LMFDB - Embedded newform 5408.2.a.bf.1.1

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			1.1
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)

$f(q)$	$=$	$q-0.414214 q^{3} +2.82843 q^{5} +1.58579 q^{7} -2.82843 q^{9} +O(q^{10})$	$q-0.414214 q^{3} +2.82843 q^{5} +1.58579 q^{7} -2.82843 q^{9} -5.24264 q^{11} -1.17157 q^{15} -0.171573 q^{17} +7.24264 q^{19} -0.656854 q^{21} +7.24264 q^{23} +3.00000 q^{25} +2.41421 q^{27} -2.65685 q^{29} -5.65685 q^{31} +2.17157 q^{33} +4.48528 q^{35} -9.48528 q^{37} +0.171573 q^{41} +10.0711 q^{43} -8.00000 q^{45} +6.00000 q^{47} -4.48528 q^{49} +0.0710678 q^{51} +2.82843 q^{53} -14.8284 q^{55} -3.00000 q^{57} +7.24264 q^{59} +7.00000 q^{61} -4.48528 q^{63} +4.75736 q^{67} -3.00000 q^{69} +1.24264 q^{71} +4.48528 q^{73} -1.24264 q^{75} -8.31371 q^{77} +6.00000 q^{79} +7.48528 q^{81} +4.00000 q^{83} -0.485281 q^{85} +1.10051 q^{87} +14.6569 q^{89} +2.34315 q^{93} +20.4853 q^{95} +9.00000 q^{97} +14.8284 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.414214	−0.239146	−0.119573	−	0.992825i	$-0.538153\pi$
$3$	−0.414214	−0.239146	−0.119573	+	0.992825i	$0.538153\pi$
$4$	0	0
$5$	2.82843	1.26491	0.632456	−	0.774597i	$-0.282047\pi$
$5$	2.82843	1.26491	0.632456	+	0.774597i	$0.282047\pi$
$6$	0	0
$7$	1.58579	0.599371	0.299685	−	0.954038i	$-0.403118\pi$
$7$	1.58579	0.599371	0.299685	+	0.954038i	$0.403118\pi$
$8$	0	0
$9$	−2.82843	−0.942809
$10$	0	0
$11$	−5.24264	−1.58072	−0.790358	−	0.612646i	$-0.790105\pi$
$11$	−5.24264	−1.58072	−0.790358	+	0.612646i	$0.790105\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−1.17157	−0.302499
$16$	0	0
$17$	−0.171573	−0.0416125	−0.0208063	−	0.999784i	$-0.506623\pi$
$17$	−0.171573	−0.0416125	−0.0208063	+	0.999784i	$0.506623\pi$
$18$	0	0
$19$	7.24264	1.66158	0.830788	−	0.556589i	$-0.187890\pi$
$19$	7.24264	1.66158	0.830788	+	0.556589i	$0.187890\pi$
$20$	0	0
$21$	−0.656854	−0.143337
$22$	0	0
$23$	7.24264	1.51019	0.755097	−	0.655613i	$-0.227590\pi$
$23$	7.24264	1.51019	0.755097	+	0.655613i	$0.227590\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	2.41421	0.464616
$28$	0	0
$29$	−2.65685	−0.493365	−0.246683	−	0.969096i	$-0.579341\pi$
$29$	−2.65685	−0.493365	−0.246683	+	0.969096i	$0.579341\pi$
$30$	0	0
$31$	−5.65685	−1.01600	−0.508001	−	0.861357i	$-0.669615\pi$
$31$	−5.65685	−1.01600	−0.508001	+	0.861357i	$0.669615\pi$
$32$	0	0
$33$	2.17157	0.378022
$34$	0	0
$35$	4.48528	0.758151
$36$	0	0
$37$	−9.48528	−1.55937	−0.779685	−	0.626172i	$-0.784621\pi$
$37$	−9.48528	−1.55937	−0.779685	+	0.626172i	$0.784621\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.171573	0.0267952	0.0133976	−	0.999910i	$-0.495735\pi$
$41$	0.171573	0.0267952	0.0133976	+	0.999910i	$0.495735\pi$
$42$	0	0
$43$	10.0711	1.53582	0.767912	−	0.640556i	$-0.221296\pi$
$43$	10.0711	1.53582	0.767912	+	0.640556i	$0.221296\pi$
$44$	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
$49$	−4.48528	−0.640754
$50$	0	0
$51$	0.0710678	0.00995148
$52$	0	0
$53$	2.82843	0.388514	0.194257	−	0.980951i	$-0.437770\pi$
$53$	2.82843	0.388514	0.194257	+	0.980951i	$0.437770\pi$
$54$	0	0
$55$	−14.8284	−1.99946
$56$	0	0
$57$	−3.00000	−0.397360
$58$	0	0
$59$	7.24264	0.942912	0.471456	−	0.881890i	$-0.343729\pi$
$59$	7.24264	0.942912	0.471456	+	0.881890i	$0.343729\pi$
$60$	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
$63$	−4.48528	−0.565092
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4.75736	0.581204	0.290602	−	0.956844i	$-0.406144\pi$
$67$	4.75736	0.581204	0.290602	+	0.956844i	$0.406144\pi$
$68$	0	0
$69$	−3.00000	−0.361158
$70$	0	0
$71$	1.24264	0.147474	0.0737372	−	0.997278i	$-0.476507\pi$
$71$	1.24264	0.147474	0.0737372	+	0.997278i	$0.476507\pi$
$72$	0	0
$73$	4.48528	0.524962	0.262481	−	0.964937i	$-0.415459\pi$
$73$	4.48528	0.524962	0.262481	+	0.964937i	$0.415459\pi$
$74$	0	0
$75$	−1.24264	−0.143488
$76$	0	0
$77$	−8.31371	−0.947435
$78$	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
$81$	7.48528	0.831698
$82$	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
$85$	−0.485281	−0.0526362
$86$	0	0
$87$	1.10051	0.117987
$88$	0	0
$89$	14.6569	1.55362	0.776812	−	0.629733i	$-0.216836\pi$
$89$	14.6569	1.55362	0.776812	+	0.629733i	$0.216836\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.34315	0.242973
$94$	0	0
$95$	20.4853	2.10175
$96$	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
$99$	14.8284	1.49031
$100$	0	0
$101$	6.17157	0.614094	0.307047	−	0.951694i	$-0.400659\pi$
$101$	6.17157	0.614094	0.307047	+	0.951694i	$0.400659\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	−1.85786	−0.181309
$106$	0	0
$107$	−10.7574	−1.03995	−0.519977	−	0.854181i	$-0.674059\pi$
$107$	−10.7574	−1.03995	−0.519977	+	0.854181i	$0.674059\pi$
$108$	0	0
$109$	−8.48528	−0.812743	−0.406371	−	0.913708i	$-0.633206\pi$
$109$	−8.48528	−0.812743	−0.406371	+	0.913708i	$0.633206\pi$
$110$	0	0
$111$	3.92893	0.372918
$112$	0	0
$113$	−5.82843	−0.548292	−0.274146	−	0.961688i	$-0.588395\pi$
$113$	−5.82843	−0.548292	−0.274146	+	0.961688i	$0.588395\pi$
$114$	0	0
$115$	20.4853	1.91026
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−0.272078	−0.0249413
$120$	0	0
$121$	16.4853	1.49866
$122$	0	0
$123$	−0.0710678	−0.00640797
$124$	0	0
$125$	−5.65685	−0.505964
$126$	0	0
$127$	−4.41421	−0.391698	−0.195849	−	0.980634i	$-0.562746\pi$
$127$	−4.41421	−0.391698	−0.195849	+	0.980634i	$0.562746\pi$
$128$	0	0
$129$	−4.17157	−0.367287
$130$	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
$133$	11.4853	0.995900
$134$	0	0
$135$	6.82843	0.587697
$136$	0	0
$137$	−20.6569	−1.76483	−0.882417	−	0.470468i	$-0.844085\pi$
$137$	−20.6569	−1.76483	−0.882417	+	0.470468i	$0.844085\pi$
$138$	0	0
$139$	−7.24264	−0.614313	−0.307156	−	0.951659i	$-0.599377\pi$
$139$	−7.24264	−0.614313	−0.307156	+	0.951659i	$0.599377\pi$
$140$	0	0
$141$	−2.48528	−0.209298
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−7.51472	−0.624063
$146$	0	0
$147$	1.85786	0.153234
$148$	0	0
$149$	14.6569	1.20074	0.600368	−	0.799724i	$-0.295021\pi$
$149$	14.6569	1.20074	0.600368	+	0.799724i	$0.295021\pi$
$150$	0	0
$151$	16.9706	1.38104	0.690522	−	0.723311i	$-0.257381\pi$
$151$	16.9706	1.38104	0.690522	+	0.723311i	$0.257381\pi$
$152$	0	0
$153$	0.485281	0.0392327
$154$	0	0
$155$	−16.0000	−1.28515
$156$	0	0
$157$	16.4853	1.31567	0.657834	−	0.753163i	$-0.271473\pi$
$157$	16.4853	1.31567	0.657834	+	0.753163i	$0.271473\pi$
$158$	0	0
$159$	−1.17157	−0.0929118
$160$	0	0
$161$	11.4853	0.905167
$162$	0	0
$163$	25.2426	1.97716	0.988578	−	0.150709i	$-0.0481558\pi$
$163$	25.2426	1.97716	0.988578	+	0.150709i	$0.0481558\pi$
$164$	0	0
$165$	6.14214	0.478165
$166$	0	0
$167$	9.72792	0.752769	0.376385	−	0.926464i	$-0.377167\pi$
$167$	9.72792	0.752769	0.376385	+	0.926464i	$0.377167\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	−20.4853	−1.56655
$172$	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
$175$	4.75736	0.359623
$176$	0	0
$177$	−3.00000	−0.225494
$178$	0	0
$179$	21.2426	1.58775	0.793875	−	0.608081i	$-0.208061\pi$
$179$	21.2426	1.58775	0.793875	+	0.608081i	$0.208061\pi$
$180$	0	0
$181$	16.4853	1.22534	0.612671	−	0.790338i	$-0.290095\pi$
$181$	16.4853	1.22534	0.612671	+	0.790338i	$0.290095\pi$
$182$	0	0
$183$	−2.89949	−0.214337
$184$	0	0
$185$	−26.8284	−1.97247
$186$	0	0
$187$	0.899495	0.0657776
$188$	0	0
$189$	3.82843	0.278477
$190$	0	0
$191$	−15.2426	−1.10292	−0.551459	−	0.834202i	$-0.685929\pi$
$191$	−15.2426	−1.10292	−0.551459	+	0.834202i	$0.685929\pi$
$192$	0	0
$193$	2.51472	0.181013	0.0905067	−	0.995896i	$-0.471151\pi$
$193$	2.51472	0.181013	0.0905067	+	0.995896i	$0.471151\pi$
$194$	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
$199$	−18.2132	−1.29110	−0.645550	−	0.763718i	$-0.723372\pi$
$199$	−18.2132	−1.29110	−0.645550	+	0.763718i	$0.723372\pi$
$200$	0	0
$201$	−1.97056	−0.138993
$202$	0	0
$203$	−4.21320	−0.295709
$204$	0	0
$205$	0.485281	0.0338935
$206$	0	0
$207$	−20.4853	−1.42383
$208$	0	0
$209$	−37.9706	−2.62648
$210$	0	0
$211$	−16.4142	−1.13000	−0.565001	−	0.825091i	$-0.691124\pi$
$211$	−16.4142	−1.13000	−0.565001	+	0.825091i	$0.691124\pi$
$212$	0	0
$213$	−0.514719	−0.0352679
$214$	0	0
$215$	28.4853	1.94268
$216$	0	0
$217$	−8.97056	−0.608961
$218$	0	0
$219$	−1.85786	−0.125543
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−12.8995	−0.863814	−0.431907	−	0.901918i	$-0.642159\pi$
$223$	−12.8995	−0.863814	−0.431907	+	0.901918i	$0.642159\pi$
$224$	0	0
$225$	−8.48528	−0.565685
$226$	0	0
$227$	−9.72792	−0.645665	−0.322832	−	0.946456i	$-0.604635\pi$
$227$	−9.72792	−0.645665	−0.322832	+	0.946456i	$0.604635\pi$
$228$	0	0
$229$	8.48528	0.560723	0.280362	−	0.959894i	$-0.409546\pi$
$229$	8.48528	0.560723	0.280362	+	0.959894i	$0.409546\pi$
$230$	0	0
$231$	3.44365	0.226576
$232$	0	0
$233$	−3.51472	−0.230257	−0.115128	−	0.993351i	$-0.536728\pi$
$233$	−3.51472	−0.230257	−0.115128	+	0.993351i	$0.536728\pi$
$234$	0	0
$235$	16.9706	1.10704
$236$	0	0
$237$	−2.48528	−0.161436
$238$	0	0
$239$	24.9706	1.61521	0.807606	−	0.589723i	$-0.200763\pi$
$239$	24.9706	1.61521	0.807606	+	0.589723i	$0.200763\pi$
$240$	0	0
$241$	20.4558	1.31768	0.658838	−	0.752285i	$-0.271048\pi$
$241$	20.4558	1.31768	0.658838	+	0.752285i	$0.271048\pi$
$242$	0	0
$243$	−10.3431	−0.663513
$244$	0	0
$245$	−12.6863	−0.810497
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−1.65685	−0.104999
$250$	0	0
$251$	7.72792	0.487782	0.243891	−	0.969803i	$-0.421576\pi$
$251$	7.72792	0.487782	0.243891	+	0.969803i	$0.421576\pi$
$252$	0	0
$253$	−37.9706	−2.38719
$254$	0	0
$255$	0.201010	0.0125877
$256$	0	0
$257$	−22.7990	−1.42216	−0.711081	−	0.703110i	$-0.751794\pi$
$257$	−22.7990	−1.42216	−0.711081	+	0.703110i	$0.751794\pi$
$258$	0	0
$259$	−15.0416	−0.934641
$260$	0	0
$261$	7.51472	0.465149
$262$	0	0
$263$	−0.757359	−0.0467008	−0.0233504	−	0.999727i	$-0.507433\pi$
$263$	−0.757359	−0.0467008	−0.0233504	+	0.999727i	$0.507433\pi$
$264$	0	0
$265$	8.00000	0.491436
$266$	0	0
$267$	−6.07107	−0.371543
$268$	0	0
$269$	−11.4853	−0.700270	−0.350135	−	0.936699i	$-0.613864\pi$
$269$	−11.4853	−0.700270	−0.350135	+	0.936699i	$0.613864\pi$
$270$	0	0
$271$	16.7574	1.01794	0.508969	−	0.860785i	$-0.330027\pi$
$271$	16.7574	1.01794	0.508969	+	0.860785i	$0.330027\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−15.7279	−0.948429
$276$	0	0
$277$	−3.48528	−0.209410	−0.104705	−	0.994503i	$-0.533390\pi$
$277$	−3.48528	−0.209410	−0.104705	+	0.994503i	$0.533390\pi$
$278$	0	0
$279$	16.0000	0.957895
$280$	0	0
$281$	−26.1421	−1.55951	−0.779755	−	0.626085i	$-0.784656\pi$
$281$	−26.1421	−1.55951	−0.779755	+	0.626085i	$0.784656\pi$
$282$	0	0
$283$	7.58579	0.450928	0.225464	−	0.974251i	$-0.427610\pi$
$283$	7.58579	0.450928	0.225464	+	0.974251i	$0.427610\pi$
$284$	0	0
$285$	−8.48528	−0.502625
$286$	0	0
$287$	0.272078	0.0160603
$288$	0	0
$289$	−16.9706	−0.998268
$290$	0	0
$291$	−3.72792	−0.218535
$292$	0	0
$293$	20.3137	1.18674	0.593370	−	0.804930i	$-0.297797\pi$
$293$	20.3137	1.18674	0.593370	+	0.804930i	$0.297797\pi$
$294$	0	0
$295$	20.4853	1.19270
$296$	0	0
$297$	−12.6569	−0.734425
$298$	0	0
$299$	0	0
$300$	0	0
$301$	15.9706	0.920528
$302$	0	0
$303$	−2.55635	−0.146858
$304$	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
$313$	3.51472	0.198664	0.0993318	−	0.995054i	$-0.468329\pi$
$313$	3.51472	0.198664	0.0993318	+	0.995054i	$0.468329\pi$
$314$	0	0
$315$	−12.6863	−0.714792
$316$	0	0
$317$	−5.31371	−0.298448	−0.149224	−	0.988803i	$-0.547677\pi$
$317$	−5.31371	−0.298448	−0.149224	+	0.988803i	$0.547677\pi$
$318$	0	0
$319$	13.9289	0.779870
$320$	0	0
$321$	4.45584	0.248701
$322$	0	0
$323$	−1.24264	−0.0691424
$324$	0	0
$325$	0	0
$326$	0	0
$327$	3.51472	0.194364
$328$	0	0
$329$	9.51472	0.524563
$330$	0	0
$331$	1.24264	0.0683017	0.0341509	−	0.999417i	$-0.489127\pi$
$331$	1.24264	0.0683017	0.0341509	+	0.999417i	$0.489127\pi$
$332$	0	0
$333$	26.8284	1.47019
$334$	0	0
$335$	13.4558	0.735171
$336$	0	0
$337$	−12.4853	−0.680117	−0.340058	−	0.940404i	$-0.610447\pi$
$337$	−12.4853	−0.680117	−0.340058	+	0.940404i	$0.610447\pi$
$338$	0	0
$339$	2.41421	0.131122
$340$	0	0
$341$	29.6569	1.60601
$342$	0	0
$343$	−18.2132	−0.983421
$344$	0	0
$345$	−8.48528	−0.456832
$346$	0	0
$347$	22.7574	1.22168	0.610840	−	0.791754i	$-0.290832\pi$
$347$	22.7574	1.22168	0.610840	+	0.791754i	$0.290832\pi$
$348$	0	0
$349$	16.4558	0.880861	0.440431	−	0.897787i	$-0.354826\pi$
$349$	16.4558	0.880861	0.440431	+	0.897787i	$0.354826\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.34315	0.177938	0.0889688	−	0.996034i	$-0.471643\pi$
$353$	3.34315	0.177938	0.0889688	+	0.996034i	$0.471643\pi$
$354$	0	0
$355$	3.51472	0.186542
$356$	0	0
$357$	0.112698	0.00596463
$358$	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
$361$	33.4558	1.76083
$362$	0	0
$363$	−6.82843	−0.358399
$364$	0	0
$365$	12.6863	0.664031
$366$	0	0
$367$	−33.7279	−1.76058	−0.880292	−	0.474433i	$-0.842653\pi$
$367$	−33.7279	−1.76058	−0.880292	+	0.474433i	$0.842653\pi$
$368$	0	0
$369$	−0.485281	−0.0252627
$370$	0	0
$371$	4.48528	0.232864
$372$	0	0
$373$	20.5147	1.06221	0.531106	−	0.847305i	$-0.321777\pi$
$373$	20.5147	1.06221	0.531106	+	0.847305i	$0.321777\pi$
$374$	0	0
$375$	2.34315	0.121000
$376$	0	0
$377$	0	0
$378$	0	0
$379$	9.38478	0.482064	0.241032	−	0.970517i	$-0.422514\pi$
$379$	9.38478	0.482064	0.241032	+	0.970517i	$0.422514\pi$
$380$	0	0
$381$	1.82843	0.0936732
$382$	0	0
$383$	−9.72792	−0.497074	−0.248537	−	0.968622i	$-0.579950\pi$
$383$	−9.72792	−0.497074	−0.248537	+	0.968622i	$0.579950\pi$
$384$	0	0
$385$	−23.5147	−1.19842
$386$	0	0
$387$	−28.4853	−1.44799
$388$	0	0
$389$	28.6274	1.45147	0.725734	−	0.687976i	$-0.241500\pi$
$389$	28.6274	1.45147	0.725734	+	0.687976i	$0.241500\pi$
$390$	0	0
$391$	−1.24264	−0.0628430
$392$	0	0
$393$	−1.65685	−0.0835772
$394$	0	0
$395$	16.9706	0.853882
$396$	0	0
$397$	−5.48528	−0.275298	−0.137649	−	0.990481i	$-0.543955\pi$
$397$	−5.48528	−0.275298	−0.137649	+	0.990481i	$0.543955\pi$
$398$	0	0
$399$	−4.75736	−0.238166
$400$	0	0
$401$	17.1421	0.856037	0.428019	−	0.903770i	$-0.359212\pi$
$401$	17.1421	0.856037	0.428019	+	0.903770i	$0.359212\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	21.1716	1.05202
$406$	0	0
$407$	49.7279	2.46492
$408$	0	0
$409$	27.4853	1.35906	0.679530	−	0.733648i	$-0.262184\pi$
$409$	27.4853	1.35906	0.679530	+	0.733648i	$0.262184\pi$
$410$	0	0
$411$	8.55635	0.422054
$412$	0	0
$413$	11.4853	0.565154
$414$	0	0
$415$	11.3137	0.555368
$416$	0	0
$417$	3.00000	0.146911
$418$	0	0
$419$	−13.7279	−0.670653	−0.335326	−	0.942102i	$-0.608847\pi$
$419$	−13.7279	−0.670653	−0.335326	+	0.942102i	$0.608847\pi$
$420$	0	0
$421$	16.4853	0.803443	0.401722	−	0.915762i	$-0.368412\pi$
$421$	16.4853	0.803443	0.401722	+	0.915762i	$0.368412\pi$
$422$	0	0
$423$	−16.9706	−0.825137
$424$	0	0
$425$	−0.514719	−0.0249675
$426$	0	0
$427$	11.1005	0.537191
$428$	0	0
$429$	0	0
$430$	0	0
$431$	24.6985	1.18968	0.594842	−	0.803843i	$-0.297215\pi$
$431$	24.6985	1.18968	0.594842	+	0.803843i	$0.297215\pi$
$432$	0	0
$433$	−19.4853	−0.936403	−0.468201	−	0.883622i	$-0.655098\pi$
$433$	−19.4853	−0.936403	−0.468201	+	0.883622i	$0.655098\pi$
$434$	0	0
$435$	3.11270	0.149242
$436$	0	0
$437$	52.4558	2.50930
$438$	0	0
$439$	13.9289	0.664792	0.332396	−	0.943140i	$-0.392143\pi$
$439$	13.9289	0.664792	0.332396	+	0.943140i	$0.392143\pi$
$440$	0	0
$441$	12.6863	0.604109
$442$	0	0
$443$	−4.97056	−0.236159	−0.118079	−	0.993004i	$-0.537674\pi$
$443$	−4.97056	−0.236159	−0.118079	+	0.993004i	$0.537674\pi$
$444$	0	0
$445$	41.4558	1.96520
$446$	0	0
$447$	−6.07107	−0.287152
$448$	0	0
$449$	−29.4853	−1.39150	−0.695748	−	0.718286i	$-0.744927\pi$
$449$	−29.4853	−1.39150	−0.695748	+	0.718286i	$0.744927\pi$
$450$	0	0
$451$	−0.899495	−0.0423556
$452$	0	0
$453$	−7.02944	−0.330272
$454$	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
$459$	−0.414214	−0.0193338
$460$	0	0
$461$	−40.7990	−1.90020	−0.950099	−	0.311948i	$-0.899019\pi$
$461$	−40.7990	−1.90020	−0.950099	+	0.311948i	$0.899019\pi$
$462$	0	0
$463$	29.6569	1.37827	0.689135	−	0.724633i	$-0.257990\pi$
$463$	29.6569	1.37827	0.689135	+	0.724633i	$0.257990\pi$
$464$	0	0
$465$	6.62742	0.307339
$466$	0	0
$467$	26.9706	1.24805	0.624024	−	0.781405i	$-0.285497\pi$
$467$	26.9706	1.24805	0.624024	+	0.781405i	$0.285497\pi$
$468$	0	0
$469$	7.54416	0.348357
$470$	0	0
$471$	−6.82843	−0.314637
$472$	0	0
$473$	−52.7990	−2.42770
$474$	0	0
$475$	21.7279	0.996945
$476$	0	0
$477$	−8.00000	−0.366295
$478$	0	0
$479$	−17.2426	−0.787836	−0.393918	−	0.919146i	$-0.628881\pi$
$479$	−17.2426	−0.787836	−0.393918	+	0.919146i	$0.628881\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−4.75736	−0.216467
$484$	0	0
$485$	25.4558	1.15589
$486$	0	0
$487$	6.89949	0.312646	0.156323	−	0.987706i	$-0.450036\pi$
$487$	6.89949	0.312646	0.156323	+	0.987706i	$0.450036\pi$
$488$	0	0
$489$	−10.4558	−0.472830
$490$	0	0
$491$	20.2132	0.912209	0.456105	−	0.889926i	$-0.349244\pi$
$491$	20.2132	0.912209	0.456105	+	0.889926i	$0.349244\pi$
$492$	0	0
$493$	0.455844	0.0205302
$494$	0	0
$495$	41.9411	1.88511
$496$	0	0
$497$	1.97056	0.0883918
$498$	0	0
$499$	−6.34315	−0.283958	−0.141979	−	0.989870i	$-0.545347\pi$
$499$	−6.34315	−0.283958	−0.141979	+	0.989870i	$0.545347\pi$
$500$	0	0
$501$	−4.02944	−0.180022
$502$	0	0
$503$	−23.2426	−1.03634	−0.518169	−	0.855278i	$-0.673386\pi$
$503$	−23.2426	−1.03634	−0.518169	+	0.855278i	$0.673386\pi$
$504$	0	0
$505$	17.4558	0.776775
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−23.1421	−1.02576	−0.512879	−	0.858461i	$-0.671421\pi$
$509$	−23.1421	−1.02576	−0.512879	+	0.858461i	$0.671421\pi$
$510$	0	0
$511$	7.11270	0.314647
$512$	0	0
$513$	17.4853	0.771994
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−31.4558	−1.38343
$518$	0	0
$519$	3.72792	0.163638
$520$	0	0
$521$	−14.8284	−0.649645	−0.324823	−	0.945775i	$-0.605305\pi$
$521$	−14.8284	−0.649645	−0.324823	+	0.945775i	$0.605305\pi$
$522$	0	0
$523$	13.5858	0.594065	0.297032	−	0.954867i	$-0.404003\pi$
$523$	13.5858	0.594065	0.297032	+	0.954867i	$0.404003\pi$
$524$	0	0
$525$	−1.97056	−0.0860024
$526$	0	0
$527$	0.970563	0.0422784
$528$	0	0
$529$	29.4558	1.28069
$530$	0	0
$531$	−20.4853	−0.888985
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−30.4264	−1.31545
$536$	0	0
$537$	−8.79899	−0.379704
$538$	0	0
$539$	23.5147	1.01285
$540$	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
$543$	−6.82843	−0.293036
$544$	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
$551$	−19.2426	−0.819764
$552$	0	0
$553$	9.51472	0.404607
$554$	0	0
$555$	11.1127	0.471708
$556$	0	0
$557$	2.65685	0.112575	0.0562873	−	0.998415i	$-0.482074\pi$
$557$	2.65685	0.112575	0.0562873	+	0.998415i	$0.482074\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	−0.372583	−0.0157305
$562$	0	0
$563$	26.6985	1.12521	0.562603	−	0.826727i	$-0.309800\pi$
$563$	26.6985	1.12521	0.562603	+	0.826727i	$0.309800\pi$
$564$	0	0
$565$	−16.4853	−0.693541
$566$	0	0
$567$	11.8701	0.498496
$568$	0	0
$569$	7.97056	0.334143	0.167072	−	0.985945i	$-0.446569\pi$
$569$	7.97056	0.334143	0.167072	+	0.985945i	$0.446569\pi$
$570$	0	0
$571$	16.2843	0.681476	0.340738	−	0.940158i	$-0.389323\pi$
$571$	16.2843	0.681476	0.340738	+	0.940158i	$0.389323\pi$
$572$	0	0
$573$	6.31371	0.263759
$574$	0	0
$575$	21.7279	0.906117
$576$	0	0
$577$	12.4853	0.519769	0.259885	−	0.965640i	$-0.416315\pi$
$577$	12.4853	0.519769	0.259885	+	0.965640i	$0.416315\pi$
$578$	0	0
$579$	−1.04163	−0.0432887
$580$	0	0
$581$	6.34315	0.263158
$582$	0	0
$583$	−14.8284	−0.614131
$584$	0	0
$585$	0	0
$586$	0	0
$587$	38.6985	1.59726	0.798629	−	0.601824i	$-0.205559\pi$
$587$	38.6985	1.59726	0.798629	+	0.601824i	$0.205559\pi$
$588$	0	0
$589$	−40.9706	−1.68816
$590$	0	0
$591$	1.24264	0.0511154
$592$	0	0
$593$	−9.17157	−0.376631	−0.188316	−	0.982109i	$-0.560303\pi$
$593$	−9.17157	−0.376631	−0.188316	+	0.982109i	$0.560303\pi$
$594$	0	0
$595$	−0.769553	−0.0315486
$596$	0	0
$597$	7.54416	0.308762
$598$	0	0
$599$	19.9411	0.814772	0.407386	−	0.913256i	$-0.366440\pi$
$599$	19.9411	0.814772	0.407386	+	0.913256i	$0.366440\pi$
$600$	0	0
$601$	−27.4853	−1.12115	−0.560574	−	0.828104i	$-0.689419\pi$
$601$	−27.4853	−1.12115	−0.560574	+	0.828104i	$0.689419\pi$
$602$	0	0
$603$	−13.4558	−0.547964
$604$	0	0
$605$	46.6274	1.89567
$606$	0	0
$607$	4.41421	0.179167	0.0895837	−	0.995979i	$-0.471446\pi$
$607$	4.41421	0.179167	0.0895837	+	0.995979i	$0.471446\pi$
$608$	0	0
$609$	1.74517	0.0707177
$610$	0	0
$611$	0	0
$612$	0	0
$613$	31.4853	1.27168	0.635839	−	0.771821i	$-0.280654\pi$
$613$	31.4853	1.27168	0.635839	+	0.771821i	$0.280654\pi$
$614$	0	0
$615$	−0.201010	−0.00810551
$616$	0	0
$617$	21.6863	0.873057	0.436529	−	0.899690i	$-0.356208\pi$
$617$	21.6863	0.873057	0.436529	+	0.899690i	$0.356208\pi$
$618$	0	0
$619$	28.9706	1.16443	0.582213	−	0.813037i	$-0.302187\pi$
$619$	28.9706	1.16443	0.582213	+	0.813037i	$0.302187\pi$
$620$	0	0
$621$	17.4853	0.701660
$622$	0	0
$623$	23.2426	0.931197
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	15.7279	0.628113
$628$	0	0
$629$	1.62742	0.0648894
$630$	0	0
$631$	1.24264	0.0494687	0.0247344	−	0.999694i	$-0.492126\pi$
$631$	1.24264	0.0494687	0.0247344	+	0.999694i	$0.492126\pi$
$632$	0	0
$633$	6.79899	0.270236
$634$	0	0
$635$	−12.4853	−0.495463
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−3.51472	−0.139040
$640$	0	0
$641$	16.7990	0.663520	0.331760	−	0.943364i	$-0.392358\pi$
$641$	16.7990	0.663520	0.331760	+	0.943364i	$0.392358\pi$
$642$	0	0
$643$	42.2132	1.66473	0.832363	−	0.554231i	$-0.186987\pi$
$643$	42.2132	1.66473	0.832363	+	0.554231i	$0.186987\pi$
$644$	0	0
$645$	−11.7990	−0.464585
$646$	0	0
$647$	−14.7574	−0.580172	−0.290086	−	0.957001i	$-0.593684\pi$
$647$	−14.7574	−0.580172	−0.290086	+	0.957001i	$0.593684\pi$
$648$	0	0
$649$	−37.9706	−1.49047
$650$	0	0
$651$	3.71573	0.145631
$652$	0	0
$653$	−5.14214	−0.201227	−0.100614	−	0.994926i	$-0.532081\pi$
$653$	−5.14214	−0.201227	−0.100614	+	0.994926i	$0.532081\pi$
$654$	0	0
$655$	11.3137	0.442063
$656$	0	0
$657$	−12.6863	−0.494939
$658$	0	0
$659$	−18.2132	−0.709486	−0.354743	−	0.934964i	$-0.615432\pi$
$659$	−18.2132	−0.709486	−0.354743	+	0.934964i	$0.615432\pi$
$660$	0	0
$661$	−26.4558	−1.02901	−0.514507	−	0.857486i	$-0.672025\pi$
$661$	−26.4558	−1.02901	−0.514507	+	0.857486i	$0.672025\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	32.4853	1.25973
$666$	0	0
$667$	−19.2426	−0.745078
$668$	0	0
$669$	5.34315	0.206578
$670$	0	0
$671$	−36.6985	−1.41673
$672$	0	0
$673$	−41.9706	−1.61785	−0.808923	−	0.587914i	$-0.799949\pi$
$673$	−41.9706	−1.61785	−0.808923	+	0.587914i	$0.799949\pi$
$674$	0	0
$675$	7.24264	0.278769
$676$	0	0
$677$	25.4558	0.978348	0.489174	−	0.872186i	$-0.337298\pi$
$677$	25.4558	0.978348	0.489174	+	0.872186i	$0.337298\pi$
$678$	0	0
$679$	14.2721	0.547712
$680$	0	0
$681$	4.02944	0.154408
$682$	0	0
$683$	−0.757359	−0.0289795	−0.0144898	−	0.999895i	$-0.504612\pi$
$683$	−0.757359	−0.0289795	−0.0144898	+	0.999895i	$0.504612\pi$
$684$	0	0
$685$	−58.4264	−2.23236
$686$	0	0
$687$	−3.51472	−0.134095
$688$	0	0
$689$	0	0
$690$	0	0
$691$	−1.92893	−0.0733800	−0.0366900	−	0.999327i	$-0.511681\pi$
$691$	−1.92893	−0.0733800	−0.0366900	+	0.999327i	$0.511681\pi$
$692$	0	0
$693$	23.5147	0.893250
$694$	0	0
$695$	−20.4853	−0.777051
$696$	0	0
$697$	−0.0294373	−0.00111502
$698$	0	0
$699$	1.45584	0.0550651
$700$	0	0
$701$	8.48528	0.320485	0.160242	−	0.987078i	$-0.448772\pi$
$701$	8.48528	0.320485	0.160242	+	0.987078i	$0.448772\pi$
$702$	0	0
$703$	−68.6985	−2.59101
$704$	0	0
$705$	−7.02944	−0.264744
$706$	0	0
$707$	9.78680	0.368070
$708$	0	0
$709$	2.51472	0.0944422	0.0472211	−	0.998884i	$-0.484963\pi$
$709$	2.51472	0.0944422	0.0472211	+	0.998884i	$0.484963\pi$
$710$	0	0
$711$	−16.9706	−0.636446
$712$	0	0
$713$	−40.9706	−1.53436
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−10.3431	−0.386272
$718$	0	0
$719$	−31.2426	−1.16515	−0.582577	−	0.812776i	$-0.697956\pi$
$719$	−31.2426	−1.16515	−0.582577	+	0.812776i	$0.697956\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−8.47309	−0.315118
$724$	0	0
$725$	−7.97056	−0.296019
$726$	0	0
$727$	12.6863	0.470509	0.235254	−	0.971934i	$-0.424408\pi$
$727$	12.6863	0.470509	0.235254	+	0.971934i	$0.424408\pi$
$728$	0	0
$729$	−18.1716	−0.673021
$730$	0	0
$731$	−1.72792	−0.0639095
$732$	0	0
$733$	23.9411	0.884286	0.442143	−	0.896945i	$-0.354218\pi$
$733$	23.9411	0.884286	0.442143	+	0.896945i	$0.354218\pi$
$734$	0	0
$735$	5.25483	0.193827
$736$	0	0
$737$	−24.9411	−0.918718
$738$	0	0
$739$	−12.2132	−0.449270	−0.224635	−	0.974443i	$-0.572119\pi$
$739$	−12.2132	−0.449270	−0.224635	+	0.974443i	$0.572119\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−1.24264	−0.0455881	−0.0227940	−	0.999740i	$-0.507256\pi$
$743$	−1.24264	−0.0455881	−0.0227940	+	0.999740i	$0.507256\pi$
$744$	0	0
$745$	41.4558	1.51882
$746$	0	0
$747$	−11.3137	−0.413947
$748$	0	0
$749$	−17.0589	−0.623318
$750$	0	0
$751$	36.8995	1.34648	0.673241	−	0.739423i	$-0.264902\pi$
$751$	36.8995	1.34648	0.673241	+	0.739423i	$0.264902\pi$
$752$	0	0
$753$	−3.20101	−0.116651
$754$	0	0
$755$	48.0000	1.74690
$756$	0	0
$757$	−6.51472	−0.236781	−0.118391	−	0.992967i	$-0.537774\pi$
$757$	−6.51472	−0.236781	−0.118391	+	0.992967i	$0.537774\pi$
$758$	0	0
$759$	15.7279	0.570887
$760$	0	0
$761$	−12.5147	−0.453658	−0.226829	−	0.973935i	$-0.572836\pi$
$761$	−12.5147	−0.453658	−0.226829	+	0.973935i	$0.572836\pi$
$762$	0	0
$763$	−13.4558	−0.487134
$764$	0	0
$765$	1.37258	0.0496258
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−39.4264	−1.42175	−0.710876	−	0.703317i	$-0.751701\pi$
$769$	−39.4264	−1.42175	−0.710876	+	0.703317i	$0.751701\pi$
$770$	0	0
$771$	9.44365	0.340105
$772$	0	0
$773$	−5.48528	−0.197292	−0.0986459	−	0.995123i	$-0.531451\pi$
$773$	−5.48528	−0.197292	−0.0986459	+	0.995123i	$0.531451\pi$
$774$	0	0
$775$	−16.9706	−0.609601
$776$	0	0
$777$	6.23045	0.223516
$778$	0	0
$779$	1.24264	0.0445222
$780$	0	0
$781$	−6.51472	−0.233115
$782$	0	0
$783$	−6.41421	−0.229225
$784$	0	0
$785$	46.6274	1.66420
$786$	0	0
$787$	−44.0122	−1.56887	−0.784433	−	0.620214i	$-0.787046\pi$
$787$	−44.0122	−1.56887	−0.784433	+	0.620214i	$0.787046\pi$
$788$	0	0
$789$	0.313708	0.0111683
$790$	0	0
$791$	−9.24264	−0.328630
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−3.31371	−0.117525
$796$	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
$799$	−1.02944	−0.0364189
$800$	0	0
$801$	−41.4558	−1.46477
$802$	0	0
$803$	−23.5147	−0.829816
$804$	0	0
$805$	32.4853	1.14496
$806$	0	0
$807$	4.75736	0.167467
$808$	0	0
$809$	−0.171573	−0.00603218	−0.00301609	−	0.999995i	$-0.500960\pi$
$809$	−0.171573	−0.00603218	−0.00301609	+	0.999995i	$0.500960\pi$
$810$	0	0
$811$	−23.6569	−0.830705	−0.415352	−	0.909661i	$-0.636342\pi$
$811$	−23.6569	−0.830705	−0.415352	+	0.909661i	$0.636342\pi$
$812$	0	0
$813$	−6.94113	−0.243436
$814$	0	0
$815$	71.3970	2.50093
$816$	0	0
$817$	72.9411	2.55189
$818$	0	0
$819$	0	0
$820$	0	0
$821$	13.9706	0.487576	0.243788	−	0.969829i	$-0.421610\pi$
$821$	13.9706	0.487576	0.243788	+	0.969829i	$0.421610\pi$
$822$	0	0
$823$	−42.2132	−1.47146	−0.735730	−	0.677275i	$-0.763161\pi$
$823$	−42.2132	−1.47146	−0.735730	+	0.677275i	$0.763161\pi$
$824$	0	0
$825$	6.51472	0.226813
$826$	0	0
$827$	−45.9411	−1.59753	−0.798765	−	0.601644i	$-0.794513\pi$
$827$	−45.9411	−1.59753	−0.798765	+	0.601644i	$0.794513\pi$
$828$	0	0
$829$	2.02944	0.0704853	0.0352426	−	0.999379i	$-0.488780\pi$
$829$	2.02944	0.0704853	0.0352426	+	0.999379i	$0.488780\pi$
$830$	0	0
$831$	1.44365	0.0500797
$832$	0	0
$833$	0.769553	0.0266634
$834$	0	0
$835$	27.5147	0.952186
$836$	0	0
$837$	−13.6569	−0.472050
$838$	0	0
$839$	−48.6985	−1.68126	−0.840629	−	0.541611i	$-0.817815\pi$
$839$	−48.6985	−1.68126	−0.840629	+	0.541611i	$0.817815\pi$
$840$	0	0
$841$	−21.9411	−0.756591
$842$	0	0
$843$	10.8284	0.372951
$844$	0	0
$845$	0	0
$846$	0	0
$847$	26.1421	0.898254
$848$	0	0
$849$	−3.14214	−0.107838
$850$	0	0
$851$	−68.6985	−2.35495
$852$	0	0
$853$	−9.45584	−0.323762	−0.161881	−	0.986810i	$-0.551756\pi$
$853$	−9.45584	−0.323762	−0.161881	+	0.986810i	$0.551756\pi$
$854$	0	0
$855$	−57.9411	−1.98154
$856$	0	0
$857$	38.8284	1.32635	0.663177	−	0.748463i	$-0.269208\pi$
$857$	38.8284	1.32635	0.663177	+	0.748463i	$0.269208\pi$
$858$	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
$861$	−0.112698	−0.00384075
$862$	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
$867$	7.02944	0.238732
$868$	0	0
$869$	−31.4558	−1.06707
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−25.4558	−0.861550
$874$	0	0
$875$	−8.97056	−0.303260
$876$	0	0
$877$	9.97056	0.336682	0.168341	−	0.985729i	$-0.446159\pi$
$877$	9.97056	0.336682	0.168341	+	0.985729i	$0.446159\pi$
$878$	0	0
$879$	−8.41421	−0.283804
$880$	0	0
$881$	23.8284	0.802800	0.401400	−	0.915903i	$-0.368524\pi$
$881$	23.8284	0.802800	0.401400	+	0.915903i	$0.368524\pi$
$882$	0	0
$883$	−51.5980	−1.73641	−0.868205	−	0.496205i	$-0.834726\pi$
$883$	−51.5980	−1.73641	−0.868205	+	0.496205i	$0.834726\pi$
$884$	0	0
$885$	−8.48528	−0.285230
$886$	0	0
$887$	18.2132	0.611540	0.305770	−	0.952105i	$-0.401086\pi$
$887$	18.2132	0.611540	0.305770	+	0.952105i	$0.401086\pi$
$888$	0	0
$889$	−7.00000	−0.234772
$890$	0	0
$891$	−39.2426	−1.31468
$892$	0	0
$893$	43.4558	1.45419
$894$	0	0
$895$	60.0833	2.00836
$896$	0	0
$897$	0	0
$898$	0	0
$899$	15.0294	0.501260
$900$	0	0
$901$	−0.485281	−0.0161671
$902$	0	0
$903$	−6.61522	−0.220141
$904$	0	0
$905$	46.6274	1.54995
$906$	0	0
$907$	6.21320	0.206306	0.103153	−	0.994665i	$-0.467107\pi$
$907$	6.21320	0.206306	0.103153	+	0.994665i	$0.467107\pi$
$908$	0	0
$909$	−17.4558	−0.578974
$910$	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
$913$	−20.9706	−0.694024
$914$	0	0
$915$	−8.20101	−0.271117
$916$	0	0
$917$	6.34315	0.209469
$918$	0	0
$919$	32.3553	1.06730	0.533652	−	0.845704i	$-0.320819\pi$
$919$	32.3553	1.06730	0.533652	+	0.845704i	$0.320819\pi$
$920$	0	0
$921$	−4.97056	−0.163786
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−28.4558	−0.935622
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−52.1127	−1.70976	−0.854881	−	0.518824i	$-0.826370\pi$
$929$	−52.1127	−1.70976	−0.854881	+	0.518824i	$0.826370\pi$
$930$	0	0
$931$	−32.4853	−1.06466
$932$	0	0
$933$	9.94113	0.325458
$934$	0	0
$935$	2.54416	0.0832028
$936$	0	0
$937$	5.45584	0.178235	0.0891173	−	0.996021i	$-0.471595\pi$
$937$	5.45584	0.178235	0.0891173	+	0.996021i	$0.471595\pi$
$938$	0	0
$939$	−1.45584	−0.0475097
$940$	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
$943$	1.24264	0.0404659
$944$	0	0
$945$	10.8284	0.352249
$946$	0	0
$947$	−5.24264	−0.170363	−0.0851815	−	0.996365i	$-0.527147\pi$
$947$	−5.24264	−0.170363	−0.0851815	+	0.996365i	$0.527147\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	2.20101	0.0713726
$952$	0	0
$953$	−14.6569	−0.474782	−0.237391	−	0.971414i	$-0.576292\pi$
$953$	−14.6569	−0.474782	−0.237391	+	0.971414i	$0.576292\pi$
$954$	0	0
$955$	−43.1127	−1.39509
$956$	0	0
$957$	−5.76955	−0.186503
$958$	0	0
$959$	−32.7574	−1.05779
$960$	0	0
$961$	1.00000	0.0322581
$962$	0	0
$963$	30.4264	0.980477
$964$	0	0
$965$	7.11270	0.228966
$966$	0	0
$967$	40.9706	1.31752	0.658762	−	0.752351i	$-0.271080\pi$
$967$	40.9706	1.31752	0.658762	+	0.752351i	$0.271080\pi$
$968$	0	0
$969$	0.514719	0.0165351
$970$	0	0
$971$	−41.2426	−1.32354	−0.661770	−	0.749707i	$-0.730194\pi$
$971$	−41.2426	−1.32354	−0.661770	+	0.749707i	$0.730194\pi$
$972$	0	0
$973$	−11.4853	−0.368201
$974$	0	0
$975$	0	0
$976$	0	0
$977$	13.2843	0.425001	0.212501	−	0.977161i	$-0.431839\pi$
$977$	13.2843	0.425001	0.212501	+	0.977161i	$0.431839\pi$
$978$	0	0
$979$	−76.8406	−2.45584
$980$	0	0
$981$	24.0000	0.766261
$982$	0	0
$983$	−32.9706	−1.05160	−0.525799	−	0.850609i	$-0.676234\pi$
$983$	−32.9706	−1.05160	−0.525799	+	0.850609i	$0.676234\pi$
$984$	0	0
$985$	−8.48528	−0.270364
$986$	0	0
$987$	−3.94113	−0.125447
$988$	0	0
$989$	72.9411	2.31939
$990$	0	0
$991$	52.1543	1.65674	0.828368	−	0.560184i	$-0.189269\pi$
$991$	52.1543	1.65674	0.828368	+	0.560184i	$0.189269\pi$
$992$	0	0
$993$	−0.514719	−0.0163341
$994$	0	0
$995$	−51.5147	−1.63313
$996$	0	0
$997$	32.9411	1.04326	0.521628	−	0.853173i	$-0.325325\pi$
$997$	32.9411	1.04326	0.521628	+	0.853173i	$0.325325\pi$
$998$	0	0
$999$	−22.8995	−0.724508

Twists

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

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

\(f(q)\)	\(=\)	\(q-0.414214 q^{3} +2.82843 q^{5} +1.58579 q^{7} -2.82843 q^{9} +O(q^{10})\)	\(q-0.414214 q^{3} +2.82843 q^{5} +1.58579 q^{7} -2.82843 q^{9} -5.24264 q^{11} -1.17157 q^{15} -0.171573 q^{17} +7.24264 q^{19} -0.656854 q^{21} +7.24264 q^{23} +3.00000 q^{25} +2.41421 q^{27} -2.65685 q^{29} -5.65685 q^{31} +2.17157 q^{33} +4.48528 q^{35} -9.48528 q^{37} +0.171573 q^{41} +10.0711 q^{43} -8.00000 q^{45} +6.00000 q^{47} -4.48528 q^{49} +0.0710678 q^{51} +2.82843 q^{53} -14.8284 q^{55} -3.00000 q^{57} +7.24264 q^{59} +7.00000 q^{61} -4.48528 q^{63} +4.75736 q^{67} -3.00000 q^{69} +1.24264 q^{71} +4.48528 q^{73} -1.24264 q^{75} -8.31371 q^{77} +6.00000 q^{79} +7.48528 q^{81} +4.00000 q^{83} -0.485281 q^{85} +1.10051 q^{87} +14.6569 q^{89} +2.34315 q^{93} +20.4853 q^{95} +9.00000 q^{97} +14.8284 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 6 q^{7}+O(q^{10}) \) 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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