$
$409$	10.5147	0.519919	0.259960	+	0.965619i	$0.416291\pi$
$410$	0	0
$411$	−22.5563	−1.11262
$412$	0	0
$413$	−5.48528	−0.269913
$414$	0	0
$415$	−11.3137	−0.555368
$416$	0	0
$417$	3.00000	0.146911
$418$	0	0
$419$	11.7279	0.572946	0.286473	−	0.958088i	$-0.407517\pi$
$419$	11.7279	0.572946	0.286473	+	0.958088i	$0.407517\pi$
$420$	0	0
$421$	−0.485281	−0.0236512	−0.0118256	−	0.999930i	$-0.503764\pi$
$421$	−0.485281	−0.0236512	−0.0118256	+	0.999930i	$0.503764\pi$
$422$	0	0
$423$	16.9706	0.825137
$424$	0	0
$425$	−17.4853	−0.848161
$426$	0	0
$427$	30.8995	1.49533
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−34.6985	−1.67137	−0.835684	−	0.549211i	$-0.814928\pi$
$431$	−34.6985	−1.67137	−0.835684	+	0.549211i	$0.814928\pi$
$432$	0	0
$433$	−2.51472	−0.120850	−0.0604248	−	0.998173i	$-0.519246\pi$
$433$	−2.51472	−0.120850	−0.0604248	+	0.998173i	$0.519246\pi$
$434$	0	0
$435$	−59.1127	−2.83424
$436$	0	0
$437$	1.54416	0.0738670
$438$	0	0
$439$	28.0711	1.33976	0.669879	−	0.742470i	$-0.266346\pi$
$439$	28.0711	1.33976	0.669879	+	0.742470i	$0.266346\pi$
$440$	0	0
$441$	35.3137	1.68161
$442$	0	0
$443$	28.9706	1.37643	0.688216	−	0.725505i	$-0.258394\pi$
$443$	28.9706	1.37643	0.688216	+	0.725505i	$0.258394\pi$
$444$	0	0
$445$	−9.45584	−0.448250
$446$	0	0
$447$	8.07107	0.381748
$448$	0	0
$449$	−12.5147	−0.590606	−0.295303	−	0.955404i	$-0.595421\pi$
$449$	−12.5147	−0.590606	−0.295303	+	0.955404i	$0.595421\pi$
$450$	0	0
$451$	18.8995	0.889942
$452$	0	0
$453$	−40.9706	−1.92496
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−31.0000	−1.45012	−0.725059	−	0.688686i	$-0.758188\pi$
$457$	−31.0000	−1.45012	−0.725059	+	0.688686i	$0.758188\pi$
$458$	0	0
$459$	2.41421	0.112686
$460$	0	0
$461$	−1.20101	−0.0559366	−0.0279683	−	0.999609i	$-0.508904\pi$
$461$	−1.20101	−0.0559366	−0.0279683	+	0.999609i	$0.508904\pi$
$462$	0	0
$463$	18.3431	0.852478	0.426239	−	0.904611i	$-0.359838\pi$
$463$	18.3431	0.852478	0.426239	+	0.904611i	$0.359838\pi$
$464$	0	0
$465$	−38.6274	−1.79130
$466$	0	0
$467$	−6.97056	−0.322559	−0.161280	−	0.986909i	$-0.551562\pi$
$467$	−6.97056	−0.322559	−0.161280	+	0.986909i	$0.551562\pi$
$468$	0	0
$469$	58.4558	2.69924
$470$	0	0
$471$	−1.17157	−0.0539832
$472$	0	0
$473$	−13.2010	−0.606983
$474$	0	0
$475$	−3.72792	−0.171049
$476$	0	0
$477$	−8.00000	−0.366295
$478$	0	0
$479$	−8.75736	−0.400134	−0.200067	−	0.979782i	$-0.564116\pi$
$479$	−8.75736	−0.400134	−0.200067	+	0.979782i	$0.564116\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−13.2426	−0.602561
$484$	0	0
$485$	−25.4558	−1.15589
$486$	0	0
$487$	−12.8995	−0.584532	−0.292266	−	0.956337i	$-0.594409\pi$
$487$	−12.8995	−0.584532	−0.292266	+	0.956337i	$0.594409\pi$
$488$	0	0
$489$	40.4558	1.82948
$490$	0	0
$491$	−22.2132	−1.00247	−0.501234	−	0.865312i	$-0.667120\pi$
$491$	−22.2132	−1.00247	−0.501234	+	0.865312i	$0.667120\pi$
$492$	0	0
$493$	−50.4558	−2.27242
$494$	0	0
$495$	−25.9411	−1.16597
$496$	0	0
$497$	−31.9706	−1.43408
$498$	0	0
$499$	−17.6569	−0.790429	−0.395215	−	0.918589i	$-0.629330\pi$
$499$	−17.6569	−0.790429	−0.395215	+	0.918589i	$0.629330\pi$
$500$	0	0
$501$	−37.9706	−1.69640
$502$	0	0
$503$	−14.7574	−0.657998	−0.328999	−	0.944330i	$-0.606711\pi$
$503$	−14.7574	−0.657998	−0.328999	+	0.944330i	$0.606711\pi$
$504$	0	0
$505$	−33.4558	−1.48877
$506$	0	0
$507$	0	0
$508$	0	0
$509$	5.14214	0.227921	0.113961	−	0.993485i	$-0.463646\pi$
$509$	5.14214	0.227921	0.113961	+	0.993485i	$0.463646\pi$
$510$	0	0
$511$	−55.1127	−2.43804
$512$	0	0
$513$	0.514719	0.0227254
$514$	0	0
$515$	0	0
$516$	0	0
$517$	19.4558	0.855667
$518$	0	0
$519$	−21.7279	−0.953750
$520$	0	0
$521$	−9.17157	−0.401814	−0.200907	−	0.979610i	$-0.564389\pi$
$521$	−9.17157	−0.401814	−0.200907	+	0.979610i	$0.564389\pi$
$522$	0	0
$523$	16.4142	0.717743	0.358872	−	0.933387i	$-0.383162\pi$
$523$	16.4142	0.717743	0.358872	+	0.933387i	$0.383162\pi$
$524$	0	0
$525$	31.9706	1.39531
$526$	0	0
$527$	−32.9706	−1.43622
$528$	0	0
$529$	−21.4558	−0.932863
$530$	0	0
$531$	−3.51472	−0.152526
$532$	0	0
$533$	0	0
$534$	0	0
$535$	54.4264	2.35306
$536$	0	0
$537$	30.7990	1.32907
$538$	0	0
$539$	40.4853	1.74382
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	0	0
$543$	−1.17157	−0.0502770
$544$	0	0
$545$	−24.0000	−1.02805
$546$	0	0
$547$	−36.0000	−1.53925	−0.769624	−	0.638497i	$-0.779557\pi$
$547$	−36.0000	−1.53925	−0.769624	+	0.638497i	$0.779557\pi$
$548$	0	0
$549$	19.7990	0.845000
$550$	0	0
$551$	−10.7574	−0.458279
$552$	0	0
$553$	26.4853	1.12627
$554$	0	0
$555$	−51.1127	−2.16961
$556$	0	0
$557$	−8.65685	−0.366803	−0.183401	−	0.983038i	$-0.558711\pi$
$557$	−8.65685	−0.366803	−0.183401	+	0.983038i	$0.558711\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−45.6274	−1.92639
$562$	0	0
$563$	−32.6985	−1.37808	−0.689038	−	0.724725i	$-0.741967\pi$
$563$	−32.6985	−1.37808	−0.689038	+	0.724725i	$0.741967\pi$
$564$	0	0
$565$	0.485281	0.0204159
$566$	0	0
$567$	−41.8701	−1.75838
$568$	0	0
$569$	−25.9706	−1.08874	−0.544371	−	0.838844i	$-0.683232\pi$
$569$	−25.9706	−1.08874	−0.544371	+	0.838844i	$0.683232\pi$
$570$	0	0
$571$	−40.2843	−1.68584	−0.842922	−	0.538036i	$-0.819167\pi$
$571$	−40.2843	−1.68584	−0.842922	+	0.538036i	$0.819167\pi$
$572$	0	0
$573$	−16.3137	−0.681515
$574$	0	0
$575$	−3.72792	−0.155465
$576$	0	0
$577$	−4.48528	−0.186725	−0.0933624	−	0.995632i	$-0.529762\pi$
$577$	−4.48528	−0.186725	−0.0933624	+	0.995632i	$0.529762\pi$
$578$	0	0
$579$	47.0416	1.95498
$580$	0	0
$581$	17.6569	0.732530
$582$	0	0
$583$	−9.17157	−0.379848
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−20.6985	−0.854318	−0.427159	−	0.904177i	$-0.640486\pi$
$587$	−20.6985	−0.854318	−0.427159	+	0.904177i	$0.640486\pi$
$588$	0	0
$589$	−7.02944	−0.289643
$590$	0	0
$591$	−7.24264	−0.297922
$592$	0	0
$593$	−14.8284	−0.608931	−0.304465	−	0.952523i	$-0.598478\pi$
$593$	−14.8284	−0.608931	−0.304465	+	0.952523i	$0.598478\pi$
$594$	0	0
$595$	72.7696	2.98326
$596$	0	0
$597$	58.4558	2.39244
$598$	0	0
$599$	−47.9411	−1.95882	−0.979411	−	0.201878i	$-0.935295\pi$
$599$	−47.9411	−1.95882	−0.979411	+	0.201878i	$0.935295\pi$
$600$	0	0
$601$	−10.5147	−0.428904	−0.214452	−	0.976734i	$-0.568797\pi$
$601$	−10.5147	−0.428904	−0.214452	+	0.976734i	$0.568797\pi$
$602$	0	0
$603$	37.4558	1.52532
$604$	0	0
$605$	1.37258	0.0558034
$606$	0	0
$607$	1.58579	0.0643651	0.0321825	−	0.999482i	$-0.489754\pi$
$607$	1.58579	0.0643651	0.0321825	+	0.999482i	$0.489754\pi$
$608$	0	0
$609$	92.2548	3.73835
$610$	0	0
$611$	0	0
$612$	0	0
$613$	14.5147	0.586244	0.293122	−	0.956075i	$-0.405306\pi$
$613$	14.5147	0.586244	0.293122	+	0.956075i	$0.405306\pi$
$614$	0	0
$615$	−39.7990	−1.60485
$616$	0	0
$617$	44.3137	1.78400	0.892001	−	0.452033i	$-0.149301\pi$
$617$	44.3137	1.78400	0.892001	+	0.452033i	$0.149301\pi$
$618$	0	0
$619$	−4.97056	−0.199784	−0.0998919	−	0.994998i	$-0.531850\pi$
$619$	−4.97056	−0.199784	−0.0998919	+	0.994998i	$0.531850\pi$
$620$	0	0
$621$	0.514719	0.0206549
$622$	0	0
$623$	14.7574	0.591241
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	−9.72792	−0.388496
$628$	0	0
$629$	−43.6274	−1.73954
$630$	0	0
$631$	−7.24264	−0.288325	−0.144162	−	0.989554i	$-0.546049\pi$
$631$	−7.24264	−0.288325	−0.144162	+	0.989554i	$0.546049\pi$
$632$	0	0
$633$	−32.7990	−1.30364
$634$	0	0
$635$	4.48528	0.177993
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−20.4853	−0.810385
$640$	0	0
$641$	−22.7990	−0.900506	−0.450253	−	0.892901i	$-0.648666\pi$
$641$	−22.7990	−0.900506	−0.450253	+	0.892901i	$0.648666\pi$
$642$	0	0
$643$	−0.213203	−0.00840792	−0.00420396	−	0.999991i	$-0.501338\pi$
$643$	−0.213203	−0.00840792	−0.00420396	+	0.999991i	$0.501338\pi$
$644$	0	0
$645$	27.7990	1.09458
$646$	0	0
$647$	−23.2426	−0.913762	−0.456881	−	0.889528i	$-0.651034\pi$
$647$	−23.2426	−0.913762	−0.456881	+	0.889528i	$0.651034\pi$
$648$	0	0
$649$	−4.02944	−0.158169
$650$	0	0
$651$	60.2843	2.36273
$652$	0	0
$653$	23.1421	0.905622	0.452811	−	0.891607i	$-0.350421\pi$
$653$	23.1421	0.905622	0.452811	+	0.891607i	$0.350421\pi$
$654$	0	0
$655$	−11.3137	−0.442063
$656$	0	0
$657$	−35.3137	−1.37772
$658$	0	0
$659$	24.2132	0.943212	0.471606	−	0.881809i	$-0.343674\pi$
$659$	24.2132	0.943212	0.471606	+	0.881809i	$0.343674\pi$
$660$	0	0
$661$	24.4558	0.951222	0.475611	−	0.879656i	$-0.342227\pi$
$661$	24.4558	0.951222	0.475611	+	0.879656i	$0.342227\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	15.5147	0.601635
$666$	0	0
$667$	−10.7574	−0.416527
$668$	0	0
$669$	16.6569	0.643991
$670$	0	0
$671$	22.6985	0.876265
$672$	0	0
$673$	−8.02944	−0.309512	−0.154756	−	0.987953i	$-0.549459\pi$
$673$	−8.02944	−0.309512	−0.154756	+	0.987953i	$0.549459\pi$
$674$	0	0
$675$	−1.24264	−0.0478293
$676$	0	0
$677$	−25.4558	−0.978348	−0.489174	−	0.872186i	$-0.662702\pi$
$677$	−25.4558	−0.978348	−0.489174	+	0.872186i	$0.662702\pi$
$678$	0	0
$679$	39.7279	1.52462
$680$	0	0
$681$	37.9706	1.45504
$682$	0	0
$683$	−9.24264	−0.353660	−0.176830	−	0.984241i	$-0.556584\pi$
$683$	−9.24264	−0.353660	−0.176830	+	0.984241i	$0.556584\pi$
$684$	0	0
$685$	26.4264	1.00970
$686$	0	0
$687$	−20.4853	−0.781562
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−16.0711	−0.611372	−0.305686	−	0.952132i	$-0.598886\pi$
$691$	−16.0711	−0.611372	−0.305686	+	0.952132i	$0.598886\pi$
$692$	0	0
$693$	40.4853	1.53791
$694$	0	0
$695$	−3.51472	−0.133321
$696$	0	0
$697$	−33.9706	−1.28673
$698$	0	0
$699$	−49.4558	−1.87059
$700$	0	0
$701$	−8.48528	−0.320485	−0.160242	−	0.987078i	$-0.551228\pi$
$701$	−8.48528	−0.320485	−0.160242	+	0.987078i	$0.551228\pi$
$702$	0	0
$703$	−9.30152	−0.350813
$704$	0	0
$705$	−40.9706	−1.54304
$706$	0	0
$707$	52.2132	1.96368
$708$	0	0
$709$	19.4853	0.731785	0.365893	−	0.930657i	$-0.380764\pi$
$709$	19.4853	0.731785	0.365893	+	0.930657i	$0.380764\pi$
$710$	0	0
$711$	16.9706	0.636446
$712$	0	0
$713$	−7.02944	−0.263254
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−21.6569	−0.808790
$718$	0	0
$719$	−22.7574	−0.848706	−0.424353	−	0.905497i	$-0.639498\pi$
$719$	−22.7574	−0.848706	−0.424353	+	0.905497i	$0.639498\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−73.5269	−2.73450
$724$	0	0
$725$	25.9706	0.964522
$726$	0	0
$727$	35.3137	1.30971	0.654856	−	0.755753i	$-0.272729\pi$
$727$	35.3137	1.30971	0.654856	+	0.755753i	$0.272729\pi$
$728$	0	0
$729$	−23.8284	−0.882534
$730$	0	0
$731$	23.7279	0.877609
$732$	0	0
$733$	−43.9411	−1.62300	−0.811501	−	0.584351i	$-0.801349\pi$
$733$	−43.9411	−1.62300	−0.811501	+	0.584351i	$0.801349\pi$
$734$	0	0
$735$	−85.2548	−3.14467
$736$	0	0
$737$	42.9411	1.58176
$738$	0	0
$739$	30.2132	1.11141	0.555705	−	0.831379i	$-0.312448\pi$
$739$	30.2132	1.11141	0.555705	+	0.831379i	$0.312448\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	7.24264	0.265707	0.132853	−	0.991136i	$-0.457586\pi$
$743$	7.24264	0.265707	0.132853	+	0.991136i	$0.457586\pi$
$744$	0	0
$745$	−9.45584	−0.346435
$746$	0	0
$747$	11.3137	0.413947
$748$	0	0
$749$	−84.9411	−3.10368
$750$	0	0
$751$	17.1005	0.624006	0.312003	−	0.950081i	$-0.399000\pi$
$751$	17.1005	0.624006	0.312003	+	0.950081i	$0.399000\pi$
$752$	0	0
$753$	−42.7990	−1.55968
$754$	0	0
$755$	48.0000	1.74690
$756$	0	0
$757$	−23.4853	−0.853587	−0.426794	−	0.904349i	$-0.640357\pi$
$757$	−23.4853	−0.853587	−0.426794	+	0.904349i	$0.640357\pi$
$758$	0	0
$759$	−9.72792	−0.353101
$760$	0	0
$761$	−29.4853	−1.06884	−0.534420	−	0.845219i	$-0.679470\pi$
$761$	−29.4853	−1.06884	−0.534420	+	0.845219i	$0.679470\pi$
$762$	0	0
$763$	37.4558	1.35599
$764$	0	0
$765$	46.6274	1.68582
$766$	0	0
$767$	0	0
$768$	0	0
$769$	45.4264	1.63812	0.819059	−	0.573709i	$-0.194496\pi$
$769$	45.4264	1.63812	0.819059	+	0.573709i	$0.194496\pi$
$770$	0	0
$771$	40.5563	1.46060
$772$	0	0
$773$	11.4853	0.413097	0.206548	−	0.978436i	$-0.433777\pi$
$773$	11.4853	0.413097	0.206548	+	0.978436i	$0.433777\pi$
$774$	0	0
$775$	16.9706	0.609601
$776$	0	0
$777$	79.7696	2.86172
$778$	0	0
$779$	−7.24264	−0.259495
$780$	0	0
$781$	−23.4853	−0.840369
$782$	0	0
$783$	−3.58579	−0.128146
$784$	0	0
$785$	1.37258	0.0489896
$786$	0	0
$787$	38.0122	1.35499	0.677494	−	0.735528i	$-0.263066\pi$
$787$	38.0122	1.35499	0.677494	+	0.735528i	$0.263066\pi$
$788$	0	0
$789$	−22.3137	−0.794389
$790$	0	0
$791$	−0.757359	−0.0269286
$792$	0	0
$793$	0	0
$794$	0	0
$795$	19.3137	0.684987
$796$	0	0
$797$	−33.0000	−1.16892	−0.584460	−	0.811423i	$-0.698694\pi$
$797$	−33.0000	−1.16892	−0.584460	+	0.811423i	$0.698694\pi$
$798$	0	0
$799$	−34.9706	−1.23717
$800$	0	0
$801$	9.45584	0.334106
$802$	0	0
$803$	−40.4853	−1.42869
$804$	0	0
$805$	15.5147	0.546822
$806$	0	0
$807$	13.2426	0.466163
$808$	0	0
$809$	−5.82843	−0.204917	−0.102458	−	0.994737i	$-0.532671\pi$
$809$	−5.82843	−0.204917	−0.102458	+	0.994737i	$0.532671\pi$
$810$	0	0
$811$	−12.3431	−0.433426	−0.216713	−	0.976235i	$-0.569534\pi$
$811$	−12.3431	−0.433426	−0.216713	+	0.976235i	$0.569534\pi$
$812$	0	0
$813$	60.9411	2.13730
$814$	0	0
$815$	−47.3970	−1.66024
$816$	0	0
$817$	5.05887	0.176988
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−19.9706	−0.696977	−0.348489	−	0.937313i	$-0.613305\pi$
$821$	−19.9706	−0.696977	−0.348489	+	0.937313i	$0.613305\pi$
$822$	0	0
$823$	0.213203	0.00743180	0.00371590	−	0.999993i	$-0.498817\pi$
$823$	0.213203	0.00743180	0.00371590	+	0.999993i	$0.498817\pi$
$824$	0	0
$825$	23.4853	0.817653
$826$	0	0
$827$	21.9411	0.762968	0.381484	−	0.924376i	$-0.375413\pi$
$827$	21.9411	0.762968	0.381484	+	0.924376i	$0.375413\pi$
$828$	0	0
$829$	35.9706	1.24931	0.624655	−	0.780901i	$-0.285240\pi$
$829$	35.9706	1.24931	0.624655	+	0.780901i	$0.285240\pi$
$830$	0	0
$831$	32.5563	1.12937
$832$	0	0
$833$	−72.7696	−2.52132
$834$	0	0
$835$	44.4853	1.53948
$836$	0	0
$837$	−2.34315	−0.0809910
$838$	0	0
$839$	10.6985	0.369353	0.184676	−	0.982799i	$-0.440876\pi$
$839$	10.6985	0.369353	0.184676	+	0.982799i	$0.440876\pi$
$840$	0	0
$841$	45.9411	1.58418
$842$	0	0
$843$	5.17157	0.178118
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−2.14214	−0.0736047
$848$	0	0
$849$	25.1421	0.862876
$850$	0	0
$851$	−9.30152	−0.318852
$852$	0	0
$853$	41.4558	1.41942	0.709711	−	0.704493i	$-0.248826\pi$
$853$	41.4558	1.41942	0.709711	+	0.704493i	$0.248826\pi$
$854$	0	0
$855$	9.94113	0.339979
$856$	0	0
$857$	33.1716	1.13312	0.566560	−	0.824021i	$-0.308274\pi$
$857$	33.1716	1.13312	0.566560	+	0.824021i	$0.308274\pi$
$858$	0	0
$859$	−36.0000	−1.22830	−0.614152	−	0.789188i	$-0.710502\pi$
$859$	−36.0000	−1.22830	−0.614152	+	0.789188i	$0.710502\pi$
$860$	0	0
$861$	62.1127	2.11680
$862$	0	0
$863$	−32.0000	−1.08929	−0.544646	−	0.838666i	$-0.683336\pi$
$863$	−32.0000	−1.08929	−0.544646	+	0.838666i	$0.683336\pi$
$864$	0	0
$865$	25.4558	0.865525
$866$	0	0
$867$	40.9706	1.39143
$868$	0	0
$869$	19.4558	0.659994
$870$	0	0
$871$	0	0
$872$	0	0
$873$	25.4558	0.861550
$874$	0	0
$875$	24.9706	0.844159
$876$	0	0
$877$	−23.9706	−0.809428	−0.404714	−	0.914443i	$-0.632629\pi$
$877$	−23.9706	−0.809428	−0.404714	+	0.914443i	$0.632629\pi$
$878$	0	0
$879$	−5.58579	−0.188404
$880$	0	0
$881$	18.1716	0.612216	0.306108	−	0.951997i	$-0.400973\pi$
$881$	18.1716	0.612216	0.306108	+	0.951997i	$0.400973\pi$
$882$	0	0
$883$	27.5980	0.928746	0.464373	−	0.885640i	$-0.346280\pi$
$883$	27.5980	0.928746	0.464373	+	0.885640i	$0.346280\pi$
$884$	0	0
$885$	8.48528	0.285230
$886$	0	0
$887$	−24.2132	−0.813000	−0.406500	−	0.913651i	$-0.633251\pi$
$887$	−24.2132	−0.813000	−0.406500	+	0.913651i	$0.633251\pi$
$888$	0	0
$889$	−7.00000	−0.234772
$890$	0	0
$891$	−30.7574	−1.03041
$892$	0	0
$893$	−7.45584	−0.249500
$894$	0	0
$895$	−36.0833	−1.20613
$896$	0	0
$897$	0	0
$898$	0	0
$899$	48.9706	1.63326
$900$	0	0
$901$	16.4853	0.549204
$902$	0	0
$903$	−43.3848	−1.44375
$904$	0	0
$905$	1.37258	0.0456262
$906$	0	0
$907$	−36.2132	−1.20244	−0.601220	−	0.799084i	$-0.705318\pi$
$907$	−36.2132	−1.20244	−0.601220	+	0.799084i	$0.705318\pi$
$908$	0	0
$909$	33.4558	1.10966
$910$	0	0
$911$	−24.0000	−0.795155	−0.397578	−	0.917568i	$-0.630149\pi$
$911$	−24.0000	−0.795155	−0.397578	+	0.917568i	$0.630149\pi$
$912$	0	0
$913$	12.9706	0.429263
$914$	0	0
$915$	−47.7990	−1.58019
$916$	0	0
$917$	17.6569	0.583081
$918$	0	0
$919$	−38.3553	−1.26523	−0.632613	−	0.774468i	$-0.718018\pi$
$919$	−38.3553	−1.26523	−0.632613	+	0.774468i	$0.718018\pi$
$920$	0	0
$921$	28.9706	0.954612
$922$	0	0
$923$	0	0
$924$	0	0
$925$	22.4558	0.738344
$926$	0	0
$927$	0	0
$928$	0	0
$929$	10.1127	0.331787	0.165893	−	0.986144i	$-0.446949\pi$
$929$	10.1127	0.331787	0.165893	+	0.986144i	$0.446949\pi$
$930$	0	0
$931$	−15.5147	−0.508474
$932$	0	0
$933$	−57.9411	−1.89691
$934$	0	0
$935$	53.4558	1.74819
$936$	0	0
$937$	−45.4558	−1.48498	−0.742489	−	0.669858i	$-0.766355\pi$
$937$	−45.4558	−1.48498	−0.742489	+	0.669858i	$0.766355\pi$
$938$	0	0
$939$	49.4558	1.61393
$940$	0	0
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	0	0
$943$	−7.24264	−0.235853
$944$	0	0
$945$	5.17157	0.168231
$946$	0	0
$947$	3.24264	0.105372	0.0526858	−	0.998611i	$-0.483222\pi$
$947$	3.24264	0.105372	0.0526858	+	0.998611i	$0.483222\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	41.7990	1.35543
$952$	0	0
$953$	−3.34315	−0.108295	−0.0541476	−	0.998533i	$-0.517244\pi$
$953$	−3.34315	−0.108295	−0.0541476	+	0.998533i	$0.517244\pi$
$954$	0	0
$955$	19.1127	0.618472
$956$	0	0
$957$	67.7696	2.19068
$958$	0	0
$959$	−41.2426	−1.33179
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	−54.4264	−1.75387
$964$	0	0
$965$	−55.1127	−1.77414
$966$	0	0
$967$	7.02944	0.226051	0.113026	−	0.993592i	$-0.463946\pi$
$967$	7.02944	0.226051	0.113026	+	0.993592i	$0.463946\pi$
$968$	0	0
$969$	17.4853	0.561708
$970$	0	0
$971$	−32.7574	−1.05123	−0.525617	−	0.850721i	$-0.676165\pi$
$971$	−32.7574	−1.05123	−0.525617	+	0.850721i	$0.676165\pi$
$972$	0	0
$973$	5.48528	0.175850
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−43.2843	−1.38479	−0.692393	−	0.721520i	$-0.743444\pi$
$977$	−43.2843	−1.38479	−0.692393	+	0.721520i	$0.743444\pi$
$978$	0	0
$979$	10.8406	0.346468
$980$	0	0
$981$	24.0000	0.766261
$982$	0	0
$983$	0.970563	0.0309561	0.0154781	−	0.999880i	$-0.495073\pi$
$983$	0.970563	0.0309561	0.0154781	+	0.999880i	$0.495073\pi$
$984$	0	0
$985$	8.48528	0.270364
$986$	0	0
$987$	63.9411	2.03527
$988$	0	0
$989$	5.05887	0.160863
$990$	0	0
$991$	−58.1543	−1.84733	−0.923667	−	0.383197i	$-0.874823\pi$
$991$	−58.1543	−1.84733	−0.923667	+	0.383197i	$0.874823\pi$
$992$	0	0
$993$	−17.4853	−0.554879
$994$	0	0
$995$	−68.4853	−2.17113
$996$	0	0
$997$	−34.9411	−1.10660	−0.553298	−	0.832983i	$-0.686631\pi$
$997$	−34.9411	−1.10660	−0.553298	+	0.832983i	$0.686631\pi$
$998$	0	0
$999$	−3.10051	−0.0980956

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5408.2.a.bf.1.2		2
4.3	odd	2		5408.2.a.n.1.1		2
13.4	even	6		416.2.i.c.289.1	✓	4
13.10	even	6		416.2.i.c.321.1	yes	4
13.12	even	2		5408.2.a.be.1.2		2
52.23	odd	6		416.2.i.f.321.2	yes	4
52.43	odd	6		416.2.i.f.289.2	yes	4
52.51	odd	2		5408.2.a.o.1.1		2
104.43	odd	6		832.2.i.k.705.1		4
104.69	even	6		832.2.i.p.705.2		4
104.75	odd	6		832.2.i.k.321.1		4
104.101	even	6		832.2.i.p.321.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.i.c.289.1	✓	4	13.4	even	6
416.2.i.c.321.1	yes	4	13.10	even	6
416.2.i.f.289.2	yes	4	52.43	odd	6
416.2.i.f.321.2	yes	4	52.23	odd	6
832.2.i.k.321.1		4	104.75	odd	6
832.2.i.k.705.1		4	104.43	odd	6
832.2.i.p.321.2		4	104.101	even	6
832.2.i.p.705.2		4	104.69	even	6
5408.2.a.n.1.1		2	4.3	odd	2
5408.2.a.o.1.1		2	52.51	odd	2
5408.2.a.be.1.2		2	13.12	even	2
5408.2.a.bf.1.2		2	1.1	even	1	trivial

Overview	Random
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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:	\( N \)	\(=\)	\( 5408 = 2^{5} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5408.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41421 q^{3} -2.82843 q^{5} +4.41421 q^{7} +2.82843 q^{9} +O(q^{10})\)	\(q+2.41421 q^{3} -2.82843 q^{5} +4.41421 q^{7} +2.82843 q^{9} +3.24264 q^{11} -6.82843 q^{15} -5.82843 q^{17} -1.24264 q^{19} +10.6569 q^{21} -1.24264 q^{23} +3.00000 q^{25} -0.414214 q^{27} +8.65685 q^{29} +5.65685 q^{31} +7.82843 q^{33} -12.4853 q^{35} +7.48528 q^{37} +5.82843 q^{41} -4.07107 q^{43} -8.00000 q^{45} +6.00000 q^{47} +12.4853 q^{49} -14.0711 q^{51} -2.82843 q^{53} -9.17157 q^{55} -3.00000 q^{57} -1.24264 q^{59} +7.00000 q^{61} +12.4853 q^{63} +13.2426 q^{67} -3.00000 q^{69} -7.24264 q^{71} -12.4853 q^{73} +7.24264 q^{75} +14.3137 q^{77} +6.00000 q^{79} -9.48528 q^{81} +4.00000 q^{83} +16.4853 q^{85} +20.8995 q^{87} +3.34315 q^{89} +13.6569 q^{93} +3.51472 q^{95} +9.00000 q^{97} +9.17157 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 6 q^{7}+O(q^{10}) \) 2 * q + 2 * q^3 + 6 * q^7	\( 2 q + 2 q^{3} + 6 q^{7} - 2 q^{11} - 8 q^{15} - 6 q^{17} + 6 q^{19} + 10 q^{21} + 6 q^{23} + 6 q^{25} + 2 q^{27} + 6 q^{29} + 10 q^{33} - 8 q^{35} - 2 q^{37} + 6 q^{41} + 6 q^{43} - 16 q^{45} + 12 q^{47} + 8 q^{49} - 14 q^{51} - 24 q^{55} - 6 q^{57} + 6 q^{59} + 14 q^{61} + 8 q^{63} + 18 q^{67} - 6 q^{69} - 6 q^{71} - 8 q^{73} + 6 q^{75} + 6 q^{77} + 12 q^{79} - 2 q^{81} + 8 q^{83} + 16 q^{85} + 22 q^{87} + 18 q^{89} + 16 q^{93} + 24 q^{95} + 18 q^{97} + 24 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 6 * q^7 - 2 * q^11 - 8 * q^15 - 6 * q^17 + 6 * q^19 + 10 * q^21 + 6 * q^23 + 6 * q^25 + 2 * q^27 + 6 * q^29 + 10 * q^33 - 8 * q^35 - 2 * q^37 + 6 * q^41 + 6 * q^43 - 16 * q^45 + 12 * q^47 + 8 * q^49 - 14 * q^51 - 24 * q^55 - 6 * q^57 + 6 * q^59 + 14 * q^61 + 8 * q^63 + 18 * q^67 - 6 * q^69 - 6 * q^71 - 8 * q^73 + 6 * q^75 + 6 * q^77 + 12 * q^79 - 2 * q^81 + 8 * q^83 + 16 * q^85 + 22 * q^87 + 18 * q^89 + 16 * q^93 + 24 * q^95 + 18 * q^97 + 24 * q^99

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