LMFDB - Embedded newform 1456.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1456,2,Mod(1,1456)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1456, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1456.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1456 = 2^{4} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1456.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:	$11.6262185343$
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 728)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	1456.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.585786 q^{3} +0.414214 q^{5} -1.00000 q^{7} -2.65685 q^{9} +O(q^{10})$	$q+0.585786 q^{3} +0.414214 q^{5} -1.00000 q^{7} -2.65685 q^{9} +6.24264 q^{11} +1.00000 q^{13} +0.242641 q^{15} +0.585786 q^{17} +6.41421 q^{19} -0.585786 q^{21} -4.65685 q^{23} -4.82843 q^{25} -3.31371 q^{27} +0.171573 q^{29} +5.58579 q^{31} +3.65685 q^{33} -0.414214 q^{35} +0.242641 q^{37} +0.585786 q^{39} +3.17157 q^{41} +11.4853 q^{43} -1.10051 q^{45} +3.24264 q^{47} +1.00000 q^{49} +0.343146 q^{51} +9.48528 q^{53} +2.58579 q^{55} +3.75736 q^{57} +11.6569 q^{59} -10.0000 q^{61} +2.65685 q^{63} +0.414214 q^{65} +3.17157 q^{67} -2.72792 q^{69} +2.58579 q^{71} -0.757359 q^{73} -2.82843 q^{75} -6.24264 q^{77} +4.65685 q^{79} +6.02944 q^{81} -14.0711 q^{83} +0.242641 q^{85} +0.100505 q^{87} +0.414214 q^{89} -1.00000 q^{91} +3.27208 q^{93} +2.65685 q^{95} +0.899495 q^{97} -16.5858 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 4 q^{3} - 2 q^{5} - 2 q^{7} + 6 q^{9}+O(q^{10}) $ 2 * q + 4 * q^3 - 2 * q^5 - 2 * q^7 + 6 * q^9	$ 2 q + 4 q^{3} - 2 q^{5} - 2 q^{7} + 6 q^{9} + 4 q^{11} + 2 q^{13} - 8 q^{15} + 4 q^{17} + 10 q^{19} - 4 q^{21} + 2 q^{23} - 4 q^{25} + 16 q^{27} + 6 q^{29} + 14 q^{31} - 4 q^{33} + 2 q^{35} - 8 q^{37} + 4 q^{39} + 12 q^{41} + 6 q^{43} - 22 q^{45} - 2 q^{47} + 2 q^{49} + 12 q^{51} + 2 q^{53} + 8 q^{55} + 16 q^{57} + 12 q^{59} - 20 q^{61} - 6 q^{63} - 2 q^{65} + 12 q^{67} + 20 q^{69} + 8 q^{71} - 10 q^{73} - 4 q^{77} - 2 q^{79} + 46 q^{81} - 14 q^{83} - 8 q^{85} + 20 q^{87} - 2 q^{89} - 2 q^{91} + 32 q^{93} - 6 q^{95} - 18 q^{97} - 36 q^{99}+O(q^{100}) $ 2 * q + 4 * q^3 - 2 * q^5 - 2 * q^7 + 6 * q^9 + 4 * q^11 + 2 * q^13 - 8 * q^15 + 4 * q^17 + 10 * q^19 - 4 * q^21 + 2 * q^23 - 4 * q^25 + 16 * q^27 + 6 * q^29 + 14 * q^31 - 4 * q^33 + 2 * q^35 - 8 * q^37 + 4 * q^39 + 12 * q^41 + 6 * q^43 - 22 * q^45 - 2 * q^47 + 2 * q^49 + 12 * q^51 + 2 * q^53 + 8 * q^55 + 16 * q^57 + 12 * q^59 - 20 * q^61 - 6 * q^63 - 2 * q^65 + 12 * q^67 + 20 * q^69 + 8 * q^71 - 10 * q^73 - 4 * q^77 - 2 * q^79 + 46 * q^81 - 14 * q^83 - 8 * q^85 + 20 * q^87 - 2 * q^89 - 2 * q^91 + 32 * q^93 - 6 * q^95 - 18 * q^97 - 36 * 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
$3$	0.585786	0.338204	0.169102	−	0.985599i	$-0.445913\pi$
$3$	0.585786	0.338204	0.169102	+	0.985599i	$0.445913\pi$
$4$	0	0
$5$	0.414214	0.185242	0.0926210	−	0.995701i	$-0.470476\pi$
$5$	0.414214	0.185242	0.0926210	+	0.995701i	$0.470476\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	−2.65685	−0.885618
$10$	0	0
$11$	6.24264	1.88223	0.941113	−	0.338091i	$-0.109781\pi$
$11$	6.24264	1.88223	0.941113	+	0.338091i	$0.109781\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	0.242641	0.0626496
$16$	0	0
$17$	0.585786	0.142074	0.0710370	−	0.997474i	$-0.477369\pi$
$17$	0.585786	0.142074	0.0710370	+	0.997474i	$0.477369\pi$
$18$	0	0
$19$	6.41421	1.47152	0.735761	−	0.677242i	$-0.236825\pi$
$19$	6.41421	1.47152	0.735761	+	0.677242i	$0.236825\pi$
$20$	0	0
$21$	−0.585786	−0.127829
$22$	0	0
$23$	−4.65685	−0.971021	−0.485511	−	0.874231i	$-0.661366\pi$
$23$	−4.65685	−0.971021	−0.485511	+	0.874231i	$0.661366\pi$
$24$	0	0
$25$	−4.82843	−0.965685
$26$	0	0
$27$	−3.31371	−0.637723
$28$	0	0
$29$	0.171573	0.0318603	0.0159301	−	0.999873i	$-0.494929\pi$
$29$	0.171573	0.0318603	0.0159301	+	0.999873i	$0.494929\pi$
$30$	0	0
$31$	5.58579	1.00324	0.501618	−	0.865089i	$-0.332738\pi$
$31$	5.58579	1.00324	0.501618	+	0.865089i	$0.332738\pi$
$32$	0	0
$33$	3.65685	0.636577
$34$	0	0
$35$	−0.414214	−0.0700149
$36$	0	0
$37$	0.242641	0.0398899	0.0199449	−	0.999801i	$-0.493651\pi$
$37$	0.242641	0.0398899	0.0199449	+	0.999801i	$0.493651\pi$
$38$	0	0
$39$	0.585786	0.0938009
$40$	0	0
$41$	3.17157	0.495316	0.247658	−	0.968847i	$-0.420339\pi$
$41$	3.17157	0.495316	0.247658	+	0.968847i	$0.420339\pi$
$42$	0	0
$43$	11.4853	1.75149	0.875744	−	0.482775i	$-0.160371\pi$
$43$	11.4853	1.75149	0.875744	+	0.482775i	$0.160371\pi$
$44$	0	0
$45$	−1.10051	−0.164054
$46$	0	0
$47$	3.24264	0.472988	0.236494	−	0.971633i	$-0.424002\pi$
$47$	3.24264	0.472988	0.236494	+	0.971633i	$0.424002\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0.343146	0.0480500
$52$	0	0
$53$	9.48528	1.30290	0.651452	−	0.758690i	$-0.274160\pi$
$53$	9.48528	1.30290	0.651452	+	0.758690i	$0.274160\pi$
$54$	0	0
$55$	2.58579	0.348667
$56$	0	0
$57$	3.75736	0.497674
$58$	0	0
$59$	11.6569	1.51759	0.758797	−	0.651328i	$-0.225788\pi$
$59$	11.6569	1.51759	0.758797	+	0.651328i	$0.225788\pi$
$60$	0	0
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	0	0
$63$	2.65685	0.334732
$64$	0	0
$65$	0.414214	0.0513769
$66$	0	0
$67$	3.17157	0.387469	0.193735	−	0.981054i	$-0.437940\pi$
$67$	3.17157	0.387469	0.193735	+	0.981054i	$0.437940\pi$
$68$	0	0
$69$	−2.72792	−0.328403
$70$	0	0
$71$	2.58579	0.306876	0.153438	−	0.988158i	$-0.450965\pi$
$71$	2.58579	0.306876	0.153438	+	0.988158i	$0.450965\pi$
$72$	0	0
$73$	−0.757359	−0.0886422	−0.0443211	−	0.999017i	$-0.514112\pi$
$73$	−0.757359	−0.0886422	−0.0443211	+	0.999017i	$0.514112\pi$
$74$	0	0
$75$	−2.82843	−0.326599
$76$	0	0
$77$	−6.24264	−0.711415
$78$	0	0
$79$	4.65685	0.523937	0.261969	−	0.965076i	$-0.415628\pi$
$79$	4.65685	0.523937	0.261969	+	0.965076i	$0.415628\pi$
$80$	0	0
$81$	6.02944	0.669937
$82$	0	0
$83$	−14.0711	−1.54450	−0.772250	−	0.635319i	$-0.780869\pi$
$83$	−14.0711	−1.54450	−0.772250	+	0.635319i	$0.780869\pi$
$84$	0	0
$85$	0.242641	0.0263181
$86$	0	0
$87$	0.100505	0.0107753
$88$	0	0
$89$	0.414214	0.0439065	0.0219533	−	0.999759i	$-0.493011\pi$
$89$	0.414214	0.0439065	0.0219533	+	0.999759i	$0.493011\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	3.27208	0.339299
$94$	0	0
$95$	2.65685	0.272587
$96$	0	0
$97$	0.899495	0.0913299	0.0456649	−	0.998957i	$-0.485459\pi$
$97$	0.899495	0.0913299	0.0456649	+	0.998957i	$0.485459\pi$
$98$	0	0
$99$	−16.5858	−1.66693
$100$	0	0
$101$	−8.24264	−0.820173	−0.410087	−	0.912047i	$-0.634502\pi$
$101$	−8.24264	−0.820173	−0.410087	+	0.912047i	$0.634502\pi$
$102$	0	0
$103$	1.65685	0.163255	0.0816274	−	0.996663i	$-0.473988\pi$
$103$	1.65685	0.163255	0.0816274	+	0.996663i	$0.473988\pi$
$104$	0	0
$105$	−0.242641	−0.0236793
$106$	0	0
$107$	−14.4853	−1.40035	−0.700173	−	0.713974i	$-0.746894\pi$
$107$	−14.4853	−1.40035	−0.700173	+	0.713974i	$0.746894\pi$
$108$	0	0
$109$	14.7279	1.41068	0.705340	−	0.708870i	$-0.250795\pi$
$109$	14.7279	1.41068	0.705340	+	0.708870i	$0.250795\pi$
$110$	0	0
$111$	0.142136	0.0134909
$112$	0	0
$113$	6.31371	0.593944	0.296972	−	0.954886i	$-0.404023\pi$
$113$	6.31371	0.593944	0.296972	+	0.954886i	$0.404023\pi$
$114$	0	0
$115$	−1.92893	−0.179874
$116$	0	0
$117$	−2.65685	−0.245626
$118$	0	0
$119$	−0.585786	−0.0536990
$120$	0	0
$121$	27.9706	2.54278
$122$	0	0
$123$	1.85786	0.167518
$124$	0	0
$125$	−4.07107	−0.364127
$126$	0	0
$127$	−18.9706	−1.68337	−0.841683	−	0.539973i	$-0.818435\pi$
$127$	−18.9706	−1.68337	−0.841683	+	0.539973i	$0.818435\pi$
$128$	0	0
$129$	6.72792	0.592361
$130$	0	0
$131$	5.17157	0.451842	0.225921	−	0.974146i	$-0.427461\pi$
$131$	5.17157	0.451842	0.225921	+	0.974146i	$0.427461\pi$
$132$	0	0
$133$	−6.41421	−0.556183
$134$	0	0
$135$	−1.37258	−0.118133
$136$	0	0
$137$	−4.24264	−0.362473	−0.181237	−	0.983440i	$-0.558010\pi$
$137$	−4.24264	−0.362473	−0.181237	+	0.983440i	$0.558010\pi$
$138$	0	0
$139$	18.3848	1.55938	0.779688	−	0.626168i	$-0.215378\pi$
$139$	18.3848	1.55938	0.779688	+	0.626168i	$0.215378\pi$
$140$	0	0
$141$	1.89949	0.159966
$142$	0	0
$143$	6.24264	0.522036
$144$	0	0
$145$	0.0710678	0.00590186
$146$	0	0
$147$	0.585786	0.0483149
$148$	0	0
$149$	0.585786	0.0479895	0.0239947	−	0.999712i	$-0.492362\pi$
$149$	0.585786	0.0479895	0.0239947	+	0.999712i	$0.492362\pi$
$150$	0	0
$151$	−3.07107	−0.249920	−0.124960	−	0.992162i	$-0.539880\pi$
$151$	−3.07107	−0.249920	−0.124960	+	0.992162i	$0.539880\pi$
$152$	0	0
$153$	−1.55635	−0.125823
$154$	0	0
$155$	2.31371	0.185842
$156$	0	0
$157$	−23.8995	−1.90739	−0.953694	−	0.300780i	$-0.902753\pi$
$157$	−23.8995	−1.90739	−0.953694	+	0.300780i	$0.902753\pi$
$158$	0	0
$159$	5.55635	0.440647
$160$	0	0
$161$	4.65685	0.367012
$162$	0	0
$163$	−10.8284	−0.848148	−0.424074	−	0.905628i	$-0.639400\pi$
$163$	−10.8284	−0.848148	−0.424074	+	0.905628i	$0.639400\pi$
$164$	0	0
$165$	1.51472	0.117921
$166$	0	0
$167$	−11.2426	−0.869982	−0.434991	−	0.900435i	$-0.643248\pi$
$167$	−11.2426	−0.869982	−0.434991	+	0.900435i	$0.643248\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−17.0416	−1.30321
$172$	0	0
$173$	21.5563	1.63890	0.819449	−	0.573151i	$-0.194279\pi$
$173$	21.5563	1.63890	0.819449	+	0.573151i	$0.194279\pi$
$174$	0	0
$175$	4.82843	0.364995
$176$	0	0
$177$	6.82843	0.513256
$178$	0	0
$179$	13.1421	0.982289	0.491145	−	0.871078i	$-0.336579\pi$
$179$	13.1421	0.982289	0.491145	+	0.871078i	$0.336579\pi$
$180$	0	0
$181$	5.41421	0.402435	0.201218	−	0.979547i	$-0.435510\pi$
$181$	5.41421	0.402435	0.201218	+	0.979547i	$0.435510\pi$
$182$	0	0
$183$	−5.85786	−0.433026
$184$	0	0
$185$	0.100505	0.00738928
$186$	0	0
$187$	3.65685	0.267416
$188$	0	0
$189$	3.31371	0.241037
$190$	0	0
$191$	−0.828427	−0.0599429	−0.0299714	−	0.999551i	$-0.509542\pi$
$191$	−0.828427	−0.0599429	−0.0299714	+	0.999551i	$0.509542\pi$
$192$	0	0
$193$	−2.48528	−0.178894	−0.0894472	−	0.995992i	$-0.528510\pi$
$193$	−2.48528	−0.178894	−0.0894472	+	0.995992i	$0.528510\pi$
$194$	0	0
$195$	0.242641	0.0173759
$196$	0	0
$197$	−25.3137	−1.80353	−0.901764	−	0.432230i	$-0.857727\pi$
$197$	−25.3137	−1.80353	−0.901764	+	0.432230i	$0.857727\pi$
$198$	0	0
$199$	−27.2132	−1.92909	−0.964546	−	0.263913i	$-0.914987\pi$
$199$	−27.2132	−1.92909	−0.964546	+	0.263913i	$0.914987\pi$
$200$	0	0
$201$	1.85786	0.131044
$202$	0	0
$203$	−0.171573	−0.0120421
$204$	0	0
$205$	1.31371	0.0917534
$206$	0	0
$207$	12.3726	0.859954
$208$	0	0
$209$	40.0416	2.76974
$210$	0	0
$211$	−11.4853	−0.790679	−0.395340	−	0.918535i	$-0.629373\pi$
$211$	−11.4853	−0.790679	−0.395340	+	0.918535i	$0.629373\pi$
$212$	0	0
$213$	1.51472	0.103787
$214$	0	0
$215$	4.75736	0.324449
$216$	0	0
$217$	−5.58579	−0.379188
$218$	0	0
$219$	−0.443651	−0.0299791
$220$	0	0
$221$	0.585786	0.0394043
$222$	0	0
$223$	4.07107	0.272619	0.136309	−	0.990666i	$-0.456476\pi$
$223$	4.07107	0.272619	0.136309	+	0.990666i	$0.456476\pi$
$224$	0	0
$225$	12.8284	0.855228
$226$	0	0
$227$	22.8284	1.51518	0.757588	−	0.652733i	$-0.226378\pi$
$227$	22.8284	1.51518	0.757588	+	0.652733i	$0.226378\pi$
$228$	0	0
$229$	−18.1421	−1.19887	−0.599433	−	0.800425i	$-0.704607\pi$
$229$	−18.1421	−1.19887	−0.599433	+	0.800425i	$0.704607\pi$
$230$	0	0
$231$	−3.65685	−0.240603
$232$	0	0
$233$	15.9706	1.04627	0.523133	−	0.852251i	$-0.324763\pi$
$233$	15.9706	1.04627	0.523133	+	0.852251i	$0.324763\pi$
$234$	0	0
$235$	1.34315	0.0876172
$236$	0	0
$237$	2.72792	0.177198
$238$	0	0
$239$	−14.1421	−0.914779	−0.457389	−	0.889267i	$-0.651215\pi$
$239$	−14.1421	−0.914779	−0.457389	+	0.889267i	$0.651215\pi$
$240$	0	0
$241$	−27.7279	−1.78611	−0.893056	−	0.449945i	$-0.851444\pi$
$241$	−27.7279	−1.78611	−0.893056	+	0.449945i	$0.851444\pi$
$242$	0	0
$243$	13.4731	0.864299
$244$	0	0
$245$	0.414214	0.0264631
$246$	0	0
$247$	6.41421	0.408127
$248$	0	0
$249$	−8.24264	−0.522356
$250$	0	0
$251$	−27.5563	−1.73934	−0.869671	−	0.493632i	$-0.835669\pi$
$251$	−27.5563	−1.73934	−0.869671	+	0.493632i	$0.835669\pi$
$252$	0	0
$253$	−29.0711	−1.82768
$254$	0	0
$255$	0.142136	0.00890088
$256$	0	0
$257$	−8.92893	−0.556971	−0.278486	−	0.960440i	$-0.589832\pi$
$257$	−8.92893	−0.556971	−0.278486	+	0.960440i	$0.589832\pi$
$258$	0	0
$259$	−0.242641	−0.0150770
$260$	0	0
$261$	−0.455844	−0.0282160
$262$	0	0
$263$	15.8284	0.976023	0.488011	−	0.872837i	$-0.337722\pi$
$263$	15.8284	0.976023	0.488011	+	0.872837i	$0.337722\pi$
$264$	0	0
$265$	3.92893	0.241352
$266$	0	0
$267$	0.242641	0.0148494
$268$	0	0
$269$	1.85786	0.113276	0.0566380	−	0.998395i	$-0.481962\pi$
$269$	1.85786	0.113276	0.0566380	+	0.998395i	$0.481962\pi$
$270$	0	0
$271$	6.34315	0.385319	0.192659	−	0.981266i	$-0.438289\pi$
$271$	6.34315	0.385319	0.192659	+	0.981266i	$0.438289\pi$
$272$	0	0
$273$	−0.585786	−0.0354534
$274$	0	0
$275$	−30.1421	−1.81764
$276$	0	0
$277$	−18.7990	−1.12952	−0.564761	−	0.825255i	$-0.691032\pi$
$277$	−18.7990	−1.12952	−0.564761	+	0.825255i	$0.691032\pi$
$278$	0	0
$279$	−14.8406	−0.888485
$280$	0	0
$281$	6.92893	0.413345	0.206673	−	0.978410i	$-0.433736\pi$
$281$	6.92893	0.413345	0.206673	+	0.978410i	$0.433736\pi$
$282$	0	0
$283$	−1.85786	−0.110439	−0.0552193	−	0.998474i	$-0.517586\pi$
$283$	−1.85786	−0.110439	−0.0552193	+	0.998474i	$0.517586\pi$
$284$	0	0
$285$	1.55635	0.0921902
$286$	0	0
$287$	−3.17157	−0.187212
$288$	0	0
$289$	−16.6569	−0.979815
$290$	0	0
$291$	0.526912	0.0308881
$292$	0	0
$293$	−23.7279	−1.38620	−0.693100	−	0.720841i	$-0.743756\pi$
$293$	−23.7279	−1.38620	−0.693100	+	0.720841i	$0.743756\pi$
$294$	0	0
$295$	4.82843	0.281122
$296$	0	0
$297$	−20.6863	−1.20034
$298$	0	0
$299$	−4.65685	−0.269313
$300$	0	0
$301$	−11.4853	−0.662001
$302$	0	0
$303$	−4.82843	−0.277386
$304$	0	0
$305$	−4.14214	−0.237178
$306$	0	0
$307$	−16.4142	−0.936809	−0.468404	−	0.883514i	$-0.655171\pi$
$307$	−16.4142	−0.936809	−0.468404	+	0.883514i	$0.655171\pi$
$308$	0	0
$309$	0.970563	0.0552134
$310$	0	0
$311$	−24.2426	−1.37467	−0.687337	−	0.726339i	$-0.741220\pi$
$311$	−24.2426	−1.37467	−0.687337	+	0.726339i	$0.741220\pi$
$312$	0	0
$313$	−9.89949	−0.559553	−0.279776	−	0.960065i	$-0.590260\pi$
$313$	−9.89949	−0.559553	−0.279776	+	0.960065i	$0.590260\pi$
$314$	0	0
$315$	1.10051	0.0620064
$316$	0	0
$317$	16.9706	0.953162	0.476581	−	0.879131i	$-0.341876\pi$
$317$	16.9706	0.953162	0.476581	+	0.879131i	$0.341876\pi$
$318$	0	0
$319$	1.07107	0.0599683
$320$	0	0
$321$	−8.48528	−0.473602
$322$	0	0
$323$	3.75736	0.209065
$324$	0	0
$325$	−4.82843	−0.267833
$326$	0	0
$327$	8.62742	0.477097
$328$	0	0
$329$	−3.24264	−0.178773
$330$	0	0
$331$	13.3137	0.731788	0.365894	−	0.930657i	$-0.380763\pi$
$331$	13.3137	0.731788	0.365894	+	0.930657i	$0.380763\pi$
$332$	0	0
$333$	−0.644661	−0.0353272
$334$	0	0
$335$	1.31371	0.0717756
$336$	0	0
$337$	−10.6569	−0.580516	−0.290258	−	0.956948i	$-0.593741\pi$
$337$	−10.6569	−0.580516	−0.290258	+	0.956948i	$0.593741\pi$
$338$	0	0
$339$	3.69848	0.200874
$340$	0	0
$341$	34.8701	1.88832
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−1.12994	−0.0608340
$346$	0	0
$347$	24.0000	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$348$	0	0
$349$	−33.2426	−1.77944	−0.889719	−	0.456509i	$-0.849100\pi$
$349$	−33.2426	−1.77944	−0.889719	+	0.456509i	$0.849100\pi$
$350$	0	0
$351$	−3.31371	−0.176873
$352$	0	0
$353$	24.4853	1.30322	0.651610	−	0.758554i	$-0.274094\pi$
$353$	24.4853	1.30322	0.651610	+	0.758554i	$0.274094\pi$
$354$	0	0
$355$	1.07107	0.0568464
$356$	0	0
$357$	−0.343146	−0.0181612
$358$	0	0
$359$	22.3848	1.18142	0.590712	−	0.806883i	$-0.298847\pi$
$359$	22.3848	1.18142	0.590712	+	0.806883i	$0.298847\pi$
$360$	0	0
$361$	22.1421	1.16538
$362$	0	0
$363$	16.3848	0.859978
$364$	0	0
$365$	−0.313708	−0.0164203
$366$	0	0
$367$	10.1005	0.527242	0.263621	−	0.964626i	$-0.415083\pi$
$367$	10.1005	0.527242	0.263621	+	0.964626i	$0.415083\pi$
$368$	0	0
$369$	−8.42641	−0.438661
$370$	0	0
$371$	−9.48528	−0.492451
$372$	0	0
$373$	27.7990	1.43938	0.719689	−	0.694297i	$-0.244285\pi$
$373$	27.7990	1.43938	0.719689	+	0.694297i	$0.244285\pi$
$374$	0	0
$375$	−2.38478	−0.123149
$376$	0	0
$377$	0.171573	0.00883645
$378$	0	0
$379$	18.7279	0.961989	0.480994	−	0.876724i	$-0.340276\pi$
$379$	18.7279	0.961989	0.480994	+	0.876724i	$0.340276\pi$
$380$	0	0
$381$	−11.1127	−0.569321
$382$	0	0
$383$	10.8284	0.553307	0.276653	−	0.960970i	$-0.410775\pi$
$383$	10.8284	0.553307	0.276653	+	0.960970i	$0.410775\pi$
$384$	0	0
$385$	−2.58579	−0.131784
$386$	0	0
$387$	−30.5147	−1.55115
$388$	0	0
$389$	−4.00000	−0.202808	−0.101404	−	0.994845i	$-0.532333\pi$
$389$	−4.00000	−0.202808	−0.101404	+	0.994845i	$0.532333\pi$
$390$	0	0
$391$	−2.72792	−0.137957
$392$	0	0
$393$	3.02944	0.152815
$394$	0	0
$395$	1.92893	0.0970551
$396$	0	0
$397$	−26.4142	−1.32569	−0.662846	−	0.748756i	$-0.730651\pi$
$397$	−26.4142	−1.32569	−0.662846	+	0.748756i	$0.730651\pi$
$398$	0	0
$399$	−3.75736	−0.188103
$400$	0	0
$401$	8.68629	0.433773	0.216886	−	0.976197i	$-0.430410\pi$
$401$	8.68629	0.433773	0.216886	+	0.976197i	$0.430410\pi$
$402$	0	0
$403$	5.58579	0.278248
$404$	0	0
$405$	2.49747	0.124101
$406$	0	0
$407$	1.51472	0.0750818
$408$	0	0
$409$	−13.9289	−0.688741	−0.344371	−	0.938834i	$-0.611908\pi$
$409$	−13.9289	−0.688741	−0.344371	+	0.938834i	$0.611908\pi$
$410$	0	0
$411$	−2.48528	−0.122590
$412$	0	0
$413$	−11.6569	−0.573596
$414$	0	0
$415$	−5.82843	−0.286106
$416$	0	0
$417$	10.7696	0.527387
$418$	0	0
$419$	5.21320	0.254682	0.127341	−	0.991859i	$-0.459356\pi$
$419$	5.21320	0.254682	0.127341	+	0.991859i	$0.459356\pi$
$420$	0	0
$421$	20.0416	0.976769	0.488385	−	0.872628i	$-0.337586\pi$
$421$	20.0416	0.976769	0.488385	+	0.872628i	$0.337586\pi$
$422$	0	0
$423$	−8.61522	−0.418887
$424$	0	0
$425$	−2.82843	−0.137199
$426$	0	0
$427$	10.0000	0.483934
$428$	0	0
$429$	3.65685	0.176555
$430$	0	0
$431$	3.65685	0.176144	0.0880722	−	0.996114i	$-0.471929\pi$
$431$	3.65685	0.176144	0.0880722	+	0.996114i	$0.471929\pi$
$432$	0	0
$433$	−33.9411	−1.63111	−0.815553	−	0.578682i	$-0.803567\pi$
$433$	−33.9411	−1.63111	−0.815553	+	0.578682i	$0.803567\pi$
$434$	0	0
$435$	0.0416306	0.00199603
$436$	0	0
$437$	−29.8701	−1.42888
$438$	0	0
$439$	−40.1421	−1.91588	−0.957940	−	0.286969i	$-0.907352\pi$
$439$	−40.1421	−1.91588	−0.957940	+	0.286969i	$0.907352\pi$
$440$	0	0
$441$	−2.65685	−0.126517
$442$	0	0
$443$	16.7990	0.798144	0.399072	−	0.916920i	$-0.369332\pi$
$443$	16.7990	0.798144	0.399072	+	0.916920i	$0.369332\pi$
$444$	0	0
$445$	0.171573	0.00813333
$446$	0	0
$447$	0.343146	0.0162302
$448$	0	0
$449$	−15.5147	−0.732185	−0.366092	−	0.930578i	$-0.619305\pi$
$449$	−15.5147	−0.732185	−0.366092	+	0.930578i	$0.619305\pi$
$450$	0	0
$451$	19.7990	0.932298
$452$	0	0
$453$	−1.79899	−0.0845239
$454$	0	0
$455$	−0.414214	−0.0194186
$456$	0	0
$457$	−9.21320	−0.430975	−0.215488	−	0.976507i	$-0.569134\pi$
$457$	−9.21320	−0.430975	−0.215488	+	0.976507i	$0.569134\pi$
$458$	0	0
$459$	−1.94113	−0.0906040
$460$	0	0
$461$	−26.4853	−1.23354	−0.616771	−	0.787142i	$-0.711560\pi$
$461$	−26.4853	−1.23354	−0.616771	+	0.787142i	$0.711560\pi$
$462$	0	0
$463$	20.1005	0.934150	0.467075	−	0.884218i	$-0.345308\pi$
$463$	20.1005	0.934150	0.467075	+	0.884218i	$0.345308\pi$
$464$	0	0
$465$	1.35534	0.0628523
$466$	0	0
$467$	14.8701	0.688104	0.344052	−	0.938951i	$-0.388200\pi$
$467$	14.8701	0.688104	0.344052	+	0.938951i	$0.388200\pi$
$468$	0	0
$469$	−3.17157	−0.146450
$470$	0	0
$471$	−14.0000	−0.645086
$472$	0	0
$473$	71.6985	3.29670
$474$	0	0
$475$	−30.9706	−1.42103
$476$	0	0
$477$	−25.2010	−1.15387
$478$	0	0
$479$	28.4142	1.29828	0.649139	−	0.760670i	$-0.275129\pi$
$479$	28.4142	1.29828	0.649139	+	0.760670i	$0.275129\pi$
$480$	0	0
$481$	0.242641	0.0110635
$482$	0	0
$483$	2.72792	0.124125
$484$	0	0
$485$	0.372583	0.0169181
$486$	0	0
$487$	−3.85786	−0.174817	−0.0874083	−	0.996173i	$-0.527858\pi$
$487$	−3.85786	−0.174817	−0.0874083	+	0.996173i	$0.527858\pi$
$488$	0	0
$489$	−6.34315	−0.286847
$490$	0	0
$491$	−23.6569	−1.06762	−0.533809	−	0.845605i	$-0.679240\pi$
$491$	−23.6569	−1.06762	−0.533809	+	0.845605i	$0.679240\pi$
$492$	0	0
$493$	0.100505	0.00452652
$494$	0	0
$495$	−6.87006	−0.308786
$496$	0	0
$497$	−2.58579	−0.115988
$498$	0	0
$499$	−21.2132	−0.949633	−0.474817	−	0.880085i	$-0.657486\pi$
$499$	−21.2132	−0.949633	−0.474817	+	0.880085i	$0.657486\pi$
$500$	0	0
$501$	−6.58579	−0.294231
$502$	0	0
$503$	7.65685	0.341402	0.170701	−	0.985323i	$-0.445397\pi$
$503$	7.65685	0.341402	0.170701	+	0.985323i	$0.445397\pi$
$504$	0	0
$505$	−3.41421	−0.151931
$506$	0	0
$507$	0.585786	0.0260157
$508$	0	0
$509$	−23.2426	−1.03021	−0.515106	−	0.857127i	$-0.672247\pi$
$509$	−23.2426	−1.03021	−0.515106	+	0.857127i	$0.672247\pi$
$510$	0	0
$511$	0.757359	0.0335036
$512$	0	0
$513$	−21.2548	−0.938424
$514$	0	0
$515$	0.686292	0.0302416
$516$	0	0
$517$	20.2426	0.890270
$518$	0	0
$519$	12.6274	0.554282
$520$	0	0
$521$	38.6274	1.69230	0.846149	−	0.532947i	$-0.178915\pi$
$521$	38.6274	1.69230	0.846149	+	0.532947i	$0.178915\pi$
$522$	0	0
$523$	−11.6569	−0.509719	−0.254859	−	0.966978i	$-0.582029\pi$
$523$	−11.6569	−0.509719	−0.254859	+	0.966978i	$0.582029\pi$
$524$	0	0
$525$	2.82843	0.123443
$526$	0	0
$527$	3.27208	0.142534
$528$	0	0
$529$	−1.31371	−0.0571178
$530$	0	0
$531$	−30.9706	−1.34401
$532$	0	0
$533$	3.17157	0.137376
$534$	0	0
$535$	−6.00000	−0.259403
$536$	0	0
$537$	7.69848	0.332214
$538$	0	0
$539$	6.24264	0.268890
$540$	0	0
$541$	−4.92893	−0.211911	−0.105956	−	0.994371i	$-0.533790\pi$
$541$	−4.92893	−0.211911	−0.105956	+	0.994371i	$0.533790\pi$
$542$	0	0
$543$	3.17157	0.136105
$544$	0	0
$545$	6.10051	0.261317
$546$	0	0
$547$	−27.6274	−1.18126	−0.590632	−	0.806941i	$-0.701121\pi$
$547$	−27.6274	−1.18126	−0.590632	+	0.806941i	$0.701121\pi$
$548$	0	0
$549$	26.5685	1.13392
$550$	0	0
$551$	1.10051	0.0468831
$552$	0	0
$553$	−4.65685	−0.198030
$554$	0	0
$555$	0.0588745	0.00249908
$556$	0	0
$557$	−19.3137	−0.818348	−0.409174	−	0.912456i	$-0.634183\pi$
$557$	−19.3137	−0.818348	−0.409174	+	0.912456i	$0.634183\pi$
$558$	0	0
$559$	11.4853	0.485776
$560$	0	0
$561$	2.14214	0.0904410
$562$	0	0
$563$	−20.0000	−0.842900	−0.421450	−	0.906852i	$-0.638479\pi$
$563$	−20.0000	−0.842900	−0.421450	+	0.906852i	$0.638479\pi$
$564$	0	0
$565$	2.61522	0.110023
$566$	0	0
$567$	−6.02944	−0.253213
$568$	0	0
$569$	−20.1127	−0.843168	−0.421584	−	0.906789i	$-0.638526\pi$
$569$	−20.1127	−0.843168	−0.421584	+	0.906789i	$0.638526\pi$
$570$	0	0
$571$	23.6274	0.988777	0.494388	−	0.869241i	$-0.335392\pi$
$571$	23.6274	0.988777	0.494388	+	0.869241i	$0.335392\pi$
$572$	0	0
$573$	−0.485281	−0.0202729
$574$	0	0
$575$	22.4853	0.937701
$576$	0	0
$577$	−20.3431	−0.846896	−0.423448	−	0.905920i	$-0.639180\pi$
$577$	−20.3431	−0.846896	−0.423448	+	0.905920i	$0.639180\pi$
$578$	0	0
$579$	−1.45584	−0.0605028
$580$	0	0
$581$	14.0711	0.583766
$582$	0	0
$583$	59.2132	2.45236
$584$	0	0
$585$	−1.10051	−0.0455003
$586$	0	0
$587$	12.0711	0.498226	0.249113	−	0.968474i	$-0.419861\pi$
$587$	12.0711	0.498226	0.249113	+	0.968474i	$0.419861\pi$
$588$	0	0
$589$	35.8284	1.47628
$590$	0	0
$591$	−14.8284	−0.609960
$592$	0	0
$593$	−11.1005	−0.455843	−0.227922	−	0.973679i	$-0.573193\pi$
$593$	−11.1005	−0.455843	−0.227922	+	0.973679i	$0.573193\pi$
$594$	0	0
$595$	−0.242641	−0.00994730
$596$	0	0
$597$	−15.9411	−0.652427
$598$	0	0
$599$	23.0000	0.939755	0.469877	−	0.882732i	$-0.344298\pi$
$599$	23.0000	0.939755	0.469877	+	0.882732i	$0.344298\pi$
$600$	0	0
$601$	6.97056	0.284335	0.142168	−	0.989843i	$-0.454593\pi$
$601$	6.97056	0.284335	0.142168	+	0.989843i	$0.454593\pi$
$602$	0	0
$603$	−8.42641	−0.343150
$604$	0	0
$605$	11.5858	0.471029
$606$	0	0
$607$	33.6985	1.36778	0.683890	−	0.729585i	$-0.260287\pi$
$607$	33.6985	1.36778	0.683890	+	0.729585i	$0.260287\pi$
$608$	0	0
$609$	−0.100505	−0.00407267
$610$	0	0
$611$	3.24264	0.131183
$612$	0	0
$613$	31.3137	1.26475	0.632374	−	0.774663i	$-0.282080\pi$
$613$	31.3137	1.26475	0.632374	+	0.774663i	$0.282080\pi$
$614$	0	0
$615$	0.769553	0.0310314
$616$	0	0
$617$	30.9706	1.24683	0.623414	−	0.781892i	$-0.285745\pi$
$617$	30.9706	1.24683	0.623414	+	0.781892i	$0.285745\pi$
$618$	0	0
$619$	−16.9706	−0.682105	−0.341052	−	0.940044i	$-0.610783\pi$
$619$	−16.9706	−0.682105	−0.341052	+	0.940044i	$0.610783\pi$
$620$	0	0
$621$	15.4315	0.619243
$622$	0	0
$623$	−0.414214	−0.0165951
$624$	0	0
$625$	22.4558	0.898234
$626$	0	0
$627$	23.4558	0.936736
$628$	0	0
$629$	0.142136	0.00566732
$630$	0	0
$631$	−18.0000	−0.716569	−0.358284	−	0.933613i	$-0.616638\pi$
$631$	−18.0000	−0.716569	−0.358284	+	0.933613i	$0.616638\pi$
$632$	0	0
$633$	−6.72792	−0.267411
$634$	0	0
$635$	−7.85786	−0.311830
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−6.87006	−0.271775
$640$	0	0
$641$	25.6863	1.01455	0.507274	−	0.861785i	$-0.330653\pi$
$641$	25.6863	1.01455	0.507274	+	0.861785i	$0.330653\pi$
$642$	0	0
$643$	−30.1421	−1.18869	−0.594345	−	0.804210i	$-0.702589\pi$
$643$	−30.1421	−1.18869	−0.594345	+	0.804210i	$0.702589\pi$
$644$	0	0
$645$	2.78680	0.109730
$646$	0	0
$647$	−40.2426	−1.58210	−0.791051	−	0.611751i	$-0.790466\pi$
$647$	−40.2426	−1.58210	−0.791051	+	0.611751i	$0.790466\pi$
$648$	0	0
$649$	72.7696	2.85645
$650$	0	0
$651$	−3.27208	−0.128243
$652$	0	0
$653$	−8.62742	−0.337617	−0.168808	−	0.985649i	$-0.553992\pi$
$653$	−8.62742	−0.337617	−0.168808	+	0.985649i	$0.553992\pi$
$654$	0	0
$655$	2.14214	0.0837002
$656$	0	0
$657$	2.01219	0.0785031
$658$	0	0
$659$	6.51472	0.253777	0.126889	−	0.991917i	$-0.459501\pi$
$659$	6.51472	0.253777	0.126889	+	0.991917i	$0.459501\pi$
$660$	0	0
$661$	25.0416	0.974007	0.487003	−	0.873400i	$-0.338090\pi$
$661$	25.0416	0.974007	0.487003	+	0.873400i	$0.338090\pi$
$662$	0	0
$663$	0.343146	0.0133267
$664$	0	0
$665$	−2.65685	−0.103028
$666$	0	0
$667$	−0.798990	−0.0309370
$668$	0	0
$669$	2.38478	0.0922008
$670$	0	0
$671$	−62.4264	−2.40994
$672$	0	0
$673$	−21.9706	−0.846903	−0.423451	−	0.905919i	$-0.639182\pi$
$673$	−21.9706	−0.846903	−0.423451	+	0.905919i	$0.639182\pi$
$674$	0	0
$675$	16.0000	0.615840
$676$	0	0
$677$	1.41421	0.0543526	0.0271763	−	0.999631i	$-0.491348\pi$
$677$	1.41421	0.0543526	0.0271763	+	0.999631i	$0.491348\pi$
$678$	0	0
$679$	−0.899495	−0.0345194
$680$	0	0
$681$	13.3726	0.512438
$682$	0	0
$683$	0.485281	0.0185688	0.00928439	−	0.999957i	$-0.497045\pi$
$683$	0.485281	0.0185688	0.00928439	+	0.999957i	$0.497045\pi$
$684$	0	0
$685$	−1.75736	−0.0671452
$686$	0	0
$687$	−10.6274	−0.405461
$688$	0	0
$689$	9.48528	0.361360
$690$	0	0
$691$	44.0122	1.67430	0.837151	−	0.546971i	$-0.184219\pi$
$691$	44.0122	1.67430	0.837151	+	0.546971i	$0.184219\pi$
$692$	0	0
$693$	16.5858	0.630042
$694$	0	0
$695$	7.61522	0.288862
$696$	0	0
$697$	1.85786	0.0703716
$698$	0	0
$699$	9.35534	0.353851
$700$	0	0
$701$	36.7990	1.38988	0.694939	−	0.719068i	$-0.255431\pi$
$701$	36.7990	1.38988	0.694939	+	0.719068i	$0.255431\pi$
$702$	0	0
$703$	1.55635	0.0586988
$704$	0	0
$705$	0.786797	0.0296325
$706$	0	0
$707$	8.24264	0.309996
$708$	0	0
$709$	7.41421	0.278447	0.139223	−	0.990261i	$-0.455539\pi$
$709$	7.41421	0.278447	0.139223	+	0.990261i	$0.455539\pi$
$710$	0	0
$711$	−12.3726	−0.464008
$712$	0	0
$713$	−26.0122	−0.974164
$714$	0	0
$715$	2.58579	0.0967029
$716$	0	0
$717$	−8.28427	−0.309382
$718$	0	0
$719$	24.7279	0.922196	0.461098	−	0.887349i	$-0.347456\pi$
$719$	24.7279	0.922196	0.461098	+	0.887349i	$0.347456\pi$
$720$	0	0
$721$	−1.65685	−0.0617045
$722$	0	0
$723$	−16.2426	−0.604070
$724$	0	0
$725$	−0.828427	−0.0307670
$726$	0	0
$727$	8.62742	0.319973	0.159987	−	0.987119i	$-0.448855\pi$
$727$	8.62742	0.319973	0.159987	+	0.987119i	$0.448855\pi$
$728$	0	0
$729$	−10.1960	−0.377628
$730$	0	0
$731$	6.72792	0.248841
$732$	0	0
$733$	24.7574	0.914434	0.457217	−	0.889355i	$-0.348846\pi$
$733$	24.7574	0.914434	0.457217	+	0.889355i	$0.348846\pi$
$734$	0	0
$735$	0.242641	0.00894994
$736$	0	0
$737$	19.7990	0.729305
$738$	0	0
$739$	−9.75736	−0.358930	−0.179465	−	0.983764i	$-0.557437\pi$
$739$	−9.75736	−0.358930	−0.179465	+	0.983764i	$0.557437\pi$
$740$	0	0
$741$	3.75736	0.138030
$742$	0	0
$743$	−12.0000	−0.440237	−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000	−0.440237	−0.220119	+	0.975473i	$0.570644\pi$
$744$	0	0
$745$	0.242641	0.00888967
$746$	0	0
$747$	37.3848	1.36784
$748$	0	0
$749$	14.4853	0.529281
$750$	0	0
$751$	20.6569	0.753779	0.376890	−	0.926258i	$-0.376994\pi$
$751$	20.6569	0.753779	0.376890	+	0.926258i	$0.376994\pi$
$752$	0	0
$753$	−16.1421	−0.588252
$754$	0	0
$755$	−1.27208	−0.0462957
$756$	0	0
$757$	−41.1421	−1.49534	−0.747668	−	0.664073i	$-0.768827\pi$
$757$	−41.1421	−1.49534	−0.747668	+	0.664073i	$0.768827\pi$
$758$	0	0
$759$	−17.0294	−0.618129
$760$	0	0
$761$	−15.3848	−0.557698	−0.278849	−	0.960335i	$-0.589953\pi$
$761$	−15.3848	−0.557698	−0.278849	+	0.960335i	$0.589953\pi$
$762$	0	0
$763$	−14.7279	−0.533187
$764$	0	0
$765$	−0.644661	−0.0233078
$766$	0	0
$767$	11.6569	0.420905
$768$	0	0
$769$	−7.58579	−0.273550	−0.136775	−	0.990602i	$-0.543674\pi$
$769$	−7.58579	−0.273550	−0.136775	+	0.990602i	$0.543674\pi$
$770$	0	0
$771$	−5.23045	−0.188370
$772$	0	0
$773$	12.8284	0.461406	0.230703	−	0.973024i	$-0.425897\pi$
$773$	12.8284	0.461406	0.230703	+	0.973024i	$0.425897\pi$
$774$	0	0
$775$	−26.9706	−0.968811
$776$	0	0
$777$	−0.142136	−0.00509909
$778$	0	0
$779$	20.3431	0.728869
$780$	0	0
$781$	16.1421	0.577611
$782$	0	0
$783$	−0.568542	−0.0203181
$784$	0	0
$785$	−9.89949	−0.353328
$786$	0	0
$787$	4.61522	0.164515	0.0822575	−	0.996611i	$-0.473787\pi$
$787$	4.61522	0.164515	0.0822575	+	0.996611i	$0.473787\pi$
$788$	0	0
$789$	9.27208	0.330095
$790$	0	0
$791$	−6.31371	−0.224490
$792$	0	0
$793$	−10.0000	−0.355110
$794$	0	0
$795$	2.30152	0.0816263
$796$	0	0
$797$	50.2843	1.78116	0.890580	−	0.454826i	$-0.150299\pi$
$797$	50.2843	1.78116	0.890580	+	0.454826i	$0.150299\pi$
$798$	0	0
$799$	1.89949	0.0671993
$800$	0	0
$801$	−1.10051	−0.0388844
$802$	0	0
$803$	−4.72792	−0.166845
$804$	0	0
$805$	1.92893	0.0679859
$806$	0	0
$807$	1.08831	0.0383104
$808$	0	0
$809$	3.54416	0.124606	0.0623029	−	0.998057i	$-0.480156\pi$
$809$	3.54416	0.124606	0.0623029	+	0.998057i	$0.480156\pi$
$810$	0	0
$811$	47.3137	1.66141	0.830705	−	0.556714i	$-0.187938\pi$
$811$	47.3137	1.66141	0.830705	+	0.556714i	$0.187938\pi$
$812$	0	0
$813$	3.71573	0.130316
$814$	0	0
$815$	−4.48528	−0.157113
$816$	0	0
$817$	73.6690	2.57735
$818$	0	0
$819$	2.65685	0.0928380
$820$	0	0
$821$	−27.1716	−0.948294	−0.474147	−	0.880446i	$-0.657244\pi$
$821$	−27.1716	−0.948294	−0.474147	+	0.880446i	$0.657244\pi$
$822$	0	0
$823$	−21.1127	−0.735942	−0.367971	−	0.929837i	$-0.619948\pi$
$823$	−21.1127	−0.735942	−0.367971	+	0.929837i	$0.619948\pi$
$824$	0	0
$825$	−17.6569	−0.614733
$826$	0	0
$827$	17.8579	0.620979	0.310489	−	0.950577i	$-0.399507\pi$
$827$	17.8579	0.620979	0.310489	+	0.950577i	$0.399507\pi$
$828$	0	0
$829$	−5.41421	−0.188043	−0.0940217	−	0.995570i	$-0.529972\pi$
$829$	−5.41421	−0.188043	−0.0940217	+	0.995570i	$0.529972\pi$
$830$	0	0
$831$	−11.0122	−0.382009
$832$	0	0
$833$	0.585786	0.0202963
$834$	0	0
$835$	−4.65685	−0.161157
$836$	0	0
$837$	−18.5097	−0.639788
$838$	0	0
$839$	47.1127	1.62651	0.813255	−	0.581907i	$-0.197693\pi$
$839$	47.1127	1.62651	0.813255	+	0.581907i	$0.197693\pi$
$840$	0	0
$841$	−28.9706	−0.998985
$842$	0	0
$843$	4.05887	0.139795
$844$	0	0
$845$	0.414214	0.0142494
$846$	0	0
$847$	−27.9706	−0.961080
$848$	0	0
$849$	−1.08831	−0.0373508
$850$	0	0
$851$	−1.12994	−0.0387339
$852$	0	0
$853$	−22.4142	−0.767448	−0.383724	−	0.923448i	$-0.625359\pi$
$853$	−22.4142	−0.767448	−0.383724	+	0.923448i	$0.625359\pi$
$854$	0	0
$855$	−7.05887	−0.241408
$856$	0	0
$857$	−24.1421	−0.824680	−0.412340	−	0.911030i	$-0.635288\pi$
$857$	−24.1421	−0.824680	−0.412340	+	0.911030i	$0.635288\pi$
$858$	0	0
$859$	−14.0000	−0.477674	−0.238837	−	0.971060i	$-0.576766\pi$
$859$	−14.0000	−0.477674	−0.238837	+	0.971060i	$0.576766\pi$
$860$	0	0
$861$	−1.85786	−0.0633158
$862$	0	0
$863$	9.65685	0.328723	0.164362	−	0.986400i	$-0.447444\pi$
$863$	9.65685	0.328723	0.164362	+	0.986400i	$0.447444\pi$
$864$	0	0
$865$	8.92893	0.303593
$866$	0	0
$867$	−9.75736	−0.331377
$868$	0	0
$869$	29.0711	0.986168
$870$	0	0
$871$	3.17157	0.107465
$872$	0	0
$873$	−2.38983	−0.0808834
$874$	0	0
$875$	4.07107	0.137627
$876$	0	0
$877$	31.0711	1.04920	0.524598	−	0.851350i	$-0.324216\pi$
$877$	31.0711	1.04920	0.524598	+	0.851350i	$0.324216\pi$
$878$	0	0
$879$	−13.8995	−0.468818
$880$	0	0
$881$	19.4558	0.655484	0.327742	−	0.944767i	$-0.393712\pi$
$881$	19.4558	0.655484	0.327742	+	0.944767i	$0.393712\pi$
$882$	0	0
$883$	−18.9706	−0.638410	−0.319205	−	0.947686i	$-0.603416\pi$
$883$	−18.9706	−0.638410	−0.319205	+	0.947686i	$0.603416\pi$
$884$	0	0
$885$	2.82843	0.0950765
$886$	0	0
$887$	−48.4853	−1.62798	−0.813988	−	0.580881i	$-0.802708\pi$
$887$	−48.4853	−1.62798	−0.813988	+	0.580881i	$0.802708\pi$
$888$	0	0
$889$	18.9706	0.636252
$890$	0	0
$891$	37.6396	1.26097
$892$	0	0
$893$	20.7990	0.696012
$894$	0	0
$895$	5.44365	0.181961
$896$	0	0
$897$	−2.72792	−0.0910827
$898$	0	0
$899$	0.958369	0.0319634
$900$	0	0
$901$	5.55635	0.185109
$902$	0	0
$903$	−6.72792	−0.223891
$904$	0	0
$905$	2.24264	0.0745479
$906$	0	0
$907$	−29.8284	−0.990437	−0.495218	−	0.868769i	$-0.664912\pi$
$907$	−29.8284	−0.990437	−0.495218	+	0.868769i	$0.664912\pi$
$908$	0	0
$909$	21.8995	0.726360
$910$	0	0
$911$	41.4853	1.37447	0.687234	−	0.726436i	$-0.258825\pi$
$911$	41.4853	1.37447	0.687234	+	0.726436i	$0.258825\pi$
$912$	0	0
$913$	−87.8406	−2.90710
$914$	0	0
$915$	−2.42641	−0.0802145
$916$	0	0
$917$	−5.17157	−0.170780
$918$	0	0
$919$	20.2843	0.669116	0.334558	−	0.942375i	$-0.391413\pi$
$919$	20.2843	0.669116	0.334558	+	0.942375i	$0.391413\pi$
$920$	0	0
$921$	−9.61522	−0.316832
$922$	0	0
$923$	2.58579	0.0851122
$924$	0	0
$925$	−1.17157	−0.0385211
$926$	0	0
$927$	−4.40202	−0.144581
$928$	0	0
$929$	13.3848	0.439140	0.219570	−	0.975597i	$-0.429535\pi$
$929$	13.3848	0.439140	0.219570	+	0.975597i	$0.429535\pi$
$930$	0	0
$931$	6.41421	0.210217
$932$	0	0
$933$	−14.2010	−0.464920
$934$	0	0
$935$	1.51472	0.0495366
$936$	0	0
$937$	−16.8701	−0.551121	−0.275560	−	0.961284i	$-0.588863\pi$
$937$	−16.8701	−0.551121	−0.275560	+	0.961284i	$0.588863\pi$
$938$	0	0
$939$	−5.79899	−0.189243
$940$	0	0
$941$	51.1838	1.66854	0.834272	−	0.551354i	$-0.185889\pi$
$941$	51.1838	1.66854	0.834272	+	0.551354i	$0.185889\pi$
$942$	0	0
$943$	−14.7696	−0.480963
$944$	0	0
$945$	1.37258	0.0446501
$946$	0	0
$947$	7.17157	0.233045	0.116522	−	0.993188i	$-0.462825\pi$
$947$	7.17157	0.233045	0.116522	+	0.993188i	$0.462825\pi$
$948$	0	0
$949$	−0.757359	−0.0245849
$950$	0	0
$951$	9.94113	0.322363
$952$	0	0
$953$	26.4558	0.856989	0.428494	−	0.903544i	$-0.359044\pi$
$953$	26.4558	0.856989	0.428494	+	0.903544i	$0.359044\pi$
$954$	0	0
$955$	−0.343146	−0.0111039
$956$	0	0
$957$	0.627417	0.0202815
$958$	0	0
$959$	4.24264	0.137002
$960$	0	0
$961$	0.201010	0.00648420
$962$	0	0
$963$	38.4853	1.24017
$964$	0	0
$965$	−1.02944	−0.0331388
$966$	0	0
$967$	57.3553	1.84442	0.922212	−	0.386685i	$-0.126380\pi$
$967$	57.3553	1.84442	0.922212	+	0.386685i	$0.126380\pi$
$968$	0	0
$969$	2.20101	0.0707066
$970$	0	0
$971$	48.8284	1.56698	0.783489	−	0.621405i	$-0.213438\pi$
$971$	48.8284	1.56698	0.783489	+	0.621405i	$0.213438\pi$
$972$	0	0
$973$	−18.3848	−0.589389
$974$	0	0
$975$	−2.82843	−0.0905822
$976$	0	0
$977$	20.1005	0.643072	0.321536	−	0.946897i	$-0.395801\pi$
$977$	20.1005	0.643072	0.321536	+	0.946897i	$0.395801\pi$
$978$	0	0
$979$	2.58579	0.0826421
$980$	0	0
$981$	−39.1299	−1.24932
$982$	0	0
$983$	−27.7279	−0.884383	−0.442192	−	0.896921i	$-0.645799\pi$
$983$	−27.7279	−0.884383	−0.442192	+	0.896921i	$0.645799\pi$
$984$	0	0
$985$	−10.4853	−0.334089
$986$	0	0
$987$	−1.89949	−0.0604616
$988$	0	0
$989$	−53.4853	−1.70073
$990$	0	0
$991$	−39.6569	−1.25974	−0.629871	−	0.776700i	$-0.716892\pi$
$991$	−39.6569	−1.25974	−0.629871	+	0.776700i	$0.716892\pi$
$992$	0	0
$993$	7.79899	0.247493
$994$	0	0
$995$	−11.2721	−0.357349
$996$	0	0
$997$	36.4853	1.15550	0.577750	−	0.816214i	$-0.303931\pi$
$997$	36.4853	1.15550	0.577750	+	0.816214i	$0.303931\pi$
$998$	0	0
$999$	−0.804041	−0.0254387

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1456.2.a.s.1.1		2
4.3	odd	2		728.2.a.e.1.2	✓	2
8.3	odd	2		5824.2.a.br.1.1		2
8.5	even	2		5824.2.a.bg.1.2		2
12.11	even	2		6552.2.a.bj.1.1		2
28.27	even	2		5096.2.a.q.1.1		2
52.51	odd	2		9464.2.a.k.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
728.2.a.e.1.2	✓	2	4.3	odd	2
1456.2.a.s.1.1		2	1.1	even	1	trivial
5096.2.a.q.1.1		2	28.27	even	2
5824.2.a.bg.1.2		2	8.5	even	2
5824.2.a.br.1.1		2	8.3	odd	2
6552.2.a.bj.1.1		2	12.11	even	2
9464.2.a.k.1.2		2	52.51	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1456 = 2^{4} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1456.a (trivial)

\(f(q)\)	\(=\)	\(q+0.585786 q^{3} +0.414214 q^{5} -1.00000 q^{7} -2.65685 q^{9} +O(q^{10})\)	\(q+0.585786 q^{3} +0.414214 q^{5} -1.00000 q^{7} -2.65685 q^{9} +6.24264 q^{11} +1.00000 q^{13} +0.242641 q^{15} +0.585786 q^{17} +6.41421 q^{19} -0.585786 q^{21} -4.65685 q^{23} -4.82843 q^{25} -3.31371 q^{27} +0.171573 q^{29} +5.58579 q^{31} +3.65685 q^{33} -0.414214 q^{35} +0.242641 q^{37} +0.585786 q^{39} +3.17157 q^{41} +11.4853 q^{43} -1.10051 q^{45} +3.24264 q^{47} +1.00000 q^{49} +0.343146 q^{51} +9.48528 q^{53} +2.58579 q^{55} +3.75736 q^{57} +11.6569 q^{59} -10.0000 q^{61} +2.65685 q^{63} +0.414214 q^{65} +3.17157 q^{67} -2.72792 q^{69} +2.58579 q^{71} -0.757359 q^{73} -2.82843 q^{75} -6.24264 q^{77} +4.65685 q^{79} +6.02944 q^{81} -14.0711 q^{83} +0.242641 q^{85} +0.100505 q^{87} +0.414214 q^{89} -1.00000 q^{91} +3.27208 q^{93} +2.65685 q^{95} +0.899495 q^{97} -16.5858 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 4 q^{3} - 2 q^{5} - 2 q^{7} + 6 q^{9}+O(q^{10}) \) 2 * q + 4 * q^3 - 2 * q^5 - 2 * q^7 + 6 * q^9	\( 2 q + 4 q^{3} - 2 q^{5} - 2 q^{7} + 6 q^{9} + 4 q^{11} + 2 q^{13} - 8 q^{15} + 4 q^{17} + 10 q^{19} - 4 q^{21} + 2 q^{23} - 4 q^{25} + 16 q^{27} + 6 q^{29} + 14 q^{31} - 4 q^{33} + 2 q^{35} - 8 q^{37} + 4 q^{39} + 12 q^{41} + 6 q^{43} - 22 q^{45} - 2 q^{47} + 2 q^{49} + 12 q^{51} + 2 q^{53} + 8 q^{55} + 16 q^{57} + 12 q^{59} - 20 q^{61} - 6 q^{63} - 2 q^{65} + 12 q^{67} + 20 q^{69} + 8 q^{71} - 10 q^{73} - 4 q^{77} - 2 q^{79} + 46 q^{81} - 14 q^{83} - 8 q^{85} + 20 q^{87} - 2 q^{89} - 2 q^{91} + 32 q^{93} - 6 q^{95} - 18 q^{97} - 36 q^{99}+O(q^{100}) \) 2 * q + 4 * q^3 - 2 * q^5 - 2 * q^7 + 6 * q^9 + 4 * q^11 + 2 * q^13 - 8 * q^15 + 4 * q^17 + 10 * q^19 - 4 * q^21 + 2 * q^23 - 4 * q^25 + 16 * q^27 + 6 * q^29 + 14 * q^31 - 4 * q^33 + 2 * q^35 - 8 * q^37 + 4 * q^39 + 12 * q^41 + 6 * q^43 - 22 * q^45 - 2 * q^47 + 2 * q^49 + 12 * q^51 + 2 * q^53 + 8 * q^55 + 16 * q^57 + 12 * q^59 - 20 * q^61 - 6 * q^63 - 2 * q^65 + 12 * q^67 + 20 * q^69 + 8 * q^71 - 10 * q^73 - 4 * q^77 - 2 * q^79 + 46 * q^81 - 14 * q^83 - 8 * q^85 + 20 * q^87 - 2 * q^89 - 2 * q^91 + 32 * q^93 - 6 * q^95 - 18 * q^97 - 36 * q^99

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